{"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 (a\u2081, b\u2081) < (a\u2082, b\u2082) \u2194 (a\u2081, b\u2081) \u2264 (a\u2082, b\u2082) \u2227 \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 (a\u2081, b\u2081) < (a\u2082, b\u2082) \u2192 (a\u2081, b\u2081) \u2264 (a\u2082, b\u2082) \u2227 \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nrintro (\u27e8_, _, hlt\u27e9 | \u27e8_, hlt\u27e9)\n[GOAL]\ncase mp.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nhlt : a\u2081 < a\u2082\n\u22a2 (a\u2081, b\u2081) \u2264 (a\u2082, b\u2082) \u2227 \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nhlt : a\u2081 < a\u2082\n\u22a2 (a\u2081, b\u2081) \u2264 (a\u2082, b\u2082)\n[PROOFSTEP]\nexact left _ _ hlt\n[GOAL]\ncase mp.left.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nhlt : a\u2081 < a\u2082\n\u22a2 \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mp.left.right.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nhlt : a\u2081 < a\u2082\nh\u271d : a\u2082 < a\u2081\n\u22a2 False\n[PROOFSTEP]\napply lt_asymm hlt\n[GOAL]\ncase mp.left.right.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nhlt : a\u2081 < a\u2082\nh\u271d : a\u2082 < a\u2081\n\u22a2 a\u2082 < a\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mp.left.right.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : a\u2081 < a\u2081\nh\u271d : b\u2082 \u2264 b\u2081\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ hlt\n[GOAL]\ncase mp.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\n\u22a2 (a\u2081, b\u2081) \u2264 (a\u2081, b\u2082) \u2227 \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.right.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\n\u22a2 (a\u2081, b\u2081) \u2264 (a\u2081, b\u2082)\n[PROOFSTEP]\nright\n[GOAL]\ncase mp.right.left.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\n\u22a2 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nrw [lt_iff_le_not_le] at hlt \n[GOAL]\ncase mp.right.left.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 \u2264 b\u2082 \u2227 \u00acb\u2082 \u2264 b\u2081\n\u22a2 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nexact hlt.1\n[GOAL]\ncase mp.right.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\n\u22a2 \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase mp.right.right.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\nh\u271d : a\u2081 < a\u2081\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl a\u2081\n[GOAL]\ncase mp.right.right.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\nh\u271d : a\u2081 < a\u2081\n\u22a2 a\u2081 < a\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mp.right.right.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 < b\u2082\nh\u271d : b\u2082 \u2264 b\u2081\n\u22a2 False\n[PROOFSTEP]\nrw [lt_iff_le_not_le] at hlt \n[GOAL]\ncase mp.right.right.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 \u2264 b\u2082 \u2227 \u00acb\u2082 \u2264 b\u2081\nh\u271d : b\u2082 \u2264 b\u2081\n\u22a2 False\n[PROOFSTEP]\napply hlt.2\n[GOAL]\ncase mp.right.right.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nhlt : b\u2081 \u2264 b\u2082 \u2227 \u00acb\u2082 \u2264 b\u2081\nh\u271d : b\u2082 \u2264 b\u2081\n\u22a2 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\n\u22a2 (a\u2081, b\u2081) \u2264 (a\u2082, b\u2082) \u2227 \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081) \u2192 (a\u2081, b\u2081) < (a\u2082, b\u2082)\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e9, h\u2082r\u27e9\n[GOAL]\ncase mpr.intro.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : a\u2081 < a\u2082\n\u22a2 (a\u2081, b\u2081) < (a\u2082, b\u2082)\n[PROOFSTEP]\nleft\n[GOAL]\ncase mpr.intro.left.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2082, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : a\u2081 < a\u2082\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr.intro.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\n\u22a2 (a\u2081, b\u2081) < (a\u2081, b\u2082)\n[PROOFSTEP]\nright\n[GOAL]\ncase mpr.intro.right.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\n\u22a2 b\u2081 < b\u2082\n[PROOFSTEP]\nrw [lt_iff_le_not_le]\n[GOAL]\ncase mpr.intro.right.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\n\u22a2 b\u2081 \u2264 b\u2082 \u2227 \u00acb\u2082 \u2264 b\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.intro.right.h.left\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\n\u22a2 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr.intro.right.h.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\n\u22a2 \u00acb\u2082 \u2264 b\u2081\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr.intro.right.h.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\nh : b\u2082 \u2264 b\u2081\n\u22a2 False\n[PROOFSTEP]\napply h\u2082r\n[GOAL]\ncase mpr.intro.right.h.right\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\nh : b\u2082 \u2264 b\u2081\n\u22a2 (a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\n[PROOFSTEP]\nright\n[GOAL]\ncase mpr.intro.right.h.right.h\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nsrc\u271d\u00b9 : LE (Lex (\u03b1 \u00d7 \u03b2)) := instLE \u03b1 \u03b2\nsrc\u271d : LT (Lex (\u03b1 \u00d7 \u03b2)) := instLT \u03b1 \u03b2\nx\u2081 x\u2082 : Lex (\u03b1 \u00d7 \u03b2)\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh\u2082r : \u00ac(a\u2081, b\u2082) \u2264 (a\u2081, b\u2081)\nh\u271d : b\u2081 \u2264 b\u2082\nh : b\u2082 \u2264 b\u2081\n\u22a2 b\u2082 \u2264 b\u2081\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\n\u22a2 Monotone \u2191toLex\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 \u27e8ha, hb\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha : (a\u2081, b\u2081).fst \u2264 (a\u2082, b\u2082).fst\nhb : (a\u2081, b\u2081).snd \u2264 (a\u2082, b\u2082).snd\n\u22a2 \u2191toLex (a\u2081, b\u2081) \u2264 \u2191toLex (a\u2082, b\u2082)\n[PROOFSTEP]\nobtain rfl | ha : a\u2081 = a\u2082 \u2228 _ := ha.eq_or_lt\n[GOAL]\ncase mk.mk.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nha : (a\u2081, b\u2081).fst \u2264 (a\u2081, b\u2082).fst\nhb : (a\u2081, b\u2081).snd \u2264 (a\u2081, b\u2082).snd\n\u22a2 \u2191toLex (a\u2081, b\u2081) \u2264 \u2191toLex (a\u2081, b\u2082)\n[PROOFSTEP]\nexact right _ hb\n[GOAL]\ncase mk.mk.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u271d : (a\u2081, b\u2081).fst \u2264 (a\u2082, b\u2082).fst\nhb : (a\u2081, b\u2081).snd \u2264 (a\u2082, b\u2082).snd\nha : (a\u2081, b\u2081).fst < (a\u2082, b\u2082).fst\n\u22a2 \u2191toLex (a\u2081, b\u2081) \u2264 \u2191toLex (a\u2082, b\u2082)\n[PROOFSTEP]\nexact left _ _ ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\n\u22a2 StrictMono \u2191toLex\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 h\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh : (a\u2081, b\u2081) < (a\u2082, b\u2082)\n\u22a2 \u2191toLex (a\u2081, b\u2081) < \u2191toLex (a\u2082, b\u2082)\n[PROOFSTEP]\nobtain rfl | ha : a\u2081 = a\u2082 \u2228 _ := h.le.1.eq_or_lt\n[GOAL]\ncase mk.mk.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 b\u2082 : \u03b2\nh : (a\u2081, b\u2081) < (a\u2081, b\u2082)\n\u22a2 \u2191toLex (a\u2081, b\u2081) < \u2191toLex (a\u2081, b\u2082)\n[PROOFSTEP]\nexact right _ (Prod.mk_lt_mk_iff_right.1 h)\n[GOAL]\ncase mk.mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh : (a\u2081, b\u2081) < (a\u2082, b\u2082)\nha : (a\u2081, b\u2081).fst < (a\u2082, b\u2082).fst\n\u22a2 \u2191toLex (a\u2081, b\u2081) < \u2191toLex (a\u2082, b\u2082)\n[PROOFSTEP]\nexact left _ _ ha\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nsrc\u271d : Preorder (Lex (\u03b1 \u00d7 \u03b2)) := preorder \u03b1 \u03b2\n\u22a2 \u2200 (a b : Lex (\u03b1 \u00d7 \u03b2)), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nhaveI : IsStrictOrder \u03b1 (\u00b7 < \u00b7) := { irrefl := lt_irrefl, trans := fun _ _ _ => lt_trans }\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nsrc\u271d : Preorder (Lex (\u03b1 \u00d7 \u03b2)) := preorder \u03b1 \u03b2\nthis : IsStrictOrder \u03b1 fun x x_1 => x < x_1\n\u22a2 \u2200 (a b : Lex (\u03b1 \u00d7 \u03b2)), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nhaveI : IsAntisymm \u03b2 (\u00b7 \u2264 \u00b7) := \u27e8fun _ _ => le_antisymm\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PartialOrder \u03b2\nsrc\u271d : Preorder (Lex (\u03b1 \u00d7 \u03b2)) := preorder \u03b1 \u03b2\nthis\u271d : IsStrictOrder \u03b1 fun x x_1 => x < x_1\nthis : IsAntisymm \u03b2 fun x x_1 => x \u2264 x_1\n\u22a2 \u2200 (a b : Lex (\u03b1 \u00d7 \u03b2)), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nexact @antisymm _ (Prod.Lex _ _) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\n\u22a2 \u2200 (a\u2081 a\u2082 : Lex (\u03b1 \u00d7 \u03b2)), a\u2081 < a\u2082 \u2192 \u2203 a, a\u2081 < a \u2227 a < a\u2082\n[PROOFSTEP]\nrintro _ _ (@\u27e8a\u2081, b\u2081, a\u2082, b\u2082, h\u27e9 | @\u27e8a, b\u2081, b\u2082, h\u27e9)\n[GOAL]\ncase left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh : a\u2081 < a\u2082\n\u22a2 \u2203 a, (a\u2081, b\u2081) < a \u2227 a < (a\u2082, b\u2082)\n[PROOFSTEP]\nobtain \u27e8c, h\u2081, h\u2082\u27e9 := exists_between h\n[GOAL]\ncase left.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na\u2081 : \u03b1\nb\u2081 : \u03b2\na\u2082 : \u03b1\nb\u2082 : \u03b2\nh : a\u2081 < a\u2082\nc : \u03b1\nh\u2081 : a\u2081 < c\nh\u2082 : c < a\u2082\n\u22a2 \u2203 a, (a\u2081, b\u2081) < a \u2227 a < (a\u2082, b\u2082)\n[PROOFSTEP]\nexact \u27e8(c, b\u2081), left _ _ h\u2081, left _ _ h\u2082\u27e9\n[GOAL]\ncase right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na : \u03b1\nb\u2081 b\u2082 : \u03b2\nh : b\u2081 < b\u2082\n\u22a2 \u2203 a_1, (a, b\u2081) < a_1 \u2227 a_1 < (a, b\u2082)\n[PROOFSTEP]\nobtain \u27e8c, h\u2081, h\u2082\u27e9 := exists_between h\n[GOAL]\ncase right.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na : \u03b1\nb\u2081 b\u2082 : \u03b2\nh : b\u2081 < b\u2082\nc : \u03b2\nh\u2081 : b\u2081 < c\nh\u2082 : c < b\u2082\n\u22a2 \u2203 a_1, (a, b\u2081) < a_1 \u2227 a_1 < (a, b\u2082)\n[PROOFSTEP]\nexact \u27e8(a, c), right _ h\u2081, right _ h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 \u2200 (a : Lex (\u03b1 \u00d7 \u03b2)), \u2203 b, a < b\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 b_1, (a, b) < b_1\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := exists_gt a\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b1\na : \u03b1\nb : \u03b2\nc : \u03b1\nh : a < c\n\u22a2 \u2203 b_1, (a, b) < b_1\n[PROOFSTEP]\nexact \u27e8\u27e8c, b\u27e9, left _ _ h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b1\n\u22a2 \u2200 (a : Lex (\u03b1 \u00d7 \u03b2)), \u2203 b, b < a\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b1\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 b_1, b_1 < (a, b)\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := exists_lt a\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b1\na : \u03b1\nb : \u03b2\nc : \u03b1\nh : c < a\n\u22a2 \u2203 b_1, b_1 < (a, b)\n[PROOFSTEP]\nexact \u27e8\u27e8c, b\u27e9, left _ _ h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\n\u22a2 \u2200 (a : Lex (\u03b1 \u00d7 \u03b2)), \u2203 b, a < b\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 b_1, (a, b) < b_1\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := exists_gt b\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMaxOrder \u03b2\na : \u03b1\nb c : \u03b2\nh : b < c\n\u22a2 \u2203 b_1, (a, b) < b_1\n[PROOFSTEP]\nexact \u27e8\u27e8a, c\u27e9, right _ h\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b2\n\u22a2 \u2200 (a : Lex (\u03b1 \u00d7 \u03b2)), \u2203 b, b < a\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2203 b_1, b_1 < (a, b)\n[PROOFSTEP]\nobtain \u27e8c, h\u27e9 := exists_lt b\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : NoMinOrder \u03b2\na : \u03b1\nb c : \u03b2\nh : c < b\n\u22a2 \u2203 b_1, b_1 < (a, b)\n[PROOFSTEP]\nexact \u27e8\u27e8a, c\u27e9, right _ h\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Prod.Lex", "llama_tokens": 9650, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722394, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.5499373435701101}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\n\u22a2 \u220f x in Ico a b, f (x + c) = \u220f x in Ico (a + c) (b + c), f x\n[PROOFSTEP]\nrw [\u2190 map_add_right_Ico, prod_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\n\u22a2 \u220f x in Ico a b, f (x + c) = \u220f x in Ico a b, f (\u2191(addRightEmbedding c) x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b1 \u2192 \u03b2\na b c : \u03b1\n\u22a2 \u220f x in Ico a b, f (c + x) = \u220f x in Ico (a + c) (b + c), f x\n[PROOFSTEP]\nconvert prod_Ico_add' f a b c using 2\n[GOAL]\ncase h.e'_2.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b3 : CommMonoid \u03b2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b1 \u2192 \u03b2\na b c x\u271d : \u03b1\na\u271d : x\u271d \u2208 Ico a b\n\u22a2 f (c + x\u271d) = f (x\u271d + c)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\na b : \u2115\nhab : a \u2264 b\nf : \u2115 \u2192 \u03b2\n\u22a2 \u220f k in Ico a (b + 1), f k = (\u220f k in Ico a b, f k) * f b\n[PROOFSTEP]\nrw [Nat.Ico_succ_right_eq_insert_Ico hab, prod_insert right_not_mem_Ico, mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 \u03b2\n\u22a2 \u220f k in Ico a b, f k = f a * \u220f k in Ico (a + 1) b, f k\n[PROOFSTEP]\nhave ha : a \u2209 Ico (a + 1) b := by simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 \u03b2\n\u22a2 \u00aca \u2208 Ico (a + 1) b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\na b : \u2115\nhab : a < b\nf : \u2115 \u2192 \u03b2\nha : \u00aca \u2208 Ico (a + 1) b\n\u22a2 \u220f k in Ico a b, f k = f a * \u220f k in Ico (a + 1) b, f k\n[PROOFSTEP]\nrw [\u2190 prod_insert ha, Nat.Ico_insert_succ_left hab]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n k : \u2115\nhmn : m \u2264 n\nhnk : n \u2264 k\n\u22a2 (\u220f i in Ioc m n, f i) * \u220f i in Ioc n k, f i = \u220f i in Ioc m k, f i\n[PROOFSTEP]\nrw [\u2190 Ioc_union_Ioc_eq_Ioc hmn hnk, prod_union]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n k : \u2115\nhmn : m \u2264 n\nhnk : n \u2264 k\n\u22a2 Disjoint (Ioc m n) (Ioc n k)\n[PROOFSTEP]\napply disjoint_left.2 fun x hx h'x => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n k : \u2115\nhmn : m \u2264 n\nhnk : n \u2264 k\n\u22a2 \u2200 (x : \u2115), x \u2208 Ioc m n \u2192 x \u2208 Ioc n k \u2192 False\n[PROOFSTEP]\nintros x hx h'x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n k : \u2115\nhmn : m \u2264 n\nhnk : n \u2264 k\nx : \u2115\nhx : x \u2208 Ioc m n\nh'x : x \u2208 Ioc n k\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\na b : \u2115\nhab : a \u2264 b\nf : \u2115 \u2192 \u03b2\n\u22a2 \u220f k in Ioc a (b + 1), f k = (\u220f k in Ioc a b, f k) * f (b + 1)\n[PROOFSTEP]\nrw [\u2190 prod_Ioc_consecutive _ hab (Nat.le_succ b), Nat.Ioc_succ_singleton, prod_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b4 : Type u_1\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k in Ico m n, f k) * \u220f k in range m, f k = \u220f k in range n, f k\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b4 : Type u_1\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 (\u220f k in range m, f k) * \u220f k in Ico m n, f k = \u220f k in range n, f k\n[PROOFSTEP]\nexact prod_range_mul_prod_Ico f h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b4 : Type u_1\ninst\u271d : CommGroup \u03b4\nf : \u2115 \u2192 \u03b4\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k in Ico m n, f k = (\u220f k in range n, f k) / \u220f k in range m, f k\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using prod_Ico_eq_mul_inv f h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\u271d\na : \u03b1\u271d\ng f\u271d : \u03b1\u271d \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 (\u220f k in range m, f k) / \u220f k in range n, f k = \u220f k in filter (fun k => n \u2264 k) (range m), f k\n[PROOFSTEP]\nrw [\u2190 prod_Ico_eq_div f hnm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\u271d\na : \u03b1\u271d\ng f\u271d : \u03b1\u271d \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u220f k in Ico n m, f k = \u220f k in filter (fun k => n \u2264 k) (range m), f k\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\u271d\na : \u03b1\u271d\ng f\u271d : \u03b1\u271d \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 Ico n m = filter (fun k => n \u2264 k) (range m)\n[PROOFSTEP]\napply Finset.ext\n[GOAL]\ncase e_s.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\u271d\na : \u03b1\u271d\ng f\u271d : \u03b1\u271d \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2200 (a : \u2115), a \u2208 Ico n m \u2194 a \u2208 filter (fun k => n \u2264 k) (range m)\n[PROOFSTEP]\nsimp only [mem_Ico, mem_filter, mem_range, *]\n[GOAL]\ncase e_s.a\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\u271d\na : \u03b1\u271d\ng f\u271d : \u03b1\u271d \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\n\u03b1 : Type u_1\ninst\u271d : CommGroup \u03b1\nf : \u2115 \u2192 \u03b1\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2200 (a : \u2115), n \u2264 a \u2227 a < m \u2194 a < m \u2227 n \u2264 a\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2211 i in Ico a b, \u2211 j in Ico i b, f i j = \u2211 j in Ico a b, \u2211 i in Ico a (j + 1), f i j\n[PROOFSTEP]\nrw [Finset.sum_sigma', Finset.sum_sigma']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2211 x in Finset.sigma (Ico a b) fun i => Ico i b, f x.fst x.snd =\n    \u2211 x in Finset.sigma (Ico a b) fun j => Ico a (j + 1), f x.snd x.fst\n[PROOFSTEP]\nrefine'\n  Finset.sum_bij' (fun (x : \u03a3 _ : \u2115, \u2115) _ => (\u27e8x.2, x.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (fun _ _ => rfl)\n    (fun (x : \u03a3 _ : \u2115, \u2115) _ => (\u27e8x.2, x.1\u27e9 : \u03a3 _ : \u2115, \u2115)) _ (by (rintro \u27e8\u27e9 _; rfl)) (by (rintro \u27e8\u27e9 _; rfl))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 : (_ : \u2115) \u00d7 \u2115) (ha : a_1 \u2208 Finset.sigma (Ico a b) fun i => Ico i b),\n    (fun x x_1 => { fst := x.snd, snd := x.fst }) ((fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha)\n        (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha \u2208 ?m.28850) =\n      a_1\n[PROOFSTEP]\nrintro \u27e8\u27e9 _\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\nfst\u271d snd\u271d : \u2115\nha\u271d : { fst := fst\u271d, snd := snd\u271d } \u2208 Finset.sigma (Ico a b) fun i => Ico i b\n\u22a2 (fun x x_1 => { fst := x.snd, snd := x.fst })\n      ((fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst\u271d, snd := snd\u271d } ha\u271d)\n      (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst\u271d, snd := snd\u271d } ha\u271d \u2208 ?m.28850) =\n    { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 : (_ : \u2115) \u00d7 \u2115) (ha : a_1 \u2208 Finset.sigma (Ico a b) fun j => Ico a (j + 1)),\n    (fun x x_1 => { fst := x.snd, snd := x.fst }) ((fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha)\n        (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha \u2208 ?m.28849) =\n      a_1\n[PROOFSTEP]\nrintro \u27e8\u27e9 _\n[GOAL]\ncase mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\nfst\u271d snd\u271d : \u2115\nha\u271d : { fst := fst\u271d, snd := snd\u271d } \u2208 Finset.sigma (Ico a b) fun j => Ico a (j + 1)\n\u22a2 (fun x x_1 => { fst := x.snd, snd := x.fst })\n      ((fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst\u271d, snd := snd\u271d } ha\u271d)\n      (_ : (fun x x_1 => { fst := x.snd, snd := x.fst }) { fst := fst\u271d, snd := snd\u271d } ha\u271d \u2208 ?m.28849) =\n    { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 : (_ : \u2115) \u00d7 \u2115) (ha : a_1 \u2208 Finset.sigma (Ico a b) fun i => Ico i b),\n    (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha \u2208 Finset.sigma (Ico a b) fun j => Ico a (j + 1)\n[PROOFSTEP]\nsimp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 : (_ : \u2115) \u00d7 \u2115) (ha : a_1 \u2208 Finset.sigma (Ico a b) fun j => Ico a (j + 1)),\n    (fun x x_1 => { fst := x.snd, snd := x.fst }) a_1 ha \u2208 Finset.sigma (Ico a b) fun i => Ico i b\n[PROOFSTEP]\nsimp only [Finset.mem_Ico, Sigma.forall, Finset.mem_sigma]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 b_1 : \u2115), (a \u2264 a_1 \u2227 a_1 < b) \u2227 a_1 \u2264 b_1 \u2227 b_1 < b \u2192 (a \u2264 b_1 \u2227 b_1 < b) \u2227 a \u2264 a_1 \u2227 a_1 < b_1 + 1\n[PROOFSTEP]\nrintro a b \u27e8\u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2083, h\u2084\u27e9\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na b : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\n\u22a2 \u2200 (a_1 b_1 : \u2115), (a \u2264 a_1 \u2227 a_1 < b) \u2227 a \u2264 b_1 \u2227 b_1 < a_1 + 1 \u2192 (a \u2264 b_1 \u2227 b_1 < b) \u2227 b_1 \u2264 a_1 \u2227 a_1 < b\n[PROOFSTEP]\nrintro a b \u27e8\u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2083, h\u2084\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a \u2264 b\nh\u2084 : b < b\u271d\n\u22a2 (a\u271d \u2264 b \u2227 b < b\u271d) \u2227 a\u271d \u2264 a \u2227 a < b + 1\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a\u271d \u2264 b\nh\u2084 : b < a + 1\n\u22a2 (a\u271d \u2264 b \u2227 b < b\u271d) \u2227 b \u2264 a \u2227 a < b\u271d\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, _\u27e9, \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a \u2264 b\nh\u2084 : b < b\u271d\n\u22a2 a\u271d \u2264 b\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_1.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a \u2264 b\nh\u2084 : b < b\u271d\n\u22a2 b < b\u271d\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_1.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a \u2264 b\nh\u2084 : b < b\u271d\n\u22a2 a\u271d \u2264 a\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_1.intro.intro.intro.refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a \u2264 b\nh\u2084 : b < b\u271d\n\u22a2 a < b + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2.intro.intro.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a\u271d \u2264 b\nh\u2084 : b < a + 1\n\u22a2 a\u271d \u2264 b\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2.intro.intro.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a\u271d \u2264 b\nh\u2084 : b < a + 1\n\u22a2 b < b\u271d\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2.intro.intro.intro.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a\u271d \u2264 b\nh\u2084 : b < a + 1\n\u22a2 b \u2264 a\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2.intro.intro.intro.refine'_4\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na\u271d\u00b9 : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_1\ninst\u271d : AddCommMonoid M\na\u271d b\u271d : \u2115\nf : \u2115 \u2192 \u2115 \u2192 M\na b : \u2115\nh\u2081 : a\u271d \u2264 a\nh\u2082 : a < b\u271d\nh\u2083 : a\u271d \u2264 b\nh\u2084 : b < a + 1\n\u22a2 a < b\u271d\n[PROOFSTEP]\nlinarith\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)\n[PROOFSTEP]\nby_cases h : m \u2264 n\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)\n[PROOFSTEP]\nrw [\u2190 Nat.Ico_zero_eq_range, prod_Ico_add, zero_add, tsub_add_cancel_of_le h]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : \u00acm \u2264 n\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)\n[PROOFSTEP]\nreplace h : n \u2264 m := le_of_not_ge h\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nm n : \u2115\nh : n \u2264 m\n\u22a2 \u220f k in Ico m n, f k = \u220f k in range (n - m), f (m + k)\n[PROOFSTEP]\nrw [Ico_eq_empty_of_le h, tsub_eq_zero_iff_le.mpr h, range_zero, prod_empty, prod_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in Ico (n + 1 - m) (n + 1 - k), f j\n[PROOFSTEP]\nhave : \u2200 i < m, i \u2264 n := by\n  intro i hi\n  exact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\n\u22a2 \u2200 (i : \u2115), i < m \u2192 i \u2264 n\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\ni : \u2115\nhi : i < m\n\u22a2 i \u2264 n\n[PROOFSTEP]\nexact (add_le_add_iff_right 1).1 (le_trans (Nat.lt_iff_add_one_le.1 hi) h)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in Ico (n + 1 - m) (n + 1 - k), f j\n[PROOFSTEP]\ncases' lt_or_le k m with hkm hkm\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : k < m\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in Ico (n + 1 - m) (n + 1 - k), f j\n[PROOFSTEP]\nrw [\u2190 Nat.Ico_image_const_sub_eq_Ico (this _ hkm)]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : k < m\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in image (fun x => n - x) (Ico k m), f j\n[PROOFSTEP]\nrefine' (prod_image _).symm\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : k < m\n\u22a2 \u2200 (x : \u2115), x \u2208 Ico k m \u2192 \u2200 (y : \u2115), y \u2208 Ico k m \u2192 n - x = n - y \u2192 x = y\n[PROOFSTEP]\nsimp only [mem_Ico]\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : k < m\n\u22a2 \u2200 (x : \u2115), k \u2264 x \u2227 x < m \u2192 \u2200 (y : \u2115), k \u2264 y \u2227 y < m \u2192 n - x = n - y \u2192 x = y\n[PROOFSTEP]\nrintro i \u27e8_, im\u27e9 j \u27e8_, jm\u27e9 Hij\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : k < m\ni : \u2115\nleft\u271d\u00b9 : k \u2264 i\nim : i < m\nj : \u2115\nleft\u271d : k \u2264 j\njm : j < m\nHij : n - i = n - j\n\u22a2 i = j\n[PROOFSTEP]\nrw [\u2190 tsub_tsub_cancel_of_le (this _ im), Hij, tsub_tsub_cancel_of_le (this _ jm)]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : m \u2264 k\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in Ico (n + 1 - m) (n + 1 - k), f j\n[PROOFSTEP]\nhave : n + 1 - k \u2264 n + 1 - m := by\n  rw [tsub_le_tsub_iff_left h]\n  exact hkm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : m \u2264 k\n\u22a2 n + 1 - k \u2264 n + 1 - m\n[PROOFSTEP]\nrw [tsub_le_tsub_iff_left h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : m \u2264 k\n\u22a2 m \u2264 k\n[PROOFSTEP]\nexact hkm\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nk m n : \u2115\nh : m \u2264 n + 1\nthis\u271d : \u2200 (i : \u2115), i < m \u2192 i \u2264 n\nhkm : m \u2264 k\nthis : n + 1 - k \u2264 n + 1 - m\n\u22a2 \u220f j in Ico k m, f (n - j) = \u220f j in Ico (n + 1 - m) (n + 1 - k), f j\n[PROOFSTEP]\nsimp only [ge_iff_le, hkm, Ico_eq_empty_of_le, prod_empty, tsub_le_iff_right, Ico_eq_empty_of_le this]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nn : \u2115\n\u22a2 \u220f j in range n, f (n - 1 - j) = \u220f j in range n, f j\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\n\u22a2 \u220f j in range zero, f (zero - 1 - j) = \u220f j in range zero, f j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nn\u271d : \u2115\n\u22a2 \u220f j in range (succ n\u271d), f (succ n\u271d - 1 - j) = \u220f j in range (succ n\u271d), f j\n[PROOFSTEP]\nsimp only [\u2190 Nat.Ico_zero_eq_range, Nat.succ_sub_succ_eq_sub, tsub_zero]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nn\u271d : \u2115\n\u22a2 \u220f x in Ico 0 (succ n\u271d), f (n\u271d - x) = \u220f x in Ico 0 (succ n\u271d), f x\n[PROOFSTEP]\nrw [prod_Ico_reflect _ _ le_rfl]\n[GOAL]\ncase succ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nn\u271d : \u2115\n\u22a2 \u220f j in Ico (n\u271d + 1 - (n\u271d + 1)) (n\u271d + 1 - 0), f j = \u220f x in Ico 0 (succ n\u271d), f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\n\u22a2 \u220f x in Ico 1 (n + 1 + 1), x = (n + 1)!\n[PROOFSTEP]\nrw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ, prod_Ico_id_eq_factorial n,\n  Nat.succ_eq_add_one, mul_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\n\u22a2 \u220f x in range (n + 1), (x + 1) = (n + 1)!\n[PROOFSTEP]\nsimp [Finset.range_succ, prod_range_add_one_eq_factorial n]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\n\u22a2 (\u2211 i in range n, i) * 2 = \u2211 i in range n, i + \u2211 i in range n, (n - 1 - i)\n[PROOFSTEP]\nrw [sum_range_reflect (fun i => i) n, mul_two]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\n\u22a2 \u2211 i in range n, (n - 1) = n * (n - 1)\n[PROOFSTEP]\nrw [sum_const, card_range, Nat.nsmul_eq_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u2082 s\u2081 s : Finset \u03b1\na : \u03b1\ng f : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\n\u22a2 \u2211 i in range n, i = n * (n - 1) / 2\n[PROOFSTEP]\nrw [\u2190 sum_range_id_mul_two n, Nat.mul_div_cancel _ zero_lt_two]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\n\u22a2 \u2211 i in Ico m n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nhave h\u2081 : (\u2211 i in Ico (m + 1) n, f i \u2022 G i) = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 G (i + 1) :=\n  by\n  rw [\u2190 Nat.sub_add_cancel (Nat.one_le_of_lt hmn), \u2190 sum_Ico_add']\n  simp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero, tsub_eq_zero_iff_le,\n    add_tsub_cancel_right]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\n\u22a2 \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel (Nat.one_le_of_lt hmn), \u2190 sum_Ico_add']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\n\u22a2 \u2211 x in Ico m (n - 1), f (x + 1) \u2022 \u2211 i in range (x + 1), g i =\n    \u2211 i in Ico m (n - 1 + 1 - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nsimp only [ge_iff_le, tsub_le_iff_right, add_le_iff_nonpos_left, nonpos_iff_eq_zero, tsub_eq_zero_iff_le,\n  add_tsub_cancel_right]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\n\u22a2 \u2211 i in Ico m n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nhave h\u2082 :\n  (\u2211 i in Ico (m + 1) n, f i \u2022 G (i + 1)) =\n    (\u2211 i in Ico m (n - 1), f i \u2022 G (i + 1)) + f (n - 1) \u2022 G n - f m \u2022 G (m + 1) :=\n  by\n  rw [\u2190 sum_Ico_sub_bot _ hmn, \u2190 sum_Ico_succ_sub_top _ (Nat.le_pred_of_lt hmn), Nat.sub_add_cancel (pos_of_gt hmn),\n    sub_add_cancel]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\n\u22a2 \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\n[PROOFSTEP]\nrw [\u2190 sum_Ico_sub_bot _ hmn, \u2190 sum_Ico_succ_sub_top _ (Nat.le_pred_of_lt hmn), Nat.sub_add_cancel (pos_of_gt hmn),\n  sub_add_cancel]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\n\u22a2 \u2211 i in Ico m n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nrw [sum_eq_sum_Ico_succ_bot hmn]\n  -- porting note: the following used to be done with `conv`\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\n\u22a2 f m \u2022 g m + \u2211 k in Ico (m + 1) n, f k \u2022 g k =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nhave h\u2083 :\n  (Finset.sum (Ico (m + 1) n) fun i => f i \u2022 g i) =\n    (Finset.sum (Ico (m + 1) n) fun i =>\n      f i \u2022 ((Finset.sum (Finset.range (i + 1)) g) - (Finset.sum (Finset.range i) g))) :=\n  by congr; funext; rw [\u2190 sum_range_succ_sub_sum g]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\n\u22a2 \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\n\u22a2 (fun i => f i \u2022 g i) = fun i => f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase e_f.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nx\u271d : \u2115\n\u22a2 f x\u271d \u2022 g x\u271d = f x\u271d \u2022 (Finset.sum (range (x\u271d + 1)) g - Finset.sum (range x\u271d) g)\n[PROOFSTEP]\nrw [\u2190 sum_range_succ_sub_sum g]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n\u22a2 f m \u2022 g m + \u2211 k in Ico (m + 1) n, f k \u2022 g k =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nrw [h\u2083]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n\u22a2 f m \u2022 g m + \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g) =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nsimp_rw [smul_sub, sum_sub_distrib, h\u2082, h\u2081]\n  -- porting note: the following used to be done with `conv`\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n\u22a2 f m \u2022 g m +\n      (\u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n          f m \u2022 \u2211 i in range (m + 1), g i -\n        \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i) =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nhave h\u2084 :\n  ((((Finset.sum (Ico m (n - 1)) fun i => f i \u2022 Finset.sum (range (i + 1)) fun i => g i) +\n          f (n - 1) \u2022 Finset.sum (range n) fun i => g i) -\n        f m \u2022 Finset.sum (range (m + 1)) fun i => g i) -\n      Finset.sum (Ico m (n - 1)) fun i => f (i + 1) \u2022 Finset.sum (range (i + 1)) fun i => g i) =\n    f (n - 1) \u2022 (range n).sum g - f m \u2022 (range (m + 1)).sum g +\n      Finset.sum (Ico m (n - 1)) (fun i => f i \u2022 (range (i + 1)).sum g - f (i + 1) \u2022 (range (i + 1)).sum g) :=\n  by rw [\u2190 add_sub, add_comm, \u2190 add_sub, \u2190 sum_sub_distrib]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\n\u22a2 \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\n[PROOFSTEP]\nrw [\u2190 add_sub, add_comm, \u2190 add_sub, \u2190 sum_sub_distrib]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\n\u22a2 f m \u2022 g m +\n      (\u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n          f m \u2022 \u2211 i in range (m + 1), g i -\n        \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i) =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nrw [h\u2084]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\n\u22a2 f m \u2022 g m +\n      (f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n        \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)) =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nhave : \u2200 i, f i \u2022 G (i + 1) - f (i + 1) \u2022 G (i + 1) = -((f (i + 1) - f i) \u2022 G (i + 1)) :=\n  by\n  intro i\n  rw [sub_smul]\n  abel\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\n\u22a2 \u2200 (i : \u2115),\n    f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n      -((f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i)\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\ni : \u2115\n\u22a2 f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    -((f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i)\n[PROOFSTEP]\nrw [sub_smul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\ni : \u2115\n\u22a2 f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    -(f (i + 1) \u2022 \u2211 i in range (i + 1), g i - f i \u2022 \u2211 i in range (i + 1), g i)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\ni : \u2115\n\u22a2 f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    -(f (i + 1) \u2022 \u2211 i in range (i + 1), g i - f i \u2022 \u2211 i in range (i + 1), g i)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\nthis :\n  \u2200 (i : \u2115),\n    f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n      -((f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i)\n\u22a2 f m \u2022 g m +\n      (f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n        \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)) =\n    f (n - 1) \u2022 \u2211 i in range n, g i - f m \u2022 \u2211 i in range m, g i -\n      \u2211 i in Ico m (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nsimp_rw [this, sum_neg_distrib, sum_range_succ, smul_add]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\nthis :\n  \u2200 (i : \u2115),\n    f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n      -((f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i)\n\u22a2 f m \u2022 g m +\n      (f (n - 1) \u2022 Finset.sum (range n) g - (f m \u2022 \u2211 x in range m, g x + f m \u2022 g m) +\n        -\u2211 x in Ico m (n - 1), ((f (x + 1) - f x) \u2022 \u2211 x in range x, g x + (f (x + 1) - f x) \u2022 g x)) =\n    f (n - 1) \u2022 \u2211 x in range n, g x - f m \u2022 \u2211 x in range m, g x -\n      \u2211 x in Ico m (n - 1), ((f (x + 1) - f x) \u2022 \u2211 x in range x, g x + (f (x + 1) - f x) \u2022 g x)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhmn : m < n\nh\u2081 : \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range i, g i = \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i\nh\u2082 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 \u2211 i in range (i + 1), g i =\n    \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n      f m \u2022 \u2211 i in range (m + 1), g i\nh\u2083 :\n  \u2211 i in Ico (m + 1) n, f i \u2022 g i = \u2211 i in Ico (m + 1) n, f i \u2022 (Finset.sum (range (i + 1)) g - Finset.sum (range i) g)\nh\u2084 :\n  \u2211 i in Ico m (n - 1), f i \u2022 \u2211 i in range (i + 1), g i + f (n - 1) \u2022 \u2211 i in range n, g i -\n        f m \u2022 \u2211 i in range (m + 1), g i -\n      \u2211 i in Ico m (n - 1), f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n    f (n - 1) \u2022 Finset.sum (range n) g - f m \u2022 Finset.sum (range (m + 1)) g +\n      \u2211 i in Ico m (n - 1), (f i \u2022 Finset.sum (range (i + 1)) g - f (i + 1) \u2022 Finset.sum (range (i + 1)) g)\nthis :\n  \u2200 (i : \u2115),\n    f i \u2022 \u2211 i in range (i + 1), g i - f (i + 1) \u2022 \u2211 i in range (i + 1), g i =\n      -((f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i)\n\u22a2 f m \u2022 g m +\n      (f (n - 1) \u2022 Finset.sum (range n) g - (f m \u2022 \u2211 x in range m, g x + f m \u2022 g m) +\n        -\u2211 x in Ico m (n - 1), ((f (x + 1) - f x) \u2022 \u2211 x in range x, g x + (f (x + 1) - f x) \u2022 g x)) =\n    f (n - 1) \u2022 \u2211 x in range n, g x - f m \u2022 \u2211 x in range m, g x -\n      \u2211 x in Ico m (n - 1), ((f (x + 1) - f x) \u2022 \u2211 x in range x, g x + (f (x + 1) - f x) \u2022 g x)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\n\u22a2 \u2211 i in range n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - \u2211 i in range (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhn : n = 0\n\u22a2 \u2211 i in range n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - \u2211 i in range (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : \u2115 \u2192 R\ng : \u2115 \u2192 M\nm n : \u2115\nhn : \u00acn = 0\n\u22a2 \u2211 i in range n, f i \u2022 g i =\n    f (n - 1) \u2022 \u2211 i in range n, g i - \u2211 i in range (n - 1), (f (i + 1) - f i) \u2022 \u2211 i in range (i + 1), g i\n[PROOFSTEP]\nrw [range_eq_Ico, sum_Ico_by_parts f g (Nat.pos_of_ne_zero hn), sum_range_zero, smul_zero, sub_zero, range_eq_Ico]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Intervals", "llama_tokens": 24217, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321936479701, "lm_q2_score": 0.682573740869499, "lm_q1q2_score": 0.5499033801831955}}
{"text": "[GOAL]\nV : SemiNormedGroupCat\nv : \u2191V\n\u22a2 \u2016(fun v => \u2191\u2191V v) v\u2016 \u2264 1 * \u2016v\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\n\u22a2 \u2200 (P Q R : SemiNormedGroupCat) (f f' : P \u27f6 Q) (g : Q \u27f6 R), (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\nintros _ Q _ f f' g\n[GOAL]\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\nP\u271d Q R\u271d : SemiNormedGroupCat\nf f' : P\u271d \u27f6 Q\ng : Q \u27f6 R\u271d\n\u22a2 (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\nP\u271d Q R\u271d : SemiNormedGroupCat\nf f' : P\u271d \u27f6 Q\ng : Q \u27f6 R\u271d\nx : \u2191P\u271d\n\u22a2 \u2191((f + f') \u226b g) x = \u2191(f \u226b g + f' \u226b g) x\n[PROOFSTEP]\nrw [NormedAddGroupHom.add_apply, CategoryTheory.comp_apply, CategoryTheory.comp_apply, CategoryTheory.comp_apply,\n  @NormedAddGroupHom.add_apply _ _ (_) (_)]\n[GOAL]\ncase h\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\nP\u271d Q R\u271d : SemiNormedGroupCat\nf f' : P\u271d \u27f6 Q\ng : Q \u27f6 R\u271d\nx : \u2191P\u271d\n\u22a2 \u2191g (\u2191f x + \u2191f' x) = \u2191g (\u2191f x) + \u2191g (\u2191f' x)\n[PROOFSTEP]\nconvert map_add g (f x) (f' x)\n[GOAL]\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\n\u22a2 \u2200 (P Q R : SemiNormedGroupCat) (f : P \u27f6 Q) (g g' : Q \u27f6 R), f \u226b (g + g') = f \u226b g + f \u226b g'\n[PROOFSTEP]\nintros\n[GOAL]\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\nP\u271d Q\u271d R\u271d : SemiNormedGroupCat\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 f\u271d \u226b (g\u271d + g'\u271d) = f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d\n[PROOFSTEP]\next\n  -- Porting note: failing simps probably due to instance synthesis issues with concrete\n      -- cats; see the gymnastics below for what used to be\n      -- simp only [add_apply, comp_apply. map_add]\n[GOAL]\ncase h\nV : SemiNormedGroupCat\nW : SemiNormedGroupCat\nP\u271d Q\u271d R\u271d : SemiNormedGroupCat\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\nx\u271d : \u2191P\u271d\n\u22a2 \u2191(f\u271d \u226b (g\u271d + g'\u271d)) x\u271d = \u2191(f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d) x\u271d\n[PROOFSTEP]\nrw [NormedAddGroupHom.add_apply, CategoryTheory.comp_apply, CategoryTheory.comp_apply, CategoryTheory.comp_apply,\n  @NormedAddGroupHom.add_apply _ _ (_) (_)]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.SemiNormedGroupCat.Completion", "llama_tokens": 1051, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754471, "lm_q2_score": 0.7310585786300049, "lm_q1q2_score": 0.5497652514109203}}
{"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \u211d V\nK L : ConvexBody V\nh : K.carrier = L.carrier\n\u22a2 K = L\n[PROOFSTEP]\ncases K\n[GOAL]\ncase mk\nV : Type u_1\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \u211d V\nL : ConvexBody V\ncarrier\u271d : Set V\nconvex'\u271d : Convex \u211d carrier\u271d\nisCompact'\u271d : IsCompact carrier\u271d\nnonempty'\u271d : Set.Nonempty carrier\u271d\nh : { carrier := carrier\u271d, convex' := convex'\u271d, isCompact' := isCompact'\u271d, nonempty' := nonempty'\u271d }.carrier = L.carrier\n\u22a2 { carrier := carrier\u271d, convex' := convex'\u271d, isCompact' := isCompact'\u271d, nonempty' := nonempty'\u271d } = L\n[PROOFSTEP]\ncases L\n[GOAL]\ncase mk.mk\nV : Type u_1\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module \u211d V\ncarrier\u271d\u00b9 : Set V\nconvex'\u271d\u00b9 : Convex \u211d carrier\u271d\u00b9\nisCompact'\u271d\u00b9 : IsCompact carrier\u271d\u00b9\nnonempty'\u271d\u00b9 : Set.Nonempty carrier\u271d\u00b9\ncarrier\u271d : Set V\nconvex'\u271d : Convex \u211d carrier\u271d\nisCompact'\u271d : IsCompact carrier\u271d\nnonempty'\u271d : Set.Nonempty carrier\u271d\nh :\n  { carrier := carrier\u271d\u00b9, convex' := convex'\u271d\u00b9, isCompact' := isCompact'\u271d\u00b9, nonempty' := nonempty'\u271d\u00b9 }.carrier =\n    { carrier := carrier\u271d, convex' := convex'\u271d, isCompact' := isCompact'\u271d, nonempty' := nonempty'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, convex' := convex'\u271d\u00b9, isCompact' := isCompact'\u271d\u00b9, nonempty' := nonempty'\u271d\u00b9 } =\n    { carrier := carrier\u271d, convex' := convex'\u271d, isCompact' := isCompact'\u271d, nonempty' := nonempty'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : NormedSpace \u211d V\nK\u271d L\u271d K L : ConvexBody V\n\u22a2 EMetric.hausdorffEdist \u2191K \u2191L \u2260 \u22a4\n[PROOFSTEP]\napply_rules [Metric.hausdorffEdist_ne_top_of_nonempty_of_bounded, ConvexBody.nonempty, ConvexBody.bounded]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : NormedSpace \u211d V\nK L x\u271d\u00b9 x\u271d : ConvexBody V\n\u22a2 (fun x y => \u2191{ val := Metric.hausdorffDist \u2191x \u2191y, property := (_ : 0 \u2264 Metric.hausdorffDist \u2191x \u2191y) }) x\u271d\u00b9 x\u271d =\n    ENNReal.ofReal (dist x\u271d\u00b9 x\u271d)\n[PROOFSTEP]\nexact ENNReal.coe_nnreal_eq _\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : NormedSpace \u211d V\nK L : ConvexBody V\n\u22a2 EMetric.hausdorffEdist \u2191K \u2191L = edist K L\n[PROOFSTEP]\nrw [edist_dist]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup V\ninst\u271d : NormedSpace \u211d V\nK L : ConvexBody V\n\u22a2 EMetric.hausdorffEdist \u2191K \u2191L = ENNReal.ofReal (dist K L)\n[PROOFSTEP]\nexact (ENNReal.ofReal_toReal hausdorffEdist_ne_top).symm\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Body", "llama_tokens": 1162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.5491304872375226}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\n\u22a2 \u2200 \u2983x : \u211d\u2984, x \u2208 Icc a b \u2192 f x \u2264 B x\n[PROOFSTEP]\nchange Icc a b \u2286 {x | f x \u2264 B x}\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\n\u22a2 Icc a b \u2286 {x | f x \u2264 B x}\n[PROOFSTEP]\nset s := {x | f x \u2264 B x} \u2229 Icc a b\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\n\u22a2 Icc a b \u2286 {x | f x \u2264 B x}\n[PROOFSTEP]\nhave A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\n\u22a2 Icc a b \u2286 {x | f x \u2264 B x}\n[PROOFSTEP]\nhave : IsClosed s := by\n  simp only [inter_comm]\n  exact A.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le'\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\n\u22a2 IsClosed s\n[PROOFSTEP]\nsimp only [inter_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\n\u22a2 IsClosed (Icc a b \u2229 {x | f x \u2264 B x})\n[PROOFSTEP]\nexact A.preimage_closed_of_closed isClosed_Icc OrderClosedTopology.isClosed_le'\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\n\u22a2 Icc a b \u2286 {x | f x \u2264 B x}\n[PROOFSTEP]\napply this.Icc_subset_of_forall_exists_gt ha\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\n\u22a2 \u2200 (x : \u211d), x \u2208 {x | f x \u2264 B x} \u2229 Ico a b \u2192 \u2200 (y : \u211d), y \u2208 Ioi x \u2192 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nrintro x \u27e8hxB : f x \u2264 B x, xab\u27e9 y hy\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\ncases' hxB.lt_or_eq with hxB hxB\n[GOAL]\ncase intro.inl\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x < B x\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nrefine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, hy\u27e9))\n[GOAL]\ncase intro.inl\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x < B x\n\u22a2 {x | f x \u2264 B x} \u2208 \ud835\udcdd[Ioi x] x\n[PROOFSTEP]\nhave : \u2200\u1da0 x in \ud835\udcdd[Icc a b] x, f x < B x :=\n  A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)\n[GOAL]\ncase intro.inl\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis\u271d : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x < B x\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc a b] x, f x < B x\n\u22a2 {x | f x \u2264 B x} \u2208 \ud835\udcdd[Ioi x] x\n[PROOFSTEP]\nhave : \u2200\u1da0 x in \ud835\udcdd[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this\n[GOAL]\ncase intro.inl\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis\u271d\u00b9 : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x < B x\nthis\u271d : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Icc a b] x, f x < B x\nthis : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi x] x, f x < B x\n\u22a2 {x | f x \u2264 B x} \u2208 \ud835\udcdd[Ioi x] x\n[PROOFSTEP]\nexact this.mono fun y => le_of_lt\n[GOAL]\ncase intro.inr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nrcases exists_between (bound x xab hxB) with \u27e8r, hfr, hrB\u27e9\n[GOAL]\ncase intro.inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nspecialize hf' x xab r hfr\n[GOAL]\ncase intro.inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nhave HB : \u2200\u1da0 z in \ud835\udcdd[>] x, r < slope B x z :=\n  (hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici (Ioi_mem_nhds hrB)\n[GOAL]\ncase intro.inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nobtain \u27e8z, hfz, hzB, hz\u27e9 : \u2203 z, slope f x z < r \u2227 r < slope B x z \u2227 z \u2208 Ioc x y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\n\u22a2 \u2203 z, slope f x z < r \u2227 r < slope B x z \u2227 z \u2208 Ioc x y\ncase intro.inr.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\nz : \u211d\nhfz : slope f x z < r\nhzB : r < slope B x z\nhz : z \u2208 Ioc x y\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nexact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi \u27e8le_rfl, hy\u27e9))).exists\n[GOAL]\ncase intro.inr.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\nz : \u211d\nhfz : slope f x z < r\nhzB : r < slope B x z\nhz : z \u2208 Ioc x y\n\u22a2 Set.Nonempty ({x | f x \u2264 B x} \u2229 Ioc x y)\n[PROOFSTEP]\nrefine' \u27e8z, _, hz\u27e9\n[GOAL]\ncase intro.inr.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\nz : \u211d\nhfz : slope f x z < r\nhzB : r < slope B x z\nhz : z \u2208 Ioc x y\n\u22a2 z \u2208 {x | f x \u2264 B x}\n[PROOFSTEP]\nhave := (hfz.trans hzB).le\n[GOAL]\ncase intro.inr.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x \u2192 f' x < B' x\ns : Set \u211d := {x | f x \u2264 B x} \u2229 Icc a b\nA : ContinuousOn (fun x => (f x, B x)) (Icc a b)\nthis\u271d : IsClosed s\nx : \u211d\nhxB\u271d : f x \u2264 B x\nxab : x \u2208 Ico a b\ny : \u211d\nhy : y \u2208 Ioi x\nhxB : f x = B x\nr : \u211d\nhfr : f' x < r\nhrB : r < B' x\nhf' : \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHB : \u2200\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, r < slope B x z\nz : \u211d\nhfz : slope f x z < r\nhzB : r < slope B x z\nhz : z \u2208 Ioc x y\nthis : slope f x z \u2264 slope B x z\n\u22a2 z \u2208 {x | f x \u2264 B x}\n[PROOFSTEP]\nrwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB, sub_le_sub_iff_right] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\n\u22a2 \u2200 \u2983x : \u211d\u2984, x \u2208 Icc a b \u2192 f x \u2264 B x\n[PROOFSTEP]\nhave Hr : \u2200 x \u2208 Icc a b, \u2200 r > 0, f x \u2264 B x + r * (x - a) := fun x hx r hr =>\n  by\n  apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound\n  \u00b7 rwa [sub_self, mul_zero, add_zero]\n  \u00b7 exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))\n  \u00b7 intro x hx\n    exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)\n  \u00b7 intro x _ _\n    rw [mul_one]\n    exact (lt_add_iff_pos_right _).2 hr\n  exact hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 f x \u2264 B x + r * (x - a)\n[PROOFSTEP]\napply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound\n[GOAL]\ncase ha\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 f a \u2264 B a + r * (a - a)\n[PROOFSTEP]\nrwa [sub_self, mul_zero, add_zero]\n[GOAL]\ncase hB\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 ContinuousOn (fun x => B x + r * (x - a)) (Icc a b)\n[PROOFSTEP]\nexact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))\n[GOAL]\ncase hB'\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt (fun x => B x + r * (x - a)) (?m.12328 x) (Ici x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase hB'\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 Icc a b\nr : \u211d\nhr : r > 0\nx : \u211d\nhx : x \u2208 Ico a b\n\u22a2 HasDerivWithinAt (fun x => B x + r * (x - a)) (?m.12328 x) (Ici x) x\n[PROOFSTEP]\nexact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = B x + r * (x - a) \u2192 B' x < B' x + r * 1\n[PROOFSTEP]\nintro x _ _\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx\u271d : \u211d\nhx : x\u271d \u2208 Icc a b\nr : \u211d\nhr : r > 0\nx : \u211d\na\u271d\u00b9 : x \u2208 Ico a b\na\u271d : f x = B x + r * (x - a)\n\u22a2 B' x < B' x + r * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx\u271d : \u211d\nhx : x\u271d \u2208 Icc a b\nr : \u211d\nhr : r > 0\nx : \u211d\na\u271d\u00b9 : x \u2208 Ico a b\na\u271d : f x = B x + r * (x - a)\n\u22a2 B' x < B' x + r\n[PROOFSTEP]\nexact (lt_add_iff_pos_right _).2 hr\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nx : \u211d\nhx : x \u2208 Icc a b\nr : \u211d\nhr : r > 0\n\u22a2 x \u2208 Icc a b\n[PROOFSTEP]\nexact hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHr : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (r : \u211d), r > 0 \u2192 f x \u2264 B x + r * (x - a)\n\u22a2 \u2200 \u2983x : \u211d\u2984, x \u2208 Icc a b \u2192 f x \u2264 B x\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHr : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (r : \u211d), r > 0 \u2192 f x \u2264 B x + r * (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 f x \u2264 B x\n[PROOFSTEP]\nhave : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 :=\n  continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHr : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (r : \u211d), r > 0 \u2192 f x \u2264 B x + r * (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\nthis : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0\n\u22a2 f x \u2264 B x\n[PROOFSTEP]\nconvert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHr : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (r : \u211d), r > 0 \u2192 f x \u2264 B x + r * (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\nthis : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0\n\u22a2 B x = B x + 0 * (x - a)\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 \u211d\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nB B' : \u211d \u2192 \u211d\nha : f a \u2264 B a\nhB : ContinuousOn B (Icc a b)\nhB' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt B (B' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), B' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, slope f x z < r\nHr : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (r : \u211d), r > 0 \u2192 f x \u2264 B x + r * (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\nthis : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0\n\u22a2 0 \u2208 closure (Ioi 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nlet g x := f x - f a\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nhave hg : ContinuousOn g (Icc a b) := hf.sub continuousOn_const\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nhave hg' : \u2200 x \u2208 Ico a b, HasDerivWithinAt g (f' x) (Ici x) x :=\n  by\n  intro x hx\n  simpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nx : \u211d\nhx : x \u2208 Ico a b\n\u22a2 HasDerivWithinAt g (f' x) (Ici x) x\n[PROOFSTEP]\nsimpa using (hf' x hx).sub (hasDerivWithinAt_const _ _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nlet B x := C * (x - a)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nhave hB : \u2200 x, HasDerivAt B C x := by\n  intro x\n  simpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\n\u22a2 \u2200 (x : \u211d), HasDerivAt B C x\n[PROOFSTEP]\nintro x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\nx : \u211d\n\u22a2 HasDerivAt B C x\n[PROOFSTEP]\nsimpa using (hasDerivAt_const x C).mul ((hasDerivAt_id x).sub (hasDerivAt_const x a))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\nhB : \u2200 (x : \u211d), HasDerivAt B C x\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nconvert image_norm_le_of_norm_deriv_right_le_deriv_boundary hg hg' _ hB bound\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\nhB : \u2200 (x : \u211d), HasDerivAt B C x\n\u22a2 \u2016g a\u2016 \u2264 B a\n[PROOFSTEP]\nsimp only\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\ng : \u211d \u2192 E := fun x => f x - f a\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nB : \u211d \u2192 \u211d := fun x => C * (x - a)\nhB : \u2200 (x : \u211d), HasDerivAt B C x\n\u22a2 \u2016f a - f a\u2016 \u2264 C * (a - a)\n[PROOFSTEP]\nrw [sub_self, norm_zero, sub_self, mul_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 HasDerivWithinAt f (f' x) (Icc a b) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nrefine'\n  norm_image_sub_le_of_norm_deriv_right_le_segment (fun x hx => (hf x hx).continuousWithinAt) (fun x hx => _) bound\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 HasDerivWithinAt f (f' x) (Icc a b) x\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x\u2016 \u2264 C\nx : \u211d\nhx : x \u2208 Ico a b\n\u22a2 HasDerivWithinAt (fun x => f x) (f' x) (Ici x) x\n[PROOFSTEP]\nexact (hf x <| Ico_subset_Icc_self hx).nhdsWithin (Icc_mem_nhdsWithin_Ici hx)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b C : \u211d\nhf : DifferentiableOn \u211d f (Icc a b)\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016derivWithin f (Icc a b) x\u2016 \u2264 C\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 C * (x - a)\n[PROOFSTEP]\nrefine' norm_image_sub_le_of_norm_deriv_le_segment' _ bound\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b C : \u211d\nhf : DifferentiableOn \u211d f (Icc a b)\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016derivWithin f (Icc a b) x\u2016 \u2264 C\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 HasDerivWithinAt (fun x => f x) (derivWithin f (Icc a b) x) (Icc a b) x\n[PROOFSTEP]\nexact fun x hx => (hf x hx).hasDerivWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' : \u211d \u2192 E\nC : \u211d\nhf : \u2200 (x : \u211d), x \u2208 Icc 0 1 \u2192 HasDerivWithinAt f (f' x) (Icc 0 1) x\nbound : \u2200 (x : \u211d), x \u2208 Ico 0 1 \u2192 \u2016f' x\u2016 \u2264 C\n\u22a2 \u2016f 1 - f 0\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [sub_zero, mul_one] using\n  norm_image_sub_le_of_norm_deriv_le_segment' hf bound 1 (right_mem_Icc.2 zero_le_one)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b C : \u211d\nhf : DifferentiableOn \u211d f (Icc 0 1)\nbound : \u2200 (x : \u211d), x \u2208 Ico 0 1 \u2192 \u2016derivWithin f (Icc 0 1) x\u2016 \u2264 C\n\u22a2 \u2016f 1 - f 0\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [sub_zero, mul_one] using norm_image_sub_le_of_norm_deriv_le_segment hf bound 1 (right_mem_Icc.2 zero_le_one)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f 0 (Ici x) x\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x = f a\n[PROOFSTEP]\nhave : \u2200 x \u2208 Icc a b, \u2016f x - f a\u2016 \u2264 0 * (x - a) := fun x hx =>\n  norm_image_sub_le_of_norm_deriv_right_le_segment hcont hderiv (fun _ _ => norm_zero.le) x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nhcont : ContinuousOn f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f 0 (Ici x) x\nthis : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2016f x - f a\u2016 \u2264 0 * (x - a)\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x = f a\n[PROOFSTEP]\nsimpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nhdiff : DifferentiableOn \u211d f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 derivWithin f (Icc a b) x = 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x = f a\n[PROOFSTEP]\nhave H : \u2200 x \u2208 Ico a b, \u2016derivWithin f (Icc a b) x\u2016 \u2264 0 := by\n  simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nhdiff : DifferentiableOn \u211d f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 derivWithin f (Icc a b) x = 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016derivWithin f (Icc a b) x\u2016 \u2264 0\n[PROOFSTEP]\nsimpa only [norm_le_zero_iff] using fun x hx => hderiv x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nhdiff : DifferentiableOn \u211d f (Icc a b)\nhderiv : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 derivWithin f (Icc a b) x = 0\nH : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016derivWithin f (Icc a b) x\u2016 \u2264 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x = f a\n[PROOFSTEP]\nsimpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>\n  norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nderivf : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nderivg : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nfcont : ContinuousOn f (Icc a b)\ngcont : ContinuousOn g (Icc a b)\nhi : f a = g a\n\u22a2 \u2200 (y : \u211d), y \u2208 Icc a b \u2192 f y = g y\n[PROOFSTEP]\nsimp only [\u2190 @sub_eq_zero _ _ (f _)] at hi \u22a2\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nderivf : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nderivg : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nfcont : ContinuousOn f (Icc a b)\ngcont : ContinuousOn g (Icc a b)\nhi : f a - g a = 0\n\u22a2 \u2200 (y : \u211d), y \u2208 Icc a b \u2192 f y - g y = 0\n[PROOFSTEP]\nexact\n  hi \u25b8\n    constant_of_has_deriv_right_zero (fcont.sub gcont) fun y hy => by\n      simpa only [sub_self] using (derivf y hy).sub (derivg y hy)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nderivf : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt f (f' x) (Ici x) x\nderivg : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 HasDerivWithinAt g (f' x) (Ici x) x\nfcont : ContinuousOn f (Icc a b)\ngcont : ContinuousOn g (Icc a b)\nhi : f a - g a = 0\ny : \u211d\nhy : y \u2208 Ico a b\n\u22a2 HasDerivWithinAt (fun y => f y - g y) 0 (Ici y) y\n[PROOFSTEP]\nsimpa only [sub_self] using (derivf y hy).sub (derivg y hy)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nfdiff : DifferentiableOn \u211d f (Icc a b)\ngdiff : DifferentiableOn \u211d g (Icc a b)\nhderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)\nhi : f a = g a\n\u22a2 \u2200 (y : \u211d), y \u2208 Icc a b \u2192 f y = g y\n[PROOFSTEP]\nhave A : \u2200 y \u2208 Ico a b, HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y := fun y hy =>\n  (fdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.nhdsWithin (Icc_mem_nhdsWithin_Ici hy)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nfdiff : DifferentiableOn \u211d f (Icc a b)\ngdiff : DifferentiableOn \u211d g (Icc a b)\nhderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)\nhi : f a = g a\nA : \u2200 (y : \u211d), y \u2208 Ico a b \u2192 HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y\n\u22a2 \u2200 (y : \u211d), y \u2208 Icc a b \u2192 f y = g y\n[PROOFSTEP]\nhave B : \u2200 y \u2208 Ico a b, HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y := fun y hy =>\n  (gdiff y (mem_Icc_of_Ico hy)).hasDerivWithinAt.nhdsWithin (Icc_mem_nhdsWithin_Ici hy)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 E\na b : \u211d\nf' g : \u211d \u2192 E\nfdiff : DifferentiableOn \u211d f (Icc a b)\ngdiff : DifferentiableOn \u211d g (Icc a b)\nhderiv : EqOn (derivWithin f (Icc a b)) (derivWithin g (Icc a b)) (Ico a b)\nhi : f a = g a\nA : \u2200 (y : \u211d), y \u2208 Ico a b \u2192 HasDerivWithinAt f (derivWithin f (Icc a b) y) (Ici y) y\nB : \u2200 (y : \u211d), y \u2208 Ico a b \u2192 HasDerivWithinAt g (derivWithin g (Icc a b) y) (Ici y) y\n\u22a2 \u2200 (y : \u211d), y \u2208 Icc a b \u2192 f y = g y\n[PROOFSTEP]\nexact eq_of_has_deriv_right_eq A (fun y hy => (hderiv hy).symm \u25b8 B y hy) fdiff.continuousOn gdiff.continuousOn hi\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nletI : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nset g := (AffineMap.lineMap x y : \u211d \u2192 E)\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\ng : (a : \u211d) \u2192 (fun a => E) a := \u2191(AffineMap.lineMap x y)\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nhave segm : MapsTo g (Icc 0 1 : Set \u211d) s := hs.mapsTo_lineMap xs ys\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\ng : (a : \u211d) \u2192 (fun a => E) a := \u2191(AffineMap.lineMap x y)\nsegm : MapsTo g (Icc 0 1) s\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nhave hD : \u2200 t \u2208 Icc (0 : \u211d) 1, HasDerivWithinAt (f \u2218 g) (f' (g t) (y - x)) (Icc 0 1) t := fun t ht => by\n  simpa using ((hf (g t) (segm ht)).restrictScalars \u211d).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\ng : (a : \u211d) \u2192 (fun a => E) a := \u2191(AffineMap.lineMap x y)\nsegm : MapsTo g (Icc 0 1) s\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) (Icc 0 1) t\n[PROOFSTEP]\nsimpa using ((hf (g t) (segm ht)).restrictScalars \u211d).comp_hasDerivWithinAt _ AffineMap.hasDerivWithinAt_lineMap segm\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\ng : (a : \u211d) \u2192 (fun a => E) a := \u2191(AffineMap.lineMap x y)\nsegm : MapsTo g (Icc 0 1) s\nhD : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) (Icc 0 1) t\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nhave bound : \u2200 t \u2208 Ico (0 : \u211d) 1, \u2016f' (g t) (y - x)\u2016 \u2264 C * \u2016y - x\u2016 := fun t ht =>\n  le_of_op_norm_le _ (bound _ <| segm <| Ico_subset_Icc_self ht) _\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound\u271d : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\ng : (a : \u211d) \u2192 (fun a => E) a := \u2191(AffineMap.lineMap x y)\nsegm : MapsTo g (Icc 0 1) s\nhD : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) (Icc 0 1) t\nbound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016\u2191(f' (g t)) (y - x)\u2016 \u2264 C * \u2016y - x\u2016\n\u22a2 \u2016f y - f x\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nsimpa using norm_image_sub_le_of_norm_deriv_le_segment_01' hD bound\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC\u271d : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nC : \u211d\u22650\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016\u208a \u2264 C\nhs : Convex \u211d s\n\u22a2 LipschitzOnWith C f s\n[PROOFSTEP]\nrw [lipschitzOnWith_iff_norm_sub_le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC\u271d : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nC : \u211d\u22650\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016\u208a \u2264 C\nhs : Convex \u211d s\n\u22a2 \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2016f x - f y\u2016 \u2264 \u2191C * \u2016x - y\u2016\n[PROOFSTEP]\nintro x x_in y y_in\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC\u271d : \u211d\ns : Set E\nx\u271d y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nC : \u211d\u22650\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x\u2016\u208a \u2264 C\nhs : Convex \u211d s\nx : E\nx_in : x \u2208 s\ny : E\ny_in : y \u2208 s\n\u22a2 \u2016f x - f y\u2016 \u2264 \u2191C * \u2016x - y\u2016\n[PROOFSTEP]\nexact hs.norm_image_sub_le_of_norm_hasFDerivWithin_le hf bound y_in x_in\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 LipschitzOnWith K f t\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b50, h\u03b5\u27e9 : \u2203 \u03b5 > 0, ball x \u03b5 \u2229 s \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 ball x \u03b5 \u2229 s \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\ncase intro.intro\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2229 s \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 LipschitzOnWith K f t\n[PROOFSTEP]\nexact mem_nhdsWithin_iff.1 (hder.and <| hcont.nnnorm.eventually (gt_mem_nhds hK))\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2229 s \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 LipschitzOnWith K f t\n[PROOFSTEP]\nrw [inter_comm] at h\u03b5 \n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : s \u2229 ball x \u03b5 \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd[s] x \u2227 LipschitzOnWith K f t\n[PROOFSTEP]\nrefine' \u27e8s \u2229 ball x \u03b5, inter_mem_nhdsWithin _ (ball_mem_nhds _ \u03b50), _\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf\u271d g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nf : E \u2192 G\nhder : \u2200\u1da0 (y : E) in \ud835\udcdd[s] x, HasFDerivWithinAt f (f' y) s y\nhcont : ContinuousWithinAt f' s x\nK : \u211d\u22650\nhK : \u2016f' x\u2016\u208a < K\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : s \u2229 ball x \u03b5 \u2286 {y | HasFDerivWithinAt f (f' y) s y \u2227 \u2016f' y\u2016\u208a < K}\n\u22a2 LipschitzOnWith K f (s \u2229 ball x \u03b5)\n[PROOFSTEP]\nexact\n  (hs.inter (convex_ball _ _)).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le\n    (fun y hy => (h\u03b5 hy).1.mono (inter_subset_left _ _)) fun y hy => (h\u03b5 hy).2.le\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\n\u22a2 \u2016f y - f x - \u2191\u03c6 (y - x)\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nlet g y := f y - \u03c6 y\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\n\u22a2 \u2016f y - f x - \u2191\u03c6 (y - x)\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\nhave hg : \u2200 x \u2208 s, HasFDerivWithinAt g (f' x - \u03c6) s x := fun x xs => (hf x xs).sub \u03c6.hasFDerivWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 \u2016f y - f x - \u2191\u03c6 (y - x)\u2016 \u2264 C * \u2016y - x\u2016\n[PROOFSTEP]\ncalc\n  \u2016f y - f x - \u03c6 (y - x)\u2016 = \u2016f y - f x - (\u03c6 y - \u03c6 x)\u2016 := by simp\n  _ = \u2016f y - \u03c6 y - (f x - \u03c6 x)\u2016 := by congr 1; abel\n  _ = \u2016g y - g x\u2016 := by simp\n  _ \u2264 C * \u2016y - x\u2016 := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 \u2016f y - f x - \u2191\u03c6 (y - x)\u2016 = \u2016f y - f x - (\u2191\u03c6 y - \u2191\u03c6 x)\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 \u2016f y - f x - (\u2191\u03c6 y - \u2191\u03c6 x)\u2016 = \u2016f y - \u2191\u03c6 y - (f x - \u2191\u03c6 x)\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 f y - f x - (\u2191\u03c6 y - \u2191\u03c6 x) = f y - \u2191\u03c6 y - (f x - \u2191\u03c6 x)\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 f y - f x - (\u2191\u03c6 y - \u2191\u03c6 x) = f y - \u2191\u03c6 y - (f x - \u2191\u03c6 x)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016f' x - \u03c6\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : E \u2192 G := fun y => f y - \u2191\u03c6 y\nhg : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt g (f' x - \u03c6) s x\n\u22a2 \u2016f y - \u2191\u03c6 y - (f x - \u2191\u03c6 x)\u2016 = \u2016g y - g x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = 0\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 f x = f y\n[PROOFSTEP]\nhave bound : \u2200 x \u2208 s, \u2016fderivWithin \ud835\udd5c f s x\u2016 \u2264 0 := fun x hx => by simp only [hf' x hx, norm_zero, le_rfl]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx\u271d y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = 0\nhx\u271d : x\u271d \u2208 s\nhy : y \u2208 s\nx : E\nhx : x \u2208 s\n\u22a2 \u2016fderivWithin \ud835\udd5c f s x\u2016 \u2264 0\n[PROOFSTEP]\nsimp only [hf' x hx, norm_zero, le_rfl]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = 0\nhx : x \u2208 s\nhy : y \u2208 s\nbound : \u2200 (x : E), x \u2208 s \u2192 \u2016fderivWithin \ud835\udd5c f s x\u2016 \u2264 0\n\u22a2 f x = f y\n[PROOFSTEP]\nsimpa only [(dist_eq_norm _ _).symm, zero_mul, dist_le_zero, eq_comm] using\n  hs.norm_image_sub_le_of_norm_fderivWithin_le hf bound hx hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx\u271d\u00b2 y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : Differentiable \ud835\udd5c f\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = 0\nx\u271d\u00b9 y x : E\nx\u271d : x \u2208 univ\n\u22a2 fderivWithin \ud835\udd5c f univ x = 0\n[PROOFSTEP]\nrw [fderivWithin_univ]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx\u271d\u00b2 y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : Differentiable \ud835\udd5c f\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = 0\nx\u271d\u00b9 y x : E\nx\u271d : x \u2208 univ\n\u22a2 fderiv \ud835\udd5c f x = 0\n[PROOFSTEP]\nexact hf' x\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhg : DifferentiableOn \ud835\udd5c g s\nhs' : UniqueDiffOn \ud835\udd5c s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c g s x\nhx : x \u2208 s\nhfgx : f x = g x\ny : E\nhy : y \u2208 s\n\u22a2 f y = g y\n[PROOFSTEP]\nsuffices f x - g x = f y - g y by rwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhg : DifferentiableOn \ud835\udd5c g s\nhs' : UniqueDiffOn \ud835\udd5c s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c g s x\nhx : x \u2208 s\nhfgx : f x = g x\ny : E\nhy : y \u2208 s\nthis : f x - g x = f y - g y\n\u22a2 f y = g y\n[PROOFSTEP]\nrwa [hfgx, sub_self, eq_comm, sub_eq_zero] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhg : DifferentiableOn \ud835\udd5c g s\nhs' : UniqueDiffOn \ud835\udd5c s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c g s x\nhx : x \u2208 s\nhfgx : f x = g x\ny : E\nhy : y \u2208 s\n\u22a2 f x - g x = f y - g y\n[PROOFSTEP]\nrefine' hs.is_const_of_fderivWithin_eq_zero (hf.sub hg) (fun z hz => _) hx hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx y\u271d : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhs : Convex \u211d s\nhf : DifferentiableOn \ud835\udd5c f s\nhg : DifferentiableOn \ud835\udd5c g s\nhs' : UniqueDiffOn \ud835\udd5c s\nhf' : \u2200 (x : E), x \u2208 s \u2192 fderivWithin \ud835\udd5c f s x = fderivWithin \ud835\udd5c g s x\nhx : x \u2208 s\nhfgx : f x = g x\ny : E\nhy : y \u2208 s\nz : E\nhz : z \u2208 s\n\u22a2 fderivWithin \ud835\udd5c (fun y => f y - g y) s z = 0\n[PROOFSTEP]\nrw [fderivWithin_sub (hs' _ hz) (hf _ hz) (hg _ hz), sub_eq_zero, hf' _ hz]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 G\nC : \u211d\ns : Set E\nx\u271d\u00b2 y : E\nf' g' : E \u2192 E \u2192L[\ud835\udd5c] G\n\u03c6 : E \u2192L[\ud835\udd5c] G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx\u271d\u00b9 : E\nhfgx : f x\u271d\u00b9 = g x\u271d\u00b9\nx : E\nx\u271d : x \u2208 univ\n\u22a2 fderivWithin \ud835\udd5c f univ x = fderivWithin \ud835\udd5c g univ x\n[PROOFSTEP]\nsimpa using hf' _\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf f' : \ud835\udd5c \u2192 G\ns : Set \ud835\udd5c\nx\u271d y : \ud835\udd5c\nC : \u211d\nhf : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x\nbound : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 \u2016f' x\u2016 \u2264 C\nhs : Convex \u211d s\nxs : x\u271d \u2208 s\nys : y \u2208 s\nx : \ud835\udd5c\nhx : x \u2208 s\n\u22a2 \u2016smulRight 1 (f' x)\u2016 \u2264 \u2016f' x\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf f' : \ud835\udd5c \u2192 G\ns : Set \ud835\udd5c\nx\u271d y : \ud835\udd5c\nC : \u211d\u22650\nhs : Convex \u211d s\nhf : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x\nbound : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 \u2016f' x\u2016\u208a \u2264 C\nx : \ud835\udd5c\nhx : x \u2208 s\n\u22a2 \u2016smulRight 1 (f' x)\u2016\u208a \u2264 \u2016f' x\u2016\u208a\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf f' : \ud835\udd5c \u2192 G\ns : Set \ud835\udd5c\nx\u271d y\u271d : \ud835\udd5c\nhf : Differentiable \ud835\udd5c f\nhf' : \u2200 (x : \ud835\udd5c), deriv f x = 0\nx y z : \ud835\udd5c\n\u22a2 fderiv \ud835\udd5c f z = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf f' : \ud835\udd5c \u2192 G\ns : Set \ud835\udd5c\nx\u271d y\u271d : \ud835\udd5c\nhf : Differentiable \ud835\udd5c f\nhf' : \u2200 (x : \ud835\udd5c), deriv f x = 0\nx y z : \ud835\udd5c\n\u22a2 \u2191(fderiv \ud835\udd5c f z) 1 = \u21910 1\n[PROOFSTEP]\nsimp [\u2190 deriv_fderiv, hf']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nlet h x := (g b - g a) * f x - (f b - f a) * g x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nhave hI : h a = h b := by simp only; ring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\n\u22a2 h a = h b\n[PROOFSTEP]\nsimp only\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\n\u22a2 (g b - g a) * f a - (f b - f a) * g a = (g b - g a) * f b - (f b - f a) * g b\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nlet h' x := (g b - g a) * f' x - (f b - f a) * g' x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : \u211d \u2192 \u211d := fun x => (g b - g a) * f' x - (f b - f a) * g' x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nhave hhh' : \u2200 x \u2208 Ioo a b, HasDerivAt h (h' x) x := fun x hx =>\n  ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : \u211d \u2192 \u211d := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nhave hhc : ContinuousOn h (Icc a b) := (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : \u211d \u2192 \u211d := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nrcases exists_hasDerivAt_eq_zero hab hhc hI hhh' with \u27e8c, cmem, hc\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nh : \u211d \u2192 \u211d := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : \u211d \u2192 \u211d := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : h' c = 0\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (g b - g a) * f' c = (f b - f a) * g' c\n[PROOFSTEP]\nexact \u27e8c, cmem, sub_eq_zero.1 hc\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nlet h x := (lgb - lga) * f x - (lfb - lfa) * g x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nhave hha : Tendsto h (\ud835\udcdd[>] a) (\ud835\udcdd <| lgb * lfa - lfb * lga) :=\n  by\n  have : Tendsto h (\ud835\udcdd[>] a) (\ud835\udcdd <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=\n    (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)\n  convert this using 2\n  ring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\n\u22a2 Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\n[PROOFSTEP]\nhave : Tendsto h (\ud835\udcdd[>] a) (\ud835\udcdd <| (lgb - lga) * lfa - (lfb - lfa) * lga) :=\n  (tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd ((lgb - lga) * lfa - (lfb - lfa) * lga))\n\u22a2 Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\n[PROOFSTEP]\nconvert this using 2\n[GOAL]\ncase h.e'_5.h.e'_3\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nthis : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd ((lgb - lga) * lfa - (lfb - lfa) * lga))\n\u22a2 lgb * lfa - lfb * lga = (lgb - lga) * lfa - (lfb - lfa) * lga\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nhave hhb : Tendsto h (\ud835\udcdd[<] b) (\ud835\udcdd <| lgb * lfa - lfb * lga) :=\n  by\n  have : Tendsto h (\ud835\udcdd[<] b) (\ud835\udcdd <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=\n    (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)\n  convert this using 2\n  ring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\n\u22a2 Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\n[PROOFSTEP]\nhave : Tendsto h (\ud835\udcdd[<] b) (\ud835\udcdd <| (lgb - lga) * lfb - (lfb - lfa) * lgb) :=\n  (tendsto_const_nhds.mul hfb).sub (tendsto_const_nhds.mul hgb)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nthis : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n\u22a2 Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\n[PROOFSTEP]\nconvert this using 2\n[GOAL]\ncase h.e'_5.h.e'_3\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nthis : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd ((lgb - lga) * lfb - (lfb - lfa) * lgb))\n\u22a2 lgb * lfa - lfb * lga = (lgb - lga) * lfb - (lfb - lfa) * lgb\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nlet h' x := (lgb - lga) * f' x - (lfb - lfa) * g' x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\nh' : \u211d \u2192 \u211d := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nhave hhh' : \u2200 x \u2208 Ioo a b, HasDerivAt h (h' x) x := by\n  intro x hx\n  exact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\nh' : \u211d \u2192 \u211d := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\nh' : \u211d \u2192 \u211d := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 HasDerivAt h (h' x) x\n[PROOFSTEP]\nexact ((hff' x hx).const_mul _).sub ((hgg' x hx).const_mul _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\nh' : \u211d \u2192 \u211d := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nrcases exists_hasDerivAt_eq_zero' hab hha hhb hhh' with \u27e8c, cmem, hc\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg'\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nlfa lga lfb lgb : \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lfa)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd lga)\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd lfb)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd lgb)\nh : \u211d \u2192 \u211d := fun x => (lgb - lga) * f x - (lfb - lfa) * g x\nhha : Tendsto h (\ud835\udcdd[Ioi a] a) (\ud835\udcdd (lgb * lfa - lfb * lga))\nhhb : Tendsto h (\ud835\udcdd[Iio b] b) (\ud835\udcdd (lgb * lfa - lfb * lga))\nh' : \u211d \u2192 \u211d := fun x => (lgb - lga) * f' x - (lfb - lfa) * g' x\nhhh' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt h (h' x) x\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : h' c = 0\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 (lgb - lga) * f' c = (lfb - lfa) * g' c\n[PROOFSTEP]\nexact \u27e8c, cmem, sub_eq_zero.1 hc\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 f' c = (f b - f a) / (b - a)\n[PROOFSTEP]\nobtain \u27e8c, cmem, hc\u27e9 : \u2203 c \u2208 Ioo a b, (b - a) * f' c = (f b - f a) * 1 :=\n  exists_ratio_hasDerivAt_eq_ratio_slope f f' hab hfc hff' id 1 continuousOn_id fun x _ => hasDerivAt_id x\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n\u22a2 \u2203 c, c \u2208 Ioo a b \u2227 f' c = (f b - f a) / (b - a)\n[PROOFSTEP]\nuse c, cmem\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\na b : \u211d\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhfd : DifferentiableOn \u211d f (Ioo a b)\ng g' : \u211d \u2192 \u211d\nhgc : ContinuousOn g (Icc a b)\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhgd : DifferentiableOn \u211d g (Ioo a b)\nc : \u211d\ncmem : c \u2208 Ioo a b\nhc : (b - a) * f' c = (f b - f a) * 1\n\u22a2 f' c = (f b - f a) / (b - a)\n[PROOFSTEP]\nrwa [mul_one, mul_comm, \u2190 eq_div_iff (sub_ne_zero.2 hab.ne')] at hc \n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\n\u22a2 \u2200 (x : \u211d), x \u2208 D \u2192 \u2200 (y : \u211d), y \u2208 D \u2192 x < y \u2192 C * (y - x) < f y - f x\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\n\u22a2 C * (y - x) < f y - f x\n[PROOFSTEP]\nhave hxyD : Icc x y \u2286 D := hD.ordConnected.out hx hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\nhxyD : Icc x y \u2286 D\n\u22a2 C * (y - x) < f y - f x\n[PROOFSTEP]\nhave hxyD' : Ioo x y \u2286 interior D := subset_sUnion_of_mem \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\n\u22a2 C * (y - x) < f y - f x\n[PROOFSTEP]\nobtain \u27e8a, a_mem, ha\u27e9 : \u2203 a \u2208 Ioo x y, deriv f a = (f y - f x) / (y - x) :=\n  exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\na : \u211d\na_mem : a \u2208 Ioo x y\nha : deriv f a = (f y - f x) / (y - x)\n\u22a2 C * (y - x) < f y - f x\n[PROOFSTEP]\nhave : C < (f y - f x) / (y - x) := ha \u25b8 hf'_gt _ (hxyD' a_mem)\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 C < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\na : \u211d\na_mem : a \u2208 Ioo x y\nha : deriv f a = (f y - f x) / (y - x)\nthis : C < (f y - f x) / (y - x)\n\u22a2 C * (y - x) < f y - f x\n[PROOFSTEP]\nexact (lt_div_iff (sub_pos.2 hxy)).1 this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\n\u22a2 \u2200 (x : \u211d), x \u2208 D \u2192 \u2200 (y : \u211d), y \u2208 D \u2192 x \u2264 y \u2192 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\ncases' eq_or_lt_of_le hxy with hxy' hxy'\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x = y\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nrw [hxy', sub_self, sub_self, mul_zero]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nhave hxyD : Icc x y \u2286 D := hD.ordConnected.out hx hy\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nhave hxyD' : Ioo x y \u2286 interior D := subset_sUnion_of_mem \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD\u27e9\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nobtain \u27e8a, a_mem, ha\u27e9 : \u2203 a \u2208 Ioo x y, deriv f a = (f y - f x) / (y - x)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a = (f y - f x) / (y - x)\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\na : \u211d\na_mem : a \u2208 Ioo x y\nha : deriv f a = (f y - f x) / (y - x)\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nexact exists_deriv_eq_slope f hxy' (hf.mono hxyD) (hf'.mono hxyD')\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\na : \u211d\na_mem : a \u2208 Ioo x y\nha : deriv f a = (f y - f x) / (y - x)\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nhave : C \u2264 (f y - f x) / (y - x) := ha \u25b8 hf'_ge _ (hxyD' a_mem)\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 C \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhxy' : x < y\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\na : \u211d\na_mem : a \u2208 Ioo x y\nha : deriv f a = (f y - f x) / (y - x)\nthis : C \u2264 (f y - f x) / (y - x)\n\u22a2 C * (y - x) \u2264 f y - f x\n[PROOFSTEP]\nexact (le_div_iff (sub_pos.2 hxy')).1 this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nlt_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x < C\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x\u271d < y\nx : \u211d\nhx : x \u2208 interior D\n\u22a2 -C < deriv (fun y => -f y) x\n[PROOFSTEP]\nrw [deriv.neg, neg_lt_neg_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nlt_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x < C\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x\u271d < y\nx : \u211d\nhx : x \u2208 interior D\n\u22a2 deriv (fun y => f y) x < C\n[PROOFSTEP]\nexact lt_hf' x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nlt_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x < C\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x < y\nhf'_gt : \u2200 (x : \u211d), x \u2208 interior D \u2192 -C < deriv (fun y => -f y) x\n\u22a2 f y - f x < C * (y - x)\n[PROOFSTEP]\nlinarith [hD.mul_sub_lt_image_sub_of_lt_deriv hf.neg hf'.neg hf'_gt x hx y hy hxy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nle_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x \u2264 C\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x\u271d \u2264 y\nx : \u211d\nhx : x \u2208 interior D\n\u22a2 -C \u2264 deriv (fun y => -f y) x\n[PROOFSTEP]\nrw [deriv.neg, neg_le_neg_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nle_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x \u2264 C\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x\u271d \u2264 y\nx : \u211d\nhx : x \u2208 interior D\n\u22a2 deriv (fun y => f y) x \u2264 C\n[PROOFSTEP]\nexact le_hf' x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nC : (fun x => \u211d) 1\nle_hf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x \u2264 C\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\nhf'_ge : \u2200 (x : \u211d), x \u2208 interior D \u2192 -C \u2264 deriv (fun y => -f y) x\n\u22a2 f y - f x \u2264 C * (y - x)\n[PROOFSTEP]\nlinarith [hD.mul_sub_le_image_sub_of_le_deriv hf.neg hf'.neg hf'_ge x hx y hy hxy]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 < deriv f x\n\u22a2 StrictMonoOn f D\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\n\u22a2 x < y \u2192 f x < f y\n[PROOFSTEP]\nhave : DifferentiableOn \u211d f (interior D) := fun z hz =>\n  (differentiableAt_of_deriv_ne_zero (hf' z hz).ne').differentiableWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 < deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nthis : DifferentiableOn \u211d f (interior D)\n\u22a2 x < y \u2192 f x < f y\n[PROOFSTEP]\nsimpa only [zero_mul, sub_pos] using hD.mul_sub_lt_image_sub_of_lt_deriv hf this hf' x hx y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv f x\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nsimpa only [zero_mul, sub_nonneg] using hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x < 0\nx : \u211d\nhx : x \u2208 D\ny : \u211d\n\u22a2 y \u2208 D \u2192 x < y \u2192 f y < f x\n[PROOFSTEP]\nsimpa only [zero_mul, sub_lt_zero] using\n  hD.image_sub_lt_mul_sub_of_deriv_lt hf\n    (fun z hz => (differentiableAt_of_deriv_ne_zero (hf' z hz).ne).differentiableWithinAt) hf' x hx y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_nonpos : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv f x \u2264 0\nx : \u211d\nhx : x \u2208 D\ny : \u211d\nhy : y \u2208 D\nhxy : x \u2264 y\n\u22a2 f y \u2264 f x\n[PROOFSTEP]\nsimpa only [zero_mul, sub_nonpos] using hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\n\u22a2 \u2200 {x y z : \u211d}, x \u2208 D \u2192 z \u2208 D \u2192 x < y \u2192 y < z \u2192 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nintro x y z hx hz hxy hyz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxzD : Icc x z \u2286 D := hD.ordConnected.out hx hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxyD : Icc x y \u2286 D := (Icc_subset_Icc_right hyz.le).trans hxzD\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxyD' : Ioo x y \u2286 interior D := subset_sUnion_of_mem \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyzD : Icc y z \u2286 D := (Icc_subset_Icc_left hxy.le).trans hxzD\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyzD' : Ioo y z \u2286 interior D :=\n  subset_sUnion_of_mem\n    \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD\u27e9\n      -- Then we apply MVT to both `[x, y]` and `[y, z]`\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, hay\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x y, deriv f a = (f y - f x) / (y - x)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a = (f y - f x) / (y - x)\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nexact exists_deriv_eq_slope f hxy (hf.mono hxyD) (hf'.mono hxyD')\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nobtain \u27e8b, \u27e8hyb, hbz\u27e9, hb\u27e9 : \u2203 b \u2208 Ioo y z, deriv f b = (f z - f y) / (z - y)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 b, b \u2208 Ioo y z \u2227 deriv f b = (f z - f y) / (z - y)\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhb : deriv f b = (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nexact exists_deriv_eq_slope f hyz (hf.mono hyzD) (hf'.mono hyzD')\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhb : deriv f b = (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n\u22a2 (f y - f x) / (y - x) \u2264 (f z - f y) / (z - y)\n[PROOFSTEP]\nrw [\u2190 ha, \u2190 hb]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'_mono : MonotoneOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhb : deriv f b = (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n\u22a2 deriv f a \u2264 deriv f b\n[PROOFSTEP]\nexact hf'_mono (hxyD' \u27e8hxa, hay\u27e9) (hyzD' \u27e8hyb, hbz\u27e9) (hay.trans hyb).le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nh_anti : AntitoneOn (deriv f) (interior D)\n\u22a2 MonotoneOn (deriv (-f)) (interior D)\n[PROOFSTEP]\nsimpa only [\u2190 deriv.neg] using h_anti.neg\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nhave A : DifferentiableOn \u211d f (Ioo x y) := fun w wmem =>\n  (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, hay\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x y, deriv f a = (f y - f x) / (y - x)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a = (f y - f x) / (y - x)\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nexact exists_deriv_eq_slope f hxy hf A\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nrcases nonempty_Ioo.2 hay with \u27e8b, \u27e8hab, hby\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhab : a < b\nhby : b < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nrefine' \u27e8b, \u27e8hxa.trans hab, hby\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhab : a < b\nhby : b < y\n\u22a2 (f y - f x) / (y - x) < deriv f b\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhab : a < b\nhby : b < y\n\u22a2 deriv f a < deriv f b\n[PROOFSTEP]\nexact hf'_mono \u27e8hxa, hay\u27e9 \u27e8hxa.trans hab, hby\u27e9 hab\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nby_cases h : \u2200 w \u2208 Ioo x y, deriv f w \u2260 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\napply StrictMonoOn.exists_slope_lt_deriv_aux hf hxy hf'_mono h\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u00ac\u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2203 w, w \u2208 Ioo x y \u2227 deriv f w = 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nrcases h with \u27e8w, \u27e8hxw, hwy\u27e9, hw\u27e9\n[GOAL]\ncase neg.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, haw\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x w, (f w - f x) / (w - x) < deriv f a :=\n  by\n  apply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _\n  \u00b7 exact hf.mono (Icc_subset_Icc le_rfl hwy.le)\n  \u00b7 exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)\n  \u00b7 intro z hz\n    rw [\u2190 hw]\n    apply ne_of_lt\n    exact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x w \u2227 (f w - f x) / (w - x) < deriv f a\n[PROOFSTEP]\napply StrictMonoOn.exists_slope_lt_deriv_aux _ hxw _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 ContinuousOn (fun w => f w) (Icc x w)\n[PROOFSTEP]\nexact hf.mono (Icc_subset_Icc le_rfl hwy.le)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 StrictMonoOn (deriv fun w => f w) (Ioo x w)\n[PROOFSTEP]\nexact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo x w \u2192 deriv (fun w => f w) w_1 \u2260 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 0\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z \u2260 deriv f w\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv (fun w => f w) z < deriv f w\n[PROOFSTEP]\nexact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nobtain \u27e8b, \u27e8hwb, hby\u27e9, hb\u27e9 : \u2203 b \u2208 Ioo w y, (f y - f w) / (y - w) < deriv f b :=\n  by\n  apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _\n  \u00b7 refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)\n  \u00b7 exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)\n  \u00b7 intro z hz\n    rw [\u2190 hw]\n    apply ne_of_gt\n    exact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2203 b, b \u2208 Ioo w y \u2227 (f y - f w) / (y - w) < deriv f b\n[PROOFSTEP]\napply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 ContinuousOn (fun {y} => f y) (Icc w y)\n[PROOFSTEP]\nrefine' hf.mono (Icc_subset_Icc hxw.le le_rfl)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 StrictMonoOn (deriv fun {y} => f y) (Ioo w y)\n[PROOFSTEP]\nexact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo w y \u2192 deriv (fun {y} => f y) w_1 \u2260 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 0\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv (fun {y} => f y) z \u2260 deriv f w\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f w < deriv (fun {y} => f y) z\n[PROOFSTEP]\nexact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : (f y - f w) / (y - w) < deriv f b\nhwb : w < b\nhby : b < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nrefine' \u27e8b, \u27e8hxw.trans hwb, hby\u27e9, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : (f y - f w) / (y - w) < deriv f b\nhwb : w < b\nhby : b < y\n\u22a2 (f y - f x) / (y - x) < deriv f b\n[PROOFSTEP]\nsimp only [div_lt_iff, hxy, hxw, hwy, sub_pos] at ha hb \u22a2\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 f y - f x < deriv f b * (y - x)\n[PROOFSTEP]\nhave : deriv f a * (w - x) < deriv f b * (w - x) :=\n  by\n  apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _\n  \u00b7 exact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)\n  \u00b7 rw [\u2190 hw]\n    exact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f a * (w - x) < deriv f b * (w - x)\n[PROOFSTEP]\napply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f a < deriv f b\n[PROOFSTEP]\nexact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 0 \u2264 deriv f b\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\n\u22a2 deriv f w \u2264 deriv f b\n[PROOFSTEP]\nexact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\nthis : deriv f a * (w - x) < deriv f b * (w - x)\n\u22a2 f y - f x < deriv f b * (y - x)\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nhave A : DifferentiableOn \u211d f (Ioo x y) := fun w wmem =>\n  (differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, hay\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x y, deriv f a = (f y - f x) / (y - x)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a = (f y - f x) / (y - x)\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nexact exists_deriv_eq_slope f hxy hf A\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nrcases nonempty_Ioo.2 hxa with \u27e8b, \u27e8hxb, hba\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhxb : x < b\nhba : b < a\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nrefine' \u27e8b, \u27e8hxb, hba.trans hay\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhxb : x < b\nhba : b < a\n\u22a2 deriv f b < (f y - f x) / (y - x)\n[PROOFSTEP]\nrw [\u2190 ha]\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\nA : DifferentiableOn \u211d f (Ioo x y)\na : \u211d\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\nb : \u211d\nhxb : x < b\nhba : b < a\n\u22a2 deriv f b < deriv f a\n[PROOFSTEP]\nexact hf'_mono \u27e8hxb, hba.trans hay\u27e9 \u27e8hxa, hay\u27e9 hba\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nby_cases h : \u2200 w \u2208 Ioo x y, deriv f w \u2260 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\napply StrictMonoOn.exists_deriv_lt_slope_aux hf hxy hf'_mono h\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u00ac\u2200 (w : \u211d), w \u2208 Ioo x y \u2192 deriv f w \u2260 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : \u2203 w, w \u2208 Ioo x y \u2227 deriv f w = 0\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nrcases h with \u27e8w, \u27e8hxw, hwy\u27e9, hw\u27e9\n[GOAL]\ncase neg.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, haw\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x w, deriv f a < (f w - f x) / (w - x) :=\n  by\n  apply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _\n  \u00b7 exact hf.mono (Icc_subset_Icc le_rfl hwy.le)\n  \u00b7 exact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)\n  \u00b7 intro z hz\n    rw [\u2190 hw]\n    apply ne_of_lt\n    exact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2203 a, a \u2208 Ioo x w \u2227 deriv f a < (f w - f x) / (w - x)\n[PROOFSTEP]\napply StrictMonoOn.exists_deriv_lt_slope_aux _ hxw _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 ContinuousOn f (Icc x w)\n[PROOFSTEP]\nexact hf.mono (Icc_subset_Icc le_rfl hwy.le)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 StrictMonoOn (deriv f) (Ioo x w)\n[PROOFSTEP]\nexact hf'_mono.mono (Ioo_subset_Ioo le_rfl hwy.le)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo x w \u2192 deriv f w_1 \u2260 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv f z \u2260 0\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv f z \u2260 deriv f w\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\nz : \u211d\nhz : z \u2208 Ioo x w\n\u22a2 deriv f z < deriv f w\n[PROOFSTEP]\nexact hf'_mono \u27e8hz.1, hz.2.trans hwy\u27e9 \u27e8hxw, hwy\u27e9 hz.2\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nobtain \u27e8b, \u27e8hwb, hby\u27e9, hb\u27e9 : \u2203 b \u2208 Ioo w y, deriv f b < (f y - f w) / (y - w) :=\n  by\n  apply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _\n  \u00b7 refine' hf.mono (Icc_subset_Icc hxw.le le_rfl)\n  \u00b7 exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)\n  \u00b7 intro z hz\n    rw [\u2190 hw]\n    apply ne_of_gt\n    exact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\n\u22a2 \u2203 b, b \u2208 Ioo w y \u2227 deriv f b < (f y - f w) / (y - w)\n[PROOFSTEP]\napply StrictMonoOn.exists_deriv_lt_slope_aux _ hwy _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\n\u22a2 ContinuousOn f (Icc w y)\n[PROOFSTEP]\nrefine' hf.mono (Icc_subset_Icc hxw.le le_rfl)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\n\u22a2 StrictMonoOn (deriv f) (Ioo w y)\n[PROOFSTEP]\nexact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\n\u22a2 \u2200 (w_1 : \u211d), w_1 \u2208 Ioo w y \u2192 deriv f w_1 \u2260 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f z \u2260 0\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f z \u2260 deriv f w\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nz : \u211d\nhz : z \u2208 Ioo w y\n\u22a2 deriv f w < deriv f z\n[PROOFSTEP]\nexact hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hz.1, hz.2\u27e9 hz.1\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : deriv f b < (f y - f w) / (y - w)\nhwb : w < b\nhby : b < y\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nrefine' \u27e8a, \u27e8hxa, haw.trans hwy\u27e9, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nha : deriv f a < (f w - f x) / (w - x)\nhxa : x < a\nhaw : a < w\nb : \u211d\nhb : deriv f b < (f y - f w) / (y - w)\nhwb : w < b\nhby : b < y\n\u22a2 deriv f a < (f y - f x) / (y - x)\n[PROOFSTEP]\nsimp only [lt_div_iff, hxy, hxw, hwy, sub_pos] at ha hb \u22a2\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n\u22a2 deriv f a * (y - x) < f y - f x\n[PROOFSTEP]\nhave : deriv f a * (y - w) < deriv f b * (y - w) :=\n  by\n  apply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _\n  \u00b7 exact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)\n  \u00b7 rw [\u2190 hw]\n    exact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n\u22a2 deriv f a * (y - w) < deriv f b * (y - w)\n[PROOFSTEP]\napply mul_lt_mul _ le_rfl (sub_pos.2 hwy) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n\u22a2 deriv f a < deriv f b\n[PROOFSTEP]\nexact hf'_mono \u27e8hxa, haw.trans hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 (haw.trans hwb)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n\u22a2 0 \u2264 deriv f b\n[PROOFSTEP]\nrw [\u2190 hw]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\n\u22a2 deriv f w \u2264 deriv f b\n[PROOFSTEP]\nexact (hf'_mono \u27e8hxw, hwy\u27e9 \u27e8hxw.trans hwb, hby\u27e9 hwb).le\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : \u211d\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : \u211d\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : \u211d\nhxa : x < a\nhaw : a < w\nb : \u211d\nhwb : w < b\nhby : b < y\nha : deriv f a * (w - x) < f w - f x\nhb : deriv f b * (y - w) < f y - f w\nthis : deriv f a * (y - w) < deriv f b * (y - w)\n\u22a2 deriv f a * (y - x) < f y - f x\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxzD : Icc x z \u2286 D := hD.ordConnected.out hx hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxyD : Icc x y \u2286 D := (Icc_subset_Icc_right hyz.le).trans hxzD\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hxyD' : Ioo x y \u2286 interior D := subset_sUnion_of_mem \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyzD : Icc y z \u2286 D := (Icc_subset_Icc_left hxy.le).trans hxzD\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nhave hyzD' : Ioo y z \u2286 interior D :=\n  subset_sUnion_of_mem\n    \u27e8isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD\u27e9\n      -- Then we get points `a` and `b` in each interval `[x, y]` and `[y, z]` where the derivatives\n          -- can be compared to the slopes between `x, y` and `y, z` respectively.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nobtain \u27e8a, \u27e8hxa, hay\u27e9, ha\u27e9 : \u2203 a \u2208 Ioo x y, (f y - f x) / (y - x) < deriv f a\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\n\u22a2 \u2203 a, a \u2208 Ioo x y \u2227 (f y - f x) / (y - x) < deriv f a\n[PROOFSTEP]\nexact StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\nobtain \u27e8b, \u27e8hyb, hbz\u27e9, hb\u27e9 : \u2203 b \u2208 Ioo y z, deriv f b < (f z - f y) / (z - y)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n\u22a2 \u2203 b, b \u2208 Ioo y z \u2227 deriv f b < (f z - f y) / (z - y)\n[PROOFSTEP]\nexact StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD')\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : \u211d\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n\u22a2 (f y - f x) / (y - x) < (f z - f y) / (z - y)\n[PROOFSTEP]\napply ha.trans (lt_trans _ hb)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : \u211d\nhx : x \u2208 D\nhz : z \u2208 D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z \u2286 D\nhxyD : Icc x y \u2286 D\nhxyD' : Ioo x y \u2286 interior D\nhyzD : Icc y z \u2286 D\nhyzD' : Ioo y z \u2286 interior D\na : \u211d\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : \u211d\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n\u22a2 deriv f a < deriv f b\n[PROOFSTEP]\nexact hf' (hxyD' \u27e8hxa, hay\u27e9) (hyzD' \u27e8hyb, hbz\u27e9) (hay.trans hyb)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nh_anti : StrictAntiOn (deriv f) (interior D)\n\u22a2 StrictMonoOn (deriv (-f)) (interior D)\n[PROOFSTEP]\nsimpa only [\u2190 deriv.neg] using h_anti.neg\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x\n\u22a2 DifferentiableOn \u211d (deriv f) (interior (interior D))\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonneg : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 \u2264 deriv^[2] f x\n\u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 0 \u2264 deriv (deriv f) x\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonpos : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv^[2] f x \u2264 0\n\u22a2 DifferentiableOn \u211d (deriv f) (interior (interior D))\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf' : DifferentiableOn \u211d f (interior D)\nhf'' : DifferentiableOn \u211d (deriv f) (interior D)\nhf''_nonpos : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv^[2] f x \u2264 0\n\u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 deriv (deriv f) x \u2264 0\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf'' : \u2200 (x : \u211d), x \u2208 interior D \u2192 0 < deriv^[2] f x\n\u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 0 < deriv (deriv f) x\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nD : Set \u211d\nhD : Convex \u211d D\nf : \u211d \u2192 \u211d\nhf : ContinuousOn f D\nhf'' : \u2200 (x : \u211d), x \u2208 interior D \u2192 deriv^[2] f x < 0\n\u22a2 \u2200 (x : \u211d), x \u2208 interior (interior D) \u2192 deriv (deriv f) x < 0\n[PROOFSTEP]\nrwa [interior_interior]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nset g : \u211d \u2192 E := fun t => AffineMap.lineMap x y t\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nset I := Icc (0 : \u211d) 1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nhave hsub : Ioo (0 : \u211d) 1 \u2286 I := Ioo_subset_Icc_self\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nhave hmaps : MapsTo g I s := hs.mapsTo_lineMap xs ys\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nhave hfg : \u2200 t \u2208 I, HasDerivWithinAt (f \u2218 g) (f' (g t) (y - x)) I t := fun t ht =>\n  (hf _ (hmaps ht)).comp_hasDerivWithinAt t AffineMap.hasDerivWithinAt_lineMap hmaps\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nhave hMVT : \u2203 t \u2208 Ioo (0 : \u211d) 1, f' (g t) (y - x) = (f (g 1) - f (g 0)) / (1 - 0) :=\n  by\n  refine' exists_hasDerivAt_eq_slope (f \u2218 g) _ (by norm_num) _ _\n  \u00b7 exact fun t Ht => (hfg t Ht).continuousWithinAt\n  \u00b7\n    exact fun t Ht =>\n      (hfg t <| hsub Ht).hasDerivAt\n        (Icc_mem_nhds Ht.1 Ht.2)\n          -- reinterpret on domain\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\n\u22a2 \u2203 t, t \u2208 Ioo 0 1 \u2227 \u2191(f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)\n[PROOFSTEP]\nrefine' exists_hasDerivAt_eq_slope (f \u2218 g) _ (by norm_num) _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\n\u22a2 ContinuousOn (f \u2218 g) (Icc 0 1)\n[PROOFSTEP]\nexact fun t Ht => (hfg t Ht).continuousWithinAt\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\n\u22a2 \u2200 (x_1 : \u211d), x_1 \u2208 Ioo 0 1 \u2192 HasDerivAt (f \u2218 g) (\u2191(f' (g x_1)) (y - x)) x_1\n[PROOFSTEP]\nexact fun t Ht =>\n  (hfg t <| hsub Ht).hasDerivAt\n    (Icc_mem_nhds Ht.1 Ht.2)\n      -- reinterpret on domain\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\nhMVT : \u2203 t, t \u2208 Ioo 0 1 \u2227 \u2191(f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nrcases hMVT with \u27e8t, Ht, hMVT'\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\nt : \u211d\nHt : t \u2208 Ioo 0 1\nhMVT' : \u2191(f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)\n\u22a2 \u2203 z, z \u2208 segment \u211d x y \u2227 f y - f x = \u2191(f' z) (y - x)\n[PROOFSTEP]\nrw [segment_eq_image_lineMap, bex_image_iff]\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\nt : \u211d\nHt : t \u2208 Ioo 0 1\nhMVT' : \u2191(f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)\n\u22a2 \u2203 x_1, x_1 \u2208 Icc 0 1 \u2227 f y - f x = \u2191(f' (\u2191(AffineMap.lineMap x y) x_1)) (y - x)\n[PROOFSTEP]\nrefine \u27e8t, hsub Ht, ?_\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 \u211d\ns : Set E\nx y : E\nf' : E \u2192 E \u2192L[\u211d] \u211d\nhf : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nhs : Convex \u211d s\nxs : x \u2208 s\nys : y \u2208 s\ng : \u211d \u2192 E := fun t => \u2191(AffineMap.lineMap x y) t\nI : Set \u211d := Icc 0 1\nhsub : Ioo 0 1 \u2286 I\nhmaps : MapsTo g I s\nhfg : \u2200 (t : \u211d), t \u2208 I \u2192 HasDerivWithinAt (f \u2218 g) (\u2191(f' (g t)) (y - x)) I t\nt : \u211d\nHt : t \u2208 Ioo 0 1\nhMVT' : \u2191(f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)\n\u22a2 f y - f x = \u2191(f' (\u2191(AffineMap.lineMap x y) t)) (y - x)\n[PROOFSTEP]\nsimpa using hMVT'.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\n\u22a2 HasStrictFDerivAt f (f' x) x\n[PROOFSTEP]\nrefine'\n  isLittleO_iff.mpr fun c hc =>\n    Metric.eventually_nhds_iff_ball.mpr\n      _\n        -- the correct \u03b5 is the modulus of continuity of f'\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : G \u00d7 G), y \u2208 ball (x, x) \u03b5 \u2192 \u2016f y.fst - f y.snd - \u2191(f' x) (y.fst - y.snd)\u2016 \u2264 c * \u2016y.fst - y.snd\u2016\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.mp (inter_mem hder (hcont <| ball_mem_nhds _ hc)) with \u27e8\u03b5, \u03b50, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (y : G \u00d7 G), y \u2208 ball (x, x) \u03b5 \u2192 \u2016f y.fst - f y.snd - \u2191(f' x) (y.fst - y.snd)\u2016 \u2264 c * \u2016y.fst - y.snd\u2016\n[PROOFSTEP]\nrefine'\n  \u27e8\u03b5, \u03b50, _\u27e9\n    -- simplify formulas involving the product E \u00d7 E\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\n\u22a2 \u2200 (y : G \u00d7 G), y \u2208 ball (x, x) \u03b5 \u2192 \u2016f y.fst - f y.snd - \u2191(f' x) (y.fst - y.snd)\u2016 \u2264 c * \u2016y.fst - y.snd\u2016\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9 h\n[GOAL]\ncase intro.intro.mk\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : (a, b) \u2208 ball (x, x) \u03b5\n\u22a2 \u2016f (a, b).fst - f (a, b).snd - \u2191(f' x) ((a, b).fst - (a, b).snd)\u2016 \u2264 c * \u2016(a, b).fst - (a, b).snd\u2016\n[PROOFSTEP]\nrw [\u2190 ball_prod_same, prod_mk_mem_set_prod_eq] at h \n[GOAL]\ncase intro.intro.mk\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\n\u22a2 \u2016f (a, b).fst - f (a, b).snd - \u2191(f' x) ((a, b).fst - (a, b).snd)\u2016 \u2264 c * \u2016(a, b).fst - (a, b).snd\u2016\n[PROOFSTEP]\nhave hf' : \u2200 x' \u2208 ball x \u03b5, \u2016f' x' - f' x\u2016 \u2264 c := fun x' H' =>\n  by\n  rw [\u2190 dist_eq_norm]\n  exact\n    le_of_lt\n      (h\u03b5 H').2\n        -- apply mean value theorem\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\nx' : G\nH' : x' \u2208 ball x \u03b5\n\u22a2 \u2016f' x' - f' x\u2016 \u2264 c\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\nx' : G\nH' : x' \u2208 ball x \u03b5\n\u22a2 dist (f' x') (f' x) \u2264 c\n[PROOFSTEP]\nexact\n  le_of_lt\n    (h\u03b5 H').2\n      -- apply mean value theorem\n[GOAL]\ncase intro.intro.mk\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\nhf' : \u2200 (x' : G), x' \u2208 ball x \u03b5 \u2192 \u2016f' x' - f' x\u2016 \u2264 c\n\u22a2 \u2016f (a, b).fst - f (a, b).snd - \u2191(f' x) ((a, b).fst - (a, b).snd)\u2016 \u2264 c * \u2016(a, b).fst - (a, b).snd\u2016\n[PROOFSTEP]\nletI : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\n[GOAL]\ncase intro.intro.mk\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\nhf' : \u2200 (x' : G), x' \u2208 ball x \u03b5 \u2192 \u2016f' x' - f' x\u2016 \u2264 c\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\n\u22a2 \u2016f (a, b).fst - f (a, b).snd - \u2191(f' x) ((a, b).fst - (a, b).snd)\u2016 \u2264 c * \u2016(a, b).fst - (a, b).snd\u2016\n[PROOFSTEP]\nrefine' (convex_ball _ _).norm_image_sub_le_of_norm_hasFDerivWithin_le' _ hf' h.2 h.1\n[GOAL]\ncase intro.intro.mk\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nH : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nf : G \u2192 H\nf' : G \u2192 G \u2192L[\ud835\udd5c] H\nx : G\nhder : \u2200\u1da0 (y : G) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : \u211d\nhc : 0 < c\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x} \u2229 f' \u207b\u00b9' ball (f' x) c\na b : G\nh : a \u2208 ball x \u03b5 \u2227 b \u2208 ball x \u03b5\nhf' : \u2200 (x' : G), x' \u2208 ball x \u03b5 \u2192 \u2016f' x' - f' x\u2016 \u2264 c\nthis : NormedSpace \u211d G := RestrictScalars.normedSpace \u211d \ud835\udd5c G\n\u22a2 \u2200 (x_1 : G), x_1 \u2208 ball x \u03b5 \u2192 HasFDerivWithinAt f (f' x_1) (ball x \u03b5) x_1\n[PROOFSTEP]\nexact fun y hy => (h\u03b5 hy).1.hasFDerivWithinAt\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.MeanValue", "llama_tokens": 94477, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619393159451, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.548605679705705}}
{"text": "[GOAL]\nb x y : \u211d\n\u22a2 logb b 0 = 0\n[PROOFSTEP]\nsimp [logb]\n[GOAL]\nb x y : \u211d\n\u22a2 logb b 1 = 0\n[PROOFSTEP]\nsimp [logb]\n[GOAL]\nb x\u271d y x : \u211d\n\u22a2 logb b |x| = logb b x\n[PROOFSTEP]\nrw [logb, logb, log_abs]\n[GOAL]\nb x\u271d y x : \u211d\n\u22a2 logb b (-x) = logb b x\n[PROOFSTEP]\nrw [\u2190 logb_abs x, \u2190 logb_abs (-x), abs_neg]\n[GOAL]\nb x y : \u211d\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 logb b (x * y) = logb b x + logb b y\n[PROOFSTEP]\nsimp_rw [logb, log_mul hx hy, add_div]\n[GOAL]\nb x y : \u211d\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 logb b (x / y) = logb b x - logb b y\n[PROOFSTEP]\nsimp_rw [logb, log_div hx hy, sub_div]\n[GOAL]\nb x\u271d y x : \u211d\n\u22a2 logb b x\u207b\u00b9 = -logb b x\n[PROOFSTEP]\nsimp [logb, neg_div]\n[GOAL]\nb\u271d x y a b : \u211d\n\u22a2 (logb a b)\u207b\u00b9 = logb b a\n[PROOFSTEP]\nsimp_rw [logb, inv_div]\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 (logb (a * b) c)\u207b\u00b9 = (logb a c)\u207b\u00b9 + (logb b c)\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [inv_logb]\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 logb c (a * b) = logb c a + logb c b\n[PROOFSTEP]\nexact logb_mul h\u2081 h\u2082\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 (logb (a / b) c)\u207b\u00b9 = (logb a c)\u207b\u00b9 - (logb b c)\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [inv_logb]\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 logb c (a / b) = logb c a - logb c b\n[PROOFSTEP]\nexact logb_div h\u2081 h\u2082\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 logb (a * b) c = ((logb a c)\u207b\u00b9 + (logb b c)\u207b\u00b9)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 inv_logb_mul_base h\u2081 h\u2082 c, inv_inv]\n[GOAL]\nb\u271d x y a b : \u211d\nh\u2081 : a \u2260 0\nh\u2082 : b \u2260 0\nc : \u211d\n\u22a2 logb (a / b) c = ((logb a c)\u207b\u00b9 - (logb b c)\u207b\u00b9)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 inv_logb_div_base h\u2081 h\u2082 c, inv_inv]\n[GOAL]\nb\u271d x y a b c : \u211d\nh\u2081 : b \u2260 0\nh\u2082 : b \u2260 1\nh\u2083 : b \u2260 -1\n\u22a2 logb a b * logb b c = logb a c\n[PROOFSTEP]\nunfold logb\n[GOAL]\nb\u271d x y a b c : \u211d\nh\u2081 : b \u2260 0\nh\u2082 : b \u2260 1\nh\u2083 : b \u2260 -1\n\u22a2 log b / log a * (log c / log b) = log c / log a\n[PROOFSTEP]\nrw [mul_comm, div_mul_div_cancel _ (log_ne_zero.mpr \u27e8h\u2081, h\u2082, h\u2083\u27e9)]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\n\u22a2 log b \u2260 0\n[PROOFSTEP]\nhave b_ne_zero : b \u2260 0\n[GOAL]\ncase b_ne_zero\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\n\u22a2 b \u2260 0\nb x y : \u211d b_pos : 0 < b b_ne_one : b \u2260 1 b_ne_zero : b \u2260 0 \u22a2 log b \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nb_ne_zero : b \u2260 0\n\u22a2 log b \u2260 0\n[PROOFSTEP]\nhave b_ne_minus_one : b \u2260 -1\n[GOAL]\ncase b_ne_minus_one\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nb_ne_zero : b \u2260 0\n\u22a2 b \u2260 -1\nb x y : \u211d b_pos : 0 < b b_ne_one : b \u2260 1 b_ne_zero : b \u2260 0 b_ne_minus_one : b \u2260 -1 \u22a2 log b \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nb_ne_zero : b \u2260 0\nb_ne_minus_one : b \u2260 -1\n\u22a2 log b \u2260 0\n[PROOFSTEP]\nsimp [b_ne_one, b_ne_zero, b_ne_minus_one]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\n\u22a2 logb b (b ^ x) = x\n[PROOFSTEP]\nrw [logb, div_eq_iff, log_rpow b_pos]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\n\u22a2 log b \u2260 0\n[PROOFSTEP]\nexact log_b_ne_zero b_pos b_ne_one\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 b ^ logb b x = |x|\n[PROOFSTEP]\napply log_injOn_pos\n[GOAL]\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 b ^ logb b x \u2208 Ioi 0\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 |x| \u2208 Ioi 0\ncase a b x y : \u211d b_pos : 0 < b b_ne_one : b \u2260 1 hx : x \u2260 0 \u22a2 log (b ^ logb b x) = log |x|\n[PROOFSTEP]\nsimp only [Set.mem_Ioi]\n[GOAL]\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 0 < b ^ logb b x\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 |x| \u2208 Ioi 0\ncase a b x y : \u211d b_pos : 0 < b b_ne_one : b \u2260 1 hx : x \u2260 0 \u22a2 log (b ^ logb b x) = log |x|\n[PROOFSTEP]\napply rpow_pos_of_pos b_pos\n[GOAL]\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 |x| \u2208 Ioi 0\ncase a b x y : \u211d b_pos : 0 < b b_ne_one : b \u2260 1 hx : x \u2260 0 \u22a2 log (b ^ logb b x) = log |x|\n[PROOFSTEP]\nsimp only [abs_pos, mem_Ioi, Ne.def, hx, not_false_iff]\n[GOAL]\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 log (b ^ logb b x) = log |x|\n[PROOFSTEP]\nrw [log_rpow b_pos, logb, log_abs]\n[GOAL]\ncase a\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x \u2260 0\n\u22a2 log x / log b * log b = log x\n[PROOFSTEP]\nfield_simp [log_b_ne_zero b_pos b_ne_one]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : 0 < x\n\u22a2 b ^ logb b x = x\n[PROOFSTEP]\nrw [rpow_logb_eq_abs b_pos b_ne_one hx.ne']\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : 0 < x\n\u22a2 |x| = x\n[PROOFSTEP]\nexact abs_of_pos hx\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x < 0\n\u22a2 b ^ logb b x = -x\n[PROOFSTEP]\nrw [rpow_logb_eq_abs b_pos b_ne_one (ne_of_lt hx)]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nhx : x < 0\n\u22a2 |x| = -x\n[PROOFSTEP]\nexact abs_of_neg hx\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\n\u22a2 SurjOn (logb b) (Iio 0) univ\n[PROOFSTEP]\nintro x _\n[GOAL]\nb x\u271d y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nx : \u211d\na\u271d : x \u2208 univ\n\u22a2 x \u2208 logb b '' Iio 0\n[PROOFSTEP]\nuse-b ^ x\n[GOAL]\ncase h\nb x\u271d y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nx : \u211d\na\u271d : x \u2208 univ\n\u22a2 -b ^ x \u2208 Iio 0 \u2227 logb b (-b ^ x) = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nb x\u271d y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nx : \u211d\na\u271d : x \u2208 univ\n\u22a2 -b ^ x \u2208 Iio 0\n[PROOFSTEP]\nsimp only [Right.neg_neg_iff, Set.mem_Iio]\n[GOAL]\ncase h.left\nb x\u271d y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nx : \u211d\na\u271d : x \u2208 univ\n\u22a2 0 < b ^ x\n[PROOFSTEP]\napply rpow_pos_of_pos b_pos\n[GOAL]\ncase h.right\nb x\u271d y : \u211d\nb_pos : 0 < b\nb_ne_one : b \u2260 1\nx : \u211d\na\u271d : x \u2208 univ\n\u22a2 logb b (-b ^ x) = x\n[PROOFSTEP]\nrw [logb_neg_eq_logb, logb_rpow b_pos b_ne_one]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\n\u22a2 0 < b\n[PROOFSTEP]\nlinarith\n  -- Porting note: prime added to avoid clashing with `b_ne_one` further down the file\n[GOAL]\nb x y : \u211d\nhb : 1 < b\n\u22a2 b \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nh : 0 < x\nh\u2081 : 0 < y\n\u22a2 logb b x \u2264 logb b y \u2194 x \u2264 y\n[PROOFSTEP]\nrw [logb, logb, div_le_div_right (log_pos hb), log_le_log h h\u2081]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\nhxy : x < y\n\u22a2 logb b x < logb b y\n[PROOFSTEP]\nrw [logb, logb, div_lt_div_right (log_pos hb)]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\nhxy : x < y\n\u22a2 log x < log y\n[PROOFSTEP]\nexact log_lt_log hx hxy\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\nhy : 0 < y\n\u22a2 logb b x < logb b y \u2194 x < y\n[PROOFSTEP]\nrw [logb, logb, div_lt_div_right (log_pos hb)]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\nhy : 0 < y\n\u22a2 log x < log y \u2194 x < y\n[PROOFSTEP]\nexact log_lt_log_iff hx hy\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 logb b x \u2264 y \u2194 x \u2264 b ^ y\n[PROOFSTEP]\nrw [\u2190 rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 logb b x < y \u2194 x < b ^ y\n[PROOFSTEP]\nrw [\u2190 rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hx]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhy : 0 < y\n\u22a2 x \u2264 logb b y \u2194 b ^ x \u2264 y\n[PROOFSTEP]\nrw [\u2190 rpow_le_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hy]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhy : 0 < y\n\u22a2 x < logb b y \u2194 b ^ x < y\n[PROOFSTEP]\nrw [\u2190 rpow_lt_rpow_left_iff hb, rpow_logb (b_pos hb) (b_ne_one' hb) hy]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 0 < logb b x \u2194 1 < x\n[PROOFSTEP]\nrw [\u2190 @logb_one b]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 logb b 1 < logb b x \u2194 1 < x\n[PROOFSTEP]\nrw [logb_lt_logb_iff hb zero_lt_one hx]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 1 < x\n\u22a2 0 < logb b x\n[PROOFSTEP]\nrw [logb_pos_iff hb (lt_trans zero_lt_one hx)]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 1 < x\n\u22a2 1 < x\n[PROOFSTEP]\nexact hx\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nh : 0 < x\n\u22a2 logb b x < 0 \u2194 x < 1\n[PROOFSTEP]\nrw [\u2190 logb_one]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nh : 0 < x\n\u22a2 logb b x < logb ?m.14910 1 \u2194 x < 1\nb x y : \u211d hb : 1 < b h : 0 < x \u22a2 \u211d\n[PROOFSTEP]\nexact logb_lt_logb_iff hb h zero_lt_one\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 0 \u2264 logb b x \u2194 1 \u2264 x\n[PROOFSTEP]\nrw [\u2190 not_lt, logb_neg_iff hb hx, not_lt]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 < x\n\u22a2 logb b x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nrw [\u2190 not_lt, logb_pos_iff hb hx, not_lt]\n[GOAL]\nb x y : \u211d\nhb : 1 < b\nhx : 0 \u2264 x\n\u22a2 logb b x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hx)\n[GOAL]\ncase inl\nb y : \u211d\nhb : 1 < b\nhx : 0 \u2264 0\n\u22a2 logb b 0 \u2264 0 \u2194 0 \u2264 1\n[PROOFSTEP]\nsimp [le_refl, zero_le_one]\n[GOAL]\ncase inr\nb x y : \u211d\nhb : 1 < b\nhx\u271d : 0 \u2264 x\nhx : 0 < x\n\u22a2 logb b x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nexact logb_nonpos_iff hb hx\n[GOAL]\nb x y : \u211d\nhb : 1 < b\n\u22a2 StrictAntiOn (logb b) (Iio 0)\n[PROOFSTEP]\nrintro x (hx : x < 0) y (hy : y < 0) hxy\n[GOAL]\nb x\u271d y\u271d : \u211d\nhb : 1 < b\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 logb b y < logb b x\n[PROOFSTEP]\nrw [\u2190 logb_abs y, \u2190 logb_abs x]\n[GOAL]\nb x\u271d y\u271d : \u211d\nhb : 1 < b\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 logb b |y| < logb b |x|\n[PROOFSTEP]\nrefine' logb_lt_logb hb (abs_pos.2 hy.ne) _\n[GOAL]\nb x\u271d y\u271d : \u211d\nhb : 1 < b\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 |y| < |x|\n[PROOFSTEP]\nrwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\n\u22a2 b \u2260 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nh : 0 < x\nh\u2081 : 0 < y\n\u22a2 logb b x \u2264 logb b y \u2194 y \u2264 x\n[PROOFSTEP]\nrw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log h\u2081 h]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhxy : x < y\n\u22a2 logb b y < logb b x\n[PROOFSTEP]\nrw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhxy : x < y\n\u22a2 log x < log y\n[PROOFSTEP]\nexact log_lt_log hx hxy\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhy : 0 < y\n\u22a2 logb b x < logb b y \u2194 y < x\n[PROOFSTEP]\nrw [logb, logb, div_lt_div_right_of_neg (log_neg b_pos b_lt_one)]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhy : 0 < y\n\u22a2 log y < log x \u2194 y < x\n[PROOFSTEP]\nexact log_lt_log_iff hy hx\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n\u22a2 logb b x \u2264 y \u2194 b ^ y \u2264 x\n[PROOFSTEP]\nrw [\u2190 rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n\u22a2 logb b x < y \u2194 b ^ y < x\n[PROOFSTEP]\nrw [\u2190 rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hx]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n\u22a2 x \u2264 logb b y \u2194 y \u2264 b ^ x\n[PROOFSTEP]\nrw [\u2190 rpow_le_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n\u22a2 x < logb b y \u2194 y < b ^ x\n[PROOFSTEP]\nrw [\u2190 rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n\u22a2 0 < logb b x \u2194 x < 1\n[PROOFSTEP]\nrw [\u2190 @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one zero_lt_one hx]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhx' : x < 1\n\u22a2 0 < logb b x\n[PROOFSTEP]\nrw [logb_pos_iff_of_base_lt_one b_pos b_lt_one hx]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhx' : x < 1\n\u22a2 x < 1\n[PROOFSTEP]\nexact hx'\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nh : 0 < x\n\u22a2 logb b x < 0 \u2194 1 < x\n[PROOFSTEP]\nrw [\u2190 @logb_one b, logb_lt_logb_iff_of_base_lt_one b_pos b_lt_one h zero_lt_one]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n\u22a2 0 \u2264 logb b x \u2194 x \u2264 1\n[PROOFSTEP]\nrw [\u2190 not_lt, logb_neg_iff_of_base_lt_one b_pos b_lt_one hx, not_lt]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 0 \u2264 logb b x\n[PROOFSTEP]\nrw [logb_nonneg_iff_of_base_lt_one b_pos b_lt_one hx]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\nhx' : x \u2264 1\n\u22a2 x \u2264 1\n[PROOFSTEP]\nexact hx'\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nhx : 0 < x\n\u22a2 logb b x \u2264 0 \u2194 1 \u2264 x\n[PROOFSTEP]\nrw [\u2190 not_lt, logb_pos_iff_of_base_lt_one b_pos b_lt_one hx, not_lt]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\n\u22a2 StrictMonoOn (logb b) (Iio 0)\n[PROOFSTEP]\nrintro x (hx : x < 0) y (hy : y < 0) hxy\n[GOAL]\nb x\u271d y\u271d : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 logb b x < logb b y\n[PROOFSTEP]\nrw [\u2190 logb_abs y, \u2190 logb_abs x]\n[GOAL]\nb x\u271d y\u271d : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 logb b |x| < logb b |y|\n[PROOFSTEP]\nrefine' logb_lt_logb_of_base_lt_one b_pos b_lt_one (abs_pos.2 hy.ne) _\n[GOAL]\nb x\u271d y\u271d : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 |y| < |x|\n[PROOFSTEP]\nrwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\n\u22a2 Tendsto (logb b) atTop atBot\n[PROOFSTEP]\nrw [tendsto_atTop_atBot]\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\n\u22a2 \u2200 (b_1 : \u211d), \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 logb b a \u2264 b_1\n[PROOFSTEP]\nintro e\n[GOAL]\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne : \u211d\n\u22a2 \u2203 i, \u2200 (a : \u211d), i \u2264 a \u2192 logb b a \u2264 e\n[PROOFSTEP]\nuse 1 \u2294 b ^ e\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne : \u211d\n\u22a2 \u2200 (a : \u211d), 1 \u2294 b ^ e \u2264 a \u2192 logb b a \u2264 e\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne a : \u211d\n\u22a2 1 \u2294 b ^ e \u2264 a \u2192 logb b a \u2264 e\n[PROOFSTEP]\nsimp only [and_imp, sup_le_iff]\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne a : \u211d\n\u22a2 1 \u2264 a \u2192 b ^ e \u2264 a \u2192 logb b a \u2264 e\n[PROOFSTEP]\nintro ha\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne a : \u211d\nha : 1 \u2264 a\n\u22a2 b ^ e \u2264 a \u2192 logb b a \u2264 e\n[PROOFSTEP]\nrw [logb_le_iff_le_rpow_of_base_lt_one b_pos b_lt_one]\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne a : \u211d\nha : 1 \u2264 a\n\u22a2 b ^ e \u2264 a \u2192 b ^ e \u2264 a\ncase h b x y : \u211d b_pos : 0 < b b_lt_one : b < 1 e a : \u211d ha : 1 \u2264 a \u22a2 0 < a\n[PROOFSTEP]\ntauto\n[GOAL]\ncase h\nb x y : \u211d\nb_pos : 0 < b\nb_lt_one : b < 1\ne a : \u211d\nha : 1 \u2264 a\n\u22a2 0 < a\n[PROOFSTEP]\nexact lt_of_lt_of_le zero_lt_one ha\n[GOAL]\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr : 0 \u2264 r\n\u22a2 \u230alogb (\u2191b) r\u230b = Int.log b r\n[PROOFSTEP]\nobtain rfl | hr := hr.eq_or_lt\n[GOAL]\ncase inl\nb\u271d x y : \u211d\nb : \u2115\nhb : 1 < b\nhr : 0 \u2264 0\n\u22a2 \u230alogb (\u2191b) 0\u230b = Int.log b 0\n[PROOFSTEP]\nrw [logb_zero, Int.log_zero_right, Int.floor_zero]\n[GOAL]\ncase inr\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\n\u22a2 \u230alogb (\u2191b) r\u230b = Int.log b r\n[PROOFSTEP]\nhave hb1' : 1 < (b : \u211d) := Nat.one_lt_cast.mpr hb\n[GOAL]\ncase inr\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u230alogb (\u2191b) r\u230b = Int.log b r\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u230alogb (\u2191b) r\u230b \u2264 Int.log b r\n[PROOFSTEP]\nrw [\u2190 Int.zpow_le_iff_le_log hb hr, \u2190 rpow_int_cast b]\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2191b ^ \u2191\u230alogb (\u2191b) r\u230b \u2264 r\n[PROOFSTEP]\nrefine' le_of_le_of_eq _ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2191b ^ \u2191\u230alogb (\u2191b) r\u230b \u2264 \u2191b ^ logb (\u2191b) r\n[PROOFSTEP]\nexact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _)\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 Int.log b r \u2264 \u230alogb (\u2191b) r\u230b\n[PROOFSTEP]\nrw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_int_cast]\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2191b ^ Int.log b r \u2264 r\n[PROOFSTEP]\nexact Int.zpow_log_le_self hb hr\n[GOAL]\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr : 0 \u2264 r\n\u22a2 \u2308logb (\u2191b) r\u2309 = Int.clog b r\n[PROOFSTEP]\nobtain rfl | hr := hr.eq_or_lt\n[GOAL]\ncase inl\nb\u271d x y : \u211d\nb : \u2115\nhb : 1 < b\nhr : 0 \u2264 0\n\u22a2 \u2308logb (\u2191b) 0\u2309 = Int.clog b 0\n[PROOFSTEP]\nrw [logb_zero, Int.clog_zero_right, Int.ceil_zero]\n[GOAL]\ncase inr\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\n\u22a2 \u2308logb (\u2191b) r\u2309 = Int.clog b r\n[PROOFSTEP]\nhave hb1' : 1 < (b : \u211d) := Nat.one_lt_cast.mpr hb\n[GOAL]\ncase inr\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2308logb (\u2191b) r\u2309 = Int.clog b r\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2308logb (\u2191b) r\u2309 \u2264 Int.clog b r\n[PROOFSTEP]\nrw [Int.ceil_le, logb_le_iff_le_rpow hb1' hr, rpow_int_cast]\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 r \u2264 \u2191b ^ Int.clog b r\n[PROOFSTEP]\nrefine' Int.self_le_zpow_clog hb r\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 Int.clog b r \u2264 \u2308logb (\u2191b) r\u2309\n[PROOFSTEP]\nrw [\u2190 Int.le_zpow_iff_clog_le hb hr, \u2190 rpow_int_cast b]\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 r \u2264 \u2191b ^ \u2191\u2308logb (\u2191b) r\u2309\n[PROOFSTEP]\nrefine' (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr).symm.trans_le _\n[GOAL]\ncase inr.a\nb\u271d x y : \u211d\nb : \u2115\nr : \u211d\nhb : 1 < b\nhr\u271d : 0 \u2264 r\nhr : 0 < r\nhb1' : 1 < \u2191b\n\u22a2 \u2191b ^ logb (\u2191b) r \u2264 \u2191b ^ \u2191\u2308logb (\u2191b) r\u2309\n[PROOFSTEP]\nexact rpow_le_rpow_of_exponent_le hb1'.le (Int.le_ceil _)\n[GOAL]\nb x y : \u211d\n\u22a2 logb b x = 0 \u2194 b = 0 \u2228 b = 1 \u2228 b = -1 \u2228 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nsimp_rw [logb, div_eq_zero_iff, log_eq_zero]\n[GOAL]\nb x y : \u211d\n\u22a2 (x = 0 \u2228 x = 1 \u2228 x = -1) \u2228 b = 0 \u2228 b = 1 \u2228 b = -1 \u2194 b = 0 \u2228 b = 1 \u2228 b = -1 \u2228 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\ntauto\n[GOAL]\nb x y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 logb b (\u220f i in s, f i) = \u2211 i in s, logb b (f i)\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction_on with a s ha ih\n\u00b7 simp\nsimp only [Finset.mem_insert, forall_eq_or_imp] at hf \nsimp [ha, ih hf.2, logb_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)]\n[GOAL]\nb x y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 logb b (\u220f i in s, f i) = \u2211 i in s, logb b (f i)\n[PROOFSTEP]\ninduction' s using Finset.induction_on with a s ha ih\n[GOAL]\ncase empty\nb x y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nhf : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 f x \u2260 0\n\u22a2 logb b (\u220f i in \u2205, f i) = \u2211 i in \u2205, logb b (f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nb x y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 logb b (\u220f i in s, f i) = \u2211 i in s, logb b (f i)\nhf : \u2200 (x : \u03b1), x \u2208 insert a s \u2192 f x \u2260 0\n\u22a2 logb b (\u220f i in insert a s, f i) = \u2211 i in insert a s, logb b (f i)\n[PROOFSTEP]\nsimp only [Finset.mem_insert, forall_eq_or_imp] at hf \n[GOAL]\ncase insert\nb x y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 logb b (\u220f i in s, f i) = \u2211 i in s, logb b (f i)\nhf : f a \u2260 0 \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2260 0\n\u22a2 logb b (\u220f i in insert a s, f i) = \u2211 i in insert a s, logb b (f i)\n[PROOFSTEP]\nsimp [ha, ih hf.2, logb_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Log.Base", "llama_tokens": 11375, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.841825635346563, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5483272284809338}}
{"text": "[GOAL]\na b : PosNum\n\u22a2 Decidable ((fun x x_1 => x < x_1) a b)\n[PROOFSTEP]\ndsimp [LT.lt]\n[GOAL]\na b : PosNum\n\u22a2 Decidable (cmp a b = lt)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b : PosNum\n\u22a2 Decidable ((fun x x_1 => x \u2264 x_1) a b)\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\na b : PosNum\n\u22a2 Decidable \u00acb < a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b : Num\n\u22a2 Decidable ((fun x x_1 => x < x_1) a b)\n[PROOFSTEP]\ndsimp [LT.lt]\n[GOAL]\na b : Num\n\u22a2 Decidable (cmp a b = lt)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b : Num\n\u22a2 Decidable ((fun x x_1 => x \u2264 x_1) a b)\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\na b : Num\n\u22a2 Decidable \u00acb < a\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b : ZNum\n\u22a2 Decidable ((fun x x_1 => x < x_1) a b)\n[PROOFSTEP]\ndsimp [LT.lt]\n[GOAL]\na b : ZNum\n\u22a2 Decidable (cmp a b = lt)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\na b : ZNum\n\u22a2 Decidable ((fun x x_1 => x \u2264 x_1) a b)\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\na b : ZNum\n\u22a2 Decidable \u00acb < a\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Data.Num.Basic", "llama_tokens": 488, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256393148981, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.5483272196710117}}
{"text": "[GOAL]\nx y : \u2124\nd : \u2115\nhd : Int.gcd x y = d\nh : Nat.beq d 1 = false\n\u22a2 \u00acIsCoprime x y\n[PROOFSTEP]\nrw [Int.isCoprime_iff_gcd_eq_one, hd]\n[GOAL]\nx y : \u2124\nd : \u2115\nhd : Int.gcd x y = d\nh : Nat.beq d 1 = false\n\u22a2 \u00acd = 1\n[PROOFSTEP]\nexact Nat.ne_of_beq_eq_false h\n", "meta": {"mathlib_filename": "Mathlib.Tactic.NormNum.IsCoprime", "llama_tokens": 150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8705972549785201, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.5482800384119677}}
{"text": "[GOAL]\nI\u271d : Type u\nf : I\u271d \u2192 Type v\nI : Type u_1\nJ : Type u_2\n\u03b1 : Type u_3\nx : I \u2192 \u03b1\ny : J \u2192 \u03b1\ninst\u271d : Star \u03b1\n\u22a2 star (Sum.elim x y) = Sum.elim (star x) (star y)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nI\u271d : Type u\nf : I\u271d \u2192 Type v\nI : Type u_1\nJ : Type u_2\n\u03b1 : Type u_3\nx\u271d : I \u2192 \u03b1\ny : J \u2192 \u03b1\ninst\u271d : Star \u03b1\nx : I \u2295 J\n\u22a2 star (Sum.elim x\u271d y) x = Sum.elim (star x\u271d) (star y) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase h.inl\nI\u271d : Type u\nf : I\u271d \u2192 Type v\nI : Type u_1\nJ : Type u_2\n\u03b1 : Type u_3\nx : I \u2192 \u03b1\ny : J \u2192 \u03b1\ninst\u271d : Star \u03b1\nval\u271d : I\n\u22a2 star (Sum.elim x y) (Sum.inl val\u271d) = Sum.elim (star x) (star y) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp only [Pi.star_apply, Sum.elim_inl, Sum.elim_inr]\n[GOAL]\ncase h.inr\nI\u271d : Type u\nf : I\u271d \u2192 Type v\nI : Type u_1\nJ : Type u_2\n\u03b1 : Type u_3\nx : I \u2192 \u03b1\ny : J \u2192 \u03b1\ninst\u271d : Star \u03b1\nval\u271d : J\n\u22a2 star (Sum.elim x y) (Sum.inr val\u271d) = Sum.elim (star x) (star y) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp only [Pi.star_apply, Sum.elim_inl, Sum.elim_inr]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Star.Pi", "llama_tokens": 525, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929104825007, "lm_q2_score": 0.7025300636233416, "lm_q1q2_score": 0.5479684690270267}}
{"text": "[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j : n\n\u22a2 transvection i j 0 = 1\n[PROOFSTEP]\nsimp [transvection]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) = transvection i j c\n[PROOFSTEP]\ncases nonempty_fintype n\n[GOAL]\ncase intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) = transvection i j c\n[PROOFSTEP]\next a b\n[GOAL]\ncase intro.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\n[PROOFSTEP]\nby_cases ha : i = a\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\nha : i = a\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\nha : \u00aci = a\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\n[PROOFSTEP]\nby_cases hb : j = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\nha : i = a\nhb : j = b\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\n[PROOFSTEP]\nsimp only [updateRow_self, transvection, ha, hb, Pi.add_apply, StdBasisMatrix.apply_same, one_apply_eq, Pi.smul_apply,\n  mul_one, Algebra.id.smul_eq_mul, add_apply]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\nha : i = a\nhb : \u00acj = b\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\n[PROOFSTEP]\nsimp only [updateRow_self, transvection, ha, hb, StdBasisMatrix.apply_of_ne, Pi.add_apply, Ne.def, not_false_iff,\n  Pi.smul_apply, and_false_iff, one_apply_ne, Algebra.id.smul_eq_mul, mul_zero, add_apply]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Finite n\nc : R\nval\u271d : Fintype n\na b : n\nha : \u00aci = a\n\u22a2 updateRow 1 i (OfNat.ofNat 1 i + c \u2022 OfNat.ofNat 1 j) a b = transvection i j c a b\n[PROOFSTEP]\nsimp only [updateRow_ne, transvection, ha, Ne.symm ha, StdBasisMatrix.apply_of_ne, add_zero, Algebra.id.smul_eq_mul,\n  Ne.def, not_false_iff, DMatrix.add_apply, Pi.smul_apply, mul_zero, false_and_iff, add_apply]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nh : i \u2260 j\nc d : R\n\u22a2 transvection i j c * transvection i j d = transvection i j (c + d)\n[PROOFSTEP]\nsimp [transvection, Matrix.add_mul, Matrix.mul_add, h, h.symm, add_smul, add_assoc, stdBasisMatrix_add]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nb : n\nc : R\nM : Matrix n n R\n\u22a2 (transvection i j c * M) i b = M i b + c * M j b\n[PROOFSTEP]\nsimp [transvection, Matrix.add_mul]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\na : n\nc : R\nM : Matrix n n R\n\u22a2 (M * transvection i j c) a j = M a j + c * M a i\n[PROOFSTEP]\nsimp [transvection, Matrix.mul_add, mul_comm]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\na b : n\nha : a \u2260 i\nc : R\nM : Matrix n n R\n\u22a2 (transvection i j c * M) a b = M a b\n[PROOFSTEP]\nsimp [transvection, Matrix.add_mul, ha]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\na b : n\nhb : b \u2260 j\nc : R\nM : Matrix n n R\n\u22a2 (M * transvection i j c) a b = M a b\n[PROOFSTEP]\nsimp [transvection, Matrix.mul_add, hb]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nh : i \u2260 j\nc : R\n\u22a2 det (transvection i j c) = 1\n[PROOFSTEP]\nrw [\u2190 updateRow_eq_transvection i j, det_updateRow_add_smul_self _ h, det_one]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Nontrivial n\n\u22a2 Nonempty (TransvectionStruct n R)\n[PROOFSTEP]\nchoose x y hxy using exists_pair_ne n\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Nontrivial n\nx y : n\nhxy : x \u2260 y\n\u22a2 Nonempty (TransvectionStruct n R)\n[PROOFSTEP]\nexact \u27e8\u27e8x, y, hxy, 0\u27e9\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nL : List (TransvectionStruct n \ud835\udd5c)\n\u22a2 det (List.prod (List.map toMatrix L)) = 1\n[PROOFSTEP]\ninduction' L with t L IH\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\n\u22a2 det (List.prod (List.map toMatrix [])) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n \ud835\udd5c\nL : List (TransvectionStruct n \ud835\udd5c)\nIH : det (List.prod (List.map toMatrix L)) = 1\n\u22a2 det (List.prod (List.map toMatrix (t :: L))) = 1\n[PROOFSTEP]\nsimp [IH]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\n\u22a2 toMatrix (TransvectionStruct.inv t) * toMatrix t = 1\n[PROOFSTEP]\nrcases t with \u27e8_, _, t_hij\u27e9\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\ni\u271d j\u271d : n\nt_hij : i\u271d \u2260 j\u271d\nc\u271d : R\n\u22a2 toMatrix (TransvectionStruct.inv { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }) *\n      toMatrix { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d } =\n    1\n[PROOFSTEP]\nsimp [toMatrix, transvection_mul_transvection_same, t_hij]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\n\u22a2 toMatrix t * toMatrix (TransvectionStruct.inv t) = 1\n[PROOFSTEP]\nrcases t with \u27e8_, _, t_hij\u27e9\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\ni\u271d j\u271d : n\nt_hij : i\u271d \u2260 j\u271d\nc\u271d : R\n\u22a2 toMatrix { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d } *\n      toMatrix (TransvectionStruct.inv { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }) =\n    1\n[PROOFSTEP]\nsimp [toMatrix, transvection_mul_transvection_same, t_hij]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nL : List (TransvectionStruct n R)\n\u22a2 List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) = 1\n[PROOFSTEP]\ninduction' L with t L IH\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\n\u22a2 List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse [])) * List.prod (List.map toMatrix []) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) = 1\n\u22a2 List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) *\n      List.prod (List.map toMatrix (t :: L)) =\n    1\n[PROOFSTEP]\nsuffices\n  (L.reverse.map (toMatrix \u2218 TransvectionStruct.inv)).prod * (t.inv.toMatrix * t.toMatrix) * (L.map toMatrix).prod = 1\n  by simpa [Matrix.mul_assoc]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) = 1\nthis :\n  List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) *\n        (toMatrix (TransvectionStruct.inv t) * toMatrix t) *\n      List.prod (List.map toMatrix L) =\n    1\n\u22a2 List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) *\n      List.prod (List.map toMatrix (t :: L)) =\n    1\n[PROOFSTEP]\nsimpa [Matrix.mul_assoc]\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) = 1\n\u22a2 List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) *\n        (toMatrix (TransvectionStruct.inv t) * toMatrix t) *\n      List.prod (List.map toMatrix L) =\n    1\n[PROOFSTEP]\nsimpa [inv_mul] using IH\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nL : List (TransvectionStruct n R)\n\u22a2 List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n[PROOFSTEP]\ninduction' L with t L IH\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\n\u22a2 List.prod (List.map toMatrix []) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse [])) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 List.prod (List.map toMatrix (t :: L)) *\n      List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) =\n    1\n[PROOFSTEP]\nsuffices\n  t.toMatrix * ((L.map toMatrix).prod * (L.reverse.map (toMatrix \u2218 TransvectionStruct.inv)).prod) * t.inv.toMatrix = 1\n  by simpa [Matrix.mul_assoc]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\nthis :\n  toMatrix t *\n        (List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L))) *\n      toMatrix (TransvectionStruct.inv t) =\n    1\n\u22a2 List.prod (List.map toMatrix (t :: L)) *\n      List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse (t :: L))) =\n    1\n[PROOFSTEP]\nsimpa [Matrix.mul_assoc]\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) = 1\n\u22a2 toMatrix t *\n        (List.prod (List.map toMatrix L) * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L))) *\n      toMatrix (TransvectionStruct.inv t) =\n    1\n[PROOFSTEP]\nsimp_rw [IH, Matrix.mul_one, t.mul_inv]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j : n\nt : TransvectionStruct n R\n\u22a2 inl t.i \u2260 inl t.j\n[PROOFSTEP]\nsimp [t.hij]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j : n\nt : TransvectionStruct n R\n\u22a2 toMatrix (sumInl p t) = fromBlocks (toMatrix t) 0 0 1\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1\n[PROOFSTEP]\next a b\n[GOAL]\ncase mk.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : n \u2295 p\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) a b =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 a b\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase mk.a.h.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\nb : n \u2295 p\na : n\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inl a) b =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inl a) b\n[PROOFSTEP]\ncases' b with b b\n[GOAL]\ncase mk.a.h.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\nb : n \u2295 p\na : p\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inr a) b =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inr a) b\n[PROOFSTEP]\ncases' b with b b\n[GOAL]\ncase mk.a.h.inl.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : n\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inl a) (inl b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inl a) (inl b)\n[PROOFSTEP]\nby_cases h : a = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : n\nh : a = b\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inl a) (inl b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inl a) (inl b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection, h, stdBasisMatrix]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : n\nh : \u00aca = b\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inl a) (inl b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inl a) (inl b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection, h, stdBasisMatrix]\n[GOAL]\ncase mk.a.h.inl.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na : n\nb : p\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inl a) (inr b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inl a) (inr b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection]\n[GOAL]\ncase mk.a.h.inr.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na : p\nb : n\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inr a) (inl b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inr a) (inl b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection]\n[GOAL]\ncase mk.a.h.inr.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : p\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inr a) (inr b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inr a) (inr b)\n[PROOFSTEP]\nby_cases h : a = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : p\nh : a = b\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inr a) (inr b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inr a) (inr b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection, h]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j i\u271d j\u271d : n\nhij\u271d : i\u271d \u2260 j\u271d\nc\u271d : R\na b : p\nh : \u00aca = b\n\u22a2 toMatrix (sumInl p { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) (inr a) (inr b) =\n    fromBlocks (toMatrix { i := i\u271d, j := j\u271d, hij := hij\u271d, c := c\u271d }) 0 0 1 (inr a) (inr b)\n[PROOFSTEP]\nsimp [TransvectionStruct.sumInl, transvection, h]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n R\nL : List (TransvectionStruct n R)\nN : Matrix p p R\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl p) L) * fromBlocks M 0 0 N =\n    fromBlocks (List.prod (List.map toMatrix L) * M) 0 0 N\n[PROOFSTEP]\ninduction' L with t L IH\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n R\nN : Matrix p p R\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl p) []) * fromBlocks M 0 0 N =\n    fromBlocks (List.prod (List.map toMatrix []) * M) 0 0 N\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n R\nN : Matrix p p R\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH :\n  List.prod (List.map (toMatrix \u2218 sumInl p) L) * fromBlocks M 0 0 N =\n    fromBlocks (List.prod (List.map toMatrix L) * M) 0 0 N\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl p) (t :: L)) * fromBlocks M 0 0 N =\n    fromBlocks (List.prod (List.map toMatrix (t :: L)) * M) 0 0 N\n[PROOFSTEP]\nsimp [Matrix.mul_assoc, IH, toMatrix_sumInl, fromBlocks_multiply]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n R\nL : List (TransvectionStruct n R)\nN : Matrix p p R\n\u22a2 fromBlocks M 0 0 N * List.prod (List.map (toMatrix \u2218 sumInl p) L) =\n    fromBlocks (M * List.prod (List.map toMatrix L)) 0 0 N\n[PROOFSTEP]\ninduction' L with t L IH generalizing M N\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM\u271d : Matrix n n R\nN\u271d : Matrix p p R\nM : Matrix n n R\nN : Matrix p p R\n\u22a2 fromBlocks M 0 0 N * List.prod (List.map (toMatrix \u2218 sumInl p) []) =\n    fromBlocks (M * List.prod (List.map toMatrix [])) 0 0 N\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM\u271d : Matrix n n R\nN\u271d : Matrix p p R\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH :\n  \u2200 (M : Matrix n n R) (N : Matrix p p R),\n    fromBlocks M 0 0 N * List.prod (List.map (toMatrix \u2218 sumInl p) L) =\n      fromBlocks (M * List.prod (List.map toMatrix L)) 0 0 N\nM : Matrix n n R\nN : Matrix p p R\n\u22a2 fromBlocks M 0 0 N * List.prod (List.map (toMatrix \u2218 sumInl p) (t :: L)) =\n    fromBlocks (M * List.prod (List.map toMatrix (t :: L))) 0 0 N\n[PROOFSTEP]\nsimp [IH, toMatrix_sumInl, fromBlocks_multiply]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\ni j : n\ne : n \u2243 p\nt : TransvectionStruct n R\n\u22a2 \u2191e t.i \u2260 \u2191e t.j\n[PROOFSTEP]\nsimp [t.hij]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\n\u22a2 toMatrix (reindexEquiv e t) = \u2191(reindexAlgEquiv R e) (toMatrix t)\n[PROOFSTEP]\nrcases t with \u27e8t_i, t_j, _\u27e9\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\n\u22a2 toMatrix (reindexEquiv e { i := t_i, j := t_j, hij := hij\u271d, c := c\u271d }) =\n    \u2191(reindexAlgEquiv R e) (toMatrix { i := t_i, j := t_j, hij := hij\u271d, c := c\u271d })\n[PROOFSTEP]\next a b\n[GOAL]\ncase mk.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\n\u22a2 toMatrix (reindexEquiv e { i := t_i, j := t_j, hij := hij\u271d, c := c\u271d }) a b =\n    \u2191(reindexAlgEquiv R e) (toMatrix { i := t_i, j := t_j, hij := hij\u271d, c := c\u271d }) a b\n[PROOFSTEP]\nsimp only [reindexEquiv, transvection, mul_boole, Algebra.id.smul_eq_mul, toMatrix_mk, submatrix_apply, reindex_apply,\n  DMatrix.add_apply, Pi.smul_apply, reindexAlgEquiv_apply]\n[GOAL]\ncase mk.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases ha : e t_i = a\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hb : e t_j = b\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hb : e t_j = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u2191e t_j = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u2191e t_j = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u2191e t_j = b\nhab : a = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u2191e t_j = b\nhab : \u00aca = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\nhab : a = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\nhab : \u00aca = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u2191e t_j = b\nhab : a = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u2191e t_j = b\nhab : \u00aca = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\nhab : a = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt_i t_j : n\nhij\u271d : t_i \u2260 t_j\nc\u271d : R\na b : p\nha : \u00ac\u2191e t_i = a\nhb : \u00ac\u2191e t_j = b\nhab : \u00aca = b\n\u22a2 (1 + stdBasisMatrix (\u2191e t_i) (\u2191e t_j) c\u271d) a b = (1 + stdBasisMatrix t_i t_j c\u271d) (\u2191e.symm a) (\u2191e.symm b)\n[PROOFSTEP]\nsimp [ha, hb, hab, \u2190 e.apply_eq_iff_eq_symm_apply, stdBasisMatrix]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nL : List (TransvectionStruct n R)\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n[PROOFSTEP]\ninduction' L with t L IH\n[GOAL]\ncase nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) []) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix []))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) (t :: L)) =\n    \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix (t :: L)))\n[PROOFSTEP]\nsimp only [toMatrix_reindexEquiv, IH, Function.comp_apply, List.prod_cons, reindexAlgEquiv_apply, List.map]\n[GOAL]\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 \u2191(reindex e e) (toMatrix t) * \u2191(reindex e e) (List.prod (List.map toMatrix L)) =\n    \u2191(reindex e e) (toMatrix t * List.prod (List.map toMatrix L))\n[PROOFSTEP]\nexact (reindexAlgEquiv_mul _ _ _ _).symm\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 (List.prod (List.drop k (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nrefine' Nat.decreasingInduction' _ hk _\n[GOAL]\ncase refine'_1\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 \u2200 (k_1 : \u2115),\n    k_1 < r \u2192\n      k \u2264 k_1 \u2192\n        (List.prod (List.drop (k_1 + 1) (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i \u2192\n          (List.prod (List.drop k_1 (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nintro n hn _ IH\n[GOAL]\ncase refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n\u22a2 (List.prod (List.drop n (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nhave hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n\u22a2 n < List.length (listTransvecCol M)\n[PROOFSTEP]\nsimpa [listTransvecCol] using hn\n[GOAL]\ncase refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n\u22a2 (List.prod (List.drop n (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nrw [\u2190 @List.cons_get_drop_succ _ _ \u27e8n, hn'\u27e9]\n[GOAL]\ncase refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\nn : \u2115\nhn : n < r\na\u271d : k \u2264 n\nIH : (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\nhn' : n < List.length (listTransvecCol M)\n\u22a2 (List.prod\n          (List.get (listTransvecCol M) { val := n, isLt := hn' } ::\n            List.drop (\u2191{ val := n, isLt := hn' } + 1) (listTransvecCol M)) *\n        M)\n      (inr ()) i =\n    M (inr ()) i\n[PROOFSTEP]\nsimpa [listTransvecCol, Matrix.mul_assoc]\n[GOAL]\ncase refine'_2\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 (List.prod (List.drop r (listTransvecCol M)) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nsimp only [listTransvecCol, List.length_ofFn, le_refl, List.drop_eq_nil_of_le, List.prod_nil, Matrix.one_mul]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\n\u22a2 (List.prod (listTransvecCol M) * M) (inr ()) i = M (inr ()) i\n[PROOFSTEP]\nsimpa using listTransvecCol_mul_last_row_drop M i (zero_le _)\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 (List.prod (listTransvecCol M) * M) (inl i) (inr ()) = 0\n[PROOFSTEP]\nsuffices H :\n  \u2200 k : \u2115,\n    k \u2264 r \u2192 (((listTransvecCol M).drop k).prod * M) (inl i) (inr unit) = if k \u2264 i then 0 else M (inl i) (inr unit)\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (List.prod (List.drop k (listTransvecCol M)) * M) (inl i) (inr ()) = if k \u2264 \u2191i then 0 else M (inl i) (inr ())\n\u22a2 (List.prod (listTransvecCol M) * M) (inl i) (inr ()) = 0\n[PROOFSTEP]\nsimpa only [List.drop, _root_.zero_le, ite_true] using H 0 (zero_le _)\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (List.prod (List.drop k (listTransvecCol M)) * M) (inl i) (inr ()) = if k \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nintro k hk\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 (List.prod (List.drop k (listTransvecCol M)) * M) (inl i) (inr ()) = if k \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrefine' Nat.decreasingInduction' _ hk _\n[GOAL]\ncase H.refine'_1\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 \u2200 (k_1 : \u2115),\n    k_1 < r \u2192\n      k \u2264 k_1 \u2192\n        ((List.prod (List.drop (k_1 + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n            if k_1 + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())) \u2192\n          (List.prod (List.drop k_1 (listTransvecCol M)) * M) (inl i) (inr ()) =\n            if k_1 \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nintro n hn hk IH\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\n\u22a2 (List.prod (List.drop n (listTransvecCol M)) * M) (inl i) (inr ()) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nhave hn' : n < (listTransvecCol M).length := by simpa [listTransvecCol] using hn\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\n\u22a2 n < List.length (listTransvecCol M)\n[PROOFSTEP]\nsimpa [listTransvecCol] using hn\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\n\u22a2 (List.prod (List.drop n (listTransvecCol M)) * M) (inl i) (inr ()) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nlet n' : Fin r :=\n  \u27e8n, hn\u27e9\n    -- porting note: after changing from `nthLe` to `get`, we need to provide all arguments\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\n\u22a2 (List.prod (List.drop n (listTransvecCol M)) * M) (inl i) (inr ()) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrw [\u2190 @List.cons_get_drop_succ _ _ \u27e8n, hn'\u27e9]\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\n\u22a2 (List.prod\n          (List.get (listTransvecCol M) { val := n, isLt := hn' } ::\n            List.drop (\u2191{ val := n, isLt := hn' } + 1) (listTransvecCol M)) *\n        M)\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nhave A :\n  (listTransvecCol M).get \u27e8n, hn'\u27e9 =\n    transvection (inl n') (inr unit) (-M (inl n') (inr unit) / M (inr unit) (inr unit)) :=\n  by simp [listTransvecCol]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\n\u22a2 List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\n[PROOFSTEP]\nsimp [listTransvecCol]\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\n\u22a2 (List.prod\n          (List.get (listTransvecCol M) { val := n, isLt := hn' } ::\n            List.drop (\u2191{ val := n, isLt := hn' } + 1) (listTransvecCol M)) *\n        M)\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nsimp only [Matrix.mul_assoc, A, List.prod_cons]\n[GOAL]\ncase H.refine'_1\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\n\u22a2 (transvection (inl { val := n, isLt := hn }) (inr ())\n          (-M (inl { val := n, isLt := hn }) (inr ()) / M (inr ()) (inr ())) *\n        (List.prod (List.drop (n + 1) (listTransvecCol M)) * M))\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nby_cases h : n' = i\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : n' = i\n\u22a2 (transvection (inl { val := n, isLt := hn }) (inr ())\n          (-M (inl { val := n, isLt := hn }) (inr ()) / M (inr ()) (inr ())) *\n        (List.prod (List.drop (n + 1) (listTransvecCol M)) * M))\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nhave hni : n = i := by\n  cases i\n  simp only [Fin.mk_eq_mk] at h \n  simp [h]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : n' = i\n\u22a2 n = \u2191i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl { val := val\u271d, isLt := isLt\u271d }) (inr ()) =\n    if n + 1 \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then 0 else M (inl { val := val\u271d, isLt := isLt\u271d }) (inr ())\nh : n' = { val := val\u271d, isLt := isLt\u271d }\n\u22a2 n = \u2191{ val := val\u271d, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp only [Fin.mk_eq_mk] at h \n[GOAL]\ncase mk\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl { val := val\u271d, isLt := isLt\u271d }) (inr ()) =\n    if n + 1 \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then 0 else M (inl { val := val\u271d, isLt := isLt\u271d }) (inr ())\nh : n = val\u271d\n\u22a2 n = \u2191{ val := val\u271d, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : n' = i\nhni : n = \u2191i\n\u22a2 (transvection (inl { val := n, isLt := hn }) (inr ())\n          (-M (inl { val := n, isLt := hn }) (inr ()) / M (inr ()) (inr ())) *\n        (List.prod (List.drop (n + 1) (listTransvecCol M)) * M))\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nsimp only [h, transvection_mul_apply_same, IH, \u2190 hni, add_le_iff_nonpos_right, listTransvecCol_mul_last_row_drop _ _ hn]\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : n' = i\nhni : n = \u2191i\n\u22a2 (if False then 0 else M (inl i) (inr ())) + -M (inl i) (inr ()) / M (inr ()) (inr ()) * M (inr ()) (inr ()) =\n    if n \u2264 n then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nfield_simp [hM]\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\n\u22a2 (transvection (inl { val := n, isLt := hn }) (inr ())\n          (-M (inl { val := n, isLt := hn }) (inr ()) / M (inr ()) (inr ())) *\n        (List.prod (List.drop (n + 1) (listTransvecCol M)) * M))\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nhave hni : n \u2260 i := by\n  rintro rfl\n  cases i\n  simp at h \n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\n\u22a2 n \u2260 \u2191i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nhn : \u2191i < r\nhk : k \u2264 \u2191i\nIH :\n  (List.prod (List.drop (\u2191i + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if \u2191i + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : \u2191i < List.length (listTransvecCol M)\nn' : Fin r := { val := \u2191i, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := \u2191i, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\n\u22a2 False\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nhn : \u2191{ val := val\u271d, isLt := isLt\u271d } < r\nhk : k \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d }\nIH :\n  (List.prod (List.drop (\u2191{ val := val\u271d, isLt := isLt\u271d } + 1) (listTransvecCol M)) * M)\n      (inl { val := val\u271d, isLt := isLt\u271d }) (inr ()) =\n    if \u2191{ val := val\u271d, isLt := isLt\u271d } + 1 \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then 0\n    else M (inl { val := val\u271d, isLt := isLt\u271d }) (inr ())\nhn' : \u2191{ val := val\u271d, isLt := isLt\u271d } < List.length (listTransvecCol M)\nn' : Fin r := { val := \u2191{ val := val\u271d, isLt := isLt\u271d }, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := \u2191{ val := val\u271d, isLt := isLt\u271d }, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = { val := val\u271d, isLt := isLt\u271d }\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\n\u22a2 (transvection (inl { val := n, isLt := hn }) (inr ())\n          (-M (inl { val := n, isLt := hn }) (inr ()) / M (inr ()) (inr ())) *\n        (List.prod (List.drop (n + 1) (listTransvecCol M)) * M))\n      (inl i) (inr ()) =\n    if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nsimp only [ne_eq, inl.injEq, Ne.symm h, not_false_eq_true, transvection_mul_apply_of_ne]\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\n\u22a2 (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrw [IH]\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\n\u22a2 (if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrcases le_or_lt (n + 1) i with (hi | hi)\n[GOAL]\ncase neg.inl\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : n + 1 \u2264 \u2191i\n\u22a2 (if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nsimp only [hi, n.le_succ.trans hi, if_true]\n[GOAL]\ncase neg.inr\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 (if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())) = if n \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrw [if_neg, if_neg]\n[GOAL]\ncase neg.inr.hnc\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 \u00acn \u2264 \u2191i\n[PROOFSTEP]\nsimpa only [hni.symm, not_le, or_false_iff] using Nat.lt_succ_iff_lt_or_eq.1 hi\n[GOAL]\ncase neg.inr.hnc\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhn : n < r\nhk : k \u2264 n\nIH :\n  (List.prod (List.drop (n + 1) (listTransvecCol M)) * M) (inl i) (inr ()) =\n    if n + 1 \u2264 \u2191i then 0 else M (inl i) (inr ())\nhn' : n < List.length (listTransvecCol M)\nn' : Fin r := { val := n, isLt := hn }\nA :\n  List.get (listTransvecCol M) { val := n, isLt := hn' } =\n    transvection (inl n') (inr ()) (-M (inl n') (inr ()) / M (inr ()) (inr ()))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 \u00acn + 1 \u2264 \u2191i\n[PROOFSTEP]\nsimpa only [not_le] using hi\n[GOAL]\ncase H.refine'_2\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 (List.prod (List.drop r (listTransvecCol M)) * M) (inl i) (inr ()) = if r \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nsimp only [listTransvecCol, List.length_ofFn, le_refl, List.drop_eq_nil_of_le, List.prod_nil, Matrix.one_mul]\n[GOAL]\ncase H.refine'_2\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 M (inl i) (inr ()) = if r \u2264 \u2191i then 0 else M (inl i) (inr ())\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase H.refine'_2.hnc\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 \u00acr \u2264 \u2191i\n[PROOFSTEP]\nsimpa only [not_le] using i.2\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk : k \u2264 r\n\u22a2 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk : \u2115\nhk\u271d : k \u2264 r\nhk : Nat.zero \u2264 r\n\u22a2 (M * List.prod (List.take Nat.zero (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nsimp only [Matrix.mul_one, List.take_zero, List.prod_nil, List.take, Matrix.mul_one]\n[GOAL]\ncase succ\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\n\u22a2 (M * List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nhave hkr : k < r := hk\n[GOAL]\ncase succ\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\n\u22a2 (M * List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nlet k' : Fin r := \u27e8k, hkr\u27e9\n[GOAL]\ncase succ\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\n\u22a2 (M * List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nhave :\n  (listTransvecRow M).get? k =\n    \u2191(transvection (inr Unit.unit) (inl k') (-M (inr Unit.unit) (inl k') / M (inr Unit.unit) (inr Unit.unit))) :=\n  by simp only [listTransvecRow, List.ofFnNthVal, hkr, dif_pos, List.get?_ofFn]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\n\u22a2 List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n[PROOFSTEP]\nsimp only [listTransvecRow, List.ofFnNthVal, hkr, dif_pos, List.get?_ofFn]\n[GOAL]\ncase succ\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n  List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n\u22a2 (M * List.prod (List.take (Nat.succ k) (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nsimp only [List.take_succ, \u2190 Matrix.mul_assoc, this, List.prod_append, Matrix.mul_one, List.prod_cons, List.prod_nil,\n  Option.to_list_some]\n[GOAL]\ncase succ\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n  List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n\u22a2 (M * List.prod (List.take k (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := k, isLt := hkr })\n          (-M (inr ()) (inl { val := k, isLt := hkr }) / M (inr ()) (inr ())))\n      i (inr ()) =\n    M i (inr ())\n[PROOFSTEP]\nrw [mul_transvection_apply_of_ne, IH hkr.le]\n[GOAL]\ncase succ.hb\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nk\u271d : \u2115\nhk\u271d : k\u271d \u2264 r\nk : \u2115\nIH : k \u2264 r \u2192 (M * List.prod (List.take k (listTransvecRow M))) i (inr ()) = M i (inr ())\nhk : Nat.succ k \u2264 r\nhkr : k < r\nk' : Fin r := { val := k, isLt := hkr }\nthis :\n  List.get? (listTransvecRow M) k = some (transvection (inr ()) (inl k') (-M (inr ()) (inl k') / M (inr ()) (inr ())))\n\u22a2 inr () \u2260 inl { val := k, isLt := hkr }\n[PROOFSTEP]\nsimp only [Ne.def, not_false_iff]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\n\u22a2 (M * List.prod (listTransvecRow M)) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nhave A : (listTransvecRow M).length = r := by simp [listTransvecRow]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\n\u22a2 List.length (listTransvecRow M) = r\n[PROOFSTEP]\nsimp [listTransvecRow]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nA : List.length (listTransvecRow M) = r\n\u22a2 (M * List.prod (listTransvecRow M)) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nrw [\u2190 List.take_length (listTransvecRow M), A]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ni : Fin r \u2295 Unit\nA : List.length (listTransvecRow M) = r\n\u22a2 (M * List.prod (List.take r (listTransvecRow M))) i (inr ()) = M i (inr ())\n[PROOFSTEP]\nsimpa using mul_listTransvecRow_last_col_take M i le_rfl\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 (M * List.prod (listTransvecRow M)) (inr ()) (inl i) = 0\n[PROOFSTEP]\nsuffices H :\n  \u2200 k : \u2115,\n    k \u2264 r \u2192 (M * ((listTransvecRow M).take k).prod) (inr unit) (inl i) = if k \u2264 i then M (inr unit) (inl i) else 0\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\n\u22a2 (M * List.prod (listTransvecRow M)) (inr ()) (inl i) = 0\n[PROOFSTEP]\nhave A : (listTransvecRow M).length = r := by simp [listTransvecRow]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\n\u22a2 List.length (listTransvecRow M) = r\n[PROOFSTEP]\nsimp [listTransvecRow]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\nA : List.length (listTransvecRow M) = r\n\u22a2 (M * List.prod (listTransvecRow M)) (inr ()) (inl i) = 0\n[PROOFSTEP]\nrw [\u2190 List.take_length (listTransvecRow M), A]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\nA : List.length (listTransvecRow M) = r\n\u22a2 (M * List.prod (List.take r (listTransvecRow M))) (inr ()) (inl i) = 0\n[PROOFSTEP]\nhave : \u00acr \u2264 i := by simp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\nA : List.length (listTransvecRow M) = r\n\u22a2 \u00acr \u2264 \u2191i\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nH :\n  \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\nA : List.length (listTransvecRow M) = r\nthis : \u00acr \u2264 \u2191i\n\u22a2 (M * List.prod (List.take r (listTransvecRow M))) (inr ()) (inl i) = 0\n[PROOFSTEP]\nsimpa only [this, ite_eq_right_iff] using H r le_rfl\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 \u2200 (k : \u2115),\n    k \u2264 r \u2192\n      (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nintro k hk\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk : k \u2264 r\n\u22a2 (M * List.prod (List.take k (listTransvecRow M))) (inr ()) (inl i) = if k \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\ninduction' k with n IH\n[GOAL]\ncase H.zero\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nhk : Nat.zero \u2264 r\n\u22a2 (M * List.prod (List.take Nat.zero (listTransvecRow M))) (inr ()) (inl i) =\n    if Nat.zero \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nsimp only [if_true, Matrix.mul_one, List.take_zero, zero_le', List.prod_nil, Nat.zero_eq]\n[GOAL]\ncase H.succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\n\u22a2 (M * List.prod (List.take (Nat.succ n) (listTransvecRow M))) (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nhave hnr : n < r := hk\n[GOAL]\ncase H.succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\n\u22a2 (M * List.prod (List.take (Nat.succ n) (listTransvecRow M))) (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nlet n' : Fin r := \u27e8n, hnr\u27e9\n[GOAL]\ncase H.succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\n\u22a2 (M * List.prod (List.take (Nat.succ n) (listTransvecRow M))) (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nhave A :\n  (listTransvecRow M).get? n = \u2191(transvection (inr unit) (inl n') (-M (inr unit) (inl n') / M (inr unit) (inr unit))) :=\n  by simp only [listTransvecRow, List.ofFnNthVal, hnr, dif_pos, List.get?_ofFn]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\n\u22a2 List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\n[PROOFSTEP]\nsimp only [listTransvecRow, List.ofFnNthVal, hnr, dif_pos, List.get?_ofFn]\n[GOAL]\ncase H.succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\n\u22a2 (M * List.prod (List.take (Nat.succ n) (listTransvecRow M))) (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nsimp only [List.take_succ, A, \u2190 Matrix.mul_assoc, List.prod_append, Matrix.mul_one, List.prod_cons, List.prod_nil,\n  Option.to_list_some]\n[GOAL]\ncase H.succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nby_cases h : n' = i\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nhave hni : n = i := by\n  cases i\n  simp only [Fin.mk_eq_mk] at h \n  simp only [h]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\n\u22a2 n = \u2191i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nIH :\n  n \u2264 r \u2192\n    (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl { val := val\u271d, isLt := isLt\u271d }) =\n      if n \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then M (inr ()) (inl { val := val\u271d, isLt := isLt\u271d }) else 0\nh : n' = { val := val\u271d, isLt := isLt\u271d }\n\u22a2 n = \u2191{ val := val\u271d, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp only [Fin.mk_eq_mk] at h \n[GOAL]\ncase mk\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nIH :\n  n \u2264 r \u2192\n    (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl { val := val\u271d, isLt := isLt\u271d }) =\n      if n \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then M (inr ()) (inl { val := val\u271d, isLt := isLt\u271d }) else 0\nh : n = val\u271d\n\u22a2 n = \u2191{ val := val\u271d, isLt := isLt\u271d }\n[PROOFSTEP]\nsimp only [h]\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\nhni : n = \u2191i\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nhave : \u00acn.succ \u2264 i := by simp only [\u2190 hni, n.lt_succ_self, not_le]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\nhni : n = \u2191i\n\u22a2 \u00acNat.succ n \u2264 \u2191i\n[PROOFSTEP]\nsimp only [\u2190 hni, n.lt_succ_self, not_le]\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\nhni : n = \u2191i\nthis : \u00acNat.succ n \u2264 \u2191i\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nsimp only [h, mul_transvection_apply_same, List.take, if_false, mul_listTransvecRow_last_col_take _ _ hnr.le, hni.le,\n  this, if_true, IH hnr.le]\n[GOAL]\ncase pos\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : n' = i\nhni : n = \u2191i\nthis : \u00acNat.succ n \u2264 \u2191i\n\u22a2 M (inr ()) (inl i) + -M (inr ()) (inl i) / M (inr ()) (inr ()) * M (inr ()) (inr ()) = 0\n[PROOFSTEP]\nfield_simp [hM]\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nhave hni : n \u2260 i := by\n  rintro rfl\n  cases i\n  tauto\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\n\u22a2 n \u2260 \u2191i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nIH :\n  \u2191i \u2264 r \u2192\n    (M * List.prod (List.take (\u2191i) (listTransvecRow M))) (inr ()) (inl i) = if \u2191i \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ \u2191i \u2264 r\nhnr : \u2191i < r\nn' : Fin r := { val := \u2191i, isLt := hnr }\nA :\n  List.get? (listTransvecRow M) \u2191i = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\n\u22a2 False\n[PROOFSTEP]\ncases i\n[GOAL]\ncase mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nk : \u2115\nhk\u271d : k \u2264 r\nval\u271d : \u2115\nisLt\u271d : val\u271d < r\nIH :\n  \u2191{ val := val\u271d, isLt := isLt\u271d } \u2264 r \u2192\n    (M * List.prod (List.take (\u2191{ val := val\u271d, isLt := isLt\u271d }) (listTransvecRow M))) (inr ())\n        (inl { val := val\u271d, isLt := isLt\u271d }) =\n      if \u2191{ val := val\u271d, isLt := isLt\u271d } \u2264 \u2191{ val := val\u271d, isLt := isLt\u271d } then\n        M (inr ()) (inl { val := val\u271d, isLt := isLt\u271d })\n      else 0\nhk : Nat.succ \u2191{ val := val\u271d, isLt := isLt\u271d } \u2264 r\nhnr : \u2191{ val := val\u271d, isLt := isLt\u271d } < r\nn' : Fin r := { val := \u2191{ val := val\u271d, isLt := isLt\u271d }, isLt := hnr }\nA :\n  List.get? (listTransvecRow M) \u2191{ val := val\u271d, isLt := isLt\u271d } =\n    some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = { val := val\u271d, isLt := isLt\u271d }\n\u22a2 False\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\n\u22a2 (M * List.prod (List.take n (listTransvecRow M)) *\n        transvection (inr ()) (inl { val := n, isLt := hnr })\n          (-M (inr ()) (inl { val := n, isLt := hnr }) / M (inr ()) (inr ())))\n      (inr ()) (inl i) =\n    if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nsimp only [IH hnr.le, Ne.def, mul_transvection_apply_of_ne, Ne.symm h, inl.injEq]\n[GOAL]\ncase neg\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\n\u22a2 (if n \u2264 \u2191i then M (inr ()) (inl i) else 0) = if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nrcases le_or_lt (n + 1) i with (hi | hi)\n[GOAL]\ncase neg.inl\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : n + 1 \u2264 \u2191i\n\u22a2 (if n \u2264 \u2191i then M (inr ()) (inl i) else 0) = if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nsimp [hi, n.le_succ.trans hi, if_true]\n[GOAL]\ncase neg.inr\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 (if n \u2264 \u2191i then M (inr ()) (inl i) else 0) = if Nat.succ n \u2264 \u2191i then M (inr ()) (inl i) else 0\n[PROOFSTEP]\nrw [if_neg, if_neg]\n[GOAL]\ncase neg.inr.hnc\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 \u00acNat.succ n \u2264 \u2191i\n[PROOFSTEP]\nsimpa only [not_le] using hi\n[GOAL]\ncase neg.inr.hnc\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nk : \u2115\nhk\u271d : k \u2264 r\nn : \u2115\nIH :\n  n \u2264 r \u2192 (M * List.prod (List.take n (listTransvecRow M))) (inr ()) (inl i) = if n \u2264 \u2191i then M (inr ()) (inl i) else 0\nhk : Nat.succ n \u2264 r\nhnr : n < r\nn' : Fin r := { val := n, isLt := hnr }\nA : List.get? (listTransvecRow M) n = some (transvection (inr ()) (inl n') (-M (inr ()) (inl n') / M (inr ()) (inr ())))\nh : \u00acn' = i\nhni : n \u2260 \u2191i\nhi : \u2191i < n + 1\n\u22a2 \u00acn \u2264 \u2191i\n[PROOFSTEP]\nsimpa only [hni.symm, not_le, or_false_iff] using Nat.lt_succ_iff_lt_or_eq.1 hi\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) (inr ()) (inl i) = 0\n[PROOFSTEP]\nhave : listTransvecRow M = listTransvecRow ((listTransvecCol M).prod * M) := by\n  simp [listTransvecRow, listTransvecCol_mul_last_row]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) * M)\n[PROOFSTEP]\nsimp [listTransvecRow, listTransvecCol_mul_last_row]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) * M)\n\u22a2 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) (inr ()) (inl i) = 0\n[PROOFSTEP]\nrw [this]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) * M)\n\u22a2 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow (List.prod (listTransvecCol M) * M))) (inr ())\n      (inl i) =\n    0\n[PROOFSTEP]\napply mul_listTransvecRow_last_row\n[GOAL]\ncase hM\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecRow M = listTransvecRow (List.prod (listTransvecCol M) * M)\n\u22a2 (List.prod (listTransvecCol M) * M) (inr ()) (inr ()) \u2260 0\n[PROOFSTEP]\nsimpa [listTransvecCol_mul_last_row] using hM\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) (inl i) (inr ()) = 0\n[PROOFSTEP]\nhave : listTransvecCol M = listTransvecCol (M * (listTransvecRow M).prod) := by\n  simp [listTransvecCol, mul_listTransvecRow_last_col]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 listTransvecCol M = listTransvecCol (M * List.prod (listTransvecRow M))\n[PROOFSTEP]\nsimp [listTransvecCol, mul_listTransvecRow_last_col]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M * List.prod (listTransvecRow M))\n\u22a2 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) (inl i) (inr ()) = 0\n[PROOFSTEP]\nrw [this, Matrix.mul_assoc]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M * List.prod (listTransvecRow M))\n\u22a2 (List.prod (listTransvecCol (M * List.prod (listTransvecRow M))) * (M * List.prod (listTransvecRow M))) (inl i)\n      (inr ()) =\n    0\n[PROOFSTEP]\napply listTransvecCol_mul_last_col\n[GOAL]\ncase hM\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nthis : listTransvecCol M = listTransvecCol (M * List.prod (listTransvecRow M))\n\u22a2 (M * List.prod (listTransvecRow M)) (inr ()) (inr ()) \u2260 0\n[PROOFSTEP]\nsimpa [mul_listTransvecRow_last_col] using hM\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 IsTwoBlockDiagonal (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) = 0\n[PROOFSTEP]\next i j\n[GOAL]\ncase left.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nj : Unit\n\u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nhave : j = unit := by simp only [eq_iff_true_of_subsingleton]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nj : Unit\n\u22a2 j = ()\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\ncase left.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\nj : Unit\nthis : j = ()\n\u22a2 toBlocks\u2081\u2082 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nsimp [toBlocks\u2081\u2082, this, listTransvecCol_mul_mul_listTransvecRow_last_row M hM]\n[GOAL]\ncase right\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) = 0\n[PROOFSTEP]\next i j\n[GOAL]\ncase right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nhave : i = unit := by simp only [eq_iff_true_of_subsingleton]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\n\u22a2 i = ()\n[PROOFSTEP]\nsimp only [eq_iff_true_of_subsingleton]\n[GOAL]\ncase right.a.h\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Unit\nj : Fin r\nthis : i = ()\n\u22a2 toBlocks\u2082\u2081 (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nsimp [toBlocks\u2082\u2081, this, listTransvecCol_mul_mul_listTransvecRow_last_col M hM]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nlet L : List (TransvectionStruct (Sum (Fin r) Unit) \ud835\udd5c) :=\n  List.ofFn fun i : Fin r => \u27e8inl i, inr unit, by simp, -M (inl i) (inr unit) / M (inr unit) (inr unit)\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\ni : Fin r\n\u22a2 inl i \u2260 inr ()\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nlet L' : List (TransvectionStruct (Sum (Fin r) Unit) \ud835\udd5c) :=\n  List.ofFn fun i : Fin r => \u27e8inr unit, inl i, by simp, -M (inr unit) (inl i) / M (inr unit) (inr unit)\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\ni : Fin r\n\u22a2 inr () \u2260 inl i\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrefine' \u27e8L, L', _\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\n\u22a2 IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nhave A : L.map toMatrix = listTransvecCol M := by simp [listTransvecCol, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\n\u22a2 List.map toMatrix L = listTransvecCol M\n[PROOFSTEP]\nsimp [listTransvecCol, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\nA : List.map toMatrix L = listTransvecCol M\n\u22a2 IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nhave B : L'.map toMatrix = listTransvecRow M := by simp [listTransvecRow, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\nA : List.map toMatrix L = listTransvecCol M\n\u22a2 List.map toMatrix L' = listTransvecRow M\n[PROOFSTEP]\nsimp [listTransvecRow, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\nA : List.map toMatrix L = listTransvecCol M\nB : List.map toMatrix L' = listTransvecRow M\n\u22a2 IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrw [A, B]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) \u2260 0\nL : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := -M (inl i) (inr ()) / M (inr ()) (inr ()) }\nL' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c) :=\n  List.ofFn fun i =>\n    { i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := -M (inr ()) (inl i) / M (inr ()) (inr ()) }\nA : List.map toMatrix L = listTransvecCol M\nB : List.map toMatrix L' = listTransvecRow M\n\u22a2 IsTwoBlockDiagonal (List.prod (listTransvecCol M) * M * List.prod (listTransvecRow M))\n[PROOFSTEP]\nexact isTwoBlockDiagonal_listTransvecCol_mul_mul_listTransvecRow M hM\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nby_cases H : IsTwoBlockDiagonal M\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : IsTwoBlockDiagonal M\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrefine'\n  \u27e8List.nil, List.nil, by simpa using H\u27e9\n    -- we have already proved this when the last coefficient is nonzero\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : IsTwoBlockDiagonal M\n\u22a2 IsTwoBlockDiagonal (List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix []))\n[PROOFSTEP]\nsimpa using H\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : \u00acIsTwoBlockDiagonal M\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nby_cases hM : M (inr unit) (inr unit) \u2260 0\n[GOAL]\ncase pos\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : \u00acIsTwoBlockDiagonal M\nhM : M (inr ()) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nexact exists_isTwoBlockDiagonal_of_ne_zero M hM\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : \u00acIsTwoBlockDiagonal M\nhM : \u00acM (inr ()) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\npush_neg at hM \n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nH : \u00acIsTwoBlockDiagonal M\nhM : M (inr ()) (inr ()) = 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nsimp only [not_and_or, IsTwoBlockDiagonal, toBlocks\u2081\u2082, toBlocks\u2082\u2081, \u2190 Matrix.ext_iff] at H \n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nhave : \u2203 i : Fin r, M (inl i) (inr unit) \u2260 0 \u2228 M (inr unit) (inl i) \u2260 0 :=\n  by\n  cases' H with H H\n  \u00b7 contrapose! H\n    rintro i \u27e8\u27e9\n    exact (H i).1\n  \u00b7 contrapose! H\n    rintro \u27e8\u27e9 j\n    exact (H j).2\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\n\u22a2 \u2203 i, M (inl i) (inr ()) \u2260 0 \u2228 M (inr ()) (inl i) \u2260 0\n[PROOFSTEP]\ncases' H with H H\n[GOAL]\ncase inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j\n\u22a2 \u2203 i, M (inl i) (inr ()) \u2260 0 \u2228 M (inr ()) (inl i) \u2260 0\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\ncase inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u2200 (i : Fin r), M (inl i) (inr ()) = 0 \u2227 M (inr ()) (inl i) = 0\n\u22a2 \u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nrintro i \u27e8\u27e9\n[GOAL]\ncase inl.unit\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u2200 (i : Fin r), M (inl i) (inr ()) = 0 \u2227 M (inr ()) (inl i) = 0\ni : Fin r\n\u22a2 \u2191of (fun i j => M (inl i) (inr j)) i PUnit.unit = OfNat.ofNat 0 i PUnit.unit\n[PROOFSTEP]\nexact (H i).1\n[GOAL]\ncase inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\n\u22a2 \u2203 i, M (inl i) (inr ()) \u2260 0 \u2228 M (inr ()) (inl i) \u2260 0\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\ncase inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u2200 (i : Fin r), M (inl i) (inr ()) = 0 \u2227 M (inr ()) (inl i) = 0\n\u22a2 \u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\n[PROOFSTEP]\nrintro \u27e8\u27e9 j\n[GOAL]\ncase inr.unit\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH : \u2200 (i : Fin r), M (inl i) (inr ()) = 0 \u2227 M (inr ()) (inl i) = 0\nj : Fin r\n\u22a2 \u2191of (fun i j => M (inr i) (inl j)) PUnit.unit j = OfNat.ofNat 0 PUnit.unit j\n[PROOFSTEP]\nexact (H j).2\n[GOAL]\ncase neg\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\nthis : \u2203 i, M (inl i) (inr ()) \u2260 0 \u2228 M (inr ()) (inl i) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrcases this with \u27e8i, h | h\u27e9\n[GOAL]\ncase neg.intro.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nlet M' := transvection (inr Unit.unit) (inl i) 1 * M\n[GOAL]\ncase neg.intro.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nhave hM' : M' (inr unit) (inr unit) \u2260 0 := by simpa [hM]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\n\u22a2 M' (inr ()) (inr ()) \u2260 0\n[PROOFSTEP]\nsimpa [hM]\n[GOAL]\ncase neg.intro.inl\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrcases exists_isTwoBlockDiagonal_of_ne_zero M' hM' with \u27e8L, L', hLL'\u27e9\n[GOAL]\ncase neg.intro.inl.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrw [Matrix.mul_assoc] at hLL' \n[GOAL]\ncase neg.intro.inl.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * (M' * List.prod (List.map toMatrix L')))\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrefine' \u27e8L ++ [\u27e8inr unit, inl i, by simp, 1\u27e9], L', _\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * (M' * List.prod (List.map toMatrix L')))\n\u22a2 inr () \u2260 inl i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.intro.inl.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * (M' * List.prod (List.map toMatrix L')))\n\u22a2 IsTwoBlockDiagonal\n    (List.prod (List.map toMatrix (L ++ [{ i := inr (), j := inl i, hij := (_ : \u00acinr () = inl i), c := 1 }])) * M *\n      List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nsimp only [List.map_append, List.prod_append, Matrix.mul_one, toMatrix_mk, List.prod_cons, List.prod_nil, List.map,\n  Matrix.mul_assoc (L.map toMatrix).prod]\n[GOAL]\ncase neg.intro.inl.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inl i) (inr ()) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := transvection (inr ()) (inl i) 1 * M\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * (M' * List.prod (List.map toMatrix L')))\n\u22a2 IsTwoBlockDiagonal\n    (List.prod (List.map toMatrix L) * (transvection (inr ()) (inl i) 1 * M * List.prod (List.map toMatrix L')))\n[PROOFSTEP]\nexact hLL'\n[GOAL]\ncase neg.intro.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nlet M' := M * transvection (inl i) (inr unit) 1\n[GOAL]\ncase neg.intro.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nhave hM' : M' (inr unit) (inr unit) \u2260 0 := by simpa [hM]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\n\u22a2 M' (inr ()) (inr ()) \u2260 0\n[PROOFSTEP]\nsimpa [hM]\n[GOAL]\ncase neg.intro.inr\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrcases exists_isTwoBlockDiagonal_of_ne_zero M' hM' with \u27e8L, L', hLL'\u27e9\n[GOAL]\ncase neg.intro.inr.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 \u2203 L L', IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrefine' \u27e8L, \u27e8inl i, inr unit, by simp, 1\u27e9 :: L', _\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 inl i \u2260 inr ()\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.intro.inr.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 IsTwoBlockDiagonal\n    (List.prod (List.map toMatrix L) * M *\n      List.prod (List.map toMatrix ({ i := inl i, j := inr (), hij := (_ : \u00acinl i = inr ()), c := 1 } :: L')))\n[PROOFSTEP]\nsimp only [\u2190 Matrix.mul_assoc, toMatrix_mk, List.prod_cons, List.map]\n[GOAL]\ncase neg.intro.inr.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 IsTwoBlockDiagonal\n    (List.prod (List.map toMatrix L) * M * transvection (inl i) (inr ()) 1 * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nrw [Matrix.mul_assoc (L.map toMatrix).prod]\n[GOAL]\ncase neg.intro.inr.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d M : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nhM : M (inr ()) (inr ()) = 0\nH :\n  (\u00ac\u2200 (i : Fin r) (j : Unit), \u2191of (fun i j => M (inl i) (inr j)) i j = OfNat.ofNat 0 i j) \u2228\n    \u00ac\u2200 (i : Unit) (j : Fin r), \u2191of (fun i j => M (inr i) (inl j)) i j = OfNat.ofNat 0 i j\ni : Fin r\nh : M (inr ()) (inl i) \u2260 0\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := M * transvection (inl i) (inr ()) 1\nhM' : M' (inr ()) (inr ()) \u2260 0\nL L' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhLL' : IsTwoBlockDiagonal (List.prod (List.map toMatrix L) * M' * List.prod (List.map toMatrix L'))\n\u22a2 IsTwoBlockDiagonal\n    (List.prod (List.map toMatrix L) * (M * transvection (inl i) (inr ()) 1) * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\nexact hLL'\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrcases exists_isTwoBlockDiagonal_list_transvec_mul_mul_list_transvec M with \u27e8L\u2081, L\u2081', hM\u27e9\n[GOAL]\ncase intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nlet M' := (L\u2081.map toMatrix).prod * M * (L\u2081'.map toMatrix).prod\n[GOAL]\ncase intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nlet M'' := toBlocks\u2081\u2081 M'\n[GOAL]\ncase intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrcases IH M'' with \u27e8L\u2080, L\u2080', D\u2080, h\u2080\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nset c := M' (inr unit) (inr unit)\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrefine' \u27e8L\u2080.map (sumInl Unit) ++ L\u2081, L\u2081' ++ L\u2080'.map (sumInl Unit), Sum.elim D\u2080 fun _ => M' (inr unit) (inr unit), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\n\u22a2 List.prod (List.map toMatrix (List.map (sumInl Unit) L\u2080 ++ L\u2081)) * M *\n      List.prod (List.map toMatrix (L\u2081' ++ List.map (sumInl Unit) L\u2080')) =\n    diagonal (Sum.elim D\u2080 fun x => M' (inr ()) (inr ()))\n[PROOFSTEP]\nsuffices\n  (L\u2080.map (toMatrix \u2218 sumInl Unit)).prod * M' * (L\u2080'.map (toMatrix \u2218 sumInl Unit)).prod =\n    diagonal (Sum.elim D\u2080 fun _ => c)\n  by simpa [Matrix.mul_assoc]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\nthis :\n  List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080) * M' * List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080') =\n    diagonal (Sum.elim D\u2080 fun x => c)\n\u22a2 List.prod (List.map toMatrix (List.map (sumInl Unit) L\u2080 ++ L\u2081)) * M *\n      List.prod (List.map toMatrix (L\u2081' ++ List.map (sumInl Unit) L\u2080')) =\n    diagonal (Sum.elim D\u2080 fun x => M' (inr ()) (inr ()))\n[PROOFSTEP]\nsimpa [Matrix.mul_assoc]\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080) * M' * List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080') =\n    diagonal (Sum.elim D\u2080 fun x => c)\n[PROOFSTEP]\nhave : M' = fromBlocks M'' 0 0 (diagonal fun _ => c) := by\n  -- porting note: simplified proof, because `congr` didn't work anymore\n  rw [\u2190 fromBlocks_toBlocks M', hM.1, hM.2]\n  rfl\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\n\u22a2 M' = fromBlocks M'' 0 0 (diagonal fun x => c)\n[PROOFSTEP]\nrw [\u2190 fromBlocks_toBlocks M', hM.1, hM.2]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\n\u22a2 fromBlocks (toBlocks\u2081\u2081 M') 0 0 (toBlocks\u2082\u2082 M') = fromBlocks M'' 0 0 (diagonal fun x => c)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\nthis : M' = fromBlocks M'' 0 0 (diagonal fun x => c)\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080) * M' * List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080') =\n    diagonal (Sum.elim D\u2080 fun x => c)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro.intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : DecidableEq p\ninst\u271d : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nIH :\n  \u2200 (M : Matrix (Fin r) (Fin r) \ud835\udd5c),\n    \u2203 L\u2080 L\u2080' D\u2080, List.prod (List.map toMatrix L\u2080) * M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nM : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\nL\u2081 L\u2081' : List (TransvectionStruct (Fin r \u2295 Unit) \ud835\udd5c)\nhM : IsTwoBlockDiagonal (List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081'))\nM' : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c := List.prod (List.map toMatrix L\u2081) * M * List.prod (List.map toMatrix L\u2081')\nM'' : Matrix (Fin r) (Fin r) \ud835\udd5c := toBlocks\u2081\u2081 M'\nL\u2080 L\u2080' : List (TransvectionStruct (Fin r) \ud835\udd5c)\nD\u2080 : Fin r \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * M'' * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nc : \ud835\udd5c := M' (inr ()) (inr ())\nthis : M' = fromBlocks M'' 0 0 (diagonal fun x => c)\n\u22a2 List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080) * fromBlocks M'' 0 0 (diagonal fun x => c) *\n      List.prod (List.map (toMatrix \u2218 sumInl Unit) L\u2080') =\n    diagonal (Sum.elim D\u2080 fun x => c)\n[PROOFSTEP]\nsimp [h\u2080]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nH : \u2203 L L' D, List.prod (List.map toMatrix L) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L') = diagonal D\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrcases H with \u27e8L\u2080, L\u2080', D\u2080, h\u2080\u27e9\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrefine' \u27e8L\u2080.map (reindexEquiv e.symm), L\u2080'.map (reindexEquiv e.symm), D\u2080 \u2218 e, _\u27e9\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\n\u22a2 List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080)) * M *\n      List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080')) =\n    diagonal (D\u2080 \u2218 \u2191e)\n[PROOFSTEP]\nhave : M = reindexAlgEquiv \ud835\udd5c e.symm (reindexAlgEquiv \ud835\udd5c e M) := by\n  simp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id, Equiv.symm_comp_self,\n    reindexAlgEquiv_apply]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\n\u22a2 M = \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M)\n[PROOFSTEP]\nsimp only [Equiv.symm_symm, submatrix_submatrix, reindex_apply, submatrix_id_id, Equiv.symm_comp_self,\n  reindexAlgEquiv_apply]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nthis : M = \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M)\n\u22a2 List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080)) * M *\n      List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080')) =\n    diagonal (D\u2080 \u2218 \u2191e)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nthis : M = \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M)\n\u22a2 List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080)) *\n        \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M) *\n      List.prod (List.map toMatrix (List.map (reindexEquiv e.symm) L\u2080')) =\n    diagonal (D\u2080 \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [toMatrix_reindexEquiv_prod, List.map_map, reindexAlgEquiv_apply]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nthis : M = \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M)\n\u22a2 \u2191(reindex e.symm e.symm) (List.prod (List.map toMatrix L\u2080)) * \u2191(reindex e.symm e.symm) (\u2191(reindex e e) M) *\n      \u2191(reindex e.symm e.symm) (List.prod (List.map toMatrix L\u2080')) =\n    diagonal (D\u2080 \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [\u2190 reindexAlgEquiv_apply, \u2190 reindexAlgEquiv_mul, h\u2080]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix p p \ud835\udd5c\ne : p \u2243 n\nL\u2080 L\u2080' : List (TransvectionStruct n \ud835\udd5c)\nD\u2080 : n \u2192 \ud835\udd5c\nh\u2080 : List.prod (List.map toMatrix L\u2080) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L\u2080') = diagonal D\u2080\nthis : M = \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (\u2191(reindexAlgEquiv \ud835\udd5c e) M)\n\u22a2 \u2191(reindexAlgEquiv \ud835\udd5c e.symm) (diagonal D\u2080) = diagonal (D\u2080 \u2218 \u2191e)\n[PROOFSTEP]\nsimp only [Equiv.symm_symm, reindex_apply, submatrix_diagonal_equiv, reindexAlgEquiv_apply]\n[GOAL]\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\ninduction' hn : Fintype.card n with r IH generalizing n M\n[GOAL]\ncase zero\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr : \u2115\nM\u271d\u00b9 : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.zero\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nrefine' \u27e8List.nil, List.nil, fun _ => 1, _\u27e9\n[GOAL]\ncase zero\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr : \u2115\nM\u271d\u00b9 : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.zero\n\u22a2 List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix []) = diagonal fun x => 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase zero.a.h\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr : \u2115\nM\u271d\u00b9 : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.zero\ni j : n\n\u22a2 (List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix [])) i j = diagonal (fun x => 1) i j\n[PROOFSTEP]\nrw [Fintype.card_eq_zero_iff] at hn \n[GOAL]\ncase zero.a.h\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr : \u2115\nM\u271d\u00b9 : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : IsEmpty n\ni j : n\n\u22a2 (List.prod (List.map toMatrix []) * M * List.prod (List.map toMatrix [])) i j = diagonal (fun x => 1) i j\n[PROOFSTEP]\nexact hn.elim' i\n[GOAL]\ncase succ\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nhave e : n \u2243 Sum (Fin r) Unit := by\n  refine' Fintype.equivOfCardEq _\n  rw [hn]\n  rw [@Fintype.card_sum (Fin r) Unit _ _]\n  simp\n[GOAL]\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\n\u22a2 n \u2243 Fin r \u2295 Unit\n[PROOFSTEP]\nrefine' Fintype.equivOfCardEq _\n[GOAL]\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\n\u22a2 Fintype.card n = Fintype.card (Fin r \u2295 Unit)\n[PROOFSTEP]\nrw [hn]\n[GOAL]\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\n\u22a2 Nat.succ r = Fintype.card (Fin r \u2295 Unit)\n[PROOFSTEP]\nrw [@Fintype.card_sum (Fin r) Unit _ _]\n[GOAL]\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\n\u22a2 Nat.succ r = Fintype.card (Fin r) + Fintype.card Unit\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\napply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e\n[GOAL]\ncase succ\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\napply exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction fun N => IH (Fin r) N (by simp)\n[GOAL]\nn\u271d\u00b9 : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : Field \ud835\udd5c\ninst\u271d\u2078 : DecidableEq n\u271d\u00b9\ninst\u271d\u2077 : DecidableEq p\ninst\u271d\u2076 : CommRing R\nr\u271d : \u2115\nM\u271d\u00b9 : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u2075 : Fintype n\u271d\u00b9\ninst\u271d\u2074 : Fintype p\nn\u271d : Type\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq n\u271d\nM\u271d : Matrix n\u271d n\u271d \ud835\udd5c\nx\u271d : \u2115\nhn\u271d : Fintype.card n\u271d = x\u271d\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = Nat.succ r\ne : n \u2243 Fin r \u2295 Unit\nN : Matrix (Fin r) (Fin r) \ud835\udd5c\n\u22a2 Fintype.card (Fin r) = r\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\nhave e : n \u2243 Fin (Fintype.card n) := Fintype.equivOfCardEq (by simp)\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\n\u22a2 Fintype.card n = Fintype.card (Fin (Fintype.card n))\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\ne : n \u2243 Fin (Fintype.card n)\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\napply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\ne : n \u2243 Fin (Fintype.card n)\n\u22a2 \u2203 L L' D, List.prod (List.map toMatrix L) * \u2191(reindexAlgEquiv \ud835\udd5c e) M * List.prod (List.map toMatrix L') = diagonal D\n[PROOFSTEP]\napply exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\n\u22a2 \u2203 L L' D, M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\n[PROOFSTEP]\nrcases exists_list_transvec_mul_mul_list_transvec_eq_diagonal M with \u27e8L, L', D, h\u27e9\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n\u22a2 \u2203 L L' D, M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\n[PROOFSTEP]\nrefine' \u27e8L.reverse.map TransvectionStruct.inv, L'.reverse.map TransvectionStruct.inv, D, _\u27e9\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n\u22a2 M =\n    List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L))) * diagonal D *\n      List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L')))\n[PROOFSTEP]\nsuffices\n  M =\n    (L.reverse.map (toMatrix \u2218 TransvectionStruct.inv)).prod * (L.map toMatrix).prod * M *\n      ((L'.map toMatrix).prod * (L'.reverse.map (toMatrix \u2218 TransvectionStruct.inv)).prod)\n  by simpa [\u2190 h, Matrix.mul_assoc]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\nthis :\n  M =\n    List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) * M *\n      (List.prod (List.map toMatrix L') * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L')))\n\u22a2 M =\n    List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L))) * diagonal D *\n      List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L')))\n[PROOFSTEP]\nsimpa [\u2190 h, Matrix.mul_assoc]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\nM : Matrix n n \ud835\udd5c\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : List.prod (List.map toMatrix L) * M * List.prod (List.map toMatrix L') = diagonal D\n\u22a2 M =\n    List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L)) * List.prod (List.map toMatrix L) * M *\n      (List.prod (List.map toMatrix L') * List.prod (List.map (toMatrix \u2218 TransvectionStruct.inv) (List.reverse L')))\n[PROOFSTEP]\nrw [reverse_inv_prod_mul_prod, prod_mul_reverse_inv_prod, Matrix.one_mul, Matrix.mul_one]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\n\u22a2 P M\n[PROOFSTEP]\nrcases exists_list_transvec_mul_diagonal_mul_list_transvec M with \u27e8L, L', D, h\u27e9\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\n\u22a2 P M\n[PROOFSTEP]\nhave PD : P (diagonal D) := hdiag D (by simp [h])\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\n\u22a2 det (diagonal D) = det M\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\n\u22a2 P M\n[PROOFSTEP]\nsuffices H :\n  \u2200 (L\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)) (E : Matrix n n \ud835\udd5c),\n    P E \u2192 P ((L\u2081.map toMatrix).prod * E * (L\u2082.map toMatrix).prod)\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nH :\n  \u2200 (L\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)) (E : Matrix n n \ud835\udd5c),\n    P E \u2192 P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n\u22a2 P M\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase intro.intro.intro\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nH :\n  \u2200 (L\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)) (E : Matrix n n \ud835\udd5c),\n    P E \u2192 P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n\u22a2 P (List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L'))\n[PROOFSTEP]\napply H L L'\n[GOAL]\ncase intro.intro.intro.a\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nH :\n  \u2200 (L\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)) (E : Matrix n n \ud835\udd5c),\n    P E \u2192 P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n\u22a2 P (diagonal D)\n[PROOFSTEP]\nexact PD\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\n\u22a2 \u2200 (L\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)) (E : Matrix n n \ud835\udd5c),\n    P E \u2192 P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\nintro L\u2081 L\u2082 E PE\n[GOAL]\ncase H\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nL\u2081 L\u2082 : List (TransvectionStruct n \ud835\udd5c)\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\ninduction' L\u2081 with t L\u2081 IH\n[GOAL]\ncase H.nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (List.prod (List.map toMatrix []) * E * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\nsimp only [Matrix.one_mul, List.prod_nil, List.map]\n[GOAL]\ncase H.nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (E * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\ninduction' L\u2082 with t L\u2082 IH generalizing E\n[GOAL]\ncase H.nil.nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nE\u271d : Matrix n n \ud835\udd5c\nPE\u271d : P E\u271d\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (E * List.prod (List.map toMatrix []))\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase H.nil.cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nE\u271d : Matrix n n \ud835\udd5c\nPE\u271d : P E\u271d\nt : TransvectionStruct n \ud835\udd5c\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nIH : \u2200 (E : Matrix n n \ud835\udd5c), P E \u2192 P (E * List.prod (List.map toMatrix L\u2082))\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (E * List.prod (List.map toMatrix (t :: L\u2082)))\n[PROOFSTEP]\nsimp only [\u2190 Matrix.mul_assoc, List.prod_cons, List.map]\n[GOAL]\ncase H.nil.cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nE\u271d : Matrix n n \ud835\udd5c\nPE\u271d : P E\u271d\nt : TransvectionStruct n \ud835\udd5c\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nIH : \u2200 (E : Matrix n n \ud835\udd5c), P E \u2192 P (E * List.prod (List.map toMatrix L\u2082))\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (E * toMatrix t * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\napply IH\n[GOAL]\ncase H.nil.cons.PE\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nE\u271d : Matrix n n \ud835\udd5c\nPE\u271d : P E\u271d\nt : TransvectionStruct n \ud835\udd5c\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nIH : \u2200 (E : Matrix n n \ud835\udd5c), P E \u2192 P (E * List.prod (List.map toMatrix L\u2082))\nE : Matrix n n \ud835\udd5c\nPE : P E\n\u22a2 P (E * toMatrix t)\n[PROOFSTEP]\nexact hmul _ _ PE (htransvec _)\n[GOAL]\ncase H.cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nE : Matrix n n \ud835\udd5c\nPE : P E\nt : TransvectionStruct n \ud835\udd5c\nL\u2081 : List (TransvectionStruct n \ud835\udd5c)\nIH : P (List.prod (List.map toMatrix L\u2081) * E * List.prod (List.map toMatrix L\u2082))\n\u22a2 P (List.prod (List.map toMatrix (t :: L\u2081)) * E * List.prod (List.map toMatrix L\u2082))\n[PROOFSTEP]\nsimp only [Matrix.mul_assoc, List.prod_cons, List.map] at IH \u22a2\n[GOAL]\ncase H.cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), P A \u2192 P B \u2192 P (A * B)\nL L' : List (TransvectionStruct n \ud835\udd5c)\nD : n \u2192 \ud835\udd5c\nh : M = List.prod (List.map toMatrix L) * diagonal D * List.prod (List.map toMatrix L')\nPD : P (diagonal D)\nL\u2082 : List (TransvectionStruct n \ud835\udd5c)\nE : Matrix n n \ud835\udd5c\nPE : P E\nt : TransvectionStruct n \ud835\udd5c\nL\u2081 : List (TransvectionStruct n \ud835\udd5c)\nIH : P (List.prod (List.map toMatrix L\u2081) * (E * List.prod (List.map toMatrix L\u2082)))\n\u22a2 P (toMatrix t * (List.prod (List.map toMatrix L\u2081) * (E * List.prod (List.map toMatrix L\u2082))))\n[PROOFSTEP]\nexact hmul _ _ (htransvec _) IH\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\n\u22a2 P M\n[PROOFSTEP]\nlet Q : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n\u22a2 P M\n[PROOFSTEP]\nhave : Q M := by\n  apply diagonal_transvection_induction Q M\n  \u00b7 intro D hD\n    have detD : det (diagonal D) \u2260 0 := by\n      rw [hD]\n      exact hMdet\n    exact \u27e8detD, hdiag _ detD\u27e9\n  \u00b7 intro t\n    exact \u27e8by simp, htransvec t\u27e9\n  \u00b7 intro A B QA QB\n    exact \u27e8by simp [QA.1, QB.1], hmul A B QA.1 QB.1 QA.2 QB.2\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n\u22a2 Q M\n[PROOFSTEP]\napply diagonal_transvection_induction Q M\n[GOAL]\ncase hdiag\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n\u22a2 \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) = det M \u2192 Q (diagonal D)\n[PROOFSTEP]\nintro D hD\n[GOAL]\ncase hdiag\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nD : n \u2192 \ud835\udd5c\nhD : det (diagonal D) = det M\n\u22a2 Q (diagonal D)\n[PROOFSTEP]\nhave detD : det (diagonal D) \u2260 0 := by\n  rw [hD]\n  exact hMdet\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nD : n \u2192 \ud835\udd5c\nhD : det (diagonal D) = det M\n\u22a2 det (diagonal D) \u2260 0\n[PROOFSTEP]\nrw [hD]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nD : n \u2192 \ud835\udd5c\nhD : det (diagonal D) = det M\n\u22a2 det M \u2260 0\n[PROOFSTEP]\nexact hMdet\n[GOAL]\ncase hdiag\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nD : n \u2192 \ud835\udd5c\nhD : det (diagonal D) = det M\ndetD : det (diagonal D) \u2260 0\n\u22a2 Q (diagonal D)\n[PROOFSTEP]\nexact \u27e8detD, hdiag _ detD\u27e9\n[GOAL]\ncase htransvec\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n\u22a2 \u2200 (t : TransvectionStruct n \ud835\udd5c), Q (toMatrix t)\n[PROOFSTEP]\nintro t\n[GOAL]\ncase htransvec\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nt : TransvectionStruct n \ud835\udd5c\n\u22a2 Q (toMatrix t)\n[PROOFSTEP]\nexact \u27e8by simp, htransvec t\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nt : TransvectionStruct n \ud835\udd5c\n\u22a2 det (toMatrix t) \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hmul\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\n\u22a2 \u2200 (A B : Matrix n n \ud835\udd5c), Q A \u2192 Q B \u2192 Q (A * B)\n[PROOFSTEP]\nintro A B QA QB\n[GOAL]\ncase hmul\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nA B : Matrix n n \ud835\udd5c\nQA : Q A\nQB : Q B\n\u22a2 Q (A * B)\n[PROOFSTEP]\nexact \u27e8by simp [QA.1, QB.1], hmul A B QA.1 QB.1 QA.2 QB.2\u27e9\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nA B : Matrix n n \ud835\udd5c\nQA : Q A\nQB : Q B\n\u22a2 det (A * B) \u2260 0\n[PROOFSTEP]\nsimp [QA.1, QB.1]\n[GOAL]\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype n\nP : Matrix n n \ud835\udd5c \u2192 Prop\nM : Matrix n n \ud835\udd5c\nhMdet : det M \u2260 0\nhdiag : \u2200 (D : n \u2192 \ud835\udd5c), det (diagonal D) \u2260 0 \u2192 P (diagonal D)\nhtransvec : \u2200 (t : TransvectionStruct n \ud835\udd5c), P (toMatrix t)\nhmul : \u2200 (A B : Matrix n n \ud835\udd5c), det A \u2260 0 \u2192 det B \u2260 0 \u2192 P A \u2192 P B \u2192 P (A * B)\nQ : Matrix n n \ud835\udd5c \u2192 Prop := fun N => det N \u2260 0 \u2227 P N\nthis : Q M\n\u22a2 P M\n[PROOFSTEP]\nexact this.2\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Transvection", "llama_tokens": 91700, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680199891789, "lm_q2_score": 0.7371581568543044, "lm_q1q2_score": 0.547832367848286}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 (x\u2081 ^ 2 + x\u2082 ^ 2) * (y\u2081 ^ 2 + y\u2082 ^ 2) = (x\u2081 * y\u2081 - x\u2082 * y\u2082) ^ 2 + (x\u2081 * y\u2082 + x\u2082 * y\u2081) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 (x\u2081 ^ 2 + n * x\u2082 ^ 2) * (y\u2081 ^ 2 + n * y\u2082 ^ 2) = (x\u2081 * y\u2081 - n * x\u2082 * y\u2082) ^ 2 + n * (x\u2081 * y\u2082 + x\u2082 * y\u2081) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 a ^ 4 + 4 * b ^ 4 = ((a - b) ^ 2 + b ^ 2) * ((a + b) ^ 2 + b ^ 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 a ^ 4 + 4 * b ^ 4 = (a ^ 2 - 2 * a * b + 2 * b ^ 2) * (a ^ 2 + 2 * a * b + 2 * b ^ 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 (x\u2081 ^ 2 + x\u2082 ^ 2 + x\u2083 ^ 2 + x\u2084 ^ 2) * (y\u2081 ^ 2 + y\u2082 ^ 2 + y\u2083 ^ 2 + y\u2084 ^ 2) =\n    (x\u2081 * y\u2081 - x\u2082 * y\u2082 - x\u2083 * y\u2083 - x\u2084 * y\u2084) ^ 2 + (x\u2081 * y\u2082 + x\u2082 * y\u2081 + x\u2083 * y\u2084 - x\u2084 * y\u2083) ^ 2 +\n        (x\u2081 * y\u2083 - x\u2082 * y\u2084 + x\u2083 * y\u2081 + x\u2084 * y\u2082) ^ 2 +\n      (x\u2081 * y\u2084 + x\u2082 * y\u2083 - x\u2083 * y\u2082 + x\u2084 * y\u2081) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x\u2081 x\u2082 x\u2083 x\u2084 x\u2085 x\u2086 x\u2087 x\u2088 y\u2081 y\u2082 y\u2083 y\u2084 y\u2085 y\u2086 y\u2087 y\u2088 n : R\n\u22a2 (x\u2081 ^ 2 + x\u2082 ^ 2 + x\u2083 ^ 2 + x\u2084 ^ 2 + x\u2085 ^ 2 + x\u2086 ^ 2 + x\u2087 ^ 2 + x\u2088 ^ 2) *\n      (y\u2081 ^ 2 + y\u2082 ^ 2 + y\u2083 ^ 2 + y\u2084 ^ 2 + y\u2085 ^ 2 + y\u2086 ^ 2 + y\u2087 ^ 2 + y\u2088 ^ 2) =\n    (x\u2081 * y\u2081 - x\u2082 * y\u2082 - x\u2083 * y\u2083 - x\u2084 * y\u2084 - x\u2085 * y\u2085 - x\u2086 * y\u2086 - x\u2087 * y\u2087 - x\u2088 * y\u2088) ^ 2 +\n                  (x\u2081 * y\u2082 + x\u2082 * y\u2081 + x\u2083 * y\u2084 - x\u2084 * y\u2083 + x\u2085 * y\u2086 - x\u2086 * y\u2085 - x\u2087 * y\u2088 + x\u2088 * y\u2087) ^ 2 +\n                (x\u2081 * y\u2083 - x\u2082 * y\u2084 + x\u2083 * y\u2081 + x\u2084 * y\u2082 + x\u2085 * y\u2087 + x\u2086 * y\u2088 - x\u2087 * y\u2085 - x\u2088 * y\u2086) ^ 2 +\n              (x\u2081 * y\u2084 + x\u2082 * y\u2083 - x\u2083 * y\u2082 + x\u2084 * y\u2081 + x\u2085 * y\u2088 - x\u2086 * y\u2087 + x\u2087 * y\u2086 - x\u2088 * y\u2085) ^ 2 +\n            (x\u2081 * y\u2085 - x\u2082 * y\u2086 - x\u2083 * y\u2087 - x\u2084 * y\u2088 + x\u2085 * y\u2081 + x\u2086 * y\u2082 + x\u2087 * y\u2083 + x\u2088 * y\u2084) ^ 2 +\n          (x\u2081 * y\u2086 + x\u2082 * y\u2085 - x\u2083 * y\u2088 + x\u2084 * y\u2087 - x\u2085 * y\u2082 + x\u2086 * y\u2081 - x\u2087 * y\u2084 + x\u2088 * y\u2083) ^ 2 +\n        (x\u2081 * y\u2087 + x\u2082 * y\u2088 + x\u2083 * y\u2085 - x\u2084 * y\u2086 - x\u2085 * y\u2083 + x\u2086 * y\u2084 + x\u2087 * y\u2081 - x\u2088 * y\u2082) ^ 2 +\n      (x\u2081 * y\u2088 - x\u2082 * y\u2087 + x\u2083 * y\u2086 + x\u2084 * y\u2085 - x\u2085 * y\u2084 - x\u2086 * y\u2083 + x\u2087 * y\u2082 + x\u2088 * y\u2081) ^ 2\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupPower.Identities", "llama_tokens": 1369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117983401363, "lm_q2_score": 0.6619228891883799, "lm_q1q2_score": 0.5472194620834243}}
{"text": "[GOAL]\n\u03b1 \u03b2 : LinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191e \u226b \u2191(OrderIso.symm e) = \ud835\udfd9 \u03b1\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : LinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget LinOrdCat).obj \u03b1\n\u22a2 \u2191(\u2191e \u226b \u2191(OrderIso.symm e)) x = \u2191(\ud835\udfd9 \u03b1) x\n[PROOFSTEP]\nexact e.symm_apply_apply x\n[GOAL]\n\u03b1 \u03b2 : LinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191(OrderIso.symm e) \u226b \u2191e = \ud835\udfd9 \u03b2\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : LinOrdCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget LinOrdCat).obj \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm e) \u226b \u2191e) x = \u2191(\ud835\udfd9 \u03b2) x\n[PROOFSTEP]\nexact e.apply_symm_apply x\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.LinOrdCat", "llama_tokens": 306, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267118026095991, "lm_q2_score": 0.661922862511608, "lm_q1q2_score": 0.5472194428554773}}
{"text": "[GOAL]\np : \u2115\n\u22a2 padicValNat p 0 = 0\n[PROOFSTEP]\nsimp [padicValNat]\n[GOAL]\np : \u2115\n\u22a2 padicValNat p 1 = 0\n[PROOFSTEP]\nunfold padicValNat\n[GOAL]\np : \u2115\n\u22a2 (if h : p \u2260 1 \u2227 0 < 1 then Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) else 0) = 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\np : \u2115\nh\u271d : p \u2260 1 \u2227 0 < 1\n\u22a2 Part.get (multiplicity p 1) (_ : multiplicity.Finite p 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\np : \u2115\nh\u271d : \u00ac(p \u2260 1 \u2227 0 < 1)\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : 1 < p\n\u22a2 padicValNat p p = 1\n[PROOFSTEP]\nhave neq_one : \u00acp = 1 \u2194 True := iff_of_true hp.ne' trivial\n[GOAL]\np : \u2115\nhp : 1 < p\nneq_one : \u00acp = 1 \u2194 True\n\u22a2 padicValNat p p = 1\n[PROOFSTEP]\nhave eq_zero_false : p = 0 \u2194 False := iff_false_intro (zero_lt_one.trans hp).ne'\n[GOAL]\np : \u2115\nhp : 1 < p\nneq_one : \u00acp = 1 \u2194 True\neq_zero_false : p = 0 \u2194 False\n\u22a2 padicValNat p p = 1\n[PROOFSTEP]\nsimp [padicValNat, neq_one, eq_zero_false]\n[GOAL]\np n : \u2115\n\u22a2 padicValNat p n = 0 \u2194 p = 1 \u2228 n = 0 \u2228 \u00acp \u2223 n\n[PROOFSTEP]\nsimp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero, multiplicity_eq_zero, and_imp,\n  pos_iff_ne_zero, Ne.def, \u2190 or_iff_not_imp_left]\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\n\u22a2 \u2191(maxPowDiv p n) = multiplicity p n\n[PROOFSTEP]\napply multiplicity.unique <| pow_dvd p n\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\n\u22a2 \u00acp ^ (maxPowDiv p n + 1) \u2223 n\n[PROOFSTEP]\nintro h\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\nh : p ^ (maxPowDiv p n + 1) \u2223 n\n\u22a2 False\n[PROOFSTEP]\napply Nat.not_lt.mpr <| le_of_dvd hp hn h\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\nh : p ^ (maxPowDiv p n + 1) \u2223 n\n\u22a2 maxPowDiv p n < maxPowDiv p n + 1\n[PROOFSTEP]\nsimp\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\nh : multiplicity.Finite p n\n\u22a2 maxPowDiv p n = Part.get (multiplicity p n) h\n[PROOFSTEP]\nrw [PartENat.get_eq_iff_eq_coe.mpr]\n[GOAL]\np\u271d p n : \u2115\nhp : 1 < p\nhn : 0 < n\nh : multiplicity.Finite p n\n\u22a2 multiplicity p n = \u2191(maxPowDiv p n)\n[PROOFSTEP]\napply maxPowDiv_eq_multiplicity hp hn |>.symm\n[GOAL]\np : \u2115\n\u22a2 padicValNat = maxPowDiv\n[PROOFSTEP]\next p n\n[GOAL]\ncase h.h\np\u271d p n : \u2115\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nby_cases (1 < p \u2227 0 < n)\n[GOAL]\ncase h.h\np\u271d p n : \u2115\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nby_cases (1 < p \u2227 0 < n)\n[GOAL]\ncase pos\np\u271d p n : \u2115\nh : 1 < p \u2227 0 < n\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\ndsimp [padicValNat]\n[GOAL]\ncase pos\np\u271d p n : \u2115\nh : 1 < p \u2227 0 < n\n\u22a2 (if h : \u00acp = 1 \u2227 0 < n then Part.get (multiplicity p n) (_ : multiplicity.Finite p n) else 0) = maxPowDiv p n\n[PROOFSTEP]\nrw [dif_pos \u27e8Nat.ne_of_gt h.1, h.2\u27e9, maxPowDiv_eq_multiplicity_get h.1 h.2]\n[GOAL]\ncase neg\np\u271d p n : \u2115\nh : \u00ac(1 < p \u2227 0 < n)\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nsimp only [not_and_or, not_gt_eq, le_zero_iff] at h \n[GOAL]\ncase neg\np\u271d p n : \u2115\nh : p \u2264 1 \u2228 n = 0\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\napply h.elim\n[GOAL]\ncase neg.left\np\u271d p n : \u2115\nh : p \u2264 1 \u2228 n = 0\n\u22a2 p \u2264 1 \u2192 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg.left\np\u271d p n : \u2115\nh\u271d : p \u2264 1 \u2228 n = 0\nh : p \u2264 1\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\ninterval_cases p\n[GOAL]\ncase neg.left.\u00ab0\u00bb\np\u271d p n : \u2115\nh\u271d : 0 \u2264 1 \u2228 n = 0\nh : 0 \u2264 1\n\u22a2 padicValNat 0 n = maxPowDiv 0 n\n[PROOFSTEP]\nsimp [Classical.em]\n[GOAL]\ncase neg.left.\u00ab1\u00bb\np\u271d p n : \u2115\nh\u271d : 1 \u2264 1 \u2228 n = 0\nh : 1 \u2264 1\n\u22a2 padicValNat 1 n = maxPowDiv 1 n\n[PROOFSTEP]\ndsimp [padicValNat, maxPowDiv]\n[GOAL]\ncase neg.left.\u00ab1\u00bb\np\u271d p n : \u2115\nh\u271d : 1 \u2264 1 \u2228 n = 0\nh : 1 \u2264 1\n\u22a2 (if h : \u00ac1 = 1 \u2227 0 < n then Part.get (multiplicity 1 n) (_ : multiplicity.Finite 1 n) else 0) = go 0 1 n\n[PROOFSTEP]\nrw [go_eq, if_neg, dif_neg]\n[GOAL]\ncase neg.left.\u00ab1\u00bb.hnc\np\u271d p n : \u2115\nh\u271d : 1 \u2264 1 \u2228 n = 0\nh : 1 \u2264 1\n\u22a2 \u00ac(\u00ac1 = 1 \u2227 0 < n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.left.\u00ab1\u00bb.hnc\np\u271d p n : \u2115\nh\u271d : 1 \u2264 1 \u2228 n = 0\nh : 1 \u2264 1\n\u22a2 \u00ac(1 < 1 \u2227 0 < n \u2227 n % 1 = 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.right\np\u271d p n : \u2115\nh : p \u2264 1 \u2228 n = 0\n\u22a2 n = 0 \u2192 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase neg.right\np\u271d p n : \u2115\nh\u271d : p \u2264 1 \u2228 n = 0\nh : n = 0\n\u22a2 padicValNat p n = maxPowDiv p n\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : \u2115\nz : \u2124\nhp : p \u2260 1\nhz : z \u2260 0\n\u22a2 (multiplicity (\u2191p) z).Dom\n[PROOFSTEP]\napply multiplicity.finite_int_iff.2\n[GOAL]\np : \u2115\nz : \u2124\nhp : p \u2260 1\nhz : z \u2260 0\n\u22a2 Int.natAbs \u2191p \u2260 1 \u2227 z \u2260 0\n[PROOFSTEP]\nsimp [hp, hz]\n[GOAL]\np : \u2115\nz : \u2124\nhp : p \u2260 1\nhz : z \u2260 0\n\u22a2 padicValInt p z = Part.get (multiplicity (\u2191p) z) (_ : multiplicity.Finite (\u2191p) z)\n[PROOFSTEP]\nrw [padicValInt, padicValNat, dif_pos (And.intro hp (Int.natAbs_pos.mpr hz))]\n[GOAL]\np : \u2115\nz : \u2124\nhp : p \u2260 1\nhz : z \u2260 0\n\u22a2 Part.get (multiplicity p (Int.natAbs z)) (_ : multiplicity.Finite p (Int.natAbs z)) =\n    Part.get (multiplicity (\u2191p) z) (_ : multiplicity.Finite (\u2191p) z)\n[PROOFSTEP]\nsimp only [multiplicity.Int.natAbs p z]\n[GOAL]\np : \u2115\n\u22a2 padicValInt p 0 = 0\n[PROOFSTEP]\nsimp [padicValInt]\n[GOAL]\np : \u2115\n\u22a2 padicValInt p 1 = 0\n[PROOFSTEP]\nsimp [padicValInt]\n[GOAL]\np n : \u2115\n\u22a2 padicValInt p \u2191n = padicValNat p n\n[PROOFSTEP]\nsimp [padicValInt]\n[GOAL]\np : \u2115\nhp : 1 < p\n\u22a2 padicValInt p \u2191p = 1\n[PROOFSTEP]\nsimp [padicValNat.self hp]\n[GOAL]\np : \u2115\nz : \u2124\nh : \u00ac\u2191p \u2223 z\n\u22a2 padicValInt p z = 0\n[PROOFSTEP]\nrw [padicValInt, padicValNat]\n[GOAL]\np : \u2115\nz : \u2124\nh : \u00ac\u2191p \u2223 z\n\u22a2 (if h : p \u2260 1 \u2227 0 < Int.natAbs z then\n      Part.get (multiplicity p (Int.natAbs z)) (_ : multiplicity.Finite p (Int.natAbs z))\n    else 0) =\n    0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\np : \u2115\nz : \u2124\nh : \u00ac\u2191p \u2223 z\nh\u271d : p \u2260 1 \u2227 0 < Int.natAbs z\n\u22a2 Part.get (multiplicity p (Int.natAbs z)) (_ : multiplicity.Finite p (Int.natAbs z)) = 0\n[PROOFSTEP]\nsimp [multiplicity.Int.natAbs, multiplicity_eq_zero.2 h]\n[GOAL]\ncase neg\np : \u2115\nz : \u2124\nh : \u00ac\u2191p \u2223 z\nh\u271d : \u00ac(p \u2260 1 \u2227 0 < Int.natAbs z)\n\u22a2 0 = 0\n[PROOFSTEP]\nsimp [multiplicity.Int.natAbs, multiplicity_eq_zero.2 h]\n[GOAL]\np : \u2115\nq : \u211a\n\u22a2 padicValRat p (-q) = padicValRat p q\n[PROOFSTEP]\nsimp [padicValRat, padicValInt]\n[GOAL]\np : \u2115\n\u22a2 padicValRat p 0 = 0\n[PROOFSTEP]\nsimp [padicValRat]\n[GOAL]\np : \u2115\n\u22a2 padicValRat p 1 = 0\n[PROOFSTEP]\nsimp [padicValRat]\n[GOAL]\np : \u2115\nz : \u2124\n\u22a2 padicValRat p \u2191z = \u2191(padicValInt p z)\n[PROOFSTEP]\nsimp [padicValRat]\n[GOAL]\np : \u2115\nz : \u2124\nhp : p \u2260 1\nhz : z \u2260 0\n\u22a2 padicValRat p \u2191z = \u2191(Part.get (multiplicity (\u2191p) z) (_ : multiplicity.Finite (\u2191p) z))\n[PROOFSTEP]\nrw [of_int, padicValInt.of_ne_one_ne_zero hp hz]\n[GOAL]\np : \u2115\nq : \u211a\nhp : p \u2260 1\nhq : q \u2260 0\n\u22a2 (multiplicity p q.den).Dom\n[PROOFSTEP]\nrw [\u2190 finite_iff_dom, finite_nat_iff]\n[GOAL]\np : \u2115\nq : \u211a\nhp : p \u2260 1\nhq : q \u2260 0\n\u22a2 p \u2260 1 \u2227 0 < q.den\n[PROOFSTEP]\nexact \u27e8hp, q.pos\u27e9\n[GOAL]\np : \u2115\nq : \u211a\nhp : p \u2260 1\nhq : q \u2260 0\n\u22a2 padicValRat p q =\n    \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom))\n[PROOFSTEP]\nrw [padicValRat, padicValInt.of_ne_one_ne_zero hp, padicValNat, dif_pos]\n[GOAL]\ncase hc\np : \u2115\nq : \u211a\nhp : p \u2260 1\nhq : q \u2260 0\n\u22a2 p \u2260 1 \u2227 0 < q.den\n[PROOFSTEP]\nexact \u27e8hp, q.pos\u27e9\n[GOAL]\np : \u2115\nq : \u211a\nhp : p \u2260 1\nhq : q \u2260 0\n\u22a2 q.num \u2260 0\n[PROOFSTEP]\nexact Rat.num_ne_zero_of_ne_zero hq\n[GOAL]\np n : \u2115\n\u22a2 padicValRat p \u2191n = \u2191(padicValNat p n)\n[PROOFSTEP]\nsimp [padicValRat]\n[GOAL]\np : \u2115\nhp : 1 < p\n\u22a2 padicValRat p \u2191p = 1\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\np n : \u2115\n\u22a2 0 \u2264 padicValRat p \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115\n\u22a2 \u2191(padicValNat p n) = padicValRat p \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115\nhp : p \u2260 1\nhn : 0 < n\n\u22a2 \u2191(padicValNat p n) = multiplicity p n\n[PROOFSTEP]\nsimp [padicValNat, hp, hn]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 padicValNat p p = 1\n[PROOFSTEP]\nrw [padicValNat_def (@Fact.out p.Prime).pos]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Part.get (multiplicity p p) (_ : multiplicity.Finite p p) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : 0 < n\ndiv : p \u2223 n\n\u22a2 1 \u2264 padicValNat p n\n[PROOFSTEP]\nrwa [\u2190 PartENat.coe_le_coe, padicValNat_def' hp.out.ne_one hn, \u2190 pow_dvd_iff_le_multiplicity, pow_one]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\n\u22a2 multiplicity.Finite (\u2191p) a \u2194 a \u2260 0\n[PROOFSTEP]\nsimp [finite_int_iff, hp.1.ne_one]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhn : n = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\n\u22a2 padicValRat p q =\n    \u2191(Part.get (multiplicity (\u2191p) n) (_ : multiplicity.Finite (\u2191p) n)) -\n      \u2191(Part.get (multiplicity (\u2191p) d) (_ : multiplicity.Finite (\u2191p) d))\n[PROOFSTEP]\nhave hd : d \u2260 0 := Rat.mk_denom_ne_zero_of_ne_zero hqz qdf\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\n\u22a2 padicValRat p q =\n    \u2191(Part.get (multiplicity (\u2191p) n) (_ : multiplicity.Finite (\u2191p) n)) -\n      \u2191(Part.get (multiplicity (\u2191p) d) (_ : multiplicity.Finite (\u2191p) d))\n[PROOFSTEP]\nlet \u27e8c, hc1, hc2\u27e9 := Rat.num_den_mk hd qdf\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 padicValRat p q =\n    \u2191(Part.get (multiplicity (\u2191p) n) (_ : multiplicity.Finite (\u2191p) n)) -\n      \u2191(Part.get (multiplicity (\u2191p) d) (_ : multiplicity.Finite (\u2191p) d))\n[PROOFSTEP]\nrw [padicValRat.multiplicity_sub_multiplicity hp.1.ne_one hqz]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)) =\n    \u2191(Part.get (multiplicity (\u2191p) n) (_ : multiplicity.Finite (\u2191p) n)) -\n      \u2191(Part.get (multiplicity (\u2191p) d) (_ : multiplicity.Finite (\u2191p) d))\n[PROOFSTEP]\nsimp only [Nat.isUnit_iff, hc1, hc2]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)) =\n    \u2191(Part.get (multiplicity (\u2191p) (c * q.num)) (_ : (multiplicity (\u2191p) (c * q.num)).Dom)) -\n      \u2191(Part.get (multiplicity (\u2191p) (c * \u2191q.den)) (_ : (multiplicity (\u2191p) (c * \u2191q.den)).Dom))\n[PROOFSTEP]\nrw [multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1), multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1)]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)) =\n    \u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c) +\n          Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c) +\n          Part.get (multiplicity \u2191p \u2191q.den) (_ : multiplicity.Finite \u2191p \u2191q.den))\n[PROOFSTEP]\nrw [Nat.cast_add, Nat.cast_add]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)) =\n    \u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c)) +\n        \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      (\u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c)) +\n        \u2191(Part.get (multiplicity \u2191p \u2191q.den) (_ : multiplicity.Finite \u2191p \u2191q.den)))\n[PROOFSTEP]\nsimp_rw [Int.coe_nat_multiplicity p q.den]\n[GOAL]\np\u271d : \u2115\nhp\u271d : Fact (Nat.Prime p\u271d)\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nn d : \u2124\nhqz : q \u2260 0\nqdf : q = n /. d\nhd : d \u2260 0\nc : \u2124\nhc1 : n = c * q.num\nhc2 : d = c * \u2191q.den\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)) =\n    \u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c)) +\n        \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n      (\u2191(Part.get (multiplicity (\u2191p) c) (_ : multiplicity.Finite (\u2191p) c)) +\n        \u2191(Part.get (multiplicity p q.den) (_ : (multiplicity p q.den).Dom)))\n[PROOFSTEP]\nring\n  -- Porting note: was\n    -- simp only [hc1, hc2, multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1),\n    --   hp.1.ne_one, hqz, pos_iff_ne_zero, Int.coe_nat_multiplicity p q.den\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\n\u22a2 padicValRat p (q * r) = padicValRat p q + padicValRat p r\n[PROOFSTEP]\nhave : q * r = q.num * r.num /. (q.den * r.den) := by rw_mod_cast [Rat.mul_num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\n\u22a2 q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\n[PROOFSTEP]\nrw_mod_cast [Rat.mul_num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\n\u22a2 padicValRat p (q * r) = padicValRat p q + padicValRat p r\n[PROOFSTEP]\nhave hq' : q.num /. q.den \u2260 0 := by rwa [Rat.num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\n\u22a2 q.num /. \u2191q.den \u2260 0\n[PROOFSTEP]\nrwa [Rat.num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\n\u22a2 padicValRat p (q * r) = padicValRat p q + padicValRat p r\n[PROOFSTEP]\nhave hr' : r.num /. r.den \u2260 0 := by rwa [Rat.num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\n\u22a2 r.num /. \u2191r.den \u2260 0\n[PROOFSTEP]\nrwa [Rat.num_den]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\n\u22a2 padicValRat p (q * r) = padicValRat p q + padicValRat p r\n[PROOFSTEP]\nhave hp' : Prime (p : \u2124) := Nat.prime_iff_prime_int.1 hp.1\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n\u22a2 padicValRat p (q * r) = padicValRat p q + padicValRat p r\n[PROOFSTEP]\nrw [padicValRat.defn p (mul_ne_zero hq hr) this]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) (q.num * r.num)) (_ : multiplicity.Finite (\u2191p) (q.num * r.num))) -\n      \u2191(Part.get (multiplicity (\u2191p) (\u2191q.den * \u2191r.den)) (_ : multiplicity.Finite (\u2191p) (\u2191q.den * \u2191r.den))) =\n    padicValRat p q + padicValRat p r\n[PROOFSTEP]\nconv_rhs => rw [\u2190 @Rat.num_den q, padicValRat.defn p hq', \u2190 @Rat.num_den r, padicValRat.defn p hr']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n| padicValRat p q + padicValRat p r\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q, padicValRat.defn p hq', \u2190 @Rat.num_den r, padicValRat.defn p hr']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n| padicValRat p q + padicValRat p r\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q, padicValRat.defn p hq', \u2190 @Rat.num_den r, padicValRat.defn p hr']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n| padicValRat p q + padicValRat p r\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q, padicValRat.defn p hq', \u2190 @Rat.num_den r, padicValRat.defn p hr']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) (q.num * r.num)) (_ : multiplicity.Finite (\u2191p) (q.num * r.num))) -\n      \u2191(Part.get (multiplicity (\u2191p) (\u2191q.den * \u2191r.den)) (_ : multiplicity.Finite (\u2191p) (\u2191q.den * \u2191r.den))) =\n    \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n        \u2191(Part.get (multiplicity \u2191p \u2191q.den) (_ : multiplicity.Finite \u2191p \u2191q.den)) +\n      (\u2191(Part.get (multiplicity (\u2191p) r.num) (_ : multiplicity.Finite (\u2191p) r.num)) -\n        \u2191(Part.get (multiplicity \u2191p \u2191r.den) (_ : multiplicity.Finite \u2191p \u2191r.den)))\n[PROOFSTEP]\nrw [multiplicity.mul' hp', multiplicity.mul' hp', Nat.cast_add, Nat.cast_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\nthis : q * r = q.num * r.num /. (\u2191q.den * \u2191r.den)\nhq' : q.num /. \u2191q.den \u2260 0\nhr' : r.num /. \u2191r.den \u2260 0\nhp' : _root_.Prime \u2191p\n\u22a2 \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) +\n        \u2191(Part.get (multiplicity (\u2191p) r.num) (_ : multiplicity.Finite (\u2191p) r.num)) -\n      (\u2191(Part.get (multiplicity \u2191p \u2191q.den) (_ : multiplicity.Finite \u2191p \u2191q.den)) +\n        \u2191(Part.get (multiplicity \u2191p \u2191r.den) (_ : multiplicity.Finite \u2191p \u2191r.den))) =\n    \u2191(Part.get (multiplicity (\u2191p) q.num) (_ : multiplicity.Finite (\u2191p) q.num)) -\n        \u2191(Part.get (multiplicity \u2191p \u2191q.den) (_ : multiplicity.Finite \u2191p \u2191q.den)) +\n      (\u2191(Part.get (multiplicity (\u2191p) r.num) (_ : multiplicity.Finite (\u2191p) r.num)) -\n        \u2191(Part.get (multiplicity \u2191p \u2191r.den) (_ : multiplicity.Finite \u2191p \u2191r.den)))\n[PROOFSTEP]\nring\n  -- Porting note: was\n    -- simp [add_comm, add_left_comm, sub_eq_add_neg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q \u2260 0\nk : \u2115\n\u22a2 padicValRat p (q ^ k) = \u2191k * padicValRat p q\n[PROOFSTEP]\ninduction k\n[GOAL]\ncase zero\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q \u2260 0\n\u22a2 padicValRat p (q ^ zero) = \u2191zero * padicValRat p q\n[PROOFSTEP]\nsimp [*, padicValRat.mul hq (pow_ne_zero _ hq), _root_.pow_succ, add_mul, add_comm]\n[GOAL]\ncase succ\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q \u2260 0\nn\u271d : \u2115\nn_ih\u271d : padicValRat p (q ^ n\u271d) = \u2191n\u271d * padicValRat p q\n\u22a2 padicValRat p (q ^ succ n\u271d) = \u2191(succ n\u271d) * padicValRat p q\n[PROOFSTEP]\nsimp [*, padicValRat.mul hq (pow_ne_zero _ hq), _root_.pow_succ, add_mul, add_comm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\n\u22a2 padicValRat p q\u207b\u00b9 = -padicValRat p q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q = 0\n\u22a2 padicValRat p q\u207b\u00b9 = -padicValRat p q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : \u00acq = 0\n\u22a2 padicValRat p q\u207b\u00b9 = -padicValRat p q\n[PROOFSTEP]\nrw [eq_neg_iff_add_eq_zero, \u2190 padicValRat.mul (inv_ne_zero hq) hq, inv_mul_cancel hq, padicValRat.one]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhq : q \u2260 0\nhr : r \u2260 0\n\u22a2 padicValRat p (q / r) = padicValRat p q - padicValRat p r\n[PROOFSTEP]\nrw [div_eq_mul_inv, padicValRat.mul hq (inv_ne_zero hr), padicValRat.inv r, sub_eq_add_neg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\n\u22a2 padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\nhave hf1 : Finite (p : \u2124) (n\u2081 * d\u2082) := finite_int_prime_iff.2 (mul_ne_zero hn\u2081 hd\u2082)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\n\u22a2 padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\nhave hf2 : Finite (p : \u2124) (n\u2082 * d\u2081) := finite_int_prime_iff.2 (mul_ne_zero hn\u2082 hd\u2081)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n\u22a2 padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\nconv =>\n  lhs\n  rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2081 hd\u2081) rfl,\n    padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2082 hd\u2082) rfl, sub_le_iff_le_add', \u2190 add_sub_assoc,\n    _root_.le_sub_iff_add_le]\n  norm_cast\n  rw [\u2190 multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm, \u2190\n    multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf2, PartENat.get_le_get, multiplicity_le_multiplicity_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\n  lhs\n  rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2081 hd\u2081) rfl,\n    padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2082 hd\u2082) rfl, sub_le_iff_le_add', \u2190 add_sub_assoc,\n    _root_.le_sub_iff_add_le]\n  norm_cast\n  rw [\u2190 multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm, \u2190\n    multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf2, PartENat.get_le_get, multiplicity_le_multiplicity_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\n  lhs\n  rw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2081 hd\u2081) rfl,\n    padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2082 hd\u2082) rfl, sub_le_iff_le_add', \u2190 add_sub_assoc,\n    _root_.le_sub_iff_add_le]\n  norm_cast\n  rw [\u2190 multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm, \u2190\n    multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf2, PartENat.get_le_get, multiplicity_le_multiplicity_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082) \u2194 \u2200 (n : \u2115), \u2191p ^ n \u2223 n\u2081 * d\u2082 \u2192 \u2191p ^ n \u2223 n\u2082 * d\u2081\n[PROOFSTEP]\nlhs\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| padicValRat p (n\u2081 /. d\u2081) \u2264 padicValRat p (n\u2082 /. d\u2082)\n[PROOFSTEP]\nrw [padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2081 hd\u2081) rfl,\n  padicValRat.defn p (Rat.divInt_ne_zero_of_ne_zero hn\u2082 hd\u2082) rfl, sub_le_iff_le_add', \u2190 add_sub_assoc,\n  _root_.le_sub_iff_add_le]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| \u2191(Part.get (multiplicity (\u2191p) n\u2081) (_ : multiplicity.Finite (\u2191p) n\u2081)) +\n      \u2191(Part.get (multiplicity (\u2191p) d\u2082) (_ : multiplicity.Finite (\u2191p) d\u2082)) \u2264\n    \u2191(Part.get (multiplicity (\u2191p) d\u2081) (_ : multiplicity.Finite (\u2191p) d\u2081)) +\n      \u2191(Part.get (multiplicity (\u2191p) n\u2082) (_ : multiplicity.Finite (\u2191p) n\u2082))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn\u2081 n\u2082 d\u2081 d\u2082 : \u2124\nhn\u2081 : n\u2081 \u2260 0\nhn\u2082 : n\u2082 \u2260 0\nhd\u2081 : d\u2081 \u2260 0\nhd\u2082 : d\u2082 \u2260 0\nhf1 : multiplicity.Finite (\u2191p) (n\u2081 * d\u2082)\nhf2 : multiplicity.Finite (\u2191p) (n\u2082 * d\u2081)\n| Part.get (multiplicity (\u2191p) n\u2081) (_ : multiplicity.Finite (\u2191p) n\u2081) +\n      Part.get (multiplicity (\u2191p) d\u2082) (_ : multiplicity.Finite (\u2191p) d\u2082) \u2264\n    Part.get (multiplicity (\u2191p) d\u2081) (_ : multiplicity.Finite (\u2191p) d\u2081) +\n      Part.get (multiplicity (\u2191p) n\u2082) (_ : multiplicity.Finite (\u2191p) n\u2082)\n[PROOFSTEP]\nrw [\u2190 multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf1, add_comm, \u2190\n  multiplicity.mul' (Nat.prime_iff_prime_int.1 hp.1) hf2, PartENat.get_le_get, multiplicity_le_multiplicity_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : q = 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nsimpa [hq] using h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : r = 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hqn : q.num \u2260 0 := Rat.num_ne_zero_of_ne_zero hq\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hqd : (q.den : \u2124) \u2260 0 := by exact_mod_cast Rat.den_nz _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\n\u22a2 \u2191q.den \u2260 0\n[PROOFSTEP]\nexact_mod_cast Rat.den_nz _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hrn : r.num \u2260 0 := Rat.num_ne_zero_of_ne_zero hr\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hrd : (r.den : \u2124) \u2260 0 := by exact_mod_cast Rat.den_nz _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\n\u22a2 \u2191r.den \u2260 0\n[PROOFSTEP]\nexact_mod_cast Rat.den_nz _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hqreq : q + r = (q.num * r.den + q.den * r.num) /. (q.den * r.den) := Rat.add_num_den _ _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nhave hqrd : q.num * r.den + q.den * r.num \u2260 0 := Rat.mk_num_ne_zero_of_ne_zero hqr hqreq\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 @Rat.num_den q]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n| padicValRat p q\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n| padicValRat p q\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n| padicValRat p q\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 padicValRat p (q.num /. \u2191q.den) \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nrw [hqreq, padicValRat_le_padicValRat_iff hqn hqrd hqd (mul_ne_zero hqd hrd), \u2190 multiplicity_le_multiplicity_iff,\n  mul_left_comm, multiplicity.mul (Nat.prime_iff_prime_int.1 hp.1), add_mul]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (q.num * \u2191r.den) \u2264\n    multiplicity (\u2191p) (q.num * \u2191r.den * \u2191q.den + \u2191q.den * r.num * \u2191q.den)\n[PROOFSTEP]\nrw [\u2190 @Rat.num_den q, \u2190 @Rat.num_den r, padicValRat_le_padicValRat_iff hqn hrn hqd hrd, \u2190\n  multiplicity_le_multiplicity_iff] at h \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh\u271d : \u2200 (n : \u2115), \u2191p ^ n \u2223 q.num * \u2191r.den \u2192 \u2191p ^ n \u2223 r.num * \u2191q.den\nh : multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (r.num * \u2191q.den)\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (q.num * \u2191r.den) \u2264\n    multiplicity (\u2191p) (q.num * \u2191r.den * \u2191q.den + \u2191q.den * r.num * \u2191q.den)\n[PROOFSTEP]\ncalc\n  _ \u2264 min (multiplicity (\u2191p) (q.num * r.den * q.den)) (multiplicity (\u2191p) (\u2191q.den * r.num * \u2191q.den)) :=\n    le_min (by rw [@multiplicity.mul _ _ _ _ (_ * _) _ (Nat.prime_iff_prime_int.1 hp.1), add_comm])\n      (by\n        rw [mul_assoc, @multiplicity.mul _ _ _ _ (q.den : \u2124) (_ * _) (Nat.prime_iff_prime_int.1 hp.1)]\n        exact add_le_add_left h _)\n  _ \u2264 _ := min_le_multiplicity_add\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh\u271d : \u2200 (n : \u2115), \u2191p ^ n \u2223 q.num * \u2191r.den \u2192 \u2191p ^ n \u2223 r.num * \u2191q.den\nh : multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (r.num * \u2191q.den)\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (q.num * \u2191r.den * \u2191q.den)\n[PROOFSTEP]\nrw [@multiplicity.mul _ _ _ _ (_ * _) _ (Nat.prime_iff_prime_int.1 hp.1), add_comm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh\u271d : \u2200 (n : \u2115), \u2191p ^ n \u2223 q.num * \u2191r.den \u2192 \u2191p ^ n \u2223 r.num * \u2191q.den\nh : multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (r.num * \u2191q.den)\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (\u2191q.den * r.num * \u2191q.den)\n[PROOFSTEP]\nrw [mul_assoc, @multiplicity.mul _ _ _ _ (q.den : \u2124) (_ * _) (Nat.prime_iff_prime_int.1 hp.1)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh\u271d : \u2200 (n : \u2115), \u2191p ^ n \u2223 q.num * \u2191r.den \u2192 \u2191p ^ n \u2223 r.num * \u2191q.den\nh : multiplicity (\u2191p) (q.num * \u2191r.den) \u2264 multiplicity (\u2191p) (r.num * \u2191q.den)\nhq : \u00acq = 0\nhr : \u00acr = 0\nhqn : q.num \u2260 0\nhqd : \u2191q.den \u2260 0\nhrn : r.num \u2260 0\nhrd : \u2191r.den \u2260 0\nhqreq : q + r = (q.num * \u2191r.den + \u2191q.den * r.num) /. (\u2191q.den * \u2191r.den)\nhqrd : q.num * \u2191r.den + \u2191q.den * r.num \u2260 0\n\u22a2 multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (q.num * \u2191r.den) \u2264\n    multiplicity \u2191p \u2191q.den + multiplicity (\u2191p) (r.num * \u2191q.den)\n[PROOFSTEP]\nexact add_le_add_left h _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\n\u22a2 min (padicValRat p q) (padicValRat p r) \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nrw [min_eq_left h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p q \u2264 padicValRat p r\n\u22a2 padicValRat p q \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nexact le_padicValRat_add_of_le hqr h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p r \u2264 padicValRat p q\n\u22a2 min (padicValRat p q) (padicValRat p r) \u2264 padicValRat p (q + r)\n[PROOFSTEP]\nrw [min_eq_right h, add_comm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p r \u2264 padicValRat p q\n\u22a2 padicValRat p r \u2264 padicValRat p (r + q)\n[PROOFSTEP]\nexact le_padicValRat_add_of_le (by rwa [add_comm]) h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a\nhqr : q + r \u2260 0\nh : padicValRat p r \u2264 padicValRat p q\n\u22a2 r + q \u2260 0\n[PROOFSTEP]\nrwa [add_comm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 i in Finset.range n, F i \u2260 0\n\u22a2 0 < padicValRat p (\u2211 i in Finset.range n, F i)\n[PROOFSTEP]\ninduction' n with d hd\n[GOAL]\ncase zero\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nhF : \u2200 (i : \u2115), i < zero \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 i in Finset.range zero, F i \u2260 0\n\u22a2 0 < padicValRat p (\u2211 i in Finset.range zero, F i)\n[PROOFSTEP]\nexact False.elim (hn0 rfl)\n[GOAL]\ncase succ\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 i in Finset.range (succ d), F i \u2260 0\n\u22a2 0 < padicValRat p (\u2211 i in Finset.range (succ d), F i)\n[PROOFSTEP]\nrw [Finset.sum_range_succ] at hn0 \u22a2\n[GOAL]\ncase succ\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\n\u22a2 0 < padicValRat p (\u2211 x in Finset.range d, F x + F d)\n[PROOFSTEP]\nby_cases h : \u2211 x : \u2115 in Finset.range d, F x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\nh : \u2211 x in Finset.range d, F x = 0\n\u22a2 0 < padicValRat p (\u2211 x in Finset.range d, F x + F d)\n[PROOFSTEP]\nrw [h, zero_add]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\nh : \u2211 x in Finset.range d, F x = 0\n\u22a2 0 < padicValRat p (F d)\n[PROOFSTEP]\nexact hF d (lt_add_one _)\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\nh : \u00ac\u2211 x in Finset.range d, F x = 0\n\u22a2 0 < padicValRat p (\u2211 x in Finset.range d, F x + F d)\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (min_le_padicValRat_add hn0)\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\nh : \u00ac\u2211 x in Finset.range d, F x = 0\n\u22a2 0 < min (padicValRat p (\u2211 x in Finset.range d, F x)) (padicValRat p (F d))\n[PROOFSTEP]\nrefine' lt_min (hd (fun i hi => _) h) (hF d (lt_add_one _))\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nF : \u2115 \u2192 \u211a\nhF\u271d : \u2200 (i : \u2115), i < n \u2192 0 < padicValRat p (F i)\nhn0\u271d : \u2211 i in Finset.range n, F i \u2260 0\nd : \u2115\nhd :\n  (\u2200 (i : \u2115), i < d \u2192 0 < padicValRat p (F i)) \u2192\n    \u2211 i in Finset.range d, F i \u2260 0 \u2192 0 < padicValRat p (\u2211 i in Finset.range d, F i)\nhF : \u2200 (i : \u2115), i < succ d \u2192 0 < padicValRat p (F i)\nhn0 : \u2211 x in Finset.range d, F x + F d \u2260 0\nh : \u00ac\u2211 x in Finset.range d, F x = 0\ni : \u2115\nhi : i < d\n\u22a2 0 < padicValRat p (F i)\n[PROOFSTEP]\nexact hF _ (lt_trans hi (lt_add_one _))\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 a \u2260 0 \u2192 b \u2260 0 \u2192 padicValNat p (a * b) = padicValNat p a + padicValNat p b\n[PROOFSTEP]\nexact_mod_cast @padicValRat.mul p _ a b\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\nh : b \u2223 a\n\u22a2 padicValNat p (a / b) = padicValNat p a - padicValNat p b\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\np b : \u2115\nhp : Fact (Nat.Prime p)\nh : b \u2223 0\n\u22a2 padicValNat p (0 / b) = padicValNat p 0 - padicValNat p b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np a b : \u2115\nhp : Fact (Nat.Prime p)\nh : b \u2223 a\nha : a \u2260 0\n\u22a2 padicValNat p (a / b) = padicValNat p a - padicValNat p b\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := h\n[GOAL]\ncase inr.intro\np b : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2115\nha : b * k \u2260 0\n\u22a2 padicValNat p (b * k / b) = padicValNat p (b * k) - padicValNat p b\n[PROOFSTEP]\nobtain \u27e8hb, hk\u27e9 := mul_ne_zero_iff.mp ha\n[GOAL]\ncase inr.intro.intro\np b : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2115\nha : b * k \u2260 0\nhb : b \u2260 0\nhk : k \u2260 0\n\u22a2 padicValNat p (b * k / b) = padicValNat p (b * k) - padicValNat p b\n[PROOFSTEP]\nrw [mul_comm, k.mul_div_cancel hb.bot_lt, padicValNat.mul hk hb, Nat.add_sub_cancel]\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\ndvd : p \u2223 b\n\u22a2 padicValNat p (b / p) = padicValNat p b - 1\n[PROOFSTEP]\nrw [padicValNat.div_of_dvd dvd, padicValNat_self]\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nha : a \u2260 0\n\u22a2 padicValNat p (a ^ n) = n * padicValNat p a\n[PROOFSTEP]\nsimpa only [\u2190 @Nat.cast_inj \u2124, push_cast] using padicValRat.pow (Nat.cast_ne_zero.mpr ha)\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 padicValNat p (p ^ n) = n\n[PROOFSTEP]\nrw [padicValNat.pow _ (@Fact.out p.Prime).ne_zero, padicValNat_self, mul_one]\n[GOAL]\np a b : \u2115\nhp : Fact (Nat.Prime p)\ndvd : p ^ a \u2223 b\n\u22a2 padicValNat p (b / p ^ a) = padicValNat p b - a\n[PROOFSTEP]\nrw [padicValNat.div_of_dvd dvd, padicValNat.prime_pow]\n[GOAL]\np a b\u271d : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2115\ncpm : coprime p m\nb : \u2115\ndvd : m \u2223 b\n\u22a2 padicValNat p (b / m) = padicValNat p b\n[PROOFSTEP]\nrw [padicValNat.div_of_dvd dvd, eq_zero_of_not_dvd (hp.out.coprime_iff_not_dvd.mp cpm), Nat.sub_zero]\n[GOAL]\np n : \u2115\nhp : 1 \u2264 padicValNat p n\n\u22a2 p \u2223 n\n[PROOFSTEP]\nby_contra h\n[GOAL]\np n : \u2115\nhp : 1 \u2264 padicValNat p n\nh : \u00acp \u2223 n\n\u22a2 False\n[PROOFSTEP]\nrw [padicValNat.eq_zero_of_not_dvd h] at hp \n[GOAL]\np n : \u2115\nhp : 1 \u2264 0\nh : \u00acp \u2223 n\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl 0 (lt_of_lt_of_le zero_lt_one hp)\n[GOAL]\np n : \u2115\n\u22a2 p ^ padicValNat p n \u2223 n\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\n\u22a2 p ^ padicValNat p 0 \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np n : \u2115\nhn : n > 0\n\u22a2 p ^ padicValNat p n \u2223 n\n[PROOFSTEP]\nrcases eq_or_ne p 1 with (rfl | hp)\n[GOAL]\ncase inr.inl\nn : \u2115\nhn : n > 0\n\u22a2 1 ^ padicValNat 1 n \u2223 n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\np n : \u2115\nhn : n > 0\nhp : p \u2260 1\n\u22a2 p ^ padicValNat p n \u2223 n\n[PROOFSTEP]\nrw [multiplicity.pow_dvd_iff_le_multiplicity, padicValNat_def']\n[GOAL]\ncase inr.inr.hp\np n : \u2115\nhn : n > 0\nhp : p \u2260 1\n\u22a2 p \u2260 1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.hn\np n : \u2115\nhn : n > 0\nhp : p \u2260 1\n\u22a2 0 < n\n[PROOFSTEP]\nassumption\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na n : \u2115\nha : a \u2260 0\n\u22a2 p ^ n \u2223 a \u2194 n \u2264 padicValNat p a\n[PROOFSTEP]\nrw [pow_dvd_iff_le_multiplicity, \u2190 padicValNat_def' hp.out.ne_one ha.bot_lt, PartENat.coe_le_coe]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\na : \u2115\n\u22a2 p ^ n \u2223 a \u2194 a = 0 \u2228 n \u2264 padicValNat p a\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 p ^ n \u2223 0 \u2194 0 = 0 \u2228 n \u2264 padicValNat p 0\n[PROOFSTEP]\nexact iff_of_true (dvd_zero _) (Or.inl rfl)\n[GOAL]\ncase inr\np n : \u2115\nhp : Fact (Nat.Prime p)\na : \u2115\nha : a \u2260 0\n\u22a2 p ^ n \u2223 a \u2194 a = 0 \u2228 n \u2264 padicValNat p a\n[PROOFSTEP]\nrw [padicValNat_dvd_iff_le ha, or_iff_right ha]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\n\u22a2 \u00acp ^ (padicValNat p n + 1) \u2223 n\n[PROOFSTEP]\nrw [padicValNat_dvd_iff_le hn, not_le]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\n\u22a2 padicValNat p n < padicValNat p n + 1\n[PROOFSTEP]\nexact Nat.lt_succ_self _\n[GOAL]\np n : \u2115\nhn : n \u2260 0\n\u22a2 Finset.image (fun x => p ^ x) (Finset.range (padicValNat p n + 1)) \u2286 divisors n\n[PROOFSTEP]\nintro t ht\n[GOAL]\np n : \u2115\nhn : n \u2260 0\nt : \u2115\nht : t \u2208 Finset.image (fun x => p ^ x) (Finset.range (padicValNat p n + 1))\n\u22a2 t \u2208 divisors n\n[PROOFSTEP]\nsimp only [exists_prop, Finset.mem_image, Finset.mem_range] at ht \n[GOAL]\np n : \u2115\nhn : n \u2260 0\nt : \u2115\nht : \u2203 a, a < padicValNat p n + 1 \u2227 p ^ a = t\n\u22a2 t \u2208 divisors n\n[PROOFSTEP]\nobtain \u27e8k, hk, rfl\u27e9 := ht\n[GOAL]\ncase intro.intro\np n : \u2115\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n + 1\n\u22a2 p ^ k \u2208 divisors n\n[PROOFSTEP]\nrw [Nat.mem_divisors]\n[GOAL]\ncase intro.intro\np n : \u2115\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n + 1\n\u22a2 p ^ k \u2223 n \u2227 n \u2260 0\n[PROOFSTEP]\nexact \u27e8(pow_dvd_pow p <| by linarith).trans pow_padicValNat_dvd, hn\u27e9\n[GOAL]\np n : \u2115\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n + 1\n\u22a2 k \u2264 padicValNat p n\n[PROOFSTEP]\nlinarith\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Finset.image (fun t => p ^ (t + 1)) (Finset.range (padicValNat p n)) \u2286 Finset.erase (divisors n) 1\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Finset.image (fun t => p ^ (t + 1)) (Finset.range (padicValNat p 0)) \u2286 Finset.erase (divisors 0) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\n\u22a2 Finset.image (fun t => p ^ (t + 1)) (Finset.range (padicValNat p n)) \u2286 Finset.erase (divisors n) 1\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase inr\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\nt : \u2115\nht : t \u2208 Finset.image (fun t => p ^ (t + 1)) (Finset.range (padicValNat p n))\n\u22a2 t \u2208 Finset.erase (divisors n) 1\n[PROOFSTEP]\nsimp only [exists_prop, Finset.mem_image, Finset.mem_range] at ht \n[GOAL]\ncase inr\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\nt : \u2115\nht : \u2203 a, a < padicValNat p n \u2227 p ^ (a + 1) = t\n\u22a2 t \u2208 Finset.erase (divisors n) 1\n[PROOFSTEP]\nobtain \u27e8k, hk, rfl\u27e9 := ht\n[GOAL]\ncase inr.intro.intro\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n\n\u22a2 p ^ (k + 1) \u2208 Finset.erase (divisors n) 1\n[PROOFSTEP]\nrw [Finset.mem_erase, Nat.mem_divisors]\n[GOAL]\ncase inr.intro.intro\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n\n\u22a2 p ^ (k + 1) \u2260 1 \u2227 p ^ (k + 1) \u2223 n \u2227 n \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, (pow_dvd_pow p <| succ_le_iff.2 hk).trans pow_padicValNat_dvd, hn\u27e9\n[GOAL]\ncase inr.intro.intro\np n : \u2115\nhp : Fact (Nat.Prime p)\nhn : n \u2260 0\nk : \u2115\nhk : k < padicValNat p n\n\u22a2 p ^ (k + 1) \u2260 1\n[PROOFSTEP]\nexact (Nat.one_lt_pow _ _ k.succ_pos hp.out.one_lt).ne'\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 padicValNat p (p * n)! = padicValNat p n ! + n\n[PROOFSTEP]\nrefine' PartENat.natCast_inj.mp _\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191(padicValNat p (p * n)!) = \u2191(padicValNat p n ! + n)\n[PROOFSTEP]\nrw [padicValNat_def' (Nat.Prime.ne_one hp.out) <| factorial_pos (p * n), Nat.cast_add,\n  padicValNat_def' (Nat.Prime.ne_one hp.out) <| factorial_pos n]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 multiplicity p (p * n)! = multiplicity p n ! + \u2191n\n[PROOFSTEP]\nexact Prime.multiplicity_factorial_mul hp.out\n[GOAL]\np n m : \u2115\nhp : Fact (Nat.Prime p)\nh : n < p\n\u22a2 padicValNat p (p * m + n)! = padicValNat p (p * m)!\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\np n m : \u2115\nhp : Fact (Nat.Prime p)\nh\u271d : n < p\nh : zero < p\n\u22a2 padicValNat p (p * m + zero)! = padicValNat p (p * m)!\n[PROOFSTEP]\nrw [zero_eq, add_zero]\n[GOAL]\ncase succ\np n\u271d m : \u2115\nhp : Fact (Nat.Prime p)\nh\u271d : n\u271d < p\nn : \u2115\nhn : n < p \u2192 padicValNat p (p * m + n)! = padicValNat p (p * m)!\nh : succ n < p\n\u22a2 padicValNat p (p * m + succ n)! = padicValNat p (p * m)!\n[PROOFSTEP]\nrw [add_succ, factorial_succ, padicValNat.mul (succ_ne_zero (p * m + n)) <| factorial_ne_zero (p * m + _),\n  hn <| lt_of_succ_lt h, \u2190 add_succ,\n  padicValNat_eq_zero_of_mem_Ioo\n    \u27e8(Nat.lt_add_of_pos_right <| succ_pos n),\n      (Nat.mul_add _ _ _ \u25b8 Nat.mul_one _ \u25b8 ((add_lt_add_iff_left (p * m)).mpr h))\u27e9,\n  zero_add]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 padicValNat p (p * (n / p))! = padicValNat p n !\n[PROOFSTEP]\nnth_rw 2 [\u2190 div_add_mod n p]\n[GOAL]\np n : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 padicValNat p (p * (n / p))! = padicValNat p (p * (n / p) + n % p)!\n[PROOFSTEP]\nexact (padicValNat_factorial_mul_add (n / p) <| mod_lt n <| Prime.pos hp.out).symm\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 (p - 1) * padicValNat p n ! = n - List.sum (digits p n)\n[PROOFSTEP]\nrw [padicValNat_factorial <| lt_succ_of_lt <| lt.base (log p n), \u2190 Finset.sum_Ico_add' _ 0 _ 1, Ico_zero_eq_range, \u2190\n  sub_one_mul_sum_log_div_pow_eq_sub_sum_digits]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\na : \u2124\n\u22a2 \u2191p ^ n \u2223 a \u2194 a = 0 \u2228 n \u2264 padicValInt p a\n[PROOFSTEP]\nrw [padicValInt, \u2190 Int.natAbs_eq_zero, \u2190 padicValNat_dvd_iff, \u2190 Int.coe_nat_dvd_left, Int.coe_nat_pow]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\n\u22a2 \u2191p ^ padicValInt p a \u2223 a\n[PROOFSTEP]\nrw [padicValInt_dvd_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\n\u22a2 a = 0 \u2228 padicValInt p a \u2264 padicValInt p a\n[PROOFSTEP]\nexact Or.inr le_rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na b : \u2124\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 padicValInt p (a * b) = padicValInt p a + padicValInt p b\n[PROOFSTEP]\nsimp_rw [padicValInt]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na b : \u2124\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 padicValNat p (Int.natAbs (a * b)) = padicValNat p (Int.natAbs a) + padicValNat p (Int.natAbs b)\n[PROOFSTEP]\nrw [Int.natAbs_mul, padicValNat.mul]\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\na b : \u2124\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 Int.natAbs a \u2260 0\n[PROOFSTEP]\nrwa [Int.natAbs_ne_zero]\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\na b : \u2124\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 Int.natAbs b \u2260 0\n[PROOFSTEP]\nrwa [Int.natAbs_ne_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nha : a \u2260 0\n\u22a2 padicValInt p (a * \u2191p) = padicValInt p a + 1\n[PROOFSTEP]\nrw [padicValInt.mul ha (Int.coe_nat_ne_zero.mpr hp.out.ne_zero)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : \u2124\nha : a \u2260 0\n\u22a2 padicValInt p a + padicValInt p \u2191p = padicValInt p a + 1\n[PROOFSTEP]\nsimp only [eq_self_iff_true, padicValInt.of_nat, padicValNat_self]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Padics.PadicVal", "llama_tokens": 26118, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117940706734, "lm_q2_score": 0.661922862511608, "lm_q1q2_score": 0.5472194372033672}}
{"text": "[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\n\u22a2 kernel.\u03b9 (prod.lift f g) \u226b f = kernel.\u03b9 (prod.lift f g) \u226b prod.lift f g \u226b prod.fst\n[PROOFSTEP]\nrw [prod.lift_fst]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\n\u22a2 kernel.\u03b9 (prod.lift f g) \u226b prod.lift f g \u226b prod.fst = 0 \u226b prod.fst\n[PROOFSTEP]\nrw [kernel.condition_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\n\u22a2 kernel.\u03b9 (prod.lift f g) \u226b g = kernel.\u03b9 (prod.lift f g) \u226b prod.lift f g \u226b prod.snd\n[PROOFSTEP]\nrw [prod.lift_snd]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\n\u22a2 kernel.\u03b9 (prod.lift f g) \u226b prod.lift f g \u226b prod.snd = 0 \u226b prod.snd\n[PROOFSTEP]\nrw [kernel.condition_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\n\u22a2 a' \u226b a = b' \u226b b\n[PROOFSTEP]\nsimp at ha' hb' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b a = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b b = kernel.\u03b9 (prod.lift f g)\n\u22a2 a' \u226b a = b' \u226b b\n[PROOFSTEP]\nrw [ha', hb']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((PullbackCone.snd s \u226b b) \u226b prod.lift f g) \u226b prod.fst = PullbackCone.snd s \u226b b \u226b f\n[PROOFSTEP]\nsimp only [prod.lift_fst, Category.assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 PullbackCone.snd s \u226b b \u226b f = PullbackCone.fst s \u226b a \u226b f\n[PROOFSTEP]\nrw [PullbackCone.condition_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 PullbackCone.fst s \u226b a \u226b f = PullbackCone.fst s \u226b 0\n[PROOFSTEP]\nrw [haf]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 PullbackCone.fst s \u226b 0 = 0 \u226b prod.fst\n[PROOFSTEP]\nrw [comp_zero, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((PullbackCone.snd s \u226b b) \u226b prod.lift f g) \u226b prod.snd = PullbackCone.snd s \u226b b \u226b g\n[PROOFSTEP]\nsimp only [prod.lift_snd, Category.assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 PullbackCone.snd s \u226b b \u226b g = PullbackCone.snd s \u226b 0\n[PROOFSTEP]\nrw [hbg]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 PullbackCone.snd s \u226b 0 = 0 \u226b prod.snd\n[PROOFSTEP]\nrw [comp_zero, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((fun s => kernel.lift (prod.lift f g) (PullbackCone.snd s \u226b b) (_ : (PullbackCone.snd s \u226b b) \u226b prod.lift f g = 0))\n          s \u226b\n        a') \u226b\n      a =\n    PullbackCone.fst s \u226b a\n[PROOFSTEP]\nrw [KernelFork.\u03b9_of\u03b9] at ha' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b a = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((fun s => kernel.lift (prod.lift f g) (PullbackCone.snd s \u226b b) (_ : (PullbackCone.snd s \u226b b) \u226b prod.lift f g = 0))\n          s \u226b\n        a') \u226b\n      a =\n    PullbackCone.fst s \u226b a\n[PROOFSTEP]\nsimp [ha', PullbackCone.condition s]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((fun s => kernel.lift (prod.lift f g) (PullbackCone.snd s \u226b b) (_ : (PullbackCone.snd s \u226b b) \u226b prod.lift f g = 0))\n          s \u226b\n        b') \u226b\n      b =\n    PullbackCone.snd s \u226b b\n[PROOFSTEP]\nrw [KernelFork.\u03b9_of\u03b9] at hb' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b b = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\n\u22a2 ((fun s => kernel.lift (prod.lift f g) (PullbackCone.snd s \u226b b) (_ : (PullbackCone.snd s \u226b b) \u226b prod.lift f g = 0))\n          s \u226b\n        b') \u226b\n      b =\n    PullbackCone.snd s \u226b b\n[PROOFSTEP]\nsimp [hb']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\nm : s.pt \u27f6 kernel (prod.lift f g)\nh\u2081 : m \u226b a' = PullbackCone.fst s\nx\u271d : m \u226b b' = PullbackCone.snd s\n\u22a2 m \u226b kernel.\u03b9 (prod.lift f g) = m \u226b a' \u226b a\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\nm : s.pt \u27f6 kernel (prod.lift f g)\nh\u2081 : m \u226b a' = PullbackCone.fst s\nx\u271d : m \u226b b' = PullbackCone.snd s\n\u22a2 kernel.\u03b9 (prod.lift f g) = a' \u226b a\n[PROOFSTEP]\nexact ha'.symm\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\nm : s.pt \u27f6 kernel (prod.lift f g)\nh\u2081 : m \u226b a' = PullbackCone.fst s\nx\u271d : m \u226b b' = PullbackCone.snd s\n\u22a2 m \u226b a' \u226b a = PullbackCone.fst s \u226b a\n[PROOFSTEP]\nrw [\u2190 Category.assoc, h\u2081]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.100, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : HasKernels C\ninst\u271d\u00b2 : NormalMonoCategory C\nX Y Z : C\na : X \u27f6 Z\nb : Y \u27f6 Z\ninst\u271d\u00b9 : Mono a\ninst\u271d : Mono b\nP : C\nf : Z \u27f6 P\nhaf : a \u226b f = 0\ni : IsLimit (KernelFork.of\u03b9 a haf)\nQ : C\ng : Z \u27f6 Q\nhbg : b \u226b g = 0\ni' : IsLimit (KernelFork.of\u03b9 b hbg)\na' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 a haf).pt\nha' : a' \u226b Fork.\u03b9 (KernelFork.of\u03b9 a haf) = kernel.\u03b9 (prod.lift f g)\nb' : kernel (prod.lift f g) \u27f6 (KernelFork.of\u03b9 b hbg).pt\nhb' : b' \u226b Fork.\u03b9 (KernelFork.of\u03b9 b hbg) = kernel.\u03b9 (prod.lift f g)\ns : PullbackCone a b\nm : s.pt \u27f6 kernel (prod.lift f g)\nh\u2081 : m \u226b a' = PullbackCone.fst s\nx\u271d : m \u226b b' = PullbackCone.snd s\n\u22a2 PullbackCone.snd s \u226b b =\n    kernel.lift (prod.lift f g) (PullbackCone.snd s \u226b b) (_ : (PullbackCone.snd s \u226b b) \u226b prod.lift f g = 0) \u226b\n      kernel.\u03b9 (prod.lift f g)\n[PROOFSTEP]\nrw [kernel.lift_\u03b9]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 pullback.fst \u226b \ud835\udfd9 X = pullback.fst \u226b prod.lift (\ud835\udfd9 X) f \u226b prod.fst\n[PROOFSTEP]\nrw [prod.lift_fst]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 pullback.fst \u226b prod.lift (\ud835\udfd9 X) f \u226b prod.fst = pullback.snd \u226b prod.lift (\ud835\udfd9 X) g \u226b prod.fst\n[PROOFSTEP]\nrw [pullback.condition_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 pullback.snd \u226b prod.lift (\ud835\udfd9 X) g \u226b prod.fst = pullback.snd\n[PROOFSTEP]\nrw [prod.lift_fst, Category.comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\n\u22a2 pullback.fst \u226b f = pullback.fst \u226b prod.lift (\ud835\udfd9 X) f \u226b prod.snd\n[PROOFSTEP]\nrw [prod.lift_snd]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\n\u22a2 pullback.fst \u226b prod.lift (\ud835\udfd9 X) f \u226b prod.snd = pullback.snd \u226b prod.lift (\ud835\udfd9 X) g \u226b prod.snd\n[PROOFSTEP]\nrw [pullback.condition_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\n\u22a2 pullback.snd \u226b prod.lift (\ud835\udfd9 X) g \u226b prod.snd = pullback.snd \u226b g\n[PROOFSTEP]\nrw [prod.lift_snd]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\n\u22a2 pullback.fst \u226b f = pullback.fst \u226b g\n[PROOFSTEP]\nrw [hvu, \u2190 huv]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\nhuu : pullback.fst \u226b f = pullback.fst \u226b g\ns : Fork f g\n\u22a2 (Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) f) \u226b prod.fst = (Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) g) \u226b prod.fst\n[PROOFSTEP]\nsimp only [prod.lift_fst, Category.assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\nhuu : pullback.fst \u226b f = pullback.fst \u226b g\ns : Fork f g\n\u22a2 (Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) f) \u226b prod.snd = (Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) g) \u226b prod.snd\n[PROOFSTEP]\nsimp only [prod.comp_lift, Fork.condition s]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\nhuu : pullback.fst \u226b f = pullback.fst \u226b g\ns : Fork f g\n\u22a2 (fun s => pullback.lift (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) f = Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) g)) s \u226b\n      Fork.\u03b9 (Fork.of\u03b9 pullback.fst huu) =\n    Fork.\u03b9 s\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\nhuu : pullback.fst \u226b f = pullback.fst \u226b g\ns : Fork f g\nm : s.pt \u27f6 (Fork.of\u03b9 pullback.fst huu).pt\nh : m \u226b Fork.\u03b9 (Fork.of\u03b9 pullback.fst huu) = Fork.\u03b9 s\n\u22a2 m \u226b pullback.fst =\n    (fun s => pullback.lift (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) f = Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) g)) s \u226b\n      pullback.fst\n[PROOFSTEP]\nsimpa only [pullback.lift_fst] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.45510, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pullback.fst = pullback.snd\nhvu : pullback.fst \u226b f = pullback.snd \u226b g\nhuu : pullback.fst \u226b f = pullback.fst \u226b g\ns : Fork f g\nm : s.pt \u27f6 (Fork.of\u03b9 pullback.fst huu).pt\nh : m \u226b Fork.\u03b9 (Fork.of\u03b9 pullback.fst huu) = Fork.\u03b9 s\n\u22a2 m \u226b pullback.snd =\n    (fun s => pullback.lift (Fork.\u03b9 s) (Fork.\u03b9 s) (_ : Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) f = Fork.\u03b9 s \u226b prod.lift (\ud835\udfd9 X) g)) s \u226b\n      pullback.snd\n[PROOFSTEP]\nsimpa only [huv.symm, pullback.lift_fst] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.76598, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 f \u226b 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8W, w, hw, hl\u27e9 := normalMonoOfMono (equalizer.\u03b9 u v)\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := equalizer.lift' f huv\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\n\u22a2 u = v\n[PROOFSTEP]\nhave hwf : f \u226b w = 0 := by rw [\u2190 hm, Category.assoc, hw, comp_zero]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\n\u22a2 f \u226b w = 0\n[PROOFSTEP]\nrw [\u2190 hm, Category.assoc, hw, comp_zero]\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := CokernelCofork.IsColimit.desc' l _ hwf\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\nn : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 W\nhn : Cofork.\u03c0 (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)) \u226b n = w\n\u22a2 u = v\n[PROOFSTEP]\nrw [Cofork.\u03c0_of\u03c0, zero_comp] at hn \n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\nn : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 W\nhn : 0 = w\n\u22a2 u = v\n[PROOFSTEP]\nhave : IsIso (equalizer.\u03b9 u v) := by apply isIso_limit_cone_parallelPair_of_eq hn.symm hl\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\nn : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 W\nhn : 0 = w\n\u22a2 IsIso (equalizer.\u03b9 u v)\n[PROOFSTEP]\napply isIso_limit_cone_parallelPair_of_eq hn.symm hl\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\nn : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 W\nhn : 0 = w\nthis : IsIso (equalizer.\u03b9 u v)\n\u22a2 u = v\n[PROOFSTEP]\napply (cancel_epi (equalizer.\u03b9 u v)).1\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteProducts C\ninst\u271d\u00b9 : HasKernels C\ninst\u271d : NormalMonoCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsColimit (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0))\nZ\u271d : C\nu v : Y \u27f6 Z\u271d\nhuv : f \u226b u = f \u226b v\nW : C\nw : Y \u27f6 W\nhw : equalizer.\u03b9 u v \u226b w = 0\nhl : IsLimit (KernelFork.of\u03b9 (equalizer.\u03b9 u v) hw)\nm : X \u27f6 equalizer u v\nhm : m \u226b equalizer.\u03b9 u v = f\nhwf : f \u226b w = 0\nn : (CokernelCofork.of\u03c0 0 (_ : f \u226b 0 = 0)).pt \u27f6 W\nhn : 0 = w\nthis : IsIso (equalizer.\u03b9 u v)\n\u22a2 equalizer.\u03b9 u v \u226b u = equalizer.\u03b9 u v \u226b v\n[PROOFSTEP]\nexact equalizer.condition _ _\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\n\u22a2 f \u226b cokernel.\u03c0 (coprod.desc f g) = coprod.inl \u226b coprod.desc f g \u226b cokernel.\u03c0 (coprod.desc f g)\n[PROOFSTEP]\nrw [coprod.inl_desc_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\n\u22a2 coprod.inl \u226b coprod.desc f g \u226b cokernel.\u03c0 (coprod.desc f g) = coprod.inl \u226b 0\n[PROOFSTEP]\nrw [cokernel.condition]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\n\u22a2 g \u226b cokernel.\u03c0 (coprod.desc f g) = coprod.inr \u226b coprod.desc f g \u226b cokernel.\u03c0 (coprod.desc f g)\n[PROOFSTEP]\nrw [coprod.inr_desc_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\n\u22a2 coprod.inr \u226b coprod.desc f g \u226b cokernel.\u03c0 (coprod.desc f g) = coprod.inr \u226b 0\n[PROOFSTEP]\nrw [cokernel.condition]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\n\u22a2 a \u226b a' = b \u226b b'\n[PROOFSTEP]\nsimp only [Cofork.\u03c0_of\u03c0] at ha' hb' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : a \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : b \u226b b' = cokernel.\u03c0 (coprod.desc f g)\n\u22a2 a \u226b a' = b \u226b b'\n[PROOFSTEP]\nrw [ha', hb']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 coprod.inl \u226b coprod.desc f g \u226b b \u226b PushoutCocone.inr s = f \u226b b \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [coprod.inl_desc_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 f \u226b b \u226b PushoutCocone.inr s = f \u226b a \u226b PushoutCocone.inl s\n[PROOFSTEP]\nrw [PushoutCocone.condition]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 f \u226b a \u226b PushoutCocone.inl s = 0 \u226b PushoutCocone.inl s\n[PROOFSTEP]\nrw [\u2190 Category.assoc, eq_whisker hfa]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 0 \u226b PushoutCocone.inl s = coprod.inl \u226b 0\n[PROOFSTEP]\nrw [comp_zero, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 coprod.inr \u226b coprod.desc f g \u226b b \u226b PushoutCocone.inr s = g \u226b b \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [coprod.inr_desc_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 g \u226b b \u226b PushoutCocone.inr s = 0 \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [\u2190 Category.assoc, eq_whisker hgb]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 0 \u226b PushoutCocone.inr s = coprod.inr \u226b 0\n[PROOFSTEP]\nrw [comp_zero, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 a \u226b\n      a' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    a \u226b PushoutCocone.inl s\n[PROOFSTEP]\nrw [CokernelCofork.\u03c0_of\u03c0] at ha' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : a \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 a \u226b\n      a' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    a \u226b PushoutCocone.inl s\n[PROOFSTEP]\nhave reassoced {W : C} (h : cokernel (coprod.desc f g) \u27f6 W) : a \u226b a' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h := by\n  rw [\u2190 Category.assoc, eq_whisker ha']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : a \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nW : C\nh : cokernel (coprod.desc f g) \u27f6 W\n\u22a2 a \u226b a' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h\n[PROOFSTEP]\nrw [\u2190 Category.assoc, eq_whisker ha']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : a \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nreassoced : \u2200 {W : C} (h : cokernel (coprod.desc f g) \u27f6 W), a \u226b a' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h\n\u22a2 a \u226b\n      a' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    a \u226b PushoutCocone.inl s\n[PROOFSTEP]\nsimp [reassoced, PushoutCocone.condition s]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 b \u226b\n      b' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    b \u226b PushoutCocone.inr s\n[PROOFSTEP]\nrw [CokernelCofork.\u03c0_of\u03c0] at hb' \n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : b \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\n\u22a2 b \u226b\n      b' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    b \u226b PushoutCocone.inr s\n[PROOFSTEP]\nhave reassoced' {W : C} (h : cokernel (coprod.desc f g) \u27f6 W) : b \u226b b' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h := by\n  rw [\u2190 Category.assoc, eq_whisker hb']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : b \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nW : C\nh : cokernel (coprod.desc f g) \u27f6 W\n\u22a2 b \u226b b' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h\n[PROOFSTEP]\nrw [\u2190 Category.assoc, eq_whisker hb']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : b \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nreassoced' : \u2200 {W : C} (h : cokernel (coprod.desc f g) \u27f6 W), b \u226b b' \u226b h = cokernel.\u03c0 (coprod.desc f g) \u226b h\n\u22a2 b \u226b\n      b' \u226b\n        (fun s =>\n            cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s)\n              (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0))\n          s =\n    b \u226b PushoutCocone.inr s\n[PROOFSTEP]\nsimp [reassoced']\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nm : cokernel (coprod.desc f g) \u27f6 s.pt\nh\u2081 : a' \u226b m = PushoutCocone.inl s\nx\u271d : b' \u226b m = PushoutCocone.inr s\n\u22a2 cokernel.\u03c0 (coprod.desc f g) \u226b m = (a \u226b a') \u226b m\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nm : cokernel (coprod.desc f g) \u27f6 s.pt\nh\u2081 : a' \u226b m = PushoutCocone.inl s\nx\u271d : b' \u226b m = PushoutCocone.inr s\n\u22a2 cokernel.\u03c0 (coprod.desc f g) = a \u226b a'\n[PROOFSTEP]\nexact ha'.symm\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nm : cokernel (coprod.desc f g) \u27f6 s.pt\nh\u2081 : a' \u226b m = PushoutCocone.inl s\nx\u271d : b' \u226b m = PushoutCocone.inr s\n\u22a2 (a \u226b a') \u226b m = a \u226b PushoutCocone.inl s\n[PROOFSTEP]\nrw [Category.assoc, h\u2081]\n[GOAL]\nC : Type u_1\ninst\u271d\u2076 : Category.{?u.80661, u_1} C\ninst\u271d\u2075 : HasZeroMorphisms C\ninst\u271d\u2074 : HasFiniteCoproducts C\ninst\u271d\u00b3 : HasCokernels C\ninst\u271d\u00b2 : NormalEpiCategory C\nX Y Z : C\na : X \u27f6 Y\nb : X \u27f6 Z\ninst\u271d\u00b9 : Epi a\ninst\u271d : Epi b\nP : C\nf : P \u27f6 X\nhfa : f \u226b a = 0\ni : IsColimit (CokernelCofork.of\u03c0 a hfa)\nQ : C\ng : Q \u27f6 X\nhgb : g \u226b b = 0\ni' : IsColimit (CokernelCofork.of\u03c0 b hgb)\na' : (CokernelCofork.of\u03c0 a hfa).pt \u27f6 cokernel (coprod.desc f g)\nha' : Cofork.\u03c0 (CokernelCofork.of\u03c0 a hfa) \u226b a' = cokernel.\u03c0 (coprod.desc f g)\nb' : (CokernelCofork.of\u03c0 b hgb).pt \u27f6 cokernel (coprod.desc f g)\nhb' : Cofork.\u03c0 (CokernelCofork.of\u03c0 b hgb) \u226b b' = cokernel.\u03c0 (coprod.desc f g)\ns : PushoutCocone a b\nm : cokernel (coprod.desc f g) \u27f6 s.pt\nh\u2081 : a' \u226b m = PushoutCocone.inl s\nx\u271d : b' \u226b m = PushoutCocone.inr s\n\u22a2 b \u226b PushoutCocone.inr s =\n    cokernel.\u03c0 (coprod.desc f g) \u226b\n      cokernel.desc (coprod.desc f g) (b \u226b PushoutCocone.inr s) (_ : coprod.desc f g \u226b b \u226b PushoutCocone.inr s = 0)\n[PROOFSTEP]\nrw [cokernel.\u03c0_desc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 \ud835\udfd9 Y \u226b pushout.inl = (coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) f) \u226b pushout.inl\n[PROOFSTEP]\nrw [coprod.inl_desc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 (coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) f) \u226b pushout.inl = (coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) g) \u226b pushout.inr\n[PROOFSTEP]\nsimp only [Category.assoc, pushout.condition]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\n\u22a2 (coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) g) \u226b pushout.inr = pushout.inr\n[PROOFSTEP]\nrw [coprod.inl_desc, Category.id_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\n\u22a2 f \u226b pushout.inl = (coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) f) \u226b pushout.inl\n[PROOFSTEP]\nrw [coprod.inr_desc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\n\u22a2 (coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) f) \u226b pushout.inl = (coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) g) \u226b pushout.inr\n[PROOFSTEP]\nsimp only [Category.assoc, pushout.condition]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\n\u22a2 (coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) g) \u226b pushout.inr = g \u226b pushout.inr\n[PROOFSTEP]\nrw [coprod.inr_desc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\n\u22a2 f \u226b pushout.inl = g \u226b pushout.inl\n[PROOFSTEP]\nrw [hvu, huv]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\nhuu : f \u226b pushout.inl = g \u226b pushout.inl\ns : Cofork f g\n\u22a2 coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) f \u226b Cofork.\u03c0 s = coprod.inl \u226b coprod.desc (\ud835\udfd9 Y) g \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp only [coprod.inl_desc_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\nhuu : f \u226b pushout.inl = g \u226b pushout.inl\ns : Cofork f g\n\u22a2 coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) f \u226b Cofork.\u03c0 s = coprod.inr \u226b coprod.desc (\ud835\udfd9 Y) g \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp only [coprod.desc_comp, Cofork.condition s]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\nhuu : f \u226b pushout.inl = g \u226b pushout.inl\ns : Cofork f g\n\u22a2 Cofork.\u03c0 (Cofork.of\u03c0 pushout.inl huu) \u226b\n      (fun s =>\n          pushout.desc (Cofork.\u03c0 s) (Cofork.\u03c0 s)\n            (_ : coprod.desc (\ud835\udfd9 Y) f \u226b Cofork.\u03c0 s = coprod.desc (\ud835\udfd9 Y) g \u226b Cofork.\u03c0 s))\n        s =\n    Cofork.\u03c0 s\n[PROOFSTEP]\nsimp only [pushout.inl_desc, Cofork.\u03c0_of\u03c0]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\nhuu : f \u226b pushout.inl = g \u226b pushout.inl\ns : Cofork f g\nm : (Cofork.of\u03c0 pushout.inl huu).pt \u27f6 s.pt\nh : Cofork.\u03c0 (Cofork.of\u03c0 pushout.inl huu) \u226b m = Cofork.\u03c0 s\n\u22a2 pushout.inl \u226b m =\n    pushout.inl \u226b\n      (fun s =>\n          pushout.desc (Cofork.\u03c0 s) (Cofork.\u03c0 s)\n            (_ : coprod.desc (\ud835\udfd9 Y) f \u226b Cofork.\u03c0 s = coprod.desc (\ud835\udfd9 Y) g \u226b Cofork.\u03c0 s))\n        s\n[PROOFSTEP]\nsimpa only [pushout.inl_desc] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.141722, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf g : X \u27f6 Y\nhuv : pushout.inl = pushout.inr\nhvu : f \u226b pushout.inl = g \u226b pushout.inr\nhuu : f \u226b pushout.inl = g \u226b pushout.inl\ns : Cofork f g\nm : (Cofork.of\u03c0 pushout.inl huu).pt \u27f6 s.pt\nh : Cofork.\u03c0 (Cofork.of\u03c0 pushout.inl huu) \u226b m = Cofork.\u03c0 s\n\u22a2 pushout.inr \u226b m =\n    pushout.inr \u226b\n      (fun s =>\n          pushout.desc (Cofork.\u03c0 s) (Cofork.\u03c0 s)\n            (_ : coprod.desc (\ud835\udfd9 Y) f \u226b Cofork.\u03c0 s = coprod.desc (\ud835\udfd9 Y) g \u226b Cofork.\u03c0 s))\n        s\n[PROOFSTEP]\nsimpa only [huv.symm, pushout.inl_desc] using h\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{?u.173668, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 0 \u226b f = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8W, w, hw, hl\u27e9 := normalEpiOfEpi (coequalizer.\u03c0 u v)\n[GOAL]\ncase mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := coequalizer.desc' f huv\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\n\u22a2 u = v\n[PROOFSTEP]\nhave reassoced {W : C} (h : coequalizer u v \u27f6 W) : w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h := by\n  rw [\u2190 Category.assoc, eq_whisker hw]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW\u271d : C\nw : W\u271d \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nW : C\nh : coequalizer u v \u27f6 W\n\u22a2 w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\n[PROOFSTEP]\nrw [\u2190 Category.assoc, eq_whisker hw]\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\n\u22a2 u = v\n[PROOFSTEP]\nhave hwf : w \u226b f = 0 := by rw [\u2190 hm, reassoced, zero_comp]\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\n\u22a2 w \u226b f = 0\n[PROOFSTEP]\nrw [\u2190 hm, reassoced, zero_comp]\n[GOAL]\ncase mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\n\u22a2 u = v\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := KernelFork.IsLimit.lift' l _ hwf\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nhn : n \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)) = w\n\u22a2 u = v\n[PROOFSTEP]\nrw [Fork.\u03b9_of\u03b9, HasZeroMorphisms.comp_zero] at hn \n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nhn : 0 = w\n\u22a2 u = v\n[PROOFSTEP]\nhave : IsIso (coequalizer.\u03c0 u v) := by apply isIso_colimit_cocone_parallelPair_of_eq hn.symm hl\n[GOAL]\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nhn : 0 = w\n\u22a2 IsIso (coequalizer.\u03c0 u v)\n[PROOFSTEP]\napply isIso_colimit_cocone_parallelPair_of_eq hn.symm hl\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u = v\n[PROOFSTEP]\napply (cancel_mono (coequalizer.\u03c0 u v)).1\n[GOAL]\ncase mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0))\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 (_ : 0 \u226b f = 0)).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u \u226b coequalizer.\u03c0 u v = v \u226b coequalizer.\u03c0 u v\n[PROOFSTEP]\nexact coequalizer.condition _ _\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Equalizers", "llama_tokens": 31475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117769928211, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.5472194314126467}}
{"text": "[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\n\u22a2 expUnitary (a + b) = expUnitary a * expUnitary b\n[PROOFSTEP]\next\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\n\u22a2 \u2191(expUnitary (a + b)) = \u2191(expUnitary a * expUnitary b)\n[PROOFSTEP]\nhave hcomm : Commute (I \u2022 (a : A)) (I \u2022 (b : A)) :=\n  by\n  unfold Commute SemiconjBy\n  simp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\n\u22a2 Commute (I \u2022 \u2191a) (I \u2022 \u2191b)\n[PROOFSTEP]\nunfold Commute SemiconjBy\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\n\u22a2 I \u2022 \u2191a * I \u2022 \u2191b = I \u2022 \u2191b * I \u2022 \u2191a\n[PROOFSTEP]\nsimp only [h.eq, Algebra.smul_mul_assoc, Algebra.mul_smul_comm]\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\nhcomm : Commute (I \u2022 \u2191a) (I \u2022 \u2191b)\n\u22a2 \u2191(expUnitary (a + b)) = \u2191(expUnitary a * expUnitary b)\n[PROOFSTEP]\nsimpa only [expUnitary_coe, AddSubgroup.coe_add, smul_add] using exp_add_of_commute hcomm\n[GOAL]\nA : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : StarRing A\ninst\u271d\u00b2 : ContinuousStar A\ninst\u271d\u00b9 : CompleteSpace A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 selfAdjoint A }\nh : Commute \u2191a \u2191b\n\u22a2 selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a\n[PROOFSTEP]\nrw [\u2190 h.expUnitary_add, \u2190 h.symm.expUnitary_add, add_comm]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Star.Exponential", "llama_tokens": 1040, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891218080991, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5470933174825152}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => inner (\u2191f i) (\u2191g i)\n[PROOFSTEP]\nrefine' summable_of_norm_bounded (fun i => \u2016f i\u2016 * \u2016g i\u2016) (lp.summable_mul _ f g) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\n\u22a2 Real.IsConjugateExponent (ENNReal.toReal 2) (ENNReal.toReal 2)\n[PROOFSTEP]\nrw [Real.isConjugateExponent_iff]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\n\u22a2 ENNReal.toReal 2 = ENNReal.toReal 2 / (ENNReal.toReal 2 - 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\n\u22a2 1 < ENNReal.toReal 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\n\u22a2 \u2200 (i : \u03b9), \u2016inner (\u2191f i) (\u2191g i)\u2016 \u2264 (fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) i\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf g : { x // x \u2208 lp G 2 }\ni : \u03b9\n\u22a2 \u2016inner (\u2191f i) (\u2191g i)\u2016 \u2264 (fun i => \u2016\u2191f i\u2016 * \u2016\u2191g i\u2016) i\n[PROOFSTEP]\nexact norm_inner_le_norm (\ud835\udd5c := \ud835\udd5c) _ _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2016f\u2016 ^ 2 = \u2191re (inner f f)\n[PROOFSTEP]\ncalc\n  \u2016f\u2016 ^ 2 = \u2016f\u2016 ^ (2 : \u211d\u22650\u221e).toReal := by norm_cast\n  _ = \u2211' i, \u2016f i\u2016 ^ (2 : \u211d\u22650\u221e).toReal := (lp.norm_rpow_eq_tsum ?_ f)\n  _ = \u2211' i, \u2016f i\u2016 ^ (2 : \u2115) := by norm_cast\n  _ = \u2211' i, re \u27eaf i, f i\u27eb := by\n    congr\n    funext i\n    rw [norm_sq_eq_inner (\ud835\udd5c := \ud835\udd5c)]\n      -- porting note: `simp` couldn't do this anymore\n  _ = re (\u2211' i, \u27eaf i, f i\u27eb) := (IsROrC.reClm.map_tsum ?_).symm\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2016f\u2016 ^ 2 = \u2016f\u2016 ^ ENNReal.toReal 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), \u2016\u2191f i\u2016 ^ ENNReal.toReal 2 = \u2211' (i : \u03b9), \u2016\u2191f i\u2016 ^ 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), \u2016\u2191f i\u2016 ^ 2 = \u2211' (i : \u03b9), \u2191re (inner (\u2191f i) (\u2191f i))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 (fun i => \u2016\u2191f i\u2016 ^ 2) = fun i => \u2191re (inner (\u2191f i) (\u2191f i))\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase e_f.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\ni : \u03b9\n\u22a2 \u2016\u2191f i\u2016 ^ 2 = \u2191re (inner (\u2191f i) (\u2191f i))\n[PROOFSTEP]\nrw [norm_sq_eq_inner (\ud835\udd5c := \ud835\udd5c)]\n  -- porting note: `simp` couldn't do this anymore\n[GOAL]\ncase calc_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 0 < ENNReal.toReal 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase calc_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => inner (\u2191f i) (\u2191f i)\n[PROOFSTEP]\nexact summable_inner f f\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner g f) = inner f g\n[PROOFSTEP]\ncalc\n  conj _ = conj (\u2211' i, \u27eag i, f i\u27eb) := by congr\n  _ = \u2211' i, conj \u27eag i, f i\u27eb := IsROrC.conjCle.map_tsum\n  _ = \u2211' i, \u27eaf i, g i\u27eb := by simp only [inner_conj_symm]\n  _ = _ := by congr\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner g f) = \u2191(starRingEnd \ud835\udd5c) (\u2211' (i : \u03b9), inner (\u2191g i) (\u2191f i))\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), \u2191(starRingEnd \ud835\udd5c) (inner (\u2191g i) (\u2191f i)) = \u2211' (i : \u03b9), inner (\u2191f i) (\u2191g i)\n[PROOFSTEP]\nsimp only [inner_conj_symm]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), inner (\u2191f i) (\u2191g i) = inner f g\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 inner (f\u2081 + f\u2082) g = inner f\u2081 g + inner f\u2082 g\n[PROOFSTEP]\ncalc\n  _ = \u2211' i, \u27ea(f\u2081 + f\u2082) i, g i\u27eb := ?_\n  _ = \u2211' i, (\u27eaf\u2081 i, g i\u27eb + \u27eaf\u2082 i, g i\u27eb) := by simp only [inner_add_left, Pi.add_apply, coeFn_add]\n  _ = (\u2211' i, \u27eaf\u2081 i, g i\u27eb) + \u2211' i, \u27eaf\u2082 i, g i\u27eb := (tsum_add ?_ ?_)\n  _ = _ := by congr\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), inner (\u2191(f\u2081 + f\u2082) i) (\u2191g i) = \u2211' (i : \u03b9), (inner (\u2191f\u2081 i) (\u2191g i) + inner (\u2191f\u2082 i) (\u2191g i))\n[PROOFSTEP]\nsimp only [inner_add_left, Pi.add_apply, coeFn_add]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 \u2211' (i : \u03b9), inner (\u2191f\u2081 i) (\u2191g i) + \u2211' (i : \u03b9), inner (\u2191f\u2082 i) (\u2191g i) = inner f\u2081 g + inner f\u2082 g\n[PROOFSTEP]\ncongr\n[GOAL]\ncase calc_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 inner (f\u2081 + f\u2082) g = \u2211' (i : \u03b9), inner (\u2191(f\u2081 + f\u2082) i) (\u2191g i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase calc_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => inner (\u2191f\u2081 i) (\u2191g i)\n[PROOFSTEP]\nexact summable_inner f\u2081 g\n[GOAL]\ncase calc_3\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf\u2081 f\u2082 g : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => inner (\u2191f\u2082 i) (\u2191g i)\n[PROOFSTEP]\nexact summable_inner f\u2082 g\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\nc : \ud835\udd5c\n\u22a2 inner (c \u2022 f) g = \u2191(starRingEnd \ud835\udd5c) c * inner f g\n[PROOFSTEP]\ncalc\n  _ = \u2211' i, \u27eac \u2022 f i, g i\u27eb := ?_\n  _ = \u2211' i, conj c * \u27eaf i, g i\u27eb := by simp only [inner_smul_left]\n  _ = conj c * \u2211' i, \u27eaf i, g i\u27eb := tsum_mul_left\n  _ = _ := ?_\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\nc : \ud835\udd5c\n\u22a2 \u2211' (i : \u03b9), inner (c \u2022 \u2191f i) (\u2191g i) = \u2211' (i : \u03b9), \u2191(starRingEnd \ud835\udd5c) c * inner (\u2191f i) (\u2191g i)\n[PROOFSTEP]\nsimp only [inner_smul_left]\n[GOAL]\ncase calc_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\nc : \ud835\udd5c\n\u22a2 inner (c \u2022 f) g = \u2211' (i : \u03b9), inner (c \u2022 \u2191f i) (\u2191g i)\n[PROOFSTEP]\nsimp only [coeFn_smul, Pi.smul_apply]\n[GOAL]\ncase calc_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nsrc\u271d : NormedAddCommGroup { x // x \u2208 lp G 2 } := normedAddCommGroup\nf g : { x // x \u2208 lp G 2 }\nc : \ud835\udd5c\n\u22a2 \u2191(starRingEnd \ud835\udd5c) c * \u2211' (i : \u03b9), inner (\u2191f i) (\u2191g i) = \u2191(starRingEnd \ud835\udd5c) c * inner f g\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\n\u22a2 inner (lp.single 2 i a) f = inner a (\u2191f i)\n[PROOFSTEP]\nrefine' (hasSum_inner (lp.single 2 i a) f).unique _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\n\u22a2 HasSum (fun i_1 => inner (\u2191(lp.single 2 i a) i_1) (\u2191f i_1)) (inner a (\u2191f i))\n[PROOFSTEP]\nconvert hasSum_ite_eq i \u27eaa, f i\u27eb using 1\n[GOAL]\ncase h.e'_5\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\n\u22a2 (fun i_1 => inner (\u2191(lp.single 2 i a) i_1) (\u2191f i_1)) = fun b' => if b' = i then inner a (\u2191f i) else 0\n[PROOFSTEP]\next j\n[GOAL]\ncase h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\nj : \u03b9\n\u22a2 inner (\u2191(lp.single 2 i a) j) (\u2191f j) = if j = i then inner a (\u2191f i) else 0\n[PROOFSTEP]\nrw [lp.single_apply]\n[GOAL]\ncase h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\nj : \u03b9\n\u22a2 inner (if h : j = i then (_ : i = j) \u25b8 a else 0) (\u2191f j) = if j = i then inner a (\u2191f i) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\nj : \u03b9\nh : j = i\n\u22a2 inner ((_ : i = j) \u25b8 a) (\u2191f j) = inner a (\u2191f i)\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nf : { x // x \u2208 lp G 2 }\nj : \u03b9\na : G j\n\u22a2 inner ((_ : j = j) \u25b8 a) (\u2191f j) = inner a (\u2191f j)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\nj : \u03b9\nh : \u00acj = i\n\u22a2 inner 0 (\u2191f j) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ni : \u03b9\na : G i\nf : { x // x \u2208 lp G 2 }\n\u22a2 inner f (lp.single 2 i a) = inner (\u2191f i) a\n[PROOFSTEP]\nsimpa [inner_conj_symm] using congr_arg conj (@inner_single_left _ \ud835\udd5c _ _ _ _ i a f)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => \u2191(V i) (\u2191f i)\n[PROOFSTEP]\nrw [hV.summable_iff_norm_sq_summable]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 Summable fun i => \u2016\u2191f i\u2016 ^ 2\n[PROOFSTEP]\nconvert (lp.mem\u2113p f).summable _\n[GOAL]\ncase h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nx\u271d : \u03b9\n\u22a2 \u2016\u2191f x\u271d\u2016 ^ 2 = \u2016\u2191f x\u271d\u2016 ^ ENNReal.toReal 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 0 < ENNReal.toReal 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf g : { x // x \u2208 lp G 2 }\n\u22a2 (fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i)) (f + g) =\n    (fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i)) f + (fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i)) g\n[PROOFSTEP]\nsimp only [tsum_add (hV.summable_of_lp f) (hV.summable_of_lp g), lp.coeFn_add, Pi.add_apply, LinearIsometry.map_add]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nc : \ud835\udd5c\nf : { x // x \u2208 lp G 2 }\n\u22a2 AddHom.toFun\n      { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n        map_add' :=\n          (_ :\n            \u2200 (f g : { x // x \u2208 lp G 2 }),\n              \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n      (c \u2022 f) =\n    \u2191(RingHom.id \ud835\udd5c) c \u2022\n      AddHom.toFun\n        { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n          map_add' :=\n            (_ :\n              \u2200 (f g : { x // x \u2208 lp G 2 }),\n                \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n        f\n[PROOFSTEP]\nsimpa only [LinearIsometry.map_smul, Pi.smul_apply, lp.coeFn_smul] using tsum_const_smul c (hV.summable_of_lp f)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : { x // x \u2208 lp G 2 }),\n                      \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : \ud835\udd5c) (f : { x // x \u2208 lp G 2 }),\n                  AddHom.toFun\n                      { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : { x // x \u2208 lp G 2 }),\n                              \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id \ud835\udd5c) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : { x // x \u2208 lp G 2 }),\n                                \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                  \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                        f) }\n        f\u2016 =\n    \u2016f\u2016\n[PROOFSTEP]\nclassical\n  -- needed for lattice instance on `Finset \u03b9`, for `Filter.atTop_neBot`\nhave H : 0 < (2 : \u211d\u22650\u221e).toReal := by norm_num\nsuffices \u2016\u2211' i : \u03b9, V i (f i)\u2016 ^ (2 : \u211d\u22650\u221e).toReal = \u2016f\u2016 ^ (2 : \u211d\u22650\u221e).toReal by\n  exact Real.rpow_left_injOn H.ne' (norm_nonneg _) (norm_nonneg _) this\nrefine' tendsto_nhds_unique _ (lp.hasSum_norm H f)\nconvert (hV.summable_of_lp f).hasSum.norm.rpow_const (Or.inr H.le) using 1\next s\nexact_mod_cast (hV.norm_sum f s).symm\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : { x // x \u2208 lp G 2 }),\n                      \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : \ud835\udd5c) (f : { x // x \u2208 lp G 2 }),\n                  AddHom.toFun\n                      { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : { x // x \u2208 lp G 2 }),\n                              \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id \ud835\udd5c) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : { x // x \u2208 lp G 2 }),\n                                \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                  \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                        f) }\n        f\u2016 =\n    \u2016f\u2016\n[PROOFSTEP]\nhave H : 0 < (2 : \u211d\u22650\u221e).toReal := by norm_num\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\n\u22a2 0 < ENNReal.toReal 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : { x // x \u2208 lp G 2 }),\n                      \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : \ud835\udd5c) (f : { x // x \u2208 lp G 2 }),\n                  AddHom.toFun\n                      { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : { x // x \u2208 lp G 2 }),\n                              \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id \ud835\udd5c) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : { x // x \u2208 lp G 2 }),\n                                \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                  \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                        f) }\n        f\u2016 =\n    \u2016f\u2016\n[PROOFSTEP]\nsuffices \u2016\u2211' i : \u03b9, V i (f i)\u2016 ^ (2 : \u211d\u22650\u221e).toReal = \u2016f\u2016 ^ (2 : \u211d\u22650\u221e).toReal by\n  exact Real.rpow_left_injOn H.ne' (norm_nonneg _) (norm_nonneg _) this\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\nthis : \u2016\u2211' (i : \u03b9), \u2191(V i) (\u2191f i)\u2016 ^ ENNReal.toReal 2 = \u2016f\u2016 ^ ENNReal.toReal 2\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : { x // x \u2208 lp G 2 }),\n                      \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) = \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : \ud835\udd5c) (f : { x // x \u2208 lp G 2 }),\n                  AddHom.toFun\n                      { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : { x // x \u2208 lp G 2 }),\n                              \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id \ud835\udd5c) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => \u2211' (i : \u03b9), \u2191(V i) (\u2191f i),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : { x // x \u2208 lp G 2 }),\n                                \u2211' (x : \u03b9), \u2191(V x) (\u2191f x + \u2191g x) =\n                                  \u2211' (b : \u03b9), \u2191(V b) (\u2191f b) + \u2211' (b : \u03b9), \u2191(V b) (\u2191g b)) }\n                        f) }\n        f\u2016 =\n    \u2016f\u2016\n[PROOFSTEP]\nexact Real.rpow_left_injOn H.ne' (norm_nonneg _) (norm_nonneg _) this\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\n\u22a2 \u2016\u2211' (i : \u03b9), \u2191(V i) (\u2191f i)\u2016 ^ ENNReal.toReal 2 = \u2016f\u2016 ^ ENNReal.toReal 2\n[PROOFSTEP]\nrefine' tendsto_nhds_unique _ (lp.hasSum_norm H f)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\n\u22a2 Tendsto (fun s => \u2211 b in s, (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal 2) b) atTop\n    (\ud835\udcdd (\u2016\u2211' (i : \u03b9), \u2191(V i) (\u2191f i)\u2016 ^ ENNReal.toReal 2))\n[PROOFSTEP]\nconvert (hV.summable_of_lp f).hasSum.norm.rpow_const (Or.inr H.le) using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\n\u22a2 (fun s => \u2211 b in s, (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal 2) b) = fun a =>\n    \u2016\u2211 b in a, (fun i => \u2191(V i) (\u2191f i)) b\u2016 ^ ENNReal.toReal 2\n[PROOFSTEP]\next s\n[GOAL]\ncase h.e'_3.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : { x // x \u2208 lp G 2 }\nH : 0 < ENNReal.toReal 2\ns : Finset \u03b9\n\u22a2 \u2211 b in s, (fun i => \u2016\u2191f i\u2016 ^ ENNReal.toReal 2) b = \u2016\u2211 b in s, (fun i => \u2191(V i) (\u2191f i)) b\u2016 ^ ENNReal.toReal 2\n[PROOFSTEP]\nexact_mod_cast (hV.norm_sum f s).symm\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\n\u22a2 \u2191(OrthogonalFamily.linearIsometry hV) (lp.single 2 i x) = \u2191(V i) x\n[PROOFSTEP]\nrw [hV.linearIsometry_apply, \u2190 tsum_ite_eq i (V i x)]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\n\u22a2 \u2211' (i_1 : \u03b9), \u2191(V i_1) (\u2191(lp.single 2 i x) i_1) = \u2211' (b' : \u03b9), if b' = i then \u2191(V i) x else 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\n\u22a2 (fun i_1 => \u2191(V i_1) (\u2191(lp.single 2 i x) i_1)) = fun b' => if b' = i then \u2191(V i) x else 0\n[PROOFSTEP]\next j\n[GOAL]\ncase e_f.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\nj : \u03b9\n\u22a2 \u2191(V j) (\u2191(lp.single 2 i x) j) = if j = i then \u2191(V i) x else 0\n[PROOFSTEP]\nrw [lp.single_apply]\n[GOAL]\ncase e_f.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\nj : \u03b9\n\u22a2 \u2191(V j) (if h : j = i then (_ : i = j) \u25b8 x else 0) = if j = i then \u2191(V i) x else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\nj : \u03b9\nh : j = i\n\u22a2 \u2191(V j) (if h : j = i then (_ : i = j) \u25b8 x else 0) = \u2191(V i) x\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nj : \u03b9\nx : G j\n\u22a2 \u2191(V j) (if h : j = j then (_ : j = j) \u25b8 x else 0) = \u2191(V j) x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ni : \u03b9\nx : G i\nj : \u03b9\nh : \u00acj = i\n\u22a2 \u2191(V j) (if h : j = i then (_ : i = j) \u25b8 x else 0) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\n\u22a2 \u2191(OrthogonalFamily.linearIsometry hV) (DFinsupp.sum W\u2080 (lp.single 2)) = DFinsupp.sum W\u2080 fun i => \u2191(V i)\n[PROOFSTEP]\nhave :\n  hV.linearIsometry (\u2211 i in W\u2080.support, lp.single 2 i (W\u2080 i)) =\n    \u2211 i in W\u2080.support, hV.linearIsometry (lp.single 2 i (W\u2080 i)) :=\n  hV.linearIsometry.toLinearMap.map_sum\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\nthis :\n  \u2191(OrthogonalFamily.linearIsometry hV) (\u2211 i in DFinsupp.support W\u2080, lp.single 2 i (\u2191W\u2080 i)) =\n    \u2211 i in DFinsupp.support W\u2080, \u2191(OrthogonalFamily.linearIsometry hV) (lp.single 2 i (\u2191W\u2080 i))\n\u22a2 \u2191(OrthogonalFamily.linearIsometry hV) (DFinsupp.sum W\u2080 (lp.single 2)) = DFinsupp.sum W\u2080 fun i => \u2191(V i)\n[PROOFSTEP]\nsimp (config := { contextual := true }) [DFinsupp.sum, this]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap =\n    topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap \u2264\n    topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n[PROOFSTEP]\nrintro x \u27e8f, rfl\u27e9\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2191(OrthogonalFamily.linearIsometry hV).toLinearMap f \u2208\n    topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n[PROOFSTEP]\nrefine' mem_closure_of_tendsto (hV.hasSum_linearIsometry f) (eventually_of_forall _)\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\n\u22a2 \u2200 (x : Finset \u03b9), \u2211 b in x, (fun i => \u2191(V i) (\u2191f i)) b \u2208 \u2191(\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n[PROOFSTEP]\nintro s\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\ns : Finset \u03b9\n\u22a2 \u2211 b in s, (fun i => \u2191(V i) (\u2191f i)) b \u2208 \u2191(\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n[PROOFSTEP]\nrw [SetLike.mem_coe]\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\ns : Finset \u03b9\n\u22a2 \u2211 b in s, (fun i => \u2191(V i) (\u2191f i)) b \u2208 \u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap\n[PROOFSTEP]\nrefine' sum_mem _\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\ns : Finset \u03b9\n\u22a2 \u2200 (c : \u03b9), c \u2208 s \u2192 (fun i => \u2191(V i) (\u2191f i)) c \u2208 \u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\ns : Finset \u03b9\ni : \u03b9\na\u271d : i \u2208 s\n\u22a2 (fun i => \u2191(V i) (\u2191f i)) i \u2208 \u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap\n[PROOFSTEP]\nrefine' mem_iSup_of_mem i _\n[GOAL]\ncase refine'_1.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nf : { x // x \u2208 lp G 2 }\ns : Finset \u03b9\ni : \u03b9\na\u271d : i \u2208 s\n\u22a2 (fun i => \u2191(V i) (\u2191f i)) i \u2208 LinearMap.range (V i).toLinearMap\n[PROOFSTEP]\nexact LinearMap.mem_range_self _ (f i)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap) \u2264\n    LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap\n[PROOFSTEP]\napply topologicalClosure_minimal\n[GOAL]\ncase refine'_2.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 \u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap \u2264 LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap\n[PROOFSTEP]\nrefine' iSup_le _\n[GOAL]\ncase refine'_2.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 \u2200 (i : \u03b9), LinearMap.range (V i).toLinearMap \u2264 LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap\n[PROOFSTEP]\nrintro i x \u27e8x, rfl\u27e9\n[GOAL]\ncase refine'_2.h.intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\ni : \u03b9\nx : G i\n\u22a2 \u2191(V i).toLinearMap x \u2208 LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap\n[PROOFSTEP]\nuse lp.single 2 i x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\ni : \u03b9\nx : G i\n\u22a2 \u2191(OrthogonalFamily.linearIsometry hV).toLinearMap (lp.single 2 i x) = \u2191(V i).toLinearMap x\n[PROOFSTEP]\nexact hV.linearIsometry_apply_single x\n[GOAL]\ncase refine'_2.ht\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\n\u22a2 IsClosed \u2191(LinearMap.range (OrthogonalFamily.linearIsometry hV).toLinearMap)\n[PROOFSTEP]\nexact hV.linearIsometry.isometry.uniformInducing.isComplete_range.isClosed\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nhVortho : OrthogonalFamily \ud835\udd5c G V\nhVtotal : \u22a4 \u2264 topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n\u22a2 Function.Surjective \u2191(OrthogonalFamily.linearIsometry hVortho)\n[PROOFSTEP]\nrw [\u2190 LinearIsometry.coe_toLinearMap]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i)\nhVortho : OrthogonalFamily \ud835\udd5c G V\nhVtotal : \u22a4 \u2264 topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (V i).toLinearMap)\n\u22a2 Function.Surjective \u2191(OrthogonalFamily.linearIsometry hVortho).toLinearMap\n[PROOFSTEP]\nexact LinearMap.range_eq_top.mp (eq_top_iff.mpr <| hVtotal.trans_eq hVortho.range_linearIsometry.symm)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 F i }\nhFortho : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 F i }) fun i => subtype\u2097\u1d62 (F i)\nhFtotal : \u22a4 \u2264 topologicalClosure (\u2a06 (i : \u03b9), F i)\n\u22a2 \u22a4 \u2264 topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (subtype\u2097\u1d62 (F i)).toLinearMap)\n[PROOFSTEP]\nsimpa [subtype\u2097\u1d62_toLinearMap, range_subtype] using hFtotal\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nw : { x // x \u2208 lp G 2 }\n\u22a2 \u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) w = \u2211' (i : \u03b9), \u2191(V i) (\u2191w i)\n[PROOFSTEP]\nsimp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nw : { x // x \u2208 lp G 2 }\n\u22a2 HasSum (fun i => \u2191(V i) (\u2191w i)) (\u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) w)\n[PROOFSTEP]\nsimp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.hasSum_linearIsometry]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\ni : \u03b9\nx : G i\n\u22a2 \u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (lp.single 2 i x) = \u2191(V i) x\n[PROOFSTEP]\nsimp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply_single]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\n\u22a2 \u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (DFinsupp.sum W\u2080 (lp.single 2)) = DFinsupp.sum W\u2080 fun i => \u2191(V i)\n[PROOFSTEP]\nsimp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply_dfinsupp_sum_single]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\n\u22a2 \u2191(\u2191(linearIsometryEquiv hV) (DFinsupp.sum W\u2080 fun i => \u2191(V i))) = \u2191W\u2080\n[PROOFSTEP]\nrw [\u2190 hV.linearIsometryEquiv_symm_apply_dfinsupp_sum_single]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\n\u22a2 \u2191(\u2191(linearIsometryEquiv hV) (\u2191(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (DFinsupp.sum W\u2080 (lp.single 2)))) =\n    \u2191W\u2080\n[PROOFSTEP]\nrw [LinearIsometryEquiv.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\n\u22a2 \u2191(DFinsupp.sum W\u2080 (lp.single 2)) = \u2191W\u2080\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsHilbertSum \ud835\udd5c G V\nW\u2080 : \u03a0\u2080 (i : \u03b9), G i\ni : \u03b9\n\u22a2 \u2191(DFinsupp.sum W\u2080 (lp.single 2)) i = \u2191W\u2080 i\n[PROOFSTEP]\nsimp (config := { contextual := true }) [DFinsupp.sum, lp.single_apply]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range v))\n\u22a2 \u22a4 \u2264 topologicalClosure (\u2a06 (i : \u03b9), LinearMap.range (LinearIsometry.toSpanSingleton \ud835\udd5c E (_ : \u2016v i\u2016 = 1)).toLinearMap)\n[PROOFSTEP]\nconvert hsp\n[GOAL]\ncase h.e'_4.h.e'_9\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range v))\n\u22a2 \u2a06 (i : \u03b9), LinearMap.range (LinearIsometry.toSpanSingleton \ud835\udd5c E (_ : \u2016v i\u2016 = 1)).toLinearMap = span \ud835\udd5c (Set.range v)\n[PROOFSTEP]\nsimp [\u2190 LinearMap.span_singleton_eq_range, \u2190 Submodule.span_iUnion]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 IsHilbertSum \ud835\udd5c (fun b => { x // x \u2208 bif b then K else K\u15ee }) fun b => subtype\u2097\u1d62 (bif b then K else K\u15ee)\n[PROOFSTEP]\nhave : \u2200 b, CompleteSpace (\u21a5(cond b K K\u15ee)) := by\n  intro b\n  cases b <;>\n    first\n    | exact instOrthogonalCompleteSpace K\n    | assumption\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n[PROOFSTEP]\nintro b\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nb : Bool\n\u22a2 CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 CompleteSpace { x // x \u2208 bif false then K else K\u15ee }\n[PROOFSTEP]\nfirst\n| exact instOrthogonalCompleteSpace K\n| assumption\n[GOAL]\ncase false\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 CompleteSpace { x // x \u2208 bif false then K else K\u15ee }\n[PROOFSTEP]\nexact instOrthogonalCompleteSpace K\n[GOAL]\ncase true\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 CompleteSpace { x // x \u2208 bif true then K else K\u15ee }\n[PROOFSTEP]\nfirst\n| exact instOrthogonalCompleteSpace K\n| assumption\n[GOAL]\ncase true\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 CompleteSpace { x // x \u2208 bif true then K else K\u15ee }\n[PROOFSTEP]\nexact instOrthogonalCompleteSpace K\n[GOAL]\ncase true\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\n\u22a2 CompleteSpace { x // x \u2208 bif true then K else K\u15ee }\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nthis : \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n\u22a2 IsHilbertSum \ud835\udd5c (fun b => { x // x \u2208 bif b then K else K\u15ee }) fun b => subtype\u2097\u1d62 (bif b then K else K\u15ee)\n[PROOFSTEP]\nrefine' IsHilbertSum.mkInternal _ K.orthogonalFamily_self _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nthis : \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n\u22a2 \u22a4 \u2264 topologicalClosure (\u2a06 (i : Bool), bif i then K else K\u15ee)\n[PROOFSTEP]\nrefine' le_trans _ (Submodule.le_topologicalClosure _)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nthis : \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n\u22a2 \u22a4 \u2264 \u2a06 (i : Bool), bif i then K else K\u15ee\n[PROOFSTEP]\nrw [iSup_bool_eq, cond, cond]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nthis : \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n\u22a2 \u22a4 \u2264\n    (match true with\n      | true => K\n      | false => K\u15ee) \u2294\n      match false with\n      | true => K\n      | false => K\u15ee\n[PROOFSTEP]\nrefine' Codisjoint.top_le _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nF : \u03b9 \u2192 Submodule \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nhK : CompleteSpace { x // x \u2208 K }\nthis : \u2200 (b : Bool), CompleteSpace { x // x \u2208 bif b then K else K\u15ee }\n\u22a2 Codisjoint\n    (match true with\n    | true => K\n    | false => K\u15ee)\n    (match false with\n    | true => K\n    | false => K\u15ee)\n[PROOFSTEP]\nexact Submodule.isCompl_orthogonal_of_completeSpace.codisjoint\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 \u2191b.repr ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) = lp.single 2 i 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nv : E\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr v) i = inner ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) v\n[PROOFSTEP]\nrw [\u2190 b.repr.inner_map_map (b i) v, b.repr_self, lp.inner_single_left]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nv : E\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr v) i = inner 1 (\u2191(\u2191b.repr v) i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 Orthonormal \ud835\udd5c fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)\n[PROOFSTEP]\nrw [orthonormal_iff_ite]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2200 (i j : \u03b9),\n    inner (\u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n        (\u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n      if i = j then 1 else 0\n[PROOFSTEP]\nintro i j\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 inner (\u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) (\u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n    if i = j then 1 else 0\n[PROOFSTEP]\nrw [\u2190 b.repr.inner_map_map (b i) (b j), b.repr_self, b.repr_self, lp.inner_single_left, lp.single_apply]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 inner 1 (if h : i = j then (_ : j = i) \u25b8 1 else 0) = if i = j then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\n\u22a2 HasSum (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)\n    (\u2191(LinearIsometryEquiv.symm b.repr) f)\n[PROOFSTEP]\nsuffices H :\n  (fun i : \u03b9 => f i \u2022 b i) = fun b_1 : \u03b9 =>\n    b.repr.symm.toContinuousLinearEquiv <| (fun i : \u03b9 => lp.single 2 i (f i) (E := (fun _ : \u03b9 => \ud835\udd5c))) b_1\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\nH :\n  (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) = fun b_1 =>\n    \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n      ((fun i => lp.single 2 i (\u2191f i)) b_1)\n\u22a2 HasSum (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)\n    (\u2191(LinearIsometryEquiv.symm b.repr) f)\n[PROOFSTEP]\nrw [H]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\nH :\n  (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) = fun b_1 =>\n    \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n      ((fun i => lp.single 2 i (\u2191f i)) b_1)\n\u22a2 HasSum\n    (fun b_1 =>\n      \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) b_1))\n    (\u2191(LinearIsometryEquiv.symm b.repr) f)\n[PROOFSTEP]\nhave : HasSum (fun i : \u03b9 => lp.single 2 i (f i)) f := lp.hasSum_single ENNReal.two_ne_top f\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\nH :\n  (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) = fun b_1 =>\n    \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n      ((fun i => lp.single 2 i (\u2191f i)) b_1)\nthis : HasSum (fun i => lp.single 2 i (\u2191f i)) f\n\u22a2 HasSum\n    (fun b_1 =>\n      \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) b_1))\n    (\u2191(LinearIsometryEquiv.symm b.repr) f)\n[PROOFSTEP]\nexact (\u2191b.repr.symm.toContinuousLinearEquiv : \u2113\u00b2(\u03b9, \ud835\udd5c) \u2192L[\ud835\udd5c] E).hasSum this\n[GOAL]\ncase H\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\n\u22a2 (fun i => \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) = fun b_1 =>\n    \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n      ((fun i => lp.single 2 i (\u2191f i)) b_1)\n[PROOFSTEP]\next i\n[GOAL]\ncase H.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\n\u22a2 \u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i =\n    \u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr)) ((fun i => lp.single 2 i (\u2191f i)) i)\n[PROOFSTEP]\napply b.repr.injective\n[GOAL]\ncase H.h.a\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\n\u22a2 \u2191b.repr (\u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) =\n    \u2191b.repr\n      (\u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) i))\n[PROOFSTEP]\nletI : NormedSpace \ud835\udd5c (lp (fun _i : \u03b9 => \ud835\udd5c) 2) := by infer_instance\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\n\u22a2 NormedSpace \ud835\udd5c { x // x \u2208 lp (fun _i => \ud835\udd5c) 2 }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase H.h.a\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\nthis : NormedSpace \ud835\udd5c { x // x \u2208 lp (fun _i => \ud835\udd5c) 2 } := inferInstance\n\u22a2 \u2191b.repr (\u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) =\n    \u2191b.repr\n      (\u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) i))\n[PROOFSTEP]\nhave : lp.single (E := (fun _ : \u03b9 => \ud835\udd5c)) 2 i (f i * 1) = f i \u2022 lp.single 2 i 1 :=\n  lp.single_smul (E := (fun _ : \u03b9 => \ud835\udd5c)) 2 i (1 : \ud835\udd5c) (f i)\n[GOAL]\ncase H.h.a\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\nthis\u271d : NormedSpace \ud835\udd5c { x // x \u2208 lp (fun _i => \ud835\udd5c) 2 } := inferInstance\nthis : lp.single 2 i (\u2191f i * 1) = \u2191f i \u2022 lp.single 2 i 1\n\u22a2 \u2191b.repr (\u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) =\n    \u2191b.repr\n      (\u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) i))\n[PROOFSTEP]\nrw [mul_one] at this \n[GOAL]\ncase H.h.a\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\nthis\u271d : NormedSpace \ud835\udd5c { x // x \u2208 lp (fun _i => \ud835\udd5c) 2 } := inferInstance\nthis : lp.single 2 i (\u2191f i) = \u2191f i \u2022 lp.single 2 i 1\n\u22a2 \u2191b.repr (\u2191f i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) =\n    \u2191b.repr\n      (\u2191(LinearIsometryEquiv.toContinuousLinearEquiv (LinearIsometryEquiv.symm b.repr))\n        ((fun i => lp.single 2 i (\u2191f i)) i))\n[PROOFSTEP]\nrw [LinearIsometryEquiv.map_smul, b.repr_self, \u2190 this, LinearIsometryEquiv.coe_toContinuousLinearEquiv]\n[GOAL]\ncase H.h.a\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nf : { x // x \u2208 lp (fun i => \ud835\udd5c) 2 }\ni : \u03b9\nthis\u271d : NormedSpace \ud835\udd5c { x // x \u2208 lp (fun _i => \ud835\udd5c) 2 } := inferInstance\nthis : lp.single 2 i (\u2191f i) = \u2191f i \u2022 lp.single 2 i 1\n\u22a2 lp.single 2 i (\u2191f i) = \u2191b.repr (\u2191(LinearIsometryEquiv.symm b.repr) ((fun i => lp.single 2 i (\u2191f i)) i))\n[PROOFSTEP]\nexact (b.repr.apply_symm_apply (lp.single 2 i (f i))).symm\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\n\u22a2 HasSum (fun i => \u2191(\u2191b.repr x) i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) x\n[PROOFSTEP]\nsimpa using b.hasSum_repr_symm (b.repr x)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))) = \u22a4\n[PROOFSTEP]\nclassical\nrw [eq_top_iff]\nrintro x -\nrefine' mem_closure_of_tendsto (b.hasSum_repr x) (eventually_of_forall _)\nintro s\nsimp only [SetLike.mem_coe]\nrefine' sum_mem _\nrintro i -\nrefine' smul_mem _ _ _\nexact subset_span \u27e8i, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)))\n[PROOFSTEP]\nrintro x -\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\n\u22a2 x \u2208 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)))\n[PROOFSTEP]\nrefine' mem_closure_of_tendsto (b.hasSum_repr x) (eventually_of_forall _)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\n\u22a2 \u2200 (x_1 : Finset \u03b9),\n    \u2211 b_1 in x_1, (fun i => \u2191(\u2191b.repr x) i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) b_1 \u2208\n      \u2191(span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)))\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\ns : Finset \u03b9\n\u22a2 \u2211 b_1 in s, (fun i => \u2191(\u2191b.repr x) i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) b_1 \u2208\n    \u2191(span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)))\n[PROOFSTEP]\nsimp only [SetLike.mem_coe]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\ns : Finset \u03b9\n\u22a2 \u2211 x_1 in s, \u2191(\u2191b.repr x) x_1 \u2022 \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 x_1 1) \u2208\n    span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n[PROOFSTEP]\nrefine' sum_mem _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\ns : Finset \u03b9\n\u22a2 \u2200 (c : \u03b9),\n    c \u2208 s \u2192\n      \u2191(\u2191b.repr x) c \u2022 \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 c 1) \u2208\n        span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n[PROOFSTEP]\nrintro i -\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\ns : Finset \u03b9\ni : \u03b9\n\u22a2 \u2191(\u2191b.repr x) i \u2022 \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1) \u2208\n    span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n[PROOFSTEP]\nrefine' smul_mem _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx : E\ns : Finset \u03b9\ni : \u03b9\n\u22a2 \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1) \u2208\n    span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n[PROOFSTEP]\nexact subset_span \u27e8i, rfl\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx y : E\n\u22a2 HasSum\n    (fun i =>\n      inner x ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) *\n        inner ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) y)\n    (inner x y)\n[PROOFSTEP]\nconvert (b.hasSum_repr y).mapL (innerSL _ x) using 1\n[GOAL]\ncase h.e'_5\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx y : E\n\u22a2 (fun i =>\n      inner x ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) *\n        inner ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) y) =\n    fun b_1 =>\n    \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191(\u2191b.repr y) b_1 \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) b_1)\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\nx y : E\ni : \u03b9\n\u22a2 inner x ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) *\n      inner ((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i) y =\n    \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191(\u2191b.repr y) i \u2022 (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)\n[PROOFSTEP]\nrw [innerSL_apply, b.repr_apply_apply, inner_smul_right, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ninst\u271d : Fintype \u03b9\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 \u22a4 \u2264 span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n[PROOFSTEP]\nrefine' Eq.ge _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ninst\u271d : Fintype \u03b9\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) = \u22a4\n[PROOFSTEP]\nhave := (span \ud835\udd5c (Finset.univ.image b : Set E)).closed_of_finiteDimensional\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\ninst\u271d : Fintype \u03b9\nb : HilbertBasis \u03b9 \ud835\udd5c E\nthis : IsClosed \u2191(span \ud835\udd5c \u2191(Finset.image (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) Finset.univ))\n\u22a2 span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) = \u22a4\n[PROOFSTEP]\nsimpa only [Finset.coe_image, Finset.coe_univ, Set.image_univ, HilbertBasis.dense_span] using\n  this.submodule_topologicalClosure_eq.symm\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\nb : HilbertBasis \u03b9 \ud835\udd5c { x // x \u2208 U }\nx : E\n\u22a2 HasSum\n    (fun i =>\n      inner (\u2191((fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)) x \u2022\n        (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)\n    (\u2191(orthogonalProjection U) x)\n[PROOFSTEP]\nsimpa only [b.repr_apply_apply, inner_orthogonalProjection_eq_of_mem_left] using\n  b.hasSum_repr (orthogonalProjection U x)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))) \u2264\n    topologicalClosure\n      (\u2a06 (J : Finset \u03b9), span \ud835\udd5c \u2191(Finset.image (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) J))\n[PROOFSTEP]\nsimp_rw [\u2190 Submodule.span_iUnion]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nb : HilbertBasis \u03b9 \ud835\udd5c E\n\u22a2 topologicalClosure (span \ud835\udd5c (Set.range fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))) \u2264\n    topologicalClosure\n      (span \ud835\udd5c (\u22c3 (i : Finset \u03b9), \u2191(Finset.image (fun i => \u2191(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) i)))\n[PROOFSTEP]\nexact\n  topologicalClosure_mono\n    (span_mono <|\n      Set.range_subset_iff.mpr fun i =>\n        Set.mem_iUnion_of_mem { i } <| Finset.mem_coe.mpr <| Finset.mem_image_of_mem _ <| Finset.mem_singleton_self i)\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nh : \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range v))\ni : \u03b9\n\u22a2 \u2191(LinearIsometryEquiv.symm\n          (IsHilbertSum.linearIsometryEquiv\n            (_ : IsHilbertSum \ud835\udd5c (fun x => \ud835\udd5c) fun i => LinearIsometry.toSpanSingleton \ud835\udd5c E (_ : \u2016v i\u2016 = 1))))\n      (lp.single 2 i 1) =\n    v i\n[PROOFSTEP]\nrw [IsHilbertSum.linearIsometryEquiv_symm_apply_single, LinearIsometry.toSpanSingleton_apply, one_smul]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range v))\n\u22a2 (fun i => \u2191(LinearIsometryEquiv.symm (HilbertBasis.mk hv hsp).repr) (lp.single 2 i 1)) = v\n[PROOFSTEP]\napply funext <| Orthonormal.linearIsometryEquiv_symm_apply_single_one hv hsp\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhsp : (span \ud835\udd5c (Set.range v))\u15ee = \u22a5\n\u22a2 \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range v))\n[PROOFSTEP]\nrw [\u2190 orthogonal_orthogonal_eq_closure, \u2190 eq_top_iff, orthogonal_eq_top_iff, hsp]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u22a4 \u2264 topologicalClosure (span \ud835\udd5c (Set.range \u2191b))\n[PROOFSTEP]\nsimpa only [\u2190 OrthonormalBasis.coe_toBasis, b.toBasis.span_eq, eq_top_iff] using @subset_closure E _ _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nw : Set E\nhws : w \u2287 s\nhw_ortho : Orthonormal \ud835\udd5c Subtype.val\nhw_max : \u2200 (u : Set E), u \u2287 w \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = w\n\u22a2 (span \ud835\udd5c (Set.range Subtype.val))\u15ee = \u22a5\n[PROOFSTEP]\nsimpa [maximal_orthonormal_iff_orthogonalComplement_eq_bot hw_ortho] using hw_max\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.l2Space", "llama_tokens": 40302, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891218080991, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5470933174825152}}
{"text": "[GOAL]\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u22a2 \u2200 {\u03b1 : TypeVec n} (x : Sigma F \u03b1), (fun {\u03b1} => Sigma.abs F) ((fun {\u03b1} => Sigma.repr F) x) = x\n[PROOFSTEP]\nrintro \u03b1 \u27e8x, f\u27e9\n[GOAL]\ncase mk\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 : TypeVec n\nx : A\nf : F x \u03b1\n\u22a2 (fun {\u03b1} => Sigma.abs F) ((fun {\u03b1} => Sigma.repr F) { fst := x, snd := f }) = { fst := x, snd := f }\n[PROOFSTEP]\nsimp only [Sigma.abs, Sigma.repr, Sigma.eta, abs_repr]\n[GOAL]\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u22a2 \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (p : MvPFunctor.Obj (Sigma.P F) \u03b1),\n    (fun {\u03b1} => Sigma.abs F) (f <$$> p) = f <$$> (fun {\u03b1} => Sigma.abs F) p\n[PROOFSTEP]\nrintro \u03b1 \u03b2 f \u27e8x, g\u27e9\n[GOAL]\ncase mk\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nx : (Sigma.P F).A\ng : MvPFunctor.B (Sigma.P F) x \u27f9 \u03b1\n\u22a2 (fun {\u03b1} => Sigma.abs F) (f <$$> { fst := x, snd := g }) = f <$$> (fun {\u03b1} => Sigma.abs F) { fst := x, snd := g }\n[PROOFSTEP]\nsimp only [Sigma.abs, MvPFunctor.map_eq]\n[GOAL]\ncase mk\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nx : (Sigma.P F).A\ng : MvPFunctor.B (Sigma.P F) x \u27f9 \u03b1\n\u22a2 { fst := x.fst, snd := abs { fst := x.snd, snd := f \u229a g } } =\n    f <$$> { fst := x.fst, snd := abs { fst := x.snd, snd := g } }\n[PROOFSTEP]\nsimp only [(\u00b7 <$$> \u00b7), \u2190 abs_map, \u2190 MvPFunctor.map_eq]\n[GOAL]\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u22a2 \u2200 {\u03b1 : TypeVec n} (x : Pi F \u03b1), Pi.abs F (Pi.repr F x) = x\n[PROOFSTEP]\nrintro \u03b1 f\n[GOAL]\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 : TypeVec n\nf : Pi F \u03b1\n\u22a2 Pi.abs F (Pi.repr F f) = f\n[PROOFSTEP]\nsimp only [Pi.abs, Pi.repr, Sigma.eta, abs_repr]\n[GOAL]\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u22a2 \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (p : MvPFunctor.Obj (Pi.P F) \u03b1), Pi.abs F (f <$$> p) = f <$$> Pi.abs F p\n[PROOFSTEP]\nrintro \u03b1 \u03b2 f \u27e8x, g\u27e9\n[GOAL]\ncase mk\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nx : (Pi.P F).A\ng : MvPFunctor.B (Pi.P F) x \u27f9 \u03b1\n\u22a2 Pi.abs F (f <$$> { fst := x, snd := g }) = f <$$> Pi.abs F { fst := x, snd := g }\n[PROOFSTEP]\nsimp only [Pi.abs, (\u00b7 <$$> \u00b7), \u2190 abs_map]\n[GOAL]\ncase mk\nn : \u2115\nA : Type u\nF : A \u2192 TypeVec n \u2192 Type u\ninst\u271d\u00b9 : (\u03b1 : A) \u2192 MvFunctor (F \u03b1)\ninst\u271d : (\u03b1 : A) \u2192 MvQPF (F \u03b1)\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nx : (Pi.P F).A\ng : MvPFunctor.B (Pi.P F) x \u27f9 \u03b1\n\u22a2 (match MvPFunctor.map (Pi.P F) f { fst := x, snd := g } with\n    | { fst := a, snd := f } => fun x => abs { fst := a x, snd := fun i y => f i { fst := x, snd := y } }) =\n    fun a => abs (MvPFunctor.map (P (F a)) f { fst := x a, snd := fun i y => g i { fst := a, snd := y } })\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.QPF.Multivariate.Constructions.Sigma", "llama_tokens": 1728, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8539127417985636, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.547047119300804}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\n\u22a2 \u2200 (s : Set (Tropical R)), \u00acBddAbove s \u2192 sSup s = sSup Set.univ\n[PROOFSTEP]\nintro s hs\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddAbove s\n\u22a2 sSup s = sSup Set.univ\n[PROOFSTEP]\nhave : Set.range untrop = (Set.univ : Set R) := Equiv.range_eq_univ tropEquiv.symm\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddAbove s\nthis : Set.range untrop = Set.univ\n\u22a2 sSup s = sSup Set.univ\n[PROOFSTEP]\nsimp [sSup, this]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddAbove s\nthis : Set.range untrop = Set.univ\n\u22a2 sSup (untrop '' s) = sSup Set.univ\n[PROOFSTEP]\napply csSup_of_not_bddAbove\n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddAbove s\nthis : Set.range untrop = Set.univ\n\u22a2 \u00acBddAbove (untrop '' s)\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddAbove (untrop '' s)\n\u22a2 BddAbove s\n[PROOFSTEP]\nchange BddAbove (tropOrderIso.symm '' s) at hs \n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddAbove (\u2191(OrderIso.symm tropOrderIso) '' s)\n\u22a2 BddAbove s\n[PROOFSTEP]\nexact tropOrderIso.symm.bddAbove_image.1 hs\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\n\u22a2 \u2200 (s : Set (Tropical R)), \u00acBddBelow s \u2192 sInf s = sInf Set.univ\n[PROOFSTEP]\nintro s hs\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddBelow s\n\u22a2 sInf s = sInf Set.univ\n[PROOFSTEP]\nhave : Set.range untrop = (Set.univ : Set R) := Equiv.range_eq_univ tropEquiv.symm\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddBelow s\nthis : Set.range untrop = Set.univ\n\u22a2 sInf s = sInf Set.univ\n[PROOFSTEP]\nsimp [sInf, this]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddBelow s\nthis : Set.range untrop = Set.univ\n\u22a2 sInf (untrop '' s) = sInf Set.univ\n[PROOFSTEP]\napply csInf_of_not_bddBelow\n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nhs : \u00acBddBelow s\nthis : Set.range untrop = Set.univ\n\u22a2 \u00acBddBelow (untrop '' s)\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddBelow (untrop '' s)\n\u22a2 BddBelow s\n[PROOFSTEP]\nchange BddBelow (tropOrderIso.symm '' s) at hs \n[GOAL]\ncase hs\nR : Type u_1\nS : Type u_2\ninst\u271d : ConditionallyCompleteLinearOrder R\nsrc\u271d\u00b9 : ConditionallyCompleteLattice (Tropical R) := instConditionallyCompleteLatticeTropical\nsrc\u271d : LinearOrder (Tropical R) := instLinearOrderTropical\ns : Set (Tropical R)\nthis : Set.range untrop = Set.univ\nhs : BddBelow (\u2191(OrderIso.symm tropOrderIso) '' s)\n\u22a2 BddBelow s\n[PROOFSTEP]\nexact tropOrderIso.symm.bddBelow_image.1 hs\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Tropical.Lattice", "llama_tokens": 2090, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.721743206297598, "lm_q1q2_score": 0.5469329392380315}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : IsTrans \u03b1 r\nf : Fin (n + 1) \u2192 \u03b1\na : \u03b1\n\u22a2 ((fun x x_1 => x < x_1) \u21d2 r) (vecCons a f) (vecCons a f) \u2194 r a (f 0) \u2227 ((fun x x_1 => x < x_1) \u21d2 r) f f\n[PROOFSTEP]\nsimp only [liftFun_iff_succ r, forall_fin_succ, cons_val_succ, cons_val_zero, \u2190 succ_castSucc, castSucc_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\na : \u03b1\n\u22a2 Monotone (vecCons a f) \u2194 a \u2264 f 0 \u2227 Monotone f\n[PROOFSTEP]\nsimpa only [monotone_iff_forall_lt] using @liftFun_vecCons \u03b1 n (\u00b7 \u2264 \u00b7) _ f a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\na : \u03b1\n\u22a2 Monotone ![1, 2, 2, 3]\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Fin.Tuple.Monotone", "llama_tokens": 344, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744761936435, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5468594231126113}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nxs : List (FreeMonoid \u03b1)\n\u22a2 \u2191toList (List.prod xs) = List.join (List.map (\u2191toList) xs)\n[PROOFSTEP]\ninduction xs\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\n\u22a2 \u2191toList (List.prod []) = List.join (List.map \u2191toList [])\n[PROOFSTEP]\nsimp [*, List.join]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nhead\u271d : FreeMonoid \u03b1\ntail\u271d : List (FreeMonoid \u03b1)\ntail_ih\u271d : \u2191toList (List.prod tail\u271d) = List.join (List.map (\u2191toList) tail\u271d)\n\u22a2 \u2191toList (List.prod (head\u271d :: tail\u271d)) = List.join (List.map (\u2191toList) (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [*, List.join]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nxs : List (List \u03b1)\n\u22a2 \u2191toList (\u2191ofList (List.join xs)) = \u2191toList (List.prod (List.map (\u2191ofList) xs))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nf g : FreeMonoid \u03b1 \u2192* M\nh : \u2200 (x : \u03b1), \u2191f (of x) = \u2191g (of x)\nl : FreeMonoid \u03b1\nx : \u03b1\nxs : FreeMonoid \u03b1\nhxs : \u2191f xs = \u2191g xs\n\u22a2 \u2191f (of x * xs) = \u2191g (of x * xs)\n[PROOFSTEP]\nsimp only [h, hxs, MonoidHom.map_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\nf : \u03b1 \u2192 M\nx\u271d\u00b9 x\u271d : FreeMonoid \u03b1\n\u22a2 OneHom.toFun\n      { toFun := fun l => prodAux (List.map f (\u2191toList l)),\n        map_one' :=\n          (_ : (fun l => prodAux (List.map f (\u2191toList l))) 1 = (fun l => prodAux (List.map f (\u2191toList l))) 1) }\n      (x\u271d\u00b9 * x\u271d) =\n    OneHom.toFun\n        { toFun := fun l => prodAux (List.map f (\u2191toList l)),\n          map_one' :=\n            (_ : (fun l => prodAux (List.map f (\u2191toList l))) 1 = (fun l => prodAux (List.map f (\u2191toList l))) 1) }\n        x\u271d\u00b9 *\n      OneHom.toFun\n        { toFun := fun l => prodAux (List.map f (\u2191toList l)),\n          map_one' :=\n            (_ : (fun l => prodAux (List.map f (\u2191toList l))) 1 = (fun l => prodAux (List.map f (\u2191toList l))) 1) }\n        x\u271d\n[PROOFSTEP]\nsimp only [prodAux_eq, toList_mul, List.map_append, List.prod_append]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\ng : M \u2192* N\nf : \u03b1 \u2192 M\n\u22a2 MonoidHom.comp g (\u2191lift f) = \u2191lift (\u2191g \u2218 f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : Monoid M\nN : Type u_5\ninst\u271d : Monoid N\ng : M \u2192* N\nf : \u03b1 \u2192 M\nx\u271d : \u03b1\n\u22a2 \u2191(MonoidHom.comp g (\u2191lift f)) (of x\u271d) = \u2191(\u2191lift (\u2191g \u2218 f)) (of x\u271d)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Algebra.FreeMonoid.Basic", "llama_tokens": 1323, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.793105951184112, "lm_q2_score": 0.6893056231680121, "lm_q1q2_score": 0.5466923919192234}}
{"text": "[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\na : \u2124\nb : \u2115\nhb : b \u2260 0\n\u22a2 toNat a < b \u2194 a < \u2191b\n[PROOFSTEP]\nrw [\u2190 toNat_lt_toNat, toNat_coe_nat]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\na : \u2124\nb : \u2115\nhb : b \u2260 0\n\u22a2 0 < \u2191b\n[PROOFSTEP]\nexact coe_nat_pos.2 hb.bot_lt\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : NonAssocRing \u03b1\nm : \u2124\n\u22a2 \u2200 (n : \u2124), \u2191(0 * n) = \u21910 * \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : NonAssocRing \u03b1\nm k : \u2124\nx\u271d : 0 \u2264 k\nih : \u2200 (n : \u2124), \u2191(k * n) = \u2191k * \u2191n\nn : \u2124\n\u22a2 \u2191((k + 1) * n) = \u2191(k + 1) * \u2191n\n[PROOFSTEP]\nsimp [add_mul, ih]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : NonAssocRing \u03b1\nm k : \u2124\nx\u271d : k \u2264 0\nih : \u2200 (n : \u2124), \u2191(k * n) = \u2191k * \u2191n\nn : \u2124\n\u22a2 \u2191((k - 1) * n) = \u2191(k - 1) * \u2191n\n[PROOFSTEP]\nsimp [sub_mul, ih]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : NonAssocRing \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 Commute (\u2191\u2191n) x\n[PROOFSTEP]\nsimpa using n.cast_commute x\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : NonAssocRing \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 Commute (\u2191-[n+1]) x\n[PROOFSTEP]\nsimpa only [cast_negSucc, Commute.neg_left_iff, Commute.neg_right_iff] using (n + 1).cast_commute (-x)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : OrderedRing \u03b1\n\u22a2 Monotone fun x => \u2191x\n[PROOFSTEP]\nintro m n h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : OrderedRing \u03b1\nm n : \u2124\nh : m \u2264 n\n\u22a2 (fun x => \u2191x) m \u2264 (fun x => \u2191x) n\n[PROOFSTEP]\nrw [\u2190 sub_nonneg] at h \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : OrderedRing \u03b1\nm n : \u2124\nh\u271d : m \u2264 n\nh : 0 \u2264 n - m\n\u22a2 (fun x => \u2191x) m \u2264 (fun x => \u2191x) n\n[PROOFSTEP]\nlift n - m to \u2115 using h with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : OrderedRing \u03b1\nm n : \u2124\nh : m \u2264 n\nk : \u2115\nhk : \u2191k = n - m\n\u22a2 \u2191m \u2264 \u2191n\n[PROOFSTEP]\nrw [\u2190 sub_nonneg, \u2190 cast_sub, \u2190 hk, cast_ofNat]\n[GOAL]\ncase intro\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : OrderedRing \u03b1\nm n : \u2124\nh : m \u2264 n\nk : \u2115\nhk : \u2191k = n - m\n\u22a2 0 \u2264 \u2191k\n[PROOFSTEP]\nexact k.cast_nonneg\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2115\n\u22a2 0 \u2264 \u2191\u2191n \u2194 0 \u2264 \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2115\n\u22a2 0 \u2264 \u2191-[n+1] \u2194 0 \u2264 -[n+1]\n[PROOFSTEP]\nhave : -(n : \u03b1) < 1 := lt_of_le_of_lt (by simp) zero_lt_one\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2115\n\u22a2 -\u2191n \u2264 0\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2115\nthis : -\u2191n < 1\n\u22a2 0 \u2264 \u2191-[n+1] \u2194 0 \u2264 -[n+1]\n[PROOFSTEP]\nsimpa [(negSucc_lt_zero n).not_le, \u2190 sub_eq_add_neg, le_neg] using this.not_le\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nm n : \u2124\n\u22a2 \u2191m \u2264 \u2191n \u2194 m \u2264 n\n[PROOFSTEP]\nrw [\u2190 sub_nonneg, \u2190 cast_sub, cast_nonneg, sub_nonneg]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2124\n\u22a2 \u2191n \u2264 0 \u2194 n \u2264 0\n[PROOFSTEP]\nrw [\u2190 cast_zero, cast_le]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2124\n\u22a2 0 < \u2191n \u2194 0 < n\n[PROOFSTEP]\nrw [\u2190 cast_zero, cast_lt]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : OrderedRing \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u2124\n\u22a2 \u2191n < 0 \u2194 n < 0\n[PROOFSTEP]\nrw [\u2190 cast_zero, cast_lt]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\n\u22a2 \u2191|a| = |\u2191a|\n[PROOFSTEP]\nsimp [abs_eq_max_neg]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nh : 0 < a\n\u22a2 1 \u2264 \u2191a\n[PROOFSTEP]\nexact_mod_cast Int.add_one_le_of_lt h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nh : a < 0\n\u22a2 \u2191a \u2264 -1\n[PROOFSTEP]\nrw [\u2190 Int.cast_one, \u2190 Int.cast_neg, cast_le]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nh : a < 0\n\u22a2 a \u2264 -1\n[PROOFSTEP]\nexact Int.le_sub_one_of_lt h\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n\n[PROOFSTEP]\nhave hnx : 0 < n \u2192 0 \u2264 x + n := fun hn =>\n  by\n  have := _root_.add_le_add (neg_le_of_abs_le hx) (cast_one_le_of_pos hn)\n  rwa [add_left_neg] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhn : 0 < n\n\u22a2 0 \u2264 x + \u2191n\n[PROOFSTEP]\nhave := _root_.add_le_add (neg_le_of_abs_le hx) (cast_one_le_of_pos hn)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhn : 0 < n\nthis : -1 + 1 \u2264 x + \u2191n\n\u22a2 0 \u2264 x + \u2191n\n[PROOFSTEP]\nrwa [add_left_neg] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n\n[PROOFSTEP]\nhave hnx' : n < 0 \u2192 x + n \u2264 0 := fun hn =>\n  by\n  have := _root_.add_le_add (le_of_abs_le hx) (cast_le_neg_one_of_neg hn)\n  rwa [add_right_neg] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhn : n < 0\n\u22a2 x + \u2191n \u2264 0\n[PROOFSTEP]\nhave := _root_.add_le_add (le_of_abs_le hx) (cast_le_neg_one_of_neg hn)\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhn : n < 0\nthis : x + \u2191n \u2264 1 + -1\n\u22a2 x + \u2191n \u2264 0\n[PROOFSTEP]\nrwa [add_right_neg] at this \n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n * x + \u2191n * \u2191n\n[PROOFSTEP]\nrw [\u2190 mul_add, mul_nonneg_iff]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0\n[PROOFSTEP]\nrcases lt_trichotomy n 0 with (h | rfl | h)\n[GOAL]\ncase inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : n < 0\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0\n[PROOFSTEP]\nexact Or.inr \u27e8by exact_mod_cast h.le, hnx' h\u27e9\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : n < 0\n\u22a2 \u2191n \u2264 0\n[PROOFSTEP]\nexact_mod_cast h.le\n[GOAL]\ncase inr.inl\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < 0 \u2192 0 \u2264 x + \u21910\nhnx' : 0 < 0 \u2192 x + \u21910 \u2264 0\n\u22a2 0 \u2264 \u21910 \u2227 0 \u2264 x + \u21910 \u2228 \u21910 \u2264 0 \u2227 x + \u21910 \u2264 0\n[PROOFSTEP]\nsimp [le_total 0 x]\n[GOAL]\ncase inr.inr\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : 0 < n\n\u22a2 0 \u2264 \u2191n \u2227 0 \u2264 x + \u2191n \u2228 \u2191n \u2264 0 \u2227 x + \u2191n \u2264 0\n[PROOFSTEP]\nexact Or.inl \u27e8by exact_mod_cast h.le, hnx h\u27e9\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\nx : \u03b1\nhx : |x| \u2264 1\nhnx : 0 < n \u2192 0 \u2264 x + \u2191n\nhnx' : n < 0 \u2192 x + \u2191n \u2264 0\nh : 0 < n\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast h.le\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b n : \u2124\n\u22a2 \u2191(natAbs n) = \u2191|n|\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b : \u2124\na\u271d : \u2115\n\u22a2 \u2191(natAbs (ofNat a\u271d)) = \u2191|ofNat a\u271d|\n[PROOFSTEP]\nsimp\n[GOAL]\ncase negSucc\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d : LinearOrderedRing \u03b1\na b : \u2124\na\u271d : \u2115\n\u22a2 \u2191(natAbs -[a\u271d+1]) = \u2191|-[a\u271d+1]|\n[PROOFSTEP]\nrw [abs_eq_natAbs, natAbs_negSucc, cast_succ, cast_ofNat, cast_succ]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nA : Type u_5\ninst\u271d : AddGroupWithOne A\nf : \u2124 \u2192+ A\nh1 : \u2191f 1 = 1\n\u22a2 \u2191f 1 = \u2191(castAddHom A) 1\n[PROOFSTEP]\nsimp [h1]\n[GOAL]\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM : Type u_5\ninst\u271d : Monoid M\nf g : \u2124 \u2192* M\nh_neg_one : \u2191f (-1) = \u2191g (-1)\nh_nat : comp f \u2191ofNatHom = comp g \u2191ofNatHom\n\u22a2 f = g\n[PROOFSTEP]\next (x | x)\n[GOAL]\ncase h.ofNat\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM : Type u_5\ninst\u271d : Monoid M\nf g : \u2124 \u2192* M\nh_neg_one : \u2191f (-1) = \u2191g (-1)\nh_nat : comp f \u2191ofNatHom = comp g \u2191ofNatHom\nx : \u2115\n\u22a2 \u2191f (ofNat x) = \u2191g (ofNat x)\n[PROOFSTEP]\nexact (FunLike.congr_fun h_nat x : _)\n[GOAL]\ncase h.negSucc\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM : Type u_5\ninst\u271d : Monoid M\nf g : \u2124 \u2192* M\nh_neg_one : \u2191f (-1) = \u2191g (-1)\nh_nat : comp f \u2191ofNatHom = comp g \u2191ofNatHom\nx : \u2115\n\u22a2 \u2191f -[x+1] = \u2191g -[x+1]\n[PROOFSTEP]\nrw [Int.negSucc_eq, \u2190 neg_one_mul, f.map_mul, g.map_mul]\n[GOAL]\ncase h.negSucc\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM : Type u_5\ninst\u271d : Monoid M\nf g : \u2124 \u2192* M\nh_neg_one : \u2191f (-1) = \u2191g (-1)\nh_nat : comp f \u2191ofNatHom = comp g \u2191ofNatHom\nx : \u2115\n\u22a2 \u2191f (-1) * \u2191f (\u2191x + 1) = \u2191g (-1) * \u2191g (\u2191x + 1)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.negSucc.e_a\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nM : Type u_5\ninst\u271d : Monoid M\nf g : \u2124 \u2192* M\nh_neg_one : \u2191f (-1) = \u2191g (-1)\nh_nat : comp f \u2191ofNatHom = comp g \u2191ofNatHom\nx : \u2115\n\u22a2 \u2191f (\u2191x + 1) = \u2191g (\u2191x + 1)\n[PROOFSTEP]\nexact_mod_cast (FunLike.congr_fun h_nat (x + 1) : _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Cast.Lemmas", "llama_tokens": 5647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059609645724, "lm_q2_score": 0.6893056104028799, "lm_q1q2_score": 0.5466923885368472}}
{"text": "[GOAL]\n\u03b1 \u03b2 : Type u\nf g : \u03b1 \u27f6 \u03b2\nh : \u2200 (a : \u03b1), f a = g a\n\u22a2 f = g\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\n\u03b1 \u03b2 : Type u\nf g : \u03b1 \u27f6 \u03b2\nh : \u2200 (a : \u03b1), f a = g a\nx : \u03b1\n\u22a2 f x = g x\n[PROOFSTEP]\nexact h x\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ninst\u271d : IsIso (\u21bef)\n\u22a2 Mono (\u21bef)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 \u03b2 \u03b3 : Type u\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\ninst\u271d : IsIso (\u21bef)\n\u22a2 \u21bef \u226b inv (\u21bef) = \ud835\udfd9 \u03b1\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G H : C \u2964 Type w\nX Y Z : C\n\u03c3 : F \u27f6 G\n\u03c4 : G \u27f6 H\nf : X \u27f6 Y\ng : Y \u27f6 Z\na : F.obj X\n\u22a2 F.map (f \u226b g) a = F.map g (F.map f a)\n[PROOFSTEP]\nsimp [types_comp]\n[GOAL]\nC : Type u\ninst\u271d : Category.{v, u} C\nF G H : C \u2964 Type w\nX Y Z : C\n\u03c3 : F \u27f6 G\n\u03c4 : G \u27f6 H\na : F.obj X\n\u22a2 F.map (\ud835\udfd9 X) a = a\n[PROOFSTEP]\nsimp [types_id]\n[GOAL]\nX : Type u\nx y : X\n\u22a2 x = y \u2192 homOfElement x = homOfElement y\n[PROOFSTEP]\naesop\n[GOAL]\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Mono f \u2194 Function.Injective f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Mono f \u2192 Function.Injective f\n[PROOFSTEP]\nintro H x x' h\n[GOAL]\ncase mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'\n[PROOFSTEP]\nskip\n[GOAL]\ncase mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : f x = f x'\n\u22a2 x = x'\n[PROOFSTEP]\nrw [\u2190 homOfElement_eq_iff] at h \u22a2\n[GOAL]\ncase mp\nX Y : Type u\nf : X \u27f6 Y\nH : Mono f\nx x' : X\nh : homOfElement (f x) = homOfElement (f x')\n\u22a2 homOfElement x = homOfElement x'\n[PROOFSTEP]\nexact (cancel_mono f).mp h\n[GOAL]\ncase mpr\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Function.Injective f \u2192 Mono f\n[PROOFSTEP]\nexact fun H => \u27e8fun g g' h => H.comp_left h\u27e9\n[GOAL]\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Epi f \u2194 Function.Surjective f\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Epi f \u2192 Function.Surjective f\n[PROOFSTEP]\nrintro \u27e8H\u27e9\n[GOAL]\ncase mp.mk\nX Y : Type u\nf : X \u27f6 Y\nH : \u2200 {Z : Type u} (g h : Y \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\n\u22a2 Function.Surjective f\n[PROOFSTEP]\nrefine' Function.surjective_of_right_cancellable_Prop fun g\u2081 g\u2082 hg => _\n[GOAL]\ncase mp.mk\nX Y : Type u\nf : X \u27f6 Y\nH : \u2200 {Z : Type u} (g h : Y \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\ng\u2081 g\u2082 : Y \u2192 Prop\nhg : g\u2081 \u2218 f = g\u2082 \u2218 f\n\u22a2 g\u2081 = g\u2082\n[PROOFSTEP]\nrw [\u2190 Equiv.ulift.symm.injective.comp_left.eq_iff]\n[GOAL]\ncase mp.mk\nX Y : Type u\nf : X \u27f6 Y\nH : \u2200 {Z : Type u} (g h : Y \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\ng\u2081 g\u2082 : Y \u2192 Prop\nhg : g\u2081 \u2218 f = g\u2082 \u2218 f\n\u22a2 (fun x x_1 => x \u2218 x_1) (\u2191Equiv.ulift.symm) g\u2081 = (fun x x_1 => x \u2218 x_1) (\u2191Equiv.ulift.symm) g\u2082\n[PROOFSTEP]\napply H\n[GOAL]\ncase mp.mk.a\nX Y : Type u\nf : X \u27f6 Y\nH : \u2200 {Z : Type u} (g h : Y \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\ng\u2081 g\u2082 : Y \u2192 Prop\nhg : g\u2081 \u2218 f = g\u2082 \u2218 f\n\u22a2 f \u226b (fun x x_1 => x \u2218 x_1) (\u2191Equiv.ulift.symm) g\u2081 = f \u226b (fun x x_1 => x \u2218 x_1) (\u2191Equiv.ulift.symm) g\u2082\n[PROOFSTEP]\nchange ULift.up \u2218 g\u2081 \u2218 f = ULift.up \u2218 g\u2082 \u2218 f\n[GOAL]\ncase mp.mk.a\nX Y : Type u\nf : X \u27f6 Y\nH : \u2200 {Z : Type u} (g h : Y \u27f6 Z), f \u226b g = f \u226b h \u2192 g = h\ng\u2081 g\u2082 : Y \u2192 Prop\nhg : g\u2081 \u2218 f = g\u2082 \u2218 f\n\u22a2 ULift.up \u2218 g\u2081 \u2218 f = ULift.up \u2218 g\u2082 \u2218 f\n[PROOFSTEP]\nrw [hg]\n[GOAL]\ncase mpr\nX Y : Type u\nf : X \u27f6 Y\n\u22a2 Function.Surjective f \u2192 Epi f\n[PROOFSTEP]\nexact fun H => \u27e8fun g g' h => H.injective_comp_right h\u27e9\n[GOAL]\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : _root_.Functor m\ninst\u271d : LawfulFunctor m\n\u03b1 : Type u\n\u22a2 { obj := m, map := fun {X Y} f => Functor.map f }.map (\ud835\udfd9 \u03b1) =\n    \ud835\udfd9 ({ obj := m, map := fun {X Y} f => Functor.map f }.obj \u03b1)\n[PROOFSTEP]\nfunext X\n[GOAL]\ncase h\nm : Type u \u2192 Type v\ninst\u271d\u00b9 : _root_.Functor m\ninst\u271d : LawfulFunctor m\n\u03b1 : Type u\nX : { obj := m, map := fun {X Y} f => Functor.map f }.obj \u03b1\n\u22a2 { obj := m, map := fun {X Y} f => Functor.map f }.map (\ud835\udfd9 \u03b1) X =\n    \ud835\udfd9 ({ obj := m, map := fun {X Y} f => Functor.map f }.obj \u03b1) X\n[PROOFSTEP]\napply id_map\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Types", "llama_tokens": 1929, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059560743421, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5466923801039326}}
{"text": "[GOAL]\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nx x' : B\nh : x = x'\nb : E x\n\u22a2 mk' F x' (cast (_ : E x = E x') b) = { proj := x, snd := b }\n[PROOFSTEP]\nsubst h\n[GOAL]\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nx : B\nb : E x\n\u22a2 mk' F x (cast (_ : E x = E x) b) = { proj := x, snd := b }\n[PROOFSTEP]\nrfl\n[GOAL]\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\ny y' : E b\n\u22a2 mk' F b y = mk' F b y' \u2194 y = y'\n[PROOFSTEP]\nsimp [TotalSpace.ext_iff]\n[GOAL]\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\n\u22a2 range (mk b) = proj \u207b\u00b9' {b}\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\n\u22a2 range (mk b) \u2286 proj \u207b\u00b9' {b}\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9\n[GOAL]\ncase h\u2081.intro\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\nx : E b\n\u22a2 { proj := b, snd := x } \u2208 proj \u207b\u00b9' {b}\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\u2082\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nb : B\n\u22a2 proj \u207b\u00b9' {b} \u2286 range (mk b)\n[PROOFSTEP]\nrintro \u27e8_, x\u27e9 rfl\n[GOAL]\ncase h\u2082.mk\nB : Type u_1\nF : Type u_2\nE : B \u2192 Type u_3\nproj\u271d : B\nx : E proj\u271d\n\u22a2 { proj := proj\u271d, snd := x } \u2208 range (mk { proj := proj\u271d, snd := x }.proj)\n[PROOFSTEP]\nexact \u27e8x, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Bundle", "llama_tokens": 632, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.5466893480096755}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhaveI : Encodable s := s_count.toEncodable\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave h' :\n  \u2200 p q,\n    \u2203 u v,\n      MeasurableSet u \u2227\n        MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u03bc (u \u2229 v) = 0) :=\n  by\n  intro p q\n  by_cases H : p \u2208 s \u2227 q \u2208 s \u2227 p < q\n  \u00b7 rcases h p H.1 q H.2.1 H.2.2 with \u27e8u, v, hu, hv, h'u, h'v, h\u03bc\u27e9\n    exact \u27e8u, v, hu, hv, h'u, h'v, fun _ _ _ => h\u03bc\u27e9\n  \u00b7 refine' \u27e8univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _, fun ps qs pq => _\u27e9\n    simp only [not_and] at H \n    exact (H ps qs pq).elim\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\n\u22a2 \u2200 (p q : \u03b2),\n    \u2203 u v,\n      MeasurableSet u \u2227\n        MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n[PROOFSTEP]\nintro p q\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\n\u22a2 \u2203 u v,\n    MeasurableSet u \u2227\n      MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n[PROOFSTEP]\nby_cases H : p \u2208 s \u2227 q \u2208 s \u2227 p < q\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\nH : p \u2208 s \u2227 q \u2208 s \u2227 p < q\n\u22a2 \u2203 u v,\n    MeasurableSet u \u2227\n      MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n[PROOFSTEP]\nrcases h p H.1 q H.2.1 H.2.2 with \u27e8u, v, hu, hv, h'u, h'v, h\u03bc\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\nH : p \u2208 s \u2227 q \u2208 s \u2227 p < q\nu v : Set \u03b1\nhu : MeasurableSet u\nhv : MeasurableSet v\nh'u : {x | f x < p} \u2286 u\nh'v : {x | q < f x} \u2286 v\nh\u03bc : \u2191\u2191\u03bc (u \u2229 v) = 0\n\u22a2 \u2203 u v,\n    MeasurableSet u \u2227\n      MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n[PROOFSTEP]\nexact \u27e8u, v, hu, hv, h'u, h'v, fun _ _ _ => h\u03bc\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\nH : \u00ac(p \u2208 s \u2227 q \u2208 s \u2227 p < q)\n\u22a2 \u2203 u v,\n    MeasurableSet u \u2227\n      MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n[PROOFSTEP]\nrefine' \u27e8univ, univ, MeasurableSet.univ, MeasurableSet.univ, subset_univ _, subset_univ _, fun ps qs pq => _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\nH : \u00ac(p \u2208 s \u2227 q \u2208 s \u2227 p < q)\nps : p \u2208 s\nqs : q \u2208 s\npq : p < q\n\u22a2 \u2191\u2191\u03bc (univ \u2229 univ) = 0\n[PROOFSTEP]\nsimp only [not_and] at H \n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\np q : \u03b2\nps : p \u2208 s\nqs : q \u2208 s\npq : p < q\nH : p \u2208 s \u2192 q \u2208 s \u2192 \u00acp < q\n\u22a2 \u2191\u2191\u03bc (univ \u2229 univ) = 0\n[PROOFSTEP]\nexact (H ps qs pq).elim\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nh' :\n  \u2200 (p q : \u03b2),\n    \u2203 u v,\n      MeasurableSet u \u2227\n        MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u \u2229 v) = 0)\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nchoose! u v huv using h'\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nlet u' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 q \u2208 s \u2229 Ioi p, u p q\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave u'_meas : \u2200 i, MeasurableSet (u' i) := by\n  intro i\n  exact MeasurableSet.biInter (s_count.mono (inter_subset_left _ _)) fun b _ => (huv i b).1\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\n\u22a2 \u2200 (i : \u03b2), MeasurableSet (u' i)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\ni : \u03b2\n\u22a2 MeasurableSet (u' i)\n[PROOFSTEP]\nexact MeasurableSet.biInter (s_count.mono (inter_subset_left _ _)) fun b _ => (huv i b).1\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nlet f' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 i : s, piecewise (u' i) (fun _ => (i : \u03b2)) (fun _ => (\u22a4 : \u03b2)) x\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave f'_meas : Measurable f' := by\n  apply measurable_iInf\n  exact fun i => Measurable.piecewise (u'_meas i) measurable_const measurable_const\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\n\u22a2 Measurable f'\n[PROOFSTEP]\napply measurable_iInf\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\n\u22a2 \u2200 (i : \u2191s), Measurable fun b => piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) b\n[PROOFSTEP]\nexact fun i => Measurable.piecewise (u'_meas i) measurable_const measurable_const\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nlet t := \u22c3 (p : s) (q : \u21a5(s \u2229 Ioi p)), u' p \u2229 v p q\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave \u03bct : \u03bc t \u2264 0 :=\n  calc\n    \u03bc t \u2264 \u2211' (p : s) (q : \u21a5(s \u2229 Ioi p)), \u03bc (u' p \u2229 v p q) :=\n      by\n      refine (measure_iUnion_le _).trans ?_\n      refine ENNReal.tsum_le_tsum fun p => ?_\n      refine @measure_iUnion_le _ _ _ _ ?_ _\n      exact (s_count.mono (inter_subset_left _ _)).to_subtype\n    _ \u2264 \u2211' (p : s) (q : \u21a5(s \u2229 Ioi p)), \u03bc (u p q \u2229 v p q) :=\n      by\n      refine ENNReal.tsum_le_tsum fun p => ?_\n      refine ENNReal.tsum_le_tsum fun q => measure_mono ?_\n      exact inter_subset_inter_left _ (biInter_subset_of_mem q.2)\n    _ = \u2211' (p : s) (_ : \u21a5(s \u2229 Ioi p)), (0 : \u211d\u22650\u221e) := by\n      congr\n      ext1 p\n      congr\n      ext1 q\n      exact (huv p q).2.2.2.2 p.2 q.2.1 q.2.2\n    _ = 0 := by simp only [tsum_zero]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 \u2191\u2191\u03bc t \u2264 \u2211' (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u' \u2191p \u2229 v \u2191p \u2191q)\n[PROOFSTEP]\nrefine (measure_iUnion_le _).trans ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 \u2211' (i : \u2191s), \u2191\u2191\u03bc (\u22c3 (q : \u2191(s \u2229 Ioi \u2191i)), u' \u2191i \u2229 v \u2191i \u2191q) \u2264 \u2211' (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u' \u2191p \u2229 v \u2191p \u2191q)\n[PROOFSTEP]\nrefine ENNReal.tsum_le_tsum fun p => ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\n\u22a2 \u2191\u2191\u03bc (\u22c3 (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q) \u2264 \u2211' (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u' \u2191p \u2229 v \u2191p \u2191q)\n[PROOFSTEP]\nrefine @measure_iUnion_le _ _ _ _ ?_ _\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\n\u22a2 Countable \u2191(s \u2229 Ioi \u2191p)\n[PROOFSTEP]\nexact (s_count.mono (inter_subset_left _ _)).to_subtype\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 \u2211' (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u' \u2191p \u2229 v \u2191p \u2191q) \u2264 \u2211' (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q)\n[PROOFSTEP]\nrefine ENNReal.tsum_le_tsum fun p => ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\n\u22a2 \u2211' (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u' \u2191p \u2229 v \u2191p \u2191q) \u2264 \u2211' (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q)\n[PROOFSTEP]\nrefine ENNReal.tsum_le_tsum fun q => measure_mono ?_\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\nq : \u2191(s \u2229 Ioi \u2191p)\n\u22a2 u' \u2191p \u2229 v \u2191p \u2191q \u2286 u \u2191p \u2191q \u2229 v \u2191p \u2191q\n[PROOFSTEP]\nexact inter_subset_inter_left _ (biInter_subset_of_mem q.2)\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 \u2211' (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q) = \u2211' (p : \u2191s) (x : \u2191(s \u2229 Ioi \u2191p)), 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 (fun p => \u2211' (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q)) = fun p => \u2211' (x : \u2191(s \u2229 Ioi \u2191p)), 0\n[PROOFSTEP]\next1 p\n[GOAL]\ncase e_f.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\n\u22a2 \u2211' (q : \u2191(s \u2229 Ioi \u2191p)), \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q) = \u2211' (x : \u2191(s \u2229 Ioi \u2191p)), 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f.h.e_f\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\n\u22a2 (fun q => \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q)) = fun x => 0\n[PROOFSTEP]\next1 q\n[GOAL]\ncase e_f.h.e_f.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\np : \u2191s\nq : \u2191(s \u2229 Ioi \u2191p)\n\u22a2 \u2191\u2191\u03bc (u \u2191p \u2191q \u2229 v \u2191p \u2191q) = 0\n[PROOFSTEP]\nexact (huv p q).2.2.2.2 p.2 q.2.1 q.2.2\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u22a2 \u2211' (p : \u2191s) (x : \u2191(s \u2229 Ioi \u2191p)), 0 = 0\n[PROOFSTEP]\nsimp only [tsum_zero]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave ff' : \u2200\u1d50 x \u2202\u03bc, f x = f' x :=\n  by\n  have : \u2200\u1d50 x \u2202\u03bc, x \u2209 t := by\n    have : \u03bc t = 0 := le_antisymm \u03bct bot_le\n    change \u03bc _ = 0\n    convert this\n    ext y\n    simp only [not_exists, exists_prop, mem_setOf_eq, mem_compl_iff, not_not_mem]\n  filter_upwards [this] with x hx\n  apply (iInf_eq_of_forall_ge_of_forall_gt_exists_lt _ _).symm\n  \u00b7 intro i\n    by_cases H : x \u2208 u' i\n    swap\n    \u00b7 simp only [H, le_top, not_false_iff, piecewise_eq_of_not_mem]\n    simp only [H, piecewise_eq_of_mem]\n    contrapose! hx\n    obtain \u27e8r, \u27e8xr, rq\u27e9, rs\u27e9 : \u2203 r, r \u2208 Ioo (i : \u03b2) (f x) \u2229 s :=\n      dense_iff_inter_open.1 s_dense (Ioo i (f x)) isOpen_Ioo (nonempty_Ioo.2 hx)\n    have A : x \u2208 v i r := (huv i r).2.2.2.1 rq\n    refine mem_iUnion.2 \u27e8i, ?_\u27e9\n    refine mem_iUnion.2 \u27e8\u27e8r, \u27e8rs, xr\u27e9\u27e9, ?_\u27e9\n    exact \u27e8H, A\u27e9\n  \u00b7 intro q hq\n    obtain \u27e8r, \u27e8xr, rq\u27e9, rs\u27e9 : \u2203 r, r \u2208 Ioo (f x) q \u2229 s :=\n      dense_iff_inter_open.1 s_dense (Ioo (f x) q) isOpen_Ioo (nonempty_Ioo.2 hq)\n    refine' \u27e8\u27e8r, rs\u27e9, _\u27e9\n    have A : x \u2208 u' r := mem_biInter fun i _ => (huv r i).2.2.1 xr\n    simp only [A, rq, piecewise_eq_of_mem, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = f' x\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, x \u2209 t := by\n  have : \u03bc t = 0 := le_antisymm \u03bct bot_le\n  change \u03bc _ = 0\n  convert this\n  ext y\n  simp only [not_exists, exists_prop, mem_setOf_eq, mem_compl_iff, not_not_mem]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\n[PROOFSTEP]\nhave : \u03bc t = 0 := le_antisymm \u03bct bot_le\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2191\u2191\u03bc t = 0\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\n[PROOFSTEP]\nchange \u03bc _ = 0\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2191\u2191\u03bc t = 0\n\u22a2 \u2191\u2191\u03bc {x | (fun x => \u00acx \u2208 t) x}\u1d9c = 0\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2191\u2191\u03bc t = 0\n\u22a2 {x | (fun x => \u00acx \u2208 t) x}\u1d9c = t\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2191\u2191\u03bc t = 0\ny : \u03b1\n\u22a2 y \u2208 {x | (fun x => \u00acx \u2208 t) x}\u1d9c \u2194 y \u2208 t\n[PROOFSTEP]\nsimp only [not_exists, exists_prop, mem_setOf_eq, mem_compl_iff, not_not_mem]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = f' x\n[PROOFSTEP]\nfilter_upwards [this] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n\u22a2 f x = \u2a05 (i : \u2191s), piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\napply (iInf_eq_of_forall_ge_of_forall_gt_exists_lt _ _).symm\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n\u22a2 \u2200 (i : \u2191s), f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\n\u22a2 f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\nby_cases H : x \u2208 u' i\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\nH : x \u2208 u' \u2191i\n\u22a2 f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\nH : \u00acx \u2208 u' \u2191i\n\u22a2 f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\nH : \u00acx \u2208 u' \u2191i\n\u22a2 f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\nsimp only [H, le_top, not_false_iff, piecewise_eq_of_not_mem]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\nH : x \u2208 u' \u2191i\n\u22a2 f x \u2264 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x\n[PROOFSTEP]\nsimp only [H, piecewise_eq_of_mem]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\ni : \u2191s\nH : x \u2208 u' \u2191i\n\u22a2 f x \u2264 \u2191i\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\ni : \u2191s\nH : x \u2208 u' \u2191i\nhx : \u2191i < f x\n\u22a2 x \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n[PROOFSTEP]\nobtain \u27e8r, \u27e8xr, rq\u27e9, rs\u27e9 : \u2203 r, r \u2208 Ioo (i : \u03b2) (f x) \u2229 s :=\n  dense_iff_inter_open.1 s_dense (Ioo i (f x)) isOpen_Ioo (nonempty_Ioo.2 hx)\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\ni : \u2191s\nH : x \u2208 u' \u2191i\nhx : \u2191i < f x\nr : \u03b2\nrs : r \u2208 s\nxr : \u2191i < r\nrq : r < f x\n\u22a2 x \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n[PROOFSTEP]\nhave A : x \u2208 v i r := (huv i r).2.2.2.1 rq\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\ni : \u2191s\nH : x \u2208 u' \u2191i\nhx : \u2191i < f x\nr : \u03b2\nrs : r \u2208 s\nxr : \u2191i < r\nrq : r < f x\nA : x \u2208 v (\u2191i) r\n\u22a2 x \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n[PROOFSTEP]\nrefine mem_iUnion.2 \u27e8i, ?_\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\ni : \u2191s\nH : x \u2208 u' \u2191i\nhx : \u2191i < f x\nr : \u03b2\nrs : r \u2208 s\nxr : \u2191i < r\nrq : r < f x\nA : x \u2208 v (\u2191i) r\n\u22a2 x \u2208 \u22c3 (q : \u2191(s \u2229 Ioi \u2191i)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) \u2229 v \u2191i \u2191q\n[PROOFSTEP]\nrefine mem_iUnion.2 \u27e8\u27e8r, \u27e8rs, xr\u27e9\u27e9, ?_\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\ni : \u2191s\nH : x \u2208 u' \u2191i\nhx : \u2191i < f x\nr : \u03b2\nrs : r \u2208 s\nxr : \u2191i < r\nrq : r < f x\nA : x \u2208 v (\u2191i) r\n\u22a2 x \u2208 (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) \u2229 v \u2191i \u2191{ val := r, property := (_ : r \u2208 s \u2227 r \u2208 Ioi \u2191i) }\n[PROOFSTEP]\nexact \u27e8H, A\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\n\u22a2 \u2200 (w : \u03b2), f x < w \u2192 \u2203 i, piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x < w\n[PROOFSTEP]\nintro q hq\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\nq : \u03b2\nhq : f x < q\n\u22a2 \u2203 i, piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x < q\n[PROOFSTEP]\nobtain \u27e8r, \u27e8xr, rq\u27e9, rs\u27e9 : \u2203 r, r \u2208 Ioo (f x) q \u2229 s :=\n  dense_iff_inter_open.1 s_dense (Ioo (f x) q) isOpen_Ioo (nonempty_Ioo.2 hq)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\nq : \u03b2\nhq : f x < q\nr : \u03b2\nrs : r \u2208 s\nxr : f x < r\nrq : r < q\n\u22a2 \u2203 i, piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191i), u (\u2191i) q) (fun x => \u2191i) (fun x => \u22a4) x < q\n[PROOFSTEP]\nrefine' \u27e8\u27e8r, rs\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\nq : \u03b2\nhq : f x < q\nr : \u03b2\nrs : r \u2208 s\nxr : f x < r\nrq : r < q\n\u22a2 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191{ val := r, property := rs }), u (\u2191{ val := r, property := rs }) q)\n      (fun x => \u2191{ val := r, property := rs }) (fun x => \u22a4) x <\n    q\n[PROOFSTEP]\nhave A : x \u2208 u' r := mem_biInter fun i _ => (huv r i).2.2.1 xr\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis\u271d : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nthis : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 t\nx : \u03b1\nhx : \u00acx \u2208 \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191p), u (\u2191p) q) \u2229 v \u2191p \u2191q\nq : \u03b2\nhq : f x < q\nr : \u03b2\nrs : r \u2208 s\nxr : f x < r\nrq : r < q\nA : x \u2208 u' r\n\u22a2 piecewise (\u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi \u2191{ val := r, property := rs }), u (\u2191{ val := r, property := rs }) q)\n      (fun x => \u2191{ val := r, property := rs }) (fun x => \u22a4) x <\n    q\n[PROOFSTEP]\nsimp only [A, rq, piecewise_eq_of_mem, Subtype.coe_mk]\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\u03b2 : Type u_2\ninst\u271d\u2076 : CompleteLinearOrder \u03b2\ninst\u271d\u2075 : DenselyOrdered \u03b2\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : OrderTopology \u03b2\ninst\u271d\u00b2 : SecondCountableTopology \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b2\ns_count : Set.Countable s\ns_dense : Dense s\nf : \u03b1 \u2192 \u03b2\nh :\n  \u2200 (p : \u03b2),\n    p \u2208 s \u2192\n      \u2200 (q : \u03b2),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\nthis : Encodable \u2191s\nu v : \u03b2 \u2192 \u03b2 \u2192 Set \u03b1\nhuv :\n  \u2200 (p q : \u03b2),\n    MeasurableSet (u p q) \u2227\n      MeasurableSet (v p q) \u2227\n        {x | f x < p} \u2286 u p q \u2227 {x | q < f x} \u2286 v p q \u2227 (p \u2208 s \u2192 q \u2208 s \u2192 p < q \u2192 \u2191\u2191\u03bc (u p q \u2229 v p q) = 0)\nu' : \u03b2 \u2192 Set \u03b1 := fun p => \u22c2 (q : \u03b2) (_ : q \u2208 s \u2229 Ioi p), u p q\nu'_meas : \u2200 (i : \u03b2), MeasurableSet (u' i)\nf' : \u03b1 \u2192 \u03b2 := fun x => \u2a05 (i : \u2191s), piecewise (u' \u2191i) (fun x => \u2191i) (fun x => \u22a4) x\nf'_meas : Measurable f'\nt : Set \u03b1 := \u22c3 (p : \u2191s) (q : \u2191(s \u2229 Ioi \u2191p)), u' \u2191p \u2229 v \u2191p \u2191q\n\u03bct : \u2191\u2191\u03bc t \u2264 0\nff' : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x = f' x\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nexact \u27e8f', f'_meas, ff'\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nobtain \u27e8s, s_count, s_dense, _, s_top\u27e9 : \u2203 s : Set \u211d\u22650\u221e, s.Countable \u2227 Dense s \u2227 0 \u2209 s \u2227 \u221e \u2209 s :=\n  ENNReal.exists_countable_dense_no_zero_top\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\n\u22a2 AEMeasurable f\n[PROOFSTEP]\nhave I : \u2200 x \u2208 s, x \u2260 \u221e := fun x xs hx => s_top (hx \u25b8 xs)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 AEMeasurable f\n[PROOFSTEP]\napply MeasureTheory.aemeasurable_of_exist_almost_disjoint_supersets \u03bc s s_count s_dense _\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\n\u22a2 \u2200 (p : \u211d\u22650\u221e),\n    p \u2208 s \u2192\n      \u2200 (q : \u211d\u22650\u221e),\n        q \u2208 s \u2192\n          p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n[PROOFSTEP]\nrintro p hp q hq hpq\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\u221e\nhp : p \u2208 s\nq : \u211d\u22650\u221e\nhq : q \u2208 s\nhpq : p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n[PROOFSTEP]\nlift p to \u211d\u22650 using I p hp\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\nq : \u211d\u22650\u221e\nhq : q \u2208 s\np : \u211d\u22650\nhp : \u2191p \u2208 s\nhpq : \u2191p < q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n[PROOFSTEP]\nlift q to \u211d\u22650 using I q hq\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nh :\n  \u2200 (p q : \u211d\u22650),\n    p < q \u2192 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\ns : Set \u211d\u22650\u221e\ns_count : Set.Countable s\ns_dense : Dense s\nleft\u271d : \u00ac0 \u2208 s\ns_top : \u00ac\u22a4 \u2208 s\nI : \u2200 (x : \u211d\u22650\u221e), x \u2208 s \u2192 x \u2260 \u22a4\np : \u211d\u22650\nhp : \u2191p \u2208 s\nq : \u211d\u22650\nhq : \u2191q \u2208 s\nhpq : \u2191p < \u2191q\n\u22a2 \u2203 u v, MeasurableSet u \u2227 MeasurableSet v \u2227 {x | f x < \u2191p} \u2286 u \u2227 {x | \u2191q < f x} \u2286 v \u2227 \u2191\u2191\u03bc (u \u2229 v) = 0\n[PROOFSTEP]\nexact h p q (ENNReal.coe_lt_coe.1 hpq)\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.AEMeasurableOrder", "llama_tokens": 39542, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.6959583250334526, "lm_q1q2_score": 0.5465420445890418}}
{"text": "[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1 : Type u_20\nR : Type u_21\nM : Type u_22\nM\u2082 : Type u_23\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R M\u2082\nv : \u03b1 \u2192\u2080 M\nc : R\nh : \u03b1 \u2192 M \u2192\u2097[R] M\u2082\n\u22a2 (sum (c \u2022 v) fun a => \u2191(h a)) = c \u2022 sum v fun a => \u2191(h a)\n[PROOFSTEP]\nrw [Finsupp.sum_smul_index', Finsupp.smul_sum]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1 : Type u_20\nR : Type u_21\nM : Type u_22\nM\u2082 : Type u_23\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R M\u2082\nv : \u03b1 \u2192\u2080 M\nc : R\nh : \u03b1 \u2192 M \u2192\u2097[R] M\u2082\n\u22a2 (sum v fun i c_1 => \u2191(h i) (c \u2022 c_1)) = sum v fun a b => c \u2022 \u2191(h a) b\n[PROOFSTEP]\nsimp only [map_smul]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1 : Type u_20\nR : Type u_21\nM : Type u_22\nM\u2082 : Type u_23\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R M\u2082\nv : \u03b1 \u2192\u2080 M\nc : R\nh : \u03b1 \u2192 M \u2192\u2097[R] M\u2082\n\u22a2 \u2200 (i : \u03b1), \u2191(h i) 0 = 0\n[PROOFSTEP]\nintro i\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1 : Type u_20\nR : Type u_21\nM : Type u_22\nM\u2082 : Type u_23\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R M\u2082\nv : \u03b1 \u2192\u2080 M\nc : R\nh : \u03b1 \u2192 M \u2192\u2097[R] M\u2082\ni : \u03b1\n\u22a2 \u2191(h i) 0 = 0\n[PROOFSTEP]\nexact (h i).map_zero\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1\u271d : Type u_20\ninst\u271d\u2074 : Finite \u03b1\u271d\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R M\n\u03b1 : Type u_21\ninst\u271d : Unique \u03b1\nm : M\n\u22a2 \u2191(LinearEquiv.symm (finsuppUnique R M \u03b1)) m = single default m\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b1\u271d : Type u_20\ninst\u271d\u2074 : Finite \u03b1\u271d\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R M\n\u03b1 : Type u_21\ninst\u271d : Unique \u03b1\nm : M\n\u22a2 \u2191(\u2191(LinearEquiv.symm (finsuppUnique R M \u03b1)) m) default = \u2191(single default m) default\n[PROOFSTEP]\nsimp [LinearEquiv.finsuppUnique, Equiv.funUnique, single, Pi.single, equivFunOnFinite, Function.update]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b9 : Type u_20\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nR : Type u_21\ninst\u271d : Semiring R\nx : \u03b9 \u2192 R\n\u22a2 x = \u2211 i : \u03b9, x i \u2022 fun j => if i = j then 1 else 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\n\u03b9 : Type u_20\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\nR : Type u_21\ninst\u271d : Semiring R\nx : \u03b9 \u2192 R\nx\u271d : \u03b9\n\u22a2 x x\u271d = Finset.sum Finset.univ (fun i => x i \u2022 fun j => if i = j then 1 else 0) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\n\u22a2 M \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 p }\n[PROOFSTEP]\nrefine' { toFun := fun c => \u27e8f c, h c\u27e9 .. }\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\n\u22a2 \u2200 (x y : M),\n    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n      (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n        (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\n\u22a2 \u2200 (r : R) (x : M),\n    AddHom.toFun\n        { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : M),\n                (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n        (r \u2022 x) =\n      \u2191\u03c3\u2081\u2082 r \u2022\n        AddHom.toFun\n          { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n            map_add' :=\n              (_ :\n                \u2200 (x y : M),\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                      (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n          x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\nx\u271d y\u271d : M\n\u22a2 (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x\u271d + y\u271d) =\n    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x\u271d +\n      (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y\u271d\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\nr\u271d : R\nx\u271d : M\n\u22a2 AddHom.toFun\n      { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n        map_add' :=\n          (_ :\n            \u2200 (x y : M),\n              (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n      (r\u271d \u2022 x\u271d) =\n    \u2191\u03c3\u2081\u2082 r\u271d \u2022\n      AddHom.toFun\n        { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : M),\n                (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n        x\u271d\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase refine'_1.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\nx\u271d y\u271d : M\n\u22a2 \u2191((fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x\u271d + y\u271d)) =\n    \u2191((fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x\u271d +\n        (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R\u2082 M\u2082\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : \u2200 (c : M), \u2191f c \u2208 p\nr\u271d : R\nx\u271d : M\n\u22a2 \u2191(AddHom.toFun\n        { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : M),\n                (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n        (r\u271d \u2022 x\u271d)) =\n    \u2191(\u2191\u03c3\u2081\u2082 r\u271d \u2022\n        AddHom.toFun\n          { toFun := fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) },\n            map_add' :=\n              (_ :\n                \u2200 (x y : M),\n                  (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) (x + y) =\n                    (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) x +\n                      (fun c => { val := \u2191f c, property := (_ : \u2191f c \u2208 p) }) y) }\n          x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : Semiring R\u2082\ninst\u271d\u00b9\u2076 : Semiring R\u2083\ninst\u271d\u00b9\u2075 : Semiring R\u2084\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module R\u2082 M\u2082\ninst\u271d\u2076 : Module R\u2083 M\u2083\ninst\u271d\u2075 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d : Subsingleton M\nsrc\u271d : Inhabited (M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) := inferInstanceAs (Inhabited (M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082))\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nx : M\n\u22a2 \u2191f x = \u2191default x\n[PROOFSTEP]\nrw [Subsingleton.elim x 0, map_zero, map_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\n\u22a2 toAddMonoidHom 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nx\u271d : M\n\u22a2 \u2191(toAddMonoidHom 0) x\u271d = \u21910 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\n\u22a2 \u2200 (x y : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082),\n    ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } (x + y) =\n      ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } x +\n        ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } y\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nx\u271d y\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\n\u22a2 ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } (x\u271d + y\u271d) =\n    ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } x\u271d +\n      ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nx\u271d\u00b9 y\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\n\u22a2 \u2191(ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } (x\u271d\u00b9 + y\u271d)) x\u271d =\n    \u2191(ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } x\u271d\u00b9 +\n          ZeroHom.toFun { toFun := toAddMonoidHom, map_zero' := (_ : toAddMonoidHom 0 = 0) } y\u271d)\n      x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2\u00b9 : Semiring R\ninst\u271d\u00b2\u2070 : Semiring R\u2082\ninst\u271d\u00b9\u2079 : Semiring R\u2083\ninst\u271d\u00b9\u2078 : Semiring R\u2084\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\ninst\u271d\u2079 : Module R\u2083 M\u2083\ninst\u271d\u2078 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : Module R S\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nf : M\u2081 \u2192\u2097[R] S\nx\u271d : M\nx y : M\u2081\n\u22a2 (fun b => \u2191f b \u2022 x\u271d) (x + y) = (fun b => \u2191f b \u2022 x\u271d) x + (fun b => \u2191f b \u2022 x\u271d) y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2\u00b9 : Semiring R\ninst\u271d\u00b2\u2070 : Semiring R\u2082\ninst\u271d\u00b9\u2079 : Semiring R\u2083\ninst\u271d\u00b9\u2078 : Semiring R\u2084\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\ninst\u271d\u2079 : Module R\u2083 M\u2083\ninst\u271d\u2078 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : Module R S\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nf : M\u2081 \u2192\u2097[R] S\nx\u271d : M\nx y : M\u2081\n\u22a2 \u2191f (x + y) \u2022 x\u271d = \u2191f x \u2022 x\u271d + \u2191f y \u2022 x\u271d\n[PROOFSTEP]\nrw [f.map_add, add_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2\u00b9 : Semiring R\ninst\u271d\u00b2\u2070 : Semiring R\u2082\ninst\u271d\u00b9\u2079 : Semiring R\u2083\ninst\u271d\u00b9\u2078 : Semiring R\u2084\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\ninst\u271d\u2079 : Module R\u2083 M\u2083\ninst\u271d\u2078 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : Module R S\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nf : M\u2081 \u2192\u2097[R] S\nx : M\nb : R\ny : M\u2081\n\u22a2 AddHom.toFun\n      { toFun := fun b => \u2191f b \u2022 x,\n        map_add' :=\n          (_ : \u2200 (x_1 y : M\u2081), (fun b => \u2191f b \u2022 x) (x_1 + y) = (fun b => \u2191f b \u2022 x) x_1 + (fun b => \u2191f b \u2022 x) y) }\n      (b \u2022 y) =\n    \u2191(RingHom.id R) b \u2022\n      AddHom.toFun\n        { toFun := fun b => \u2191f b \u2022 x,\n          map_add' :=\n            (_ : \u2200 (x_1 y : M\u2081), (fun b => \u2191f b \u2022 x) (x_1 + y) = (fun b => \u2191f b \u2022 x) x_1 + (fun b => \u2191f b \u2022 x) y) }\n        y\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2\u00b9 : Semiring R\ninst\u271d\u00b2\u2070 : Semiring R\u2082\ninst\u271d\u00b9\u2079 : Semiring R\u2083\ninst\u271d\u00b9\u2078 : Semiring R\u2084\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b2 : Module R M\ninst\u271d\u00b9\u00b9 : Module R M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\ninst\u271d\u2079 : Module R\u2083 M\u2083\ninst\u271d\u2078 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2077 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b3 : Semiring S\ninst\u271d\u00b2 : Module R S\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nf : M\u2081 \u2192\u2097[R] S\nx : M\nb : R\ny : M\u2081\n\u22a2 \u2191f (b \u2022 y) \u2022 x = b \u2022 \u2191f y \u2022 x\n[PROOFSTEP]\nrw [map_smul, smul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : Semiring R\u2082\ninst\u271d\u00b9\u2076 : Semiring R\u2083\ninst\u271d\u00b9\u2075 : Semiring R\u2084\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module R\u2082 M\u2082\ninst\u271d\u2076 : Module R\u2083 M\u2083\ninst\u271d\u2075 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d : Nontrivial M\n\u22a2 Nontrivial (Module.End R M)\n[PROOFSTEP]\nobtain \u27e8m, ne\u27e9 := (nontrivial_iff_exists_ne (0 : M)).mp inferInstance\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : Semiring R\u2082\ninst\u271d\u00b9\u2076 : Semiring R\u2083\ninst\u271d\u00b9\u2075 : Semiring R\u2084\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2084\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M\u2081\ninst\u271d\u2077 : Module R\u2082 M\u2082\ninst\u271d\u2076 : Module R\u2083 M\u2083\ninst\u271d\u2075 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d : Nontrivial M\nm : M\nne : m \u2260 0\n\u22a2 Nontrivial (Module.End R M)\n[PROOFSTEP]\nexact nontrivial_of_ne 1 0 fun p => ne (LinearMap.congr_fun p m)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf : Module.End R M\nm : M\nk l : \u2115\nhk : k \u2264 l\nhm : \u2191(f ^ k) m = 0\n\u22a2 \u2191(f ^ l) m = 0\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le hk, pow_add, mul_apply, hm, map_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : Module.End R M\ng\u2082 : Module.End R\u2082 M\u2082\nh : comp g\u2082 f = comp f g\nk : \u2115\n\u22a2 comp (g\u2082 ^ k) f = comp f (g ^ k)\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : Module.End R M\ng\u2082 : Module.End R\u2082 M\u2082\nh : comp g\u2082 f = comp f g\n\u22a2 comp (g\u2082 ^ Nat.zero) f = comp f (g ^ Nat.zero)\n[PROOFSTEP]\nsimp only [pow_zero, Nat.zero_eq]\n[GOAL]\ncase zero\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : Module.End R M\ng\u2082 : Module.End R\u2082 M\u2082\nh : comp g\u2082 f = comp f g\n\u22a2 comp 1 f = comp f 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : Module.End R M\ng\u2082 : Module.End R\u2082 M\u2082\nh : comp g\u2082 f = comp f g\nk : \u2115\nih : comp (g\u2082 ^ k) f = comp f (g ^ k)\n\u22a2 comp (g\u2082 ^ Nat.succ k) f = comp f (g ^ Nat.succ k)\n[PROOFSTEP]\nrw [pow_succ, pow_succ, LinearMap.mul_eq_comp, LinearMap.comp_assoc, ih, \u2190 LinearMap.comp_assoc, h,\n  LinearMap.comp_assoc, LinearMap.mul_eq_comp]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nN : Submodule R M\ng : Module.End R { x // x \u2208 N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : \u2115\nhG : G ^ k = 0\n\u22a2 g ^ k = 0\n[PROOFSTEP]\next m\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nN : Submodule R M\ng : Module.End R { x // x \u2208 N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : \u2115\nhG : G ^ k = 0\nm : { x // x \u2208 N }\n\u22a2 \u2191(\u2191(g ^ k) m) = \u2191(\u21910 m)\n[PROOFSTEP]\nhave hg : N.subtype.comp (g ^ k) m = 0 := by rw [\u2190 commute_pow_left_of_commute h, hG, zero_comp, zero_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nN : Submodule R M\ng : Module.End R { x // x \u2208 N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : \u2115\nhG : G ^ k = 0\nm : { x // x \u2208 N }\n\u22a2 \u2191(comp (Submodule.subtype N) (g ^ k)) m = 0\n[PROOFSTEP]\nrw [\u2190 commute_pow_left_of_commute h, hG, zero_comp, zero_apply]\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nN : Submodule R M\ng : Module.End R { x // x \u2208 N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : \u2115\nhG : G ^ k = 0\nm : { x // x \u2208 N }\nhg : \u2191(comp (Submodule.subtype N) (g ^ k)) m = 0\n\u22a2 \u2191(\u2191(g ^ k) m) = \u2191(\u21910 m)\n[PROOFSTEP]\nsimpa using hg\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nn : \u2115\n\u22a2 f' ^ (n + 1) = comp (f' ^ n) f'\n[PROOFSTEP]\nrw [pow_succ', mul_eq_comp]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Surjective \u2191f'\nn : \u2115\n\u22a2 Surjective \u2191(f' ^ (n + 1))\n[PROOFSTEP]\nrw [iterate_succ]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Surjective \u2191f'\nn : \u2115\n\u22a2 Surjective \u2191(comp (f' ^ n) f')\n[PROOFSTEP]\nexact (iterate_surjective h n).comp h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Injective \u2191f'\nn : \u2115\n\u22a2 Injective \u2191(f' ^ (n + 1))\n[PROOFSTEP]\nrw [iterate_succ]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Injective \u2191f'\nn : \u2115\n\u22a2 Injective \u2191(comp (f' ^ n) f')\n[PROOFSTEP]\nexact (iterate_injective h n).comp h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Bijective \u2191f'\nn : \u2115\n\u22a2 Bijective \u2191(f' ^ (n + 1))\n[PROOFSTEP]\nrw [iterate_succ]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nh : Bijective \u2191f'\nn : \u2115\n\u22a2 Bijective \u2191(comp (f' ^ n) f')\n[PROOFSTEP]\nexact (iterate_bijective h n).comp h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nn : \u2115\nhn : n \u2260 0\nh : Injective \u2191(f' ^ n)\n\u22a2 Injective \u2191f'\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos (pos_iff_ne_zero.mpr hn), iterate_succ, coe_comp] at h \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nn : \u2115\nhn : n \u2260 0\nh : Injective (\u2191(f' ^ Nat.pred n) \u2218 \u2191f')\n\u22a2 Injective \u2191f'\n[PROOFSTEP]\nexact h.of_comp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nn : \u2115\nhn : n \u2260 0\nh : Surjective \u2191(f' ^ n)\n\u22a2 Surjective \u2191f'\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos (pos_iff_ne_zero.mpr hn), Nat.succ_eq_add_one, add_comm, pow_add] at h \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\nn : \u2115\nhn : n \u2260 0\nh : Surjective \u2191(f' ^ 1 * f' ^ Nat.pred n)\n\u22a2 Surjective \u2191f'\n[PROOFSTEP]\nexact Surjective.of_comp h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nn : \u2115\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nx : M\nhx : x \u2208 p\n\u22a2 \u2191(f' ^ n) x \u2208 p\n[PROOFSTEP]\ninduction' n with n ih generalizing x\n[GOAL]\ncase zero\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nx\u271d : M\nhx\u271d : x\u271d \u2208 p\nx : M\nhx : x \u2208 p\n\u22a2 \u2191(f' ^ Nat.zero) x \u2208 p\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase succ\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nx\u271d : M\nhx\u271d : x\u271d \u2208 p\nn : \u2115\nih : \u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p\nx : M\nhx : x \u2208 p\n\u22a2 \u2191(f' ^ Nat.succ n) x \u2208 p\n[PROOFSTEP]\nsimpa only [iterate_succ, coe_comp, Function.comp_apply, restrict_apply] using ih _ (h _ hx)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nn : \u2115\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nh' : optParam (\u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p) (_ : \u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p)\n\u22a2 restrict f' h ^ n = restrict (f' ^ n) h'\n[PROOFSTEP]\next x\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nn : \u2115\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nh' : optParam (\u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p) (_ : \u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p)\nx : { x // x \u2208 p }\n\u22a2 \u2191(\u2191(restrict f' h ^ n) x) = \u2191(\u2191(restrict (f' ^ n) h') x)\n[PROOFSTEP]\nhave : Semiconj (\u2191) (f'.restrict h) f' := fun _ \u21a6 rfl\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : Semiring R\u2082\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : Semiring R\u2084\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2083\ninst\u271d\u2079 : AddCommMonoid M\u2084\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2081\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : Module R\u2083 M\u2083\ninst\u271d\u2074 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\nf' : M \u2192\u2097[R] M\np : Submodule R M\nn : \u2115\nh : \u2200 (x : M), x \u2208 p \u2192 \u2191f' x \u2208 p\nh' : optParam (\u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p) (_ : \u2200 (x : M), x \u2208 p \u2192 \u2191(f' ^ n) x \u2208 p)\nx : { x // x \u2208 p }\nthis : Semiconj Subtype.val \u2191(restrict f' h) \u2191f'\n\u22a2 \u2191(\u2191(restrict f' h ^ n) x) = \u2191(\u2191(restrict (f' ^ n) h') x)\n[PROOFSTEP]\nsimp [coe_pow, this.iterate_right _ _]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\n\u22a2 \u2191f x = \u2211 i : \u03b9, x i \u2022 \u2191f fun j => if i = j then 1 else 0\n[PROOFSTEP]\nconv_lhs => rw [pi_eq_sum_univ x, f.map_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\n| \u2191f x\n[PROOFSTEP]\nrw [pi_eq_sum_univ x, f.map_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\n| \u2191f x\n[PROOFSTEP]\nrw [pi_eq_sum_univ x, f.map_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\n| \u2191f x\n[PROOFSTEP]\nrw [pi_eq_sum_univ x, f.map_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\n\u22a2 \u2211 i : \u03b9, \u2191f (x i \u2022 fun j => if i = j then 1 else 0) = \u2211 i : \u03b9, x i \u2022 \u2191f fun j => if i = j then 1 else 0\n[PROOFSTEP]\nrefine Finset.sum_congr rfl (fun _ _ => ?_)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : Semiring R\u2082\ninst\u271d\u00b9\u2077 : Semiring R\u2083\ninst\u271d\u00b9\u2076 : Semiring R\u2084\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : Module R M\u2081\ninst\u271d\u2078 : Module R\u2082 M\u2082\ninst\u271d\u2077 : Module R\u2083 M\u2083\ninst\u271d\u2076 : Module R\u2084 M\u2084\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084\n\u03c3\u2081\u2084 : R \u2192+* R\u2084\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nf : (\u03b9 \u2192 R) \u2192\u2097[R] M\nx : \u03b9 \u2192 R\nx\u271d\u00b9 : \u03b9\nx\u271d : x\u271d\u00b9 \u2208 Finset.univ\n\u22a2 \u2191f (x x\u271d\u00b9 \u2022 fun j => if x\u271d\u00b9 = j then 1 else 0) = x x\u271d\u00b9 \u2022 \u2191f fun j => if x\u271d\u00b9 = j then 1 else 0\n[PROOFSTEP]\nrw [map_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2082\ninst\u271d\u2076 : Module R M\u2083\ninst\u271d\u2075 : Module S M\u2082\ninst\u271d\u2074 : Module S M\u2083\ninst\u271d\u00b3 : SMulCommClass R S M\u2082\ninst\u271d\u00b2 : SMulCommClass R S M\u2083\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\ninst\u271d : SMulCommClass R S M\nsrc\u271d : (fun x => (R \u2192\u2097[R] M) \u2192\u2097[S] M) 1 := \u2191(apply\u2097' S) 1\nf : R \u2192\u2097[R] M\n\u22a2 smulRight 1\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f => \u2191f 1,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : R \u2192\u2097[R] M),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : S) (x : R \u2192\u2097[R] M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id S) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2082\ninst\u271d\u2076 : Module R M\u2083\ninst\u271d\u2075 : Module S M\u2082\ninst\u271d\u2074 : Module S M\u2083\ninst\u271d\u00b3 : SMulCommClass R S M\u2082\ninst\u271d\u00b2 : SMulCommClass R S M\u2083\nf\u271d : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\ninst\u271d : SMulCommClass R S M\nsrc\u271d : (fun x => (R \u2192\u2097[R] M) \u2192\u2097[S] M) 1 := \u2191(apply\u2097' S) 1\nf : R \u2192\u2097[R] M\n\u22a2 \u2191(smulRight 1\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => \u2191f 1,\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : R \u2192\u2097[R] M),\n                          AddHom.toFun src\u271d.toAddHom (x + y) =\n                            AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : S) (x : R \u2192\u2097[R] M),\n                      AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id S) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n            f))\n      1 =\n    \u2191f 1\n[PROOFSTEP]\nsimp only [coe_smulRight, one_apply, smul_eq_mul, \u2190 map_smul f, mul_one]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring S\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2082\ninst\u271d\u2076 : Module R M\u2083\ninst\u271d\u2075 : Module S M\u2082\ninst\u271d\u2074 : Module S M\u2083\ninst\u271d\u00b3 : SMulCommClass R S M\u2082\ninst\u271d\u00b2 : SMulCommClass R S M\u2083\nf : M \u2192\u2097[R] M\u2082\ninst\u271d\u00b9 : Module S M\ninst\u271d : SMulCommClass R S M\nsrc\u271d : (fun x => (R \u2192\u2097[R] M) \u2192\u2097[S] M) 1 := \u2191(apply\u2097' S) 1\nx : M\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f => \u2191f 1,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : R \u2192\u2097[R] M),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : S) (x : R \u2192\u2097[R] M),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id S) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (smulRight 1 x) =\n    x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf g : M \u2192\u2097[R] M\u2082\np : Submodule R M\n\u22a2 \u2200 (x y : M \u2192\u2097[R] M\u2082), (fun \u03c6 => domRestrict \u03c6 p) (x + y) = (fun \u03c6 => domRestrict \u03c6 p) x + (fun \u03c6 => domRestrict \u03c6 p) y\n[PROOFSTEP]\nsimp [LinearMap.ext_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf g : M \u2192\u2097[R] M\u2082\np : Submodule R M\n\u22a2 \u2200 (r : R) (x : M \u2192\u2097[R] M\u2082),\n    AddHom.toFun\n        { toFun := fun \u03c6 => domRestrict \u03c6 p,\n          map_add' := (_ : \u2200 (a a_1 : M \u2192\u2097[R] M\u2082), domRestrict (a + a_1) p = domRestrict a p + domRestrict a_1 p) }\n        (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022\n        AddHom.toFun\n          { toFun := fun \u03c6 => domRestrict \u03c6 p,\n            map_add' := (_ : \u2200 (a a_1 : M \u2192\u2097[R] M\u2082), domRestrict (a + a_1) p = domRestrict a p + domRestrict a_1 p) }\n          x\n[PROOFSTEP]\nsimp [LinearMap.ext_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf : M\u2082 \u2192\u2097[R] R\nm m' : M\n\u22a2 smulRight f (m + m') = smulRight f m + smulRight f m'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf : M\u2082 \u2192\u2097[R] R\nm m' : M\nx\u271d : M\u2082\n\u22a2 \u2191(smulRight f (m + m')) x\u271d = \u2191(smulRight f m + smulRight f m') x\u271d\n[PROOFSTEP]\napply smul_add\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf : M\u2082 \u2192\u2097[R] R\nc : R\nm : M\n\u22a2 AddHom.toFun\n      { toFun := smulRight f, map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n      (c \u2022 m) =\n    \u2191(RingHom.id R) c \u2022\n      AddHom.toFun\n        { toFun := smulRight f, map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n        m\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf : M\u2082 \u2192\u2097[R] R\nc : R\nm : M\nx\u271d : M\u2082\n\u22a2 \u2191(AddHom.toFun\n          { toFun := smulRight f,\n            map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n          (c \u2022 m))\n      x\u271d =\n    \u2191(\u2191(RingHom.id R) c \u2022\n          AddHom.toFun\n            { toFun := smulRight f,\n              map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n            m)\n      x\u271d\n[PROOFSTEP]\napply smul_comm\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf f' : M\u2082 \u2192\u2097[R] R\n\u22a2 (fun f =>\n        {\n          toAddHom :=\n            { toFun := smulRight f,\n              map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (m : M),\n                AddHom.toFun\n                    { toFun := smulRight f,\n                      map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                    (c \u2022 m) =\n                  \u2191(RingHom.id R) c \u2022\n                    AddHom.toFun\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                      m) })\n      (f + f') =\n    (fun f =>\n          {\n            toAddHom :=\n              { toFun := smulRight f,\n                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (m : M),\n                  AddHom.toFun\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                      (c \u2022 m) =\n                    \u2191(RingHom.id R) c \u2022\n                      AddHom.toFun\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                        m) })\n        f +\n      (fun f =>\n          {\n            toAddHom :=\n              { toFun := smulRight f,\n                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (m : M),\n                  AddHom.toFun\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                      (c \u2022 m) =\n                    \u2191(RingHom.id R) c \u2022\n                      AddHom.toFun\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                        m) })\n        f'\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nf f' : M\u2082 \u2192\u2097[R] R\nx\u271d\u00b9 : M\nx\u271d : M\u2082\n\u22a2 \u2191(\u2191((fun f =>\n                {\n                  toAddHom :=\n                    { toFun := smulRight f,\n                      map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R) (m : M),\n                        AddHom.toFun\n                            { toFun := smulRight f,\n                              map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                            (c \u2022 m) =\n                          \u2191(RingHom.id R) c \u2022\n                            AddHom.toFun\n                              { toFun := smulRight f,\n                                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                              m) })\n              (f + f'))\n          x\u271d\u00b9)\n      x\u271d =\n    \u2191(\u2191((fun f =>\n                  {\n                    toAddHom :=\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                    map_smul' :=\n                      (_ :\n                        \u2200 (c : R) (m : M),\n                          AddHom.toFun\n                              { toFun := smulRight f,\n                                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                              (c \u2022 m) =\n                            \u2191(RingHom.id R) c \u2022\n                              AddHom.toFun\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                m) })\n                f +\n              (fun f =>\n                  {\n                    toAddHom :=\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                    map_smul' :=\n                      (_ :\n                        \u2200 (c : R) (m : M),\n                          AddHom.toFun\n                              { toFun := smulRight f,\n                                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                              (c \u2022 m) =\n                            \u2191(RingHom.id R) c \u2022\n                              AddHom.toFun\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                m) })\n                f')\n          x\u271d\u00b9)\n      x\u271d\n[PROOFSTEP]\napply add_smul\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nc : R\nf : M\u2082 \u2192\u2097[R] R\n\u22a2 AddHom.toFun\n      {\n        toFun := fun f =>\n          {\n            toAddHom :=\n              { toFun := smulRight f,\n                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (m : M),\n                  AddHom.toFun\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                      (c \u2022 m) =\n                    \u2191(RingHom.id R) c \u2022\n                      AddHom.toFun\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                        m) },\n        map_add' :=\n          (_ :\n            \u2200 (f f' : M\u2082 \u2192\u2097[R] R),\n              (fun f =>\n                    {\n                      toAddHom :=\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                      map_smul' :=\n                        (_ :\n                          \u2200 (c : R) (m : M),\n                            AddHom.toFun\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                (c \u2022 m) =\n                              \u2191(RingHom.id R) c \u2022\n                                AddHom.toFun\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                  m) })\n                  (f + f') =\n                (fun f =>\n                      {\n                        toAddHom :=\n                          { toFun := smulRight f,\n                            map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                        map_smul' :=\n                          (_ :\n                            \u2200 (c : R) (m : M),\n                              AddHom.toFun\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                  (c \u2022 m) =\n                                \u2191(RingHom.id R) c \u2022\n                                  AddHom.toFun\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                    m) })\n                    f +\n                  (fun f =>\n                      {\n                        toAddHom :=\n                          { toFun := smulRight f,\n                            map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                        map_smul' :=\n                          (_ :\n                            \u2200 (c : R) (m : M),\n                              AddHom.toFun\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                  (c \u2022 m) =\n                                \u2191(RingHom.id R) c \u2022\n                                  AddHom.toFun\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                    m) })\n                    f') }\n      (c \u2022 f) =\n    \u2191(RingHom.id R) c \u2022\n      AddHom.toFun\n        {\n          toFun := fun f =>\n            {\n              toAddHom :=\n                { toFun := smulRight f,\n                  map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n              map_smul' :=\n                (_ :\n                  \u2200 (c : R) (m : M),\n                    AddHom.toFun\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                        (c \u2022 m) =\n                      \u2191(RingHom.id R) c \u2022\n                        AddHom.toFun\n                          { toFun := smulRight f,\n                            map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                          m) },\n          map_add' :=\n            (_ :\n              \u2200 (f f' : M\u2082 \u2192\u2097[R] R),\n                (fun f =>\n                      {\n                        toAddHom :=\n                          { toFun := smulRight f,\n                            map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                        map_smul' :=\n                          (_ :\n                            \u2200 (c : R) (m : M),\n                              AddHom.toFun\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                  (c \u2022 m) =\n                                \u2191(RingHom.id R) c \u2022\n                                  AddHom.toFun\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                    m) })\n                    (f + f') =\n                  (fun f =>\n                        {\n                          toAddHom :=\n                            { toFun := smulRight f,\n                              map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                          map_smul' :=\n                            (_ :\n                              \u2200 (c : R) (m : M),\n                                AddHom.toFun\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                    (c \u2022 m) =\n                                  \u2191(RingHom.id R) c \u2022\n                                    AddHom.toFun\n                                      { toFun := smulRight f,\n                                        map_add' :=\n                                          (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                      m) })\n                      f +\n                    (fun f =>\n                        {\n                          toAddHom :=\n                            { toFun := smulRight f,\n                              map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                          map_smul' :=\n                            (_ :\n                              \u2200 (c : R) (m : M),\n                                AddHom.toFun\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                    (c \u2022 m) =\n                                  \u2191(RingHom.id R) c \u2022\n                                    AddHom.toFun\n                                      { toFun := smulRight f,\n                                        map_add' :=\n                                          (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                      m) })\n                      f') }\n        f\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\nf\u271d g : M \u2192\u2097[R] M\u2082\nc : R\nf : M\u2082 \u2192\u2097[R] R\nx\u271d\u00b9 : M\nx\u271d : M\u2082\n\u22a2 \u2191(\u2191(AddHom.toFun\n              {\n                toFun := fun f =>\n                  {\n                    toAddHom :=\n                      { toFun := smulRight f,\n                        map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                    map_smul' :=\n                      (_ :\n                        \u2200 (c : R) (m : M),\n                          AddHom.toFun\n                              { toFun := smulRight f,\n                                map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                              (c \u2022 m) =\n                            \u2191(RingHom.id R) c \u2022\n                              AddHom.toFun\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                m) },\n                map_add' :=\n                  (_ :\n                    \u2200 (f f' : M\u2082 \u2192\u2097[R] R),\n                      (fun f =>\n                            {\n                              toAddHom :=\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                              map_smul' :=\n                                (_ :\n                                  \u2200 (c : R) (m : M),\n                                    AddHom.toFun\n                                        { toFun := smulRight f,\n                                          map_add' :=\n                                            (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                        (c \u2022 m) =\n                                      \u2191(RingHom.id R) c \u2022\n                                        AddHom.toFun\n                                          { toFun := smulRight f,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                          m) })\n                          (f + f') =\n                        (fun f =>\n                              {\n                                toAddHom :=\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                                map_smul' :=\n                                  (_ :\n                                    \u2200 (c : R) (m : M),\n                                      AddHom.toFun\n                                          { toFun := smulRight f,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                          (c \u2022 m) =\n                                        \u2191(RingHom.id R) c \u2022\n                                          AddHom.toFun\n                                            { toFun := smulRight f,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                            m) })\n                            f +\n                          (fun f =>\n                              {\n                                toAddHom :=\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                                map_smul' :=\n                                  (_ :\n                                    \u2200 (c : R) (m : M),\n                                      AddHom.toFun\n                                          { toFun := smulRight f,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                          (c \u2022 m) =\n                                        \u2191(RingHom.id R) c \u2022\n                                          AddHom.toFun\n                                            { toFun := smulRight f,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                            m) })\n                            f') }\n              (c \u2022 f))\n          x\u271d\u00b9)\n      x\u271d =\n    \u2191(\u2191(\u2191(RingHom.id R) c \u2022\n              AddHom.toFun\n                {\n                  toFun := fun f =>\n                    {\n                      toAddHom :=\n                        { toFun := smulRight f,\n                          map_add' := (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                      map_smul' :=\n                        (_ :\n                          \u2200 (c : R) (m : M),\n                            AddHom.toFun\n                                { toFun := smulRight f,\n                                  map_add' :=\n                                    (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                (c \u2022 m) =\n                              \u2191(RingHom.id R) c \u2022\n                                AddHom.toFun\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                  m) },\n                  map_add' :=\n                    (_ :\n                      \u2200 (f f' : M\u2082 \u2192\u2097[R] R),\n                        (fun f =>\n                              {\n                                toAddHom :=\n                                  { toFun := smulRight f,\n                                    map_add' :=\n                                      (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                                map_smul' :=\n                                  (_ :\n                                    \u2200 (c : R) (m : M),\n                                      AddHom.toFun\n                                          { toFun := smulRight f,\n                                            map_add' :=\n                                              (_ :\n                                                \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                          (c \u2022 m) =\n                                        \u2191(RingHom.id R) c \u2022\n                                          AddHom.toFun\n                                            { toFun := smulRight f,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                            m) })\n                            (f + f') =\n                          (fun f =>\n                                {\n                                  toAddHom :=\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (c : R) (m : M),\n                                        AddHom.toFun\n                                            { toFun := smulRight f,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                            (c \u2022 m) =\n                                          \u2191(RingHom.id R) c \u2022\n                                            AddHom.toFun\n                                              { toFun := smulRight f,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (m m' : M),\n                                                      smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                              m) })\n                              f +\n                            (fun f =>\n                                {\n                                  toAddHom :=\n                                    { toFun := smulRight f,\n                                      map_add' :=\n                                        (_ : \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') },\n                                  map_smul' :=\n                                    (_ :\n                                      \u2200 (c : R) (m : M),\n                                        AddHom.toFun\n                                            { toFun := smulRight f,\n                                              map_add' :=\n                                                (_ :\n                                                  \u2200 (m m' : M), smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                            (c \u2022 m) =\n                                          \u2191(RingHom.id R) c \u2022\n                                            AddHom.toFun\n                                              { toFun := smulRight f,\n                                                map_add' :=\n                                                  (_ :\n                                                    \u2200 (m m' : M),\n                                                      smulRight f (m + m') = smulRight f m + smulRight f m') }\n                                              m) })\n                              f') }\n                f)\n          x\u271d\u00b9)\n      x\u271d\n[PROOFSTEP]\napply mul_smul\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\n\u22a2 \u2200 (x y : A \u2192+ B), AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y\n[PROOFSTEP]\nintros\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nx\u271d y\u271d : A \u2192+ B\n\u22a2 AddMonoidHom.toNatLinearMap (x\u271d + y\u271d) = AddMonoidHom.toNatLinearMap x\u271d + AddMonoidHom.toNatLinearMap y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nx\u271d\u00b9 y\u271d : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(AddMonoidHom.toNatLinearMap (x\u271d\u00b9 + y\u271d)) x\u271d = \u2191(AddMonoidHom.toNatLinearMap x\u271d\u00b9 + AddMonoidHom.toNatLinearMap y\u271d) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\n\u22a2 \u2200 (r : R) (x : A \u2192+ B),\n    AddHom.toFun\n        { toFun := AddMonoidHom.toNatLinearMap,\n          map_add' :=\n            (_ :\n              \u2200 (x y : A \u2192+ B),\n                AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n        (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022\n        AddHom.toFun\n          { toFun := AddMonoidHom.toNatLinearMap,\n            map_add' :=\n              (_ :\n                \u2200 (x y : A \u2192+ B),\n                  AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n          x\n[PROOFSTEP]\nintros\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nr\u271d : R\nx\u271d : A \u2192+ B\n\u22a2 AddHom.toFun\n      { toFun := AddMonoidHom.toNatLinearMap,\n        map_add' :=\n          (_ :\n            \u2200 (x y : A \u2192+ B),\n              AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n      (r\u271d \u2022 x\u271d) =\n    \u2191(RingHom.id R) r\u271d \u2022\n      AddHom.toFun\n        { toFun := AddMonoidHom.toNatLinearMap,\n          map_add' :=\n            (_ :\n              \u2200 (x y : A \u2192+ B),\n                AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n        x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nr\u271d : R\nx\u271d\u00b9 : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(AddHom.toFun\n          { toFun := AddMonoidHom.toNatLinearMap,\n            map_add' :=\n              (_ :\n                \u2200 (x y : A \u2192+ B),\n                  AddMonoidHom.toNatLinearMap (x + y) = AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n          (r\u271d \u2022 x\u271d\u00b9))\n      x\u271d =\n    \u2191(\u2191(RingHom.id R) r\u271d \u2022\n          AddHom.toFun\n            { toFun := AddMonoidHom.toNatLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toNatLinearMap (x + y) =\n                      AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n            x\u271d\u00b9)\n      x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\n\u22a2 LeftInverse LinearMap.toAddMonoidHom\n    {\n          toAddHom :=\n            { toFun := AddMonoidHom.toNatLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toNatLinearMap (x + y) =\n                      AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toNatLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toNatLinearMap (x + y) =\n                              AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toNatLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toNatLinearMap (x + y) =\n                                AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                      x) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nf : A \u2192+ B\n\u22a2 LinearMap.toAddMonoidHom\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := AddMonoidHom.toNatLinearMap,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A \u2192+ B),\n                      AddMonoidHom.toNatLinearMap (x + y) =\n                        AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : A \u2192+ B),\n                  AddHom.toFun\n                      { toFun := AddMonoidHom.toNatLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toNatLinearMap (x + y) =\n                                AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                      (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := AddMonoidHom.toNatLinearMap,\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : A \u2192+ B),\n                                AddMonoidHom.toNatLinearMap (x + y) =\n                                  AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                        x) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nf : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(LinearMap.toAddMonoidHom\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := AddMonoidHom.toNatLinearMap,\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : A \u2192+ B),\n                          AddMonoidHom.toNatLinearMap (x + y) =\n                            AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : A \u2192+ B),\n                      AddHom.toFun\n                          { toFun := AddMonoidHom.toNatLinearMap,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : A \u2192+ B),\n                                  AddMonoidHom.toNatLinearMap (x + y) =\n                                    AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                          (r \u2022 x) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := AddMonoidHom.toNatLinearMap,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : A \u2192+ B),\n                                    AddMonoidHom.toNatLinearMap (x + y) =\n                                      AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                            x) }.toAddHom\n            f))\n      x\u271d =\n    \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\n\u22a2 Function.RightInverse LinearMap.toAddMonoidHom\n    {\n          toAddHom :=\n            { toFun := AddMonoidHom.toNatLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toNatLinearMap (x + y) =\n                      AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toNatLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toNatLinearMap (x + y) =\n                              AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toNatLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toNatLinearMap (x + y) =\n                                AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                      x) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nf : A \u2192\u2097[\u2115] B\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := AddMonoidHom.toNatLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toNatLinearMap (x + y) =\n                      AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toNatLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toNatLinearMap (x + y) =\n                              AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toNatLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toNatLinearMap (x + y) =\n                                AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                      x) }.toAddHom\n      (LinearMap.toAddMonoidHom f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommMonoid A\ninst\u271d\u00b9 : AddCommMonoid B\ninst\u271d : Module R B\nf : A \u2192\u2097[\u2115] B\nx\u271d : A\n\u22a2 \u2191(AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := AddMonoidHom.toNatLinearMap,\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : A \u2192+ B),\n                        AddMonoidHom.toNatLinearMap (x + y) =\n                          AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : A \u2192+ B),\n                    AddHom.toFun\n                        { toFun := AddMonoidHom.toNatLinearMap,\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : A \u2192+ B),\n                                AddMonoidHom.toNatLinearMap (x + y) =\n                                  AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                        (r \u2022 x) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := AddMonoidHom.toNatLinearMap,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : A \u2192+ B),\n                                  AddMonoidHom.toNatLinearMap (x + y) =\n                                    AddMonoidHom.toNatLinearMap x + AddMonoidHom.toNatLinearMap y) }\n                          x) }.toAddHom\n          (LinearMap.toAddMonoidHom f))\n      x\u271d =\n    \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\n\u22a2 \u2200 (x y : A \u2192+ B), AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y\n[PROOFSTEP]\nintros\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nx\u271d y\u271d : A \u2192+ B\n\u22a2 AddMonoidHom.toIntLinearMap (x\u271d + y\u271d) = AddMonoidHom.toIntLinearMap x\u271d + AddMonoidHom.toIntLinearMap y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nx\u271d\u00b9 y\u271d : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(AddMonoidHom.toIntLinearMap (x\u271d\u00b9 + y\u271d)) x\u271d = \u2191(AddMonoidHom.toIntLinearMap x\u271d\u00b9 + AddMonoidHom.toIntLinearMap y\u271d) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\n\u22a2 \u2200 (r : R) (x : A \u2192+ B),\n    AddHom.toFun\n        { toFun := AddMonoidHom.toIntLinearMap,\n          map_add' :=\n            (_ :\n              \u2200 (x y : A \u2192+ B),\n                AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n        (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022\n        AddHom.toFun\n          { toFun := AddMonoidHom.toIntLinearMap,\n            map_add' :=\n              (_ :\n                \u2200 (x y : A \u2192+ B),\n                  AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n          x\n[PROOFSTEP]\nintros\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nr\u271d : R\nx\u271d : A \u2192+ B\n\u22a2 AddHom.toFun\n      { toFun := AddMonoidHom.toIntLinearMap,\n        map_add' :=\n          (_ :\n            \u2200 (x y : A \u2192+ B),\n              AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n      (r\u271d \u2022 x\u271d) =\n    \u2191(RingHom.id R) r\u271d \u2022\n      AddHom.toFun\n        { toFun := AddMonoidHom.toIntLinearMap,\n          map_add' :=\n            (_ :\n              \u2200 (x y : A \u2192+ B),\n                AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n        x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nr\u271d : R\nx\u271d\u00b9 : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(AddHom.toFun\n          { toFun := AddMonoidHom.toIntLinearMap,\n            map_add' :=\n              (_ :\n                \u2200 (x y : A \u2192+ B),\n                  AddMonoidHom.toIntLinearMap (x + y) = AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n          (r\u271d \u2022 x\u271d\u00b9))\n      x\u271d =\n    \u2191(\u2191(RingHom.id R) r\u271d \u2022\n          AddHom.toFun\n            { toFun := AddMonoidHom.toIntLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toIntLinearMap (x + y) =\n                      AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n            x\u271d\u00b9)\n      x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\n\u22a2 LeftInverse LinearMap.toAddMonoidHom\n    {\n          toAddHom :=\n            { toFun := AddMonoidHom.toIntLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toIntLinearMap (x + y) =\n                      AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toIntLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toIntLinearMap (x + y) =\n                              AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toIntLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toIntLinearMap (x + y) =\n                                AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                      x) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nf : A \u2192+ B\n\u22a2 LinearMap.toAddMonoidHom\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := AddMonoidHom.toIntLinearMap,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A \u2192+ B),\n                      AddMonoidHom.toIntLinearMap (x + y) =\n                        AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : A \u2192+ B),\n                  AddHom.toFun\n                      { toFun := AddMonoidHom.toIntLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toIntLinearMap (x + y) =\n                                AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                      (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022\n                      AddHom.toFun\n                        { toFun := AddMonoidHom.toIntLinearMap,\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : A \u2192+ B),\n                                AddMonoidHom.toIntLinearMap (x + y) =\n                                  AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                        x) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nf : A \u2192+ B\nx\u271d : A\n\u22a2 \u2191(LinearMap.toAddMonoidHom\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := AddMonoidHom.toIntLinearMap,\n                    map_add' :=\n                      (_ :\n                        \u2200 (x y : A \u2192+ B),\n                          AddMonoidHom.toIntLinearMap (x + y) =\n                            AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : A \u2192+ B),\n                      AddHom.toFun\n                          { toFun := AddMonoidHom.toIntLinearMap,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : A \u2192+ B),\n                                  AddMonoidHom.toIntLinearMap (x + y) =\n                                    AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                          (r \u2022 x) =\n                        \u2191(RingHom.id R) r \u2022\n                          AddHom.toFun\n                            { toFun := AddMonoidHom.toIntLinearMap,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (x y : A \u2192+ B),\n                                    AddMonoidHom.toIntLinearMap (x + y) =\n                                      AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                            x) }.toAddHom\n            f))\n      x\u271d =\n    \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\n\u22a2 Function.RightInverse LinearMap.toAddMonoidHom\n    {\n          toAddHom :=\n            { toFun := AddMonoidHom.toIntLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toIntLinearMap (x + y) =\n                      AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toIntLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toIntLinearMap (x + y) =\n                              AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toIntLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toIntLinearMap (x + y) =\n                                AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                      x) }.toAddHom.toFun\n[PROOFSTEP]\nintro f\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nf : A \u2192\u2097[\u2124] B\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := AddMonoidHom.toIntLinearMap,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A \u2192+ B),\n                    AddMonoidHom.toIntLinearMap (x + y) =\n                      AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : A \u2192+ B),\n                AddHom.toFun\n                    { toFun := AddMonoidHom.toIntLinearMap,\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : A \u2192+ B),\n                            AddMonoidHom.toIntLinearMap (x + y) =\n                              AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                    (r \u2022 x) =\n                  \u2191(RingHom.id R) r \u2022\n                    AddHom.toFun\n                      { toFun := AddMonoidHom.toIntLinearMap,\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : A \u2192+ B),\n                              AddMonoidHom.toIntLinearMap (x + y) =\n                                AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                      x) }.toAddHom\n      (LinearMap.toAddMonoidHom f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nA : Type u_20\nB : Type u_21\nR : Type u_22\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : AddCommGroup A\ninst\u271d\u00b9 : AddCommGroup B\ninst\u271d : Module R B\nf : A \u2192\u2097[\u2124] B\nx\u271d : A\n\u22a2 \u2191(AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := AddMonoidHom.toIntLinearMap,\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : A \u2192+ B),\n                        AddMonoidHom.toIntLinearMap (x + y) =\n                          AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : A \u2192+ B),\n                    AddHom.toFun\n                        { toFun := AddMonoidHom.toIntLinearMap,\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : A \u2192+ B),\n                                AddMonoidHom.toIntLinearMap (x + y) =\n                                  AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                        (r \u2022 x) =\n                      \u2191(RingHom.id R) r \u2022\n                        AddHom.toFun\n                          { toFun := AddMonoidHom.toIntLinearMap,\n                            map_add' :=\n                              (_ :\n                                \u2200 (x y : A \u2192+ B),\n                                  AddMonoidHom.toIntLinearMap (x + y) =\n                                    AddMonoidHom.toIntLinearMap x + AddMonoidHom.toIntLinearMap y) }\n                          x) }.toAddHom\n          (LinearMap.toAddMonoidHom f))\n      x\u271d =\n    \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq\u271d q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\np q : Submodule R M\nh : p \u2264 q\n\u22a2 LinearMap.comp (Submodule.subtype q) (ofLe h) = Submodule.subtype p\n[PROOFSTEP]\next \u27e8b, hb\u27e9\n[GOAL]\ncase h.mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq\u271d q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\np q : Submodule R M\nh : p \u2264 q\nb : M\nhb : b \u2208 p\n\u22a2 \u2191(LinearMap.comp (Submodule.subtype q) (ofLe h)) { val := b, property := hb } =\n    \u2191(Submodule.subtype p) { val := b, property := hb }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\n\u22a2 Subsingleton (Submodule R M) \u2194 Subsingleton (AddSubmonoid M)\n[PROOFSTEP]\nrw [\u2190 subsingleton_iff_bot_eq_top, \u2190 subsingleton_iff_bot_eq_top, \u2190 toAddSubmonoid_eq]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\n\u22a2 \u22a5.toAddSubmonoid = \u22a4.toAddSubmonoid \u2194 \u22a5 = \u22a4\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : Subsingleton R\n\u22a2 Unique (Submodule R M)\n[PROOFSTEP]\nhaveI := Module.subsingleton R M\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : Subsingleton R\nthis : Subsingleton M\n\u22a2 Unique (Submodule R M)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nsrc\u271d : AddSubmonoid M\u2082 := AddSubmonoid.map f p.toAddSubmonoid\n\u22a2 \u2200 (c : R\u2082) {x : M\u2082},\n    x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := \u2191f '' \u2191p,\n                  add_mem' := (_ : \u2200 {a b : M\u2082}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n              zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier \u2192\n      c \u2022 x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := \u2191f '' \u2191p,\n                  add_mem' := (_ : \u2200 {a b : M\u2082}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n              zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro c x \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y\u271d : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nsrc\u271d : AddSubmonoid M\u2082 := AddSubmonoid.map f p.toAddSubmonoid\nc : R\u2082\ny : M\nhy : y \u2208 \u2191p\n\u22a2 c \u2022 \u2191f y \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f '' \u2191p,\n              add_mem' := (_ : \u2200 {a b : M\u2082}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n          zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := \u03c3\u2081\u2082.surjective c\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y\u271d : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nsrc\u271d : AddSubmonoid M\u2082 := AddSubmonoid.map f p.toAddSubmonoid\ny : M\nhy : y \u2208 \u2191p\na : R\n\u22a2 \u2191\u03c3\u2081\u2082 a \u2022 \u2191f y \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f '' \u2191p,\n              add_mem' := (_ : \u2200 {a b : M\u2082}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n          zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact \u27e8_, p.smul_mem a hy, map_smul\u209b\u2097 f _ _\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\na : M\n\u22a2 a \u2208 map LinearMap.id p \u2194 a \u2208 p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : Semiring R\u2082\ninst\u271d\u00b9\u2074 : Semiring R\u2083\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R M'\ninst\u271d\u2077 : Module R\u2082 M\u2082\ninst\u271d\u2076 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u2075 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u2074 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b3 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d\u00b2 : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\u2082\u2083\ninst\u271d : RingHomSurjective \u03c3\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np : Submodule R M\n\u22a2 \u2191(map (LinearMap.comp g f) p) = \u2191(map g (map f p))\n[PROOFSTEP]\nsimp only [\u2190 image_comp, map_coe, LinearMap.coe_comp, comp_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nthis : \u2203 x, x \u2208 p\n\u22a2 \u2200 (x : M\u2082), x \u2208 map 0 p \u2194 x \u2208 \u22a5\n[PROOFSTEP]\nsimp [this, eq_comm]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf g : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\n\u22a2 map (f + g) p \u2264 map f p \u2294 map g p\n[PROOFSTEP]\nrintro x \u27e8m, hm, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf g : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nm : M\nhm : m \u2208 \u2191p\n\u22a2 \u2191(f + g) m \u2208 map f p \u2294 map g p\n[PROOFSTEP]\nexact add_mem_sup (mem_map_of_mem hm) (mem_map_of_mem hm)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\n\u22a2 \u2200 (x y : { x // x \u2208 p }), Equiv.toFun src\u271d (x + y) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d y\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\nx\u271d y\u271d : { x // x \u2208 p }\n\u22a2 Equiv.toFun src\u271d (x\u271d + y\u271d) = Equiv.toFun src\u271d x\u271d + Equiv.toFun src\u271d y\u271d\n[PROOFSTEP]\nsimp only [coe_add, map_add, Equiv.toFun_as_coe, Equiv.Set.image_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\nx\u271d y\u271d : { x // x \u2208 p }\n\u22a2 { val := \u2191f \u2191x\u271d + \u2191f \u2191y\u271d, property := (_ : (fun x => x \u2208 \u2191f '' \u2191p) (\u2191f \u2191x\u271d + \u2191f \u2191y\u271d)) } =\n    { val := \u2191f \u2191x\u271d, property := (_ : \u2191f \u2191x\u271d \u2208 \u2191f '' \u2191p) } + { val := \u2191f \u2191y\u271d, property := (_ : \u2191f \u2191y\u271d \u2208 \u2191f '' \u2191p) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\n\u22a2 \u2200 (r : R) (x : { x // x \u2208 p }),\n    AddHom.toFun\n        { toFun := src\u271d.toFun,\n          map_add' :=\n            (_ : \u2200 (x y : { x // x \u2208 p }), Equiv.toFun src\u271d (x + y) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d y) }\n        (r \u2022 x) =\n      \u2191\u03c3\u2081\u2082 r \u2022\n        AddHom.toFun\n          { toFun := src\u271d.toFun,\n            map_add' :=\n              (_ : \u2200 (x y : { x // x \u2208 p }), Equiv.toFun src\u271d (x + y) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d y) }\n          x\n[PROOFSTEP]\nintros\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\nr\u271d : R\nx\u271d : { x // x \u2208 p }\n\u22a2 AddHom.toFun\n      { toFun := src\u271d.toFun,\n        map_add' := (_ : \u2200 (x y : { x // x \u2208 p }), Equiv.toFun src\u271d (x + y) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d y) }\n      (r\u271d \u2022 x\u271d) =\n    \u2191\u03c3\u2081\u2082 r\u271d \u2022\n      AddHom.toFun\n        { toFun := src\u271d.toFun,\n          map_add' :=\n            (_ : \u2200 (x y : { x // x \u2208 p }), Equiv.toFun src\u271d (x + y) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d y) }\n        x\u271d\n[PROOFSTEP]\nsimp only [coe_smul_of_tower, map_smul\u209b\u2097, Equiv.toFun_as_coe, Equiv.Set.image_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\ni : Injective \u2191f\np : Submodule R M\nsrc\u271d : \u2191\u2191p \u2243 \u2191(\u2191f '' \u2191p) := Equiv.Set.image (\u2191f) (\u2191p) i\nr\u271d : R\nx\u271d : { x // x \u2208 p }\n\u22a2 { val := \u2191\u03c3\u2081\u2082 r\u271d \u2022 \u2191f \u2191x\u271d, property := (_ : (fun x => x \u2208 \u2191f '' \u2191p) (\u2191\u03c3\u2081\u2082 r\u271d \u2022 \u2191f \u2191x\u271d)) } =\n    \u2191\u03c3\u2081\u2082 r\u271d \u2022 { val := \u2191f \u2191x\u271d, property := (_ : \u2191f \u2191x\u271d \u2208 \u2191f '' \u2191p) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\np : Submodule R\u2082 M\u2082\nsrc\u271d : AddSubmonoid M := AddSubmonoid.comap f p.toAddSubmonoid\na : R\nx : M\nh :\n  x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f \u207b\u00b9' \u2191p,\n              add_mem' := (_ : \u2200 {a b : M}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n          zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n\u22a2 a \u2022 x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f \u207b\u00b9' \u2191p,\n              add_mem' := (_ : \u2200 {a b : M}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a + b \u2208 src\u271d.carrier) },\n          zero_mem' := (_ : 0 \u2208 src\u271d.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimp [p.smul_mem (\u03c3\u2081\u2082 a) h]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\np : Submodule R M\nf : M \u2192\u2097[R] M\nh : p \u2264 comap f p\nk : \u2115\n\u22a2 p \u2264 comap (f ^ k) p\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\np : Submodule R M\nf : M \u2192\u2097[R] M\nh : p \u2264 comap f p\n\u22a2 p \u2264 comap (f ^ Nat.zero) p\n[PROOFSTEP]\nsimp [LinearMap.one_eq_id]\n[GOAL]\ncase succ\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\np : Submodule R M\nf : M \u2192\u2097[R] M\nh : p \u2264 comap f p\nk : \u2115\nih : p \u2264 comap (f ^ k) p\n\u22a2 p \u2264 comap (f ^ Nat.succ k) p\n[PROOFSTEP]\nsimp [LinearMap.iterate_succ, comap_comp, h.trans (comap_mono ih)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u22a2 \u2200 (x : M), x \u2208 comap 0 q \u2194 x \u2208 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\nhf : Surjective \u2191f\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nS : Submodule R\u2082 M\u2082\nx : M\u2082\nhx : x \u2208 S\n\u22a2 x \u2208 map f (comap f S)\n[PROOFSTEP]\nrcases hf x with \u27e8y, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y\u271d : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\nhf : Surjective \u2191f\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nS : Submodule R\u2082 M\u2082\ny : M\nhx : \u2191f y \u2208 S\n\u22a2 \u2191f y \u2208 map f (comap f S)\n[PROOFSTEP]\nsimp only [mem_map, mem_comap]\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y\u271d : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nf : F\nhf : Surjective \u2191f\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nS : Submodule R\u2082 M\u2082\ny : M\nhx : \u2191f y \u2208 S\n\u22a2 \u2203 y_1, \u2191f y_1 \u2208 S \u2227 \u2191f y_1 = \u2191f y\n[PROOFSTEP]\nexact \u27e8y, hx, rfl\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nhf : Injective \u2191f\nS : Submodule R M\nx : M\n\u22a2 x \u2208 comap f (map f S) \u2192 x \u2208 S\n[PROOFSTEP]\nsimp [mem_comap, mem_map, forall_exists_index, and_imp]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nhf : Injective \u2191f\nS : Submodule R M\nx : M\n\u22a2 \u2200 (x_1 : M), x_1 \u2208 S \u2192 \u2191f x_1 = \u2191f x \u2192 x \u2208 S\n[PROOFSTEP]\nintro y hy hxy\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y\u271d : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nhf : Injective \u2191f\nS : Submodule R M\nx y : M\nhy : y \u2208 S\nhxy : \u2191f y = \u2191f x\n\u22a2 x \u2208 S\n[PROOFSTEP]\nrw [hf.eq_iff] at hxy \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y\u271d : M\nF : Type u_20\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\nhf : Injective \u2191f\nS : Submodule R M\nx y : M\nhy : y \u2208 S\nhxy : y = x\n\u22a2 x \u2208 S\n[PROOFSTEP]\nrwa [\u2190 hxy]\n[GOAL]\nF : Type u_20\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p'\u271d : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\np : Submodule R M\np' : Submodule R\u2082 M\u2082\n\u22a2 map f p \u2293 p' \u2264 map f (p \u2293 comap f p')\n[PROOFSTEP]\nrintro _ \u27e8\u27e8x, h\u2081, rfl\u27e9, h\u2082\u27e9\n[GOAL]\ncase intro.intro.intro\nF : Type u_20\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p'\u271d : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c3\u2081\u2082\nf : F\np : Submodule R M\np' : Submodule R\u2082 M\u2082\nx : M\nh\u2081 : x \u2208 \u2191p\nh\u2082 : \u2191f x \u2208 \u2191p'\n\u22a2 \u2191f x \u2208 map f (p \u2293 comap f p')\n[PROOFSTEP]\nexact \u27e8_, \u27e8h\u2081, h\u2082\u27e9, rfl\u27e9\n[GOAL]\nF : Type ?u.972060\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx\u271d y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nx : M\n\u22a2 x \u2208 map (Submodule.subtype p) (comap (Submodule.subtype p) p') \u2192 x \u2208 p \u2293 p'\n[PROOFSTEP]\nrintro \u27e8\u27e8_, h\u2081\u27e9, h\u2082, rfl\u27e9\n[GOAL]\ncase intro.mk.intro\nF : Type ?u.972060\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b3 : Semiring R\ninst\u271d\u00b9\u00b2 : Semiring R\u2082\ninst\u271d\u00b9\u00b9 : Semiring R\u2083\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2083\ninst\u271d\u2077 : AddCommMonoid M'\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\nval\u271d : M\nh\u2081 : val\u271d \u2208 p\nh\u2082 : { val := val\u271d, property := h\u2081 } \u2208 \u2191(comap (Submodule.subtype p) p')\n\u22a2 \u2191(Submodule.subtype p) { val := val\u271d, property := h\u2081 } \u2208 p \u2293 p'\n[PROOFSTEP]\nexact \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\n\u22a2 \u2200 (v : M), v \u2208 iInf p \u2192 \u2191f v \u2208 iInf p\n[PROOFSTEP]\nhave : \u2200 i, (p i).map f \u2264 p i := by\n  rintro i - \u27e8v, hv, rfl\u27e9\n  exact hf i v hv\n[GOAL]\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\n\u22a2 \u2200 (i : \u03b9), map f (p i) \u2264 p i\n[PROOFSTEP]\nrintro i - \u27e8v, hv, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\ni : \u03b9\nv : M\nhv : v \u2208 \u2191(p i)\n\u22a2 \u2191f v \u2208 p i\n[PROOFSTEP]\nexact hf i v hv\n[GOAL]\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\nthis : \u2200 (i : \u03b9), map f (p i) \u2264 p i\n\u22a2 \u2200 (v : M), v \u2208 iInf p \u2192 \u2191f v \u2208 iInf p\n[PROOFSTEP]\nsuffices (iInf p).map f \u2264 iInf p by exact fun v hv => this \u27e8v, hv, rfl\u27e9\n[GOAL]\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\nthis\u271d : \u2200 (i : \u03b9), map f (p i) \u2264 p i\nthis : map f (iInf p) \u2264 iInf p\n\u22a2 \u2200 (v : M), v \u2208 iInf p \u2192 \u2191f v \u2208 iInf p\n[PROOFSTEP]\nexact fun v hv => this \u27e8v, hv, rfl\u27e9\n[GOAL]\nF : Type ?u.976655\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9\u271d : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2074 : Semiring R\ninst\u271d\u00b9\u00b3 : Semiring R\u2082\ninst\u271d\u00b9\u00b2 : Semiring R\u2083\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R M'\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq q' : Submodule R\u2082 M\u2082\nq\u2081 q\u2081' : Submodule R M'\nr : R\nx y : M\nsc : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\n\u03c3 : R \u2192+* R\ninst\u271d : RingHomSurjective \u03c3\n\u03b9 : Sort u_20\nf : M \u2192\u209b\u2097[\u03c3] M\np : \u03b9 \u2192 Submodule R M\nhf : \u2200 (i : \u03b9) (v : M), v \u2208 p i \u2192 \u2191f v \u2208 p i\nthis : \u2200 (i : \u03b9), map f (p i) \u2264 p i\n\u22a2 map f (iInf p) \u2264 iInf p\n[PROOFSTEP]\nexact le_iInf fun i => (Submodule.map_mono (iInf_le p i)).trans (this i)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\nh : a \u2260 0\n\u22a2 comap (a \u2022 f) p = comap f p\n[PROOFSTEP]\next b\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\nh : a \u2260 0\nb : V\n\u22a2 b \u2208 comap (a \u2022 f) p \u2194 b \u2208 comap f p\n[PROOFSTEP]\nsimp only [Submodule.mem_comap, p.smul_mem_iff h, LinearMap.smul_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\nh : a \u2260 0\n\u22a2 map (a \u2022 f) p \u2264 map f p\n[PROOFSTEP]\nrw [map_le_iff_le_comap, comap_smul f _ a h, \u2190 map_le_iff_le_comap]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\nh : a \u2260 0\n\u22a2 map f p \u2264 map (a \u2022 f) p\n[PROOFSTEP]\nrw [map_le_iff_le_comap, \u2190 comap_smul f _ a h, \u2190 map_le_iff_le_comap]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p\n[PROOFSTEP]\nclassical by_cases h : a = 0 <;> simp [h, comap_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\nh : a = 0\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p\n[PROOFSTEP]\nsimp [h, comap_smul]\n[GOAL]\ncase neg\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\u2082\na : K\nh : \u00aca = 0\n\u22a2 comap (a \u2022 f) p = \u2a05 (_ : a \u2260 0), comap f p\n[PROOFSTEP]\nsimp [h, comap_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\n\u22a2 map (a \u2022 f) p = \u2a06 (_ : a \u2260 0), map f p\n[PROOFSTEP]\nclassical by_cases h : a = 0 <;> simp [h, Submodule.map_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\n\u22a2 map (a \u2022 f) p = \u2a06 (_ : a \u2260 0), map f p\n[PROOFSTEP]\nby_cases h : a = 0\n[GOAL]\ncase pos\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\nh : a = 0\n\u22a2 map (a \u2022 f) p = \u2a06 (_ : a \u2260 0), map f p\n[PROOFSTEP]\nsimp [h, Submodule.map_smul]\n[GOAL]\ncase neg\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semifield K\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\np : Submodule K V\na : K\nh : \u00aca = 0\n\u22a2 map (a \u2022 f) p = \u2a06 (_ : a \u2260 0), map f p\n[PROOFSTEP]\nsimp [h, Submodule.map_smul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\ninst\u271d : RingHomSurjective \u03c3\u2082\u2081\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\nh : \u2200 (c : M\u2082), \u2191f c \u2208 p\np' : Submodule R\u2082 M\u2082\nx\u271d : { x // x \u2208 p }\nx : M\nhx : x \u2208 p\n\u22a2 { val := x, property := hx } \u2208 map (codRestrict p f h) p' \u2194\n    { val := x, property := hx } \u2208 comap (Submodule.subtype p) (map f p')\n[PROOFSTEP]\nsimp [Subtype.ext_iff_val]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\nhf : \u2200 (c : M\u2082), \u2191f c \u2208 p\np' : Submodule R { x // x \u2208 p }\nx : M\u2082\n\u22a2 x \u2208 comap f (map (Submodule.subtype p) p') \u2192 x \u2208 comap (codRestrict p f hf) p'\n[PROOFSTEP]\nrintro \u27e8\u27e8_, _\u27e9, h, \u27e8\u27e9\u27e9\n[GOAL]\ncase intro.mk.intro.refl\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\nhf : \u2200 (c : M\u2082), \u2191f c \u2208 p\np' : Submodule R { x // x \u2208 p }\nx : M\u2082\nproperty\u271d : f.toAddHom.1 x \u2208 p\nh : { val := f.toAddHom.1 x, property := property\u271d } \u2208 \u2191p'\n\u22a2 x \u2208 comap (codRestrict p f hf) p'\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\n\u22a2 range f = map f \u22a4\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nx\u271d : M\u2082\n\u22a2 x\u271d \u2208 range f \u2194 x\u271d \u2208 map f \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\n\u22a2 range f = \u22a4 \u2194 Surjective \u2191f\n[PROOFSTEP]\nrw [SetLike.ext'_iff, range_coe, top_coe, Set.range_iff_surjective]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\np : Submodule R\u2082 M\u2082\n\u22a2 range f \u2264 p \u2194 comap f p = \u22a4\n[PROOFSTEP]\nrw [range_eq_map, map_le_iff_le_comap, eq_top_iff]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082\u271d : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\u271d\ninst\u271d\u00b9\u2076 : Semiring R\u2082\u271d\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\u271d\n\u03c3\u2082\u2083 : R\u2082\u271d \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u271d \u2192+* R\u2083\ninst\u271d\u00b9\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2070 : Module R\u271d M\u271d\ninst\u271d\u2079 : Module R\u2082\u271d M\u2082\u271d\ninst\u271d\u2078 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082\u271d \u2192+* R\u271d\n\u03c4\u2081\u2082\u271d : R\u271d \u2192+* R\u2082\u271d\n\u03c4\u2082\u2083 : R\u2082\u271d \u2192+* R\u2083\n\u03c4\u2081\u2083 : R\u271d \u2192+* R\u2083\ninst\u271d\u2077 : RingHomCompTriple \u03c4\u2081\u2082\u271d \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082\u271d M\u271d M\u2082\u271d\nR : Type u_21\nR\u2082 : Type u_22\nM : Type u_23\nM\u2082 : Type u_24\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Ring R\u2082\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 range (-f) = range f\n[PROOFSTEP]\nchange range ((-LinearMap.id : M\u2082 \u2192\u2097[R\u2082] M\u2082).comp f) = _\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082\u271d : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2077 : Semiring R\u271d\ninst\u271d\u00b9\u2076 : Semiring R\u2082\u271d\ninst\u271d\u00b9\u2075 : Semiring R\u2083\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R\u271d \u2192+* R\u2082\u271d\n\u03c3\u2082\u2083 : R\u2082\u271d \u2192+* R\u2083\n\u03c3\u2081\u2083 : R\u271d \u2192+* R\u2083\ninst\u271d\u00b9\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2070 : Module R\u271d M\u271d\ninst\u271d\u2079 : Module R\u2082\u271d M\u2082\u271d\ninst\u271d\u2078 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082\u271d \u2192+* R\u271d\n\u03c4\u2081\u2082\u271d : R\u271d \u2192+* R\u2082\u271d\n\u03c4\u2082\u2083 : R\u2082\u271d \u2192+* R\u2083\n\u03c4\u2081\u2083 : R\u271d \u2192+* R\u2083\ninst\u271d\u2077 : RingHomCompTriple \u03c4\u2081\u2082\u271d \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082\u271d M\u271d M\u2082\u271d\nR : Type u_21\nR\u2082 : Type u_22\nM : Type u_23\nM\u2082 : Type u_24\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Ring R\u2082\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 range (comp (-id) f) = range f\n[PROOFSTEP]\nrw [range_comp, Submodule.map_neg, Submodule.map_id]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf g : F\nsrc\u271d : AddSubmonoid M := AddMonoidHom.eqLocusM \u2191f \u2191g\nr : R\nx : M\nhx : \u2191f x = \u2191g x\n\u22a2 \u2191f (r \u2022 x) = \u2191g (r \u2022 x)\n[PROOFSTEP]\nsimpa only [map_smul\u209b\u2097] using congr_arg ((\u00b7 \u2022 \u00b7) (\u03c4\u2081\u2082 r)) hx\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf g : F\n\u22a2 eqLocus f g = \u22a4 \u2194 f = g\n[PROOFSTEP]\nsimp [SetLike.ext_iff, FunLike.ext_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf g : F\nS T : Submodule R M\nhS : Set.EqOn \u2191f \u2191g \u2191S\nhT : Set.EqOn \u2191f \u2191g \u2191T\n\u22a2 Set.EqOn \u2191f \u2191g \u2191(S \u2294 T)\n[PROOFSTEP]\nrw [\u2190 le_eqLocus] at hS hT \u22a2\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS\u271d : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf g : F\nS T : Submodule R M\nhS : S \u2264 eqLocus f g\nhT : T \u2264 eqLocus f g\n\u22a2 S \u2294 T \u2264 eqLocus f g\n[PROOFSTEP]\nexact sup_le hS hT\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn m : \u2115\nw : n \u2264 m\nx : M\nh : x \u2208 (fun n => range (f ^ n)) m\n\u22a2 x \u2208 (fun n => range (f ^ n)) n\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := le_iff_exists_add.mp w\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn : \u2115\nx : M\nc : \u2115\nw : n \u2264 n + c\nh : x \u2208 (fun n => range (f ^ n)) (n + c)\n\u22a2 x \u2208 (fun n => range (f ^ n)) n\n[PROOFSTEP]\nrw [LinearMap.mem_range] at h \n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn : \u2115\nx : M\nc : \u2115\nw : n \u2264 n + c\nh : \u2203 y, \u2191(f ^ (n + c)) y = x\n\u22a2 x \u2208 (fun n => range (f ^ n)) n\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := h\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn c : \u2115\nw : n \u2264 n + c\nm : M\n\u22a2 \u2191(f ^ (n + c)) m \u2208 (fun n => range (f ^ n)) n\n[PROOFSTEP]\nrw [LinearMap.mem_range]\n[GOAL]\ncase intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn c : \u2115\nw : n \u2264 n + c\nm : M\n\u22a2 \u2203 y, \u2191(f ^ n) y = \u2191(f ^ (n + c)) m\n[PROOFSTEP]\nuse(f ^ c) m\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u2097[R] M\nn c : \u2115\nw : n \u2264 n + c\nm : M\n\u22a2 \u2191(f ^ n) (\u2191(f ^ c) m) = \u2191(f ^ (n + c)) m\n[PROOFSTEP]\nrw [pow_add, LinearMap.mul_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\n\u22a2 ker f \u2264 ker (comp g f)\n[PROOFSTEP]\nrw [ker_comp]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\n\u22a2 ker f \u2264 comap f (ker g)\n[PROOFSTEP]\nexact comap_mono bot_le\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\n\u22a2 Disjoint p (ker f) \u2194 \u2200 (x : M), x \u2208 p \u2192 \u2191f x = 0 \u2192 x = 0\n[PROOFSTEP]\nsimp [disjoint_def]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\n\u22a2 ker f = \u22a5 \u2194 \u2200 (m : M), \u2191f m = 0 \u2192 m = 0\n[PROOFSTEP]\nsimpa [disjoint_iff_inf_le] using @disjoint_ker _ _ _ _ _ _ _ _ _ _ _ _ _ f \u22a4\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2081] M\nh : comp g f = id\nm : M\nhm : \u2191f m = 0\n\u22a2 m = 0\n[PROOFSTEP]\nrw [\u2190 id_apply (R := R) m, \u2190 h, comp_apply, hm, g.map_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\np : Submodule R M\n\u22a2 p \u2264 ker f \u2194 map f p = \u22a5\n[PROOFSTEP]\nrw [ker, eq_bot_iff, map_le_iff_le_comap]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2081] M\nhf : \u2200 (c : M\u2082), \u2191f c \u2208 p\n\u22a2 ker (codRestrict p f hf) = ker f\n[PROOFSTEP]\nrw [ker, comap_codRestrict, Submodule.map_bot]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2081] M\nhf : \u2200 (c : M\u2082), \u2191f c \u2208 p\n\u22a2 comap f \u22a5 = ker f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d : RingHomSurjective \u03c4\u2082\u2081\np : Submodule R M\nf : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2081] M\nhf : \u2200 (c : M\u2082), \u2191f c \u2208 p\n\u22a2 range (codRestrict p f hf) = comap (Submodule.subtype p) (range f)\n[PROOFSTEP]\nsimpa only [range_eq_map] using map_codRestrict _ _ _ _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : Semiring R\u2082\ninst\u271d\u00b9\u2070 : Semiring R\u2083\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2076 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2081\ninst\u271d : Module R M\u2081\np : Submodule R M\nq : Submodule R M\u2081\nf : M \u2192\u2097[R] M\u2081\nhf : \u2200 (x : M), x \u2208 p \u2192 \u2191f x \u2208 q\n\u22a2 ker (restrict f hf) = ker (domRestrict f p)\n[PROOFSTEP]\nrw [restrict_eq_codRestrict_domRestrict, ker_codRestrict]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\n\u22a2 range f \u2293 q \u2264 map f (comap f q)\n[PROOFSTEP]\nrintro _ \u27e8\u27e8x, _, rfl\u27e9, hx\u27e9\n[GOAL]\ncase intro.intro.refl\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\nx : M\nhx : \u2191f x \u2208 \u2191q\n\u22a2 \u2191f x \u2208 map f (comap f q)\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nq : Submodule R\u2082 M\u2082\nh : q \u2264 range f\n\u22a2 map f (comap f q) = q\n[PROOFSTEP]\nrwa [Submodule.map_comap_eq, inf_eq_right]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nx : M\n\u22a2 x \u2208 ker 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\n\u22a2 range 0 = \u22a5\n[PROOFSTEP]\nsimpa only [range_eq_map] using Submodule.map_zero _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 range f \u2264 \u22a5 \u2194 f = 0\n[PROOFSTEP]\nrw [range_le_iff_comap]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 comap f \u22a5 = \u22a4 \u2194 f = 0\n[PROOFSTEP]\nexact ker_eq_top\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 range f = \u22a5 \u2194 f = 0\n[PROOFSTEP]\nrw [\u2190 range_le_bot_iff, le_bot_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\nh : comp g f = 0\nx : M\u2082\nhx : x \u2208 range f\ny : M\nhy : \u2191f y = x\n\u22a2 \u2191g x = 0\n[PROOFSTEP]\nrw [\u2190 hy, \u2190 comp_apply, h, zero_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nhf : range f = \u22a4\np p' : Submodule R\u2082 M\u2082\nH : comap f p \u2264 comap f p'\nx : M\u2082\nhx : x \u2208 p\n\u22a2 x \u2208 p'\n[PROOFSTEP]\nrcases range_eq_top.1 hf x with \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.refl\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\nf : F\nhf : range f = \u22a4\np p' : Submodule R\u2082 M\u2082\nH : comap f p \u2264 comap f p'\ny : M\nhx : \u2191f y \u2208 p\n\u22a2 \u2191f y \u2208 p'\n[PROOFSTEP]\nexact H hx\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nhave : Disjoint \u22a4 (ker f) := by\n  -- Porting note: `\u2190 map_zero f` should work here, but it needs to be directly applied to H.\n  rw [disjoint_ker]\n  intros _ _ H\n  rw [\u2190 map_zero f] at H \n  exact hf H\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\n\u22a2 Disjoint \u22a4 (ker f)\n[PROOFSTEP]\nrw [disjoint_ker]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\n\u22a2 \u2200 (x : M), x \u2208 \u22a4 \u2192 \u2191f x = 0 \u2192 x = 0\n[PROOFSTEP]\nintros _ _ H\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\nx\u271d : M\na\u271d : x\u271d \u2208 \u22a4\nH : \u2191f x\u271d = 0\n\u22a2 x\u271d = 0\n[PROOFSTEP]\nrw [\u2190 map_zero f] at H \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\nx\u271d : M\na\u271d : x\u271d \u2208 \u22a4\nH : \u2191f x\u271d = \u2191f 0\n\u22a2 x\u271d = 0\n[PROOFSTEP]\nexact hf H\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\nthis : Disjoint \u22a4 (ker f)\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, top_inf_eq, le_bot_iff] at this \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nhf : Injective \u2191f\nthis : ker f = \u22a5\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u2097[R] M\nn m : \u2115\nw : n \u2264 m\nx : M\nh : x \u2208 (fun n => ker (f ^ n)) n\n\u22a2 x \u2208 (fun n => ker (f ^ n)) m\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := le_iff_exists_add.mp w\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u2097[R] M\nn : \u2115\nx : M\nh : x \u2208 (fun n => ker (f ^ n)) n\nc : \u2115\nw : n \u2264 n + c\n\u22a2 x \u2208 (fun n => ker (f ^ n)) (n + c)\n[PROOFSTEP]\nrw [LinearMap.mem_ker] at h \n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u2074 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : M \u2192\u2097[R] M\nn : \u2115\nx : M\nh : \u2191(f ^ n) x = 0\nc : \u2115\nw : n \u2264 n + c\n\u22a2 x \u2208 (fun n => ker (f ^ n)) (n + c)\n[PROOFSTEP]\nrw [LinearMap.mem_ker, add_comm, pow_add, LinearMap.mul_apply, h, LinearMap.map_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : Ring R\u2082\ninst\u271d\u2077 : Ring R\u2083\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\nx y : M\n\u22a2 x - y \u2208 ker f \u2194 \u2191f x = \u2191f y\n[PROOFSTEP]\nrw [mem_ker, map_sub, sub_eq_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : Ring R\u2082\ninst\u271d\u2077 : Ring R\u2083\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nH : \u2200 (x : M), x \u2208 p \u2192 \u2191f x = 0 \u2192 x = 0\nx : M\nhx : x \u2208 p\ny : M\nhy : y \u2208 p\nh : \u2191f x = \u2191f y\n\u22a2 \u2191f (x - y) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : Ring R\u2082\ninst\u271d\u2077 : Ring R\u2083\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\np : Submodule R M\nH : \u2200 (x : M), x \u2208 p \u2192 \u2200 (y : M), y \u2208 p \u2192 \u2191f x = \u2191f y \u2192 x = y\nx : M\nh\u2081 : x \u2208 p\nh\u2082 : \u2191f x = 0\n\u22a2 \u2191f x = \u2191f 0\n[PROOFSTEP]\nsimpa using h\u2082\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : Ring R\u2082\ninst\u271d\u2077 : Ring R\u2083\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\ninst\u271d\u2074 : AddCommGroup M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\n\u22a2 ker f = \u22a5 \u2194 Injective \u2191f\n[PROOFSTEP]\nsimpa [disjoint_iff_inf_le] using @disjoint_ker' _ _ _ _ _ _ _ _ _ _ _ _ _ f \u22a4\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\n\u22a2 ker f \u2264 p \u2194 \u2203 y, y \u2208 range f \u2227 \u2191f \u207b\u00b9' {y} \u2286 \u2191p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\n\u22a2 ker f \u2264 p \u2192 \u2203 y, y \u2208 range f \u2227 \u2191f \u207b\u00b9' {y} \u2286 \u2191p\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\nh : ker f \u2264 p\n\u22a2 \u2203 y, y \u2208 range f \u2227 \u2191f \u207b\u00b9' {y} \u2286 \u2191p\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\nh : ker f \u2264 p\n\u22a2 0 \u2208 range f \u2227 \u2191f \u207b\u00b9' {0} \u2286 \u2191p\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, range_coe]\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\nh : ker f \u2264 p\n\u22a2 0 \u2208 Set.range \u2191f \u2227 \u2191f \u207b\u00b9' {0} \u2286 \u2191p\n[PROOFSTEP]\nexact \u27e8\u27e80, map_zero f\u27e9, h\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\n\u22a2 (\u2203 y, y \u2208 range f \u2227 \u2191f \u207b\u00b9' {y} \u2286 \u2191p) \u2192 ker f \u2264 p\n[PROOFSTEP]\nrintro \u27e8y, h\u2081, h\u2082\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2081 : y \u2208 range f\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\n\u22a2 ker f \u2264 p\n[PROOFSTEP]\nrw [SetLike.le_def]\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2081 : y \u2208 range f\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\n\u22a2 \u2200 \u2983x : M\u2984, x \u2208 ker f \u2192 x \u2208 p\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2081 : y \u2208 range f\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : z \u2208 ker f\n\u22a2 z \u2208 p\n[PROOFSTEP]\nsimp only [mem_ker, SetLike.mem_coe] at hz \n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2081 : y \u2208 range f\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\n\u22a2 z \u2208 p\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, range_coe, Set.mem_range] at h\u2081 \n[GOAL]\ncase mpr.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2081 : \u2203 y_1, \u2191f y_1 = y\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\n\u22a2 z \u2208 p\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := h\u2081\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\n\u22a2 z \u2208 p\n[PROOFSTEP]\nhave hx' : x \u2208 p := h\u2082 hx\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\n\u22a2 z \u2208 p\n[PROOFSTEP]\nhave hxz : z + x \u2208 p := by\n  apply h\u2082\n  simp [hx, hz]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\n\u22a2 z + x \u2208 p\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\n\u22a2 z + x \u2208 \u2191f \u207b\u00b9' {y}\n[PROOFSTEP]\nsimp [hx, hz]\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\nhxz : z + x \u2208 p\n\u22a2 z \u2208 p\n[PROOFSTEP]\nsuffices z + x - x \u2208 p by simpa only [this, add_sub_cancel]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\nhxz : z + x \u2208 p\nthis : z + x - x \u2208 p\n\u22a2 z \u2208 p\n[PROOFSTEP]\nsimpa only [this, add_sub_cancel]\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : Ring R\u2082\ninst\u271d\u2078 : Ring R\u2083\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R\u2082 M\u2082\ninst\u271d\u00b2 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nf : F\ninst\u271d : RingHomSurjective \u03c4\u2081\u2082\np : Submodule R M\ny : M\u2082\nh\u2082 : \u2191f \u207b\u00b9' {y} \u2286 \u2191p\nz : M\nhz : \u2191f z = 0\nx : M\nhx : \u2191f x = y\nhx' : x \u2208 p\nhxz : z + x \u2208 p\n\u22a2 z + x - x \u2208 p\n[PROOFSTEP]\nexact p.sub_mem hxz hx'\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semifield K\ninst\u271d\u2074 : Semifield K\u2082\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\na : K\nh : a \u2260 0\n\u22a2 range (a \u2022 f) = range f\n[PROOFSTEP]\nsimpa only [range_eq_map] using Submodule.map_smul f _ a h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semifield K\ninst\u271d\u2074 : Semifield K\u2082\ninst\u271d\u00b3 : AddCommMonoid V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : AddCommMonoid V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\na : K\n\u22a2 range (a \u2022 f) = \u2a06 (_ : a \u2260 0), range f\n[PROOFSTEP]\nsimpa only [range_eq_map] using Submodule.map_smul' f _ a\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 IsLinearMap R fun x => x.fst + x.snd\n[PROOFSTEP]\napply IsLinearMap.mk\n[GOAL]\ncase map_add\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 \u2200 (x y : M \u00d7 M), (x + y).fst + (x + y).snd = x.fst + x.snd + (y.fst + y.snd)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase map_add\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 (x + y).fst + (x + y).snd = x.fst + x.snd + (y.fst + y.snd)\n[PROOFSTEP]\nsimp only [Prod.fst_add, Prod.snd_add]\n[GOAL]\ncase map_add\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 x.fst + y.fst + (x.snd + y.snd) = x.fst + x.snd + (y.fst + y.snd)\n[PROOFSTEP]\nabel\n  -- Porting Note: was cc\n[GOAL]\ncase map_add\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 x.fst + y.fst + (x.snd + y.snd) = x.fst + x.snd + (y.fst + y.snd)\n[PROOFSTEP]\nabel\n  -- Porting Note: was cc\n[GOAL]\ncase map_smul\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 \u2200 (c : R) (x : M \u00d7 M), (c \u2022 x).fst + (c \u2022 x).snd = c \u2022 (x.fst + x.snd)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase map_smul\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nx : R\ny : M \u00d7 M\n\u22a2 (x \u2022 y).fst + (x \u2022 y).snd = x \u2022 (y.fst + y.snd)\n[PROOFSTEP]\nsimp [smul_add]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 IsLinearMap R fun x => x.fst - x.snd\n[PROOFSTEP]\napply IsLinearMap.mk\n[GOAL]\ncase map_add\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u2200 (x y : M \u00d7 M), (x + y).fst - (x + y).snd = x.fst - x.snd + (y.fst - y.snd)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase map_add\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 (x + y).fst - (x + y).snd = x.fst - x.snd + (y.fst - y.snd)\n[PROOFSTEP]\nrw [Prod.fst_add, Prod.snd_add]\n[GOAL]\ncase map_add\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 x.fst + y.fst - (x.snd + y.snd) = x.fst - x.snd + (y.fst - y.snd)\n[PROOFSTEP]\nabel\n[GOAL]\ncase map_add\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx y : M \u00d7 M\n\u22a2 x.fst + y.fst - (x.snd + y.snd) = x.fst - x.snd + (y.fst - y.snd)\n[PROOFSTEP]\nabel\n[GOAL]\ncase map_smul\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 \u2200 (c : R) (x : M \u00d7 M), (c \u2022 x).fst - (c \u2022 x).snd = c \u2022 (x.fst - x.snd)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase map_smul\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM\u271d : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\nR : Type u_20\nM : Type u_21\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx : R\ny : M \u00d7 M\n\u22a2 (x \u2022 y).fst - (x \u2022 y).snd = x \u2022 (y.fst - y.snd)\n[PROOFSTEP]\nsimp [smul_sub]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p' : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u22a2 range (Submodule.subtype p) = p\n[PROOFSTEP]\nsimpa using map_comap_subtype p \u22a4\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np' : Submodule R { x // x \u2208 p }\n\u22a2 map (Submodule.subtype p) p' \u2264 p\n[PROOFSTEP]\nsimpa using (map_le_range : map p.subtype p' \u2264 range p.subtype)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p' : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\n\u22a2 map (Submodule.subtype p) \u22a4 = p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np p' : Submodule R M\n\u22a2 map (Submodule.subtype p) \u22a4 \u2264 p' \u2194 p \u2264 p'\n[PROOFSTEP]\nrw [map_subtype_top]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np p' : Submodule R M\nh : p \u2264 p'\n\u22a2 ker (ofLe h) = \u22a5\n[PROOFSTEP]\nrw [ofLe, ker_codRestrict, ker_subtype]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p' : Submodule R M\nq\u271d : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np q : Submodule R M\nh : p \u2264 q\n\u22a2 range (ofLe h) = comap (Submodule.subtype q) p\n[PROOFSTEP]\nrw [\u2190 map_top, ofLe, LinearMap.map_codRestrict, map_top, range_subtype]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np p' : Submodule R M\nh : p \u2264 p'\n\u22a2 map (Submodule.subtype p') (range (ofLe h)) = p\n[PROOFSTEP]\nsimp [range_ofLe, map_comap_eq, h]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np\u271d p' : Submodule R M\nq\u271d : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np q : Submodule R M\n\u22a2 Disjoint p q \u2194 comap (Submodule.subtype p) q = \u22a5\n[PROOFSTEP]\nrw [\u2190 (map_injective_of_injective (show Injective p.subtype from Subtype.coe_injective)).eq_iff, map_comap_subtype,\n  map_bot, disjoint_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p'\u271d : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np' : Submodule R { x // x \u2208 p }\n\u22a2 Injective \u2191(Submodule.subtype p)\n[PROOFSTEP]\nexact Subtype.val_injective\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p' : Submodule R M\nq\u271d : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\nx\u271d : { p' // p' \u2264 p }\nq : Submodule R M\nhq : q \u2264 p\n\u22a2 \u2191((fun p' => { val := map (Submodule.subtype p) p', property := (_ : map (Submodule.subtype p) p' \u2264 p) })\n        ((fun q => comap (Submodule.subtype p) \u2191q) { val := q, property := hq })) =\n    \u2191{ val := q, property := hq }\n[PROOFSTEP]\nsimp [map_comap_subtype p, inf_of_le_right hq]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p' : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np\u2081 p\u2082 : Submodule R { x // x \u2208 p }\n\u22a2 \u2191(\u2191{ toFun := fun p' => { val := map (Submodule.subtype p) p', property := (_ : map (Submodule.subtype p) p' \u2264 p) },\n              invFun := fun q => comap (Submodule.subtype p) \u2191q,\n              left_inv :=\n                (_ :\n                  \u2200 (p' : Submodule R { x // x \u2208 p }), comap (Submodule.subtype p) (map (Submodule.subtype p) p') = p'),\n              right_inv :=\n                (_ :\n                  \u2200 (x : { p' // p' \u2264 p }),\n                    (fun p' =>\n                          { val := map (Submodule.subtype p) p', property := (_ : map (Submodule.subtype p) p' \u2264 p) })\n                        ((fun q => comap (Submodule.subtype p) \u2191q) x) =\n                      x) }\n          p\u2081) \u2264\n      \u2191(\u2191{\n              toFun := fun p' =>\n                { val := map (Submodule.subtype p) p', property := (_ : map (Submodule.subtype p) p' \u2264 p) },\n              invFun := fun q => comap (Submodule.subtype p) \u2191q,\n              left_inv :=\n                (_ :\n                  \u2200 (p' : Submodule R { x // x \u2208 p }), comap (Submodule.subtype p) (map (Submodule.subtype p) p') = p'),\n              right_inv :=\n                (_ :\n                  \u2200 (x : { p' // p' \u2264 p }),\n                    (fun p' =>\n                          { val := map (Submodule.subtype p) p', property := (_ : map (Submodule.subtype p) p' \u2264 p) })\n                        ((fun q => comap (Submodule.subtype p) \u2191q) x) =\n                      x) }\n          p\u2082) \u2194\n    p\u2081 \u2264 p\u2082\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : Semiring R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\np p' : Submodule R M\nq : Submodule R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\nF : Type u_20\nsc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082\np\u2081 p\u2082 : Submodule R { x // x \u2208 p }\n\u22a2 map (Submodule.subtype p) p\u2081 \u2264 map (Submodule.subtype p) p\u2082 \u2194 p\u2081 \u2264 p\u2082\n[PROOFSTEP]\nrw [map_le_iff_le_comap, comap_map_eq_of_injective (show Injective p.subtype from Subtype.coe_injective) p\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nh : \u2200 (u v : { x // x \u2208 ker f } \u2192\u2097[R] M), comp f u = comp f v \u2192 u = v\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nhave h\u2081 : f.comp (0 : ker f \u2192\u2097[R] M) = 0 := comp_zero _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nh : \u2200 (u v : { x // x \u2208 ker f } \u2192\u2097[R] M), comp f u = comp f v \u2192 u = v\nh\u2081 : comp f 0 = 0\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nrw [\u2190 Submodule.range_subtype (ker f), \u2190 h 0 f.ker.subtype (Eq.trans h\u2081 (comp_ker_subtype f).symm)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nh : \u2200 (u v : { x // x \u2208 ker f } \u2192\u2097[R] M), comp f u = comp f v \u2192 u = v\nh\u2081 : comp f 0 = 0\n\u22a2 range 0 = \u22a5\n[PROOFSTEP]\nexact range_zero\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : Semiring R\u2082\ninst\u271d\u00b9\u2070 : Semiring R\u2083\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2083\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R\u2082 M\u2082\ninst\u271d\u2074 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b3 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\ninst\u271d\u00b2 : RingHomSurjective \u03c4\u2081\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c4\u2082\u2083\ninst\u271d : RingHomSurjective \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\nhf : range f = \u22a4\n\u22a2 range (comp g f) = range g\n[PROOFSTEP]\nrw [range_comp, hf, Submodule.map_top]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\ninst\u271d\u00b9 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nf : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083\nhg : ker g = \u22a5\n\u22a2 ker (comp g f) = ker f\n[PROOFSTEP]\nrw [ker_comp, hg, Submodule.comap_bot]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\n\u22a2 x \u2208 submoduleImage \u03d5 N \u2194 \u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x\n[PROOFSTEP]\nrefine' Submodule.mem_map.trans \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\n\u22a2 (\u2203 y, y \u2208 Submodule.comap (Submodule.subtype O) N \u2227 \u2191\u03d5 y = x) \u2192 \u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x\n[PROOFSTEP]\nsimp_rw [Submodule.mem_comap]\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\n\u22a2 (\u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x) \u2192 \u2203 y, y \u2208 Submodule.comap (Submodule.subtype O) N \u2227 \u2191\u03d5 y = x\n[PROOFSTEP]\nsimp_rw [Submodule.mem_comap]\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\n\u22a2 (\u2203 y, \u2191(Submodule.subtype O) y \u2208 N \u2227 \u2191\u03d5 y = x) \u2192 \u2203 y h h_1, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n[PROOFSTEP]\nrintro \u27e8\u27e8y, yO\u27e9, yN : y \u2208 N, h\u27e9\n[GOAL]\ncase refine'_1.intro.mk.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\ny : M\nyO : y \u2208 O\nyN : y \u2208 N\nh : \u2191\u03d5 { val := y, property := yO } = x\n\u22a2 \u2203 y h h_1, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n[PROOFSTEP]\nexact \u27e8y, yO, yN, h\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\n\u22a2 (\u2203 y h h_1, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x) \u2192 \u2203 y, \u2191(Submodule.subtype O) y \u2208 N \u2227 \u2191\u03d5 y = x\n[PROOFSTEP]\nrintro \u27e8y, yO, yN, h\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nx : M'\ny : M\nyO : y \u2208 O\nyN : y \u2208 N\nh : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n\u22a2 \u2203 y, \u2191(Submodule.subtype O) y \u2208 N \u2227 \u2191\u03d5 y = x\n[PROOFSTEP]\nexact \u27e8\u27e8y, yO\u27e9, yN, h\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\nx : M'\n\u22a2 x \u2208 submoduleImage \u03d5 N \u2194 \u2203 y yN, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n[PROOFSTEP]\nrefine' mem_submoduleImage.trans \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\nx : M'\n\u22a2 (\u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x) \u2192 \u2203 y yN, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n[PROOFSTEP]\nrintro \u27e8y, yO, yN, h\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\nx : M'\ny : M\nyO : y \u2208 O\nyN : y \u2208 N\nh : \u2191\u03d5 { val := y, property := yO } = x\n\u22a2 \u2203 y yN, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n[PROOFSTEP]\nexact \u27e8y, yN, h\u27e9\n[GOAL]\ncase refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\nx : M'\n\u22a2 (\u2203 y yN, \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x) \u2192 \u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x\n[PROOFSTEP]\nrintro \u27e8y, yN, h\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\nx : M'\ny : M\nyN : y \u2208 N\nh : \u2191\u03d5 { val := y, property := (_ : y \u2208 O) } = x\n\u22a2 \u2203 y yO x_1, \u2191\u03d5 { val := y, property := yO } = x\n[PROOFSTEP]\nexact \u27e8y, hNO yN, yN, h\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM'\u271d : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c4\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083\nM' : Type u_20\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nO : Submodule R M\n\u03d5 : { x // x \u2208 O } \u2192\u2097[R] M'\nN : Submodule R M\nhNO : N \u2264 O\n\u22a2 submoduleImage \u03d5 N = range (comp \u03d5 (Submodule.ofLe hNO))\n[PROOFSTEP]\nrw [submoduleImage, range_comp, Submodule.range_ofLe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\nf : M \u2192\u2097[R] M\u2082\n\u22a2 range (rangeRestrict f) = \u22a4\n[PROOFSTEP]\nsimp [f.range_codRestrict _]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : Subsingleton R\ninst\u271d : Subsingleton R\u2082\n\u22a2 Unique (M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n[PROOFSTEP]\nhaveI := Module.subsingleton R M\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : Subsingleton R\ninst\u271d : Subsingleton R\u2082\nthis : Subsingleton M\n\u22a2 Unique (M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n[PROOFSTEP]\nhaveI := Module.subsingleton R\u2082 M\u2082\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ninst\u271d\u00b9 : Subsingleton R\ninst\u271d : Subsingleton R\u2082\nthis\u271d : Subsingleton M\nthis : Subsingleton M\u2082\n\u22a2 Unique (M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\n\u22a2 \u2191(Submodule.map (\u2191e) p) = \u2191(Submodule.comap (\u2191(symm e)) p)\n[PROOFSTEP]\nsimp [e.image_eq_preimage]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nx : { x // x \u2208 p }\n\u22a2 \u2191x \u2208 \u2191p \u2227 \u2191\u2191e \u2191x = \u2191(LinearMap.domRestrict (\u2191e) p) x\n[PROOFSTEP]\nsimp only [LinearMap.domRestrict_apply, eq_self_iff_true, and_true_iff, SetLike.coe_mem, SetLike.mem_coe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny : { x // x \u2208 Submodule.map (\u2191e) p }\n\u22a2 \u2191\u2191(symm e) \u2191y \u2208 p\n[PROOFSTEP]\nrcases y with \u27e8y', hy\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny' : M\u2082\nhy : y' \u2208 Submodule.map (\u2191e) p\n\u22a2 \u2191\u2191(symm e) \u2191{ val := y', property := hy } \u2208 p\n[PROOFSTEP]\nrw [Submodule.mem_map] at hy \n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny' : M\u2082\nhy\u271d : y' \u2208 Submodule.map (\u2191e) p\nhy : \u2203 y, y \u2208 p \u2227 \u2191\u2191e y = y'\n\u22a2 \u2191\u2191(symm e) \u2191{ val := y', property := hy\u271d } \u2208 p\n[PROOFSTEP]\nrcases hy with \u27e8x, hx, hxy\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny' : M\u2082\nhy : y' \u2208 Submodule.map (\u2191e) p\nx : M\nhx : x \u2208 p\nhxy : \u2191\u2191e x = y'\n\u22a2 \u2191\u2191(symm e) \u2191{ val := y', property := hy } \u2208 p\n[PROOFSTEP]\nsubst hxy\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\nx : M\nhx : x \u2208 p\nhy : \u2191\u2191e x \u2208 Submodule.map (\u2191e) p\n\u22a2 \u2191\u2191(symm e) \u2191{ val := \u2191\u2191e x, property := hy } \u2208 p\n[PROOFSTEP]\nsimp only [symm_apply_apply, Submodule.coe_mk, coe_coe, hx]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\nx : { x // x \u2208 p }\n\u22a2 (fun y => { val := \u2191\u2191(symm e) \u2191y, property := (_ : \u2191\u2191(symm e) \u2191y \u2208 p) })\n      (AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : { x // x \u2208 p }),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191\u03c3\u2081\u2082 r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\nsimp only [LinearMap.domRestrict_apply, LinearMap.codRestrict_apply, LinearMap.toFun_eq_coe, LinearEquiv.coe_coe,\n  LinearEquiv.symm_apply_apply, SetLike.eta]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny : { x // x \u2208 Submodule.map (\u2191e) p }\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : { x // x \u2208 p }),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191\u03c3\u2081\u2082 r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      ((fun y => { val := \u2191\u2191(symm e) \u2191y, property := (_ : \u2191\u2191(symm e) \u2191y \u2208 p) }) y) =\n    y\n[PROOFSTEP]\napply SetCoe.ext\n[GOAL]\ncase a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ne e' : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\np : Submodule R M\nsrc\u271d : { x // x \u2208 p } \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 Submodule.map (\u2191e) p } :=\n  LinearMap.codRestrict (Submodule.map (\u2191e) p) (LinearMap.domRestrict (\u2191e) p)\n    (_ : \u2200 (x : { x // x \u2208 p }), \u2203 a, a \u2208 \u2191p \u2227 \u2191\u2191e a = \u2191(LinearMap.domRestrict (\u2191e) p) x)\ny : { x // x \u2208 Submodule.map (\u2191e) p }\n\u22a2 \u2191(AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : { x // x \u2208 p }),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191\u03c3\u2081\u2082 r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        ((fun y => { val := \u2191\u2191(symm e) \u2191y, property := (_ : \u2191\u2191(symm e) \u2191y \u2208 p) }) y)) =\n    \u2191y\n[PROOFSTEP]\nsimp only [LinearMap.domRestrict_apply, LinearMap.codRestrict_apply, LinearMap.toFun_eq_coe, LinearEquiv.coe_coe,\n  LinearEquiv.apply_symm_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nsrc\u271d : (V \u00d7 V\u2082 \u2192 R) \u2243 (V \u2192 V\u2082 \u2192 R) := Equiv.curry V V\u2082 R\nx\u271d\u00b9 x\u271d : V \u00d7 V\u2082 \u2192 R\n\u22a2 Equiv.toFun src\u271d (x\u271d\u00b9 + x\u271d) = Equiv.toFun src\u271d x\u271d\u00b9 + Equiv.toFun src\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nsrc\u271d : (V \u00d7 V\u2082 \u2192 R) \u2243 (V \u2192 V\u2082 \u2192 R) := Equiv.curry V V\u2082 R\nx\u271d\u00b3 x\u271d\u00b2 : V \u00d7 V\u2082 \u2192 R\nx\u271d\u00b9 : V\nx\u271d : V\u2082\n\u22a2 Equiv.toFun src\u271d (x\u271d\u00b3 + x\u271d\u00b2) x\u271d\u00b9 x\u271d = (Equiv.toFun src\u271d x\u271d\u00b3 + Equiv.toFun src\u271d x\u271d\u00b2) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nsrc\u271d : (V \u00d7 V\u2082 \u2192 R) \u2243 (V \u2192 V\u2082 \u2192 R) := Equiv.curry V V\u2082 R\nx\u271d\u00b9 : R\nx\u271d : V \u00d7 V\u2082 \u2192 R\n\u22a2 AddHom.toFun\n      { toFun := src\u271d.toFun,\n        map_add' :=\n          (_ : \u2200 (x x_1 : V \u00d7 V\u2082 \u2192 R), Equiv.toFun src\u271d (x + x_1) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d x_1) }\n      (x\u271d\u00b9 \u2022 x\u271d) =\n    \u2191(RingHom.id R) x\u271d\u00b9 \u2022\n      AddHom.toFun\n        { toFun := src\u271d.toFun,\n          map_add' :=\n            (_ : \u2200 (x x_1 : V \u00d7 V\u2082 \u2192 R), Equiv.toFun src\u271d (x + x_1) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d x_1) }\n        x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\ninst\u271d\u2074 : Semiring R\u2084\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : AddCommMonoid M\u2084\nsrc\u271d : (V \u00d7 V\u2082 \u2192 R) \u2243 (V \u2192 V\u2082 \u2192 R) := Equiv.curry V V\u2082 R\nx\u271d\u00b3 : R\nx\u271d\u00b2 : V \u00d7 V\u2082 \u2192 R\nx\u271d\u00b9 : V\nx\u271d : V\u2082\n\u22a2 AddHom.toFun\n      { toFun := src\u271d.toFun,\n        map_add' :=\n          (_ : \u2200 (x x_1 : V \u00d7 V\u2082 \u2192 R), Equiv.toFun src\u271d (x + x_1) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d x_1) }\n      (x\u271d\u00b3 \u2022 x\u271d\u00b2) x\u271d\u00b9 x\u271d =\n    (\u2191(RingHom.id R) x\u271d\u00b3 \u2022\n        AddHom.toFun\n          { toFun := src\u271d.toFun,\n            map_add' :=\n              (_ : \u2200 (x x_1 : V \u00d7 V\u2082 \u2192 R), Equiv.toFun src\u271d (x + x_1) = Equiv.toFun src\u271d x + Equiv.toFun src\u271d x_1) }\n          x\u271d\u00b2)\n      x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring R\u2082\ninst\u271d\u2076 : Semiring R\u2083\ninst\u271d\u2075 : Semiring R\u2084\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\n\u22a2 ofEq p p (_ : p = p) = refl R { x // x \u2208 p }\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : Semiring R\u2082\ninst\u271d\u2076 : Semiring R\u2083\ninst\u271d\u2075 : Semiring R\u2084\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\nx\u271d : { x // x \u2208 p }\n\u22a2 \u2191(\u2191(ofEq p p (_ : p = p)) x\u271d) = \u2191(\u2191(refl R { x // x \u2208 p }) x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : Semiring R\u2084\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\nf : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nU : Submodule R\u2082 M\u2082\n\u22a2 \u2191(ofSubmodule' f U) =\n    LinearMap.codRestrict U (LinearMap.domRestrict (\u2191f) (Submodule.comap (\u2191f) U))\n      (_ : \u2200 (x : { x // x \u2208 Submodule.comap (\u2191f) U }), \u2191x \u2208 Submodule.comap (\u2191f) U)\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : Semiring R\u2084\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R\u2082 M\u2082\nf : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nU : Submodule R\u2082 M\u2082\nx\u271d : { x // x \u2208 Submodule.comap (\u2191f) U }\n\u22a2 \u2191(\u2191\u2191(ofSubmodule' f U) x\u271d) =\n    \u2191(\u2191(LinearMap.codRestrict U (LinearMap.domRestrict (\u2191f) (Submodule.comap (\u2191f) U))\n            (_ : \u2200 (x : { x // x \u2208 Submodule.comap (\u2191f) U }), \u2191x \u2208 Submodule.comap (\u2191f) U))\n        x\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : Semiring R\u2084\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne\u271d : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d : Module R\u2082 M\u2082\ne : { x // x \u2208 p } \u2243\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 \u22a5 }\n\u22a2 p = \u22a5\n[PROOFSTEP]\nrefine' bot_unique (SetLike.le_def.2 fun b hb => (Submodule.mem_bot R).2 _)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : Semiring R\u2084\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne\u271d : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d : Module R\u2082 M\u2082\ne : { x // x \u2208 p } \u2243\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 \u22a5 }\nb : M\nhb : b \u2208 p\n\u22a2 b = 0\n[PROOFSTEP]\nrw [\u2190 p.mk_eq_zero hb, \u2190 e.map_eq_zero_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\ninst\u271d\u2076 : Semiring R\u2084\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b9 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne\u271d : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d : Module R\u2082 M\u2082\ne : { x // x \u2208 p } \u2243\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 \u22a5 }\nb : M\nhb : b \u2208 p\n\u22a2 \u2191e { val := b, property := hb } = 0\n[PROOFSTEP]\napply Submodule.eq_zero_of_bot_submodule\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : Semiring R\u2082\ninst\u271d\u2078 : Semiring R\u2083\ninst\u271d\u2077 : Semiring R\u2084\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2084\nmodule_M : Module R M\nmodule_M\u2082 : Module R\u2082 M\u2082\nmodule_M\u2083 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\nre\u2081\u2082 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\nre\u2082\u2081 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\nre\u2082\u2083 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\nre\u2083\u2082 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\nf : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\ng\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2081] M\ne : M \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nh\u271d : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083\ne'' : M\u2082 \u2243\u209b\u2097[\u03c3\u2082\u2083] M\u2083\np q : Submodule R M\ninst\u271d\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\ng : M\u2082 \u2192 M\nh : LeftInverse g \u2191f\nsrc\u271d : M \u2192\u209b\u2097[\u03c3\u2081\u2082] { x // x \u2208 LinearMap.range f } := LinearMap.rangeRestrict f\nx : { x // x \u2208 LinearMap.range f }\nx' : M\nhx' : \u2191f x' = \u2191x\n\u22a2 \u2191f (g \u2191x) = \u2191x\n[PROOFSTEP]\nrw [\u2190 hx', h x']\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nx : M\n\u22a2 \u2191(neg R) x = -x\n[PROOFSTEP]\nsimp\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf g : M\u2081 \u2192\u2097[R] M\u2082\u2081\n\u22a2 (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf g : M\u2081 \u2192\u2097[R] M\u2082\u2081\nx : M\u2082\n\u22a2 \u2191((fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g)) x =\n    \u2191((fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n          (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g)\n      x\n[PROOFSTEP]\nsimp only [map_add, add_apply, Function.comp_apply, coe_comp, coe_coe]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nc : R\nf : M\u2081 \u2192\u2097[R] M\u2082\u2081\n\u22a2 AddHom.toFun\n      { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n        map_add' :=\n          (_ :\n            \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n              (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n      (c \u2022 f) =\n    \u2191(RingHom.id R) c \u2022\n      AddHom.toFun\n        { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n          map_add' :=\n            (_ :\n              \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n        f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nc : R\nf : M\u2081 \u2192\u2097[R] M\u2082\u2081\nx : M\u2082\n\u22a2 \u2191(AddHom.toFun\n          { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n            map_add' :=\n              (_ :\n                \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n          (c \u2022 f))\n      x =\n    \u2191(\u2191(RingHom.id R) c \u2022\n          AddHom.toFun\n            { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n              map_add' :=\n                (_ :\n                  \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                        (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n            f)\n      x\n[PROOFSTEP]\nsimp only [smul_apply, Function.comp_apply, coe_comp, map_smul\u209b\u2097 e\u2082, coe_coe]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\u2081\n\u22a2 (fun f => LinearMap.comp (\u2191(symm e\u2082)) (LinearMap.comp f \u2191e\u2081))\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                        (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                          (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (f : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                  AddHom.toFun\n                      { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                              (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id R) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                        f) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\u2081\nx : M\u2081\n\u22a2 \u2191((fun f => LinearMap.comp (\u2191(symm e\u2082)) (LinearMap.comp f \u2191e\u2081))\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                    map_add' :=\n                      (_ :\n                        \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                          (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                            (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                              (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R) (f : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                      AddHom.toFun\n                          { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                          (c \u2022 f) =\n                        \u2191(RingHom.id R) c \u2022\n                          AddHom.toFun\n                            { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                        (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                            f) }.toAddHom\n            f))\n      x =\n    \u2191f x\n[PROOFSTEP]\nsimp only [symm_apply_apply, Function.comp_apply, coe_comp, coe_coe]\n[GOAL]\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf : M\u2082 \u2192\u2097[R] M\u2082\u2082\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n              map_add' :=\n                (_ :\n                  \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                        (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (f : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                AddHom.toFun\n                    { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                            (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                              (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                    (c \u2022 f) =\n                  \u2191(RingHom.id R) c \u2022\n                    AddHom.toFun\n                      { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                              (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                      f) }.toAddHom\n      ((fun f => LinearMap.comp (\u2191(symm e\u2082)) (LinearMap.comp f \u2191e\u2081)) f) =\n    f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR\u271d : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081\u271d : Type u_11\nM\u2082\u271d : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u2075 : CommSemiring R\u271d\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\u271d\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b9 : Module R\u271d M\ninst\u271d\u00b9\u2070 : Module R\u271d M\u2082\u271d\ninst\u271d\u2079 : Module R\u271d M\u2083\nR : Type ?u.2478657\nM\u2081 : Type ?u.2478660\nM\u2082 : Type ?u.2478663\nM\u2082\u2081 : Type ?u.2478666\nM\u2082\u2082 : Type ?u.2478669\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2082\u2081\ninst\u271d : Module R M\u2082\u2082\ne\u2081 : M\u2081 \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082\u2081 \u2243\u2097[R] M\u2082\u2082\nf : M\u2082 \u2192\u2097[R] M\u2082\u2082\nx : M\u2082\n\u22a2 \u2191(AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                  map_add' :=\n                    (_ :\n                      \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                        (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                          (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                            (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) },\n              map_smul' :=\n                (_ :\n                  \u2200 (c : R) (f : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                    AddHom.toFun\n                        { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                                (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                        (c \u2022 f) =\n                      \u2191(RingHom.id R) c \u2022\n                        AddHom.toFun\n                          { toFun := fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081)),\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : M\u2081 \u2192\u2097[R] M\u2082\u2081),\n                                  (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) (f + g) =\n                                    (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) f +\n                                      (fun f => LinearMap.comp (\u2191e\u2082) (LinearMap.comp f \u2191(symm e\u2081))) g) }\n                          f) }.toAddHom\n          ((fun f => LinearMap.comp (\u2191(symm e\u2082)) (LinearMap.comp f \u2191e\u2081)) f))\n      x =\n    \u2191f x\n[PROOFSTEP]\nsimp only [Function.comp_apply, apply_symm_apply, coe_comp, coe_coe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082\u271d : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2082\ninst\u271d\u2076 : Module R M\u2083\nN : Type u_20\nN\u2082 : Type u_21\nN\u2083 : Type u_22\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : AddCommMonoid N\u2083\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module R N\u2082\ninst\u271d : Module R N\u2083\ne\u2081 : M \u2243\u2097[R] N\ne\u2082 : M\u2082 \u2243\u2097[R] N\u2082\ne\u2083 : M\u2083 \u2243\u2097[R] N\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M\u2082 \u2192\u2097[R] M\u2083\n\u22a2 \u2191(arrowCongr e\u2081 e\u2083) (LinearMap.comp g f) = LinearMap.comp (\u2191(arrowCongr e\u2082 e\u2083) g) (\u2191(arrowCongr e\u2081 e\u2082) f)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN\u271d : Type u_15\nN\u2082\u271d : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2083\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R M\u2082\ninst\u271d\u2076 : Module R M\u2083\nN : Type u_20\nN\u2082 : Type u_21\nN\u2083 : Type u_22\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : AddCommMonoid N\u2083\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module R N\u2082\ninst\u271d : Module R N\u2083\ne\u2081 : M \u2243\u2097[R] N\ne\u2082 : M\u2082 \u2243\u2097[R] N\u2082\ne\u2083 : M\u2083 \u2243\u2097[R] N\u2083\nf : M \u2192\u2097[R] M\u2082\ng : M\u2082 \u2192\u2097[R] M\u2083\nx\u271d : N\n\u22a2 \u2191(\u2191(arrowCongr e\u2081 e\u2083) (LinearMap.comp g f)) x\u271d = \u2191(LinearMap.comp (\u2191(arrowCongr e\u2082 e\u2083) g) (\u2191(arrowCongr e\u2081 e\u2082) f)) x\u271d\n[PROOFSTEP]\nsimp only [symm_apply_apply, arrowCongr_apply, LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\ne\u2081 : M \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082 \u2243\u2097[R] M\u2083\n\u22a2 trans (conj e\u2081) (conj e\u2082) = conj (trans e\u2081 e\u2082)\n[PROOFSTEP]\next f x\n[GOAL]\ncase h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\ne\u2081 : M \u2243\u2097[R] M\u2082\ne\u2082 : M\u2082 \u2243\u2097[R] M\u2083\nf : Module.End R M\nx : M\u2083\n\u22a2 \u2191(\u2191(trans (conj e\u2081) (conj e\u2082)) f) x = \u2191(\u2191(conj (trans e\u2081 e\u2082)) f) x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\ne : M \u2243\u2097[R] M\u2082\n\u22a2 \u2191(conj e) LinearMap.id = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M\u2082\ninst\u271d : Module R M\u2083\ne : M \u2243\u2097[R] M\u2082\nx\u271d : M\u2082\n\u22a2 \u2191(\u2191(conj e) LinearMap.id) x\u271d = \u2191LinearMap.id x\u271d\n[PROOFSTEP]\nsimp [conj_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\n\u22a2 \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nx : { x // x \u2208 p }\nhx : x \u2208 q\n\u22a2 \u2191(LinearMap.domRestrict (Submodule.subtype p) q) { val := x, property := hx } \u2208 map (Submodule.subtype p) q\n[PROOFSTEP]\nexact \u27e8x, hx, rfl\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\n\u22a2 { x // x \u2208 map (Submodule.subtype p) q } \u2192 { x // x \u2208 q }\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nx : M\nhx : x \u2208 map (Submodule.subtype p) q\n\u22a2 { x // x \u2208 q }\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, _\u27e9, _\u27e9\n[GOAL]\ncase mk.refine'_1\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nx : M\nhx : x \u2208 map (Submodule.subtype p) q\n\u22a2 x \u2208 p\n[PROOFSTEP]\nrcases hx with \u27e8\u27e8_, h\u27e9, _, rfl\u27e9\n[GOAL]\ncase mk.refine'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nx : M\nhx : x \u2208 map (Submodule.subtype p) q\n\u22a2 { val := x, property := (_ : x \u2208 p) } \u2208 q\n[PROOFSTEP]\nrcases hx with \u27e8\u27e8_, h\u27e9, _, rfl\u27e9\n[GOAL]\ncase mk.refine'_1.intro.mk.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nval\u271d : M\nh : val\u271d \u2208 p\nleft\u271d : { val := val\u271d, property := h } \u2208 \u2191q\n\u22a2 \u2191(Submodule.subtype p) { val := val\u271d, property := h } \u2208 p\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.refine'_2.intro.mk.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nval\u271d : M\nh : val\u271d \u2208 p\nleft\u271d : { val := val\u271d, property := h } \u2208 \u2191q\n\u22a2 { val := \u2191(Submodule.subtype p) { val := val\u271d, property := h },\n      property := (_ : \u2191(Submodule.subtype p) { val := val\u271d, property := h } \u2208 p) } \u2208\n    q\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nx\u271d : { x // x \u2208 map (Submodule.subtype p) q }\nx : M\nh : x \u2208 p\nleft\u271d : { val := x, property := h } \u2208 \u2191q\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : { x // x \u2208 q }),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (Subtype.casesOn { val := x, property := (_ : \u2203 a, a \u2208 \u2191q \u2227 \u2191(Submodule.subtype p) a = x) } fun x hx =>\n        { val := { val := x, property := (_ : x \u2208 p) }, property := (_ : { val := x, property := (_ : x \u2208 p) } \u2208 q) }) =\n    { val := x, property := (_ : \u2203 a, a \u2208 \u2191q \u2227 \u2191(Submodule.subtype p) a = x) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nsrc\u271d : { x // x \u2208 q } \u2192\u2097[R] { x // x \u2208 map (Submodule.subtype p) q } :=\n  LinearMap.codRestrict (map (Submodule.subtype p) q) (LinearMap.domRestrict (Submodule.subtype p) q)\n    (_ : \u2200 (c : { x // x \u2208 q }), \u2191(LinearMap.domRestrict (Submodule.subtype p) q) c \u2208 map (Submodule.subtype p) q)\nx\u271d : { x // x \u2208 map (Submodule.subtype p) q }\nx : M\nh : x \u2208 p\nleft\u271d : { val := x, property := h } \u2208 \u2191q\n\u22a2 \u2191(AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : { x // x \u2208 q }),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        (Subtype.casesOn { val := x, property := (_ : \u2203 a, a \u2208 \u2191q \u2227 \u2191(Submodule.subtype p) a = x) } fun x hx =>\n          { val := { val := x, property := (_ : x \u2208 p) },\n            property := (_ : { val := x, property := (_ : x \u2208 p) } \u2208 q) })) =\n    \u2191{ val := x, property := (_ : \u2203 a, a \u2208 \u2191q \u2227 \u2191(Submodule.subtype p) a = x) }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nx : { x // x \u2208 map (Submodule.subtype p) q }\n\u22a2 \u2191\u2191(\u2191(LinearEquiv.symm (equivSubtypeMap p q)) x) = \u2191x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np : Submodule R M\nq : Submodule R { x // x \u2208 p }\nval\u271d : M\nproperty\u271d : val\u271d \u2208 map (Submodule.subtype p) q\n\u22a2 \u2191\u2191(\u2191(LinearEquiv.symm (equivSubtypeMap p q)) { val := val\u271d, property := property\u271d }) =\n    \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np q : Submodule R M\nhpq : p \u2264 q\nx : { x // x \u2208 comap (Submodule.subtype q) p }\n\u22a2 (fun x => { val := { val := \u2191x, property := (_ : \u2191x \u2208 q) }, property := (_ : \u2191x \u2208 p) })\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                      (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                        (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y)) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (x : { x // x \u2208 comap (Submodule.subtype q) p }),\n                  AddHom.toFun\n                      { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                              (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                                (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) })\n                                  (x + y)) }\n                      (c \u2022 x) =\n                    AddHom.toFun\n                      { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                              (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                                (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) })\n                                  (x + y)) }\n                      (c \u2022 x)) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\nsimp only [coe_mk, SetLike.eta, LinearEquiv.coe_coe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\np q : Submodule R M\nhpq : p \u2264 q\nx : { x // x \u2208 p }\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n              map_add' :=\n                (_ :\n                  \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                    (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                      (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y)) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (x : { x // x \u2208 comap (Submodule.subtype q) p }),\n                AddHom.toFun\n                    { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                            (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                              (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y)) }\n                    (c \u2022 x) =\n                  AddHom.toFun\n                    { toFun := fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) },\n                      map_add' :=\n                        (_ :\n                          \u2200 (x y : { x // x \u2208 comap (Submodule.subtype q) p }),\n                            (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y) =\n                              (fun x => { val := \u2191\u2191x, property := (_ : \u2191x \u2208 comap (Submodule.subtype q) p) }) (x + y)) }\n                    (c \u2022 x)) }.toAddHom\n      ((fun x => { val := { val := \u2191x, property := (_ : \u2191x \u2208 q) }, property := (_ : \u2191x \u2208 p) }) x) =\n    x\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, SetLike.eta, LinearEquiv.coe_coe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\n\u22a2 x \u2208 map (\u2191e) p \u2194 \u2191(LinearEquiv.symm e) x \u2208 p\n[PROOFSTEP]\nrw [Submodule.mem_map]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2191\u2191e y = x) \u2194 \u2191(LinearEquiv.symm e) x \u2208 p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\n\u22a2 (\u2203 y, y \u2208 p \u2227 \u2191\u2191e y = x) \u2192 \u2191(LinearEquiv.symm e) x \u2208 p\n[PROOFSTEP]\nrintro \u27e8y, hy, hx\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\ny : M\nhy : y \u2208 p\nhx : \u2191\u2191e y = x\n\u22a2 \u2191(LinearEquiv.symm e) x \u2208 p\n[PROOFSTEP]\nsimp [\u2190 hx, hy]\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\n\u22a2 \u2191(LinearEquiv.symm e) x \u2208 p \u2192 \u2203 y, y \u2208 p \u2227 \u2191\u2191e y = x\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\nhx : \u2191(LinearEquiv.symm e) x \u2208 p\n\u22a2 \u2203 y, y \u2208 p \u2227 \u2191\u2191e y = x\n[PROOFSTEP]\nrefine' \u27e8e.symm x, hx, by simp\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nx : M\u2082\nhx : \u2191(LinearEquiv.symm e) x \u2208 p\n\u22a2 \u2191\u2191e (\u2191(LinearEquiv.symm e) x) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK\u271d : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R M\nx\u271d : M\u2082\n\u22a2 x\u271d \u2208 map (\u2191e) K \u2194 x\u271d \u2208 comap (\u2191(LinearEquiv.symm e)) K\n[PROOFSTEP]\nrw [mem_map_equiv, mem_comap, LinearEquiv.coe_coe]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK\u271d : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R\u2082 M\u2082\n\u22a2 map (LinearEquiv.symm e) K = p \u2194 map e p = K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK\u271d : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R\u2082 M\u2082\n\u22a2 map (LinearEquiv.symm e) K = p \u2192 map e p = K\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK\u271d : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R\u2082 M\u2082\n\u22a2 map e p = K \u2192 map (LinearEquiv.symm e) K = p\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK\u271d : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R\u2082 M\u2082\n\u22a2 map e (map (LinearEquiv.symm e) K) = K\n[PROOFSTEP]\ncalc\n  map e (map e.symm K) = comap e.symm (map e.symm K) := map_equiv_eq_comap_symm _ _\n  _ = K := comap_map_eq_of_injective e.symm.injective _\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 map (LinearEquiv.symm e) (map e p) = p\n[PROOFSTEP]\ncalc\n  map e.symm (map e p) = comap e (map e p) := (comap_equiv_eq_map_symm _ _).symm\n  _ = p := comap_map_eq_of_injective e.injective _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2097 : N \u2192\u2097[R] N\u2082\nc : R\n\u22a2 comap f\u2097 q\u2097 \u2264 comap (c \u2022 f\u2097) q\u2097\n[PROOFSTEP]\nrw [SetLike.le_def]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2097 : N \u2192\u2097[R] N\u2082\nc : R\n\u22a2 \u2200 \u2983x : N\u2984, x \u2208 comap f\u2097 q\u2097 \u2192 x \u2208 comap (c \u2022 f\u2097) q\u2097\n[PROOFSTEP]\nintro m h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2097 : N \u2192\u2097[R] N\u2082\nc : R\nm : N\nh : m \u2208 comap f\u2097 q\u2097\n\u22a2 m \u2208 comap (c \u2022 f\u2097) q\u2097\n[PROOFSTEP]\nchange c \u2022 f\u2097 m \u2208 q\u2097\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2097 : N \u2192\u2097[R] N\u2082\nc : R\nm : N\nh : m \u2208 comap f\u2097 q\u2097\n\u22a2 c \u2022 \u2191f\u2097 m \u2208 q\u2097\n[PROOFSTEP]\nreplace h : f\u2097 m \u2208 q\u2097 := h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2097 : N \u2192\u2097[R] N\u2082\nc : R\nm : N\nh : \u2191f\u2097 m \u2208 q\u2097\n\u22a2 c \u2022 \u2191f\u2097 m \u2208 q\u2097\n[PROOFSTEP]\napply q\u2097.smul_mem _ h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 comap f\u2081 q \u2293 comap f\u2082 q \u2264 comap (f\u2081 + f\u2082) q\n[PROOFSTEP]\nrw [SetLike.le_def]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\n\u22a2 \u2200 \u2983x : M\u2984, x \u2208 comap f\u2081 q \u2293 comap f\u2082 q \u2192 x \u2208 comap (f\u2081 + f\u2082) q\n[PROOFSTEP]\nintro m h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nm : M\nh : m \u2208 comap f\u2081 q \u2293 comap f\u2082 q\n\u22a2 m \u2208 comap (f\u2081 + f\u2082) q\n[PROOFSTEP]\nchange f\u2081 m + f\u2082 m \u2208 q\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nm : M\nh : m \u2208 comap f\u2081 q \u2293 comap f\u2082 q\n\u22a2 \u2191f\u2081 m + \u2191f\u2082 m \u2208 q\n[PROOFSTEP]\nreplace h : f\u2081 m \u2208 q \u2227 f\u2082 m \u2208 q := h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nm : M\nh : \u2191f\u2081 m \u2208 q \u2227 \u2191f\u2082 m \u2208 q\n\u22a2 \u2191f\u2081 m + \u2191f\u2082 m \u2208 q\n[PROOFSTEP]\napply q.add_mem h.1 h.2\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : N \u2192\u2097[R] N\u2082\nh\u2081 : f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\nh\u2082 : f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\n\u22a2 f\u2081 + f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\n[PROOFSTEP]\napply le_trans _ (inf_comap_le_comap_add q\u2097 f\u2081 f\u2082)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : N \u2192\u2097[R] N\u2082\nh\u2081 : f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\nh\u2082 : f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\n\u22a2 p\u2097 \u2264 comap f\u2081 q\u2097 \u2293 comap f\u2082 q\u2097\n[PROOFSTEP]\nrw [le_inf_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nf\u2081 f\u2082 : N \u2192\u2097[R] N\u2082\nh\u2081 : f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\nh\u2082 : f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097}\n\u22a2 p\u2097 \u2264 comap f\u2081 q\u2097 \u2227 p\u2097 \u2264 comap f\u2082 q\u2097\n[PROOFSTEP]\nexact \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\n\u22a2 0 \u2208\n    { carrier := {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097},\n        add_mem' :=\n          (_ :\n            \u2200 {f\u2081 f\u2082 : N \u2192\u2097[R] N\u2082},\n              f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 p\u2097 \u2264 comap (f\u2081 + f\u2082) q\u2097) }.carrier\n[PROOFSTEP]\nchange p\u2097 \u2264 comap (0 : N \u2192\u2097[R] N\u2082) q\u2097\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\n\u22a2 p\u2097 \u2264 comap 0 q\u2097\n[PROOFSTEP]\nrw [comap_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\n\u22a2 p\u2097 \u2264 \u22a4\n[PROOFSTEP]\nrefine' le_top\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nc : R\nf\u2097 : N \u2192\u2097[R] N\u2082\nh :\n  f\u2097 \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097},\n              add_mem' :=\n                (_ :\n                  \u2200 {f\u2081 f\u2082 : N \u2192\u2097[R] N\u2082},\n                    f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 p\u2097 \u2264 comap (f\u2081 + f\u2082) q\u2097) },\n          zero_mem' := (_ : p\u2097 \u2264 comap 0 q\u2097) }.toAddSubsemigroup.carrier\n\u22a2 c \u2022 f\u2097 \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097},\n              add_mem' :=\n                (_ :\n                  \u2200 {f\u2081 f\u2082 : N \u2192\u2097[R] N\u2082},\n                    f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 p\u2097 \u2264 comap (f\u2081 + f\u2082) q\u2097) },\n          zero_mem' := (_ : p\u2097 \u2264 comap 0 q\u2097) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\ndsimp at h \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : AddCommMonoid N\u2082\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R N\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\np\u2097 : Submodule R N\nq\u2097 : Submodule R N\u2082\nc : R\nf\u2097 : N \u2192\u2097[R] N\u2082\nh : p\u2097 \u2264 comap f\u2097 q\u2097\n\u22a2 c \u2022 f\u2097 \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097},\n              add_mem' :=\n                (_ :\n                  \u2200 {f\u2081 f\u2082 : N \u2192\u2097[R] N\u2082},\n                    f\u2081 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 f\u2082 \u2208 {f\u2097 | p\u2097 \u2264 comap f\u2097 q\u2097} \u2192 p\u2097 \u2264 comap (f\u2081 + f\u2082) q\u2097) },\n          zero_mem' := (_ : p\u2097 \u2264 comap 0 q\u2097) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact le_trans h (comap_le_comap_smul q\u2097 f\u2097 c)\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\n\u22a2 Surjective \u2191(funLeft R M f)\n[PROOFSTEP]\nclassical\nintro g\nrefine' \u27e8fun x => if h : \u2203 y, f y = x then g h.choose else 0, _\u27e9\n\u00b7 ext\n  dsimp only [funLeft_apply]\n  split_ifs with w\n  \u00b7 congr\n    exact hf w.choose_spec\n  \u00b7 simp only [not_true, exists_apply_eq_apply] at w \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\n\u22a2 Surjective \u2191(funLeft R M f)\n[PROOFSTEP]\nintro g\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\n\u22a2 \u2203 a, \u2191(funLeft R M f) a = g\n[PROOFSTEP]\nrefine' \u27e8fun x => if h : \u2203 y, f y = x then g h.choose else 0, _\u27e9\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\n\u22a2 (\u2191(funLeft R M f) fun x => if h : \u2203 y, f y = x then g (Exists.choose h) else 0) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\nx\u271d : m\n\u22a2 \u2191(funLeft R M f) (fun x => if h : \u2203 y, f y = x then g (Exists.choose h) else 0) x\u271d = g x\u271d\n[PROOFSTEP]\ndsimp only [funLeft_apply]\n[GOAL]\ncase h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\nx\u271d : m\n\u22a2 (if h : \u2203 y, f y = f x\u271d then g (Exists.choose h) else 0) = g x\u271d\n[PROOFSTEP]\nsplit_ifs with w\n[GOAL]\ncase pos\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\nx\u271d : m\nw : \u2203 y, f y = f x\u271d\n\u22a2 g (Exists.choose w) = g x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase pos.e_a\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\nx\u271d : m\nw : \u2203 y, f y = f x\u271d\n\u22a2 Exists.choose w = x\u271d\n[PROOFSTEP]\nexact hf w.choose_spec\n[GOAL]\ncase neg\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Injective f\ng : m \u2192 M\nx\u271d : m\nw : \u00ac\u2203 y, f y = f x\u271d\n\u22a2 0 = g x\u271d\n[PROOFSTEP]\nsimp only [not_true, exists_apply_eq_apply] at w \n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Surjective f\n\u22a2 Injective \u2191(funLeft R M f)\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := hf.hasRightInverse\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Surjective f\ng : n \u2192 m\nhg : Function.RightInverse g f\n\u22a2 Injective \u2191(funLeft R M f)\n[PROOFSTEP]\nsuffices LeftInverse (funLeft R M g) (funLeft R M f) by exact this.injective\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Surjective f\ng : n \u2192 m\nhg : Function.RightInverse g f\nthis : LeftInverse \u2191(funLeft R M g) \u2191(funLeft R M f)\n\u22a2 Injective \u2191(funLeft R M f)\n[PROOFSTEP]\nexact this.injective\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Surjective f\ng : n \u2192 m\nhg : Function.RightInverse g f\n\u22a2 LeftInverse \u2191(funLeft R M g) \u2191(funLeft R M f)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\nf : m \u2192 n\nhf : Surjective f\ng : n \u2192 m\nhg : Function.RightInverse g f\nx : n \u2192 M\n\u22a2 \u2191(funLeft R M g) (\u2191(funLeft R M f) x) = x\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_apply, \u2190 funLeft_comp, hg.id, funLeft_id]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\ne : m \u2243 n\nx : m \u2192 M\ni : m\n\u22a2 \u2191(LinearMap.comp (funLeft R M \u2191e) (funLeft R M \u2191e.symm)) x i = \u2191LinearMap.id x i\n[PROOFSTEP]\nrw [id_apply, \u2190 funLeft_comp, Equiv.symm_comp_self, LinearMap.funLeft_id]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nR\u2084 : Type u_5\nS : Type u_6\nK : Type u_7\nK\u2082 : Type u_8\nM : Type u_9\nM' : Type u_10\nM\u2081 : Type u_11\nM\u2082 : Type u_12\nM\u2083 : Type u_13\nM\u2084 : Type u_14\nN : Type u_15\nN\u2082 : Type u_16\n\u03b9 : Type u_17\nV : Type u_18\nV\u2082 : Type u_19\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nm : Type u_20\nn : Type u_21\np : Type u_22\ne : m \u2243 n\nx : n \u2192 M\ni : n\n\u22a2 \u2191(LinearMap.comp (funLeft R M \u2191e.symm) (funLeft R M \u2191e)) x i = \u2191LinearMap.id x i\n[PROOFSTEP]\nrw [id_apply, \u2190 funLeft_comp, Equiv.self_comp_symm, LinearMap.funLeft_id]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Basic", "llama_tokens": 192002, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.6825737344123242, "lm_q1q2_score": 0.5465313299847161}}
{"text": "[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\n\u22a2 rank 1 = Fintype.card n\n[PROOFSTEP]\nrw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Nontrivial R\n\u22a2 rank 0 = 0\n[PROOFSTEP]\nrw [rank, mulVecLin_zero, LinearMap.range_zero, finrank_bot]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\n\u22a2 rank A \u2264 Fintype.card n\n[PROOFSTEP]\nhaveI : Module.Finite R (n \u2192 R) := Module.Finite.pi\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nthis : Module.Finite R (n \u2192 R)\n\u22a2 rank A \u2264 Fintype.card n\n[PROOFSTEP]\nhaveI : Module.Free R (n \u2192 R) := Module.Free.pi _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nthis\u271d : Module.Finite R (n \u2192 R)\nthis : Module.Free R (n \u2192 R)\n\u22a2 rank A \u2264 Fintype.card n\n[PROOFSTEP]\nexact A.mulVecLin.finrank_range_le.trans_eq (finrank_pi _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nB : Matrix n o R\n\u22a2 rank (A * B) \u2264 rank A\n[PROOFSTEP]\nrw [rank, rank, mulVecLin_mul]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nB : Matrix n o R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (LinearMap.comp (mulVecLin A) (mulVecLin B)) } \u2264\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }\n[PROOFSTEP]\nexact Cardinal.toNat_le_of_le_of_lt_aleph0 (rank_lt_aleph0 _ _) (LinearMap.rank_comp_le_left _ _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix l m R\nB : Matrix m n R\n\u22a2 rank (A * B) \u2264 rank B\n[PROOFSTEP]\nrw [rank, rank, mulVecLin_mul]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix l m R\nB : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (LinearMap.comp (mulVecLin A) (mulVecLin B)) } \u2264\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin B) }\n[PROOFSTEP]\nexact finrank_le_finrank_of_rank_le_rank (LinearMap.lift_rank_comp_le_right _ _) (rank_lt_aleph0 _ _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\nA : (Matrix n n R)\u02e3\n\u22a2 rank \u2191A = Fintype.card n\n[PROOFSTEP]\nrefine' le_antisymm (rank_le_card_width A) _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\nA : (Matrix n n R)\u02e3\n\u22a2 Fintype.card n \u2264 rank \u2191A\n[PROOFSTEP]\nhave := rank_mul_le_left (A : Matrix n n R) (\u2191A\u207b\u00b9 : Matrix n n R)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\nA : (Matrix n n R)\u02e3\nthis : rank (\u2191A * \u2191A\u207b\u00b9) \u2264 rank \u2191A\n\u22a2 Fintype.card n \u2264 rank \u2191A\n[PROOFSTEP]\nrwa [\u2190 Units.val_mul, mul_inv_self, Units.val_one, rank_one] at this \n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nh : IsUnit A\n\u22a2 rank A = Fintype.card n\n[PROOFSTEP]\nobtain \u27e8A, rfl\u27e9 := h\n[GOAL]\ncase intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : DecidableEq n\nA : (Matrix n n R)\u02e3\n\u22a2 rank \u2191A = Fintype.card n\n[PROOFSTEP]\nexact rank_unit A\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit (det A)\n\u22a2 rank (B * A) = rank B\n[PROOFSTEP]\nsuffices : Function.Surjective A.mulVecLin\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit (det A)\nthis : Function.Surjective \u2191(mulVecLin A)\n\u22a2 rank (B * A) = rank B\n[PROOFSTEP]\nrw [rank, mulVecLin_mul, LinearMap.range_comp_of_range_eq_top _ (LinearMap.range_eq_top.mpr this), \u2190 rank]\n[GOAL]\ncase this\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit (det A)\n\u22a2 Function.Surjective \u2191(mulVecLin A)\n[PROOFSTEP]\nintro v\n[GOAL]\ncase this\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit (det A)\nv : n \u2192 R\n\u22a2 \u2203 a, \u2191(mulVecLin A) a = v\n[PROOFSTEP]\nexact \u27e8(A\u207b\u00b9).mulVecLin v, by simp [mul_nonsing_inv _ hA]\u27e9\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq n\nA : Matrix n n R\nB : Matrix m n R\nhA : IsUnit (det A)\nv : n \u2192 R\n\u22a2 \u2191(mulVecLin A) (\u2191(mulVecLin A\u207b\u00b9) v) = v\n[PROOFSTEP]\nsimp [mul_nonsing_inv _ hA]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nhA : IsUnit (det A)\n\u22a2 rank (A * B) = rank B\n[PROOFSTEP]\nlet b : Basis m R (m \u2192 R) := Pi.basisFun R m\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nhA : IsUnit (det A)\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\n\u22a2 rank (A * B) = rank B\n[PROOFSTEP]\nreplace hA : IsUnit (LinearMap.toMatrix b b A.mulVecLin).det := by convert hA; rw [\u2190 LinearEquiv.eq_symm_apply]; rfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nhA : IsUnit (det A)\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\n\u22a2 IsUnit (det (\u2191(LinearMap.toMatrix b b) (mulVecLin A)))\n[PROOFSTEP]\nconvert hA\n[GOAL]\ncase h.e'_3.h.e'_6\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nhA : IsUnit (det A)\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\n\u22a2 \u2191(LinearMap.toMatrix b b) (mulVecLin A) = A\n[PROOFSTEP]\nrw [\u2190 LinearEquiv.eq_symm_apply]\n[GOAL]\ncase h.e'_3.h.e'_6\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nhA : IsUnit (det A)\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\n\u22a2 mulVecLin A = \u2191(LinearEquiv.symm (LinearMap.toMatrix b b)) A\n[PROOFSTEP]\nrfl\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\nhA : IsUnit (det (\u2191(LinearMap.toMatrix b b) (mulVecLin A)))\n\u22a2 rank (A * B) = rank B\n[PROOFSTEP]\nhave hAB : mulVecLin (A * B) = (LinearEquiv.ofIsUnitDet hA).comp (mulVecLin B) := by ext; simp\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\nhA : IsUnit (det (\u2191(LinearMap.toMatrix b b) (mulVecLin A)))\n\u22a2 mulVecLin (A * B) = LinearMap.comp (\u2191(LinearEquiv.ofIsUnitDet hA)) (mulVecLin B)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\nhA : IsUnit (det (\u2191(LinearMap.toMatrix b b) (mulVecLin A)))\nx\u271d\u00b9 : n \u2192 R\nx\u271d : m\n\u22a2 \u2191(mulVecLin (A * B)) x\u271d\u00b9 x\u271d = \u2191(LinearMap.comp (\u2191(LinearEquiv.ofIsUnitDet hA)) (mulVecLin B)) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq m\nA : Matrix m m R\nB : Matrix m n R\nb : Basis m R (m \u2192 R) := Pi.basisFun R m\nhA : IsUnit (det (\u2191(LinearMap.toMatrix b b) (mulVecLin A)))\nhAB : mulVecLin (A * B) = LinearMap.comp (\u2191(LinearEquiv.ofIsUnitDet hA)) (mulVecLin B)\n\u22a2 rank (A * B) = rank B\n[PROOFSTEP]\nrw [rank, rank, hAB, LinearMap.range_comp, LinearEquiv.finrank_map_eq]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : Fintype m\nf : n \u2192 m\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 rank (submatrix A f \u2191e) \u2264 rank A\n[PROOFSTEP]\nrw [rank, rank, mulVecLin_submatrix, LinearMap.range_comp, LinearMap.range_comp,\n  show LinearMap.funLeft R R e.symm = LinearEquiv.funCongrLeft R R e.symm from rfl, LinearEquiv.range,\n  Submodule.map_top]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : StrongRankCondition R\ninst\u271d : Fintype m\nf : n \u2192 m\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 finrank R { x // x \u2208 Submodule.map (LinearMap.funLeft R R f) (LinearMap.range (mulVecLin A)) } \u2264\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }\n[PROOFSTEP]\nexact Submodule.finrank_map_le _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype m\ne\u2081 e\u2082 : m \u2243 n\nA : Matrix m m R\n\u22a2 rank (\u2191(reindex e\u2081 e\u2082) A) = rank A\n[PROOFSTEP]\nrw [rank, rank, mulVecLin_reindex, LinearMap.range_comp, LinearMap.range_comp, LinearEquiv.range, Submodule.map_top,\n  LinearEquiv.finrank_map_eq]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Fintype m\nA : Matrix m m R\ne\u2081 e\u2082 : n \u2243 m\n\u22a2 rank (submatrix A \u2191e\u2081 \u2191e\u2082) = rank A\n[PROOFSTEP]\nsimpa only [reindex_apply] using rank_reindex e\u2081.symm e\u2082.symm A\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\n\u22a2 rank A = finrank R { x // x \u2208 LinearMap.range (\u2191(toLin v\u2082 v\u2081) A) }\n[PROOFSTEP]\nlet e\u2081 := (Pi.basisFun R m).equiv v\u2081 (Equiv.refl _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\n\u22a2 rank A = finrank R { x // x \u2208 LinearMap.range (\u2191(toLin v\u2082 v\u2081) A) }\n[PROOFSTEP]\nlet e\u2082 := (Pi.basisFun R n).equiv v\u2082 (Equiv.refl _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\n\u22a2 rank A = finrank R { x // x \u2208 LinearMap.range (\u2191(toLin v\u2082 v\u2081) A) }\n[PROOFSTEP]\nhave range_e\u2082 : LinearMap.range e\u2082 = \u22a4 := by\n  rw [LinearMap.range_eq_top]\n  exact e\u2082.surjective\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\n\u22a2 LinearMap.range e\u2082 = \u22a4\n[PROOFSTEP]\nrw [LinearMap.range_eq_top]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\n\u22a2 Function.Surjective \u2191e\u2082\n[PROOFSTEP]\nexact e\u2082.surjective\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\n\u22a2 rank A = finrank R { x // x \u2208 LinearMap.range (\u2191(toLin v\u2082 v\u2081) A) }\n[PROOFSTEP]\nrefine' LinearEquiv.finrank_eq (e\u2081.ofSubmodules _ _ _)\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\n\u22a2 Submodule.map (\u2191e\u2081) (LinearMap.range (mulVecLin A)) = LinearMap.range (\u2191(toLin v\u2082 v\u2081) A)\n[PROOFSTEP]\nrw [\u2190 LinearMap.range_comp, \u2190 LinearMap.range_comp_of_range_eq_top (toLin v\u2082 v\u2081 A) range_e\u2082]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\n\u22a2 LinearMap.range (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) = LinearMap.range (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_f\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\n\u22a2 LinearMap.comp (\u2191e\u2081) (mulVecLin A) = LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082\n[PROOFSTEP]\napply LinearMap.pi_ext'\n[GOAL]\ncase e_f.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\n\u22a2 \u2200 (i : n),\n    LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i) =\n      LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)\n[PROOFSTEP]\nrintro i\n[GOAL]\ncase e_f.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\n\u22a2 LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i) =\n    LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)\n[PROOFSTEP]\napply LinearMap.ext_ring\n[GOAL]\ncase e_f.h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)) 1\n[PROOFSTEP]\nhave aux\u2081 := toLin_self (Pi.basisFun R n) (Pi.basisFun R m) A i\n[GOAL]\ncase e_f.h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\naux\u2081 :\n  \u2191(\u2191(toLin (Pi.basisFun R n) (Pi.basisFun R m)) A) (\u2191(Pi.basisFun R n) i) =\n    Finset.sum Finset.univ fun j => A j i \u2022 \u2191(Pi.basisFun R m) j\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)) 1\n[PROOFSTEP]\nhave aux\u2082 := Basis.equiv_apply (Pi.basisFun R n) i v\u2082\n[GOAL]\ncase e_f.h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\naux\u2081 :\n  \u2191(\u2191(toLin (Pi.basisFun R n) (Pi.basisFun R m)) A) (\u2191(Pi.basisFun R n) i) =\n    Finset.sum Finset.univ fun j => A j i \u2022 \u2191(Pi.basisFun R m) j\naux\u2082 : \u2200 (e : n \u2243 n), \u2191(Basis.equiv (Pi.basisFun R n) v\u2082 e) (\u2191(Pi.basisFun R n) i) = \u2191v\u2082 (\u2191e i)\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)) 1\n[PROOFSTEP]\nrw [toLin_eq_toLin', toLin'_apply'] at aux\u2081 \n[GOAL]\ncase e_f.h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\naux\u2081 : \u2191(mulVecLin A) (\u2191(Pi.basisFun R n) i) = Finset.sum Finset.univ fun j => A j i \u2022 \u2191(Pi.basisFun R m) j\naux\u2082 : \u2200 (e : n \u2243 n), \u2191(Basis.equiv (Pi.basisFun R n) v\u2082 e) (\u2191(Pi.basisFun R n) i) = \u2191v\u2082 (\u2191e i)\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)) 1\n[PROOFSTEP]\nrw [Pi.basisFun_apply, LinearMap.coe_stdBasis] at aux\u2081 aux\u2082 \n[GOAL]\ncase e_f.h.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype o\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_6\nM\u2082 : Type u_7\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nA : Matrix m n R\nv\u2081 : Basis m R M\u2081\nv\u2082 : Basis n R M\u2082\ne\u2081 : (m \u2192 R) \u2243\u2097[R] M\u2081 := Basis.equiv (Pi.basisFun R m) v\u2081 (Equiv.refl m)\ne\u2082 : (n \u2192 R) \u2243\u2097[R] M\u2082 := Basis.equiv (Pi.basisFun R n) v\u2082 (Equiv.refl n)\nrange_e\u2082 : LinearMap.range e\u2082 = \u22a4\ni : n\naux\u2081 : \u2191(mulVecLin A) (Pi.single i 1) = Finset.sum Finset.univ fun j => A j i \u2022 \u2191(Pi.basisFun R m) j\naux\u2082 : \u2200 (e : n \u2243 n), \u2191(Basis.equiv (Pi.basisFun R n) v\u2082 e) (Pi.single i 1) = \u2191v\u2082 (\u2191e i)\n\u22a2 \u2191(LinearMap.comp (LinearMap.comp (\u2191e\u2081) (mulVecLin A)) (LinearMap.single i)) 1 =\n    \u2191(LinearMap.comp (LinearMap.comp (\u2191(toLin v\u2082 v\u2081) A) \u2191e\u2082) (LinearMap.single i)) 1\n[PROOFSTEP]\nsimp only [LinearMap.comp_apply, LinearEquiv.coe_coe, Equiv.refl_apply, aux\u2081, aux\u2082, LinearMap.coe_single, toLin_self,\n  LinearEquiv.map_sum, LinearEquiv.map_smul, Basis.equiv_apply]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\n\u22a2 rank A \u2264 Fintype.card m\n[PROOFSTEP]\nhaveI : Module.Finite R (m \u2192 R) := Module.Finite.pi\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nthis : Module.Finite R (m \u2192 R)\n\u22a2 rank A \u2264 Fintype.card m\n[PROOFSTEP]\nhaveI : Module.Free R (m \u2192 R) := Module.Free.pi _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : CommRing R\ninst\u271d : StrongRankCondition R\nA : Matrix m n R\nthis\u271d : Module.Finite R (m \u2192 R)\nthis : Module.Free R (m \u2192 R)\n\u22a2 rank A \u2264 Fintype.card m\n[PROOFSTEP]\nexact (Submodule.finrank_le _).trans (finrank_pi R).le\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : CommRing R\nA : Matrix m n R\n\u22a2 rank A = finrank R { x // x \u2208 Submodule.span R (Set.range A\u1d40) }\n[PROOFSTEP]\nrw [rank, Matrix.range_mulVecLin]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : Fintype o\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq R\nw : m \u2192 R\n\u22a2 rank (diagonal w) = Fintype.card { i // w i \u2260 0 }\n[PROOFSTEP]\nrw [Matrix.rank, \u2190 Matrix.toLin'_apply', FiniteDimensional.finrank, \u2190 LinearMap.rank, LinearMap.rank_diagonal,\n  Cardinal.toNat_cast]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 LinearMap.ker (mulVecLin (A\u1d34 * A)) = LinearMap.ker (mulVecLin A)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 x \u2208 LinearMap.ker (mulVecLin (A\u1d34 * A)) \u2194 x \u2208 LinearMap.ker (mulVecLin A)\n[PROOFSTEP]\nsimp only [LinearMap.mem_ker, mulVecLin_apply, conjTranspose_mul_self_mulVec_eq_zero]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 rank (A\u1d34 * A) = rank A\n[PROOFSTEP]\ndsimp only [rank]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } = finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }\n[PROOFSTEP]\nrefine' add_left_injective (finrank R (LinearMap.ker (mulVecLin A))) _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 (fun x => x + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) })\n      (finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) }) =\n    (fun x => x + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) })\n      (finrank R { x // x \u2208 LinearMap.range (mulVecLin A) })\n[PROOFSTEP]\ndsimp only\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) }\n[PROOFSTEP]\ntrans\n  finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } +\n    finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d34 * A)) }\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } +\n      finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d34 * A)) }\n[PROOFSTEP]\nrw [ker_mulVecLin_conjTranspose_mul_self]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d34 * A)) } +\n      finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d34 * A)) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) }\n[PROOFSTEP]\nsimp only [LinearMap.finrank_range_add_finrank_ker]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Field R\ninst\u271d\u00b9 : PartialOrder R\ninst\u271d : StarOrderedRing R\nA : Matrix m n R\n\u22a2 rank (A * A\u1d34) = rank A\n[PROOFSTEP]\nsimpa only [rank_conjTranspose, conjTranspose_conjTranspose] using rank_conjTranspose_mul_self A\u1d34\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 LinearMap.ker (mulVecLin (A\u1d40 * A)) = LinearMap.ker (mulVecLin A)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 x \u2208 LinearMap.ker (mulVecLin (A\u1d40 * A)) \u2194 x \u2208 LinearMap.ker (mulVecLin A)\n[PROOFSTEP]\nsimp only [LinearMap.mem_ker, mulVecLin_apply, \u2190 mulVec_mulVec]\n[GOAL]\ncase h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 mulVec A\u1d40 (mulVec A x) = 0 \u2194 mulVec A x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 mulVec A\u1d40 (mulVec A x) = 0 \u2192 mulVec A x = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\nh : mulVec A\u1d40 (mulVec A x) = 0\n\u22a2 mulVec A x = 0\n[PROOFSTEP]\nreplace h := congr_arg (dotProduct x) h\n[GOAL]\ncase h.mp\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\nh : x \u2b1d\u1d65 mulVec A\u1d40 (mulVec A x) = x \u2b1d\u1d65 0\n\u22a2 mulVec A x = 0\n[PROOFSTEP]\nrwa [dotProduct_mulVec, dotProduct_zero, vecMul_transpose, dotProduct_self_eq_zero] at h \n[GOAL]\ncase h.mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\n\u22a2 mulVec A x = 0 \u2192 mulVec A\u1d40 (mulVec A x) = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mpr\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\nx : n \u2192 R\nh : mulVec A x = 0\n\u22a2 mulVec A\u1d40 (mulVec A x) = 0\n[PROOFSTEP]\nrw [h, mulVec_zero]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 rank (A\u1d40 * A) = rank A\n[PROOFSTEP]\ndsimp only [rank]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } = finrank R { x // x \u2208 LinearMap.range (mulVecLin A) }\n[PROOFSTEP]\nrefine' add_left_injective (finrank R <| LinearMap.ker A.mulVecLin) _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 (fun x => x + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) })\n      (finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) }) =\n    (fun x => x + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) })\n      (finrank R { x // x \u2208 LinearMap.range (mulVecLin A) })\n[PROOFSTEP]\ndsimp only\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) }\n[PROOFSTEP]\ntrans\n  finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } +\n    finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d40 * A)) }\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } +\n      finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d40 * A)) }\n[PROOFSTEP]\nrw [ker_mulVecLin_transpose_mul_self]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 finrank R { x // x \u2208 LinearMap.range (mulVecLin (A\u1d40 * A)) } +\n      finrank R { x // x \u2208 LinearMap.ker (mulVecLin (A\u1d40 * A)) } =\n    finrank R { x // x \u2208 LinearMap.range (mulVecLin A) } + finrank R { x // x \u2208 LinearMap.ker (mulVecLin A) }\n[PROOFSTEP]\nsimp only [LinearMap.finrank_range_add_finrank_ker]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : Fintype m\ninst\u271d : LinearOrderedField R\nA : Matrix m n R\n\u22a2 rank (A * A\u1d40) = rank A\n[PROOFSTEP]\nsimpa only [rank_transpose, transpose_transpose] using rank_transpose_mul_self A\u1d40\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Finite m\nA : Matrix m n R\n\u22a2 rank A = finrank R { x // x \u2208 Submodule.span R (Set.range A) }\n[PROOFSTEP]\ncases nonempty_fintype m\n[GOAL]\ncase intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\nm_fin : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : LinearOrderedField R\ninst\u271d : Finite m\nA : Matrix m n R\nval\u271d : Fintype m\n\u22a2 rank A = finrank R { x // x \u2208 Submodule.span R (Set.range A) }\n[PROOFSTEP]\nrw [\u2190 rank_transpose, rank_eq_finrank_span_cols, transpose_transpose]\n", "meta": {"mathlib_filename": "Mathlib.Data.Matrix.Rank", "llama_tokens": 17397, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.6825737344123242, "lm_q1q2_score": 0.5465313299847161}}
{"text": "[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\n\u22a2 r \u2208 annihilator (span R s) \u2194 \u2200 (n : \u2191s), r \u2022 \u2191n = 0\n[PROOFSTEP]\nrw [Submodule.mem_annihilator]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\n\u22a2 (\u2200 (n : M), n \u2208 span R s \u2192 r \u2022 n = 0) \u2194 \u2200 (n : \u2191s), r \u2022 \u2191n = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\n\u22a2 (\u2200 (n : M), n \u2208 span R s \u2192 r \u2022 n = 0) \u2192 \u2200 (n : \u2191s), r \u2022 \u2191n = 0\n[PROOFSTEP]\nintro h n\n[GOAL]\ncase mp\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : M), n \u2208 span R s \u2192 r \u2022 n = 0\nn : \u2191s\n\u22a2 r \u2022 \u2191n = 0\n[PROOFSTEP]\nexact h _ (Submodule.subset_span n.prop)\n[GOAL]\ncase mpr\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\n\u22a2 (\u2200 (n : \u2191s), r \u2022 \u2191n = 0) \u2192 \u2200 (n : M), n \u2208 span R s \u2192 r \u2022 n = 0\n[PROOFSTEP]\nintro h n hn\n[GOAL]\ncase mpr\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\n\u22a2 r \u2022 n = 0\n[PROOFSTEP]\nrefine Submodule.span_induction hn ?_ ?_ ?_ ?_\n[GOAL]\ncase mpr.refine_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\n\u22a2 \u2200 (x : M), x \u2208 s \u2192 r \u2022 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase mpr.refine_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\nx : M\nhx : x \u2208 s\n\u22a2 r \u2022 x = 0\n[PROOFSTEP]\nexact h \u27e8x, hx\u27e9\n[GOAL]\ncase mpr.refine_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\n\u22a2 r \u2022 0 = 0\n[PROOFSTEP]\nexact smul_zero _\n[GOAL]\ncase mpr.refine_3\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\n\u22a2 \u2200 (x y : M), r \u2022 x = 0 \u2192 r \u2022 y = 0 \u2192 r \u2022 (x + y) = 0\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase mpr.refine_3\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\nx y : M\nhx : r \u2022 x = 0\nhy : r \u2022 y = 0\n\u22a2 r \u2022 (x + y) = 0\n[PROOFSTEP]\nrw [smul_add, hx, hy, zero_add]\n[GOAL]\ncase mpr.refine_4\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\n\u22a2 \u2200 (a : R) (x : M), r \u2022 x = 0 \u2192 r \u2022 a \u2022 x = 0\n[PROOFSTEP]\nintro a x hx\n[GOAL]\ncase mpr.refine_4\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ns : Set M\nr : R\nh : \u2200 (n : \u2191s), r \u2022 \u2191n = 0\nn : M\nhn : n \u2208 span R s\na : R\nx : M\nhx : r \u2022 x = 0\n\u22a2 r \u2022 a \u2022 x = 0\n[PROOFSTEP]\nrw [smul_comm, hx, smul_zero]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\ng : M\nr : R\n\u22a2 r \u2208 annihilator (span R {g}) \u2194 r \u2022 g = 0\n[PROOFSTEP]\nsimp [mem_annihilator_span]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\n\u22a2 p x\n[PROOFSTEP]\nhave H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\n\u22a2 p 0\n[PROOFSTEP]\nsimpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH0 : p 0\n\u22a2 p x\n[PROOFSTEP]\nrefine Submodule.iSup_induction (x := x) _ H ?_ H0 H1\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH0 : p 0\n\u22a2 \u2200 (i : { x // x \u2208 I }) (x : M), x \u2208 map (\u2191(LinearMap.lsmul R M) \u2191i) N \u2192 p x\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9 m \u27e8j, hj, hj'\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH0 : p 0\ni : R\nhi : i \u2208 I\nm j : M\nhj : j \u2208 \u2191N\nhj' : \u2191(\u2191(LinearMap.lsmul R M) \u2191{ val := i, property := hi }) j = m\n\u22a2 p m\n[PROOFSTEP]\nrw [\u2190 hj']\n[GOAL]\ncase mk.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\np : M \u2192 Prop\nx : M\nH : x \u2208 I \u2022 N\nHb : \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : M), n \u2208 N \u2192 p (r \u2022 n)\nH1 : \u2200 (x y : M), p x \u2192 p y \u2192 p (x + y)\nH0 : p 0\ni : R\nhi : i \u2208 I\nm j : M\nhj : j \u2208 \u2191N\nhj' : \u2191(\u2191(LinearMap.lsmul R M) \u2191{ val := i, property := hi }) j = m\n\u22a2 p (\u2191(\u2191(LinearMap.lsmul R M) \u2191{ val := i, property := hi }) j)\n[PROOFSTEP]\nexact Hb _ hi _ hj\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nx : M\nhx : x \u2208 I \u2022 N\np : (x : M) \u2192 x \u2208 I \u2022 N \u2192 Prop\nHb : \u2200 (r : R) (hr : r \u2208 I) (n : M) (hn : n \u2208 N), p (r \u2022 n) (_ : r \u2022 n \u2208 I \u2022 N)\nH1 : \u2200 (x : M) (hx : x \u2208 I \u2022 N) (y : M) (hy : y \u2208 I \u2022 N), p x hx \u2192 p y hy \u2192 p (x + y) (_ : x + y \u2208 I \u2022 N)\n\u22a2 p x hx\n[PROOFSTEP]\nrefine' Exists.elim _ fun (h : x \u2208 I \u2022 N) (H : p x h) => H\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nx : M\nhx : x \u2208 I \u2022 N\np : (x : M) \u2192 x \u2208 I \u2022 N \u2192 Prop\nHb : \u2200 (r : R) (hr : r \u2208 I) (n : M) (hn : n \u2208 N), p (r \u2022 n) (_ : r \u2022 n \u2208 I \u2022 N)\nH1 : \u2200 (x : M) (hx : x \u2208 I \u2022 N) (y : M) (hy : y \u2208 I \u2022 N), p x hx \u2192 p y hy \u2192 p (x + y) (_ : x + y \u2208 I \u2022 N)\n\u22a2 \u2203 x_1, p x x_1\n[PROOFSTEP]\nexact smul_induction_on hx (fun a ha x hx => \u27e8_, Hb _ ha _ hx\u27e9) fun x y \u27e8_, hx\u27e9 \u27e8_, hy\u27e9 => \u27e8_, H1 _ _ _ _ hx hy\u27e9\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nI : Ideal R\nm x : M\nhx : x \u2208 I \u2022 span R {m}\nm1 m2 : M\nx\u271d\u00b9 : \u2203 y, y \u2208 I \u2227 y \u2022 m = m1\nx\u271d : \u2203 y, y \u2208 I \u2227 y \u2022 m = m2\ny1 : R\nhyi1 : y1 \u2208 I\nhy1 : y1 \u2022 m = m1\ny2 : R\nhyi2 : y2 \u2208 I\nhy2 : y2 \u2022 m = m2\n\u22a2 (y1 + y2) \u2022 m = m1 + m2\n[PROOFSTEP]\nrw [add_smul, hy1, hy2]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nI : Ideal R\nf : R \u2192\u2097[R] M\n\u22a2 map f I \u2264 I \u2022 \u22a4\n[PROOFSTEP]\nrintro _ \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nI : Ideal R\nf : R \u2192\u2097[R] M\ny : R\nhy : y \u2208 \u2191I\n\u22a2 \u2191f y \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrw [\u2190 mul_one y, \u2190 smul_eq_mul, f.map_smul]\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nI : Ideal R\nf : R \u2192\u2097[R] M\ny : R\nhy : y \u2208 \u2191I\n\u22a2 y \u2022 \u2191f 1 \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact smul_mem_smul hy mem_top\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nI : Ideal R\n\u22a2 I * annihilator I = \u22a5\n[PROOFSTEP]\nrw [mul_comm, annihilator_mul]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\n\u22a2 Ideal.span {r} \u2022 N = r \u2022 N\n[PROOFSTEP]\nhave : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N :=\n  by\n  convert span_eq (r \u2022 N)\n  exact (Set.image_eq_iUnion _ (N : Set M)).symm\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\n\u22a2 span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n[PROOFSTEP]\nconvert span_eq (r \u2022 N)\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\n\u22a2 \u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t} = \u2191(r \u2022 N)\n[PROOFSTEP]\nexact (Set.image_eq_iUnion _ (N : Set M)).symm\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\nthis : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n\u22a2 Ideal.span {r} \u2022 N = r \u2022 N\n[PROOFSTEP]\nconv_lhs => rw [\u2190 span_eq N, span_smul_span]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\nthis : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n| Ideal.span {r} \u2022 N\n[PROOFSTEP]\nrw [\u2190 span_eq N, span_smul_span]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\nthis : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n| Ideal.span {r} \u2022 N\n[PROOFSTEP]\nrw [\u2190 span_eq N, span_smul_span]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\nthis : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n| Ideal.span {r} \u2022 N\n[PROOFSTEP]\nrw [\u2190 span_eq N, span_smul_span]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nr : R\nN : Submodule R M\nthis : span R (\u22c3 (t : M) (_ : t \u2208 N), {r \u2022 t}) = r \u2022 N\n\u22a2 span R (\u22c3 (s : R) (_ : s \u2208 {r}) (t : M) (_ : t \u2208 \u2191N), {s \u2022 t}) = r \u2022 N\n[PROOFSTEP]\nsimpa\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2191r \u2022 x \u2208 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nsuffices (\u22a4 : Ideal R) \u2022 span R ({ x } : Set M) \u2264 M'\n  by\n  rw [top_smul] at this \n  exact this (subset_span (Set.mem_singleton x))\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2191r \u2022 x \u2208 M'\nthis : \u22a4 \u2022 span R {x} \u2264 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nrw [top_smul] at this \n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2191r \u2022 x \u2208 M'\nthis : span R {x} \u2264 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nexact this (subset_span (Set.mem_singleton x))\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2191r \u2022 x \u2208 M'\n\u22a2 \u22a4 \u2022 span R {x} \u2264 M'\n[PROOFSTEP]\nrw [\u2190 hs, span_smul_span, span_le]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2191r \u2022 x \u2208 M'\n\u22a2 \u22c3 (s_1 : R) (_ : s_1 \u2208 s) (t : M) (_ : t \u2208 {x}), {s_1 \u2022 t} \u2286 \u2191M'\n[PROOFSTEP]\nsimpa using H\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2203 n, \u2191r ^ n \u2022 x \u2208 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nobtain \u27e8s', hs\u2081, hs\u2082\u27e9 := (Ideal.span_eq_top_iff_finite _).mp hs\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\nH : \u2200 (r : \u2191s), \u2203 n, \u2191r ^ n \u2022 x \u2208 M'\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nreplace H : \u2200 r : s', \u2203 n : \u2115, ((r : R) ^ n : R) \u2022 x \u2208 M' := fun r => H \u27e8_, hs\u2081 r.2\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nH : \u2200 (r : { x // x \u2208 s' }), \u2203 n, \u2191r ^ n \u2022 x \u2208 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nchoose n\u2081 n\u2082 using H\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nlet N := s'.attach.sup n\u2081\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\n\u22a2 x \u2208 M'\n[PROOFSTEP]\nhave hs' := Ideal.span_pow_eq_top (s' : Set R) hs\u2082 N\n[GOAL]\ncase intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\nhs' : Ideal.span ((fun x => x ^ N) '' \u2191s') = \u22a4\n\u22a2 x \u2208 M'\n[PROOFSTEP]\napply M'.mem_of_span_top_of_smul_mem _ hs'\n[GOAL]\ncase intro.intro.H\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\nhs' : Ideal.span ((fun x => x ^ N) '' \u2191s') = \u22a4\n\u22a2 \u2200 (r : \u2191((fun x => x ^ N) '' \u2191s')), \u2191r \u2022 x \u2208 M'\n[PROOFSTEP]\nrintro \u27e8_, r, hr, rfl\u27e9\n[GOAL]\ncase intro.intro.H.mk.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\nhs' : Ideal.span ((fun x => x ^ N) '' \u2191s') = \u22a4\nr : R\nhr : r \u2208 \u2191s'\n\u22a2 \u2191{ val := (fun x => x ^ N) r, property := (_ : \u2203 a, a \u2208 \u2191s' \u2227 (fun x => x ^ N) a = (fun x => x ^ N) r) } \u2022 x \u2208 M'\n[PROOFSTEP]\nconvert M'.smul_mem (r ^ (N - n\u2081 \u27e8r, hr\u27e9)) (n\u2082 \u27e8r, hr\u27e9) using 1\n[GOAL]\ncase h.e'_4\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\nhs' : Ideal.span ((fun x => x ^ N) '' \u2191s') = \u22a4\nr : R\nhr : r \u2208 \u2191s'\n\u22a2 \u2191{ val := (fun x => x ^ N) r, property := (_ : \u2203 a, a \u2208 \u2191s' \u2227 (fun x => x ^ N) a = (fun x => x ^ N) r) } \u2022 x =\n    r ^ (N - n\u2081 { val := r, property := hr }) \u2022 \u2191{ val := r, property := hr } ^ n\u2081 { val := r, property := hr } \u2022 x\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, smul_smul, \u2190 pow_add]\n[GOAL]\ncase h.e'_4\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = \u22a4\nx : M\ns' : Finset R\nhs\u2081 : \u2191s' \u2286 s\nhs\u2082 : Ideal.span \u2191s' = \u22a4\nn\u2081 : { x // x \u2208 s' } \u2192 \u2115\nn\u2082 : \u2200 (r : { x // x \u2208 s' }), \u2191r ^ n\u2081 r \u2022 x \u2208 M'\nN : \u2115 := Finset.sup (Finset.attach s') n\u2081\nhs' : Ideal.span ((fun x => x ^ N) '' \u2191s') = \u22a4\nr : R\nhr : r \u2208 \u2191s'\n\u22a2 r ^ Finset.sup (Finset.attach s') n\u2081 \u2022 x =\n    r ^ (Finset.sup (Finset.attach s') n\u2081 - n\u2081 { val := r, property := hr } + n\u2081 { val := r, property := hr }) \u2022 x\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n\u2081 \u27e8r, hr\u27e9 \u2264 N)]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\ns : Set M\nx : M\n\u22a2 x \u2208 I \u2022 span R s \u2194 x \u2208 span R (\u22c3 (a : R) (_ : a \u2208 I) (b : M) (_ : b \u2208 s), {a \u2022 b})\n[PROOFSTEP]\nrw [\u2190 I.span_eq, Submodule.span_smul_span, I.span_eq]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\ns : Set M\nx : M\n\u22a2 x \u2208 span R (\u22c3 (s_1 : R) (_ : s_1 \u2208 \u2191I) (t : M) (_ : t \u2208 s), {s_1 \u2022 t}) \u2194\n    x \u2208 span R (\u22c3 (a : R) (_ : a \u2208 I) (b : M) (_ : b \u2208 s), {a \u2022 b})\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\n\u22a2 x \u2208 I \u2022 span R (Set.range f) \u2194 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\n\u22a2 x \u2208 I \u2022 span R (Set.range f) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x\ncase mpr\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\n\u22a2 (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 x \u2208 I \u2022 span R (Set.range f)\n[PROOFSTEP]\nswap\n[GOAL]\ncase mpr\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\n\u22a2 (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 x \u2208 I \u2022 span R (Set.range f)\n[PROOFSTEP]\nrintro \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun i c => c \u2022 f i) \u2208 I \u2022 span R (Set.range f)\n[PROOFSTEP]\nexact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _\n[GOAL]\ncase mp\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\n\u22a2 x \u2208 I \u2022 span R (Set.range f) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x\n[PROOFSTEP]\nrefine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _\n[GOAL]\ncase mp.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\n\u22a2 \u2200 (x : M),\n    x \u2208 \u22c3 (a : R) (_ : a \u2208 I) (b : M) (_ : b \u2208 Set.range f), {a \u2022 b} \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff]\n[GOAL]\ncase mp.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\n\u22a2 \u2200 (x : M), (\u2203 i h i_1 h, x = i \u2022 i_1) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x\n[PROOFSTEP]\nrintro x \u27e8y, hy, x, \u27e8i, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni : \u03b9\n\u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = y \u2022 f i\n[PROOFSTEP]\nrefine' \u27e8Finsupp.single i y, fun j => _, _\u27e9\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni j : \u03b9\n\u22a2 \u2191(Finsupp.single i y) j \u2208 I\n[PROOFSTEP]\nletI := Classical.decEq \u03b9\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni j : \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2191(Finsupp.single i y) j \u2208 I\n[PROOFSTEP]\nrw [Finsupp.single_apply]\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni j : \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 (if i = j then y else 0) \u2208 I\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni j : \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nh\u271d : i = j\n\u22a2 y \u2208 I\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni j : \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\nh\u271d : \u00aci = j\n\u22a2 0 \u2208 I\n[PROOFSTEP]\nexact I.zero_mem\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro.refine'_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni : \u03b9\n\u22a2 (Finsupp.sum (Finsupp.single i y) fun i c => c \u2022 f i) = y \u2022 f i\n[PROOFSTEP]\nrefine' @Finsupp.sum_single_index \u03b9 R M _ _ i _ (fun i y => y \u2022 f i) _\n[GOAL]\ncase mp.refine'_1.intro.intro.intro.intro.intro.refine'_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\ny : R\nhy : y \u2208 I\ni : \u03b9\n\u22a2 (fun i y => y \u2022 f i) i 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mp.refine'_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\n\u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = 0\n[PROOFSTEP]\nexact \u27e80, fun _ => I.zero_mem, Finsupp.sum_zero_index\u27e9\n[GOAL]\ncase mp.refine'_3\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\n\u22a2 \u2200 (x y : M),\n    (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192\n      (\u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = y) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x + y\n[PROOFSTEP]\nrintro x y \u27e8ax, hax, rfl\u27e9 \u27e8ay, hay, rfl\u27e9\n[GOAL]\ncase mp.refine'_3.intro.intro.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nax : \u03b9 \u2192\u2080 R\nhax : \u2200 (i : \u03b9), \u2191ax i \u2208 I\nay : \u03b9 \u2192\u2080 R\nhay : \u2200 (i : \u03b9), \u2191ay i \u2208 I\n\u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = (Finsupp.sum ax fun i c => c \u2022 f i) + Finsupp.sum ay fun i c => c \u2022 f i\n[PROOFSTEP]\nrefine' \u27e8ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _\u27e9\n[GOAL]\ncase mp.refine'_3.intro.intro.intro.intro.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nax : \u03b9 \u2192\u2080 R\nhax : \u2200 (i : \u03b9), \u2191ax i \u2208 I\nay : \u03b9 \u2192\u2080 R\nhay : \u2200 (i : \u03b9), \u2191ay i \u2208 I\n\u22a2 \u2200 (a : \u03b9), 0 \u2022 f a = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.refine'_3.intro.intro.intro.intro.refine'_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nax : \u03b9 \u2192\u2080 R\nhax : \u2200 (i : \u03b9), \u2191ax i \u2208 I\nay : \u03b9 \u2192\u2080 R\nhay : \u2200 (i : \u03b9), \u2191ay i \u2208 I\n\u22a2 \u2200 (a : \u03b9) (b\u2081 b\u2082 : R), (b\u2081 + b\u2082) \u2022 f a = b\u2081 \u2022 f a + b\u2082 \u2022 f a\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.refine'_3.intro.intro.intro.intro.refine'_1\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nax : \u03b9 \u2192\u2080 R\nhax : \u2200 (i : \u03b9), \u2191ax i \u2208 I\nay : \u03b9 \u2192\u2080 R\nhay : \u2200 (i : \u03b9), \u2191ay i \u2208 I\na\u271d : \u03b9\n\u22a2 0 \u2022 f a\u271d = 0\n[PROOFSTEP]\nsimp only [zero_smul, add_smul]\n[GOAL]\ncase mp.refine'_3.intro.intro.intro.intro.refine'_2\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nax : \u03b9 \u2192\u2080 R\nhax : \u2200 (i : \u03b9), \u2191ax i \u2208 I\nay : \u03b9 \u2192\u2080 R\nhay : \u2200 (i : \u03b9), \u2191ay i \u2208 I\na\u271d : \u03b9\nb\u2081\u271d b\u2082\u271d : R\n\u22a2 (b\u2081\u271d + b\u2082\u271d) \u2022 f a\u271d = b\u2081\u271d \u2022 f a\u271d + b\u2082\u271d \u2022 f a\u271d\n[PROOFSTEP]\nsimp only [zero_smul, add_smul]\n[GOAL]\ncase mp.refine'_4\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\n\u22a2 \u2200 (a : R) (x : M),\n    (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 \u2203 a_2 x_1, (Finsupp.sum a_2 fun i c => c \u2022 f i) = a \u2022 x\n[PROOFSTEP]\nrintro c x \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 \u2203 a_1 x, (Finsupp.sum a_1 fun i c => c \u2022 f i) = c \u2022 Finsupp.sum a fun i c => c \u2022 f i\n[PROOFSTEP]\nrefine' \u27e8c \u2022 a, fun i => I.mul_mem_left c (ha i), _\u27e9\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum (c \u2022 a) fun i c => c \u2022 f i) = c \u2022 Finsupp.sum a fun i c => c \u2022 f i\n[PROOFSTEP]\nrw [Finsupp.sum_smul_index, Finsupp.smul_sum]\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun i a => (c * a) \u2022 f i) = Finsupp.sum a fun a b => c \u2022 b \u2022 f a\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 \u2200 (i : \u03b9), 0 \u2022 f i = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun i a => (c * a) \u2022 f i) = Finsupp.sum a fun a b => c \u2022 b \u2022 f a\n[PROOFSTEP]\nsimp only [zero_smul, mul_smul]\n[GOAL]\ncase mp.refine'_4.intro.intro\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\nf : \u03b9 \u2192 M\nx : M\nhx : x \u2208 I \u2022 span R (Set.range f)\nc : R\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\ni\u271d : \u03b9\n\u22a2 0 \u2022 f i\u271d = 0\n[PROOFSTEP]\nsimp only [zero_smul, mul_smul]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\n\u03b9 : Type u_3\ns : Set \u03b9\nf : \u03b9 \u2192 M\nx : M\n\u22a2 x \u2208 I \u2022 span R (f '' s) \u2194 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f \u2191i) = x\n[PROOFSTEP]\nrw [\u2190 Submodule.mem_ideal_smul_span_iff_exists_sum, \u2190 Set.image_eq_range]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : { x // x \u2208 N }\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191x \u2208 I \u2022 N\n[PROOFSTEP]\nchange _ \u2194 N.subtype x \u2208 I \u2022 N\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : { x // x \u2208 N }\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 I \u2022 N\n[PROOFSTEP]\nhave : Submodule.map N.subtype (I \u2022 \u22a4) = I \u2022 N := by\n  rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : { x // x \u2208 N }\n\u22a2 map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N\n[PROOFSTEP]\nrw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : { x // x \u2208 N }\nthis : map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 I \u2022 N\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : { x // x \u2208 N }\nthis : map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 map (Submodule.subtype N) (I \u2022 \u22a4)\n[PROOFSTEP]\nexact (Function.Injective.mem_set_image N.injective_subtype).symm\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nS\u271d : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nS : Submodule R M'\nI : Ideal R\n\u22a2 I \u2022 comap f S \u2264 comap f (I \u2022 S)\n[PROOFSTEP]\nrefine' Submodule.smul_le.mpr fun r hr x hx => _\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nS\u271d : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nS : Submodule R M'\nI : Ideal R\nr : R\nhr : r \u2208 I\nx : M\nhx : x \u2208 comap f S\n\u22a2 r \u2022 x \u2208 comap f (I \u2022 S)\n[PROOFSTEP]\nrw [Submodule.mem_comap] at hx \u22a2\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nS\u271d : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nS : Submodule R M'\nI : Ideal R\nr : R\nhr : r \u2208 I\nx : M\nhx : \u2191f x \u2208 S\n\u22a2 \u2191f (r \u2022 x) \u2208 I \u2022 S\n[PROOFSTEP]\nrw [f.map_smul]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nI\u271d J : Ideal R\nN P : Submodule R M\nS\u271d : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nS : Submodule R M'\nI : Ideal R\nr : R\nhr : r \u2208 I\nx : M\nhx : \u2191f x \u2208 S\n\u22a2 r \u2022 \u2191f x \u2208 I \u2022 S\n[PROOFSTEP]\nexact Submodule.smul_mem_smul hr hx\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 r \u2208 colon N (span R {x}) \u2194 \u2200 (a : R), r \u2022 a \u2022 x \u2208 N\n[PROOFSTEP]\nsimp [Submodule.mem_colon, Submodule.mem_span_singleton]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 (\u2200 (a : R), r \u2022 a \u2022 x \u2208 N) \u2194 r \u2022 x \u2208 N\n[PROOFSTEP]\nsimp_rw [fun (a : R) \u21a6 smul_comm r a x]\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN\u271d N\u2081 N\u2082 P P\u2081 P\u2082 N : Submodule R M\nx : M\nr : R\n\u22a2 (\u2200 (a : R), a \u2022 r \u2022 x \u2208 N) \u2194 r \u2022 x \u2208 N\n[PROOFSTEP]\nexact SetLike.forall_smul_mem_iff\n[GOAL]\nR : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN N\u2081 N\u2082 P P\u2081 P\u2082 : Submodule R M\nI : Ideal R\nx r : R\n\u22a2 r \u2208 colon I (Ideal.span {x}) \u2194 r * x \u2208 I\n[PROOFSTEP]\nsimp only [\u2190 Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 1 = \u22a4\n[PROOFSTEP]\nerw [Submodule.one_eq_range, LinearMap.range_id]\n[GOAL]\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i\n[PROOFSTEP]\nclassical\nrefine Finset.induction_on s ?_ ?_\n\u00b7 intro\n  rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]\n  exact Submodule.mem_top\n\u00b7 intro a s ha IH h\n  rw [Finset.prod_insert ha, Finset.prod_insert ha]\n  exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi)\n[GOAL]\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i\n[PROOFSTEP]\nrefine Finset.induction_on s ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 \u2205 \u2192 x i \u2208 I i) \u2192 \u220f i in \u2205, x i \u2208 \u220f i in \u2205, I i\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine_1\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\na\u271d : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 x i \u2208 I i\n\u22a2 \u220f i in \u2205, x i \u2208 \u220f i in \u2205, I i\n[PROOFSTEP]\nrw [Finset.prod_empty, Finset.prod_empty, one_eq_top]\n[GOAL]\ncase refine_1\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\na\u271d : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 x i \u2208 I i\n\u22a2 1 \u2208 \u22a4\n[PROOFSTEP]\nexact Submodule.mem_top\n[GOAL]\ncase refine_2\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 x i \u2208 I i) \u2192 \u220f i in insert a s, x i \u2208 \u220f i in insert a s, I i\n[PROOFSTEP]\nintro a s ha IH h\n[GOAL]\ncase refine_2\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns\u271d : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 x i \u2208 I i\n\u22a2 \u220f i in insert a s, x i \u2208 \u220f i in insert a s, I i\n[PROOFSTEP]\nrw [Finset.prod_insert ha, Finset.prod_insert ha]\n[GOAL]\ncase refine_2\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns\u271d : Finset \u03b9\nI : \u03b9 \u2192 Ideal R\nx : \u03b9 \u2192 R\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nIH : (\u2200 (i : \u03b9), i \u2208 s \u2192 x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i\nh : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 x i \u2208 I i\n\u22a2 x a * \u220f x_1 in s, x x_1 \u2208 I a * \u220f x in s, I x\n[PROOFSTEP]\nexact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi)\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nS T : Set R\n\u22a2 span S * span T = span (S * T)\n[PROOFSTEP]\nunfold span\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nS T : Set R\n\u22a2 Submodule.span R S * Submodule.span R T = Submodule.span R (S * T)\n[PROOFSTEP]\nrw [Submodule.span_mul_span]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nr s : R\n\u22a2 span {r} * span {s} = span {r * s}\n[PROOFSTEP]\nunfold span\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nr s : R\n\u22a2 Submodule.span R {r} * Submodule.span R {s} = Submodule.span R {r * s}\n[PROOFSTEP]\nrw [Submodule.span_mul_span, Set.singleton_mul_singleton]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : R\nn : \u2115\n\u22a2 span {s} ^ n = span {s ^ n}\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : R\n\u22a2 span {s} ^ Nat.zero = span {s ^ Nat.zero}\n[PROOFSTEP]\nsimp [Set.singleton_one]\n[GOAL]\ncase succ\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : R\nn : \u2115\nih : span {s} ^ n = span {s ^ n}\n\u22a2 span {s} ^ Nat.succ n = span {s ^ Nat.succ n}\n[PROOFSTEP]\nsimp only [pow_succ, ih, span_singleton_mul_span_singleton]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\nx y : R\nI : Ideal R\n\u22a2 x \u2208 span {y} * I \u2194 \u2203 z, z \u2208 I \u2227 y * z = x\n[PROOFSTEP]\nsimp only [mul_comm, mem_mul_span_singleton]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 (\u2200 {zI : R}, zI \u2208 I \u2192 zI \u2208 span {x} * J) \u2194 \u2200 (zI : R), zI \u2208 I \u2192 \u2203 zJ, zJ \u2208 J \u2227 x * zJ = zI\n[PROOFSTEP]\nsimp only [mem_span_singleton_mul]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 span {x} * I \u2264 J \u2194 \u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J\n[PROOFSTEP]\nsimp only [mul_le, mem_span_singleton_mul, mem_span_singleton]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 (\u2200 (r : R), x \u2223 r \u2192 \u2200 (s : R), s \u2208 I \u2192 r * s \u2208 J) \u2194 \u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 (\u2200 (r : R), x \u2223 r \u2192 \u2200 (s : R), s \u2208 I \u2192 r * s \u2208 J) \u2192 \u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J\n[PROOFSTEP]\nintro h zI hzI\n[GOAL]\ncase mp\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\nh : \u2200 (r : R), x \u2223 r \u2192 \u2200 (s : R), s \u2208 I \u2192 r * s \u2208 J\nzI : R\nhzI : zI \u2208 I\n\u22a2 x * zI \u2208 J\n[PROOFSTEP]\nexact h x (dvd_refl x) zI hzI\n[GOAL]\ncase mpr\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 (\u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J) \u2192 \u2200 (r : R), x \u2223 r \u2192 \u2200 (s : R), s \u2208 I \u2192 r * s \u2208 J\n[PROOFSTEP]\nrintro h _ \u27e8z, rfl\u27e9 zI hzI\n[GOAL]\ncase mpr.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\nh : \u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J\nz zI : R\nhzI : zI \u2208 I\n\u22a2 x * z * zI \u2208 J\n[PROOFSTEP]\nrw [mul_comm x z, mul_assoc]\n[GOAL]\ncase mpr.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\nh : \u2200 (z : R), z \u2208 I \u2192 x * z \u2208 J\nz zI : R\nhzI : zI \u2208 I\n\u22a2 z * (x * zI) \u2208 J\n[PROOFSTEP]\nexact J.mul_mem_left _ (h zI hzI)\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx y : R\nI J : Ideal R\n\u22a2 span {x} * I \u2264 span {y} * J \u2194 \u2200 (zI : R), zI \u2208 I \u2192 \u2203 zJ, zJ \u2208 J \u2227 x * zI = y * zJ\n[PROOFSTEP]\nsimp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI J K L : Ideal R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2260 0\n\u22a2 span {x} * I \u2264 span {x} * J \u2194 I \u2264 J\n[PROOFSTEP]\nsimp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI J K L : Ideal R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2260 0\n\u22a2 I * span {x} \u2264 J * span {x} \u2194 I \u2264 J\n[PROOFSTEP]\nsimpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI J K L : Ideal R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2260 0\n\u22a2 span {x} * I = span {x} * J \u2194 I = J\n[PROOFSTEP]\nsimp only [le_antisymm_iff, span_singleton_mul_right_mono hx]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI J K L : Ideal R\ninst\u271d : IsDomain R\nx : R\nhx : x \u2260 0\n\u22a2 I * span {x} = J * span {x} \u2194 I = J\n[PROOFSTEP]\nsimp only [le_antisymm_iff, span_singleton_mul_left_mono hx]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx : R\nI J : Ideal R\n\u22a2 I = span {x} * J \u2194 (\u2200 (zI : R), zI \u2208 I \u2192 \u2203 zJ, zJ \u2208 J \u2227 x * zJ = zI) \u2227 \u2200 (z : R), z \u2208 J \u2192 x * z \u2208 I\n[PROOFSTEP]\nsimp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L : Ideal R\nx y : R\nI J : Ideal R\n\u22a2 span {x} * I = span {y} * J \u2194\n    (\u2200 (zI : R), zI \u2208 I \u2192 \u2203 zJ, zJ \u2208 J \u2227 x * zI = y * zJ) \u2227 \u2200 (zJ : R), zJ \u2208 J \u2192 \u2203 zI, zI \u2208 I \u2227 x * zI = y * zJ\n[PROOFSTEP]\nsimp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nm : Multiset R\n\u22a2 Multiset.prod (Multiset.map (fun x => span {x}) 0) = span {Multiset.prod 0}\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nm\u271d : Multiset R\na : R\nm : Multiset R\nih : Multiset.prod (Multiset.map (fun x => span {x}) m) = span {Multiset.prod m}\n\u22a2 Multiset.prod (Multiset.map (fun x => span {x}) (a ::\u2098 m)) = span {Multiset.prod (a ::\u2098 m)}\n[PROOFSTEP]\nsimp only [Multiset.map_cons, Multiset.prod_cons, ih, \u2190 Ideal.span_singleton_mul_span_singleton]\n[GOAL]\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\n\u22a2 (Finset.inf s fun i => span {I i}) = span {\u220f i in s, I i}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (x \u2208 Finset.inf s fun i => span {I i}) \u2194 x \u2208 span {\u220f i in s, I i}\n[PROOFSTEP]\nsimp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]\n[GOAL]\ncase h\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ns : Finset \u03b9\nI : \u03b9 \u2192 R\nhI : Set.Pairwise (\u2191s) (IsCoprime on I)\nx : R\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 I i \u2223 x) \u2194 \u220f i in s, I i \u2223 x\n[PROOFSTEP]\nexact \u27e8Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h\u27e9\n[GOAL]\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nI : \u03b9 \u2192 R\nhI : \u2200 (i j : \u03b9), i \u2260 j \u2192 IsCoprime (I i) (I j)\n\u22a2 \u2a05 (i : \u03b9), span {I i} = span {\u220f i : \u03b9, I i}\n[PROOFSTEP]\nrw [\u2190 Finset.inf_univ_eq_iInf, finset_inf_span_singleton]\n[GOAL]\ncase hI\nR : Type u\n\u03b9\u271d : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nI\u271d J K L : Ideal R\n\u03b9 : Type u_2\ninst\u271d : Fintype \u03b9\nI : \u03b9 \u2192 R\nhI : \u2200 (i j : \u03b9), i \u2260 j \u2192 IsCoprime (I i) (I j)\n\u22a2 Set.Pairwise (\u2191Finset.univ) (IsCoprime on fun i => I i)\n[PROOFSTEP]\nrwa [Finset.coe_univ, Set.pairwise_univ]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 span {x} \u2294 span {y} = \u22a4 \u2194 IsCoprime x y\n[PROOFSTEP]\nrw [eq_top_iff_one, Submodule.mem_sup]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 (\u2203 y_1, y_1 \u2208 span {x} \u2227 \u2203 z, z \u2208 span {y} \u2227 y_1 + z = 1) \u2194 IsCoprime x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 (\u2203 y_1, y_1 \u2208 span {x} \u2227 \u2203 z, z \u2208 span {y} \u2227 y_1 + z = 1) \u2192 IsCoprime x y\n[PROOFSTEP]\nrintro \u27e8u, hu, v, hv, h1\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y u : R\nhu : u \u2208 span {x}\nv : R\nhv : v \u2208 span {y}\nh1 : u + v = 1\n\u22a2 IsCoprime x y\n[PROOFSTEP]\nrw [mem_span_singleton'] at hu hv \n[GOAL]\ncase mp.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y u : R\nhu : \u2203 a, a * x = u\nv : R\nhv : \u2203 a, a * y = v\nh1 : u + v = 1\n\u22a2 IsCoprime x y\n[PROOFSTEP]\nrw [\u2190 hu.choose_spec, \u2190 hv.choose_spec] at h1 \n[GOAL]\ncase mp.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y u : R\nhu : \u2203 a, a * x = u\nv : R\nhv : \u2203 a, a * y = v\nh1 : Exists.choose hu * x + Exists.choose hv * y = 1\n\u22a2 IsCoprime x y\n[PROOFSTEP]\nexact \u27e8_, _, h1\u27e9\n[GOAL]\ncase mpr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 IsCoprime x y \u2192 \u2203 y_1, y_1 \u2208 span {x} \u2227 \u2203 z, z \u2208 span {y} \u2227 y_1 + z = 1\n[PROOFSTEP]\nexact fun \u27e8u, v, h1\u27e9 => \u27e8_, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, _, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, h1\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n\u22a2 Multiset.prod s \u2264 Multiset.inf s\n[PROOFSTEP]\nclassical\nrefine' s.induction_on _ _\n\u00b7 rw [Multiset.inf_zero]\n  exact le_top\nintro a s ih\nrw [Multiset.prod_cons, Multiset.inf_cons]\nexact le_trans mul_le_inf (inf_le_inf le_rfl ih)\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n\u22a2 Multiset.prod s \u2264 Multiset.inf s\n[PROOFSTEP]\nrefine' s.induction_on _ _\n[GOAL]\ncase refine'_1\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n\u22a2 Multiset.prod 0 \u2264 Multiset.inf 0\n[PROOFSTEP]\nrw [Multiset.inf_zero]\n[GOAL]\ncase refine'_1\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n\u22a2 Multiset.prod 0 \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n\u22a2 \u2200 \u2983a : Ideal R\u2984 {s : Multiset (Ideal R)},\n    Multiset.prod s \u2264 Multiset.inf s \u2192 Multiset.prod (a ::\u2098 s) \u2264 Multiset.inf (a ::\u2098 s)\n[PROOFSTEP]\nintro a s ih\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns\u271d : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s \u2264 Multiset.inf s\n\u22a2 Multiset.prod (a ::\u2098 s) \u2264 Multiset.inf (a ::\u2098 s)\n[PROOFSTEP]\nrw [Multiset.prod_cons, Multiset.inf_cons]\n[GOAL]\ncase refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns\u271d : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s \u2264 Multiset.inf s\n\u22a2 a * Multiset.prod s \u2264 a \u2293 Multiset.inf s\n[PROOFSTEP]\nexact le_trans mul_le_inf (inf_le_inf le_rfl ih)\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 J = \u22a4\ni : R\nhi : i \u2208 I \u2294 K\n\u22a2 i \u2208 I \u2294 J * K\n[PROOFSTEP]\nrw [eq_top_iff_one] at h \n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : 1 \u2208 I \u2294 J\ni : R\nhi : i \u2208 I \u2294 K\n\u22a2 i \u2208 I \u2294 J * K\n[PROOFSTEP]\nrw [Submodule.mem_sup] at h hi \u22a2\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 J \u2227 y + z = 1\ni : R\nhi : \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 K \u2227 y + z = i\n\u22a2 \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 J * K \u2227 y + z = i\n[PROOFSTEP]\nobtain \u27e8i1, hi1, j, hj, h\u27e9 := h\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ni : R\nhi : \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 K \u2227 y + z = i\ni1 : R\nhi1 : i1 \u2208 I\nj : R\nhj : j \u2208 J\nh : i1 + j = 1\n\u22a2 \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 J * K \u2227 y + z = i\n[PROOFSTEP]\nobtain \u27e8i', hi', k, hk, hi\u27e9 := hi\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 \u2208 I\nj : R\nhj : j \u2208 J\nh : i1 + j = 1\ni' : R\nhi' : i' \u2208 I\nk : R\nhk : k \u2208 K\nhi : i' + k = i\n\u22a2 \u2203 y, y \u2208 I \u2227 \u2203 z, z \u2208 J * K \u2227 y + z = i\n[PROOFSTEP]\nrefine' \u27e8_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 \u2208 I\nj : R\nhj : j \u2208 J\nh : i1 + j = 1\ni' : R\nhi' : i' \u2208 I\nk : R\nhk : k \u2208 K\nhi : i' + k = i\n\u22a2 i' + i1 * k + j * k = i\n[PROOFSTEP]\nrw [add_assoc, \u2190 add_mul, h, one_mul, hi]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 K = \u22a4\n\u22a2 I \u2294 J * K = I \u2294 J\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 K = \u22a4\n\u22a2 I \u2294 K * J = I \u2294 J\n[PROOFSTEP]\nexact sup_mul_eq_of_coprime_left h\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : I \u2294 J = \u22a4\n\u22a2 I * K \u2294 J = K \u2294 J\n[PROOFSTEP]\nrw [sup_comm] at h \n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : J \u2294 I = \u22a4\n\u22a2 I * K \u2294 J = K \u2294 J\n[PROOFSTEP]\nrw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : K \u2294 J = \u22a4\n\u22a2 I * K \u2294 J = I \u2294 J\n[PROOFSTEP]\nrw [sup_comm] at h \n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nh : J \u2294 K = \u22a4\n\u22a2 I * K \u2294 J = I \u2294 J\n[PROOFSTEP]\nrw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J\u271d K L : Ideal R\ns : Finset \u03b9\nJ : \u03b9 \u2192 Ideal R\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 I \u2294 J i = \u22a4\n\u22a2 (fun J => I \u2294 J = \u22a4) 1\n[PROOFSTEP]\nsimp_rw [one_eq_top, sup_top_eq]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 I \u2294 J ^ n = \u22a4\n[PROOFSTEP]\nrw [\u2190 Finset.card_range n, \u2190 Finset.prod_const]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 I \u2294 \u220f _x in Finset.range n, J = \u22a4\n[PROOFSTEP]\nexact sup_prod_eq_top fun _ _ => h\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 I ^ n \u2294 J = \u22a4\n[PROOFSTEP]\nrw [\u2190 Finset.card_range n, \u2190 Finset.prod_const]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nh : I \u2294 J = \u22a4\n\u22a2 (\u220f _x in Finset.range n, I) \u2294 J = \u22a4\n[PROOFSTEP]\nexact prod_sup_eq_top fun _ _ => h\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 I * \u22a5 = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 \u22a5 * I = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nm n : \u2115\nh : m \u2264 n\n\u22a2 I ^ n \u2264 I ^ m\n[PROOFSTEP]\ncases' Nat.exists_eq_add_of_le h with k hk\n[GOAL]\ncase intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nm n : \u2115\nh : m \u2264 n\nk : \u2115\nhk : n = m + k\n\u22a2 I ^ n \u2264 I ^ m\n[PROOFSTEP]\nrw [hk, pow_add]\n[GOAL]\ncase intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nm n : \u2115\nh : m \u2264 n\nk : \u2115\nhk : n = m + k\n\u22a2 I ^ m * I ^ k \u2264 I ^ m\n[PROOFSTEP]\nexact le_trans mul_le_inf inf_le_left\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L I J : Ideal R\ne : I \u2264 J\nn : \u2115\n\u22a2 I ^ n \u2264 J ^ n\n[PROOFSTEP]\ninduction' n with _ hn\n[GOAL]\ncase zero\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L I J : Ideal R\ne : I \u2264 J\n\u22a2 I ^ Nat.zero \u2264 J ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero, pow_zero]\n[GOAL]\ncase succ\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L I J : Ideal R\ne : I \u2264 J\nn\u271d : \u2115\nhn : I ^ n\u271d \u2264 J ^ n\u271d\n\u22a2 I ^ Nat.succ n\u271d \u2264 J ^ Nat.succ n\u271d\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J\u271d K L I J : Ideal R\ne : I \u2264 J\nn\u271d : \u2115\nhn : I ^ n\u271d \u2264 J ^ n\u271d\n\u22a2 I * I ^ n\u271d \u2264 J * J ^ n\u271d\n[PROOFSTEP]\nexact Ideal.mul_mono e hn\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b2 : CommSemiring R\u271d\nI\u271d J\u271d K L : Ideal R\u271d\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nI J : Ideal R\nh : I = \u22a5 \u2228 J = \u22a5\n\u22a2 I * J = \u22a5\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b2 : CommSemiring R\u271d\nI\u271d J\u271d K L : Ideal R\u271d\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nI J : Ideal R\nh : I = \u22a5\n\u22a2 I * J = \u22a5\n[PROOFSTEP]\nrw [\u2190 Ideal.mul_bot, h, Ideal.mul_comm]\n[GOAL]\ncase inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b2 : CommSemiring R\u271d\nI\u271d J\u271d K L : Ideal R\u271d\nR : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : NoZeroDivisors R\nI J : Ideal R\nh : J = \u22a5\n\u22a2 I * J = \u22a5\n[PROOFSTEP]\nrw [\u2190 Ideal.mul_bot, h, Ideal.mul_comm]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b2 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ns : Multiset (Ideal R)\n\u22a2 Multiset.prod s = \u22a5 \u2194 \u2203 I, I \u2208 s \u2227 I = \u22a5\n[PROOFSTEP]\nrw [bot_eq_zero, prod_zero_iff_exists_zero]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b2 : CommSemiring R\u271d\nI J K L : Ideal R\u271d\nR : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\ns : Multiset (Ideal R)\n\u22a2 (\u2203 r x, r = 0) \u2194 \u2203 I, I \u2208 s \u2227 I = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nw x y z : R\n\u22a2 span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z}\n[PROOFSTEP]\nsimp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr s : R\n\u22a2 s \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := {r | \u2203 n, r ^ n \u2208 I},\n                add_mem' :=\n                  (_ :\n                    \u2200 {x y : R}, x \u2208 {r | \u2203 n, r ^ n \u2208 I} \u2192 y \u2208 {r | \u2203 n, r ^ n \u2208 I} \u2192 x + y \u2208 {r | \u2203 n, r ^ n \u2208 I}) },\n            zero_mem' := (_ : \u2203 n, 0 ^ n \u2208 I) }.toAddSubsemigroup.carrier \u2192\n    r \u2022 s \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := {r | \u2203 n, r ^ n \u2208 I},\n                add_mem' :=\n                  (_ :\n                    \u2200 {x y : R}, x \u2208 {r | \u2203 n, r ^ n \u2208 I} \u2192 y \u2208 {r | \u2203 n, r ^ n \u2208 I} \u2192 x + y \u2208 {r | \u2203 n, r ^ n \u2208 I}) },\n            zero_mem' := (_ : \u2203 n, 0 ^ n \u2208 I) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nexact fun \u27e8n, h\u27e9 \u21a6 \u27e8n, (mul_pow r s n).symm \u25b8 I.mul_mem_left (r ^ n) h\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 radical I = I \u2194 IsRadical I\n[PROOFSTEP]\nrw [le_antisymm_iff, and_iff_left le_radical, IsRadical]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 radical (I * J) = radical I \u2293 radical J\n[PROOFSTEP]\nrefine le_antisymm ?_ fun r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9 => \u27e8m + n, (pow_add r m n).symm \u25b8 mul_mem_mul hrm hrn\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\n\u22a2 radical (I * J) \u2264 radical I \u2293 radical J\n[PROOFSTEP]\nhave := radical_mono <| @mul_le_inf _ _ I J\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nthis : radical (I * J) \u2264 radical (I \u2293 J)\n\u22a2 radical (I * J) \u2264 radical I \u2293 radical J\n[PROOFSTEP]\nsimp_rw [radical_inf I J] at this \n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nthis : radical (I * J) \u2264 radical I \u2293 radical J\n\u22a2 radical (I * J) \u2264 radical I \u2293 radical J\n[PROOFSTEP]\nassumption\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr : R\nhr : r \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nhri : \u00acr \u2208 radical I\nm : Ideal R\nhrm : \u00acr \u2208 radical m\nhim : I \u2264 m\nhm : \u2200 (z : Ideal R), z \u2208 {K | \u00acr \u2208 radical K} \u2192 m \u2264 z \u2192 z = m\nthis : \u2200 (x : R), \u00acx \u2208 m \u2192 r \u2208 radical (m \u2294 span {x})\n\u22a2 m \u2260 \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr : R\nhr : r \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nhri : \u00acr \u2208 radical I\nhrm : \u00acr \u2208 radical \u22a4\nhim : I \u2264 \u22a4\nhm : \u2200 (z : Ideal R), z \u2208 {K | \u00acr \u2208 radical K} \u2192 \u22a4 \u2264 z \u2192 z = \u22a4\nthis : \u2200 (x : R), \u00acx \u2208 \u22a4 \u2192 r \u2208 radical (\u22a4 \u2294 span {x})\n\u22a2 False\n[PROOFSTEP]\nrw [radical_top] at hrm \n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr : R\nhr : r \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nhri : \u00acr \u2208 radical I\nhrm : \u00acr \u2208 \u22a4\nhim : I \u2264 \u22a4\nhm : \u2200 (z : Ideal R), z \u2208 {K | \u00acr \u2208 radical K} \u2192 \u22a4 \u2264 z \u2192 z = \u22a4\nthis : \u2200 (x : R), \u00acx \u2208 \u22a4 \u2192 r \u2208 radical (\u22a4 \u2294 span {x})\n\u22a2 False\n[PROOFSTEP]\nexact hrm trivial\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr : R\nhr : r \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nhri : \u00acr \u2208 radical I\nm : Ideal R\nhrm : \u00acr \u2208 radical m\nhim : I \u2264 m\nhm : \u2200 (z : Ideal R), z \u2208 {K | \u00acr \u2208 radical K} \u2192 m \u2264 z \u2192 z = m\nthis : \u2200 (x : R), \u00acx \u2208 m \u2192 r \u2208 radical (m \u2294 span {x})\nx y : R\nhxym : x * y \u2208 m\nhxm : \u00acx \u2208 m\nhym : \u00acy \u2208 m\nn : \u2115\nhrn : r ^ n \u2208 m \u2294 span {x}\np : R\nhpm : p \u2208 m\nq : R\nhq : q \u2208 span {x}\nhpqrn : p + q = r ^ n\nc : R\nhcxq : c * x = q\nk : \u2115\nhrk : r ^ k \u2208 m \u2294 span {y}\nf : R\nhfm : f \u2208 m\ng : R\nhg : g \u2208 span {y}\nhfgrk : f + g = r ^ k\nd : R\nhdyg : d * y = g\n\u22a2 r ^ (n + k) \u2208 m\n[PROOFSTEP]\nrw [pow_add, \u2190 hpqrn, \u2190 hcxq, \u2190 hfgrk, \u2190 hdyg, add_mul, mul_add (c * x), mul_assoc c x (d * y), mul_left_comm x, \u2190\n  mul_assoc]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nr : R\nhr : r \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nhri : \u00acr \u2208 radical I\nm : Ideal R\nhrm : \u00acr \u2208 radical m\nhim : I \u2264 m\nhm : \u2200 (z : Ideal R), z \u2208 {K | \u00acr \u2208 radical K} \u2192 m \u2264 z \u2192 z = m\nthis : \u2200 (x : R), \u00acx \u2208 m \u2192 r \u2208 radical (m \u2294 span {x})\nx y : R\nhxym : x * y \u2208 m\nhxm : \u00acx \u2208 m\nhym : \u00acy \u2208 m\nn : \u2115\nhrn : r ^ n \u2208 m \u2294 span {x}\np : R\nhpm : p \u2208 m\nq : R\nhq : q \u2208 span {x}\nhpqrn : p + q = r ^ n\nc : R\nhcxq : c * x = q\nk : \u2115\nhrk : r ^ k \u2208 m \u2294 span {y}\nf : R\nhfm : f \u2208 m\ng : R\nhg : g \u2208 span {y}\nhfgrk : f + g = r ^ k\nd : R\nhdyg : d * y = g\n\u22a2 p * (f + d * y) + (c * x * f + c * d * (x * y)) \u2208 m\n[PROOFSTEP]\nrefine' m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm) (m.mul_mem_left _ hxym))\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn\u271d n : \u2115\nih : \u22a4 ^ n = \u22a4\n\u22a2 \u22a4 ^ Nat.succ n = \u22a4\n[PROOFSTEP]\nrw [pow_succ, ih, top_mul]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn : \u2115\nH : n > 0\n\u22a2 \u00acNat.zero > 0\n[PROOFSTEP]\ndecide\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn\u271d : \u2115\nH\u271d\u00b9 : n\u271d > 0\nn : \u2115\nih : n > 0 \u2192 radical (I ^ n) = radical I\nH\u271d : Nat.succ n > 0\nH : 0 < n\n\u22a2 radical (I ^ (n + 1)) = radical I \u2293 radical (I ^ n)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn\u271d : \u2115\nH\u271d\u00b9 : n\u271d > 0\nn : \u2115\nih : n > 0 \u2192 radical (I ^ n) = radical I\nH\u271d : Nat.succ n > 0\nH : 0 < n\n\u22a2 radical (I * I ^ n) = radical I \u2293 radical (I ^ n)\n[PROOFSTEP]\nexact radical_mul _ _\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nn\u271d : \u2115\nH\u271d\u00b9 : n\u271d > 0\nn : \u2115\nih : n > 0 \u2192 radical (I ^ n) = radical I\nH\u271d : Nat.succ n > 0\nH : 0 < n\n\u22a2 radical I \u2293 radical (I ^ n) = radical I \u2293 radical I\n[PROOFSTEP]\nrw [ih H]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\n\u22a2 Multiset.prod s \u2264 P \u2194 \u2203 I, I \u2208 s \u2227 I \u2264 P\n[PROOFSTEP]\nsuffices s.prod \u2264 P \u2192 \u2203 I \u2208 s, I \u2264 P from\n  \u27e8this, fun \u27e8i, his, hip\u27e9 => le_trans multiset_prod_le_inf <| le_trans (Multiset.inf_le his) hip\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\n\u22a2 Multiset.prod s \u2264 P \u2192 \u2203 I, I \u2208 s \u2227 I \u2264 P\n[PROOFSTEP]\nclassical\nobtain \u27e8b, hb\u27e9 : \u2203 b, b \u2208 s := Multiset.exists_mem_of_ne_zero hne\nobtain \u27e8t, rfl\u27e9 : \u2203 t, s = b ::\u2098 t\nexact \u27e8s.erase b, (Multiset.cons_erase hb).symm\u27e9\nrefine' t.induction_on _ _\n\u00b7 simp only [exists_prop, Multiset.cons_zero, Multiset.prod_singleton, Multiset.mem_singleton, exists_eq_left, imp_self]\nintro a s ih h\nrw [Multiset.cons_swap, Multiset.prod_cons, hp.mul_le] at h \nrw [Multiset.cons_swap]\ncases' h with h h\n\u00b7 exact \u27e8a, Multiset.mem_cons_self a _, h\u27e9\nobtain \u27e8I, hI, ih\u27e9 : \u2203 I \u2208 b ::\u2098 s, I \u2264 P := ih h\nexact \u27e8I, Multiset.mem_cons_of_mem hI, ih\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\n\u22a2 Multiset.prod s \u2264 P \u2192 \u2203 I, I \u2208 s \u2227 I \u2264 P\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 : \u2203 b, b \u2208 s := Multiset.exists_mem_of_ne_zero hne\n[GOAL]\ncase intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\nb : Ideal R\nhb : b \u2208 s\n\u22a2 Multiset.prod s \u2264 P \u2192 \u2203 I, I \u2208 s \u2227 I \u2264 P\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 : \u2203 t, s = b ::\u2098 t\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\nb : Ideal R\nhb : b \u2208 s\n\u22a2 \u2203 t, s = b ::\u2098 t\ncase intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\n\u22a2 Multiset.prod (b ::\u2098 t) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 t \u2227 I \u2264 P\n[PROOFSTEP]\nexact \u27e8s.erase b, (Multiset.cons_erase hb).symm\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\n\u22a2 Multiset.prod (b ::\u2098 t) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 t \u2227 I \u2264 P\n[PROOFSTEP]\nrefine' t.induction_on _ _\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\n\u22a2 Multiset.prod (b ::\u2098 0) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 0 \u2227 I \u2264 P\n[PROOFSTEP]\nsimp only [exists_prop, Multiset.cons_zero, Multiset.prod_singleton, Multiset.mem_singleton, exists_eq_left, imp_self]\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\n\u22a2 \u2200 \u2983a : Ideal R\u2984 {s : Multiset (Ideal R)},\n    (Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P) \u2192\n      Multiset.prod (b ::\u2098 a ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 a ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nintro a s ih h\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : Multiset.prod (b ::\u2098 a ::\u2098 s) \u2264 P\n\u22a2 \u2203 I, I \u2208 b ::\u2098 a ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nrw [Multiset.cons_swap, Multiset.prod_cons, hp.mul_le] at h \n[GOAL]\ncase intro.intro.refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : a \u2264 P \u2228 Multiset.prod (b ::\u2098 s) \u2264 P\n\u22a2 \u2203 I, I \u2208 b ::\u2098 a ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nrw [Multiset.cons_swap]\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : a \u2264 P \u2228 Multiset.prod (b ::\u2098 s) \u2264 P\n\u22a2 \u2203 I, I \u2208 a ::\u2098 b ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase intro.intro.refine'_2.inl\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : a \u2264 P\n\u22a2 \u2203 I, I \u2208 a ::\u2098 b ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nexact \u27e8a, Multiset.mem_cons_self a _, h\u27e9\n[GOAL]\ncase intro.intro.refine'_2.inr\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : Multiset.prod (b ::\u2098 s) \u2264 P\n\u22a2 \u2203 I, I \u2208 a ::\u2098 b ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nobtain \u27e8I, hI, ih\u27e9 : \u2203 I \u2208 b ::\u2098 s, I \u2264 P := ih h\n[GOAL]\ncase intro.intro.refine'_2.inr.intro.intro\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L P : Ideal R\nhp : IsPrime P\nb : Ideal R\nt : Multiset (Ideal R)\nhne : b ::\u2098 t \u2260 0\nhb : b \u2208 b ::\u2098 t\na : Ideal R\ns : Multiset (Ideal R)\nih\u271d : Multiset.prod (b ::\u2098 s) \u2264 P \u2192 \u2203 I, I \u2208 b ::\u2098 s \u2227 I \u2264 P\nh : Multiset.prod (b ::\u2098 s) \u2264 P\nI : Ideal R\nhI : I \u2208 b ::\u2098 s\nih : I \u2264 P\n\u22a2 \u2203 I, I \u2208 a ::\u2098 b ::\u2098 s \u2227 I \u2264 P\n[PROOFSTEP]\nexact \u27e8I, Multiset.mem_cons_of_mem hI, ih\u27e9\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset \u03b9\nf : \u03b9 \u2192 Ideal R\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\n\u22a2 Multiset.prod (Multiset.map f s) \u2264 P \u2194 \u2203 i, i \u2208 s \u2227 f i \u2264 P\n[PROOFSTEP]\nrw [hp.multiset_prod_le (mt Multiset.map_eq_zero.mp hne)]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\ns : Multiset \u03b9\nf : \u03b9 \u2192 Ideal R\nP : Ideal R\nhp : IsPrime P\nhne : s \u2260 0\n\u22a2 (\u2203 I, I \u2208 Multiset.map f s \u2227 I \u2264 P) \u2194 \u2203 i, i \u2208 s \u2227 f i \u2264 P\n[PROOFSTEP]\nsimp_rw [Multiset.mem_map, exists_exists_and_eq_and]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2194 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nsuffices ((I : Set R) \u2286 f a \u222a f b \u222a \u22c3 i \u2208 (\u2191s : Set \u03b9), f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i from\n  \u27e8this, fun h =>\n    Or.casesOn h\n      (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_left _ _) (Set.subset_union_left _ _)) fun h =>\n      Or.casesOn h\n        (fun h => Set.Subset.trans h <| Set.Subset.trans (Set.subset_union_right _ _) (Set.subset_union_left _ _))\n        fun \u27e8i, his, hi\u27e9 => by refine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _;\n        exact Set.subset_biUnion_of_mem (u := fun x \u21a6 (f x : Set R)) (Finset.mem_coe.2 his)\u27e9\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nthis : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nh\u271d : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nh : I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nx\u271d : \u2203 i, i \u2208 s \u2227 I \u2264 f i\ni : \u03b9\nhis : i \u2208 s\nhi : I \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n[PROOFSTEP]\nrefine' Set.Subset.trans hi <| Set.Subset.trans _ <| Set.subset_union_right _ _\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nthis : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nh\u271d : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nh : I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\nx\u271d : \u2203 i, i \u2208 s \u2227 I \u2264 f i\ni : \u03b9\nhis : i \u2208 s\nhi : I \u2264 f i\n\u22a2 \u2191(f i) \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n[PROOFSTEP]\nexact Set.subset_biUnion_of_mem (u := fun x \u21a6 (f x : Set R)) (Finset.mem_coe.2 his)\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\ngeneralize hn : s.card = n\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn : Finset.card s = n\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nintro h\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn : Finset.card s = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\ninduction' n with n ih generalizing a b s\n[GOAL]\ncase zero\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn\u271d : Finset.card s\u271d = n\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\ns : Finset \u03b9\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nhn : Finset.card s = Nat.zero\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nclear hp\n[GOAL]\ncase zero\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn\u271d : Finset.card s\u271d = n\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\ns : Finset \u03b9\na b : \u03b9\nhn : Finset.card s = Nat.zero\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.card_eq_zero] at hn \n[GOAL]\ncase zero\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn\u271d : Finset.card s\u271d = n\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\ns : Finset \u03b9\na b : \u03b9\nhn : s = \u2205\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase zero\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn : Finset.card s = n\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\na b : \u03b9\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191\u2205), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h \n[GOAL]\ncase zero\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn : \u2115\nhn : Finset.card s = n\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\na b : \u03b9\nh : I \u2264 f a \u2228 I \u2264 f b\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i\n[PROOFSTEP]\nsimpa only [exists_prop, Finset.not_mem_empty, false_and_iff, exists_false, or_false_iff]\n[GOAL]\ncase succ\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\ns : Finset \u03b9\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nhn : Finset.card s = Nat.succ n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nclassical\nreplace hn : \u2203 (i : \u03b9) (t : Finset \u03b9), i \u2209 t \u2227 insert i t = s \u2227 t.card = n := Finset.card_eq_succ.1 hn\nrcases hn with \u27e8i, t, hit, rfl, hn\u27e9\nreplace hp : IsPrime (f i) \u2227 \u2200 x \u2208 t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp\nby_cases Ht : \u2203 j \u2208 t, f j \u2264 f i\n\u00b7 obtain \u27e8j, hjt, hfji\u27e9 : \u2203 j \u2208 t, f j \u2264 f i := Ht\n  obtain \u27e8u, hju, rfl\u27e9 : \u2203 u, j \u2209 u \u2227 insert j u = t := \u27e8t.erase j, t.not_mem_erase j, Finset.insert_erase hjt\u27e9\n  have hp' : \u2200 k \u2208 insert i u, IsPrime (f k) :=\n    by\n    rw [Finset.forall_mem_insert] at hp \u22a2\n    exact \u27e8hp.1, hp.2.2\u27e9\n  have hiu : i \u2209 u := mt Finset.mem_insert_of_mem hit\n  have hn' : (insert i u).card = n := by\n    rwa [Finset.card_insert_of_not_mem] at hn \u22a2\n    exacts [hiu, hju]\n  have h' : (I : Set R) \u2286 f a \u222a f b \u222a \u22c3 k \u2208 (\u2191(insert i u) : Set \u03b9), f k :=\n    by\n    rw [Finset.coe_insert] at h \u22a2\n    rw [Finset.coe_insert] at h \n    simp only [Set.biUnion_insert] at h \u22a2\n    rw [\u2190 Set.union_assoc (f i : Set R)] at h \n    erw [Set.union_eq_self_of_subset_right hfji] at h \n    exact h\n  specialize ih hp' hn' h'\n  refine' ih.imp id (Or.imp id (Exists.imp fun k => _))\n  exact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id\nby_cases Ha : f a \u2264 f i\n\u00b7 have h' : (I : Set R) \u2286 f i \u222a f b \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j :=\n    by\n    rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_right_comm (f a : Set R)] at h \n    erw [Set.union_eq_self_of_subset_left Ha] at h \n    exact h\n  specialize ih hp.2 hn h'\n  right\n  rcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n  \u00b7 exact Or.inr \u27e8i, Finset.mem_insert_self i t, ih\u27e9\n  \u00b7 exact Or.inl ih\n  \u00b7 exact Or.inr \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9\nby_cases Hb : f b \u2264 f i\n\u00b7 have h' : (I : Set R) \u2286 f a \u222a f i \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j :=\n    by\n    rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_assoc (f a : Set R)] at h \n    erw [Set.union_eq_self_of_subset_left Hb] at h \n    exact h\n  specialize ih hp.2 hn h'\n  rcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n  \u00b7 exact Or.inl ih\n  \u00b7 exact Or.inr (Or.inr \u27e8i, Finset.mem_insert_self i t, ih\u27e9)\n  \u00b7 exact Or.inr (Or.inr \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9)\nby_cases Hi : I \u2264 f i\n\u00b7 exact Or.inr (Or.inr \u27e8i, Finset.mem_insert_self i t, Hi\u27e9)\nhave : \u00acI \u2293 f a \u2293 f b \u2293 t.inf f \u2264 f i :=\n  by\n  rcases t.eq_empty_or_nonempty with (rfl | hsne)\n  \u00b7 rw [Finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or, not_or]\n    exact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9\n  simp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or]\n  exact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9\nrcases Set.not_subset.1 this with \u27e8r, \u27e8\u27e8\u27e8hrI, hra\u27e9, hrb\u27e9, hr\u27e9, hri\u27e9\nby_cases HI : (I : Set R) \u2286 f a \u222a f b \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j\n\u00b7 specialize ih hp.2 hn HI\n  rcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n  \u00b7 left\n    exact ih\n  \u00b7 right\n    left\n    exact ih\n  \u00b7 right\n    right\n    exact \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9\nexfalso\nrcases Set.not_subset.1 HI with \u27e8s, hsI, hs\u27e9\nrw [Finset.coe_insert, Set.biUnion_insert] at h \nhave hsi : s \u2208 f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs)\nrcases h (I.add_mem hrI hsI) with (\u27e8ha | hb\u27e9 | hi | ht)\n\u00b7 exact hs (Or.inl <| Or.inl <| add_sub_cancel' r s \u25b8 (f a).sub_mem ha hra)\n\u00b7 exact hs (Or.inl <| Or.inr <| add_sub_cancel' r s \u25b8 (f b).sub_mem hb hrb)\n\u00b7 exact hri (add_sub_cancel r s \u25b8 (f i).sub_mem hi hsi)\n\u00b7 rw [Set.mem_iUnion\u2082] at ht \n  rcases ht with \u27e8j, hjt, hj\u27e9\n  simp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr \n  exact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s \u25b8 (f j).sub_mem hj <| hr j hjt)\n[GOAL]\ncase succ\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\ns : Finset \u03b9\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nhn : Finset.card s = Nat.succ n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nreplace hn : \u2203 (i : \u03b9) (t : Finset \u03b9), i \u2209 t \u2227 insert i t = s \u2227 t.card = n := Finset.card_eq_succ.1 hn\n[GOAL]\ncase succ\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\ns : Finset \u03b9\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhn : \u2203 i t, \u00aci \u2208 t \u2227 insert i t = s \u2227 Finset.card t = n\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nrcases hn with \u27e8i, t, hit, rfl, hn\u27e9\n[GOAL]\ncase succ.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nreplace hp : IsPrime (f i) \u2227 \u2200 x \u2208 t, IsPrime (f x) := (t.forall_mem_insert _ _).1 hp\n[GOAL]\ncase succ.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nby_cases Ht : \u2203 j \u2208 t, f j \u2264 f i\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u2203 j, j \u2208 t \u2227 f j \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nobtain \u27e8j, hjt, hfji\u27e9 : \u2203 j \u2208 t, f j \u2264 f i := Ht\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nj : \u03b9\nhjt : j \u2208 t\nhfji : f j \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nobtain \u27e8u, hju, rfl\u27e9 : \u2203 u, j \u2209 u \u2227 insert j u = t := \u27e8t.erase j, t.not_mem_erase j, Finset.insert_erase hjt\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave hp' : \u2200 k \u2208 insert i u, IsPrime (f k) :=\n  by\n  rw [Finset.forall_mem_insert] at hp \u22a2\n  exact \u27e8hp.1, hp.2.2\u27e9\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\n\u22a2 \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\n[PROOFSTEP]\nrw [Finset.forall_mem_insert] at hp \u22a2\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 IsPrime (f j) \u2227 \u2200 (x : \u03b9), x \u2208 u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\n\u22a2 IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 u \u2192 IsPrime (f x)\n[PROOFSTEP]\nexact \u27e8hp.1, hp.2.2\u27e9\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave hiu : i \u2209 u := mt Finset.mem_insert_of_mem hit\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave hn' : (insert i u).card = n := by\n  rwa [Finset.card_insert_of_not_mem] at hn \u22a2\n  exacts [hiu, hju]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\n\u22a2 Finset.card (insert i u) = n\n[PROOFSTEP]\nrwa [Finset.card_insert_of_not_mem] at hn \u22a2\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card u + 1 = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\n\u22a2 \u00aci \u2208 u\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\n\u22a2 \u00acj \u2208 u\n[PROOFSTEP]\nexacts [hiu, hju]\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave h' : (I : Set R) \u2286 f a \u222a f b \u222a \u22c3 k \u2208 (\u2191(insert i u) : Set \u03b9), f k :=\n  by\n  rw [Finset.coe_insert] at h \u22a2\n  rw [Finset.coe_insert] at h \n  simp only [Set.biUnion_insert] at h \u22a2\n  rw [\u2190 Set.union_assoc (f i : Set R)] at h \n  erw [Set.union_eq_self_of_subset_right hfji] at h \n  exact h\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 \u2191(insert i u)), \u2191(f k)\n[PROOFSTEP]\nrw [Finset.coe_insert] at h \u22a2\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i \u2191(insert j u)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 insert i \u2191u), \u2191(f k)\n[PROOFSTEP]\nrw [Finset.coe_insert] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i (insert j \u2191u)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 insert i \u2191u), \u2191(f k)\n[PROOFSTEP]\nsimp only [Set.biUnion_insert] at h \u22a2\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a (\u2191(f j) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x)))\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x))\n[PROOFSTEP]\nrw [\u2190 Set.union_assoc (f i : Set R)] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u2191(f j) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x))\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x))\n[PROOFSTEP]\nerw [Set.union_eq_self_of_subset_right hfji] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x))\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191u), \u2191(f x))\n[PROOFSTEP]\nexact h\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 \u2191(insert i u)), \u2191(f k)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nspecialize ih hp' hn' h'\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 \u2191(insert i u)), \u2191(f k)\nih : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i u \u2227 I \u2264 f i_1\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i (insert j u) \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrefine' ih.imp id (Or.imp id (Exists.imp fun k => _))\n[GOAL]\ncase pos.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i j : \u03b9\nhfji : f j \u2264 f i\nu : Finset \u03b9\nhju : \u00acj \u2208 u\nhit : \u00aci \u2208 insert j u\nhn : Finset.card (insert j u) = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert j u))), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 insert j u \u2192 IsPrime (f x)\nhjt : j \u2208 insert j u\nhp' : \u2200 (k : \u03b9), k \u2208 insert i u \u2192 IsPrime (f k)\nhiu : \u00aci \u2208 u\nhn' : Finset.card (insert i u) = n\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (k : \u03b9) (_ : k \u2208 \u2191(insert i u)), \u2191(f k)\nih : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i u \u2227 I \u2264 f i_1\nk : \u03b9\n\u22a2 k \u2208 insert i u \u2227 I \u2264 f k \u2192 k \u2208 insert i (insert j u) \u2227 I \u2264 f k\n[PROOFSTEP]\nexact And.imp (fun hk => Finset.insert_subset_insert i (Finset.subset_insert j u) hk) id\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nby_cases Ha : f a \u2264 f i\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave h' : (I : Set R) \u2286 f i \u222a f b \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j :=\n  by\n  rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_right_comm (f a : Set R)] at h \n  erw [Set.union_eq_self_of_subset_left Ha] at h \n  exact h\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_right_comm (f a : Set R)] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nerw [Set.union_eq_self_of_subset_left Ha] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nexact h\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nspecialize ih hp.2 hn h'\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f i \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f i \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n[GOAL]\ncase pos.h.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f i\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inr \u27e8i, Finset.mem_insert_self i t, ih\u27e9\n[GOAL]\ncase pos.h.inr.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f b\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inl ih\n[GOAL]\ncase pos.h.inr.inr.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : f a \u2264 f i\nh' : \u2191I \u2286 \u2191(f i) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nk : \u03b9\nhkt : k \u2208 t\nih : I \u2264 f k\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inr \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nby_cases Hb : f b \u2264 f i\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave h' : (I : Set R) \u2286 f a \u222a f i \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j :=\n  by\n  rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_assoc (f a : Set R)] at h \n  erw [Set.union_eq_self_of_subset_left Hb] at h \n  exact h\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_assoc, Set.union_assoc (f a : Set R)] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a (\u2191(f b) \u222a \u2191(f i)) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nerw [Set.union_eq_self_of_subset_left Hb] at h \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\n\u22a2 \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n[PROOFSTEP]\nexact h\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nspecialize ih hp.2 hn h'\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f a \u2228 I \u2264 f i \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n[GOAL]\ncase pos.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f a\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inl ih\n[GOAL]\ncase pos.inr.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8i, Finset.mem_insert_self i t, ih\u27e9)\n[GOAL]\ncase pos.inr.inr.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : f b \u2264 f i\nh' : \u2191I \u2286 \u2191(f a) \u222a \u2191(f i) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nk : \u03b9\nhkt : k \u2208 t\nih : I \u2264 f k\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9)\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nby_cases Hi : I \u2264 f i\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : I \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8i, Finset.mem_insert_self i t, Hi\u27e9)\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave : \u00acI \u2293 f a \u2293 f b \u2293 t.inf f \u2264 f i :=\n  by\n  rcases t.eq_empty_or_nonempty with (rfl | hsne)\n  \u00b7 rw [Finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or, not_or]\n    exact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9\n  simp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or]\n  exact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\n\u22a2 \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\n[PROOFSTEP]\nrcases t.eq_empty_or_nonempty with (rfl | hsne)\n[GOAL]\ncase inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nhit : \u00aci \u2208 \u2205\nhn : Finset.card \u2205 = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i \u2205)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 \u2205 \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 \u2205 \u2227 f j \u2264 f i\n\u22a2 \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf \u2205 f \u2264 f i\n[PROOFSTEP]\nrw [Finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or, not_or]\n[GOAL]\ncase inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nhit : \u00aci \u2208 \u2205\nhn : Finset.card \u2205 = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i \u2205)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 \u2205 \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 \u2205 \u2227 f j \u2264 f i\n\u22a2 (\u00acI \u2264 f i \u2227 \u00acf a \u2264 f i) \u2227 \u00acf b \u2264 f i\n[PROOFSTEP]\nexact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9\n[GOAL]\ncase inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nhsne : Finset.Nonempty t\n\u22a2 \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\n[PROOFSTEP]\nsimp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or]\n[GOAL]\ncase inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nhsne : Finset.Nonempty t\n\u22a2 ((\u00acI \u2264 f i \u2227 \u00acf a \u2264 f i) \u2227 \u00acf b \u2264 f i) \u2227 \u00ac\u2203 i_1, i_1 \u2208 t \u2227 f i_1 \u2264 f i\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrcases Set.not_subset.1 this with \u27e8r, \u27e8\u27e8\u27e8hrI, hra\u27e9, hrb\u27e9, hr\u27e9, hri\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nby_cases HI : (I : Set R) \u2286 f a \u222a f b \u222a \u22c3 j \u2208 (\u2191t : Set \u03b9), f j\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nspecialize ih hp.2 hn HI\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrcases ih with (ih | ih | \u27e8k, hkt, ih\u27e9)\n[GOAL]\ncase pos.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f a\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.inl.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f a\n\u22a2 I \u2264 f a\n[PROOFSTEP]\nexact ih\n[GOAL]\ncase pos.inr.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f b\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.inr.inl.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f b\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.inr.inl.h.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nih : I \u2264 f b\n\u22a2 I \u2264 f b\n[PROOFSTEP]\nexact ih\n[GOAL]\ncase pos.inr.inr.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nk : \u03b9\nhkt : k \u2208 t\nih : I \u2264 f k\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.inr.inr.intro.intro.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nk : \u03b9\nhkt : k \u2208 t\nih : I \u2264 f k\n\u22a2 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.inr.inr.intro.intro.h.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nk : \u03b9\nhkt : k \u2208 t\nih : I \u2264 f k\n\u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexact \u27e8k, Finset.mem_insert_of_mem hkt, ih\u27e9\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase neg.h\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 False\n[PROOFSTEP]\nrcases Set.not_subset.1 HI with \u27e8s, hsI, hs\u27e9\n[GOAL]\ncase neg.h.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 False\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert] at h \n[GOAL]\ncase neg.h.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\n\u22a2 False\n[PROOFSTEP]\nhave hsi : s \u2208 f i := ((h hsI).resolve_left (mt Or.inl hs)).resolve_right (mt Or.inr hs)\n[GOAL]\ncase neg.h.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\n\u22a2 False\n[PROOFSTEP]\nrcases h (I.add_mem hrI hsI) with (\u27e8ha | hb\u27e9 | hi | ht)\n[GOAL]\ncase neg.h.intro.intro.inl.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nha : r + s \u2208 \u2191(f a)\n\u22a2 False\n[PROOFSTEP]\nexact hs (Or.inl <| Or.inl <| add_sub_cancel' r s \u25b8 (f a).sub_mem ha hra)\n[GOAL]\ncase neg.h.intro.intro.inl.inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nhb : r + s \u2208 \u2191(f b)\n\u22a2 False\n[PROOFSTEP]\nexact hs (Or.inl <| Or.inr <| add_sub_cancel' r s \u25b8 (f b).sub_mem hb hrb)\n[GOAL]\ncase neg.h.intro.intro.inr.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nhi : r + s \u2208 \u2191(f i)\n\u22a2 False\n[PROOFSTEP]\nexact hri (add_sub_cancel r s \u25b8 (f i).sub_mem hi hsi)\n[GOAL]\ncase neg.h.intro.intro.inr.inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nht : r + s \u2208 \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x)\n\u22a2 False\n[PROOFSTEP]\nrw [Set.mem_iUnion\u2082] at ht \n[GOAL]\ncase neg.h.intro.intro.inr.inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nht : \u2203 i j, r + s \u2208 \u2191(f i)\n\u22a2 False\n[PROOFSTEP]\nrcases ht with \u27e8j, hjt, hj\u27e9\n[GOAL]\ncase neg.h.intro.intro.inr.inr.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhr : r \u2208 \u2191(Finset.inf t f)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nj : \u03b9\nhjt : j \u2208 \u2191t\nhj : r + s \u2208 \u2191(f j)\n\u22a2 False\n[PROOFSTEP]\nsimp only [Finset.inf_eq_iInf, SetLike.mem_coe, Submodule.mem_iInf] at hr \n[GOAL]\ncase neg.h.intro.intro.inr.inr.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na\u271d b\u271d : \u03b9\nhp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsPrime (f i)\nI : Ideal R\nn\u271d : \u2115\nhn\u271d : Finset.card s\u271d = n\u271d\nh\u271d : \u2191I \u2286 \u2191(f a\u271d) \u222a \u2191(f b\u271d) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s\u271d), \u2191(f i)\nn : \u2115\nih :\n  \u2200 {s : Finset \u03b9} {a b : \u03b9},\n    (\u2200 (i : \u03b9), i \u2208 s \u2192 IsPrime (f i)) \u2192\n      Finset.card s = n \u2192\n        \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 s \u2227 I \u2264 f i\na b i : \u03b9\nt : Finset \u03b9\nhit : \u00aci \u2208 t\nhn : Finset.card t = n\nh : \u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a (\u2191(f i) \u222a \u22c3 (x : \u03b9) (_ : x \u2208 \u2191t), \u2191(f x))\nhp : IsPrime (f i) \u2227 \u2200 (x : \u03b9), x \u2208 t \u2192 IsPrime (f x)\nHt : \u00ac\u2203 j, j \u2208 t \u2227 f j \u2264 f i\nHa : \u00acf a \u2264 f i\nHb : \u00acf b \u2264 f i\nHi : \u00acI \u2264 f i\nthis : \u00acI \u2293 f a \u2293 f b \u2293 Finset.inf t f \u2264 f i\nr : R\nhri : \u00acr \u2208 \u2191(f i)\nhrb : r \u2208 \u2191(f b)\nhrI : r \u2208 \u2191I\nhra : r \u2208 \u2191(f a)\nHI : \u00ac\u2191I \u2286 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\ns : R\nhsI : s \u2208 \u2191I\nhs : \u00acs \u2208 \u2191(f a) \u222a \u2191(f b) \u222a \u22c3 (j : \u03b9) (_ : j \u2208 \u2191t), \u2191(f j)\nhsi : s \u2208 f i\nj : \u03b9\nhjt : j \u2208 \u2191t\nhj : r + s \u2208 \u2191(f j)\nhr : \u2200 (i : \u03b9), i \u2208 t \u2192 r \u2208 f i\n\u22a2 False\n[PROOFSTEP]\nexact hs (Or.inr <| Set.mem_biUnion hjt <| add_sub_cancel' r s \u25b8 (f j).sub_mem hj <| hr j hjt)\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nclassical\nby_cases has : a \u2208 s\n\u00b7 obtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s := \u27e8s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has\u27e9\n  by_cases hbt : b \u2208 t\n  \u00b7 obtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t := \u27e8t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt\u27e9\n    have hp' : \u2200 i \u2208 u, IsPrime (f i) := by\n      intro i hiu\n      refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>\n        solve_by_elim only [Finset.mem_insert_of_mem, *]\n    rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, \u2190 Set.union_assoc,\n      subset_union_prime' hp'] at h \n    rwa [Finset.exists_mem_insert, Finset.exists_mem_insert]\n  \u00b7 have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n      intro j hj\n      refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n    rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f a : Set R), subset_union_prime' hp', \u2190 or_assoc,\n      or_self_iff] at h \n    rwa [Finset.exists_mem_insert]\n\u00b7 by_cases hbs : b \u2208 s\n  \u00b7 obtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s := \u27e8s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs\u27e9\n    have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n      intro j hj\n      refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n    rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f b : Set R), subset_union_prime' hp', \u2190 or_assoc,\n      or_self_iff] at h \n    rwa [Finset.exists_mem_insert]\n  cases' s.eq_empty_or_nonempty with hse hsne\n  \u00b7 subst hse\n    rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h \n    have : (I : Set R) \u2260 \u2205 := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)\n    exact absurd h this\n  \u00b7 cases' hsne.bex with i his\n    obtain \u27e8t, _, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s := \u27e8s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his\u27e9\n    have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n      intro j hj\n      refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n    rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f i : Set R), subset_union_prime' hp', \u2190 or_assoc,\n      or_self_iff] at h \n    rwa [Finset.exists_mem_insert]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nby_cases has : a \u2208 s\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : a \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nobtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s := \u27e8s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has\u27e9\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\n\u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i\n[PROOFSTEP]\nby_cases hbt : b \u2208 t\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : b \u2208 t\n\u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i\n[PROOFSTEP]\nobtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t := \u27e8t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt\u27e9\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\n\u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i\n[PROOFSTEP]\nhave hp' : \u2200 i \u2208 u, IsPrime (f i) := by\n  intro i hiu\n  refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;> rintro rfl <;>\n    solve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\n\u22a2 \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i)\n[PROOFSTEP]\nintro i hiu\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\ni : \u03b9\nhiu : i \u2208 u\n\u22a2 IsPrime (f i)\n[PROOFSTEP]\nrefine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\ni : \u03b9\nhiu : i \u2208 u\n\u22a2 i \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\ni : \u03b9\nhiu : i \u2208 u\n\u22a2 i \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\nb : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhbt : b \u2208 insert b u\ni : \u03b9\nhiu : i \u2208 u\nhat : \u00aci \u2208 insert b u\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i (insert b u) \u2192 i_1 \u2260 i \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i (insert b u))), \u2191(f i_1)\nhas : i \u2208 insert i (insert b u)\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na : \u03b9\nI : Ideal R\nu : Finset \u03b9\ni : \u03b9\nhiu : i \u2208 u\nhbu : \u00aci \u2208 u\nhat : \u00aca \u2208 insert i u\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert a (insert i u) \u2192 i_1 \u2260 a \u2192 i_1 \u2260 i \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert a (insert i u))), \u2191(f i_1)\nhas : a \u2208 insert a (insert i u)\nhbt : i \u2208 insert i u\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a (insert b u))), \u2191(f i)\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\nhp' : \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i)\n\u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, \u2190 Set.union_assoc,\n  subset_union_prime' hp'] at h \n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nu : Finset \u03b9\nhbu : \u00acb \u2208 u\nhat : \u00aca \u2208 insert b u\nhp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 u \u2227 I \u2264 f i\nhas : a \u2208 insert a (insert b u)\nhbt : b \u2208 insert b u\nhp' : \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i)\n\u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i\n[PROOFSTEP]\nrwa [Finset.exists_mem_insert, Finset.exists_mem_insert]\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\n\u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i\n[PROOFSTEP]\nhave hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n  intro j hj\n  refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\n\u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n[PROOFSTEP]\nintro j hj\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 IsPrime (f j)\n[PROOFSTEP]\nrefine' hp j (Finset.mem_insert_of_mem hj) _ _\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\nb : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nj : \u03b9\nhj : j \u2208 t\nhat : \u00acj \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert j t \u2192 i \u2260 j \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert j t)), \u2191(f i)\nhas : j \u2208 insert j t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nj : \u03b9\nhj : j \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 j \u2192 IsPrime (f i)\nhbt : \u00acj \u2208 t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert a t)), \u2191(f i)\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f a : Set R), subset_union_prime' hp', \u2190 or_assoc,\n  or_self_iff] at h \n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhat : \u00aca \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : I \u2264 f a \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\nhas : a \u2208 insert a t\nhbt : \u00acb \u2208 t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i\n[PROOFSTEP]\nrwa [Finset.exists_mem_insert]\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nby_cases hbs : b \u2208 s\n[GOAL]\ncase pos\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\nhbs : b \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nobtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s := \u27e8s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs\u27e9\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\n\u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i\n[PROOFSTEP]\nhave hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n  intro j hj\n  refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\n\u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n[PROOFSTEP]\nintro j hj\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 IsPrime (f j)\n[PROOFSTEP]\nrefine' hp j (Finset.mem_insert_of_mem hj) _ _\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\nb : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhbs : b \u2208 insert b t\nj : \u03b9\nhj : j \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 j \u2192 i \u2260 b \u2192 IsPrime (f i)\nhas : \u00acj \u2208 insert b t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na : \u03b9\nI : Ideal R\nt : Finset \u03b9\nj : \u03b9\nhj : j \u2208 t\nhbt : \u00acj \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert j t \u2192 i \u2260 a \u2192 i \u2260 j \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert j t)), \u2191(f i)\nhas : \u00aca \u2208 insert j t\nhbs : j \u2208 insert j t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191(insert b t)), \u2191(f i)\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f b : Set R), subset_union_prime' hp', \u2190 or_assoc,\n  or_self_iff] at h \n[GOAL]\ncase pos.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nt : Finset \u03b9\nhbt : \u00acb \u2208 t\nhp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : I \u2264 f b \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\nhas : \u00aca \u2208 insert b t\nhbs : b \u2208 insert b t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i\n[PROOFSTEP]\nrwa [Finset.exists_mem_insert]\n[GOAL]\ncase neg\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\nhbs : \u00acb \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hse hsne\n[GOAL]\ncase neg.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\nhbs : \u00acb \u2208 s\nhse : s = \u2205\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nsubst hse\n[GOAL]\ncase neg.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nhp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191\u2205), \u2191(f i)\nhas : \u00aca \u2208 \u2205\nhbs : \u00acb \u2208 \u2205\n\u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i\n[PROOFSTEP]\nrw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h \n[GOAL]\ncase neg.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nhp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I = \u2205\nhas : \u00aca \u2208 \u2205\nhbs : \u00acb \u2208 \u2205\n\u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i\n[PROOFSTEP]\nhave : (I : Set R) \u2260 \u2205 := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem)\n[GOAL]\ncase neg.inl\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\nhp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nh : \u2191I = \u2205\nhas : \u00aca \u2208 \u2205\nhbs : \u00acb \u2208 \u2205\nthis : \u2191I \u2260 \u2205\n\u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i\n[PROOFSTEP]\nexact absurd h this\n[GOAL]\ncase neg.inr\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\nhbs : \u00acb \u2208 s\nhsne : Finset.Nonempty s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\ncases' hsne.bex with i his\n[GOAL]\ncase neg.inr.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nh : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\nhas : \u00aca \u2208 s\nhbs : \u00acb \u2208 s\nhsne : Finset.Nonempty s\ni : \u03b9\nhis : i \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nobtain \u27e8t, _, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s := \u27e8s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his\u27e9\n[GOAL]\ncase neg.inr.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\n\u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nhave hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n  intro j hj\n  refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;> solve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\n\u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n[PROOFSTEP]\nintro j hj\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 IsPrime (f j)\n[PROOFSTEP]\nrefine' hp j (Finset.mem_insert_of_mem hj) _ _\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 a\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nj : \u03b9\nhj : j \u2208 t\n\u22a2 j \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\nb : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nj : \u03b9\nhj : j \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 j \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nhas : \u00acj \u2208 insert i t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase refine'_2\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nj : \u03b9\nhj : j \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 j \u2192 IsPrime (f i_1)\nhbs : \u00acj \u2208 insert i t\n\u22a2 False\n[PROOFSTEP]\nsolve_by_elim only [Finset.mem_insert_of_mem, *]\n[GOAL]\ncase neg.inr.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : \u2191I \u2286 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 \u2191(insert i t)), \u2191(f i_1)\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f i : Set R), subset_union_prime' hp', \u2190 or_assoc,\n  or_self_iff] at h \n[GOAL]\ncase neg.inr.intro.intro.intro\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nI : Ideal R\ni : \u03b9\nt : Finset \u03b9\nleft\u271d : \u00aci \u2208 t\nhp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1)\nh : I \u2264 f i \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i\nhas : \u00aca \u2208 insert i t\nhbs : \u00acb \u2208 insert i t\nhsne : Finset.Nonempty (insert i t)\nhis : i \u2208 insert i t\nhp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j)\n\u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1\n[PROOFSTEP]\nrwa [Finset.exists_mem_insert]\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nthis : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n\u22a2 \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nhave aux := fun h => (bex_def.2 <| this h)\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nthis : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i\naux : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 \u2203 x x_1, I \u2264 f x\n\u22a2 \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nsimp_rw [exists_prop] at aux \n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nthis aux : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 \u2203 x, x \u2208 s \u2227 I \u2264 f x\n\u22a2 \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i\n[PROOFSTEP]\nrefine \u27e8aux, fun \u27e8i, his, hi\u27e9 \u21a6 Set.Subset.trans hi ?_\u27e9\n[GOAL]\nR\u271d : Type u\n\u03b9 : Type u_1\ninst\u271d\u00b9 : CommSemiring R\u271d\nI\u271d J K L : Ideal R\u271d\nR : Type u\ninst\u271d : CommRing R\ns : Finset \u03b9\nf : \u03b9 \u2192 Ideal R\na b : \u03b9\nhp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i)\nI : Ideal R\nthis aux : \u2191I \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i) \u2192 \u2203 x, x \u2208 s \u2227 I \u2264 f x\nx\u271d : \u2203 i, i \u2208 s \u2227 I \u2264 f i\ni : \u03b9\nhis : i \u2208 s\nhi : I \u2264 f i\n\u22a2 \u2191(f i) \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(f i)\n[PROOFSTEP]\napply Set.subset_biUnion_of_mem (show i \u2208 (\u2191s : Set \u03b9) from his)\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI\u271d J K L I : Ideal R\nh : I = \u22a4\n\u22a2 \u22a4 = I * \u22a4\n[PROOFSTEP]\nrw [mul_top, h]\n[GOAL]\nR : Type u\n\u03b9 : Type u_1\ninst\u271d : CommSemiring R\nI J K L : Ideal R\nu : (Ideal R)\u02e3\n\u22a2 \u2191u = 1\n[PROOFSTEP]\nrw [isUnit_iff.mp u.isUnit, one_eq_top]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L I : Ideal S\nx y : R\nhx : x \u2208 \u2191f \u207b\u00b9' \u2191I\nhy : y \u2208 \u2191f \u207b\u00b9' \u2191I\n\u22a2 x + y \u2208 \u2191f \u207b\u00b9' \u2191I\n[PROOFSTEP]\nsimp only [Set.mem_preimage, SetLike.mem_coe, map_add] at hx hy \u22a2\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L I : Ideal S\nx y : R\nhx : \u2191f x \u2208 I\nhy : \u2191f y \u2208 I\n\u22a2 \u2191f x + \u2191f y \u2208 I\n[PROOFSTEP]\nexact add_mem hx hy\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L I : Ideal S\n\u22a2 0 \u2208 { carrier := \u2191f \u207b\u00b9' \u2191I, add_mem' := (_ : \u2200 {x y : R}, x \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 y \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 x + y \u2208 \u2191f \u207b\u00b9' \u2191I) }.carrier\n[PROOFSTEP]\nsimp only [Set.mem_preimage, map_zero, SetLike.mem_coe, Submodule.zero_mem]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L I : Ideal S\nc x : R\nhx :\n  x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f \u207b\u00b9' \u2191I, add_mem' := (_ : \u2200 {x y : R}, x \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 y \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 x + y \u2208 \u2191f \u207b\u00b9' \u2191I) },\n          zero_mem' := (_ : 0 \u2208 \u2191f \u207b\u00b9' \u2191I) }.toAddSubsemigroup.carrier\n\u22a2 c \u2022 x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191f \u207b\u00b9' \u2191I, add_mem' := (_ : \u2200 {x y : R}, x \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 y \u2208 \u2191f \u207b\u00b9' \u2191I \u2192 x + y \u2208 \u2191f \u207b\u00b9' \u2191I) },\n          zero_mem' := (_ : 0 \u2208 \u2191f \u207b\u00b9' \u2191I) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [smul_eq_mul, Set.mem_preimage, map_mul, SetLike.mem_coe] at *\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L I : Ideal S\nc x : R\nhx : \u2191f x \u2208 I\n\u22a2 \u2191f c * \u2191f x \u2208 I\n[PROOFSTEP]\nexact mul_mem_left I _ hx\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nhK : K \u2260 \u22a4\n\u22a2 \u00ac1 \u2208 comap f K\n[PROOFSTEP]\nrw [mem_comap, map_one]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nhK : K \u2260 \u22a4\n\u22a2 \u00ac1 \u2208 K\n[PROOFSTEP]\nexact (ne_top_iff_one _).1 hK\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\ng : G\nI : Ideal R\nhf : Set.LeftInvOn \u2191g \u2191f \u2191I\n\u22a2 map f I \u2264 comap g I\n[PROOFSTEP]\nrefine' Ideal.span_le.2 _\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\ng : G\nI : Ideal R\nhf : Set.LeftInvOn \u2191g \u2191f \u2191I\n\u22a2 \u2191f '' \u2191I \u2286 \u2191(comap g I)\n[PROOFSTEP]\nrintro x \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\ng : G\nI : Ideal R\nhf : Set.LeftInvOn \u2191g \u2191f \u2191I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2191f x \u2208 \u2191(comap g I)\n[PROOFSTEP]\nrw [SetLike.mem_coe, mem_comap, hf hx]\n[GOAL]\ncase intro.intro\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\ng : G\nI : Ideal R\nhf : Set.LeftInvOn \u2191g \u2191f \u2191I\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2208 I\n[PROOFSTEP]\nexact hx\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nhK : IsPrime K\nx y : R\n\u22a2 x * y \u2208 Ideal.comap f K \u2192 x \u2208 Ideal.comap f K \u2228 y \u2208 Ideal.comap f K\n[PROOFSTEP]\nsimp only [mem_comap, map_mul]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nhK : IsPrime K\nx y : R\n\u22a2 \u2191f x * \u2191f y \u2208 K \u2192 \u2191f x \u2208 K \u2228 \u2191f y \u2208 K\n[PROOFSTEP]\napply hK.2\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nf : F\ns : Set R\n\u22a2 map f (span s) = span (\u2191f '' s)\n[PROOFSTEP]\nrefine (Submodule.span_eq_of_le _ ?_ ?_).symm\n[GOAL]\ncase refine_1\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nf : F\ns : Set R\n\u22a2 \u2191f '' s \u2286 \u2191(map f (span s))\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase refine_1.intro.intro\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nf : F\ns : Set R\nx : R\nhx : x \u2208 s\n\u22a2 \u2191f x \u2208 \u2191(map f (span s))\n[PROOFSTEP]\nexact mem_map_of_mem f (subset_span hx)\n[GOAL]\ncase refine_2\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nf : F\ns : Set R\n\u22a2 map f (span s) \u2264 Submodule.span S (\u2191f '' s)\n[PROOFSTEP]\nrw [map_le_iff_le_comap, span_le, coe_comap, \u2190 Set.image_subset_iff]\n[GOAL]\ncase refine_2\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nf : F\ns : Set R\n\u22a2 \u2191f '' s \u2286 \u2191(Submodule.span S (\u2191f '' s))\n[PROOFSTEP]\nexact subset_span\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI\u271d J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\nI : Ideal S\nh : I = \u22a4\n\u22a2 comap f I = \u22a4\n[PROOFSTEP]\nrw [h, comap_top]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\n\u03b9 : Sort u_3\ns : Set (Ideal S)\n\u22a2 \u2a05 (I : Ideal S) (_ : I \u2208 s), comap f I = \u2a05 (I : Ideal R) (_ : I \u2208 comap f '' s), I\n[PROOFSTEP]\nrw [iInf_image]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\n\u03b9 : Sort u_3\nH : IsPrime K\nx y : R\nh : x * y \u2208 comap f K\n\u22a2 \u2191f x * \u2191f y \u2208 K\n[PROOFSTEP]\nrwa [mem_comap, map_mul] at h \n[GOAL]\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\n\u22a2 I \u2022 \u22a4 = Submodule.restrictScalars R (map (algebraMap R S) I)\n[PROOFSTEP]\nrefine' le_antisymm (Submodule.smul_le.mpr fun r hr y _ => _) fun x hx => Submodule.span_induction hx _ _ _ _\n[GOAL]\ncase refine'_1\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nr : R\nhr : r \u2208 I\ny : S\nx\u271d : y \u2208 \u22a4\n\u22a2 r \u2022 y \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n[PROOFSTEP]\nrw [Algebra.smul_def]\n[GOAL]\ncase refine'_1\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nr : R\nhr : r \u2208 I\ny : S\nx\u271d : y \u2208 \u22a4\n\u22a2 \u2191(algebraMap R S) r * y \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n[PROOFSTEP]\nexact mul_mem_right _ _ (mem_map_of_mem _ hr)\n[GOAL]\ncase refine'_2\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx : S\nhx : x \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n\u22a2 \u2200 (x : S), x \u2208 \u2191(algebraMap R S) '' \u2191I \u2192 x \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\nx : R\nhx : x \u2208 \u2191I\n\u22a2 \u2191(algebraMap R S) x \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrw [\u2190 mul_one (algebraMap R S x), \u2190 Algebra.smul_def]\n[GOAL]\ncase refine'_2.intro.intro\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\nx : R\nhx : x \u2208 \u2191I\n\u22a2 x \u2022 1 \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.smul_mem_smul hx Submodule.mem_top\n[GOAL]\ncase refine'_3\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx : S\nhx : x \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n\u22a2 0 \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.zero_mem _\n[GOAL]\ncase refine'_4\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx : S\nhx : x \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n\u22a2 \u2200 (x y : S), x \u2208 I \u2022 \u22a4 \u2192 y \u2208 I \u2022 \u22a4 \u2192 x + y \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase refine'_4\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\nx y : S\n\u22a2 x \u2208 I \u2022 \u22a4 \u2192 y \u2208 I \u2022 \u22a4 \u2192 x + y \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.add_mem _\n[GOAL]\ncase refine'_5\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx : S\nhx : x \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\n\u22a2 \u2200 (a x : S), x \u2208 I \u2022 \u22a4 \u2192 a \u2022 x \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nintro a x hx\n[GOAL]\ncase refine'_5\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x : S\nhx : x \u2208 I \u2022 \u22a4\n\u22a2 a \u2022 x \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrefine' Submodule.smul_induction_on hx _ _\n[GOAL]\ncase refine'_5.refine'_1\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x : S\nhx : x \u2208 I \u2022 \u22a4\n\u22a2 \u2200 (r : R), r \u2208 I \u2192 \u2200 (n : S), n \u2208 \u22a4 \u2192 a \u2022 r \u2022 n \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nintro r hr s _\n[GOAL]\ncase refine'_5.refine'_1\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x : S\nhx : x \u2208 I \u2022 \u22a4\nr : R\nhr : r \u2208 I\ns : S\na\u271d : s \u2208 \u22a4\n\u22a2 a \u2022 r \u2022 s \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrw [smul_comm]\n[GOAL]\ncase refine'_5.refine'_1\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x : S\nhx : x \u2208 I \u2022 \u22a4\nr : R\nhr : r \u2208 I\ns : S\na\u271d : s \u2208 \u22a4\n\u22a2 r \u2022 a \u2022 s \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.smul_mem_smul hr Submodule.mem_top\n[GOAL]\ncase refine'_5.refine'_2\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d : S\nhx\u271d : x\u271d \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x : S\nhx : x \u2208 I \u2022 \u22a4\n\u22a2 \u2200 (x y : S), a \u2022 x \u2208 I \u2022 \u22a4 \u2192 a \u2022 y \u2208 I \u2022 \u22a4 \u2192 a \u2022 (x + y) \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine'_5.refine'_2\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d\u00b9 : S\nhx\u271d\u00b9 : x\u271d\u00b9 \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x\u271d : S\nhx\u271d : x\u271d \u2208 I \u2022 \u22a4\nx y : S\nhx : a \u2022 x \u2208 I \u2022 \u22a4\nhy : a \u2022 y \u2208 I \u2022 \u22a4\n\u22a2 a \u2022 (x + y) \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nrw [smul_add]\n[GOAL]\ncase refine'_5.refine'_2\nR\u271d : Type u\nS\u271d : Type v\nF : Type u_1\ninst\u271d\u2074 : Semiring R\u271d\ninst\u271d\u00b3 : Semiring S\u271d\nrc : RingHomClass F R\u271d S\u271d\nf : F\nI\u271d J : Ideal R\u271d\nK L : Ideal S\u271d\nG : Type u_2\nrcg : RingHomClass G S\u271d R\u271d\n\u03b9 : Sort u_3\nR : Type u_4\nS : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra R S\nI : Ideal R\nx\u271d\u00b9 : S\nhx\u271d\u00b9 : x\u271d\u00b9 \u2208 Submodule.restrictScalars R (map (algebraMap R S) I)\na x\u271d : S\nhx\u271d : x\u271d \u2208 I \u2022 \u22a4\nx y : S\nhx : a \u2022 x \u2208 I \u2022 \u22a4\nhy : a \u2022 y \u2208 I \u2022 \u22a4\n\u22a2 a \u2022 x + a \u2022 y \u2208 I \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.add_mem _ hx hy\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\n\u03b9 : Sort u_3\nhf : Function.Injective \u2191f\n\u22a2 comap f \u22a5 \u2264 I\n[PROOFSTEP]\nrefine' le_trans (fun x hx => _) bot_le\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\n\u03b9 : Sort u_3\nhf : Function.Injective \u2191f\nx : R\nhx : x \u2208 comap f \u22a5\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_comap, Submodule.mem_bot, \u2190 map_zero f] at hx \n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\nrcg : RingHomClass G S R\n\u03b9 : Sort u_3\nhf : Function.Injective \u2191f\nx : R\nhx : \u2191f x = \u2191f 0\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexact Eq.symm (hf hx) \u25b8 Submodule.zero_mem \u22a5\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Surjective \u2191f\nI : Ideal R\nr : R\nh : \u2191f r \u2208 map f I\ns : R\nhsi : s \u2208 \u2191I\nhfsr : \u2191f s = \u2191f r\n\u22a2 \u2191f (r - s) = 0\n[PROOFSTEP]\nrw [map_sub, hfsr, sub_self]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsMaximal I\n\u22a2 map f I = \u22a4 \u2228 IsMaximal (map f I)\n[PROOFSTEP]\nrefine' or_iff_not_imp_left.2 fun ne_top => \u27e8\u27e8fun h => ne_top h, fun J hJ => _\u27e9\u27e9\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsMaximal I\nne_top : \u00acmap f I = \u22a4\nJ : Ideal S\nhJ : map f I < J\n\u22a2 J = \u22a4\n[PROOFSTEP]\nrefine'\n  (relIsoOfSurjective f hf).injective\n    (Subtype.ext_iff.2 (Eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm))\n[GOAL]\ncase refine'_1\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsMaximal I\nne_top : \u00acmap f I = \u22a4\nJ : Ideal S\nhJ : map f I < J\n\u22a2 I \u2264 comap f J\n[PROOFSTEP]\nexact map_le_iff_le_comap.1 (le_of_lt hJ)\n[GOAL]\ncase refine'_2\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsMaximal I\nne_top : \u00acmap f I = \u22a4\nJ : Ideal S\nhJ : map f I < J\n\u22a2 I \u2260 comap f J\n[PROOFSTEP]\nexact fun h => hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm))\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\n\u22a2 IsMaximal (comap f K)\n[PROOFSTEP]\nrefine' \u27e8\u27e8comap_ne_top _ H.1.1, fun J hJ => _\u27e9\u27e9\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\n\u22a2 J = \u22a4\n[PROOFSTEP]\nsuffices map f J = \u22a4 by\n  have := congr_arg (comap f) this\n  rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this \n  rw [eq_top_iff]\n  exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono bot_le) (le_of_lt hJ)))\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nthis : map f J = \u22a4\n\u22a2 J = \u22a4\n[PROOFSTEP]\nhave := congr_arg (comap f) this\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nthis\u271d : map f J = \u22a4\nthis : comap f (map f J) = comap f \u22a4\n\u22a2 J = \u22a4\n[PROOFSTEP]\nrw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this \n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nthis\u271d : map f J = \u22a4\nthis : \u22a4 \u2264 J \u2294 comap f \u22a5\n\u22a2 J = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nthis\u271d : map f J = \u22a4\nthis : \u22a4 \u2264 J \u2294 comap f \u22a5\n\u22a2 \u22a4 \u2264 J\n[PROOFSTEP]\nexact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono bot_le) (le_of_lt hJ)))\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\n\u22a2 map f J = \u22a4\n[PROOFSTEP]\nrefine'\n  H.1.2 (map f J)\n    (lt_of_le_of_ne (le_map_of_comap_le_of_surjective _ hf (le_of_lt hJ)) fun h =>\n      ne_of_lt hJ (_root_.trans (congr_arg (comap f) h) _))\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nh : K = map f J\n\u22a2 comap f (map f J) = J\n[PROOFSTEP]\nrw [comap_map_of_surjective _ hf, sup_eq_left]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI : Ideal R\nhf : Function.Surjective \u2191f\nK : Ideal S\nH : IsMaximal K\nJ : Ideal R\nhJ : comap f K < J\nh : K = map f J\n\u22a2 comap f \u22a5 \u2264 J\n[PROOFSTEP]\nexact le_trans (comap_mono bot_le) (le_of_lt hJ)\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf\u271d : F\nI\u271d I : Ideal R\nf : R \u2243+* S\n\u22a2 map (\u2191(RingEquiv.symm f)) (map (\u2191f) I) = I\n[PROOFSTEP]\nrw [\u2190 RingEquiv.toRingHom_eq_coe, \u2190 RingEquiv.toRingHom_eq_coe, map_map, RingEquiv.toRingHom_eq_coe,\n  RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp, map_id]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf\u271d : F\nI\u271d I : Ideal R\nf : R \u2243+* S\n\u22a2 comap (\u2191f) (comap (\u2191(RingEquiv.symm f)) I) = I\n[PROOFSTEP]\nrw [\u2190 RingEquiv.toRingHom_eq_coe, \u2190 RingEquiv.toRingHom_eq_coe, comap_comap, RingEquiv.toRingHom_eq_coe,\n  RingEquiv.toRingHom_eq_coe, RingEquiv.symm_comp, comap_id]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Bijective \u2191f\nI : Ideal R\nH : IsMaximal I\n\u22a2 IsMaximal (map f I)\n[PROOFSTEP]\nrefine' or_iff_not_imp_left.1 (map_eq_top_or_isMaximal_of_surjective f hf.right H) fun h => H.1.1 _\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Bijective \u2191f\nI : Ideal R\nH : IsMaximal I\nh : map f I = \u22a4\n\u22a2 I = \u22a4\n[PROOFSTEP]\ncalc\n  I = comap f (map f I) := ((relIsoOfBijective f hf).right_inv I).symm\n  _ = comap f \u22a4 := by rw [h]\n  _ = \u22a4 := by rw [comap_top]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Bijective \u2191f\nI : Ideal R\nH : IsMaximal I\nh : map f I = \u22a4\n\u22a2 comap f (map f I) = comap f \u22a4\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : RingHomClass F R S\nf : F\nI\u271d : Ideal R\nhf : Function.Bijective \u2191f\nI : Ideal R\nH : IsMaximal I\nh : map f I = \u22a4\n\u22a2 comap f \u22a4 = \u22a4\n[PROOFSTEP]\nrw [comap_top]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nr : R\nhri : r \u2208 I\ns : R\nhsj : s \u2208 J\n\u22a2 \u2191f (r * s) \u2208 map f I * map f J\n[PROOFSTEP]\nrw [_root_.map_mul]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nr : R\nhri : r \u2208 I\ns : R\nhsj : s \u2208 J\n\u22a2 \u2191f r * \u2191f s \u2208 map f I * map f J\n[PROOFSTEP]\nexact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj)\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\ni : S\nx\u271d\u00b9 : i \u2208 \u2191f '' \u2191I\nr : R\nhri : r \u2208 \u2191I\nhfri : \u2191f r = i\nj : S\nx\u271d : j \u2208 \u2191f '' \u2191J\ns : R\nhsj : s \u2208 \u2191J\nhfsj : \u2191f s = j\n\u22a2 \u2191f r * \u2191f s \u2208 \u2191(map f (I * J))\n[PROOFSTEP]\nrw [\u2190 _root_.map_mul]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\ni : S\nx\u271d\u00b9 : i \u2208 \u2191f '' \u2191I\nr : R\nhri : r \u2208 \u2191I\nhfri : \u2191f r = i\nj : S\nx\u271d : j \u2208 \u2191f '' \u2191J\ns : R\nhsj : s \u2208 \u2191J\nhfsj : \u2191f s = j\n\u22a2 \u2191f (r * s) \u2208 \u2191(map f (I * J))\n[PROOFSTEP]\nexact mem_map_of_mem f (mul_mem_mul hri hsj)\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n\u22a2 ZeroHom.toFun { toFun := map f, map_zero' := (_ : map f \u22a5 = \u22a5) } 1 = 1\n[PROOFSTEP]\nsimp only [one_eq_top]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n\u22a2 map f \u22a4 = \u22a4\n[PROOFSTEP]\nexact Ideal.map_top f\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n\u22a2 comap f (radical K) = radical (comap f K)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nx\u271d : R\n\u22a2 x\u271d \u2208 comap f (radical K) \u2194 x\u271d \u2208 radical (comap f K)\n[PROOFSTEP]\nsimp [radical]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nhK : IsRadical K\n\u22a2 IsRadical (Ideal.comap f K)\n[PROOFSTEP]\nrw [\u2190 hK.radical, comap_radical]\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nhK : IsRadical K\n\u22a2 IsRadical (Ideal.radical (Ideal.comap f K))\n[PROOFSTEP]\napply radical_isRadical\n[GOAL]\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nn : \u2115\n\u22a2 comap f K ^ n \u2264 comap f (K ^ n)\n[PROOFSTEP]\ninduction' n with n n_ih\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n\u22a2 comap f K ^ Nat.zero \u2264 comap f (K ^ Nat.zero)\n[PROOFSTEP]\nrw [pow_zero, pow_zero, Ideal.one_eq_top, Ideal.one_eq_top]\n[GOAL]\ncase zero\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\n\u22a2 \u22a4 \u2264 comap f \u22a4\n[PROOFSTEP]\nexact rfl.le\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nn : \u2115\nn_ih : comap f K ^ n \u2264 comap f (K ^ n)\n\u22a2 comap f K ^ Nat.succ n \u2264 comap f (K ^ Nat.succ n)\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\nR : Type u\nS : Type v\nF : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nn : \u2115\nn_ih : comap f K ^ n \u2264 comap f (K ^ n)\n\u22a2 comap f K * comap f K ^ n \u2264 comap f (K * K ^ n)\n[PROOFSTEP]\nexact (Ideal.mul_mono_right n_ih).trans (Ideal.le_comap_mul f)\n[GOAL]\nR : Type u\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx\u271d : x * y \u2208 radical I\nm : \u2115\nhxy : (x * y) ^ m \u2208 I\n\u22a2 x \u2208 radical I \u2228 y \u2208 radical I\n[PROOFSTEP]\nrw [mul_pow] at hxy \n[GOAL]\nR : Type u\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx\u271d : x * y \u2208 radical I\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\n\u22a2 x \u2208 radical I \u2228 y \u2208 radical I\n[PROOFSTEP]\ncases' hi.2 hxy with h h\n[GOAL]\ncase inl\nR : Type u\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx\u271d : x * y \u2208 radical I\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : x ^ m \u2208 I\n\u22a2 x \u2208 radical I \u2228 y \u2208 radical I\n[PROOFSTEP]\nexact Or.inl \u27e8m, h\u27e9\n[GOAL]\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI : Ideal R\nhi : IsPrimary I\nx y : R\nx\u271d : x * y \u2208 radical I\nm : \u2115\nhxy : x ^ m * y ^ m \u2208 I\nh : y ^ m \u2208 radical I\n\u22a2 x \u2208 radical I \u2228 y \u2208 radical I\n[PROOFSTEP]\nexact Or.inr (mem_radical_of_pow_mem h)\n[GOAL]\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical (I \u2293 J)\n[PROOFSTEP]\nrw [radical_inf, hij, inf_idem]\n[GOAL]\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\ncases' hi.2 hxyi with hxi hyi\n[GOAL]\ncase inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\ncases' hj.2 hxyj with hxj hyj\n[GOAL]\ncase inl.inl\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhxj : x \u2208 J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\nexact Or.inl \u27e8hxi, hxj\u27e9\n[GOAL]\ncase inl.inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhxi : x \u2208 I\nhyj : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\nexact Or.inr hyj\n[GOAL]\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical I\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\nrw [hij] at hyi \n[GOAL]\ncase inr\nR : Type u\ninst\u271d : CommSemiring R\nI J : Ideal R\nhi : IsPrimary I\nhj : IsPrimary J\nhij : radical I = radical J\nx y : R\nx\u271d : x * y \u2208 I \u2293 J\nhxyi : x * y \u2208 \u2191I\nhxyj : x * y \u2208 \u2191J\nhyi : y \u2208 radical J\n\u22a2 x \u2208 I \u2293 J \u2228 y \u2208 radical J\n[PROOFSTEP]\nexact Or.inr hyi\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 { x // x \u2208 I }\n\u22a2 \u2191(finsuppTotal \u03b9 M I v) f = Finsupp.sum f fun i x => \u2191x \u2022 v i\n[PROOFSTEP]\ndsimp [finsuppTotal]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 { x // x \u2208 I }\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (Finsupp.mapRange Subtype.val (_ : \u2191(Submodule.subtype I) 0 = 0) f) =\n    Finsupp.sum f fun i x => \u2191x \u2022 v i\n[PROOFSTEP]\nrw [Finsupp.total_apply, Finsupp.sum_mapRange_index]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 { x // x \u2208 I }\n\u22a2 \u2200 (a : \u03b9), 0 \u2022 v a = 0\n[PROOFSTEP]\nexact fun _ => zero_smul _ _\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192\u2080 { x // x \u2208 I }\n\u22a2 \u2191(finsuppTotal \u03b9 M I v) f = \u2211 i : \u03b9, \u2191(\u2191f i) \u2022 v i\n[PROOFSTEP]\nrw [finsuppTotal_apply, Finsupp.sum_fintype]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\ninst\u271d : Fintype \u03b9\nf : \u03b9 \u2192\u2080 { x // x \u2208 I }\n\u22a2 \u2200 (i : \u03b9), \u21910 \u2022 v i = 0\n[PROOFSTEP]\nexact fun _ => zero_smul _ _\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\n\u22a2 LinearMap.range (finsuppTotal \u03b9 M I v) = I \u2022 Submodule.span R (Set.range v)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nx\u271d : M\n\u22a2 x\u271d \u2208 LinearMap.range (finsuppTotal \u03b9 M I v) \u2194 x\u271d \u2208 I \u2022 Submodule.span R (Set.range v)\n[PROOFSTEP]\nrw [Submodule.mem_ideal_smul_span_iff_exists_sum]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nx\u271d : M\n\u22a2 x\u271d \u2208 LinearMap.range (finsuppTotal \u03b9 M I v) \u2194 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 v i) = x\u271d\n[PROOFSTEP]\nrefine' \u27e8fun \u27e8f, h\u27e9 => \u27e8Finsupp.mapRange.linearMap I.subtype f, fun i => (f i).2, h\u27e9, _\u27e9\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nx\u271d : M\n\u22a2 (\u2203 a x, (Finsupp.sum a fun i c => c \u2022 v i) = x\u271d) \u2192 x\u271d \u2208 LinearMap.range (finsuppTotal \u03b9 M I v)\n[PROOFSTEP]\nrintro \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun i c => c \u2022 v i) \u2208 LinearMap.range (finsuppTotal \u03b9 M I v)\n[PROOFSTEP]\nclassical\nrefine' \u27e8a.mapRange (fun r => if h : r \u2208 I then \u27e8r, h\u27e9 else 0) (by simp), _\u27e9\nrw [finsuppTotal_apply, Finsupp.sum_mapRange_index]\n\u00b7 apply Finsupp.sum_congr\n  intro i _\n  rw [dif_pos (ha i)]\n\u00b7 exact fun _ => zero_smul _ _\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun i c => c \u2022 v i) \u2208 LinearMap.range (finsuppTotal \u03b9 M I v)\n[PROOFSTEP]\nrefine' \u27e8a.mapRange (fun r => if h : r \u2208 I then \u27e8r, h\u27e9 else 0) (by simp), _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (fun r => if h : r \u2208 I then { val := r, property := h } else 0) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 \u2191(finsuppTotal \u03b9 M I v)\n      (Finsupp.mapRange (fun r => if h : r \u2208 I then { val := r, property := h } else 0)\n        (_ : (if h : 0 \u2208 I then { val := 0, property := h } else 0) = 0) a) =\n    Finsupp.sum a fun i c => c \u2022 v i\n[PROOFSTEP]\nrw [finsuppTotal_apply, Finsupp.sum_mapRange_index]\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 (Finsupp.sum a fun a b => \u2191(if h : b \u2208 I then { val := b, property := h } else 0) \u2022 v a) =\n    Finsupp.sum a fun i c => c \u2022 v i\n[PROOFSTEP]\napply Finsupp.sum_congr\n[GOAL]\ncase h.intro.intro.h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 \u2200 (x : \u03b9), x \u2208 a.support \u2192 \u2191(if h : \u2191a x \u2208 I then { val := \u2191a x, property := h } else 0) \u2022 v x = \u2191a x \u2022 v x\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase h.intro.intro.h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\ni : \u03b9\na\u271d : i \u2208 a.support\n\u22a2 \u2191(if h : \u2191a i \u2208 I then { val := \u2191a i, property := h } else 0) \u2022 v i = \u2191a i \u2022 v i\n[PROOFSTEP]\nrw [dif_pos (ha i)]\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\na : \u03b9 \u2192\u2080 R\nha : \u2200 (i : \u03b9), \u2191a i \u2208 I\n\u22a2 \u2200 (a : \u03b9), \u21910 \u2022 v a = 0\n[PROOFSTEP]\nexact fun _ => zero_smul _ _\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\n\u22a2 LinearMap.range (\u2191(Algebra.lmul R S) x) = Submodule.restrictScalars R (span {x})\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\nx\u271d : S\n\u22a2 x\u271d \u2208 LinearMap.range (\u2191(Algebra.lmul R S) x) \u2194 x\u271d \u2208 Submodule.restrictScalars R (span {x})\n[PROOFSTEP]\nsimp [mem_span_singleton', mul_comm]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(\u2191(basisSpanSingleton b hx) i) = x * \u2191b i\n[PROOFSTEP]\nsimp only [basisSpanSingleton, Basis.map_apply, LinearEquiv.trans_apply, Submodule.restrictScalarsEquiv_apply,\n  LinearEquiv.ofInjective_apply, LinearEquiv.coe_ofEq_apply, LinearEquiv.restrictScalars_apply, Algebra.coe_lmul_eq_mul,\n  LinearMap.mul_apply']\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R S\nN : Type u_4\ninst\u271d\u00b2 : Semiring N\ninst\u271d\u00b9 : Module N S\ninst\u271d : SMulCommClass R N S\nb : Basis \u03b9 R S\nx : S\nhx : x \u2260 0\ni : \u03b9\n\u22a2 \u2191(AddHom.toFun (\u2191(Basis.constr b N)).toAddHom (Subtype.val \u2218 \u2191(basisSpanSingleton b hx))) (\u2191b i) =\n    \u2191(\u2191(Algebra.lmul R S) x) (\u2191b i)\n[PROOFSTEP]\nerw [Basis.constr_basis, Function.comp_apply, basisSpanSingleton_apply, LinearMap.mul_apply']\n[GOAL]\nR : Type u_1\ninst\u271d : CommSemiring R\nr : R\n\u22a2 Associates.mk (Ideal.span {r}) \u2260 0 \u2194 r \u2260 0\n[PROOFSTEP]\nrw [Associates.mk_ne_zero, Ideal.zero_eq_bot, Ne.def, Ideal.span_singleton_eq_bot]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nrcf : RingHomClass F R S\nrcg : RingHomClass G T S\nf : F\ng : G\nr : R\n\u22a2 r \u2208 ker f \u2194 \u2191f r = 0\n[PROOFSTEP]\nrw [ker, Ideal.mem_comap, Submodule.mem_bot]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Semiring T\nrcf : RingHomClass F R S\nrcg : RingHomClass G T S\nf\u271d : F\ng\u271d : G\nf : S \u2192+* R\ng : T \u2192+* S\n\u22a2 Ideal.comap g (ker f) = ker (comp f g)\n[PROOFSTEP]\nrw [RingHom.ker_eq_comap_bot, Ideal.comap_comap, RingHom.ker_eq_comap_bot]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Semiring T\nrcf : RingHomClass F R S\nrcg : RingHomClass G T S\nf\u271d : F\ng : G\ninst\u271d : Nontrivial S\nf : F\n\u22a2 \u00ac1 \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker, map_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Semiring T\nrcf : RingHomClass F R S\nrcg : RingHomClass G T S\nf\u271d : F\ng : G\ninst\u271d : Nontrivial S\nf : F\n\u22a2 \u00ac1 = 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 Function.Injective \u2191f \u2194 ker f = \u22a5\n[PROOFSTEP]\nrw [SetLike.ext'_iff, ker_eq, Set.ext_iff]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 Function.Injective \u2191f \u2194 \u2200 (x : R), x \u2208 \u2191f \u207b\u00b9' {0} \u2194 x \u2208 \u2191\u22a5\n[PROOFSTEP]\nexact injective_iff_map_eq_zero' f\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf : F\n\u22a2 ker f = \u22a5 \u2194 \u2200 (x : R), \u2191f x = 0 \u2192 x = 0\n[PROOFSTEP]\nrw [\u2190 injective_iff_map_eq_zero f, injective_iff_ker_eq_bot]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nf : R \u2243+* S\n\u22a2 ker \u2191f = \u22a5\n[PROOFSTEP]\nsimpa only [\u2190 injective_iff_ker_eq_bot] using EquivLike.injective f\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Semiring S\nrc : RingHomClass F R S\nf\u271d : F\nF' : Type u_2\ninst\u271d : RingEquivClass F' R S\nf : F'\n\u22a2 ker f = \u22a5\n[PROOFSTEP]\nsimpa only [\u2190 injective_iff_ker_eq_bot] using EquivLike.injective f\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nx y : R\n\u22a2 x - y \u2208 ker f \u2194 \u2191f x = \u2191f y\n[PROOFSTEP]\nrw [mem_ker, map_sub, sub_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\n\u22a2 ker f \u2260 \u22a4\n[PROOFSTEP]\nrw [Ne.def, Ideal.eq_top_iff_one]\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\n\u22a2 \u00ac1 \u2208 ker f\n[PROOFSTEP]\nexact not_one_mem_ker f\n[GOAL]\nR : Type u\nS : Type v\nT : Type w\nF : Type u_1\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : RingHomClass F R S\nf : F\nx y : R\n\u22a2 x * y \u2208 ker f \u2192 x \u2208 ker f \u2228 y \u2208 ker f\n[PROOFSTEP]\nsimpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\n\u22a2 Ideal.IsMaximal (ker f)\n[PROOFSTEP]\nrefine' Ideal.isMaximal_iff.mpr \u27e8fun h1 => one_ne_zero' K <| map_one f \u25b8 (mem_ker f).mp h1, fun J x hJ hxf hxJ => _\u27e9\n[GOAL]\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\n\u22a2 1 \u2208 J\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hf (f x)\u207b\u00b9\n[GOAL]\ncase intro\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\ny : R\nhy : \u2191f y = (\u2191f x)\u207b\u00b9\n\u22a2 1 \u2208 J\n[PROOFSTEP]\nhave H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm\n[GOAL]\ncase intro\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\ny : R\nhy : \u2191f y = (\u2191f x)\u207b\u00b9\nH : 1 = y * x - (y * x - 1)\n\u22a2 1 \u2208 J\n[PROOFSTEP]\nrw [H]\n[GOAL]\ncase intro\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\ny : R\nhy : \u2191f y = (\u2191f x)\u207b\u00b9\nH : 1 = y * x - (y * x - 1)\n\u22a2 y * x - (y * x - 1) \u2208 J\n[PROOFSTEP]\nrefine' J.sub_mem (J.mul_mem_left _ hxJ) (hJ _)\n[GOAL]\ncase intro\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\ny : R\nhy : \u2191f y = (\u2191f x)\u207b\u00b9\nH : 1 = y * x - (y * x - 1)\n\u22a2 y * x - 1 \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker]\n[GOAL]\ncase intro\nR\u271d : Type u\nS : Type v\nT : Type w\nR : Type u_1\nK : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Field K\ninst\u271d : RingHomClass F R K\nf : F\nhf : Function.Surjective \u2191f\nJ : Ideal R\nx : R\nhJ : ker f \u2264 J\nhxf : \u00acx \u2208 ker f\nhxJ : x \u2208 J\ny : R\nhy : \u2191f y = (\u2191f x)\u207b\u00b9\nH : 1 = y * x - (y * x - 1)\n\u22a2 \u2191f (y * x - 1) = 0\n[PROOFSTEP]\nsimp only [hy, map_sub, map_one, map_mul, inv_mul_cancel (mt (mem_ker f).mpr hxf), sub_self]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nrc : RingHomClass F R S\nI : Ideal R\nf : F\n\u22a2 map f I = \u22a5 \u2194 I \u2264 RingHom.ker f\n[PROOFSTEP]\nrw [RingHom.ker, eq_bot_iff, map_le_iff_le_comap]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\n\u22a2 (\u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J) \u2192 map f (sInf A) = sInf (map f '' A)\n[PROOFSTEP]\nrefine' fun h => le_antisymm (le_sInf _) _\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\n\u22a2 \u2200 (b : Ideal S), b \u2208 map f '' A \u2192 map f (sInf A) \u2264 b\n[PROOFSTEP]\nintro j hj y hy\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' A\ny : S\nhy : y \u2208 map f (sInf A)\n\u22a2 y \u2208 j\n[PROOFSTEP]\ncases' (mem_map_iff_of_surjective f hf).1 hy with x hx\n[GOAL]\ncase refine'_1.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' A\ny : S\nhy : y \u2208 map f (sInf A)\nx : R\nhx : x \u2208 sInf A \u2227 \u2191f x = y\n\u22a2 y \u2208 j\n[PROOFSTEP]\ncases' (Set.mem_image _ _ _).mp hj with J hJ\n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' A\ny : S\nhy : y \u2208 map f (sInf A)\nx : R\nhx : x \u2208 sInf A \u2227 \u2191f x = y\nJ : Ideal R\nhJ : J \u2208 A \u2227 map f J = j\n\u22a2 y \u2208 j\n[PROOFSTEP]\nrw [\u2190 hJ.right, \u2190 hx.right]\n[GOAL]\ncase refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : j \u2208 map f '' A\ny : S\nhy : y \u2208 map f (sInf A)\nx : R\nhx : x \u2208 sInf A \u2227 \u2191f x = y\nJ : Ideal R\nhJ : J \u2208 A \u2227 map f J = j\n\u22a2 \u2191f x \u2208 map f J\n[PROOFSTEP]\nexact mem_map_of_mem f (sInf_le_of_le hJ.left (le_of_eq rfl) hx.left)\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\n\u22a2 sInf (map f '' A) \u2264 map f (sInf A)\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\n\u22a2 y \u2208 map f (sInf A)\n[PROOFSTEP]\ncases' hf y with x hx\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\n\u22a2 y \u2208 map f (sInf A)\n[PROOFSTEP]\nrefine' hx \u25b8 mem_map_of_mem f _\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\n\u22a2 x \u2208 sInf A\n[PROOFSTEP]\nhave : \u2200 I \u2208 A, y \u2208 map f I := by simpa using hy\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\n\u22a2 \u2200 (I : Ideal R), I \u2208 A \u2192 y \u2208 map f I\n[PROOFSTEP]\nsimpa using hy\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 y \u2208 map f I\n\u22a2 x \u2208 sInf A\n[PROOFSTEP]\nrw [Submodule.mem_sInf]\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 y \u2208 map f I\n\u22a2 \u2200 (p : Submodule R R), p \u2208 A \u2192 x \u2208 p\n[PROOFSTEP]\nintro J hJ\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\ny : S\nhy : y \u2208 sInf (map f '' A)\nx : R\nhx : \u2191f x = y\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 y \u2208 map f I\nJ : Submodule R R\nhJ : J \u2208 A\n\u22a2 x \u2208 J\n[PROOFSTEP]\nrcases(mem_map_iff_of_surjective f hf).1 (this J hJ) with \u27e8x', hx', rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nx : R\nJ : Submodule R R\nhJ : J \u2208 A\nx' : R\nhx' : x' \u2208 J\nhy : \u2191f x' \u2208 sInf (map f '' A)\nhx : \u2191f x = \u2191f x'\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 \u2191f x' \u2208 map f I\n\u22a2 x \u2208 J\n[PROOFSTEP]\nhave : x - x' \u2208 J := by\n  apply h J hJ\n  rw [RingHom.mem_ker, map_sub, hx, sub_self]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nx : R\nJ : Submodule R R\nhJ : J \u2208 A\nx' : R\nhx' : x' \u2208 J\nhy : \u2191f x' \u2208 sInf (map f '' A)\nhx : \u2191f x = \u2191f x'\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 \u2191f x' \u2208 map f I\n\u22a2 x - x' \u2208 J\n[PROOFSTEP]\napply h J hJ\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nx : R\nJ : Submodule R R\nhJ : J \u2208 A\nx' : R\nhx' : x' \u2208 J\nhy : \u2191f x' \u2208 sInf (map f '' A)\nhx : \u2191f x = \u2191f x'\nthis : \u2200 (I : Ideal R), I \u2208 A \u2192 \u2191f x' \u2208 map f I\n\u22a2 x - x' \u2208 RingHom.ker f\n[PROOFSTEP]\nrw [RingHom.mem_ker, map_sub, hx, sub_self]\n[GOAL]\ncase refine'_2.intro.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective \u2191f\nh : \u2200 (J : Ideal R), J \u2208 A \u2192 RingHom.ker f \u2264 J\nx : R\nJ : Submodule R R\nhJ : J \u2208 A\nx' : R\nhx' : x' \u2208 J\nhy : \u2191f x' \u2208 sInf (map f '' A)\nhx : \u2191f x = \u2191f x'\nthis\u271d : \u2200 (I : Ideal R), I \u2208 A \u2192 \u2191f x' \u2208 map f I\nthis : x - x' \u2208 J\n\u22a2 x \u2208 J\n[PROOFSTEP]\nsimpa only [sub_add_cancel] using J.add_mem this hx'\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\n\u22a2 IsPrime (map f I)\n[PROOFSTEP]\nrefine' \u27e8fun h => H.ne_top (eq_top_iff.2 _), fun {x y} => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : map f I = \u22a4\n\u22a2 \u22a4 \u2264 I\n[PROOFSTEP]\nreplace h := congr_arg (comap f) h\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : comap f (map f I) = comap f \u22a4\n\u22a2 \u22a4 \u2264 I\n[PROOFSTEP]\nrw [comap_map_of_surjective _ hf, comap_top] at h \n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nh : I \u2294 comap f \u22a5 = \u22a4\n\u22a2 \u22a4 \u2264 I\n[PROOFSTEP]\nexact h \u25b8 sup_le (le_of_eq rfl) hk\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\n\u22a2 x * y \u2208 map f I \u2192 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nrefine' fun hxy => (hf x).recOn fun a ha => (hf y).recOn fun b hb => _\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\nhxy : x * y \u2208 map f I\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nrw [\u2190 ha, \u2190 hb, \u2190 _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy \n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhxy : \u2203 x, x \u2208 I \u2227 \u2191f x = \u2191f (a * b)\nhb : \u2191f b = y\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nrcases hxy with \u27e8c, hc, hc'\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc' : \u2191f c = \u2191f (a * b)\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 map_sub] at hc' \n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nhave : a * b \u2208 I := by\n  convert I.sub_mem hc (hk (hc' : c - a * b \u2208 RingHom.ker f)) using 1\n  abel\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b \u2208 I\n[PROOFSTEP]\nconvert I.sub_mem hc (hk (hc' : c - a * b \u2208 RingHom.ker f)) using 1\n[GOAL]\ncase h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b = c - (c - a * b)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\n\u22a2 a * b = c - (c - a * b)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\nI : Ideal R\nH : IsPrime I\nhk : RingHom.ker f \u2264 I\nx y : S\na : R\nha : \u2191f a = x\nb : R\nhb : \u2191f b = y\nc : R\nhc : c \u2208 I\nhc'\u271d : \u2191f c = \u2191f (a * b)\nhc' : \u2191f (c - a * b) = 0\nthis : a * b \u2208 I\n\u22a2 x \u2208 map f I \u2228 y \u2208 map f I\n[PROOFSTEP]\nexact (H.mem_or_mem this).imp (fun h => ha \u25b8 mem_map_of_mem f h) fun h => hb \u25b8 mem_map_of_mem f h\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nrc : RingHomClass F R S\nI : Ideal R\nf : F\nhf : Function.Injective \u2191f\n\u22a2 map f I = \u22a5 \u2194 I = \u22a5\n[PROOFSTEP]\nrw [map_eq_bot_iff_le_ker, (RingHom.injective_iff_ker_eq_bot f).mp hf, le_bot_iff]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Ring S\nrc : RingHomClass F R S\nF' : Type u_4\ninst\u271d\u00b9 : RingEquivClass F' R S\nf : F'\nI : Ideal R\ninst\u271d : IsPrime I\n\u22a2 RingHom.ker f \u2264 I\n[PROOFSTEP]\nsimp only [RingHom.ker_equiv, bot_le]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\n\u22a2 map f I = map f J \u2194 I \u2294 RingHom.ker f = J \u2294 RingHom.ker f\n[PROOFSTEP]\nrw [\u2190 (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf, comap_map_of_surjective f hf,\n  RingHom.ker_eq_comap_bot]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\n\u22a2 map f (radical I) = radical (map f I)\n[PROOFSTEP]\nrw [radical_eq_sInf, radical_eq_sInf]\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\n\u22a2 map f (sInf {J | I \u2264 J \u2227 IsPrime J}) = sInf {J | map f I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nhave : \u2200 J \u2208 {J : Ideal R | I \u2264 J \u2227 J.IsPrime}, RingHom.ker f \u2264 J := fun J hJ => h.trans hJ.left\n[GOAL]\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\n\u22a2 map f (sInf {J | I \u2264 J \u2227 IsPrime J}) = sInf {J | map f I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nconvert map_sInf hf this\n[GOAL]\ncase h.e'_3.h.e'_3\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\n\u22a2 {J | map f I \u2264 J \u2227 IsPrime J} = map f '' {J | I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nrefine' funext fun j => propext \u27e8_, _\u27e9\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\n\u22a2 setOf (fun J => map f I \u2264 J \u2227 IsPrime J) j \u2192 (map f '' {J | I \u2264 J \u2227 IsPrime J}) j\n[PROOFSTEP]\nrintro \u27e8hj, hj'\u27e9\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_1.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : map f I \u2264 j\nhj' : IsPrime j\n\u22a2 (map f '' {J | I \u2264 J \u2227 IsPrime J}) j\n[PROOFSTEP]\nhaveI : j.IsPrime := hj'\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_1.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis\u271d : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nhj : map f I \u2264 j\nhj' this : IsPrime j\n\u22a2 (map f '' {J | I \u2264 J \u2227 IsPrime J}) j\n[PROOFSTEP]\nexact \u27e8comap f j, \u27e8\u27e8map_le_iff_le_comap.1 hj, comap_isPrime f j\u27e9, map_comap_of_surjective f hf j\u27e9\u27e9\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\n\u22a2 (map f '' {J | I \u2264 J \u2227 IsPrime J}) j \u2192 setOf (fun J => map f I \u2264 J \u2227 IsPrime J) j\n[PROOFSTEP]\nrintro \u27e8J, \u27e8hJ, hJ'\u27e9\u27e9\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsPrime J}\nhJ' : map f J = j\n\u22a2 setOf (fun J => map f I \u2264 J \u2227 IsPrime J) j\n[PROOFSTEP]\nhaveI : J.IsPrime := hJ.right\n[GOAL]\ncase h.e'_3.h.e'_3.refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal R\nh : RingHom.ker f \u2264 I\nthis\u271d : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 RingHom.ker f \u2264 J\nj : Ideal S\nJ : Ideal R\nhJ : J \u2208 {J | I \u2264 J \u2227 IsPrime J}\nhJ' : map f J = j\nthis : IsPrime J\n\u22a2 setOf (fun J => map f I \u2264 J \u2227 IsPrime J) j\n[PROOFSTEP]\nrefine' \u27e8hJ' \u25b8 map_mono hJ.left, hJ' \u25b8 map_isPrime_of_surjective hf (le_trans h hJ.left)\u27e9\n[GOAL]\nR : Type u\nM : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\n\u22a2 \u2200 (b : Submodule R M), 1 \u2022 b = b\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\n\u22a2 (fun b => \u2191g (f_inv b)) 1 = 1\n[PROOFSTEP]\nrw [\u2190 map_one g, \u2190 sub_eq_zero, \u2190 map_sub g, \u2190 mem_ker g]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\n\u22a2 f_inv 1 - 1 \u2208 ker g\n[PROOFSTEP]\napply hg\n[GOAL]\ncase a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\n\u22a2 f_inv 1 - 1 \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker f, map_sub f, sub_eq_zero, map_one f]\n[GOAL]\ncase a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\n\u22a2 \u2191f (f_inv 1) = 1\n[PROOFSTEP]\nexact hf 1\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\n\u22a2 \u2200 (x y : B),\n    OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } (x * y) =\n      OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } x *\n        OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } y\n[PROOFSTEP]\nintro x y\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\nx y : B\n\u22a2 OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } (x * y) =\n    OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } x *\n      OneHom.toFun { toFun := fun b => \u2191g (f_inv b), map_one' := (_ : (fun b => \u2191g (f_inv b)) 1 = 1) } y\n[PROOFSTEP]\nrw [\u2190 map_mul g, \u2190 sub_eq_zero, \u2190 map_sub g, \u2190 mem_ker g]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\nx y : B\n\u22a2 f_inv (x * y) - f_inv x * f_inv y \u2208 ker g\n[PROOFSTEP]\napply hg\n[GOAL]\ncase a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\nx y : B\n\u22a2 f_inv (x * y) - f_inv x * f_inv y \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker f, map_sub f, sub_eq_zero, map_mul f]\n[GOAL]\ncase a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nsrc\u271d : (fun x => B \u2192+ C) { val := toAddMonoidHom g, property := hg } :=\n  \u2191(AddMonoidHom.liftOfRightInverse (toAddMonoidHom f) f_inv hf) { val := toAddMonoidHom g, property := hg }\nx y : B\n\u22a2 \u2191f (f_inv (x * y)) = \u2191f (f_inv x) * \u2191f (f_inv y)\n[PROOFSTEP]\nsimp only [hf _]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\n\u03c6 : B \u2192+* C\nx : A\nhx : x \u2208 ker f\n\u22a2 \u2191(comp \u03c6 f) x = 0\n[PROOFSTEP]\nsimp [(mem_ker _).mp hx]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : { g // ker f \u2264 ker g }\n\u22a2 (fun \u03c6 => { val := comp \u03c6 f, property := (_ : \u2200 (x : A), x \u2208 ker f \u2192 x \u2208 ker (comp \u03c6 f)) })\n      ((fun g => liftOfRightInverseAux f f_inv hf \u2191g (_ : ker f \u2264 ker \u2191g)) g) =\n    g\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : { g // ker f \u2264 ker g }\nx\u271d : A\n\u22a2 \u2191\u2191((fun \u03c6 => { val := comp \u03c6 f, property := (_ : \u2200 (x : A), x \u2208 ker f \u2192 x \u2208 ker (comp \u03c6 f)) })\n            ((fun g => liftOfRightInverseAux f f_inv hf \u2191g (_ : ker f \u2264 ker \u2191g)) g))\n      x\u271d =\n    \u2191\u2191g x\u271d\n[PROOFSTEP]\nsimp only [comp_apply, liftOfRightInverseAux_comp_apply, Subtype.coe_mk]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\n\u03c6 : B \u2192+* C\n\u22a2 (fun g => liftOfRightInverseAux f f_inv hf \u2191g (_ : ker f \u2264 ker \u2191g))\n      ((fun \u03c6 => { val := comp \u03c6 f, property := (_ : \u2200 (x : A), x \u2208 ker f \u2192 x \u2208 ker (comp \u03c6 f)) }) \u03c6) =\n    \u03c6\n[PROOFSTEP]\next b\n[GOAL]\ncase a\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\n\u03c6 : B \u2192+* C\nb : B\n\u22a2 \u2191((fun g => liftOfRightInverseAux f f_inv hf \u2191g (_ : ker f \u2264 ker \u2191g))\n          ((fun \u03c6 => { val := comp \u03c6 f, property := (_ : \u2200 (x : A), x \u2208 ker f \u2192 x \u2208 ker (comp \u03c6 f)) }) \u03c6))\n      b =\n    \u2191\u03c6 b\n[PROOFSTEP]\nsimp [liftOfRightInverseAux, hf b]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nh : B \u2192+* C\nhh : comp h f = g\n\u22a2 h = \u2191(liftOfRightInverse f f_inv hf) { val := g, property := hg }\n[PROOFSTEP]\nsimp_rw [\u2190 hh]\n[GOAL]\nA : Type u_1\nB : Type u_2\nC : Type u_3\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Ring B\ninst\u271d : Ring C\nf : A \u2192+* B\nf_inv : B \u2192 A\nhf : Function.RightInverse f_inv \u2191f\ng : A \u2192+* C\nhg : ker f \u2264 ker g\nh : B \u2192+* C\nhh : comp h f = g\n\u22a2 h = \u2191(liftOfRightInverse f f_inv hf) { val := comp h f, property := (_ : (fun g => ker f \u2264 ker g) (comp h f)) }\n[PROOFSTEP]\nexact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.Operations", "llama_tokens": 141617, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006919925839875, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.5465313183218892}}
{"text": "[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np q : MvPolynomial \u03c3 k\nhp : p \u2208 {p | \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) p = 0}\nhq : q \u2208 {p | \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) p = 0}\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 \u2191(eval x) (p + q) = 0\n[PROOFSTEP]\nsimp only [hq x hx, hp x hx, add_zero, RingHom.map_add]\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np q : MvPolynomial \u03c3 k\nhq :\n  q \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {p | \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) p = 0},\n              add_mem' :=\n                (_ :\n                  \u2200 {p q : MvPolynomial \u03c3 k},\n                    p \u2208 {p | \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) p = 0} \u2192\n                      q \u2208 {p | \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) p = 0} \u2192\n                        \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) (p + q) = 0) },\n          zero_mem' := (_ : \u2200 (x : \u03c3 \u2192 k), x \u2208 V \u2192 \u2191(eval x) 0 = 0) }.toAddSubsemigroup.carrier\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 \u2191(eval x) (p \u2022 q) = 0\n[PROOFSTEP]\nsimp only [hq x hx, Algebra.id.smul_eq_mul, mul_zero, RingHom.map_mul]\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\n\u22a2 IsMaximal (vanishingIdeal {x})\n[PROOFSTEP]\nhave : MvPolynomial \u03c3 k \u29f8 vanishingIdeal { x } \u2243+* k :=\n  RingEquiv.ofBijective (Ideal.Quotient.lift _ (eval x) fun p h => (mem_vanishingIdeal_singleton_iff x p).mp h)\n    (by\n      refine'\n        \u27e8(injective_iff_map_eq_zero _).mpr fun p hp => _, fun z =>\n          \u27e8(Ideal.Quotient.mk (vanishingIdeal { x } : Ideal (MvPolynomial \u03c3 k))) (C z), by simp\u27e9\u27e9\n      obtain \u27e8q, rfl\u27e9 := Quotient.mk_surjective p\n      rwa [Ideal.Quotient.lift_mk, \u2190 mem_vanishingIdeal_singleton_iff, \u2190 Quotient.eq_zero_iff_mem] at hp )\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\n\u22a2 Function.Bijective\n    \u2191(Ideal.Quotient.lift (vanishingIdeal {x}) (eval x)\n        (_ : \u2200 (p : MvPolynomial \u03c3 k), p \u2208 vanishingIdeal {x} \u2192 \u2191(eval x) p = 0))\n[PROOFSTEP]\nrefine'\n  \u27e8(injective_iff_map_eq_zero _).mpr fun p hp => _, fun z =>\n    \u27e8(Ideal.Quotient.mk (vanishingIdeal { x } : Ideal (MvPolynomial \u03c3 k))) (C z), by simp\u27e9\u27e9\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\nz : k\n\u22a2 \u2191(Ideal.Quotient.lift (vanishingIdeal {x}) (eval x)\n          (_ : \u2200 (p : MvPolynomial \u03c3 k), p \u2208 vanishingIdeal {x} \u2192 \u2191(eval x) p = 0))\n      (\u2191(Ideal.Quotient.mk (vanishingIdeal {x})) (\u2191C z)) =\n    z\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\np : MvPolynomial \u03c3 k \u29f8 vanishingIdeal {x}\nhp :\n  \u2191(Ideal.Quotient.lift (vanishingIdeal {x}) (eval x)\n          (_ : \u2200 (p : MvPolynomial \u03c3 k), p \u2208 vanishingIdeal {x} \u2192 \u2191(eval x) p = 0))\n      p =\n    0\n\u22a2 p = 0\n[PROOFSTEP]\nobtain \u27e8q, rfl\u27e9 := Quotient.mk_surjective p\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\nq : MvPolynomial \u03c3 k\nhp :\n  \u2191(Ideal.Quotient.lift (vanishingIdeal {x}) (eval x)\n          (_ : \u2200 (p : MvPolynomial \u03c3 k), p \u2208 vanishingIdeal {x} \u2192 \u2191(eval x) p = 0))\n      (\u2191(Ideal.Quotient.mk (vanishingIdeal {x})) q) =\n    0\n\u22a2 \u2191(Ideal.Quotient.mk (vanishingIdeal {x})) q = 0\n[PROOFSTEP]\nrwa [Ideal.Quotient.lift_mk, \u2190 mem_vanishingIdeal_singleton_iff, \u2190 Quotient.eq_zero_iff_mem] at hp \n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\nthis : MvPolynomial \u03c3 k \u29f8 vanishingIdeal {x} \u2243+* k\n\u22a2 IsMaximal (vanishingIdeal {x})\n[PROOFSTEP]\nrw [\u2190 bot_quotient_isMaximal_iff, RingEquiv.bot_maximal_iff this]\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\nthis : MvPolynomial \u03c3 k \u29f8 vanishingIdeal {x} \u2243+* k\n\u22a2 IsMaximal \u22a5\n[PROOFSTEP]\nexact bot_isMaximal\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nI : Ideal (MvPolynomial \u03c3 k)\n\u22a2 radical I \u2264 vanishingIdeal (zeroLocus I)\n[PROOFSTEP]\nintro p hp x hx\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nI : Ideal (MvPolynomial \u03c3 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 radical I\nx : \u03c3 \u2192 k\nhx : x \u2208 zeroLocus I\n\u22a2 \u2191(eval x) p = 0\n[PROOFSTEP]\nrw [\u2190 mem_vanishingIdeal_singleton_iff]\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nI : Ideal (MvPolynomial \u03c3 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 radical I\nx : \u03c3 \u2192 k\nhx : x \u2208 zeroLocus I\n\u22a2 p \u2208 vanishingIdeal {x}\n[PROOFSTEP]\nrw [radical_eq_sInf] at hp \n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nI : Ideal (MvPolynomial \u03c3 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\nx : \u03c3 \u2192 k\nhx : x \u2208 zeroLocus I\n\u22a2 p \u2208 vanishingIdeal {x}\n[PROOFSTEP]\nrefine'\n  (mem_sInf.mp hp)\n    \u27e8le_trans (le_vanishingIdeal_zeroLocus I) (vanishingIdeal_anti_mono fun y hy => hy.symm \u25b8 hx), IsMaximal.isPrime' _\u27e9\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nx : \u03c3 \u2192 k\n\u22a2 IsPrime (vanishingIdeal {x})\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 PrimeSpectrum.vanishingIdeal (pointToPoint '' V)\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 IsPrime (vanishingIdeal {x})\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk : Type u_1\ninst\u271d : Field k\n\u03c3 : Type u_2\nV : Set (\u03c3 \u2192 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 PrimeSpectrum.vanishingIdeal (pointToPoint '' V)\nx : \u03c3 \u2192 k\nhx : x \u2208 V\n\u22a2 { asIdeal := vanishingIdeal {x}, IsPrime := (_ : IsPrime (vanishingIdeal {x})) } \u2208 pointToPoint '' V\n[PROOFSTEP]\nexact \u27e8x, \u27e8hx, rfl\u27e9\u27e9\n[GOAL]\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\n\u22a2 IsMaximal I \u2194 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\ncases nonempty_fintype \u03c3\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\n\u22a2 IsMaximal I \u2194 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nrefine'\n  \u27e8fun hI => _, fun h =>\n    let \u27e8x, hx\u27e9 := h\n    hx.symm \u25b8 MvPolynomial.vanishingIdeal_singleton_isMaximal\u27e9\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nletI : I.IsMaximal := hI\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis : IsMaximal I := hI\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nletI : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nlet \u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := (Ideal.Quotient.mk I).comp C\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nhave h\u03d5 : Function.Bijective \u03d5 :=\n  \u27e8quotient_mk_comp_C_injective _ _ I hI.ne_top,\n    IsAlgClosed.algebraMap_surjective_of_isIntegral' \u03d5\n      (MvPolynomial.comp_C_integral_of_surjective_of_jacobson _ Quotient.mk_surjective)\u27e9\n[GOAL]\ncase intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nobtain \u27e8\u03c6, h\u03c6\u27e9 := Function.Surjective.hasRightInverse h\u03d5.2\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nlet x : \u03c3 \u2192 k := fun s => \u03c6 ((Ideal.Quotient.mk I) (X s))\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nhave hx : \u2200 s : \u03c3, \u03d5 (x s) = (Ideal.Quotient.mk I) (X s) := fun s => h\u03c6 ((Ideal.Quotient.mk I) (X s))\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\n\u22a2 \u2203 x, I = vanishingIdeal {x}\n[PROOFSTEP]\nrefine' \u27e8x, (IsMaximal.eq_of_le (by infer_instance) hI.ne_top _).symm\u27e9\n[GOAL]\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\n\u22a2 IsMaximal (vanishingIdeal {x})\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\n\u22a2 vanishingIdeal {x} \u2264 I\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\np : MvPolynomial \u03c3 k\nhp : p \u2208 vanishingIdeal {x}\n\u22a2 p \u2208 I\n[PROOFSTEP]\nrw [\u2190 Quotient.eq_zero_iff_mem, map_mvPolynomial_eq_eval\u2082 (Ideal.Quotient.mk I) p, eval\u2082_eq']\n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\np : MvPolynomial \u03c3 k\nhp : p \u2208 vanishingIdeal {x}\n\u22a2 (Finset.sum (support p) fun d =>\n      \u2191(RingHom.comp (Ideal.Quotient.mk I) C) (coeff d p) *\n        Finset.prod Finset.univ fun i => \u2191(Ideal.Quotient.mk I) (X i) ^ \u2191d i) =\n    0\n[PROOFSTEP]\nrw [mem_vanishingIdeal_singleton_iff, eval_eq'] at hp \n[GOAL]\ncase intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\nval\u271d : Fintype \u03c3\nhI : IsMaximal I\nthis\u271d : IsMaximal I := hI\nthis : Field (MvPolynomial \u03c3 k \u29f8 I) := Quotient.field I\n\u03d5 : k \u2192+* MvPolynomial \u03c3 k \u29f8 I := RingHom.comp (Ideal.Quotient.mk I) C\nh\u03d5 : Function.Bijective \u2191\u03d5\n\u03c6 : MvPolynomial \u03c3 k \u29f8 I \u2192 k\nh\u03c6 : Function.RightInverse \u03c6 \u2191\u03d5\nx : \u03c3 \u2192 k := fun s => \u03c6 (\u2191(Ideal.Quotient.mk I) (X s))\nhx : \u2200 (s : \u03c3), \u2191\u03d5 (x s) = \u2191(Ideal.Quotient.mk I) (X s)\np : MvPolynomial \u03c3 k\nhp\u271d : \u2191(eval x) p = 0\nhp : (Finset.sum (support p) fun d => coeff d p * Finset.prod Finset.univ fun i => x i ^ \u2191d i) = 0\n\u22a2 (Finset.sum (support p) fun d =>\n      \u2191(RingHom.comp (Ideal.Quotient.mk I) C) (coeff d p) *\n        Finset.prod Finset.univ fun i => \u2191(Ideal.Quotient.mk I) (X i) ^ \u2191d i) =\n    0\n[PROOFSTEP]\nsimpa only [\u03d5.map_sum, \u03d5.map_mul, \u03d5.map_prod, \u03d5.map_pow, \u03d5.map_zero, hx] using congr_arg \u03d5 hp\n[GOAL]\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\n\u22a2 vanishingIdeal (zeroLocus I) = radical I\n[PROOFSTEP]\nrw [I.radical_eq_jacobson]\n[GOAL]\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\n\u22a2 vanishingIdeal (zeroLocus I) = jacobson I\n[PROOFSTEP]\nrefine' le_antisymm (le_sInf _) fun p hp x hx => _\n[GOAL]\ncase refine'_1\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\n\u22a2 \u2200 (b : Ideal (MvPolynomial \u03c3 k)), b \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 vanishingIdeal (zeroLocus I) \u2264 b\n[PROOFSTEP]\nrintro J \u27e8hJI, hJ\u27e9\n[GOAL]\ncase refine'_1.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI J : Ideal (MvPolynomial \u03c3 k)\nhJI : I \u2264 J\nhJ : IsMaximal J\n\u22a2 vanishingIdeal (zeroLocus I) \u2264 J\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := (isMaximal_iff_eq_vanishingIdeal_singleton J).1 hJ\n[GOAL]\ncase refine'_1.intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI J : Ideal (MvPolynomial \u03c3 k)\nhJI : I \u2264 J\nhJ : IsMaximal J\nx : \u03c3 \u2192 k\nhx : J = vanishingIdeal {x}\n\u22a2 vanishingIdeal (zeroLocus I) \u2264 J\n[PROOFSTEP]\nrefine' hx.symm \u25b8 vanishingIdeal_anti_mono fun y hy p hp => _\n[GOAL]\ncase refine'_1.intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI J : Ideal (MvPolynomial \u03c3 k)\nhJI : I \u2264 J\nhJ : IsMaximal J\nx : \u03c3 \u2192 k\nhx : J = vanishingIdeal {x}\ny : \u03c3 \u2192 k\nhy : y \u2208 {x}\np : MvPolynomial \u03c3 k\nhp : p \u2208 I\n\u22a2 \u2191(eval y) p = 0\n[PROOFSTEP]\nrw [\u2190 mem_vanishingIdeal_singleton_iff, Set.mem_singleton_iff.1 hy, \u2190 hx]\n[GOAL]\ncase refine'_1.intro.intro\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI J : Ideal (MvPolynomial \u03c3 k)\nhJI : I \u2264 J\nhJ : IsMaximal J\nx : \u03c3 \u2192 k\nhx : J = vanishingIdeal {x}\ny : \u03c3 \u2192 k\nhy : y \u2208 {x}\np : MvPolynomial \u03c3 k\nhp : p \u2208 I\n\u22a2 p \u2208 J\n[PROOFSTEP]\nrefine' hJI hp\n[GOAL]\ncase refine'_2\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 jacobson I\nx : \u03c3 \u2192 k\nhx : x \u2208 zeroLocus I\n\u22a2 \u2191(eval x) p = 0\n[PROOFSTEP]\nrw [\u2190 mem_vanishingIdeal_singleton_iff x p]\n[GOAL]\ncase refine'_2\nk : Type u_1\ninst\u271d\u00b2 : Field k\n\u03c3 : Type u_2\ninst\u271d\u00b9 : IsAlgClosed k\ninst\u271d : Finite \u03c3\nI : Ideal (MvPolynomial \u03c3 k)\np : MvPolynomial \u03c3 k\nhp : p \u2208 jacobson I\nx : \u03c3 \u2192 k\nhx : x \u2208 zeroLocus I\n\u22a2 p \u2208 vanishingIdeal {x}\n[PROOFSTEP]\nrefine'\n  (mem_sInf.mp hp)\n    \u27e8le_trans (le_vanishingIdeal_zeroLocus I) (vanishingIdeal_anti_mono fun y hy => hy.symm \u25b8 hx),\n      MvPolynomial.vanishingIdeal_singleton_isMaximal\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Nullstellensatz", "llama_tokens": 7895, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673269042767, "lm_q2_score": 0.6723317123102956, "lm_q1q2_score": 0.5465164817786452}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalSpace M\nx : tsze R M\n\u22a2 nhds x = Filter.prod (nhds (fst x)) (nhds (snd x))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nS : Type u_2\nR : Type u_3\nM : Type u_4\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalSpace M\nfst\u271d : R\nsnd\u271d : M\n\u22a2 nhds (fst\u271d, snd\u271d) = Filter.prod (nhds (fst (fst\u271d, snd\u271d))) (nhds (snd (fst\u271d, snd\u271d)))\n[PROOFSTEP]\nexact nhds_prod_eq\n", "meta": {"mathlib_filename": "Mathlib.Topology.Instances.TrivSqZeroExt", "llama_tokens": 251, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8128673269042767, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.5465164764417608}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\nf\u271d : \u03b9 \u2192\u2080 \u03b1\ni\u271d : \u03b9\na : \u03b1\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\n\u22a2 i \u2208 f.support \u2194 (fun i => {\u2191f i}) i \u2260 0\n[PROOFSTEP]\nrw [\u2190 not_iff_not, not_mem_support_iff, not_ne_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\nf\u271d : \u03b9 \u2192\u2080 \u03b1\ni\u271d : \u03b9\na : \u03b1\nf : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\n\u22a2 \u2191f i = 0 \u2194 (fun i => {\u2191f i}) i = 0\n[PROOFSTEP]\nexact singleton_injective.eq_iff.symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf\u271d g\u271d : \u03b9 \u2192\u2080 \u03b1\ni\u271d : \u03b9\na : \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\n\u22a2 i \u2208 f.support \u222a g.support \u2194 (fun i => Icc (\u2191f i) (\u2191g i)) i \u2260 0\n[PROOFSTEP]\nrw [mem_union, \u2190 not_iff_not, not_or, not_mem_support_iff, not_mem_support_iff, not_ne_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf\u271d g\u271d : \u03b9 \u2192\u2080 \u03b1\ni\u271d : \u03b9\na : \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\ni : \u03b9\n\u22a2 \u2191f i = 0 \u2227 \u2191g i = 0 \u2194 (fun i => Icc (\u2191f i) (\u2191g i)) i = 0\n[PROOFSTEP]\nexact Icc_eq_singleton_iff.symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf\u271d g\u271d f g x : \u03b9 \u2192\u2080 \u03b1\n\u22a2 x \u2208 (fun f g => Finset.finsupp (f.support \u222a g.support) \u2191(rangeIcc f g)) f g \u2194 f \u2264 x \u2227 x \u2264 g\n[PROOFSTEP]\nrefine' (mem_finsupp_iff_of_support_subset <| Finset.subset_of_eq <| rangeIcc_support _ _).trans _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf\u271d g\u271d f g x : \u03b9 \u2192\u2080 \u03b1\n\u22a2 (\u2200 (i : \u03b9), \u2191x i \u2208 \u2191(rangeIcc f g) i) \u2194 f \u2264 x \u2227 x \u2264 g\n[PROOFSTEP]\nsimp_rw [mem_rangeIcc_apply_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf\u271d g\u271d f g x : \u03b9 \u2192\u2080 \u03b1\n\u22a2 (\u2200 (i : \u03b9), \u2191f i \u2264 \u2191x i \u2227 \u2191x i \u2264 \u2191g i) \u2194 f \u2264 x \u2227 x \u2264 g\n[PROOFSTEP]\nexact forall_and\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Icc f g) = \u220f i in f.support \u222a g.support, card (Icc (\u2191f i) (\u2191g i))\n[PROOFSTEP]\nsimp_rw [Icc_eq, card_finsupp, coe_rangeIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Ico f g) = \u220f i in f.support \u222a g.support, card (Icc (\u2191f i) (\u2191g i)) - 1\n[PROOFSTEP]\nrw [card_Ico_eq_card_Icc_sub_one, card_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Ioc f g) = \u220f i in f.support \u222a g.support, card (Icc (\u2191f i) (\u2191g i)) - 1\n[PROOFSTEP]\nrw [card_Ioc_eq_card_Icc_sub_one, card_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Ioo f g) = \u220f i in f.support \u222a g.support, card (Icc (\u2191f i) (\u2191g i)) - 2\n[PROOFSTEP]\nrw [card_Ioo_eq_card_Icc_sub_two, card_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (uIcc f g) = \u220f i in f.support \u222a g.support, card (uIcc (\u2191f i) (\u2191g i))\n[PROOFSTEP]\nrw [\u2190 support_inf_union_support_sup]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Zero \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf g : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (uIcc f g) = \u220f i in (f \u2293 g).support \u222a (f \u2294 g).support, card (uIcc (\u2191f i) (\u2191g i))\n[PROOFSTEP]\nexact card_Icc (_ : \u03b9 \u2192\u2080 \u03b1) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Iic f) = \u220f i in f.support, card (Iic (\u2191f i))\n[PROOFSTEP]\nclassical simp_rw [Iic_eq_Icc, card_Icc, Finsupp.bot_eq_zero, support_zero, empty_union, zero_apply, bot_eq_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Iic f) = \u220f i in f.support, card (Iic (\u2191f i))\n[PROOFSTEP]\nsimp_rw [Iic_eq_Icc, card_Icc, Finsupp.bot_eq_zero, support_zero, empty_union, zero_apply, bot_eq_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\nf : \u03b9 \u2192\u2080 \u03b1\n\u22a2 card (Iio f) = \u220f i in f.support, card (Iic (\u2191f i)) - 1\n[PROOFSTEP]\nrw [card_Iio_eq_card_Iic_sub_one, card_Iic]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.Interval", "llama_tokens": 2156, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527631, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.5460632128864651}}
{"text": "[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 \u2191(aeval (zeta n A B)) (cyclotomic (\u2191n) A) = 0\n[PROOFSTEP]\nrw [aeval_def, \u2190 eval_map, \u2190 IsRoot.def, map_cyclotomic, isRoot_cyclotomic_iff]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 IsPrimitiveRoot (zeta n A B) \u2191n\n[PROOFSTEP]\nexact zeta_spec n A B\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 IsRoot (cyclotomic (\u2191n) B) (zeta n A B)\n[PROOFSTEP]\nconvert aeval_zeta n A B using 0\n[GOAL]\ncase a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b9 : IsDomain B\ninst\u271d : NeZero \u2191\u2191n\n\u22a2 IsRoot (cyclotomic (\u2191n) B) (zeta n A B) \u2194 \u2191(aeval (zeta n A B)) (cyclotomic (\u2191n) A) = 0\n[PROOFSTEP]\nrw [IsRoot.def, aeval_def, eval\u2082_eq_eval_map, map_cyclotomic]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 (IsPrimitiveRoot.powerBasis K h\u03b6).gen \u2208 adjoin K {\u03b6 - 1}\n[PROOFSTEP]\nrw [powerBasis_gen, adjoin_singleton_eq_range_aeval, AlgHom.mem_range]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2203 x, \u2191(aeval (\u03b6 - 1)) x = \u03b6\n[PROOFSTEP]\nexact \u27e8X + 1, by simp\u27e9\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 \u2191(aeval (\u03b6 - 1)) (X + 1) = \u03b6\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\n\u22a2 { x // x \u2208 primitiveRoots (\u2191n) C }\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\nthis : NeZero \u2191\u2191n\n\u22a2 { x // x \u2208 primitiveRoots (\u2191n) C }\n[PROOFSTEP]\nhaveI hn := NeZero.of_noZeroSMulDivisors K C n\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\n\u22a2 { x // x \u2208 primitiveRoots (\u2191n) C }\n[PROOFSTEP]\nrefine' \u27e8x.1, _\u27e9\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\n\u22a2 \u2191x \u2208 primitiveRoots (\u2191n) C\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\nval\u271d : C\nproperty\u271d : \u2191(aeval val\u271d) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0\n\u22a2 \u2191{ val := val\u271d, property := property\u271d } \u2208 primitiveRoots (\u2191n) C\n[PROOFSTEP]\nrwa [mem_primitiveRoots n.pos, \u2190 isRoot_cyclotomic_iff, IsRoot.def, \u2190 map_cyclotomic _ (algebraMap K C),\n  h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr, \u2190 eval\u2082_eq_eval_map, \u2190 aeval_def]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { x // x \u2208 primitiveRoots (\u2191n) C }\n\u22a2 { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { x // x \u2208 primitiveRoots (\u2191n) C }\nthis : NeZero \u2191\u2191n\n\u22a2 { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\n[PROOFSTEP]\nhaveI hn := NeZero.of_noZeroSMulDivisors K C n\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { x // x \u2208 primitiveRoots (\u2191n) C }\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\n\u22a2 { y // \u2191(aeval y) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0 }\n[PROOFSTEP]\nrefine' \u27e8x.1, _\u27e9\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nx : { x // x \u2208 primitiveRoots (\u2191n) C }\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\n\u22a2 \u2191(aeval \u2191x) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC\u271d : Type w\ninst\u271d\u00b9\u00b9 : CommRing A\ninst\u271d\u00b9\u2070 : CommRing B\ninst\u271d\u2079 : Algebra A B\ninst\u271d\u2078 : IsCyclotomicExtension {n} A B\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CommRing L\ninst\u271d\u2075 : IsDomain L\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nC : Type u_1\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : IsDomain C\ninst\u271d : Algebra K C\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nhn : NeZero \u2191\u2191n\nval\u271d : C\nproperty\u271d : val\u271d \u2208 primitiveRoots (\u2191n) C\n\u22a2 \u2191(aeval \u2191{ val := val\u271d, property := property\u271d }) (minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen) = 0\n[PROOFSTEP]\nrwa [aeval_def, eval\u2082_eq_eval_map, h\u03b6.powerBasis_gen K, \u2190 h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr, map_cyclotomic,\n  \u2190 IsRoot.def, isRoot_cyclotomic_iff, \u2190 mem_primitiveRoots n.pos]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 FiniteDimensional.finrank K L = \u03c6 \u2191n\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 FiniteDimensional.finrank K L = \u03c6 \u2191n\n[PROOFSTEP]\nrw [((zeta_spec n K L).powerBasis K).finrank, IsPrimitiveRoot.powerBasis_dim, \u2190\n  (zeta_spec n K L).minpoly_eq_cyclotomic_of_irreducible hirr, natDegree_cyclotomic]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : CommRing L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nh\u03b6 : IsPrimitiveRoot \u03b6 2\ninst\u271d : IsDomain L\n\u22a2 \u2191(Algebra.norm K) \u03b6 = (-1) ^ FiniteDimensional.finrank K L\n[PROOFSTEP]\nrw [h\u03b6.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : CommRing L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nh\u03b6 : IsPrimitiveRoot \u03b6 2\ninst\u271d : IsDomain L\n\u22a2 -1 = \u2191(algebraMap K L) (-1)\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nby_cases h1 : n = 1\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : n = 1\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nrw [h1, one_coe, one_right_iff] at h\u03b6 \n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : \u03b6 = 1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : n = 1\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nrw [h\u03b6, show 1 = algebraMap K L 1 by simp, Algebra.norm_algebraMap, one_pow]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : \u03b6 = 1\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : n = 1\n\u22a2 1 = \u2191(algebraMap K L) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : \u00acn = 1\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nreplace h1 : 2 \u2264 n\n[GOAL]\ncase h1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : \u00acn = 1\n\u22a2 2 \u2264 n\n[PROOFSTEP]\nby_contra' h\n[GOAL]\ncase h1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : \u00acn = 1\nh : n < 2\n\u22a2 False\n[PROOFSTEP]\nexact\n  h1\n    (PNat.eq_one_of_lt_two h)\n      -- Porting note: specyfing the type of `cyclotomic_coeff_zero K h1` was not needed.\n[GOAL]\ncase neg\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : 2 \u2264 n\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nrw [\u2190 h\u03b6.powerBasis_gen K, PowerBasis.norm_gen_eq_coeff_zero_minpoly, h\u03b6.powerBasis_gen K, \u2190\n  h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr, (cyclotomic_coeff_zero K h1 : coeff (cyclotomic n K) 0 = 1), mul_one,\n  h\u03b6.powerBasis_dim K, \u2190 h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr, natDegree_cyclotomic]\n[GOAL]\ncase neg\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhn : n \u2260 2\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nh1 : 2 \u2264 n\n\u22a2 (-1) ^ \u03c6 \u2191n = 1\n[PROOFSTEP]\nexact (totient_even <| h1.lt_of_ne hn.symm).neg_one_pow\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK\u271d : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\u271d\ninst\u271d\u00b2 : Algebra K\u271d L\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : Algebra K L\nhodd : Odd \u2191n\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nhave hz := congr_arg (norm K) ((IsPrimitiveRoot.iff_def _ n).1 h\u03b6).1\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK\u271d : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\u271d\ninst\u271d\u00b2 : Algebra K\u271d L\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : Algebra K L\nhodd : Odd \u2191n\nhz : \u2191(Algebra.norm K) (\u03b6 ^ \u2191n) = \u2191(Algebra.norm K) 1\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nrw [\u2190 (algebraMap K L).map_one, Algebra.norm_algebraMap, one_pow, map_pow, \u2190 one_pow \u2191n] at hz \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK\u271d : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\u271d\ninst\u271d\u00b2 : Algebra K\u271d L\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedField K\ninst\u271d : Algebra K L\nhodd : Odd \u2191n\nhz : \u2191(Algebra.norm K) \u03b6 ^ \u2191n = 1 ^ \u2191n\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nexact StrictMono.injective hodd.strictMono_pow hz\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 \u2191(Algebra.norm K) \u03b6 = if n = 2 then -1 else 1\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nhn : n = 2\n\u22a2 \u2191(Algebra.norm K) \u03b6 = -1\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase pos\np : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : CommRing L\n\u03b6 : L\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : IsCyclotomicExtension {2} A B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u21912\ninst\u271d : IsCyclotomicExtension {2} K L\nhirr : Irreducible (cyclotomic (\u21912) K)\n\u22a2 \u2191(Algebra.norm K) \u03b6 = -1\n[PROOFSTEP]\nconvert norm_eq_neg_one_pow (K := K) h\u03b6\n[GOAL]\ncase h.e'_3\np : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : CommRing L\n\u03b6 : L\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : IsDomain L\ninst\u271d\u00b9 : IsCyclotomicExtension {2} A B\nh\u03b6 : IsPrimitiveRoot \u03b6 \u21912\ninst\u271d : IsCyclotomicExtension {2} K L\nhirr : Irreducible (cyclotomic (\u21912) K)\n\u22a2 -1 = (-1) ^ FiniteDimensional.finrank K L\n[PROOFSTEP]\nerw [IsCyclotomicExtension.finrank _ hirr, totient_two, pow_one]\n[GOAL]\ncase neg\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : CommRing B\ninst\u271d\u2076 : Algebra A B\ninst\u271d\u2075 : IsCyclotomicExtension {n} A B\ninst\u271d\u2074 : CommRing L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : IsDomain L\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nhn : \u00acn = 2\n\u22a2 \u2191(Algebra.norm K) \u03b6 = 1\n[PROOFSTEP]\nexact h\u03b6.norm_eq_one hn hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nlet E := AlgebraicClosure L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 := IsAlgClosed.exists_root _ (degree_cyclotomic_pos n E n.pos).ne.symm\n[GOAL]\ncase intro\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\napply (algebraMap K E).injective\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.norm K) (\u03b6 - 1)) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nletI := IsCyclotomicExtension.finiteDimensional { n } K L\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis : FiniteDimensional K L := finiteDimensional {n} K L\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.norm K) (\u03b6 - 1)) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nletI := IsCyclotomicExtension.isGalois n K L\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.norm K) (\u03b6 - 1)) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nrw [norm_eq_prod_embeddings]\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 (Finset.prod Finset.univ fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  rfl\n  ext\n  rw [\u2190 neg_sub, AlgHom.map_neg, AlgHom.map_sub, AlgHom.map_one, neg_eq_neg_one_mul]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| Finset.prod Finset.univ fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [\u2190 neg_sub, AlgHom.map_neg, AlgHom.map_sub, AlgHom.map_one, neg_eq_neg_one_mul]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| Finset.prod Finset.univ fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [\u2190 neg_sub, AlgHom.map_neg, AlgHom.map_sub, AlgHom.map_one, neg_eq_neg_one_mul]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| Finset.prod Finset.univ fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| Finset.univ\ncase f\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n| fun \u03c3 => \u2191\u03c3 (\u03b6 - 1)\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\nx\u271d : L \u2192\u2090[K] E\n| \u2191x\u271d (\u03b6 - 1)\n[PROOFSTEP]\nrw [\u2190 neg_sub, AlgHom.map_neg, AlgHom.map_sub, AlgHom.map_one, neg_eq_neg_one_mul]\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 (Finset.prod Finset.univ fun x => -1 * (1 - \u2191x \u03b6)) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nrw [prod_mul_distrib, prod_const, card_univ, AlgHom.card, IsCyclotomicExtension.finrank L hirr,\n  (totient_even h).neg_one_pow, one_mul]\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 (Finset.prod Finset.univ fun x => 1 - \u2191x \u03b6) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nhave Hprod : (Finset.univ.prod fun \u03c3 : L \u2192\u2090[K] E => 1 - \u03c3 \u03b6) = eval 1 (cyclotomic' n E) :=\n  by\n  rw [cyclotomic', eval_prod, \u2190 @Finset.prod_attach E E, \u2190 univ_eq_attach]\n  refine' Fintype.prod_equiv (h\u03b6.embeddingsEquivPrimitiveRoots E hirr) _ _ fun \u03c3 => _\n  simp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 (Finset.prod Finset.univ fun \u03c3 => 1 - \u2191\u03c3 \u03b6) = eval 1 (cyclotomic' (\u2191n) E)\n[PROOFSTEP]\nrw [cyclotomic', eval_prod, \u2190 @Finset.prod_attach E E, \u2190 univ_eq_attach]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u22a2 (Finset.prod Finset.univ fun \u03c3 => 1 - \u2191\u03c3 \u03b6) = Finset.prod Finset.univ fun x => eval 1 (X - \u2191Polynomial.C \u2191x)\n[PROOFSTEP]\nrefine' Fintype.prod_equiv (h\u03b6.embeddingsEquivPrimitiveRoots E hirr) _ _ fun \u03c3 => _\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\n\u03c3 : L \u2192\u2090[K] E\n\u22a2 1 - \u2191\u03c3 \u03b6 = eval 1 (X - \u2191Polynomial.C \u2191(\u2191(embeddingsEquivPrimitiveRoots h\u03b6 E hirr) \u03c3))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b9 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d : FiniteDimensional K L := finiteDimensional {n} K L\nthis : IsGalois K L := isGalois n K L\nHprod : (Finset.prod Finset.univ fun \u03c3 => 1 - \u2191\u03c3 \u03b6) = eval 1 (cyclotomic' (\u2191n) E)\n\u22a2 (Finset.prod Finset.univ fun x => 1 - \u2191x \u03b6) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nhaveI : NeZero ((n : \u2115) : E) := NeZero.of_noZeroSMulDivisors K _ (n : \u2115)\n[GOAL]\ncase intro.a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : 2 < \u2191n\nhirr : Irreducible (cyclotomic (\u2191n) K)\nthis\u271d\u00b2 : NeZero \u2191\u2191n\nE : Type v := AlgebraicClosure L\nz : E\nhz : IsRoot (cyclotomic (\u2191n) E) z\nthis\u271d\u00b9 : FiniteDimensional K L := finiteDimensional {n} K L\nthis\u271d : IsGalois K L := isGalois n K L\nHprod : (Finset.prod Finset.univ fun \u03c3 => 1 - \u2191\u03c3 \u03b6) = eval 1 (cyclotomic' (\u2191n) E)\nthis : NeZero \u2191\u2191n\n\u22a2 (Finset.prod Finset.univ fun x => 1 - \u2191x \u03b6) = \u2191(algebraMap K E) \u2191(eval 1 (cyclotomic \u2191n \u2124))\n[PROOFSTEP]\nrw [Hprod, cyclotomic', \u2190 cyclotomic_eq_prod_X_sub_primitiveRoots (isRoot_cyclotomic_iff.1 hz), \u2190 map_cyclotomic_int,\n  _root_.map_intCast, \u2190 Int.cast_one, eval_int_cast_map, eq_intCast, Int.cast_id]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nhave :=\n  (coe_lt_coe 2 _).1\n    (lt_of_le_of_ne (succ_le_of_lt (IsPrimePow.one_lt hn)) (Function.Injective.ne PNat.coe_injective h).symm)\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\nthis : 2 < n\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nletI hprime : Fact (n : \u2115).minFac.Prime := \u27e8minFac_prime (IsPrimePow.ne_one hn)\u27e9\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\nthis : 2 < n\nhprime : Fact (Nat.Prime (minFac \u2191n)) := { out := minFac_prime (IsPrimePow.ne_one hn) }\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nrw [sub_one_norm_eq_eval_cyclotomic h\u03b6 this hirr]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\nthis : 2 < n\nhprime : Fact (Nat.Prime (minFac \u2191n)) := { out := minFac_prime (IsPrimePow.ne_one hn) }\n\u22a2 \u2191(eval 1 (cyclotomic \u2191n \u2124)) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nnth_rw 1 [\u2190 IsPrimePow.minFac_pow_factorization_eq hn]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\nthis : 2 < n\nhprime : Fact (Nat.Prime (minFac \u2191n)) := { out := minFac_prime (IsPrimePow.ne_one hn) }\n\u22a2 \u2191(eval 1 (cyclotomic (minFac \u2191n ^ \u2191(Nat.factorization \u2191n) (minFac \u2191n)) \u2124)) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 : \u2203 k, (n : \u2115).factorization (n : \u2115).minFac = k + 1 :=\n  exists_eq_succ_of_ne_zero\n    (((n : \u2115).factorization.mem_support_toFun (n : \u2115).minFac).1 <|\n      factor_iff_mem_factorization.2 <| (mem_factors (IsPrimePow.ne_zero hn)).2 \u27e8hprime.out, minFac_dvd _\u27e9)\n[GOAL]\ncase intro\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nhn : IsPrimePow \u2191n\ninst\u271d : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic (\u2191n) K)\nh : n \u2260 2\nthis : 2 < n\nhprime : Fact (Nat.Prime (minFac \u2191n)) := { out := minFac_prime (IsPrimePow.ne_one hn) }\nk : \u2115\nhk : \u2191(Nat.factorization \u2191n) (minFac \u2191n) = k + 1\n\u22a2 \u2191(eval 1 (cyclotomic (minFac \u2191n ^ \u2191(Nat.factorization \u2191n) (minFac \u2191n)) \u2124)) = \u2191(minFac \u2191n)\n[PROOFSTEP]\nsimp [hk, sub_one_norm_eq_eval_cyclotomic h\u03b6 this hirr]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsCyclotomicExtension {n} A B\ninst\u271d\u2075 : Field L\n\u03b6\u271d : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6\u271d \u2191n\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K A\ninst\u271d\u00b9 : IsDomain A\n\u03b6 : A\ninst\u271d : IsCyclotomicExtension {n} K A\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nh : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 minpoly K (\u03b6 - 1) = comp (cyclotomic (\u2191n) K) (X + 1)\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K A\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsCyclotomicExtension {n} A B\ninst\u271d\u2075 : Field L\n\u03b6\u271d : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6\u271d \u2191n\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K A\ninst\u271d\u00b9 : IsDomain A\n\u03b6 : A\ninst\u271d : IsCyclotomicExtension {n} K A\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nh : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 minpoly K (\u03b6 - 1) = comp (cyclotomic (\u2191n) K) (X + 1)\n[PROOFSTEP]\nrw [show \u03b6 - 1 = \u03b6 + algebraMap K A (-1) by simp [sub_eq_add_neg],\n  minpoly.add_algebraMap (IsCyclotomicExtension.integral { n } K A \u03b6), h\u03b6.minpoly_eq_cyclotomic_of_irreducible h]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsCyclotomicExtension {n} A B\ninst\u271d\u2075 : Field L\n\u03b6\u271d : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6\u271d \u2191n\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K A\ninst\u271d\u00b9 : IsDomain A\n\u03b6 : A\ninst\u271d : IsCyclotomicExtension {n} K A\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nh : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 \u03b6 - 1 = \u03b6 + \u2191(algebraMap K A) (-1)\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : IsCyclotomicExtension {n} A B\ninst\u271d\u2075 : Field L\n\u03b6\u271d : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6\u271d \u2191n\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K A\ninst\u271d\u00b9 : IsDomain A\n\u03b6 : A\ninst\u271d : IsCyclotomicExtension {n} K A\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nh : Irreducible (cyclotomic (\u2191n) K)\nthis : NeZero \u2191\u2191n\n\u22a2 comp (minpoly K \u03b6) (X - \u2191Polynomial.C (-1)) = comp (minpoly K \u03b6) (X + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhave hirr\u2081 : Irreducible (cyclotomic ((p : \u2115) ^ (k - s + 1)) K) :=\n  cyclotomic_irreducible_pow_of_irreducible_pow hpri.1 (by simp) hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\n\u22a2 k - s + 1 \u2264 k + 1\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191p ^ (k - s + 1)) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nrw [\u2190 PNat.pow_coe] at hirr\u2081 \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nset \u03b7 := \u03b6 ^ (p : \u2115) ^ s - 1 with \u03b7def\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nlet \u03b7\u2081 : K\u27ee\u03b7\u27ef := IntermediateField.AdjoinSimple.gen K \u03b7\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhave h\u03b7 : IsPrimitiveRoot (\u03b7 + 1) ((p : \u2115) ^ (k + 1 - s)) :=\n  by\n  rw [sub_add_cancel]\n  refine' IsPrimitiveRoot.pow (p ^ (k + 1)).pos h\u03b6 _\n  rw [PNat.pow_coe, \u2190 pow_add, add_comm s, Nat.sub_add_cancel (le_trans hs (Nat.le_succ k))]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\n\u22a2 IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\n[PROOFSTEP]\nrw [sub_add_cancel]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\n\u22a2 IsPrimitiveRoot (\u03b6 ^ \u2191p ^ s) (\u2191p ^ (k + 1 - s))\n[PROOFSTEP]\nrefine' IsPrimitiveRoot.pow (p ^ (k + 1)).pos h\u03b6 _\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\n\u22a2 \u2191(p ^ (k + 1)) = \u2191p ^ s * \u2191p ^ (k + 1 - s)\n[PROOFSTEP]\nrw [PNat.pow_coe, \u2190 pow_add, add_comm s, Nat.sub_add_cancel (le_trans hs (Nat.le_succ k))]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhave : IsCyclotomicExtension {p ^ (k - s + 1)} K K\u27ee\u03b7\u27ef :=\n  by\n  suffices IsCyclotomicExtension {p ^ (k - s + 1)} K K\u27ee\u03b7 + 1\u27ef.toSubalgebra\n    by\n    have H : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = K\u27ee\u03b7\u27ef.toSubalgebra :=\n      by\n      simp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral\n          (IsCyclotomicExtension.integral {p ^ (k + 1)} K L _)]\n      refine' Subalgebra.ext fun x => \u27e8fun hx => adjoin_le _ hx, fun hx => adjoin_le _ hx\u27e9\n      \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe]\n        exact\n          Subalgebra.add_mem _ (subset_adjoin (mem_singleton \u03b7))\n            (Subalgebra.one_mem _)\n              -- Porting note `\u03b7def` was not needed.\n      \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe, \u2190 \u03b7def]\n        nth_rw 1 [\u2190 add_sub_cancel \u03b7 1]\n        refine'\n          Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _))\n            (Subalgebra.one_mem _)\n              -- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.\n    exact\n      IsCyclotomicExtension.equiv _ _ _\n        (Subalgebra.equivOfEq _ _ H)\n          -- Porting note: the next `refine` was `rw [H]`, abusing defeq, and it now fails.\n  have H :=\n    IntermediateField.adjoin_simple_toSubalgebra_of_integral (IsCyclotomicExtension.integral {p ^ (k + 1)} K L (\u03b7 + 1))\n  refine @IsCyclotomicExtension.equiv _ _ _ _ _ _ _ _ _ ?_ (Subalgebra.equivOfEq _ _ H).symm\n  have h\u03b7' : IsPrimitiveRoot (\u03b7 + 1) \u2191(p ^ (k + 1 - s)) := by simpa using h\u03b7\n  convert h\u03b7'.adjoin_isCyclotomicExtension K using 1\n  rw [Nat.sub_add_comm hs]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n[PROOFSTEP]\nsuffices IsCyclotomicExtension {p ^ (k - s + 1)} K K\u27ee\u03b7 + 1\u27ef.toSubalgebra\n  by\n  have H : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = K\u27ee\u03b7\u27ef.toSubalgebra :=\n    by\n    simp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral\n        (IsCyclotomicExtension.integral {p ^ (k + 1)} K L _)]\n    refine' Subalgebra.ext fun x => \u27e8fun hx => adjoin_le _ hx, fun hx => adjoin_le _ hx\u27e9\n    \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe]\n      exact\n        Subalgebra.add_mem _ (subset_adjoin (mem_singleton \u03b7))\n          (Subalgebra.one_mem _)\n            -- Porting note `\u03b7def` was not needed.\n    \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe, \u2190 \u03b7def]\n      nth_rw 1 [\u2190 add_sub_cancel \u03b7 1]\n      refine'\n        Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _))\n          (Subalgebra.one_mem _)\n            -- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.\n  exact\n    IsCyclotomicExtension.equiv _ _ _\n      (Subalgebra.equivOfEq _ _ H)\n        -- Porting note: the next `refine` was `rw [H]`, abusing defeq, and it now fails.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n[PROOFSTEP]\nhave H : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = K\u27ee\u03b7\u27ef.toSubalgebra :=\n  by\n  simp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral\n      (IsCyclotomicExtension.integral {p ^ (k + 1)} K L _)]\n  refine' Subalgebra.ext fun x => \u27e8fun hx => adjoin_le _ hx, fun hx => adjoin_le _ hx\u27e9\n  \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe]\n    exact\n      Subalgebra.add_mem _ (subset_adjoin (mem_singleton \u03b7))\n        (Subalgebra.one_mem _)\n          -- Porting note `\u03b7def` was not needed.\n  \u00b7 simp only [Set.singleton_subset_iff, SetLike.mem_coe, \u2190 \u03b7def]\n    nth_rw 1 [\u2190 add_sub_cancel \u03b7 1]\n    refine'\n      Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _))\n        (Subalgebra.one_mem _)\n          -- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\n\u22a2 K\u27ee\u03b7 + 1\u27ef.toSubalgebra = K\u27ee\u03b7\u27ef.toSubalgebra\n[PROOFSTEP]\nsimp only [IntermediateField.adjoin_simple_toSubalgebra_of_integral\n    (IsCyclotomicExtension.integral {p ^ (k + 1)} K L _)]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\n\u22a2 adjoin K {\u03b6 ^ \u2191p ^ s - 1 + 1} = adjoin K {\u03b6 ^ \u2191p ^ s - 1}\n[PROOFSTEP]\nrefine' Subalgebra.ext fun x => \u27e8fun hx => adjoin_le _ hx, fun hx => adjoin_le _ hx\u27e9\n[GOAL]\ncase refine'_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nx : L\nhx : x \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1 + 1}\n\u22a2 {\u03b6 ^ \u2191p ^ s - 1 + 1} \u2286 \u2191(adjoin K {\u03b6 ^ \u2191p ^ s - 1})\n[PROOFSTEP]\nsimp only [Set.singleton_subset_iff, SetLike.mem_coe]\n[GOAL]\ncase refine'_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nx : L\nhx : x \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1 + 1}\n\u22a2 \u03b6 ^ \u2191p ^ s - 1 + 1 \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1}\n[PROOFSTEP]\nexact\n  Subalgebra.add_mem _ (subset_adjoin (mem_singleton \u03b7))\n    (Subalgebra.one_mem _)\n      -- Porting note `\u03b7def` was not needed.\n[GOAL]\ncase refine'_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nx : L\nhx : x \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1}\n\u22a2 {\u03b6 ^ \u2191p ^ s - 1} \u2286 \u2191(adjoin K {\u03b6 ^ \u2191p ^ s - 1 + 1})\n[PROOFSTEP]\nsimp only [Set.singleton_subset_iff, SetLike.mem_coe, \u2190 \u03b7def]\n[GOAL]\ncase refine'_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nx : L\nhx : x \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1}\n\u22a2 \u03b7 \u2208 adjoin K {\u03b7 + 1}\n[PROOFSTEP]\nnth_rw 1 [\u2190 add_sub_cancel \u03b7 1]\n[GOAL]\ncase refine'_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nx : L\nhx : x \u2208 adjoin K {\u03b6 ^ \u2191p ^ s - 1}\n\u22a2 \u03b7 + 1 - 1 \u2208 adjoin K {\u03b7 + 1}\n[PROOFSTEP]\nrefine'\n  Subalgebra.sub_mem _ (subset_adjoin (mem_singleton _))\n    (Subalgebra.one_mem _)\n      -- Porting note: the previous proof was `rw [H] at this; exact this` but it now fails.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = K\u27ee\u03b7\u27ef.toSubalgebra\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n[PROOFSTEP]\nexact\n  IsCyclotomicExtension.equiv _ _ _\n    (Subalgebra.equivOfEq _ _ H)\n      -- Porting note: the next `refine` was `rw [H]`, abusing defeq, and it now fails.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\n[PROOFSTEP]\nhave H :=\n  IntermediateField.adjoin_simple_toSubalgebra_of_integral (IsCyclotomicExtension.integral {p ^ (k + 1)} K L (\u03b7 + 1))\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = adjoin K {\u03b7 + 1}\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7 + 1\u27ef.toSubalgebra }\n[PROOFSTEP]\nrefine @IsCyclotomicExtension.equiv _ _ _ _ _ _ _ _ _ ?_ (Subalgebra.equivOfEq _ _ H).symm\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = adjoin K {\u03b7 + 1}\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 adjoin K {\u03b7 + 1} }\n[PROOFSTEP]\nhave h\u03b7' : IsPrimitiveRoot (\u03b7 + 1) \u2191(p ^ (k + 1 - s)) := by simpa using h\u03b7\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = adjoin K {\u03b7 + 1}\n\u22a2 IsPrimitiveRoot (\u03b7 + 1) \u2191(p ^ (k + 1 - s))\n[PROOFSTEP]\nsimpa using h\u03b7\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = adjoin K {\u03b7 + 1}\nh\u03b7' : IsPrimitiveRoot (\u03b7 + 1) \u2191(p ^ (k + 1 - s))\n\u22a2 IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 adjoin K {\u03b7 + 1} }\n[PROOFSTEP]\nconvert h\u03b7'.adjoin_isCyclotomicExtension K using 1\n[GOAL]\ncase h.e'_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nH : K\u27ee\u03b7 + 1\u27ef.toSubalgebra = adjoin K {\u03b7 + 1}\nh\u03b7' : IsPrimitiveRoot (\u03b7 + 1) \u2191(p ^ (k + 1 - s))\n\u22a2 {p ^ (k - s + 1)} = {p ^ (k + 1 - s)}\n[PROOFSTEP]\nrw [Nat.sub_add_comm hs]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nreplace h\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\n[GOAL]\ncase h\u03b7\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n\u22a2 IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\n[PROOFSTEP]\napply coe_submonoidClass_iff.1\n[GOAL]\ncase h\u03b7\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n\u22a2 IsPrimitiveRoot \u2191(\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\n[PROOFSTEP]\nconvert h\u03b7 using 1\n[GOAL]\ncase h.e'_4\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nh\u03b7 : IsPrimitiveRoot (\u03b7 + 1) (\u2191p ^ (k + 1 - s))\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\n\u22a2 \u2191(p ^ (k - s + 1)) = \u2191p ^ (k + 1 - s)\n[PROOFSTEP]\nrw [Nat.sub_add_comm hs, pow_coe]\n  -- Porting note: the following `haveI` were not needed because the locale `cyclotomic` set them\n  -- as instances.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.finiteDimensional {p ^ (k + 1)} K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis : FiniteDimensional K L\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.isGalois (p ^ (k + 1)) K L\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\n\u22a2 \u2191(Algebra.norm K) \u03b7 = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nrw [norm_eq_norm_adjoin K]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\n\u22a2 \u2191(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K \u03b7) ^ FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L =\n    \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhave H := h\u03b7.sub_one_norm_isPrimePow ?_ hirr\u2081 htwo\n[GOAL]\ncase refine_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) (\u03b7\u2081 + 1 - 1) = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 \u2191(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K \u03b7) ^ FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L =\n    \u2191\u2191p ^ \u2191p ^ s\ncase refine_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\n\u22a2 IsPrimePow \u2191(p ^ (k - s + 1))\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\n\u22a2 IsPrimePow \u2191(p ^ (k - s + 1))\n[PROOFSTEP]\nrw [PNat.pow_coe]\n[GOAL]\ncase refine_1\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\n\u22a2 IsPrimePow (\u2191p ^ (k - s + 1))\n[PROOFSTEP]\nexact hpri.1.isPrimePow.pow (Nat.succ_ne_zero _)\n[GOAL]\ncase refine_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) (\u03b7\u2081 + 1 - 1) = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 \u2191(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K \u03b7) ^ FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L =\n    \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nrw [add_sub_cancel] at H \n[GOAL]\ncase refine_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 \u2191(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K \u03b7) ^ FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L =\n    \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nrw [H]\n[GOAL]\ncase refine_2\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 \u2191(minFac \u2191(p ^ (k - s + 1))) ^ FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine_2.e_a.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 minFac \u2191(p ^ (k - s + 1)) = \u2191p\n[PROOFSTEP]\nrw [PNat.pow_coe, Nat.pow_minFac, hpri.1.minFac_eq]\n[GOAL]\ncase refine_2.e_a.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 k - s + 1 \u2260 0\n[PROOFSTEP]\nexact Nat.succ_ne_zero _\n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b9 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d : FiniteDimensional K L\nthis : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nhave := FiniteDimensional.finrank_mul_finrank K K\u27ee\u03b7\u27ef L\n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis :\n  FiniteDimensional.finrank K { x // x \u2208 K\u27ee\u03b7\u27ef } * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L =\n    FiniteDimensional.finrank K L\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nrw [IsCyclotomicExtension.finrank L hirr, IsCyclotomicExtension.finrank K\u27ee\u03b7\u27ef hirr\u2081, PNat.pow_coe, PNat.pow_coe,\n  Nat.totient_prime_pow hpri.out (k - s).succ_pos, Nat.totient_prime_pow hpri.out k.succ_pos, mul_comm _ ((p : \u2115) - 1),\n  mul_assoc, mul_comm ((p : \u2115) ^ (k.succ - 1))] at this \n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis :\n  (\u2191p - 1) * (\u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L) = (\u2191p - 1) * \u2191p ^ (succ k - 1)\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nreplace this := mul_left_cancel\u2080 (tsub_pos_iff_lt.2 hpri.out.one_lt).ne' this\n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis : \u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ (succ k - 1)\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nhave Hex : k.succ - 1 = (k - s).succ - 1 + s :=\n  by\n  simp only [Nat.succ_sub_succ_eq_sub, tsub_zero]\n  exact (Nat.sub_add_cancel hs).symm\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis : \u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ (succ k - 1)\n\u22a2 succ k - 1 = succ (k - s) - 1 + s\n[PROOFSTEP]\nsimp only [Nat.succ_sub_succ_eq_sub, tsub_zero]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis : \u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ (succ k - 1)\n\u22a2 k = k - s + s\n[PROOFSTEP]\nexact (Nat.sub_add_cancel hs).symm\n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis : \u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ (succ k - 1)\nHex : succ k - 1 = succ (k - s) - 1 + s\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nrw [Hex, pow_add] at this \n[GOAL]\ncase refine_2.e_a\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhtwo : p ^ (k - s + 1) \u2260 2\nhirr\u2081 : Irreducible (cyclotomic (\u2191(p ^ (k - s + 1))) K)\n\u03b7 : L := \u03b6 ^ \u2191p ^ s - 1\n\u03b7def : \u03b7 = \u03b6 ^ \u2191p ^ s - 1\n\u03b7\u2081 : { x // x \u2208 K\u27ee\u03b7\u27ef } := IntermediateField.AdjoinSimple.gen K \u03b7\nthis\u271d\u00b2 : IsCyclotomicExtension {p ^ (k - s + 1)} K { x // x \u2208 K\u27ee\u03b7\u27ef }\nh\u03b7 : IsPrimitiveRoot (\u03b7\u2081 + 1) \u2191(p ^ (k - s + 1))\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsGalois K L\nH : \u2191(Algebra.norm K) \u03b7\u2081 = \u2191(minFac \u2191(p ^ (k - s + 1)))\nthis : \u2191p ^ (succ (k - s) - 1) * FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ (succ (k - s) - 1) * \u2191p ^ s\nHex : succ k - 1 = succ (k - s) - 1 + s\n\u22a2 FiniteDimensional.finrank { x // x \u2208 K\u27ee\u03b7\u27ef } L = \u2191p ^ s\n[PROOFSTEP]\nexact mul_left_cancel\u2080 (pow_ne_zero _ hpri.out.ne_zero) this\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nrefine' h\u03b6.pow_sub_one_norm_prime_pow_ne_two hirr hs fun h => _\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\nh : p ^ (k - s + 1) = 2\n\u22a2 False\n[PROOFSTEP]\nhave coe_two : ((2 : \u2115+) : \u2115) = 2 := by norm_cast\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\nh : p ^ (k - s + 1) = 2\n\u22a2 \u21912 = 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\nh : p ^ (k - s + 1) = 2\ncoe_two : \u21912 = 2\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 PNat.coe_inj, coe_two, PNat.pow_coe, \u2190 pow_one 2] at h \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\nh\u271d : \u2191p ^ (k - s + 1) = 2\nh : \u2191p ^ (k - s + 1) = 2 ^ 1\ncoe_two : \u21912 = 2\n\u22a2 False\n[PROOFSTEP]\nreplace h := eq_of_prime_pow_eq (prime_iff.1 hpri.out) (prime_iff.1 Nat.prime_two) (k - s).succ_pos h\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\ns : \u2115\nhs : s \u2264 k\nhodd : p \u2260 2\nh\u271d : \u2191p ^ (k - s + 1) = 2\ncoe_two : \u21912 = 2\nh : \u2191p = 2\n\u22a2 False\n[PROOFSTEP]\nexact hodd (PNat.coe_injective h)\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nh : p \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p\n[PROOFSTEP]\nsimpa using h\u03b6.pow_sub_one_norm_prime_ne_two hirr k.zero_le h\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nh : p \u2260 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p\n[PROOFSTEP]\nreplace hirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1)) : \u2115) K) := by simp [hirr]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nh : p \u2260 2\n\u22a2 Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n[PROOFSTEP]\nsimp [hirr]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p\n[PROOFSTEP]\nreplace h\u03b6 : IsPrimitiveRoot \u03b6 (\u2191(p ^ (0 + 1)) : \u2115) := by simp [h\u03b6]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\n\u22a2 IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n[PROOFSTEP]\nsimp [h\u03b6]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p\n[PROOFSTEP]\nhaveI : IsCyclotomicExtension {p ^ (0 + 1)} K L := by simp [hcyc]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 IsCyclotomicExtension {p ^ (0 + 1)} K L\n[PROOFSTEP]\nsimp [hcyc]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nhpri : Fact (Nat.Prime \u2191p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p \u2260 2\nhirr : Irreducible (cyclotomic (\u2191(p ^ (0 + 1))) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = \u2191\u2191p\n[PROOFSTEP]\nsimpa using sub_one_norm_prime_ne_two h\u03b6 hirr h\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nhave := h\u03b6.pow_of_dvd (fun h => two_ne_zero (pow_eq_zero h)) (pow_dvd_pow 2 (le_succ k))\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) (2 ^ (k + 1) / 2 ^ k)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nrw [Nat.pow_div (le_succ k) zero_lt_two, Nat.succ_sub (le_refl k), Nat.sub_self, pow_one] at this \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nhave H : (-1 : L) - (1 : L) = algebraMap K L (-2) :=\n  by\n  simp only [map_neg, map_ofNat]\n  ring\n    -- Porting note: `simpa using hirr` was `simp [hirr]`.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\n\u22a2 -1 - 1 = \u2191(algebraMap K L) (-2)\n[PROOFSTEP]\nsimp only [map_neg, map_ofNat]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\n\u22a2 -1 - 1 = -2\n[PROOFSTEP]\nring\n  -- Porting note: `simpa using hirr` was `simp [hirr]`.\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\nH : -1 - 1 = \u2191(algebraMap K L) (-2)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nreplace hirr : Irreducible (cyclotomic ((2 : \u2115+) ^ (k + 1) : \u2115+) K) := by simpa using hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\nH : -1 - 1 = \u2191(algebraMap K L) (-2)\n\u22a2 Irreducible (cyclotomic (\u2191(2 ^ (k + 1))) K)\n[PROOFSTEP]\nsimpa using hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\nH : -1 - 1 = \u2191(algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (k + 1))) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nrw [this.eq_neg_one_of_two_right, H, Algebra.norm_algebraMap, IsCyclotomicExtension.finrank L hirr, pow_coe,\n  show ((2 : \u2115+) : \u2115) = 2 from rfl, totient_prime_pow Nat.prime_two (zero_lt_succ k), succ_sub_succ_eq_sub, tsub_zero]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nthis : IsPrimitiveRoot (\u03b6 ^ 2 ^ k) 2\nH : -1 - 1 = \u2191(algebraMap K L) (-2)\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (k + 1))) K)\n\u22a2 (-2) ^ (2 ^ k * (2 - 1)) = (-2) ^ 2 ^ k\n[PROOFSTEP]\nsimp\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = 2\n[PROOFSTEP]\nhave : 2 < (2 : \u2115+) ^ k := by\n  simp only [\u2190 coe_lt_coe, one_coe, pow_coe]\n  nth_rw 1 [\u2190 pow_one 2]\n  exact\n    pow_lt_pow one_lt_two\n      (lt_of_lt_of_le one_lt_two hk)\n        -- Porting note: `simpa using hirr` was `simp [hirr]`_\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\n\u22a2 2 < 2 ^ k\n[PROOFSTEP]\nsimp only [\u2190 coe_lt_coe, one_coe, pow_coe]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\n\u22a2 \u21912 < \u21912 ^ k\n[PROOFSTEP]\nnth_rw 1 [\u2190 pow_one 2]\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\n\u22a2 \u2191(2 ^ 1) < \u21912 ^ k\n[PROOFSTEP]\nexact\n  pow_lt_pow one_lt_two\n    (lt_of_lt_of_le one_lt_two hk)\n      -- Porting note: `simpa using hirr` was `simp [hirr]`_\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nthis : 2 < 2 ^ k\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = 2\n[PROOFSTEP]\nreplace hirr : Irreducible (cyclotomic ((2 : \u2115+) ^ k : \u2115+) K) := by simpa using hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nthis : 2 < 2 ^ k\n\u22a2 Irreducible (cyclotomic (\u2191(2 ^ k)) K)\n[PROOFSTEP]\nsimpa using hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (\u2191(2 ^ k)) K)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = 2\n[PROOFSTEP]\nreplace h\u03b6 : IsPrimitiveRoot \u03b6 ((2 : \u2115+) ^ k) := by simpa using h\u03b6\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ k)\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (\u2191(2 ^ k)) K)\n\u22a2 IsPrimitiveRoot \u03b6 \u2191(2 ^ k)\n[PROOFSTEP]\nsimpa using h\u03b6\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (\u2191(2 ^ k)) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ k)\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = 2\n[PROOFSTEP]\nobtain \u27e8k\u2081, hk\u2081\u27e9 := exists_eq_succ_of_ne_zero (lt_of_lt_of_le zero_lt_two hk).ne.symm\n[GOAL]\ncase intro\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk : \u2115\nhk : 2 \u2264 k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (\u2191(2 ^ k)) K)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ k)\nk\u2081 : \u2115\nhk\u2081 : k = succ k\u2081\n\u22a2 \u2191(Algebra.norm K) (\u03b6 - 1) = 2\n[PROOFSTEP]\nsimpa [hk\u2081, show ((2 : \u2115+) : \u2115) = 2 from rfl] using sub_one_norm_eq_eval_cyclotomic h\u03b6 this hirr\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nby_cases htwo : p ^ (k - s + 1) = 2\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nhave hp : p = 2 := by\n  rw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at htwo \n  replace htwo := eq_of_prime_pow_eq (prime_iff.1 hpri.out) (prime_iff.1 Nat.prime_two) (succ_pos _) htwo\n  rwa [show 2 = ((2 : \u2115+) : \u2115) by simp, PNat.coe_inj] at htwo \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\n\u22a2 p = 2\n[PROOFSTEP]\nrw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at htwo \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u2191p ^ (k - s + 1) = \u21912\nhtwo : \u2191p ^ (k - s + 1) = \u2191(2 ^ 1)\n\u22a2 p = 2\n[PROOFSTEP]\nreplace htwo := eq_of_prime_pow_eq (prime_iff.1 hpri.out) (prime_iff.1 Nat.prime_two) (succ_pos _) htwo\n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u2191p ^ (k - s + 1) = \u21912\nhtwo : \u2191p = 2\n\u22a2 p = 2\n[PROOFSTEP]\nrwa [show 2 = ((2 : \u2115+) : \u2115) by simp, PNat.coe_inj] at htwo \n[GOAL]\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u2191p ^ (k - s + 1) = \u21912\nhtwo : \u2191p = 2\n\u22a2 2 = \u21912\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nreplace hs : s = k\n[GOAL]\ncase hs\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\n\u22a2 s = k\n[PROOFSTEP]\nrw [hp, \u2190 PNat.coe_inj, PNat.pow_coe] at htwo \n[GOAL]\ncase hs\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : \u21912 ^ (k - s + 1) = \u21912\nhp : p = 2\n\u22a2 s = k\n[PROOFSTEP]\nnth_rw 2 [\u2190 pow_one 2] at htwo \n[GOAL]\ncase hs\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u21912 ^ (k - s + 1) = \u21912\nhtwo : \u21912 ^ (k - s + 1) = \u2191(2 ^ 1)\nhp : p = 2\n\u22a2 s = k\n[PROOFSTEP]\nreplace htwo := Nat.pow_right_injective rfl.le htwo\n[GOAL]\ncase hs\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u21912 ^ (k - s + 1) = \u21912\nhp : p = 2\nhtwo : k - s + 1 = 1\n\u22a2 s = k\n[PROOFSTEP]\nrw [add_left_eq_self, Nat.sub_eq_zero_iff_le] at htwo \n[GOAL]\ncase hs\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo\u271d : \u21912 ^ (k - s + 1) = \u21912\nhp : p = 2\nhtwo : k \u2264 s\n\u22a2 s = k\n[PROOFSTEP]\nrefine' le_antisymm hs htwo\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\nhs : s = k\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nsimp only [hs, hp, one_coe, cast_one, pow_coe, show ((2 : \u2115+) : \u2115) = 2 from rfl] at h\u03b6 hirr hcycl \u22a2\n[GOAL]\ncase pos\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nhpri : Fact (Nat.Prime \u2191p)\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\nhs : s = k\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nhcycl : IsCyclotomicExtension {2 ^ (k + 1)} K L\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = \u21912 ^ 2 ^ k\n[PROOFSTEP]\nobtain \u27e8k\u2081, hk\u2081\u27e9 := Nat.exists_eq_succ_of_ne_zero hk\n[GOAL]\ncase pos.intro\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nhpri : Fact (Nat.Prime \u2191p)\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\nhs : s = k\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nhcycl : IsCyclotomicExtension {2 ^ (k + 1)} K L\nk\u2081 : \u2115\nhk\u2081 : k = succ k\u2081\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ 2 ^ k - 1) = \u21912 ^ 2 ^ k\n[PROOFSTEP]\nrw [h\u03b6.pow_sub_one_norm_two hirr, hk\u2081, _root_.pow_succ, pow_mul, neg_eq_neg_one_mul, mul_pow, neg_one_sq, one_mul, \u2190\n  pow_mul, \u2190 _root_.pow_succ]\n[GOAL]\ncase pos.intro\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nhpri : Fact (Nat.Prime \u2191p)\nhk : k \u2260 0\nhtwo : p ^ (k - s + 1) = 2\nhp : p = 2\nhs : s = k\nh\u03b6 : IsPrimitiveRoot \u03b6 (2 ^ (k + 1))\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nhcycl : IsCyclotomicExtension {2 ^ (k + 1)} K L\nk\u2081 : \u2115\nhk\u2081 : k = succ k\u2081\n\u22a2 2 ^ 2 ^ (k\u2081 + 1) = \u21912 ^ 2 ^ (k\u2081 + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\np n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b2 : Field L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra K L\nk s : \u2115\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhpri : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhs : s \u2264 k\nhk : k \u2260 0\nhtwo : \u00acp ^ (k - s + 1) = 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ s - 1) = \u2191\u2191p ^ \u2191p ^ s\n[PROOFSTEP]\nexact h\u03b6.pow_sub_one_norm_prime_pow_ne_two hirr hs htwo\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots", "llama_tokens": 62008, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519528057272544, "lm_q2_score": 0.640635847978761, "lm_q1q2_score": 0.5457915081349644}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Prime p)\nhp : p % 4 \u2260 3\n\u22a2 \u2203 a b, a ^ 2 + b ^ 2 = p\n[PROOFSTEP]\napply sq_add_sq_of_nat_prime_of_not_irreducible p\n[GOAL]\np : \u2115\ninst\u271d : Fact (Prime p)\nhp : p % 4 \u2260 3\n\u22a2 \u00acIrreducible \u2191p\n[PROOFSTEP]\nrwa [PrincipalIdealRing.irreducible_iff_prime, prime_iff_mod_four_eq_three_of_nat_prime p]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x y u v : R\nha : a = x ^ 2 + y ^ 2\nhb : b = u ^ 2 + v ^ 2\n\u22a2 a * b = (x * u - y * v) ^ 2 + (x * v + y * u) ^ 2\n[PROOFSTEP]\nrw [ha, hb]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\na b x y u v : R\nha : a = x ^ 2 + y ^ 2\nhb : b = u ^ 2 + v ^ 2\n\u22a2 (x ^ 2 + y ^ 2) * (u ^ 2 + v ^ 2) = (x * u - y * v) ^ 2 + (x * v + y * u) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\na b x y u v : \u2115\nha : a = x ^ 2 + y ^ 2\nhb : b = u ^ 2 + v ^ 2\n\u22a2 \u2203 r s, a * b = r ^ 2 + s ^ 2\n[PROOFSTEP]\nzify at ha hb \u22a2\n[GOAL]\na b x y u v : \u2115\nha : \u2191a = \u2191x ^ 2 + \u2191y ^ 2\nhb : \u2191b = \u2191u ^ 2 + \u2191v ^ 2\n\u22a2 \u2203 r s, \u2191a * \u2191b = \u2191r ^ 2 + \u2191s ^ 2\n[PROOFSTEP]\nobtain \u27e8r, s, h\u27e9 := _root_.sq_add_sq_mul ha hb\n[GOAL]\ncase intro.intro\na b x y u v : \u2115\nha : \u2191a = \u2191x ^ 2 + \u2191y ^ 2\nhb : \u2191b = \u2191u ^ 2 + \u2191v ^ 2\nr s : \u2124\nh : \u2191a * \u2191b = r ^ 2 + s ^ 2\n\u22a2 \u2203 r s, \u2191a * \u2191b = \u2191r ^ 2 + \u2191s ^ 2\n[PROOFSTEP]\nrefine' \u27e8r.natAbs, s.natAbs, _\u27e9\n[GOAL]\ncase intro.intro\na b x y u v : \u2115\nha : \u2191a = \u2191x ^ 2 + \u2191y ^ 2\nhb : \u2191b = \u2191u ^ 2 + \u2191v ^ 2\nr s : \u2124\nh : \u2191a * \u2191b = r ^ 2 + s ^ 2\n\u22a2 \u2191a * \u2191b = \u2191(Int.natAbs r) ^ 2 + \u2191(Int.natAbs s) ^ 2\n[PROOFSTEP]\nsimpa only [Int.coe_natAbs, sq_abs]\n[GOAL]\nm n : \u2115\nhd : m \u2223 n\nhs : IsSquare (-1)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nlet f : ZMod n \u2192+* ZMod m := ZMod.castHom hd _\n[GOAL]\nm n : \u2115\nhd : m \u2223 n\nhs : IsSquare (-1)\nf : ZMod n \u2192+* ZMod m := castHom hd (ZMod m)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nrw [\u2190 RingHom.map_one f, \u2190 RingHom.map_neg]\n[GOAL]\nm n : \u2115\nhd : m \u2223 n\nhs : IsSquare (-1)\nf : ZMod n \u2192+* ZMod m := castHom hd (ZMod m)\n\u22a2 IsSquare (\u2191f (-1))\n[PROOFSTEP]\nexact hs.map f\n[GOAL]\nm n : \u2115\nhc : Nat.coprime m n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nhave : IsSquare (-1 : ZMod m \u00d7 ZMod n) :=\n  by\n  rw [show (-1 : ZMod m \u00d7 ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl]\n  obtain \u27e8x, hx\u27e9 := hm\n  obtain \u27e8y, hy\u27e9 := hn\n  rw [hx, hy]\n  exact \u27e8(x, y), rfl\u27e9\n[GOAL]\nm n : \u2115\nhc : Nat.coprime m n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nrw [show (-1 : ZMod m \u00d7 ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl]\n[GOAL]\nm n : \u2115\nhc : Nat.coprime m n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\n\u22a2 IsSquare (-1, -1)\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := hm\n[GOAL]\ncase intro\nm n : \u2115\nhc : Nat.coprime m n\nhn : IsSquare (-1)\nx : ZMod m\nhx : -1 = x * x\n\u22a2 IsSquare (-1, -1)\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := hn\n[GOAL]\ncase intro.intro\nm n : \u2115\nhc : Nat.coprime m n\nx : ZMod m\nhx : -1 = x * x\ny : ZMod n\nhy : -1 = y * y\n\u22a2 IsSquare (-1, -1)\n[PROOFSTEP]\nrw [hx, hy]\n[GOAL]\ncase intro.intro\nm n : \u2115\nhc : Nat.coprime m n\nx : ZMod m\nhx : -1 = x * x\ny : ZMod n\nhy : -1 = y * y\n\u22a2 IsSquare (x * x, y * y)\n[PROOFSTEP]\nexact \u27e8(x, y), rfl\u27e9\n[GOAL]\nm n : \u2115\nhc : Nat.coprime m n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\nthis : IsSquare (-1)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nsimpa only [RingEquiv.map_neg_one] using this.map (ZMod.chineseRemainder hc).symm\n[GOAL]\np n : \u2115\nhpp : Prime p\nhp : p \u2223 n\nhs : IsSquare (-1)\n\u22a2 p % 4 \u2260 3\n[PROOFSTEP]\nobtain \u27e8y, h\u27e9 := ZMod.isSquare_neg_one_of_dvd hp hs\n[GOAL]\ncase intro\np n : \u2115\nhpp : Prime p\nhp : p \u2223 n\nhs : IsSquare (-1)\ny : ZMod p\nh : -1 = y * y\n\u22a2 p % 4 \u2260 3\n[PROOFSTEP]\nrw [\u2190 sq, eq_comm, show (-1 : ZMod p) = -1 ^ 2 by ring] at h \n[GOAL]\np n : \u2115\nhpp : Prime p\nhp : p \u2223 n\nhs : IsSquare (-1)\ny : ZMod p\nh : y ^ 2 = -1\n\u22a2 -1 = -1 ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase intro\np n : \u2115\nhpp : Prime p\nhp : p \u2223 n\nhs : IsSquare (-1)\ny : ZMod p\nh : y ^ 2 = -1 ^ 2\n\u22a2 p % 4 \u2260 3\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8hpp\u27e9\n[GOAL]\ncase intro\np n : \u2115\nhpp : Prime p\nhp : p \u2223 n\nhs : IsSquare (-1)\ny : ZMod p\nh : y ^ 2 = -1 ^ 2\nthis : Fact (Prime p)\n\u22a2 p % 4 \u2260 3\n[PROOFSTEP]\nexact ZMod.mod_four_ne_three_of_sq_eq_neg_sq' one_ne_zero h\n[GOAL]\nn : \u2115\nhn : Squarefree n\n\u22a2 IsSquare (-1) \u2194 \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nrefine' \u27e8fun H q hqp hqd => hqp.mod_four_ne_three_of_dvd_isSquare_neg_one hqd H, fun H => _\u27e9\n[GOAL]\nn : \u2115\nhn : Squarefree n\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\ninduction' n using induction_on_primes with p n hpp ih\n[GOAL]\ncase h\u2080\nn : \u2115\nhn\u271d : Squarefree n\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\nhn : Squarefree 0\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 0 \u2192 q % 4 \u2260 3\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nexact False.elim (hn.ne_zero rfl)\n[GOAL]\ncase h\u2081\nn : \u2115\nhn\u271d : Squarefree n\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\nhn : Squarefree 1\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 1 \u2192 q % 4 \u2260 3\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nexact \u27e80, by simp only [Fin.zero_mul, neg_eq_zero, Fin.one_eq_zero_iff]\u27e9\n[GOAL]\nn : \u2115\nhn\u271d : Squarefree n\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\nhn : Squarefree 1\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 1 \u2192 q % 4 \u2260 3\n\u22a2 -1 = 0 * 0\n[PROOFSTEP]\nsimp only [Fin.zero_mul, neg_eq_zero, Fin.one_eq_zero_iff]\n[GOAL]\ncase h\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8hpp\u27e9\n[GOAL]\ncase h\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\nthis : Fact (Nat.Prime p)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nhave hcp : p.coprime n := by\n  by_contra hc\n  exact hpp.not_unit (hn p <| mul_dvd_mul_left p <| hpp.dvd_iff_not_coprime.mpr hc)\n[GOAL]\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\nthis : Fact (Nat.Prime p)\n\u22a2 Nat.coprime p n\n[PROOFSTEP]\nby_contra hc\n[GOAL]\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\nthis : Fact (Nat.Prime p)\nhc : \u00acNat.coprime p n\n\u22a2 False\n[PROOFSTEP]\nexact hpp.not_unit (hn p <| mul_dvd_mul_left p <| hpp.dvd_iff_not_coprime.mpr hc)\n[GOAL]\ncase h\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\nthis : Fact (Nat.Prime p)\nhcp : Nat.coprime p n\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nhave hp\u2081 := ZMod.exists_sq_eq_neg_one_iff.mpr (H hpp (dvd_mul_right p n))\n[GOAL]\ncase h\nn\u271d : \u2115\nhn\u271d : Squarefree n\u271d\nH\u271d : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n\u271d \u2192 q % 4 \u2260 3\np n : \u2115\nhpp : Nat.Prime p\nih : Squarefree n \u2192 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 IsSquare (-1)\nhn : Squarefree (p * n)\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 p * n \u2192 q % 4 \u2260 3\nthis : Fact (Nat.Prime p)\nhcp : Nat.coprime p n\nhp\u2081 : IsSquare (-1)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nexact ZMod.isSquare_neg_one_mul hcp hp\u2081 (ih hn.of_mul_right fun hqp hqd => H hqp <| dvd_mul_of_dvd_right hqd _)\n[GOAL]\nn : \u2115\nhn : Squarefree n\n\u22a2 IsSquare (-1) \u2194 \u2200 {q : \u2115}, q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nhave help : \u2200 a b : ZMod 4, a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3 := by decide\n[GOAL]\nn : \u2115\nhn : Squarefree n\n\u22a2 \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\n[PROOFSTEP]\ndecide\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\n\u22a2 IsSquare (-1) \u2194 \u2200 {q : \u2115}, q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nrw [ZMod.isSquare_neg_one_iff hn]\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\n\u22a2 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2194 \u2200 {q : \u2115}, q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nrefine' \u27e8_, fun H q _ => H\u27e9\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\n\u22a2 (\u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3) \u2192 \u2200 {q : \u2115}, q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nintro H\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\n\u22a2 \u2200 {q : \u2115}, q \u2223 n \u2192 q % 4 \u2260 3\n[PROOFSTEP]\nrefine' @induction_on_primes _ _ _ (fun p q hp hq hpq => _)\n[GOAL]\ncase refine'_1\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\n\u22a2 0 \u2223 n \u2192 0 % 4 \u2260 3\n[PROOFSTEP]\nexact fun _ => by norm_num\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\nx\u271d : 0 \u2223 n\n\u22a2 0 % 4 \u2260 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\n\u22a2 1 \u2223 n \u2192 1 % 4 \u2260 3\n[PROOFSTEP]\nexact fun _ => by norm_num\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\nx\u271d : 1 \u2223 n\n\u22a2 1 % 4 \u2260 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_3\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhp : Nat.Prime p\nhq : q \u2223 n \u2192 q % 4 \u2260 3\nhpq : p * q \u2223 n\n\u22a2 p * q % 4 \u2260 3\n[PROOFSTEP]\nreplace hp := H hp (dvd_of_mul_right_dvd hpq)\n[GOAL]\ncase refine'_3\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhq : q \u2223 n \u2192 q % 4 \u2260 3\nhpq : p * q \u2223 n\nhp : p % 4 \u2260 3\n\u22a2 p * q % 4 \u2260 3\n[PROOFSTEP]\nreplace hq := hq (dvd_of_mul_left_dvd hpq)\n[GOAL]\ncase refine'_3\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : p % 4 \u2260 3\nhq : q % 4 \u2260 3\n\u22a2 p * q % 4 \u2260 3\n[PROOFSTEP]\nrw [show 3 = 3 % 4 by norm_num, Ne.def, \u2190 ZMod.nat_cast_eq_nat_cast_iff'] at hp hq \u22a2\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : p % 4 \u2260 3\nhq : q % 4 \u2260 3\n\u22a2 3 = 3 % 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : p % 4 \u2260 3 % 4\nhq : q % 4 \u2260 3\n\u22a2 3 = 3 % 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : p % 4 \u2260 3 % 4\nhq : q % 4 \u2260 3 % 4\n\u22a2 3 = 3 % 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_3\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : \u00ac\u2191p = \u21913\nhq : \u00ac\u2191q = \u21913\n\u22a2 \u00ac\u2191(p * q) = \u21913\n[PROOFSTEP]\nrw [Nat.cast_mul]\n[GOAL]\ncase refine'_3\nn : \u2115\nhn : Squarefree n\nhelp : \u2200 (a b : ZMod 4), a \u2260 3 \u2192 b \u2260 3 \u2192 a * b \u2260 3\nH : \u2200 {q : \u2115}, Nat.Prime q \u2192 q \u2223 n \u2192 q % 4 \u2260 3\np q : \u2115\nhpq : p * q \u2223 n\nhp : \u00ac\u2191p = \u21913\nhq : \u00ac\u2191q = \u21913\n\u22a2 \u00ac\u2191p * \u2191q = \u21913\n[PROOFSTEP]\nexact help p q hp hq\n[GOAL]\nn : \u2115\nh : IsSquare (-1)\n\u22a2 \u2203 x y, n = x ^ 2 + y ^ 2\n[PROOFSTEP]\ninduction' n using induction_on_primes with p n hpp ih\n[GOAL]\ncase h\u2080\nn : \u2115\nh\u271d : IsSquare (-1)\nh : IsSquare (-1)\n\u22a2 \u2203 x y, 0 = x ^ 2 + y ^ 2\n[PROOFSTEP]\nexact \u27e80, 0, rfl\u27e9\n[GOAL]\ncase h\u2081\nn : \u2115\nh\u271d : IsSquare (-1)\nh : IsSquare (-1)\n\u22a2 \u2203 x y, 1 = x ^ 2 + y ^ 2\n[PROOFSTEP]\nexact \u27e80, 1, rfl\u27e9\n[GOAL]\ncase h\nn\u271d : \u2115\nh\u271d : IsSquare (-1)\np n : \u2115\nhpp : Prime p\nih : IsSquare (-1) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\nh : IsSquare (-1)\n\u22a2 \u2203 x y, p * n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8hpp\u27e9\n[GOAL]\ncase h\nn\u271d : \u2115\nh\u271d : IsSquare (-1)\np n : \u2115\nhpp : Prime p\nih : IsSquare (-1) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\nh : IsSquare (-1)\nthis : Fact (Prime p)\n\u22a2 \u2203 x y, p * n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nhave hp : IsSquare (-1 : ZMod p) := ZMod.isSquare_neg_one_of_dvd \u27e8n, rfl\u27e9 h\n[GOAL]\ncase h\nn\u271d : \u2115\nh\u271d : IsSquare (-1)\np n : \u2115\nhpp : Prime p\nih : IsSquare (-1) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\nh : IsSquare (-1)\nthis : Fact (Prime p)\nhp : IsSquare (-1)\n\u22a2 \u2203 x y, p * n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nobtain \u27e8u, v, huv\u27e9 := Nat.Prime.sq_add_sq (ZMod.exists_sq_eq_neg_one_iff.mp hp)\n[GOAL]\ncase h.intro.intro\nn\u271d : \u2115\nh\u271d : IsSquare (-1)\np n : \u2115\nhpp : Prime p\nih : IsSquare (-1) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\nh : IsSquare (-1)\nthis : Fact (Prime p)\nhp : IsSquare (-1)\nu v : \u2115\nhuv : u ^ 2 + v ^ 2 = p\n\u22a2 \u2203 x y, p * n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nobtain \u27e8x, y, hxy\u27e9 := ih (ZMod.isSquare_neg_one_of_dvd \u27e8p, mul_comm _ _\u27e9 h)\n[GOAL]\ncase h.intro.intro.intro.intro\nn\u271d : \u2115\nh\u271d : IsSquare (-1)\np n : \u2115\nhpp : Prime p\nih : IsSquare (-1) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\nh : IsSquare (-1)\nthis : Fact (Prime p)\nhp : IsSquare (-1)\nu v : \u2115\nhuv : u ^ 2 + v ^ 2 = p\nx y : \u2115\nhxy : n = x ^ 2 + y ^ 2\n\u22a2 \u2203 x y, p * n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nexact Nat.sq_add_sq_mul huv.symm hxy\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nobtain \u27e8u, v, huv\u27e9 : IsCoprime x n :=\n  by\n  have hc2 : IsCoprime (x ^ 2) (y ^ 2) := hc.pow\n  rw [show y ^ 2 = n + -1 * x ^ 2 by rw [h]; ring] at hc2 \n  exact (IsCoprime.pow_left_iff zero_lt_two).mp hc2.of_add_mul_right_right\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\n\u22a2 IsCoprime x n\n[PROOFSTEP]\nhave hc2 : IsCoprime (x ^ 2) (y ^ 2) := hc.pow\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nhc2 : IsCoprime (x ^ 2) (y ^ 2)\n\u22a2 IsCoprime x n\n[PROOFSTEP]\nrw [show y ^ 2 = n + -1 * x ^ 2 by rw [h]; ring] at hc2 \n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nhc2 : IsCoprime (x ^ 2) (y ^ 2)\n\u22a2 y ^ 2 = n + -1 * x ^ 2\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nhc2 : IsCoprime (x ^ 2) (y ^ 2)\n\u22a2 y ^ 2 = x ^ 2 + y ^ 2 + -1 * x ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nhc2 : IsCoprime (x ^ 2) (n + -1 * x ^ 2)\n\u22a2 IsCoprime x n\n[PROOFSTEP]\nexact (IsCoprime.pow_left_iff zero_lt_two).mp hc2.of_add_mul_right_right\n[GOAL]\ncase intro.intro\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nhave H : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v) := by\n  linear_combination -u ^ 2 * h + (n * v - u * x - 1) * huv\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\n\u22a2 u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n[PROOFSTEP]\nlinear_combination -u ^ 2 * h + (n * v - u * x - 1) * huv\n[GOAL]\ncase intro.intro\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nrefine' \u27e8u * y, _\u27e9\n[GOAL]\ncase intro.intro\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 -1 = \u2191u * \u2191y * (\u2191u * \u2191y)\n[PROOFSTEP]\nconv_rhs => tactic => norm_cast\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n| \u2191u * \u2191y * (\u2191u * \u2191y)\n[PROOFSTEP]\ntactic => norm_cast\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n| \u2191u * \u2191y * (\u2191u * \u2191y)\n[PROOFSTEP]\ntactic => norm_cast\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n| \u2191u * \u2191y * (\u2191u * \u2191y)\n[PROOFSTEP]\ntactic => norm_cast\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 \u2191u * \u2191y * (\u2191u * \u2191y) = ?m.37150\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 -1 = \u2191(u * y * (u * y))\n[PROOFSTEP]\nrw [(by norm_cast : (-1 : ZMod n.natAbs) = (-1 : \u2124))]\n[GOAL]\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 -1 = \u2191(-1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro\nn x y : \u2124\nh : n = x ^ 2 + y ^ 2\nhc : IsCoprime x y\nu v : \u2124\nhuv : u * x + v * n = 1\nH : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)\n\u22a2 \u2191(-1) = \u2191(u * y * (u * y))\n[PROOFSTEP]\nexact (ZMod.int_cast_eq_int_cast_iff_dvd_sub _ _ _).mpr (Int.natAbs_dvd.mpr \u27e8_, H\u27e9)\n[GOAL]\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhc : Nat.coprime x y\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nzify at h \n[GOAL]\nn x y : \u2115\nhc : Nat.coprime x y\nh : \u2191n = \u2191x ^ 2 + \u2191y ^ 2\n\u22a2 IsSquare (-1)\n[PROOFSTEP]\nexact ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime h hc.isCoprime\n[GOAL]\nn : \u2115\n\u22a2 (\u2203 x y, n = x ^ 2 + y ^ 2) \u2194 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115\n\u22a2 (\u2203 x y, n = x ^ 2 + y ^ 2) \u2192 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nrintro \u27e8x, y, h\u27e9\n[GOAL]\ncase mp.intro.intro\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nby_cases hxy : x = 0 \u2227 y = 0\n[GOAL]\ncase pos\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : x = 0 \u2227 y = 0\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nexact\n  \u27e80, 1, by rw [h, hxy.1, hxy.2, zero_pow zero_lt_two, add_zero, zero_mul],\n    \u27e80, by rw [zero_mul, neg_eq_zero, Fin.one_eq_zero_iff]\u27e9\u27e9\n[GOAL]\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : x = 0 \u2227 y = 0\n\u22a2 n = 0 ^ 2 * 1\n[PROOFSTEP]\nrw [h, hxy.1, hxy.2, zero_pow zero_lt_two, add_zero, zero_mul]\n[GOAL]\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : x = 0 \u2227 y = 0\n\u22a2 -1 = 0 * 0\n[PROOFSTEP]\nrw [zero_mul, neg_eq_zero, Fin.one_eq_zero_iff]\n[GOAL]\ncase neg\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : \u00ac(x = 0 \u2227 y = 0)\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nhave hg := Nat.pos_of_ne_zero (mt Nat.gcd_eq_zero_iff.mp hxy)\n[GOAL]\ncase neg\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : \u00ac(x = 0 \u2227 y = 0)\nhg : 0 < gcd x y\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nobtain \u27e8g, x\u2081, y\u2081, _, h\u2082, h\u2083, h\u2084\u27e9 := Nat.exists_coprime' hg\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : \u00ac(x = 0 \u2227 y = 0)\nhg : 0 < gcd x y\ng : \u2115\nx\u2081 y\u2081 : \u2115\nleft\u271d : 0 < g\nh\u2082 : coprime x\u2081 y\u2081\nh\u2083 : x = x\u2081 * g\nh\u2084 : y = y\u2081 * g\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nexact \u27e8g, x\u2081 ^ 2 + y\u2081 ^ 2, by rw [h, h\u2083, h\u2084]; ring, ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_coprime rfl h\u2082\u27e9\n[GOAL]\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : \u00ac(x = 0 \u2227 y = 0)\nhg : 0 < gcd x y\ng : \u2115\nx\u2081 y\u2081 : \u2115\nleft\u271d : 0 < g\nh\u2082 : coprime x\u2081 y\u2081\nh\u2083 : x = x\u2081 * g\nh\u2084 : y = y\u2081 * g\n\u22a2 n = g ^ 2 * (x\u2081 ^ 2 + y\u2081 ^ 2)\n[PROOFSTEP]\nrw [h, h\u2083, h\u2084]\n[GOAL]\nn x y : \u2115\nh : n = x ^ 2 + y ^ 2\nhxy : \u00ac(x = 0 \u2227 y = 0)\nhg : 0 < gcd x y\ng : \u2115\nx\u2081 y\u2081 : \u2115\nleft\u271d : 0 < g\nh\u2082 : coprime x\u2081 y\u2081\nh\u2083 : x = x\u2081 * g\nh\u2084 : y = y\u2081 * g\n\u22a2 (x\u2081 * g) ^ 2 + (y\u2081 * g) ^ 2 = g ^ 2 * (x\u2081 ^ 2 + y\u2081 ^ 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase mpr\nn : \u2115\n\u22a2 (\u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)) \u2192 \u2203 x y, n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nrintro \u27e8a, b, h\u2081, h\u2082\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nn a b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\n\u22a2 \u2203 x y, n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nobtain \u27e8x', y', h\u27e9 := Nat.eq_sq_add_sq_of_isSquare_mod_neg_one h\u2082\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\nn a b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nx' y' : \u2115\nh : b = x' ^ 2 + y' ^ 2\n\u22a2 \u2203 x y, n = x ^ 2 + y ^ 2\n[PROOFSTEP]\nexact \u27e8a * x', a * y', by rw [h\u2081, h]; ring\u27e9\n[GOAL]\nn a b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nx' y' : \u2115\nh : b = x' ^ 2 + y' ^ 2\n\u22a2 n = (a * x') ^ 2 + (a * y') ^ 2\n[PROOFSTEP]\nrw [h\u2081, h]\n[GOAL]\nn a b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nx' y' : \u2115\nh : b = x' ^ 2 + y' ^ 2\n\u22a2 a ^ 2 * (x' ^ 2 + y' ^ 2) = (a * x') ^ 2 + (a * y') ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 (\u2203 x y, n = x ^ 2 + y ^ 2) \u2194 \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn\u2080)\n[GOAL]\ncase inl\n\u22a2 (\u2203 x y, 0 = x ^ 2 + y ^ 2) \u2194 \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q 0)\n[PROOFSTEP]\nexact\n  \u27e8fun _ q _ _ => (@padicValNat.zero q).symm \u25b8 even_zero, fun _ => \u27e80, 0, rfl\u27e9\u27e9\n    -- now `0 < n`\n[GOAL]\ncase inr\nn : \u2115\nhn\u2080 : n > 0\n\u22a2 (\u2203 x y, n = x ^ 2 + y ^ 2) \u2194 \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\n[PROOFSTEP]\nrw [Nat.eq_sq_add_sq_iff_eq_sq_mul]\n[GOAL]\ncase inr\nn : \u2115\nhn\u2080 : n > 0\n\u22a2 (\u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)) \u2194 \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\n[PROOFSTEP]\nrefine' \u27e8fun H q hq h => _, fun H => _\u27e9\n[GOAL]\ncase inr.refine'_1\nn : \u2115\nhn\u2080 : n > 0\nH : \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nobtain \u27e8a, b, h\u2081, h\u2082\u27e9 := H\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhave hqb := padicValNat.eq_zero_of_not_dvd fun hf => (hq.mod_four_ne_three_of_dvd_isSquare_neg_one hf h\u2082) h\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhave hab : a ^ 2 * b \u2260 0 := h\u2081 \u25b8 hn\u2080.ne'\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhave ha\u2082 := left_ne_zero_of_mul hab\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\nha\u2082 : a ^ 2 \u2260 0\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhave ha := mt sq_eq_zero_iff.mpr ha\u2082\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\nha\u2082 : a ^ 2 \u2260 0\nha : \u00aca = 0\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhave hb := right_ne_zero_of_mul hab\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\nha\u2082 : a ^ 2 \u2260 0\nha : \u00aca = 0\nhb : b \u2260 0\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nhaveI hqi : Fact q.Prime := \u27e8hq\u27e9\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\nha\u2082 : a ^ 2 \u2260 0\nha : \u00aca = 0\nhb : b \u2260 0\nhqi : Fact (Prime q)\n\u22a2 Even (padicValNat q n)\n[PROOFSTEP]\nsimp_rw [h\u2081, padicValNat.mul ha\u2082 hb, padicValNat.pow 2 ha, hqb, add_zero]\n[GOAL]\ncase inr.refine'_1.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nq : \u2115\nhq : Prime q\nh : q % 4 = 3\na b : \u2115\nh\u2081 : n = a ^ 2 * b\nh\u2082 : IsSquare (-1)\nhqb : padicValNat q b = 0\nhab : a ^ 2 * b \u2260 0\nha\u2082 : a ^ 2 \u2260 0\nha : \u00aca = 0\nhb : b \u2260 0\nhqi : Fact (Prime q)\n\u22a2 Even (2 * padicValNat q a)\n[PROOFSTEP]\nexact even_two_mul _\n[GOAL]\ncase inr.refine'_2\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nobtain \u27e8b, a, hb\u2080, ha\u2080, hab, hb\u27e9 := Nat.sq_mul_squarefree_of_pos hn\u2080\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\n\u22a2 \u2203 a b, n = a ^ 2 * b \u2227 IsSquare (-1)\n[PROOFSTEP]\nrefine' \u27e8a, b, hab.symm, (ZMod.isSquare_neg_one_iff hb).mpr fun {q} hqp hqb hq4 => _\u27e9\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : \u2115\nhqp : Prime q\nhqb : q \u2223 b\nhq4 : q % 4 = 3\n\u22a2 False\n[PROOFSTEP]\nrefine' Nat.odd_iff_not_even.mp _ (H hqp hq4)\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : \u2115\nhqp : Prime q\nhqb : q \u2223 b\nhq4 : q % 4 = 3\n\u22a2 Odd (padicValNat q n)\n[PROOFSTEP]\nhave hqb' : padicValNat q b = 1 :=\n  b.factorization_def hqp \u25b8\n    le_antisymm (Nat.Squarefree.factorization_le_one _ hb) ((hqp.dvd_iff_one_le_factorization hb\u2080.ne').mp hqb)\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : \u2115\nhqp : Prime q\nhqb : q \u2223 b\nhq4 : q % 4 = 3\nhqb' : padicValNat q b = 1\n\u22a2 Odd (padicValNat q n)\n[PROOFSTEP]\nhaveI hqi : Fact q.Prime := \u27e8hqp\u27e9\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : \u2115\nhqp : Prime q\nhqb : q \u2223 b\nhq4 : q % 4 = 3\nhqb' : padicValNat q b = 1\nhqi : Fact (Prime q)\n\u22a2 Odd (padicValNat q n)\n[PROOFSTEP]\nsimp_rw [\u2190 hab, padicValNat.mul (pow_ne_zero 2 ha\u2080.ne') hb\u2080.ne', hqb', padicValNat.pow 2 ha\u2080.ne']\n[GOAL]\ncase inr.refine'_2.intro.intro.intro.intro.intro\nn : \u2115\nhn\u2080 : n > 0\nH : \u2200 {q : \u2115}, Prime q \u2192 q % 4 = 3 \u2192 Even (padicValNat q n)\nb a : \u2115\nhb\u2080 : 0 < b\nha\u2080 : 0 < a\nhab : a ^ 2 * b = n\nhb : Squarefree b\nq : \u2115\nhqp : Prime q\nhqb : q \u2223 b\nhq4 : q % 4 = 3\nhqb' : padicValNat q b = 1\nhqi : Fact (Prime q)\n\u22a2 Odd (2 * padicValNat q a + 1)\n[PROOFSTEP]\nexact odd_two_mul_add_one _\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.SumTwoSquares", "llama_tokens": 14858, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933447152497, "lm_q2_score": 0.665410572017153, "lm_q1q2_score": 0.5455656995000311}}
{"text": "[GOAL]\nF : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Mul N\na x y : M\ninst\u271d : MulHomClass F M N\nh : SemiconjBy a x y\nf : F\n\u22a2 SemiconjBy (\u2191f a) (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nsimpa only [SemiconjBy, map_mul] using congr_arg f h\n[GOAL]\nF : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Mul N\na x y : M\ninst\u271d : MulHomClass F M N\nf : F\nhf : Function.Injective \u2191f\nh : SemiconjBy (\u2191f a) (\u2191f x) (\u2191f y)\n\u22a2 \u2191f (a * x) = \u2191f (y * a)\n[PROOFSTEP]\nsimpa only [SemiconjBy, map_mul] using h\n[GOAL]\nF : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Mul N\na x y : M\ninst\u271d : MulHomClass F M N\nf : F\nhf : Function.Injective \u2191f\nh : Commute (\u2191f x) (\u2191f y)\n\u22a2 \u2191f (x * y) = \u2191f (y * x)\n[PROOFSTEP]\nsimpa only [map_mul] using h.eq\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Commute", "llama_tokens": 397, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737869342623, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.5454985173905238}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : AddSubmonoid M\nhA : \u2200 (a : A) {m : M}, m \u2208 p \u2192 a \u2022 m \u2208 p\nhB : \u2200 (b : B) {m : M}, m \u2208 p \u2192 b \u2022 m \u2208 p\nab : A \u2297[R] B\nm : M\nx\u271d :\n  m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n\u22a2 0 \u2022 m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimpa only [zero_smul] using p.zero_mem\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : AddSubmonoid M\nhA : \u2200 (a : A) {m : M}, m \u2208 p \u2192 a \u2022 m \u2208 p\nhB : \u2200 (b : B) {m : M}, m \u2208 p \u2192 b \u2022 m \u2208 p\nab : A \u2297[R] B\nm : M\na : A\nb : B\nhm :\n  m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n\u22a2 a \u2297\u209c[R] b \u2022 m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimpa only [TensorProduct.Algebra.smul_def] using hA a (hB b hm)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : AddSubmonoid M\nhA : \u2200 (a : A) {m : M}, m \u2208 p \u2192 a \u2022 m \u2208 p\nhB : \u2200 (b : B) {m : M}, m \u2208 p \u2192 b \u2022 m \u2208 p\nab : A \u2297[R] B\nm : M\nz w : A \u2297[R] B\nhz :\n  m \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n            zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier \u2192\n    z \u2022 m \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n            zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\nhw :\n  m \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n            zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier \u2192\n    w \u2022 m \u2208\n      {\n            toAddSubsemigroup :=\n              { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n            zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\nhm :\n  m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n\u22a2 (z + w) \u2022 m \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191p, add_mem' := (_ : \u2200 {a b : M}, a \u2208 p.carrier \u2192 b \u2208 p.carrier \u2192 a + b \u2208 p.carrier) },\n          zero_mem' := (_ : 0 \u2208 p.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nsimpa only [add_smul] using p.add_mem (hz hm) (hw hm)\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\na : A\nm : M\nhm : m \u2208 p\n\u22a2 a \u2022 m \u2208 p\n[PROOFSTEP]\nsuffices a \u2022 m = a \u2297\u209c[R] (1 : B) \u2022 m by exact this.symm \u25b8 p.smul_mem _ hm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\na : A\nm : M\nhm : m \u2208 p\nthis : a \u2022 m = a \u2297\u209c[R] 1 \u2022 m\n\u22a2 a \u2022 m \u2208 p\n[PROOFSTEP]\nexact this.symm \u25b8 p.smul_mem _ hm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\na : A\nm : M\nhm : m \u2208 p\n\u22a2 a \u2022 m = a \u2297\u209c[R] 1 \u2022 m\n[PROOFSTEP]\nsimp [TensorProduct.Algebra.smul_def]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\nb : B\nm : M\nhm : m \u2208 p\n\u22a2 b \u2022 m \u2208 p\n[PROOFSTEP]\nsuffices b \u2022 m = (1 : A) \u2297\u209c[R] b \u2022 m by exact this.symm \u25b8 p.smul_mem _ hm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\nb : B\nm : M\nhm : m \u2208 p\nthis : b \u2022 m = 1 \u2297\u209c[R] b \u2022 m\n\u22a2 b \u2022 m \u2208 p\n[PROOFSTEP]\nexact this.symm \u25b8 p.smul_mem _ hm\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring R\ninst\u271d\u00b9\u2070 : AddCommMonoid M\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Semiring A\ninst\u271d\u2077 : Semiring B\ninst\u271d\u2076 : Module A M\ninst\u271d\u2075 : Module B M\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R B\ninst\u271d\u00b2 : IsScalarTower R A M\ninst\u271d\u00b9 : IsScalarTower R B M\ninst\u271d : SMulCommClass A B M\np : Submodule (A \u2297[R] B) M\nb : B\nm : M\nhm : m \u2208 p\n\u22a2 b \u2022 m = 1 \u2297\u209c[R] b \u2022 m\n[PROOFSTEP]\nsimp [TensorProduct.Algebra.smul_def]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Module.Bimodule", "llama_tokens": 3413, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256393148981, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.545333148974587}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ns : Set \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nhs : Set.Finite s\n\u22a2 Set.Finite \u2191(upperClosure s)\n[PROOFSTEP]\nrw [coe_upperClosure]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ns : Set \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\nhs : Set.Finite s\n\u22a2 Set.Finite (\u22c3 (a : \u03b1) (_ : a \u2208 s), Ici a)\n[PROOFSTEP]\nexact hs.biUnion fun _ _ => finite_Ici _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ns : Set \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nhs : Set.Finite s\n\u22a2 Set.Finite \u2191(lowerClosure s)\n[PROOFSTEP]\nrw [coe_lowerClosure]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ns : Set \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nhs : Set.Finite s\n\u22a2 Set.Finite (\u22c3 (a : \u03b1) (_ : a \u2208 s), Iic a)\n[PROOFSTEP]\nexact hs.biUnion fun _ _ => finite_Iic _\n", "meta": {"mathlib_filename": "Mathlib.Order.UpperLower.LocallyFinite", "llama_tokens": 360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152325073083131, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5452929371323201}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 MulActionHom.toFun\n      { toFun := src\u271d.toFun,\n        map_smul' :=\n          (_ :\n            \u2200 (r : R) (x : A), AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n      0 =\n    0\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na : A\n\u22a2 \u2191(MulActionHom.toFun\n          { toFun := src\u271d.toFun,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : A),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n          0)\n      a =\n    \u21910 a\n[PROOFSTEP]\nexact zero_mul a\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 \u2200 (x y : A),\n    MulActionHom.toFun\n        {\n            toMulActionHom :=\n              { toFun := src\u271d.toFun,\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : A),\n                      AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n            map_zero' :=\n              (_ :\n                MulActionHom.toFun\n                    { toFun := src\u271d.toFun,\n                      map_smul' :=\n                        (_ :\n                          \u2200 (r : R) (x : A),\n                            AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : A),\n                  AddHom.toFun src\u271d.toAddHom (x + y) =\n                    AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n        (x * y) =\n      MulActionHom.toFun\n          {\n              toMulActionHom :=\n                { toFun := src\u271d.toFun,\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (x : A),\n                        AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n              map_zero' :=\n                (_ :\n                  MulActionHom.toFun\n                      { toFun := src\u271d.toFun,\n                        map_smul' :=\n                          (_ :\n                            \u2200 (r : R) (x : A),\n                              AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A),\n                    AddHom.toFun src\u271d.toAddHom (x + y) =\n                      AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n          x *\n        MulActionHom.toFun\n          {\n              toMulActionHom :=\n                { toFun := src\u271d.toFun,\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (x : A),\n                        AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n              map_zero' :=\n                (_ :\n                  MulActionHom.toFun\n                      { toFun := src\u271d.toFun,\n                        map_smul' :=\n                          (_ :\n                            \u2200 (r : R) (x : A),\n                              AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A),\n                    AddHom.toFun src\u271d.toAddHom (x + y) =\n                      AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n          y\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na b : A\n\u22a2 MulActionHom.toFun\n      {\n          toMulActionHom :=\n            { toFun := src\u271d.toFun,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : A),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n          map_zero' :=\n            (_ :\n              MulActionHom.toFun\n                  { toFun := src\u271d.toFun,\n                    map_smul' :=\n                      (_ :\n                        \u2200 (r : R) (x : A),\n                          AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                  0 =\n                0),\n          map_add' :=\n            (_ :\n              \u2200 (x y : A),\n                AddHom.toFun src\u271d.toAddHom (x + y) =\n                  AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n      (a * b) =\n    MulActionHom.toFun\n        {\n            toMulActionHom :=\n              { toFun := src\u271d.toFun,\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : A),\n                      AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n            map_zero' :=\n              (_ :\n                MulActionHom.toFun\n                    { toFun := src\u271d.toFun,\n                      map_smul' :=\n                        (_ :\n                          \u2200 (r : R) (x : A),\n                            AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : A),\n                  AddHom.toFun src\u271d.toAddHom (x + y) =\n                    AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n        a *\n      MulActionHom.toFun\n        {\n            toMulActionHom :=\n              { toFun := src\u271d.toFun,\n                map_smul' :=\n                  (_ :\n                    \u2200 (r : R) (x : A),\n                      AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n            map_zero' :=\n              (_ :\n                MulActionHom.toFun\n                    { toFun := src\u271d.toFun,\n                      map_smul' :=\n                        (_ :\n                          \u2200 (r : R) (x : A),\n                            AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : A),\n                  AddHom.toFun src\u271d.toAddHom (x + y) =\n                    AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n        b\n[PROOFSTEP]\next c\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na b c : A\n\u22a2 \u2191(MulActionHom.toFun\n          {\n              toMulActionHom :=\n                { toFun := src\u271d.toFun,\n                  map_smul' :=\n                    (_ :\n                      \u2200 (r : R) (x : A),\n                        AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n              map_zero' :=\n                (_ :\n                  MulActionHom.toFun\n                      { toFun := src\u271d.toFun,\n                        map_smul' :=\n                          (_ :\n                            \u2200 (r : R) (x : A),\n                              AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A),\n                    AddHom.toFun src\u271d.toAddHom (x + y) =\n                      AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n          (a * b))\n      c =\n    \u2191(MulActionHom.toFun\n            {\n                toMulActionHom :=\n                  { toFun := src\u271d.toFun,\n                    map_smul' :=\n                      (_ :\n                        \u2200 (r : R) (x : A),\n                          AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n                map_zero' :=\n                  (_ :\n                    MulActionHom.toFun\n                        { toFun := src\u271d.toFun,\n                          map_smul' :=\n                            (_ :\n                              \u2200 (r : R) (x : A),\n                                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                        0 =\n                      0),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n            a *\n          MulActionHom.toFun\n            {\n                toMulActionHom :=\n                  { toFun := src\u271d.toFun,\n                    map_smul' :=\n                      (_ :\n                        \u2200 (r : R) (x : A),\n                          AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) },\n                map_zero' :=\n                  (_ :\n                    MulActionHom.toFun\n                        { toFun := src\u271d.toFun,\n                          map_smul' :=\n                            (_ :\n                              \u2200 (r : R) (x : A),\n                                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }\n                        0 =\n                      0),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) }.toMulActionHom\n            b)\n      c\n[PROOFSTEP]\nexact mul_assoc a b c\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b : A\n\u22a2 Commute (mulLeft R a) (mulRight R b)\n[PROOFSTEP]\next c\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b c : A\n\u22a2 \u2191(mulLeft R a * mulRight R b) c = \u2191(mulRight R b * mulLeft R a) c\n[PROOFSTEP]\nexact (mul_assoc a c b).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b : A\n\u22a2 mulLeft R (a * b) = comp (mulLeft R a) (mulLeft R b)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b x\u271d : A\n\u22a2 \u2191(mulLeft R (a * b)) x\u271d = \u2191(comp (mulLeft R a) (mulLeft R b)) x\u271d\n[PROOFSTEP]\nsimp only [mulLeft_apply, comp_apply, mul_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b : A\n\u22a2 mulRight R (a * b) = comp (mulRight R b) (mulRight R a)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b x\u271d : A\n\u22a2 \u2191(mulRight R (a * b)) x\u271d = \u2191(comp (mulRight R b) (mulRight R a)) x\u271d\n[PROOFSTEP]\nsimp only [mulRight_apply, comp_apply, mul_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 AddHom.toFun src\u271d.toAddHom 1 = 1\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na : A\n\u22a2 \u2191(AddHom.toFun src\u271d.toAddHom 1) a = \u21911 a\n[PROOFSTEP]\nexact one_mul a\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 \u2200 (x y : A),\n    OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (x * y) =\n      OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } x *\n        OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } y\n[PROOFSTEP]\nintro a b\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na b : A\n\u22a2 OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (a * b) =\n    OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n      OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b\n[PROOFSTEP]\next c\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na b c : A\n\u22a2 \u2191(OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (a * b)) c =\n    \u2191(OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n          OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b)\n      c\n[PROOFSTEP]\nexact mul_assoc a b c\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (a b : A),\n                OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (a * b) =\n                  OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                    OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b) })\n      0 =\n    0\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\na : A\n\u22a2 \u2191(OneHom.toFun\n          (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n              map_mul' :=\n                (_ :\n                  \u2200 (a b : A),\n                    OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (a * b) =\n                      OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                        OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b) })\n          0)\n      a =\n    \u21910 a\n[PROOFSTEP]\nexact zero_mul a\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\n\u22a2 \u2200 (r : R),\n    OneHom.toFun\n        (\u2191\u2191{\n              toMonoidHom :=\n                { toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                  map_mul' :=\n                    (_ :\n                      \u2200 (a b : A),\n                        OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                            (a * b) =\n                          OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                            OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                              b) },\n              map_zero' :=\n                (_ :\n                  OneHom.toFun\n                      (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                          map_mul' :=\n                            (_ :\n                              \u2200 (a b : A),\n                                OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                    (a * b) =\n                                  OneHom.toFun\n                                      { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                                    OneHom.toFun\n                                      { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b) })\n                      0 =\n                    0),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : A),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) })\n        (\u2191(algebraMap R A) r) =\n      \u2191(algebraMap R (End R A)) r\n[PROOFSTEP]\nintro r\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\nr : R\n\u22a2 OneHom.toFun\n      (\u2191\u2191{\n            toMonoidHom :=\n              { toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (a b : A),\n                      OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } (a * b) =\n                        OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                          OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b) },\n            map_zero' :=\n              (_ :\n                OneHom.toFun\n                    (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                        map_mul' :=\n                          (_ :\n                            \u2200 (a b : A),\n                              OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                  (a * b) =\n                                OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                    a *\n                                  OneHom.toFun\n                                    { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } b) })\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : A),\n                  AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) })\n      (\u2191(algebraMap R A) r) =\n    \u2191(algebraMap R (End R A)) r\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nsrc\u271d : A \u2192\u2097[R] A \u2192\u2097[R] A := mul R A\nr : R\na : A\n\u22a2 \u2191(OneHom.toFun\n          (\u2191\u2191{\n                toMonoidHom :=\n                  { toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                    map_mul' :=\n                      (_ :\n                        \u2200 (a b : A),\n                          OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                              (a * b) =\n                            OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                              OneHom.toFun { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                b) },\n                map_zero' :=\n                  (_ :\n                    OneHom.toFun\n                        (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) },\n                            map_mul' :=\n                              (_ :\n                                \u2200 (a b : A),\n                                  OneHom.toFun\n                                      { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                      (a * b) =\n                                    OneHom.toFun\n                                        { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) } a *\n                                      OneHom.toFun\n                                        { toFun := src\u271d.toFun, map_one' := (_ : AddHom.toFun src\u271d.toAddHom 1 = 1) }\n                                        b) })\n                        0 =\n                      0),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : A),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) })\n          (\u2191(algebraMap R A) r))\n      a =\n    \u2191(\u2191(algebraMap R (End R A)) r) a\n[PROOFSTEP]\nexact (Algebra.smul_def r a).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 mulLeft R a = 0 \u2194 a = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 mulLeft R a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 a = 0 \u2192 mulLeft R a = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : mulLeft R a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 mul_one a, \u2190 @mulLeft_apply R _ _ _ _ _ _ a 1, h, LinearMap.zero_apply]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : a = 0\n\u22a2 mulLeft R a = 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : a = 0\n\u22a2 mulLeft R 0 = 0\n[PROOFSTEP]\nexact mulLeft_zero_eq_zero\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 mulRight R a = 0 \u2194 a = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 mulRight R a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 a = 0 \u2192 mulRight R a = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : mulRight R a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 one_mul a, \u2190 @mulRight_apply R _ _ _ _ _ _ a 1, h, LinearMap.zero_apply]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : a = 0\n\u22a2 mulRight R a = 0\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nh : a = 0\n\u22a2 mulRight R 0 = 0\n[PROOFSTEP]\nexact mulRight_zero_eq_zero\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 mulLeft R 1 = id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx\u271d : A\n\u22a2 \u2191(mulLeft R 1) x\u271d = \u2191id x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.id_coe, one_mul, id.def, mulLeft_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 mulRight R 1 = id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\nx\u271d : A\n\u22a2 \u2191(mulRight R 1) x\u271d = \u2191id x\u271d\n[PROOFSTEP]\nsimp only [LinearMap.id_coe, mul_one, id.def, mulRight_apply]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\n\u22a2 mulLeft R a ^ n = mulLeft R (a ^ n)\n[PROOFSTEP]\nsimpa only [mulLeft, \u2190 Algebra.coe_lmul_eq_mul] using ((Algebra.lmul R A).map_pow a n).symm\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\n\u22a2 mulRight R a ^ n = mulRight R (a ^ n)\n[PROOFSTEP]\nsimp only [mulRight, \u2190 Algebra.coe_lmul_eq_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\n\u22a2 \u2191(flip (mul R A)) a ^ n = \u2191(flip (mul R A)) (a ^ n)\n[PROOFSTEP]\nexact LinearMap.coe_injective (((mulRight R a).coe_pow n).symm \u25b8 mul_right_iterate a n)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\n\u22a2 Function.Injective \u2191(mulLeft R x)\n[PROOFSTEP]\nletI : Nontrivial A := \u27e8\u27e8x, 0, hx\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\n\u22a2 Function.Injective \u2191(mulLeft R x)\n[PROOFSTEP]\nletI := NoZeroDivisors.to_isDomain A\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis\u271d : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\nthis : IsDomain A := NoZeroDivisors.to_isDomain A\n\u22a2 Function.Injective \u2191(mulLeft R x)\n[PROOFSTEP]\nexact mul_right_injective\u2080 hx\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\n\u22a2 Function.Injective \u2191(mulRight R x)\n[PROOFSTEP]\nletI : Nontrivial A := \u27e8\u27e8x, 0, hx\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\n\u22a2 Function.Injective \u2191(mulRight R x)\n[PROOFSTEP]\nletI := NoZeroDivisors.to_isDomain A\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis\u271d : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\nthis : IsDomain A := NoZeroDivisors.to_isDomain A\n\u22a2 Function.Injective \u2191(mulRight R x)\n[PROOFSTEP]\nexact mul_left_injective\u2080 hx\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\n\u22a2 Function.Injective \u2191(\u2191(mul R A) x)\n[PROOFSTEP]\nletI : Nontrivial A := \u27e8\u27e8x, 0, hx\u27e9\u27e9\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\n\u22a2 Function.Injective \u2191(\u2191(mul R A) x)\n[PROOFSTEP]\nletI := NoZeroDivisors.to_isDomain A\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : NoZeroDivisors A\nx : A\nhx : x \u2260 0\nthis\u271d : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\nthis : IsDomain A := NoZeroDivisors.to_isDomain A\n\u22a2 Function.Injective \u2191(\u2191(mul R A) x)\n[PROOFSTEP]\nexact mul_right_injective\u2080 hx\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.Bilinear", "llama_tokens": 11347, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511469672594, "lm_q2_score": 0.6370307806984443, "lm_q1q2_score": 0.5452035243142123}}
{"text": "[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\n\u22a2 adE R A (i, i) = 0\n[PROOFSTEP]\nhave h' : (-2 : \u2124).toNat = 0 := by rfl\n[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\n\u22a2 Int.toNat (-2) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\nh' : Int.toNat (-2) = 0\n\u22a2 adE R A (i, i) = 0\n[PROOFSTEP]\nsimp [adE, h, h']\n[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\n\u22a2 adF R A (i, i) = 0\n[PROOFSTEP]\nhave h' : (-2 : \u2124).toNat = 0 := by rfl\n[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\n\u22a2 Int.toNat (-2) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nB : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq B\ninst\u271d : Fintype B\nA : Matrix B B \u2124\ni : B\nh : A i i = 2\nh' : Int.toNat (-2) = 0\n\u22a2 adF R A (i, i) = 0\n[PROOFSTEP]\nsimp [adF, h, h']\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.CartanMatrix", "llama_tokens": 636, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.682573734412324, "lm_q1q2_score": 0.5448213296402729}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nR : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedCommRing R\ninst\u271d\u00b3 : NormedAlgebra \ud835\udd5c R\ninst\u271d\u00b2 : TopologicalRing R\ninst\u271d\u00b9 : CompleteSpace R\ninst\u271d : T2Space R\nr : R\n\u22a2 exp \ud835\udd5c (r \u2022 \u03b5) = \u21911 + r \u2022 \u03b5\n[PROOFSTEP]\nrw [eps, \u2190 inr_smul, exp_inr, Nat.cast_one]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.DualNumber", "llama_tokens": 146, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8354835289107309, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5441962665588023}}
{"text": "[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 log x = \u2191(OrderIso.symm expOrderIso) { val := x, property := hx }\n[PROOFSTEP]\nrw [log_of_ne_zero hx.ne']\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 \u2191(OrderIso.symm expOrderIso) { val := |x|, property := (_ : 0 < |x|) } =\n    \u2191(OrderIso.symm expOrderIso) { val := x, property := hx }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_6.h.e_val\nx y : \u211d\nhx : 0 < x\n\u22a2 |x| = x\n[PROOFSTEP]\nexact abs_of_pos hx\n[GOAL]\nx y : \u211d\nhx : x \u2260 0\n\u22a2 exp (log x) = |x|\n[PROOFSTEP]\nrw [log_of_ne_zero hx, \u2190 coe_expOrderIso_apply, OrderIso.apply_symm_apply, Subtype.coe_mk]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 exp (log x) = x\n[PROOFSTEP]\nrw [exp_log_eq_abs hx.ne']\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 |x| = x\n[PROOFSTEP]\nexact abs_of_pos hx\n[GOAL]\nx y : \u211d\nhx : x < 0\n\u22a2 exp (log x) = -x\n[PROOFSTEP]\nrw [exp_log_eq_abs (ne_of_lt hx)]\n[GOAL]\nx y : \u211d\nhx : x < 0\n\u22a2 |x| = -x\n[PROOFSTEP]\nexact abs_of_neg hx\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 x \u2264 exp (log x)\n[PROOFSTEP]\nby_cases h_zero : x = 0\n[GOAL]\ncase pos\nx\u271d y x : \u211d\nh_zero : x = 0\n\u22a2 x \u2264 exp (log x)\n[PROOFSTEP]\nrw [h_zero, log, dif_pos rfl, exp_zero]\n[GOAL]\ncase pos\nx\u271d y x : \u211d\nh_zero : x = 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\ncase neg\nx\u271d y x : \u211d\nh_zero : \u00acx = 0\n\u22a2 x \u2264 exp (log x)\n[PROOFSTEP]\nrw [exp_log_eq_abs h_zero]\n[GOAL]\ncase neg\nx\u271d y x : \u211d\nh_zero : \u00acx = 0\n\u22a2 x \u2264 |x|\n[PROOFSTEP]\nexact le_abs_self _\n[GOAL]\nx y : \u211d\n\u22a2 exp (log 1) = exp 0\n[PROOFSTEP]\nrw [exp_log zero_lt_one, exp_zero]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 log |x| = log x\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\nx\u271d y x : \u211d\nh : x = 0\n\u22a2 log |x| = log x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nx\u271d y x : \u211d\nh : \u00acx = 0\n\u22a2 log |x| = log x\n[PROOFSTEP]\nrw [\u2190 exp_eq_exp, exp_log_eq_abs h, exp_log_eq_abs (abs_pos.2 h).ne', abs_abs]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 log (-x) = log x\n[PROOFSTEP]\nrw [\u2190 log_abs x, \u2190 log_abs (-x), abs_neg]\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 < x\n\u22a2 sinh (log x) = (x - x\u207b\u00b9) / 2\n[PROOFSTEP]\nrw [sinh_eq, exp_neg, exp_log hx]\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 < x\n\u22a2 cosh (log x) = (x + x\u207b\u00b9) / 2\n[PROOFSTEP]\nrw [cosh_eq, exp_neg, exp_log hx]\n[GOAL]\nx\u271d\u00b9 y x : \u211d\nx\u271d : x \u2208 univ\n\u22a2 log (-exp x) = x\n[PROOFSTEP]\nrw [log_neg_eq_log, log_exp]\n[GOAL]\nx y : \u211d\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 exp (log (x * y)) = exp (log x + log y)\n[PROOFSTEP]\nrw [exp_log_eq_abs (mul_ne_zero hx hy), exp_add, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_mul]\n[GOAL]\nx y : \u211d\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 exp (log (x / y)) = exp (log x - log y)\n[PROOFSTEP]\nrw [exp_log_eq_abs (div_ne_zero hx hy), exp_sub, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_div]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 log x\u207b\u00b9 = -log x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nx\u271d y x : \u211d\nhx : x = 0\n\u22a2 log x\u207b\u00b9 = -log x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nx\u271d y x : \u211d\nhx : \u00acx = 0\n\u22a2 log x\u207b\u00b9 = -log x\n[PROOFSTEP]\nrw [\u2190 exp_eq_exp, exp_log_eq_abs (inv_ne_zero hx), exp_neg, exp_log_eq_abs hx, abs_inv]\n[GOAL]\nx y : \u211d\nh : 0 < x\nh\u2081 : 0 < y\n\u22a2 log x \u2264 log y \u2194 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 exp_le_exp, exp_log h, exp_log h\u2081]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 x < y \u2192 log x < log y\n[PROOFSTEP]\nintro h\n[GOAL]\nx y : \u211d\nhx : 0 < x\nh : x < y\n\u22a2 log x < log y\n[PROOFSTEP]\nrwa [\u2190 exp_lt_exp, exp_log hx, exp_log (lt_trans hx h)]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 x \u2264 y \u2192 log x \u2264 log y\n[PROOFSTEP]\nintro hxy\n[GOAL]\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\ncases hxy.eq_or_lt with\n| inl h_eq => simp [h_eq]\n| inr hlt => exact le_of_lt <| log_lt_log hx hlt\n[GOAL]\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\nx\u271d : x = y \u2228 x < y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\ncases hxy.eq_or_lt with\n| inl h_eq => simp [h_eq]\n| inr hlt => exact le_of_lt <| log_lt_log hx hlt\n[GOAL]\ncase inl\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\nh_eq : x = y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\n\n| inl h_eq => simp [h_eq]\n[GOAL]\ncase inl\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\nh_eq : x = y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\nsimp [h_eq]\n[GOAL]\ncase inr\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\nhlt : x < y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\n\n| inr hlt => exact le_of_lt <| log_lt_log hx hlt\n[GOAL]\ncase inr\nx y : \u211d\nhx : 0 < x\nhxy : x \u2264 y\nhlt : x < y\n\u22a2 log x \u2264 log y\n[PROOFSTEP]\nexact le_of_lt <| log_lt_log hx hlt\n[GOAL]\nx y : \u211d\nhx : 0 < x\nhy : 0 < y\n\u22a2 log x < log y \u2194 x < y\n[PROOFSTEP]\nrw [\u2190 exp_lt_exp, exp_log hx, exp_log hy]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 log x \u2264 y \u2194 x \u2264 exp y\n[PROOFSTEP]\nrw [\u2190 exp_le_exp, exp_log hx]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 log x < y \u2194 x < exp y\n[PROOFSTEP]\nrw [\u2190 exp_lt_exp, exp_log hx]\n[GOAL]\nx y : \u211d\nhy : 0 < y\n\u22a2 x \u2264 log y \u2194 exp x \u2264 y\n[PROOFSTEP]\nrw [\u2190 exp_le_exp, exp_log hy]\n[GOAL]\nx y : \u211d\nhy : 0 < y\n\u22a2 x < log y \u2194 exp x < y\n[PROOFSTEP]\nrw [\u2190 exp_lt_exp, exp_log hy]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 0 < log x \u2194 1 < x\n[PROOFSTEP]\nrw [\u2190 log_one]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 log 1 < log x \u2194 1 < x\n[PROOFSTEP]\nexact log_lt_log_iff zero_lt_one hx\n[GOAL]\nx y : \u211d\nhx : x < -1\n\u22a2 0 < log x\n[PROOFSTEP]\nrw [\u2190 neg_neg x, log_neg_eq_log]\n[GOAL]\nx y : \u211d\nhx : x < -1\n\u22a2 0 < log (-x)\n[PROOFSTEP]\nhave : 1 < -x := by linarith\n[GOAL]\nx y : \u211d\nhx : x < -1\n\u22a2 1 < -x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nhx : x < -1\nthis : 1 < -x\n\u22a2 0 < log (-x)\n[PROOFSTEP]\nexact log_pos this\n[GOAL]\nx y : \u211d\nh : 0 < x\n\u22a2 log x < 0 \u2194 x < 1\n[PROOFSTEP]\nrw [\u2190 log_one]\n[GOAL]\nx y : \u211d\nh : 0 < x\n\u22a2 log x < log 1 \u2194 x < 1\n[PROOFSTEP]\nexact log_lt_log_iff h zero_lt_one\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\n\u22a2 log x < 0\n[PROOFSTEP]\nrw [\u2190 neg_neg x, log_neg_eq_log]\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\n\u22a2 log (-x) < 0\n[PROOFSTEP]\nhave h0' : 0 < -x := by linarith\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\n\u22a2 0 < -x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\nh0' : 0 < -x\n\u22a2 log (-x) < 0\n[PROOFSTEP]\nhave h1' : -x < 1 := by linarith\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\nh0' : 0 < -x\n\u22a2 -x < 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y : \u211d\nh0 : x < 0\nh1 : -1 < x\nh0' : 0 < -x\nh1' : -x < 1\n\u22a2 log (-x) < 0\n[PROOFSTEP]\nexact log_neg h0' h1'\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 0 \u2264 log x \u2194 1 \u2264 x\n[PROOFSTEP]\nrw [\u2190 not_lt, log_neg_iff hx, not_lt]\n[GOAL]\nx y : \u211d\nhx : 0 < x\n\u22a2 log x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nrw [\u2190 not_lt, log_pos_iff hx, not_lt]\n[GOAL]\nx y : \u211d\nhx : 0 \u2264 x\n\u22a2 log x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nrcases hx.eq_or_lt with (rfl | hx)\n[GOAL]\ncase inl\ny : \u211d\nhx : 0 \u2264 0\n\u22a2 log 0 \u2264 0 \u2194 0 \u2264 1\n[PROOFSTEP]\nsimp [le_refl, zero_le_one]\n[GOAL]\ncase inr\nx y : \u211d\nhx\u271d : 0 \u2264 x\nhx : 0 < x\n\u22a2 log x \u2264 0 \u2194 x \u2264 1\n[PROOFSTEP]\nexact log_nonpos_iff hx\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nx y : \u211d\nn : \u2115\nhn : n = 0\n\u22a2 0 \u2264 log \u2191n\ncase neg x y : \u211d n : \u2115 hn : \u00acn = 0 \u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncase pos => simp [hn]\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : n = 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncase pos => simp [hn]\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : n = 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase neg\nx y : \u211d\nn : \u2115\nhn : \u00acn = 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncase neg =>\n  have : (1 : \u211d) \u2264 n := by exact_mod_cast Nat.one_le_of_lt <| Nat.pos_of_ne_zero hn\n  exact log_nonneg this\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : \u00acn = 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncase neg =>\n  have : (1 : \u211d) \u2264 n := by exact_mod_cast Nat.one_le_of_lt <| Nat.pos_of_ne_zero hn\n  exact log_nonneg this\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : \u00acn = 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nhave : (1 : \u211d) \u2264 n := by exact_mod_cast Nat.one_le_of_lt <| Nat.pos_of_ne_zero hn\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : \u00acn = 0\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast Nat.one_le_of_lt <| Nat.pos_of_ne_zero hn\n[GOAL]\nx y : \u211d\nn : \u2115\nhn : \u00acn = 0\nthis : 1 \u2264 \u2191n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nexact log_nonneg this\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 0 \u2264 log (-\u2191n)\n[PROOFSTEP]\nrw [\u2190 log_neg_eq_log, neg_neg]\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nexact log_nat_cast_nonneg _\n[GOAL]\nx y : \u211d\nn : \u2124\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncases lt_trichotomy 0 n with\n| inl hn =>\n  have : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n  exact log_nonneg this\n| inr hn =>\n  cases hn with\n  | inl hn => simp [hn.symm]\n  | inr hn =>\n    have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n    rw [\u2190 log_neg_eq_log]\n    exact log_nonneg this\n[GOAL]\nx y : \u211d\nn : \u2124\nx\u271d : 0 < n \u2228 0 = n \u2228 n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncases lt_trichotomy 0 n with\n| inl hn =>\n  have : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n  exact log_nonneg this\n| inr hn =>\n  cases hn with\n  | inl hn => simp [hn.symm]\n  | inr hn =>\n    have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n    rw [\u2190 log_neg_eq_log]\n    exact log_nonneg this\n[GOAL]\ncase inl\nx y : \u211d\nn : \u2124\nhn : 0 < n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\n\n| inl hn =>\n  have : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n  exact log_nonneg this\n[GOAL]\ncase inl\nx y : \u211d\nn : \u2124\nhn : 0 < n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nhave : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n[GOAL]\nx y : \u211d\nn : \u2124\nhn : 0 < n\n\u22a2 1 \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase inl\nx y : \u211d\nn : \u2124\nhn : 0 < n\nthis : 1 \u2264 \u2191n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nexact log_nonneg this\n[GOAL]\ncase inr\nx y : \u211d\nn : \u2124\nhn : 0 = n \u2228 n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\n\n| inr hn =>\n  cases hn with\n  | inl hn => simp [hn.symm]\n  | inr hn =>\n    have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n    rw [\u2190 log_neg_eq_log]\n    exact log_nonneg this\n[GOAL]\ncase inr\nx y : \u211d\nn : \u2124\nhn : 0 = n \u2228 n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncases hn with\n| inl hn => simp [hn.symm]\n| inr hn =>\n  have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n  rw [\u2190 log_neg_eq_log]\n  exact log_nonneg this\n[GOAL]\ncase inr\nx y : \u211d\nn : \u2124\nhn : 0 = n \u2228 n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\ncases hn with\n| inl hn => simp [hn.symm]\n| inr hn =>\n  have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n  rw [\u2190 log_neg_eq_log]\n  exact log_nonneg this\n[GOAL]\ncase inr.inl\nx y : \u211d\nn : \u2124\nhn : 0 = n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\n\n| inl hn => simp [hn.symm]\n[GOAL]\ncase inr.inl\nx y : \u211d\nn : \u2124\nhn : 0 = n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nsimp [hn.symm]\n[GOAL]\ncase inr.inr\nx y : \u211d\nn : \u2124\nhn : n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\n\n| inr hn =>\n  have : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n  rw [\u2190 log_neg_eq_log]\n  exact log_nonneg this\n[GOAL]\ncase inr.inr\nx y : \u211d\nn : \u2124\nhn : n < 0\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nhave : (1 : \u211d) \u2264 -n := by rw [\u2190 neg_zero, \u2190 lt_neg] at hn ; exact_mod_cast hn\n[GOAL]\nx y : \u211d\nn : \u2124\nhn : n < 0\n\u22a2 1 \u2264 -\u2191n\n[PROOFSTEP]\nrw [\u2190 neg_zero, \u2190 lt_neg] at hn \n[GOAL]\nx y : \u211d\nn : \u2124\nhn : 0 < -n\n\u22a2 1 \u2264 -\u2191n\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase inr.inr\nx y : \u211d\nn : \u2124\nhn : n < 0\nthis : 1 \u2264 -\u2191n\n\u22a2 0 \u2264 log \u2191n\n[PROOFSTEP]\nrw [\u2190 log_neg_eq_log]\n[GOAL]\ncase inr.inr\nx y : \u211d\nn : \u2124\nhn : n < 0\nthis : 1 \u2264 -\u2191n\n\u22a2 0 \u2264 log (-\u2191n)\n[PROOFSTEP]\nexact log_nonneg this\n[GOAL]\nx y : \u211d\n\u22a2 StrictAntiOn log (Iio 0)\n[PROOFSTEP]\nrintro x (hx : x < 0) y (hy : y < 0) hxy\n[GOAL]\nx\u271d y\u271d x : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 log y < log x\n[PROOFSTEP]\nrw [\u2190 log_abs y, \u2190 log_abs x]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 log |y| < log |x|\n[PROOFSTEP]\nrefine' log_lt_log (abs_pos.2 hy.ne) _\n[GOAL]\nx\u271d y\u271d x : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x < y\n\u22a2 |y| < |x|\n[PROOFSTEP]\nrwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]\n[GOAL]\nx y : \u211d\nhx1 : 0 < x\nhx2 : x \u2260 1\n\u22a2 log x < x - 1\n[PROOFSTEP]\nhave h : log x \u2260 0\n[GOAL]\ncase h\nx y : \u211d\nhx1 : 0 < x\nhx2 : x \u2260 1\n\u22a2 log x \u2260 0\n[PROOFSTEP]\nrwa [\u2190 log_one, log_injOn_pos.ne_iff hx1]\n[GOAL]\ncase h\nx y : \u211d\nhx1 : 0 < x\nhx2 : x \u2260 1\n\u22a2 1 \u2208 Ioi 0\n[PROOFSTEP]\nexact mem_Ioi.mpr zero_lt_one\n[GOAL]\nx y : \u211d\nhx1 : 0 < x\nhx2 : x \u2260 1\nh : log x \u2260 0\n\u22a2 log x < x - 1\n[PROOFSTEP]\nlinarith [add_one_lt_exp_of_nonzero h, exp_log hx1]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 log x = 0 \u2194 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nx\u271d y x : \u211d\n\u22a2 log x = 0 \u2192 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nx\u271d y x : \u211d\nh : log x = 0\n\u22a2 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nrcases lt_trichotomy x 0 with (x_lt_zero | rfl | x_gt_zero)\n[GOAL]\ncase mp.inl\nx\u271d y x : \u211d\nh : log x = 0\nx_lt_zero : x < 0\n\u22a2 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nrefine' Or.inr (Or.inr (neg_eq_iff_eq_neg.mp _))\n[GOAL]\ncase mp.inl\nx\u271d y x : \u211d\nh : log x = 0\nx_lt_zero : x < 0\n\u22a2 -x = 1\n[PROOFSTEP]\nrw [\u2190 log_neg_eq_log x] at h \n[GOAL]\ncase mp.inl\nx\u271d y x : \u211d\nh : log (-x) = 0\nx_lt_zero : x < 0\n\u22a2 -x = 1\n[PROOFSTEP]\nexact eq_one_of_pos_of_log_eq_zero (neg_pos.mpr x_lt_zero) h\n[GOAL]\ncase mp.inr.inl\nx y : \u211d\nh : log 0 = 0\n\u22a2 0 = 0 \u2228 0 = 1 \u2228 0 = -1\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\ncase mp.inr.inr\nx\u271d y x : \u211d\nh : log x = 0\nx_gt_zero : 0 < x\n\u22a2 x = 0 \u2228 x = 1 \u2228 x = -1\n[PROOFSTEP]\nexact Or.inr (Or.inl (eq_one_of_pos_of_log_eq_zero x_gt_zero h))\n[GOAL]\ncase mpr\nx\u271d y x : \u211d\n\u22a2 x = 0 \u2228 x = 1 \u2228 x = -1 \u2192 log x = 0\n[PROOFSTEP]\nrintro (rfl | rfl | rfl)\n[GOAL]\ncase mpr.inl\nx y : \u211d\n\u22a2 log 0 = 0\n[PROOFSTEP]\nsimp only [log_one, log_zero, log_neg_eq_log]\n[GOAL]\ncase mpr.inr.inl\nx y : \u211d\n\u22a2 log 1 = 0\n[PROOFSTEP]\nsimp only [log_one, log_zero, log_neg_eq_log]\n[GOAL]\ncase mpr.inr.inr\nx y : \u211d\n\u22a2 log (-1) = 0\n[PROOFSTEP]\nsimp only [log_one, log_zero, log_neg_eq_log]\n[GOAL]\nx\u271d y x : \u211d\n\u22a2 log x \u2260 0 \u2194 x \u2260 0 \u2227 x \u2260 1 \u2227 x \u2260 -1\n[PROOFSTEP]\nsimpa only [not_or] using log_eq_zero.not\n[GOAL]\nx\u271d y x : \u211d\nn : \u2115\n\u22a2 log (x ^ n) = \u2191n * log x\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nx\u271d y x : \u211d\n\u22a2 log (x ^ Nat.zero) = \u2191Nat.zero * log x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nx\u271d y x : \u211d\nn : \u2115\nih : log (x ^ n) = \u2191n * log x\n\u22a2 log (x ^ Nat.succ n) = \u2191(Nat.succ n) * log x\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase succ.inl\nx y : \u211d\nn : \u2115\nih : log (0 ^ n) = \u2191n * log 0\n\u22a2 log (0 ^ Nat.succ n) = \u2191(Nat.succ n) * log 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.inr\nx\u271d y x : \u211d\nn : \u2115\nih : log (x ^ n) = \u2191n * log x\nhx : x \u2260 0\n\u22a2 log (x ^ Nat.succ n) = \u2191(Nat.succ n) * log x\n[PROOFSTEP]\nrw [pow_succ', log_mul (pow_ne_zero _ hx) hx, ih, Nat.cast_succ, add_mul, one_mul]\n[GOAL]\nx\u271d y x : \u211d\nn : \u2124\n\u22a2 log (x ^ n) = \u2191n * log x\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase ofNat\nx\u271d y x : \u211d\na\u271d : \u2115\n\u22a2 log (x ^ Int.ofNat a\u271d) = \u2191(Int.ofNat a\u271d) * log x\n[PROOFSTEP]\nrw [Int.ofNat_eq_coe, zpow_ofNat, log_pow, Int.cast_ofNat]\n[GOAL]\ncase negSucc\nx\u271d y x : \u211d\na\u271d : \u2115\n\u22a2 log (x ^ Int.negSucc a\u271d) = \u2191(Int.negSucc a\u271d) * log x\n[PROOFSTEP]\nrw [zpow_negSucc, log_inv, log_pow, Int.cast_negSucc, Nat.cast_add_one, neg_mul_eq_neg_mul]\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 \u2264 x\n\u22a2 log (sqrt x) = log x / 2\n[PROOFSTEP]\nrw [eq_div_iff, mul_comm, \u2190 Nat.cast_two, \u2190 log_pow, sq_sqrt hx]\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 \u2264 x\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 < x\n\u22a2 log x \u2264 x - 1\n[PROOFSTEP]\nrw [le_sub_iff_add_le]\n[GOAL]\nx\u271d y x : \u211d\nhx : 0 < x\n\u22a2 log x + 1 \u2264 x\n[PROOFSTEP]\nconvert add_one_le_exp (log x)\n[GOAL]\ncase h.e'_4\nx\u271d y x : \u211d\nhx : 0 < x\n\u22a2 x = exp (log x)\n[PROOFSTEP]\nrw [exp_log hx]\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nhave : 0 < 1 / x := by simpa only [one_div, inv_pos] using h1\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\n\u22a2 0 < 1 / x\n[PROOFSTEP]\nsimpa only [one_div, inv_pos] using h1\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : 0 < 1 / x\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nreplace := log_le_sub_one_of_pos this\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : log (1 / x) \u2264 1 / x - 1\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nreplace : log (1 / x) < 1 / x := by linarith\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : log (1 / x) \u2264 1 / x - 1\n\u22a2 log (1 / x) < 1 / x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : log (1 / x) < 1 / x\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nrw [log_div one_ne_zero h1.ne', log_one, zero_sub, lt_div_iff h1] at this \n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nhave aux : 0 \u2264 -log x * x := by\n  refine' mul_nonneg _ h1.le\n  rw [\u2190 log_inv]\n  apply log_nonneg\n  rw [\u2190 le_inv h1 zero_lt_one, inv_one]\n  exact h2\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 0 \u2264 -log x * x\n[PROOFSTEP]\nrefine' mul_nonneg _ h1.le\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 0 \u2264 -log x\n[PROOFSTEP]\nrw [\u2190 log_inv]\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 0 \u2264 log x\u207b\u00b9\n[PROOFSTEP]\napply log_nonneg\n[GOAL]\ncase hx\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 1 \u2264 x\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 le_inv h1 zero_lt_one, inv_one]\n[GOAL]\ncase hx\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\n\u22a2 x \u2264 1\n[PROOFSTEP]\nexact h2\n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : -log x * x < 1\naux : 0 \u2264 -log x * x\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nrw [\u2190 abs_of_nonneg aux, neg_mul, abs_neg] at this \n[GOAL]\nx\u271d y x : \u211d\nh1 : 0 < x\nh2 : x \u2264 1\nthis : |log x * x| < 1\naux : 0 \u2264 -log x * x\n\u22a2 |log x * x| < 1\n[PROOFSTEP]\nexact this\n[GOAL]\nx y : \u211d\n\u22a2 Tendsto (fun x => log (exp x)) atTop atTop\n[PROOFSTEP]\nsimpa only [log_exp] using tendsto_id\n[GOAL]\nx y : \u211d\n\u22a2 Tendsto log (\ud835\udcdd[{0}\u1d9c] 0) atBot\n[PROOFSTEP]\nrw [\u2190 show _ = log from funext log_abs]\n[GOAL]\nx y : \u211d\n\u22a2 Tendsto (fun x => log |x|) (\ud835\udcdd[{0}\u1d9c] 0) atBot\n[PROOFSTEP]\nrefine' Tendsto.comp (g := log) _ tendsto_abs_nhdsWithin_zero\n[GOAL]\nx y : \u211d\n\u22a2 Tendsto log (\ud835\udcdd[Ioi 0] 0) atBot\n[PROOFSTEP]\nsimpa [\u2190 tendsto_comp_exp_atBot] using tendsto_id\n[GOAL]\nx y : \u211d\n\u22a2 ContinuousOn log {0}\u1d9c\n[PROOFSTEP]\nsimp only [continuousOn_iff_continuous_restrict, restrict]\n[GOAL]\nx y : \u211d\n\u22a2 Continuous fun x => log \u2191x\n[PROOFSTEP]\nconv in log _ => rw [log_of_ne_zero (show (x : \u211d) \u2260 0 from x.2)]\n[GOAL]\nx\u271d y : \u211d\nx : \u2191{0}\u1d9c\n| log \u2191x\n[PROOFSTEP]\nrw [log_of_ne_zero (show (x : \u211d) \u2260 0 from x.2)]\n[GOAL]\nx\u271d y : \u211d\nx : \u2191{0}\u1d9c\n| log \u2191x\n[PROOFSTEP]\nrw [log_of_ne_zero (show (x : \u211d) \u2260 0 from x.2)]\n[GOAL]\nx\u271d y : \u211d\nx : \u2191{0}\u1d9c\n| log \u2191x\n[PROOFSTEP]\nrw [log_of_ne_zero (show (x : \u211d) \u2260 0 from x.2)]\n[GOAL]\nx y : \u211d\n\u22a2 Continuous fun x => \u2191(OrderIso.symm expOrderIso) { val := |\u2191x|, property := (_ : 0 < |\u2191x|) }\n[PROOFSTEP]\nexact expOrderIso.symm.continuous.comp (continuous_subtype_val.norm.subtype_mk _)\n[GOAL]\nx y : \u211d\n\u22a2 ContinuousAt log x \u2194 x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, continuousAt_log\u27e9\n[GOAL]\nx y : \u211d\n\u22a2 ContinuousAt log x \u2192 x \u2260 0\n[PROOFSTEP]\nrintro h rfl\n[GOAL]\ny : \u211d\nh : ContinuousAt log 0\n\u22a2 False\n[PROOFSTEP]\nexact not_tendsto_nhds_of_tendsto_atBot tendsto_log_nhdsWithin_zero _ (h.tendsto.mono_left inf_le_left)\n[GOAL]\nx y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\n[PROOFSTEP]\ninduction' s using Finset.cons_induction_on with a s ha ih\n[GOAL]\ncase h\u2081\nx y : \u211d\n\u03b1 : Type u_1\ns : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\nhf : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 f x \u2260 0\n\u22a2 log (\u220f i in \u2205, f i) = \u2211 i in \u2205, log (f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : \u2200 (x : \u03b1), x \u2208 Finset.cons a s ha \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)\n[PROOFSTEP]\nrw [Finset.forall_mem_cons] at hf \n[GOAL]\ncase h\u2082\nx y : \u211d\n\u03b1 : Type u_1\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u211d\nhf\u271d : \u2200 (x : \u03b1), x \u2208 s\u271d \u2192 f x \u2260 0\na : \u03b1\ns : Finset \u03b1\nha : \u00aca \u2208 s\nih : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) \u2192 log (\u220f i in s, f i) = \u2211 i in s, log (f i)\nhf : f a \u2260 0 \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0\n\u22a2 log (\u220f i in Finset.cons a s ha, f i) = \u2211 i in Finset.cons a s ha, log (f i)\n[PROOFSTEP]\nsimp [ih hf.2, log_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)]\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 log \u2191n = Finsupp.sum (Nat.factorization n) fun p t => \u2191t * log \u2191p\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nx y : \u211d\n\u22a2 log \u21910 = Finsupp.sum (Nat.factorization 0) fun p t => \u2191t * log \u2191p\n[PROOFSTEP]\nsimp\n  -- relies on junk values of `log` and `Nat.factorization`\n[GOAL]\ncase inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = Finsupp.sum (Nat.factorization n) fun p t => \u2191t * log \u2191p\n[PROOFSTEP]\nsimp only [\u2190 log_pow, \u2190 Nat.cast_pow]\n[GOAL]\ncase inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = Finsupp.sum (Nat.factorization n) fun p t => log \u2191(p ^ t)\n[PROOFSTEP]\nrw [\u2190 Finsupp.log_prod, \u2190 Nat.cast_finsupp_prod, Nat.factorization_prod_pow_eq_self hn]\n[GOAL]\ncase inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (a : \u2115), \u2191(a ^ \u2191(Nat.factorization n) a) = 0 \u2192 \u2191(Nat.factorization n) a = 0\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\np : \u2115\nhp : \u2191(p ^ \u2191(Nat.factorization n) p) = 0\n\u22a2 \u2191(Nat.factorization n) p = 0\n[PROOFSTEP]\nrw [pow_eq_zero (Nat.cast_eq_zero.1 hp), Nat.factorization_zero_right]\n[GOAL]\nx y a b : \u211d\nn : \u2115\nha : a \u2260 0\n\u22a2 (fun x => x ^ n / (a * exp x + b)) \u2218 log =\u1da0[atTop] fun x => log x ^ n / (a * x + b)\n[PROOFSTEP]\nfilter_upwards [eventually_gt_atTop (0 : \u211d)] with x hx using by simp [exp_log hx]\n[GOAL]\nx\u271d y a b : \u211d\nn : \u2115\nha : a \u2260 0\nx : \u211d\nhx : 0 < x\n\u22a2 ((fun x => x ^ n / (a * exp x + b)) \u2218 log) x = log x ^ n / (a * x + b)\n[PROOFSTEP]\nsimp [exp_log hx]\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 (fun x => log x ^ n) =o[atTop] id\n[PROOFSTEP]\nrw [Asymptotics.isLittleO_iff_tendsto']\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 Tendsto (fun x => log x ^ n / id x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using tendsto_pow_log_div_mul_add_atTop 1 0 n one_ne_zero\n[GOAL]\nx y : \u211d\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, id x = 0 \u2192 log x ^ n = 0\n[PROOFSTEP]\nfilter_upwards [eventually_ne_atTop (0 : \u211d)] with x h\u2081 h\u2082 using (h\u2081 h\u2082).elim\n[GOAL]\nx y c : \u211d\n\u22a2 (fun x => c) =o[atTop] log\n[PROOFSTEP]\nrefine Asymptotics.isLittleO_of_tendsto' ?_ <| Tendsto.div_atTop (a := c) (by simp) tendsto_log_atTop\n[GOAL]\nx y c : \u211d\n\u22a2 Tendsto (fun x => c) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nsimp\n[GOAL]\nx y c : \u211d\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop, log x = 0 \u2192 c = 0\n[PROOFSTEP]\nfilter_upwards [eventually_gt_atTop 1] with x hx\n[GOAL]\ncase h\nx\u271d y c x : \u211d\nhx : 1 < x\n\u22a2 log x = 0 \u2192 c = 0\n[PROOFSTEP]\naesop (add safe forward log_pos)\n[GOAL]\ny : \u211d\n\u22a2 Tendsto (fun x => log (x + y) - log x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun x \u21a6 1 + y / x) atTop (\ud835\udcdd (1 + 0))\n[GOAL]\ncase this\ny : \u211d\n\u22a2 Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\n[PROOFSTEP]\nexact tendsto_const_nhds.add (tendsto_const_nhds.div_atTop tendsto_id)\n[GOAL]\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\n\u22a2 Tendsto (fun x => log (x + y) - log x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 comap_exp_nhds_exp, exp_zero, tendsto_comap_iff, \u2190 add_zero (1 : \u211d)]\n[GOAL]\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\n\u22a2 Tendsto (exp \u2218 fun x => log (x + y) - log x) atTop (\ud835\udcdd (1 + 0))\n[PROOFSTEP]\nrefine' this.congr' _\n[GOAL]\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\n\u22a2 (fun x => 1 + y / x) =\u1da0[atTop] exp \u2218 fun x => log (x + y) - log x\n[PROOFSTEP]\nfilter_upwards [eventually_gt_atTop (0 : \u211d), eventually_gt_atTop (-y)] with x hx\u2080 hxy\n[GOAL]\ncase h\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\nx : \u211d\nhx\u2080 : 0 < x\nhxy : -y < x\n\u22a2 1 + y / x = (exp \u2218 fun x => log (x + y) - log x) x\n[PROOFSTEP]\nrw [comp_apply, exp_sub, exp_log, exp_log, one_add_div]\n[GOAL]\ncase h\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\nx : \u211d\nhx\u2080 : 0 < x\nhxy : -y < x\n\u22a2 x \u2260 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\nx : \u211d\nhx\u2080 : 0 < x\nhxy : -y < x\n\u22a2 0 < x\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\ny : \u211d\nthis : Tendsto (fun x => 1 + y / x) atTop (\ud835\udcdd (1 + 0))\nx : \u211d\nhx\u2080 : 0 < x\nhxy : -y < x\n\u22a2 0 < x + y\n[PROOFSTEP]\nlinarith\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsNat e n\n\u22a2 0 \u2264 Real.log e\n[PROOFSTEP]\nrw [NormNum.IsNat.to_eq h rfl]\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsNat e n\n\u22a2 0 \u2264 Real.log \u2191n\n[PROOFSTEP]\nexact Real.log_nat_cast_nonneg _\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsNat e n\nw : Nat.blt 1 n = true\n\u22a2 0 < Real.log e\n[PROOFSTEP]\nrw [NormNum.IsNat.to_eq h rfl]\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsNat e n\nw : Nat.blt 1 n = true\n\u22a2 0 < Real.log \u2191n\n[PROOFSTEP]\napply Real.log_pos\n[GOAL]\ncase hx\ne : \u211d\nn : \u2115\nh : NormNum.IsNat e n\nw : Nat.blt 1 n = true\n\u22a2 1 < \u2191n\n[PROOFSTEP]\nsimpa using w\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\n\u22a2 0 \u2264 Real.log e\n[PROOFSTEP]\nrw [NormNum.IsInt.neg_to_eq h rfl]\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\n\u22a2 0 \u2264 Real.log (-\u2191n)\n[PROOFSTEP]\nexact Real.log_neg_nat_cast_nonneg _\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.blt 1 n = true\n\u22a2 0 < Real.log e\n[PROOFSTEP]\nrw [NormNum.IsInt.neg_to_eq h rfl]\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.blt 1 n = true\n\u22a2 0 < Real.log (-\u2191n)\n[PROOFSTEP]\nrw [Real.log_neg_eq_log]\n[GOAL]\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.blt 1 n = true\n\u22a2 0 < Real.log \u2191n\n[PROOFSTEP]\napply Real.log_pos\n[GOAL]\ncase hx\ne : \u211d\nn : \u2115\nh : NormNum.IsInt e (Int.negOfNat n)\nw : Nat.blt 1 n = true\n\u22a2 1 < \u2191n\n[PROOFSTEP]\nsimpa using w\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (1 < \u2191n / \u2191d) = true\n\u22a2 0 < Real.log e\n[PROOFSTEP]\nrw [eq, invOf_eq_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (1 < \u2191n / \u2191d) = true\n\u22a2 0 < Real.log (\u2191n / \u2191d)\n[PROOFSTEP]\nhave : 1 < (n : \u211d) / d := by exact_mod_cast of_decide_eq_true h\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (1 < \u2191n / \u2191d) = true\n\u22a2 1 < \u2191n / \u2191d\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (1 < \u2191n / \u2191d) = true\nthis : 1 < \u2191n / \u2191d\n\u22a2 0 < Real.log (\u2191n / \u2191d)\n[PROOFSTEP]\nexact Real.log_pos this\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (\u2191n / \u2191d < -1) = true\n\u22a2 0 < Real.log e\n[PROOFSTEP]\nrw [eq, invOf_eq_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (\u2191n / \u2191d < -1) = true\n\u22a2 0 < Real.log (\u2191n / \u2191d)\n[PROOFSTEP]\nhave : (n : \u211d) / d < -1 := by exact_mod_cast of_decide_eq_true h\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (\u2191n / \u2191d < -1) = true\n\u22a2 \u2191n / \u2191d < -1\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh : decide (\u2191n / \u2191d < -1) = true\nthis : \u2191n / \u2191d < -1\n\u22a2 0 < Real.log (\u2191n / \u2191d)\n[PROOFSTEP]\nexact Real.log_pos_of_lt_neg_one this\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\n\u22a2 Real.log e \u2260 0\n[PROOFSTEP]\nrw [eq, invOf_eq_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nhave h\u2081' : 0 < (n : \u211d) / d := by exact_mod_cast of_decide_eq_true h\u2081\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\n\u22a2 0 < \u2191n / \u2191d\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\u2081\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\nh\u2081' : 0 < \u2191n / \u2191d\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nhave h\u2082' : (n : \u211d) / d < 1 := by exact_mod_cast of_decide_eq_true h\u2082\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\nh\u2081' : 0 < \u2191n / \u2191d\n\u22a2 \u2191n / \u2191d < 1\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\u2082\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (0 < \u2191n / \u2191d) = true\nh\u2082 : decide (\u2191n / \u2191d < 1) = true\nh\u2081' : 0 < \u2191n / \u2191d\nh\u2082' : \u2191n / \u2191d < 1\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nexact ne_of_lt <| Real.log_neg h\u2081' h\u2082'\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\n\u22a2 Real.log e \u2260 0\n[PROOFSTEP]\nrw [eq, invOf_eq_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nhave h\u2081' : (n : \u211d) / d < 0 := by exact_mod_cast of_decide_eq_true h\u2081\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\n\u22a2 \u2191n / \u2191d < 0\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\u2081\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\nh\u2081' : \u2191n / \u2191d < 0\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nhave h\u2082' : -1 < (n : \u211d) / d := by exact_mod_cast of_decide_eq_true h\u2082\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\nh\u2081' : \u2191n / \u2191d < 0\n\u22a2 -1 < \u2191n / \u2191d\n[PROOFSTEP]\nexact_mod_cast of_decide_eq_true h\u2082\n[GOAL]\ne : \u211d\nn : \u2124\nd : \u2115\ninv : Invertible \u2191d\neq : e = \u2191n * \u215f\u2191d\nh\u2081 : decide (\u2191n / \u2191d < 0) = true\nh\u2082 : decide (-1 < \u2191n / \u2191d) = true\nh\u2081' : \u2191n / \u2191d < 0\nh\u2082' : -1 < \u2191n / \u2191d\n\u22a2 Real.log (\u2191n / \u2191d) \u2260 0\n[PROOFSTEP]\nexact ne_of_lt <| Real.log_neg_of_lt_zero h\u2081' h\u2082'\n[GOAL]\n$x\u271d : Type x\u271d\n\u00ab$_z\u03b1\u00bb : Zero $x\u271d\n\u00ab$_p\u03b1\u00bb : PartialOrder $x\u271d\n\u00ab$e\u00bb : $x\u271d\n\u00ab$f\u00bb : \u211d \u2192 \u211d\n\u00ab$a\u00bb : \u211d\n\u00ab$n\u00bb : \u2124\n\u00ab$d\u00bb : \u2115\n\u00ab$i\u00bb : DivisionRing \u211d := Real.instDivisionRingReal\n\u00ab$p\u00bb : NormNum.IsRat \u00ab$a\u00bb \u00ab$n\u00bb \u00ab$d\u00bb\n\u22a2 Sort ?u.471336\n[PROOFSTEP]\nclear! \u00ab$i\u00bb\n[GOAL]\n$x\u271d : Type x\u271d\n\u00ab$_z\u03b1\u00bb : Zero $x\u271d\n\u00ab$_p\u03b1\u00bb : PartialOrder $x\u271d\n\u00ab$e\u00bb : $x\u271d\n\u00ab$f\u00bb : \u211d \u2192 \u211d\n\u00ab$a\u00bb : \u211d\n\u00ab$n\u00bb : \u2124\n\u00ab$d\u00bb : \u2115\n\u22a2 Sort ?u.471336\n[PROOFSTEP]\nexact NormNum.IsRat \u00ab$a\u00bb \u00ab$n\u00bb \u00ab$d\u00bb\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Log.Basic", "llama_tokens": 16598, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.7025300511670689, "lm_q1q2_score": 0.5441681118219207}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a + a = 0\n[PROOFSTEP]\nhave : a + a = a + a + (a + a) :=\n  calc\n    a + a = (a + a) * (a + a) := by rw [mul_self]\n    _ = a * a + a * a + (a * a + a * a) := by rw [add_mul, mul_add]\n    _ = a + a + (a + a) := by rw [mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a + a = (a + a) * (a + a)\n[PROOFSTEP]\nrw [mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 (a + a) * (a + a) = a * a + a * a + (a * a + a * a)\n[PROOFSTEP]\nrw [add_mul, mul_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a * a + a * a + (a * a + a * a) = a + a + (a + a)\n[PROOFSTEP]\nrw [mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\nthis : a + a = a + a + (a + a)\n\u22a2 a + a = 0\n[PROOFSTEP]\nrwa [self_eq_add_left] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 -a = -a + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 -a + 0 = -a + -a + a\n[PROOFSTEP]\nrw [\u2190 neg_add_self, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 -a + -a + a = a\n[PROOFSTEP]\nrw [add_self, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a = -b \u2194 a = b\n[PROOFSTEP]\nrw [neg_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a * b + b * a = 0\n[PROOFSTEP]\nhave : a + b = a + b + (a * b + b * a) :=\n  calc\n    a + b = (a + b) * (a + b) := by rw [mul_self]\n    _ = a * a + a * b + (b * a + b * b) := by rw [add_mul, mul_add, mul_add]\n    _ = a + a * b + (b * a + b) := by simp only [mul_self]\n    _ = a + b + (a * b + b * a) := by abel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a + b = (a + b) * (a + b)\n[PROOFSTEP]\nrw [mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 (a + b) * (a + b) = a * a + a * b + (b * a + b * b)\n[PROOFSTEP]\nrw [add_mul, mul_add, mul_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a * a + a * b + (b * a + b * b) = a + a * b + (b * a + b)\n[PROOFSTEP]\nsimp only [mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a + a * b + (b * a + b) = a + b + (a * b + b * a)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a + a * b + (b * a + b) = a + b + (a * b + b * a)\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\nthis : a + b = a + b + (a * b + b * a)\n\u22a2 a * b + b * a = 0\n[PROOFSTEP]\nrwa [self_eq_add_right] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a - b = a + b\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_right_inj, neg_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na b : \u03b1\n\u22a2 a * (1 + a) = 0\n[PROOFSTEP]\nrw [mul_add, mul_one, mul_self, add_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : BooleanRing \u03b1\na\u271d b\u271d : \u03b1\nsrc\u271d : BooleanRing \u03b1 := inferInstance\na b : \u03b1\n\u22a2 a * b = b * a\n[PROOFSTEP]\nrw [\u2190 add_eq_zero', mul_add_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a \u2294 b = b \u2294 a\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2294 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a + b + a * b = b + a + b * a\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a \u2293 b = b \u2293 a\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2293 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a * b = b * a\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2294 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a + b + a * b + c + (a + b + a * b) * c = a + (b + c + b * c) + a * (b + c + b * c)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2293 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a * b * c = a * (b * c)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a \u2294 a \u2293 b = a\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2294 \u00b7), (\u00b7 \u2293 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a + a * b + a * (a * b) = a\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_self, add_assoc, add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a \u2293 (a \u2294 b) = a\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2294 \u00b7), (\u00b7 \u2293 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a * (a + b + a * b) = a\n[PROOFSTEP]\nrw [mul_add, mul_add, mul_self, \u2190 mul_assoc, mul_self, add_assoc, add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 (a + b + a * b) * (a + c + a * c) =\n    a * a + b * c + a * (b * c) + (a * b + a * a * b) + (a * c + a * a * c) + (a * b * c + a * a * b * c)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 a * a + b * c + a * (b * c) + (a * b + a * a * b) + (a * c + a * a * c) + (a * b * c + a * a * b * c) =\n    a + b * c + a * (b * c)\n[PROOFSTEP]\nsimp only [mul_self, add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) \u2294 (a \u2294 b \u2293 c) = a \u2294 b \u2293 c\n[PROOFSTEP]\ndsimp only [(\u00b7 \u2294 \u00b7), (\u00b7 \u2293 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b c : \u03b1\n\u22a2 (a + b + a * b) * (a + c + a * c) + (a + b * c + a * (b * c)) +\n      (a + b + a * b) * (a + c + a * c) * (a + b * c + a * (b * c)) =\n    a + b * c + a * (b * c)\n[PROOFSTEP]\nrw [le_sup_inf_aux, add_self, mul_self, zero_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 a * (1 + a) + 0 + a * (1 + a) * 0 = 0\n[PROOFSTEP]\nnorm_num [mul_add, mul_self, add_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 \u22a4 \u2264 a \u2294 a\u1d9c\n[PROOFSTEP]\nchange 1 + (a + (1 + a) + a * (1 + a)) + 1 * (a + (1 + a) + a * (1 + a)) = a + (1 + a) + a * (1 + a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 1 + (a + (1 + a) + a * (1 + a)) + 1 * (a + (1 + a) + a * (1 + a)) = a + (1 + a) + a * (1 + a)\n[PROOFSTEP]\nnorm_num [mul_add, mul_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 1 + (a + (1 + a)) = 0\n[PROOFSTEP]\nrw [\u2190 add_assoc, add_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 a + 1 + a * 1 = 1\n[PROOFSTEP]\nrw [mul_one, (add_comm a 1), add_assoc, add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nsrc\u271d : Lattice \u03b1 :=\n  Lattice.mk' (_ : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a) (_ : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c))\n    (_ : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a) (_ : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)) (_ : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a)\n    (_ : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a)\na : \u03b1\n\u22a2 0 + a + 0 * a = a\n[PROOFSTEP]\nrw [zero_mul, zero_add, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 (a + b + a * b) * (1 + a * b) = a + b + (a * b + a * b * (a * b)) + (a * (b * b) + a * a * b)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a + b + (a * b + a * b * (a * b)) + (a * (b * b) + a * a * b) = a + b\n[PROOFSTEP]\nsimp only [mul_self, add_self, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : AsBoolAlg \u03b1\n\u22a2 \u2191ofBoolAlg (a \u2206 b) = \u2191ofBoolAlg a + \u2191ofBoolAlg b\n[PROOFSTEP]\nrw [symmDiff_eq_sup_sdiff_inf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : AsBoolAlg \u03b1\n\u22a2 \u2191ofBoolAlg ((a \u2294 b) \\ (a \u2293 b)) = \u2191ofBoolAlg a + \u2191ofBoolAlg b\n[PROOFSTEP]\nexact of_boolalg_symmDiff_aux _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nf : \u03b1 \u2192+* \u03b2\na b : AsBoolAlg \u03b1\n\u22a2 (\u2191toBoolAlg \u2218 \u2191f \u2218 \u2191ofBoolAlg) (a \u2294 b) = (\u2191toBoolAlg \u2218 \u2191f \u2218 \u2191ofBoolAlg) a \u2294 (\u2191toBoolAlg \u2218 \u2191f \u2218 \u2191ofBoolAlg) b\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\nf : \u03b1 \u2192+* \u03b2\na b : AsBoolAlg \u03b1\n\u22a2 \u2191toBoolAlg (\u2191f (\u2191ofBoolAlg a + \u2191ofBoolAlg b + \u2191ofBoolAlg a * \u2191ofBoolAlg b)) =\n    \u2191toBoolAlg (\u2191f (\u2191ofBoolAlg a)) \u2294 \u2191toBoolAlg (\u2191f (\u2191ofBoolAlg b))\n[PROOFSTEP]\nsimp_rw [map_add f, map_mul f, toBoolAlg_add_add_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : Bool\n\u22a2 a * 0 = 0\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 false * 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u22a2 true * 0 = 0\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.BooleanRing", "llama_tokens": 6475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744584140004, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5440226717656387}}
{"text": "[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d\u00b9 : CommRing R\np m : \u2115\ninst\u271d : CharP R p\nn : \u2115\n\u22a2 \u2191n = 0 \u2194 p \u2223 n\n[PROOFSTEP]\nrw [\u2190 C_eq_coe_nat, \u2190 C_0, C_inj, CharP.cast_eq_zero_iff R p]\n[GOAL]\n\u03c3 : Type u\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\np\u271d m : \u2115\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\np : MvPolynomial \u03c3 R\nf : R \u2192+* S\n\u22a2 Finsupp.mapRange \u2191f (_ : \u2191f 0 = 0) p = \u2191(map f) p\n[PROOFSTEP]\nchange Finsupp.mapRange (f : R \u2192+ S) (f : R \u2192+ S).map_zero p = map f p\n[GOAL]\n\u03c3 : Type u\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\np\u271d m : \u2115\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\np : MvPolynomial \u03c3 R\nf : R \u2192+* S\n\u22a2 Finsupp.mapRange \u2191\u2191f (_ : \u2191\u2191f 0 = 0) p = \u2191(map f) p\n[PROOFSTEP]\nrw [p.as_sum, Finsupp.mapRange_finset_sum, (map f).map_sum]\n[GOAL]\n\u03c3 : Type u\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\np\u271d m : \u2115\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\np : MvPolynomial \u03c3 R\nf : R \u2192+* S\n\u22a2 \u2211 x in support p, Finsupp.mapRange \u2191\u2191f (_ : \u2191\u2191f 0 = 0) (\u2191(monomial x) (coeff x p)) =\n    \u2211 x in support p, \u2191(map f) (\u2191(monomial x) (coeff x p))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun n _ => _\n[GOAL]\n\u03c3 : Type u\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\np\u271d m : \u2115\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\np : MvPolynomial \u03c3 R\nf : R \u2192+* S\nn : \u03c3 \u2192\u2080 \u2115\nx\u271d : n \u2208 support p\n\u22a2 Finsupp.mapRange \u2191\u2191f (_ : \u2191\u2191f 0 = 0) (\u2191(monomial n) (coeff n p)) = \u2191(map f) (\u2191(monomial n) (coeff n p))\n[PROOFSTEP]\nrw [map_monomial, \u2190 single_eq_monomial, Finsupp.mapRange_single, single_eq_monomial]\n[GOAL]\n\u03c3 : Type u\nR\u271d : Type v\ninst\u271d\u00b2 : CommRing R\u271d\np\u271d m : \u2115\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\np : MvPolynomial \u03c3 R\nf : R \u2192+* S\nn : \u03c3 \u2192\u2080 \u2115\nx\u271d : n \u2208 support p\n\u22a2 \u2191(monomial n) (\u2191\u2191f (coeff n p)) = \u2191(monomial n) (\u2191f (coeff n p))\n[PROOFSTEP]\nsimp_all only [AddMonoidHom.coe_coe]\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommRing R\np\u271d m : \u2115\np : MvPolynomial \u03c3 R\n\u22a2 p \u2208 restrictTotalDegree \u03c3 R m \u2194 totalDegree p \u2264 m\n[PROOFSTEP]\nrw [totalDegree, Finset.sup_le_iff]\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommRing R\np\u271d m : \u2115\np : MvPolynomial \u03c3 R\n\u22a2 p \u2208 restrictTotalDegree \u03c3 R m \u2194 \u2200 (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 (Finsupp.sum b fun x e => e) \u2264 m\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommRing R\np\u271d m : \u2115\np : MvPolynomial \u03c3 R\nn : \u2115\n\u22a2 p \u2208 restrictDegree \u03c3 R n \u2194 \u2200 (s : \u03c3 \u2192\u2080 \u2115), s \u2208 support p \u2192 \u2200 (i : \u03c3), \u2191s i \u2264 n\n[PROOFSTEP]\nrw [restrictDegree, Finsupp.mem_supported]\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d : CommRing R\np\u271d m : \u2115\np : MvPolynomial \u03c3 R\nn : \u2115\n\u22a2 \u2191p.support \u2286 {n_1 | \u2200 (i : \u03c3), \u2191n_1 i \u2264 n} \u2194 \u2200 (s : \u03c3 \u2192\u2080 \u2115), s \u2208 support p \u2192 \u2200 (i : \u03c3), \u2191s i \u2264 n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d\u00b9 : CommRing R\np\u271d m : \u2115\ninst\u271d : DecidableEq \u03c3\np : MvPolynomial \u03c3 R\nn : \u2115\n\u22a2 p \u2208 restrictDegree \u03c3 R n \u2194 \u2200 (i : \u03c3), Multiset.count i (degrees p) \u2264 n\n[PROOFSTEP]\nsimp only [mem_restrictDegree, degrees_def, Multiset.count_finset_sup, Finsupp.count_toMultiset, Finset.sup_le_iff]\n[GOAL]\n\u03c3 : Type u\nR : Type v\ninst\u271d\u00b9 : CommRing R\np\u271d m : \u2115\ninst\u271d : DecidableEq \u03c3\np : MvPolynomial \u03c3 R\nn : \u2115\n\u22a2 (\u2200 (s : \u03c3 \u2192\u2080 \u2115), s \u2208 support p \u2192 \u2200 (i : \u03c3), \u2191s i \u2264 n) \u2194 \u2200 (i : \u03c3) (b : \u03c3 \u2192\u2080 \u2115), b \u2208 support p \u2192 \u2191b i \u2264 n\n[PROOFSTEP]\nexact \u27e8fun h n s hs => h s hs n, fun h s hs n => h n s hs\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.MvPolynomial.Basic", "llama_tokens": 1775, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.6926419894793248, "lm_q1q2_score": 0.5439377047818381}}
{"text": "[GOAL]\nr : \u211d\nhr : r < 0\n\u22a2 sign r = -1\n[PROOFSTEP]\nrw [sign, if_pos hr]\n[GOAL]\nr : \u211d\nhr : 0 < r\n\u22a2 sign r = 1\n[PROOFSTEP]\nrw [sign, if_pos hr, if_neg hr.not_lt]\n[GOAL]\n\u22a2 sign 0 = 0\n[PROOFSTEP]\nrw [sign, if_neg (lt_irrefl _), if_neg (lt_irrefl _)]\n[GOAL]\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nr : \u211d\n\u22a2 sign r = -1 \u2228 sign r = 0 \u2228 sign r = 1\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)\n[GOAL]\ncase inl\nr : \u211d\nhn : r < 0\n\u22a2 sign r = -1 \u2228 sign r = 0 \u2228 sign r = 1\n[PROOFSTEP]\nexact Or.inl <| sign_of_neg hn\n[GOAL]\ncase inr.inl\n\u22a2 sign 0 = -1 \u2228 sign 0 = 0 \u2228 sign 0 = 1\n[PROOFSTEP]\nexact Or.inr <| Or.inl <| sign_zero\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 sign r = -1 \u2228 sign r = 0 \u2228 sign r = 1\n[PROOFSTEP]\nexact Or.inr <| Or.inr <| sign_of_pos hp\n[GOAL]\nr : \u211d\n\u22a2 sign r = 0 \u2194 r = 0\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.symm \u25b8 sign_zero\u27e9\n[GOAL]\nr : \u211d\nh : sign r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)\n[GOAL]\ncase inl\nr : \u211d\nh : sign r = 0\nhn : r < 0\n\u22a2 r = 0\n[PROOFSTEP]\nrw [sign_of_neg hn, neg_eq_zero] at h \n[GOAL]\ncase inl\nr : \u211d\nh : 1 = 0\nhn : r < 0\n\u22a2 r = 0\n[PROOFSTEP]\nexact (one_ne_zero h).elim\n[GOAL]\ncase inr.inl\nh : sign 0 = 0\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.inr\nr : \u211d\nh : sign r = 0\nhp : 0 < r\n\u22a2 r = 0\n[PROOFSTEP]\nrw [sign_of_pos hp] at h \n[GOAL]\ncase inr.inr\nr : \u211d\nh : 1 = 0\nhp : 0 < r\n\u22a2 r = 0\n[PROOFSTEP]\nexact (one_ne_zero h).elim\n[GOAL]\nz : \u2124\n\u22a2 sign \u2191z = \u2191(Int.sign z)\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy z (0 : \u2124)\n[GOAL]\ncase inl\nz : \u2124\nhn : z < 0\n\u22a2 sign \u2191z = \u2191(Int.sign z)\n[PROOFSTEP]\nrw [sign_of_neg (Int.cast_lt_zero.mpr hn), Int.sign_eq_neg_one_of_neg hn, Int.cast_neg, Int.cast_one]\n[GOAL]\ncase inr.inl\n\u22a2 sign \u21910 = \u2191(Int.sign 0)\n[PROOFSTEP]\nrw [Int.cast_zero, sign_zero, Int.sign_zero, Int.cast_zero]\n[GOAL]\ncase inr.inr\nz : \u2124\nhp : 0 < z\n\u22a2 sign \u2191z = \u2191(Int.sign z)\n[PROOFSTEP]\nrw [sign_of_pos (Int.cast_pos.mpr hp), Int.sign_eq_one_of_pos hp, Int.cast_one]\n[GOAL]\nr : \u211d\n\u22a2 sign (-r) = -sign r\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)\n[GOAL]\ncase inl\nr : \u211d\nhn : r < 0\n\u22a2 sign (-r) = -sign r\n[PROOFSTEP]\nrw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]\n[GOAL]\ncase inr.inl\n\u22a2 sign (-0) = -sign 0\n[PROOFSTEP]\nrw [sign_zero, neg_zero, sign_zero]\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 sign (-r) = -sign r\n[PROOFSTEP]\nrw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]\n[GOAL]\nr : \u211d\n\u22a2 0 \u2264 sign r * r\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)\n[GOAL]\ncase inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 sign r * r\n[PROOFSTEP]\nrw [sign_of_neg hn]\n[GOAL]\ncase inl\nr : \u211d\nhn : r < 0\n\u22a2 0 \u2264 -1 * r\n[PROOFSTEP]\nexact mul_nonneg_of_nonpos_of_nonpos (by norm_num) hn.le\n[GOAL]\nr : \u211d\nhn : r < 0\n\u22a2 -1 \u2264 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inl\n\u22a2 0 \u2264 sign 0 * 0\n[PROOFSTEP]\nrw [mul_zero]\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 sign r * r\n[PROOFSTEP]\nrw [sign_of_pos hp, one_mul]\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 0 \u2264 r\n[PROOFSTEP]\nexact hp.le\n[GOAL]\nr : \u211d\nhr : r \u2260 0\n\u22a2 0 < sign r * r\n[PROOFSTEP]\nrefine' lt_of_le_of_ne (sign_mul_nonneg r) fun h => hr _\n[GOAL]\nr : \u211d\nhr : r \u2260 0\nh : 0 = sign r * r\n\u22a2 r = 0\n[PROOFSTEP]\nhave hs0 := (zero_eq_mul.mp h).resolve_right hr\n[GOAL]\nr : \u211d\nhr : r \u2260 0\nh : 0 = sign r * r\nhs0 : sign r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nexact sign_eq_zero_iff.mp hs0\n[GOAL]\nr : \u211d\n\u22a2 (sign r)\u207b\u00b9 = sign r\n[PROOFSTEP]\nobtain hn | hz | hp := sign_apply_eq r\n[GOAL]\ncase inl\nr : \u211d\nhn : sign r = -1\n\u22a2 (sign r)\u207b\u00b9 = sign r\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase inl\nr : \u211d\nhn : sign r = -1\n\u22a2 (-1)\u207b\u00b9 = -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inl\nr : \u211d\nhz : sign r = 0\n\u22a2 (sign r)\u207b\u00b9 = sign r\n[PROOFSTEP]\nrw [hz]\n[GOAL]\ncase inr.inl\nr : \u211d\nhz : sign r = 0\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nexact inv_zero\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : sign r = 1\n\u22a2 (sign r)\u207b\u00b9 = sign r\n[PROOFSTEP]\nrw [hp]\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : sign r = 1\n\u22a2 1\u207b\u00b9 = 1\n[PROOFSTEP]\nexact inv_one\n[GOAL]\nr : \u211d\n\u22a2 sign r\u207b\u00b9 = sign r\n[PROOFSTEP]\nobtain hn | rfl | hp := lt_trichotomy r (0 : \u211d)\n[GOAL]\ncase inl\nr : \u211d\nhn : r < 0\n\u22a2 sign r\u207b\u00b9 = sign r\n[PROOFSTEP]\nrw [sign_of_neg hn, sign_of_neg (inv_lt_zero.mpr hn)]\n[GOAL]\ncase inr.inl\n\u22a2 sign 0\u207b\u00b9 = sign 0\n[PROOFSTEP]\nrw [sign_zero, inv_zero, sign_zero]\n[GOAL]\ncase inr.inr\nr : \u211d\nhp : 0 < r\n\u22a2 sign r\u207b\u00b9 = sign r\n[PROOFSTEP]\nrw [sign_of_pos hp, sign_of_pos (inv_pos.mpr hp)]\n", "meta": {"mathlib_filename": "Mathlib.Data.Real.Sign", "llama_tokens": 2426, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438951182587158, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5436584224235487}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 card s * card t \u2264 multiplicativeEnergy s t\n[PROOFSTEP]\nrw [\u2190 card_product]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 card (s \u00d7\u02e2 t) \u2264 multiplicativeEnergy s t\n[PROOFSTEP]\nrefine'\n  card_le_card_of_inj_on (@fun x => ((x.1, x.1), x.2, x.2))\n    (by\n      -- porting note: changed this from a `simp` proof without `only` because of a timeoutsimp only [\u2190 and_imp,\n        mem_product, and_imp, Prod.forall, mem_filter, and_self, and_true, imp_self, implies_true])\n    fun a _ b _ => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 \u2200 (a : \u03b1 \u00d7 \u03b1),\n    a \u2208 s \u00d7\u02e2 t \u2192\n      (fun x => ((x.fst, x.fst), x.snd, x.snd)) a \u2208\n        filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) ((s \u00d7\u02e2 s) \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nsimp only [\u2190 and_imp, mem_product, and_imp, Prod.forall, mem_filter, and_self, and_true, imp_self, implies_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na : \u03b1 \u00d7 \u03b1\nx\u271d\u00b9 : a \u2208 s \u00d7\u02e2 t\nb : \u03b1 \u00d7 \u03b1\nx\u271d : b \u2208 s \u00d7\u02e2 t\n\u22a2 (fun x => ((x.fst, x.fst), x.snd, x.snd)) a = (fun x => ((x.fst, x.fst), x.snd, x.snd)) b \u2192 a = b\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff, and_self_iff, and_imp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\na : \u03b1 \u00d7 \u03b1\nx\u271d\u00b9 : a \u2208 s \u00d7\u02e2 t\nb : \u03b1 \u00d7 \u03b1\nx\u271d : b \u2208 s \u00d7\u02e2 t\n\u22a2 a.fst = b.fst \u2192 a.snd = b.snd \u2192 a = b\n[PROOFSTEP]\nexact Prod.ext\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 multiplicativeEnergy \u2205 t = 0\n[PROOFSTEP]\nsimp [multiplicativeEnergy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 multiplicativeEnergy s \u2205 = 0\n[PROOFSTEP]\nsimp [multiplicativeEnergy]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\nh : 0 < multiplicativeEnergy s t\nH : \u00ac(Finset.Nonempty s \u2227 Finset.Nonempty t)\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [not_and_or, not_nonempty_iff_eq_empty] at H \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\nh : 0 < multiplicativeEnergy s t\nH : s = \u2205 \u2228 t = \u2205\n\u22a2 False\n[PROOFSTEP]\nobtain rfl | rfl := H\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\nh : 0 < multiplicativeEnergy \u2205 t\n\u22a2 False\n[PROOFSTEP]\nsimp [Nat.not_lt_zero] at h \n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t\u2081 t\u2082 : Finset \u03b1\nh : 0 < multiplicativeEnergy s \u2205\n\u22a2 False\n[PROOFSTEP]\nsimp [Nat.not_lt_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Mul \u03b1\ns s\u2081 s\u2082 t t\u2081 t\u2082 : Finset \u03b1\n\u22a2 multiplicativeEnergy s t = 0 \u2194 s = \u2205 \u2228 t = \u2205\n[PROOFSTEP]\nsimp [\u2190 (Nat.zero_le _).not_gt_iff_eq, not_and_or, imp_iff_or_not, or_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 multiplicativeEnergy s t = multiplicativeEnergy t s\n[PROOFSTEP]\nrw [multiplicativeEnergy, \u2190 Finset.card_map (Equiv.prodComm _ _).toEmbedding, map_filter]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommMonoid \u03b1\ns t : Finset \u03b1\n\u22a2 card\n      (filter ((fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) \u2218 \u2191(Equiv.prodComm (\u03b1 \u00d7 \u03b1) (\u03b1 \u00d7 \u03b1)).symm)\n        (map (Equiv.toEmbedding (Equiv.prodComm (\u03b1 \u00d7 \u03b1) (\u03b1 \u00d7 \u03b1))) ((s \u00d7\u02e2 s) \u00d7\u02e2 t \u00d7\u02e2 t))) =\n    multiplicativeEnergy t s\n[PROOFSTEP]\nsimp [-Finset.card_map, eq_comm, multiplicativeEnergy, mul_comm, map_eq_image, Function.comp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 multiplicativeEnergy univ t = Fintype.card \u03b1 * card t ^ 2\n[PROOFSTEP]\nsimp only [multiplicativeEnergy, univ_product_univ, Fintype.card, sq, \u2190 card_product]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 card (filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nlet f : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.1 * x.2.2, x.1 * x.2.1), x.2)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\n\u22a2 card (filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nhave : (\u2191((univ : Finset \u03b1) \u00d7\u02e2 t \u00d7\u02e2 t) : Set (\u03b1 \u00d7 \u03b1 \u00d7 \u03b1)).InjOn f :=\n  by\n  rintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h\n  simp_rw [Prod.ext_iff] at h \n  obtain \u27e8h, rfl, rfl\u27e9 := h\n  rw [mul_right_cancel h.1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\n\u22a2 Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081, c\u2081\u27e9 _ \u27e8a\u2082, b\u2082, c\u2082\u27e9 h\u2082 h\n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : f (a\u2081, b\u2081, c\u2081) = f (a\u2082, b\u2082, c\u2082)\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)\n[PROOFSTEP]\nsimp_rw [Prod.ext_iff] at h \n[GOAL]\ncase mk.mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 b\u2082 c\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2082, c\u2082) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : (a\u2081 * c\u2081 = a\u2082 * c\u2082 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2082) \u2227 b\u2081 = b\u2082 \u2227 c\u2081 = c\u2082\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2082, c\u2082)\n[PROOFSTEP]\nobtain \u27e8h, rfl, rfl\u27e9 := h\n[GOAL]\ncase mk.mk.mk.mk.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\na\u2081 b\u2081 c\u2081 : \u03b1\na\u271d : (a\u2081, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na\u2082 : \u03b1\nh\u2082 : (a\u2082, b\u2081, c\u2081) \u2208 \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nh : a\u2081 * c\u2081 = a\u2082 * c\u2081 \u2227 a\u2081 * b\u2081 = a\u2082 * b\u2081\n\u22a2 (a\u2081, b\u2081, c\u2081) = (a\u2082, b\u2081, c\u2081)\n[PROOFSTEP]\nrw [mul_right_cancel h.1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) = card (univ \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [\u2190 card_image_of_injOn this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\n\u22a2 card (filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) (univ \u00d7\u02e2 t \u00d7\u02e2 t)) =\n    card (image f (univ \u00d7\u02e2 t \u00d7\u02e2 t))\n[PROOFSTEP]\ncongr with a\n[GOAL]\ncase e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 a \u2208 filter (fun x => x.fst.fst * x.snd.fst = x.fst.snd * x.snd.snd) (univ \u00d7\u02e2 t \u00d7\u02e2 t) \u2194 a \u2208 image f (univ \u00d7\u02e2 t \u00d7\u02e2 t)\n[PROOFSTEP]\nsimp only [mem_filter, mem_product, mem_univ, true_and_iff, mem_image, exists_prop, Prod.exists]\n[GOAL]\ncase e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (a.snd.fst \u2208 t \u2227 a.snd.snd \u2208 t) \u2227 a.fst.fst * a.snd.fst = a.fst.snd * a.snd.snd \u2194\n    \u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8a.1.1 * a.2.2\u207b\u00b9, _, _, h.1, by simp [mul_right_comm, h.2]\u27e9, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\nh : (a.snd.fst \u2208 t \u2227 a.snd.snd \u2208 t) \u2227 a.fst.fst * a.snd.fst = a.fst.snd * a.snd.snd\n\u22a2 ((a.fst.fst * a.snd.snd\u207b\u00b9 * a.snd.snd, a.fst.fst * a.snd.snd\u207b\u00b9 * a.snd.fst), a.snd.fst, a.snd.snd) = a\n[PROOFSTEP]\nsimp [mul_right_comm, h.2]\n[GOAL]\ncase e_s.a\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\na : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1\n\u22a2 (\u2203 a_1 a_2 b, (a_2 \u2208 t \u2227 b \u2208 t) \u2227 ((a_1 * b, a_1 * a_2), a_2, b) = a) \u2192\n    (a.snd.fst \u2208 t \u2227 a.snd.snd \u2208 t) \u2227 a.fst.fst * a.snd.fst = a.fst.snd * a.snd.snd\n[PROOFSTEP]\nrintro \u27e8b, c, d, hcd, rfl\u27e9\n[GOAL]\ncase e_s.a.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\nf : \u03b1 \u00d7 \u03b1 \u00d7 \u03b1 \u2192 (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 := fun x => ((x.fst * x.snd.snd, x.fst * x.snd.fst), x.snd)\nthis : Set.InjOn f \u2191(univ \u00d7\u02e2 t \u00d7\u02e2 t)\nb c d : \u03b1\nhcd : c \u2208 t \u2227 d \u2208 t\n\u22a2 (((b * d, b * c), c, d).snd.fst \u2208 t \u2227 ((b * d, b * c), c, d).snd.snd \u2208 t) \u2227\n    ((b * d, b * c), c, d).fst.fst * ((b * d, b * c), c, d).snd.fst =\n      ((b * d, b * c), c, d).fst.snd * ((b * d, b * c), c, d).snd.snd\n[PROOFSTEP]\nsimpa [mul_right_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : Fintype \u03b1\ns t : Finset \u03b1\n\u22a2 multiplicativeEnergy s univ = Fintype.card \u03b1 * card s ^ 2\n[PROOFSTEP]\nrw [multiplicativeEnergy_comm, multiplicativeEnergy_univ_left]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Additive.Energy", "llama_tokens": 5303, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.7090191337850932, "lm_q1q2_score": 0.5433168550458305}}
{"text": "[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nr : \u211d\nhr : 0 \u2264 r\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016 \u2264 r \u2194 \u2200 (i : m) (j : n), \u2016A i j\u2016 \u2264 r\n[PROOFSTEP]\nsimp_rw [norm_def, pi_norm_le_iff_of_nonneg hr]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nr : \u211d\u22650\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016\u208a \u2264 r \u2194 \u2200 (i : m) (j : n), \u2016A i j\u2016\u208a \u2264 r\n[PROOFSTEP]\nsimp_rw [nnnorm_def, pi_nnnorm_le_iff]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nr : \u211d\nhr : 0 < r\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016 < r \u2194 \u2200 (i : m) (j : n), \u2016A i j\u2016 < r\n[PROOFSTEP]\nsimp_rw [norm_def, pi_norm_lt_iff hr]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nr : \u211d\u22650\nhr : 0 < r\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016\u208a < r \u2194 \u2200 (i : m) (j : n), \u2016A i j\u2016\u208a < r\n[PROOFSTEP]\nsimp_rw [nnnorm_def, pi_nnnorm_lt_iff hr]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), \u2016f a\u2016\u208a = \u2016a\u2016\u208a\n\u22a2 \u2016map A f\u2016\u208a = \u2016A\u2016\u208a\n[PROOFSTEP]\nsimp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : m \u2192 \u03b1\n\u22a2 \u2016col v\u2016\u208a = \u2016v\u2016\u208a\n[PROOFSTEP]\nsimp [nnnorm_def, Pi.nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : n \u2192 \u03b1\n\u22a2 \u2016row v\u2016\u208a = \u2016v\u2016\u208a\n[PROOFSTEP]\nsimp [nnnorm_def, Pi.nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 \u2016diagonal v\u2016\u208a = \u2016v\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_def, Pi.nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 (Finset.sup Finset.univ fun b => Finset.sup Finset.univ fun b_1 => \u2016diagonal v b b_1\u2016\u208a) =\n    Finset.sup Finset.univ fun b => \u2016v b\u2016\u208a\n[PROOFSTEP]\ncongr 1 with i : 1\n[GOAL]\ncase e_f.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni : n\n\u22a2 (Finset.sup Finset.univ fun b => \u2016diagonal v i b\u2016\u208a) = \u2016v i\u2016\u208a\n[PROOFSTEP]\nrefine' le_antisymm (Finset.sup_le fun j hj => _) _\n[GOAL]\ncase e_f.h.refine'_1\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni j : n\nhj : j \u2208 Finset.univ\n\u22a2 \u2016diagonal v i j\u2016\u208a \u2264 \u2016v i\u2016\u208a\n[PROOFSTEP]\nobtain rfl | hij := eq_or_ne i j\n[GOAL]\ncase e_f.h.refine'_1.inl\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni : n\nhj : i \u2208 Finset.univ\n\u22a2 \u2016diagonal v i i\u2016\u208a \u2264 \u2016v i\u2016\u208a\n[PROOFSTEP]\nrw [diagonal_apply_eq]\n[GOAL]\ncase e_f.h.refine'_1.inr\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni j : n\nhj : j \u2208 Finset.univ\nhij : i \u2260 j\n\u22a2 \u2016diagonal v i j\u2016\u208a \u2264 \u2016v i\u2016\u208a\n[PROOFSTEP]\nrw [diagonal_apply_ne _ hij, nnnorm_zero]\n[GOAL]\ncase e_f.h.refine'_1.inr\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni j : n\nhj : j \u2208 Finset.univ\nhij : i \u2260 j\n\u22a2 0 \u2264 \u2016v i\u2016\u208a\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase e_f.h.refine'_2\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni : n\n\u22a2 \u2016v i\u2016\u208a \u2264 Finset.sup Finset.univ fun b => \u2016diagonal v i b\u2016\u208a\n[PROOFSTEP]\nrefine' Eq.trans_le _ (Finset.le_sup (Finset.mem_univ i))\n[GOAL]\ncase e_f.h.refine'_2\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ni : n\n\u22a2 \u2016v i\u2016\u208a = \u2016diagonal v i i\u2016\u208a\n[PROOFSTEP]\nrw [diagonal_apply_eq]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\n\u22a2 SeminormedAddCommGroup (m \u2192 PiLp 1 fun j => \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NormedAddCommGroup \u03b1\n\u22a2 NormedAddCommGroup (m \u2192 PiLp 1 fun j => \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : NormedField R\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : NormedSpace R \u03b1\n\u22a2 NormedSpace R (m \u2192 PiLp 1 fun j => \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016 = \u2191(Finset.sup Finset.univ fun i => \u2211 j : n, \u2016A i j\u2016\u208a)\n[PROOFSTEP]\nchange \u2016fun i => (PiLp.equiv 1 _).symm (A i)\u2016 = _\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nA : Matrix m n \u03b1\n\u22a2 \u2016fun i => \u2191(PiLp.equiv 1 fun i => \u03b1).symm (A i)\u2016 = \u2191(Finset.sup Finset.univ fun i => \u2211 j : n, \u2016A i j\u2016\u208a)\n[PROOFSTEP]\nsimp [Pi.norm_def, PiLp.nnnorm_eq_sum ENNReal.one_ne_top]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nv : m \u2192 \u03b1\n\u22a2 \u2016col v\u2016\u208a = \u2016v\u2016\u208a\n[PROOFSTEP]\nrw [linfty_op_nnnorm_def, Pi.nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nv : m \u2192 \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 j : Unit, \u2016col v i j\u2016\u208a) = Finset.sup Finset.univ fun b => \u2016v b\u2016\u208a\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nv : n \u2192 \u03b1\n\u22a2 \u2016row v\u2016\u208a = \u2211 i : n, \u2016v i\u2016\u208a\n[PROOFSTEP]\nsimp [linfty_op_nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : SeminormedAddCommGroup \u03b1\nv : n \u2192 \u03b1\n\u22a2 \u2191(\u2211 i : n, \u2016v i\u2016\u208a) = \u2211 i : n, \u2016v i\u2016\n[PROOFSTEP]\nsimp [NNReal.coe_sum]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : DecidableEq m\nv : m \u2192 \u03b1\n\u22a2 \u2016diagonal v\u2016\u208a = \u2016v\u2016\u208a\n[PROOFSTEP]\nrw [linfty_op_nnnorm_def, Pi.nnnorm_def]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : DecidableEq m\nv : m \u2192 \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016diagonal v i j\u2016\u208a) = Finset.sup Finset.univ fun b => \u2016v b\u2016\u208a\n[PROOFSTEP]\ncongr 1 with i : 1\n[GOAL]\ncase e_f.h\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : DecidableEq m\nv : m \u2192 \u03b1\ni : m\n\u22a2 \u2211 j : m, \u2016diagonal v i j\u2016\u208a = \u2016v i\u2016\u208a\n[PROOFSTEP]\nrefine' (Finset.sum_eq_single_of_mem _ (Finset.mem_univ i) fun j _hj hij => _).trans _\n[GOAL]\ncase e_f.h.refine'_1\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : DecidableEq m\nv : m \u2192 \u03b1\ni j : m\n_hj : j \u2208 Finset.univ\nhij : j \u2260 i\n\u22a2 \u2016diagonal v i j\u2016\u208a = 0\n[PROOFSTEP]\nrw [diagonal_apply_ne' _ hij, nnnorm_zero]\n[GOAL]\ncase e_f.h.refine'_2\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : DecidableEq m\nv : m \u2192 \u03b1\ni : m\n\u22a2 \u2016diagonal v i i\u2016\u208a = \u2016v i\u2016\u208a\n[PROOFSTEP]\nrw [diagonal_apply_eq]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 \u2016A * B\u2016\u208a \u2264 \u2016A\u2016\u208a * \u2016B\u2016\u208a\n[PROOFSTEP]\nsimp_rw [linfty_op_nnnorm_def, Matrix.mul_apply]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 x : n, \u2016\u2211 j : m, A i j * B j x\u2016\u208a) \u2264\n    (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a) * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\ncalc\n  (Finset.univ.sup fun i => \u2211 k, \u2016\u2211 j, A i j * B j k\u2016\u208a) \u2264 Finset.univ.sup fun i => \u2211 k, \u2211 j, \u2016A i j\u2016\u208a * \u2016B j k\u2016\u208a :=\n    Finset.sup_mono_fun fun i _hi => Finset.sum_le_sum fun k _hk => nnnorm_sum_le_of_le _ fun j _hj => nnnorm_mul_le _ _\n  _ = Finset.univ.sup fun i => \u2211 j, \u2016A i j\u2016\u208a * \u2211 k, \u2016B j k\u2016\u208a := by simp_rw [@Finset.sum_comm _ m n, Finset.mul_sum]\n  _ \u2264 Finset.univ.sup fun i => \u2211 j, \u2016A i j\u2016\u208a * Finset.univ.sup fun i => \u2211 j, \u2016B i j\u2016\u208a :=\n    by\n    refine Finset.sup_mono_fun fun i _hi => ?_\n    gcongr with j hj\n    exact Finset.le_sup (f := fun i \u21a6 \u2211 k : n, \u2016B i k\u2016\u208a) hj\n  _ \u2264 (Finset.univ.sup fun i => \u2211 j, \u2016A i j\u2016\u208a) * Finset.univ.sup fun i => \u2211 j, \u2016B i j\u2016\u208a :=\n    by\n    simp_rw [\u2190 Finset.sum_mul, \u2190 NNReal.finset_sup_mul]\n    rfl\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 k : n, \u2211 j : m, \u2016A i j\u2016\u208a * \u2016B j k\u2016\u208a) =\n    Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a * \u2211 k : n, \u2016B j k\u2016\u208a\n[PROOFSTEP]\nsimp_rw [@Finset.sum_comm _ m n, Finset.mul_sum]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a * \u2211 k : n, \u2016B j k\u2016\u208a) \u2264\n    Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\nrefine Finset.sup_mono_fun fun i _hi => ?_\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\n_hi : i \u2208 Finset.univ\n\u22a2 \u2211 j : m, \u2016A i j\u2016\u208a * \u2211 k : n, \u2016B j k\u2016\u208a \u2264 \u2211 j : m, \u2016A i j\u2016\u208a * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\ngcongr with j hj\n[GOAL]\ncase h.bc\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\n_hi : i \u2208 Finset.univ\nj : m\nhj : j \u2208 Finset.univ\n\u22a2 \u2211 k : n, \u2016B j k\u2016\u208a \u2264 Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\nexact Finset.le_sup (f := fun i \u21a6 \u2211 k : n, \u2016B i k\u2016\u208a) hj\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a) \u2264\n    (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a) * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\nsimp_rw [\u2190 Finset.sum_mul, \u2190 NNReal.finset_sup_mul]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 ((Finset.sup Finset.univ fun a => \u2211 x : m, \u2016A a x\u2016\u208a) * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a) \u2264\n    (Finset.sup Finset.univ fun i => \u2211 j : m, \u2016A i j\u2016\u208a) * Finset.sup Finset.univ fun i => \u2211 j : n, \u2016B i j\u2016\u208a\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nv : m \u2192 \u03b1\n\u22a2 \u2016mulVec A v\u2016\u208a \u2264 \u2016A\u2016\u208a * \u2016v\u2016\u208a\n[PROOFSTEP]\nrw [\u2190 linfty_op_nnnorm_col (A.mulVec v), \u2190 linfty_op_nnnorm_col v]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NonUnitalSeminormedRing \u03b1\nA : Matrix l m \u03b1\nv : m \u2192 \u03b1\n\u22a2 \u2016col (mulVec A v)\u2016\u208a \u2264 \u2016A\u2016\u208a * \u2016col v\u2016\u208a\n[PROOFSTEP]\nexact linfty_op_nnnorm_mul A (col v)\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : NormedAddCommGroup \u03b1\n\u22a2 NormedAddCommGroup (PiLp 2 fun i => PiLp 2 fun j => \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : NormedField R\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : NormedSpace R \u03b1\n\u22a2 NormedSpace R (PiLp 2 fun i => PiLp 2 fun j => \u03b1)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u2016\u208a = (\u2211 i : m, \u2211 j : n, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nchange \u2016(PiLp.equiv 2 _).symm <| fun i => (PiLp.equiv 2 _).symm <| fun j => A i j\u2016\u208a = _\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\n\u22a2 \u2016\u2191(PiLp.equiv 2 fun i => (fun x => PiLp 2 fun j => \u03b1) fun j => A i j).symm fun i =>\n        \u2191(PiLp.equiv 2 fun j => \u03b1).symm fun j => A i j\u2016\u208a =\n    (\u2211 i : m, \u2211 j : n, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp_rw [PiLp.nnnorm_eq_of_L2, NNReal.sq_sqrt, NNReal.sqrt_eq_rpow, NNReal.rpow_two, PiLp.equiv_symm_apply]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\n\u22a2 \u2191((\u2211 i : m, \u2211 j : n, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2)) = (\u2211 i : m, \u2211 j : n, \u2016A i j\u2016 ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp [NNReal.coe_sum]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), \u2016f a\u2016\u208a = \u2016a\u2016\u208a\n\u22a2 \u2016map A f\u2016\u208a = \u2016A\u2016\u208a\n[PROOFSTEP]\nsimp_rw [frobenius_nnnorm_def, Matrix.map_apply, hf]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\n\u22a2 \u2016A\u1d40\u2016\u208a = \u2016A\u2016\u208a\n[PROOFSTEP]\nrw [frobenius_nnnorm_def, frobenius_nnnorm_def, Finset.sum_comm]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nA : Matrix m n \u03b1\n\u22a2 (\u2211 y : m, \u2211 x : n, \u2016A\u1d40 x y\u2016\u208a ^ 2) ^ (1 / 2) = (\u2211 i : m, \u2211 j : n, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp_rw [Matrix.transpose_apply]\n  -- porting note: added\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : m \u2192 \u03b1\n\u22a2 \u2016row v\u2016 = \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v\u2016\n[PROOFSTEP]\nrw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : m \u2192 \u03b1\n\u22a2 (\u2211 j : m, \u2016row v default j\u2016 ^ 2) ^ (1 / 2) = (\u2211 i : m, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016 ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp only [row_apply, Real.rpow_two, PiLp.equiv_symm_apply]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : n \u2192 \u03b1\n\u22a2 \u2016col v\u2016 = \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v\u2016\n[PROOFSTEP]\nsimp_rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2074 : Fintype l\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nv : n \u2192 \u03b1\n\u22a2 (\u2211 x : n, \u2016col v x default\u2016 ^ 2) ^ (1 / 2) = (\u2211 i : n, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016 ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp only [col_apply, Real.rpow_two, PiLp.equiv_symm_apply]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 \u2016diagonal v\u2016\u208a = \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v\u2016\u208a\n[PROOFSTEP]\nsimp_rw [frobenius_nnnorm_def, \u2190 Finset.sum_product', Finset.univ_product_univ, PiLp.nnnorm_eq_of_L2]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.fst x.snd\u2016\u208a ^ 2) ^ (1 / 2) =\n    \u2191NNReal.sqrt (\u2211 i : n, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016\u208a ^ 2)\n[PROOFSTEP]\nlet s := (Finset.univ : Finset n).map \u27e8fun i : n => (i, i), fun i j h => congr_arg Prod.fst h\u27e9\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\n\u22a2 (\u2211 x : n \u00d7 n, \u2016diagonal v x.fst x.snd\u2016\u208a ^ 2) ^ (1 / 2) =\n    \u2191NNReal.sqrt (\u2211 i : n, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016\u208a ^ 2)\n[PROOFSTEP]\nrw [\u2190 Finset.sum_subset (Finset.subset_univ s) fun i _hi his => ?_]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\n\u22a2 (\u2211 x in s, \u2016diagonal v x.fst x.snd\u2016\u208a ^ 2) ^ (1 / 2) =\n    \u2191NNReal.sqrt (\u2211 i : n, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016\u208a ^ 2)\n[PROOFSTEP]\nrw [Finset.sum_map, NNReal.sqrt_eq_rpow]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\n\u22a2 (\u2211 x : n,\n        \u2016diagonal v\n              (\u2191{ toFun := fun i => (i, i),\n                      inj' :=\n                        (_ :\n                          \u2200 (i j : n),\n                            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192\n                              ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n                  x).fst\n              (\u2191{ toFun := fun i => (i, i),\n                      inj' :=\n                        (_ :\n                          \u2200 (i j : n),\n                            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192\n                              ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n                  x).snd\u2016\u208a ^\n          2) ^\n      (1 / 2) =\n    (\u2211 i : n, \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm v i\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\n\u22a2 (\u2211 x : n, \u2016diagonal v x x\u2016\u208a ^ 2) ^ (1 / 2) = (\u2211 i : n, \u2016v i\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimp_rw [diagonal_apply_eq, NNReal.rpow_two]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : \u00aci \u2208 s\n\u22a2 \u2016diagonal v i.fst i.snd\u2016\u208a ^ 2 = 0\n[PROOFSTEP]\nsuffices i.1 \u2260 i.2 by rw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : \u00aci \u2208 s\nthis : i.fst \u2260 i.snd\n\u22a2 \u2016diagonal v i.fst i.snd\u2016\u208a ^ 2 = 0\n[PROOFSTEP]\nrw [diagonal_apply_ne _ this, nnnorm_zero, NNReal.zero_rpow two_ne_zero]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : \u00aci \u2208 s\n\u22a2 i.fst \u2260 i.snd\n[PROOFSTEP]\nintro h\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : SeminormedAddCommGroup \u03b1\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b2\ninst\u271d : DecidableEq n\nv : n \u2192 \u03b1\ns : Finset (n \u00d7 n) :=\n  Finset.map\n    { toFun := fun i => (i, i),\n      inj' :=\n        (_ :\n          \u2200 (i j : n),\n            (fun i => (i, i)) i = (fun i => (i, i)) j \u2192 ((fun i => (i, i)) i).fst = ((fun i => (i, i)) j).fst) }\n    Finset.univ\ni : n \u00d7 n\n_hi : i \u2208 Finset.univ\nhis : \u00aci \u2208 s\nh : i.fst = i.snd\n\u22a2 False\n[PROOFSTEP]\nexact Finset.mem_map.not.mp his \u27e8i.1, Finset.mem_univ _, Prod.ext rfl h\u27e9\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : One \u03b1\n\u22a2 \u20161\u2016\u208a = \u2191NNReal.sqrt \u2191(Fintype.card n) * \u20161\u2016\u208a\n[PROOFSTEP]\nrefine'\n  (frobenius_nnnorm_diagonal _).trans\n    _\n      -- porting note: change to erw, since `fun x => 1` no longer matches `Function.const`\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : One \u03b1\n\u22a2 \u2016\u2191(PiLp.equiv 2 fun i => \u03b1).symm fun x => 1\u2016\u208a = \u2191NNReal.sqrt \u2191(Fintype.card n) * \u20161\u2016\u208a\n[PROOFSTEP]\nerw [PiLp.nnnorm_equiv_symm_const ENNReal.two_ne_top]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : One \u03b1\n\u22a2 \u2191(Fintype.card n) ^ ENNReal.toReal (1 / 2) * \u20161\u2016\u208a = \u2191NNReal.sqrt \u2191(Fintype.card n) * \u20161\u2016\u208a\n[PROOFSTEP]\nsimp_rw [NNReal.sqrt_eq_rpow]\n  -- porting note: added `ENNReal.toReal_ofNat`\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u2075 : Fintype l\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : One \u03b1\n\u22a2 \u2191(Fintype.card n) ^ ENNReal.toReal (1 / 2) * \u20161\u2016\u208a = \u2191(Fintype.card n) ^ (1 / 2) * \u20161\u2016\u208a\n[PROOFSTEP]\nsimp only [ENNReal.toReal_div, ENNReal.one_toReal, ENNReal.toReal_ofNat]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 \u2016A * B\u2016\u208a \u2264 \u2016A\u2016\u208a * \u2016B\u2016\u208a\n[PROOFSTEP]\nsimp_rw [frobenius_nnnorm_def, Matrix.mul_apply]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (\u2211 x : l, \u2211 x_1 : n, \u2016\u2211 j : m, A x j * B j x_1\u2016\u208a ^ 2) ^ (1 / 2) \u2264\n    (\u2211 i : l, \u2211 j : m, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2) * (\u2211 i : m, \u2211 j : n, \u2016B i j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nrw [\u2190 NNReal.mul_rpow, @Finset.sum_comm _ n m, Finset.sum_mul_sum, Finset.sum_product]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 (\u2211 x : l, \u2211 x_1 : n, \u2016\u2211 j : m, A x j * B j x_1\u2016\u208a ^ 2) ^ (1 / 2) \u2264\n    (\u2211 x : l, \u2211 y : n, (\u2211 j : m, \u2016A (x, y).fst j\u2016\u208a ^ 2) * \u2211 x_1 : m, \u2016B x_1 (x, y).snd\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nrefine' NNReal.rpow_le_rpow _ one_half_pos.le\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\n\u22a2 \u2211 x : l, \u2211 x_1 : n, \u2016\u2211 j : m, A x j * B j x_1\u2016\u208a ^ 2 \u2264\n    \u2211 x : l, \u2211 y : n, (\u2211 j : m, \u2016A (x, y).fst j\u2016\u208a ^ 2) * \u2211 x_1 : m, \u2016B x_1 (x, y).snd\u2016\u208a ^ 2\n[PROOFSTEP]\nrefine' Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => _\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\nx\u271d\u00b9 : i \u2208 Finset.univ\nj : n\nx\u271d : j \u2208 Finset.univ\n\u22a2 \u2016\u2211 j_1 : m, A i j_1 * B j_1 j\u2016\u208a ^ 2 \u2264 (\u2211 j_1 : m, \u2016A (i, j).fst j_1\u2016\u208a ^ 2) * \u2211 x : m, \u2016B x (i, j).snd\u2016\u208a ^ 2\n[PROOFSTEP]\nrw [\u2190 NNReal.rpow_le_rpow_iff one_half_pos, \u2190 NNReal.rpow_mul, mul_div_cancel' (1 : \u211d) two_ne_zero, NNReal.rpow_one,\n  NNReal.mul_rpow]\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\nx\u271d\u00b9 : i \u2208 Finset.univ\nj : n\nx\u271d : j \u2208 Finset.univ\n\u22a2 \u2016\u2211 j_1 : m, A i j_1 * B j_1 j\u2016\u208a \u2264\n    (\u2211 j_1 : m, \u2016A (i, j).fst j_1\u2016\u208a ^ 2) ^ (1 / 2) * (\u2211 x : m, \u2016B x (i, j).snd\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\nx\u271d\u00b9 : i \u2208 Finset.univ\nj : n\nx\u271d : j \u2208 Finset.univ\n\u22a2 \u2016\u2211 j_1 : m, A i j_1 * B j_1 j\u2016\u208a \u2264 (\u2211 j : m, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2) * (\u2211 x : m, \u2016B x j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nhave :=\n  @nnnorm_inner_le_nnnorm \u03b1 _ _ _ _ ((PiLp.equiv 2 fun _ => \u03b1).symm fun j => star (A i j))\n    ((PiLp.equiv 2 fun _ => \u03b1).symm fun k => B k j)\n[GOAL]\nR : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\n\u03b1 : Type u_5\n\u03b2 : Type u_6\ninst\u271d\u00b3 : Fintype l\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : IsROrC \u03b1\nA : Matrix l m \u03b1\nB : Matrix m n \u03b1\ni : l\nx\u271d\u00b9 : i \u2208 Finset.univ\nj : n\nx\u271d : j \u2208 Finset.univ\nthis :\n  \u2016inner (\u2191(PiLp.equiv 2 fun x => \u03b1).symm fun j => star (A i j)) (\u2191(PiLp.equiv 2 fun x => \u03b1).symm fun k => B k j)\u2016\u208a \u2264\n    \u2016\u2191(PiLp.equiv 2 fun x => \u03b1).symm fun j => star (A i j)\u2016\u208a * \u2016\u2191(PiLp.equiv 2 fun x => \u03b1).symm fun k => B k j\u2016\u208a\n\u22a2 \u2016\u2211 j_1 : m, A i j_1 * B j_1 j\u2016\u208a \u2264 (\u2211 j : m, \u2016A i j\u2016\u208a ^ 2) ^ (1 / 2) * (\u2211 x : m, \u2016B x j\u2016\u208a ^ 2) ^ (1 / 2)\n[PROOFSTEP]\nsimpa only [PiLp.equiv_symm_apply, PiLp.inner_apply, IsROrC.inner_apply, starRingEnd_apply, Pi.nnnorm_def,\n  PiLp.nnnorm_eq_of_L2, star_star, nnnorm_star, NNReal.sqrt_eq_rpow, NNReal.rpow_two] using this\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Matrix", "llama_tokens": 16718, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.6825737279551493, "lm_q1q2_score": 0.5430953155378183}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2191F \u2286 S \u2192 \u22c2\u2080 \u2191F \u2208 S\n[PROOFSTEP]\nclassical\nrefine' Finset.induction_on F (fun _ => _) _\n\u00b7 simp [cond.univ_mem]\n\u00b7 intro a s _ h1 h2\n  suffices a \u2229 \u22c2\u2080 \u2191s \u2208 S by simpa\n  exact cond.inter_mem (h2 (Finset.mem_insert_self a s)) (h1 fun x hx => h2 <| Finset.mem_insert_of_mem hx)\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2191F \u2286 S \u2192 \u22c2\u2080 \u2191F \u2208 S\n[PROOFSTEP]\nrefine' Finset.induction_on F (fun _ => _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\nx\u271d : \u2191\u2205 \u2286 S\n\u22a2 \u22c2\u2080 \u2191\u2205 \u2208 S\n[PROOFSTEP]\nsimp [cond.univ_mem]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\n\u22a2 \u2200 \u2983a : Set \u03b1\u2984 {s : Finset (Set \u03b1)}, \u00aca \u2208 s \u2192 (\u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S) \u2192 \u2191(insert a s) \u2286 S \u2192 \u22c2\u2080 \u2191(insert a s) \u2208 S\n[PROOFSTEP]\nintro a s _ h1 h2\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : \u00aca \u2208 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 \u22c2\u2080 \u2191(insert a s) \u2208 S\n[PROOFSTEP]\nsuffices a \u2229 \u22c2\u2080 \u2191s \u2208 S by simpa\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : \u00aca \u2208 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\nthis : a \u2229 \u22c2\u2080 \u2191s \u2208 S\n\u22a2 \u22c2\u2080 \u2191(insert a s) \u2208 S\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\ncond : FiniteInter S\nF : Finset (Set \u03b1)\na : Set \u03b1\ns : Finset (Set \u03b1)\na\u271d : \u00aca \u2208 s\nh1 : \u2191s \u2286 S \u2192 \u22c2\u2080 \u2191s \u2208 S\nh2 : \u2191(insert a s) \u2286 S\n\u22a2 a \u2229 \u22c2\u2080 \u2191s \u2208 S\n[PROOFSTEP]\nexact cond.inter_mem (h2 (Finset.mem_insert_self a s)) (h1 fun x hx => h2 <| Finset.mem_insert_of_mem hx)\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP : Set \u03b1\nH : P \u2208 finiteInterClosure (insert A S)\n\u22a2 P \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 P = A \u2229 Q\n[PROOFSTEP]\ninduction' H with S h T1 T2 _ _ h1 h2\n[GOAL]\ncase basic\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh : S \u2208 insert A S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q, Q \u2208 S\u271d \u2227 S = A \u2229 Q\n[PROOFSTEP]\ncases h\n[GOAL]\ncase basic.inl\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S = A\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q, Q \u2208 S\u271d \u2227 S = A \u2229 Q\n[PROOFSTEP]\nexact Or.inr \u27e8Set.univ, cond.univ_mem, by simpa\u27e9\n[GOAL]\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S = A\n\u22a2 S = A \u2229 Set.univ\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase basic.inr\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S \u2208 S\u271d\n\u22a2 S \u2208 S\u271d \u2228 \u2203 Q, Q \u2208 S\u271d \u2227 S = A \u2229 Q\n[PROOFSTEP]\nexact Or.inl (by assumption)\n[GOAL]\n\u03b1 : Type u_1\nS\u271d : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\u271d\nP S : Set \u03b1\nh\u271d : S \u2208 S\u271d\n\u22a2 S \u2208 S\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\ncase univ\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP : Set \u03b1\n\u22a2 Set.univ \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 Set.univ = A \u2229 Q\n[PROOFSTEP]\nexact Or.inl cond.univ_mem\n[GOAL]\ncase inter\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh1 : T1 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T1 = A \u2229 Q\nh2 : T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T2 = A \u2229 Q\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T1 \u2229 T2 = A \u2229 Q\n[PROOFSTEP]\nrcases h1 with (h | \u27e8Q, hQ, rfl\u27e9)\n[GOAL]\ncase inter.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh2 : T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T2 = A \u2229 Q\nh : T1 \u2208 S\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T1 \u2229 T2 = A \u2229 Q\n[PROOFSTEP]\nrcases h2 with (i | \u27e8R, hR, rfl\u27e9)\n[GOAL]\ncase inter.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nh2 : T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T2 = A \u2229 Q\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\n\u22a2 A \u2229 Q \u2229 T2 \u2208 S \u2228 \u2203 Q_1, Q_1 \u2208 S \u2227 A \u2229 Q \u2229 T2 = A \u2229 Q_1\n[PROOFSTEP]\nrcases h2 with (i | \u27e8R, hR, rfl\u27e9)\n[GOAL]\ncase inter.inl.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\na\u271d : finiteInterClosure (insert A S) T2\nh : T1 \u2208 S\ni : T2 \u2208 S\n\u22a2 T1 \u2229 T2 \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T1 \u2229 T2 = A \u2229 Q\n[PROOFSTEP]\nexact Or.inl (cond.inter_mem h i)\n[GOAL]\ncase inter.inl.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\nh : T1 \u2208 S\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 T1 \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q, Q \u2208 S \u2227 T1 \u2229 (A \u2229 R) = A \u2229 Q\n[PROOFSTEP]\nexact Or.inr \u27e8T1 \u2229 R, cond.inter_mem h hR, by simp only [\u2190 Set.inter_assoc, Set.inter_comm _ A]\u27e9\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T1 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T1\nh : T1 \u2208 S\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 T1 \u2229 (A \u2229 R) = A \u2229 (T1 \u2229 R)\n[PROOFSTEP]\nsimp only [\u2190 Set.inter_assoc, Set.inter_comm _ A]\n[GOAL]\ncase inter.inr.intro.intro.inl\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\ni : T2 \u2208 S\n\u22a2 A \u2229 Q \u2229 T2 \u2208 S \u2228 \u2203 Q_1, Q_1 \u2208 S \u2227 A \u2229 Q \u2229 T2 = A \u2229 Q_1\n[PROOFSTEP]\nexact Or.inr \u27e8Q \u2229 T2, cond.inter_mem hQ i, by simp only [Set.inter_assoc]\u27e9\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP T2 : Set \u03b1\na\u271d\u00b9 : finiteInterClosure (insert A S) T2\nQ : Set \u03b1\nhQ : Q \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 Q)\ni : T2 \u2208 S\n\u22a2 A \u2229 Q \u2229 T2 = A \u2229 (Q \u2229 T2)\n[PROOFSTEP]\nsimp only [Set.inter_assoc]\n[GOAL]\ncase inter.inr.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 A \u2229 Q \u2229 (A \u2229 R) \u2208 S \u2228 \u2203 Q_1, Q_1 \u2208 S \u2227 A \u2229 Q \u2229 (A \u2229 R) = A \u2229 Q_1\n[PROOFSTEP]\nexact\n  Or.inr\n    \u27e8Q \u2229 R, cond.inter_mem hQ hR, by\n      ext x\n      constructor <;> simp (config := { contextual := true })\u27e9\n[GOAL]\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\n\u22a2 A \u2229 Q \u2229 (A \u2229 R) = A \u2229 (Q \u2229 R)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\nx : \u03b1\n\u22a2 x \u2208 A \u2229 Q \u2229 (A \u2229 R) \u2194 x \u2208 A \u2229 (Q \u2229 R)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\nx : \u03b1\n\u22a2 x \u2208 A \u2229 Q \u2229 (A \u2229 R) \u2192 x \u2208 A \u2229 (Q \u2229 R)\n[PROOFSTEP]\nsimp (config := { contextual := true })\n[GOAL]\ncase h.mpr\n\u03b1 : Type u_1\nS : Set (Set \u03b1)\nA : Set \u03b1\ncond : FiniteInter S\nP Q : Set \u03b1\nhQ : Q \u2208 S\na\u271d\u00b9 : finiteInterClosure (insert A S) (A \u2229 Q)\nR : Set \u03b1\nhR : R \u2208 S\na\u271d : finiteInterClosure (insert A S) (A \u2229 R)\nx : \u03b1\n\u22a2 x \u2208 A \u2229 (Q \u2229 R) \u2192 x \u2208 A \u2229 Q \u2229 (A \u2229 R)\n[PROOFSTEP]\nsimp (config := { contextual := true })\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Constructions", "llama_tokens": 3892, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580903722561, "lm_q2_score": 0.6825737279551493, "lm_q1q2_score": 0.5430953089230659}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\n\u22a2 g \u2022 s = s\n[PROOFSTEP]\nsuffices \u2200 {g' : G} (_ : g' ^ n ^ j = 1), g' \u2022 s \u2286 s\n  by\n  refine' le_antisymm (this hg) _\n  conv_lhs => rw [\u2190 smul_inv_smul g s]\n  replace hg : g\u207b\u00b9 ^ n ^ j = 1\n  \u00b7 rw [inv_zpow, hg, inv_one]\n  simpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n\u22a2 g \u2022 s = s\n[PROOFSTEP]\nrefine' le_antisymm (this hg) _\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n\u22a2 s \u2264 g \u2022 s\n[PROOFSTEP]\nconv_lhs => rw [\u2190 smul_inv_smul g s]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n| s\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g s]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n| s\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g s]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n| s\n[PROOFSTEP]\nrw [\u2190 smul_inv_smul g s]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n\u22a2 g \u2022 g\u207b\u00b9 \u2022 s \u2264 g \u2022 s\n[PROOFSTEP]\nreplace hg : g\u207b\u00b9 ^ n ^ j = 1\n[GOAL]\ncase hg\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n\u22a2 g\u207b\u00b9 ^ n ^ j = 1\n[PROOFSTEP]\nrw [inv_zpow, hg, inv_one]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nthis : \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\nhg : g\u207b\u00b9 ^ n ^ j = 1\n\u22a2 g \u2022 g\u207b\u00b9 \u2022 s \u2264 g \u2022 s\n[PROOFSTEP]\nsimpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\n\u22a2 \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 s \u2286 s\n[PROOFSTEP]\nrw [(IsFixedPt.preimage_iterate hs j : (zpowGroupHom n)^[j] \u207b\u00b9' s = s).symm]\n[GOAL]\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\n\u22a2 \u2200 {g' : G}, g' ^ n ^ j = 1 \u2192 g' \u2022 (fun x => x ^ n)^[j] \u207b\u00b9' s \u2286 (fun x => x ^ n)^[j] \u207b\u00b9' s\n[PROOFSTEP]\nrintro g' hg' - \u27e8y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y \u2208 (fun x => x ^ n)^[j] \u207b\u00b9' s\n\u22a2 (fun x => g' \u2022 x) y \u2208 (fun x => x ^ n)^[j] \u207b\u00b9' s\n[PROOFSTEP]\nchange (zpowGroupHom n)^[j] (g' * y) \u2208 s\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y \u2208 (fun x => x ^ n)^[j] \u207b\u00b9' s\n\u22a2 (\u2191(zpowGroupHom n))^[j] (g' * y) \u2208 s\n[PROOFSTEP]\nreplace hg' : (zpowGroupHom n)^[j] g' = 1\n[GOAL]\ncase hg'\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\ng' : G\nhg' : g' ^ n ^ j = 1\ny : G\nhy : y \u2208 (fun x => x ^ n)^[j] \u207b\u00b9' s\n\u22a2 (\u2191(zpowGroupHom n))^[j] g' = 1\n[PROOFSTEP]\nsimpa [zpowGroupHom]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d : CommGroup G\nn : \u2124\ns : Set G\nhs : (fun x => x ^ n) \u207b\u00b9' s = s\ng : G\nj : \u2115\nhg : g ^ n ^ j = 1\ng' y : G\nhy : y \u2208 (fun x => x ^ n)^[j] \u207b\u00b9' s\nhg' : (\u2191(zpowGroupHom n))^[j] g' = 1\n\u22a2 (\u2191(zpowGroupHom n))^[j] (g' * y) \u2208 s\n[PROOFSTEP]\nrwa [iterate_map_mul, hg', one_mul]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Pointwise.Iterate", "llama_tokens": 2242, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.7057850340255387, "lm_q1q2_score": 0.5428053212893332}}
{"text": "[GOAL]\nb : \u2115\nh : 2 \u2264 b\nn : \u2115\n\u22a2 (invImage (fun a => sizeOf a) instWellFoundedRelation).1 ((n + 1) / b) (succ n)\n[PROOFSTEP]\nexact Nat.div_lt_self (Nat.succ_pos _) h\n[GOAL]\nn b : \u2115\nh : 2 \u2264 b\n\u22a2 digitsAux b h 0 = []\n[PROOFSTEP]\nrw [digitsAux]\n[GOAL]\nn\u271d b : \u2115\nh : 2 \u2264 b\nn : \u2115\nw : 0 < n\n\u22a2 digitsAux b h n = n % b :: digitsAux b h (n / b)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nn b : \u2115\nh : 2 \u2264 b\nw : 0 < zero\n\u22a2 digitsAux b h zero = zero % b :: digitsAux b h (zero / b)\n[PROOFSTEP]\ncases w\n[GOAL]\ncase succ\nn b : \u2115\nh : 2 \u2264 b\nn\u271d : \u2115\nw : 0 < succ n\u271d\n\u22a2 digitsAux b h (succ n\u271d) = succ n\u271d % b :: digitsAux b h (succ n\u271d / b)\n[PROOFSTEP]\nrw [digitsAux]\n[GOAL]\nn b : \u2115\n\u22a2 2 \u2264 b + 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn b : \u2115\n\u22a2 digits b 0 = []\n[PROOFSTEP]\nrcases b with (_ | \u27e8_ | \u27e8_\u27e9\u27e9)\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 digits zero 0 = []\n[PROOFSTEP]\nsimp [digits, digitsAux0, digitsAux1]\n[GOAL]\ncase succ.zero\nn : \u2115\n\u22a2 digits (succ zero) 0 = []\n[PROOFSTEP]\nsimp [digits, digitsAux0, digitsAux1]\n[GOAL]\ncase succ.succ\nn n\u271d : \u2115\n\u22a2 digits (succ (succ n\u271d)) 0 = []\n[PROOFSTEP]\nsimp [digits, digitsAux0, digitsAux1]\n[GOAL]\nn\u271d b n : \u2115\n\u22a2 digits (b + 2) (n + 1) = (n + 1) % (b + 2) :: digits (b + 2) ((n + 1) / (b + 2))\n[PROOFSTEP]\nsimp [digits, digitsAux_def]\n[GOAL]\nn n\u271d : \u2115\nh : 1 < 0\n\u22a2 \u00ac1 < 0\n[PROOFSTEP]\ndecide\n[GOAL]\nn n\u271d : \u2115\nh : 1 < 1\n\u22a2 \u00ac1 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\nn n\u271d b : \u2115\nx\u271d : 1 < b + 2\n\u22a2 2 \u2264 Nat.add b 0 + 2\n[PROOFSTEP]\nsimp\n[GOAL]\nn b x : \u2115\nhx : x \u2260 0\nhxb : x < b\n\u22a2 digits b x = [x]\n[PROOFSTEP]\nrcases exists_eq_succ_of_ne_zero hx with \u27e8x, rfl\u27e9\n[GOAL]\ncase intro\nn b x : \u2115\nhx : succ x \u2260 0\nhxb : succ x < b\n\u22a2 digits b (succ x) = [succ x]\n[PROOFSTEP]\nrcases exists_eq_add_of_le' ((Nat.le_add_left 1 x).trans_lt hxb) with \u27e8b, rfl\u27e9\n[GOAL]\ncase intro.intro\nn x : \u2115\nhx : succ x \u2260 0\nb : \u2115\nhxb : succ x < b + succ 1\n\u22a2 digits (b + succ 1) (succ x) = [succ x]\n[PROOFSTEP]\nrw [digits_add_two_add_one, div_eq_of_lt hxb, digits_zero, mod_eq_of_lt hxb]\n[GOAL]\nn b : \u2115\nh : 1 < b\nx y : \u2115\nhxb : x < b\nhxy : x \u2260 0 \u2228 y \u2260 0\n\u22a2 digits b (x + b * y) = x :: digits b y\n[PROOFSTEP]\nrcases exists_eq_add_of_le' h with \u27e8b, rfl : _ = _ + 2\u27e9\n[GOAL]\ncase intro\nn x y : \u2115\nhxy : x \u2260 0 \u2228 y \u2260 0\nb : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\n\u22a2 digits (b + 2) (x + (b + 2) * y) = x :: digits (b + 2) y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase intro.zero\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nhxy : x \u2260 0 \u2228 zero \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * zero) = x :: digits (b + 2) zero\n[PROOFSTEP]\nsimp [hxb, hxy.resolve_right (absurd rfl)]\n[GOAL]\ncase intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digits (b + 2) (x + (b + 2) * succ n\u271d) = x :: digits (b + 2) (succ n\u271d)\n[PROOFSTEP]\ndsimp [digits]\n[GOAL]\ncase intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 digitsAux (b + 2) (_ : 2 \u2264 b + 2) (x + (b + 2) * succ n\u271d) = x :: digitsAux (b + 2) (_ : 2 \u2264 b + 2) (succ n\u271d)\n[PROOFSTEP]\nrw [digitsAux_def]\n[GOAL]\ncase intro.succ\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 2) :: digitsAux (b + 2) (_ : 2 \u2264 b + 2) ((x + (b + 2) * succ n\u271d) / (b + 2)) =\n    x :: digitsAux (b + 2) (_ : 2 \u2264 b + 2) (succ n\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.succ.e_head\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) % (b + 2) = x\n[PROOFSTEP]\nsimp [Nat.add_mod, mod_eq_of_lt hxb]\n[GOAL]\ncase intro.succ.e_tail.e_a\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 (x + (b + 2) * succ n\u271d) / (b + 2) = succ n\u271d\n[PROOFSTEP]\nsimp [add_mul_div_left, div_eq_of_lt hxb]\n[GOAL]\ncase intro.succ.w\nn x b : \u2115\nh : 1 < b + 2\nhxb : x < b + 2\nn\u271d : \u2115\nhxy : x \u2260 0 \u2228 succ n\u271d \u2260 0\n\u22a2 0 < x + (b + 2) * succ n\u271d\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u03b1\nL : List \u2115\n\u22a2 ofDigits b L = List.foldr (fun x y => \u2191x + b * y) 0 L\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u03b1\n\u22a2 ofDigits b [] = List.foldr (fun x y => \u2191x + b * y) 0 []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u03b1\nd : \u2115\nL : List \u2115\nih : ofDigits b L = List.foldr (fun x y => \u2191x + b * y) 0 L\n\u22a2 ofDigits b (d :: L) = List.foldr (fun x y => \u2191x + b * y) 0 (d :: L)\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u03b1\nd : \u2115\nL : List \u2115\nih : ofDigits b L = List.foldr (fun x y => \u2191x + b * y) 0 L\n\u22a2 \u2191d + b * ofDigits b L = \u2191d + b * List.foldr (fun x y => \u2191x + b * y) 0 L\n[PROOFSTEP]\nrw [ih]\n[GOAL]\nn b : \u2115\nl : List \u2115\n\u22a2 List.sum (List.zipWith ((fun i a => a * b ^ i) \u2218 succ) (List.range (List.length l)) l) =\n    b * List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length l)) l)\n[PROOFSTEP]\nsuffices\n  (List.range l.length).zipWith ((fun i a : \u2115 => a * b ^ i) \u2218 succ) l =\n    (List.range l.length).zipWith (fun i a => b * (a * b ^ i)) l\n  by simp [this]\n[GOAL]\nn b : \u2115\nl : List \u2115\nthis :\n  List.zipWith ((fun i a => a * b ^ i) \u2218 succ) (List.range (List.length l)) l =\n    List.zipWith (fun i a => b * (a * b ^ i)) (List.range (List.length l)) l\n\u22a2 List.sum (List.zipWith ((fun i a => a * b ^ i) \u2218 succ) (List.range (List.length l)) l) =\n    b * List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length l)) l)\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nn b : \u2115\nl : List \u2115\n\u22a2 List.zipWith ((fun i a => a * b ^ i) \u2218 succ) (List.range (List.length l)) l =\n    List.zipWith (fun i a => b * (a * b ^ i)) (List.range (List.length l)) l\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nn b : \u2115\nl : List \u2115\n\u22a2 (fun i a => a * b ^ i) \u2218 succ = fun i a => b * (a * b ^ i)\n[PROOFSTEP]\next\n[GOAL]\ncase e_f.h.h\nn b : \u2115\nl : List \u2115\nx\u271d\u00b9 x\u271d : \u2115\n\u22a2 ((fun i a => a * b ^ i) \u2218 succ) x\u271d\u00b9 x\u271d = b * (x\u271d * b ^ x\u271d\u00b9)\n[PROOFSTEP]\nsimp [pow_succ]\n[GOAL]\ncase e_f.h.h\nn b : \u2115\nl : List \u2115\nx\u271d\u00b9 x\u271d : \u2115\n\u22a2 x\u271d * (b ^ x\u271d\u00b9 * b) = b * (x\u271d * b ^ x\u271d\u00b9)\n[PROOFSTEP]\nring\n[GOAL]\nn b : \u2115\nL : List \u2115\n\u22a2 ofDigits b L = List.sum (List.mapIdx (fun i a => a * b ^ i) L)\n[PROOFSTEP]\nrw [List.mapIdx_eq_enum_map, List.enum_eq_zip_range, List.map_uncurry_zip_eq_zipWith, ofDigits_eq_foldr]\n[GOAL]\nn b : \u2115\nL : List \u2115\n\u22a2 List.foldr (fun x y => \u2191x + b * y) 0 L = List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length L)) L)\n[PROOFSTEP]\ninduction' L with hd tl hl\n[GOAL]\ncase nil\nn b : \u2115\n\u22a2 List.foldr (fun x y => \u2191x + b * y) 0 [] =\n    List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length [])) [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn b hd : \u2115\ntl : List \u2115\nhl :\n  List.foldr (fun x y => \u2191x + b * y) 0 tl =\n    List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length tl)) tl)\n\u22a2 List.foldr (fun x y => \u2191x + b * y) 0 (hd :: tl) =\n    List.sum (List.zipWith (fun i a => a * b ^ i) (List.range (List.length (hd :: tl))) (hd :: tl))\n[PROOFSTEP]\nsimpa [List.range_succ_eq_map, List.zipWith_map_left, ofDigits_eq_sum_map_with_index_aux] using Or.inl hl\n[GOAL]\nn\u271d b n : \u2115\n\u22a2 ofDigits b [n] = n\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : \u2115\nL : List \u2115\n\u22a2 ofDigits 1 (h :: L) = \u2191h + ofDigits 1 L\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\nn b : \u2115\nl1 l2 : List \u2115\n\u22a2 ofDigits b (l1 ++ l2) = ofDigits b l1 + b ^ List.length l1 * ofDigits b l2\n[PROOFSTEP]\ninduction' l1 with hd tl IH\n[GOAL]\ncase nil\nn b : \u2115\nl2 : List \u2115\n\u22a2 ofDigits b ([] ++ l2) = ofDigits b [] + b ^ List.length [] * ofDigits b l2\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase cons\nn b : \u2115\nl2 : List \u2115\nhd : \u2115\ntl : List \u2115\nIH : ofDigits b (tl ++ l2) = ofDigits b tl + b ^ List.length tl * ofDigits b l2\n\u22a2 ofDigits b (hd :: tl ++ l2) = ofDigits b (hd :: tl) + b ^ List.length (hd :: tl) * ofDigits b l2\n[PROOFSTEP]\nrw [ofDigits, List.cons_append, ofDigits, IH, List.length_cons, pow_succ']\n[GOAL]\ncase cons\nn b : \u2115\nl2 : List \u2115\nhd : \u2115\ntl : List \u2115\nIH : ofDigits b (tl ++ l2) = ofDigits b tl + b ^ List.length tl * ofDigits b l2\n\u22a2 \u2191hd + b * (ofDigits b tl + b ^ List.length tl * ofDigits b l2) =\n    \u2191hd + b * ofDigits b tl + b * b ^ List.length tl * ofDigits b l2\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u2115\nL : List \u2115\n\u22a2 \u2191(ofDigits b L) = ofDigits (\u2191b) L\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb : \u2115\n\u22a2 \u2191(ofDigits b []) = ofDigits \u2191b []\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb d : \u2115\nL : List \u2115\nih : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(ofDigits b (d :: L)) = ofDigits (\u2191b) (d :: L)\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb d : \u2115\nL : List \u2115\nih : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(d + b * ofDigits b L) = \u2191d + \u2191b * ofDigits (\u2191b) L\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nb d : \u2115\nL : List \u2115\nih : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191d + \u2191b * \u2191(ofDigits b L) = \u2191d + \u2191b * ofDigits (\u2191b) L\n[PROOFSTEP]\nrw [ih]\n[GOAL]\nn b : \u2115\nL : List \u2115\n\u22a2 \u2191(ofDigits b L) = ofDigits (\u2191b) L\n[PROOFSTEP]\ninduction' L with d L _\n[GOAL]\ncase nil\nn b : \u2115\n\u22a2 \u2191(ofDigits b []) = ofDigits \u2191b []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(ofDigits b (d :: L)) = ofDigits (\u2191b) (d :: L)\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191(d + b * ofDigits b L) = \u2191d + \u2191b * ofDigits (\u2191b) L\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase cons\nn b d : \u2115\nL : List \u2115\ntail_ih\u271d : \u2191(ofDigits b L) = ofDigits (\u2191b) L\n\u22a2 \u2191d + \u2191b * ofDigits (\u2191b) L = \u2191d + \u2191b * ofDigits (\u2191b) L\n[PROOFSTEP]\nsimp only\n[GOAL]\nn b : \u2115\nh : 1 < b\nL : List \u2115\nw\u2081 : \u2200 (l : \u2115), l \u2208 L \u2192 l < b\nw\u2082 : \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\n\u22a2 digits b (ofDigits b L) = L\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn b : \u2115\nh : 1 < b\nL : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < b\nw\u2082\u271d : \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\nw\u2081 : \u2200 (l : \u2115), l \u2208 [] \u2192 l < b\nw\u2082 : \u2200 (h : [] \u2260 []), List.getLast [] h \u2260 0\n\u22a2 digits b (ofDigits b []) = []\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase nil\nn b : \u2115\nh : 1 < b\nL : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < b\nw\u2082\u271d : \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\nw\u2081 : \u2200 (l : \u2115), l \u2208 [] \u2192 l < b\nw\u2082 : \u2200 (h : [] \u2260 []), List.getLast [] h \u2260 0\n\u22a2 digits b 0 = []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : \u2200 (h : d :: L \u2260 []), List.getLast (d :: L) h \u2260 0\n\u22a2 digits b (ofDigits b (d :: L)) = d :: L\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : \u2200 (h : d :: L \u2260 []), List.getLast (d :: L) h \u2260 0\n\u22a2 digits b (d + b * ofDigits b L) = d :: L\n[PROOFSTEP]\nreplace w\u2082 := w\u2082 (by simp)\n[GOAL]\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : \u2200 (h : d :: L \u2260 []), List.getLast (d :: L) h \u2260 0\n\u22a2 d :: L \u2260 []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 digits b (d + b * ofDigits b L) = d :: L\n[PROOFSTEP]\nrw [digits_add b h]\n[GOAL]\ncase cons\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 d :: digits b (ofDigits b L) = d :: L\n[PROOFSTEP]\nrw [ih]\n[GOAL]\ncase cons.w\u2081\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 \u2200 (l : \u2115), l \u2208 L \u2192 l < b\n[PROOFSTEP]\nintro l m\n[GOAL]\ncase cons.w\u2081\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nl : \u2115\nm : l \u2208 L\n\u22a2 l < b\n[PROOFSTEP]\napply w\u2081\n[GOAL]\ncase cons.w\u2081.a\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nl : \u2115\nm : l \u2208 L\n\u22a2 l \u2208 d :: L\n[PROOFSTEP]\nexact List.mem_cons_of_mem _ m\n[GOAL]\ncase cons.w\u2082\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons.w\u2082\nn b : \u2115\nh\u271d : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nh : L \u2260 []\n\u22a2 List.getLast L h \u2260 0\n[PROOFSTEP]\nrw [List.getLast_cons h] at w\u2082 \n[GOAL]\ncase cons.w\u2082\nn b : \u2115\nh\u271d : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nh : L \u2260 []\nw\u2082 : List.getLast L h \u2260 0\n\u22a2 List.getLast L h \u2260 0\n[PROOFSTEP]\nconvert w\u2082\n[GOAL]\ncase cons.hxb\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 d < b\n[PROOFSTEP]\nexact w\u2081 d (List.mem_cons_self _ _)\n[GOAL]\ncase cons.hxy\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\n\u22a2 d \u2260 0 \u2228 ofDigits b L \u2260 0\n[PROOFSTEP]\nby_cases h' : L = []\n[GOAL]\ncase pos\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nh' : L = []\n\u22a2 d \u2260 0 \u2228 ofDigits b L \u2260 0\n[PROOFSTEP]\nrcases h' with rfl\n[GOAL]\ncase pos\nn b : \u2115\nh : 1 < b\nL : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < b\nw\u2082\u271d : \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\nd : \u2115\nih : (\u2200 (l : \u2115), l \u2208 [] \u2192 l < b) \u2192 (\u2200 (h : [] \u2260 []), List.getLast [] h \u2260 0) \u2192 digits b (ofDigits b []) = []\nw\u2081 : \u2200 (l : \u2115), l \u2208 [d] \u2192 l < b\nw\u2082 : List.getLast [d] (_ : \u00ac[d] = []) \u2260 0\n\u22a2 d \u2260 0 \u2228 ofDigits b [] \u2260 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\nn b : \u2115\nh : 1 < b\nL : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < b\nw\u2082\u271d : \u2200 (h : L \u2260 []), List.getLast L h \u2260 0\nd : \u2115\nih : (\u2200 (l : \u2115), l \u2208 [] \u2192 l < b) \u2192 (\u2200 (h : [] \u2260 []), List.getLast [] h \u2260 0) \u2192 digits b (ofDigits b []) = []\nw\u2081 : \u2200 (l : \u2115), l \u2208 [d] \u2192 l < b\nw\u2082 : List.getLast [d] (_ : \u00ac[d] = []) \u2260 0\n\u22a2 d \u2260 0\n[PROOFSTEP]\nsimpa using w\u2082\n[GOAL]\ncase neg\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nh' : \u00acL = []\n\u22a2 d \u2260 0 \u2228 ofDigits b L \u2260 0\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nw\u2082 : List.getLast (d :: L) (_ : \u00acd :: L = []) \u2260 0\nh' : \u00acL = []\n\u22a2 ofDigits b L \u2260 0\n[PROOFSTEP]\ncontrapose! w\u2082\n[GOAL]\ncase neg.h\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nh' : \u00acL = []\nw\u2082 : ofDigits b L = 0\n\u22a2 List.getLast (d :: L) (_ : \u00acd :: L = []) = 0\n[PROOFSTEP]\nrefine' digits_zero_of_eq_zero h.ne_bot w\u2082 _ _\n[GOAL]\ncase neg.h\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nh' : \u00acL = []\nw\u2082 : ofDigits b L = 0\n\u22a2 List.getLast (d :: L) (_ : \u00acd :: L = []) \u2208 L\n[PROOFSTEP]\nrw [List.getLast_cons h']\n[GOAL]\ncase neg.h\nn b : \u2115\nh : 1 < b\nL\u271d : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 L\u271d \u2192 l < b\nw\u2082\u271d : \u2200 (h : L\u271d \u2260 []), List.getLast L\u271d h \u2260 0\nd : \u2115\nL : List \u2115\nih : (\u2200 (l : \u2115), l \u2208 L \u2192 l < b) \u2192 (\u2200 (h : L \u2260 []), List.getLast L h \u2260 0) \u2192 digits b (ofDigits b L) = L\nw\u2081 : \u2200 (l : \u2115), l \u2208 d :: L \u2192 l < b\nh' : \u00acL = []\nw\u2082 : ofDigits b L = 0\n\u22a2 List.getLast L h' \u2208 L\n[PROOFSTEP]\nexact List.getLast_mem h'\n[GOAL]\nn\u271d b n : \u2115\n\u22a2 ofDigits b (digits b n) = n\n[PROOFSTEP]\ncases' b with b\n[GOAL]\ncase zero\nn\u271d n : \u2115\n\u22a2 ofDigits zero (digits zero n) = n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero.zero\nn : \u2115\n\u22a2 ofDigits zero (digits zero zero) = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero.succ\nn\u271d n : \u2115\n\u22a2 ofDigits zero (digits zero (succ n)) = succ n\n[PROOFSTEP]\nchange ofDigits 0 [n + 1] = n + 1\n[GOAL]\ncase zero.succ\nn\u271d n : \u2115\n\u22a2 ofDigits 0 [n + 1] = n + 1\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase succ\nn\u271d n b : \u2115\n\u22a2 ofDigits (succ b) (digits (succ b) n) = n\n[PROOFSTEP]\ncases' b with b\n[GOAL]\ncase succ.zero\nn\u271d n : \u2115\n\u22a2 ofDigits (succ zero) (digits (succ zero) n) = n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase succ.zero.zero\nn : \u2115\n\u22a2 ofDigits (succ zero) (digits (succ zero) zero) = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.zero.succ\nn\u271d n : \u2115\nih : ofDigits (succ zero) (digits (succ zero) n) = n\n\u22a2 ofDigits (succ zero) (digits (succ zero) (succ n)) = succ n\n[PROOFSTEP]\nrw [show succ zero = 1 by rfl] at ih \u22a2\n[GOAL]\nn\u271d n : \u2115\nih : ofDigits (succ zero) (digits (succ zero) n) = n\n\u22a2 succ zero = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nn\u271d n : \u2115\nih : ofDigits 1 (digits 1 n) = n\n\u22a2 succ zero = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.zero.succ\nn\u271d n : \u2115\nih : ofDigits 1 (digits 1 n) = n\n\u22a2 ofDigits 1 (digits 1 (succ n)) = succ n\n[PROOFSTEP]\nsimp only [ih, add_comm 1, ofDigits_one_cons, Nat.cast_id, digits_one_succ]\n[GOAL]\ncase succ.succ\nn\u271d n b : \u2115\n\u22a2 ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n\n[PROOFSTEP]\napply Nat.strongInductionOn n _\n[GOAL]\nn\u271d n b : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115), m < n \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) \u2192\n      ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n\n[PROOFSTEP]\nclear n\n[GOAL]\nn b : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115), m < n \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m) \u2192\n      ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n\n[PROOFSTEP]\nintro n h\n[GOAL]\nn\u271d b n : \u2115\nh : \u2200 (m : \u2115), m < n \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 ofDigits (succ (succ b)) (digits (succ (succ b)) n) = n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nn b : \u2115\nh : \u2200 (m : \u2115), m < zero \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 ofDigits (succ (succ b)) (digits (succ (succ b)) zero) = zero\n[PROOFSTEP]\nrw [digits_zero]\n[GOAL]\ncase zero\nn b : \u2115\nh : \u2200 (m : \u2115), m < zero \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 ofDigits (succ (succ b)) [] = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn b n\u271d : \u2115\nh : \u2200 (m : \u2115), m < succ n\u271d \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 ofDigits (succ (succ b)) (digits (succ (succ b)) (succ n\u271d)) = succ n\u271d\n[PROOFSTEP]\nsimp only [Nat.succ_eq_add_one, digits_add_two_add_one]\n[GOAL]\ncase succ\nn b n\u271d : \u2115\nh : \u2200 (m : \u2115), m < succ n\u271d \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 ofDigits (b + 1 + 1) ((n\u271d + 1) % (b + 2) :: digits (b + 2) ((n\u271d + 1) / (b + 2))) = n\u271d + 1\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase succ\nn b n\u271d : \u2115\nh : \u2200 (m : \u2115), m < succ n\u271d \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 (n\u271d + 1) % (b + 2) + (b + 1 + 1) * ofDigits (b + 1 + 1) (digits (b + 2) ((n\u271d + 1) / (b + 2))) = n\u271d + 1\n[PROOFSTEP]\nrw [h _ (Nat.div_lt_self' _ b)]\n[GOAL]\ncase succ\nn b n\u271d : \u2115\nh : \u2200 (m : \u2115), m < succ n\u271d \u2192 ofDigits (succ (succ b)) (digits (succ (succ b)) m) = m\n\u22a2 (n\u271d + 1) % (b + 2) + (b + 1 + 1) * ((n\u271d + 1) / (b + 2)) = n\u271d + 1\n[PROOFSTEP]\nrw [Nat.mod_add_div]\n[GOAL]\nn : \u2115\nL : List \u2115\n\u22a2 ofDigits 1 L = List.sum L\n[PROOFSTEP]\ninduction' L with _ _ ih\n[GOAL]\ncase nil\nn : \u2115\n\u22a2 ofDigits 1 [] = List.sum []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn head\u271d : \u2115\ntail\u271d : List \u2115\nih : ofDigits 1 tail\u271d = List.sum tail\u271d\n\u22a2 ofDigits 1 (head\u271d :: tail\u271d) = List.sum (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [ofDigits, List.sum_cons, ih]\n[GOAL]\nn\u271d b n : \u2115\n\u22a2 digits b n = [] \u2194 n = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn\u271d b n : \u2115\n\u22a2 digits b n = [] \u2192 n = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nn\u271d b n : \u2115\nh : digits b n = []\n\u22a2 n = 0\n[PROOFSTEP]\nhave : ofDigits b (digits b n) = ofDigits b [] := by rw [h]\n[GOAL]\nn\u271d b n : \u2115\nh : digits b n = []\n\u22a2 ofDigits b (digits b n) = ofDigits b []\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mp\nn\u271d b n : \u2115\nh : digits b n = []\nthis : ofDigits b (digits b n) = ofDigits b []\n\u22a2 n = 0\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2\nn\u271d b n : \u2115\nh : digits b n = []\nthis : ofDigits b (digits b n) = ofDigits b []\n\u22a2 n = ofDigits b (digits b n)\n[PROOFSTEP]\nrw [ofDigits_digits]\n[GOAL]\ncase mpr\nn\u271d b n : \u2115\n\u22a2 n = 0 \u2192 digits b n = []\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nn b : \u2115\n\u22a2 digits b 0 = []\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u271d b n : \u2115\nh : 1 < b\nw : n \u2260 0\n\u22a2 digits b n = n % b :: digits b (n / b)\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn\u271d n : \u2115\nw : n \u2260 0\nh : 1 < zero\n\u22a2 digits zero n = n % zero :: digits zero (n / zero)\n[PROOFSTEP]\nrw [digits_zero_succ' w, Nat.mod_zero, Nat.div_zero, Nat.digits_zero_zero]\n[GOAL]\ncase succ.zero\nn\u271d n : \u2115\nw : n \u2260 0\nh : 1 < succ zero\n\u22a2 digits (succ zero) n = n % succ zero :: digits (succ zero) (n / succ zero)\n[PROOFSTEP]\nnorm_num at h \n[GOAL]\ncase succ.succ\nn\u271d n : \u2115\nw : n \u2260 0\nb : \u2115\nh : 1 < succ (succ b)\n\u22a2 digits (succ (succ b)) n = n % succ (succ b) :: digits (succ (succ b)) (n / succ (succ b))\n[PROOFSTEP]\nrcases n with (_ | n)\n[GOAL]\ncase succ.succ.zero\nn b : \u2115\nh : 1 < succ (succ b)\nw : zero \u2260 0\n\u22a2 digits (succ (succ b)) zero = zero % succ (succ b) :: digits (succ (succ b)) (zero / succ (succ b))\n[PROOFSTEP]\nnorm_num at w \n[GOAL]\ncase succ.succ.succ\nn\u271d b : \u2115\nh : 1 < succ (succ b)\nn : \u2115\nw : succ n \u2260 0\n\u22a2 digits (succ (succ b)) (succ n) = succ n % succ (succ b) :: digits (succ (succ b)) (succ n / succ (succ b))\n[PROOFSTEP]\nsimp only [digits_add_two_add_one, ne_eq]\n[GOAL]\nn b m : \u2115\nh : 1 < b\np : digits b m \u2260 []\nq : digits b (m / b) \u2260 []\n\u22a2 List.getLast (digits b m) p = List.getLast (digits b (m / b)) q\n[PROOFSTEP]\nby_cases hm : m = 0\n[GOAL]\ncase pos\nn b m : \u2115\nh : 1 < b\np : digits b m \u2260 []\nq : digits b (m / b) \u2260 []\nhm : m = 0\n\u22a2 List.getLast (digits b m) p = List.getLast (digits b (m / b)) q\n[PROOFSTEP]\nsimp [hm]\n[GOAL]\ncase neg\nn b m : \u2115\nh : 1 < b\np : digits b m \u2260 []\nq : digits b (m / b) \u2260 []\nhm : \u00acm = 0\n\u22a2 List.getLast (digits b m) p = List.getLast (digits b (m / b)) q\n[PROOFSTEP]\nsimp only [digits_eq_cons_digits_div h hm]\n[GOAL]\ncase neg\nn b m : \u2115\nh : 1 < b\np : digits b m \u2260 []\nq : digits b (m / b) \u2260 []\nhm : \u00acm = 0\n\u22a2 List.getLast (m % b :: digits b (m / b)) (_ : m % b :: digits b (m / b) \u2260 []) = List.getLast (digits b (m / b)) q\n[PROOFSTEP]\nrw [List.getLast_cons]\n[GOAL]\nn\u271d b n : \u2115\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 List.length (digits b n) = log b n + 1\n[PROOFSTEP]\ninduction' n using Nat.strong_induction_on with n IH\n[GOAL]\ncase h\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\n\u22a2 List.length (digits b n) = log b n + 1\n[PROOFSTEP]\nrw [digits_eq_cons_digits_div hb hn, List.length]\n[GOAL]\ncase h\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\n\u22a2 List.length (digits b (n / b)) + 1 = log b n + 1\n[PROOFSTEP]\nby_cases h : n / b = 0\n[GOAL]\ncase pos\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : n / b = 0\n\u22a2 List.length (digits b (n / b)) + 1 = log b n + 1\n[PROOFSTEP]\nhave hb0 : b \u2260 0 := (Nat.succ_le_iff.1 hb).ne_bot\n[GOAL]\ncase pos\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : n / b = 0\nhb0 : b \u2260 0\n\u22a2 List.length (digits b (n / b)) + 1 = log b n + 1\n[PROOFSTEP]\nsimp [h, log_eq_zero_iff, \u2190 Nat.div_eq_zero_iff hb0.bot_lt]\n[GOAL]\ncase neg\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : \u00acn / b = 0\n\u22a2 List.length (digits b (n / b)) + 1 = log b n + 1\n[PROOFSTEP]\nhave : n / b < n := div_lt_self (Nat.pos_of_ne_zero hn) hb\n[GOAL]\ncase neg\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : \u00acn / b = 0\nthis : n / b < n\n\u22a2 List.length (digits b (n / b)) + 1 = log b n + 1\n[PROOFSTEP]\nrw [IH _ this h, log_div_base, tsub_add_cancel_of_le]\n[GOAL]\ncase neg\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : \u00acn / b = 0\nthis : n / b < n\n\u22a2 1 \u2264 log b n\n[PROOFSTEP]\nrefine' Nat.succ_le_of_lt (log_pos hb _)\n[GOAL]\ncase neg\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nh : \u00acn / b = 0\nthis : n / b < n\n\u22a2 b \u2264 n\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase neg\nn\u271d\u00b9 b n\u271d : \u2115\nhb : 1 < b\nhn\u271d : n\u271d \u2260 0\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 m \u2260 0 \u2192 List.length (digits b m) = log b m + 1\nhn : n \u2260 0\nthis : n / b < n\nh : n < b\n\u22a2 n / b = 0\n[PROOFSTEP]\nexact div_eq_of_lt h\n[GOAL]\nn b m : \u2115\nhm : m \u2260 0\n\u22a2 List.getLast (digits b m) (_ : digits b m \u2260 []) \u2260 0\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn m : \u2115\nhm : m \u2260 0\n\u22a2 List.getLast (digits zero m) (_ : digits zero m \u2260 []) \u2260 0\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero.zero\nn : \u2115\nhm : zero \u2260 0\n\u22a2 List.getLast (digits zero zero) (_ : digits zero zero \u2260 []) \u2260 0\n[PROOFSTEP]\ncases hm rfl\n[GOAL]\ncase zero.succ\nn n\u271d : \u2115\nhm : succ n\u271d \u2260 0\n\u22a2 List.getLast (digits zero (succ n\u271d)) (_ : digits zero (succ n\u271d) \u2260 []) \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.zero\nn m : \u2115\nhm : m \u2260 0\n\u22a2 List.getLast (digits (succ zero) m) (_ : digits (succ zero) m \u2260 []) \u2260 0\n[PROOFSTEP]\ncases m\n[GOAL]\ncase succ.zero.zero\nn : \u2115\nhm : zero \u2260 0\n\u22a2 List.getLast (digits (succ zero) zero) (_ : digits (succ zero) zero \u2260 []) \u2260 0\n[PROOFSTEP]\ncases hm rfl\n[GOAL]\ncase succ.zero.succ\nn n\u271d : \u2115\nhm : succ n\u271d \u2260 0\n\u22a2 List.getLast (digits (succ zero) (succ n\u271d)) (_ : digits (succ zero) (succ n\u271d) \u2260 []) \u2260 0\n[PROOFSTEP]\nrename \u2115 => m\n[GOAL]\ncase succ.zero.succ\nn m : \u2115\nhm : succ m \u2260 0\n\u22a2 List.getLast (digits (succ zero) (succ m)) (_ : digits (succ zero) (succ m) \u2260 []) \u2260 0\n[PROOFSTEP]\nsimp only [digits_one, List.getLast_replicate_succ m 1]\n[GOAL]\ncase succ.succ\nn m : \u2115\nhm : m \u2260 0\nb : \u2115\n\u22a2 List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\n[PROOFSTEP]\nrevert hm\n[GOAL]\ncase succ.succ\nn m b : \u2115\n\u22a2 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\n[PROOFSTEP]\napply Nat.strongInductionOn m\n[GOAL]\ncase succ.succ\nn m b : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115),\n        m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0) \u2192\n      \u2200 (hm : n \u2260 0), List.getLast (digits (succ (succ b)) n) (_ : digits (succ (succ b)) n \u2260 []) \u2260 0\n[PROOFSTEP]\nintro n IH hn\n[GOAL]\ncase succ.succ\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\n\u22a2 List.getLast (digits (succ (succ b)) n) (_ : digits (succ (succ b)) n \u2260 []) \u2260 0\n[PROOFSTEP]\nby_cases hnb : n < b + 2\n[GOAL]\ncase pos\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : n < b + 2\n\u22a2 List.getLast (digits (succ (succ b)) n) (_ : digits (succ (succ b)) n \u2260 []) \u2260 0\n[PROOFSTEP]\nsimpa only [digits_of_lt (b + 2) n hn hnb]\n[GOAL]\ncase neg\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : \u00acn < b + 2\n\u22a2 List.getLast (digits (succ (succ b)) n) (_ : digits (succ (succ b)) n \u2260 []) \u2260 0\n[PROOFSTEP]\nrw [digits_getLast n (le_add_left 2 b)]\n[GOAL]\ncase neg\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : \u00acn < b + 2\n\u22a2 List.getLast (digits (b + 2) (n / (b + 2))) ?neg.q\u271d \u2260 0\ncase neg.q\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : \u00acn < b + 2\n\u22a2 digits (b + 2) (n / (b + 2)) \u2260 []\n[PROOFSTEP]\nrefine' IH _ (Nat.div_lt_self hn.bot_lt (one_lt_succ_succ b)) _\n[GOAL]\ncase neg\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : \u00acn < b + 2\n\u22a2 n / succ (succ b) \u2260 0\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero]\n[GOAL]\ncase neg\nn\u271d m b n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 (hm : m \u2260 0), List.getLast (digits (succ (succ b)) m) (_ : digits (succ (succ b)) m \u2260 []) \u2260 0\nhn : n \u2260 0\nhnb : \u00acn < b + 2\n\u22a2 0 < n / succ (succ b)\n[PROOFSTEP]\nexact Nat.div_pos (le_of_not_lt hnb) (zero_lt_succ (succ b))\n[GOAL]\nn b m : \u2115\n\u22a2 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\n[PROOFSTEP]\napply Nat.strongInductionOn m\n[GOAL]\nn b m : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115), m < n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2) \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) n \u2192 d < b + 2\n[PROOFSTEP]\nintro n IH d hd\n[GOAL]\nn\u271d b m n : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\nd : \u2115\nhd : d \u2208 digits (b + 2) n\n\u22a2 d < b + 2\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nn b m d : \u2115\nIH : \u2200 (m : \u2115), m < zero \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\nhd : d \u2208 digits (b + 2) zero\n\u22a2 d < b + 2\n[PROOFSTEP]\nrw [digits_zero] at hd \n[GOAL]\ncase zero\nn b m d : \u2115\nIH : \u2200 (m : \u2115), m < zero \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\nhd : d \u2208 []\n\u22a2 d < b + 2\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase succ\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\nhd : d \u2208 digits (b + 2) (succ n)\n\u22a2 d < b + 2\n[PROOFSTEP]\nrw [digits_add_two_add_one] at hd \n[GOAL]\ncase succ\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\nhd : d \u2208 (n + 1) % (b + 2) :: digits (b + 2) ((n + 1) / (b + 2))\n\u22a2 d < b + 2\n[PROOFSTEP]\ncases hd\n[GOAL]\ncase succ.head\nn\u271d b m n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\n\u22a2 (n + 1) % (b + 2) < b + 2\n[PROOFSTEP]\nexact\n  n.succ.mod_lt\n    (by simp)\n      -- Porting note: Previous code (single line) contained linarith.\n        -- . exact IH _ (Nat.div_lt_self (Nat.succ_pos _) (by linarith)) hd\n[GOAL]\nn\u271d b m n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\n\u22a2 b + 2 > 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.tail\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\na\u271d : List.Mem d (digits (b + 2) ((n + 1) / (b + 2)))\n\u22a2 d < b + 2\n[PROOFSTEP]\napply IH ((n + 1) / (b + 2))\n[GOAL]\ncase succ.tail.a\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\na\u271d : List.Mem d (digits (b + 2) ((n + 1) / (b + 2)))\n\u22a2 (n + 1) / (b + 2) < succ n\n[PROOFSTEP]\napply Nat.div_lt_self\n[GOAL]\ncase succ.tail.a.hLtN\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\na\u271d : List.Mem d (digits (b + 2) ((n + 1) / (b + 2)))\n\u22a2 0 < n + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.tail.a.hLtK\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\na\u271d : List.Mem d (digits (b + 2) ((n + 1) / (b + 2)))\n\u22a2 1 < b + 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.tail.a\nn\u271d b m d n : \u2115\nIH : \u2200 (m : \u2115), m < succ n \u2192 \u2200 {d : \u2115}, d \u2208 digits (b + 2) m \u2192 d < b + 2\na\u271d : List.Mem d (digits (b + 2) ((n + 1) / (b + 2)))\n\u22a2 d \u2208 digits (b + 2) ((n + 1) / (b + 2))\n[PROOFSTEP]\nassumption\n[GOAL]\nn b m d : \u2115\nhb : 1 < b\nhd : d \u2208 digits b m\n\u22a2 d < b\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn m d : \u2115\nhb : 1 < zero\nhd : d \u2208 digits zero m\n\u22a2 d < zero\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase zero\nn m d : \u2115\nhb : 1 < zero\nhd : d \u2208 digits zero m\n\u22a2 d < zero\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.zero\nn m d : \u2115\nhb : 1 < succ zero\nhd : d \u2208 digits (succ zero) m\n\u22a2 d < succ zero\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.zero\nn m d : \u2115\nhb : 1 < succ zero\nhd : d \u2208 digits (succ zero) m\n\u22a2 d < succ zero\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m d b : \u2115\nhb : 1 < succ (succ b)\nhd : d \u2208 digits (succ (succ b)) m\n\u22a2 d < succ (succ b)\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.succ\nn m d b : \u2115\nhb : 1 < succ (succ b)\nhd : d \u2208 digits (succ (succ b)) m\n\u22a2 d < succ (succ b)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m d b : \u2115\nhd : d \u2208 digits (succ (succ b)) m\n\u22a2 d < succ (succ b)\n[PROOFSTEP]\nexact digits_lt_base' hd\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\n\u22a2 ofDigits (b + 2) l < (b + 2) ^ List.length l\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhl : \u2200 (x : \u2115), x \u2208 [] \u2192 x < b + 2\n\u22a2 ofDigits (b + 2) [] < (b + 2) ^ List.length []\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase cons\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\n\u22a2 ofDigits (b + 2) (hd :: tl) < (b + 2) ^ List.length (hd :: tl)\n[PROOFSTEP]\nrw [ofDigits, List.length_cons, pow_succ]\n[GOAL]\ncase cons\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\n\u22a2 \u2191hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)\n[PROOFSTEP]\nhave : (ofDigits (b + 2) tl + 1) * (b + 2) \u2264 (b + 2) ^ tl.length * (b + 2) :=\n  mul_le_mul (IH fun x hx => hl _ (List.mem_cons_of_mem _ hx)) (by rfl) (by simp only [zero_le]) (Nat.zero_le _)\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\n\u22a2 b + 2 \u2264 b + 2\n[PROOFSTEP]\nrfl\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\n\u22a2 0 \u2264 b + 2\n[PROOFSTEP]\nsimp only [zero_le]\n[GOAL]\ncase cons\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) \u2264 (b + 2) ^ List.length tl * (b + 2)\n\u22a2 \u2191hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)\n[PROOFSTEP]\nsuffices \u2191hd < b + 2 by linarith\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\nthis\u271d : (ofDigits (b + 2) tl + 1) * (b + 2) \u2264 (b + 2) ^ List.length tl * (b + 2)\nthis : hd < b + 2\n\u22a2 \u2191hd + (b + 2) * ofDigits (b + 2) tl < (b + 2) ^ List.length tl * (b + 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase cons\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) \u2264 (b + 2) ^ List.length tl * (b + 2)\n\u22a2 hd < b + 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase cons\nn b : \u2115\nl : List \u2115\nhl\u271d : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + 2\nhd : \u2115\ntl : List \u2115\nIH : (\u2200 (x : \u2115), x \u2208 tl \u2192 x < b + 2) \u2192 ofDigits (b + 2) tl < (b + 2) ^ List.length tl\nhl : \u2200 (x : \u2115), x \u2208 hd :: tl \u2192 x < b + 2\nthis : (ofDigits (b + 2) tl + 1) * (b + 2) \u2264 (b + 2) ^ List.length tl * (b + 2)\n\u22a2 hd < b + 2\n[PROOFSTEP]\nexact hl hd (List.mem_cons_self _ _)\n[GOAL]\nn b : \u2115\nl : List \u2115\nhb : 1 < b\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < b\n\u22a2 ofDigits b l < b ^ List.length l\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn : \u2115\nl : List \u2115\nhb : 1 < zero\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < zero\n\u22a2 ofDigits zero l < zero ^ List.length l\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase zero\nn : \u2115\nl : List \u2115\nhb : 1 < zero\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < zero\n\u22a2 ofDigits zero l < zero ^ List.length l\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.zero\nn : \u2115\nl : List \u2115\nhb : 1 < succ zero\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < succ zero\n\u22a2 ofDigits (succ zero) l < succ zero ^ List.length l\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.zero\nn : \u2115\nl : List \u2115\nhb : 1 < succ zero\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < succ zero\n\u22a2 ofDigits (succ zero) l < succ zero ^ List.length l\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn : \u2115\nl : List \u2115\nb : \u2115\nhb : 1 < succ (succ b)\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < succ (succ b)\n\u22a2 ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.succ\nn : \u2115\nl : List \u2115\nb : \u2115\nhb : 1 < succ (succ b)\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < succ (succ b)\n\u22a2 ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn : \u2115\nl : List \u2115\nb : \u2115\nhl : \u2200 (x : \u2115), x \u2208 l \u2192 x < succ (succ b)\n\u22a2 ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l\n[PROOFSTEP]\nexact ofDigits_lt_base_pow_length' hl\n[GOAL]\nn b m : \u2115\n\u22a2 m < (b + 2) ^ List.length (digits (b + 2) m)\n[PROOFSTEP]\nconvert @ofDigits_lt_base_pow_length' b (digits (b + 2) m) fun _ => digits_lt_base'\n[GOAL]\ncase h.e'_3\nn b m : \u2115\n\u22a2 m = ofDigits (b + 2) (digits (b + 2) m)\n[PROOFSTEP]\nrw [ofDigits_digits (b + 2) m]\n[GOAL]\nn b m : \u2115\nhb : 1 < b\n\u22a2 m < b ^ List.length (digits b m)\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn m : \u2115\nhb : 1 < zero\n\u22a2 m < zero ^ List.length (digits zero m)\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase zero\nn m : \u2115\nhb : 1 < zero\n\u22a2 m < zero ^ List.length (digits zero m)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.zero\nn m : \u2115\nhb : 1 < succ zero\n\u22a2 m < succ zero ^ List.length (digits (succ zero) m)\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.zero\nn m : \u2115\nhb : 1 < succ zero\n\u22a2 m < succ zero ^ List.length (digits (succ zero) m)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\nhb : 1 < succ (succ b)\n\u22a2 m < succ (succ b) ^ List.length (digits (succ (succ b)) m)\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\nhb : 1 < succ (succ b)\n\u22a2 m < succ (succ b) ^ List.length (digits (succ (succ b)) m)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\n\u22a2 m < succ (succ b) ^ List.length (digits (succ (succ b)) m)\n[PROOFSTEP]\nexact lt_base_pow_length_digits'\n[GOAL]\nn\u271d b m n : \u2115\n\u22a2 ofDigits b (digits b n ++ digits b m) = n + b ^ List.length (digits b n) * m\n[PROOFSTEP]\nrw [ofDigits_append, ofDigits_digits, ofDigits_digits]\n[GOAL]\nn\u271d b n : \u2115\n\u22a2 List.length (digits b n) \u2264 List.length (digits b (n + 1))\n[PROOFSTEP]\nrcases Decidable.eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nn b : \u2115\n\u22a2 List.length (digits b 0) \u2264 List.length (digits b (0 + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn\u271d b n : \u2115\nhn : n \u2260 0\n\u22a2 List.length (digits b n) \u2264 List.length (digits b (n + 1))\n[PROOFSTEP]\ncases' le_or_lt b 1 with hb hb\n[GOAL]\ncase inr.inl\nn\u271d b n : \u2115\nhn : n \u2260 0\nhb : b \u2264 1\n\u22a2 List.length (digits b n) \u2264 List.length (digits b (n + 1))\n[PROOFSTEP]\ninterval_cases b\n[GOAL]\ncase inr.inl.\u00ab0\u00bb\nn\u271d b n : \u2115\nhn : n \u2260 0\nhb : 0 \u2264 1\n\u22a2 List.length (digits 0 n) \u2264 List.length (digits 0 (n + 1))\n[PROOFSTEP]\nsimp_arith [digits_zero_succ', hn]\n[GOAL]\ncase inr.inl.\u00ab1\u00bb\nn\u271d b n : \u2115\nhn : n \u2260 0\nhb : 1 \u2264 1\n\u22a2 List.length (digits 1 n) \u2264 List.length (digits 1 (n + 1))\n[PROOFSTEP]\nsimp_arith [digits_zero_succ', hn]\n[GOAL]\ncase inr.inr\nn\u271d b n : \u2115\nhn : n \u2260 0\nhb : 1 < b\n\u22a2 List.length (digits b n) \u2264 List.length (digits b (n + 1))\n[PROOFSTEP]\nsimpa [digits_len, hb, hn] using log_mono_right (le_succ _)\n[GOAL]\nn p q : \u2115\nL : List \u2115\nh : p \u2264 q\n\u22a2 ofDigits p L \u2264 ofDigits q L\n[PROOFSTEP]\ninduction' L with _ _ hi\n[GOAL]\ncase nil\nn p q : \u2115\nh : p \u2264 q\n\u22a2 ofDigits p [] \u2264 ofDigits q []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn p q : \u2115\nh : p \u2264 q\nhead\u271d : \u2115\ntail\u271d : List \u2115\nhi : ofDigits p tail\u271d \u2264 ofDigits q tail\u271d\n\u22a2 ofDigits p (head\u271d :: tail\u271d) \u2264 ofDigits q (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [ofDigits, cast_id, add_le_add_iff_left]\n[GOAL]\ncase cons\nn p q : \u2115\nh : p \u2264 q\nhead\u271d : \u2115\ntail\u271d : List \u2115\nhi : ofDigits p tail\u271d \u2264 ofDigits q tail\u271d\n\u22a2 p * ofDigits p tail\u271d \u2264 q * ofDigits q tail\u271d\n[PROOFSTEP]\nexact Nat.mul_le_mul h hi\n[GOAL]\nn\u271d p n : \u2115\n\u22a2 List.sum (digits p n) \u2264 n\n[PROOFSTEP]\ninduction' n with n\n[GOAL]\ncase zero\nn p : \u2115\n\u22a2 List.sum (digits p zero) \u2264 zero\n[PROOFSTEP]\nexact digits_zero _ \u25b8 Nat.le_refl (List.sum [])\n[GOAL]\ncase succ\nn\u271d p n : \u2115\nn_ih\u271d : List.sum (digits p n) \u2264 n\n\u22a2 List.sum (digits p (succ n)) \u2264 succ n\n[PROOFSTEP]\ninduction' p with p\n[GOAL]\ncase succ.zero\nn\u271d p n : \u2115\nn_ih\u271d\u00b9 : List.sum (digits p n) \u2264 n\nn_ih\u271d : List.sum (digits zero n) \u2264 n\n\u22a2 List.sum (digits zero (succ n)) \u2264 succ n\n[PROOFSTEP]\nrw [digits_zero_succ, List.sum_cons, List.sum_nil, add_zero]\n[GOAL]\ncase succ.succ\nn\u271d p\u271d n : \u2115\nn_ih\u271d\u00b2 : List.sum (digits p\u271d n) \u2264 n\np : \u2115\nn_ih\u271d\u00b9 : List.sum (digits p n) \u2264 n \u2192 List.sum (digits p (succ n)) \u2264 succ n\nn_ih\u271d : List.sum (digits (succ p) n) \u2264 n\n\u22a2 List.sum (digits (succ p) (succ n)) \u2264 succ n\n[PROOFSTEP]\nnth_rw 2 [\u2190 ofDigits_digits p.succ n.succ]\n[GOAL]\ncase succ.succ\nn\u271d p\u271d n : \u2115\nn_ih\u271d\u00b2 : List.sum (digits p\u271d n) \u2264 n\np : \u2115\nn_ih\u271d\u00b9 : List.sum (digits p n) \u2264 n \u2192 List.sum (digits p (succ n)) \u2264 succ n\nn_ih\u271d : List.sum (digits (succ p) n) \u2264 n\n\u22a2 List.sum (digits (succ p) (succ n)) \u2264 ofDigits (succ p) (digits (succ p) (succ n))\n[PROOFSTEP]\nrw [\u2190 ofDigits_one <| digits p.succ n.succ]\n[GOAL]\ncase succ.succ\nn\u271d p\u271d n : \u2115\nn_ih\u271d\u00b2 : List.sum (digits p\u271d n) \u2264 n\np : \u2115\nn_ih\u271d\u00b9 : List.sum (digits p n) \u2264 n \u2192 List.sum (digits p (succ n)) \u2264 succ n\nn_ih\u271d : List.sum (digits (succ p) n) \u2264 n\n\u22a2 ofDigits 1 (digits (succ p) (succ n)) \u2264 ofDigits (succ p) (digits (succ p) (succ n))\n[PROOFSTEP]\nexact ofDigits_monotone (digits p.succ n.succ) <| Nat.succ_pos p\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 (b + 2) ^ List.length l \u2264 (b + 2) * ofDigits (b + 2) l\n[PROOFSTEP]\nrw [\u2190 List.dropLast_append_getLast hl]\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 (b + 2) ^ List.length (List.dropLast l ++ [List.getLast l hl]) \u2264\n    (b + 2) * ofDigits (b + 2) (List.dropLast l ++ [List.getLast l hl])\n[PROOFSTEP]\nsimp only [List.length_append, List.length, zero_add, List.length_dropLast, ofDigits_append, List.length_dropLast,\n  ofDigits_singleton, add_comm (l.length - 1), pow_add, pow_one]\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 (b + 2) * (b + 2) ^ (List.length l - 1) \u2264\n    (b + 2) * (ofDigits (b + 2) (List.dropLast l) + (b + 2) ^ (List.length l - 1) * List.getLast l hl)\n[PROOFSTEP]\napply Nat.mul_le_mul_left\n[GOAL]\ncase h\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 (b + 2) ^ (List.length l - 1) \u2264 ofDigits (b + 2) (List.dropLast l) + (b + 2) ^ (List.length l - 1) * List.getLast l hl\n[PROOFSTEP]\nrefine' le_trans _ (Nat.le_add_left _ _)\n[GOAL]\ncase h\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 (b + 2) ^ (List.length l - 1) \u2264 (b + 2) ^ (List.length l - 1) * List.getLast l hl\n[PROOFSTEP]\nhave : 0 < l.getLast hl := by rwa [pos_iff_ne_zero]\n[GOAL]\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\n\u22a2 0 < List.getLast l hl\n[PROOFSTEP]\nrwa [pos_iff_ne_zero]\n[GOAL]\ncase h\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\nthis : 0 < List.getLast l hl\n\u22a2 (b + 2) ^ (List.length l - 1) \u2264 (b + 2) ^ (List.length l - 1) * List.getLast l hl\n[PROOFSTEP]\nconvert Nat.mul_le_mul_left ((b + 2) ^ (l.length - 1)) this using 1\n[GOAL]\ncase h.e'_3\nn b : \u2115\nl : List \u2115\nhl : l \u2260 []\nhl2 : List.getLast l hl \u2260 0\nthis : 0 < List.getLast l hl\n\u22a2 (b + 2) ^ (List.length l - 1) = (b + 2) ^ (List.length l - 1) * succ 0\n[PROOFSTEP]\nrw [Nat.mul_one]\n[GOAL]\nn b m : \u2115\nhm : m \u2260 0\n\u22a2 (b + 2) ^ List.length (digits (b + 2) m) \u2264 (b + 2) * m\n[PROOFSTEP]\nhave : digits (b + 2) m \u2260 [] := digits_ne_nil_iff_ne_zero.mpr hm\n[GOAL]\nn b m : \u2115\nhm : m \u2260 0\nthis : digits (b + 2) m \u2260 []\n\u22a2 (b + 2) ^ List.length (digits (b + 2) m) \u2264 (b + 2) * m\n[PROOFSTEP]\nconvert @pow_length_le_mul_ofDigits b (digits (b + 2) m) this (getLast_digit_ne_zero _ hm)\n[GOAL]\ncase h.e'_4.h.e'_6\nn b m : \u2115\nhm : m \u2260 0\nthis : digits (b + 2) m \u2260 []\n\u22a2 m = ofDigits (b + 2) (digits (b + 2) m)\n[PROOFSTEP]\nrw [ofDigits_digits]\n[GOAL]\nn b m : \u2115\nhb : 1 < b\n\u22a2 m \u2260 0 \u2192 b ^ List.length (digits b m) \u2264 b * m\n[PROOFSTEP]\nrcases b with (_ | _ | b)\n[GOAL]\ncase zero\nn m : \u2115\nhb : 1 < zero\n\u22a2 m \u2260 0 \u2192 zero ^ List.length (digits zero m) \u2264 zero * m\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase zero\nn m : \u2115\nhb : 1 < zero\n\u22a2 m \u2260 0 \u2192 zero ^ List.length (digits zero m) \u2264 zero * m\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.zero\nn m : \u2115\nhb : 1 < succ zero\n\u22a2 m \u2260 0 \u2192 succ zero ^ List.length (digits (succ zero) m) \u2264 succ zero * m\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.zero\nn m : \u2115\nhb : 1 < succ zero\n\u22a2 m \u2260 0 \u2192 succ zero ^ List.length (digits (succ zero) m) \u2264 succ zero * m\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\nhb : 1 < succ (succ b)\n\u22a2 m \u2260 0 \u2192 succ (succ b) ^ List.length (digits (succ (succ b)) m) \u2264 succ (succ b) * m\n[PROOFSTEP]\ntry simp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\nhb : 1 < succ (succ b)\n\u22a2 m \u2260 0 \u2192 succ (succ b) ^ List.length (digits (succ (succ b)) m) \u2264 succ (succ b) * m\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase succ.succ\nn m b : \u2115\n\u22a2 \u00acm = 0 \u2192 succ (succ b) ^ List.length (digits (succ (succ b)) m) \u2264 succ (succ b) * m\n[PROOFSTEP]\nexact base_pow_length_digits_le' b m\n[GOAL]\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081 : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\n\u22a2 ofDigits p digits / p = ofDigits p (List.tail digits)\n[PROOFSTEP]\ninduction' digits with hd tl\n[GOAL]\ncase nil\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\nw\u2081 : \u2200 (l : \u2115), l \u2208 [] \u2192 l < p\n\u22a2 ofDigits p [] / p = ofDigits p (List.tail [])\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase cons\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\nhd : \u2115\ntl : List \u2115\ntail_ih\u271d : (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192 ofDigits p tl / p = ofDigits p (List.tail tl)\nw\u2081 : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 ofDigits p (hd :: tl) / p = ofDigits p (List.tail (hd :: tl))\n[PROOFSTEP]\nrefine' Eq.trans (add_mul_div_left hd _ hpos) _\n[GOAL]\ncase cons\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\nhd : \u2115\ntl : List \u2115\ntail_ih\u271d : (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192 ofDigits p tl / p = ofDigits p (List.tail tl)\nw\u2081 : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 hd / p + ofDigits p tl = ofDigits p (List.tail (hd :: tl))\n[PROOFSTEP]\nrw [Nat.div_eq_zero <| w\u2081 _ <| List.mem_cons_self _ _, zero_add]\n[GOAL]\ncase cons\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081\u271d : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\nhd : \u2115\ntl : List \u2115\ntail_ih\u271d : (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192 ofDigits p tl / p = ofDigits p (List.tail tl)\nw\u2081 : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 ofDigits p tl = ofDigits p (List.tail (hd :: tl))\n[PROOFSTEP]\nrfl\n[GOAL]\nn p i : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081 : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\n\u22a2 ofDigits p digits / p ^ i = ofDigits p (List.drop i digits)\n[PROOFSTEP]\ninduction' i with i hi\n[GOAL]\ncase zero\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081 : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\n\u22a2 ofDigits p digits / p ^ zero = ofDigits p (List.drop zero digits)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn p : \u2115\nhpos : 0 < p\ndigits : List \u2115\nw\u2081 : \u2200 (l : \u2115), l \u2208 digits \u2192 l < p\ni : \u2115\nhi : ofDigits p digits / p ^ i = ofDigits p (List.drop i digits)\n\u22a2 ofDigits p digits / p ^ succ i = ofDigits p (List.drop (succ i) digits)\n[PROOFSTEP]\nrw [Nat.pow_succ, \u2190 Nat.div_div_eq_div_mul, hi,\n  ofDigits_div_eq_ofDigits_tail hpos (List.drop i digits) <| fun x hx \u21a6 w\u2081 x <| List.mem_of_mem_drop hx, \u2190\n  List.drop_one, List.drop_drop, add_comm]\n[GOAL]\nn\u271d p i n : \u2115\nh : 2 \u2264 p\n\u22a2 n / p ^ i = ofDigits p (List.drop i (digits p n))\n[PROOFSTEP]\nconvert ofDigits_div_pow_eq_ofDigits_drop i (zero_lt_of_lt h) (p.digits n) (fun l hl \u21a6 digits_lt_base h hl)\n[GOAL]\ncase h.e'_2.h.e'_5\nn\u271d p i n : \u2115\nh : 2 \u2264 p\n\u22a2 n = ofDigits p (digits p n)\n[PROOFSTEP]\nexact (ofDigits_digits p n).symm\n[GOAL]\nn p : \u2115\nL : List \u2115\nh_nonempty : L \u2260 []\nh_ne_zero : List.getLast L h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\n\u22a2 (p - 1) * \u2211 i in range (List.length L), ofDigits p L / p ^ succ i = ofDigits p L - List.sum L\n[PROOFSTEP]\nobtain h | rfl | h : 1 < p \u2228 1 = p \u2228 p < 1 := trichotomous 1 p\n[GOAL]\ncase inl\nn p : \u2115\nL : List \u2115\nh_nonempty : L \u2260 []\nh_ne_zero : List.getLast L h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\n\u22a2 (p - 1) * \u2211 i in range (List.length L), ofDigits p L / p ^ succ i = ofDigits p L - List.sum L\n[PROOFSTEP]\ninduction' L with hd tl ih\n[GOAL]\ncase inl.nil\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nh_nonempty : [] \u2260 []\nh_ne_zero : List.getLast [] h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 [] \u2192 l < p\n\u22a2 (p - 1) * \u2211 i in range (List.length []), ofDigits p [] / p ^ succ i = ofDigits p [] - List.sum []\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase inl.cons\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 (p - 1) * \u2211 i in range (List.length (hd :: tl)), ofDigits p (hd :: tl) / p ^ succ i =\n    ofDigits p (hd :: tl) - List.sum (hd :: tl)\n[PROOFSTEP]\nsimp only [List.length_cons, List.sum_cons, self_div_pow_eq_ofDigits_drop _ _ h,\n  digits_ofDigits p h (hd :: tl) h_lt (fun _ => h_ne_zero)]\n[GOAL]\ncase inl.cons\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 (p - 1) * \u2211 x in range (succ (List.length tl)), ofDigits p (List.drop (succ x) (hd :: tl)) =\n    ofDigits p (hd :: tl) - (hd + List.sum tl)\n[PROOFSTEP]\nsimp only [ofDigits]\n[GOAL]\ncase inl.cons\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 (p - 1) * \u2211 x in range (succ (List.length tl)), ofDigits p (List.drop (succ x) (hd :: tl)) =\n    \u2191hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nrw [sum_range_succ, Nat.cast_id]\n[GOAL]\ncase inl.cons\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 (p - 1) *\n      (\u2211 x in range (List.length tl), ofDigits p (List.drop (succ x) (hd :: tl)) +\n        ofDigits p (List.drop (succ (List.length tl)) (hd :: tl))) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nsimp only [List.drop, List.drop_length]\n[GOAL]\ncase inl.cons\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nobtain rfl | h' := em <| tl = []\n[GOAL]\ncase inl.cons.inl\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\nih :\n  \u2200 {h_nonempty : [] \u2260 []},\n    List.getLast [] h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 [] \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length []), ofDigits p [] / p ^ succ i = ofDigits p [] - List.sum []\nh_nonempty : [hd] \u2260 []\nh_ne_zero : List.getLast [hd] h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 [hd] \u2192 l < p\n\u22a2 (p - 1) * (\u2211 x in range (List.length []), ofDigits p (List.drop x []) + ofDigits p []) =\n    hd + p * ofDigits p [] - (hd + List.sum [])\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nhave w\u2081' := fun l hl \u21a6 h_lt l <| List.mem_cons_of_mem hd hl\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nhave w\u2082' := fun (h : tl \u2260 []) \u21a6 (List.getLast_cons h) \u25b8 h_ne_zero\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nhave ih := ih (w\u2082' h') w\u2081'\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nsimp only [self_div_pow_eq_ofDigits_drop _ _ h, digits_ofDigits p h tl w\u2081' w\u2082', succ_eq_one_add] at ih \n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nhave := @sum_singleton _ _ tl.length (fun x => ofDigits p <| tl.drop x) _\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\nthis : \u2211 x in {List.length tl}, ofDigits p (List.drop x tl) = ofDigits p (List.drop (List.length tl) tl)\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nrw [\u2190 Ico_succ_singleton, List.drop_length, ofDigits] at this \n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\nthis : \u2211 x in Ico (List.length tl) (List.length tl + 1), ofDigits p (List.drop x tl) = 0\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nhave h\u2081 : 1 \u2264 tl.length := List.length_pos.mpr h'\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\nthis : \u2211 x in Ico (List.length tl) (List.length tl + 1), ofDigits p (List.drop x tl) = 0\nh\u2081 : 1 \u2264 List.length tl\n\u22a2 (p - 1) * (\u2211 x in range (List.length tl), ofDigits p (List.drop x tl) + ofDigits p []) =\n    hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nrw [\u2190 sum_range_add_sum_Ico _ <| h\u2081, \u2190 add_zero (\u2211 x in Ico _ _, ofDigits p (tl.drop x)), \u2190 this,\n  sum_Ico_consecutive _ h\u2081 <| le_succ tl.length, \u2190 sum_Ico_add _ 0 tl.length 1, Ico_zero_eq_range, mul_add, mul_add, ih,\n  range_one, sum_singleton, List.drop, ofDigits, mul_zero, add_zero, \u2190\n  Nat.add_sub_assoc <| sum_le_ofDigits _ <| Nat.le_of_lt h]\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\nthis : \u2211 x in Ico (List.length tl) (List.length tl + 1), ofDigits p (List.drop x tl) = 0\nh\u2081 : 1 \u2264 List.length tl\n\u22a2 (p - 1) * ofDigits p tl + ofDigits p tl - List.sum tl = hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_mul <| ofDigits p tl]\n[GOAL]\ncase inl.cons.inr\nn p : \u2115\nL : List \u2115\nh_nonempty\u271d : L \u2260 []\nh_ne_zero\u271d : List.getLast L h_nonempty\u271d \u2260 0\nh_lt\u271d : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : 1 < p\nhd : \u2115\ntl : List \u2115\nih\u271d :\n  \u2200 {h_nonempty : tl \u2260 []},\n    List.getLast tl h_nonempty \u2260 0 \u2192\n      (\u2200 (l : \u2115), l \u2208 tl \u2192 l < p) \u2192\n        (p - 1) * \u2211 i in range (List.length tl), ofDigits p tl / p ^ succ i = ofDigits p tl - List.sum tl\nh_nonempty : hd :: tl \u2260 []\nh_ne_zero : List.getLast (hd :: tl) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 hd :: tl \u2192 l < p\nh' : \u00actl = []\nw\u2081' : \u2200 (l : \u2115), l \u2208 tl \u2192 l < p\nw\u2082' : \u2200 (h : tl \u2260 []), List.getLast tl h \u2260 0\nih : (p - 1) * \u2211 x in range (List.length tl), ofDigits p (List.drop (1 + x) tl) = ofDigits p tl - List.sum tl\nthis : \u2211 x in Ico (List.length tl) (List.length tl + 1), ofDigits p (List.drop x tl) = 0\nh\u2081 : 1 \u2264 List.length tl\n\u22a2 (p - 1) * ofDigits p tl + 1 * ofDigits p tl - List.sum tl = hd + p * ofDigits p tl - (hd + List.sum tl)\n[PROOFSTEP]\nrw [\u2190 add_mul, one_eq_succ_zero, Nat.sub_add_cancel <| zero_lt_of_lt h, Nat.add_sub_add_left]\n[GOAL]\ncase inr.inl\nn : \u2115\nL : List \u2115\nh_nonempty : L \u2260 []\nh_ne_zero : List.getLast L h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 L \u2192 l < 1\n\u22a2 (1 - 1) * \u2211 i in range (List.length L), ofDigits 1 L / 1 ^ succ i = ofDigits 1 L - List.sum L\n[PROOFSTEP]\nsimp [ofDigits_one]\n[GOAL]\ncase inr.inr\nn p : \u2115\nL : List \u2115\nh_nonempty : L \u2260 []\nh_ne_zero : List.getLast L h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : p < 1\n\u22a2 (p - 1) * \u2211 i in range (List.length L), ofDigits p L / p ^ succ i = ofDigits p L - List.sum L\n[PROOFSTEP]\nsimp [lt_one_iff.mp h]\n[GOAL]\ncase inr.inr\nn p : \u2115\nL : List \u2115\nh_nonempty : L \u2260 []\nh_ne_zero : List.getLast L h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 L \u2192 l < p\nh : p < 1\n\u22a2 0 = ofDigits 0 L - List.sum L\n[PROOFSTEP]\ncases L\n[GOAL]\ncase inr.inr.nil\nn p : \u2115\nh : p < 1\nh_nonempty : [] \u2260 []\nh_ne_zero : List.getLast [] h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 [] \u2192 l < p\n\u22a2 0 = ofDigits 0 [] - List.sum []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.cons\nn p : \u2115\nh : p < 1\nhead\u271d : \u2115\ntail\u271d : List \u2115\nh_nonempty : head\u271d :: tail\u271d \u2260 []\nh_ne_zero : List.getLast (head\u271d :: tail\u271d) h_nonempty \u2260 0\nh_lt : \u2200 (l : \u2115), l \u2208 head\u271d :: tail\u271d \u2192 l < p\n\u22a2 0 = ofDigits 0 (head\u271d :: tail\u271d) - List.sum (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp [ofDigits]\n[GOAL]\nn\u271d p n : \u2115\n\u22a2 (p - 1) * \u2211 i in range (succ (log p n)), n / p ^ succ i = n - List.sum (digits p n)\n[PROOFSTEP]\nobtain h | rfl | h : 1 < p \u2228 1 = p \u2228 p < 1 := trichotomous 1 p\n[GOAL]\ncase inl\nn\u271d p n : \u2115\nh : 1 < p\n\u22a2 (p - 1) * \u2211 i in range (succ (log p n)), n / p ^ succ i = n - List.sum (digits p n)\n[PROOFSTEP]\nrcases eq_or_ne n 0 with rfl | hn\n[GOAL]\ncase inl.inl\nn p : \u2115\nh : 1 < p\n\u22a2 (p - 1) * \u2211 i in range (succ (log p 0)), 0 / p ^ succ i = 0 - List.sum (digits p 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr\nn\u271d p n : \u2115\nh : 1 < p\nhn : n \u2260 0\n\u22a2 (p - 1) * \u2211 i in range (succ (log p n)), n / p ^ succ i = n - List.sum (digits p n)\n[PROOFSTEP]\nconvert\n  sub_one_mul_sum_div_pow_eq_sub_sum_digits (p.digits n) (getLast_digit_ne_zero p hn) <| (fun l a \u21a6 digits_lt_base h a)\n[GOAL]\ncase h.e'_2.h.e'_6.h.h.e'_1\nn\u271d p n : \u2115\nh : 1 < p\nhn : n \u2260 0\n\u22a2 succ (log p n) = List.length (digits p n)\n[PROOFSTEP]\nrefine' (digits_len p n h hn).symm\n[GOAL]\ncase h.e'_2.h.e'_6.a.h.e'_5\nn\u271d p n : \u2115\nh : 1 < p\nhn : n \u2260 0\nx\u271d : \u2115\na\u271d : x\u271d \u2208 range (List.length (digits p n))\n\u22a2 n = ofDigits p (digits p n)\ncase h.e'_3.h.e'_5 n\u271d p n : \u2115 h : 1 < p hn : n \u2260 0 \u22a2 n = ofDigits p (digits p n)\n[PROOFSTEP]\nall_goals exact (ofDigits_digits p n).symm\n[GOAL]\ncase h.e'_2.h.e'_6.a.h.e'_5\nn\u271d p n : \u2115\nh : 1 < p\nhn : n \u2260 0\nx\u271d : \u2115\na\u271d : x\u271d \u2208 range (List.length (digits p n))\n\u22a2 n = ofDigits p (digits p n)\n[PROOFSTEP]\nexact (ofDigits_digits p n).symm\n[GOAL]\ncase h.e'_3.h.e'_5\nn\u271d p n : \u2115\nh : 1 < p\nhn : n \u2260 0\n\u22a2 n = ofDigits p (digits p n)\n[PROOFSTEP]\nexact (ofDigits_digits p n).symm\n[GOAL]\ncase inr.inl\nn\u271d n : \u2115\n\u22a2 (1 - 1) * \u2211 i in range (succ (log 1 n)), n / 1 ^ succ i = n - List.sum (digits 1 n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nn\u271d p n : \u2115\nh : p < 1\n\u22a2 (p - 1) * \u2211 i in range (succ (log p n)), n / p ^ succ i = n - List.sum (digits p n)\n[PROOFSTEP]\nsimp [lt_one_iff.mp h]\n[GOAL]\ncase inr.inr\nn\u271d p n : \u2115\nh : p < 1\n\u22a2 0 = n - List.sum (digits 0 n)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase inr.inr.zero\nn p : \u2115\nh : p < 1\n\u22a2 0 = zero - List.sum (digits 0 zero)\ncase inr.inr.succ n p : \u2115 h : p < 1 n\u271d : \u2115 \u22a2 0 = succ n\u271d - List.sum (digits 0 (succ n\u271d))\n[PROOFSTEP]\nall_goals simp\n[GOAL]\ncase inr.inr.zero\nn p : \u2115\nh : p < 1\n\u22a2 0 = zero - List.sum (digits 0 zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.succ\nn p : \u2115\nh : p < 1\nn\u271d : \u2115\n\u22a2 0 = succ n\u271d - List.sum (digits 0 (succ n\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u271d n : \u2115\n\u22a2 digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n[PROOFSTEP]\ninduction' n using Nat.binaryRecFromOne with b n h ih\n[GOAL]\ncase z\u2080\nn : \u2115\n\u22a2 digits 2 0 = List.map (fun b => bif b then 1 else 0) (bits 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase z\u2081\nn : \u2115\n\u22a2 digits 2 1 = List.map (fun b => bif b then 1 else 0) (bits 1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase f\nn\u271d : \u2115\nb : Bool\nn : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 digits 2 (bit b n) = List.map (fun b => bif b then 1 else 0) (bits (bit b n))\n[PROOFSTEP]\nrw [bits_append_bit _ _ fun hn => absurd hn h]\n[GOAL]\ncase f\nn\u271d : \u2115\nb : Bool\nn : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 digits 2 (bit b n) = List.map (fun b => bif b then 1 else 0) (b :: bits n)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase f.false\nn\u271d n : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 digits 2 (bit false n) = List.map (fun b => bif b then 1 else 0) (false :: bits n)\n[PROOFSTEP]\nrw [digits_def' one_lt_two]\n[GOAL]\ncase f.false\nn\u271d n : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 bit false n % 2 :: digits 2 (bit false n / 2) = List.map (fun b => bif b then 1 else 0) (false :: bits n)\n[PROOFSTEP]\nsimpa [Nat.bit, Nat.bit0_val n]\n[GOAL]\ncase f.false\nn\u271d n : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 0 < bit false n\n[PROOFSTEP]\nsimpa [pos_iff_ne_zero, bit_eq_zero_iff]\n[GOAL]\ncase f.true\nn\u271d n : \u2115\nh : n \u2260 0\nih : digits 2 n = List.map (fun b => bif b then 1 else 0) (bits n)\n\u22a2 digits 2 (bit true n) = List.map (fun b => bif b then 1 else 0) (true :: bits n)\n[PROOFSTEP]\nsimpa [Nat.bit, Nat.bit1_val n, add_comm, digits_add 2 one_lt_two 1 n]\n[GOAL]\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\na b k : \u03b1\nh : k \u2223 a - b\nL : List \u2115\n\u22a2 k \u2223 ofDigits a L - ofDigits b L\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\na b k : \u03b1\nh : k \u2223 a - b\n\u22a2 k \u2223 ofDigits a [] - ofDigits b []\n[PROOFSTEP]\nchange k \u2223 0 - 0\n[GOAL]\ncase nil\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\na b k : \u03b1\nh : k \u2223 a - b\n\u22a2 k \u2223 0 - 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\na b k : \u03b1\nh : k \u2223 a - b\nd : \u2115\nL : List \u2115\nih : k \u2223 ofDigits a L - ofDigits b L\n\u22a2 k \u2223 ofDigits a (d :: L) - ofDigits b (d :: L)\n[PROOFSTEP]\nsimp only [ofDigits, add_sub_add_left_eq_sub]\n[GOAL]\ncase cons\nn : \u2115\n\u03b1 : Type u_1\ninst\u271d : CommRing \u03b1\na b k : \u03b1\nh : k \u2223 a - b\nd : \u2115\nL : List \u2115\nih : k \u2223 ofDigits a L - ofDigits b L\n\u22a2 k \u2223 a * ofDigits a L - b * ofDigits b L\n[PROOFSTEP]\nexact dvd_mul_sub_mul h ih\n[GOAL]\nn b b' k : \u2115\nh : b \u2261 b' [MOD k]\nL : List \u2115\n\u22a2 ofDigits b L \u2261 ofDigits b' L [MOD k]\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn b b' k : \u2115\nh : b \u2261 b' [MOD k]\n\u22a2 ofDigits b [] \u2261 ofDigits b' [] [MOD k]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn b b' k : \u2115\nh : b \u2261 b' [MOD k]\nd : \u2115\nL : List \u2115\nih : ofDigits b L \u2261 ofDigits b' L [MOD k]\n\u22a2 ofDigits b (d :: L) \u2261 ofDigits b' (d :: L) [MOD k]\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn b b' k : \u2115\nh : b \u2261 b' [MOD k]\nd : \u2115\nL : List \u2115\nih : ofDigits b L \u2261 ofDigits b' L [MOD k]\n\u22a2 d + b * ofDigits b L \u2261 d + b' * ofDigits b' L [MOD k]\n[PROOFSTEP]\ndsimp [Nat.ModEq] at *\n[GOAL]\ncase cons\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n\u22a2 (d + b * ofDigits b L) % k = (d + b' * ofDigits b' L) % k\n[PROOFSTEP]\nconv_lhs => rw [Nat.add_mod, Nat.mul_mod, h, ih]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b * ofDigits b L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod, h, ih]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b * ofDigits b L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod, h, ih]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b * ofDigits b L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod, h, ih]\n[GOAL]\ncase cons\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n\u22a2 (d % k + b' % k * (ofDigits b' L % k) % k) % k = (d + b' * ofDigits b' L) % k\n[PROOFSTEP]\nconv_rhs => rw [Nat.add_mod, Nat.mul_mod]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b' * ofDigits b' L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b' * ofDigits b' L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod]\n[GOAL]\nn b b' k : \u2115\nh : b % k = b' % k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % k = ofDigits b' L % k\n| (d + b' * ofDigits b' L) % k\n[PROOFSTEP]\nrw [Nat.add_mod, Nat.mul_mod]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b \u2261 b' [ZMOD \u2191k]\nL : List \u2115\n\u22a2 ofDigits b L \u2261 ofDigits b' L [ZMOD \u2191k]\n[PROOFSTEP]\ninduction' L with d L ih\n[GOAL]\ncase nil\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b \u2261 b' [ZMOD \u2191k]\n\u22a2 ofDigits b [] \u2261 ofDigits b' [] [ZMOD \u2191k]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b \u2261 b' [ZMOD \u2191k]\nd : \u2115\nL : List \u2115\nih : ofDigits b L \u2261 ofDigits b' L [ZMOD \u2191k]\n\u22a2 ofDigits b (d :: L) \u2261 ofDigits b' (d :: L) [ZMOD \u2191k]\n[PROOFSTEP]\ndsimp [ofDigits]\n[GOAL]\ncase cons\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b \u2261 b' [ZMOD \u2191k]\nd : \u2115\nL : List \u2115\nih : ofDigits b L \u2261 ofDigits b' L [ZMOD \u2191k]\n\u22a2 \u2191d + b * ofDigits b L \u2261 \u2191d + b' * ofDigits b' L [ZMOD \u2191k]\n[PROOFSTEP]\ndsimp [Int.ModEq] at *\n[GOAL]\ncase cons\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n\u22a2 (\u2191d + b * ofDigits b L) % \u2191k = (\u2191d + b' * ofDigits b' L) % \u2191k\n[PROOFSTEP]\nconv_lhs => rw [Int.add_emod, Int.mul_emod, h, ih]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b * ofDigits b L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod, h, ih]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b * ofDigits b L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod, h, ih]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b * ofDigits b L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod, h, ih]\n[GOAL]\ncase cons\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n\u22a2 (\u2191d % \u2191k + b' % \u2191k * (ofDigits b' L % \u2191k) % \u2191k) % \u2191k = (\u2191d + b' * ofDigits b' L) % \u2191k\n[PROOFSTEP]\nconv_rhs => rw [Int.add_emod, Int.mul_emod]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b' * ofDigits b' L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b' * ofDigits b' L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod]\n[GOAL]\nn : \u2115\nb b' : \u2124\nk : \u2115\nh : b % \u2191k = b' % \u2191k\nd : \u2115\nL : List \u2115\nih : ofDigits b L % \u2191k = ofDigits b' L % \u2191k\n| (\u2191d + b' * ofDigits b' L) % \u2191k\n[PROOFSTEP]\nrw [Int.add_emod, Int.mul_emod]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 n \u2261 List.sum (digits b' n) [MOD b]\n[PROOFSTEP]\nrw [\u2190 ofDigits_one]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 n \u2261 ofDigits 1 (digits b' n) [MOD b]\n[PROOFSTEP]\nconv =>\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n \u2261 ofDigits 1 (digits b' n) [MOD b]\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n \u2261 ofDigits 1 (digits b' n) [MOD b]\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n \u2261 ofDigits 1 (digits b' n) [MOD b]\n[PROOFSTEP]\ncongr\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b\ncase a\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n\ncase b n\u271d b b' : \u2115 h : b' % b = 1 n : \u2115 | ofDigits 1 (digits b' n)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b\n[PROOFSTEP]\nskip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b\n[PROOFSTEP]\nskip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n\ncase b n\u271d b b' : \u2115 h : b' % b = 1 n : \u2115 | ofDigits 1 (digits b' n)\n[PROOFSTEP]\n\u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 ofDigits b' (digits b' n) \u2261 ofDigits 1 (digits b' n) [MOD b]\n[PROOFSTEP]\nconvert ofDigits_modEq b' b (digits b' n)\n[GOAL]\ncase h.e'_3.h.e'_3\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 1 = b' % b\n[PROOFSTEP]\nexact h.symm\n[GOAL]\nn\u271d n : \u2115\n\u22a2 10 % 3 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn\u271d n : \u2115\n\u22a2 10 % 9 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n\u22a2 \u2191n \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\nconv =>\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\n  congr\n  \u00b7skip\n  \u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\ncongr\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191b\ncase a\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n\ncase b n\u271d b b' : \u2115 c : \u2124 h : \u2191b' \u2261 c [ZMOD \u2191b] n : \u2115 | ofDigits c (digits b' n)\n[PROOFSTEP]\n\u00b7skip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191b\n[PROOFSTEP]\nskip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191b\n[PROOFSTEP]\nskip\n[GOAL]\ncase n\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191b\n[PROOFSTEP]\nskip\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n\ncase b n\u271d b b' : \u2115 c : \u2124 h : \u2191b' \u2261 c [ZMOD \u2191b] n : \u2115 | ofDigits c (digits b' n)\n[PROOFSTEP]\n\u00b7rw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\ncase a\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n| \u2191n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n\u22a2 \u2191(ofDigits b' (digits b' n)) \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\nrw [coe_int_ofDigits]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b' \u2261 c [ZMOD \u2191b]\nn : \u2115\n\u22a2 ofDigits (\u2191b') (digits b' n) \u2261 ofDigits c (digits b' n) [ZMOD \u2191b]\n[PROOFSTEP]\napply ofDigits_zmodeq' _ _ _ h\n[GOAL]\nn\u271d n : \u2115\n\u22a2 ofDigits (-1) [n] = List.alternatingSum (List.map (fun n => \u2191n) [n])\n[PROOFSTEP]\nsimp [ofDigits, List.alternatingSum]\n[GOAL]\nn a b : \u2115\nt : List \u2115\n\u22a2 ofDigits (-1) (a :: b :: t) = List.alternatingSum (List.map (fun n => \u2191n) (a :: b :: t))\n[PROOFSTEP]\nsimp only [ofDigits, List.alternatingSum, List.map_cons, ofDigits_neg_one t]\n[GOAL]\nn a b : \u2115\nt : List \u2115\n\u22a2 \u2191a + -1 * (\u2191b + -1 * List.alternatingSum (List.map (fun n => \u2191n) t)) =\n    \u2191a + -\u2191b + List.alternatingSum (List.map (fun n => \u2191n) t)\n[PROOFSTEP]\nring\n[GOAL]\nn\u271d n : \u2115\n\u22a2 \u2191n \u2261 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n)) [ZMOD 11]\n[PROOFSTEP]\nhave t := zmodeq_ofDigits_digits 11 10 (-1 : \u2124) (by unfold Int.ModEq; norm_num) n\n[GOAL]\nn\u271d n : \u2115\n\u22a2 \u219110 \u2261 -1 [ZMOD \u219111]\n[PROOFSTEP]\nunfold Int.ModEq\n[GOAL]\nn\u271d n : \u2115\n\u22a2 \u219110 % \u219111 = -1 % \u219111\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn\u271d n : \u2115\nt : \u2191n \u2261 ofDigits (-1) (digits 10 n) [ZMOD \u219111]\n\u22a2 \u2191n \u2261 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n)) [ZMOD 11]\n[PROOFSTEP]\nrwa [ofDigits_neg_one] at t \n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 b \u2223 n \u2194 b \u2223 List.sum (digits b' n)\n[PROOFSTEP]\nrw [\u2190 ofDigits_one]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 b \u2223 n \u2194 b \u2223 ofDigits 1 (digits b' n)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b \u2223 n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b \u2223 n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n| b \u2223 n\n[PROOFSTEP]\nrw [\u2190 ofDigits_digits b' n]\n[GOAL]\nn\u271d b b' : \u2115\nh : b' % b = 1\nn : \u2115\n\u22a2 b \u2223 ofDigits b' (digits b' n) \u2194 b \u2223 ofDigits 1 (digits b' n)\n[PROOFSTEP]\nrw [Nat.dvd_iff_mod_eq_zero, Nat.dvd_iff_mod_eq_zero, ofDigits_mod, h]\n[GOAL]\nn\u271d n : \u2115\n\u22a2 10 % 3 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn\u271d n : \u2115\n\u22a2 10 % 9 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b \u2223 \u2191b' - c\nn : \u2115\n\u22a2 b \u2223 n \u2194 \u2191b \u2223 ofDigits c (digits b' n)\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd]\n[GOAL]\nn\u271d b b' : \u2115\nc : \u2124\nh : \u2191b \u2223 \u2191b' - c\nn : \u2115\n\u22a2 \u2191b \u2223 \u2191n \u2194 \u2191b \u2223 ofDigits c (digits b' n)\n[PROOFSTEP]\nexact dvd_iff_dvd_of_dvd_sub (zmodeq_ofDigits_digits b b' c (Int.modEq_iff_dvd.2 h).symm _).symm.dvd\n[GOAL]\nn : \u2115\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n[PROOFSTEP]\nhave t := dvd_iff_dvd_ofDigits 11 10 (-1 : \u2124) (by norm_num) n\n[GOAL]\nn : \u2115\n\u22a2 \u219111 \u2223 \u219110 - -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 ofDigits (-1) (digits 10 n)\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n[PROOFSTEP]\nrw [ofDigits_neg_one] at t \n[GOAL]\nn : \u2115\nt : 11 \u2223 n \u2194 \u219111 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n\u22a2 11 \u2223 n \u2194 11 \u2223 List.alternatingSum (List.map (fun n => \u2191n) (digits 10 n))\n[PROOFSTEP]\nexact t\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\nh : Even (List.length (digits 10 n))\n\u22a2 11 \u2223 n\n[PROOFSTEP]\nlet dig := (digits 10 n).map (Coe.coe : \u2115 \u2192 \u2124)\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\nh : Even (List.length (digits 10 n))\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\n\u22a2 11 \u2223 n\n[PROOFSTEP]\nreplace h : Even dig.length := by rwa [List.length_map]\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\nh : Even (List.length (digits 10 n))\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\n\u22a2 Even (List.length dig)\n[PROOFSTEP]\nrwa [List.length_map]\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\nh : Even (List.length dig)\n\u22a2 11 \u2223 n\n[PROOFSTEP]\nrefine' eleven_dvd_iff.2 \u27e80, (_ : dig.alternatingSum = 0)\u27e9\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\nh : Even (List.length dig)\n\u22a2 List.alternatingSum dig = 0\n[PROOFSTEP]\nhave := dig.alternatingSum_reverse\n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\nh : Even (List.length dig)\nthis : List.alternatingSum (List.reverse dig) = (-1) ^ (List.length dig + 1) \u2022 List.alternatingSum dig\n\u22a2 List.alternatingSum dig = 0\n[PROOFSTEP]\nrw [(p.map _).reverse_eq, _root_.pow_succ, h.neg_one_pow, mul_one, neg_one_zsmul] at this \n[GOAL]\nn : \u2115\np : List.Palindrome (digits 10 n)\ndig : List \u2124 := List.map Coe.coe (digits 10 n)\nh : Even (List.length dig)\nthis : List.alternatingSum (List.map Coe.coe (digits 10 n)) = -List.alternatingSum dig\n\u22a2 List.alternatingSum dig = 0\n[PROOFSTEP]\nexact eq_zero_of_neg_eq this.symm\n[GOAL]\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l \u2227 1 < b \u2227 0 < m\n\u22a2 digits b n = r :: l \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nrcases h with \u27e8h, b2, m0\u27e9\n[GOAL]\ncase intro.intro\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\n\u22a2 digits b n = r :: l \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nhave b0 : 0 < b := by linarith\n[GOAL]\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\n\u22a2 0 < b\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro.intro\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\nb0 : 0 < b\n\u22a2 digits b n = r :: l \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nhave n0 : 0 < n := by linarith [mul_pos b0 m0]\n[GOAL]\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\nb0 : 0 < b\n\u22a2 0 < n\n[PROOFSTEP]\nlinarith [mul_pos b0 m0]\n[GOAL]\ncase intro.intro\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\nb0 : 0 < b\nn0 : 0 < n\n\u22a2 digits b n = r :: l \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nrefine' \u27e8_, b2, n0\u27e9\n[GOAL]\ncase intro.intro\nn\u271d b n m r : \u2115\nl : List \u2115\ne : r + b * m = n\nhr : r < b\nh : digits b m = l\nb2 : 1 < b\nm0 : 0 < m\nb0 : 0 < b\nn0 : 0 < n\n\u22a2 digits b n = r :: l\n[PROOFSTEP]\nobtain \u27e8rfl, rfl\u27e9 := (Nat.div_mod_unique b0).2 \u27e8e, hr\u27e9\n[GOAL]\ncase intro.intro.intro\nn\u271d b n : \u2115\nl : List \u2115\nb2 : 1 < b\nb0 : 0 < b\nn0 : 0 < n\nh : digits b (n / b) = l\nm0 : 0 < n / b\nhr : n % b < b\ne : n % b + b * (n / b) = n\n\u22a2 digits b n = n % b :: l\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase intro.intro.intro\nn\u271d b n : \u2115\nb2 : 1 < b\nb0 : 0 < b\nn0 : 0 < n\nm0 : 0 < n / b\nhr : n % b < b\ne : n % b + b * (n / b) = n\n\u22a2 digits b n = n % b :: digits b (n / b)\n[PROOFSTEP]\nexact Nat.digits_def' b2 n0\n[GOAL]\nn\u271d b n : \u2115\nn0 : 0 < n\nnb : n < b\n\u22a2 digits b n = [n] \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nhave b2 : 1 < b := lt_iff_add_one_le.mpr (le_trans (add_le_add_right (lt_iff_add_one_le.mp n0) 1) nb)\n[GOAL]\nn\u271d b n : \u2115\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n\u22a2 digits b n = [n] \u2227 1 < b \u2227 0 < n\n[PROOFSTEP]\nrefine' \u27e8_, b2, n0\u27e9\n[GOAL]\nn\u271d b n : \u2115\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n\u22a2 digits b n = [n]\n[PROOFSTEP]\nrw [Nat.digits_def' b2 n0, Nat.mod_eq_of_lt nb, (Nat.div_eq_zero_iff ((zero_le n).trans_lt nb)).2 nb, Nat.digits_zero]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Digits", "llama_tokens": 46969, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245870332531, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5427900292497644}}
{"text": "[GOAL]\nn : \u2115\ni : Fin2 n\n\u22a2 \u2200 {\u03b1 : TypeVec n} (x : Prj i \u03b1), abs i (repr i x) = x\n[PROOFSTEP]\nintros\n[GOAL]\nn : \u2115\ni : Fin2 n\n\u03b1\u271d : TypeVec n\nx\u271d : Prj i \u03b1\u271d\n\u22a2 abs i (repr i x\u271d) = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\ni : Fin2 n\n\u22a2 \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (p : MvPFunctor.Obj (P i) \u03b1), abs i (f <$$> p) = f <$$> abs i p\n[PROOFSTEP]\nintros \u03b1 \u03b2 f P\n[GOAL]\nn : \u2115\ni : Fin2 n\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nP : MvPFunctor.Obj (MvQPF.Prj.P i) \u03b1\n\u22a2 abs i (f <$$> P) = f <$$> abs i P\n[PROOFSTEP]\ncases P\n[GOAL]\ncase mk\nn : \u2115\ni : Fin2 n\n\u03b1 \u03b2 : TypeVec n\nf : \u03b1 \u27f9 \u03b2\nfst\u271d : (P i).A\nsnd\u271d : MvPFunctor.B (P i) fst\u271d \u27f9 \u03b1\n\u22a2 abs i (f <$$> { fst := fst\u271d, snd := snd\u271d }) = f <$$> abs i { fst := fst\u271d, snd := snd\u271d }\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.QPF.Multivariate.Constructions.Prj", "llama_tokens": 436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.833324587033253, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.5427900179701157}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\n\u22a2 padicNorm p (\u2191f m) = padicNorm p (\u2191f n)\n[PROOFSTEP]\nhave : padicNorm p (f n - f m) < \u03b5 := hN2 _ (max_le_iff.1 hn).2 _ (max_le_iff.1 hm).2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis : padicNorm p (\u2191f n - \u2191f m) < \u03b5\n\u22a2 padicNorm p (\u2191f m) = padicNorm p (\u2191f n)\n[PROOFSTEP]\nhave : padicNorm p (f n - f m) < padicNorm p (f n) := lt_of_lt_of_le this <| hN1 _ (max_le_iff.1 hn).1\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b9 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\n\u22a2 padicNorm p (\u2191f m) = padicNorm p (\u2191f n)\n[PROOFSTEP]\nhave : padicNorm p (f n - f m) < max (padicNorm p (f n)) (padicNorm p (f m)) := lt_max_iff.2 (Or.inl this)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (\u2191f n - \u2191f m) < max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\n\u22a2 padicNorm p (\u2191f m) = padicNorm p (\u2191f n)\n[PROOFSTEP]\nby_contra hne\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (\u2191f n - \u2191f m) < max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\nhne : \u00acpadicNorm p (\u2191f m) = padicNorm p (\u2191f n)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 padicNorm.neg (f m)] at hne \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (\u2191f n - \u2191f m) < max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\nhne : \u00acpadicNorm p (-\u2191f m) = padicNorm p (\u2191f n)\n\u22a2 False\n[PROOFSTEP]\nhave hnam := add_eq_max_of_ne hne\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (\u2191f n - \u2191f m) < max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\nhne : \u00acpadicNorm p (-\u2191f m) = padicNorm p (\u2191f n)\nhnam : padicNorm p (-\u2191f m + \u2191f n) = max (padicNorm p (-\u2191f m)) (padicNorm p (\u2191f n))\n\u22a2 False\n[PROOFSTEP]\nrw [padicNorm.neg, max_comm] at hnam \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (\u2191f n - \u2191f m) < max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\nhne : \u00acpadicNorm p (-\u2191f m) = padicNorm p (\u2191f n)\nhnam : padicNorm p (-\u2191f m + \u2191f n) = max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 hnam, sub_eq_add_neg, add_comm] at this \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a (padicNorm p)\nhf : \u00acf \u2248 0\nthis\u271d\u00b2 : \u2203 \u03b5, \u03b5 > 0 \u2227 \u2203 N1, \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN1 : \u2115\nhN1 : \u2200 (j : \u2115), j \u2265 N1 \u2192 \u03b5 \u2264 padicNorm p (\u2191f j)\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\nn m : \u2115\nhn : max N1 N2 \u2264 n\nhm : max N1 N2 \u2264 m\nthis\u271d\u00b9 : padicNorm p (\u2191f n - \u2191f m) < \u03b5\nthis\u271d : padicNorm p (\u2191f n - \u2191f m) < padicNorm p (\u2191f n)\nthis : padicNorm p (-\u2191f m + \u2191f n) < padicNorm p (-\u2191f m + \u2191f n)\nhne : \u00acpadicNorm p (-\u2191f m) = padicNorm p (\u2191f n)\nhnam : padicNorm p (-\u2191f m + \u2191f n) = max (padicNorm p (\u2191f n)) (padicNorm p (\u2191f m))\n\u22a2 False\n[PROOFSTEP]\napply _root_.lt_irrefl _ this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 norm f = 0 \u2194 f \u2248 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 norm f = 0 \u2192 f \u2248 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\n\u22a2 f \u2248 0\n[PROOFSTEP]\nby_contra hf\n[GOAL]\ncase mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : norm f = 0\nhf : \u00acf \u2248 0\n\u22a2 False\n[PROOFSTEP]\nunfold norm at h \n[GOAL]\ncase mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) = 0\nhf : \u00acf \u2248 0\n\u22a2 False\n[PROOFSTEP]\nsplit_ifs at h \n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh\u271d : f \u2248 0\nh : 0 = 0\n\u22a2 False\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u22a2 False\n[PROOFSTEP]\napply hf\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u22a2 f \u2248 0\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (\u2191(f - 0) j) < \u03b5\n[PROOFSTEP]\nexists stationaryPoint hf\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200 (j : \u2115), j \u2265 stationaryPoint hf \u2192 padicNorm p (\u2191(f - 0) j) < \u03b5\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5\n[PROOFSTEP]\nhave heq := stationaryPoint_spec hf le_rfl hj\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf h\u271d : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint h\u271d)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\nheq : padicNorm p (\u2191f j) = padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5\n[PROOFSTEP]\nsimpa [h, heq]\n[GOAL]\ncase mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f \u2248 0 \u2192 norm f = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f \u2248 0\n\u22a2 norm f = 0\n[PROOFSTEP]\nsimp [norm, h]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : f \u2248 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (\u2191(f - 0) j) < \u03b5\nj : \u2115\nhj : j \u2265 i\n\u22a2 padicNorm p (\u2191(g - 0) j) < \u03b5\n[PROOFSTEP]\nsimpa [h] using hi _ hj\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\n\u22a2 norm f = padicNorm p (\u2191f (stationaryPoint hf))\n[PROOFSTEP]\nsimp [norm, hf]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nheq : norm f = padicNorm p (\u2191f (stationaryPoint hf))\nh : \u2191f (stationaryPoint hf) = 0\n\u22a2 norm f = 0\n[PROOFSTEP]\nsimpa [h] using heq\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nq : \u211a\nhq : q \u2260 0\nh : LimZero (const (padicNorm p) q - 0)\n\u22a2 LimZero (const (padicNorm (?m.10166 hq h)) q)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : f \u2248 0\n\u22a2 0 \u2264 norm f\n[PROOFSTEP]\nsimp [hf, norm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\n\u22a2 0 \u2264 norm f\n[PROOFSTEP]\nsimp [norm, hf, padicNorm.nonneg]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv2 v3 : \u2115\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f (max (stationaryPoint hf) (max v2 v3)))\n[PROOFSTEP]\napply stationaryPoint_spec hf\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv2 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 max (stationaryPoint hf) (max v2 v3)\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv2 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f (max v1 (max (stationaryPoint hf) v3)))\n[PROOFSTEP]\napply stationaryPoint_spec hf\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 max v1 (max (stationaryPoint hf) v3)\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 ?a.b\u271d\n[PROOFSTEP]\napply le_max_left _ v3\n[GOAL]\ncase a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 max (stationaryPoint hf) v3 \u2264 max v1 (max (stationaryPoint hf) v3)\n[PROOFSTEP]\napply le_max_right\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v3 : \u2115\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v2 : \u2115\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f (max v1 (max v2 (stationaryPoint hf))))\n[PROOFSTEP]\napply stationaryPoint_spec hf\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v2 : \u2115\n\u22a2 stationaryPoint hf \u2264 max v1 (max v2 (stationaryPoint hf))\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v2 : \u2115\n\u22a2 stationaryPoint hf \u2264 ?a.b\u271d\n[PROOFSTEP]\napply le_max_right v2\n[GOAL]\ncase a.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v2 : \u2115\n\u22a2 max v2 (stationaryPoint hf) \u2264 max v1 (max v2 (stationaryPoint hf))\n[PROOFSTEP]\napply le_max_right\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nv1 v2 : \u2115\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\n\u22a2 norm f = \u2191p ^ (-valuation f)\n[PROOFSTEP]\nrw [norm, valuation, dif_neg hf, dif_neg hf, padicNorm, if_neg]\n[GOAL]\ncase hnc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\n\u22a2 \u00ac\u2191f (stationaryPoint hf) = 0\n[PROOFSTEP]\nintro H\n[GOAL]\ncase hnc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u22a2 False\n[PROOFSTEP]\napply CauSeq.not_limZero_of_not_congr_zero hf\n[GOAL]\ncase hnc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u22a2 LimZero f\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase hnc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (\u2191f j) < \u03b5\n[PROOFSTEP]\nuse stationaryPoint hf\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200 (j : \u2115), j \u2265 stationaryPoint hf \u2192 padicNorm p (\u2191f j) < \u03b5\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : n \u2265 stationaryPoint hf\n\u22a2 padicNorm p (\u2191f n) < \u03b5\n[PROOFSTEP]\nrw [stationaryPoint_spec hf le_rfl hn]\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nH : \u2191f (stationaryPoint hf) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nn : \u2115\nhn : n \u2265 stationaryPoint hf\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) < \u03b5\n[PROOFSTEP]\nsimpa [H] using h\u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 valuation f = valuation g \u2194 norm f = norm g\n[PROOFSTEP]\nrw [norm_eq_pow_val hf, norm_eq_pow_val hg, \u2190 neg_inj, zpow_inj]\n[GOAL]\ncase h\u2080\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nexact_mod_cast (Fact.out : p.Prime).pos\n[GOAL]\ncase h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2191p \u2260 1\n[PROOFSTEP]\nexact_mod_cast (Fact.out : p.Prime).ne_one\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : f \u2248 0\n\u22a2 norm (f * g) = norm f * norm g\n[PROOFSTEP]\nhave hg : f * g \u2248 0 := mul_equiv_zero' _ hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : f \u2248 0\nhg : f * g \u2248 0\n\u22a2 norm (f * g) = norm f * norm g\n[PROOFSTEP]\nsimp only [hf, hg, norm, dif_pos, zero_mul]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : g \u2248 0\n\u22a2 norm (f * g) = norm f * norm g\n[PROOFSTEP]\nhave hf : f * g \u2248 0 := mul_equiv_zero _ hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf\u271d : \u00acf \u2248 0\nhg : g \u2248 0\nhf : f * g \u2248 0\n\u22a2 norm (f * g) = norm f * norm g\n[PROOFSTEP]\nsimp only [hf, hg, norm, dif_pos, mul_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 norm (f * g) = norm f * norm g\n[PROOFSTEP]\nunfold norm\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 (if hf : f * g \u2248 0 then 0 else padicNorm p (\u2191(f * g) (stationaryPoint hf))) =\n    (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) *\n      if hf : g \u2248 0 then 0 else padicNorm p (\u2191g (stationaryPoint hf))\n[PROOFSTEP]\nsplit_ifs with hfg\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f * g \u2248 0\n\u22a2 0 = padicNorm p (\u2191f (stationaryPoint hf)) * padicNorm p (\u2191g (stationaryPoint hg))\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 padicNorm p (\u2191(f * g) (stationaryPoint hfg)) =\n    padicNorm p (\u2191f (stationaryPoint hf)) * padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\nexact (mul_not_equiv_zero hf hg hfg).elim\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 padicNorm p (\u2191(f * g) (stationaryPoint hfg)) =\n    padicNorm p (\u2191f (stationaryPoint hf)) * padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\nrw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 padicNorm p (\u2191(f * g) (max (stationaryPoint hfg) (max ?neg.v2\u271d ?neg.v3\u271d))) =\n    padicNorm p (\u2191f (max ?neg.v1\u271d (max (stationaryPoint hf) ?neg.v3\u271d))) *\n      padicNorm p (\u2191g (max ?neg.v1\u271d (max ?neg.v2\u271d (stationaryPoint hg))))\ncase neg.v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : \u00acf * g \u2248 0\n\u22a2 \u2115\ncase neg.v3 p : \u2115 hp : Fact (Nat.Prime p) f g : PadicSeq p hf : \u00acf \u2248 0 hg : \u00acg \u2248 0 hfg : \u00acf * g \u2248 0 \u22a2 \u2115\n[PROOFSTEP]\napply padicNorm.mul\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q = 0\n\u22a2 norm (const (padicNorm p) q) = padicNorm p q\n[PROOFSTEP]\nhave : const (padicNorm p) q \u2248 0 := by simp [hq]; apply Setoid.refl (const (padicNorm p) 0)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q = 0\n\u22a2 const (padicNorm p) q \u2248 0\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q = 0\n\u22a2 0 \u2248 0\n[PROOFSTEP]\napply Setoid.refl (const (padicNorm p) 0)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : q = 0\nthis : const (padicNorm p) q \u2248 0\n\u22a2 norm (const (padicNorm p) q) = padicNorm p q\n[PROOFSTEP]\nsubst hq\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nthis : const (padicNorm p) 0 \u2248 0\n\u22a2 norm (const (padicNorm p) 0) = padicNorm p 0\n[PROOFSTEP]\nsimp [norm, this]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : \u00acq = 0\n\u22a2 norm (const (padicNorm p) q) = padicNorm p q\n[PROOFSTEP]\nhave : \u00acconst (padicNorm p) q \u2248 0 := not_equiv_zero_const_of_nonzero hq\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\nhq : \u00acq = 0\nthis : \u00acconst (padicNorm p) q \u2248 0\n\u22a2 norm (const (padicNorm p) q) = padicNorm p q\n[PROOFSTEP]\nsimp [norm, this]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : PadicSeq p\nha : \u00aca \u2248 0\n\u22a2 \u2203 z, norm a = \u2191p ^ (-z)\n[PROOFSTEP]\nlet \u27e8k, hk, hk'\u27e9 := norm_eq_norm_app_of_nonzero ha\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : PadicSeq p\nha : \u00aca \u2248 0\nk : \u211a\nhk : norm a = padicNorm p k\nhk' : k \u2260 0\n\u22a2 \u2203 z, norm a = \u2191p ^ (-z)\n[PROOFSTEP]\nsimpa [hk] using padicNorm.values_discrete hk'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 norm 1 = 1\n[PROOFSTEP]\nhave h1 : \u00ac(1 : PadicSeq p) \u2248 0 := one_not_equiv_zero _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nh1 : \u00ac1 \u2248 0\n\u22a2 norm 1 = 1\n[PROOFSTEP]\nsimp [h1, norm, hp.1.one_lt]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 False\n[PROOFSTEP]\nhave hpn : 0 < padicNorm p (f (stationaryPoint hf)) - padicNorm p (g (stationaryPoint hg)) := sub_pos_of_lt hlt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 False\n[PROOFSTEP]\ncases' hfg _ hpn with N hN\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 False\n[PROOFSTEP]\nlet i := max N (max (stationaryPoint hf) (stationaryPoint hg))\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\n\u22a2 False\n[PROOFSTEP]\nhave hi : N \u2264 i := le_max_left _ _\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\n\u22a2 False\n[PROOFSTEP]\nhave hN' := hN _ hi\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' : padicNorm p (\u2191(f - g) i) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 False\n[PROOFSTEP]\nrw [lift_index_left hf N (stationaryPoint hg), lift_index_right hg N (stationaryPoint hf)] at hN' h hlt \n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191(f - g) i) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n\u22a2 False\n[PROOFSTEP]\nhave hpne : padicNorm p (f i) \u2260 padicNorm p (-g i) := by rwa [\u2190 padicNorm.neg (g i)] at h \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191(f - g) i) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n\u22a2 padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n[PROOFSTEP]\nrwa [\u2190 padicNorm.neg (g i)] at h \n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191(f - g) i) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n\u22a2 False\n[PROOFSTEP]\nrw [CauSeq.sub_apply, sub_eq_add_neg, add_eq_max_of_ne hpne, padicNorm.neg, max_eq_left_of_lt hlt] at hN' \n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n\u22a2 False\n[PROOFSTEP]\nhave : padicNorm p (f i) < padicNorm p (f i) :=\n  by\n  apply lt_of_lt_of_le hN'\n  apply sub_le_self\n  apply padicNorm.nonneg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n\u22a2 padicNorm p (\u2191f i) < padicNorm p (\u2191f i)\n[PROOFSTEP]\napply lt_of_lt_of_le hN'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n\u22a2 padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2264\n    padicNorm p (\u2191f i)\n[PROOFSTEP]\napply sub_le_self\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\n\u22a2 0 \u2264 padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\n[PROOFSTEP]\napply padicNorm.nonneg\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nhpn : 0 < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\nN : \u2115\nhlt :\n  padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nh :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) \u2260\n    padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhN :\n  \u2200 (j : \u2115),\n    j \u2265 N \u2192 padicNorm p (\u2191(f - g) j) < padicNorm p (\u2191f (stationaryPoint hf)) - padicNorm p (\u2191g (stationaryPoint hg))\ni : \u2115 := max N (max (stationaryPoint hf) (stationaryPoint hg))\nhi : N \u2264 i\nhN' :\n  padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) <\n    padicNorm p (\u2191f (max N (max (stationaryPoint hf) (stationaryPoint hg)))) -\n      padicNorm p (\u2191g (max N (max (stationaryPoint hf) (stationaryPoint hg))))\nhpne : padicNorm p (\u2191f i) \u2260 padicNorm p (-\u2191g i)\nthis : padicNorm p (\u2191f i) < padicNorm p (\u2191f i)\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\nby_contra h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 False\n[PROOFSTEP]\ncases' Decidable.em (padicNorm p (g (stationaryPoint hg)) < padicNorm p (f (stationaryPoint hf))) with hlt hnlt\n[GOAL]\ncase inl\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhlt : padicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 False\n[PROOFSTEP]\nexact norm_eq_of_equiv_aux hf hg hfg h hlt\n[GOAL]\ncase inr\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhnlt : \u00acpadicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 False\n[PROOFSTEP]\napply norm_eq_of_equiv_aux hg hf (Setoid.symm hfg) (Ne.symm h)\n[GOAL]\ncase inr\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhnlt : \u00acpadicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) < padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\napply lt_of_le_of_ne\n[GOAL]\ncase inr.a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhnlt : \u00acpadicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) \u2264 padicNorm p (\u2191g (stationaryPoint hg))\ncase inr.a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhnlt : \u00acpadicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\napply le_of_not_gt hnlt\n[GOAL]\ncase inr.a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfg : f \u2248 g\nh : \u00acpadicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\nhnlt : \u00acpadicNorm p (\u2191g (stationaryPoint hg)) < padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\napply h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : f \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave hg : g \u2248 0 := Setoid.trans (Setoid.symm hfg) hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : f \u2248 0\nhg : g \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nsimp [norm, hf, hg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : \u00acf \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave hg : \u00acg \u2248 0 := hf \u2218 Setoid.trans hfg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nunfold norm\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) =\n    if hf : g \u2248 0 then 0 else padicNorm p (\u2191g (stationaryPoint hf))\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f \u2248 g\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191g (stationaryPoint hg))\n[PROOFSTEP]\nexact norm_eq_of_equiv hf hg hfg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nunfold norm\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 (if hf : f + g \u2248 0 then 0 else padicNorm p (\u2191(f + g) (stationaryPoint hf))) \u2264\n    max (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf)))\n      (if hf : g \u2248 0 then 0 else padicNorm p (\u2191g (stationaryPoint hf)))\n[PROOFSTEP]\nsplit_ifs\n  -- Porting note: originally `padic_index_simp [hfg, hf, hg]`\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 padicNorm p (\u2191(f + g) (stationaryPoint hfg)) \u2264\n    max (padicNorm p (\u2191f (stationaryPoint hf))) (padicNorm p (\u2191g (stationaryPoint hg)))\n[PROOFSTEP]\nrw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 padicNorm p (\u2191(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) \u2264\n    max (padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))))\n      (padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg)))))\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 \u2115\ncase v3 p : \u2115 hp : Fact (Nat.Prime p) f g : PadicSeq p hfg : \u00acf + g \u2248 0 hf : \u00acf \u2248 0 hg : \u00acg \u2248 0 \u22a2 \u2115\n[PROOFSTEP]\napply padicNorm.nonarchimedean\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f + g \u2248 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave : 0 \u2264 max f.norm g.norm := le_max_of_le_left (norm_nonneg _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : f + g \u2248 0\nthis : 0 \u2264 max (norm f) (norm g)\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nsimpa only [hfg, norm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hfg' : f + g \u2248 g := by\n  change LimZero (f - 0) at hf \n  show LimZero (f + g - g); \u00b7 simpa only [sub_zero, add_sub_cancel] using hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\n\u22a2 f + g \u2248 g\n[PROOFSTEP]\nchange LimZero (f - 0) at hf \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : LimZero (f - 0)\n\u22a2 f + g \u2248 g\n[PROOFSTEP]\nshow LimZero (f + g - g)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : LimZero (f - 0)\n\u22a2 LimZero (f + g - g)\n[PROOFSTEP]\nsimpa only [sub_zero, add_sub_cancel] using hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hcfg : (f + g).norm = g.norm := norm_equiv hfg'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\nhcfg : norm (f + g) = norm g\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hcl : f.norm = 0 := (norm_zero_iff f).2 hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\nhcfg : norm (f + g) = norm g\nhcl : norm f = 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave : max f.norm g.norm = g.norm := by rw [hcl]; exact max_eq_right (norm_nonneg _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\nhcfg : norm (f + g) = norm g\nhcl : norm f = 0\n\u22a2 max (norm f) (norm g) = norm g\n[PROOFSTEP]\nrw [hcl]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\nhcfg : norm (f + g) = norm g\nhcl : norm f = 0\n\u22a2 max 0 (norm g) = norm g\n[PROOFSTEP]\nexact max_eq_right (norm_nonneg _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nhfg' : f + g \u2248 g\nhcfg : norm (f + g) = norm g\nhcl : norm f = 0\nthis : max (norm f) (norm g) = norm g\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nrw [this, hcfg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hfg' : f + g \u2248 f := by\n  change LimZero (g - 0) at hg \n  show LimZero (f + g - f); \u00b7 simpa only [add_sub_cancel', sub_zero] using hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\n\u22a2 f + g \u2248 f\n[PROOFSTEP]\nchange LimZero (g - 0) at hg \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : LimZero (g - 0)\n\u22a2 f + g \u2248 f\n[PROOFSTEP]\nshow LimZero (f + g - f)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : LimZero (g - 0)\n\u22a2 LimZero (f + g - f)\n[PROOFSTEP]\nsimpa only [add_sub_cancel', sub_zero] using hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hcfg : (f + g).norm = f.norm := norm_equiv hfg'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\nhcfg : norm (f + g) = norm f\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave hcl : g.norm = 0 := (norm_zero_iff g).2 hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\nhcfg : norm (f + g) = norm f\nhcl : norm g = 0\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nhave : max f.norm g.norm = f.norm := by rw [hcl]; exact max_eq_left (norm_nonneg _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\nhcfg : norm (f + g) = norm f\nhcl : norm g = 0\n\u22a2 max (norm f) (norm g) = norm f\n[PROOFSTEP]\nrw [hcl]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\nhcfg : norm (f + g) = norm f\nhcl : norm g = 0\n\u22a2 max (norm f) 0 = norm f\n[PROOFSTEP]\nexact max_eq_left (norm_nonneg _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nhfg' : f + g \u2248 f\nhcfg : norm (f + g) = norm f\nhcl : norm g = 0\nthis : max (norm f) (norm g) = norm f\n\u22a2 norm (f + g) \u2264 max (norm f) (norm g)\n[PROOFSTEP]\nrw [this, hcfg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : f \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave hg : g \u2248 0 := equiv_zero_of_val_eq_of_equiv_zero h hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : f \u2248 0\nhg : g \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nsimp only [hf, hg, norm, dif_pos]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave hg : \u00acg \u2248 0 := fun hg \u21a6\n  hf <| equiv_zero_of_val_eq_of_equiv_zero (by simp only [h, forall_const, eq_self_iff_true]) hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : g \u2248 0\n\u22a2 \u2200 (k : \u2115), padicNorm p (\u2191g k) = padicNorm p (\u2191f k)\n[PROOFSTEP]\nsimp only [h, forall_const, eq_self_iff_true]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nsimp only [hg, hf, norm, dif_neg, not_false_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 padicNorm p (\u2191f (stationaryPoint (_ : \u00acf \u2248 0))) = padicNorm p (\u2191g (stationaryPoint (_ : \u00acg \u2248 0)))\n[PROOFSTEP]\nlet i := max (stationaryPoint hf) (stationaryPoint hg)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 padicNorm p (\u2191f (stationaryPoint (_ : \u00acf \u2248 0))) = padicNorm p (\u2191g (stationaryPoint (_ : \u00acg \u2248 0)))\n[PROOFSTEP]\nhave hpf : padicNorm p (f (stationaryPoint hf)) = padicNorm p (f i) :=\n  by\n  apply stationaryPoint_spec\n  apply le_max_left\n  exact le_rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n[PROOFSTEP]\napply stationaryPoint_spec\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 stationaryPoint ?hf \u2264 i\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 stationaryPoint ?hf \u2264 stationaryPoint hf\ncase hf\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 \u00acf \u2248 0\n[PROOFSTEP]\napply le_max_left\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\n\u22a2 stationaryPoint hf \u2264 stationaryPoint hf\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 padicNorm p (\u2191f (stationaryPoint (_ : \u00acf \u2248 0))) = padicNorm p (\u2191g (stationaryPoint (_ : \u00acg \u2248 0)))\n[PROOFSTEP]\nhave hpg : padicNorm p (g (stationaryPoint hg)) = padicNorm p (g i) :=\n  by\n  apply stationaryPoint_spec\n  apply le_max_right\n  exact le_rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 padicNorm p (\u2191g (stationaryPoint hg)) = padicNorm p (\u2191g i)\n[PROOFSTEP]\napply stationaryPoint_spec\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 stationaryPoint ?hf \u2264 i\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 stationaryPoint ?hf \u2264 stationaryPoint hg\ncase hf\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 \u00acg \u2248 0\n[PROOFSTEP]\napply le_max_right\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\n\u22a2 stationaryPoint hg \u2264 stationaryPoint hg\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : \u2200 (k : \u2115), padicNorm p (\u2191f k) = padicNorm p (\u2191g k)\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\ni : \u2115 := max (stationaryPoint hf) (stationaryPoint hg)\nhpf : padicNorm p (\u2191f (stationaryPoint hf)) = padicNorm p (\u2191f i)\nhpg : padicNorm p (\u2191g (stationaryPoint hg)) = padicNorm p (\u2191g i)\n\u22a2 padicNorm p (\u2191f (stationaryPoint (_ : \u00acf \u2248 0))) = padicNorm p (\u2191g (stationaryPoint (_ : \u00acg \u2248 0)))\n[PROOFSTEP]\nrw [hpf, hpg, h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\na : PadicSeq p\n\u22a2 \u2200 (k : \u2115), padicNorm p (\u2191(-a) k) = padicNorm p (\u2191a k)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g \u2248 0\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave : LimZero (f + g - 0) := h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g \u2248 0\nthis : LimZero (f + g - 0)\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave : f \u2248 -g := show LimZero (f - -g) by simpa only [sub_zero, sub_neg_eq_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g \u2248 0\nthis : LimZero (f + g - 0)\n\u22a2 LimZero (f - -g)\n[PROOFSTEP]\nsimpa only [sub_zero, sub_neg_eq_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g \u2248 0\nthis\u271d : LimZero (f + g - 0)\nthis : f \u2248 -g\n\u22a2 norm f = norm g\n[PROOFSTEP]\nhave : f.norm = (-g).norm := norm_equiv this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nh : f + g \u2248 0\nthis\u271d\u00b9 : LimZero (f + g - 0)\nthis\u271d : f \u2248 -g\nthis : norm f = norm (-g)\n\u22a2 norm f = norm g\n[PROOFSTEP]\nsimpa only [norm_neg] using this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave : LimZero (f - 0) := hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nthis : LimZero (f - 0)\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave : f + g \u2248 g := show LimZero (f + g - g) by simpa only [sub_zero, add_sub_cancel]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nthis : LimZero (f - 0)\n\u22a2 LimZero (f + g - g)\n[PROOFSTEP]\nsimpa only [sub_zero, add_sub_cancel]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nthis\u271d : LimZero (f - 0)\nthis : f + g \u2248 g\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave h1 : (f + g).norm = g.norm := norm_equiv this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nthis\u271d : LimZero (f - 0)\nthis : f + g \u2248 g\nh1 : norm (f + g) = norm g\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave h2 : f.norm = 0 := (norm_zero_iff _).2 hf\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : f \u2248 0\nthis\u271d : LimZero (f - 0)\nthis : f + g \u2248 g\nh1 : norm (f + g) = norm g\nh2 : norm f = 0\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nrw [h1, h2, max_eq_right (norm_nonneg _)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave : LimZero (g - 0) := hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis : LimZero (g - 0)\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave : f + g \u2248 f := show LimZero (f + g - f) by rw [add_sub_cancel']; simpa only [sub_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis : LimZero (g - 0)\n\u22a2 LimZero (f + g - f)\n[PROOFSTEP]\nrw [add_sub_cancel']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis : LimZero (g - 0)\n\u22a2 LimZero g\n[PROOFSTEP]\nsimpa only [sub_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis\u271d : LimZero (g - 0)\nthis : f + g \u2248 f\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave h1 : (f + g).norm = f.norm := norm_equiv this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis\u271d : LimZero (g - 0)\nthis : f + g \u2248 f\nh1 : norm (f + g) = norm f\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nhave h2 : g.norm = 0 := (norm_zero_iff _).2 hg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : g \u2248 0\nthis\u271d : LimZero (g - 0)\nthis : f + g \u2248 f\nh1 : norm (f + g) = norm f\nh2 : norm g = 0\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nrw [h1, h2, max_eq_left (norm_nonneg _)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne : norm f \u2260 norm g\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 norm (f + g) = max (norm f) (norm g)\n[PROOFSTEP]\nunfold norm at hfgne \u22a2\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfgne :\n  (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) \u2260\n    if hf : g \u2248 0 then 0 else padicNorm p (\u2191g (stationaryPoint hf))\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 (if hf : f + g \u2248 0 then 0 else padicNorm p (\u2191(f + g) (stationaryPoint hf))) =\n    max (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf)))\n      (if hf : g \u2248 0 then 0 else padicNorm p (\u2191g (stationaryPoint hf)))\n[PROOFSTEP]\nsplit_ifs at hfgne \u22a2\n  -- Porting note: originally `padic_index_simp [hfg, hf, hg] at hfgne \u22a2`\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 padicNorm p (\u2191(f + g) (stationaryPoint hfg)) =\n    max (padicNorm p (\u2191f (stationaryPoint hf))) (padicNorm p (\u2191g (stationaryPoint hg)))\n[PROOFSTEP]\nrw [lift_index_left hf, lift_index_right hg] at hfgne \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 padicNorm p (\u2191(f + g) (stationaryPoint hfg)) =\n    max (padicNorm p (\u2191f (stationaryPoint hf))) (padicNorm p (\u2191g (stationaryPoint hg)))\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\n[PROOFSTEP]\nrw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 padicNorm p (\u2191(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) =\n    max (padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))))\n      (padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg)))))\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d\u00b9 : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne\u271d : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne :\n  padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (max ?v1 (max ?v2 (stationaryPoint hg))))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v2\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne\u271d : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\nhfgne : padicNorm p (\u2191f (max ?v1 (max (stationaryPoint hf) ?v3))) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v1\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\ncase v3\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : \u00acf + g \u2248 0\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\nhfgne : padicNorm p (\u2191f (stationaryPoint hf)) \u2260 padicNorm p (\u2191g (stationaryPoint hg))\n\u22a2 \u2115\n[PROOFSTEP]\nexact padicNorm.add_eq_max_of_ne hfgne\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Zero \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 One \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Add \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Mul \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Sub \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Neg \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Div \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 AddCommGroup \u211a_[p]\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nq r : \u211a\nheq : q = r\n\u22a2 const (padicNorm p) q \u2248 const (padicNorm p) r\n[PROOFSTEP]\nrw [heq]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nq r : \u211a\nh : q = r\n\u22a2 \u2191q = \u2191r\n[PROOFSTEP]\nrw [h]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nm n : \u2115\n\u22a2 \u2191m = \u2191n \u2192 m = n\n[PROOFSTEP]\nrw [\u2190 Rat.cast_coe_nat]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nm n : \u2115\n\u22a2 \u2191\u2191m = \u2191n \u2192 m = n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nm n : \u2115\n\u22a2 m = n \u2192 m = n\n[PROOFSTEP]\nexact id\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a_[p]\nr : CauSeq \u211a (padicNorm p)\n\u22a2 MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        (Quotient.mk equiv r) =\n      0 \u2194\n    Quotient.mk equiv r = 0\n[PROOFSTEP]\nrw [Padic.zero_def, Quotient.eq]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a_[p]\nr : CauSeq \u211a (padicNorm p)\n\u22a2 MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        (Quotient.mk equiv r) =\n      0 \u2194\n    r \u2248 0\n[PROOFSTEP]\nexact PadicSeq.norm_zero_iff r\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 MulHom.toFun\n      { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n        map_mul' :=\n          (_ :\n            \u2200 (q r : \u211a_[p]),\n              Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                  (q * r) =\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r) }\n      (q + r) \u2264\n    MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        q +\n      MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        r\n[PROOFSTEP]\ntrans\n  max ((Quotient.lift PadicSeq.norm <| @PadicSeq.norm_equiv _ _) q)\n    ((Quotient.lift PadicSeq.norm <| @PadicSeq.norm_equiv _ _) r)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 MulHom.toFun\n      { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n        map_mul' :=\n          (_ :\n            \u2200 (q r : \u211a_[p]),\n              Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                  (q * r) =\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r) }\n      (q + r) \u2264\n    max (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q)\n      (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r)\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 max (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q)\n      (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r) \u2264\n    MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        q +\n      MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        r\n[PROOFSTEP]\nexact Quotient.inductionOn\u2082 q r <| PadicSeq.norm_nonarchimedean\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 max (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q)\n      (Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r) \u2264\n    MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        q +\n      MulHom.toFun\n        { toFun := Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g),\n          map_mul' :=\n            (_ :\n              \u2200 (q r : \u211a_[p]),\n                Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                    (q * r) =\n                  Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) q *\n                    Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n                      r) }\n        r\n[PROOFSTEP]\nrefine' max_le_add_of_nonneg (Quotient.inductionOn q <| PadicSeq.norm_nonneg) _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 0 \u2264 Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g) r\n[PROOFSTEP]\nexact Quotient.inductionOn r <| PadicSeq.norm_nonneg\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (Padic.mk f - \u2191(\u2191f i)) < \u03b5\n[PROOFSTEP]\ndsimp [padicNormE]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 N,\n    \u2200 (i : \u2115),\n      i \u2265 N \u2192\n        Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n            (Padic.mk f - \u2191(\u2191f i)) <\n          \u03b5\n[PROOFSTEP]\nchange \u2203 N, \u2200 i \u2265 N, (f - const _ (f i)).norm < \u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 PadicSeq.norm (f - const (padicNorm p) (\u2191f i)) < \u03b5\n[PROOFSTEP]\nby_contra' h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\n\u22a2 False\n[PROOFSTEP]\ncases' cauchy\u2082 f h\u03b5 with N hN\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\n\u22a2 False\n[PROOFSTEP]\nrcases h N with \u27e8i, hi, hge\u27e9\n[GOAL]\ncase intro.intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge : \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\n\u22a2 False\n[PROOFSTEP]\nhave hne : \u00acf - const (padicNorm p) (f i) \u2248 0 := by\n  intro h\n  unfold PadicSeq.norm at hge ; split_ifs at hge \n  exact not_lt_of_ge hge h\u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge : \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\n\u22a2 \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\n[PROOFSTEP]\nintro h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh\u271d : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge : \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nh : f - const (padicNorm p) (\u2191f i) \u2248 0\n\u22a2 False\n[PROOFSTEP]\nunfold PadicSeq.norm at hge \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh\u271d : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge :\n  \u03b5 \u2264\n    if hf : f - const (padicNorm p) (\u2191f i) \u2248 0 then 0\n    else padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint hf))\nh : f - const (padicNorm p) (\u2191f i) \u2248 0\n\u22a2 False\n[PROOFSTEP]\nsplit_ifs at hge \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh\u271d : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nh : f - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 0\n\u22a2 False\n[PROOFSTEP]\nexact not_lt_of_ge hge h\u03b5\n[GOAL]\ncase intro.intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge : \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nhne : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\n\u22a2 False\n[PROOFSTEP]\nunfold PadicSeq.norm at hge \n[GOAL]\ncase intro.intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhge :\n  \u03b5 \u2264\n    if hf : f - const (padicNorm p) (\u2191f i) \u2248 0 then 0\n    else padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint hf))\nhne : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\n\u22a2 False\n[PROOFSTEP]\nsplit_ifs at hge \n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nh\u271d : f - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 0\n\u22a2 False\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n\u22a2 False\n[PROOFSTEP]\nexact not_le_of_gt h\u03b5 hge\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n\u22a2 False\n[PROOFSTEP]\napply not_le_of_gt _ hge\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n\u22a2 \u03b5 > padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n[PROOFSTEP]\ncases' _root_.em (N \u2264 stationaryPoint hne) with hgen hngen\n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\nhgen : N \u2264 stationaryPoint hne\n\u22a2 \u03b5 > padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n[PROOFSTEP]\napply hN _ hgen _ hi\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\nhngen : \u00acN \u2264 stationaryPoint hne\n\u22a2 \u03b5 > padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n[PROOFSTEP]\nhave := stationaryPoint_spec hne le_rfl (le_of_not_le hngen)\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\nhngen : \u00acN \u2264 stationaryPoint hne\nthis :\n  padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) N) =\n    padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint hne))\n\u22a2 \u03b5 > padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nh : \u2200 (N : \u2115), \u2203 i, i \u2265 N \u2227 \u03b5 \u2264 PadicSeq.norm (f - const (padicNorm p) (\u2191f i))\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2200 (k : \u2115), k \u2265 N \u2192 padicNorm p (\u2191f j - \u2191f k) < \u03b5\ni : \u2115\nhi : i \u2265 N\nhne h\u271d : \u00acf - const (padicNorm p) (\u2191f i) \u2248 0\nhge : \u03b5 \u2264 padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint h\u271d))\nhngen : \u00acN \u2264 stationaryPoint hne\nthis :\n  padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) N) =\n    padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) (stationaryPoint hne))\n\u22a2 \u03b5 > padicNorm p (\u2191(f - const (padicNorm p) (\u2191f i)) N)\n[PROOFSTEP]\nexact hN _ le_rfl _ hi\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\n\u22a2 \u2191padicNormE (Quotient.mk equiv q' - \u2191(\u2191q' N)) < \u03b5\n[PROOFSTEP]\ndsimp [padicNormE]\n  -- Porting note: `change` \u2192 `convert_to` (`change` times out!)\n        -- and add `PadicSeq p` type annotation\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\n\u22a2 Quotient.lift PadicSeq.norm (_ : \u2200 {f g : PadicSeq p}, f \u2248 g \u2192 PadicSeq.norm f = PadicSeq.norm g)\n      (Completion.mk q' - \u2191(\u2191q' N)) <\n    \u03b5\n[PROOFSTEP]\nconvert_to PadicSeq.norm (q' - const _ (q' N) : PadicSeq p) < \u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\n\u22a2 PadicSeq.norm (q' - const (padicNorm p) (\u2191q' N)) < \u03b5\n[PROOFSTEP]\ncases' Decidable.em (q' - const (padicNorm p) (q' N) \u2248 0) with heq hne'\n[GOAL]\ncase inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nheq : q' - const (padicNorm p) (\u2191q' N) \u2248 0\n\u22a2 PadicSeq.norm (q' - const (padicNorm p) (\u2191q' N)) < \u03b5\n[PROOFSTEP]\nsimpa only [heq, PadicSeq.norm, dif_pos]\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\n\u22a2 PadicSeq.norm (q' - const (padicNorm p) (\u2191q' N)) < \u03b5\n[PROOFSTEP]\nsimp only [PadicSeq.norm, dif_neg hne']\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\n\u22a2 padicNorm p (\u2191(q' - const (padicNorm p) (\u2191q' N)) (stationaryPoint hne')) < \u03b5\n[PROOFSTEP]\nchange padicNorm p (q' _ - q' _) < \u03b5\n[GOAL]\ncase inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\n\u22a2 padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) < \u03b5\n[PROOFSTEP]\ncases' Decidable.em (stationaryPoint hne' \u2264 N) with hle hle\n[GOAL]\ncase inr.inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\nhle : stationaryPoint hne' \u2264 N\n\u22a2 padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) < \u03b5\n[PROOFSTEP]\nhave := (stationaryPoint_spec hne' le_rfl hle).symm\n[GOAL]\ncase inr.inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis\u271d : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\nhle : stationaryPoint hne' \u2264 N\nthis :\n  padicNorm p (\u2191(q' - const (padicNorm p) (\u2191q' N)) (stationaryPoint hne')) =\n    padicNorm p (\u2191(q' - const (padicNorm p) (\u2191q' N)) N)\n\u22a2 padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) < \u03b5\n[PROOFSTEP]\nsimp only [const_apply, sub_apply, padicNorm.zero, sub_self] at this \n[GOAL]\ncase inr.inl\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis\u271d : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\nhle : stationaryPoint hne' \u2264 N\nthis : padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) = 0\n\u22a2 padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) < \u03b5\n[PROOFSTEP]\nsimpa only [this]\n[GOAL]\ncase inr.inr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nq : \u211a_[p]\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nq' : CauSeq \u211a (padicNorm p)\nthis : \u2203 N, \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nN : \u2115\nhN : \u2200 (m : \u2115), m \u2265 N \u2192 \u2200 (n : \u2115), n \u2265 N \u2192 padicNorm p (\u2191q' m - \u2191q' n) < \u03b5\nhne' : \u00acq' - const (padicNorm p) (\u2191q' N) \u2248 0\nhle : \u00acstationaryPoint hne' \u2264 N\n\u22a2 padicNorm p (\u2191q' (stationaryPoint hne') - \u2191q' N) < \u03b5\n[PROOFSTEP]\nexact hN _ (lt_of_not_ge hle).le _ le_rfl\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nn : \u2115\n\u22a2 0 < \u2191n + 1\n[PROOFSTEP]\nexact_mod_cast succ_pos _\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5\n[PROOFSTEP]\nrefine' (exists_nat_gt (1 / \u03b5)).imp fun N hN i hi \u21a6 _\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\n\u22a2 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5\n[PROOFSTEP]\nhave h := Classical.choose_spec (rat_dense' (f i) (div_nat_pos i))\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5\n[PROOFSTEP]\nrefine' lt_of_lt_of_le h ((div_le_iff' <| by exact_mod_cast succ_pos _).mpr _)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 0 < \u2191i + 1\n[PROOFSTEP]\nexact_mod_cast succ_pos _\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 1 \u2264 (\u2191i + 1) * \u03b5\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 1 \u2264 \u2191i * \u03b5 + 1 * \u03b5\n[PROOFSTEP]\napply le_add_of_le_of_nonneg\n[GOAL]\ncase hbc\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 1 \u2264 \u2191i * \u03b5\n[PROOFSTEP]\nexact (div_le_iff h\u03b5).mp (le_trans (le_of_lt hN) (by exact_mod_cast hi))\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 \u2191N \u2264 \u2191i\n[PROOFSTEP]\nexact_mod_cast hi\n[GOAL]\ncase ha\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 0 \u2264 1 * \u03b5\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase ha.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nN : \u2115\nhN : 1 / \u03b5 < \u2191N\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191f i - \u2191(Classical.choose (_ : \u2203 r, \u2191padicNormE (\u2191f i - \u2191r) < 1 / (\u2191i + 1)))) < 1 / (\u2191i + 1)\n\u22a2 0 < 1 * \u03b5\n[PROOFSTEP]\nsimpa\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5\n[PROOFSTEP]\nhave h\u03b53 : 0 < \u03b5 / 3 := div_pos h\u03b5 (by norm_num)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 0 < 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5\n[PROOFSTEP]\nlet \u27e8N, hN\u27e9 := exi_rat_seq_conv f h\u03b53\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5\n[PROOFSTEP]\nlet \u27e8N2, hN2\u27e9 := f.cauchy\u2082 h\u03b53\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 padicNorm p (limSeq f j - limSeq f i) < \u03b5\n[PROOFSTEP]\nexists max N N2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\n\u22a2 \u2200 (j : \u2115), j \u2265 max N N2 \u2192 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5\n[PROOFSTEP]\nintro j hj\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5\n[PROOFSTEP]\nsuffices padicNormE (limSeq f j - f (max N N2) + (f (max N N2) - limSeq f (max N N2)) : \u211a_[p]) < \u03b5\n  by\n  ring_nf at this \u22a2\n  rw [\u2190 padicNormE.eq_padic_norm']\n  exact_mod_cast this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) < \u03b5\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5\n[PROOFSTEP]\nring_nf at this \u22a2\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 padicNorm p (limSeq f j - limSeq f (max N N2)) < \u03b5\n[PROOFSTEP]\nrw [\u2190 padicNormE.eq_padic_norm']\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191(limSeq f (max N N2))) < \u03b5\n\u22a2 \u2191padicNormE \u2191(limSeq f j - limSeq f (max N N2)) < \u03b5\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) < \u03b5\n[PROOFSTEP]\napply lt_of_le_of_lt\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2) + (\u2191f (max N N2) - \u2191(limSeq f (max N N2)))) \u2264 ?b\n[PROOFSTEP]\napply padicNormE.add_le\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) + \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) < \u03b5\n[PROOFSTEP]\nrw [\u2190 add_thirds \u03b5]\n[GOAL]\ncase a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) + \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) <\n    \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\napply _root_.add_lt_add\n[GOAL]\ncase a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\nsuffices padicNormE (limSeq f j - f j + (f j - f (max N N2)) : \u211a_[p]) < \u03b5 / 3 + \u03b5 / 3 by simpa only [sub_add_sub_cancel]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\nthis : \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) < \u03b5 / 3 + \u03b5 / 3\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\nsimpa only [sub_add_sub_cancel]\n[GOAL]\ncase a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) < \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\napply lt_of_le_of_lt\n[GOAL]\ncase a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j + (\u2191f j - \u2191f (max N N2))) \u2264 ?a.h\u2081.b\u271d\n[PROOFSTEP]\napply padicNormE.add_le\n[GOAL]\ncase a.h\u2081.a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j) + \u2191padicNormE (\u2191f j - \u2191f (max N N2)) < \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\napply _root_.add_lt_add\n[GOAL]\ncase a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(limSeq f j) - \u2191f j) < \u03b5 / 3\n[PROOFSTEP]\nrw [padicNormE.map_sub]\n[GOAL]\ncase a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191(limSeq f j)) < \u03b5 / 3\n[PROOFSTEP]\napply_mod_cast hN j\n[GOAL]\ncase a.h\u2081.a.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 j \u2265 N\n[PROOFSTEP]\nexact le_of_max_le_left hj\n[GOAL]\ncase a.h\u2081.a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f j - \u2191f (max N N2)) < \u03b5 / 3\n[PROOFSTEP]\nexact hN2 _ (le_of_max_le_right hj) _ (le_max_right _ _)\n[GOAL]\ncase a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f (max N N2) - \u2191(limSeq f (max N N2))) < \u03b5 / 3\n[PROOFSTEP]\napply_mod_cast hN (max N N2)\n[GOAL]\ncase a.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh\u03b53 : 0 < \u03b5 / 3\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 3\nN2 : \u2115\nhN2 : \u2200 (j : \u2115), j \u2265 N2 \u2192 \u2200 (k : \u2115), k \u2265 N2 \u2192 \u2191padicNormE (\u2191f j - \u2191f k) < \u03b5 / 3\nj : \u2115\nhj : j \u2265 max N N2\n\u22a2 max N N2 \u2265 N\n[PROOFSTEP]\napply le_max_left\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (Padic.lim f - \u2191f i) < \u03b5\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 := exi_rat_seq_conv f (half_pos h\u03b5)\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (Padic.lim f - \u2191f i) < \u03b5\n[PROOFSTEP]\nobtain \u27e8N2, hN2\u27e9 := padicNormE.defn (lim' f) (half_pos h\u03b5)\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (Padic.lim f - \u2191f i) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8max N N2, fun i hi \u21a6 _\u27e9\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (Padic.lim f - \u2191f i) < \u03b5\n[PROOFSTEP]\nrw [\u2190 sub_add_sub_cancel _ (lim' f i : \u211a_[p]) _]\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (Padic.lim f - \u2191(\u2191(Padic.lim' f) i) + (\u2191(\u2191(Padic.lim' f) i) - \u2191f i)) < \u03b5\n[PROOFSTEP]\nrefine' (padicNormE.add_le _ _).trans_lt _\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (Padic.lim f - \u2191(\u2191(Padic.lim' f) i)) + \u2191padicNormE (\u2191(\u2191(Padic.lim' f) i) - \u2191f i) < \u03b5\n[PROOFSTEP]\nrw [\u2190 add_halves \u03b5]\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (Padic.lim f - \u2191(\u2191(Padic.lim' f) i)) + \u2191padicNormE (\u2191(\u2191(Padic.lim' f) i) - \u2191f i) < \u03b5 / 2 + \u03b5 / 2\n[PROOFSTEP]\napply _root_.add_lt_add\n[GOAL]\ncase intro.intro.h\u2081\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (Padic.lim f - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\n[PROOFSTEP]\napply hN2 _ (le_of_max_le_right hi)\n[GOAL]\ncase intro.intro.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191(\u2191(Padic.lim' f) i) - \u2191f i) < \u03b5 / 2\n[PROOFSTEP]\nrw [padicNormE.map_sub]\n[GOAL]\ncase intro.intro.h\u2082\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - \u2191(limSeq f i)) < \u03b5 / 2\nN2 : \u2115\nhN2 : \u2200 (i : \u2115), i \u2265 N2 \u2192 \u2191padicNormE (mk (Padic.lim' f) - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\ni : \u2115\nhi : i \u2265 max N N2\n\u22a2 \u2191padicNormE (\u2191f i - \u2191(\u2191(Padic.lim' f) i)) < \u03b5 / 2\n[PROOFSTEP]\nexact hN _ (le_of_max_le_left hi)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\n\u22a2 \u2203 q, \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - q) < \u03b5\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := complete' f\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nx : \u211a_[p]\nhx : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u22a2 \u2203 q, \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - q) < \u03b5\n[PROOFSTEP]\nrefine \u27e8x, fun \u03b5 h\u03b5 => ?_\u27e9\n[GOAL]\ncase intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nx : \u211a_[p]\nhx : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - x) < \u03b5\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 := hx \u03b5 h\u03b5\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nx : \u211a_[p]\nhx : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u22a2 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191f i - x) < \u03b5\n[PROOFSTEP]\nrefine \u27e8N, fun i hi => ?_\u27e9\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nx : \u211a_[p]\nhx : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\ni : \u2115\nhi : i \u2265 N\n\u22a2 \u2191padicNormE (\u2191f i - x) < \u03b5\n[PROOFSTEP]\nrw [padicNormE.map_sub]\n[GOAL]\ncase intro.intro\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] \u2191padicNormE\nx : \u211a_[p]\nhx : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (x - \u2191f i) < \u03b5\ni : \u2115\nhi : i \u2265 N\n\u22a2 \u2191padicNormE (x - \u2191f i) < \u03b5\n[PROOFSTEP]\nexact hN i hi\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 (x : \u211a_[p]), dist x x = 0\n[PROOFSTEP]\nsimp [dist]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx y : \u211a_[p]\n\u22a2 dist x y = dist y x\n[PROOFSTEP]\nsimp [dist, \u2190 padicNormE.map_neg (x - y : \u211a_[p])]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx y z : \u211a_[p]\n\u22a2 dist x z \u2264 dist x y + dist y z\n[PROOFSTEP]\ndsimp [dist]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx y z : \u211a_[p]\n\u22a2 \u2191(\u2191padicNormE (x - z)) \u2264 \u2191(\u2191padicNormE (x - y)) + \u2191(\u2191padicNormE (y - z))\n[PROOFSTEP]\nexact_mod_cast padicNormE.sub_le x y z\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 (x y : \u211a_[p]),\n    (fun x y => \u2191{ val := \u2191(\u2191padicNormE (x - y)), property := (_ : 0 \u2264 \u2191(\u2191padicNormE (x - y))) }) x y =\n      ENNReal.ofReal (dist x y)\n[PROOFSTEP]\nintros\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx\u271d y\u271d : \u211a_[p]\n\u22a2 (fun x y => \u2191{ val := \u2191(\u2191padicNormE (x - y)), property := (_ : 0 \u2264 \u2191(\u2191padicNormE (x - y))) }) x\u271d y\u271d =\n    ENNReal.ofReal (dist x\u271d y\u271d)\n[PROOFSTEP]\nexact (ENNReal.ofReal_eq_coe_nnreal _).symm\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 {x y : \u211a_[p]}, dist x y = 0 \u2192 x = y\n[PROOFSTEP]\ndsimp [dist]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 {x y : \u211a_[p]}, \u2191(\u2191padicNormE (x - y)) = 0 \u2192 x = y\n[PROOFSTEP]\nintro _ _ h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx\u271d y\u271d : \u211a_[p]\nh : \u2191(\u2191padicNormE (x\u271d - y\u271d)) = 0\n\u22a2 x\u271d = y\u271d\n[PROOFSTEP]\napply eq_of_sub_eq_zero\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx\u271d y\u271d : \u211a_[p]\nh : \u2191(\u2191padicNormE (x\u271d - y\u271d)) = 0\n\u22a2 x\u271d - y\u271d = 0\n[PROOFSTEP]\napply padicNormE.eq_zero.1\n[GOAL]\ncase h\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx\u271d y\u271d : \u211a_[p]\nh : \u2191(\u2191padicNormE (x\u271d - y\u271d)) = 0\n\u22a2 \u2191padicNormE (x\u271d - y\u271d) = 0\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nsrc\u271d\u00b9 : Field \u211a_[p] := field\nsrc\u271d : MetricSpace \u211a_[p] := metricSpace p\n\u22a2 \u2200 (a b : \u211a_[p]), \u2016a * b\u2016 = \u2016a\u2016 * \u2016b\u2016\n[PROOFSTEP]\nsimp [Norm.norm, map_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 (x y : \u211a_[p]), \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp [Norm.norm, map_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nq : \u211a_[p]\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u211a\nh\u03b5'l : 0 < \u2191\u03b5'\nh\u03b5'r : \u2191\u03b5' < \u03b5\n\u22a2 0 < \u03b5'\n[PROOFSTEP]\nsimpa using h\u03b5'l\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nq : \u211a_[p]\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u03b5' : \u211a\nh\u03b5'l : 0 < \u2191\u03b5'\nh\u03b5'r : \u2191\u03b5' < \u03b5\nr : \u211a\nhr : \u2191padicNormE (q - \u2191r) < \u03b5'\n\u22a2 \u2016q - \u2191r\u2016 < \u2191\u03b5'\n[PROOFSTEP]\nsimpa [Norm.norm] using hr\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 \u2016q * r\u2016 = \u2016q\u2016 * \u2016r\u2016\n[PROOFSTEP]\nsimp [Norm.norm, map_mul]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 \u2016q + r\u2016 \u2264 max \u2016q\u2016 \u2016r\u2016\n[PROOFSTEP]\ndsimp [norm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\n\u22a2 \u2191(\u2191padicNormE (q + r)) \u2264 max \u2191(\u2191padicNormE q) \u2191(\u2191padicNormE r)\n[PROOFSTEP]\nexact_mod_cast nonarchimedean' _ _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\nh : \u2016q\u2016 \u2260 \u2016r\u2016\n\u22a2 \u2016q + r\u2016 = max \u2016q\u2016 \u2016r\u2016\n[PROOFSTEP]\ndsimp [norm] at h \u22a2\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\nh : \u00ac\u2191(\u2191padicNormE q) = \u2191(\u2191padicNormE r)\n\u22a2 \u2191(\u2191padicNormE (q + r)) = max \u2191(\u2191padicNormE q) \u2191(\u2191padicNormE r)\n[PROOFSTEP]\nhave : padicNormE q \u2260 padicNormE r := by exact_mod_cast h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\nh : \u00ac\u2191(\u2191padicNormE q) = \u2191(\u2191padicNormE r)\n\u22a2 \u2191padicNormE q \u2260 \u2191padicNormE r\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq r : \u211a_[p]\nh : \u00ac\u2191(\u2191padicNormE q) = \u2191(\u2191padicNormE r)\nthis : \u2191padicNormE q \u2260 \u2191padicNormE r\n\u22a2 \u2191(\u2191padicNormE (q + r)) = max \u2191(\u2191padicNormE q) \u2191(\u2191padicNormE r)\n[PROOFSTEP]\nexact_mod_cast add_eq_max_of_ne' this\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\n\u22a2 \u2016\u2191q\u2016 = \u2191(padicNorm p q)\n[PROOFSTEP]\ndsimp [norm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a\n\u22a2 \u2191(\u2191padicNormE \u2191q) = \u2191(padicNorm p q)\n[PROOFSTEP]\nrw [\u2190 padicNormE.eq_padic_norm']\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2016\u2191p\u2016 = (\u2191p)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 @Rat.cast_coe_nat \u211d _ p]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2016\u2191p\u2016 = (\u2191\u2191p)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 @Rat.cast_coe_nat \u211a_[p] _ p]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2016\u2191\u2191p\u2016 = (\u2191\u2191p)\u207b\u00b9\n[PROOFSTEP]\nsimp [hp.1.ne_zero, hp.1.ne_one, norm, padicNorm, padicValRat, padicValInt, zpow_neg, -Rat.cast_coe_nat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2016\u2191p\u2016 < 1\n[PROOFSTEP]\nrw [norm_p]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 (\u2191p)\u207b\u00b9 < 1\n[PROOFSTEP]\napply inv_lt_one\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\n\u22a2 \u2016\u2191p ^ n\u2016 = \u2191p ^ (-n)\n[PROOFSTEP]\nrw [norm_zpow, norm_p, zpow_neg, inv_zpow]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2016\u2191p ^ n\u2016 = \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nrw [\u2190 norm_p_zpow, zpow_ofNat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nsrc\u271d : NormedField \u211a_[p] := Padic.normedField p\n\u22a2 1 < \u2016(\u2191p)\u207b\u00b9\u2016\n[PROOFSTEP]\nrw [norm_inv, norm_p, inv_inv]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nsrc\u271d : NormedField \u211a_[p] := Padic.normedField p\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk equiv f \u2260 0\nthis : \u00acf \u2248 0\nn : \u2124\nhn : PadicSeq.norm f = \u2191p ^ (-n)\n\u22a2 \u2016Quotient.mk equiv f\u2016 = \u2191(\u2191p ^ (-n))\n[PROOFSTEP]\nrw [\u2190 hn]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk equiv f \u2260 0\nthis : \u00acf \u2248 0\nn : \u2124\nhn : PadicSeq.norm f = \u2191p ^ (-n)\n\u22a2 \u2016Quotient.mk equiv f\u2016 = \u2191(PadicSeq.norm f)\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nq : \u211a_[p]\nh : q = 0\n\u22a2 \u2016q\u2016 = \u21910\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : n = 0\n\u22a2 \u2016\u2191(Rat.mk' n d)\u2016 \u2264 1\n[PROOFSTEP]\nhave : (\u27e8n, d, hn, hd\u27e9 : \u211a) = 0 := Rat.zero_iff_num_zero.mpr hnz\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : n = 0\nthis : Rat.mk' n d = 0\n\u22a2 \u2016\u2191(Rat.mk' n d)\u2016 \u2264 1\n[PROOFSTEP]\nrw [this]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : n = 0\nthis : Rat.mk' n d = 0\n\u22a2 \u2016\u21910\u2016 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\n\u22a2 \u2016\u2191(Rat.mk' n d)\u2016 \u2264 1\n[PROOFSTEP]\nhave hnz' : (\u27e8n, d, hn, hd\u27e9 : \u211a) \u2260 0 := mt Rat.zero_iff_num_zero.1 hnz\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 \u2016\u2191(Rat.mk' n d)\u2016 \u2264 1\n[PROOFSTEP]\nrw [padicNormE.eq_padicNorm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 \u2191(padicNorm p (Rat.mk' n d)) \u2264 1\n[PROOFSTEP]\nnorm_cast\n  -- Porting note: `Nat.cast_zero` instead of another `norm_cast` call\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 padicNorm p (Rat.mk' n d) \u2264 1\n[PROOFSTEP]\nrw [padicNorm.eq_zpow_of_nonzero hnz', padicValRat, neg_sub, padicValNat.eq_zero_of_not_dvd hq, Nat.cast_zero, zero_sub,\n  zpow_neg, zpow_ofNat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 (\u2191p ^ padicValInt p (Rat.mk' n d).num)\u207b\u00b9 \u2264 1\n[PROOFSTEP]\napply inv_le_one\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 1 \u2264 \u2191p ^ padicValInt p (Rat.mk' n d).num\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 1 \u2264 p ^ padicValInt p (Rat.mk' n d).num\n[PROOFSTEP]\napply one_le_pow\n[GOAL]\ncase ha.h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2124\nd : \u2115\nhn : d \u2260 0\nhd : coprime (Int.natAbs n) d\nhq : \u00acp \u2223 d\nhnz : \u00acn = 0\nhnz' : Rat.mk' n d \u2260 0\n\u22a2 0 < p\n[PROOFSTEP]\nexact hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124\n\u22a2 \u00acp \u2223 (\u2191z).den\n[PROOFSTEP]\nsimp [hp.1.ne_one]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124\nthis : \u2016\u2191\u2191z\u2016 \u2264 1\n\u22a2 \u2016\u2191z\u2016 \u2264 1\n[PROOFSTEP]\nsimpa\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2016\u2191k\u2016 < 1 \u2194 \u2191p \u2223 k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2016\u2191k\u2016 < 1 \u2192 \u2191p \u2223 k\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u2016\u2191k\u2016 < 1\n\u22a2 \u2191p \u2223 k\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase mp\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 \u2264 \u2016\u2191k\u2016\n[PROOFSTEP]\napply le_of_eq\n[GOAL]\ncase mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 1 = \u2016\u2191k\u2016\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase mp.a\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 = 1\n[PROOFSTEP]\ncalc\n  \u2016(k : \u211a_[p])\u2016 = \u2016((k : \u211a) : \u211a_[p])\u2016 := by norm_cast\n  _ = padicNorm p k := (padicNormE.eq_padicNorm _)\n  _ = 1 := by exact_mod_cast (int_eq_one_iff k).mpr h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 = \u2016\u2191\u2191k\u2016\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nh : \u00ac\u2191p \u2223 k\n\u22a2 \u2191(padicNorm p \u2191k) = 1\n[PROOFSTEP]\nexact_mod_cast (int_eq_one_iff k).mpr h\n[GOAL]\ncase mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\n\u22a2 \u2191p \u2223 k \u2192 \u2016\u2191k\u2016 < 1\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191(\u2191p * x)\u2016 < 1\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p * \u2191x\u2016 < 1\n[PROOFSTEP]\nrw [padicNormE.mul]\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p\u2016 * \u2016\u2191x\u2016 < 1\n[PROOFSTEP]\ncalc\n  _ \u2264 \u2016(p : \u211a_[p])\u2016 * 1 := mul_le_mul le_rfl (by simpa using norm_int_le_one _) (norm_nonneg _) (norm_nonneg _)\n  _ < 1 := by\n    rw [mul_one, padicNormE.norm_p]\n    apply inv_lt_one\n    exact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191x\u2016 \u2264 1\n[PROOFSTEP]\nsimpa using norm_int_le_one _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 \u2016\u2191p\u2016 * 1 < 1\n[PROOFSTEP]\nrw [mul_one, padicNormE.norm_p]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 (\u2191p)\u207b\u00b9 < 1\n[PROOFSTEP]\napply inv_lt_one\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\n\u22a2 \u2016\u2191k\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191(p ^ n) \u2223 k\n[PROOFSTEP]\nhave : (p : \u211d) ^ (-n : \u2124) = (p : \u211a) ^ (-n : \u2124) := by simp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\n\u22a2 \u2191p ^ (-\u2191n) = \u2191(\u2191p ^ (-\u2191n))\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\nthis : \u2191p ^ (-\u2191n) = \u2191(\u2191p ^ (-\u2191n))\n\u22a2 \u2016\u2191k\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191(p ^ n) \u2223 k\n[PROOFSTEP]\nrw [show (k : \u211a_[p]) = ((k : \u211a) : \u211a_[p]) by norm_cast, eq_padicNorm, this]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\nthis : \u2191p ^ (-\u2191n) = \u2191(\u2191p ^ (-\u2191n))\n\u22a2 \u2191k = \u2191\u2191k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\nthis : \u2191p ^ (-\u2191n) = \u2191(\u2191p ^ (-\u2191n))\n\u22a2 \u2191(padicNorm p \u2191k) \u2264 \u2191(\u2191p ^ (-\u2191n)) \u2194 \u2191(p ^ n) \u2223 k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\nthis : \u2191p ^ (-\u2191n) = \u2191(\u2191p ^ (-\u2191n))\n\u22a2 padicNorm p \u2191k \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191(p ^ n) \u2223 k\n[PROOFSTEP]\nrw [\u2190 padicNorm.dvd_iff_norm_le]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z2\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 \u2016z1 + z2\u2016 \u2265 \u2016z2\u2016\n[PROOFSTEP]\nrw [padicNormE.add_eq_max_of_ne hne]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z2\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z2\u2016\n[PROOFSTEP]\napply le_max_right\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 \u2016z1 + z2\u2016 \u2265 \u2016z1\u2016\n[PROOFSTEP]\nrw [padicNormE.add_eq_max_of_ne hne]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u211a_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z1\u2016\n[PROOFSTEP]\napply le_max_left\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\n\u22a2 \u2203 b, f \u2248 const norm b\n[PROOFSTEP]\nhave cau_seq_norm_e : IsCauSeq padicNormE f := fun \u03b5 h\u03b5 =>\n  by\n  have h := isCauSeq f \u03b5 (by exact_mod_cast h\u03b5)\n  dsimp [norm] at h \n  exact_mod_cast h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2191padicNormE (\u2191f j - \u2191f i) < \u03b5\n[PROOFSTEP]\nhave h := isCauSeq f \u03b5 (by exact_mod_cast h\u03b5)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2191\u03b5 > 0\n[PROOFSTEP]\nexact_mod_cast h\u03b5\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191f j - \u2191f i\u2016 < \u2191\u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2191padicNormE (\u2191f j - \u2191f i) < \u03b5\n[PROOFSTEP]\ndsimp [norm] at h \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nh : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2191(\u2191padicNormE (\u2191f j - \u2191f i)) < \u2191\u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2191padicNormE (\u2191f j - \u2191f i) < \u03b5\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\n\u22a2 \u2203 b, f \u2248 const norm b\n[PROOFSTEP]\ncases' Padic.complete'' \u27e8f, cau_seq_norm_e\u27e9 with q hq\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u22a2 \u2203 b, f \u2248 const norm b\n[PROOFSTEP]\nexists q\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u22a2 f \u2248 const norm q\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(f - const norm q) j\u2016 < \u03b5\n[PROOFSTEP]\ncases' exists_rat_btwn h\u03b5 with \u03b5' h\u03b5'\n[GOAL]\ncase intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u2191\u03b5' \u2227 \u2191\u03b5' < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(f - const norm q) j\u2016 < \u03b5\n[PROOFSTEP]\nnorm_cast at h\u03b5' \n[GOAL]\ncase intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(f - const norm q) j\u2016 < \u03b5\n[PROOFSTEP]\ncases' hq \u03b5' h\u03b5'.1 with N hN\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(f - const norm q) j\u2016 < \u03b5\n[PROOFSTEP]\nexists N\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(f - const norm q) j\u2016 < \u03b5\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\ni : \u2115\nhi : i \u2265 N\n\u22a2 \u2016\u2191(f - const norm q) i\u2016 < \u03b5\n[PROOFSTEP]\nhave h := hN i hi\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2016\u2191(f - const norm q) i\u2016 < \u03b5\n[PROOFSTEP]\nchange norm (f i - q) < \u03b5\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2016\u2191f i - q\u2016 < \u03b5\n[PROOFSTEP]\nrefine lt_trans ?_ h\u03b5'.2\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2016\u2191f i - q\u2016 < \u2191\u03b5'\n[PROOFSTEP]\ndsimp [norm]\n[GOAL]\ncase intro.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\ncau_seq_norm_e : IsCauSeq \u2191padicNormE \u2191f\nq : \u211a_[p]\nhq : \u2200 (\u03b5 : \u211a), \u03b5 > 0 \u2192 \u2203 N, \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u03b5' : \u211a\nh\u03b5' : 0 < \u03b5' \u2227 \u2191\u03b5' < \u03b5\nN : \u2115\nhN : \u2200 (i : \u2115), i \u2265 N \u2192 \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\ni : \u2115\nhi : i \u2265 N\nh : \u2191padicNormE (\u2191{ val := \u2191f, property := cau_seq_norm_e } i - q) < \u03b5'\n\u22a2 \u2191(\u2191padicNormE (\u2191f i - q)) < \u2191\u03b5'\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\na : \u211d\nha : 0 < a\nhf : \u2200 (i : \u2115), \u2016\u2191f i\u2016 \u2264 a\n\u22a2 \u2016CauSeq.lim f\u2016 \u2264 a\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 := (CauSeq.equiv_lim f) _ ha\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\na : \u211d\nha : 0 < a\nhf : \u2200 (i : \u2115), \u2016\u2191f i\u2016 \u2264 a\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(f - const norm (CauSeq.lim f)) j\u2016 < a\n\u22a2 \u2016CauSeq.lim f\u2016 \u2264 a\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel f.lim (f N)]\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\na : \u211d\nha : 0 < a\nhf : \u2200 (i : \u2115), \u2016\u2191f i\u2016 \u2264 a\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(f - const norm (CauSeq.lim f)) j\u2016 < a\n\u22a2 \u2016CauSeq.lim f - \u2191f N + \u2191f N\u2016 \u2264 a\n[PROOFSTEP]\nrefine le_trans (padicNormE.nonarchimedean _ _) ?_\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\na : \u211d\nha : 0 < a\nhf : \u2200 (i : \u2115), \u2016\u2191f i\u2016 \u2264 a\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(f - const norm (CauSeq.lim f)) j\u2016 < a\n\u22a2 max \u2016CauSeq.lim f - \u2191f N\u2016 \u2016\u2191f N\u2016 \u2264 a\n[PROOFSTEP]\nrw [norm_sub_rev]\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nf : CauSeq \u211a_[p] norm\na : \u211d\nha : 0 < a\nhf : \u2200 (i : \u2115), \u2016\u2191f i\u2016 \u2264 a\nN : \u2115\nhN : \u2200 (j : \u2115), j \u2265 N \u2192 \u2016\u2191(f - const norm (CauSeq.lim f)) j\u2016 < a\n\u22a2 max \u2016\u2191f N - CauSeq.lim f\u2016 \u2016\u2191f N\u2016 \u2264 a\n[PROOFSTEP]\nexact\n  max_le (le_of_lt (hN _ le_rfl))\n    (hf _)\n      -- Porting note: the following nice `calc` block does not work\n        -- exact calc\n        --   \u2016f.lim\u2016 = \u2016f.lim - f N + f N\u2016 := sorry\n        --   \u2016f.lim - f N + f N\u2016 \u2264 max \u2016f.lim - f N\u2016 \u2016f N\u2016 := sorry -- (padicNormE.nonarchimedean _ _)\n        --   max \u2016f.lim - f N\u2016 \u2016f N\u2016 = max \u2016f N - f.lim\u2016 \u2016f N\u2016 := sorry -- by congr; rw [norm_sub_rev]\n        --   max \u2016f N - f.lim\u2016 \u2016f N\u2016 \u2264 a := sorry -- max_le (le_of_lt (hN _ le_rfl)) (hf _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 CompleteSpace \u211a_[p]\n[PROOFSTEP]\napply complete_of_cauchySeq_tendsto\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (u : \u2115 \u2192 \u211a_[p]), CauchySeq u \u2192 \u2203 a, Tendsto u atTop (nhds a)\n[PROOFSTEP]\nintro u hu\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\n\u22a2 \u2203 a, Tendsto u atTop (nhds a)\n[PROOFSTEP]\nlet c : CauSeq \u211a_[p] norm := \u27e8u, Metric.cauchySeq_iff'.mp hu\u27e9\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\nc : CauSeq \u211a_[p] norm := { val := u, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < \u03b5) }\n\u22a2 \u2203 a, Tendsto u atTop (nhds a)\n[PROOFSTEP]\nrefine' \u27e8c.lim, fun s h \u21a6 _\u27e9\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\nc : CauSeq \u211a_[p] norm := { val := u, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < \u03b5) }\ns : Set \u211a_[p]\nh : s \u2208 nhds (CauSeq.lim c)\n\u22a2 s \u2208 map u atTop\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.1 h with \u27e8\u03b5, \u03b50, h\u03b5\u27e9\n[GOAL]\ncase a.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\nc : CauSeq \u211a_[p] norm := { val := u, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < \u03b5) }\ns : Set \u211a_[p]\nh : s \u2208 nhds (CauSeq.lim c)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball (CauSeq.lim c) \u03b5 \u2286 s\n\u22a2 s \u2208 map u atTop\n[PROOFSTEP]\nhave := c.equiv_lim \u03b5 \u03b50\n[GOAL]\ncase a.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\nc : CauSeq \u211a_[p] norm := { val := u, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < \u03b5) }\ns : Set \u211a_[p]\nh : s \u2208 nhds (CauSeq.lim c)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball (CauSeq.lim c) \u03b5 \u2286 s\nthis : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(c - const norm (CauSeq.lim c)) j\u2016 < \u03b5\n\u22a2 s \u2208 map u atTop\n[PROOFSTEP]\nsimp only [mem_map, mem_atTop_sets, mem_setOf_eq]\n[GOAL]\ncase a.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2115 \u2192 \u211a_[p]\nhu : CauchySeq u\nc : CauSeq \u211a_[p] norm := { val := u, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (u n) (u N) < \u03b5) }\ns : Set \u211a_[p]\nh : s \u2208 nhds (CauSeq.lim c)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\nh\u03b5 : ball (CauSeq.lim c) \u03b5 \u2286 s\nthis : \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016\u2191(c - const norm (CauSeq.lim c)) j\u2016 < \u03b5\n\u22a2 \u2203 a, \u2200 (b : \u2115), b \u2265 a \u2192 b \u2208 u \u207b\u00b9' s\n[PROOFSTEP]\nexact this.imp fun N hN n hn \u21a6 h\u03b5 (hN n hn)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\n\u22a2 PadicSeq.valuation f = PadicSeq.valuation g\n[PROOFSTEP]\nby_cases hf : f \u2248 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\nhf : f \u2248 0\n\u22a2 PadicSeq.valuation f = PadicSeq.valuation g\n[PROOFSTEP]\nhave hg : g \u2248 0 := Setoid.trans (Setoid.symm h) hf\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\nhf : f \u2248 0\nhg : g \u2248 0\n\u22a2 PadicSeq.valuation f = PadicSeq.valuation g\n[PROOFSTEP]\nsimp [hf, hg, PadicSeq.valuation]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\nhf : \u00acf \u2248 0\n\u22a2 PadicSeq.valuation f = PadicSeq.valuation g\n[PROOFSTEP]\nhave hg : \u00acg \u2248 0 := fun hg \u21a6 hf (Setoid.trans h hg)\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 PadicSeq.valuation f = PadicSeq.valuation g\n[PROOFSTEP]\nrw [PadicSeq.val_eq_iff_norm_eq hf hg]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nf g : CauSeq \u211a (padicNorm p)\nh : f \u2248 g\nhf : \u00acf \u2248 0\nhg : \u00acg \u2248 0\n\u22a2 PadicSeq.norm f = PadicSeq.norm g\n[PROOFSTEP]\nexact PadicSeq.norm_equiv h\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 valuation 1 = 0\n[PROOFSTEP]\nchange dite (CauSeq.const (padicNorm p) 1 \u2248 _) _ _ = _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 (if h : const (padicNorm p) 1 \u2248 0 then (fun hf => 0) h\n    else (fun hf => padicValRat p (\u2191(const (padicNorm p) 1) (PadicSeq.stationaryPoint hf))) h) =\n    0\n[PROOFSTEP]\nhave h : \u00acCauSeq.const (padicNorm p) 1 \u2248 0 := by\n  intro H\n  erw [const_equiv p] at H \n  exact one_ne_zero H\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u00acconst (padicNorm p) 1 \u2248 0\n[PROOFSTEP]\nintro H\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nH : const (padicNorm p) 1 \u2248 0\n\u22a2 False\n[PROOFSTEP]\nerw [const_equiv p] at H \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nH : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero H\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nh : \u00acconst (padicNorm p) 1 \u2248 0\n\u22a2 (if h : const (padicNorm p) 1 \u2248 0 then (fun hf => 0) h\n    else (fun hf => padicValRat p (\u2191(const (padicNorm p) 1) (PadicSeq.stationaryPoint hf))) h) =\n    0\n[PROOFSTEP]\nrw [dif_neg h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nh : \u00acconst (padicNorm p) 1 \u2248 0\n\u22a2 (fun hf => padicValRat p (\u2191(const (padicNorm p) 1) (PadicSeq.stationaryPoint hf))) h = 0\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\n\u22a2 x \u2260 0 \u2192 \u2016x\u2016 = \u2191p ^ (-valuation x)\n[PROOFSTEP]\nrefine Quotient.inductionOn' x fun f hf => ?_\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u2016Quotient.mk'' f\u2016 = \u2191p ^ (-valuation (Quotient.mk'' f))\n[PROOFSTEP]\nchange (PadicSeq.norm _ : \u211d) = (p : \u211d) ^ (-PadicSeq.valuation _)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u2191(PadicSeq.norm f) = \u2191p ^ (-PadicSeq.valuation f)\n[PROOFSTEP]\nrw [PadicSeq.norm_eq_pow_val]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u2191(\u2191p ^ (-PadicSeq.valuation f)) = \u2191p ^ (-PadicSeq.valuation f)\np : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u00acf \u2248 0\n[PROOFSTEP]\nchange \u2191((p : \u211a) ^ (-PadicSeq.valuation f)) = (p : \u211d) ^ (-PadicSeq.valuation f)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u2191(\u2191p ^ (-PadicSeq.valuation f)) = \u2191p ^ (-PadicSeq.valuation f)\n[PROOFSTEP]\nrw [Rat.cast_zpow, Rat.cast_coe_nat]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u00acf \u2248 0\n[PROOFSTEP]\napply CauSeq.not_limZero_of_not_congr_zero\n[GOAL]\ncase hf\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\n\u22a2 \u00acf - 0 \u2248 0\n[PROOFSTEP]\nintro hf'\n[GOAL]\ncase hf\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\nhf' : f - 0 \u2248 0\n\u22a2 False\n[PROOFSTEP]\napply hf\n[GOAL]\ncase hf\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\nhf' : f - 0 \u2248 0\n\u22a2 Quotient.mk'' f = 0\n[PROOFSTEP]\napply Quotient.sound\n[GOAL]\ncase hf.a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nf : CauSeq \u211a (padicNorm p)\nhf : Quotient.mk'' f \u2260 0\nhf' : f - 0 \u2248 0\n\u22a2 f \u2248 const (padicNorm p) 0\n[PROOFSTEP]\nsimpa using hf'\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 valuation \u2191p = 1\n[PROOFSTEP]\nhave h : (1 : \u211d) < p := by exact_mod_cast (Fact.out : p.Prime).one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast (Fact.out : p.Prime).one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nh : 1 < \u2191p\n\u22a2 valuation \u2191p = 1\n[PROOFSTEP]\nrefine' neg_injective ((zpow_strictMono h).injective <| (norm_eq_pow_val _).symm.trans _)\n[GOAL]\ncase refine'_1\np : \u2115\nhp : Fact (Nat.Prime p)\nh : 1 < \u2191p\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast (Fact.out : p.Prime).ne_zero\n[GOAL]\ncase refine'_2\np : \u2115\nhp : Fact (Nat.Prime p)\nh : 1 < \u2191p\n\u22a2 \u2016\u2191p\u2016 = (fun x x_1 => x ^ x_1) (\u2191p) (-1)\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : x = 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nrw [hx, zero_add]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : x = 0\n\u22a2 min (valuation 0) (valuation y) \u2264 valuation y\n[PROOFSTEP]\nexact min_le_right _ _\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nrw [hy, add_zero]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 min (valuation x) (valuation 0) \u2264 valuation x\n[PROOFSTEP]\nexact min_le_left _ _\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nhave h_norm : \u2016x + y\u2016 \u2264 max \u2016x\u2016 \u2016y\u2016 := padicNormE.nonarchimedean x y\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\nh_norm : \u2016x + y\u2016 \u2264 max \u2016x\u2016 \u2016y\u2016\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nhave hp_one : (1 : \u211d) < p := by\n  rw [\u2190 Nat.cast_one, Nat.cast_lt]\n  exact Nat.Prime.one_lt hp.elim\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\nh_norm : \u2016x + y\u2016 \u2264 max \u2016x\u2016 \u2016y\u2016\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, Nat.cast_lt]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\nh_norm : \u2016x + y\u2016 \u2264 max \u2016x\u2016 \u2016y\u2016\n\u22a2 1 < p\n[PROOFSTEP]\nexact Nat.Prime.one_lt hp.elim\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y \u2260 0\nhx : \u00acx = 0\nhy : \u00acy = 0\nh_norm : \u2016x + y\u2016 \u2264 max \u2016x\u2016 \u2016y\u2016\nhp_one : 1 < \u2191p\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nrwa [norm_eq_pow_val hx, norm_eq_pow_val hy, norm_eq_pow_val hxy, zpow_le_max_iff_min_le hp_one] at h_norm \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 valuation (x * y) = valuation x + valuation y\n[PROOFSTEP]\nhave h_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016 := norm_mul x y\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\n\u22a2 valuation (x * y) = valuation x + valuation y\n[PROOFSTEP]\nhave hp_ne_one : (p : \u211d) \u2260 1 := by\n  rw [\u2190 Nat.cast_one, Ne.def, Nat.cast_inj]\n  exact Nat.Prime.ne_one hp.elim\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\n\u22a2 \u2191p \u2260 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, Ne.def, Nat.cast_inj]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\n\u22a2 \u00acp = 1\n[PROOFSTEP]\nexact Nat.Prime.ne_one hp.elim\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhp_ne_one : \u2191p \u2260 1\n\u22a2 valuation (x * y) = valuation x + valuation y\n[PROOFSTEP]\nhave hp_pos : (0 : \u211d) < p := by\n  rw [\u2190 Nat.cast_zero, Nat.cast_lt]\n  exact Nat.Prime.pos hp.elim\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhp_ne_one : \u2191p \u2260 1\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nrw [\u2190 Nat.cast_zero, Nat.cast_lt]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhp_ne_one : \u2191p \u2260 1\n\u22a2 0 < p\n[PROOFSTEP]\nexact Nat.Prime.pos hp.elim\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : \u2016x * y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhp_ne_one : \u2191p \u2260 1\nhp_pos : 0 < \u2191p\n\u22a2 valuation (x * y) = valuation x + valuation y\n[PROOFSTEP]\nrw [norm_eq_pow_val hx, norm_eq_pow_val hy, norm_eq_pow_val (mul_ne_zero hx hy), \u2190 zpow_add\u2080 (ne_of_gt hp_pos),\n  zpow_inj hp_pos hp_ne_one, \u2190 neg_add, neg_inj] at h_norm \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x \u2260 0\nhy : y \u2260 0\nh_norm : valuation (x * y) = valuation x + valuation y\nhp_ne_one : \u2191p \u2260 1\nhp_pos : 0 < \u2191p\n\u22a2 valuation (x * y) = valuation x + valuation y\n[PROOFSTEP]\nexact h_norm\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 addValuationDef 0 = \u22a4\n[PROOFSTEP]\nrw [addValuationDef, if_pos (Eq.refl _)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 addValuationDef 1 = 0\n[PROOFSTEP]\nrw [addValuationDef, if_neg one_ne_zero, valuation_one, WithTop.coe_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\n\u22a2 addValuationDef (x * y) = addValuationDef x + addValuationDef y\n[PROOFSTEP]\nsimp only [addValuationDef]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(valuation (x * y))) =\n    (if x = 0 then \u22a4 else \u2191(valuation x)) + if y = 0 then \u22a4 else \u2191(valuation y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : x = 0\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(valuation (x * y))) =\n    (if x = 0 then \u22a4 else \u2191(valuation x)) + if y = 0 then \u22a4 else \u2191(valuation y)\n[PROOFSTEP]\nrw [hx, if_pos (Eq.refl _), zero_mul, if_pos (Eq.refl _), WithTop.top_add]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : \u00acx = 0\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(valuation (x * y))) =\n    (if x = 0 then \u22a4 else \u2191(valuation x)) + if y = 0 then \u22a4 else \u2191(valuation y)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(valuation (x * y))) =\n    (if x = 0 then \u22a4 else \u2191(valuation x)) + if y = 0 then \u22a4 else \u2191(valuation y)\n[PROOFSTEP]\nrw [hy, if_pos (Eq.refl _), mul_zero, if_pos (Eq.refl _), WithTop.add_top]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (if x * y = 0 then \u22a4 else \u2191(valuation (x * y))) =\n    (if x = 0 then \u22a4 else \u2191(valuation x)) + if y = 0 then \u22a4 else \u2191(valuation y)\n[PROOFSTEP]\nrw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), \u2190 WithTop.coe_add, WithTop.coe_eq_coe, valuation_map_mul hx hy]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\n\u22a2 min (addValuationDef x) (addValuationDef y) \u2264 addValuationDef (x + y)\n[PROOFSTEP]\nsimp only [addValuationDef]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nby_cases hxy : x + y = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nrw [hxy, if_pos (Eq.refl _)]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : x + y = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : x = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nrw [hx, if_pos (Eq.refl _), min_eq_right, zero_add]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : x = 0\n\u22a2 (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : \u00acx = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nrw [hy, if_pos (Eq.refl _), min_eq_left, add_zero]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 (if x = 0 then \u22a4 else \u2191(valuation x)) \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (if x = 0 then \u22a4 else \u2191(valuation x)) (if y = 0 then \u22a4 else \u2191(valuation y)) \u2264\n    if x + y = 0 then \u22a4 else \u2191(valuation (x + y))\n[PROOFSTEP]\nrw [if_neg hx, if_neg hy, if_neg hxy, \u2190 WithTop.coe_min, WithTop.coe_le_coe]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx y : \u211a_[p]\nhxy : \u00acx + y = 0\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 min (valuation x) (valuation y) \u2264 valuation (x + y)\n[PROOFSTEP]\nexact valuation_map_add hxy\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : x \u2260 0\n\u22a2 \u2191addValuation x = \u2191(valuation x)\n[PROOFSTEP]\nsimp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ n \u2194 \u2016x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nhave aux : \u2200 n : \u2124, 0 < ((p : \u211d) ^ n) := by\n  apply Nat.zpow_pos_of_pos\n  exact hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\n\u22a2 \u2200 (n : \u2124), 0 < \u2191p ^ n\n[PROOFSTEP]\napply Nat.zpow_pos_of_pos\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\n\u22a2 0 < p\n[PROOFSTEP]\nexact hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ n \u2194 \u2016x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nby_cases hx0 : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : x = 0\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ n \u2194 \u2016x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nsimp [hx0, norm_zero, aux, le_of_lt (aux _)]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ n \u2194 \u2016x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nrw [norm_eq_pow_val hx0]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : \u00acx = 0\n\u22a2 \u2191p ^ (-valuation x) \u2264 \u2191p ^ n \u2194 \u2191p ^ (-valuation x) < \u2191p ^ (n + 1)\n[PROOFSTEP]\nhave h1p : 1 < (p : \u211d) := by exact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : \u00acx = 0\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : \u00acx = 0\nh1p : 1 < \u2191p\n\u22a2 \u2191p ^ (-valuation x) \u2264 \u2191p ^ n \u2194 \u2191p ^ (-valuation x) < \u2191p ^ (n + 1)\n[PROOFSTEP]\nhave H := zpow_strictMono h1p\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\naux : \u2200 (n : \u2124), 0 < \u2191p ^ n\nhx0 : \u00acx = 0\nh1p : 1 < \u2191p\nH : StrictMono ((fun x x_1 => x ^ x_1) \u2191p)\n\u22a2 \u2191p ^ (-valuation x) \u2264 \u2191p ^ n \u2194 \u2191p ^ (-valuation x) < \u2191p ^ (n + 1)\n[PROOFSTEP]\nrw [H.le_iff_le, H.lt_iff_lt, Int.lt_add_one_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nn : \u2124\n\u22a2 \u2016x\u2016 < \u2191p ^ n \u2194 \u2016x\u2016 \u2264 \u2191p ^ (n - 1)\n[PROOFSTEP]\nrw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\n\u22a2 \u2016x\u2016 \u2264 1 \u2194 0 \u2264 valuation x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : x = 0\n\u22a2 \u2016x\u2016 \u2264 1 \u2194 0 \u2264 valuation x\n[PROOFSTEP]\nsimp only [hx, norm_zero, valuation_zero, zero_le_one, le_refl]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 1 \u2194 0 \u2264 valuation x\n[PROOFSTEP]\nrw [norm_eq_pow_val hx, \u2190 zpow_zero (p : \u211d), zpow_le_iff_le, Right.neg_nonpos_iff]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00acx = 0\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact Nat.one_lt_cast.2 (Nat.Prime.one_lt' p).1\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Padics.PadicNumbers", "llama_tokens": 78742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8824278540866547, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5427706564178049}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\n\u03b2 : Type u_2\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 (i : \u03b9) \u2192 \u03b1 i\n\u22a2 UniformContinuous f \u2194 \u2200 (i : \u03b9), UniformContinuous fun x => f x i\n[PROOFSTEP]\nsimp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\n\u22a2 \u2200 {f : Filter ((i : \u03b9) \u2192 \u03b1 i)}, Cauchy f \u2192 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nintro f hf\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nhaveI := hf.1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nhave : \u2200 i, \u2203 x : \u03b1 i, Filter.map (fun a : \u2200 i, \u03b1 i => a i) f \u2264 \ud835\udcdd x :=\n  by\n  intro i\n  have key : Cauchy (map (fun a : \u2200 i : \u03b9, \u03b1 i => a i) f) := hf.map (Pi.uniformContinuous_proj \u03b1 i)\n  exact cauchy_iff_exists_le_nhds.1 key\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\n\u22a2 \u2200 (i : \u03b9), \u2203 x, map (fun a => a i) f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\ni : \u03b9\n\u22a2 \u2203 x, map (fun a => a i) f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nhave key : Cauchy (map (fun a : \u2200 i : \u03b9, \u03b1 i => a i) f) := hf.map (Pi.uniformContinuous_proj \u03b1 i)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\ni : \u03b9\nkey : Cauchy (map (fun a => a i) f)\n\u22a2 \u2203 x, map (fun a => a i) f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nexact cauchy_iff_exists_le_nhds.1 key\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis\u271d : NeBot f\nthis : \u2200 (i : \u03b9), \u2203 x, map (fun a => a i) f \u2264 \ud835\udcdd x\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nchoose x hx using this\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\nx : (i : \u03b9) \u2192 \u03b1 i\nhx : \u2200 (i : \u03b9), map (fun a => a i) f \u2264 \ud835\udcdd (x i)\n\u22a2 \u2203 x, f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace (\u03b1 i)\nf : Filter ((i : \u03b9) \u2192 \u03b1 i)\nhf : Cauchy f\nthis : NeBot f\nx : (i : \u03b9) \u2192 \u03b1 i\nhx : \u2200 (i : \u03b9), map (fun a => a i) f \u2264 \ud835\udcdd (x i)\n\u22a2 f \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrwa [nhds_pi, le_pi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SeparatedSpace (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\nH : \u2200 (r : Set (((i : \u03b9) \u2192 \u03b1 i) \u00d7 ((i : \u03b9) \u2192 \u03b1 i))), r \u2208 \ud835\udce4 ((i : \u03b9) \u2192 \u03b1 i) \u2192 (x, y) \u2208 r\n\u22a2 x = y\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SeparatedSpace (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\nH : \u2200 (r : Set (((i : \u03b9) \u2192 \u03b1 i) \u00d7 ((i : \u03b9) \u2192 \u03b1 i))), r \u2208 \ud835\udce4 ((i : \u03b9) \u2192 \u03b1 i) \u2192 (x, y) \u2208 r\ni : \u03b9\n\u22a2 x i = y i\n[PROOFSTEP]\napply eq_of_separated_of_uniformContinuous (Pi.uniformContinuous_proj \u03b1 i)\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u\nU : (i : \u03b9) \u2192 UniformSpace (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), SeparatedSpace (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\nH : \u2200 (r : Set (((i : \u03b9) \u2192 \u03b1 i) \u00d7 ((i : \u03b9) \u2192 \u03b1 i))), r \u2208 \ud835\udce4 ((i : \u03b9) \u2192 \u03b1 i) \u2192 (x, y) \u2208 r\ni : \u03b9\n\u22a2 (fun i => x i) \u2248 fun i => y i\n[PROOFSTEP]\napply H\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.Pi", "llama_tokens": 1946, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933315126792, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.5427061464170785}}
{"text": "[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\n\u22a2 lift (Module.rank F K) * lift (Module.rank K A) = lift (Module.rank F A)\n[PROOFSTEP]\nobtain \u27e8_, b\u27e9 := Module.Free.exists_basis (R := F) (M := K)\n[GOAL]\ncase intro.mk\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\nfst\u271d : Type v\nb : Basis fst\u271d F K\n\u22a2 lift (Module.rank F K) * lift (Module.rank K A) = lift (Module.rank F A)\n[PROOFSTEP]\nobtain \u27e8_, c\u27e9 := Module.Free.exists_basis (R := K) (M := A)\n[GOAL]\ncase intro.mk.intro.mk\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\nfst\u271d\u00b9 : Type v\nb : Basis fst\u271d\u00b9 F K\nfst\u271d : Type w\nc : Basis fst\u271d K A\n\u22a2 lift (Module.rank F K) * lift (Module.rank K A) = lift (Module.rank F A)\n[PROOFSTEP]\nrw [\u2190 (Module.rank F K).lift_id, \u2190 b.mk_eq_rank, \u2190 (Module.rank K A).lift_id, \u2190 c.mk_eq_rank, \u2190 lift_umax.{w, v}, \u2190\n  (b.smul c).mk_eq_rank, mk_prod, lift_mul, lift_lift, lift_lift, lift_lift, lift_lift, lift_umax.{v, w}]\n[GOAL]\nF\u271d : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u00b2\u00b9 : CommRing F\u271d\ninst\u271d\u00b2\u2070 : Ring K\u271d\ninst\u271d\u00b9\u2079 : AddCommGroup A\u271d\ninst\u271d\u00b9\u2078 : Algebra F\u271d K\u271d\ninst\u271d\u00b9\u2077 : Module K\u271d A\u271d\ninst\u271d\u00b9\u2076 : Module F\u271d A\u271d\ninst\u271d\u00b9\u2075 : IsScalarTower F\u271d K\u271d A\u271d\ninst\u271d\u00b9\u2074 : StrongRankCondition F\u271d\ninst\u271d\u00b9\u00b3 : StrongRankCondition K\u271d\ninst\u271d\u00b9\u00b2 : Module.Free F\u271d K\u271d\ninst\u271d\u00b9\u00b9 : Module.Free K\u271d A\u271d\nF : Type u\nK A : Type v\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\n\u22a2 Module.rank F K * Module.rank K A = Module.rank F A\n[PROOFSTEP]\nconvert lift_rank_mul_lift_rank F K A\n[GOAL]\ncase h.e'_2.h.e'_5\nF\u271d : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u00b2\u00b9 : CommRing F\u271d\ninst\u271d\u00b2\u2070 : Ring K\u271d\ninst\u271d\u00b9\u2079 : AddCommGroup A\u271d\ninst\u271d\u00b9\u2078 : Algebra F\u271d K\u271d\ninst\u271d\u00b9\u2077 : Module K\u271d A\u271d\ninst\u271d\u00b9\u2076 : Module F\u271d A\u271d\ninst\u271d\u00b9\u2075 : IsScalarTower F\u271d K\u271d A\u271d\ninst\u271d\u00b9\u2074 : StrongRankCondition F\u271d\ninst\u271d\u00b9\u00b3 : StrongRankCondition K\u271d\ninst\u271d\u00b9\u00b2 : Module.Free F\u271d K\u271d\ninst\u271d\u00b9\u00b9 : Module.Free K\u271d A\u271d\nF : Type u\nK A : Type v\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\n\u22a2 Module.rank F K = lift (Module.rank F K)\n[PROOFSTEP]\nrw [lift_id]\n[GOAL]\ncase h.e'_2.h.e'_6\nF\u271d : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u00b2\u00b9 : CommRing F\u271d\ninst\u271d\u00b2\u2070 : Ring K\u271d\ninst\u271d\u00b9\u2079 : AddCommGroup A\u271d\ninst\u271d\u00b9\u2078 : Algebra F\u271d K\u271d\ninst\u271d\u00b9\u2077 : Module K\u271d A\u271d\ninst\u271d\u00b9\u2076 : Module F\u271d A\u271d\ninst\u271d\u00b9\u2075 : IsScalarTower F\u271d K\u271d A\u271d\ninst\u271d\u00b9\u2074 : StrongRankCondition F\u271d\ninst\u271d\u00b9\u00b3 : StrongRankCondition K\u271d\ninst\u271d\u00b9\u00b2 : Module.Free F\u271d K\u271d\ninst\u271d\u00b9\u00b9 : Module.Free K\u271d A\u271d\nF : Type u\nK A : Type v\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\n\u22a2 Module.rank K A = lift (Module.rank K A)\n[PROOFSTEP]\nrw [lift_id]\n[GOAL]\ncase h.e'_3\nF\u271d : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u00b2\u00b9 : CommRing F\u271d\ninst\u271d\u00b2\u2070 : Ring K\u271d\ninst\u271d\u00b9\u2079 : AddCommGroup A\u271d\ninst\u271d\u00b9\u2078 : Algebra F\u271d K\u271d\ninst\u271d\u00b9\u2077 : Module K\u271d A\u271d\ninst\u271d\u00b9\u2076 : Module F\u271d A\u271d\ninst\u271d\u00b9\u2075 : IsScalarTower F\u271d K\u271d A\u271d\ninst\u271d\u00b9\u2074 : StrongRankCondition F\u271d\ninst\u271d\u00b9\u00b3 : StrongRankCondition K\u271d\ninst\u271d\u00b9\u00b2 : Module.Free F\u271d K\u271d\ninst\u271d\u00b9\u00b9 : Module.Free K\u271d A\u271d\nF : Type u\nK A : Type v\ninst\u271d\u00b9\u2070 : CommRing F\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup A\ninst\u271d\u2077 : Algebra F K\ninst\u271d\u2076 : Module K A\ninst\u271d\u2075 : Module F A\ninst\u271d\u2074 : IsScalarTower F K A\ninst\u271d\u00b3 : StrongRankCondition F\ninst\u271d\u00b2 : StrongRankCondition K\ninst\u271d\u00b9 : Module.Free F K\ninst\u271d : Module.Free K A\n\u22a2 Module.rank F A = lift (Module.rank F A)\n[PROOFSTEP]\nrw [lift_id]\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u00b2 : CommRing F\ninst\u271d\u00b9\u00b9 : Ring K\ninst\u271d\u00b9\u2070 : AddCommGroup A\ninst\u271d\u2079 : Algebra F K\ninst\u271d\u2078 : Module K A\ninst\u271d\u2077 : Module F A\ninst\u271d\u2076 : IsScalarTower F K A\ninst\u271d\u2075 : StrongRankCondition F\ninst\u271d\u2074 : StrongRankCondition K\ninst\u271d\u00b3 : Module.Free F K\ninst\u271d\u00b2 : Module.Free K A\ninst\u271d\u00b9 : Module.Finite F K\ninst\u271d : Module.Finite K A\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nletI := nontrivial_of_invariantBasisNumber F\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u00b2 : CommRing F\ninst\u271d\u00b9\u00b9 : Ring K\ninst\u271d\u00b9\u2070 : AddCommGroup A\ninst\u271d\u2079 : Algebra F K\ninst\u271d\u2078 : Module K A\ninst\u271d\u2077 : Module F A\ninst\u271d\u2076 : IsScalarTower F K A\ninst\u271d\u2075 : StrongRankCondition F\ninst\u271d\u2074 : StrongRankCondition K\ninst\u271d\u00b3 : Module.Free F K\ninst\u271d\u00b2 : Module.Free K A\ninst\u271d\u00b9 : Module.Finite F K\ninst\u271d : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nlet b := Module.Free.chooseBasis F K\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u00b2 : CommRing F\ninst\u271d\u00b9\u00b9 : Ring K\ninst\u271d\u00b9\u2070 : AddCommGroup A\ninst\u271d\u2079 : Algebra F K\ninst\u271d\u2078 : Module K A\ninst\u271d\u2077 : Module F A\ninst\u271d\u2076 : IsScalarTower F K A\ninst\u271d\u2075 : StrongRankCondition F\ninst\u271d\u2074 : StrongRankCondition K\ninst\u271d\u00b3 : Module.Free F K\ninst\u271d\u00b2 : Module.Free K A\ninst\u271d\u00b9 : Module.Finite F K\ninst\u271d : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nlet c := Module.Free.chooseBasis K A\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u00b9\u00b2 : CommRing F\ninst\u271d\u00b9\u00b9 : Ring K\ninst\u271d\u00b9\u2070 : AddCommGroup A\ninst\u271d\u2079 : Algebra F K\ninst\u271d\u2078 : Module K A\ninst\u271d\u2077 : Module F A\ninst\u271d\u2076 : IsScalarTower F K A\ninst\u271d\u2075 : StrongRankCondition F\ninst\u271d\u2074 : StrongRankCondition K\ninst\u271d\u00b3 : Module.Free F K\ninst\u271d\u00b2 : Module.Free K A\ninst\u271d\u00b9 : Module.Finite F K\ninst\u271d : Module.Finite K A\nthis : Nontrivial F := nontrivial_of_invariantBasisNumber F\nb : Basis (Module.Free.ChooseBasisIndex F K) F K := Module.Free.chooseBasis F K\nc : Basis (Module.Free.ChooseBasisIndex K A) K A := Module.Free.chooseBasis K A\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nrw [finrank_eq_card_basis b, finrank_eq_card_basis c, finrank_eq_card_basis (b.smul c), Fintype.card_prod]\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Module K A\ninst\u271d\u00b9 : Module F A\ninst\u271d : IsScalarTower F K A\nhf : FiniteDimensional F A\nb : Finset A\nhb : span F \u2191b = \u22a4\n\u22a2 restrictScalars F (span K \u2191b) = restrictScalars F \u22a4\n[PROOFSTEP]\nrw [Submodule.restrictScalars_top, eq_top_iff, \u2190 hb, Submodule.span_le]\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Module K A\ninst\u271d\u00b9 : Module F A\ninst\u271d : IsScalarTower F K A\nhf : FiniteDimensional F A\nb : Finset A\nhb : span F \u2191b = \u22a4\n\u22a2 \u2191b \u2286 \u2191(restrictScalars F (span K \u2191b))\n[PROOFSTEP]\nexact Submodule.subset_span\n[GOAL]\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup A\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Module K A\ninst\u271d\u00b2 : Module F A\ninst\u271d\u00b9 : IsScalarTower F K A\ninst\u271d : FiniteDimensional F K\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nby_cases hA : FiniteDimensional K A\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup A\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Module K A\ninst\u271d\u00b2 : Module F A\ninst\u271d\u00b9 : IsScalarTower F K A\ninst\u271d : FiniteDimensional F K\nhA : FiniteDimensional K A\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nreplace hA : FiniteDimensional K A := hA\n[GOAL]\ncase pos\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup A\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Module K A\ninst\u271d\u00b2 : Module F A\ninst\u271d\u00b9 : IsScalarTower F K A\ninst\u271d : FiniteDimensional F K\nhA : FiniteDimensional K A\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nrw [finrank_mul_finrank']\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup A\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Module K A\ninst\u271d\u00b2 : Module F A\ninst\u271d\u00b9 : IsScalarTower F K A\ninst\u271d : FiniteDimensional F K\nhA : \u00acFiniteDimensional K A\n\u22a2 finrank F K * finrank K A = finrank F A\n[PROOFSTEP]\nrw [finrank_of_infinite_dimensional hA, mul_zero, finrank_of_infinite_dimensional]\n[GOAL]\ncase neg\nF : Type u\nK : Type v\nA : Type w\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : DivisionRing K\ninst\u271d\u2075 : AddCommGroup A\ninst\u271d\u2074 : Algebra F K\ninst\u271d\u00b3 : Module K A\ninst\u271d\u00b2 : Module F A\ninst\u271d\u00b9 : IsScalarTower F K A\ninst\u271d : FiniteDimensional F K\nhA : \u00acFiniteDimensional K A\n\u22a2 \u00acFiniteDimensional F A\n[PROOFSTEP]\nexact mt (@right F K A _ _ _ _ _ _ _) hA\n[GOAL]\nF : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : DivisionRing K\u271d\ninst\u271d\u2077 : AddCommGroup A\u271d\ninst\u271d\u2076 : Algebra F K\u271d\ninst\u271d\u2075 : Module K\u271d A\u271d\ninst\u271d\u2074 : Module F A\u271d\ninst\u271d\u00b3 : IsScalarTower F K\u271d A\u271d\nA : Type u_1\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\n\u22a2 K = \u22a5 \u2228 K = \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional _ _ := finiteDimensional_of_finrank hp.pos\n[GOAL]\nF : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : DivisionRing K\u271d\ninst\u271d\u2077 : AddCommGroup A\u271d\ninst\u271d\u2076 : Algebra F K\u271d\ninst\u271d\u2075 : Module K\u271d A\u271d\ninst\u271d\u2074 : Module F A\u271d\ninst\u271d\u00b3 : IsScalarTower F K\u271d A\u271d\nA : Type u_1\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis : FiniteDimensional F A\n\u22a2 K = \u22a5 \u2228 K = \u22a4\n[PROOFSTEP]\nletI := divisionRingOfFiniteDimensional F K\n[GOAL]\nF : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : DivisionRing K\u271d\ninst\u271d\u2077 : AddCommGroup A\u271d\ninst\u271d\u2076 : Algebra F K\u271d\ninst\u271d\u2075 : Module K\u271d A\u271d\ninst\u271d\u2074 : Module F A\u271d\ninst\u271d\u00b3 : IsScalarTower F K\u271d A\u271d\nA : Type u_1\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis\u271d : FiniteDimensional F A\nthis : DivisionRing { x // x \u2208 K } := divisionRingOfFiniteDimensional F { x // x \u2208 K }\n\u22a2 K = \u22a5 \u2228 K = \u22a4\n[PROOFSTEP]\nrefine' (hp.eq_one_or_self_of_dvd _ \u27e8_, (finrank_mul_finrank F K A).symm\u27e9).imp _ fun h => _\n[GOAL]\ncase refine'_1\nF : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : DivisionRing K\u271d\ninst\u271d\u2077 : AddCommGroup A\u271d\ninst\u271d\u2076 : Algebra F K\u271d\ninst\u271d\u2075 : Module K\u271d A\u271d\ninst\u271d\u2074 : Module F A\u271d\ninst\u271d\u00b3 : IsScalarTower F K\u271d A\u271d\nA : Type u_1\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis\u271d : FiniteDimensional F A\nthis : DivisionRing { x // x \u2208 K } := divisionRingOfFiniteDimensional F { x // x \u2208 K }\n\u22a2 finrank F { x // x \u2208 K } = 1 \u2192 K = \u22a5\n[PROOFSTEP]\nexact Subalgebra.eq_bot_of_finrank_one\n[GOAL]\ncase refine'_2\nF : Type u\nK\u271d : Type v\nA\u271d : Type w\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : DivisionRing K\u271d\ninst\u271d\u2077 : AddCommGroup A\u271d\ninst\u271d\u2076 : Algebra F K\u271d\ninst\u271d\u2075 : Module K\u271d A\u271d\ninst\u271d\u2074 : Module F A\u271d\ninst\u271d\u00b3 : IsScalarTower F K\u271d A\u271d\nA : Type u_1\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : Algebra F A\nhp : Nat.Prime (finrank F A)\nK : Subalgebra F A\nthis\u271d : FiniteDimensional F A\nthis : DivisionRing { x // x \u2208 K } := divisionRingOfFiniteDimensional F { x // x \u2208 K }\nh : finrank F { x // x \u2208 K } = finrank F A\n\u22a2 K = \u22a4\n[PROOFSTEP]\nexact Algebra.toSubmodule_eq_top.1 (eq_top_of_finrank_eq <| K.finrank_toSubmodule.trans h)\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Tower", "llama_tokens": 5895, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.815232489352, "lm_q2_score": 0.665410572017153, "lm_q1q2_score": 0.542464317066682}}
{"text": "[GOAL]\nr n : \u2115\n\u22a2 choose (2 * n) (2 * n / 2) = choose (2 * n) n\n[PROOFSTEP]\nrw [Nat.mul_div_cancel_left n zero_lt_two]\n[GOAL]\nn : \u2115\n\u22a2 choose (2 * n + 2) (n + 1) * (n + 1) = choose (2 * n + 1) n * (2 * n + 2)\n[PROOFSTEP]\nrw [choose_succ_right_eq, choose_mul_succ_eq]\n[GOAL]\nn : \u2115\n\u22a2 choose (2 * n + 1) n * (2 * n + 2) = 2 * (choose (2 * n + 1) n * (n + 1))\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 2 * (choose (2 * n + 1) n * (n + 1)) = 2 * (choose (2 * n + 1) n * (2 * n + 1 - n))\n[PROOFSTEP]\nrw [two_mul n, add_assoc, Nat.add_sub_cancel_left]\n[GOAL]\nn : \u2115\n\u22a2 2 * (choose (2 * n + 1) n * (2 * n + 1 - n)) = 2 * (choose (2 * n) n * (2 * n + 1))\n[PROOFSTEP]\nrw [choose_mul_succ_eq]\n[GOAL]\nn : \u2115\n\u22a2 2 * (choose (2 * n) n * (2 * n + 1)) = 2 * (2 * n + 1) * choose (2 * n) n\n[PROOFSTEP]\nrw [mul_assoc, mul_comm (2 * n + 1)]\n[GOAL]\nn : \u2115\nn_big : 4 \u2264 n\n\u22a2 4 ^ n < n * centralBinom n\n[PROOFSTEP]\ninduction' n using Nat.strong_induction_on with n IH\n[GOAL]\ncase h\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n\n\u22a2 4 ^ n < n * centralBinom n\n[PROOFSTEP]\nrcases lt_trichotomy n 4 with (hn | rfl | hn)\n[GOAL]\ncase h.inl\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n\nhn : n < 4\n\u22a2 4 ^ n < n * centralBinom n\n[PROOFSTEP]\nclear IH\n[GOAL]\ncase h.inl\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nn_big : 4 \u2264 n\nhn : n < 4\n\u22a2 4 ^ n < n * centralBinom n\n[PROOFSTEP]\nexact False.elim ((not_lt.2 n_big) hn)\n[GOAL]\ncase h.inr.inl\nn : \u2115\nn_big\u271d : 4 \u2264 n\nIH : \u2200 (m : \u2115), m < 4 \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 4\n\u22a2 4 ^ 4 < 4 * centralBinom 4\n[PROOFSTEP]\nnorm_num [centralBinom, choose]\n[GOAL]\ncase h.inr.inr\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n\nhn : 4 < n\n\u22a2 4 ^ n < n * centralBinom n\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 : \u2203 m, n = m + 1 := Nat.exists_eq_succ_of_ne_zero (Nat.not_eq_zero_of_lt hn)\n[GOAL]\ncase h.inr.inr.intro\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n + 1 \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n + 1\nhn : 4 < n + 1\n\u22a2 4 ^ (n + 1) < (n + 1) * centralBinom (n + 1)\n[PROOFSTEP]\ncalc\n  4 ^ (n + 1) < 4 * (n * centralBinom n) :=\n    lt_of_eq_of_lt (pow_succ'' n 4) $\n      (mul_lt_mul_left <| zero_lt_four' \u2115).mpr (IH n n.lt_succ_self (Nat.le_of_lt_succ hn))\n  _ \u2264 2 * (2 * n + 1) * centralBinom n := by rw [\u2190 mul_assoc]; linarith\n  _ = (n + 1) * centralBinom (n + 1) := (succ_mul_centralBinom_succ n).symm\n[GOAL]\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n + 1 \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n + 1\nhn : 4 < n + 1\n\u22a2 4 * (n * centralBinom n) \u2264 2 * (2 * n + 1) * centralBinom n\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\nn\u271d : \u2115\nn_big\u271d : 4 \u2264 n\u271d\nn : \u2115\nIH : \u2200 (m : \u2115), m < n + 1 \u2192 4 \u2264 m \u2192 4 ^ m < m * centralBinom m\nn_big : 4 \u2264 n + 1\nhn : 4 < n + 1\n\u22a2 4 * n * centralBinom n \u2264 2 * (2 * n + 1) * centralBinom n\n[PROOFSTEP]\nlinarith\n[GOAL]\nx\u271d : 0 < 1\n\u22a2 4 ^ 1 \u2264 2 * 1 * centralBinom 1\n[PROOFSTEP]\nnorm_num [centralBinom, choose]\n[GOAL]\nx\u271d : 0 < 2\n\u22a2 4 ^ 2 \u2264 2 * 2 * centralBinom 2\n[PROOFSTEP]\nnorm_num [centralBinom, choose]\n[GOAL]\nx\u271d : 0 < 3\n\u22a2 4 ^ 3 \u2264 2 * 3 * centralBinom 3\n[PROOFSTEP]\nnorm_num [centralBinom, choose]\n[GOAL]\nn : \u2115\nx\u271d : 0 < n + 4\n\u22a2 (n + 4) * centralBinom (n + 4) \u2264 2 * (n + 4) * centralBinom (n + 4)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nn : \u2115\nx\u271d : 0 < n + 4\n\u22a2 (n + 4) * centralBinom (n + 4) \u2264 2 * ((n + 4) * centralBinom (n + 4))\n[PROOFSTEP]\nrefine' le_mul_of_pos_left zero_lt_two\n[GOAL]\nn : \u2115\n\u22a2 2 \u2223 centralBinom (n + 1)\n[PROOFSTEP]\nuse(n + 1 + n).choose n\n[GOAL]\ncase h\nn : \u2115\n\u22a2 centralBinom (n + 1) = 2 * choose (n + 1 + n) n\n[PROOFSTEP]\nrw [centralBinom_eq_two_mul_choose, two_mul, \u2190 add_assoc, choose_succ_succ' (n + 1 + n) n, choose_symm_add, \u2190 two_mul]\n[GOAL]\nn : \u2115\nh : 0 < n\n\u22a2 2 \u2223 centralBinom n\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos h]\n[GOAL]\nn : \u2115\nh : 0 < n\n\u22a2 2 \u2223 centralBinom (succ (pred n))\n[PROOFSTEP]\nexact two_dvd_centralBinom_succ n.pred\n[GOAL]\nn : \u2115\n\u22a2 n + 1 \u2223 centralBinom n\n[PROOFSTEP]\nhave h_s : (n + 1).coprime (2 * n + 1) :=\n  by\n  rw [two_mul, add_assoc, coprime_add_self_right, coprime_self_add_left]\n  exact coprime_one_left n\n[GOAL]\nn : \u2115\n\u22a2 coprime (n + 1) (2 * n + 1)\n[PROOFSTEP]\nrw [two_mul, add_assoc, coprime_add_self_right, coprime_self_add_left]\n[GOAL]\nn : \u2115\n\u22a2 coprime 1 n\n[PROOFSTEP]\nexact coprime_one_left n\n[GOAL]\nn : \u2115\nh_s : coprime (n + 1) (2 * n + 1)\n\u22a2 n + 1 \u2223 centralBinom n\n[PROOFSTEP]\napply h_s.dvd_of_dvd_mul_left\n[GOAL]\nn : \u2115\nh_s : coprime (n + 1) (2 * n + 1)\n\u22a2 n + 1 \u2223 (2 * n + 1) * centralBinom n\n[PROOFSTEP]\napply Nat.dvd_of_mul_dvd_mul_left zero_lt_two\n[GOAL]\nn : \u2115\nh_s : coprime (n + 1) (2 * n + 1)\n\u22a2 2 * (n + 1) \u2223 2 * ((2 * n + 1) * centralBinom n)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 succ_mul_centralBinom_succ, mul_comm]\n[GOAL]\nn : \u2115\nh_s : coprime (n + 1) (2 * n + 1)\n\u22a2 (n + 1) * 2 \u2223 (n + 1) * centralBinom (n + 1)\n[PROOFSTEP]\nexact mul_dvd_mul_left _ (two_dvd_centralBinom_succ n)\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Choose.Central", "llama_tokens": 2860, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867681382279, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.5421114508831326}}
{"text": "[GOAL]\nm n a b c d : \u2115\n\u22a2 a \u2261 0 [MOD n] \u2194 n \u2223 a\n[PROOFSTEP]\nrw [ModEq, zero_mod, dvd_iff_mod_eq_zero]\n[GOAL]\nm n a b c d : \u2115\n\u22a2 a \u2261 b [MOD n] \u2194 \u2191n \u2223 \u2191b - \u2191a\n[PROOFSTEP]\nrw [ModEq, eq_comm, \u2190 Int.coe_nat_inj', Int.coe_nat_mod, Int.coe_nat_mod, Int.emod_eq_emod_iff_emod_sub_eq_zero,\n  Int.dvd_iff_emod_eq_zero]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2264 b\n\u22a2 a \u2261 b [MOD n] \u2194 n \u2223 b - a\n[PROOFSTEP]\nrw [modEq_iff_dvd, \u2190 Int.coe_nat_dvd, Int.ofNat_sub h]\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 c * a \u2261 c * b [MOD c * n]\n[PROOFSTEP]\nunfold ModEq at *\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a % n = b % n\n\u22a2 c * a % (c * n) = c * b % (c * n)\n[PROOFSTEP]\nrw [mul_mod_mul_left, mul_mod_mul_left, h]\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a * c \u2261 b * c [MOD n * c]\n[PROOFSTEP]\nrw [mul_comm a, mul_comm b, mul_comm n]\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 c * a \u2261 c * b [MOD c * n]\n[PROOFSTEP]\nexact h.mul_left' c\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a * c \u2261 b * c [MOD n]\n[PROOFSTEP]\nrw [mul_comm a, mul_comm b]\n[GOAL]\nm n a b c\u271d d c : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 c * a \u2261 c * b [MOD n]\n[PROOFSTEP]\nexact h.mul_left c\n[GOAL]\nm\u271d n a b c d m : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a ^ m \u2261 b ^ m [MOD n]\n[PROOFSTEP]\ninduction m with\n| zero => rfl\n| succ d hd =>\n  rw [pow_succ, pow_succ]\n  exact hd.mul h\n[GOAL]\nm\u271d n a b c d m : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a ^ m \u2261 b ^ m [MOD n]\n[PROOFSTEP]\ninduction m with\n| zero => rfl\n| succ d hd =>\n  rw [pow_succ, pow_succ]\n  exact hd.mul h\n[GOAL]\ncase zero\nm n a b c d : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a ^ zero \u2261 b ^ zero [MOD n]\n[PROOFSTEP]\n\n| zero => rfl\n[GOAL]\ncase zero\nm n a b c d : \u2115\nh : a \u2261 b [MOD n]\n\u22a2 a ^ zero \u2261 b ^ zero [MOD n]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nm n a b c d\u271d : \u2115\nh : a \u2261 b [MOD n]\nd : \u2115\nhd : a ^ d \u2261 b ^ d [MOD n]\n\u22a2 a ^ succ d \u2261 b ^ succ d [MOD n]\n[PROOFSTEP]\n\n| succ d hd =>\n  rw [pow_succ, pow_succ]\n  exact hd.mul h\n[GOAL]\ncase succ\nm n a b c d\u271d : \u2115\nh : a \u2261 b [MOD n]\nd : \u2115\nhd : a ^ d \u2261 b ^ d [MOD n]\n\u22a2 a ^ succ d \u2261 b ^ succ d [MOD n]\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\nm n a b c d\u271d : \u2115\nh : a \u2261 b [MOD n]\nd : \u2115\nhd : a ^ d \u2261 b ^ d [MOD n]\n\u22a2 a ^ d * a \u2261 b ^ d * b [MOD n]\n[PROOFSTEP]\nexact hd.mul h\n[GOAL]\nm n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 a + c \u2261 b + d [MOD n]\n[PROOFSTEP]\nrw [modEq_iff_dvd, Int.ofNat_add, Int.ofNat_add, add_sub_add_comm]\n[GOAL]\nm n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : c \u2261 d [MOD n]\n\u22a2 \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)\n[PROOFSTEP]\nexact dvd_add h\u2081.dvd h\u2082.dvd\n[GOAL]\nm n a b c d : \u2115\nh\u2081 : a \u2261 b [MOD n]\nh\u2082 : a + c \u2261 b + d [MOD n]\n\u22a2 c \u2261 d [MOD n]\n[PROOFSTEP]\nsimp only [modEq_iff_dvd, Int.ofNat_add] at *\n[GOAL]\nm n a b c d : \u2115\nh\u2081 : \u2191n \u2223 \u2191b - \u2191a\nh\u2082 : \u2191n \u2223 \u2191b + \u2191d - (\u2191a + \u2191c)\n\u22a2 \u2191n \u2223 \u2191d - \u2191c\n[PROOFSTEP]\nrw [add_sub_add_comm] at h\u2082 \n[GOAL]\nm n a b c d : \u2115\nh\u2081 : \u2191n \u2223 \u2191b - \u2191a\nh\u2082 : \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)\n\u22a2 \u2191n \u2223 \u2191d - \u2191c\n[PROOFSTEP]\nconvert _root_.dvd_sub h\u2082 h\u2081 using 1\n[GOAL]\ncase h.e'_4\nm n a b c d : \u2115\nh\u2081 : \u2191n \u2223 \u2191b - \u2191a\nh\u2082 : \u2191n \u2223 \u2191b - \u2191a + (\u2191d - \u2191c)\n\u22a2 \u2191d - \u2191c = \u2191b - \u2191a + (\u2191d - \u2191c) - (\u2191b - \u2191a)\n[PROOFSTEP]\nrw [add_sub_cancel']\n[GOAL]\nm n a b c d : \u2115\nh\u2081 : c \u2261 d [MOD n]\nh\u2082 : a + c \u2261 b + d [MOD n]\n\u22a2 a \u2261 b [MOD n]\n[PROOFSTEP]\nrw [add_comm a, add_comm b] at h\u2082 \n[GOAL]\nm n a b c d : \u2115\nh\u2081 : c \u2261 d [MOD n]\nh\u2082 : c + a \u2261 d + b [MOD n]\n\u22a2 a \u2261 b [MOD n]\n[PROOFSTEP]\nexact h\u2081.add_left_cancel h\u2082\n[GOAL]\nm\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 c * a \u2261 c * b [MOD c * m] \u2192 a \u2261 b [MOD m]\n[PROOFSTEP]\nsimp [modEq_iff_dvd, \u2190 mul_sub, mul_dvd_mul_iff_left (by simp [hc] : (c : \u2124) \u2260 0)]\n[GOAL]\nm\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\nm\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 a * c \u2261 b * c [MOD m * c] \u2192 a \u2261 b [MOD m]\n[PROOFSTEP]\nsimp [modEq_iff_dvd, \u2190 sub_mul, mul_dvd_mul_iff_right (by simp [hc] : (c : \u2124) \u2260 0)]\n[GOAL]\nm\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\nm\u271d n a b c d m : \u2115\nh : a \u2261 b [MOD m * n]\n\u22a2 a \u2261 b [MOD n]\n[PROOFSTEP]\nrw [modEq_iff_dvd] at *\n[GOAL]\nm\u271d n a b c d m : \u2115\nh : \u2191(m * n) \u2223 \u2191b - \u2191a\n\u22a2 \u2191n \u2223 \u2191b - \u2191a\n[PROOFSTEP]\nexact (dvd_mul_left (n : \u2124) (m : \u2124)).trans h\n[GOAL]\nm n a b c d : \u2115\nh : a / c \u2261 b / c [MOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 a \u2261 b [MOD m]\n[PROOFSTEP]\nconvert h.mul_left' c\n[GOAL]\ncase h.e'_1\nm n a b c d : \u2115\nh : a / c \u2261 b / c [MOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 m = c * (m / c)\n[PROOFSTEP]\nrwa [Nat.mul_div_cancel']\n[GOAL]\ncase h.e'_2\nm n a b c d : \u2115\nh : a / c \u2261 b / c [MOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 a = c * (a / c)\n[PROOFSTEP]\nrwa [Nat.mul_div_cancel']\n[GOAL]\ncase h.e'_3\nm n a b c d : \u2115\nh : a / c \u2261 b / c [MOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 b = c * (b / c)\n[PROOFSTEP]\nrwa [Nat.mul_div_cancel']\n[GOAL]\nm n a b c d : \u2115\nh : b \u2264 a\n\u22a2 \u2191(a - b) \u2223 \u2191a - \u2191b\n[PROOFSTEP]\nrw [Int.ofNat_sub h]\n[GOAL]\nm n a b c d : \u2115\n\u22a2 a \u2261 b [MOD 0] \u2194 a = b\n[PROOFSTEP]\nrw [ModEq, mod_zero, mod_zero]\n[GOAL]\nm n a b c d : \u2115\n\u22a2 n + a \u2261 a [MOD n]\n[PROOFSTEP]\nrw [ModEq, add_mod_left]\n[GOAL]\nm n a b c d : \u2115\n\u22a2 a + n \u2261 a [MOD n]\n[PROOFSTEP]\nrw [ModEq, add_mod_right]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nhdm : d \u2223 m\n\u22a2 d \u2223 a \u2194 d \u2223 b\n[PROOFSTEP]\nsimp only [\u2190 modEq_zero_iff_dvd]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nhdm : d \u2223 m\n\u22a2 a \u2261 0 [MOD d] \u2194 b \u2261 0 [MOD d]\n[PROOFSTEP]\nreplace h := h.of_dvd hdm\n[GOAL]\nm n a b c d : \u2115\nhdm : d \u2223 m\nh : a \u2261 b [MOD d]\n\u22a2 a \u2261 0 [MOD d] \u2194 b \u2261 0 [MOD d]\n[PROOFSTEP]\nexact \u27e8h.symm.trans, h.trans\u27e9\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\n\u22a2 gcd a m = gcd b m\n[PROOFSTEP]\nhave h1 := gcd_dvd_right a m\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nh1 : gcd a m \u2223 m\n\u22a2 gcd a m = gcd b m\n[PROOFSTEP]\nhave h2 := gcd_dvd_right b m\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nh1 : gcd a m \u2223 m\nh2 : gcd b m \u2223 m\n\u22a2 gcd a m = gcd b m\n[PROOFSTEP]\nexact\n  dvd_antisymm (dvd_gcd ((h.dvd_iff h1).mp (gcd_dvd_left a m)) h1) (dvd_gcd ((h.dvd_iff h2).mpr (gcd_dvd_left b m)) h2)\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nh2 : |\u2191b - \u2191a| < \u2191m\n\u22a2 a = b\n[PROOFSTEP]\napply Int.ofNat.inj\n[GOAL]\ncase x\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nh2 : |\u2191b - \u2191a| < \u2191m\n\u22a2 Int.ofNat a = Int.ofNat b\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero]\n[GOAL]\ncase x\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nh2 : |\u2191b - \u2191a| < \u2191m\n\u22a2 Int.ofNat b - Int.ofNat a = 0\n[PROOFSTEP]\nexact Int.eq_zero_of_abs_lt_dvd h.dvd h2\n[GOAL]\nm n a b c d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\n\u22a2 a \u2261 b [MOD m / gcd m c]\n[PROOFSTEP]\nlet d := gcd m c\n[GOAL]\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\n\u22a2 a \u2261 b [MOD m / gcd m c]\n[PROOFSTEP]\nhave hmd := gcd_dvd_left m c\n[GOAL]\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\n\u22a2 a \u2261 b [MOD m / gcd m c]\n[PROOFSTEP]\nhave hcd := gcd_dvd_right m c\n[GOAL]\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 a \u2261 b [MOD m / gcd m c]\n[PROOFSTEP]\nrw [modEq_iff_dvd]\n[GOAL]\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191(m / gcd m c) \u2223 \u2191b - \u2191a\n[PROOFSTEP]\nrefine' @Int.dvd_of_dvd_mul_right_of_gcd_one (m / d) (c / d) (b - a) _ _\n[GOAL]\ncase refine'_1\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191m / \u2191d \u2223 \u2191c / \u2191d * (\u2191b - \u2191a)\ncase refine'_2\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 Int.gcd (\u2191m / \u2191d) (\u2191c / \u2191d) = 1\n[PROOFSTEP]\nshow (m / d : \u2124) \u2223 c / d * (b - a)\n[GOAL]\ncase refine'_1\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191m / \u2191d \u2223 \u2191c / \u2191d * (\u2191b - \u2191a)\n[PROOFSTEP]\nrw [mul_comm, \u2190 Int.mul_ediv_assoc (b - a) (Int.coe_nat_dvd.mpr hcd), mul_comm]\n[GOAL]\ncase refine'_1\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191m / \u2191d \u2223 \u2191c * (\u2191b - \u2191a) / \u2191(gcd m c)\n[PROOFSTEP]\napply Int.ediv_dvd_ediv (Int.coe_nat_dvd.mpr hmd)\n[GOAL]\ncase refine'_1\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191m \u2223 \u2191c * (\u2191b - \u2191a)\n[PROOFSTEP]\nrw [mul_sub]\n[GOAL]\ncase refine'_1\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 \u2191m \u2223 \u2191c * \u2191b - \u2191c * \u2191a\n[PROOFSTEP]\nexact modEq_iff_dvd.mp h\n[GOAL]\ncase refine'_2\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 Int.gcd (\u2191m / \u2191d) (\u2191c / \u2191d) = 1\n[PROOFSTEP]\nshow Int.gcd (m / d) (c / d) = 1\n[GOAL]\ncase refine'_2\nm n a b c d\u271d : \u2115\nhm : 0 < m\nh : c * a \u2261 c * b [MOD m]\nd : \u2115 := gcd m c\nhmd : gcd m c \u2223 m\nhcd : gcd m c \u2223 c\n\u22a2 Int.gcd (\u2191m / \u2191d) (\u2191c / \u2191d) = 1\n[PROOFSTEP]\nsimp only [\u2190 Int.coe_nat_div, Int.coe_nat_gcd (m / d) (c / d), gcd_div hmd hcd, Nat.div_self (gcd_pos_of_pos_left c hm)]\n[GOAL]\nm n a b c d : \u2115\nhm : 0 < m\nh : a * c \u2261 b * c [MOD m]\n\u22a2 a \u2261 b [MOD m / gcd m c]\n[PROOFSTEP]\napply cancel_left_div_gcd hm\n[GOAL]\nm n a b c d : \u2115\nhm : 0 < m\nh : a * c \u2261 b * c [MOD m]\n\u22a2 c * a \u2261 c * b [MOD m]\n[PROOFSTEP]\nsimpa [mul_comm] using h\n[GOAL]\nm n a b c d : \u2115\nhmc : gcd m c = 1\nh : c * a \u2261 c * b [MOD m]\n\u22a2 a \u2261 b [MOD m]\n[PROOFSTEP]\nrcases m.eq_zero_or_pos with (rfl | hm)\n[GOAL]\ncase inl\nn a b c d : \u2115\nhmc : gcd 0 c = 1\nh : c * a \u2261 c * b [MOD 0]\n\u22a2 a \u2261 b [MOD 0]\n[PROOFSTEP]\nsimp only [gcd_zero_left] at hmc \n[GOAL]\ncase inl\nn a b c d : \u2115\nhmc : c = 1\nh : c * a \u2261 c * b [MOD 0]\n\u22a2 a \u2261 b [MOD 0]\n[PROOFSTEP]\nsimp only [gcd_zero_left, hmc, one_mul, modEq_zero_iff] at h \n[GOAL]\ncase inl\nn a b c d : \u2115\nhmc : c = 1\nh : a = b\n\u22a2 a \u2261 b [MOD 0]\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inl\nn a c d : \u2115\nhmc : c = 1\n\u22a2 a \u2261 a [MOD 0]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nm n a b c d : \u2115\nhmc : gcd m c = 1\nh : c * a \u2261 c * b [MOD m]\nhm : m > 0\n\u22a2 a \u2261 b [MOD m]\n[PROOFSTEP]\nsimpa [hmc] using h.cancel_left_div_gcd hm\n[GOAL]\nm n a b c d : \u2115\nhmc : gcd m c = 1\nh : a * c \u2261 b * c [MOD m]\n\u22a2 c * a \u2261 c * b [MOD m]\n[PROOFSTEP]\nsimpa [mul_comm] using h\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : n = 0\n\u22a2 a \u2261 a [MOD n] \u2227 a \u2261 b [MOD m]\n[PROOFSTEP]\nrw [hn, gcd_zero_left] at h \n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nhn : n = 0\n\u22a2 a \u2261 a [MOD n] \u2227 a \u2261 b [MOD m]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nhn : n = 0\n\u22a2 a \u2261 a [MOD n]\ncase right m n a b c d : \u2115 h : a \u2261 b [MOD m] hn : n = 0 \u22a2 a \u2261 b [MOD m]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nm n a b c d : \u2115\nh : a \u2261 b [MOD m]\nhn : n = 0\n\u22a2 a \u2261 b [MOD m]\n[PROOFSTEP]\nexact h\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 b \u2261 a [MOD n] \u2227 b \u2261 b [MOD m]\n[PROOFSTEP]\nrw [hm, gcd_zero_right] at h \n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD n]\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 b \u2261 a [MOD n] \u2227 b \u2261 b [MOD m]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nm n a b c d : \u2115\nh : a \u2261 b [MOD n]\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 b \u2261 a [MOD n]\ncase right m n a b c d : \u2115 h : a \u2261 b [MOD n] hn : \u00acn = 0 hm : m = 0 \u22a2 b \u2261 b [MOD m]\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase right\nm n a b c d : \u2115\nh : a \u2261 b [MOD n]\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 b \u2261 b [MOD m]\n[PROOFSTEP]\nrfl\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 (match xgcd n m with\n      | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n      a [MOD n] \u2227\n    (match xgcd n m with\n      | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n      b [MOD m]\n[PROOFSTEP]\nrw [xgcd_val]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 (match (gcdA n m, gcdB n m) with\n      | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n      a [MOD n] \u2227\n    (match (gcdA n m, gcdB n m) with\n      | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n      b [MOD m]\n[PROOFSTEP]\ndsimp\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 Int.toNat ((\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)) \u2261 a [MOD n] \u2227\n    Int.toNat ((\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)) \u2261 b [MOD m]\n[PROOFSTEP]\nrw [modEq_iff_dvd, modEq_iff_dvd, Int.toNat_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero.2 (lcm_ne_zero hn hm)))]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m) \u2227\n    \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nhave hnonzero : (gcd n m : \u2124) \u2260 0 := by\n  norm_cast\n  rw [Nat.gcd_eq_zero_iff, not_and]\n  exact fun _ => hm\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 \u2191(gcd n m) \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 \u00acgcd n m = 0\n[PROOFSTEP]\nrw [Nat.gcd_eq_zero_iff, not_and]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 n = 0 \u2192 \u00acm = 0\n[PROOFSTEP]\nexact fun _ => hm\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m) \u2227\n    \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nhave hcoedvd : \u2200 t, (gcd n m : \u2124) \u2223 t * (b - a) := fun t => h.dvd.mul_left _\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m) \u2227\n    \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nhave := gcd_eq_gcd_ab n m\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m) \u2227\n    \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nrw [Int.emod_def, \u2190 sub_add]\n[GOAL]\ncase right\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)\n[PROOFSTEP]\nrw [Int.emod_def, \u2190 sub_add]\n[GOAL]\ncase left\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223\n    \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) +\n      \u2191(lcm n m) * ((\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) / \u2191(lcm n m))\n[PROOFSTEP]\nrefine' dvd_add _ (dvd_mul_of_dvd_left _ _)\n[GOAL]\ncase right\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223\n    \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) +\n      \u2191(lcm n m) * ((\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m) / \u2191(lcm n m))\n[PROOFSTEP]\nrefine' dvd_add _ (dvd_mul_of_dvd_left _ _)\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\ntry norm_cast\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase left.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191(lcm n m)\n[PROOFSTEP]\ntry norm_cast\n[GOAL]\ncase left.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191(lcm n m)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase right.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\ntry norm_cast\n[GOAL]\ncase right.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase right.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191(lcm n m)\n[PROOFSTEP]\ntry norm_cast\n[GOAL]\ncase right.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191(lcm n m)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nrw [\u2190 sub_eq_iff_eq_add'] at this \n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191n * gcdA n m = \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191a - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nrw [\u2190 this, sub_mul, \u2190 add_sub_assoc, add_comm, add_sub_assoc, \u2190 mul_sub, Int.add_ediv_of_dvd_left,\n  Int.mul_ediv_cancel_left _ hnonzero, Int.mul_ediv_assoc _ h.dvd, \u2190 sub_sub, sub_self, zero_sub, dvd_neg, mul_assoc]\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191n * gcdA n m = \u2191m * gcdB n m\n\u22a2 \u2191n \u2223 \u2191n * (gcdA n m * ((\u2191b - \u2191a) / \u2191(gcd n m)))\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191n * gcdA n m = \u2191m * gcdB n m\n\u22a2 \u2191(gcd n m) \u2223 \u2191(gcd n m) * \u2191a\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191n * gcdA n m = \u2191m * gcdB n m\n\u22a2 \u2191(gcd n m) \u2223 \u2191(gcd n m) * \u2191a\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase left.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191n * gcdA n m = \u2191m * gcdB n m\n\u22a2 gcd n m \u2223 gcd n m * a\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\ncase left.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 n \u2223 lcm n m\n[PROOFSTEP]\nexact dvd_lcm_left n m\n[GOAL]\ncase right.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nrw [\u2190 sub_eq_iff_eq_add] at this \n[GOAL]\ncase right.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191m * gcdB n m = \u2191n * gcdA n m\n\u22a2 \u2191m \u2223 \u2191b - (\u2191n * gcdA n m * \u2191b + \u2191m * gcdB n m * \u2191a) / \u2191(gcd n m)\n[PROOFSTEP]\nrw [\u2190 this, sub_mul, sub_add, \u2190 mul_sub, Int.sub_ediv_of_dvd, Int.mul_ediv_cancel_left _ hnonzero,\n  Int.mul_ediv_assoc _ h.dvd, \u2190 sub_add, sub_self, zero_add, mul_assoc]\n[GOAL]\ncase right.refine'_1\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191m * gcdB n m = \u2191n * gcdA n m\n\u22a2 \u2191m \u2223 \u2191m * (gcdB n m * ((\u2191b - \u2191a) / \u2191(gcd n m)))\ncase right.refine'_1.hcb\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191m * gcdB n m = \u2191n * gcdA n m\n\u22a2 \u2191(gcd n m) \u2223 \u2191m * gcdB n m * (\u2191b - \u2191a)\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\ncase right.refine'_1.hcb\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) - \u2191m * gcdB n m = \u2191n * gcdA n m\n\u22a2 \u2191(gcd n m) \u2223 \u2191m * gcdB n m * (\u2191b - \u2191a)\n[PROOFSTEP]\nexact hcoedvd _\n[GOAL]\ncase right.refine'_2\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : \u00acn = 0\nhm : \u00acm = 0\nhnonzero : \u2191(gcd n m) \u2260 0\nhcoedvd : \u2200 (t : \u2124), \u2191(gcd n m) \u2223 t * (\u2191b - \u2191a)\nthis : \u2191(gcd n m) = \u2191n * gcdA n m + \u2191m * gcdB n m\n\u22a2 m \u2223 lcm n m\n[PROOFSTEP]\nexact dvd_lcm_right n m\n[GOAL]\nm n a\u271d b\u271d c d : \u2115\nco : coprime n m\na b : \u2115\n\u22a2 a \u2261 b [MOD gcd n m]\n[PROOFSTEP]\nconvert @modEq_one a b\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : n \u2260 0\nhm : m \u2260 0\n\u22a2 \u2191(chineseRemainder' h) < lcm n m\n[PROOFSTEP]\ndsimp only [chineseRemainder']\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : n \u2260 0\nhm : m \u2260 0\n\u22a2 \u2191(if hn : n = 0 then { val := a, property := (_ : a \u2261 a [MOD n] \u2227 a \u2261 b [MOD m]) }\n      else\n        if hm : m = 0 then { val := b, property := (_ : b \u2261 a [MOD n] \u2227 b \u2261 b [MOD m]) }\n        else\n          { val := Int.toNat ((\u2191n * (xgcd n m).fst * \u2191b + \u2191m * (xgcd n m).snd * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m)),\n            property :=\n              (_ :\n                (match xgcd n m with\n                    | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n                    a [MOD n] \u2227\n                  (match xgcd n m with\n                    | (c, d) => Int.toNat ((\u2191n * c * \u2191b + \u2191m * d * \u2191a) / \u2191(gcd n m) % \u2191(lcm n m))) \u2261\n                    b [MOD m]) }) <\n    lcm n m\n[PROOFSTEP]\nrw [dif_neg hn, dif_neg hm, Subtype.coe_mk, xgcd_val, \u2190 Int.toNat_coe_nat (lcm n m)]\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : n \u2260 0\nhm : m \u2260 0\n\u22a2 Int.toNat\n      ((\u2191n * (gcdA n m, gcdB n m).fst * \u2191b + \u2191m * (gcdA n m, gcdB n m).snd * \u2191a) / \u2191(gcd n m) %\n        \u2191(Int.toNat \u2191(lcm n m))) <\n    Int.toNat \u2191(lcm n m)\n[PROOFSTEP]\nhave lcm_pos := Int.coe_nat_pos.mpr (Nat.pos_of_ne_zero (lcm_ne_zero hn hm))\n[GOAL]\nm n a b c d : \u2115\nh : a \u2261 b [MOD gcd n m]\nhn : n \u2260 0\nhm : m \u2260 0\nlcm_pos : 0 < \u2191(lcm n m)\n\u22a2 Int.toNat\n      ((\u2191n * (gcdA n m, gcdB n m).fst * \u2191b + \u2191m * (gcdA n m, gcdB n m).snd * \u2191a) / \u2191(gcd n m) %\n        \u2191(Int.toNat \u2191(lcm n m))) <\n    Int.toNat \u2191(lcm n m)\n[PROOFSTEP]\nexact (Int.toNat_lt_toNat lcm_pos).mpr (Int.emod_lt_of_pos _ lcm_pos)\n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : coprime m n\nh : a \u2261 b [MOD m] \u2227 a \u2261 b [MOD n]\n\u22a2 a \u2261 b [MOD m * n]\n[PROOFSTEP]\nrw [Nat.modEq_iff_dvd, Nat.modEq_iff_dvd, \u2190 Int.dvd_natAbs, Int.coe_nat_dvd, \u2190 Int.dvd_natAbs, Int.coe_nat_dvd] at h \n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : coprime m n\nh : m \u2223 Int.natAbs (\u2191b - \u2191a) \u2227 n \u2223 Int.natAbs (\u2191b - \u2191a)\n\u22a2 a \u2261 b [MOD m * n]\n[PROOFSTEP]\nrw [Nat.modEq_iff_dvd, \u2190 Int.dvd_natAbs, Int.coe_nat_dvd]\n[GOAL]\nm\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : coprime m n\nh : m \u2223 Int.natAbs (\u2191b - \u2191a) \u2227 n \u2223 Int.natAbs (\u2191b - \u2191a)\n\u22a2 m * n \u2223 Int.natAbs (\u2191b - \u2191a)\n[PROOFSTEP]\nexact hmn.mul_dvd_of_dvd_of_dvd h.1 h.2\n[GOAL]\nm n\u271d a\u271d b\u271d c d b a n : \u2115\nh : a * b \u2261 1 [MOD n]\n\u22a2 coprime a n\n[PROOFSTEP]\nobtain \u27e8g, hh\u27e9 := Nat.gcd_dvd_right a n\n[GOAL]\ncase intro\nm n\u271d a\u271d b\u271d c d b a n : \u2115\nh : a * b \u2261 1 [MOD n]\ng : \u2115\nhh : n = gcd a n * g\n\u22a2 coprime a n\n[PROOFSTEP]\nrw [Nat.coprime_iff_gcd_eq_one, \u2190 Nat.dvd_one, \u2190 Nat.modEq_zero_iff_dvd]\n[GOAL]\ncase intro\nm n\u271d a\u271d b\u271d c d b a n : \u2115\nh : a * b \u2261 1 [MOD n]\ng : \u2115\nhh : n = gcd a n * g\n\u22a2 1 \u2261 0 [MOD gcd a n]\n[PROOFSTEP]\ncalc\n  1 \u2261 a * b [MOD a.gcd n] := (hh \u25b8 h).symm.of_mul_right g\n  _ \u2261 0 * b [MOD a.gcd n] := ((Nat.modEq_zero_iff_dvd.mpr (Nat.gcd_dvd_left _ _)).mul_right b)\n  _ = 0 := by rw [zero_mul]\n[GOAL]\nm n\u271d a\u271d b\u271d c d b a n : \u2115\nh : a * b \u2261 1 [MOD n]\ng : \u2115\nhh : n = gcd a n * g\n\u22a2 0 * b = 0\n[PROOFSTEP]\nrw [zero_mul]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhb0 : b = 0\n\u22a2 a / b % c = a % (b * c) / b\n[PROOFSTEP]\nsimp [hb0]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhb0 : \u00acb = 0\n\u22a2 a / b % c = a % (b * c) / b\n[PROOFSTEP]\nrw [\u2190 @add_right_cancel_iff _ _ _ (c * (a / b / c)), mod_add_div, Nat.div_div_eq_div_mul, \u2190 mul_right_inj' hb0, \u2190\n  @add_left_cancel_iff _ _ _ (a % b), mod_add_div, mul_add, \u2190 @add_left_cancel_iff _ _ _ (a % (b * c) % b),\n  add_left_comm, \u2190 add_assoc (a % (b * c) % b), mod_add_div, \u2190 mul_assoc, mod_add_div, mod_mul_right_mod]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : c = 0\n\u22a2 ((a + b) % c + if c \u2264 a % c + b % c then c else 0) = a % c + b % c\n[PROOFSTEP]\nsimp [hc0, Nat.mod_zero]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\n\u22a2 ((a + b) % c + if c \u2264 a % c + b % c then c else 0) = a % c + b % c\n[PROOFSTEP]\nrw [this]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\n\u22a2 ((a % c + b % c) % c + if c \u2264 a % c + b % c then c else 0) = a % c + b % c\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : c \u2264 a % c + b % c\n\u22a2 (a % c + b % c) % c + c = a % c + b % c\n[PROOFSTEP]\nhave h2 : (a % c + b % c) / c < 2 :=\n  Nat.div_lt_of_lt_mul\n    (by\n      rw [mul_two]\n      exact add_lt_add (Nat.mod_lt _ (Nat.pos_of_ne_zero hc0)) (Nat.mod_lt _ (Nat.pos_of_ne_zero hc0)))\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : c \u2264 a % c + b % c\n\u22a2 a % c + b % c < c * 2\n[PROOFSTEP]\nrw [mul_two]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : c \u2264 a % c + b % c\n\u22a2 a % c + b % c < c + c\n[PROOFSTEP]\nexact add_lt_add (Nat.mod_lt _ (Nat.pos_of_ne_zero hc0)) (Nat.mod_lt _ (Nat.pos_of_ne_zero hc0))\n[GOAL]\ncase pos\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : c \u2264 a % c + b % c\nh2 : (a % c + b % c) / c < 2\n\u22a2 (a % c + b % c) % c + c = a % c + b % c\n[PROOFSTEP]\nhave h0 : 0 < (a % c + b % c) / c := Nat.div_pos h (Nat.pos_of_ne_zero hc0)\n[GOAL]\ncase pos\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : c \u2264 a % c + b % c\nh2 : (a % c + b % c) / c < 2\nh0 : 0 < (a % c + b % c) / c\n\u22a2 (a % c + b % c) % c + c = a % c + b % c\n[PROOFSTEP]\nrw [\u2190 @add_right_cancel_iff _ _ _ (c * ((a % c + b % c) / c)), add_comm _ c, add_assoc, mod_add_div,\n  le_antisymm (le_of_lt_succ h2) h0, mul_one, add_comm]\n[GOAL]\ncase neg\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nthis : (a + b) % c = (a % c + b % c) % c\nhc0 : \u00acc = 0\nh : \u00acc \u2264 a % c + b % c\n\u22a2 (a % c + b % c) % c + 0 = a % c + b % c\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (lt_of_not_ge h), add_zero]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc : a % c + b % c < c\n\u22a2 (a + b) % c = a % c + b % c\n[PROOFSTEP]\nrw [\u2190 add_mod_add_ite, if_neg (not_le_of_lt hc), add_zero]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc : c \u2264 a % c + b % c\n\u22a2 (a + b) % c + c = a % c + b % c\n[PROOFSTEP]\nrw [\u2190 add_mod_add_ite, if_pos hc]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 (a + b) / c = a / c + b / c + if c \u2264 a % c + b % c then 1 else 0\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' hc0.ne', \u2190 @add_left_cancel_iff _ _ _ ((a + b) % c + a % c + b % c)]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 (a + b) % c + a % c + b % c + c * ((a + b) / c) =\n    (a + b) % c + a % c + b % c + c * (a / c + b / c + if c \u2264 a % c + b % c then 1 else 0)\n[PROOFSTEP]\nsuffices\n  (a + b) % c + c * ((a + b) / c) + a % c + b % c =\n    (a % c + c * (a / c) + (b % c + c * (b / c)) + c * if c \u2264 a % c + b % c then 1 else 0) + (a + b) % c\n  by simpa only [mul_add, add_comm, add_left_comm, add_assoc]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\nthis :\n  (a + b) % c + c * ((a + b) / c) + a % c + b % c =\n    (a % c + c * (a / c) + (b % c + c * (b / c)) + c * if c \u2264 a % c + b % c then 1 else 0) + (a + b) % c\n\u22a2 (a + b) % c + a % c + b % c + c * ((a + b) / c) =\n    (a + b) % c + a % c + b % c + c * (a / c + b / c + if c \u2264 a % c + b % c then 1 else 0)\n[PROOFSTEP]\nsimpa only [mul_add, add_comm, add_left_comm, add_assoc]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 (a + b) % c + c * ((a + b) / c) + a % c + b % c =\n    (a % c + c * (a / c) + (b % c + c * (b / c)) + c * if c \u2264 a % c + b % c then 1 else 0) + (a + b) % c\n[PROOFSTEP]\nrw [mod_add_div, mod_add_div, mod_add_div, mul_ite, add_assoc, add_assoc]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 a + (b + (a % c + b % c)) = (a + b + if c \u2264 a % c + b % c then c * 1 else c * 0) + (a + b) % c\n[PROOFSTEP]\nconv_lhs => rw [\u2190 add_mod_add_ite]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n| a + (b + (a % c + b % c))\n[PROOFSTEP]\nrw [\u2190 add_mod_add_ite]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n| a + (b + (a % c + b % c))\n[PROOFSTEP]\nrw [\u2190 add_mod_add_ite]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n| a + (b + (a % c + b % c))\n[PROOFSTEP]\nrw [\u2190 add_mod_add_ite]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 a + (b + ((a + b) % c + if c \u2264 a % c + b % c then c else 0)) =\n    (a + b + if c \u2264 a % c + b % c then c * 1 else c * 0) + (a + b) % c\n[PROOFSTEP]\nsimp\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : 0 < c\n\u22a2 a + (b + ((a + b) % c + if c \u2264 a % c + b % c then c else 0)) =\n    (a + b + if c \u2264 a % c + b % c then c else 0) + (a + b) % c\n[PROOFSTEP]\nac_rfl\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc : a % c + b % c < c\nhc0 : c = 0\n\u22a2 (a + b) / c = a / c + b / c\n[PROOFSTEP]\nsimp [hc0]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc : a % c + b % c < c\nhc0 : \u00acc = 0\n\u22a2 (a + b) / c = a / c + b / c\n[PROOFSTEP]\nrw [add_div (Nat.pos_of_ne_zero hc0), if_neg (not_le_of_lt hc), add_zero]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhca : c \u2223 a\nh : c = 0\n\u22a2 (a + b) / c = a / c + b / c\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhca : c \u2223 a\nh : \u00acc = 0\n\u22a2 a % c + b % c < c\n[PROOFSTEP]\nrw [Nat.mod_eq_zero_of_dvd hca, zero_add]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhca : c \u2223 a\nh : \u00acc = 0\n\u22a2 b % c < c\n[PROOFSTEP]\nexact Nat.mod_lt _ (pos_iff_ne_zero.mpr h)\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhca : c \u2223 b\n\u22a2 (a + b) / c = a / c + b / c\n[PROOFSTEP]\nrwa [add_comm, Nat.add_div_of_dvd_right, add_comm]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc : c \u2264 a % c + b % c\nhc0 : 0 < c\n\u22a2 (a + b) / c = a / c + b / c + 1\n[PROOFSTEP]\nrw [add_div hc0, if_pos hc]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : c = 0\n\u22a2 a / c + b / c \u2264 (a + b) / c\n[PROOFSTEP]\nsimp [hc0]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : \u00acc = 0\n\u22a2 a / c + b / c \u2264 (a + b) / c\n[PROOFSTEP]\nrw [Nat.add_div (Nat.pos_of_ne_zero hc0)]\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nhc0 : \u00acc = 0\n\u22a2 a / c + b / c \u2264 a / c + b / c + if c \u2264 a % c + b % c then 1 else 0\n[PROOFSTEP]\nexact Nat.le_add_right _ _\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nh : c \u2223 a + b\nha : \u00acc \u2223 a\nhc : \u00acc \u2264 a % c + b % c\n\u22a2 False\n[PROOFSTEP]\nhave : (a + b) % c = a % c + b % c := add_mod_of_add_mod_lt (lt_of_not_ge hc)\n[GOAL]\nm n a\u271d b\u271d c\u271d d a b c : \u2115\nh : c \u2223 a + b\nha : \u00acc \u2223 a\nhc : \u00acc \u2264 a % c + b % c\nthis : (a + b) % c = a % c + b % c\n\u22a2 False\n[PROOFSTEP]\nsimp_all [dvd_iff_mod_eq_zero]\n[GOAL]\nm\u271d n\u271d a b c d n m : \u2115\n\u22a2 n % 2 = 1 \u2192 m % 2 = 1 \u2192 n * m % 2 = 1\n[PROOFSTEP]\nsimpa [Nat.ModEq] using @ModEq.mul 2 n 1 m 1\n[GOAL]\nm\u271d n\u271d a b c d m n : \u2115\nhm1 : m % 2 = 1\nhn1 : n % 2 = 1\nh : m = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\nm\u271d n\u271d a b c d m n : \u2115\nhm1 : m % 2 = 1\nhn1 : n % 2 = 1\nhm0 : 0 < m\nh : n = 0\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\nm\u271d n\u271d a b c d m n : \u2115\nhm1 : m % 2 = 1\nhn1 : n % 2 = 1\nhm0 : 0 < m\nhn0 : 0 < n\n\u22a2 (fun x x_1 => x * x_1) 2 (m * n / 2) = (fun x x_1 => x * x_1) 2 (m * (n / 2) + m / 2)\n[PROOFSTEP]\ndsimp\n[GOAL]\nm\u271d n\u271d a b c d m n : \u2115\nhm1 : m % 2 = 1\nhn1 : n % 2 = 1\nhm0 : 0 < m\nhn0 : 0 < n\n\u22a2 2 * (m * n / 2) = 2 * (m * (n / 2) + m / 2)\n[PROOFSTEP]\nrw [mul_add, two_mul_odd_div_two hm1, mul_left_comm, two_mul_odd_div_two hn1,\n  two_mul_odd_div_two (Nat.odd_mul_odd hm1 hn1), mul_tsub, mul_one, \u2190 add_tsub_assoc_of_le (succ_le_of_lt hm0),\n  tsub_add_cancel_of_le (le_mul_of_one_le_right (Nat.zero_le _) hn0)]\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 n % 4 = 1 \u2192 n % 2 = 1\n[PROOFSTEP]\nsimpa [ModEq, show 2 * 2 = 4 by norm_num] using @ModEq.of_mul_left 2 n 1 2\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 2 * 2 = 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 n % 4 = 3 \u2192 n % 2 = 1\n[PROOFSTEP]\nsimpa [ModEq, show 2 * 2 = 4 by norm_num, show 3 % 4 = 3 by norm_num] using @ModEq.of_mul_left 2 n 3 2\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 2 * 2 = 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 3 % 4 = 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n\u271d a b c d n : \u2115\n\u22a2 \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3\n[PROOFSTEP]\ndecide\n[GOAL]\nm n\u271d a b c d n : \u2115\nhelp : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3\nhn : n % 2 = 1\n\u22a2 4 > 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n\u271d a b c d n : \u2115\nhelp : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3\nhn : n % 2 = 1\n\u22a2 2 \u2223 4\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.ModEq", "llama_tokens": 20646, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424373085145, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.541941068078741}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\n\u22a2 \u2191(powerset univ) = \u2191univ\n[PROOFSTEP]\nsimp [-coe_eq_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Finset \u03b1\n\u22a2 powerset s = univ \u2194 s = univ\n[PROOFSTEP]\nrw [\u2190 Finset.powerset_univ, powerset_inj]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Fintype \u03b1\nk : \u2115\n\u22a2 filter (fun s => card s = k) univ = powersetLen k univ\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Fintype \u03b1\nk : \u2115\na\u271d : Finset \u03b1\n\u22a2 a\u271d \u2208 filter (fun s => card s = k) univ \u2194 a\u271d \u2208 powersetLen k univ\n[PROOFSTEP]\nsimp [Finset.mem_powersetLen]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nk : \u2115\n\u22a2 card { s // Finset.card s = k } = Nat.choose (card \u03b1) k\n[PROOFSTEP]\nsimp [Fintype.subtype_card, Finset.card_univ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 map { toFun := toSet, inj' := (_ : Function.Injective toSet) } (Finset.powerset Finset.univ)\n[PROOFSTEP]\nclassical\nrefine' mem_map.2 \u27e8Finset.univ.filter s, Finset.mem_powerset.2 (Finset.subset_univ _), _\u27e9\napply (coe_filter _ _).trans\nsimp\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 map { toFun := toSet, inj' := (_ : Function.Injective toSet) } (Finset.powerset Finset.univ)\n[PROOFSTEP]\nrefine' mem_map.2 \u27e8Finset.univ.filter s, Finset.mem_powerset.2 (Finset.subset_univ _), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Set \u03b1\n\u22a2 \u2191{ toFun := toSet, inj' := (_ : Function.Injective toSet) } (filter s Finset.univ) = s\n[PROOFSTEP]\napply (coe_filter _ _).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Set \u03b1\n\u22a2 {x | x \u2208 Finset.univ \u2227 s x} = s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\ns : Set \u03b1\n\u22a2 {x | s x} = s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Finite \u03b1\n\u22a2 Finite (Set \u03b1)\n[PROOFSTEP]\ncases nonempty_fintype \u03b1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : Finite \u03b1\nval\u271d : Fintype \u03b1\n\u22a2 Finite (Set \u03b1)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Data.Fintype.Powerset", "llama_tokens": 896, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506418255928, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.5417598634239853}}
{"text": "[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 #((n : \u2115) \u00d7 Functions (Language.sum L (skolem\u2081 L)) n) = #((n : \u2115) \u00d7 BoundedFormula L Empty (n + 1))\n[PROOFSTEP]\nsimp only [card_functions_sum, skolem\u2081_Functions, mk_sigma, sum_add_distrib']\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 ((sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))) =\n    sum fun i => #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nconv_lhs => enter [2, 1, i]; rw [lift_id'.{u, v}]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nenter [2, 1, i]; rw [lift_id'.{u, v}]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nenter [2, 1, i]; rw [lift_id'.{u, v}]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n| (sum fun i => lift #(Functions L i)) + sum fun i => lift #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nenter [2, 1, i]\n[GOAL]\ncase h\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ni : \u2115\n| lift #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nrw [lift_id'.{u, v}]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 ((sum fun i => lift #(Functions L i)) + sum fun i => #(BoundedFormula L Empty (i + 1))) =\n    sum fun i => #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nrw [add_comm, add_eq_max, max_eq_left]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 (sum fun i => lift #(Functions L i)) \u2264 sum fun i => #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nrefine' sum_le_sum _ _ fun n => _\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nn : \u2115\n\u22a2 lift #(Functions L n) \u2264 #(BoundedFormula L Empty (n + 1))\n[PROOFSTEP]\nrw [\u2190 lift_le.{_, max u v}, lift_lift, lift_mk_le.{v}]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nn : \u2115\n\u22a2 Nonempty (Functions L n \u21aa BoundedFormula L Empty (n + 1))\n[PROOFSTEP]\nrefine' \u27e8\u27e8fun f => (func f default).bdEqual (func f default), fun f g h => _\u27e9\u27e9\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nn : \u2115\nf g : Functions L n\nh : (fun f => func f default =' func f default) f = (fun f => func f default =' func f default) g\n\u22a2 f = g\n[PROOFSTEP]\nrcases h with \u27e8rfl, \u27e8rfl\u27e9\u27e9\n[GOAL]\ncase refl\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nn : \u2115\nf : Functions L n\n\u22a2 f = f\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 \u2135\u2080 \u2264 sum fun i => #(BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nrw [\u2190 mk_sigma]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 \u2135\u2080 \u2264 #((i : \u2115) \u00d7 BoundedFormula L Empty (i + 1))\n[PROOFSTEP]\nexact infinite_iff.1 (Infinite.of_injective (fun n => \u27e8n, \u22a5\u27e9) fun x y xy => (Sigma.mk.inj_iff.1 xy).1)\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 #((n : \u2115) \u00d7 Functions (Language.sum L (skolem\u2081 L)) n) \u2264 max \u2135\u2080 (card L)\n[PROOFSTEP]\nrw [card_functions_sum_skolem\u2081]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 #((n : \u2115) \u00d7 BoundedFormula L Empty (n + 1)) \u2264 max \u2135\u2080 (card L)\n[PROOFSTEP]\ntrans #(\u03a3 n, L.BoundedFormula Empty n)\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 #((n : \u2115) \u00d7 BoundedFormula L Empty (n + 1)) \u2264 #((n : \u2115) \u00d7 BoundedFormula L Empty n)\n[PROOFSTEP]\nexact \u27e8\u27e8Sigma.map Nat.succ fun _ => id, Nat.succ_injective.sigma_map fun _ => Function.injective_id\u27e9\u27e9\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 #((n : \u2115) \u00d7 BoundedFormula L Empty n) \u2264 max \u2135\u2080 (card L)\n[PROOFSTEP]\nrefine' _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 _)\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 lift (max \u2135\u2080 (lift #Empty + lift (card L))) \u2264 lift (max \u2135\u2080 (card L))\n[PROOFSTEP]\nsimp only [mk_empty, lift_zero, lift_uzero, zero_add]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 lift (max \u2135\u2080 (card L)) \u2264 lift (max \u2135\u2080 (card L))\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\n\u22a2 IsElementary (\u2191(LHom.substructureReduct LHom.sumInl) S)\n[PROOFSTEP]\napply (LHom.sumInl.substructureReduct S).isElementary_of_exists\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\n\u22a2 \u2200 (n : \u2115) (\u03c6 : BoundedFormula L Empty (n + 1)) (x : Fin n \u2192 { x // x \u2208 \u2191(LHom.substructureReduct LHom.sumInl) S })\n    (a : M),\n    BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a) \u2192\n      \u2203 b, BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) \u2191b)\n[PROOFSTEP]\nintro n \u03c6 x a h\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\nn : \u2115\n\u03c6 : BoundedFormula L Empty (n + 1)\nx : Fin n \u2192 { x // x \u2208 \u2191(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)\n\u22a2 \u2203 b, BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) \u2191b)\n[PROOFSTEP]\nlet \u03c6' : (L.sum L.skolem\u2081).Functions n := LHom.sumInr.onFunction \u03c6\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\nn : \u2115\n\u03c6 : BoundedFormula L Empty (n + 1)\nx : Fin n \u2192 { x // x \u2208 \u2191(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)\n\u03c6' : Functions (Language.sum L (skolem\u2081 L)) n := LHom.onFunction LHom.sumInr \u03c6\n\u22a2 \u2203 b, BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) \u2191b)\n[PROOFSTEP]\nexact\n  \u27e8\u27e8funMap \u03c6' ((\u2191) \u2218 x), S.fun_mem (LHom.sumInr.onFunction \u03c6) ((\u2191) \u2218 x) (by exact fun i => (x i).2)\u27e9, by\n    exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)) \u27e8a, h\u27e9\u27e9\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\nn : \u2115\n\u03c6 : BoundedFormula L Empty (n + 1)\nx : Fin n \u2192 { x // x \u2208 \u2191(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)\n\u03c6' : Functions (Language.sum L (skolem\u2081 L)) n := LHom.onFunction LHom.sumInr \u03c6\n\u22a2 \u2200 (i : Fin n), (Subtype.val \u2218 x) i \u2208 \u2191S\n[PROOFSTEP]\nexact fun i => (x i).2\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\nS : Substructure (Language.sum L (skolem\u2081 L)) M\nn : \u2115\n\u03c6 : BoundedFormula L Empty (n + 1)\nx : Fin n \u2192 { x // x \u2208 \u2191(LHom.substructureReduct LHom.sumInl) S }\na : M\nh : BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)\n\u03c6' : Functions (Language.sum L (skolem\u2081 L)) n := LHom.onFunction LHom.sumInr \u03c6\n\u22a2 BoundedFormula.Realize \u03c6 default\n    (Fin.snoc (Subtype.val \u2218 x)\n      \u2191{ val := funMap \u03c6' (Subtype.val \u2218 x),\n          property := (_ : funMap (LHom.onFunction LHom.sumInr \u03c6) (Subtype.val \u2218 x) \u2208 \u2191S) })\n[PROOFSTEP]\nexact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize \u03c6 default (Fin.snoc (Subtype.val \u2218 x) a)) \u27e8a, h\u27e9\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 Small.{?u.38330, w} { x // x \u2208 elementarySkolem\u2081Reduct \u22a5 }\n[PROOFSTEP]\nrw [coeSort_elementarySkolem\u2081Reduct]\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\n\u22a2 Small.{?u.38330, w} { x // x \u2208 \u22a5 }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh1 : \u2135\u2080 \u2264 \u03ba\nh2 : lift #\u2191s \u2264 lift \u03ba\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\n\u22a2 \u2203 S, s \u2286 \u2191S \u2227 lift #{ x // x \u2208 S } = lift \u03ba\n[PROOFSTEP]\nobtain \u27e8s', hs'\u27e9 := Cardinal.le_mk_iff_exists_set.1 h4\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh1 : \u2135\u2080 \u2264 \u03ba\nh2 : lift #\u2191s \u2264 lift \u03ba\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nhs' : #\u2191s' = lift \u03ba\n\u22a2 \u2203 S, s \u2286 \u2191S \u2227 lift #{ x // x \u2208 S } = lift \u03ba\n[PROOFSTEP]\nrw [\u2190 aleph0_le_lift.{_, w}] at h1 \n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh1 : \u2135\u2080 \u2264 lift \u03ba\nh2 : lift #\u2191s \u2264 lift \u03ba\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nhs' : #\u2191s' = lift \u03ba\n\u22a2 \u2203 S, s \u2286 \u2191S \u2227 lift #{ x // x \u2208 S } = lift \u03ba\n[PROOFSTEP]\nrw [\u2190 hs'] at h1 h2 \u22a2\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\n\u22a2 \u2203 S, s \u2286 \u2191S \u2227 lift #{ x // x \u2208 S } = #\u2191s'\n[PROOFSTEP]\nrefine'\n  \u27e8elementarySkolem\u2081Reduct (closure (L.sum L.skolem\u2081) (s \u222a Equiv.ulift '' s')),\n    (s.subset_union_left _).trans subset_closure, _\u27e9\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\n\u22a2 lift\n      #{ x //\n          x \u2208\n            elementarySkolem\u2081Reduct\n              (LowerAdjoint.toFun (closure (Language.sum L (skolem\u2081 L))) (s \u222a \u2191Equiv.ulift '' s')) } =\n    #\u2191s'\n[PROOFSTEP]\nhave h := mk_image_eq_lift _ s' Equiv.ulift.injective\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = lift #\u2191s'\n\u22a2 lift\n      #{ x //\n          x \u2208\n            elementarySkolem\u2081Reduct\n              (LowerAdjoint.toFun (closure (Language.sum L (skolem\u2081 L))) (s \u222a \u2191Equiv.ulift '' s')) } =\n    #\u2191s'\n[PROOFSTEP]\nrw [lift_umax.{w, w'}, lift_id'.{w, w'}] at h \n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift\n      #{ x //\n          x \u2208\n            elementarySkolem\u2081Reduct\n              (LowerAdjoint.toFun (closure (Language.sum L (skolem\u2081 L))) (s \u222a \u2191Equiv.ulift '' s')) } =\n    #\u2191s'\n[PROOFSTEP]\nrw [coeSort_elementarySkolem\u2081Reduct, \u2190 h, lift_inj]\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 #{ x // x \u2208 LowerAdjoint.toFun (closure (Language.sum L (skolem\u2081 L))) (s \u222a \u2191Equiv.ulift '' s') } =\n    #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrefine'\n  le_antisymm (lift_le.1 (lift_card_closure_le.trans _))\n    (mk_le_mk_of_subset ((Set.subset_union_right _ _).trans subset_closure))\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 max \u2135\u2080 (lift #\u2191(s \u222a \u2191Equiv.ulift '' s') + lift #((i : \u2115) \u00d7 Functions (Language.sum L (skolem\u2081 L)) i)) \u2264\n    lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [max_le_iff, aleph0_le_lift, \u2190 aleph0_le_lift.{_, w'}, h, add_eq_max, max_le_iff, lift_le]\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 \u2135\u2080 \u2264 #\u2191s' \u2227\n    #\u2191(s \u222a \u2191Equiv.ulift '' s') \u2264 #\u2191(\u2191Equiv.ulift '' s') \u2227\n      lift #((i : \u2115) \u00d7 Functions (Language.sum L (skolem\u2081 L)) i) \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 \u2135\u2080 \u2264 lift #\u2191(s \u222a \u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrefine' \u27e8h1, (mk_union_le _ _).trans _, (lift_le.2 card_functions_sum_skolem\u2081_le).trans _\u27e9\n[GOAL]\ncase intro.refine'_1\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 #\u2191s + #\u2191(\u2191Equiv.ulift '' s') \u2264 #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [\u2190 lift_le, lift_add, h, add_comm, add_eq_max h1]\n[GOAL]\ncase intro.refine'_1\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 max (#\u2191s') (lift #\u2191s) \u2264 #\u2191s'\n[PROOFSTEP]\nexact max_le le_rfl h2\n[GOAL]\ncase intro.refine'_2\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift (max \u2135\u2080 (card L)) \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [lift_max, lift_aleph0, max_le_iff, aleph0_le_lift, and_comm, \u2190 lift_le.{w'}, lift_lift, lift_lift, \u2190 aleph0_le_lift,\n  h]\n[GOAL]\ncase intro.refine'_2\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift (card L) \u2264 lift #\u2191(\u2191Equiv.ulift '' s') \u2227 \u2135\u2080 \u2264 #\u2191s'\n[PROOFSTEP]\nrefine' \u27e8_, h1\u27e9\n[GOAL]\ncase intro.refine'_2\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift (card L) \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [\u2190 lift_lift.{w', w}]\n[GOAL]\ncase intro.refine'_2\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift (lift (card L)) \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrefine' _root_.trans (lift_le.{w}.2 h3) _\n[GOAL]\ncase intro.refine'_2\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 lift (lift \u03ba) \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [lift_lift, \u2190 lift_lift.{w, max u v}, \u2190 hs', \u2190 h, lift_lift]\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 \u2135\u2080 \u2264 lift #\u2191(s \u222a \u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrefine' _root_.trans _ (lift_le.2 (mk_le_mk_of_subset (Set.subset_union_right _ _)))\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 \u2135\u2080 \u2264 lift #\u2191(\u2191Equiv.ulift '' s')\n[PROOFSTEP]\nrw [aleph0_le_lift, \u2190 aleph0_le_lift, h]\n[GOAL]\ncase intro\nL : Language\nM : Type w\ninst\u271d\u00b9 : Nonempty M\ninst\u271d : Structure L M\ns : Set M\n\u03ba : Cardinal.{w'}\nh3 : lift (card L) \u2264 lift \u03ba\nh4 : lift \u03ba \u2264 lift #M\ns' : Set (ULift M)\nh2 : lift #\u2191s \u2264 #\u2191s'\nh1 : \u2135\u2080 \u2264 #\u2191s'\nhs' : #\u2191s' = lift \u03ba\nh : lift #\u2191(\u2191Equiv.ulift '' s') = #\u2191s'\n\u22a2 \u2135\u2080 \u2264 #\u2191s'\n[PROOFSTEP]\nexact h1\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Skolem", "llama_tokens": 8038, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.835483553488848, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.5412247627593008}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nlet \u03b4 := \u2a05 w : K, \u2016u - w\u2016\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nletI : Nonempty K := ne.to_subtype\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave zero_le_\u03b4 : 0 \u2264 \u03b4 := le_ciInf fun _ => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave \u03b4_le : \u2200 w : K, \u03b4 \u2264 \u2016u - w\u2016 := ciInf_le \u27e80, Set.forall_range_iff.2 fun _ => norm_nonneg _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave \u03b4_le' : \u2200 w \u2208 K, \u03b4 \u2264 \u2016u - w\u2016 := fun w hw =>\n  \u03b4_le\n    \u27e8w, hw\u27e9\n      -- Step 1: since `\u03b4` is the infimum, can find a sequence `w : \u2115 \u2192 K` in `K`\n        -- such that `\u2016u - w n\u2016 < \u03b4 + 1 / (n + 1)` (which implies `\u2016u - w n\u2016 --> \u03b4`);\n        -- maybe this should be a separate lemma\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave exists_seq : \u2203 w : \u2115 \u2192 K, \u2200 n, \u2016u - w n\u2016 < \u03b4 + 1 / (n + 1) :=\n  by\n  have h\u03b4 : \u2200 n : \u2115, \u03b4 < \u03b4 + 1 / (n + 1) := fun n => lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat\n  have h := fun n => exists_lt_of_ciInf_lt (h\u03b4 n)\n  let w : \u2115 \u2192 K := fun n => Classical.choose (h n)\n  exact \u27e8w, fun n => Classical.choose_spec (h n)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u22a2 \u2203 w, \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n[PROOFSTEP]\nhave h\u03b4 : \u2200 n : \u2115, \u03b4 < \u03b4 + 1 / (n + 1) := fun n => lt_add_of_le_of_pos le_rfl Nat.one_div_pos_of_nat\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nh\u03b4 : \u2200 (n : \u2115), \u03b4 < \u03b4 + 1 / (\u2191n + 1)\n\u22a2 \u2203 w, \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n[PROOFSTEP]\nhave h := fun n => exists_lt_of_ciInf_lt (h\u03b4 n)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nh\u03b4 : \u2200 (n : \u2115), \u03b4 < \u03b4 + 1 / (\u2191n + 1)\nh : \u2200 (n : \u2115), \u2203 i, \u2016u - \u2191i\u2016 < \u03b4 + 1 / (\u2191n + 1)\n\u22a2 \u2203 w, \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n[PROOFSTEP]\nlet w : \u2115 \u2192 K := fun n => Classical.choose (h n)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nh\u03b4 : \u2200 (n : \u2115), \u03b4 < \u03b4 + 1 / (\u2191n + 1)\nh : \u2200 (n : \u2115), \u2203 i, \u2016u - \u2191i\u2016 < \u03b4 + 1 / (\u2191n + 1)\nw : \u2115 \u2192 \u2191K := fun n => Classical.choose (_ : \u2203 i, \u2016u - \u2191i\u2016 < \u03b4 + 1 / (\u2191n + 1))\n\u22a2 \u2203 w, \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n[PROOFSTEP]\nexact \u27e8w, fun n => Classical.choose_spec (h n)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nexists_seq : \u2203 w, \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nrcases exists_seq with \u27e8w, hw\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave norm_tendsto : Tendsto (fun n => \u2016u - w n\u2016) atTop (nhds \u03b4) :=\n  by\n  have h : Tendsto (fun _ : \u2115 => \u03b4) atTop (nhds \u03b4) := tendsto_const_nhds\n  have h' : Tendsto (fun n : \u2115 => \u03b4 + 1 / (n + 1)) atTop (nhds \u03b4) :=\n    by\n    convert h.add tendsto_one_div_add_atTop_nhds_0_nat\n    simp only [add_zero]\n  exact\n    tendsto_of_tendsto_of_tendsto_of_le_of_le h h' (fun x => \u03b4_le _) fun x =>\n      le_of_lt\n        (hw _)\n          -- Step 2: Prove that the sequence `w : \u2115 \u2192 K` is a Cauchy sequence\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\n\u22a2 Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n[PROOFSTEP]\nhave h : Tendsto (fun _ : \u2115 => \u03b4) atTop (nhds \u03b4) := tendsto_const_nhds\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nh : Tendsto (fun x => \u03b4) atTop (\ud835\udcdd \u03b4)\n\u22a2 Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n[PROOFSTEP]\nhave h' : Tendsto (fun n : \u2115 => \u03b4 + 1 / (n + 1)) atTop (nhds \u03b4) :=\n  by\n  convert h.add tendsto_one_div_add_atTop_nhds_0_nat\n  simp only [add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nh : Tendsto (fun x => \u03b4) atTop (\ud835\udcdd \u03b4)\n\u22a2 Tendsto (fun n => \u03b4 + 1 / (\u2191n + 1)) atTop (\ud835\udcdd \u03b4)\n[PROOFSTEP]\nconvert h.add tendsto_one_div_add_atTop_nhds_0_nat\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nh : Tendsto (fun x => \u03b4) atTop (\ud835\udcdd \u03b4)\n\u22a2 \u03b4 = \u03b4 + 0\n[PROOFSTEP]\nsimp only [add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nh : Tendsto (fun x => \u03b4) atTop (\ud835\udcdd \u03b4)\nh' : Tendsto (fun n => \u03b4 + 1 / (\u2191n + 1)) atTop (\ud835\udcdd \u03b4)\n\u22a2 Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n[PROOFSTEP]\nexact\n  tendsto_of_tendsto_of_tendsto_of_le_of_le h h' (fun x => \u03b4_le _) fun x =>\n    le_of_lt\n      (hw _)\n        -- Step 2: Prove that the sequence `w : \u2115 \u2192 K` is a Cauchy sequence\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave seq_is_cauchy : CauchySeq fun n => (w n : F) :=\n  by\n  rw [cauchySeq_iff_le_tendsto_0]\n    -- splits into three goals\n  let b := fun n : \u2115 => 8 * \u03b4 * (1 / (n + 1)) + 4 * (1 / (n + 1)) * (1 / (n + 1))\n  use fun n => sqrt (b n)\n  constructor\n    -- first goal :  `\u2200 (n : \u2115), 0 \u2264 sqrt (b n)`\n  intro n\n  exact sqrt_nonneg _\n  constructor\n    -- second goal : `\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)`\n  intro p q N hp hq\n  let wp := (w p : F)\n  let wq := (w q : F)\n  let a := u - wq\n  let b := u - wp\n  let half := 1 / (2 : \u211d)\n  let div := 1 / ((N : \u211d) + 1)\n  have : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) :=\n    calc\n      4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n          2 * \u2016u - half \u2022 (wq + wp)\u2016 * (2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 :=\n        by ring\n      _ = absR (2 : \u211d) * \u2016u - half \u2022 (wq + wp)\u2016 * (absR (2 : \u211d) * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 :=\n        by\n        rw [_root_.abs_of_nonneg]\n        exact zero_le_two\n      _ = \u2016(2 : \u211d) \u2022 (u - half \u2022 (wq + wp))\u2016 * \u2016(2 : \u211d) \u2022 (u - half \u2022 (wq + wp))\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 := by\n        simp [norm_smul]\n      _ = \u2016a + b\u2016 * \u2016a + b\u2016 + \u2016a - b\u2016 * \u2016a - b\u2016 :=\n        by\n        rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : \u211d) \u2260 0), \u2190 one_add_one_eq_two, add_smul]\n        simp only [one_smul]\n        have eq\u2081 : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm\n        have eq\u2082 : u + u - (wq + wp) = a + b\n        show u + u - (wq + wp) = u - wq + (u - wp)\n        abel\n        rw [eq\u2081, eq\u2082]\n      _ = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) := parallelogram_law_with_norm \u211d _ _\n  have eq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016 := by\n    rw [smul_add]\n    apply \u03b4_le'\n    apply h\u2082\n    repeat' exact Subtype.mem _\n    repeat' exact le_of_lt one_half_pos\n    exact add_halves 1\n  have eq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 :=\n    by\n    simp_rw [mul_assoc]\n    gcongr\n  have eq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div := le_trans (le_of_lt <| hw q) (add_le_add_left (Nat.one_div_le_one_div hq) _)\n  have eq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div := le_trans (le_of_lt <| hw p) (add_le_add_left (Nat.one_div_le_one_div hp) _)\n  rw [dist_eq_norm]\n  apply nonneg_le_nonneg_of_sq_le_sq\n  \u00b7 exact sqrt_nonneg _\n  rw [mul_self_sqrt]\n  calc\n    \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 := by\n      simp [\u2190 this]\n    _ \u2264 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u03b4 * \u03b4 := by gcongr\n    _ \u2264 2 * ((\u03b4 + div) * (\u03b4 + div) + (\u03b4 + div) * (\u03b4 + div)) - 4 * \u03b4 * \u03b4 := by gcongr\n    _ = 8 * \u03b4 * div + 4 * div * div := by ring\n  positivity\n    -- third goal : `Tendsto (fun (n : \u2115) => sqrt (b n)) atTop (\ud835\udcdd 0)`\n  apply Tendsto.comp (f := b) (g := sqrt)\n  \u00b7 have : Tendsto sqrt (nhds 0) (nhds (sqrt 0)) := continuous_sqrt.continuousAt\n    convert this\n    exact sqrt_zero.symm\n  have eq\u2081 : Tendsto (fun n : \u2115 => 8 * \u03b4 * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n    by\n    convert (@tendsto_const_nhds _ _ _ (8 * \u03b4) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n    simp only [mul_zero]\n  have : Tendsto (fun n : \u2115 => (4 : \u211d) * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n    by\n    convert (@tendsto_const_nhds _ _ _ (4 : \u211d) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n    simp only [mul_zero]\n  have eq\u2082 : Tendsto (fun n : \u2115 => (4 : \u211d) * (1 / (n + 1)) * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n    by\n    convert this.mul tendsto_one_div_add_atTop_nhds_0_nat\n    simp only [mul_zero]\n  convert eq\u2081.add eq\u2082\n  simp only [add_zero]\n    -- Step 3: By completeness of `K`, let `w : \u2115 \u2192 K` converge to some `v : K`.\n      -- Prove that it satisfies all requirements.\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n\u22a2 CauchySeq fun n => \u2191(w n)\n[PROOFSTEP]\nrw [cauchySeq_iff_le_tendsto_0]\n  -- splits into three goals\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet b := fun n : \u2115 => 8 * \u03b4 * (1 / (n + 1)) + 4 * (1 / (n + 1)) * (1 / (n + 1))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 \u2203 b, (\u2200 (n : \u2115), 0 \u2264 b n) \u2227 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 b N) \u2227 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nuse fun n => sqrt (b n)\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 (\u2200 (n : \u2115), 0 \u2264 sqrt (b n)) \u2227\n    (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)) \u2227 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconstructor\n  -- first goal :  `\u2200 (n : \u2115), 0 \u2264 sqrt (b n)`\n[GOAL]\ncase h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 \u2200 (n : \u2115), 0 \u2264 sqrt (b n)\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)) \u2227 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\nn : \u2115\n\u22a2 0 \u2264 sqrt (b n)\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)) \u2227 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact sqrt_nonneg _\n[GOAL]\ncase h.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)) \u2227 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconstructor\n  -- second goal : `\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)`\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 \u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist \u2191(w n) \u2191(w m) \u2264 sqrt (b N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nintro p q N hp hq\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet wp := (w p : F)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet wq := (w q : F)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet a := u - wq\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet b := u - wp\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet half := 1 / (2 : \u211d)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nlet div := 1 / ((N : \u211d) + 1)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) :=\n  calc\n    4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n        2 * \u2016u - half \u2022 (wq + wp)\u2016 * (2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 :=\n      by ring\n    _ = absR (2 : \u211d) * \u2016u - half \u2022 (wq + wp)\u2016 * (absR (2 : \u211d) * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 :=\n      by\n      rw [_root_.abs_of_nonneg]\n      exact zero_le_two\n    _ = \u2016(2 : \u211d) \u2022 (u - half \u2022 (wq + wp))\u2016 * \u2016(2 : \u211d) \u2022 (u - half \u2022 (wq + wp))\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 := by\n      simp [norm_smul]\n    _ = \u2016a + b\u2016 * \u2016a + b\u2016 + \u2016a - b\u2016 * \u2016a - b\u2016 :=\n      by\n      rw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : \u211d) \u2260 0), \u2190 one_add_one_eq_two, add_smul]\n      simp only [one_smul]\n      have eq\u2081 : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm\n      have eq\u2082 : u + u - (wq + wp) = a + b\n      show u + u - (wq + wp) = u - wq + (u - wp)\n      abel\n      rw [eq\u2081, eq\u2082]\n    _ = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) := parallelogram_law_with_norm \u211d _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n    2 * \u2016u - half \u2022 (wq + wp)\u2016 * (2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 2 * \u2016u - half \u2022 (wq + wp)\u2016 * (2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n    absR 2 * \u2016u - half \u2022 (wq + wp)\u2016 * (absR 2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016\n[PROOFSTEP]\nrw [_root_.abs_of_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nexact zero_le_two\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 absR 2 * \u2016u - half \u2022 (wq + wp)\u2016 * (absR 2 * \u2016u - half \u2022 (wq + wp)\u2016) + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n    \u20162 \u2022 (u - half \u2022 (wq + wp))\u2016 * \u20162 \u2022 (u - half \u2022 (wq + wp))\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016\n[PROOFSTEP]\nsimp [norm_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 \u20162 \u2022 (u - half \u2022 (wq + wp))\u2016 * \u20162 \u2022 (u - half \u2022 (wq + wp))\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n    \u2016a + b\u2016 * \u2016a + b\u2016 + \u2016a - b\u2016 * \u2016a - b\u2016\n[PROOFSTEP]\nrw [smul_sub, smul_smul, mul_one_div_cancel (_root_.two_ne_zero : (2 : \u211d) \u2260 0), \u2190 one_add_one_eq_two, add_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 \u20161 \u2022 u + 1 \u2022 u - 1 \u2022 (wq + wp)\u2016 * \u20161 \u2022 u + 1 \u2022 u - 1 \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 =\n    \u2016a + b\u2016 * \u2016a + b\u2016 + \u2016a - b\u2016 * \u2016a - b\u2016\n[PROOFSTEP]\nsimp only [one_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nhave eq\u2081 : wp - wq = a - b := (sub_sub_sub_cancel_left _ _ _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nhave eq\u2082 : u + u - (wq + wp) = a + b\n[GOAL]\ncase eq\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\n\u22a2 u + u - (wq + wp) = a + b\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\neq\u2082 : u + u - (wq + wp) = a + b\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nshow u + u - (wq + wp) = u - wq + (u - wp)\n[GOAL]\ncase eq\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\n\u22a2 u + u - (wq + wp) = u - wq + (u - wp)\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\neq\u2082 : u + u - (wq + wp) = a + b\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nabel\n[GOAL]\ncase eq\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\n\u22a2 u + u - (wq + wp) = u - wq + (u - wp)\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\neq\u2082 : u + u - (wq + wp) = a + b\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\neq\u2081 : wp - wq = a - b\neq\u2082 : u + u - (wq + wp) = a + b\n\u22a2 \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 * \u2016u + u - (\u2191(w q) + \u2191(w p))\u2016 + \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 =\n    \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) + (u - \u2191(w p))\u2016 +\n      \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016 * \u2016u - \u2191(w q) - (u - \u2191(w p))\u2016\n[PROOFSTEP]\nrw [eq\u2081, eq\u2082]\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016 := by\n  rw [smul_add]\n  apply \u03b4_le'\n  apply h\u2082\n  repeat' exact Subtype.mem _\n  repeat' exact le_of_lt one_half_pos\n  exact add_halves 1\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\n[PROOFSTEP]\nrw [smul_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 \u03b4 \u2264 \u2016u - (half \u2022 wq + half \u2022 wp)\u2016\n[PROOFSTEP]\napply \u03b4_le'\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half \u2022 wq + half \u2022 wp \u2208 K\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 wq \u2208 K\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 wp \u2208 K\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half + half = 1\n[PROOFSTEP]\nrepeat' exact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 wq \u2208 K\n[PROOFSTEP]\nexact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 wp \u2208 K\n[PROOFSTEP]\nexact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\n[PROOFSTEP]\nexact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\n[PROOFSTEP]\nexact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half + half = 1\n[PROOFSTEP]\nexact Subtype.mem _\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half + half = 1\n[PROOFSTEP]\nrepeat' exact le_of_lt one_half_pos\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\n[PROOFSTEP]\nexact le_of_lt one_half_pos\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 0 \u2264 half\n[PROOFSTEP]\nexact le_of_lt one_half_pos\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half + half = 1\n[PROOFSTEP]\nexact le_of_lt one_half_pos\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\n\u22a2 half + half = 1\n[PROOFSTEP]\nexact add_halves 1\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 :=\n  by\n  simp_rw [mul_assoc]\n  gcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\n\u22a2 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\n[PROOFSTEP]\nsimp_rw [mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\n\u22a2 4 * ((\u2a05 (w : \u2191K), \u2016u - \u2191w\u2016) * \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016) \u2264\n    4 * (\u2016u - (1 / 2) \u2022 (\u2191(w q) + \u2191(w p))\u2016 * \u2016u - (1 / 2) \u2022 (\u2191(w q) + \u2191(w p))\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div := le_trans (le_of_lt <| hw q) (add_le_add_left (Nat.one_div_le_one_div hq) _)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div := le_trans (le_of_lt <| hw p) (add_le_add_left (Nat.one_div_le_one_div hp) _)\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 dist \u2191(w p) \u2191(w q) \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\ncase h.right.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 \u2016\u2191(w p) - \u2191(w q)\u2016 \u2264 sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply nonneg_le_nonneg_of_sq_le_sq\n[GOAL]\ncase h.right.left.hb\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 0 \u2264 sqrt (b\u271d N)\n[PROOFSTEP]\nexact sqrt_nonneg _\n[GOAL]\ncase h.right.left.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 \u2264 sqrt (b\u271d N) * sqrt (b\u271d N)\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [mul_self_sqrt]\n[GOAL]\ncase h.right.left.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 \u2016\u2191(w p) - \u2191(w q)\u2016 * \u2016\u2191(w p) - \u2191(w q)\u2016 \u2264 b\u271d N\ncase h.right.left.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 0 \u2264 b\u271d N\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\ncalc\n  \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 := by\n    simp [\u2190 this]\n  _ \u2264 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u03b4 * \u03b4 := by gcongr\n  _ \u2264 2 * ((\u03b4 + div) * (\u03b4 + div) + (\u03b4 + div) * (\u03b4 + div)) - 4 * \u03b4 * \u03b4 := by gcongr\n  _ = 8 * \u03b4 * div + 4 * div * div := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\n[PROOFSTEP]\nsimp [\u2190 this]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 \u2264\n    2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u03b4 * \u03b4\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016) - 4 * \u03b4 * \u03b4 \u2264 2 * ((\u03b4 + div) * (\u03b4 + div) + (\u03b4 + div) * (\u03b4 + div)) - 4 * \u03b4 * \u03b4\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 2 * ((\u03b4 + div) * (\u03b4 + div) + (\u03b4 + div) * (\u03b4 + div)) - 4 * \u03b4 * \u03b4 = 8 * \u03b4 * div + 4 * div * div\n[PROOFSTEP]\nring\n[GOAL]\ncase h.right.left.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb\u271d : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\np q N : \u2115\nhp : N \u2264 p\nhq : N \u2264 q\nwp : F := \u2191(w p)\nwq : F := \u2191(w q)\na : F := u - wq\nb : F := u - wp\nhalf : \u211d := 1 / 2\ndiv : \u211d := 1 / (\u2191N + 1)\nthis : 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016 + \u2016wp - wq\u2016 * \u2016wp - wq\u2016 = 2 * (\u2016a\u2016 * \u2016a\u2016 + \u2016b\u2016 * \u2016b\u2016)\neq : \u03b4 \u2264 \u2016u - half \u2022 (wq + wp)\u2016\neq\u2081 : 4 * \u03b4 * \u03b4 \u2264 4 * \u2016u - half \u2022 (wq + wp)\u2016 * \u2016u - half \u2022 (wq + wp)\u2016\neq\u2082 : \u2016a\u2016 \u2264 \u03b4 + div\neq\u2082' : \u2016b\u2016 \u2264 \u03b4 + div\n\u22a2 0 \u2264 b\u271d N\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\npositivity\n  -- third goal : `Tendsto (fun (n : \u2115) => sqrt (b n)) atTop (\ud835\udcdd 0)`\n[GOAL]\ncase h.right.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => sqrt (b n)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply Tendsto.comp (f := b) (g := sqrt)\n[GOAL]\ncase h.right.right.hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto sqrt ?m.169624 (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto sqrt (nhds 0) (nhds (sqrt 0)) := continuous_sqrt.continuousAt\n[GOAL]\ncase h.right.right.hg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\nthis : Tendsto sqrt (\ud835\udcdd 0) (\ud835\udcdd (sqrt 0))\n\u22a2 Tendsto sqrt ?m.169624 (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\nthis : Tendsto sqrt (\ud835\udcdd 0) (\ud835\udcdd (sqrt 0))\n\u22a2 0 = sqrt 0\n[PROOFSTEP]\nexact sqrt_zero.symm\n[GOAL]\ncase h.right.right.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq\u2081 : Tendsto (fun n : \u2115 => 8 * \u03b4 * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n  by\n  convert (@tendsto_const_nhds _ _ _ (8 * \u03b4) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n  simp only [mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert (@tendsto_const_nhds _ _ _ (8 * \u03b4) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\n\u22a2 0 = 8 * \u03b4 * 0\n[PROOFSTEP]\nsimp only [mul_zero]\n[GOAL]\ncase h.right.right.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun n : \u2115 => (4 : \u211d) * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n  by\n  convert (@tendsto_const_nhds _ _ _ (4 : \u211d) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n  simp only [mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert (@tendsto_const_nhds _ _ _ (4 : \u211d) _).mul tendsto_one_div_add_atTop_nhds_0_nat\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 0 = 4 * 0\n[PROOFSTEP]\nsimp only [mul_zero]\n[GOAL]\ncase h.right.right.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave eq\u2082 : Tendsto (fun n : \u2115 => (4 : \u211d) * (1 / (n + 1)) * (1 / (n + 1))) atTop (nhds (0 : \u211d)) :=\n  by\n  convert this.mul tendsto_one_div_add_atTop_nhds_0_nat\n  simp only [mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert this.mul tendsto_one_div_add_atTop_nhds_0_nat\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 0 = 0 * 0\n[PROOFSTEP]\nsimp only [mul_zero]\n[GOAL]\ncase h.right.right.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\neq\u2082 : Tendsto (fun n => 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto b atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert eq\u2081.add eq\u2082\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nb : \u2115 \u2192 \u211d := fun n => 8 * \u03b4 * (1 / (\u2191n + 1)) + 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))\neq\u2081 : Tendsto (fun n => 8 * \u03b4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => 4 * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\neq\u2082 : Tendsto (fun n => 4 * (1 / (\u2191n + 1)) * (1 / (\u2191n + 1))) atTop (\ud835\udcdd 0)\n\u22a2 0 = 0 + 0\n[PROOFSTEP]\nsimp only [add_zero]\n  -- Step 3: By completeness of `K`, let `w : \u2115 \u2192 K` converge to some `v : K`.\n    -- Prove that it satisfies all requirements.\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nrcases cauchySeq_tendsto_of_isComplete h\u2081 (fun n => Subtype.mem _) seq_is_cauchy with \u27e8v, hv, w_tendsto\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\n\u22a2 \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nuse v\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\n\u22a2 v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nuse hv\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave h_cont : Continuous fun v => \u2016u - v\u2016 :=\n  Continuous.comp continuous_norm (Continuous.sub continuous_const continuous_id)\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\nh_cont : Continuous fun v => \u2016u - v\u2016\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave : Tendsto (fun n => \u2016u - w n\u2016) atTop (nhds \u2016u - v\u2016)\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\nh_cont : Continuous fun v => \u2016u - v\u2016\n\u22a2 Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u2016u - v\u2016)\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\nh_cont : Continuous fun v => \u2016u - v\u2016\nthis : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u2016u - v\u2016)\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nconvert Tendsto.comp h_cont.continuousAt w_tendsto\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nne : Set.Nonempty K\nh\u2081 : IsComplete K\nh\u2082 : Convex \u211d K\nu : F\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nthis\u271d : Nonempty \u2191K := Set.Nonempty.to_subtype ne\nzero_le_\u03b4 : 0 \u2264 \u03b4\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nw : \u2115 \u2192 \u2191K\nhw : \u2200 (n : \u2115), \u2016u - \u2191(w n)\u2016 < \u03b4 + 1 / (\u2191n + 1)\nnorm_tendsto : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u03b4)\nseq_is_cauchy : CauchySeq fun n => \u2191(w n)\nv : F\nhv : v \u2208 K\nw_tendsto : Tendsto (fun n => \u2191(w n)) atTop (\ud835\udcdd v)\nh_cont : Continuous fun v => \u2016u - v\u2016\nthis : Tendsto (fun n => \u2016u - \u2191(w n)\u2016) atTop (\ud835\udcdd \u2016u - v\u2016)\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nexact tendsto_nhds_unique this norm_tendsto\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nletI : Nonempty K := \u27e8\u27e8v, hv\u27e9\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2192 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nintro eq w hw\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nlet \u03b4 := \u2a05 w : K, \u2016u - w\u2016\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nlet p := \u27eau - v, w - v\u27eb_\u211d\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nlet q := \u2016w - v\u2016 ^ 2\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nhave \u03b4_le (w : K) : \u03b4 \u2264 \u2016u - w\u2016 := ciInf_le \u27e80, fun _ \u27e8_, h\u27e9 => h \u25b8 norm_nonneg _\u27e9 _\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nhave \u03b4_le' (w) (hw : w \u2208 K) : \u03b4 \u2264 \u2016u - w\u2016 := \u03b4_le \u27e8w, hw\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nhave (\u03b8 : \u211d) (h\u03b8\u2081 : 0 < \u03b8) (h\u03b8\u2082 : \u03b8 \u2264 1) : 2 * p \u2264 \u03b8 * q :=\n  by\n  have : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * \u27eau - v, w - v\u27eb_\u211d + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 :=\n    calc\n      \u2016u - v\u2016 ^ 2\n      _ \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 ^ 2 := by\n        simp only [sq]; apply mul_self_le_mul_self (norm_nonneg _)\n        rw [eq]; apply \u03b4_le'\n        apply h hw hv\n        exacts [le_of_lt h\u03b8\u2081, sub_nonneg.2 h\u03b8\u2082, add_sub_cancel'_right _ _]\n      _ = \u2016u - v - \u03b8 \u2022 (w - v)\u2016 ^ 2 :=\n        by\n        have : u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v) = u - v - \u03b8 \u2022 (w - v) :=\n          by\n          rw [smul_sub, sub_smul, one_smul]\n          simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev]\n        rw [this]\n      _ = \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 :=\n        by\n        rw [@norm_sub_sq \u211d, inner_smul_right, norm_smul]\n        simp only [sq]\n        show\n          \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * (\u03b8 * inner (u - v) (w - v)) + absR \u03b8 * \u2016w - v\u2016 * (absR \u03b8 * \u2016w - v\u2016) =\n            \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n        rw [abs_of_pos h\u03b8\u2081]; ring\n  have eq\u2081 :\n    \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n      \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)) :=\n    by abel\n  rw [eq\u2081, le_add_iff_nonneg_right] at this \n  have eq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n  ring\n  rw [eq\u2082] at this \n  have := le_of_sub_nonneg (nonneg_of_mul_nonneg_right this h\u03b8\u2081)\n  exact this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nhave : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * \u27eau - v, w - v\u27eb_\u211d + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 :=\n  calc\n    \u2016u - v\u2016 ^ 2\n    _ \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 ^ 2 := by\n      simp only [sq]; apply mul_self_le_mul_self (norm_nonneg _)\n      rw [eq]; apply \u03b4_le'\n      apply h hw hv\n      exacts [le_of_lt h\u03b8\u2081, sub_nonneg.2 h\u03b8\u2082, add_sub_cancel'_right _ _]\n    _ = \u2016u - v - \u03b8 \u2022 (w - v)\u2016 ^ 2 :=\n      by\n      have : u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v) = u - v - \u03b8 \u2022 (w - v) :=\n        by\n        rw [smul_sub, sub_smul, one_smul]\n        simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev]\n      rw [this]\n    _ = \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 :=\n      by\n      rw [@norm_sub_sq \u211d, inner_smul_right, norm_smul]\n      simp only [sq]\n      show\n        \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * (\u03b8 * inner (u - v) (w - v)) + absR \u03b8 * \u2016w - v\u2016 * (absR \u03b8 * \u2016w - v\u2016) =\n          \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n      rw [abs_of_pos h\u03b8\u2081]; ring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 ^ 2 \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 ^ 2\n[PROOFSTEP]\nsimp only [sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 * \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016\n[PROOFSTEP]\napply mul_self_le_mul_self (norm_nonneg _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016\n[PROOFSTEP]\nrw [eq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2264 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016\n[PROOFSTEP]\napply \u03b4_le'\n[GOAL]\ncase hw\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u03b8 \u2022 w + (1 - \u03b8) \u2022 v \u2208 K\n[PROOFSTEP]\napply h hw hv\n[GOAL]\ncase hw.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 0 \u2264 \u03b8\ncase hw.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 0 \u2264 1 - \u03b8\ncase hw.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u03b8 + (1 - \u03b8) = 1\n[PROOFSTEP]\nexacts [le_of_lt h\u03b8\u2081, sub_nonneg.2 h\u03b8\u2082, add_sub_cancel'_right _ _]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 ^ 2 = \u2016u - v - \u03b8 \u2022 (w - v)\u2016 ^ 2\n[PROOFSTEP]\nhave : u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v) = u - v - \u03b8 \u2022 (w - v) :=\n  by\n  rw [smul_sub, sub_smul, one_smul]\n  simp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v) = u - v - \u03b8 \u2022 (w - v)\n[PROOFSTEP]\nrw [smul_sub, sub_smul, one_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 u - (\u03b8 \u2022 w + (v - \u03b8 \u2022 v)) = u - v - (\u03b8 \u2022 w - \u03b8 \u2022 v)\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, add_comm, add_left_comm, add_assoc, neg_add_rev]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v) = u - v - \u03b8 \u2022 (w - v)\n\u22a2 \u2016u - (\u03b8 \u2022 w + (1 - \u03b8) \u2022 v)\u2016 ^ 2 = \u2016u - v - \u03b8 \u2022 (w - v)\u2016 ^ 2\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v - \u03b8 \u2022 (w - v)\u2016 ^ 2 = \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\n[PROOFSTEP]\nrw [@norm_sub_sq \u211d, inner_smul_right, norm_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 ^ 2 - 2 * \u2191re (\u03b8 * inner (u - v) (w - v)) + (\u2016\u03b8\u2016 * \u2016w - v\u2016) ^ 2 =\n    \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\n[PROOFSTEP]\nsimp only [sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u2191re (\u03b8 * inner (u - v) (w - v)) + \u2016\u03b8\u2016 * \u2016w - v\u2016 * (\u2016\u03b8\u2016 * \u2016w - v\u2016) =\n    \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n[PROOFSTEP]\nshow\n  \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * (\u03b8 * inner (u - v) (w - v)) + absR \u03b8 * \u2016w - v\u2016 * (absR \u03b8 * \u2016w - v\u2016) =\n    \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * (\u03b8 * inner (u - v) (w - v)) + absR \u03b8 * \u2016w - v\u2016 * (absR \u03b8 * \u2016w - v\u2016) =\n    \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n[PROOFSTEP]\nrw [abs_of_pos h\u03b8\u2081]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * (\u03b8 * inner (u - v) (w - v)) + \u03b8 * \u2016w - v\u2016 * (\u03b8 * \u2016w - v\u2016) =\n    \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * (\u2016w - v\u2016 * \u2016w - v\u2016)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nhave eq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)) :=\n  by abel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\n\u22a2 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\n\u22a2 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : \u2016u - v\u2016 ^ 2 \u2264 \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nrw [eq\u2081, le_add_iff_nonneg_right] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : 0 \u2264 \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nhave eq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n[GOAL]\ncase eq\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : 0 \u2264 \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\n\u22a2 \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : 0 \u2264 \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\neq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : 0 \u2264 \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v)\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\neq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nrw [eq\u2082] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis : 0 \u2264 \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\neq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nhave := le_of_sub_nonneg (nonneg_of_mul_nonneg_right this h\u03b8\u2081)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d\u00b9 : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\n\u03b8 : \u211d\nh\u03b8\u2081 : 0 < \u03b8\nh\u03b8\u2082 : \u03b8 \u2264 1\nthis\u271d : 0 \u2264 \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\neq\u2081 :\n  \u2016u - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) + \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 =\n    \u2016u - v\u2016 ^ 2 + (\u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v))\neq\u2082 : \u03b8 * \u03b8 * \u2016w - v\u2016 ^ 2 - 2 * \u03b8 * inner (u - v) (w - v) = \u03b8 * (\u03b8 * \u2016w - v\u2016 ^ 2 - 2 * inner (u - v) (w - v))\nthis : 2 * inner (u - v) (w - v) \u2264 \u03b8 * \u2016w - v\u2016 ^ 2\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nexact this\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : q = 0\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nrw [hq] at this \n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nhave : p \u2264 0 := by\n  have := this (1 : \u211d) (by norm_num) (by norm_num)\n  linarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\n\u22a2 p \u2264 0\n[PROOFSTEP]\nhave := this (1 : \u211d) (by norm_num) (by norm_num)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\n\u22a2 1 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d\u00b9 : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis\u271d : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\nthis : 2 * p \u2264 1 * 0\n\u22a2 p \u2264 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d\u00b9 : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis\u271d : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * 0\nhq : q = 0\nthis : p \u2264 0\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nexact this\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nhave q_pos : 0 < q := lt_of_le_of_ne (sq_nonneg _) fun h \u21a6 hq h.symm\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\n\u22a2 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nby_contra hp\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : \u00acinner (u - v) (w - v) \u2264 0\n\u22a2 False\n[PROOFSTEP]\nrw [not_le] at hp \n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u22a2 False\n[PROOFSTEP]\nlet \u03b8 := min (1 : \u211d) (p / q)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\n\u22a2 False\n[PROOFSTEP]\nhave eq\u2081 : \u03b8 * q \u2264 p :=\n  calc\n    \u03b8 * q \u2264 p / q * q := mul_le_mul_of_nonneg_right (min_le_right _ _) (sq_nonneg _)\n    _ = p := div_mul_cancel _ hq\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\neq\u2081 : \u03b8 * q \u2264 p\n\u22a2 False\n[PROOFSTEP]\nhave : 2 * p \u2264 p :=\n  calc\n    2 * p \u2264 \u03b8 * q := by refine' this \u03b8 (lt_min (by norm_num) (div_pos hp q_pos)) (by norm_num)\n    _ \u2264 p := eq\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\neq\u2081 : \u03b8 * q \u2264 p\n\u22a2 2 * p \u2264 \u03b8 * q\n[PROOFSTEP]\nrefine' this \u03b8 (lt_min (by norm_num) (div_pos hp q_pos)) (by norm_num)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\neq\u2081 : \u03b8 * q \u2264 p\n\u22a2 0 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\neq\u2081 : \u03b8 * q \u2264 p\n\u22a2 \u03b8 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d\u00b9 : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\neq : \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\nw : F\nhw : w \u2208 K\n\u03b4 : \u211d := \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\np : \u211d := inner (u - v) (w - v)\nq : \u211d := \u2016w - v\u2016 ^ 2\n\u03b4_le : \u2200 (w : \u2191K), \u03b4 \u2264 \u2016u - \u2191w\u2016\n\u03b4_le' : \u2200 (w : F), w \u2208 K \u2192 \u03b4 \u2264 \u2016u - w\u2016\nthis\u271d : \u2200 (\u03b8 : \u211d), 0 < \u03b8 \u2192 \u03b8 \u2264 1 \u2192 2 * p \u2264 \u03b8 * q\nhq : \u00acq = 0\nq_pos : 0 < q\nhp : 0 < inner (u - v) (w - v)\n\u03b8 : \u211d := min 1 (p / q)\neq\u2081 : \u03b8 * q \u2264 p\nthis : 2 * p \u2264 p\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\n\u22a2 (\u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0) \u2192 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase mpr.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 \u2264 \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\napply le_ciInf\n[GOAL]\ncase mpr.a.H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2200 (x : \u2191K), \u2016u - v\u2016 \u2264 \u2016u - \u2191x\u2016\n[PROOFSTEP]\nintro w\n[GOAL]\ncase mpr.a.H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\n\u22a2 \u2016u - v\u2016 \u2264 \u2016u - \u2191w\u2016\n[PROOFSTEP]\napply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _)\n[GOAL]\ncase mpr.a.H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 \u2264 \u2016u - \u2191w\u2016 * \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave := h w w.2\n[GOAL]\ncase mpr.a.H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 \u2264 \u2016u - \u2191w\u2016 * \u2016u - \u2191w\u2016\n[PROOFSTEP]\ncalc\n  \u2016u - v\u2016 * \u2016u - v\u2016 \u2264 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * inner (u - v) ((w : F) - v) := by linarith\n  _ \u2264 \u2016u - v\u2016 ^ 2 - 2 * inner (u - v) ((w : F) - v) + \u2016(w : F) - v\u2016 ^ 2 :=\n    by\n    rw [sq]\n    refine' le_add_of_nonneg_right _\n    exact sq_nonneg _\n  _ = \u2016u - v - (w - v)\u2016 ^ 2 := (@norm_sub_sq \u211d _ _ _ _ _ _).symm\n  _ = \u2016u - w\u2016 * \u2016u - w\u2016 := by\n    have : u - v - (w - v) = u - w := by abel\n    rw [this, sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 \u2264 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * inner (u - v) (\u2191w - v)\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * inner (u - v) (\u2191w - v) \u2264 \u2016u - v\u2016 ^ 2 - 2 * inner (u - v) (\u2191w - v) + \u2016\u2191w - v\u2016 ^ 2\n[PROOFSTEP]\nrw [sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * inner (u - v) (\u2191w - v) \u2264 \u2016u - v\u2016 * \u2016u - v\u2016 - 2 * inner (u - v) (\u2191w - v) + \u2016\u2191w - v\u2016 ^ 2\n[PROOFSTEP]\nrefine' le_add_of_nonneg_right _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 0 \u2264 \u2016\u2191w - v\u2016 ^ 2\n[PROOFSTEP]\nexact sq_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 \u2016u - v - (\u2191w - v)\u2016 ^ 2 = \u2016u - \u2191w\u2016 * \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave : u - v - (w - v) = u - w := by abel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 u - v - (\u2191w - v) = u - \u2191w\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis : inner (u - v) (\u2191w - v) \u2264 0\n\u22a2 u - v - (\u2191w - v) = u - \u2191w\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis\u271d\u00b9 : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : \u2191K\nthis\u271d : inner (u - v) (\u2191w - v) \u2264 0\nthis : u - v - (\u2191w - v) = u - \u2191w\n\u22a2 \u2016u - v - (\u2191w - v)\u2016 ^ 2 = \u2016u - \u2191w\u2016 * \u2016u - \u2191w\u2016\n[PROOFSTEP]\nrw [this, sq]\n[GOAL]\ncase mpr.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2264 \u2016u - v\u2016\n[PROOFSTEP]\nshow \u2a05 w : K, \u2016u - w\u2016 \u2264 (fun w : K => \u2016u - w\u2016) \u27e8v, hv\u27e9\n[GOAL]\ncase mpr.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2a05 (w : \u2191K), \u2016u - \u2191w\u2016 \u2264 (fun w => \u2016u - \u2191w\u2016) { val := v, property := hv }\n[PROOFSTEP]\napply ciInf_le\n[GOAL]\ncase mpr.a.H\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 BddBelow (Set.range fun w => \u2016u - \u2191w\u2016)\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 0 \u2208 lowerBounds (Set.range fun w => \u2016u - \u2191w\u2016)\n[PROOFSTEP]\nrintro y \u27e8z, rfl\u27e9\n[GOAL]\ncase h.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Set F\nh\u271d : Convex \u211d K\nu v : F\nhv : v \u2208 K\nthis : Nonempty \u2191K := Nonempty.intro { val := v, property := hv }\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nz : \u2191K\n\u22a2 0 \u2264 (fun w => \u2016u - \u2191w\u2016) z\n[PROOFSTEP]\nexact norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nh : IsComplete \u2191K\n\u22a2 \u2200 (u : E), \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nletI : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nh : IsComplete \u2191K\nthis : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\n\u22a2 \u2200 (u : E), \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nletI : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nh : IsComplete \u2191K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n\u22a2 \u2200 (u : E), \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nlet K' : Submodule \u211d E := Submodule.restrictScalars \u211d K\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nh : IsComplete \u2191K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\n\u22a2 \u2200 (u : E), \u2203 v, v \u2208 K \u2227 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nexact exists_norm_eq_iInf_of_complete_convex \u27e80, K'.zero_mem\u27e9 h K'.convex\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016 \u2192 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nintro h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n\u22a2 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nhave h : \u2200 w \u2208 K, \u27eau - v, w - v\u27eb_\u211d \u2264 0 :=\n  by\n  rwa [norm_eq_iInf_iff_real_inner_le_zero] at h \n  exacts [K.convex, hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n\u22a2 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n[PROOFSTEP]\nrwa [norm_eq_iInf_iff_real_inner_le_zero] at h \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n\u22a2 Convex \u211d \u2191K\ncase hv\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n\u22a2 v \u2208 \u2191K\n[PROOFSTEP]\nexacts [K.convex, hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave le : \u27eau - v, w\u27eb_\u211d \u2264 0\n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nlet w' := w + v\n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave : w' \u2208 K := Submodule.add_mem _ hw hv\n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave h\u2081 := h w' this\n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) (w' - v) \u2264 0\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave h\u2082 : w' - v = w\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) (w' - v) \u2264 0\n\u22a2 w' - v = w\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) (w' - v) \u2264 0\nh\u2082 : w' - v = w\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nsimp only [add_neg_cancel_right, sub_eq_add_neg]\n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) (w' - v) \u2264 0\nh\u2082 : w' - v = w\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nrw [h\u2082] at h\u2081 \n[GOAL]\ncase le\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nw' : F := w + v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) w \u2264 0\nh\u2082 : w' - v = w\n\u22a2 inner (u - v) w \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nexact h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave ge : \u27eau - v, w\u27eb_\u211d \u2265 0\n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nlet w'' := -w + v\n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave : w'' \u2208 K := Submodule.add_mem _ (Submodule.neg_mem _ hw) hv\n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave h\u2081 := h w'' this\n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\nh\u2081 : inner (u - v) (w'' - v) \u2264 0\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nhave h\u2082 : w'' - v = -w\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\nh\u2081 : inner (u - v) (w'' - v) \u2264 0\n\u22a2 w'' - v = -w\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\nh\u2081 : inner (u - v) (w'' - v) \u2264 0\nh\u2082 : w'' - v = -w\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nsimp only [neg_inj, add_neg_cancel_right, sub_eq_add_neg]\n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\nh\u2081 : inner (u - v) (w'' - v) \u2264 0\nh\u2082 : w'' - v = -w\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nrw [h\u2082, inner_neg_right] at h\u2081 \n[GOAL]\ncase ge\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nw'' : F := -w + v\nthis : w'' \u2208 K\nh\u2081 : -inner (u - v) w \u2264 0\nh\u2082 : w'' - v = -w\n\u22a2 inner (u - v) w \u2265 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh\u271d : \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\nw : F\nhw : w \u2208 K\nle : inner (u - v) w \u2264 0\nge : inner (u - v) w \u2265 0\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nexact le_antisymm le ge\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\n\u22a2 (\u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0) \u2192 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nintro h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave : \u2200 w \u2208 K, \u27eau - v, w - v\u27eb_\u211d \u2264 0\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nintro w hw\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nw : F\nhw : w \u2208 K\n\u22a2 inner (u - v) (w - v) \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nlet w' := w - v\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nw : F\nhw : w \u2208 K\nw' : F := w - v\n\u22a2 inner (u - v) (w - v) \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave : w' \u2208 K := Submodule.sub_mem _ hw hv\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nw : F\nhw : w \u2208 K\nw' : F := w - v\nthis : w' \u2208 K\n\u22a2 inner (u - v) (w - v) \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave h\u2081 := h w' this\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nw : F\nhw : w \u2208 K\nw' : F := w - v\nthis : w' \u2208 K\nh\u2081 : inner (u - v) w' = 0\n\u22a2 inner (u - v) (w - v) \u2264 0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nexact le_of_eq h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : \u2191\u2191K), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nrwa [norm_eq_iInf_iff_real_inner_le_zero]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 Convex \u211d \u2191K\ncase hv\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\nK : Submodule \u211d F\nu v : F\nhv : v \u2208 K\nh : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : F), w \u2208 K \u2192 inner (u - v) (w - v) \u2264 0\n\u22a2 v \u2208 \u2191K\n[PROOFSTEP]\nexacts [Submodule.convex _, hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nletI : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nletI : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nlet K' : Submodule \u211d E := K.restrictScalars \u211d\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016 \u2194 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016 \u2192 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nhave A : \u2200 w \u2208 K, re \u27eau - v, w\u27eb = 0 := (norm_eq_iInf_iff_real_inner_eq_zero K' hv).1 H\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nintro w hw\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\napply ext\n[GOAL]\ncase mp.hre\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 \u2191re (inner (u - v) w) = \u2191re 0\n[PROOFSTEP]\nsimp [A w hw]\n[GOAL]\ncase mp.him\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 \u2191im (inner (u - v) w) = \u2191im 0\n[PROOFSTEP]\nsymm\n[GOAL]\ncase mp.him\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 \u2191im 0 = \u2191im (inner (u - v) w)\n[PROOFSTEP]\ncalc\n  im (0 : \ud835\udd5c) = 0 := im.map_zero\n  _ = re \u27eau - v, (-I : \ud835\udd5c) \u2022 w\u27eb := (A _ (K.smul_mem (-I) hw)).symm\n  _ = re (-I * \u27eau - v, w\u27eb) := by rw [inner_smul_right]\n  _ = im \u27eau - v, w\u27eb := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 \u2191re (inner (u - v) (-I \u2022 w)) = \u2191re (-I * inner (u - v) w)\n[PROOFSTEP]\nrw [inner_smul_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\nA : \u2200 (w : E), w \u2208 K \u2192 \u2191re (inner (u - v) w) = 0\nw : E\nhw : w \u2208 K\n\u22a2 \u2191re (-I * inner (u - v) w) = \u2191im (inner (u - v) w)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\n\u22a2 (\u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0) \u2192 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nhave : \u2200 w \u2208 K', \u27eau - v, w\u27eb_\u211d = 0 := by\n  intro w hw\n  rw [real_inner_eq_re_inner, H w hw]\n  exact zero_re'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 \u2200 (w : E), w \u2208 K' \u2192 inner (u - v) w = 0\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nw : E\nhw : w \u2208 K'\n\u22a2 inner (u - v) w = 0\n[PROOFSTEP]\nrw [real_inner_eq_re_inner, H w hw]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nw : E\nhw : w \u2208 K'\n\u22a2 \u2191re 0 = 0\n[PROOFSTEP]\nexact zero_re'\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : v \u2208 K\nthis\u271d\u00b9 : InnerProductSpace \u211d E := InnerProductSpace.isROrCToReal \ud835\udd5c E\nthis\u271d : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nK' : Submodule \u211d E := Submodule.restrictScalars \u211d K\nH : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nthis : \u2200 (w : E), w \u2208 K' \u2192 inner (u - v) w = 0\n\u22a2 \u2016u - v\u2016 = \u2a05 (w : { x // x \u2208 K }), \u2016u - \u2191w\u2016\n[PROOFSTEP]\nexact (norm_eq_iInf_iff_real_inner_eq_zero K' hv).2 this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 K }\nv : E\n\u22a2 \u2203 w, w \u2208 K \u2227 v - w \u2208 K\u15ee\n[PROOFSTEP]\nrcases exists_norm_eq_iInf_of_complete_subspace K (completeSpace_coe_iff_isComplete.mp \u2039_\u203a) v with \u27e8w, hwK, hw\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 K }\nv w : E\nhwK : w \u2208 K\nhw : \u2016v - w\u2016 = \u2a05 (w : \u2191\u2191K), \u2016v - \u2191w\u2016\n\u22a2 \u2203 w, w \u2208 K \u2227 v - w \u2208 K\u15ee\n[PROOFSTEP]\nrefine \u27e8w, hwK, (K.mem_orthogonal' _).2 ?_\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 K }\nv w : E\nhwK : w \u2208 K\nhw : \u2016v - w\u2016 = \u2a05 (w : \u2191\u2191K), \u2016v - \u2191w\u2016\n\u22a2 \u2200 (u : E), u \u2208 K \u2192 inner (v - w) u = 0\n[PROOFSTEP]\nrwa [\u2190 norm_eq_iInf_iff_inner_eq_zero K hwK]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 \u2203 w, w \u2208 K\u15ee \u2227 v - w \u2208 K\u15ee\u15ee\n[PROOFSTEP]\nrcases HasOrthogonalProjection.exists_orthogonal (K := K) v with \u27e8w, hwK, hw\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhwK : w \u2208 K\nhw : v - w \u2208 K\u15ee\n\u22a2 \u2203 w, w \u2208 K\u15ee \u2227 v - w \u2208 K\u15ee\u15ee\n[PROOFSTEP]\nrefine \u27e8_, hw, ?_\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhwK : w \u2208 K\nhw : v - w \u2208 K\u15ee\n\u22a2 v - (v - w) \u2208 K\u15ee\u15ee\n[PROOFSTEP]\nrw [sub_sub_cancel]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhwK : w \u2208 K\nhw : v - w \u2208 K\u15ee\n\u22a2 w \u2208 K\u15ee\u15ee\n[PROOFSTEP]\nexact K.le_orthogonal_orthogonal hwK\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : HasOrthogonalProjection K\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2243\u2097\u1d62[\ud835\udd5c] E'\nv : E'\n\u22a2 \u2203 w, w \u2208 Submodule.map (\u2191f.toLinearEquiv) K \u2227 v - w \u2208 (Submodule.map (\u2191f.toLinearEquiv) K)\u15ee\n[PROOFSTEP]\nrcases HasOrthogonalProjection.exists_orthogonal (K := K) (f.symm v) with \u27e8w, hwK, hw\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : HasOrthogonalProjection K\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2243\u2097\u1d62[\ud835\udd5c] E'\nv : E'\nw : E\nhwK : w \u2208 K\nhw : \u2191(LinearIsometryEquiv.symm f) v - w \u2208 K\u15ee\n\u22a2 \u2203 w, w \u2208 Submodule.map (\u2191f.toLinearEquiv) K \u2227 v - w \u2208 (Submodule.map (\u2191f.toLinearEquiv) K)\u15ee\n[PROOFSTEP]\nrefine \u27e8f w, Submodule.mem_map_of_mem hwK, Set.ball_image_iff.2 fun u hu \u21a6 ?_\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : HasOrthogonalProjection K\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2243\u2097\u1d62[\ud835\udd5c] E'\nv : E'\nw : E\nhwK : w \u2208 K\nhw : \u2191(LinearIsometryEquiv.symm f) v - w \u2208 K\u15ee\nu : E\nhu : u \u2208 \u2191K\n\u22a2 inner (\u2191\u2191f.toLinearEquiv u) (v - \u2191f w) = 0\n[PROOFSTEP]\nerw [\u2190 f.symm.inner_map_map, f.symm_apply_apply, map_sub, f.symm_apply_apply, hw u hu]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : E\n\u22a2 v - v \u2208 \u22a4\u15ee\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 orthogonalProjectionFn K u = v\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 @inner_self_eq_zero \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\n\u22a2 inner (orthogonalProjectionFn K u - v) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nhave hvs : orthogonalProjectionFn K u - v \u2208 K := Submodule.sub_mem K (orthogonalProjectionFn_mem u) hvm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nhvs : orthogonalProjectionFn K u - v \u2208 K\n\u22a2 inner (orthogonalProjectionFn K u - v) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nhave huo : \u27eau - orthogonalProjectionFn K u, orthogonalProjectionFn K u - v\u27eb = 0 :=\n  orthogonalProjectionFn_inner_eq_zero u _ hvs\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nhvs : orthogonalProjectionFn K u - v \u2208 K\nhuo : inner (u - orthogonalProjectionFn K u) (orthogonalProjectionFn K u - v) = 0\n\u22a2 inner (orthogonalProjectionFn K u - v) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nhave huv : \u27eau - v, orthogonalProjectionFn K u - v\u27eb = 0 := hvo _ hvs\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nhvs : orthogonalProjectionFn K u - v \u2208 K\nhuo : inner (u - orthogonalProjectionFn K u) (orthogonalProjectionFn K u - v) = 0\nhuv : inner (u - v) (orthogonalProjectionFn K u - v) = 0\n\u22a2 inner (orthogonalProjectionFn K u - v) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nhave houv : \u27eau - v - (u - orthogonalProjectionFn K u), orthogonalProjectionFn K u - v\u27eb = 0 := by\n  rw [inner_sub_left, huo, huv, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nhvs : orthogonalProjectionFn K u - v \u2208 K\nhuo : inner (u - orthogonalProjectionFn K u) (orthogonalProjectionFn K u - v) = 0\nhuv : inner (u - v) (orthogonalProjectionFn K u - v) = 0\n\u22a2 inner (u - v - (u - orthogonalProjectionFn K u)) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nrw [inner_sub_left, huo, huv, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\nhvm : v \u2208 K\nhvo : \u2200 (w : E), w \u2208 K \u2192 inner (u - v) w = 0\nhvs : orthogonalProjectionFn K u - v \u2208 K\nhuo : inner (u - orthogonalProjectionFn K u) (orthogonalProjectionFn K u - v) = 0\nhuv : inner (u - v) (orthogonalProjectionFn K u - v) = 0\nhouv : inner (u - v - (u - orthogonalProjectionFn K u)) (orthogonalProjectionFn K u - v) = 0\n\u22a2 inner (orthogonalProjectionFn K u - v) (orthogonalProjectionFn K u - v) = 0\n[PROOFSTEP]\nrwa [sub_sub_sub_cancel_left] at houv \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 \u2016v\u2016 * \u2016v\u2016 =\n    \u2016v - orthogonalProjectionFn K v\u2016 * \u2016v - orthogonalProjectionFn K v\u2016 +\n      \u2016orthogonalProjectionFn K v\u2016 * \u2016orthogonalProjectionFn K v\u2016\n[PROOFSTEP]\nset p := orthogonalProjectionFn K v\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\np : E := orthogonalProjectionFn K v\n\u22a2 \u2016v\u2016 * \u2016v\u2016 = \u2016v - p\u2016 * \u2016v - p\u2016 + \u2016p\u2016 * \u2016p\u2016\n[PROOFSTEP]\nhave h' : \u27eav - p, p\u27eb = 0 := orthogonalProjectionFn_inner_eq_zero _ _ (orthogonalProjectionFn_mem v)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\np : E := orthogonalProjectionFn K v\nh' : inner (v - p) p = 0\n\u22a2 \u2016v\u2016 * \u2016v\u2016 = \u2016v - p\u2016 * \u2016v - p\u2016 + \u2016p\u2016 * \u2016p\u2016\n[PROOFSTEP]\nconvert norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2\n[GOAL]\ncase h.e'_2.h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\np : E := orthogonalProjectionFn K v\nh' : inner (v - p) p = 0\n\u22a2 \u2016v\u2016 = \u2016v - p + p\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_6\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\np : E := orthogonalProjectionFn K v\nh' : inner (v - p) p = 0\n\u22a2 \u2016v\u2016 = \u2016v - p + p\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\n\u22a2 (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) (x + y) =\n    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) y\n[PROOFSTEP]\nhave hm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K :=\n  Submodule.add_mem K (orthogonalProjectionFn_mem x) (orthogonalProjectionFn_mem y)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\nhm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K\n\u22a2 (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) (x + y) =\n    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) y\n[PROOFSTEP]\nhave ho : \u2200 w \u2208 K, \u27eax + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y), w\u27eb = 0 :=\n  by\n  intro w hw\n  rw [add_sub_add_comm, inner_add_left, orthogonalProjectionFn_inner_eq_zero _ w hw,\n    orthogonalProjectionFn_inner_eq_zero _ w hw, add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\nhm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (x + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y)) w = 0\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\nhm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K\nw : E\nhw : w \u2208 K\n\u22a2 inner (x + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y)) w = 0\n[PROOFSTEP]\nrw [add_sub_add_comm, inner_add_left, orthogonalProjectionFn_inner_eq_zero _ w hw,\n  orthogonalProjectionFn_inner_eq_zero _ w hw, add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\nhm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K\nho : \u2200 (w : E), w \u2208 K \u2192 inner (x + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y)) w = 0\n\u22a2 (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) (x + y) =\n    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) y\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx y : E\nhm : orthogonalProjectionFn K x + orthogonalProjectionFn K y \u2208 K\nho : \u2200 (w : E), w \u2208 K \u2192 inner (x + y - (orthogonalProjectionFn K x + orthogonalProjectionFn K y)) w = 0\n\u22a2 \u2191((fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) (x + y)) =\n    \u2191((fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n        (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) y)\n[PROOFSTEP]\nsimp [eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hm ho]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\n\u22a2 AddHom.toFun\n      { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n        map_add' :=\n          (_ :\n            \u2200 (x y : E),\n              (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                  (x + y) =\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    y) }\n      (c \u2022 x) =\n    \u2191(RingHom.id \ud835\udd5c) c \u2022\n      AddHom.toFun\n        { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : E),\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    (x + y) =\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      y) }\n        x\n[PROOFSTEP]\nhave hm : c \u2022 orthogonalProjectionFn K x \u2208 K := Submodule.smul_mem K _ (orthogonalProjectionFn_mem x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\nhm : c \u2022 orthogonalProjectionFn K x \u2208 K\n\u22a2 AddHom.toFun\n      { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n        map_add' :=\n          (_ :\n            \u2200 (x y : E),\n              (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                  (x + y) =\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    y) }\n      (c \u2022 x) =\n    \u2191(RingHom.id \ud835\udd5c) c \u2022\n      AddHom.toFun\n        { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : E),\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    (x + y) =\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      y) }\n        x\n[PROOFSTEP]\nhave ho : \u2200 w \u2208 K, \u27eac \u2022 x - c \u2022 orthogonalProjectionFn K x, w\u27eb = 0 :=\n  by\n  intro w hw\n  rw [\u2190 smul_sub, inner_smul_left, orthogonalProjectionFn_inner_eq_zero _ w hw, mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\nhm : c \u2022 orthogonalProjectionFn K x \u2208 K\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (c \u2022 x - c \u2022 orthogonalProjectionFn K x) w = 0\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\nhm : c \u2022 orthogonalProjectionFn K x \u2208 K\nw : E\nhw : w \u2208 K\n\u22a2 inner (c \u2022 x - c \u2022 orthogonalProjectionFn K x) w = 0\n[PROOFSTEP]\nrw [\u2190 smul_sub, inner_smul_left, orthogonalProjectionFn_inner_eq_zero _ w hw, mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\nhm : c \u2022 orthogonalProjectionFn K x \u2208 K\nho : \u2200 (w : E), w \u2208 K \u2192 inner (c \u2022 x - c \u2022 orthogonalProjectionFn K x) w = 0\n\u22a2 AddHom.toFun\n      { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n        map_add' :=\n          (_ :\n            \u2200 (x y : E),\n              (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                  (x + y) =\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    y) }\n      (c \u2022 x) =\n    \u2191(RingHom.id \ud835\udd5c) c \u2022\n      AddHom.toFun\n        { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : E),\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    (x + y) =\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      y) }\n        x\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nc : \ud835\udd5c\nx : E\nhm : c \u2022 orthogonalProjectionFn K x \u2208 K\nho : \u2200 (w : E), w \u2208 K \u2192 inner (c \u2022 x - c \u2022 orthogonalProjectionFn K x) w = 0\n\u22a2 \u2191(AddHom.toFun\n        { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : E),\n                (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                    (x + y) =\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) }) x +\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      y) }\n        (c \u2022 x)) =\n    \u2191(\u2191(RingHom.id \ud835\udd5c) c \u2022\n        AddHom.toFun\n          { toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n            map_add' :=\n              (_ :\n                \u2200 (x y : E),\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      (x + y) =\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                        x +\n                      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                        y) }\n          x)\n[PROOFSTEP]\nsimp [eq_orthogonalProjectionFn_of_mem_of_inner_eq_zero hm ho]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2016\u2191{\n            toAddHom :=\n              {\n                toFun := fun v =>\n                  { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : E),\n                      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                          (x + y) =\n                        (fun v =>\n                              { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                            x +\n                          (fun v =>\n                              { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                            y) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : \ud835\udd5c) (x : E),\n                  AddHom.toFun\n                      {\n                        toFun := fun v =>\n                          { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n                        map_add' :=\n                          (_ :\n                            \u2200 (x y : E),\n                              (fun v =>\n                                    { val := orthogonalProjectionFn K v,\n                                      property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                  (x + y) =\n                                (fun v =>\n                                      { val := orthogonalProjectionFn K v,\n                                        property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                    x +\n                                  (fun v =>\n                                      { val := orthogonalProjectionFn K v,\n                                        property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                    y) }\n                      (c \u2022 x) =\n                    \u2191(RingHom.id \ud835\udd5c) c \u2022\n                      AddHom.toFun\n                        {\n                          toFun := fun v =>\n                            { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n                          map_add' :=\n                            (_ :\n                              \u2200 (x y : E),\n                                (fun v =>\n                                      { val := orthogonalProjectionFn K v,\n                                        property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                    (x + y) =\n                                  (fun v =>\n                                        { val := orthogonalProjectionFn K v,\n                                          property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                      x +\n                                    (fun v =>\n                                        { val := orthogonalProjectionFn K v,\n                                          property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                                      y) }\n                        x) }\n        x\u2016 \u2264\n    1 * \u2016x\u2016\n[PROOFSTEP]\nsimp only [one_mul, LinearMap.coe_mk]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2016\u2191{ toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n            map_add' :=\n              (_ :\n                \u2200 (x y : E),\n                  (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                      (x + y) =\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                        x +\n                      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                        y) }\n        x\u2016 \u2264\n    \u2016x\u2016\n[PROOFSTEP]\nrefine' le_of_pow_le_pow 2 (norm_nonneg _) (by norm_num) _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2016\u2191{ toFun := fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) },\n              map_add' :=\n                (_ :\n                  \u2200 (x y : E),\n                    (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                        (x + y) =\n                      (fun v => { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                          x +\n                        (fun v =>\n                            { val := orthogonalProjectionFn K v, property := (_ : orthogonalProjectionFn K v \u2208 K) })\n                          y) }\n          x\u2016 ^\n      2 \u2264\n    \u2016x\u2016 ^ 2\n[PROOFSTEP]\nchange \u2016orthogonalProjectionFn K x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2016orthogonalProjectionFn K x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nnlinarith [orthogonalProjectionFn_norm_sq K x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 v - \u2191(\u2191(orthogonalProjection K) v) \u2208 K\u15ee\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhw : w \u2208 K\n\u22a2 inner w (v - \u2191(\u2191(orthogonalProjection K) v)) = 0\n[PROOFSTEP]\nrw [inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhw : w \u2208 K\n\u22a2 inner (v - \u2191(\u2191(orthogonalProjection K) v)) w = 0\n[PROOFSTEP]\nexact orthogonalProjection_inner_eq_zero _ _ hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 inst\u271d : HasOrthogonalProjection K\nu v z : E\nhv : v \u2208 K\nhz : z \u2208 K\u15ee\nhu : u = v + z\n\u22a2 u - v \u2208 K\u15ee\n[PROOFSTEP]\nsimpa [hu]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 inst\u271d : HasOrthogonalProjection K\nu : E\n\u22a2 u = u - \u2191(\u2191(orthogonalProjection K) u) + \u2191(\u2191(orthogonalProjection K) u)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection K\nU : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection U\ny : E\n\u22a2 \u2016y - \u2191(\u2191(orthogonalProjection U) y)\u2016 = \u2a05 (x : { x // x \u2208 U }), \u2016y - \u2191x\u2016\n[PROOFSTEP]\nrw [norm_eq_iInf_iff_inner_eq_zero _ (Submodule.coe_mem _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection K\nU : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection U\ny : E\n\u22a2 \u2200 (w : E), w \u2208 U \u2192 inner (y - \u2191(\u2191(orthogonalProjection U) y)) w = 0\n[PROOFSTEP]\nexact orthogonalProjection_inner_eq_zero _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection K\nK' : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K'\nh : K = K'\nu : E\n\u22a2 \u2191(\u2191(orthogonalProjection K) u) = \u2191(\u2191(orthogonalProjection K') u)\n[PROOFSTEP]\nsubst h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection K\nu : E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 \u2191(\u2191(orthogonalProjection K) u) = \u2191(\u2191(orthogonalProjection K) u)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : { x // x \u2208 K }\n\u22a2 \u2191(orthogonalProjection K) \u2191v = v\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : { x // x \u2208 K }\n\u22a2 \u2191(\u2191(orthogonalProjection K) \u2191v) = \u2191v\n[PROOFSTEP]\napply eq_orthogonalProjection_of_mem_of_inner_eq_zero\n[GOAL]\ncase a.hvm\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : { x // x \u2208 K }\n\u22a2 \u2191v \u2208 K\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.hvo\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : { x // x \u2208 K }\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (\u2191v - \u2191v) w = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 \u2191(\u2191(orthogonalProjection K) v) = v \u2194 v \u2208 K\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => eq_orthogonalProjection_of_mem_of_inner_eq_zero h _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191(\u2191(orthogonalProjection K) v) = v\n\u22a2 v \u2208 K\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : \u2191(\u2191(orthogonalProjection K) v) = v\n\u22a2 \u2191(\u2191(orthogonalProjection K) v) \u2208 K\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nh : v \u2208 K\n\u22a2 \u2200 (w : E), w \u2208 K \u2192 inner (v - v) w = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\u271d\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2077 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2076 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection p\ninst\u271d : HasOrthogonalProjection (Submodule.map f.toLinearMap p)\nx : E\n\u22a2 \u2191f \u2191(\u2191(orthogonalProjection p) x) = \u2191(\u2191(orthogonalProjection (Submodule.map f.toLinearMap p)) (\u2191f x))\n[PROOFSTEP]\nrefine' (eq_orthogonalProjection_of_mem_of_inner_eq_zero _ fun y hy => _).symm\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\u271d\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2077 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2076 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection p\ninst\u271d : HasOrthogonalProjection (Submodule.map f.toLinearMap p)\nx : E\n\u22a2 \u2191f \u2191(\u2191(orthogonalProjection p) x) \u2208 Submodule.map f.toLinearMap p\ncase refine'_2\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\u271d\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2077 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2076 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection p\ninst\u271d : HasOrthogonalProjection (Submodule.map f.toLinearMap p)\nx : E\ny : (fun x => E') \u2191(\u2191(orthogonalProjection p) x)\nhy : y \u2208 Submodule.map f.toLinearMap p\n\u22a2 inner (\u2191f x - \u2191f \u2191(\u2191(orthogonalProjection p) x)) y = 0\n[PROOFSTEP]\nrefine' Submodule.apply_coe_mem_map _ _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\u271d\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2077 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2076 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection p\ninst\u271d : HasOrthogonalProjection (Submodule.map f.toLinearMap p)\nx : E\ny : (fun x => E') \u2191(\u2191(orthogonalProjection p) x)\nhy : y \u2208 Submodule.map f.toLinearMap p\n\u22a2 inner (\u2191f x - \u2191f \u2191(\u2191(orthogonalProjection p) x)) y = 0\n[PROOFSTEP]\nrcases hy with \u27e8x', hx', rfl : f x' = y\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\u271d\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2077 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2076 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection p\ninst\u271d : HasOrthogonalProjection (Submodule.map f.toLinearMap p)\nx x' : E\nhx' : x' \u2208 \u2191p\n\u22a2 inner (\u2191f x - \u2191f \u2191(\u2191(orthogonalProjection p) x)) (\u2191f x') = 0\n[PROOFSTEP]\nrw [\u2190 f.map_sub, f.inner_map_map, orthogonalProjection_inner_eq_zero x x' hx']\n[GOAL]\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2076 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2075 : HasOrthogonalProjection K\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E'\nf : E \u2243\u2097\u1d62[\ud835\udd5c] E'\np : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection p\nx : E'\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.map (\u2191f.toLinearEquiv) p)) x) =\n    \u2191f \u2191(\u2191(orthogonalProjection p) (\u2191(LinearIsometryEquiv.symm f) x))\n[PROOFSTEP]\nsimpa only [f.coe_toLinearIsometry, f.apply_symm_apply] using\n  (f.toLinearIsometry.map_orthogonalProjection' p (f.symm x)).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 orthogonalProjection \u22a5 = 0\n[PROOFSTEP]\next\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 \u2191(\u2016v\u2016 ^ 2) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = inner v w \u2022 v\n[PROOFSTEP]\nsuffices ((orthogonalProjection (\ud835\udd5c \u2219 v) (((\u2016v\u2016 : \ud835\udd5c) ^ 2) \u2022 w)) : E) = \u27eav, w\u27eb \u2022 v by simpa using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nthis : \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) (\u2191\u2016v\u2016 ^ 2 \u2022 w)) = inner v w \u2022 v\n\u22a2 \u2191(\u2016v\u2016 ^ 2) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = inner v w \u2022 v\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) (\u2191\u2016v\u2016 ^ 2 \u2022 w)) = inner v w \u2022 v\n[PROOFSTEP]\napply eq_orthogonalProjection_of_mem_of_inner_eq_zero\n[GOAL]\ncase hvm\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 inner v w \u2022 v \u2208 Submodule.span \ud835\udd5c {v}\n[PROOFSTEP]\nrw [Submodule.mem_span_singleton]\n[GOAL]\ncase hvm\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 \u2203 a, a \u2022 v = inner v w \u2022 v\n[PROOFSTEP]\nuse\u27eav, w\u27eb\n[GOAL]\ncase hvo\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 \u2200 (w_1 : E), w_1 \u2208 Submodule.span \ud835\udd5c {v} \u2192 inner (\u2191\u2016v\u2016 ^ 2 \u2022 w - inner v w \u2022 v) w_1 = 0\n[PROOFSTEP]\nrw [\u2190 Submodule.mem_orthogonal', Submodule.mem_orthogonal_singleton_iff_inner_left]\n[GOAL]\ncase hvo\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 inner (\u2191\u2016v\u2016 ^ 2 \u2022 w - inner v w \u2022 v) v = 0\n[PROOFSTEP]\nsimp [inner_sub_left, inner_smul_left, inner_self_eq_norm_sq_to_K, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v\n[PROOFSTEP]\nby_cases hv : v = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : v = 0\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v\n[PROOFSTEP]\nrw [hv, eq_orthogonalProjection_of_eq_submodule (Submodule.span_zero_singleton \ud835\udd5c)]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : v = 0\n\u22a2 \u2191(\u2191(orthogonalProjection \u22a5) w) = (inner 0 w / \u2191(\u20160\u2016 ^ 2)) \u2022 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v\n[PROOFSTEP]\nhave hv' : \u2016v\u2016 \u2260 0 := ne_of_gt (norm_pos_iff.mpr hv)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\nhv' : \u2016v\u2016 \u2260 0\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v\n[PROOFSTEP]\nhave key :\n  (((\u2016v\u2016 ^ 2 : \u211d) : \ud835\udd5c)\u207b\u00b9 * ((\u2016v\u2016 ^ 2 : \u211d) : \ud835\udd5c)) \u2022 ((orthogonalProjection (\ud835\udd5c \u2219 v) w) : E) =\n    (((\u2016v\u2016 ^ 2 : \u211d) : \ud835\udd5c)\u207b\u00b9 * \u27eav, w\u27eb) \u2022 v :=\n  by simp [mul_smul, smul_orthogonalProjection_singleton \ud835\udd5c w, -ofReal_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\nhv' : \u2016v\u2016 \u2260 0\n\u22a2 ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * \u2191(\u2016v\u2016 ^ 2)) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * inner v w) \u2022 v\n[PROOFSTEP]\nsimp [mul_smul, smul_orthogonalProjection_singleton \ud835\udd5c w, -ofReal_pow]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\nhv' : \u2016v\u2016 \u2260 0\nkey :\n  ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * \u2191(\u2016v\u2016 ^ 2)) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * inner v w) \u2022 v\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v\n[PROOFSTEP]\nconvert key using 1\n[GOAL]\ncase h.e'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\nhv' : \u2016v\u2016 \u2260 0\nkey :\n  ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * \u2191(\u2016v\u2016 ^ 2)) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * inner v w) \u2022 v\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) =\n    ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * \u2191(\u2016v\u2016 ^ 2)) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w)\n[PROOFSTEP]\nfield_simp [hv']\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv w : E\nhv : \u00acv = 0\nhv' : \u2016v\u2016 \u2260 0\nkey :\n  ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * \u2191(\u2016v\u2016 ^ 2)) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * inner v w) \u2022 v\n\u22a2 (inner v w / \u2191(\u2016v\u2016 ^ 2)) \u2022 v = ((\u2191(\u2016v\u2016 ^ 2))\u207b\u00b9 * inner v w) \u2022 v\n[PROOFSTEP]\nfield_simp [hv']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : \u2016v\u2016 = 1\nw : E\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = inner v w \u2022 v\n[PROOFSTEP]\nrw [\u2190 smul_orthogonalProjection_singleton \ud835\udd5c w]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : \u2016v\u2016 = 1\nw : E\n\u22a2 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w) = \u2191(\u2016v\u2016 ^ 2) \u2022 \u2191(\u2191(orthogonalProjection (Submodule.span \ud835\udd5c {v})) w)\n[PROOFSTEP]\nsimp [hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2191(2 \u2022 LinearMap.comp (Submodule.subtype K) \u2191(orthogonalProjection K) - LinearMap.id)\n      (\u2191(2 \u2022 LinearMap.comp (Submodule.subtype K) \u2191(orthogonalProjection K) - LinearMap.id) x) =\n    x\n[PROOFSTEP]\nsimp [two_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\n\u22a2 \u2200 (x : E),\n    \u2016\u2191{ toLinearMap := \u2191src\u271d, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n              right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n          x\u2016 =\n      \u2016x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\n\u22a2 \u2016\u2191{ toLinearMap := \u2191src\u271d, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        x\u2016 =\n    \u2016x\u2016\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\n\u22a2 \u2016\u2191{ toLinearMap := \u2191(reflectionLinearEquiv K), invFun := (reflectionLinearEquiv K).invFun,\n            left_inv :=\n              (_ : Function.LeftInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun),\n            right_inv :=\n              (_ : Function.RightInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun) }\n        x\u2016 =\n    \u2016x\u2016\n[PROOFSTEP]\nlet w : K := orthogonalProjection K x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\n\u22a2 \u2016\u2191{ toLinearMap := \u2191(reflectionLinearEquiv K), invFun := (reflectionLinearEquiv K).invFun,\n            left_inv :=\n              (_ : Function.LeftInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun),\n            right_inv :=\n              (_ : Function.RightInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun) }\n        x\u2016 =\n    \u2016x\u2016\n[PROOFSTEP]\nlet v := x - w\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\n\u22a2 \u2016\u2191{ toLinearMap := \u2191(reflectionLinearEquiv K), invFun := (reflectionLinearEquiv K).invFun,\n            left_inv :=\n              (_ : Function.LeftInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun),\n            right_inv :=\n              (_ : Function.RightInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun) }\n        x\u2016 =\n    \u2016x\u2016\n[PROOFSTEP]\nhave : \u27eav, w\u27eb = 0 := orthogonalProjection_inner_eq_zero x w w.2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 \u2016\u2191{ toLinearMap := \u2191(reflectionLinearEquiv K), invFun := (reflectionLinearEquiv K).invFun,\n            left_inv :=\n              (_ : Function.LeftInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun),\n            right_inv :=\n              (_ : Function.RightInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun) }\n        x\u2016 =\n    \u2016x\u2016\n[PROOFSTEP]\nconvert norm_sub_eq_norm_add this using 2\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 \u2191{ toLinearMap := \u2191(reflectionLinearEquiv K), invFun := (reflectionLinearEquiv K).invFun,\n          left_inv :=\n            (_ : Function.LeftInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun),\n          right_inv :=\n            (_ : Function.RightInverse (reflectionLinearEquiv K).invFun (\u2191(reflectionLinearEquiv K)).toAddHom.toFun) }\n      x =\n    \u2191w - v\n[PROOFSTEP]\nrw [LinearEquiv.coe_mk, reflectionLinearEquiv, LinearEquiv.toFun_eq_coe, LinearEquiv.coe_ofInvolutive,\n  LinearMap.sub_apply, LinearMap.id_apply, two_smul, LinearMap.add_apply, LinearMap.comp_apply, Submodule.subtype_apply,\n  ContinuousLinearMap.coe_coe]\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 \u2191(\u2191(orthogonalProjection K) x) + \u2191(\u2191(orthogonalProjection K) x) - x = \u2191w - v\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 \u2191(\u2191(orthogonalProjection K) x) + \u2191(\u2191(orthogonalProjection K) x) - x =\n    \u2191(\u2191(orthogonalProjection K) x) - (x - \u2191(\u2191(orthogonalProjection K) x))\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 \u2191(\u2191(orthogonalProjection K) x) + \u2191(\u2191(orthogonalProjection K) x) - x =\n    \u2191(\u2191(orthogonalProjection K) x) - (x - \u2191(\u2191(orthogonalProjection K) x))\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nsrc\u271d : E \u2243\u2097[\ud835\udd5c] E := reflectionLinearEquiv K\nx : E\nw : { x // x \u2208 K } := \u2191(orthogonalProjection K) x\nv : E := x - \u2191w\nthis : inner v \u2191w = 0\n\u22a2 x = \u2191w + v\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right, eq_self_iff_true]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 \u2191(reflection K\u15ee) v = -\u2191(reflection K) v\n[PROOFSTEP]\nsimp [reflection_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 2 \u2022 v - 2 \u2022 \u2191(\u2191(orthogonalProjection K) v) - v = v - 2 \u2022 \u2191(\u2191(orthogonalProjection K) v)\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 2 \u2022 v - 2 \u2022 \u2191(\u2191(orthogonalProjection K) v) - v = v - 2 \u2022 \u2191(\u2191(orthogonalProjection K) v)\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 reflection K\u15ee = LinearIsometryEquiv.trans (reflection K) (LinearIsometryEquiv.neg \ud835\udd5c)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx\u271d : E\n\u22a2 \u2191(reflection K\u15ee) x\u271d = \u2191(LinearIsometryEquiv.trans (reflection K) (LinearIsometryEquiv.neg \ud835\udd5c)) x\u271d\n[PROOFSTEP]\napply reflection_orthogonal_apply\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\n\u22a2 \u2191(reflection (Submodule.span \ud835\udd5c {u})) v = 2 \u2022 (inner u v / \u2191\u2016u\u2016 ^ 2) \u2022 u - v\n[PROOFSTEP]\nrw [reflection_apply, orthogonalProjection_singleton, ofReal_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2191(reflection K) x = x \u2194 x \u2208 K\n[PROOFSTEP]\nrw [\u2190 orthogonalProjection_eq_self_iff, reflection_apply, sub_eq_iff_eq_add', \u2190 two_smul \ud835\udd5c, two_smul \u2115, \u2190 two_smul \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 2 \u2022 \u2191(\u2191(orthogonalProjection K) x) = 2 \u2022 x \u2194 \u2191(\u2191(orthogonalProjection K) x) = x\n[PROOFSTEP]\nrefine' (smul_right_injective E _).eq_iff\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nE\u271d : Type u_2\nF : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u2076 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\u271d\ninst\u271d\u2075 : HasOrthogonalProjection K\u271d\nE : Type u_4\nE' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E'\nf : E \u2243\u2097\u1d62[\ud835\udd5c] E'\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E'\n\u22a2 \u2191(reflection (Submodule.map (\u2191f.toLinearEquiv) K)) x = \u2191f (\u2191(reflection K) (\u2191(LinearIsometryEquiv.symm f) x))\n[PROOFSTEP]\nsimp [two_smul, reflection_apply, orthogonalProjection_map_apply f K x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 reflection \u22a5 = LinearIsometryEquiv.neg \ud835\udd5c\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx\u271d : E\n\u22a2 \u2191(reflection \u22a5) x\u271d = \u2191(LinearIsometryEquiv.neg \ud835\udd5c) x\u271d\n[PROOFSTEP]\nsimp [reflection_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\n\u22a2 K\u2081 \u2294 K\u2081\u15ee \u2293 K\u2082 = K\u2082\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\n\u22a2 x \u2208 K\u2081 \u2294 K\u2081\u15ee \u2293 K\u2082 \u2194 x \u2208 K\u2082\n[PROOFSTEP]\nrw [Submodule.mem_sup]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\n\u22a2 (\u2203 y, y \u2208 K\u2081 \u2227 \u2203 z, z \u2208 K\u2081\u15ee \u2293 K\u2082 \u2227 y + z = x) \u2194 x \u2208 K\u2082\n[PROOFSTEP]\nlet v : K\u2081 := orthogonalProjection K\u2081 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\nv : { x // x \u2208 K\u2081 } := \u2191(orthogonalProjection K\u2081) x\n\u22a2 (\u2203 y, y \u2208 K\u2081 \u2227 \u2203 z, z \u2208 K\u2081\u15ee \u2293 K\u2082 \u2227 y + z = x) \u2194 x \u2208 K\u2082\n[PROOFSTEP]\nhave hvm : x - v \u2208 K\u2081\u15ee := sub_orthogonalProjection_mem_orthogonal x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\nv : { x // x \u2208 K\u2081 } := \u2191(orthogonalProjection K\u2081) x\nhvm : x - \u2191v \u2208 K\u2081\u15ee\n\u22a2 (\u2203 y, y \u2208 K\u2081 \u2227 \u2203 z, z \u2208 K\u2081\u15ee \u2293 K\u2082 \u2227 y + z = x) \u2194 x \u2208 K\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\nv : { x // x \u2208 K\u2081 } := \u2191(orthogonalProjection K\u2081) x\nhvm : x - \u2191v \u2208 K\u2081\u15ee\n\u22a2 (\u2203 y, y \u2208 K\u2081 \u2227 \u2203 z, z \u2208 K\u2081\u15ee \u2293 K\u2082 \u2227 y + z = x) \u2192 x \u2208 K\u2082\n[PROOFSTEP]\nrintro \u27e8y, hy, z, hz, rfl\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\ny : E\nhy : y \u2208 K\u2081\nz : E\nhz : z \u2208 K\u2081\u15ee \u2293 K\u2082\nv : { x // x \u2208 K\u2081 } := \u2191(orthogonalProjection K\u2081) (y + z)\nhvm : y + z - \u2191v \u2208 K\u2081\u15ee\n\u22a2 y + z \u2208 K\u2082\n[PROOFSTEP]\nexact K\u2082.add_mem (h hy) hz.2\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\nh : K\u2081 \u2264 K\u2082\ninst\u271d : HasOrthogonalProjection K\u2081\nx : E\nv : { x // x \u2208 K\u2081 } := \u2191(orthogonalProjection K\u2081) x\nhvm : x - \u2191v \u2208 K\u2081\u15ee\n\u22a2 x \u2208 K\u2082 \u2192 \u2203 y, y \u2208 K\u2081 \u2227 \u2203 z, z \u2208 K\u2081\u15ee \u2293 K\u2082 \u2227 y + z = x\n[PROOFSTEP]\nexact fun hx => \u27e8v, v.prop, x - v, \u27e8hvm, K\u2082.sub_mem hx (h v.prop)\u27e9, add_sub_cancel'_right _ _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K \u2294 K\u15ee = \u22a4\n[PROOFSTEP]\nconvert Submodule.sup_orthogonal_inf_of_completeSpace (le_top : K \u2264 \u22a4) using 2\n[GOAL]\ncase h.e'_2.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee = K\u15ee \u2293 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 v = \u2191(\u2191(orthogonalProjection K) v) + (v - \u2191(\u2191(orthogonalProjection K) v))\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee\u15ee = K\n[PROOFSTEP]\next v\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 v \u2208 K\u15ee\u15ee \u2194 v \u2208 K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 v \u2208 K\u15ee\u15ee \u2192 v \u2208 K\n[PROOFSTEP]\nobtain \u27e8y, hy, z, hz, rfl\u27e9 := K.exists_add_mem_mem_orthogonal v\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\n\u22a2 y + z \u2208 K\u15ee\u15ee \u2192 y + z \u2208 K\n[PROOFSTEP]\nintro hv\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\nhv : y + z \u2208 K\u15ee\u15ee\n\u22a2 y + z \u2208 K\n[PROOFSTEP]\nhave hz' : z = 0 := by\n  have hyz : \u27eaz, y\u27eb = 0 := by simp [hz y hy, inner_eq_zero_symm]\n  simpa [inner_add_right, hyz] using hv z hz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\nhv : y + z \u2208 K\u15ee\u15ee\n\u22a2 z = 0\n[PROOFSTEP]\nhave hyz : \u27eaz, y\u27eb = 0 := by simp [hz y hy, inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\nhv : y + z \u2208 K\u15ee\u15ee\n\u22a2 inner z y = 0\n[PROOFSTEP]\nsimp [hz y hy, inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\nhv : y + z \u2208 K\u15ee\u15ee\nhyz : inner z y = 0\n\u22a2 z = 0\n[PROOFSTEP]\nsimpa [inner_add_right, hyz] using hv z hz\n[GOAL]\ncase h.mp.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\ny : E\nhy : y \u2208 K\nz : E\nhz : z \u2208 K\u15ee\nhv : y + z \u2208 K\u15ee\u15ee\nhz' : z = 0\n\u22a2 y + z \u2208 K\n[PROOFSTEP]\nsimp [hy, hz']\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\n\u22a2 v \u2208 K \u2192 v \u2208 K\u15ee\u15ee\n[PROOFSTEP]\nintro hv w hw\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\nw : E\nhw : w \u2208 K\u15ee\n\u22a2 inner w v = 0\n[PROOFSTEP]\nrw [inner_eq_zero_symm]\n[GOAL]\ncase h.mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\nw : E\nhw : w \u2208 K\u15ee\n\u22a2 inner v w = 0\n[PROOFSTEP]\nexact hw v hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee\u15ee = topologicalClosure K\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee\u15ee \u2264 topologicalClosure K\n[PROOFSTEP]\nconvert Submodule.orthogonal_orthogonal_monotone K.le_topologicalClosure using 1\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 topologicalClosure K = (topologicalClosure K)\u15ee\u15ee\n[PROOFSTEP]\nrw [K.topologicalClosure.orthogonal_orthogonal]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 topologicalClosure K \u2264 K\u15ee\u15ee\n[PROOFSTEP]\nexact K.topologicalClosure_minimal K.le_orthogonal_orthogonal K\u15ee.isClosed_orthogonal\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee = \u22a5 \u2194 K = \u22a4\n[PROOFSTEP]\nrefine' \u27e8_, fun h => by rw [h, Submodule.top_orthogonal_eq_bot]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nh : K = \u22a4\n\u22a2 K\u15ee = \u22a5\n[PROOFSTEP]\nrw [h, Submodule.top_orthogonal_eq_bot]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 K\u15ee = \u22a5 \u2192 K = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nh : K\u15ee = \u22a5\n\u22a2 K = \u22a4\n[PROOFSTEP]\nhave : K \u2294 K\u15ee = \u22a4 := Submodule.sup_orthogonal_of_completeSpace\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nh : K\u15ee = \u22a5\nthis : K \u2294 K\u15ee = \u22a4\n\u22a2 K = \u22a4\n[PROOFSTEP]\nrwa [h, sup_comm, bot_sup_eq] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(orthogonalProjection K) v = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(\u2191(orthogonalProjection K) v) = \u21910\n[PROOFSTEP]\nconvert eq_orthogonalProjection_of_mem_orthogonal (K := K) _ _\n[GOAL]\ncase a.convert_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u21910 \u2208 K\n[PROOFSTEP]\nsimp [hv]\n[GOAL]\ncase a.convert_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 v - \u21910 \u2208 K\u15ee\n[PROOFSTEP]\nsimp [hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK U V : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection U\nh : ContinuousLinearMap.comp (orthogonalProjection U) (Submodule.subtypeL V) = 0\nu : E\nhu : u \u2208 U\nv : E\nhv : v \u2208 V\n\u22a2 inner v u = 0\n[PROOFSTEP]\nconvert orthogonalProjection_inner_eq_zero v u hu using 2\n[GOAL]\ncase h.e'_2.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK U V : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection U\nh : ContinuousLinearMap.comp (orthogonalProjection U) (Submodule.subtypeL V) = 0\nu : E\nhu : u \u2208 U\nv : E\nhv : v \u2208 V\n\u22a2 v = v - \u2191(\u2191(orthogonalProjection U) v)\n[PROOFSTEP]\nhave : orthogonalProjection U v = 0 := FunLike.congr_fun h (\u27e8_, hv\u27e9 : V)\n[GOAL]\ncase h.e'_2.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK U V : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection U\nh : ContinuousLinearMap.comp (orthogonalProjection U) (Submodule.subtypeL V) = 0\nu : E\nhu : u \u2208 U\nv : E\nhv : v \u2208 V\nthis : \u2191(orthogonalProjection U) v = 0\n\u22a2 v = v - \u2191(\u2191(orthogonalProjection U) v)\n[PROOFSTEP]\nrw [this, Submodule.coe_zero, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\n\u22a2 \u2191(orthogonalProjection K) x = \u2191(Submodule.linearProjOfIsCompl K K\u15ee (_ : IsCompl K K\u15ee)) x\n[PROOFSTEP]\nhave : IsCompl K K\u15ee := Submodule.isCompl_orthogonal_of_completeSpace\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\nthis : IsCompl K K\u15ee\n\u22a2 \u2191(orthogonalProjection K) x = \u2191(Submodule.linearProjOfIsCompl K K\u15ee (_ : IsCompl K K\u15ee)) x\n[PROOFSTEP]\nconv_lhs => rw [\u2190 Submodule.linear_proj_add_linearProjOfIsCompl_eq_self this x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\nthis : IsCompl K K\u15ee\n| \u2191(orthogonalProjection K) x\n[PROOFSTEP]\nrw [\u2190 Submodule.linear_proj_add_linearProjOfIsCompl_eq_self this x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\nthis : IsCompl K K\u15ee\n| \u2191(orthogonalProjection K) x\n[PROOFSTEP]\nrw [\u2190 Submodule.linear_proj_add_linearProjOfIsCompl_eq_self this x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\nthis : IsCompl K K\u15ee\n| \u2191(orthogonalProjection K) x\n[PROOFSTEP]\nrw [\u2190 Submodule.linear_proj_add_linearProjOfIsCompl_eq_self this x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nx : E\nthis : IsCompl K K\u15ee\n\u22a2 \u2191(orthogonalProjection K)\n      (\u2191(\u2191(Submodule.linearProjOfIsCompl K K\u15ee this) x) +\n        \u2191(\u2191(Submodule.linearProjOfIsCompl K\u15ee K (_ : IsCompl K\u15ee K)) x)) =\n    \u2191(Submodule.linearProjOfIsCompl K K\u15ee (_ : IsCompl K K\u15ee)) x\n[PROOFSTEP]\nrw [map_add, orthogonalProjection_mem_subspace_eq_self,\n  orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero (Submodule.coe_mem _), add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nv : E\nhv : v \u2208 K\u15ee\n\u22a2 \u2191(reflection K) v = -v\n[PROOFSTEP]\nsimp [reflection_apply, orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK U V : Submodule \ud835\udd5c E\ninst\u271d\u00b9 : HasOrthogonalProjection U\ninst\u271d : HasOrthogonalProjection V\nh : U \u2264 V\nx : E\n\u22a2 \u2191(orthogonalProjection U) x = \u2191(orthogonalProjection U) \u2191(\u2191(orthogonalProjection V) x)\n[PROOFSTEP]\nsimpa only [sub_eq_zero, map_sub] using\n  orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero\n    (Submodule.orthogonal_le h (sub_orthogonalProjection_mem_orthogonal x))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b9\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : IsEmpty \u03b9\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\nexact tendsto_of_isEmpty\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\nlet y := (orthogonalProjection (\u2a06 i, U i).topologicalClosure x : E)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\nhave proj_x : \u2200 i, orthogonalProjection (U i) x = orthogonalProjection (U i) y := fun i =>\n  (orthogonalProjection_orthogonalProjection_of_le ((le_iSup U i).trans (iSup U).le_topologicalClosure) _).symm\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\nsuffices \u2200 \u03b5 > 0, \u2203 I, \u2200 i \u2265 I, \u2016(orthogonalProjection (U i) y : E) - y\u2016 < \u03b5 by\n  simpa only [proj_x, NormedAddCommGroup.tendsto_atTop] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\nthis : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n\u22a2 Tendsto (fun i => \u2191(\u2191(orthogonalProjection (U i)) x)) atTop\n    (\ud835\udcdd \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x))\n[PROOFSTEP]\nsimpa only [proj_x, NormedAddCommGroup.tendsto_atTop] using this\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8a, ha, hay\u27e9 : \u2203 a \u2208 \u2a06 i, U i, dist y a < \u03b5 :=\n  by\n  have y_mem : y \u2208 (\u2a06 i, U i).topologicalClosure := Submodule.coe_mem _\n  rw [\u2190 SetLike.mem_coe, Submodule.topologicalClosure_coe, Metric.mem_closure_iff] at y_mem \n  exact y_mem \u03b5 h\u03b5\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 a, a \u2208 \u2a06 (i : \u03b9), U i \u2227 dist y a < \u03b5\n[PROOFSTEP]\nhave y_mem : y \u2208 (\u2a06 i, U i).topologicalClosure := Submodule.coe_mem _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ny_mem : y \u2208 Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i)\n\u22a2 \u2203 a, a \u2208 \u2a06 (i : \u03b9), U i \u2227 dist y a < \u03b5\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, Submodule.topologicalClosure_coe, Metric.mem_closure_iff] at y_mem \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\ny_mem : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 b, b \u2208 \u2191(\u2a06 (i : \u03b9), U i) \u2227 dist y b < \u03b5\n\u22a2 \u2203 a, a \u2208 \u2a06 (i : \u03b9), U i \u2227 dist y a < \u03b5\n[PROOFSTEP]\nexact y_mem \u03b5 h\u03b5\n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : dist y a < \u03b5\n\u22a2 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nrw [dist_eq_norm] at hay \n[GOAL]\ncase inr.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\n\u22a2 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8I, hI\u27e9 : \u2203 I, a \u2208 U I := by rwa [Submodule.mem_iSup_of_directed _ hU.directed_le] at ha \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\n\u22a2 \u2203 I, a \u2208 U I\n[PROOFSTEP]\nrwa [Submodule.mem_iSup_of_directed _ hU.directed_le] at ha \n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\nI : \u03b9\nhI : a \u2208 U I\n\u22a2 \u2203 I, \u2200 (i : \u03b9), i \u2265 I \u2192 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nrefine' \u27e8I, fun i (hi : I \u2264 i) => _\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\nI : \u03b9\nhI : a \u2208 U I\ni : \u03b9\nhi : I \u2264 i\n\u22a2 \u2016\u2191(\u2191(orthogonalProjection (U i)) y) - y\u2016 < \u03b5\n[PROOFSTEP]\nrw [norm_sub_rev, orthogonalProjection_minimal]\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\nI : \u03b9\nhI : a \u2208 U I\ni : \u03b9\nhi : I \u2264 i\n\u22a2 \u2a05 (x : { x // x \u2208 U i }), \u2016y - \u2191x\u2016 < \u03b5\n[PROOFSTEP]\nrefine' lt_of_le_of_lt _ hay\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\nI : \u03b9\nhI : a \u2208 U I\ni : \u03b9\nhi : I \u2264 i\n\u22a2 \u2a05 (x : { x // x \u2208 U i }), \u2016y - \u2191x\u2016 \u2264 \u2016y - a\u2016\n[PROOFSTEP]\nchange _ \u2264 \u2016y - (\u27e8a, hU hi hI\u27e9 : U i)\u2016\n[GOAL]\ncase inr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 U i }\nhU : Monotone U\nx : E\nh\u271d : Nonempty \u03b9\ny : E := \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\nproj_x : \u2200 (i : \u03b9), \u2191(orthogonalProjection (U i)) x = \u2191(orthogonalProjection (U i)) y\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : E\nha : a \u2208 \u2a06 (i : \u03b9), U i\nhay : \u2016y - a\u2016 < \u03b5\nI : \u03b9\nhI : a \u2208 U I\ni : \u03b9\nhi : I \u2264 i\n\u22a2 \u2a05 (x : { x // x \u2208 U i }), \u2016y - \u2191x\u2016 \u2264 \u2016y - \u2191{ val := a, property := (_ : a \u2208 U i) }\u2016\n[PROOFSTEP]\nexact ciInf_le \u27e80, Set.forall_range_iff.mpr fun _ => norm_nonneg _\u27e9 _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (t : \u03b9), CompleteSpace { x // x \u2208 U t }\nhU : Monotone U\nx : E\nhU' : \u22a4 \u2264 Submodule.topologicalClosure (\u2a06 (t : \u03b9), U t)\n\u22a2 Tendsto (fun t => \u2191(\u2191(orthogonalProjection (U t)) x)) atTop (\ud835\udcdd x)\n[PROOFSTEP]\nrw [\u2190 eq_top_iff] at hU' \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (t : \u03b9), CompleteSpace { x // x \u2208 U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (\u2a06 (t : \u03b9), U t) = \u22a4\n\u22a2 Tendsto (fun t => \u2191(\u2191(orthogonalProjection (U t)) x)) atTop (\ud835\udcdd x)\n[PROOFSTEP]\nconvert orthogonalProjection_tendsto_closure_iSup U hU x\n[GOAL]\ncase h.e'_5.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (t : \u03b9), CompleteSpace { x // x \u2208 U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (\u2a06 (t : \u03b9), U t) = \u22a4\n\u22a2 x = \u2191(\u2191(orthogonalProjection (Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i))) x)\n[PROOFSTEP]\nrw [orthogonalProjection_eq_self_iff.mpr _]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (t : \u03b9), CompleteSpace { x // x \u2208 U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (\u2a06 (t : \u03b9), U t) = \u22a4\n\u22a2 x \u2208 Submodule.topologicalClosure (\u2a06 (i : \u03b9), U i)\n[PROOFSTEP]\nrw [hU']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : SemilatticeSup \u03b9\nU : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (t : \u03b9), CompleteSpace { x // x \u2208 U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (\u2a06 (t : \u03b9), U t) = \u22a4\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee\u15ee\u15ee = K\u15ee\n[PROOFSTEP]\nrw [K\u15ee.orthogonal_orthogonal_eq_closure]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 topologicalClosure K\u15ee = K\u15ee\n[PROOFSTEP]\nexact K.isClosed_orthogonal.submodule_topologicalClosure_eq\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 topologicalClosure K = \u22a4 \u2194 K\u15ee = \u22a5\n[PROOFSTEP]\nrw [\u2190 Submodule.orthogonal_orthogonal_eq_closure]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee\u15ee = \u22a4 \u2194 K\u15ee = \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee\u15ee = \u22a4 \u2192 K\u15ee = \u22a5\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\n\u22a2 K\u15ee = \u22a5 \u2192 K\u15ee\u15ee = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\nh : K\u15ee\u15ee = \u22a4\n\u22a2 K\u15ee = \u22a5\n[PROOFSTEP]\nrw [\u2190 Submodule.triorthogonal_eq_orthogonal, h, Submodule.top_orthogonal_eq_bot]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace E\nh : K\u15ee = \u22a5\n\u22a2 K\u15ee\u15ee = \u22a4\n[PROOFSTEP]\nrw [h, Submodule.bot_orthogonal_eq_top]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx y : E\nhK : Dense \u2191K\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = 0\n\u22a2 x = 0\n[PROOFSTEP]\nhave : (\u27eax, \u00b7\u27eb) = 0 := (continuous_const.inner continuous_id).ext_on hK continuous_const (Subtype.forall.1 h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx y : E\nhK : Dense \u2191K\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = 0\nthis : (fun x_1 => inner x x_1) = 0\n\u22a2 x = 0\n[PROOFSTEP]\nsimpa using congr_fun this x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx y : E\nhK : Dense \u2191K\nh : \u2200 (v : { x // x \u2208 K }), inner (\u2191v) x = 0\nv : { x // x \u2208 K }\n\u22a2 inner (\u2191v) x = inner (\u2191v) 0\n[PROOFSTEP]\nrw [inner_zero_right, h v]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\n\u22a2 \u2191(reflection (Submodule.span \u211d {v - w})\u15ee) v = w\n[PROOFSTEP]\nset R : F \u2243\u2097\u1d62[\u211d] F := reflection (\u211d \u2219 v - w)\u15ee\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\n\u22a2 \u2191R v = w\n[PROOFSTEP]\nsuffices R v + R v = w + w by\n  apply smul_right_injective F (by norm_num : (2 : \u211d) \u2260 0)\n  simpa [two_smul] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nthis : \u2191R v + \u2191R v = w + w\n\u22a2 \u2191R v = w\n[PROOFSTEP]\napply smul_right_injective F (by norm_num : (2 : \u211d) \u2260 0)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nthis : \u2191R v + \u2191R v = w + w\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nthis : \u2191R v + \u2191R v = w + w\n\u22a2 (fun x x_1 => x \u2022 x_1) 2 (\u2191R v) = (fun x x_1 => x \u2022 x_1) 2 w\n[PROOFSTEP]\nsimpa [two_smul] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\n\u22a2 \u2191R v + \u2191R v = w + w\n[PROOFSTEP]\nhave h\u2081 : R (v - w) = -(v - w) := reflection_orthogonalComplement_singleton_eq_neg (v - w)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\n\u22a2 \u2191R v + \u2191R v = w + w\n[PROOFSTEP]\nhave h\u2082 : R (v + w) = v + w := by\n  apply reflection_mem_subspace_eq_self\n  rw [Submodule.mem_orthogonal_singleton_iff_inner_left]\n  rw [real_inner_add_sub_eq_zero_iff]\n  exact h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\n\u22a2 \u2191R (v + w) = v + w\n[PROOFSTEP]\napply reflection_mem_subspace_eq_self\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\n\u22a2 v + w \u2208 (Submodule.span \u211d {v - w})\u15ee\n[PROOFSTEP]\nrw [Submodule.mem_orthogonal_singleton_iff_inner_left]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\n\u22a2 inner (v + w) (v - w) = 0\n[PROOFSTEP]\nrw [real_inner_add_sub_eq_zero_iff]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\n\u22a2 \u2016v\u2016 = \u2016w\u2016\n[PROOFSTEP]\nexact h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\nh\u2082 : \u2191R (v + w) = v + w\n\u22a2 \u2191R v + \u2191R v = w + w\n[PROOFSTEP]\nconvert congr_arg\u2082 (\u00b7 + \u00b7) h\u2082 h\u2081 using 1\n[GOAL]\ncase h.e'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\nh\u2082 : \u2191R (v + w) = v + w\n\u22a2 \u2191R v + \u2191R v = \u2191R (v + w) + \u2191R (v - w)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\nh\u2082 : \u2191R (v + w) = v + w\n\u22a2 w + w = v + w + -(v - w)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv w : F\nh : \u2016v\u2016 = \u2016w\u2016\nR : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {v - w})\u15ee\nh\u2081 : \u2191R (v - w) = -(v - w)\nh\u2082 : \u2191R (v + w) = v + w\n\u22a2 w + w = v + w + -(v - w)\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nw : E\n\u22a2 \u2191(\u2191(orthogonalProjection K) w) + \u2191(\u2191(orthogonalProjection K\u15ee) w) = w\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx : E\nS : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection S\n\u22a2 \u2016x\u2016 ^ 2 = \u2016\u2191(\u2191(orthogonalProjection S) x) + \u2191(\u2191(orthogonalProjection S\u15ee) x)\u2016 ^ 2\n[PROOFSTEP]\nrw [orthogonalProjection_add_orthogonalProjection_orthogonal]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx : E\nS : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection S\n\u22a2 \u2016\u2191(\u2191(orthogonalProjection S) x) + \u2191(\u2191(orthogonalProjection S\u15ee) x)\u2016 ^ 2 =\n    \u2016\u2191(orthogonalProjection S) x\u2016 ^ 2 + \u2016\u2191(orthogonalProjection S\u15ee) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nx : E\nS : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection S\n\u22a2 \u2016\u2191(\u2191(orthogonalProjection S) x) + \u2191(\u2191(orthogonalProjection S\u15ee) x)\u2016 *\n      \u2016\u2191(\u2191(orthogonalProjection S) x) + \u2191(\u2191(orthogonalProjection S\u15ee) x)\u2016 =\n    \u2016\u2191(orthogonalProjection S) x\u2016 * \u2016\u2191(orthogonalProjection S) x\u2016 +\n      \u2016\u2191(orthogonalProjection S\u15ee) x\u2016 * \u2016\u2191(orthogonalProjection S\u15ee) x\u2016\n[PROOFSTEP]\nexact\n  norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero _ _ <|\n    (S.mem_orthogonal _).1 (orthogonalProjection S\u15ee x).2 _ (orthogonalProjection S x).2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\n\u22a2 ContinuousLinearMap.id \ud835\udd5c E =\n    ContinuousLinearMap.comp (Submodule.subtypeL K) (orthogonalProjection K) +\n      ContinuousLinearMap.comp (Submodule.subtypeL K\u15ee) (orthogonalProjection K\u15ee)\n[PROOFSTEP]\next w\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nw : E\n\u22a2 \u2191(ContinuousLinearMap.id \ud835\udd5c E) w =\n    \u2191(ContinuousLinearMap.comp (Submodule.subtypeL K) (orthogonalProjection K) +\n          ContinuousLinearMap.comp (Submodule.subtypeL K\u15ee) (orthogonalProjection K\u15ee))\n      w\n[PROOFSTEP]\nexact (orthogonalProjection_add_orthogonalProjection_orthogonal K w).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu : { x // x \u2208 K }\nv : E\n\u22a2 inner \u2191(\u2191(orthogonalProjection K) v) \u2191u =\n    inner \u2191(\u2191(orthogonalProjection K) v) \u2191u + inner (v - \u2191(\u2191(orthogonalProjection K) v)) \u2191u\n[PROOFSTEP]\nrw [orthogonalProjection_inner_eq_zero _ _ (Submodule.coe_mem _), add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu : { x // x \u2208 K }\nv : E\n\u22a2 inner \u2191(\u2191(orthogonalProjection K) v) \u2191u + inner (v - \u2191(\u2191(orthogonalProjection K) v)) \u2191u = inner v \u2191u\n[PROOFSTEP]\nrw [\u2190 inner_add_left, add_sub_cancel'_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu : { x // x \u2208 K }\nv : E\n\u22a2 inner u (\u2191(orthogonalProjection K) v) = inner (\u2191u) v\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, \u2190 inner_conj_symm (u : E), inner_orthogonalProjection_eq_of_mem_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : HasOrthogonalProjection K\nu v : E\n\u22a2 inner (\u2191(\u2191(orthogonalProjection K) u)) v = inner u \u2191(\u2191(orthogonalProjection K) v)\n[PROOFSTEP]\nrw [\u2190 inner_orthogonalProjection_eq_of_mem_left, inner_orthogonalProjection_eq_of_mem_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nhaveI : FiniteDimensional \ud835\udd5c K\u2081 := Submodule.finiteDimensional_of_le h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2081 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nhaveI := proper_isROrC \ud835\udd5c K\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2081 }\nthis : ProperSpace { x // x \u2208 K\u2081 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nhave hd := Submodule.finrank_sup_add_finrank_inf_eq K\u2081 (K\u2081\u15ee \u2293 K\u2082)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2081 }\nthis : ProperSpace { x // x \u2208 K\u2081 }\nhd :\n  finrank \ud835\udd5c { x // x \u2208 K\u2081 \u2294 K\u2081\u15ee \u2293 K\u2082 } + finrank \ud835\udd5c { x // x \u2208 K\u2081 \u2293 (K\u2081\u15ee \u2293 K\u2082) } =\n    finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nrw [\u2190 inf_assoc, (Submodule.orthogonal_disjoint K\u2081).eq_bot, bot_inf_eq, finrank_bot,\n  Submodule.sup_orthogonal_inf_of_completeSpace h] at hd \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2081 }\nthis : ProperSpace { x // x \u2208 K\u2081 }\nhd : finrank \ud835\udd5c { x // x \u2208 K\u2082 } + 0 = finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nrw [add_zero] at hd \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2081 }\nthis : ProperSpace { x // x \u2208 K\u2081 }\nhd : finrank \ud835\udd5c { x // x \u2208 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n[PROOFSTEP]\nexact hd.symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nn : \u2115\nh_dim : finrank \ud835\udd5c { x // x \u2208 K\u2081 } + n = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = n\n[PROOFSTEP]\nrw [\u2190 add_right_inj (finrank \ud835\udd5c K\u2081)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK K\u2081 K\u2082 : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 K\u2082 }\nh : K\u2081 \u2264 K\u2082\nn : \u2115\nh_dim : finrank \ud835\udd5c { x // x \u2208 K\u2081 } + n = finrank \ud835\udd5c { x // x \u2208 K\u2082 }\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u2081 } + finrank \ud835\udd5c { x // x \u2208 K\u2081\u15ee \u2293 K\u2082 } = finrank \ud835\udd5c { x // x \u2208 K\u2081 } + n\n[PROOFSTEP]\nsimp [Submodule.finrank_add_inf_finrank_orthogonal h, h_dim]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nK : Submodule \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K } + finrank \ud835\udd5c { x // x \u2208 K\u15ee } = finrank \ud835\udd5c E\n[PROOFSTEP]\nconvert Submodule.finrank_add_inf_finrank_orthogonal (le_top : K \u2264 \u22a4) using 1\n[GOAL]\ncase h.e'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nK : Submodule \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K } + finrank \ud835\udd5c { x // x \u2208 K\u15ee } = finrank \ud835\udd5c { x // x \u2208 K } + finrank \ud835\udd5c { x // x \u2208 K\u15ee \u2293 \u22a4 }\n[PROOFSTEP]\nrw [inf_top_eq]\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nK : Submodule \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c E = finrank \ud835\udd5c { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nn : \u2115\nh_dim : finrank \ud835\udd5c { x // x \u2208 K } + n = finrank \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K\u15ee } = n\n[PROOFSTEP]\nrw [\u2190 add_right_inj (finrank \ud835\udd5c K)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK\u271d : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nn : \u2115\nh_dim : finrank \ud835\udd5c { x // x \u2208 K } + n = finrank \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 K } + finrank \ud835\udd5c { x // x \u2208 K\u15ee } = finrank \ud835\udd5c { x // x \u2208 K } + n\n[PROOFSTEP]\nsimp [Submodule.finrank_add_finrank_orthogonal, h_dim]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nn : \u2115\n_i : Fact (finrank \ud835\udd5c E = n + 1)\nv : E\nhv : v \u2260 0\n\u22a2 finrank \ud835\udd5c { x // x \u2208 (Submodule.span \ud835\udd5c {v})\u15ee } = n\n[PROOFSTEP]\nhaveI : FiniteDimensional \ud835\udd5c E := fact_finiteDimensional_of_finrank_eq_succ n\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nn : \u2115\n_i : Fact (finrank \ud835\udd5c E = n + 1)\nv : E\nhv : v \u2260 0\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 (Submodule.span \ud835\udd5c {v})\u15ee } = n\n[PROOFSTEP]\nexact Submodule.finrank_add_finrank_orthogonal' <| by simp [finrank_span_singleton hv, _i.elim, add_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nn : \u2115\n_i : Fact (finrank \ud835\udd5c E = n + 1)\nv : E\nhv : v \u2260 0\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 finrank \ud835\udd5c { x // x \u2208 Submodule.span \ud835\udd5c {v} } + n = finrank \ud835\udd5c E\n[PROOFSTEP]\nsimp [finrank_span_singleton hv, _i.elim, add_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n\n\u22a2 \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\ninduction' n with n IH generalizing \u03c6\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\n\u22a2 \u2203 l, List.length l \u2264 Nat.zero \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nrefine' \u27e8[], rfl.le, show \u03c6 = 1 from _\u27e9\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\n\u22a2 \u03c6 = 1\n[PROOFSTEP]\nhave : ker (ContinuousLinearMap.id \u211d F - \u03c6) = \u22a4 := by\n  rwa [Nat.zero_eq, le_zero_iff, finrank_eq_zero, Submodule.orthogonal_eq_bot_iff] at hn \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\n\u22a2 ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) = \u22a4\n[PROOFSTEP]\nrwa [Nat.zero_eq, le_zero_iff, finrank_eq_zero, Submodule.orthogonal_eq_bot_iff] at hn \n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\nthis : ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) = \u22a4\n\u22a2 \u03c6 = 1\n[PROOFSTEP]\nsymm\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\nthis : ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) = \u22a4\n\u22a2 1 = \u03c6\n[PROOFSTEP]\next x\n[GOAL]\ncase zero.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\nthis : ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) = \u22a4\nx : F\n\u22a2 \u21911 x = \u2191\u03c6 x\n[PROOFSTEP]\nhave := LinearMap.congr_fun (LinearMap.ker_eq_top.mp this) x\n[GOAL]\ncase zero.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.zero\nthis\u271d : ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) = \u22a4\nx : F\nthis : \u2191\u2191(ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)) x = \u21910 x\n\u22a2 \u21911 x = \u2191\u03c6 x\n[PROOFSTEP]\nsimpa only [sub_eq_zero, ContinuousLinearMap.coe_sub, LinearMap.sub_apply, LinearMap.zero_apply] using this\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nlet W := ker (ContinuousLinearMap.id \u211d F - \u03c6)\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave hW : \u2200 w \u2208 W, \u03c6 w = w := fun w hw => (sub_eq_zero.mp hw).symm\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nby_cases hn' : finrank \u211d W\u15ee \u2264 n\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nobtain \u27e8V, hV\u2081, hV\u2082\u27e9 := IH \u03c6 hn'\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nV : List F\nhV\u2081 : List.length V \u2264 n\nhV\u2082 : \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) V)\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nexact\n  \u27e8V, hV\u2081.trans n.le_succ, hV\u2082\u27e9\n    -- Take a nonzero element `v` of the orthogonal complement of `W`.\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhaveI : Nontrivial W\u15ee := nontrivial_of_finrank_pos (by linarith [zero_le n] : 0 < finrank \u211d W\u15ee)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\n\u22a2 0 < finrank \u211d { x // x \u2208 W\u15ee }\n[PROOFSTEP]\nlinarith [zero_le n]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nobtain \u27e8v, hv\u27e9 := exists_ne (0 : W\u15ee)\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave h\u03c6v : \u03c6 v \u2208 W\u15ee := by\n  intro w hw\n  rw [\u2190 hW w hw, LinearIsometryEquiv.inner_map_map]\n  exact v.prop w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\n\u22a2 \u2191\u03c6 \u2191v \u2208 W\u15ee\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nw : F\nhw : w \u2208 W\n\u22a2 inner w (\u2191\u03c6 \u2191v) = 0\n[PROOFSTEP]\nrw [\u2190 hW w hw, LinearIsometryEquiv.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nw : F\nhw : w \u2208 W\n\u22a2 inner w \u2191v = 0\n[PROOFSTEP]\nexact v.prop w hw\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave hv' : (v : F) \u2209 W := by\n  intro h\n  exact\n    hv\n      ((Submodule.mem_left_iff_eq_zero_of_disjoint W.orthogonal_disjoint).mp h)\n        -- Let `\u03c1` be the reflection in `v - \u03c6 v`; this is designed to swap `v` and `\u03c6 v`\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\n\u22a2 \u00ac\u2191v \u2208 W\n[PROOFSTEP]\nintro h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nh : \u2191v \u2208 W\n\u22a2 False\n[PROOFSTEP]\nexact\n  hv\n    ((Submodule.mem_left_iff_eq_zero_of_disjoint W.orthogonal_disjoint).mp h)\n      -- Let `\u03c1` be the reflection in `v - \u03c6 v`; this is designed to swap `v` and `\u03c6 v`\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nlet x : F := v - \u03c6 v\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nlet \u03c1 :=\n  reflection\n    (\u211d \u2219 x)\u15ee\n      -- Notation: Let `V` be the fixed subspace of `\u03c6.trans \u03c1`\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nlet V := ker (ContinuousLinearMap.id \u211d F - \u03c6.trans \u03c1)\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave hV : \u2200 w, \u03c1 (\u03c6 w) = w \u2192 w \u2208 V := by\n  intro w hw\n  change w - \u03c1 (\u03c6 w) = 0\n  rw [sub_eq_zero, hw]\n    -- Everything fixed by `\u03c6` is fixed by `\u03c6.trans \u03c1`\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\n\u22a2 \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nw : F\nhw : \u2191\u03c1 (\u2191\u03c6 w) = w\n\u22a2 w \u2208 V\n[PROOFSTEP]\nchange w - \u03c1 (\u03c6 w) = 0\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nw : F\nhw : \u2191\u03c1 (\u2191\u03c6 w) = w\n\u22a2 w - \u2191\u03c1 (\u2191\u03c6 w) = 0\n[PROOFSTEP]\nrw [sub_eq_zero, hw]\n  -- Everything fixed by `\u03c6` is fixed by `\u03c6.trans \u03c1`\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave H\u2082V : W \u2264 V := by\n  intro w hw\n  apply hV\n  rw [hW w hw]\n  refine' reflection_mem_subspace_eq_self _\n  rw [Submodule.mem_orthogonal_singleton_iff_inner_left]\n  exact Submodule.sub_mem _ v.prop h\u03c6v _ hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\n\u22a2 W \u2264 V\n[PROOFSTEP]\nintro w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nw : F\nhw : w \u2208 W\n\u22a2 w \u2208 V\n[PROOFSTEP]\napply hV\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nw : F\nhw : w \u2208 W\n\u22a2 \u2191\u03c1 (\u2191\u03c6 w) = w\n[PROOFSTEP]\nrw [hW w hw]\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nw : F\nhw : w \u2208 W\n\u22a2 \u2191\u03c1 w = w\n[PROOFSTEP]\nrefine' reflection_mem_subspace_eq_self _\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nw : F\nhw : w \u2208 W\n\u22a2 w \u2208 (Submodule.span \u211d {x})\u15ee\n[PROOFSTEP]\nrw [Submodule.mem_orthogonal_singleton_iff_inner_left]\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nw : F\nhw : w \u2208 W\n\u22a2 inner w x = 0\n[PROOFSTEP]\nexact Submodule.sub_mem _ v.prop h\u03c6v _ hw\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave H\u2081V : (v : F) \u2208 V := by\n  apply hV\n  have : \u03c1 v = \u03c6 v := reflection_sub (\u03c6.norm_map v).symm\n  rw [\u2190 this]\n  exact\n    reflection_reflection _\n      _\n        -- By dimension-counting, the complement of the fixed subspace of `\u03c6.trans \u03c1` has dimension at\n            -- most `n`\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\n\u22a2 \u2191v \u2208 V\n[PROOFSTEP]\napply hV\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\n\u22a2 \u2191\u03c1 (\u2191\u03c6 \u2191v) = \u2191v\n[PROOFSTEP]\nhave : \u03c1 v = \u03c6 v := reflection_sub (\u03c6.norm_map v).symm\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nthis : \u2191\u03c1 \u2191v = \u2191\u03c6 \u2191v\n\u22a2 \u2191\u03c1 (\u2191\u03c6 \u2191v) = \u2191v\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nthis : \u2191\u03c1 \u2191v = \u2191\u03c6 \u2191v\n\u22a2 \u2191\u03c1 (\u2191\u03c1 \u2191v) = \u2191v\n[PROOFSTEP]\nexact\n  reflection_reflection _\n    _\n      -- By dimension-counting, the complement of the fixed subspace of `\u03c6.trans \u03c1` has dimension at\n          -- most `n`\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave : finrank \u211d V\u15ee \u2264 n := by\n  change finrank \u211d W\u15ee \u2264 n + 1 at hn \n  have : finrank \u211d W + 1 \u2264 finrank \u211d V :=\n    Submodule.finrank_lt_finrank_of_lt (SetLike.lt_iff_le_and_exists.2 \u27e8H\u2082V, v, H\u2081V, hv'\u27e9)\n  have : finrank \u211d V + finrank \u211d V\u15ee = finrank \u211d F := V.finrank_add_finrank_orthogonal\n  have : finrank \u211d W + finrank \u211d W\u15ee = finrank \u211d F := W.finrank_add_finrank_orthogonal\n  linarith\n    -- So apply the inductive hypothesis to `\u03c6.trans \u03c1`\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\n\u22a2 finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n[PROOFSTEP]\nchange finrank \u211d W\u15ee \u2264 n + 1 at hn \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nhn : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n + 1\n\u22a2 finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n[PROOFSTEP]\nhave : finrank \u211d W + 1 \u2264 finrank \u211d V :=\n  Submodule.finrank_lt_finrank_of_lt (SetLike.lt_iff_le_and_exists.2 \u27e8H\u2082V, v, H\u2081V, hv'\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nhn : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n + 1\nthis : finrank \u211d { x // x \u2208 W } + 1 \u2264 finrank \u211d { x // x \u2208 V }\n\u22a2 finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n[PROOFSTEP]\nhave : finrank \u211d V + finrank \u211d V\u15ee = finrank \u211d F := V.finrank_add_finrank_orthogonal\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d\u00b9 : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nhn : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n + 1\nthis\u271d : finrank \u211d { x // x \u2208 W } + 1 \u2264 finrank \u211d { x // x \u2208 V }\nthis : finrank \u211d { x // x \u2208 V } + finrank \u211d { x // x \u2208 V\u15ee } = finrank \u211d F\n\u22a2 finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n[PROOFSTEP]\nhave : finrank \u211d W + finrank \u211d W\u15ee = finrank \u211d F := W.finrank_add_finrank_orthogonal\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d\u00b2 : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nhn : finrank \u211d { x // x \u2208 W\u15ee } \u2264 n + 1\nthis\u271d\u00b9 : finrank \u211d { x // x \u2208 W } + 1 \u2264 finrank \u211d { x // x \u2208 V }\nthis\u271d : finrank \u211d { x // x \u2208 V } + finrank \u211d { x // x \u2208 V\u15ee } = finrank \u211d F\nthis : finrank \u211d { x // x \u2208 W } + finrank \u211d { x // x \u2208 W\u15ee } = finrank \u211d F\n\u22a2 finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n[PROOFSTEP]\nlinarith\n  -- So apply the inductive hypothesis to `\u03c6.trans \u03c1`\n[GOAL]\ncase neg.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nobtain \u27e8l, hl, h\u03c6l\u27e9 := IH (\u03c1 * \u03c6) this\n[GOAL]\ncase neg.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\nl : List F\nhl : List.length l \u2264 n\nh\u03c6l : \u03c1 * \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u22a2 \u2203 l, List.length l \u2264 Nat.succ n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nrefine' \u27e8x :: l, Nat.succ_le_succ hl, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\nl : List F\nhl : List.length l \u2264 n\nh\u03c6l : \u03c1 * \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u22a2 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) (x :: l))\n[PROOFSTEP]\nrw [List.map_cons, List.prod_cons]\n[GOAL]\ncase neg.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\nl : List F\nhl : List.length l \u2264 n\nh\u03c6l : \u03c1 * \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u22a2 \u03c6 = reflection (Submodule.span \u211d {x})\u15ee * List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nhave := congr_arg ((\u00b7 * \u00b7) \u03c1) h\u03c6l\n[GOAL]\ncase neg.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d\u00b9 : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis\u271d : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\nl : List F\nhl : List.length l \u2264 n\nh\u03c6l : \u03c1 * \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\nthis :\n  (fun x x_1 => x * x_1) \u03c1 (\u03c1 * \u03c6) =\n    (fun x x_1 => x * x_1) \u03c1 (List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l))\n\u22a2 \u03c6 = reflection (Submodule.span \u211d {x})\u15ee * List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\ndsimp only at this \n[GOAL]\ncase neg.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nn\u271d : \u2115\n\u03c6\u271d : F \u2243\u2097\u1d62[\u211d] F\nhn\u271d : finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6\u271d.toLinearEquiv)))\u15ee } \u2264 n\u271d\nn : \u2115\nIH :\n  \u2200 (\u03c6 : F \u2243\u2097\u1d62[\u211d] F),\n    finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 n \u2192\n      \u2203 l, List.length l \u2264 n \u2227 \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\nhn :\n  finrank \u211d { x // x \u2208 (ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv)))\u15ee } \u2264 Nat.succ n\nW : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk \u03c6.toLinearEquiv))\nhW : \u2200 (w : F), w \u2208 W \u2192 \u2191\u03c6 w = w\nhn' : \u00acfinrank \u211d { x // x \u2208 W\u15ee } \u2264 n\nthis\u271d\u00b9 : Nontrivial { x // x \u2208 W\u15ee }\nv : { x // x \u2208 W\u15ee }\nhv : v \u2260 0\nh\u03c6v : \u2191\u03c6 \u2191v \u2208 W\u15ee\nhv' : \u00ac\u2191v \u2208 W\nx : F := \u2191v - \u2191\u03c6 \u2191v\n\u03c1 : F \u2243\u2097\u1d62[\u211d] F := reflection (Submodule.span \u211d {x})\u15ee\nV : Submodule \u211d F := ker (ContinuousLinearMap.id \u211d F - \u2191(ContinuousLinearEquiv.mk (trans \u03c6 \u03c1).toLinearEquiv))\nhV : \u2200 (w : F), \u2191\u03c1 (\u2191\u03c6 w) = w \u2192 w \u2208 V\nH\u2082V : W \u2264 V\nH\u2081V : \u2191v \u2208 V\nthis\u271d : finrank \u211d { x // x \u2208 V\u15ee } \u2264 n\nl : List F\nhl : List.length l \u2264 n\nh\u03c6l : \u03c1 * \u03c6 = List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\nthis :\n  reflection (Submodule.span \u211d {\u2191v - \u2191\u03c6 \u2191v})\u15ee * (reflection (Submodule.span \u211d {\u2191v - \u2191\u03c6 \u2191v})\u15ee * \u03c6) =\n    reflection (Submodule.span \u211d {\u2191v - \u2191\u03c6 \u2191v})\u15ee * List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n\u22a2 \u03c6 = reflection (Submodule.span \u211d {x})\u15ee * List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l)\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, reflection_mul_reflection, one_mul] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\n\u22a2 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee) = \u22a4\n[PROOFSTEP]\nrw [Subgroup.eq_top_iff']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\n\u22a2 \u2200 (x : F \u2243\u2097\u1d62[\u211d] F), x \u2208 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\nintro \u03c6\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\n\u03c6 : F \u2243\u2097\u1d62[\u211d] F\n\u22a2 \u03c6 \u2208 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\nrcases \u03c6.reflections_generate_dim with \u27e8l, _, rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nl : List F\nleft\u271d : List.length l \u2264 finrank \u211d F\n\u22a2 List.prod (List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l) \u2208\n    Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\napply (Subgroup.closure _).list_prod_mem\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nl : List F\nleft\u271d : List.length l \u2264 finrank \u211d F\n\u22a2 \u2200 (x : F \u2243\u2097\u1d62[\u211d] F),\n    x \u2208 List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l \u2192\n      x \u2208 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nl : List F\nleft\u271d : List.length l \u2264 finrank \u211d F\nx : F \u2243\u2097\u1d62[\u211d] F\nhx : x \u2208 List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l\n\u22a2 x \u2208 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\nrcases List.mem_map.mp hx with \u27e8a, _, hax\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \u211d F\nl : List F\nleft\u271d\u00b9 : List.length l \u2264 finrank \u211d F\nx : F \u2243\u2097\u1d62[\u211d] F\nhx : x \u2208 List.map (fun v => reflection (Submodule.span \u211d {v})\u15ee) l\na : F\nleft\u271d : a \u2208 l\nhax : reflection (Submodule.span \u211d {a})\u15ee = x\n\u22a2 x \u2208 Subgroup.closure (Set.range fun v => reflection (Submodule.span \u211d {v})\u15ee)\n[PROOFSTEP]\nexact Subgroup.subset_closure \u27e8a, hax\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhc : IsComplete \u2191(iSup V)\n\u22a2 DirectSum.IsInternal V \u2194 (iSup V)\u15ee = \u22a5\n[PROOFSTEP]\nhaveI : CompleteSpace (\u21a5(iSup V)) := hc.completeSpace_coe\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhc : IsComplete \u2191(iSup V)\nthis : CompleteSpace { x // x \u2208 iSup V }\n\u22a2 DirectSum.IsInternal V \u2194 (iSup V)\u15ee = \u22a5\n[PROOFSTEP]\nsimp only [DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top, hV.independent, true_and_iff,\n  Submodule.orthogonal_eq_bot_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx : E\nhx : x \u2208 iSup V\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\ninduction hx using Submodule.iSup_induction' with\n|\n  hp i x hx =>\n  refine'\n    (Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => _).trans (orthogonalProjection_eq_self_iff.mpr hx)\n  rw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.coe_zero]\n  exact hV.isOrtho hij.symm hx\n| h0 => simp_rw [map_zero, Submodule.coe_zero, Finset.sum_const_zero]\n| hadd x y _ _ hx hy =>\n  simp_rw [map_add, Submodule.coe_add, Finset.sum_add_distrib]\n  exact congr_arg\u2082 (\u00b7 + \u00b7) hx hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx : E\nhx : x \u2208 iSup V\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\ninduction hx using Submodule.iSup_induction' with\n|\n  hp i x hx =>\n  refine'\n    (Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => _).trans (orthogonalProjection_eq_self_iff.mpr hx)\n  rw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.coe_zero]\n  exact hV.isOrtho hij.symm hx\n| h0 => simp_rw [map_zero, Submodule.coe_zero, Finset.sum_const_zero]\n| hadd x y _ _ hx hy =>\n  simp_rw [map_add, Submodule.coe_add, Finset.sum_add_distrib]\n  exact congr_arg\u2082 (\u00b7 + \u00b7) hx hy\n[GOAL]\ncase hp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d : E\ni : \u03b9\nx : E\nhx : x \u2208 V i\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\n\n|\n  hp i x hx =>\n  refine'\n    (Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => _).trans (orthogonalProjection_eq_self_iff.mpr hx)\n  rw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.coe_zero]\n  exact hV.isOrtho hij.symm hx\n[GOAL]\ncase hp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d : E\ni : \u03b9\nx : E\nhx : x \u2208 V i\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\nrefine'\n  (Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => _).trans (orthogonalProjection_eq_self_iff.mpr hx)\n[GOAL]\ncase hp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d\u00b9 : E\ni : \u03b9\nx : E\nhx : x \u2208 V i\nj : \u03b9\nx\u271d : j \u2208 Finset.univ\nhij : j \u2260 i\n\u22a2 \u2191(\u2191(orthogonalProjection (V j)) x) = 0\n[PROOFSTEP]\nrw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.coe_zero]\n[GOAL]\ncase hp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d\u00b9 : E\ni : \u03b9\nx : E\nhx : x \u2208 V i\nj : \u03b9\nx\u271d : j \u2208 Finset.univ\nhij : j \u2260 i\n\u22a2 x \u2208 (V j)\u15ee\n[PROOFSTEP]\nexact hV.isOrtho hij.symm hx\n[GOAL]\ncase h0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx : E\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) 0) = 0\n[PROOFSTEP]\n\n| h0 => simp_rw [map_zero, Submodule.coe_zero, Finset.sum_const_zero]\n[GOAL]\ncase h0\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx : E\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) 0) = 0\n[PROOFSTEP]\nsimp_rw [map_zero, Submodule.coe_zero, Finset.sum_const_zero]\n[GOAL]\ncase hadd\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d x y : E\nhx\u271d : x \u2208 \u2a06 (i : \u03b9), V i\nhy\u271d : y \u2208 \u2a06 (i : \u03b9), V i\nhx : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\nhy : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) y) = y\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) (x + y)) = x + y\n[PROOFSTEP]\n\n| hadd x y _ _ hx hy =>\n  simp_rw [map_add, Submodule.coe_add, Finset.sum_add_distrib]\n  exact congr_arg\u2082 (\u00b7 + \u00b7) hx hy\n[GOAL]\ncase hadd\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d x y : E\nhx\u271d : x \u2208 \u2a06 (i : \u03b9), V i\nhy\u271d : y \u2208 \u2a06 (i : \u03b9), V i\nhx : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\nhy : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) y) = y\n\u22a2 \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) (x + y)) = x + y\n[PROOFSTEP]\nsimp_rw [map_add, Submodule.coe_add, Finset.sum_add_distrib]\n[GOAL]\ncase hadd\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx\u271d x y : E\nhx\u271d : x \u2208 \u2a06 (i : \u03b9), V i\nhy\u271d : y \u2208 \u2a06 (i : \u03b9), V i\nhx : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) x) = x\nhy : \u2211 i : \u03b9, \u2191(\u2191(orthogonalProjection (V i)) y) = y\n\u22a2 \u2211 x_1 : \u03b9, \u2191(\u2191(orthogonalProjection (V x_1)) x) + \u2211 x : \u03b9, \u2191(\u2191(orthogonalProjection (V x)) y) = x + y\n[PROOFSTEP]\nexact congr_arg\u2082 (\u00b7 + \u00b7) hx hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nx : \u2a01 (i : \u03b9), { x // x \u2208 V i }\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\n\u22a2 \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) x) = \u2191x i\n[PROOFSTEP]\ninduction' x using DirectSum.induction_on with j x x y hx hy\n[GOAL]\ncase H_zero\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\n\u22a2 \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) 0) = \u21910 i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H_basic\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nj : \u03b9\nx : { x // x \u2208 V j }\n\u22a2 \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) (\u2191(of (fun i => { x // x \u2208 V i }) j) x)) =\n    \u2191(\u2191(of (fun i => { x // x \u2208 V i }) j) x) i\n[PROOFSTEP]\nsimp_rw [DirectSum.coeAddMonoidHom_of, DirectSum.of]\n  -- porting note: was in the previous `simp_rw`, no longer works\n[GOAL]\ncase H_basic\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nj : \u03b9\nx : { x // x \u2208 V j }\n\u22a2 \u2191(orthogonalProjection (V i)) \u2191x = \u2191(\u2191(DFinsupp.singleAddHom (fun i => { x // x \u2208 V i }) j) x) i\n[PROOFSTEP]\nrw [DFinsupp.singleAddHom_apply]\n[GOAL]\ncase H_basic\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nj : \u03b9\nx : { x // x \u2208 V j }\n\u22a2 \u2191(orthogonalProjection (V i)) \u2191x = \u2191(DFinsupp.single j x) i\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase H_basic.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nx : { x // x \u2208 V i }\n\u22a2 \u2191(orthogonalProjection (V i)) \u2191x = \u2191(DFinsupp.single i x) i\n[PROOFSTEP]\nrw [orthogonalProjection_mem_subspace_eq_self, DFinsupp.single_eq_same]\n[GOAL]\ncase H_basic.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nj : \u03b9\nx : { x // x \u2208 V j }\nhij : i \u2260 j\n\u22a2 \u2191(orthogonalProjection (V i)) \u2191x = \u2191(DFinsupp.single j x) i\n[PROOFSTEP]\nrw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, DFinsupp.single_eq_of_ne hij.symm]\n[GOAL]\ncase H_basic.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nj : \u03b9\nx : { x // x \u2208 V j }\nhij : i \u2260 j\n\u22a2 \u2191x \u2208 (V i)\u15ee\n[PROOFSTEP]\nexact hV.isOrtho hij.symm x.prop\n[GOAL]\ncase H_plus\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nx y : \u2a01 (i : \u03b9), { x // x \u2208 V i }\nhx : \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) x) = \u2191x i\nhy : \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) y) = \u2191y i\n\u22a2 \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) (x + y)) = \u2191(x + y) i\n[PROOFSTEP]\nsimp_rw [map_add]\n[GOAL]\ncase H_plus\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\ni : \u03b9\ninst\u271d : CompleteSpace { x // x \u2208 V i }\nx y : \u2a01 (i : \u03b9), { x // x \u2208 V i }\nhx : \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) x) = \u2191x i\nhy : \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) y) = \u2191y i\n\u22a2 \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) x) +\n      \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) y) =\n    \u2191(x + y) i\n[PROOFSTEP]\nexact congr_arg\u2082 (\u00b7 + \u00b7) hx hy\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\n\u22a2 \u2191(DirectSum.coeAddMonoidHom V)\n      ((fun x => \u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) x) x) =\n    x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\n\u22a2 \u2191(DirectSum.coeAddMonoidHom V) (\u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\nletI := fun i => Classical.decEq (V i)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\nthis : (i : \u03b9) \u2192 DecidableEq { x // x \u2208 V i } := fun i => Classical.decEq { x // x \u2208 V i }\n\u22a2 \u2191(DirectSum.coeAddMonoidHom V) (\u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) x) = x\n[PROOFSTEP]\nrw [DirectSum.coeAddMonoidHom, DirectSum.toAddMonoid, DFinsupp.liftAddHom_apply, DFinsupp.sumAddHom_apply,\n  DFinsupp.sum_eq_sum_fintype]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\nthis : (i : \u03b9) \u2192 DecidableEq { x // x \u2208 V i } := fun i => Classical.decEq { x // x \u2208 V i }\n\u22a2 \u2211 i : \u03b9,\n      \u2191(AddSubmonoidClass.subtype (V i))\n        (\u2191DFinsupp.equivFunOnFintype (\u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) x) i) =\n    x\n[PROOFSTEP]\nsimp_rw [Equiv.apply_symm_apply, AddSubmonoidClass.coe_subtype]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\nthis : (i : \u03b9) \u2192 DecidableEq { x // x \u2208 V i } := fun i => Classical.decEq { x // x \u2208 V i }\n\u22a2 \u2211 x_1 : \u03b9, \u2191(\u2191(orthogonalProjection (V x_1)) x) = x\n[PROOFSTEP]\nexact hV.sum_projection_of_mem_iSup _ ((h.ge : _) Submodule.mem_top)\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\nthis : (i : \u03b9) \u2192 DecidableEq { x // x \u2208 V i } := fun i => Classical.decEq { x // x \u2208 V i }\n\u22a2 \u2200 (i : \u03b9), \u2191(AddSubmonoidClass.subtype (V i)) 0 = 0\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : E\nthis : (i : \u03b9) \u2192 DecidableEq { x // x \u2208 V i } := fun i => Classical.decEq { x // x \u2208 V i }\ni : \u03b9\n\u22a2 \u2191(AddSubmonoidClass.subtype (V i)) 0 = 0\n[PROOFSTEP]\nexact map_zero _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : \u2a01 (i : \u03b9), { x // x \u2208 V i }\n\u22a2 (fun x => \u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) x)\n      (\u2191(DirectSum.coeAddMonoidHom V) x) =\n    x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\n\u03b9 : Type u_4\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 V i }\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nh : iSup V = \u22a4\nx : \u2a01 (i : \u03b9), { x // x \u2208 V i }\n\u22a2 (\u2191DFinsupp.equivFunOnFintype.symm fun i => \u2191(orthogonalProjection (V i)) (\u2191(DirectSum.coeAddMonoidHom V) x)) = x\n[PROOFSTEP]\nsimp_rw [hV.projection_directSum_coeAddHom, DFinsupp.equivFunOnFintype_symm_coe]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v) \u2194 (span \ud835\udd5c v)\u15ee = \u22a5\n[PROOFSTEP]\nrw [Submodule.eq_bot_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v) \u2194 \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v) \u2192 \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\n[PROOFSTEP]\ncontrapose!\n  -- ** direction 1: nonempty orthogonal complement implies nonmaximal\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2203 x, x \u2208 (span \ud835\udd5c v)\u15ee \u2227 x \u2260 0) \u2192 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nrintro\n  \u27e8x, hx', hx\u27e9\n      -- take a nonzero vector and normalize it\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\n\u22a2 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nlet e := (\u2016x\u2016\u207b\u00b9 : \ud835\udd5c) \u2022 x\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\n\u22a2 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nhave he : \u2016e\u2016 = 1 := by simp [norm_smul_inv_norm hx]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\n\u22a2 \u2016e\u2016 = 1\n[PROOFSTEP]\nsimp [norm_smul_inv_norm hx]\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\n\u22a2 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nhave he' : e \u2208 (span \ud835\udd5c v)\u15ee := smul_mem' _ _ hx'\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\n\u22a2 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nhave he'' : e \u2209 v := by\n  intro hev\n  have : e = 0 := by\n    have : e \u2208 span \ud835\udd5c v \u2293 (span \ud835\udd5c v)\u15ee := \u27e8subset_span hev, he'\u27e9\n    simpa [(span \ud835\udd5c v).inf_orthogonal_eq_bot] using this\n  have : e \u2260 0 := hv.ne_zero \u27e8e, hev\u27e9\n  contradiction\n    -- put this together with `v` to provide a candidate orthonormal basis for the whole space\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\n\u22a2 \u00ace \u2208 v\n[PROOFSTEP]\nintro hev\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhev : e \u2208 v\n\u22a2 False\n[PROOFSTEP]\nhave : e = 0 := by\n  have : e \u2208 span \ud835\udd5c v \u2293 (span \ud835\udd5c v)\u15ee := \u27e8subset_span hev, he'\u27e9\n  simpa [(span \ud835\udd5c v).inf_orthogonal_eq_bot] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhev : e \u2208 v\n\u22a2 e = 0\n[PROOFSTEP]\nhave : e \u2208 span \ud835\udd5c v \u2293 (span \ud835\udd5c v)\u15ee := \u27e8subset_span hev, he'\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhev : e \u2208 v\nthis : e \u2208 span \ud835\udd5c v \u2293 (span \ud835\udd5c v)\u15ee\n\u22a2 e = 0\n[PROOFSTEP]\nsimpa [(span \ud835\udd5c v).inf_orthogonal_eq_bot] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhev : e \u2208 v\nthis : e = 0\n\u22a2 False\n[PROOFSTEP]\nhave : e \u2260 0 := hv.ne_zero \u27e8e, hev\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhev : e \u2208 v\nthis\u271d : e = 0\nthis : e \u2260 0\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n  -- put this together with `v` to provide a candidate orthonormal basis for the whole space\n[GOAL]\ncase mp.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\n\u22a2 \u2203 u, u \u2287 v \u2227 Orthonormal \ud835\udd5c Subtype.val \u2227 u \u2260 v\n[PROOFSTEP]\nrefine' \u27e8insert e v, v.subset_insert e, \u27e8_, _\u27e9, (ne_insert_of_not_mem v he'').symm\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\n\u22a2 \u2200 (i : { x // x \u2208 insert e v }), \u2016\u2191i\u2016 = 1\n[PROOFSTEP]\nrintro \u27e8a, ha'\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_1.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\na : E\nha' : a \u2208 insert e v\n\u22a2 \u2016\u2191{ val := a, property := ha' }\u2016 = 1\n[PROOFSTEP]\ncases' eq_or_mem_of_mem_insert ha' with ha ha\n[GOAL]\ncase mp.intro.intro.refine'_1.mk.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\na : E\nha' : a \u2208 insert e v\nha : a = e\n\u22a2 \u2016\u2191{ val := a, property := ha' }\u2016 = 1\n[PROOFSTEP]\nsimp [ha, he]\n[GOAL]\ncase mp.intro.intro.refine'_1.mk.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\n\u22a2 \u2016\u2191{ val := a, property := ha' }\u2016 = 1\n[PROOFSTEP]\nexact hv.1 \u27e8a, ha\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\n\u22a2 \u2200 {i j : { x // x \u2208 insert e v }}, i \u2260 j \u2192 inner \u2191i \u2191j = 0\n[PROOFSTEP]\nhave h_end : \u2200 a \u2208 v, \u27eaa, e\u27eb = 0 := by\n  intro a ha\n  exact he' a (Submodule.subset_span ha)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\n\u22a2 \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\na : E\nha : a \u2208 v\n\u22a2 inner a e = 0\n[PROOFSTEP]\nexact he' a (Submodule.subset_span ha)\n[GOAL]\ncase mp.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\n\u22a2 \u2200 {i j : { x // x \u2208 insert e v }}, i \u2260 j \u2192 inner \u2191i \u2191j = 0\n[PROOFSTEP]\nrintro \u27e8a, ha'\u27e9\n[GOAL]\ncase mp.intro.intro.refine'_2.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\n\u22a2 \u2200 {j : { x // x \u2208 insert e v }}, { val := a, property := ha' } \u2260 j \u2192 inner \u2191{ val := a, property := ha' } \u2191j = 0\n[PROOFSTEP]\ncases' eq_or_mem_of_mem_insert ha' with ha ha\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\n\u22a2 \u2200 {j : { x // x \u2208 insert e v }}, { val := a, property := ha' } \u2260 j \u2192 inner \u2191{ val := a, property := ha' } \u2191j = 0\n[PROOFSTEP]\nrintro \u27e8b, hb'\u27e9 hab'\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inl.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\nhave hb : b \u2208 v := by\n  refine' mem_of_mem_insert_of_ne hb' _\n  intro hbe'\n  apply hab'\n  simp [ha, hbe']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\n\u22a2 b \u2208 v\n[PROOFSTEP]\nrefine' mem_of_mem_insert_of_ne hb' _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\n\u22a2 b \u2260 e\n[PROOFSTEP]\nintro hbe'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhbe' : b = e\n\u22a2 False\n[PROOFSTEP]\napply hab'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhbe' : b = e\n\u22a2 { val := a, property := ha' } = { val := b, property := hb' }\n[PROOFSTEP]\nsimp [ha, hbe']\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inl.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\nrw [inner_eq_zero_symm]\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inl.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a = e\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\n\u22a2 inner \u2191{ val := b, property := hb' } \u2191{ val := a, property := ha' } = 0\n[PROOFSTEP]\nsimpa [ha] using h_end b hb\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\n\u22a2 \u2200 {j : { x // x \u2208 insert e v }}, { val := a, property := ha' } \u2260 j \u2192 inner \u2191{ val := a, property := ha' } \u2191j = 0\n[PROOFSTEP]\nrintro \u27e8b, hb'\u27e9 hab'\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inr.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\ncases' eq_or_mem_of_mem_insert hb' with hb hb\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inr.mk.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b = e\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\nsimpa [hb] using h_end a ha\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inr.mk.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\nhave : (\u27e8a, ha\u27e9 : v) \u2260 \u27e8b, hb\u27e9 := by\n  intro hab''\n  apply hab'\n  simpa using hab''\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\n\u22a2 { val := a, property := ha } \u2260 { val := b, property := hb }\n[PROOFSTEP]\nintro hab''\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\nhab'' : { val := a, property := ha } = { val := b, property := hb }\n\u22a2 False\n[PROOFSTEP]\napply hab'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\nhab'' : { val := a, property := ha } = { val := b, property := hb }\n\u22a2 { val := a, property := ha' } = { val := b, property := hb' }\n[PROOFSTEP]\nsimpa using hab''\n[GOAL]\ncase mp.intro.intro.refine'_2.mk.inr.mk.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhx' : x \u2208 (span \ud835\udd5c v)\u15ee\nhx : x \u2260 0\ne : E := (\u2191\u2016x\u2016)\u207b\u00b9 \u2022 x\nhe : \u2016e\u2016 = 1\nhe' : e \u2208 (span \ud835\udd5c v)\u15ee\nhe'' : \u00ace \u2208 v\nh_end : \u2200 (a : E), a \u2208 v \u2192 inner a e = 0\na : E\nha' : a \u2208 insert e v\nha : a \u2208 v\nb : E\nhb' : b \u2208 insert e v\nhab' : { val := a, property := ha' } \u2260 { val := b, property := hb' }\nhb : b \u2208 v\nthis : { val := a, property := ha } \u2260 { val := b, property := hb }\n\u22a2 inner \u2191{ val := a, property := ha' } \u2191{ val := b, property := hb' } = 0\n[PROOFSTEP]\nexact hv.2 this\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0) \u2192 \u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v\n[PROOFSTEP]\nsimp only [Subset.antisymm_iff]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0) \u2192 \u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u \u2286 v \u2227 v \u2286 u\n[PROOFSTEP]\nrintro h u (huv : v \u2286 u) hu\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 u \u2286 v \u2227 v \u2286 u\n[PROOFSTEP]\nrefine' \u27e8_, huv\u27e9\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 u \u2286 v\n[PROOFSTEP]\nintro x hxu\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\n\u22a2 x \u2208 v\n[PROOFSTEP]\nrefine' ((mt (h x)) (hu.ne_zero \u27e8x, hxu\u27e9)).imp_symm _\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\n\u22a2 \u00acx \u2208 v \u2192 x \u2208 (span \ud835\udd5c v)\u15ee\n[PROOFSTEP]\nintro hxv y hy\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\ny : E\nhy : y \u2208 span \ud835\udd5c v\n\u22a2 inner y x = 0\n[PROOFSTEP]\nhave hxv' : (\u27e8x, hxu\u27e9 : u) \u2209 ((\u2191) \u207b\u00b9' v : Set u) := by simp [huv, hxv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\ny : E\nhy : y \u2208 span \ud835\udd5c v\n\u22a2 \u00ac{ val := x, property := hxu } \u2208 Subtype.val \u207b\u00b9' v\n[PROOFSTEP]\nsimp [huv, hxv]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\ny : E\nhy : y \u2208 span \ud835\udd5c v\nhxv' : \u00ac{ val := x, property := hxu } \u2208 Subtype.val \u207b\u00b9' v\n\u22a2 inner y x = 0\n[PROOFSTEP]\nobtain \u27e8l, hl, rfl\u27e9 : \u2203 l \u2208 Finsupp.supported \ud835\udd5c \ud835\udd5c ((\u2191) \u207b\u00b9' v : Set u), (Finsupp.total (\u21a5u) E \ud835\udd5c (\u2191)) l = y :=\n  by\n  rw [\u2190 Finsupp.mem_span_image_iff_total]\n  simp [huv, inter_eq_self_of_subset_left, hy]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\ny : E\nhy : y \u2208 span \ud835\udd5c v\nhxv' : \u00ac{ val := x, property := hxu } \u2208 Subtype.val \u207b\u00b9' v\n\u22a2 \u2203 l, l \u2208 Finsupp.supported \ud835\udd5c \ud835\udd5c (Subtype.val \u207b\u00b9' v) \u2227 \u2191(Finsupp.total (\u2191u) E \ud835\udd5c Subtype.val) l = y\n[PROOFSTEP]\nrw [\u2190 Finsupp.mem_span_image_iff_total]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\ny : E\nhy : y \u2208 span \ud835\udd5c v\nhxv' : \u00ac{ val := x, property := hxu } \u2208 Subtype.val \u207b\u00b9' v\n\u22a2 y \u2208 span \ud835\udd5c (Subtype.val '' (Subtype.val \u207b\u00b9' v))\n[PROOFSTEP]\nsimp [huv, inter_eq_self_of_subset_left, hy]\n[GOAL]\ncase mpr.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\nhv : Orthonormal \ud835\udd5c Subtype.val\nh : \u2200 (x : E), x \u2208 (span \ud835\udd5c v)\u15ee \u2192 x = 0\nu : Set E\nhuv : v \u2286 u\nhu : Orthonormal \ud835\udd5c Subtype.val\nx : E\nhxu : x \u2208 u\nhxv : \u00acx \u2208 v\nhxv' : \u00ac{ val := x, property := hxu } \u2208 Subtype.val \u207b\u00b9' v\nl : \u2191u \u2192\u2080 \ud835\udd5c\nhl : l \u2208 Finsupp.supported \ud835\udd5c \ud835\udd5c (Subtype.val \u207b\u00b9' v)\nhy : \u2191(Finsupp.total (\u2191u) E \ud835\udd5c Subtype.val) l \u2208 span \ud835\udd5c v\n\u22a2 inner (\u2191(Finsupp.total (\u2191u) E \ud835\udd5c Subtype.val) l) x = 0\n[PROOFSTEP]\nexact hu.inner_finsupp_eq_zero hxv' hl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (\u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v) \u2194 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nhaveI := proper_isROrC \ud835\udd5c (span \ud835\udd5c v)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\n\u22a2 (\u2200 (u : Set E), u \u2287 v \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = v) \u2194 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nrw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\n\u22a2 (span \ud835\udd5c v)\u15ee = \u22a5 \u2194 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nrw [Submodule.orthogonal_eq_bot_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\n\u22a2 span \ud835\udd5c v = \u22a4 \u2194 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nhave hv_coe : range ((\u2191) : v \u2192 E) = v := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\n\u22a2 Set.range Subtype.val = v\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\nhv_coe : Set.range Subtype.val = v\n\u22a2 span \ud835\udd5c v = \u22a4 \u2194 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\nhv_coe : Set.range Subtype.val = v\n\u22a2 span \ud835\udd5c v = \u22a4 \u2192 \u2203 b, \u2191b = Subtype.val\n[PROOFSTEP]\nrefine' fun h => \u27e8Basis.mk hv.linearIndependent _, Basis.coe_mk _ _\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\nhv_coe : Set.range Subtype.val = v\nh : span \ud835\udd5c v = \u22a4\n\u22a2 \u22a4 \u2264 span \ud835\udd5c (Set.range Subtype.val)\n[PROOFSTEP]\nconvert h.ge\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\nhv_coe : Set.range Subtype.val = v\n\u22a2 (\u2203 b, \u2191b = Subtype.val) \u2192 span \ud835\udd5c v = \u22a4\n[PROOFSTEP]\nrintro \u27e8h, coe_h\u27e9\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : InnerProductSpace \u211d F\nK : Submodule \ud835\udd5c E\nv : Set E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nthis : ProperSpace { x // x \u2208 span \ud835\udd5c v }\nhv_coe : Set.range Subtype.val = v\nh : Basis (\u2191v) \ud835\udd5c E\ncoe_h : \u2191h = Subtype.val\n\u22a2 span \ud835\udd5c v = \u22a4\n[PROOFSTEP]\nrw [\u2190 h.span_eq, coe_h, hv_coe]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Projection", "llama_tokens": 229258, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339556397749, "lm_q2_score": 0.6442250928250376, "lm_q1q2_score": 0.54097768552037}}
{"text": "[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : q = 0\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nhave hpq : p * q \u2260 0 := mul_ne_zero hp hq\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : p * q \u2260 0\n\u22a2 (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n      (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q\n[PROOFSTEP]\nsimp only [hpq, hp, hq, eq_self_iff_true, if_true, if_false, Polynomial.natDegree_mul hp hq, pow_add]\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\n\u22a2 0 \u2264\n    MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      p\n[PROOFSTEP]\ndsimp\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\n\u22a2 0 \u2264 if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\nh\u271d : p = 0\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\nh\u271d : \u00acp = 0\n\u22a2 0 \u2264 \u2191(Fintype.card Fq) ^ natDegree p\n[PROOFSTEP]\nexact pow_nonneg (Int.ofNat_zero_le _) _\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\nh : \u00acp = 0 \u2192 \u2191(Fintype.card Fq) ^ natDegree p = 0\n\u22a2 p = 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np : Fq[X]\nh : p \u2260 0\n\u22a2 p \u2260 0 \u2227 \u2191(Fintype.card Fq) ^ natDegree p \u2260 0\n[PROOFSTEP]\nexact \u27e8h, (pow_pos _).ne'\u27e9\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nby_cases hp : p = 0\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : q = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nby_cases hpq : p + q = 0\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : p + q = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nsimp only [hpq, hp, hq, eq_self_iff_true, if_true, if_false]\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : p + q = 0\n\u22a2 0 \u2264 \u2191(Fintype.card Fq) ^ natDegree p + \u2191(Fintype.card Fq) ^ natDegree q\n[PROOFSTEP]\nexact add_nonneg (pow_pos _).le (pow_pos _).le\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 MulHom.toFun\n      { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n        map_mul' :=\n          (_ :\n            \u2200 (p q : Fq[X]),\n              (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n      (p + q) \u2264\n    MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        p +\n      MulHom.toFun\n        { toFun := fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p,\n          map_mul' :=\n            (_ :\n              \u2200 (p q : Fq[X]),\n                (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) (p * q) =\n                  (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) p *\n                    (fun p => if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) q) }\n        q\n[PROOFSTEP]\nsimp only [hpq, hp, hq, if_false]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 \u2191(Fintype.card Fq) ^ natDegree (p + q) \u2264 \u2191(Fintype.card Fq) ^ natDegree p + \u2191(Fintype.card Fq) ^ natDegree q\n[PROOFSTEP]\nrefine' le_trans (pow_le_pow (by linarith) (Polynomial.natDegree_add_le _ _)) _\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 1 \u2264 \u2191(Fintype.card Fq)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 \u2191(Fintype.card Fq) ^ max (natDegree p) (natDegree q) \u2264\n    \u2191(Fintype.card Fq) ^ natDegree p + \u2191(Fintype.card Fq) ^ natDegree q\n[PROOFSTEP]\nrefine' le_trans (le_max_iff.mpr _) (max_le_add_of_nonneg (pow_nonneg (by linarith) _) (pow_nonneg (by linarith) _))\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 0 \u2264 \u2191(Fintype.card Fq)\n[PROOFSTEP]\nlinarith\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 0 \u2264 \u2191(Fintype.card Fq)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\n\u22a2 \u2191(Fintype.card Fq) ^ max (natDegree p) (natDegree q) \u2264 \u2191(Fintype.card Fq) ^ natDegree p \u2228\n    \u2191(Fintype.card Fq) ^ max (natDegree p) (natDegree q) \u2264 \u2191(Fintype.card Fq) ^ natDegree q\n[PROOFSTEP]\nexact (max_choice p.natDegree q.natDegree).imp (fun h => by rw [h]) fun h => by rw [h]\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\nh : max (natDegree p) (natDegree q) = natDegree p\n\u22a2 \u2191(Fintype.card Fq) ^ max (natDegree p) (natDegree q) \u2264 \u2191(Fintype.card Fq) ^ natDegree p\n[PROOFSTEP]\nrw [h]\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\nhpq : \u00acp + q = 0\nh : max (natDegree p) (natDegree q) = natDegree q\n\u22a2 \u2191(Fintype.card Fq) ^ max (natDegree p) (natDegree q) \u2264 \u2191(Fintype.card Fq) ^ natDegree q\n[PROOFSTEP]\nrw [h]\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\n\u22a2 \u2191cardPowDegree p < \u2191cardPowDegree q \u2194 EuclideanDomain.r p q\n[PROOFSTEP]\nshow cardPowDegree p < cardPowDegree q \u2194 degree p < degree q\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\n\u22a2 \u2191cardPowDegree p < \u2191cardPowDegree q \u2194 degree p < degree q\n[PROOFSTEP]\nsimp only [cardPowDegree_apply]\n[GOAL]\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\n\u22a2 ((if p = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree p) < if q = 0 then 0 else \u2191(Fintype.card Fq) ^ natDegree q) \u2194\n    degree p < degree q\n[PROOFSTEP]\nsplit_ifs with hp hq hq\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : q = 0\n\u22a2 0 < 0 \u2194 degree p < degree q\n[PROOFSTEP]\nsimp only [hp, hq, lt_self_iff_false]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : p = 0\nhq : \u00acq = 0\n\u22a2 0 < \u2191(Fintype.card Fq) ^ natDegree q \u2194 degree p < degree q\n[PROOFSTEP]\nsimp only [hp, hq, degree_zero, Ne.def, bot_lt_iff_ne_bot, degree_eq_bot, pow_pos, not_false_iff]\n[GOAL]\ncase pos\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : q = 0\n\u22a2 \u2191(Fintype.card Fq) ^ natDegree p < 0 \u2194 degree p < degree q\n[PROOFSTEP]\nsimp only [hp, hq, degree_zero, not_lt_bot, (pow_pos _).not_lt]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\n\u22a2 \u2191(Fintype.card Fq) ^ natDegree p < \u2191(Fintype.card Fq) ^ natDegree q \u2194 degree p < degree q\n[PROOFSTEP]\nrw [degree_eq_natDegree hp, degree_eq_natDegree hq, Nat.cast_withBot, Nat.cast_withBot, WithBot.coe_lt_coe,\n  pow_lt_pow_iff]\n[GOAL]\ncase neg\nFq : Type u_1\ninst\u271d\u00b9 : Field Fq\ninst\u271d : Fintype Fq\ncard_pos : 0 < Fintype.card Fq\npow_pos : \u2200 (n : \u2115), 0 < \u2191(Fintype.card Fq) ^ n\np q : Fq[X]\nhp : \u00acp = 0\nhq : \u00acq = 0\n\u22a2 1 < \u2191(Fintype.card Fq)\n[PROOFSTEP]\nexact_mod_cast @Fintype.one_lt_card Fq _ _\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Degree.CardPowDegree", "llama_tokens": 10195, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673178375735, "lm_q2_score": 0.6654105454764747, "lm_q1q2_score": 0.5408904853622987}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Semigroup (f i)\n\u22a2 \u2200 (a b c : (i : I) \u2192 f i), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Semigroup (f i)\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Semigroup (f i)\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d * b\u271d * c\u271d) x\u271d = (a\u271d * (b\u271d * c\u271d)) x\u271d\n[PROOFSTEP]\nexact mul_assoc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 CommSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\n\u22a2 \u2200 (a b : (i : I) \u2192 f i), a * b = b * a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 CommSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d * b\u271d = b\u271d * a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 CommSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d * b\u271d) x\u271d = (b\u271d * a\u271d) x\u271d\n[PROOFSTEP]\nexact mul_comm _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\n\u22a2 \u2200 (a : (i : I) \u2192 f i), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\na\u271d : (i : I) \u2192 f i\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (1 * a\u271d) x\u271d = a\u271d x\u271d\n[PROOFSTEP]\nexact one_mul _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\n\u22a2 \u2200 (a : (i : I) \u2192 f i), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\na\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d * 1) x\u271d = a\u271d x\u271d\n[PROOFSTEP]\nexact mul_one _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvOneClass (f i)\n\u22a2 1\u207b\u00b9 = 1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvOneClass (f i)\n\u22a2 1\u207b\u00b9 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvOneClass (f i)\nx\u271d : I\n\u22a2 1\u207b\u00b9 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact inv_one\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\n\u22a2 \u2200 (x : (i : I) \u2192 f i), (fun n x i => x i ^ n) 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\nx\u271d : (i : I) \u2192 f i\n\u22a2 (fun n x i => x i ^ n) 0 x\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\nx\u271d\u00b9 : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (fun n x i => x i ^ n) 0 x\u271d\u00b9 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact Monoid.npow_zero _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\n\u22a2 \u2200 (n : \u2115) (x : (i : I) \u2192 f i), (fun n x i => x i ^ n) (n + 1) x = x * (fun n x i => x i ^ n) n x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\nn\u271d : \u2115\nx\u271d : (i : I) \u2192 f i\n\u22a2 (fun n x i => x i ^ n) (n\u271d + 1) x\u271d = x\u271d * (fun n x i => x i ^ n) n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Monoid (f i)\nsrc\u271d\u00b9 : Semigroup ((i : I) \u2192 f i) := semigroup\nsrc\u271d : MulOneClass ((i : I) \u2192 f i) := mulOneClass\nn\u271d : \u2115\nx\u271d\u00b9 : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (fun n x i => x i ^ n) (n\u271d + 1) x\u271d\u00b9 x\u271d = (x\u271d\u00b9 * (fun n x i => x i ^ n) n\u271d x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nexact Monoid.npow_succ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\n\u22a2 \u2200 (a b : (i : I) \u2192 f i), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\na\u271d b\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d / b\u271d = a\u271d * b\u271d\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\na\u271d b\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d / b\u271d) x\u271d = (a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\nexact div_eq_mul_inv _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\n\u22a2 \u2200 (a : (i : I) \u2192 f i), (fun z x i => x i ^ z) 0 a = 1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\na\u271d : (i : I) \u2192 f i\n\u22a2 (fun z x i => x i ^ z) 0 a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (fun z x i => x i ^ z) 0 a\u271d x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact DivInvMonoid.zpow_zero' _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\n\u22a2 \u2200 (n : \u2115) (a : (i : I) \u2192 f i),\n    (fun z x i => x i ^ z) (Int.ofNat (Nat.succ n)) a = a * (fun z x i => x i ^ z) (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\nn\u271d : \u2115\na\u271d : (i : I) \u2192 f i\n\u22a2 (fun z x i => x i ^ z) (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d * (fun z x i => x i ^ z) (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\nn\u271d : \u2115\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (fun z x i => x i ^ z) (Int.ofNat (Nat.succ n\u271d)) a\u271d x\u271d = (a\u271d * (fun z x i => x i ^ z) (Int.ofNat n\u271d) a\u271d) x\u271d\n[PROOFSTEP]\nexact DivInvMonoid.zpow_succ' _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\n\u22a2 \u2200 (n : \u2115) (a : (i : I) \u2192 f i), (fun z x i => x i ^ z) (Int.negSucc n) a = ((fun z x i => x i ^ z) (\u2191(Nat.succ n)) a)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\nn\u271d : \u2115\na\u271d : (i : I) \u2192 f i\n\u22a2 (fun z x i => x i ^ z) (Int.negSucc n\u271d) a\u271d = ((fun z x i => x i ^ z) (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvMonoid (f i)\nsrc\u271d : Monoid ((i : I) \u2192 f i) := monoid\nn\u271d : \u2115\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (fun z x i => x i ^ z) (Int.negSucc n\u271d) a\u271d x\u271d = ((fun z x i => x i ^ z) (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9 x\u271d\n[PROOFSTEP]\nexact DivInvMonoid.zpow_neg' _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvOneMonoid (f i)\nsrc\u271d : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\n\u22a2 1\u207b\u00b9 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivInvOneMonoid (f i)\nsrc\u271d : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nx\u271d : I\n\u22a2 1\u207b\u00b9 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact inv_one\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvolutiveInv (f i)\n\u22a2 \u2200 (x : (i : I) \u2192 f i), x\u207b\u00b9\u207b\u00b9 = x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvolutiveInv (f i)\nx\u271d : (i : I) \u2192 f i\n\u22a2 x\u271d\u207b\u00b9\u207b\u00b9 = x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 InvolutiveInv (f i)\nx\u271d\u00b9 : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 x\u271d\u00b9\u207b\u00b9\u207b\u00b9 x\u271d = x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nexact inv_inv _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\n\u22a2 \u2200 (a b : (i : I) \u2192 f i), (a * b)\u207b\u00b9 = b\u207b\u00b9 * a\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\na\u271d b\u271d : (i : I) \u2192 f i\n\u22a2 (a\u271d * b\u271d)\u207b\u00b9 = b\u271d\u207b\u00b9 * a\u271d\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\na\u271d b\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d * b\u271d)\u207b\u00b9 x\u271d = (b\u271d\u207b\u00b9 * a\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\nexact mul_inv_rev _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\n\u22a2 \u2200 (a b : (i : I) \u2192 f i), a * b = 1 \u2192 a\u207b\u00b9 = b\n[PROOFSTEP]\nintros _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\na\u271d b\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = 1\n\u22a2 a\u271d\u207b\u00b9 = b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 DivisionMonoid (f i)\nsrc\u271d\u00b9 : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\nsrc\u271d : InvolutiveInv ((i : I) \u2192 f i) := involutiveInv\na\u271d b\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = 1\nx\u271d : I\n\u22a2 a\u271d\u207b\u00b9 x\u271d = b\u271d x\u271d\n[PROOFSTEP]\nexact DivisionMonoid.inv_eq_of_mul _ _ (congrFun h _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Group (f i)\nsrc\u271d : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\n\u22a2 \u2200 (a : (i : I) \u2192 f i), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Group (f i)\nsrc\u271d : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\na\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d\u207b\u00b9 * a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 Group (f i)\nsrc\u271d : DivInvMonoid ((i : I) \u2192 f i) := divInvMonoid\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d\u207b\u00b9 * a\u271d) x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nexact mul_left_inv _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 LeftCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\n\u22a2 \u2200 (a b c : (i : I) \u2192 f i), a * b = a * c \u2192 b = c\n[PROOFSTEP]\nintros _ _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 LeftCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = a\u271d * c\u271d\n\u22a2 b\u271d = c\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 LeftCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = a\u271d * c\u271d\nx\u271d : I\n\u22a2 b\u271d x\u271d = c\u271d x\u271d\n[PROOFSTEP]\nexact LeftCancelSemigroup.mul_left_cancel _ _ _ (congr_fun h _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 RightCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\n\u22a2 \u2200 (a b c : (i : I) \u2192 f i), a * b = c * b \u2192 a = c\n[PROOFSTEP]\nintros _ _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 RightCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = c\u271d * b\u271d\n\u22a2 a\u271d = c\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 RightCancelSemigroup (f i)\nsrc\u271d : Semigroup ((i : I) \u2192 f i) := semigroup\na\u271d b\u271d c\u271d : (i : I) \u2192 f i\nh : a\u271d * b\u271d = c\u271d * b\u271d\nx\u271d : I\n\u22a2 a\u271d x\u271d = c\u271d x\u271d\n[PROOFSTEP]\nexact RightCancelSemigroup.mul_right_cancel _ _ _ (congr_fun h _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\n\u22a2 \u2200 (a : (i : I) \u2192 f i), 0 * a = 0\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\na\u271d : (i : I) \u2192 f i\n\u22a2 0 * a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (0 * a\u271d) x\u271d = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nexact zero_mul _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\n\u22a2 \u2200 (a : (i : I) \u2192 f i), a * 0 = 0\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\na\u271d : (i : I) \u2192 f i\n\u22a2 a\u271d * 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d : (i : I) \u2192 MulZeroClass (f i)\na\u271d : (i : I) \u2192 f i\nx\u271d : I\n\u22a2 (a\u271d * 0) x\u271d = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nexact mul_zero _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 Mul (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : Mul \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nI : Type u_5\nx\u271d\u00b9 x\u271d : I \u2192 \u03b1\n\u22a2 (fun h => \u2191f \u2218 h) (x\u271d\u00b9 * x\u271d) = (fun h => \u2191f \u2218 h) x\u271d\u00b9 * (fun h => \u2191f \u2218 h) x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 Mul (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : Mul \u03b2\nf : \u03b1 \u2192\u2099* \u03b2\nI : Type u_5\nx\u271d\u00b2 x\u271d\u00b9 : I \u2192 \u03b1\nx\u271d : I\n\u22a2 (fun h => \u2191f \u2218 h) (x\u271d\u00b2 * x\u271d\u00b9) x\u271d = ((fun h => \u2191f \u2218 h) x\u271d\u00b2 * (fun h => \u2191f \u2218 h) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 MulOneClass (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192* \u03b2\nI : Type u_5\n\u22a2 (fun h => \u2191f \u2218 h) 1 = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 MulOneClass (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192* \u03b2\nI : Type u_5\nx\u271d : I\n\u22a2 (fun h => \u2191f \u2218 h) 1 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 MulOneClass (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192* \u03b2\nI : Type u_5\nx\u271d : I\n\u22a2 \u2191f 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 MulOneClass (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192* \u03b2\nI : Type u_5\nx\u271d\u00b9 x\u271d : I \u2192 \u03b1\n\u22a2 OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } (x\u271d\u00b9 * x\u271d) =\n    OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } x\u271d\u00b9 *\n      OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nI\u271d : Type u\nf\u271d : I\u271d \u2192 Type v\nx y : (i : I\u271d) \u2192 f\u271d i\ni j : I\u271d\ninst\u271d\u00b2 : (i : I\u271d) \u2192 MulOneClass (f\u271d i)\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b9 : MulOneClass \u03b1\ninst\u271d : MulOneClass \u03b2\nf : \u03b1 \u2192* \u03b2\nI : Type u_5\nx\u271d\u00b2 x\u271d\u00b9 : I \u2192 \u03b1\nx\u271d : I\n\u22a2 OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } (x\u271d\u00b2 * x\u271d\u00b9) x\u271d =\n    (OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } x\u271d\u00b2 *\n        OneHom.toFun { toFun := fun h => \u2191f \u2218 h, map_one' := (_ : (fun h => \u2191f \u2218 h) 1 = 1) } x\u271d\u00b9)\n      x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\n\u22a2 Pairwise fun i j => \u2200 (x : f i) (y : f j), Commute (mulSingle i x) (mulSingle j y)\n[PROOFSTEP]\nintro i j hij x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\n\u22a2 Commute (mulSingle i x) (mulSingle j y)\n[PROOFSTEP]\next k\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nk : I\n\u22a2 (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n[PROOFSTEP]\nby_cases h1 : i = k\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nk : I\nh1 : i = k\n\u22a2 (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n[PROOFSTEP]\nsubst h1\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\n\u22a2 (mulSingle i x * mulSingle j y) i = (mulSingle j y * mulSingle i x) i\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nk : I\nh1 : \u00aci = k\n\u22a2 (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n[PROOFSTEP]\nby_cases h2 : j = k\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nk : I\nh1 : \u00aci = k\nh2 : j = k\n\u22a2 (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n[PROOFSTEP]\nsubst h2\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nh1 : \u00aci = j\n\u22a2 (mulSingle i x * mulSingle j y) j = (mulSingle j y * mulSingle i x) j\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y\u271d : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\ni j : I\nhij : i \u2260 j\nx : f i\ny : f j\nk : I\nh1 : \u00aci = k\nh2 : \u00acj = k\n\u22a2 (mulSingle i x * mulSingle j y) k = (mulSingle j y * mulSingle i x) k\n[PROOFSTEP]\nsimp [h1, h2]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\nx : (i : I) \u2192 f i\ni j : I\n\u22a2 Commute (mulSingle i (x i)) (mulSingle j (x j))\n[PROOFSTEP]\nobtain rfl | hij := Decidable.eq_or_ne i j\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni\u271d j : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\nx : (i : I) \u2192 f i\ni : I\n\u22a2 Commute (mulSingle i (x i)) (mulSingle i (x i))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni\u271d j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 MulOneClass (f i)\nx : (i : I) \u2192 f i\ni j : I\nhij : i \u2260 j\n\u22a2 Commute (mulSingle i (x i)) (mulSingle j (x j))\n[PROOFSTEP]\nexact Pi.mulSingle_commute hij _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 Group (f i)\ng : (i : I) \u2192 f i\nx : f i\n\u22a2 Function.update g i x = g / mulSingle i (g i) * mulSingle i x\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 Group (f i)\ng : (i : I) \u2192 f i\nx : f i\nj : I\n\u22a2 Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j\n[PROOFSTEP]\nrcases eq_or_ne i j with (rfl | h)\n[GOAL]\ncase h.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 Group (f i)\ng : (i : I) \u2192 f i\nx : f i\n\u22a2 Function.update g i x i = (g / mulSingle i (g i) * mulSingle i x) i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx\u271d y : (i : I) \u2192 f i\ni j\u271d : I\ninst\u271d\u00b9 : DecidableEq I\ninst\u271d : (i : I) \u2192 Group (f i)\ng : (i : I) \u2192 f i\nx : f i\nj : I\nh : i \u2260 j\n\u22a2 Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j\n[PROOFSTEP]\nsimp [Function.update_noteq h.symm, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v \u2194\n    k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nrefine' \u27e8fun h => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nhave hk := congr_fun h k\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nhave hl := congr_fun h l\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nhave hm := (congr_fun h m).symm\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nhave hn := (congr_fun h n).symm\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : (mulSingle k u * mulSingle l v) k = (mulSingle m u * mulSingle n v) k\nhl : (mulSingle k u * mulSingle l v) l = (mulSingle m u * mulSingle n v) l\nhm : (mulSingle m u * mulSingle n v) m = (mulSingle k u * mulSingle l v) m\nhn : (mulSingle m u * mulSingle n v) n = (mulSingle k u * mulSingle l v) n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nsimp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn \n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nrcases eq_or_ne k m with (rfl | hkm)\n[GOAL]\ncase refine'_1.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\n\u22a2 k = k \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = k \u2228 u * v = 1 \u2227 k = l \u2227 k = n\n[PROOFSTEP]\nrefine' Or.inl \u27e8rfl, not_ne_iff.mp fun hln => (hv _).elim\u27e9\n[GOAL]\ncase refine'_1.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\n\u22a2 v = 1\n[PROOFSTEP]\nrcases eq_or_ne k l with (rfl | hkl)\n[GOAL]\ncase refine'_1.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle k v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = k then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = k then v else 1\nhln : k \u2260 n\n\u22a2 v = 1\n[PROOFSTEP]\nrwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn \n[GOAL]\ncase refine'_1.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle k u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = k then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = k then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if k = n then v else 1) = (if k = k then u else 1) * if k = l then v else 1\nhn : ((if n = k then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhln : l \u2260 n\nhkl : k \u2260 l\n\u22a2 v = 1\n[PROOFSTEP]\nrwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl \n[GOAL]\ncase refine'_1.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nrcases eq_or_ne m n with (rfl | hmn)\n[GOAL]\ncase refine'_1.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m\n[PROOFSTEP]\nrcases eq_or_ne k l with (rfl | hkl)\n[GOAL]\ncase refine'_1.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = k then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m\n[PROOFSTEP]\nrw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm \n[GOAL]\ncase refine'_1.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhm : (if True then u else 1) * v = 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1\n\u22a2 k = m \u2227 k = m \u2228 u = v \u2227 k = m \u2227 k = m \u2228 u * v = 1 \u2227 k = k \u2227 m = m\n[PROOFSTEP]\nexact Or.inr (Or.inr \u27e8hm, rfl, rfl\u27e9)\n[GOAL]\ncase refine'_1.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m\n[PROOFSTEP]\nsimp only [if_neg hkm, if_neg hkl, mul_one] at hk \n[GOAL]\ncase refine'_1.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : (if True then u else 1) = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m\n[PROOFSTEP]\ndsimp at hk \n[GOAL]\ncase refine'_1.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1\nhm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhkl : k \u2260 l\nhk : u = 1\n\u22a2 k = m \u2227 l = m \u2228 u = v \u2227 k = m \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = m\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase refine'_1.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nrw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm \n[GOAL]\ncase refine'_1.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1\nhm : (if True then u else 1) = if m = l then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1\nhkm : k \u2260 m\nhmn : m \u2260 n\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n\n[PROOFSTEP]\nobtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1\n[GOAL]\ncase refine'_1.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n\n[PROOFSTEP]\nrw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk \n[GOAL]\ncase refine'_1.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhmn : m \u2260 n\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v\nhk : (if True then u else 1) = if k = n then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1\nhm : (if True then u else 1) = if m = m then v else 1\nhn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1\n\u22a2 k = m \u2227 m = n \u2228 u = v \u2227 k = n \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = n\n[PROOFSTEP]\nobtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hk.symm hu)).1\n[GOAL]\ncase refine'_1.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nhkm : k \u2260 m\nhm : (if True then u else 1) = if m = m then v else 1\nhmn : m \u2260 k\nh : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle k v\nhk : (if True then u else 1) = if k = k then v else 1\nhl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = k then v else 1\nhn : ((if k = m then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = m then v else 1\n\u22a2 k = m \u2227 m = k \u2228 u = v \u2227 k = k \u2227 m = m \u2228 u * v = 1 \u2227 k = m \u2227 m = k\n[PROOFSTEP]\nexact Or.inr (Or.inl \u27e8hk.trans (if_pos rfl), rfl, rfl\u27e9)\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l m n : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 k = m \u2227 l = n \u2228 u = v \u2227 k = n \u2227 l = m \u2228 u * v = 1 \u2227 k = l \u2227 m = n \u2192\n    mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v\n[PROOFSTEP]\nrintro (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl, rfl\u27e9 | \u27e8h, rfl, rfl\u27e9)\n[GOAL]\ncase refine'_2.inl.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\n\u22a2 mulSingle k u * mulSingle l v = mulSingle k u * mulSingle l v\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.inr.inl.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk l : I\nu : M\nhu hv : u \u2260 1\n\u22a2 mulSingle k u * mulSingle l u = mulSingle l u * mulSingle k u\n[PROOFSTEP]\napply mul_comm\n[GOAL]\ncase refine'_2.inr.inr.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf : I \u2192 Type v\nx y : (i : I) \u2192 f i\ni j : I\ninst\u271d\u00b9 : DecidableEq I\nM : Type u_3\ninst\u271d : CommMonoid M\nk m : I\nu v : M\nhu : u \u2260 1\nhv : v \u2260 1\nh : u * v = 1\n\u22a2 mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v\n[PROOFSTEP]\nsimp_rw [\u2190 Pi.mulSingle_mul, h, mulSingle_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nI : Type u\nf\u271d : I \u2192 Type v\nx y : (i : I) \u2192 f\u271d i\ni j : I\n\u03b7 : Type v\nR : Type w\ns : \u03b9 \u2192 \u03b7\ninst\u271d : MulOneClass R\nf g : \u03b9 \u2192 R\n\u22a2 OneHom.toFun { toFun := fun f => extend s f 1, map_one' := (_ : extend s 1 1 = 1) } (f * g) =\n    OneHom.toFun { toFun := fun f => extend s f 1, map_one' := (_ : extend s 1 1 = 1) } f *\n      OneHom.toFun { toFun := fun f => extend s f 1, map_one' := (_ : extend s 1 1 = 1) } g\n[PROOFSTEP]\nsimpa using Function.extend_mul s f g 1 1\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Pi", "llama_tokens": 21789, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673087708699, "lm_q2_score": 0.6654105454764747, "lm_q1q2_score": 0.5408904793292185}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b : \u03b1\nh : MonotoneOn f s\nhl : BddBelow (f '' s)\nhu : BddAbove (f '' s)\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nclassical\n  /- The extension is defined by `f x = f a` for `x \u2264 a`, and `f x` is the supremum of the values\n        of `f` to the left of `x` for `x \u2265 a`. -/\nrcases hl with \u27e8a, ha\u27e9\nhave hu' : \u2200 x, BddAbove (f '' (Iic x \u2229 s)) := fun x => hu.mono (image_subset _ (inter_subset_right _ _))\nlet g : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhave hgs : EqOn f g s := by\n  intro x hx\n  simp only []\n  have : IsGreatest (Iic x \u2229 s) x := \u27e8\u27e8right_mem_Iic, hx\u27e9, fun y hy => hy.1\u27e9\n  rw [if_neg this.nonempty.not_disjoint, ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]\nrefine' \u27e8g, fun x y hxy => _, hgs\u27e9\nby_cases hx : Disjoint (Iic x) s <;> by_cases hy : Disjoint (Iic y) s <;>\n  simp only [if_pos, if_neg, not_false_iff, *, refl]\n\u00b7 rcases not_disjoint_iff_nonempty_inter.1 hy with \u27e8z, hz\u27e9\n  exact le_csSup_of_le (hu' _) (mem_image_of_mem _ hz) (ha <| mem_image_of_mem _ hz.2)\n\u00b7 exact (hx <| hy.mono_left <| Iic_subset_Iic.2 hxy).elim\n\u00b7 rw [not_disjoint_iff_nonempty_inter] at hx hy \n  refine' csSup_le_csSup (hu' _) (hx.image _) (image_subset _ _)\n  exact inter_subset_inter_left _ (Iic_subset_Iic.2 hxy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na b : \u03b1\nh : MonotoneOn f s\nhl : BddBelow (f '' s)\nhu : BddAbove (f '' s)\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nrcases hl with \u27e8a, ha\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nhave hu' : \u2200 x, BddAbove (f '' (Iic x \u2229 s)) := fun x => hu.mono (image_subset _ (inter_subset_right _ _))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nlet g : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nhave hgs : EqOn f g s := by\n  intro x hx\n  simp only []\n  have : IsGreatest (Iic x \u2229 s) x := \u27e8\u27e8right_mem_Iic, hx\u27e9, fun y hy => hy.1\u27e9\n  rw [if_neg this.nonempty.not_disjoint, ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\n\u22a2 EqOn f g s\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = g x\n[PROOFSTEP]\nsimp only []\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\n[PROOFSTEP]\nhave : IsGreatest (Iic x \u2229 s) x := \u27e8\u27e8right_mem_Iic, hx\u27e9, fun y hy => hy.1\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nx : \u03b1\nhx : x \u2208 s\nthis : IsGreatest (Iic x \u2229 s) x\n\u22a2 f x = if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\n[PROOFSTEP]\nrw [if_neg this.nonempty.not_disjoint, ((h.mono <| inter_subset_right _ _).map_isGreatest this).csSup_eq]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\n\u22a2 \u2203 g, Monotone g \u2227 EqOn f g s\n[PROOFSTEP]\nrefine' \u27e8g, fun x y hxy => _, hgs\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nby_cases hx : Disjoint (Iic x) s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Disjoint (Iic x) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nby_cases hy : Disjoint (Iic y) s\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : \u00acDisjoint (Iic x) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nby_cases hy : Disjoint (Iic y) s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Disjoint (Iic x) s\nhy : Disjoint (Iic y) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nsimp only [if_pos, if_neg, not_false_iff, *, refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Disjoint (Iic x) s\nhy : \u00acDisjoint (Iic y) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nsimp only [if_pos, if_neg, not_false_iff, *, refl]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : \u00acDisjoint (Iic x) s\nhy : Disjoint (Iic y) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nsimp only [if_pos, if_neg, not_false_iff, *, refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : \u00acDisjoint (Iic x) s\nhy : \u00acDisjoint (Iic y) s\n\u22a2 g x \u2264 g y\n[PROOFSTEP]\nsimp only [if_pos, if_neg, not_false_iff, *, refl]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Disjoint (Iic x) s\nhy : \u00acDisjoint (Iic y) s\n\u22a2 a \u2264 sSup (f '' (Iic y \u2229 s))\n[PROOFSTEP]\nrcases not_disjoint_iff_nonempty_inter.1 hy with \u27e8z, hz\u27e9\n[GOAL]\ncase neg.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Disjoint (Iic x) s\nhy : \u00acDisjoint (Iic y) s\nz : \u03b1\nhz : z \u2208 Iic y \u2229 s\n\u22a2 a \u2264 sSup (f '' (Iic y \u2229 s))\n[PROOFSTEP]\nexact le_csSup_of_le (hu' _) (mem_image_of_mem _ hz) (ha <| mem_image_of_mem _ hz.2)\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : \u00acDisjoint (Iic x) s\nhy : Disjoint (Iic y) s\n\u22a2 sSup (f '' (Iic x \u2229 s)) \u2264 a\n[PROOFSTEP]\nexact (hx <| hy.mono_left <| Iic_subset_Iic.2 hxy).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : \u00acDisjoint (Iic x) s\nhy : \u00acDisjoint (Iic y) s\n\u22a2 sSup (f '' (Iic x \u2229 s)) \u2264 sSup (f '' (Iic y \u2229 s))\n[PROOFSTEP]\nrw [not_disjoint_iff_nonempty_inter] at hx hy \n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Set.Nonempty (Iic x \u2229 s)\nhy : Set.Nonempty (Iic y \u2229 s)\n\u22a2 sSup (f '' (Iic x \u2229 s)) \u2264 sSup (f '' (Iic y \u2229 s))\n[PROOFSTEP]\nrefine' csSup_le_csSup (hu' _) (hx.image _) (image_subset _ _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : ConditionallyCompleteLinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\na\u271d b : \u03b1\nh : MonotoneOn f s\nhu : BddAbove (f '' s)\na : \u03b2\nha : a \u2208 lowerBounds (f '' s)\nhu' : \u2200 (x : \u03b1), BddAbove (f '' (Iic x \u2229 s))\ng : \u03b1 \u2192 \u03b2 := fun x => if Disjoint (Iic x) s then a else sSup (f '' (Iic x \u2229 s))\nhgs : EqOn f g s\nx y : \u03b1\nhxy : x \u2264 y\nhx : Set.Nonempty (Iic x \u2229 s)\nhy : Set.Nonempty (Iic y \u2229 s)\n\u22a2 Iic x \u2229 s \u2286 Iic y \u2229 s\n[PROOFSTEP]\nexact inter_subset_inter_left _ (Iic_subset_Iic.2 hxy)\n", "meta": {"mathlib_filename": "Mathlib.Order.Monotone.Extension", "llama_tokens": 5912, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527631, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5405002028647172}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : SMulCommClass R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 r \u2022 s \u2022 x = s \u2022 r \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : SMulCommClass R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 (r \u2022 s \u2022 x).re = (s \u2022 r \u2022 x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_comm]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : SMulCommClass R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 (r \u2022 s \u2022 x).im = (s \u2022 r \u2022 x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : SMul R S\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : IsScalarTower R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 (r \u2022 s) \u2022 x = r \u2022 s \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : SMul R S\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : IsScalarTower R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 ((r \u2022 s) \u2022 x).re = (r \u2022 s \u2022 x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_assoc]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : SMul R S\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul S \u211d\ninst\u271d : IsScalarTower R S \u211d\nr : R\ns : S\nx : \u2102\n\u22a2 ((r \u2022 s) \u2022 x).im = (r \u2022 s \u2022 x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R\u1d50\u1d52\u1d56 \u211d\ninst\u271d : IsCentralScalar R \u211d\nr : R\nx : \u2102\n\u22a2 MulOpposite.op r \u2022 x = r \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R\u1d50\u1d52\u1d56 \u211d\ninst\u271d : IsCentralScalar R \u211d\nr : R\nx : \u2102\n\u22a2 (MulOpposite.op r \u2022 x).re = (r \u2022 x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, op_smul_eq_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : SMul R \u211d\ninst\u271d\u00b9 : SMul R\u1d50\u1d52\u1d56 \u211d\ninst\u271d : IsCentralScalar R \u211d\nr : R\nx : \u2102\n\u22a2 (MulOpposite.op r \u2022 x).im = (r \u2022 x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, op_smul_eq_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nx : \u2102\n\u22a2 1 \u2022 x = x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nx : \u2102\n\u22a2 (1 \u2022 x).re = x.re\n[PROOFSTEP]\nsimp [smul_re, smul_im, one_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nx : \u2102\n\u22a2 (1 \u2022 x).im = x.im\n[PROOFSTEP]\nsimp [smul_re, smul_im, one_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nr s : R\nx : \u2102\n\u22a2 (r * s) \u2022 x = r \u2022 s \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nr s : R\nx : \u2102\n\u22a2 ((r * s) \u2022 x).re = (r \u2022 s \u2022 x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, mul_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Monoid R\ninst\u271d : MulAction R \u211d\nr s : R\nx : \u2102\n\u22a2 ((r * s) \u2022 x).im = (r \u2022 s \u2022 x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, mul_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\n\u22a2 r \u2022 0 = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\n\u22a2 (r \u2022 0).re = 0.re\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_zero]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\n\u22a2 (r \u2022 0).im = 0.im\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\nx y : \u2102\n\u22a2 r \u2022 (x + y) = r \u2022 x + r \u2022 y\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\nx y : \u2102\n\u22a2 (r \u2022 (x + y)).re = (r \u2022 x + r \u2022 y).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_add]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d : DistribSMul R \u211d\nr : R\nx y : \u2102\n\u22a2 (r \u2022 (x + y)).im = (r \u2022 x + r \u2022 y).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, smul_add]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr s : R\nx : \u2102\n\u22a2 (r + s) \u2022 x = r \u2022 x + s \u2022 x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr s : R\nx : \u2102\n\u22a2 ((r + s) \u2022 x).re = (r \u2022 x + s \u2022 x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, add_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr s : R\nx : \u2102\n\u22a2 ((r + s) \u2022 x).im = (r \u2022 x + s \u2022 x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, add_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr : \u2102\n\u22a2 0 \u2022 r = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr : \u2102\n\u22a2 (0 \u2022 r).re = 0.re\n[PROOFSTEP]\nsimp [smul_re, smul_im, zero_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Module R \u211d\nr : \u2102\n\u22a2 (0 \u2022 r).im = 0.im\n[PROOFSTEP]\nsimp [smul_re, smul_im, zero_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx\u271d : (fun x => \u2102) r\nxr xi : \u211d\n\u22a2 \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n        r *\n      { re := xr, im := xi } =\n    { re := xr, im := xi } *\n      \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n        r\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx\u271d : (fun x => \u2102) r\nxr xi : \u211d\n\u22a2 (\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r *\n        { re := xr, im := xi }).re =\n    ({ re := xr, im := xi } *\n        \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, Algebra.commutes]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx\u271d : (fun x => \u2102) r\nxr xi : \u211d\n\u22a2 (\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r *\n        { re := xr, im := xi }).im =\n    ({ re := xr, im := xi } *\n        \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, Algebra.commutes]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx : (fun x => \u2102) r\n\u22a2 r \u2022 x =\n    \u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n        r *\n      x\n[PROOFSTEP]\next\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx : (fun x => \u2102) r\n\u22a2 (r \u2022 x).re =\n    (\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r *\n        x).re\n[PROOFSTEP]\nsimp [smul_re, smul_im, Algebra.smul_def]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R \u211d\nsrc\u271d : R \u2192+* \u2102 := RingHom.comp ofReal (algebraMap R \u211d)\nr : R\nx : (fun x => \u2102) r\n\u22a2 (r \u2022 x).im =\n    (\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n              map_add' :=\n                (_ : \u2200 (x y : R), OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) }\n          r *\n        x).im\n[PROOFSTEP]\nsimp [smul_re, smul_im, Algebra.smul_def]\n[GOAL]\nR : Type u_1\nS : Type u_2\nr : \u211d\nx : \u2102\n\u22a2 star (r \u2022 x) = star r \u2022 star x\n[PROOFSTEP]\nsimp only [star_def, star_trivial, real_smul, map_mul, conj_ofReal]\n[GOAL]\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d A\nf g : \u2102 \u2192\u2090[\u211d] A\nh : \u2191f I = \u2191g I\n\u22a2 f = g\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase H.mk\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra \u211d A\nf g : \u2102 \u2192\u2090[\u211d] A\nh : \u2191f I = \u2191g I\nx y : \u211d\n\u22a2 \u2191f { re := x, im := y } = \u2191g { re := x, im := y }\n[PROOFSTEP]\nsimp only [mk_eq_add_mul_I, AlgHom.map_add, AlgHom.map_coe_real_complex, AlgHom.map_mul, h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nz z' : \u2102\n\u22a2 (fun z => ![z.re, z.im]) (z + z') = (fun z => ![z.re, z.im]) z + (fun z => ![z.re, z.im]) z'\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nc : \u211d\nz : \u2102\n\u22a2 AddHom.toFun\n      { toFun := fun z => ![z.re, z.im],\n        map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) }\n      (c \u2022 z) =\n    \u2191(RingHom.id \u211d) c \u2022\n      AddHom.toFun\n        { toFun := fun z => ![z.re, z.im],\n          map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) }\n        z\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nz : \u2102\n\u22a2 (fun c => \u2191(c 0) + c 1 \u2022 I)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun z => ![z.re, z.im],\n                map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) },\n            map_smul' :=\n              (_ : \u2200 (c : \u211d) (z : \u2102), ![c * z.re - 0 * z.im, c * z.im + 0 * z.re] = c \u2022 ![z.re, z.im]) }.toAddHom\n        z) =\n    z\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nc : Fin 2 \u2192 \u211d\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun z => ![z.re, z.im],\n              map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) },\n          map_smul' :=\n            (_ : \u2200 (c : \u211d) (z : \u2102), ![c * z.re - 0 * z.im, c * z.im + 0 * z.re] = c \u2022 ![z.re, z.im]) }.toAddHom\n      ((fun c => \u2191(c 0) + c 1 \u2022 I) c) =\n    c\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nc : Fin 2 \u2192 \u211d\ni : Fin 2\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun z => ![z.re, z.im],\n              map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) },\n          map_smul' :=\n            (_ : \u2200 (c : \u211d) (z : \u2102), ![c * z.re - 0 * z.im, c * z.im + 0 * z.re] = c \u2022 ![z.re, z.im]) }.toAddHom\n      ((fun c => \u2191(c 0) + c 1 \u2022 I) c) i =\n    c i\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.head\nR : Type u_1\nS : Type u_2\nc : Fin 2 \u2192 \u211d\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun z => ![z.re, z.im],\n              map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) },\n          map_smul' :=\n            (_ : \u2200 (c : \u211d) (z : \u2102), ![c * z.re - 0 * z.im, c * z.im + 0 * z.re] = c \u2022 ![z.re, z.im]) }.toAddHom\n      ((fun c => \u2191(c 0) + c 1 \u2022 I) c) { val := 0, isLt := (_ : 0 < 2) } =\n    c { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.tail.head\nR : Type u_1\nS : Type u_2\nc : Fin 2 \u2192 \u211d\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun z => ![z.re, z.im],\n              map_add' := (_ : \u2200 (z z' : \u2102), ![z.re + z'.re, z.im + z'.im] = ![z.re, z.im] + ![z'.re, z'.im]) },\n          map_smul' :=\n            (_ : \u2200 (c : \u211d) (z : \u2102), ![c * z.re - 0 * z.im, c * z.im + 0 * z.re] = c \u2022 ![z.re, z.im]) }.toAddHom\n      ((fun c => \u2191(c 0) + c 1 \u2022 I) c) { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    c { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ni j : Fin 2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] i)) j = \u2191(Finsupp.single i 1) j\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nR : Type u_1\nS : Type u_2\nj : Fin 2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) })) j =\n    \u2191(Finsupp.single { val := 0, isLt := (_ : 0 < 2) } 1) j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase tail.head\nR : Type u_1\nS : Type u_2\nj : Fin 2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) })) j =\n    \u2191(Finsupp.single { val := 1, isLt := (_ : (fun a => a < 2) 1) } 1) j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase head.head\nR : Type u_1\nS : Type u_2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) })) { val := 0, isLt := (_ : 0 < 2) } =\n    \u2191(Finsupp.single { val := 0, isLt := (_ : 0 < 2) } 1) { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp [coe_basisOneI_repr, Finsupp.single_eq_of_ne, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,\n  Fin.one_eq_zero_iff, Ne.def, not_false_iff, I_re, Nat.succ_succ_ne_one, one_im, I_im, one_re, Finsupp.single_eq_same,\n  Fin.zero_eq_one_iff]\n[GOAL]\ncase head.tail.head\nR : Type u_1\nS : Type u_2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }))\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    \u2191(Finsupp.single { val := 0, isLt := (_ : 0 < 2) } 1) { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp [coe_basisOneI_repr, Finsupp.single_eq_of_ne, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,\n  Fin.one_eq_zero_iff, Ne.def, not_false_iff, I_re, Nat.succ_succ_ne_one, one_im, I_im, one_re, Finsupp.single_eq_same,\n  Fin.zero_eq_one_iff]\n[GOAL]\ncase tail.head.head\nR : Type u_1\nS : Type u_2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }))\n      { val := 0, isLt := (_ : 0 < 2) } =\n    \u2191(Finsupp.single { val := 1, isLt := (_ : (fun a => a < 2) 1) } 1) { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp [coe_basisOneI_repr, Finsupp.single_eq_of_ne, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,\n  Fin.one_eq_zero_iff, Ne.def, not_false_iff, I_re, Nat.succ_succ_ne_one, one_im, I_im, one_re, Finsupp.single_eq_same,\n  Fin.zero_eq_one_iff]\n[GOAL]\ncase tail.head.tail.head\nR : Type u_1\nS : Type u_2\n\u22a2 \u2191(\u2191basisOneI.repr (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }))\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    \u2191(Finsupp.single { val := 1, isLt := (_ : (fun a => a < 2) 1) } 1) { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp [coe_basisOneI_repr, Finsupp.single_eq_of_ne, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons,\n  Fin.one_eq_zero_iff, Ne.def, not_false_iff, I_re, Nat.succ_succ_ne_one, one_im, I_im, one_re, Finsupp.single_eq_same,\n  Fin.zero_eq_one_iff]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u22a2 finrank \u211d \u2102 = 2\n[PROOFSTEP]\nrw [finrank_eq_card_basis basisOneI, Fintype.card_fin]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u22a2 Module.rank \u211d \u2102 = 2\n[PROOFSTEP]\nsimp [\u2190 finrank_eq_rank, finrank_real_complex]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u22a2 Cardinal.lift (Module.rank \u211d \u2102) = 2\n[PROOFSTEP]\nrw [\u2190 finrank_eq_rank, finrank_real_complex, Cardinal.lift_natCast, Nat.cast_ofNat]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \u2102 E\n\u22a2 Cardinal.lift (Module.rank \u211d E) = Cardinal.lift (2 * Module.rank \u2102 E)\n[PROOFSTEP]\nrw [\u2190 lift_rank_mul_lift_rank \u211d \u2102 E, Complex.rank_real_complex']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \u2102 E\n\u22a2 2 * Cardinal.lift (Module.rank \u2102 E) = Cardinal.lift (2 * Module.rank \u2102 E)\n[PROOFSTEP]\nsimp only [Cardinal.lift_id']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \u2102 E\n\u22a2 FiniteDimensional.finrank \u211d E = 2 * FiniteDimensional.finrank \u2102 E\n[PROOFSTEP]\nrw [\u2190 FiniteDimensional.finrank_mul_finrank \u211d \u2102 E, Complex.finrank_real_complex]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Star E\ninst\u271d\u00b9 : Module \u2102 E\ninst\u271d : StarModule \u2102 E\nr : \u211d\na : E\n\u22a2 star (r \u2022 a) = star r \u2022 star a\n[PROOFSTEP]\nrw [\u2190 smul_one_smul \u2102 r a, star_smul, star_smul, star_one, smul_one_smul]\n[GOAL]\n\u22a2 \u2200 (r : \u211d) (x : \u2102),\n    AddHom.toFun { toFun := fun x => x.re, map_add' := add_re } (r \u2022 x) =\n      \u2191(RingHom.id \u211d) r \u2022 AddHom.toFun { toFun := fun x => x.re, map_add' := add_re } x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 (r : \u211d) (x : \u2102),\n    AddHom.toFun { toFun := fun x => x.im, map_add' := add_im } (r \u2022 x) =\n      \u2191(RingHom.id \u211d) r \u2022 AddHom.toFun { toFun := fun x => x.im, map_add' := add_im } x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) = \u2191Matrix.of ![![1, 0], ![0, -1]]\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\ni j : Fin 2\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) i j = \u2191Matrix.of ![![1, 0], ![0, -1]] i j\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase a.h.head\nj : Fin 2\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 0, isLt := (_ : 0 < 2) } j =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 0, isLt := (_ : 0 < 2) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase a.h.tail.head\nj : Fin 2\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n      j =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) } j\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase a.h.head.head\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 0, isLt := (_ : 0 < 2) }\n      { val := 0, isLt := (_ : 0 < 2) } =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply]\n[GOAL]\ncase a.h.head.tail.head\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 0, isLt := (_ : 0 < 2) }\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply]\n[GOAL]\ncase a.h.tail.head.head\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n      { val := 0, isLt := (_ : 0 < 2) } =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply]\n[GOAL]\ncase a.h.tail.head.tail.head\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (AlgEquiv.toLinearMap conjAe) { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    \u2191Matrix.of ![![1, 0], ![0, -1]] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply]\n[GOAL]\nf : \u2102 \u2192\u2090[\u211d] \u2102\n\u22a2 f = AlgHom.id \u211d \u2102 \u2228 f = \u2191conjAe\n[PROOFSTEP]\nrefine' (eq_or_eq_neg_of_sq_eq_sq (f I) I <| by rw [\u2190 map_pow, I_sq, map_neg, map_one]).imp _ _\n[GOAL]\nf : \u2102 \u2192\u2090[\u211d] \u2102\n\u22a2 \u2191f I ^ 2 = I ^ 2\n[PROOFSTEP]\nrw [\u2190 map_pow, I_sq, map_neg, map_one]\n[GOAL]\ncase refine'_1\nf : \u2102 \u2192\u2090[\u211d] \u2102\n\u22a2 \u2191f I = I \u2192 f = AlgHom.id \u211d \u2102\n[PROOFSTEP]\nrefine' fun h => algHom_ext _\n[GOAL]\ncase refine'_2\nf : \u2102 \u2192\u2090[\u211d] \u2102\n\u22a2 \u2191f I = -I \u2192 f = \u2191conjAe\n[PROOFSTEP]\nrefine' fun h => algHom_ext _\n[GOAL]\ncase refine'_1\nf : \u2102 \u2192\u2090[\u211d] \u2102\nh : \u2191f I = I\n\u22a2 \u2191f I = \u2191(AlgHom.id \u211d \u2102) I\ncase refine'_2 f : \u2102 \u2192\u2090[\u211d] \u2102 h : \u2191f I = -I \u22a2 \u2191f I = \u2191\u2191conjAe I\n[PROOFSTEP]\nexacts [h, conj_I.symm \u25b8 h]\n[GOAL]\nsrc\u271d : \u2102 \u2243 \u211d \u00d7 \u211d := equivRealProd\n\u22a2 \u2200 (x y : \u2102),\n    Equiv.toFun\n        { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n          right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        (x + y) =\n      Equiv.toFun\n          { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n          x +\n        Equiv.toFun\n          { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n          y\n[PROOFSTEP]\nsimp\n[GOAL]\nsrc\u271d : \u2102 \u2243+ \u211d \u00d7 \u211d := equivRealProdAddHom\nr : \u211d\nc : \u2102\n\u22a2 AddHom.toFun\n      { toFun := src\u271d.toFun,\n        map_add' :=\n          (_ :\n            \u2200 (x y : \u2102), Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n      (r \u2022 c) =\n    \u2191(RingHom.id \u211d) r \u2022\n      AddHom.toFun\n        { toFun := src\u271d.toFun,\n          map_add' :=\n            (_ :\n              \u2200 (x y : \u2102), Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n        c\n[PROOFSTEP]\nsimp [equivRealProdAddHom, (Prod.smul_def), smul_eq_mul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\n\u22a2 \u2191(algebraMap \u211d A) 1 + 0 \u2022 I' = 1\n[PROOFSTEP]\nrw [RingHom.map_one, zero_smul, add_zero]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\n\u22a2 \u2191(algebraMap \u211d A) (x\u2081 * x\u2082 - y\u2081 * y\u2082) + (x\u2081 * y\u2082 + y\u2081 * x\u2082) \u2022 I' =\n    (\u2191(algebraMap \u211d A) x\u2081 + y\u2081 \u2022 I') * (\u2191(algebraMap \u211d A) x\u2082 + y\u2082 \u2022 I')\n[PROOFSTEP]\nrw [add_mul, mul_add, mul_add, add_comm _ (y\u2081 \u2022 I' * y\u2082 \u2022 I'), add_add_add_comm]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\n\u22a2 \u2191(algebraMap \u211d A) (x\u2081 * x\u2082 - y\u2081 * y\u2082) + (x\u2081 * y\u2082 + y\u2081 * x\u2082) \u2022 I' =\n    \u2191(algebraMap \u211d A) x\u2081 * \u2191(algebraMap \u211d A) x\u2082 + y\u2081 \u2022 I' * y\u2082 \u2022 I' +\n      (\u2191(algebraMap \u211d A) x\u2081 * y\u2082 \u2022 I' + y\u2081 \u2022 I' * \u2191(algebraMap \u211d A) x\u2082)\n[PROOFSTEP]\ncongr 1\n  -- equate \"real\" and \"imaginary\" parts\n[GOAL]\ncase e_a\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\n\u22a2 \u2191(algebraMap \u211d A) (x\u2081 * x\u2082 - y\u2081 * y\u2082) = \u2191(algebraMap \u211d A) x\u2081 * \u2191(algebraMap \u211d A) x\u2082 + y\u2081 \u2022 I' * y\u2082 \u2022 I'\n[PROOFSTEP]\nlet inst : SMulCommClass \u211d A A := by\n  infer_instance\n    -- porting note: added\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\n\u22a2 SMulCommClass \u211d A A\n[PROOFSTEP]\ninfer_instance\n  -- porting note: added\n[GOAL]\ncase e_a\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\ninst : SMulCommClass \u211d A A := inferInstance\n\u22a2 \u2191(algebraMap \u211d A) (x\u2081 * x\u2082 - y\u2081 * y\u2082) = \u2191(algebraMap \u211d A) x\u2081 * \u2191(algebraMap \u211d A) x\u2082 + y\u2081 \u2022 I' * y\u2082 \u2022 I'\n[PROOFSTEP]\nrw [smul_mul_smul, hf, smul_neg, \u2190 Algebra.algebraMap_eq_smul_one, \u2190 sub_eq_add_neg, \u2190 RingHom.map_mul, \u2190\n  RingHom.map_sub]\n[GOAL]\ncase e_a\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhf : I' * I' = -1\nx\u271d\u00b9 x\u271d : \u2102\nx\u2081 y\u2081 x\u2082 y\u2082 : \u211d\n\u22a2 (x\u2081 * y\u2082 + y\u2081 * x\u2082) \u2022 I' = \u2191(algebraMap \u211d A) x\u2081 * y\u2082 \u2022 I' + y\u2081 \u2022 I' * \u2191(algebraMap \u211d A) x\u2082\n[PROOFSTEP]\nrw [Algebra.smul_def, Algebra.smul_def, Algebra.smul_def, \u2190 Algebra.right_comm _ x\u2082, \u2190 mul_assoc, \u2190 add_mul, \u2190\n  RingHom.map_mul, \u2190 RingHom.map_mul, \u2190 RingHom.map_add]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nI' : A\nhI' : I' * I' = -1\n\u22a2 \u2191(liftAux I' hI') I = I'\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u_1\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \u211d A\nF : \u2102 \u2192\u2090[\u211d] A\n\u22a2 \u2191F I * \u2191F I = -1\n[PROOFSTEP]\nrw [\u2190 F.map_mul, I_mul_I, AlgHom.map_neg, AlgHom.map_one]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : { x // x \u2208 skewAdjoint A }\n\u22a2 -I \u2022 \u2191a \u2208 selfAdjoint A\n[PROOFSTEP]\nsimp only [neg_smul, neg_mem_iff, selfAdjoint.mem_iff, star_smul, star_def, conj_I, star_val_eq, smul_neg, neg_neg]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 skewAdjoint A }\n\u22a2 (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b) =\n    (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n      (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b\n[PROOFSTEP]\next\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na b : { x // x \u2208 skewAdjoint A }\n\u22a2 \u2191((fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b)) =\n    \u2191((fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n        (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b)\n[PROOFSTEP]\nsimp only [AddSubgroup.coe_add, smul_add, AddMemClass.mk_add_mk]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : \u211d\nb : { x // x \u2208 skewAdjoint A }\n\u22a2 AddHom.toFun\n      { toFun := fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) },\n        map_add' :=\n          (_ :\n            \u2200 (a b : { x // x \u2208 skewAdjoint A }),\n              (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b) =\n                (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n                  (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b) }\n      (a \u2022 b) =\n    \u2191(RingHom.id \u211d) a \u2022\n      AddHom.toFun\n        { toFun := fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) },\n          map_add' :=\n            (_ :\n              \u2200 (a b : { x // x \u2208 skewAdjoint A }),\n                (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b) =\n                  (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n                    (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b) }\n        b\n[PROOFSTEP]\next\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : \u211d\nb : { x // x \u2208 skewAdjoint A }\n\u22a2 \u2191(AddHom.toFun\n        { toFun := fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) },\n          map_add' :=\n            (_ :\n              \u2200 (a b : { x // x \u2208 skewAdjoint A }),\n                (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b) =\n                  (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n                    (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b) }\n        (a \u2022 b)) =\n    \u2191(\u2191(RingHom.id \u211d) a \u2022\n        AddHom.toFun\n          { toFun := fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) },\n            map_add' :=\n              (_ :\n                \u2200 (a b : { x // x \u2208 skewAdjoint A }),\n                  (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) (a + b) =\n                    (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) a +\n                      (fun a => { val := -I \u2022 \u2191a, property := (_ : -I \u2022 \u2191a \u2208 selfAdjoint A) }) b) }\n          b)\n[PROOFSTEP]\nsimp only [neg_smul, skewAdjoint.val_smul, AddSubgroup.coe_mk, RingHom.id_apply, selfAdjoint.val_smul, smul_neg,\n  neg_inj]\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : \u211d\nb : { x // x \u2208 skewAdjoint A }\n\u22a2 I \u2022 a \u2022 \u2191b = a \u2022 I \u2022 \u2191b\n[PROOFSTEP]\nrw [smul_comm]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : { x // x \u2208 skewAdjoint A }\n\u22a2 I \u2022 \u2191(\u2191negISMul a) = \u2191a\n[PROOFSTEP]\nsimp only [smul_smul, skewAdjoint.negISMul_apply_coe, neg_smul, smul_neg, I_mul_I, one_smul, neg_neg]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191\u211c a) = 2\u207b\u00b9 \u2022 (a + star a)\n[PROOFSTEP]\nunfold realPart\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191(selfAdjointPart \u211d) a) = 2\u207b\u00b9 \u2022 (a + star a)\n[PROOFSTEP]\nsimp only [selfAdjointPart_apply_coe, invOf_eq_inv]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191\u2111 a) = -I \u2022 2\u207b\u00b9 \u2022 (a - star a)\n[PROOFSTEP]\nunfold imaginaryPart\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191(LinearMap.comp skewAdjoint.negISMul (skewAdjointPart \u211d)) a) = -I \u2022 2\u207b\u00b9 \u2022 (a - star a)\n[PROOFSTEP]\nsimp only [LinearMap.coe_comp, Function.comp_apply, skewAdjoint.negISMul_apply_coe, skewAdjointPart_apply_coe,\n  invOf_eq_inv, neg_smul]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191\u211c a) + I \u2022 \u2191(\u2191\u2111 a) = a\n[PROOFSTEP]\nsimpa only [smul_smul, realPart_apply_coe, imaginaryPart_apply_coe, neg_smul, I_mul_I, one_smul, neg_sub,\n  add_add_sub_cancel, smul_sub, smul_add, neg_sub_neg, invOf_eq_inv] using invOf_two_smul_add_invOf_two_smul \u211d a\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191\u211c (I \u2022 a) = -\u2191\u2111 a\n[PROOFSTEP]\next\n  -- Porting note: was\n    -- simp [smul_comm I, smul_sub, sub_eq_add_neg, add_comm]\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191\u211c (I \u2022 a)) = \u2191(-\u2191\u2111 a)\n[PROOFSTEP]\nrw [realPart_apply_coe, AddSubgroupClass.coe_neg, imaginaryPart_apply_coe, neg_smul, neg_neg, smul_comm I, star_smul,\n  star_def, conj_I, smul_sub, neg_smul, sub_eq_add_neg]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191\u2111 (I \u2022 a) = \u2191\u211c a\n[PROOFSTEP]\next\n  -- Porting note: was\n    -- simp [smul_comm I, smul_smul I]\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 \u2191(\u2191\u2111 (I \u2022 a)) = \u2191(\u2191\u211c a)\n[PROOFSTEP]\nrw [realPart_apply_coe, imaginaryPart_apply_coe, smul_comm]\n[GOAL]\ncase a\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\na : A\n\u22a2 2\u207b\u00b9 \u2022 -I \u2022 (I \u2022 a - star (I \u2022 a)) = 2\u207b\u00b9 \u2022 (a + star a)\n[PROOFSTEP]\nsimp [\u2190 smul_assoc]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n\u22a2 \u2191\u211c (z \u2022 a) = z.re \u2022 \u2191\u211c a - z.im \u2022 \u2191\u2111 a\n[PROOFSTEP]\nconv_lhs => rw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u211c (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u211c (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u211c (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n\u22a2 \u2191\u211c ((\u2191z.re + \u2191z.im * I) \u2022 a) = z.re \u2022 \u2191\u211c a - z.im \u2022 \u2191\u2111 a\n[PROOFSTEP]\nsimp [-re_add_im, add_smul, \u2190 smul_smul, sub_eq_add_neg]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n\u22a2 \u2191\u2111 (z \u2022 a) = z.re \u2022 \u2191\u2111 a + z.im \u2022 \u2191\u211c a\n[PROOFSTEP]\nconv_lhs => rw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u2111 (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u2111 (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n| \u2191\u2111 (z \u2022 a)\n[PROOFSTEP]\nrw [\u2190 re_add_im z]\n[GOAL]\nA : Type u_1\ninst\u271d\u00b3 : AddCommGroup A\ninst\u271d\u00b2 : Module \u2102 A\ninst\u271d\u00b9 : StarAddMonoid A\ninst\u271d : StarModule \u2102 A\nz : \u2102\na : A\n\u22a2 \u2191\u2111 ((\u2191z.re + \u2191z.im * I) \u2022 a) = z.re \u2022 \u2191\u2111 a + z.im \u2022 \u2191\u211c a\n[PROOFSTEP]\nsimp [-re_add_im, add_smul, \u2190 smul_smul]\n", "meta": {"mathlib_filename": "Mathlib.Data.Complex.Module", "llama_tokens": 16395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.6791787121629466, "lm_q1q2_score": 0.5403940437229099}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nrintro u v hu hv hs \u27e8z, zs, zu\u27e9 \u27e8y, ys, yv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave xs : x \u2208 s := by\n  rcases H y ys with \u27e8t, ts, xt, -, -\u27e9\n  exact ts xt\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases H y ys with \u27e8t, ts, xt, -, -\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact ts xt\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncases hs xs\n[GOAL]\ncase intro.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nh\u271d : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nh\u271d : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncase inl xu =>\n  rcases H y ys with \u27e8t, ts, xt, yt, ht\u27e9\n  have := ht u v hu hv (ts.trans hs) \u27e8x, xt, xu\u27e9 \u27e8y, yt, yv\u27e9\n  exact this.imp fun z hz => \u27e8ts hz.1, hz.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncase inl xu =>\n  rcases H y ys with \u27e8t, ts, xt, yt, ht\u27e9\n  have := ht u v hu hv (ts.trans hs) \u27e8x, xt, xu\u27e9 \u27e8y, yt, yv\u27e9\n  exact this.imp fun z hz => \u27e8ts hz.1, hz.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases H y ys with \u27e8t, ts, xt, yt, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxu : x \u2208 u\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\nyt : y \u2208 t\nht : IsPreconnected t\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave := ht u v hu hv (ts.trans hs) \u27e8x, xt, xu\u27e9 \u27e8y, yt, yv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxu : x \u2208 u\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\nyt : y \u2208 t\nht : IsPreconnected t\nthis : Set.Nonempty (t \u2229 (u \u2229 v))\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact this.imp fun z hz => \u27e8ts hz.1, hz.2\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nh\u271d : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncase inr xv =>\n  rcases H z zs with \u27e8t, ts, xt, zt, ht\u27e9\n  have := ht v u hv hu (ts.trans <| by rwa [union_comm]) \u27e8x, xt, xv\u27e9 \u27e8z, zt, zu\u27e9\n  exact this.imp fun _ h => \u27e8ts h.1, h.2.2, h.2.1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncase inr xv =>\n  rcases H z zs with \u27e8t, ts, xt, zt, ht\u27e9\n  have := ht v u hv hu (ts.trans <| by rwa [union_comm]) \u27e8x, xt, xv\u27e9 \u27e8z, zt, zu\u27e9\n  exact this.imp fun _ h => \u27e8ts h.1, h.2.2, h.2.1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases H z zs with \u27e8t, ts, xt, zt, ht\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxv : x \u2208 v\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\nzt : z \u2208 t\nht : IsPreconnected t\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave := ht v u hv hu (ts.trans <| by rwa [union_comm]) \u27e8x, xt, xv\u27e9 \u27e8z, zt, zu\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxv : x \u2208 v\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\nzt : z \u2208 t\nht : IsPreconnected t\n\u22a2 s \u2286 v \u222a u\n[PROOFSTEP]\nrwa [union_comm]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nx : \u03b1\nH : \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nz : \u03b1\nzs : z \u2208 s\nzu : z \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nxs : x \u2208 s\nxv : x \u2208 v\nt : Set \u03b1\nts : t \u2286 s\nxt : x \u2208 t\nzt : z \u2208 t\nht : IsPreconnected t\nthis : Set.Nonempty (t \u2229 (v \u2229 u))\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact this.imp fun _ h => \u27e8ts h.1, h.2.2, h.2.1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nH : \u2200 (x : \u03b1), x \u2208 \u2205 \u2192 \u2200 (y : \u03b1), y \u2208 \u2205 \u2192 \u2203 t, t \u2286 \u2205 \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\n\u22a2 IsPreconnected \u2205\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nH : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 \u2203 t, t \u2286 s \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nexacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nc : Set (Set \u03b1)\nH1 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 x \u2208 s\nH2 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 IsPreconnected s\n\u22a2 IsPreconnected (\u22c3\u2080 c)\n[PROOFSTEP]\napply isPreconnected_of_forall x\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nc : Set (Set \u03b1)\nH1 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 x \u2208 s\nH2 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 IsPreconnected s\n\u22a2 \u2200 (y : \u03b1), y \u2208 \u22c3\u2080 c \u2192 \u2203 t, t \u2286 \u22c3\u2080 c \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\n[PROOFSTEP]\nrintro y \u27e8s, sc, ys\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nx : \u03b1\nc : Set (Set \u03b1)\nH1 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 x \u2208 s\nH2 : \u2200 (s : Set \u03b1), s \u2208 c \u2192 IsPreconnected s\ny : \u03b1\ns : Set \u03b1\nsc : s \u2208 c\nys : y \u2208 s\n\u22a2 \u2203 t, t \u2286 \u22c3\u2080 c \u2227 x \u2208 t \u2227 y \u2208 t \u2227 IsPreconnected t\n[PROOFSTEP]\nexact \u27e8s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\nH1 : x \u2208 s\nH2 : x \u2208 t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n\u22a2 \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s, t} \u2192 x \u2208 s_1\n[PROOFSTEP]\nrintro r (rfl | rfl | h)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\nx : \u03b1\nt : Set \u03b1\nH2 : x \u2208 t\nH4 : IsPreconnected t\nr : Set \u03b1\nH1 : x \u2208 r\nH3 : IsPreconnected r\n\u22a2 x \u2208 r\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\nH1 : x \u2208 s\nH2 : x \u2208 t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n\u22a2 x \u2208 t\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\nH1 : x \u2208 s\nH2 : x \u2208 t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n\u22a2 \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s, t} \u2192 IsPreconnected s_1\n[PROOFSTEP]\nrintro r (rfl | rfl | h)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\nx : \u03b1\nt : Set \u03b1\nH2 : x \u2208 t\nH4 : IsPreconnected t\nr : Set \u03b1\nH1 : x \u2208 r\nH3 : IsPreconnected r\n\u22a2 IsPreconnected r\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\nH1 : x \u2208 s\nH2 : x \u2208 t\nH3 : IsPreconnected s\nH4 : IsPreconnected t\n\u22a2 IsPreconnected t\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nH : Set.Nonempty (s \u2229 t)\nhs : IsPreconnected s\nht : IsPreconnected t\n\u22a2 IsPreconnected (s \u222a t)\n[PROOFSTEP]\nrcases H with \u27e8x, hxs, hxt\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nhs : IsPreconnected s\nht : IsPreconnected t\nx : \u03b1\nhxs : x \u2208 s\nhxt : x \u2208 t\n\u22a2 IsPreconnected (s \u222a t)\n[PROOFSTEP]\nexact hs.union x hxs hxt ht\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nH : Set.Nonempty (s \u2229 t)\nHs : IsConnected s\nHt : IsConnected t\n\u22a2 IsConnected (s \u222a t)\n[PROOFSTEP]\nrcases H with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nHs : IsConnected s\nHt : IsConnected t\nx : \u03b1\nhx : x \u2208 s \u2229 t\n\u22a2 IsConnected (s \u222a t)\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, mem_union_left t (mem_of_mem_inter_left hx)\u27e9, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nHs : IsConnected s\nHt : IsConnected t\nx : \u03b1\nhx : x \u2208 s \u2229 t\n\u22a2 IsPreconnected (s \u222a t)\n[PROOFSTEP]\nexact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx) Ht.isPreconnected\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nS : Set (Set \u03b1)\nK : DirectedOn (fun x x_1 => x \u2286 x_1) S\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsPreconnected s\n\u22a2 IsPreconnected (\u22c3\u2080 S)\n[PROOFSTEP]\nrintro u v hu hv Huv \u27e8a, \u27e8s, hsS, has\u27e9, hau\u27e9 \u27e8b, \u27e8t, htS, hbt\u27e9, hbv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d : Set \u03b1\nS : Set (Set \u03b1)\nK : DirectedOn (fun x x_1 => x \u2286 x_1) S\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsPreconnected s\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nHuv : \u22c3\u2080 S \u2286 u \u222a v\na : \u03b1\nhau : a \u2208 u\ns : Set \u03b1\nhsS : s \u2208 S\nhas : a \u2208 s\nb : \u03b1\nhbv : b \u2208 v\nt : Set \u03b1\nhtS : t \u2208 S\nhbt : b \u2208 t\n\u22a2 Set.Nonempty (\u22c3\u2080 S \u2229 (u \u2229 v))\n[PROOFSTEP]\nobtain \u27e8r, hrS, hsr, htr\u27e9 : \u2203 r \u2208 S, s \u2286 r \u2227 t \u2286 r := K s hsS t htS\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d : Set \u03b1\nS : Set (Set \u03b1)\nK : DirectedOn (fun x x_1 => x \u2286 x_1) S\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsPreconnected s\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nHuv : \u22c3\u2080 S \u2286 u \u222a v\na : \u03b1\nhau : a \u2208 u\ns : Set \u03b1\nhsS : s \u2208 S\nhas : a \u2208 s\nb : \u03b1\nhbv : b \u2208 v\nt : Set \u03b1\nhtS : t \u2208 S\nhbt : b \u2208 t\nr : Set \u03b1\nhrS : r \u2208 S\nhsr : s \u2286 r\nhtr : t \u2286 r\n\u22a2 Set.Nonempty (\u22c3\u2080 S \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave Hnuv : (r \u2229 (u \u2229 v)).Nonempty :=\n  H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) \u27e8a, hsr has, hau\u27e9 \u27e8b, htr hbt, hbv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d : Set \u03b1\nS : Set (Set \u03b1)\nK : DirectedOn (fun x x_1 => x \u2286 x_1) S\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsPreconnected s\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nHuv : \u22c3\u2080 S \u2286 u \u222a v\na : \u03b1\nhau : a \u2208 u\ns : Set \u03b1\nhsS : s \u2208 S\nhas : a \u2208 s\nb : \u03b1\nhbv : b \u2208 v\nt : Set \u03b1\nhtS : t \u2208 S\nhbt : b \u2208 t\nr : Set \u03b1\nhrS : r \u2208 S\nhsr : s \u2286 r\nhtr : t \u2286 r\nHnuv : Set.Nonempty (r \u2229 (u \u2229 v))\n\u22a2 Set.Nonempty (\u22c3\u2080 S \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave Kruv : r \u2229 (u \u2229 v) \u2286 \u22c3\u2080 S \u2229 (u \u2229 v) := inter_subset_inter_left _ (subset_sUnion_of_mem hrS)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d : Set \u03b1\nS : Set (Set \u03b1)\nK : DirectedOn (fun x x_1 => x \u2286 x_1) S\nH : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsPreconnected s\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nHuv : \u22c3\u2080 S \u2286 u \u222a v\na : \u03b1\nhau : a \u2208 u\ns : Set \u03b1\nhsS : s \u2208 S\nhas : a \u2208 s\nb : \u03b1\nhbv : b \u2208 v\nt : Set \u03b1\nhtS : t \u2208 S\nhbt : b \u2208 t\nr : Set \u03b1\nhrS : r \u2208 S\nhsr : s \u2286 r\nhtr : t \u2286 r\nHnuv : Set.Nonempty (r \u2229 (u \u2229 v))\nKruv : r \u2229 (u \u2229 v) \u2286 \u22c3\u2080 S \u2229 (u \u2229 v)\n\u22a2 Set.Nonempty (\u22c3\u2080 S \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact Hnuv.mono Kruv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\n\u22a2 IsPreconnected (\u22c3 (n : \u03b9) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nlet R := fun i j : \u03b9 => (s i \u2229 s j).Nonempty \u2227 i \u2208 t\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\n\u22a2 IsPreconnected (\u22c3 (n : \u03b9) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nhave P : \u2200 i, i \u2208 t \u2192 \u2200 j, j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 j \u2208 p, s j) :=\n  fun i hi j hj h => by\n  induction h\n  case refl =>\n    refine \u27e8{ i }, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_\u27e9\n    rw [biUnion_singleton]\n    exact H i hi\n  case tail j k _ hjk ih =>\n    obtain \u27e8p, hpt, hip, hjp, hp\u27e9 := ih hjk.2\n    refine \u27e8insert k p, insert_subset_iff.mpr \u27e8hj, hpt\u27e9, mem_insert_of_mem k hip, mem_insert k p, ?_\u27e9\n    rw [biUnion_insert]\n    refine (H k hj).union' (hjk.1.mono ?_) hp\n    rw [inter_comm]\n    exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : j \u2208 t\nh : ReflTransGen R i j\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase refl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : i \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 i \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\ncase tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj b\u271d c\u271d : \u03b9\na\u271d\u00b9 : ReflTransGen R i b\u271d\na\u271d : R b\u271d c\u271d\na_ih\u271d : b\u271d \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 b\u271d \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : c\u271d \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 c\u271d \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\ncase refl =>\n  refine \u27e8{ i }, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_\u27e9\n  rw [biUnion_singleton]\n  exact H i hi\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : i \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 i \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\ncase refl =>\n  refine \u27e8{ i }, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_\u27e9\n  rw [biUnion_singleton]\n  exact H i hi\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : i \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 i \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\nrefine \u27e8{ i }, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : i \u2208 t\n\u22a2 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 {i}), s j)\n[PROOFSTEP]\nrw [biUnion_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj : \u03b9\nhj : i \u2208 t\n\u22a2 IsPreconnected (s i)\n[PROOFSTEP]\nexact H i hi\n[GOAL]\ncase tail\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj b\u271d c\u271d : \u03b9\na\u271d\u00b9 : ReflTransGen R i b\u271d\na\u271d : R b\u271d c\u271d\na_ih\u271d : b\u271d \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 b\u271d \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : c\u271d \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 c\u271d \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\ncase tail j k _ hjk ih =>\n  obtain \u27e8p, hpt, hip, hjp, hp\u27e9 := ih hjk.2\n  refine \u27e8insert k p, insert_subset_iff.mpr \u27e8hj, hpt\u27e9, mem_insert_of_mem k hip, mem_insert k p, ?_\u27e9\n  rw [biUnion_insert]\n  refine (H k hj).union' (hjk.1.mono ?_) hp\n  rw [inter_comm]\n  exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 k \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\ncase tail j k _ hjk ih =>\n  obtain \u27e8p, hpt, hip, hjp, hp\u27e9 := ih hjk.2\n  refine \u27e8insert k p, insert_subset_iff.mpr \u27e8hj, hpt\u27e9, mem_insert_of_mem k hip, mem_insert k p, ?_\u27e9\n  rw [biUnion_insert]\n  refine (H k hj).union' (hjk.1.mono ?_) hp\n  rw [inter_comm]\n  exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 k \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\nobtain \u27e8p, hpt, hip, hjp, hp\u27e9 := ih hjk.2\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 k \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n[PROOFSTEP]\nrefine \u27e8insert k p, insert_subset_iff.mpr \u27e8hj, hpt\u27e9, mem_insert_of_mem k hip, mem_insert k p, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 insert k p), s j)\n[PROOFSTEP]\nrw [biUnion_insert]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 IsPreconnected (s k \u222a \u22c3 (x : \u03b9) (_ : x \u2208 p), s x)\n[PROOFSTEP]\nrefine (H k hj).union' (hjk.1.mono ?_) hp\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 s j \u2229 s k \u2286 s k \u2229 \u22c3 (x : \u03b9) (_ : x \u2208 p), s x\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\ni : \u03b9\nhi : i \u2208 t\nj\u271d j k : \u03b9\na\u271d : ReflTransGen R i j\nhjk : R j k\nih : j \u2208 t \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nhj : k \u2208 t\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 s k \u2229 s j \u2286 s k \u2229 \u22c3 (x : \u03b9) (_ : x \u2208 p), s x\n[PROOFSTEP]\nexact inter_subset_inter_right _ (subset_biUnion_of_mem hjp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 IsPreconnected (\u22c3 (n : \u03b9) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nrefine' isPreconnected_of_forall_pair _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 \u2200 (x : \u03b1),\n    x \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2192\n      \u2200 (y : \u03b1),\n        y \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2192\n          \u2203 t_1, t_1 \u2286 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nx : \u03b1\nhx : x \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ny : \u03b1\nhy : y \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\n\u22a2 \u2203 t_1, t_1 \u2286 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nobtain \u27e8i : \u03b9, hi : i \u2208 t, hxi : x \u2208 s i\u27e9 := mem_iUnion\u2082.1 hx\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nx : \u03b1\nhx : x \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ny : \u03b1\nhy : y \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ni : \u03b9\nhi : i \u2208 t\nhxi : x \u2208 s i\n\u22a2 \u2203 t_1, t_1 \u2286 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nobtain \u27e8j : \u03b9, hj : j \u2208 t, hyj : y \u2208 s j\u27e9 := mem_iUnion\u2082.1 hy\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nx : \u03b1\nhx : x \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ny : \u03b1\nhy : y \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ni : \u03b9\nhi : i \u2208 t\nhxi : x \u2208 s i\nj : \u03b9\nhj : j \u2208 t\nhyj : y \u2208 s j\n\u22a2 \u2203 t_1, t_1 \u2286 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nobtain \u27e8p, hpt, hip, hjp, hp\u27e9 := P i hi j hj (K i hi j hj)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\n\u03b9 : Type u_3\nt : Set \u03b9\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), i \u2208 t \u2192 IsPreconnected (s i)\nK : \u2200 (i : \u03b9), i \u2208 t \u2192 \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\nR : \u03b9 \u2192 \u03b9 \u2192 Prop := fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t\nP :\n  \u2200 (i : \u03b9),\n    i \u2208 t \u2192\n      \u2200 (j : \u03b9), j \u2208 t \u2192 ReflTransGen R i j \u2192 \u2203 p, p \u2286 t \u2227 i \u2208 p \u2227 j \u2208 p \u2227 IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\nx : \u03b1\nhx : x \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ny : \u03b1\nhy : y \u2208 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n\ni : \u03b9\nhi : i \u2208 t\nhxi : x \u2208 s i\nj : \u03b9\nhj : j \u2208 t\nhyj : y \u2208 s j\np : Set \u03b9\nhpt : p \u2286 t\nhip : i \u2208 p\nhjp : j \u2208 p\nhp : IsPreconnected (\u22c3 (j : \u03b9) (_ : j \u2208 p), s j)\n\u22a2 \u2203 t_1, t_1 \u2286 \u22c3 (n : \u03b9) (_ : n \u2208 t), s n \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nexact \u27e8\u22c3 j \u2208 p, s j, biUnion_subset_biUnion_left hpt, mem_biUnion hip hxi, mem_biUnion hjp hyj, hp\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\n\u03b9 : Type u_3\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsPreconnected (s i)\nK : \u2200 (i j : \u03b9), ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j)) i j\n\u22a2 IsPreconnected (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nrw [\u2190 biUnion_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\n\u03b9 : Type u_3\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsPreconnected (s i)\nK : \u2200 (i j : \u03b9), ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j)) i j\n\u22a2 IsPreconnected (\u22c3 (x : \u03b9) (_ : x \u2208 univ), s x)\n[PROOFSTEP]\nexact IsPreconnected.biUnion_of_reflTransGen (fun i _ => H i) fun i _ j _ => by simpa [mem_univ] using K i j\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\n\u03b9 : Type u_3\ns : \u03b9 \u2192 Set \u03b1\nH : \u2200 (i : \u03b9), IsPreconnected (s i)\nK : \u2200 (i j : \u03b9), ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j)) i j\ni : \u03b9\nx\u271d\u00b9 : i \u2208 univ\nj : \u03b9\nx\u271d : j \u2208 univ\n\u22a2 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 univ) i j\n[PROOFSTEP]\nsimpa [mem_univ] using K i j\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nH : \u2200 (n : \u03b2), IsPreconnected (s n)\nK : \u2200 (n : \u03b2), Set.Nonempty (s n \u2229 s (succ n))\ni\u271d j i : \u03b2\nx\u271d : i \u2208 Ico j i\u271d\n\u22a2 Set.Nonempty (s (succ i) \u2229 s i)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nH : \u2200 (n : \u03b2), IsPreconnected (s n)\nK : \u2200 (n : \u03b2), Set.Nonempty (s n \u2229 s (succ n))\ni\u271d j i : \u03b2\nx\u271d : i \u2208 Ico j i\u271d\n\u22a2 Set.Nonempty (s i \u2229 s (succ i))\n[PROOFSTEP]\nexact K i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : SuccOrder \u03b2\ninst\u271d\u00b9 : IsSuccArchimedean \u03b2\ninst\u271d : Nonempty \u03b2\ns : \u03b2 \u2192 Set \u03b1\nH : \u2200 (n : \u03b2), IsConnected (s n)\nK : \u2200 (n : \u03b2), Set.Nonempty (s n \u2229 s (succ n))\ni\u271d j i : \u03b2\nx\u271d : i \u2208 Ico j i\u271d\n\u22a2 Set.Nonempty (s (succ i) \u2229 s i)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : SuccOrder \u03b2\ninst\u271d\u00b9 : IsSuccArchimedean \u03b2\ninst\u271d : Nonempty \u03b2\ns : \u03b2 \u2192 Set \u03b1\nH : \u2200 (n : \u03b2), IsConnected (s n)\nK : \u2200 (n : \u03b2), Set.Nonempty (s n \u2229 s (succ n))\ni\u271d j i : \u03b2\nx\u271d : i \u2208 Ico j i\u271d\n\u22a2 Set.Nonempty (s i \u2229 s (succ i))\n[PROOFSTEP]\nexact K i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\n\u22a2 IsPreconnected (\u22c3 (n : \u03b2) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nhave h1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t := fun hi hj hk => ht.out hi hj (Ico_subset_Icc_self hk)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\n\u22a2 IsPreconnected (\u22c3 (n : \u03b2) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nhave h2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t := fun hi hj hk =>\n  ht.out hi hj \u27e8hk.1.trans <| le_succ _, succ_le_of_lt hk.2\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\nh2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t\n\u22a2 IsPreconnected (\u22c3 (n : \u03b2) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nhave h3 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 (s k \u2229 s (succ k)).Nonempty := fun hi hj hk =>\n  K _ (h1 hi hj hk) (h2 hi hj hk)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\nh2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t\nh3 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 Set.Nonempty (s k \u2229 s (succ k))\n\u22a2 IsPreconnected (\u22c3 (n : \u03b2) (_ : n \u2208 t), s n)\n[PROOFSTEP]\nrefine' IsPreconnected.biUnion_of_reflTransGen H fun i hi j hj => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\nh2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t\nh3 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 Set.Nonempty (s k \u2229 s (succ k))\ni : \u03b2\nhi : i \u2208 t\nj : \u03b2\nhj : j \u2208 t\n\u22a2 ReflTransGen (fun i j => Set.Nonempty (s i \u2229 s j) \u2227 i \u2208 t) i j\n[PROOFSTEP]\nexact\n  reflTransGen_of_succ _ (fun k hk => \u27e8h3 hi hj hk, h1 hi hj hk\u27e9) fun k hk =>\n    \u27e8by rw [inter_comm]; exact h3 hj hi hk, h2 hj hi hk\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\nh2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t\nh3 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 Set.Nonempty (s k \u2229 s (succ k))\ni : \u03b2\nhi : i \u2208 t\nj : \u03b2\nhj : j \u2208 t\nk : \u03b2\nhk : k \u2208 Ico j i\n\u22a2 Set.Nonempty (s (succ k) \u2229 s k)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : SuccOrder \u03b2\ninst\u271d : IsSuccArchimedean \u03b2\ns : \u03b2 \u2192 Set \u03b1\nt : Set \u03b2\nht : OrdConnected t\nH : \u2200 (n : \u03b2), n \u2208 t \u2192 IsPreconnected (s n)\nK : \u2200 (n : \u03b2), n \u2208 t \u2192 succ n \u2208 t \u2192 Set.Nonempty (s n \u2229 s (succ n))\nh1 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 k \u2208 t\nh2 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 succ k \u2208 t\nh3 : \u2200 {i j k : \u03b2}, i \u2208 t \u2192 j \u2208 t \u2192 k \u2208 Ico i j \u2192 Set.Nonempty (s k \u2229 s (succ k))\ni : \u03b2\nhi : i \u2208 t\nj : \u03b2\nhj : j \u2208 t\nk : \u03b2\nhk : k \u2208 Ico j i\n\u22a2 Set.Nonempty (s k \u2229 s (succ k))\n[PROOFSTEP]\nexact h3 hj hi hk\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u22a2 IsPreconnected (f '' s)\n[PROOFSTEP]\nrintro u v hu hv huv \u27e8_, \u27e8x, xs, rfl\u27e9, xu\u27e9 \u27e8_, \u27e8y, ys, rfl\u27e9, yv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nhuv : f '' s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases continuousOn_iff'.1 hf u hu with \u27e8u', hu', u'_eq\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nhuv : f '' s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases continuousOn_iff'.1 hf v hv with\n  \u27e8v', hv', v'_eq\u27e9\n    -- Reformulate `huv : f '' s \u2286 u \u222a v` in terms of `u'` and `v'`\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nhuv : f '' s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nreplace huv : s \u2286 u' \u222a v'\n[GOAL]\ncase huv\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nhuv : f '' s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\n\u22a2 s \u2286 u' \u222a v'\n[PROOFSTEP]\nrw [image_subset_iff, preimage_union] at huv \n[GOAL]\ncase huv\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 f \u207b\u00b9' u \u222a f \u207b\u00b9' v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\n\u22a2 s \u2286 u' \u222a v'\n[PROOFSTEP]\nreplace huv := subset_inter huv Subset.rfl\n[GOAL]\ncase huv\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 (f \u207b\u00b9' u \u222a f \u207b\u00b9' v) \u2229 s\n\u22a2 s \u2286 u' \u222a v'\n[PROOFSTEP]\nrw [inter_distrib_right, u'_eq, v'_eq, \u2190 inter_distrib_right] at huv \n[GOAL]\ncase huv\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 (u' \u222a v') \u2229 s\n\u22a2 s \u2286 u' \u222a v'\n[PROOFSTEP]\nexact\n  (subset_inter_iff.1 huv).1\n    -- Now `s \u2286 u' \u222a v'`, so we can apply `\u2039IsPreconnected s\u203a`\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 : (s \u2229 (u' \u2229 v')).Nonempty :=\n  by\n  refine H u' v' hu' hv' huv \u27e8x, ?_\u27e9 \u27e8y, ?_\u27e9 <;> rw [inter_comm]\n  exacts [u'_eq \u25b8 \u27e8xu, xs\u27e9, v'_eq \u25b8 \u27e8yv, ys\u27e9]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 Set.Nonempty (s \u2229 (u' \u2229 v'))\n[PROOFSTEP]\nrefine H u' v' hu' hv' huv \u27e8x, ?_\u27e9 \u27e8y, ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 x \u2208 s \u2229 u'\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 y \u2208 s \u2229 v'\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 x \u2208 u' \u2229 s\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\n\u22a2 y \u2208 v' \u2229 s\n[PROOFSTEP]\nexacts [u'_eq \u25b8 \u27e8xu, xs\u27e9, v'_eq \u25b8 \u27e8yv, ys\u27e9]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\nz : \u03b1\nhz : z \u2208 s \u2229 (u' \u2229 v')\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [\u2190 inter_self s, inter_assoc, inter_left_comm s u', \u2190 inter_assoc, inter_comm s, inter_comm s, \u2190 u'_eq, \u2190 v'_eq] at\n  hz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nH : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\nu v : Set \u03b2\nhu : IsOpen u\nhv : IsOpen v\nx : \u03b1\nxs : x \u2208 s\nxu : f x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : f y \u2208 v\nu' : Set \u03b1\nhu' : IsOpen u'\nu'_eq : f \u207b\u00b9' u \u2229 s = u' \u2229 s\nv' : Set \u03b1\nhv' : IsOpen v'\nv'_eq : f \u207b\u00b9' v \u2229 s = v' \u2229 s\nhuv : s \u2286 u' \u222a v'\nz : \u03b1\nhz : z \u2208 f \u207b\u00b9' u \u2229 s \u2229 (f \u207b\u00b9' v \u2229 s)\n\u22a2 Set.Nonempty (f '' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8f z, \u27e8z, hz.1.2, rfl\u27e9, hz.1.1, hz.2.1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 IsPreconnected s \u2192\n    \u2200 (t t' : Set \u03b1),\n      IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\n[PROOFSTEP]\nrintro h t t' ht ht' htt' \u27e8x, xs, xt\u27e9 \u27e8y, ys, yt'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\n\u22a2 Set.Nonempty (s \u2229 (t \u2229 t'))\n[PROOFSTEP]\nrw [\u2190 not_disjoint_iff_nonempty_inter, \u2190 subset_compl_iff_disjoint_right, compl_inter]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\n\u22a2 \u00acs \u2286 t\u1d9c \u222a t'\u1d9c\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\nh' : s \u2286 t\u1d9c \u222a t'\u1d9c\n\u22a2 False\n[PROOFSTEP]\nhave xt' : x \u2209 t' := (h' xs).resolve_left (absurd xt)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\nh' : s \u2286 t\u1d9c \u222a t'\u1d9c\nxt' : \u00acx \u2208 t'\n\u22a2 False\n[PROOFSTEP]\nhave yt : y \u2209 t := (h' ys).resolve_right (absurd yt')\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\nh' : s \u2286 t\u1d9c \u222a t'\u1d9c\nxt' : \u00acx \u2208 t'\nyt : \u00acy \u2208 t\n\u22a2 False\n[PROOFSTEP]\nhave := h _ _ ht.isOpen_compl ht'.isOpen_compl h' \u27e8y, ys, yt\u27e9 \u27e8x, xs, xt'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\nh' : s \u2286 t\u1d9c \u222a t'\u1d9c\nxt' : \u00acx \u2208 t'\nyt : \u00acy \u2208 t\nthis : Set.Nonempty (s \u2229 (t\u1d9c \u2229 t'\u1d9c))\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 compl_union] at this \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nh : IsPreconnected s\nt t' : Set \u03b1\nht : IsClosed t\nht' : IsClosed t'\nhtt' : s \u2286 t \u222a t'\nx : \u03b1\nxs : x \u2208 s\nxt : x \u2208 t\ny : \u03b1\nys : y \u2208 s\nyt' : y \u2208 t'\nh' : s \u2286 t\u1d9c \u222a t'\u1d9c\nxt' : \u00acx \u2208 t'\nyt : \u00acy \u2208 t\nthis : Set.Nonempty (s \u2229 (t \u222a t')\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nexact this.ne_empty htt'.disjoint_compl_right.inter_eq\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 (\u2200 (t t' : Set \u03b1),\n      IsClosed t \u2192\n        IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))) \u2192\n    IsPreconnected s\n[PROOFSTEP]\nrintro h u v hu hv huv \u27e8x, xs, xu\u27e9 \u27e8y, ys, yv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [\u2190 not_disjoint_iff_nonempty_inter, \u2190 subset_compl_iff_disjoint_right, compl_inter]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\n\u22a2 \u00acs \u2286 u\u1d9c \u222a v\u1d9c\n[PROOFSTEP]\nintro h'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nh' : s \u2286 u\u1d9c \u222a v\u1d9c\n\u22a2 False\n[PROOFSTEP]\nhave xv : x \u2209 v := (h' xs).elim (absurd xu) id\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nh' : s \u2286 u\u1d9c \u222a v\u1d9c\nxv : \u00acx \u2208 v\n\u22a2 False\n[PROOFSTEP]\nhave yu : y \u2209 u := (h' ys).elim id (absurd yv)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nh' : s \u2286 u\u1d9c \u222a v\u1d9c\nxv : \u00acx \u2208 v\nyu : \u00acy \u2208 u\n\u22a2 False\n[PROOFSTEP]\nhave := h _ _ hu.isClosed_compl hv.isClosed_compl h' \u27e8y, ys, yu\u27e9 \u27e8x, xs, xv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nh' : s \u2286 u\u1d9c \u222a v\u1d9c\nxv : \u00acx \u2208 v\nyu : \u00acy \u2208 u\nthis : Set.Nonempty (s \u2229 (u\u1d9c \u2229 v\u1d9c))\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 compl_union] at this \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nxs : x \u2208 s\nxu : x \u2208 u\ny : \u03b1\nys : y \u2208 s\nyv : y \u2208 v\nh' : s \u2286 u\u1d9c \u222a v\u1d9c\nxv : \u00acx \u2208 v\nyu : \u00acy \u2208 u\nthis : Set.Nonempty (s \u2229 (u \u222a v)\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nexact this.ne_empty huv.disjoint_compl_right.inter_eq\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\n\u22a2 IsPreconnected (f '' s) \u2194 IsPreconnected s\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.image _ hf.continuous.continuousOn\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nrintro u v hu' hv' huv \u27e8x, hxs, hxu\u27e9 \u27e8y, hys, hyv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nu v : Set \u03b1\nhu' : IsOpen u\nhv' : IsOpen v\nhuv : s \u2286 u \u222a v\nx : \u03b1\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : y \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases hf.isOpen_iff.1 hu' with \u27e8u, hu, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nv : Set \u03b1\nhv' : IsOpen v\nx : \u03b1\nhxs : x \u2208 s\ny : \u03b1\nhys : y \u2208 s\nhyv : y \u2208 v\nu : Set \u03b2\nhu : IsOpen u\nhu' : IsOpen (f \u207b\u00b9' u)\nhuv : s \u2286 f \u207b\u00b9' u \u222a v\nhxu : x \u2208 f \u207b\u00b9' u\n\u22a2 Set.Nonempty (s \u2229 (f \u207b\u00b9' u \u2229 v))\n[PROOFSTEP]\nrcases hf.isOpen_iff.1 hv' with \u27e8v, hv, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : \u03b1\nhxs : x \u2208 s\ny : \u03b1\nhys : y \u2208 s\nu : Set \u03b2\nhu : IsOpen u\nhu' : IsOpen (f \u207b\u00b9' u)\nhxu : x \u2208 f \u207b\u00b9' u\nv : Set \u03b2\nhv : IsOpen v\nhv' : IsOpen (f \u207b\u00b9' v)\nhyv : y \u2208 f \u207b\u00b9' v\nhuv : s \u2286 f \u207b\u00b9' u \u222a f \u207b\u00b9' v\n\u22a2 Set.Nonempty (s \u2229 (f \u207b\u00b9' u \u2229 f \u207b\u00b9' v))\n[PROOFSTEP]\nreplace huv : f '' s \u2286 u \u222a v\n[GOAL]\ncase huv\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : \u03b1\nhxs : x \u2208 s\ny : \u03b1\nhys : y \u2208 s\nu : Set \u03b2\nhu : IsOpen u\nhu' : IsOpen (f \u207b\u00b9' u)\nhxu : x \u2208 f \u207b\u00b9' u\nv : Set \u03b2\nhv : IsOpen v\nhv' : IsOpen (f \u207b\u00b9' v)\nhyv : y \u2208 f \u207b\u00b9' v\nhuv : s \u2286 f \u207b\u00b9' u \u222a f \u207b\u00b9' v\n\u22a2 f '' s \u2286 u \u222a v\n[PROOFSTEP]\nrwa [image_subset_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : \u03b1\nhxs : x \u2208 s\ny : \u03b1\nhys : y \u2208 s\nu : Set \u03b2\nhu : IsOpen u\nhu' : IsOpen (f \u207b\u00b9' u)\nhxu : x \u2208 f \u207b\u00b9' u\nv : Set \u03b2\nhv : IsOpen v\nhv' : IsOpen (f \u207b\u00b9' v)\nhyv : y \u2208 f \u207b\u00b9' v\nhuv : f '' s \u2286 u \u222a v\n\u22a2 Set.Nonempty (s \u2229 (f \u207b\u00b9' u \u2229 f \u207b\u00b9' v))\n[PROOFSTEP]\nrcases h u v hu hv huv \u27e8f x, mem_image_of_mem _ hxs, hxu\u27e9 \u27e8f y, mem_image_of_mem _ hys, hyv\u27e9 with\n  \u27e8_, \u27e8z, hzs, rfl\u27e9, hzuv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Inducing f\nh : IsPreconnected (f '' s)\nx : \u03b1\nhxs : x \u2208 s\ny : \u03b1\nhys : y \u2208 s\nu : Set \u03b2\nhu : IsOpen u\nhu' : IsOpen (f \u207b\u00b9' u)\nhxu : x \u2208 f \u207b\u00b9' u\nv : Set \u03b2\nhv : IsOpen v\nhv' : IsOpen (f \u207b\u00b9' v)\nhyv : y \u2208 f \u207b\u00b9' v\nhuv : f '' s \u2286 u \u222a v\nz : \u03b1\nhzs : z \u2208 s\nhzuv : f z \u2208 u \u2229 v\n\u22a2 Set.Nonempty (s \u2229 (f \u207b\u00b9' u \u2229 f \u207b\u00b9' v))\n[PROOFSTEP]\nexact \u27e8z, hzs, hzuv\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nhsf : s \u2286 range f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nreplace hsf : f '' (f \u207b\u00b9' s) = s := image_preimage_eq_of_subset hsf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nobtain \u27e8_, has, \u27e8a, hau, rfl\u27e9, hav\u27e9 : (s \u2229 (f '' u \u2229 f '' v)).Nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 (f '' u \u2229 f '' v))\n[PROOFSTEP]\nrefine hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 s \u2286 f '' u \u222a f '' v\n[PROOFSTEP]\nsimpa only [hsf, image_union] using image_subset f hsuv\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 f '' u)\n[PROOFSTEP]\nsimpa only [image_preimage_inter] using hsu.image f\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 f '' v)\n[PROOFSTEP]\nsimpa only [image_preimage_inter] using hsv.image f\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\na : \u03b1\nhau : a \u2208 u\nhas : f a \u2208 s\nhav : f a \u2208 f '' v\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8a, has, hau, hinj.mem_set_image.1 hav\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nhsf : s \u2286 range f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nreplace hsf : f '' (f \u207b\u00b9' s) = s := image_preimage_eq_of_subset hsf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nobtain \u27e8_, has, \u27e8a, hau, rfl\u27e9, hav\u27e9 : (s \u2229 (f '' u \u2229 f '' v)).Nonempty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 (f '' u \u2229 f '' v))\n[PROOFSTEP]\nrefine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 s \u2286 f '' u \u222a f '' v\n[PROOFSTEP]\nsimpa only [hsf, image_union] using image_subset f hsuv\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 f '' u)\n[PROOFSTEP]\nsimpa only [image_preimage_inter] using hsu.image f\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\n\u22a2 Set.Nonempty (s \u2229 f '' v)\n[PROOFSTEP]\nsimpa only [image_preimage_inter] using hsv.image f\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b2\nhs : IsPreconnected s\nf : \u03b1 \u2192 \u03b2\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f \u207b\u00b9' s \u2286 u \u222a v\nhsu : Set.Nonempty (f \u207b\u00b9' s \u2229 u)\nhsv : Set.Nonempty (f \u207b\u00b9' s \u2229 v)\nhsf : f '' (f \u207b\u00b9' s) = s\na : \u03b1\nhau : a \u2208 u\nhas : f a \u2208 s\nhav : f a \u2208 f '' v\n\u22a2 Set.Nonempty (f \u207b\u00b9' s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8a, has, hau, hinj.mem_set_image.1 hav\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhs : IsPreconnected s\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nspecialize hs u v hu hv hsuv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhs : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nobtain hsu | hsu := (s \u2229 u).eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhs : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhsu : s \u2229 u = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nexact Or.inr ((Set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhs : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhsu : Set.Nonempty (s \u2229 u)\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nreplace hs := mt (hs hsu)\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : \u00acSet.Nonempty (s \u2229 (u \u2229 v)) \u2192 \u00acSet.Nonempty (s \u2229 v)\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nsimp_rw [Set.not_nonempty_iff_eq_empty, \u2190 Set.disjoint_iff_inter_eq_empty, disjoint_iff_inter_eq_empty.1 huv] at hs \n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : Disjoint s \u2205 \u2192 Disjoint s v\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nexact Or.inl ((hs s.disjoint_empty).subset_left_of_subset_union hsuv)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : IsPreconnected s\n\u22a2 Disjoint s v\n[PROOFSTEP]\nby_contra hsv\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : IsPreconnected s\nhsv : \u00acDisjoint s v\n\u22a2 False\n[PROOFSTEP]\nrw [not_disjoint_iff_nonempty_inter] at hsv \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : IsPreconnected s\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, _, hx\u27e9 := hs u v hu hv hsuv hsu hsv\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhs : IsPreconnected s\nhsv : Set.Nonempty (s \u2229 v)\nx : \u03b1\nleft\u271d : x \u2208 s\nhx : x \u2208 u \u2229 v\n\u22a2 False\n[PROOFSTEP]\nexact Set.disjoint_iff.1 huv hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s t : Set \u03b1\nhs : IsPreconnected s\nht : IsClopen t\nhne : Set.Nonempty (s \u2229 t)\n\u22a2 s \u2286 t \u222a t\u1d9c\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\n\u22a2 s \u2286 u\n[PROOFSTEP]\nhave A : s \u2286 u \u222a (closure u)\u1d9c := by\n  intro x hx\n  by_cases xu : x \u2208 u\n  \u00b7 exact Or.inl xu\n  \u00b7 right\n    intro h'x\n    exact xu (h (mem_inter h'x hx))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\n\u22a2 s \u2286 u \u222a (closure u)\u1d9c\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nx : \u03b1\nhx : x \u2208 s\n\u22a2 x \u2208 u \u222a (closure u)\u1d9c\n[PROOFSTEP]\nby_cases xu : x \u2208 u\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nx : \u03b1\nhx : x \u2208 s\nxu : x \u2208 u\n\u22a2 x \u2208 u \u222a (closure u)\u1d9c\n[PROOFSTEP]\nexact Or.inl xu\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nx : \u03b1\nhx : x \u2208 s\nxu : \u00acx \u2208 u\n\u22a2 x \u2208 u \u222a (closure u)\u1d9c\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nx : \u03b1\nhx : x \u2208 s\nxu : \u00acx \u2208 u\n\u22a2 x \u2208 (closure u)\u1d9c\n[PROOFSTEP]\nintro h'x\n[GOAL]\ncase neg.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nx : \u03b1\nhx : x \u2208 s\nxu : \u00acx \u2208 u\nh'x : x \u2208 closure u\n\u22a2 False\n[PROOFSTEP]\nexact xu (h (mem_inter h'x hx))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nA : s \u2286 u \u222a (closure u)\u1d9c\n\u22a2 s \u2286 u\n[PROOFSTEP]\napply hs.subset_left_of_subset_union hu isClosed_closure.isOpen_compl _ A h'u\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nhs : IsPreconnected s\nhu : IsOpen u\nh'u : Set.Nonempty (s \u2229 u)\nh : closure u \u2229 s \u2286 u\nA : s \u2286 u \u222a (closure u)\u1d9c\n\u22a2 Disjoint u (closure u)\u1d9c\n[PROOFSTEP]\nexact disjoint_compl_right.mono_right (compl_subset_compl.2 subset_closure)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\n\u22a2 IsPreconnected (s \u00d7\u02e2 t)\n[PROOFSTEP]\napply isPreconnected_of_forall_pair\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\n\u22a2 \u2200 (x : \u03b1 \u00d7 \u03b2), x \u2208 s \u00d7\u02e2 t \u2192 \u2200 (y : \u03b1 \u00d7 \u03b2), y \u2208 s \u00d7\u02e2 t \u2192 \u2203 t_1, t_1 \u2286 s \u00d7\u02e2 t \u2227 x \u2208 t_1 \u2227 y \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8ha\u2081, hb\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 \u27e8ha\u2082, hb\u2082\u27e9\n[GOAL]\ncase H.mk.intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\na\u2081 : \u03b1\nb\u2081 : \u03b2\nha\u2081 : (a\u2081, b\u2081).fst \u2208 s\nhb\u2081 : (a\u2081, b\u2081).snd \u2208 t\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u2082 : (a\u2082, b\u2082).fst \u2208 s\nhb\u2082 : (a\u2082, b\u2082).snd \u2208 t\n\u22a2 \u2203 t_1, t_1 \u2286 s \u00d7\u02e2 t \u2227 (a\u2081, b\u2081) \u2208 t_1 \u2227 (a\u2082, b\u2082) \u2208 t_1 \u2227 IsPreconnected t_1\n[PROOFSTEP]\nrefine' \u27e8Prod.mk a\u2081 '' t \u222a flip Prod.mk b\u2082 '' s, _, .inl \u27e8b\u2081, hb\u2081, rfl\u27e9, .inr \u27e8a\u2082, ha\u2082, rfl\u27e9, _\u27e9\n[GOAL]\ncase H.mk.intro.mk.intro.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\na\u2081 : \u03b1\nb\u2081 : \u03b2\nha\u2081 : (a\u2081, b\u2081).fst \u2208 s\nhb\u2081 : (a\u2081, b\u2081).snd \u2208 t\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u2082 : (a\u2082, b\u2082).fst \u2208 s\nhb\u2082 : (a\u2082, b\u2082).snd \u2208 t\n\u22a2 Prod.mk a\u2081 '' t \u222a flip Prod.mk b\u2082 '' s \u2286 s \u00d7\u02e2 t\n[PROOFSTEP]\nrintro _ (\u27e8y, hy, rfl\u27e9 | \u27e8x, hx, rfl\u27e9)\n[GOAL]\ncase H.mk.intro.mk.intro.refine'_1.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\na\u2081 : \u03b1\nb\u2081 : \u03b2\nha\u2081 : (a\u2081, b\u2081).fst \u2208 s\nhb\u2081 : (a\u2081, b\u2081).snd \u2208 t\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u2082 : (a\u2082, b\u2082).fst \u2208 s\nhb\u2082 : (a\u2082, b\u2082).snd \u2208 t\ny : \u03b2\nhy : y \u2208 t\n\u22a2 (a\u2081, y) \u2208 s \u00d7\u02e2 t\ncase H.mk.intro.mk.intro.refine'_1.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\na\u2081 : \u03b1\nb\u2081 : \u03b2\nha\u2081 : (a\u2081, b\u2081).fst \u2208 s\nhb\u2081 : (a\u2081, b\u2081).snd \u2208 t\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u2082 : (a\u2082, b\u2082).fst \u2208 s\nhb\u2082 : (a\u2082, b\u2082).snd \u2208 t\nx : \u03b1\nhx : x \u2208 s\n\u22a2 flip Prod.mk b\u2082 x \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nexacts [\u27e8ha\u2081, hy\u27e9, \u27e8hx, hb\u2082\u27e9]\n[GOAL]\ncase H.mk.intro.mk.intro.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : IsPreconnected s\nht : IsPreconnected t\na\u2081 : \u03b1\nb\u2081 : \u03b2\nha\u2081 : (a\u2081, b\u2081).fst \u2208 s\nhb\u2081 : (a\u2081, b\u2081).snd \u2208 t\na\u2082 : \u03b1\nb\u2082 : \u03b2\nha\u2082 : (a\u2082, b\u2082).fst \u2208 s\nhb\u2082 : (a\u2082, b\u2082).snd \u2208 t\n\u22a2 IsPreconnected (Prod.mk a\u2081 '' t \u222a flip Prod.mk b\u2082 '' s)\n[PROOFSTEP]\nexact\n  (ht.image _ (Continuous.Prod.mk _).continuousOn).union (a\u2081, b\u2082) \u27e8b\u2082, hb\u2082, rfl\u27e9 \u27e8a\u2081, ha\u2081, rfl\u27e9\n    (hs.image _ (continuous_id.prod_mk continuous_const).continuousOn)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\n\u22a2 IsPreconnected (pi univ s)\n[PROOFSTEP]\nrintro u v uo vo hsuv \u27e8f, hfs, hfu\u27e9 \u27e8g, hgs, hgv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with \u27e8I, hI\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI : Finset \u03b9\nhI : Finset.piecewise I f g \u2208 u\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\ninduction' I using Finset.induction_on with i I _ ihI\n[GOAL]\ncase intro.intro.intro.intro.intro.empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI : Finset \u03b9\nhI\u271d : Finset.piecewise I f g \u2208 u\nhI : Finset.piecewise \u2205 f g \u2208 u\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrefine' \u27e8g, hgs, \u27e8_, hgv\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI : Finset \u03b9\nhI\u271d : Finset.piecewise I f g \u2208 u\nhI : Finset.piecewise \u2205 f g \u2208 u\n\u22a2 g \u2208 u\n[PROOFSTEP]\nsimpa using hI\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : Finset.piecewise (insert i I) f g \u2208 u\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [Finset.piecewise_insert] at hI \n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave := I.piecewise_mem_set_pi hfs hgs\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrefine' (hsuv this).elim ihI fun h => _\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nset S := update (I.piecewise f g) i '' s i\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hsub : S \u2286 pi univ s := by\n  refine' image_subset_iff.2 fun z hz => _\n  rwa [update_preimage_univ_pi]\n  exact fun j _ => this j trivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\n\u22a2 S \u2286 pi univ s\n[PROOFSTEP]\nrefine' image_subset_iff.2 fun z hz => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nz : \u03c0 i\nhz : z \u2208 s i\n\u22a2 z \u2208 update (Finset.piecewise I f g) i \u207b\u00b9' pi univ s\n[PROOFSTEP]\nrwa [update_preimage_univ_pi]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nz : \u03c0 i\nhz : z \u2208 s i\n\u22a2 \u2200 (j : \u03b9), j \u2260 i \u2192 Finset.piecewise I f g j \u2208 s j\n[PROOFSTEP]\nexact fun j _ => this j trivial\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nhsub : S \u2286 pi univ s\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hconn : IsPreconnected S := (hs i).image _ (continuous_const.update i continuous_id).continuousOn\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nhsub : S \u2286 pi univ s\nhconn : IsPreconnected S\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hSu : (S \u2229 u).Nonempty := \u27e8_, mem_image_of_mem _ (hfs _ trivial), hI\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nhsub : S \u2286 pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S \u2229 u)\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hSv : (S \u2229 v).Nonempty := \u27e8_, \u27e8_, this _ trivial, update_eq_self _ _\u27e9, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nhsub : S \u2286 pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S \u2229 u)\nhSv : Set.Nonempty (S \u2229 v)\n\u22a2 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrefine' (hconn u v uo vo (hsub.trans hsuv) hSu hSv).mono _\n[GOAL]\ncase intro.intro.intro.intro.intro.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhs : \u2200 (i : \u03b9), IsPreconnected (s i)\nu v : Set ((i : \u03b9) \u2192 \u03c0 i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s \u2286 u \u222a v\nf : (i : \u03b9) \u2192 \u03c0 i\nhfs : f \u2208 pi univ s\nhfu : f \u2208 u\ng : (i : \u03b9) \u2192 \u03c0 i\nhgs : g \u2208 pi univ s\nhgv : g \u2208 v\nI\u271d : Finset \u03b9\nhI\u271d : Finset.piecewise I\u271d f g \u2208 u\ni : \u03b9\nI : Finset \u03b9\na\u271d : \u00aci \u2208 I\nihI : Finset.piecewise I f g \u2208 u \u2192 Set.Nonempty (pi univ s \u2229 (u \u2229 v))\nhI : update (Finset.piecewise I f g) i (f i) \u2208 u\nthis : Finset.piecewise I f g \u2208 pi univ s\nh : Finset.piecewise I f g \u2208 v\nS : Set ((a : \u03b9) \u2192 \u03c0 a) := update (Finset.piecewise I f g) i '' s i\nhsub : S \u2286 pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S \u2229 u)\nhSv : Set.Nonempty (S \u2229 v)\n\u22a2 S \u2229 (u \u2229 v) \u2286 pi univ s \u2229 (u \u2229 v)\n[PROOFSTEP]\nexact inter_subset_inter_left _ hsub\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 IsConnected (pi univ s) \u2194 \u2200 (i : \u03b9), IsConnected (s i)\n[PROOFSTEP]\nsimp only [IsConnected, \u2190 univ_pi_nonempty_iff, forall_and, and_congr_right_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\n\u22a2 Set.Nonempty (pi univ s) \u2192 (IsPreconnected (pi univ s) \u2194 \u2200 (x : \u03b9), IsPreconnected (s x))\n[PROOFSTEP]\nrefine' fun hne => \u27e8fun hc i => _, isPreconnected_univ_pi\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : Set.Nonempty (pi univ s)\nhc : IsPreconnected (pi univ s)\ni : \u03b9\n\u22a2 IsPreconnected (s i)\n[PROOFSTEP]\nrw [\u2190 eval_image_univ_pi hne]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nhne : Set.Nonempty (pi univ s)\nhc : IsPreconnected (pi univ s)\ni : \u03b9\n\u22a2 IsPreconnected ((fun f => f i) '' pi univ s)\n[PROOFSTEP]\nexact hc.image _ (continuous_apply _).continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 IsConnected s \u2194 \u2203 i t, IsConnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nrefine' \u27e8fun hs => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nobtain \u27e8\u27e8i, x\u27e9, hx\u27e9 := hs.nonempty\n[GOAL]\ncase refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nhave : s \u2286 range (Sigma.mk i) := hs.isPreconnected.subset_clopen isClopen_range_sigmaMk \u27e8\u27e8i, x\u27e9, hx, x, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsConnected s\ni : \u03b9\nx : \u03c0 i\nhx : { fst := i, snd := x } \u2208 s\nthis : s \u2286 range (mk i)\n\u22a2 \u2203 i t, IsConnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nexact\n  \u27e8i, Sigma.mk i \u207b\u00b9' s, hs.preimage_of_openMap sigma_mk_injective isOpenMap_sigmaMk this,\n    (Set.image_preimage_eq_of_subset this).symm\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 (\u2203 i t, IsConnected t \u2227 s = mk i '' t) \u2192 IsConnected s\n[PROOFSTEP]\nrintro \u27e8i, t, ht, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\nt : Set (\u03c0 i)\nht : IsConnected t\n\u22a2 IsConnected (mk i '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_sigmaMk.continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 IsPreconnected s \u2194 \u2203 i t, IsPreconnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nrefine' \u27e8fun hs => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsPreconnected s\n\u22a2 \u2203 i t, IsPreconnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nobtain rfl | h := s.eq_empty_or_nonempty\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nhs : IsPreconnected \u2205\n\u22a2 \u2203 i t, IsPreconnected t \u2227 \u2205 = mk i '' t\n[PROOFSTEP]\nexact \u27e8Classical.choice h\u03b9, \u2205, isPreconnected_empty, (Set.image_empty _).symm\u27e9\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nhs : IsPreconnected s\nh : Set.Nonempty s\n\u22a2 \u2203 i t, IsPreconnected t \u2227 s = mk i '' t\n[PROOFSTEP]\nobtain \u27e8a, t, ht, rfl\u27e9 := Sigma.isConnected_iff.1 \u27e8h, hs\u27e9\n[GOAL]\ncase refine'_1.inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\na : \u03b9\nt : Set (\u03c0 a)\nht : IsConnected t\nhs : IsPreconnected (mk a '' t)\nh : Set.Nonempty (mk a '' t)\n\u22a2 \u2203 i t_1, IsPreconnected t_1 \u2227 mk a '' t = mk i '' t_1\n[PROOFSTEP]\nrefine' \u27e8a, t, ht.isPreconnected, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\n\u22a2 (\u2203 i t, IsPreconnected t \u2227 s = mk i '' t) \u2192 IsPreconnected s\n[PROOFSTEP]\nrintro \u27e8a, t, ht, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\nh\u03b9 : Nonempty \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\na : \u03b9\nt : Set (\u03c0 a)\nht : IsPreconnected t\n\u22a2 IsPreconnected (mk a '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_sigmaMk.continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\n\u22a2 IsConnected s \u2194 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nrefine' \u27e8fun hs => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\n\u22a2 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nobtain \u27e8x | x, hx\u27e9 := hs.nonempty\n[GOAL]\ncase refine'_1.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b1\nhx : inl x \u2208 s\n\u22a2 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nhave h : s \u2286 range Sum.inl := hs.isPreconnected.subset_clopen isClopen_range_inl \u27e8.inl x, hx, x, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b1\nhx : inl x \u2208 s\nh : s \u2286 range inl\n\u22a2 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nrefine' Or.inl \u27e8Sum.inl \u207b\u00b9' s, _, _\u27e9\n[GOAL]\ncase refine'_1.intro.inl.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b1\nhx : inl x \u2208 s\nh : s \u2286 range inl\n\u22a2 IsConnected (inl \u207b\u00b9' s)\n[PROOFSTEP]\nexact hs.preimage_of_openMap Sum.inl_injective isOpenMap_inl h\n[GOAL]\ncase refine'_1.intro.inl.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b1\nhx : inl x \u2208 s\nh : s \u2286 range inl\n\u22a2 s = inl '' (inl \u207b\u00b9' s)\n[PROOFSTEP]\nexact (image_preimage_eq_of_subset h).symm\n[GOAL]\ncase refine'_1.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b2\nhx : inr x \u2208 s\n\u22a2 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nhave h : s \u2286 range Sum.inr := hs.isPreconnected.subset_clopen isClopen_range_inr \u27e8.inr x, hx, x, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b2\nhx : inr x \u2208 s\nh : s \u2286 range inr\n\u22a2 (\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t\n[PROOFSTEP]\nrefine' Or.inr \u27e8Sum.inr \u207b\u00b9' s, _, _\u27e9\n[GOAL]\ncase refine'_1.intro.inr.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b2\nhx : inr x \u2208 s\nh : s \u2286 range inr\n\u22a2 IsConnected (inr \u207b\u00b9' s)\n[PROOFSTEP]\nexact hs.preimage_of_openMap Sum.inr_injective isOpenMap_inr h\n[GOAL]\ncase refine'_1.intro.inr.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsConnected s\nx : \u03b2\nhx : inr x \u2208 s\nh : s \u2286 range inr\n\u22a2 s = inr '' (inr \u207b\u00b9' s)\n[PROOFSTEP]\nexact (image_preimage_eq_of_subset h).symm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\n\u22a2 ((\u2203 t, IsConnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsConnected t \u2227 s = inr '' t) \u2192 IsConnected s\n[PROOFSTEP]\nrintro (\u27e8t, ht, rfl\u27e9 | \u27e8t, ht, rfl\u27e9)\n[GOAL]\ncase refine'_2.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b1\nht : IsConnected t\n\u22a2 IsConnected (inl '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_inl.continuousOn\n[GOAL]\ncase refine'_2.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b2\nht : IsConnected t\n\u22a2 IsConnected (inr '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_inr.continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\n\u22a2 IsPreconnected s \u2194 (\u2203 t, IsPreconnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsPreconnected t \u2227 s = inr '' t\n[PROOFSTEP]\nrefine' \u27e8fun hs => _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsPreconnected s\n\u22a2 (\u2203 t, IsPreconnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsPreconnected t \u2227 s = inr '' t\n[PROOFSTEP]\nobtain rfl | h := s.eq_empty_or_nonempty\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nhs : IsPreconnected \u2205\n\u22a2 (\u2203 t, IsPreconnected t \u2227 \u2205 = inl '' t) \u2228 \u2203 t, IsPreconnected t \u2227 \u2205 = inr '' t\n[PROOFSTEP]\nexact Or.inl \u27e8\u2205, isPreconnected_empty, (Set.image_empty _).symm\u27e9\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nhs : IsPreconnected s\nh : Set.Nonempty s\n\u22a2 (\u2203 t, IsPreconnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsPreconnected t \u2227 s = inr '' t\n[PROOFSTEP]\nobtain \u27e8t, ht, rfl\u27e9 | \u27e8t, ht, rfl\u27e9 := Sum.isConnected_iff.1 \u27e8h, hs\u27e9\n[GOAL]\ncase refine'_1.inr.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b1\nht : IsConnected t\nhs : IsPreconnected (inl '' t)\nh : Set.Nonempty (inl '' t)\n\u22a2 (\u2203 t_1, IsPreconnected t_1 \u2227 inl '' t = inl '' t_1) \u2228 \u2203 t_1, IsPreconnected t_1 \u2227 inl '' t = inr '' t_1\n[PROOFSTEP]\nexact Or.inl \u27e8t, ht.isPreconnected, rfl\u27e9\n[GOAL]\ncase refine'_1.inr.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b2\nht : IsConnected t\nhs : IsPreconnected (inr '' t)\nh : Set.Nonempty (inr '' t)\n\u22a2 (\u2203 t_1, IsPreconnected t_1 \u2227 inr '' t = inl '' t_1) \u2228 \u2203 t_1, IsPreconnected t_1 \u2227 inr '' t = inr '' t_1\n[PROOFSTEP]\nexact Or.inr \u27e8t, ht.isPreconnected, rfl\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\n\u22a2 ((\u2203 t, IsPreconnected t \u2227 s = inl '' t) \u2228 \u2203 t, IsPreconnected t \u2227 s = inr '' t) \u2192 IsPreconnected s\n[PROOFSTEP]\nrintro (\u27e8t, ht, rfl\u27e9 | \u27e8t, ht, rfl\u27e9)\n[GOAL]\ncase refine'_2.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b1\nht : IsPreconnected t\n\u22a2 IsPreconnected (inl '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_inl.continuousOn\n[GOAL]\ncase refine'_2.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Set \u03b2\nht : IsPreconnected t\n\u22a2 IsPreconnected (inr '' t)\n[PROOFSTEP]\nexact ht.image _ continuous_inr.continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\n\u22a2 x \u2208 connectedComponentIn F x\n[PROOFSTEP]\nsimp [connectedComponentIn_eq_image hx, mem_connectedComponent, hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 Set.Nonempty (connectedComponentIn F x) \u2194 x \u2208 F\n[PROOFSTEP]\nrw [connectedComponentIn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 Set.Nonempty (if h : x \u2208 F then Subtype.val '' connectedComponent { val := x, property := h } else \u2205) \u2194 x \u2208 F\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nh\u271d : x \u2208 F\n\u22a2 Set.Nonempty (Subtype.val '' connectedComponent { val := x, property := h\u271d }) \u2194 x \u2208 F\n[PROOFSTEP]\nsimp [connectedComponent_nonempty, *]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nh\u271d : \u00acx \u2208 F\n\u22a2 Set.Nonempty \u2205 \u2194 x \u2208 F\n[PROOFSTEP]\nsimp [connectedComponent_nonempty, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v F : Set \u03b1\nx : \u03b1\n\u22a2 connectedComponentIn F x \u2286 F\n[PROOFSTEP]\nrw [connectedComponentIn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v F : Set \u03b1\nx : \u03b1\n\u22a2 (if h : x \u2208 F then Subtype.val '' connectedComponent { val := x, property := h } else \u2205) \u2286 F\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v F : Set \u03b1\nx : \u03b1\nh\u271d : x \u2208 F\n\u22a2 Subtype.val '' connectedComponent { val := x, property := h\u271d } \u2286 F\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v F : Set \u03b1\nx : \u03b1\nh\u271d : \u00acx \u2208 F\n\u22a2 \u2205 \u2286 F\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 IsPreconnected (connectedComponentIn F x)\n[PROOFSTEP]\nrw [connectedComponentIn]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 IsPreconnected (if h : x \u2208 F then Subtype.val '' connectedComponent { val := x, property := h } else \u2205)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nh\u271d : x \u2208 F\n\u22a2 IsPreconnected (Subtype.val '' connectedComponent { val := x, property := h\u271d })\n[PROOFSTEP]\nexact inducing_subtype_val.isPreconnected_image.mpr isPreconnected_connectedComponent\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nh\u271d : \u00acx \u2208 F\n\u22a2 IsPreconnected \u2205\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\n\u22a2 IsConnected (connectedComponentIn F x) \u2194 x \u2208 F\n[PROOFSTEP]\nsimp_rw [\u2190 connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn, and_true_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\n\u22a2 s \u2286 connectedComponentIn F x\n[PROOFSTEP]\nhave : IsPreconnected (((\u2191) : F \u2192 \u03b1) \u207b\u00b9' s) :=\n  by\n  refine' inducing_subtype_val.isPreconnected_image.mp _\n  rwa [Subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\n\u22a2 IsPreconnected (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' inducing_subtype_val.isPreconnected_image.mp _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\n\u22a2 IsPreconnected (Subtype.val '' (Subtype.val \u207b\u00b9' s))\n[PROOFSTEP]\nrwa [Subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis : IsPreconnected (Subtype.val \u207b\u00b9' s)\n\u22a2 s \u2286 connectedComponentIn F x\n[PROOFSTEP]\nhave h2xs : (\u27e8x, hsF hxs\u27e9 : F) \u2208 (\u2191) \u207b\u00b9' s := by\n  rw [mem_preimage]\n  exact hxs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis : IsPreconnected (Subtype.val \u207b\u00b9' s)\n\u22a2 { val := x, property := (_ : x \u2208 F) } \u2208 Subtype.val \u207b\u00b9' s\n[PROOFSTEP]\nrw [mem_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis : IsPreconnected (Subtype.val \u207b\u00b9' s)\n\u22a2 \u2191{ val := x, property := (_ : x \u2208 F) } \u2208 s\n[PROOFSTEP]\nexact hxs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis : IsPreconnected (Subtype.val \u207b\u00b9' s)\nh2xs : { val := x, property := (_ : x \u2208 F) } \u2208 Subtype.val \u207b\u00b9' s\n\u22a2 s \u2286 connectedComponentIn F x\n[PROOFSTEP]\nhave := this.subset_connectedComponent h2xs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis\u271d : IsPreconnected (Subtype.val \u207b\u00b9' s)\nh2xs : { val := x, property := (_ : x \u2208 F) } \u2208 Subtype.val \u207b\u00b9' s\nthis : Subtype.val \u207b\u00b9' s \u2286 connectedComponent { val := x, property := (_ : x \u2208 F) }\n\u22a2 s \u2286 connectedComponentIn F x\n[PROOFSTEP]\nrw [connectedComponentIn_eq_image (hsF hxs)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis\u271d : IsPreconnected (Subtype.val \u207b\u00b9' s)\nh2xs : { val := x, property := (_ : x \u2208 F) } \u2208 Subtype.val \u207b\u00b9' s\nthis : Subtype.val \u207b\u00b9' s \u2286 connectedComponent { val := x, property := (_ : x \u2208 F) }\n\u22a2 s \u2286 Subtype.val '' connectedComponent { val := x, property := (_ : x \u2208 F) }\n[PROOFSTEP]\nrefine' Subset.trans _ (image_subset _ this)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhs : IsPreconnected s\nhxs : x \u2208 s\nhsF : s \u2286 F\nthis\u271d : IsPreconnected (Subtype.val \u207b\u00b9' s)\nh2xs : { val := x, property := (_ : x \u2208 F) } \u2208 Subtype.val \u207b\u00b9' s\nthis : Subtype.val \u207b\u00b9' s \u2286 connectedComponent { val := x, property := (_ : x \u2208 F) }\n\u22a2 s \u2286 Subtype.val '' (Subtype.val \u207b\u00b9' s)\n[PROOFSTEP]\nrw [Subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nh : y \u2208 connectedComponentIn F x\n\u22a2 connectedComponentIn F x = connectedComponentIn F y\n[PROOFSTEP]\nhave hx : x \u2208 F := connectedComponentIn_nonempty_iff.mp \u27e8y, h\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nh : y \u2208 connectedComponentIn F x\nhx : x \u2208 F\n\u22a2 connectedComponentIn F x = connectedComponentIn F y\n[PROOFSTEP]\nsimp_rw [connectedComponentIn_eq_image hx] at h \u22a2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx y : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\nh : y \u2208 Subtype.val '' connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F y\n[PROOFSTEP]\nobtain \u27e8\u27e8y, hy\u27e9, h2y, rfl\u27e9 := h\n[GOAL]\ncase intro.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF : Set \u03b1\nhx : x \u2208 F\ny : \u03b1\nhy : y \u2208 F\nh2y : { val := y, property := hy } \u2208 connectedComponent { val := x, property := hx }\n\u22a2 Subtype.val '' connectedComponent { val := x, property := hx } = connectedComponentIn F \u2191{ val := y, property := hy }\n[PROOFSTEP]\nsimp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF G : Set \u03b1\nh : F \u2286 G\n\u22a2 connectedComponentIn F x \u2286 connectedComponentIn G x\n[PROOFSTEP]\nby_cases hx : x \u2208 F\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF G : Set \u03b1\nh : F \u2286 G\nhx : x \u2208 F\n\u22a2 connectedComponentIn F x \u2286 connectedComponentIn G x\n[PROOFSTEP]\nrw [connectedComponentIn_eq_image hx, connectedComponentIn_eq_image (h hx), \u2190\n  show ((\u2191) : G \u2192 \u03b1) \u2218 inclusion h = (\u2191) from rfl, image_comp]\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF G : Set \u03b1\nh : F \u2286 G\nhx : x \u2208 F\n\u22a2 Subtype.val '' (inclusion h '' connectedComponent { val := x, property := hx }) \u2286\n    Subtype.val '' connectedComponent { val := x, property := (_ : x \u2208 G) }\n[PROOFSTEP]\nexact image_subset _ ((continuous_inclusion h).image_connectedComponent_subset \u27e8x, hx\u27e9)\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF G : Set \u03b1\nh : F \u2286 G\nhx : \u00acx \u2208 F\n\u22a2 connectedComponentIn F x \u2286 connectedComponentIn G x\n[PROOFSTEP]\nrw [connectedComponentIn_eq_empty hx]\n[GOAL]\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nF G : Set \u03b1\nh : F \u2286 G\nhx : \u00acx \u2208 F\n\u22a2 \u2205 \u2286 connectedComponentIn G x\n[PROOFSTEP]\nexact Set.empty_subset _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nhf' : Continuous f\n\u22a2 ConnectedSpace \u03b2\n[PROOFSTEP]\nrw [connectedSpace_iff_univ, \u2190 hf.range_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nhf' : Continuous f\n\u22a2 IsConnected (range f)\n[PROOFSTEP]\nexact isConnected_range hf'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 ConnectedSpace \u03b1 \u2194 \u2203 x, connectedComponent x = univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 ConnectedSpace \u03b1 \u2192 \u2203 x, connectedComponent x = univ\n[PROOFSTEP]\nrintro \u27e8\u27e8x\u27e9\u27e9\n[GOAL]\ncase mp.mk.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ntoPreconnectedSpace\u271d : PreconnectedSpace \u03b1\nx : \u03b1\n\u22a2 \u2203 x, connectedComponent x = univ\n[PROOFSTEP]\nexact \u27e8x, eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2203 x, connectedComponent x = univ) \u2192 ConnectedSpace \u03b1\n[PROOFSTEP]\nrintro \u27e8x, h\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nh : connectedComponent x = univ\n\u22a2 ConnectedSpace \u03b1\n[PROOFSTEP]\nhaveI : PreconnectedSpace \u03b1 := \u27e8by rw [\u2190 h]; exact isPreconnected_connectedComponent\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nh : connectedComponent x = univ\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nh : connectedComponent x = univ\n\u22a2 IsPreconnected (connectedComponent x)\n[PROOFSTEP]\nexact isPreconnected_connectedComponent\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\nh : connectedComponent x = univ\nthis : PreconnectedSpace \u03b1\n\u22a2 ConnectedSpace \u03b1\n[PROOFSTEP]\nexact \u27e8\u27e8x\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 PreconnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1), connectedComponent x = univ\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 PreconnectedSpace \u03b1 \u2192 \u2200 (x : \u03b1), connectedComponent x = univ\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : PreconnectedSpace \u03b1\nx : \u03b1\n\u22a2 connectedComponent x = univ\n[PROOFSTEP]\nexact eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), connectedComponent x = univ) \u2192 PreconnectedSpace \u03b1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\n\u22a2 PreconnectedSpace \u03b1\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b1 with h\u03b1 h\u03b1\n[GOAL]\ncase mpr.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : IsEmpty \u03b1\n\u22a2 PreconnectedSpace \u03b1\n[PROOFSTEP]\nexact \u27e8by rw [univ_eq_empty_iff.mpr h\u03b1]; exact isPreconnected_empty\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : IsEmpty \u03b1\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrw [univ_eq_empty_iff.mpr h\u03b1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : IsEmpty \u03b1\n\u22a2 IsPreconnected \u2205\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase mpr.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : Nonempty \u03b1\n\u22a2 PreconnectedSpace \u03b1\n[PROOFSTEP]\nexact \u27e8by rw [\u2190 h (Classical.choice h\u03b1)]; exact isPreconnected_connectedComponent\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : Nonempty \u03b1\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrw [\u2190 h (Classical.choice h\u03b1)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), connectedComponent x = univ\nh\u03b1 : Nonempty \u03b1\n\u22a2 IsPreconnected (connectedComponent (Classical.choice h\u03b1))\n[PROOFSTEP]\nexact isPreconnected_connectedComponent\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ninst\u271d : PreconnectedSpace \u03b2\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrw [\u2190 univ_prod_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ninst\u271d : PreconnectedSpace \u03b2\n\u22a2 IsPreconnected (univ \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact isPreconnected_univ.prod isPreconnected_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), PreconnectedSpace (\u03c0 i)\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrw [\u2190 pi_univ univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), PreconnectedSpace (\u03c0 i)\n\u22a2 IsPreconnected (pi univ fun i => univ)\n[PROOFSTEP]\nexact isPreconnected_univ_pi fun i => isPreconnected_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u00d7 \u03c0 i\nhf : Continuous f\n\u22a2 \u2203 i g, Continuous g \u2227 f = Sigma.mk i \u2218 g\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 : \u2203 i, range f \u2286 range (.mk i)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u00d7 \u03c0 i\nhf : Continuous f\n\u22a2 \u2203 i, range f \u2286 range (Sigma.mk i)\n[PROOFSTEP]\nrcases Sigma.isConnected_iff.1 (isConnected_range hf) with \u27e8i, s, -, hs\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u00d7 \u03c0 i\nhf : Continuous f\ni : \u03b9\ns : Set (\u03c0 i)\nhs : range f = Sigma.mk i '' s\n\u22a2 \u2203 i, range f \u2286 range (Sigma.mk i)\n[PROOFSTEP]\nexact \u27e8i, hs.trans_subset (image_subset_range _ _)\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf : \u03b1 \u2192 (i : \u03b9) \u00d7 \u03c0 i\nhf : Continuous f\ni : \u03b9\nhi : range f \u2286 range (Sigma.mk i)\n\u22a2 \u2203 i g, Continuous g \u2227 f = Sigma.mk i \u2218 g\n[PROOFSTEP]\nrcases range_subset_range_iff_exists_comp.1 hi with \u27e8g, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\ng : \u03b1 \u2192 \u03c0 i\nhf : Continuous (Sigma.mk i \u2218 g)\nhi : range (Sigma.mk i \u2218 g) \u2286 range (Sigma.mk i)\n\u22a2 \u2203 i_1 g_1, Continuous g_1 \u2227 Sigma.mk i \u2218 g = Sigma.mk i_1 \u2218 g_1\n[PROOFSTEP]\nrefine \u27e8i, g, ?_, rfl\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : ConnectedSpace \u03b1\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ni : \u03b9\ng : \u03b1 \u2192 \u03c0 i\nhf : Continuous (Sigma.mk i \u2218 g)\nhi : range (Sigma.mk i \u2218 g) \u2286 range (Sigma.mk i)\n\u22a2 Continuous g\n[PROOFSTEP]\nrwa [\u2190 embedding_sigmaMk.continuous_iff] at hf \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t\u271d u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns t : Set \u03b1\n\u22a2 IsOpen s \u2192 IsOpen t \u2192 s \u222a t = univ \u2192 Set.Nonempty s \u2192 Set.Nonempty t \u2192 Set.Nonempty (s \u2229 t)\n[PROOFSTEP]\nsimpa only [univ_inter, univ_subset_iff] using @PreconnectedSpace.isPreconnected_univ \u03b1 _ _ s t\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : Set \u03b1\nhs : IsClopen s\nh : \u00ac(s = \u2205 \u2228 s = univ)\nh2 : s\u1d9c = \u2205\n\u22a2 s = univ\n[PROOFSTEP]\nrw [\u2190 compl_compl s, h2, compl_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : Set \u03b1\n\u22a2 s = \u2205 \u2228 s = univ \u2192 IsClopen s\n[PROOFSTEP]\nrintro (rfl | rfl) <;> [exact isClopen_empty; exact isClopen_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : Set \u03b1\n\u22a2 s = \u2205 \u2228 s = univ \u2192 IsClopen s\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\n\u22a2 IsClopen \u2205\n[PROOFSTEP]\nexact isClopen_empty\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\n\u22a2 IsClopen univ\n[PROOFSTEP]\nexact isClopen_univ\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\n\u22a2 Subsingleton \u03b9\n[PROOFSTEP]\nreplace h_nonempty : \u2200 i, s i \u2260 \u2205 := by intro i; rw [\u2190 nonempty_iff_ne_empty]; exact h_nonempty i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\n\u22a2 \u2200 (i : \u03b9), s i \u2260 \u2205\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\ni : \u03b9\n\u22a2 s i \u2260 \u2205\n[PROOFSTEP]\nrw [\u2190 nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\ni : \u03b9\n\u22a2 Set.Nonempty (s i)\n[PROOFSTEP]\nexact h_nonempty i\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\n\u22a2 Subsingleton \u03b9\n[PROOFSTEP]\nrw [\u2190 not_nontrivial_iff_subsingleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\n\u22a2 \u00acNontrivial \u03b9\n[PROOFSTEP]\nby_contra contra\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ncontra : Nontrivial \u03b9\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8i, j, h_ne\u27e9 := contra\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : i \u2260 j\n\u22a2 False\n[PROOFSTEP]\nreplace h_ne : s i \u2229 s j = \u2205 := by\n  simpa only [\u2190 bot_eq_empty, eq_bot_iff, \u2190 inf_eq_inter, \u2190 disjoint_iff_inf_le] using h_disj h_ne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : i \u2260 j\n\u22a2 s i \u2229 s j = \u2205\n[PROOFSTEP]\nsimpa only [\u2190 bot_eq_empty, eq_bot_iff, \u2190 inf_eq_inter, \u2190 disjoint_iff_inf_le] using h_disj h_ne\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : s i \u2229 s j = \u2205\n\u22a2 False\n[PROOFSTEP]\ncases' isClopen_iff.mp (h_clopen i) with hi hi\n[GOAL]\ncase mk.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : s i \u2229 s j = \u2205\nhi : s i = \u2205\n\u22a2 False\n[PROOFSTEP]\nexact h_nonempty i hi\n[GOAL]\ncase mk.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : s i \u2229 s j = \u2205\nhi : s i = univ\n\u22a2 False\n[PROOFSTEP]\nrw [hi, univ_inter] at h_ne \n[GOAL]\ncase mk.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_disj : Pairwise (Disjoint on s)\nh_clopen : \u2200 (i : \u03b9), IsClopen (s i)\nh_nonempty : \u2200 (i : \u03b9), s i \u2260 \u2205\ni j : \u03b9\nh_ne : s j = \u2205\nhi : s i = univ\n\u22a2 False\n[PROOFSTEP]\nexact h_nonempty j h_ne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\n\u22a2 Subsingleton \u03b9\n[PROOFSTEP]\nrefine' subsingleton_of_disjoint_isClopen h_nonempty h_disj (fun i \u21a6 \u27e8h_open i, _\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsClosed (s i)\n[PROOFSTEP]\nrw [\u2190 isOpen_compl_iff, compl_eq_univ_diff, \u2190 h_Union, iUnion_diff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsOpen (\u22c3 (i_1 : \u03b9), s i_1 \\ s i)\n[PROOFSTEP]\nrefine' isOpen_iUnion (fun j \u21a6 _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni j : \u03b9\n\u22a2 IsOpen (s j \\ s i)\n[PROOFSTEP]\nrcases eq_or_ne i j with rfl | h_ne\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsOpen (s i \\ s i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\nh_open : \u2200 (i : \u03b9), IsOpen (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni j : \u03b9\nh_ne : i \u2260 j\n\u22a2 IsOpen (s j \\ s i)\n[PROOFSTEP]\nsimpa only [(h_disj h_ne.symm).sdiff_eq_left] using h_open j\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\n\u22a2 Subsingleton \u03b9\n[PROOFSTEP]\nrefine' subsingleton_of_disjoint_isClopen h_nonempty h_disj (fun i \u21a6 \u27e8_, h_closed i\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsOpen (s i)\n[PROOFSTEP]\nrw [\u2190 isClosed_compl_iff, compl_eq_univ_diff, \u2190 h_Union, iUnion_diff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsClosed (\u22c3 (i_1 : \u03b9), s i_1 \\ s i)\n[PROOFSTEP]\nrefine' isClosed_iUnion (fun j \u21a6 _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni j : \u03b9\n\u22a2 IsClosed (s j \\ s i)\n[PROOFSTEP]\nrcases eq_or_ne i j with rfl | h_ne\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni : \u03b9\n\u22a2 IsClosed (s i \\ s i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : PreconnectedSpace \u03b1\ns : \u03b9 \u2192 Set \u03b1\nh_nonempty : \u2200 (i : \u03b9), Set.Nonempty (s i)\nh_disj : Pairwise (Disjoint on s)\ninst\u271d : Finite \u03b9\nh_closed : \u2200 (i : \u03b9), IsClosed (s i)\nh_Union : \u22c3 (i : \u03b9), s i = univ\ni j : \u03b9\nh_ne : i \u2260 j\n\u22a2 IsClosed (s j \\ s i)\n[PROOFSTEP]\nsimpa only [(h_disj h_ne.symm).sdiff_eq_left] using h_closed j\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\ns : Set \u03b1\n\u22a2 Set.Nonempty (frontier s) \u2194 Set.Nonempty s \u2227 s \u2260 univ\n[PROOFSTEP]\nsimp only [nonempty_iff_ne_empty, Ne.def, frontier_eq_empty_iff, not_or]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nh : IsPreconnected s\n\u22a2 IsPreconnected univ\n[PROOFSTEP]\nrwa [\u2190 inducing_subtype_val.isPreconnected_image, image_univ, Subtype.range_val]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nh : PreconnectedSpace \u2191s\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nsimpa using isPreconnected_univ.image ((\u2191) : s \u2192 \u03b1) continuous_subtype_val.continuousOn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\n\u22a2 P x y\n[PROOFSTEP]\nlet u := {z | P x z}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\n\u22a2 P x y\n[PROOFSTEP]\nhave A : IsOpen u := by\n  apply isOpen_iff_mem_nhds.2 (fun z hz \u21a6 ?_)\n  filter_upwards [h z] with t ht\n  exact h' hz ht.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\n\u22a2 IsOpen u\n[PROOFSTEP]\napply isOpen_iff_mem_nhds.2 (fun z hz \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nz : \u03b1\nhz : z \u2208 u\n\u22a2 u \u2208 \ud835\udcdd z\n[PROOFSTEP]\nfilter_upwards [h z] with t ht\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nz : \u03b1\nhz : z \u2208 u\nt : \u03b1\nht : P z t \u2227 P t z\n\u22a2 P x t\n[PROOFSTEP]\nexact h' hz ht.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\n\u22a2 P x y\n[PROOFSTEP]\nhave B : IsClosed u := by\n  apply isClosed_iff_nhds.2 (fun z hz \u21a6 ?_)\n  rcases hz _ (h z) with \u27e8t, ht, h't\u27e9\n  exact h' h't ht.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\n\u22a2 IsClosed u\n[PROOFSTEP]\napply isClosed_iff_nhds.2 (fun z hz \u21a6 ?_)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nz : \u03b1\nhz : \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd z \u2192 Set.Nonempty (U \u2229 u)\n\u22a2 z \u2208 u\n[PROOFSTEP]\nrcases hz _ (h z) with \u27e8t, ht, h't\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nz : \u03b1\nhz : \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd z \u2192 Set.Nonempty (U \u2229 u)\nt : \u03b1\nht : t \u2208 {x | (fun y => P z y \u2227 P y z) x}\nh't : t \u2208 u\n\u22a2 z \u2208 u\n[PROOFSTEP]\nexact h' h't ht.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nB : IsClosed u\n\u22a2 P x y\n[PROOFSTEP]\nhave C : u.Nonempty := \u27e8x, (mem_of_mem_nhds (h x)).1\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nB : IsClosed u\nC : Set.Nonempty u\n\u22a2 P x y\n[PROOFSTEP]\nhave D : u = Set.univ := IsClopen.eq_univ \u27e8A, B\u27e9 C\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nB : IsClosed u\nC : Set.Nonempty u\nD : u = univ\n\u22a2 P x y\n[PROOFSTEP]\nshow y \u2208 u\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u\u271d v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y \u2227 P y x\nh' : Transitive P\nx y : \u03b1\nu : Set \u03b1 := {z | P x z}\nA : IsOpen u\nB : IsClosed u\nC : Set.Nonempty u\nD : u = univ\n\u22a2 y \u2208 u\n[PROOFSTEP]\nsimp [D]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y\nh' : Transitive P\nh'' : Symmetric P\nx y : \u03b1\n\u22a2 P x y\n[PROOFSTEP]\nrefine PreconnectedSpace.induction\u2082' P (fun z \u21a6 ?_) h' x y\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y\nh' : Transitive P\nh'' : Symmetric P\nx y z : \u03b1\n\u22a2 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd z, P z y \u2227 P y z\n[PROOFSTEP]\nfilter_upwards [h z] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : PreconnectedSpace \u03b1\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, P x y\nh' : Transitive P\nh'' : Symmetric P\nx y z a : \u03b1\nha : P z a\n\u22a2 P z a \u2227 P a z\n[PROOFSTEP]\nrefine \u27e8ha, h'' ha\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 P x y\n[PROOFSTEP]\nlet Q : s \u2192 s \u2192 Prop := fun a b \u21a6 P a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\n\u22a2 P x y\n[PROOFSTEP]\nshow Q \u27e8x, hx\u27e9 \u27e8y, hy\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\n\u22a2 Q { val := x, property := hx } { val := y, property := hy }\n[PROOFSTEP]\nhave : PreconnectedSpace s := Subtype.preconnectedSpace hs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\n\u22a2 Q { val := x, property := hx } { val := y, property := hy }\n[PROOFSTEP]\napply PreconnectedSpace.induction\u2082'\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\n\u22a2 \u2200 (x : { x // x \u2208 s }), \u2200\u1da0 (y : { x // x \u2208 s }) in \ud835\udcdd x, Q x y \u2227 Q y x\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx\u271d y : \u03b1\nhx\u271d : x\u271d \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 \u2200\u1da0 (y : { x // x \u2208 s }) in \ud835\udcdd { val := x, property := hx },\n    Q { val := x, property := hx } y \u2227 Q y { val := x, property := hx }\n[PROOFSTEP]\nhave Z := h x hx\n[GOAL]\ncase h.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx\u271d y : \u03b1\nhx\u271d : x\u271d \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\nx : \u03b1\nhx : x \u2208 s\nZ : \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\n\u22a2 \u2200\u1da0 (y : { x // x \u2208 s }) in \ud835\udcdd { val := x, property := hx },\n    Q { val := x, property := hx } y \u2227 Q y { val := x, property := hx }\n[PROOFSTEP]\nrwa [nhdsWithin_eq_map_subtype_coe] at Z \n[GOAL]\ncase h'\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\n\u22a2 Transitive Q\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9 \u27e8c, hc\u27e9 hab hbc\n[GOAL]\ncase h'.mk.mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y \u2227 P y x\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nQ : \u2191s \u2192 \u2191s \u2192 Prop := fun a b => P \u2191a \u2191b\nthis : PreconnectedSpace \u2191s\na : \u03b1\nha : a \u2208 s\nb : \u03b1\nhb : b \u2208 s\nc : \u03b1\nhc : c \u2208 s\nhab : Q { val := a, property := ha } { val := b, property := hb }\nhbc : Q { val := b, property := hb } { val := c, property := hc }\n\u22a2 Q { val := a, property := ha } { val := c, property := hc }\n[PROOFSTEP]\nexact h' a b c ha hb hc hab hbc\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nh'' : \u2200 (x y : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 P x y \u2192 P y x\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 P x y\n[PROOFSTEP]\napply hs.induction\u2082' P (fun z hz \u21a6 ?_) h' hx hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nh'' : \u2200 (x y : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 P x y \u2192 P y x\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nz : \u03b1\nhz : z \u2208 s\n\u22a2 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] z, P z y \u2227 P y z\n[PROOFSTEP]\nfilter_upwards [h z hz, self_mem_nhdsWithin] with a ha h'a\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsPreconnected s\nP : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd[s] x, P x y\nh' : \u2200 (x y z : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 z \u2208 s \u2192 P x y \u2192 P y z \u2192 P x z\nh'' : \u2200 (x y : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 P x y \u2192 P y x\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 s\nz : \u03b1\nhz : z \u2208 s\na : \u03b1\nha : P z a\nh'a : a \u2208 s\n\u22a2 P z a \u2227 P a z\n[PROOFSTEP]\nexact \u27e8ha, h'' z a hz h'a ha\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 IsPreconnected s \u2194 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 IsPreconnected s \u2192 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 (\u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v) \u2192 IsPreconnected s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nh : IsPreconnected s\n\u22a2 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro u v hu hv hs huv\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh : IsPreconnected s\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nspecialize h u v hu hv hs\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\ncontrapose! huv\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205\n[PROOFSTEP]\nrw [\u2190 nonempty_iff_ne_empty]\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nsimp [not_subset] at huv \n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : (\u2203 a, a \u2208 s \u2227 \u00aca \u2208 u) \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases huv with \u27e8\u27e8x, hxs, hxu\u27e9, \u27e8y, hys, hyv\u27e9\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hxv : x \u2208 v := or_iff_not_imp_left.mp (hs hxs) hxu\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hyu : y \u2208 u := or_iff_not_imp_right.mp (hs hys) hyv\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\nhyu : y \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact h \u27e8y, hys, hyu\u27e9 \u27e8x, hxs, hxv\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nintro u v hu hv hs hsu hsv\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\n\u22a2 False\n[PROOFSTEP]\nspecialize h u v hu hv hs H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\nh : s \u2286 u \u2228 s \u2286 v\n\u22a2 False\n[PROOFSTEP]\ncontrapose H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 \u00acs \u2229 (u \u2229 v) = \u2205\n[PROOFSTEP]\napply Nonempty.ne_empty\n[GOAL]\ncase mpr.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase mpr.a.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases hsv with \u27e8x, hxs, hxv\u27e9\n[GOAL]\ncase mpr.a.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nH : \u00acFalse\nh : s \u2286 u\nx : \u03b1\nhxs : x \u2208 s\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8x, hxs, \u27e8h hxs, hxv\u27e9\u27e9\n[GOAL]\ncase mpr.a.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases hsu with \u27e8x, hxs, hxu\u27e9\n[GOAL]\ncase mpr.a.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s u v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\nx : \u03b1\nhxs : x \u2208 s\nhxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8x, hxs, \u27e8hxu, h hxs\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 IsConnected s \u2194\n    \u2200 (U : Finset (Set \u03b1)),\n      (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n        (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\n[PROOFSTEP]\nrw [IsConnected, isPreconnected_iff_subset_of_disjoint]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\n\u22a2 (Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v) \u2194\n    \u2200 (U : Finset (Set \u03b1)),\n      (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n        (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\n[PROOFSTEP]\nrefine \u27e8fun \u27e8hne, h\u27e9 U hU hUo hsU => ?_, fun h => \u27e8?_, fun u v hu hv hs hsuv => ?_\u27e9\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nU : Finset (Set \u03b1)\nhU : \u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191U\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 \u2203 u, u \u2208 U \u2227 s \u2286 u\n[PROOFSTEP]\ninduction U using Finset.induction_on\n[GOAL]\ncase refine_1.empty\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhU : \u2200 (u v : Set \u03b1), u \u2208 \u2205 \u2192 v \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191\u2205\n\u22a2 \u2203 u, u \u2208 \u2205 \u2227 s \u2286 u\ncase refine_1.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d\u00b9 t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\na\u271d\u00b2 : Set \u03b1\ns\u271d : Finset (Set \u03b1)\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 s\u271d\na\u271d :\n  (\u2200 (u v : Set \u03b1), u \u2208 s\u271d \u2192 v \u2208 s\u271d \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191s\u271d \u2192 \u2203 u, u \u2208 s\u271d \u2227 s \u2286 u\nhU : \u2200 (u v : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 v \u2208 insert a\u271d\u00b2 s\u271d \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191(insert a\u271d\u00b2 s\u271d)\n\u22a2 \u2203 u, u \u2208 insert a\u271d\u00b2 s\u271d \u2227 s \u2286 u\n[PROOFSTEP]\ncase empty => exact absurd (by simpa using hsU) hne.not_subset_empty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhU : \u2200 (u v : Set \u03b1), u \u2208 \u2205 \u2192 v \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191\u2205\n\u22a2 \u2203 u, u \u2208 \u2205 \u2227 s \u2286 u\n[PROOFSTEP]\ncase empty => exact absurd (by simpa using hsU) hne.not_subset_empty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhU : \u2200 (u v : Set \u03b1), u \u2208 \u2205 \u2192 v \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191\u2205\n\u22a2 \u2203 u, u \u2208 \u2205 \u2227 s \u2286 u\n[PROOFSTEP]\nexact absurd (by simpa using hsU) hne.not_subset_empty\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhU : \u2200 (u v : Set \u03b1), u \u2208 \u2205 \u2192 v \u2208 \u2205 \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 \u2205 \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191\u2205\n\u22a2 s \u2286 \u2205\n[PROOFSTEP]\nsimpa using hsU\n[GOAL]\ncase refine_1.insert\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d\u00b9 t u v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\na\u271d\u00b2 : Set \u03b1\ns\u271d : Finset (Set \u03b1)\na\u271d\u00b9 : \u00aca\u271d\u00b2 \u2208 s\u271d\na\u271d :\n  (\u2200 (u v : Set \u03b1), u \u2208 s\u271d \u2192 v \u2208 s\u271d \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 s\u271d \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191s\u271d \u2192 \u2203 u, u \u2208 s\u271d \u2227 s \u2286 u\nhU : \u2200 (u v : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 v \u2208 insert a\u271d\u00b2 s\u271d \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v\nhUo : \u2200 (u : Set \u03b1), u \u2208 insert a\u271d\u00b2 s\u271d \u2192 IsOpen u\nhsU : s \u2286 \u22c3\u2080 \u2191(insert a\u271d\u00b2 s\u271d)\n\u22a2 \u2203 u, u \u2208 insert a\u271d\u00b2 s\u271d \u2227 s \u2286 u\n[PROOFSTEP]\ncase insert u U uU\n  IH =>\n  simp only [\u2190 ball_cond_comm, Finset.forall_mem_insert, Finset.exists_mem_insert, Finset.coe_insert, sUnion_insert,\n    implies_true, true_and] at *\n  refine (h _ hUo.1 (\u22c3\u2080 \u2191U) (isOpen_sUnion hUo.2) hsU ?_).imp_right ?_\n  \u00b7 refine subset_empty_iff.1 fun x \u27e8hxs, hxu, v, hvU, hxv\u27e9 => ?_\n    exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU \u27e8x, hxs, hxu, hxv\u27e9).symm\n  \u00b7 exact IH (fun u hu => (hU.2 u hu).2) hUo.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nIH :\n  (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU : \u2200 (u_1 v : Set \u03b1), u_1 \u2208 insert u U \u2192 v \u2208 insert u U \u2192 Set.Nonempty (s \u2229 (u_1 \u2229 v)) \u2192 u_1 = v\nhUo : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 IsOpen u_1\nhsU : s \u2286 \u22c3\u2080 \u2191(insert u U)\n\u22a2 \u2203 u_1, u_1 \u2208 insert u U \u2227 s \u2286 u_1\n[PROOFSTEP]\ncase insert u U uU\n  IH =>\n  simp only [\u2190 ball_cond_comm, Finset.forall_mem_insert, Finset.exists_mem_insert, Finset.coe_insert, sUnion_insert,\n    implies_true, true_and] at *\n  refine (h _ hUo.1 (\u22c3\u2080 \u2191U) (isOpen_sUnion hUo.2) hsU ?_).imp_right ?_\n  \u00b7 refine subset_empty_iff.1 fun x \u27e8hxs, hxu, v, hvU, hxv\u27e9 => ?_\n    exact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU \u27e8x, hxs, hxu, hxv\u27e9).symm\n  \u00b7 exact IH (fun u hu => (hU.2 u hu).2) hUo.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v s : Set \u03b1\nx\u271d : Set.Nonempty s \u2227 \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nhne : Set.Nonempty s\nh : \u2200 (u v : Set \u03b1), IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nIH :\n  (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU : \u2200 (u_1 v : Set \u03b1), u_1 \u2208 insert u U \u2192 v \u2208 insert u U \u2192 Set.Nonempty (s \u2229 (u_1 \u2229 v)) \u2192 u_1 = v\nhUo : \u2200 (u_1 : Set \u03b1), u_1 \u2208 insert u U \u2192 IsOpen u_1\nhsU : s \u2286 \u22c3\u2080 \u2191(insert u U)\n\u22a2 \u2203 u_1, u_1 \u2208 insert u U \u2227 s \u2286 u_1\n[PROOFSTEP]\nsimp only [\u2190 ball_cond_comm, Finset.forall_mem_insert, Finset.exists_mem_insert, Finset.coe_insert, sUnion_insert,\n  implies_true, true_and] at *\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v s : Set \u03b1\nhne : Set.Nonempty s\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nx\u271d : Set.Nonempty s \u2227 \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nh : \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nIH :\n  (\u2200 (a : Set \u03b1), a \u2208 U \u2192 \u2200 (b : Set \u03b1), b \u2208 U \u2192 Set.Nonempty (s \u2229 (a \u2229 b)) \u2192 a = b) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU :\n  (\u2200 (x : Set \u03b1), x \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 x)) \u2192 u = x) \u2227\n    \u2200 (x : Set \u03b1),\n      x \u2208 U \u2192 (Set.Nonempty (s \u2229 (x \u2229 u)) \u2192 x = u) \u2227 \u2200 (x_1 : Set \u03b1), x_1 \u2208 U \u2192 Set.Nonempty (s \u2229 (x \u2229 x_1)) \u2192 x = x_1\nhUo : IsOpen u \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 IsOpen x\nhsU : s \u2286 u \u222a \u22c3\u2080 \u2191U\n\u22a2 s \u2286 u \u2228 \u2203 x, x \u2208 U \u2227 s \u2286 x\n[PROOFSTEP]\nrefine (h _ hUo.1 (\u22c3\u2080 \u2191U) (isOpen_sUnion hUo.2) hsU ?_).imp_right ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v s : Set \u03b1\nhne : Set.Nonempty s\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nx\u271d : Set.Nonempty s \u2227 \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nh : \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nIH :\n  (\u2200 (a : Set \u03b1), a \u2208 U \u2192 \u2200 (b : Set \u03b1), b \u2208 U \u2192 Set.Nonempty (s \u2229 (a \u2229 b)) \u2192 a = b) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU :\n  (\u2200 (x : Set \u03b1), x \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 x)) \u2192 u = x) \u2227\n    \u2200 (x : Set \u03b1),\n      x \u2208 U \u2192 (Set.Nonempty (s \u2229 (x \u2229 u)) \u2192 x = u) \u2227 \u2200 (x_1 : Set \u03b1), x_1 \u2208 U \u2192 Set.Nonempty (s \u2229 (x \u2229 x_1)) \u2192 x = x_1\nhUo : IsOpen u \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 IsOpen x\nhsU : s \u2286 u \u222a \u22c3\u2080 \u2191U\n\u22a2 s \u2229 (u \u2229 \u22c3\u2080 \u2191U) = \u2205\n[PROOFSTEP]\nrefine subset_empty_iff.1 fun x \u27e8hxs, hxu, v, hvU, hxv\u27e9 => ?_\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhne : Set.Nonempty s\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nx\u271d\u00b9 : Set.Nonempty s \u2227 \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nh : \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nIH :\n  (\u2200 (a : Set \u03b1), a \u2208 U \u2192 \u2200 (b : Set \u03b1), b \u2208 U \u2192 Set.Nonempty (s \u2229 (a \u2229 b)) \u2192 a = b) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU :\n  (\u2200 (x : Set \u03b1), x \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 x)) \u2192 u = x) \u2227\n    \u2200 (x : Set \u03b1),\n      x \u2208 U \u2192 (Set.Nonempty (s \u2229 (x \u2229 u)) \u2192 x = u) \u2227 \u2200 (x_1 : Set \u03b1), x_1 \u2208 U \u2192 Set.Nonempty (s \u2229 (x \u2229 x_1)) \u2192 x = x_1\nhUo : IsOpen u \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 IsOpen x\nhsU : s \u2286 u \u222a \u22c3\u2080 \u2191U\nx : \u03b1\nx\u271d : x \u2208 s \u2229 (u \u2229 \u22c3\u2080 \u2191U)\nhxs : x \u2208 s\nhxu : x \u2208 u\nv : Set \u03b1\nhvU : v \u2208 \u2191U\nhxv : x \u2208 v\n\u22a2 x \u2208 \u2205\n[PROOFSTEP]\nexact ne_of_mem_of_not_mem hvU uU (hU.1 v hvU \u27e8x, hxs, hxu, hxv\u27e9).symm\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v s : Set \u03b1\nhne : Set.Nonempty s\nu : Set \u03b1\nU : Finset (Set \u03b1)\nuU : \u00acu \u2208 U\nx\u271d : Set.Nonempty s \u2227 \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nh : \u2200 (a : Set \u03b1), IsOpen a \u2192 \u2200 (b : Set \u03b1), IsOpen b \u2192 s \u2286 a \u222a b \u2192 s \u2229 (a \u2229 b) = \u2205 \u2192 s \u2286 a \u2228 s \u2286 b\nIH :\n  (\u2200 (a : Set \u03b1), a \u2208 U \u2192 \u2200 (b : Set \u03b1), b \u2208 U \u2192 Set.Nonempty (s \u2229 (a \u2229 b)) \u2192 a = b) \u2192\n    (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nhU :\n  (\u2200 (x : Set \u03b1), x \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 x)) \u2192 u = x) \u2227\n    \u2200 (x : Set \u03b1),\n      x \u2208 U \u2192 (Set.Nonempty (s \u2229 (x \u2229 u)) \u2192 x = u) \u2227 \u2200 (x_1 : Set \u03b1), x_1 \u2208 U \u2192 Set.Nonempty (s \u2229 (x \u2229 x_1)) \u2192 x = x_1\nhUo : IsOpen u \u2227 \u2200 (x : Set \u03b1), x \u2208 U \u2192 IsOpen x\nhsU : s \u2286 u \u222a \u22c3\u2080 \u2191U\n\u22a2 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 x, x \u2208 U \u2227 s \u2286 x\n[PROOFSTEP]\nexact IH (fun u hu => (hU.2 u hu).2) hUo.2\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nh :\n  \u2200 (U : Finset (Set \u03b1)),\n    (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n      (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\n\u22a2 Set.Nonempty s\n[PROOFSTEP]\nsimpa [subset_empty_iff, nonempty_iff_ne_empty] using h \u2205\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (U : Finset (Set \u03b1)),\n    (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n      (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nrw [\u2190 not_nonempty_iff_eq_empty] at hsuv \n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (U : Finset (Set \u03b1)),\n    (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n      (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsuv : \u00acSet.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nhave := hsuv\n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (U : Finset (Set \u03b1)),\n    (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n      (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsuv this : \u00acSet.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nrw [inter_comm u] at this \n[GOAL]\ncase refine_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nh :\n  \u2200 (U : Finset (Set \u03b1)),\n    (\u2200 (u v : Set \u03b1), u \u2208 U \u2192 v \u2208 U \u2192 Set.Nonempty (s \u2229 (u \u2229 v)) \u2192 u = v) \u2192\n      (\u2200 (u : Set \u03b1), u \u2208 U \u2192 IsOpen u) \u2192 s \u2286 \u22c3\u2080 \u2191U \u2192 \u2203 u, u \u2208 U \u2227 s \u2286 u\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhs : s \u2286 u \u222a v\nhsuv : \u00acSet.Nonempty (s \u2229 (u \u2229 v))\nthis : \u00acSet.Nonempty (s \u2229 (v \u2229 u))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nsimpa [*, or_imp, forall_and] using h { u, v }\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 IsPreconnected s \u2194 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 IsPreconnected s \u2192 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v) \u2192 IsPreconnected s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : IsPreconnected s\n\u22a2 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro u v hu hv hs huv\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : IsPreconnected s\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nrw [isPreconnected_closed_iff] at h \n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh :\n  \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nspecialize h u v hu hv hs\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\ncontrapose! huv\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205\n[PROOFSTEP]\nrw [\u2190 nonempty_iff_ne_empty]\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : \u00acs \u2286 u \u2227 \u00acs \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nsimp [not_subset] at huv \n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nhuv : (\u2203 a, a \u2208 s \u2227 \u00aca \u2208 u) \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases huv with \u27e8\u27e8x, hxs, hxu\u27e9, \u27e8y, hys, hyv\u27e9\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hxv : x \u2208 v := or_iff_not_imp_left.mp (hs hxs) hxu\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nhave hyu : y \u2208 u := or_iff_not_imp_right.mp (hs hys) hyv\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nh : Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\nx : \u03b1\nhxs : x \u2208 s\nhxu : \u00acx \u2208 u\ny : \u03b1\nhys : y \u2208 s\nhyv : \u00acy \u2208 v\nhxv : x \u2208 v\nhyu : y \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact h \u27e8y, hys, hyu\u27e9 \u27e8x, hxs, hxv\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nrw [isPreconnected_closed_iff]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 \u2200 (t t' : Set \u03b1),\n    IsClosed t \u2192 IsClosed t' \u2192 s \u2286 t \u222a t' \u2192 Set.Nonempty (s \u2229 t) \u2192 Set.Nonempty (s \u2229 t') \u2192 Set.Nonempty (s \u2229 (t \u2229 t'))\n[PROOFSTEP]\nintro u v hu hv hs hsu hsv\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\n\u22a2 s \u2229 (u \u2229 v) \u2260 \u2205\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d : Set \u03b1\nh : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\n\u22a2 False\n[PROOFSTEP]\nspecialize h u v hu hv hs H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : s \u2229 (u \u2229 v) = \u2205\nh : s \u2286 u \u2228 s \u2286 v\n\u22a2 False\n[PROOFSTEP]\ncontrapose H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 \u00acs \u2229 (u \u2229 v) = \u2205\n[PROOFSTEP]\napply Nonempty.ne_empty\n[GOAL]\ncase mpr.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nh : s \u2286 u \u2228 s \u2286 v\nH : \u00acFalse\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase mpr.a.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases hsv with \u27e8x, hxs, hxv\u27e9\n[GOAL]\ncase mpr.a.inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nH : \u00acFalse\nh : s \u2286 u\nx : \u03b1\nhxs : x \u2208 s\nhxv : x \u2208 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8x, hxs, \u27e8h hxs, hxv\u27e9\u27e9\n[GOAL]\ncase mpr.a.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsu : Set.Nonempty (s \u2229 u)\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nrcases hsu with \u27e8x, hxs, hxu\u27e9\n[GOAL]\ncase mpr.a.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u\u271d v\u271d u v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhs : s \u2286 u \u222a v\nhsv : Set.Nonempty (s \u2229 v)\nH : \u00acFalse\nh : s \u2286 v\nx : \u03b1\nhxs : x \u2208 s\nhxu : x \u2208 u\n\u22a2 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nexact \u27e8x, hxs, \u27e8hxu, h hxs\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsClosed s\n\u22a2 IsPreconnected s \u2194 \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nrefine isPreconnected_iff_subset_of_disjoint_closed.trans \u27e8?_, ?_\u27e9\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsClosed s\n\u22a2 (\u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v) \u2192\n    \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro H u v hu hv hss huv\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : IsClosed s\n\u22a2 (\u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v) \u2192\n    \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nintro H u v hu hv hss huv\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : Disjoint u v\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\napply H u v hu hv hss\n[GOAL]\ncase refine_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 s \u2229 (u \u2229 v) = \u2205 \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : Disjoint u v\n\u22a2 s \u2229 (u \u2229 v) = \u2205\n[PROOFSTEP]\nrw [huv.inter_eq, inter_empty]\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nhave H1 := H (u \u2229 s) (v \u2229 s)\n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2229 s \u2228 s \u2286 v \u2229 s\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nrw [subset_inter_iff, subset_inter_iff] at H1 \n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2227 s \u2286 s \u2228 s \u2286 v \u2227 s \u2286 s\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\nsimp only [Subset.refl, and_true] at H1 \n[GOAL]\ncase refine_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 s \u2286 u \u2228 s \u2286 v\n[PROOFSTEP]\napply H1 (hu.inter hs) (hv.inter hs)\n[GOAL]\ncase refine_2.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 s \u2286 u \u2229 s \u222a v \u2229 s\n[PROOFSTEP]\nrw [\u2190 inter_distrib_right]\n[GOAL]\ncase refine_2.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 s \u2286 (u \u222a v) \u2229 s\n[PROOFSTEP]\nexact subset_inter hss Subset.rfl\n[GOAL]\ncase refine_2.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : IsClosed s\nH : \u2200 (u v : Set \u03b1), IsClosed u \u2192 IsClosed v \u2192 s \u2286 u \u222a v \u2192 Disjoint u v \u2192 s \u2286 u \u2228 s \u2286 v\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhss : s \u2286 u \u222a v\nhuv : s \u2229 (u \u2229 v) = \u2205\nH1 : IsClosed (u \u2229 s) \u2192 IsClosed (v \u2229 s) \u2192 s \u2286 u \u2229 s \u222a v \u2229 s \u2192 Disjoint (u \u2229 s) (v \u2229 s) \u2192 s \u2286 u \u2228 s \u2286 v\n\u22a2 Disjoint (u \u2229 s) (v \u2229 s)\n[PROOFSTEP]\nrwa [disjoint_iff_inter_eq_empty, \u2190 inter_inter_distrib_right, inter_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\n\u22a2 IsConnected (f \u207b\u00b9' connectedComponent t)\n[PROOFSTEP]\nhave hf : Surjective f := Surjective.of_comp fun t : \u03b2 => (connected_fibers t).1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\n\u22a2 IsConnected (f \u207b\u00b9' connectedComponent t)\n[PROOFSTEP]\nrefine \u27e8Nonempty.preimage connectedComponent_nonempty hf, ?_\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\n\u22a2 IsPreconnected (f \u207b\u00b9' connectedComponent t)\n[PROOFSTEP]\nhave hT : IsClosed (f \u207b\u00b9' connectedComponent t) := (hcl (connectedComponent t)).1 isClosed_connectedComponent\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\n\u22a2 IsPreconnected (f \u207b\u00b9' connectedComponent t)\n[PROOFSTEP]\nrw [isPreconnected_iff_subset_of_fully_disjoint_closed hT]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\n\u22a2 \u2200 (u v : Set \u03b1),\n    IsClosed u \u2192\n      IsClosed v \u2192\n        f \u207b\u00b9' connectedComponent t \u2286 u \u222a v \u2192\n          Disjoint u v \u2192 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nintro u v hu hv huv uv_disj\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nlet T\u2081 := {t' \u2208 connectedComponent t | f \u207b\u00b9' { t' } \u2286 u}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nlet T\u2082 := {t' \u2208 connectedComponent t | f \u207b\u00b9' { t' } \u2286 v}\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave fiber_decomp : \u2200 t' \u2208 connectedComponent t, f \u207b\u00b9' { t' } \u2286 u \u2228 f \u207b\u00b9' { t' } \u2286 v :=\n  by\n  intro t' ht'\n  apply isPreconnected_iff_subset_of_disjoint_closed.1 (connected_fibers t').2 u v hu hv\n  \u00b7 exact Subset.trans (preimage_mono (singleton_subset_iff.2 ht')) huv\n  rw [uv_disj.inter_eq, inter_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\n\u22a2 \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n[PROOFSTEP]\nintro t' ht'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\n\u22a2 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n[PROOFSTEP]\napply isPreconnected_iff_subset_of_disjoint_closed.1 (connected_fibers t').2 u v hu hv\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\n\u22a2 f \u207b\u00b9' {t'} \u2286 u \u222a v\n[PROOFSTEP]\nexact Subset.trans (preimage_mono (singleton_subset_iff.2 ht')) huv\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\n\u22a2 f \u207b\u00b9' {t'} \u2229 (u \u2229 v) = \u2205\n[PROOFSTEP]\nrw [uv_disj.inter_eq, inter_empty]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave T\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u :=\n  by\n  apply eq_of_subset_of_subset\n  \u00b7 rw [\u2190 biUnion_preimage_singleton]\n    refine' iUnion\u2082_subset fun t' ht' => subset_inter _ ht'.2\n    rw [hf.preimage_subset_preimage_iff, singleton_subset_iff]\n    exact ht'.1\n  rintro a \u27e8hat, hau\u27e9\n  constructor\n  \u00b7 exact mem_preimage.1 hat\n  refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_right fun h => ?_\n  exact\n    uv_disj.subset_compl_right hau\n      (h rfl)\n        -- This proof is exactly the same as the above (modulo some symmetry)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n\u22a2 f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n[PROOFSTEP]\napply eq_of_subset_of_subset\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n\u22a2 f \u207b\u00b9' T\u2081 \u2286 f \u207b\u00b9' connectedComponent t \u2229 u\n[PROOFSTEP]\nrw [\u2190 biUnion_preimage_singleton]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n\u22a2 \u22c3 (y : \u03b2) (_ : y \u2208 T\u2081), f \u207b\u00b9' {y} \u2286 f \u207b\u00b9' connectedComponent t \u2229 u\n[PROOFSTEP]\nrefine' iUnion\u2082_subset fun t' ht' => subset_inter _ ht'.2\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nt' : \u03b2\nht' : t' \u2208 T\u2081\n\u22a2 f \u207b\u00b9' {t'} \u2286 f \u207b\u00b9' connectedComponent t\n[PROOFSTEP]\nrw [hf.preimage_subset_preimage_iff, singleton_subset_iff]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nt' : \u03b2\nht' : t' \u2208 T\u2081\n\u22a2 t' \u2208 connectedComponent t\n[PROOFSTEP]\nexact ht'.1\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\n\u22a2 f \u207b\u00b9' connectedComponent t \u2229 u \u2286 f \u207b\u00b9' T\u2081\n[PROOFSTEP]\nrintro a \u27e8hat, hau\u27e9\n[GOAL]\ncase a.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhau : a \u2208 u\n\u22a2 a \u2208 f \u207b\u00b9' T\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhau : a \u2208 u\n\u22a2 f a \u2208 connectedComponent t\n[PROOFSTEP]\nexact mem_preimage.1 hat\n[GOAL]\ncase a.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhau : a \u2208 u\n\u22a2 f \u207b\u00b9' {f a} \u2286 u\n[PROOFSTEP]\nrefine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_right fun h => ?_\n[GOAL]\ncase a.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhau : a \u2208 u\nh : f \u207b\u00b9' {f a} \u2286 v\n\u22a2 False\n[PROOFSTEP]\nexact\n  uv_disj.subset_compl_right hau\n    (h rfl)\n      -- This proof is exactly the same as the above (modulo some symmetry)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave T\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v :=\n  by\n  apply eq_of_subset_of_subset\n  \u00b7 rw [\u2190 biUnion_preimage_singleton]\n    refine' iUnion\u2082_subset fun t' ht' => subset_inter _ ht'.2\n    rw [hf.preimage_subset_preimage_iff, singleton_subset_iff]\n    exact ht'.1\n  rintro a \u27e8hat, hav\u27e9\n  constructor\n  \u00b7 exact mem_preimage.1 hat\n  \u00b7 refine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_left fun h => ?_\n    exact\n      uv_disj.subset_compl_left hav\n        (h rfl)\n          -- Now we show T\u2081, T\u2082 are closed, cover connectedComponent t and are disjoint.\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n\u22a2 f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\n[PROOFSTEP]\napply eq_of_subset_of_subset\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n\u22a2 f \u207b\u00b9' T\u2082 \u2286 f \u207b\u00b9' connectedComponent t \u2229 v\n[PROOFSTEP]\nrw [\u2190 biUnion_preimage_singleton]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n\u22a2 \u22c3 (y : \u03b2) (_ : y \u2208 T\u2082), f \u207b\u00b9' {y} \u2286 f \u207b\u00b9' connectedComponent t \u2229 v\n[PROOFSTEP]\nrefine' iUnion\u2082_subset fun t' ht' => subset_inter _ ht'.2\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nt' : \u03b2\nht' : t' \u2208 T\u2082\n\u22a2 f \u207b\u00b9' {t'} \u2286 f \u207b\u00b9' connectedComponent t\n[PROOFSTEP]\nrw [hf.preimage_subset_preimage_iff, singleton_subset_iff]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nt' : \u03b2\nht' : t' \u2208 T\u2082\n\u22a2 t' \u2208 connectedComponent t\n[PROOFSTEP]\nexact ht'.1\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\n\u22a2 f \u207b\u00b9' connectedComponent t \u2229 v \u2286 f \u207b\u00b9' T\u2082\n[PROOFSTEP]\nrintro a \u27e8hat, hav\u27e9\n[GOAL]\ncase a.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhav : a \u2208 v\n\u22a2 a \u2208 f \u207b\u00b9' T\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.intro.left\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhav : a \u2208 v\n\u22a2 f a \u2208 connectedComponent t\n[PROOFSTEP]\nexact mem_preimage.1 hat\n[GOAL]\ncase a.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhav : a \u2208 v\n\u22a2 f \u207b\u00b9' {f a} \u2286 v\n[PROOFSTEP]\nrefine (fiber_decomp (f a) (mem_preimage.1 hat)).resolve_left fun h => ?_\n[GOAL]\ncase a.intro.right\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\na : \u03b1\nhat : a \u2208 f \u207b\u00b9' connectedComponent t\nhav : a \u2208 v\nh : f \u207b\u00b9' {f a} \u2286 u\n\u22a2 False\n[PROOFSTEP]\nexact\n  uv_disj.subset_compl_left hav\n    (h rfl)\n      -- Now we show T\u2081, T\u2082 are closed, cover connectedComponent t and are disjoint.\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave hT\u2081 : IsClosed T\u2081 := (hcl T\u2081).2 (T\u2081_u.symm \u25b8 IsClosed.inter hT hu)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave hT\u2082 : IsClosed T\u2082 := (hcl T\u2082).2 (T\u2082_v.symm \u25b8 IsClosed.inter hT hv)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave T_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082 := fun t' ht' =>\n  by\n  rw [mem_union t' T\u2081 T\u2082]\n  cases' fiber_decomp t' ht' with htu htv\n  \u00b7 left\n    exact \u27e8ht', htu\u27e9\n  right\n  exact \u27e8ht', htv\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\n\u22a2 t' \u2208 T\u2081 \u222a T\u2082\n[PROOFSTEP]\nrw [mem_union t' T\u2081 T\u2082]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\n\u22a2 t' \u2208 T\u2081 \u2228 t' \u2208 T\u2082\n[PROOFSTEP]\ncases' fiber_decomp t' ht' with htu htv\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\nhtu : f \u207b\u00b9' {t'} \u2286 u\n\u22a2 t' \u2208 T\u2081 \u2228 t' \u2208 T\u2082\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\nhtu : f \u207b\u00b9' {t'} \u2286 u\n\u22a2 t' \u2208 T\u2081\n[PROOFSTEP]\nexact \u27e8ht', htu\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\nhtv : f \u207b\u00b9' {t'} \u2286 v\n\u22a2 t' \u2208 T\u2081 \u2228 t' \u2208 T\u2082\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nt' : \u03b2\nht' : t' \u2208 connectedComponent t\nhtv : f \u207b\u00b9' {t'} \u2286 v\n\u22a2 t' \u2208 T\u2082\n[PROOFSTEP]\nexact \u27e8ht', htv\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nhave T_disjoint : Disjoint T\u2081 T\u2082 := by\n  refine' Disjoint.of_preimage hf _\n  rw [T\u2081_u, T\u2082_v, disjoint_iff_inter_eq_empty, \u2190 inter_inter_distrib_left, uv_disj.inter_eq, inter_empty]\n    -- Now we do cases on whether (connectedComponent t) is a subset of T\u2081 or T\u2082 to show\n      -- that the preimage is a subset of u or v.\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\n\u22a2 Disjoint T\u2081 T\u2082\n[PROOFSTEP]\nrefine' Disjoint.of_preimage hf _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\n\u22a2 Disjoint (f \u207b\u00b9' T\u2081) (f \u207b\u00b9' T\u2082)\n[PROOFSTEP]\nrw [T\u2081_u, T\u2082_v, disjoint_iff_inter_eq_empty, \u2190 inter_inter_distrib_left, uv_disj.inter_eq, inter_empty]\n  -- Now we do cases on whether (connectedComponent t) is a subset of T\u2081 or T\u2082 to show\n    -- that the preimage is a subset of u or v.\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\ncases'\n  (isPreconnected_iff_subset_of_fully_disjoint_closed isClosed_connectedComponent).1 isPreconnected_connectedComponent\n    T\u2081 T\u2082 hT\u2081 hT\u2082 T_decomp T_disjoint with\n  h h\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2081\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nleft\n[GOAL]\ncase inl.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2081\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u\n[PROOFSTEP]\nrw [Subset.antisymm_iff] at T\u2081_u \n[GOAL]\ncase inl.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 \u2286 f \u207b\u00b9' connectedComponent t \u2229 u \u2227 f \u207b\u00b9' connectedComponent t \u2229 u \u2286 f \u207b\u00b9' T\u2081\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2081\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u\n[PROOFSTEP]\nsuffices f \u207b\u00b9' connectedComponent t \u2286 f \u207b\u00b9' T\u2081 from (this.trans T\u2081_u.1).trans (inter_subset_right _ _)\n[GOAL]\ncase inl.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 \u2286 f \u207b\u00b9' connectedComponent t \u2229 u \u2227 f \u207b\u00b9' connectedComponent t \u2229 u \u2286 f \u207b\u00b9' T\u2081\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2081\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 f \u207b\u00b9' T\u2081\n[PROOFSTEP]\nexact preimage_mono h\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 u \u2228 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nright\n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 = f \u207b\u00b9' connectedComponent t \u2229 v\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nrw [Subset.antisymm_iff] at T\u2082_v \n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 \u2286 f \u207b\u00b9' connectedComponent t \u2229 v \u2227 f \u207b\u00b9' connectedComponent t \u2229 v \u2286 f \u207b\u00b9' T\u2082\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 v\n[PROOFSTEP]\nsuffices f \u207b\u00b9' connectedComponent t \u2286 f \u207b\u00b9' T\u2082 from (this.trans T\u2082_v.1).trans (inter_subset_right _ _)\n[GOAL]\ncase inr.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t\u271d u\u271d v\u271d : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nconnected_fibers : \u2200 (t : \u03b2), IsConnected (f \u207b\u00b9' {t})\nhcl : \u2200 (T : Set \u03b2), IsClosed T \u2194 IsClosed (f \u207b\u00b9' T)\nt : \u03b2\nhf : Surjective f\nhT : IsClosed (f \u207b\u00b9' connectedComponent t)\nu v : Set \u03b1\nhu : IsClosed u\nhv : IsClosed v\nhuv : f \u207b\u00b9' connectedComponent t \u2286 u \u222a v\nuv_disj : Disjoint u v\nT\u2081 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 u}\nT\u2082 : Set \u03b2 := {t' | t' \u2208 connectedComponent t \u2227 f \u207b\u00b9' {t'} \u2286 v}\nfiber_decomp : \u2200 (t' : \u03b2), t' \u2208 connectedComponent t \u2192 f \u207b\u00b9' {t'} \u2286 u \u2228 f \u207b\u00b9' {t'} \u2286 v\nT\u2081_u : f \u207b\u00b9' T\u2081 = f \u207b\u00b9' connectedComponent t \u2229 u\nT\u2082_v : f \u207b\u00b9' T\u2082 \u2286 f \u207b\u00b9' connectedComponent t \u2229 v \u2227 f \u207b\u00b9' connectedComponent t \u2229 v \u2286 f \u207b\u00b9' T\u2082\nhT\u2081 : IsClosed T\u2081\nhT\u2082 : IsClosed T\u2082\nT_decomp : connectedComponent t \u2286 T\u2081 \u222a T\u2082\nT_disjoint : Disjoint T\u2081 T\u2082\nh : connectedComponent t \u2286 T\u2082\n\u22a2 f \u207b\u00b9' connectedComponent t \u2286 f \u207b\u00b9' T\u2082\n[PROOFSTEP]\nexact preimage_mono h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : QuotientMap f\nh_fibers : \u2200 (y : \u03b2), IsConnected (f \u207b\u00b9' {y})\na : \u03b1\n\u22a2 f '' connectedComponent a = connectedComponent (f a)\n[PROOFSTEP]\nrw [\u2190 hf.preimage_connectedComponent h_fibers, image_preimage_eq _ hf.surjective]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nsimp_rw [locallyConnectedSpace_iff_open_connected_basis]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 x \u2208 s \u2227 IsConnected s) id) \u2194\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nrefine forall_congr' fun _ => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2194\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id \u2192\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nintro h U hU\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nrcases h.mem_iff.mp hU with \u27e8V, hV, hVU\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\nh : Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\u271d\nV : Set \u03b1\nhV : IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\nhVU : id V \u2286 U\n\u22a2 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nexact \u27e8V, hVU, hV\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx\u271d : \u03b1\n\u22a2 (\u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd x\u271d \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x\u271d \u2208 V \u2227 IsConnected V) \u2192\n    Filter.HasBasis (\ud835\udcdd x\u271d) (fun s => IsOpen s \u2227 x\u271d \u2208 s \u2227 IsConnected s) id\n[PROOFSTEP]\nexact fun h =>\n  \u27e8fun U =>\n    \u27e8fun hU =>\n      let \u27e8V, hVU, hV\u27e9 := h U hU\n      \u27e8V, hV, hVU\u27e9,\n      fun \u27e8V, \u27e8hV, hxV, _\u27e9, hVU\u27e9 => mem_nhds_iff.mpr \u27e8V, hVU, hV, hxV\u27e9\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nh : F \u2208 \ud835\udcdd x\n\u22a2 connectedComponentIn F x \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrw [(LocallyConnectedSpace.open_connected_basis x).mem_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nh : \u2203 i, (IsOpen i \u2227 x \u2208 i \u2227 IsConnected i) \u2227 id i \u2286 F\n\u22a2 connectedComponentIn F x \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrcases h with \u27e8s, \u27e8h1s, hxs, h2s\u27e9, hsF\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\ns : Set \u03b1\nhsF : id s \u2286 F\nh1s : IsOpen s\nhxs : x \u2208 s\nh2s : IsConnected s\n\u22a2 connectedComponentIn F x \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact mem_nhds_iff.mpr \u27e8s, h2s.isPreconnected.subset_connectedComponentIn hxs hsF, h1s, hxs\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nhF : IsOpen F\n\u22a2 IsOpen (connectedComponentIn F x)\n[PROOFSTEP]\nrw [isOpen_iff_mem_nhds]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nhF : IsOpen F\n\u22a2 \u2200 (a : \u03b1), a \u2208 connectedComponentIn F x \u2192 connectedComponentIn F x \u2208 \ud835\udcdd a\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nhF : IsOpen F\ny : \u03b1\nhy : y \u2208 connectedComponentIn F x\n\u22a2 connectedComponentIn F x \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrw [connectedComponentIn_eq hy]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nF : Set \u03b1\nx : \u03b1\nhF : IsOpen F\ny : \u03b1\nhy : y \u2208 connectedComponentIn F x\n\u22a2 connectedComponentIn F y \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact connectedComponentIn_mem_nhds (hF.mem_nhds <| connectedComponentIn_subset F x hy)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nx : \u03b1\n\u22a2 IsOpen (connectedComponent x)\n[PROOFSTEP]\nrw [\u2190 connectedComponentIn_univ]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d : LocallyConnectedSpace \u03b1\nx : \u03b1\n\u22a2 IsOpen (connectedComponentIn univ x)\n[PROOFSTEP]\nexact isOpen_univ.connectedComponentIn\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2192 \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : LocallyConnectedSpace \u03b1\n\u22a2 \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n[PROOFSTEP]\nexact fun F hF x _ => hF.connectedComponentIn\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)) \u2192 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n\u22a2 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nrw [locallyConnectedSpace_iff_open_connected_subsets]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n\u22a2 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\n[PROOFSTEP]\nrefine' fun x U hU =>\n  \u27e8connectedComponentIn (interior U) x, (connectedComponentIn_subset _ _).trans interior_subset,\n    h _ isOpen_interior x _, mem_connectedComponentIn _, isConnected_connectedComponentIn_iff.mpr _\u27e9\n[GOAL]\ncase mpr.refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\nx : \u03b1\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\n\u22a2 x \u2208 interior U\n[PROOFSTEP]\nexact mem_interior_iff_mem_nhds.mpr hU\n[GOAL]\ncase mpr.refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\nx : \u03b1\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\n\u22a2 x \u2208 interior U\n[PROOFSTEP]\nexact mem_interior_iff_mem_nhds.mpr hU\n[GOAL]\ncase mpr.refine'_3\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\nx : \u03b1\nU : Set \u03b1\nhU : U \u2208 \ud835\udcdd x\n\u22a2 x \u2208 interior U\n[PROOFSTEP]\nexact mem_interior_iff_mem_nhds.mpr hU\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2192 \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n[PROOFSTEP]\nrw [locallyConnectedSpace_iff_open_connected_subsets]\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V) \u2192\n    \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n[PROOFSTEP]\nintro h x U hxU\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n[PROOFSTEP]\nrcases h x U hxU with \u27e8V, hVU, hV\u2081, hxV, hV\u2082\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2286 U \u2227 IsOpen V \u2227 x \u2208 V \u2227 IsConnected V\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nV : Set \u03b1\nhVU : V \u2286 U\nhV\u2081 : IsOpen V\nhxV : x \u2208 V\nhV\u2082 : IsConnected V\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\n[PROOFSTEP]\nexact \u27e8V, hV\u2081.mem_nhds hxV, hV\u2082.isPreconnected, hVU\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nrw [locallyConnectedSpace_iff_connectedComponentIn_open]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2192\n    \u2200 (F : Set \u03b1), IsOpen F \u2192 \u2200 (x : \u03b1), x \u2208 F \u2192 IsOpen (connectedComponentIn F x)\n[PROOFSTEP]\nrefine' fun h U hU x _ => isOpen_iff_mem_nhds.mpr fun y hy => _\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U x \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrw [connectedComponentIn_eq hy]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrcases h y U (hU.mem_nhds <| (connectedComponentIn_subset _ _) hy) with \u27e8V, hVy, hV, hVU\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U\nU : Set \u03b1\nhU : IsOpen U\nx : \u03b1\nx\u271d : x \u2208 U\ny : \u03b1\nhy : y \u2208 connectedComponentIn U x\nV : Set \u03b1\nhVy : V \u2208 \ud835\udcdd y\nhV : IsPreconnected V\nhVU : V \u2286 U\n\u22a2 connectedComponentIn U y \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact Filter.mem_of_superset hVy (hV.subset_connectedComponentIn (mem_of_mem_nhds hVy) hVU)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 LocallyConnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => s \u2208 \ud835\udcdd x \u2227 IsPreconnected s) id\n[PROOFSTEP]\nrw [locallyConnectedSpace_iff_connected_subsets]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1) (U : Set \u03b1), U \u2208 \ud835\udcdd x \u2192 \u2203 V, V \u2208 \ud835\udcdd x \u2227 IsPreconnected V \u2227 V \u2286 U) \u2194\n    \u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => s \u2208 \ud835\udcdd x \u2227 IsPreconnected s) id\n[PROOFSTEP]\nexact forall_congr' <| fun x => Filter.hasBasis_self.symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u03b9 : Type u_3\nb : \u03b1 \u2192 \u03b9 \u2192 Set \u03b1\np : \u03b1 \u2192 \u03b9 \u2192 Prop\nhbasis : \u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (p x) (b x)\nhconnected : \u2200 (x : \u03b1) (i : \u03b9), p x i \u2192 IsPreconnected (b x i)\n\u22a2 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nrw [locallyConnectedSpace_iff_connected_basis]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9\u271d : Type u_1\n\u03c0 : \u03b9\u271d \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u03b9 : Type u_3\nb : \u03b1 \u2192 \u03b9 \u2192 Set \u03b1\np : \u03b1 \u2192 \u03b9 \u2192 Prop\nhbasis : \u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (p x) (b x)\nhconnected : \u2200 (x : \u03b1) (i : \u03b9), p x i \u2192 IsPreconnected (b x i)\n\u22a2 \u2200 (x : \u03b1), Filter.HasBasis (\ud835\udcdd x) (fun s => s \u2208 \ud835\udcdd x \u2227 IsPreconnected s) id\n[PROOFSTEP]\nexact fun x =>\n  (hbasis x).to_hasBasis (fun i hi => \u27e8b x i, \u27e8(hbasis x).mem_of_mem hi, hconnected x i hi\u27e9, subset_rfl\u27e9) fun s hs =>\n    \u27e8(hbasis x).index s hs.1, \u27e8(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TotallyDisconnectedSpace \u03b1\ninst\u271d : TotallyDisconnectedSpace \u03b2\n\u22a2 TotallyDisconnectedSpace (\u03b1 \u2295 \u03b2)\n[PROOFSTEP]\nrefine' \u27e8fun s _ hs => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TotallyDisconnectedSpace \u03b1\ninst\u271d : TotallyDisconnectedSpace \u03b2\ns : Set (\u03b1 \u2295 \u03b2)\nx\u271d : s \u2286 univ\nhs : IsPreconnected s\n\u22a2 Set.Subsingleton s\n[PROOFSTEP]\nobtain \u27e8t, ht, rfl\u27e9 | \u27e8t, ht, rfl\u27e9 := Sum.isPreconnected_iff.1 hs\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TotallyDisconnectedSpace \u03b1\ninst\u271d : TotallyDisconnectedSpace \u03b2\nt : Set \u03b1\nht : IsPreconnected t\nx\u271d : Sum.inl '' t \u2286 univ\nhs : IsPreconnected (Sum.inl '' t)\n\u22a2 Set.Subsingleton (Sum.inl '' t)\n[PROOFSTEP]\nexact ht.subsingleton.image _\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TotallyDisconnectedSpace \u03b1\ninst\u271d : TotallyDisconnectedSpace \u03b2\nt : Set \u03b2\nht : IsPreconnected t\nx\u271d : Sum.inr '' t \u2286 univ\nhs : IsPreconnected (Sum.inr '' t)\n\u22a2 Set.Subsingleton (Sum.inr '' t)\n[PROOFSTEP]\nexact ht.subsingleton.image _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), TotallyDisconnectedSpace (\u03c0 i)\n\u22a2 TotallyDisconnectedSpace ((i : \u03b9) \u00d7 \u03c0 i)\n[PROOFSTEP]\nrefine' \u27e8fun s _ hs => _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), TotallyDisconnectedSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nx\u271d : s \u2286 univ\nhs : IsPreconnected s\n\u22a2 Set.Subsingleton s\n[PROOFSTEP]\nobtain rfl | h := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), TotallyDisconnectedSpace (\u03c0 i)\nx\u271d : \u2205 \u2286 univ\nhs : IsPreconnected \u2205\n\u22a2 Set.Subsingleton \u2205\n[PROOFSTEP]\nexact subsingleton_empty\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), TotallyDisconnectedSpace (\u03c0 i)\ns : Set ((i : \u03b9) \u00d7 \u03c0 i)\nx\u271d : s \u2286 univ\nhs : IsPreconnected s\nh : Set.Nonempty s\n\u22a2 Set.Subsingleton s\n[PROOFSTEP]\nobtain \u27e8a, t, ht, rfl\u27e9 := Sigma.isConnected_iff.1 \u27e8h, hs\u27e9\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t\u271d u v : Set \u03b1\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), TotallyDisconnectedSpace (\u03c0 i)\na : \u03b9\nt : Set (\u03c0 a)\nht : IsConnected t\nx\u271d : Sigma.mk a '' t \u2286 univ\nhs : IsPreconnected (Sigma.mk a '' t)\nh : Set.Nonempty (Sigma.mk a '' t)\n\u22a2 Set.Subsingleton (Sigma.mk a '' t)\n[PROOFSTEP]\nexact\n  ht.isPreconnected.subsingleton.image\n    _\n      -- porting note: reformulated using `Pairwise`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\n\u22a2 IsTotallyDisconnected univ\n[PROOFSTEP]\nrintro S - hS\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nhS : IsPreconnected S\n\u22a2 Set.Subsingleton S\n[PROOFSTEP]\nunfold Set.Subsingleton\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nhS : IsPreconnected S\n\u22a2 \u2200 \u2983x : X\u2984, x \u2208 S \u2192 \u2200 \u2983y : X\u2984, y \u2208 S \u2192 x = y\n[PROOFSTEP]\nby_contra' h_contra\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nhS : IsPreconnected S\nh_contra : Exists fun \u2983x\u2984 => x \u2208 S \u2227 Exists fun \u2983y\u2984 => y \u2208 S \u2227 x \u2260 y\n\u22a2 False\n[PROOFSTEP]\nrcases h_contra with \u27e8x, hx, y, hy, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x \u2208 S\ny : X\nhy : y \u2208 S\nhxy : x \u2260 y\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8U, h_clopen, hxU, hyU\u27e9 := hX hxy\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nhS : IsPreconnected S\nx : X\nhx : x \u2208 S\ny : X\nhy : y \u2208 S\nhxy : x \u2260 y\nU : Set X\nh_clopen : IsClopen U\nhxU : x \u2208 U\nhyU : \u00acy \u2208 U\n\u22a2 False\n[PROOFSTEP]\nspecialize hS U U\u1d9c h_clopen.1 h_clopen.compl.1 (fun a _ => em (a \u2208 U)) \u27e8x, hx, hxU\u27e9 \u27e8y, hy, hyU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nx : X\nhx : x \u2208 S\ny : X\nhy : y \u2208 S\nhxy : x \u2260 y\nU : Set X\nh_clopen : IsClopen U\nhxU : x \u2208 U\nhyU : \u00acy \u2208 U\nhS : Set.Nonempty (S \u2229 (U \u2229 U\u1d9c))\n\u22a2 False\n[PROOFSTEP]\nrw [inter_compl_self, Set.inter_empty] at hS \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nX : Type u_3\ninst\u271d : TopologicalSpace X\nhX : Pairwise fun x y => \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 \u00acy \u2208 U\nS : Set X\nx : X\nhx : x \u2208 S\ny : X\nhy : y \u2208 S\nhxy : x \u2260 y\nU : Set X\nh_clopen : IsClopen U\nhxU : x \u2208 U\nhyU : \u00acy \u2208 U\nhS : Set.Nonempty \u2205\n\u22a2 False\n[PROOFSTEP]\nexact Set.not_nonempty_empty hS\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 TotallyDisconnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 TotallyDisconnectedSpace \u03b1 \u2192 \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\n[PROOFSTEP]\nintro h x\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : TotallyDisconnectedSpace \u03b1\nx : \u03b1\n\u22a2 Set.Subsingleton (connectedComponent x)\n[PROOFSTEP]\napply h.1\n[GOAL]\ncase mp.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : TotallyDisconnectedSpace \u03b1\nx : \u03b1\n\u22a2 connectedComponent x \u2286 univ\n[PROOFSTEP]\nexact subset_univ _\n[GOAL]\ncase mp.a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : TotallyDisconnectedSpace \u03b1\nx : \u03b1\n\u22a2 IsPreconnected (connectedComponent x)\n[PROOFSTEP]\nexact isPreconnected_connectedComponent\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)) \u2192 TotallyDisconnectedSpace \u03b1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\n\u22a2 TotallyDisconnectedSpace \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.isTotallyDisconnected_univ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\n\u22a2 IsTotallyDisconnected univ\n[PROOFSTEP]\nintro s s_sub hs\n[GOAL]\ncase mpr.isTotallyDisconnected_univ\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh : \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\ns : Set \u03b1\ns_sub : s \u2286 univ\nhs : IsPreconnected s\n\u22a2 Set.Subsingleton s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8x, x_in\u27e9)\n[GOAL]\ncase mpr.isTotallyDisconnected_univ.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nh : \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\ns_sub : \u2205 \u2286 univ\nhs : IsPreconnected \u2205\n\u22a2 Set.Subsingleton \u2205\n[PROOFSTEP]\nexact subsingleton_empty\n[GOAL]\ncase mpr.isTotallyDisconnected_univ.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v : Set \u03b1\nh : \u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)\ns : Set \u03b1\ns_sub : s \u2286 univ\nhs : IsPreconnected s\nx : \u03b1\nx_in : x \u2208 s\n\u22a2 Set.Subsingleton s\n[PROOFSTEP]\nexact (h x).anti (hs.subset_connectedComponent x_in)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 TotallyDisconnectedSpace \u03b1 \u2194 \u2200 (x : \u03b1), connectedComponent x = {x}\n[PROOFSTEP]\nrw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\n\u22a2 (\u2200 (x : \u03b1), Set.Subsingleton (connectedComponent x)) \u2194 \u2200 (x : \u03b1), connectedComponent x = {x}\n[PROOFSTEP]\nrefine forall_congr' fun x => ?_\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\n\u22a2 Set.Subsingleton (connectedComponent x) \u2194 connectedComponent x = {x}\n[PROOFSTEP]\nrw [subsingleton_iff_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns t u v : Set \u03b1\nx : \u03b1\n\u22a2 x \u2208 connectedComponent x\n[PROOFSTEP]\nexact mem_connectedComponent\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nH : IsTotallySeparated s\n\u22a2 IsTotallyDisconnected s\n[PROOFSTEP]\nintro t hts ht x x_in y y_in\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nH : IsTotallySeparated s\nt : Set \u03b1\nhts : t \u2286 s\nht : IsPreconnected t\nx : \u03b1\nx_in : x \u2208 t\ny : \u03b1\ny_in : y \u2208 t\n\u22a2 x = y\n[PROOFSTEP]\nby_contra h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u v s : Set \u03b1\nH : IsTotallySeparated s\nt : Set \u03b1\nhts : t \u2286 s\nht : IsPreconnected t\nx : \u03b1\nx_in : x \u2208 t\ny : \u03b1\ny_in : y \u2208 t\nh : \u00acx = y\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8u : Set \u03b1, v : Set \u03b1, hu : IsOpen u, hv : IsOpen v, hxu : x \u2208 u, hyv : y \u2208 v, hs : s \u2286 u \u222a v, huv\u27e9 :=\n  H x (hts x_in) y (hts y_in) h\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nH : IsTotallySeparated s\nt : Set \u03b1\nhts : t \u2286 s\nht : IsPreconnected t\nx : \u03b1\nx_in : x \u2208 t\ny : \u03b1\ny_in : y \u2208 t\nh : \u00acx = y\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhxu : x \u2208 u\nhyv : y \u2208 v\nhs : s \u2286 u \u222a v\nhuv : Disjoint u v\n\u22a2 False\n[PROOFSTEP]\nrefine' (ht _ _ hu hv (hts.trans hs) \u27e8x, x_in, hxu\u27e9 \u27e8y, y_in, hyv\u27e9).ne_empty _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t\u271d u\u271d v\u271d s : Set \u03b1\nH : IsTotallySeparated s\nt : Set \u03b1\nhts : t \u2286 s\nht : IsPreconnected t\nx : \u03b1\nx_in : x \u2208 t\ny : \u03b1\ny_in : y \u2208 t\nh : \u00acx = y\nu v : Set \u03b1\nhu : IsOpen u\nhv : IsOpen v\nhxu : x \u2208 u\nhyv : y \u2208 v\nhs : s \u2286 u \u222a v\nhuv : Disjoint u v\n\u22a2 t \u2229 (u \u2229 v) = \u2205\n[PROOFSTEP]\nrw [huv.inter_eq, inter_empty]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TotallySeparatedSpace \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 y \u2208 U\u1d9c\n[PROOFSTEP]\nobtain \u27e8U, V, hU, hV, Ux, Vy, f, disj\u27e9 :=\n  TotallySeparatedSpace.isTotallySeparated_univ x (Set.mem_univ x) y (Set.mem_univ y) hxy\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TotallySeparatedSpace \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nU V : Set \u03b1\nhU : IsOpen U\nhV : IsOpen V\nUx : x \u2208 U\nVy : y \u2208 V\nf : univ \u2286 U \u222a V\ndisj : Disjoint U V\n\u22a2 \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 y \u2208 U\u1d9c\n[PROOFSTEP]\nhave clopen_U := isClopen_inter_of_disjoint_cover_clopen isClopen_univ f hU hV disj\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TotallySeparatedSpace \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nU V : Set \u03b1\nhU : IsOpen U\nhV : IsOpen V\nUx : x \u2208 U\nVy : y \u2208 V\nf : univ \u2286 U \u222a V\ndisj : Disjoint U V\nclopen_U : IsClopen (univ \u2229 U)\n\u22a2 \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 y \u2208 U\u1d9c\n[PROOFSTEP]\nrw [univ_inter _] at clopen_U \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TotallySeparatedSpace \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nU V : Set \u03b1\nhU : IsOpen U\nhV : IsOpen V\nUx : x \u2208 U\nVy : y \u2208 V\nf : univ \u2286 U \u222a V\ndisj : Disjoint U V\nclopen_U : IsClopen U\n\u22a2 \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 y \u2208 U\u1d9c\n[PROOFSTEP]\nrw [\u2190 Set.subset_compl_iff_disjoint_right, subset_compl_comm] at disj \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TotallySeparatedSpace \u03b1\nx y : \u03b1\nhxy : x \u2260 y\nU V : Set \u03b1\nhU : IsOpen U\nhV : IsOpen V\nUx : x \u2208 U\nVy : y \u2208 V\nf : univ \u2286 U \u222a V\ndisj : V \u2286 U\u1d9c\nclopen_U : IsClopen U\n\u22a2 \u2203 U, IsClopen U \u2227 x \u2208 U \u2227 y \u2208 U\u1d9c\n[PROOFSTEP]\nexact \u27e8U, clopen_U, Ux, disj Vy\u27e9\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\u271d\ns t u v : Set \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace \u03b1\nx : \u03b1\n\u22a2 connectedComponent x = connectedComponent x\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nh : Continuous f\n\u22a2 \u2200 (a b : \u03b1), Setoid.r a b \u2192 f a = f b\n[PROOFSTEP]\nconvert h.image_eq_of_connectedComponent_eq\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 ConnectedComponents.mk \u207b\u00b9' {ConnectedComponents.mk x} = connectedComponent x\n[PROOFSTEP]\next y\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 y \u2208 ConnectedComponents.mk \u207b\u00b9' {ConnectedComponents.mk x} \u2194 y \u2208 connectedComponent x\n[PROOFSTEP]\nrw [mem_preimage, mem_singleton_iff, ConnectedComponents.coe_eq_coe']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nU : Set \u03b1\n\u22a2 ConnectedComponents.mk \u207b\u00b9' (ConnectedComponents.mk '' U) = \u22c3 (x : \u03b1) (_ : x \u2208 U), connectedComponent x\n[PROOFSTEP]\nsimp only [connectedComponents_preimage_singleton, preimage_iUnion\u2082, image_eq_iUnion]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 TotallyDisconnectedSpace (ConnectedComponents \u03b1)\n[PROOFSTEP]\nrw [totallyDisconnectedSpace_iff_connectedComponent_singleton]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (x : ConnectedComponents \u03b1), connectedComponent x = {x}\n[PROOFSTEP]\nrefine' ConnectedComponents.surjective_coe.forall.2 fun x => _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 connectedComponent (mk x) = {mk x}\n[PROOFSTEP]\nrw [\u2190 ConnectedComponents.quotientMap_coe.image_connectedComponent, \u2190 connectedComponents_preimage_singleton,\n  image_preimage_eq _ ConnectedComponents.surjective_coe]\n[GOAL]\ncase h_fibers\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2200 (y : ConnectedComponents \u03b1), IsConnected (mk \u207b\u00b9' {y})\n[PROOFSTEP]\nrefine' ConnectedComponents.surjective_coe.forall.2 fun y => _\n[GOAL]\ncase h_fibers\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 IsConnected (mk \u207b\u00b9' {mk y})\n[PROOFSTEP]\nrw [connectedComponents_preimage_singleton]\n[GOAL]\ncase h_fibers\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TotallyDisconnectedSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx y : \u03b1\n\u22a2 IsConnected (connectedComponent y)\n[PROOFSTEP]\nexact isConnected_connectedComponent\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nunfold IsPreconnected\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\n\u22a2 \u2200 (u v : Set \u03b1),\n    IsOpen u \u2192 IsOpen v \u2192 s \u2286 u \u222a v \u2192 Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (s \u2229 v) \u2192 Set.Nonempty (s \u2229 (u \u2229 v))\n[PROOFSTEP]\nby_contra'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u v s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nthis :\n  \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 s \u2286 u \u222a v \u2227 Set.Nonempty (s \u2229 u) \u2227 Set.Nonempty (s \u2229 v) \u2227 \u00acSet.Nonempty (s \u2229 (u \u2229 v))\n\u22a2 False\n[PROOFSTEP]\nrcases this with \u27e8u, v, u_op, v_op, hsuv, \u27e8x, x_in_s, x_in_u\u27e9, \u27e8y, y_in_s, y_in_v\u27e9, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : \u00acSet.Nonempty (s \u2229 (u \u2229 v))\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\n\u22a2 False\n[PROOFSTEP]\nrw [not_nonempty_iff_eq_empty] at H \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\n\u22a2 False\n[PROOFSTEP]\nhave hy : y \u2209 u := fun y_in_u => eq_empty_iff_forall_not_mem.mp H y \u27e8y_in_s, \u27e8y_in_u, y_in_v\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\n\u22a2 False\n[PROOFSTEP]\nhave : ContinuousOn u.boolIndicator s :=\n  by\n  apply (continuousOn_boolIndicator_iff_clopen _ _).mpr \u27e8_, _\u27e9\n  \u00b7 exact u_op.preimage continuous_subtype_val\n  \u00b7 rw [preimage_subtype_coe_eq_compl hsuv H]\n    exact (v_op.preimage continuous_subtype_val).isClosed_compl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\n\u22a2 ContinuousOn (boolIndicator u) s\n[PROOFSTEP]\napply (continuousOn_boolIndicator_iff_clopen _ _).mpr \u27e8_, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\n\u22a2 IsOpen (Subtype.val \u207b\u00b9' u)\n[PROOFSTEP]\nexact u_op.preimage continuous_subtype_val\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\n\u22a2 IsClosed (Subtype.val \u207b\u00b9' u)\n[PROOFSTEP]\nrw [preimage_subtype_coe_eq_compl hsuv H]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\n\u22a2 IsClosed (Subtype.val \u207b\u00b9' v)\u1d9c\n[PROOFSTEP]\nexact (v_op.preimage continuous_subtype_val).isClosed_compl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d : TopologicalSpace \u03b1\ns\u271d t u\u271d v\u271d s : Set \u03b1\nhs : \u2200 (f : \u03b1 \u2192 Bool), ContinuousOn f s \u2192 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 f x = f y\nu v : Set \u03b1\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s \u2286 u \u222a v\nx : \u03b1\nx_in_s : x \u2208 s\nx_in_u : x \u2208 u\nH : s \u2229 (u \u2229 v) = \u2205\ny : \u03b1\ny_in_s : y \u2208 s\ny_in_v : y \u2208 v\nhy : \u00acy \u2208 u\nthis : ContinuousOn (boolIndicator u) s\n\u22a2 False\n[PROOFSTEP]\nsimpa [(u.mem_iff_boolIndicator _).mp x_in_u, (u.not_mem_iff_boolIndicator _).mp hy] using hs _ this x x_in_s y y_in_s\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nx y : \u03b1\nhx : x \u2208 S\nhy : y \u2208 S\n\u22a2 f x = f y\n[PROOFSTEP]\nlet F : S \u2192 T := hTm.restrict f S T\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nx y : \u03b1\nhx : x \u2208 S\nhy : y \u2208 S\nF : \u2191S \u2192 \u2191T := MapsTo.restrict f S T hTm\n\u22a2 f x = f y\n[PROOFSTEP]\nsuffices F \u27e8x, hx\u27e9 = F \u27e8y, hy\u27e9 by rwa [\u2190 Subtype.coe_inj] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nx y : \u03b1\nhx : x \u2208 S\nhy : y \u2208 S\nF : \u2191S \u2192 \u2191T := MapsTo.restrict f S T hTm\nthis : F { val := x, property := hx } = F { val := y, property := hy }\n\u22a2 f x = f y\n[PROOFSTEP]\nrwa [\u2190 Subtype.coe_inj] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nx y : \u03b1\nhx : x \u2208 S\nhy : y \u2208 S\nF : \u2191S \u2192 \u2191T := MapsTo.restrict f S T hTm\n\u22a2 F { val := x, property := hx } = F { val := y, property := hy }\n[PROOFSTEP]\nexact (isPreconnected_iff_preconnectedSpace.mp hS).constant (hc.restrict_mapsTo _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nhne : Set.Nonempty T\n\u22a2 \u2203 y, y \u2208 T \u2227 EqOn f (const \u03b1 y) S\n[PROOFSTEP]\nrcases S.eq_empty_or_nonempty with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhne : Set.Nonempty T\nhS : IsPreconnected \u2205\nhc : ContinuousOn f \u2205\nhTm : MapsTo f \u2205 T\n\u22a2 \u2203 y, y \u2208 T \u2227 EqOn f (const \u03b1 y) \u2205\n[PROOFSTEP]\nexact hne.imp fun _ hy => \u27e8hy, eqOn_empty _ _\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ns t u v : Set \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nS : Set \u03b1\nhS : IsPreconnected S\nT : Set \u03b2\ninst\u271d : DiscreteTopology \u2191T\nf : \u03b1 \u2192 \u03b2\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nhne : Set.Nonempty T\nx : \u03b1\nhx : x \u2208 S\n\u22a2 \u2203 y, y \u2208 T \u2227 EqOn f (const \u03b1 y) S\n[PROOFSTEP]\nexact \u27e8f x, hTm hx, fun x' hx' => hS.constant_of_mapsTo hc hTm hx' hx\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Connected", "llama_tokens": 134115, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333246118695629, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.5398262042109854}}
{"text": "[GOAL]\np : \u211d[X]\n\u22a2 Tendsto (fun x => eval x p / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_monomial n c =>\n  simpa [exp_neg, div_eq_mul_inv, mul_assoc] using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)\n| h_add p q hp hq => simpa [add_div] using hp.add hq\n[GOAL]\np : \u211d[X]\n\u22a2 Tendsto (fun x => eval x p / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\ninduction p using Polynomial.induction_on' with\n| h_monomial n c =>\n  simpa [exp_neg, div_eq_mul_inv, mul_assoc] using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)\n| h_add p q hp hq => simpa [add_div] using hp.add hq\n[GOAL]\ncase h_monomial\nn : \u2115\nc : \u211d\n\u22a2 Tendsto (fun x => eval x (\u2191(monomial n) c) / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\n\n| h_monomial n c =>\n  simpa [exp_neg, div_eq_mul_inv, mul_assoc] using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)\n[GOAL]\ncase h_monomial\nn : \u2115\nc : \u211d\n\u22a2 Tendsto (fun x => eval x (\u2191(monomial n) c) / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa [exp_neg, div_eq_mul_inv, mul_assoc] using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_0 n)\n[GOAL]\ncase h_add\np q : \u211d[X]\nhp : Tendsto (fun x => eval x p / exp x) atTop (\ud835\udcdd 0)\nhq : Tendsto (fun x => eval x q / exp x) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => eval x (p + q) / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\n\n| h_add p q hp hq => simpa [add_div] using hp.add hq\n[GOAL]\ncase h_add\np q : \u211d[X]\nhp : Tendsto (fun x => eval x p / exp x) atTop (\ud835\udcdd 0)\nhq : Tendsto (fun x => eval x q / exp x) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => eval x (p + q) / exp x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa [add_div] using hp.add hq\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.PolynomialExp", "llama_tokens": 777, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867873410141, "lm_q2_score": 0.6757646140788307, "lm_q1q2_score": 0.5393863863103221}}
{"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Mul \u03b1\ns t u : Set \u03b1\na : \u03b1\n\u22a2 op '' s \u2022 t = t * s\n[PROOFSTEP]\nrw [\u2190 image2_smul, \u2190 image2_mul, image2_image_left, image2_swap]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d : Mul \u03b1\ns t u : Set \u03b1\na : \u03b1\n\u22a2 image2 (fun a b => SMul.smul (op b) a) t s = image2 (fun x x_1 => x * x_1) t s\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\ninst\u271d\u00b3 : SMul \u03b1 \u03b2\ninst\u271d\u00b2 : SMul \u03b1 \u03b3\ninst\u271d\u00b9 : SMul \u03b2 \u03b3\ninst\u271d : IsScalarTower \u03b1 \u03b2 \u03b3\na : \u03b1\nb : \u03b2\nT : Set \u03b3\n\u22a2 (a \u2022 b) \u2022 T = a \u2022 b \u2022 T\n[PROOFSTEP]\nsimp only [\u2190 image_smul, image_image, smul_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns : Set \u03b2\n\u22a2 (fun x => 1 \u2022 x) '' s = s\n[PROOFSTEP]\nsimp_rw [one_smul, image_id']\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\n\u22a2 \u2200 (b : Set \u03b2), 1 \u2022 b = b\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nb\u271d : Set \u03b2\n\u22a2 1 \u2022 b\u271d = b\u271d\n[PROOFSTEP]\nsimp only [\u2190 image_smul, one_smul, image_id']\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\n\u22a2 \u2200 (x y : \u03b1) (b : Set \u03b2), (x * y) \u2022 b = x \u2022 y \u2022 b\n[PROOFSTEP]\nintros\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\nx\u271d y\u271d : \u03b1\nb\u271d : Set \u03b2\n\u22a2 (x\u271d * y\u271d) \u2022 b\u271d = x\u271d \u2022 y\u271d \u2022 b\u271d\n[PROOFSTEP]\nsimp only [\u2190 image_smul, image_image, \u2190 mul_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddMonoid \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\nx\u271d : \u03b1\n\u22a2 {x\u271d \u2022 0} = 0\n[PROOFSTEP]\nrw [smul_zero, singleton_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : MulDistribMulAction \u03b1 \u03b2\nx\u271d : \u03b1\n\u22a2 {x\u271d \u2022 1} = 1\n[PROOFSTEP]\nrw [smul_one, singleton_one]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\n\u22a2 s = 0 \u2228 t = 0\n[PROOFSTEP]\nby_contra' H\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\nH : s \u2260 0 \u2227 t \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave hst : (s \u2022 t).Nonempty := h.symm.subst zero_nonempty\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\nH : s \u2260 0 \u2227 t \u2260 0\nhst : Set.Nonempty (s \u2022 t)\n\u22a2 False\n[PROOFSTEP]\nrw [Ne.def, \u2190 hst.of_smul_left.subset_zero_iff, Ne.def, \u2190 hst.of_smul_right.subset_zero_iff] at H \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\nH : \u00acs \u2286 0 \u2227 \u00act \u2286 0\nhst : Set.Nonempty (s \u2022 t)\n\u22a2 False\n[PROOFSTEP]\nsimp only [not_subset, mem_zero] at H \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\nhst : Set.Nonempty (s \u2022 t)\nH : (\u2203 a, a \u2208 s \u2227 \u00aca = 0) \u2227 \u2203 a, a \u2208 t \u2227 \u00aca = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8a, hs, ha\u27e9, b, ht, hb\u27e9 := H\n[GOAL]\ncase intro.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt\u271d t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nh : s \u2022 t = 0\nhst : Set.Nonempty (s \u2022 t)\na : \u03b1\nhs : a \u2208 s\nha : \u00aca = 0\nb : \u03b2\nht : b \u2208 t\nhb : \u00acb = 0\n\u22a2 False\n[PROOFSTEP]\nexact (eq_zero_or_eq_zero_of_smul_eq_zero <| h.subset <| smul_mem_smul hs ht).elim ha hb\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\ns : Set \u03b2\nh : a \u2022 s = 0\n\u22a2 a = 0 \u2228 s = 0\n[PROOFSTEP]\nby_contra' H\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\ns : Set \u03b2\nh : a \u2022 s = 0\nH : a \u2260 0 \u2227 s \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave hst : (a \u2022 s).Nonempty := h.symm.subst zero_nonempty\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\ns : Set \u03b2\nh : a \u2022 s = 0\nH : a \u2260 0 \u2227 s \u2260 0\nhst : Set.Nonempty (a \u2022 s)\n\u22a2 False\n[PROOFSTEP]\nrw [Ne.def, Ne.def, \u2190 hst.of_image.subset_zero_iff, not_subset] at H \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\ns : Set \u03b2\nh : a \u2022 s = 0\nH : \u00aca = 0 \u2227 \u2203 a, a \u2208 s \u2227 \u00aca \u2208 0\nhst : Set.Nonempty (a \u2022 s)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8ha, b, ht, hb\u27e9 := H\n[GOAL]\ncase intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ns\u271d s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMul \u03b1 \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\ns : Set \u03b2\nh : a \u2022 s = 0\nhst : Set.Nonempty (a \u2022 s)\nha : \u00aca = 0\nb : \u03b2\nht : b \u2208 s\nhb : \u00acb \u2208 0\n\u22a2 False\n[PROOFSTEP]\nexact (eq_zero_or_eq_zero_of_smul_eq_zero <| h.subset <| smul_mem_smul_set ht).elim ha hb\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : SMul \u03b1\u1d50\u1d52\u1d56 \u03b2\ninst\u271d\u00b9 : SMul \u03b2 \u03b3\ninst\u271d : SMul \u03b1 \u03b3\na : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), (op a \u2022 b) \u2022 c = b \u2022 a \u2022 c\n\u22a2 (op a \u2022 s) \u2022 t = s \u2022 a \u2022 t\n[PROOFSTEP]\next\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : SMul \u03b1\u1d50\u1d52\u1d56 \u03b2\ninst\u271d\u00b9 : SMul \u03b2 \u03b3\ninst\u271d : SMul \u03b1 \u03b3\na : \u03b1\ns : Set \u03b2\nt : Set \u03b3\nh : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), (op a \u2022 b) \u2022 c = b \u2022 a \u2022 c\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 (op a \u2022 s) \u2022 t \u2194 x\u271d \u2208 s \u2022 a \u2022 t\n[PROOFSTEP]\nsimp [mem_smul, mem_smul_set, h]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns\u271d : Set \u03b1\nt : Set \u03b2\ns : Set \u03b1\n\u22a2 s \u2022 0 \u2286 0\n[PROOFSTEP]\nsimp [subset_def, mem_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt\u271d t : Set \u03b2\n\u22a2 0 \u2022 t \u2286 0\n[PROOFSTEP]\nsimp [subset_def, mem_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nhs : Set.Nonempty s\n\u22a2 0 \u2286 s \u2022 0\n[PROOFSTEP]\nsimpa [mem_smul] using hs\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\nht : Set.Nonempty t\n\u22a2 0 \u2286 0 \u2022 t\n[PROOFSTEP]\nsimpa [mem_smul] using ht\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Zero \u03b1\ninst\u271d\u00b9 : Zero \u03b2\ninst\u271d : SMulWithZero \u03b1 \u03b2\ns\u271d : Set \u03b1\nt s : Set \u03b2\nh : Set.Nonempty s\n\u22a2 0 \u2022 s = 0\n[PROOFSTEP]\nsimp only [\u2190 image_smul, image_eta, zero_smul, h.image_const, singleton_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 s \u2022 t \u2194 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 s \u2022 t \u2192 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na\u271d a : \u03b1\nb : \u03b2\nha : a \u2208 s\nhb : b \u2208 t\nh : (fun x x_1 => x \u2022 x_1) a b = 0\n\u22a2 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s\n[PROOFSTEP]\nobtain rfl | rfl := eq_zero_or_eq_zero_of_smul_eq_zero h\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\nb : \u03b2\nhb : b \u2208 t\nha : 0 \u2208 s\nh : (fun x x_1 => x \u2022 x_1) 0 b = 0\n\u22a2 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s\n[PROOFSTEP]\nexact Or.inl \u27e8ha, b, hb\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na\u271d a : \u03b1\nha : a \u2208 s\nhb : 0 \u2208 t\nh : (fun x x_1 => x \u2022 x_1) a 0 = 0\n\u22a2 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s\n[PROOFSTEP]\nexact Or.inr \u27e8hb, a, ha\u27e9\n[GOAL]\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\n\u22a2 0 \u2208 s \u2227 Set.Nonempty t \u2228 0 \u2208 t \u2227 Set.Nonempty s \u2192 0 \u2208 s \u2022 t\n[PROOFSTEP]\nrintro (\u27e8hs, b, hb\u27e9 | \u27e8ht, a, ha\u27e9)\n[GOAL]\ncase mpr.inl.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\nhs : 0 \u2208 s\nb : \u03b2\nhb : b \u2208 t\n\u22a2 0 \u2208 s \u2022 t\n[PROOFSTEP]\nexact \u27e80, b, hs, hb, zero_smul _ _\u27e9\n[GOAL]\ncase mpr.inr.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na\u271d : \u03b1\nht : 0 \u2208 t\na : \u03b1\nha : a \u2208 s\n\u22a2 0 \u2208 s \u2022 t\n[PROOFSTEP]\nexact \u27e8a, 0, ha, ht, smul_zero _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\nha : a \u2260 0\n\u22a2 0 \u2208 a \u2022 t \u2194 0 \u2208 t\n[PROOFSTEP]\nrefine' \u27e8_, zero_mem_smul_set\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\nha : a \u2260 0\n\u22a2 0 \u2208 a \u2022 t \u2192 0 \u2208 t\n[PROOFSTEP]\nrintro \u27e8b, hb, h\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b3 : Zero \u03b1\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2\ns : Set \u03b1\nt : Set \u03b2\ninst\u271d : NoZeroSMulDivisors \u03b1 \u03b2\na : \u03b1\nha : a \u2260 0\nb : \u03b2\nhb : b \u2208 t\nh : (fun x => a \u2022 x) b = 0\n\u22a2 0 \u2208 t\n[PROOFSTEP]\nrwa [(eq_zero_or_eq_zero_of_smul_eq_zero h).resolve_left ha] at hb \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns t A B : Set \u03b2\na : \u03b1\nx : \u03b2\n\u22a2 x \u2208 a\u207b\u00b9 \u2022 A \u2194 a \u2022 x \u2208 A\n[PROOFSTEP]\nsimp only [\u2190 image_smul, mem_image, inv_smul_eq_iff, exists_eq_right]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 x \u2022 s \u2229 t \u2260 \u2205 \u2194 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x\n[PROOFSTEP]\nrw [\u2190 nonempty_iff_ne_empty]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t) \u2194 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t) \u2192 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x\n[PROOFSTEP]\nrintro \u27e8a, h, ha\u27e9\n[GOAL]\ncase mp.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx a : \u03b1\nh : a \u2208 x \u2022 s\nha : a \u2208 t\n\u22a2 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x\n[PROOFSTEP]\nobtain \u27e8b, hb, rfl\u27e9 := mem_smul_set.mp h\n[GOAL]\ncase mp.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx b : \u03b1\nhb : b \u2208 s\nh : x \u2022 b \u2208 x \u2022 s\nha : x \u2022 b \u2208 t\n\u22a2 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x\n[PROOFSTEP]\nexact \u27e8x \u2022 b, b, \u27e8ha, hb\u27e9, by simp\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx b : \u03b1\nhb : b \u2208 s\nh : x \u2022 b \u2208 x \u2022 s\nha : x \u2022 b \u2208 t\n\u22a2 x \u2022 b * b\u207b\u00b9 = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 (\u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a * b\u207b\u00b9 = x) \u2192 Set.Nonempty (x \u2022 s \u2229 t)\n[PROOFSTEP]\nrintro \u27e8a, b, \u27e8ha, hb\u27e9, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx : \u03b2\ns t : Set \u03b1\na b : \u03b1\nha : a \u2208 t\nhb : b \u2208 s\n\u22a2 Set.Nonempty ((a * b\u207b\u00b9) \u2022 s \u2229 t)\n[PROOFSTEP]\nexact \u27e8a, mem_inter (mem_smul_set.mpr \u27e8b, hb, by simp\u27e9) ha\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx : \u03b2\ns t : Set \u03b1\na b : \u03b1\nha : a \u2208 t\nhb : b \u2208 s\n\u22a2 (a * b\u207b\u00b9) \u2022 b = a\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\n\u22a2 x \u2022 s \u2229 t \u2260 \u2205 \u2194 \u2203 a b, (a \u2208 t \u2227 b \u2208 s) \u2227 a / b = x\n[PROOFSTEP]\nsimp_rw [smul_inter_ne_empty_iff, div_eq_mul_inv]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 x \u2022 s \u2229 t \u2260 \u2205 \u2194 \u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x\n[PROOFSTEP]\nrw [\u2190 nonempty_iff_ne_empty]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t) \u2194 \u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t) \u2192 \u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x\n[PROOFSTEP]\nrintro \u27e8a, h, ha\u27e9\n[GOAL]\ncase mp.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\na : \u03b1\nh : a \u2208 x \u2022 s\nha : a \u2208 t\n\u22a2 \u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x\n[PROOFSTEP]\nobtain \u27e8b, hb, rfl\u27e9 := mem_smul_set.mp h\n[GOAL]\ncase mp.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\nb : \u03b1\nhb : b \u2208 s\nh : x \u2022 b \u2208 x \u2022 s\nha : x \u2022 b \u2208 t\n\u22a2 \u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x\n[PROOFSTEP]\nexact \u27e8b, x \u2022 b, \u27e8hb, ha\u27e9, by simp\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\nb : \u03b1\nhb : b \u2208 s\nh : x \u2022 b \u2208 x \u2022 s\nha : x \u2022 b \u2208 t\n\u22a2 b\u207b\u00b9 * x \u2022 b = unop x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\n\u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 t) \u2227 a\u207b\u00b9 * b = unop x) \u2192 Set.Nonempty (x \u2022 s \u2229 t)\n[PROOFSTEP]\nrintro \u27e8a, b, \u27e8ha, hb\u27e9, H\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\na b : \u03b1\nH : a\u207b\u00b9 * b = unop x\nha : a \u2208 s\nhb : b \u2208 t\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t)\n[PROOFSTEP]\nhave : MulOpposite.op (a\u207b\u00b9 * b) = x := congr_arg MulOpposite.op H\n[GOAL]\ncase mpr.intro.intro.intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\na b : \u03b1\nH : a\u207b\u00b9 * b = unop x\nha : a \u2208 s\nhb : b \u2208 t\nthis : op (a\u207b\u00b9 * b) = x\n\u22a2 Set.Nonempty (x \u2022 s \u2229 t)\n[PROOFSTEP]\nexact \u27e8b, mem_inter (mem_smul_set.mpr \u27e8a, ha, by simp [\u2190 this]\u27e9) hb\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t\u271d A B : Set \u03b2\na\u271d : \u03b1\nx\u271d : \u03b2\ns t : Set \u03b1\nx : \u03b1\u1d50\u1d52\u1d56\na b : \u03b1\nH : a\u207b\u00b9 * b = unop x\nha : a \u2208 s\nhb : b \u2208 t\nthis : op (a\u207b\u00b9 * b) = x\n\u22a2 x \u2022 a = b\n[PROOFSTEP]\nsimp [\u2190 this]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : MulAction \u03b1 \u03b2\ns\u271d t A B : Set \u03b2\na : \u03b1\nx : \u03b2\ns : Set \u03b2\n\u22a2 \u22c3 (g : \u03b1), g \u2022 s = {a | \u2203 g, g \u2022 a \u2208 s}\n[PROOFSTEP]\nsimp_rw [\u2190 iUnion_setOf, \u2190 iUnion_inv_smul, \u2190 preimage_smul, preimage]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 a \u2022 -t = -(a \u2022 t)\n[PROOFSTEP]\nsimp_rw [\u2190 image_smul, \u2190 image_neg, image_image, smul_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 -t = -(s \u2022 t)\n[PROOFSTEP]\nsimp_rw [\u2190 image_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Monoid \u03b1\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : DistribMulAction \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 s \u2022 Neg.neg '' t = Neg.neg '' (s \u2022 t)\n[PROOFSTEP]\nexact image_image2_right_comm smul_neg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module \u03b1 \u03b2\na b : \u03b1\ns : Set \u03b2\n\u22a2 (a + b) \u2022 s \u2286 a \u2022 s + b \u2022 s\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : AddCommMonoid \u03b2\ninst\u271d : Module \u03b1 \u03b2\na b : \u03b1\ns : Set \u03b2\nx : \u03b2\nhx : x \u2208 s\n\u22a2 (fun x => (a + b) \u2022 x) x \u2208 a \u2022 s + b \u2022 s\n[PROOFSTEP]\nsimpa only [add_smul] using add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hx)\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 -a \u2022 t = -(a \u2022 t)\n[PROOFSTEP]\nsimp_rw [\u2190 image_smul, \u2190 image_neg, image_image, neg_smul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 -s \u2022 t = -(s \u2022 t)\n[PROOFSTEP]\nsimp_rw [\u2190 image_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u00b2 : Ring \u03b1\ninst\u271d\u00b9 : AddCommGroup \u03b2\ninst\u271d : Module \u03b1 \u03b2\na : \u03b1\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 Neg.neg '' s \u2022 t = Neg.neg '' (s \u2022 t)\n[PROOFSTEP]\nexact image2_image_left_comm neg_smul\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Pointwise.SMul", "llama_tokens": 10974, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867729389246, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.5393863661537535}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\n\u03c0 : Y \u27f6 Z\nq : IsSplitCoequalizer f g \u03c0\nF : C \u2964 D\n\u22a2 F.map f \u226b F.map \u03c0 = F.map g \u226b F.map \u03c0\n[PROOFSTEP]\nrw [\u2190 F.map_comp, q.condition, F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\n\u03c0 : Y \u27f6 Z\nq : IsSplitCoequalizer f g \u03c0\nF : C \u2964 D\n\u22a2 F.map q.rightSection \u226b F.map \u03c0 = \ud835\udfd9 (F.obj Z)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, q.rightSection_\u03c0, F.map_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\n\u03c0 : Y \u27f6 Z\nq : IsSplitCoequalizer f g \u03c0\nF : C \u2964 D\n\u22a2 F.map q.leftSection \u226b F.map g = \ud835\udfd9 (F.obj Y)\n[PROOFSTEP]\nrw [\u2190 F.map_comp, q.leftSection_bottom, F.map_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\n\u03c0 : Y \u27f6 Z\nq : IsSplitCoequalizer f g \u03c0\nF : C \u2964 D\n\u22a2 F.map q.leftSection \u226b F.map f = F.map \u03c0 \u226b F.map q.rightSection\n[PROOFSTEP]\nrw [\u2190 F.map_comp, q.leftSection_top, F.map_comp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\nh : Y \u27f6 Z\nt : IsSplitCoequalizer f g h\ns : Cofork f g\n\u22a2 Cofork.\u03c0 (asCofork t) \u226b t.rightSection \u226b Cofork.\u03c0 s = Cofork.\u03c0 s\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\nh : Y \u27f6 Z\nt : IsSplitCoequalizer f g h\ns : Cofork f g\n\u22a2 h \u226b t.rightSection \u226b Cofork.\u03c0 s = Cofork.\u03c0 s\n[PROOFSTEP]\nrw [\u2190 t.leftSection_top_assoc, s.condition, t.leftSection_bottom_assoc]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nG : C \u2964 D\nX Y : C\nf g : X \u27f6 Y\nZ : C\nh : Y \u27f6 Z\nt : IsSplitCoequalizer f g h\ns : Cofork f g\nm\u271d :\n  ((Functor.const WalkingParallelPair).obj (asCofork t).pt).obj WalkingParallelPair.one \u27f6\n    ((Functor.const WalkingParallelPair).obj s.pt).obj WalkingParallelPair.one\nhm : Cofork.\u03c0 (asCofork t) \u226b m\u271d = Cofork.\u03c0 s\n\u22a2 m\u271d = t.rightSection \u226b Cofork.\u03c0 s\n[PROOFSTEP]\nsimp [\u2190 hm]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer", "llama_tokens": 1134, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396212, "lm_q2_score": 0.6757645879592641, "lm_q1q2_score": 0.5393863589737479}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Finset.Nonempty (Icc a b) \u2194 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 coe_nonempty, coe_Icc, Set.nonempty_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Finset.Nonempty (Ico a b) \u2194 a < b\n[PROOFSTEP]\nrw [\u2190 coe_nonempty, coe_Ico, Set.nonempty_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Finset.Nonempty (Ioc a b) \u2194 a < b\n[PROOFSTEP]\nrw [\u2190 coe_nonempty, coe_Ioc, Set.nonempty_Ioc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Finset.Nonempty (Ioo a b) \u2194 a < b\n[PROOFSTEP]\nrw [\u2190 coe_nonempty, coe_Ioo, Set.nonempty_Ioo]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Icc a b = \u2205 \u2194 \u00aca \u2264 b\n[PROOFSTEP]\nrw [\u2190 coe_eq_empty, coe_Icc, Set.Icc_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ico a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 coe_eq_empty, coe_Ico, Set.Ico_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ioc a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 coe_eq_empty, coe_Ioc, Set.Ioc_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : DenselyOrdered \u03b1\n\u22a2 Ioo a b = \u2205 \u2194 \u00aca < b\n[PROOFSTEP]\nrw [\u2190 coe_eq_empty, coe_Ioo, Set.Ioo_eq_empty_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Icc a b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only [mem_Icc, true_and_iff, le_rfl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 Ico a b \u2194 a < b\n[PROOFSTEP]\nsimp only [mem_Ico, true_and_iff, le_refl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Icc a b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp only [mem_Icc, and_true_iff, le_rfl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 Ioc a b \u2194 a < b\n[PROOFSTEP]\nsimp only [mem_Ioc, and_true_iff, le_rfl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Icc a\u2081 b\u2081 \u2286 Icc a\u2082 b\u2082\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Icc_subset_Icc ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ico_subset_Ico ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Ioc a\u2081 b\u2081 \u2286 Ioc a\u2082 b\u2082\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioc_subset_Ioc ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Ioo a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioo_subset_Ioo ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : a\u2081 < a\u2082\n\u22a2 Ico a\u2082 b \u2286 Ioo a\u2081 b\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ico, coe_Ioo]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : a\u2081 < a\u2082\n\u22a2 Set.Ico a\u2082 b \u2286 Set.Ioo a\u2081 b\n[PROOFSTEP]\nexact Set.Ico_subset_Ioo_left h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : b\u2081 < b\u2082\n\u22a2 Ioc a b\u2081 \u2286 Ioo a b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ioc, coe_Ioo]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : b\u2081 < b\u2082\n\u22a2 Set.Ioc a b\u2081 \u2286 Set.Ioo a b\u2082\n[PROOFSTEP]\nexact Set.Ioc_subset_Ioo_right h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : b\u2081 < b\u2082\n\u22a2 Icc a b\u2081 \u2286 Ico a b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Icc, coe_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : b\u2081 < b\u2082\n\u22a2 Set.Icc a b\u2081 \u2286 Set.Ico a b\u2082\n[PROOFSTEP]\nexact Set.Icc_subset_Ico_right h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ioo a b \u2286 Ico a b\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ioo, coe_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.Ioo a b \u2286 Set.Ico a b\n[PROOFSTEP]\nexact Set.Ioo_subset_Ico_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ioo a b \u2286 Ioc a b\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ioo, coe_Ioc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.Ioo a b \u2286 Set.Ioc a b\n[PROOFSTEP]\nexact Set.Ioo_subset_Ioc_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ico a b \u2286 Icc a b\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ico, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.Ico a b \u2286 Set.Icc a b\n[PROOFSTEP]\nexact Set.Ico_subset_Icc_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Ioc a b \u2286 Icc a b\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ioc, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.Ioc a b \u2286 Set.Icc a b\n[PROOFSTEP]\nexact Set.Ioc_subset_Icc_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 Icc a\u2081 b\u2081 \u2286 Icc a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Icc, coe_Icc, Set.Icc_subset_Icc_iff h\u2081]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 Icc a\u2081 b\u2081 \u2286 Ioo a\u2082 b\u2082 \u2194 a\u2082 < a\u2081 \u2227 b\u2081 < b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h\u2081]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2264 b\u2081\n\u22a2 Icc a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 < b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Icc, coe_Ico, Set.Icc_subset_Ico_iff h\u2081]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 < a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_ssubset, coe_Icc, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 < a\u2081\nhb : b\u2081 \u2264 b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082\n[PROOFSTEP]\nexact Set.Icc_ssubset_Icc_left hI ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 < b\u2082\n\u22a2 Icc a\u2081 b\u2081 \u2282 Icc a\u2082 b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_ssubset, coe_Icc, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nhI : a\u2082 \u2264 b\u2082\nha : a\u2082 \u2264 a\u2081\nhb : b\u2081 < b\u2082\n\u22a2 Set.Icc a\u2081 b\u2081 \u2282 Set.Icc a\u2082 b\u2082\n[PROOFSTEP]\nexact Set.Icc_ssubset_Icc_right hI ha hb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\n\u22a2 Set.Finite s\n[PROOFSTEP]\nlet \u27e8a, ha\u27e9 := h\u2080\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\n\u22a2 Set.Finite s\n[PROOFSTEP]\nlet \u27e8b, hb\u27e9 := h\u2081\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 Set.Finite s\n[PROOFSTEP]\nclassical exact \u27e8Set.fintypeOfMemBounds ha hb\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x : \u03b1\ns : Set \u03b1\nh\u2080 : BddBelow s\nh\u2081 : BddAbove s\na : \u03b1\nha : a \u2208 lowerBounds s\nb : \u03b1\nhb : b \u2208 upperBounds s\n\u22a2 Set.Finite s\n[PROOFSTEP]\nexact \u27e8Set.fintypeOfMemBounds ha hb\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhcb : c \u2264 b\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a c\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x\u271d : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhcb : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 filter (fun x => x < c) (Ico a b) \u2194 x \u2208 Ico a c\n[PROOFSTEP]\nrw [mem_filter, mem_Ico, mem_Ico, and_right_comm]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x\u271d : \u03b1\ninst\u271d : DecidablePred fun x => x < c\nhcb : c \u2264 b\nx : \u03b1\n\u22a2 (a \u2264 x \u2227 x < c) \u2227 x < b \u2194 a \u2264 x \u2227 x < c\n[PROOFSTEP]\nexact and_iff_left_of_imp fun h => h.2.trans_le hcb\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhac : a \u2264 c\n\u22a2 filter ((fun x x_1 => x \u2264 x_1) c) (Ico a b) = Ico c b\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x\u271d a b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhac : a \u2264 c\nx : \u03b1\n\u22a2 x \u2208 filter ((fun x x_1 => x \u2264 x_1) c) (Ico a b) \u2194 x \u2208 Ico c b\n[PROOFSTEP]\nrw [mem_filter, mem_Ico, mem_Ico, and_comm, and_left_comm]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x\u271d a b c : \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) c)\nhac : a \u2264 c\nx : \u03b1\n\u22a2 a \u2264 x \u2227 (fun x x_1 => x \u2264 x_1) c x \u2227 x < b \u2194 c \u2264 x \u2227 x < b\n[PROOFSTEP]\nexact and_iff_right_of_imp fun h => hac.trans h.1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a < j \u2227 j < b\n\u22a2 filter (fun j => a < j \u2227 j < b) univ = Ioo a b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a < j \u2227 j < b\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun j => a < j \u2227 j < b) univ \u2194 a\u271d \u2208 Ioo a b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a < j \u2227 j \u2264 b\n\u22a2 filter (fun j => a < j \u2227 j \u2264 b) univ = Ioc a b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a < j \u2227 j \u2264 b\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun j => a < j \u2227 j \u2264 b) univ \u2194 a\u271d \u2208 Ioc a b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a \u2264 j \u2227 j < b\n\u22a2 filter (fun j => a \u2264 j \u2227 j < b) univ = Ico a b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a \u2264 j \u2227 j < b\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun j => a \u2264 j \u2227 j < b) univ \u2194 a\u271d \u2208 Ico a b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a \u2264 j \u2227 j \u2264 b\n\u22a2 filter (fun j => a \u2264 j \u2227 j \u2264 b) univ = Icc a b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun j => a \u2264 j \u2227 j \u2264 b\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun j => a \u2264 j \u2227 j \u2264 b) univ \u2194 a\u271d \u2208 Icc a b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Icc a b \u2286 Ici a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Icc_subset_Ici_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Ico a b \u2286 Ici a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ico_subset_Ici_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Ioc a b \u2286 Ioi a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioc_subset_Ioi_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Ioo a b \u2286 Ioi a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioo_subset_Ioi_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Icc a b \u2286 Iic b\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Icc_subset_Iic_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Ioc a b \u2286 Iic b\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioc_subset_Iic_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Ico a b \u2286 Iio b\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ico_subset_Iio_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\n\u22a2 Ioo a b \u2286 Iio b\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioo_subset_Iio_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 Ioi a \u2286 Ici a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Ioi_subset_Ici_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x < x_1) a)\n\u22a2 filter ((fun x x_1 => x < x_1) a) univ = Ioi a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x < x_1) a)\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter ((fun x x_1 => x < x_1) a) univ \u2194 a\u271d \u2208 Ioi a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) a)\n\u22a2 filter ((fun x x_1 => x \u2264 x_1) a) univ = Ici a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred ((fun x x_1 => x \u2264 x_1) a)\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter ((fun x x_1 => x \u2264 x_1) a) univ \u2194 a\u271d \u2208 Ici a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 Iio a \u2286 Iic a\n[PROOFSTEP]\nsimpa [\u2190 coe_subset] using Set.Iio_subset_Iic_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun x => x < a\n\u22a2 filter (fun x => x < a) univ = Iio a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun x => x < a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun x => x < a) univ \u2194 a\u271d \u2208 Iio a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\n\u22a2 filter (fun x => x \u2264 a) univ = Iic a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : LocallyFiniteOrderBot \u03b1\na : \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\na\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun x => x \u2264 a) univ \u2194 a\u271d \u2208 Iic a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b c a : \u03b1\n\u22a2 Icc a a = {a}\n[PROOFSTEP]\nrw [\u2190 coe_eq_singleton, coe_Icc, Set.Icc_self]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Icc a b = {c} \u2194 a = c \u2227 b = c\n[PROOFSTEP]\nrw [\u2190 coe_eq_singleton, coe_Icc, Set.Icc_eq_singleton_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 erase (Icc a b) a = Ioc a b\n[PROOFSTEP]\nsimp [\u2190 coe_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 erase (Icc a b) b = Ico a b\n[PROOFSTEP]\nsimp [\u2190 coe_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 erase (Ico a b) a = Ioo a b\n[PROOFSTEP]\nsimp [\u2190 coe_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 erase (Ioc a b) b = Ioo a b\n[PROOFSTEP]\nsimp [\u2190 coe_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c : \u03b1\ninst\u271d : DecidableEq \u03b1\na b : \u03b1\n\u22a2 Icc a b \\ {a, b} = Ioo a b\n[PROOFSTEP]\nsimp [\u2190 coe_inj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 insert b (Ico a b) = Icc a b\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_insert, coe_Icc, coe_Ico, Set.insert_eq, Set.union_comm, Set.Ico_union_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 insert a (Ioc a b) = Icc a b\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_insert, coe_Ioc, coe_Icc, Set.insert_eq, Set.union_comm, Set.Ioc_union_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a < b\n\u22a2 insert a (Ioo a b) = Ico a b\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_insert, coe_Ioo, coe_Ico, Set.insert_eq, Set.union_comm, Set.Ioo_union_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a < b\n\u22a2 insert b (Ioo a b) = Ioc a b\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_insert, coe_Ioo, coe_Ioc, Set.insert_eq, Set.union_comm, Set.Ioo_union_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ico a b = {b}\n[PROOFSTEP]\nsimp [\u2190 coe_inj, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ioc a b = {a}\n[PROOFSTEP]\nsimp [\u2190 coe_inj, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2264 b\n\u22a2 Icc a b \\ Ioo a b = {a, b}\n[PROOFSTEP]\nsimp [\u2190 coe_inj, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a < b\n\u22a2 Ico a b \\ Ioo a b = {a}\n[PROOFSTEP]\nsimp [\u2190 coe_inj, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na b c : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a < b\n\u22a2 Ioc a b \\ Ioo a b = {b}\n[PROOFSTEP]\nsimp [\u2190 coe_inj, h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons b (Ico a b) (_ : \u00acb \u2208 Ico a b)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Ico_insert_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons b (Ico a b) (_ : \u00acb \u2208 Ico a b)\n[PROOFSTEP]\nrw [cons_eq_insert, Ico_insert_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Ioc_insert_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a \u2264 b\n\u22a2 Icc a b = cons a (Ioc a b) (_ : \u00aca \u2208 Ioc a b)\n[PROOFSTEP]\nrw [cons_eq_insert, Ioc_insert_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Ioo_insert_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ioc a b = cons b (Ioo a b) (_ : \u00acb \u2208 Ioo a b)\n[PROOFSTEP]\nrw [cons_eq_insert, Ioo_insert_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ico a b = cons a (Ioo a b) (_ : \u00aca \u2208 Ioo a b)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Ioo_insert_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\nh : a < b\n\u22a2 Ico a b = cons a (Ioo a b) (_ : \u00aca \u2208 Ioo a b)\n[PROOFSTEP]\nrw [cons_eq_insert, Ioo_insert_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\nhab : a < b\n\u22a2 filter (fun x => x \u2264 a) (Ico a b) = {a}\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\nhab : a < b\nx : \u03b1\n\u22a2 x \u2208 filter (fun x => x \u2264 a) (Ico a b) \u2194 x \u2208 {a}\n[PROOFSTEP]\nrw [mem_filter, mem_Ico, mem_singleton, and_right_comm, \u2190 le_antisymm_iff, eq_comm]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\ninst\u271d : DecidablePred fun x => x \u2264 a\nhab : a < b\nx : \u03b1\n\u22a2 x = a \u2227 x < b \u2194 x = a\n[PROOFSTEP]\nexact and_iff_left_of_imp fun h => h.le.trans_lt hab\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Ico a b) = card (Icc a b) - 1\n[PROOFSTEP]\nclassical\nby_cases h : a \u2264 b\n\u00b7 rw [Icc_eq_cons_Ico h, card_cons]\n  exact (Nat.add_sub_cancel _ _).symm\n\u00b7 rw [Ico_eq_empty fun h' => h h'.le, Icc_eq_empty h, card_empty, zero_tsub]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Ico a b) = card (Icc a b) - 1\n[PROOFSTEP]\nby_cases h : a \u2264 b\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : a \u2264 b\n\u22a2 card (Ico a b) = card (Icc a b) - 1\n[PROOFSTEP]\nrw [Icc_eq_cons_Ico h, card_cons]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : a \u2264 b\n\u22a2 card (Ico a b) = card (Ico a b) + 1 - 1\n[PROOFSTEP]\nexact (Nat.add_sub_cancel _ _).symm\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 card (Ico a b) = card (Icc a b) - 1\n[PROOFSTEP]\nrw [Ico_eq_empty fun h' => h h'.le, Icc_eq_empty h, card_empty, zero_tsub]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Ioo a b) = card (Ico a b) - 1\n[PROOFSTEP]\nclassical\nby_cases h : a < b\n\u00b7 rw [Ico_eq_cons_Ioo h, card_cons]\n  exact (Nat.add_sub_cancel _ _).symm\n\u00b7 rw [Ioo_eq_empty h, Ico_eq_empty h, card_empty, zero_tsub]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Ioo a b) = card (Ico a b) - 1\n[PROOFSTEP]\nby_cases h : a < b\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : a < b\n\u22a2 card (Ioo a b) = card (Ico a b) - 1\n[PROOFSTEP]\nrw [Ico_eq_cons_Ioo h, card_cons]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : a < b\n\u22a2 card (Ioo a b) = card (Ioo a b) + 1 - 1\n[PROOFSTEP]\nexact (Nat.add_sub_cancel _ _).symm\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\nh : \u00aca < b\n\u22a2 card (Ioo a b) = card (Ico a b) - 1\n[PROOFSTEP]\nrw [Ioo_eq_empty h, Ico_eq_empty h, card_empty, zero_tsub]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Ioo a b) = card (Icc a b) - 2\n[PROOFSTEP]\nrw [card_Ioo_eq_card_Ico_sub_one, card_Ico_eq_card_Icc_sub_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 card (Icc a b) - 1 - 1 = card (Icc a b) - 2\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 erase (Ici a) a = Ioi a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 erase (Ici a) a \u2194 a\u271d \u2208 Ioi a\n[PROOFSTEP]\nsimp_rw [Finset.mem_erase, mem_Ici, mem_Ioi, lt_iff_le_and_ne, and_comm, ne_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 insert a (Ioi a) = Ici a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : DecidableEq \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 insert a (Ioi a) \u2194 a\u271d \u2208 Ici a\n[PROOFSTEP]\nsimp_rw [Finset.mem_insert, mem_Ici, mem_Ioi, le_iff_lt_or_eq, or_comm, eq_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Ioi_insert]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 Ici a = cons a (Ioi a) (_ : \u00aca \u2208 Ioi a)\n[PROOFSTEP]\nrw [cons_eq_insert, Ioi_insert]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\na : \u03b1\n\u22a2 card (Ioi a) = card (Ici a) - 1\n[PROOFSTEP]\nrw [Ici_eq_cons_Ioi, card_cons, add_tsub_cancel_right]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\nb : \u03b1\n\u22a2 erase (Iic b) b = Iio b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\nb a\u271d : \u03b1\n\u22a2 a\u271d \u2208 erase (Iic b) b \u2194 a\u271d \u2208 Iio b\n[PROOFSTEP]\nsimp_rw [Finset.mem_erase, mem_Iic, mem_Iio, lt_iff_le_and_ne, and_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\nb : \u03b1\n\u22a2 insert b (Iio b) = Iic b\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b1\ninst\u271d : DecidableEq \u03b1\nb a\u271d : \u03b1\n\u22a2 a\u271d \u2208 insert b (Iio b) \u2194 a\u271d \u2208 Iic b\n[PROOFSTEP]\nsimp_rw [Finset.mem_insert, mem_Iic, mem_Iio, le_iff_lt_or_eq, or_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb : \u03b1\n\u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b)\n[PROOFSTEP]\nclassical rw [cons_eq_insert, Iio_insert]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\nb : \u03b1\n\u22a2 Iic b = cons b (Iio b) (_ : \u00acb \u2208 Iio b)\n[PROOFSTEP]\nrw [cons_eq_insert, Iio_insert]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 card (Iio a) = card (Iic a) - 1\n[PROOFSTEP]\nrw [Iic_eq_cons_Iio, card_cons, add_tsub_cancel_right]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b a\u2081 b\u2081 a\u2082 b\u2082 : \u03b1\nh : a\u2081 < b\u2081\n\u22a2 Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_Ico, coe_Ico, Set.Ico_subset_Ico_iff h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 Ico a b \u222a Ico b c = Ico a c\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico_eq_Ico hab hbc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh\u2081 : a \u2264 b\nh\u2082 : b \u2264 c\n\u22a2 Ioc a b \u222a Ioc b c = Ioc a c\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_union, coe_Ioc, coe_Ioc, coe_Ioc, Set.Ioc_union_Ioc_eq_Ioc h\u2081 h\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 Ico a c \u2286 Ico a b \u222a Ico b c\n[PROOFSTEP]\nrw [\u2190 coe_subset, coe_union, coe_Ico, coe_Ico, coe_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 Set.Ico a c \u2286 Set.Ico a b \u222a Set.Ico b c\n[PROOFSTEP]\nexact Set.Ico_subset_Ico_union_Ico\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c d : \u03b1\nhcb : c \u2264 b\nhad : a \u2264 d\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico' hcb had]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c d : \u03b1\nh\u2081 : min a b \u2264 max c d\nh\u2082 : min c d \u2264 max a b\n\u22a2 Ico a b \u222a Ico c d = Ico (min a c) (max b d)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_union, coe_Ico, coe_Ico, coe_Ico, Set.Ico_union_Ico h\u2081 h\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c d : \u03b1\n\u22a2 Ico a b \u2229 Ico c d = Ico (max a c) (min b d)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_inter, coe_Ico, coe_Ico, coe_Ico, \u2190 inf_eq_min, \u2190 sup_eq_max, Set.Ico_inter_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\ncases le_total b c with\n| inl h => rw [Ico_filter_lt_of_right_le h, min_eq_left h]\n| inr h => rw [Ico_filter_lt_of_le_right h, min_eq_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nx\u271d : b \u2264 c \u2228 c \u2264 b\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\ncases le_total b c with\n| inl h => rw [Ico_filter_lt_of_right_le h, min_eq_left h]\n| inr h => rw [Ico_filter_lt_of_le_right h, min_eq_right h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : b \u2264 c\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\n\n| inl h => rw [Ico_filter_lt_of_right_le h, min_eq_left h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : b \u2264 c\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\nrw [Ico_filter_lt_of_right_le h, min_eq_left h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\n\n| inr h => rw [Ico_filter_lt_of_le_right h, min_eq_right h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\n\u22a2 filter (fun x => x < c) (Ico a b) = Ico a (min b c)\n[PROOFSTEP]\nrw [Ico_filter_lt_of_le_right h, min_eq_right h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\ncases le_total a c with\n| inl h => rw [Ico_filter_le_of_left_le h, max_eq_right h]\n| inr h => rw [Ico_filter_le_of_le_left h, max_eq_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nx\u271d : a \u2264 c \u2228 c \u2264 a\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\ncases le_total a c with\n| inl h => rw [Ico_filter_le_of_left_le h, max_eq_right h]\n| inr h => rw [Ico_filter_le_of_le_left h, max_eq_left h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\n\n| inl h => rw [Ico_filter_le_of_left_le h, max_eq_right h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\nrw [Ico_filter_le_of_left_le h, max_eq_right h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 a\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\n\n| inr h => rw [Ico_filter_le_of_le_left h, max_eq_left h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 a\n\u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b\n[PROOFSTEP]\nrw [Ico_filter_le_of_le_left h, max_eq_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 filter (fun x => x < c) (Ioo a b) = Ioo a (min b c)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d\u00b9 b\u271d a b c a\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun x => x < c) (Ioo a b) \u2194 a\u271d \u2208 Ioo a (min b c)\n[PROOFSTEP]\nsimp [and_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\u271d\na\u271d b\u271d : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na b : \u03b1\n\u22a2 filter (fun x => x < b) (Iio a) = Iio (min a b)\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\u271d\ninst\u271d\u00b2 : LocallyFiniteOrder \u03b1\u271d\na\u271d\u00b9 b\u271d : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na b a\u271d : \u03b1\n\u22a2 a\u271d \u2208 filter (fun x => x < b) (Iio a) \u2194 a\u271d \u2208 Iio (min a b)\n[PROOFSTEP]\nsimp [and_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\ncases le_total a c with\n| inl h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]\n  exact and_congr_left' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8h.trans hx, fun _ => hx\u27e9\u27e9\n| inr h => rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nx\u271d : a \u2264 c \u2228 c \u2264 a\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\ncases le_total a c with\n| inl h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]\n  exact and_congr_left' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8h.trans hx, fun _ => hx\u27e9\u27e9\n| inr h => rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\n\n| inl h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]\n  exact and_congr_left' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8h.trans hx, fun _ => hx\u27e9\u27e9\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\next x\n[GOAL]\ncase inl.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\nx : \u03b1\n\u22a2 x \u2208 Ico a b \\ Ico a c \u2194 x \u2208 Ico (max a c) b\n[PROOFSTEP]\nrw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]\n[GOAL]\ncase inl.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : a \u2264 c\nx : \u03b1\n\u22a2 (a \u2264 x \u2227 (a \u2264 x \u2192 c \u2264 x)) \u2227 x < b \u2194 c \u2264 x \u2227 x < b\n[PROOFSTEP]\nexact and_congr_left' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8h.trans hx, fun _ => hx\u27e9\u27e9\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 a\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\n\n| inr h => rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 a\n\u22a2 Ico a b \\ Ico a c = Ico (max a c) b\n[PROOFSTEP]\nrw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\ncases le_total b c with\n| inl h => rw [Ico_eq_empty_of_le h, sdiff_empty, min_eq_left h]\n| inr h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, min_eq_right h, and_assoc, not_and', not_le]\n  exact and_congr_right' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8hx.trans_le h, fun _ => hx\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nx\u271d : b \u2264 c \u2228 c \u2264 b\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\ncases le_total b c with\n| inl h => rw [Ico_eq_empty_of_le h, sdiff_empty, min_eq_left h]\n| inr h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, min_eq_right h, and_assoc, not_and', not_le]\n  exact and_congr_right' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8hx.trans_le h, fun _ => hx\u27e9\u27e9\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : b \u2264 c\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\n\n| inl h => rw [Ico_eq_empty_of_le h, sdiff_empty, min_eq_left h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : b \u2264 c\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\nrw [Ico_eq_empty_of_le h, sdiff_empty, min_eq_left h]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\n\n| inr h =>\n  ext x\n  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, min_eq_right h, and_assoc, not_and', not_le]\n  exact and_congr_right' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8hx.trans_le h, fun _ => hx\u27e9\u27e9\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\n\u22a2 Ico a b \\ Ico c b = Ico a (min b c)\n[PROOFSTEP]\next x\n[GOAL]\ncase inr.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\nx : \u03b1\n\u22a2 x \u2208 Ico a b \\ Ico c b \u2194 x \u2208 Ico a (min b c)\n[PROOFSTEP]\nrw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, min_eq_right h, and_assoc, not_and', not_le]\n[GOAL]\ncase inr.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d b\u271d a b c : \u03b1\nh : c \u2264 b\nx : \u03b1\n\u22a2 a \u2264 x \u2227 x < b \u2227 (x < b \u2192 x < c) \u2194 a \u2264 x \u2227 x < c\n[PROOFSTEP]\nexact and_congr_right' \u27e8fun hx => hx.2 hx.1, fun hx => \u27e8hx.trans_le h, fun _ => hx\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : \u03b1\n\u22a2 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) = {a}\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrderTop \u03b1\ninst\u271d : LocallyFiniteOrderBot \u03b1\na a\u271d : \u03b1\n\u22a2 a\u271d \u2208 disjUnion (Ioi a) (Iio a) (_ : Disjoint (Ioi a) (Iio a)) \u2194 a\u271d \u2208 {a}\u1d9c\n[PROOFSTEP]\nsimp [eq_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : a \u2264 b\n\u22a2 [[a, b]] = Icc a b\n[PROOFSTEP]\nrw [uIcc, inf_eq_left.2 h, sup_eq_right.2 h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh : b \u2264 a\n\u22a2 [[a, b]] = Icc b a\n[PROOFSTEP]\nrw [uIcc, inf_eq_right.2 h, sup_eq_left.2 h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c x a b : \u03b1\n\u22a2 [[a, b]] = [[b, a]]\n[PROOFSTEP]\nrw [uIcc, uIcc, inf_comm, sup_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 [[a, a]] = {a}\n[PROOFSTEP]\nsimp [uIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2081 \u2208 [[a\u2082, b\u2082]]\nh\u2082 : b\u2081 \u2208 [[a\u2082, b\u2082]]\n\u22a2 [[a\u2081, b\u2081]] \u2286 [[a\u2082, b\u2082]]\n[PROOFSTEP]\nrw [mem_uIcc] at h\u2081 h\u2082 \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nh\u2081 : a\u2082 \u2293 b\u2082 \u2264 a\u2081 \u2227 a\u2081 \u2264 a\u2082 \u2294 b\u2082\nh\u2082 : a\u2082 \u2293 b\u2082 \u2264 b\u2081 \u2227 b\u2081 \u2264 a\u2082 \u2294 b\u2082\n\u22a2 [[a\u2081, b\u2081]] \u2286 [[a\u2082, b\u2082]]\n[PROOFSTEP]\nexact Icc_subset_Icc (_root_.le_inf h\u2081.1 h\u2082.1) (_root_.sup_le h\u2081.2 h\u2082.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2081 \u2208 Icc a\u2082 b\u2082\nhb : b\u2081 \u2208 Icc a\u2082 b\u2082\n\u22a2 [[a\u2081, b\u2081]] \u2286 Icc a\u2082 b\u2082\n[PROOFSTEP]\nrw [mem_Icc] at ha hb \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\nha : a\u2082 \u2264 a\u2081 \u2227 a\u2081 \u2264 b\u2082\nhb : a\u2082 \u2264 b\u2081 \u2227 b\u2081 \u2264 b\u2082\n\u22a2 [[a\u2081, b\u2081]] \u2286 Icc a\u2082 b\u2082\n[PROOFSTEP]\nexact Icc_subset_Icc (_root_.le_inf ha.1 hb.1) (_root_.sup_le ha.2 hb.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 [[b, c]] \u2192 b \u2208 [[a, c]] \u2192 a = b\n[PROOFSTEP]\nsimp_rw [mem_uIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2293 c \u2264 a \u2227 a \u2264 b \u2294 c \u2192 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a = b\n[PROOFSTEP]\nexact Set.eq_of_mem_uIcc_of_mem_uIcc\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 b \u2208 [[a, c]] \u2192 c \u2208 [[a, b]] \u2192 b = c\n[PROOFSTEP]\nsimp_rw [mem_uIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2293 c \u2264 b \u2227 b \u2264 a \u2294 c \u2192 a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b \u2192 b = c\n[PROOFSTEP]\nexact Set.eq_of_mem_uIcc_of_mem_uIcc'\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\nh : (fun b => [[b, a]]) b = (fun b => [[b, a]]) c\n\u22a2 b = c\n[PROOFSTEP]\nrw [ext_iff] at h \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 c\u271d x a b c : \u03b1\nh : \u2200 (a_1 : \u03b1), a_1 \u2208 (fun b => [[b, a]]) b \u2194 a_1 \u2208 (fun b => [[b, a]]) c\n\u22a2 b = c\n[PROOFSTEP]\nexact eq_of_mem_uIcc_of_mem_uIcc ((h _).1 left_mem_uIcc) ((h _).2 left_mem_uIcc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 c x a : \u03b1\n\u22a2 Injective (uIcc a)\n[PROOFSTEP]\nsimpa only [uIcc_comm] using uIcc_injective_right a\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 \u2191[[a, b]] = \u2191(Icc a b \u222a Icc b a)\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.uIcc a b = Set.Icc a b \u222a Set.Icc b a\n[PROOFSTEP]\nexact Set.uIcc_eq_union\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a \u2208 [[b, c]] \u2194 b \u2264 a \u2227 a \u2264 c \u2228 c \u2264 a \u2227 a \u2264 b\n[PROOFSTEP]\nsimp [uIcc_eq_union]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 c < a \u2192 c < b \u2192 \u00acc \u2208 [[a, b]]\n[PROOFSTEP]\nrw [mem_uIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 c < a \u2192 c < b \u2192 \u00ac(a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b)\n[PROOFSTEP]\nexact Set.not_mem_uIcc_of_lt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a < c \u2192 b < c \u2192 \u00acc \u2208 [[a, b]]\n[PROOFSTEP]\nrw [mem_uIcc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 a < c \u2192 b < c \u2192 \u00ac(a \u2293 b \u2264 c \u2227 c \u2264 a \u2294 b)\n[PROOFSTEP]\nexact Set.not_mem_uIcc_of_gt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 \u2191[[a, c]] \u2286 \u2191([[a, b]] \u222a [[b, c]])\n[PROOFSTEP]\npush_cast\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\n\u22a2 Set.uIcc a c \u2286 Set.uIcc a b \u222a Set.uIcc b c\n[PROOFSTEP]\nexact Set.uIcc_subset_uIcc_union_uIcc\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addLeftEmbedding c) (Icc a b) = Icc (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Icc, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Icc a b = Set.Icc (c + a) (c + b)\n[PROOFSTEP]\nexact Set.image_const_add_Icc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addRightEmbedding c) (Icc a b) = Icc (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Icc, coe_Icc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addRightEmbedding c) '' Set.Icc a b = Set.Icc (a + c) (b + c)\n[PROOFSTEP]\nexact Set.image_add_const_Icc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addLeftEmbedding c) (Ico a b) = Ico (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ico, coe_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Ico a b = Set.Ico (c + a) (c + b)\n[PROOFSTEP]\nexact Set.image_const_add_Ico _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addRightEmbedding c) (Ico a b) = Ico (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ico, coe_Ico]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addRightEmbedding c) '' Set.Ico a b = Set.Ico (a + c) (b + c)\n[PROOFSTEP]\nexact Set.image_add_const_Ico _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addLeftEmbedding c) (Ioc a b) = Ioc (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ioc, coe_Ioc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Ioc a b = Set.Ioc (c + a) (c + b)\n[PROOFSTEP]\nexact Set.image_const_add_Ioc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addRightEmbedding c) (Ioc a b) = Ioc (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ioc, coe_Ioc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addRightEmbedding c) '' Set.Ioc a b = Set.Ioc (a + c) (b + c)\n[PROOFSTEP]\nexact Set.image_add_const_Ioc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addLeftEmbedding c) (Ioo a b) = Ioo (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ioo, coe_Ioo]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addLeftEmbedding c) '' Set.Ioo a b = Set.Ioo (c + a) (c + b)\n[PROOFSTEP]\nexact Set.image_const_add_Ioo _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (addRightEmbedding c) (Ioo a b) = Ioo (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 coe_inj, coe_map, coe_Ioo, coe_Ioo]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 \u2191(addRightEmbedding c) '' Set.Ioo a b = Set.Ioo (a + c) (b + c)\n[PROOFSTEP]\nexact Set.image_add_const_Ioo _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Icc a b) = Icc (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 map_add_left_Icc, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Icc a b) = image (\u2191(addLeftEmbedding c)) (Icc a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ico a b) = Ico (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 map_add_left_Ico, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ico a b) = image (\u2191(addLeftEmbedding c)) (Ico a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ioc a b) = Ioc (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 map_add_left_Ioc, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ioc a b) = image (\u2191(addLeftEmbedding c)) (Ioc a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ioo a b) = Ioo (c + a) (c + b)\n[PROOFSTEP]\nrw [\u2190 map_add_left_Ioo, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image ((fun x x_1 => x + x_1) c) (Ioo a b) = image (\u2191(addLeftEmbedding c)) (Ioo a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Icc a b) = Icc (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 map_add_right_Icc, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Icc a b) = image (\u2191(addRightEmbedding c)) (Icc a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ico a b) = Ico (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 map_add_right_Ico, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ico a b) = image (\u2191(addRightEmbedding c)) (Ico a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ioc a b) = Ioc (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 map_add_right_Ioc, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ioc a b) = image (\u2191(addRightEmbedding c)) (Ioc a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ioo a b) = Ioo (a + c) (b + c)\n[PROOFSTEP]\nrw [\u2190 map_add_right_Ioo, map_eq_image]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b2 : ExistsAddOfLE \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\ninst\u271d : DecidableEq \u03b1\na b c : \u03b1\n\u22a2 image (fun x => x + c) (Ioo a b) = image (\u2191(addRightEmbedding c)) (Ioo a b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f i : \u03b9, \u220f j in Ioi i, f j i * f i j = \u220f i : \u03b9, \u220f j in {i}\u1d9c, f j i\n[PROOFSTEP]\nsimp_rw [\u2190 Ioi_disjUnion_Iio, prod_disjUnion, prod_mul_distrib]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 =\n    (\u220f x : \u03b9, \u220f x_1 in Ioi x, f x_1 x) * \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x : \u03b9, \u220f x_1 in Ioi x, f x x_1 = \u220f x : \u03b9, \u220f x_1 in Iio x, f x_1 x\n[PROOFSTEP]\nrw [prod_sigma', prod_sigma']\n[GOAL]\ncase e_a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u220f x in Finset.sigma univ fun x => Ioi x, f x.fst x.snd = \u220f x in Finset.sigma univ fun x => Iio x, f x.snd x.fst\n[PROOFSTEP]\nrefine' prod_bij' (fun i _ => \u27e8i.2, i.1\u27e9) _ _ (fun i _ => \u27e8i.2, i.1\u27e9) _ _ _\n[GOAL]\ncase e_a.refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : (_ : \u03b9) \u00d7 \u03b9) (ha : a \u2208 Finset.sigma univ fun x => Ioi x),\n    (fun i x => { fst := i.snd, snd := i.fst }) a ha \u2208 Finset.sigma univ fun x => Iio x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : (_ : \u03b9) \u00d7 \u03b9) (ha : a \u2208 Finset.sigma univ fun x => Ioi x),\n    f a.fst a.snd =\n      f ((fun i x => { fst := i.snd, snd := i.fst }) a ha).snd ((fun i x => { fst := i.snd, snd := i.fst }) a ha).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : (_ : \u03b9) \u00d7 \u03b9) (ha : a \u2208 Finset.sigma univ fun x => Iio x),\n    (fun i x => { fst := i.snd, snd := i.fst }) a ha \u2208 Finset.sigma univ fun x => Ioi x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.refine'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : (_ : \u03b9) \u00d7 \u03b9) (ha : a \u2208 Finset.sigma univ fun x => Ioi x),\n    (fun i x => { fst := i.snd, snd := i.fst }) ((fun i x => { fst := i.snd, snd := i.fst }) a ha)\n        (_ : { fst := a.snd, snd := a.fst } \u2208 Finset.sigma univ fun x => Iio x) =\n      a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.refine'_5\n\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Fintype \u03b9\ninst\u271d\u00b3 : LinearOrder \u03b9\ninst\u271d\u00b2 : LocallyFiniteOrderTop \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : CommMonoid \u03b1\nf : \u03b9 \u2192 \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : (_ : \u03b9) \u00d7 \u03b9) (ha : a \u2208 Finset.sigma univ fun x => Iio x),\n    (fun i x => { fst := i.snd, snd := i.fst }) ((fun i x => { fst := i.snd, snd := i.fst }) a ha)\n        (_ : { fst := a.snd, snd := a.fst } \u2208 Finset.sigma univ fun x => Ioi x) =\n      a\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.LocallyFinite", "llama_tokens": 29405, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289836, "lm_q2_score": 0.7090191276365463, "lm_q1q2_score": 0.5393158698914801}}
{"text": "[GOAL]\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\n\u22a2 FrobeniusNumber (m * n - m - n) {m, n}\n[PROOFSTEP]\nsimp_rw [FrobeniusNumber, AddSubmonoid.mem_closure_pair]\n[GOAL]\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\n\u22a2 IsGreatest {k | \u00ac\u2203 m_1 n_1, m_1 \u2022 m + n_1 \u2022 n = k} (m * n - m - n)\n[PROOFSTEP]\nhave hmn : m + n \u2264 m * n := add_le_mul hm hn\n[GOAL]\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\n\u22a2 IsGreatest {k | \u00ac\u2203 m_1 n_1, m_1 \u2022 m + n_1 \u2022 n = k} (m * n - m - n)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\n\u22a2 m * n - m - n \u2208 {k | \u00ac\u2203 m_1 n_1, m_1 \u2022 m + n_1 \u2022 n = k}\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\n\u22a2 m * n - m - n \u2208 {k | \u2200 (m_1 n_1 : \u2115), m_1 \u2022 m + n_1 \u2022 n \u2260 k}\n[PROOFSTEP]\nintro a b h\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh : a \u2022 m + b \u2022 n = m * n - m - n\n\u22a2 False\n[PROOFSTEP]\napply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh : a \u2022 m + b \u2022 n = m * n - m - n\n\u22a2 (a + 1) * m + (b + 1) * n = m * n\n[PROOFSTEP]\nsimp only [Nat.sub_sub, smul_eq_mul] at h \n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh : a * m + b * n = m * n - (m + n)\n\u22a2 (a + 1) * m + (b + 1) * n = m * n\n[PROOFSTEP]\nzify [hmn] at h \u22a2\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh : \u2191a * \u2191m + \u2191b * \u2191n = \u2191m * \u2191n - (\u2191m + \u2191n)\n\u22a2 (\u2191a + 1) * \u2191m + (\u2191b + 1) * \u2191n = \u2191m * \u2191n\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero] at h \u22a2\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh\u271d : \u2191a * \u2191m + \u2191b * \u2191n = \u2191m * \u2191n - (\u2191m + \u2191n)\nh : \u2191a * \u2191m + \u2191b * \u2191n - (\u2191m * \u2191n - (\u2191m + \u2191n)) = 0\n\u22a2 (\u2191a + 1) * \u2191m + (\u2191b + 1) * \u2191n - \u2191m * \u2191n = 0\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase left\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\na b : \u2115\nh\u271d : \u2191a * \u2191m + \u2191b * \u2191n = \u2191m * \u2191n - (\u2191m + \u2191n)\nh : \u2191a * \u2191m + \u2191b * \u2191n - (\u2191m * \u2191n - (\u2191m + \u2191n)) = 0\n\u22a2 (\u2191a + 1) * \u2191m + (\u2191b + 1) * \u2191n - \u2191m * \u2191n = \u2191a * \u2191m + \u2191b * \u2191n - (\u2191m * \u2191n - (\u2191m + \u2191n))\n[PROOFSTEP]\nring\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\n\u22a2 m * n - m - n \u2208 upperBounds {k | \u00ac\u2203 m_1 n_1, m_1 \u2022 m + n_1 \u2022 n = k}\n[PROOFSTEP]\nintro k hk\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : k \u2208 {k | \u00ac\u2203 m_1 n_1, m_1 \u2022 m + n_1 \u2022 n = k}\n\u22a2 k \u2264 m * n - m - n\n[PROOFSTEP]\ndsimp at hk \n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : \u00ac\u2203 m_1 n_1, m_1 * m + n_1 * n = k\n\u22a2 k \u2264 m * n - m - n\n[PROOFSTEP]\ncontrapose! hk\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nlet x := chineseRemainder cop 0 k\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nhave hx : x.val < m * n := chineseRemainder_lt_mul cop 0 k (ne_bot_of_gt hm) (ne_bot_of_gt hn)\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nsuffices key : x.1 \u2264 k\n[GOAL]\ncase right\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\nkey : \u2191x \u2264 k\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := modEq_zero_iff_dvd.mp x.2.1\n[GOAL]\ncase right.intro\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\nkey : \u2191x \u2264 k\na : \u2115\nha : \u2191x = m * a\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := (modEq_iff_dvd' key).mp x.2.2\n[GOAL]\ncase right.intro.intro\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\nkey : \u2191x \u2264 k\na : \u2115\nha : \u2191x = m * a\nb : \u2115\nhb : k - \u2191x = n * b\n\u22a2 \u2203 m_1 n_1, m_1 * m + n_1 * n = k\n[PROOFSTEP]\nexact \u27e8a, b, by rw [mul_comm, \u2190 ha, mul_comm, \u2190 hb, Nat.add_sub_of_le key]\u27e9\n[GOAL]\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\nkey : \u2191x \u2264 k\na : \u2115\nha : \u2191x = m * a\nb : \u2115\nhb : k - \u2191x = n * b\n\u22a2 a * m + b * n = k\n[PROOFSTEP]\nrw [mul_comm, \u2190 ha, mul_comm, \u2190 hb, Nat.add_sub_of_le key]\n[GOAL]\ncase key\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\n\u22a2 \u2191x \u2264 k\n[PROOFSTEP]\nrefine' ModEq.le_of_lt_add x.2.2 (lt_of_le_of_lt _ (add_lt_add_right hk n))\n[GOAL]\ncase key\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\n\u22a2 \u2191x \u2264 m * n - m - n + n\n[PROOFSTEP]\nrw [Nat.sub_add_cancel (le_tsub_of_add_le_left hmn)]\n[GOAL]\ncase key\nm n : \u2115\ncop : coprime m n\nhm : 1 < m\nhn : 1 < n\nhmn : m + n \u2264 m * n\nk : \u2115\nhk : m * n - m - n < k\nx : { k_1 // k_1 \u2261 0 [MOD m] \u2227 k_1 \u2261 k [MOD n] } := chineseRemainder cop 0 k\nhx : \u2191x < m * n\n\u22a2 \u2191x \u2264 m * n - m\n[PROOFSTEP]\nexact\n  ModEq.le_of_lt_add (x.2.1.trans (modEq_zero_iff_dvd.mpr (Nat.dvd_sub' (dvd_mul_right m n) dvd_rfl)).symm)\n    (lt_of_lt_of_le hx le_tsub_add)\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.FrobeniusNumber", "llama_tokens": 3561, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8705972717658209, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5387085574394672}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u22a2 AnalyticSet \u2205\n[PROOFSTEP]\nrw [AnalyticSet]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u22a2 \u2205 = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = \u2205\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\n\u22a2 AnalyticSet (range f)\n[PROOFSTEP]\ncases isEmpty_or_nonempty \u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet (range f)\n[PROOFSTEP]\nrw [range_eq_empty]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : IsEmpty \u03b2\n\u22a2 AnalyticSet \u2205\n[PROOFSTEP]\nexact analyticSet_empty\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 AnalyticSet (range f)\n[PROOFSTEP]\nrw [AnalyticSet]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f\n[PROOFSTEP]\nobtain \u27e8g, g_cont, hg\u27e9 : \u2203 g : (\u2115 \u2192 \u2115) \u2192 \u03b2, Continuous g \u2227 Surjective g := exists_nat_nat_continuous_surjective \u03b2\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range f = \u2205 \u2228 \u2203 f_1, Continuous f_1 \u2227 range f_1 = range f\n[PROOFSTEP]\nrefine' Or.inr \u27e8f \u2218 g, f_cont.comp g_cont, _\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nh\u271d : Nonempty \u03b2\ng : (\u2115 \u2192 \u2115) \u2192 \u03b2\ng_cont : Continuous g\nhg : Surjective g\n\u22a2 range (f \u2218 g) = range f\n[PROOFSTEP]\nrw [hg.range_comp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\ns : Set \u03b2\nhs : IsOpen s\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nrw [image_eq_range]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\ns : Set \u03b2\nhs : IsOpen s\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\n\u22a2 AnalyticSet (range fun x => f \u2191x)\n[PROOFSTEP]\nhaveI : PolishSpace s := hs.polishSpace\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : PolishSpace \u03b2\ns : Set \u03b2\nhs : IsOpen s\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nthis : PolishSpace \u2191s\n\u22a2 AnalyticSet (range fun x => f \u2191x)\n[PROOFSTEP]\nexact analyticSet_range_of_polishSpace (f_cont.comp continuous_subtype_val)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u22a2 AnalyticSet s \u2194 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u22a2 AnalyticSet s \u2192 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : AnalyticSet s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\nrw [AnalyticSet] at h \n[GOAL]\ncase mp\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase mp.inl\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : s = \u2205\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\nrefine' \u27e8Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, _\u27e9\n[GOAL]\ncase mp.inl\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = s\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mp.inl\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : s = \u2205\n\u22a2 range Empty.elim = \u2205\n[PROOFSTEP]\nexact range_eq_empty _\n[GOAL]\ncase mp.inr\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\nh : \u2203 f, Continuous f \u2227 range f = s\n\u22a2 \u2203 \u03b2 h x f, Continuous f \u2227 range f = s\n[PROOFSTEP]\nexact \u27e8\u2115 \u2192 \u2115, inferInstance, inferInstance, h\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u22a2 (\u2203 \u03b2 h x f, Continuous f \u2227 range f = s) \u2192 AnalyticSet s\n[PROOFSTEP]\nrintro \u27e8\u03b2, h, h', f, f_cont, f_range\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nskip\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nrw [\u2190 f_range]\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ns : Set \u03b1\n\u03b2 : Type\nh : TopologicalSpace \u03b2\nh' : PolishSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nf_cont : Continuous f\nf_range : range f = s\n\u22a2 AnalyticSet (range f)\n[PROOFSTEP]\nexact analyticSet_range_of_polishSpace f_cont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nrcases analyticSet_iff_exists_polishSpace_range.1 hs with \u27e8\u03b3, \u03b3top, \u03b3polish, g, g_cont, gs\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nhave : f '' s = range (f \u2218 g) := by rw [range_comp, gs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\n\u22a2 f '' s = range (f \u2218 g)\n[PROOFSTEP]\nrw [range_comp, gs]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\nthis : f '' s = range (f \u2218 g)\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\nthis : f '' s = range (f \u2218 g)\n\u22a2 AnalyticSet (range (f \u2218 g))\n[PROOFSTEP]\napply analyticSet_range_of_polishSpace\n[GOAL]\ncase intro.intro.intro.intro.intro.f_cont\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\nthis : f '' s = range (f \u2218 g)\n\u22a2 Continuous (f \u2218 g)\n[PROOFSTEP]\napply hf.comp_continuous g_cont fun x => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\nthis : f '' s = range (f \u2218 g)\n\u22a2 \u2200 (x : \u03b3), g x \u2208 s\n[PROOFSTEP]\nrw [\u2190 gs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : TopologicalSpace \u03b2\ns : Set \u03b1\nhs : AnalyticSet s\nf : \u03b1 \u2192 \u03b2\nhf : ContinuousOn f s\n\u03b3 : Type\n\u03b3top : TopologicalSpace \u03b3\n\u03b3polish : PolishSpace \u03b3\ng : \u03b3 \u2192 \u03b1\ng_cont : Continuous g\ngs : range g = s\nthis : f '' s = range (f \u2218 g)\n\u22a2 \u2200 (x : \u03b3), g x \u2208 range g\n[PROOFSTEP]\nexact mem_range_self\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh\u03b9 : Nonempty \u03b9\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nrcases h\u03b9 with\n  \u27e8i\u2080\u27e9\n    /- For the proof, write each `s n` as the continuous image under a map `f n` of a\n        Polish space `\u03b2 n`. The product space `\u03b3 = \u03a0 n, \u03b2 n` is also Polish, and so is the subset\n        `t` of sequences `x n` for which `f n (x n)` is independent of `n`. The set `t` is Polish, and\n        the range of `x \u21a6 f 0 (x 0)` on `t` is exactly `\u22c2 n, s n`, so this set is analytic. -/\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nchoose \u03b2 h\u03b2 h'\u03b2 f f_cont f_range using fun n => analyticSet_iff_exists_polishSpace_range.1 (hs n)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nskip\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nlet \u03b3 := \u2200 n, \u03b2 n\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nlet t : Set \u03b3 := \u22c2 n, {x | f n (x n) = f i\u2080 (x i\u2080)}\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nhave t_closed : IsClosed t := by\n  apply isClosed_iInter\n  intro n\n  exact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i\u2080).comp (continuous_apply i\u2080))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\n\u22a2 IsClosed t\n[PROOFSTEP]\napply isClosed_iInter\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\n\u22a2 \u2200 (i : \u03b9), IsClosed {x | f i (x i) = f i\u2080 (x i\u2080)}\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nn : \u03b9\n\u22a2 IsClosed {x | f n (x n) = f i\u2080 (x i\u2080)}\n[PROOFSTEP]\nexact isClosed_eq ((f_cont n).comp (continuous_apply n)) ((f_cont i\u2080).comp (continuous_apply i\u2080))\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nhaveI : PolishSpace t := t_closed.polishSpace\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nlet F : t \u2192 \u03b1 := fun x => f i\u2080 ((x : \u03b3) i\u2080)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nhave F_cont : Continuous F := (f_cont i\u2080).comp ((continuous_apply i\u2080).comp continuous_subtype_val)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nhave F_range : range F = \u22c2 n : \u03b9, s n := by\n  apply Subset.antisymm\n  \u00b7 rintro y \u27e8x, rfl\u27e9\n    refine mem_iInter.2 fun n => ?_\n    have : f n ((x : \u03b3) n) = F x := (mem_iInter.1 x.2 n : _)\n    rw [\u2190 this, \u2190 f_range n]\n    exact mem_range_self _\n  \u00b7 intro y hy\n    have A : \u2200 n, \u2203 x : \u03b2 n, f n x = y := by\n      intro n\n      rw [\u2190 mem_range, f_range n]\n      exact mem_iInter.1 hy n\n    choose x hx using A\n    have xt : x \u2208 t := by\n      refine mem_iInter.2 fun n => ?_\n      simp [hx]\n    refine' \u27e8\u27e8x, xt\u27e9, _\u27e9\n    exact hx i\u2080\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\n\u22a2 range F = \u22c2 (n : \u03b9), s n\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\n\u22a2 range F \u2286 \u22c2 (n : \u03b9), s n\n[PROOFSTEP]\nrintro y \u27e8x, rfl\u27e9\n[GOAL]\ncase h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nx : \u2191t\n\u22a2 F x \u2208 \u22c2 (n : \u03b9), s n\n[PROOFSTEP]\nrefine mem_iInter.2 fun n => ?_\n[GOAL]\ncase h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nx : \u2191t\nn : \u03b9\n\u22a2 F x \u2208 s n\n[PROOFSTEP]\nhave : f n ((x : \u03b3) n) = F x := (mem_iInter.1 x.2 n : _)\n[GOAL]\ncase h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis\u271d : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nx : \u2191t\nn : \u03b9\nthis : f n (\u2191x n) = F x\n\u22a2 F x \u2208 s n\n[PROOFSTEP]\nrw [\u2190 this, \u2190 f_range n]\n[GOAL]\ncase h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis\u271d : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nx : \u2191t\nn : \u03b9\nthis : f n (\u2191x n) = F x\n\u22a2 f n (\u2191x n) \u2208 range (f n)\n[PROOFSTEP]\nexact mem_range_self _\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\n\u22a2 \u22c2 (n : \u03b9), s n \u2286 range F\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\n\u22a2 y \u2208 range F\n[PROOFSTEP]\nhave A : \u2200 n, \u2203 x : \u03b2 n, f n x = y := by\n  intro n\n  rw [\u2190 mem_range, f_range n]\n  exact mem_iInter.1 hy n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\n\u22a2 \u2200 (n : \u03b9), \u2203 x, f n x = y\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nn : \u03b9\n\u22a2 \u2203 x, f n x = y\n[PROOFSTEP]\nrw [\u2190 mem_range, f_range n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nn : \u03b9\n\u22a2 y \u2208 s n\n[PROOFSTEP]\nexact mem_iInter.1 hy n\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nA : \u2200 (n : \u03b9), \u2203 x, f n x = y\n\u22a2 y \u2208 range F\n[PROOFSTEP]\nchoose x hx using A\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nx : (n : \u03b9) \u2192 \u03b2 n\nhx : \u2200 (n : \u03b9), f n (x n) = y\n\u22a2 y \u2208 range F\n[PROOFSTEP]\nhave xt : x \u2208 t := by\n  refine mem_iInter.2 fun n => ?_\n  simp [hx]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nx : (n : \u03b9) \u2192 \u03b2 n\nhx : \u2200 (n : \u03b9), f n (x n) = y\n\u22a2 x \u2208 t\n[PROOFSTEP]\nrefine mem_iInter.2 fun n => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nx : (n : \u03b9) \u2192 \u03b2 n\nhx : \u2200 (n : \u03b9), f n (x n) = y\nn : \u03b9\n\u22a2 x \u2208 {x | f n (x n) = f i\u2080 (x i\u2080)}\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nx : (n : \u03b9) \u2192 \u03b2 n\nhx : \u2200 (n : \u03b9), f n (x n) = y\nxt : x \u2208 t\n\u22a2 y \u2208 range F\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, xt\u27e9, _\u27e9\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\ny : \u03b1\nhy : y \u2208 \u22c2 (n : \u03b9), s n\nx : (n : \u03b9) \u2192 \u03b2 n\nhx : \u2200 (n : \u03b9), f n (x n) = y\nxt : x \u2208 t\n\u22a2 F { val := x, property := xt } = y\n[PROOFSTEP]\nexact hx i\u2080\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nF_range : range F = \u22c2 (n : \u03b9), s n\n\u22a2 AnalyticSet (\u22c2 (n : \u03b9), s n)\n[PROOFSTEP]\nrw [\u2190 F_range]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\ninst\u271d : T2Space \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\ni\u2080 : \u03b9\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u2192 \u03b2 n\nt : Set \u03b3 := \u22c2 (n : \u03b9), {x | f n (x n) = f i\u2080 (x i\u2080)}\nt_closed : IsClosed t\nthis : PolishSpace \u2191t\nF : \u2191t \u2192 \u03b1 := fun x => f i\u2080 (\u2191x i\u2080)\nF_cont : Continuous F\nF_range : range F = \u22c2 (n : \u03b9), s n\n\u22a2 AnalyticSet (range F)\n[PROOFSTEP]\nexact analyticSet_range_of_polishSpace F_cont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nchoose \u03b2 h\u03b2 h'\u03b2 f f_cont f_range using fun n => analyticSet_iff_exists_polishSpace_range.1 (hs n)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nskip\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nlet \u03b3 := \u03a3 n, \u03b2 n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nlet F : \u03b3 \u2192 \u03b1 := by\n  rintro \u27e8n, x\u27e9\n  exact f n x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\n\u22a2 \u03b3 \u2192 \u03b1\n[PROOFSTEP]\nrintro \u27e8n, x\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nn : \u03b9\nx : \u03b2 n\n\u22a2 \u03b1\n[PROOFSTEP]\nexact f n x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nhave F_cont : Continuous F := continuous_sigma f_cont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nhave F_range : range F = \u22c3 n, s n := by\n  rw [range_sigma_eq_iUnion_range]\n  apply congr_arg\n  ext1 n\n  rw [\u2190 f_range n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\n\u22a2 range F = \u22c3 (n : \u03b9), s n\n[PROOFSTEP]\nrw [range_sigma_eq_iUnion_range]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\n\u22a2 (\u22c3 (a : \u03b9), range fun b => F { fst := a, snd := b }) = \u22c3 (n : \u03b9), s n\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\n\u22a2 (fun a => range fun b => F { fst := a, snd := b }) = fun n => s n\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\nn : \u03b9\n\u22a2 (range fun b => F { fst := n, snd := b }) = s n\n[PROOFSTEP]\nrw [\u2190 f_range n]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 (n : \u03b9), s n\n\u22a2 AnalyticSet (\u22c3 (n : \u03b9), s n)\n[PROOFSTEP]\nrw [\u2190 F_range]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\ns : \u03b9 \u2192 Set \u03b1\nhs : \u2200 (n : \u03b9), AnalyticSet (s n)\n\u03b2 : \u03b9 \u2192 Type\nh\u03b2 : (n : \u03b9) \u2192 TopologicalSpace (\u03b2 n)\nh'\u03b2 : \u2200 (n : \u03b9), PolishSpace (\u03b2 n)\nf : (n : \u03b9) \u2192 \u03b2 n \u2192 \u03b1\nf_cont : \u2200 (n : \u03b9), Continuous (f n)\nf_range : \u2200 (n : \u03b9), range (f n) = s n\n\u03b3 : Type u_2 := (n : \u03b9) \u00d7 \u03b2 n\nF : \u03b3 \u2192 \u03b1 := fun a => Sigma.casesOn a fun n x => f n x\nF_cont : Continuous F\nF_range : range F = \u22c3 (n : \u03b9), s n\n\u22a2 AnalyticSet (range F)\n[PROOFSTEP]\nexact analyticSet_range_of_polishSpace F_cont\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nhaveI : PolishSpace s := hs.polishSpace\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis : PolishSpace \u2191s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nrw [\u2190 @Subtype.range_val \u03b1 s]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d : PolishSpace \u03b1\ns : Set \u03b1\nhs : IsClosed s\nthis : PolishSpace \u2191s\n\u22a2 AnalyticSet (range Subtype.val)\n[PROOFSTEP]\nexact analyticSet_range_of_polishSpace continuous_subtype_val\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 IsClopenable s\n[PROOFSTEP]\nrevert s\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 {s : Set \u03b1}, MeasurableSet s \u2192 IsClopenable s\n[PROOFSTEP]\napply MeasurableSet.induction_on_open\n[GOAL]\ncase h_open\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 IsClopenable U\n[PROOFSTEP]\nexact fun u hu => hu.isClopenable\n[GOAL]\ncase h_compl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 IsClopenable t \u2192 IsClopenable t\u1d9c\n[PROOFSTEP]\nexact fun u _ h'u => h'u.compl\n[GOAL]\ncase h_union\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1),\n    Pairwise (Disjoint on f) \u2192\n      (\u2200 (i : \u2115), MeasurableSet (f i)) \u2192 (\u2200 (i : \u2115), IsClopenable (f i)) \u2192 IsClopenable (\u22c3 (i : \u2115), f i)\n[PROOFSTEP]\nexact fun f _ _ hf => IsClopenable.iUnion hf\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nobtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b1, t' \u2264 t \u2227 @PolishSpace \u03b1 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s := hs.isClopenable\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nhave A := @IsClosed.analyticSet \u03b1 t' t'_polish s s_closed\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nconvert @AnalyticSet.image_of_continuous \u03b1 t' \u03b1 t s A id (continuous_id_of_le t't)\n[GOAL]\ncase h.e'_3\n\u03b1\u271d : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\nt : TopologicalSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\nt' : TopologicalSpace \u03b1\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\ns_closed : IsClosed s\nright\u271d : IsOpen s\nA : AnalyticSet s\n\u22a2 s = id '' s\n[PROOFSTEP]\nsimp only [id.def, image_id']\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nobtain \u27e8b, b_count, -, hb\u27e9 : \u2203 b : Set (Set (range f)), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b :=\n  exists_countable_basis (range f)\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nhaveI : Countable b := b_count.to_subtype\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nhave : \u2200 s : b, IsClopenable (rangeFactorization f \u207b\u00b9' s) := fun s \u21a6\n  by\n  apply MeasurableSet.isClopenable\n  exact hf.subtype_mk (hb.isOpen s.2).measurableSet\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\ns : \u2191b\n\u22a2 IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s)\n[PROOFSTEP]\napply MeasurableSet.isClopenable\n[GOAL]\ncase hs\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\ns : \u2191b\n\u22a2 MeasurableSet (rangeFactorization f \u207b\u00b9' \u2191s)\n[PROOFSTEP]\nexact hf.subtype_mk (hb.isOpen s.2).measurableSet\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nthis : \u2200 (s : \u2191b), IsClopenable (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nchoose T Tt Tpolish _ Topen using this\n[GOAL]\ncase intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nobtain \u27e8t', t'T, t't, t'_polish\u27e9 : \u2203 t' : TopologicalSpace \u03b1, (\u2200 i, t' \u2264 T i) \u2227 t' \u2264 t \u2227 @PolishSpace \u03b1 t' :=\n  exists_polishSpace_forall_le T Tt Tpolish\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 \u2203 t', t' \u2264 t \u2227 Continuous f \u2227 PolishSpace \u03b1\n[PROOFSTEP]\nrefine' \u27e8t', t't, _, t'_polish\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\n\u22a2 Continuous f\n[PROOFSTEP]\nhave : Continuous[t', _] (rangeFactorization f) := hb.continuous _ fun s hs => t'T \u27e8s, hs\u27e9 _ (Topen \u27e8s, hs\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\nt : TopologicalSpace \u03b1\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : OpensMeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nb : Set (Set \u2191(range f))\nb_count : Set.Countable b\nhb : IsTopologicalBasis b\nthis\u271d : Countable \u2191b\nT : \u2191b \u2192 TopologicalSpace \u03b1\nTt : \u2200 (s : \u2191b), T s \u2264 t\nTpolish : \u2200 (s : \u2191b), PolishSpace \u03b1\nh\u271d : \u2200 (s : \u2191b), IsClosed (rangeFactorization f \u207b\u00b9' \u2191s)\nTopen : \u2200 (s : \u2191b), IsOpen (rangeFactorization f \u207b\u00b9' \u2191s)\nt' : TopologicalSpace \u03b1\nt'T : \u2200 (i : \u2191b), t' \u2264 T i\nt't : t' \u2264 t\nt'_polish : PolishSpace \u03b1\nthis : Continuous (rangeFactorization f)\n\u22a2 Continuous f\n[PROOFSTEP]\nexact continuous_subtype_val.comp this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2078 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2077 : TopologicalSpace X\ninst\u271d\u2076 : PolishSpace X\ninst\u271d\u2075 : MeasurableSpace X\ninst\u271d\u2074 : BorelSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d : SecondCountableTopology \u2191(range f)\ns : Set X\nhs : MeasurableSet s\nhf : Measurable f\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nborelize X\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : SecondCountableTopology \u2191(range f)\ns : Set X\ninst\u271d : BorelSpace X\nhs : MeasurableSet s\nhf : Measurable f\nthis\u271d : MeasurableSpace X := borel X\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nrcases hf.exists_continuous with \u27e8\u03c4', hle, hfc, h\u03c4'\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : SecondCountableTopology \u2191(range f)\ns : Set X\ninst\u271d : BorelSpace X\nhs : MeasurableSet s\nhf : Measurable f\nthis\u271d : MeasurableSpace X := borel X\n\u03c4' : TopologicalSpace X\nhle : \u03c4' \u2264 inst\u271d\u2076\nhfc : Continuous f\nh\u03c4' : PolishSpace X\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nletI m' : MeasurableSpace X := @borel _ \u03c4'\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : SecondCountableTopology \u2191(range f)\ns : Set X\ninst\u271d : BorelSpace X\nhs : MeasurableSet s\nhf : Measurable f\nthis\u271d : MeasurableSpace X := borel X\n\u03c4' : TopologicalSpace X\nhle : \u03c4' \u2264 inst\u271d\u2076\nhfc : Continuous f\nh\u03c4' : PolishSpace X\nm' : MeasurableSpace X := borel X\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nhaveI b' : BorelSpace X := \u27e8rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : SecondCountableTopology \u2191(range f)\ns : Set X\ninst\u271d : BorelSpace X\nhs : MeasurableSet s\nhf : Measurable f\nthis\u271d : MeasurableSpace X := borel X\n\u03c4' : TopologicalSpace X\nhle : \u03c4' \u2264 inst\u271d\u2076\nhfc : Continuous f\nh\u03c4' : PolishSpace X\nm' : MeasurableSpace X := borel X\nb' : BorelSpace X\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nhave hle := borel_anti hle\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : SecondCountableTopology \u2191(range f)\ns : Set X\ninst\u271d : BorelSpace X\nhs : MeasurableSet s\nhf : Measurable f\nthis\u271d : MeasurableSpace X := borel X\n\u03c4' : TopologicalSpace X\nhle\u271d : \u03c4' \u2264 inst\u271d\u2076\nhfc : Continuous f\nh\u03c4' : PolishSpace X\nm' : MeasurableSpace X := borel X\nb' : BorelSpace X\nhle : borel X \u2264 borel X\n\u22a2 AnalyticSet (f '' s)\n[PROOFSTEP]\nexact (hle _ hs).analyticSet.image_of_continuous hfc\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nh : \u2200 (m n : \u03b9), MeasurablySeparable (s m) (t n)\n\u22a2 MeasurablySeparable (\u22c3 (n : \u03b9), s n) (\u22c3 (m : \u03b9), t m)\n[PROOFSTEP]\nchoose u hsu htu hu using h\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurablySeparable (\u22c3 (n : \u03b9), s n) (\u22c3 (m : \u03b9), t m)\n[PROOFSTEP]\nrefine' \u27e8\u22c3 m, \u22c2 n, u m n, _, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u22c3 (n : \u03b9), s n \u2286 \u22c3 (m : \u03b9), \u22c2 (n : \u03b9), u m n\n[PROOFSTEP]\nrefine' iUnion_subset fun m => subset_iUnion_of_subset m _\n[GOAL]\ncase refine'_1\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 s m \u2286 \u22c2 (n : \u03b9), u m n\n[PROOFSTEP]\nexact subset_iInter fun n => hsu m n\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 Disjoint (\u22c3 (m : \u03b9), t m) (\u22c3 (m : \u03b9), \u22c2 (n : \u03b9), u m n)\n[PROOFSTEP]\nsimp_rw [disjoint_iUnion_left, disjoint_iUnion_right]\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 \u2200 (i i_1 : \u03b9), Disjoint (t i) (\u22c2 (n : \u03b9), u i_1 n)\n[PROOFSTEP]\nintro n m\n[GOAL]\ncase refine'_2\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 Disjoint (t n) (\u22c2 (n : \u03b9), u m n)\n[PROOFSTEP]\napply Disjoint.mono_right _ (htu m n)\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nn m : \u03b9\n\u22a2 \u22c2 (n : \u03b9), u m n \u2264 u m n\n[PROOFSTEP]\napply iInter_subset\n[GOAL]\ncase refine'_3\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\n\u22a2 MeasurableSet (\u22c3 (m : \u03b9), \u22c2 (n : \u03b9), u m n)\n[PROOFSTEP]\nrefine' MeasurableSet.iUnion fun m => _\n[GOAL]\ncase refine'_3\n\u03b1\u271d : Type u_1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Countable \u03b9\n\u03b1 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\ns t : \u03b9 \u2192 Set \u03b1\nu : \u03b9 \u2192 \u03b9 \u2192 Set \u03b1\nhsu : \u2200 (m n : \u03b9), s m \u2286 u m n\nhtu : \u2200 (m n : \u03b9), Disjoint (t n) (u m n)\nhu : \u2200 (m n : \u03b9), MeasurableSet (u m n)\nm : \u03b9\n\u22a2 MeasurableSet (\u22c2 (n : \u03b9), u m n)\n[PROOFSTEP]\nexact MeasurableSet.iInter fun n => hu m n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\n\u22a2 MeasurablySeparable (range f) (range g)\n[PROOFSTEP]\nby_contra hfg\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\n\u22a2 False\n[PROOFSTEP]\nhave I :\n  \u2200 n x y,\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) :=\n  by\n  intro n x y\n  contrapose!\n  intro H\n  rw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion]\n  refine' MeasurablySeparable.iUnion fun i j => _\n  exact\n    H _ _ (update_mem_cylinder _ _ _)\n      (update_mem_cylinder _ _ _)\n        -- consider the set of pairs of cylinders of some length whose images are not Borel-separated\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\n\u22a2 \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n[PROOFSTEP]\nintro n x y\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n    \u2203 x' y',\n      x' \u2208 cylinder x n \u2227 y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n[PROOFSTEP]\ncontrapose!\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\n\u22a2 (\u2200 (x' y' : \u2115 \u2192 \u2115),\n      x' \u2208 cylinder x n \u2192\n        y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) \u2192\n    MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n[PROOFSTEP]\nintro H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n[PROOFSTEP]\nrw [\u2190 iUnion_cylinder_update x n, \u2190 iUnion_cylinder_update y n, image_iUnion, image_iUnion]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 MeasurablySeparable (\u22c3 (i : \u2115), f '' cylinder (update x n i) (n + 1))\n    (\u22c3 (i : \u2115), g '' cylinder (update y n i) (n + 1))\n[PROOFSTEP]\nrefine' MeasurablySeparable.iUnion fun i j => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nn : \u2115\nx y : \u2115 \u2192 \u2115\nH :\n  \u2200 (x' y' : \u2115 \u2192 \u2115),\n    x' \u2208 cylinder x n \u2192 y' \u2208 cylinder y n \u2192 MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\ni j : \u2115\n\u22a2 MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))\n[PROOFSTEP]\nexact\n  H _ _ (update_mem_cylinder _ _ _)\n    (update_mem_cylinder _ _ _)\n      -- consider the set of pairs of cylinders of some length whose images are not Borel-separated\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 False\n[PROOFSTEP]\nlet A :=\n  { p : \u2115 \u00d7 (\u2115 \u2192 \u2115) \u00d7 (\u2115 \u2192 \u2115) // \u00acMeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }\n    -- for each such pair, one can find longer cylinders whose images are not Borel-separated either\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\n\u22a2 False\n[PROOFSTEP]\nhave : \u2200 p : A, \u2203 q : A, q.1.1 = p.1.1 + 1 \u2227 q.1.2.1 \u2208 cylinder p.1.2.1 p.1.1 \u2227 q.1.2.2 \u2208 cylinder p.1.2.2 p.1.1 :=\n  by\n  rintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9\n  rcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9\n  exact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\n\u22a2 \u2200 (p : A),\n    \u2203 q,\n      (\u2191q).fst = (\u2191p).fst + 1 \u2227\n        (\u2191q).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst \u2227 (\u2191q).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\n[PROOFSTEP]\nrintro \u27e8\u27e8n, x, y\u27e9, hp\u27e9\n[GOAL]\ncase mk.mk.mk\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp :\n  \u00acMeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\n\u22a2 \u2203 q,\n    (\u2191q).fst = (\u2191{ val := (n, x, y), property := hp }).fst + 1 \u2227\n      (\u2191q).snd.fst \u2208\n          cylinder (\u2191{ val := (n, x, y), property := hp }).snd.fst (\u2191{ val := (n, x, y), property := hp }).fst \u2227\n        (\u2191q).snd.snd \u2208\n          cylinder (\u2191{ val := (n, x, y), property := hp }).snd.snd (\u2191{ val := (n, x, y), property := hp }).fst\n[PROOFSTEP]\nrcases I n x y hp with \u27e8x', y', hx', hy', h'\u27e9\n[GOAL]\ncase mk.mk.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : \u2115\nx y : \u2115 \u2192 \u2115\nhp :\n  \u00acMeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\nx' y' : \u2115 \u2192 \u2115\nhx' : x' \u2208 cylinder x n\nhy' : y' \u2208 cylinder y n\nh' : \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n\u22a2 \u2203 q,\n    (\u2191q).fst = (\u2191{ val := (n, x, y), property := hp }).fst + 1 \u2227\n      (\u2191q).snd.fst \u2208\n          cylinder (\u2191{ val := (n, x, y), property := hp }).snd.fst (\u2191{ val := (n, x, y), property := hp }).fst \u2227\n        (\u2191q).snd.snd \u2208\n          cylinder (\u2191{ val := (n, x, y), property := hp }).snd.snd (\u2191{ val := (n, x, y), property := hp }).fst\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8n + 1, x', y'\u27e9, h'\u27e9, rfl, hx', hy'\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nthis :\n  \u2200 (p : A),\n    \u2203 q,\n      (\u2191q).fst = (\u2191p).fst + 1 \u2227\n        (\u2191q).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst \u2227 (\u2191q).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\n\u22a2 False\n[PROOFSTEP]\nchoose F hFn hFx hFy using this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\n\u22a2 False\n[PROOFSTEP]\nlet p0 : A :=\n  \u27e8\u27e80, fun _ => 0, fun _ => 0\u27e9, by simp [hfg]\u27e9\n    -- construct inductively decreasing sequences of cylinders whose images are not separated\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\n\u22a2 \u00acMeasurablySeparable (f '' cylinder (0, fun x => 0, fun x => 0).snd.fst (0, fun x => 0, fun x => 0).fst)\n      (g '' cylinder (0, fun x => 0, fun x => 0).snd.snd (0, fun x => 0, fun x => 0).fst)\n[PROOFSTEP]\nsimp [hfg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\n\u22a2 False\n[PROOFSTEP]\nlet p : \u2115 \u2192 A := fun n => F^[n] p0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\n\u22a2 False\n[PROOFSTEP]\nhave prec : \u2200 n, p (n + 1) = F (p n) := fun n => by\n  simp only [iterate_succ', Function.comp]\n    -- check that at the `n`-th step we deal with cylinders of length `n`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nn : \u2115\n\u22a2 p (n + 1) = F (p n)\n[PROOFSTEP]\nsimp only [iterate_succ', Function.comp]\n  -- check that at the `n`-th step we deal with cylinders of length `n`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 False\n[PROOFSTEP]\nhave pn_fst : \u2200 n, (p n).1.1 = n := by\n  intro n\n  induction' n with n IH\n  \u00b7 rfl\n  \u00b7\n    simp only [prec, hFn, IH]\n      -- check that the cylinders we construct are indeed decreasing, by checking that the coordinates\n        -- are stationary.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 \u2200 (n : \u2115), (\u2191(p n)).fst = n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\n\u22a2 (\u2191(p n)).fst = n\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\n\u22a2 (\u2191(p Nat.zero)).fst = Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\nn : \u2115\nIH : (\u2191(p n)).fst = n\n\u22a2 (\u2191(p (Nat.succ n))).fst = Nat.succ n\n[PROOFSTEP]\nsimp only [prec, hFn, IH]\n  -- check that the cylinders we construct are indeed decreasing, by checking that the coordinates\n    -- are stationary.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\n\u22a2 False\n[PROOFSTEP]\nhave Ix : \u2200 m n, m + 1 \u2264 n \u2192 (p n).1.2.1 m = (p (m + 1)).1.2.1 m :=\n  by\n  intro m\n  apply Nat.le_induction\n  \u00b7 rfl\n  intro n hmn IH\n  have I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m :=\n    by\n    apply hFx (p n) m\n    rw [pn_fst]\n    exact hmn\n  rw [prec, I, IH]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\n\u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nintro m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\napply Nat.le_induction\n[GOAL]\ncase base\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm : \u2115\n\u22a2 Prod.fst (\u2191(p (m + 1))).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm : \u2115\n\u22a2 \u2200 (n : \u2115),\n    m + 1 \u2264 n \u2192\n      Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m \u2192\n        Prod.fst (\u2191(p (n + 1))).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nintro n hmn IH\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 Prod.fst (\u2191(p (n + 1))).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nhave I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m :=\n  by\n  apply hFx (p n) m\n  rw [pn_fst]\n  exact hmn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 Prod.fst (\u2191(F (p n))).snd m = Prod.fst (\u2191(p n)).snd m\n[PROOFSTEP]\napply hFx (p n) m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 m < (\u2191(p n)).fst\n[PROOFSTEP]\nrw [pn_fst]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 m < n\n[PROOFSTEP]\nexact hmn\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nI : Prod.fst (\u2191(F (p n))).snd m = Prod.fst (\u2191(p n)).snd m\n\u22a2 Prod.fst (\u2191(p (n + 1))).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nrw [prec, I, IH]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 False\n[PROOFSTEP]\nhave Iy : \u2200 m n, m + 1 \u2264 n \u2192 (p n).1.2.2 m = (p (m + 1)).1.2.2 m :=\n  by\n  intro m\n  apply Nat.le_induction\n  \u00b7 rfl\n  intro n hmn IH\n  have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m :=\n    by\n    apply hFy (p n) m\n    rw [pn_fst]\n    exact hmn\n  rw [prec, I, IH]\n    -- denote by `x` and `y` the limit points of these two sequences of cylinders.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\n\u22a2 \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nintro m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm : \u2115\n\u22a2 \u2200 (n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\napply Nat.le_induction\n[GOAL]\ncase base\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm : \u2115\n\u22a2 Prod.snd (\u2191(p (m + 1))).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm : \u2115\n\u22a2 \u2200 (n : \u2115),\n    m + 1 \u2264 n \u2192\n      Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m \u2192\n        Prod.snd (\u2191(p (n + 1))).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nintro n hmn IH\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n\u22a2 Prod.snd (\u2191(p (n + 1))).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nhave I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m :=\n  by\n  apply hFy (p n) m\n  rw [pn_fst]\n  exact hmn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n\u22a2 Prod.snd (\u2191(F (p n))).snd m = Prod.snd (\u2191(p n)).snd m\n[PROOFSTEP]\napply hFy (p n) m\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n\u22a2 m < (\u2191(p n)).fst\n[PROOFSTEP]\nrw [pn_fst]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n\u22a2 m < n\n[PROOFSTEP]\nexact hmn\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI\u271d :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nm n : \u2115\nhmn : m + 1 \u2264 n\nIH : Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nI : Prod.snd (\u2191(F (p n))).snd m = Prod.snd (\u2191(p n)).snd m\n\u22a2 Prod.snd (\u2191(p (n + 1))).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n[PROOFSTEP]\nrw [prec, I, IH]\n  -- denote by `x` and `y` the limit points of these two sequences of cylinders.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\n\u22a2 False\n[PROOFSTEP]\nset x : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.1 n with hx\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\n\u22a2 False\n[PROOFSTEP]\nset y : \u2115 \u2192 \u2115 := fun n => (p (n + 1)).1.2.2 n with hy\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\n\u22a2 False\n[PROOFSTEP]\nhave M : \u2200 n, \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) :=\n  by\n  intro n\n  convert (p n).2 using 3\n  \u00b7 rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]\n    intro i hi\n    rw [hx]\n    exact (Ix i n hi).symm\n  \u00b7 rw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]\n    intro i hi\n    rw [hy]\n    exact (Iy i n hi).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\n\u22a2 \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn : \u2115\n\u22a2 \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n[PROOFSTEP]\nconvert (p n).2 using 3\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn : \u2115\n\u22a2 cylinder x n = cylinder (\u2191(p n)).snd.fst (\u2191(p n)).fst\n[PROOFSTEP]\nrw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 x i = Prod.fst (\u2191(p n)).snd i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn i : \u2115\nhi : i < n\n\u22a2 x i = Prod.fst (\u2191(p n)).snd i\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h.e'_1.h.e'_3.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => Prod.fst (\u2191(p (n + 1))).snd n) i = Prod.fst (\u2191(p n)).snd i\n[PROOFSTEP]\nexact (Ix i n hi).symm\n[GOAL]\ncase h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn : \u2115\n\u22a2 cylinder y n = cylinder (\u2191(p n)).snd.snd (\u2191(p n)).fst\n[PROOFSTEP]\nrw [pn_fst, \u2190 mem_cylinder_iff_eq, mem_cylinder_iff]\n[GOAL]\ncase h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i < n \u2192 y i = Prod.snd (\u2191(p n)).snd i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn i : \u2115\nhi : i < n\n\u22a2 y i = Prod.snd (\u2191(p n)).snd i\n[PROOFSTEP]\nrw [hy]\n[GOAL]\ncase h.e'_1.h.e'_4.h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nn i : \u2115\nhi : i < n\n\u22a2 (fun n => Prod.snd (\u2191(p (n + 1))).snd n) i = Prod.snd (\u2191(p n)).snd i\n[PROOFSTEP]\nexact (Iy i n hi).symm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8u, v, u_open, v_open, xu, yv, huv\u27e9 : \u2203 u v : Set \u03b1, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v :=\n  by\n  apply t2_separation\n  exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 \u2203 u v, IsOpen u \u2227 IsOpen v \u2227 f x \u2208 u \u2227 g y \u2208 v \u2227 Disjoint u v\n[PROOFSTEP]\napply t2_separation\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 f x \u2260 g y\n[PROOFSTEP]\nexact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\n\u22a2 False\n[PROOFSTEP]\nletI : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u03b5x, \u03b5xpos, h\u03b5x\u27e9 : \u2203 (\u03b5x : \u211d), \u03b5x > 0 \u2227 Metric.ball x \u03b5x \u2286 f \u207b\u00b9' u :=\n  by\n  apply Metric.mem_nhds_iff.1\n  exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 \u2203 \u03b5x, \u03b5x > 0 \u2227 ball x \u03b5x \u2286 f \u207b\u00b9' u\n[PROOFSTEP]\napply Metric.mem_nhds_iff.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u22a2 f \u207b\u00b9' u \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u03b5y, \u03b5ypos, h\u03b5y\u27e9 : \u2203 (\u03b5y : \u211d), \u03b5y > 0 \u2227 Metric.ball y \u03b5y \u2286 g \u207b\u00b9' v :=\n  by\n  apply Metric.mem_nhds_iff.1\n  exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 \u2203 \u03b5y, \u03b5y > 0 \u2227 ball y \u03b5y \u2286 g \u207b\u00b9' v\n[PROOFSTEP]\napply Metric.mem_nhds_iff.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u22a2 g \u207b\u00b9' v \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n : \u2115, (1 / 2 : \u211d) ^ n < min \u03b5x \u03b5y :=\n  exists_pow_lt_of_lt_one (lt_min \u03b5xpos \u03b5ypos)\n    (by norm_num)\n      -- for large enough `n`, these open sets separate the images of long cylinders around `x` and `y`\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 False\n[PROOFSTEP]\nhave B : MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) :=\n  by\n  refine' \u27e8u, _, _, u_open.measurableSet\u27e9\n  \u00b7 rw [image_subset_iff]\n    apply Subset.trans _ h\u03b5x\n    intro z hz\n    rw [mem_cylinder_iff_dist_le] at hz \n    exact hz.trans_lt (hn.trans_le (min_le_left _ _))\n  \u00b7 refine' Disjoint.mono_left _ huv.symm\n    change g '' cylinder y n \u2286 v\n    rw [image_subset_iff]\n    apply Subset.trans _ h\u03b5y\n    intro z hz\n    rw [mem_cylinder_iff_dist_le] at hz \n    exact\n      hz.trans_lt\n        (hn.trans_le (min_le_right _ _))\n          -- this is a contradiction.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n[PROOFSTEP]\nrefine' \u27e8u, _, _, u_open.measurableSet\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 f '' cylinder x n \u2286 u\n[PROOFSTEP]\nrw [image_subset_iff]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 f \u207b\u00b9' u\n[PROOFSTEP]\napply Subset.trans _ h\u03b5x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder x n \u2286 ball x \u03b5x\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder x n\n\u22a2 z \u2208 ball x \u03b5x\n[PROOFSTEP]\nrw [mem_cylinder_iff_dist_le] at hz \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z x \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball x \u03b5x\n[PROOFSTEP]\nexact hz.trans_lt (hn.trans_le (min_le_left _ _))\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 Disjoint (g '' cylinder y n) u\n[PROOFSTEP]\nrefine' Disjoint.mono_left _ huv.symm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2264 v\n[PROOFSTEP]\nchange g '' cylinder y n \u2286 v\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 g '' cylinder y n \u2286 v\n[PROOFSTEP]\nrw [image_subset_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 g \u207b\u00b9' v\n[PROOFSTEP]\napply Subset.trans _ h\u03b5y\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\n\u22a2 cylinder y n \u2286 ball y \u03b5y\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : z \u2208 cylinder y n\n\u22a2 z \u2208 ball y \u03b5y\n[PROOFSTEP]\nrw [mem_cylinder_iff_dist_le] at hz \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nz : \u2115 \u2192 \u2115\nhz : dist z y \u2264 (1 / 2) ^ n\n\u22a2 z \u2208 ball y \u03b5y\n[PROOFSTEP]\nexact\n  hz.trans_lt\n    (hn.trans_le (min_le_right _ _))\n      -- this is a contradiction.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf g : (\u2115 \u2192 \u2115) \u2192 \u03b1\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : \u00acMeasurablySeparable (range f) (range g)\nI :\n  \u2200 (n : \u2115) (x y : \u2115 \u2192 \u2115),\n    \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) \u2192\n      \u2203 x' y',\n        x' \u2208 cylinder x n \u2227\n          y' \u2208 cylinder y n \u2227 \u00acMeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // \u00acMeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A \u2192 A\nhFn : \u2200 (p : A), (\u2191(F p)).fst = (\u2191p).fst + 1\nhFx : \u2200 (p : A), (\u2191(F p)).snd.fst \u2208 cylinder (\u2191p).snd.fst (\u2191p).fst\nhFy : \u2200 (p : A), (\u2191(F p)).snd.snd \u2208 cylinder (\u2191p).snd.snd (\u2191p).fst\np0 : A :=\n  { val := (0, fun x => 0, fun x => 0),\n    property := (_ : \u00acMeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : \u2115 \u2192 A := fun n => F^[n] p0\nprec : \u2200 (n : \u2115), p (n + 1) = F (p n)\npn_fst : \u2200 (n : \u2115), (\u2191(p n)).fst = n\nIx : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.fst (\u2191(p n)).snd m = Prod.fst (\u2191(p (m + 1))).snd m\nIy : \u2200 (m n : \u2115), m + 1 \u2264 n \u2192 Prod.snd (\u2191(p n)).snd m = Prod.snd (\u2191(p (m + 1))).snd m\nx : \u2115 \u2192 \u2115 := fun n => Prod.fst (\u2191(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (\u2191(p (n + 1))).snd n\ny : \u2115 \u2192 \u2115 := fun n => Prod.snd (\u2191(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (\u2191(p (n + 1))).snd n\nM : \u2200 (n : \u2115), \u00acMeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set \u03b1\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x \u2208 u\nyv : g y \u2208 v\nhuv : Disjoint u v\nthis : MetricSpace (\u2115 \u2192 \u2115) := metricSpaceNatNat\n\u03b5x : \u211d\n\u03b5xpos : \u03b5x > 0\nh\u03b5x : ball x \u03b5x \u2286 f \u207b\u00b9' u\n\u03b5y : \u211d\n\u03b5ypos : \u03b5y > 0\nh\u03b5y : ball y \u03b5y \u2286 g \u207b\u00b9' v\nn : \u2115\nhn : (1 / 2) ^ n < min \u03b5x \u03b5y\nB : MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n\u22a2 False\n[PROOFSTEP]\nexact M n B\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\ns t : Set \u03b1\nhs : AnalyticSet s\nht : AnalyticSet t\nh : Disjoint s t\n\u22a2 MeasurablySeparable s t\n[PROOFSTEP]\nrw [AnalyticSet] at hs ht \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\ns t : Set \u03b1\nhs : s = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = s\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint s t\n\u22a2 MeasurablySeparable s t\n[PROOFSTEP]\nrcases hs with (rfl | \u27e8f, f_cont, rfl\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint \u2205 t\n\u22a2 MeasurablySeparable \u2205 t\n[PROOFSTEP]\nrefine' \u27e8\u2205, Subset.refl _, by simp, MeasurableSet.empty\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nh : Disjoint \u2205 t\n\u22a2 Disjoint t \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nt : Set \u03b1\nht : t = \u2205 \u2228 \u2203 f, Continuous f \u2227 range f = t\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) t\n\u22a2 MeasurablySeparable (range f) t\n[PROOFSTEP]\nrcases ht with (rfl | \u27e8g, g_cont, rfl\u27e9)\n[GOAL]\ncase inr.intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) \u2205\n\u22a2 MeasurablySeparable (range f) \u2205\n[PROOFSTEP]\nexact \u27e8univ, subset_univ _, by simp, MeasurableSet.univ\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\nh : Disjoint (range f) \u2205\n\u22a2 Disjoint \u2205 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.intro.inr.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\nf : (\u2115 \u2192 \u2115) \u2192 \u03b1\nf_cont : Continuous f\ng : (\u2115 \u2192 \u2115) \u2192 \u03b1\ng_cont : Continuous g\nh : Disjoint (range f) (range g)\n\u22a2 MeasurablySeparable (range f) (range g)\n[PROOFSTEP]\nexact measurablySeparable_range_of_disjoint f_cont g_cont h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\ns : Set \u03b1\nhs : AnalyticSet s\nhsc : AnalyticSet s\u1d9c\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nrcases hs.measurablySeparable hsc disjoint_compl_right with \u27e8u, hsu, hdu, hmu\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\ns : Set \u03b1\nhs : AnalyticSet s\nhsc : AnalyticSet s\u1d9c\nu : Set \u03b1\nhsu : s \u2286 u\nhdu : Disjoint s\u1d9c u\nhmu : MeasurableSet u\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nobtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : T2Space \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : OpensMeasurableSpace \u03b1\ns : Set \u03b1\nhs : AnalyticSet s\nhsc : AnalyticSet s\u1d9c\nhsu : s \u2286 s\nhdu : Disjoint s\u1d9c s\nhmu : MeasurableSet s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nexact hmu\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\n\u22a2 MeasurableSet (f \u207b\u00b9' s) \u2194 MeasurableSet s\n[PROOFSTEP]\nrefine \u27e8fun h => ?_, fun h => hf h\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 MeasurableSet s\n[PROOFSTEP]\napply AnalyticSet.measurableSet_of_compl\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s\n[PROOFSTEP]\nrw [\u2190 image_preimage_eq s hsurj]\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s))\n[PROOFSTEP]\nexact h.analyticSet_image hf\n[GOAL]\ncase hsc\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet s\u1d9c\n[PROOFSTEP]\nrw [\u2190 image_preimage_eq s\u1d9c hsurj]\n[GOAL]\ncase hsc\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\ns : Set Y\nh : MeasurableSet (f \u207b\u00b9' s)\n\u22a2 AnalyticSet (f '' (f \u207b\u00b9' s\u1d9c))\n[PROOFSTEP]\nexact h.compl.analyticSet_image hf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\n\u22a2 MeasurableSpace.map f inst\u271d\u2077 = borel Y\n[PROOFSTEP]\nhave d := hf.mono le_rfl OpensMeasurableSpace.borel_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\n\u22a2 MeasurableSpace.map f inst\u271d\u2077 = borel Y\n[PROOFSTEP]\nletI := borel Y\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis : MeasurableSpace Y := borel Y\n\u22a2 MeasurableSpace.map f inst\u271d\u2077 = borel Y\n[PROOFSTEP]\nhaveI : BorelSpace Y := \u27e8rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\nthis\u271d : MeasurableSpace Y := borel Y\nthis : BorelSpace Y\n\u22a2 MeasurableSpace.map f inst\u271d\u2077 = borel Y\n[PROOFSTEP]\nexact d.map_measurableSpace_eq hsurj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2079 : TopologicalSpace X\ninst\u271d\u2078 : PolishSpace X\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : TopologicalSpace Y\ninst\u271d\u2074 : T2Space Y\ninst\u271d\u00b3 : MeasurableSpace Y\ninst\u271d\u00b2 : OpensMeasurableSpace Y\ninst\u271d\u00b9 : MeasurableSpace \u03b2\nf : X \u2192 Y\ninst\u271d : SecondCountableTopology \u2191(range f)\nhf : Measurable f\nhr : MeasurableSet (range f)\ns : Set Y\n\u22a2 MeasurableSet (f \u207b\u00b9' s) \u2194 MeasurableSet (s \u2229 range f)\n[PROOFSTEP]\nrw [hf.measurableSet_preimage_iff_preimage_val, \u2190 (MeasurableEmbedding.subtype_coe hr).measurableSet_image,\n  Subtype.image_preimage_coe]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Continuous f\nhsurj : Surjective f\n\u22a2 MeasurableSpace.map f inst\u271d\u2074 = borel Y\n[PROOFSTEP]\nborelize Y\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2076 : TopologicalSpace X\ninst\u271d\u2075 : PolishSpace X\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Continuous f\nhsurj : Surjective f\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\n\u22a2 MeasurableSpace.map f inst\u271d\u2074 = borel Y\n[PROOFSTEP]\nexact hf.measurable.map_measurableSpace_eq hsurj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : PolishSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Continuous f\nhsurj : Surjective f\n\u22a2 MeasurableSpace.map f (borel X) = borel Y\n[PROOFSTEP]\nborelize X\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nX : Type u_3\nY : Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : PolishSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : T2Space Y\ninst\u271d : SecondCountableTopology Y\nf : X \u2192 Y\nhf : Continuous f\nhsurj : Surjective f\nthis\u271d\u00b9 : MeasurableSpace X := borel X\nthis\u271d : BorelSpace X\n\u22a2 MeasurableSpace.map f (borel X) = borel Y\n[PROOFSTEP]\nexact hf.map_eq_borel hsurj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b3\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nobtain \u27e8b, b_count, b_nonempty, hb\u27e9 : \u2203 b : Set (Set \u03b3), b.Countable \u2227 \u2205 \u2209 b \u2227 IsTopologicalBasis b :=\n  exists_countable_basis \u03b3\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nhaveI : Encodable b := b_count.toEncodable\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nlet A :=\n  { p : b \u00d7 b // Disjoint (p.1 : Set \u03b3) p.2 }\n    -- for each pair of disjoint sets in the topological basis `b`, consider Borel sets separating\n      -- their images, by injectivity of `f` and the Lusin separation theorem.\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nhave : \u2200 p : A, \u2203 q : Set \u03b2, f '' (p.1.1 : Set \u03b3) \u2286 q \u2227 Disjoint (f '' (p.1.2 : Set \u03b3)) q \u2227 MeasurableSet q :=\n  by\n  intro p\n  apply\n    AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n      ((hb.isOpen p.1.2.2).analyticSet_image f_cont)\n  exact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\n\u22a2 \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).fst \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).snd) q \u2227 MeasurableSet q\n[PROOFSTEP]\nintro p\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\np : A\n\u22a2 \u2203 q, f '' \u2191(\u2191p).fst \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).snd) q \u2227 MeasurableSet q\n[PROOFSTEP]\napply\n  AnalyticSet.measurablySeparable ((hb.isOpen p.1.1.2).analyticSet_image f_cont)\n    ((hb.isOpen p.1.2.2).analyticSet_image f_cont)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\np : A\n\u22a2 Disjoint (f '' \u2191(\u2191p).fst) (f '' \u2191(\u2191p).snd)\n[PROOFSTEP]\nexact Disjoint.image p.2 (f_inj.injOn univ) (subset_univ _) (subset_univ _)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nthis : \u2200 (p : A), \u2203 q, f '' \u2191(\u2191p).fst \u2286 q \u2227 Disjoint (f '' \u2191(\u2191p).snd) q \u2227 MeasurableSet q\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nchoose q hq1 hq2 q_meas using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nlet E : b \u2192 Set \u03b2 := fun s => closure (f '' s) \u2229 \u22c2 (t : b) (ht : Disjoint s.1 t.1), q \u27e8(s, t), ht\u27e9 \\ q \u27e8(t, s), ht.symm\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nobtain \u27e8u, u_anti, u_pos, u_lim\u27e9 : \u2203 u : \u2115 \u2192 \u211d, StrictAnti u \u2227 (\u2200 n : \u2115, 0 < u n) \u2227 Tendsto u atTop (\ud835\udcdd 0) :=\n  exists_seq_strictAnti_tendsto (0 : \u211d)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nlet F : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : b) (_ : Bounded s.1 \u2227 diam s.1 \u2264 u n), E s\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nsuffices range f = \u22c2 n, F n\n  by\n  have E_meas : \u2200 s : b, MeasurableSet (E s) := by\n    intro b\n    refine' isClosed_closure.measurableSet.inter _\n    refine' MeasurableSet.iInter fun s => _\n    exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)\n  have F_meas : \u2200 n, MeasurableSet (F n) := by\n    intro n\n    refine' MeasurableSet.iUnion fun s => _\n    exact MeasurableSet.iUnion fun _ => E_meas _\n  rw [this]\n  exact MeasurableSet.iInter fun n => F_meas n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nhave E_meas : \u2200 s : b, MeasurableSet (E s) := by\n  intro b\n  refine' isClosed_closure.measurableSet.inter _\n  refine' MeasurableSet.iInter fun s => _\n  exact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\n\u22a2 \u2200 (s : \u2191b), MeasurableSet (E s)\n[PROOFSTEP]\nintro b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b\u271d) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b\u271d) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet (E b)\n[PROOFSTEP]\nrefine' isClosed_closure.measurableSet.inter _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b\u271d) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b\u271d) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nb : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 (t : \u2191b\u271d) (ht : Disjoint \u2191b \u2191t),\n      q { val := (b, t), property := ht } \\ q { val := (t, b), property := (_ : Disjoint \u2191t \u2191b) })\n[PROOFSTEP]\nrefine' MeasurableSet.iInter fun s => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb\u271d : Set (Set \u03b3)\nb_count : Set.Countable b\u271d\nb_nonempty : \u00ac\u2205 \u2208 b\u271d\nhb : IsTopologicalBasis b\u271d\nthis\u271d : Encodable \u2191b\u271d\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b\u271d \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b\u271d) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b\u271d) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nb s : \u2191b\u271d\n\u22a2 MeasurableSet\n    (\u22c2 (ht : Disjoint \u2191b \u2191s),\n      q { val := (b, s), property := ht } \\ q { val := (s, b), property := (_ : Disjoint \u2191s \u2191b) })\n[PROOFSTEP]\nexact MeasurableSet.iInter fun hs => (q_meas _).diff (q_meas _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nhave F_meas : \u2200 n, MeasurableSet (F n) := by\n  intro n\n  refine' MeasurableSet.iUnion fun s => _\n  exact MeasurableSet.iUnion fun _ => E_meas _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\n\u22a2 \u2200 (n : \u2115), MeasurableSet (F n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\n\u22a2 MeasurableSet (F n)\n[PROOFSTEP]\nrefine' MeasurableSet.iUnion fun s => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nn : \u2115\ns : \u2191b\n\u22a2 MeasurableSet (\u22c3 (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s)\n[PROOFSTEP]\nexact MeasurableSet.iUnion fun _ => E_meas _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (range f)\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nthis : range f = \u22c2 (n : \u2115), F n\nE_meas : \u2200 (s : \u2191b), MeasurableSet (E s)\nF_meas : \u2200 (n : \u2115), MeasurableSet (F n)\n\u22a2 MeasurableSet (\u22c2 (n : \u2115), F n)\n[PROOFSTEP]\nexact MeasurableSet.iInter fun n => F_meas n\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f = \u22c2 (n : \u2115), F n\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 range f \u2286 \u22c2 (n : \u2115), F n\n[PROOFSTEP]\nrintro x \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\n\u22a2 f y \u2208 \u22c2 (n : \u2115), F n\n[PROOFSTEP]\nrefine mem_iInter.2 fun n => ?_\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 f y \u2208 F n\n[PROOFSTEP]\nobtain \u27e8s, sb, ys, hs\u27e9 : \u2203 (s : Set \u03b3), s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2) :=\n  by\n  apply hb.mem_nhds_iff.1\n  exact ball_mem_nhds _ (half_pos (u_pos n))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 \u2203 s, s \u2208 b \u2227 y \u2208 s \u2227 s \u2286 ball y (u n / 2)\n[PROOFSTEP]\napply hb.mem_nhds_iff.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\n\u22a2 ball y (u n / 2) \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact ball_mem_nhds _ (half_pos (u_pos n))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 f y \u2208 F n\n[PROOFSTEP]\nhave diam_s : diam s \u2264 u n := by\n  apply (diam_mono hs bounded_ball).trans\n  convert diam_ball (x := y) (half_pos (u_pos n)).le\n  ring\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 diam s \u2264 u n\n[PROOFSTEP]\napply (diam_mono hs bounded_ball).trans\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 diam (ball y (u n / 2)) \u2264 u n\n[PROOFSTEP]\nconvert diam_ball (x := y) (half_pos (u_pos n)).le\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\n\u22a2 u n = 2 * (u n / 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 F n\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8\u27e8s, sb\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c3 (_ : Metric.Bounded \u2191{ val := s, property := sb } \u2227 diam \u2191{ val := s, property := sb } \u2264 u n),\n      E { val := s, property := sb }\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8\u27e8Metric.Bounded.mono hs bounded_ball, diam_s\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208 E { val := s, property := sb }\n[PROOFSTEP]\napply mem_inter (subset_closure (mem_image_of_mem _ ys))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\n\u22a2 f y \u2208\n    \u22c2 (t : \u2191b) (ht : Disjoint \u2191{ val := s, property := sb } \u2191t),\n      q { val := ({ val := s, property := sb }, t), property := ht } \\\n        q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }\n[PROOFSTEP]\nrefine' mem_iInter.2 fun t => mem_iInter.2 fun ht => \u27e8_, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 q { val := ({ val := s, property := sb }, t), property := ht }\n[PROOFSTEP]\napply hq1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_1.a\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208 f '' \u2191(\u2191{ val := ({ val := s, property := sb }, t), property := ht }).fst\n[PROOFSTEP]\nexact mem_image_of_mem _ ys\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 \u00acf y \u2208 q { val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }\n[PROOFSTEP]\napply disjoint_left.1 (hq2 \u27e8(t, \u27e8s, sb\u27e9), ht.symm\u27e9)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2081.intro.intro.intro.intro.refine'_2.a\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\ny : \u03b3\nn : \u2115\ns : Set \u03b3\nsb : s \u2208 b\nys : y \u2208 s\nhs : s \u2286 ball y (u n / 2)\ndiam_s : diam s \u2264 u n\nt : \u2191b\nht : Disjoint \u2191{ val := s, property := sb } \u2191t\n\u22a2 f y \u2208\n    f ''\n      \u2191(\u2191{ val := (t, { val := s, property := sb }), property := (_ : Disjoint \u2191t \u2191{ val := s, property := sb }) }).snd\n[PROOFSTEP]\nexact mem_image_of_mem _ ys\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\n\u22a2 \u22c2 (n : \u2115), F n \u2286 range f\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhave C1 : \u2200 n, \u2203 (s : b) (_ : Bounded s.1 \u2227 diam s.1 \u2264 u n), x \u2208 E s := fun n => by\n  simpa only [mem_iUnion] using mem_iInter.1 hx n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\nn : \u2115\n\u22a2 \u2203 s x_1, x \u2208 E s\n[PROOFSTEP]\nsimpa only [mem_iUnion] using mem_iInter.1 hx n\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\nC1 : \u2200 (n : \u2115), \u2203 s x_1, x \u2208 E s\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nchoose s hs hxs using C1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhave C2 : \u2200 n, (s n).1.Nonempty := by\n  intro n\n  rw [nonempty_iff_ne_empty]\n  intro hn\n  have := (s n).2\n  rw [hn] at this \n  exact b_nonempty this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\n\u22a2 \u2200 (n : \u2115), Set.Nonempty \u2191(s n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 Set.Nonempty \u2191(s n)\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\n\u22a2 \u2191(s n) \u2260 \u2205\n[PROOFSTEP]\nintro hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\n\u22a2 False\n[PROOFSTEP]\nhave := (s n).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2191(s n) \u2208 b\n\u22a2 False\n[PROOFSTEP]\nrw [hn] at this \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nn : \u2115\nhn : \u2191(s n) = \u2205\nthis : \u2205 \u2208 b\n\u22a2 False\n[PROOFSTEP]\nexact b_nonempty this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\nC2 : \u2200 (n : \u2115), Set.Nonempty \u2191(s n)\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nchoose y hy using C2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhave I : \u2200 m n, ((s m).1 \u2229 (s n).1).Nonempty := by\n  intro m n\n  rw [\u2190 not_disjoint_iff_nonempty_inter]\n  by_contra' h\n  have A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n    haveI := mem_iInter.1 (hxs m).2 (s n)\n    (mem_iInter.1 this h : _)\n  have B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n    haveI := mem_iInter.1 (hxs n).2 (s m)\n    (mem_iInter.1 this h.symm : _)\n  exact\n    A.2\n      B.1\n        -- the points `y n` are nearby, and therefore they form a Cauchy sequence.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\n\u22a2 \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n[PROOFSTEP]\nintro m n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n[PROOFSTEP]\nrw [\u2190 not_disjoint_iff_nonempty_inter]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\n\u22a2 \u00acDisjoint \u2191(s m) \u2191(s n)\n[PROOFSTEP]\nby_contra' h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\n\u22a2 False\n[PROOFSTEP]\nhave A : x \u2208 q \u27e8(s m, s n), h\u27e9 \\ q \u27e8(s n, s m), h.symm\u27e9 :=\n  haveI := mem_iInter.1 (hxs m).2 (s n)\n  (mem_iInter.1 this h : _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\n\u22a2 False\n[PROOFSTEP]\nhave B : x \u2208 q \u27e8(s n, s m), h.symm\u27e9 \\ q \u27e8(s m, s n), h\u27e9 :=\n  haveI := mem_iInter.1 (hxs n).2 (s m)\n  (mem_iInter.1 this h.symm : _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA\u271d : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A\u271d \u2192 Set \u03b2\nhq1 : \u2200 (p : A\u271d), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A\u271d), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A\u271d), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nm n : \u2115\nh : Disjoint \u2191(s m) \u2191(s n)\nA : x \u2208 q { val := (s m, s n), property := h } \\ q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) }\nB : x \u2208 q { val := (s n, s m), property := (_ : Disjoint \u2191(s n) \u2191(s m)) } \\ q { val := (s m, s n), property := h }\n\u22a2 False\n[PROOFSTEP]\nexact\n  A.2\n    B.1\n      -- the points `y n` are nearby, and therefore they form a Cauchy sequence.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhave cauchy_y : CauchySeq y :=\n  by\n  have : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by simpa only [mul_zero] using u_lim.const_mul 2\n  refine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this\n  rcases I m n with \u27e8z, zsm, zsn\u27e9\n  calc\n    dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n    _ \u2264 u m + u n :=\n      (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n        ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n    _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 CauchySeq y\n[PROOFSTEP]\nhave : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0) := by simpa only [mul_zero] using u_lim.const_mul 2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\n\u22a2 Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [mul_zero] using u_lim.const_mul 2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\n\u22a2 CauchySeq y\n[PROOFSTEP]\nrefine cauchySeq_of_le_tendsto_0' (fun n => 2 * u n) (fun m n hmn => ?_) this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m\n[PROOFSTEP]\nrcases I m n with \u27e8z, zsm, zsn\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\nz : \u03b3\nzsm : z \u2208 \u2191(s m)\nzsn : z \u2208 \u2191(s n)\n\u22a2 dist (y m) (y n) \u2264 (fun n => 2 * u n) m\n[PROOFSTEP]\ncalc\n  dist (y m) (y n) \u2264 dist (y m) z + dist z (y n) := dist_triangle _ _ _\n  _ \u2264 u m + u n :=\n    (add_le_add ((dist_le_diam_of_mem (hs m).1 (hy m) zsm).trans (hs m).2)\n      ((dist_le_diam_of_mem (hs n).1 zsn (hy n)).trans (hs n).2))\n  _ \u2264 2 * u m := by linarith [u_anti.antitone hmn]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\nthis : Tendsto (fun n => 2 * u n) atTop (\ud835\udcdd 0)\nm n : \u2115\nhmn : m \u2264 n\nz : \u03b3\nzsm : z \u2208 \u2191(s m)\nzsn : z \u2208 \u2191(s n)\n\u22a2 u m + u n \u2264 2 * u m\n[PROOFSTEP]\nlinarith [u_anti.antitone hmn]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhaveI : Nonempty \u03b3 :=\n  \u27e8y 0\u27e9\n    -- let `z` be its limit.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nlet z := limUnder atTop y\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nhave y_lim : Tendsto y atTop (\ud835\udcdd z) := cauchy_y.tendsto_limUnder\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nsuffices f z = x by\n  rw [\u2190 this]\n  exact\n    mem_range_self\n      _\n        -- assume for a contradiction that `f z \u2260 x`.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nthis : f z = x\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nthis : f z = x\n\u22a2 f z \u2208 range f\n[PROOFSTEP]\nexact\n  mem_range_self\n    _\n      -- assume for a contradiction that `f z \u2260 x`.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\n\u22a2 f z = x\n[PROOFSTEP]\nby_contra' hne\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8v, w, v_open, w_open, fzv, xw, hvw\u27e9 := t2_separation hne\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 > (0 : \u211d), ball z \u03b4 \u2286 f \u207b\u00b9' v :=\n  by\n  apply Metric.mem_nhds_iff.1\n  exact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 ball z \u03b4 \u2286 f \u207b\u00b9' v\n[PROOFSTEP]\napply Metric.mem_nhds_iff.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u22a2 f \u207b\u00b9' v \u2208 \ud835\udcdd z\n[PROOFSTEP]\nexact f_cont.continuousAt.preimage_mem_nhds (v_open.mem_nhds fzv)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, u n + dist (y n) z < \u03b4 :=\n  haveI : Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0) := by\n    simpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim)\n  ((tendsto_order.1 this).2 _ \u03b4pos).exists\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\n\u22a2 Tendsto (fun n => u n + dist (y n) z) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [add_zero] using u_lim.add (tendsto_iff_dist_tendsto_zero.1 y_lim)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 False\n[PROOFSTEP]\nhave fsnv : f '' s n \u2286 v := by\n  rw [image_subset_iff]\n  apply Subset.trans _ h\u03b4\n  intro a ha\n  calc\n    dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n    _ \u2264 u n + dist (y n) z := (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n    _ < \u03b4 := hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 f '' \u2191(s n) \u2286 v\n[PROOFSTEP]\nrw [image_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 f \u207b\u00b9' v\n[PROOFSTEP]\napply Subset.trans _ h\u03b4\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\n\u22a2 \u2191(s n) \u2286 ball z \u03b4\n[PROOFSTEP]\nintro a ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\na : \u03b3\nha : a \u2208 \u2191(s n)\n\u22a2 a \u2208 ball z \u03b4\n[PROOFSTEP]\ncalc\n  dist a z \u2264 dist a (y n) + dist (y n) z := dist_triangle _ _ _\n  _ \u2264 u n + dist (y n) z := (add_le_add_right ((dist_le_diam_of_mem (hs n).1 ha (hy n)).trans (hs n).2) _)\n  _ < \u03b4 := hn\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b9 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\n\u22a2 False\n[PROOFSTEP]\nhave : x \u2208 closure v :=\n  closure_mono fsnv\n    (hxs n).1\n      -- this is a contradiction, as `x` is supposed to belong to `w`, which is disjoint from\n          -- the closure of `v`.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.h\u2082.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\nf : \u03b3 \u2192 \u03b2\nf_cont : Continuous f\nf_inj : Injective f\nthis\u271d\u00b2 : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nb : Set (Set \u03b3)\nb_count : Set.Countable b\nb_nonempty : \u00ac\u2205 \u2208 b\nhb : IsTopologicalBasis b\nthis\u271d\u00b9 : Encodable \u2191b\nA : Type u_3 := { p // Disjoint \u2191p.fst \u2191p.snd }\nq : A \u2192 Set \u03b2\nhq1 : \u2200 (p : A), f '' \u2191(\u2191p).fst \u2286 q p\nhq2 : \u2200 (p : A), Disjoint (f '' \u2191(\u2191p).snd) (q p)\nq_meas : \u2200 (p : A), MeasurableSet (q p)\nE : \u2191b \u2192 Set \u03b2 :=\n  fun s =>\n    closure (f '' \u2191s) \u2229\n      \u22c2 (t : \u2191b) (ht : Disjoint \u2191s \u2191t),\n        q { val := (s, t), property := ht } \\ q { val := (t, s), property := (_ : Disjoint \u2191t \u2191s) }\nu : \u2115 \u2192 \u211d\nu_anti : StrictAnti u\nu_pos : \u2200 (n : \u2115), 0 < u n\nu_lim : Tendsto u atTop (\ud835\udcdd 0)\nF : \u2115 \u2192 Set \u03b2 := fun n => \u22c3 (s : \u2191b) (_ : Metric.Bounded \u2191s \u2227 diam \u2191s \u2264 u n), E s\nx : \u03b2\nhx : x \u2208 \u22c2 (n : \u2115), F n\ns : \u2115 \u2192 \u2191b\nhs : \u2200 (n : \u2115), Metric.Bounded \u2191(s n) \u2227 diam \u2191(s n) \u2264 u n\nhxs : \u2200 (n : \u2115), x \u2208 E (s n)\ny : \u2115 \u2192 \u03b3\nhy : \u2200 (n : \u2115), y n \u2208 \u2191(s n)\nI : \u2200 (m n : \u2115), Set.Nonempty (\u2191(s m) \u2229 \u2191(s n))\ncauchy_y : CauchySeq y\nthis\u271d : Nonempty \u03b3\nz : \u03b3 := limUnder atTop y\ny_lim : Tendsto y atTop (\ud835\udcdd z)\nhne : f (limUnder atTop y) \u2260 x\nv w : Set \u03b2\nv_open : IsOpen v\nw_open : IsOpen w\nfzv : f (limUnder atTop y) \u2208 v\nxw : x \u2208 w\nhvw : Disjoint v w\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball z \u03b4 \u2286 f \u207b\u00b9' v\nn : \u2115\nhn : u n + dist (y n) z < \u03b4\nfsnv : f '' \u2191(s n) \u2286 v\nthis : x \u2208 closure v\n\u22a2 False\n[PROOFSTEP]\nexact disjoint_left.1 (hvw.closure_left w_open) this xw\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nrw [image_eq_range]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (range fun x => f \u2191x)\n[PROOFSTEP]\nhaveI : PolishSpace s := IsClosed.polishSpace hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 MeasurableSet (range fun x => f \u2191x)\n[PROOFSTEP]\napply measurableSet_range_of_continuous_injective\n[GOAL]\ncase f_cont\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Continuous fun x => f \u2191x\n[PROOFSTEP]\nrwa [continuousOn_iff_continuous_restrict] at f_cont \n[GOAL]\ncase f_inj\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nhs : IsClosed s\nf : \u03b3 \u2192 \u03b2\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nthis : PolishSpace \u2191s\n\u22a2 Injective fun x => f \u2191x\n[PROOFSTEP]\nrwa [injOn_iff_injective] at f_inj \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nobtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @PolishSpace \u03b3 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s := hs.isClopenable\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nexact\n  @IsClosed.measurableSet_image_of_continuousOn_injOn \u03b3 t' t'_polish \u03b2 _ _ _ _ s s_closed f (f_cont.mono_dom t't) f_inj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2075 : PolishSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : T2Space \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nobtain \u27e8t', t't, f_cont, t'_polish\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @Continuous \u03b3 \u03b2 t' t\u03b2 f \u2227 @PolishSpace \u03b3 t' := f_meas.exists_continuous\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2075 : PolishSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : T2Space \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nhave M : MeasurableSet[@borel \u03b3 t'] s :=\n  @Continuous.measurable \u03b3 \u03b3 t' (@borel \u03b3 t')\n    (@BorelSpace.opensMeasurable \u03b3 t' (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl)) t\u03b3 _ _ _\n    (continuous_id_of_le t't) s hs\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2076 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2075 : PolishSpace \u03b3\ninst\u271d\u2074 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : T2Space \u03b2\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nhs : MeasurableSet s\nf_meas : Measurable f\nf_inj : InjOn f s\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nf_cont : Continuous f\nt'_polish : PolishSpace \u03b3\nM : MeasurableSet s\n\u22a2 MeasurableSet (f '' s)\n[PROOFSTEP]\nexact\n  @MeasurableSet.image_of_continuousOn_injOn \u03b3 t' t'_polish (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl) \u03b2 _ _ _ _ s\n    f M (@Continuous.continuousOn \u03b3 \u03b2 t' t\u03b2 f s f_cont) f_inj\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\n\u22a2 \u2200 \u2983s_1 : Set \u2191s\u2984, MeasurableSet s_1 \u2192 MeasurableSet (restrict s f '' s_1)\n[PROOFSTEP]\nintro u hu\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nu : Set \u2191s\nhu : MeasurableSet u\n\u22a2 MeasurableSet (restrict s f '' u)\n[PROOFSTEP]\nhave A : MeasurableSet (((\u2191) : s \u2192 \u03b3) '' u) := (MeasurableEmbedding.subtype_coe hs).measurableSet_image.2 hu\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nu : Set \u2191s\nhu : MeasurableSet u\nA : MeasurableSet (Subtype.val '' u)\n\u22a2 MeasurableSet (restrict s f '' u)\n[PROOFSTEP]\nhave B : MeasurableSet (f '' (((\u2191) : s \u2192 \u03b3) '' u)) :=\n  A.image_of_continuousOn_injOn (f_cont.mono (Subtype.coe_image_subset s u)) (f_inj.mono (Subtype.coe_image_subset s u))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : MeasurableSet s\nf_cont : ContinuousOn f s\nf_inj : InjOn f s\nu : Set \u2191s\nhu : MeasurableSet u\nA : MeasurableSet (Subtype.val '' u)\nB : MeasurableSet (f '' (Subtype.val '' u))\n\u22a2 MeasurableSet (restrict s f '' u)\n[PROOFSTEP]\nrwa [\u2190 image_comp] at B \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\n\u22a2 IsClopenable s \u2194 MeasurableSet s\n[PROOFSTEP]\nrefine'\n  \u27e8fun hs => _, fun hs => hs.isClopenable\u27e9\n    -- consider a finer topology `t'` in which `s` is open and closed.\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nhs : IsClopenable s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nobtain \u27e8t', t't, t'_polish, s_closed, _\u27e9 :\n  \u2203 t' : TopologicalSpace \u03b3, t' \u2264 t\u03b3 \u2227 @PolishSpace \u03b3 t' \u2227 IsClosed[t'] s \u2227 IsOpen[t'] s := hs\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nhave C : @Continuous \u03b3 \u03b3 t' t\u03b3 id := continuous_id_of_le t't\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\nC : Continuous id\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nhave E :=\n  @Continuous.measurableEmbedding \u03b3 t' t'_polish (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl) \u03b3 t\u03b3\n    (@PolishSpace.t2Space \u03b3 t\u03b3 _) _ _ id C injective_id\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\nC : Continuous id\nE : MeasurableEmbedding id\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nhave M : @MeasurableSet \u03b3 (@borel \u03b3 t') s :=\n  @IsClosed.measurableSet \u03b3 s t' (@borel \u03b3 t')\n    (@BorelSpace.opensMeasurable \u03b3 t' (@borel \u03b3 t') (@BorelSpace.mk _ _ (borel \u03b3) rfl)) s_closed\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\nC : Continuous id\nE : MeasurableEmbedding id\nM : MeasurableSet s\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nconvert E.measurableSet_image.2 M\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2074 : PolishSpace \u03b3\ninst\u271d\u00b3 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u00b2 : T2Space \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : BorelSpace \u03b2\ns : Set \u03b3\nf : \u03b3 \u2192 \u03b2\nt' : TopologicalSpace \u03b3\nt't : t' \u2264 t\u03b3\nt'_polish : PolishSpace \u03b3\ns_closed : IsClosed s\nright\u271d : IsOpen s\nC : Continuous id\nE : MeasurableEmbedding id\nM : MeasurableSet s\n\u22a2 s = id '' s\n[PROOFSTEP]\nsimp only [id.def, image_id']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}\n[PROOFSTEP]\nrcases l.eq_or_neBot with rfl | hl\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\ninst\u271d : IsCountablyGenerated \u22a5\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) \u22a5 (\ud835\udcdd c)}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}\n[PROOFSTEP]\nletI := upgradePolishSpace \u03b3\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}\n[PROOFSTEP]\nrcases l.exists_antitone_basis with \u27e8u, hu\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | \u2203 c, Tendsto (fun n => f n x) l (\ud835\udcdd c)}\n[PROOFSTEP]\nsimp_rw [\u2190 cauchy_map_iff_exists_tendsto]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | Cauchy (map (fun n => f n x) l)}\n[PROOFSTEP]\nchange MeasurableSet {x | _ \u2227 _}\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}\n[PROOFSTEP]\nhave :\n  \u2200 x,\n    (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l).HasAntitoneBasis fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n) :=\n  fun x => hu.map.prod hu.map\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet {x | NeBot (map (fun n => f n x) l) \u2227 map (fun n => f n x) l \u00d7\u02e2 map (fun n => f n x) l \u2264 uniformity \u03b3}\n[PROOFSTEP]\nsimp_rw [and_iff_right (hl.map _),\n  Filter.HasBasis.le_basis_iff (this _).toHasBasis Metric.uniformity_basis_dist_inv_nat_succ, Set.setOf_forall]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\n\u22a2 MeasurableSet\n    (\u22c2 (i : \u2115) (_ : True),\n      {x |\n        \u2203 i_2,\n          True \u2227 ((fun n => f n x) '' u i_2) \u00d7\u02e2 ((fun n => f n x) '' u i_2) \u2286 {p | dist p.fst p.snd < 1 / (\u2191i + 1)}})\n[PROOFSTEP]\nrefine' MeasurableSet.biInter Set.countable_univ fun K _ => _\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet\n    {x | \u2203 i, True \u2227 ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.fst p.snd < 1 / (\u2191K + 1)}}\n[PROOFSTEP]\nsimp_rw [Set.setOf_exists, true_and]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\n\u22a2 MeasurableSet\n    (\u22c3 (i : \u2115), {x | ((fun n => f n x) '' u i) \u00d7\u02e2 ((fun n => f n x) '' u i) \u2286 {p | dist p.fst p.snd < 1 / (\u2191K + 1)}})\n[PROOFSTEP]\nrefine' MeasurableSet.iUnion fun N => _\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet {x | ((fun n => f n x) '' u N) \u00d7\u02e2 ((fun n => f n x) '' u N) \u2286 {p | dist p.fst p.snd < 1 / (\u2191K + 1)}}\n[PROOFSTEP]\nsimp_rw [prod_image_image_eq, image_subset_iff, prod_subset_iff, Set.setOf_forall]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nt\u03b3 : TopologicalSpace \u03b3\ninst\u271d\u2076 : PolishSpace \u03b3\ninst\u271d\u2075 : MeasurableSpace \u03b3\nh\u03b3b : BorelSpace \u03b3\n\u03b2 : Type u_4\nt\u03b2 : TopologicalSpace \u03b2\ninst\u271d\u2074 : T2Space \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b2\ninst\u271d\u00b2 : BorelSpace \u03b2\ns : Set \u03b3\nf\u271d : \u03b3 \u2192 \u03b2\nh\u03b3 : OpensMeasurableSpace \u03b3\ninst\u271d\u00b9 : Countable \u03b9\nl : Filter \u03b9\ninst\u271d : IsCountablyGenerated l\nf : \u03b9 \u2192 \u03b2 \u2192 \u03b3\nhf : \u2200 (i : \u03b9), Measurable (f i)\nhl : NeBot l\nthis\u271d : UpgradedPolishSpace \u03b3 := upgradePolishSpace \u03b3\nu : \u2115 \u2192 Set \u03b9\nhu : HasAntitoneBasis l u\nthis :\n  \u2200 (x : \u03b2),\n    HasAntitoneBasis (map (fun i => f i x) l \u00d7\u02e2 map (fun i => f i x) l) fun n =>\n      ((fun i => f i x) '' u n) \u00d7\u02e2 ((fun i => f i x) '' u n)\nK : \u2115\nx\u271d : K \u2208 fun i => True\nN : \u2115\n\u22a2 MeasurableSet\n    (\u22c2 (i : \u03b9) (_ : i \u2208 u N) (i_1 : \u03b9) (_ : i_1 \u2208 u N),\n      {x | (i, i_1) \u2208 (fun p => (f p.fst x, f p.snd x)) \u207b\u00b9' {p | dist p.fst p.snd < 1 / (\u2191K + 1)}})\n[PROOFSTEP]\nexact\n  MeasurableSet.biInter (to_countable (u N)) fun i _ =>\n    MeasurableSet.biInter (to_countable (u N)) fun j _ =>\n      measurableSet_lt (Measurable.dist (hf i) (hf j)) measurable_const\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh : Countable \u03b1\ninst\u271d : DiscreteTopology \u03b1\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nobtain \u27e8f, hf\u27e9 := h.exists_injective_nat\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh : Countable \u03b1\ninst\u271d : DiscreteTopology \u03b1\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nhave : ClosedEmbedding f :=\n  by\n  apply closedEmbedding_of_continuous_injective_closed continuous_of_discreteTopology hf\n  exact fun t _ => isClosed_discrete _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh : Countable \u03b1\ninst\u271d : DiscreteTopology \u03b1\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 ClosedEmbedding f\n[PROOFSTEP]\napply closedEmbedding_of_continuous_injective_closed continuous_of_discreteTopology hf\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh : Countable \u03b1\ninst\u271d : DiscreteTopology \u03b1\nf : \u03b1 \u2192 \u2115\nhf : Injective f\n\u22a2 IsClosedMap f\n[PROOFSTEP]\nexact fun t _ => isClosed_discrete _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\nh : Countable \u03b1\ninst\u271d : DiscreteTopology \u03b1\nf : \u03b1 \u2192 \u2115\nhf : Injective f\nthis : ClosedEmbedding f\n\u22a2 PolishSpace \u03b1\n[PROOFSTEP]\nexact this.polishSpace\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\nh : \u00acCountable \u03b1\n\u22a2 \u03b1 \u2243\u1d50 (\u2115 \u2192 Bool)\n[PROOFSTEP]\napply Nonempty.some\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\nh : \u00acCountable \u03b1\n\u22a2 Nonempty (\u03b1 \u2243\u1d50 (\u2115 \u2192 Bool))\n[PROOFSTEP]\nobtain \u27e8f, -, fcts, finj\u27e9 :=\n  isClosed_univ.exists_nat_bool_injection_of_not_countable\n    (by rwa [\u2190 countable_coe_iff, (Equiv.Set.univ _).countable_iff])\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\nh : \u00acCountable \u03b1\n\u22a2 \u00acSet.Countable univ\n[PROOFSTEP]\nrwa [\u2190 countable_coe_iff, (Equiv.Set.univ _).countable_iff]\n[GOAL]\ncase h.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\nh : \u00acCountable \u03b1\nf : (\u2115 \u2192 Bool) \u2192 \u03b1\nfcts : Continuous f\nfinj : Injective f\n\u22a2 Nonempty (\u03b1 \u2243\u1d50 (\u2115 \u2192 Bool))\n[PROOFSTEP]\nobtain \u27e8g, gmeas, ginj\u27e9 := MeasurableSpace.measurable_injection_nat_bool_of_countablyGenerated \u03b1\n[GOAL]\ncase h.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\nh : \u00acCountable \u03b1\nf : (\u2115 \u2192 Bool) \u2192 \u03b1\nfcts : Continuous f\nfinj : Injective f\ng : \u03b1 \u2192 \u2115 \u2192 Bool\ngmeas : Measurable g\nginj : Injective g\n\u22a2 Nonempty (\u03b1 \u2243\u1d50 (\u2115 \u2192 Bool))\n[PROOFSTEP]\nexact \u27e8borelSchroederBernstein gmeas ginj fcts.measurable finj\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\n\u22a2 \u03b1 \u2243\u1d50 \u03b2\n[PROOFSTEP]\nby_cases h : Countable \u03b1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : Countable \u03b1\n\u22a2 \u03b1 \u2243\u1d50 \u03b2\n[PROOFSTEP]\nletI := Countable.of_equiv \u03b1 e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : Countable \u03b1\nthis : Countable \u03b2 := Countable.of_equiv \u03b1 e\n\u22a2 \u03b1 \u2243\u1d50 \u03b2\n[PROOFSTEP]\nrefine \u27e8e, ?_, ?_\u27e9\n[GOAL]\ncase pos.refine_1\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : Countable \u03b1\nthis : Countable \u03b2 := Countable.of_equiv \u03b1 e\n\u22a2 Measurable \u2191e\n[PROOFSTEP]\napply measurable_of_countable\n[GOAL]\ncase pos.refine_2\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : Countable \u03b1\nthis : Countable \u03b2 := Countable.of_equiv \u03b1 e\n\u22a2 Measurable \u2191e.symm\n[PROOFSTEP]\napply measurable_of_countable\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : \u00acCountable \u03b1\n\u22a2 \u03b1 \u2243\u1d50 \u03b2\n[PROOFSTEP]\nrefine' measurableEquivOfNotCountable h _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u2077 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : PolishSpace \u03b1\ninst\u271d\u2074 : PolishSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : BorelSpace \u03b2\ne : \u03b1 \u2243 \u03b2\nh : \u00acCountable \u03b1\n\u22a2 \u00acCountable \u03b2\n[PROOFSTEP]\nrwa [e.countable_iff] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : Finite \u03b1\n\u22a2 \u2203 n, Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n[PROOFSTEP]\nobtain \u27e8n, \u27e8n_equiv\u27e9\u27e9 := Finite.exists_equiv_fin \u03b1\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : Finite \u03b1\nn : \u2115\nn_equiv : \u03b1 \u2243 Fin n\n\u22a2 \u2203 n, Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n[PROOFSTEP]\nrefine' \u27e8n, \u27e8PolishSpace.Equiv.measurableEquiv (n_equiv.trans _)\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : PolishSpace \u03b1\ninst\u271d\u00b9 : BorelSpace \u03b1\ninst\u271d : Finite \u03b1\nn : \u2115\nn_equiv : \u03b1 \u2243 Fin n\n\u22a2 Fin n \u2243 \u2191(range fun x => \u2191\u2191x)\n[PROOFSTEP]\nexact Equiv.ofInjective _ (Nat.cast_injective.comp Fin.val_injective)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PolishSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : Infinite \u03b1\ninst\u271d : Countable \u03b1\n\u22a2 Nonempty (\u03b1 \u2243\u1d50 \u2191(range Nat.cast))\n[PROOFSTEP]\nhave : PolishSpace (range ((\u2191) : \u2115 \u2192 \u211d)) := Nat.closedEmbedding_coe_real.isClosedMap.closed_range.polishSpace\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PolishSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : Infinite \u03b1\ninst\u271d : Countable \u03b1\nthis : PolishSpace \u2191(range Nat.cast)\n\u22a2 Nonempty (\u03b1 \u2243\u1d50 \u2191(range Nat.cast))\n[PROOFSTEP]\nrefine' \u27e8PolishSpace.Equiv.measurableEquiv _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PolishSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : Infinite \u03b1\ninst\u271d : Countable \u03b1\nthis : PolishSpace \u2191(range Nat.cast)\n\u22a2 \u03b1 \u2243 \u2191(range Nat.cast)\n[PROOFSTEP]\nrefine' (nonempty_equiv_of_countable.some : \u03b1 \u2243 \u2115).trans _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u2074 : MeasurableSpace \u03b1\ninst\u271d\u00b3 : PolishSpace \u03b1\ninst\u271d\u00b2 : BorelSpace \u03b1\ninst\u271d\u00b9 : Infinite \u03b1\ninst\u271d : Countable \u03b1\nthis : PolishSpace \u2191(range Nat.cast)\n\u22a2 \u2115 \u2243 \u2191(range Nat.cast)\n[PROOFSTEP]\nexact Equiv.ofInjective ((\u2191) : \u2115 \u2192 \u211d) Nat.cast_injective\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\nby_cases h\u03b1 : Countable \u03b1\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\ncases finite_or_infinite \u03b1\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\nobtain \u27e8n, h_nonempty_equiv\u27e9 := exists_nat_measurableEquiv_range_coe_fin_of_finite \u03b1\n[GOAL]\ncase pos.inl.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\nrefine' \u27e8_, _, h_nonempty_equiv\u27e9\n[GOAL]\ncase pos.inl.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)\n[PROOFSTEP]\nletI : MeasurableSpace (Fin n) := borel (Fin n)\n[GOAL]\ncase pos.inl.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis : MeasurableSpace (Fin n) := borel (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)\n[PROOFSTEP]\nhaveI : BorelSpace (Fin n) := \u27e8rfl\u27e9\n[GOAL]\ncase pos.inl.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSet (range fun x => \u2191\u2191x)\n[PROOFSTEP]\nrefine' MeasurableEmbedding.measurableSet_range _\n[GOAL]\ncase pos.inl.intro.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableSpace (Fin n)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase pos.inl.intro.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Finite \u03b1\nn : \u2115\nh_nonempty_equiv : Nonempty (\u03b1 \u2243\u1d50 \u2191(range fun x => \u2191\u2191x))\nthis\u271d : MeasurableSpace (Fin n) := borel (Fin n)\nthis : BorelSpace (Fin n)\n\u22a2 MeasurableEmbedding fun x => \u2191\u2191x\n[PROOFSTEP]\nexact continuous_of_discreteTopology.measurableEmbedding (Nat.cast_injective.comp Fin.val_injective)\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\nrefine' \u27e8_, _, measurableEquiv_range_coe_nat_of_infinite_of_countable \u03b1\u27e9\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSet (range Nat.cast)\n[PROOFSTEP]\nrefine' MeasurableEmbedding.measurableSet_range _\n[GOAL]\ncase pos.inr.refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableSpace \u2115\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase pos.inr.refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : Countable \u03b1\nh\u271d : Infinite \u03b1\n\u22a2 MeasurableEmbedding Nat.cast\n[PROOFSTEP]\nexact continuous_of_discreteTopology.measurableEmbedding Nat.cast_injective\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u2203 s, MeasurableSet s \u2227 Nonempty (\u03b1 \u2243\u1d50 \u2191s)\n[PROOFSTEP]\nrefine' \u27e8univ, MeasurableSet.univ, \u27e8(PolishSpace.measurableEquivOfNotCountable h\u03b1 _ : \u03b1 \u2243\u1d50 (univ : Set \u211d))\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acCountable \u2191univ\n[PROOFSTEP]\nrw [countable_coe_iff]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\nh\u03b1 : \u00acCountable \u03b1\n\u22a2 \u00acSet.Countable univ\n[PROOFSTEP]\nexact Cardinal.not_countable_real\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\n\u22a2 \u2203 f, MeasurableEmbedding f\n[PROOFSTEP]\nobtain \u27e8s, hs, \u27e8e\u27e9\u27e9 := exists_subset_real_measurableEquiv \u03b1\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b9 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : PolishSpace \u03b1\ninst\u271d : BorelSpace \u03b1\ns : Set \u211d\nhs : MeasurableSet s\ne : \u03b1 \u2243\u1d50 \u2191s\n\u22a2 \u2203 f, MeasurableEmbedding f\n[PROOFSTEP]\nexact \u27e8(\u2191) \u2218 e, (MeasurableEmbedding.subtype_coe hs).comp e.measurableEmbedding\u27e9\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Constructions.Polish", "llama_tokens": 168583, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.5386820050082752}}
{"text": "[GOAL]\nm : \u2115\nxs : Bitvec m\nb : Bool\n\u22a2 Bitvec.toNat (xs++\u209cb ::\u1d65 Vector.nil) = Bitvec.toNat xs * 2 + Bitvec.toNat (b ::\u1d65 Vector.nil)\n[PROOFSTEP]\ncases' xs with xs P\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nP : List.length xs = m\n\u22a2 Bitvec.toNat ({ val := xs, property := P }++\u209cb ::\u1d65 Vector.nil) =\n    Bitvec.toNat { val := xs, property := P } * 2 + Bitvec.toNat (b ::\u1d65 Vector.nil)\n[PROOFSTEP]\nsimp [bitsToNat_toList]\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nP : List.length xs = m\n\u22a2 bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2\n[PROOFSTEP]\nclear P\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\n\u22a2 bitsToNat (xs ++ [b]) = bitsToNat [b] + bitsToNat xs * 2\n[PROOFSTEP]\nunfold bitsToNat\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\n\u22a2 List.foldl addLsb 0 (xs ++ [b]) = List.foldl addLsb 0 [b] + List.foldl addLsb 0 xs * 2\n[PROOFSTEP]\nrw [List.foldl, List.foldl]\n  -- generalize the accumulator of foldl\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\n\u22a2 List.foldl addLsb 0 (xs ++ [b]) = addLsb 0 b + List.foldl addLsb 0 xs * 2\n[PROOFSTEP]\ngeneralize h : 0 = x\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\nh : 0 = x\n\u22a2 List.foldl addLsb x (xs ++ [b]) = addLsb x b + List.foldl addLsb x xs * 2\n[PROOFSTEP]\nconv in addLsb x b => rw [\u2190 h]\n[GOAL]\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\nh : 0 = x\n| addLsb x b\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\nh : 0 = x\n| addLsb x b\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\nh : 0 = x\n| addLsb x b\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\nh : 0 = x\n\u22a2 List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2\n[PROOFSTEP]\nclear h\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\n\u22a2 List.foldl addLsb x (xs ++ [b]) = addLsb 0 b + List.foldl addLsb x xs * 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk\nm : \u2115\nb : Bool\nxs : List Bool\nx : \u2115\n\u22a2 addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\n[PROOFSTEP]\ninduction' xs with x xs xs_ih generalizing x\n[GOAL]\ncase mk.nil\nm : \u2115\nb : Bool\nx\u271d x : \u2115\n\u22a2 addLsb (List.foldl addLsb x []) b = addLsb 0 b + List.foldl addLsb x [] * 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.nil\nm : \u2115\nb : Bool\nx\u271d x : \u2115\n\u22a2 addLsb x b = addLsb 0 b + x * 2\n[PROOFSTEP]\nunfold addLsb\n[GOAL]\ncase mk.nil\nm : \u2115\nb : Bool\nx\u271d x : \u2115\n\u22a2 (x + x + bif b then 1 else 0) = (0 + 0 + bif b then 1 else 0) + x * 2\n[PROOFSTEP]\nsimp [Nat.mul_succ]\n[GOAL]\ncase mk.cons\nm : \u2115\nb : Bool\nx\u271d\u00b9 : \u2115\nx\u271d : Bool\nxs : List Bool\nxs_ih : \u2200 (x : \u2115), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : \u2115\n\u22a2 addLsb (List.foldl addLsb x (x\u271d :: xs)) b = addLsb 0 b + List.foldl addLsb x (x\u271d :: xs) * 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.cons\nm : \u2115\nb : Bool\nx\u271d\u00b9 : \u2115\nx\u271d : Bool\nxs : List Bool\nxs_ih : \u2200 (x : \u2115), addLsb (List.foldl addLsb x xs) b = addLsb 0 b + List.foldl addLsb x xs * 2\nx : \u2115\n\u22a2 addLsb (List.foldl addLsb (addLsb x x\u271d) xs) b = addLsb 0 b + List.foldl addLsb (addLsb x x\u271d) xs * 2\n[PROOFSTEP]\napply xs_ih\n[GOAL]\nn : \u2115\n\u22a2 Bitvec.toNat (decide (n % 2 = 1) ::\u1d65 Vector.nil) = n % 2\n[PROOFSTEP]\nsimp [bitsToNat_toList]\n[GOAL]\nn : \u2115\n\u22a2 bitsToNat [decide (n % 2 = 1)] = n % 2\n[PROOFSTEP]\nunfold bitsToNat addLsb List.foldl\n[GOAL]\nn : \u2115\n\u22a2 List.foldl (fun r b => r + r + bif b then 1 else 0) (0 + 0 + bif decide (n % 2 = 1) then 1 else 0) [] = n % 2\n[PROOFSTEP]\nsimp [Nat.cond_decide_mod_two, -Bool.cond_decide]\n[GOAL]\nk n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\n[PROOFSTEP]\ninduction' k with k ih generalizing n\n[GOAL]\ncase zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat zero n) = n % 2 ^ zero\n[PROOFSTEP]\nsimp [Nat.mod_one]\n[GOAL]\ncase zero\nn\u271d n : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat 0 n) = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn\u271d k : \u2115\nih : \u2200 {n : \u2115}, Bitvec.toNat (Bitvec.ofNat k n) = n % 2 ^ k\nn : \u2115\n\u22a2 Bitvec.toNat (Bitvec.ofNat (succ k) n) = n % 2 ^ succ k\n[PROOFSTEP]\nrw [ofNat_succ, toNat_append, ih, bits_toNat_decide, mod_pow_succ, Nat.mul_comm]\n[GOAL]\nn : \u2115\ni : Fin (2 ^ n)\n\u22a2 Bitvec.toNat (ofFin i) = \u2191i\n[PROOFSTEP]\nrw [ofFin, toNat_ofNat, Nat.mod_eq_of_lt]\n[GOAL]\nn : \u2115\ni : Fin (2 ^ n)\n\u22a2 \u2191i < 2 ^ n\n[PROOFSTEP]\napply i.is_lt\n[GOAL]\nx : \u2115\nb : Bool\n\u22a2 addLsb x b = 2 * x + bif b then 1 else 0\n[PROOFSTEP]\nsimp [addLsb, two_mul]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.toNat v = List.foldr (flip addLsb) 0 (List.reverse (Vector.toList v))\n[PROOFSTEP]\nrw [List.foldr_reverse]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.toNat v = List.foldl (fun x y => flip addLsb y x) 0 (Vector.toList v)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.toNat v < 2 ^ n\n[PROOFSTEP]\nsuffices : v.toNat + 1 \u2264 2 ^ n\n[GOAL]\nn : \u2115\nv : Bitvec n\nthis : Bitvec.toNat v + 1 \u2264 2 ^ n\n\u22a2 Bitvec.toNat v < 2 ^ n\ncase this n : \u2115 v : Bitvec n \u22a2 Bitvec.toNat v + 1 \u2264 2 ^ n\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase this\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.toNat v + 1 \u2264 2 ^ n\n[PROOFSTEP]\nrw [toNat_eq_foldr_reverse]\n[GOAL]\ncase this\nn : \u2115\nv : Bitvec n\n\u22a2 List.foldr (flip addLsb) 0 (List.reverse (Vector.toList v)) + 1 \u2264 2 ^ n\n[PROOFSTEP]\ncases' v with xs h\n[GOAL]\ncase this.mk\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 List.foldr (flip addLsb) 0 (List.reverse (Vector.toList { val := xs, property := h })) + 1 \u2264 2 ^ n\n[PROOFSTEP]\ndsimp [Bitvec.toNat, bitsToNat]\n[GOAL]\ncase this.mk\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 List.foldr (flip addLsb) 0 (List.reverse xs) + 1 \u2264 2 ^ n\n[PROOFSTEP]\nrw [\u2190 List.length_reverse] at h \n[GOAL]\ncase this.mk\nn : \u2115\nxs : List Bool\nh : List.length (List.reverse xs) = n\n\u22a2 List.foldr (flip addLsb) 0 (List.reverse xs) + 1 \u2264 2 ^ n\n[PROOFSTEP]\ngeneralize xs.reverse = ys at h \n[GOAL]\ncase this.mk\nn : \u2115\nxs ys : List Bool\nh : List.length ys = n\n\u22a2 List.foldr (flip addLsb) 0 ys + 1 \u2264 2 ^ n\n[PROOFSTEP]\ninduction' ys with head tail ih generalizing n\n[GOAL]\ncase this.mk.nil\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nn : \u2115\nh : List.length [] = n\n\u22a2 List.foldr (flip addLsb) 0 [] + 1 \u2264 2 ^ n\n[PROOFSTEP]\nsimp [\u2190 h]\n[GOAL]\ncase this.mk.cons\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 List.foldr (flip addLsb) 0 (head :: tail) + 1 \u2264 2 ^ n\n[PROOFSTEP]\nsimp only [\u2190 h, pow_add, flip, List.length, List.foldr, pow_one]\n[GOAL]\ncase this.mk.cons\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 addLsb (List.foldr (fun b a => addLsb a b) 0 tail) head + 1 \u2264 2 ^ List.length tail * 2\n[PROOFSTEP]\nrw [addLsb_eq_twice_add_one]\n[GOAL]\ncase this.mk.cons\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 (2 * List.foldr (fun b a => addLsb a b) 0 tail + bif head then 1 else 0) + 1 \u2264 2 ^ List.length tail * 2\n[PROOFSTEP]\ntrans\n  2 * List.foldr (fun (x : Bool) (y : \u2115) => addLsb y x) 0 tail +\n    2 *\n      1\n        -- Porting note: removed `ac_mono`, `mono` calls\n[GOAL]\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 (2 * List.foldr (fun b a => addLsb a b) 0 tail + bif head then 1 else 0) + 1 \u2264\n    2 * List.foldr (fun x y => addLsb y x) 0 tail + 2 * 1\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 2 * List.foldr (fun b a => addLsb a b) 0 tail + ((bif head then 1 else 0) + 1) \u2264\n    2 * List.foldr (fun x y => addLsb y x) 0 tail + 2 * 1\n[PROOFSTEP]\napply Nat.add_le_add_left\n[GOAL]\ncase h\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 (bif head then 1 else 0) + 1 \u2264 2 * 1\n[PROOFSTEP]\ncases head\n[GOAL]\ncase h.false\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (false :: tail) = n\n\u22a2 (bif false then 1 else 0) + 1 \u2264 2 * 1\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h.true\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (true :: tail) = n\n\u22a2 (bif true then 1 else 0) + 1 \u2264 2 * 1\n[PROOFSTEP]\nsimp only\n[GOAL]\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 2 * List.foldr (fun x y => addLsb y x) 0 tail + 2 * 1 \u2264 2 ^ List.length tail * 2\n[PROOFSTEP]\nrw [\u2190 left_distrib]\n[GOAL]\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 2 * (List.foldr (fun x y => addLsb y x) 0 tail + 1) \u2264 2 ^ List.length tail * 2\n[PROOFSTEP]\nrw [mul_comm _ 2]\n[GOAL]\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 2 * (List.foldr (fun x y => addLsb y x) 0 tail + 1) \u2264 2 * 2 ^ List.length tail\n[PROOFSTEP]\napply Nat.mul_le_mul_left\n[GOAL]\ncase h\nn\u271d : \u2115\nxs ys : List Bool\nh\u271d : List.length ys = n\u271d\nhead : Bool\ntail : List Bool\nih : \u2200 {n : \u2115}, List.length tail = n \u2192 List.foldr (flip addLsb) 0 tail + 1 \u2264 2 ^ n\nn : \u2115\nh : List.length (head :: tail) = n\n\u22a2 List.foldr (fun x y => addLsb y x) 0 tail + 1 \u2264 2 ^ List.length tail\n[PROOFSTEP]\nexact ih rfl\n[GOAL]\nx : \u2115\nb : Bool\n\u22a2 addLsb x b / 2 = x\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\nx : \u2115\n\u22a2 addLsb x false / 2 = x\n[PROOFSTEP]\nsimp only [Nat.add_mul_div_left, addLsb, \u2190 two_mul, add_comm, Nat.succ_pos', Nat.mul_div_right, gt_iff_lt, zero_add,\n  cond]\n[GOAL]\ncase true\nx : \u2115\n\u22a2 addLsb x true / 2 = x\n[PROOFSTEP]\nsimp only [Nat.add_mul_div_left, addLsb, \u2190 two_mul, add_comm, Nat.succ_pos', Nat.mul_div_right, gt_iff_lt, zero_add,\n  cond]\n[GOAL]\ncase true\nx : \u2115\n\u22a2 x + 1 / 2 = x\n[PROOFSTEP]\nnorm_num\n[GOAL]\nx : \u2115\nb : Bool\n\u22a2 decide (addLsb x b % 2 = 1) = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\nx : \u2115\n\u22a2 decide (addLsb x false % 2 = 1) = false\n[PROOFSTEP]\nsimp only [Bool.decide_iff, Nat.add_mul_mod_self_left, addLsb, \u2190 two_mul, add_comm, Bool.decide_False,\n  Nat.mul_mod_right, zero_add, cond, zero_ne_one]\n[GOAL]\ncase true\nx : \u2115\n\u22a2 decide (addLsb x true % 2 = 1) = true\n[PROOFSTEP]\nsimp only [Bool.decide_iff, Nat.add_mul_mod_self_left, addLsb, \u2190 two_mul, add_comm, Bool.decide_False,\n  Nat.mul_mod_right, zero_add, cond, zero_ne_one]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.ofNat n (Bitvec.toNat v) = v\n[PROOFSTEP]\ncases' v with xs h\n[GOAL]\ncase mk\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h }) = { val := xs, property := h }\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase mk.a\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 \u2191(Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = \u2191{ val := xs, property := h }\n[PROOFSTEP]\nchange Vector.toList _ = xs\n[GOAL]\ncase mk.a\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 Vector.toList (Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = xs\n[PROOFSTEP]\ndsimp [Bitvec.toNat, bitsToNat]\n[GOAL]\ncase mk.a\nn : \u2115\nxs : List Bool\nh : List.length xs = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs\n[PROOFSTEP]\nrw [\u2190 List.length_reverse] at h \n[GOAL]\ncase mk.a\nn : \u2115\nxs : List Bool\nh : List.length (List.reverse xs) = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs\n[PROOFSTEP]\nrw [\u2190 List.reverse_reverse xs, List.foldl_reverse]\n[GOAL]\ncase mk.a\nn : \u2115\nxs : List Bool\nh : List.length (List.reverse xs) = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (List.reverse xs))) =\n    List.reverse (List.reverse xs)\n[PROOFSTEP]\ngeneralize xs.reverse = ys at h \u22a2\n[GOAL]\ncase mk.a\nn : \u2115\nxs ys : List Bool\nh : List.length ys = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys\n[PROOFSTEP]\nclear xs\n[GOAL]\ncase mk.a\nn : \u2115\nys : List Bool\nh : List.length ys = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys\n[PROOFSTEP]\ninduction' ys with ys_head ys_tail ys_ih generalizing n\n[GOAL]\ncase mk.a.nil\nn\u271d : \u2115\nys : List Bool\nh\u271d : List.length ys = n\u271d\nn : \u2115\nh : List.length [] = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []\n[PROOFSTEP]\ncases h\n[GOAL]\ncase mk.a.nil.refl\nn : \u2115\nys : List Bool\nh : List.length ys = n\n\u22a2 Vector.toList (Bitvec.ofNat (List.length []) (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []\n[PROOFSTEP]\nsimp [Bitvec.ofNat]\n[GOAL]\ncase mk.a.cons\nn\u271d : \u2115\nys : List Bool\nh\u271d : List.length ys = n\u271d\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : \u2115\nh : List.length (ys_head :: ys_tail) = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n    List.reverse (ys_head :: ys_tail)\n[PROOFSTEP]\nsimp only [\u2190 Nat.succ_eq_add_one, List.length] at h \n[GOAL]\ncase mk.a.cons\nn\u271d : \u2115\nys : List Bool\nh\u271d : List.length ys = n\u271d\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : \u2115\nh : succ (List.length ys_tail) = n\n\u22a2 Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n    List.reverse (ys_head :: ys_tail)\n[PROOFSTEP]\nsubst n\n[GOAL]\ncase mk.a.cons\nn : \u2115\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n\u22a2 Vector.toList\n      (Bitvec.ofNat (succ (List.length ys_tail)) (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n    List.reverse (ys_head :: ys_tail)\n[PROOFSTEP]\nsimp only [Bitvec.ofNat, Vector.toList_cons, Vector.toList_nil, List.reverse_cons, Vector.toList_append, List.foldr]\n[GOAL]\ncase mk.a.cons\nn : \u2115\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n\u22a2 Vector.toList\n        (Bitvec.ofNat (List.length ys_tail) (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head / 2)) ++\n      [decide (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head % 2 = 1)] =\n    List.reverse ys_tail ++ [ys_head]\n[PROOFSTEP]\nerw [addLsb_div_two, decide_addLsb_mod_two]\n[GOAL]\ncase mk.a.cons\nn : \u2115\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n\u22a2 Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) ++ [ys_head] =\n    List.reverse ys_tail ++ [ys_head]\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.a.cons.e_a\nn : \u2115\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n\u22a2 Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) =\n    List.reverse ys_tail\n[PROOFSTEP]\napply ys_ih\n[GOAL]\ncase mk.a.cons.e_a.h\nn : \u2115\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n  \u2200 {n : \u2115},\n    List.length ys_tail = n \u2192\n      Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n\u22a2 List.length ys_tail = List.length ys_tail\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 \u2191(toFin v) = Bitvec.toNat v\n[PROOFSTEP]\nrw [toFin, Fin.coe_ofNat_eq_mod, Nat.mod_eq_of_lt]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.toNat v < 2 ^ n\n[PROOFSTEP]\napply toNat_lt\n[GOAL]\nn : \u2115\nv\u2080 v\u2081 : Bitvec n\nh : v\u2080 \u2264 v\u2081\n\u22a2 \u2191(toFin v\u2080) \u2264 \u2191(toFin v\u2081)\n[PROOFSTEP]\nrw [toFin_val, toFin_val]\n[GOAL]\nn : \u2115\nv\u2080 v\u2081 : Bitvec n\nh : v\u2080 \u2264 v\u2081\n\u22a2 Bitvec.toNat v\u2080 \u2264 Bitvec.toNat v\u2081\n[PROOFSTEP]\nexact h\n[GOAL]\nn : \u2115\ni j : Fin (2 ^ n)\nh : i \u2264 j\n\u22a2 Bitvec.toNat (Bitvec.ofNat n \u2191i) \u2264 Bitvec.toNat (Bitvec.ofNat n \u2191j)\n[PROOFSTEP]\nsimp only [toNat_ofNat, Nat.mod_eq_of_lt, Fin.is_lt]\n[GOAL]\nn : \u2115\ni j : Fin (2 ^ n)\nh : i \u2264 j\n\u22a2 \u2191i \u2264 \u2191j\n[PROOFSTEP]\nexact h\n[GOAL]\nn : \u2115\ni : Fin (2 ^ n)\n\u22a2 \u2191(toFin (ofFin i)) = \u2191i\n[PROOFSTEP]\nsimp [toFin_val, ofFin, toNat_ofNat, Nat.mod_eq_of_lt, i.is_lt]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 ofFin (toFin v) = v\n[PROOFSTEP]\ndsimp [ofFin]\n[GOAL]\nn : \u2115\nv : Bitvec n\n\u22a2 Bitvec.ofNat n \u2191(toFin v) = v\n[PROOFSTEP]\nrw [toFin_val, ofNat_toNat]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bitvec.Lemmas", "llama_tokens": 8521, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.5386819981149742}}
{"text": "[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nhaveI : IsAddLeftInvariant StieltjesFunction.id.measure :=\n  \u27e8fun a =>\n    Eq.symm <|\n      Real.measure_ext_Ioo_rat fun p q => by\n        simp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo, sub_sub_sub_cancel_right,\n          StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim, StieltjesFunction.id_apply, id.def,\n          preimage_const_add_Ioo]\u27e9\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\np q : \u211a\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (Ioo \u2191p \u2191q) =\n    \u2191\u2191(Measure.map (fun x => a + x) (StieltjesFunction.measure StieltjesFunction.id)) (Ioo \u2191p \u2191q)\n[PROOFSTEP]\nsimp only [Measure.map_apply (measurable_const_add a) measurableSet_Ioo, sub_sub_sub_cancel_right,\n  StieltjesFunction.measure_Ioo, StieltjesFunction.id_leftLim, StieltjesFunction.id_apply, id.def,\n  preimage_const_add_Ioo]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nhave A : StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped = 1 :=\n  by\n  change StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1\n  rcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H) <;>\n    simp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n      StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n[PROOFSTEP]\nchange StieltjesFunction.id.measure (parallelepiped (stdOrthonormalBasis \u211d \u211d)) = 1\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1\n[PROOFSTEP]\nrcases parallelepiped_orthonormalBasis_one_dim (stdOrthonormalBasis \u211d \u211d) with (H | H)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nH : parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d) = Icc 0 1\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1\n[PROOFSTEP]\nsimp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n  StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nH : parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d) = Icc (-1) 0\n\u22a2 \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) (parallelepiped \u2191(stdOrthonormalBasis \u211d \u211d)) = 1\n[PROOFSTEP]\nsimp only [H, StieltjesFunction.measure_Icc, StieltjesFunction.id_apply, id.def, tsub_zero,\n  StieltjesFunction.id_leftLim, sub_neg_eq_add, zero_add, ENNReal.ofReal_one]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nconv_rhs => rw [addHaarMeasure_unique StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n| StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nrw [addHaarMeasure_unique StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n| StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nrw [addHaarMeasure_unique StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n| StieltjesFunction.measure StieltjesFunction.id\n[PROOFSTEP]\nrw [addHaarMeasure_unique StieltjesFunction.id.measure (stdOrthonormalBasis \u211d \u211d).toBasis.parallelepiped, A]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nthis : IsAddLeftInvariant (StieltjesFunction.measure StieltjesFunction.id)\nA :\n  \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id)\n      \u2191(Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d))) =\n    1\n\u22a2 volume = 1 \u2022 addHaarMeasure (Basis.parallelepiped (OrthonormalBasis.toBasis (stdOrthonormalBasis \u211d \u211d)))\n[PROOFSTEP]\nsimp only [volume, Basis.addHaar, one_smul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\n\u22a2 \u2191\u2191volume s = \u2191\u2191(StieltjesFunction.measure StieltjesFunction.id) s\n[PROOFSTEP]\nsimp [volume_eq_stieltjes_id]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u211d\n\u22a2 \u2191\u2191volume (Ico a b) = ofReal (b - a)\n[PROOFSTEP]\nsimp [volume_val]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u211d\n\u22a2 \u2191\u2191volume (Icc a b) = ofReal (b - a)\n[PROOFSTEP]\nsimp [volume_val]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u211d\n\u22a2 \u2191\u2191volume (Ioo a b) = ofReal (b - a)\n[PROOFSTEP]\nsimp [volume_val]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u211d\n\u22a2 \u2191\u2191volume (Ioc a b) = ofReal (b - a)\n[PROOFSTEP]\nsimp [volume_val]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume {a} = 0\n[PROOFSTEP]\nsimp [volume_val]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nr : \u211d\u22650\n\u22a2 \u2191r = \u2191\u2191volume (Icc 0 \u2191r)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na r : \u211d\n\u22a2 \u2191\u2191volume (Metric.ball a r) = ofReal (2 * r)\n[PROOFSTEP]\nrw [ball_eq_Ioo, volume_Ioo, \u2190 sub_add, add_sub_cancel', two_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na r : \u211d\n\u22a2 \u2191\u2191volume (Metric.closedBall a r) = ofReal (2 * r)\n[PROOFSTEP]\nrw [closedBall_eq_Icc, volume_Icc, \u2190 sub_add, add_sub_cancel', two_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\n\u22a2 \u2191\u2191volume (EMetric.ball a r) = 2 * r\n[PROOFSTEP]\nrcases eq_or_ne r \u221e with (rfl | hr)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (EMetric.ball a \u22a4) = 2 * \u22a4\n[PROOFSTEP]\nrw [Metric.emetric_ball_top, volume_univ, two_mul, _root_.top_add]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u2191\u2191volume (EMetric.ball a r) = 2 * r\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\n\u22a2 \u2191\u2191volume (EMetric.ball a \u2191r) = 2 * \u2191r\n[PROOFSTEP]\nrw [Metric.emetric_ball_nnreal, volume_ball, two_mul, \u2190 NNReal.coe_add, ENNReal.ofReal_coe_nnreal, ENNReal.coe_add,\n  two_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\n\u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r\n[PROOFSTEP]\nrcases eq_or_ne r \u221e with (rfl | hr)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u22a4) = 2 * \u22a4\n[PROOFSTEP]\nrw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\u221e\nhr : r \u2260 \u22a4\n\u22a2 \u2191\u2191volume (EMetric.closedBall a r) = 2 * r\n[PROOFSTEP]\nlift r to \u211d\u22650 using hr\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nr : \u211d\u22650\n\u22a2 \u2191\u2191volume (EMetric.closedBall a \u2191r) = 2 * \u2191r\n[PROOFSTEP]\nrw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, \u2190 NNReal.coe_add, ENNReal.ofReal_coe_nnreal,\n  ENNReal.coe_add, two_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u211d\n\u22a2 \u2191\u2191volume (uIcc a b) = ofReal |b - a|\n[PROOFSTEP]\nrw [\u2190 Icc_min_max, volume_Icc, max_sub_min_eq_abs]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nn : \u2115\n\u22a2 \u2191n = \u2191\u2191volume (Ioo a (a + \u2191n))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (Ici a) = \u22a4\n[PROOFSTEP]\nrw [\u2190 measure_congr Ioi_ae_eq_Ici]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (Ioi a) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nn : \u2115\n\u22a2 \u2191n = \u2191\u2191volume (Ioo (a - \u2191n) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (Iic a) = \u22a4\n[PROOFSTEP]\nrw [\u2190 measure_congr Iio_ae_eq_Iic]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\n\u22a2 \u2191\u2191volume (Iio a) = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nx : \u211d\n\u22a2 \u2191\u2191volume (Ioo (x - 1) (x + 1)) < \u22a4\n[PROOFSTEP]\nsimp only [Real.volume_Ioo, ENNReal.ofReal_lt_top]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nx y : \u211d\n\u22a2 \u2191\u2191(Measure.restrict volume (Icc x y)) univ < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nx y : \u211d\n\u22a2 \u2191\u2191(Measure.restrict volume (Ico x y)) univ < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nx y : \u211d\n\u22a2 \u2191\u2191(Measure.restrict volume (Ioc x y)) univ < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nx y : \u211d\n\u22a2 \u2191\u2191(Measure.restrict volume (Ioo x y)) univ < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\n\u22a2 \u2191\u2191volume s \u2264 EMetric.diam s\n[PROOFSTEP]\nby_cases hs : Metric.Bounded s\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\nhs : Metric.Bounded s\n\u22a2 \u2191\u2191volume s \u2264 EMetric.diam s\n[PROOFSTEP]\nrw [Real.ediam_eq hs, \u2190 volume_Icc]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\nhs : Metric.Bounded s\n\u22a2 \u2191\u2191volume s \u2264 \u2191\u2191volume (Icc (sInf s) (sSup s))\n[PROOFSTEP]\nexact volume.mono (Real.subset_Icc_sInf_sSup_of_bounded hs)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\nhs : \u00acMetric.Bounded s\n\u22a2 \u2191\u2191volume s \u2264 EMetric.diam s\n[PROOFSTEP]\nrw [Metric.ediam_of_unbounded hs]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set \u211d\nhs : \u00acMetric.Bounded s\n\u22a2 \u2191\u2191volume s \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\np : \u211d \u2192 Prop\na : \u211d\nh : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, p x\n\u22a2 0 < \u2191\u2191volume {x | p x}\n[PROOFSTEP]\nrcases h.exists_Ioo_subset with \u27e8l, u, hx, hs\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\np : \u211d \u2192 Prop\na : \u211d\nh : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, p x\nl u : \u211d\nhx : a \u2208 Ioo l u\nhs : Ioo l u \u2286 {x | p x}\n\u22a2 0 < \u2191\u2191volume {x | p x}\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ (measure_mono hs)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\np : \u211d \u2192 Prop\na : \u211d\nh : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, p x\nl u : \u211d\nhx : a \u2208 Ioo l u\nhs : Ioo l u \u2286 {x | p x}\n\u22a2 0 < \u2191\u2191volume (Ioo l u)\n[PROOFSTEP]\nsimpa [-mem_Ioo] using hx.1.trans hx.2\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\n\u22a2 \u2191\u2191volume (Icc a b) = \u220f i : \u03b9, ofReal (b i - a i)\n[PROOFSTEP]\nrw [\u2190 pi_univ_Icc, volume_pi_pi]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume (Icc (a i) (b i)) = \u220f i : \u03b9, ofReal (b i - a i)\n[PROOFSTEP]\nsimp only [Real.volume_Icc]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Icc a b)) = \u220f i : \u03b9, (b i - a i)\n[PROOFSTEP]\nsimp only [volume_Icc_pi, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ioo (a i) (b i))) = \u220f i : \u03b9, (b i - a i)\n[PROOFSTEP]\nsimp only [volume_pi_Ioo, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ioc (a i) (b i))) = \u220f i : \u03b9, (b i - a i)\n[PROOFSTEP]\nsimp only [volume_pi_Ioc, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211d\nh : a \u2264 b\n\u22a2 ENNReal.toReal (\u2191\u2191volume (Set.pi univ fun i => Ico (a i) (b i))) = \u220f i : \u03b9, (b i - a i)\n[PROOFSTEP]\nsimp only [volume_pi_Ico, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 \u2191\u2191volume (Metric.ball a r) = ofReal ((2 * r) ^ Fintype.card \u03b9)\n[PROOFSTEP]\nsimp only [MeasureTheory.volume_pi_ball a hr, volume_ball, Finset.prod_const]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 < r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)\n[PROOFSTEP]\nexact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr.le) _).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2191volume (Metric.closedBall a r) = ofReal ((2 * r) ^ Fintype.card \u03b9)\n[PROOFSTEP]\nsimp only [MeasureTheory.volume_pi_closedBall a hr, volume_closedBall, Finset.prod_const]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u03b9 \u2192 \u211d\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 ofReal (2 * r) ^ Finset.card Finset.univ = ofReal ((2 * r) ^ Fintype.card \u03b9)\n[PROOFSTEP]\nexact (ENNReal.ofReal_pow (mul_nonneg zero_le_two hr) _).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\ns : Set (\u03b9 \u2192 \u211d)\n\u22a2 \u220f _i : \u03b9, \u21911 * EMetric.diam s = EMetric.diam s ^ Fintype.card \u03b9\n[PROOFSTEP]\nsimp only [ENNReal.coe_one, one_mul, Finset.prod_const, Fintype.card]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n\u22a2 ofReal |a| \u2022 Measure.map (fun x => a * x) volume = volume\n[PROOFSTEP]\nrefine' (Real.measure_ext_Ioo_rat fun p q => _).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\np q : \u211a\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)\n[PROOFSTEP]\ncases' lt_or_gt_of_ne h with h h\n[GOAL]\ncase inl\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a < 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)\n[PROOFSTEP]\nsimp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,\n  preimage_const_mul_Ioo_of_neg _ _ h, abs_of_neg h, mul_sub, smul_eq_mul, mul_div_cancel' _ (ne_of_lt h)]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh\u271d : a \u2260 0\np q : \u211a\nh : a > 0\n\u22a2 \u2191\u2191volume (Ioo \u2191p \u2191q) = \u2191\u2191(ofReal |a| \u2022 Measure.map (fun x => a * x) volume) (Ioo \u2191p \u2191q)\n[PROOFSTEP]\nsimp only [Real.volume_Ioo, Measure.smul_apply, \u2190 ENNReal.ofReal_mul (le_of_lt h),\n  Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, preimage_const_mul_Ioo _ _ h, abs_of_pos h, mul_sub,\n  mul_div_cancel' _ (ne_of_gt h), smul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n\u22a2 Measure.map (fun x => a * x) volume = ofReal |a\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nconv_rhs =>\n  rw [\u2190 Real.smul_map_volume_mul_left h, smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel h,\n    abs_one, ENNReal.ofReal_one, one_smul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n| ofReal |a\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 Real.smul_map_volume_mul_left h, smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel h,\n    abs_one, ENNReal.ofReal_one, one_smul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n| ofReal |a\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 Real.smul_map_volume_mul_left h, smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel h,\n    abs_one, ENNReal.ofReal_one, one_smul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n| ofReal |a\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 Real.smul_map_volume_mul_left h, smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel h,\n  abs_one, ENNReal.ofReal_one, one_smul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(Measure.map (fun x => a * x) volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s\n[PROOFSTEP]\nrw [map_volume_mul_left h]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(ofReal |a\u207b\u00b9| \u2022 volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n\u22a2 ofReal |a| \u2022 Measure.map (fun x => x * a) volume = volume\n[PROOFSTEP]\nsimpa only [mul_comm] using Real.smul_map_volume_mul_left h\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\n\u22a2 Measure.map (fun x => x * a) volume = ofReal |a\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nsimpa only [mul_comm] using Real.map_volume_mul_left h\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(Measure.map (fun x => x * a) volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s\n[PROOFSTEP]\nrw [map_volume_mul_right h]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\na : \u211d\nh : a \u2260 0\ns : Set \u211d\n\u22a2 \u2191\u2191(ofReal |a\u207b\u00b9| \u2022 volume) s = ofReal |a\u207b\u00b9| * \u2191\u2191volume s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\n\u22a2 ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = volume\n[PROOFSTEP]\nrefine' (Measure.pi_eq fun s hs => _).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 \u2191\u2191(ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f i : \u03b9, \u2191\u2191volume (s i)\n[PROOFSTEP]\nsimp only [det_diagonal, Measure.coe_smul, Algebra.id.smul_eq_mul, Pi.smul_apply]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191(Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume) (Set.pi univ s) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n[PROOFSTEP]\nrw [Measure.map_apply _ (MeasurableSet.univ_pi hs)]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))\n[PROOFSTEP]\nswap\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 Measurable \u2191(\u2191toLin' (Matrix.diagonal D))\n[PROOFSTEP]\nexact Continuous.measurable (LinearMap.continuous_on_pi _)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n[PROOFSTEP]\nhave :\n  (Matrix.toLin' (diagonal D) \u207b\u00b9' Set.pi Set.univ fun i : \u03b9 => s i) = Set.pi Set.univ fun i : \u03b9 => (D i * \u00b7) \u207b\u00b9' s i :=\n  by\n  ext f\n  simp only [LinearMap.coe_proj, Algebra.id.smul_eq_mul, LinearMap.smul_apply, mem_univ_pi, mem_preimage,\n    LinearMap.pi_apply, diagonal_toLin']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\n\u22a2 (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\n[PROOFSTEP]\next f\n[GOAL]\ncase h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nf : \u03b9 \u2192 \u211d\n\u22a2 (f \u2208 \u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) \u2194\n    f \u2208 Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\n[PROOFSTEP]\nsimp only [LinearMap.coe_proj, Algebra.id.smul_eq_mul, LinearMap.smul_apply, mem_univ_pi, mem_preimage,\n  LinearMap.pi_apply, diagonal_toLin']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n[PROOFSTEP]\nhave B : \u2200 i, ofReal (abs (D i)) * volume ((D i * \u00b7) \u207b\u00b9' s i) = volume (s i) :=\n  by\n  intro i\n  have A : D i \u2260 0 := by\n    simp only [det_diagonal, Ne.def] at h \n    exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)\n  rw [volume_preimage_mul_left A, \u2190 mul_assoc, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, mul_inv_cancel A,\n    abs_one, ENNReal.ofReal_one, one_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\n\u22a2 \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n[PROOFSTEP]\nhave A : D i \u2260 0 := by\n  simp only [det_diagonal, Ne.def] at h \n  exact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\n\u22a2 D i \u2260 0\n[PROOFSTEP]\nsimp only [det_diagonal, Ne.def] at h \n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nh : \u00ac\u220f i : \u03b9, D i = 0\n\u22a2 D i \u2260 0\n[PROOFSTEP]\nexact Finset.prod_ne_zero_iff.1 h i (Finset.mem_univ i)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\ni : \u03b9\nA : D i \u2260 0\n\u22a2 ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n[PROOFSTEP]\nrw [volume_preimage_mul_left A, \u2190 mul_assoc, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, mul_inv_cancel A, abs_one,\n  ENNReal.ofReal_one, one_mul]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\nB : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n\u22a2 ofReal |\u220f i : \u03b9, D i| * \u2191\u2191volume (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) =\n    \u220f x : \u03b9, \u2191\u2191volume (s x)\n[PROOFSTEP]\nrw [this, volume_pi_pi, Finset.abs_prod, ENNReal.ofReal_prod_of_nonneg fun i _ => abs_nonneg (D i), \u2190\n  Finset.prod_mul_distrib]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nD : \u03b9 \u2192 \u211d\nh : det (Matrix.diagonal D) \u2260 0\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), MeasurableSet (s i)\nthis : (\u2191(\u2191toLin' (Matrix.diagonal D)) \u207b\u00b9' Set.pi univ fun i => s i) = Set.pi univ fun i => (fun x => D i * x) \u207b\u00b9' s i\nB : \u2200 (i : \u03b9), ofReal |D i| * \u2191\u2191volume ((fun x => D i * x) \u207b\u00b9' s i) = \u2191\u2191volume (s i)\n\u22a2 \u220f x : \u03b9, ofReal |D x| * \u2191\u2191volume ((fun x_1 => D x * x_1) \u207b\u00b9' s x) = \u220f x : \u03b9, \u2191\u2191volume (s x)\n[PROOFSTEP]\nsimp only [B]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet p : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet \u03b1 : Type _ := { x // p x }\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet \u03b2 : Type _ := { x // \u00acp x }\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet g : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun _ => t.c * a \u27e8t.j, t.hij.symm\u27e9) + b\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet F : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.1, g p.1 p.2)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nlet e : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nhave : (toLin' t.toMatrix : (\u03b9 \u2192 \u211d) \u2192 \u03b9 \u2192 \u211d) = e.symm \u2218 F \u2218 e := by\n  cases t with\n  | mk t_i t_j t_hij t_c =>\n    ext f k\n    simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not, Algebra.id.smul_eq_mul, one_mul,\n      dotProduct, stdBasisMatrix, MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply,\n      zero_mul, LinearMap.smul_apply, Function.comp_apply, MeasurableEquiv.piEquivPiSubtypeProd_apply,\n      Matrix.TransvectionStruct.toMatrix_mk, Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply,\n      Pi.smul_apply, Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]\n    by_cases h : t_i = k\n    \u00b7\n      simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq, one_apply, boole_mul,\n        add_comm]\n    \u00b7 simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff, mul_zero]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n[PROOFSTEP]\ncases t with\n| mk t_i t_j t_hij t_c =>\n  ext f k\n  simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not, Algebra.id.smul_eq_mul, one_mul, dotProduct,\n    stdBasisMatrix, MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply, zero_mul,\n    LinearMap.smul_apply, Function.comp_apply, MeasurableEquiv.piEquivPiSubtypeProd_apply,\n    Matrix.TransvectionStruct.toMatrix_mk, Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply,\n    Pi.smul_apply, Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]\n  by_cases h : t_i = k\n  \u00b7\n    simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq, one_apply, boole_mul,\n      add_comm]\n  \u00b7 simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff, mul_zero]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n[PROOFSTEP]\ncases t with\n| mk t_i t_j t_hij t_c =>\n  ext f k\n  simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not, Algebra.id.smul_eq_mul, one_mul, dotProduct,\n    stdBasisMatrix, MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply, zero_mul,\n    LinearMap.smul_apply, Function.comp_apply, MeasurableEquiv.piEquivPiSubtypeProd_apply,\n    Matrix.TransvectionStruct.toMatrix_mk, Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply,\n    Pi.smul_apply, Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]\n  by_cases h : t_i = k\n  \u00b7\n    simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq, one_apply, boole_mul,\n      add_comm]\n  \u00b7 simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff, mul_zero]\n[GOAL]\ncase mk\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) =\n    \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n[PROOFSTEP]\n\n| mk t_i t_j t_hij t_c =>\n  ext f k\n  simp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not, Algebra.id.smul_eq_mul, one_mul, dotProduct,\n    stdBasisMatrix, MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply, zero_mul,\n    LinearMap.smul_apply, Function.comp_apply, MeasurableEquiv.piEquivPiSubtypeProd_apply,\n    Matrix.TransvectionStruct.toMatrix_mk, Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply,\n    Pi.smul_apply, Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]\n  by_cases h : t_i = k\n  \u00b7\n    simp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq, one_apply, boole_mul,\n      add_comm]\n  \u00b7 simp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff, mul_zero]\n[GOAL]\ncase mk\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) =\n    \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n[PROOFSTEP]\next f k\n[GOAL]\ncase mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 \u2191(\u2191toLin' (TransvectionStruct.toMatrix { i := t_i, j := t_j, hij := t_hij, c := t_c })) f k =\n    (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e) f k\n[PROOFSTEP]\nsimp only [LinearEquiv.map_smul, dite_eq_ite, LinearMap.id_coe, ite_not, Algebra.id.smul_eq_mul, one_mul, dotProduct,\n  stdBasisMatrix, MeasurableEquiv.piEquivPiSubtypeProd_symm_apply, id.def, transvection, Pi.add_apply, zero_mul,\n  LinearMap.smul_apply, Function.comp_apply, MeasurableEquiv.piEquivPiSubtypeProd_apply,\n  Matrix.TransvectionStruct.toMatrix_mk, Matrix.mulVec, LinearEquiv.map_add, ite_mul, Matrix.toLin'_apply,\n  Pi.smul_apply, Subtype.coe_mk, LinearMap.add_apply, Finset.sum_congr, Matrix.toLin'_one]\n[GOAL]\ncase mk.h.h\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k\n[PROOFSTEP]\nby_cases h : t_i = k\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : t_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k\n[PROOFSTEP]\nsimp only [h, true_and_iff, Finset.mem_univ, if_true, eq_self_iff_true, Finset.sum_ite_eq, one_apply, boole_mul,\n  add_comm]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt_i t_j : \u03b9\nt_hij : t_i \u2260 t_j\nt_c : \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 { i := t_i, j := t_j, hij := t_hij, c := t_c }.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d :=\n  fun a b =>\n    (fun x =>\n        { i := t_i, j := t_j, hij := t_hij, c := t_c }.c *\n          a\n            { val := { i := t_i, j := t_j, hij := t_hij, c := t_c }.j,\n              property :=\n                (_ :\n                  { i := t_i, j := t_j, hij := t_hij, c := t_c }.j \u2260\n                    { i := t_i, j := t_j, hij := t_hij, c := t_c }.i) }) +\n      b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nf : \u03b9 \u2192 \u211d\nk : \u03b9\nh : \u00act_i = k\n\u22a2 (f k + \u2211 x : \u03b9, if t_i = k \u2227 t_j = x then t_c * f x else 0) = if k = t_i then t_c * f t_j + f k else f k\n[PROOFSTEP]\nsimp only [h, Ne.symm h, add_zero, if_false, Finset.sum_const_zero, false_and_iff, mul_zero]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving \u2191(\u2191toLin' (TransvectionStruct.toMatrix t))\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)\n[PROOFSTEP]\nhave A : MeasurePreserving e := by convert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\n\u22a2 MeasurePreserving \u2191e\n[PROOFSTEP]\nconvert volume_preserving_piEquivPiSubtypeProd (fun _ : \u03b9 => \u211d) p\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)\n[PROOFSTEP]\nhave B : MeasurePreserving F :=\n  haveI g_meas : Measurable (Function.uncurry g) :=\n    by\n    have : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 := measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9\n    refine Measurable.add ?_ measurable_snd\n    refine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _\n    exact this.comp measurable_fst\n  (MeasurePreserving.id _).skew_product g_meas\n    (eventually_of_forall fun a =>\n      map_add_left_eq_self (Measure.pi fun _ => (stdOrthonormalBasis \u211d \u211d).toBasis.addHaar) _)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\n\u22a2 Measurable (Function.uncurry g)\n[PROOFSTEP]\nhave : Measurable fun c : \u03b1 \u2192 \u211d => c \u27e8t.j, t.hij.symm\u27e9 := measurable_pi_apply \u27e8t.j, t.hij.symm\u27e9\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable (Function.uncurry g)\n[PROOFSTEP]\nrefine Measurable.add ?_ measurable_snd\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\n\u22a2 Measurable fun a x => t.c * Prod.fst a { val := t.j, property := (_ : t.j \u2260 t.i) }\n[PROOFSTEP]\nrefine measurable_pi_lambda _ fun _ => Measurable.const_mul ?_ _\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis\u271d : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nthis : Measurable fun c => c { val := t.j, property := (_ : t.j \u2260 t.i) }\nx\u271d : \u03b2\n\u22a2 Measurable fun c => Prod.fst c { val := t.j, property := (_ : t.j \u2260 t.i) }\n[PROOFSTEP]\nexact this.comp measurable_fst\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nt : TransvectionStruct \u03b9 \u211d\np : \u03b9 \u2192 Prop := fun i => i \u2260 t.i\n\u03b1 : Type u_1 := { x // p x }\n\u03b2 : Type u_1 := { x // \u00acp x }\ng : (\u03b1 \u2192 \u211d) \u2192 (\u03b2 \u2192 \u211d) \u2192 \u03b2 \u2192 \u211d := fun a b => (fun x => t.c * a { val := t.j, property := (_ : t.j \u2260 t.i) }) + b\nF : (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) \u2192 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := fun p => (id p.fst, g p.fst p.snd)\ne : (\u03b9 \u2192 \u211d) \u2243\u1d50 (\u03b1 \u2192 \u211d) \u00d7 (\u03b2 \u2192 \u211d) := MeasurableEquiv.piEquivPiSubtypeProd (fun x => \u211d) p\nthis : \u2191(\u2191toLin' (TransvectionStruct.toMatrix t)) = \u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e\nA : MeasurePreserving \u2191e\nB : MeasurePreserving F\n\u22a2 MeasurePreserving (\u2191(MeasurableEquiv.symm e) \u2218 F \u2218 \u2191e)\n[PROOFSTEP]\nexact ((A.symm e).comp B).comp A\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\napply diagonal_transvection_induction_of_det_ne_zero _ M hM\n[GOAL]\ncase hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (D : \u03b9 \u2192 \u211d),\n    det (Matrix.diagonal D) \u2260 0 \u2192\n      Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nintro D hD\n[GOAL]\ncase hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume = ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nconv_rhs => rw [\u2190 smul_map_diagonal_volume_pi hD]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n| ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 smul_map_diagonal_volume_pi hD]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n| ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 smul_map_diagonal_volume_pi hD]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n| ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [\u2190 smul_map_diagonal_volume_pi hD]\n[GOAL]\ncase hdiag\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nD : \u03b9 \u2192 \u211d\nhD : det (Matrix.diagonal D) \u2260 0\n\u22a2 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume =\n    ofReal |(det (Matrix.diagonal D))\u207b\u00b9| \u2022\n      ofReal |det (Matrix.diagonal D)| \u2022 Measure.map (\u2191(\u2191toLin' (Matrix.diagonal D))) volume\n[PROOFSTEP]\nrw [smul_smul, \u2190 ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, inv_mul_cancel hD, abs_one, ENNReal.ofReal_one, one_smul]\n[GOAL]\ncase htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (t : TransvectionStruct \u03b9 \u211d),\n    Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n      ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nintro t\n[GOAL]\ncase htransvec\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nt : TransvectionStruct \u03b9 \u211d\n\u22a2 Measure.map (\u2191(\u2191toLin' (TransvectionStruct.toMatrix t))) volume =\n    ofReal |(det (TransvectionStruct.toMatrix t))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nsimp only [Matrix.TransvectionStruct.det, ENNReal.ofReal_one, (volume_preserving_transvectionStruct _).map_eq, one_smul,\n  _root_.inv_one, abs_one]\n[GOAL]\ncase hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\n\u22a2 \u2200 (A B : Matrix \u03b9 \u03b9 \u211d),\n    det A \u2260 0 \u2192\n      det B \u2260 0 \u2192\n        Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume \u2192\n          Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume \u2192\n            Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nintro A B _ _ IHA IHB\n[GOAL]\ncase hmul\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measure.map (\u2191(\u2191toLin' (A * B))) volume = ofReal |(det (A * B))\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [toLin'_mul, det_mul, LinearMap.coe_comp, \u2190 Measure.map_map, IHB, Measure.map_smul, IHA, smul_smul, \u2190\n  ENNReal.ofReal_mul (abs_nonneg _), \u2190 abs_mul, mul_comm, mul_inv]\n[GOAL]\ncase hmul.hg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' A)\n[PROOFSTEP]\napply Continuous.measurable\n[GOAL]\ncase hmul.hg.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' A)\n[PROOFSTEP]\napply LinearMap.continuous_on_pi\n[GOAL]\ncase hmul.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Measurable \u2191(\u2191toLin' B)\n[PROOFSTEP]\napply Continuous.measurable\n[GOAL]\ncase hmul.hf.hf\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nM : Matrix \u03b9 \u03b9 \u211d\nhM : det M \u2260 0\nA B : Matrix \u03b9 \u03b9 \u211d\na\u271d\u00b9 : det A \u2260 0\na\u271d : det B \u2260 0\nIHA : Measure.map (\u2191(\u2191toLin' A)) volume = ofReal |(det A)\u207b\u00b9| \u2022 volume\nIHB : Measure.map (\u2191(\u2191toLin' B)) volume = ofReal |(det B)\u207b\u00b9| \u2022 volume\n\u22a2 Continuous \u2191(\u2191toLin' B)\n[PROOFSTEP]\napply LinearMap.continuous_on_pi\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nclassical\n  -- this is deduced from the matrix case\nlet M := LinearMap.toMatrix' f\nhave A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']\nhave B : f = toLin' M := by simp only [toLin'_toMatrix']\nrw [A, B]\napply map_matrix_volume_pi_eq_smul_volume_pi\nrwa [A] at hf \n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nlet M := LinearMap.toMatrix' f\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nhave A : LinearMap.det f = det M := by simp only [LinearMap.det_toMatrix']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\n\u22a2 \u2191LinearMap.det f = det M\n[PROOFSTEP]\nsimp only [LinearMap.det_toMatrix']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nhave B : f = toLin' M := by simp only [toLin'_toMatrix']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\n\u22a2 f = \u2191toLin' M\n[PROOFSTEP]\nsimp only [toLin'_toMatrix']\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191f) volume = ofReal |(\u2191LinearMap.det f)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\nrw [A, B]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 Measure.map (\u2191(\u2191toLin' M)) volume = ofReal |(det M)\u207b\u00b9| \u2022 volume\n[PROOFSTEP]\napply map_matrix_volume_pi_eq_smul_volume_pi\n[GOAL]\ncase hM\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nf : (\u03b9 \u2192 \u211d) \u2192\u2097[\u211d] \u03b9 \u2192 \u211d\nhf : \u2191LinearMap.det f \u2260 0\nM : (fun x => Matrix \u03b9 \u03b9 \u211d) f := \u2191LinearMap.toMatrix' f\nA : \u2191LinearMap.det f = det M\nB : f = \u2191toLin' M\n\u22a2 det M \u2260 0\n[PROOFSTEP]\nrwa [A] at hf \n[GOAL]\n\u03b1 : Type u_1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\n\u22a2 regionBetween f g s \u2286 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_univ, regionBetween, Set.preimage, setOf_subset_setOf] using fun a => And.left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet (regionBetween f g s)\n[PROOFSTEP]\ndsimp only [regionBetween, Ioo, mem_setOf_eq, setOf_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.fst \u2208 s} \u2229 {a | a.snd \u2208 {a_1 | f a.fst < a_1} \u2229 {a_1 | a_1 < g a.fst}})\n[PROOFSTEP]\nrefine'\n  MeasurableSet.inter _\n    ((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter\n      (measurableSet_lt measurable_snd (hg.comp measurable_fst)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.fst \u2208 s}\n[PROOFSTEP]\nexact measurable_fst hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.fst \u2208 s \u2227 p.snd \u2208 Ioc (f p.fst) (g p.fst)}\n[PROOFSTEP]\ndsimp only [regionBetween, Ioc, mem_setOf_eq, setOf_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.fst \u2208 s} \u2229 {a | a.snd \u2208 {a_1 | f a.fst < a_1} \u2229 {a_1 | a_1 \u2264 g a.fst}})\n[PROOFSTEP]\nrefine'\n  MeasurableSet.inter _\n    ((measurableSet_lt (hf.comp measurable_fst) measurable_snd).inter\n      (measurableSet_le measurable_snd (hg.comp measurable_fst)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.fst \u2208 s}\n[PROOFSTEP]\nexact measurable_fst hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.fst \u2208 s \u2227 p.snd \u2208 Ico (f p.fst) (g p.fst)}\n[PROOFSTEP]\ndsimp only [regionBetween, Ico, mem_setOf_eq, setOf_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.fst \u2208 s} \u2229 {a | a.snd \u2208 {a_1 | f a.fst \u2264 a_1} \u2229 {a_1 | a_1 < g a.fst}})\n[PROOFSTEP]\nrefine'\n  MeasurableSet.inter _\n    ((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter\n      (measurableSet_lt measurable_snd (hg.comp measurable_fst)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.fst \u2208 s}\n[PROOFSTEP]\nexact measurable_fst hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {p | p.fst \u2208 s \u2227 p.snd \u2208 Icc (f p.fst) (g p.fst)}\n[PROOFSTEP]\ndsimp only [regionBetween, Icc, mem_setOf_eq, setOf_and]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet ({a | a.fst \u2208 s} \u2229 {a | a.snd \u2208 {a_1 | f a.fst \u2264 a_1} \u2229 {a_1 | a_1 \u2264 g a.fst}})\n[PROOFSTEP]\nrefine'\n  MeasurableSet.inter _\n    ((measurableSet_le (hf.comp measurable_fst) measurable_snd).inter\n      (measurableSet_le measurable_snd (hg.comp measurable_fst)))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet {a | a.fst \u2208 s}\n[PROOFSTEP]\nexact measurable_fst hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\n\u22a2 MeasurableSet {p | p.snd = f p.fst}\n[PROOFSTEP]\nsimpa using measurableSet_region_between_cc hf hf MeasurableSet.univ\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nclassical\nrw [Measure.prod_apply]\n\u00b7 have h : (fun x => volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = s.indicator fun x => ENNReal.ofReal (g x - f x) :=\n    by\n    funext x\n    rw [indicator_apply]\n    split_ifs with h\n    \u00b7 have hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x) := by simp [h, Ioo]\n      simp only [hx, Real.volume_Ioo, sub_zero]\n    \u00b7 have hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205 := by simp [h]\n      simp only [hx, measure_empty]\n  dsimp only [regionBetween, preimage_setOf_eq]\n  rw [h, lintegral_indicator] <;> simp only [hs, Pi.sub_apply]\n\u00b7 exact measurableSet_regionBetween hf hg hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nrw [Measure.prod_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume (Prod.mk x \u207b\u00b9' regionBetween f g s) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nhave h : (fun x => volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = s.indicator fun x => ENNReal.ofReal (g x - f x) :=\n  by\n  funext x\n  rw [indicator_apply]\n  split_ifs with h\n  \u00b7 have hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x) := by simp [h, Ioo]\n    simp only [hx, Real.volume_Ioo, sub_zero]\n  \u00b7 have hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205 := by simp [h]\n    simp only [hx, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n[PROOFSTEP]\nfunext x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = indicator s (fun x => ofReal (g x - f x)) x\n[PROOFSTEP]\nrw [indicator_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = if x \u2208 s then ofReal (g x - f x) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)\n[PROOFSTEP]\nhave hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x) := by simp [h, Ioo]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\n\u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x)\n[PROOFSTEP]\nsimp [h, Ioo]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : x \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = Ioo (f x) (g x)\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = ofReal (g x - f x)\n[PROOFSTEP]\nsimp only [hx, Real.volume_Ioo, sub_zero]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0\n[PROOFSTEP]\nhave hx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205 := by simp [h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\n\u22a2 {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nx : \u03b1\nh : \u00acx \u2208 s\nhx : {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = \u2205\n\u22a2 \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} = 0\n[PROOFSTEP]\nsimp only [hx, measure_empty]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume (Prod.mk x \u207b\u00b9' regionBetween f g s) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\ndsimp only [regionBetween, preimage_setOf_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (x : \u03b1), \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)} \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nrw [h, lintegral_indicator]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 \u222b\u207b (a : \u03b1) in s, ofReal (g a - f a) \u2202\u03bc = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nsimp only [hs, Pi.sub_apply]\n[GOAL]\ncase hs\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x => \u2191\u2191volume {a | x \u2208 s \u2227 a \u2208 Ioo (f x) (g x)}) = indicator s fun x => ofReal (g x - f x)\n\u22a2 MeasurableSet s\n[PROOFSTEP]\nsimp only [hs, Pi.sub_apply]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\n\u22a2 MeasurableSet (regionBetween f g s)\n[PROOFSTEP]\nexact measurableSet_regionBetween hf hg hs\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nhave h\u2081 :\n  (fun y => ENNReal.ofReal ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n    ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n  (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nhave h\u2082 :\n  (\u03bc.restrict s).prod volume (regionBetween f g s) =\n    (\u03bc.restrict s).prod volume (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) :=\n  by\n  apply measure_congr\n  apply EventuallyEq.rfl.inter\n  exact\n    ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n      (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n[PROOFSTEP]\napply measure_congr\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]\n    regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s\n[PROOFSTEP]\napply EventuallyEq.rfl.inter\n[GOAL]\ncase H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.fst) (g p.fst) p.snd) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>\n    Ioo (AEMeasurable.mk f hf p.fst) (AEMeasurable.mk g hg p.fst) p.snd\n[PROOFSTEP]\nexact\n  ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n    (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae h\u2081, \u2190 volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n[PROOFSTEP]\nconvert h\u2082 using 1\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s)\n[PROOFSTEP]\nrw [Measure.restrict_prod_eq_prod_univ]\n[GOAL]\ncase h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s)\n[PROOFSTEP]\nexact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n[PROOFSTEP]\nrw [Measure.restrict_prod_eq_prod_univ]\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n  (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n    ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n  \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n    \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n    \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))\n      (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n[PROOFSTEP]\nexact (Measure.restrict_eq_self _ (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nlet T : s \u00d7 s \u2192 Set \u211d := fun p => Ioo p.1 p.2\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nlet u := \u22c3 i : \u21a5s \u00d7 \u21a5s, T i\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nhave hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo'\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nobtain \u27e8A, A_count, hA\u27e9 : \u2203 A : Set (\u21a5s \u00d7 \u21a5s), A.Countable \u2227 \u22c3 i \u2208 A, T i = \u22c3 i : \u21a5s \u00d7 \u21a5s, T i :=\n  isOpen_iUnion_countable _ fun p => isOpen_Ioo\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\nhave : s \u2286 s \\ u \u222a \u22c3 p \u2208 A, s \u2229 T p := by\n  intro x hx\n  by_cases h'x : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i\n  \u00b7 rw [\u2190 hA] at h'x \n    obtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n      simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x\n    right\n    exact mem_biUnion pA \u27e8hx, xp\u27e9\n  \u00b7 exact Or.inl \u27e8hx, h'x\u27e9\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\n\u22a2 x \u2208 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nby_cases h'x : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i\n[GOAL]\ncase pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nrw [\u2190 hA] at h'x \n[GOAL]\ncase pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nobtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n  simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\n\u22a2 \u2203 p, p \u2208 A \u2227 x \u2208 T p\n[PROOFSTEP]\nsimpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using h'x\n[GOAL]\ncase pos.intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.intro.intro.h\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 x \u2208 \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nexact mem_biUnion pA \u27e8hx, xp\u27e9\n[GOAL]\ncase neg\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nx : \u211d\nhx : x \u2208 s\nh'x : \u00acx \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 x \u2208 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n[PROOFSTEP]\nexact Or.inl \u27e8hx, h'x\u27e9\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc s, p x\n[PROOFSTEP]\napply ae_restrict_of_ae_restrict_of_subset this\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p), p x\n[PROOFSTEP]\nrw [ae_restrict_union_iff, ae_restrict_biUnion_iff _ A_count]\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 (\u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x) \u2227 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x\n[PROOFSTEP]\nhave : \u03bc.restrict (s \\ u) = 0 := by simp only [restrict_eq_zero, hfinite.measure_zero]\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 Measure.restrict \u03bc (s \\ u) = 0\n[PROOFSTEP]\nsimp only [restrict_eq_zero, hfinite.measure_zero]\n[GOAL]\ncase intro.intro.left\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis\u271d : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\nthis : Measure.restrict \u03bc (s \\ u) = 0\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \\ u), p x\n[PROOFSTEP]\nsimp only [this, ae_zero, eventually_bot]\n[GOAL]\ncase intro.intro.right\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\n\u22a2 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T i), p x\n[PROOFSTEP]\nrintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 -\n[GOAL]\ncase intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 T ({ val := a, property := as }, { val := b, property := bs })), p x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase intro.intro.right.mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n[PROOFSTEP]\nrcases le_or_lt b a with (hba | hab)\n[GOAL]\ncase intro.intro.right.mk.mk.mk.inl\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhba : b \u2264 a\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n[PROOFSTEP]\nsimp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot]\n[GOAL]\ncase intro.intro.right.mk.mk.mk.inr\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nthis : s \u2286 s \\ u \u222a \u22c3 (p : \u2191s \u00d7 \u2191s) (_ : p \u2208 A), s \u2229 T p\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhab : a < b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202Measure.restrict \u03bc (s \u2229 Ioo a b), p x\n[PROOFSTEP]\nexact h a b as bs hab\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nlet T : s \u00d7 s \u2192 Set \u211d := fun p => Ioo p.1 p.2\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nlet u := \u22c3 i : \u21a5s \u00d7 \u21a5s, T i\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nhave hfinite : (s \\ u).Finite := s.finite_diff_iUnion_Ioo'\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nobtain \u27e8A, A_count, hA\u27e9 : \u2203 A : Set (\u21a5s \u00d7 \u21a5s), A.Countable \u2227 \u22c3 i \u2208 A, T i = \u22c3 i : \u21a5s \u00d7 \u21a5s, T i :=\n  isOpen_iUnion_countable _ fun p => isOpen_Ioo\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nhave M : \u2200\u1d50 x \u2202\u03bc, x \u2209 s \\ u := hfinite.countable.ae_not_mem _\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nhave M' : \u2200\u1d50 x \u2202\u03bc, \u2200 (i : \u21a5s \u00d7 \u21a5s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x :=\n  by\n  rw [ae_ball_iff A_count]\n  rintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 -\n  change \u2200\u1d50 x : \u211d \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n  rcases le_or_lt b a with (hba | hab)\n  \u00b7 simp only [Ioo_eq_empty_of_le hba, inter_empty, IsEmpty.forall_iff, eventually_true, mem_empty_iff_false]\n  \u00b7 exact h a b as bs hab\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\n[PROOFSTEP]\nrw [ae_ball_iff A_count]\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\n\u22a2 \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 T i \u2192 p x\n[PROOFSTEP]\nrintro \u27e8\u27e8a, as\u27e9, \u27e8b, bs\u27e9\u27e9 -\n[GOAL]\ncase mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 T ({ val := a, property := as }, { val := b, property := bs }) \u2192 p x\n[PROOFSTEP]\nchange \u2200\u1d50 x : \u211d \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n[GOAL]\ncase mk.mk.mk\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n[PROOFSTEP]\nrcases le_or_lt b a with (hba | hab)\n[GOAL]\ncase mk.mk.mk.inl\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhba : b \u2264 a\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n[PROOFSTEP]\nsimp only [Ioo_eq_empty_of_le hba, inter_empty, IsEmpty.forall_iff, eventually_true, mem_empty_iff_false]\n[GOAL]\ncase mk.mk.mk.inr\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\na : \u211d\nas : a \u2208 s\nb : \u211d\nbs : b \u2208 s\nhab : a < b\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\n[PROOFSTEP]\nexact h a b as bs hab\n[GOAL]\ncase intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\n\u22a2 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2192 p x\n[PROOFSTEP]\nfilter_upwards [M, M'] with x hx h'x\n[GOAL]\ncase h\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\n\u22a2 x \u2208 s \u2192 p x\n[PROOFSTEP]\nintro xs\n[GOAL]\ncase h\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\nxs : x \u2208 s\n\u22a2 p x\n[PROOFSTEP]\nby_cases Hx : x \u2208 \u22c3 i : \u21a5s \u00d7 \u21a5s, T i\n[GOAL]\ncase pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\nxs : x \u2208 s\nHx : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 p x\n[PROOFSTEP]\nrw [\u2190 hA] at Hx \n[GOAL]\ncase pos\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\nxs : x \u2208 s\nHx : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\n\u22a2 p x\n[PROOFSTEP]\nobtain \u27e8p, pA, xp\u27e9 : \u2203 p : \u21a5s \u00d7 \u21a5s, p \u2208 A \u2227 x \u2208 T p := by\n  simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx\n[GOAL]\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\nxs : x \u2208 s\nHx : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\n\u22a2 \u2203 p, p \u2208 A \u2227 x \u2208 T p\n[PROOFSTEP]\nsimpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx\n[GOAL]\ncase pos.intro.intro\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np\u271d : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p\u271d x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p\u271d x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p\u271d x\nxs : x \u2208 s\nHx : x \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i\np : \u2191s \u00d7 \u2191s\npA : p \u2208 A\nxp : x \u2208 T p\n\u22a2 p\u271d x\n[PROOFSTEP]\napply h'x p pA \u27e8xs, xp\u27e9\n[GOAL]\ncase neg\n\u03bc : Measure \u211d\ninst\u271d : NoAtoms \u03bc\ns : Set \u211d\np : \u211d \u2192 Prop\nh : \u2200 (a b : \u211d), a \u2208 s \u2192 b \u2208 s \u2192 a < b \u2192 \u2200\u1d50 (x : \u211d) \u2202\u03bc, x \u2208 s \u2229 Ioo a b \u2192 p x\nT : \u2191s \u00d7 \u2191s \u2192 Set \u211d := fun p => Ioo \u2191p.fst \u2191p.snd\nu : Set \u211d := \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nhfinite : Set.Finite (s \\ u)\nA : Set (\u2191s \u00d7 \u2191s)\nA_count : Set.Countable A\nhA : \u22c3 (i : \u2191s \u00d7 \u2191s) (_ : i \u2208 A), T i = \u22c3 (i : \u2191s \u00d7 \u2191s), T i\nM : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u00acx \u2208 s \\ u\nM' : \u2200\u1d50 (x : \u211d) \u2202\u03bc, \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 T i \u2192 p x\nx : \u211d\nhx : \u00acx \u2208 s \\ \u22c3 (i : \u2191s \u00d7 \u2191s), Ioo \u2191i.fst \u2191i.snd\nh'x : \u2200 (i : \u2191s \u00d7 \u2191s), i \u2208 A \u2192 x \u2208 s \u2229 Ioo \u2191i.fst \u2191i.snd \u2192 p x\nxs : x \u2208 s\nHx : \u00acx \u2208 \u22c3 (i : \u2191s \u00d7 \u2191s), T i\n\u22a2 p x\n[PROOFSTEP]\nexact False.elim (hx \u27e8xs, Hx\u27e9)\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Lebesgue.Basic", "llama_tokens": 49437, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.6859494550081926, "lm_q1q2_score": 0.5386819930732148}}
{"text": "[GOAL]\nS : Type u_1\ninst\u271d : Semigroup S\na b x y z x' y' : S\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 SemiconjBy a (x * x') (y * y')\n[PROOFSTEP]\nunfold SemiconjBy\n[GOAL]\nS : Type u_1\ninst\u271d : Semigroup S\na b x y z x' y' : S\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 a * (x * x') = y * y' * a\n[PROOFSTEP]\nrw [\u2190 mul_assoc, h.eq, mul_assoc, h'.eq, \u2190 mul_assoc]\n[GOAL]\nS : Type u_1\ninst\u271d : Semigroup S\na b x y z x' y' : S\nha : SemiconjBy a y z\nhb : SemiconjBy b x y\n\u22a2 SemiconjBy (a * b) x z\n[PROOFSTEP]\nunfold SemiconjBy\n[GOAL]\nS : Type u_1\ninst\u271d : Semigroup S\na b x y z x' y' : S\nha : SemiconjBy a y z\nhb : SemiconjBy b x y\n\u22a2 a * b * x = z * (a * b)\n[PROOFSTEP]\nrw [mul_assoc, hb.eq, \u2190 mul_assoc, ha.eq, mul_assoc]\n[GOAL]\nM : Type u_1\ninst\u271d : MulOneClass M\na : M\n\u22a2 SemiconjBy a 1 1\n[PROOFSTEP]\nrw [SemiconjBy, mul_one, one_mul]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\na : M\nx y : M\u02e3\nh : SemiconjBy a \u2191x \u2191y\n\u22a2 a * \u2191x\u207b\u00b9 = \u2191y\u207b\u00b9 * (\u2191y * a) * \u2191x\u207b\u00b9\n[PROOFSTEP]\nrw [Units.inv_mul_cancel_left]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\na : M\nx y : M\u02e3\nh : SemiconjBy a \u2191x \u2191y\n\u22a2 \u2191y\u207b\u00b9 * (\u2191y * a) * \u2191x\u207b\u00b9 = \u2191y\u207b\u00b9 * a\n[PROOFSTEP]\nrw [\u2190 h.eq, mul_assoc, Units.mul_inv_cancel_right]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\na : M\u02e3\nx y : M\nh : SemiconjBy (\u2191a) x y\n\u22a2 \u2191a\u207b\u00b9 * y = \u2191a\u207b\u00b9 * (y * \u2191a * \u2191a\u207b\u00b9)\n[PROOFSTEP]\nrw [Units.mul_inv_cancel_right]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\na : M\u02e3\nx y : M\nh : SemiconjBy (\u2191a) x y\n\u22a2 \u2191a\u207b\u00b9 * (y * \u2191a * \u2191a\u207b\u00b9) = x * \u2191a\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 h.eq, \u2190 mul_assoc, Units.inv_mul_cancel_left]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ n) (y ^ n)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : Monoid M\na x y : M\nh : SemiconjBy a x y\n\u22a2 SemiconjBy a (x ^ Nat.zero) (y ^ Nat.zero)\n[PROOFSTEP]\nrw [pow_zero, pow_zero]\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : Monoid M\na x y : M\nh : SemiconjBy a x y\n\u22a2 SemiconjBy a 1 1\n[PROOFSTEP]\nexact SemiconjBy.one_right _\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : \u2115\nih : SemiconjBy a (x ^ n) (y ^ n)\n\u22a2 SemiconjBy a (x ^ Nat.succ n) (y ^ Nat.succ n)\n[PROOFSTEP]\nrw [pow_succ, pow_succ]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : Monoid M\na x y : M\nh : SemiconjBy a x y\nn : \u2115\nih : SemiconjBy a (x ^ n) (y ^ n)\n\u22a2 SemiconjBy a (x * x ^ n) (y * y ^ n)\n[PROOFSTEP]\nexact h.mul_right ih\n[GOAL]\nG : Type u_1\ninst\u271d : DivisionMonoid G\na x y : G\n\u22a2 (a\u207b\u00b9 * x\u207b\u00b9)\u207b\u00b9 = (y\u207b\u00b9 * a\u207b\u00b9)\u207b\u00b9 \u2194 SemiconjBy a y x\n[PROOFSTEP]\nrw [mul_inv_rev, mul_inv_rev, inv_inv, inv_inv, inv_inv, eq_comm, SemiconjBy]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na\u271d x\u271d y a x : G\n\u22a2 SemiconjBy a x (a * x * a\u207b\u00b9)\n[PROOFSTEP]\nunfold SemiconjBy\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\na\u271d x\u271d y a x : G\n\u22a2 a * x = a * x * a\u207b\u00b9 * a\n[PROOFSTEP]\nrw [mul_assoc, inv_mul_self, mul_one]\n[GOAL]\nM : Type u_1\ninst\u271d : CancelCommMonoid M\na x y : M\nh : x = y\n\u22a2 SemiconjBy a x y\n[PROOFSTEP]\nrw [h, SemiconjBy, mul_comm]\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nu : M\u02e3\nx : M\n\u22a2 SemiconjBy (\u2191u) x (\u2191u * x * \u2191u\u207b\u00b9)\n[PROOFSTEP]\nunfold SemiconjBy\n[GOAL]\nM : Type u_1\ninst\u271d : Monoid M\nu : M\u02e3\nx : M\n\u22a2 \u2191u * x = \u2191u * x * \u2191u\u207b\u00b9 * \u2191u\n[PROOFSTEP]\nrw [Units.inv_mul_cancel_right]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Group.Semiconj", "llama_tokens": 1722, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.7217432003123989, "lm_q1q2_score": 0.538520744707885}}
{"text": "[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 \u2191(pointReflection R (midpoint R x y)) x = y\n[PROOFSTEP]\nrw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, \u2190 add_smul, \u2190 two_mul, mul_invOf_self,\n  one_smul, vsub_vadd]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 \u2191(pointReflection (midpoint R x y)) x = y\n[PROOFSTEP]\nrw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, \u2190 add_smul, \u2190 two_mul, mul_invOf_self,\n  one_smul, vsub_vadd]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 midpoint R x y = midpoint R y x\n[PROOFSTEP]\nrw [midpoint, \u2190 lineMap_apply_one_sub, one_sub_invOf_two, midpoint]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 \u2191(pointReflection R (midpoint R x y)) y = x\n[PROOFSTEP]\nrw [midpoint_comm, AffineEquiv.pointReflection_midpoint_left]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 \u2191(pointReflection (midpoint R x y)) y = x\n[PROOFSTEP]\nrw [midpoint_comm, Equiv.pointReflection_midpoint_left]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y z p\u2081 p\u2082 : P\n\u22a2 midpoint R p\u2081 p\u2082 -\u1d65 p\u2082 = \u215f2 \u2022 (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [midpoint_comm, midpoint_vsub_left]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y z p\u2081 p\u2082 : P\n\u22a2 p\u2082 -\u1d65 midpoint R p\u2081 p\u2082 = \u215f2 \u2022 (p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [midpoint_comm, left_vsub_midpoint]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y z p\u2081 p\u2082 p : P\n\u22a2 midpoint R p\u2081 p\u2082 -\u1d65 p = \u215f2 \u2022 (p\u2081 -\u1d65 p) + \u215f2 \u2022 (p\u2082 -\u1d65 p)\n[PROOFSTEP]\nrw [\u2190 vsub_sub_vsub_cancel_right p\u2081 p p\u2082, smul_sub, sub_eq_add_neg, \u2190 smul_neg, neg_vsub_eq_vsub_rev, add_assoc,\n  invOf_two_smul_add_invOf_two_smul, \u2190 vadd_vsub_assoc, midpoint_comm, midpoint, lineMap_apply]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx y z p\u2081 p\u2082 p : P\n\u22a2 p -\u1d65 midpoint R p\u2081 p\u2082 = \u215f2 \u2022 (p -\u1d65 p\u2081) + \u215f2 \u2022 (p -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev, midpoint_vsub, neg_add, \u2190 smul_neg, \u2190 smul_neg, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 midpoint R x y = x \u2194 x = y\n[PROOFSTEP]\nrw [midpoint_eq_iff, pointReflection_self]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 x = midpoint R x y \u2194 x = y\n[PROOFSTEP]\nrw [eq_comm, midpoint_eq_left_iff]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 midpoint R x y = y \u2194 x = y\n[PROOFSTEP]\nrw [midpoint_comm, midpoint_eq_left_iff, eq_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x y : P\n\u22a2 y = midpoint R x y \u2194 x = y\n[PROOFSTEP]\nrw [eq_comm, midpoint_eq_right_iff]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z x x' y y' : P\n\u22a2 midpoint R x y = midpoint R x' y' \u2194 x -\u1d65 x' = y' -\u1d65 y\n[PROOFSTEP]\nrw [\u2190 @vsub_eq_zero_iff_eq V, midpoint_vsub_midpoint, midpoint_eq_iff, pointReflection_apply, vsub_eq_sub, zero_sub,\n  vadd_eq_add, add_zero, neg_eq_iff_eq_neg, neg_vsub_eq_vsub_rev]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x y +\u1d65 midpoint R x y = midpoint R x y +\u1d65 midpoint R y x\n[PROOFSTEP]\nrw [midpoint_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x y +\u1d65 midpoint R y x = x + y\n[PROOFSTEP]\nrw [midpoint_vadd_midpoint, vadd_eq_add, vadd_eq_add, add_comm, midpoint_self]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 0 -\u1d65 x = y -\u1d65 (x + y)\n[PROOFSTEP]\nsimp [sub_add_eq_sub_sub_swap]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x y = \u215f2 \u2022 (x + y)\n[PROOFSTEP]\nrw [midpoint_eq_iff, pointReflection_apply, vsub_eq_sub, vadd_eq_add, sub_add_eq_add_sub, \u2190 two_smul R, smul_smul,\n  mul_invOf_self, one_smul, add_sub_cancel']\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y z : P\nx : V\n\u22a2 midpoint R x (-x) = 0\n[PROOFSTEP]\nrw [midpoint_eq_smul_add, add_neg_self, smul_zero]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y z : P\nx : V\n\u22a2 midpoint R (-x) x = 0\n[PROOFSTEP]\nsimpa using midpoint_self_neg R (-x)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R (x - y) (x + y) = x\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 vadd_eq_add, \u2190 vadd_eq_add, \u2190 midpoint_vadd_midpoint]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R x x +\u1d65 midpoint R (-y) y = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R (x + y) (x - y) = x\n[PROOFSTEP]\nrw [midpoint_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\nx\u271d y\u271d z : P\nx y : V\n\u22a2 midpoint R (x - y) (x + y) = x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Ring R'\ninst\u271d\u00b2 : Invertible 2\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module R' F\nf : E \u2192 F\nh0 : f 0 = 0\nhm : \u2200 (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)\nx y : E\n\u22a2 f (x + y) = f 0 + f (x + y)\n[PROOFSTEP]\nrw [h0, zero_add]\n[GOAL]\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Ring R'\ninst\u271d\u00b2 : Invertible 2\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module R' F\nf : E \u2192 F\nh0 : f 0 = 0\nhm : \u2200 (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)\nx y : E\n\u22a2 midpoint R' (f 0) (f (x + y)) + midpoint R' (f 0) (f (x + y)) = f (midpoint R x y) + f (midpoint R x y)\n[PROOFSTEP]\nrw [\u2190 hm, midpoint_zero_add]\n[GOAL]\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Invertible 2\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module R E\ninst\u271d\u00b3 : Ring R'\ninst\u271d\u00b2 : Invertible 2\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module R' F\nf : E \u2192 F\nh0 : f 0 = 0\nhm : \u2200 (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)\nx y : E\n\u22a2 f (midpoint R x y) + f (midpoint R x y) = f x + f y\n[PROOFSTEP]\nrw [hm, midpoint_add_self]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Midpoint", "llama_tokens": 5685, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.5384473278545552}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u22a2 IsComplement' K H\n[PROOFSTEP]\nlet \u03d5 : H \u00d7 K \u2243 K \u00d7 H :=\n  Equiv.mk (fun x => \u27e8x.2\u207b\u00b9, x.1\u207b\u00b9\u27e9) (fun x => \u27e8x.2\u207b\u00b9, x.1\u207b\u00b9\u27e9) (fun x => Prod.ext (inv_inv _) (inv_inv _)) fun x =>\n    Prod.ext (inv_inv _) (inv_inv _)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u22a2 IsComplement' K H\n[PROOFSTEP]\nlet \u03c8 : G \u2243 G := Equiv.mk (fun g : G => g\u207b\u00b9) (fun g : G => g\u207b\u00b9) inv_inv inv_inv\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u03c8 : G \u2243 G :=\n  { toFun := fun g => g\u207b\u00b9, invFun := fun g => g\u207b\u00b9, left_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a),\n    right_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a) }\n\u22a2 IsComplement' K H\n[PROOFSTEP]\nsuffices hf : (\u03c8 \u2218 fun x : H \u00d7 K => x.1.1 * x.2.1) = (fun x : K \u00d7 H => x.1.1 * x.2.1) \u2218 \u03d5\n  by\n  rw [isComplement'_def, IsComplement, \u2190 Equiv.bijective_comp \u03d5]\n  apply (congr_arg Function.Bijective hf).mp\n  rwa [\u03c8.comp_bijective]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u03c8 : G \u2243 G :=\n  { toFun := fun g => g\u207b\u00b9, invFun := fun g => g\u207b\u00b9, left_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a),\n    right_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a) }\nhf : (\u2191\u03c8 \u2218 fun x => \u2191x.fst * \u2191x.snd) = (fun x => \u2191x.fst * \u2191x.snd) \u2218 \u2191\u03d5\n\u22a2 IsComplement' K H\n[PROOFSTEP]\nrw [isComplement'_def, IsComplement, \u2190 Equiv.bijective_comp \u03d5]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u03c8 : G \u2243 G :=\n  { toFun := fun g => g\u207b\u00b9, invFun := fun g => g\u207b\u00b9, left_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a),\n    right_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a) }\nhf : (\u2191\u03c8 \u2218 fun x => \u2191x.fst * \u2191x.snd) = (fun x => \u2191x.fst * \u2191x.snd) \u2218 \u2191\u03d5\n\u22a2 Function.Bijective ((fun x => \u2191x.fst * \u2191x.snd) \u2218 \u2191\u03d5)\n[PROOFSTEP]\napply (congr_arg Function.Bijective hf).mp\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u03c8 : G \u2243 G :=\n  { toFun := fun g => g\u207b\u00b9, invFun := fun g => g\u207b\u00b9, left_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a),\n    right_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a) }\nhf : (\u2191\u03c8 \u2218 fun x => \u2191x.fst * \u2191x.snd) = (fun x => \u2191x.fst * \u2191x.snd) \u2218 \u2191\u03d5\n\u22a2 Function.Bijective (\u2191\u03c8 \u2218 fun x => \u2191x.fst * \u2191x.snd)\n[PROOFSTEP]\nrwa [\u03c8.comp_bijective]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u03d5 : { x // x \u2208 H } \u00d7 { x // x \u2208 K } \u2243 { x // x \u2208 K } \u00d7 { x // x \u2208 H } :=\n  { toFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9), invFun := fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9),\n    left_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 H } \u00d7 { x // x \u2208 K }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x),\n    right_inv :=\n      (_ :\n        \u2200 (x : { x // x \u2208 K } \u00d7 { x // x \u2208 H }), (fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) ((fun x => (x.snd\u207b\u00b9, x.fst\u207b\u00b9)) x) = x) }\n\u03c8 : G \u2243 G :=\n  { toFun := fun g => g\u207b\u00b9, invFun := fun g => g\u207b\u00b9, left_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a),\n    right_inv := (_ : \u2200 (a : G), a\u207b\u00b9\u207b\u00b9 = a) }\n\u22a2 (\u2191\u03c8 \u2218 fun x => \u2191x.fst * \u2191x.snd) = (fun x => \u2191x.fst * \u2191x.snd) \u2218 \u2191\u03d5\n[PROOFSTEP]\nexact funext fun x => mul_inv_rev _ _\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 IsComplement {g} S \u2194 S = \u22a4\n[PROOFSTEP]\nrefine' \u27e8fun h => top_le_iff.mp fun x _ => _, fun h => (congr_arg _ h).mpr isComplement_singleton_top\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement {g} S\nx : G\nx\u271d : x \u2208 \u22a4\n\u22a2 x \u2208 S\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8z, rfl : z = g\u27e9, y, _\u27e9, hy\u27e9 := h.2 (g * x)\n[GOAL]\ncase intro.mk.mk.mk\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nx : G\nx\u271d : x \u2208 \u22a4\nz : G\nh : IsComplement {z} S\ny : G\nproperty\u271d : y \u2208 S\nhy : (fun x => \u2191x.fst * \u2191x.snd) ({ val := z, property := (_ : z = z) }, { val := y, property := property\u271d }) = z * x\n\u22a2 x \u2208 S\n[PROOFSTEP]\nrwa [\u2190 mul_left_cancel hy]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 IsComplement S {g} \u2194 S = \u22a4\n[PROOFSTEP]\nrefine' \u27e8fun h => top_le_iff.mp fun x _ => _, fun h => h \u25b8 isComplement_top_singleton\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\n\u22a2 x \u2208 S\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := h.2 (x * g)\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * g\n\u22a2 x \u2208 S\n[PROOFSTEP]\nconv_rhs at hy => rw [\u2190 show y.2.1 = g from y.2.2]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * g\n| x * g\n[PROOFSTEP]\nrw [\u2190 show y.2.1 = g from y.2.2]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * g\n| x * g\n[PROOFSTEP]\nrw [\u2190 show y.2.1 = g from y.2.2]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * g\n| x * g\n[PROOFSTEP]\nrw [\u2190 show y.2.1 = g from y.2.2]\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * \u2191y.snd\n\u22a2 x \u2208 S\n[PROOFSTEP]\nrw [\u2190 mul_right_cancel hy]\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx\u271d : x \u2208 \u22a4\ny : \u2191S \u00d7 \u2191{g}\nhy : (fun x => \u2191x.fst * \u2191x.snd) y = x * \u2191y.snd\n\u22a2 \u2191y.fst \u2208 S\n[PROOFSTEP]\nexact y.1.2\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 IsComplement \u22a4 S \u2194 \u2203 g, S = {g}\n[PROOFSTEP]\nrefine' \u27e8fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr \u27e8_, fun a ha b hb => _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement \u22a4 S\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nobtain \u27e8a, _\u27e9 := h.2 1\n[GOAL]\ncase refine'_1.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement \u22a4 S\na : \u2191\u22a4 \u00d7 \u2191S\nh\u271d : (fun x => \u2191x.fst * \u2191x.snd) a = 1\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nexact \u27e8a.2.1, a.2.2\u27e9\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement \u22a4 S\na : G\nha : a \u2208 S\nb : G\nhb : b \u2208 S\n\u22a2 a = b\n[PROOFSTEP]\nhave : (\u27e8\u27e8_, mem_top a\u207b\u00b9\u27e9, \u27e8a, ha\u27e9\u27e9 : (\u22a4 : Set G) \u00d7 S) = \u27e8\u27e8_, mem_top b\u207b\u00b9\u27e9, \u27e8b, hb\u27e9\u27e9 :=\n  h.1 ((inv_mul_self a).trans (inv_mul_self b).symm)\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement \u22a4 S\na : G\nha : a \u2208 S\nb : G\nhb : b \u2208 S\nthis :\n  ({ val := a\u207b\u00b9, property := (_ : a\u207b\u00b9 \u2208 \u22a4) }, { val := a, property := ha }) =\n    ({ val := b\u207b\u00b9, property := (_ : b\u207b\u00b9 \u2208 \u22a4) }, { val := b, property := hb })\n\u22a2 a = b\n[PROOFSTEP]\nexact Subtype.ext_iff.mp (Prod.ext_iff.mp this).2\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2203 g, S = {g}) \u2192 IsComplement \u22a4 S\n[PROOFSTEP]\nrintro \u27e8g, rfl\u27e9\n[GOAL]\ncase refine'_3.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nT : Set G\ng : G\n\u22a2 IsComplement \u22a4 {g}\n[PROOFSTEP]\nexact isComplement_top_singleton\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 IsComplement S \u22a4 \u2194 \u2203 g, S = {g}\n[PROOFSTEP]\nrefine' \u27e8fun h => Set.exists_eq_singleton_iff_nonempty_subsingleton.mpr \u27e8_, fun a ha b hb => _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S \u22a4\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nobtain \u27e8a, _\u27e9 := h.2 1\n[GOAL]\ncase refine'_1.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S \u22a4\na : \u2191S \u00d7 \u2191\u22a4\nh\u271d : (fun x => \u2191x.fst * \u2191x.snd) a = 1\n\u22a2 Set.Nonempty S\n[PROOFSTEP]\nexact \u27e8a.1.1, a.1.2\u27e9\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S \u22a4\na : G\nha : a \u2208 S\nb : G\nhb : b \u2208 S\n\u22a2 a = b\n[PROOFSTEP]\nhave : (\u27e8\u27e8a, ha\u27e9, \u27e8_, mem_top a\u207b\u00b9\u27e9\u27e9 : S \u00d7 (\u22a4 : Set G)) = \u27e8\u27e8b, hb\u27e9, \u27e8_, mem_top b\u207b\u00b9\u27e9\u27e9 :=\n  h.1 ((mul_inv_self a).trans (mul_inv_self b).symm)\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement S \u22a4\na : G\nha : a \u2208 S\nb : G\nhb : b \u2208 S\nthis :\n  ({ val := a, property := ha }, { val := a\u207b\u00b9, property := (_ : a\u207b\u00b9 \u2208 \u22a4) }) =\n    ({ val := b, property := hb }, { val := b\u207b\u00b9, property := (_ : b\u207b\u00b9 \u2208 \u22a4) })\n\u22a2 a = b\n[PROOFSTEP]\nexact Subtype.ext_iff.mp (Prod.ext_iff.mp this).1\n[GOAL]\ncase refine'_3\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2203 g, S = {g}) \u2192 IsComplement S \u22a4\n[PROOFSTEP]\nrintro \u27e8g, rfl\u27e9\n[GOAL]\ncase refine'_3.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nT : Set G\ng : G\n\u22a2 IsComplement {g} \u22a4\n[PROOFSTEP]\nexact isComplement_singleton_top\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 S \u2208 leftTransversals T \u2194 \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\n[PROOFSTEP]\nrw [leftTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g) \u2194 \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\n[PROOFSTEP]\nrefine' \u27e8fun h g => _, fun h g => _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g\ng : G\n\u22a2 \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\n[PROOFSTEP]\nobtain \u27e8x, h1, h2\u27e9 := h g\n[GOAL]\ncase refine'_1.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g\ng : G\nx : \u2191S \u00d7 \u2191T\nh1 : \u2191x.fst * \u2191x.snd = g\nh2 : \u2200 (y : \u2191S \u00d7 \u2191T), (fun x => \u2191x.fst * \u2191x.snd = g) y \u2192 y = x\n\u22a2 \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\n[PROOFSTEP]\nexact\n  \u27e8x.1, (congr_arg (\u00b7 \u2208 T) (eq_inv_mul_of_mul_eq h1)).mp x.2.2, fun y hy =>\n    (Prod.ext_iff.mp (h2 \u27e8y, (\u2191y)\u207b\u00b9 * g, hy\u27e9 (mul_inv_cancel_left (\u2191y) g))).1\u27e9\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\ng : G\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nobtain \u27e8x, h1, h2\u27e9 := h g\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\ng : G\nx : \u2191S\nh1 : (\u2191x)\u207b\u00b9 * g \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => (\u2191s)\u207b\u00b9 * g \u2208 T) y \u2192 y = x\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nrefine' \u27e8\u27e8x, (\u2191x)\u207b\u00b9 * g, h1\u27e9, mul_inv_cancel_left (\u2191x) g, fun y hy => _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\ng : G\nx : \u2191S\nh1 : (\u2191x)\u207b\u00b9 * g \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => (\u2191s)\u207b\u00b9 * g \u2208 T) y \u2192 y = x\ny : \u2191S \u00d7 \u2191T\nhy : (fun x => \u2191x.fst * \u2191x.snd = g) y\n\u22a2 y = (x, { val := (\u2191x)\u207b\u00b9 * g, property := h1 })\n[PROOFSTEP]\nhave hf := h2 y.1 ((congr_arg (\u00b7 \u2208 T) (eq_inv_mul_of_mul_eq hy)).mp y.2.2)\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 T\ng : G\nx : \u2191S\nh1 : (\u2191x)\u207b\u00b9 * g \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => (\u2191s)\u207b\u00b9 * g \u2208 T) y \u2192 y = x\ny : \u2191S \u00d7 \u2191T\nhy : (fun x => \u2191x.fst * \u2191x.snd = g) y\nhf : y.fst = x\n\u22a2 y = (x, { val := (\u2191x)\u207b\u00b9 * g, property := h1 })\n[PROOFSTEP]\nexact Prod.ext hf (Subtype.ext (eq_inv_mul_of_mul_eq (hf \u25b8 hy)))\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 S \u2208 rightTransversals T \u2194 \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\n[PROOFSTEP]\nrw [rightTransversals, Set.mem_setOf_eq, isComplement_iff_existsUnique]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g) \u2194 \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\n[PROOFSTEP]\nrefine' \u27e8fun h g => _, fun h g => _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g\ng : G\n\u22a2 \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\n[PROOFSTEP]\nobtain \u27e8x, h1, h2\u27e9 := h g\n[GOAL]\ncase refine'_1.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! x, \u2191x.fst * \u2191x.snd = g\ng : G\nx : \u2191T \u00d7 \u2191S\nh1 : \u2191x.fst * \u2191x.snd = g\nh2 : \u2200 (y : \u2191T \u00d7 \u2191S), (fun x => \u2191x.fst * \u2191x.snd = g) y \u2192 y = x\n\u22a2 \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\n[PROOFSTEP]\nexact\n  \u27e8x.2, (congr_arg (\u00b7 \u2208 T) (eq_mul_inv_of_mul_eq h1)).mp x.1.2, fun y hy =>\n    (Prod.ext_iff.mp (h2 \u27e8\u27e8g * (\u2191y)\u207b\u00b9, hy\u27e9, y\u27e9 (inv_mul_cancel_right g y))).2\u27e9\n[GOAL]\ncase refine'_2\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\ng : G\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nobtain \u27e8x, h1, h2\u27e9 := h g\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\ng : G\nx : \u2191S\nh1 : g * (\u2191x)\u207b\u00b9 \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => g * (\u2191s)\u207b\u00b9 \u2208 T) y \u2192 y = x\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8g * (\u2191x)\u207b\u00b9, h1\u27e9, x\u27e9, inv_mul_cancel_right g x, fun y hy => _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\ng : G\nx : \u2191S\nh1 : g * (\u2191x)\u207b\u00b9 \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => g * (\u2191s)\u207b\u00b9 \u2208 T) y \u2192 y = x\ny : \u2191T \u00d7 \u2191S\nhy : (fun x => \u2191x.fst * \u2191x.snd = g) y\n\u22a2 y = ({ val := g * (\u2191x)\u207b\u00b9, property := h1 }, x)\n[PROOFSTEP]\nhave hf := h2 y.2 ((congr_arg (\u00b7 \u2208 T) (eq_mul_inv_of_mul_eq hy)).mp y.1.2)\n[GOAL]\ncase refine'_2.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : \u2200 (g : G), \u2203! s, g * (\u2191s)\u207b\u00b9 \u2208 T\ng : G\nx : \u2191S\nh1 : g * (\u2191x)\u207b\u00b9 \u2208 T\nh2 : \u2200 (y : \u2191S), (fun s => g * (\u2191s)\u207b\u00b9 \u2208 T) y \u2192 y = x\ny : \u2191T \u00d7 \u2191S\nhy : (fun x => \u2191x.fst * \u2191x.snd = g) y\nhf : y.snd = x\n\u22a2 y = ({ val := g * (\u2191x)\u207b\u00b9, property := h1 }, x)\n[PROOFSTEP]\nexact Prod.ext (Subtype.ext (eq_mul_inv_of_mul_eq (hf \u25b8 hy))) hf\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 S \u2208 leftTransversals \u2191H \u2194 \u2200 (q : Quotient (QuotientGroup.leftRel H)), \u2203! s, Quotient.mk'' \u2191s = q\n[PROOFSTEP]\nsimp_rw [mem_leftTransversals_iff_existsUnique_inv_mul_mem, SetLike.mem_coe, \u2190 QuotientGroup.eq']\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! s, \u2191\u2191s = \u2191g) \u2194 \u2200 (q : Quotient (QuotientGroup.leftRel H)), \u2203! s, Quotient.mk'' \u2191s = q\n[PROOFSTEP]\nexact \u27e8fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 S \u2208 rightTransversals \u2191H \u2194 \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q\n[PROOFSTEP]\nsimp_rw [mem_rightTransversals_iff_existsUnique_mul_inv_mem, SetLike.mem_coe, \u2190 QuotientGroup.rightRel_apply, \u2190\n  Quotient.eq'']\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\n\u22a2 (\u2200 (g : G), \u2203! s, Quotient.mk'' \u2191s = Quotient.mk'' g) \u2194\n    \u2200 (q : Quotient (QuotientGroup.rightRel H)), \u2203! s, Quotient.mk'' \u2191s = q\n[PROOFSTEP]\nexact \u27e8fun h q => Quotient.inductionOn' q h, fun h g => h (Quotient.mk'' g)\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : G \u29f8 H \u2192 G\nhf : \u2200 (q : G \u29f8 H), \u2191(f q) = q\n\u22a2 Function.Injective (Set.restrict (Set.range f) Quotient.mk'')\n[PROOFSTEP]\nrintro \u27e8-, q\u2081, rfl\u27e9 \u27e8-, q\u2082, rfl\u27e9 h\n[GOAL]\ncase mk.intro.mk.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : G \u29f8 H \u2192 G\nhf : \u2200 (q : G \u29f8 H), \u2191(f q) = q\nq\u2081 q\u2082 : G \u29f8 H\nh :\n  Set.restrict (Set.range f) Quotient.mk'' { val := f q\u2081, property := (_ : \u2203 y, f y = f q\u2081) } =\n    Set.restrict (Set.range f) Quotient.mk'' { val := f q\u2082, property := (_ : \u2203 y, f y = f q\u2082) }\n\u22a2 { val := f q\u2081, property := (_ : \u2203 y, f y = f q\u2081) } = { val := f q\u2082, property := (_ : \u2203 y, f y = f q\u2082) }\n[PROOFSTEP]\nexact Subtype.ext $ congr_arg f $ ((hf q\u2081).symm.trans h).trans (hf q\u2082)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\n\u22a2 Function.Injective (Set.restrict (Set.range f) Quotient.mk'')\n[PROOFSTEP]\nrintro \u27e8-, q\u2081, rfl\u27e9 \u27e8-, q\u2082, rfl\u27e9 h\n[GOAL]\ncase mk.intro.mk.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq\u2081 q\u2082 : Quotient (QuotientGroup.rightRel H)\nh :\n  Set.restrict (Set.range f) Quotient.mk'' { val := f q\u2081, property := (_ : \u2203 y, f y = f q\u2081) } =\n    Set.restrict (Set.range f) Quotient.mk'' { val := f q\u2082, property := (_ : \u2203 y, f y = f q\u2082) }\n\u22a2 { val := f q\u2081, property := (_ : \u2203 y, f y = f q\u2081) } = { val := f q\u2082, property := (_ : \u2203 y, f y = f q\u2082) }\n[PROOFSTEP]\nexact Subtype.ext $ congr_arg f $ ((hf q\u2081).symm.trans h).trans (hf q\u2082)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 \u2203 S, S \u2208 leftTransversals \u2191H \u2227 g \u2208 S\n[PROOFSTEP]\nclassical\nrefine'\n  \u27e8Set.range (Function.update Quotient.out' _ g), range_mem_leftTransversals fun q => _, Quotient.mk'' g,\n    Function.update_same (Quotient.mk'' g) g Quotient.out'\u27e9\nby_cases hq : q = Quotient.mk'' g\n\u00b7 exact hq.symm \u25b8 congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out')\n\u00b7 refine' (Function.update_noteq _ g Quotient.out') \u25b8 q.out_eq'\n  exact hq\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 \u2203 S, S \u2208 leftTransversals \u2191H \u2227 g \u2208 S\n[PROOFSTEP]\nrefine'\n  \u27e8Set.range (Function.update Quotient.out' _ g), range_mem_leftTransversals fun q => _, Quotient.mk'' g,\n    Function.update_same (Quotient.mk'' g) g Quotient.out'\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : G \u29f8 H\n\u22a2 \u2191(Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nby_cases hq : q = Quotient.mk'' g\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : G \u29f8 H\nhq : q = Quotient.mk'' g\n\u22a2 \u2191(Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nexact hq.symm \u25b8 congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out')\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : G \u29f8 H\nhq : \u00acq = Quotient.mk'' g\n\u22a2 \u2191(Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nrefine' (Function.update_noteq _ g Quotient.out') \u25b8 q.out_eq'\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : G \u29f8 H\nhq : \u00acq = Quotient.mk'' g\n\u22a2 q \u2260 Quotient.mk'' g\n[PROOFSTEP]\nexact hq\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 \u2203 S, S \u2208 rightTransversals \u2191H \u2227 g \u2208 S\n[PROOFSTEP]\nclassical\nrefine'\n  \u27e8Set.range (Function.update Quotient.out' _ g), range_mem_rightTransversals fun q => _, Quotient.mk'' g,\n    Function.update_same (Quotient.mk'' g) g Quotient.out'\u27e9\nby_cases hq : q = Quotient.mk'' g\n\u00b7 exact hq.symm \u25b8 congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out')\n\u00b7 exact Eq.trans (congr_arg _ (Function.update_noteq hq g Quotient.out')) q.out_eq'\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\n\u22a2 \u2203 S, S \u2208 rightTransversals \u2191H \u2227 g \u2208 S\n[PROOFSTEP]\nrefine'\n  \u27e8Set.range (Function.update Quotient.out' _ g), range_mem_rightTransversals fun q => _, Quotient.mk'' g,\n    Function.update_same (Quotient.mk'' g) g Quotient.out'\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 Quotient.mk'' (Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nby_cases hq : q = Quotient.mk'' g\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : Quotient (QuotientGroup.rightRel H)\nhq : q = Quotient.mk'' g\n\u22a2 Quotient.mk'' (Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nexact hq.symm \u25b8 congr_arg _ (Function.update_same (Quotient.mk'' g) g Quotient.out')\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\ng : G\nq : Quotient (QuotientGroup.rightRel H)\nhq : \u00acq = Quotient.mk'' g\n\u22a2 Quotient.mk'' (Function.update Quotient.out' (Quotient.mk'' g) g q) = q\n[PROOFSTEP]\nexact Eq.trans (congr_arg _ (Function.update_noteq hq g Quotient.out')) q.out_eq'\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : G \u29f8 H \u2192 G\nhf : \u2200 (q : G \u29f8 H), \u2191(f q) = q\nq : G \u29f8 H\n\u22a2 \u2191(\u2191(toEquiv (_ : (Set.range fun q => f q) \u2208 leftTransversals \u2191H)) q) = f q\n[PROOFSTEP]\nrefine' (Subtype.ext_iff.mp _).trans (Subtype.coe_mk (f q) \u27e8q, rfl\u27e9)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : G \u29f8 H \u2192 G\nhf : \u2200 (q : G \u29f8 H), \u2191(f q) = q\nq : G \u29f8 H\n\u22a2 \u2191(toEquiv (_ : (Set.range fun q => f q) \u2208 leftTransversals \u2191H)) q =\n    { val := f q, property := (_ : \u2203 y, (fun q => f q) y = f q) }\n[PROOFSTEP]\nexact (toEquiv (range_mem_leftTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nhS : S \u2208 leftTransversals \u2191H\ng : G\n\u22a2 ((\u2191(toFun hS g))\u207b\u00b9 * g)\u207b\u00b9 = g\u207b\u00b9 * \u2191(toFun hS g)\n[PROOFSTEP]\nrw [mul_inv_rev, inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 \u2191(\u2191(toEquiv (_ : (Set.range fun q => f q) \u2208 rightTransversals \u2191H)) q) = f q\n[PROOFSTEP]\nrefine' (Subtype.ext_iff.mp _).trans (Subtype.coe_mk (f q) \u27e8q, rfl\u27e9)\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) \u2192 G\nhf : \u2200 (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n\u22a2 \u2191(toEquiv (_ : (Set.range fun q => f q) \u2208 rightTransversals \u2191H)) q =\n    { val := f q, property := (_ : \u2203 y, (fun q => f q) y = f q) }\n[PROOFSTEP]\nexact (toEquiv (range_mem_rightTransversals hf)).apply_eq_iff_eq_symm_apply.mpr (hf q).symm\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nhS : S \u2208 rightTransversals \u2191H\ng : G\n\u22a2 (g * (\u2191(toFun hS g))\u207b\u00b9)\u207b\u00b9 = \u2191(toFun hS g) * g\u207b\u00b9\n[PROOFSTEP]\nrw [mul_inv_rev, inv_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\n\u22a2 f \u2022 \u2191T \u2208 leftTransversals \u2191H\n[PROOFSTEP]\nrefine' mem_leftTransversals_iff_existsUnique_inv_mul_mem.mpr fun g => _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\n\u22a2 \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 \u2191H\n[PROOFSTEP]\nobtain \u27e8t, ht1, ht2\u27e9 := mem_leftTransversals_iff_existsUnique_inv_mul_mem.mp T.2 (f\u207b\u00b9 \u2022 g)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\n\u22a2 \u2203! s, (\u2191s)\u207b\u00b9 * g \u2208 \u2191H\n[PROOFSTEP]\nrefine' \u27e8\u27e8f \u2022 (t : G), Set.smul_mem_smul_set t.2\u27e9, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\n\u22a2 (fun s => (\u2191s)\u207b\u00b9 * g \u2208 \u2191H) { val := f \u2022 \u2191t, property := (_ : f \u2022 \u2191t \u2208 f \u2022 \u2191T) }\n[PROOFSTEP]\nexact smul_inv_smul f g \u25b8 QuotientAction.inv_mul_mem f ht1\n[GOAL]\ncase intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\n\u22a2 \u2200 (y : \u2191(f \u2022 \u2191T)), (fun s => (\u2191s)\u207b\u00b9 * g \u2208 \u2191H) y \u2192 y = { val := f \u2022 \u2191t, property := (_ : f \u2022 \u2191t \u2208 f \u2022 \u2191T) }\n[PROOFSTEP]\nrintro \u27e8-, t', ht', rfl\u27e9 h\n[GOAL]\ncase intro.intro.refine'_2.mk.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\nt' : G\nht' : t' \u2208 \u2191T\nh :\n  (\u2191{ val := (fun x => f \u2022 x) t', property := (_ : \u2203 a, a \u2208 \u2191T \u2227 (fun x => f \u2022 x) a = (fun x => f \u2022 x) t') })\u207b\u00b9 * g \u2208 \u2191H\n\u22a2 { val := (fun x => f \u2022 x) t', property := (_ : \u2203 a, a \u2208 \u2191T \u2227 (fun x => f \u2022 x) a = (fun x => f \u2022 x) t') } =\n    { val := f \u2022 \u2191t, property := (_ : f \u2022 \u2191t \u2208 f \u2022 \u2191T) }\n[PROOFSTEP]\nreplace h := QuotientAction.inv_mul_mem f\u207b\u00b9 h\n[GOAL]\ncase intro.intro.refine'_2.mk.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\nt' : G\nht' : t' \u2208 \u2191T\nh :\n  (f\u207b\u00b9 \u2022 \u2191{ val := (fun x => f \u2022 x) t', property := (_ : \u2203 a, a \u2208 \u2191T \u2227 (fun x => f \u2022 x) a = (fun x => f \u2022 x) t') })\u207b\u00b9 *\n      f\u207b\u00b9 \u2022 g \u2208\n    H\n\u22a2 { val := (fun x => f \u2022 x) t', property := (_ : \u2203 a, a \u2208 \u2191T \u2227 (fun x => f \u2022 x) a = (fun x => f \u2022 x) t') } =\n    { val := f \u2022 \u2191t, property := (_ : f \u2022 \u2191t \u2208 f \u2022 \u2191T) }\n[PROOFSTEP]\nsimp only [Subtype.ext_iff, Subtype.coe_mk, smul_left_cancel_iff, inv_smul_smul] at h \u22a2\n[GOAL]\ncase intro.intro.refine'_2.mk.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\ng : G\nt : \u2191\u2191T\nht1 : (\u2191t)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H\nht2 : \u2200 (y : \u2191\u2191T), (fun s => (\u2191s)\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 \u2191H) y \u2192 y = t\nt' : G\nht' : t' \u2208 \u2191T\nh : t'\u207b\u00b9 * f\u207b\u00b9 \u2022 g \u2208 H\n\u22a2 t' = \u2191t\n[PROOFSTEP]\nexact Subtype.ext_iff.mp (ht2 \u27e8t', ht'\u27e9 h)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\nH K : Subgroup G\nS T\u271d : Set G\nF : Type u_2\ninst\u271d\u00b2 : Group F\ninst\u271d\u00b9 : MulAction F G\ninst\u271d : QuotientAction F H\nf : F\nT : \u2191(leftTransversals \u2191H)\nq : G \u29f8 H\n\u22a2 \u2191(\u2191(toEquiv (_ : \u2191(f \u2022 T) \u2208 leftTransversals \u2191H)) q) = f \u2022 \u2191(\u2191(toEquiv (_ : \u2191T \u2208 leftTransversals \u2191H)) (f\u207b\u00b9 \u2022 q))\n[PROOFSTEP]\nrw [smul_toEquiv, smul_inv_smul]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\n\u22a2 IsCompl H K\n[PROOFSTEP]\nrefine'\n  \u27e8disjoint_iff_inf_le.mpr fun g \u27e8p, q\u27e9 =>\n      let x : H \u00d7 K := \u27e8\u27e8g, p\u27e9, 1\u27e9\n      let y : H \u00d7 K := \u27e81, g, q\u27e9\n      Subtype.ext_iff.mp (Prod.ext_iff.mp (show x = y from h.1 ((mul_one g).trans (one_mul g).symm))).1,\n    codisjoint_iff_le_sup.mpr fun g _ => _\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh : IsComplement' H K\ng : G\nx\u271d : g \u2208 \u22a4\n\u22a2 g \u2208 H \u2294 K\n[PROOFSTEP]\nobtain \u27e8\u27e8h, k\u27e9, rfl\u27e9 := h.2 g\n[GOAL]\ncase intro.mk\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh\u271d : IsComplement' H K\nh : \u2191\u2191H\nk : \u2191\u2191K\nx\u271d : (fun x => \u2191x.fst * \u2191x.snd) (h, k) \u2208 \u22a4\n\u22a2 (fun x => \u2191x.fst * \u2191x.snd) (h, k) \u2208 H \u2294 K\n[PROOFSTEP]\nexact Subgroup.mul_mem_sup h.2 k.2\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh1 : Disjoint H K\nh2 : \u2191H * \u2191K = Set.univ\n\u22a2 IsComplement' H K\n[PROOFSTEP]\nrefine' \u27e8mul_injective_of_disjoint h1, fun g => _\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh1 : Disjoint H K\nh2 : \u2191H * \u2191K = Set.univ\ng : G\n\u22a2 \u2203 a, (fun x => \u2191x.fst * \u2191x.snd) a = g\n[PROOFSTEP]\nobtain \u27e8h, k, hh, hk, hg\u27e9 := Set.eq_univ_iff_forall.mp h2 g\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nh1 : Disjoint H K\nh2 : \u2191H * \u2191K = Set.univ\ng h k : G\nhh : h \u2208 \u2191H\nhk : k \u2208 \u2191K\nhg : (fun x x_1 => x * x_1) h k = g\n\u22a2 \u2203 a, (fun x => \u2191x.fst * \u2191x.snd) a = g\n[PROOFSTEP]\nexact \u27e8(\u27e8h, hh\u27e9, \u27e8k, hk\u27e9), hg\u27e9\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\n\u22a2 IsComplement' H (MulAction.stabilizer G a)\n[PROOFSTEP]\nrefine' isComplement_iff_existsUnique.mpr fun g => _\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\ng : G\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nobtain \u27e8h, hh\u27e9 := h2 g\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\ng : G\nh : { x // x \u2208 H }\nhh : h \u2022 g \u2022 a = a\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nhave hh' : (\u2191h * g) \u2022 a = a := by rwa [mul_smul]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\ng : G\nh : { x // x \u2208 H }\nhh : h \u2022 g \u2022 a = a\n\u22a2 (\u2191h * g) \u2022 a = a\n[PROOFSTEP]\nrwa [mul_smul]\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\ng : G\nh : { x // x \u2208 H }\nhh : h \u2022 g \u2022 a = a\nhh' : (\u2191h * g) \u2022 a = a\n\u22a2 \u2203! x, \u2191x.fst * \u2191x.snd = g\n[PROOFSTEP]\nrefine' \u27e8\u27e8h\u207b\u00b9, h * g, hh'\u27e9, inv_mul_cancel_left (\u2191h) g, _\u27e9\n[GOAL]\ncase intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\ng : G\nh : { x // x \u2208 H }\nhh : h \u2022 g \u2022 a = a\nhh' : (\u2191h * g) \u2022 a = a\n\u22a2 \u2200 (y : \u2191\u2191H \u00d7 \u2191\u2191(MulAction.stabilizer G a)),\n    (fun x => \u2191x.fst * \u2191x.snd = g) y \u2192 y = (h\u207b\u00b9, { val := \u2191h * g, property := hh' })\n[PROOFSTEP]\nrintro \u27e8h', g, hg : g \u2022 a = a\u27e9 rfl\n[GOAL]\ncase intro.mk.mk\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\nh : { x // x \u2208 H }\nh' : \u2191\u2191H\ng : G\nhg : g \u2022 a = a\nhh : h \u2022 (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd) \u2022 a = a\nhh' : (\u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd)) \u2022 a = a\n\u22a2 (h', { val := g, property := hg }) =\n    (h\u207b\u00b9,\n      { val := \u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd),\n        property := hh' })\n[PROOFSTEP]\nspecialize h1 (h * h') (by rwa [mul_smul, smul_def h', \u2190 hg, \u2190 mul_smul, hg])\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh1 : \u2200 (h : { x // x \u2208 H }), h \u2022 a = a \u2192 h = 1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\nh : { x // x \u2208 H }\nh' : \u2191\u2191H\ng : G\nhg : g \u2022 a = a\nhh : h \u2022 (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd) \u2022 a = a\nhh' : (\u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd)) \u2022 a = a\n\u22a2 (h * h') \u2022 a = a\n[PROOFSTEP]\nrwa [mul_smul, smul_def h', \u2190 hg, \u2190 mul_smul, hg]\n[GOAL]\ncase intro.mk.mk\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\nh : { x // x \u2208 H }\nh' : \u2191\u2191H\ng : G\nhg : g \u2022 a = a\nhh : h \u2022 (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd) \u2022 a = a\nhh' : (\u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd)) \u2022 a = a\nh1 : h * h' = 1\n\u22a2 (h', { val := g, property := hg }) =\n    (h\u207b\u00b9,\n      { val := \u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd),\n        property := hh' })\n[PROOFSTEP]\nrefine' Prod.ext (eq_inv_of_mul_eq_one_right h1) (Subtype.ext _)\n[GOAL]\ncase intro.mk.mk\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH K : Subgroup G\nS T : Set G\n\u03b1 : Type u_2\ninst\u271d : MulAction G \u03b1\na : \u03b1\nh2 : \u2200 (g : G), \u2203 h, h \u2022 g \u2022 a = a\nh : { x // x \u2208 H }\nh' : \u2191\u2191H\ng : G\nhg : g \u2022 a = a\nhh : h \u2022 (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd) \u2022 a = a\nhh' : (\u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd)) \u2022 a = a\nh1 : h * h' = 1\n\u22a2 \u2191(h', { val := g, property := hg }).snd =\n    \u2191(h\u207b\u00b9,\n          { val := \u2191h * (\u2191(h', { val := g, property := hg }).fst * \u2191(h', { val := g, property := hg }).snd),\n            property := hh' }).snd\n[PROOFSTEP]\nrwa [Subtype.ext_iff, coe_one, coe_mul, \u2190 self_eq_mul_left, mul_assoc (\u2191h) (\u2191h') g] at h1 \n[GOAL]\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : \u2124\n\u22a2 \u2191(quotientEquivSigmaZMod H g) (g ^ k \u2022 Quotient.out' q) = { fst := q, snd := \u2191k }\n[PROOFSTEP]\nrw [apply_eq_iff_eq_symm_apply, quotientEquivSigmaZMod_symm_apply, ZMod.coe_int_cast, zpow_smul_mod_minimalPeriod]\n[GOAL]\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : G \u29f8 H\n\u22a2 \u2191(transferFunction H g q) = q\n[PROOFSTEP]\nrw [transferFunction_apply, \u2190 smul_eq_mul, Quotient.coe_smul_out', \u2190 quotientEquivSigmaZMod_symm_apply, Sigma.eta,\n  symm_apply_apply]\n[GOAL]\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : ZMod (minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q))\n\u22a2 \u2191(\u2191(toEquiv (_ : \u2191(transferTransversal H g) \u2208 leftTransversals \u2191H)) (g ^ \u2191k \u2022 Quotient.out' q)) =\n    g ^ \u2191k * Quotient.out' (Quotient.out' q)\n[PROOFSTEP]\nrw [transferTransversal_apply, transferFunction_apply, \u2190 quotientEquivSigmaZMod_symm_apply, apply_symm_apply]\n[GOAL]\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : ZMod (minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q))\n\u22a2 \u2191(\u2191(toEquiv (_ : \u2191(g \u2022 transferTransversal H g) \u2208 leftTransversals \u2191H)) (g ^ \u2191k \u2022 Quotient.out' q)) =\n    if k = 0 then g ^ minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) * Quotient.out' (Quotient.out' q)\n    else g ^ \u2191k * Quotient.out' (Quotient.out' q)\n[PROOFSTEP]\nrw [smul_apply_eq_smul_apply_inv_smul, transferTransversal_apply, transferFunction_apply, \u2190 mul_smul, \u2190 zpow_neg_one, \u2190\n  zpow_add, quotientEquivSigmaZMod_apply, smul_eq_mul, \u2190 mul_assoc, \u2190 zpow_one_add, Int.cast_add, Int.cast_neg,\n  Int.cast_one, int_cast_cast, cast_id', id.def, \u2190 sub_eq_neg_add, cast_sub_one, add_sub_cancel'_right]\n[GOAL]\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : ZMod (minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q))\n\u22a2 (g ^\n        if k = 0 then \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' { fst := q, snd := k - 1 }.fst))\n        else \u2191k) *\n      Quotient.out' (Quotient.out' { fst := q, snd := k - 1 }.fst) =\n    if k = 0 then g ^ minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) * Quotient.out' (Quotient.out' q)\n    else g ^ \u2191k * Quotient.out' (Quotient.out' q)\n[PROOFSTEP]\nby_cases hk : k = 0\n[GOAL]\ncase pos\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : ZMod (minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q))\nhk : k = 0\n\u22a2 (g ^\n        if k = 0 then \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' { fst := q, snd := k - 1 }.fst))\n        else \u2191k) *\n      Quotient.out' (Quotient.out' { fst := q, snd := k - 1 }.fst) =\n    if k = 0 then g ^ minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) * Quotient.out' (Quotient.out' q)\n    else g ^ \u2191k * Quotient.out' (Quotient.out' q)\n[PROOFSTEP]\nrw [if_pos hk, if_pos hk, zpow_ofNat]\n[GOAL]\ncase neg\nG : Type u\ninst\u271d : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient { x // x \u2208 zpowers g } (G \u29f8 H)\nk : ZMod (minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q))\nhk : \u00ack = 0\n\u22a2 (g ^\n        if k = 0 then \u2191(minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' { fst := q, snd := k - 1 }.fst))\n        else \u2191k) *\n      Quotient.out' (Quotient.out' { fst := q, snd := k - 1 }.fst) =\n    if k = 0 then g ^ minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) * Quotient.out' (Quotient.out' q)\n    else g ^ \u2191k * Quotient.out' (Quotient.out' q)\n[PROOFSTEP]\nrw [if_neg hk, if_neg hk]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Complement", "llama_tokens": 19595, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8499711908591638, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.5383804643568145}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\nl : Filter \u03b1\na : \u03b1\nt\u271d : Set \u03b1\nl' : Filter \u03b1\nt : Set \u03b4\ng : \u03b4 \u2192 \u03b1\nb : \u03b4\nhf : IsMinOn f s a\nhg : MapsTo g t s\nha : g b = a\ny : \u03b4\nhy : y \u2208 t\n\u22a2 y \u2208 {x | (fun x => (f \u2218 g) b \u2264 (f \u2218 g) x) x}\n[PROOFSTEP]\nsimpa only [ha, (\u00b7 \u2218 \u00b7)] using hf (hg hy)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d : OrderedAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b1\nhf : IsMinFilter f l a\nhg : IsMaxFilter g l a\n\u22a2 IsMinFilter (fun x => f x - g x) l a\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d : OrderedAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b1\nhf : IsMaxFilter f l a\nhg : IsMinFilter g l a\n\u22a2 IsMaxFilter (fun x => f x - g x) l a\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d : OrderedAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b1\nhf : IsMinOn f s a\nhg : IsMaxOn g s a\n\u22a2 IsMinOn (fun x => f x - g x) s a\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type x\ninst\u271d : OrderedAddCommGroup \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b1\nhf : IsMaxOn f s a\nhg : IsMinOn g s a\n\u22a2 IsMaxOn (fun x => f x - g x) s a\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using hf.add hg.neg\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nhle : g \u2264\u1da0[l] f\nhfga : f a = g a\nh : IsMaxFilter f l a\n\u22a2 IsMaxFilter g l a\n[PROOFSTEP]\nrefine' hle.mp (h.mono fun x hf hgf => _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nhle : g \u2264\u1da0[l] f\nhfga : f a = g a\nh : IsMaxFilter f l a\nx : \u03b1\nhf : f x \u2264 f a\nhgf : g x \u2264 f x\n\u22a2 g x \u2264 g a\n[PROOFSTEP]\nrw [\u2190 hfga]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nhle : g \u2264\u1da0[l] f\nhfga : f a = g a\nh : IsMaxFilter f l a\nx : \u03b1\nhf : f x \u2264 f a\nhgf : g x \u2264 f x\n\u22a2 g x \u2264 f a\n[PROOFSTEP]\nexact le_trans hgf hf\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nh : IsExtrFilter f l a\nheq : f =\u1da0[l] g\nhfga : f a = g a\n\u22a2 IsExtrFilter g l a\n[PROOFSTEP]\nrw [IsExtrFilter] at *\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b4 : Type x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf g : \u03b1 \u2192 \u03b2\na : \u03b1\nl : Filter \u03b1\nh : IsMinFilter f l a \u2228 IsMaxFilter f l a\nheq : f =\u1da0[l] g\nhfga : f a = g a\n\u22a2 IsMinFilter g l a \u2228 IsMaxFilter g l a\n[PROOFSTEP]\nrwa [\u2190 heq.isMaxFilter_iff hfga, \u2190 heq.isMinFilter_iff hfga]\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.Extr", "llama_tokens": 1448, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.672331705744791, "lm_q1q2_score": 0.5383306195453847}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nn : \u2115\nx\u271d : 0 < n + 1\n\u22a2 0 ^ (n + 1) = 0\n[PROOFSTEP]\nrw [pow_succ, zero_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nk : \u2115\nx\u271d : k + 1 \u2260 0\n\u22a2 0 ^ (k + 1) = 0\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nk : \u2115\nx\u271d : k + 1 \u2260 0\n\u22a2 0 * 0 ^ k = 0\n[PROOFSTEP]\nexact zero_mul _\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nn : \u2115\n\u22a2 0 ^ n = if n = 0 then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nn : \u2115\nh : n = 0\n\u22a2 0 ^ n = 1\n[PROOFSTEP]\nrw [h, pow_zero]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nn : \u2115\nh : \u00acn = 0\n\u22a2 0 ^ n = 0\n[PROOFSTEP]\nrw [zero_pow (Nat.pos_of_ne_zero h)]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nx : M\nn m : \u2115\nhn : n \u2264 m\nhx : x ^ n = 0\n\u22a2 x ^ m = 0\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le hn, pow_add, hx, mul_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nx : M\nn : \u2115\nH : x ^ n = 0\n\u22a2 x = 0\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nx : M\nn : \u2115\nH\u271d : x ^ n = 0\nH : x ^ Nat.zero = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [pow_zero] at H \n[GOAL]\ncase zero\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nx : M\nn : \u2115\nH\u271d : x ^ n = 0\nH : 1 = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 mul_one x, H, mul_zero]\n[GOAL]\ncase succ\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nx : M\nn\u271d : \u2115\nH\u271d : x ^ n\u271d = 0\nn : \u2115\nih : x ^ n = 0 \u2192 x = 0\nH : x ^ Nat.succ n = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [pow_succ] at H \n[GOAL]\ncase succ\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nx : M\nn\u271d : \u2115\nH\u271d : x ^ n\u271d = 0\nn : \u2115\nih : x ^ n = 0 \u2192 x = 0\nH : x * x ^ n = 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact Or.casesOn (mul_eq_zero.1 H) id ih\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\na : M\nn : \u2115\nhn : 0 < n\n\u22a2 a ^ n = 0 \u2194 a = 0\n[PROOFSTEP]\nrefine' \u27e8pow_eq_zero, _\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\na : M\nn : \u2115\nhn : 0 < n\n\u22a2 a = 0 \u2192 a ^ n = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : NoZeroDivisors M\nn : \u2115\nhn : 0 < n\n\u22a2 0 ^ n = 0\n[PROOFSTEP]\nexact zero_pow hn\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : MonoidWithZero M\ninst\u271d\u00b9 : NoZeroDivisors M\ninst\u271d : Nontrivial M\na : M\nn : \u2115\n\u22a2 a ^ n = 0 \u2194 a = 0 \u2227 n \u2260 0\n[PROOFSTEP]\ncases (zero_le n).eq_or_gt\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : MonoidWithZero M\ninst\u271d\u00b9 : NoZeroDivisors M\ninst\u271d : Nontrivial M\na : M\nn : \u2115\nh\u271d : n = 0\n\u22a2 a ^ n = 0 \u2194 a = 0 \u2227 n \u2260 0\n[PROOFSTEP]\nsimp [*, ne_of_gt]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b2 : MonoidWithZero M\ninst\u271d\u00b9 : NoZeroDivisors M\ninst\u271d : Nontrivial M\na : M\nn : \u2115\nh\u271d : 0 < n\n\u22a2 a ^ n = 0 \u2194 a = 0 \u2227 n \u2260 0\n[PROOFSTEP]\nsimp [*, ne_of_gt]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\na : M\nn : \u2115\nhn : n \u2260 0\n\u22a2 a ^ n \u2260 0 \u2192 a \u2260 0\n[PROOFSTEP]\ncontrapose!\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\na : M\nn : \u2115\nhn : n \u2260 0\n\u22a2 a = 0 \u2192 a ^ n = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nn : \u2115\nhn : n \u2260 0\n\u22a2 0 ^ n = 0\n[PROOFSTEP]\nexact zero_pow' n hn\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\n\u22a2 0 ^ n = 0 \u2194 0 < n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\n\u22a2 0 ^ n = 0 \u2192 0 < n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\n\u22a2 0 < n \u2192 0 ^ n = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\nh : 0 ^ n = 0\n\u22a2 0 < n\n[PROOFSTEP]\nrw [pos_iff_ne_zero]\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\nh : 0 ^ n = 0\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nh : 0 ^ 0 = 0\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : MonoidWithZero M\ninst\u271d : Nontrivial M\nn : \u2115\nh : 0 < n\n\u22a2 0 ^ n = 0\n[PROOFSTEP]\nexact zero_pow' n h.ne.symm\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nr : M\n\u22a2 inverse r ^ 0 = inverse (r ^ 0)\n[PROOFSTEP]\nrw [pow_zero, pow_zero, Ring.inverse_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : MonoidWithZero M\nr : M\nn : \u2115\n\u22a2 inverse r ^ (n + 1) = inverse (r ^ (n + 1))\n[PROOFSTEP]\nrw [pow_succ, pow_succ', Ring.mul_inverse_rev' ((Commute.refl r).pow_left n), Ring.inverse_pow r n]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\n\u22a2 x ^ n \u2223 x ^ m \u2194 n \u2264 m\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\n\u22a2 x ^ n \u2223 x ^ m \u2192 n \u2264 m\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\n\u22a2 n \u2264 m\n[PROOFSTEP]\nrw [\u2190 not_lt]\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\n\u22a2 \u00acm < n\n[PROOFSTEP]\nintro hmn\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\n\u22a2 False\n[PROOFSTEP]\napply h1\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\n\u22a2 IsUnit x\n[PROOFSTEP]\nhave : x ^ m * x \u2223 x ^ m * 1 := by\n  rw [\u2190 pow_succ', mul_one]\n  exact (pow_dvd_pow _ (Nat.succ_le_of_lt hmn)).trans h\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\n\u22a2 x ^ m * x \u2223 x ^ m * 1\n[PROOFSTEP]\nrw [\u2190 pow_succ', mul_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\n\u22a2 x ^ (m + 1) \u2223 x ^ m\n[PROOFSTEP]\nexact (pow_dvd_pow _ (Nat.succ_le_of_lt hmn)).trans h\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\nthis : x ^ m * x \u2223 x ^ m * 1\n\u22a2 IsUnit x\n[PROOFSTEP]\nrwa [mul_dvd_mul_iff_left, \u2190 isUnit_iff_dvd_one] at this \n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\nh : x ^ n \u2223 x ^ m\nhmn : m < n\nthis : x ^ m * x \u2223 x ^ m * 1\n\u22a2 x ^ m \u2260 0\n[PROOFSTEP]\napply pow_ne_zero m h0\n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CancelCommMonoidWithZero R\nx : R\nn m : \u2115\nh0 : x \u2260 0\nh1 : \u00acIsUnit x\n\u22a2 n \u2264 m \u2192 x ^ n \u2223 x ^ m\n[PROOFSTEP]\napply pow_dvd_pow\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nn m : \u2115\na b c : R\nha : c ^ n \u2223 a\nhb : c ^ m \u2223 b\n\u22a2 c ^ min n m \u2223 a + b\n[PROOFSTEP]\nreplace ha := (pow_dvd_pow c (min_le_left n m)).trans ha\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nn m : \u2115\na b c : R\nhb : c ^ m \u2223 b\nha : c ^ min n m \u2223 a\n\u22a2 c ^ min n m \u2223 a + b\n[PROOFSTEP]\nreplace hb := (pow_dvd_pow c (min_le_right n m)).trans hb\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nn m : \u2115\na b c : R\nha : c ^ min n m \u2223 a\nhb : c ^ min n m \u2223 b\n\u22a2 c ^ min n m \u2223 a + b\n[PROOFSTEP]\nexact dvd_add ha hb\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CommSemiring R\na b : R\n\u22a2 (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2\n[PROOFSTEP]\nsimp only [sq, add_mul_self_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CommSemiring R\na b : R\n\u22a2 (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b\n[PROOFSTEP]\nrw [add_sq, add_assoc, add_comm _ (b ^ 2), add_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn : \u2115\nh : (-1) ^ n = -1\n\u22a2 (-1) ^ (n + 1) = 1\n[PROOFSTEP]\nrw [pow_succ, h, neg_one_mul, neg_neg]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\nn : \u2115\nh : (-1) ^ n = 1\n\u22a2 (-1) ^ (n + 1) = -1\n[PROOFSTEP]\nrw [pow_succ, h, mul_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\na : R\nn : \u2115\n\u22a2 (-a) ^ bit0 n = a ^ bit0 n\n[PROOFSTEP]\nrw [pow_bit0', neg_mul_neg, pow_bit0']\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\na : R\nn : \u2115\n\u22a2 (-a) ^ bit1 n = -a ^ bit1 n\n[PROOFSTEP]\nsimp only [bit1, pow_succ, neg_pow_bit0, neg_mul_eq_neg_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\na : R\n\u22a2 (-a) ^ 2 = a ^ 2\n[PROOFSTEP]\nsimp [sq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Monoid R\ninst\u271d : HasDistribNeg R\n\u22a2 (-1) ^ 2 = 1\n[PROOFSTEP]\nsimp [neg_sq, one_pow]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nh : Commute a b\n\u22a2 a ^ 2 - b ^ 2 = (a + b) * (a - b)\n[PROOFSTEP]\nrw [sq, sq, h.mul_self_sub_mul_self_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\n\u22a2 (-1) ^ n * r = 0 \u2194 r = 0\n[PROOFSTEP]\nrcases neg_one_pow_eq_or R n with h | h\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\nh : (-1) ^ n = 1\n\u22a2 (-1) ^ n * r = 0 \u2194 r = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\nh : (-1) ^ n = -1\n\u22a2 (-1) ^ n * r = 0 \u2194 r = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\n\u22a2 r * (-1) ^ n = 0 \u2194 r = 0\n[PROOFSTEP]\nrcases neg_one_pow_eq_or R n with h | h\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\nh : (-1) ^ n = 1\n\u22a2 r * (-1) ^ n = 0 \u2194 r = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nn : \u2115\nr : R\nh : (-1) ^ n = -1\n\u22a2 r * (-1) ^ n = 0 \u2194 r = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Ring R\na b : R\ninst\u271d : NoZeroDivisors R\nh : Commute a b\n\u22a2 a ^ 2 = b ^ 2 \u2194 a = b \u2228 a = -b\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, h.sq_sub_sq, mul_eq_zero, add_eq_zero_iff_eq_neg, sub_eq_zero, or_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : Ring R\na b : R\ninst\u271d : NoZeroDivisors R\n\u22a2 a ^ 2 = 1 \u2194 a = 1 \u2228 a = -1\n[PROOFSTEP]\nrw [\u2190 (Commute.one_right a).sq_eq_sq_iff_eq_or_eq_neg, one_pow]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CommRing R\na b : R\n\u22a2 (a - b) ^ 2 = a ^ 2 - 2 * a * b + b ^ 2\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_sq, neg_sq, mul_neg, \u2190 sub_eq_add_neg]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : CommRing R\na b : R\n\u22a2 (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_sq', neg_sq, mul_neg, \u2190 sub_eq_add_neg]\n[GOAL]\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : NoZeroDivisors R\na\u271d b\u271d : R\na b : R\u02e3\n\u22a2 a ^ 2 = b ^ 2 \u2194 a = b \u2228 a = -b\n[PROOFSTEP]\nsimp_rw [ext_iff, val_pow_eq_pow_val, sq_eq_sq_iff_eq_or_eq_neg, Units.val_neg]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupPower.Ring", "llama_tokens": 6500, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835289107307, "lm_q2_score": 0.6442251133170357, "lm_q1q2_score": 0.5382394710870324}}
{"text": "[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2079 : CommSemiring R\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2076 : CommRing R\u2083\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d : Fintype n\nM : Matrix n n R\u2082\nx y z : n \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) (x + y) z =\n    (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x z + (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) y z\n[PROOFSTEP]\nsimp only [Pi.add_apply, add_mul, sum_add_distrib]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2079 : CommSemiring R\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2076 : CommRing R\u2083\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d : Fintype n\nM : Matrix n n R\u2082\na : R\u2082\nx y : n \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) (a \u2022 x) y = a * (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x y\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_eq_mul, mul_assoc, mul_sum]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2079 : CommSemiring R\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2076 : CommRing R\u2083\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d : Fintype n\nM : Matrix n n R\u2082\nx y z : n \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x (y + z) =\n    (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x y + (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x z\n[PROOFSTEP]\nsimp only [Pi.add_apply, mul_add, sum_add_distrib]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2075 : Semiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u00b3 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : Ring R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2079 : CommSemiring R\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2076 : CommRing R\u2083\ninst\u271d\u2075 : AddCommGroup M\u2083\ninst\u271d\u2074 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d : Fintype n\nM : Matrix n n R\u2082\na : R\u2082\nx y : n \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x (a \u2022 y) = a * (fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j) x y\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_eq_mul, mul_assoc, mul_left_comm, mul_sum]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 bilin (toBilin'Aux M) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) =\n    M i j\n[PROOFSTEP]\nrw [Matrix.toBilin'Aux]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 bilin\n      { bilin := fun v w => \u2211 i : n, \u2211 j : n, v i * M i j * w j,\n        bilin_add_left :=\n          (_ :\n            \u2200 (x y z : n \u2192 R\u2082),\n              \u2211 i : n, \u2211 j : n, (x + y) i * M i j * z j =\n                \u2211 x_1 : n, \u2211 x_2 : n, x x_1 * M x_1 x_2 * z x_2 + \u2211 x : n, \u2211 x_1 : n, y x * M x x_1 * z x_1),\n        bilin_smul_left :=\n          (_ :\n            \u2200 (a : R\u2082) (x y : n \u2192 R\u2082),\n              \u2211 i : n, \u2211 j : n, (a \u2022 x) i * M i j * y j = a * \u2211 i : n, \u2211 j : n, x i * M i j * y j),\n        bilin_add_right :=\n          (_ :\n            \u2200 (x y z : n \u2192 R\u2082),\n              \u2211 i : n, \u2211 j : n, x i * M i j * (y + z) j =\n                \u2211 x_1 : n, \u2211 x_2 : n, x x_1 * M x_1 x_2 * y x_2 + \u2211 x_1 : n, \u2211 x_2 : n, x x_1 * M x_1 x_2 * z x_2),\n        bilin_smul_right :=\n          (_ :\n            \u2200 (a : R\u2082) (x y : n \u2192 R\u2082),\n              \u2211 i : n, \u2211 j : n, x i * M i j * (a \u2022 y) j = a * \u2211 i : n, \u2211 j : n, x i * M i j * y j) }\n      (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) =\n    M i j\n[PROOFSTEP]\ndsimp only\n  -- Porting note: had to add `dsimp only` to get rid of the projections\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u2211 i_1 : n,\n      \u2211 j_1 : n,\n        \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i_1 * M i_1 j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 =\n    M i j\n[PROOFSTEP]\nrw [sum_eq_single i, sum_eq_single j]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j = M i j\n[PROOFSTEP]\nsimp only [stdBasis_same, stdBasis_same, one_mul, mul_one]\n[GOAL]\ncase h\u2080\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u2200 (b : n),\n    b \u2208 univ \u2192\n      b \u2260 j \u2192 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i b * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 b = 0\n[PROOFSTEP]\nrintro j' - hj'\n[GOAL]\ncase h\u2080\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j j' : n\nhj' : j' \u2260 j\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j' * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j' = 0\n[PROOFSTEP]\napply mul_eq_zero_of_right\n[GOAL]\ncase h\u2080.h\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j j' : n\nhj' : j' \u2260 j\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j' = 0\n[PROOFSTEP]\nexact stdBasis_ne R\u2082 (fun _ => R\u2082) _ _ hj' 1\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u00acj \u2208 univ \u2192 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\na\u271d : \u00acj \u2208 univ\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j = 0\n[PROOFSTEP]\nhave := Finset.mem_univ j\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\na\u271d : \u00acj \u2208 univ\nthis : j \u2208 univ\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase h\u2080\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u2200 (b : n),\n    b \u2208 univ \u2192\n      b \u2260 i \u2192\n        \u2211 j_1 : n,\n            \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 b * M b j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 =\n          0\n[PROOFSTEP]\nrintro i' - hi'\n[GOAL]\ncase h\u2080\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j i' : n\nhi' : i' \u2260 i\n\u22a2 \u2211 j_1 : n, \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i' * M i' j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 =\n    0\n[PROOFSTEP]\nrefine' Finset.sum_eq_zero fun j _ => _\n[GOAL]\ncase h\u2080\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j\u271d i' : n\nhi' : i' \u2260 i\nj : n\nx\u271d : j \u2208 univ\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i' * M i' j * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j\u271d) 1 j = 0\n[PROOFSTEP]\napply mul_eq_zero_of_left\n[GOAL]\ncase h\u2080.h\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j\u271d i' : n\nhi' : i' \u2260 i\nj : n\nx\u271d : j \u2208 univ\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i' * M i' j = 0\n[PROOFSTEP]\napply mul_eq_zero_of_left\n[GOAL]\ncase h\u2080.h.h\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j\u271d i' : n\nhi' : i' \u2260 i\nj : n\nx\u271d : j \u2208 univ\n\u22a2 \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i' = 0\n[PROOFSTEP]\nexact stdBasis_ne R\u2082 (fun _ => R\u2082) _ _ hi' 1\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\n\u22a2 \u00aci \u2208 univ \u2192\n    \u2211 j_1 : n, \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 =\n      0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\na\u271d : \u00aci \u2208 univ\n\u22a2 \u2211 j_1 : n, \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 = 0\n[PROOFSTEP]\nhave := Finset.mem_univ i\n[GOAL]\ncase h\u2081\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2074 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n R\u2082\ni j : n\na\u271d : \u00aci \u2208 univ\nthis : i \u2208 univ\n\u22a2 \u2211 j_1 : n, \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1 i * M i j_1 * \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1 = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nB\u2082 : BilinForm R\u2082 (n \u2192 R\u2082)\n\u22a2 toBilin'Aux (\u2191(toMatrixAux fun j => \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) B\u2082) = B\u2082\n[PROOFSTEP]\nrefine' ext_basis (Pi.basisFun R\u2082 n) fun i j => _\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082\u271d : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nB\u2082 : BilinForm R\u2082 (n \u2192 R\u2082)\ni j : n\n\u22a2 bilin (toBilin'Aux (\u2191(toMatrixAux fun j => \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) B\u2082)) (\u2191(Pi.basisFun R\u2082 n) i)\n      (\u2191(Pi.basisFun R\u2082 n) j) =\n    bilin B\u2082 (\u2191(Pi.basisFun R\u2082 n) i) (\u2191(Pi.basisFun R\u2082 n) j)\n[PROOFSTEP]\nrw [Pi.basisFun_apply, Pi.basisFun_apply, Matrix.toBilin'Aux_stdBasis, BilinForm.toMatrixAux_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nsrc\u271d : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1) \u2192\u2097[R\u2082] Matrix n n R\u2082 :=\n  toMatrixAux fun j => \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1\nB : BilinForm R\u2082 (n \u2192 R\u2082)\n\u22a2 toBilin'Aux\n      (AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R\u2082) (x : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1)),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R\u2082) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        B) =\n    B\n[PROOFSTEP]\nconvert toBilin'Aux_toMatrixAux B\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nsrc\u271d : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1) \u2192\u2097[R\u2082] Matrix n n R\u2082 :=\n  toMatrixAux fun j => \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1\nM : Matrix n n R\u2082\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R\u2082) (x : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1)),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R\u2082) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (toBilin'Aux M) =\n    M\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nsrc\u271d : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1) \u2192\u2097[R\u2082] Matrix n n R\u2082 :=\n  toMatrixAux fun j => \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1\nM : Matrix n n R\u2082\ni j : n\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R\u2082) (x : BilinForm R\u2082 ((fun x => n \u2192 R\u2082) 1)),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R\u2082) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (toBilin'Aux M) i j =\n    M i j\n[PROOFSTEP]\nsimp only [toFun_eq_coe, BilinForm.toMatrixAux_apply, Matrix.toBilin'Aux_stdBasis]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nv w : n \u2192 R\u2082\n\u22a2 bilin (\u2191toBilin' M) v w = v \u2b1d\u1d65 mulVec M w\n[PROOFSTEP]\nsimp_rw [Matrix.toBilin'_apply, Matrix.dotProduct, Matrix.mulVec, Matrix.dotProduct]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nv w : n \u2192 R\u2082\n\u22a2 \u2211 i : n, \u2211 j : n, v i * M i j * w j = \u2211 x : n, v x * \u2211 x_1 : n, M x x_1 * w x_1\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun _ _ => _\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nv w : n \u2192 R\u2082\nx\u271d\u00b9 : n\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 \u2211 j : n, v x\u271d\u00b9 * M x\u271d\u00b9 j * w j = v x\u271d\u00b9 * \u2211 x : n, M x\u271d\u00b9 x * w x\n[PROOFSTEP]\nrw [Finset.mul_sum]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nv w : n \u2192 R\u2082\nx\u271d\u00b9 : n\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 \u2211 j : n, v x\u271d\u00b9 * M x\u271d\u00b9 j * w j = \u2211 x : n, v x\u271d\u00b9 * (M x\u271d\u00b9 x * w x)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun _ _ => _\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nv w : n \u2192 R\u2082\nx\u271d\u00b3 : n\nx\u271d\u00b2 : x\u271d\u00b3 \u2208 univ\nx\u271d\u00b9 : n\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 v x\u271d\u00b3 * M x\u271d\u00b3 x\u271d\u00b9 * w x\u271d\u00b9 = v x\u271d\u00b3 * (M x\u271d\u00b3 x\u271d\u00b9 * w x\u271d\u00b9)\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\n\u22a2 \u2191toMatrix' (comp B l r) = (\u2191LinearMap.toMatrix' l)\u1d40 * \u2191toMatrix' B * \u2191LinearMap.toMatrix' r\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 \u2191toMatrix' (comp B l r) i j = ((\u2191LinearMap.toMatrix' l)\u1d40 * \u2191toMatrix' B * \u2191LinearMap.toMatrix' r) i j\n[PROOFSTEP]\nsimp only [BilinForm.toMatrix'_apply, BilinForm.comp_apply, transpose_apply, Matrix.mul_apply, LinearMap.toMatrix',\n  LinearEquiv.coe_mk, sum_mul]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 bilin B (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1)) (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1)) =\n    \u2211 x : n,\n      \u2211 x_1 : n,\n        \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x_1 i *\n            bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x_1) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\nrw [sum_comm]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 bilin B (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1)) (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1)) =\n    \u2211 y : n,\n      \u2211 x : n,\n        \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) y i *\n            bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) y) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\nconv_lhs => rw [\u2190 BilinForm.sum_repr_mul_repr_mul (Pi.basisFun R\u2082 n) (l _) (r _)]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n| bilin B (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1)) (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul (Pi.basisFun R\u2082 n) (l _) (r _)]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n| bilin B (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1)) (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul (Pi.basisFun R\u2082 n) (l _) (r _)]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n| bilin B (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1)) (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul (Pi.basisFun R\u2082 n) (l _) (r _)]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) fun i xi =>\n      Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n        xi \u2022 yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i) (\u2191(Pi.basisFun R\u2082 n) j)) =\n    \u2211 y : n,\n      \u2211 x : n,\n        \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) y i *\n            bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) y) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 (\u2211 i_1 : n,\n      Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n        \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i_1 \u2022\n          yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i_1) (\u2191(Pi.basisFun R\u2082 n) j)) =\n    \u2211 y : n,\n      \u2211 x : n,\n        \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) y i *\n            bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) y) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      (Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n          \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) x \u2022\n            yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) x) (\u2191(Pi.basisFun R\u2082 n) j)) =\n        \u2211 x_1 : n,\n          \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x i *\n              bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x_1) 1) *\n            \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x_1 j\n[PROOFSTEP]\nrintro i' -\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' : n\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n      \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n        yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) j)) =\n    \u2211 x : n,\n      \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) i' i *\n          bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i') 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n        \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' : n\n\u22a2 \u2211 i_1 : n,\n      \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n        \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) i_1 \u2022\n          bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) i_1) =\n    \u2211 x : n,\n      \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) i' i *\n          bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i') 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n        \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' : n\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n          \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) x \u2022\n            bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) x) =\n        \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) i' i *\n            bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i') 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) x j\n[PROOFSTEP]\nrintro j' -\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' j' : n\n\u22a2 \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n      \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) j' \u2022\n        bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) j') =\n    \u2191of (fun i j => \u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) i' i *\n        bilin B (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i') 1) (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j') 1) *\n      \u2191of (fun i j => \u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) i) j' j\n[PROOFSTEP]\nsimp only [smul_eq_mul, Pi.basisFun_repr, mul_assoc, mul_comm, mul_left_comm, Pi.basisFun_apply, of_apply]\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' : n\n\u22a2 \u2200 (i_1 : n),\n    \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n        0 \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) i_1) =\n      0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni' i\u271d : n\n\u22a2 \u2191(\u2191(Pi.basisFun R\u2082 n).repr (\u2191l (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) i) 1))) i' \u2022\n      0 \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i') (\u2191(Pi.basisFun R\u2082 n) i\u271d) =\n    0\n[PROOFSTEP]\nsimp only [zero_smul, smul_zero]\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\n\u22a2 \u2200 (i : n),\n    (Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n        0 \u2022 yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i) (\u2191(Pi.basisFun R\u2082 n) j)) =\n      0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nl r : (o \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\ni j : o\ni\u271d : n\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R\u2082 n).repr (\u2191r (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1))) fun j yj =>\n      0 \u2022 yj \u2022 bilin B (\u2191(Pi.basisFun R\u2082 n) i\u271d) (\u2191(Pi.basisFun R\u2082 n) j)) =\n    0\n[PROOFSTEP]\nsimp only [zero_smul, Finsupp.sum_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nf : (n \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\n\u22a2 \u2191toMatrix' (compLeft B f) = (\u2191LinearMap.toMatrix' f)\u1d40 * \u2191toMatrix' B\n[PROOFSTEP]\nsimp only [BilinForm.compLeft, BilinForm.toMatrix'_comp, toMatrix'_id, Matrix.mul_one]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nf : (n \u2192 R\u2082) \u2192\u2097[R\u2082] n \u2192 R\u2082\n\u22a2 \u2191toMatrix' (compRight B f) = \u2191toMatrix' B * \u2191LinearMap.toMatrix' f\n[PROOFSTEP]\nsimp only [BilinForm.compRight, BilinForm.toMatrix'_comp, toMatrix'_id, transpose_one, Matrix.one_mul]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nM : Matrix o n R\u2082\nN : Matrix n o R\u2082\n\u22a2 M * \u2191toMatrix' B * N = \u2191toMatrix' (comp B (\u2191Matrix.toLin' M\u1d40) (\u2191Matrix.toLin' N))\n[PROOFSTEP]\nsimp only [B.toMatrix'_comp, transpose_transpose, toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nM : Matrix n n R\u2082\n\u22a2 M * \u2191toMatrix' B = \u2191toMatrix' (compLeft B (\u2191Matrix.toLin' M\u1d40))\n[PROOFSTEP]\nsimp only [toMatrix'_compLeft, transpose_transpose, toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 (n \u2192 R\u2082)\nM : Matrix n n R\u2082\n\u22a2 \u2191toMatrix' B * M = \u2191toMatrix' (compRight B (\u2191Matrix.toLin' M))\n[PROOFSTEP]\nsimp only [toMatrix'_compRight, toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2076 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2075 : Ring R\u2081\ninst\u271d\u00b9\u2074 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b3 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u2070 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2079 : CommRing R\u2083\ninst\u271d\u2078 : AddCommGroup M\u2083\ninst\u271d\u2077 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype o\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nP Q : Matrix n o R\u2082\n\u22a2 \u2191BilinForm.toMatrix' (BilinForm.comp (\u2191toBilin' M) (\u2191toLin' P) (\u2191toLin' Q)) =\n    \u2191BilinForm.toMatrix' (\u2191toBilin' (P\u1d40 * M * Q))\n[PROOFSTEP]\nsimp only [BilinForm.toMatrix'_comp, BilinForm.toMatrix'_toBilin', toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nB : BilinForm R\u2082 M\u2082\ni j : n\n\u22a2 \u2191(toMatrix b) B i j = bilin B (\u2191b i) (\u2191b j)\n[PROOFSTEP]\nrw [BilinForm.toMatrix, LinearEquiv.trans_apply, BilinForm.toMatrix'_apply, congr_apply, b.equivFun_symm_stdBasis,\n  b.equivFun_symm_stdBasis]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2075 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM : Matrix n n R\u2082\nx y : M\u2082\n\u22a2 bilin (\u2191(toBilin b) M) x y = \u2211 i : n, \u2211 j : n, \u2191(\u2191b.repr x) i * M i j * \u2191(\u2191b.repr y) j\n[PROOFSTEP]\nrw [Matrix.toBilin, BilinForm.toMatrix, LinearEquiv.symm_trans_apply, \u2190 Matrix.toBilin']\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2075 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM : Matrix n n R\u2082\nx y : M\u2082\n\u22a2 bilin (\u2191(LinearEquiv.symm (BilinForm.congr (Basis.equivFun b))) (\u2191toBilin' M)) x y =\n    \u2211 i : n, \u2211 j : n, \u2191(\u2191b.repr x) i * M i j * \u2191(\u2191b.repr y) j\n[PROOFSTEP]\nsimp only [congr_symm, congr_apply, LinearEquiv.symm_symm, Matrix.toBilin'_apply, Basis.equivFun_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nB : BilinForm R\u2082 M\u2082\ni j : n\n\u22a2 \u2191(toMatrixAux \u2191b) B i j = \u2191(BilinForm.toMatrix b) B i j\n[PROOFSTEP]\nrw [BilinForm.toMatrix_apply, BilinForm.toMatrixAux_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\n\u22a2 toBilin (Pi.basisFun R\u2082 n) = toBilin'\n[PROOFSTEP]\next M\n[GOAL]\ncase h.H\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2075 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM : Matrix n n R\u2082\nx\u271d y\u271d : n \u2192 R\u2082\n\u22a2 bilin (\u2191(toBilin (Pi.basisFun R\u2082 n)) M) x\u271d y\u271d = bilin (\u2191toBilin' M) x\u271d y\u271d\n[PROOFSTEP]\nsimp only [Matrix.toBilin_apply, Matrix.toBilin'_apply, Pi.basisFun_repr]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\n\u22a2 toMatrix (Pi.basisFun R\u2082 n) = toMatrix'\n[PROOFSTEP]\next B\n[GOAL]\ncase h.a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2077 : Semiring R\ninst\u271d\u00b9\u2076 : AddCommMonoid M\ninst\u271d\u00b9\u2075 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2074 : Ring R\u2081\ninst\u271d\u00b9\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b2 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b9 : CommSemiring R\u2082\ninst\u271d\u00b9\u2070 : AddCommMonoid M\u2082\ninst\u271d\u2079 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2078 : CommRing R\u2083\ninst\u271d\u2077 : AddCommGroup M\u2083\ninst\u271d\u2076 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq n\nb : Basis n R\u2082 M\u2082\nB : BilinForm R\u2082 (n \u2192 R\u2082)\ni\u271d x\u271d : n\n\u22a2 \u2191(toMatrix (Pi.basisFun R\u2082 n)) B i\u271d x\u271d = \u2191toMatrix' B i\u271d x\u271d\n[PROOFSTEP]\nrw [BilinForm.toMatrix_apply, BilinForm.toMatrix'_apply, Pi.basisFun_apply, Pi.basisFun_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\n\u22a2 \u2191(toMatrix c) (comp B l r) = (\u2191(LinearMap.toMatrix c b) l)\u1d40 * \u2191(toMatrix b) B * \u2191(LinearMap.toMatrix c b) r\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 \u2191(toMatrix c) (comp B l r) i j = ((\u2191(LinearMap.toMatrix c b) l)\u1d40 * \u2191(toMatrix b) B * \u2191(LinearMap.toMatrix c b) r) i j\n[PROOFSTEP]\nsimp only [BilinForm.toMatrix_apply, BilinForm.comp_apply, transpose_apply, Matrix.mul_apply, LinearMap.toMatrix',\n  LinearEquiv.coe_mk, sum_mul]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 bilin B (\u2191l (\u2191c i)) (\u2191r (\u2191c j)) =\n    \u2211 x : n, \u2211 x_1 : n, \u2191(LinearMap.toMatrix c b) l x_1 i * bilin B (\u2191b x_1) (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\nrw [sum_comm]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 bilin B (\u2191l (\u2191c i)) (\u2191r (\u2191c j)) =\n    \u2211 y : n, \u2211 x : n, \u2191(LinearMap.toMatrix c b) l y i * bilin B (\u2191b y) (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\nconv_lhs => rw [\u2190 BilinForm.sum_repr_mul_repr_mul b]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n| bilin B (\u2191l (\u2191c i)) (\u2191r (\u2191c j))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul b]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n| bilin B (\u2191l (\u2191c i)) (\u2191r (\u2191c j))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul b]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n| bilin B (\u2191l (\u2191c i)) (\u2191r (\u2191c j))\n[PROOFSTEP]\nrw [\u2190 BilinForm.sum_repr_mul_repr_mul b]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 (Finsupp.sum (\u2191b.repr (\u2191l (\u2191c i))) fun i xi =>\n      Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => xi \u2022 yj \u2022 bilin B (\u2191b i) (\u2191b j)) =\n    \u2211 y : n, \u2211 x : n, \u2191(LinearMap.toMatrix c b) l y i * bilin B (\u2191b y) (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 (\u2211 i_1 : n, Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => \u2191(\u2191b.repr (\u2191l (\u2191c i))) i_1 \u2022 yj \u2022 bilin B (\u2191b i_1) (\u2191b j)) =\n    \u2211 y : n, \u2211 x : n, \u2191(LinearMap.toMatrix c b) l y i * bilin B (\u2191b y) (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      (Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => \u2191(\u2191b.repr (\u2191l (\u2191c i))) x \u2022 yj \u2022 bilin B (\u2191b x) (\u2191b j)) =\n        \u2211 x_1 : n, \u2191(LinearMap.toMatrix c b) l x i * bilin B (\u2191b x) (\u2191b x_1) * \u2191(LinearMap.toMatrix c b) r x_1 j\n[PROOFSTEP]\nrintro i' -\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' : n\n\u22a2 (Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 yj \u2022 bilin B (\u2191b i') (\u2191b j)) =\n    \u2211 x : n, \u2191(LinearMap.toMatrix c b) l i' i * bilin B (\u2191b i') (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' : n\n\u22a2 \u2211 i_1 : n, \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 \u2191(\u2191b.repr (\u2191r (\u2191c j))) i_1 \u2022 bilin B (\u2191b i') (\u2191b i_1) =\n    \u2211 x : n, \u2191(LinearMap.toMatrix c b) l i' i * bilin B (\u2191b i') (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' : n\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 \u2191(\u2191b.repr (\u2191r (\u2191c j))) x \u2022 bilin B (\u2191b i') (\u2191b x) =\n        \u2191(LinearMap.toMatrix c b) l i' i * bilin B (\u2191b i') (\u2191b x) * \u2191(LinearMap.toMatrix c b) r x j\n[PROOFSTEP]\nrintro j' -\n[GOAL]\ncase a.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' j' : n\n\u22a2 \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 \u2191(\u2191b.repr (\u2191r (\u2191c j))) j' \u2022 bilin B (\u2191b i') (\u2191b j') =\n    \u2191(LinearMap.toMatrix c b) l i' i * bilin B (\u2191b i') (\u2191b j') * \u2191(LinearMap.toMatrix c b) r j' j\n[PROOFSTEP]\nsimp only [smul_eq_mul, LinearMap.toMatrix_apply, Basis.equivFun_apply, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' : n\n\u22a2 \u2200 (i_1 : n), \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 0 \u2022 bilin B (\u2191b i') (\u2191b i_1) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni' i\u271d : n\n\u22a2 \u2191(\u2191b.repr (\u2191l (\u2191c i))) i' \u2022 0 \u2022 bilin B (\u2191b i') (\u2191b i\u271d) = 0\n[PROOFSTEP]\nsimp only [zero_smul, smul_zero]\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\n\u22a2 \u2200 (i : n), (Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => 0 \u2022 yj \u2022 bilin B (\u2191b i) (\u2191b j)) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nl r : M\u2082' \u2192\u2097[R\u2082] M\u2082\ni j : o\ni\u271d : n\n\u22a2 (Finsupp.sum (\u2191b.repr (\u2191r (\u2191c j))) fun j yj => 0 \u2022 yj \u2022 bilin B (\u2191b i\u271d) (\u2191b j)) = 0\n[PROOFSTEP]\nsimp only [zero_smul, Finsupp.sum_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nf : M\u2082 \u2192\u2097[R\u2082] M\u2082\n\u22a2 \u2191(toMatrix b) (compLeft B f) = (\u2191(LinearMap.toMatrix b b) f)\u1d40 * \u2191(toMatrix b) B\n[PROOFSTEP]\nsimp only [compLeft, BilinForm.toMatrix_comp b b, toMatrix_id, Matrix.mul_one]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nf : M\u2082 \u2192\u2097[R\u2082] M\u2082\n\u22a2 \u2191(toMatrix b) (compRight B f) = \u2191(toMatrix b) B * \u2191(LinearMap.toMatrix b b) f\n[PROOFSTEP]\nsimp only [BilinForm.compRight, BilinForm.toMatrix_comp b b, toMatrix_id, transpose_one, Matrix.one_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\ninst\u271d\u00b9\u2078 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc\u271d : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nc : Basis o R\u2082 M\u2082\nB : BilinForm R\u2082 M\u2082\n\u22a2 (Basis.toMatrix b \u2191c)\u1d40 * \u2191(toMatrix b) B * Basis.toMatrix b \u2191c = \u2191(toMatrix c) B\n[PROOFSTEP]\nrw [\u2190 LinearMap.toMatrix_id_eq_basis_toMatrix, \u2190 BilinForm.toMatrix_comp, BilinForm.comp_id_id]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2078 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nM : Matrix o n R\u2082\nN : Matrix n o R\u2082\n\u22a2 M * \u2191(toMatrix b) B * N = \u2191(toMatrix c) (comp B (\u2191(Matrix.toLin c b) M\u1d40) (\u2191(Matrix.toLin c b) N))\n[PROOFSTEP]\nsimp only [B.toMatrix_comp b c, toMatrix_toLin, transpose_transpose]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2078 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nM : Matrix n n R\u2082\n\u22a2 M * \u2191(toMatrix b) B = \u2191(toMatrix b) (compLeft B (\u2191(Matrix.toLin b b) M\u1d40))\n[PROOFSTEP]\nrw [B.toMatrix_compLeft b, toMatrix_toLin, transpose_transpose]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2078 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB\u271d : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nB : BilinForm R\u2082 M\u2082\nM : Matrix n n R\u2082\n\u22a2 \u2191(toMatrix b) B * M = \u2191(toMatrix b) (compRight B (\u2191(Matrix.toLin b b) M))\n[PROOFSTEP]\nrw [B.toMatrix_compRight b, toMatrix_toLin]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b2\u2070 : Semiring R\ninst\u271d\u00b9\u2079 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2078 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2077 : Ring R\u2081\ninst\u271d\u00b9\u2076 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2075 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2074 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : CommRing R\u2083\ninst\u271d\u00b9\u2070 : AddCommGroup M\u2083\ninst\u271d\u2079 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : AddCommGroup V\ninst\u271d\u2076 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\no : Type u_12\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : Fintype o\ninst\u271d\u00b3 : DecidableEq n\nb : Basis n R\u2082 M\u2082\nM\u2082' : Type u_13\ninst\u271d\u00b2 : AddCommMonoid M\u2082'\ninst\u271d\u00b9 : Module R\u2082 M\u2082'\nc : Basis o R\u2082 M\u2082'\ninst\u271d : DecidableEq o\nM : Matrix n n R\u2082\nP Q : Matrix n o R\u2082\n\u22a2 \u2191(BilinForm.toMatrix c) (BilinForm.comp (\u2191(toBilin b) M) (\u2191(toLin c b) P) (\u2191(toLin c b) Q)) =\n    \u2191(BilinForm.toMatrix c) (\u2191(toBilin c) (P\u1d40 * M * Q))\n[PROOFSTEP]\nsimp only [BilinForm.toMatrix_comp b c, BilinForm.toMatrix_toBilin, toMatrix_toLin]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.IsAdjointPair (\u2191toBilin' J) (\u2191toBilin' J\u2083) (\u2191toLin' A) (\u2191toLin' A') \u2194 IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrw [BilinForm.isAdjointPair_iff_compLeft_eq_compRight]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.compLeft (\u2191toBilin' J\u2083) (\u2191toLin' A) = BilinForm.compRight (\u2191toBilin' J) (\u2191toLin' A') \u2194\n    IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nhave h : \u2200 B B' : BilinForm R\u2083 (n \u2192 R\u2083), B = B' \u2194 BilinForm.toMatrix' B = BilinForm.toMatrix' B' :=\n  by\n  intro B B'\n  constructor <;> intro h\n  \u00b7 rw [h]\n  \u00b7 exact BilinForm.toMatrix'.injective h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 \u2200 (B B' : BilinForm R\u2083 (n \u2192 R\u2083)), B = B' \u2194 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n[PROOFSTEP]\nintro B B'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 (n \u2192 R\u2083)\n\u22a2 B = B' \u2194 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 (n \u2192 R\u2083)\n\u22a2 B = B' \u2192 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 (n \u2192 R\u2083)\n\u22a2 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B' \u2192 B = B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 (n \u2192 R\u2083)\nh : B = B'\n\u22a2 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 (n \u2192 R\u2083)\nh : \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n\u22a2 B = B'\n[PROOFSTEP]\nexact BilinForm.toMatrix'.injective h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 (n \u2192 R\u2083)), B = B' \u2194 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n\u22a2 BilinForm.compLeft (\u2191toBilin' J\u2083) (\u2191toLin' A) = BilinForm.compRight (\u2191toBilin' J) (\u2191toLin' A') \u2194\n    IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrw [h, BilinForm.toMatrix'_compLeft, BilinForm.toMatrix'_compRight, LinearMap.toMatrix'_toLin',\n  LinearMap.toMatrix'_toLin', BilinForm.toMatrix'_toBilin', BilinForm.toMatrix'_toBilin']\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 (n \u2192 R\u2083)), B = B' \u2194 \u2191BilinForm.toMatrix' B = \u2191BilinForm.toMatrix' B'\n\u22a2 A\u1d40 * J\u2083 = J * A' \u2194 IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.IsAdjointPair (\u2191(toBilin b) J) (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrw [BilinForm.isAdjointPair_iff_compLeft_eq_compRight]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 BilinForm.compLeft (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) = BilinForm.compRight (\u2191(toBilin b) J) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nhave h : \u2200 B B' : BilinForm R\u2083 M\u2083, B = B' \u2194 BilinForm.toMatrix b B = BilinForm.toMatrix b B' :=\n  by\n  intro B B'\n  constructor <;> intro h\n  \u00b7 rw [h]\n  \u00b7 exact (BilinForm.toMatrix b).injective h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n[PROOFSTEP]\nintro B B'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\n\u22a2 B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\n\u22a2 B = B' \u2192 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\n\u22a2 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B' \u2192 B = B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : B = B'\n\u22a2 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nB B' : BilinForm R\u2083 M\u2083\nh : \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 B = B'\n[PROOFSTEP]\nexact (BilinForm.toMatrix b).injective h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 BilinForm.compLeft (\u2191(toBilin b) J\u2083) (\u2191(toLin b b) A) = BilinForm.compRight (\u2191(toBilin b) J) (\u2191(toLin b b) A') \u2194\n    IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrw [h, BilinForm.toMatrix_compLeft, BilinForm.toMatrix_compRight, LinearMap.toMatrix_toLin, LinearMap.toMatrix_toLin,\n  BilinForm.toMatrix_toBilin, BilinForm.toMatrix_toBilin]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nh : \u2200 (B B' : BilinForm R\u2083 M\u2083), B = B' \u2194 \u2191(BilinForm.toMatrix b) B = \u2191(BilinForm.toMatrix b) B'\n\u22a2 A\u1d40 * J\u2083 = J * A' \u2194 IsAdjointPair J J\u2083 A A'\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nhave h' : IsUnit P.det := P.isUnit_iff_isUnit_det.mp h\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nlet u := P.nonsingInvUnit h'\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nhave coe_u : (u : Matrix n n R\u2083) = P := rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nhave coe_u_inv : (\u2191u\u207b\u00b9 : Matrix n n R\u2083) = P\u207b\u00b9 := rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nlet v := P\u1d40.nonsingInvUnit (P.isUnit_det_transpose h')\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nhave coe_v : (v : Matrix n n R\u2083) = P\u1d40 := rfl\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nhave coe_v_inv : (\u2191v\u207b\u00b9 : Matrix n n R\u2083) = P\u207b\u00b9\u1d40 := P.transpose_nonsing_inv.symm\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nset x := A\u1d40 * P\u1d40 * J with x_def\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nset y := J * P * A' with y_def\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A A' \u2194 IsAdjointPair J J (P * A * P\u207b\u00b9) (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nsimp only [Matrix.IsAdjointPair]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 A\u1d40 * (P\u1d40 * J * P) = P\u1d40 * J * P * A' \u2194 (P * A * P\u207b\u00b9)\u1d40 * J = J * (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\ncalc\n  (A\u1d40 * (P\u1d40 * J * P) = P\u1d40 * J * P * A') \u2194 (x * \u2191u = \u2191v * y) := ?_\n  _ \u2194 (\u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9) := ?_\n  _ \u2194 ((P * A * P\u207b\u00b9)\u1d40 * J = J * (P * A' * P\u207b\u00b9)) := ?_\n[GOAL]\ncase calc_1\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 A\u1d40 * (P\u1d40 * J * P) = P\u1d40 * J * P * A' \u2194 x * \u2191u = \u2191v * y\n[PROOFSTEP]\nsimp only [mul_assoc, x_def, y_def, coe_u, coe_v]\n[GOAL]\ncase calc_2\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [Units.eq_mul_inv_iff_mul_eq, mul_assoc (\u2191v\u207b\u00b9) x, Units.inv_mul_eq_iff_eq_mul]\n[GOAL]\ncase calc_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9 \u2194 (P * A * P\u207b\u00b9)\u1d40 * J = J * (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nrw [x_def, y_def, coe_u_inv, coe_v_inv]\n[GOAL]\ncase calc_3\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\nP : Matrix n n R\u2083\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P h'\ncoe_u : \u2191u = P\ncoe_u_inv : \u2191u\u207b\u00b9 = P\u207b\u00b9\nv : (Matrix n n R\u2083)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\ncoe_v : \u2191v = P\u1d40\ncoe_v_inv : \u2191v\u207b\u00b9 = P\u207b\u00b9\u1d40\nx : Matrix n n R\u2083 := A\u1d40 * P\u1d40 * J\nx_def : x = A\u1d40 * P\u1d40 * J\ny : Matrix n n R\u2083 := J * P * A'\ny_def : y = J * P * A'\n\u22a2 P\u207b\u00b9\u1d40 * (A\u1d40 * P\u1d40 * J) = J * P * A' * P\u207b\u00b9 \u2194 (P * A * P\u207b\u00b9)\u1d40 * J = J * (P * A' * P\u207b\u00b9)\n[PROOFSTEP]\nsimp only [Matrix.mul_assoc, Matrix.transpose_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 A \u2208 pairSelfAdjointMatricesSubmodule J J\u2083 \u2194 IsAdjointPair J J\u2083 A A\n[PROOFSTEP]\nsimp only [mem_pairSelfAdjointMatricesSubmodule]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 A \u2208 selfAdjointMatricesSubmodule J \u2194 Matrix.IsSelfAdjoint J A\n[PROOFSTEP]\nsimp only [mem_selfAdjointMatricesSubmodule]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2076 : Semiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : Ring R\u2081\ninst\u271d\u00b9\u00b2 : AddCommGroup M\u2081\ninst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u2070 : CommSemiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\u2082\ninst\u271d\u2078 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : CommRing R\u2083\ninst\u271d\u2076 : AddCommGroup M\u2083\ninst\u271d\u2075 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nB : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nn : Type u_11\ninst\u271d\u00b9 : Fintype n\nb : Basis n R\u2083 M\u2083\nJ J\u2083 A A' : Matrix n n R\u2083\ninst\u271d : DecidableEq n\n\u22a2 A \u2208 skewAdjointMatricesSubmodule J \u2194 IsSkewAdjoint J A\n[PROOFSTEP]\nsimp only [mem_skewAdjointMatricesSubmodule]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2077 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_11\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nM : Matrix \u03b9 \u03b9 R\u2083\nh : Matrix.Nondegenerate M\nx : \u03b9 \u2192 R\u2083\nhx : \u2200 (n : \u03b9 \u2192 R\u2083), bilin (\u2191Matrix.toBilin' M) x n = 0\ny : \u03b9 \u2192 R\u2083\n\u22a2 x \u2b1d\u1d65 mulVec M y = 0\n[PROOFSTEP]\nsimpa only [toBilin'_apply'] using hx y\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2077 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_11\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nM : Matrix \u03b9 \u03b9 R\u2083\nb : Basis \u03b9 R\u2083 M\u2083\n\u22a2 Nondegenerate (\u2191(toBilin b) M) \u2194 Matrix.Nondegenerate M\n[PROOFSTEP]\nrw [\u2190 Matrix.nondegenerate_toBilin'_iff_nondegenerate_toBilin, Matrix.nondegenerate_toBilin'_iff]\n[GOAL]\nR : Type u_1\nM\u271d : Type u_2\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\u271d\ninst\u271d\u00b9\u2077 : Module R M\u271d\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB : BilinForm R M\u271d\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_11\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nM : Matrix \u03b9 \u03b9 A\n\u22a2 Nondegenerate (\u2191toBilin' M) \u2194 det M \u2260 0\n[PROOFSTEP]\nrw [Matrix.nondegenerate_toBilin'_iff, Matrix.nondegenerate_iff_det_ne_zero]\n[GOAL]\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2079 : Semiring R\ninst\u271d\u00b9\u2078 : AddCommMonoid M\ninst\u271d\u00b9\u2077 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u2076 : Ring R\u2081\ninst\u271d\u00b9\u2075 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2074 : Module R\u2081 M\u2081\nR\u2082 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b9\u00b3 : CommSemiring R\u2082\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b9 : Module R\u2082 M\u2082\nR\u2083 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u2070 : CommRing R\u2083\ninst\u271d\u2079 : AddCommGroup M\u2083\ninst\u271d\u2078 : Module R\u2083 M\u2083\nV : Type u_9\nK : Type u_10\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module K V\nB\u271d : BilinForm R M\nB\u2081 : BilinForm R\u2081 M\u2081\nB\u2082 : BilinForm R\u2082 M\u2082\nA : Type u_11\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : IsDomain A\ninst\u271d\u00b2 : Module A M\u2083\nB\u2083 : BilinForm A M\u2083\n\u03b9 : Type u_12\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : BilinForm A M\u2083\nb : Basis \u03b9 A M\u2083\n\u22a2 Nondegenerate B \u2194 det (\u2191(toMatrix b) B) \u2260 0\n[PROOFSTEP]\nrw [\u2190 Matrix.nondegenerate_iff_det_ne_zero, nondegenerate_toMatrix_iff]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.BilinearForm", "llama_tokens": 66498, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8856314798554444, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.5381656323527542}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\na\u271d b\u271d : E\nhx : a\u271d \u2208 {v | \u2200 (u : E), u \u2208 K \u2192 inner u v = 0}\nhy : b\u271d \u2208 {v | \u2200 (u : E), u \u2208 K \u2192 inner u v = 0}\nu : E\nhu : u \u2208 K\n\u22a2 inner u (a\u271d + b\u271d) = 0\n[PROOFSTEP]\nrw [inner_add_right, hx u hu, hy u hu, add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nc : \ud835\udd5c\nx : E\nhx :\n  x \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {v | \u2200 (u : E), u \u2208 K \u2192 inner u v = 0},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : E},\n                    a \u2208 {v | \u2200 (u : E), u \u2208 K \u2192 inner u v = 0} \u2192\n                      b \u2208 {v | \u2200 (u : E), u \u2208 K \u2192 inner u v = 0} \u2192 \u2200 (u : E), u \u2208 K \u2192 inner u (a + b) = 0) },\n          zero_mem' := (_ : \u2200 (x : E), x \u2208 K \u2192 inner x 0 = 0) }.toAddSubsemigroup.carrier\nu : E\nhu : u \u2208 K\n\u22a2 inner u (c \u2022 x) = 0\n[PROOFSTEP]\nrw [inner_smul_right, hx u hu, mul_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nv : E\n\u22a2 v \u2208 K\u15ee \u2194 \u2200 (u : E), u \u2208 K \u2192 inner v u = 0\n[PROOFSTEP]\nsimp_rw [mem_orthogonal, inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhu : u \u2208 K\nhv : v \u2208 K\u15ee\n\u22a2 inner v u = 0\n[PROOFSTEP]\nrw [inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhu : u \u2208 K\nhv : v \u2208 K\u15ee\n\u22a2 inner u v = 0\n[PROOFSTEP]\nexact inner_right_of_mem_orthogonal hu hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 v \u2208 (span \ud835\udd5c {u})\u15ee \u2194 inner u v = 0\n[PROOFSTEP]\nrefine' \u27e8inner_right_of_mem_orthogonal (mem_span_singleton_self u), _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 inner u v = 0 \u2192 v \u2208 (span \ud835\udd5c {u})\u15ee\n[PROOFSTEP]\nintro hv w hw\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : w \u2208 span \ud835\udd5c {u}\n\u22a2 inner w v = 0\n[PROOFSTEP]\nrw [mem_span_singleton] at hw \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nw : E\nhw : \u2203 a, a \u2022 u = w\n\u22a2 inner w v = 0\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := hw\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\nhv : inner u v = 0\nc : \ud835\udd5c\n\u22a2 inner (c \u2022 u) v = 0\n[PROOFSTEP]\nsimp [inner_smul_left, hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nu v : E\n\u22a2 v \u2208 (span \ud835\udd5c {u})\u15ee \u2194 inner v u = 0\n[PROOFSTEP]\nrw [mem_orthogonal_singleton_iff_inner_right, inner_eq_zero_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = inner y \u2191v\n\u22a2 x - y \u2208 K\u15ee\n[PROOFSTEP]\nrw [mem_orthogonal']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = inner y \u2191v\n\u22a2 \u2200 (u : E), u \u2208 K \u2192 inner (x - y) u = 0\n[PROOFSTEP]\nintro u hu\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = inner y \u2191v\nu : E\nhu : u \u2208 K\n\u22a2 inner (x - y) u = 0\n[PROOFSTEP]\nrw [inner_sub_left, sub_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner x \u2191v = inner y \u2191v\nu : E\nhu : u \u2208 K\n\u22a2 inner x u = inner y u\n[PROOFSTEP]\nexact h \u27e8u, hu\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner (\u2191v) x = inner (\u2191v) y\n\u22a2 x - y \u2208 K\u15ee\n[PROOFSTEP]\nintro u hu\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner (\u2191v) x = inner (\u2191v) y\nu : E\nhu : u \u2208 K\n\u22a2 inner u (x - y) = 0\n[PROOFSTEP]\nrw [inner_sub_right, sub_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx y : E\nh : \u2200 (v : { x // x \u2208 K }), inner (\u2191v) x = inner (\u2191v) y\nu : E\nhu : u \u2208 K\n\u22a2 inner u x = inner u y\n[PROOFSTEP]\nexact h \u27e8u, hu\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K \u2293 K\u15ee = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K \u2293 K\u15ee \u2264 \u22a5\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x \u2208 K \u2293 K\u15ee \u2192 x \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_inf]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x \u2208 K \u2227 x \u2208 K\u15ee \u2192 x \u2208 \u22a5\n[PROOFSTEP]\nexact fun \u27e8hx, ho\u27e9 => inner_self_eq_zero.1 (ho x hx)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 Disjoint K K\u15ee\n[PROOFSTEP]\nsimp [disjoint_iff, K.inf_orthogonal_eq_bot]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K\u15ee = \u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K\u15ee \u2264 \u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v)\n[PROOFSTEP]\nrw [le_iInf_iff]\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u2200 (i : { x // x \u2208 K }), K\u15ee \u2264 LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191i)\n[PROOFSTEP]\nrintro \u27e8v, hv\u27e9 w hw\n[GOAL]\ncase a.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nv : E\nhv : v \u2208 K\nw : E\nhw : w \u2208 K\u15ee\n\u22a2 w \u2208 LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191{ val := v, property := hv })\n[PROOFSTEP]\nsimpa using hw _ hv\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v) \u2264 K\u15ee\n[PROOFSTEP]\nintro v hv w hw\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nv : E\nhv : v \u2208 \u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v)\nw : E\nhw : w \u2208 K\n\u22a2 inner w v = 0\n[PROOFSTEP]\nsimp only [mem_iInf] at hv \n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nv w : E\nhw : w \u2208 K\nhv : \u2200 (i : { x // x \u2208 K }), v \u2208 LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191i)\n\u22a2 inner w v = 0\n[PROOFSTEP]\nexact hv \u27e8w, hw\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 IsClosed \u2191K\u15ee\n[PROOFSTEP]\nrw [orthogonal_eq_inter K]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 IsClosed \u2191(\u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v))\n[PROOFSTEP]\nhave := fun v : K => ContinuousLinearMap.isClosed_ker (innerSL \ud835\udd5c (v : E))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nthis : \u2200 (v : { x // x \u2208 K }), IsClosed \u2191(LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v))\n\u22a2 IsClosed \u2191(\u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v))\n[PROOFSTEP]\nconvert isClosed_iInter this\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nthis : \u2200 (v : { x // x \u2208 K }), IsClosed \u2191(LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v))\n\u22a2 \u2191(\u2a05 (v : { x // x \u2208 K }), LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191v)) =\n    \u22c2 (i : { x // x \u2208 K }), \u2191(LinearMap.ker (\u2191(innerSL \ud835\udd5c) \u2191i))\n[PROOFSTEP]\nsimp only [iInf_coe]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u22a4\u15ee = \u22a5\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x \u2208 \u22a4\u15ee \u2194 x \u2208 \u22a5\n[PROOFSTEP]\nrw [mem_bot, mem_orthogonal]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 (\u2200 (u : E), u \u2208 \u22a4 \u2192 inner u x = 0) \u2194 x = 0\n[PROOFSTEP]\nexact\n  \u27e8fun h => inner_self_eq_zero.mp (h x mem_top), by\n    rintro rfl\n    simp\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nx : E\n\u22a2 x = 0 \u2192 \u2200 (u : E), u \u2208 \u22a4 \u2192 inner u x = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u2200 (u : E), u \u2208 \u22a4 \u2192 inner u 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u22a5\u15ee = \u22a4\n[PROOFSTEP]\nrw [\u2190 top_orthogonal_eq_bot, eq_top_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 \u22a4 \u2264 \u22a4\u15ee\u15ee\n[PROOFSTEP]\nexact le_orthogonal_orthogonal \u22a4\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K\u15ee = \u22a4 \u2194 K = \u22a5\n[PROOFSTEP]\nrefine'\n  \u27e8_, by\n    rintro rfl\n    exact bot_orthogonal_eq_top\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K = \u22a5 \u2192 K\u15ee = \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\n\u22a2 \u22a5\u15ee = \u22a4\n[PROOFSTEP]\nexact bot_orthogonal_eq_top\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\n\u22a2 K\u15ee = \u22a4 \u2192 K = \u22a5\n[PROOFSTEP]\nintro h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nh : K\u15ee = \u22a4\n\u22a2 K = \u22a5\n[PROOFSTEP]\nhave : K \u2293 K\u15ee = \u22a5 := K.orthogonal_disjoint.eq_bot\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nK : Submodule \ud835\udd5c E\nh : K\u15ee = \u22a4\nthis : K \u2293 K\u15ee = \u22a5\n\u22a2 K = \u22a5\n[PROOFSTEP]\nrwa [h, inf_comm, top_inf_eq] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nU : Submodule \ud835\udd5c E\nV : Set (Submodule \ud835\udd5c E)\n\u22a2 (\u2200 (U\u1d62 : Submodule \ud835\udd5c E), U\u1d62 \u2208 V \u2192 U\u1d62 \u27c2 U) \u2194 \u2200 (V\u1d62 : Submodule \ud835\udd5c E), V\u1d62 \u2208 V \u2192 U \u27c2 V\u1d62\n[PROOFSTEP]\nsimp_rw [isOrtho_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\n\u03b9 : Sort u_4\nU : Submodule \ud835\udd5c E\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\n\u22a2 (\u2200 (i : \u03b9), V i \u27c2 U) \u2194 \u2200 (i : \u03b9), U \u27c2 V i\n[PROOFSTEP]\nsimp_rw [isOrtho_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\ns t : Set E\n\u22a2 span \ud835\udd5c s \u27c2 span \ud835\udd5c t \u2194 \u2200 \u2983u : E\u2984, u \u2208 s \u2192 \u2200 \u2983v : E\u2984, v \u2208 t \u2192 inner u v = 0\n[PROOFSTEP]\nsimp_rw [span_eq_iSup_of_singleton_spans s, span_eq_iSup_of_singleton_spans t, isOrtho_iSup_left, isOrtho_iSup_right,\n  isOrtho_iff_le, span_le, Set.subset_def, SetLike.mem_coe, mem_orthogonal_singleton_iff_inner_left,\n  Set.mem_singleton_iff, forall_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c E\nh : U \u27c2 V\n\u22a2 Submodule.map f U \u27c2 Submodule.map f V\n[PROOFSTEP]\nrw [isOrtho_iff_inner_eq] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c E\nh : \u2200 (u : E), u \u2208 U \u2192 \u2200 (v : E), v \u2208 V \u2192 inner u v = 0\n\u22a2 \u2200 (u : F), u \u2208 Submodule.map f U \u2192 \u2200 (v : F), v \u2208 Submodule.map f V \u2192 inner u v = 0\n[PROOFSTEP]\nsimp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082, LinearIsometry.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c E\nh : \u2200 (u : E), u \u2208 U \u2192 \u2200 (v : E), v \u2208 V \u2192 inner u v = 0\n\u22a2 \u2200 (a : E), a \u2208 U \u2192 \u2200 (a_2 : E), a_2 \u2208 V \u2192 inner a a_2 = 0\n[PROOFSTEP]\nexact h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : U \u27c2 V\n\u22a2 Submodule.comap f U \u27c2 Submodule.comap f V\n[PROOFSTEP]\nrw [isOrtho_iff_inner_eq] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : \u2200 (u : F), u \u2208 U \u2192 \u2200 (v : F), v \u2208 V \u2192 inner u v = 0\n\u22a2 \u2200 (u : E), u \u2208 Submodule.comap f U \u2192 \u2200 (v : E), v \u2208 Submodule.comap f V \u2192 inner u v = 0\n[PROOFSTEP]\nsimp_rw [mem_comap, \u2190 f.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : \u2200 (u : F), u \u2208 U \u2192 \u2200 (v : F), v \u2208 V \u2192 inner u v = 0\n\u22a2 \u2200 (u : E), \u2191f u \u2208 U \u2192 \u2200 (v : E), \u2191f v \u2208 V \u2192 inner (\u2191f u) (\u2191f v) = 0\n[PROOFSTEP]\nintro u hu v hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2192\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : \u2200 (u : F), u \u2208 U \u2192 \u2200 (v : F), v \u2208 V \u2192 inner u v = 0\nu : E\nhu : \u2191f u \u2208 U\nv : E\nhv : \u2191f v \u2208 V\n\u22a2 inner (\u2191f u) (\u2191f v) = 0\n[PROOFSTEP]\nexact h _ hu _ hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c E\nh : Submodule.map f U \u27c2 Submodule.map f V\n\u22a2 U \u27c2 V\n[PROOFSTEP]\nhave hf : \u2200 p : Submodule \ud835\udd5c E, (p.map f).comap f.toLinearIsometry = p := comap_map_eq_of_injective f.injective\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c E\nh : Submodule.map f U \u27c2 Submodule.map f V\nhf : \u2200 (p : Submodule \ud835\udd5c E), Submodule.comap (LinearIsometryEquiv.toLinearIsometry f) (Submodule.map f p) = p\n\u22a2 U \u27c2 V\n[PROOFSTEP]\nsimpa only [hf] using h.comap f.toLinearIsometry\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : Submodule.comap f U \u27c2 Submodule.comap f V\n\u22a2 U \u27c2 V\n[PROOFSTEP]\nhave hf : \u2200 p : Submodule \ud835\udd5c F, (p.comap f).map f.toLinearIsometry = p := map_comap_eq_of_surjective f.surjective\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \ud835\udd5c F\nf : E \u2243\u2097\u1d62[\ud835\udd5c] F\nU V : Submodule \ud835\udd5c F\nh : Submodule.comap f U \u27c2 Submodule.comap f V\nhf : \u2200 (p : Submodule \ud835\udd5c F), Submodule.map (LinearIsometryEquiv.toLinearIsometry f) (Submodule.comap f p) = p\n\u22a2 U \u27c2 V\n[PROOFSTEP]\nsimpa only [hf] using h.map f.toLinearIsometry\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Orthogonal", "llama_tokens": 10313, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339516289533, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.5379636894629595}}
{"text": "[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Iic b = Iic (\u2191(symm e) b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u2191e \u207b\u00b9' Iic b \u2194 x \u2208 Iic (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 e.le_iff_le]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Ici b = Ici (\u2191(symm e) b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u2191e \u207b\u00b9' Ici b \u2194 x \u2208 Ici (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 e.le_iff_le]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Iio b = Iio (\u2191(symm e) b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u2191e \u207b\u00b9' Iio b \u2194 x \u2208 Iio (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 e.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Ioi b = Ioi (\u2191(symm e) b)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\nb : \u03b2\nx : \u03b1\n\u22a2 x \u2208 \u2191e \u207b\u00b9' Ioi b \u2194 x \u2208 Ioi (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 e.lt_iff_lt]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Icc a b = Icc (\u2191(symm e) a) (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Ico a b = Ico (\u2191(symm e) a) (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Ioc a b = Ioc (\u2191(symm e) a) (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b2\n\u22a2 \u2191e \u207b\u00b9' Ioo a b = Ioo (\u2191(symm e) a) (\u2191(symm e) b)\n[PROOFSTEP]\nsimp [\u2190 Ioi_inter_Iio]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191e '' Iic a = Iic (\u2191e a)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na : \u03b1\n\u22a2 \u2191e '' Iio a = Iio (\u2191e a)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b1\n\u22a2 \u2191e '' Ioo a b = Ioo (\u2191e a) (\u2191e b)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b1\n\u22a2 \u2191e '' Ioc a b = Ioc (\u2191e a) (\u2191e b)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b1\n\u22a2 \u2191e '' Ico a b = Ico (\u2191e a) (\u2191e b)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Ico, e.symm_symm]\n[GOAL]\n\u03b1 : Type u_2\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\na b : \u03b1\n\u22a2 \u2191e '' Icc a b = Icc (\u2191e a) (\u2191e b)\n[PROOFSTEP]\nrw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]\n[GOAL]\n\u03b1 : Type ?u.15052\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderTop \u03b1\nsrc\u271d : Subtype (Iic \u22a4) \u2243 \u03b1 := Equiv.subtypeUnivEquiv (_ : \u2200 (x : \u03b1), x \u2264 \u22a4)\nx y : \u2191(Iic \u22a4)\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        x \u2264\n      \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        y \u2194\n    x \u2264 y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.15647\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderBot \u03b1\nsrc\u271d : Subtype (Ici \u22a5) \u2243 \u03b1 := Equiv.subtypeUnivEquiv (_ : \u2200 (x : \u03b1), \u22a5 \u2264 x)\nx y : \u2191(Ici \u22a5)\n\u22a2 \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        x \u2264\n      \u2191{ toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        y \u2194\n    x \u2264 y\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.OrderIso", "llama_tokens": 2445, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.6825737214979745, "lm_q1q2_score": 0.5378211216961064}}
{"text": "[GOAL]\nx : \u211d\nhx : x \u2260 0\n\u22a2 HasStrictDerivAt sqrt (1 / (2 * sqrt x)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\ncases' hx.lt_or_lt with hx hx\n[GOAL]\ncase inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\n\u22a2 HasStrictDerivAt sqrt (1 / (2 * sqrt x)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nrw [sqrt_eq_zero_of_nonpos hx.le, mul_zero, div_zero]\n[GOAL]\ncase inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\n\u22a2 HasStrictDerivAt sqrt 0 x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nhave : sqrt =\u1da0[\ud835\udcdd x] fun _ => 0 := (gt_mem_nhds hx).mono fun x hx => sqrt_eq_zero_of_nonpos hx.le\n[GOAL]\ncase inl\nx : \u211d\nhx\u271d : x \u2260 0\nhx : x < 0\nthis : sqrt =\u1da0[\ud835\udcdd x] fun x => 0\n\u22a2 HasStrictDerivAt sqrt 0 x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nexact\n  \u27e8(hasStrictDerivAt_const x (0 : \u211d)).congr_of_eventuallyEq this.symm, fun n =>\n    contDiffAt_const.congr_of_eventuallyEq this\u27e9\n[GOAL]\ncase inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\n\u22a2 HasStrictDerivAt sqrt (1 / (2 * sqrt x)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nhave : \u21912 * sqrt x ^ (2 - 1) \u2260 0 := by simp [(sqrt_pos.2 hx).ne', @two_ne_zero \u211d]\n[GOAL]\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\n\u22a2 2 * sqrt x ^ (2 - 1) \u2260 0\n[PROOFSTEP]\nsimp [(sqrt_pos.2 hx).ne', @two_ne_zero \u211d]\n[GOAL]\ncase inr\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * sqrt x ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt sqrt (1 / (2 * sqrt x)) x \u2227 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase inr.left\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * sqrt x ^ (2 - 1) \u2260 0\n\u22a2 HasStrictDerivAt sqrt (1 / (2 * sqrt x)) x\n[PROOFSTEP]\nsimpa using sqLocalHomeomorph.hasStrictDerivAt_symm hx this (hasStrictDerivAt_pow 2 _)\n[GOAL]\ncase inr.right\nx : \u211d\nhx\u271d : x \u2260 0\nhx : 0 < x\nthis : 2 * sqrt x ^ (2 - 1) \u2260 0\n\u22a2 \u2200 (n : \u2115\u221e), ContDiffAt \u211d n sqrt x\n[PROOFSTEP]\nexact fun n => sqLocalHomeomorph.contDiffAt_symm_deriv this hx (hasDerivAt_pow 2 (sqrt x)) (contDiffAt_id.pow 2)\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nf' x : \u211d\nhf : HasDerivWithinAt f f' s x\nhx : f x \u2260 0\n\u22a2 HasDerivWithinAt (fun y => Real.sqrt (f y)) (f' / (2 * Real.sqrt (f x))) s x\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), div_eq_inv_mul, mul_one] using (hasDerivAt_sqrt hx).comp_hasDerivWithinAt x hf\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nf' x : \u211d\nhf : HasDerivAt f f' x\nhx : f x \u2260 0\n\u22a2 HasDerivAt (fun y => Real.sqrt (f y)) (f' / (2 * Real.sqrt (f x))) x\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), div_eq_inv_mul, mul_one] using (hasDerivAt_sqrt hx).comp x hf\n[GOAL]\nf : \u211d \u2192 \u211d\ns : Set \u211d\nf' x : \u211d\nhf : HasStrictDerivAt f f' x\nhx : f x \u2260 0\n\u22a2 HasStrictDerivAt (fun t => Real.sqrt (f t)) (f' / (2 * Real.sqrt (f x))) x\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), div_eq_inv_mul, mul_one] using (hasStrictDerivAt_sqrt hx).comp x hf\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Sqrt", "llama_tokens": 1411, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.6992544335934766, "lm_q1q2_score": 0.5377827655608287}}
{"text": "[GOAL]\nK : Type v\nV : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\ninst\u271d\u00b2 : IsAlgClosed K\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : Nontrivial V\nf : End K V\n\u22a2 \u2203 c, HasEigenvalue f c\n[PROOFSTEP]\nsimp_rw [hasEigenvalue_iff_mem_spectrum]\n[GOAL]\nK : Type v\nV : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\ninst\u271d\u00b2 : IsAlgClosed K\ninst\u271d\u00b9 : FiniteDimensional K V\ninst\u271d : Nontrivial V\nf : End K V\n\u22a2 \u2203 c, c \u2208 spectrum K f\n[PROOFSTEP]\nexact spectrum.nonempty_of_isAlgClosed_of_finiteDimensional K f\n[GOAL]\nK : Type v\nV : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : IsAlgClosed K\ninst\u271d : FiniteDimensional K V\nf : End K V\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\ninduction' h_dim : finrank K V using Nat.strong_induction_on with n ih generalizing V\n[GOAL]\ncase h\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nh_dim : finrank K V = n\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase h.zero\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nih :\n  \u2200 (m : \u2115),\n    m < Nat.zero \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.zero\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nrw [\u2190 top_le_iff]\n[GOAL]\ncase h.zero\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nih :\n  \u2200 (m : \u2115),\n    m < Nat.zero \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.zero\n\u22a2 \u22a4 \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n[PROOFSTEP]\nsimp only [finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le]\n  -- Otherwise the vector space is nontrivial.\n[GOAL]\ncase h.succ\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhaveI : Nontrivial V :=\n  finrank_pos_iff.1\n    (by rw [h_dim]; apply Nat.zero_lt_succ)\n      -- Hence, `f` has an eigenvalue `\u03bc\u2080`.\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\n\u22a2 0 < finrank ?m.10065 V\n[PROOFSTEP]\nrw [h_dim]\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\n\u22a2 0 < Nat.succ n\n[PROOFSTEP]\napply Nat.zero_lt_succ\n[GOAL]\ncase h.succ\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nobtain \u27e8\u03bc\u2080, h\u03bc\u2080\u27e9 : \u2203 \u03bc\u2080, f.HasEigenvalue \u03bc\u2080 := exists_eigenvalue f\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nlet ES :=\n  f.generalizedEigenspace \u03bc\u2080\n    (finrank K V)\n      -- and `ER` to be the generalized eigenrange.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nlet ER :=\n  f.generalizedEigenrange \u03bc\u2080\n    (finrank K V)\n      -- `f` maps `ER` into itself.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave h_f_ER : \u2200 x : V, x \u2208 ER \u2192 f x \u2208 ER := fun x hx =>\n  map_generalizedEigenrange_le\n    (Submodule.mem_map_of_mem hx)\n      -- Therefore, we can define the restriction `f'` of `f` to `ER`.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nlet f' : End K ER := f.restrict h_f_ER\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave h_dim_ES_pos : 0 < finrank K ES := by\n  dsimp only\n  rw [h_dim]\n  apply\n    pos_finrank_generalizedEigenspace_of_hasEigenvalue h\u03bc\u2080\n      (Nat.zero_lt_succ n)\n        -- and the dimensions of `ES` and `ER` add up to `finrank K V`.\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\n\u22a2 0 < finrank K { x // x \u2208 ES }\n[PROOFSTEP]\ndsimp only\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\n\u22a2 0 < finrank K { x // x \u2208 \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V) }\n[PROOFSTEP]\nrw [h_dim]\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\n\u22a2 0 < finrank K { x // x \u2208 \u2191(generalizedEigenspace f \u03bc\u2080) (Nat.succ n) }\n[PROOFSTEP]\napply\n  pos_finrank_generalizedEigenspace_of_hasEigenvalue h\u03bc\u2080\n    (Nat.zero_lt_succ n)\n      -- and the dimensions of `ES` and `ER` add up to `finrank K V`.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave h_dim_add : finrank K ER + finrank K ES = finrank K V := by apply LinearMap.finrank_range_add_finrank_ker\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\n\u22a2 finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\n[PROOFSTEP]\napply LinearMap.finrank_range_add_finrank_ker\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave h_dim_ER : finrank K ER < n.succ := by\n  linarith\n    -- This allows us to apply the induction hypothesis on `ER`:\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\n\u22a2 finrank K { x // x \u2208 ER } < Nat.succ n\n[PROOFSTEP]\nlinarith\n  -- This allows us to apply the induction hypothesis on `ER`:\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave ih_ER : \u2a06 (\u03bc : K) (k : \u2115), f'.generalizedEigenspace \u03bc k = \u22a4 := ih (finrank K ER) h_dim_ER f' rfl\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave ih_ER' : \u2a06 (\u03bc : K) (k : \u2115), (f'.generalizedEigenspace \u03bc k).map ER.subtype = ER := by\n  simp only [(Submodule.map_iSup _ _).symm, ih_ER, Submodule.map_subtype_top ER]\n    -- Moreover, every generalized eigenspace of `f'` is contained in the corresponding generalized\n        -- eigenspace of `f`.\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\n[PROOFSTEP]\nsimp only [(Submodule.map_iSup _ _).symm, ih_ER, Submodule.map_subtype_top ER]\n  -- Moreover, every generalized eigenspace of `f'` is contained in the corresponding generalized\n      -- eigenspace of `f`.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave hff' : \u2200 \u03bc k, (f'.generalizedEigenspace \u03bc k).map ER.subtype \u2264 f.generalizedEigenspace \u03bc k :=\n  by\n  intros\n  rw [generalizedEigenspace_restrict]\n  apply Submodule.map_comap_le\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\n\u22a2 \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\n[PROOFSTEP]\nintros\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\n\u03bc\u271d : K\nk\u271d : \u2115\n\u22a2 Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc\u271d) k\u271d) \u2264 \u2191(generalizedEigenspace f \u03bc\u271d) k\u271d\n[PROOFSTEP]\nrw [generalizedEigenspace_restrict]\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\n\u03bc\u271d : K\nk\u271d : \u2115\n\u22a2 Submodule.map (Submodule.subtype ER) (Submodule.comap (Submodule.subtype ER) (\u2191(generalizedEigenspace f \u03bc\u271d) k\u271d)) \u2264\n    \u2191(generalizedEigenspace f \u03bc\u271d) k\u271d\n[PROOFSTEP]\napply Submodule.map_comap_le\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave hER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), f.generalizedEigenspace \u03bc k :=\n  by\n  rw [\u2190 ih_ER']\n  exact iSup\u2082_mono hff'\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\n\u22a2 ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n[PROOFSTEP]\nrw [\u2190 ih_ER']\n[GOAL]\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264\n    \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n[PROOFSTEP]\nexact iSup\u2082_mono hff'\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\nhER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave hES : ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), f.generalizedEigenspace \u03bc k :=\n  le_trans (le_iSup (fun k => f.generalizedEigenspace \u03bc\u2080 k) (finrank K V))\n    (le_iSup (fun \u03bc : K => \u2a06 k : \u2115, f.generalizedEigenspace \u03bc k) \u03bc\u2080)\n      -- Moreover, we know that `ER` and `ES` are disjoint.\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\nhER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nhES : ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nhave h_disjoint : Disjoint ER ES := generalized_eigenvec_disjoint_range_ker f \u03bc\u2080\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\nhER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nhES : ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nh_disjoint : Disjoint ER ES\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nshow \u2a06 (\u03bc : K) (k : \u2115), f.generalizedEigenspace \u03bc k = \u22a4\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\nhER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nhES : ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nh_disjoint : Disjoint ER ES\n\u22a2 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\n[PROOFSTEP]\nrw [\u2190 top_le_iff, \u2190 Submodule.eq_top_of_disjoint ER ES h_dim_add h_disjoint]\n[GOAL]\ncase h.succ.intro\nK : Type v\nV\u271d : Type w\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : AddCommGroup V\u271d\ninst\u271d\u2075 : Module K V\u271d\ninst\u271d\u2074 : IsAlgClosed K\ninst\u271d\u00b3 : FiniteDimensional K V\u271d\nf\u271d : End K V\u271d\nx\u271d : \u2115\nh_dim\u271d : finrank K V\u271d = x\u271d\nV : Type w\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : FiniteDimensional K V\nf : End K V\nn : \u2115\nih :\n  \u2200 (m : \u2115),\n    m < Nat.succ n \u2192\n      \u2200 {V : Type w} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] (f : End K V),\n        finrank K V = m \u2192 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k = \u22a4\nh_dim : finrank K V = Nat.succ n\nthis : Nontrivial V\n\u03bc\u2080 : K\nh\u03bc\u2080 : HasEigenvalue f \u03bc\u2080\nES : Submodule K V := \u2191(generalizedEigenspace f \u03bc\u2080) (finrank K V)\nER : Submodule K V := generalizedEigenrange f \u03bc\u2080 (finrank K V)\nh_f_ER : \u2200 (x : V), x \u2208 ER \u2192 \u2191f x \u2208 ER\nf' : End K { x // x \u2208 ER } := LinearMap.restrict f h_f_ER\nh_dim_ES_pos : 0 < finrank K { x // x \u2208 ES }\nh_dim_add : finrank K { x // x \u2208 ER } + finrank K { x // x \u2208 ES } = finrank K V\nh_dim_ER : finrank K { x // x \u2208 ER } < Nat.succ n\nih_ER : \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f' \u03bc) k = \u22a4\nih_ER' : \u2a06 (\u03bc : K) (k : \u2115), Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) = ER\nhff' :\n  \u2200 (\u03bc : K) (k : \u2115),\n    Submodule.map (Submodule.subtype ER) (\u2191(generalizedEigenspace f' \u03bc) k) \u2264 \u2191(generalizedEigenspace f \u03bc) k\nhER : ER \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nhES : ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\nh_disjoint : Disjoint ER ES\n\u22a2 ER \u2294 ES \u2264 \u2a06 (\u03bc : K) (k : \u2115), \u2191(generalizedEigenspace f \u03bc) k\n[PROOFSTEP]\napply sup_le hER hES\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Eigenspace.IsAlgClosed", "llama_tokens": 19866, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802476562641, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.5377827585025055}}
{"text": "[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\n\u22a2 \u2200 (s : Set \u2124) (a : \u2124), BddAbove s \u2192 a \u2208 s \u2192 a \u2264 sSup s\n[PROOFSTEP]\nintro s n hs hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddAbove s\nhns : n \u2208 s\n\u22a2 n \u2264 sSup s\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddAbove s :=\n  \u27e8\u27e8n, hns\u27e9, hs\u27e9\n    -- Porting note: this was `rw [dif_pos this]`\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddAbove s\nhns : n \u2208 s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 n \u2264 sSup s\n[PROOFSTEP]\nsimp only [this, and_self, dite_true, ge_iff_le]\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddAbove s\nhns : n \u2208 s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 n \u2264 \u2191(greatestOfBdd (choose (_ : BddAbove s)) (_ : choose (_ : BddAbove s) \u2208 upperBounds s) (_ : Set.Nonempty s))\n[PROOFSTEP]\nexact (greatestOfBdd _ _ _).2.2 n hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\n\u22a2 \u2200 (s : Set \u2124) (a : \u2124), Set.Nonempty s \u2192 a \u2208 upperBounds s \u2192 sSup s \u2264 a\n[PROOFSTEP]\nintro s n hs hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 upperBounds s\n\u22a2 sSup s \u2264 n\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddAbove s :=\n  \u27e8hs, \u27e8n, hns\u27e9\u27e9\n    -- Porting note: this was `rw [dif_pos this]`\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 upperBounds s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 sSup s \u2264 n\n[PROOFSTEP]\nsimp only [this, and_self, dite_true, ge_iff_le]\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 upperBounds s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 \u2191(greatestOfBdd (choose (_ : BddAbove s)) (_ : choose (_ : BddAbove s) \u2208 upperBounds s) (_ : Set.Nonempty s)) \u2264 n\n[PROOFSTEP]\nexact hns (greatestOfBdd _ (Classical.choose_spec this.2) _).2.1\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\n\u22a2 \u2200 (s : Set \u2124) (a : \u2124), BddBelow s \u2192 a \u2208 s \u2192 sInf s \u2264 a\n[PROOFSTEP]\nintro s n hs hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddBelow s\nhns : n \u2208 s\n\u22a2 sInf s \u2264 n\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddBelow s :=\n  \u27e8\u27e8n, hns\u27e9, hs\u27e9\n    -- Porting note: this was `rw [dif_pos this]`\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddBelow s\nhns : n \u2208 s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 sInf s \u2264 n\n[PROOFSTEP]\nsimp only [this, and_self, dite_true, ge_iff_le]\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : BddBelow s\nhns : n \u2208 s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 \u2191(leastOfBdd (choose (_ : BddBelow s)) (_ : choose (_ : BddBelow s) \u2208 lowerBounds s) (_ : Set.Nonempty s)) \u2264 n\n[PROOFSTEP]\nexact (leastOfBdd _ _ _).2.2 n hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\n\u22a2 \u2200 (s : Set \u2124) (a : \u2124), Set.Nonempty s \u2192 a \u2208 lowerBounds s \u2192 a \u2264 sInf s\n[PROOFSTEP]\nintro s n hs hns\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 lowerBounds s\n\u22a2 n \u2264 sInf s\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddBelow s :=\n  \u27e8hs, \u27e8n, hns\u27e9\u27e9\n    -- Porting note: this was `rw [dif_pos this]`\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 lowerBounds s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 n \u2264 sInf s\n[PROOFSTEP]\nsimp only [this, and_self, dite_true, ge_iff_le]\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nn : \u2124\nhs : Set.Nonempty s\nhns : n \u2208 lowerBounds s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 n \u2264 \u2191(leastOfBdd (choose (_ : BddBelow s)) (_ : choose (_ : BddBelow s) \u2208 lowerBounds s) (_ : Set.Nonempty s))\n[PROOFSTEP]\nexact hns (leastOfBdd _ (Classical.choose_spec this.2) _).2.1\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nhs : \u00acBddAbove s\n\u22a2 sSup s = sSup Set.univ\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\nsrc\u271d\u00b9 : LinearOrderedCommRing \u2124 := linearOrderedCommRing\nsrc\u271d : Lattice \u2124 := LinearOrder.toLattice\ns : Set \u2124\nhs : \u00acBddBelow s\n\u22a2 sInf s = sInf Set.univ\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 z \u2264 b\nHinh : \u2203 z, z \u2208 s\n\u22a2 sSup s = \u2191(greatestOfBdd b Hb Hinh)\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddAbove s := \u27e8Hinh, b, Hb\u27e9\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 z \u2264 b\nHinh : \u2203 z, z \u2208 s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 sSup s = \u2191(greatestOfBdd b Hb Hinh)\n[PROOFSTEP]\nsimp only [sSup, this, and_self, dite_true]\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 z \u2264 b\nHinh : \u2203 z, z \u2208 s\nthis : Set.Nonempty s \u2227 BddAbove s\n\u22a2 \u2191(greatestOfBdd (choose (_ : BddAbove s)) (_ : choose (_ : BddAbove s) \u2208 upperBounds s) (_ : Set.Nonempty s)) =\n    \u2191(greatestOfBdd b Hb Hinh)\n[PROOFSTEP]\nconvert (coe_greatestOfBdd_eq Hb (Classical.choose_spec (\u27e8b, Hb\u27e9 : BddAbove s)) Hinh).symm\n[GOAL]\n\u22a2 \u00ac(Set.Nonempty \u2205 \u2227 BddAbove \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ns : Set \u2124\nh : \u00acBddAbove s\n\u22a2 \u00ac(Set.Nonempty s \u2227 BddAbove s)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 b \u2264 z\nHinh : \u2203 z, z \u2208 s\n\u22a2 sInf s = \u2191(leastOfBdd b Hb Hinh)\n[PROOFSTEP]\nhave : s.Nonempty \u2227 BddBelow s := \u27e8Hinh, b, Hb\u27e9\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 b \u2264 z\nHinh : \u2203 z, z \u2208 s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 sInf s = \u2191(leastOfBdd b Hb Hinh)\n[PROOFSTEP]\nsimp only [sInf, this, and_self, dite_true]\n[GOAL]\ns : Set \u2124\ninst\u271d : DecidablePred fun x => x \u2208 s\nb : \u2124\nHb : \u2200 (z : \u2124), z \u2208 s \u2192 b \u2264 z\nHinh : \u2203 z, z \u2208 s\nthis : Set.Nonempty s \u2227 BddBelow s\n\u22a2 \u2191(leastOfBdd (choose (_ : BddBelow s)) (_ : choose (_ : BddBelow s) \u2208 lowerBounds s) (_ : Set.Nonempty s)) =\n    \u2191(leastOfBdd b Hb Hinh)\n[PROOFSTEP]\nconvert (coe_leastOfBdd_eq Hb (Classical.choose_spec (\u27e8b, Hb\u27e9 : BddBelow s)) Hinh).symm\n[GOAL]\n\u22a2 \u00ac(Set.Nonempty \u2205 \u2227 BddBelow \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ns : Set \u2124\nh : \u00acBddBelow s\n\u22a2 \u00ac(Set.Nonempty s \u2227 BddBelow s)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ns : Set \u2124\nh1 : Set.Nonempty s\nh2 : BddAbove s\n\u22a2 sSup s \u2208 s\n[PROOFSTEP]\nconvert (greatestOfBdd _ (Classical.choose_spec h2) h1).2.1\n[GOAL]\ncase h.e'_4\ns : Set \u2124\nh1 : Set.Nonempty s\nh2 : BddAbove s\n\u22a2 sSup s = \u2191(greatestOfBdd (choose h2) (_ : choose h2 \u2208 upperBounds s) h1)\n[PROOFSTEP]\nexact dif_pos \u27e8h1, h2\u27e9\n[GOAL]\ns : Set \u2124\nh1 : Set.Nonempty s\nh2 : BddBelow s\n\u22a2 sInf s \u2208 s\n[PROOFSTEP]\nconvert (leastOfBdd _ (Classical.choose_spec h2) h1).2.1\n[GOAL]\ncase h.e'_4\ns : Set \u2124\nh1 : Set.Nonempty s\nh2 : BddBelow s\n\u22a2 sInf s = \u2191(leastOfBdd (choose h2) (_ : choose h2 \u2208 lowerBounds s) h1)\n[PROOFSTEP]\nexact dif_pos \u27e8h1, h2\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.ConditionallyCompleteOrder", "llama_tokens": 3618, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527631, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.5376964560751252}}
{"text": "[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng h : b \u27f6 c\n\u03b7 : g = h\n\u22a2 f \u25c1 eqToHom \u03b7 = eqToHom (_ : f \u226b g = f \u226b h)\n[PROOFSTEP]\ncases \u03b7\n[GOAL]\ncase refl\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\ng : b \u27f6 c\n\u22a2 f \u25c1 eqToHom (_ : g = g) = eqToHom (_ : f \u226b g = f \u226b g)\n[PROOFSTEP]\nsimp only [whiskerLeft_id, eqToHom_refl]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf g : a \u27f6 b\n\u03b7 : f = g\nh : b \u27f6 c\n\u22a2 eqToHom \u03b7 \u25b7 h = eqToHom (_ : f \u226b h = g \u226b h)\n[PROOFSTEP]\ncases \u03b7\n[GOAL]\ncase refl\nB : Type u\ninst\u271d : Bicategory B\na b c : B\nf : a \u27f6 b\nh : b \u27f6 c\n\u22a2 eqToHom (_ : f = f) \u25b7 h = eqToHom (_ : f \u226b h = f \u226b h)\n[PROOFSTEP]\nsimp only [id_whiskerRight, eqToHom_refl]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Bicategory.Strict", "llama_tokens": 374, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.6893056167854461, "lm_q1q2_score": 0.537653487197926}}
{"text": "[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nt : Term L \u03b1\ng : \u03b1 \u2192 \u03b2\nv : \u03b2 \u2192 M\n\u22a2 realize v (relabel g t) = realize (v \u2218 g) t\n[PROOFSTEP]\ninduction' t with _ n f ts ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : \u03b1 \u2192 \u03b2\nv : \u03b2 \u2192 M\n_a\u271d : \u03b1\n\u22a2 realize v (relabel g (var _a\u271d)) = realize (v \u2218 g) (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : \u03b1 \u2192 \u03b2\nv : \u03b2 \u2192 M\nn : \u2115\nf : Functions L n\nts : Fin n \u2192 Term L \u03b1\nih : \u2200 (a : Fin n), realize v (relabel g (ts a)) = realize (v \u2218 g) (ts a)\n\u22a2 realize v (relabel g (func f ts)) = realize (v \u2218 g) (func f ts)\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 1\nt : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 realize v (Functions.apply\u2081 f t) = funMap f ![realize v t]\n[PROOFSTEP]\nrw [Functions.apply\u2081, Term.realize]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 1\nt : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 (funMap f fun i => realize v (Matrix.vecCons t ![] i)) = funMap f ![realize v t]\n[PROOFSTEP]\nrefine' congr rfl (funext fun i => _)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 1\nt : Term L \u03b1\nv : \u03b1 \u2192 M\ni : Fin 1\n\u22a2 realize v (Matrix.vecCons t ![] i) = Matrix.vecCons (realize v t) ![] i\n[PROOFSTEP]\nsimp only [Matrix.cons_val_fin_one]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 2\nt\u2081 t\u2082 : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 realize v (Functions.apply\u2082 f t\u2081 t\u2082) = funMap f ![realize v t\u2081, realize v t\u2082]\n[PROOFSTEP]\nrw [Functions.apply\u2082, Term.realize]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 2\nt\u2081 t\u2082 : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 (funMap f fun i => realize v (Matrix.vecCons t\u2081 ![t\u2082] i)) = funMap f ![realize v t\u2081, realize v t\u2082]\n[PROOFSTEP]\nrefine' congr rfl (funext (Fin.cases _ _))\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 2\nt\u2081 t\u2082 : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 realize v (Matrix.vecCons t\u2081 ![t\u2082] 0) = Matrix.vecCons (realize v t\u2081) ![realize v t\u2082] 0\n[PROOFSTEP]\nsimp only [Matrix.cons_val_zero]\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : Functions L 2\nt\u2081 t\u2082 : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 \u2200 (i : Fin 1), realize v (Matrix.vecCons t\u2081 ![t\u2082] (succ i)) = Matrix.vecCons (realize v t\u2081) ![realize v t\u2082] (succ i)\n[PROOFSTEP]\nsimp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nt : Term L \u03b1\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\n\u22a2 realize v (subst t tf) = realize (fun a => realize v (tf a)) t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\n_a\u271d : \u03b1\n\u22a2 realize v (subst (var _a\u271d) tf) = realize (fun a => realize v (tf a)) (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), realize v (subst (_ts\u271d a) tf) = realize (fun a => realize v (tf a)) (_ts\u271d a)\n\u22a2 realize v (subst (func _f\u271d _ts\u271d) tf) = realize (fun a => realize v (tf a)) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\nt : Term L \u03b1\ns : Set \u03b1\nh : \u2191(varFinset t) \u2286 s\nv : \u03b1 \u2192 M\n\u22a2 realize (v \u2218 Subtype.val) (restrictVar t (Set.inclusion h)) = realize v t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\nt : Term L \u03b1\ns : Set \u03b1\nh\u271d : \u2191(varFinset t) \u2286 s\nv : \u03b1 \u2192 M\n_a\u271d : \u03b1\nh : \u2191(varFinset (var _a\u271d)) \u2286 s\n\u22a2 realize (v \u2218 Subtype.val) (restrictVar (var _a\u271d) (Set.inclusion h)) = realize v (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\nt : Term L \u03b1\ns : Set \u03b1\nh\u271d : \u2191(varFinset t) \u2286 s\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih :\n  \u2200 (a : Fin l\u271d) (h : \u2191(varFinset (_ts\u271d a)) \u2286 s),\n    realize (v \u2218 Subtype.val) (restrictVar (_ts\u271d a) (Set.inclusion h)) = realize v (_ts\u271d a)\nh : \u2191(varFinset (func _f\u271d _ts\u271d)) \u2286 s\n\u22a2 realize (v \u2218 Subtype.val) (restrictVar (func _f\u271d _ts\u271d) (Set.inclusion h)) = realize v (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp_rw [varFinset, Finset.coe_biUnion, Set.iUnion_subset_iff] at h \n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\nt : Term L \u03b1\ns : Set \u03b1\nh\u271d\u00b9 : \u2191(varFinset t) \u2286 s\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih :\n  \u2200 (a : Fin l\u271d) (h : \u2191(varFinset (_ts\u271d a)) \u2286 s),\n    realize (v \u2218 Subtype.val) (restrictVar (_ts\u271d a) (Set.inclusion h)) = realize v (_ts\u271d a)\nh\u271d : \u2191(Finset.biUnion Finset.univ fun i => varFinset (_ts\u271d i)) \u2286 s\nh : \u2200 (i : Fin l\u271d), i \u2208 \u2191Finset.univ \u2192 \u2191(varFinset (_ts\u271d i)) \u2286 s\n\u22a2 realize (v \u2218 Subtype.val) (restrictVar (func _f\u271d _ts\u271d) (Set.inclusion h\u271d)) = realize v (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nexact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft t (Set.inclusion h)) = realize (Sum.elim v xs) t\n[PROOFSTEP]\ninduction' t with a _ _ _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh\u271d : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\na : \u03b1 \u2295 \u03b3\nh : \u2191(varFinsetLeft (var a)) \u2286 s\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (var a) (Set.inclusion h)) = realize (Sum.elim v xs) (var a)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase var.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh\u271d : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\nval\u271d : \u03b1\nh : \u2191(varFinsetLeft (var (Sum.inl val\u271d))) \u2286 s\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (var (Sum.inl val\u271d)) (Set.inclusion h)) =\n    realize (Sum.elim v xs) (var (Sum.inl val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase var.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh\u271d : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\nval\u271d : \u03b3\nh : \u2191(varFinsetLeft (var (Sum.inr val\u271d))) \u2286 s\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (var (Sum.inr val\u271d)) (Set.inclusion h)) =\n    realize (Sum.elim v xs) (var (Sum.inr val\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh\u271d : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 \u03b3)\nih :\n  \u2200 (a : Fin l\u271d) (h : \u2191(varFinsetLeft (_ts\u271d a)) \u2286 s),\n    realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (_ts\u271d a) (Set.inclusion h)) =\n      realize (Sum.elim v xs) (_ts\u271d a)\nh : \u2191(varFinsetLeft (func _f\u271d _ts\u271d)) \u2286 s\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (func _f\u271d _ts\u271d) (Set.inclusion h)) =\n    realize (Sum.elim v xs) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h \n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d : DecidableEq \u03b1\n\u03b3 : Type u_3\nt : Term L (\u03b1 \u2295 \u03b3)\ns : Set \u03b1\nh\u271d\u00b9 : \u2191(varFinsetLeft t) \u2286 s\nv : \u03b1 \u2192 M\nxs : \u03b3 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 \u03b3)\nih :\n  \u2200 (a : Fin l\u271d) (h : \u2191(varFinsetLeft (_ts\u271d a)) \u2286 s),\n    realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (_ts\u271d a) (Set.inclusion h)) =\n      realize (Sum.elim v xs) (_ts\u271d a)\nh\u271d : \u2191(Finset.biUnion Finset.univ fun i => varFinsetLeft (_ts\u271d i)) \u2286 s\nh : \u2200 (i : Fin l\u271d), i \u2208 \u2191Finset.univ \u2192 \u2191(varFinsetLeft (_ts\u271d i)) \u2286 s\n\u22a2 realize (Sum.elim (v \u2218 Subtype.val) xs) (restrictVarLeft (func _f\u271d _ts\u271d) (Set.inclusion h\u271d)) =\n    realize (Sum.elim v xs) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nexact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nt : Term (L[[\u03b1]]) \u03b2\nv : \u03b2 \u2192 M\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars t) = realize v t\n[PROOFSTEP]\ninduction' t with _ n f ts ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\n_a\u271d : \u03b2\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (var _a\u271d)) = realize v (var _a\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn : \u2115\nf : Functions (L[[\u03b1]]) n\nts : Fin n \u2192 Term (L[[\u03b1]]) \u03b2\nih : \u2200 (a : Fin n), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func f ts)) = realize v (func f ts)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase func.zero\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nf : Functions (L[[\u03b1]]) Nat.zero\nts : Fin Nat.zero \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin Nat.zero), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func f ts)) = realize v (func f ts)\n[PROOFSTEP]\ncases f\n[GOAL]\ncase func.zero.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nts : Fin Nat.zero \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin Nat.zero), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions L Nat.zero\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func (Sum.inl val\u271d) ts)) =\n    realize v (func (Sum.inl val\u271d) ts)\n[PROOFSTEP]\nsimp only [realize, ih, Nat.zero_eq, constantsOn, mk\u2082_Functions]\n  --Porting note: below lemma does not work with simp for some reason\n[GOAL]\ncase func.zero.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nts : Fin Nat.zero \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin Nat.zero), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions L Nat.zero\n\u22a2 (funMap val\u271d fun i => realize v (ts i)) = funMap (Sum.inl val\u271d) fun i => realize v (ts i)\n[PROOFSTEP]\nrw [withConstants_funMap_sum_inl]\n[GOAL]\ncase func.zero.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nts : Fin Nat.zero \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin Nat.zero), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions (constantsOn \u03b1) Nat.zero\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func (Sum.inr val\u271d) ts)) =\n    realize v (func (Sum.inr val\u271d) ts)\n[PROOFSTEP]\nsimp only [realize, constantsToVars, Sum.elim_inl, funMap_eq_coe_constants]\n[GOAL]\ncase func.zero.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nts : Fin Nat.zero \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin Nat.zero), realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions (constantsOn \u03b1) Nat.zero\n\u22a2 \u2191(Language.con L val\u271d) = \u2191(Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func.succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn\u271d : \u2115\nf : Functions (L[[\u03b1]]) (Nat.succ n\u271d)\nts : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin (Nat.succ n\u271d)),\n    realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func f ts)) = realize v (func f ts)\n[PROOFSTEP]\ncases' f with _ f\n[GOAL]\ncase func.succ.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn\u271d : \u2115\nts : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin (Nat.succ n\u271d)),\n    realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions L (Nat.succ n\u271d)\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func (Sum.inl val\u271d) ts)) =\n    realize v (func (Sum.inl val\u271d) ts)\n[PROOFSTEP]\nsimp only [realize, ih, constantsOn, mk\u2082_Functions]\n  --Porting note: below lemma does not work with simp for some reason\n[GOAL]\ncase func.succ.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn\u271d : \u2115\nts : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin (Nat.succ n\u271d)),\n    realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nval\u271d : Functions L (Nat.succ n\u271d)\n\u22a2 (funMap val\u271d fun i => realize v (ts i)) = funMap (Sum.inl val\u271d) fun i => realize v (ts i)\n[PROOFSTEP]\nrw [withConstants_funMap_sum_inl]\n[GOAL]\ncase func.succ.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn\u271d : \u2115\nts : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b1]]) \u03b2\nih :\n  \u2200 (a : Fin (Nat.succ n\u271d)),\n    realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (ts a)) = realize v (ts a)\nf : Functions (constantsOn \u03b1) (Nat.succ n\u271d)\n\u22a2 realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (constantsToVars (func (Sum.inr f) ts)) =\n    realize v (func (Sum.inr f) ts)\n[PROOFSTEP]\nexact isEmptyElim f\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nt : Term L (\u03b1 \u2295 \u03b2)\nv : \u03b2 \u2192 M\n\u22a2 realize v (varsToConstants t) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) t\n[PROOFSTEP]\ninduction' t with ab n f ts ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nab : \u03b1 \u2295 \u03b2\n\u22a2 realize v (varsToConstants (var ab)) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (var ab)\n[PROOFSTEP]\ncases' ab with a b\n[GOAL]\ncase var.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\na : \u03b1\n\u22a2 realize v (varsToConstants (var (Sum.inl a))) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (var (Sum.inl a))\n[PROOFSTEP]\nsimp [Language.con, realize, constantMap, funMap_eq_coe_constants]\n[GOAL]\ncase var.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nb : \u03b2\n\u22a2 realize v (varsToConstants (var (Sum.inr b))) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (var (Sum.inr b))\n[PROOFSTEP]\nsimp [realize, constantMap]\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn : \u2115\nf : Functions L n\nts : Fin n \u2192 Term L (\u03b1 \u2295 \u03b2)\nih : \u2200 (a : Fin n), realize v (varsToConstants (ts a)) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (ts a)\n\u22a2 realize v (varsToConstants (func f ts)) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (func f ts)\n[PROOFSTEP]\nsimp only [realize, constantsOn, mk\u2082_Functions, ih]\n  --Porting note: below lemma does not work with simp for some reason\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nv : \u03b2 \u2192 M\nn : \u2115\nf : Functions L n\nts : Fin n \u2192 Term L (\u03b1 \u2295 \u03b2)\nih : \u2200 (a : Fin n), realize v (varsToConstants (ts a)) = realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (ts a)\n\u22a2 (funMap (Sum.inl f) fun i => realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (ts i)) =\n    funMap f fun i => realize (Sum.elim (fun a => \u2191(Language.con L a)) v) (ts i)\n[PROOFSTEP]\nrw [withConstants_funMap_sum_inl]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 realize (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs) (\u2191constantsVarsEquivLeft t) =\n    realize (Sum.elim v xs) t\n[PROOFSTEP]\nsimp only [constantsVarsEquivLeft, realize_relabel, Equiv.coe_trans, Function.comp_apply, constantsVarsEquiv_apply,\n  relabelEquiv_symm_apply]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 realize (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm)\n      (constantsToVars t) =\n    realize (Sum.elim v xs) t\n[PROOFSTEP]\nrefine' _root_.trans _ realize_constantsToVars\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 realize (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm)\n      (constantsToVars t) =\n    realize (Sum.elim (fun a => \u2191(Language.con L a)) (Sum.elim v xs)) (constantsToVars t)\n[PROOFSTEP]\nrcongr x\n[GOAL]\ncase e_v.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\nx : \u03b1 \u2295 \u03b2 \u2295 Fin n\n\u22a2 (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm) x =\n    Sum.elim (fun a => \u2191(Language.con L a)) (Sum.elim v xs) x\n[PROOFSTEP]\nrcases x with (a | (b | i))\n[GOAL]\ncase e_v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\na : \u03b1\n\u22a2 (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm) (Sum.inl a) =\n    Sum.elim (fun a => \u2191(Language.con L a)) (Sum.elim v xs) (Sum.inl a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_v.h.inr.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\nb : \u03b2\n\u22a2 (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm) (Sum.inr (Sum.inl b)) =\n    Sum.elim (fun a => \u2191(Language.con L a)) (Sum.elim v xs) (Sum.inr (Sum.inl b))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_v.h.inr.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\nt : Term (L[[\u03b1]]) (\u03b2 \u2295 Fin n)\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\ni : Fin n\n\u22a2 (Sum.elim (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2218 \u2191(Equiv.sumAssoc \u03b1 \u03b2 (Fin n)).symm) (Sum.inr (Sum.inr i)) =\n    Sum.elim (fun a => \u2191(Language.con L a)) (Sum.elim v xs) (Sum.inr (Sum.inr i))\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nt : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 Term.realize v (onTerm \u03c6 t) = Term.realize v t\n[PROOFSTEP]\ninduction' t with _ n f ts ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nv : \u03b1 \u2192 M\n_a\u271d : \u03b1\n\u22a2 Term.realize v (onTerm \u03c6 (var _a\u271d)) = Term.realize v (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nv : \u03b1 \u2192 M\nn : \u2115\nf : Functions L n\nts : Fin n \u2192 Term L \u03b1\nih : \u2200 (a : Fin n), Term.realize v (onTerm \u03c6 (ts a)) = Term.realize v (ts a)\n\u22a2 Term.realize v (onTerm \u03c6 (func f ts)) = Term.realize v (func f ts)\n[PROOFSTEP]\nsimp only [Term.realize, LHom.onTerm, LHom.map_onFunction, ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nt : Term L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 Term.realize (\u2191g \u2218 v) t = \u2191g (Term.realize v t)\n[PROOFSTEP]\ninduction t\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nv : \u03b1 \u2192 M\n_a\u271d : \u03b1\n\u22a2 Term.realize (\u2191g \u2218 v) (var _a\u271d) = \u2191g (Term.realize v (var _a\u271d))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\n_ts_ih\u271d : \u2200 (a : Fin l\u271d), Term.realize (\u2191g \u2218 v) (_ts\u271d a) = \u2191g (Term.realize v (_ts\u271d a))\n\u22a2 Term.realize (\u2191g \u2218 v) (func _f\u271d _ts\u271d) = \u2191g (Term.realize v (func _f\u271d _ts\u271d))\n[PROOFSTEP]\nrw [Term.realize, Term.realize, g.map_fun]\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\n_ts_ih\u271d : \u2200 (a : Fin l\u271d), Term.realize (\u2191g \u2218 v) (_ts\u271d a) = \u2191g (Term.realize v (_ts\u271d a))\n\u22a2 (funMap _f\u271d fun i => Term.realize (\u2191g \u2218 v) (_ts\u271d i)) = funMap _f\u271d (\u2191g \u2218 fun i => Term.realize v (_ts\u271d i))\n[PROOFSTEP]\nrefine' congr rfl _\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\n_ts_ih\u271d : \u2200 (a : Fin l\u271d), Term.realize (\u2191g \u2218 v) (_ts\u271d a) = \u2191g (Term.realize v (_ts\u271d a))\n\u22a2 (fun i => Term.realize (\u2191g \u2218 v) (_ts\u271d i)) = \u2191g \u2218 fun i => Term.realize v (_ts\u271d i)\n[PROOFSTEP]\next x\n[GOAL]\ncase func.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\ng : M \u2192[L] N\nv : \u03b1 \u2192 M\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\n_ts_ih\u271d : \u2200 (a : Fin l\u271d), Term.realize (\u2191g \u2218 v) (_ts\u271d a) = \u2191g (Term.realize v (_ts\u271d a))\nx : Fin l\u271d\n\u22a2 Term.realize (\u2191g \u2218 v) (_ts\u271d x) = (\u2191g \u2218 fun i => Term.realize v (_ts\u271d i)) x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize \u22a4 v xs \u2194 True\n[PROOFSTEP]\nsimp [Top.top]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u03c6 \u2293 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2227 Realize \u03c8 v xs\n[PROOFSTEP]\nsimp [Inf.inf, Realize]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 (Realize \u03c6 v xs \u2192 Realize \u03c8 v xs \u2192 False) \u2192 False \u2194 Realize \u03c6 v xs \u2227 Realize \u03c8 v xs\n[PROOFSTEP]\ntauto\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l\u271d : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\u271d\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l\u271d)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l\u271d \u2192 M\nl : List (BoundedFormula L \u03b1 n)\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 l) v xs \u2194 \u2200 (\u03c6 : BoundedFormula L \u03b1 n), \u03c6 \u2208 l \u2192 Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' l with \u03c6 l ih\n[GOAL]\ncase nil\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 []) v xs \u2194 \u2200 (\u03c6 : BoundedFormula L \u03b1 n), \u03c6 \u2208 [] \u2192 Realize \u03c6 v xs\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l\u271d : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\u271d\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l\u271d)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l\u271d \u2192 M\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\nl : List (BoundedFormula L \u03b1 n)\nih : Realize (List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 l) v xs \u2194 \u2200 (\u03c6 : BoundedFormula L \u03b1 n), \u03c6 \u2208 l \u2192 Realize \u03c6 v xs\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2293 x_1) \u22a4 (\u03c6 :: l)) v xs \u2194\n    \u2200 (\u03c6_1 : BoundedFormula L \u03b1 n), \u03c6_1 \u2208 \u03c6 :: l \u2192 Realize \u03c6_1 v xs\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u03c6 \u27f9 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nsimp only [Realize]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 1\nt : Term L (\u03b1 \u2295 Fin l)\n\u22a2 Realize (Relations.boundedFormula\u2081 R t) v xs \u2194 RelMap R ![realize (Sum.elim v xs) t]\n[PROOFSTEP]\nrw [Relations.boundedFormula\u2081, realize_rel, iff_eq_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 1\nt : Term L (\u03b1 \u2295 Fin l)\n\u22a2 (RelMap R fun i => realize (Sum.elim v xs) (Matrix.vecCons t ![] i)) = RelMap R ![realize (Sum.elim v xs) t]\n[PROOFSTEP]\nrefine' congr rfl (funext fun _ => _)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 1\nt : Term L (\u03b1 \u2295 Fin l)\nx\u271d : Fin 1\n\u22a2 realize (Sum.elim v xs) (Matrix.vecCons t ![] x\u271d) = Matrix.vecCons (realize (Sum.elim v xs) t) ![] x\u271d\n[PROOFSTEP]\nsimp only [Matrix.cons_val_fin_one]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L (\u03b1 \u2295 Fin l)\n\u22a2 Realize (Relations.boundedFormula\u2082 R t\u2081 t\u2082) v xs \u2194 RelMap R ![realize (Sum.elim v xs) t\u2081, realize (Sum.elim v xs) t\u2082]\n[PROOFSTEP]\nrw [Relations.boundedFormula\u2082, realize_rel, iff_eq_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L (\u03b1 \u2295 Fin l)\n\u22a2 (RelMap R fun i => realize (Sum.elim v xs) (Matrix.vecCons t\u2081 ![t\u2082] i)) =\n    RelMap R ![realize (Sum.elim v xs) t\u2081, realize (Sum.elim v xs) t\u2082]\n[PROOFSTEP]\nrefine' congr rfl (funext (Fin.cases _ _))\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L (\u03b1 \u2295 Fin l)\n\u22a2 realize (Sum.elim v xs) (Matrix.vecCons t\u2081 ![t\u2082] 0) =\n    Matrix.vecCons (realize (Sum.elim v xs) t\u2081) ![realize (Sum.elim v xs) t\u2082] 0\n[PROOFSTEP]\nsimp only [Matrix.cons_val_zero]\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L (\u03b1 \u2295 Fin l)\n\u22a2 \u2200 (i : Fin 1),\n    realize (Sum.elim v xs) (Matrix.vecCons t\u2081 ![t\u2082] (succ i)) =\n      Matrix.vecCons (realize (Sum.elim v xs) t\u2081) ![realize (Sum.elim v xs) t\u2082] (succ i)\n[PROOFSTEP]\nsimp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u03c6 \u2294 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs\n[PROOFSTEP]\nsimp only [realize, Sup.sup, realize_not, eq_iff_iff]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u223c\u03c6 \u27f9 \u03c8) v xs \u2194 Realize \u03c6 v xs \u2228 Realize \u03c8 v xs\n[PROOFSTEP]\ntauto\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l\u271d : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\u271d\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l\u271d)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l\u271d \u2192 M\nl : List (BoundedFormula L \u03b1 n)\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 l) v xs \u2194 \u2203 \u03c6, \u03c6 \u2208 l \u2227 Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' l with \u03c6 l ih\n[GOAL]\ncase nil\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 []) v xs \u2194 \u2203 \u03c6, \u03c6 \u2208 [] \u2227 Realize \u03c6 v xs\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l\u271d : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\u271d\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l\u271d)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l\u271d \u2192 M\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\nl : List (BoundedFormula L \u03b1 n)\nih : Realize (List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 l) v xs \u2194 \u2203 \u03c6, \u03c6 \u2208 l \u2227 Realize \u03c6 v xs\n\u22a2 Realize (List.foldr (fun x x_1 => x \u2294 x_1) \u22a5 (\u03c6 :: l)) v xs \u2194 \u2203 \u03c6_1, \u03c6_1 \u2208 \u03c6 :: l \u2227 Realize \u03c6_1 v xs\n[PROOFSTEP]\nsimp_rw [List.foldr_cons, realize_sup, ih, List.mem_cons, or_and_right, exists_or, exists_eq_left]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u2203'\u03b8) v xs \u2194 \u2203 a, Realize \u03b8 v (snoc xs a)\n[PROOFSTEP]\nrw [BoundedFormula.ex, realize_not, realize_all, not_forall]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 (\u2203 x, \u00acRealize (\u223c\u03b8) v (snoc xs x)) \u2194 \u2203 a, Realize \u03b8 v (snoc xs a)\n[PROOFSTEP]\nsimp_rw [realize_not, Classical.not_not]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u22a2 Realize (\u03c6 \u21d4 \u03c8) v xs \u2194 (Realize \u03c6 v xs \u2194 Realize \u03c8 v xs)\n[PROOFSTEP]\nsimp only [BoundedFormula.iff, realize_inf, realize_imp, and_imp, \u2190 iff_def]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\nh : m = n\nh' : m \u2264 n\n\u03c6 : BoundedFormula L \u03b1 m\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (castLE h' \u03c6) v xs \u2194 Realize \u03c6 v (xs \u2218 \u2191(castIso h))\n[PROOFSTEP]\nsubst h\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm : \u2115\n\u03c6 : BoundedFormula L \u03b1 m\nv : \u03b1 \u2192 M\nh' : m \u2264 m\nxs : Fin m \u2192 M\n\u22a2 Realize (castLE h' \u03c6) v xs \u2194 Realize \u03c6 v (xs \u2218 \u2191(castIso (_ : m = m)))\n[PROOFSTEP]\nsimp only [castLE_rfl, castIso_refl, OrderIso.coe_refl, Function.comp.right_id]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\n\u22a2 Realize (mapTermRel ft fr (fun x => id) \u03c6) v' xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nn\u271d : \u2115\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun x => id) falsum) v' xs \u2194 Realize falsum v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun x => id) (equal t\u2081\u271d t\u2082\u271d)) v' xs \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nsimp [mapTermRel, Realize, h1]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun x => id) (rel R\u271d ts\u271d)) v' xs \u2194 Realize (rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nsimp [mapTermRel, Realize, h1, h2]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (mapTermRel ft fr (fun x => id) f\u2081\u271d) v' xs \u2194 Realize f\u2081\u271d v xs\nih2 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (mapTermRel ft fr (fun x => id) f\u2082\u271d) v' xs \u2194 Realize f\u2082\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun x => id) (f\u2081\u271d \u27f9 f\u2082\u271d)) v' xs \u2194 Realize (f\u2081\u271d \u27f9 f\u2082\u271d) v xs\n[PROOFSTEP]\nsimp [mapTermRel, Realize, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin n \u2192 M\nh1 : \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs : Fin n \u2192 M), realize (Sum.elim v' xs) (ft n t) = realize (Sum.elim v xs) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (mapTermRel ft fr (fun x => id) f\u271d) v' xs \u2194 Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun x => id) (\u2200'f\u271d)) v' xs \u2194 Realize (\u2200'f\u271d) v xs\n[PROOFSTEP]\nsimp only [mapTermRel, Realize, ih, id.def]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) \u03c6) v' xs \u2194 Realize \u03c6 (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\nn\u271d : \u2115\nxs : Fin (k + n\u271d) \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) falsum) v' xs \u2194\n    Realize falsum (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin (k + n\u271d) \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) (equal t\u2081\u271d t\u2082\u271d)) v' xs \u2194\n    Realize (equal t\u2081\u271d t\u2082\u271d) (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\nsimp [mapTermRel, Realize, h1]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin (k + n\u271d) \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) (rel R\u271d ts\u271d)) v' xs \u2194\n    Realize (rel R\u271d ts\u271d) (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\nsimp [mapTermRel, Realize, h1, h2]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 :\n  \u2200 (xs : Fin (k + n\u271d) \u2192 M),\n    Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) f\u2081\u271d) v' xs \u2194\n      Realize f\u2081\u271d (v xs) (xs \u2218 natAdd k)\nih2 :\n  \u2200 (xs : Fin (k + n\u271d) \u2192 M),\n    Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) f\u2082\u271d) v' xs \u2194\n      Realize f\u2082\u271d (v xs) (xs \u2218 natAdd k)\nxs : Fin (k + n\u271d) \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) (f\u2081\u271d \u27f9 f\u2082\u271d)) v' xs \u2194\n    Realize (f\u2081\u271d \u27f9 f\u2082\u271d) (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\nsimp [mapTermRel, Realize, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Structure L' M\nk : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin (k + n))\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nn : \u2115\nv : {n : \u2115} \u2192 (Fin (k + n) \u2192 M) \u2192 \u03b1 \u2192 M\nv' : \u03b2 \u2192 M\nxs\u271d : Fin (k + n) \u2192 M\nh1 :\n  \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (k + n) \u2192 M),\n    realize (Sum.elim v' xs') (ft n t) = realize (Sum.elim (v xs') (xs' \u2218 natAdd k)) t\nh2 : \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (fr n R) x = RelMap R x\nhv : \u2200 (n : \u2115) (xs : Fin (k + n) \u2192 M) (x : M), v (snoc xs x) = v xs\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih :\n  \u2200 (xs : Fin (k + (n\u271d + 1)) \u2192 M),\n    Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) f\u271d) v' xs \u2194\n      Realize f\u271d (v xs) (xs \u2218 natAdd k)\nxs : Fin (k + n\u271d) \u2192 M\n\u22a2 Realize (mapTermRel ft fr (fun n => castLE (_ : k + (n + 1) \u2264 k + n + 1)) (\u2200'f\u271d)) v' xs \u2194\n    Realize (\u2200'f\u271d) (v xs) (xs \u2218 natAdd k)\n[PROOFSTEP]\nsimp [mapTermRel, Realize, ih, hv]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\n\u22a2 Realize (relabel g \u03c6) v xs \u2194 Realize \u03c6 (Sum.elim v (xs \u2218 castAdd n) \u2218 g) (xs \u2218 natAdd m)\n[PROOFSTEP]\nrw [relabel, realize_mapTermRel_add_castLe]\n[GOAL]\ncase h1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\n\u22a2 \u2200 (n : \u2115) (t : Term L (\u03b1 \u2295 Fin n)) (xs' : Fin (m + n) \u2192 M),\n    realize (Sum.elim v xs') (Term.relabel (relabelAux g n) t) =\n      realize (Sum.elim (Sum.elim v (xs' \u2218 castAdd n) \u2218 g) (xs' \u2218 natAdd m)) t\n[PROOFSTEP]\nintros\n[GOAL]\ncase h2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\n\u22a2 \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (id R) x = RelMap R x\n[PROOFSTEP]\nintros\n[GOAL]\ncase hv\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\n\u22a2 \u2200 (n : \u2115) (xs : Fin (m + n) \u2192 M) (x : M),\n    Sum.elim v (snoc xs x \u2218 castAdd (n + 1)) \u2218 g = Sum.elim v (xs \u2218 castAdd n) \u2218 g\n[PROOFSTEP]\nintros\n[GOAL]\ncase h1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\nn\u271d : \u2115\nt\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs'\u271d : Fin (m + n\u271d) \u2192 M\n\u22a2 realize (Sum.elim v xs'\u271d) (Term.relabel (relabelAux g n\u271d) t\u271d) =\n    realize (Sum.elim (Sum.elim v (xs'\u271d \u2218 castAdd n\u271d) \u2218 g) (xs'\u271d \u2218 natAdd m)) t\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\nn\u271d : \u2115\nR\u271d : Relations L n\u271d\nx\u271d : Fin n\u271d \u2192 M\n\u22a2 RelMap (id R\u271d) x\u271d = RelMap R\u271d x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hv\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nm n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ng : \u03b1 \u2192 \u03b2 \u2295 Fin m\nv : \u03b2 \u2192 M\nxs : Fin (m + n) \u2192 M\nn\u271d : \u2115\nxs\u271d : Fin (m + n\u271d) \u2192 M\nx\u271d : M\n\u22a2 Sum.elim v (snoc xs\u271d x\u271d \u2218 castAdd (n\u271d + 1)) \u2218 g = Sum.elim v (xs\u271d \u2218 castAdd n\u271d) \u2218 g\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn n' m : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + n') \u2192 M\nhmn : m + n' \u2264 n + 1\n\u22a2 Realize (liftAt n' m \u03c6) v xs \u2194 Realize \u03c6 v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nrw [liftAt]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn n' m : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + n') \u2192 M\nhmn : m + n' \u2264 n + 1\n\u22a2 Realize (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) \u03c6) v\n      xs \u2194\n    Realize \u03c6 v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 k _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nn\u271d : \u2115\nxs : Fin (n\u271d + n') \u2192 M\nhmn : m + n' \u2264 n\u271d + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) falsum)\n      v xs \u2194\n    Realize falsum v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsimp [mapTermRel, Realize]\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin (n\u271d + n') \u2192 M\nhmn : m + n' \u2264 n\u271d + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1))\n        (equal t\u2081\u271d t\u2082\u271d))\n      v xs \u2194\n    Realize (equal t\u2081\u271d t\u2082\u271d) v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsimp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin (n\u271d + n') \u2192 M\nhmn : m + n' \u2264 n\u271d + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1))\n        (rel R\u271d ts\u271d))\n      v xs \u2194\n    Realize (rel R\u271d ts\u271d) v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsimp [mapTermRel, Realize, realize_rel, realize_liftAt, Sum.elim_comp_map]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 :\n  \u2200 {xs : Fin (n\u271d + n') \u2192 M},\n    m + n' \u2264 n\u271d + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u2081\u271d)\n          v xs \u2194\n        Realize f\u2081\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nih2 :\n  \u2200 {xs : Fin (n\u271d + n') \u2192 M},\n    m + n' \u2264 n\u271d + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u2082\u271d)\n          v xs \u2194\n        Realize f\u2082\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (n\u271d + n') \u2192 M\nhmn : m + n' \u2264 n\u271d + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1))\n        (f\u2081\u271d \u27f9 f\u2082\u271d))\n      v xs \u2194\n    Realize (f\u2081\u271d \u27f9 f\u2082\u271d) v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsimp only [mapTermRel, Realize, ih1 hmn, ih2 hmn]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) (\u2200'f\u271d))\n      v xs \u2194\n    Realize (\u2200'f\u271d) v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nhave h : k + 1 + n' = k + n' + 1 := by rw [add_assoc, add_comm 1 n', \u2190 add_assoc]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\n\u22a2 k + 1 + n' = k + n' + 1\n[PROOFSTEP]\nrw [add_assoc, add_comm 1 n', \u2190 add_assoc]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\n\u22a2 Realize\n      (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) (\u2200'f\u271d))\n      v xs \u2194\n    Realize (\u2200'f\u271d) v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsimp only [mapTermRel, Realize, realize_castLE_of_eq h, ih3 (hmn.trans k.succ.le_succ)]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\n\u22a2 (\u2200 (x : M), Realize f\u271d v ((snoc xs x \u2218 \u2191(castIso h)) \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n')) \u2194\n    \u2200 (x : M), Realize f\u271d v (snoc (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n') x)\n[PROOFSTEP]\nrefine' forall_congr' fun x => iff_eq_eq.mpr (congr rfl (funext (Fin.lastCases _ fun i => _)))\n[GOAL]\ncase all.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\n\u22a2 ((snoc xs x \u2218 \u2191(castIso h)) \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n') (last k) =\n    snoc (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n') x (last k)\n[PROOFSTEP]\nsimp only [Function.comp_apply, val_last, snoc_last]\n[GOAL]\ncase all.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\n\u22a2 snoc xs x (\u2191(castIso h) (if k < m then castAdd n' (last k) else addNat (last k) n')) = x\n[PROOFSTEP]\nby_cases h : k < m\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : k < m\n\u22a2 snoc xs x (\u2191(castIso h\u271d) (if k < m then castAdd n' (last k) else addNat (last k) n')) = x\n[PROOFSTEP]\nrw [if_pos h]\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : k < m\n\u22a2 snoc xs x (\u2191(castIso h\u271d) (castAdd n' (last k))) = x\n[PROOFSTEP]\nrefine' (congr rfl (ext _)).trans (snoc_last _ _)\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : k < m\n\u22a2 \u2191(\u2191(castIso h\u271d) (castAdd n' (last k))) = \u2191(last (k + n'))\n[PROOFSTEP]\nsimp only [coe_orderIso_apply, coe_castAdd, val_last, self_eq_add_right]\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : k < m\n\u22a2 n' = 0\n[PROOFSTEP]\nrefine' le_antisymm (le_of_add_le_add_left ((hmn.trans (Nat.succ_le_of_lt h)).trans _)) n'.zero_le\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : k < m\n\u22a2 m \u2264 m + 0\n[PROOFSTEP]\nrw [add_zero]\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : \u00ack < m\n\u22a2 snoc xs x (\u2191(castIso h\u271d) (if k < m then castAdd n' (last k) else addNat (last k) n')) = x\n[PROOFSTEP]\nrw [if_neg h]\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : \u00ack < m\n\u22a2 snoc xs x (\u2191(castIso h\u271d) (addNat (last k) n')) = x\n[PROOFSTEP]\nrefine' (congr rfl (ext _)).trans (snoc_last _ _)\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh\u271d : k + 1 + n' = k + n' + 1\nx : M\nh : \u00ack < m\n\u22a2 \u2191(\u2191(castIso h\u271d) (addNat (last k) n')) = \u2191(last (k + n'))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\n\u22a2 ((snoc xs x \u2218 \u2191(castIso h)) \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n') (castSucc i) =\n    snoc (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n') x (castSucc i)\n[PROOFSTEP]\nsimp only [Function.comp_apply, Fin.snoc_castSucc]\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\n\u22a2 snoc xs x (\u2191(castIso h) (if \u2191(castSucc i) < m then castAdd n' (castSucc i) else addNat (castSucc i) n')) =\n    xs (if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nrefine' (congr rfl (ext _)).trans (snoc_castSucc _ _ _)\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\n\u22a2 \u2191(\u2191(castIso h) (if \u2191(castSucc i) < m then castAdd n' (castSucc i) else addNat (castSucc i) n')) =\n    \u2191(castSucc (if \u2191i < m then castAdd n' i else addNat i n'))\n[PROOFSTEP]\nsimp only [coe_castSucc, coe_orderIso_apply]\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\n\u22a2 \u2191(if \u2191i < m then castAdd n' (castSucc i) else addNat (castSucc i) n') =\n    \u2191(if \u2191i < m then castAdd n' i else addNat i n')\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\nh\u271d : \u2191i < m\n\u22a2 \u2191(castAdd n' (castSucc i)) = \u2191(castAdd n' i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\nn n' m : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin (n + n') \u2192 M\nhmn\u271d : m + n' \u2264 n + 1\nk : \u2115\nf\u271d : BoundedFormula L \u03b1 (k + 1)\nih3 :\n  \u2200 {xs : Fin (k + 1 + n') \u2192 M},\n    m + n' \u2264 k + 1 + 1 \u2192\n      (Realize\n          (mapTermRel (fun k t => Term.liftAt n' m t) (fun x => id) (fun x => castLE (_ : x + 1 + n' \u2264 x + n' + 1)) f\u271d)\n          v xs \u2194\n        Realize f\u271d v (xs \u2218 fun i => if \u2191i < m then castAdd n' i else addNat i n'))\nxs : Fin (k + n') \u2192 M\nhmn : m + n' \u2264 k + 1\nh : k + 1 + n' = k + n' + 1\nx : M\ni : Fin k\nh\u271d : \u00ac\u2191i < m\n\u22a2 \u2191(addNat (castSucc i) n') = \u2191(addNat i n')\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn m : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + 1) \u2192 M\nhmn : m \u2264 n\n\u22a2 Realize (liftAt 1 m \u03c6) v xs \u2194 Realize \u03c6 v (xs \u2218 fun i => if \u2191i < m then castSucc i else succ i)\n[PROOFSTEP]\nsimp [realize_liftAt (add_le_add_right hmn 1), castSucc]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + 1) \u2192 M\n\u22a2 Realize (liftAt 1 n \u03c6) v xs \u2194 Realize \u03c6 v (xs \u2218 castSucc)\n[PROOFSTEP]\nrw [realize_liftAt_one (refl n), iff_eq_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + 1) \u2192 M\n\u22a2 Realize \u03c6 v (xs \u2218 fun i => if \u2191i < n then castSucc i else succ i) = Realize \u03c6 v (xs \u2218 castSucc)\n[PROOFSTEP]\nrefine' congr rfl (congr rfl (funext fun i => _))\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin (n + 1) \u2192 M\ni : Fin n\n\u22a2 (if \u2191i < n then castSucc i else succ i) = castSucc i\n[PROOFSTEP]\nrw [if_pos i.is_lt]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\u271d\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n\u271d \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nx : Fin n \u2192 M\n\u22a2 realize (Sum.elim v x) (Term.subst t (Sum.elim (Term.relabel Sum.inl \u2218 tf) (var \u2218 Sum.inr))) =\n    realize (Sum.elim (fun a => realize v (tf a)) x) t\n[PROOFSTEP]\nrw [Term.realize_subst]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\u271d\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n\u271d \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nx : Fin n \u2192 M\n\u22a2 realize (fun a => realize (Sum.elim v x) (Sum.elim (Term.relabel Sum.inl \u2218 tf) (var \u2218 Sum.inr) a)) t =\n    realize (Sum.elim (fun a => realize v (tf a)) x) t\n[PROOFSTEP]\nrcongr a\n[GOAL]\ncase e_v.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\u271d\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n\u271d \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nx : Fin n \u2192 M\na : \u03b1 \u2295 Fin n\n\u22a2 realize (Sum.elim v x) (Sum.elim (Term.relabel Sum.inl \u2218 tf) (var \u2218 Sum.inr) a) =\n    Sum.elim (fun a => realize v (tf a)) x a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase e_v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\u271d\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n\u271d \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nx : Fin n \u2192 M\nval\u271d : \u03b1\n\u22a2 realize (Sum.elim v x) (Sum.elim (Term.relabel Sum.inl \u2218 tf) (var \u2218 Sum.inr) (Sum.inl val\u271d)) =\n    Sum.elim (fun a => realize v (tf a)) x (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp only [Sum.elim_inl, Function.comp_apply, Term.realize_relabel, Sum.elim_comp_inl]\n[GOAL]\ncase e_v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\u271d\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n\u271d \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nx : Fin n \u2192 M\nval\u271d : Fin n\n\u22a2 realize (Sum.elim v x) (Sum.elim (Term.relabel Sum.inl \u2218 tf) (var \u2218 Sum.inr) (Sum.inr val\u271d)) =\n    Sum.elim (fun a => realize v (tf a)) x (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 n\ntf : \u03b1 \u2192 Term L \u03b2\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 \u2200 (n : \u2115) (R : Relations L n) (x : Fin n \u2192 M), RelMap (id R) x = RelMap R x\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (restrictFreeVar \u03c6 (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh\u271d : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nh : \u2191(freeVarFinset falsum) \u2286 s\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (restrictFreeVar falsum (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize falsum v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh\u271d : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nh : \u2191(freeVarFinset (equal t\u2081\u271d t\u2082\u271d)) \u2286 s\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (restrictFreeVar (equal t\u2081\u271d t\u2082\u271d) (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nsimp [restrictFreeVar, Realize]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh\u271d : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nh : \u2191(freeVarFinset (rel R\u271d ts\u271d)) \u2286 s\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (restrictFreeVar (rel R\u271d ts\u271d) (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize (rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nsimp [restrictFreeVar, Realize]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh\u271d : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 :\n  \u2200 (h : \u2191(freeVarFinset f\u2081\u271d) \u2286 s) {xs : Fin n\u271d \u2192 M},\n    Realize (restrictFreeVar f\u2081\u271d (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize f\u2081\u271d v xs\nih2 :\n  \u2200 (h : \u2191(freeVarFinset f\u2082\u271d) \u2286 s) {xs : Fin n\u271d \u2192 M},\n    Realize (restrictFreeVar f\u2082\u271d (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize f\u2082\u271d v xs\nh : \u2191(freeVarFinset (f\u2081\u271d \u27f9 f\u2082\u271d)) \u2286 s\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (restrictFreeVar (f\u2081\u271d \u27f9 f\u2082\u271d) (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize (f\u2081\u271d \u27f9 f\u2082\u271d) v xs\n[PROOFSTEP]\nsimp [restrictFreeVar, Realize, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : DecidableEq \u03b1\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\ns : Set \u03b1\nh\u271d : \u2191(freeVarFinset \u03c6) \u2286 s\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 :\n  \u2200 (h : \u2191(freeVarFinset f\u271d) \u2286 s) {xs : Fin (n\u271d + 1) \u2192 M},\n    Realize (restrictFreeVar f\u271d (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize f\u271d v xs\nh : \u2191(freeVarFinset (\u2200'f\u271d)) \u2286 s\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (restrictFreeVar (\u2200'f\u271d) (Set.inclusion h)) (v \u2218 Subtype.val) xs \u2194 Realize (\u2200'f\u271d) v xs\n[PROOFSTEP]\nsimp [restrictFreeVar, Realize, ih3]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\nv : \u03b2 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (\u2191constantsVarsEquiv \u03c6) (Sum.elim (fun a => \u2191(Language.con L a)) v) xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\nrefine' realize_mapTermRel_id (fun n t xs => realize_constantsVarsEquivLeft) fun n R xs => _\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn\u271d : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\u271d\nv : \u03b2 \u2192 M\nxs\u271d : Fin n\u271d \u2192 M\nn : \u2115\nR : Relations (L[[\u03b1]]) n\nxs : Fin n \u2192 M\n\u22a2 RelMap (\u2191((fun x => Equiv.sumEmpty (Relations L x) (Relations (constantsOn \u03b1) x)) n) R) xs = RelMap R xs\n[PROOFSTEP]\nrw [\u2190 (lhomWithConstants L \u03b1).map_onRelation (Equiv.sumEmpty (L.Relations n) ((constantsOn \u03b1).Relations n) R) xs]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn\u271d : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\u271d\nv : \u03b2 \u2192 M\nxs\u271d : Fin n\u271d \u2192 M\nn : \u2115\nR : Relations (L[[\u03b1]]) n\nxs : Fin n \u2192 M\n\u22a2 RelMap (LHom.onRelation (lhomWithConstants L \u03b1) (\u2191(Equiv.sumEmpty (Relations L n) (Relations (constantsOn \u03b1) n)) R))\n      xs =\n    RelMap R xs\n[PROOFSTEP]\nrcongr\n[GOAL]\ncase a\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn\u271d : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\u271d\nv : \u03b2 \u2192 M\nxs\u271d : Fin n\u271d \u2192 M\nn : \u2115\nR : Relations (L[[\u03b1]]) n\nxs : Fin n \u2192 M\n\u22a2 RelMap (LHom.onRelation (lhomWithConstants L \u03b1) (\u2191(Equiv.sumEmpty (Relations L n) (Relations (constantsOn \u03b1) n)) R))\n      xs \u2194\n    RelMap R xs\n[PROOFSTEP]\ncases' R with R R\n[GOAL]\ncase a.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn\u271d : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\u271d\nv : \u03b2 \u2192 M\nxs\u271d : Fin n\u271d \u2192 M\nn : \u2115\nxs : Fin n \u2192 M\nR : Relations L n\n\u22a2 RelMap\n      (LHom.onRelation (lhomWithConstants L \u03b1)\n        (\u2191(Equiv.sumEmpty (Relations L n) (Relations (constantsOn \u03b1) n)) (Sum.inl R)))\n      xs \u2194\n    RelMap (Sum.inl R) xs\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\nn\u271d : \u2115\n\u03c6 : BoundedFormula (L[[\u03b1]]) \u03b2 n\u271d\nv : \u03b2 \u2192 M\nxs\u271d : Fin n\u271d \u2192 M\nn : \u2115\nxs : Fin n \u2192 M\nR : Relations (constantsOn \u03b1) n\n\u22a2 RelMap\n      (LHom.onRelation (lhomWithConstants L \u03b1)\n        (\u2191(Equiv.sumEmpty (Relations L n) (Relations (constantsOn \u03b1) n)) (Sum.inr R)))\n      xs \u2194\n    RelMap (Sum.inr R) xs\n[PROOFSTEP]\nexact isEmptyElim R\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs : Fin k \u2192 M\n\u22a2 Realize (\u2191(relabelEquiv g) \u03c6) v xs \u2194 Realize \u03c6 (v \u2218 \u2191g) xs\n[PROOFSTEP]\nsimp only [relabelEquiv, mapTermRelEquiv_apply, Equiv.coe_refl]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs : Fin k \u2192 M\n\u22a2 Realize\n      (mapTermRel (fun n => \u2191(Term.relabelEquiv (Equiv.sumCongr g (_root_.Equiv.refl (Fin n))))) (fun n => id)\n        (fun x => id) \u03c6)\n      v xs \u2194\n    Realize \u03c6 (v \u2218 \u2191g) xs\n[PROOFSTEP]\nrefine' realize_mapTermRel_id (fun n t xs => _) fun _ _ _ => rfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs\u271d : Fin k \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nxs : Fin n \u2192 M\n\u22a2 realize (Sum.elim v xs) (\u2191(Term.relabelEquiv (Equiv.sumCongr g (_root_.Equiv.refl (Fin n)))) t) =\n    realize (Sum.elim (v \u2218 \u2191g) xs) t\n[PROOFSTEP]\nsimp only [relabelEquiv_apply, Term.realize_relabel]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs\u271d : Fin k \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nxs : Fin n \u2192 M\n\u22a2 realize (Sum.elim v xs \u2218 \u2191(Equiv.sumCongr g (_root_.Equiv.refl (Fin n)))) t = realize (Sum.elim (v \u2218 \u2191g) xs) t\n[PROOFSTEP]\nrefine' congr (congr rfl _) rfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs\u271d : Fin k \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nxs : Fin n \u2192 M\n\u22a2 Sum.elim v xs \u2218 \u2191(Equiv.sumCongr g (_root_.Equiv.refl (Fin n))) = Sum.elim (v \u2218 \u2191g) xs\n[PROOFSTEP]\next (i | i)\n[GOAL]\ncase h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs\u271d : Fin k \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nxs : Fin n \u2192 M\ni : \u03b1\n\u22a2 (Sum.elim v xs \u2218 \u2191(Equiv.sumCongr g (_root_.Equiv.refl (Fin n)))) (Sum.inl i) = Sum.elim (v \u2218 \u2191g) xs (Sum.inl i)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ng : \u03b1 \u2243 \u03b2\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\nv : \u03b2 \u2192 M\nxs\u271d : Fin k \u2192 M\nn : \u2115\nt : Term L (\u03b1 \u2295 Fin n)\nxs : Fin n \u2192 M\ni : Fin n\n\u22a2 (Sum.elim v xs \u2218 \u2191(Equiv.sumCongr g (_root_.Equiv.refl (Fin n)))) (Sum.inr i) = Sum.elim (v \u2218 \u2191g) xs (Sum.inr i)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (\u2200'liftAt 1 n \u03c6) v xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\ninhabit M\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\n\u22a2 Realize (\u2200'liftAt 1 n \u03c6) v xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\nsimp only [realize_all, realize_liftAt_one_self]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\n\u22a2 (\u2200 (a : M), Realize \u03c6 v (snoc xs a \u2218 castSucc)) \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h a => _\u27e9\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\nh : \u2200 (a : M), Realize \u03c6 v (snoc xs a \u2218 castSucc)\n\u22a2 Realize \u03c6 v xs\n[PROOFSTEP]\nrefine' (congr rfl (funext fun i => _)).mp (h default)\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\nh : \u2200 (a : M), Realize \u03c6 v (snoc xs a \u2218 castSucc)\ni : Fin n\n\u22a2 (snoc xs default \u2218 castSucc) i = xs i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\nh : Realize \u03c6 v xs\na : M\n\u22a2 Realize \u03c6 v (snoc xs a \u2218 castSucc)\n[PROOFSTEP]\nrefine' (congr rfl (funext fun i => _)).mp h\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\ninhabited_h : Inhabited M\nh : Realize \u03c6 v xs\na : M\ni : Fin n\n\u22a2 xs i = (snoc xs a \u2218 castSucc) i\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6 : IsQF \u03c6\nh\u03c8 : IsPrenex \u03c8\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (toPrenexImpRight \u03c6 \u03c8) v xs \u2194 Realize (\u03c6 \u27f9 \u03c8) v xs\n[PROOFSTEP]\ninduction' h\u03c8 with _ _ h\u03c8 _ _ _h\u03c8 ih _ _ _h\u03c8 ih\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsQF \u03c6\u271d\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n[PROOFSTEP]\nrw [h\u03c8.toPrenexImpRight]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (toPrenexImpRight \u03c6 (\u2200'\u03c6\u271d)) v xs \u2194 Realize (\u03c6 \u27f9 \u2200'\u03c6\u271d) v xs\n[PROOFSTEP]\nrefine' _root_.trans (forall_congr' fun _ => ih h\u03c6.liftAt) _\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M), Realize (liftAt 1 n\u271d \u03c6 \u27f9 \u03c6\u271d) v (snoc xs a)) \u2194 Realize (\u03c6 \u27f9 \u2200'\u03c6\u271d) v xs\n[PROOFSTEP]\nsimp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M), Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)) \u2194 Realize \u03c6 v xs \u2192 \u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nexact \u27e8fun h1 a h2 => h1 h2 a, fun h1 h2 a => h1 a h2\u27e9\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (toPrenexImpRight \u03c6 (\u2203'\u03c6\u271d)) v xs \u2194 Realize (\u03c6 \u27f9 \u2203'\u03c6\u271d) v xs\n[PROOFSTEP]\nunfold toPrenexImpRight\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (\u2203'toPrenexImpRight (liftAt 1 n\u271d \u03c6) \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u2203'\u03c6\u271d) v xs\n[PROOFSTEP]\nrw [realize_ex]\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2203 a, Realize (toPrenexImpRight (liftAt 1 n\u271d \u03c6) \u03c6\u271d) v (snoc xs a)) \u2194 Realize (\u03c6 \u27f9 \u2203'\u03c6\u271d) v xs\n[PROOFSTEP]\nrefine' _root_.trans (exists_congr fun _ => ih h\u03c6.liftAt) _\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2203 a, Realize (liftAt 1 n\u271d \u03c6 \u27f9 \u03c6\u271d) v (snoc xs a)) \u2194 Realize (\u03c6 \u27f9 \u2203'\u03c6\u271d) v xs\n[PROOFSTEP]\nsimp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_ex]\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)) \u2194 Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nrefine' \u27e8_, fun h' => _\u27e9\n[GOAL]\ncase ex.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 h\n[GOAL]\ncase ex.refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\na : M\nha : Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\nh : Realize \u03c6 v xs\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nexact \u27e8a, ha h\u27e9\n[GOAL]\ncase ex.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nby_cases \u03c6.Realize v xs\n[GOAL]\ncase ex.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nby_cases \u03c6.Realize v xs\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\nh : Realize \u03c6 v xs\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := h' h\n[GOAL]\ncase pos.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\nh : Realize \u03c6 v xs\na : M\nha : Realize \u03c6\u271d v (snoc xs a)\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nexact \u27e8a, fun _ => ha\u27e9\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\nh : \u00acRealize \u03c6 v xs\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\ninhabit M\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c8 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsQF \u03c6 \u2192 \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenexImpRight \u03c6 \u03c6\u271d) v xs \u2194 Realize (\u03c6 \u27f9 \u03c6\u271d) v xs\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\nxs : Fin n\u271d \u2192 M\nh' : Realize \u03c6 v xs \u2192 \u2203 a, Realize \u03c6\u271d v (snoc xs a)\nh : \u00acRealize \u03c6 v xs\ninhabited_h : Inhabited M\n\u22a2 \u2203 a, Realize \u03c6 v xs \u2192 Realize \u03c6\u271d v (snoc xs a)\n[PROOFSTEP]\nexact \u27e8default, fun h'' => (h h'').elim\u27e9\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6 : IsPrenex \u03c6\nh\u03c8 : IsPrenex \u03c8\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (toPrenexImp \u03c6 \u03c8) v xs \u2194 Realize (\u03c6 \u27f9 \u03c8) v xs\n[PROOFSTEP]\nrevert \u03c8\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nh\u03c6 : IsPrenex \u03c6\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 \u2200 {\u03c8 : BoundedFormula L \u03b1 n}, IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6 \u03c8) v xs \u2194 Realize (\u03c6 \u27f9 \u03c8) v xs)\n[PROOFSTEP]\ninduction' h\u03c6 with _ _ h\u03c6 _ _ _h\u03c6 ih _ _ _h\u03c6 ih\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\u271d\nxs : Fin n\u271d \u2192 M\n\u22a2 \u2200 {\u03c8 : BoundedFormula L \u03b1 n\u271d}, IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\n[PROOFSTEP]\nintro \u03c8 h\u03c8\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u22a2 \u2200 {\u03c8 : BoundedFormula L \u03b1 n\u271d}, IsPrenex \u03c8 \u2192 (Realize (toPrenexImp (\u2200'\u03c6\u271d) \u03c8) v xs \u2194 Realize (\u2200'\u03c6\u271d \u27f9 \u03c8) v xs)\n[PROOFSTEP]\nintro \u03c8 h\u03c8\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u22a2 \u2200 {\u03c8 : BoundedFormula L \u03b1 n\u271d}, IsPrenex \u03c8 \u2192 (Realize (toPrenexImp (\u2203'\u03c6\u271d) \u03c8) v xs \u2194 Realize (\u2203'\u03c6\u271d \u27f9 \u03c8) v xs)\n[PROOFSTEP]\nintro \u03c8 h\u03c8\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\u271d\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nrw [h\u03c6.toPrenexImp]\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\u271d\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 Realize (toPrenexImpRight \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nexact realize_toPrenexImpRight h\u03c6 h\u03c8\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 Realize (toPrenexImp (\u2200'\u03c6\u271d) \u03c8) v xs \u2194 Realize (\u2200'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nunfold toPrenexImp\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 Realize (\u2203'toPrenexImp \u03c6\u271d (liftAt 1 n\u271d \u03c8)) v xs \u2194 Realize (\u2200'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nrw [realize_ex]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 (\u2203 a, Realize (toPrenexImp \u03c6\u271d (liftAt 1 n\u271d \u03c8)) v (snoc xs a)) \u2194 Realize (\u2200'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nrefine' _root_.trans (exists_congr fun _ => ih h\u03c8.liftAt) _\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 (\u2203 a, Realize (\u03c6\u271d \u27f9 liftAt 1 n\u271d \u03c8) v (snoc xs a)) \u2194 Realize (\u2200'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nsimp only [realize_imp, realize_liftAt_one_self, snoc_comp_castSucc, realize_all]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 (\u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs) \u2194 (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nrefine' \u27e8_, fun h' => _\u27e9\n[GOAL]\ncase all.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 (\u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs) \u2192 (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 h\n[GOAL]\ncase all.refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\na : M\nha : Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\nh : \u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)\n\u22a2 Realize \u03c8 v xs\n[PROOFSTEP]\nexact ha (h a)\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nby_cases \u03c8.Realize v xs\n[GOAL]\ncase all.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nby_cases \u03c8.Realize v xs\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\nh : Realize \u03c8 v xs\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\ninhabit M\n[GOAL]\ncase pos\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\nh : Realize \u03c8 v xs\ninhabited_h : Inhabited M\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nexact \u27e8default, fun _h'' => h\u27e9\n[GOAL]\ncase neg\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\nh : \u00acRealize \u03c8 v xs\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := not_forall.1 (h \u2218 h')\n[GOAL]\ncase neg.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\nh' : (\u2200 (a : M), Realize \u03c6\u271d v (snoc xs a)) \u2192 Realize \u03c8 v xs\nh : \u00acRealize \u03c8 v xs\na : M\nha : \u00acRealize \u03c6\u271d v (snoc xs a)\n\u22a2 \u2203 a, Realize \u03c6\u271d v (snoc xs a) \u2192 Realize \u03c8 v xs\n[PROOFSTEP]\nexact \u27e8a, fun h => (ha h).elim\u27e9\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 Realize (toPrenexImp (\u2203'\u03c6\u271d) \u03c8) v xs \u2194 Realize (\u2203'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nrefine' _root_.trans (forall_congr' fun _ => ih h\u03c8.liftAt) _\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs\u271d\u00b9 : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\n_h\u03c6 : IsPrenex \u03c6\u271d\nih :\n  \u2200 {xs : Fin (n\u271d + 1) \u2192 M} {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)},\n    IsPrenex \u03c8 \u2192 (Realize (toPrenexImp \u03c6\u271d \u03c8) v xs \u2194 Realize (\u03c6\u271d \u27f9 \u03c8) v xs)\nxs : Fin n\u271d \u2192 M\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 (\u2200 (a : M), Realize (\u03c6\u271d \u27f9 liftAt 1 n\u271d \u03c8) v (snoc xs a)) \u2194 Realize (\u2203'\u03c6\u271d \u27f9 \u03c8) v xs\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\n\u22a2 \u2200 {xs : Fin n \u2192 M}, Realize (toPrenex \u03c6) v xs \u2194 Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex falsum) v xs \u2194 Realize falsum v xs\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex (equal t\u2081\u271d t\u2082\u271d)) v xs \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex (rel R\u271d ts\u271d)) v xs \u2194 Realize (rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : BoundedFormula L \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex f1) v xs \u2194 Realize f1 v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex f2) v xs \u2194 Realize f2 v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex (f1 \u27f9 f2)) v xs \u2194 Realize (f1 \u27f9 f2) v xs\n[PROOFSTEP]\nintros\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : BoundedFormula L \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex f1) v xs \u2194 Realize f1 v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex f2) v xs \u2194 Realize f2 v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 Realize (toPrenex (f1 \u27f9 f2)) v xs\u271d \u2194 Realize (f1 \u27f9 f2) v xs\u271d\n[PROOFSTEP]\nrw [toPrenex, realize_toPrenexImp f1.toPrenex_isPrenex f2.toPrenex_isPrenex, realize_imp, realize_imp, h1, h2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenex f\u271d) v xs \u2194 Realize f\u271d v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, Realize (toPrenex (\u2200'f\u271d)) v xs \u2194 Realize (\u2200'f\u271d) v xs\n[PROOFSTEP]\nintros\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenex f\u271d) v xs \u2194 Realize f\u271d v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 Realize (toPrenex (\u2200'f\u271d)) v xs\u271d \u2194 Realize (\u2200'f\u271d) v xs\u271d\n[PROOFSTEP]\nrw [realize_all, toPrenex, realize_all]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn l : \u2115\n\u03c6 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (toPrenex f\u271d) v xs \u2194 Realize f\u271d v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M), Realize (toPrenex f\u271d) v (snoc xs\u271d a)) \u2194 \u2200 (a : M), Realize f\u271d v (snoc xs\u271d a)\n[PROOFSTEP]\nexact forall_congr' fun a => h\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\n\u03c8 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 \u03c8) v xs \u2194 Realize \u03c8 v xs\n[PROOFSTEP]\ninduction' \u03c8 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 falsum) v xs \u2194 Realize falsum v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 (equal t\u2081\u271d t\u2082\u271d)) v xs \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nsimp only [onBoundedFormula, realize_bdEqual, realize_onTerm]\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 Term.realize (Sum.elim v xs) t\u2081\u271d = Term.realize (Sum.elim v xs) t\u2082\u271d \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 (rel R\u271d ts\u271d)) v xs \u2194 Realize (rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nsimp only [onBoundedFormula, realize_rel, LHom.map_onRelation, Function.comp_apply, realize_onTerm]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 (RelMap R\u271d fun i => Term.realize (Sum.elim v xs) (ts\u271d i)) \u2194 Realize (rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (onBoundedFormula \u03c6 f\u2081\u271d) v xs \u2194 Realize f\u2081\u271d v xs\nih2 : \u2200 {xs : Fin n\u271d \u2192 M}, Realize (onBoundedFormula \u03c6 f\u2082\u271d) v xs \u2194 Realize f\u2082\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 (f\u2081\u271d \u27f9 f\u2082\u271d)) v xs \u2194 Realize (f\u2081\u271d \u27f9 f\u2082\u271d) v xs\n[PROOFSTEP]\nsimp only [onBoundedFormula, ih1, ih2, realize_imp]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d\u00b9 : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nn : \u2115\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, Realize (onBoundedFormula \u03c6 f\u271d) v xs \u2194 Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 Realize (onBoundedFormula \u03c6 (\u2200'f\u271d)) v xs \u2194 Realize (\u2200'f\u271d) v xs\n[PROOFSTEP]\nsimp only [onBoundedFormula, ih3, realize_all]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nk : \u2115\nR : Relations L k\nts : Fin k \u2192 Term L \u03b1\n\u22a2 (RelMap R fun i => Term.realize (Sum.elim v default) (Term.relabel Sum.inl (ts i))) \u2194\n    RelMap R fun i => Term.realize v (ts i)\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 1\nt : Term L \u03b1\n\u22a2 Realize (Relations.formula\u2081 R t) v \u2194 RelMap R ![Term.realize v t]\n[PROOFSTEP]\nrw [Relations.formula\u2081, realize_rel, iff_eq_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 1\nt : Term L \u03b1\n\u22a2 (RelMap R fun i => Term.realize v (Matrix.vecCons t ![] i)) = RelMap R ![Term.realize v t]\n[PROOFSTEP]\nrefine' congr rfl (funext fun _ => _)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 1\nt : Term L \u03b1\nx\u271d : Fin 1\n\u22a2 Term.realize v (Matrix.vecCons t ![] x\u271d) = Matrix.vecCons (Term.realize v t) ![] x\u271d\n[PROOFSTEP]\nsimp only [Matrix.cons_val_fin_one]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L \u03b1\n\u22a2 Realize (Relations.formula\u2082 R t\u2081 t\u2082) v \u2194 RelMap R ![Term.realize v t\u2081, Term.realize v t\u2082]\n[PROOFSTEP]\nrw [Relations.formula\u2082, realize_rel, iff_eq_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L \u03b1\n\u22a2 (RelMap R fun i => Term.realize v (Matrix.vecCons t\u2081 ![t\u2082] i)) = RelMap R ![Term.realize v t\u2081, Term.realize v t\u2082]\n[PROOFSTEP]\nrefine' congr rfl (funext (Fin.cases _ _))\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L \u03b1\n\u22a2 Term.realize v (Matrix.vecCons t\u2081 ![t\u2082] 0) = Matrix.vecCons (Term.realize v t\u2081) ![Term.realize v t\u2082] 0\n[PROOFSTEP]\nsimp only [Matrix.cons_val_zero]\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nR : Relations L 2\nt\u2081 t\u2082 : Term L \u03b1\n\u22a2 \u2200 (i : Fin 1),\n    Term.realize v (Matrix.vecCons t\u2081 ![t\u2082] (succ i)) = Matrix.vecCons (Term.realize v t\u2081) ![Term.realize v t\u2082] (succ i)\n[PROOFSTEP]\nsimp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6\u271d \u03c8 : Formula L \u03b1\nv\u271d : \u03b1 \u2192 M\n\u03c6 : Formula L \u03b1\ng : \u03b1 \u2192 \u03b2\nv : \u03b2 \u2192 M\n\u22a2 Realize (relabel g \u03c6) v \u2194 Realize \u03c6 (v \u2218 g)\n[PROOFSTEP]\nrw [Realize, Realize, relabel, BoundedFormula.realize_relabel, iff_eq_eq, Fin.castAdd_zero]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6\u271d \u03c8 : Formula L \u03b1\nv\u271d : \u03b1 \u2192 M\n\u03c6 : Formula L \u03b1\ng : \u03b1 \u2192 \u03b2\nv : \u03b2 \u2192 M\n\u22a2 BoundedFormula.Realize \u03c6 (Sum.elim v (default \u2218 Fin.cast (_ : 0 = 0)) \u2218 Sum.inl \u2218 g) (default \u2218 natAdd 0) =\n    BoundedFormula.Realize \u03c6 (v \u2218 g) default\n[PROOFSTEP]\nexact congr rfl (funext finZeroElim)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6\u271d \u03c8 : Formula L \u03b1\nv\u271d : \u03b1 \u2192 M\n\u03c6 : Formula L (Fin n)\nv : Empty \u2192 M\nx : Fin n \u2192 M\n\u22a2 BoundedFormula.Realize (BoundedFormula.relabel Sum.inr \u03c6) v x \u2194 Realize \u03c6 x\n[PROOFSTEP]\nrw [BoundedFormula.realize_relabel, Formula.Realize, Sum.elim_comp_inr, Fin.castAdd_zero, cast_refl,\n  Function.comp.right_id, Subsingleton.elim (x \u2218 (natAdd n : Fin 0 \u2192 Fin n)) default]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nt\u2081 t\u2082 : Term L \u03b1\nx : \u03b1 \u2192 M\n\u22a2 Realize (Term.equal t\u2081 t\u2082) x \u2194 Term.realize x t\u2081 = Term.realize x t\u2082\n[PROOFSTEP]\nsimp [Term.equal, Realize]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nf : Functions L n\nx : Fin n \u2192 M\ny : M\n\u22a2 Realize (graph f) (cons y x) \u2194 funMap f x = y\n[PROOFSTEP]\nsimp only [Formula.graph, Term.realize, realize_equal, Fin.cons_zero, Fin.cons_succ]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u03c6 \u03c8 : Formula L \u03b1\nv : \u03b1 \u2192 M\nf : Functions L n\nx : Fin n \u2192 M\ny : M\n\u22a2 (y = funMap f fun i => x i) \u2194 funMap f x = y\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\n\u03c8 : Formula L \u03b1\n\u22a2 setOf (Formula.Realize (onFormula \u03c6 \u03c8)) = setOf (Formula.Realize \u03c8)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\n\u03c8 : Formula L \u03b1\nx\u271d : \u03b1 \u2192 M\n\u22a2 x\u271d \u2208 setOf (Formula.Realize (onFormula \u03c6 \u03c8)) \u2194 x\u271d \u2208 setOf (Formula.Realize \u03c8)\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\n\u22a2 (Realize (\u2191equivSentence.symm \u03c6) fun a => \u2191(Language.con L a)) \u2194 M \u22a8 \u03c6\n[PROOFSTEP]\nsimp only [equivSentence, Equiv.symm_symm, Equiv.coe_trans, Realize, BoundedFormula.realize_relabelEquiv, Function.comp]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\n\u22a2 BoundedFormula.Realize (\u2191BoundedFormula.constantsVarsEquiv \u03c6)\n      (fun x => \u2191(Language.con L (\u2191(Equiv.sumEmpty \u03b1 Empty) x))) default \u2194\n    M \u22a8 \u03c6\n[PROOFSTEP]\nrefine' _root_.trans _ BoundedFormula.realize_constantsVarsEquiv\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\n\u22a2 BoundedFormula.Realize (\u2191BoundedFormula.constantsVarsEquiv \u03c6)\n      (fun x => \u2191(Language.con L (\u2191(Equiv.sumEmpty \u03b1 Empty) x))) default \u2194\n    BoundedFormula.Realize (\u2191BoundedFormula.constantsVarsEquiv \u03c6) (Sum.elim (fun a => \u2191(Language.con L a)) default)\n      default\n[PROOFSTEP]\nrw [iff_iff_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\n\u22a2 BoundedFormula.Realize (\u2191BoundedFormula.constantsVarsEquiv \u03c6)\n      (fun x => \u2191(Language.con L (\u2191(Equiv.sumEmpty \u03b1 Empty) x))) default =\n    BoundedFormula.Realize (\u2191BoundedFormula.constantsVarsEquiv \u03c6) (Sum.elim (fun a => \u2191(Language.con L a)) default)\n      default\n[PROOFSTEP]\ncongr with (_ | a)\n[GOAL]\ncase e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\nval\u271d : \u03b1\n\u22a2 \u2191(Language.con L (\u2191(Equiv.sumEmpty \u03b1 Empty) (Sum.inl val\u271d))) =\n    Sum.elim (fun a => \u2191(Language.con L a)) default (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Sentence (L[[\u03b1]])\na : Empty\n\u22a2 \u2191(Language.con L (\u2191(Equiv.sumEmpty \u03b1 Empty) (Sum.inr a))) =\n    Sum.elim (fun a => \u2191(Language.con L a)) default (Sum.inr a)\n[PROOFSTEP]\ncases a\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\ninst\u271d\u00b9 : Structure (L[[\u03b1]]) M\ninst\u271d : LHom.IsExpansionOn (lhomWithConstants L \u03b1) M\n\u03c6 : Formula L \u03b1\n\u22a2 M \u22a8 \u2191equivSentence \u03c6 \u2194 Realize \u03c6 fun a => \u2191(Language.con L a)\n[PROOFSTEP]\nrw [\u2190 realize_equivSentence_symm_con M (equivSentence \u03c6), _root_.Equiv.symm_apply_apply]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\n\u22a2 M \u2245[L] N \u2194 \u2200 (\u03c6 : Sentence L), M \u22a8 \u03c6 \u2194 N \u22a8 \u03c6\n[PROOFSTEP]\nsimp only [ElementarilyEquivalent, Set.ext_iff, completeTheory, Set.mem_setOf_eq]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u2074 : Structure L M\ninst\u271d\u00b3 : Structure L N\ninst\u271d\u00b2 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT\u271d : Theory L\ninst\u271d\u00b9 : Structure L' M\n\u03c6 : L \u2192\u1d38 L'\ninst\u271d : IsExpansionOn \u03c6 M\nT : Theory L\n\u22a2 M \u22a8 onTheory \u03c6 T \u2194 M \u22a8 T\n[PROOFSTEP]\nsimp [Theory.model_iff, LHom.onTheory]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT T' : Theory L\nh : M \u22a8 T\nh' : M \u22a8 T'\n\u22a2 M \u22a8 T \u222a T'\n[PROOFSTEP]\nsimp only [model_iff, Set.mem_union] at *\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT T' : Theory L\nh : \u2200 (\u03c6 : Sentence L), \u03c6 \u2208 T \u2192 M \u22a8 \u03c6\nh' : \u2200 (\u03c6 : Sentence L), \u03c6 \u2208 T' \u2192 M \u22a8 \u03c6\n\u22a2 \u2200 (\u03c6 : Sentence L), \u03c6 \u2208 T \u2228 \u03c6 \u2208 T' \u2192 M \u22a8 \u03c6\n[PROOFSTEP]\nexact fun \u03c6 h\u03c6 => h\u03c6.elim (h _) (h' _)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6 : Sentence L\n\u22a2 M \u22a8 {\u03c6} \u2194 M \u22a8 \u03c6\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ninst\u271d : N \u22a8 completeTheory L M\n\u03c6 : Sentence L\n\u22a2 N \u22a8 \u03c6 \u2194 M \u22a8 \u03c6\n[PROOFSTEP]\nrefine' \u27e8fun h => _, (L.completeTheory M).realize_sentence_of_mem\u27e9\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ninst\u271d : N \u22a8 completeTheory L M\n\u03c6 : Sentence L\nh : N \u22a8 \u03c6\n\u22a2 M \u22a8 \u03c6\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ninst\u271d : N \u22a8 completeTheory L M\n\u03c6 : Sentence L\nh : \u00acM \u22a8 \u03c6\n\u22a2 \u00acN \u22a8 \u03c6\n[PROOFSTEP]\nrw [\u2190 Sentence.realize_not] at *\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ninst\u271d : N \u22a8 completeTheory L M\n\u03c6 : Sentence L\nh : M \u22a8 Formula.not \u03c6\n\u22a2 N \u22a8 Formula.not \u03c6\n[PROOFSTEP]\nexact (L.completeTheory M).realize_sentence_of_mem (mem_completeTheory.2 h)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\n\u22a2 Formula.Realize (alls \u03c6) v \u2194 \u2200 (xs : Fin n \u2192 M), Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\n\u03c6 : BoundedFormula L \u03b1 Nat.zero\n\u22a2 Formula.Realize (alls \u03c6) v \u2194 \u2200 (xs : Fin Nat.zero \u2192 M), Realize \u03c6 v xs\n[PROOFSTEP]\nexact Unique.forall_iff.symm\n[GOAL]\ncase succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (alls \u03c6) v \u2194 \u2200 (xs : Fin n \u2192 M), Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 Formula.Realize (alls \u03c6) v \u2194 \u2200 (xs : Fin (Nat.succ n) \u2192 M), Realize \u03c6 v xs\n[PROOFSTEP]\nsimp only [alls, ih, Realize]\n[GOAL]\ncase succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (alls \u03c6) v \u2194 \u2200 (xs : Fin n \u2192 M), Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 (\u2200 (xs : Fin n \u2192 M) (x : M), Realize \u03c6 v (snoc xs x)) \u2194 \u2200 (xs : Fin (Nat.succ n) \u2192 M), Realize \u03c6 v xs\n[PROOFSTEP]\nexact \u27e8fun h xs => Fin.snoc_init_self xs \u25b8 h _ _, fun h xs x => h (Fin.snoc xs x)\u27e9\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\n\u22a2 Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\n\u03c6 : BoundedFormula L \u03b1 Nat.zero\n\u22a2 Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\nexact Unique.exists_iff.symm\n[GOAL]\ncase succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\nsimp only [BoundedFormula.exs, ih, realize_ex]\n[GOAL]\ncase succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 (\u2203 xs a, Realize \u03c6 v (snoc xs a)) \u2194 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase succ.mp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 (\u2203 xs a, Realize \u03c6 v (snoc xs a)) \u2192 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\nrintro \u27e8xs, x, h\u27e9\n[GOAL]\ncase succ.mp.intro.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\nxs : Fin n \u2192 M\nx : M\nh : Realize \u03c6 v (snoc xs x)\n\u22a2 \u2203 xs, Realize \u03c6 v xs\n[PROOFSTEP]\nexact \u27e8_, h\u27e9\n[GOAL]\ncase succ.mpr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\n\u22a2 (\u2203 xs, Realize \u03c6 v xs) \u2192 \u2203 xs a, Realize \u03c6 v (snoc xs a)\n[PROOFSTEP]\nrintro \u27e8xs, h\u27e9\n[GOAL]\ncase succ.mpr.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\nxs : Fin (Nat.succ n) \u2192 M\nh : Realize \u03c6 v xs\n\u22a2 \u2203 xs a, Realize \u03c6 v (snoc xs a)\n[PROOFSTEP]\nrw [\u2190 Fin.snoc_init_self xs] at h \n[GOAL]\ncase succ.mpr.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nv : \u03b1 \u2192 M\nn : \u2115\nih : \u2200 {\u03c6 : BoundedFormula L \u03b1 n}, Formula.Realize (exs \u03c6) v \u2194 \u2203 xs, Realize \u03c6 v xs\n\u03c6 : BoundedFormula L \u03b1 (Nat.succ n)\nxs : Fin (Nat.succ n) \u2192 M\nh : Realize \u03c6 v (snoc (init xs) (xs (last n)))\n\u22a2 \u2203 xs a, Realize \u03c6 v (snoc xs a)\n[PROOFSTEP]\nexact \u27e8_, _, h\u27e9\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2295 Fin n \u2192 M\n\u22a2 Formula.Realize (toFormula \u03c6) v \u2194 Realize \u03c6 (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3 a8 a9 a0\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 Formula.Realize (toFormula falsum) v \u2194 Realize falsum (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 Formula.Realize (toFormula (equal t\u2081\u271d t\u2082\u271d)) v \u2194 Realize (equal t\u2081\u271d t\u2082\u271d) (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\nsimp [BoundedFormula.Realize]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 Formula.Realize (toFormula (rel R\u271d ts\u271d)) v \u2194 Realize (rel R\u271d ts\u271d) (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\nsimp [BoundedFormula.Realize]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 (v : \u03b1 \u2295 Fin n\u271d \u2192 M), Formula.Realize (toFormula f\u2081\u271d) v \u2194 Realize f\u2081\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nih2 : \u2200 (v : \u03b1 \u2295 Fin n\u271d \u2192 M), Formula.Realize (toFormula f\u2082\u271d) v \u2194 Realize f\u2082\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 Formula.Realize (toFormula (f\u2081\u271d \u27f9 f\u2082\u271d)) v \u2194 Realize (f\u2081\u271d \u27f9 f\u2082\u271d) (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\nrw [toFormula, Formula.Realize, realize_imp, \u2190 Formula.Realize, ih1, \u2190 Formula.Realize, ih2, realize_imp]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 Formula.Realize (toFormula (\u2200'f\u271d)) v \u2194 Realize (\u2200'f\u271d) (v \u2218 Sum.inl) (v \u2218 Sum.inr)\n[PROOFSTEP]\nrw [toFormula, Formula.Realize, realize_all, realize_all]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M),\n      Realize (relabel (Sum.elim (Sum.inl \u2218 Sum.inl) (Sum.map Sum.inr id \u2218 \u2191finSumFinEquiv.symm)) (toFormula f\u271d)) v\n        (snoc default a)) \u2194\n    \u2200 (a : M), Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n[PROOFSTEP]\nrefine' forall_congr' fun a => _\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\n\u22a2 Realize (relabel (Sum.elim (Sum.inl \u2218 Sum.inl) (Sum.map Sum.inr id \u2218 \u2191finSumFinEquiv.symm)) (toFormula f\u271d)) v\n      (snoc default a) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n[PROOFSTEP]\nhave h := ih3 (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a))\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a) \u2218 Sum.inl)\n      (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a) \u2218 Sum.inr)\n\u22a2 Realize (relabel (Sum.elim (Sum.inl \u2218 Sum.inl) (Sum.map Sum.inr id \u2218 \u2191finSumFinEquiv.symm)) (toFormula f\u271d)) v\n      (snoc default a) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n[PROOFSTEP]\nsimp only [Sum.elim_comp_inl, Sum.elim_comp_inr] at h \n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n\u22a2 Realize (relabel (Sum.elim (Sum.inl \u2218 Sum.inl) (Sum.map Sum.inr id \u2218 \u2191finSumFinEquiv.symm)) (toFormula f\u271d)) v\n      (snoc default a) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n[PROOFSTEP]\nrw [\u2190 h, realize_relabel, Formula.Realize, iff_iff_eq]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n\u22a2 Realize (toFormula f\u271d)\n      (Sum.elim v (snoc default a \u2218 castAdd 0) \u2218\n        Sum.elim (Sum.inl \u2218 Sum.inl) (Sum.map Sum.inr id \u2218 \u2191finSumFinEquiv.symm))\n      (snoc default a \u2218 natAdd 1) =\n    Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) default\n[PROOFSTEP]\nsimp only [Function.comp]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\n\u22a2 (Realize (toFormula f\u271d)\n      (fun x =>\n        Sum.elim v (fun x => snoc default a (castAdd 0 x))\n          (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x)) x))\n      fun x => snoc default a (natAdd 1 x)) =\n    Realize (toFormula f\u271d) (Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a)) default\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase all.e__v.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : \u03b1 \u2295 Fin (n\u271d + 1)\n\u22a2 Sum.elim v (fun x => snoc default a (castAdd 0 x))\n      (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x)) x) =\n    Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) x\n[PROOFSTEP]\ncases' x with _ x\n[GOAL]\ncase all.e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nval\u271d : \u03b1\n\u22a2 Sum.elim v (fun x => snoc default a (castAdd 0 x))\n      (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x)) (Sum.inl val\u271d)) =\n    Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase all.e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 Sum.elim v (fun x => snoc default a (castAdd 0 x))\n      (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x)) (Sum.inr x)) =\n    Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr x)\n[PROOFSTEP]\nrefine' Fin.lastCases _ _ x\n[GOAL]\ncase all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 Sum.elim v (fun x => snoc default a (castAdd 0 x))\n      (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x))\n        (Sum.inr (last n\u271d))) =\n    Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (last n\u271d))\n[PROOFSTEP]\nrw [Sum.elim_inr, Sum.elim_inr, finSumFinEquiv_symm_last, Sum.map_inr, Sum.elim_inr]\n[GOAL]\ncase all.e__v.h.inr.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 snoc default a (castAdd 0 (id 0)) = snoc (fun x => v (Sum.inr x)) a (last n\u271d)\n[PROOFSTEP]\nsimp [Fin.snoc]\n[GOAL]\ncase all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 \u2200 (i : Fin n\u271d),\n    Sum.elim v (fun x => snoc default a (castAdd 0 x))\n        (Sum.elim (fun x => Sum.inl (Sum.inl x)) (fun x => Sum.map Sum.inr id (\u2191finSumFinEquiv.symm x))\n          (Sum.inr (castSucc i))) =\n      Sum.elim (fun x => v (Sum.inl x)) (snoc (fun x => v (Sum.inr x)) a) (Sum.inr (castSucc i))\n[PROOFSTEP]\nsimp only [castSucc, Function.comp_apply, Sum.elim_inr, finSumFinEquiv_symm_apply_castAdd, Sum.map_inl, Sum.elim_inl]\n[GOAL]\ncase all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 \u2200 (i : Fin n\u271d), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (castAdd 1 i)\n[PROOFSTEP]\nrw [\u2190 castSucc]\n[GOAL]\ncase all.e__v.h.inr.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin (n\u271d + 1)\n\u22a2 \u2200 (i : Fin n\u271d), v (Sum.inr i) = snoc (fun x => v (Sum.inr x)) a (castSucc i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase all.e__xs.h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nv\u271d : \u03b1 \u2295 Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (v : \u03b1 \u2295 Fin (n\u271d + 1) \u2192 M), Formula.Realize (toFormula f\u271d) v \u2194 Realize f\u271d (v \u2218 Sum.inl) (v \u2218 Sum.inr)\nv : \u03b1 \u2295 Fin n\u271d \u2192 M\na : M\nh :\n  Formula.Realize (toFormula f\u271d) (Sum.elim (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)) \u2194\n    Realize f\u271d (v \u2218 Sum.inl) (snoc (v \u2218 Sum.inr) a)\nx : Fin 0\n\u22a2 snoc default a (natAdd 1 x) = default x\n[PROOFSTEP]\nexact Fin.elim0 x\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\n\u03c6 : BoundedFormula L \u03b1 n\nv : \u03b1 \u2192 M\nxs : Fin n \u2192 M\n\u22a2 BoundedFormula.Realize \u03c6 (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize \u03c6 v xs\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nxs : Fin n\u271d \u2192 M\n\u22a2 BoundedFormula.Realize BoundedFormula.falsum (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize BoundedFormula.falsum v xs\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 BoundedFormula.Realize (BoundedFormula.equal t\u2081\u271d t\u2082\u271d) (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194\n    BoundedFormula.Realize (BoundedFormula.equal t\u2081\u271d t\u2082\u271d) v xs\n[PROOFSTEP]\nsimp only [BoundedFormula.Realize, \u2190 Sum.comp_elim, Equiv.realize_term, g.injective.eq_iff]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 BoundedFormula.Realize (BoundedFormula.rel R\u271d ts\u271d) (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194\n    BoundedFormula.Realize (BoundedFormula.rel R\u271d ts\u271d) v xs\n[PROOFSTEP]\nsimp only [BoundedFormula.Realize, \u2190 Sum.comp_elim, Equiv.realize_term]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nxs : Fin n\u271d \u2192 M\n\u22a2 (RelMap R\u271d fun i => \u2191g (Term.realize (Sum.elim v xs) (ts\u271d i))) \u2194\n    RelMap R\u271d fun i => Term.realize (Sum.elim v xs) (ts\u271d i)\n[PROOFSTEP]\nexact g.map_rel _ _\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 {xs : Fin n\u271d \u2192 M}, BoundedFormula.Realize f\u2081\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u2081\u271d v xs\nih2 : \u2200 {xs : Fin n\u271d \u2192 M}, BoundedFormula.Realize f\u2082\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u2082\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 BoundedFormula.Realize (f\u2081\u271d \u27f9 f\u2082\u271d) (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize (f\u2081\u271d \u27f9 f\u2082\u271d) v xs\n[PROOFSTEP]\nrw [BoundedFormula.Realize, ih1, ih2, BoundedFormula.Realize]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 BoundedFormula.Realize (\u2200'f\u271d) (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize (\u2200'f\u271d) v xs\n[PROOFSTEP]\nrw [BoundedFormula.Realize, BoundedFormula.Realize]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)) \u2194\n    \u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase all.mp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)) \u2192\n    \u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)\n[PROOFSTEP]\nintro h a\n[GOAL]\ncase all.mp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)\na : M\n\u22a2 BoundedFormula.Realize f\u271d v (snoc xs a)\n[PROOFSTEP]\nhave h' := h (g a)\n[GOAL]\ncase all.mp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)\na : M\nh' : BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) (\u2191g a))\n\u22a2 BoundedFormula.Realize f\u271d v (snoc xs a)\n[PROOFSTEP]\nrw [\u2190 Fin.comp_snoc, ih3] at h' \n[GOAL]\ncase all.mp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)\na : M\nh' : BoundedFormula.Realize f\u271d v (snoc xs a)\n\u22a2 BoundedFormula.Realize f\u271d v (snoc xs a)\n[PROOFSTEP]\nexact h'\n[GOAL]\ncase all.mpr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)) \u2192\n    \u2200 (x : N), BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) x)\n[PROOFSTEP]\nintro h a\n[GOAL]\ncase all.mpr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)\na : N\n\u22a2 BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) a)\n[PROOFSTEP]\nhave h' := h (g.symm a)\n[GOAL]\ncase all.mpr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)\na : N\nh' : BoundedFormula.Realize f\u271d v (snoc xs (\u2191(symm g) a))\n\u22a2 BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) a)\n[PROOFSTEP]\nrw [\u2190 ih3, Fin.comp_snoc, g.apply_symm_apply] at h' \n[GOAL]\ncase all.mpr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\nv : \u03b1 \u2192 M\nxs\u271d : Fin n \u2192 M\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (\u2191g \u2218 xs) \u2194 BoundedFormula.Realize f\u271d v xs\nxs : Fin n\u271d \u2192 M\nh : \u2200 (x : M), BoundedFormula.Realize f\u271d v (snoc xs x)\na : N\nh' : BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) a)\n\u22a2 BoundedFormula.Realize f\u271d (\u2191g \u2218 v) (snoc (\u2191g \u2218 xs) a)\n[PROOFSTEP]\nexact h'\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\n\u03c6 : Formula L \u03b1\nv : \u03b1 \u2192 M\n\u22a2 Formula.Realize \u03c6 (\u2191g \u2218 v) \u2194 Formula.Realize \u03c6 v\n[PROOFSTEP]\nrw [Formula.Realize, Formula.Realize, \u2190 g.realize_boundedFormula \u03c6, iff_eq_eq, Unique.eq_default (g \u2218 default)]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\ng : M \u2243[L] N\n\u03c6 : Sentence L\n\u22a2 M \u22a8 \u03c6 \u2194 N \u22a8 \u03c6\n[PROOFSTEP]\nrw [Sentence.Realize, Sentence.Realize, \u2190 g.realize_formula, Unique.eq_default (g \u2218 default)]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\n\u22a2 M \u22a8 Sentence.cardGe L n \u2194 \u2191n \u2264 #M\n[PROOFSTEP]\nrw [\u2190 lift_mk_fin, \u2190 lift_le.{0}, lift_lift, lift_mk_le, Sentence.cardGe, Sentence.Realize, BoundedFormula.realize_exs]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\n\u22a2 (\u2203 xs,\n      BoundedFormula.Realize\n        (List.foldr (fun x x_1 => x \u2293 x_1) \u22a4\n          (List.map (fun ij => \u223c((var \u2218 Sum.inr) ij.fst =' (var \u2218 Sum.inr) ij.snd))\n            (List.filter (fun ij => decide (ij.fst \u2260 ij.snd)) (List.finRange n \u00d7\u02e2 List.finRange n))))\n        default xs) \u2194\n    Nonempty (Fin n \u21aa M)\n[PROOFSTEP]\nsimp_rw [BoundedFormula.realize_foldr_inf]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\n\u22a2 (\u2203 xs,\n      \u2200 (\u03c6 : BoundedFormula L Empty n),\n        \u03c6 \u2208\n            List.map (fun ij => \u223c((var \u2218 Sum.inr) ij.fst =' (var \u2218 Sum.inr) ij.snd))\n              (List.filter (fun ij => decide (ij.fst \u2260 ij.snd)) (List.finRange n \u00d7\u02e2 List.finRange n)) \u2192\n          BoundedFormula.Realize \u03c6 default xs) \u2194\n    Nonempty (Fin n \u21aa M)\n[PROOFSTEP]\nsimp only [Function.comp_apply, List.mem_map, Prod.exists, Ne.def, List.mem_product, List.mem_finRange,\n  forall_exists_index, and_imp, List.mem_filter, true_and_iff]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\n\u22a2 (\u2203 xs,\n      \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n        (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs) \u2194\n    Nonempty (Fin n \u21aa M)\n[PROOFSTEP]\nrefine' \u27e8_, fun xs => \u27e8xs.some, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\n\u22a2 (\u2203 xs,\n      \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n        (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs) \u2192\n    Nonempty (Fin n \u21aa M)\n[PROOFSTEP]\nrintro \u27e8xs, h\u27e9\n[GOAL]\ncase refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\n\u22a2 Nonempty (Fin n \u21aa M)\n[PROOFSTEP]\nrefine' \u27e8\u27e8xs, fun i j ij => _\u27e9\u27e9\n[GOAL]\ncase refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\ni j : Fin n\nij : xs i = xs j\n\u22a2 i = j\n[PROOFSTEP]\ncontrapose! ij\n[GOAL]\ncase refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\ni j : Fin n\nij : i \u2260 j\n\u22a2 xs i \u2260 xs j\n[PROOFSTEP]\nhave hij := h _ i j (by simpa using ij) rfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\ni j : Fin n\nij : i \u2260 j\n\u22a2 (decide \u00aci = j) = true\n[PROOFSTEP]\nsimpa using ij\n[GOAL]\ncase refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\ni j : Fin n\nij : i \u2260 j\nhij : BoundedFormula.Realize (\u223c(var (Sum.inr i) =' var (Sum.inr j))) default xs\n\u22a2 xs i \u2260 xs j\n[PROOFSTEP]\nsimp only [BoundedFormula.realize_not, Term.realize, BoundedFormula.realize_bdEqual, Sum.elim_inr] at hij \n[GOAL]\ncase refine'_1.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Fin n \u2192 M\nh :\n  \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192 \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default xs\ni j : Fin n\nij : i \u2260 j\nhij : \u00acxs i = xs j\n\u22a2 xs i \u2260 xs j\n[PROOFSTEP]\nexact hij\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Nonempty (Fin n \u21aa M)\n\u22a2 \u2200 (\u03c6 : BoundedFormula L Empty n) (x x_1 : Fin n),\n    (decide \u00acx = x_1) = true \u2192\n      \u223c(var (Sum.inr x) =' var (Sum.inr x_1)) = \u03c6 \u2192 BoundedFormula.Realize \u03c6 default \u2191(Nonempty.some xs)\n[PROOFSTEP]\nrintro _ i j ij rfl\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn\u271d : \u2115\nT : Theory L\nn : \u2115\nxs : Nonempty (Fin n \u21aa M)\ni j : Fin n\nij : (decide \u00aci = j) = true\n\u22a2 BoundedFormula.Realize (\u223c(var (Sum.inr i) =' var (Sum.inr j))) default \u2191(Nonempty.some xs)\n[PROOFSTEP]\nsimpa using ij\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u22a2 M \u22a8 infiniteTheory L \u2194 Infinite M\n[PROOFSTEP]\nsimp [infiniteTheory, infinite_iff, aleph0_le]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\n\u22a2 M \u22a8 nonemptyTheory L \u2194 Nonempty M\n[PROOFSTEP]\nsimp only [nonemptyTheory, Theory.model_iff, Set.mem_singleton_iff, forall_eq, Sentence.realize_cardGe, Nat.cast_one,\n  one_le_iff_ne_zero, mk_ne_zero_iff]\n[GOAL]\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\n\u22a2 M \u22a8 distinctConstantsTheory L s \u2194 Set.InjOn (fun i => \u2191(Language.con L i)) s\n[PROOFSTEP]\nsimp only [distinctConstantsTheory, Theory.model_iff, Set.mem_image, Set.mem_inter, Set.mem_prod, Set.mem_compl,\n  Prod.exists, forall_exists_index, and_imp]\n[GOAL]\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\n\u22a2 (\u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n      (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n        Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192\n          M \u22a8 \u03c6) \u2194\n    Set.InjOn (fun i => \u2191(Language.con L i)) s\n[PROOFSTEP]\nrefine' \u27e8fun h a as b bs ab => _, _\u27e9\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh :\n  \u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n    (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n      Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192 M \u22a8 \u03c6\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nab : (fun i => \u2191(Language.con L i)) a = (fun i => \u2191(Language.con L i)) b\n\u22a2 a = b\n[PROOFSTEP]\ncontrapose! ab\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh :\n  \u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n    (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n      Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192 M \u22a8 \u03c6\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nab : a \u2260 b\n\u22a2 \u2191(Language.con L a) \u2260 \u2191(Language.con L b)\n[PROOFSTEP]\nhave h' := h _ a b \u27e8\u27e8as, bs\u27e9, ab\u27e9 rfl\n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh :\n  \u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n    (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n      Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192 M \u22a8 \u03c6\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nab : a \u2260 b\nh' : M \u22a8 Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))\n\u22a2 \u2191(Language.con L a) \u2260 \u2191(Language.con L b)\n[PROOFSTEP]\nsimp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal, Term.realize_constants] at h' \n[GOAL]\ncase refine'_1\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh :\n  \u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n    (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n      Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192 M \u22a8 \u03c6\na : \u03b1\nas : a \u2208 s\nb : \u03b1\nbs : b \u2208 s\nab : a \u2260 b\nh' : \u00ac\u2191(Language.con L a) = \u2191(Language.con L b)\n\u22a2 \u2191(Language.con L a) \u2260 \u2191(Language.con L b)\n[PROOFSTEP]\nexact h'\n[GOAL]\ncase refine'_2\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\n\u22a2 Set.InjOn (fun i => \u2191(Language.con L i)) s \u2192\n    \u2200 (\u03c6 : Sentence (L[[\u03b1]])) (x x_1 : \u03b1),\n      (x, x_1) \u2208 s \u00d7\u02e2 s \u2229 (Set.diagonal \u03b1)\u1d9c \u2192\n        Formula.not (Term.equal (Constants.term (Language.con L x)) (Constants.term (Language.con L x_1))) = \u03c6 \u2192 M \u22a8 \u03c6\n[PROOFSTEP]\nrintro h \u03c6 a b \u27e8\u27e8as, bs\u27e9, ab\u27e9 rfl\n[GOAL]\ncase refine'_2.intro.intro\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh : Set.InjOn (fun i => \u2191(Language.con L i)) s\na b : \u03b1\nab : (a, b) \u2208 (Set.diagonal \u03b1)\u1d9c\nas : (a, b).fst \u2208 s\nbs : (a, b).snd \u2208 s\n\u22a2 M \u22a8 Formula.not (Term.equal (Constants.term (Language.con L a)) (Constants.term (Language.con L b)))\n[PROOFSTEP]\nsimp only [Sentence.Realize, Formula.realize_not, Formula.realize_equal, Term.realize_constants]\n[GOAL]\ncase refine'_2.intro.intro\nL : Language\nL' : Language\nM\u271d : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : Structure L M\u271d\ninst\u271d\u00b2 : Structure L N\ninst\u271d\u00b9 : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nM : Type w\ninst\u271d : Structure (L[[\u03b1]]) M\ns : Set \u03b1\nh : Set.InjOn (fun i => \u2191(Language.con L i)) s\na b : \u03b1\nab : (a, b) \u2208 (Set.diagonal \u03b1)\u1d9c\nas : (a, b).fst \u2208 s\nbs : (a, b).snd \u2208 s\n\u22a2 \u00ac\u2191(Language.con L a) = \u2191(Language.con L b)\n[PROOFSTEP]\nexact fun contra => ab (h as bs contra)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\nn : \u2115\nT : Theory L\nh : M \u2245[L] N\n\u22a2 M \u22a8 T \u2194 N \u22a8 T\n[PROOFSTEP]\nrw [Theory.model_iff_subset_completeTheory, Theory.model_iff_subset_completeTheory, h.completeTheory_eq]\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Semantics", "llama_tokens": 97807, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257126, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.5376534801474364}}
{"text": "[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 a * a = b * b\n[PROOFSTEP]\nrw [\u2190 abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs]\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a = natAbs b \u2194 \u2191(natAbs a) = \u2191(natAbs b)\n[PROOFSTEP]\nexact Int.coe_nat_inj'.symm\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 a * a < b * b\n[PROOFSTEP]\nrw [\u2190 abs_lt_iff_mul_self_lt, abs_eq_natAbs, abs_eq_natAbs]\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a < natAbs b \u2194 \u2191(natAbs a) < \u2191(natAbs b)\n[PROOFSTEP]\nexact Int.ofNat_lt.symm\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 a * a \u2264 b * b\n[PROOFSTEP]\nrw [\u2190 abs_le_iff_mul_self_le, abs_eq_natAbs, abs_eq_natAbs]\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\n\u22a2 natAbs a \u2264 natAbs b \u2194 \u2191(natAbs a) \u2264 \u2191(natAbs b)\n[PROOFSTEP]\nexact Int.ofNat_le.symm\n[GOAL]\na\u271d b\u271d : \u2124\nn : \u2115\na b c : \u2124\nh : a * b \u2223 c\n\u22a2 b \u2223 c / a\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\na b\u271d : \u2124\nn : \u2115\nb c : \u2124\nh : 0 * b \u2223 c\n\u22a2 b \u2223 c / 0\n[PROOFSTEP]\nsimp only [Int.ediv_zero, dvd_zero]\n[GOAL]\ncase inr\na\u271d b\u271d : \u2124\nn : \u2115\na b c : \u2124\nh : a * b \u2223 c\nha : a \u2260 0\n\u22a2 b \u2223 c / a\n[PROOFSTEP]\nrcases h with \u27e8d, rfl\u27e9\n[GOAL]\ncase inr.intro\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : a \u2260 0\nd : \u2124\n\u22a2 b \u2223 a * b * d / a\n[PROOFSTEP]\nrefine' \u27e8d, _\u27e9\n[GOAL]\ncase inr.intro\na\u271d b\u271d : \u2124\nn : \u2115\na b : \u2124\nha : a \u2260 0\nd : \u2124\n\u22a2 a * b * d / a = b * d\n[PROOFSTEP]\nrw [mul_assoc, Int.mul_ediv_cancel_left _ ha]\n[GOAL]\na b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\n\u22a2 x = 0\n[PROOFSTEP]\nby_cases hm : m = 0\n[GOAL]\ncase pos\na b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\nhm : m = 0\n\u22a2 x = 0\n[PROOFSTEP]\nsubst m\n[GOAL]\ncase pos\na b : \u2124\nn : \u2115\nx : \u2124\nh1 : 0 \u2223 x\nh2 : |x| < 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact zero_dvd_iff.mp h1\n[GOAL]\ncase neg\na b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\nhm : \u00acm = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrcases h1 with \u27e8d, rfl\u27e9\n[GOAL]\ncase neg.intro\na b : \u2124\nn : \u2115\nm : \u2124\nhm : \u00acm = 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 m * d = 0\n[PROOFSTEP]\napply mul_eq_zero_of_right\n[GOAL]\ncase neg.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : \u00acm = 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 d = 0\n[PROOFSTEP]\nrw [\u2190 abs_lt_one_iff, \u2190 mul_lt_iff_lt_one_right (abs_pos.mpr hm), \u2190 abs_mul]\n[GOAL]\ncase neg.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : \u00acm = 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 |m * d| < |m|\n[PROOFSTEP]\nexact lt_of_lt_of_le h2 (le_abs_self m)\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.Order.Lemmas", "llama_tokens": 1435, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746407, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.5372907133676993}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\n\u22a2 \u2200 (r\u2081 r\u2082 : List (List \u03b1)),\n    sublists'Aux a r\u2081 r\u2082 =\n      Array.toList\n        (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r\u2082) (toArray r\u2081) 0 (Array.size (toArray r\u2081)))\n[PROOFSTEP]\nintro r\u2081 r\u2082\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081 r\u2082 : List (List \u03b1)\n\u22a2 sublists'Aux a r\u2081 r\u2082 =\n    Array.toList (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r\u2082) (toArray r\u2081) 0 (Array.size (toArray r\u2081)))\n[PROOFSTEP]\nrw [sublists'Aux, Array.foldl_eq_foldl_data]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081 r\u2082 : List (List \u03b1)\n\u22a2 foldl (fun r l => r ++ [a :: l]) r\u2082 r\u2081 =\n    Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r\u2082) (toArray r\u2081).data)\n[PROOFSTEP]\nhave := List.foldl_hom Array.toList (fun r l => r.push (a :: l)) (fun r l => r ++ [a :: l]) r\u2081 r\u2082.toArray (by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081 r\u2082 : List (List \u03b1)\n\u22a2 \u2200 (x : Array (List \u03b1)) (y : List \u03b1),\n    (fun r l => r ++ [a :: l]) (Array.toList x) y = Array.toList ((fun r l => Array.push r (a :: l)) x y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081 r\u2082 : List (List \u03b1)\nthis :\n  foldl (fun r l => r ++ [a :: l]) (Array.toList (toArray r\u2082)) r\u2081 =\n    Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r\u2082) r\u2081)\n\u22a2 foldl (fun r l => r ++ [a :: l]) r\u2082 r\u2081 =\n    Array.toList (foldl (fun r l => Array.push r (a :: l)) (toArray r\u2082) (toArray r\u2081).data)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists' l = foldr (fun a r => sublists'Aux a r r) [[]] l\n[PROOFSTEP]\nsimp only [sublists', sublists'Aux_eq_array_foldl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 Array.toList\n      (foldr (fun a arr => Array.foldl (fun r l => Array.push r (a :: l)) arr arr 0 (Array.size arr)) #[[]] l) =\n    foldr\n      (fun a r =>\n        Array.toList\n          (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n      [[]] l\n[PROOFSTEP]\nrw [\u2190 List.foldr_hom Array.toList]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 foldr ?g\u2082 (Array.toList #[[]]) l =\n    foldr\n      (fun a r =>\n        Array.toList\n          (Array.foldl (fun r l => Array.push r (a :: l)) (toArray r) (toArray r) 0 (Array.size (toArray r))))\n      [[]] l\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (x : \u03b1) (y : Array (List \u03b1)),\n    Array.toList\n        (Array.foldl (fun r l => Array.push r (x :: l)) (toArray (Array.toList y)) (toArray (Array.toList y)) 0\n          (Array.size (toArray (Array.toList y)))) =\n      Array.toList (Array.foldl (fun r l => Array.push r (x :: l)) y y 0 (Array.size y))\n[PROOFSTEP]\nintros _ _\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 Array.toList\n      (Array.foldl (fun r l => Array.push r (x\u271d :: l)) (toArray (Array.toList y\u271d)) (toArray (Array.toList y\u271d)) 0\n        (Array.size (toArray (Array.toList y\u271d)))) =\n    Array.toList (Array.foldl (fun r l => Array.push r (x\u271d :: l)) y\u271d y\u271d 0 (Array.size y\u271d))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase H.e_as.h.e_4.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 toArray (Array.toList y\u271d) = y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.e_as.h.e_5.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 toArray (Array.toList y\u271d) = y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.e_as.h.e_7.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 toArray (Array.toList y\u271d) = y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081 x\u271d : List (List \u03b1)\n\u22a2 sublists'Aux a [] x\u271d = x\u271d ++ map (cons a) []\n[PROOFSTEP]\nsimp [sublists'Aux]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081\u271d r\u2081 : List (List \u03b1)\nl : List \u03b1\nih : \u2200 (r\u2082 : List (List \u03b1)), sublists'Aux a r\u2081 r\u2082 = r\u2082 ++ map (cons a) r\u2081\nr\u2082 : List (List \u03b1)\n\u22a2 sublists'Aux a (r\u2081 ++ [l]) r\u2082 = r\u2082 ++ map (cons a) (r\u2081 ++ [l])\n[PROOFSTEP]\nrw [map_append, map_singleton, \u2190 append_assoc, \u2190 ih, sublists'Aux, foldl_append, foldl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u2081\u271d r\u2081 : List (List \u03b1)\nl : List \u03b1\nih : \u2200 (r\u2082 : List (List \u03b1)), sublists'Aux a r\u2081 r\u2082 = r\u2082 ++ map (cons a) r\u2081\nr\u2082 : List (List \u03b1)\n\u22a2 foldl (fun r l => r ++ [a :: l]) (foldl (fun r l => r ++ [a :: l]) r\u2082 r\u2081 ++ [a :: l]) [] =\n    sublists'Aux a r\u2081 r\u2082 ++ [a :: l]\n[PROOFSTEP]\nsimp [sublists'Aux]\n  -- Porting note: simp can prove `sublists'_singleton`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l)\n[PROOFSTEP]\nsimp [sublists'_eq_sublists'Aux, foldr_cons, sublists'Aux_eq_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : List \u03b1\n\u22a2 s \u2208 sublists' t \u2194 s <+ t\n[PROOFSTEP]\ninduction' t with a t IH generalizing s\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : List \u03b1\n\u22a2 s \u2208 sublists' [] \u2194 s <+ []\n[PROOFSTEP]\nsimp only [sublists'_nil, mem_singleton]\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : List \u03b1\n\u22a2 s = [] \u2194 s <+ []\n[PROOFSTEP]\nexact \u27e8fun h => by rw [h], eq_nil_of_sublist_nil\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d s : List \u03b1\nh : s = []\n\u22a2 s <+ []\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\n\u22a2 s \u2208 sublists' (a :: t) \u2194 s <+ a :: t\n[PROOFSTEP]\nsimp only [sublists'_cons, mem_append, IH, mem_map]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\n\u22a2 (s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s) \u2194 s <+ a :: t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\n\u22a2 (s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s) \u2192 s <+ a :: t\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\n\u22a2 s <+ a :: t \u2192 s <+ t \u2228 \u2203 a_2, a_2 <+ t \u2227 a :: a_2 = s\n[PROOFSTEP]\nintro h\n[GOAL]\ncase cons.mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s\n\u22a2 s <+ a :: t\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ a :: t\n\u22a2 s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s\n[PROOFSTEP]\nrcases h with (h | \u27e8s, h, rfl\u27e9)\n[GOAL]\ncase cons.mp.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ t\n\u22a2 s <+ a :: t\n[PROOFSTEP]\nexact sublist_cons_of_sublist _ h\n[GOAL]\ncase cons.mp.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ t\n\u22a2 a :: s <+ a :: t\n[PROOFSTEP]\nexact h.cons_cons _\n[GOAL]\ncase cons.mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ a :: t\n\u22a2 s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s\n[PROOFSTEP]\ncases' h with _ _ _ h s _ _ h\n[GOAL]\ncase cons.mpr.cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ t\n\u22a2 s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = s\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\ncase cons.mpr.cons\u2082\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns\u271d : List \u03b1\na : \u03b1\nt : List \u03b1\nIH : \u2200 {s : List \u03b1}, s \u2208 sublists' t \u2194 s <+ t\ns : List \u03b1\nh : s <+ t\n\u22a2 a :: s <+ t \u2228 \u2203 a_1, a_1 <+ t \u2227 a :: a_1 = a :: s\n[PROOFSTEP]\nexact Or.inr \u27e8s, h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 length (sublists' (a :: l)) = 2 ^ length (a :: l)\n[PROOFSTEP]\nsimp_arith only [sublists'_cons, length_append, length_sublists' l, length_map, length, Nat.pow_succ', mul_succ,\n  mul_zero, zero_add]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 sublistsAux = fun a r =>\n    Array.toList\n      (Array.foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r) 0 (Array.size (toArray r)))\n[PROOFSTEP]\nfunext a r\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr : List (List \u03b1)\n\u22a2 sublistsAux a r =\n    Array.toList\n      (Array.foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r) 0 (Array.size (toArray r)))\n[PROOFSTEP]\nsimp only [sublistsAux, Array.foldl_eq_foldl_data, Array.mkEmpty]\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr : List (List \u03b1)\n\u22a2 foldl (fun r l => r ++ [l, a :: l]) [] r =\n    Array.toList (foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r).data)\n[PROOFSTEP]\nhave :=\n  foldl_hom Array.toList (fun r l => (r.push l).push (a :: l)) (fun (r : List (List \u03b1)) l => r ++ [l, a :: l]) r #[]\n    (by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr : List (List \u03b1)\n\u22a2 \u2200 (x : Array (List \u03b1)) (y : List \u03b1),\n    (fun r l => r ++ [l, a :: l]) (Array.toList x) y =\n      Array.toList ((fun r l => Array.push (Array.push r l) (a :: l)) x y)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr : List (List \u03b1)\nthis :\n  foldl (fun r l => r ++ [l, a :: l]) (Array.toList #[]) r =\n    Array.toList (foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] r)\n\u22a2 foldl (fun r l => r ++ [l, a :: l]) [] r =\n    Array.toList (foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r).data)\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr : List (List \u03b1)\n\u22a2 sublistsAux a [] = List.bind [] fun l => [l, a :: l]\n[PROOFSTEP]\nsimp [sublistsAux]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u271d r : List (List \u03b1)\nl : List \u03b1\nih : sublistsAux a r = List.bind r fun l => [l, a :: l]\n\u22a2 sublistsAux a (r ++ [l]) = List.bind (r ++ [l]) fun l => [l, a :: l]\n[PROOFSTEP]\nrw [append_bind, \u2190 ih, bind_singleton, sublistsAux, foldl_append]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nr\u271d r : List (List \u03b1)\nl : List \u03b1\nih : sublistsAux a r = List.bind r fun l => [l, a :: l]\n\u22a2 foldl (fun r l => r ++ [l, a :: l]) (foldl (fun r l => r ++ [l, a :: l]) [] r) [l] = sublistsAux a r ++ [l, a :: l]\n[PROOFSTEP]\nsimp [sublistsAux]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists l = foldr sublistsAux [[]] l\n[PROOFSTEP]\nsimp only [sublists, sublistsAux_eq_array_foldl, Array.foldr_eq_foldr_data]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 Array.toList\n      (foldr\n        (fun a arr =>\n          Array.foldl (fun r l => Array.push (Array.push r l) (a :: l)) (Array.mkEmpty (Array.size arr * 2)) arr 0\n            (Array.size arr))\n        #[[]] l) =\n    foldr\n      (fun a r =>\n        Array.toList\n          (Array.foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r) 0 (Array.size (toArray r))))\n      [[]] l\n[PROOFSTEP]\nrw [\u2190 foldr_hom Array.toList]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 foldr ?g\u2082 (Array.toList #[[]]) l =\n    foldr\n      (fun a r =>\n        Array.toList\n          (Array.foldl (fun r l => Array.push (Array.push r l) (a :: l)) #[] (toArray r) 0 (Array.size (toArray r))))\n      [[]] l\n[PROOFSTEP]\nrfl\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (x : \u03b1) (y : Array (List \u03b1)),\n    Array.toList\n        (Array.foldl (fun r l => Array.push (Array.push r l) (x :: l)) #[] (toArray (Array.toList y)) 0\n          (Array.size (toArray (Array.toList y)))) =\n      Array.toList\n        (Array.foldl (fun r l => Array.push (Array.push r l) (x :: l)) (Array.mkEmpty (Array.size y * 2)) y 0\n          (Array.size y))\n[PROOFSTEP]\nintros _ _\n[GOAL]\ncase H\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 Array.toList\n      (Array.foldl (fun r l => Array.push (Array.push r l) (x\u271d :: l)) #[] (toArray (Array.toList y\u271d)) 0\n        (Array.size (toArray (Array.toList y\u271d)))) =\n    Array.toList\n      (Array.foldl (fun r l => Array.push (Array.push r l) (x\u271d :: l)) (Array.mkEmpty (Array.size y\u271d * 2)) y\u271d 0\n        (Array.size y\u271d))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase H.e_as.h.e_5.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 toArray (Array.toList y\u271d) = y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.e_as.h.e_7.e_a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nx\u271d : \u03b1\ny\u271d : Array (List \u03b1)\n\u22a2 toArray (Array.toList y\u271d) = y\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\n\u22a2 sublists (l\u2081 ++ l\u2082) = do\n    let x \u2190 sublists l\u2082\n    map (fun x_1 => x_1 ++ x) (sublists l\u2081)\n[PROOFSTEP]\nsimp only [sublists_eq_sublistsAux, foldr_append, sublistsAux_eq_bind]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) l\u2081 =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2081)\n[PROOFSTEP]\ninduction l\u2081\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) [] =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] [])\n[PROOFSTEP]\ncase nil => simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) [] =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] [])\n[PROOFSTEP]\ncase nil => simp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) [] =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d :\n  foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082)\n      tail\u271d =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] tail\u271d)\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082)\n      (head\u271d :: tail\u271d) =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] (head\u271d :: tail\u271d))\n[PROOFSTEP]\ncase cons a l\u2081 ih =>\n  rw [foldr_cons, ih]\n  simp [List.bind, join_join, Function.comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nih :\n  foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) l\u2081 =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2081)\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082)\n      (a :: l\u2081) =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] (a :: l\u2081))\n[PROOFSTEP]\ncase cons a l\u2081 ih =>\n  rw [foldr_cons, ih]\n  simp [List.bind, join_join, Function.comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nih :\n  foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) l\u2081 =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2081)\n\u22a2 foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082)\n      (a :: l\u2081) =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] (a :: l\u2081))\n[PROOFSTEP]\nrw [foldr_cons, ih]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2082 : List \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nih :\n  foldr (fun a r => List.bind r fun l => [l, a :: l]) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082) l\u2081 =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2081)\n\u22a2 (List.bind\n      (do\n        let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n        map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2081))\n      fun l => [l, a :: l]) =\n    do\n    let x \u2190 foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] l\u2082\n    map (fun x_1 => x_1 ++ x) (foldr (fun a r => List.bind r fun l => [l, a :: l]) [[]] (a :: l\u2081))\n[PROOFSTEP]\nsimp [List.bind, join_join, Function.comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists ([a] ++ l) = do\n    let x \u2190 sublists l\n    [x, a :: x]\n[PROOFSTEP]\nrw [sublists_append]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 (do\n      let x \u2190 sublists l\n      map (fun x_1 => x_1 ++ x) (sublists [a])) =\n    do\n    let x \u2190 sublists l\n    [x, a :: x]\n[PROOFSTEP]\nsimp only [sublists_singleton, map_cons, bind_eq_bind, nil_append, cons_append, map_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\na : \u03b1\n\u22a2 sublists (l ++ [a]) = sublists l ++ map (fun x => x ++ [a]) (sublists l)\n[PROOFSTEP]\nrw [sublists_append, sublists_singleton, bind_eq_bind, cons_bind, cons_bind, nil_bind, map_id' append_nil, append_nil]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists (reverse l) = map reverse (sublists' l)\n[PROOFSTEP]\ninduction' l with hd tl ih <;> [rfl;\n  simp only [reverse_cons, sublists_append, sublists'_cons, map_append, ih, sublists_singleton, map_eq_map,\n    bind_eq_bind, map_map, cons_bind, append_nil, nil_bind, (\u00b7 \u2218 \u00b7)]]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists (reverse l) = map reverse (sublists' l)\n[PROOFSTEP]\ninduction' l with hd tl ih\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 sublists (reverse []) = map reverse (sublists' [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nhd : \u03b1\ntl : List \u03b1\nih : sublists (reverse tl) = map reverse (sublists' tl)\n\u22a2 sublists (reverse (hd :: tl)) = map reverse (sublists' (hd :: tl))\n[PROOFSTEP]\nsimp only [reverse_cons, sublists_append, sublists'_cons, map_append, ih, sublists_singleton, map_eq_map, bind_eq_bind,\n  map_map, cons_bind, append_nil, nil_bind, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists l = map reverse (sublists' (reverse l))\n[PROOFSTEP]\nrw [\u2190 sublists_reverse, reverse_reverse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists' (reverse l) = map reverse (sublists l)\n[PROOFSTEP]\nsimp only [sublists_eq_sublists', map_map, map_id' reverse_reverse, Function.comp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists' l = map reverse (sublists (reverse l))\n[PROOFSTEP]\nrw [\u2190 sublists'_reverse, reverse_reverse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ns t : List \u03b1\n\u22a2 s \u2208 sublists t \u2194 s <+ t\n[PROOFSTEP]\nrw [\u2190 reverse_sublist, \u2190 mem_sublists', sublists'_reverse, mem_map_of_injective reverse_injective]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 length (sublists l) = 2 ^ length l\n[PROOFSTEP]\nsimp only [sublists_eq_sublists', length_map, length_sublists', length_reverse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 map List.ret l <+ sublists l\n[PROOFSTEP]\ninduction' l using reverseRecOn with l a ih\n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 map List.ret [] <+ sublists []\n[PROOFSTEP]\nsimp only [map, map_append, sublists_concat]\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\na : \u03b1\nih : map List.ret l <+ sublists l\n\u22a2 map List.ret (l ++ [a]) <+ sublists (l ++ [a])\n[PROOFSTEP]\nsimp only [map, map_append, sublists_concat]\n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 [] <+ sublists []\n[PROOFSTEP]\nsimp only [sublists_nil, sublist_cons]\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\na : \u03b1\nih : map List.ret l <+ sublists l\n\u22a2 map List.ret l ++ [List.ret a] <+ sublists l ++ map (fun x => x ++ [a]) (sublists l)\n[PROOFSTEP]\nexact\n  ((append_sublist_append_left _).2 <|\n        singleton_sublist.2 <| mem_map.2 \u27e8[], mem_sublists.2 (nil_sublist _), by rfl\u27e9).trans\n    ((append_sublist_append_right _).2 ih)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\na : \u03b1\nih : map List.ret l <+ sublists l\n\u22a2 [] ++ [a] = List.ret a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3\u271d : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nr : List \u03b2\ns : List \u03b3\n\u22a2 sublistsLenAux 0 l (g \u2218 f) (map g r ++ s) = map g (sublistsLenAux 0 l f r) ++ s\n[PROOFSTEP]\nunfold sublistsLenAux\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3\u271d : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nr : List \u03b2\ns : List \u03b3\n\u22a2 (g \u2218 f) [] :: (map g r ++ s) = map g (f [] :: r) ++ s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3\u271d : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\na : \u03b1\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nr : List \u03b2\ns : List \u03b3\n\u22a2 sublistsLenAux (n + 1) (a :: l) (g \u2218 f) (map g r ++ s) = map g (sublistsLenAux (n + 1) (a :: l) f r) ++ s\n[PROOFSTEP]\nunfold sublistsLenAux\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3\u271d : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\na : \u03b1\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nr : List \u03b2\ns : List \u03b3\n\u22a2 sublistsLenAux (Nat.add n 0 + 1) l (g \u2218 f) (sublistsLenAux (Nat.add n 0) l ((g \u2218 f) \u2218 cons a) (map g r ++ s)) =\n    map g (sublistsLenAux (Nat.add n 0 + 1) l f (sublistsLenAux (Nat.add n 0) l (f \u2218 cons a) r)) ++ s\n[PROOFSTEP]\nsimp only [show (g \u2218 f) \u2218 List.cons a = g \u2218 f \u2218 List.cons a by rfl, sublistsLenAux_append, sublistsLenAux_append]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3\u271d : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\na : \u03b1\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nr : List \u03b2\ns : List \u03b3\n\u22a2 (g \u2218 f) \u2218 cons a = g \u2218 f \u2218 cons a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nn : \u2115\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b2\n\u22a2 sublistsLenAux n l f r = map f (sublistsLen n l) ++ r\n[PROOFSTEP]\nrw [sublistsLen, \u2190 sublistsLenAux_append]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nn : \u2115\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b2\n\u22a2 sublistsLenAux n l f r = sublistsLenAux n l (f \u2218 id) (map f [] ++ r)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl : List \u03b1\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b2\n\u22a2 sublistsLenAux 0 l f r = f [] :: r\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b2\n\u22a2 sublistsLenAux 0 [] f r = f [] :: r\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nf : List \u03b1 \u2192 \u03b2\nr : List \u03b2\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\n\u22a2 sublistsLenAux 0 (head\u271d :: tail\u271d) f r = f [] :: r\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 sublistsLen (n + 1) (a :: l) = sublistsLen (n + 1) l ++ map (cons a) (sublistsLen n l)\n[PROOFSTEP]\nrw [sublistsLen, sublistsLenAux, sublistsLenAux_eq, sublistsLenAux_eq, map_id, append_nil]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 sublistsLen (n + 1) l ++ map (id \u2218 cons a) (sublistsLen n l) = sublistsLen (n + 1) l ++ map (cons a) (sublistsLen n l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 length (sublistsLen 0 l) = Nat.choose (length l) 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn\u271d : \u2115\n\u22a2 length (sublistsLen (n\u271d + 1) []) = Nat.choose (length []) (n\u271d + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 length (sublistsLen (n + 1) (a :: l)) = Nat.choose (length (a :: l)) (n + 1)\n[PROOFSTEP]\nrw [sublistsLen_succ_cons, length_append, length_sublistsLen (n + 1) l, length_map, length_sublistsLen n l, length_cons,\n  Nat.choose_succ_succ, add_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 sublistsLen 0 l <+ sublists' l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 sublistsLen (n + 1) (a :: l) <+ sublists' (a :: l)\n[PROOFSTEP]\nrw [sublistsLen_succ_cons, sublists'_cons]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 sublistsLen (n + 1) l ++ map (cons a) (sublistsLen n l) <+ sublists' l ++ map (cons a) (sublists' l)\n[PROOFSTEP]\nexact (sublistsLen_sublist_sublists' _ _).append ((sublistsLen_sublist_sublists' _ _).map _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\n[PROOFSTEP]\ninduction' n with n IHn generalizing l\u2081 l\u2082\n[GOAL]\ncase zero\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nh\u271d : l\u2081\u271d <+ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 sublistsLen zero l\u2081 <+ sublistsLen zero l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nh\u271d : l\u2081\u271d <+ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n[PROOFSTEP]\ninduction' h with l\u2081 l\u2082 a _ IH l\u2081 l\u2082 a s IH\n[GOAL]\ncase succ.slnil\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\nh : l\u2081\u271d <+ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081 l\u2082 : List \u03b1\n\u22a2 sublistsLen (succ n) [] <+ sublistsLen (succ n) []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nh : l\u2081\u271d\u00b9 <+ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\na\u271d : l\u2081 <+ l\u2082\nIH : sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n\u22a2 sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) (a :: l\u2082)\n[PROOFSTEP]\nrefine' IH.trans _\n[GOAL]\ncase succ.cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nh : l\u2081\u271d\u00b9 <+ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\na\u271d : l\u2081 <+ l\u2082\nIH : sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n\u22a2 sublistsLen (succ n) l\u2082 <+ sublistsLen (succ n) (a :: l\u2082)\n[PROOFSTEP]\nrw [sublistsLen_succ_cons]\n[GOAL]\ncase succ.cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nh : l\u2081\u271d\u00b9 <+ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\na\u271d : l\u2081 <+ l\u2082\nIH : sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n\u22a2 sublistsLen (succ n) l\u2082 <+ sublistsLen (n + 1) l\u2082 ++ map (cons a) (sublistsLen n l\u2082)\n[PROOFSTEP]\napply sublist_append_left\n[GOAL]\ncase succ.cons\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nh : l\u2081\u271d\u00b9 <+ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n\u22a2 sublistsLen (succ n) (a :: l\u2081) <+ sublistsLen (succ n) (a :: l\u2082)\n[PROOFSTEP]\nsimp [sublistsLen_succ_cons]\n[GOAL]\ncase succ.cons\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\nh : l\u2081\u271d\u00b9 <+ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 sublistsLen n l\u2081 <+ sublistsLen n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : sublistsLen (succ n) l\u2081 <+ sublistsLen (succ n) l\u2082\n\u22a2 sublistsLen (n + 1) l\u2081 ++ map (cons a) (sublistsLen n l\u2081) <+ sublistsLen (n + 1) l\u2082 ++ map (cons a) (sublistsLen n l\u2082)\n[PROOFSTEP]\nexact IH.append ((IHn s).map _)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' : List \u03b1\nh : l' \u2208 sublistsLen 0 l\n\u22a2 length l' = 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl l' : List \u03b1\nh : l' \u2208 sublistsLen (n + 1) (a :: l)\n\u22a2 length l' = n + 1\n[PROOFSTEP]\nrw [sublistsLen_succ_cons, mem_append, mem_map] at h \n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl l' : List \u03b1\nh : l' \u2208 sublistsLen (n + 1) l \u2228 \u2203 a_1, a_1 \u2208 sublistsLen n l \u2227 a :: a_1 = l'\n\u22a2 length l' = n + 1\n[PROOFSTEP]\nrcases h with (h | \u27e8l', h, rfl\u27e9)\n[GOAL]\ncase inl\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl l' : List \u03b1\nh : l' \u2208 sublistsLen (n + 1) l\n\u22a2 length l' = n + 1\n[PROOFSTEP]\nexact length_of_sublistsLen h\n[GOAL]\ncase inr.intro.intro\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl l' : List \u03b1\nh : l' \u2208 sublistsLen n l\n\u22a2 length (a :: l') = n + 1\n[PROOFSTEP]\nexact congr_arg (\u00b7 + 1) (length_of_sublistsLen h)\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' : List \u03b1\nh : l' <+ l\n\u22a2 l' \u2208 sublistsLen (length l') l\n[PROOFSTEP]\ninduction' h with l\u2081 l\u2082 a s IH l\u2081 l\u2082 a s IH\n[GOAL]\ncase slnil\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' : List \u03b1\n\u22a2 [] \u2208 sublistsLen (length []) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : l\u2081 \u2208 sublistsLen (length l\u2081) l\u2082\n\u22a2 l\u2081 \u2208 sublistsLen (length l\u2081) (a :: l\u2082)\n[PROOFSTEP]\ncases' l\u2081 with b l\u2081\n[GOAL]\ncase cons.nil\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2082 : List \u03b1\na : \u03b1\ns : [] <+ l\u2082\nIH : [] \u2208 sublistsLen (length []) l\u2082\n\u22a2 [] \u2208 sublistsLen (length []) (a :: l\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2082 : List \u03b1\na b : \u03b1\nl\u2081 : List \u03b1\ns : b :: l\u2081 <+ l\u2082\nIH : b :: l\u2081 \u2208 sublistsLen (length (b :: l\u2081)) l\u2082\n\u22a2 b :: l\u2081 \u2208 sublistsLen (length (b :: l\u2081)) (a :: l\u2082)\n[PROOFSTEP]\nrw [length, sublistsLen_succ_cons]\n[GOAL]\ncase cons.cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2082 : List \u03b1\na b : \u03b1\nl\u2081 : List \u03b1\ns : b :: l\u2081 <+ l\u2082\nIH : b :: l\u2081 \u2208 sublistsLen (length (b :: l\u2081)) l\u2082\n\u22a2 b :: l\u2081 \u2208 sublistsLen (length l\u2081 + 1) l\u2082 ++ map (cons a) (sublistsLen (length l\u2081) l\u2082)\n[PROOFSTEP]\nexact mem_append_left _ IH\n[GOAL]\ncase cons\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : l\u2081 \u2208 sublistsLen (length l\u2081) l\u2082\n\u22a2 a :: l\u2081 \u2208 sublistsLen (length (a :: l\u2081)) (a :: l\u2082)\n[PROOFSTEP]\nrw [length, sublistsLen_succ_cons]\n[GOAL]\ncase cons\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl l' l\u2081 l\u2082 : List \u03b1\na : \u03b1\ns : l\u2081 <+ l\u2082\nIH : l\u2081 \u2208 sublistsLen (length l\u2081) l\u2082\n\u22a2 a :: l\u2081 \u2208 sublistsLen (length l\u2081 + 1) l\u2082 ++ map (cons a) (sublistsLen (length l\u2081) l\u2082)\n[PROOFSTEP]\nexact mem_append_right _ (mem_map.2 \u27e8_, IH, rfl\u27e9)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublistsLen (length (a :: l)) (a :: l) = [a :: l]\n[PROOFSTEP]\nsimp only [length, sublistsLen_succ_cons, sublistsLen_length, map, sublistsLen_of_length_lt (lt_succ_self _),\n  nil_append]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\n\u22a2 Pairwise (Lex (swap R)) (List.sublists' (a :: l))\n[PROOFSTEP]\nsimp only [sublists'_cons, pairwise_append, pairwise_map, mem_sublists', mem_map, exists_imp, and_imp]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\n\u22a2 Pairwise (Lex (swap R)) (List.sublists' l) \u2227\n    Pairwise (fun a_1 b => Lex (swap R) (a :: a_1) (a :: b)) (List.sublists' l) \u2227\n      \u2200 (a_1 : List \u03b1), a_1 <+ l \u2192 \u2200 (b x : List \u03b1), x <+ l \u2192 a :: x = b \u2192 Lex (swap R) a_1 b\n[PROOFSTEP]\nrefine' \u27e8H\u2082.sublists', H\u2082.sublists'.imp fun l\u2081 => Lex.cons l\u2081, _\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\n\u22a2 \u2200 (a_1 : List \u03b1), a_1 <+ l \u2192 \u2200 (b x : List \u03b1), x <+ l \u2192 a :: x = b \u2192 Lex (swap R) a_1 b\n[PROOFSTEP]\nrintro l\u2081 sl\u2081 x l\u2082 _ rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\nl\u2081 : List \u03b1\nsl\u2081 : l\u2081 <+ l\nl\u2082 : List \u03b1\na\u271d : l\u2082 <+ l\n\u22a2 Lex (swap R) l\u2081 (a :: l\u2082)\n[PROOFSTEP]\ncases' l\u2081 with b l\u2081\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\nl\u2082 : List \u03b1\na\u271d : l\u2082 <+ l\nsl\u2081 : [] <+ l\n\u22a2 Lex (swap R) [] (a :: l\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH\u2081 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nH\u2082 : Pairwise R l\nl\u2082 : List \u03b1\na\u271d : l\u2082 <+ l\nb : \u03b1\nl\u2081 : List \u03b1\nsl\u2081 : b :: l\u2081 <+ l\n\u22a2 Lex (swap R) (b :: l\u2081) (a :: l\u2082)\n[PROOFSTEP]\nexact Lex.rel (H\u2081 _ <| sl\u2081.subset <| mem_cons_self _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : Pairwise R l\n\u22a2 Pairwise (fun l\u2081 l\u2082 => Lex R (reverse l\u2081) (reverse l\u2082)) (sublists l)\n[PROOFSTEP]\nhave := (pairwise_reverse.2 H).sublists'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : Pairwise R l\nthis : Pairwise (Lex (swap fun b a => R a b)) (sublists' (reverse l))\n\u22a2 Pairwise (fun l\u2081 l\u2082 => Lex R (reverse l\u2081) (reverse l\u2082)) (sublists l)\n[PROOFSTEP]\nrwa [sublists'_reverse, pairwise_map] at this \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\nh\u271d : Nodup l\nl\u2081 l\u2082 : List \u03b1\nh : Lex (fun x x_1 => x \u2260 x_1) (reverse l\u2081) (reverse l\u2082)\n\u22a2 l\u2081 \u2260 l\u2082\n[PROOFSTEP]\nsimpa using h.to_ne\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 Nodup (sublists' l) \u2194 Nodup l\n[PROOFSTEP]\nrw [sublists'_eq_sublists, nodup_map_iff reverse_injective, nodup_sublists, nodup_reverse]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nn : \u2115\nl : List \u03b1\nh : Nodup l\n\u22a2 Nodup (sublistsLen n l)\n[PROOFSTEP]\nhave : Pairwise (\u00b7 \u2260 \u00b7) l.sublists' :=\n  Pairwise.imp (fun h => Lex.to_ne (by convert h using 3; simp [swap, eq_comm])) h.sublists'\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nn : \u2115\nl : List \u03b1\nh\u271d : Nodup l\na\u271d b\u271d : List \u03b1\nh : Lex (swap fun x x_1 => x \u2260 x_1) a\u271d b\u271d\n\u22a2 Lex (fun x x_1 => x \u2260 x_1) a\u271d b\u271d\n[PROOFSTEP]\nconvert h using 3\n[GOAL]\ncase h.e'_2.h.e'_4.h.h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nn : \u2115\nl : List \u03b1\nh\u271d : Nodup l\na\u271d b\u271d : List \u03b1\nh : Lex (swap fun x x_1 => x \u2260 x_1) a\u271d b\u271d\nx\u271d : \u03b1\n\u22a2 (fun x => \u00acx = x\u271d) = fun x => x\u271d \u2260 x\n[PROOFSTEP]\nsimp [swap, eq_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nn : \u2115\nl : List \u03b1\nh : Nodup l\nthis : Pairwise (fun x x_1 => x \u2260 x_1) (sublists' l)\n\u22a2 Nodup (sublistsLen n l)\n[PROOFSTEP]\nexact this.sublist (sublistsLen_sublist_sublists' _ _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\n\u22a2 sublists (map f []) = map (map f) (sublists [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 sublists (map f (a :: l)) = map (map f) (sublists (a :: l))\n[PROOFSTEP]\nrw [map_cons, sublists_cons, bind_eq_bind, sublists_map f l, sublists_cons, bind_eq_bind, map_eq_bind, map_eq_bind]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 (List.bind (List.bind (sublists l) fun x => [map f x]) fun x => [x, f a :: x]) =\n    List.bind (List.bind (sublists l) fun x => [x, a :: x]) fun x => [map f x]\n[PROOFSTEP]\ninduction sublists l\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 (List.bind (List.bind [] fun x => [map f x]) fun x => [x, f a :: x]) =\n    List.bind (List.bind [] fun x => [x, a :: x]) fun x => [map f x]\n[PROOFSTEP]\nsimp [*]\n  --Porting note: new theorem\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl head\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d :\n  (List.bind (List.bind tail\u271d fun x => [map f x]) fun x => [x, f a :: x]) =\n    List.bind (List.bind tail\u271d fun x => [x, a :: x]) fun x => [map f x]\n\u22a2 (List.bind (List.bind (head\u271d :: tail\u271d) fun x => [map f x]) fun x => [x, f a :: x]) =\n    List.bind (List.bind (head\u271d :: tail\u271d) fun x => [x, a :: x]) fun x => [map f x]\n[PROOFSTEP]\nsimp [*]\n  --Porting note: new theorem\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\n\u22a2 sublists' (map f []) = map (map f) (sublists' [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' (map f (a :: l)) = map (map f) (sublists' (a :: l))\n[PROOFSTEP]\nsimp [map_cons, sublists'_cons, sublists'_map f l, Function.comp]\n  --Porting note: moved because it is now used to prove `sublists_cons_perm_append`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists l ~ sublists' l\n[PROOFSTEP]\nrw [\u2190 finRange_map_get l, sublists_map, sublists'_map]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 map (map (get l)) (sublists (finRange (length l))) ~ map (map (get l)) (sublists' (finRange (length l)))\n[PROOFSTEP]\nrefine' Perm.map _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 sublists (finRange (length l)) ~ sublists' (finRange (length l))\n[PROOFSTEP]\nexact (perm_ext (nodup_sublists.2 (nodup_finRange _)) (nodup_sublists'.2 (nodup_finRange _))).2 (by simp)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (a : List (Fin (length l))), a \u2208 sublists (finRange (length l)) \u2194 a \u2208 sublists' (finRange (length l))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' (a :: l) ~ sublists l ++ map (cons a) (sublists l)\n[PROOFSTEP]\nrw [sublists'_cons]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' l ++ map (cons a) (sublists' l) ~ sublists l ++ map (cons a) (sublists l)\n[PROOFSTEP]\nexact Perm.append (sublists_perm_sublists' _).symm (Perm.map _ (sublists_perm_sublists' _).symm)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 revzip (sublists l) \u2192 l\u2081 ++ l\u2082 ~ l\n[PROOFSTEP]\nrw [revzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l) (reverse (sublists l)) \u2192 l\u2081 ++ l\u2082 ~ l\n[PROOFSTEP]\ninduction' l using List.reverseRecOn with l' a ih\n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists []) (reverse (sublists [])) \u2192 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nintro l\u2081 l\u2082 h\n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists []) (reverse (sublists []))\n\u22a2 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase H0\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 = [] \u2227 l\u2082 = []\n\u22a2 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists (l' ++ [a])) (reverse (sublists (l' ++ [a]))) \u2192 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nintro l\u2081 l\u2082 h\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists (l' ++ [a])) (reverse (sublists (l' ++ [a])))\n\u22a2 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nrw [sublists_concat, reverse_append, zip_append, \u2190 map_reverse, zip_map_right, zip_map_left] at * <;> [skip; simp]\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists (l' ++ [a])) (reverse (sublists (l' ++ [a])))\n\u22a2 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nrw [sublists_concat, reverse_append, zip_append, \u2190 map_reverse, zip_map_right, zip_map_left] at *\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh :\n  (l\u2081, l\u2082) \u2208\n    map (Prod.map id fun x => x ++ [a]) (zip (sublists l') (reverse (sublists l'))) ++\n      map (Prod.map (fun x => x ++ [a]) id) (zip (sublists l') (reverse (sublists l')))\n\u22a2 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nskip\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh :\n  (l\u2081, l\u2082) \u2208\n    zip (sublists l' ++ map (fun x => x ++ [a]) (sublists l'))\n      (reverse (map (fun x => x ++ [a]) (sublists l')) ++ reverse (sublists l'))\n\u22a2 length (sublists l') = length (reverse (map (fun x => x ++ [a]) (sublists l')))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh :\n  (l\u2081, l\u2082) \u2208\n    map (Prod.map id fun x => x ++ [a]) (zip (sublists l') (reverse (sublists l'))) ++\n      map (Prod.map (fun x => x ++ [a]) id) (zip (sublists l') (reverse (sublists l')))\n\u22a2 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nsimp only [Prod.mk.inj_iff, mem_map, mem_append, Prod.map_mk, Prod.exists] at h \n[GOAL]\ncase H1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082 : List \u03b1\nh :\n  (\u2203 a_1 b, (a_1, b) \u2208 zip (sublists l') (reverse (sublists l')) \u2227 id a_1 = l\u2081 \u2227 b ++ [a] = l\u2082) \u2228\n    \u2203 a_1 b, (a_1, b) \u2208 zip (sublists l') (reverse (sublists l')) \u2227 a_1 ++ [a] = l\u2081 \u2227 id b = l\u2082\n\u22a2 l\u2081 ++ l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nrcases h with (\u27e8l\u2081, l\u2082', h, rfl, rfl\u27e9 | \u27e8l\u2081', l\u2082, h, rfl, rfl\u27e9)\n[GOAL]\ncase H1.inl.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082' : List \u03b1\nh : (l\u2081, l\u2082') \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 id l\u2081 ++ (l\u2082' ++ [a]) ~ l' ++ [a]\n[PROOFSTEP]\nrw [\u2190 append_assoc]\n[GOAL]\ncase H1.inl.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081 l\u2082' : List \u03b1\nh : (l\u2081, l\u2082') \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 id l\u2081 ++ l\u2082' ++ [a] ~ l' ++ [a]\n[PROOFSTEP]\nexact (ih _ _ h).append_right _\n[GOAL]\ncase H1.inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081' l\u2082 : List \u03b1\nh : (l\u2081', l\u2082) \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 l\u2081' ++ [a] ++ id l\u2082 ~ l' ++ [a]\n[PROOFSTEP]\nrw [append_assoc]\n[GOAL]\ncase H1.inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081' l\u2082 : List \u03b1\nh : (l\u2081', l\u2082) \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 l\u2081' ++ ([a] ++ id l\u2082) ~ l' ++ [a]\n[PROOFSTEP]\napply (perm_append_comm.append_left _).trans\n[GOAL]\ncase H1.inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081' l\u2082 : List \u03b1\nh : (l\u2081', l\u2082) \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 l\u2081' ++ (id l\u2082 ++ [a]) ~ l' ++ [a]\n[PROOFSTEP]\nrw [\u2190 append_assoc]\n[GOAL]\ncase H1.inr.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl' : List \u03b1\na : \u03b1\nih : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists l') (reverse (sublists l')) \u2192 l\u2081 ++ l\u2082 ~ l'\nl\u2081' l\u2082 : List \u03b1\nh : (l\u2081', l\u2082) \u2208 zip (sublists l') (reverse (sublists l'))\n\u22a2 l\u2081' ++ id l\u2082 ++ [a] ~ l' ++ [a]\n[PROOFSTEP]\nexact (ih _ _ h).append_right _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 revzip (sublists' l) \u2192 l\u2081 ++ l\u2082 ~ l\n[PROOFSTEP]\nrw [revzip]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl : List \u03b1\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\n[PROOFSTEP]\ninduction' l with a l IH\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' []) (reverse (sublists' [])) \u2192 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nintro l\u2081 l\u2082 h\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\n\u22a2 \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' (a :: l)) (reverse (sublists' (a :: l))) \u2192 l\u2081 ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nintro l\u2081 l\u2082 h\n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists' []) (reverse (sublists' []))\n\u22a2 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase nil\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 = [] \u2227 l\u2082 = []\n\u22a2 l\u2081 ++ l\u2082 ~ []\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists' (a :: l)) (reverse (sublists' (a :: l)))\n\u22a2 l\u2081 ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nrw [sublists'_cons, reverse_append, zip_append, \u2190 map_reverse, zip_map_right, zip_map_left] at * <;> [simp at h ; simp]\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 zip (sublists' (a :: l)) (reverse (sublists' (a :: l)))\n\u22a2 l\u2081 ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nrw [sublists'_cons, reverse_append, zip_append, \u2190 map_reverse, zip_map_right, zip_map_left] at *\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082 : List \u03b1\nh :\n  (l\u2081, l\u2082) \u2208\n    map (Prod.map id (cons a)) (zip (sublists' l) (reverse (sublists' l))) ++\n      map (Prod.map (cons a) id) (zip (sublists' l) (reverse (sublists' l)))\n\u22a2 l\u2081 ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082 : List \u03b1\nh :\n  (l\u2081, l\u2082) \u2208\n    zip (sublists' l ++ map (cons a) (sublists' l)) (reverse (map (cons a) (sublists' l)) ++ reverse (sublists' l))\n\u22a2 length (sublists' l) = length (reverse (map (cons a) (sublists' l)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082 : List \u03b1\nh :\n  (\u2203 a_1 b, (a_1, b) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2227 a_1 = l\u2081 \u2227 a :: b = l\u2082) \u2228\n    \u2203 a_1, (a_1, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2227 a :: a_1 = l\u2081\n\u22a2 l\u2081 ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nrcases h with (\u27e8l\u2081, l\u2082', h, rfl, rfl\u27e9 | \u27e8l\u2081', h, rfl\u27e9)\n[GOAL]\ncase cons.inl.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2081 l\u2082' : List \u03b1\nh : (l\u2081, l\u2082') \u2208 zip (sublists' l) (reverse (sublists' l))\n\u22a2 l\u2081 ++ a :: l\u2082' ~ a :: l\n[PROOFSTEP]\nexact perm_middle.trans ((IH _ _ h).cons _)\n[GOAL]\ncase cons.inr.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\nIH : \u2200 (l\u2081 l\u2082 : List \u03b1), (l\u2081, l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l)) \u2192 l\u2081 ++ l\u2082 ~ l\nl\u2082 l\u2081' : List \u03b1\nh : (l\u2081', l\u2082) \u2208 zip (sublists' l) (reverse (sublists' l))\n\u22a2 a :: l\u2081' ++ l\u2082 ~ a :: l\n[PROOFSTEP]\nexact (IH _ _ h).cons _\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 (List.bind (range (length l + 1)) fun n => sublistsLen n l) ~ sublists' l\n[PROOFSTEP]\ninduction' l with h tl l_ih\n[GOAL]\ncase nil\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\n\u22a2 (List.bind (range (length [] + 1)) fun n => sublistsLen n []) ~ sublists' []\n[PROOFSTEP]\nsimp [range_succ]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 (List.bind (range (length (h :: tl) + 1)) fun n => sublistsLen n (h :: tl)) ~ sublists' (h :: tl)\n[PROOFSTEP]\nsimp_rw [range_succ_eq_map, length, cons_bind, map_bind, sublistsLen_succ_cons, sublists'_cons, List.sublistsLen_zero,\n  List.singleton_append]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 ([] :: List.bind (range (length tl + 1)) fun a => sublistsLen (a + 1) tl ++ map (cons h) (sublistsLen a tl)) ~\n    sublists' tl ++ map (cons h) (sublists' tl)\n[PROOFSTEP]\nrefine' ((bind_append_perm (range (tl.length + 1)) _ _).symm.cons _).trans _\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 [] ::\n      ((List.bind (range (length tl + 1)) fun a => sublistsLen (a + 1) tl) ++\n        List.bind (range (length tl + 1)) fun a => map (cons h) (sublistsLen a tl)) ~\n    sublists' tl ++ map (cons h) (sublists' tl)\n[PROOFSTEP]\nsimp_rw [\u2190 List.bind_map, \u2190 cons_append]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 ([] :: List.bind (range (length tl + 1)) fun a => sublistsLen (a + 1) tl) ++\n      map (cons h) (List.bind (range (length tl + 1)) fun a => sublistsLen a tl) ~\n    sublists' tl ++ map (cons h) (sublists' tl)\n[PROOFSTEP]\nrw [\u2190 List.singleton_append, \u2190 List.sublistsLen_zero tl]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 (sublistsLen 0 tl ++ List.bind (range (length tl + 1)) fun a => sublistsLen (a + 1) tl) ++\n      map (cons h) (List.bind (range (length tl + 1)) fun a => sublistsLen a tl) ~\n    sublists' tl ++ map (cons h) (sublists' tl)\n[PROOFSTEP]\nrefine' Perm.append _ (l_ih.map _)\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 (sublistsLen 0 tl ++ List.bind (range (length tl + 1)) fun a => sublistsLen (a + 1) tl) ~ sublists' tl\n[PROOFSTEP]\nrw [List.range_succ, append_bind, bind_singleton, sublistsLen_of_length_lt (Nat.lt_succ_self _), append_nil, \u2190\n  List.map_bind (fun n => sublistsLen n tl) Nat.succ, \u2190 cons_bind 0 _ fun n => sublistsLen n tl, \u2190 range_succ_eq_map]\n[GOAL]\ncase cons\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b1 : Type u_1\nh : \u03b1\ntl : List \u03b1\nl_ih : (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n\u22a2 (List.bind (range (length tl + 1)) fun n => sublistsLen n tl) ~ sublists' tl\n[PROOFSTEP]\nexact l_ih\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Sublists", "llama_tokens": 23853, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943603346811, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.5372907056110586}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\nm : Multiset \u03b1\na : \u03b1\nx\u271d : List \u03b1\n\u22a2 count a (dedup (Quot.mk Setoid.r x\u271d)) = if a \u2208 Quot.mk Setoid.r x\u271d then 1 else 0\n[PROOFSTEP]\nsimp only [quot_mk_to_coe'', coe_dedup, mem_coe, List.mem_dedup, coe_nodup, coe_count]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\nm : Multiset \u03b1\na : \u03b1\nx\u271d : List \u03b1\n\u22a2 List.count a (List.dedup x\u271d) = if a \u2208 x\u271d then 1 else 0\n[PROOFSTEP]\napply List.count_dedup _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\n\u22a2 dedup (bind (dedup m) f) = dedup (bind m f)\n[PROOFSTEP]\next x\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\nx : \u03b2\n\u22a2 count x (dedup (bind (dedup m) f)) = count x (dedup (bind m f))\n[PROOFSTEP]\nsimp_rw [count_dedup]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\nx : \u03b2\n\u22a2 (if x \u2208 bind (dedup m) f then 1 else 0) = if x \u2208 bind m f then 1 else 0\n[PROOFSTEP]\nrefine if_congr ?_ rfl rfl\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nm : Multiset \u03b1\nf : \u03b1 \u2192 Multiset \u03b2\nx : \u03b2\n\u22a2 x \u2208 bind (dedup m) f \u2194 x \u2208 bind m f\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\n\u22a2 s \u2264 dedup s \u2194 Nodup s\n[PROOFSTEP]\nrw [le_dedup, and_iff_right le_rfl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\n\u22a2 dedup s = dedup t \u2194 \u2200 (a : \u03b1), a \u2208 s \u2194 a \u2208 t\n[PROOFSTEP]\nsimp [Nodup.ext]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Multiset \u03b1\n\u22a2 dedup (map f (dedup s)) = dedup (map f s)\n[PROOFSTEP]\nsimp [dedup_ext]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\n\u22a2 dedup (n \u2022 s) = dedup s\n[PROOFSTEP]\next a\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\na : \u03b1\n\u22a2 count a (dedup (n \u2022 s)) = count a (dedup s)\n[PROOFSTEP]\nby_cases h : a \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\na : \u03b1\nh : a \u2208 s\n\u22a2 count a (dedup (n \u2022 s)) = count a (dedup s)\n[PROOFSTEP]\nsimp [h, h0]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\na : \u03b1\nh : \u00aca \u2208 s\n\u22a2 count a (dedup (n \u2022 s)) = count a (dedup s)\n[PROOFSTEP]\nsimp [h, h0]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns t : Multiset \u03b1\nhno : Nodup s\n\u22a2 s \u2264 Multiset.dedup t \u2194 s \u2264 t\n[PROOFSTEP]\nsimp [le_dedup, hno]\n[GOAL]\n\u03b1 : Type u_1\ns t : Multiset \u03b1\nn : \u2115\nh : Nodup s\nhn : n \u2260 0\n\u22a2 s \u2264 n \u2022 t \u2194 s \u2264 t\n[PROOFSTEP]\nclassical\nrw [\u2190 h.le_dedup_iff_le, Iff.comm, \u2190 h.le_dedup_iff_le]\nsimp [hn]\n[GOAL]\n\u03b1 : Type u_1\ns t : Multiset \u03b1\nn : \u2115\nh : Nodup s\nhn : n \u2260 0\n\u22a2 s \u2264 n \u2022 t \u2194 s \u2264 t\n[PROOFSTEP]\nrw [\u2190 h.le_dedup_iff_le, Iff.comm, \u2190 h.le_dedup_iff_le]\n[GOAL]\n\u03b1 : Type u_1\ns t : Multiset \u03b1\nn : \u2115\nh : Nodup s\nhn : n \u2260 0\n\u22a2 s \u2264 Multiset.dedup t \u2194 s \u2264 Multiset.dedup (n \u2022 t)\n[PROOFSTEP]\nsimp [hn]\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Dedup", "llama_tokens": 1643, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.7090191214879991, "lm_q1q2_score": 0.5372906916330557}}
{"text": "[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : Type u\ninst\u271d : TopologicalSpace X\nx : X\n\u22a2 Group (FundamentalGroup X x)\n[PROOFSTEP]\ndsimp only [FundamentalGroup]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : Type u\ninst\u271d : TopologicalSpace X\nx : X\n\u22a2 Group (Aut x)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : Type u\ninst\u271d : TopologicalSpace X\nx : X\n\u22a2 Inhabited (FundamentalGroup X x)\n[PROOFSTEP]\ndsimp only [FundamentalGroup]\n[GOAL]\nX\u271d : Type u\nY : Type v\ninst\u271d\u00b2 : TopologicalSpace X\u271d\ninst\u271d\u00b9 : TopologicalSpace Y\nx\u2080 x\u2081 : X\u271d\nX : Type u\ninst\u271d : TopologicalSpace X\nx : X\n\u22a2 Inhabited (Aut x)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup", "llama_tokens": 386, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.537227108633237}}
{"text": "[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 leastGE f r (\u03c0 \u03c9) \u03c9 = min (\u03c0 \u03c9) (leastGE f r n \u03c9)\n[PROOFSTEP]\nclassical\nrefine' le_antisymm (le_min (leastGE_le _) (leastGE_mono (h\u03c0n \u03c9) r \u03c9)) _\nby_cases hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u00b7 rw [min_eq_left hle, leastGE]\n  by_cases h : \u2203 j \u2208 Set.Icc 0 (\u03c0 \u03c9), f j \u03c9 \u2208 Set.Ici r\n  \u00b7 refine' hle.trans (Eq.le _)\n    rw [leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) h]\n  \u00b7 simp only [hitting, if_neg h, le_rfl]\n\u00b7 rw [min_eq_right (not_le.1 hle).le, leastGE, leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) _]\n  rw [not_le, leastGE, hitting_lt_iff _ (h\u03c0n \u03c9)] at hle \n  exact\n    let \u27e8j, hj\u2081, hj\u2082\u27e9 := hle\n    \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 leastGE f r (\u03c0 \u03c9) \u03c9 = min (\u03c0 \u03c9) (leastGE f r n \u03c9)\n[PROOFSTEP]\nrefine' le_antisymm (le_min (leastGE_le _) (leastGE_mono (h\u03c0n \u03c9) r \u03c9)) _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nby_cases hle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nrw [min_eq_left hle, leastGE]\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nby_cases h : \u2203 j \u2208 Set.Icc 0 (\u03c0 \u03c9), f j \u03c9 \u2208 Set.Ici r\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nrefine' hle.trans (Eq.le _)\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 leastGE f r n \u03c9 = hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nrw [leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) h]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u03c0 \u03c9 \u2264 leastGE f r n \u03c9\nh : \u00ac\u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u03c0 \u03c9 \u2264 hitting f (Set.Ici r) 0 (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nsimp only [hitting, if_neg h, le_rfl]\n[GOAL]\ncase neg\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 min (\u03c0 \u03c9) (leastGE f r n \u03c9) \u2264 leastGE f r (\u03c0 \u03c9) \u03c9\n[PROOFSTEP]\nrw [min_eq_right (not_le.1 hle).le, leastGE, leastGE, \u2190 hitting_eq_hitting_of_exists (h\u03c0n \u03c9) _]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u00ac\u03c0 \u03c9 \u2264 leastGE f r n \u03c9\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n[PROOFSTEP]\nrw [not_le, leastGE, hitting_lt_iff _ (h\u03c0n \u03c9)] at hle \n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\n\u03c9 : \u03a9\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\nhle : \u2203 j, j \u2208 Set.Ico 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n\u22a2 \u2203 j, j \u2208 Set.Icc 0 (\u03c0 \u03c9) \u2227 f j \u03c9 \u2208 Set.Ici r\n[PROOFSTEP]\nexact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hle\n  \u27e8j, \u27e8hj\u2081.1, hj\u2081.2.le\u27e9, hj\u2082\u27e9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0\n[PROOFSTEP]\next1 \u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 \u03c9 = stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 \u03c9\n[PROOFSTEP]\nsimp_rw [stoppedProcess, stoppedValue]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf\u271d : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c0 : \u03a9 \u2192 \u2115\nr : \u211d\nn : \u2115\nh\u03c0n : \u2200 (\u03c9 : \u03a9), \u03c0 \u03c9 \u2264 n\n\u03c9 : \u03a9\n\u22a2 f (leastGE f r (\u03c0 \u03c9) \u03c9) \u03c9 = f (min (\u03c0 \u03c9) (leastGE f r n \u03c9)) \u03c9\n[PROOFSTEP]\nrw [leastGE_eq_min _ _ _ h\u03c0n]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u22a2 Submartingale (fun i => stoppedValue f (leastGE f r i)) \u2131 \u03bc\n[PROOFSTEP]\nrw [submartingale_iff_expected_stoppedValue_mono]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u22a2 \u2200 (\u03c4 \u03c0 : \u03a9 \u2192 \u2115),\n    IsStoppingTime \u2131 \u03c4 \u2192\n      IsStoppingTime \u2131 \u03c0 \u2192\n        \u03c4 \u2264 \u03c0 \u2192\n          (\u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N) \u2192\n            \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c4 x \u2202\u03bc \u2264\n              \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 x \u2202\u03bc\n[PROOFSTEP]\nintro \u03c3 \u03c0 h\u03c3 h\u03c0 h\u03c3_le_\u03c0 h\u03c0_bdd\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nh\u03c0_bdd : \u2203 N, \u2200 (x : \u03a9), \u03c0 x \u2264 N\n\u22a2 \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c3 x \u2202\u03bc \u2264\n    \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 x \u2202\u03bc\n[PROOFSTEP]\nobtain \u27e8n, h\u03c0_le_n\u27e9 := h\u03c0_bdd\n[GOAL]\ncase intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c3 x \u2202\u03bc \u2264\n    \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [stoppedValue_stoppedValue_leastGE f \u03c3 r fun i => (h\u03c3_le_\u03c0 i).trans (h\u03c0_le_n i)]\n[GOAL]\ncase intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 \u222b (x : \u03a9), stoppedValue (stoppedProcess f (leastGE f r n)) \u03c3 x \u2202\u03bc \u2264\n    \u222b (x : \u03a9), stoppedValue (fun i => stoppedValue f (leastGE f r i)) \u03c0 x \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [stoppedValue_stoppedValue_leastGE f \u03c0 r h\u03c0_le_n]\n[GOAL]\ncase intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 \u222b (x : \u03a9), stoppedValue (stoppedProcess f (leastGE f r n)) \u03c3 x \u2202\u03bc \u2264\n    \u222b (x : \u03a9), stoppedValue (stoppedProcess f (leastGE f r n)) \u03c0 x \u2202\u03bc\n[PROOFSTEP]\nrefine' hf.expected_stoppedValue_mono _ _ _ fun \u03c9 => (min_le_left _ _).trans (h\u03c0_le_n \u03c9)\n[GOAL]\ncase intro.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 IsStoppingTime \u2131 fun x => min (\u03c3 x) (leastGE f r n x)\n[PROOFSTEP]\nexact h\u03c3.min (hf.adapted.isStoppingTime_leastGE _ _)\n[GOAL]\ncase intro.refine'_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 IsStoppingTime \u2131 fun x => min (\u03c0 x) (leastGE f r n x)\n[PROOFSTEP]\nexact h\u03c0.min (hf.adapted.isStoppingTime_leastGE _ _)\n[GOAL]\ncase intro.refine'_3\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u03c3 \u03c0 : \u03a9 \u2192 \u2115\nh\u03c3 : IsStoppingTime \u2131 \u03c3\nh\u03c0 : IsStoppingTime \u2131 \u03c0\nh\u03c3_le_\u03c0 : \u03c3 \u2264 \u03c0\nn : \u2115\nh\u03c0_le_n : \u2200 (x : \u03a9), \u03c0 x \u2264 n\n\u22a2 (fun x => min (\u03c3 x) (leastGE f r n x)) \u2264 fun x => min (\u03c0 x) (leastGE f r n x)\n[PROOFSTEP]\nexact fun \u03c9 => min_le_min (h\u03c3_le_\u03c0 \u03c9) le_rfl\n[GOAL]\ncase hadp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u22a2 Adapted \u2131 fun i => stoppedValue f (leastGE f r i)\n[PROOFSTEP]\nexact fun i =>\n  stronglyMeasurable_stoppedValue_of_le hf.adapted.progMeasurable_of_discrete (hf.adapted.isStoppingTime_leastGE _ _)\n    leastGE_le\n[GOAL]\ncase hint\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nr : \u211d\n\u22a2 \u2200 (i : \u2115), Integrable (stoppedValue f (leastGE f r i))\n[PROOFSTEP]\nexact fun i => integrable_stoppedValue _ (hf.adapted.isStoppingTime_leastGE _ _) hf.integrable leastGE_le\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nfilter_upwards [hbdd] with \u03c9 hbdd\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 stoppedValue f (leastGE f r i) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nchange f (leastGE f r i \u03c9) \u03c9 \u2264 r + R\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nby_cases heq : leastGE f r i \u03c9 = 0\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nrw [heq, hf0, Pi.zero_apply]\n[GOAL]\ncase pos\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : leastGE f r i \u03c9 = 0\n\u22a2 0 \u2264 r + \u2191R\n[PROOFSTEP]\nexact add_nonneg hr R.coe_nonneg\n[GOAL]\ncase neg\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := Nat.exists_eq_succ_of_ne_zero heq\n[GOAL]\ncase neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (leastGE f r i \u03c9) \u03c9 \u2264 r + \u2191R\n[PROOFSTEP]\nrw [hk, add_comm, \u2190 sub_le_iff_le_add]\n[GOAL]\ncase neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R\n[PROOFSTEP]\nhave := not_mem_of_lt_hitting (hk.symm \u25b8 k.lt_succ_self : k < leastGE f r i \u03c9) (zero_le _)\n[GOAL]\ncase neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : \u00acf k \u03c9 \u2208 Set.Ici r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R\n[PROOFSTEP]\nsimp only [Set.mem_union, Set.mem_Iic, Set.mem_Ici, not_or, not_le] at this \n[GOAL]\ncase neg.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u03c9 : \u03a9\nhbdd\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nheq : \u00acleastGE f r i \u03c9 = 0\nk : \u2115\nhk : leastGE f r i \u03c9 = Nat.succ k\nthis : f k \u03c9 < r\n\u22a2 f (Nat.succ k) \u03c9 - r \u2264 \u2191R\n[PROOFSTEP]\nexact (sub_lt_sub_left this _).le.trans ((le_abs_self _).trans (hbdd\u03c9 _))\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 snorm (stoppedValue f (leastGE f r i)) 1 \u03bc \u2264 2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R)\n[PROOFSTEP]\nrefine' snorm_one_le_of_le' ((hf.stoppedValue_leastGE r).integrable _) _ (norm_stoppedValue_leastGE_le hr hf0 hbdd i)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 0 \u2264 \u222b (x : \u03a9), stoppedValue f (leastGE f r i) x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_univ]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 0 \u2264 \u222b (x : \u03a9) in Set.univ, stoppedValue f (leastGE f r i) x \u2202\u03bc\n[PROOFSTEP]\nrefine' le_trans _ ((hf.stoppedValue_leastGE r).set_integral_le (zero_le _) MeasurableSet.univ)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in Set.univ, stoppedValue f (leastGE f r 0) \u03c9 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [stoppedValue, leastGE, hitting_of_le le_rfl, hf0, integral_zero', le_rfl]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 snorm (stoppedValue f (leastGE f r i)) 1 \u03bc \u2264 \u2191(ENNReal.toNNReal (2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R)))\n[PROOFSTEP]\nrefine' (hf.stoppedValue_leastGE_snorm_le hr hf0 hbdd i).trans _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhr : 0 \u2264 r\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R) \u2264 \u2191(ENNReal.toNNReal (2 * \u2191\u2191\u03bc Set.univ * ENNReal.ofReal (r + \u2191R)))\n[PROOFSTEP]\nsimp [ENNReal.coe_toNNReal (measure_ne_top \u03bc _), ENNReal.coe_toNNReal]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave ht : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f i n) \u03c9) atTop (\ud835\udcdd c) :=\n  by\n  rw [ae_all_iff]\n  exact fun i =>\n    Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n      (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrw [ae_all_iff]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200 (i : \u2115), \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) a) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nexact fun i =>\n  Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedValue_leastGE i)\n    (hf.stoppedValue_leastGE_snorm_le' i.cast_nonneg hf0 hbdd)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nfilter_upwards [ht] with \u03c9 h\u03c9 h\u03c9b\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : BddAbove (Set.range fun n => f n \u03c9)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrw [BddAbove] at h\u03c9b \n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := exists_nat_gt h\u03c9b.some\n[GOAL]\ncase h.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave hib : \u2200 n, f n \u03c9 < i := by\n  intro n\n  exact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\n\u22a2 \u2200 (n : \u2115), f n \u03c9 < \u2191i\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nn : \u2115\n\u22a2 f n \u03c9 < \u2191i\n[PROOFSTEP]\nexact lt_of_le_of_lt ((mem_upperBounds.1 h\u03c9b.some_mem) _ \u27e8n, rfl\u27e9) hi\n[GOAL]\ncase h.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave heq : \u2200 n, stoppedValue f (leastGE f i n) \u03c9 = f n \u03c9 :=\n  by\n  intro n\n  rw [leastGE]; unfold hitting; rw [stoppedValue]\n  rw [if_neg]\n  simp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici]\n  push_neg\n  exact fun j _ => hib j\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\n\u22a2 \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9\n[PROOFSTEP]\nrw [leastGE]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f (hitting f (Set.Ici \u2191i) 0 n) \u03c9 = f n \u03c9\n[PROOFSTEP]\nunfold hitting\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 stoppedValue f\n      (fun x =>\n        if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j x \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 x \u2208 Set.Ici \u2191i}) else n)\n      \u03c9 =\n    f n \u03c9\n[PROOFSTEP]\nrw [stoppedValue]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 f (if \u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i then sInf (Set.Icc 0 n \u2229 {i_1 | f i_1 \u03c9 \u2208 Set.Ici \u2191i}) else n) \u03c9 =\n    f n \u03c9\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase hnc\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, j \u2208 Set.Icc 0 n \u2227 f j \u03c9 \u2208 Set.Ici \u2191i\n[PROOFSTEP]\nsimp only [Set.mem_Icc, Set.mem_union, Set.mem_Ici]\n[GOAL]\ncase hnc\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u00ac\u2203 j, (0 \u2264 j \u2227 j \u2264 n) \u2227 \u2191i \u2264 f j \u03c9\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase hnc\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nn : \u2115\n\u22a2 \u2200 (j : \u2115), 0 \u2264 j \u2227 j \u2264 n \u2192 f j \u03c9 < \u2191i\n[PROOFSTEP]\nexact fun j _ => hib j\n[GOAL]\ncase h.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nht : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), \u2203 c, Tendsto (fun n => stoppedValue f (leastGE f (\u2191i) n) \u03c9) atTop (\ud835\udcdd c)\nh\u03c9b : Set.Nonempty (upperBounds (Set.range fun n => f n \u03c9))\ni : \u2115\nhi : Set.Nonempty.some h\u03c9b < \u2191i\nhib : \u2200 (n : \u2115), f n \u03c9 < \u2191i\nheq : \u2200 (n : \u2115), stoppedValue f (leastGE f (\u2191i) n) \u03c9 = f n \u03c9\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nsimp only [\u2190 heq, h\u03c9 i]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhf0 : f 0 = 0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nfilter_upwards [hf.exists_tendsto_of_abs_bddAbove_aux hf0 hbdd] with \u03c9 h\u03c9 using \u27e8h\u03c9, fun \u27e8c, hc\u27e9 => hc.bddAbove_range\u27e9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nset g : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave hg : Submartingale g \u2131 \u03bc := hf.sub_martingale (martingale_const_fun _ _ (hf.adapted 0) (hf.integrable 0))\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave hg0 : g 0 = 0 := by\n  ext \u03c9\n  simp only [sub_self, Pi.zero_apply]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u22a2 g 0 = 0\n[PROOFSTEP]\next \u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\n\u03c9 : \u03a9\n\u22a2 g 0 \u03c9 = OfNat.ofNat 0 \u03c9\n[PROOFSTEP]\nsimp only [sub_self, Pi.zero_apply]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nhave hgbdd : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i : \u2115, |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R := by simpa only [sub_sub_sub_cancel_right]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nsimpa only [sub_sub_sub_cancel_right]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nfilter_upwards [hg.bddAbove_iff_exists_tendsto_aux hg0 hgbdd] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nconvert h\u03c9 using 1\n[GOAL]\ncase h.e'_1.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase h.e'_1.a.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nh : BddAbove (Set.range fun n => f n \u03c9)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9)\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := h\n[GOAL]\ncase h.e'_1.a.refine'_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nh : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 := h\n[GOAL]\ncase h.e'_1.a.refine'_1.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9)\n[PROOFSTEP]\nrefine' \u27e8b + |f 0 \u03c9|, fun y hy => _\u27e9\n[GOAL]\ncase h.e'_1.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9 - f 0 \u03c9)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nrefine' \u27e8b + |f 0 \u03c9|, fun y hy => _\u27e9\n[GOAL]\ncase h.e'_1.a.refine'_1.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\ny : \u211d\nhy : y \u2208 Set.range fun n => f n \u03c9 - f 0 \u03c9\n\u22a2 y \u2264 b + |f 0 \u03c9|\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hy\n[GOAL]\ncase h.e'_1.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9 - f 0 \u03c9)\ny : \u211d\nhy : y \u2208 Set.range fun n => f n \u03c9\n\u22a2 y \u2264 b + |f 0 \u03c9|\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hy\n[GOAL]\ncase h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 (fun n => f n \u03c9 - f 0 \u03c9) n \u2264 b + |f 0 \u03c9|\n[PROOFSTEP]\nsimp_rw [sub_eq_add_neg]\n[GOAL]\ncase h.e'_1.a.refine'_1.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\nn : \u2115\n\u22a2 f n \u03c9 + -f 0 \u03c9 \u2264 b + |f 0 \u03c9|\n[PROOFSTEP]\nexact add_le_add (hb \u27e8n, rfl\u27e9) (neg_le_abs_self _)\n[GOAL]\ncase h.e'_1.a.refine'_2.intro.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nb : \u211d\nhb : b \u2208 upperBounds (Set.range fun n => f n \u03c9 - f 0 \u03c9)\nn : \u2115\n\u22a2 (fun n => f n \u03c9) n \u2264 b + |f 0 \u03c9|\n[PROOFSTEP]\nexact sub_le_iff_le_add.1 (le_trans (sub_le_sub_left (le_abs_self _) _) (hb \u27e8n, rfl\u27e9))\n[GOAL]\ncase h.e'_2.a\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase h.e'_2.a.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nh : \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := h\n[GOAL]\ncase h.e'_2.a.refine'_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nh : \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := h\n[GOAL]\ncase h.e'_2.a.refine'_1.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nexact \u27e8c - f 0 \u03c9, hc.sub_const _\u27e9\n[GOAL]\ncase h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e8c + f 0 \u03c9, _\u27e9\n[GOAL]\ncase h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))\n[PROOFSTEP]\nhave := hc.add_const (f 0 \u03c9)\n[GOAL]\ncase h.e'_2.a.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Submartingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ng : \u2115 \u2192 \u03a9 \u2192 \u211d := fun n \u03c9 => f n \u03c9 - f 0 \u03c9\nhg : Submartingale g \u2131 \u03bc\nhg0 : g 0 = 0\nhgbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |g (i + 1) \u03c9 - g i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : BddAbove (Set.range fun n => f n \u03c9 - f 0 \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9 - f 0 \u03c9) atTop (\ud835\udcdd c)\nthis : Tendsto (fun k => f k \u03c9 - f 0 \u03c9 + f 0 \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (c + f 0 \u03c9))\n[PROOFSTEP]\nsimpa only [sub_add_cancel]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nhave hbdd' : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 i, |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 R :=\n  by\n  filter_upwards [hbdd] with \u03c9 h\u03c9 i\n  erw [\u2190 abs_neg, neg_sub, sub_neg_eq_add, neg_add_eq_sub]\n  exact h\u03c9 i\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nfilter_upwards [hbdd] with \u03c9 h\u03c9 i\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nerw [\u2190 abs_neg, neg_sub, sub_neg_eq_add, neg_add_eq_sub]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nexact h\u03c9 i\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nhave hup := hf.submartingale.bddAbove_iff_exists_tendsto hbdd\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nhave hdown := hf.neg.submartingale.bddAbove_iff_exists_tendsto hbdd'\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nfilter_upwards [hup, hdown] with \u03c9 h\u03c9\u2081 h\u03c9\u2082\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nhave : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c) :=\n  by\n  constructor <;> rintro \u27e8c, hc\u27e9\n  \u00b7 exact \u27e8-c, hc.neg\u27e9\n  \u00b7 refine' \u27e8-c, _\u27e9\n    convert hc.neg\n    simp only [neg_neg, Pi.neg_apply]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2192 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase mpr\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 (\u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)) \u2192 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase mp.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nexact \u27e8-c, hc.neg\u27e9\n[GOAL]\ncase mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e8-c, _\u27e9\n[GOAL]\ncase mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd (-c))\n[PROOFSTEP]\nconvert hc.neg\n[GOAL]\ncase h.e'_3.h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nx\u271d : \u2115\n\u22a2 f x\u271d \u03c9 = -(-f) x\u271d \u03c9\n[PROOFSTEP]\nsimp only [neg_neg, Pi.neg_apply]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => f n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nrw [h\u03c9\u2081, this, \u2190 h\u03c9\u2082]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddAbove (Set.range fun n => (-f) n \u03c9) \u2192 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase h.mpr\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u22a2 BddBelow (Set.range fun n => f n \u03c9) \u2192 BddAbove (Set.range fun n => (-f) n \u03c9)\n[PROOFSTEP]\nrintro \u27e8c, hc\u27e9\n[GOAL]\ncase h.mp.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : c \u2208 upperBounds (Set.range fun n => (-f) n \u03c9)\n\u22a2 BddBelow (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nrefine' \u27e8-c, fun \u03c9 h\u03c9 => _\u27e9\n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9 : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\nc : \u211d\nhc : c \u2208 lowerBounds (Set.range fun n => f n \u03c9)\n\u22a2 BddAbove (Set.range fun n => (-f) n \u03c9)\n[PROOFSTEP]\nrefine' \u27e8-c, fun \u03c9 h\u03c9 => _\u27e9\n[GOAL]\ncase h.mp.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : c \u2208 upperBounds (Set.range fun n => (-f) n \u03c9\u271d)\n\u03c9 : \u211d\nh\u03c9 : \u03c9 \u2208 Set.range fun n => f n \u03c9\u271d\n\u22a2 -c \u2264 \u03c9\n[PROOFSTEP]\nrw [mem_upperBounds] at hc \n[GOAL]\ncase h.mp.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : \u2200 (x : \u211d), (x \u2208 Set.range fun n => (-f) n \u03c9\u271d) \u2192 x \u2264 c\n\u03c9 : \u211d\nh\u03c9 : \u03c9 \u2208 Set.range fun n => f n \u03c9\u271d\n\u22a2 -c \u2264 \u03c9\n[PROOFSTEP]\nrefine' neg_le.2 (hc _ _)\n[GOAL]\ncase h.mp.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : \u2200 (x : \u211d), (x \u2208 Set.range fun n => (-f) n \u03c9\u271d) \u2192 x \u2264 c\n\u03c9 : \u211d\nh\u03c9 : \u03c9 \u2208 Set.range fun n => f n \u03c9\u271d\n\u22a2 -\u03c9 \u2208 Set.range fun n => (-f) n \u03c9\u271d\n[PROOFSTEP]\nsimpa only [Pi.neg_apply, Set.mem_range, neg_inj]\n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : c \u2208 lowerBounds (Set.range fun n => f n \u03c9\u271d)\n\u03c9 : \u211d\nh\u03c9 : \u03c9 \u2208 Set.range fun n => (-f) n \u03c9\u271d\n\u22a2 \u03c9 \u2264 -c\n[PROOFSTEP]\nrw [mem_lowerBounds] at hc \n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : \u2200 (x : \u211d), (x \u2208 Set.range fun n => f n \u03c9\u271d) \u2192 c \u2264 x\n\u03c9 : \u211d\nh\u03c9 : \u03c9 \u2208 Set.range fun n => (-f) n \u03c9\u271d\n\u22a2 \u03c9 \u2264 -c\n[PROOFSTEP]\nsimp_rw [Set.mem_range, Pi.neg_apply, neg_eq_iff_eq_neg] at h\u03c9 \n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : \u2200 (x : \u211d), (x \u2208 Set.range fun n => f n \u03c9\u271d) \u2192 c \u2264 x\n\u03c9 : \u211d\nh\u03c9 : \u2203 y, f y \u03c9\u271d = -\u03c9\n\u22a2 \u03c9 \u2264 -c\n[PROOFSTEP]\nrefine' le_neg.1 (hc _ _)\n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d\u00b9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nhbdd' : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(-f) (i + 1) \u03c9 - (-f) i \u03c9| \u2264 \u2191R\nhup : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => f n \u03c9) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9) atTop (\ud835\udcdd c)\nhdown : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, BddAbove (Set.range fun n => (-f) n \u03c9) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9) atTop (\ud835\udcdd c)\n\u03c9\u271d : \u03a9\nh\u03c9\u2081 : BddAbove (Set.range fun n => f n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)\nh\u03c9\u2082 : BddAbove (Set.range fun n => (-f) n \u03c9\u271d) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nthis : (\u2203 c, Tendsto (fun n => f n \u03c9\u271d) atTop (\ud835\udcdd c)) \u2194 \u2203 c, Tendsto (fun n => (-f) n \u03c9\u271d) atTop (\ud835\udcdd c)\nc : \u211d\nhc : \u2200 (x : \u211d), (x \u2208 Set.range fun n => f n \u03c9\u271d) \u2192 c \u2264 x\n\u03c9 : \u211d\nh\u03c9 : \u2203 y, f y \u03c9\u271d = -\u03c9\n\u22a2 -\u03c9 \u2208 Set.range fun n => f n \u03c9\u271d\n[PROOFSTEP]\nsimpa only [Set.mem_range]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => f n \u03c9) atTop atTop\n[PROOFSTEP]\nfilter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atTop htop (h\u03c9.2 <| bddBelow_range_of_tendsto_atTop_atTop htop)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhf : Martingale f \u2131 \u03bc\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => f n \u03c9) atTop atBot\n[PROOFSTEP]\nfilter_upwards [hf.bddAbove_range_iff_bddBelow_range hbdd] with \u03c9 h\u03c9 htop using\n  unbounded_of_tendsto_atBot htop (h\u03c9.1 <| bddAbove_range_of_tendsto_atTop_atBot htop)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns : \u2115 \u2192 Set \u03a9\n\u22a2 process s 0 = 0\n[PROOFSTEP]\nrw [process, Finset.range_zero, Finset.sum_empty]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 martingalePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])\n[PROOFSTEP]\nsimp only [martingalePart_eq_sum, process_zero, zero_add]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 \u2211 i in Finset.range n, (process s (i + 1) - process s i - \u03bc[process s (i + 1) - process s i|\u2191\u2131 i]) =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun k _ => _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn k : \u2115\nx\u271d : k \u2208 Finset.range n\n\u22a2 process s (k + 1) - process s k - \u03bc[process s (k + 1) - process s k|\u2191\u2131 k] =\n    Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]\n[PROOFSTEP]\nsimp only [process, Finset.sum_range_succ_sub_sum]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]\n[PROOFSTEP]\nhave := martingalePart_process_ae_eq \u2131 \u03bc s n\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  martingalePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, (Set.indicator (s (k + 1)) 1 - \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k])\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]\n[PROOFSTEP]\nsimp_rw [martingalePart, process, Finset.sum_sub_distrib] at this \n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc\u271d : Measure \u03a9\n\u2131\u271d : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d : \u2115 \u2192 Set \u03a9\n\u2131 : Filtration \u2115 m0\n\u03bc : Measure \u03a9\ns : \u2115 \u2192 Set \u03a9\nn : \u2115\nthis :\n  \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - predictablePart (process s) \u2131 \u03bc n =\n    \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 - \u2211 x in Finset.range n, \u03bc[Set.indicator (s (x + 1)) 1|\u2191\u2131 x]\n\u22a2 predictablePart (process s) \u2131 \u03bc n = \u2211 k in Finset.range n, \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]\n[PROOFSTEP]\nexact sub_right_injective this\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d s : \u2115 \u2192 Set \u03a9\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 |process s (n + 1) \u03c9 - process s n \u03c9| \u2264 \u21911\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d s : \u2115 \u2192 Set \u03a9\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 |process s (n + 1) \u03c9 - process s n \u03c9| \u2264 1\n[PROOFSTEP]\nrw [process, process, Finset.sum_apply, Finset.sum_apply, Finset.sum_range_succ_sub_sum, \u2190 Real.norm_eq_abs,\n  norm_indicator_eq_indicator_norm]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d s : \u2115 \u2192 Set \u03a9\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 Set.indicator (s (n + 1)) (fun a => \u2016OfNat.ofNat 1 a\u2016) \u03c9 \u2264 1\n[PROOFSTEP]\nrefine' Set.indicator_le' (fun _ _ => _) (fun _ _ => zero_le_one) _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ns\u271d s : \u2115 \u2192 Set \u03a9\n\u03c9 : \u03a9\nn : \u2115\nx\u271d\u00b9 : \u03a9\nx\u271d : x\u271d\u00b9 \u2208 s (n + 1)\n\u22a2 \u2016OfNat.ofNat 1 x\u271d\u00b9\u2016 \u2264 1\n[PROOFSTEP]\nrw [Pi.one_apply, norm_one]\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nhave h\u2081 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atTop (martingalePart_bdd_difference \u2131 hbdd)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nhave h\u2082 := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atBot (martingalePart_bdd_difference \u2131 hbdd)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nhave h\u2083 : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, 0 \u2264 (\u03bc[f (n + 1) - f n|\u2131 n]) \u03c9 :=\n  by\n  refine' ae_all_iff.2 fun n => condexp_nonneg _\n  filter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\u03c9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n[PROOFSTEP]\nrefine' ae_all_iff.2 fun n => condexp_nonneg _\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nn : \u2115\n\u22a2 0 \u2264\u1d50[\u03bc] f (n + 1) - f n\n[PROOFSTEP]\nfilter_upwards [ae_all_iff.1 hfmono n] with \u03c9 h\u03c9 using sub_nonneg.2 h\u03c9\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nfilter_upwards [h\u2081, h\u2082, h\u2083, hfmono] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 h\u03c9\u2083 h\u03c9\u2084\n[GOAL]\ncase h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop \u2194 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop \u2192 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nintro ht\n[GOAL]\ncase h.mpr\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\n\u22a2 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop \u2192 Tendsto (fun n => f n \u03c9) atTop atTop\n[PROOFSTEP]\nintro ht\n[GOAL]\ncase h.mp\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n[PROOFSTEP]\nrefine' tendsto_atTop_atTop_of_monotone' _ _\n[GOAL]\ncase h.mp.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 Monotone fun n => predictablePart f \u2131 \u03bc n \u03c9\n[PROOFSTEP]\nintro n m hnm\n[GOAL]\ncase h.mp.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 (fun n => predictablePart f \u2131 \u03bc n \u03c9) n \u2264 (fun n => predictablePart f \u2131 \u03bc n \u03c9) m\n[PROOFSTEP]\nsimp only [predictablePart, Finset.sum_apply]\n[GOAL]\ncase h.mp.refine'_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nn m : \u2115\nhnm : n \u2264 m\n\u22a2 \u2211 c in Finset.range n, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9 \u2264 \u2211 c in Finset.range m, (\u03bc[f (c + 1) - f c|\u2191\u2131 c]) \u03c9\n[PROOFSTEP]\nrefine' Finset.sum_mono_set_of_nonneg h\u03c9\u2083 (Finset.range_mono hnm)\n[GOAL]\ncase h.mp.refine'_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n[PROOFSTEP]\nrintro \u27e8b, hbdd\u27e9\n[GOAL]\ncase h.mp.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => f n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 tendsto_neg_atBot_iff] at ht \n[GOAL]\ncase h.mp.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\n\u22a2 False\n[PROOFSTEP]\nsimp only [martingalePart, sub_eq_add_neg] at h\u03c9\u2081 \n[GOAL]\ncase h.mp.refine'_2.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun x => -f x \u03c9) atTop atBot\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => predictablePart f \u2131 \u03bc n \u03c9)\nh\u03c9\u2081 : \u00acTendsto (fun n => (f n + -predictablePart f \u2131 \u03bc n) \u03c9) atTop atTop\n\u22a2 False\n[PROOFSTEP]\nexact h\u03c9\u2081 (tendsto_atTop_add_right_of_le _ (-b) (tendsto_neg_atBot_iff.1 ht) fun n => neg_le_neg (hbdd \u27e8n, rfl\u27e9))\n[GOAL]\ncase h.mpr\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 Tendsto (fun n => f n \u03c9) atTop atTop\n[PROOFSTEP]\nrefine' tendsto_atTop_atTop_of_monotone' (monotone_nat_of_le_succ h\u03c9\u2084) _\n[GOAL]\ncase h.mpr\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u00acBddAbove (Set.range fun n => f n \u03c9)\n[PROOFSTEP]\nrintro \u27e8b, hbdd\u27e9\n[GOAL]\ncase h.mpr.intro\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\nhfmono : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nhf : Adapted \u2131 f\nhint : \u2200 (n : \u2115), Integrable (f n)\nhbdd\u271d : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), |f (n + 1) \u03c9 - f n \u03c9| \u2264 \u2191R\nh\u2081 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u2082 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u2083 : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atTop\nh\u03c9\u2082 : \u00acTendsto (fun n => martingalePart f \u2131 \u03bc n \u03c9) atTop atBot\nh\u03c9\u2083 : \u2200 (n : \u2115), 0 \u2264 (\u03bc[f (n + 1) - f n|\u2191\u2131 n]) \u03c9\nh\u03c9\u2084 : \u2200 (n : \u2115), f n \u03c9 \u2264 f (n + 1) \u03c9\nht : Tendsto (fun n => predictablePart f \u2131 \u03bc n \u03c9) atTop atTop\nb : \u211d\nhbdd : b \u2208 upperBounds (Set.range fun n => f n \u03c9)\n\u22a2 False\n[PROOFSTEP]\nexact h\u03c9\u2082 ((tendsto_atBot_add_left_of_ge _ b fun n => hbdd \u27e8n, rfl\u27e9) <| tendsto_neg_atBot_iff.2 ht)\n[GOAL]\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n[PROOFSTEP]\nhave :=\n  tendsto_sum_indicator_atTop_iff (eventually_of_forall fun \u03c9 n => ?_) (adapted_process hs) (integrable_process \u03bc hs)\n    (eventually_of_forall <| process_difference_le s)\n[GOAL]\ncase refine_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\ncase refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 process (fun n => s n) n \u03c9 \u2264 process (fun n => s n) (n + 1) \u03c9\n[PROOFSTEP]\nrw [process, process, \u2190 sub_nonneg, Finset.sum_apply, Finset.sum_apply, Finset.sum_range_succ_sub_sum]\n[GOAL]\ncase refine_1\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9\u271d : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\n\u03c9 : \u03a9\nn : \u2115\n\u22a2 0 \u2264 Set.indicator (s (n + 1)) 1 \u03c9\n[PROOFSTEP]\nexact Set.indicator_nonneg (fun _ _ => zero_le_one) _\n[GOAL]\ncase refine_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => process (fun n => s n) n \u03c9) atTop atTop \u2194\n      Tendsto (fun n => predictablePart (process fun n => s n) \u2131 \u03bc n \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n[PROOFSTEP]\nsimp_rw [process, predictablePart_process_ae_eq] at this \n[GOAL]\ncase refine_2\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\u2131 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c9 : \u03a9\nr : \u211d\nR : \u211d\u22650\ninst\u271d : IsFiniteMeasure \u03bc\ns : \u2115 \u2192 Set \u03a9\nhs : \u2200 (n : \u2115), MeasurableSet (s n)\nthis :\n  \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => Finset.sum (Finset.range n) (fun k => Set.indicator (s (k + 1)) 1) \u03c9) atTop atTop \u2194\n      Tendsto (fun n => Finset.sum (Finset.range n) (fun k => \u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc,\n    Tendsto (fun n => \u2211 k in Finset.range n, Set.indicator (s (k + 1)) 1 \u03c9) atTop atTop \u2194\n      Tendsto (fun n => \u2211 k in Finset.range n, (\u03bc[Set.indicator (s (k + 1)) 1|\u2191\u2131 k]) \u03c9) atTop atTop\n[PROOFSTEP]\nsimpa using this\n", "meta": {"mathlib_filename": "Mathlib.Probability.Martingale.BorelCantelli", "llama_tokens": 55307, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677506936878, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.5371114882933217}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 Lex r s a b \u2194 r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 Lex r s a b \u2192 r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd\n[PROOFSTEP]\nrintro (\u27e8a, b, hij\u27e9 | \u27e8a, b, hab\u27e9)\n[GOAL]\ncase mp.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nhij : r i\u271d j\u271d\n\u22a2 r { fst := i\u271d, snd := a }.fst { fst := j\u271d, snd := b }.fst \u2228\n    \u2203 h, s { fst := j\u271d, snd := b }.fst (h \u25b8 { fst := i\u271d, snd := a }.snd) { fst := j\u271d, snd := b }.snd\n[PROOFSTEP]\nexact Or.inl hij\n[GOAL]\ncase mp.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\n\u22a2 r { fst := i\u271d, snd := a }.fst { fst := i\u271d, snd := b }.fst \u2228\n    \u2203 h, s { fst := i\u271d, snd := b }.fst (h \u25b8 { fst := i\u271d, snd := a }.snd) { fst := i\u271d, snd := b }.snd\n[PROOFSTEP]\nexact Or.inr \u27e8rfl, hab\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 (r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd) \u2192 Lex r s a b\n[PROOFSTEP]\nobtain \u27e8i, a\u27e9 := a\n[GOAL]\ncase mpr.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nb : (i : \u03b9) \u00d7 \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 (r { fst := i, snd := a }.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 { fst := i, snd := a }.snd) b.snd) \u2192\n    Lex r s { fst := i, snd := a } b\n[PROOFSTEP]\nobtain \u27e8j, b\u27e9 := b\n[GOAL]\ncase mpr.mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 (r { fst := i, snd := a }.fst { fst := j, snd := b }.fst \u2228\n      \u2203 h, s { fst := j, snd := b }.fst (h \u25b8 { fst := i, snd := a }.snd) { fst := j, snd := b }.snd) \u2192\n    Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mpr.mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 (r i j \u2228 \u2203 h, s j (h \u25b8 a) b) \u2192 Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\nrintro (h | \u27e8rfl, h\u27e9)\n[GOAL]\ncase mpr.mk.mk.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : r i j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ h\n[GOAL]\ncase mpr.mk.mk.inr.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na b : \u03b1 i\nh : s i ((_ : i = i) \u25b8 a) b\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b }\n[PROOFSTEP]\nexact Lex.right _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\na b : (i : \u03b9) \u00d7 \u03b1 i\nh : Lex r\u2081 s\u2081 a b\n\u22a2 Lex r\u2082 s\u2082 a b\n[PROOFSTEP]\nobtain \u27e8a, b, hij\u27e9 | \u27e8a, b, hab\u27e9 := h\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nhij : r\u2081 i\u271d j\u271d\n\u22a2 Lex r\u2082 s\u2082 { fst := i\u271d, snd := a } { fst := j\u271d, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ (hr _ _ hij)\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s\u2081 i\u271d a b\n\u22a2 Lex r\u2082 s\u2082 { fst := i\u271d, snd := a } { fst := i\u271d, snd := b }\n[PROOFSTEP]\nexact Lex.right _ _ (hs _ _ _ hab)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 Lex (Function.swap r) s a b \u2194 Lex r (fun i => Function.swap (s i)) b a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 Lex (Function.swap r) s a b \u2192 Lex r (fun i => Function.swap (s i)) b a\n[PROOFSTEP]\nrintro (\u27e8a, b, h\u27e9 | \u27e8a, b, h\u27e9)\n[GOAL]\ncase mp.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nh : Function.swap r i\u271d j\u271d\n\u22a2 Lex r (fun i => Function.swap (s i)) { fst := j\u271d, snd := b } { fst := i\u271d, snd := a }\ncase mp.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nh : s i\u271d a b\n\u22a2 Lex r (fun i => Function.swap (s i)) { fst := i\u271d, snd := b } { fst := i\u271d, snd := a }\n[PROOFSTEP]\nexacts [Lex.left _ _ h, Lex.right _ _ h]\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 Lex r (fun i => Function.swap (s i)) b a \u2192 Lex (Function.swap r) s a b\n[PROOFSTEP]\nrintro (\u27e8a, b, h\u27e9 | \u27e8a, b, h\u27e9)\n[GOAL]\ncase mpr.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nh : r i\u271d j\u271d\n\u22a2 Lex (Function.swap r) s { fst := j\u271d, snd := b } { fst := i\u271d, snd := a }\ncase mpr.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nh : Function.swap (s i\u271d) a b\n\u22a2 Lex (Function.swap r) s { fst := i\u271d, snd := b } { fst := i\u271d, snd := a }\n[PROOFSTEP]\nexacts [Lex.left _ _ h, Lex.right _ _ h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsIrrefl \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsIrrefl (\u03b1 i) (s i)\n\u22a2 \u2200 (a : (i : \u03b9) \u00d7 \u03b1 i), \u00acLex r s a a\n[PROOFSTEP]\nrintro _ (\u27e8a, b, hi\u27e9 | \u27e8a, b, ha\u27e9)\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsIrrefl \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsIrrefl (\u03b1 i) (s i)\ni\u271d : \u03b9\na : \u03b1 i\u271d\nhi : r i\u271d i\u271d\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ hi\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsIrrefl \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsIrrefl (\u03b1 i) (s i)\ni\u271d : \u03b9\na : \u03b1 i\u271d\nha : s i\u271d a a\n\u22a2 False\n[PROOFSTEP]\nexact irrefl _ ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrans \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrans (\u03b1 i) (s i)\n\u22a2 \u2200 (a b c : (i : \u03b9) \u00d7 \u03b1 i), Lex r s a b \u2192 Lex r s b c \u2192 Lex r s a c\n[PROOFSTEP]\nrintro _ _ _ (\u27e8a, b, hij\u27e9 | \u27e8a, b, hab\u27e9) (\u27e8_, c, hk\u27e9 | \u27e8_, c, hc\u27e9)\n[GOAL]\ncase left.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrans \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrans (\u03b1 i) (s i)\ni\u271d j\u271d\u00b9 : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\u00b9\nhij : r i\u271d j\u271d\u00b9\nj\u271d : \u03b9\nc : \u03b1 j\u271d\nhk : r j\u271d\u00b9 j\u271d\n\u22a2 Lex r s { fst := i\u271d, snd := a } { fst := j\u271d, snd := c }\n[PROOFSTEP]\nexact Lex.left _ _ (_root_.trans hij hk)\n[GOAL]\ncase left.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrans \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrans (\u03b1 i) (s i)\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nhij : r i\u271d j\u271d\nc : \u03b1 j\u271d\nhc : s j\u271d b c\n\u22a2 Lex r s { fst := i\u271d, snd := a } { fst := j\u271d, snd := c }\n[PROOFSTEP]\nexact Lex.left _ _ hij\n[GOAL]\ncase right.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrans \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrans (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\nj\u271d : \u03b9\nc : \u03b1 j\u271d\nhk : r i\u271d j\u271d\n\u22a2 Lex r s { fst := i\u271d, snd := a } { fst := j\u271d, snd := c }\n[PROOFSTEP]\nexact Lex.left _ _ hk\n[GOAL]\ncase right.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrans \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrans (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\nc : \u03b1 i\u271d\nhc : s i\u271d b c\n\u22a2 Lex r s { fst := i\u271d, snd := a } { fst := i\u271d, snd := c }\n[PROOFSTEP]\nexact Lex.right _ _ (_root_.trans hab hc)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsSymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsSymm (\u03b1 i) (s i)\n\u22a2 \u2200 (a b : (i : \u03b9) \u00d7 \u03b1 i), Lex r s a b \u2192 Lex r s b a\n[PROOFSTEP]\nrintro _ _ (\u27e8a, b, hij\u27e9 | \u27e8a, b, hab\u27e9)\n[GOAL]\ncase left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsSymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsSymm (\u03b1 i) (s i)\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nhij : r i\u271d j\u271d\n\u22a2 Lex r s { fst := j\u271d, snd := b } { fst := i\u271d, snd := a }\n[PROOFSTEP]\nexact Lex.left _ _ (symm hij)\n[GOAL]\ncase right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsSymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsSymm (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\n\u22a2 Lex r s { fst := i\u271d, snd := b } { fst := i\u271d, snd := a }\n[PROOFSTEP]\nexact Lex.right _ _ (symm hab)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsAsymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsAntisymm (\u03b1 i) (s i)\n\u22a2 \u2200 (a b : (i : \u03b9) \u00d7 \u03b1 i), Lex r s a b \u2192 Lex r s b a \u2192 a = b\n[PROOFSTEP]\nrintro _ _ (\u27e8a, b, hij\u27e9 | \u27e8a, b, hab\u27e9) (\u27e8_, _, hji\u27e9 | \u27e8_, _, hba\u27e9)\n[GOAL]\ncase left.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsAsymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsAntisymm (\u03b1 i) (s i)\ni\u271d j\u271d : \u03b9\na : \u03b1 i\u271d\nb : \u03b1 j\u271d\nhij : r i\u271d j\u271d\nhji : r j\u271d i\u271d\n\u22a2 { fst := i\u271d, snd := a } = { fst := j\u271d, snd := b }\n[PROOFSTEP]\nexact (asymm hij hji).elim\n[GOAL]\ncase left.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsAsymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsAntisymm (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhij : r i\u271d i\u271d\nhba : s i\u271d b a\n\u22a2 { fst := i\u271d, snd := a } = { fst := i\u271d, snd := b }\n[PROOFSTEP]\nexact (irrefl _ hij).elim\n[GOAL]\ncase right.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsAsymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsAntisymm (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\nhji : r i\u271d i\u271d\n\u22a2 { fst := i\u271d, snd := a } = { fst := i\u271d, snd := b }\n[PROOFSTEP]\nexact (irrefl _ hji).elim\n[GOAL]\ncase right.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsAsymm \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsAntisymm (\u03b1 i) (s i)\ni\u271d : \u03b9\na b : \u03b1 i\u271d\nhab : s i\u271d a b\nhba : s i\u271d b a\n\u22a2 { fst := i\u271d, snd := a } = { fst := i\u271d, snd := b }\n[PROOFSTEP]\nexact ext rfl (heq_of_eq $ antisymm hab hba)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\n\u22a2 \u2200 (a b : (i : \u03b9) \u00d7 \u03b1 i), Lex r s a b \u2228 Lex r s b a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nobtain hij | rfl | hji := trichotomous_of r i j\n[GOAL]\ncase mk.mk.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhij : r i j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.left _ _ hij)\n[GOAL]\ncase mk.mk.inr.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nobtain hab | hba := total_of (s i) a b\n[GOAL]\ncase mk.mk.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\nhab : s i a b\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.right _ _ hab)\n[GOAL]\ncase mk.mk.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\nhba : s i b a\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Lex.right _ _ hba)\n[GOAL]\ncase mk.mk.inr.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTotal (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhji : r j i\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Lex.left _ _ hji)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\n\u22a2 \u2200 (a b : (i : \u03b9) \u00d7 \u03b1 i), Lex r s a b \u2228 a = b \u2228 Lex r s b a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228\n    { fst := i, snd := a } = { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nobtain hij | rfl | hji := trichotomous_of r i j\n[GOAL]\ncase mk.mk.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhij : r i j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228\n    { fst := i, snd := a } = { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.left _ _ hij)\n[GOAL]\ncase mk.mk.inr.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228\n    { fst := i, snd := a } = { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nobtain hab | rfl | hba := trichotomous_of (s i) a b\n[GOAL]\ncase mk.mk.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\nhab : s i a b\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228\n    { fst := i, snd := a } = { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.right _ _ hab)\n[GOAL]\ncase mk.mk.inr.inl.inr.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := a } \u2228\n    { fst := i, snd := a } = { fst := i, snd := a } \u2228 Lex r s { fst := i, snd := a } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Or.inl rfl)\n[GOAL]\ncase mk.mk.inr.inl.inr.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na b : \u03b1 i\nhba : s i b a\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b } \u2228\n    { fst := i, snd := a } = { fst := i, snd := b } \u2228 Lex r s { fst := i, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Or.inr $ Lex.right _ _ hba)\n[GOAL]\ncase mk.mk.inr.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u271d b\u271d : (i : \u03b9) \u00d7 \u03b1 i\ninst\u271d\u00b9 : IsTrichotomous \u03b9 r\ninst\u271d : \u2200 (i : \u03b9), IsTrichotomous (\u03b1 i) (s i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhji : r j i\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b } \u2228\n    { fst := i, snd := a } = { fst := j, snd := b } \u2228 Lex r s { fst := j, snd := b } { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Or.inr $ Lex.left _ _ hji)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7' \u03b1 i\n\u22a2 Lex r s a b \u2194 r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7' \u03b1 i\n\u22a2 Lex r s a b \u2192 r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd\n[PROOFSTEP]\nrintro (\u27e8a, b, hij\u27e9 | \u27e8i, hab\u27e9)\n[GOAL]\ncase mp.left\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na\u2081\u271d : \u03b9\na : \u03b1 a\u2081\u271d\na\u2082\u271d : \u03b9\nb : \u03b1 a\u2082\u271d\nhij : r a\u2081\u271d a\u2082\u271d\n\u22a2 r { fst := a\u2081\u271d, snd := a }.fst { fst := a\u2082\u271d, snd := b }.fst \u2228\n    \u2203 h, s { fst := a\u2082\u271d, snd := b }.fst (h \u25b8 { fst := a\u2081\u271d, snd := a }.snd) { fst := a\u2082\u271d, snd := b }.snd\n[PROOFSTEP]\nexact Or.inl hij\n[GOAL]\ncase mp.right\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhab : s i b\u2081\u271d b\u2082\u271d\n\u22a2 r { fst := i, snd := b\u2081\u271d }.fst { fst := i, snd := b\u2082\u271d }.fst \u2228\n    \u2203 h, s { fst := i, snd := b\u2082\u271d }.fst (h \u25b8 { fst := i, snd := b\u2081\u271d }.snd) { fst := i, snd := b\u2082\u271d }.snd\n[PROOFSTEP]\nexact Or.inr \u27e8rfl, hab\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\na b : (i : \u03b9) \u00d7' \u03b1 i\n\u22a2 (r a.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 a.snd) b.snd) \u2192 Lex r s a b\n[PROOFSTEP]\nobtain \u27e8i, a\u27e9 := a\n[GOAL]\ncase mpr.mk\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nb : (i : \u03b9) \u00d7' \u03b1 i\ni : \u03b9\na : \u03b1 i\n\u22a2 (r { fst := i, snd := a }.fst b.fst \u2228 \u2203 h, s b.fst (h \u25b8 { fst := i, snd := a }.snd) b.snd) \u2192\n    Lex r s { fst := i, snd := a } b\n[PROOFSTEP]\nobtain \u27e8j, b\u27e9 := b\n[GOAL]\ncase mpr.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 (r { fst := i, snd := a }.fst { fst := j, snd := b }.fst \u2228\n      \u2203 h, s { fst := j, snd := b }.fst (h \u25b8 { fst := i, snd := a }.snd) { fst := j, snd := b }.snd) \u2192\n    Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mpr.mk.mk\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 (r i j \u2228 \u2203 h, s j (h \u25b8 a) b) \u2192 Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\nrintro (h | \u27e8rfl, h\u27e9)\n[GOAL]\ncase mpr.mk.mk.inl\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : r i j\n\u22a2 Lex r s { fst := i, snd := a } { fst := j, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ h\n[GOAL]\ncase mpr.mk.mk.inr.intro\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\ni : \u03b9\na b : \u03b1 i\nh : s i ((_ : i = i) \u25b8 a) b\n\u22a2 Lex r s { fst := i, snd := a } { fst := i, snd := b }\n[PROOFSTEP]\nexact Lex.right _ h\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081\u271d r\u2082\u271d : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nr\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\na b : (i : \u03b9) \u00d7' \u03b1 i\nh : Lex r\u2081 s\u2081 a b\n\u22a2 Lex r\u2082 s\u2082 a b\n[PROOFSTEP]\nobtain \u27e8a, b, hij\u27e9 | \u27e8i, hab\u27e9 := h\n[GOAL]\ncase left\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081\u271d r\u2082\u271d : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nr\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\na\u2081\u271d : \u03b9\na : \u03b1 a\u2081\u271d\na\u2082\u271d : \u03b9\nb : \u03b1 a\u2082\u271d\nhij : r\u2081 a\u2081\u271d a\u2082\u271d\n\u22a2 Lex r\u2082 s\u2082 { fst := a\u2081\u271d, snd := a } { fst := a\u2082\u271d, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ (hr _ _ hij)\n[GOAL]\ncase right\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nr r\u2081\u271d r\u2082\u271d : \u03b9 \u2192 \u03b9 \u2192 Prop\ns s\u2081\u271d s\u2082\u271d : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nr\u2081 r\u2082 : \u03b9 \u2192 \u03b9 \u2192 Prop\ns\u2081 s\u2082 : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b1 i \u2192 Prop\nhr : \u2200 (a b : \u03b9), r\u2081 a b \u2192 r\u2082 a b\nhs : \u2200 (i : \u03b9) (a b : \u03b1 i), s\u2081 i a b \u2192 s\u2082 i a b\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhab : s\u2081 i b\u2081\u271d b\u2082\u271d\n\u22a2 Lex r\u2082 s\u2082 { fst := i, snd := b\u2081\u271d } { fst := i, snd := b\u2082\u271d }\n[PROOFSTEP]\nexact Lex.right _ (hs _ _ _ hab)\n", "meta": {"mathlib_filename": "Mathlib.Data.Sigma.Lex", "llama_tokens": 12368, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619350028204, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5370172976759184}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nlet u := {x | f =\u1da0[\ud835\udcdd x] 0}\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nsuffices main : closure u \u2229 U \u2286 u\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nmain : closure u \u2229 U \u2286 u\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nhave Uu : U \u2286 u := hU.subset_of_closure_inter_subset isOpen_setOf_eventually_nhds \u27e8z\u2080, h\u2080, hfz\u2080\u27e9 main\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nmain : closure u \u2229 U \u2286 u\nUu : U \u2286 u\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nmain : closure u \u2229 U \u2286 u\nUu : U \u2286 u\nz : E\nhz : z \u2208 U\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nsimpa using\n  mem_of_mem_nhds\n    (Uu hz)\n      /- Take a limit point `x`, then a ball `B (x, r)` on which it has a power series expansion, and\n          then `y \u2208 B (x, r/2) \u2229 u`. Then `f` has a power series expansion on `B (y, r/2)` as it is\n          contained in `B (x, r)`. All the coefficients in this series expansion vanish, as `f` is zero\n          on a neighborhood of `y`. Therefore, `f` is zero on `B (y, r/2)`. As this ball contains `x`,\n          it follows that `f` vanishes on a neighborhood of `x`, proving the claim. -/\n[GOAL]\ncase main\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\n\u22a2 closure u \u2229 U \u2286 u\n[PROOFSTEP]\nrintro x \u27e8xu, xU\u27e9\n[GOAL]\ncase main.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\n\u22a2 x \u2208 u\n[PROOFSTEP]\nrcases hf x xU with \u27e8p, r, hp\u27e9\n[GOAL]\ncase main.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\n\u22a2 x \u2208 u\n[PROOFSTEP]\nobtain \u27e8y, yu, hxy\u27e9 : \u2203 y \u2208 u, edist x y < r / 2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\n\u22a2 \u2203 y, y \u2208 u \u2227 edist x y < r / 2\ncase main.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\n\u22a2 x \u2208 u\n[PROOFSTEP]\nexact EMetric.mem_closure_iff.1 xu (r / 2) (ENNReal.half_pos hp.r_pos.ne')\n[GOAL]\ncase main.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\n\u22a2 x \u2208 u\n[PROOFSTEP]\nlet q := p.changeOrigin (y - x)\n[GOAL]\ncase main.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\n\u22a2 x \u2208 u\n[PROOFSTEP]\nhave has_series : HasFPowerSeriesOnBall f q y (r / 2) :=\n  by\n  have A : (\u2016y - x\u2016\u208a : \u211d\u22650\u221e) < r / 2 := by rwa [edist_comm, edist_eq_coe_nnnorm_sub] at hxy \n  have := hp.changeOrigin (A.trans_le ENNReal.half_le_self)\n  simp only [add_sub_cancel'_right] at this \n  apply this.mono (ENNReal.half_pos hp.r_pos.ne')\n  apply ENNReal.le_sub_of_add_le_left ENNReal.coe_ne_top\n  apply (add_le_add A.le (le_refl (r / 2))).trans (le_of_eq _)\n  exact ENNReal.add_halves _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\n\u22a2 HasFPowerSeriesOnBall f q y (r / 2)\n[PROOFSTEP]\nhave A : (\u2016y - x\u2016\u208a : \u211d\u22650\u221e) < r / 2 := by rwa [edist_comm, edist_eq_coe_nnnorm_sub] at hxy \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\n\u22a2 \u2191\u2016y - x\u2016\u208a < r / 2\n[PROOFSTEP]\nrwa [edist_comm, edist_eq_coe_nnnorm_sub] at hxy \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\n\u22a2 HasFPowerSeriesOnBall f q y (r / 2)\n[PROOFSTEP]\nhave := hp.changeOrigin (A.trans_le ENNReal.half_le_self)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\nthis : HasFPowerSeriesOnBall f (FormalMultilinearSeries.changeOrigin p (y - x)) (x + (y - x)) (r - \u2191\u2016y - x\u2016\u208a)\n\u22a2 HasFPowerSeriesOnBall f q y (r / 2)\n[PROOFSTEP]\nsimp only [add_sub_cancel'_right] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\nthis : HasFPowerSeriesOnBall f (FormalMultilinearSeries.changeOrigin p (y - x)) y (r - \u2191\u2016y - x\u2016\u208a)\n\u22a2 HasFPowerSeriesOnBall f q y (r / 2)\n[PROOFSTEP]\napply this.mono (ENNReal.half_pos hp.r_pos.ne')\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\nthis : HasFPowerSeriesOnBall f (FormalMultilinearSeries.changeOrigin p (y - x)) y (r - \u2191\u2016y - x\u2016\u208a)\n\u22a2 r / 2 \u2264 r - \u2191\u2016y - x\u2016\u208a\n[PROOFSTEP]\napply ENNReal.le_sub_of_add_le_left ENNReal.coe_ne_top\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\nthis : HasFPowerSeriesOnBall f (FormalMultilinearSeries.changeOrigin p (y - x)) y (r - \u2191\u2016y - x\u2016\u208a)\n\u22a2 \u2191\u2016y - x\u2016\u208a + r / 2 \u2264 r\n[PROOFSTEP]\napply (add_le_add A.le (le_refl (r / 2))).trans (le_of_eq _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nA : \u2191\u2016y - x\u2016\u208a < r / 2\nthis : HasFPowerSeriesOnBall f (FormalMultilinearSeries.changeOrigin p (y - x)) y (r - \u2191\u2016y - x\u2016\u208a)\n\u22a2 r / 2 + r / 2 = r\n[PROOFSTEP]\nexact ENNReal.add_halves _\n[GOAL]\ncase main.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\n\u22a2 x \u2208 u\n[PROOFSTEP]\nhave M : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x := EMetric.isOpen_ball.mem_nhds hxy\n[GOAL]\ncase main.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\n\u22a2 x \u2208 u\n[PROOFSTEP]\nfilter_upwards [M] with z hz\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nhave A : HasSum (fun n : \u2115 => q n fun _ : Fin n => z - y) (f z) := has_series.hasSum_sub hz\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nhave B : HasSum (fun n : \u2115 => q n fun _ : Fin n => z - y) 0 :=\n  by\n  have : HasFPowerSeriesAt 0 q y := has_series.hasFPowerSeriesAt.congr yu\n  convert hasSum_zero (\u03b1 := F) using 2\n  ext n\n  exact this.apply_eq_zero n _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\n\u22a2 HasSum (fun n => \u2191(q n) fun x => z - y) 0\n[PROOFSTEP]\nhave : HasFPowerSeriesAt 0 q y := has_series.hasFPowerSeriesAt.congr yu\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\nthis : HasFPowerSeriesAt 0 q y\n\u22a2 HasSum (fun n => \u2191(q n) fun x => z - y) 0\n[PROOFSTEP]\nconvert hasSum_zero (\u03b1 := F) using 2\n[GOAL]\ncase h.e.h.e'_5\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\nthis : HasFPowerSeriesAt 0 q y\n\u22a2 (fun n => \u2191(q n) fun x => z - y) = fun x => 0\n[PROOFSTEP]\next n\n[GOAL]\ncase h.e.h.e'_5.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\nthis : HasFPowerSeriesAt 0 q y\nn : \u2115\n\u22a2 (\u2191(q n) fun x => z - y) = 0\n[PROOFSTEP]\nexact this.apply_eq_zero n _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nu : Set E := {x | f =\u1da0[\ud835\udcdd x] 0}\nx : E\nxu : x \u2208 closure u\nxU : x \u2208 U\np : FormalMultilinearSeries \ud835\udd5c E F\nr : \u211d\u22650\u221e\nhp : HasFPowerSeriesOnBall f p x r\ny : E\nyu : y \u2208 u\nhxy : edist x y < r / 2\nq : FormalMultilinearSeries \ud835\udd5c E F := FormalMultilinearSeries.changeOrigin p (y - x)\nhas_series : HasFPowerSeriesOnBall f q y (r / 2)\nM : EMetric.ball y (r / 2) \u2208 \ud835\udcdd x\nz : E\nhz : z \u2208 EMetric.ball y (r / 2)\nA : HasSum (fun n => \u2191(q n) fun x => z - y) (f z)\nB : HasSum (fun n => \u2191(q n) fun x => z - y) 0\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nexact HasSum.unique A B\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nlet F' := UniformSpace.Completion F\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nset e : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nhave : AnalyticOn \ud835\udd5c (e \u2218 f) U := fun x hx => (e.analyticAt _).comp (hf x hx)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nhave A : EqOn (e \u2218 f) 0 U :=\n  by\n  apply eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux this hU h\u2080\n  filter_upwards [hfz\u2080] with x hx\n  simp only [hx, Function.comp_apply, Pi.zero_apply, map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\n\u22a2 EqOn (\u2191e \u2218 f) 0 U\n[PROOFSTEP]\napply eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux this hU h\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\n\u22a2 \u2191e \u2218 f =\u1da0[\ud835\udcdd z\u2080] 0\n[PROOFSTEP]\nfilter_upwards [hfz\u2080] with x hx\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\nx : E\nhx : f x = OfNat.ofNat 0 x\n\u22a2 (\u2191UniformSpace.Completion.toComplL \u2218 f) x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp only [hx, Function.comp_apply, Pi.zero_apply, map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\nA : EqOn (\u2191e \u2218 f) 0 U\n\u22a2 EqOn f 0 U\n[PROOFSTEP]\nintro z hz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\nA : EqOn (\u2191e \u2218 f) 0 U\nz : E\nhz : z \u2208 U\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nhave : e (f z) = e 0 := by simpa only using A hz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\nA : EqOn (\u2191e \u2218 f) 0 U\nz : E\nhz : z \u2208 U\n\u22a2 \u2191e (f z) = \u2191e 0\n[PROOFSTEP]\nsimpa only using A hz\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfz\u2080 : f =\u1da0[\ud835\udcdd z\u2080] 0\nF' : Type u_3 := UniformSpace.Completion F\ne : F \u2192L[\ud835\udd5c] F' := UniformSpace.Completion.toComplL\nthis\u271d : AnalyticOn \ud835\udd5c (\u2191e \u2218 f) U\nA : EqOn (\u2191e \u2218 f) 0 U\nz : E\nhz : z \u2208 U\nthis : \u2191e (f z) = \u2191e 0\n\u22a2 f z = OfNat.ofNat 0 z\n[PROOFSTEP]\nexact UniformSpace.Completion.coe_injective F this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf g : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfg : f =\u1da0[\ud835\udcdd z\u2080] g\n\u22a2 EqOn f g U\n[PROOFSTEP]\nhave hfg' : f - g =\u1da0[\ud835\udcdd z\u2080] 0 := hfg.mono fun z h => by simp [h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf g : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfg : f =\u1da0[\ud835\udcdd z\u2080] g\nz : E\nh : f z = g z\n\u22a2 (f - g) z = OfNat.ofNat 0 z\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nf g : E \u2192 F\nU : Set E\nhf : AnalyticOn \ud835\udd5c f U\nhg : AnalyticOn \ud835\udd5c g U\nhU : IsPreconnected U\nz\u2080 : E\nh\u2080 : z\u2080 \u2208 U\nhfg : f =\u1da0[\ud835\udcdd z\u2080] g\nhfg' : f - g =\u1da0[\ud835\udcdd z\u2080] 0\n\u22a2 EqOn f g U\n[PROOFSTEP]\nsimpa [sub_eq_zero] using fun z hz => (hf.sub hg).eqOn_zero_of_preconnected_of_eventuallyEq_zero hU h\u2080 hfg' hz\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Analytic.Uniqueness", "llama_tokens": 14160, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388083214156, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.536985088458386}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211 k in Ico 1 (2 ^ Nat.zero), f k \u2264 \u2211 k in range Nat.zero, 2 ^ k \u2022 f (2 ^ k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\n\u22a2 \u2211 k in Ico 1 (2 ^ Nat.succ n), f k \u2264 \u2211 k in range (Nat.succ n), 2 ^ k \u2022 f (2 ^ k)\n[PROOFSTEP]\nsuffices (\u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k) \u2264 2 ^ n \u2022 f (2 ^ n)\n  by\n  rw [sum_range_succ, \u2190 sum_Ico_consecutive]\n  exact add_le_add ihn this\n  exacts [n.one_le_two_pow, Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 \u2211 k in Ico 1 (2 ^ Nat.succ n), f k \u2264 \u2211 k in range (Nat.succ n), 2 ^ k \u2022 f (2 ^ k)\n[PROOFSTEP]\nrw [sum_range_succ, \u2190 sum_Ico_consecutive]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 \u2211 i in Ico 1 ?n, f i + \u2211 i in Ico ?n (2 ^ Nat.succ n), f i \u2264 \u2211 x in range n, 2 ^ x \u2022 f (2 ^ x) + 2 ^ n \u2022 f (2 ^ n)\ncase n\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 \u2115\ncase hmn\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 1 \u2264 ?n\ncase hnk\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 ?n \u2264 2 ^ Nat.succ n\n[PROOFSTEP]\nexact add_le_add ihn this\n[GOAL]\ncase hmn\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 1 \u2264 2 ^ n\ncase hnk\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n\u22a2 2 ^ n \u2264 2 ^ Nat.succ n\n[PROOFSTEP]\nexacts [n.one_le_two_pow, Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\n\u22a2 \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n[PROOFSTEP]\nhave : \u2200 k \u2208 Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 f (2 ^ n) := fun k hk => hf (pow_pos zero_lt_two _) (mem_Ico.mp hk).1\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2200 (k : \u2115), k \u2208 Ico (2 ^ n) (2 ^ (n + 1)) \u2192 f k \u2264 f (2 ^ n)\n\u22a2 \u2211 k in Ico (2 ^ n) (2 ^ (n + 1)), f k \u2264 2 ^ n \u2022 f (2 ^ n)\n[PROOFSTEP]\nconvert sum_le_sum this\n[GOAL]\ncase h.e'_4\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in Ico 1 (2 ^ n), f k \u2264 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\nthis : \u2200 (k : \u2115), k \u2208 Ico (2 ^ n) (2 ^ (n + 1)) \u2192 f k \u2264 f (2 ^ n)\n\u22a2 2 ^ n \u2022 f (2 ^ n) = \u2211 i in Ico (2 ^ n) (2 ^ (n + 1)), f (2 ^ n)\n[PROOFSTEP]\nsimp [pow_succ, two_mul]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in range (2 ^ n), f k \u2264 f 0 + \u2211 k in range n, 2 ^ k \u2022 f (2 ^ k)\n[PROOFSTEP]\nconvert add_le_add_left (le_sum_condensed' hf n) (f 0)\n[GOAL]\ncase h.e'_3\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in range (2 ^ n), f k = f 0 + \u2211 k in Ico 1 (2 ^ n), f k\n[PROOFSTEP]\nrw [\u2190 sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ, sum_range_zero, zero_add]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211 k in range Nat.zero, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ Nat.zero + 1), f k\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\n\u22a2 \u2211 k in range (Nat.succ n), 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ Nat.succ n + 1), f k\n[PROOFSTEP]\nsuffices 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n  by\n  rw [sum_range_succ, \u2190 sum_Ico_consecutive]\n  exacts [add_le_add ihn this, (add_le_add_right n.one_le_two_pow _ : 1 + 1 \u2264 2 ^ n + 1),\n    add_le_add_right (Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ) _]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n\u22a2 \u2211 k in range (Nat.succ n), 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ Nat.succ n + 1), f k\n[PROOFSTEP]\nrw [sum_range_succ, \u2190 sum_Ico_consecutive]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n\u22a2 \u2211 x in range n, 2 ^ x \u2022 f (2 ^ (x + 1)) + 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264\n    \u2211 i in Ico 2 ?n, f i + \u2211 i in Ico ?n (2 ^ Nat.succ n + 1), f i\ncase n\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n\u22a2 \u2115\ncase hmn\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n\u22a2 2 \u2264 ?n\ncase hnk\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n\u22a2 ?n \u2264 2 ^ Nat.succ n + 1\n[PROOFSTEP]\nexacts [add_le_add ihn this, (add_le_add_right n.one_le_two_pow _ : 1 + 1 \u2264 2 ^ n + 1),\n  add_le_add_right (Nat.pow_le_pow_of_le_right zero_lt_two n.le_succ) _]\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\n\u22a2 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n[PROOFSTEP]\nhave : \u2200 k \u2208 Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f (2 ^ (n + 1)) \u2264 f k := fun k hk =>\n  hf (n.one_le_two_pow.trans_lt <| (Nat.lt_succ_of_le le_rfl).trans_le (mem_Ico.mp hk).1)\n    (Nat.le_of_lt_succ <| (mem_Ico.mp hk).2)\n[GOAL]\ncase succ\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : \u2200 (k : \u2115), k \u2208 Ico (2 ^ n + 1) (2 ^ (n + 1) + 1) \u2192 f (2 ^ (n + 1)) \u2264 f k\n\u22a2 2 ^ n \u2022 f (2 ^ (n + 1)) \u2264 \u2211 k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k\n[PROOFSTEP]\nconvert sum_le_sum this\n[GOAL]\ncase h.e'_3\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\nihn : \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1)) \u2264 \u2211 k in Ico 2 (2 ^ n + 1), f k\nthis : \u2200 (k : \u2115), k \u2208 Ico (2 ^ n + 1) (2 ^ (n + 1) + 1) \u2192 f (2 ^ (n + 1)) \u2264 f k\n\u22a2 2 ^ n \u2022 f (2 ^ (n + 1)) = \u2211 i in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f (2 ^ (n + 1))\n[PROOFSTEP]\nsimp [pow_succ, two_mul]\n[GOAL]\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in range (n + 1), 2 ^ k \u2022 f (2 ^ k) \u2264 f 1 + 2 \u2022 \u2211 k in Ico 2 (2 ^ n + 1), f k\n[PROOFSTEP]\nconvert add_le_add_left (nsmul_le_nsmul_of_le_right (sum_condensed_le' hf n) 2) (f 1)\n[GOAL]\ncase h.e'_3\nM : Type u_1\ninst\u271d : OrderedAddCommMonoid M\nf : \u2115 \u2192 M\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in range (n + 1), 2 ^ k \u2022 f (2 ^ k) = f 1 + 2 \u2022 \u2211 k in range n, 2 ^ k \u2022 f (2 ^ (k + 1))\n[PROOFSTEP]\nsimp [sum_range_succ', add_comm, pow_succ, mul_nsmul', sum_nsmul]\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211' (k : \u2115), f k \u2264 f 0 + \u2211' (k : \u2115), \u2191(2 ^ k) * f (2 ^ k)\n[PROOFSTEP]\nrw [ENNReal.tsum_eq_iSup_nat' (Nat.tendsto_pow_atTop_atTop_of_one_lt _root_.one_lt_two)]\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2a06 (i : \u2115), \u2211 a in Finset.range (2 ^ i), f a \u2264 f 0 + \u2211' (k : \u2115), \u2191(2 ^ k) * f (2 ^ k)\n[PROOFSTEP]\nrefine' iSup_le fun n => (Finset.le_sum_condensed hf n).trans (add_le_add_left _ _)\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 k in Finset.range n, 2 ^ k \u2022 f (2 ^ k) \u2264 \u2211' (k : \u2115), \u2191(2 ^ k) * f (2 ^ k)\n[PROOFSTEP]\nsimp only [nsmul_eq_mul, Nat.cast_pow, Nat.cast_two]\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 x in Finset.range n, 2 ^ x * f (2 ^ x) \u2264 \u2211' (k : \u2115), 2 ^ k * f (2 ^ k)\n[PROOFSTEP]\napply ENNReal.sum_le_tsum\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211' (k : \u2115), \u2191(2 ^ k) * f (2 ^ k) \u2264 f 1 + 2 * \u2211' (k : \u2115), f k\n[PROOFSTEP]\nrw [ENNReal.tsum_eq_iSup_nat' (tendsto_atTop_mono Nat.le_succ tendsto_id), two_mul, \u2190 two_nsmul]\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2a06 (i : \u2115), \u2211 a in Finset.range (Nat.succ i), \u2191(2 ^ a) * f (2 ^ a) \u2264 f 1 + 2 \u2022 \u2211' (k : \u2115), f k\n[PROOFSTEP]\nrefine'\n  iSup_le fun n =>\n    le_trans _ (add_le_add_left (nsmul_le_nsmul_of_le_right (ENNReal.sum_le_tsum <| Finset.Ico 2 (2 ^ n + 1)) _) _)\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\u221e\nhf : \u2200 \u2983m n : \u2115\u2984, 1 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nn : \u2115\n\u22a2 \u2211 a in Finset.range (Nat.succ n), \u2191(2 ^ a) * f (2 ^ a) \u2264 f 1 + 2 \u2022 \u2211 x in Finset.Ico 2 (2 ^ n + 1), f x\n[PROOFSTEP]\nsimpa using Finset.sum_condensed_le hf n\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 (Summable fun k => 2 ^ k * f (2 ^ k)) \u2194 Summable f\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.tsum_coe_ne_top_iff_summable, Ne.def, not_iff_not, ENNReal.coe_mul, ENNReal.coe_pow,\n  ENNReal.coe_two]\n[GOAL]\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4 \u2194 \u2211' (b : \u2115), \u2191(f b) = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4 \u2192 \u2211' (b : \u2115), \u2191(f b) = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 \u2211' (b : \u2115), \u2191(f b) = \u22a4 \u2192 \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nh : \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4\n\u22a2 \u2211' (b : \u2115), \u2191(f b) = \u22a4\n[PROOFSTEP]\nreplace hf : \u2200 m n, 1 < m \u2192 m \u2264 n \u2192 (f n : \u211d\u22650\u221e) \u2264 f m := fun m n hm hmn =>\n  ENNReal.coe_le_coe.2 (hf (zero_lt_one.trans hm) hmn)\n[GOAL]\ncase mp\nf : \u2115 \u2192 \u211d\u22650\nh : \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4\nhf : \u2200 (m n : \u2115), 1 < m \u2192 m \u2264 n \u2192 \u2191(f n) \u2264 \u2191(f m)\n\u22a2 \u2211' (b : \u2115), \u2191(f b) = \u22a4\n[PROOFSTEP]\nsimpa [h, ENNReal.add_eq_top, ENNReal.mul_eq_top] using ENNReal.tsum_condensed_le hf\n[GOAL]\ncase mpr\nf : \u2115 \u2192 \u211d\u22650\nhf : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\nh : \u2211' (b : \u2115), \u2191(f b) = \u22a4\n\u22a2 \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4\n[PROOFSTEP]\nreplace hf : \u2200 m n, 0 < m \u2192 m \u2264 n \u2192 (f n : \u211d\u22650\u221e) \u2264 f m := fun m n hm hmn => ENNReal.coe_le_coe.2 (hf hm hmn)\n[GOAL]\ncase mpr\nf : \u2115 \u2192 \u211d\u22650\nh : \u2211' (b : \u2115), \u2191(f b) = \u22a4\nhf : \u2200 (m n : \u2115), 0 < m \u2192 m \u2264 n \u2192 \u2191(f n) \u2264 \u2191(f m)\n\u22a2 \u2211' (b : \u2115), 2 ^ b * \u2191(f (2 ^ b)) = \u22a4\n[PROOFSTEP]\nsimpa [h, ENNReal.add_eq_top] using ENNReal.le_tsum_condensed hf\n[GOAL]\nf : \u2115 \u2192 \u211d\nh_nonneg : \u2200 (n : \u2115), 0 \u2264 f n\nh_mono : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 (Summable fun k => 2 ^ k * f (2 ^ k)) \u2194 Summable f\n[PROOFSTEP]\nlift f to \u2115 \u2192 \u211d\u22650 using h_nonneg\n[GOAL]\ncase intro\nf : \u2115 \u2192 \u211d\u22650\nh_mono : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 (fun i => \u2191(f i)) n \u2264 (fun i => \u2191(f i)) m\n\u22a2 (Summable fun k => 2 ^ k * (fun i => \u2191(f i)) (2 ^ k)) \u2194 Summable fun i => \u2191(f i)\n[PROOFSTEP]\nsimp only [NNReal.coe_le_coe] at *\n[GOAL]\ncase intro\nf : \u2115 \u2192 \u211d\u22650\nh_mono : \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 f n \u2264 f m\n\u22a2 (Summable fun k => 2 ^ k * \u2191(f (2 ^ k))) \u2194 Summable fun i => \u2191(f i)\n[PROOFSTEP]\nexact_mod_cast NNReal.summable_condensed_iff h_mono\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\ncases' le_or_lt 0 p with hp hp\n[GOAL]\ncase inl\np : \u211d\nhp : 0 \u2264 p\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nrw [\u2190 summable_condensed_iff_of_nonneg]\n[GOAL]\ncase inl\np : \u211d\nhp : 0 \u2264 p\n\u22a2 (Summable fun k => 2 ^ k * (\u2191(2 ^ k) ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nsimp_rw [Nat.cast_pow, Nat.cast_two, \u2190 rpow_nat_cast, \u2190 rpow_mul zero_lt_two.le, mul_comm _ p, rpow_mul zero_lt_two.le,\n  rpow_nat_cast, \u2190 inv_pow, \u2190 mul_pow, summable_geometric_iff_norm_lt_1]\n[GOAL]\ncase inl\np : \u211d\nhp : 0 \u2264 p\n\u22a2 \u20162 * (2 ^ p)\u207b\u00b9\u2016 < 1 \u2194 1 < p\n[PROOFSTEP]\nnth_rw 1 [\u2190 rpow_one 2]\n[GOAL]\ncase inl\np : \u211d\nhp : 0 \u2264 p\n\u22a2 \u20162 ^ 1 * (2 ^ p)\u207b\u00b9\u2016 < 1 \u2194 1 < p\n[PROOFSTEP]\nrw [\u2190 division_def, \u2190 rpow_sub zero_lt_two, norm_eq_abs, abs_of_pos (rpow_pos_of_pos zero_lt_two _),\n  rpow_lt_one_iff zero_lt_two.le]\n[GOAL]\ncase inl\np : \u211d\nhp : 0 \u2264 p\n\u22a2 2 = 0 \u2227 1 - p \u2260 0 \u2228 1 < 2 \u2227 1 - p < 0 \u2228 2 < 1 \u2227 0 < 1 - p \u2194 1 < p\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inl.h_nonneg\np : \u211d\nhp : 0 \u2264 p\n\u22a2 \u2200 (n : \u2115), 0 \u2264 (\u2191n ^ p)\u207b\u00b9\n[PROOFSTEP]\nintro n\n[GOAL]\ncase inl.h_nonneg\np : \u211d\nhp : 0 \u2264 p\nn : \u2115\n\u22a2 0 \u2264 (\u2191n ^ p)\u207b\u00b9\n[PROOFSTEP]\nexact inv_nonneg.2 (rpow_nonneg_of_nonneg n.cast_nonneg _)\n[GOAL]\ncase inl.h_mono\np : \u211d\nhp : 0 \u2264 p\n\u22a2 \u2200 \u2983m n : \u2115\u2984, 0 < m \u2192 m \u2264 n \u2192 (\u2191n ^ p)\u207b\u00b9 \u2264 (\u2191m ^ p)\u207b\u00b9\n[PROOFSTEP]\nintro m n hm hmn\n[GOAL]\ncase inl.h_mono\np : \u211d\nhp : 0 \u2264 p\nm n : \u2115\nhm : 0 < m\nhmn : m \u2264 n\n\u22a2 (\u2191n ^ p)\u207b\u00b9 \u2264 (\u2191m ^ p)\u207b\u00b9\n[PROOFSTEP]\nexact\n  inv_le_inv_of_le (rpow_pos_of_pos (Nat.cast_pos.2 hm) _)\n    (rpow_le_rpow m.cast_nonneg (Nat.cast_le.2 hmn) hp)\n      -- If `p < 0`, then `1 / n ^ p` tends to infinity, thus the series diverges.\n[GOAL]\ncase inr\np : \u211d\nhp : p < 0\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nsuffices \u00acSummable (fun n => ((n : \u211d) ^ p)\u207b\u00b9 : \u2115 \u2192 \u211d)\n  by\n  have : \u00ac1 < p := fun hp\u2081 => hp.not_le (zero_le_one.trans hp\u2081.le)\n  simpa only [this, iff_false]\n[GOAL]\np : \u211d\nhp : p < 0\nthis : \u00acSummable fun n => (\u2191n ^ p)\u207b\u00b9\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nhave : \u00ac1 < p := fun hp\u2081 => hp.not_le (zero_le_one.trans hp\u2081.le)\n[GOAL]\np : \u211d\nhp : p < 0\nthis\u271d : \u00acSummable fun n => (\u2191n ^ p)\u207b\u00b9\nthis : \u00ac1 < p\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nsimpa only [this, iff_false]\n[GOAL]\ncase inr\np : \u211d\nhp : p < 0\n\u22a2 \u00acSummable fun n => (\u2191n ^ p)\u207b\u00b9\n[PROOFSTEP]\nintro h\n[GOAL]\ncase inr\np : \u211d\nhp : p < 0\nh : Summable fun n => (\u2191n ^ p)\u207b\u00b9\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8k : \u2115, hk\u2081 : ((k : \u211d) ^ p)\u207b\u00b9 < 1, hk\u2080 : k \u2260 0\u27e9 :=\n  ((h.tendsto_cofinite_zero.eventually (gt_mem_nhds zero_lt_one)).and (eventually_cofinite_ne 0)).exists\n[GOAL]\ncase inr.intro.intro\np : \u211d\nhp : p < 0\nh : Summable fun n => (\u2191n ^ p)\u207b\u00b9\nk : \u2115\nhk\u2081 : (\u2191k ^ p)\u207b\u00b9 < 1\nhk\u2080 : k \u2260 0\n\u22a2 False\n[PROOFSTEP]\napply hk\u2080\n[GOAL]\ncase inr.intro.intro\np : \u211d\nhp : p < 0\nh : Summable fun n => (\u2191n ^ p)\u207b\u00b9\nk : \u2115\nhk\u2081 : (\u2191k ^ p)\u207b\u00b9 < 1\nhk\u2080 : k \u2260 0\n\u22a2 k = 0\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero, \u2190 @Nat.cast_pos \u211d] at hk\u2080 \n[GOAL]\ncase inr.intro.intro\np : \u211d\nhp : p < 0\nh : Summable fun n => (\u2191n ^ p)\u207b\u00b9\nk : \u2115\nhk\u2081 : (\u2191k ^ p)\u207b\u00b9 < 1\nhk\u2080 : 0 < \u2191k\n\u22a2 k = 0\n[PROOFSTEP]\nsimpa [inv_lt_one_iff_of_pos (rpow_pos_of_pos hk\u2080 _), one_lt_rpow_iff_of_pos hk\u2080, hp, hp.not_lt, hk\u2080] using hk\u2081\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => \u2191n ^ p) \u2194 p < -1\n[PROOFSTEP]\nrcases neg_surjective p with \u27e8p, rfl\u27e9\n[GOAL]\ncase intro\np : \u211d\n\u22a2 (Summable fun n => \u2191n ^ (-p)) \u2194 -p < -1\n[PROOFSTEP]\nsimp [rpow_neg]\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => 1 / \u2191n ^ p) \u2194 1 < p\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nsimp only [\u2190 rpow_nat_cast, Real.summable_nat_rpow_inv, Nat.one_lt_cast]\n[GOAL]\np : \u2115\n\u22a2 (Summable fun n => 1 / \u2191n ^ p) \u2194 1 < p\n[PROOFSTEP]\nsimp only [one_div, Real.summable_nat_pow_inv]\n[GOAL]\np : \u2115\n\u22a2 (Summable fun n => 1 / \u2191n ^ p) \u2194 1 < p\n[PROOFSTEP]\nrefine'\n  \u27e8fun h => Real.summable_one_div_nat_pow.mp (h.comp_injective Nat.cast_injective), fun h =>\n    summable_int_of_summable_nat (Real.summable_one_div_nat_pow.mpr h)\n      (((Real.summable_one_div_nat_pow.mpr h).mul_left <| 1 / (-1 : \u211d) ^ p).congr fun n => _)\u27e9\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n\u22a2 1 / (-1) ^ p * (1 / \u2191n ^ p) = 1 / \u2191(-\u2191n) ^ p\n[PROOFSTEP]\nconv_rhs => rw [Int.cast_neg, neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / \u2191(-\u2191n) ^ p\n[PROOFSTEP]\nrw [Int.cast_neg, neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / \u2191(-\u2191n) ^ p\n[PROOFSTEP]\nrw [Int.cast_neg, neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / \u2191(-\u2191n) ^ p\n[PROOFSTEP]\nrw [Int.cast_neg, neg_eq_neg_one_mul, mul_pow, \u2190 div_div]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n\u22a2 1 / (-1) ^ p * (1 / \u2191n ^ p) = 1 / (-1) ^ p / \u2191\u2191n ^ p\n[PROOFSTEP]\nconv_lhs => rw [mul_div, mul_one]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / (-1) ^ p * (1 / \u2191n ^ p)\n[PROOFSTEP]\nrw [mul_div, mul_one]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / (-1) ^ p * (1 / \u2191n ^ p)\n[PROOFSTEP]\nrw [mul_div, mul_one]\n[GOAL]\np : \u2115\nh : 1 < p\nn : \u2115\n| 1 / (-1) ^ p * (1 / \u2191n ^ p)\n[PROOFSTEP]\nrw [mul_div, mul_one]\n[GOAL]\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191n| ^ (-b)\n[PROOFSTEP]\napply summable_int_of_summable_nat\n[GOAL]\ncase hp\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\ncase hn b : \u211d hb : 1 < b \u22a2 Summable fun n => |\u2191(-\u2191n)| ^ (-b)\n[PROOFSTEP]\non_goal 2 => simp_rw [Int.cast_neg, abs_neg]\n[GOAL]\ncase hp\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\ncase hn b : \u211d hb : 1 < b \u22a2 Summable fun n => |\u2191(-\u2191n)| ^ (-b)\n[PROOFSTEP]\non_goal 2 => simp_rw [Int.cast_neg, abs_neg]\n[GOAL]\ncase hn\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191(-\u2191n)| ^ (-b)\n[PROOFSTEP]\nsimp_rw [Int.cast_neg, abs_neg]\n[GOAL]\ncase hp\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\ncase hn b : \u211d hb : 1 < b \u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\n[PROOFSTEP]\nall_goals\n  simp_rw [Int.cast_ofNat, fun n : \u2115 => abs_of_nonneg (n.cast_nonneg : 0 \u2264 (n : \u211d))]\n  rwa [Real.summable_nat_rpow, neg_lt_neg_iff]\n[GOAL]\ncase hp\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\n[PROOFSTEP]\nsimp_rw [Int.cast_ofNat, fun n : \u2115 => abs_of_nonneg (n.cast_nonneg : 0 \u2264 (n : \u211d))]\n[GOAL]\ncase hp\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => \u2191n ^ (-b)\n[PROOFSTEP]\nrwa [Real.summable_nat_rpow, neg_lt_neg_iff]\n[GOAL]\ncase hn\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => |\u2191\u2191n| ^ (-b)\n[PROOFSTEP]\nsimp_rw [Int.cast_ofNat, fun n : \u2115 => abs_of_nonneg (n.cast_nonneg : 0 \u2264 (n : \u211d))]\n[GOAL]\ncase hn\nb : \u211d\nhb : 1 < b\n\u22a2 Summable fun n => \u2191n ^ (-b)\n[PROOFSTEP]\nrwa [Real.summable_nat_rpow, neg_lt_neg_iff]\n[GOAL]\n\u22a2 \u00acSummable fun n => (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nhave : \u00acSummable (fun n => ((n : \u211d) ^ 1)\u207b\u00b9 : \u2115 \u2192 \u211d) := mt (Real.summable_nat_pow_inv (p := 1)).1 (lt_irrefl 1)\n[GOAL]\nthis : \u00acSummable fun n => (\u2191n ^ 1)\u207b\u00b9\n\u22a2 \u00acSummable fun n => (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u22a2 \u00acSummable fun n => 1 / \u2191n\n[PROOFSTEP]\nsimpa only [inv_eq_one_div] using Real.not_summable_nat_cast_inv\n[GOAL]\n\u22a2 Tendsto (fun n => \u2211 i in Finset.range n, 1 / (\u2191i + 1)) atTop atTop\n[PROOFSTEP]\nrw [\u2190 not_summable_iff_tendsto_nat_atTop_of_nonneg]\n[GOAL]\n\u22a2 \u00acSummable fun i => 1 / (\u2191i + 1)\n[PROOFSTEP]\nexact_mod_cast mt (_root_.summable_nat_add_iff 1).1 Real.not_summable_one_div_nat_cast\n[GOAL]\n\u22a2 \u2200 (n : \u2115), 0 \u2264 1 / (\u2191n + 1)\n[PROOFSTEP]\nexact fun i => div_nonneg zero_le_one i.cast_add_one_pos.le\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => (\u2191n ^ p)\u207b\u00b9) \u2194 1 < p\n[PROOFSTEP]\nsimp [\u2190 NNReal.summable_coe]\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => \u2191n ^ p) \u2194 p < -1\n[PROOFSTEP]\nsimp [\u2190 NNReal.summable_coe]\n[GOAL]\np : \u211d\n\u22a2 (Summable fun n => 1 / \u2191n ^ p) \u2194 1 < p\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nhk : k \u2260 0\nh : k \u2264 n\n\u22a2 \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrefine' Nat.le_induction _ _ n h\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nhk : k \u2260 0\nh : k \u2264 n\n\u22a2 \u2211 i in Ioc k k, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191k)\u207b\u00b9\n[PROOFSTEP]\nsimp only [Ioc_self, sum_empty, sub_self, le_refl]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nhk : k \u2260 0\nh : k \u2264 n\n\u22a2 \u2200 (n : \u2115),\n    k \u2264 n \u2192 \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u2192 \u2211 i in Ioc k (n + 1), (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9\n[PROOFSTEP]\nintro n hn IH\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 \u2211 i in Ioc k (n + 1), (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9\n[PROOFSTEP]\nrw [sum_Ioc_succ_top hn]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 \u2211 k in Ioc k n, (\u2191k ^ 2)\u207b\u00b9 + (\u2191(n + 1) ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9\n[PROOFSTEP]\napply (add_le_add IH le_rfl).trans\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 + (\u2191(n + 1) ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, add_assoc, Nat.cast_add, Nat.cast_one, le_add_neg_iff_add_le, add_le_iff_nonpos_right,\n  neg_add_le_iff_le_add, add_zero]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nhave A : 0 < (n : \u03b1) := by simpa using hk.bot_lt.trans_le hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nsimpa using hk.bot_lt.trans_le hn\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\n\u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nhave B : 0 < (n : \u03b1) + 1 := by linarith\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\n\u22a2 0 < \u2191n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nfield_simp [B.ne']\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 (\u2191n + 1 + (\u2191n + 1) ^ 2) / ((\u2191n + 1) ^ 2 * (\u2191n + 1)) \u2264 1 / \u2191n\n[PROOFSTEP]\nrw [div_le_div_iff _ A, \u2190 sub_nonneg]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 0 \u2264 1 * ((\u2191n + 1) ^ 2 * (\u2191n + 1)) - (\u2191n + 1 + (\u2191n + 1) ^ 2) * \u2191n\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 0 \u2264 1 + \u2191n\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 0 \u2264 \u2191n + 1\n[PROOFSTEP]\nexact B.le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n\u271d : \u2115\nhk : k \u2260 0\nh : k \u2264 n\u271d\nn : \u2115\nhn : k \u2264 n\nIH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9\nA : 0 < \u2191n\nB : 0 < \u2191n + 1\n\u22a2 0 < (\u2191n + 1) ^ 2 * (\u2191n + 1)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 \u2211 i in Ioo k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 \u2211 i in Ioc k (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9\n[PROOFSTEP]\napply sum_le_sum_of_subset_of_nonneg\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 Ioo k n \u2286 Ioc k (max (k + 1) n)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n x : \u2115\nhx : x \u2208 Ioo k n\n\u22a2 x \u2208 Ioc k (max (k + 1) n)\n[PROOFSTEP]\nsimp only [mem_Ioo] at hx \n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n x : \u2115\nhx : k < x \u2227 x < n\n\u22a2 x \u2208 Ioc k (max (k + 1) n)\n[PROOFSTEP]\nsimp only [hx, hx.2.le, mem_Ioc, le_max_iff, or_true_iff, and_self_iff]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 \u2200 (i : \u2115), i \u2208 Ioc k (max (k + 1) n) \u2192 \u00aci \u2208 Ioo k n \u2192 0 \u2264 (\u2191i ^ 2)\u207b\u00b9\n[PROOFSTEP]\nintro i _hi _hident\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n i : \u2115\n_hi : i \u2208 Ioc k (max (k + 1) n)\n_hident : \u00aci \u2208 Ioo k n\n\u22a2 0 \u2264 (\u2191i ^ 2)\u207b\u00b9\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 \u2211 i in Ioc k (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9 \u2264 ((\u2191k + 1) ^ 2)\u207b\u00b9 + \u2211 i in Ioc (Nat.succ k) (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 Nat.Icc_succ_left, \u2190 Nat.Ico_succ_right, sum_eq_sum_Ico_succ_bot]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 (\u2191(Nat.succ k) ^ 2)\u207b\u00b9 + \u2211 k in Ico (Nat.succ k + 1) (Nat.succ (max (k + 1) n)), (\u2191k ^ 2)\u207b\u00b9 \u2264\n    ((\u2191k + 1) ^ 2)\u207b\u00b9 + \u2211 i in Ioc (Nat.succ k) (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9\ncase hab \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n : \u2115 \u22a2 Nat.succ k < Nat.succ (max (k + 1) n)\n[PROOFSTEP]\nswap\n[GOAL]\ncase hab\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 Nat.succ k < Nat.succ (max (k + 1) n)\n[PROOFSTEP]\nexact Nat.succ_lt_succ ((Nat.lt_succ_self k).trans_le (le_max_left _ _))\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 (\u2191(Nat.succ k) ^ 2)\u207b\u00b9 + \u2211 k in Ico (Nat.succ k + 1) (Nat.succ (max (k + 1) n)), (\u2191k ^ 2)\u207b\u00b9 \u2264\n    ((\u2191k + 1) ^ 2)\u207b\u00b9 + \u2211 i in Ioc (Nat.succ k) (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9\n[PROOFSTEP]\nrw [Nat.Ico_succ_right, Nat.Icc_succ_left, Nat.cast_succ]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 ((\u2191k + 1) ^ 2)\u207b\u00b9 + \u2211 i in Ioc (Nat.succ k) (max (k + 1) n), (\u2191i ^ 2)\u207b\u00b9 \u2264 ((\u2191k + 1) ^ 2)\u207b\u00b9 + (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nrefine' add_le_add le_rfl ((sum_Ioc_inv_sq_le_sub _ (le_max_left _ _)).trans _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 Nat.succ k \u2260 0\n[PROOFSTEP]\nsimp only [Ne.def, Nat.succ_ne_zero, not_false_iff]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 (\u2191(Nat.succ k))\u207b\u00b9 - (\u2191(max (Nat.succ k) n))\u207b\u00b9 \u2264 (\u2191k + 1)\u207b\u00b9\n[PROOFSTEP]\nsimp only [Nat.cast_succ, one_div, sub_le_self_iff, inv_nonneg, Nat.cast_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 ((\u2191k + 1) ^ 2)\u207b\u00b9 + (\u2191k + 1)\u207b\u00b9 \u2264 1 / (\u2191k + 1) + 1 / (\u2191k + 1)\n[PROOFSTEP]\nhave A : (1 : \u03b1) \u2264 k + 1 := by simp only [le_add_iff_nonneg_left, Nat.cast_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 1 \u2264 \u2191k + 1\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_left, Nat.cast_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nA : 1 \u2264 \u2191k + 1\n\u22a2 ((\u2191k + 1) ^ 2)\u207b\u00b9 + (\u2191k + 1)\u207b\u00b9 \u2264 1 / (\u2191k + 1) + 1 / (\u2191k + 1)\n[PROOFSTEP]\nsimp_rw [\u2190 one_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nA : 1 \u2264 \u2191k + 1\n\u22a2 1 / (\u2191k + 1) ^ 2 + 1 / (\u2191k + 1) \u2264 1 / (\u2191k + 1) + 1 / (\u2191k + 1)\n[PROOFSTEP]\napply add_le_add_right\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nA : 1 \u2264 \u2191k + 1\n\u22a2 1 / (\u2191k + 1) ^ 2 \u2264 1 / (\u2191k + 1)\n[PROOFSTEP]\nrefine' div_le_div zero_le_one le_rfl (zero_lt_one.trans_le A) _\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\nA : 1 \u2264 \u2191k + 1\n\u22a2 \u2191k + 1 \u2264 (\u2191k + 1) ^ 2\n[PROOFSTEP]\nsimpa using pow_le_pow A one_le_two\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedField \u03b1\nk n : \u2115\n\u22a2 1 / (\u2191k + 1) + 1 / (\u2191k + 1) = 2 / (\u2191k + 1)\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.Analysis.PSeries", "llama_tokens": 17182, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388125473629, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.5369850855595202}}
{"text": "[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nrcases coercive with \u27e8C, C_ge_0, coercivity\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nrefine' \u27e8C, C_ge_0, _\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nintro v\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\n\u22a2 C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nby_cases h : 0 < \u2016v\u2016\n[GOAL]\ncase pos\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\nh : 0 < \u2016v\u2016\n\u22a2 C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nrefine' (mul_le_mul_right h).mp _\n[GOAL]\ncase pos\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\nh : 0 < \u2016v\u2016\n\u22a2 C * \u2016v\u2016 * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016 * \u2016v\u2016\n[PROOFSTEP]\ncalc\n  C * \u2016v\u2016 * \u2016v\u2016 \u2264 B v v := coercivity v\n  _ = \u27eaB\u266f v, v\u27eb_\u211d := (continuousLinearMapOfBilin_apply B v v).symm\n  _ \u2264 \u2016B\u266f v\u2016 * \u2016v\u2016 := real_inner_le_norm (B\u266f v) v\n[GOAL]\ncase neg\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\nh : \u00ac0 < \u2016v\u2016\n\u22a2 C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nhave : v = 0 := by simpa using h\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\nh : \u00ac0 < \u2016v\u2016\n\u22a2 v = 0\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase neg\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nC : \u211d\nC_ge_0 : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nv : V\nh : \u00ac0 < \u2016v\u2016\nthis : v = 0\n\u22a2 C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 \u2203 C, 0 < C \u2227 AntilipschitzWith C \u2191(continuousLinearMapOfBilin B)\n[PROOFSTEP]\nrcases coercive.bounded_below with \u27e8C, C_pos, below_bound\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nC : \u211d\nC_pos : 0 < C\nbelow_bound : \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n\u22a2 \u2203 C, 0 < C \u2227 AntilipschitzWith C \u2191(continuousLinearMapOfBilin B)\n[PROOFSTEP]\nrefine' \u27e8C\u207b\u00b9.toNNReal, Real.toNNReal_pos.mpr (inv_pos.mpr C_pos), _\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nC : \u211d\nC_pos : 0 < C\nbelow_bound : \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n\u22a2 AntilipschitzWith (Real.toNNReal C\u207b\u00b9) \u2191(continuousLinearMapOfBilin B)\n[PROOFSTEP]\nrefine' ContinuousLinearMap.antilipschitz_of_bound B\u266f _\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nC : \u211d\nC_pos : 0 < C\nbelow_bound : \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n\u22a2 \u2200 (x : V), \u2016x\u2016 \u2264 \u2191(Real.toNNReal C\u207b\u00b9) * \u2016\u2191(continuousLinearMapOfBilin B) x\u2016\n[PROOFSTEP]\nsimp_rw [Real.coe_toNNReal', max_eq_left_of_lt (inv_pos.mpr C_pos), \u2190 inv_mul_le_iff (inv_pos.mpr C_pos)]\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nC : \u211d\nC_pos : 0 < C\nbelow_bound : \u2200 (v : V), C * \u2016v\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) v\u2016\n\u22a2 \u2200 (x : V), C\u207b\u00b9\u207b\u00b9 * \u2016x\u2016 \u2264 \u2016\u2191(continuousLinearMapOfBilin B) x\u2016\n[PROOFSTEP]\nsimpa using below_bound\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 ker (continuousLinearMapOfBilin B) = \u22a5\n[PROOFSTEP]\nrw [LinearMapClass.ker_eq_bot]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)\n[PROOFSTEP]\nrcases coercive.antilipschitz with \u27e8_, _, antilipschitz\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nw\u271d : \u211d\u22650\nleft\u271d : 0 < w\u271d\nantilipschitz : AntilipschitzWith w\u271d \u2191(continuousLinearMapOfBilin B)\n\u22a2 Function.Injective \u2191(continuousLinearMapOfBilin B)\n[PROOFSTEP]\nexact antilipschitz.injective\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 IsClosed \u2191(range (continuousLinearMapOfBilin B))\n[PROOFSTEP]\nrcases coercive.antilipschitz with \u27e8_, _, antilipschitz\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nw\u271d : \u211d\u22650\nleft\u271d : 0 < w\u271d\nantilipschitz : AntilipschitzWith w\u271d \u2191(continuousLinearMapOfBilin B)\n\u22a2 IsClosed \u2191(range (continuousLinearMapOfBilin B))\n[PROOFSTEP]\nexact antilipschitz.isClosed_range B\u266f.uniformContinuous\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\n\u22a2 range (continuousLinearMapOfBilin B) = \u22a4\n[PROOFSTEP]\nhaveI := coercive.closed_range.completeSpace_coe\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\n\u22a2 range (continuousLinearMapOfBilin B) = \u22a4\n[PROOFSTEP]\nrw [\u2190 (range B\u266f).orthogonal_orthogonal]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\n\u22a2 (range (continuousLinearMapOfBilin B))\u15ee\u15ee = \u22a4\n[PROOFSTEP]\nrw [Submodule.eq_top_iff']\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\n\u22a2 \u2200 (x : V), x \u2208 (range (continuousLinearMapOfBilin B))\u15ee\u15ee\n[PROOFSTEP]\nintro v w mem_w_orthogonal\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\ncoercive : IsCoercive B\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\n\u22a2 inner w v = 0\n[PROOFSTEP]\nrcases coercive with \u27e8C, C_pos, coercivity\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 inner w v = 0\n[PROOFSTEP]\nobtain rfl : w = 0 :=\n  by\n  rw [\u2190 norm_eq_zero, \u2190 mul_self_eq_zero, \u2190 mul_right_inj' C_pos.ne', mul_zero, \u2190 mul_assoc]\n  apply le_antisymm\n  \u00b7\n    calc\n      C * \u2016w\u2016 * \u2016w\u2016 \u2264 B w w := coercivity w\n      _ = \u27eaB\u266f w, w\u27eb_\u211d := (continuousLinearMapOfBilin_apply B w w).symm\n      _ = 0 := mem_w_orthogonal _ \u27e8w, rfl\u27e9\n  \u00b7 exact mul_nonneg (mul_nonneg C_pos.le (norm_nonneg w)) (norm_nonneg w)\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 w = 0\n[PROOFSTEP]\nrw [\u2190 norm_eq_zero, \u2190 mul_self_eq_zero, \u2190 mul_right_inj' C_pos.ne', mul_zero, \u2190 mul_assoc]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 C * \u2016w\u2016 * \u2016w\u2016 = 0\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 C * \u2016w\u2016 * \u2016w\u2016 \u2264 0\n[PROOFSTEP]\ncalc\n  C * \u2016w\u2016 * \u2016w\u2016 \u2264 B w w := coercivity w\n  _ = \u27eaB\u266f w, w\u27eb_\u211d := (continuousLinearMapOfBilin_apply B w w).symm\n  _ = 0 := mem_w_orthogonal _ \u27e8w, rfl\u27e9\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv w : V\nmem_w_orthogonal : w \u2208 (range (continuousLinearMapOfBilin B))\u15ee\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\n\u22a2 0 \u2264 C * \u2016w\u2016 * \u2016w\u2016\n[PROOFSTEP]\nexact mul_nonneg (mul_nonneg C_pos.le (norm_nonneg w)) (norm_nonneg w)\n[GOAL]\ncase intro.intro\nV : Type u\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : CompleteSpace V\nB : V \u2192L[\u211d] V \u2192L[\u211d] \u211d\nthis : CompleteSpace \u2191\u2191(range (continuousLinearMapOfBilin B))\nv : V\nC : \u211d\nC_pos : 0 < C\ncoercivity : \u2200 (u : V), C * \u2016u\u2016 * \u2016u\u2016 \u2264 \u2191(\u2191B u) u\nmem_w_orthogonal : 0 \u2208 (range (continuousLinearMapOfBilin B))\u15ee\n\u22a2 inner 0 v = 0\n[PROOFSTEP]\nexact inner_zero_left _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.LaxMilgram", "llama_tokens": 5406, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933315126792, "lm_q2_score": 0.6548947223065754, "lm_q1q2_score": 0.536943815662009}}
{"text": "[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nrefine'\n  IsLittleO.trans_isBigO (isLittleO_iff.2 fun \u03b5 \u03b5pos => _)\n    (isBigO_const_mul_self ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) _ _)\n      -- consider a ball of radius `\u03b4` around `x` in which the Taylor approximation for `f''` is\n        -- good up to `\u03b4`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2016f (x + x_1 \u2022 v + x_1 \u2022 w) - f (x + x_1 \u2022 v) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n          (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n      \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * x_1 ^ 2\u2016\n[PROOFSTEP]\nrw [HasFDerivWithinAt, HasFDerivAtFilter, isLittleO_iff] at hx \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2016f (x + x_1 \u2022 v + x_1 \u2022 w) - f (x + x_1 \u2022 v) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n          (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n      \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * x_1 ^ 2\u2016\n[PROOFSTEP]\nrcases Metric.mem_nhdsWithin_iff.1 (hx \u03b5pos) with \u27e8\u03b4, \u03b4pos, s\u03b4\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2016f (x + x_1 \u2022 v + x_1 \u2022 w) - f (x + x_1 \u2022 v) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n          (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n      \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * x_1 ^ 2\u2016\n[PROOFSTEP]\nhave E1 : \u2200\u1da0 h in \ud835\udcdd[>] (0 : \u211d), h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4 :=\n  by\n  have : Filter.Tendsto (fun h => h * (\u2016v\u2016 + \u2016w\u2016)) (\ud835\udcdd[>] (0 : \u211d)) (\ud835\udcdd (0 * (\u2016v\u2016 + \u2016w\u2016))) :=\n    (continuous_id.mul continuous_const).continuousWithinAt\n  apply (tendsto_order.1 this).2 \u03b4\n  simpa only [zero_mul] using \u03b4pos\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\n\u22a2 \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\n[PROOFSTEP]\nhave : Filter.Tendsto (fun h => h * (\u2016v\u2016 + \u2016w\u2016)) (\ud835\udcdd[>] (0 : \u211d)) (\ud835\udcdd (0 * (\u2016v\u2016 + \u2016w\u2016))) :=\n  (continuous_id.mul continuous_const).continuousWithinAt\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nthis : Filter.Tendsto (fun h => h * (\u2016v\u2016 + \u2016w\u2016)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2016v\u2016 + \u2016w\u2016)))\n\u22a2 \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\n[PROOFSTEP]\napply (tendsto_order.1 this).2 \u03b4\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nthis : Filter.Tendsto (fun h => h * (\u2016v\u2016 + \u2016w\u2016)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (0 * (\u2016v\u2016 + \u2016w\u2016)))\n\u22a2 \u03b4 > 0 * (\u2016v\u2016 + \u2016w\u2016)\n[PROOFSTEP]\nsimpa only [zero_mul] using \u03b4pos\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2016f (x + x_1 \u2022 v + x_1 \u2022 w) - f (x + x_1 \u2022 v) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n          (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n      \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * x_1 ^ 2\u2016\n[PROOFSTEP]\nhave E2 : \u2200\u1da0 h in \ud835\udcdd[>] (0 : \u211d), (h : \u211d) < 1 :=\n  mem_nhdsWithin_Ioi_iff_exists_Ioo_subset.2 \u27e8(1 : \u211d), by simp only [mem_Ioi, zero_lt_one], fun x hx => hx.2\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\n\u22a2 1 \u2208 Ioi 0\n[PROOFSTEP]\nsimp only [mem_Ioi, zero_lt_one]\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\n\u22a2 \u2200\u1da0 (x_1 : \u211d) in \ud835\udcdd[Ioi 0] 0,\n    \u2016f (x + x_1 \u2022 v + x_1 \u2022 w) - f (x + x_1 \u2022 v) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n          (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n      \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * x_1 ^ 2\u2016\n[PROOFSTEP]\nfilter_upwards [E1, E2, self_mem_nhdsWithin] with h h\u03b4 h_lt_1 hpos\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : h \u2208 Ioi 0\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nreplace hpos : 0 < h := hpos\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nhave xt_mem : \u2200 t \u2208 Icc (0 : \u211d) 1, x + h \u2022 v + (t * h) \u2022 w \u2208 interior s :=\n  by\n  intro t ht\n  have : x + h \u2022 v \u2208 interior s := s_conv.add_smul_mem_interior xs hv \u27e8hpos, h_lt_1.le\u27e9\n  rw [\u2190 smul_smul]\n  apply s_conv.interior.add_smul_mem this _ ht\n  rw [add_assoc] at hw \n  rw [add_assoc, \u2190 smul_add]\n  exact\n    s_conv.add_smul_mem_interior xs hw\n      \u27e8hpos, h_lt_1.le\u27e9\n        -- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the\n          -- quantity to be estimated. We will check that its derivative is given by an explicit\n          -- expression `g'`, that we can bound. Then the desired bound for `g 1 - g 0` follows from the\n          -- mean value inequality.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\n[PROOFSTEP]\nintro t ht\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\n[PROOFSTEP]\nhave : x + h \u2022 v \u2208 interior s := s_conv.add_smul_mem_interior xs hv \u27e8hpos, h_lt_1.le\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\nthis : x + h \u2022 v \u2208 interior s\n\u22a2 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\n[PROOFSTEP]\nrw [\u2190 smul_smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\nthis : x + h \u2022 v \u2208 interior s\n\u22a2 x + h \u2022 v + t \u2022 h \u2022 w \u2208 interior s\n[PROOFSTEP]\napply s_conv.interior.add_smul_mem this _ ht\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\nthis : x + h \u2022 v \u2208 interior s\n\u22a2 x + h \u2022 v + h \u2022 w \u2208 interior s\n[PROOFSTEP]\nrw [add_assoc] at hw \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + (v + w) \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\nthis : x + h \u2022 v \u2208 interior s\n\u22a2 x + h \u2022 v + h \u2022 w \u2208 interior s\n[PROOFSTEP]\nrw [add_assoc, \u2190 smul_add]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + (v + w) \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nt : \u211d\nht : t \u2208 Icc 0 1\nthis : x + h \u2022 v \u2208 interior s\n\u22a2 x + h \u2022 (v + w) \u2208 interior s\n[PROOFSTEP]\nexact\n  s_conv.add_smul_mem_interior xs hw\n    \u27e8hpos, h_lt_1.le\u27e9\n      -- define a function `g` on `[0,1]` (identified with `[v, v + w]`) such that `g 1 - g 0` is the\n        -- quantity to be estimated. We will check that its derivative is given by an explicit\n        -- expression `g'`, that we can bound. Then the desired bound for `g 1 - g 0` follows from the\n        -- mean value inequality.\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nlet g t := f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 f' x w - (t * h ^ 2) \u2022 f'' v w - ((t * h) ^ 2 / 2) \u2022 f'' w w\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nset g' := fun t => f' (x + h \u2022 v + (t * h) \u2022 w) (h \u2022 w) - h \u2022 f' x w - h ^ 2 \u2022 f'' v w - (t * h ^ 2) \u2022 f'' w w with hg'\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nhave g_deriv : \u2200 t \u2208 Icc (0 : \u211d) 1, HasDerivWithinAt g (g' t) (Icc 0 1) t :=\n  by\n  intro t ht\n  apply_rules [HasDerivWithinAt.sub, HasDerivWithinAt.add]\n  \u00b7 refine' (hf _ _).comp_hasDerivWithinAt _ _\n    \u00b7 exact xt_mem t ht\n    apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.const_add, HasDerivAt.smul_const, hasDerivAt_mul_const]\n  \u00b7 apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]\n  \u00b7 apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]\n  \u00b7 suffices H :\n      HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) \u2022 f'' w w)\n        ((((2 : \u2115) : \u211d) * (t * h) ^ (2 - 1) * (1 * h) / 2) \u2022 f'' w w) (Icc 0 1) t\n    \u00b7 convert H using 2\n      ring\n    apply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_id', HasDerivAt.pow,\n      HasDerivAt.mul_const]\n      -- check that `g'` is uniformly bounded, with a suitable bound `\u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h^2`.\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\n[PROOFSTEP]\nintro t ht\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt g (g' t) (Icc 0 1) t\n[PROOFSTEP]\napply_rules [HasDerivWithinAt.sub, HasDerivWithinAt.add]\n[GOAL]\ncase hf.hf.hf\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun x_1 => f (x + h \u2022 v + (x_1 * h) \u2022 w)) (\u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)) (Icc 0 1) t\n[PROOFSTEP]\nrefine' (hf _ _).comp_hasDerivWithinAt _ _\n[GOAL]\ncase hf.hf.hf.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\n[PROOFSTEP]\nexact xt_mem t ht\n[GOAL]\ncase hf.hf.hf.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun x_1 => x + h \u2022 v + (x_1 * h) \u2022 w) (h \u2022 w) (Icc 0 1) t\n[PROOFSTEP]\napply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.const_add, HasDerivAt.smul_const, hasDerivAt_mul_const]\n[GOAL]\ncase hf.hf.hg\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun x_1 => (x_1 * h) \u2022 \u2191(f' x) w) (h \u2022 \u2191(f' x) w) (Icc 0 1) t\n[PROOFSTEP]\napply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]\n[GOAL]\ncase hf.hg\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun x => (x * h ^ 2) \u2022 \u2191(\u2191f'' v) w) (h ^ 2 \u2022 \u2191(\u2191f'' v) w) (Icc 0 1) t\n[PROOFSTEP]\napply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_mul_const]\n[GOAL]\ncase hg\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun x => ((x * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) ((t * h ^ 2) \u2022 \u2191(\u2191f'' w) w) (Icc 0 1) t\n[PROOFSTEP]\nsuffices H :\n  HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) \u2022 f'' w w) ((((2 : \u2115) : \u211d) * (t * h) ^ (2 - 1) * (1 * h) / 2) \u2022 f'' w w)\n    (Icc 0 1) t\n[GOAL]\ncase hg\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\nH :\n  HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) ((\u21912 * (t * h) ^ (2 - 1) * (1 * h) / 2) \u2022 \u2191(\u2191f'' w) w)\n    (Icc 0 1) t\n\u22a2 HasDerivWithinAt (fun x => ((x * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) ((t * h ^ 2) \u2022 \u2191(\u2191f'' w) w) (Icc 0 1) t\n[PROOFSTEP]\nconvert H using 2\n[GOAL]\ncase h.e'_7.h.e'_5\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\nH :\n  HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) ((\u21912 * (t * h) ^ (2 - 1) * (1 * h) / 2) \u2022 \u2191(\u2191f'' w) w)\n    (Icc 0 1) t\n\u22a2 t * h ^ 2 = \u21912 * (t * h) ^ (2 - 1) * (1 * h) / 2\n[PROOFSTEP]\nring\n[GOAL]\ncase H\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nt : \u211d\nht : t \u2208 Icc 0 1\n\u22a2 HasDerivWithinAt (fun u => ((u * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) ((\u21912 * (t * h) ^ (2 - 1) * (1 * h) / 2) \u2022 \u2191(\u2191f'' w) w)\n    (Icc 0 1) t\n[PROOFSTEP]\napply_rules [HasDerivAt.hasDerivWithinAt, HasDerivAt.smul_const, hasDerivAt_id', HasDerivAt.pow, HasDerivAt.mul_const]\n  -- check that `g'` is uniformly bounded, with a suitable bound `\u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h^2`.\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nhave g'_bound : \u2200 t \u2208 Ico (0 : \u211d) 1, \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2 :=\n  by\n  intro t ht\n  have I : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016) :=\n    calc\n      \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 \u2016h \u2022 v\u2016 + \u2016(t * h) \u2022 w\u2016 := norm_add_le _ _\n      _ = h * \u2016v\u2016 + t * (h * \u2016w\u2016) := by\n        simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left, mul_assoc]\n      _ \u2264 h * \u2016v\u2016 + 1 * (h * \u2016w\u2016) := by gcongr; exact ht.2.le\n      _ = h * (\u2016v\u2016 + \u2016w\u2016) := by ring\n  calc\n    \u2016g' t\u2016 = \u2016(f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - f'' (h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)\u2016 :=\n      by\n      rw [hg']\n      have : h * (t * h) = t * (h * h) := by ring\n      simp only [ContinuousLinearMap.coe_sub', ContinuousLinearMap.map_add, pow_two, ContinuousLinearMap.add_apply,\n        Pi.smul_apply, smul_sub, smul_add, smul_smul, \u2190 sub_sub, ContinuousLinearMap.coe_smul', Pi.sub_apply,\n        ContinuousLinearMap.map_smul, this]\n    _ \u2264 \u2016f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - f'' (h \u2022 v + (t * h) \u2022 w)\u2016 * \u2016h \u2022 w\u2016 :=\n      (ContinuousLinearMap.le_op_norm _ _)\n    _ \u2264 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016 * \u2016h \u2022 w\u2016 :=\n      by\n      apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n      have H : x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4 \u2229 interior s :=\n        by\n        refine' \u27e8_, xt_mem t \u27e8ht.1, ht.2.le\u27e9\u27e9\n        rw [add_assoc, add_mem_ball_iff_norm]\n        exact I.trans_lt h\u03b4\n      simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel'] using s\u03b4 H\n    _ \u2264 \u03b5 * (\u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016) * \u2016h \u2022 w\u2016 := by\n      gcongr\n      apply (norm_add_le _ _).trans\n      gcongr\n      simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]\n      exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le\n    _ = \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2 := by simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le];\n      ring\n        -- conclude using the mean value inequality\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\n\u22a2 \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nintro t ht\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\n\u22a2 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nhave I : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016) :=\n  calc\n    \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 \u2016h \u2022 v\u2016 + \u2016(t * h) \u2022 w\u2016 := norm_add_le _ _\n    _ = h * \u2016v\u2016 + t * (h * \u2016w\u2016) := by\n      simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left, mul_assoc]\n    _ \u2264 h * \u2016v\u2016 + 1 * (h * \u2016w\u2016) := by gcongr; exact ht.2.le\n    _ = h * (\u2016v\u2016 + \u2016w\u2016) := by ring\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\n\u22a2 \u2016h \u2022 v\u2016 + \u2016(t * h) \u2022 w\u2016 = h * \u2016v\u2016 + t * (h * \u2016w\u2016)\n[PROOFSTEP]\nsimp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left, mul_assoc]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\n\u22a2 h * \u2016v\u2016 + t * (h * \u2016w\u2016) \u2264 h * \u2016v\u2016 + 1 * (h * \u2016w\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\n\u22a2 t \u2264 1\n[PROOFSTEP]\nexact ht.2.le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\n\u22a2 h * \u2016v\u2016 + 1 * (h * \u2016w\u2016) = h * (\u2016v\u2016 + \u2016w\u2016)\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\ncalc\n  \u2016g' t\u2016 = \u2016(f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - f'' (h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)\u2016 :=\n    by\n    rw [hg']\n    have : h * (t * h) = t * (h * h) := by ring\n    simp only [ContinuousLinearMap.coe_sub', ContinuousLinearMap.map_add, pow_two, ContinuousLinearMap.add_apply,\n      Pi.smul_apply, smul_sub, smul_add, smul_smul, \u2190 sub_sub, ContinuousLinearMap.coe_smul', Pi.sub_apply,\n      ContinuousLinearMap.map_smul, this]\n  _ \u2264 \u2016f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - f'' (h \u2022 v + (t * h) \u2022 w)\u2016 * \u2016h \u2022 w\u2016 :=\n    (ContinuousLinearMap.le_op_norm _ _)\n  _ \u2264 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016 * \u2016h \u2022 w\u2016 :=\n    by\n    apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n    have H : x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4 \u2229 interior s :=\n      by\n      refine' \u27e8_, xt_mem t \u27e8ht.1, ht.2.le\u27e9\u27e9\n      rw [add_assoc, add_mem_ball_iff_norm]\n      exact I.trans_lt h\u03b4\n    simpa only [mem_setOf_eq, add_assoc x, add_sub_cancel'] using s\u03b4 H\n  _ \u2264 \u03b5 * (\u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016) * \u2016h \u2022 w\u2016 := by\n    gcongr\n    apply (norm_add_le _ _).trans\n    gcongr\n    simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]\n    exact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le\n  _ = \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2 := by simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le];\n    ring\n      -- conclude using the mean value inequality\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016g' t\u2016 = \u2016\u2191(f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)\u2016\n[PROOFSTEP]\nrw [hg']\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016(fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w)\n        t\u2016 =\n    \u2016\u2191(f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)\u2016\n[PROOFSTEP]\nhave : h * (t * h) = t * (h * h) := by ring\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 h * (t * h) = t * (h * h)\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\nthis : h * (t * h) = t * (h * h)\n\u22a2 \u2016(fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w)\n        t\u2016 =\n    \u2016\u2191(f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)) (h \u2022 w)\u2016\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.coe_sub', ContinuousLinearMap.map_add, pow_two, ContinuousLinearMap.add_apply,\n  Pi.smul_apply, smul_sub, smul_add, smul_smul, \u2190 sub_sub, ContinuousLinearMap.coe_smul', Pi.sub_apply,\n  ContinuousLinearMap.map_smul, this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)\u2016 * \u2016h \u2022 w\u2016 \u2264 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016 * \u2016h \u2022 w\u2016\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)\u2016 \u2264 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016\n[PROOFSTEP]\nhave H : x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4 \u2229 interior s :=\n  by\n  refine' \u27e8_, xt_mem t \u27e8ht.1, ht.2.le\u27e9\u27e9\n  rw [add_assoc, add_mem_ball_iff_norm]\n  exact I.trans_lt h\u03b4\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4 \u2229 interior s\n[PROOFSTEP]\nrefine' \u27e8_, xt_mem t \u27e8ht.1, ht.2.le\u27e9\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4\n[PROOFSTEP]\nrw [add_assoc, add_mem_ball_iff_norm]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016h \u2022 v + (t * h) \u2022 w\u2016 < \u03b4\n[PROOFSTEP]\nexact I.trans_lt h\u03b4\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\nH : x + h \u2022 v + (t * h) \u2022 w \u2208 Metric.ball x \u03b4 \u2229 interior s\n\u22a2 \u2016f' (x + h \u2022 v + (t * h) \u2022 w) - f' x - \u2191f'' (h \u2022 v + (t * h) \u2022 w)\u2016 \u2264 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016\n[PROOFSTEP]\nsimpa only [mem_setOf_eq, add_assoc x, add_sub_cancel'] using s\u03b4 H\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u03b5 * \u2016h \u2022 v + (t * h) \u2022 w\u2016 * \u2016h \u2022 w\u2016 \u2264 \u03b5 * (\u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016) * \u2016h \u2022 w\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 \u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016\n[PROOFSTEP]\napply (norm_add_le _ _).trans\n[GOAL]\ncase h.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016h \u2022 v\u2016 + \u2016(t * h) \u2022 w\u2016 \u2264 \u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.h.bc\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u2016(t * h) \u2022 w\u2016 \u2264 \u2016h \u2022 w\u2016\n[PROOFSTEP]\nsimp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]\n[GOAL]\ncase h.h.bc\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 t * (h * \u2016w\u2016) \u2264 h * \u2016w\u2016\n[PROOFSTEP]\nexact mul_le_of_le_one_left (mul_nonneg hpos.le (norm_nonneg _)) ht.2.le\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u03b5 * (\u2016h \u2022 v\u2016 + \u2016h \u2022 w\u2016) * \u2016h \u2022 w\u2016 = \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nsimp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, hpos.le]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\nt : \u211d\nht : t \u2208 Ico 0 1\nI : \u2016h \u2022 v + (t * h) \u2022 w\u2016 \u2264 h * (\u2016v\u2016 + \u2016w\u2016)\n\u22a2 \u03b5 * (h * \u2016v\u2016 + h * \u2016w\u2016) * (h * \u2016w\u2016) = \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nring\n  -- conclude using the mean value inequality\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nhave I : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2 := by\n  simpa only [mul_one, sub_zero] using\n    norm_image_sub_le_of_norm_deriv_le_segment' g_deriv g'_bound 1 (right_mem_Icc.2 zero_le_one)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nsimpa only [mul_one, sub_zero] using\n  norm_image_sub_le_of_norm_deriv_le_segment' g_deriv g'_bound 1 (right_mem_Icc.2 zero_le_one)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 \u2264\n    \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016\n[PROOFSTEP]\nconvert I using 1\n[GOAL]\ncase h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 \u2016f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\u2016 =\n    \u2016g 1 - g 0\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_3.e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w = g 1 - g 0\n[PROOFSTEP]\nsimp only [Nat.one_ne_zero, add_zero, one_mul, zero_div, zero_mul, sub_zero, zero_smul, Ne.def, not_false_iff,\n  bit0_eq_zero, zero_pow']\n[GOAL]\ncase h.e'_3.e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w =\n    f (x + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w - f (x + h \u2022 v)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w =\n    f (x + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w - f (x + h \u2022 v)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : \u2200 \u2983c : \u211d\u2984, 0 < c \u2192 \u2200\u1da0 (x_1 : E) in \ud835\udcdd[interior s] x, \u2016f' x_1 - f' x - \u2191f'' (x_1 - x)\u2016 \u2264 c * \u2016x_1 - x\u2016\nv w : E\nhv : x + v \u2208 interior s\nhw : x + v + w \u2208 interior s\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\ns\u03b4 : Metric.ball x \u03b4 \u2229 interior s \u2286 {x_1 | (fun x_2 => \u2016f' x_2 - f' x - \u2191f'' (x_2 - x)\u2016 \u2264 \u03b5 * \u2016x_2 - x\u2016) x_1}\nE1 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nE2 : \u2200\u1da0 (h : \u211d) in \ud835\udcdd[Ioi 0] 0, h < 1\nh : \u211d\nh\u03b4 : h * (\u2016v\u2016 + \u2016w\u2016) < \u03b4\nh_lt_1 : h < 1\nhpos : 0 < h\nxt_mem : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 x + h \u2022 v + (t * h) \u2022 w \u2208 interior s\ng : \u211d \u2192 F :=\n  fun t =>\n    f (x + h \u2022 v + (t * h) \u2022 w) - (t * h) \u2022 \u2191(f' x) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' v) w - ((t * h) ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w\ng' : \u211d \u2192 (fun x => F) (h \u2022 w) :=\n  fun t => \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\nhg' :\n  g' = fun t =>\n    \u2191(f' (x + h \u2022 v + (t * h) \u2022 w)) (h \u2022 w) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' v) w - (t * h ^ 2) \u2022 \u2191(\u2191f'' w) w\ng_deriv : \u2200 (t : \u211d), t \u2208 Icc 0 1 \u2192 HasDerivWithinAt g (g' t) (Icc 0 1) t\ng'_bound : \u2200 (t : \u211d), t \u2208 Ico 0 1 \u2192 \u2016g' t\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\nI : \u2016g 1 - g 0\u2016 \u2264 \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n\u22a2 \u03b5 * \u2016(\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016 * h ^ 2\u2016 = \u03b5 * ((\u2016v\u2016 + \u2016w\u2016) * \u2016w\u2016) * h ^ 2\n[PROOFSTEP]\nsimp only [Real.norm_eq_abs, abs_mul, add_nonneg (norm_nonneg v) (norm_nonneg w), abs_of_nonneg, hpos.le, mul_assoc,\n  pow_bit0_abs, norm_nonneg, abs_pow]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave A : (1 : \u211d) / 2 \u2208 Ioc (0 : \u211d) 1 := \u27e8by norm_num, by norm_num\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 1 / 2 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave B : (1 : \u211d) / 2 \u2208 Icc (0 : \u211d) 1 := \u27e8by norm_num, by norm_num\u27e9\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\n\u22a2 1 / 2 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave C : \u2200 w : E, (2 : \u211d) \u2022 w = 2 \u2022 w := fun w => by simp only [two_smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w\u271d : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w\u271d \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nw : E\n\u22a2 2 \u2022 w = 2 \u2022 w\n[PROOFSTEP]\nsimp only [two_smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave h2v2w : x + (2 : \u211d) \u2022 v + (2 : \u211d) \u2022 w \u2208 interior s :=\n  by\n  convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1\n  simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]\n  norm_num\n  simp only [show (4 : \u211d) = (2 : \u211d) + (2 : \u211d) by norm_num, _root_.add_smul]\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.interior.add_smul_sub_mem h4v h4w B using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w = x + 4 \u2022 v + (1 / 2) \u2022 (x + 4 \u2022 w - (x + 4 \u2022 v))\n[PROOFSTEP]\nsimp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w = x + 4 \u2022 v + ((2\u207b\u00b9 * 4) \u2022 w - (2\u207b\u00b9 * 4) \u2022 v)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w = x + 4 \u2022 v + (2 \u2022 w - 2 \u2022 v)\n[PROOFSTEP]\nsimp only [show (4 : \u211d) = (2 : \u211d) + (2 : \u211d) by norm_num, _root_.add_smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 4 = 2 + 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w = x + (2 \u2022 v + 2 \u2022 v) + (2 \u2022 w - 2 \u2022 v)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\n\u22a2 x + 2 \u2022 v + 2 \u2022 w = x + (2 \u2022 v + 2 \u2022 v) + (2 \u2022 w - 2 \u2022 v)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave h2vww : x + (2 \u2022 v + w) + w \u2208 interior s := by\n  convert h2v2w using 1\n  simp only [two_smul]\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) + w \u2208 interior s\n[PROOFSTEP]\nconvert h2v2w using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) + w = x + 2 \u2022 v + 2 \u2022 w\n[PROOFSTEP]\nsimp only [two_smul]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n\u22a2 x + (v + v + w) + w = x + (v + v) + (w + w)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\n\u22a2 x + (v + v + w) + w = x + (v + v) + (w + w)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave h2v : x + (2 : \u211d) \u2022 v \u2208 interior s :=\n  by\n  convert s_conv.add_smul_sub_mem_interior xs h4v A using 1\n  simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n  norm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\n\u22a2 x + 2 \u2022 v \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.add_smul_sub_mem_interior xs h4v A using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\n\u22a2 x + 2 \u2022 v = x + (1 / 2) \u2022 (x + 4 \u2022 v - x)\n[PROOFSTEP]\nsimp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\n\u22a2 2 \u2022 v = (2\u207b\u00b9 * 4) \u2022 v\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave h2w : x + (2 : \u211d) \u2022 w \u2208 interior s :=\n  by\n  convert s_conv.add_smul_sub_mem_interior xs h4w A using 1\n  simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n  norm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\n\u22a2 x + 2 \u2022 w \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.add_smul_sub_mem_interior xs h4w A using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\n\u22a2 x + 2 \u2022 w = x + (1 / 2) \u2022 (x + 4 \u2022 w - x)\n[PROOFSTEP]\nsimp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\n\u22a2 2 \u2022 w = (2\u207b\u00b9 * 4) \u2022 w\n[PROOFSTEP]\nnorm_num\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave hvw : x + (v + w) \u2208 interior s :=\n  by\n  convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1\n  simp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]\n  norm_num\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 x + (v + w) \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 x + (v + w) = x + (1 / 2) \u2022 (x + 2 \u2022 v + 2 \u2022 w - x)\n[PROOFSTEP]\nsimp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 v + w = 2\u207b\u00b9 \u2022 x + (2\u207b\u00b9 * 2) \u2022 v + (2\u207b\u00b9 * 2) \u2022 w - 2\u207b\u00b9 \u2022 x\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 v + w = (1 / 2) \u2022 x + v + w - (1 / 2) \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\n\u22a2 v + w = (1 / 2) \u2022 x + v + w - (1 / 2) \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave h2vw : x + (2 \u2022 v + w) \u2208 interior s :=\n  by\n  convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1\n  simp only [smul_add, smul_sub, smul_smul, \u2190 C]\n  norm_num\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) = x + 2 \u2022 v + (1 / 2) \u2022 (x + 2 \u2022 v + 2 \u2022 w - (x + 2 \u2022 v))\n[PROOFSTEP]\nsimp only [smul_add, smul_sub, smul_smul, \u2190 C]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) = x + 2 \u2022 v + ((1 / 2) \u2022 x + (1 / 2 * 2) \u2022 v + (1 / 2 * 2) \u2022 w - ((1 / 2) \u2022 x + (1 / 2 * 2) \u2022 v))\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) = x + 2 \u2022 v + w\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\n\u22a2 x + (2 \u2022 v + w) = x + 2 \u2022 v + w\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave hvww : x + (v + w) + w \u2208 interior s :=\n  by\n  convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1\n  rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel', inv_smul_smul\u2080 two_ne_zero, two_smul]\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\n\u22a2 x + (v + w) + w \u2208 interior s\n[PROOFSTEP]\nconvert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\n\u22a2 x + (v + w) + w = x + 2 \u2022 w + (1 / 2) \u2022 (x + 2 \u2022 v + 2 \u2022 w - (x + 2 \u2022 w))\n[PROOFSTEP]\nrw [one_div, add_sub_add_right_eq_sub, add_sub_cancel', inv_smul_smul\u2080 two_ne_zero, two_smul]\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\n\u22a2 x + (v + w) + w = x + (w + w) + v\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\n\u22a2 x + (v + w) + w = x + (w + w) + v\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nhave TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n[PROOFSTEP]\nconvert TA1.sub TA2 using 1\n[GOAL]\ncase h.e'_7\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\n\u22a2 (fun h =>\n      f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w) =\n    fun x_1 =>\n    f (x + x_1 \u2022 (2 \u2022 v + w) + x_1 \u2022 w) - f (x + x_1 \u2022 (2 \u2022 v + w)) - x_1 \u2022 \u2191(f' x) w -\n          x_1 ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w -\n      (f (x + x_1 \u2022 (v + w) + x_1 \u2022 w) - f (x + x_1 \u2022 (v + w)) - x_1 \u2022 \u2191(f' x) w - x_1 ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (x_1 ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w)\n[PROOFSTEP]\next h\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nh : \u211d\n\u22a2 f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n      h ^ 2 \u2022 \u2191(\u2191f'' v) w =\n    f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w -\n      (f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w)\n[PROOFSTEP]\nsimp only [two_smul, smul_add, \u2190 add_assoc, ContinuousLinearMap.map_add, ContinuousLinearMap.add_apply, Pi.smul_apply,\n  ContinuousLinearMap.coe_smul', ContinuousLinearMap.map_smul]\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nh : \u211d\n\u22a2 f (x + h \u2022 v + h \u2022 v + h \u2022 w + h \u2022 w) + f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v + h \u2022 v + h \u2022 w) -\n        f (x + h \u2022 v + h \u2022 w + h \u2022 w) -\n      h ^ 2 \u2022 \u2191(\u2191f'' v) w =\n    f (x + h \u2022 v + h \u2022 v + h \u2022 w + h \u2022 w) - f (x + h \u2022 v + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w -\n          (h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' w) w) -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w -\n      (f (x + h \u2022 v + h \u2022 w + h \u2022 w) - f (x + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w -\n          (h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' w) w) -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : 1 / 2 \u2208 Ioc 0 1\nB : 1 / 2 \u2208 Icc 0 1\nC : \u2200 (w : E), 2 \u2022 w = 2 \u2022 w\nh2v2w : x + 2 \u2022 v + 2 \u2022 w \u2208 interior s\nh2vww : x + (2 \u2022 v + w) + w \u2208 interior s\nh2v : x + 2 \u2022 v \u2208 interior s\nh2w : x + 2 \u2022 w \u2208 interior s\nhvw : x + (v + w) \u2208 interior s\nh2vw : x + (2 \u2022 v + w) \u2208 interior s\nhvww : x + (v + w) + w \u2208 interior s\nTA1 :\n  (fun h =>\n      f (x + h \u2022 (2 \u2022 v + w) + h \u2022 w) - f (x + h \u2022 (2 \u2022 v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (2 \u2022 v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h \u2022 (v + w) + h \u2022 w) - f (x + h \u2022 (v + w)) - h \u2022 \u2191(f' x) w - h ^ 2 \u2022 \u2191(\u2191f'' (v + w)) w -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w) =o[\ud835\udcdd[Ioi 0] 0]\n    fun h => h ^ 2\nh : \u211d\n\u22a2 f (x + h \u2022 v + h \u2022 v + h \u2022 w + h \u2022 w) + f (x + h \u2022 v + h \u2022 w) - f (x + h \u2022 v + h \u2022 v + h \u2022 w) -\n        f (x + h \u2022 v + h \u2022 w + h \u2022 w) -\n      h ^ 2 \u2022 \u2191(\u2191f'' v) w =\n    f (x + h \u2022 v + h \u2022 v + h \u2022 w + h \u2022 w) - f (x + h \u2022 v + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w -\n          (h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' w) w) -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w -\n      (f (x + h \u2022 v + h \u2022 w + h \u2022 w) - f (x + h \u2022 v + h \u2022 w) - h \u2022 \u2191(f' x) w -\n          (h ^ 2 \u2022 \u2191(\u2191f'' v) w + h ^ 2 \u2022 \u2191(\u2191f'' w) w) -\n        (h ^ 2 / 2) \u2022 \u2191(\u2191f'' w) w)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 \u2191(\u2191f'' w) v = \u2191(\u2191f'' v) w\n[PROOFSTEP]\nhave A : (fun h : \u211d => h ^ 2 \u2022 (f'' w v - f'' v w)) =o[\ud835\udcdd[>] 0] fun h => h ^ 2 :=\n  by\n  convert\n    (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub\n      (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using\n    1\n  ext h\n  simp only [add_comm, smul_add, smul_sub]\n  abel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\n[PROOFSTEP]\nconvert\n  (s_conv.isLittleO_alternate_sum_square hf xs hx h4v h4w).sub\n    (s_conv.isLittleO_alternate_sum_square hf xs hx h4w h4v) using\n  1\n[GOAL]\ncase h.e'_7\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\n\u22a2 (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) = fun x_1 =>\n    f (x + x_1 \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + x_1 \u2022 (v + w)) - f (x + x_1 \u2022 (2 \u2022 v + w)) - f (x + x_1 \u2022 (v + 2 \u2022 w)) -\n        x_1 ^ 2 \u2022 \u2191(\u2191f'' v) w -\n      (f (x + x_1 \u2022 (2 \u2022 w + 2 \u2022 v)) + f (x + x_1 \u2022 (w + v)) - f (x + x_1 \u2022 (2 \u2022 w + v)) - f (x + x_1 \u2022 (w + 2 \u2022 v)) -\n        x_1 ^ 2 \u2022 \u2191(\u2191f'' w) v)\n[PROOFSTEP]\next h\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nh : \u211d\n\u22a2 h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) =\n    f (x + h \u2022 (2 \u2022 v + 2 \u2022 w)) + f (x + h \u2022 (v + w)) - f (x + h \u2022 (2 \u2022 v + w)) - f (x + h \u2022 (v + 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w -\n      (f (x + h \u2022 (2 \u2022 w + 2 \u2022 v)) + f (x + h \u2022 (w + v)) - f (x + h \u2022 (2 \u2022 w + v)) - f (x + h \u2022 (w + 2 \u2022 v)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' w) v)\n[PROOFSTEP]\nsimp only [add_comm, smul_add, smul_sub]\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nh : \u211d\n\u22a2 h ^ 2 \u2022 \u2191(\u2191f'' w) v - h ^ 2 \u2022 \u2191(\u2191f'' v) w =\n    f (x + (h \u2022 v + h \u2022 w)) + f (x + (h \u2022 2 \u2022 v + h \u2022 2 \u2022 w)) - f (x + (h \u2022 w + h \u2022 2 \u2022 v)) -\n          f (x + (h \u2022 v + h \u2022 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w -\n      (f (x + (h \u2022 v + h \u2022 w)) + f (x + (h \u2022 2 \u2022 v + h \u2022 2 \u2022 w)) - f (x + (h \u2022 v + h \u2022 2 \u2022 w)) -\n          f (x + (h \u2022 w + h \u2022 2 \u2022 v)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' w) v)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nh : \u211d\n\u22a2 h ^ 2 \u2022 \u2191(\u2191f'' w) v - h ^ 2 \u2022 \u2191(\u2191f'' v) w =\n    f (x + (h \u2022 v + h \u2022 w)) + f (x + (h \u2022 2 \u2022 v + h \u2022 2 \u2022 w)) - f (x + (h \u2022 w + h \u2022 2 \u2022 v)) -\n          f (x + (h \u2022 v + h \u2022 2 \u2022 w)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' v) w -\n      (f (x + (h \u2022 v + h \u2022 w)) + f (x + (h \u2022 2 \u2022 v + h \u2022 2 \u2022 w)) - f (x + (h \u2022 v + h \u2022 2 \u2022 w)) -\n          f (x + (h \u2022 w + h \u2022 2 \u2022 v)) -\n        h ^ 2 \u2022 \u2191(\u2191f'' w) v)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\n\u22a2 \u2191(\u2191f'' w) v = \u2191(\u2191f'' v) w\n[PROOFSTEP]\nhave B : (fun _ : \u211d => f'' w v - f'' v w) =o[\ud835\udcdd[>] 0] fun _ => (1 : \u211d) :=\n  by\n  have : (fun h : \u211d => 1 / h ^ 2) =O[\ud835\udcdd[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _\n  have C := this.smul_isLittleO A\n  apply C.congr' _ _\n  \u00b7 filter_upwards [self_mem_nhdsWithin]\n    intro h (hpos : 0 < h)\n    rw [\u2190 one_smul \u211d (f'' w v - f'' v w), smul_smul, smul_smul]\n    congr 1\n    field_simp [LT.lt.ne' hpos]\n  \u00b7 filter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)\n    field_simp [LT.lt.ne' hpos, SMul.smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\n\u22a2 (fun x => \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0] fun x => 1\n[PROOFSTEP]\nhave : (fun h : \u211d => 1 / h ^ 2) =O[\ud835\udcdd[>] 0] fun h => 1 / h ^ 2 := isBigO_refl _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\n\u22a2 (fun x => \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0] fun x => 1\n[PROOFSTEP]\nhave C := this.smul_isLittleO A\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\n\u22a2 (fun x => \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0] fun x => 1\n[PROOFSTEP]\napply C.congr' _ _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\n\u22a2 (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\n\u22a2 \u2200 (a : \u211d), a \u2208 Ioi 0 \u2192 (1 / a ^ 2) \u2022 a ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) = \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w\n[PROOFSTEP]\nintro h (hpos : 0 < h)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\nh : \u211d\nhpos : 0 < h\n\u22a2 (1 / h ^ 2) \u2022 h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) = \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w\n[PROOFSTEP]\nrw [\u2190 one_smul \u211d (f'' w v - f'' v w), smul_smul, smul_smul]\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\nh : \u211d\nhpos : 0 < h\n\u22a2 (1 / h ^ 2 * h ^ 2 * 1) \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) = 1 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\nh : \u211d\nhpos : 0 < h\n\u22a2 1 / h ^ 2 * h ^ 2 * 1 = 1\n[PROOFSTEP]\nfield_simp [LT.lt.ne' hpos]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\n\u22a2 (fun x => (1 / x ^ 2) \u2022 x ^ 2) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => 1\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with h (hpos : 0 < h)\n[GOAL]\ncase h\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nthis : (fun h => 1 / h ^ 2) =O[\ud835\udcdd[Ioi 0] 0] fun h => 1 / h ^ 2\nC : (fun x => (1 / x ^ 2) \u2022 x ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun x => (1 / x ^ 2) \u2022 x ^ 2\nh : \u211d\nhpos : 0 < h\n\u22a2 (1 / h ^ 2) \u2022 h ^ 2 = 1\n[PROOFSTEP]\nfield_simp [LT.lt.ne' hpos, SMul.smul]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 \u2022 v \u2208 interior s\nh4w : x + 4 \u2022 w \u2208 interior s\nA : (fun h => h ^ 2 \u2022 (\u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w)) =o[\ud835\udcdd[Ioi 0] 0] fun h => h ^ 2\nB : (fun x => \u2191(\u2191f'' w) v - \u2191(\u2191f'' v) w) =o[\ud835\udcdd[Ioi 0] 0] fun x => 1\n\u22a2 \u2191(\u2191f'' w) v = \u2191(\u2191f'' v) w\n[PROOFSTEP]\nsimpa only [sub_eq_zero] using isLittleO_const_const_iff.1 B\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nhne : Set.Nonempty (interior s)\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nrcases hne with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 : \u2203 z, z = ((1 : \u211d) / 4) \u2022 (y - x) := \u27e8((1 : \u211d) / 4) \u2022 (y - x), rfl\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nhave A : \u2200 m : E, Filter.Tendsto (fun t : \u211d => x + (4 : \u211d) \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y) :=\n  by\n  intro m\n  have : x + (4 : \u211d) \u2022 (z + (0 : \u211d) \u2022 m) = y := by simp [hz]\n  rw [\u2190 this]\n  refine' tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add _\n  exact continuousAt_id.smul continuousAt_const\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\n\u22a2 \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\n[PROOFSTEP]\nintro m\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nm : E\n\u22a2 Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\n[PROOFSTEP]\nhave : x + (4 : \u211d) \u2022 (z + (0 : \u211d) \u2022 m) = y := by simp [hz]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nm : E\n\u22a2 x + 4 \u2022 (z + 0 \u2022 m) = y\n[PROOFSTEP]\nsimp [hz]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nm : E\nthis : x + 4 \u2022 (z + 0 \u2022 m) = y\n\u22a2 Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nm : E\nthis : x + 4 \u2022 (z + 0 \u2022 m) = y\n\u22a2 Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd (x + 4 \u2022 (z + 0 \u2022 m)))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add _\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nm : E\nthis : x + 4 \u2022 (z + 0 \u2022 m) = y\n\u22a2 Filter.Tendsto (fun t => t \u2022 m) (\ud835\udcdd 0) (\ud835\udcdd (0 \u2022 m))\n[PROOFSTEP]\nexact continuousAt_id.smul continuousAt_const\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nhave B : \u2200 m : E, \u2200\u1da0 t in \ud835\udcdd[>] (0 : \u211d), x + (4 : \u211d) \u2022 (z + t \u2022 m) \u2208 interior s :=\n  by\n  intro m\n  apply nhdsWithin_le_nhds\n  apply A m\n  rw [mem_interior_iff_mem_nhds] at hy \n  exact interior_mem_nhds.2 hy\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\n\u22a2 \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\n[PROOFSTEP]\nintro m\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nm : E\n\u22a2 \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\n[PROOFSTEP]\napply nhdsWithin_le_nhds\n[GOAL]\ncase a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nm : E\n\u22a2 {x_1 | (fun t => x + 4 \u2022 (z + t \u2022 m) \u2208 interior s) x_1} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\napply A m\n[GOAL]\ncase a.a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nm : E\n\u22a2 interior s \u2208 \ud835\udcdd y\n[PROOFSTEP]\nrw [mem_interior_iff_mem_nhds] at hy \n[GOAL]\ncase a.a\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : s \u2208 \ud835\udcdd y\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nm : E\n\u22a2 interior s \u2208 \ud835\udcdd y\n[PROOFSTEP]\nexact interior_mem_nhds.2 hy\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nchoose t ts tpos using fun m => ((B m).and self_mem_nhdsWithin).exists\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nhave C : \u2200 m : E, f'' m z = f'' z m := by\n  intro m\n  have : f'' (z + t m \u2022 m) (z + t 0 \u2022 (0 : E)) = f'' (z + t 0 \u2022 (0 : E)) (z + t m \u2022 m) :=\n    s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)\n  simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj, ContinuousLinearMap.add_apply,\n    Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero, ContinuousLinearMap.zero_apply, smul_zero,\n    ContinuousLinearMap.map_zero] at this \n  exact smul_right_injective F (tpos m).ne' this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\n\u22a2 \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\n[PROOFSTEP]\nintro m\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nm : E\n\u22a2 \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\n[PROOFSTEP]\nhave : f'' (z + t m \u2022 m) (z + t 0 \u2022 (0 : E)) = f'' (z + t 0 \u2022 (0 : E)) (z + t m \u2022 m) :=\n  s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nm : E\nthis : \u2191(\u2191f'' (z + t m \u2022 m)) (z + t 0 \u2022 0) = \u2191(\u2191f'' (z + t 0 \u2022 0)) (z + t m \u2022 m)\n\u22a2 \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj, ContinuousLinearMap.add_apply,\n  Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero, ContinuousLinearMap.zero_apply, smul_zero,\n  ContinuousLinearMap.map_zero] at this \n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nm : E\nthis : t m \u2022 \u2191(\u2191f'' m) z = t m \u2022 \u2191(\u2191f'' z) m\n\u22a2 \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\n[PROOFSTEP]\nexact smul_right_injective F (tpos m).ne' this\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nC : \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nhave : f'' (z + t v \u2022 v) (z + t w \u2022 w) = f'' (z + t w \u2022 w) (z + t v \u2022 v) :=\n  s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nC : \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\nthis : \u2191(\u2191f'' (z + t v \u2022 v)) (z + t w \u2022 w) = \u2191(\u2191f'' (z + t w \u2022 w)) (z + t v \u2022 v)\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, smul_add, smul_smul,\n  ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', C] at this \n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nC : \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\nthis :\n  \u2191(\u2191f'' z) z + t v \u2022 \u2191(\u2191f'' z) v + (t w \u2022 \u2191(\u2191f'' z) w + (t w * t v) \u2022 \u2191(\u2191f'' v) w) =\n    \u2191(\u2191f'' z) z + t w \u2022 \u2191(\u2191f'' z) w + (t v \u2022 \u2191(\u2191f'' z) v + (t v * t w) \u2022 \u2191(\u2191f'' w) v)\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nrw [add_assoc, add_assoc, add_right_inj, add_left_comm, add_right_inj, add_right_inj, mul_comm] at this \n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nC : \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\nthis : (t v * t w) \u2022 \u2191(\u2191f'' v) w = (t v * t w) \u2022 \u2191(\u2191f'' w) v\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\napply smul_right_injective F _ this\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns\u271d : Set E\ns_conv\u271d : Convex \u211d s\u271d\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s\u271d \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx\u271d : E\nxs\u271d : x\u271d \u2208 s\u271d\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s\u271d) x\u271d\ns : Set E\ns_conv : Convex \u211d s\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f (f' x) x\nx : E\nxs : x \u2208 s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w y : E\nhy : y \u2208 interior s\nz : E\nhz : z = (1 / 4) \u2022 (y - x)\nA : \u2200 (m : E), Filter.Tendsto (fun t => x + 4 \u2022 (z + t \u2022 m)) (\ud835\udcdd 0) (\ud835\udcdd y)\nB : \u2200 (m : E), \u2200\u1da0 (t : \u211d) in \ud835\udcdd[Ioi 0] 0, x + 4 \u2022 (z + t \u2022 m) \u2208 interior s\nt : E \u2192 \u211d\nts : \u2200 (m : E), x + 4 \u2022 (z + t m \u2022 m) \u2208 interior s\ntpos : \u2200 (m : E), 0 < t m\nC : \u2200 (m : E), \u2191(\u2191f'' m) z = \u2191(\u2191f'' z) m\nthis : (t v * t w) \u2022 \u2191(\u2191f'' v) w = (t v * t w) \u2022 \u2191(\u2191f'' w) v\n\u22a2 t v * t w \u2260 0\n[PROOFSTEP]\nsimp [(tpos v).ne', (tpos w).ne']\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx : E\nxs : x \u2208 s\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s) x\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200\u1da0 (y : E) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.1 hf with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx : E\nxs : x \u2208 s\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s) x\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200\u1da0 (y : E) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : Metric.ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x}\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nhave A : (interior (Metric.ball x \u03b5)).Nonempty := by rwa [Metric.isOpen_ball.interior_eq, Metric.nonempty_ball]\n[GOAL]\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx : E\nxs : x \u2208 s\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s) x\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200\u1da0 (y : E) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : Metric.ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x}\n\u22a2 Set.Nonempty (interior (Metric.ball x \u03b5))\n[PROOFSTEP]\nrwa [Metric.isOpen_ball.interior_eq, Metric.nonempty_ball]\n[GOAL]\ncase intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ns : Set E\ns_conv : Convex \u211d s\nf\u271d : E \u2192 F\nf'\u271d : E \u2192 E \u2192L[\u211d] F\nf''\u271d : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf\u271d : \u2200 (x : E), x \u2208 interior s \u2192 HasFDerivAt f\u271d (f'\u271d x) x\nx : E\nxs : x \u2208 s\nhx\u271d : HasFDerivWithinAt f'\u271d f''\u271d (interior s) x\nf : E \u2192 F\nf' : E \u2192 E \u2192L[\u211d] F\nf'' : E \u2192L[\u211d] E \u2192L[\u211d] F\nhf : \u2200\u1da0 (y : E) in \ud835\udcdd x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : Metric.ball x \u03b5 \u2286 {x | (fun y => HasFDerivAt f (f' y) y) x}\nA : Set.Nonempty (interior (Metric.ball x \u03b5))\n\u22a2 \u2191(\u2191f'' v) w = \u2191(\u2191f'' w) v\n[PROOFSTEP]\nexact\n  Convex.second_derivative_within_at_symmetric (convex_ball x \u03b5) A (fun y hy => h\u03b5 (interior_subset hy))\n    (Metric.mem_ball_self \u03b5pos) hx.hasFDerivWithinAt v w\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Symmetric", "llama_tokens": 104944, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8198933359135361, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.5369438130216727}}
{"text": "[GOAL]\nb n : \u2115\n\u22a2 log b n = 0 \u2194 n < b \u2228 b \u2264 1\n[PROOFSTEP]\nrw [log, dite_eq_right_iff]\n[GOAL]\nb n : \u2115\n\u22a2 (\u2200 (h : b \u2264 n \u2227 1 < b),\n      (let_fun this := (_ : n / b < n);\n        log b (n / b) + 1) =\n        0) \u2194\n    n < b \u2228 b \u2264 1\n[PROOFSTEP]\nsimp only [Nat.succ_ne_zero, imp_false, not_and_or, not_le, not_lt]\n[GOAL]\nb n : \u2115\n\u22a2 0 < log b n \u2194 b \u2264 n \u2227 1 < b\n[PROOFSTEP]\nrw [pos_iff_ne_zero, Ne.def, log_eq_zero_iff, not_or, not_lt, not_le]\n[GOAL]\nb n : \u2115\nh : 1 < b\nhn : b \u2264 n\n\u22a2 log b n = log b (n / b) + 1\n[PROOFSTEP]\nrw [log]\n[GOAL]\nb n : \u2115\nh : 1 < b\nhn : b \u2264 n\n\u22a2 (if h : b \u2264 n \u2227 1 < b then\n      let_fun this := (_ : n / b < n);\n      log b (n / b) + 1\n    else 0) =\n    log b (n / b) + 1\n[PROOFSTEP]\nexact if_pos \u27e8hn, h\u27e9\n[GOAL]\nb : \u2115\nhb : 1 < b\nx y : \u2115\nhy : y \u2260 0\n\u22a2 b ^ x \u2264 y \u2194 x \u2264 log b y\n[PROOFSTEP]\ninduction' y using Nat.strong_induction_on with y ih generalizing x\n[GOAL]\ncase h\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nx : \u2115\nhy : y \u2260 0\n\u22a2 b ^ x \u2264 y \u2194 x \u2264 log b y\n[PROOFSTEP]\ncases x with\n| zero => exact iff_of_true hy.bot_lt (zero_le _)\n| succ x =>\n  rw [log]; split_ifs with h\n  \u00b7 have b_pos : 0 < b := zero_le_one.trans_lt hb\n    rw [succ_eq_add_one, add_le_add_iff_right, \u2190 ih (y / b) (div_lt_self hy.bot_lt hb) (Nat.div_pos h.1 b_pos).ne',\n      le_div_iff_mul_le b_pos, pow_succ', mul_comm]\n  \u00b7 exact iff_of_false (fun hby => h \u27e8(le_self_pow x.succ_ne_zero _).trans hby, hb\u27e9) (not_succ_le_zero _)\n[GOAL]\ncase h\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nx : \u2115\nhy : y \u2260 0\n\u22a2 b ^ x \u2264 y \u2194 x \u2264 log b y\n[PROOFSTEP]\ncases x with\n| zero => exact iff_of_true hy.bot_lt (zero_le _)\n| succ x =>\n  rw [log]; split_ifs with h\n  \u00b7 have b_pos : 0 < b := zero_le_one.trans_lt hb\n    rw [succ_eq_add_one, add_le_add_iff_right, \u2190 ih (y / b) (div_lt_self hy.bot_lt hb) (Nat.div_pos h.1 b_pos).ne',\n      le_div_iff_mul_le b_pos, pow_succ', mul_comm]\n  \u00b7 exact iff_of_false (fun hby => h \u27e8(le_self_pow x.succ_ne_zero _).trans hby, hb\u27e9) (not_succ_le_zero _)\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\nx y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\n\u22a2 b ^ zero \u2264 y \u2194 zero \u2264 log b y\n[PROOFSTEP]\n\n| zero => exact iff_of_true hy.bot_lt (zero_le _)\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\nx y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\n\u22a2 b ^ zero \u2264 y \u2194 zero \u2264 log b y\n[PROOFSTEP]\nexact iff_of_true hy.bot_lt (zero_le _)\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\n\u22a2 b ^ succ x \u2264 y \u2194 succ x \u2264 log b y\n[PROOFSTEP]\n\n| succ x =>\n  rw [log]; split_ifs with h\n  \u00b7 have b_pos : 0 < b := zero_le_one.trans_lt hb\n    rw [succ_eq_add_one, add_le_add_iff_right, \u2190 ih (y / b) (div_lt_self hy.bot_lt hb) (Nat.div_pos h.1 b_pos).ne',\n      le_div_iff_mul_le b_pos, pow_succ', mul_comm]\n  \u00b7 exact iff_of_false (fun hby => h \u27e8(le_self_pow x.succ_ne_zero _).trans hby, hb\u27e9) (not_succ_le_zero _)\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\n\u22a2 b ^ succ x \u2264 y \u2194 succ x \u2264 log b y\n[PROOFSTEP]\nrw [log]\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\n\u22a2 b ^ succ x \u2264 y \u2194\n    succ x \u2264\n      if h : b \u2264 y \u2227 1 < b then\n        let_fun this := (_ : y / b < y);\n        log b (y / b) + 1\n      else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\nh : b \u2264 y \u2227 1 < b\n\u22a2 b ^ succ x \u2264 y \u2194\n    succ x \u2264\n      let_fun this := (_ : y / b < y);\n      log b (y / b) + 1\n[PROOFSTEP]\nhave b_pos : 0 < b := zero_le_one.trans_lt hb\n[GOAL]\ncase pos\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\nh : b \u2264 y \u2227 1 < b\nb_pos : 0 < b\n\u22a2 b ^ succ x \u2264 y \u2194\n    succ x \u2264\n      let_fun this := (_ : y / b < y);\n      log b (y / b) + 1\n[PROOFSTEP]\nrw [succ_eq_add_one, add_le_add_iff_right, \u2190 ih (y / b) (div_lt_self hy.bot_lt hb) (Nat.div_pos h.1 b_pos).ne',\n  le_div_iff_mul_le b_pos, pow_succ', mul_comm]\n[GOAL]\ncase neg\nb : \u2115\nhb : 1 < b\nx\u271d y\u271d : \u2115\nhy\u271d : y\u271d \u2260 0\ny : \u2115\nih : \u2200 (m : \u2115), m < y \u2192 \u2200 {x : \u2115}, m \u2260 0 \u2192 (b ^ x \u2264 m \u2194 x \u2264 log b m)\nhy : y \u2260 0\nx : \u2115\nh : \u00ac(b \u2264 y \u2227 1 < b)\n\u22a2 b ^ succ x \u2264 y \u2194 succ x \u2264 0\n[PROOFSTEP]\nexact iff_of_false (fun hby => h \u27e8(le_self_pow x.succ_ne_zero _).trans hby, hb\u27e9) (not_succ_le_zero _)\n[GOAL]\nb x y : \u2115\nhy : y \u2260 0\nh : x \u2264 log b y\n\u22a2 b ^ x \u2264 y\n[PROOFSTEP]\nrefine' (le_or_lt b 1).elim (fun hb => _) fun hb => (pow_le_iff_le_log hb hy).2 h\n[GOAL]\nb x y : \u2115\nhy : y \u2260 0\nh : x \u2264 log b y\nhb : b \u2264 1\n\u22a2 b ^ x \u2264 y\n[PROOFSTEP]\nrw [log_of_left_le_one hb, nonpos_iff_eq_zero] at h \n[GOAL]\nb x y : \u2115\nhy : y \u2260 0\nh : x = 0\nhb : b \u2264 1\n\u22a2 b ^ x \u2264 y\n[PROOFSTEP]\nrwa [h, pow_zero, one_le_iff_ne_zero]\n[GOAL]\nb x y : \u2115\nhb : 1 < b\nh : b ^ x \u2264 y\n\u22a2 x \u2264 log b y\n[PROOFSTEP]\nrcases ne_or_eq y 0 with (hy | rfl)\n[GOAL]\ncase inl\nb x y : \u2115\nhb : 1 < b\nh : b ^ x \u2264 y\nhy : y \u2260 0\n\u22a2 x \u2264 log b y\ncase inr b x : \u2115 hb : 1 < b h : b ^ x \u2264 0 \u22a2 x \u2264 log b 0\n[PROOFSTEP]\nexacts [(pow_le_iff_le_log hb hy).1 h, (h.not_lt (pow_pos (zero_lt_one.trans hb) _)).elim]\n[GOAL]\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nrcases em (1 < b \u2227 n \u2260 0) with (\u27e8hb, hn\u27e9 | hbn)\n[GOAL]\ncase inl.intro\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nrw [le_antisymm_iff, \u2190 lt_succ_iff, \u2190 pow_le_iff_le_log, \u2190 lt_pow_iff_log_lt, and_comm]\n[GOAL]\ncase inl.intro.hb\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 1 < b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inl.intro.hy\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 n \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inl.intro.hb\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 1 < b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inl.intro.hy\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 n \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhbn : \u00ac(1 < b \u2227 n \u2260 0)\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nhave hm : m \u2260 0 := h.resolve_right hbn\n[GOAL]\ncase inr\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhbn : \u00ac(1 < b \u2227 n \u2260 0)\nhm : m \u2260 0\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nrw [not_and_or, not_lt, Ne.def, not_not] at hbn \n[GOAL]\ncase inr\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhbn : b \u2264 1 \u2228 n = 0\nhm : m \u2260 0\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nrcases hbn with (hb | rfl)\n[GOAL]\ncase inr.inl\nb m n : \u2115\nh : m \u2260 0 \u2228 1 < b \u2227 n \u2260 0\nhm : m \u2260 0\nhb : b \u2264 1\n\u22a2 log b n = m \u2194 b ^ m \u2264 n \u2227 n < b ^ (m + 1)\n[PROOFSTEP]\nsimpa only [log_of_left_le_one hb, hm.symm, false_iff_iff, not_and, not_lt] using\n  le_trans (pow_le_pow_of_le_one' hb m.le_succ)\n[GOAL]\ncase inr.inr\nb m : \u2115\nhm : m \u2260 0\nh : m \u2260 0 \u2228 1 < b \u2227 0 \u2260 0\n\u22a2 log b 0 = m \u2194 b ^ m \u2264 0 \u2227 0 < b ^ (m + 1)\n[PROOFSTEP]\nsimpa only [log_zero_right, hm.symm, nonpos_iff_eq_zero, false_iff, not_and, not_lt, add_pos_iff, or_true,\n  pow_eq_zero_iff] using pow_eq_zero\n[GOAL]\nb m n : \u2115\nh\u2081 : b ^ m \u2264 n\nh\u2082 : n < b ^ (m + 1)\n\u22a2 log b n = m\n[PROOFSTEP]\nrcases eq_or_ne m 0 with (rfl | hm)\n[GOAL]\ncase inl\nb n : \u2115\nh\u2081 : b ^ 0 \u2264 n\nh\u2082 : n < b ^ (0 + 1)\n\u22a2 log b n = 0\n[PROOFSTEP]\nrw [pow_one] at h\u2082 \n[GOAL]\ncase inl\nb n : \u2115\nh\u2081 : b ^ 0 \u2264 n\nh\u2082 : n < b\n\u22a2 log b n = 0\n[PROOFSTEP]\nexact log_of_lt h\u2082\n[GOAL]\ncase inr\nb m n : \u2115\nh\u2081 : b ^ m \u2264 n\nh\u2082 : n < b ^ (m + 1)\nhm : m \u2260 0\n\u22a2 log b n = m\n[PROOFSTEP]\nexact (log_eq_iff (Or.inl hm)).2 \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\nb n : \u2115\n\u22a2 log b n = 1 \u2194 b \u2264 n \u2227 n < b * b\n[PROOFSTEP]\nrw [log_eq_iff (Or.inl one_ne_zero), pow_add, pow_one]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 log b (n * b) = log b n + 1\n[PROOFSTEP]\napply log_eq_of_pow_le_of_lt_pow\n[GOAL]\ncase h\u2081\nb n : \u2115\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 b ^ (log b n + 1) \u2264 n * b\n[PROOFSTEP]\nrw [pow_succ', mul_comm b]\n[GOAL]\ncase h\u2082\nb n : \u2115\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 n * b < b ^ (log b n + 1 + 1)\n[PROOFSTEP]\nrw [pow_succ', mul_comm b]\n[GOAL]\ncase h\u2081\nb n : \u2115\nhb : 1 < b\nhn : n \u2260 0\n\u22a2 b ^ log b n * b \u2264 n * b\ncase h\u2082 b n : \u2115 hb : 1 < b hn : n \u2260 0 \u22a2 n * b < b ^ (log b n + 1) * b\n[PROOFSTEP]\nexacts [mul_le_mul_right' (pow_log_le_self _ hn) _,\n  (mul_lt_mul_right (zero_lt_one.trans hb)).2 (lt_pow_succ_log_self hb _)]\n[GOAL]\nb : \u2115\n\u22a2 b ^ log b 0 \u2264 0 + 1\n[PROOFSTEP]\nrw [log_zero_right, pow_zero]\n[GOAL]\nb : \u2115\n\u22a2 Monotone (log b)\n[PROOFSTEP]\nrefine' monotone_nat_of_le_succ fun n => _\n[GOAL]\nb n : \u2115\n\u22a2 log b n \u2264 log b (n + 1)\n[PROOFSTEP]\ncases' le_or_lt b 1 with hb hb\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b n \u2264 log b (n + 1)\n[PROOFSTEP]\nrw [log_of_left_le_one hb]\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 0 \u2264 log b (n + 1)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b n \u2264 log b (n + 1)\n[PROOFSTEP]\nexact le_log_of_pow_le hb (pow_log_le_add_one _ _)\n[GOAL]\nb c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 log b n \u2264 log c n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nb c : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 log b 0 \u2264 log c 0\n[PROOFSTEP]\nrw [log_zero_right, log_zero_right]\n[GOAL]\ncase inr\nb c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\nhn : n \u2260 0\n\u22a2 log b n \u2264 log c n\n[PROOFSTEP]\napply le_log_of_pow_le hc\n[GOAL]\ncase inr\nb c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\nhn : n \u2260 0\n\u22a2 c ^ log b n \u2264 n\n[PROOFSTEP]\ncalc\n  c ^ log b n \u2264 b ^ log b n := pow_le_pow_of_le_left' hb _\n  _ \u2264 n := pow_log_le_self _ hn\n[GOAL]\nb n : \u2115\n\u22a2 log b (n / b) = log b n - 1\n[PROOFSTEP]\ncases' le_or_lt b 1 with hb hb\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b) = log b n - 1\n[PROOFSTEP]\nrw [log_of_left_le_one hb, log_of_left_le_one hb, Nat.zero_sub]\n[GOAL]\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b) = log b n - 1\n[PROOFSTEP]\ncases' lt_or_le n b with h h\n[GOAL]\ncase inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b) = log b n - 1\n[PROOFSTEP]\nrw [div_eq_of_lt h, log_of_lt h, log_zero_right]\n[GOAL]\ncase inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b) = log b n - 1\n[PROOFSTEP]\nrw [log_of_one_lt_of_le hb h, add_tsub_cancel_right]\n[GOAL]\nb n : \u2115\n\u22a2 log b (n / b * b) = log b n\n[PROOFSTEP]\ncases' le_or_lt b 1 with hb hb\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b (n / b * b) = log b n\n[PROOFSTEP]\nrw [log_of_left_le_one hb, log_of_left_le_one hb]\n[GOAL]\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b (n / b * b) = log b n\n[PROOFSTEP]\ncases' lt_or_le n b with h h\n[GOAL]\ncase inr.inl\nb n : \u2115\nhb : 1 < b\nh : n < b\n\u22a2 log b (n / b * b) = log b n\n[PROOFSTEP]\nrw [div_eq_of_lt h, zero_mul, log_zero_right, log_of_lt h]\n[GOAL]\ncase inr.inr\nb n : \u2115\nhb : 1 < b\nh : b \u2264 n\n\u22a2 log b (n / b * b) = log b n\n[PROOFSTEP]\nrw [log_mul_base hb (Nat.div_pos h (zero_le_one.trans_lt hb)).ne', log_div_base,\n  tsub_add_cancel_of_le (succ_le_iff.2 <| log_pos hb h)]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : 2 \u2264 n\n\u22a2 (n + b - 1) / b < n\n[PROOFSTEP]\nrw [div_lt_iff_lt_mul (zero_lt_one.trans hb), \u2190 succ_le_iff, \u2190 pred_eq_sub_one,\n  succ_pred_eq_of_pos (add_pos (zero_lt_one.trans hn) (zero_lt_one.trans hb))]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : 2 \u2264 n\n\u22a2 n + b \u2264 n * b\n[PROOFSTEP]\nexact add_le_mul hn hb\n[GOAL]\nb : \u2115\nhb : b \u2264 1\nn : \u2115\n\u22a2 clog b n = 0\n[PROOFSTEP]\nrw [clog, dif_neg fun h : 1 < b \u2227 1 < n => h.1.not_le hb]\n[GOAL]\nn : \u2115\nhn : n \u2264 1\nb : \u2115\n\u22a2 clog b n = 0\n[PROOFSTEP]\nrw [clog, dif_neg fun h : 1 < b \u2227 1 < n => h.2.not_le hn]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : 2 \u2264 n\n\u22a2 clog b n = clog b ((n + b - 1) / b) + 1\n[PROOFSTEP]\nrw [clog, dif_pos (\u27e8hb, hn\u27e9 : 1 < b \u2227 1 < n)]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : 2 \u2264 n\n\u22a2 0 < clog b n\n[PROOFSTEP]\nrw [clog_of_two_le hb hn]\n[GOAL]\nb n : \u2115\nhb : 1 < b\nhn : 2 \u2264 n\n\u22a2 0 < clog b ((n + b - 1) / b) + 1\n[PROOFSTEP]\nexact zero_lt_succ _\n[GOAL]\nb n : \u2115\nhn : 2 \u2264 n\nh : n \u2264 b\n\u22a2 clog b n = 1\n[PROOFSTEP]\nrw [clog_of_two_le (hn.trans h) hn, clog_of_right_le_one]\n[GOAL]\ncase hn\nb n : \u2115\nhn : 2 \u2264 n\nh : n \u2264 b\n\u22a2 (n + b - 1) / b \u2264 1\n[PROOFSTEP]\nhave n_pos : 0 < n := (zero_lt_two' \u2115).trans_le hn\n[GOAL]\ncase hn\nb n : \u2115\nhn : 2 \u2264 n\nh : n \u2264 b\nn_pos : 0 < n\n\u22a2 (n + b - 1) / b \u2264 1\n[PROOFSTEP]\nrw [\u2190 lt_succ_iff, Nat.div_lt_iff_lt_mul (n_pos.trans_le h), \u2190 succ_le_iff, \u2190 pred_eq_sub_one,\n  succ_pred_eq_of_pos (add_pos n_pos (n_pos.trans_le h)), succ_mul, one_mul]\n[GOAL]\ncase hn\nb n : \u2115\nhn : 2 \u2264 n\nh : n \u2264 b\nn_pos : 0 < n\n\u22a2 n + b \u2264 b + b\n[PROOFSTEP]\nexact add_le_add_right h _\n[GOAL]\nb : \u2115\nhb : 1 < b\nx y : \u2115\n\u22a2 x \u2264 b ^ y \u2194 clog b x \u2264 y\n[PROOFSTEP]\ninduction' x using Nat.strong_induction_on with x ih generalizing y\n[GOAL]\ncase h\nb : \u2115\nhb : 1 < b\ny\u271d x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\ny : \u2115\n\u22a2 x \u2264 b ^ y \u2194 clog b x \u2264 y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\n\u22a2 x \u2264 b ^ zero \u2194 clog b x \u2264 zero\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\n\u22a2 x \u2264 1 \u2194 clog b x \u2264 zero\n[PROOFSTEP]\nrefine' \u27e8fun h => (clog_of_right_le_one h b).le, _\u27e9\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\n\u22a2 clog b x \u2264 zero \u2192 x \u2264 1\n[PROOFSTEP]\nsimp_rw [\u2190 not_lt]\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\n\u22a2 \u00aczero < clog b x \u2192 \u00ac1 < x\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase h.zero\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\n\u22a2 1 < x \u2192 zero < clog b x\n[PROOFSTEP]\nexact clog_pos hb\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\nn\u271d : \u2115\n\u22a2 x \u2264 b ^ succ n\u271d \u2194 clog b x \u2264 succ n\u271d\n[PROOFSTEP]\nhave b_pos : 0 < b := (zero_lt_one' \u2115).trans hb\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\nn\u271d : \u2115\nb_pos : 0 < b\n\u22a2 x \u2264 b ^ succ n\u271d \u2194 clog b x \u2264 succ n\u271d\n[PROOFSTEP]\nrw [clog]\n[GOAL]\ncase h.succ\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\nn\u271d : \u2115\nb_pos : 0 < b\n\u22a2 x \u2264 b ^ succ n\u271d \u2194\n    (if h : 1 < b \u2227 1 < x then\n        let_fun this := (_ : (x + b - 1) / b < x);\n        clog b ((x + b - 1) / b) + 1\n      else 0) \u2264\n      succ n\u271d\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\nn\u271d : \u2115\nb_pos : 0 < b\nh : 1 < b \u2227 1 < x\n\u22a2 x \u2264 b ^ succ n\u271d \u2194\n    (let_fun this := (_ : (x + b - 1) / b < x);\n      clog b ((x + b - 1) / b) + 1) \u2264\n      succ n\u271d\n[PROOFSTEP]\nrw [succ_eq_add_one, add_le_add_iff_right, \u2190 ih ((x + b - 1) / b) (add_pred_div_lt hb h.2),\n  Nat.div_le_iff_le_mul_add_pred b_pos, mul_comm b, \u2190 pow_succ, add_tsub_assoc_of_le (Nat.succ_le_of_lt b_pos),\n  add_le_add_iff_right]\n[GOAL]\ncase neg\nb : \u2115\nhb : 1 < b\ny x : \u2115\nih : \u2200 (m : \u2115), m < x \u2192 \u2200 {y : \u2115}, m \u2264 b ^ y \u2194 clog b m \u2264 y\nn\u271d : \u2115\nb_pos : 0 < b\nh : \u00ac(1 < b \u2227 1 < x)\n\u22a2 x \u2264 b ^ succ n\u271d \u2194 0 \u2264 succ n\u271d\n[PROOFSTEP]\nexact iff_of_true ((not_lt.1 (not_and.1 h hb)).trans <| succ_le_of_lt <| pow_pos b_pos _) (zero_le _)\n[GOAL]\nb x : \u2115\nhb : 1 < b\nz : \u2115\n\u22a2 clog b (b ^ x) \u2264 z \u2194 x \u2264 z\n[PROOFSTEP]\nrw [\u2190 le_pow_iff_clog_le hb]\n[GOAL]\nb x : \u2115\nhb : 1 < b\nz : \u2115\n\u22a2 b ^ x \u2264 b ^ z \u2194 x \u2264 z\n[PROOFSTEP]\nexact (pow_right_strictMono hb).le_iff_le\n[GOAL]\nb : \u2115\nhb : 1 < b\nx : \u2115\nhx : 1 < x\n\u22a2 b ^ pred (clog b x) < x\n[PROOFSTEP]\nrw [\u2190 not_le, le_pow_iff_clog_le hb, not_le]\n[GOAL]\nb : \u2115\nhb : 1 < b\nx : \u2115\nhx : 1 < x\n\u22a2 pred (clog b x) < clog b x\n[PROOFSTEP]\nexact pred_lt (clog_pos hb hx).ne'\n[GOAL]\nb n m : \u2115\nh : n \u2264 m\n\u22a2 clog b n \u2264 clog b m\n[PROOFSTEP]\ncases' le_or_lt b 1 with hb hb\n[GOAL]\ncase inl\nb n m : \u2115\nh : n \u2264 m\nhb : b \u2264 1\n\u22a2 clog b n \u2264 clog b m\n[PROOFSTEP]\nrw [clog_of_left_le_one hb]\n[GOAL]\ncase inl\nb n m : \u2115\nh : n \u2264 m\nhb : b \u2264 1\n\u22a2 0 \u2264 clog b m\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr\nb n m : \u2115\nh : n \u2264 m\nhb : 1 < b\n\u22a2 clog b n \u2264 clog b m\n[PROOFSTEP]\nrw [\u2190 le_pow_iff_clog_le hb]\n[GOAL]\ncase inr\nb n m : \u2115\nh : n \u2264 m\nhb : 1 < b\n\u22a2 n \u2264 b ^ clog b m\n[PROOFSTEP]\nexact h.trans (le_pow_clog hb _)\n[GOAL]\nb c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 clog b n \u2264 clog c n\n[PROOFSTEP]\nrw [\u2190 le_pow_iff_clog_le (lt_of_lt_of_le hc hb)]\n[GOAL]\nb c n : \u2115\nhc : 1 < c\nhb : c \u2264 b\n\u22a2 n \u2264 b ^ clog c n\n[PROOFSTEP]\ncalc\n  n \u2264 c ^ clog c n := le_pow_clog hc _\n  _ \u2264 b ^ clog c n := pow_le_pow_of_le_left hb _\n[GOAL]\nb n : \u2115\n\u22a2 log b n \u2264 clog b n\n[PROOFSTEP]\nobtain hb | hb := le_or_lt b 1\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 log b n \u2264 clog b n\n[PROOFSTEP]\nrw [log_of_left_le_one hb]\n[GOAL]\ncase inl\nb n : \u2115\nhb : b \u2264 1\n\u22a2 0 \u2264 clog b n\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b n \u2264 clog b n\n[PROOFSTEP]\ncases n with\n| zero =>\n  rw [log_zero_right]\n  exact zero_le _\n| succ n => exact (pow_right_strictMono hb).le_iff_le.1 ((pow_log_le_self b n.succ_ne_zero).trans <| le_pow_clog hb _)\n[GOAL]\ncase inr\nb n : \u2115\nhb : 1 < b\n\u22a2 log b n \u2264 clog b n\n[PROOFSTEP]\ncases n with\n| zero =>\n  rw [log_zero_right]\n  exact zero_le _\n| succ n => exact (pow_right_strictMono hb).le_iff_le.1 ((pow_log_le_self b n.succ_ne_zero).trans <| le_pow_clog hb _)\n[GOAL]\ncase inr.zero\nb : \u2115\nhb : 1 < b\n\u22a2 log b zero \u2264 clog b zero\n[PROOFSTEP]\n\n| zero =>\n  rw [log_zero_right]\n  exact zero_le _\n[GOAL]\ncase inr.zero\nb : \u2115\nhb : 1 < b\n\u22a2 log b zero \u2264 clog b zero\n[PROOFSTEP]\nrw [log_zero_right]\n[GOAL]\ncase inr.zero\nb : \u2115\nhb : 1 < b\n\u22a2 0 \u2264 clog b zero\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr.succ\nb : \u2115\nhb : 1 < b\nn : \u2115\n\u22a2 log b (succ n) \u2264 clog b (succ n)\n[PROOFSTEP]\n\n| succ n => exact (pow_right_strictMono hb).le_iff_le.1 ((pow_log_le_self b n.succ_ne_zero).trans <| le_pow_clog hb _)\n[GOAL]\ncase inr.succ\nb : \u2115\nhb : 1 < b\nn : \u2115\n\u22a2 log b (succ n) \u2264 clog b (succ n)\n[PROOFSTEP]\nexact (pow_right_strictMono hb).le_iff_le.1 ((pow_log_le_self b n.succ_ne_zero).trans <| le_pow_clog hb _)\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Log", "llama_tokens": 10475, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289836, "lm_q2_score": 0.7057850154599562, "lm_q1q2_score": 0.5368558403184296}}
{"text": "[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nn : \u2115\n\u22a2 wittPolynomial p R n = \u2211 i in range (n + 1), \u2191C (\u2191p ^ i) * X i ^ p ^ (n - i)\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 range (n + 1) \u2192 \u2191(monomial (single x (p ^ (n - x)))) (\u2191p ^ x) = \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x)\n[PROOFSTEP]\nrintro i -\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nn i : \u2115\n\u22a2 \u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i) = \u2191C (\u2191p ^ i) * X i ^ p ^ (n - i)\n[PROOFSTEP]\nrw [monomial_eq, Finsupp.prod_single_index]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nn i : \u2115\n\u22a2 X i ^ 0 = 1\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 \u2191(map f) (W_ R n) = W_ S n\n[PROOFSTEP]\nrw [wittPolynomial, map_sum, wittPolynomial]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 \u2211 x in range (n + 1), \u2191(map f) (\u2191(monomial (single x (p ^ (n - x)))) (\u2191p ^ x)) =\n    \u2211 i in range (n + 1), \u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)\n[PROOFSTEP]\nrefine sum_congr rfl fun i _ => ?_\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nn i : \u2115\nx\u271d : i \u2208 range (n + 1)\n\u22a2 \u2191(map f) (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = \u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)\n[PROOFSTEP]\nrw [map_monomial, RingHom.map_pow, map_natCast]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2191constantCoeff (W_ R n) = 0\n[PROOFSTEP]\nsimp only [wittPolynomial, map_sum, constantCoeff_monomial]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 (\u2211 x in range (n + 1), if single x (p ^ (n - x)) = 0 then \u2191p ^ x else 0) = 0\n[PROOFSTEP]\nrw [sum_eq_zero]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 range (n + 1) \u2192 (if single x (p ^ (n - x)) = 0 then \u2191p ^ x else 0) = 0\n[PROOFSTEP]\nrintro i _\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn i : \u2115\na\u271d : i \u2208 range (n + 1)\n\u22a2 (if single i (p ^ (n - i)) = 0 then \u2191p ^ i else 0) = 0\n[PROOFSTEP]\nrw [if_neg]\n[GOAL]\ncase hnc\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn i : \u2115\na\u271d : i \u2208 range (n + 1)\n\u22a2 \u00acsingle i (p ^ (n - i)) = 0\n[PROOFSTEP]\nrw [Finsupp.single_eq_zero]\n[GOAL]\ncase hnc\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : Fact (Nat.Prime p)\nn i : \u2115\na\u271d : i \u2208 range (n + 1)\n\u22a2 \u00acp ^ (n - i) = 0\n[PROOFSTEP]\nexact ne_of_gt (pow_pos hp.1.pos _)\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\n\u22a2 W_ R 0 = X 0\n[PROOFSTEP]\nsimp only [wittPolynomial, X, sum_singleton, range_one, pow_zero, zero_add, tsub_self]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\n\u22a2 W_ R 1 = \u2191C \u2191p * X 1 + X 0 ^ p\n[PROOFSTEP]\nsimp only [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, range_one, sum_singleton, one_mul, pow_one, C_1,\n  pow_zero, tsub_self, tsub_zero]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b2 : CommRing S\nA : Type u_3\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nf : \u2115 \u2192 A\nn : \u2115\n\u22a2 \u2191(aeval f) (W_ R n) = \u2211 i in range (n + 1), \u2191p ^ i * f i ^ p ^ (n - i)\n[PROOFSTEP]\nsimp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 W_ (ZMod (p ^ (n + 1))) (n + 1) = \u2191(expand p) (W_ (ZMod (p ^ (n + 1))) n)\n[PROOFSTEP]\nsimp only [wittPolynomial_eq_sum_C_mul_X_pow]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2211 x in range (n + 1 + 1), \u2191C (\u2191p ^ x) * X x ^ p ^ (n + 1 - x) =\n    \u2191(expand p) (\u2211 x in range (n + 1), \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x))\n[PROOFSTEP]\nrw [sum_range_succ, \u2190 Nat.cast_pow, CharP.cast_eq_zero (ZMod (p ^ (n + 1))) (p ^ (n + 1)), C_0, zero_mul, add_zero,\n  AlgHom.map_sum, sum_congr rfl]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2208 range (n + 1) \u2192 \u2191C (\u2191p ^ x) * X x ^ p ^ (n + 1 - x) = \u2191(expand p) (\u2191C (\u2191p ^ x) * X x ^ p ^ (n - x))\n[PROOFSTEP]\nintro k hk\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn k : \u2115\nhk : k \u2208 range (n + 1)\n\u22a2 \u2191C (\u2191p ^ k) * X k ^ p ^ (n + 1 - k) = \u2191(expand p) (\u2191C (\u2191p ^ k) * X k ^ p ^ (n - k))\n[PROOFSTEP]\nrw [AlgHom.map_mul, AlgHom.map_pow, expand_X, algHom_C, \u2190 pow_mul, \u2190 pow_succ]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn k : \u2115\nhk : k \u2208 range (n + 1)\n\u22a2 \u2191C (\u2191p ^ k) * X k ^ p ^ (n + 1 - k) = \u2191C (\u2191p ^ k) * X k ^ p ^ (n - k + 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a.e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn k : \u2115\nhk : k \u2208 range (n + 1)\n\u22a2 n + 1 - k = n - k + 1\n[PROOFSTEP]\nrw [mem_range] at hk \n[GOAL]\ncase e_a.e_a.e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nn k : \u2115\nhk : k < n + 1\n\u22a2 n + 1 - k = n - k + 1\n[PROOFSTEP]\nrw [add_comm, add_tsub_assoc_of_le (Nat.lt_succ_iff.mp hk), \u2190 add_comm]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\n\u22a2 vars (W_ R n) = range (n + 1)\n[PROOFSTEP]\nhave : \u2200 i, (monomial (Finsupp.single i (p ^ (n - i))) ((p : R) ^ i)).vars = { i } :=\n  by\n  intro i\n  refine' vars_monomial_single i (pow_ne_zero _ hp.1) _\n  rw [\u2190 Nat.cast_pow, Nat.cast_ne_zero]\n  exact pow_ne_zero i hp.1\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\n\u22a2 \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n[PROOFSTEP]\nintro i\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n[PROOFSTEP]\nrefine' vars_monomial_single i (pow_ne_zero _ hp.1) _\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 \u2191p ^ i \u2260 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_pow, Nat.cast_ne_zero]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn i : \u2115\n\u22a2 p ^ i \u2260 0\n[PROOFSTEP]\nexact pow_ne_zero i hp.1\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n\u22a2 vars (W_ R n) = range (n + 1)\n[PROOFSTEP]\nrw [wittPolynomial, vars_sum_of_disjoint]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n\u22a2 (Finset.biUnion (range (n + 1)) fun i => vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i))) = range (n + 1)\n[PROOFSTEP]\nsimp only [this, biUnion_singleton_eq_self]\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n\u22a2 Pairwise (Disjoint on fun i => vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)))\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\n\u22a2 Pairwise (Disjoint on fun i => {i})\n[PROOFSTEP]\nintro a b h\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 (Disjoint on fun i => {i}) a b\n[PROOFSTEP]\napply disjoint_singleton_left.mpr\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : DecidableEq R\nS : Type u_2\ninst\u271d\u00b9 : CommRing S\nhp : NeZero p\ninst\u271d : CharZero R\nn : \u2115\nthis : \u2200 (i : \u2115), vars (\u2191(monomial (single i (p ^ (n - i)))) (\u2191p ^ i)) = {i}\na b : \u2115\nh : a \u2260 b\n\u22a2 \u00aca \u2208 (fun i => {i}) b\n[PROOFSTEP]\nrwa [mem_singleton]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : NeZero p\nn : \u2115\n\u22a2 vars (W_ R n) \u2286 range (n + 1)\n[PROOFSTEP]\nrw [\u2190 map_wittPolynomial p (Int.castRingHom R), \u2190 wittPolynomial_vars p \u2124]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nS : Type u_2\ninst\u271d : CommRing S\nhp : NeZero p\nn : \u2115\n\u22a2 vars (\u2191(map (Int.castRingHom R)) (W_ \u2124 n)) \u2286 vars (W_ \u2124 n)\n[PROOFSTEP]\napply vars_map\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 xInTermsOfW p R n = (X n - \u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i)) * \u2191C (\u215f\u2191p ^ n)\n[PROOFSTEP]\nrw [xInTermsOfW, \u2190 Fin.sum_univ_eq_sum_range]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2191constantCoeff (xInTermsOfW p R n) = 0\n[PROOFSTEP]\napply Nat.strongInductionOn n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2200 (n : \u2115), (\u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0) \u2192 \u2191constantCoeff (xInTermsOfW p R n) = 0\n[PROOFSTEP]\nclear n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\n\u22a2 \u2200 (n : \u2115), (\u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0) \u2192 \u2191constantCoeff (xInTermsOfW p R n) = 0\n[PROOFSTEP]\nintro n IH\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\n\u22a2 \u2191constantCoeff (xInTermsOfW p R n) = 0\n[PROOFSTEP]\nrw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C, constantCoeff_X, zero_sub,\n  mul_neg, neg_eq_zero]\n  -- porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that\n    -- is created is not what we expect, and the sum is not replaced by zero...\n    -- is it a bug in `rw` tactic?\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\n\u22a2 \u215f\u2191p ^ n * \u2211 x in range n, \u2191constantCoeff (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0\n[PROOFSTEP]\nrefine' Eq.trans (_ : _ = ((\u215f\u2191p : R) ^ n) * 0) (mul_zero _)\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\n\u22a2 \u215f\u2191p ^ n * \u2211 x in range n, \u2191constantCoeff (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = \u215f\u2191p ^ n * 0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\n\u22a2 \u2211 x in range n, \u2191constantCoeff (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0\n[PROOFSTEP]\nrw [sum_eq_zero]\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\n\u22a2 \u2200 (x : \u2115), x \u2208 range n \u2192 \u2191constantCoeff (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0\n[PROOFSTEP]\nintro m H\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\nm : \u2115\nH : m \u2208 range n\n\u22a2 \u2191constantCoeff (\u2191C (\u2191p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0\n[PROOFSTEP]\nrw [mem_range] at H \n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\nm : \u2115\nH : m < n\n\u22a2 \u2191constantCoeff (\u2191C (\u2191p ^ m) * xInTermsOfW p R m ^ p ^ (n - m)) = 0\n[PROOFSTEP]\nsimp only [RingHom.map_mul, RingHom.map_pow, map_natCast, IH m H]\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\nm : \u2115\nH : m < n\n\u22a2 \u2191p ^ m * 0 ^ p ^ (n - m) = 0\n[PROOFSTEP]\nrw [zero_pow, mul_zero]\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\nhp : Fact (Nat.Prime p)\ninst\u271d : Invertible \u2191p\nn : \u2115\nIH : \u2200 (m : \u2115), m < n \u2192 \u2191constantCoeff (xInTermsOfW p R m) = 0\nm : \u2115\nH : m < n\n\u22a2 0 < p ^ (n - m)\n[PROOFSTEP]\napply pow_pos hp.1.pos\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\n\u22a2 xInTermsOfW p R 0 = X 0\n[PROOFSTEP]\nrw [xInTermsOfW_eq, range_zero, sum_empty, pow_zero, C_1, mul_one, sub_zero]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 n \u2208 vars (xInTermsOfW p \u211a n) \u2227 vars (xInTermsOfW p \u211a n) \u2286 range (n + 1)\n[PROOFSTEP]\napply Nat.strongInductionOn n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)) \u2192\n      n \u2208 vars (xInTermsOfW p \u211a n) \u2227 vars (xInTermsOfW p \u211a n) \u2286 range (n + 1)\n[PROOFSTEP]\nclear n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (n : \u2115),\n    (\u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)) \u2192\n      n \u2208 vars (xInTermsOfW p \u211a n) \u2227 vars (xInTermsOfW p \u211a n) \u2286 range (n + 1)\n[PROOFSTEP]\nintro n ih\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 n \u2208 vars (xInTermsOfW p \u211a n) \u2227 vars (xInTermsOfW p \u211a n) \u2286 range (n + 1)\n[PROOFSTEP]\nrw [xInTermsOfW_eq, mul_comm, vars_C_mul _ (nonzero_of_invertible _), vars_sub_of_disjoint, vars_X, range_succ,\n  insert_eq]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2227\n    {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2286 {n} \u222a range n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 Disjoint (vars (X n)) (vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)))\n[PROOFSTEP]\non_goal 1 =>\n  simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]\n  intro i\n  rw [mem_union, mem_union]\n  apply Or.imp id\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2227\n    {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2286 {n} \u222a range n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 Disjoint (vars (X n)) (vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)))\n[PROOFSTEP]\non_goal 1 =>\n  simp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]\n  intro i\n  rw [mem_union, mem_union]\n  apply Or.imp id\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 n \u2208 {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2227\n    {n} \u222a vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)) \u2286 {n} \u222a range n\n[PROOFSTEP]\nsimp only [true_and_iff, true_or_iff, eq_self_iff_true, mem_union, mem_singleton]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 {n} \u222a vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2286 {n} \u222a range n\n[PROOFSTEP]\nintro i\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u222a vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 {n} \u222a range n\n[PROOFSTEP]\nrw [mem_union, mem_union]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 {n} \u2228 i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 {n} \u2228 i \u2208 range n\n[PROOFSTEP]\napply Or.imp id\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 range n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 Disjoint (vars (X n)) (vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)))\n[PROOFSTEP]\non_goal 2 => rw [vars_X, disjoint_singleton_left]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 range n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 Disjoint (vars (X n)) (vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)))\n[PROOFSTEP]\non_goal 2 => rw [vars_X, disjoint_singleton_left]\n[GOAL]\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 Disjoint (vars (X n)) (vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i)))\n[PROOFSTEP]\nrw [vars_X, disjoint_singleton_left]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 range n\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 \u00acn \u2208 vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i))\n[PROOFSTEP]\nall_goals\n  intro H\n  replace H := vars_sum_subset _ _ H\n  rw [mem_biUnion] at H \n  rcases H with \u27e8j, hj, H\u27e9\n  rw [vars_C_mul] at H \n  swap\n  \u00b7 apply pow_ne_zero\n    exact_mod_cast hp.1.ne_zero\n  rw [mem_range] at hj \n  replace H := (ih j hj).2 (vars_pow _ _ H)\n  rw [mem_range] at H \n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\n\u22a2 i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x)) \u2192 i \u2208 range n\n[PROOFSTEP]\nintro H\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\nH : i \u2208 vars (\u2211 x in range n, \u2191C (\u2191p ^ x) * xInTermsOfW p \u211a x ^ p ^ (n - x))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nreplace H := vars_sum_subset _ _ H\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\nH : i \u2208 Finset.biUnion (range n) fun i => vars (\u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrw [mem_biUnion] at H \n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni : \u2115\nH : \u2203 a, a \u2208 range n \u2227 i \u2208 vars (\u2191C (\u2191p ^ a) * xInTermsOfW p \u211a a ^ p ^ (n - a))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrcases H with \u27e8j, hj, H\u27e9\n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrw [vars_C_mul] at H \n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 i \u2208 range n\ncase intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p ^ j \u2260 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p ^ j \u2260 0\n[PROOFSTEP]\napply pow_ne_zero\n[GOAL]\ncase intro.intro.ha.h\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j \u2208 range n\nH : i \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrw [mem_range] at hj \n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nreplace H := (ih j hj).2 (vars_pow _ _ H)\n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i \u2208 range (j + 1)\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrw [mem_range] at H \n[GOAL]\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\n\u22a2 \u00acn \u2208 vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i))\n[PROOFSTEP]\nintro H\n[GOAL]\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nH : n \u2208 vars (\u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i))\n\u22a2 False\n[PROOFSTEP]\nreplace H := vars_sum_subset _ _ H\n[GOAL]\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nH : n \u2208 Finset.biUnion (range n) fun i => vars (\u2191C (\u2191p ^ i) * xInTermsOfW p \u211a i ^ p ^ (n - i))\n\u22a2 False\n[PROOFSTEP]\nrw [mem_biUnion] at H \n[GOAL]\ncase hpq\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nH : \u2203 a, a \u2208 range n \u2227 n \u2208 vars (\u2191C (\u2191p ^ a) * xInTermsOfW p \u211a a ^ p ^ (n - a))\n\u22a2 False\n[PROOFSTEP]\nrcases H with \u27e8j, hj, H\u27e9\n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 False\n[PROOFSTEP]\nrw [vars_C_mul] at H \n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 False\ncase hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p ^ j \u2260 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase hpq.intro.intro.ha\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p ^ j \u2260 0\n[PROOFSTEP]\napply pow_ne_zero\n[GOAL]\ncase hpq.intro.intro.ha.h\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (\u2191C (\u2191p ^ j) * xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j \u2208 range n\nH : n \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 False\n[PROOFSTEP]\nrw [mem_range] at hj \n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 vars (xInTermsOfW p \u211a j ^ p ^ (n - j))\n\u22a2 False\n[PROOFSTEP]\nreplace H := (ih j hj).2 (vars_pow _ _ H)\n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n \u2208 range (j + 1)\n\u22a2 False\n[PROOFSTEP]\nrw [mem_range] at H \n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i < j + 1\n\u22a2 i \u2208 range n\n[PROOFSTEP]\nrw [mem_range]\n[GOAL]\ncase intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\ni j : \u2115\nhj : j < n\nH : i < j + 1\n\u22a2 i < n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hpq.intro.intro\np : \u2115\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : DecidableEq R\nhp : Fact (Nat.Prime p)\nn : \u2115\nih : \u2200 (m : \u2115), m < n \u2192 m \u2208 vars (xInTermsOfW p \u211a m) \u2227 vars (xInTermsOfW p \u211a m) \u2286 range (m + 1)\nj : \u2115\nhj : j < n\nH : n < j + 1\n\u22a2 False\n[PROOFSTEP]\nlinarith\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 xInTermsOfW p R n * \u2191C (\u2191p ^ n) = X n - \u2211 i in range n, \u2191C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i)\n[PROOFSTEP]\nrw [xInTermsOfW_eq, mul_assoc, \u2190 C_mul, \u2190 mul_pow, invOf_mul_self, one_pow, C_1, mul_one]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nk : \u2115\n\u22a2 \u2191(bind\u2081 (xInTermsOfW p R)) (W_ R k) = X k\n[PROOFSTEP]\nrw [wittPolynomial_eq_sum_C_mul_X_pow, AlgHom.map_sum]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nk : \u2115\n\u22a2 \u2211 x in range (k + 1), \u2191(bind\u2081 (xInTermsOfW p R)) (\u2191C (\u2191p ^ x) * X x ^ p ^ (k - x)) = X k\n[PROOFSTEP]\nsimp only [Nat.cast_pow, AlgHom.map_pow, C_pow, AlgHom.map_mul, algHom_C]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nk : \u2115\n\u22a2 \u2211 x in range (k + 1), \u2191C \u2191p ^ x * \u2191(bind\u2081 (xInTermsOfW p R)) (X x) ^ p ^ (k - x) = X k\n[PROOFSTEP]\nrw [sum_range_succ_comm, tsub_self, pow_zero, pow_one, bind\u2081_X_right, mul_comm, \u2190 C_pow, xInTermsOfW_aux]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nk : \u2115\n\u22a2 X k - \u2211 i in range k, \u2191C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (k - i) +\n      \u2211 x in range k, \u2191C \u2191p ^ x * \u2191(bind\u2081 (xInTermsOfW p R)) (X x) ^ p ^ (k - x) =\n    X k\n[PROOFSTEP]\nsimp only [Nat.cast_pow, C_pow, bind\u2081_X_right, sub_add_cancel]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n\n[PROOFSTEP]\napply Nat.strongInductionOn n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\n\u22a2 \u2200 (n : \u2115), (\u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n\n[PROOFSTEP]\nclear n\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\n\u22a2 \u2200 (n : \u2115), (\u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m) \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n\n[PROOFSTEP]\nintro n H\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n\n[PROOFSTEP]\nrw [xInTermsOfW_eq, AlgHom.map_mul, AlgHom.map_sub, bind\u2081_X_right, algHom_C, AlgHom.map_sum,\n  show X n = (X n * C ((p : R) ^ n)) * C ((\u215fp : R) ^ n) by\n    rw [mul_assoc, \u2190 C_mul, \u2190 mul_pow, mul_invOf_self, one_pow, map_one, mul_one]]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 X n = X n * \u2191C (\u2191p ^ n) * \u2191C (\u215f\u2191p ^ n)\n[PROOFSTEP]\nrw [mul_assoc, \u2190 C_mul, \u2190 mul_pow, mul_invOf_self, one_pow, map_one, mul_one]\n[GOAL]\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 (W_ R n - \u2211 x in range n, \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))) * \u2191C (\u215f\u2191p ^ n) =\n    X n * \u2191C (\u2191p ^ n) * \u2191C (\u215f\u2191p ^ n)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 W_ R n - \u2211 x in range n, \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = X n * \u2191C (\u2191p ^ n)\n[PROOFSTEP]\nrw [wittPolynomial_eq_sum_C_mul_X_pow, sum_range_succ_comm, tsub_self, pow_zero, pow_one, mul_comm (X n), add_sub_assoc,\n  add_right_eq_self, sub_eq_zero]\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2211 x in range n, \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x) =\n    \u2211 x in range n, \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 range n \u2192 \u2191C (\u2191p ^ x) * X x ^ p ^ (n - x) = \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ x) * xInTermsOfW p R x ^ p ^ (n - x))\n[PROOFSTEP]\nintro i h\n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i \u2208 range n\n\u22a2 \u2191C (\u2191p ^ i) * X i ^ p ^ (n - i) = \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))\n[PROOFSTEP]\nrw [mem_range] at h \n[GOAL]\ncase e_a\np : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : Invertible \u2191p\nn : \u2115\nH : \u2200 (m : \u2115), m < n \u2192 \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R m) = X m\ni : \u2115\nh : i < n\n\u22a2 \u2191C (\u2191p ^ i) * X i ^ p ^ (n - i) = \u2191(bind\u2081 (W_ R)) (\u2191C (\u2191p ^ i) * xInTermsOfW p R i ^ p ^ (n - i))\n[PROOFSTEP]\nrw [AlgHom.map_mul, AlgHom.map_pow, algHom_C, H i h]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.WittPolynomial", "llama_tokens": 18767, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333246118695629, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5368486368193818}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 legendreSym p 2 = \u2191\u03c7\u2088 \u2191p\n[PROOFSTEP]\nhave : (2 : ZMod p) = (2 : \u2124) := by norm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 2 = \u21912\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nthis : 2 = \u21912\n\u22a2 legendreSym p 2 = \u2191\u03c7\u2088 \u2191p\n[PROOFSTEP]\nrw [legendreSym, \u2190 this, quadraticChar_two ((ringChar_zmod_n p).substr hp), card p]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 legendreSym p (-2) = \u2191\u03c7\u2088' \u2191p\n[PROOFSTEP]\nhave : (-2 : ZMod p) = (-2 : \u2124) := by norm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 -2 = \u2191(-2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nthis : -2 = \u2191(-2)\n\u22a2 legendreSym p (-2) = \u2191\u03c7\u2088' \u2191p\n[PROOFSTEP]\nrw [legendreSym, \u2190 this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 IsSquare 2 \u2194 p % 8 = 1 \u2228 p % 8 = 7\n[PROOFSTEP]\nrw [FiniteField.isSquare_two_iff, card p]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 p % 8 \u2260 3 \u2227 p % 8 \u2260 5 \u2194 p % 8 = 1 \u2228 p % 8 = 7\n[PROOFSTEP]\nhave h\u2081 := Prime.mod_two_eq_one_iff_ne_two.mpr hp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 2 = 1\n\u22a2 p % 8 \u2260 3 \u2227 p % 8 \u2260 5 \u2194 p % 8 = 1 \u2228 p % 8 = 7\n[PROOFSTEP]\nrw [\u2190 mod_mod_of_dvd p (by norm_num : 2 \u2223 8)] at h\u2081 \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 2 = 1\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\n\u22a2 p % 8 \u2260 3 \u2227 p % 8 \u2260 5 \u2194 p % 8 = 1 \u2228 p % 8 = 7\n[PROOFSTEP]\nhave h\u2082 := mod_lt p (by norm_num : 0 < 8)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\n\u22a2 0 < 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\nh\u2082 : p % 8 < 8\n\u22a2 p % 8 \u2260 3 \u2227 p % 8 \u2260 5 \u2194 p % 8 = 1 \u2228 p % 8 = 7\n[PROOFSTEP]\nrevert h\u2082 h\u2081\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 p % 8 % 2 = 1 \u2192 p % 8 < 8 \u2192 (p % 8 \u2260 3 \u2227 p % 8 \u2260 5 \u2194 p % 8 = 1 \u2228 p % 8 = 7)\n[PROOFSTEP]\ngeneralize p % 8 = m\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nm : \u2115\n\u22a2 m % 2 = 1 \u2192 m < 8 \u2192 (m \u2260 3 \u2227 m \u2260 5 \u2194 m = 1 \u2228 m = 7)\n[PROOFSTEP]\nclear! p\n[GOAL]\nm : \u2115\n\u22a2 m % 2 = 1 \u2192 m < 8 \u2192 (m \u2260 3 \u2227 m \u2260 5 \u2194 m = 1 \u2228 m = 7)\n[PROOFSTEP]\nintros\n[GOAL]\nm : \u2115\nh\u2081\u271d : m % 2 = 1\nh\u2082\u271d : m < 8\n\u22a2 m \u2260 3 \u2227 m \u2260 5 \u2194 m = 1 \u2228 m = 7\n[PROOFSTEP]\ninterval_cases m\n[GOAL]\ncase \u00ab0\u00bb\nm : \u2115\nh\u2081\u271d : 0 % 2 = 1\nh\u2082\u271d : 0 < 8\n\u22a2 0 \u2260 3 \u2227 0 \u2260 5 \u2194 0 = 1 \u2228 0 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab1\u00bb\nm : \u2115\nh\u2081\u271d : 1 % 2 = 1\nh\u2082\u271d : 1 < 8\n\u22a2 1 \u2260 3 \u2227 1 \u2260 5 \u2194 1 = 1 \u2228 1 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab2\u00bb\nm : \u2115\nh\u2081\u271d : 2 % 2 = 1\nh\u2082\u271d : 2 < 8\n\u22a2 2 \u2260 3 \u2227 2 \u2260 5 \u2194 2 = 1 \u2228 2 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab3\u00bb\nm : \u2115\nh\u2081\u271d : 3 % 2 = 1\nh\u2082\u271d : 3 < 8\n\u22a2 3 \u2260 3 \u2227 3 \u2260 5 \u2194 3 = 1 \u2228 3 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab4\u00bb\nm : \u2115\nh\u2081\u271d : 4 % 2 = 1\nh\u2082\u271d : 4 < 8\n\u22a2 4 \u2260 3 \u2227 4 \u2260 5 \u2194 4 = 1 \u2228 4 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab5\u00bb\nm : \u2115\nh\u2081\u271d : 5 % 2 = 1\nh\u2082\u271d : 5 < 8\n\u22a2 5 \u2260 3 \u2227 5 \u2260 5 \u2194 5 = 1 \u2228 5 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab6\u00bb\nm : \u2115\nh\u2081\u271d : 6 % 2 = 1\nh\u2082\u271d : 6 < 8\n\u22a2 6 \u2260 3 \u2227 6 \u2260 5 \u2194 6 = 1 \u2228 6 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab7\u00bb\nm : \u2115\nh\u2081\u271d : 7 % 2 = 1\nh\u2082\u271d : 7 < 8\n\u22a2 7 \u2260 3 \u2227 7 \u2260 5 \u2194 7 = 1 \u2228 7 = 7\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 IsSquare (-2) \u2194 p % 8 = 1 \u2228 p % 8 = 3\n[PROOFSTEP]\nrw [FiniteField.isSquare_neg_two_iff, card p]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 p % 8 \u2260 5 \u2227 p % 8 \u2260 7 \u2194 p % 8 = 1 \u2228 p % 8 = 3\n[PROOFSTEP]\nhave h\u2081 := Prime.mod_two_eq_one_iff_ne_two.mpr hp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 2 = 1\n\u22a2 p % 8 \u2260 5 \u2227 p % 8 \u2260 7 \u2194 p % 8 = 1 \u2228 p % 8 = 3\n[PROOFSTEP]\nrw [\u2190 mod_mod_of_dvd p (by norm_num : 2 \u2223 8)] at h\u2081 \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 2 = 1\n\u22a2 2 \u2223 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\n\u22a2 p % 8 \u2260 5 \u2227 p % 8 \u2260 7 \u2194 p % 8 = 1 \u2228 p % 8 = 3\n[PROOFSTEP]\nhave h\u2082 := mod_lt p (by norm_num : 0 < 8)\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\n\u22a2 0 < 8\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nh\u2081 : p % 8 % 2 = 1\nh\u2082 : p % 8 < 8\n\u22a2 p % 8 \u2260 5 \u2227 p % 8 \u2260 7 \u2194 p % 8 = 1 \u2228 p % 8 = 3\n[PROOFSTEP]\nrevert h\u2082 h\u2081\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\n\u22a2 p % 8 % 2 = 1 \u2192 p % 8 < 8 \u2192 (p % 8 \u2260 5 \u2227 p % 8 \u2260 7 \u2194 p % 8 = 1 \u2228 p % 8 = 3)\n[PROOFSTEP]\ngeneralize p % 8 = m\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nhp : p \u2260 2\nm : \u2115\n\u22a2 m % 2 = 1 \u2192 m < 8 \u2192 (m \u2260 5 \u2227 m \u2260 7 \u2194 m = 1 \u2228 m = 3)\n[PROOFSTEP]\nclear! p\n[GOAL]\nm : \u2115\n\u22a2 m % 2 = 1 \u2192 m < 8 \u2192 (m \u2260 5 \u2227 m \u2260 7 \u2194 m = 1 \u2228 m = 3)\n[PROOFSTEP]\nintros\n[GOAL]\nm : \u2115\nh\u2081\u271d : m % 2 = 1\nh\u2082\u271d : m < 8\n\u22a2 m \u2260 5 \u2227 m \u2260 7 \u2194 m = 1 \u2228 m = 3\n[PROOFSTEP]\ninterval_cases m\n[GOAL]\ncase \u00ab0\u00bb\nm : \u2115\nh\u2081\u271d : 0 % 2 = 1\nh\u2082\u271d : 0 < 8\n\u22a2 0 \u2260 5 \u2227 0 \u2260 7 \u2194 0 = 1 \u2228 0 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab1\u00bb\nm : \u2115\nh\u2081\u271d : 1 % 2 = 1\nh\u2082\u271d : 1 < 8\n\u22a2 1 \u2260 5 \u2227 1 \u2260 7 \u2194 1 = 1 \u2228 1 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab2\u00bb\nm : \u2115\nh\u2081\u271d : 2 % 2 = 1\nh\u2082\u271d : 2 < 8\n\u22a2 2 \u2260 5 \u2227 2 \u2260 7 \u2194 2 = 1 \u2228 2 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab3\u00bb\nm : \u2115\nh\u2081\u271d : 3 % 2 = 1\nh\u2082\u271d : 3 < 8\n\u22a2 3 \u2260 5 \u2227 3 \u2260 7 \u2194 3 = 1 \u2228 3 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab4\u00bb\nm : \u2115\nh\u2081\u271d : 4 % 2 = 1\nh\u2082\u271d : 4 < 8\n\u22a2 4 \u2260 5 \u2227 4 \u2260 7 \u2194 4 = 1 \u2228 4 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab5\u00bb\nm : \u2115\nh\u2081\u271d : 5 % 2 = 1\nh\u2082\u271d : 5 < 8\n\u22a2 5 \u2260 5 \u2227 5 \u2260 7 \u2194 5 = 1 \u2228 5 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab6\u00bb\nm : \u2115\nh\u2081\u271d : 6 % 2 = 1\nh\u2082\u271d : 6 < 8\n\u22a2 6 \u2260 5 \u2227 6 \u2260 7 \u2194 6 = 1 \u2228 6 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\ncase \u00ab7\u00bb\nm : \u2115\nh\u2081\u271d : 7 % 2 = 1\nh\u2082\u271d : 7 < 8\n\u22a2 7 \u2260 5 \u2227 7 \u2260 7 \u2194 7 = 1 \u2228 7 = 3\n[PROOFSTEP]\nsimp_all\n  -- Porting note: was `decide!`\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave hp\u2081 := (Prime.eq_two_or_odd <| @Fact.out p.Prime _).resolve_left hp\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave hq\u2081 := (Prime.eq_two_or_odd <| @Fact.out q.Prime _).resolve_left hq\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave hq\u2082 : ringChar (ZMod q) \u2260 2 := (ringChar_zmod_n q).substr hq\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave h := quadraticChar_odd_prime ((ringChar_zmod_n p).substr hp) hq ((ringChar_zmod_n p).substr hpq)\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh :\n  \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card (ZMod p))) * \u2191(Fintype.card (ZMod p)))\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nrw [card p] at h \n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh : \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191p) * \u2191p)\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave nc : \u2200 n r : \u2115, ((n : \u2124) : ZMod r) = n := fun n r => by norm_cast\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh : \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191p) * \u2191p)\nn r : \u2115\n\u22a2 \u2191\u2191n = \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh : \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191p) * \u2191p)\nnc : \u2200 (n r : \u2115), \u2191\u2191n = \u2191n\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nhave nc' : (((-1) ^ (p / 2) : \u2124) : ZMod q) = (-1) ^ (p / 2) := by norm_cast\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh : \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191p) * \u2191p)\nnc : \u2200 (n r : \u2115), \u2191\u2191n = \u2191n\n\u22a2 \u2191((-1) ^ (p / 2)) = (-1) ^ (p / 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nhpq : p \u2260 q\nhp\u2081 : p % 2 = 1\nhq\u2081 : q % 2 = 1\nhq\u2082 : ringChar (ZMod q) \u2260 2\nh : \u2191(quadraticChar (ZMod p)) \u2191q = \u2191(quadraticChar (ZMod q)) (\u2191(\u2191\u03c7\u2084 \u2191p) * \u2191p)\nnc : \u2200 (n r : \u2115), \u2191\u2191n = \u2191n\nnc' : \u2191((-1) ^ (p / 2)) = (-1) ^ (p / 2)\n\u22a2 legendreSym q \u2191p * legendreSym p \u2191q = (-1) ^ (p / 2 * (q / 2))\n[PROOFSTEP]\nrw [legendreSym, legendreSym, nc, nc, h, map_mul, mul_rotate', mul_comm (p / 2), \u2190 pow_two,\n  quadraticChar_sq_one (prime_ne_zero q p hpq.symm), mul_one, pow_mul, \u03c7\u2084_eq_neg_one_pow hp\u2081, nc', map_pow,\n  quadraticChar_neg_one hq\u2082, card q, \u03c7\u2084_eq_neg_one_pow hq\u2081]\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\n\u22a2 legendreSym q \u2191p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p \u2191q\n[PROOFSTEP]\ncases' eq_or_ne p q with h h\n[GOAL]\ncase inl\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nh : p = q\n\u22a2 legendreSym q \u2191p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p \u2191q\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase inl\nq : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime q)\nhq : q \u2260 2\ninst\u271d : Fact (Nat.Prime q)\nhp : q \u2260 2\n\u22a2 legendreSym q \u2191q = (-1) ^ (q / 2 * (q / 2)) * legendreSym q \u2191q\n[PROOFSTEP]\nrw [(eq_zero_iff q q).mpr (by exact_mod_cast nat_cast_self q), mul_zero]\n[GOAL]\nq : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime q)\nhq : q \u2260 2\ninst\u271d : Fact (Nat.Prime q)\nhp : q \u2260 2\n\u22a2 \u2191\u2191q = 0\n[PROOFSTEP]\nexact_mod_cast nat_cast_self q\n[GOAL]\ncase inr\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nh : p \u2260 q\n\u22a2 legendreSym q \u2191p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p \u2191q\n[PROOFSTEP]\nhave qr := congr_arg (\u00b7 * legendreSym p q) (quadratic_reciprocity hp hq h)\n[GOAL]\ncase inr\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nh : p \u2260 q\nqr :\n  (fun x => x * legendreSym p \u2191q) (legendreSym q \u2191p * legendreSym p \u2191q) =\n    (fun x => x * legendreSym p \u2191q) ((-1) ^ (p / 2 * (q / 2)))\n\u22a2 legendreSym q \u2191p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p \u2191q\n[PROOFSTEP]\nhave : ((q : \u2124) : ZMod p) \u2260 0 := by exact_mod_cast prime_ne_zero p q h\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nh : p \u2260 q\nqr :\n  (fun x => x * legendreSym p \u2191q) (legendreSym q \u2191p * legendreSym p \u2191q) =\n    (fun x => x * legendreSym p \u2191q) ((-1) ^ (p / 2 * (q / 2)))\n\u22a2 \u2191\u2191q \u2260 0\n[PROOFSTEP]\nexact_mod_cast prime_ne_zero p q h\n[GOAL]\ncase inr\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p \u2260 2\nhq : q \u2260 2\nh : p \u2260 q\nqr :\n  (fun x => x * legendreSym p \u2191q) (legendreSym q \u2191p * legendreSym p \u2191q) =\n    (fun x => x * legendreSym p \u2191q) ((-1) ^ (p / 2 * (q / 2)))\nthis : \u2191\u2191q \u2260 0\n\u22a2 legendreSym q \u2191p = (-1) ^ (p / 2 * (q / 2)) * legendreSym p \u2191q\n[PROOFSTEP]\nsimpa only [mul_assoc, \u2190 pow_two, sq_one p this, mul_one] using qr\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p % 4 = 1\nhq : q \u2260 2\n\u22a2 legendreSym q \u2191p = legendreSym p \u2191q\n[PROOFSTEP]\nrw [quadratic_reciprocity' (Prime.mod_two_eq_one_iff_ne_two.mp (odd_of_mod_four_eq_one hp)) hq, pow_mul,\n  neg_one_pow_div_two_of_one_mod_four hp, one_pow, one_mul]\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p % 4 = 3\nhq : q % 4 = 3\n\u22a2 legendreSym q \u2191p = -legendreSym p \u2191q\n[PROOFSTEP]\nlet nop := @neg_one_pow_div_two_of_three_mod_four\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p % 4 = 3\nhq : q % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 legendreSym q \u2191p = -legendreSym p \u2191q\n[PROOFSTEP]\nrw [quadratic_reciprocity', pow_mul, nop hp, nop hq, neg_one_mul]\n[GOAL]\ncase hp\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p % 4 = 3\nhq : q % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 p \u2260 2\n[PROOFSTEP]\nrwa [\u2190 Prime.mod_two_eq_one_iff_ne_two, odd_of_mod_four_eq_three]\n[GOAL]\ncase hq\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp : p % 4 = 3\nhq : q % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 q \u2260 2\n[PROOFSTEP]\nrwa [\u2190 Prime.mod_two_eq_one_iff_ne_two, odd_of_mod_four_eq_three]\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp1 : p % 4 = 1\nhq1 : q \u2260 2\n\u22a2 IsSquare \u2191q \u2194 IsSquare \u2191p\n[PROOFSTEP]\ncases' eq_or_ne p q with h h\n[GOAL]\ncase inl\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp1 : p % 4 = 1\nhq1 : q \u2260 2\nh : p = q\n\u22a2 IsSquare \u2191q \u2194 IsSquare \u2191p\n[PROOFSTEP]\nsubst p\n[GOAL]\ncase inl\nq : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime q)\nhq1 : q \u2260 2\ninst\u271d : Fact (Nat.Prime q)\nhp1 : q % 4 = 1\n\u22a2 IsSquare \u2191q \u2194 IsSquare \u2191q\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp1 : p % 4 = 1\nhq1 : q \u2260 2\nh : p \u2260 q\n\u22a2 IsSquare \u2191q \u2194 IsSquare \u2191p\n[PROOFSTEP]\nrw [\u2190 eq_one_iff' p (prime_ne_zero p q h), \u2190 eq_one_iff' q (prime_ne_zero q p h.symm),\n  quadratic_reciprocity_one_mod_four hp1 hq1]\n[GOAL]\np q : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : Fact (Nat.Prime q)\nhp3 : p % 4 = 3\nhq3 : q % 4 = 3\nhpq : p \u2260 q\n\u22a2 IsSquare \u2191q \u2194 \u00acIsSquare \u2191p\n[PROOFSTEP]\nrw [\u2190 eq_one_iff' p (prime_ne_zero p q hpq), \u2190 eq_neg_one_iff' q, quadratic_reciprocity_three_mod_four hp3 hq3, neg_inj]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity", "llama_tokens": 8665, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152325073083131, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.5367633443743485}}
{"text": "[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : LinearIndependent R b\n\u22a2 LinearIndependent R\u209b b\n[PROOFSTEP]\nrw [linearIndependent_iff'] at hli \u22a2\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n\u22a2 \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R\u209b), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s g hg i hi\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nchoose! a g' hg' using IsLocalization.exist_integer_multiples S s g\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\n\u22a2 g i = 0\n[PROOFSTEP]\nspecialize hli s g' _ i hi\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\n\u22a2 \u2211 i in s, g' i \u2022 b i = 0\n[PROOFSTEP]\nrw [\u2190 @smul_zero _ M _ _ (a : R), \u2190 hg, Finset.smul_sum]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\n\u22a2 \u2211 i in s, g' i \u2022 b i = \u2211 x in s, \u2191a \u2022 g x \u2022 b x\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\nhli : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 b i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni\u271d : \u03b9\nhi\u271d : i\u271d \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g' i \u2022 b i = \u2191a \u2022 g i \u2022 b i\n[PROOFSTEP]\nrw [\u2190 IsScalarTower.algebraMap_smul R\u209b, hg' i hi, smul_assoc]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhli : g' i = 0\n\u22a2 g i = 0\n[PROOFSTEP]\nrefine' (IsLocalization.map_units R\u209b a).mul_right_eq_zero.mp _\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u209b\ninst\u271d\u2074 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nM : Type u_3\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R\u209b M\ninst\u271d : IsScalarTower R R\u209b M\n\u03b9 : Type u_4\nb : \u03b9 \u2192 M\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 b i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhli : g' i = 0\n\u22a2 \u2191(algebraMap R R\u209b) \u2191a * g i = 0\n[PROOFSTEP]\nrw [\u2190 Algebra.smul_def, \u2190 map_zero (algebraMap R R\u209b), \u2190 hli, hg' i hi]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : LinearIndependent R v\n\u22a2 LinearIndependent R\u209b (\u2191(algebraMap A A\u209b) \u2218 v)\n[PROOFSTEP]\nrw [linearIndependent_iff'] at hv \u22a2\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n\u22a2 \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R\u209b), \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s g hg i hi\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nchoose! a g' hg' using IsLocalization.exist_integer_multiples S s g\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\n\u22a2 g i = 0\n[PROOFSTEP]\nhave h0 : algebraMap A A\u209b (\u2211 i in s, g' i \u2022 v i) = 0 :=\n  by\n  apply_fun (\u00b7 \u2022 \u00b7) (a : R) at hg \n  rw [smul_zero, Finset.smul_sum] at hg \n  rw [map_sum, \u2190 hg]\n  refine' Finset.sum_congr rfl fun i hi => _\n  rw [\u2190 smul_assoc, \u2190 hg' i hi, Algebra.smul_def, map_mul, \u2190 IsScalarTower.algebraMap_apply, \u2190 Algebra.smul_def,\n    algebraMap_smul, Function.comp_apply]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\n\u22a2 \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\n[PROOFSTEP]\napply_fun (\u00b7 \u2022 \u00b7) (a : R) at hg \n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhg : \u2191a \u2022 \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = \u2191a \u2022 0\n\u22a2 \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\n[PROOFSTEP]\nrw [smul_zero, Finset.smul_sum] at hg \n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhg : \u2211 x in s, \u2191a \u2022 g x \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) x = 0\n\u22a2 \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\n[PROOFSTEP]\nrw [map_sum, \u2190 hg]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhg : \u2211 x in s, \u2191a \u2022 g x \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) x = 0\n\u22a2 \u2211 x in s, \u2191(algebraMap A A\u209b) (g' x \u2022 v x) = \u2211 x in s, \u2191a \u2022 g x \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) x\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\ni\u271d : \u03b9\nhi\u271d : i\u271d \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nhg : \u2211 x in s, \u2191a \u2022 g x \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) x = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(algebraMap A A\u209b) (g' i \u2022 v i) = \u2191a \u2022 g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i\n[PROOFSTEP]\nrw [\u2190 smul_assoc, \u2190 hg' i hi, Algebra.smul_def, map_mul, \u2190 IsScalarTower.algebraMap_apply, \u2190 Algebra.smul_def,\n  algebraMap_smul, Function.comp_apply]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nh0 : \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\n\u22a2 g i = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8_, r, hrS, rfl\u27e9, hr : algebraMap R A r * _ = 0\u27e9 :=\n  (IsLocalization.map_eq_zero_iff (Algebra.algebraMapSubmonoid A S) _ _).1 h0\n[GOAL]\ncase intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nh0 : \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\nr : R\nhrS : r \u2208 \u2191S\nhr : \u2191(algebraMap R A) r * \u2211 i in s, g' i \u2022 v i = 0\n\u22a2 g i = 0\n[PROOFSTEP]\nsimp_rw [Finset.mul_sum, \u2190 Algebra.smul_def, smul_smul] at hr \n[GOAL]\ncase intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nh0 : \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\nr : R\nhrS : r \u2208 \u2191S\nhr : \u2211 x in s, (r * g' x) \u2022 v x = 0\n\u22a2 g i = 0\n[PROOFSTEP]\nspecialize hv s _ hr i hi\n[GOAL]\ncase intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nh0 : \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\nr : R\nhrS : r \u2208 \u2191S\nhr : \u2211 x in s, (r * g' x) \u2022 v x = 0\nhv : r * g' i = 0\n\u22a2 g i = 0\n[PROOFSTEP]\nrw [\u2190 (IsLocalization.map_units R\u209b a).mul_right_eq_zero, \u2190 Algebra.smul_def, \u2190 hg' i hi]\n[GOAL]\ncase intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nv : \u03b9 \u2192 A\ns : Finset \u03b9\ng : \u03b9 \u2192 R\u209b\nhg : \u2211 i in s, g i \u2022 (\u2191(algebraMap A A\u209b) \u2218 v) i = 0\ni : \u03b9\nhi : i \u2208 s\na : { x // x \u2208 S }\ng' : \u03b9 \u2192 R\nhg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R R\u209b) (g' i) = \u2191a \u2022 g i\nh0 : \u2191(algebraMap A A\u209b) (\u2211 i in s, g' i \u2022 v i) = 0\nr : R\nhrS : r \u2208 \u2191S\nhr : \u2211 x in s, (r * g' x) \u2022 v x = 0\nhv : r * g' i = 0\n\u22a2 \u2191(algebraMap R R\u209b) (g' i) = 0\n[PROOFSTEP]\nexact (IsLocalization.map_eq_zero_iff S _ _).2 \u27e8\u27e8r, hrS\u27e9, hv\u27e9\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\n\u22a2 span R\u209b (\u2191(algebraMap A A\u209b) '' v) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\n\u22a2 \u22a4 \u2264 span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nrintro a' -\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na' : A\u209b\n\u22a2 a' \u2208 span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nobtain \u27e8a, \u27e8_, s, hs, rfl\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective (Algebra.algebraMapSubmonoid A S) a'\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 IsLocalization.mk' A\u209b a\n      { val := \u2191(algebraMap R A) s, property := (_ : \u2203 a, a \u2208 \u2191S \u2227 \u2191(algebraMap R A) a = \u2191(algebraMap R A) s) } \u2208\n    span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nrw [IsLocalization.mk'_eq_mul_mk'_one, mul_comm, \u2190 map_one (algebraMap R A)]\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 IsLocalization.mk' A\u209b (\u2191(algebraMap R A) 1)\n        { val := \u2191(algebraMap R A) s, property := (_ : \u2203 a, a \u2208 \u2191S \u2227 \u2191(algebraMap R A) a = \u2191(algebraMap R A) s) } *\n      \u2191(algebraMap A A\u209b) a \u2208\n    span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nerw [\u2190 IsLocalization.algebraMap_mk' A R\u209b A\u209b (1 : R) \u27e8s, hs\u27e9]\n  -- `erw` needed to unify `\u27e8s, hs\u27e9`\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 \u2191(algebraMap R\u209b A\u209b) (IsLocalization.mk' R\u209b 1 { val := s, property := hs }) * \u2191(algebraMap A A\u209b) a \u2208\n    span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nrw [\u2190 Algebra.smul_def]\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 IsLocalization.mk' R\u209b 1 { val := s, property := hs } \u2022 \u2191(algebraMap A A\u209b) a \u2208 span R\u209b (\u2191(algebraMap A A\u209b) '' v)\n[PROOFSTEP]\nrefine' smul_mem _ _ (span_subset_span R R\u209b _ _)\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 \u2191(algebraMap A A\u209b) a \u2208 \u2191(span R (\u2191(algebraMap A A\u209b) '' v))\n[PROOFSTEP]\nrw [\u2190 Algebra.coe_linearMap, \u2190 LinearMap.coe_restrictScalars R, \u2190 LinearMap.map_span]\n[GOAL]\ncase intro.intro.mk.intro.intro\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\nv : Set A\nhv : span R v = \u22a4\na : A\ns : R\nhs : s \u2208 \u2191S\n\u22a2 \u2191(\u2191R (Algebra.linearMap A A\u209b)) a \u2208 \u2191(map (\u2191R (Algebra.linearMap A A\u209b)) (span R v))\n[PROOFSTEP]\nexact mem_map_of_mem (hv.symm \u25b8 mem_top)\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\n\u22a2 \u22a4 \u2264 span R\u209b (Set.range (\u2191(algebraMap A A\u209b) \u2218 \u2191b))\n[PROOFSTEP]\nrw [Set.range_comp, SpanEqTop.localization_localization R\u209b S A\u209b b.span_eq]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx : A\ni : \u03b9\n\u22a2 \u2191(\u2191(localizationLocalization R\u209b S A\u209b b).repr (\u2191(algebraMap A A\u209b) x)) i =\n    \u2191(\u2191(localizationLocalization R\u209b S A\u209b b).repr\n          (Finsupp.sum (\u2191b.repr x) fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(algebraMap A A\u209b) (\u2191b j)))\n      i\n[PROOFSTEP]\nsimp_rw [IsScalarTower.algebraMap_smul, Algebra.smul_def, IsScalarTower.algebraMap_apply R A A\u209b, \u2190 _root_.map_mul, \u2190\n  map_finsupp_sum, \u2190 Algebra.smul_def, \u2190 Finsupp.total_apply, Basis.total_repr]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx : A\ni : \u03b9\n\u22a2 \u2191(\u2191(localizationLocalization R\u209b S A\u209b b).repr\n          (Finsupp.sum (\u2191b.repr x) fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(algebraMap A A\u209b) (\u2191b j)))\n      i =\n    Finsupp.sum (\u2191b.repr x) fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(Finsupp.single j 1) i\n[PROOFSTEP]\nsimp_rw [\u2190 b.localizationLocalization_apply R\u209b S A\u209b, map_finsupp_sum, LinearEquiv.map_smul, Basis.repr_self,\n  Finsupp.sum_apply, Finsupp.smul_apply]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx : A\ni j : \u03b9\nx\u271d : j \u2208 (\u2191b.repr x).support\nhj : j \u2260 i\n\u22a2 (fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(Finsupp.single j 1) i) j (\u2191(\u2191b.repr x) j) = 0\n[PROOFSTEP]\nsimp [hj]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx : A\ni : \u03b9\nhi : \u00aci \u2208 (\u2191b.repr x).support\n\u22a2 (fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(Finsupp.single j 1) i) i (\u2191(\u2191b.repr x) i) = 0\n[PROOFSTEP]\nsimp [Finsupp.not_mem_support_iff.mp hi]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx : A\ni : \u03b9\n\u22a2 (fun j c => \u2191(algebraMap R R\u209b) c \u2022 \u2191(Finsupp.single j 1) i) i (\u2191(\u2191b.repr x) i) = \u2191(algebraMap R R\u209b) (\u2191(\u2191b.repr x) i)\n[PROOFSTEP]\nsimp [Algebra.smul_def]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\n\u22a2 span R (Set.range \u2191(localizationLocalization R\u209b S A\u209b b)) = span R (\u2191(IsScalarTower.toAlgHom R A A\u209b) '' Set.range \u2191b)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_s\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\n\u22a2 Set.range \u2191(localizationLocalization R\u209b S A\u209b b) = \u2191(IsScalarTower.toAlgHom R A A\u209b) '' Set.range \u2191b\n[PROOFSTEP]\next\n[GOAL]\ncase e_s.h\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\nx\u271d : A\u209b\n\u22a2 x\u271d \u2208 Set.range \u2191(localizationLocalization R\u209b S A\u209b b) \u2194 x\u271d \u2208 \u2191(IsScalarTower.toAlgHom R A A\u209b) '' Set.range \u2191b\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\n\u22a2 span R (\u2191(IsScalarTower.toAlgHom R A A\u209b) '' Set.range \u2191b) =\n    map (IsScalarTower.toAlgHom R A A\u209b) (span R (Set.range \u2191b))\n[PROOFSTEP]\nrw [Submodule.map_span]\n[GOAL]\nR : Type u_1\nR\u209b : Type u_2\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u209b\ninst\u271d\u2078 : Algebra R R\u209b\nS : Submonoid R\nhT : IsLocalization S R\u209b\nA : Type u_3\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : Algebra R A\nA\u209b : Type u_4\ninst\u271d\u2075 : CommRing A\u209b\ninst\u271d\u2074 : Algebra A A\u209b\ninst\u271d\u00b3 : Algebra R\u209b A\u209b\ninst\u271d\u00b2 : Algebra R A\u209b\ninst\u271d\u00b9 : IsScalarTower R R\u209b A\u209b\ninst\u271d : IsScalarTower R A A\u209b\nhA : IsLocalization (Algebra.algebraMapSubmonoid A S) A\u209b\n\u03b9 : Type u_5\nb : Basis \u03b9 R A\n\u22a2 map (IsScalarTower.toAlgHom R A A\u209b) (span R (Set.range \u2191b)) = LinearMap.range (IsScalarTower.toAlgHom R A A\u209b)\n[PROOFSTEP]\nrw [b.span_eq, Submodule.map_top]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\n\u22a2 \u2200 (r : A) (x : M),\n    AddHom.toFun { toFun := \u2191f, map_add' := (_ : \u2200 (x y : M), \u2191f (x + y) = \u2191f x + \u2191f y) } (r \u2022 x) =\n      \u2191(RingHom.id A) r \u2022 AddHom.toFun { toFun := \u2191f, map_add' := (_ : \u2200 (x y : M), \u2191f (x + y) = \u2191f x + \u2191f y) } x\n[PROOFSTEP]\nintro r m\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nr : A\nm : M\n\u22a2 AddHom.toFun { toFun := \u2191f, map_add' := (_ : \u2200 (x y : M), \u2191f (x + y) = \u2191f x + \u2191f y) } (r \u2022 m) =\n    \u2191(RingHom.id A) r \u2022 AddHom.toFun { toFun := \u2191f, map_add' := (_ : \u2200 (x y : M), \u2191f (x + y) = \u2191f x + \u2191f y) } m\n[PROOFSTEP]\nsimp only [RingHom.id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nr : A\nm : M\n\u22a2 \u2191f (r \u2022 m) = r \u2022 \u2191f m\n[PROOFSTEP]\nrcases mk'_surjective S r with \u27e8r, s, rfl\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 \u2191f (mk' A r s \u2022 m) = mk' A r s \u2022 \u2191f m\n[PROOFSTEP]\ncalc\n  f (mk' A r s \u2022 m) = ((s : R) \u2022 mk' A 1 s) \u2022 f (mk' A r s \u2022 m) := by simp\n  _ = (mk' A 1 s) \u2022 (s : R) \u2022 f (mk' A r s \u2022 m) := by rw [smul_comm, smul_assoc]\n  _ = (mk' A 1 s) \u2022 f ((s : R) \u2022 mk' A r s \u2022 m) := by simp\n  _ = (mk' A 1 s) \u2022 f (r \u2022 m) := by rw [\u2190 smul_assoc, smul_mk'_self, algebraMap_smul]\n  _ = (mk' A 1 s) \u2022 r \u2022 f m := by simp\n  _ = mk' A r s \u2022 f m := by rw [smul_comm, \u2190 smul_assoc, smul_mk'_one]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 \u2191f (mk' A r s \u2022 m) = (\u2191s \u2022 mk' A 1 s) \u2022 \u2191f (mk' A r s \u2022 m)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 (\u2191s \u2022 mk' A 1 s) \u2022 \u2191f (mk' A r s \u2022 m) = mk' A 1 s \u2022 \u2191s \u2022 \u2191f (mk' A r s \u2022 m)\n[PROOFSTEP]\nrw [smul_comm, smul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 mk' A 1 s \u2022 \u2191s \u2022 \u2191f (mk' A r s \u2022 m) = mk' A 1 s \u2022 \u2191f (\u2191s \u2022 mk' A r s \u2022 m)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 mk' A 1 s \u2022 \u2191f (\u2191s \u2022 mk' A r s \u2022 m) = mk' A 1 s \u2022 \u2191f (r \u2022 m)\n[PROOFSTEP]\nrw [\u2190 smul_assoc, smul_mk'_self, algebraMap_smul]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 mk' A 1 s \u2022 \u2191f (r \u2022 m) = mk' A 1 s \u2022 r \u2022 \u2191f m\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommRing R\nS : Submonoid R\nA : Type u_2\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : IsLocalization S A\nM : Type u_3\nN : Type u_4\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module A M\ninst\u271d\u2074 : IsScalarTower R A M\ninst\u271d\u00b3 : AddCommMonoid N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : Module A N\ninst\u271d : IsScalarTower R A N\nf : M \u2192\u2097[R] N\nm : M\nr : R\ns : { x // x \u2208 S }\n\u22a2 mk' A 1 s \u2022 r \u2022 \u2191f m = mk' A r s \u2022 \u2191f m\n[PROOFSTEP]\nrw [smul_comm, \u2190 smul_assoc, smul_mk'_one]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.Module", "llama_tokens": 20168, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789178257653, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.536474541508262}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : \u03b1\n\u22a2 x \\ y \u2294 x \u2293 y = x\n[PROOFSTEP]\nrw [sup_comm, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : \u03b1\n\u22a2 x \\ y \u2293 (x \u2293 y) = \u22a5\n[PROOFSTEP]\nrw [inf_comm, inf_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d : Bot \u03b1 := toBot\na : \u03b1\n\u22a2 \u22a5 \u2264 a\n[PROOFSTEP]\nrw [\u2190 inf_inf_sdiff a a, inf_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d : Bot \u03b1 := toBot\na : \u03b1\n\u22a2 a \u2293 (a \u2293 a \\ a) \u2264 a\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : x \u2293 y \u2294 z = x\ni : x \u2293 y \u2293 z = \u22a5\n\u22a2 x \\ y = z\n[PROOFSTEP]\nconv_rhs at s => rw [\u2190 sup_inf_sdiff x y, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : x \u2293 y \u2294 z = x\ni : x \u2293 y \u2293 z = \u22a5\n| x\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff x y, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : x \u2293 y \u2294 z = x\ni : x \u2293 y \u2293 z = \u22a5\n| x\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff x y, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : x \u2293 y \u2294 z = x\ni : x \u2293 y \u2293 z = \u22a5\n| x\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff x y, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : x \u2293 y \u2294 z = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = \u22a5\n\u22a2 x \\ y = z\n[PROOFSTEP]\nrw [sup_comm] at s \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = \u22a5\n\u22a2 x \\ y = z\n[PROOFSTEP]\nconv_rhs at i => rw [\u2190 inf_inf_sdiff x y, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = \u22a5\n| \u22a5\n[PROOFSTEP]\nrw [\u2190 inf_inf_sdiff x y, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = \u22a5\n| \u22a5\n[PROOFSTEP]\nrw [\u2190 inf_inf_sdiff x y, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = \u22a5\n| \u22a5\n[PROOFSTEP]\nrw [\u2190 inf_inf_sdiff x y, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : x \u2293 y \u2293 z = x \\ y \u2293 (x \u2293 y)\n\u22a2 x \\ y = z\n[PROOFSTEP]\nrw [inf_comm] at i \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\ns : z \u2294 x \u2293 y = x \\ y \u2294 x \u2293 y\ni : z \u2293 (x \u2293 y) = x \\ y \u2293 (x \u2293 y)\n\u22a2 x \\ y = z\n[PROOFSTEP]\nexact (eq_of_inf_eq_sup_eq i s).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \\ x \u2294 x = y \\ x \u2294 (x \u2294 x \u2293 y)\n[PROOFSTEP]\nrw [sup_inf_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \\ x \u2294 (x \u2294 x \u2293 y) = y \u2293 x \u2294 y \\ x \u2294 x\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \u2293 x \u2294 y \\ x \u2294 x = y \u2294 x\n[PROOFSTEP]\nrw [sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 \u22a5 = x \u2293 y \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 x \\ y = x \u2293 (y \u2293 x \u2294 y \\ x) \u2293 x \\ y\n[PROOFSTEP]\nrw [sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \u2293 x \u2294 y \\ x) \u2293 x \\ y = (x \u2293 (y \u2293 x) \u2294 x \u2293 y \\ x) \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_sup_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2293 (y \u2293 x) \u2294 x \u2293 y \\ x) \u2293 x \\ y = (y \u2293 (x \u2293 x) \u2294 x \u2293 y \\ x) \u2293 x \\ y\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 (x \u2293 x) \u2294 x \u2293 y \\ x) \u2293 x \\ y = (y \u2293 x \u2294 x \u2293 y \\ x) \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 x \u2294 x \u2293 y \\ x) \u2293 x \\ y = x \u2293 y \u2293 x \\ y \u2294 x \u2293 y \\ x \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_sup_right, @inf_comm _ _ x y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 x \\ y \u2294 x \u2293 y \\ x \u2293 x \\ y = x \u2293 y \\ x \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_inf_sdiff, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ x \u2293 x \\ y = x \u2293 x \\ y \u2293 y \\ x\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 x \\ y \u2293 y \\ x = x \\ y \u2293 y \\ x\n[PROOFSTEP]\nrw [inf_of_le_right sdiff_le']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ x = (x \u2293 y \u2294 x \\ y) \u2293 y \\ x\n[PROOFSTEP]\nrw [sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2293 y \u2294 x \\ y) \u2293 y \\ x = x \u2293 y \u2293 y \\ x \u2294 x \\ y \u2293 y \\ x\n[PROOFSTEP]\nrw [inf_sup_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 y \\ x \u2294 x \\ y \u2293 y \\ x = \u22a5\n[PROOFSTEP]\nrw [@inf_comm _ _ x y, inf_inf_sdiff, sdiff_inf_sdiff, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \\ x \u2293 x = \u22a5\n[PROOFSTEP]\nrw [inf_comm, inf_sdiff_self_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z\u271d : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : OrderBot \u03b1 := toOrderBot\ny x z : \u03b1\nh : y \\ x \u2264 z\n\u22a2 y \\ x = x \u2293 y \\ x \u2294 z \u2293 y \\ x\n[PROOFSTEP]\nrw [inf_eq_right.2 h, inf_sdiff_self_right, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z\u271d : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : OrderBot \u03b1 := toOrderBot\ny x z : \u03b1\nh : y \\ x \u2264 z\n\u22a2 y \u2294 (x \u2294 z) = y \\ x \u2294 x \u2294 z\n[PROOFSTEP]\nrw [\u2190 sup_assoc, \u2190 @sdiff_sup_self' _ x y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z\u271d : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : OrderBot \u03b1 := toOrderBot\ny x z : \u03b1\nh : y \\ x \u2264 z\n\u22a2 y \\ x \u2294 x \u2294 z = x \u2294 z \u2294 y \\ x\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z\u271d : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nsrc\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : OrderBot \u03b1 := toOrderBot\ny x z : \u03b1\nh : y \u2264 x \u2294 z\n\u22a2 x \u2294 z \u2294 x \u2264 z \u2294 x\n[PROOFSTEP]\nrw [sup_assoc, sup_comm, sup_assoc, sup_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x \u2264 y \u2227 Disjoint x z\n\u22a2 x \u2264 y \\ z\n[PROOFSTEP]\nrw [\u2190 h.2.sdiff_eq_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x \u2264 y \u2227 Disjoint x z\n\u22a2 x \\ z \u2264 y \\ z\n[PROOFSTEP]\nexact sdiff_le_sdiff_right h.1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhi : Disjoint x z\nhs : x \u2294 z = y\nh : y \u2293 x = x\n\u22a2 y \u2293 x \u2294 z = y\n[PROOFSTEP]\nrw [h, hs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhi : Disjoint x z\nhs : x \u2294 z = y\nh : y \u2293 x = x\n\u22a2 y \u2293 x \u2293 z = \u22a5\n[PROOFSTEP]\nrw [h, hi.eq_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : y \u2264 x \u2294 z\n\u22a2 y \u2293 x \u2294 z = y\n[PROOFSTEP]\nrw [\u2190 inf_eq_right] at hs \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : (x \u2294 z) \u2293 y = y\n\u22a2 y \u2293 x \u2294 z = y\n[PROOFSTEP]\nrwa [sup_inf_right, inf_sup_right, @sup_comm _ _ x, inf_sup_self, inf_comm, @sup_comm _ _ z, hs, sup_eq_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhd : Disjoint x z\nhz : z \u2264 y\nhs : y \u2264 x \u2294 z\n\u22a2 y \u2293 x \u2293 z = \u22a5\n[PROOFSTEP]\nrw [inf_assoc, hd.eq_bot, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : Disjoint z (y \\ x)\n\u22a2 z \u2294 y \\ x \u2264 x \u2294 y \\ x\n[PROOFSTEP]\nrw [sup_sdiff_cancel_right hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : Disjoint z (y \\ x)\n\u22a2 z \u2294 y \\ x \u2264 y\n[PROOFSTEP]\nrefine' le_trans (sup_le_sup_left sdiff_le z) _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : Disjoint z (y \\ x)\n\u22a2 z \u2294 y \u2264 y\n[PROOFSTEP]\nrw [sup_eq_right.2 hz]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\n\u22a2 z \u2293 y \\ x = \u22a5 \u2194 z \u2264 x\n[PROOFSTEP]\nrw [\u2190 disjoint_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\n\u22a2 Disjoint z (y \\ x) \u2194 z \u2264 x\n[PROOFSTEP]\nexact disjoint_sdiff_iff_le hz hx\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 y = z \u2294 y \\ x\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 y \u2264 z \u2294 y \\ x\n[PROOFSTEP]\nconv_lhs => rw [\u2190 sup_inf_sdiff y x]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n| y\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff y x]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n| y\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff y x]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n| y\n[PROOFSTEP]\nrw [\u2190 sup_inf_sdiff y x]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 y \u2293 x \u2294 y \\ x \u2264 z \u2294 y \\ x\n[PROOFSTEP]\napply sup_le_sup_right\n[GOAL]\ncase a.h\u2081\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 y \u2293 x \u2264 z\n[PROOFSTEP]\nrwa [inf_eq_right.2 hx]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 z \u2294 y \\ x \u2264 y\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 z \u2294 y \\ x \u2264 ?a.b\u271d\n[PROOFSTEP]\napply sup_le_sup_right hz\n[GOAL]\ncase a.a\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2264 z\n\u22a2 y \u2294 y \\ x \u2264 y\n[PROOFSTEP]\nrw [sup_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : y = z \u2294 y \\ x\n\u22a2 x \u2264 z\n[PROOFSTEP]\nconv_lhs at H => rw [\u2190 sup_sdiff_cancel_right hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : y = z \u2294 y \\ x\n| y\n[PROOFSTEP]\nrw [\u2190 sup_sdiff_cancel_right hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : y = z \u2294 y \\ x\n| y\n[PROOFSTEP]\nrw [\u2190 sup_sdiff_cancel_right hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : y = z \u2294 y \\ x\n| y\n[PROOFSTEP]\nrw [\u2190 sup_sdiff_cancel_right hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2294 y \\ x = z \u2294 y \\ x\n\u22a2 x \u2264 z\n[PROOFSTEP]\nrefine' le_of_inf_le_sup_le _ H.le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2294 y \\ x = z \u2294 y \\ x\n\u22a2 x \u2293 y \\ x \u2264 z \u2293 y \\ x\n[PROOFSTEP]\nrw [inf_sdiff_self_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhz : z \u2264 y\nhx : x \u2264 y\nH : x \u2294 y \\ x = z \u2294 y \\ x\n\u22a2 \u22a5 \u2264 z \u2293 y \\ x\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \u2293 (x \u2294 z) \u2294 y \\ x \u2293 y \\ z = (y \u2293 (x \u2294 z) \u2294 y \\ x) \u2293 (y \u2293 (x \u2294 z) \u2294 y \\ z)\n[PROOFSTEP]\nrw [sup_inf_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 (x \u2294 z) \u2294 y \\ x) \u2293 (y \u2293 (x \u2294 z) \u2294 y \\ z) = (y \u2293 x \u2294 y \u2293 z \u2294 y \\ x) \u2293 (y \u2293 x \u2294 y \u2293 z \u2294 y \\ z)\n[PROOFSTEP]\nrw [@inf_sup_left _ _ y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 x \u2294 y \u2293 z \u2294 y \\ x) \u2293 (y \u2293 x \u2294 y \u2293 z \u2294 y \\ z) = (y \u2293 z \u2294 (y \u2293 x \u2294 y \\ x)) \u2293 (y \u2293 x \u2294 (y \u2293 z \u2294 y \\ z))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 z \u2294 (y \u2293 x \u2294 y \\ x)) \u2293 (y \u2293 x \u2294 (y \u2293 z \u2294 y \\ z)) = (y \u2293 z \u2294 y) \u2293 (y \u2293 x \u2294 y)\n[PROOFSTEP]\nrw [sup_inf_sdiff, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 z \u2294 y) \u2293 (y \u2293 x \u2294 y) = (y \u2294 y \u2293 z) \u2293 (y \u2294 y \u2293 x)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2294 y \u2293 z) \u2293 (y \u2294 y \u2293 x) = y\n[PROOFSTEP]\nrw [sup_inf_self, sup_inf_self, inf_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \u2293 (x \u2294 z) \u2293 (y \\ x \u2293 y \\ z) = (y \u2293 x \u2294 y \u2293 z) \u2293 (y \\ x \u2293 y \\ z)\n[PROOFSTEP]\nrw [inf_sup_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 x \u2294 y \u2293 z) \u2293 (y \\ x \u2293 y \\ z) = y \u2293 x \u2293 (y \\ x \u2293 y \\ z) \u2294 y \u2293 z \u2293 (y \\ x \u2293 y \\ z)\n[PROOFSTEP]\nrw [inf_sup_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \u2293 x \u2293 (y \\ x \u2293 y \\ z) \u2294 y \u2293 z \u2293 (y \\ x \u2293 y \\ z) = y \u2293 x \u2293 y \\ x \u2293 y \\ z \u2294 y \\ x \u2293 (y \\ z \u2293 (y \u2293 z))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 y \u2293 x \u2293 y \\ x \u2293 y \\ z \u2294 y \\ x \u2293 (y \\ z \u2293 (y \u2293 z)) = \u22a5\n[PROOFSTEP]\nrw [inf_inf_sdiff, bot_inf_eq, bot_sup_eq, @inf_comm _ _ (y \\ z), inf_inf_sdiff, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y \\ x = y \\ z\n\u22a2 y \u2293 x \u2293 ?m.20593 h = y \u2293 z \u2293 ?m.20593 h\n[PROOFSTEP]\nrw [inf_inf_sdiff, h, inf_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y \\ x = y \\ z\n\u22a2 y \u2293 x \u2294 y \\ x = y \u2293 z \u2294 y \\ x\n[PROOFSTEP]\nrw [sup_inf_sdiff, h, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y \u2293 x = y \u2293 z\n\u22a2 y \\ x = y \\ z\n[PROOFSTEP]\nrw [\u2190 sdiff_inf_self_right, \u2190 sdiff_inf_self_right z y, inf_comm, h, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ y = x \u2194 x \\ y = x \\ \u22a5\n[PROOFSTEP]\nrw [sdiff_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y = x \u2293 \u22a5 \u2194 Disjoint y x\n[PROOFSTEP]\nrw [inf_bot_eq, inf_comm, disjoint_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ y = x \u2194 Disjoint x y\n[PROOFSTEP]\nrw [sdiff_eq_self_iff_disjoint, disjoint_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\n\u22a2 x \\ y < x\n[PROOFSTEP]\nrefine' sdiff_le.lt_of_ne fun h => hy _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \\ y = x\n\u22a2 y = \u22a5\n[PROOFSTEP]\nrw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhx : y \u2264 x\nhy : y \u2260 \u22a5\nh : x \u2293 y = \u22a5\n\u22a2 y = \u22a5\n[PROOFSTEP]\nrw [\u2190 h, inf_eq_right.mpr hx]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 z \u2294 y \\ z = x \u2293 (y \u2293 z) \u2294 y \\ z\n[PROOFSTEP]\nrw [inf_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \u2293 z) \u2294 y \\ z = (x \u2294 y \\ z) \u2293 y\n[PROOFSTEP]\nrw [sup_inf_right, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2294 y \\ z) \u2293 y = x \u2293 y \u2294 y \\ z\n[PROOFSTEP]\nrw [inf_sup_right, inf_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ (y \\ z) = x \\ y \u2294 x \u2293 y \u2293 z\n[PROOFSTEP]\nrw [sup_comm, inf_comm, \u2190 inf_assoc, sup_inf_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ (y \\ z) = z \u2293 x \u2294 x \\ y\n[PROOFSTEP]\napply sdiff_unique\n[GOAL]\ncase s\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2294 (z \u2293 x \u2294 x \\ y) = x\n[PROOFSTEP]\ncalc\n  x \u2293 y \\ z \u2294 (z \u2293 x \u2294 x \\ y) = (x \u2294 (z \u2293 x \u2294 x \\ y)) \u2293 (y \\ z \u2294 (z \u2293 x \u2294 x \\ y)) := by rw [sup_inf_right]\n  _ = (x \u2294 x \u2293 z \u2294 x \\ y) \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \\ y)) := by ac_rfl\n  _ = x \u2293 (y \\ z \u2294 x \u2293 z \u2294 x \\ y) := by rw [sup_inf_self, sup_sdiff_left, \u2190 sup_assoc]\n  _ = x \u2293 (y \\ z \u2293 (z \u2294 y) \u2294 x \u2293 (z \u2294 y) \u2294 x \\ y) := by\n    rw [sup_inf_left, sdiff_sup_self', inf_sup_right, @sup_comm _ _ y]\n  _ = x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \u2293 y) \u2294 x \\ y) := by rw [inf_sdiff_sup_right, @inf_sup_left _ _ x z y]\n  _ = x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 (x \u2293 y \u2294 x \\ y))) := by ac_rfl\n  _ = x \u2293 (y \\ z \u2294 (x \u2294 x \u2293 z)) := by rw [sup_inf_sdiff, @sup_comm _ _ (x \u2293 z)]\n  _ = x := by rw [sup_inf_self, sup_comm, inf_sup_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2294 (z \u2293 x \u2294 x \\ y) = (x \u2294 (z \u2293 x \u2294 x \\ y)) \u2293 (y \\ z \u2294 (z \u2293 x \u2294 x \\ y))\n[PROOFSTEP]\nrw [sup_inf_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2294 (z \u2293 x \u2294 x \\ y)) \u2293 (y \\ z \u2294 (z \u2293 x \u2294 x \\ y)) = (x \u2294 x \u2293 z \u2294 x \\ y) \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \\ y))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2294 x \u2293 z \u2294 x \\ y) \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \\ y)) = x \u2293 (y \\ z \u2294 x \u2293 z \u2294 x \\ y)\n[PROOFSTEP]\nrw [sup_inf_self, sup_sdiff_left, \u2190 sup_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2294 x \u2293 z \u2294 x \\ y) = x \u2293 (y \\ z \u2293 (z \u2294 y) \u2294 x \u2293 (z \u2294 y) \u2294 x \\ y)\n[PROOFSTEP]\nrw [sup_inf_left, sdiff_sup_self', inf_sup_right, @sup_comm _ _ y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2293 (z \u2294 y) \u2294 x \u2293 (z \u2294 y) \u2294 x \\ y) = x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \u2293 y) \u2294 x \\ y)\n[PROOFSTEP]\nrw [inf_sdiff_sup_right, @inf_sup_left _ _ x z y]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 x \u2293 y) \u2294 x \\ y) = x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 (x \u2293 y \u2294 x \\ y)))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2294 (x \u2293 z \u2294 (x \u2293 y \u2294 x \\ y))) = x \u2293 (y \\ z \u2294 (x \u2294 x \u2293 z))\n[PROOFSTEP]\nrw [sup_inf_sdiff, @sup_comm _ _ (x \u2293 z)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2294 (x \u2294 x \u2293 z)) = x\n[PROOFSTEP]\nrw [sup_inf_self, sup_comm, inf_sup_self]\n[GOAL]\ncase i\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2293 (z \u2293 x \u2294 x \\ y) = \u22a5\n[PROOFSTEP]\ncalc\n  x \u2293 y \\ z \u2293 (z \u2293 x \u2294 x \\ y) = x \u2293 y \\ z \u2293 (z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y := by rw [inf_sup_left]\n  _ = x \u2293 (y \\ z \u2293 z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y := by ac_rfl\n  _ = x \u2293 y \\ z \u2293 x \\ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]\n  _ = x \u2293 (y \\ z \u2293 y) \u2293 x \\ y := by conv_lhs => rw [\u2190 inf_sdiff_left]\n  _ = x \u2293 (y \\ z \u2293 (y \u2293 x \\ y)) := by ac_rfl\n  _ = \u22a5 := by rw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2293 (z \u2293 x \u2294 x \\ y) = x \u2293 y \\ z \u2293 (z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_sup_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2293 (z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y = x \u2293 (y \\ z \u2293 z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2293 z \u2293 x) \u2294 x \u2293 y \\ z \u2293 x \\ y = x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nrw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \\ z \u2293 x \\ y = x \u2293 (y \\ z \u2293 y) \u2293 x \\ y\n[PROOFSTEP]\nconv_lhs => rw [\u2190 inf_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n| x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nrw [\u2190 inf_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n| x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nrw [\u2190 inf_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n| x \u2293 y \\ z \u2293 x \\ y\n[PROOFSTEP]\nrw [\u2190 inf_sdiff_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2293 y) \u2293 x \\ y = x \u2293 (y \\ z \u2293 (y \u2293 x \\ y))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \\ z \u2293 (y \u2293 x \\ y)) = \u22a5\n[PROOFSTEP]\nrw [inf_sdiff_self_right, inf_bot_eq, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ y \u2294 x \u2293 y \u2293 z = z \u2293 x \u2293 y \u2294 x \\ y\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \u2293 x \u2293 y \u2294 x \\ y = x \\ y \u2294 x \u2293 z\n[PROOFSTEP]\nrw [sup_inf_inf_sdiff, sup_comm, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : z \u2264 x\n\u22a2 x \\ (y \\ z) = x \\ y \u2294 z\n[PROOFSTEP]\nrw [sdiff_sdiff_right', inf_eq_right.2 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ (x \\ y) = x \u2293 y\n[PROOFSTEP]\nrw [sdiff_sdiff_right, inf_idem, sdiff_self, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y \u2264 x\n\u22a2 x \\ (x \\ y) = y\n[PROOFSTEP]\nrw [sdiff_sdiff_right_self, inf_of_le_right h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhy : y \u2264 x\nh : x \\ y = z\n\u22a2 x \\ z = y\n[PROOFSTEP]\nrw [\u2190 h, sdiff_sdiff_eq_self hy]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nhxz : x \u2264 z\nhyz : y \u2264 z\nh : z \\ x = z \\ y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 sdiff_sdiff_eq_self hxz, h, sdiff_sdiff_eq_self hyz]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \\ y) \\ z = x \\ y \u2293 x \\ z\n[PROOFSTEP]\nrw [sdiff_sdiff_left, sdiff_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \\ (x \\ y \u2294 y \\ x) = (z \\ x \u2294 z \u2293 x \u2293 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x)\n[PROOFSTEP]\nrw [sdiff_sup, sdiff_sdiff_right, sdiff_sdiff_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (z \\ x \u2294 z \u2293 x \u2293 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x) = z \u2293 (z \\ x \u2294 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x)\n[PROOFSTEP]\nrw [sup_inf_left, sup_comm, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \u2293 (z \\ x \u2294 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x) = z \u2293 (z \\ x \u2294 y) \u2293 (z \u2293 (z \\ y \u2294 x))\n[PROOFSTEP]\nrw [sup_inf_left, @sup_comm _ _ (z \\ y), sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \u2293 (z \\ x \u2294 y) \u2293 (z \u2293 (z \\ y \u2294 x)) = z \u2293 z \u2293 (z \\ x \u2294 y) \u2293 (z \\ y \u2294 x)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \u2293 z \u2293 (z \\ x \u2294 y) \u2293 (z \\ y \u2294 x) = z \u2293 (z \\ x \u2294 y) \u2293 (z \\ y \u2294 x)\n[PROOFSTEP]\nrw [inf_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \\ (x \\ y) \u2293 z \\ (y \\ x) = (z \\ x \u2294 z \u2293 x \u2293 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x)\n[PROOFSTEP]\nrw [sdiff_sdiff_right, sdiff_sdiff_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (z \\ x \u2294 z \u2293 x \u2293 y) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x) = (z \\ x \u2294 z \u2293 y \u2293 x) \u2293 (z \\ y \u2294 z \u2293 y \u2293 x)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 z \\ x \u2293 z \\ y \u2294 z \u2293 y \u2293 x = z \u2293 x \u2293 y \u2294 z \\ x \u2293 z \\ y\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 z \u2294 x \\ z \u2293 y \\ z = (x \u2293 y \u2293 z \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z)\n[PROOFSTEP]\nrw [sup_inf_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2293 y \u2293 z \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z) = (x \u2293 y \u2293 (z \u2294 x) \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z)\n[PROOFSTEP]\nrw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2293 y \u2293 (z \u2294 x) \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z) = (y \u2293 (x \u2293 (x \u2294 z)) \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 (x \u2293 (x \u2294 z)) \u2294 x \\ z) \u2293 (x \u2293 y \u2293 z \u2294 y \\ z) = (y \u2293 x \u2294 x \\ z) \u2293 (x \u2293 y \u2294 y \\ z)\n[PROOFSTEP]\nrw [inf_sup_self, sup_inf_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (y \u2293 x \u2294 x \\ z) \u2293 (x \u2293 y \u2294 y \\ z) = x \u2293 y \u2294 x \\ z \u2293 y \\ z\n[PROOFSTEP]\nrw [@inf_comm _ _ y, sup_inf_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 z \u2293 (x \\ z \u2293 y \\ z) = x \u2293 y \u2293 (z \u2293 x \\ z) \u2293 y \\ z\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 (z \u2293 x \\ z) \u2293 y \\ z = \u22a5\n[PROOFSTEP]\nrw [inf_sdiff_self_right, inf_bot_eq, bot_inf_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 z \u2294 x \u2293 y \\ z = x \u2293 (y \u2293 z) \u2294 x \u2293 y \\ z\n[PROOFSTEP]\nrw [inf_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 (y \u2293 z \u2294 y \\ z) = x \u2293 y\n[PROOFSTEP]\nrw [sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2293 z \u2293 (x \u2293 y \\ z) = x \u2293 x \u2293 (y \u2293 z \u2293 y \\ z)\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \u2293 x \u2293 (y \u2293 z \u2293 y \\ z) = \u22a5\n[PROOFSTEP]\nrw [inf_inf_sdiff, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ z \u2293 y = (x \u2293 y) \\ z\n[PROOFSTEP]\nrw [@inf_comm _ _ x, inf_comm, inf_sdiff_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\na b c : \u03b1\n\u22a2 a \u2293 b \\ c = (a \u2293 b) \\ (a \u2293 c)\n[PROOFSTEP]\nrw [sdiff_inf, sdiff_eq_bot_iff.2 inf_le_left, bot_sup_eq, inf_sdiff_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\na b c : \u03b1\n\u22a2 a \\ b \u2293 c = (a \u2293 c) \\ (b \u2293 c)\n[PROOFSTEP]\nsimp_rw [@inf_comm _ _ _ c, inf_sdiff_distrib_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 Disjoint (x \\ z) y \u2194 Disjoint x (y \\ z)\n[PROOFSTEP]\nsimp_rw [disjoint_iff, inf_sdiff_right_comm, inf_sdiff_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 x \\ y \u2294 y \\ x \u2294 x \u2293 y = (x \\ y \u2294 y \\ x \u2294 x) \u2293 (x \\ y \u2294 y \\ x \u2294 y)\n[PROOFSTEP]\nrw [sup_inf_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \\ y \u2294 y \\ x \u2294 x) \u2293 (x \\ y \u2294 y \\ x \u2294 y) = (x \\ y \u2294 x \u2294 y \\ x) \u2293 (x \\ y \u2294 (y \\ x \u2294 y))\n[PROOFSTEP]\nac_rfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \\ y \u2294 x \u2294 y \\ x) \u2293 (x \\ y \u2294 (y \\ x \u2294 y)) = (x \u2294 y \\ x) \u2293 (x \\ y \u2294 y)\n[PROOFSTEP]\nrw [sup_sdiff_right, sup_sdiff_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\n\u22a2 (x \u2294 y \\ x) \u2293 (x \\ y \u2294 y) = x \u2294 y\n[PROOFSTEP]\nrw [sup_sdiff_self_right, sup_sdiff_self_left, inf_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y < z \\ x\nhxz : x \u2264 z\n\u22a2 x \u2294 y < z\n[PROOFSTEP]\nrw [\u2190 sup_sdiff_cancel_right hxz]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y < z \\ x\nhxz : x \u2264 z\n\u22a2 x \u2294 y < x \u2294 z \\ x\n[PROOFSTEP]\nrefine' (sup_le_sup_left h.le _).lt_of_not_le fun h' => h.not_le _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y < z \\ x\nhxz : x \u2264 z\nh' : x \u2294 z \\ x \u2264 x \u2294 y\n\u22a2 z \\ x \u2264 y\n[PROOFSTEP]\nrw [\u2190 sdiff_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : y < z \\ x\nhxz : x \u2264 z\nh' : x \u2294 z \\ x \u2264 x \u2294 y\n\u22a2 (z \\ x) \\ x \u2264 y\n[PROOFSTEP]\nexact (sdiff_le_sdiff_of_sup_le_sup_left h').trans sdiff_le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x < z \\ y\nhyz : y \u2264 z\n\u22a2 x \u2294 y < z\n[PROOFSTEP]\nrw [\u2190 sdiff_sup_cancel hyz]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x < z \\ y\nhyz : y \u2264 z\n\u22a2 x \u2294 y < z \\ y \u2294 y\n[PROOFSTEP]\nrefine' (sup_le_sup_right h.le _).lt_of_not_le fun h' => h.not_le _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x < z \\ y\nhyz : y \u2264 z\nh' : z \\ y \u2294 y \u2264 x \u2294 y\n\u22a2 z \\ y \u2264 x\n[PROOFSTEP]\nrw [\u2190 sdiff_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nh : x < z \\ y\nhyz : y \u2264 z\nh' : z \\ y \u2294 y \u2264 x \u2294 y\n\u22a2 (z \\ y) \\ y \u2264 x\n[PROOFSTEP]\nexact (sdiff_le_sdiff_of_sup_le_sup_right h').trans sdiff_le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1\ninst\u271d : OrderTop \u03b1\nsrc\u271d\u00b2 : GeneralizedBooleanAlgebra \u03b1 := inst\u271d\u00b9\nsrc\u271d\u00b9 : OrderBot \u03b1 := toOrderBot\nsrc\u271d : OrderTop \u03b1 := inst\u271d\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 x\u271d\u00b9 \\ x\u271d = x\u271d\u00b9 \u2293 x\u271d\u1d9c\n[PROOFSTEP]\nerw [\u2190 inf_sdiff_assoc, inf_top_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nsrc\u271d : BooleanAlgebra \u03b1 := inst\u271d\na b : \u03b1\n\u22a2 a \u2293 b \u2294 a \\ b = a\n[PROOFSTEP]\nrw [sdiff_eq, \u2190 inf_sup_left, sup_compl_eq_top, inf_top_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nsrc\u271d : BooleanAlgebra \u03b1 := inst\u271d\na b : \u03b1\n\u22a2 a \u2293 b \u2293 a \\ b = \u22a5\n[PROOFSTEP]\nrw [sdiff_eq, \u2190 inf_inf_distrib_left, inf_compl_eq_bot', inf_bot_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nsrc\u271d\u00b9 : BooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : GeneralizedCoheytingAlgebra \u03b1 := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra\na b c : \u03b1\n\u22a2 a \u2264 b \u21e8 c \u2194 a \u2293 b \u2264 c\n[PROOFSTEP]\nrw [himp_eq, isCompl_compl.le_sup_right_iff_inf_left_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nsrc\u271d\u00b9 : BooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : GeneralizedCoheytingAlgebra \u03b1 := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra\na : \u03b1\n\u22a2 \u22a4 \\ a = \uffe2a\n[PROOFSTEP]\nrw [sdiff_eq, top_inf_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nsrc\u271d\u00b9 : BooleanAlgebra \u03b1 := inst\u271d\nsrc\u271d : GeneralizedCoheytingAlgebra \u03b1 := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra\na : \u03b1\n\u22a2 a\u1d9c = \uffe2a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : x = y\u1d9c\n\u22a2 IsCompl x y\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : x = y\u1d9c\n\u22a2 IsCompl y\u1d9c y\n[PROOFSTEP]\nexact isCompl_compl.symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : x\u1d9c = y\n\u22a2 IsCompl x y\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : x\u1d9c = y\n\u22a2 IsCompl x x\u1d9c\n[PROOFSTEP]\nexact isCompl_compl\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x\u1d9c = y \u2194 y\u1d9c = x\n[PROOFSTEP]\nrw [eq_comm, compl_eq_iff_isCompl, eq_compl_iff_isCompl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x = y\u1d9c \u2194 y = x\u1d9c\n[PROOFSTEP]\nrw [eq_comm, compl_eq_iff_isCompl, eq_compl_iff_isCompl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : y\u1d9c \u2264 x\u1d9c\n\u22a2 x \u2264 y\n[PROOFSTEP]\nhave h := compl_le_compl h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh\u271d : y\u1d9c \u2264 x\u1d9c\nh : x\u1d9c\u1d9c \u2264 y\u1d9c\u1d9c\n\u22a2 x \u2264 y\n[PROOFSTEP]\nsimp at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh\u271d : y\u1d9c \u2264 x\u1d9c\nh : x \u2264 y\n\u22a2 x \u2264 y\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\nh : y\u1d9c \u2264 x\n\u22a2 x\u1d9c \u2264 y\n[PROOFSTEP]\nsimpa only [compl_compl] using compl_le_compl h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x\u1d9c \u2264 x \u2194 x = \u22a4\n[PROOFSTEP]\nsimpa using le_compl_self (a := x\u1d9c)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 x\u1d9c < x \u2194 x = \u22a4\n[PROOFSTEP]\nsimpa using lt_compl_self (a := x\u1d9c)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x \\ y\u1d9c = x \u2293 y\n[PROOFSTEP]\nrw [sdiff_eq, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x \u2293 y \u2294 x \u2293 y\u1d9c = x\n[PROOFSTEP]\nrw [\u2190 sdiff_eq, sup_inf_sdiff _ _]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 (x \\ y)\u1d9c = x \u21e8 y\n[PROOFSTEP]\nrw [sdiff_eq, himp_eq, compl_inf, compl_compl, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 x\u1d9c \\ y\u1d9c = y \\ x\n[PROOFSTEP]\nrw [sdiff_compl, sdiff_eq, inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 Disjoint x\u1d9c y \u2194 y \u2264 x\n[PROOFSTEP]\nrw [\u2190 le_compl_iff_disjoint_left, compl_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d : BooleanAlgebra \u03b1\n\u22a2 Disjoint x y\u1d9c \u2194 x \u2264 y\n[PROOFSTEP]\nrw [\u2190 le_compl_iff_disjoint_right, compl_compl]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 x \u2293 x\u1d9c \u2264 \u22a5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 (x \u2293 x\u1d9c).fst \u2264 \u22a5.fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 (x \u2293 x\u1d9c).snd \u2264 \u22a5.snd\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u22a4 \u2264 x \u2294 x\u1d9c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u22a4.fst \u2264 (x \u2294 x\u1d9c).fst\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx : \u03b1 \u00d7 \u03b2\n\u22a2 \u22a4.snd \u2264 (x \u2294 x\u1d9c).snd\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 x \\ y = x \u2293 y\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (x \\ y).fst = (x \u2293 y\u1d9c).fst\n[PROOFSTEP]\nsimp [sdiff_eq]\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (x \\ y).snd = (x \u2293 y\u1d9c).snd\n[PROOFSTEP]\nsimp [sdiff_eq]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 x \u21e8 y = y \u2294 x\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (x \u21e8 y).fst = (y \u2294 x\u1d9c).fst\n[PROOFSTEP]\nsimp [himp_eq]\n[GOAL]\ncase h\u2082\n\u03b1\u271d : Type u\n\u03b2\u271d : Type u_1\nw x\u271d y\u271d z : \u03b1\u271d\n\u03b1 : Type ?u.67933\n\u03b2 : Type ?u.67936\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nsrc\u271d\u00b9 : HeytingAlgebra (\u03b1 \u00d7 \u03b2) := heytingAlgebra\nsrc\u271d : DistribLattice (\u03b1 \u00d7 \u03b2) := distribLattice \u03b1 \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 (x \u21e8 y).snd = (y \u2294 x\u1d9c).snd\n[PROOFSTEP]\nsimp [himp_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d\u00b9 : LinearOrder Bool := Bool.linearOrder\nsrc\u271d : BoundedOrder Bool := Bool.boundedOrder\n\u22a2 \u2200 (x y z : Bool), (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y \u2293 z\n[PROOFSTEP]\ndecide\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2074 : Sup \u03b1\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_bot : f \u22a5 = \u22a5\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d\u00b9 : GeneralizedCoheytingAlgebra \u03b1 := Injective.generalizedCoheytingAlgebra f hf map_sup map_inf map_bot map_sdiff\nsrc\u271d : DistribLattice \u03b1 := Injective.distribLattice f hf map_sup map_inf\na b : \u03b1\n\u22a2 f (a \u2293 b \u2294 a \\ b) = f a\n[PROOFSTEP]\nerw [map_sup, map_sdiff, map_inf, sup_inf_sdiff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2074 : Sup \u03b1\ninst\u271d\u00b3 : Inf \u03b1\ninst\u271d\u00b2 : Bot \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_bot : f \u22a5 = \u22a5\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d\u00b9 : GeneralizedCoheytingAlgebra \u03b1 := Injective.generalizedCoheytingAlgebra f hf map_sup map_inf map_bot map_sdiff\nsrc\u271d : DistribLattice \u03b1 := Injective.distribLattice f hf map_sup map_inf\na b : \u03b1\n\u22a2 f (a \u2293 b \u2293 a \\ b) = f \u22a5\n[PROOFSTEP]\nerw [map_inf, map_sdiff, map_inf, inf_inf_sdiff, map_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2076 : Sup \u03b1\ninst\u271d\u2075 : Inf \u03b1\ninst\u271d\u2074 : Top \u03b1\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : HasCompl \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_top : f \u22a4 = \u22a4\nmap_bot : f \u22a5 = \u22a5\nmap_compl : \u2200 (a : \u03b1), f a\u1d9c = (f a)\u1d9c\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d : GeneralizedBooleanAlgebra \u03b1 := Injective.generalizedBooleanAlgebra f hf map_sup map_inf map_bot map_sdiff\na : \u03b1\n\u22a2 f a \u2293 f a\u1d9c = f \u22a5\n[PROOFSTEP]\nrw [map_compl, inf_compl_eq_bot, map_bot]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2076 : Sup \u03b1\ninst\u271d\u2075 : Inf \u03b1\ninst\u271d\u2074 : Top \u03b1\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : HasCompl \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_top : f \u22a4 = \u22a4\nmap_bot : f \u22a5 = \u22a5\nmap_compl : \u2200 (a : \u03b1), f a\u1d9c = (f a)\u1d9c\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d : GeneralizedBooleanAlgebra \u03b1 := Injective.generalizedBooleanAlgebra f hf map_sup map_inf map_bot map_sdiff\na : \u03b1\n\u22a2 f a \u2294 f a\u1d9c = f \u22a4\n[PROOFSTEP]\nrw [map_compl, sup_compl_eq_top, map_top]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2076 : Sup \u03b1\ninst\u271d\u2075 : Inf \u03b1\ninst\u271d\u2074 : Top \u03b1\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : HasCompl \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_top : f \u22a4 = \u22a4\nmap_bot : f \u22a5 = \u22a5\nmap_compl : \u2200 (a : \u03b1), f a\u1d9c = (f a)\u1d9c\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d : GeneralizedBooleanAlgebra \u03b1 := Injective.generalizedBooleanAlgebra f hf map_sup map_inf map_bot map_sdiff\na b : \u03b1\n\u22a2 a \\ b = a \u2293 b\u1d9c\n[PROOFSTEP]\nrefine hf ((map_sdiff _ _).trans (sdiff_eq.trans ?_))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\ninst\u271d\u2076 : Sup \u03b1\ninst\u271d\u2075 : Inf \u03b1\ninst\u271d\u2074 : Top \u03b1\ninst\u271d\u00b3 : Bot \u03b1\ninst\u271d\u00b2 : HasCompl \u03b1\ninst\u271d\u00b9 : SDiff \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nmap_top : f \u22a4 = \u22a4\nmap_bot : f \u22a5 = \u22a5\nmap_compl : \u2200 (a : \u03b1), f a\u1d9c = (f a)\u1d9c\nmap_sdiff : \u2200 (a b : \u03b1), f (a \\ b) = f a \\ f b\nsrc\u271d : GeneralizedBooleanAlgebra \u03b1 := Injective.generalizedBooleanAlgebra f hf map_sup map_inf map_bot map_sdiff\na b : \u03b1\n\u22a2 f a \u2293 (f b)\u1d9c = f (a \u2293 b\u1d9c)\n[PROOFSTEP]\nrw [map_inf, map_compl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\n\u22a2 BooleanAlgebra PUnit\n[PROOFSTEP]\nrefine' { PUnit.biheytingAlgebra with .. }\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\n\u22a2 \u2200 (x y z : PUnit), (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y \u2293 z\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\nx\u271d y\u271d z\u271d : PUnit\n\u22a2 (x\u271d \u2294 y\u271d) \u2293 (x\u271d \u2294 z\u271d) \u2264 x\u271d \u2294 y\u271d \u2293 z\u271d\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\n\u22a2 \u2200 (x : PUnit), x \u2293 x\u1d9c \u2264 \u22a5\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\nx\u271d : PUnit\n\u22a2 x\u271d \u2293 x\u271d\u1d9c \u2264 \u22a5\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\n\u22a2 \u2200 (x : PUnit), \u22a4 \u2264 x \u2294 x\u1d9c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\nx\u271d : PUnit\n\u22a2 \u22a4 \u2264 x\u271d \u2294 x\u271d\u1d9c\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\n\u22a2 \u2200 (x y : PUnit), x \\ y = x \u2293 y\u1d9c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\nx\u271d y\u271d : PUnit\n\u22a2 x\u271d \\ y\u271d = x\u271d \u2293 y\u271d\u1d9c\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase refine'_5\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\n\u22a2 \u2200 (x y : PUnit), x \u21e8 y = y \u2294 x\u1d9c\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\n\u03b1 : Type u\n\u03b2 : Type u_1\nw x y z : \u03b1\nsrc\u271d : BiheytingAlgebra PUnit := biheytingAlgebra\nx\u271d y\u271d : PUnit\n\u22a2 x\u271d \u21e8 y\u271d = y\u271d \u2294 x\u271d\u1d9c\n[PROOFSTEP]\ntrivial\n", "meta": {"mathlib_filename": "Mathlib.Order.BooleanAlgebra", "llama_tokens": 23922, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5362688750125704}}
{"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - 2 * \u2016x\u2016 * \u2016y\u2016 * Real.cos (angle x y)\n[PROOFSTEP]\nrw [show 2 * \u2016x\u2016 * \u2016y\u2016 * Real.cos (angle x y) = 2 * (Real.cos (angle x y) * (\u2016x\u2016 * \u2016y\u2016)) by ring,\n  cos_angle_mul_norm_mul_norm, \u2190 real_inner_self_eq_norm_mul_norm, \u2190 real_inner_self_eq_norm_mul_norm, \u2190\n  real_inner_self_eq_norm_mul_norm, real_inner_sub_sub_self, sub_add_eq_add_sub]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 2 * \u2016x\u2016 * \u2016y\u2016 * Real.cos (angle x y) = 2 * (Real.cos (angle x y) * (\u2016x\u2016 * \u2016y\u2016))\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 angle x (x - y) = angle y (y - x)\n[PROOFSTEP]\nrefine' Real.injOn_cos \u27e8angle_nonneg _ _, angle_le_pi _ _\u27e9 \u27e8angle_nonneg _ _, angle_le_pi _ _\u27e9 _\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 Real.cos (angle x (x - y)) = Real.cos (angle y (y - x))\n[PROOFSTEP]\nrw [cos_angle, cos_angle, h, \u2190 neg_sub, norm_neg, neg_sub, inner_sub_right, inner_sub_right,\n  real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm, h, real_inner_comm x y]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : angle x (x - y) = angle y (y - x)\nhpi : angle x y \u2260 \u03c0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nreplace h :=\n  Real.arccos_injOn (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x (x - y)))\n    (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one y (y - x))) h\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nh : inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) = inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nby_cases hxy : x = y\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nh : inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) = inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)\nhxy : x = y\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nh : inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) = inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)\nhxy : \u00acx = y\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 norm_neg (y - x), neg_sub, mul_comm, mul_comm \u2016y\u2016, div_eq_mul_inv, div_eq_mul_inv, mul_inv_rev, mul_inv_rev, \u2190\n  mul_assoc, \u2190 mul_assoc] at h \n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nh : inner x (x - y) * \u2016x\u2016\u207b\u00b9 * \u2016x - y\u2016\u207b\u00b9 = inner y (y - x) * \u2016y\u2016\u207b\u00b9 * \u2016x - y\u2016\u207b\u00b9\nhxy : \u00acx = y\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nreplace h := mul_right_cancel\u2080 (inv_ne_zero fun hz => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 hz))) h\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : inner x (x - y) * \u2016x\u2016\u207b\u00b9 = inner y (y - x) * \u2016y\u2016\u207b\u00b9\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [inner_sub_right, inner_sub_right, real_inner_comm x y, real_inner_self_eq_norm_mul_norm,\n  real_inner_self_eq_norm_mul_norm, mul_sub_right_distrib, mul_sub_right_distrib, mul_self_mul_inv, mul_self_mul_inv,\n  sub_eq_sub_iff_sub_eq_sub, \u2190 mul_sub_left_distrib] at h \n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 - \u2016y\u2016 = inner x y * (\u2016x\u2016\u207b\u00b9 - \u2016y\u2016\u207b\u00b9)\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nby_cases hx0 : x = 0\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 - \u2016y\u2016 = inner x y * (\u2016x\u2016\u207b\u00b9 - \u2016y\u2016\u207b\u00b9)\nhx0 : x = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [hx0, norm_zero, inner_zero_left, zero_mul, zero_sub, neg_eq_zero] at h \n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016y\u2016 = 0\nhx0 : x = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [hx0, norm_zero, h]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 - \u2016y\u2016 = inner x y * (\u2016x\u2016\u207b\u00b9 - \u2016y\u2016\u207b\u00b9)\nhx0 : \u00acx = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nby_cases hy0 : y = 0\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 - \u2016y\u2016 = inner x y * (\u2016x\u2016\u207b\u00b9 - \u2016y\u2016\u207b\u00b9)\nhx0 : \u00acx = 0\nhy0 : y = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [hy0, norm_zero, inner_zero_right, zero_mul, sub_zero] at h \n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 = 0\nhx0 : \u00acx = 0\nhy0 : y = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [hy0, norm_zero, h]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : \u2016x\u2016 - \u2016y\u2016 = inner x y * (\u2016x\u2016\u207b\u00b9 - \u2016y\u2016\u207b\u00b9)\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [inv_sub_inv (fun hz => hx0 (norm_eq_zero.1 hz)) fun hz => hy0 (norm_eq_zero.1 hz), \u2190 neg_sub, \u2190 mul_div_assoc,\n  mul_comm, mul_div_assoc, \u2190 mul_neg_one] at h \n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : (\u2016y\u2016 - \u2016x\u2016) * -1 = (\u2016y\u2016 - \u2016x\u2016) * (inner x y / (\u2016x\u2016 * \u2016y\u2016))\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nsymm\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : (\u2016y\u2016 - \u2016x\u2016) * -1 = (\u2016y\u2016 - \u2016x\u2016) * (inner x y / (\u2016x\u2016 * \u2016y\u2016))\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\n\u22a2 \u2016y\u2016 = \u2016x\u2016\n[PROOFSTEP]\nby_contra hyx\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nh : (\u2016y\u2016 - \u2016x\u2016) * -1 = (\u2016y\u2016 - \u2016x\u2016) * (inner x y / (\u2016x\u2016 * \u2016y\u2016))\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\nhyx : \u00ac\u2016y\u2016 = \u2016x\u2016\n\u22a2 False\n[PROOFSTEP]\nreplace h := (mul_left_cancel\u2080 (sub_ne_zero_of_ne hyx) h).symm\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\nhyx : \u00ac\u2016y\u2016 = \u2016x\u2016\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = -1\n\u22a2 False\n[PROOFSTEP]\nrw [real_inner_div_norm_mul_norm_eq_neg_one_iff, \u2190 angle_eq_pi_iff] at h \n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhpi : angle x y \u2260 \u03c0\nhxy : \u00acx = y\nhx0 : \u00acx = 0\nhy0 : \u00acy = 0\nhyx : \u00ac\u2016y\u2016 = \u2016x\u2016\nh : angle x y = \u03c0\n\u22a2 False\n[PROOFSTEP]\nexact hpi h\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y)\n[PROOFSTEP]\nby_cases hxy : x = y\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : x = y\n\u22a2 Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y)\n[PROOFSTEP]\nrw [hxy, angle_self hy]\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : x = y\n\u22a2 Real.cos (angle y (y - y) + angle y (y - y)) = -Real.cos 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\n\u22a2 Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y)\n[PROOFSTEP]\nrw [Real.cos_add, cos_angle, cos_angle, cos_angle]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\n\u22a2 inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016))\n[PROOFSTEP]\nhave hxn : \u2016x\u2016 \u2260 0 := fun h => hx (norm_eq_zero.1 h)\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\n\u22a2 inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016))\n[PROOFSTEP]\nhave hyn : \u2016y\u2016 \u2260 0 := fun h => hy (norm_eq_zero.1 h)\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\n\u22a2 inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016))\n[PROOFSTEP]\nhave hxyn : \u2016x - y\u2016 \u2260 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016))\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n        Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n      \u2016x\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n          Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n        \u2016x\u2016 *\n      \u2016y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n            Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n          \u2016x\u2016 *\n        \u2016y\u2016 *\n      \u2016x - y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016)) :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H2 :\n  \u27eax, x\u27eb * (\u27eax, x\u27eb - \u27eax, y\u27eb - (\u27eax, y\u27eb - \u27eay, y\u27eb)) - (\u27eax, x\u27eb - \u27eax, y\u27eb) * (\u27eax, x\u27eb - \u27eax, y\u27eb) =\n    \u27eax, x\u27eb * \u27eay, y\u27eb - \u27eax, y\u27eb * \u27eax, y\u27eb :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\n\u22a2 inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\nH2 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H3 :\n  \u27eay, y\u27eb * (\u27eay, y\u27eb - \u27eax, y\u27eb - (\u27eax, y\u27eb - \u27eax, x\u27eb)) - (\u27eay, y\u27eb - \u27eax, y\u27eb) * (\u27eay, y\u27eb - \u27eax, y\u27eb) =\n    \u27eax, x\u27eb * \u27eay, y\u27eb - \u27eax, y\u27eb * \u27eax, y\u27eb :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\nH2 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\nH2 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH3 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 (inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) -\n              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    -(inner x y / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nrw [mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib, H1,\n  sin_angle_mul_norm_mul_norm, norm_sub_rev x y, sin_angle_mul_norm_mul_norm, norm_sub_rev y x, inner_sub_left,\n  inner_sub_left, inner_sub_right, inner_sub_right, inner_sub_right, inner_sub_right, real_inner_comm x y, H2, H3,\n  Real.mul_self_sqrt (sub_nonneg_of_le (real_inner_mul_inner_self_le x y)), real_inner_self_eq_norm_mul_norm,\n  real_inner_self_eq_norm_mul_norm, real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\nH2 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH3 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 (\u2016x\u2016 * \u2016x\u2016 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2) / (\u2016x\u2016 * \u2016x - y\u2016) *\n                ((\u2016y\u2016 * \u2016y\u2016 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2) / (\u2016y\u2016 * \u2016x - y\u2016)) *\n              \u2016x\u2016 *\n            \u2016y\u2016 *\n          \u2016x - y\u2016 *\n        \u2016x - y\u2016 -\n      (\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) -\n        (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2 * ((\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2)) =\n    -((\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2 / (\u2016x\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nfield_simp [hxn, hyn, hxyn]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016x - y\u2016))\nH2 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH3 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 ((\u2016x\u2016 * \u2016x\u2016 * 2 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) *\n                    (\u2016y\u2016 * \u2016y\u2016 * 2 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) *\n                  \u2016x\u2016 *\n                \u2016y\u2016 *\n              \u2016x - y\u2016 *\n            \u2016x - y\u2016 *\n          (2 * 2) -\n        2 * (\u2016x\u2016 * \u2016x - y\u2016) * (2 * (\u2016y\u2016 * \u2016x - y\u2016)) *\n          (\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) * (2 * 2) -\n            (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) * (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016))) *\n      (2 * (\u2016x\u2016 * \u2016y\u2016)) =\n    (\u2016x - y\u2016 * \u2016x - y\u2016 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016 *\n      (2 * (\u2016x\u2016 * \u2016x - y\u2016) * (2 * (\u2016y\u2016 * \u2016x - y\u2016)) * (2 * 2))\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)\n[PROOFSTEP]\nby_cases hxy : x = y\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : x = y\n\u22a2 Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)\n[PROOFSTEP]\nrw [hxy, angle_self hy]\n[GOAL]\ncase pos\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : x = y\n\u22a2 Real.sin (angle y (y - y) + angle y (y - y)) = Real.sin 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\n\u22a2 Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)\n[PROOFSTEP]\nrw [Real.sin_add, cos_angle, cos_angle]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\n\u22a2 Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n      inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x)) =\n    Real.sin (angle x y)\n[PROOFSTEP]\nhave hxn : \u2016x\u2016 \u2260 0 := fun h => hx (norm_eq_zero.1 h)\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\n\u22a2 Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n      inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x)) =\n    Real.sin (angle x y)\n[PROOFSTEP]\nhave hyn : \u2016y\u2016 \u2260 0 := fun h => hy (norm_eq_zero.1 h)\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\n\u22a2 Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n      inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x)) =\n    Real.sin (angle x y)\n[PROOFSTEP]\nhave hxyn : \u2016x - y\u2016 \u2260 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n      inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x)) =\n    Real.sin (angle x y)\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n        inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n      \u2016x\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n          inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n        \u2016x\u2016 *\n      \u2016y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n            inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n          \u2016x\u2016 *\n        \u2016y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\napply mul_right_cancel\u2080 hxyn\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n              inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H1 :\n  Real.sin (angle x (x - y)) * (\u27eay, y - x\u27eb / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (\u27eay, y - x\u27eb / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016 :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\n\u22a2 Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n              inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H2 :\n  \u27eax, x - y\u27eb / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    \u27eax, x - y\u27eb / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\n\u22a2 inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n              inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H3 :\n  \u27eax, x\u27eb * (\u27eax, x\u27eb - \u27eax, y\u27eb - (\u27eax, y\u27eb - \u27eay, y\u27eb)) - (\u27eax, x\u27eb - \u27eax, y\u27eb) * (\u27eax, x\u27eb - \u27eax, y\u27eb) =\n    \u27eax, x\u27eb * \u27eay, y\u27eb - \u27eax, y\u27eb * \u27eax, y\u27eb :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\n\u22a2 inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\nH3 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n              inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nhave H4 :\n  \u27eay, y\u27eb * (\u27eay, y\u27eb - \u27eax, y\u27eb - (\u27eax, y\u27eb - \u27eax, x\u27eb)) - (\u27eay, y\u27eb - \u27eax, y\u27eb) * (\u27eay, y\u27eb - \u27eax, y\u27eb) =\n    \u27eax, x\u27eb * \u27eay, y\u27eb - \u27eax, y\u27eb * \u27eax, y\u27eb :=\n  by ring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\nH3 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\nH3 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH4 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 (Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) +\n              inner x (x - y) / (\u2016x\u2016 * \u2016x - y\u2016) * Real.sin (angle y (y - x))) *\n            \u2016x\u2016 *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016 =\n    Real.sin (angle x y) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 * \u2016x - y\u2016\n[PROOFSTEP]\nrw [right_distrib, right_distrib, right_distrib, right_distrib, H1, sin_angle_mul_norm_mul_norm, norm_sub_rev x y, H2,\n  sin_angle_mul_norm_mul_norm, norm_sub_rev y x, mul_assoc (Real.sin (angle x y)), sin_angle_mul_norm_mul_norm,\n  inner_sub_left, inner_sub_left, inner_sub_right, inner_sub_right, inner_sub_right, inner_sub_right,\n  real_inner_comm x y, H3, H4, real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm,\n  real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\nH3 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH4 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 Real.sqrt\n              (\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) -\n                (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2 * ((\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2)) *\n            ((\u2016y\u2016 * \u2016y\u2016 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2) / (\u2016y\u2016 * \u2016x - y\u2016)) *\n          \u2016y\u2016 *\n        \u2016x - y\u2016 +\n      (\u2016x\u2016 * \u2016x\u2016 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2) / (\u2016x\u2016 * \u2016x - y\u2016) *\n            Real.sqrt\n              (\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) -\n                (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2 * ((\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2)) *\n          \u2016x\u2016 *\n        \u2016x - y\u2016 =\n    Real.sqrt\n          (\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) -\n            (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2 * ((\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2)) *\n        \u2016x - y\u2016 *\n      \u2016x - y\u2016\n[PROOFSTEP]\nfield_simp [hxn, hyn, hxyn]\n[GOAL]\ncase neg\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhxy : \u00acx = y\nhxn : \u2016x\u2016 \u2260 0\nhyn : \u2016y\u2016 \u2260 0\nhxyn : \u2016x - y\u2016 \u2260 0\nH1 :\n  Real.sin (angle x (x - y)) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016 * \u2016y\u2016 * \u2016x - y\u2016 =\n    Real.sin (angle x (x - y)) * (\u2016x\u2016 * \u2016x - y\u2016) * (inner y (y - x) / (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016y\u2016\nH2 :\n  inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * Real.sin (angle y (y - x)) * \u2016x\u2016 * \u2016y\u2016 * \u2016y - x\u2016 =\n    inner x (x - y) / (\u2016x\u2016 * \u2016y - x\u2016) * (Real.sin (angle y (y - x)) * (\u2016y\u2016 * \u2016y - x\u2016)) * \u2016x\u2016\nH3 :\n  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\nH4 :\n  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =\n    inner x x * inner y y - inner x y * inner x y\n\u22a2 Real.sqrt\n                ((\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) * (2 * 2) -\n                    (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) * (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) /\n                  (2 * 2)) *\n              (\u2016y\u2016 * \u2016y\u2016 * 2 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) *\n            \u2016y\u2016 *\n          \u2016x - y\u2016 *\n        (2 * (\u2016x\u2016 * \u2016x - y\u2016)) +\n      (\u2016x\u2016 * \u2016x\u2016 * 2 - (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) *\n              Real.sqrt\n                ((\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) * (2 * 2) -\n                    (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) * (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) /\n                  (2 * 2)) *\n            \u2016x\u2016 *\n          \u2016x - y\u2016 *\n        (2 * (\u2016y\u2016 * \u2016x - y\u2016)) =\n    Real.sqrt\n            ((\u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) * (2 * 2) -\n                (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) * (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016)) /\n              (2 * 2)) *\n          \u2016x - y\u2016 *\n        \u2016x - y\u2016 *\n      (2 * (\u2016y\u2016 * \u2016x - y\u2016) * (2 * (\u2016x\u2016 * \u2016x - y\u2016)))\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\n[PROOFSTEP]\nrw [add_assoc, Real.cos_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,\n  sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy, mul_neg, \u2190 neg_add', add_comm, \u2190 sq, \u2190 sq, Real.sin_sq_add_cos_sq]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0\n[PROOFSTEP]\nrw [add_assoc, Real.sin_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,\n  sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 Real.sin (angle x y) * -Real.cos (angle x y) + Real.cos (angle x y) * Real.sin (angle x y) = 0\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave hcos := cos_angle_add_angle_sub_add_angle_sub_eq_neg_one hx hy\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave hsin := sin_angle_add_angle_sub_add_angle_sub_eq_zero hx hy\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nhsin : Real.sin (angle x y + angle x (x - y) + angle y (y - x)) = 0\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nrw [Real.sin_eq_zero_iff] at hsin \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nhsin : \u2203 n, \u2191n * \u03c0 = angle x y + angle x (x - y) + angle y (y - x)\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\ncases' hsin with n hn\n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : \u2191n * \u03c0 = angle x y + angle x (x - y) + angle y (y - x)\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nsymm at hn \n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave h0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x) :=\n  add_nonneg (add_nonneg (angle_nonneg _ _) (angle_nonneg _ _)) (angle_nonneg _ _)\n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave h3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0 :=\n  by\n  by_contra hnlt\n  have hxy : angle x y = \u03c0 := by\n    by_contra hxy\n    exact\n      hnlt\n        (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne (angle_le_pi _ _) hxy) (angle_le_pi _ _))\n          (angle_le_pi _ _))\n  rw [hxy] at hnlt \n  rw [angle_eq_pi_iff] at hxy \n  rcases hxy with \u27e8hx, \u27e8r, \u27e8hr, hxr\u27e9\u27e9\u27e9\n  rw [hxr, \u2190 one_smul \u211d x, \u2190 mul_smul, mul_one, \u2190 sub_smul, one_smul, sub_eq_add_neg,\n    angle_smul_right_of_pos _ _ (add_pos zero_lt_one (neg_pos_of_neg hr)), angle_self hx, add_zero] at hnlt \n  apply hnlt\n  rw [add_assoc]\n  exact add_lt_add_left (lt_of_le_of_lt (angle_le_pi _ _) (lt_add_of_pos_right \u03c0 Real.pi_pos)) _\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n[PROOFSTEP]\nby_contra hnlt\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00acangle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n\u22a2 False\n[PROOFSTEP]\nhave hxy : angle x y = \u03c0 := by\n  by_contra hxy\n  exact\n    hnlt\n      (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne (angle_le_pi _ _) hxy) (angle_le_pi _ _))\n        (angle_le_pi _ _))\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00acangle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n\u22a2 angle x y = \u03c0\n[PROOFSTEP]\nby_contra hxy\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00acangle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhxy : \u00acangle x y = \u03c0\n\u22a2 False\n[PROOFSTEP]\nexact\n  hnlt\n    (add_lt_add_of_lt_of_le (add_lt_add_of_lt_of_le (lt_of_le_of_ne (angle_le_pi _ _) hxy) (angle_le_pi _ _))\n      (angle_le_pi _ _))\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00acangle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhxy : angle x y = \u03c0\n\u22a2 False\n[PROOFSTEP]\nrw [hxy] at hnlt \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00ac\u03c0 + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhxy : angle x y = \u03c0\n\u22a2 False\n[PROOFSTEP]\nrw [angle_eq_pi_iff] at hxy \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00ac\u03c0 + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhxy : x \u2260 0 \u2227 \u2203 r, r < 0 \u2227 y = r \u2022 x\n\u22a2 False\n[PROOFSTEP]\nrcases hxy with \u27e8hx, \u27e8r, \u27e8hr, hxr\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx\u271d : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhnlt : \u00ac\u03c0 + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhx : x \u2260 0\nr : \u211d\nhr : r < 0\nhxr : y = r \u2022 x\n\u22a2 False\n[PROOFSTEP]\nrw [hxr, \u2190 one_smul \u211d x, \u2190 mul_smul, mul_one, \u2190 sub_smul, one_smul, sub_eq_add_neg,\n  angle_smul_right_of_pos _ _ (add_pos zero_lt_one (neg_pos_of_neg hr)), angle_self hx, add_zero] at hnlt \n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx\u271d : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhx : x \u2260 0\nr : \u211d\nhnlt : \u00ac\u03c0 + angle (r \u2022 x) (r \u2022 x - x) < \u03c0 + \u03c0 + \u03c0\nhr : r < 0\nhxr : y = r \u2022 x\n\u22a2 False\n[PROOFSTEP]\napply hnlt\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx\u271d : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhx : x \u2260 0\nr : \u211d\nhnlt : \u00ac\u03c0 + angle (r \u2022 x) (r \u2022 x - x) < \u03c0 + \u03c0 + \u03c0\nhr : r < 0\nhxr : y = r \u2022 x\n\u22a2 \u03c0 + angle (r \u2022 x) (r \u2022 x - x) < \u03c0 + \u03c0 + \u03c0\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\ncase intro.intro.intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx\u271d : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhx : x \u2260 0\nr : \u211d\nhnlt : \u00ac\u03c0 + angle (r \u2022 x) (r \u2022 x - x) < \u03c0 + \u03c0 + \u03c0\nhr : r < 0\nhxr : y = r \u2022 x\n\u22a2 \u03c0 + angle (r \u2022 x) (r \u2022 x - x) < \u03c0 + (\u03c0 + \u03c0)\n[PROOFSTEP]\nexact add_lt_add_left (lt_of_le_of_lt (angle_le_pi _ _) (lt_add_of_pos_right \u03c0 Real.pi_pos)) _\n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave hn0 : 0 \u2264 n := by\n  rw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0 \n  norm_cast at h0 \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n\u22a2 0 \u2264 n\n[PROOFSTEP]\nrw [hn, mul_nonneg_iff_left_nonneg_of_pos Real.pi_pos] at h0 \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 \u2191n\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\n\u22a2 0 \u2264 n\n[PROOFSTEP]\nnorm_cast at h0 \n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn0 : 0 \u2264 n\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nhave hn3 : n < 3 := by\n  rw [hn, show \u03c0 + \u03c0 + \u03c0 = 3 * \u03c0 by ring] at h3lt \n  replace h3lt := lt_of_mul_lt_mul_right h3lt (le_of_lt Real.pi_pos)\n  norm_cast at h3lt \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn0 : 0 \u2264 n\n\u22a2 n < 3\n[PROOFSTEP]\nrw [hn, show \u03c0 + \u03c0 + \u03c0 = 3 * \u03c0 by ring] at h3lt \n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : \u2191n * \u03c0 < \u03c0 + \u03c0 + \u03c0\nhn0 : 0 \u2264 n\n\u22a2 \u03c0 + \u03c0 + \u03c0 = 3 * \u03c0\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : \u2191n * \u03c0 < 3 * \u03c0\nhn0 : 0 \u2264 n\n\u22a2 n < 3\n[PROOFSTEP]\nreplace h3lt := lt_of_mul_lt_mul_right h3lt (le_of_lt Real.pi_pos)\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nhn0 : 0 \u2264 n\nh3lt : \u2191n < 3\n\u22a2 n < 3\n[PROOFSTEP]\nnorm_cast at h3lt \n[GOAL]\ncase intro\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u2191n * \u03c0\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn0 : 0 \u2264 n\nhn3 : n < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\ninterval_cases n\n[GOAL]\ncase intro.\u00ab0\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21910 * \u03c0\nhn0 : 0 \u2264 0\nhn3 : 0 < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nrw [hn] at hcos \n[GOAL]\ncase intro.\u00ab0\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (\u21910 * \u03c0) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21910 * \u03c0\nhn0 : 0 \u2264 0\nhn3 : 0 < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nsimp at hcos \n[GOAL]\ncase intro.\u00ab0\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21910 * \u03c0\nhn0 : 0 \u2264 0\nhn3 : 0 < 3\nhcos : 1 = -1\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nnorm_num at hcos \n[GOAL]\ncase intro.\u00ab1\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21911 * \u03c0\nhn0 : 0 \u2264 1\nhn3 : 1 < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase intro.\u00ab1\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21911 * \u03c0\nhn0 : 0 \u2264 1\nhn3 : 1 < 3\n\u22a2 \u21911 * \u03c0 = \u03c0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.\u00ab2\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21912 * \u03c0\nhn0 : 0 \u2264 2\nhn3 : 2 < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nrw [hn] at hcos \n[GOAL]\ncase intro.\u00ab2\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhcos : Real.cos (\u21912 * \u03c0) = -1\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21912 * \u03c0\nhn0 : 0 \u2264 2\nhn3 : 2 < 3\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nsimp at hcos \n[GOAL]\ncase intro.\u00ab2\u00bb\nV : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nn : \u2124\nh0 : 0 \u2264 angle x y + angle x (x - y) + angle y (y - x)\nh3lt : angle x y + angle x (x - y) + angle y (y - x) < \u03c0 + \u03c0 + \u03c0\nhn : angle x y + angle x (x - y) + angle y (y - x) = \u21912 * \u03c0\nhn0 : 0 \u2264 2\nhn3 : 2 < 3\nhcos : 1 = -1\n\u22a2 angle x y + angle x (x - y) + angle y (y - x) = \u03c0\n[PROOFSTEP]\nnorm_num at hcos \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\n\u22a2 dist p1 p3 * dist p1 p3 =\n    dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 - 2 * dist p1 p2 * dist p3 p2 * Real.cos (\u2220 p1 p2 p3)\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V p1 p3, dist_eq_norm_vsub V p1 p2, dist_eq_norm_vsub V p3 p2]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\n\u22a2 \u2016p1 -\u1d65 p3\u2016 * \u2016p1 -\u1d65 p3\u2016 =\n    \u2016p1 -\u1d65 p2\u2016 * \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016 * \u2016p3 -\u1d65 p2\u2016 - 2 * \u2016p1 -\u1d65 p2\u2016 * \u2016p3 -\u1d65 p2\u2016 * Real.cos (\u2220 p1 p2 p3)\n[PROOFSTEP]\nunfold angle\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\n\u22a2 \u2016p1 -\u1d65 p3\u2016 * \u2016p1 -\u1d65 p3\u2016 =\n    \u2016p1 -\u1d65 p2\u2016 * \u2016p1 -\u1d65 p2\u2016 + \u2016p3 -\u1d65 p2\u2016 * \u2016p3 -\u1d65 p2\u2016 -\n      2 * \u2016p1 -\u1d65 p2\u2016 * \u2016p3 -\u1d65 p2\u2016 * Real.cos (InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2))\n[PROOFSTEP]\nconvert norm_sub_sq_eq_norm_sq_add_norm_sq_sub_two_mul_norm_mul_norm_mul_cos_angle (p1 -\u1d65 p2 : V) (p3 -\u1d65 p2 : V)\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p3 = p1 -\u1d65 p2 - (p3 -\u1d65 p2)\n[PROOFSTEP]\nexact (vsub_sub_vsub_cancel_right p1 p3 p2).symm\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\n\u22a2 p1 -\u1d65 p3 = p1 -\u1d65 p2 - (p3 -\u1d65 p2)\n[PROOFSTEP]\nexact (vsub_sub_vsub_cancel_right p1 p3 p2).symm\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : dist p1 p2 = dist p1 p3\n\u22a2 \u2220 p1 p2 p3 = \u2220 p1 p3 p2\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V p1 p2, dist_eq_norm_vsub V p1 p3] at h \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n\u22a2 \u2220 p1 p2 p3 = \u2220 p1 p3 p2\n[PROOFSTEP]\nunfold angle\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n\u22a2 InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p3) (p2 -\u1d65 p3)\n[PROOFSTEP]\nconvert angle_sub_eq_angle_sub_rev_of_norm_eq h\n[GOAL]\ncase h.e'_2.h.e'_5\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n\u22a2 p3 -\u1d65 p2 = p1 -\u1d65 p2 - (p1 -\u1d65 p3)\n[PROOFSTEP]\nexact (vsub_sub_vsub_cancel_left p3 p2 p1).symm\n[GOAL]\ncase h.e'_3.h.e'_5\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n\u22a2 p2 -\u1d65 p3 = p1 -\u1d65 p3 - (p1 -\u1d65 p2)\n[PROOFSTEP]\nexact (vsub_sub_vsub_cancel_left p2 p3 p1).symm\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : \u2220 p1 p2 p3 = \u2220 p1 p3 p2\nhpi : \u2220 p2 p1 p3 \u2260 \u03c0\n\u22a2 dist p1 p2 = dist p1 p3\n[PROOFSTEP]\nunfold angle at h hpi \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p3) (p2 -\u1d65 p3)\nhpi : InnerProductGeometry.angle (p2 -\u1d65 p1) (p3 -\u1d65 p1) \u2260 \u03c0\n\u22a2 dist p1 p2 = dist p1 p3\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V p1 p2, dist_eq_norm_vsub V p1 p3]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p3) (p2 -\u1d65 p3)\nhpi : InnerProductGeometry.angle (p2 -\u1d65 p1) (p3 -\u1d65 p1) \u2260 \u03c0\n\u22a2 \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n[PROOFSTEP]\nrw [\u2190 angle_neg_neg, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] at hpi \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh : InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) = InnerProductGeometry.angle (p1 -\u1d65 p3) (p2 -\u1d65 p3)\nhpi : InnerProductGeometry.angle (p1 -\u1d65 p2) (p1 -\u1d65 p3) \u2260 \u03c0\n\u22a2 \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n[PROOFSTEP]\nrw [\u2190 vsub_sub_vsub_cancel_left p3 p2 p1, \u2190 vsub_sub_vsub_cancel_left p2 p3 p1] at h \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh :\n  InnerProductGeometry.angle (p1 -\u1d65 p2) (p1 -\u1d65 p2 - (p1 -\u1d65 p3)) =\n    InnerProductGeometry.angle (p1 -\u1d65 p3) (p1 -\u1d65 p3 - (p1 -\u1d65 p2))\nhpi : InnerProductGeometry.angle (p1 -\u1d65 p2) (p1 -\u1d65 p3) \u2260 \u03c0\n\u22a2 \u2016p1 -\u1d65 p2\u2016 = \u2016p1 -\u1d65 p3\u2016\n[PROOFSTEP]\nexact norm_eq_of_angle_sub_eq_angle_sub_rev_of_angle_ne_pi h hpi\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh2 : p2 \u2260 p1\nh3 : p3 \u2260 p1\n\u22a2 \u2220 p1 p2 p3 + \u2220 p2 p3 p1 + \u2220 p3 p1 p2 = \u03c0\n[PROOFSTEP]\nrw [add_assoc, add_comm, add_comm (\u2220 p2 p3 p1), angle_comm p2 p3 p1]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh2 : p2 \u2260 p1\nh3 : p3 \u2260 p1\n\u22a2 \u2220 p3 p1 p2 + \u2220 p1 p3 p2 + \u2220 p1 p2 p3 = \u03c0\n[PROOFSTEP]\nunfold angle\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh2 : p2 \u2260 p1\nh3 : p3 \u2260 p1\n\u22a2 InnerProductGeometry.angle (p3 -\u1d65 p1) (p2 -\u1d65 p1) + InnerProductGeometry.angle (p1 -\u1d65 p3) (p2 -\u1d65 p3) +\n      InnerProductGeometry.angle (p1 -\u1d65 p2) (p3 -\u1d65 p2) =\n    \u03c0\n[PROOFSTEP]\nrw [\u2190 angle_neg_neg (p1 -\u1d65 p3), \u2190 angle_neg_neg (p1 -\u1d65 p2), neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev,\n  neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev, \u2190 vsub_sub_vsub_cancel_right p3 p2 p1, \u2190\n  vsub_sub_vsub_cancel_right p2 p3 p1]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\np1 p2 p3 : P\nh2 : p2 \u2260 p1\nh3 : p3 \u2260 p1\n\u22a2 InnerProductGeometry.angle (p3 -\u1d65 p1) (p2 -\u1d65 p1) + InnerProductGeometry.angle (p3 -\u1d65 p1) (p3 -\u1d65 p1 - (p2 -\u1d65 p1)) +\n      InnerProductGeometry.angle (p2 -\u1d65 p1) (p2 -\u1d65 p1 - (p3 -\u1d65 p1)) =\n    \u03c0\n[PROOFSTEP]\nexact\n  angle_add_angle_sub_add_angle_sub_eq_pi (fun he => h3 (vsub_eq_zero_iff_eq.1 he)) fun he =>\n    h2 (vsub_eq_zero_iff_eq.1 he)\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : InnerProductSpace \u211d V\ninst\u271d\u00b3 : MetricSpace P\ninst\u271d\u00b2 : NormedAddTorsor V P\ninst\u271d\u00b9 : Module.Oriented \u211d V (Fin 2)\ninst\u271d : Fact (FiniteDimensional.finrank \u211d V = 2)\np1 p2 p3 : P\nh21 : p2 \u2260 p1\nh32 : p3 \u2260 p2\nh13 : p1 \u2260 p3\n\u22a2 \u2221 p1 p2 p3 + \u2221 p2 p3 p1 + \u2221 p3 p1 p2 = \u2191\u03c0\n[PROOFSTEP]\nsimpa only [neg_vsub_eq_vsub_rev] using\n  positiveOrientation.oangle_add_cyc3_neg_left (vsub_ne_zero.mpr h21) (vsub_ne_zero.mpr h32) (vsub_ne_zero.mpr h13)\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c p : P\nh : \u2220 b p c = \u03c0\n\u22a2 dist a b ^ 2 * dist c p + dist a c ^ 2 * dist b p = dist b c * (dist a p ^ 2 + dist b p * dist c p)\n[PROOFSTEP]\nrw [pow_two, pow_two, law_cos a p b, law_cos a p c, eq_sub_of_add_eq (angle_add_angle_eq_pi_of_angle_eq_pi a h),\n  Real.cos_pi_sub, dist_eq_add_dist_of_angle_eq_pi h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c p : P\nh : \u2220 b p c = \u03c0\n\u22a2 (dist a p * dist a p + dist b p * dist b p - 2 * dist a p * dist b p * -Real.cos (\u2220 a p c)) * dist c p +\n      (dist a p * dist a p + dist c p * dist c p - 2 * dist a p * dist c p * Real.cos (\u2220 a p c)) * dist b p =\n    (dist b p + dist c p) * (dist a p ^ 2 + dist b p * dist c p)\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nby_cases hbc : b = c\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : b = c\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nsimp [hbc, midpoint_self, dist_self, two_mul]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nlet m := midpoint \u211d b c\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nhave : dist b c \u2260 0 := (dist_pos.mpr hbc).ne'\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nhave hm := dist_sq_mul_dist_add_dist_sq_mul_dist a b c m (angle_midpoint_eq_pi b c hbc)\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm : dist a b ^ 2 * dist c m + dist a c ^ 2 * dist b m = dist b c * (dist a m ^ 2 + dist b m * dist c m)\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nsimp only [dist_left_midpoint, dist_right_midpoint, Real.norm_two] at hm \n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\ncalc\n  dist a b ^ 2 + dist a c ^ 2 =\n      2 / dist b c * (dist a b ^ 2 * ((2 : \u211d)\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c)) :=\n    by field_simp; ring\n  _ = 2 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2) := by rw [hm]; field_simp; ring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 dist a b ^ 2 + dist a c ^ 2 = \u21912 / dist b c * (dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c))\n[PROOFSTEP]\nfield_simp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 (dist a b ^ 2 + dist a c ^ 2) * (dist b c * 2) = 2 * (dist a b ^ 2 * dist b c + dist a c ^ 2 * dist b c)\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 \u21912 / dist b c * (dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c)) =\n    \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nrw [hm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 \u21912 / dist b c * (dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))) =\n    \u21912 * (dist a (midpoint \u211d b c) ^ 2 + (dist b c / 2) ^ 2)\n[PROOFSTEP]\nfield_simp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c : P\nhbc : \u00acb = c\nm : P := midpoint \u211d b c\nthis : dist b c \u2260 0\nhm :\n  dist a b ^ 2 * (2\u207b\u00b9 * dist b c) + dist a c ^ 2 * (2\u207b\u00b9 * dist b c) =\n    dist b c * (dist a (midpoint \u211d b c) ^ 2 + 2\u207b\u00b9 * dist b c * (2\u207b\u00b9 * dist b c))\n\u22a2 2 * (dist b c * (dist a (midpoint \u211d b c) ^ 2 * (2 * 2) + dist b c * dist b c)) * 2 ^ 2 =\n    2 * (dist a (midpoint \u211d b c) ^ 2 * 2 ^ 2 + dist b c ^ 2) * (dist b c * (2 * 2))\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nhave h' : dist a' c' ^ 2 = (r * dist a c) ^ 2\n[GOAL]\ncase h'\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 dist a' c' ^ 2 = (r * dist a c) ^ 2\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\ncalc\n  dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - 2 * dist a' b' * dist c' b' * Real.cos (\u2220 a' b' c') := by\n    simp [pow_two, law_cos a' b' c']\n  _ = r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * Real.cos (\u2220 a b c)) := by rw [h, hab, hcb]; ring\n  _ = (r * dist a c) ^ 2 := by simp [pow_two, \u2190 law_cos a b c, mul_pow]; ring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 dist a' c' ^ 2 = dist a' b' ^ 2 + dist c' b' ^ 2 - 2 * dist a' b' * dist c' b' * Real.cos (\u2220 a' b' c')\n[PROOFSTEP]\nsimp [pow_two, law_cos a' b' c']\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 dist a' b' ^ 2 + dist c' b' ^ 2 - 2 * dist a' b' * dist c' b' * Real.cos (\u2220 a' b' c') =\n    r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * Real.cos (\u2220 a b c))\n[PROOFSTEP]\nrw [h, hab, hcb]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 (r * dist a b) ^ 2 + (r * dist c b) ^ 2 - 2 * (r * dist a b) * (r * dist c b) * Real.cos (\u2220 a b c) =\n    r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * Real.cos (\u2220 a b c))\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 r ^ 2 * (dist a b ^ 2 + dist c b ^ 2 - 2 * dist a b * dist c b * Real.cos (\u2220 a b c)) = (r * dist a c) ^ 2\n[PROOFSTEP]\nsimp [pow_two, \u2190 law_cos a b c, mul_pow]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\n\u22a2 r * r * (dist a c * dist a c) = r * dist a c * (r * dist a c)\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nby_cases hab\u2081 : a = b\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : a = b\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nhave hab'\u2081 : a' = b' := by rw [\u2190 dist_eq_zero, hab, dist_eq_zero.mpr hab\u2081, mul_zero r]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : a = b\n\u22a2 a' = b'\n[PROOFSTEP]\nrw [\u2190 dist_eq_zero, hab, dist_eq_zero.mpr hab\u2081, mul_zero r]\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : a = b\nhab'\u2081 : a' = b'\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nrw [hab\u2081, hab'\u2081, dist_comm b' c', dist_comm b c, hcb]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : \u00aca = b\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nhave h1 : 0 \u2264 r * dist a b := by rw [\u2190 hab]; exact dist_nonneg\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : \u00aca = b\n\u22a2 0 \u2264 r * dist a b\n[PROOFSTEP]\nrw [\u2190 hab]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : \u00aca = b\n\u22a2 0 \u2264 dist a' b'\n[PROOFSTEP]\nexact dist_nonneg\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : \u00aca = b\nh1 : 0 \u2264 r * dist a b\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nhave h2 : 0 \u2264 r := nonneg_of_mul_nonneg_left h1 (dist_pos.mpr hab\u2081)\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\na b c a' b' c' : P\nr : \u211d\nh : \u2220 a' b' c' = \u2220 a b c\nhab : dist a' b' = r * dist a b\nhcb : dist c' b' = r * dist c b\nh' : dist a' c' ^ 2 = (r * dist a c) ^ 2\nhab\u2081 : \u00aca = b\nh1 : 0 \u2264 r * dist a b\nh2 : 0 \u2264 r\n\u22a2 dist a' c' = r * dist a c\n[PROOFSTEP]\nexact (sq_eq_sq dist_nonneg (mul_nonneg h2 dist_nonneg)).mp h'\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Triangle", "llama_tokens": 38286, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6825737408694988, "lm_q1q2_score": 0.5360310157934745}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b2 : DecidablePred (r a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\ninst\u271d : (a : \u03b1) \u2192 (b : \u03b2) \u2192 Decidable (r a b)\n\u22a2 \u2211 a in s, card (bipartiteAbove r t a) = \u2211 b in t, card (bipartiteBelow r s b)\n[PROOFSTEP]\nsimp_rw [card_eq_sum_ones, bipartiteAbove, bipartiteBelow, sum_filter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b2 : DecidablePred (r a)\ninst\u271d\u00b9 : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\ninst\u271d : (a : \u03b1) \u2192 (b : \u03b2) \u2192 Decidable (r a b)\n\u22a2 (\u2211 x in s, \u2211 a in t, if r x a then 1 else 0) = \u2211 x in t, \u2211 a in s, if r a x then 1 else 0\n[PROOFSTEP]\nexact sum_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s \u2264 card t\n[PROOFSTEP]\nclassical\nrw [\u2190 mul_one s.card, \u2190 mul_one t.card]\nexact\n  card_mul_le_card_mul r\n    (fun a h \u21a6\n      card_pos.2\n        (by\n          rw [\u2190 coe_nonempty, coe_bipartiteAbove]\n          exact hs _ h : (t.bipartiteAbove r a).Nonempty))\n    (fun b h \u21a6\n      card_le_one.2\n        (by\n          simp_rw [mem_bipartiteBelow]\n          exact ht _ h))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s \u2264 card t\n[PROOFSTEP]\nrw [\u2190 mul_one s.card, \u2190 mul_one t.card]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\n\u22a2 card s * 1 \u2264 card t * 1\n[PROOFSTEP]\nexact\n  card_mul_le_card_mul r\n    (fun a h \u21a6\n      card_pos.2\n        (by\n          rw [\u2190 coe_nonempty, coe_bipartiteAbove]\n          exact hs _ h : (t.bipartiteAbove r a).Nonempty))\n    (fun b h \u21a6\n      card_le_one.2\n        (by\n          simp_rw [mem_bipartiteBelow]\n          exact ht _ h))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a\u271d)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\na : \u03b1\nh : a \u2208 s\n\u22a2 Finset.Nonempty (bipartiteAbove r t a)\n[PROOFSTEP]\nrw [\u2190 coe_nonempty, coe_bipartiteAbove]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na\u271d a' : \u03b1\nb b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a\u271d)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\na : \u03b1\nh : a \u2208 s\n\u22a2 Set.Nonempty {b | b \u2208 t \u2227 r a b}\n[PROOFSTEP]\nexact hs _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb\u271d b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b\u271d)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2200 (a : \u03b1), a \u2208 bipartiteBelow r s b \u2192 \u2200 (b_1 : \u03b1), b_1 \u2208 bipartiteBelow r s b \u2192 a = b_1\n[PROOFSTEP]\nsimp_rw [mem_bipartiteBelow]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\ns : Finset \u03b1\nt : Finset \u03b2\na a' : \u03b1\nb\u271d b' : \u03b2\ninst\u271d\u00b9 : DecidablePred (r a)\ninst\u271d : (a : \u03b1) \u2192 Decidable (r a b\u271d)\nm n : \u2115\nhs : \u2200 (a : \u03b1), a \u2208 s \u2192 \u2203 b, b \u2208 t \u2227 r a b\nht : \u2200 (b : \u03b2), b \u2208 t \u2192 Set.Subsingleton {a | a \u2208 s \u2227 r a b}\nb : \u03b2\nh : b \u2208 t\n\u22a2 \u2200 (a : \u03b1), a \u2208 s \u2227 r a b \u2192 \u2200 (b_1 : \u03b1), b_1 \u2208 s \u2227 r b_1 b \u2192 a = b_1\n[PROOFSTEP]\nexact ht _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\nh\u2081 : LeftTotal r\nh\u2082 : LeftUnique r\n\u22a2 \u2200 (a : \u03b1), a \u2208 univ \u2192 \u2203 b, b \u2208 univ \u2227 r a b\n[PROOFSTEP]\nsimpa using h\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u03b2\nr : \u03b1 \u2192 \u03b2 \u2192 Prop\nh\u2081 : RightTotal r\nh\u2082 : RightUnique r\n\u22a2 \u2200 (b : \u03b2), b \u2208 univ \u2192 \u2203 a, a \u2208 univ \u2227 r a b\n[PROOFSTEP]\nsimpa using h\u2081\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.DoubleCounting", "llama_tokens": 2312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.6992544335934766, "lm_q1q2_score": 0.5358342273623545}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\n\u22a2 completion (id G) = id (Completion G)\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nx : Completion G\n\u22a2 \u2191(completion (id G)) x = \u2191(id (Completion G)) x\n[PROOFSTEP]\nrw [NormedAddGroupHom.completion_def, NormedAddGroupHom.coe_id, Completion.map_id]\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nx : Completion G\n\u22a2 _root_.id x = \u2191(id (Completion G)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\n\u22a2 NormedAddGroupHom.comp (completion g) (completion f) = completion (NormedAddGroupHom.comp g f)\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n\u22a2 \u2191(NormedAddGroupHom.comp (completion g) (completion f)) x = \u2191(completion (NormedAddGroupHom.comp g f)) x\n[PROOFSTEP]\nrw [NormedAddGroupHom.coe_comp, NormedAddGroupHom.completion_def, NormedAddGroupHom.completion_coe_to_fun,\n  NormedAddGroupHom.completion_coe_to_fun, Completion.map_comp g.uniformContinuous f.uniformContinuous]\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom H K\nx : Completion G\n\u22a2 Completion.map (\u2191g \u2218 \u2191f) x = Completion.map (\u2191(NormedAddGroupHom.comp g f)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\n\u22a2 \u2200 (v : G), \u2016\u2191G v\u2016 \u2264 1 * \u2016v\u2016\n[PROOFSTEP]\nsimp [le_refl]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\n\u22a2 NormedAddGroupHom.comp (completion f) toCompl = NormedAddGroupHom.comp toCompl f\n[PROOFSTEP]\next x\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nx : G\n\u22a2 \u2191(NormedAddGroupHom.comp (completion f) toCompl) x = \u2191(NormedAddGroupHom.comp toCompl f) x\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nx : G\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016completion f\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimpa using f.completion.le_opNorm x\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\n\u22a2 range (NormedAddGroupHom.comp toCompl (incl (ker f))) \u2264 ker (completion f)\n[PROOFSTEP]\nrintro _ \u27e8\u27e8g, h\u2080 : f g = 0\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\ng : G\nh\u2080 : \u2191f g = 0\n\u22a2 \u2191(toAddMonoidHom (NormedAddGroupHom.comp toCompl (incl (ker f)))) { val := g, property := h\u2080 } \u2208 ker (completion f)\n[PROOFSTEP]\nsimp [h\u2080, mem_ker, Completion.coe_zero]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\n\u22a2 \u2191(ker (completion f)) = closure \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f))))\n[PROOFSTEP]\nrefine le_antisymm ?_ (closure_minimal f.ker_le_ker_completion f.completion.isClosed_ker)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\n\u22a2 \u2191(ker (completion f)) \u2264 closure \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f))))\n[PROOFSTEP]\nrintro hatg (hatg_in : f.completion hatg = 0)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u22a2 hatg \u2208 closure \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f))))\n[PROOFSTEP]\nrw [SeminormedAddCommGroup.mem_closure_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5_pos\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nrcases h.exists_pos with \u27e8C', C'_pos, hC'\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nrcases exists_pos_mul_lt \u03b5_pos (1 + C' * \u2016f\u2016) with \u27e8\u03b4, \u03b4_pos, h\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8_, \u27e8g : G, rfl\u27e9, hg : \u2016hatg - g\u2016 < \u03b4\u27e9 :=\n  SeminormedAddCommGroup.mem_closure_iff.mp (Completion.denseInducing_coe.dense hatg) \u03b4 \u03b4_pos\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8g' : G, hgg' : f g' = f g, hfg : \u2016g'\u2016 \u2264 C' * \u2016f g\u2016\u27e9 := hC' (f g) (mem_range_self _ g)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nhave mem_ker : g - g' \u2208 f.ker := by rw [f.mem_ker, map_sub, sub_eq_zero.mpr hgg'.symm]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\n\u22a2 g - g' \u2208 ker f\n[PROOFSTEP]\nrw [f.mem_ker, map_sub, sub_eq_zero.mpr hgg'.symm]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\n\u22a2 \u2203 b, b \u2208 \u2191(range (NormedAddGroupHom.comp toCompl (incl (ker f)))) \u2227 \u2016hatg - b\u2016 < \u03b5\n[PROOFSTEP]\nrefine \u27e8_, \u27e8\u27e8g - g', mem_ker\u27e9, rfl\u27e9, ?_\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\n\u22a2 \u2016hatg - \u2191(toAddMonoidHom (NormedAddGroupHom.comp toCompl (incl (ker f)))) { val := g - g', property := mem_ker }\u2016 < \u03b5\n[PROOFSTEP]\nhave : \u2016f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\n\u22a2 \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u2016hatg - \u2191(toAddMonoidHom (NormedAddGroupHom.comp toCompl (incl (ker f)))) { val := g - g', property := mem_ker }\u2016 < \u03b5\n[PROOFSTEP]\ncalc\n  \u2016f g\u2016 \u2264 \u2016f\u2016 * \u2016hatg - g\u2016 := by simpa [hatg_in] using f.completion.le_opNorm (hatg - g)\n  _ \u2264 \u2016f\u2016 * \u03b4 := by gcongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\n\u22a2 \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u2016hatg - \u2191G g\u2016\n[PROOFSTEP]\nsimpa [hatg_in] using f.completion.le_opNorm (hatg - g)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\n\u22a2 \u2016f\u2016 * \u2016hatg - \u2191G g\u2016 \u2264 \u2016f\u2016 * \u03b4\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u2016hatg - \u2191(toAddMonoidHom (NormedAddGroupHom.comp toCompl (incl (ker f)))) { val := g - g', property := mem_ker }\u2016 < \u03b5\n[PROOFSTEP]\ncalc\n  \u2016hatg - \u2191(g - g')\u2016 = \u2016hatg - g + g'\u2016 := by rw [Completion.coe_sub, sub_add]\n  _ \u2264 \u2016hatg - g\u2016 + \u2016(g' : Completion G)\u2016 := (norm_add_le _ _)\n  _ = \u2016hatg - g\u2016 + \u2016g'\u2016 := by rw [Completion.norm_coe]\n  _ < \u03b4 + C' * \u2016f g\u2016 := (add_lt_add_of_lt_of_le hg hfg)\n  _ \u2264 \u03b4 + C' * (\u2016f\u2016 * \u03b4) := by gcongr\n  _ < \u03b5 := by simpa only [add_mul, one_mul, mul_assoc] using h\u03b4\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u2016hatg - \u2191G (g - g')\u2016 = \u2016hatg - \u2191G g + \u2191G g'\u2016\n[PROOFSTEP]\nrw [Completion.coe_sub, sub_add]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u2016hatg - \u2191G g\u2016 + \u2016\u2191G g'\u2016 = \u2016hatg - \u2191G g\u2016 + \u2016g'\u2016\n[PROOFSTEP]\nrw [Completion.norm_coe]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u03b4 + C' * \u2016\u2191f g\u2016 \u2264 \u03b4 + C' * (\u2016f\u2016 * \u03b4)\n[PROOFSTEP]\ngcongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b9 : SeminormedAddCommGroup H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nf : NormedAddGroupHom G H\nC : \u211d\nh : SurjectiveOnWith f (range f) C\nhatg : Completion G\nhatg_in : \u2191(completion f) hatg = 0\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nC' : \u211d\nC'_pos : C' > 0\nhC' : SurjectiveOnWith f (range f) C'\n\u03b4 : \u211d\n\u03b4_pos : 0 < \u03b4\nh\u03b4 : (1 + C' * \u2016f\u2016) * \u03b4 < \u03b5\ng : G\nhg : \u2016hatg - \u2191G g\u2016 < \u03b4\ng' : G\nhgg' : \u2191f g' = \u2191f g\nhfg : \u2016g'\u2016 \u2264 C' * \u2016\u2191f g\u2016\nmem_ker : g - g' \u2208 ker f\nthis : \u2016\u2191f g\u2016 \u2264 \u2016f\u2016 * \u03b4\n\u22a2 \u03b4 + C' * (\u2016f\u2016 * \u03b4) < \u03b5\n[PROOFSTEP]\nsimpa only [add_mul, one_mul, mul_assoc] using h\u03b4\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup H\ninst\u271d\u00b9 : SeparatedSpace H\ninst\u271d : CompleteSpace H\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom (Completion G) H\nhg : \u2200 (v : G), \u2191f v = \u2191g (\u2191G v)\n\u22a2 extension f = g\n[PROOFSTEP]\next v\n[GOAL]\ncase H\nG : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup G\nH : Type u_2\ninst\u271d\u00b2 : SeminormedAddCommGroup H\ninst\u271d\u00b9 : SeparatedSpace H\ninst\u271d : CompleteSpace H\nf : NormedAddGroupHom G H\ng : NormedAddGroupHom (Completion G) H\nhg : \u2200 (v : G), \u2191f v = \u2191g (\u2191G v)\nv : Completion G\n\u22a2 \u2191(extension f) v = \u2191g v\n[PROOFSTEP]\nrw [NormedAddGroupHom.extension_coe_to_fun,\n  Completion.extension_unique f.uniformContinuous g.uniformContinuous fun a => hg a]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.HomCompletion", "llama_tokens": 8639, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.7662936430859597, "lm_q1q2_score": 0.5358342273623545}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Distrib R\na x y x' y' : R\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 SemiconjBy a (x + x') (y + y')\n[PROOFSTEP]\nsimp only [SemiconjBy, left_distrib, right_distrib, h.eq, h'.eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Distrib R\na b x y : R\nha : SemiconjBy a x y\nhb : SemiconjBy b x y\n\u22a2 SemiconjBy (a + b) x y\n[PROOFSTEP]\nsimp only [SemiconjBy, left_distrib, right_distrib, ha.eq, hb.eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Mul R\ninst\u271d : HasDistribNeg R\na x y : R\nh : SemiconjBy a x y\n\u22a2 SemiconjBy a (-x) (-y)\n[PROOFSTEP]\nsimp only [SemiconjBy, h.eq, neg_mul, mul_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Mul R\ninst\u271d : HasDistribNeg R\na x y : R\nh : SemiconjBy a x y\n\u22a2 SemiconjBy (-a) x y\n[PROOFSTEP]\nsimp only [SemiconjBy, h.eq, neg_mul, mul_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing R\na b x y x' y' : R\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n\u22a2 SemiconjBy a (x - x') (y - y')\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using h.add_right h'.neg_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : NonUnitalNonAssocRing R\na b x y x' y' : R\nha : SemiconjBy a x y\nhb : SemiconjBy b x y\n\u22a2 SemiconjBy (a - b) x y\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using ha.add_left hb.neg_left\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Semiconj", "llama_tokens": 671, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5357971503119263}}
{"text": "[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhx : x \u2260 c\nhy : y \u2260 c\n\u22a2 inversion c R x \u2208 perpBisector c (inversion c R y) \u2194 dist x y = dist y c\n[PROOFSTEP]\nrw [mem_perpBisector_iff_dist_eq, dist_inversion_inversion hx hy, dist_inversion_center]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhx : x \u2260 c\nhy : y \u2260 c\n\u22a2 R ^ 2 / dist x c = R ^ 2 / (dist x c * dist y c) * dist x y \u2194 dist x y = dist y c\n[PROOFSTEP]\nhave hx' := dist_ne_zero.2 hx\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhx : x \u2260 c\nhy : y \u2260 c\nhx' : dist x c \u2260 0\n\u22a2 R ^ 2 / dist x c = R ^ 2 / (dist x c * dist y c) * dist x y \u2194 dist x y = dist y c\n[PROOFSTEP]\nhave hy' := dist_ne_zero.2 hy\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhx : x \u2260 c\nhy : y \u2260 c\nhx' : dist x c \u2260 0\nhy' : dist y c \u2260 0\n\u22a2 R ^ 2 / dist x c = R ^ 2 / (dist x c * dist y c) * dist x y \u2194 dist x y = dist y c\n[PROOFSTEP]\nfield_simp [mul_assoc, mul_comm, hx, hx.symm, eq_comm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R x \u2208 perpBisector c (inversion c R y) \u2194 dist x y = dist y c \u2227 x \u2260 c\n[PROOFSTEP]\nrcases eq_or_ne x c with rfl | hx\n[GOAL]\ncase inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nx y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 x\n\u22a2 inversion x R x \u2208 perpBisector x (inversion x R y) \u2194 dist x y = dist y x \u2227 x \u2260 x\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nhx : x \u2260 c\n\u22a2 inversion c R x \u2208 perpBisector c (inversion c R y) \u2194 dist x y = dist y c \u2227 x \u2260 c\n[PROOFSTEP]\nsimp [inversion_mem_perpBisector_inversion_iff hR hx hy, hx]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R \u207b\u00b9' \u2191(perpBisector c y) = sphere (inversion c R y) (R ^ 2 / dist y c) \\ {c}\n[PROOFSTEP]\nrw [\u2190 dist_inversion_center, \u2190 preimage_inversion_perpBisector_inversion hR, inversion_inversion]\n[GOAL]\ncase hR\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 R \u2260 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R y \u2260 c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R '' \u2191(perpBisector c y) = sphere (inversion c R y) (R ^ 2 / dist y c) \\ {c}\n[PROOFSTEP]\nrw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),\n  preimage_inversion_perpBisector hR hy]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R \u207b\u00b9' sphere y (dist y c) = insert c \u2191(perpBisector c (inversion c R y))\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x\u271d y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nx : P\n\u22a2 x \u2208 inversion c R \u207b\u00b9' sphere y (dist y c) \u2194 x \u2208 insert c \u2191(perpBisector c (inversion c R y))\n[PROOFSTEP]\nrcases eq_or_ne x c with rfl | hx\n[GOAL]\ncase h.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nx\u271d y : P\nR : \u211d\nhR : R \u2260 0\nx : P\nhy : y \u2260 x\n\u22a2 x \u2208 inversion x R \u207b\u00b9' sphere y (dist y x) \u2194 x \u2208 insert x \u2191(perpBisector x (inversion x R y))\n[PROOFSTEP]\nsimp [dist_comm]\n[GOAL]\ncase h.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x\u271d y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nx : P\nhx : x \u2260 c\n\u22a2 x \u2208 inversion c R \u207b\u00b9' sphere y (dist y c) \u2194 x \u2208 insert c \u2191(perpBisector c (inversion c R y))\n[PROOFSTEP]\nrw [mem_preimage, mem_sphere, \u2190 inversion_mem_perpBisector_inversion_iff hR]\n[GOAL]\ncase h.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x\u271d y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nx : P\nhx : x \u2260 c\n\u22a2 inversion c R (inversion c R x) \u2208 perpBisector c (inversion c R y) \u2194 x \u2208 insert c \u2191(perpBisector c (inversion c R y))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.inr.hx\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x\u271d y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nx : P\nhx : x \u2260 c\n\u22a2 inversion c R x \u2260 c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase h.inr.hy\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x\u271d y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\nx : P\nhx : x \u2260 c\n\u22a2 y \u2260 c\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nc x y : P\nR : \u211d\nhR : R \u2260 0\nhy : y \u2260 c\n\u22a2 inversion c R '' sphere y (dist y c) = insert c \u2191(perpBisector c (inversion c R y))\n[PROOFSTEP]\nrw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),\n  preimage_inversion_sphere_dist_center hR hy]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Inversion.ImageHyperplane", "llama_tokens": 3147, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891435927268, "lm_q2_score": 0.6513548714339144, "lm_q1q2_score": 0.5355369039192007}}
{"text": "[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedSemiring k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : 0 < c\n\u22a2 c \u2022 a < 0 \u2194 a < 0\n[PROOFSTEP]\nrw [\u2190 neg_neg a, smul_neg, neg_neg_iff_pos, neg_neg_iff_pos]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedSemiring k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : 0 < c\n\u22a2 0 < c \u2022 -a \u2194 0 < -a\n[PROOFSTEP]\nexact smul_pos_iff_of_pos hc\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n\u22a2 c \u2022 b < c \u2022 a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a < b\nhc : c < 0\n\u22a2 -c \u2022 a < -c \u2022 b\n[PROOFSTEP]\nexact smul_lt_smul_of_pos h (neg_pos_of_neg hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a \u2264 b\nhc : c \u2264 0\n\u22a2 c \u2022 b \u2264 c \u2022 a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : a \u2264 b\nhc : c \u2264 0\n\u22a2 -c \u2022 a \u2264 -c \u2022 b\n[PROOFSTEP]\nexact smul_le_smul_of_nonneg h (neg_nonneg_of_nonpos hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhab : c \u2022 a = c \u2022 b\nhc : c < 0\nh : a \u2264 b\n\u22a2 a = b\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_inj] at hab \n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhab : -c \u2022 a = -c \u2022 b\nhc : c < 0\nh : a \u2264 b\n\u22a2 a = b\n[PROOFSTEP]\nexact eq_of_smul_eq_smul_of_pos_of_le hab (neg_pos_of_neg hc) h\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : c \u2022 a < c \u2022 b\nhc : c \u2264 0\n\u22a2 b < a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff] at h \n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : -c \u2022 b < -c \u2022 a\nhc : c \u2264 0\n\u22a2 b < a\n[PROOFSTEP]\nexact lt_of_smul_lt_smul_of_nonneg h (neg_nonneg_of_nonpos hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 c \u2022 a < c \u2022 b \u2194 b < a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 -c \u2022 b < -c \u2022 a \u2194 b < a\n[PROOFSTEP]\nexact smul_lt_smul_iff_of_pos (neg_pos_of_neg hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 c \u2022 a < 0 \u2194 0 < a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_neg_iff_pos]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 0 < -c \u2022 a \u2194 0 < a\n[PROOFSTEP]\nexact smul_pos_iff_of_pos (neg_pos_of_neg hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 0 < c \u2022 a \u2194 a < 0\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_pos]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : OrderedRing k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 -c \u2022 a < 0 \u2194 a < 0\n[PROOFSTEP]\nexact smul_neg_iff_of_pos (neg_pos_of_neg hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : k\nc d : M\nhab : a \u2264 b\nhcd : c \u2264 d\n\u22a2 a \u2022 d + b \u2022 c \u2264 a \u2022 c + b \u2022 d\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 := exists_add_of_le hab\n[GOAL]\ncase intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na : k\nc d : M\nhcd : c \u2264 d\nb : k\nhab : a \u2264 a + b\n\u22a2 a \u2022 d + (a + b) \u2022 c \u2264 a \u2022 c + (a + b) \u2022 d\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := exists_add_of_le hcd\n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na : k\nc : M\nb : k\nhab : a \u2264 a + b\nd : M\nhcd : c \u2264 c + d\n\u22a2 a \u2022 (c + d) + (a + b) \u2022 c \u2264 a \u2022 c + (a + b) \u2022 (c + d)\n[PROOFSTEP]\nrw [smul_add, add_right_comm, smul_add, \u2190 add_assoc, add_smul _ _ d]\n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na : k\nc : M\nb : k\nhab : a \u2264 a + b\nd : M\nhcd : c \u2264 c + d\n\u22a2 a \u2022 c + (a + b) \u2022 c + a \u2022 d \u2264 a \u2022 c + (a + b) \u2022 c + (a \u2022 d + b \u2022 d)\n[PROOFSTEP]\nrw [le_add_iff_nonneg_right] at hab hcd \n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na : k\nc : M\nb : k\nhab : 0 \u2264 b\nd : M\nhcd : 0 \u2264 d\n\u22a2 a \u2022 c + (a + b) \u2022 c + a \u2022 d \u2264 a \u2022 c + (a + b) \u2022 c + (a \u2022 d + b \u2022 d)\n[PROOFSTEP]\nexact add_le_add_left (le_add_of_nonneg_right <| smul_nonneg hab hcd) _\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : k\nc d : M\nhba : b \u2264 a\nhdc : d \u2264 c\n\u22a2 a \u2022 d + b \u2022 c \u2264 a \u2022 c + b \u2022 d\n[PROOFSTEP]\nrw [add_comm (a \u2022 d), add_comm (a \u2022 c)]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : OrderedRing k\ninst\u271d\u00b3 : OrderedAddCommGroup M\ninst\u271d\u00b2 : Module k M\ninst\u271d\u00b9 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : k\nc d : M\nhba : b \u2264 a\nhdc : d \u2264 c\n\u22a2 b \u2022 c + a \u2022 d \u2264 b \u2022 d + a \u2022 c\n[PROOFSTEP]\nexact smul_add_smul_le_smul_add_smul hba hdc\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : k\nc d : M\nhab : a < b\nhcd : c < d\n\u22a2 a \u2022 d + b \u2022 c < a \u2022 c + b \u2022 d\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 := exists_add_of_le hab.le\n[GOAL]\ncase intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : k\nc d : M\nhcd : c < d\nb : k\nhab : a < a + b\n\u22a2 a \u2022 d + (a + b) \u2022 c < a \u2022 c + (a + b) \u2022 d\n[PROOFSTEP]\nobtain \u27e8d, rfl\u27e9 := exists_add_of_le hcd.le\n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : k\nc : M\nb : k\nhab : a < a + b\nd : M\nhcd : c < c + d\n\u22a2 a \u2022 (c + d) + (a + b) \u2022 c < a \u2022 c + (a + b) \u2022 (c + d)\n[PROOFSTEP]\nrw [smul_add, add_right_comm, smul_add, \u2190 add_assoc, add_smul _ _ d]\n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : k\nc : M\nb : k\nhab : a < a + b\nd : M\nhcd : c < c + d\n\u22a2 a \u2022 c + (a + b) \u2022 c + a \u2022 d < a \u2022 c + (a + b) \u2022 c + (a \u2022 d + b \u2022 d)\n[PROOFSTEP]\nrw [lt_add_iff_pos_right] at hab hcd \n[GOAL]\ncase intro.intro\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na : k\nc : M\nb : k\nhab : 0 < b\nd : M\nhcd : 0 < d\n\u22a2 a \u2022 c + (a + b) \u2022 c + a \u2022 d < a \u2022 c + (a + b) \u2022 c + (a \u2022 d + b \u2022 d)\n[PROOFSTEP]\nexact add_lt_add_left (lt_add_of_pos_right _ <| smul_pos hab hcd) _\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : k\nc d : M\nhba : b < a\nhdc : d < c\n\u22a2 a \u2022 d + b \u2022 c < a \u2022 c + b \u2022 d\n[PROOFSTEP]\nrw [add_comm (a \u2022 d), add_comm (a \u2022 c)]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2075 : OrderedRing k\ninst\u271d\u2074 : OrderedAddCommGroup M\ninst\u271d\u00b3 : Module k M\ninst\u271d\u00b2 : OrderedSMul k M\na\u271d b\u271d : M\nc\u271d : k\ninst\u271d\u00b9 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\ninst\u271d : ContravariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1\na b : k\nc d : M\nhba : b < a\nhdc : d < c\n\u22a2 b \u2022 c + a \u2022 d < b \u2022 d + a \u2022 c\n[PROOFSTEP]\nexact smul_add_smul_lt_smul_add_smul hba hdc\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 c \u2022 a \u2264 c \u2022 b \u2194 b \u2264 a\n[PROOFSTEP]\nrw [\u2190 neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nhc : c < 0\n\u22a2 -c \u2022 b \u2264 -c \u2022 a \u2194 b \u2264 a\n[PROOFSTEP]\nexact smul_le_smul_iff_of_pos (neg_pos_of_neg hc)\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : c < 0\n\u22a2 c\u207b\u00b9 \u2022 a \u2264 b \u2194 c \u2022 b \u2264 a\n[PROOFSTEP]\nrw [\u2190 smul_le_smul_iff_of_neg h, smul_inv_smul\u2080 h.ne]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : c < 0\n\u22a2 c\u207b\u00b9 \u2022 a < b \u2194 c \u2022 b < a\n[PROOFSTEP]\nrw [\u2190 smul_lt_smul_iff_of_neg h, smul_inv_smul\u2080 h.ne]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : c < 0\n\u22a2 a \u2264 c\u207b\u00b9 \u2022 b \u2194 b \u2264 c \u2022 a\n[PROOFSTEP]\nrw [\u2190 smul_le_smul_iff_of_neg h, smul_inv_smul\u2080 h.ne]\n[GOAL]\nk : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u00b3 : LinearOrderedField k\ninst\u271d\u00b2 : OrderedAddCommGroup M\ninst\u271d\u00b9 : Module k M\ninst\u271d : OrderedSMul k M\na b : M\nc : k\nh : c < 0\n\u22a2 a < c\u207b\u00b9 \u2022 b \u2194 b < c \u2022 a\n[PROOFSTEP]\nrw [\u2190 smul_lt_smul_iff_of_neg h, smul_inv_smul\u2080 h.ne]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Module", "llama_tokens": 6448, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744761936437, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.5354252095517802}}
{"text": "[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype (ConjClasses G)\ninst\u271d\u00b9 : Fintype G\ninst\u271d : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\n\u22a2 \u2211 x : ConjClasses G, Finset.card (Set.toFinset (carrier x)) = Fintype.card G\n[PROOFSTEP]\nsuffices : (\u03a3 x : ConjClasses G, x.carrier) \u2243 G\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype (ConjClasses G)\ninst\u271d\u00b9 : Fintype G\ninst\u271d : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\nthis : (x : ConjClasses G) \u00d7 \u2191(carrier x) \u2243 G\n\u22a2 \u2211 x : ConjClasses G, Finset.card (Set.toFinset (carrier x)) = Fintype.card G\n[PROOFSTEP]\nsimpa using (Fintype.card_congr this)\n[GOAL]\ncase this\nG : Type u\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Fintype (ConjClasses G)\ninst\u271d\u00b9 : Fintype G\ninst\u271d : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\n\u22a2 (x : ConjClasses G) \u00d7 \u2191(carrier x) \u2243 G\n[PROOFSTEP]\nsimpa [carrier_eq_preimage_mk] using Equiv.sigmaFiberEquiv ConjClasses.mk\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\n\u22a2 \u2211\u1da0 (x : ConjClasses G), Set.ncard (carrier x) = Nat.card G\n[PROOFSTEP]\nclassical\ncases nonempty_fintype G\nrw [Nat.card_eq_fintype_card, \u2190 sum_conjClasses_card_eq_card, finsum_eq_sum_of_fintype]\nsimp [Set.ncard_eq_toFinset_card']\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\n\u22a2 \u2211\u1da0 (x : ConjClasses G), Set.ncard (carrier x) = Nat.card G\n[PROOFSTEP]\ncases nonempty_fintype G\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2211\u1da0 (x : ConjClasses G), Set.ncard (carrier x) = Nat.card G\n[PROOFSTEP]\nrw [Nat.card_eq_fintype_card, \u2190 sum_conjClasses_card_eq_card, finsum_eq_sum_of_fintype]\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2211 i : ConjClasses G, Set.ncard (carrier i) = \u2211 x : ConjClasses G, Finset.card (Set.toFinset (carrier x))\n[PROOFSTEP]\nsimp [Set.ncard_eq_toFinset_card']\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\n\u22a2 Nat.card { x // x \u2208 Subgroup.center G } + \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Nat.card \u2191(carrier x) =\n    Nat.card G\n[PROOFSTEP]\nclassical\ncases nonempty_fintype G\nrw [@Nat.card_eq_fintype_card G, \u2190 sum_conjClasses_card_eq_card, \u2190\n  Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]\nsimp only [Nat.card_eq_fintype_card, Set.toFinset_card]\ncongr 1\nswap\n\u00b7 convert finsum_cond_eq_sum_of_cond_iff _ _\n  simp [Set.mem_toFinset]\ncalc\n  Fintype.card (Subgroup.center G) = Fintype.card ((noncenter G)\u1d9c : Set _) := Fintype.card_congr ((mk_bijOn G).equiv _)\n  _ = Finset.card (Finset.univ \\ (noncenter G).toFinset) := by\n    rw [\u2190 Set.toFinset_card, Set.toFinset_compl, Finset.compl_eq_univ_sdiff]\n  _ = _ := ?_\nrw [Finset.card_eq_sum_ones]\nrefine Finset.sum_congr rfl ?_\nrintro \u27e8g\u27e9 hg\nsimp only [noncenter, Set.not_subsingleton_iff, Set.toFinset_setOf, Finset.mem_univ, true_and, forall_true_left,\n  Finset.mem_sdiff, Finset.mem_filter, Set.not_nontrivial_iff] at hg \nrw [eq_comm, \u2190 Set.toFinset_card, Finset.card_eq_one]\nexact \u27e8g, Finset.coe_injective <| by simpa using hg.eq_singleton_of_mem mem_carrier_mk\u27e9\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\n\u22a2 Nat.card { x // x \u2208 Subgroup.center G } + \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Nat.card \u2191(carrier x) =\n    Nat.card G\n[PROOFSTEP]\ncases nonempty_fintype G\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Nat.card { x // x \u2208 Subgroup.center G } + \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Nat.card \u2191(carrier x) =\n    Nat.card G\n[PROOFSTEP]\nrw [@Nat.card_eq_fintype_card G, \u2190 sum_conjClasses_card_eq_card, \u2190\n  Finset.sum_sdiff (ConjClasses.noncenter G).toFinset.subset_univ]\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Nat.card { x // x \u2208 Subgroup.center G } + \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Nat.card \u2191(carrier x) =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Finset.card (Set.toFinset (carrier x)) +\n      \u2211 x in Set.toFinset (noncenter G), Finset.card (Set.toFinset (carrier x))\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, Set.toFinset_card]\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Fintype.card { x // x \u2208 Subgroup.center G } +\n      \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Fintype.card \u2191(carrier x) =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Fintype.card \u2191(carrier x) +\n      \u2211 x in Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Fintype.card { x // x \u2208 Subgroup.center G } =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Fintype.card \u2191(carrier x) =\n    \u2211 x in Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Fintype.card \u2191(carrier x) =\n    \u2211 x in Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\nconvert finsum_cond_eq_sum_of_cond_iff _ _\n[GOAL]\ncase intro.e_a.convert_7\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2200 {x : ConjClasses G}, Fintype.card \u2191(carrier x) \u2260 0 \u2192 (x \u2208 noncenter G \u2194 x \u2208 Set.toFinset (noncenter G))\n[PROOFSTEP]\nsimp [Set.mem_toFinset]\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Fintype.card { x // x \u2208 Subgroup.center G } =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\ncalc\n  Fintype.card (Subgroup.center G) = Fintype.card ((noncenter G)\u1d9c : Set _) := Fintype.card_congr ((mk_bijOn G).equiv _)\n  _ = Finset.card (Finset.univ \\ (noncenter G).toFinset) := by\n    rw [\u2190 Set.toFinset_card, Set.toFinset_compl, Finset.compl_eq_univ_sdiff]\n  _ = _ := ?_\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Fintype.card \u2191(noncenter G)\u1d9c = Finset.card (Finset.univ \\ Set.toFinset (noncenter G))\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_card, Set.toFinset_compl, Finset.compl_eq_univ_sdiff]\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 Finset.card (Finset.univ \\ Set.toFinset (noncenter G)) =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\nrw [Finset.card_eq_sum_ones]\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), 1 =\n    \u2211 x in Finset.univ \\ Set.toFinset (noncenter G), Fintype.card \u2191(carrier x)\n[PROOFSTEP]\nrefine Finset.sum_congr rfl ?_\n[GOAL]\ncase intro.e_a\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\n\u22a2 \u2200 (x : ConjClasses G), x \u2208 Finset.univ \\ Set.toFinset (noncenter G) \u2192 1 = Fintype.card \u2191(carrier x)\n[PROOFSTEP]\nrintro \u27e8g\u27e9 hg\n[GOAL]\ncase intro.e_a.mk\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\nx\u271d : ConjClasses G\ng : G\nhg : Quot.mk Setoid.r g \u2208 Finset.univ \\ Set.toFinset (noncenter G)\n\u22a2 1 = Fintype.card \u2191(carrier (Quot.mk Setoid.r g))\n[PROOFSTEP]\nsimp only [noncenter, Set.not_subsingleton_iff, Set.toFinset_setOf, Finset.mem_univ, true_and, forall_true_left,\n  Finset.mem_sdiff, Finset.mem_filter, Set.not_nontrivial_iff] at hg \n[GOAL]\ncase intro.e_a.mk\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\nx\u271d : ConjClasses G\ng : G\nhg : Set.Subsingleton (carrier (Quot.mk Setoid.r g))\n\u22a2 1 = Fintype.card \u2191(carrier (Quot.mk Setoid.r g))\n[PROOFSTEP]\nrw [eq_comm, \u2190 Set.toFinset_card, Finset.card_eq_one]\n[GOAL]\ncase intro.e_a.mk\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\nx\u271d : ConjClasses G\ng : G\nhg : Set.Subsingleton (carrier (Quot.mk Setoid.r g))\n\u22a2 \u2203 a, Set.toFinset (carrier (Quot.mk Setoid.r g)) = {a}\n[PROOFSTEP]\nexact \u27e8g, Finset.coe_injective <| by simpa using hg.eq_singleton_of_mem mem_carrier_mk\u27e9\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nval\u271d : Fintype G\nx\u271d : ConjClasses G\ng : G\nhg : Set.Subsingleton (carrier (Quot.mk Setoid.r g))\n\u22a2 \u2191(Set.toFinset (carrier (Quot.mk Setoid.r g))) = \u2191{g}\n[PROOFSTEP]\nsimpa using hg.eq_singleton_of_mem mem_carrier_mk\n[GOAL]\nG\u271d : Type u\ninst\u271d\u2075 : Group G\u271d\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\ninst\u271d\u00b2 : Fintype G\ninst\u271d\u00b9 : Fintype { x // x \u2208 Subgroup.center G }\ninst\u271d : Fintype \u2191(noncenter G)\n\u22a2 Fintype.card { x // x \u2208 Subgroup.center G } +\n      \u2211 x in Set.toFinset (noncenter G), Finset.card (Set.toFinset (carrier x)) =\n    Fintype.card G\n[PROOFSTEP]\nconvert Group.nat_card_center_add_sum_card_noncenter_eq_card G using 2\n[GOAL]\ncase h.e'_2.h.e'_5\nG\u271d : Type u\ninst\u271d\u2075 : Group G\u271d\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\ninst\u271d\u00b2 : Fintype G\ninst\u271d\u00b9 : Fintype { x // x \u2208 Subgroup.center G }\ninst\u271d : Fintype \u2191(noncenter G)\n\u22a2 Fintype.card { x // x \u2208 Subgroup.center G } = Nat.card { x // x \u2208 Subgroup.center G }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_6\nG\u271d : Type u\ninst\u271d\u2075 : Group G\u271d\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\ninst\u271d\u00b2 : Fintype G\ninst\u271d\u00b9 : Fintype { x // x \u2208 Subgroup.center G }\ninst\u271d : Fintype \u2191(noncenter G)\n\u22a2 \u2211 x in Set.toFinset (noncenter G), Finset.card (Set.toFinset (carrier x)) =\n    \u2211\u1da0 (x : ConjClasses G) (_ : x \u2208 noncenter G), Nat.card \u2191(carrier x)\n[PROOFSTEP]\nrw [\u2190 finsum_set_coe_eq_finsum_mem (noncenter G), finsum_eq_sum_of_fintype, \u2190 Finset.sum_set_coe]\n[GOAL]\ncase h.e'_2.h.e'_6\nG\u271d : Type u\ninst\u271d\u2075 : Group G\u271d\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\ninst\u271d\u00b2 : Fintype G\ninst\u271d\u00b9 : Fintype { x // x \u2208 Subgroup.center G }\ninst\u271d : Fintype \u2191(noncenter G)\n\u22a2 \u2211 i : \u2191(noncenter G), Finset.card (Set.toFinset (carrier \u2191i)) = \u2211 i : \u2191(noncenter G), Nat.card \u2191(carrier \u2191i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nG\u271d : Type u\ninst\u271d\u2075 : Group G\u271d\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : (x : ConjClasses G) \u2192 Fintype \u2191(carrier x)\ninst\u271d\u00b2 : Fintype G\ninst\u271d\u00b9 : Fintype { x // x \u2208 Subgroup.center G }\ninst\u271d : Fintype \u2191(noncenter G)\n\u22a2 Fintype.card G = Nat.card G\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.ClassEquation", "llama_tokens": 4770, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.5351460742631544}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\n\u22a2 IsGenericPoint x S \u2194 \u2200 (y : \u03b1), x \u2933 y \u2194 y \u2208 S\n[PROOFSTEP]\nsimp only [specializes_iff_mem_closure, IsGenericPoint, Set.ext_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nh : IsGenericPoint x S\nhU : IsOpen U\n\u22a2 Disjoint S U \u2194 \u00acx \u2208 U\n[PROOFSTEP]\nrw [h.mem_open_set_iff hU, \u2190 not_disjoint_iff_nonempty_inter, Classical.not_not]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nh : IsGenericPoint x S\nhZ : IsClosed Z\n\u22a2 x \u2208 Z \u2194 S \u2286 Z\n[PROOFSTEP]\nrw [\u2190 h.def, hZ.closure_subset_iff, singleton_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nh : IsGenericPoint x S\nf : \u03b1 \u2192 \u03b2\nhf : Continuous f\n\u22a2 IsGenericPoint (f x) (closure (f '' S))\n[PROOFSTEP]\nrw [isGenericPoint_def, \u2190 h.def, \u2190 image_singleton, closure_image_closure hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nhS : IsClosed S\nhxS : x \u2208 S\n\u22a2 IsGenericPoint x S \u2194 \u2200 (Z : Set \u03b1), IsClosed Z \u2192 x \u2208 Z \u2192 S \u2286 Z\n[PROOFSTEP]\nhave : closure { x } \u2286 S := closure_minimal (singleton_subset_iff.2 hxS) hS\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nx y : \u03b1\nS U Z : Set \u03b1\nhS : IsClosed S\nhxS : x \u2208 S\nthis : closure {x} \u2286 S\n\u22a2 IsGenericPoint x S \u2194 \u2200 (Z : Set \u03b1), IsClosed Z \u2192 x \u2208 Z \u2192 S \u2286 Z\n[PROOFSTEP]\nsimp_rw [IsGenericPoint, subset_antisymm_iff, this, true_and_iff, closure, subset_sInter_iff, mem_setOf_eq, and_imp,\n  singleton_subset_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : IrreducibleSpace \u03b1\n\u22a2 IsGenericPoint (genericPoint \u03b1) \u22a4\n[PROOFSTEP]\nsimpa using (IrreducibleSpace.isIrreducible_univ \u03b1).genericPoint_spec\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : IrreducibleSpace \u03b1\nx : \u03b1\n\u22a2 x \u2208 closure univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : T0Space \u03b1\nx : \u03b1\n\u22a2 IsGenericPoint x (closure (closure {x}))\n[PROOFSTEP]\nrw [closure_closure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : T0Space \u03b1\nx : \u03b1\n\u22a2 IsGenericPoint x (closure {x})\n[PROOFSTEP]\nexact isGenericPoint_closure\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : T0Space \u03b1\n\u22a2 \u2200 {a b : \u2191{s | IsIrreducible s \u2227 IsClosed s}},\n    \u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n              invFun := fun x =>\n                { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n              left_inv :=\n                (_ :\n                  \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                    (fun x =>\n                          { val := closure {x},\n                            property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                        ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                      s),\n              right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n          a \u2264\n        \u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n              invFun := fun x =>\n                { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n              left_inv :=\n                (_ :\n                  \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                    (fun x =>\n                          { val := closure {x},\n                            property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                        ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                      s),\n              right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n          b \u2194\n      a \u2264 b\n[PROOFSTEP]\nrintro \u27e8s, hs\u27e9 \u27e8t, ht\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : T0Space \u03b1\ns : Set \u03b1\nhs : s \u2208 {s | IsIrreducible s \u2227 IsClosed s}\nt : Set \u03b1\nht : t \u2208 {s | IsIrreducible s \u2227 IsClosed s}\n\u22a2 \u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n            invFun := fun x =>\n              { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n            left_inv :=\n              (_ :\n                \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                  (fun x =>\n                        { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                      ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                    s),\n            right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n        { val := s, property := hs } \u2264\n      \u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n            invFun := fun x =>\n              { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n            left_inv :=\n              (_ :\n                \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                  (fun x =>\n                        { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                      ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                    s),\n            right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n        { val := t, property := ht } \u2194\n    { val := s, property := hs } \u2264 { val := t, property := ht }\n[PROOFSTEP]\nrefine specializes_iff_closure_subset.trans ?_\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : QuasiSober \u03b1\ninst\u271d : T0Space \u03b1\ns : Set \u03b1\nhs : s \u2208 {s | IsIrreducible s \u2227 IsClosed s}\nt : Set \u03b1\nht : t \u2208 {s | IsIrreducible s \u2227 IsClosed s}\n\u22a2 closure\n        {\u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n                invFun := fun x =>\n                  { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n                left_inv :=\n                  (_ :\n                    \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                      (fun x =>\n                            { val := closure {x},\n                              property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                          ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                        s),\n                right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n            { val := s, property := hs }} \u2286\n      closure\n        {\u2191{ toFun := fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s),\n                invFun := fun x =>\n                  { val := closure {x}, property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) },\n                left_inv :=\n                  (_ :\n                    \u2200 (s : \u2191{s | IsIrreducible s \u2227 IsClosed s}),\n                      (fun x =>\n                            { val := closure {x},\n                              property := (_ : IsIrreducible (closure {x}) \u2227 IsClosed (closure {x})) })\n                          ((fun s => IsIrreducible.genericPoint (_ : IsIrreducible \u2191s)) s) =\n                        s),\n                right_inv := (_ : \u2200 (x : \u03b1), IsIrreducible.genericPoint (_ : IsIrreducible (closure {x})) = x) }\n            { val := t, property := ht }} \u2194\n    { val := s, property := hs } \u2264 { val := t, property := ht }\n[PROOFSTEP]\nsimp [hs.2.closure_eq, ht.2.closure_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nhave hS'' := hS.image f hf.continuous.continuousOn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := QuasiSober.sober hS'' (hf.isClosedMap _ hS')\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\nx : \u03b2\nhx : IsGenericPoint x (f '' S\u271d)\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nobtain \u27e8y, -, rfl\u27e9 := hx.mem\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\ny : \u03b1\nhx : IsGenericPoint (f y) (f '' S\u271d)\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\ny : \u03b1\nhx : IsGenericPoint (f y) (f '' S\u271d)\n\u22a2 IsGenericPoint y S\u271d\n[PROOFSTEP]\napply image_injective.mpr hf.inj\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : ClosedEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\ny : \u03b1\nhx : IsGenericPoint (f y) (f '' S\u271d)\n\u22a2 f '' closure {y} = f '' S\u271d\n[PROOFSTEP]\nrw [\u2190 hx.def, \u2190 hf.closure_image_eq, image_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nhave hS'' := hS.image f hf.continuous.continuousOn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := QuasiSober.sober hS''.closure isClosed_closure\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nS\u271d : Set \u03b1\nhS : IsIrreducible S\u271d\nhS' : IsClosed S\u271d\nhS'' : IsIrreducible (f '' S\u271d)\nx : \u03b2\nhx : IsGenericPoint x (closure (f '' S\u271d))\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nobtain \u27e8T, hT, rfl\u27e9 := hf.toInducing.isClosed_iff.mp hS'\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (f '' (f \u207b\u00b9' T))\nhx : IsGenericPoint x (closure (f '' (f \u207b\u00b9' T)))\n\u22a2 \u2203 x, IsGenericPoint x (f \u207b\u00b9' T)\n[PROOFSTEP]\nrw [image_preimage_eq_inter_range] at hx hS'' \n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\n\u22a2 \u2203 x, IsGenericPoint x (f \u207b\u00b9' T)\n[PROOFSTEP]\nhave hxT : x \u2208 T := by\n  rw [\u2190 hT.closure_eq]\n  exact closure_mono (inter_subset_left _ _) hx.mem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\n\u22a2 x \u2208 T\n[PROOFSTEP]\nrw [\u2190 hT.closure_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\n\u22a2 x \u2208 closure T\n[PROOFSTEP]\nexact closure_mono (inter_subset_left _ _) hx.mem\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\nhxT : x \u2208 T\n\u22a2 \u2203 x, IsGenericPoint x (f \u207b\u00b9' T)\n[PROOFSTEP]\nobtain \u27e8y, rfl\u27e9 : x \u2208 range f := by\n  rw [hx.mem_open_set_iff hf.open_range]\n  refine' Nonempty.mono _ hS''.1\n  simpa using subset_closure\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\nhxT : x \u2208 T\n\u22a2 x \u2208 range f\n[PROOFSTEP]\nrw [hx.mem_open_set_iff hf.open_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\nhxT : x \u2208 T\n\u22a2 Set.Nonempty (closure (T \u2229 range f) \u2229 range f)\n[PROOFSTEP]\nrefine' Nonempty.mono _ hS''.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nx : \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\nhx : IsGenericPoint x (closure (T \u2229 range f))\nhxT : x \u2208 T\n\u22a2 T \u2229 range f \u2286 closure (T \u2229 range f) \u2229 range f\n[PROOFSTEP]\nsimpa using subset_closure\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\n\u22a2 \u2203 x, IsGenericPoint x (f \u207b\u00b9' T)\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\n\u22a2 IsGenericPoint y (f \u207b\u00b9' T)\n[PROOFSTEP]\nchange _ = _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\n\u22a2 closure {y} = f \u207b\u00b9' T\n[PROOFSTEP]\nrw [hf.toEmbedding.closure_eq_preimage_closure_image, image_singleton, show _ = _ from hx]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\n\u22a2 f \u207b\u00b9' closure (T \u2229 range f) = f \u207b\u00b9' T\n[PROOFSTEP]\napply image_injective.mpr hf.inj\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\n\u22a2 f '' (f \u207b\u00b9' closure (T \u2229 range f)) = f '' (f \u207b\u00b9' T)\n[PROOFSTEP]\next z\n[GOAL]\ncase h.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\nz : \u03b2\n\u22a2 z \u2208 f '' (f \u207b\u00b9' closure (T \u2229 range f)) \u2194 z \u2208 f '' (f \u207b\u00b9' T)\n[PROOFSTEP]\nsimp only [image_preimage_eq_inter_range, mem_inter_iff, and_congr_left_iff]\n[GOAL]\ncase h.a.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : OpenEmbedding f\ninst\u271d : QuasiSober \u03b2\nT : Set \u03b2\nhT : IsClosed T\nhS : IsIrreducible (f \u207b\u00b9' T)\nhS' : IsClosed (f \u207b\u00b9' T)\nhS'' : IsIrreducible (T \u2229 range f)\ny : \u03b1\nhx : IsGenericPoint (f y) (closure (T \u2229 range f))\nhxT : f y \u2208 T\nz : \u03b2\n\u22a2 z \u2208 range f \u2192 (z \u2208 closure (T \u2229 range f) \u2194 z \u2208 T)\n[PROOFSTEP]\nexact fun hy => \u27e8fun h => hT.closure_eq \u25b8 closure_mono (inter_subset_left _ _) h, fun h => subset_closure \u27e8h, hy\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\n\u22a2 QuasiSober \u03b1\n[PROOFSTEP]\nrw [quasiSober_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\n\u22a2 \u2200 {S : Set \u03b1}, IsIrreducible S \u2192 IsClosed S \u2192 \u2203 x, IsGenericPoint x S\n[PROOFSTEP]\nintro t h h'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := h.1\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nobtain \u27e8U, hU, hU'\u27e9 : x \u2208 \u22c3\u2080 S := by\n  rw [hS'']\n  trivial\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22c3\u2080 S\n[PROOFSTEP]\nrw [hS'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\n\u22a2 x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nhaveI : QuasiSober U := hS' \u27e8U, hU\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis : QuasiSober \u2191U\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nhave H : IsPreirreducible ((\u2191) \u207b\u00b9' t : Set U) := h.2.preimage (hS \u27e8U, hU\u27e9).openEmbedding_subtype_val\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis : QuasiSober \u2191U\nH : IsPreirreducible (Subtype.val \u207b\u00b9' t)\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nreplace H : IsIrreducible ((\u2191) \u207b\u00b9' t : Set U) := \u27e8\u27e8\u27e8x, hU'\u27e9, by simpa using hx\u27e9, H\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis : QuasiSober \u2191U\nH : IsPreirreducible (Subtype.val \u207b\u00b9' t)\n\u22a2 { val := x, property := hU' } \u2208 Subtype.val \u207b\u00b9' t\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\n\u22a2 \u2203 x, IsGenericPoint x t\n[PROOFSTEP]\nuse H.genericPoint\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\n\u22a2 IsGenericPoint (\u2191(IsIrreducible.genericPoint H)) t\n[PROOFSTEP]\nhave := continuous_subtype_val.closure_preimage_subset _ H.genericPoint_spec.mem\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' closure t\n\u22a2 IsGenericPoint (\u2191(IsIrreducible.genericPoint H)) t\n[PROOFSTEP]\nrw [h'.closure_eq] at this \n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 IsGenericPoint (\u2191(IsIrreducible.genericPoint H)) t\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 closure {\u2191(IsIrreducible.genericPoint H)} \u2264 t\n[PROOFSTEP]\napply h'.closure_subset_iff.mpr\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 {\u2191(IsIrreducible.genericPoint H)} \u2286 t\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 t \u2264 closure {\u2191(IsIrreducible.genericPoint H)}\n[PROOFSTEP]\nrw [\u2190 image_singleton, \u2190 closure_image_closure continuous_subtype_val, H.genericPoint_spec.def]\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 t \u2264 closure (Subtype.val '' closure (Subtype.val \u207b\u00b9' t))\n[PROOFSTEP]\nrefine' (subset_closure_inter_of_isPreirreducible_of_isOpen h.2 (hS \u27e8U, hU\u27e9) \u27e8x, hx, hU'\u27e9).trans (closure_mono _)\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 t \u2229 \u2191{ val := U, property := hU } \u2286 Subtype.val '' closure (Subtype.val \u207b\u00b9' t)\n[PROOFSTEP]\nrw [\u2190 Subtype.image_preimage_coe]\n[GOAL]\ncase h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nS : Set (Set \u03b1)\nhS : \u2200 (s : \u2191S), IsOpen \u2191s\nhS' : \u2200 (s : \u2191S), QuasiSober \u2191\u2191s\nhS'' : \u22c3\u2080 S = \u22a4\nt : Set \u03b1\nh : IsIrreducible t\nh' : IsClosed t\nx : \u03b1\nhx : x \u2208 t\nU : Set \u03b1\nhU : U \u2208 S\nhU' : x \u2208 U\nthis\u271d : QuasiSober \u2191U\nH : IsIrreducible (Subtype.val \u207b\u00b9' t)\nthis : IsIrreducible.genericPoint H \u2208 Subtype.val \u207b\u00b9' t\n\u22a2 Subtype.val '' (Subtype.val \u207b\u00b9' t) \u2286 Subtype.val '' closure (Subtype.val \u207b\u00b9' t)\n[PROOFSTEP]\nexact Set.image_subset _ subset_closure\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b1\nS\u271d : Set \u03b1\nh : IsIrreducible S\u271d\nx\u271d : IsClosed S\u271d\n\u22a2 \u2203 x, IsGenericPoint x S\u271d\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := isIrreducible_iff_singleton.mp h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T2Space \u03b1\nx : \u03b1\nh : IsIrreducible {x}\nx\u271d : IsClosed {x}\n\u22a2 \u2203 x_1, IsGenericPoint x_1 {x}\n[PROOFSTEP]\nexact \u27e8x, closure_singleton\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sober", "llama_tokens": 12438, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.672331705744791, "lm_q1q2_score": 0.5349461676051199}}
{"text": "[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d : OpenNhds x\n\u22a2 x\u271d \u2264 x\u271d\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d : OpenNhds x\n\u22a2 x\u271d.obj \u2264 x\u271d.obj\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : OpenNhds x\n\u22a2 x\u271d\u00b2 \u2264 x\u271d\u00b9 \u2192 x\u271d\u00b9 \u2264 x\u271d \u2192 x\u271d\u00b2 \u2264 x\u271d\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d\u00b2 x\u271d\u00b9 x\u271d : OpenNhds x\n\u22a2 x\u271d\u00b2.obj \u2264 x\u271d\u00b9.obj \u2192 x\u271d\u00b9.obj \u2264 x\u271d.obj \u2192 x\u271d\u00b2.obj \u2264 x\u271d.obj\n[PROOFSTEP]\nexact le_trans\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d : OpenNhds x\n\u22a2 x\u271d \u2264 \u22a4\n[PROOFSTEP]\ndsimp [LE.le]\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nx\u271d : OpenNhds x\n\u22a2 x\u271d.obj \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 (map (\ud835\udfd9 X) x).obj U.unop = U.unop\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : (OpenNhds x)\u1d52\u1d56\n\u22a2 (map (\ud835\udfd9 X) x).op.obj U = U\n[PROOFSTEP]\nsimp\n[GOAL]\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U \u2245 (map f x \u22d9 inclusion x).obj U\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase hom_inv_id\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 autoParam (?hom \u226b ?inv = \ud835\udfd9 ((inclusion (\u2191f x) \u22d9 Opens.map f).obj U)) _auto\u271d\ncase inv_hom_id\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 autoParam (?inv \u226b ?hom = \ud835\udfd9 ((map f x \u22d9 inclusion x).obj U)) _auto\u271d\ncase hom\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U \u27f6 (map f x \u22d9 inclusion x).obj U\ncase inv\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (map f x \u22d9 inclusion x).obj U \u27f6 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inv_hom_id\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 autoParam (?inv \u226b ?hom = \ud835\udfd9 ((map f x \u22d9 inclusion x).obj U)) _auto\u271d\ncase hom\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U \u27f6 (map f x \u22d9 inclusion x).obj U\ncase inv\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (map f x \u22d9 inclusion x).obj U \u27f6 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U\n[PROOFSTEP]\nrfl\n[GOAL]\ncase hom\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U \u27f6 (map f x \u22d9 inclusion x).obj U\ncase inv\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (map f x \u22d9 inclusion x).obj U \u27f6 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U\n[PROOFSTEP]\nexact \ud835\udfd9 _\n[GOAL]\ncase inv\nX Y : TopCat\nf : X \u27f6 Y\nx : \u2191X\nU : OpenNhds (\u2191f x)\n\u22a2 (map f x \u22d9 inclusion x).obj U \u27f6 (inclusion (\u2191f x) \u22d9 Opens.map f).obj U\n[PROOFSTEP]\nexact \ud835\udfd9 _\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.TopCat.OpenNhds", "llama_tokens": 1470, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.808067204308405, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.534878162366728}}
{"text": "[GOAL]\n\u22a2 StableUnderComposition fun {R S} [CommRing R] [CommRing S] f => IsIntegral f\n[PROOFSTEP]\nintrov R hf hg\n[GOAL]\nR S T : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ng : S \u2192+* T\nhf : IsIntegral f\nhg : IsIntegral g\n\u22a2 IsIntegral (comp g f)\n[PROOFSTEP]\nexact RingHom.isIntegral_trans _ _ hf hg\n[GOAL]\n\u22a2 RespectsIso fun {R S} [CommRing R] [CommRing S] f => IsIntegral f\n[PROOFSTEP]\napply isIntegral_stableUnderComposition.respectsIso\n[GOAL]\n\u22a2 \u2200 {R S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), IsIntegral (RingEquiv.toRingHom e)\n[PROOFSTEP]\nintrov x\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\nx : S\n\u22a2 IsIntegralElem (RingEquiv.toRingHom e) x\n[PROOFSTEP]\nrw [\u2190 e.apply_symm_apply x]\n[GOAL]\nR S : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\ne : R \u2243+* S\nx : S\n\u22a2 IsIntegralElem (RingEquiv.toRingHom e) (\u2191e (\u2191(RingEquiv.symm e) x))\n[PROOFSTEP]\napply RingHom.is_integral_map\n[GOAL]\n\u22a2 StableUnderBaseChange fun {R S} [CommRing R] [CommRing S] f => IsIntegral f\n[PROOFSTEP]\nrefine' StableUnderBaseChange.mk _ isIntegral_respectsIso _\n[GOAL]\n\u22a2 \u2200 \u2983R S T : Type u_1\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], IsIntegral (algebraMap R T) \u2192 IsIntegral includeLeftRingHom\n[PROOFSTEP]\nintrov h x\n[GOAL]\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx : S \u2297[R] T\n\u22a2 IsIntegralElem includeLeftRingHom x\n[PROOFSTEP]\nrefine' TensorProduct.induction_on x _ _ _\n[GOAL]\ncase refine'_1\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx : S \u2297[R] T\n\u22a2 IsIntegralElem includeLeftRingHom 0\n[PROOFSTEP]\napply isIntegral_zero\n[GOAL]\ncase refine'_2\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx : S \u2297[R] T\n\u22a2 \u2200 (x : S) (y : T), IsIntegralElem includeLeftRingHom (x \u2297\u209c[R] y)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase refine'_2\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx\u271d : S \u2297[R] T\nx : S\ny : T\n\u22a2 IsIntegralElem includeLeftRingHom (x \u2297\u209c[R] y)\n[PROOFSTEP]\nexact IsIntegral.tmul x (h y)\n[GOAL]\ncase refine'_3\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx : S \u2297[R] T\n\u22a2 \u2200 (x y : S \u2297[R] T),\n    IsIntegralElem includeLeftRingHom x \u2192\n      IsIntegralElem includeLeftRingHom y \u2192 IsIntegralElem includeLeftRingHom (x + y)\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine'_3\nR S T : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing S\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : Algebra R S\ninst\u271d : Algebra R T\nh : IsIntegral (algebraMap R T)\nx\u271d x y : S \u2297[R] T\nhx : IsIntegralElem includeLeftRingHom x\nhy : IsIntegralElem includeLeftRingHom y\n\u22a2 IsIntegralElem includeLeftRingHom (x + y)\n[PROOFSTEP]\nexact isIntegral_add hx hy\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.RingHom.Integral", "llama_tokens": 1528, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.5346112159099375}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\n\u22a2 \ud835\udcdd\u02e2 (diagonal \u03b1) = \ud835\udce4 \u03b1\n[PROOFSTEP]\nrefine' nhdsSet_diagonal_le_uniformity.antisymm _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\n\u22a2 \ud835\udce4 \u03b1 \u2264 \ud835\udcdd\u02e2 (diagonal \u03b1)\n[PROOFSTEP]\nhave :\n  (\ud835\udce4 (\u03b1 \u00d7 \u03b1)).HasBasis (fun U => U \u2208 \ud835\udce4 \u03b1) fun U =>\n    (fun p : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 => ((p.1.1, p.2.1), p.1.2, p.2.2)) \u207b\u00b9' U \u00d7\u02e2 U :=\n  by\n  rw [uniformity_prod_eq_comap_prod]\n  exact (\ud835\udce4 \u03b1).basis_sets.prod_self.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\n\u22a2 HasBasis (\ud835\udce4 (\u03b1 \u00d7 \u03b1)) (fun U => U \u2208 \ud835\udce4 \u03b1) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) \u207b\u00b9' U \u00d7\u02e2 U\n[PROOFSTEP]\nrw [uniformity_prod_eq_comap_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\n\u22a2 HasBasis (Filter.comap (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) (\ud835\udce4 \u03b1 \u00d7\u02e2 \ud835\udce4 \u03b1)) (fun U => U \u2208 \ud835\udce4 \u03b1)\n    fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) \u207b\u00b9' U \u00d7\u02e2 U\n[PROOFSTEP]\nexact (\ud835\udce4 \u03b1).basis_sets.prod_self.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\nthis :\n  HasBasis (\ud835\udce4 (\u03b1 \u00d7 \u03b1)) (fun U => U \u2208 \ud835\udce4 \u03b1) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) \u207b\u00b9' U \u00d7\u02e2 U\n\u22a2 \ud835\udce4 \u03b1 \u2264 \ud835\udcdd\u02e2 (diagonal \u03b1)\n[PROOFSTEP]\nrefine' (isCompact_diagonal.nhdsSet_basis_uniformity this).ge_iff.2 fun U hU => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\nthis :\n  HasBasis (\ud835\udce4 (\u03b1 \u00d7 \u03b1)) (fun U => U \u2208 \ud835\udce4 \u03b1) fun U => (fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) \u207b\u00b9' U \u00d7\u02e2 U\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u22c3 (x : \u03b1 \u00d7 \u03b1) (_ : x \u2208 diagonal \u03b1), ball x ((fun p => ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd)) \u207b\u00b9' U \u00d7\u02e2 U) \u2208\n    \ud835\udce4 \u03b1\n[PROOFSTEP]\nexact mem_of_superset hU fun \u27e8x, y\u27e9 hxy => mem_iUnion\u2082.2 \u27e8(x, x), rfl, refl_mem_uniformity hU, hxy\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\n\u22a2 u = u'\n[PROOFSTEP]\nrefine uniformSpace_eq ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\n\u22a2 \ud835\udce4 \u03b3 = \ud835\udce4 \u03b3\n[PROOFSTEP]\nhave : @CompactSpace \u03b3 u.toTopologicalSpace := by rwa [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\n\u22a2 CompactSpace \u03b3\n[PROOFSTEP]\nrwa [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\nthis : CompactSpace \u03b3\n\u22a2 \ud835\udce4 \u03b3 = \ud835\udce4 \u03b3\n[PROOFSTEP]\nhave : @CompactSpace \u03b3 u'.toTopologicalSpace := by rwa [h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\nthis : CompactSpace \u03b3\n\u22a2 CompactSpace \u03b3\n[PROOFSTEP]\nrwa [h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nt : TopologicalSpace \u03b3\ninst\u271d : CompactSpace \u03b3\nu u' : UniformSpace \u03b3\nh : toTopologicalSpace = t\nh' : toTopologicalSpace = t\nthis\u271d : CompactSpace \u03b3\nthis : CompactSpace \u03b3\n\u22a2 \ud835\udce4 \u03b3 = \ud835\udce4 \u03b3\n[PROOFSTEP]\nrw [@compactSpace_uniformity _ u, compactSpace_uniformity, h, h']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\u22a2 (Filter.lift' (\ud835\udcdd\u02e2 (diagonal \u03b3)) fun s => s \u25cb s) \u2264 \ud835\udcdd\u02e2 (diagonal \u03b3)\n[PROOFSTEP]\nset \ud835\udcdd\u0394 :=\n  \ud835\udcdd\u02e2\n    (diagonal \u03b3)\n      -- The filter of neighborhoods of \u0394\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\n\u22a2 (Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s) \u2264 \ud835\udcdd\u0394\n[PROOFSTEP]\nset F := \ud835\udcdd\u0394.lift' fun s : Set (\u03b3 \u00d7 \u03b3) => s \u25cb s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\n\u22a2 F \u2264 \ud835\udcdd\u0394\n[PROOFSTEP]\nrw [le_iff_forall_inf_principal_compl]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\n\u22a2 \u2200 (V : Set (\u03b3 \u00d7 \u03b3)), V \u2208 \ud835\udcdd\u0394 \u2192 F \u2293 \ud835\udcdf V\u1d9c = \u22a5\n[PROOFSTEP]\nintro V V_in\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\n\u22a2 F \u2293 \ud835\udcdf V\u1d9c = \u22a5\n[PROOFSTEP]\nby_contra H\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\n\u22a2 False\n[PROOFSTEP]\nhaveI : NeBot (F \u2293 \ud835\udcdf V\u1d9c) :=\n  \u27e8H\u27e9\n    -- Hence compactness would give us a cluster point (x, y) for F \u2293 \ud835\udcdf V\u1d9c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8x, y\u27e9, hxy\u27e9 : \u2203 p : \u03b3 \u00d7 \u03b3, ClusterPt p (F \u2293 \ud835\udcdf V\u1d9c) :=\n  cluster_point_of_compact\n    _\n      -- In particular (x, y) is a cluster point of \ud835\udcdf V\u1d9c, hence is not in the interior of V,\n          -- and a fortiori not in \u0394, so x \u2260 y\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nhave clV : ClusterPt (x, y) (\ud835\udcdf <| V\u1d9c) := hxy.of_inf_right\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\n\u22a2 False\n[PROOFSTEP]\nhave : (x, y) \u2209 interior V :=\n  by\n  have : (x, y) \u2208 closure V\u1d9c := by rwa [mem_closure_iff_clusterPt]\n  rwa [closure_compl] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\n\u22a2 \u00ac(x, y) \u2208 interior V\n[PROOFSTEP]\nhave : (x, y) \u2208 closure V\u1d9c := by rwa [mem_closure_iff_clusterPt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\n\u22a2 (x, y) \u2208 closure V\u1d9c\n[PROOFSTEP]\nrwa [mem_closure_iff_clusterPt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis : (x, y) \u2208 closure V\u1d9c\n\u22a2 \u00ac(x, y) \u2208 interior V\n[PROOFSTEP]\nrwa [closure_compl] at this \n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis : \u00ac(x, y) \u2208 interior V\n\u22a2 False\n[PROOFSTEP]\nhave diag_subset : diagonal \u03b3 \u2286 interior V := subset_interior_iff_mem_nhdsSet.2 V_in\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\n\u22a2 False\n[PROOFSTEP]\nhave x_ne_y : x \u2260 y := mt (@diag_subset (x, y)) this\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\n\u22a2 False\n[PROOFSTEP]\nhaveI : NormalSpace \u03b3 := normalOfCompactT2\n[GOAL]\ncase intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8U\u2081, _, V\u2081, V\u2081_in, U\u2082, _, V\u2082, V\u2082_in, V\u2081_cl, V\u2082_cl, U\u2081_op, U\u2082_op, VU\u2081, VU\u2082, hU\u2081\u2082\u27e9 := disjoint_nested_nhds x_ne_y\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\n\u22a2 False\n[PROOFSTEP]\nlet U\u2083 := (V\u2081 \u222a V\u2082)\u1d9c\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\n\u22a2 False\n[PROOFSTEP]\nhave U\u2083_op : IsOpen U\u2083 := (V\u2081_cl.union V\u2082_cl).isOpen_compl\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\n\u22a2 False\n[PROOFSTEP]\nlet W := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\n\u22a2 False\n[PROOFSTEP]\nhave W_in : W \u2208 \ud835\udcdd\u0394 := by\n  rw [mem_nhdsSet_iff_forall]\n  rintro \u27e8z, z'\u27e9 (rfl : z = z')\n  refine' IsOpen.mem_nhds _ _\n  \u00b7 apply_rules [IsOpen.union, IsOpen.prod]\n  \u00b7 simp only [mem_union, mem_prod, and_self_iff]\n    exact (_root_.em _).imp_left fun h => union_subset_union VU\u2081 VU\u2082 h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\n\u22a2 W \u2208 \ud835\udcdd\u0394\n[PROOFSTEP]\nrw [mem_nhdsSet_iff_forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\n\u22a2 \u2200 (x : \u03b3 \u00d7 \u03b3), x \u2208 diagonal \u03b3 \u2192 W \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrintro \u27e8z, z'\u27e9 (rfl : z = z')\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nz : \u03b3\n\u22a2 W \u2208 \ud835\udcdd (z, z)\n[PROOFSTEP]\nrefine' IsOpen.mem_nhds _ _\n[GOAL]\ncase mk.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nz : \u03b3\n\u22a2 IsOpen W\n[PROOFSTEP]\napply_rules [IsOpen.union, IsOpen.prod]\n[GOAL]\ncase mk.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nz : \u03b3\n\u22a2 (z, z) \u2208 W\n[PROOFSTEP]\nsimp only [mem_union, mem_prod, and_self_iff]\n[GOAL]\ncase mk.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nz : \u03b3\n\u22a2 (z \u2208 U\u2081 \u2228 z \u2208 U\u2082) \u2228 z \u2208 (V\u2081 \u222a V\u2082)\u1d9c\n[PROOFSTEP]\nexact (_root_.em _).imp_left fun h => union_subset_union VU\u2081 VU\u2082 h\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b9 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\n\u22a2 False\n[PROOFSTEP]\nhave : W \u25cb W \u2208 F := @mem_lift' _ _ _ (fun s => s \u25cb s) _ W_in\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b2 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d\u00b9 : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis\u271d : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\nthis : W \u25cb W \u2208 F\n\u22a2 False\n[PROOFSTEP]\nhave hV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y) := prod_mem_nhds V\u2081_in V\u2082_in\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b2 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d\u00b9 : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis\u271d : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\nthis : W \u25cb W \u2208 F\nhV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8\u27e8u, v\u27e9, \u27e8u_in, v_in\u27e9, w, huw, hwv\u27e9 := clusterPt_iff.mp hxy.of_inf_left hV\u2081\u2082 this\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b2 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d\u00b9 : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis\u271d : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\nthis : W \u25cb W \u2208 F\nhV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y)\nu v : \u03b3\nu_in : (u, v).fst \u2208 V\u2081\nv_in : (u, v).snd \u2208 V\u2082\nw : \u03b3\nhuw : ((u, v).fst, w) \u2208 W\nhwv : (w, (u, v).snd) \u2208 W\n\u22a2 False\n[PROOFSTEP]\nhave uw_in : (u, w) \u2208 U\u2081 \u00d7\u02e2 U\u2081 :=\n  (huw.resolve_right fun h => h.1 <| Or.inl u_in).resolve_right fun h =>\n    hU\u2081\u2082.le_bot\n      \u27e8VU\u2081 u_in, h.1\u27e9\n        -- Similarly, because v \u2208 V\u2082, (w ,v) \u2208 W forces (w, v) \u2208 U\u2082 \u00d7\u02e2 U\u2082.\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b2 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d\u00b9 : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis\u271d : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\nthis : W \u25cb W \u2208 F\nhV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y)\nu v : \u03b3\nu_in : (u, v).fst \u2208 V\u2081\nv_in : (u, v).snd \u2208 V\u2082\nw : \u03b3\nhuw : ((u, v).fst, w) \u2208 W\nhwv : (w, (u, v).snd) \u2208 W\nuw_in : (u, w) \u2208 U\u2081 \u00d7\u02e2 U\u2081\n\u22a2 False\n[PROOFSTEP]\nhave wv_in : (w, v) \u2208 U\u2082 \u00d7\u02e2 U\u2082 :=\n  (hwv.resolve_right fun h => h.2 <| Or.inr v_in).resolve_left fun h =>\n    hU\u2081\u2082.le_bot\n      \u27e8h.2, VU\u2082 v_in\u27e9\n        -- Hence w \u2208 U\u2081 \u2229 U\u2082 which is empty.\n            -- So we have a contradiction\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.mk.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\ud835\udcdd\u0394 : Filter (\u03b3 \u00d7 \u03b3) := \ud835\udcdd\u02e2 (diagonal \u03b3)\nF : Filter (\u03b3 \u00d7 \u03b3) := Filter.lift' \ud835\udcdd\u0394 fun s => s \u25cb s\nV : Set (\u03b3 \u00d7 \u03b3)\nV_in : V \u2208 \ud835\udcdd\u0394\nH : \u00acF \u2293 \ud835\udcdf V\u1d9c = \u22a5\nthis\u271d\u00b2 : NeBot (F \u2293 \ud835\udcdf V\u1d9c)\nx y : \u03b3\nhxy : ClusterPt (x, y) (F \u2293 \ud835\udcdf V\u1d9c)\nclV : ClusterPt (x, y) (\ud835\udcdf V\u1d9c)\nthis\u271d\u00b9 : \u00ac(x, y) \u2208 interior V\ndiag_subset : diagonal \u03b3 \u2286 interior V\nx_ne_y : x \u2260 y\nthis\u271d : NormalSpace \u03b3\nU\u2081 : Set \u03b3\nleft\u271d\u00b9 : U\u2081 \u2208 \ud835\udcdd x\nV\u2081 : Set \u03b3\nV\u2081_in : V\u2081 \u2208 \ud835\udcdd x\nU\u2082 : Set \u03b3\nleft\u271d : U\u2082 \u2208 \ud835\udcdd y\nV\u2082 : Set \u03b3\nV\u2082_in : V\u2082 \u2208 \ud835\udcdd y\nV\u2081_cl : IsClosed V\u2081\nV\u2082_cl : IsClosed V\u2082\nU\u2081_op : IsOpen U\u2081\nU\u2082_op : IsOpen U\u2082\nVU\u2081 : V\u2081 \u2286 U\u2081\nVU\u2082 : V\u2082 \u2286 U\u2082\nhU\u2081\u2082 : Disjoint U\u2081 U\u2082\nU\u2083 : Set \u03b3 := (V\u2081 \u222a V\u2082)\u1d9c\nU\u2083_op : IsOpen U\u2083\nW : Set (\u03b3 \u00d7 \u03b3) := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\nW_in : W \u2208 \ud835\udcdd\u0394\nthis : W \u25cb W \u2208 F\nhV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y)\nu v : \u03b3\nu_in : (u, v).fst \u2208 V\u2081\nv_in : (u, v).snd \u2208 V\u2082\nw : \u03b3\nhuw : ((u, v).fst, w) \u2208 W\nhwv : (w, (u, v).snd) \u2208 W\nuw_in : (u, w) \u2208 U\u2081 \u00d7\u02e2 U\u2081\nwv_in : (w, v) \u2208 U\u2082 \u00d7\u02e2 U\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hU\u2081\u2082.le_bot \u27e8uw_in.2, wv_in.1\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\u22a2 \u2200 (s : Set \u03b3),\n    IsOpen s \u2194\n      \u2200 (x : \u03b3),\n        x \u2208 s \u2192\n          {p | p.fst = x \u2192 p.snd \u2208 s} \u2208\n            { uniformity := \ud835\udcdd\u02e2 (diagonal \u03b3), refl := (_ : \ud835\udcdf idRel \u2264 \ud835\udcdd\u02e2 idRel),\n                symm := (_ : Tendsto Prod.swap (\ud835\udcdd\u02e2 (diagonal \u03b3)) (\ud835\udcdd\u02e2 (diagonal \u03b3))),\n                comp := (_ : (Filter.lift' (\ud835\udcdd\u02e2 (diagonal \u03b3)) fun s => s \u25cb s) \u2264 \ud835\udcdd\u02e2 (diagonal \u03b3)) }.uniformity\n[PROOFSTEP]\nsuffices \u2200 x : \u03b3, Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\n  by\n  intro s\n  simp_rw [isOpen_iff_mem_nhds, \u2190 mem_comap_prod_mk, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nthis : \u2200 (x : \u03b3), Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\n\u22a2 \u2200 (s : Set \u03b3),\n    IsOpen s \u2194\n      \u2200 (x : \u03b3),\n        x \u2208 s \u2192\n          {p | p.fst = x \u2192 p.snd \u2208 s} \u2208\n            { uniformity := \ud835\udcdd\u02e2 (diagonal \u03b3), refl := (_ : \ud835\udcdf idRel \u2264 \ud835\udcdd\u02e2 idRel),\n                symm := (_ : Tendsto Prod.swap (\ud835\udcdd\u02e2 (diagonal \u03b3)) (\ud835\udcdd\u02e2 (diagonal \u03b3))),\n                comp := (_ : (Filter.lift' (\ud835\udcdd\u02e2 (diagonal \u03b3)) fun s => s \u25cb s) \u2264 \ud835\udcdd\u02e2 (diagonal \u03b3)) }.uniformity\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nthis : \u2200 (x : \u03b3), Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\ns : Set \u03b3\n\u22a2 IsOpen s \u2194\n    \u2200 (x : \u03b3),\n      x \u2208 s \u2192\n        {p | p.fst = x \u2192 p.snd \u2208 s} \u2208\n          { uniformity := \ud835\udcdd\u02e2 (diagonal \u03b3), refl := (_ : \ud835\udcdf idRel \u2264 \ud835\udcdd\u02e2 idRel),\n              symm := (_ : Tendsto Prod.swap (\ud835\udcdd\u02e2 (diagonal \u03b3)) (\ud835\udcdd\u02e2 (diagonal \u03b3))),\n              comp := (_ : (Filter.lift' (\ud835\udcdd\u02e2 (diagonal \u03b3)) fun s => s \u25cb s) \u2264 \ud835\udcdd\u02e2 (diagonal \u03b3)) }.uniformity\n[PROOFSTEP]\nsimp_rw [isOpen_iff_mem_nhds, \u2190 mem_comap_prod_mk, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\n\u22a2 \u2200 (x : \u03b3), Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx : \u03b3\n\u22a2 Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\n[PROOFSTEP]\nsimp_rw [nhdsSet_diagonal, comap_iSup, nhds_prod_eq, comap_prod, (\u00b7 \u2218 \u00b7), comap_id']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx : \u03b3\n\u22a2 \u2a06 (i : \u03b3), Filter.comap (fun x_1 => x) (\ud835\udcdd i) \u2293 \ud835\udcdd i = \ud835\udcdd x\n[PROOFSTEP]\nrw [iSup_split_single _ x, comap_const_of_mem fun V => mem_of_mem_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx : \u03b3\n\u22a2 \u22a4 \u2293 \ud835\udcdd x \u2294 \u2a06 (i : \u03b3) (_ : i \u2260 x), Filter.comap (fun x_2 => x) (\ud835\udcdd i) \u2293 \ud835\udcdd i = \ud835\udcdd x\n[PROOFSTEP]\nsuffices \u2200 (y) (_ : y \u2260 x), comap (fun _ : \u03b3 => x) (\ud835\udcdd y) \u2293 \ud835\udcdd y \u2264 \ud835\udcdd x by simpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx : \u03b3\nthis : \u2200 (y : \u03b3), y \u2260 x \u2192 Filter.comap (fun x_2 => x) (\ud835\udcdd y) \u2293 \ud835\udcdd y \u2264 \ud835\udcdd x\n\u22a2 \u22a4 \u2293 \ud835\udcdd x \u2294 \u2a06 (i : \u03b3) (_ : i \u2260 x), Filter.comap (fun x_2 => x) (\ud835\udcdd i) \u2293 \ud835\udcdd i = \ud835\udcdd x\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx : \u03b3\n\u22a2 \u2200 (y : \u03b3), y \u2260 x \u2192 Filter.comap (fun x_2 => x) (\ud835\udcdd y) \u2293 \ud835\udcdd y \u2264 \ud835\udcdd x\n[PROOFSTEP]\nintro y hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b3\ninst\u271d\u00b9 : CompactSpace \u03b3\ninst\u271d : T2Space \u03b3\nx y : \u03b3\nhxy : y \u2260 x\n\u22a2 Filter.comap (fun x_1 => x) (\ud835\udcdd y) \u2293 \ud835\udcdd y \u2264 \ud835\udcdd x\n[PROOFSTEP]\nsimp [comap_const_of_not_mem (compl_singleton_mem_nhds hxy) (Classical.not_not.2 rfl)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\ninst\u271d : CompactSpace \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Continuous f\n\u22a2 map (Prod.map f f) (\ud835\udce4 \u03b1) = map (Prod.map f f) (\ud835\udcdd\u02e2 (diagonal \u03b1))\n[PROOFSTEP]\nrw [nhdsSet_diagonal_eq_uniformity]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\n\u22a2 UniformContinuousOn f s\n[PROOFSTEP]\nrw [uniformContinuousOn_iff_restrict]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : IsCompact s\nhf : ContinuousOn f s\n\u22a2 UniformContinuous (restrict s f)\n[PROOFSTEP]\nrw [isCompact_iff_compactSpace] at hs \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : CompactSpace \u2191s\nhf : ContinuousOn f s\n\u22a2 UniformContinuous (restrict s f)\n[PROOFSTEP]\nrw [continuousOn_iff_continuous_restrict] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhs : CompactSpace \u2191s\nhf : Continuous (restrict s f)\n\u22a2 UniformContinuous (restrict s f)\n[PROOFSTEP]\nexact CompactSpace.uniformContinuous_of_continuous hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\n\u22a2 {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\nobtain \u27e8t, ht, htsymm, htr\u27e9 := comp_symm_mem_uniformity_sets hr\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\n\u22a2 {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\nchoose U hU T hT hb using fun a ha =>\n  exists_mem_nhds_ball_subset_of_mem_nhds ((hf a ha).preimage_mem_nhds <| mem_nhds_left _ ht)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\n\u22a2 {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\nobtain \u27e8fs, hsU\u27e9 := hs.elim_nhds_subcover' U hU\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\n\u22a2 {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\napply mem_of_superset ((biInter_finset_mem fs).2 fun a _ => hT a a.2)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\n\u22a2 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s) \u2286 {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nrintro \u27e8a\u2081, a\u2082\u27e9 h h\u2081\n[GOAL]\ncase intro.intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\na\u2081 a\u2082 : \u03b1\nh : (a\u2081, a\u2082) \u2208 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s)\nh\u2081 : (a\u2081, a\u2082).fst \u2208 s\n\u22a2 (f (a\u2081, a\u2082).fst, f (a\u2081, a\u2082).snd) \u2208 r\n[PROOFSTEP]\nobtain \u27e8a, ha, haU\u27e9 := Set.mem_iUnion\u2082.1 (hsU h\u2081)\n[GOAL]\ncase intro.intro.intro.intro.mk.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\na\u2081 a\u2082 : \u03b1\nh : (a\u2081, a\u2082) \u2208 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s)\nh\u2081 : (a\u2081, a\u2082).fst \u2208 s\na : \u2191s\nha : a \u2208 fs\nhaU : (a\u2081, a\u2082).fst \u2208 U \u2191a (_ : \u2191a \u2208 s)\n\u22a2 (f (a\u2081, a\u2082).fst, f (a\u2081, a\u2082).snd) \u2208 r\n[PROOFSTEP]\napply htr\n[GOAL]\ncase intro.intro.intro.intro.mk.intro.intro.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\na\u2081 a\u2082 : \u03b1\nh : (a\u2081, a\u2082) \u2208 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s)\nh\u2081 : (a\u2081, a\u2082).fst \u2208 s\na : \u2191s\nha : a \u2208 fs\nhaU : (a\u2081, a\u2082).fst \u2208 U \u2191a (_ : \u2191a \u2208 s)\n\u22a2 (f (a\u2081, a\u2082).fst, f (a\u2081, a\u2082).snd) \u2208 t \u25cb t\n[PROOFSTEP]\nrefine' \u27e8f a, htsymm.mk_mem_comm.1 (hb _ _ _ haU _), hb _ _ _ haU _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.mk.intro.intro.a.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\na\u2081 a\u2082 : \u03b1\nh : (a\u2081, a\u2082) \u2208 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s)\nh\u2081 : (a\u2081, a\u2082).fst \u2208 s\na : \u2191s\nha : a \u2208 fs\nhaU : (a\u2081, a\u2082).fst \u2208 U \u2191a (_ : \u2191a \u2208 s)\n\u22a2 (a\u2081, a\u2082).fst \u2208 ball (a\u2081, a\u2082).fst (T \u2191a (_ : \u2191a \u2208 s))\ncase intro.intro.intro.intro.mk.intro.intro.a.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nr : Set (\u03b2 \u00d7 \u03b2)\ns : Set \u03b1\nhs : IsCompact s\nf : \u03b1 \u2192 \u03b2\nhf : \u2200 (a : \u03b1), a \u2208 s \u2192 ContinuousAt f a\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\nU : (a : \u03b1) \u2192 a \u2208 s \u2192 Set \u03b1\nhU : \u2200 (a : \u03b1) (ha : a \u2208 s), U a ha \u2208 \ud835\udcdd a\nT : (a : \u03b1) \u2192 a \u2208 s \u2192 Set (\u03b1 \u00d7 \u03b1)\nhT : \u2200 (a : \u03b1) (ha : a \u2208 s), T a ha \u2208 \ud835\udce4 \u03b1\nhb : \u2200 (a : \u03b1) (ha : a \u2208 s) (a' : \u03b1), a' \u2208 U a ha \u2192 ball a' (T a ha) \u2286 f \u207b\u00b9' {y | (f a, y) \u2208 t}\nfs : Finset \u2191s\nhsU : s \u2286 \u22c3 (x : \u2191s) (_ : x \u2208 fs), U \u2191x (_ : \u2191x \u2208 s)\na\u2081 a\u2082 : \u03b1\nh : (a\u2081, a\u2082) \u2208 \u22c2 (i : \u2191s) (_ : i \u2208 fs), T \u2191i (_ : \u2191i \u2208 s)\nh\u2081 : (a\u2081, a\u2082).fst \u2208 s\na : \u2191s\nha : a \u2208 fs\nhaU : (a\u2081, a\u2082).fst \u2208 U \u2191a (_ : \u2191a \u2208 s)\n\u22a2 (a\u2081, a\u2082).snd \u2208 ball (a\u2081, a\u2082).fst (T \u2191a (_ : \u2191a \u2208 s))\n[PROOFSTEP]\nexacts [mem_ball_self _ (hT a a.2), mem_iInter\u2082.1 h a ha]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\n\u22a2 {x | (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\nobtain \u27e8t, ht, htsymm, htr\u27e9 := comp_symm_mem_uniformity_sets hr\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\n\u22a2 {x | (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\nobtain \u27e8s, hs, hst\u27e9 := mem_cocompact.1 (hx <| mem_nhds_left _ ht)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\n\u22a2 {x | (f x.fst, f x.snd) \u2208 r} \u2208 \ud835\udce4 \u03b1\n[PROOFSTEP]\napply\n  mem_of_superset\n    (symmetrize_mem_uniformity <|\n      (hs.uniformContinuousAt_of_continuousAt f fun _ _ => h_cont.continuousAt) <| symmetrize_mem_uniformity hr)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\n\u22a2 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r} \u2286 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nrintro \u27e8b\u2081, b\u2082\u27e9 h\n[GOAL]\ncase intro.intro.intro.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\n\u22a2 (b\u2081, b\u2082) \u2208 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nby_cases h\u2081 : b\u2081 \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\nh\u2081 : b\u2081 \u2208 s\n\u22a2 (b\u2081, b\u2082) \u2208 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nexact (h.1 h\u2081).1\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\nh\u2081 : \u00acb\u2081 \u2208 s\n\u22a2 (b\u2081, b\u2082) \u2208 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nby_cases h\u2082 : b\u2082 \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\nh\u2081 : \u00acb\u2081 \u2208 s\nh\u2082 : b\u2082 \u2208 s\n\u22a2 (b\u2081, b\u2082) \u2208 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\nexact (h.2 h\u2082).2\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\nh\u2081 : \u00acb\u2081 \u2208 s\nh\u2082 : \u00acb\u2082 \u2208 s\n\u22a2 (b\u2081, b\u2082) \u2208 {x | (f x.fst, f x.snd) \u2208 r}\n[PROOFSTEP]\napply htr\n[GOAL]\ncase neg.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\nh_cont : Continuous f\nhx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)\nr : Set (\u03b2 \u00d7 \u03b2)\nhr : r \u2208 \ud835\udce4 \u03b2\nt : Set (\u03b2 \u00d7 \u03b2)\nht : t \u2208 \ud835\udce4 \u03b2\nhtsymm : SymmetricRel t\nhtr : t \u25cb t \u2286 r\ns : Set \u03b1\nhs : IsCompact s\nhst : s\u1d9c \u2286 f \u207b\u00b9' {y | (x, y) \u2208 t}\nb\u2081 b\u2082 : \u03b1\nh : (b\u2081, b\u2082) \u2208 symmetrizeRel {x | x.fst \u2208 s \u2192 (f x.fst, f x.snd) \u2208 symmetrizeRel r}\nh\u2081 : \u00acb\u2081 \u2208 s\nh\u2082 : \u00acb\u2082 \u2208 s\n\u22a2 (f (b\u2081, b\u2082).fst, f (b\u2081, b\u2082).snd) \u2208 t \u25cb t\n[PROOFSTEP]\nexact \u27e8x, htsymm.mk_mem_comm.1 (hst h\u2081), hst h\u2082\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\n\u22a2 Tendsto f (cocompact \u03b1) (\ud835\udcdd 1)\n[PROOFSTEP]\nintro N hN\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\n\u22a2 N \u2208 map f (cocompact \u03b1)\n[PROOFSTEP]\nrw [mem_map, mem_cocompact']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\n\u22a2 \u2203 t, IsCompact t \u2227 (f \u207b\u00b9' N)\u1d9c \u2286 t\n[PROOFSTEP]\nrefine' \u27e8mulTSupport f, h.isCompact, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\n\u22a2 (f \u207b\u00b9' N)\u1d9c \u2286 mulTSupport f\n[PROOFSTEP]\nrw [compl_subset_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\n\u22a2 (mulTSupport f)\u1d9c \u2286 f \u207b\u00b9' N\n[PROOFSTEP]\nintro v hv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\nv : \u03b1\nhv : v \u2208 (mulTSupport f)\u1d9c\n\u22a2 v \u2208 f \u207b\u00b9' N\n[PROOFSTEP]\nrw [mem_preimage, image_eq_one_of_nmem_mulTSupport hv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformSpace \u03b2\nf : \u03b1 \u2192 \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : One \u03b3\nh : HasCompactMulSupport f\nN : Set \u03b3\nhN : N \u2208 \ud835\udcdd 1\nv : \u03b1\nhv : v \u2208 (mulTSupport f)\u1d9c\n\u22a2 1 \u2208 N\n[PROOFSTEP]\nexact mem_of_mem_nhds hN\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nh : ContinuousOn (\u21bff) (U \u00d7\u02e2 univ)\n\u22a2 TendstoUniformly f (f x) (\ud835\udcdd x)\n[PROOFSTEP]\nrcases LocallyCompactSpace.local_compact_nhds _ _ hxU with \u27e8K, hxK, hKU, hK\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nh : ContinuousOn (\u21bff) (U \u00d7\u02e2 univ)\nK : Set \u03b1\nhxK : K \u2208 \ud835\udcdd x\nhKU : K \u2286 U\nhK : IsCompact K\n\u22a2 TendstoUniformly f (f x) (\ud835\udcdd x)\n[PROOFSTEP]\nhave : UniformContinuousOn (\u21bff) (K \u00d7\u02e2 univ) :=\n  IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ) (h.mono <| prod_mono hKU Subset.rfl)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : LocallyCompactSpace \u03b1\ninst\u271d\u00b9 : CompactSpace \u03b2\ninst\u271d : UniformSpace \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nx : \u03b1\nU : Set \u03b1\nhxU : U \u2208 \ud835\udcdd x\nh : ContinuousOn (\u21bff) (U \u00d7\u02e2 univ)\nK : Set \u03b1\nhxK : K \u2208 \ud835\udcdd x\nhKU : K \u2286 U\nhK : IsCompact K\nthis : UniformContinuousOn (\u21bff) (K \u00d7\u02e2 univ)\n\u22a2 TendstoUniformly f (f x) (\ud835\udcdd x)\n[PROOFSTEP]\nexact this.tendstoUniformly hxK\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b9 : Type u_4\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ninst\u271d : CompactSpace \u03b2\nh : Equicontinuous F\n\u22a2 UniformEquicontinuous F\n[PROOFSTEP]\nrw [equicontinuous_iff_continuous] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b9 : Type u_4\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ninst\u271d : CompactSpace \u03b2\nh : Continuous (\u2191UniformFun.ofFun \u2218 Function.swap F)\n\u22a2 UniformEquicontinuous F\n[PROOFSTEP]\nrw [uniformEquicontinuous_iff_uniformContinuous]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\n\u03b9 : Type u_4\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ninst\u271d : CompactSpace \u03b2\nh : Continuous (\u2191UniformFun.ofFun \u2218 Function.swap F)\n\u22a2 UniformContinuous (\u2191UniformFun.ofFun \u2218 Function.swap F)\n[PROOFSTEP]\nexact CompactSpace.uniformContinuous_of_continuous h\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.Compact", "llama_tokens": 28659, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.5346112061158785}}
{"text": "[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\n\u22a2 (g\u2081 * g\u2082).support \u2286 g\u2081.support \u2229 g\u2082.support\n[PROOFSTEP]\nintro a h\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : a \u2208 (g\u2081 * g\u2082).support\n\u22a2 a \u2208 g\u2081.support \u2229 g\u2082.support\n[PROOFSTEP]\nsimp only [mul_apply, mem_support_iff] at h \n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\n\u22a2 a \u2208 g\u2081.support \u2229 g\u2082.support\n[PROOFSTEP]\nsimp only [mem_support_iff, mem_inter, Ne.def]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\n\u22a2 \u00ac\u2191g\u2081 a = 0 \u2227 \u00ac\u2191g\u2082 a = 0\n[PROOFSTEP]\nrw [\u2190 not_or]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\n\u22a2 \u00ac(\u2191g\u2081 a = 0 \u2228 \u2191g\u2082 a = 0)\n[PROOFSTEP]\nintro w\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2081 a = 0 \u2228 \u2191g\u2082 a = 0\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2081 a = 0 \u2228 \u2191g\u2082 a = 0\n\u22a2 \u2191g\u2081 a * \u2191g\u2082 a = 0\n[PROOFSTEP]\ncases' w with w w\n[GOAL]\ncase inl\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2081 a = 0\n\u22a2 \u2191g\u2081 a * \u2191g\u2082 a = 0\n[PROOFSTEP]\nrw [w]\n[GOAL]\ncase inl\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2081 a = 0\n\u22a2 0 * \u2191g\u2082 a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2082 a = 0\n\u22a2 \u2191g\u2081 a * \u2191g\u2082 a = 0\n[PROOFSTEP]\nrw [w]\n[GOAL]\ncase inr\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d\u00b9 : MulZeroClass \u03b2\ninst\u271d : DecidableEq \u03b1\ng\u2081 g\u2082 : \u03b1 \u2192\u2080 \u03b2\na : \u03b1\nh : \u2191g\u2081 a * \u2191g\u2082 a \u2260 0\nw : \u2191g\u2082 a = 0\n\u22a2 \u2191g\u2081 a * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d : Semiring \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\u2080 \u03b2\n\u22a2 Set.Finite (Function.support fun a => f a \u2022 \u2191g a)\n[PROOFSTEP]\napply Set.Finite.subset g.finite_support\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d : Semiring \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\u2080 \u03b2\n\u22a2 (Function.support fun a => f a \u2022 \u2191g a) \u2286 Function.support \u2191g\n[PROOFSTEP]\nsimp only [Function.support_subset_iff, Finsupp.mem_support_iff, Ne.def, Finsupp.fun_support_eq, Finset.mem_coe]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d : Semiring \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\u2080 \u03b2\n\u22a2 \u2200 (x : \u03b1), \u00acf x \u2022 \u2191g x = 0 \u2192 \u00ac\u2191g x = 0\n[PROOFSTEP]\nintro x hx h\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d : Semiring \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\u2080 \u03b2\nx : \u03b1\nhx : \u00acf x \u2022 \u2191g x = 0\nh : \u2191g x = 0\n\u22a2 False\n[PROOFSTEP]\napply hx\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\n\u03b3 : Type u\u2083\n\u03b4 : Type u\u2084\n\u03b9 : Type u\u2085\ninst\u271d : Semiring \u03b2\nf : \u03b1 \u2192 \u03b2\ng : \u03b1 \u2192\u2080 \u03b2\nx : \u03b1\nhx : \u00acf x \u2022 \u2191g x = 0\nh : \u2191g x = 0\n\u22a2 f x \u2022 \u2191g x = 0\n[PROOFSTEP]\nrw [h, smul_zero]\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.Pointwise", "llama_tokens": 1965, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.6926419704455588, "lm_q1q2_score": 0.5346112012188491}}
{"text": "[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\n\u22a2 (\u2211 x in s, f x) ^ (n + 1) / \u2191(card s) ^ n \u2264 \u2211 x in s, f x ^ (n + 1)\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | hs)\n[GOAL]\ncase inl\n\u03b9 : Type u\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 \u2205 \u2192 0 \u2264 f a\n\u22a2 (\u2211 x in \u2205, f x) ^ (n + 1) / \u2191(card \u2205) ^ n \u2264 \u2211 x in \u2205, f x ^ (n + 1)\n[PROOFSTEP]\nsimp_rw [Finset.sum_empty, zero_pow' _ (Nat.succ_ne_zero n), zero_div]\n[GOAL]\ncase inl\n\u03b9 : Type u\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 \u2205 \u2192 0 \u2264 f a\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\n\u22a2 (\u2211 x in s, f x) ^ (n + 1) / \u2191(card s) ^ n \u2264 \u2211 x in s, f x ^ (n + 1)\n[PROOFSTEP]\nhave hs0 : 0 < (s.card : \u211d) := Nat.cast_pos.2 hs.card_pos\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\n\u22a2 (\u2211 x in s, f x) ^ (n + 1) / \u2191(card s) ^ n \u2264 \u2211 x in s, f x ^ (n + 1)\n[PROOFSTEP]\nsuffices (\u2211 x in s, f x / s.card) ^ (n + 1) \u2264 \u2211 x in s, f x ^ (n + 1) / s.card by\n  rwa [\u2190 Finset.sum_div, \u2190 Finset.sum_div, div_pow, pow_succ' (s.card : \u211d), \u2190 div_div, div_le_iff hs0, div_mul,\n    div_self hs0.ne', div_one] at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\nthis : (\u2211 x in s, f x / \u2191(card s)) ^ (n + 1) \u2264 \u2211 x in s, f x ^ (n + 1) / \u2191(card s)\n\u22a2 (\u2211 x in s, f x) ^ (n + 1) / \u2191(card s) ^ n \u2264 \u2211 x in s, f x ^ (n + 1)\n[PROOFSTEP]\nrwa [\u2190 Finset.sum_div, \u2190 Finset.sum_div, div_pow, pow_succ' (s.card : \u211d), \u2190 div_div, div_le_iff hs0, div_mul,\n  div_self hs0.ne', div_one] at this \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\n\u22a2 (\u2211 x in s, f x / \u2191(card s)) ^ (n + 1) \u2264 \u2211 x in s, f x ^ (n + 1) / \u2191(card s)\n[PROOFSTEP]\nhave :=\n  @ConvexOn.map_sum_le \u211d \u211d \u211d \u03b9 _ _ _ _ _ _ (Set.Ici 0) (fun x => x ^ (n + 1)) s (fun _ => 1 / s.card) ((\u2191) \u2218 f)\n    (convexOn_pow (n + 1)) ?_ ?_ fun i hi => Set.mem_Ici.2 (hf i hi)\n[GOAL]\ncase inr.refine_3\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\nthis :\n  (fun x => x ^ (n + 1)) (\u2211 i in s, (fun x => 1 / \u2191(card s)) i \u2022 ((fun x => x) \u2218 f) i) \u2264\n    \u2211 i in s, (fun x => 1 / \u2191(card s)) i \u2022 (fun x => x ^ (n + 1)) (((fun x => x) \u2218 f) i)\n\u22a2 (\u2211 x in s, f x / \u2191(card s)) ^ (n + 1) \u2264 \u2211 x in s, f x ^ (n + 1) / \u2191(card s)\n[PROOFSTEP]\nsimpa only [inv_mul_eq_div, one_div, Algebra.id.smul_eq_mul] using this\n[GOAL]\ncase inr.refine_1\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 (fun x => 1 / \u2191(card s)) i\n[PROOFSTEP]\nsimp only [one_div, inv_nonneg, Nat.cast_nonneg, imp_true_iff]\n[GOAL]\ncase inr.refine_2\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\nn : \u2115\nhf : \u2200 (a : \u03b9), a \u2208 s \u2192 0 \u2264 f a\nhs : Finset.Nonempty s\nhs0 : 0 < \u2191(card s)\n\u22a2 \u2211 i in s, (fun x => 1 / \u2191(card s)) i = 1\n[PROOFSTEP]\nsimpa only [one_div, Finset.sum_const, nsmul_eq_mul] using mul_inv_cancel hs0.ne'\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, w i * z i \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave : 0 < p := by positivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\n\u22a2 0 < p\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 \u2211 i in s, w i * z i \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [\u2190 rpow_le_rpow_iff _ _ this, \u2190 rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i ^ p\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\nexact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i ^ p\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\nall_goals\n  apply_rules [sum_nonneg, rpow_nonneg_of_nonneg]\n  intro i hi\n  apply_rules [mul_nonneg, rpow_nonneg_of_nonneg, hw i hi, hz i hi]\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\napply_rules [sum_nonneg, rpow_nonneg_of_nonneg]\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i * z i ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 w i * z i ^ p\n[PROOFSTEP]\napply_rules [mul_nonneg, rpow_nonneg_of_nonneg, hw i hi, hz i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\napply_rules [sum_nonneg, rpow_nonneg_of_nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i * z i\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 w i * z i\n[PROOFSTEP]\napply_rules [mul_nonneg, rpow_nonneg_of_nonneg, hw i hi, hz i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\n\u22a2 0 \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\napply_rules [sum_nonneg, rpow_nonneg_of_nonneg]\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i * z i ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\np : \u211d\nhp : 1 \u2264 p\nthis : 0 < p\nhw'_symm : 1 = \u2211 i in s, w i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 0 \u2264 w i * z i ^ p\n[PROOFSTEP]\napply_rules [mul_nonneg, rpow_nonneg_of_nonneg, hw i hi, hz i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\nn : \u2115\n\u22a2 (\u2211 i in s, w i * z i) ^ n \u2264 \u2211 i in s, w i * z i ^ n\n[PROOFSTEP]\nexact_mod_cast\n  Real.pow_arith_mean_le_arith_mean_pow s _ _ (fun i _ => (w i).coe_nonneg) (by exact_mod_cast hw')\n    (fun i _ => (z i).coe_nonneg) n\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\nn : \u2115\n\u22a2 \u2211 i in s, \u2191(w i) = 1\n[PROOFSTEP]\nexact_mod_cast hw'\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\nn : \u2115\n\u22a2 \u2191((\u2211 x in s, f x) ^ (n + 1)) / \u2191(card s) ^ n \u2264 \u2191(\u2211 x in s, f x ^ (n + 1))\n[PROOFSTEP]\nsimpa only [\u2190 NNReal.coe_le_coe, NNReal.coe_sum, Nonneg.coe_div, NNReal.coe_pow] using\n  @Real.pow_sum_div_card_le_sum_pow \u03b9 s (((\u2191) : \u211d\u22650 \u2192 \u211d) \u2218 f) n fun _ _ => NNReal.coe_nonneg _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nexact_mod_cast\n  Real.rpow_arith_mean_le_arith_mean_rpow s _ _ (fun i _ => (w i).coe_nonneg) (by exact_mod_cast hw')\n    (fun i _ => (z i).coe_nonneg) hp\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, \u2191(w i) = 1\n[PROOFSTEP]\nexact_mod_cast hw'\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (w\u2081 * z\u2081 + w\u2082 * z\u2082) ^ p \u2264 w\u2081 * z\u2081 ^ p + w\u2082 * z\u2082 ^ p\n[PROOFSTEP]\nhave h := rpow_arith_mean_le_arith_mean_rpow univ ![w\u2081, w\u2082] ![z\u2081, z\u2082] ?_ hp\n[GOAL]\ncase refine_2\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\nh :\n  (\u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i * Matrix.vecCons z\u2081 ![z\u2082] i) ^ p \u2264\n    \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i * Matrix.vecCons z\u2081 ![z\u2082] i ^ p\n\u22a2 (w\u2081 * z\u2081 + w\u2082 * z\u2082) ^ p \u2264 w\u2081 * z\u2081 ^ p + w\u2082 * z\u2082 ^ p\n[PROOFSTEP]\nsimpa [Fin.sum_univ_succ] using h\n[GOAL]\ncase refine_1\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i = 1\n[PROOFSTEP]\nsimp [hw', Fin.sum_univ_succ]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (z\u2081 + z\u2082) ^ p \u2264 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p)\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp with (rfl | h'p)\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\nhp : 1 \u2264 1\n\u22a2 (z\u2081 + z\u2082) ^ 1 \u2264 2 ^ (1 - 1) * (z\u2081 ^ 1 + z\u2082 ^ 1)\n[PROOFSTEP]\nsimp only [rpow_one, sub_self, rpow_zero, one_mul]\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\nhp : 1 \u2264 1\n\u22a2 z\u2081 + z\u2082 \u2264 z\u2081 + z\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 (z\u2081 + z\u2082) ^ p \u2264 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p)\n[PROOFSTEP]\nconvert rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z\u2081) (2 * z\u2082) (add_halves 1) hp using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 (z\u2081 + z\u2082) ^ p = (1 / 2 * (2 * z\u2081) + 1 / 2 * (2 * z\u2082)) ^ p\n[PROOFSTEP]\nsimp only [one_div, inv_mul_cancel_left\u2080, Ne.def, mul_eq_zero, two_ne_zero, one_ne_zero, not_false_iff]\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p) = 1 / 2 * (2 * z\u2081) ^ p + 1 / 2 * (2 * z\u2082) ^ p\n[PROOFSTEP]\nhave A : p - 1 \u2260 0 := ne_of_gt (sub_pos.2 h'p)\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\nA : p - 1 \u2260 0\n\u22a2 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p) = 1 / 2 * (2 * z\u2081) ^ p + 1 / 2 * (2 * z\u2082) ^ p\n[PROOFSTEP]\nsimp only [mul_rpow, rpow_sub' _ A, div_eq_inv_mul, rpow_one, mul_one]\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\nA : p - 1 \u2260 0\n\u22a2 2\u207b\u00b9 * 2 ^ p * (z\u2081 ^ p + z\u2082 ^ p) = 2\u207b\u00b9 * (2 ^ p * z\u2081 ^ p) + 2\u207b\u00b9 * (2 ^ p * z\u2082 ^ p)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, w i * z i \u2264 (\u2211 i in s, w i * z i ^ p) ^ (1 / p)\n[PROOFSTEP]\nexact_mod_cast\n  Real.arith_mean_le_rpow_mean s _ _ (fun i _ => (w i).coe_nonneg) (by exact_mod_cast hw') (fun i _ => (z i).coe_nonneg)\n    hp\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i in s, \u2191(w i) = 1\n[PROOFSTEP]\nexact_mod_cast hw'\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhab : a + b \u2264 1\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 1\n[PROOFSTEP]\nhave h_le_one : \u2200 x : \u211d\u22650, x \u2264 1 \u2192 x ^ p \u2264 x := fun x hx => rpow_le_self_of_le_one hx hp1\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhab : a + b \u2264 1\nhp1 : 1 \u2264 p\nh_le_one : \u2200 (x : \u211d\u22650), x \u2264 1 \u2192 x ^ p \u2264 x\n\u22a2 a ^ p + b ^ p \u2264 1\n[PROOFSTEP]\nhave ha : a \u2264 1 := (self_le_add_right a b).trans hab\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhab : a + b \u2264 1\nhp1 : 1 \u2264 p\nh_le_one : \u2200 (x : \u211d\u22650), x \u2264 1 \u2192 x ^ p \u2264 x\nha : a \u2264 1\n\u22a2 a ^ p + b ^ p \u2264 1\n[PROOFSTEP]\nhave hb : b \u2264 1 := (self_le_add_left b a).trans hab\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhab : a + b \u2264 1\nhp1 : 1 \u2264 p\nh_le_one : \u2200 (x : \u211d\u22650), x \u2264 1 \u2192 x ^ p \u2264 x\nha : a \u2264 1\nhb : b \u2264 1\n\u22a2 a ^ p + b ^ p \u2264 1\n[PROOFSTEP]\nexact (add_le_add (h_le_one a ha) (h_le_one b hb)).trans hab\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave hp_pos : 0 < p := by positivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 0 < p\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nby_cases h_zero : a + b = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : a + b = 0\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nsimp [add_eq_zero_iff.mp h_zero, hp_pos.ne']\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave h_nonzero : \u00ac(a = 0 \u2227 b = 0) := by rwa [add_eq_zero_iff] at h_zero \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\n\u22a2 \u00ac(a = 0 \u2227 b = 0)\n[PROOFSTEP]\nrwa [add_eq_zero_iff] at h_zero \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave h_add : a / (a + b) + b / (a + b) = 1 := by rw [div_add_div_same, div_self h_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\n\u22a2 a / (a + b) + b / (a + b) = 1\n[PROOFSTEP]\nrw [div_add_div_same, div_self h_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave h := add_rpow_le_one_of_add_le_one (a / (a + b)) (b / (a + b)) h_add.le hp1\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : (a / (a + b)) ^ p + (b / (a + b)) ^ p \u2264 1\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nrw [div_rpow a (a + b), div_rpow b (a + b)] at h \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave hab_0 : (a + b) ^ p \u2260 0 := by simp [hp_pos, h_nonzero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\n\u22a2 (a + b) ^ p \u2260 0\n[PROOFSTEP]\nsimp [hp_pos, h_nonzero]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\nhab_0 : (a + b) ^ p \u2260 0\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave hab_0' : 0 < (a + b) ^ p := zero_lt_iff.mpr hab_0\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\nhab_0 : (a + b) ^ p \u2260 0\nhab_0' : 0 < (a + b) ^ p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) \u2264 (a + b) ^ p :=\n  by\n  nth_rw 4 [\u2190 mul_one ((a + b) ^ p)]\n  exact (mul_le_mul_left hab_0').mpr h\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\nhab_0 : (a + b) ^ p \u2260 0\nhab_0' : 0 < (a + b) ^ p\n\u22a2 (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) \u2264 (a + b) ^ p\n[PROOFSTEP]\nnth_rw 4 [\u2190 mul_one ((a + b) ^ p)]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\nhab_0 : (a + b) ^ p \u2260 0\nhab_0' : 0 < (a + b) ^ p\n\u22a2 (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) \u2264 (a + b) ^ p * 1\n[PROOFSTEP]\nexact (mul_le_mul_left hab_0').mpr h\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_zero : \u00aca + b = 0\nh_nonzero : \u00ac(a = 0 \u2227 b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p \u2264 1\nhab_0 : (a + b) ^ p \u2260 0\nhab_0' : 0 < (a + b) ^ p\nh_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) \u2264 (a + b) ^ p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nrwa [div_eq_mul_inv, div_eq_mul_inv, mul_add, mul_comm (a ^ p), mul_comm (b ^ p), \u2190 mul_assoc, \u2190 mul_assoc,\n  mul_inv_cancel hab_0, one_mul, one_mul] at h_mul \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 (a ^ p + b ^ p) ^ (1 / p) \u2264 a + b\n[PROOFSTEP]\nrw [\u2190 @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 0 < 1 / p\n[PROOFSTEP]\nsimp [lt_of_lt_of_le zero_lt_one hp1]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ (1 / (1 / p))\n[PROOFSTEP]\nrw [one_div_one_div]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nexact add_rpow_le_rpow_add _ _ hp1\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave h_rpow : \u2200 a : \u211d\u22650, a ^ q = (a ^ p) ^ (q / p) := fun a => by\n  rw [\u2190 NNReal.rpow_mul, div_eq_inv_mul, \u2190 mul_assoc, _root_.mul_inv_cancel hp_pos.ne.symm, one_mul]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na\u271d b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\na : \u211d\u22650\n\u22a2 a ^ q = (a ^ p) ^ (q / p)\n[PROOFSTEP]\nrw [\u2190 NNReal.rpow_mul, div_eq_inv_mul, \u2190 mul_assoc, _root_.mul_inv_cancel hp_pos.ne.symm, one_mul]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave h_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p :=\n  by\n  refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n  rwa [one_le_div hp_pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrefine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 1 \u2264 q / p\n[PROOFSTEP]\nrwa [one_le_div hp_pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [h_rpow a, h_rpow b, NNReal.le_rpow_one_div_iff hp_pos, \u2190 NNReal.rpow_mul, mul_comm, mul_one_div]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n\u22a2 ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrwa [one_div_div] at h_rpow_add_rpow_le_add \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrcases hp.eq_or_lt with (rfl | hp_pos)\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\nhp : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (a + b) ^ 0 \u2264 a ^ 0 + b ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nhave h := rpow_add_rpow_le a b hp_pos hp1\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ (1 / 1) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [one_div_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrepeat' rw [NNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [NNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a ^ 1 + b ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [NNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [NNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [NNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nexact (NNReal.le_rpow_one_div_iff hp_pos).mp h\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nhave hp_pos : 0 < p\n[GOAL]\ncase hp_pos\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 0 < p\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nhave hp_nonneg : 0 \u2264 p\n[GOAL]\ncase hp_nonneg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 0 \u2264 p\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nhave hp_not_neg : \u00acp < 0 := by simp [hp_nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\n\u22a2 \u00acp < 0\n[PROOFSTEP]\nsimp [hp_nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nhave h_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4) := by\n  simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\n[PROOFSTEP]\nsimp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2264 \u2211 i in s, w i * z i ^ p\n[PROOFSTEP]\nrefine' le_of_top_imp_top_of_toNNReal_le _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\n\u22a2 (\u2211 i in s, w i * z i) ^ p = \u22a4 \u2192 \u2211 i in s, w i * z i ^ p = \u22a4\n[PROOFSTEP]\nrw [rpow_eq_top_iff, sum_eq_top_iff, sum_eq_top_iff]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\n\u22a2 \u2211 i in s, w i * z i = 0 \u2227 p < 0 \u2228 (\u2203 a, a \u2208 s \u2227 w a * z a = \u22a4) \u2227 0 < p \u2192 \u2203 a, a \u2208 s \u2227 w a * z a ^ p = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh : \u2211 i in s, w i * z i = 0 \u2227 p < 0 \u2228 (\u2203 a, a \u2208 s \u2227 w a * z a = \u22a4) \u2227 0 < p\n\u22a2 \u2203 a, a \u2208 s \u2227 w a * z a ^ p = \u22a4\n[PROOFSTEP]\nsimp only [and_false_iff, hp_not_neg, false_or_iff] at h \n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh : (\u2203 a, a \u2208 s \u2227 w a * z a = \u22a4) \u2227 0 < p\n\u22a2 \u2203 a, a \u2208 s \u2227 w a * z a ^ p = \u22a4\n[PROOFSTEP]\nrcases h.left with \u27e8a, H, ha\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh : (\u2203 a, a \u2208 s \u2227 w a * z a = \u22a4) \u2227 0 < p\na : \u03b9\nH : a \u2208 s\nha : w a * z a = \u22a4\n\u22a2 \u2203 a, a \u2208 s \u2227 w a * z a ^ p = \u22a4\n[PROOFSTEP]\nuse a, H\n[GOAL]\ncase right\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh : (\u2203 a, a \u2208 s \u2227 w a * z a = \u22a4) \u2227 0 < p\na : \u03b9\nH : a \u2208 s\nha : w a * z a = \u22a4\n\u22a2 w a * z a ^ p = \u22a4\n[PROOFSTEP]\nrwa [\u2190 h_top_iff_rpow_top a H]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\n\u22a2 (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4 \u2192\n    \u2211 i in s, w i * z i ^ p \u2260 \u22a4 \u2192\n      ENNReal.toNNReal ((\u2211 i in s, w i * z i) ^ p) \u2264 ENNReal.toNNReal (\u2211 i in s, w i * z i ^ p)\n[PROOFSTEP]\nintro h_top_rpow_sum\n  _\n    -- show hypotheses needed to put the `.toNNReal` inside the sums.\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\n\u22a2 ENNReal.toNNReal ((\u2211 i in s, w i * z i) ^ p) \u2264 ENNReal.toNNReal (\u2211 i in s, w i * z i ^ p)\n[PROOFSTEP]\nhave h_top : \u2200 a : \u03b9, a \u2208 s \u2192 w a * z a \u2260 \u22a4 :=\n  haveI h_top_sum : \u2211 i : \u03b9 in s, w i * z i \u2260 \u22a4 := by\n    intro h\n    rw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum \n    exact h_top_rpow_sum rfl\n  fun a ha => (lt_top_of_sum_ne_top h_top_sum ha).ne\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\n\u22a2 \u2211 i in s, w i * z i \u2260 \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh : \u2211 i in s, w i * z i = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrw [h, top_rpow_of_pos hp_pos] at h_top_rpow_sum \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : \u22a4 \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh : \u2211 i in s, w i * z i = \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact h_top_rpow_sum rfl\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\n\u22a2 ENNReal.toNNReal ((\u2211 i in s, w i * z i) ^ p) \u2264 ENNReal.toNNReal (\u2211 i in s, w i * z i ^ p)\n[PROOFSTEP]\nhave h_top_rpow : \u2200 a : \u03b9, a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4 :=\n  by\n  intro i hi\n  specialize h_top i hi\n  rwa [Ne.def, \u2190 h_top_iff_rpow_top i hi]\n    -- put the `.toNNReal` inside the sums.\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\n\u22a2 \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i * z i ^ p \u2260 \u22a4\n[PROOFSTEP]\nspecialize h_top i hi\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\ni : \u03b9\nhi : i \u2208 s\nh_top : w i * z i \u2260 \u22a4\n\u22a2 w i * z i ^ p \u2260 \u22a4\n[PROOFSTEP]\nrwa [Ne.def, \u2190 h_top_iff_rpow_top i hi]\n  -- put the `.toNNReal` inside the sums.\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n\u22a2 ENNReal.toNNReal ((\u2211 i in s, w i * z i) ^ p) \u2264 ENNReal.toNNReal (\u2211 i in s, w i * z i ^ p)\n[PROOFSTEP]\nsimp_rw [toNNReal_sum h_top_rpow, \u2190 toNNReal_rpow, toNNReal_sum h_top, toNNReal_mul, \u2190 toNNReal_rpow]\n  -- use corresponding nnreal result\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n\u22a2 (\u2211 x in s, ENNReal.toNNReal (w x) * ENNReal.toNNReal (z x)) ^ p \u2264\n    \u2211 x in s, ENNReal.toNNReal (w x) * ENNReal.toNNReal (z x) ^ p\n[PROOFSTEP]\nrefine' NNReal.rpow_arith_mean_le_arith_mean_rpow s (fun i => (w i).toNNReal) (fun i => (z i).toNNReal) _ hp\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n\u22a2 \u2211 i in s, (fun i => ENNReal.toNNReal (w i)) i = 1\n[PROOFSTEP]\nhave h_sum_nnreal : \u2211 i in s, w i = \u2191(\u2211 i in s, (w i).toNNReal) :=\n  by\n  rw [coe_finset_sum]\n  refine' sum_congr rfl fun i hi => (coe_toNNReal _).symm\n  refine' (lt_top_of_sum_ne_top _ hi).ne\n  exact hw'.symm \u25b8 ENNReal.one_ne_top\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n\u22a2 \u2211 i in s, w i = \u2191(\u2211 i in s, ENNReal.toNNReal (w i))\n[PROOFSTEP]\nrw [coe_finset_sum]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\n\u22a2 \u2211 i in s, w i = \u2211 a in s, \u2191(ENNReal.toNNReal (w a))\n[PROOFSTEP]\nrefine' sum_congr rfl fun i hi => (coe_toNNReal _).symm\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i \u2260 \u22a4\n[PROOFSTEP]\nrefine' (lt_top_of_sum_ne_top _ hi).ne\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 x in s, w x \u2260 \u22a4\n[PROOFSTEP]\nexact hw'.symm \u25b8 ENNReal.one_ne_top\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\u221e\nhw' : \u2211 i in s, w i = 1\np : \u211d\nhp : 1 \u2264 p\nhp_pos : 0 < p\nhp_nonneg : 0 \u2264 p\nhp_not_neg : \u00acp < 0\nh_top_iff_rpow_top : \u2200 (i : \u03b9), i \u2208 s \u2192 (w i * z i = \u22a4 \u2194 w i * z i ^ p = \u22a4)\nh_top_rpow_sum : (\u2211 i in s, w i * z i) ^ p \u2260 \u22a4\na\u271d : \u2211 i in s, w i * z i ^ p \u2260 \u22a4\nh_top : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a \u2260 \u22a4\nh_top_rpow : \u2200 (a : \u03b9), a \u2208 s \u2192 w a * z a ^ p \u2260 \u22a4\nh_sum_nnreal : \u2211 i in s, w i = \u2191(\u2211 i in s, ENNReal.toNNReal (w i))\n\u22a2 \u2211 i in s, (fun i => ENNReal.toNNReal (w i)) i = 1\n[PROOFSTEP]\nrwa [\u2190 coe_eq_coe, \u2190 h_sum_nnreal]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\u221e\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (w\u2081 * z\u2081 + w\u2082 * z\u2082) ^ p \u2264 w\u2081 * z\u2081 ^ p + w\u2082 * z\u2082 ^ p\n[PROOFSTEP]\nhave h := rpow_arith_mean_le_arith_mean_rpow univ ![w\u2081, w\u2082] ![z\u2081, z\u2082] ?_ hp\n[GOAL]\ncase refine_2\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\u221e\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\nh :\n  (\u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i * Matrix.vecCons z\u2081 ![z\u2082] i) ^ p \u2264\n    \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i * Matrix.vecCons z\u2081 ![z\u2082] i ^ p\n\u22a2 (w\u2081 * z\u2081 + w\u2082 * z\u2082) ^ p \u2264 w\u2081 * z\u2081 ^ p + w\u2082 * z\u2082 ^ p\n[PROOFSTEP]\nsimpa [Fin.sum_univ_succ] using h\n[GOAL]\ncase refine_1\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 z\u2081 z\u2082 : \u211d\u22650\u221e\nhw' : w\u2081 + w\u2082 = 1\np : \u211d\nhp : 1 \u2264 p\n\u22a2 \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w\u2081 ![w\u2082] i = 1\n[PROOFSTEP]\nsimp [hw', Fin.sum_univ_succ]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (z\u2081 + z\u2082) ^ p \u2264 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p)\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp with (rfl | h'p)\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\nhp : 1 \u2264 1\n\u22a2 (z\u2081 + z\u2082) ^ 1 \u2264 2 ^ (1 - 1) * (z\u2081 ^ 1 + z\u2082 ^ 1)\n[PROOFSTEP]\nsimp only [rpow_one, sub_self, rpow_zero, one_mul, le_refl]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 (z\u2081 + z\u2082) ^ p \u2264 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p)\n[PROOFSTEP]\nconvert rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z\u2081) (2 * z\u2082) (ENNReal.add_halves 1) hp using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 (z\u2081 + z\u2082) ^ p = (1 / 2 * (2 * z\u2081) + 1 / 2 * (2 * z\u2082)) ^ p\n[PROOFSTEP]\nsimp [\u2190 mul_assoc, ENNReal.inv_mul_cancel two_ne_zero two_ne_top]\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\n\u22a2 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p) = 1 / 2 * (2 * z\u2081) ^ p + 1 / 2 * (2 * z\u2082) ^ p\n[PROOFSTEP]\nhave _ : p - 1 \u2260 0 := ne_of_gt (sub_pos.2 h'p)\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\nx\u271d : p - 1 \u2260 0\n\u22a2 2 ^ (p - 1) * (z\u2081 ^ p + z\u2082 ^ p) = 1 / 2 * (2 * z\u2081) ^ p + 1 / 2 * (2 * z\u2082) ^ p\n[PROOFSTEP]\nsimp only [mul_rpow_of_nonneg _ _ (zero_le_one.trans hp), rpow_sub _ _ two_ne_zero two_ne_top, ENNReal.div_eq_inv_mul,\n  rpow_one, mul_one]\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nz\u2081 z\u2082 : \u211d\u22650\u221e\np : \u211d\nhp : 1 \u2264 p\nh'p : 1 < p\nx\u271d : p - 1 \u2260 0\n\u22a2 2\u207b\u00b9 * 2 ^ p * (z\u2081 ^ p + z\u2082 ^ p) = 2\u207b\u00b9 * (2 ^ p * z\u2081 ^ p) + 2\u207b\u00b9 * (2 ^ p * z\u2082 ^ p)\n[PROOFSTEP]\nring\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nhave hp_pos : 0 < p := by positivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 0 < p\n[PROOFSTEP]\npositivity\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nby_cases h_top : a + b = \u22a4\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_top : a + b = \u22a4\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nrw [\u2190 @ENNReal.rpow_eq_top_iff_of_pos (a + b) p hp_pos] at h_top \n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_top : (a + b) ^ p = \u22a4\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nrw [h_top]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_top : (a + b) ^ p = \u22a4\n\u22a2 a ^ p + b ^ p \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_top : \u00aca + b = \u22a4\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nobtain \u27e8ha_top, hb_top\u27e9 := add_ne_top.mp h_top\n[GOAL]\ncase neg.intro\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nh_top : \u00aca + b = \u22a4\nha_top : a \u2260 \u22a4\nhb_top : b \u2260 \u22a4\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nlift a to \u211d\u22650 using ha_top\n[GOAL]\ncase neg.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\nb : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\nhb_top : b \u2260 \u22a4\na : \u211d\u22650\nh_top : \u00ac\u2191a + b = \u22a4\n\u22a2 \u2191a ^ p + b ^ p \u2264 (\u2191a + b) ^ p\n[PROOFSTEP]\nlift b to \u211d\u22650 using hb_top\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\nhp1 : 1 \u2264 p\nhp_pos : 0 < p\na b : \u211d\u22650\nh_top : \u00ac\u2191a + \u2191b = \u22a4\n\u22a2 \u2191a ^ p + \u2191b ^ p \u2264 (\u2191a + \u2191b) ^ p\n[PROOFSTEP]\nsimpa [\u2190 ENNReal.coe_rpow_of_nonneg _ hp_pos.le] using ENNReal.coe_le_coe.2 (NNReal.add_rpow_le_rpow_add a b hp1)\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 (a ^ p + b ^ p) ^ (1 / p) \u2264 a + b\n[PROOFSTEP]\nrw [\u2190 @ENNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 0 < 1 / p\n[PROOFSTEP]\nsimp [lt_of_lt_of_le zero_lt_one hp1]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ (1 / (1 / p))\n[PROOFSTEP]\nrw [one_div_one_div]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp1 : 1 \u2264 p\n\u22a2 a ^ p + b ^ p \u2264 (a + b) ^ p\n[PROOFSTEP]\nexact add_rpow_le_rpow_add _ _ hp1\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave h_rpow : \u2200 a : \u211d\u22650\u221e, a ^ q = (a ^ p) ^ (q / p) := fun a => by\n  rw [\u2190 ENNReal.rpow_mul, _root_.mul_div_cancel' _ hp_pos.ne']\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na\u271d b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\na : \u211d\u22650\u221e\n\u22a2 a ^ q = (a ^ p) ^ (q / p)\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul, _root_.mul_div_cancel' _ hp_pos.ne']\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650\u221e), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave h_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p :=\n  by\n  refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n  rwa [one_le_div hp_pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650\u221e), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrefine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650\u221e), a ^ q = (a ^ p) ^ (q / p)\n\u22a2 1 \u2264 q / p\n[PROOFSTEP]\nrwa [one_le_div hp_pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650\u221e), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n\u22a2 (a ^ q + b ^ q) ^ (1 / q) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [h_rpow a, h_rpow b, ENNReal.le_rpow_one_div_iff hp_pos, \u2190 ENNReal.rpow_mul, mul_comm, mul_one_div]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\na b : \u211d\u22650\u221e\nhp_pos : 0 < p\nhpq : p \u2264 q\nh_rpow : \u2200 (a : \u211d\u22650\u221e), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) \u2264 a ^ p + b ^ p\n\u22a2 ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrwa [one_div_div] at h_rpow_add_rpow_le_add \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrcases hp.eq_or_lt with (rfl | hp_pos)\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\nhp : 0 \u2264 0\nhp1 : 0 \u2264 1\n\u22a2 (a + b) ^ 0 \u2264 a ^ 0 + b ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nhave h := rpow_add_rpow_le a b hp_pos hp1\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ (1 / 1) \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [one_div_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrepeat' rw [ENNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : (a ^ 1 + b ^ 1) ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [ENNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a ^ 1 + b ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [ENNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b ^ 1 \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [ENNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nrw [ENNReal.rpow_one] at h \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\np : \u211d\na b : \u211d\u22650\u221e\nhp : 0 \u2264 p\nhp1 : p \u2264 1\nhp_pos : 0 < p\nh : a + b \u2264 (a ^ p + b ^ p) ^ (1 / p)\n\u22a2 (a + b) ^ p \u2264 a ^ p + b ^ p\n[PROOFSTEP]\nexact (ENNReal.le_rpow_one_div_iff hp_pos).mp h\n", "meta": {"mathlib_filename": "Mathlib.Analysis.MeanInequalitiesPow", "llama_tokens": 26616, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950947024555, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.534531802772157}}
{"text": "[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nh : IsNilpotent x\n\u22a2 IsNilpotent (-x)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := h\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 IsNilpotent (-x)\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR S : Type u\nx y : R\ninst\u271d : Ring R\nn : \u2115\nhn : x ^ n = 0\n\u22a2 (-x) ^ n = 0\n[PROOFSTEP]\nrw [neg_pow, hn, mul_zero]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d\u00b2 : MonoidWithZero R\ninst\u271d\u00b9 : MonoidWithZero S\nr : R\nF : Type u_1\ninst\u271d : MonoidWithZeroHomClass F R S\nhr : IsNilpotent r\nf : F\n\u22a2 IsNilpotent (\u2191f r)\n[PROOFSTEP]\nuse hr.choose\n[GOAL]\ncase h\nR S : Type u\nx y : R\ninst\u271d\u00b2 : MonoidWithZero R\ninst\u271d\u00b9 : MonoidWithZero S\nr : R\nF : Type u_1\ninst\u271d : MonoidWithZeroHomClass F R S\nhr : IsNilpotent r\nf : F\n\u22a2 \u2191f r ^ Exists.choose hr = 0\n[PROOFSTEP]\nrw [\u2190 map_pow, hr.choose_spec, map_zero]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nhnil : IsNilpotent r\n\u22a2 IsUnit (r - 1)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := hnil\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nn : \u2115\nhn : r ^ n = 0\n\u22a2 IsUnit (r - 1)\n[PROOFSTEP]\nrefine' \u27e8\u27e8r - 1, -\u2211 i in Finset.range n, r ^ i, _, _\u27e9, rfl\u27e9\n[GOAL]\ncase intro.refine'_1\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nn : \u2115\nhn : r ^ n = 0\n\u22a2 (r - 1) * -\u2211 i in Finset.range n, r ^ i = 1\n[PROOFSTEP]\nrw [mul_neg, mul_geom_sum, hn]\n[GOAL]\ncase intro.refine'_1\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nn : \u2115\nhn : r ^ n = 0\n\u22a2 -(0 - 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.refine'_2\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nn : \u2115\nhn : r ^ n = 0\n\u22a2 (-\u2211 i in Finset.range n, r ^ i) * (r - 1) = 1\n[PROOFSTEP]\nrw [neg_mul, geom_sum_mul, hn]\n[GOAL]\ncase intro.refine'_2\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nn : \u2115\nhn : r ^ n = 0\n\u22a2 -(0 - 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nu : R\u02e3\nhnil : IsNilpotent r\nhru : Commute r \u2191u\u207b\u00b9\n\u22a2 IsUnit (\u2191u + r)\n[PROOFSTEP]\nrw [\u2190 Units.isUnit_mul_units _ u\u207b\u00b9, add_mul, Units.mul_inv, \u2190 IsUnit.neg_iff, add_comm, neg_add, \u2190 sub_eq_add_neg]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nu : R\u02e3\nhnil : IsNilpotent r\nhru : Commute r \u2191u\u207b\u00b9\n\u22a2 IsUnit (-(r * \u2191u\u207b\u00b9) - 1)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := hnil\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nu : R\u02e3\nhru : Commute r \u2191u\u207b\u00b9\nn : \u2115\nhn : r ^ n = 0\n\u22a2 IsUnit (-(r * \u2191u\u207b\u00b9) - 1)\n[PROOFSTEP]\nrefine' IsNilpotent.sub_one_isUnit \u27e8n, _\u27e9\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nu : R\u02e3\nhru : Commute r \u2191u\u207b\u00b9\nn : \u2115\nhn : r ^ n = 0\n\u22a2 (-(r * \u2191u\u207b\u00b9)) ^ n = 0\n[PROOFSTEP]\nrw [neg_pow, hru.mul_pow, hn]\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Ring R\nr : R\nu : R\u02e3\nhru : Commute r \u2191u\u207b\u00b9\nn : \u2115\nhn : r ^ n = 0\n\u22a2 (-1) ^ n * (0 * \u2191u\u207b\u00b9 ^ n) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d\u00b3 : MonoidWithZero R\ninst\u271d\u00b2 : MonoidWithZero S\nF : Type u_1\ninst\u271d\u00b9 : MonoidWithZeroHomClass F R S\nf : F\nhf : Function.Injective \u2191f\ninst\u271d : IsReduced S\n\u22a2 IsReduced R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase eq_zero\nR S : Type u\nx y : R\ninst\u271d\u00b3 : MonoidWithZero R\ninst\u271d\u00b2 : MonoidWithZero S\nF : Type u_1\ninst\u271d\u00b9 : MonoidWithZeroHomClass F R S\nf : F\nhf : Function.Injective \u2191f\ninst\u271d : IsReduced S\n\u22a2 \u2200 (x : R), IsNilpotent x \u2192 x = 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase eq_zero\nR S : Type u\nx\u271d y : R\ninst\u271d\u00b3 : MonoidWithZero R\ninst\u271d\u00b2 : MonoidWithZero S\nF : Type u_1\ninst\u271d\u00b9 : MonoidWithZeroHomClass F R S\nf : F\nhf : Function.Injective \u2191f\ninst\u271d : IsReduced S\nx : R\nhx : IsNilpotent x\n\u22a2 x = 0\n[PROOFSTEP]\napply hf\n[GOAL]\ncase eq_zero.a\nR S : Type u\nx\u271d y : R\ninst\u271d\u00b3 : MonoidWithZero R\ninst\u271d\u00b2 : MonoidWithZero S\nF : Type u_1\ninst\u271d\u00b9 : MonoidWithZeroHomClass F R S\nf : F\nhf : Function.Injective \u2191f\ninst\u271d : IsReduced S\nx : R\nhx : IsNilpotent x\n\u22a2 \u2191f x = \u2191f 0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\ncase eq_zero.a\nR S : Type u\nx\u271d y : R\ninst\u271d\u00b3 : MonoidWithZero R\ninst\u271d\u00b2 : MonoidWithZero S\nF : Type u_1\ninst\u271d\u00b9 : MonoidWithZeroHomClass F R S\nf : F\nhf : Function.Injective \u2191f\ninst\u271d : IsReduced S\nx : R\nhx : IsNilpotent x\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\nexact (hx.map f).eq_zero\n[GOAL]\nR S\u271d : Type u\nx y : R\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\n\u22a2 Ideal.IsRadical (ker f) \u2194 IsReduced S\n[PROOFSTEP]\nsimp_rw [isReduced_iff, hf.forall, IsNilpotent, \u2190 map_pow, \u2190 RingHom.mem_ker]\n[GOAL]\nR S\u271d : Type u\nx y : R\nS : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R S\nf : F\nhf : Function.Surjective \u2191f\n\u22a2 Ideal.IsRadical (ker f) \u2194 \u2200 (x : R), (\u2203 n, x ^ n \u2208 ker f) \u2192 x \u2208 ker f\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : MonoidWithZero R\n\u22a2 IsRadical 0 \u2194 IsReduced R\n[PROOFSTEP]\nsimp_rw [isReduced_iff, IsNilpotent, exists_imp, \u2190 zero_dvd_iff]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : MonoidWithZero R\n\u22a2 IsRadical 0 \u2194 \u2200 (x : R) (x_1 : \u2115), 0 \u2223 x ^ x_1 \u2192 0 \u2223 x\n[PROOFSTEP]\nexact forall_swap\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : CommSemiring R\n\u22a2 IsRadical y \u2194 Ideal.IsRadical (Ideal.span {y})\n[PROOFSTEP]\nsimp_rw [IsRadical, \u2190 Ideal.mem_span_singleton]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : CommSemiring R\n\u22a2 (\u2200 (n : \u2115) (x : R), x ^ n \u2208 Ideal.span {y} \u2192 x \u2208 Ideal.span {y}) \u2194 Ideal.IsRadical (Ideal.span {y})\n[PROOFSTEP]\nexact forall_swap.trans (forall_congr' fun r => exists_imp.symm)\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : MonoidWithZero R\nk : \u2115\nhk : 1 < k\n\u22a2 IsReduced R \u2194 \u2200 (x : R), x ^ k = 0 \u2192 x = 0\n[PROOFSTEP]\nsimp_rw [\u2190 zero_isRadical_iff, isRadical_iff_pow_one_lt k hk, zero_dvd_iff]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhx : IsNilpotent x\nhy : IsNilpotent y\n\u22a2 IsNilpotent (x + y)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := hx\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhy : IsNilpotent y\nn : \u2115\nhn : x ^ n = 0\n\u22a2 IsNilpotent (x + y)\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hy\n[GOAL]\ncase intro.intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\n\u22a2 IsNilpotent (x + y)\n[PROOFSTEP]\nuse n + m - 1\n[GOAL]\ncase h\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\n\u22a2 (x + y) ^ (n + m - 1) = 0\n[PROOFSTEP]\nrw [h_comm.add_pow']\n[GOAL]\ncase h\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\n\u22a2 \u2211 m_1 in Finset.Nat.antidiagonal (n + m - 1), Nat.choose (n + m - 1) m_1.fst \u2022 (x ^ m_1.fst * y ^ m_1.snd) = 0\n[PROOFSTEP]\napply Finset.sum_eq_zero\n[GOAL]\ncase h.h\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\n\u22a2 \u2200 (x_1 : \u2115 \u00d7 \u2115),\n    x_1 \u2208 Finset.Nat.antidiagonal (n + m - 1) \u2192 Nat.choose (n + m - 1) x_1.fst \u2022 (x ^ x_1.fst * y ^ x_1.snd) = 0\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9 hij\n[GOAL]\ncase h.h.mk\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (n + m - 1)\n\u22a2 Nat.choose (n + m - 1) (i, j).fst \u2022 (x ^ (i, j).fst * y ^ (i, j).snd) = 0\n[PROOFSTEP]\nsuffices x ^ i * y ^ j = 0 by simp only [this, nsmul_eq_mul, mul_zero]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (n + m - 1)\nthis : x ^ i * y ^ j = 0\n\u22a2 Nat.choose (n + m - 1) (i, j).fst \u2022 (x ^ (i, j).fst * y ^ (i, j).snd) = 0\n[PROOFSTEP]\nsimp only [this, nsmul_eq_mul, mul_zero]\n[GOAL]\ncase h.h.mk\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (n + m - 1)\n\u22a2 x ^ i * y ^ j = 0\n[PROOFSTEP]\ncases' Nat.le_or_le_of_add_eq_add_pred (Finset.Nat.mem_antidiagonal.mp hij) with hi hj\n[GOAL]\ncase h.h.mk.inl\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (n + m - 1)\nhi : n \u2264 (i, j).fst\n\u22a2 x ^ i * y ^ j = 0\n[PROOFSTEP]\nrw [pow_eq_zero_of_le hi hn, zero_mul]\n[GOAL]\ncase h.h.mk.inr\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\nm : \u2115\nhm : y ^ m = 0\ni j : \u2115\nhij : (i, j) \u2208 Finset.Nat.antidiagonal (n + m - 1)\nhj : m \u2264 (i, j).snd\n\u22a2 x ^ i * y ^ j = 0\n[PROOFSTEP]\nrw [pow_eq_zero_of_le hj hm, mul_zero]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d : Commute x y\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)\n\u22a2 IsNilpotent (\u2211 i in s, f i)\n[PROOFSTEP]\nclassical\ninduction' s using Finset.induction with j s hj ih; simp\nrw [Finset.sum_insert hj]\napply Commute.isNilpotent_add\n\u00b7 exact Commute.sum_right _ _ _ (fun i hi \u21a6 h_comm _ _ (by simp) (by simp [hi]))\n\u00b7 apply hnp; simp\n\u00b7 exact ih (fun i hi \u21a6 hnp i (by simp [hi])) (fun i j hi hj \u21a6 h_comm i j (by simp [hi]) (by simp [hj]))\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d : Commute x y\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp : \u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)\n\u22a2 IsNilpotent (\u2211 i in s, f i)\n[PROOFSTEP]\ninduction' s using Finset.induction with j s hj ih\n[GOAL]\ncase empty\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)\nhnp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j : \u03b9), i \u2208 \u2205 \u2192 j \u2208 \u2205 \u2192 Commute (f i) (f j)\n\u22a2 IsNilpotent (\u2211 i in \u2205, f i)\ncase insert\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 IsNilpotent (\u2211 i in insert j s, f i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 IsNilpotent (\u2211 i in insert j s, f i)\n[PROOFSTEP]\nrw [Finset.sum_insert hj]\n[GOAL]\ncase insert\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 IsNilpotent (f j + \u2211 x in s, f x)\n[PROOFSTEP]\napply Commute.isNilpotent_add\n[GOAL]\ncase insert.h_comm\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 Commute (f j) (\u2211 x in s, f x)\n[PROOFSTEP]\nexact Commute.sum_right _ _ _ (fun i hi \u21a6 h_comm _ _ (by simp) (by simp [hi]))\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 j \u2208 insert j s\n[PROOFSTEP]\nsimp\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 i \u2208 insert j s\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase insert.hx\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 IsNilpotent (f j)\n[PROOFSTEP]\napply hnp\n[GOAL]\ncase insert.hx.a\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 j \u2208 insert j s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert.hy\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\n\u22a2 IsNilpotent (\u2211 x in s, f x)\n[PROOFSTEP]\nexact ih (fun i hi \u21a6 hnp i (by simp [hi])) (fun i j hi hj \u21a6 h_comm i j (by simp [hi]) (by simp [hj]))\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj : \u03b9\ns : Finset \u03b9\nhj : \u00acj \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j_1 : \u03b9), i \u2208 insert j s \u2192 j_1 \u2208 insert j s \u2192 Commute (f i) (f j_1)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 i \u2208 insert j s\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj\u271d : \u03b9\ns : Finset \u03b9\nhj\u271d : \u00acj\u271d \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j\u271d s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j : \u03b9), i \u2208 insert j\u271d s \u2192 j \u2208 insert j\u271d s \u2192 Commute (f i) (f j)\ni j : \u03b9\nhi : i \u2208 s\nhj : j \u2208 s\n\u22a2 i \u2208 insert j\u271d s\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm\u271d\u00b9 : Commute x y\n\u03b9 : Type u_1\ns\u271d : Finset \u03b9\nf : \u03b9 \u2192 R\nhnp\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 IsNilpotent (f i)\nh_comm\u271d : \u2200 (i j : \u03b9), i \u2208 s\u271d \u2192 j \u2208 s\u271d \u2192 Commute (f i) (f j)\nj\u271d : \u03b9\ns : Finset \u03b9\nhj\u271d : \u00acj\u271d \u2208 s\nih :\n  (\u2200 (i : \u03b9), i \u2208 s \u2192 IsNilpotent (f i)) \u2192\n    (\u2200 (i j : \u03b9), i \u2208 s \u2192 j \u2208 s \u2192 Commute (f i) (f j)) \u2192 IsNilpotent (\u2211 i in s, f i)\nhnp : \u2200 (i : \u03b9), i \u2208 insert j\u271d s \u2192 IsNilpotent (f i)\nh_comm : \u2200 (i j : \u03b9), i \u2208 insert j\u271d s \u2192 j \u2208 insert j\u271d s \u2192 Commute (f i) (f j)\ni j : \u03b9\nhi : i \u2208 s\nhj : j \u2208 s\n\u22a2 j \u2208 insert j\u271d s\n[PROOFSTEP]\nsimp [hj]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nh : IsNilpotent x\n\u22a2 IsNilpotent (x * y)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := h\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\n\u22a2 IsNilpotent (x * y)\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nn : \u2115\nhn : x ^ n = 0\n\u22a2 (x * y) ^ n = 0\n[PROOFSTEP]\nrw [h_comm.mul_pow, hn, zero_mul]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhy : y \u2208 nonZeroDivisorsLeft R\n\u22a2 IsNilpotent (x * y) \u2194 IsNilpotent x\n[PROOFSTEP]\nrefine' \u27e8_, h_comm.isNilpotent_mul_left\u27e9\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhy : y \u2208 nonZeroDivisorsLeft R\n\u22a2 IsNilpotent (x * y) \u2192 IsNilpotent x\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhy : y \u2208 nonZeroDivisorsLeft R\nk : \u2115\nhk : (x * y) ^ k = 0\n\u22a2 IsNilpotent x\n[PROOFSTEP]\nrw [mul_pow h_comm] at hk \n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhy : y \u2208 nonZeroDivisorsLeft R\nk : \u2115\nhk : x ^ k * y ^ k = 0\n\u22a2 IsNilpotent x\n[PROOFSTEP]\nexact \u27e8k, (nonZeroDivisorsLeft R).pow_mem hy k _ hk\u27e9\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nh : IsNilpotent y\n\u22a2 IsNilpotent (x * y)\n[PROOFSTEP]\nrw [h_comm.eq]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nh : IsNilpotent y\n\u22a2 IsNilpotent (y * x)\n[PROOFSTEP]\nexact h_comm.symm.isNilpotent_mul_left h\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhx : x \u2208 nonZeroDivisorsRight R\n\u22a2 IsNilpotent (x * y) \u2194 IsNilpotent y\n[PROOFSTEP]\nrefine' \u27e8_, h_comm.isNilpotent_mul_right\u27e9\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhx : x \u2208 nonZeroDivisorsRight R\n\u22a2 IsNilpotent (x * y) \u2192 IsNilpotent y\n[PROOFSTEP]\nrintro \u27e8k, hk\u27e9\n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhx : x \u2208 nonZeroDivisorsRight R\nk : \u2115\nhk : (x * y) ^ k = 0\n\u22a2 IsNilpotent y\n[PROOFSTEP]\nrw [mul_pow h_comm] at hk \n[GOAL]\ncase intro\nR S : Type u\nx y : R\ninst\u271d : Semiring R\nh_comm : Commute x y\nhx : x \u2208 nonZeroDivisorsRight R\nk : \u2115\nhk : x ^ k * y ^ k = 0\n\u22a2 IsNilpotent y\n[PROOFSTEP]\nexact \u27e8k, (nonZeroDivisorsRight R).pow_mem hx k _ hk\u27e9\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nh_comm : Commute x y\nhx : IsNilpotent x\nhy : IsNilpotent y\n\u22a2 IsNilpotent (x - y)\n[PROOFSTEP]\nrw [\u2190 neg_right_iff] at h_comm \n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nh_comm\u271d : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent y\n\u22a2 IsNilpotent (x - y)\n[PROOFSTEP]\nrw [\u2190 isNilpotent_neg_iff] at hy \n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nh_comm\u271d : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n\u22a2 IsNilpotent (x - y)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nR S : Type u\nx y : R\ninst\u271d : Ring R\nh_comm\u271d : Commute x y\nh_comm : Commute x (-y)\nhx : IsNilpotent x\nhy : IsNilpotent (-y)\n\u22a2 IsNilpotent (x + -y)\n[PROOFSTEP]\nexact h_comm.isNilpotent_add hx hy\n[GOAL]\nR\u271d S : Type u\nx\u271d y\u271d : R\u271d\ninst\u271d\u00b9 : CommSemiring R\u271d\nx y : R\u271d\nR : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 sInf {J | \u22a5 \u2264 J \u2227 Ideal.IsPrime J} = sInf {J | Ideal.IsPrime J}\n[PROOFSTEP]\nsimp_rw [and_iff_right bot_le]\n[GOAL]\nR S : Type u\nx\u271d y\u271d : R\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 IsNilpotent x \u2194 \u2200 (J : Ideal R), Ideal.IsPrime J \u2192 x \u2208 J\n[PROOFSTEP]\nrw [\u2190 mem_nilradical, nilradical_eq_sInf, Submodule.mem_sInf]\n[GOAL]\nR S : Type u\nx\u271d y\u271d : R\ninst\u271d : CommSemiring R\nx y : R\n\u22a2 (\u2200 (p : Submodule R R), p \u2208 {J | Ideal.IsPrime J} \u2192 x \u2208 p) \u2194 \u2200 (J : Ideal R), Ideal.IsPrime J \u2192 x \u2208 J\n[PROOFSTEP]\nrfl\n[GOAL]\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent (mulLeft R a) \u2194 IsNilpotent a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent (mulLeft R a) \u2192 IsNilpotent a\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase mpr\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent a \u2192 IsNilpotent (mulLeft R a)\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase mp.intro\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : mulLeft R a ^ n = 0\n\u22a2 IsNilpotent a\n[PROOFSTEP]\nuse n\n[GOAL]\ncase mpr.intro\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 IsNilpotent (mulLeft R a)\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : mulLeft R a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nsimp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn \u22a2\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 mulLeft R a ^ n = 0\n[PROOFSTEP]\nsimp only [mulLeft_eq_zero_iff, pow_mulLeft] at hn \u22a2\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nexact hn\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nexact hn\n[GOAL]\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent (mulRight R a) \u2194 IsNilpotent a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent (mulRight R a) \u2192 IsNilpotent a\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase mpr\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\n\u22a2 IsNilpotent a \u2192 IsNilpotent (mulRight R a)\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase mp.intro\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : mulRight R a ^ n = 0\n\u22a2 IsNilpotent a\n[PROOFSTEP]\nuse n\n[GOAL]\ncase mpr.intro\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 IsNilpotent (mulRight R a)\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : mulRight R a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nsimp only [mulRight_eq_zero_iff, pow_mulRight] at hn \u22a2\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 mulRight R a ^ n = 0\n[PROOFSTEP]\nsimp only [mulRight_eq_zero_iff, pow_mulRight] at hn \u22a2\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nexact hn\n[GOAL]\ncase h\nR S : Type u\nx y : R\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\na : A\nn : \u2115\nhn : a ^ n = 0\n\u22a2 a ^ n = 0\n[PROOFSTEP]\nexact hn\n[GOAL]\nR S : Type u\nx y : R\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : End R M\np : Submodule R M\nhp : p \u2264 Submodule.comap f p\nhnp : IsNilpotent f\n\u22a2 IsNilpotent (Submodule.mapQ p p f hp)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hnp\n[GOAL]\ncase intro\nR S : Type u\nx y : R\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : End R M\np : Submodule R M\nhp : p \u2264 Submodule.comap f p\nk : \u2115\nhk : f ^ k = 0\n\u22a2 IsNilpotent (Submodule.mapQ p p f hp)\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\nR S : Type u\nx y : R\nM : Type v\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nf : End R M\np : Submodule R M\nhp : p \u2264 Submodule.comap f p\nk : \u2115\nhk : f ^ k = 0\n\u22a2 Submodule.mapQ p p f hp ^ k = 0\n[PROOFSTEP]\nsimp [\u2190 p.mapQ_pow, hk]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Nilpotent", "llama_tokens": 13075, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031738057795403, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5344403255453746}}
{"text": "[GOAL]\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 Exact f g \u2194 AddMonoidHom.range f = AddMonoidHom.ker g\n[PROOFSTEP]\nrw [Abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]\n[GOAL]\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 f \u226b g = 0 \u2227 Fork.\u03b9 (kernelCone g) \u226b Cofork.\u03c0 (cokernelCocone f) = 0 \u2194 AddMonoidHom.range f = AddMonoidHom.ker g\n[PROOFSTEP]\nexact\n  \u27e8fun h =>\n    ((AddMonoidHom.range_le_ker_iff _ _).mpr h.left).antisymm ((QuotientAddGroup.ker_le_range_iff _ _).mpr h.right),\n    fun h => \u27e8(AddMonoidHom.range_le_ker_iff _ _).mp h.le, (QuotientAddGroup.ker_le_range_iff _ _).mp h.symm.le\u27e9\u27e9\n[GOAL]\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\n\u22a2 PreservesFiniteLimits colim\n[PROOFSTEP]\nrefine Functor.preservesFiniteLimitsOfMapExact _ fun F G H \u03b7 \u03b3 h => (exact_iff _ _).mpr (le_antisymm ?_ ?_)\n[GOAL]\ncase refine_1\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : Exact \u03b7 \u03b3\n\u22a2 AddMonoidHom.range (colim.map \u03b7) \u2264 AddMonoidHom.ker (colim.map \u03b3)\ncase refine_2\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : Exact \u03b7 \u03b3\n\u22a2 AddMonoidHom.ker (colim.map \u03b3) \u2264 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nall_goals replace h : \u2200 j : J, Exact (\u03b7.app j) (\u03b3.app j) := fun j => Functor.map_exact ((evaluation _ _).obj j) \u03b7 \u03b3 h\n[GOAL]\ncase refine_1\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : Exact \u03b7 \u03b3\n\u22a2 AddMonoidHom.range (colim.map \u03b7) \u2264 AddMonoidHom.ker (colim.map \u03b3)\n[PROOFSTEP]\nreplace h : \u2200 j : J, Exact (\u03b7.app j) (\u03b3.app j) := fun j => Functor.map_exact ((evaluation _ _).obj j) \u03b7 \u03b3 h\n[GOAL]\ncase refine_2\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : Exact \u03b7 \u03b3\n\u22a2 AddMonoidHom.ker (colim.map \u03b3) \u2264 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nreplace h : \u2200 j : J, Exact (\u03b7.app j) (\u03b3.app j) := fun j => Functor.map_exact ((evaluation _ _).obj j) \u03b7 \u03b3 h\n[GOAL]\ncase refine_1\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\n\u22a2 AddMonoidHom.range (colim.map \u03b7) \u2264 AddMonoidHom.ker (colim.map \u03b3)\n[PROOFSTEP]\nrw [AddMonoidHom.range_le_ker_iff, \u2190 comp_def]\n[GOAL]\ncase refine_1\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\n\u22a2 colim.map \u03b7 \u226b colim.map \u03b3 = 0\n[PROOFSTEP]\nexact colimit.hom_ext fun j => by simp [reassoc_of% (h j).w]\n[GOAL]\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\n\u22a2 colimit.\u03b9 F j \u226b colim.map \u03b7 \u226b colim.map \u03b3 = colimit.\u03b9 F j \u226b 0\n[PROOFSTEP]\nsimp [reassoc_of% (h j).w]\n[GOAL]\ncase refine_2\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\n\u22a2 AddMonoidHom.ker (colim.map \u03b3) \u2264 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nintro x (hx : _ = _)\n[GOAL]\ncase refine_2\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nx : \u2191(colim.obj G)\nhx : \u2191(colim.map \u03b3) x = 0\n\u22a2 x \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nrcases Concrete.colimit_exists_rep G x with \u27e8j, y, rfl\u27e9\n[GOAL]\ncase refine_2.intro.intro\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : \u2191(colim.map \u03b3) (\u2191(colimit.\u03b9 G j) y) = 0\n\u22a2 \u2191(colimit.\u03b9 G j) y \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nerw [\u2190 comp_apply, colimit.\u03b9_map, comp_apply, \u2190 map_zero (by exact colimit.\u03b9 H j : H.obj j \u2192+ \u2191(colimit H))] at hx \n[GOAL]\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\n\u22a2 \u2191(H.obj j) \u2192+ \u2191(colimit H)\n[PROOFSTEP]\nexact colimit.\u03b9 H j\n[GOAL]\ncase refine_2.intro.intro\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\n\u22a2 \u2191(colimit.\u03b9 G j) y \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nrcases Concrete.colimit_exists_of_rep_eq H _ _ hx with \u27e8k, e\u2081, e\u2082, hk : _ = H.map e\u2082 0\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\nk : J\ne\u2081 e\u2082 : j \u27f6 k\nhk : \u2191(H.map e\u2081) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(H.map e\u2082) 0\n\u22a2 \u2191(colimit.\u03b9 G j) y \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nrw [map_zero, \u2190 comp_apply, \u2190 NatTrans.naturality, comp_apply] at hk \n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\nk : J\ne\u2081 e\u2082 : j \u27f6 k\nhk : \u2191(NatTrans.app \u03b3 k) (\u2191(G.map e\u2081) y) = 0\n\u22a2 \u2191(colimit.\u03b9 G j) y \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nrcases((exact_iff _ _).mp <| h k).ge hk with \u27e8t, ht\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro.intro.intro.intro\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\nk : J\ne\u2081 e\u2082 : j \u27f6 k\nhk : \u2191(NatTrans.app \u03b3 k) (\u2191(G.map e\u2081) y) = 0\nt : \u2191(F.obj k)\nht : \u2191(NatTrans.app \u03b7 k) t = \u2191(G.map e\u2081) y\n\u22a2 \u2191(colimit.\u03b9 G j) y \u2208 AddMonoidHom.range (colim.map \u03b7)\n[PROOFSTEP]\nuse colimit.\u03b9 F k t\n[GOAL]\ncase h\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\nk : J\ne\u2081 e\u2082 : j \u27f6 k\nhk : \u2191(NatTrans.app \u03b3 k) (\u2191(G.map e\u2081) y) = 0\nt : \u2191(F.obj k)\nht : \u2191(NatTrans.app \u03b7 k) t = \u2191(G.map e\u2081) y\n\u22a2 \u2191(colim.map \u03b7) (\u2191(colimit.\u03b9 F k) t) = \u2191(colimit.\u03b9 G j) y\n[PROOFSTEP]\nerw [\u2190 comp_apply, colimit.\u03b9_map, comp_apply, ht]\n[GOAL]\ncase h\nX Y Z : AddCommGroupCat\nf : X \u27f6 Y\ng : Y \u27f6 Z\nJ : Type u\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsFiltered J\nF G H : J \u2964 AddCommGroupCat\n\u03b7 : F \u27f6 G\n\u03b3 : G \u27f6 H\nh : \u2200 (j : J), Exact (NatTrans.app \u03b7 j) (NatTrans.app \u03b3 j)\nj : J\ny : (forget AddCommGroupCat).obj (G.obj j)\nhx\u271d : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = 0\nhx : \u2191(colimit.\u03b9 H j) (\u2191(NatTrans.app \u03b3 j) y) = \u2191(colimit.\u03b9 H j) 0\nk : J\ne\u2081 e\u2082 : j \u27f6 k\nhk : \u2191(NatTrans.app \u03b3 k) (\u2191(G.map e\u2081) y) = 0\nt : \u2191(F.obj k)\nht : \u2191(NatTrans.app \u03b7 k) t = \u2191(G.map e\u2081) y\n\u22a2 \u2191(colimit.\u03b9 G k) (\u2191(G.map e\u2081) y) = \u2191(colimit.\u03b9 G j) y\n[PROOFSTEP]\nexact colimit.w_apply G e\u2081 y\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.GroupCat.Abelian", "llama_tokens": 4481, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303285397349, "lm_q2_score": 0.6757645944891558, "lm_q1q2_score": 0.5342124068970331}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs_nonempty : Finset.Nonempty s\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 f (\u220f i in s, g i) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrefine' le_trans (Multiset.le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul _ _ _) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs_nonempty : Finset.Nonempty s\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 Multiset.map (fun i => g i) s.val \u2260 \u2205\n[PROOFSTEP]\nsimp [hs_nonempty.ne_empty]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs_nonempty : Finset.Nonempty s\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 \u2200 (a : M), a \u2208 Multiset.map (fun i => g i) s.val \u2192 p a\n[PROOFSTEP]\nexact Multiset.forall_mem_map_iff.mpr hs\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs_nonempty : Finset.Nonempty s\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 Multiset.prod (Multiset.map f (Multiset.map (fun i => g i) s.val)) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrw [Multiset.map_map]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs_nonempty : Finset.Nonempty s\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 Multiset.prod (Multiset.map (f \u2218 fun i => g i) s.val) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\n\u22a2 f (\u220f i in s, g i) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | hs_nonempty)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\nhs : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 p (g i)\n\u22a2 f (\u220f i in \u2205, g i) \u2264 \u220f i in \u2205, f (g i)\n[PROOFSTEP]\nsimp [h_one]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\np : M \u2192 Prop\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), p x \u2192 p y \u2192 f (x * y) \u2264 f x * f y\nhp_mul : \u2200 (x y : M), p x \u2192 p y \u2192 p (x * y)\ng : \u03b9 \u2192 M\ns : Finset \u03b9\nhs : \u2200 (i : \u03b9), i \u2208 s \u2192 p (g i)\nhs_nonempty : Finset.Nonempty s\n\u22a2 f (\u220f i in s, g i) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nexact le_prod_nonempty_of_submultiplicative_on_pred f p h_mul hp_mul g s hs_nonempty hs\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), f (x * y) \u2264 f x * f y\ns : Finset \u03b9\ng : \u03b9 \u2192 M\n\u22a2 f (\u220f i in s, g i) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrefine' le_trans (Multiset.le_prod_of_submultiplicative f h_one h_mul _) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), f (x * y) \u2264 f x * f y\ns : Finset \u03b9\ng : \u03b9 \u2192 M\n\u22a2 Multiset.prod (Multiset.map f (Multiset.map (fun i => g i) s.val)) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrw [Multiset.map_map]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf : M \u2192 N\nh_one : f 1 = 1\nh_mul : \u2200 (x y : M), f (x * y) \u2264 f x * f y\ns : Finset \u03b9\ng : \u03b9 \u2192 M\n\u22a2 Multiset.prod (Multiset.map (f \u2218 fun i => g i) s.val) \u2264 \u220f i in s, f (g i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i\n\u22a2 1 \u2264 \u220f i in s, 1\n[PROOFSTEP]\nrw [prod_const_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\n\u22a2 \u220f i in s, 1 = 1\n[PROOFSTEP]\nrw [prod_const_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : s \u2286 t\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 \u00aci \u2208 s \u2192 1 \u2264 f i\n\u22a2 \u220f i in s, f i \u2264 \u220f i in t, f i\n[PROOFSTEP]\nclassical calc\n  \u220f i in s, f i \u2264 (\u220f i in t \\ s, f i) * \u220f i in s, f i :=\n    le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]\n  _ = \u220f i in t \\ s \u222a s, f i := (prod_union sdiff_disjoint).symm\n  _ = \u220f i in t, f i := by rw [sdiff_union_of_subset h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : s \u2286 t\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 \u00aci \u2208 s \u2192 1 \u2264 f i\n\u22a2 \u220f i in s, f i \u2264 \u220f i in t, f i\n[PROOFSTEP]\ncalc\n  \u220f i in s, f i \u2264 (\u220f i in t \\ s, f i) * \u220f i in s, f i :=\n    le_mul_of_one_le_left' <| one_le_prod' <| by simpa only [mem_sdiff, and_imp]\n  _ = \u220f i in t \\ s \u222a s, f i := (prod_union sdiff_disjoint).symm\n  _ = \u220f i in t, f i := by rw [sdiff_union_of_subset h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : s \u2286 t\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 \u00aci \u2208 s \u2192 1 \u2264 f i\n\u22a2 \u2200 (i : \u03b9), i \u2208 t \\ s \u2192 1 \u2264 f i\n[PROOFSTEP]\nsimpa only [mem_sdiff, and_imp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\nh : s \u2286 t\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 \u00aci \u2208 s \u2192 1 \u2264 f i\n\u22a2 \u220f i in t \\ s \u222a s, f i = \u220f i in t, f i\n[PROOFSTEP]\nrw [sdiff_union_of_subset h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i) \u2192 (\u220f i in s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1)\n[PROOFSTEP]\nclassical\nrefine Finset.induction_on s (fun _ \u21a6 \u27e8fun _ _ h \u21a6 False.elim (Finset.not_mem_empty _ h), fun _ \u21a6 rfl\u27e9) ?_\nintro a s ha ih H\nhave : \u2200 i \u2208 s, 1 \u2264 f i := fun _ \u21a6 H _ \u2218 mem_insert_of_mem\nrw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this), forall_mem_insert, ih this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i) \u2192 (\u220f i in s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1)\n[PROOFSTEP]\nrefine Finset.induction_on s (fun _ \u21a6 \u27e8fun _ _ h \u21a6 False.elim (Finset.not_mem_empty _ h), fun _ \u21a6 rfl\u27e9) ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns t : Finset \u03b9\n\u22a2 \u2200 \u2983a : \u03b9\u2984 {s : Finset \u03b9},\n    \u00aca \u2208 s \u2192\n      ((\u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i) \u2192 (\u220f i in s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1)) \u2192\n        (\u2200 (i : \u03b9), i \u2208 insert a s \u2192 1 \u2264 f i) \u2192 (\u220f i in insert a s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i = 1)\n[PROOFSTEP]\nintro a s ha ih H\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns\u271d t : Finset \u03b9\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i) \u2192 (\u220f i in s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1)\nH : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 1 \u2264 f i\n\u22a2 \u220f i in insert a s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i = 1\n[PROOFSTEP]\nhave : \u2200 i \u2208 s, 1 \u2264 f i := fun _ \u21a6 H _ \u2218 mem_insert_of_mem\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf g : \u03b9 \u2192 N\ns\u271d t : Finset \u03b9\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i) \u2192 (\u220f i in s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1)\nH : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 1 \u2264 f i\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 1 \u2264 f i\n\u22a2 \u220f i in insert a s, f i = 1 \u2194 \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i = 1\n[PROOFSTEP]\nrw [prod_insert ha, mul_eq_one_iff' (H _ <| mem_insert_self _ _) (one_le_prod' this), forall_mem_insert, ih this]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf\u271d g : \u03b9 \u2192 N\ns\u271d t s : Finset \u03b9\nf : \u03b9 \u2192 N\nn : N\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 n\n\u22a2 Finset.prod s f \u2264 n ^ card s\n[PROOFSTEP]\nrefine' (Multiset.prod_le_pow_card (s.val.map f) n _).trans _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf\u271d g : \u03b9 \u2192 N\ns\u271d t s : Finset \u03b9\nf : \u03b9 \u2192 N\nn : N\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 n\n\u22a2 \u2200 (x : N), x \u2208 Multiset.map f s.val \u2192 x \u2264 n\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf\u271d g : \u03b9 \u2192 N\ns\u271d t s : Finset \u03b9\nf : \u03b9 \u2192 N\nn : N\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 n\n\u22a2 n ^ \u2191Multiset.card (Multiset.map f s.val) \u2264 n ^ card s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : OrderedCommMonoid N\nf\u271d g : \u03b9 \u2192 N\ns\u271d t s : Finset \u03b9\nf : \u03b9 \u2192 N\nn : N\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2264 n\n\u22a2 n ^ \u2191Multiset.card s.val \u2264 n ^ card s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG\u271d : Type u_6\nk : Type u_7\nR : Type u_8\nG : Type u_9\ninst\u271d : LinearOrderedAddCommGroup G\nf : \u03b9 \u2192 G\ns : Finset \u03b9\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 |\u2211 i in s, f i| = \u2211 i in s, f i\n[PROOFSTEP]\nrw [abs_of_nonneg (Finset.sum_nonneg hf)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG\u271d : Type u_6\nk : Type u_7\nR : Type u_8\nG : Type u_9\ninst\u271d : LinearOrderedAddCommGroup G\nf : \u03b9 \u2192 G\ns : Finset \u03b9\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\n\u22a2 |\u2211 i in s, f i| = \u2211 i in s, f i\n[PROOFSTEP]\nrw [abs_of_nonneg (Finset.sum_nonneg' hf)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nHf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nn : \u2115\nhn : \u2200 (a : \u03b2), a \u2208 t \u2192 card (filter (fun x => f x = a) s) \u2264 n\n\u22a2 \u2211 _a in t, n = n * card t\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nHf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nn : \u2115\nhn : \u2200 (a : \u03b2), a \u2208 t \u2192 n \u2264 card (filter (fun x => f x = a) s)\n\u22a2 n * card t = \u2211 _a in t, n\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nHf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t\nn : \u2115\nhn : \u2200 (a : \u03b2), a \u2208 t \u2192 n \u2264 card (filter (fun x => f x = a) s)\n\u22a2 \u2211 a in t, card (filter (fun x => f x = a) s) = card s\n[PROOFSTEP]\nrw [\u2190 card_eq_sum_card_fiberwise Hf]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n\n\u22a2 \u2211 t in B, card (s \u2229 t) \u2264 card s * n\n[PROOFSTEP]\nrefine' le_trans _ (s.sum_le_card_nsmul _ _ h)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n\n\u22a2 \u2211 t in B, card (s \u2229 t) \u2264 \u2211 x in s, card (filter ((fun x x_1 => x \u2208 x_1) x) B)\n[PROOFSTEP]\nsimp_rw [\u2190 filter_mem_eq_inter, card_eq_sum_ones, sum_filter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n\n\u22a2 (\u2211 x in B, \u2211 a in s, if a \u2208 x then 1 else 0) \u2264 \u2211 x in s, \u2211 a in B, if x \u2208 a then 1 else 0\n[PROOFSTEP]\nexact sum_comm.le\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ninst\u271d : Fintype \u03b1\nh : \u2200 (a : \u03b1), card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n\n\u22a2 \u2211 s in B, card s = \u2211 s in B, card (univ \u2229 s)\n[PROOFSTEP]\nsimp_rw [univ_inter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 n \u2264 card (filter ((fun x x_1 => x \u2208 x_1) a) B)\n\u22a2 card s * n \u2264 \u2211 t in B, card (s \u2229 t)\n[PROOFSTEP]\napply (s.card_nsmul_le_sum _ _ h).trans\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 n \u2264 card (filter ((fun x x_1 => x \u2208 x_1) a) B)\n\u22a2 \u2211 x in s, card (filter ((fun x x_1 => x \u2208 x_1) x) B) \u2264 \u2211 t in B, card (s \u2229 t)\n[PROOFSTEP]\nsimp_rw [\u2190 filter_mem_eq_inter, card_eq_sum_ones, sum_filter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\nh : \u2200 (a : \u03b1), a \u2208 s \u2192 n \u2264 card (filter ((fun x x_1 => x \u2208 x_1) a) B)\n\u22a2 (\u2211 x in s, \u2211 a in B, if x \u2208 a then 1 else 0) \u2264 \u2211 x in B, \u2211 a in s, if a \u2208 x then 1 else 0\n[PROOFSTEP]\nexact sum_comm.le\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ninst\u271d : Fintype \u03b1\nh : \u2200 (a : \u03b1), n \u2264 card (filter ((fun x x_1 => x \u2208 x_1) a) B)\n\u22a2 \u2211 s in B, card (univ \u2229 s) = \u2211 s in B, card s\n[PROOFSTEP]\nsimp_rw [univ_inter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b1\ns : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ninst\u271d : Fintype \u03b1\nh : \u2200 (a : \u03b1), card (filter ((fun x x_1 => x \u2208 x_1) a) B) = n\n\u22a2 \u2211 s in B, card s = Fintype.card \u03b1 * n\n[PROOFSTEP]\nsimp_rw [Fintype.card, \u2190 sum_card_inter fun a _ \u21a6 h a, univ_inter]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Finset.Nonempty (f i)\n\u22a2 card s \u2264 card (Finset.biUnion s f)\n[PROOFSTEP]\nrw [card_biUnion hs, card_eq_sum_ones]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 Finset.Nonempty (f i)\n\u22a2 \u2211 x in s, 1 \u2264 \u2211 u in s, card (f u)\n[PROOFSTEP]\nexact sum_le_sum fun i hi \u21a6 (hf i hi).card_pos\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\n\u22a2 card s \u2264 card (Finset.biUnion s f) + card (filter (fun i => f i = \u2205) s)\n[PROOFSTEP]\nrw [\u2190 Finset.filter_card_add_filter_neg_card_eq_card fun i \u21a6 f i = \u2205, add_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b1\ns\u271d : Finset \u03b1\nB : Finset (Finset \u03b1)\nn : \u2115\ns : Finset \u03b9\nf : \u03b9 \u2192 Finset \u03b1\nhs : Set.PairwiseDisjoint (\u2191s) f\n\u22a2 card (filter (fun a => \u00acf a = \u2205) s) + card (filter (fun i => f i = \u2205) s) \u2264\n    card (Finset.biUnion s f) + card (filter (fun i => f i = \u2205) s)\n[PROOFSTEP]\nexact\n  add_le_add_right\n    ((card_le_card_biUnion (hs.subset <| filter_subset _ _) fun i hi \u21a6\n          nonempty_of_ne_empty <| (mem_filter.1 hi).2).trans <|\n      card_le_of_subset <| biUnion_subset_biUnion_of_subset_left _ <| filter_subset _ _)\n    _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 \u220f x in s, f x \u2264 \u220f x in t, f x\n[PROOFSTEP]\nclassical calc\n  \u220f x in s, f x = (\u220f x in s.filter fun x \u21a6 f x = 1, f x) * \u220f x in s.filter fun x \u21a6 f x \u2260 1, f x :=\n    by\n    rw [\u2190 prod_union, filter_union_filter_neg_eq]\n    exact disjoint_filter.2 fun _ _ h n_h \u21a6 n_h h\n  _ \u2264 \u220f x in t, f x :=\n    mul_le_of_le_one_of_le (prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ \u21a6 le_of_eq)\n      (prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp])\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 \u220f x in s, f x \u2264 \u220f x in t, f x\n[PROOFSTEP]\ncalc\n  \u220f x in s, f x = (\u220f x in s.filter fun x \u21a6 f x = 1, f x) * \u220f x in s.filter fun x \u21a6 f x \u2260 1, f x :=\n    by\n    rw [\u2190 prod_union, filter_union_filter_neg_eq]\n    exact disjoint_filter.2 fun _ _ h n_h \u21a6 n_h h\n  _ \u2264 \u220f x in t, f x :=\n    mul_le_of_le_one_of_le (prod_le_one' <| by simp only [mem_filter, and_imp]; exact fun _ _ \u21a6 le_of_eq)\n      (prod_le_prod_of_subset' <| by simpa only [subset_iff, mem_filter, and_imp])\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 \u220f x in s, f x = (\u220f x in filter (fun x => f x = 1) s, f x) * \u220f x in filter (fun x => f x \u2260 1) s, f x\n[PROOFSTEP]\nrw [\u2190 prod_union, filter_union_filter_neg_eq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 Disjoint (filter (fun x => f x = 1) s) (filter (fun x => f x \u2260 1) s)\n[PROOFSTEP]\nexact disjoint_filter.2 fun _ _ h n_h \u21a6 n_h h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 filter (fun x => f x = 1) s \u2192 f i \u2264 1\n[PROOFSTEP]\nsimp only [mem_filter, and_imp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 1 \u2192 f i \u2264 1\n[PROOFSTEP]\nexact fun _ _ \u21a6 le_of_eq\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nf : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (x : \u03b9), x \u2208 s \u2192 f x \u2260 1 \u2192 x \u2208 t\n\u22a2 filter (fun x => f x \u2260 1) s \u2286 t\n[PROOFSTEP]\nsimpa only [subset_iff, mem_filter, and_imp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHle : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\nHlt : \u2203 i, i \u2208 s \u2227 f i < g i\n\u22a2 \u220f i in s, f i < \u220f i in s, g i\n[PROOFSTEP]\nclassical\nrcases Hlt with \u27e8i, hi, hlt\u27e9\nrw [\u2190 insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]\nexact mul_lt_mul_of_lt_of_le hlt (prod_le_prod' fun j hj \u21a6 Hle j <| mem_of_mem_erase hj)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHle : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\nHlt : \u2203 i, i \u2208 s \u2227 f i < g i\n\u22a2 \u220f i in s, f i < \u220f i in s, g i\n[PROOFSTEP]\nrcases Hlt with \u27e8i, hi, hlt\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHle : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\ni : \u03b9\nhi : i \u2208 s\nhlt : f i < g i\n\u22a2 \u220f i in s, f i < \u220f i in s, g i\n[PROOFSTEP]\nrw [\u2190 insert_erase hi, prod_insert (not_mem_erase _ _), prod_insert (not_mem_erase _ _)]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHle : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\ni : \u03b9\nhi : i \u2208 s\nhlt : f i < g i\n\u22a2 f i * \u220f x in erase s i, f x < g i * \u220f x in erase s i, g x\n[PROOFSTEP]\nexact mul_lt_mul_of_lt_of_le hlt (prod_le_prod' fun j hj \u21a6 Hle j <| mem_of_mem_erase hj)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i\n\u22a2 \u220f i in s, f i < \u220f i in s, g i\n[PROOFSTEP]\napply prod_lt_prod'\n[GOAL]\ncase Hle\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase Hle\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i \u2264 g i\n[PROOFSTEP]\napply le_of_lt (Hlt i hi)\n[GOAL]\ncase Hlt\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i\n\u22a2 \u2203 i, i \u2208 s \u2227 f i < g i\n[PROOFSTEP]\ncases' hs with i hi\n[GOAL]\ncase Hlt.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2203 i, i \u2208 s \u2227 f i < g i\n[PROOFSTEP]\nexact \u27e8i, hi, Hlt i hi\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u220f j in s, f j < \u220f j in t, f j\n[PROOFSTEP]\nclassical calc\n  \u220f j in s, f j < \u220f j in insert i s, f j := by\n    rw [prod_insert hs]\n    exact lt_mul_of_one_lt_left' (\u220f j in s, f j) hlt\n  _ \u2264 \u220f j in t, f j := by\n    apply prod_le_prod_of_subset_of_one_le'\n    \u00b7 simp [Finset.insert_subset_iff, h, ht]\n    \u00b7 intro x hx h'x\n      simp only [mem_insert, not_or] at h'x \n      exact hle x hx h'x.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u220f j in s, f j < \u220f j in t, f j\n[PROOFSTEP]\ncalc\n  \u220f j in s, f j < \u220f j in insert i s, f j := by\n    rw [prod_insert hs]\n    exact lt_mul_of_one_lt_left' (\u220f j in s, f j) hlt\n  _ \u2264 \u220f j in t, f j := by\n    apply prod_le_prod_of_subset_of_one_le'\n    \u00b7 simp [Finset.insert_subset_iff, h, ht]\n    \u00b7 intro x hx h'x\n      simp only [mem_insert, not_or] at h'x \n      exact hle x hx h'x.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u220f j in s, f j < \u220f j in insert i s, f j\n[PROOFSTEP]\nrw [prod_insert hs]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u220f j in s, f j < f i * \u220f x in s, f x\n[PROOFSTEP]\nexact lt_mul_of_one_lt_left' (\u220f j in s, f j) hlt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u220f j in insert i s, f j \u2264 \u220f j in t, f j\n[PROOFSTEP]\napply prod_le_prod_of_subset_of_one_le'\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 insert i s \u2286 t\n[PROOFSTEP]\nsimp [Finset.insert_subset_iff, h, ht]\n[GOAL]\ncase hf\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 t \u2192 \u00aci_1 \u2208 insert i s \u2192 1 \u2264 f i_1\n[PROOFSTEP]\nintro x hx h'x\n[GOAL]\ncase hf\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\nx : \u03b9\nhx : x \u2208 t\nh'x : \u00acx \u2208 insert i s\n\u22a2 1 \u2264 f x\n[PROOFSTEP]\nsimp only [mem_insert, not_or] at h'x \n[GOAL]\ncase hf\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : s \u2286 t\ni : \u03b9\nht : i \u2208 t\nhs : \u00aci \u2208 s\nhlt : 1 < f i\nhle : \u2200 (j : \u03b9), j \u2208 t \u2192 \u00acj \u2208 s \u2192 1 \u2264 f j\nx : \u03b9\nhx : x \u2208 t\nh'x : \u00acx = i \u2227 \u00acx \u2208 s\n\u22a2 1 \u2264 f x\n[PROOFSTEP]\nexact hle x hx h'x.2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 1 < f i\nhs : Finset.Nonempty s\n\u22a2 1 \u2264 \u220f i in s, 1\n[PROOFSTEP]\nrw [prod_const_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < 1\nhs : Finset.Nonempty s\n\u22a2 \u220f i in s, 1 \u2264 1\n[PROOFSTEP]\nrw [prod_const_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns t : Finset \u03b9\nf g : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\n\u22a2 \u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i\n[PROOFSTEP]\nclassical\nrevert h\nrefine\n  Finset.induction_on s (fun _ \u21a6 \u27e8fun _ _ h \u21a6 False.elim (Finset.not_mem_empty _ h), fun _ \u21a6 rfl\u27e9) fun a s ha ih H \u21a6 ?_\nspecialize ih fun i \u21a6 H i \u2218 Finset.mem_insert_of_mem\nrw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, \u2190 ih]\nexact mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a)) (Finset.prod_le_prod' fun i \u21a6 H i \u2218 Finset.mem_insert_of_mem)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns t : Finset \u03b9\nf g : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\n\u22a2 \u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i\n[PROOFSTEP]\nrevert h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns t : Finset \u03b9\nf g : \u03b9 \u2192 M\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 (\u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i)\n[PROOFSTEP]\nrefine\n  Finset.induction_on s (fun _ \u21a6 \u27e8fun _ _ h \u21a6 False.elim (Finset.not_mem_empty _ h), fun _ \u21a6 rfl\u27e9) fun a s ha ih H \u21a6 ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns\u271d t : Finset \u03b9\nf g : \u03b9 \u2192 M\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 (\u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i)\nH : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 \u220f i in insert a s, f i = \u220f i in insert a s, g i \u2194 \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i = g i\n[PROOFSTEP]\nspecialize ih fun i \u21a6 H i \u2218 Finset.mem_insert_of_mem\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns\u271d t : Finset \u03b9\nf g : \u03b9 \u2192 M\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nH : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\nih : \u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i\n\u22a2 \u220f i in insert a s, f i = \u220f i in insert a s, g i \u2194 \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i = g i\n[PROOFSTEP]\nrw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, \u2190 ih]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCancelCommMonoid M\nf\u271d g\u271d : \u03b9 \u2192 M\ns\u271d t : Finset \u03b9\nf g : \u03b9 \u2192 M\na : \u03b9\ns : Finset \u03b9\nha : \u00aca \u2208 s\nH : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\nih : \u220f i in s, f i = \u220f i in s, g i \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i = g i\n\u22a2 f a * \u220f x in s, f x = g a * \u220f x in s, g x \u2194 f a = g a \u2227 \u220f i in s, f i = \u220f i in s, g i\n[PROOFSTEP]\nexact mul_eq_mul_iff_eq_and_eq (H a (s.mem_insert_self a)) (Finset.prod_le_prod' fun i \u21a6 H i \u2218 Finset.mem_insert_of_mem)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHlt : \u220f i in s, f i < \u220f i in s, g i\n\u22a2 \u2203 i, i \u2208 s \u2227 f i < g i\n[PROOFSTEP]\ncontrapose! Hlt with Hle\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nHle : \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2264 f i\n\u22a2 \u220f i in s, g i \u2264 \u220f i in s, f i\n[PROOFSTEP]\nexact prod_le_prod' Hle\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHle : \u220f i in s, f i \u2264 \u220f i in s, g i\n\u22a2 \u2203 i, i \u2208 s \u2227 f i \u2264 g i\n[PROOFSTEP]\ncontrapose! Hle with Hlt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf g : \u03b9 \u2192 M\ns t : Finset \u03b9\nhs : Finset.Nonempty s\nHlt : \u2200 (i : \u03b9), i \u2208 s \u2192 g i < f i\n\u22a2 \u220f i in s, g i < \u220f i in s, f i\n[PROOFSTEP]\nexact prod_lt_prod_of_nonempty' hs Hlt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf\u271d g : \u03b9 \u2192 M\ns t : Finset \u03b9\nf : \u03b9 \u2192 M\nh\u2081 : \u220f i in s, f i = 1\nh\u2082 : \u2203 i, i \u2208 s \u2227 f i \u2260 1\n\u22a2 \u2203 i, i \u2208 s \u2227 1 < f i\n[PROOFSTEP]\ncontrapose! h\u2081\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf\u271d g : \u03b9 \u2192 M\ns t : Finset \u03b9\nf : \u03b9 \u2192 M\nh\u2082 : \u2203 i, i \u2208 s \u2227 f i \u2260 1\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\n\u22a2 \u220f i in s, f i \u2260 1\n[PROOFSTEP]\nobtain \u27e8i, m, i_ne\u27e9 : \u2203 i \u2208 s, f i \u2260 1 := h\u2082\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf\u271d g : \u03b9 \u2192 M\ns t : Finset \u03b9\nf : \u03b9 \u2192 M\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\ni : \u03b9\nm : i \u2208 s\ni_ne : f i \u2260 1\n\u22a2 \u220f i in s, f i \u2260 1\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase intro.intro.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : LinearOrderedCancelCommMonoid M\nf\u271d g : \u03b9 \u2192 M\ns t : Finset \u03b9\nf : \u03b9 \u2192 M\nh\u2081 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\ni : \u03b9\nm : i \u2208 s\ni_ne : f i \u2260 1\n\u22a2 \u220f i in s, f i < 1\n[PROOFSTEP]\ncalc\n  \u220f j in s, f j < \u220f j in s, 1 := prod_lt_prod' h\u2081 \u27e8i, m, (h\u2081 i m).lt_of_ne i_ne\u27e9\n  _ = 1 := prod_const_one\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns t : Finset \u03b9\nh0 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\n\u22a2 \u220f i in s, f i \u2264 \u220f i in s, g i\n[PROOFSTEP]\ninduction' s using Finset.induction with a s has ih h\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i\nh0 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 f i \u2264 g i\n\u22a2 \u220f i in \u2205, f i \u2264 \u220f i in \u2205, g i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 \u220f i in insert a s, f i \u2264 \u220f i in insert a s, g i\n[PROOFSTEP]\nsimp only [prod_insert has]\n[GOAL]\ncase insert\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 f a * \u220f i in s, f i \u2264 g a * \u220f i in s, g i\n[PROOFSTEP]\napply mul_le_mul\n[GOAL]\ncase insert.h\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 f a \u2264 g a\n[PROOFSTEP]\nexact h1 a (mem_insert_self a s)\n[GOAL]\ncase insert.h\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 \u220f i in s, f i \u2264 \u220f i in s, g i\n[PROOFSTEP]\nrefine ih (fun x H \u21a6 h0 _ ?_) (fun x H \u21a6 h1 _ ?_)\n[GOAL]\ncase insert.h\u2082.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\nx : \u03b9\nH : x \u2208 s\n\u22a2 x \u2208 insert a s\n[PROOFSTEP]\nexact mem_insert_of_mem H\n[GOAL]\ncase insert.h\u2082.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\nx : \u03b9\nH : x \u2208 s\n\u22a2 x \u2208 insert a s\n[PROOFSTEP]\nexact mem_insert_of_mem H\n[GOAL]\ncase insert.c0\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 0 \u2264 \u220f i in s, f i\n[PROOFSTEP]\napply prod_nonneg fun x H \u21a6 h0 x (mem_insert_of_mem H)\n[GOAL]\ncase insert.b0\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns\u271d t : Finset \u03b9\nh0\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 0 \u2264 f i\nh1\u271d : \u2200 (i : \u03b9), i \u2208 s\u271d \u2192 f i \u2264 g i\na : \u03b9\ns : Finset \u03b9\nhas : \u00aca \u2208 s\nih : (\u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i) \u2192 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i) \u2192 \u220f i in s, f i \u2264 \u220f i in s, g i\nh0 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 insert a s \u2192 f i \u2264 g i\n\u22a2 0 \u2264 g a\n[PROOFSTEP]\napply le_trans (h0 a (mem_insert_self a s)) (h1 a (mem_insert_self a s))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns t : Finset \u03b9\nh0 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\n\u22a2 \u220f i in s, f i \u2264 1\n[PROOFSTEP]\nconvert \u2190 prod_le_prod h0 h1\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf g : \u03b9 \u2192 R\ns t : Finset \u03b9\nh0 : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nh1 : \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 1\n\u22a2 \u220f i in s, 1 = 1\n[PROOFSTEP]\nexact Finset.prod_const_one\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u220f i in s, g i + \u220f i in s, h i \u2264 \u220f i in s, f i\n[PROOFSTEP]\nsimp_rw [prod_eq_mul_prod_diff_singleton hi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 f i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine le_trans ?_ (mul_le_mul_of_nonneg_right h2i ?_)\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 (g i + h i) * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\ncase refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 g i * \u220f i in s \\ {i}, f i + h i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine add_le_add ?_ ?_\n[GOAL]\ncase refine_1.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 g i * \u220f i in s \\ {i}, g i \u2264 g i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine mul_le_mul_of_nonneg_left ?_ ?_\n[GOAL]\ncase refine_1.refine_1.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u220f i in s \\ {i}, g i \u2264 \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine prod_le_prod ?_ ?_\n[GOAL]\ncase refine_1.refine_1.refine_1.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 0 \u2264 g i_1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [*]\n[GOAL]\ncase refine_1.refine_1.refine_1.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 g i_1 \u2264 f i_1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [*]\n[GOAL]\ncase refine_1.refine_1.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 0 \u2264 g i\n[PROOFSTEP]\ntry apply_assumption\n[GOAL]\ncase refine_1.refine_1.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 0 \u2264 g i\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase refine_1.refine_1.refine_2.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 i \u2208 s\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine_1.refine_1.refine_2.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 i \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine_1.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 h i * \u220f i in s \\ {i}, h i \u2264 h i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine mul_le_mul_of_nonneg_left ?_ ?_\n[GOAL]\ncase refine_1.refine_2.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u220f i in s \\ {i}, h i \u2264 \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine prod_le_prod ?_ ?_\n[GOAL]\ncase refine_1.refine_2.refine_1.refine_1\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 0 \u2264 h i_1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [*]\n[GOAL]\ncase refine_1.refine_2.refine_1.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 h i_1 \u2264 f i_1\n[PROOFSTEP]\nsimp (config := { contextual := true }) [*]\n[GOAL]\ncase refine_1.refine_2.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 0 \u2264 h i\n[PROOFSTEP]\ntry apply_assumption\n[GOAL]\ncase refine_1.refine_2.refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 0 \u2264 h i\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase refine_1.refine_2.refine_2.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 i \u2208 s\n[PROOFSTEP]\ntry assumption\n[GOAL]\ncase refine_1.refine_2.refine_2.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 i \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase refine_2\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 0 \u2264 \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply prod_nonneg\n[GOAL]\ncase refine_2.h0\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 0 \u2264 f i_1\n[PROOFSTEP]\nsimp only [and_imp, mem_sdiff, mem_singleton]\n[GOAL]\ncase refine_2.h0\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \u2192 \u00aci_1 = i \u2192 0 \u2264 f i_1\n[PROOFSTEP]\nintro j h1j h2j\n[GOAL]\ncase refine_2.h0\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommSemiring R\nf\u271d g\u271d : \u03b9 \u2192 R\ns t : Finset \u03b9\ni : \u03b9\nf g h : \u03b9 \u2192 R\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\nhh : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 h i\nj : \u03b9\nh1j : j \u2208 s\nh2j : \u00acj = i\n\u22a2 0 \u2264 f j\n[PROOFSTEP]\nexact le_trans (hg j h1j) (hgf j h1j h2j)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u220f i in s, g i + \u220f i in s, h i \u2264 \u220f i in s, f i\n[PROOFSTEP]\nclassical\nsimp_rw [prod_eq_mul_prod_diff_singleton hi]\nrefine' le_trans _ (mul_le_mul_right' h2i _)\nrw [right_distrib]\napply add_le_add <;> apply mul_le_mul_left' <;> apply prod_le_prod' <;>\n        simp only [and_imp, mem_sdiff, mem_singleton] <;>\n      intros <;>\n    apply_assumption <;>\n  assumption\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u220f i in s, g i + \u220f i in s, h i \u2264 \u220f i in s, f i\n[PROOFSTEP]\nsimp_rw [prod_eq_mul_prod_diff_singleton hi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 f i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrefine' le_trans _ (mul_le_mul_right' h2i _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 (g i + h i) * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\nrw [right_distrib]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 g i * \u220f i in s \\ {i}, g i + h i * \u220f i in s \\ {i}, h i \u2264 g i * \u220f i in s \\ {i}, f i + h i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 g i * \u220f i in s \\ {i}, g i \u2264 g i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply mul_le_mul_left'\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 h i * \u220f i in s \\ {i}, h i \u2264 h i * \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply mul_le_mul_left'\n[GOAL]\ncase h\u2081.bc\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u220f i in s \\ {i}, g i \u2264 \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply prod_le_prod'\n[GOAL]\ncase h\u2082.bc\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u220f i in s \\ {i}, h i \u2264 \u220f i in s \\ {i}, f i\n[PROOFSTEP]\napply prod_le_prod'\n[GOAL]\ncase h\u2081.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 g i_1 \u2264 f i_1\n[PROOFSTEP]\nsimp only [and_imp, mem_sdiff, mem_singleton]\n[GOAL]\ncase h\u2082.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \\ {i} \u2192 h i_1 \u2264 f i_1\n[PROOFSTEP]\nsimp only [and_imp, mem_sdiff, mem_singleton]\n[GOAL]\ncase h\u2081.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \u2192 \u00aci_1 = i \u2192 g i_1 \u2264 f i_1\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2082.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\n\u22a2 \u2200 (i_1 : \u03b9), i_1 \u2208 s \u2192 \u00aci_1 = i \u2192 h i_1 \u2264 f i_1\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 g i\u271d \u2264 f i\u271d\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase h\u2082.bc.h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 h i\u271d \u2264 f i\u271d\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase h\u2081.bc.h.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 i\u271d \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2081.bc.h.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 i\u271d \u2260 i\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2082.bc.h.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 i\u271d \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase h\u2082.bc.h.a\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedCommSemiring R\nf g h : \u03b9 \u2192 R\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nh2i : g i + h i \u2264 f i\nhgf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 g j \u2264 f j\nhhf : \u2200 (j : \u03b9), j \u2208 s \u2192 j \u2260 i \u2192 h j \u2264 f j\ni\u271d : \u03b9\na\u271d\u00b9 : i\u271d \u2208 s\na\u271d : \u00aci\u271d = i\n\u22a2 i\u271d \u2260 i\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : AddCommMonoid M\ns : Finset \u03b9\nf : \u03b9 \u2192 WithTop M\n\u22a2 \u2211 i in s, f i = \u22a4 \u2194 \u2203 i, i \u2208 s \u2227 f i = \u22a4\n[PROOFSTEP]\ninduction s using Finset.cons_induction\n[GOAL]\ncase empty\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : AddCommMonoid M\nf : \u03b9 \u2192 WithTop M\n\u22a2 \u2211 i in \u2205, f i = \u22a4 \u2194 \u2203 i, i \u2208 \u2205 \u2227 f i = \u22a4\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : AddCommMonoid M\nf : \u03b9 \u2192 WithTop M\na\u271d\u00b9 : \u03b9\ns\u271d : Finset \u03b9\nh\u271d : \u00aca\u271d\u00b9 \u2208 s\u271d\na\u271d : \u2211 i in s\u271d, f i = \u22a4 \u2194 \u2203 i, i \u2208 s\u271d \u2227 f i = \u22a4\n\u22a2 \u2211 i in cons a\u271d\u00b9 s\u271d h\u271d, f i = \u22a4 \u2194 \u2203 i, i \u2208 cons a\u271d\u00b9 s\u271d h\u271d \u2227 f i = \u22a4\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : LT M\ns : Finset \u03b9\nf : \u03b9 \u2192 WithTop M\n\u22a2 \u2211 i in s, f i < \u22a4 \u2194 \u2200 (i : \u03b9), i \u2208 s \u2192 f i < \u22a4\n[PROOFSTEP]\nsimp only [WithTop.lt_top_iff_ne_top, ne_eq, sum_eq_top_iff, not_exists, not_and]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Order", "llama_tokens": 34395, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289387914176258, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5340231811717026}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimits C\nX : TopCat\nF : J \u2964 Presheaf C X\nH : \u2200 (j : J), Presheaf.IsSheaf (F.obj j)\nc : Cone F\nhc : IsLimit c\n\u22a2 Presheaf.IsSheaf c.pt\n[PROOFSTEP]\nlet F' : J \u2964 Sheaf C X :=\n  { obj := fun j => \u27e8F.obj j, H j\u27e9\n    map := fun f => \u27e8F.map f\u27e9 }\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimits C\nX : TopCat\nF : J \u2964 Presheaf C X\nH : \u2200 (j : J), Presheaf.IsSheaf (F.obj j)\nc : Cone F\nhc : IsLimit c\nF' : J \u2964 Sheaf C X :=\n  CategoryTheory.Functor.mk\n    { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) },\n      map := fun {X_1 Y} f => { val := F.map f } }\n\u22a2 Presheaf.IsSheaf c.pt\n[PROOFSTEP]\nlet e : F' \u22d9 Sheaf.forget C X \u2245 F := NatIso.ofComponents fun _ => Iso.refl _\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : HasLimits C\nX : TopCat\nF : J \u2964 Presheaf C X\nH : \u2200 (j : J), Presheaf.IsSheaf (F.obj j)\nc : Cone F\nhc : IsLimit c\nF' : J \u2964 Sheaf C X :=\n  CategoryTheory.Functor.mk\n    { obj := fun j => { val := F.obj j, cond := (_ : Presheaf.IsSheaf (F.obj j)) },\n      map := fun {X_1 Y} f => { val := F.map f } }\ne : F' \u22d9 Sheaf.forget C X \u2245 F := NatIso.ofComponents fun x => Iso.refl ((F' \u22d9 Sheaf.forget C X).obj x)\n\u22a2 Presheaf.IsSheaf c.pt\n[PROOFSTEP]\nexact\n  Presheaf.isSheaf_of_iso ((isLimitOfPreserves (Sheaf.forget C X) (limit.isLimit F')).conePointsIsoOfNatIso hc e)\n    (limit F').2\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sheaves.Limits", "llama_tokens": 720, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867729389246, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5338913765080829}}
{"text": "[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\n\u22a2 b \u2022 stdBasisMatrix i j a = stdBasisMatrix i j (b \u2022 a)\n[PROOFSTEP]\nunfold stdBasisMatrix\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\n\u22a2 (b \u2022 fun i' j' => if i = i' \u2227 j = j' then a else 0) = fun i' j' => if i = i' \u2227 j = j' then b \u2022 a else 0\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\ni\u271d : m\nx\u271d : n\n\u22a2 (b \u2022 fun i' j' => if i = i' \u2227 j = j' then a else 0) i\u271d x\u271d = if i = i\u271d \u2227 j = x\u271d then b \u2022 a else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\n\u22a2 stdBasisMatrix i j 0 = 0\n[PROOFSTEP]\nunfold stdBasisMatrix\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\n\u22a2 (fun i' j' => if i = i' \u2227 j = j' then 0 else 0) = 0\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\ni\u271d : m\nx\u271d : n\n\u22a2 (if i = i\u271d \u2227 j = x\u271d then 0 else 0) = OfNat.ofNat 0 i\u271d x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\n\u22a2 stdBasisMatrix i j (a + b) = stdBasisMatrix i j a + stdBasisMatrix i j b\n[PROOFSTEP]\nunfold stdBasisMatrix\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\n\u22a2 (fun i' j' => if i = i' \u2227 j = j' then a + b else 0) =\n    (fun i' j' => if i = i' \u2227 j = j' then a else 0) + fun i' j' => if i = i' \u2227 j = j' then b else 0\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\ni\u271d : m\nx\u271d : n\n\u22a2 (if i = i\u271d \u2227 j = x\u271d then a + b else 0) =\n    ((fun i' j' => if i = i' \u2227 j = j' then a else 0) + fun i' j' => if i = i' \u2227 j = j' then b else 0) i\u271d x\u271d\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\ni\u271d : m\nx\u271d : n\nh : i = i\u271d \u2227 j = x\u271d\n\u22a2 a + b = ((fun i' j' => if i = i' \u2227 j = j' then a else 0) + fun i' j' => if i = i' \u2227 j = j' then b else 0) i\u271d x\u271d\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\na b : \u03b1\ni\u271d : m\nx\u271d : n\nh : \u00ac(i = i\u271d \u2227 j = x\u271d)\n\u22a2 0 = ((fun i' j' => if i = i' \u2227 j = j' then a else 0) + fun i' j' => if i = i' \u2227 j = j' then b else 0) i\u271d x\u271d\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\n\u22a2 x = \u2211 i : m, \u2211 j : n, stdBasisMatrix i j (x i j)\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 x i j = Finset.sum Finset.univ (fun i => \u2211 j : n, stdBasisMatrix i j (x i j)) i j\n[PROOFSTEP]\nsymm\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun i => \u2211 j : n, stdBasisMatrix i j (x i j)) i j = x i j\n[PROOFSTEP]\niterate 2\n  rw [Finset.sum_apply]\n    -- Porting note: was `convert`\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun i => \u2211 j : n, stdBasisMatrix i j (x i j)) i j = x i j\n[PROOFSTEP]\nrw [Finset.sum_apply]\n  -- Porting note: was `convert`\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun c => Finset.sum Finset.univ (fun j => stdBasisMatrix c j (x c j)) i) j = x i j\n[PROOFSTEP]\nrw [Finset.sum_apply]\n  -- Porting note: was `convert`\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 \u2211 c : m, Finset.sum Finset.univ (fun j => stdBasisMatrix c j (x c j)) i j = x i j\n[PROOFSTEP]\nrefine (Fintype.sum_eq_single i ?_).trans ?_\n[GOAL]\ncase a.h.refine_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 \u2200 (x_1 : m), x_1 \u2260 i \u2192 Finset.sum Finset.univ (fun j => stdBasisMatrix x_1 j (x x_1 j)) i j = 0\ncase a.h.refine_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun j => stdBasisMatrix i j (x i j)) i j = x i j\n[PROOFSTEP]\nswap\n[GOAL]\ncase a.h.refine_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun j => stdBasisMatrix i j (x i j)) i j = x i j\n[PROOFSTEP]\nsimp only [stdBasisMatrix]\n[GOAL]\ncase a.h.refine_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 Finset.sum Finset.univ (fun x_1 i' j' => if i = i' \u2227 x_1 = j' then x i x_1 else 0) i j = x i j\n[PROOFSTEP]\nrw [Fintype.sum_apply, Fintype.sum_apply]\n[GOAL]\ncase a.h.refine_2\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 (\u2211 c : n, if i = i \u2227 c = j then x i c else 0) = x i j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.refine_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\n\u22a2 \u2200 (x_1 : m), x_1 \u2260 i \u2192 Finset.sum Finset.univ (fun j => stdBasisMatrix x_1 j (x x_1 j)) i j = 0\n[PROOFSTEP]\nintro j' hj'\n[GOAL]\ncase a.h.refine_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun j => stdBasisMatrix j' j (x j' j)) i j = 0\n[PROOFSTEP]\nsimp only [stdBasisMatrix]\n[GOAL]\ncase a.h.refine_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 Finset.sum Finset.univ (fun x_1 i' j'_1 => if j' = i' \u2227 x_1 = j'_1 then x j' x_1 else 0) i j = 0\n[PROOFSTEP]\nrw [Fintype.sum_apply, Fintype.sum_apply]\n[GOAL]\ncase a.h.refine_1\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nx : Matrix m n \u03b1\ni : m\nj : n\nj' : m\nhj' : j' \u2260 i\n\u22a2 (\u2211 c : n, if j' = i \u2227 c = j then x j' c else 0) = 0\n[PROOFSTEP]\nsimp [hj']\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\n\u22a2 stdBasisMatrix i j 1 = vecMulVec (fun i' => if i = i' then 1 else 0) fun j' => if j = j' then 1 else 0\n[PROOFSTEP]\next i' j'\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\ni' : m\nj' : n\n\u22a2 stdBasisMatrix i j 1 i' j' = vecMulVec (fun i' => if i = i' then 1 else 0) (fun j' => if j = j' then 1 else 0) i' j'\n[PROOFSTEP]\nsimp only [stdBasisMatrix, vecMulVec, mul_ite, mul_one, mul_zero, of_apply]\n  -- Porting note: added next line\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\ni' : m\nj' : n\n\u22a2 (if i = i' \u2227 j = j' then 1 else 0) = if j = j' then if i = i' then 1 else 0 else 0\n[PROOFSTEP]\nsimp_rw [@and_comm (i = i')]\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\ni' : m\nj' : n\n\u22a2 (if j = j' \u2227 i = i' then 1 else 0) = if j = j' then if i = i' then 1 else 0 else 0\n[PROOFSTEP]\nexact ite_and _ _ _ _\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_zero : P 0\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\n\u22a2 P M\n[PROOFSTEP]\nrw [matrix_eq_sum_std_basis M, \u2190 Finset.sum_product']\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_zero : P 0\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\n\u22a2 P (\u2211 x in Finset.univ \u00d7\u02e2 Finset.univ, stdBasisMatrix x.fst x.snd (M x.fst x.snd))\n[PROOFSTEP]\napply Finset.sum_induction _ _ h_add h_zero\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_zero : P 0\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\n\u22a2 \u2200 (x : m \u00d7 n), x \u2208 Finset.univ \u00d7\u02e2 Finset.univ \u2192 P (stdBasisMatrix x.fst x.snd (M x.fst x.snd))\n[PROOFSTEP]\nintros\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2075 : DecidableEq l\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : Fintype n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_zero : P 0\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\nx\u271d : m \u00d7 n\na\u271d : x\u271d \u2208 Finset.univ \u00d7\u02e2 Finset.univ\n\u22a2 P (stdBasisMatrix x\u271d.fst x\u271d.snd (M x\u271d.fst x\u271d.snd))\n[PROOFSTEP]\napply h_std_basis\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2077 : DecidableEq l\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Semiring \u03b1\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Nonempty m\ninst\u271d : Nonempty n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\n\u22a2 P 0\n[PROOFSTEP]\ninhabit m\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2077 : DecidableEq l\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Semiring \u03b1\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Nonempty m\ninst\u271d : Nonempty n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\ninhabited_h : Inhabited m\n\u22a2 P 0\n[PROOFSTEP]\ninhabit n\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2077 : DecidableEq l\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Semiring \u03b1\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : Nonempty m\ninst\u271d : Nonempty n\nP : Matrix m n \u03b1 \u2192 Prop\nM : Matrix m n \u03b1\nh_add : \u2200 (p q : Matrix m n \u03b1), P p \u2192 P q \u2192 P (p + q)\nh_std_basis : \u2200 (i : m) (j : n) (x : \u03b1), P (stdBasisMatrix i j x)\ninhabited_h\u271d : Inhabited m\ninhabited_h : Inhabited n\n\u22a2 P 0\n[PROOFSTEP]\nsimpa using h_std_basis default default 0\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\nc : \u03b1\ni' : m\nj' : n\nh : \u00ac(i = i' \u2227 j = j')\n\u22a2 stdBasisMatrix i j c i' j' = 0\n[PROOFSTEP]\nsimp only [stdBasisMatrix, and_imp, ite_eq_right_iff]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni : m\nj : n\nc : \u03b1\ni' : m\nj' : n\nh : \u00ac(i = i' \u2227 j = j')\n\u22a2 i = i' \u2192 j = j' \u2192 c = 0\n[PROOFSTEP]\ntauto\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni\u271d : m\nj\u271d : n\nc : \u03b1\ni'\u271d : m\nj'\u271d : n\ni i' : m\nhi : i \u2260 i'\nj j' : n\na : \u03b1\n\u22a2 stdBasisMatrix i j a i' j' = 0\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni\u271d : m\nj\u271d : n\nc : \u03b1\ni'\u271d : m\nj'\u271d : n\ni i' : m\nj j' : n\nhj : j \u2260 j'\na : \u03b1\n\u22a2 stdBasisMatrix i j a i' j' = 0\n[PROOFSTEP]\nsimp [hj]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\n\u22a2 diag (stdBasisMatrix i i c) = Pi.single i c\n[PROOFSTEP]\next j\n[GOAL]\ncase h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\nby_cases hij : i = j\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : i = j\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\ntry rw [hij]\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : i = j\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\nrw [hij]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : \u00aci = j\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\ntry rw [hij]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : \u00aci = j\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\nrw [hij]\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : i = j\n\u22a2 diag (stdBasisMatrix j j c) j = Pi.single j c j\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u00b3 : DecidableEq l\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Semiring \u03b1\ni j\u271d : n\nc : \u03b1\ni' j' j : n\nhij : \u00aci = j\n\u22a2 diag (stdBasisMatrix i i c) j = Pi.single i c j\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nh : j \u2260 i\n\u22a2 trace (stdBasisMatrix i j c) = 0\n[PROOFSTEP]\nsimp [trace, -diag_apply, h]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\n\u22a2 trace (stdBasisMatrix i i c) = c\n[PROOFSTEP]\nsimp [trace, -diag_apply]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nb : n\nM : Matrix n n \u03b1\n\u22a2 (stdBasisMatrix i j c * M) i b = c * M j b\n[PROOFSTEP]\nsimp [mul_apply, stdBasisMatrix]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\na : n\nM : Matrix n n \u03b1\n\u22a2 (M * stdBasisMatrix i j c) a j = M a i * c\n[PROOFSTEP]\nsimp [mul_apply, stdBasisMatrix, mul_comm]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\na b : n\nh : a \u2260 i\nM : Matrix n n \u03b1\n\u22a2 (stdBasisMatrix i j c * M) a b = 0\n[PROOFSTEP]\nsimp [mul_apply, h.symm]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\na b : n\nhbj : b \u2260 j\nM : Matrix n n \u03b1\n\u22a2 (M * stdBasisMatrix i j c) a b = 0\n[PROOFSTEP]\nsimp [mul_apply, hbj.symm]\n[GOAL]\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\n\u22a2 stdBasisMatrix i j c * stdBasisMatrix j k d = stdBasisMatrix i k (c * d)\n[PROOFSTEP]\next a b\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\n\u22a2 (stdBasisMatrix i j c * stdBasisMatrix j k d) a b = stdBasisMatrix i k (c * d) a b\n[PROOFSTEP]\nsimp only [mul_apply, stdBasisMatrix, boole_mul]\n[GOAL]\ncase a.h\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nby_cases h\u2081 : i = a\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : i = a\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nby_cases h\u2082 : k = b\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : \u00aci = a\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nby_cases h\u2082 : k = b\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : i = a\nh\u2082 : k = b\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : i = a\nh\u2082 : \u00ack = b\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase pos\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : \u00aci = a\nh\u2082 : k = b\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\ncase neg\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk : n\nd : \u03b1\na b : n\nh\u2081 : \u00aci = a\nh\u2082 : \u00ack = b\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if j = j_1 \u2227 k = b then d else 0) =\n    if i = a \u2227 k = b then c * d else 0\n[PROOFSTEP]\nsimp [h\u2081, h\u2082]\n[GOAL]\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\n\u22a2 stdBasisMatrix i j c * stdBasisMatrix k l d = 0\n[PROOFSTEP]\next a b\n[GOAL]\ncase a.h\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\n\u22a2 (stdBasisMatrix i j c * stdBasisMatrix k l d) a b = OfNat.ofNat 0 a b\n[PROOFSTEP]\nsimp only [mul_apply, boole_mul, stdBasisMatrix]\n[GOAL]\ncase a.h\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if k = j_1 \u2227 l = b then d else 0) = OfNat.ofNat 0 a b\n[PROOFSTEP]\nby_cases h\u2081 : i = a\n[GOAL]\ncase pos\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\nh\u2081 : i = a\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if k = j_1 \u2227 l = b then d else 0) = OfNat.ofNat 0 a b\n[PROOFSTEP]\nsimp only [h\u2081, true_and, mul_ite, ite_mul, zero_mul, mul_zero, \u2190 ite_and, zero_apply]\n[GOAL]\ncase pos\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\nh\u2081 : i = a\n\u22a2 (\u2211 x : n, if (k = x \u2227 l = b) \u2227 j = x then c * d else 0) = 0\n[PROOFSTEP]\nrefine Finset.sum_eq_zero (fun x _ => ?_)\n[GOAL]\ncase pos\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\nh\u2081 : i = a\nx : n\nx\u271d : x \u2208 Finset.univ\n\u22a2 (if (k = x \u2227 l = b) \u2227 j = x then c * d else 0) = 0\n[PROOFSTEP]\napply if_neg\n[GOAL]\ncase pos.hnc\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\nh\u2081 : i = a\nx : n\nx\u271d : x \u2208 Finset.univ\n\u22a2 \u00ac((k = x \u2227 l = b) \u2227 j = x)\n[PROOFSTEP]\nrintro \u27e8\u27e8rfl, rfl\u27e9, h\u27e9\n[GOAL]\ncase pos.hnc.intro.intro\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh\u271d : j \u2260 k\nd : \u03b1\na : n\nh\u2081 : i = a\nx\u271d : k \u2208 Finset.univ\nh : j = k\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nl\u271d : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\n\u03b1 : Type u_5\ninst\u271d\u2074 : DecidableEq l\u271d\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Semiring \u03b1\ni j : n\nc : \u03b1\ni' j' : n\ninst\u271d : Fintype n\nk l : n\nh : j \u2260 k\nd : \u03b1\na b : n\nh\u2081 : \u00aci = a\n\u22a2 (\u2211 j_1 : n, (if i = a \u2227 j = j_1 then c else 0) * if k = j_1 \u2227 l = b then d else 0) = OfNat.ofNat 0 a b\n[PROOFSTEP]\nsimp only [h\u2081, false_and, ite_false, mul_ite, zero_mul, mul_zero, ite_self, Finset.sum_const_zero, zero_apply]\n", "meta": {"mathlib_filename": "Mathlib.Data.Matrix.Basis", "llama_tokens": 13753, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.810478926981208, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.5336335093608818}}
{"text": "[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne\u2081 e\u2082 : \u03b1 \u2243 \u03b2\nh\u2081 : e\u2081.toFun = e\u2082.toFun\nh\u2082 : e\u2081.invFun = e\u2082.invFun\n\u22a2 e\u2081 = e\u2082\n[PROOFSTEP]\ncases e\u2081\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne\u2082 : \u03b1 \u2243 \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninvFun\u271d : \u03b2 \u2192 \u03b1\nleft_inv\u271d : LeftInverse invFun\u271d toFun\u271d\nright_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d\nh\u2081 : { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.toFun = e\u2082.toFun\nh\u2082 : { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.invFun = e\u2082.invFun\n\u22a2 { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } = e\u2082\n[PROOFSTEP]\ncases e\u2082\n[GOAL]\ncase mk.mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\ninvFun\u271d\u00b9 : \u03b2 \u2192 \u03b1\nleft_inv\u271d\u00b9 : LeftInverse invFun\u271d\u00b9 toFun\u271d\u00b9\nright_inv\u271d\u00b9 : Function.RightInverse invFun\u271d\u00b9 toFun\u271d\u00b9\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninvFun\u271d : \u03b2 \u2192 \u03b1\nleft_inv\u271d : LeftInverse invFun\u271d toFun\u271d\nright_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d\nh\u2081 :\n  { toFun := toFun\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 }.toFun =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.toFun\nh\u2082 :\n  { toFun := toFun\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 }.invFun =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.invFun\n\u22a2 { toFun := toFun\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 \u03b2 : Sort ?u.16583\nh : \u03b1 = \u03b2\nx\u271d : \u03b1\n\u22a2 cast (_ : \u03b2 = \u03b1) (cast h x\u271d) = x\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1\u271d : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort ?u.16583\nx\u271d : \u03b1\n\u22a2 cast (_ : \u03b1 = \u03b1) (cast (_ : \u03b1 = \u03b1) x\u271d) = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 \u03b2 : Sort ?u.16583\nh : \u03b1 = \u03b2\nx\u271d : \u03b2\n\u22a2 cast h (cast (_ : \u03b2 = \u03b1) x\u271d) = x\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b1\u271d : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort ?u.16583\nx\u271d : \u03b1\n\u22a2 cast (_ : \u03b1 = \u03b1) (cast (_ : \u03b1 = \u03b1) x\u271d) = x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1 : Type u_1\ninst\u271d : Subsingleton \u03b1\ne : Perm \u03b1\n\u22a2 \u2191e = id\n[PROOFSTEP]\nrw [Perm.subsingleton_eq_refl e, coe_refl]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nx : \u03b1\ny : (fun x => \u03b2) x\nf : \u03b1 \u2243 \u03b2\n\u22a2 \u2191f x = y \u2194 x = \u2191f.symm y\n[PROOFSTEP]\nconv_lhs => rw [\u2190 apply_symm_apply f y]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nx : \u03b1\ny : (fun x => \u03b2) x\nf : \u03b1 \u2243 \u03b2\n| \u2191f x = y\n[PROOFSTEP]\nrw [\u2190 apply_symm_apply f y]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nx : \u03b1\ny : (fun x => \u03b2) x\nf : \u03b1 \u2243 \u03b2\n| \u2191f x = y\n[PROOFSTEP]\nrw [\u2190 apply_symm_apply f y]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nx : \u03b1\ny : (fun x => \u03b2) x\nf : \u03b1 \u2243 \u03b2\n| \u2191f x = y\n[PROOFSTEP]\nrw [\u2190 apply_symm_apply f y]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nx : \u03b1\ny : (fun x => \u03b2) x\nf : \u03b1 \u2243 \u03b2\n\u22a2 \u2191f x = \u2191f (\u2191f.symm y) \u2194 x = \u2191f.symm y\n[PROOFSTEP]\nrw [apply_eq_iff_eq]\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3\u271d : Sort w\n\u03b1 \u03b2 \u03b3 : Sort u_1\nh : \u03b1 = \u03b2\nh2 : \u03b2 = \u03b3\nx : \u03b1\n\u22a2 \u2191((Equiv.cast h).trans (Equiv.cast h2)) x = \u2191(Equiv.cast (_ : \u03b1 = \u03b3)) x\n[PROOFSTEP]\nsubsts h h2\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\nx : \u03b1\n\u22a2 \u2191((Equiv.cast (_ : \u03b1 = \u03b1)).trans (Equiv.cast (_ : \u03b1 = \u03b1))) x = \u2191(Equiv.cast (_ : \u03b1 = \u03b1)) x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 \u03b2 : Sort u_1\nh : \u03b1 = \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 \u2191(Equiv.cast h) a = b \u2194 HEq a b\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\na b : \u03b1\n\u22a2 \u2191(Equiv.cast (_ : \u03b1 = \u03b1)) a = b \u2194 HEq a b\n[PROOFSTEP]\nsimp [coe_refl]\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx : \u03b2\ny : (fun x => \u03b1) x\nH : \u2191e.symm x = y\n\u22a2 x = \u2191e y\n[PROOFSTEP]\nsimp [H.symm]\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\ne : \u03b1 \u2243 \u03b2\nx : \u03b2\ny : (fun x => \u03b1) x\nH : x = \u2191e y\n\u22a2 \u2191e.symm x = y\n[PROOFSTEP]\nsimp [H]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\n\u22a2 e.symm.symm = e\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninvFun\u271d : \u03b2 \u2192 \u03b1\nleft_inv\u271d : LeftInverse invFun\u271d toFun\u271d\nright_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d\n\u22a2 { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.symm.symm =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\n\u22a2 e.trans (Equiv.refl \u03b2) = e\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninvFun\u271d : \u03b2 \u2192 \u03b1\nleft_inv\u271d : LeftInverse invFun\u271d toFun\u271d\nright_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d\n\u22a2 { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }.trans (Equiv.refl \u03b2) =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\n\u22a2 (Equiv.refl \u03b1).trans e = e\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ntoFun\u271d : \u03b1 \u2192 \u03b2\ninvFun\u271d : \u03b2 \u2192 \u03b1\nleft_inv\u271d : LeftInverse invFun\u271d toFun\u271d\nright_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d\n\u22a2 (Equiv.refl \u03b1).trans { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } =\n    { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\n\u22a2 \u2200 (x : \u03b2), \u2191(e.symm.trans e) x = \u2191(Equiv.refl \u03b2) x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\n\u22a2 \u2200 (x : \u03b1), \u2191(e.trans e.symm) x = \u2191(Equiv.refl \u03b1) x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort ?u.23940\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\nac : \u03b1 \u2243 \u03b3\n\u22a2 (fun bd => ab.trans (bd.trans cd.symm)) ((fun ac => (ab.symm.trans ac).trans cd) ac) = ac\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort ?u.23940\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\nac : \u03b1 \u2243 \u03b3\nx : \u03b1\n\u22a2 \u2191((fun bd => ab.trans (bd.trans cd.symm)) ((fun ac => (ab.symm.trans ac).trans cd) ac)) x = \u2191ac x\n[PROOFSTEP]\nsimp only [trans_apply, comp_apply, symm_apply_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort ?u.23940\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\nac : \u03b2 \u2243 \u03b4\n\u22a2 (fun ac => (ab.symm.trans ac).trans cd) ((fun bd => ab.trans (bd.trans cd.symm)) ac) = ac\n[PROOFSTEP]\next x\n[GOAL]\ncase H\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort ?u.23940\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\nac : \u03b2 \u2243 \u03b4\nx : \u03b2\n\u22a2 \u2191((fun ac => (ab.symm.trans ac).trans cd) ((fun bd => ab.trans (bd.trans cd.symm)) ac)) x = \u2191ac x\n[PROOFSTEP]\nsimp only [trans_apply, comp_apply, apply_symm_apply]\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u22a2 equivCongr (Equiv.refl \u03b1) (Equiv.refl \u03b2) = Equiv.refl (\u03b1 \u2243 \u03b2)\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\nx\u271d\u00b9 : \u03b1 \u2243 \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(\u2191(equivCongr (Equiv.refl \u03b1) (Equiv.refl \u03b2)) x\u271d\u00b9) x\u271d = \u2191(\u2191(Equiv.refl (\u03b1 \u2243 \u03b2)) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort u_1\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\n\u22a2 (equivCongr ab cd).symm = equivCongr ab.symm cd.symm\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort u_1\nab : \u03b1 \u2243 \u03b2\ncd : \u03b3 \u2243 \u03b4\nx\u271d\u00b9 : \u03b2 \u2243 \u03b4\nx\u271d : \u03b1\n\u22a2 \u2191(\u2191(equivCongr ab cd).symm x\u271d\u00b9) x\u271d = \u2191(\u2191(equivCongr ab.symm cd.symm) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort u_1\n\u03b5 : Sort u_2\n\u03b6 : Sort u_3\nab : \u03b1 \u2243 \u03b2\nde : \u03b4 \u2243 \u03b5\nbc : \u03b2 \u2243 \u03b3\nef : \u03b5 \u2243 \u03b6\n\u22a2 (equivCongr ab de).trans (equivCongr bc ef) = equivCongr (ab.trans bc) (de.trans ef)\n[PROOFSTEP]\next\n[GOAL]\ncase H.H\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b4 : Sort u_1\n\u03b5 : Sort u_2\n\u03b6 : Sort u_3\nab : \u03b1 \u2243 \u03b2\nde : \u03b4 \u2243 \u03b5\nbc : \u03b2 \u2243 \u03b3\nef : \u03b5 \u2243 \u03b6\nx\u271d\u00b9 : \u03b1 \u2243 \u03b4\nx\u271d : \u03b3\n\u22a2 \u2191(\u2191((equivCongr ab de).trans (equivCongr bc ef)) x\u271d\u00b9) x\u271d = \u2191(\u2191(equivCongr (ab.trans bc) (de.trans ef)) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1' : Type u_1\n\u03b2' : Type u_2\ne : \u03b1' \u2243 \u03b2'\n\u22a2 \u2191(permCongr e) (Equiv.refl \u03b1') = Equiv.refl \u03b2'\n[PROOFSTEP]\nsimp [permCongr_def]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1' : Type u_1\n\u03b2' : Type u_2\ne : \u03b1' \u2243 \u03b2'\np p' : Perm \u03b1'\n\u22a2 (\u2191(permCongr e) p).trans (\u2191(permCongr e) p') = \u2191(permCongr e) (p.trans p')\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1' : Type u_1\n\u03b2' : Type u_2\ne : \u03b1' \u2243 \u03b2'\np p' : Perm \u03b1'\nx\u271d : \u03b2'\n\u22a2 \u2191((\u2191(permCongr e) p).trans (\u2191(permCongr e) p')) x\u271d = \u2191(\u2191(permCongr e) (p.trans p')) x\u271d\n[PROOFSTEP]\nsimp only [trans_apply, comp_apply, permCongr_apply, symm_apply_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b1\u2082 : Sort u_3\n\u03b2\u2082 : Sort u_4\ne\u2081 : \u03b1\u2081 \u2243 \u03b1\u2082\ne\u2082 : \u03b2\u2081 \u2243 \u03b2\u2082\nf : \u03b1\u2081 \u2192 \u03b2\u2081\nx : \u03b1\u2081\n\u22a2 (fun f => \u2191e\u2082.symm \u2218 f \u2218 \u2191e\u2081) ((fun f => \u2191e\u2082 \u2218 f \u2218 \u2191e\u2081.symm) f) x = f x\n[PROOFSTEP]\nsimp only [comp_apply, symm_apply_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b1\u2082 : Sort u_3\n\u03b2\u2082 : Sort u_4\ne\u2081 : \u03b1\u2081 \u2243 \u03b1\u2082\ne\u2082 : \u03b2\u2081 \u2243 \u03b2\u2082\nf : \u03b1\u2082 \u2192 \u03b2\u2082\nx : \u03b1\u2082\n\u22a2 (fun f => \u2191e\u2082 \u2218 f \u2218 \u2191e\u2081.symm) ((fun f => \u2191e\u2082.symm \u2218 f \u2218 \u2191e\u2081) f) x = f x\n[PROOFSTEP]\nsimp only [comp_apply, apply_symm_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b3\u2081 : Sort u_3\n\u03b1\u2082 : Sort u_4\n\u03b2\u2082 : Sort u_5\n\u03b3\u2082 : Sort u_6\nea : \u03b1\u2081 \u2243 \u03b1\u2082\neb : \u03b2\u2081 \u2243 \u03b2\u2082\nec : \u03b3\u2081 \u2243 \u03b3\u2082\nf : \u03b1\u2081 \u2192 \u03b2\u2081\ng : \u03b2\u2081 \u2192 \u03b3\u2081\n\u22a2 \u2191(arrowCongr ea ec) (g \u2218 f) = \u2191(arrowCongr eb ec) g \u2218 \u2191(arrowCongr ea eb) f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u03b1\u2081 : Sort u_1\n\u03b2\u2081 : Sort u_2\n\u03b3\u2081 : Sort u_3\n\u03b1\u2082 : Sort u_4\n\u03b2\u2082 : Sort u_5\n\u03b3\u2082 : Sort u_6\nea : \u03b1\u2081 \u2243 \u03b1\u2082\neb : \u03b2\u2081 \u2243 \u03b2\u2082\nec : \u03b3\u2081 \u2243 \u03b3\u2082\nf : \u03b1\u2081 \u2192 \u03b2\u2081\ng : \u03b2\u2081 \u2192 \u03b3\u2081\nx\u271d : \u03b1\u2082\n\u22a2 \u2191(arrowCongr ea ec) (g \u2218 f) x\u271d = (\u2191(arrowCongr eb ec) g \u2218 \u2191(arrowCongr ea eb) f) x\u271d\n[PROOFSTEP]\nsimp only [comp, arrowCongr_apply, eb.symm_apply_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\ne : \u03b1 \u2243 \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b1\n\u22a2 \u2191(conj e) (f\u2081 \u2218 f\u2082) = \u2191(conj e) f\u2081 \u2218 \u2191(conj e) f\u2082\n[PROOFSTEP]\napply arrowCongr_comp\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : Prop\n\u22a2 (fun b => b = true) ((fun p => decide p) p) = p\n[PROOFSTEP]\nsimp [@Bool.decide_iff p (Classical.propDecidable _)]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nb : Bool\n\u22a2 (fun p => decide p) ((fun b => b = true) b) = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u22a2 (fun p => decide p) ((fun b => b = true) false) = false\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\n\u22a2 (fun p => decide p) ((fun b => b = true) true) = true\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b2 : \u03b1 \u2192 Sort u_1\ninst\u271d : Subsingleton \u03b1\na : \u03b1\nx\u271d : (a' : \u03b1) \u2192 \u03b2 a'\nb : \u03b1\n\u22a2 (fun x b => cast (_ : \u03b2 a = \u03b2 b) x) (eval a x\u271d) b = x\u271d b\n[PROOFSTEP]\nrw [Subsingleton.elim b a]\n[GOAL]\n\u03b1 : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b2 : \u03b1 \u2192 Sort u_1\ninst\u271d : Subsingleton \u03b1\na : \u03b1\nx\u271d : (a' : \u03b1) \u2192 \u03b2 a'\nb : \u03b1\n\u22a2 (fun x b => cast (_ : \u03b2 a = \u03b2 b) x) (eval a x\u271d) a = x\u271d a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2203! x, p x) \u2194 \u2203! y, q y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2203! x, p x) \u2192 \u2203! y, q y\n[PROOFSTEP]\nrintro \u27e8a, ha\u2081, ha\u2082\u27e9\n[GOAL]\ncase mp.intro.intro\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\na : \u03b1\nha\u2081 : p a\nha\u2082 : \u2200 (y : \u03b1), (fun x => p x) y \u2192 y = a\n\u22a2 \u2203! y, q y\n[PROOFSTEP]\nexact \u27e8f a, h.1 ha\u2081, fun b hb => f.symm_apply_eq.1 (ha\u2082 (f.symm b) (h.2 (by simpa using hb)))\u27e9\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\na : \u03b1\nha\u2081 : p a\nha\u2082 : \u2200 (y : \u03b1), (fun x => p x) y \u2192 y = a\nb : \u03b2\nhb : (fun y => q y) b\n\u22a2 q (\u2191f (\u2191f.symm b))\n[PROOFSTEP]\nsimpa using hb\n[GOAL]\ncase mpr\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2203! y, q y) \u2192 \u2203! x, p x\n[PROOFSTEP]\nrintro \u27e8b, hb\u2081, hb\u2082\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nb : \u03b2\nhb\u2081 : q b\nhb\u2082 : \u2200 (y : \u03b2), (fun y => q y) y \u2192 y = b\n\u22a2 \u2203! x, p x\n[PROOFSTEP]\nexact \u27e8f.symm b, h.2 (by simpa using hb\u2081), fun y hy => (eq_symm_apply f).2 (hb\u2082 _ (h.1 hy))\u27e9\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nb : \u03b2\nhb\u2081 : q b\nhb\u2082 : \u2200 (y : \u03b2), (fun y => q y) y \u2192 y = b\n\u22a2 q (\u2191f (\u2191f.symm b))\n[PROOFSTEP]\nsimpa using hb\u2081\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nx\u271d : \u03b1\n\u22a2 p x\u271d \u2194 p (\u2191f.symm (\u2191f x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2200 (x : \u03b1), p x) \u2194 \u2200 (y : \u03b2), q y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2200 (x : \u03b1), p x) \u2192 \u2200 (y : \u03b2), q y\n[PROOFSTEP]\nintro h\u2082 x\n[GOAL]\ncase mpr\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\n\u22a2 (\u2200 (y : \u03b2), q y) \u2192 \u2200 (x : \u03b1), p x\n[PROOFSTEP]\nintro h\u2082 x\n[GOAL]\ncase mp\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), p x\nx : \u03b2\n\u22a2 q x\n[PROOFSTEP]\nrw [\u2190 f.right_inv x]\n[GOAL]\ncase mp\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), p x\nx : \u03b2\n\u22a2 q (toFun f (invFun f x))\n[PROOFSTEP]\napply h.mp\n[GOAL]\ncase mp\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nh\u2082 : \u2200 (x : \u03b1), p x\nx : \u03b2\n\u22a2 p (invFun f x)\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase mpr\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nh\u2082 : \u2200 (y : \u03b2), q y\nx : \u03b1\n\u22a2 p x\n[PROOFSTEP]\napply h.mpr\n[GOAL]\ncase mpr\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\nh : \u2200 {x : \u03b1}, p x \u2194 q (\u2191f x)\nh\u2082 : \u2200 (y : \u03b2), q y\nx : \u03b1\n\u22a2 q (\u2191f x)\n[PROOFSTEP]\napply h\u2082\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nf : \u03b1 \u2243 \u03b2\n\u22a2 \u2200 {x : \u03b1}, p x \u2194 p (\u2191f.symm (\u2191f x))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\nf : \u03b1 \u2243 \u03b2\np : \u03b1 \u2192 Prop\nx\u271d : \u2203 a, p a\na : \u03b1\nh : p a\n\u22a2 p (\u2191f.symm (\u2191f a))\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nra : \u03b1 \u2192 \u03b1 \u2192 Prop\nrb : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\neq : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2194 rb (\u2191e a\u2081) (\u2191e a\u2082)\n\u22a2 LeftInverse (Quot.map \u2191e.symm (_ : \u2200 (b\u2081 b\u2082 : \u03b2), rb b\u2081 b\u2082 \u2192 ra (\u2191e.symm b\u2081) (\u2191e.symm b\u2082)))\n    (Quot.map \u2191e (_ : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2192 rb (\u2191e a\u2081) (\u2191e a\u2082)))\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nra : \u03b1 \u2192 \u03b1 \u2192 Prop\nrb : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\neq : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2194 rb (\u2191e a\u2081) (\u2191e a\u2082)\nx\u271d : Quot ra\na : \u03b1\n\u22a2 Quot.map \u2191e.symm (_ : \u2200 (b\u2081 b\u2082 : \u03b2), rb b\u2081 b\u2082 \u2192 ra (\u2191e.symm b\u2081) (\u2191e.symm b\u2082))\n      (Quot.map \u2191e (_ : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2192 rb (\u2191e a\u2081) (\u2191e a\u2082)) (mk ra a)) =\n    mk ra a\n[PROOFSTEP]\nsimp only [Quot.map, Equiv.symm_apply_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nra : \u03b1 \u2192 \u03b1 \u2192 Prop\nrb : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\neq : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2194 rb (\u2191e a\u2081) (\u2191e a\u2082)\n\u22a2 Function.RightInverse (Quot.map \u2191e.symm (_ : \u2200 (b\u2081 b\u2082 : \u03b2), rb b\u2081 b\u2082 \u2192 ra (\u2191e.symm b\u2081) (\u2191e.symm b\u2082)))\n    (Quot.map \u2191e (_ : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2192 rb (\u2191e a\u2081) (\u2191e a\u2082)))\n[PROOFSTEP]\nrintro \u27e8a\u27e9\n[GOAL]\ncase mk\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nra : \u03b1 \u2192 \u03b1 \u2192 Prop\nrb : \u03b2 \u2192 \u03b2 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\neq : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2194 rb (\u2191e a\u2081) (\u2191e a\u2082)\nx\u271d : Quot rb\na : \u03b2\n\u22a2 Quot.map \u2191e (_ : \u2200 (a\u2081 a\u2082 : \u03b1), ra a\u2081 a\u2082 \u2192 rb (\u2191e a\u2081) (\u2191e a\u2082))\n      (Quot.map \u2191e.symm (_ : \u2200 (b\u2081 b\u2082 : \u03b2), rb b\u2081 b\u2082 \u2192 ra (\u2191e.symm b\u2081) (\u2191e.symm b\u2082)) (mk rb a)) =\n    mk rb a\n[PROOFSTEP]\nsimp only [Quot.map, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 r x\u271d\u00b9 x\u271d \u2194 r (\u2191e.symm (\u2191e x\u271d\u00b9)) (\u2191e.symm (\u2191e x\u271d))\n[PROOFSTEP]\nsimp only [e.symm_apply_apply]\n", "meta": {"mathlib_filename": "Mathlib.Logic.Equiv.Defs", "llama_tokens": 8790, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199714402812, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.5335897158792382}}
{"text": "[GOAL]\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 @LocallyOfFiniteType = affineLocally @RingHom.FiniteType\n[PROOFSTEP]\next X Y f\n[GOAL]\ncase h.h.h.a\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 LocallyOfFiniteType f \u2194 affineLocally (@RingHom.FiniteType) f\n[PROOFSTEP]\nrw [LocallyOfFiniteType_iff, affineLocally_iff_affineOpens_le]\n[GOAL]\ncase h.h.h.a.hP\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y : Scheme\nf : X \u27f6 Y\n\u22a2 RingHom.RespectsIso @RingHom.FiniteType\n[PROOFSTEP]\nexact RingHom.finiteType_respectsIso\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\nhf : LocallyOfFiniteType (f \u226b g)\n\u22a2 LocallyOfFiniteType f\n[PROOFSTEP]\nrevert hf\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 [hf : LocallyOfFiniteType (f \u226b g)], LocallyOfFiniteType f\n[PROOFSTEP]\nrw [locallyOfFiniteType_eq]\n[GOAL]\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 [hf : affineLocally (@RingHom.FiniteType) (f \u226b g)], affineLocally (@RingHom.FiniteType) f\n[PROOFSTEP]\napply RingHom.finiteType_is_local.affineLocally_of_comp\n[GOAL]\ncase H\nX\u271d Y\u271d : Scheme\nf\u271d : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 \u2200 {R S T : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R \u2192+* S) (g : S \u2192+* T),\n    RingHom.FiniteType (RingHom.comp g f) \u2192 RingHom.FiniteType g\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase H\nX\u271d Y\u271d : Scheme\nf\u271d\u00b9 : X\u271d \u27f6 Y\u271d\nX Y Z : Scheme\nf\u271d : X \u27f6 Y\ng\u271d : Y \u27f6 Z\nR S T : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ng : S \u2192+* T\nH : RingHom.FiniteType (RingHom.comp g f)\n\u22a2 RingHom.FiniteType g\n[PROOFSTEP]\nexact RingHom.FiniteType.of_comp_finiteType H\n", "meta": {"mathlib_filename": "Mathlib.AlgebraicGeometry.Morphisms.FiniteType", "llama_tokens": 902, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936324115011, "lm_q2_score": 0.6959583187272712, "lm_q1q2_score": 0.5333084280645218}}
{"text": "[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 indicator u (fun t => \u2016L\u2016 * \u2016f t\u2016 * C) t\n[PROOFSTEP]\nrefine' le_indicator (f := fun t \u21a6 \u2016L (f t) (g (x - t))\u2016) (fun t _ => _) (fun t ht => _) t\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d\u00b9 x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nx\u271d : t \u2208 u\n\u22a2 (fun t => \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016) t \u2264 \u2016L\u2016 * \u2016f t\u2016 * C\n[PROOFSTEP]\napply_rules [L.le_of_op_norm\u2082_le_of_le, le_rfl]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nht : \u00act \u2208 u\n\u22a2 (fun t => \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016) t \u2264 0\n[PROOFSTEP]\nhave : x - t \u2209 support g := by\n  refine mt (fun hxt => hu ?_) ht\n  refine' \u27e8_, _, Set.neg_mem_neg.mpr (subset_closure hxt), hx, _\u27e9\n  simp only [neg_sub, sub_add_cancel]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nht : \u00act \u2208 u\n\u22a2 \u00acx - t \u2208 support g\n[PROOFSTEP]\nrefine mt (fun hxt => hu ?_) ht\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nht : \u00act \u2208 u\nhxt : x - t \u2208 support g\n\u22a2 t \u2208 -tsupport g + s\n[PROOFSTEP]\nrefine' \u27e8_, _, Set.neg_mem_neg.mpr (subset_closure hxt), hx, _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nht : \u00act \u2208 u\nhxt : x - t \u2208 support g\n\u22a2 (fun x x_1 => x + x_1) (-(x - t)) x = t\n[PROOFSTEP]\nsimp only [neg_sub, sub_add_cancel]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nC : \u211d\nhC : \u2200 (i : G), \u2016g i\u2016 \u2264 C\nx t\u271d : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\nt : G\nht : \u00act \u2208 u\nthis : \u00acx - t \u2208 support g\n\u22a2 (fun t => \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016) t \u2264 0\n[PROOFSTEP]\nsimp only [nmem_support.mp this, (L _).map_zero, norm_zero, le_rfl]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nhcg : HasCompactSupport g\nhg : Continuous g\nx t : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 indicator u (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016) t\n[PROOFSTEP]\nrefine convolution_integrand_bound_right_of_le_of_subset _ (fun i => ?_) hx hu\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nhcg : HasCompactSupport g\nhg : Continuous g\nx t : G\ns u : Set G\nhx : x \u2208 s\nhu : -tsupport g + s \u2286 u\ni : G\n\u22a2 \u2016g i\u2016 \u2264 \u2a06 (i : G), \u2016g i\u2016\n[PROOFSTEP]\nexact le_ciSup (hg.norm.bddAbove_range_of_hasCompactSupport hcg.norm) _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nhcf : HasCompactSupport f\nhf : Continuous f\nx t : G\ns : Set G\nhx : x \u2208 s\n\u22a2 \u2016\u2191(\u2191L (f (x - t))) (g t)\u2016 \u2264 indicator (-tsupport f + s) (fun t => (\u2016L\u2016 * \u2a06 (i : G), \u2016f i\u2016) * \u2016g t\u2016) t\n[PROOFSTEP]\nconvert hcf.convolution_integrand_bound_right L.flip hf hx using 1\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : TopologicalSpace G\nhcf : HasCompactSupport f\nhf : Continuous f\nx t : G\ns : Set G\nhx : x \u2208 s\n\u22a2 indicator (-tsupport f + s) (fun t => (\u2016L\u2016 * \u2a06 (i : G), \u2016f i\u2016) * \u2016g t\u2016) t =\n    indicator (-tsupport f + s) (fun t => \u2016ContinuousLinearMap.flip L\u2016 * \u2016g t\u2016 * \u2a06 (i : G), \u2016f i\u2016) t\n[PROOFSTEP]\nsimp_rw [L.op_norm_flip, mul_right_comm]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => -t + x\u2080) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nunfold ConvolutionExistsAt\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => -t + x\u2080) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\n\u22a2 Integrable fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\n[PROOFSTEP]\nrw [\u2190 integrableOn_iff_integrable_of_support_subset h2s]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => -t + x\u2080) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\n\u22a2 IntegrableOn (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) s\n[PROOFSTEP]\nset s' := (fun t => -t + x\u2080) \u207b\u00b9' s\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\n\u22a2 IntegrableOn (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) s\n[PROOFSTEP]\nhave : \u2200\u1d50 t : G \u2202\u03bc.restrict s, \u2016L (f t) (g (x\u2080 - t))\u2016 \u2264 s.indicator (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 i : s', \u2016g i\u2016) t\n[GOAL]\ncase this\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\n\u22a2 \u2200\u1d50 (t : G) \u2202Measure.restrict \u03bc s,\n    \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) t\n[PROOFSTEP]\nrefine' eventually_of_forall _\n[GOAL]\ncase this\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\n\u22a2 \u2200 (x : G), \u2016\u2191(\u2191L (f x)) (g (x\u2080 - x))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) x\n[PROOFSTEP]\nrefine' le_indicator (fun t ht => _) fun t ht => _\n[GOAL]\ncase this.refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nt : G\nht : t \u2208 s\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016\n[PROOFSTEP]\napply_rules [L.le_of_op_norm\u2082_le_of_le, le_rfl]\n[GOAL]\ncase this.refine'_1.hy\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nt : G\nht : t \u2208 s\n\u22a2 \u2016g (x\u2080 - t)\u2016 \u2264 \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016\n[PROOFSTEP]\nrefine' (le_ciSup_set hbg <| mem_preimage.mpr _)\n[GOAL]\ncase this.refine'_1.hy\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nt : G\nht : t \u2208 s\n\u22a2 -(x\u2080 - t) + x\u2080 \u2208 s\n[PROOFSTEP]\nrwa [neg_sub, sub_add_cancel]\n[GOAL]\ncase this.refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nt : G\nht : \u00act \u2208 s\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 0\n[PROOFSTEP]\nhave : t \u2209 support fun t => L (f t) (g (x\u2080 - t)) := mt (fun h => h2s h) ht\n[GOAL]\ncase this.refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nt : G\nht : \u00act \u2208 s\nthis : \u00act \u2208 support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 0\n[PROOFSTEP]\nrw [nmem_support.mp this, norm_zero]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nthis :\n  \u2200\u1d50 (t : G) \u2202Measure.restrict \u03bc s,\n    \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) t\n\u22a2 IntegrableOn (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) s\n[PROOFSTEP]\nrefine' Integrable.mono' _ _ this\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nthis :\n  \u2200\u1d50 (t : G) \u2202Measure.restrict \u03bc s,\n    \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) t\n\u22a2 Integrable fun a => indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) a\n[PROOFSTEP]\nrw [integrable_indicator_iff hs]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nthis :\n  \u2200\u1d50 (t : G) \u2202Measure.restrict \u03bc s,\n    \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) t\n\u22a2 IntegrableOn (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) s\n[PROOFSTEP]\nexact ((hf.norm.const_mul _).mul_const _).integrableOn\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\ns : Set G\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\ns' : Set G := (fun t => -t + x\u2080) \u207b\u00b9' s\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' s')\nthis :\n  \u2200\u1d50 (t : G) \u2202Measure.restrict \u03bc s,\n    \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t))\u2016 \u2264 indicator s (fun t => \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : \u2191s'), \u2016g \u2191i\u2016) t\n\u22a2 AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) (Measure.restrict \u03bc s)\n[PROOFSTEP]\nexact hf.aestronglyMeasurable.convolution_integrand_snd' L hmg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\nh : ConvolutionExistsAt (fun x => \u2016f x\u2016) (fun x => \u2016g x\u2016) x\u2080 (mul \u211d \u211d)\nhmf : AEStronglyMeasurable f \u03bc\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) \u03bc)\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nrefine' (h.const_mul \u2016L\u2016).mono' (hmf.convolution_integrand_snd' L hmg) (eventually_of_forall fun x => _)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\nh : ConvolutionExistsAt (fun x => \u2016f x\u2016) (fun x => \u2016g x\u2016) x\u2080 (mul \u211d \u211d)\nhmf : AEStronglyMeasurable f \u03bc\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) \u03bc)\nx : G\n\u22a2 \u2016\u2191(\u2191L (f x)) (g (x\u2080 - x))\u2016 \u2264 \u2016L\u2016 * \u2191(\u2191(mul \u211d \u211d) ((fun x => \u2016f x\u2016) x)) ((fun x => \u2016g x\u2016) (x\u2080 - x))\n[PROOFSTEP]\nrw [mul_apply', \u2190 mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : AddGroup G\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : MeasurableNeg G\nx\u2080 : G\nh : ConvolutionExistsAt (fun x => \u2016f x\u2016) (fun x => \u2016g x\u2016) x\u2080 (mul \u211d \u211d)\nhmf : AEStronglyMeasurable f \u03bc\nhmg : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) \u03bc)\nx : G\n\u22a2 \u2016\u2191(\u2191L (f x)) (g (x\u2080 - x))\u2016 \u2264 \u2016L\u2016 * (fun x => \u2016f x\u2016) x * (fun x => \u2016g x\u2016) (x\u2080 - x)\n[PROOFSTEP]\napply L.le_op_norm\u2082\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))\n[PROOFSTEP]\nhave h_meas : AEStronglyMeasurable (fun p : G \u00d7 G => L (f p.2) (g (p.1 - p.2))) (\u03bc.prod \u03bd) :=\n  hf.aestronglyMeasurable.convolution_integrand L hg.aestronglyMeasurable\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))\n[PROOFSTEP]\nhave h2_meas : AEStronglyMeasurable (fun y : G => \u222b x : G, \u2016L (f y) (g (x - y))\u2016 \u2202\u03bc) \u03bd :=\n  h_meas.prod_swap.norm.integral_prod_right'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\n\u22a2 Integrable fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))\n[PROOFSTEP]\nsimp_rw [integrable_prod_iff' h_meas]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\n\u22a2 (\u2200\u1d50 (y : G) \u2202\u03bd, Integrable fun x => \u2191(\u2191L (f y)) (g (x - y))) \u2227\n    Integrable fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' \u27e8eventually_of_forall fun t => (L (f t)).integrable_comp (hg.comp_sub_right t), _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\n\u22a2 Integrable fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' Integrable.mono' _ h2_meas (eventually_of_forall fun t => (_ : _ \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u222b x, \u2016g (x - t)\u2016 \u2202\u03bc))\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\n\u22a2 Integrable fun t => \u2016L\u2016 * \u2016f t\u2016 * \u222b (x : G), \u2016g (x - t)\u2016 \u2202\u03bc\n[PROOFSTEP]\nsimp only [integral_sub_right_eq_self (\u2016g \u00b7\u2016)]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\n\u22a2 Integrable fun t => \u2016L\u2016 * \u2016f t\u2016 * \u222b (x : G), \u2016g x\u2016 \u2202\u03bc\n[PROOFSTEP]\nexact (hf.norm.const_mul _).mul_const _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\nt : G\n\u22a2 \u2016\u222b (x : G), \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2202\u03bc\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u222b (x : G), \u2016g (x - t)\u2016 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 integral_mul_left]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\nt : G\n\u22a2 \u2016\u222b (x : G), \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2202\u03bc\u2016 \u2264 \u222b (a : G), \u2016L\u2016 * \u2016f t\u2016 * \u2016g (a - t)\u2016 \u2202\u03bc\n[PROOFSTEP]\nrw [Real.norm_of_nonneg]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\nt : G\n\u22a2 \u222b (x : G), \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2202\u03bc \u2264 \u222b (a : G), \u2016L\u2016 * \u2016f t\u2016 * \u2016g (a - t)\u2016 \u2202\u03bc\n[PROOFSTEP]\nexact\n  integral_mono_of_nonneg (eventually_of_forall fun t => norm_nonneg _) ((hg.comp_sub_right t).norm.const_mul _)\n    (eventually_of_forall fun t => L.le_op_norm\u2082 _ _)\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : MeasurableAdd\u2082 G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : SigmaFinite \u03bd\nhf : Integrable f\nhg : Integrable g\nh_meas : AEStronglyMeasurable (fun p => \u2191(\u2191L (f p.snd)) (g (p.fst - p.snd))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L (f y)) (g (x - y))\u2016 \u2202\u03bc) \u03bd\nt : G\n\u22a2 0 \u2264 \u222b (x : G), \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2202\u03bc\n[PROOFSTEP]\nexact integral_nonneg fun x => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nlet u := (Homeomorph.neg G).trans (Homeomorph.addRight x\u2080)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nlet v := (Homeomorph.neg G).trans (Homeomorph.addLeft x\u2080)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\napply\n  ((u.isCompact_preimage.mpr h).bddAbove_image hg.norm.continuousOn).convolutionExistsAt' L\n    isClosed_closure.measurableSet subset_closure (hf.integrableOn_isCompact h)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\n\u22a2 AEStronglyMeasurable (fun x => g x)\n    (Measure.map (fun t => \u2191(\u2191toAddUnits x\u2080) - t) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t)))))\n[PROOFSTEP]\nhave A : AEStronglyMeasurable (g \u2218 v) (\u03bc.restrict (tsupport fun t : G => L (f t) (g (x\u2080 - t))))\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\n\u22a2 AEStronglyMeasurable (g \u2218 \u2191v) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))))\n[PROOFSTEP]\napply (hg.comp v.continuous).continuousOn.aestronglyMeasurable_of_isCompact h\n[GOAL]\ncase A\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\n\u22a2 MeasurableSet (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t)))\n[PROOFSTEP]\nexact (isClosed_tsupport _).measurableSet\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\nA : AEStronglyMeasurable (g \u2218 \u2191v) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))))\n\u22a2 AEStronglyMeasurable (fun x => g x)\n    (Measure.map (fun t => \u2191(\u2191toAddUnits x\u2080) - t) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t)))))\n[PROOFSTEP]\nconvert\n  ((v.continuous.measurable.measurePreserving\n            (\u03bc.restrict (tsupport fun t => L (f t) (g (x\u2080 - t))))).aestronglyMeasurable_comp_iff\n        v.toMeasurableEquiv.measurableEmbedding).1\n    A\n[GOAL]\ncase h.e'_6.h.e'_5.h.h.e\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\nA : AEStronglyMeasurable (g \u2218 \u2191v) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))))\nx\u271d : G\n\u22a2 HSub.hSub \u2191(\u2191toAddUnits x\u2080) = \u2191v\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_6.h.e'_5.h.h.e.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d\u00b9 x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nx\u2080 : G\nh : HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\nhf : LocallyIntegrable f\nhg : Continuous g\nu : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addRight x\u2080)\nv : G \u2243\u209c G := Homeomorph.trans (Homeomorph.neg G) (Homeomorph.addLeft x\u2080)\nA : AEStronglyMeasurable (g \u2218 \u2191v) (Measure.restrict \u03bc (tsupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))))\nx\u271d x : G\n\u22a2 \u2191(\u2191toAddUnits x\u2080) - x = \u2191v x\n[PROOFSTEP]\nsimp only [Homeomorph.neg, sub_eq_add_neg, coe_toAddUnits, Homeomorph.trans_apply, Equiv.neg_apply, Equiv.toFun_as_coe,\n  Homeomorph.homeomorph_mk_coe, Equiv.coe_fn_mk, Homeomorph.coe_addLeft]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\n\u22a2 ConvolutionExists f g L\n[PROOFSTEP]\nintro x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nrefine' HasCompactSupport.convolutionExistsAt L _ hf hg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\n[PROOFSTEP]\nrefine' (hcg.comp_homeomorph (Homeomorph.subLeft x\u2080)).mono _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 support (g \u2218 \u2191(Homeomorph.subLeft x\u2080))\n[PROOFSTEP]\nrefine' fun t => mt fun ht : g (x\u2080 - t) = 0 => _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 t : G\nht : g (x\u2080 - t) = 0\n\u22a2 (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) t = 0\n[PROOFSTEP]\nsimp_rw [ht, (L _).map_zero]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcf : HasCompactSupport f\nhf : LocallyIntegrable f\nhg : Continuous g\n\u22a2 ConvolutionExists f g L\n[PROOFSTEP]\nintro x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcf : HasCompactSupport f\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nrefine' HasCompactSupport.convolutionExistsAt L _ hf hg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcf : HasCompactSupport f\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 HasCompactSupport fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\n[PROOFSTEP]\nrefine' hcf.mono _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcf : HasCompactSupport f\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 support f\n[PROOFSTEP]\nrefine' fun t => mt fun ht : f t = 0 => _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : AddGroup G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalAddGroup G\ninst\u271d : BorelSpace G\nhcf : HasCompactSupport f\nhf : LocallyIntegrable f\nhg : Continuous g\nx\u2080 t : G\nht : f t = 0\n\u22a2 (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) t = 0\n[PROOFSTEP]\nsimp_rw [ht, L.map_zero\u2082]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd\u2082 G\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g \u03bc\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nrefine' BddAbove.convolutionExistsAt' L _ hs h2s hf _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd\u2082 G\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g \u03bc\n\u22a2 BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => -t + x\u2080) \u207b\u00b9' s))\n[PROOFSTEP]\nsimp_rw [\u2190 sub_eq_neg_add, hbg]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd\u2082 G\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g \u03bc\n\u22a2 AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\n[PROOFSTEP]\nhave : AEStronglyMeasurable g (map (fun t : G => x\u2080 - t) \u03bc) :=\n  hmg.mono' (quasiMeasurePreserving_sub_left_of_right_invariant \u03bc x\u2080).absolutelyContinuous\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd\u2082 G\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g \u03bc\nthis : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) \u03bc)\n\u22a2 AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s))\n[PROOFSTEP]\napply this.mono_measure\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd\u2082 G\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\ns : Set G\nhbg : BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' s))\nhs : MeasurableSet s\nh2s : (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 s\nhf : IntegrableOn f s\nhmg : AEStronglyMeasurable g \u03bc\nthis : AEStronglyMeasurable g (Measure.map (fun t => x\u2080 - t) \u03bc)\n\u22a2 Measure.map (fun t => x\u2080 - t) (Measure.restrict \u03bc s) \u2264 Measure.map (fun t => x\u2080 - t) \u03bc\n[PROOFSTEP]\nexact map_mono restrict_le_self (measurable_const.sub measurable_id')\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : IsNegInvariant \u03bc\n\u22a2 ConvolutionExistsAt g f x (ContinuousLinearMap.flip L) \u2194 ConvolutionExistsAt f g x L\n[PROOFSTEP]\nsimp_rw [ConvolutionExistsAt,\n  -- porting note: added `(\u03bc := \u03bc)`\u2190 integrable_comp_sub_left (\u03bc := \u03bc) (fun t => L (f t) (g (x - t))) x, sub_sub_cancel,\n  flip_apply]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : IsNegInvariant \u03bc\nh : ConvolutionExistsAt f g x L\n\u22a2 Integrable fun t => \u2191(\u2191L (f (x - t))) (g t)\n[PROOFSTEP]\nconvert h.comp_sub_left x\n[GOAL]\ncase h.e'_5.h.h.e'_6.h.e'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : MeasurableNeg G\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : MeasurableAdd G\ninst\u271d : IsNegInvariant \u03bc\nh : ConvolutionExistsAt f g x L\nx\u271d : G\n\u22a2 x\u271d = x - (x - x\u271d)\n[PROOFSTEP]\nsimp_rw [sub_sub_self]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\ny : \ud835\udd5c\n\u22a2 convolution (y \u2022 f) g L = y \u2022 convolution f g L\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\ny : \ud835\udd5c\nx\u271d : G\n\u22a2 y \u2022 f \u22c6[L, x\u271d] g = (y \u2022 convolution f g L) x\u271d\n[PROOFSTEP]\nsimp only [Pi.smul_apply, convolution_def, \u2190 integral_smul, L.map_smul\u2082]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\ny : \ud835\udd5c\n\u22a2 convolution f (y \u2022 g) L = y \u2022 convolution f g L\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\ny : \ud835\udd5c\nx\u271d : G\n\u22a2 f \u22c6[L, x\u271d] y \u2022 g = (y \u2022 convolution f g L) x\u271d\n[PROOFSTEP]\nsimp only [Pi.smul_apply, convolution_def, \u2190 integral_smul, (L _).map_smul]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\n\u22a2 convolution 0 g L = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 0 \u22c6[L, x\u271d] g = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nsimp_rw [convolution_def, Pi.zero_apply, L.map_zero\u2082, integral_zero]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\n\u22a2 convolution f 0 L = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 f \u22c6[L, x\u271d] 0 = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nsimp_rw [convolution_def, Pi.zero_apply, (L _).map_zero, integral_zero]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nhfg : ConvolutionExistsAt f g x L\nhfg' : ConvolutionExistsAt f g' x L\n\u22a2 f \u22c6[L, x] (g + g') = f \u22c6[L, x] g + f \u22c6[L, x] g'\n[PROOFSTEP]\nsimp only [convolution_def, (L _).map_add, Pi.add_apply, integral_add hfg hfg']\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nhfg : ConvolutionExists f g L\nhfg' : ConvolutionExists f g' L\n\u22a2 convolution f (g + g') L = convolution f g L + convolution f g' L\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nhfg : ConvolutionExists f g L\nhfg' : ConvolutionExists f g' L\nx : G\n\u22a2 f \u22c6[L, x] (g + g') = (convolution f g L + convolution f g' L) x\n[PROOFSTEP]\nexact (hfg x).distrib_add (hfg' x)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nhfg : ConvolutionExistsAt f g x L\nhfg' : ConvolutionExistsAt f' g x L\n\u22a2 (f + f') \u22c6[L, x] g = f \u22c6[L, x] g + f' \u22c6[L, x] g\n[PROOFSTEP]\nsimp only [convolution_def, L.map_add\u2082, Pi.add_apply, integral_add hfg hfg']\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nhfg : ConvolutionExists f g L\nhfg' : ConvolutionExists f' g L\n\u22a2 convolution (f + f') g L = convolution f g L + convolution f' g L\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nhfg : ConvolutionExists f g L\nhfg' : ConvolutionExists f' g L\nx : G\n\u22a2 (f + f') \u22c6[L, x] g = (convolution f g L + convolution f' g L) x\n[PROOFSTEP]\nexact (hfg x).add_distrib (hfg' x)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg : ConvolutionExistsAt f g x (lsmul \u211d \u211d)\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\n\u22a2 f \u22c6[lsmul \u211d \u211d, x] g \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\napply integral_mono hfg hfg'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg : ConvolutionExistsAt f g x (lsmul \u211d \u211d)\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\n\u22a2 (fun t => \u2191(\u2191(lsmul \u211d \u211d) (f t)) (g (x - t))) \u2264 fun t => \u2191(\u2191(lsmul \u211d \u211d) (f t)) (g' (x - t))\n[PROOFSTEP]\nsimp only [lsmul_apply, Algebra.id.smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg : ConvolutionExistsAt f g x (lsmul \u211d \u211d)\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\n\u22a2 (fun t => f t * g (x - t)) \u2264 fun t => f t * g' (x - t)\n[PROOFSTEP]\nintro t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg : ConvolutionExistsAt f g x (lsmul \u211d \u211d)\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nt : G\n\u22a2 (fun t => f t * g (x - t)) t \u2264 (fun t => f t * g' (x - t)) t\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left (hg _) (hf _)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nhg' : \u2200 (x : G), 0 \u2264 g' x\n\u22a2 f \u22c6[lsmul \u211d \u211d, x] g \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\nby_cases H : ConvolutionExistsAt f g x (lsmul \u211d \u211d) \u03bc\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nhg' : \u2200 (x : G), 0 \u2264 g' x\nH : ConvolutionExistsAt f g x (lsmul \u211d \u211d)\n\u22a2 f \u22c6[lsmul \u211d \u211d, x] g \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\nexact convolution_mono_right H hfg' hf hg\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nhg' : \u2200 (x : G), 0 \u2264 g' x\nH : \u00acConvolutionExistsAt f g x (lsmul \u211d \u211d)\n\u22a2 f \u22c6[lsmul \u211d \u211d, x] g \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\nhave : (f \u22c6[lsmul \u211d \u211d, \u03bc] g) x = 0 := integral_undef H\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nhg' : \u2200 (x : G), 0 \u2264 g' x\nH : \u00acConvolutionExistsAt f g x (lsmul \u211d \u211d)\nthis : f \u22c6[lsmul \u211d \u211d, x] g = 0\n\u22a2 f \u22c6[lsmul \u211d \u211d, x] g \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nf g g' : G \u2192 \u211d\nhfg' : ConvolutionExistsAt f g' x (lsmul \u211d \u211d)\nhf : \u2200 (x : G), 0 \u2264 f x\nhg : \u2200 (x : G), g x \u2264 g' x\nhg' : \u2200 (x : G), 0 \u2264 g' x\nH : \u00acConvolutionExistsAt f g x (lsmul \u211d \u211d)\nthis : f \u22c6[lsmul \u211d \u211d, x] g = 0\n\u22a2 0 \u2264 f \u22c6[lsmul \u211d \u211d, x] g'\n[PROOFSTEP]\nexact integral_nonneg fun y => mul_nonneg (hf y) (hg' (x - y))\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : MeasurableAdd\u2082 G\ninst\u271d\u00b2 : MeasurableNeg G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddRightInvariant \u03bc\nh1 : f =\u1da0[ae \u03bc] f'\nh2 : g =\u1da0[ae \u03bc] g'\n\u22a2 convolution f g L = convolution f' g' L\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : MeasurableAdd\u2082 G\ninst\u271d\u00b2 : MeasurableNeg G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddRightInvariant \u03bc\nh1 : f =\u1da0[ae \u03bc] f'\nh2 : g =\u1da0[ae \u03bc] g'\nx : G\n\u22a2 f \u22c6[L, x] g = f' \u22c6[L, x] g'\n[PROOFSTEP]\napply integral_congr_ae\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddGroup G\ninst\u271d\u00b3 : MeasurableAdd\u2082 G\ninst\u271d\u00b2 : MeasurableNeg G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddRightInvariant \u03bc\nh1 : f =\u1da0[ae \u03bc] f'\nh2 : g =\u1da0[ae \u03bc] g'\nx : G\n\u22a2 (fun a => \u2191(\u2191L (f a)) (g (x - a))) =\u1da0[ae \u03bc] fun a => \u2191(\u2191L (f' a)) (g' (x - a))\n[PROOFSTEP]\nexact\n  (h1.prod_mk <| h2.comp_tendsto (quasiMeasurePreserving_sub_left_of_right_invariant \u03bc x).tendsto_ae).fun_comp\n    \u21bffun x y => L x y\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\n\u22a2 support (convolution f g L) \u2286 support g + support f\n[PROOFSTEP]\nintro x h2x\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\n\u22a2 x \u2208 support g + support f\n[PROOFSTEP]\nby_contra hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u00acx \u2208 support g + support f\n\u22a2 False\n[PROOFSTEP]\napply h2x\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u00acx \u2208 support g + support f\n\u22a2 f \u22c6[L, x] g = 0\n[PROOFSTEP]\nsimp_rw [Set.mem_add, not_exists, not_and_or, nmem_support] at hx \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\n\u22a2 f \u22c6[L, x] g = 0\n[PROOFSTEP]\nrw [convolution_def]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\n\u22a2 \u222b (t : G), \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bc = 0\n[PROOFSTEP]\nconvert integral_zero G F using 2\n[GOAL]\ncase h.e'_2.h.e'_7\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\n\u22a2 (fun t => \u2191(\u2191L (f t)) (g (x - t))) = fun x => 0\n[PROOFSTEP]\next t\n[GOAL]\ncase h.e'_2.h.e'_7.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\nt : G\n\u22a2 \u2191(\u2191L (f t)) (g (x - t)) = 0\n[PROOFSTEP]\nrcases hx (x - t) t with (h | h | h)\n[GOAL]\ncase h.e'_2.h.e'_7.h.inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\nt : G\nh : g (x - t) = 0\n\u22a2 \u2191(\u2191L (f t)) (g (x - t)) = 0\n[PROOFSTEP]\nrw [h, (L _).map_zero]\n[GOAL]\ncase h.e'_2.h.e'_7.h.inr.inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\nt : G\nh : f t = 0\n\u22a2 \u2191(\u2191L (f t)) (g (x - t)) = 0\n[PROOFSTEP]\nrw [h, L.map_zero\u2082]\n[GOAL]\ncase h.e'_2.h.e'_7.h.inr.inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : AddGroup G\nx : G\nh2x : x \u2208 support (convolution f g L)\nhx : \u2200 (x_1 x_2 : G), g x_1 = 0 \u2228 f x_2 = 0 \u2228 \u00acx_1 + x_2 = x\nt : G\nh : \u00acx - t + t = x\n\u22a2 \u2191(\u2191L (f t)) (g (x - t)) = 0\n[PROOFSTEP]\nexact (h <| sub_add_cancel x t).elim\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nintro q\u2080 hq\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080 \u2208 s \u00d7\u02e2 univ\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nreplace hq\u2080 : q\u2080.1 \u2208 s\n[GOAL]\ncase hq\u2080\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080 \u2208 s \u00d7\u02e2 univ\n\u22a2 q\u2080.fst \u2208 s\n[PROOFSTEP]\nsimpa only [mem_prod, mem_univ, and_true] using hq\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave A : \u2200 p \u2208 s, Continuous (g p) := fun p hp \u21a6\n  by\n  refine hg.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_\n  simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\np : P\nhp : p \u2208 s\n\u22a2 Continuous (g p)\n[PROOFSTEP]\nrefine hg.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\np : P\nhp : p \u2208 s\nx : G\n\u22a2 (p, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave B : \u2200 p \u2208 s, tsupport (g p) \u2286 k := fun p hp =>\n  closure_minimal (support_subset_iff'.2 fun z hz => hgs _ _ hp hz) h'k\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nobtain \u27e8w, C, w_open, q\u2080w, hw\u27e9 : \u2203 w C, IsOpen w \u2227 q\u2080.1 \u2208 w \u2227 \u2200 p x, p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C :=\n  by\n  have A : IsCompact ({q\u2080.1} \u00d7\u02e2 k) := isCompact_singleton.prod hk\n  obtain \u27e8t, kt, t_open, ht\u27e9 : \u2203 t, {q\u2080.1} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ)) :=\n    by\n    apply exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn A _ hg\n    simp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n  obtain \u27e8C, Cpos, hC\u27e9 : \u2203 C, 0 < C \u2227 \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall (0 : E') C := ht.subset_ball_lt 0 0\n  obtain \u27e8w, w_open, q\u2080w, hw\u27e9 : \u2203 w, IsOpen w \u2227 q\u2080.1 \u2208 w \u2227 w \u00d7\u02e2 k \u2286 t\n  \u00b7 obtain \u27e8w, v, w_open, -, hw, hv, hvw\u27e9 :\n      \u2203 (w : Set P) (v : Set G), IsOpen w \u2227 IsOpen v \u2227 { q\u2080.fst } \u2286 w \u2227 k \u2286 v \u2227 w \u00d7\u02e2 v \u2286 t\n    exact generalized_tube_lemma isCompact_singleton hk t_open kt\n    exact \u27e8w, w_open, singleton_subset_iff.1 hw, Subset.trans (Set.prod_mono Subset.rfl hv) hvw\u27e9\n  refine' \u27e8w, C, w_open, q\u2080w, _\u27e9\n  rintro p x \u27e8hp, hps\u27e9\n  by_cases hx : x \u2208 k\n  \u00b7 have H : (p, x) \u2208 t := by\n      apply hw\n      simp only [prod_mk_mem_set_prod_eq, hp, hx, and_true_iff]\n    have H' : (p, x) \u2208 (s \u00d7\u02e2 univ : Set (P \u00d7 G)) := by\n      simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hps\n    have : g p x \u2208 closedBall (0 : E') C := hC (mem_image_of_mem _ (mem_inter H H'))\n    rwa [mem_closedBall_zero_iff] at this \n  \u00b7 have : g p x = 0 := hgs _ _ hps hx\n    rw [this]\n    simpa only [norm_zero] using Cpos.le\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\n\u22a2 \u2203 w C, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nhave A : IsCompact ({q\u2080.1} \u00d7\u02e2 k) := isCompact_singleton.prod hk\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\n\u22a2 \u2203 w C, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8t, kt, t_open, ht\u27e9 : \u2203 t, {q\u2080.1} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ)) :=\n  by\n  apply exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn A _ hg\n  simp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\n\u22a2 \u2203 t, {q\u2080.fst} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\n[PROOFSTEP]\napply exists_isOpen_bounded_image_inter_of_isCompact_of_continuousOn A _ hg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\n\u22a2 {q\u2080.fst} \u00d7\u02e2 k \u2286 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\n\u22a2 \u2203 w C, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8C, Cpos, hC\u27e9 : \u2203 C, 0 < C \u2227 \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall (0 : E') C := ht.subset_ball_lt 0 0\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\n\u22a2 \u2203 w C, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8w, w_open, q\u2080w, hw\u27e9 : \u2203 w, IsOpen w \u2227 q\u2080.1 \u2208 w \u2227 w \u00d7\u02e2 k \u2286 t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\n\u22a2 \u2203 w, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 w \u00d7\u02e2 k \u2286 t\n[PROOFSTEP]\nobtain \u27e8w, v, w_open, -, hw, hv, hvw\u27e9 :\n  \u2203 (w : Set P) (v : Set G), IsOpen w \u2227 IsOpen v \u2227 { q\u2080.fst } \u2286 w \u2227 k \u2286 v \u2227 w \u00d7\u02e2 v \u2286 t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\n\u22a2 \u2203 w v, IsOpen w \u2227 IsOpen v \u2227 {q\u2080.fst} \u2286 w \u2227 k \u2286 v \u2227 w \u00d7\u02e2 v \u2286 t\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nv : Set G\nw_open : IsOpen w\nhw : {q\u2080.fst} \u2286 w\nhv : k \u2286 v\nhvw : w \u00d7\u02e2 v \u2286 t\n\u22a2 \u2203 w, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 w \u00d7\u02e2 k \u2286 t\n[PROOFSTEP]\nexact generalized_tube_lemma isCompact_singleton hk t_open kt\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nv : Set G\nw_open : IsOpen w\nhw : {q\u2080.fst} \u2286 w\nhv : k \u2286 v\nhvw : w \u00d7\u02e2 v \u2286 t\n\u22a2 \u2203 w, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 w \u00d7\u02e2 k \u2286 t\n[PROOFSTEP]\nexact \u27e8w, w_open, singleton_subset_iff.1 hw, Subset.trans (Set.prod_mono Subset.rfl hv) hvw\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\n\u22a2 \u2203 w C, IsOpen w \u2227 q\u2080.fst \u2208 w \u2227 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nrefine' \u27e8w, C, w_open, q\u2080w, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\n\u22a2 \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nrintro p x \u27e8hp, hps\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nby_cases hx : x \u2208 k\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nhave H : (p, x) \u2208 t := by\n  apply hw\n  simp only [prod_mk_mem_set_prod_eq, hp, hx, and_true_iff]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 (p, x) \u2208 t\n[PROOFSTEP]\napply hw\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 (p, x) \u2208 w \u00d7\u02e2 k\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, hp, hx, and_true_iff]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nhave H' : (p, x) \u2208 (s \u00d7\u02e2 univ : Set (P \u00d7 G)) := by\n  simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hps\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\n\u22a2 (p, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hps\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\nH' : (p, x) \u2208 s \u00d7\u02e2 univ\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nhave : g p x \u2208 closedBall (0 : E') C := hC (mem_image_of_mem _ (mem_inter H H'))\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\nH' : (p, x) \u2208 s \u00d7\u02e2 univ\nthis : g p x \u2208 closedBall 0 C\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nrwa [mem_closedBall_zero_iff] at this \n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nhave : g p x = 0 := hgs _ _ hps hx\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : \u00acx \u2208 k\nthis : g p x = 0\n\u22a2 \u2016g p x\u2016 \u2264 C\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (\u21bfg '' (t \u2229 s \u00d7\u02e2 univ))\nC : \u211d\nCpos : 0 < C\nhC : \u21bfg '' (t \u2229 s \u00d7\u02e2 univ) \u2286 closedBall 0 C\nw : Set P\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : w \u00d7\u02e2 k \u2286 t\np : P\nx : G\nhp : p \u2208 w\nhps : p \u2208 s\nhx : \u00acx \u2208 k\nthis : g p x = 0\n\u22a2 \u20160\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [norm_zero] using Cpos.le\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave I1 : \u2200\u1da0 q : P \u00d7 G in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a : G => L (f a) (g q.1 (q.2 - a))) \u03bc :=\n  by\n  filter_upwards [self_mem_nhdsWithin]\n  rintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n  refine' (HasCompactSupport.convolutionExists_right L _ hf (A _ hp) _).1\n  exact isCompact_of_isClosed_subset hk (isClosed_tsupport _) (B p hp)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n\u22a2 \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\n\u22a2 \u2200 (a : P \u00d7 G), a \u2208 s \u00d7\u02e2 univ \u2192 AEStronglyMeasurable (fun a_2 => \u2191(\u2191L (f a_2)) (g a.fst (a.snd - a_2))) \u03bc\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g (p, x).fst ((p, x).snd - a))) \u03bc\n[PROOFSTEP]\nrefine' (HasCompactSupport.convolutionExists_right L _ hf (A _ hp) _).1\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 HasCompactSupport (g (p, x).fst)\n[PROOFSTEP]\nexact isCompact_of_isClosed_subset hk (isClosed_tsupport _) (B p hp)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nlet K' := -k + {q\u2080.2}\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave hK' : IsCompact K' := hk.neg.add isCompact_singleton\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nobtain \u27e8U, U_open, K'U, hU\u27e9 : \u2203 U, IsOpen U \u2227 K' \u2286 U \u2227 IntegrableOn f U \u03bc := hf.integrableOn_nhds_isCompact hK'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nlet bound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave I2 : \u2200\u1da0 q : P \u00d7 G in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 a \u2202\u03bc, \u2016L (f a) (g q.1 (q.2 - a))\u2016 \u2264 bound a :=\n  by\n  obtain \u27e8V, V_mem, hV\u27e9 : \u2203 V \u2208 \ud835\udcdd (0 : G), K' + V \u2286 U := compact_open_separated_add_right hK' U_open K'U\n  have : ((w \u2229 s) \u00d7\u02e2 ({q\u2080.2} + V) : Set (P \u00d7 G)) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080 :=\n    by\n    conv_rhs => rw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n    refine' Filter.prod_mem_prod _ (singleton_add_mem_nhds_of_nhds_zero q\u2080.2 V_mem)\n    exact mem_nhdsWithin_iff_exists_mem_nhds_inter.2 \u27e8w, w_open.mem_nhds q\u2080w, Subset.rfl\u27e9\n  filter_upwards [this]\n  rintro \u27e8p, x\u27e9 hpx\n  simp only [prod_mk_mem_set_prod_eq] at hpx \n  refine eventually_of_forall fun a => ?_\n  apply convolution_integrand_bound_right_of_le_of_subset _ _ hpx.2 _\n  \u00b7 intro x\n    exact hw _ _ hpx.1\n  \u00b7 rw [\u2190 add_assoc]\n    apply Subset.trans (add_subset_add_right (add_subset_add_right _)) hV\n    rw [neg_subset_neg]\n    exact B p hpx.1.2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\n\u22a2 \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\nobtain \u27e8V, V_mem, hV\u27e9 : \u2203 V \u2208 \ud835\udcdd (0 : G), K' + V \u2286 U := compact_open_separated_add_right hK' U_open K'U\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u22a2 \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\nhave : ((w \u2229 s) \u00d7\u02e2 ({q\u2080.2} + V) : Set (P \u00d7 G)) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080 :=\n  by\n  conv_rhs => rw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n  refine' Filter.prod_mem_prod _ (singleton_add_mem_nhds_of_nhds_zero q\u2080.2 V_mem)\n  exact mem_nhdsWithin_iff_exists_mem_nhds_inter.2 \u27e8w, w_open.mem_nhds q\u2080w, Subset.rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u22a2 (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n[PROOFSTEP]\nconv_rhs => rw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n| \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n| \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n| \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ q\u2080, nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u22a2 (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s] q\u2080.fst \u00d7\u02e2 \ud835\udcdd q\u2080.snd\n[PROOFSTEP]\nrefine' Filter.prod_mem_prod _ (singleton_add_mem_nhds_of_nhds_zero q\u2080.2 V_mem)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u22a2 w \u2229 s \u2208 \ud835\udcdd[s] q\u2080.fst\n[PROOFSTEP]\nexact mem_nhdsWithin_iff_exists_mem_nhds_inter.2 \u27e8w, w_open.mem_nhds q\u2080w, Subset.rfl\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n\u22a2 \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\nfilter_upwards [this]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\n\u22a2 \u2200 (a : P \u00d7 G),\n    a \u2208 (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2192\n      \u2200\u1d50 (a_2 : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a_2)) (g a.fst (a.snd - a_2))\u2016 \u2264 indicator U (fun a => \u2016L\u2016 * \u2016f a\u2016 * C) a_2\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 hpx\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : (p, x) \u2208 (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V)\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g (p, x).fst ((p, x).snd - a))\u2016 \u2264 indicator U (fun a => \u2016L\u2016 * \u2016f a\u2016 * C) a\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq] at hpx \n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g (p, x).fst ((p, x).snd - a))\u2016 \u2264 indicator U (fun a => \u2016L\u2016 * \u2016f a\u2016 * C) a\n[PROOFSTEP]\nrefine eventually_of_forall fun a => ?_\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 \u2016\u2191(\u2191L (f a)) (g (p, x).fst ((p, x).snd - a))\u2016 \u2264 indicator U (fun a => \u2016L\u2016 * \u2016f a\u2016 * C) a\n[PROOFSTEP]\napply convolution_integrand_bound_right_of_le_of_subset _ _ hpx.2 _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 \u2200 (i : G), \u2016g (p, x).fst i\u2016 \u2264 C\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d\u00b9 x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx\u271d : G\nhpx : p \u2208 w \u2229 s \u2227 x\u271d \u2208 {q\u2080.snd} + V\na x : G\n\u22a2 \u2016g (p, x\u271d).fst x\u2016 \u2264 C\n[PROOFSTEP]\nexact hw _ _ hpx.1\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 -tsupport (g (p, x).fst) + ({q\u2080.snd} + V) \u2286 U\n[PROOFSTEP]\nrw [\u2190 add_assoc]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 -tsupport (g (p, x).fst) + {q\u2080.snd} + V \u2286 U\n[PROOFSTEP]\napply Subset.trans (add_subset_add_right (add_subset_add_right _)) hV\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 -tsupport (g (p, x).fst) \u2286 -k\n[PROOFSTEP]\nrw [neg_subset_neg]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\nthis : (w \u2229 s) \u00d7\u02e2 ({q\u2080.snd} + V) \u2208 \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080\np : P\nx : G\nhpx : p \u2208 w \u2229 s \u2227 x \u2208 {q\u2080.snd} + V\na : G\n\u22a2 tsupport (g (p, x).fst) \u2286 k\n[PROOFSTEP]\nexact B p hpx.1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave I3 : Integrable bound \u03bc := by\n  rw [integrable_indicator_iff U_open.measurableSet]\n  exact (hU.norm.const_mul _).mul_const _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n\u22a2 Integrable bound\n[PROOFSTEP]\nrw [integrable_indicator_iff U_open.measurableSet]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\n\u22a2 IntegrableOn (fun a => \u2016L\u2016 * \u2016f a\u2016 * C) U\n[PROOFSTEP]\nexact (hU.norm.const_mul _).mul_const _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave I4 : \u2200\u1d50 a : G \u2202\u03bc, ContinuousWithinAt (fun q : P \u00d7 G => L (f a) (g q.1 (q.2 - a))) (s \u00d7\u02e2 univ) q\u2080 :=\n  by\n  refine eventually_of_forall fun a => ?_\n  suffices H : ContinuousWithinAt (fun q : P \u00d7 G => (f a, g q.1 (q.2 - a))) (s \u00d7\u02e2 univ) q\u2080\n  exact L.continuous\u2082.continuousAt.comp_continuousWithinAt H\n  apply continuousWithinAt_const.prod\n  change ContinuousWithinAt (fun q : P \u00d7 G => (\u21bfg) (q.1, q.2 - a)) (s \u00d7\u02e2 univ) q\u2080\n  have : ContinuousAt (fun q : P \u00d7 G => (q.1, q.2 - a)) (q\u2080.1, q\u2080.2) :=\n    (continuous_fst.prod_mk (continuous_snd.sub continuous_const)).continuousAt\n  rw [\u2190 @Prod.mk.eta _ _ q\u2080]\n  have h'q\u2080 : (q\u2080.1, q\u2080.2 - a) \u2208 (s \u00d7\u02e2 univ : Set (P \u00d7 G)) := \u27e8hq\u2080, mem_univ _\u27e9\n  refine' ContinuousWithinAt.comp (hg _ h'q\u2080) this.continuousWithinAt _\n  rintro \u27e8q, x\u27e9 \u27e8hq, -\u27e9\n  exact \u27e8hq, mem_univ _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, ContinuousWithinAt (fun q => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nrefine eventually_of_forall fun a => ?_\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\n\u22a2 ContinuousWithinAt (fun q => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nsuffices H : ContinuousWithinAt (fun q : P \u00d7 G => (f a, g q.1 (q.2 - a))) (s \u00d7\u02e2 univ) q\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nH : ContinuousWithinAt (fun q => (f a, g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n\u22a2 ContinuousWithinAt (fun q => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\n\u22a2 ContinuousWithinAt (fun q => (f a, g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nexact L.continuous\u2082.continuousAt.comp_continuousWithinAt H\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\n\u22a2 ContinuousWithinAt (fun q => (f a, g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\napply continuousWithinAt_const.prod\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\n\u22a2 ContinuousWithinAt (fun x => g x.fst (x.snd - a)) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nchange ContinuousWithinAt (fun q : P \u00d7 G => (\u21bfg) (q.1, q.2 - a)) (s \u00d7\u02e2 univ) q\u2080\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\n\u22a2 ContinuousWithinAt (fun q => (\u21bfg) (q.fst, q.snd - a)) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nhave : ContinuousAt (fun q : P \u00d7 G => (q.1, q.2 - a)) (q\u2080.1, q\u2080.2) :=\n  (continuous_fst.prod_mk (continuous_snd.sub continuous_const)).continuousAt\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nthis : ContinuousAt (fun q => (q.fst, q.snd - a)) (q\u2080.fst, q\u2080.snd)\n\u22a2 ContinuousWithinAt (fun q => (\u21bfg) (q.fst, q.snd - a)) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ q\u2080]\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nthis : ContinuousAt (fun q => (q.fst, q.snd - a)) (q\u2080.fst, q\u2080.snd)\n\u22a2 ContinuousWithinAt (fun q => (\u21bfg) (q.fst, q.snd - a)) (s \u00d7\u02e2 univ) (q\u2080.fst, q\u2080.snd)\n[PROOFSTEP]\nhave h'q\u2080 : (q\u2080.1, q\u2080.2 - a) \u2208 (s \u00d7\u02e2 univ : Set (P \u00d7 G)) := \u27e8hq\u2080, mem_univ _\u27e9\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nthis : ContinuousAt (fun q => (q.fst, q.snd - a)) (q\u2080.fst, q\u2080.snd)\nh'q\u2080 : (q\u2080.fst, q\u2080.snd - a) \u2208 s \u00d7\u02e2 univ\n\u22a2 ContinuousWithinAt (fun q => (\u21bfg) (q.fst, q.snd - a)) (s \u00d7\u02e2 univ) (q\u2080.fst, q\u2080.snd)\n[PROOFSTEP]\nrefine' ContinuousWithinAt.comp (hg _ h'q\u2080) this.continuousWithinAt _\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nthis : ContinuousAt (fun q => (q.fst, q.snd - a)) (q\u2080.fst, q\u2080.snd)\nh'q\u2080 : (q\u2080.fst, q\u2080.snd - a) \u2208 s \u00d7\u02e2 univ\n\u22a2 MapsTo (fun q => (q.fst, q.snd - a)) (s \u00d7\u02e2 univ) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrintro \u27e8q, x\u27e9 \u27e8hq, -\u27e9\n[GOAL]\ncase H.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\na : G\nthis : ContinuousAt (fun q => (q.fst, q.snd - a)) (q\u2080.fst, q\u2080.snd)\nh'q\u2080 : (q\u2080.fst, q\u2080.snd - a) \u2208 s \u00d7\u02e2 univ\nq : P\nx : G\nhq : (q, x).fst \u2208 s\n\u22a2 (fun q => (q.fst, q.snd - a)) (q, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nexact \u27e8hq, mem_univ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nB : \u2200 (p : P), p \u2208 s \u2192 tsupport (g p) \u2286 k\nw : Set P\nC : \u211d\nw_open : IsOpen w\nq\u2080w : q\u2080.fst \u2208 w\nhw : \u2200 (p : P) (x : G), p \u2208 w \u2229 s \u2192 \u2016g p x\u2016 \u2264 C\nI1 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nbound : G \u2192 \u211d := indicator U fun a => \u2016L\u2016 * \u2016f a\u2016 * C\nI2 : \u2200\u1da0 (q : P \u00d7 G) in \ud835\udcdd[s \u00d7\u02e2 univ] q\u2080, \u2200\u1d50 (a : G) \u2202\u03bc, \u2016\u2191(\u2191L (f a)) (g q.fst (q.snd - a))\u2016 \u2264 bound a\nI3 : Integrable bound\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, ContinuousWithinAt (fun q => \u2191(\u2191L (f a)) (g q.fst (q.snd - a))) (s \u00d7\u02e2 univ) q\u2080\n\u22a2 ContinuousWithinAt (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) q\u2080\n[PROOFSTEP]\nexact continuousWithinAt_of_dominated I1 I2 I3 I4\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ns : Set P\nv : P \u2192 G\nhv : ContinuousOn v s\ng : P \u2192 G \u2192 E'\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (fun x => f \u22c6[L, v x] g x) s\n[PROOFSTEP]\napply (continuousOn_convolution_right_with_param' L hk h'k hgs hf hg).comp (continuousOn_id.prod hv)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ns : Set P\nv : P \u2192 G\nhv : ContinuousOn v s\ng : P \u2192 G \u2192 E'\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 MapsTo (fun x => (_root_.id x, v x)) s (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\ns : Set P\nv : P \u2192 G\nhv : ContinuousOn v s\ng : P \u2192 G \u2192 E'\nk : Set G\nhk : IsCompact k\nh'k : IsClosed k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\nx : P\nhx : x \u2208 s\n\u22a2 (fun x => (_root_.id x, v x)) x \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [hx, prod_mk_mem_set_prod_eq, mem_univ, and_self_iff, id.def]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\n\u22a2 Continuous (convolution f g L)\n[PROOFSTEP]\nrw [continuous_iff_continuousOn_univ]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\n\u22a2 ContinuousOn (convolution f g L) univ\n[PROOFSTEP]\nlet g' : G \u2192 G \u2192 E' := fun _ q => g q\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\ng' : G \u2192 G \u2192 E' := fun x q => g q\n\u22a2 ContinuousOn (convolution f g L) univ\n[PROOFSTEP]\nhave : ContinuousOn (\u21bfg') (univ \u00d7\u02e2 univ) := (hg.comp continuous_snd).continuousOn\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : TopologicalAddGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : FirstCountableTopology G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : FirstCountableTopology P\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : Continuous g\ng' : G \u2192 G \u2192 E' := fun x q => g q\nthis : ContinuousOn (\u21bfg') (univ \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (convolution f g L) univ\n[PROOFSTEP]\nexact\n  continuousOn_convolution_right_with_param_comp' L (continuous_iff_continuousOn_univ.1 continuous_id) hcg\n    (isClosed_tsupport _) (fun p x _ hx => image_eq_zero_of_nmem_tsupport hx) hf this\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\n\u22a2 Continuous (convolution f g L)\n[PROOFSTEP]\nrefine' continuous_iff_continuousAt.mpr fun x\u2080 => _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 ContinuousAt (convolution f g L) x\u2080\n[PROOFSTEP]\nhave : \u2200\u1da0 x in \ud835\udcdd x\u2080, \u2200\u1d50 t : G \u2202\u03bc, \u2016L (f t) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 i, \u2016g i\u2016 :=\n  by\n  refine' eventually_of_forall fun x => eventually_of_forall fun t => _\n  apply_rules [L.le_of_op_norm\u2082_le_of_le, le_rfl, le_ciSup hbg (x - t)]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\n\u22a2 \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, \u2200\u1d50 (t : G) \u2202\u03bc, \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n[PROOFSTEP]\nrefine' eventually_of_forall fun x => eventually_of_forall fun t => _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 x t : G\n\u22a2 \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n[PROOFSTEP]\napply_rules [L.le_of_op_norm\u2082_le_of_le, le_rfl, le_ciSup hbg (x - t)]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\nthis : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, \u2200\u1d50 (t : G) \u2202\u03bc, \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n\u22a2 ContinuousAt (convolution f g L) x\u2080\n[PROOFSTEP]\nrefine' continuousAt_of_dominated _ this _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\nthis : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, \u2200\u1d50 (t : G) \u2202\u03bc, \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n\u22a2 \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g (x - a))) \u03bc\n[PROOFSTEP]\nexact eventually_of_forall fun x => hf.aestronglyMeasurable.convolution_integrand_snd' L hg.aestronglyMeasurable\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\nthis : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, \u2200\u1d50 (t : G) \u2202\u03bc, \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n\u22a2 Integrable fun a => \u2016L\u2016 * \u2016f a\u2016 * \u2a06 (i : G), \u2016g i\u2016\n[PROOFSTEP]\nexact (hf.norm.const_mul _).mul_const _\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : AddGroup G\ninst\u271d\u2076 : TopologicalSpace G\ninst\u271d\u2075 : TopologicalAddGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : FirstCountableTopology G\ninst\u271d\u00b2 : TopologicalSpace P\ninst\u271d\u00b9 : FirstCountableTopology P\ninst\u271d : SecondCountableTopology G\nhbg : BddAbove (range fun x => \u2016g x\u2016)\nhf : Integrable f\nhg : Continuous g\nx\u2080 : G\nthis : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, \u2200\u1d50 (t : G) \u2202\u03bc, \u2016\u2191(\u2191L (f t)) (g (x - t))\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016g i\u2016\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, ContinuousAt (fun x => \u2191(\u2191L (f a)) (g (x - a))) x\u2080\n[PROOFSTEP]\nexact\n  eventually_of_forall fun t =>\n    (L.continuous\u2082.comp\u2082 continuous_const <| hg.comp <| continuous_id.sub continuous_const).continuousAt\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\n\u22a2 convolution g f (ContinuousLinearMap.flip L) = convolution f g L\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nx : G\n\u22a2 g \u22c6[ContinuousLinearMap.flip L, x] f = f \u22c6[L, x] g\n[PROOFSTEP]\nsimp_rw [convolution_def]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nx : G\n\u22a2 \u222b (t : G), \u2191(\u2191(ContinuousLinearMap.flip L) (g t)) (f (x - t)) \u2202\u03bc = \u222b (t : G), \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_sub_left_eq_self _ \u03bc x]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nx : G\n\u22a2 \u222b (x_1 : G), \u2191(\u2191(ContinuousLinearMap.flip L) (g (x - x_1))) (f (x - (x - x_1))) \u2202\u03bc =\n    \u222b (t : G), \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [sub_sub_self, flip_apply]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\n\u22a2 f \u22c6[L, x] g = \u222b (t : G), \u2191(\u2191L (f (x - t))) (g t) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\n\u22a2 g \u22c6[ContinuousLinearMap.flip L, x] f = \u222b (t : G), \u2191(\u2191L (f (x - t))) (g t) \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nh1 : \u2200\u1d50 (x : G) \u2202\u03bc, f (-x) = f x\nh2 : \u2200\u1d50 (x : G) \u2202\u03bc, g (-x) = g x\n\u22a2 \u222b (t : G), \u2191(\u2191L (f t)) (g (-x - t)) \u2202\u03bc = \u222b (t : G), \u2191(\u2191L (f (-t))) (g (x + t)) \u2202\u03bc\n[PROOFSTEP]\napply integral_congr_ae\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nh1 : \u2200\u1d50 (x : G) \u2202\u03bc, f (-x) = f x\nh2 : \u2200\u1d50 (x : G) \u2202\u03bc, g (-x) = g x\n\u22a2 (fun a => \u2191(\u2191L (f a)) (g (-x - a))) =\u1da0[ae \u03bc] fun a => \u2191(\u2191L (f (-a))) (g (x + a))\n[PROOFSTEP]\nfilter_upwards [h1, (eventually_add_left_iff \u03bc x).2 h2] with t ht h't\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nh1 : \u2200\u1d50 (x : G) \u2202\u03bc, f (-x) = f x\nh2 : \u2200\u1d50 (x : G) \u2202\u03bc, g (-x) = g x\nt : G\nht : f (-t) = f t\nh't : g (-(x + t)) = g (x + t)\n\u22a2 \u2191(\u2191L (f t)) (g (-x - t)) = \u2191(\u2191L (f (-t))) (g (x + t))\n[PROOFSTEP]\nsimp_rw [ht, \u2190 h't, neg_add']\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nh1 : \u2200\u1d50 (x : G) \u2202\u03bc, f (-x) = f x\nh2 : \u2200\u1d50 (x : G) \u2202\u03bc, g (-x) = g x\n\u22a2 \u222b (t : G), \u2191(\u2191L (f (-t))) (g (x + t)) \u2202\u03bc = \u222b (t : G), \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_neg_eq_self]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b2 : IsNegInvariant \u03bc\ninst\u271d\u00b9 : MeasurableNeg G\ninst\u271d : MeasurableAdd G\nh1 : \u2200\u1d50 (x : G) \u2202\u03bc, f (-x) = f x\nh2 : \u2200\u1d50 (x : G) \u2202\u03bc, g (-x) = g x\n\u22a2 \u222b (x_1 : G), \u2191(\u2191L (f (- -x_1))) (g (x + -x_1)) \u2202\u03bc = \u222b (t : G), \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bc\n[PROOFSTEP]\nsimp only [neg_neg, \u2190 sub_eq_add_neg]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddCommGroup G\ninst\u271d\u2075 : IsAddLeftInvariant \u03bc\ninst\u271d\u2074 : IsNegInvariant \u03bc\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FirstCountableTopology G\nhcf : HasCompactSupport f\nhf : Continuous f\nhg : LocallyIntegrable g\n\u22a2 Continuous (convolution f g L)\n[PROOFSTEP]\nrw [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddCommGroup G\ninst\u271d\u2075 : IsAddLeftInvariant \u03bc\ninst\u271d\u2074 : IsNegInvariant \u03bc\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : FirstCountableTopology G\nhcf : HasCompactSupport f\nhf : Continuous f\nhg : LocallyIntegrable g\n\u22a2 Continuous (convolution g f (ContinuousLinearMap.flip L))\n[PROOFSTEP]\nexact hcf.continuous_convolution_right L.flip hg hf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddCommGroup G\ninst\u271d\u2075 : IsAddLeftInvariant \u03bc\ninst\u271d\u2074 : IsNegInvariant \u03bc\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : SecondCountableTopology G\nhbf : BddAbove (range fun x => \u2016f x\u2016)\nhf : Continuous f\nhg : Integrable g\n\u22a2 Continuous (convolution f g L)\n[PROOFSTEP]\nrw [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2079 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : AddCommGroup G\ninst\u271d\u2075 : IsAddLeftInvariant \u03bc\ninst\u271d\u2074 : IsNegInvariant \u03bc\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalAddGroup G\ninst\u271d\u00b9 : BorelSpace G\ninst\u271d : SecondCountableTopology G\nhbf : BddAbove (range fun x => \u2016f x\u2016)\nhf : Continuous f\nhg : Integrable g\n\u22a2 Continuous (convolution g f (ContinuousLinearMap.flip L))\n[PROOFSTEP]\nexact hbf.continuous_convolution_right_of_integrable L.flip hg hf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\n\u22a2 f \u22c6[L, x\u2080] g = \u222b (t : G), \u2191(\u2191L (f t)) (g x\u2080) \u2202\u03bc\n[PROOFSTEP]\nhave h2 : \u2200 t, L (f t) (g (x\u2080 - t)) = L (f t) (g x\u2080) := fun t \u21a6\n  by\n  by_cases ht : t \u2208 support f\n  \u00b7 have h2t := hf ht\n    rw [mem_ball_zero_iff] at h2t \n    specialize hg (x\u2080 - t)\n    rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, \u2190 sub_eq_add_neg] at hg \n    rw [hg h2t]\n  \u00b7 rw [nmem_support] at ht \n    simp_rw [ht, L.map_zero\u2082]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nby_cases ht : t \u2208 support f\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\nht : t \u2208 support f\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nhave h2t := hf ht\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\nht : t \u2208 support f\nh2t : t \u2208 ball 0 R\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nrw [mem_ball_zero_iff] at h2t \n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nspecialize hg (x\u2080 - t)\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\nhg : x\u2080 - t \u2208 ball x\u2080 R \u2192 g (x\u2080 - t) = g x\u2080\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, \u2190 sub_eq_add_neg] at hg \n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\nhg : \u2016t\u2016 < R \u2192 g (x\u2080 - t) = g x\u2080\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nrw [hg h2t]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\nht : \u00act \u2208 support f\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nrw [nmem_support] at ht \n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nt : G\nht : f t = 0\n\u22a2 \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n[PROOFSTEP]\nsimp_rw [ht, L.map_zero\u2082]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : SeminormedAddCommGroup G\nx\u2080 : G\nR : \u211d\nhf : support f \u2286 ball 0 R\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 g x = g x\u2080\nh2 : \u2200 (t : G), \u2191(\u2191L (f t)) (g (x\u2080 - t)) = \u2191(\u2191L (f t)) (g x\u2080)\n\u22a2 f \u22c6[L, x\u2080] g = \u222b (t : G), \u2191(\u2191L (f t)) (g x\u2080) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [convolution_def, h2]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 dist (f \u22c6[L, x\u2080] g) (\u222b (t : G), \u2191(\u2191L (f t)) z\u2080 \u2202\u03bc) \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nhave hfg : ConvolutionExistsAt f g x\u2080 L \u03bc :=\n  by\n  refine' BddAbove.convolutionExistsAt L _ Metric.isOpen_ball.measurableSet (Subset.trans _ hf) hif.integrableOn hmg\n  swap; \u00b7 refine' fun t => mt fun ht : f t = 0 => _; simp_rw [ht, L.map_zero\u2082]\n  rw [bddAbove_def]\n  refine' \u27e8\u2016z\u2080\u2016 + \u03b5, _\u27e9\n  rintro _ \u27e8x, hx, rfl\u27e9\n  refine' norm_le_norm_add_const_of_dist_le (hg x _)\n  rwa [mem_ball_iff_norm, norm_sub_rev, \u2190 mem_ball_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 ConvolutionExistsAt f g x\u2080 L\n[PROOFSTEP]\nrefine' BddAbove.convolutionExistsAt L _ Metric.isOpen_ball.measurableSet (Subset.trans _ hf) hif.integrableOn hmg\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' ball 0 R))\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 support f\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 (support fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) \u2286 support f\n[PROOFSTEP]\nrefine' fun t => mt fun ht : f t = 0 => _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nt : G\nht : f t = 0\n\u22a2 (fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))) t = 0\n[PROOFSTEP]\nsimp_rw [ht, L.map_zero\u2082]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 BddAbove ((fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' ball 0 R))\n[PROOFSTEP]\nrw [bddAbove_def]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 \u2203 x, \u2200 (y : \u211d), y \u2208 (fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' ball 0 R) \u2192 y \u2264 x\n[PROOFSTEP]\nrefine' \u27e8\u2016z\u2080\u2016 + \u03b5, _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 \u2200 (y : \u211d), y \u2208 (fun i => \u2016g i\u2016) '' ((fun t => x\u2080 - t) \u207b\u00b9' ball 0 R) \u2192 y \u2264 \u2016z\u2080\u2016 + \u03b5\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nx : G\nhx : x \u2208 (fun t => x\u2080 - t) \u207b\u00b9' ball 0 R\n\u22a2 (fun i => \u2016g i\u2016) x \u2264 \u2016z\u2080\u2016 + \u03b5\n[PROOFSTEP]\nrefine' norm_le_norm_add_const_of_dist_le (hg x _)\n[GOAL]\ncase refine'_1.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nx : G\nhx : x \u2208 (fun t => x\u2080 - t) \u207b\u00b9' ball 0 R\n\u22a2 x \u2208 ball x\u2080 R\n[PROOFSTEP]\nrwa [mem_ball_iff_norm, norm_sub_rev, \u2190 mem_ball_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\n\u22a2 dist (f \u22c6[L, x\u2080] g) (\u222b (t : G), \u2191(\u2191L (f t)) z\u2080 \u2202\u03bc) \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nhave h2 : \u2200 t, dist (L (f t) (g (x\u2080 - t))) (L (f t) z\u2080) \u2264 \u2016L (f t)\u2016 * \u03b5 :=\n  by\n  intro t; by_cases ht : t \u2208 support f\n  \u00b7 have h2t := hf ht\n    rw [mem_ball_zero_iff] at h2t \n    specialize hg (x\u2080 - t)\n    rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, \u2190 sub_eq_add_neg] at hg \n    refine' ((L (f t)).dist_le_op_norm _ _).trans _\n    exact mul_le_mul_of_nonneg_left (hg h2t) (norm_nonneg _)\n  \u00b7 rw [nmem_support] at ht \n    simp_rw [ht, L.map_zero\u2082, L.map_zero, norm_zero, zero_mul, dist_self]\n    rfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\n\u22a2 \u2200 (t : G), dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nintro t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nby_cases ht : t \u2208 support f\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nhave h2t := hf ht\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\nh2t : t \u2208 ball 0 R\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nrw [mem_ball_zero_iff] at h2t \n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nspecialize hg (x\u2080 - t)\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\nhg : x\u2080 - t \u2208 ball x\u2080 R \u2192 dist (g (x\u2080 - t)) z\u2080 \u2264 \u03b5\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nrw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, \u2190 sub_eq_add_neg] at hg \n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\nhg : \u2016t\u2016 < R \u2192 dist (g (x\u2080 - t)) z\u2080 \u2264 \u03b5\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nrefine' ((L (f t)).dist_le_op_norm _ _).trans _\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : t \u2208 support f\nh2t : \u2016t\u2016 < R\nhg : \u2016t\u2016 < R \u2192 dist (g (x\u2080 - t)) z\u2080 \u2264 \u03b5\n\u22a2 \u2016\u2191L (f t)\u2016 * dist (g (x\u2080 - t)) z\u2080 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (hg h2t) (norm_nonneg _)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : \u00act \u2208 support f\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nrw [nmem_support] at ht \n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : f t = 0\n\u22a2 dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n[PROOFSTEP]\nsimp_rw [ht, L.map_zero\u2082, L.map_zero, norm_zero, zero_mul, dist_self]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nt : G\nht : f t = 0\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 dist (f \u22c6[L, x\u2080] g) (\u222b (t : G), \u2191(\u2191L (f t)) z\u2080 \u2202\u03bc) \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nsimp_rw [convolution_def]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), dist (\u2191(\u2191L (f t)) (g (x\u2080 - t))) (\u2191(\u2191L (f t)) z\u2080) \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 dist (\u222b (t : G), \u2191(\u2191L (f t)) (g (x\u2080 - t)) \u2202\u03bc) (\u222b (t : G), \u2191(\u2191L (f t)) z\u2080 \u2202\u03bc) \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nsimp_rw [dist_eq_norm] at h2 \u22a2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u2016\u222b (t : G), \u2191(\u2191L (f t)) (g (x\u2080 - t)) \u2202\u03bc - \u222b (t : G), \u2191(\u2191L (f t)) z\u2080 \u2202\u03bc\u2016 \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nrw [\u2190 integral_sub hfg.integrable]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u2016\u222b (a : G), \u2191(\u2191L (f a)) (g (x\u2080 - a)) - \u2191(\u2191L (f a)) z\u2080 \u2202\u03bc\u2016 \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 Integrable fun t => \u2191(\u2191L (f t)) z\u2080\n[PROOFSTEP]\nswap\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 Integrable fun t => \u2191(\u2191L (f t)) z\u2080\n[PROOFSTEP]\nexact (L.flip z\u2080).integrable_comp hif\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u2016\u222b (a : G), \u2191(\u2191L (f a)) (g (x\u2080 - a)) - \u2191(\u2191L (f a)) z\u2080 \u2202\u03bc\u2016 \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nrefine' (norm_integral_le_of_norm_le ((L.integrable_comp hif).norm.mul_const \u03b5) (eventually_of_forall h2)).trans _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u222b (x : G), \u2016\u2191L (f x)\u2016 * \u03b5 \u2202\u03bc \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nrw [integral_mul_right]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 (\u222b (a : G), \u2016\u2191L (f a)\u2016 \u2202\u03bc) * \u03b5 \u2264 (\u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right _ h\u03b5\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u222b (a : G), \u2016\u2191L (f a)\u2016 \u2202\u03bc \u2264 \u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc\n[PROOFSTEP]\nhave h3 : \u2200 t, \u2016L (f t)\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016 := by\n  intro t\n  exact L.le_op_norm (f t)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\n\u22a2 \u2200 (t : G), \u2016\u2191L (f t)\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016\n[PROOFSTEP]\nintro t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\nt : G\n\u22a2 \u2016\u2191L (f t)\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016\n[PROOFSTEP]\nexact L.le_op_norm (f t)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\nh3 : \u2200 (t : G), \u2016\u2191L (f t)\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016\n\u22a2 \u222b (a : G), \u2016\u2191L (f a)\u2016 \u2202\u03bc \u2264 \u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' (integral_mono (L.integrable_comp hif).norm (hif.norm.const_mul _) h3).trans_eq _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2077 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : CompleteSpace F\ninst\u271d\u2074 : SeminormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : SecondCountableTopology G\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhif : Integrable f\nhf : support f \u2286 ball 0 R\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhfg : ConvolutionExistsAt f g x\u2080 L\nh2 : \u2200 (t : G), \u2016\u2191(\u2191L (f t)) (g (x\u2080 - t)) - \u2191(\u2191L (f t)) z\u2080\u2016 \u2264 \u2016\u2191L (f t)\u2016 * \u03b5\nh3 : \u2200 (t : G), \u2016\u2191L (f t)\u2016 \u2264 \u2016L\u2016 * \u2016f t\u2016\n\u22a2 \u222b (a : G), \u2016L\u2016 * \u2016f a\u2016 \u2202\u03bc = \u2016L\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc\n[PROOFSTEP]\nrw [integral_mul_left]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\nf : G \u2192 \u211d\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhf : support f \u2286 ball 0 R\nhnf : \u2200 (x : G), 0 \u2264 f x\nhintf : \u222b (x : G), f x \u2202\u03bc = 1\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\n\u22a2 dist (f \u22c6[lsmul \u211d \u211d, x\u2080] g) z\u2080 \u2264 \u03b5\n[PROOFSTEP]\nhave hif : Integrable f \u03bc := integrable_of_integral_eq_one hintf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\nf : G \u2192 \u211d\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhf : support f \u2286 ball 0 R\nhnf : \u2200 (x : G), 0 \u2264 f x\nhintf : \u222b (x : G), f x \u2202\u03bc = 1\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhif : Integrable f\n\u22a2 dist (f \u22c6[lsmul \u211d \u211d, x\u2080] g) z\u2080 \u2264 \u03b5\n[PROOFSTEP]\nconvert (dist_convolution_le' (lsmul \u211d \u211d) h\u03b5 hif hf hmg hg).trans _\n[GOAL]\ncase h.e'_3.h.e'_4\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\nf : G \u2192 \u211d\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhf : support f \u2286 ball 0 R\nhnf : \u2200 (x : G), 0 \u2264 f x\nhintf : \u222b (x : G), f x \u2202\u03bc = 1\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhif : Integrable f\n\u22a2 z\u2080 = \u222b (t : G), \u2191(\u2191(lsmul \u211d \u211d) (f t)) z\u2080 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [lsmul_apply, integral_smul_const, hintf, one_smul]\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\nf : G \u2192 \u211d\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhf : support f \u2286 ball 0 R\nhnf : \u2200 (x : G), 0 \u2264 f x\nhintf : \u222b (x : G), f x \u2202\u03bc = 1\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhif : Integrable f\n\u22a2 (\u2016lsmul \u211d \u211d\u2016 * \u222b (x : G), \u2016f x\u2016 \u2202\u03bc) * \u03b5 \u2264 \u03b5\n[PROOFSTEP]\nsimp_rw [Real.norm_of_nonneg (hnf _), hintf, mul_one]\n[GOAL]\ncase convert_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\nf : G \u2192 \u211d\nx\u2080 : G\nR \u03b5 : \u211d\nz\u2080 : E'\nh\u03b5 : 0 \u2264 \u03b5\nhf : support f \u2286 ball 0 R\nhnf : \u2200 (x : G), 0 \u2264 f x\nhintf : \u222b (x : G), f x \u2202\u03bc = 1\nhmg : AEStronglyMeasurable g \u03bc\nhg : \u2200 (x : G), x \u2208 ball x\u2080 R \u2192 dist (g x) z\u2080 \u2264 \u03b5\nhif : Integrable f\n\u22a2 \u2016lsmul \u211d \u211d\u2016 * \u03b5 \u2264 \u03b5\n[PROOFSTEP]\nexact (mul_le_mul_of_nonneg_right op_norm_lsmul_le h\u03b5).trans_eq (one_mul \u03b5)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nh\u03c6 : Tendsto (fun n => support (\u03c6 n)) l (smallSets (\ud835\udcdd 0))\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhcg : Tendsto (uncurry g) (l \u00d7\u02e2 \ud835\udcdd x\u2080) (\ud835\udcdd z\u2080)\nhk : Tendsto k l (\ud835\udcdd x\u2080)\n\u22a2 Tendsto (fun i => \u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) l (\ud835\udcdd z\u2080)\n[PROOFSTEP]\nsimp_rw [tendsto_smallSets_iff] at h\u03c6 \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhcg : Tendsto (uncurry g) (l \u00d7\u02e2 \ud835\udcdd x\u2080) (\ud835\udcdd z\u2080)\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\n\u22a2 Tendsto (fun i => \u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) l (\ud835\udcdd z\u2080)\n[PROOFSTEP]\nrw [Metric.tendsto_nhds] at hcg \u22a2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhcg : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9 \u00d7 G) in l \u00d7\u02e2 \ud835\udcdd x\u2080, dist (uncurry g x) z\u2080 < \u03b5\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nsimp_rw [Metric.eventually_prod_nhds_iff] at hcg \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nhave h2\u03b5 : 0 < \u03b5 / 3 := div_pos h\u03b5 (by norm_num)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 0 < 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nobtain \u27e8p, hp, \u03b4, h\u03b4, hg\u03b4\u27e9 := hcg _ h2\u03b5\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5 / 3\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\ndsimp only [uncurry] at hg\u03b4 \n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nhave h2k := hk.eventually (ball_mem_nhds x\u2080 <| half_pos h\u03b4)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nhave h2\u03c6 := h\u03c6 (ball (0 : G) _) <| ball_mem_nhds _ (half_pos h\u03b4)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\n\u22a2 \u2200\u1da0 (x : \u03b9) in l, dist (\u03c6 x \u22c6[lsmul \u211d \u211d, k x] g x) z\u2080 < \u03b5\n[PROOFSTEP]\nfilter_upwards [hp, h2k, h2\u03c6, hn\u03c6, hi\u03c6, hmg] with i hpi hki h\u03c6i hn\u03c6i hi\u03c6i hmgi\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\n\u22a2 dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) z\u2080 < \u03b5\n[PROOFSTEP]\nhave hgi : dist (g i (k i)) z\u2080 < \u03b5 / 3 := hg\u03b4 hpi (hki.trans <| half_lt_self h\u03b4)\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\n\u22a2 dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) z\u2080 < \u03b5\n[PROOFSTEP]\nhave h1 : \u2200 x' \u2208 ball (k i) (\u03b4 / 2), dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3 :=\n  by\n  intro x' hx'\n  refine' (dist_triangle_right _ _ _).trans (add_le_add (hg\u03b4 hpi _).le hgi.le)\n  exact ((dist_triangle _ _ _).trans_lt (add_lt_add hx'.out hki)).trans_eq (add_halves \u03b4)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\n\u22a2 \u2200 (x' : G), x' \u2208 ball (k i) (\u03b4 / 2) \u2192 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\nintro x' hx'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x'\u271d : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nx' : G\nhx' : x' \u2208 ball (k i) (\u03b4 / 2)\n\u22a2 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n[PROOFSTEP]\nrefine' (dist_triangle_right _ _ _).trans (add_le_add (hg\u03b4 hpi _).le hgi.le)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x'\u271d : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nx' : G\nhx' : x' \u2208 ball (k i) (\u03b4 / 2)\n\u22a2 dist x' x\u2080 < \u03b4\n[PROOFSTEP]\nexact ((dist_triangle _ _ _).trans_lt (add_lt_add hx'.out hki)).trans_eq (add_halves \u03b4)\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nh1 : \u2200 (x' : G), x' \u2208 ball (k i) (\u03b4 / 2) \u2192 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n\u22a2 dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) z\u2080 < \u03b5\n[PROOFSTEP]\nhave := dist_convolution_le (add_pos h2\u03b5 h2\u03b5).le h\u03c6i hn\u03c6i hi\u03c6i hmgi h1\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nh1 : \u2200 (x' : G), x' \u2208 ball (k i) (\u03b4 / 2) \u2192 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\nthis : dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n\u22a2 dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) z\u2080 < \u03b5\n[PROOFSTEP]\nrefine' ((dist_triangle _ _ _).trans_lt (add_lt_add_of_le_of_lt this hgi)).trans_eq _\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nh1 : \u2200 (x' : G), x' \u2208 ball (k i) (\u03b4 / 2) \u2192 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\nthis : dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n\u22a2 \u03b5 / 3 + \u03b5 / 3 + \u03b5 / 3 = \u03b5\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2070 : MeasurableSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : SeminormedAddCommGroup G\ninst\u271d\u2076 : BorelSpace G\ninst\u271d\u2075 : SecondCountableTopology G\ninst\u271d\u2074 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : CompleteSpace E'\n\u03b9 : Type u_1\ng : \u03b9 \u2192 G \u2192 E'\nl : Filter \u03b9\nx\u2080 : G\nz\u2080 : E'\n\u03c6 : \u03b9 \u2192 G \u2192 \u211d\nk : \u03b9 \u2192 G\nhn\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6 : \u2200\u1da0 (i : \u03b9) in l, \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmg : \u2200\u1da0 (i : \u03b9) in l, AEStronglyMeasurable (g i) \u03bc\nhk : Tendsto k l (\ud835\udcdd x\u2080)\nh\u03c6 : \u2200 (t : Set G), t \u2208 \ud835\udcdd 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 t\nhcg :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 pa,\n        (\u2200\u1da0 (i : \u03b9) in l, pa i) \u2227\n          \u2203 \u03b5_1, \u03b5_1 > 0 \u2227 \u2200 {i : \u03b9}, pa i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b5_1 \u2192 dist (uncurry g (i, x)) z\u2080 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh2\u03b5 : 0 < \u03b5 / 3\np : \u03b9 \u2192 Prop\nhp : \u2200\u1da0 (i : \u03b9) in l, p i\n\u03b4 : \u211d\nh\u03b4 : \u03b4 > 0\nhg\u03b4 : \u2200 {i : \u03b9}, p i \u2192 \u2200 {x : G}, dist x x\u2080 < \u03b4 \u2192 dist (g i x) z\u2080 < \u03b5 / 3\nh2k : \u2200\u1da0 (x : \u03b9) in l, dist (k x) x\u2080 < \u03b4 / 2\nh2\u03c6 : \u2200\u1da0 (x : \u03b9) in l, support (\u03c6 x) \u2286 ball 0 (\u03b4 / 2)\ni : \u03b9\nhpi : p i\nhki : dist (k i) x\u2080 < \u03b4 / 2\nh\u03c6i : support (\u03c6 i) \u2286 ball 0 (\u03b4 / 2)\nhn\u03c6i : \u2200 (x : G), 0 \u2264 \u03c6 i x\nhi\u03c6i : \u222b (x : G), \u03c6 i x \u2202\u03bc = 1\nhmgi : AEStronglyMeasurable (g i) \u03bc\nhgi : dist (g i (k i)) z\u2080 < \u03b5 / 3\nh1 : \u2200 (x' : G), x' \u2208 ball (k i) (\u03b4 / 2) \u2192 dist (g i x') (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\nthis : dist (\u03c6 i \u22c6[lsmul \u211d \u211d, k i] g i) (g i (k i)) \u2264 \u03b5 / 3 + \u03b5 / 3\n\u22a2 \u03b5 + \u03b5 + \u03b5 = \u03b5 * 3\n[PROOFSTEP]\nring_nf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (x : G), f \u22c6[L, x] g \u2202\u03bc = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)\n[PROOFSTEP]\nrefine' (integral_integral_swap (by apply hf.convolution_integrand L hg)).trans _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 Integrable (uncurry fun x t => \u2191(\u2191L (f t)) (g (x - t)))\n[PROOFSTEP]\napply hf.convolution_integrand L hg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u222b (x : G), \u2191(\u2191L (f y)) (g (x - y)) \u2202\u03bc \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)\n[PROOFSTEP]\nsimp_rw [integral_comp_comm _ (hg.comp_sub_right _), integral_sub_right_eq_self]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2079 : NormedAddCommGroup E\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2076 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2079 : CompleteSpace F\ninst\u271d\u00b9\u2078 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2077 : NormedAddCommGroup F'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d F'\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u2074 : CompleteSpace F'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F''\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d F''\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F''\ninst\u271d\u00b9\u2070 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2079 : AddGroup G\ninst\u271d\u2078 : SigmaFinite \u03bc\ninst\u271d\u2077 : SigmaFinite \u03bd\ninst\u271d\u2076 : IsAddRightInvariant \u03bc\ninst\u271d\u2075 : MeasurableAdd\u2082 G\ninst\u271d\u2074 : MeasurableNeg G\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace E'\nhf : Integrable f\nhg : Integrable g\n\u22a2 \u222b (y : G), \u2191(\u2191L (f y)) (\u222b (x : G), g x \u2202\u03bc) \u2202\u03bd = \u2191(\u2191L (\u222b (x : G), f x \u2202\u03bd)) (\u222b (x : G), g x \u2202\u03bc)\n[PROOFSTEP]\nexact (L.flip (\u222b x, g x \u2202\u03bc)).integral_comp_comm hf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 \u222b (t : G), \u222b (s : G), \u2191(\u2191L\u2082 (\u2191(\u2191L (f s)) (g (t - s)))) (k (x\u2080 - t)) \u2202\u03bd \u2202\u03bc =\n    \u222b (t : G), \u222b (s : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g (t - s))) (k (x\u2080 - t))) \u2202\u03bd \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [hL]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 \u222b (t : G), \u222b (s : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g (t - s))) (k (x\u2080 - t))) \u2202\u03bd \u2202\u03bc =\n    \u222b (s : G), \u222b (t : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g (t - s))) (k (x\u2080 - t))) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\nrw [integral_integral_swap hi]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 \u222b (s : G), \u222b (t : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g (t - s))) (k (x\u2080 - t))) \u2202\u03bc \u2202\u03bd =\n    \u222b (s : G), \u222b (u : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u))) \u2202\u03bc \u2202\u03bd\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 (fun s => \u222b (t : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g (t - s))) (k (x\u2080 - t))) \u2202\u03bc) = fun s =>\n    \u222b (u : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u))) \u2202\u03bc\n[PROOFSTEP]\next t\n[GOAL]\ncase e_f.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\nt : G\n\u22a2 \u222b (t_1 : G), \u2191(\u2191L\u2083 (f t)) (\u2191(\u2191L\u2084 (g (t_1 - t))) (k (x\u2080 - t_1))) \u2202\u03bc =\n    \u222b (u : G), \u2191(\u2191L\u2083 (f t)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - t - u))) \u2202\u03bc\n[PROOFSTEP]\nrw [eq_comm, \u2190 integral_sub_right_eq_self _ t]\n[GOAL]\ncase e_f.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\nt : G\n\u22a2 \u222b (x : G), \u2191(\u2191L\u2083 (f t)) (\u2191(\u2191L\u2084 (g (x - t))) (k (x\u2080 - t - (x - t)))) \u2202\u03bc =\n    \u222b (t_1 : G), \u2191(\u2191L\u2083 (f t)) (\u2191(\u2191L\u2084 (g (t_1 - t))) (k (x\u2080 - t_1))) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [sub_sub_sub_cancel_right]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 \u222b (s : G), \u222b (u : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u))) \u2202\u03bc \u2202\u03bd =\n    \u222b (s : G), \u2191(\u2191L\u2083 (f s)) (\u222b (u : G), \u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u)) \u2202\u03bc) \u2202\u03bd\n[PROOFSTEP]\nrefine' integral_congr_ae _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n\u22a2 (fun s => \u222b (u : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u))) \u2202\u03bc) =\u1da0[ae \u03bd] fun s =>\n    \u2191(\u2191L\u2083 (f s)) (\u222b (u : G), \u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u)) \u2202\u03bc)\n[PROOFSTEP]\nrefine' ((quasiMeasurePreserving_sub_left_of_right_invariant \u03bd x\u2080).ae hgk).mono fun t ht => _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt g k x L\u2084\nhi : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\nt : G\nht : ConvolutionExistsAt g k (x\u2080 - t) L\u2084\n\u22a2 (fun s => \u222b (u : G), \u2191(\u2191L\u2083 (f s)) (\u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u))) \u2202\u03bc) t =\n    (fun s => \u2191(\u2191L\u2083 (f s)) (\u222b (u : G), \u2191(\u2191L\u2084 (g u)) (k (x\u2080 - s - u)) \u2202\u03bc)) t\n[PROOFSTEP]\nexact (L\u2083 (f t)).integral_comp_comm ht\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 convolution f g L \u22c6[L\u2082, x\u2080] k = f \u22c6[L\u2083, x\u2080] convolution g k L\u2084\n[PROOFSTEP]\nrefine'\n  convolution_assoc' L L\u2082 L\u2083 L\u2084 hL hfg (hgk.mono fun x hx => hx.ofNorm L\u2084 hg hk)\n    _\n      -- the following is similar to `integrable.convolution_integrand`\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nhave h_meas : AEStronglyMeasurable (uncurry fun x y => L\u2083 (f y) (L\u2084 (g x) (k (x\u2080 - y - x)))) (\u03bc.prod \u03bd) :=\n  by\n  refine' L\u2083.aestronglyMeasurable_comp\u2082 hf.snd _\n  refine' L\u2084.aestronglyMeasurable_comp\u2082 hg.fst _\n  refine' (hk.mono' _).comp_measurable ((measurable_const.sub measurable_snd).sub measurable_fst)\n  refine' QuasiMeasurePreserving.absolutelyContinuous _\n  refine'\n    QuasiMeasurePreserving.prod_of_left ((measurable_const.sub measurable_snd).sub measurable_fst)\n      (eventually_of_forall fun y => _)\n  dsimp only\n  exact quasiMeasurePreserving_sub_left_of_right_invariant \u03bc _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nrefine' L\u2083.aestronglyMeasurable_comp\u2082 hf.snd _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 AEStronglyMeasurable (fun x => \u2191(\u2191L\u2084 (g x.fst)) (k (x\u2080 - x.snd - x.fst))) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nrefine' L\u2084.aestronglyMeasurable_comp\u2082 hg.fst _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 AEStronglyMeasurable (fun x => k (x\u2080 - x.snd - x.fst)) (Measure.prod \u03bc \u03bd)\n[PROOFSTEP]\nrefine' (hk.mono' _).comp_measurable ((measurable_const.sub measurable_snd).sub measurable_fst)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 Measure.map (fun x => x\u2080 - x.snd - x.fst) (Measure.prod \u03bc \u03bd) \u226a \u03bc\n[PROOFSTEP]\nrefine' QuasiMeasurePreserving.absolutelyContinuous _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\n\u22a2 QuasiMeasurePreserving fun x => x\u2080 - x.snd - x.fst\n[PROOFSTEP]\nrefine'\n  QuasiMeasurePreserving.prod_of_left ((measurable_const.sub measurable_snd).sub measurable_fst)\n    (eventually_of_forall fun y => _)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\ny : G\n\u22a2 QuasiMeasurePreserving fun x => x\u2080 - (x, y).snd - (x, y).fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny\u271d y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\ny : G\n\u22a2 QuasiMeasurePreserving fun x => x\u2080 - y - x\n[PROOFSTEP]\nexact quasiMeasurePreserving_sub_left_of_right_invariant \u03bc _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nhave h2_meas : AEStronglyMeasurable (fun y => \u222b x, \u2016L\u2083 (f y) (L\u2084 (g x) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd :=\n  h_meas.prod_swap.norm.integral_prod_right'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nhave h3 : map (fun z : G \u00d7 G => (z.1 - z.2, z.2)) (\u03bc.prod \u03bd) = \u03bc.prod \u03bd := (measurePreserving_sub_prod \u03bc \u03bd).map_eq\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nsuffices Integrable (uncurry fun x y => L\u2083 (f y) (L\u2084 (g x) (k (x\u2080 - y - x)))) (\u03bc.prod \u03bd)\n  by\n  rw [\u2190 h3] at this \n  convert this.comp_measurable (measurable_sub.prod_mk measurable_snd)\n  ext \u27e8x, y\u27e9\n  simp_rw [uncurry, Function.comp_apply, sub_sub_sub_cancel_right]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nthis : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x))))\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nrw [\u2190 h3] at this \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nthis : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x))))\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x))))\n[PROOFSTEP]\nconvert this.comp_measurable (measurable_sub.prod_mk measurable_snd)\n[GOAL]\ncase h.e'_5\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nthis : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x))))\n\u22a2 (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x)))) =\n    (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) \u2218 fun z => (z.fst - z.snd, z.snd)\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.e'_5.h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny\u271d y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nthis : Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x))))\nx y : G\n\u22a2 uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g (x - y))) (k (x\u2080 - x)))) (x, y) =\n    ((uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) \u2218 fun z => (z.fst - z.snd, z.snd)) (x, y)\n[PROOFSTEP]\nsimp_rw [uncurry, Function.comp_apply, sub_sub_sub_cancel_right]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\n\u22a2 Integrable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x))))\n[PROOFSTEP]\nsimp_rw [integrable_prod_iff' h_meas]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\n\u22a2 (\u2200\u1d50 (y : G) \u2202\u03bd, Integrable fun x => uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, y)) \u2227\n    Integrable fun y => \u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, y)\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine'\n  \u27e8((quasiMeasurePreserving_sub_left_of_right_invariant \u03bd x\u2080).ae hgk).mono fun t ht =>\n      (L\u2083 (f t)).integrable_comp <| ht.ofNorm L\u2084 hg hk,\n    _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\n\u22a2 Integrable fun y => \u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, y)\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine'\n  (hfgk.const_mul (\u2016L\u2083\u2016 * \u2016L\u2084\u2016)).mono' h2_meas\n    (((quasiMeasurePreserving_sub_left_of_right_invariant \u03bd x\u2080).ae hgk).mono fun t ht => _)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\n\u22a2 \u2016\u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, t)\u2016 \u2202\u03bc\u2016 \u2264\n    \u2016L\u2083\u2016 * \u2016L\u2084\u2016 * \u2191(\u2191(mul \u211d \u211d) ((fun x => \u2016f x\u2016) t)) ((fun x => \u2016g x\u2016) \u22c6[mul \u211d \u211d, (x\u2080 - t)] fun x => \u2016k x\u2016)\n[PROOFSTEP]\nsimp_rw [convolution_def, mul_apply', mul_mul_mul_comm \u2016L\u2083\u2016 \u2016L\u2084\u2016, \u2190 integral_mul_left]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\n\u22a2 \u2016\u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, t)\u2016 \u2202\u03bc\u2016 \u2264\n    \u222b (a : G), \u2016L\u2083\u2016 * \u2016f t\u2016 * (\u2016L\u2084\u2016 * (\u2016g a\u2016 * \u2016k (x\u2080 - t - a)\u2016)) \u2202\u03bc\n[PROOFSTEP]\nrw [Real.norm_of_nonneg]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\n\u22a2 \u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, t)\u2016 \u2202\u03bc \u2264\n    \u222b (a : G), \u2016L\u2083\u2016 * \u2016f t\u2016 * (\u2016L\u2084\u2016 * (\u2016g a\u2016 * \u2016k (x\u2080 - t - a)\u2016)) \u2202\u03bc\n[PROOFSTEP]\nrefine'\n  integral_mono_of_nonneg (eventually_of_forall fun t => norm_nonneg _) ((ht.const_mul _).const_mul _)\n    (eventually_of_forall fun s => _)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\ns : G\n\u22a2 (fun x => \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, t)\u2016) s \u2264\n    (fun a => \u2016L\u2083\u2016 * \u2016f t\u2016 * (\u2016L\u2084\u2016 * (\u2016g a\u2016 * \u2016k (x\u2080 - t - a)\u2016))) s\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc \u2016L\u2084\u2016]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\ns : G\n\u22a2 \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (s, t)\u2016 \u2264\n    \u2016L\u2083\u2016 * \u2016f t\u2016 * (\u2016L\u2084\u2016 * \u2016g s\u2016 * \u2016k (x\u2080 - t - s)\u2016)\n[PROOFSTEP]\napply_rules [ContinuousLinearMap.le_of_op_norm\u2082_le_of_le, le_rfl]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E\ninst\u271d\u00b2\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b2\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2078 : NormedSpace \u211d F\ninst\u271d\u00b9\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b9\u2076 : CompleteSpace F\ninst\u271d\u00b9\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\ninst\u271d\u00b9\u00b9 : CompleteSpace F'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F''\ninst\u271d\u2079 : NormedSpace \u211d F''\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F''\ninst\u271d\u2077 : CompleteSpace F''\nk : G \u2192 E''\nL\u2082 : F \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F'\nL\u2083 : E \u2192L[\ud835\udd5c] F'' \u2192L[\ud835\udd5c] F'\nL\u2084 : E' \u2192L[\ud835\udd5c] E'' \u2192L[\ud835\udd5c] F''\ninst\u271d\u2076 : AddGroup G\ninst\u271d\u2075 : SigmaFinite \u03bc\ninst\u271d\u2074 : SigmaFinite \u03bd\ninst\u271d\u00b3 : IsAddRightInvariant \u03bc\ninst\u271d\u00b2 : MeasurableAdd\u2082 G\ninst\u271d\u00b9 : IsAddRightInvariant \u03bd\ninst\u271d : MeasurableNeg G\nhL : \u2200 (x : E) (y : E') (z : E''), \u2191(\u2191L\u2082 (\u2191(\u2191L x) y)) z = \u2191(\u2191L\u2083 x) (\u2191(\u2191L\u2084 y) z)\nx\u2080 : G\nhf : AEStronglyMeasurable f \u03bd\nhg : AEStronglyMeasurable g \u03bc\nhk : AEStronglyMeasurable k \u03bc\nhfg : \u2200\u1d50 (y : G) \u2202\u03bc, ConvolutionExistsAt f g y L\nhgk : \u2200\u1d50 (x : G) \u2202\u03bd, ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) x (mul \u211d \u211d)\nhfgk : ConvolutionExistsAt (fun x => \u2016f x\u2016) (convolution (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (mul \u211d \u211d)) x\u2080 (mul \u211d \u211d)\nh_meas : AEStronglyMeasurable (uncurry fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (Measure.prod \u03bc \u03bd)\nh2_meas : AEStronglyMeasurable (fun y => \u222b (x : G), \u2016\u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))\u2016 \u2202\u03bc) \u03bd\nh3 : Measure.map (fun z => (z.fst - z.snd, z.snd)) (Measure.prod \u03bc \u03bd) = Measure.prod \u03bc \u03bd\nt : G\nht : ConvolutionExistsAt (fun x => \u2016g x\u2016) (fun x => \u2016k x\u2016) (x\u2080 - t) (mul \u211d \u211d)\n\u22a2 0 \u2264 \u222b (x : G), \u2016uncurry (fun x y => \u2191(\u2191L\u2083 (f y)) (\u2191(\u2191L\u2084 (g x)) (k (x\u2080 - y - x)))) (x, t)\u2016 \u2202\u03bc\n[PROOFSTEP]\nexact integral_nonneg fun x => norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b3 : CompleteSpace F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : BorelSpace G\ng : G \u2192 E'' \u2192L[\ud835\udd5c] E'\nhf : LocallyIntegrable f\nhcg : HasCompactSupport g\nhg : Continuous g\nx\u2080 : G\nx : E''\n\u22a2 \u2191(f \u22c6[precompR E'' L, x\u2080] g) x = f \u22c6[L, x\u2080] fun a => \u2191(g a) x\n[PROOFSTEP]\nhave := hcg.convolutionExists_right (L.precompR E'' : _) hf hg x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b3 : CompleteSpace F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : BorelSpace G\ng : G \u2192 E'' \u2192L[\ud835\udd5c] E'\nhf : LocallyIntegrable f\nhcg : HasCompactSupport g\nhg : Continuous g\nx\u2080 : G\nx : E''\nthis : ConvolutionExistsAt f g x\u2080 (precompR E'' L)\n\u22a2 \u2191(f \u22c6[precompR E'' L, x\u2080] g) x = f \u22c6[L, x\u2080] fun a => \u2191(g a) x\n[PROOFSTEP]\nsimp_rw [convolution_def, ContinuousLinearMap.integral_apply this]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u00b3 : CompleteSpace F\ninst\u271d\u00b2 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : BorelSpace G\ng : G \u2192 E'' \u2192L[\ud835\udd5c] E'\nhf : LocallyIntegrable f\nhcg : HasCompactSupport g\nhg : Continuous g\nx\u2080 : G\nx : E''\nthis : ConvolutionExistsAt f g x\u2080 (precompR E'' L)\n\u22a2 \u222b (a : G), \u2191(\u2191(\u2191(precompR E'' L) (f a)) (g (x\u2080 - a))) x \u2202\u03bc = \u222b (t : G), \u2191(\u2191L (f t)) (\u2191(g (x\u2080 - t)) x) \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[precompR G L, x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nrcases hcg.eq_zero_or_finiteDimensional \ud835\udd5c hg.continuous with (rfl | fin_dim)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhf : LocallyIntegrable f\nx\u2080 : G\nhcg : HasCompactSupport 0\nhg : ContDiff \ud835\udd5c 1 0\n\u22a2 HasFDerivAt (convolution f 0 L) (f \u22c6[precompR G L, x\u2080] fderiv \ud835\udd5c 0) x\u2080\n[PROOFSTEP]\nhave : fderiv \ud835\udd5c (0 : G \u2192 E') = 0 := fderiv_const (0 : E')\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhf : LocallyIntegrable f\nx\u2080 : G\nhcg : HasCompactSupport 0\nhg : ContDiff \ud835\udd5c 1 0\nthis : fderiv \ud835\udd5c 0 = 0\n\u22a2 HasFDerivAt (convolution f 0 L) (f \u22c6[precompR G L, x\u2080] fderiv \ud835\udd5c 0) x\u2080\n[PROOFSTEP]\nsimp only [this, convolution_zero, Pi.zero_apply]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhf : LocallyIntegrable f\nx\u2080 : G\nhcg : HasCompactSupport 0\nhg : ContDiff \ud835\udd5c 1 0\nthis : fderiv \ud835\udd5c 0 = 0\n\u22a2 HasFDerivAt 0 0 x\u2080\n[PROOFSTEP]\nexact hasFDerivAt_const (0 : F) x\u2080\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[precompR G L, x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nhave : ProperSpace G := FiniteDimensional.proper_isROrC \ud835\udd5c G\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[precompR G L, x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nset L' := L.precompR G\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nhave h1 : \u2200\u1da0 x in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => L (f t) (g (x - t))) \u03bc :=\n  eventually_of_forall (hf.aestronglyMeasurable.convolution_integrand_snd L hg.continuous.aestronglyMeasurable)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nhave h2 : \u2200 x, AEStronglyMeasurable (fun t => L' (f t) (fderiv \ud835\udd5c g (x - t))) \u03bc :=\n  hf.aestronglyMeasurable.convolution_integrand_snd L' (hg.continuous_fderiv le_rfl).aestronglyMeasurable\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nhave h3 : \u2200 x t, HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x := fun x t \u21a6 by\n  simpa using\n    (hg.differentiable le_rfl).differentiableAt.hasFDerivAt.comp x ((hasFDerivAt_id x).sub (hasFDerivAt_const t x))\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nx t : G\n\u22a2 HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\n[PROOFSTEP]\nsimpa using\n  (hg.differentiable le_rfl).differentiableAt.hasFDerivAt.comp x ((hasFDerivAt_id x).sub (hasFDerivAt_const t x))\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nlet K' := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\nhave hK' : IsCompact K' :=\n  (hcg.fderiv \ud835\udd5c).neg.add\n    (isCompact_closedBall x\u2080 1)\n      -- porting note: was\n        -- `refine' hasFDerivAt_integral_of_dominated_of_fderiv_le zero_lt_one h1 _ (h2 x\u2080) _ _ _`\n        -- but it failed; surprisingly, `apply` works\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 HasFDerivAt (convolution f g L) (f \u22c6[L', x\u2080] fderiv \ud835\udd5c g) x\u2080\n[PROOFSTEP]\napply hasFDerivAt_integral_of_dominated_of_fderiv_le zero_lt_one h1 _ (h2 x\u2080)\n[GOAL]\ncase inr.h_bound\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : G), x \u2208 ball x\u2080 1 \u2192 \u2016\u2191(\u2191L' (f a)) (fderiv \ud835\udd5c g (x - a))\u2016 \u2264 ?m.2141475 a\n[PROOFSTEP]\nrefine'\n  eventually_of_forall fun t x hx =>\n    (hcg.fderiv \ud835\udd5c).convolution_integrand_bound_right L' (hg.continuous_fderiv le_rfl) (ball_subset_closedBall hx)\n[GOAL]\ncase inr.bound_integrable\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 Integrable fun t =>\n    indicator (-tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1) (fun t => \u2016L'\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016fderiv \ud835\udd5c g i\u2016) t\n[PROOFSTEP]\nrw [integrable_indicator_iff hK'.measurableSet]\n[GOAL]\ncase inr.bound_integrable\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 IntegrableOn (fun t => \u2016L'\u2016 * \u2016f t\u2016 * \u2a06 (i : G), \u2016fderiv \ud835\udd5c g i\u2016) K'\n[PROOFSTEP]\nexact ((hf.integrableOn_isCompact hK').norm.const_mul _).mul_const _\n[GOAL]\ncase inr.h_diff\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc,\n    \u2200 (x : G), x \u2208 ball x\u2080 1 \u2192 HasFDerivAt (fun x => \u2191(\u2191L (f a)) (g (x - a))) (\u2191(\u2191L' (f a)) (fderiv \ud835\udd5c g (x - a))) x\n[PROOFSTEP]\nexact eventually_of_forall fun t x _ => (L _).hasFDerivAt.comp x (h3 x t)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsAddLeftInvariant \u03bc\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c 1 g\nx\u2080 : G\nfin_dim : FiniteDimensional \ud835\udd5c G\nthis : ProperSpace G\nL' : E \u2192L[\ud835\udd5c] (G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] G \u2192L[\ud835\udd5c] F := precompR G L\nh1 : \u2200\u1da0 (x : G) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun t => \u2191(\u2191L (f t)) (g (x - t))) \u03bc\nh2 : \u2200 (x : G), AEStronglyMeasurable (fun t => \u2191(\u2191L' (f t)) (fderiv \ud835\udd5c g (x - t))) \u03bc\nh3 : \u2200 (x t : G), HasFDerivAt (fun x => g (x - t)) (fderiv \ud835\udd5c g (x - t)) x\nK' : Set G := -tsupport (fderiv \ud835\udd5c g) + closedBall x\u2080 1\nhK' : IsCompact K'\n\u22a2 Integrable fun t => \u2191(\u2191L (f t)) (g (x\u2080 - t))\n[PROOFSTEP]\nexact hcg.convolutionExists_right L hf hg.continuous x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nhcf : HasCompactSupport f\nhf : ContDiff \ud835\udd5c 1 f\nhg : LocallyIntegrable g\nx\u2080 : G\n\u22a2 HasFDerivAt (convolution f g L) (fderiv \ud835\udd5c f \u22c6[precompL G L, x\u2080] g) x\u2080\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2074 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nn : \u2115\u221e\ninst\u271d\u2077 : CompleteSpace F\ninst\u271d\u2076 : MeasurableSpace G\n\u03bc \u03bd : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : BorelSpace G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b2 : SigmaFinite \u03bc\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nhcf : HasCompactSupport f\nhf : ContDiff \ud835\udd5c 1 f\nhg : LocallyIntegrable g\nx\u2080 : G\n\u22a2 HasFDerivAt (convolution g f (ContinuousLinearMap.flip L))\n    (g \u22c6[ContinuousLinearMap.flip (precompL G L), x\u2080] fderiv \ud835\udd5c f) x\u2080\n[PROOFSTEP]\nexact hcf.hasFDerivAt_convolution_right L.flip hg hf x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u2080 : \ud835\udd5c \u2192 E\ng\u2080 : \ud835\udd5c \u2192 E'\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \ud835\udd5c\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nhf : LocallyIntegrable f\u2080\nhcg : HasCompactSupport g\u2080\nhg : ContDiff \ud835\udd5c 1 g\u2080\nx\u2080 : \ud835\udd5c\n\u22a2 HasDerivAt (convolution f\u2080 g\u2080 L) (f\u2080 \u22c6[L, x\u2080] deriv g\u2080) x\u2080\n[PROOFSTEP]\nconvert (hcg.hasFDerivAt_convolution_right L hf hg x\u2080).hasDerivAt using 1\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u2080 : \ud835\udd5c \u2192 E\ng\u2080 : \ud835\udd5c \u2192 E'\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \ud835\udd5c\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nhf : LocallyIntegrable f\u2080\nhcg : HasCompactSupport g\u2080\nhg : ContDiff \ud835\udd5c 1 g\u2080\nx\u2080 : \ud835\udd5c\n\u22a2 f\u2080 \u22c6[L, x\u2080] deriv g\u2080 = \u2191(f\u2080 \u22c6[precompR \ud835\udd5c L, x\u2080] fderiv \ud835\udd5c g\u2080) 1\n[PROOFSTEP]\nrw [convolution_precompR_apply L hf (hcg.fderiv \ud835\udd5c) (hg.continuous_fderiv le_rfl)]\n[GOAL]\ncase h.e'_7\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u2080 : \ud835\udd5c \u2192 E\ng\u2080 : \ud835\udd5c \u2192 E'\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\n\u03bc : Measure \ud835\udd5c\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : SigmaFinite \u03bc\nhf : LocallyIntegrable f\u2080\nhcg : HasCompactSupport g\u2080\nhg : ContDiff \ud835\udd5c 1 g\u2080\nx\u2080 : \ud835\udd5c\n\u22a2 f\u2080 \u22c6[L, x\u2080] deriv g\u2080 = f\u2080 \u22c6[L, x\u2080] fun a => \u2191(fderiv \ud835\udd5c g\u2080 a) 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nf\u2080 : \ud835\udd5c \u2192 E\ng\u2080 : \ud835\udd5c \u2192 E'\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : CompleteSpace F\n\u03bc : Measure \ud835\udd5c\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsNegInvariant \u03bc\nhcf : HasCompactSupport f\u2080\nhf : ContDiff \ud835\udd5c 1 f\u2080\nhg : LocallyIntegrable g\u2080\nx\u2080 : \ud835\udd5c\n\u22a2 HasDerivAt (convolution f\u2080 g\u2080 L) (deriv f\u2080 \u22c6[L, x\u2080] g\u2080) x\u2080\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E''\ninst\u271d\u2079 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nf\u2080 : \ud835\udd5c \u2192 E\ng\u2080 : \ud835\udd5c \u2192 E'\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b3 : CompleteSpace F\n\u03bc : Measure \ud835\udd5c\ninst\u271d\u00b2 : IsAddLeftInvariant \u03bc\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : IsNegInvariant \u03bc\nhcf : HasCompactSupport f\u2080\nhf : ContDiff \ud835\udd5c 1 f\u2080\nhg : LocallyIntegrable g\u2080\nx\u2080 : \ud835\udd5c\n\u22a2 HasDerivAt (convolution g\u2080 f\u2080 (ContinuousLinearMap.flip L)) (g\u2080 \u22c6[ContinuousLinearMap.flip L, x\u2080] deriv f\u2080) x\u2080\n[PROOFSTEP]\nexact hcf.hasDerivAt_convolution_right L.flip hg hf x\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nlet g' := fderiv \ud835\udd5c \u21bfg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave A : \u2200 p \u2208 s, Continuous (g p) := fun p hp \u21a6\n  by\n  refine hg.continuousOn.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_\n  simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\np : P\nhp : p \u2208 s\n\u22a2 Continuous (g p)\n[PROOFSTEP]\nrefine hg.continuousOn.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\np : P\nhp : p \u2208 s\nx : G\n\u22a2 (p, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave A' : \u2200 q : P \u00d7 G, q.1 \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q := fun q hq \u21a6\n  by\n  apply (hs.prod isOpen_univ).mem_nhds\n  simpa only [mem_prod, mem_univ, and_true_iff] using hq\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nq : P \u00d7 G\nhq : q.fst \u2208 s\n\u22a2 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\n[PROOFSTEP]\napply (hs.prod isOpen_univ).mem_nhds\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nq : P \u00d7 G\nhq : q.fst \u2208 s\n\u22a2 q \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [mem_prod, mem_univ, and_true_iff] using hq\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave g'_zero : \u2200 p x, p \u2208 s \u2192 x \u2209 k \u2192 g' (p, x) = 0 :=\n  by\n  intro p x hp hx\n  refine' (hasFDerivAt_zero_of_eventually_const 0 _).fderiv\n  have M2 : k\u1d9c \u2208 \ud835\udcdd x := hk.isClosed.isOpen_compl.mem_nhds hx\n  have M1 : s \u2208 \ud835\udcdd p := hs.mem_nhds hp\n  rw [nhds_prod_eq]\n  filter_upwards [prod_mem_prod M1 M2]\n  rintro \u27e8p, y\u27e9 \u27e8hp, hy\u27e9\n  exact hgs p y hp hy\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\n\u22a2 \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n[PROOFSTEP]\nintro p x hp hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 g' (p, x) = 0\n[PROOFSTEP]\nrefine' (hasFDerivAt_zero_of_eventually_const 0 _).fderiv\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 \u21bfg =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nhave M2 : k\u1d9c \u2208 \ud835\udcdd x := hk.isClosed.isOpen_compl.mem_nhds hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\n\u22a2 \u21bfg =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nhave M1 : s \u2208 \ud835\udcdd p := hs.mem_nhds hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 \u21bfg =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nrw [nhds_prod_eq]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 \u21bfg =\u1da0[\ud835\udcdd p \u00d7\u02e2 \ud835\udcdd x] fun x => 0\n[PROOFSTEP]\nfilter_upwards [prod_mem_prod M1 M2]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 \u2200 (a : P \u00d7 G), a \u2208 s \u00d7\u02e2 k\u1d9c \u2192 (\u21bfg) a = 0\n[PROOFSTEP]\nrintro \u27e8p, y\u27e9 \u27e8hp, hy\u27e9\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny\u271d y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\np\u271d : P\nx : G\nhp\u271d : p\u271d \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\u271d\np : P\ny : G\nhp : (p, y).fst \u2208 s\nhy : (p, y).snd \u2208 k\u1d9c\n\u22a2 (\u21bfg) (p, y) = 0\n[PROOFSTEP]\nexact hgs p y hp hy\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nobtain \u27e8\u03b5, C, \u03b5pos, h\u2080\u03b5, h\u03b5\u27e9 : \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.1 \u03b5 \u2286 s \u2227 \u2200 p x, \u2016p - q\u2080.1\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C :=\n  by\n  have A : IsCompact ({q\u2080.1} \u00d7\u02e2 k) := isCompact_singleton.prod hk\n  obtain \u27e8t, kt, t_open, ht\u27e9 : \u2203 t, {q\u2080.1} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Bounded (g' '' t) :=\n    by\n    have B : ContinuousOn g' (s \u00d7\u02e2 univ) := hg.continuousOn_fderiv_of_open (hs.prod isOpen_univ) le_rfl\n    apply exists_isOpen_bounded_image_of_isCompact_of_continuousOn A (hs.prod isOpen_univ) _ B\n    simp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n  obtain \u27e8\u03b5, \u03b5pos, h\u03b5, h'\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 thickening \u03b5 ({ q\u2080.fst } \u00d7\u02e2 k) \u2286 t \u2227 ball q\u2080.1 \u03b5 \u2286 s :=\n    by\n    obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 thickening \u03b5 (({ q\u2080.fst } : Set P) \u00d7\u02e2 k) \u2286 t\n    \u00b7 exact A.exists_thickening_subset_open t_open kt\n    obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d, 0 < \u03b4 \u2227 ball q\u2080.1 \u03b4 \u2286 s\n    \u00b7 exact Metric.isOpen_iff.1 hs _ hq\u2080\n    refine' \u27e8min \u03b5 \u03b4, lt_min \u03b5pos \u03b4pos, _, _\u27e9\n    \u00b7 exact Subset.trans (thickening_mono (min_le_left _ _) _) h\u03b5\n    \u00b7 exact Subset.trans (ball_subset_ball (min_le_right _ _)) h\u03b4\n  obtain \u27e8C, Cpos, hC\u27e9 : \u2203 C, 0 < C \u2227 g' '' t \u2286 closedBall 0 C; exact ht.subset_ball_lt 0 0\n  refine' \u27e8\u03b5, C, \u03b5pos, h'\u03b5, fun p x hp => _\u27e9\n  have hps : p \u2208 s := h'\u03b5 (mem_ball_iff_norm.2 hp)\n  by_cases hx : x \u2208 k\n  \u00b7 have H : (p, x) \u2208 t := by\n      apply h\u03b5\n      refine' mem_thickening_iff.2 \u27e8(q\u2080.1, x), _, _\u27e9\n      \u00b7 simp only [hx, singleton_prod, mem_image, Prod.mk.inj_iff, eq_self_iff_true, true_and_iff, exists_eq_right]\n      \u00b7 rw [\u2190 dist_eq_norm] at hp \n        simpa only [Prod.dist_eq, \u03b5pos, dist_self, max_lt_iff, and_true_iff] using hp\n    have : g' (p, x) \u2208 closedBall (0 : P \u00d7 G \u2192L[\ud835\udd5c] E') C := hC (mem_image_of_mem _ H)\n    rwa [mem_closedBall_zero_iff] at this \n  \u00b7 have : g' (p, x) = 0 := g'_zero _ _ hps hx\n    rw [this]\n    simpa only [norm_zero] using Cpos.le\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nhave A : IsCompact ({q\u2080.1} \u00d7\u02e2 k) := isCompact_singleton.prod hk\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8t, kt, t_open, ht\u27e9 : \u2203 t, {q\u2080.1} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Bounded (g' '' t) :=\n  by\n  have B : ContinuousOn g' (s \u00d7\u02e2 univ) := hg.continuousOn_fderiv_of_open (hs.prod isOpen_univ) le_rfl\n  apply exists_isOpen_bounded_image_of_isCompact_of_continuousOn A (hs.prod isOpen_univ) _ B\n  simp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\n\u22a2 \u2203 t, {q\u2080.fst} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Metric.Bounded (g' '' t)\n[PROOFSTEP]\nhave B : ContinuousOn g' (s \u00d7\u02e2 univ) := hg.continuousOn_fderiv_of_open (hs.prod isOpen_univ) le_rfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nB : ContinuousOn g' (s \u00d7\u02e2 univ)\n\u22a2 \u2203 t, {q\u2080.fst} \u00d7\u02e2 k \u2286 t \u2227 IsOpen t \u2227 Metric.Bounded (g' '' t)\n[PROOFSTEP]\napply exists_isOpen_bounded_image_of_isCompact_of_continuousOn A (hs.prod isOpen_univ) _ B\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nB : ContinuousOn g' (s \u00d7\u02e2 univ)\n\u22a2 {q\u2080.fst} \u00d7\u02e2 k \u2286 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [prod_subset_prod_iff, hq\u2080, singleton_subset_iff, subset_univ, and_self_iff, true_or_iff]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5, h'\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 thickening \u03b5 ({ q\u2080.fst } \u00d7\u02e2 k) \u2286 t \u2227 ball q\u2080.1 \u03b5 \u2286 s :=\n  by\n  obtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 thickening \u03b5 (({ q\u2080.fst } : Set P) \u00d7\u02e2 k) \u2286 t\n  \u00b7 exact A.exists_thickening_subset_open t_open kt\n  obtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d, 0 < \u03b4 \u2227 ball q\u2080.1 \u03b4 \u2286 s\n  \u00b7 exact Metric.isOpen_iff.1 hs _ hq\u2080\n  refine' \u27e8min \u03b5 \u03b4, lt_min \u03b5pos \u03b4pos, _, _\u27e9\n  \u00b7 exact Subset.trans (thickening_mono (min_le_left _ _) _) h\u03b5\n  \u00b7 exact Subset.trans (ball_subset_ball (min_le_right _ _)) h\u03b4\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t \u2227 ball q\u2080.fst \u03b5 \u2286 s\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9 : \u2203 \u03b5 : \u211d, 0 < \u03b5 \u2227 thickening \u03b5 (({ q\u2080.fst } : Set P) \u00d7\u02e2 k) \u2286 t\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n[PROOFSTEP]\nexact A.exists_thickening_subset_open t_open kt\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t \u2227 ball q\u2080.fst \u03b5 \u2286 s\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d, 0 < \u03b4 \u2227 ball q\u2080.1 \u03b4 \u2286 s\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 ball q\u2080.fst \u03b4 \u2286 s\n[PROOFSTEP]\nexact Metric.isOpen_iff.1 hs _ hq\u2080\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball q\u2080.fst \u03b4 \u2286 s\n\u22a2 \u2203 \u03b5, 0 < \u03b5 \u2227 thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t \u2227 ball q\u2080.fst \u03b5 \u2286 s\n[PROOFSTEP]\nrefine' \u27e8min \u03b5 \u03b4, lt_min \u03b5pos \u03b4pos, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball q\u2080.fst \u03b4 \u2286 s\n\u22a2 thickening (min \u03b5 \u03b4) ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n[PROOFSTEP]\nexact Subset.trans (thickening_mono (min_le_left _ _) _) h\u03b5\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\nh\u03b4 : ball q\u2080.fst \u03b4 \u2286 s\n\u22a2 ball q\u2080.fst (min \u03b5 \u03b4) \u2286 s\n[PROOFSTEP]\nexact Subset.trans (ball_subset_ball (min_le_right _ _)) h\u03b4\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8C, Cpos, hC\u27e9 : \u2203 C, 0 < C \u2227 g' '' t \u2286 closedBall 0 C\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\n\u22a2 \u2203 C, 0 < C \u2227 g' '' t \u2286 closedBall 0 C\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nexact ht.subset_ball_lt 0 0\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\n\u22a2 \u2203 \u03b5 C, 0 < \u03b5 \u2227 ball q\u2080.fst \u03b5 \u2286 s \u2227 \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nrefine' \u27e8\u03b5, C, \u03b5pos, h'\u03b5, fun p x hp => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nhave hps : p \u2208 s := h'\u03b5 (mem_ball_iff_norm.2 hp)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nby_cases hx : x \u2208 k\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nhave H : (p, x) \u2208 t := by\n  apply h\u03b5\n  refine' mem_thickening_iff.2 \u27e8(q\u2080.1, x), _, _\u27e9\n  \u00b7 simp only [hx, singleton_prod, mem_image, Prod.mk.inj_iff, eq_self_iff_true, true_and_iff, exists_eq_right]\n  \u00b7 rw [\u2190 dist_eq_norm] at hp \n    simpa only [Prod.dist_eq, \u03b5pos, dist_self, max_lt_iff, and_true_iff] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 (p, x) \u2208 t\n[PROOFSTEP]\napply h\u03b5\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 (p, x) \u2208 thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k)\n[PROOFSTEP]\nrefine' mem_thickening_iff.2 \u27e8(q\u2080.1, x), _, _\u27e9\n[GOAL]\ncase a.refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 (q\u2080.fst, x) \u2208 {q\u2080.fst} \u00d7\u02e2 k\n[PROOFSTEP]\nsimp only [hx, singleton_prod, mem_image, Prod.mk.inj_iff, eq_self_iff_true, true_and_iff, exists_eq_right]\n[GOAL]\ncase a.refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 dist (p, x) (q\u2080.fst, x) < \u03b5\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm] at hp \n[GOAL]\ncase a.refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : dist p q\u2080.fst < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\n\u22a2 dist (p, x) (q\u2080.fst, x) < \u03b5\n[PROOFSTEP]\nsimpa only [Prod.dist_eq, \u03b5pos, dist_self, max_lt_iff, and_true_iff] using hp\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nhave : g' (p, x) \u2208 closedBall (0 : P \u00d7 G \u2192L[\ud835\udd5c] E') C := hC (mem_image_of_mem _ H)\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : x \u2208 k\nH : (p, x) \u2208 t\nthis : g' (p, x) \u2208 closedBall 0 C\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nrwa [mem_closedBall_zero_iff] at this \n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nhave : g' (p, x) = 0 := g'_zero _ _ hps hx\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : \u00acx \u2208 k\nthis : g' (p, x) = 0\n\u22a2 \u2016g' (p, x)\u2016 \u2264 C\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA\u271d : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\nA : IsCompact ({q\u2080.fst} \u00d7\u02e2 k)\nt : Set (P \u00d7 G)\nkt : {q\u2080.fst} \u00d7\u02e2 k \u2286 t\nt_open : IsOpen t\nht : Metric.Bounded (g' '' t)\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nh\u03b5 : thickening \u03b5 ({q\u2080.fst} \u00d7\u02e2 k) \u2286 t\nh'\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nC : \u211d\nCpos : 0 < C\nhC : g' '' t \u2286 closedBall 0 C\np : P\nx : G\nhp : \u2016p - q\u2080.fst\u2016 < \u03b5\nhps : p \u2208 s\nhx : \u00acx \u2208 k\nthis : g' (p, x) = 0\n\u22a2 \u20160\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [norm_zero] using Cpos.le\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I1 : \u2200\u1da0 x : P \u00d7 G in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a : G => L (f a) (g x.1 (x.2 - a))) \u03bc :=\n  by\n  filter_upwards [A' q\u2080 hq\u2080]\n  rintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n  refine' (HasCompactSupport.convolutionExists_right L _ hf (A _ hp) _).1\n  apply isCompact_of_isClosed_subset hk (isClosed_tsupport _)\n  exact closure_minimal (support_subset_iff'.2 fun z hz => hgs _ _ hp hz) hk.isClosed\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n\u22a2 \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\n[PROOFSTEP]\nfilter_upwards [A' q\u2080 hq\u2080]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\n\u22a2 \u2200 (a : P \u00d7 G), a \u2208 s \u00d7\u02e2 univ \u2192 AEStronglyMeasurable (fun a_2 => \u2191(\u2191L (f a_2)) (g a.fst (a.snd - a_2))) \u03bc\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g (p, x).fst ((p, x).snd - a))) \u03bc\n[PROOFSTEP]\nrefine' (HasCompactSupport.convolutionExists_right L _ hf (A _ hp) _).1\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 HasCompactSupport (g (p, x).fst)\n[PROOFSTEP]\napply isCompact_of_isClosed_subset hk (isClosed_tsupport _)\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 tsupport (g (p, x).fst) \u2286 k\n[PROOFSTEP]\nexact closure_minimal (support_subset_iff'.2 fun z hz => hgs _ _ hp hz) hk.isClosed\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I2 : Integrable (fun a : G => L (f a) (g q\u2080.1 (q\u2080.2 - a))) \u03bc :=\n  by\n  have M : HasCompactSupport (g q\u2080.1) := HasCompactSupport.intro hk fun x hx => hgs q\u2080.1 x hq\u2080 hx\n  apply M.convolutionExists_right L hf (A q\u2080.1 hq\u2080) q\u2080.2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\n\u22a2 Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\n[PROOFSTEP]\nhave M : HasCompactSupport (g q\u2080.1) := HasCompactSupport.intro hk fun x hx => hgs q\u2080.1 x hq\u2080 hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nM : HasCompactSupport (g q\u2080.fst)\n\u22a2 Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\n[PROOFSTEP]\napply M.convolutionExists_right L hf (A q\u2080.1 hq\u2080) q\u2080.2\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I3 : AEStronglyMeasurable (fun a : G => (L (f a)).comp (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc :=\n  by\n  have T : HasCompactSupport fun y => g' (q\u2080.1, y) := HasCompactSupport.intro hk fun x hx => g'_zero q\u2080.1 x hq\u2080 hx\n  apply (HasCompactSupport.convolutionExists_right (L.precompR (P \u00d7 G) : _) T hf _ q\u2080.2).1\n  have : ContinuousOn g' (s \u00d7\u02e2 univ) := hg.continuousOn_fderiv_of_open (hs.prod isOpen_univ) le_rfl\n  apply this.comp_continuous (continuous_const.prod_mk continuous_id')\n  intro x\n  simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hq\u2080\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\n\u22a2 AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\n[PROOFSTEP]\nhave T : HasCompactSupport fun y => g' (q\u2080.1, y) := HasCompactSupport.intro hk fun x hx => g'_zero q\u2080.1 x hq\u2080 hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nT : HasCompactSupport fun y => g' (q\u2080.fst, y)\n\u22a2 AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\n[PROOFSTEP]\napply (HasCompactSupport.convolutionExists_right (L.precompR (P \u00d7 G) : _) T hf _ q\u2080.2).1\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nT : HasCompactSupport fun y => g' (q\u2080.fst, y)\n\u22a2 Continuous fun y => g' (q\u2080.fst, y)\n[PROOFSTEP]\nhave : ContinuousOn g' (s \u00d7\u02e2 univ) := hg.continuousOn_fderiv_of_open (hs.prod isOpen_univ) le_rfl\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nT : HasCompactSupport fun y => g' (q\u2080.fst, y)\nthis : ContinuousOn g' (s \u00d7\u02e2 univ)\n\u22a2 Continuous fun y => g' (q\u2080.fst, y)\n[PROOFSTEP]\napply this.comp_continuous (continuous_const.prod_mk continuous_id')\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nT : HasCompactSupport fun y => g' (q\u2080.fst, y)\nthis : ContinuousOn g' (s \u00d7\u02e2 univ)\n\u22a2 \u2200 (x : G), (q\u2080.fst, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nT : HasCompactSupport fun y => g' (q\u2080.fst, y)\nthis : ContinuousOn g' (s \u00d7\u02e2 univ)\nx : G\n\u22a2 (q\u2080.fst, x) \u2208 s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hq\u2080\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nset K' := (-k + {q\u2080.2} : Set G) with K'_def\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave hK' : IsCompact K' := hk.neg.add isCompact_singleton\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nobtain \u27e8U, U_open, K'U, hU\u27e9 : \u2203 U, IsOpen U \u2227 K' \u2286 U \u2227 IntegrableOn f U \u03bc\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\n\u22a2 \u2203 U, IsOpen U \u2227 K' \u2286 U \u2227 IntegrableOn f U\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nexact hf.integrableOn_nhds_isCompact hK'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, \u03b4\u03b5, h\u03b4\u27e9 : \u2203 \u03b4, (0 : \u211d) < \u03b4 \u2227 \u03b4 \u2264 \u03b5 \u2227 K' + ball 0 \u03b4 \u2286 U :=\n  by\n  obtain \u27e8V, V_mem, hV\u27e9 : \u2203 V \u2208 \ud835\udcdd (0 : G), K' + V \u2286 U := compact_open_separated_add_right hK' U_open K'U\n  rcases Metric.mem_nhds_iff.1 V_mem with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9\n  refine' \u27e8min \u03b4 \u03b5, lt_min \u03b4pos \u03b5pos, min_le_right \u03b4 \u03b5, _\u27e9\n  exact (add_subset_add_left ((ball_subset_ball (min_le_left _ _)).trans h\u03b4)).trans hV\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 \u2264 \u03b5 \u2227 K' + ball 0 \u03b4 \u2286 U\n[PROOFSTEP]\nobtain \u27e8V, V_mem, hV\u27e9 : \u2203 V \u2208 \ud835\udcdd (0 : G), K' + V \u2286 U := compact_open_separated_add_right hK' U_open K'U\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 \u2264 \u03b5 \u2227 K' + ball 0 \u03b4 \u2286 U\n[PROOFSTEP]\nrcases Metric.mem_nhds_iff.1 V_mem with \u27e8\u03b4, \u03b4pos, h\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball 0 \u03b4 \u2286 V\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u03b4 \u2264 \u03b5 \u2227 K' + ball 0 \u03b4 \u2286 U\n[PROOFSTEP]\nrefine' \u27e8min \u03b4 \u03b5, lt_min \u03b4pos \u03b5pos, min_le_right \u03b4 \u03b5, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\nV : Set G\nV_mem : V \u2208 \ud835\udcdd 0\nhV : K' + V \u2286 U\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\nh\u03b4 : ball 0 \u03b4 \u2286 V\n\u22a2 K' + ball 0 (min \u03b4 \u03b5) \u2286 U\n[PROOFSTEP]\nexact (add_subset_add_left ((ball_subset_ball (min_le_left _ _)).trans h\u03b4)).trans hV\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nletI :=\n  ContinuousLinearMap.hasOpNorm (\ud835\udd5c := \ud835\udd5c) (\ud835\udd5c\u2082 := \ud835\udd5c) (E := E) (F := (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) (\u03c3\u2081\u2082 :=\n    RingHom.id \ud835\udd5c)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nlet bound : G \u2192 \u211d := indicator U fun t => \u2016(L.precompR (P \u00d7 G))\u2016 * \u2016f t\u2016 * C\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I4 : \u2200\u1d50 a : G \u2202\u03bc, \u2200 x : P \u00d7 G, dist x q\u2080 < \u03b4 \u2192 \u2016L.precompR (P \u00d7 G) (f a) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a :=\n  by\n  apply eventually_of_forall\n  intro a x hx\n  rw [Prod.dist_eq, dist_eq_norm, dist_eq_norm] at hx \n  have : (-tsupport fun a => g' (x.1, a)) + ball q\u2080.2 \u03b4 \u2286 U :=\n    by\n    apply Subset.trans _ h\u03b4\n    rw [K'_def, add_assoc]\n    apply add_subset_add\n    \u00b7 rw [neg_subset_neg]\n      refine closure_minimal (support_subset_iff'.2 fun z hz => ?_) hk.isClosed\n      apply g'_zero x.1 z (h\u2080\u03b5 _) hz\n      rw [mem_ball_iff_norm]\n      exact ((le_max_left _ _).trans_lt hx).trans_le \u03b4\u03b5\n    \u00b7 simp only [add_ball, thickening_singleton, zero_vadd, subset_rfl]\n  apply convolution_integrand_bound_right_of_le_of_subset _ _ _ this\n  \u00b7 intro y\n    exact h\u03b5 _ _ (((le_max_left _ _).trans_lt hx).trans_le \u03b4\u03b5)\n  \u00b7 rw [mem_ball_iff_norm]\n    exact (le_max_right _ _).trans_lt hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\n\u22a2 \u2200 (x : G) (x_1 : P \u00d7 G), dist x_1 q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f x)) (g' (x_1.fst, x_1.snd - x))\u2016 \u2264 bound x\n[PROOFSTEP]\nintro a x hx\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\nrw [Prod.dist_eq, dist_eq_norm, dist_eq_norm] at hx \n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\nhave : (-tsupport fun a => g' (x.1, a)) + ball q\u2080.2 \u03b4 \u2286 U :=\n  by\n  apply Subset.trans _ h\u03b4\n  rw [K'_def, add_assoc]\n  apply add_subset_add\n  \u00b7 rw [neg_subset_neg]\n    refine closure_minimal (support_subset_iff'.2 fun z hz => ?_) hk.isClosed\n    apply g'_zero x.1 z (h\u2080\u03b5 _) hz\n    rw [mem_ball_iff_norm]\n    exact ((le_max_left _ _).trans_lt hx).trans_le \u03b4\u03b5\n  \u00b7 simp only [add_ball, thickening_singleton, zero_vadd, subset_rfl]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\n[PROOFSTEP]\napply Subset.trans _ h\u03b4\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 K' + ball 0 \u03b4\n[PROOFSTEP]\nrw [K'_def, add_assoc]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 -k + ({q\u2080.snd} + ball 0 \u03b4)\n[PROOFSTEP]\napply add_subset_add\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 (-tsupport fun a => g' (x.fst, a)) \u2286 -k\n[PROOFSTEP]\nrw [neg_subset_neg]\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 (tsupport fun a => g' (x.fst, a)) \u2286 k\n[PROOFSTEP]\nrefine closure_minimal (support_subset_iff'.2 fun z hz => ?_) hk.isClosed\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nz : G\nhz : \u00acz \u2208 k\n\u22a2 g' (x.fst, z) = 0\n[PROOFSTEP]\napply g'_zero x.1 z (h\u2080\u03b5 _) hz\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nz : G\nhz : \u00acz \u2208 k\n\u22a2 x.fst \u2208 ball q\u2080.fst \u03b5\n[PROOFSTEP]\nrw [mem_ball_iff_norm]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nz : G\nhz : \u00acz \u2208 k\n\u22a2 \u2016x.fst - q\u2080.fst\u2016 < \u03b5\n[PROOFSTEP]\nexact ((le_max_left _ _).trans_lt hx).trans_le \u03b4\u03b5\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n\u22a2 ball q\u2080.snd \u03b4 \u2286 {q\u2080.snd} + ball 0 \u03b4\n[PROOFSTEP]\nsimp only [add_ball, thickening_singleton, zero_vadd, subset_rfl]\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nthis : (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\n\u22a2 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n[PROOFSTEP]\napply convolution_integrand_bound_right_of_le_of_subset _ _ _ this\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nthis : (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\n\u22a2 \u2200 (i : G), \u2016g' (x.fst, i)\u2016 \u2264 C\n[PROOFSTEP]\nintro y\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny\u271d y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nthis : (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\ny : G\n\u22a2 \u2016g' (x.fst, y)\u2016 \u2264 C\n[PROOFSTEP]\nexact h\u03b5 _ _ (((le_max_left _ _).trans_lt hx).trans_le \u03b4\u03b5)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nthis : (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\n\u22a2 x.snd \u2208 ball q\u2080.snd \u03b4\n[PROOFSTEP]\nrw [mem_ball_iff_norm]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\na : G\nx : P \u00d7 G\nhx : max \u2016x.fst - q\u2080.fst\u2016 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\nthis : (-tsupport fun a => g' (x.fst, a)) + ball q\u2080.snd \u03b4 \u2286 U\n\u22a2 \u2016x.snd - q\u2080.snd\u2016 < \u03b4\n[PROOFSTEP]\nexact (le_max_right _ _).trans_lt hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I5 : Integrable bound \u03bc := by\n  rw [integrable_indicator_iff U_open.measurableSet]\n  exact (hU.norm.const_mul _).mul_const _\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n\u22a2 Integrable bound\n[PROOFSTEP]\nrw [integrable_indicator_iff U_open.measurableSet]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\n\u22a2 IntegrableOn (fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C) U\n[PROOFSTEP]\nexact (hU.norm.const_mul _).mul_const _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nhave I6 :\n  \u2200\u1d50 a : G \u2202\u03bc,\n    \u2200 x : P \u00d7 G,\n      dist x q\u2080 < \u03b4 \u2192\n        HasFDerivAt (fun x : P \u00d7 G => L (f a) (g x.1 (x.2 - a))) ((L (f a)).comp (g' (x.fst, x.snd - a))) x :=\n  by\n  apply eventually_of_forall\n  intro a x hx\n  apply (L _).hasFDerivAt.comp x\n  have N : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.1, x.2 - a) := by\n    apply A'\n    apply h\u2080\u03b5\n    rw [Prod.dist_eq] at hx \n    exact lt_of_lt_of_le (lt_of_le_of_lt (le_max_left _ _) hx) \u03b4\u03b5\n  have Z := ((hg.differentiableOn le_rfl).differentiableAt N).hasFDerivAt\n  have Z' : HasFDerivAt (fun x : P \u00d7 G => (x.1, x.2 - a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x :=\n    by\n    have : (fun x : P \u00d7 G => (x.1, x.2 - a)) = _root_.id - fun x => (0, a) := by\n      ext x <;> simp only [Pi.sub_apply, id.def, Prod.fst_sub, sub_zero, Prod.snd_sub]\n    rw [this]\n    exact (hasFDerivAt_id x).sub_const (0, a)\n  exact Z.comp x Z'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\n\u22a2 \u2200\u1d50 (a : G) \u2202\u03bc,\n    \u2200 (x : P \u00d7 G),\n      dist x q\u2080 < \u03b4 \u2192\n        HasFDerivAt (fun x => \u2191(\u2191L (f a)) (g x.fst (x.snd - a)))\n          (ContinuousLinearMap.comp (\u2191L (f a)) (g' (x.fst, x.snd - a))) x\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\n\u22a2 \u2200 (x : G) (x_1 : P \u00d7 G),\n    dist x_1 q\u2080 < \u03b4 \u2192\n      HasFDerivAt (fun x_2 => \u2191(\u2191L (f x)) (g x_2.fst (x_2.snd - x)))\n        (ContinuousLinearMap.comp (\u2191L (f x)) (g' (x_1.fst, x_1.snd - x))) x_1\n[PROOFSTEP]\nintro a x hx\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 HasFDerivAt (fun x => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) (ContinuousLinearMap.comp (\u2191L (f a)) (g' (x.fst, x.snd - a)))\n    x\n[PROOFSTEP]\napply (L _).hasFDerivAt.comp x\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 HasFDerivAt (fun x => g x.fst (x.snd - a)) (g' (x.fst, x.snd - a)) x\n[PROOFSTEP]\nhave N : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.1, x.2 - a) := by\n  apply A'\n  apply h\u2080\u03b5\n  rw [Prod.dist_eq] at hx \n  exact lt_of_lt_of_le (lt_of_le_of_lt (le_max_left _ _) hx) \u03b4\u03b5\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\n[PROOFSTEP]\napply A'\n[GOAL]\ncase a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 (x.fst, x.snd - a).fst \u2208 s\n[PROOFSTEP]\napply h\u2080\u03b5\n[GOAL]\ncase a.a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\n\u22a2 (x.fst, x.snd - a).fst \u2208 ball q\u2080.fst \u03b5\n[PROOFSTEP]\nrw [Prod.dist_eq] at hx \n[GOAL]\ncase a.a\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : max (dist x.fst q\u2080.fst) (dist x.snd q\u2080.snd) < \u03b4\n\u22a2 (x.fst, x.snd - a).fst \u2208 ball q\u2080.fst \u03b5\n[PROOFSTEP]\nexact lt_of_lt_of_le (lt_of_le_of_lt (le_max_left _ _) hx) \u03b4\u03b5\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\n\u22a2 HasFDerivAt (fun x => g x.fst (x.snd - a)) (g' (x.fst, x.snd - a)) x\n[PROOFSTEP]\nhave Z := ((hg.differentiableOn le_rfl).differentiableAt N).hasFDerivAt\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\n\u22a2 HasFDerivAt (fun x => g x.fst (x.snd - a)) (g' (x.fst, x.snd - a)) x\n[PROOFSTEP]\nhave Z' : HasFDerivAt (fun x : P \u00d7 G => (x.1, x.2 - a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x :=\n  by\n  have : (fun x : P \u00d7 G => (x.1, x.2 - a)) = _root_.id - fun x => (0, a) := by\n    ext x <;> simp only [Pi.sub_apply, id.def, Prod.fst_sub, sub_zero, Prod.snd_sub]\n  rw [this]\n  exact (hasFDerivAt_id x).sub_const (0, a)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\n\u22a2 HasFDerivAt (fun x => (x.fst, x.snd - a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x\n[PROOFSTEP]\nhave : (fun x : P \u00d7 G => (x.1, x.2 - a)) = _root_.id - fun x => (0, a) := by\n  ext x <;> simp only [Pi.sub_apply, id.def, Prod.fst_sub, sub_zero, Prod.snd_sub]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\n\u22a2 (fun x => (x.fst, x.snd - a)) = _root_.id - fun x => (0, a)\n[PROOFSTEP]\next x\n[GOAL]\ncase h.h\u2081\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d\u00b9 x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx\u271d : P \u00d7 G\nhx : dist x\u271d q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x\u271d.fst, x\u271d.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x\u271d.fst, x\u271d.snd - a)) (x\u271d.fst, x\u271d.snd - a)\nx : P \u00d7 G\n\u22a2 (x.fst, x.snd - a).fst = ((_root_.id - fun x => (0, a)) x).fst\n[PROOFSTEP]\nsimp only [Pi.sub_apply, id.def, Prod.fst_sub, sub_zero, Prod.snd_sub]\n[GOAL]\ncase h.h\u2082\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d\u00b9 x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx\u271d : P \u00d7 G\nhx : dist x\u271d q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x\u271d.fst, x\u271d.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x\u271d.fst, x\u271d.snd - a)) (x\u271d.fst, x\u271d.snd - a)\nx : P \u00d7 G\n\u22a2 (x.fst, x.snd - a).snd = ((_root_.id - fun x => (0, a)) x).snd\n[PROOFSTEP]\nsimp only [Pi.sub_apply, id.def, Prod.fst_sub, sub_zero, Prod.snd_sub]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\nthis : (fun x => (x.fst, x.snd - a)) = _root_.id - fun x => (0, a)\n\u22a2 HasFDerivAt (fun x => (x.fst, x.snd - a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis\u271d : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\nthis : (fun x => (x.fst, x.snd - a)) = _root_.id - fun x => (0, a)\n\u22a2 HasFDerivAt (_root_.id - fun x => (0, a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x\n[PROOFSTEP]\nexact (hasFDerivAt_id x).sub_const (0, a)\n[GOAL]\ncase hp\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\na : G\nx : P \u00d7 G\nhx : dist x q\u2080 < \u03b4\nN : s \u00d7\u02e2 univ \u2208 \ud835\udcdd (x.fst, x.snd - a)\nZ : HasFDerivAt (\u21bfg) (fderiv \ud835\udd5c (\u21bfg) (x.fst, x.snd - a)) (x.fst, x.snd - a)\nZ' : HasFDerivAt (fun x => (x.fst, x.snd - a)) (ContinuousLinearMap.id \ud835\udd5c (P \u00d7 G)) x\n\u22a2 HasFDerivAt (fun x => g x.fst (x.snd - a)) (g' (x.fst, x.snd - a)) x\n[PROOFSTEP]\nexact Z.comp x Z'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng\u271d g'\u271d : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c 1 (\u21bfg) (s \u00d7\u02e2 univ)\nq\u2080 : P \u00d7 G\nhq\u2080 : q\u2080.fst \u2208 s\ng' : P \u00d7 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] E' := fderiv \ud835\udd5c \u21bfg\nA : \u2200 (p : P), p \u2208 s \u2192 Continuous (g p)\nA' : \u2200 (q : P \u00d7 G), q.fst \u2208 s \u2192 s \u00d7\u02e2 univ \u2208 \ud835\udcdd q\ng'_zero : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g' (p, x) = 0\n\u03b5 C : \u211d\n\u03b5pos : 0 < \u03b5\nh\u2080\u03b5 : ball q\u2080.fst \u03b5 \u2286 s\nh\u03b5 : \u2200 (p : P) (x : G), \u2016p - q\u2080.fst\u2016 < \u03b5 \u2192 \u2016g' (p, x)\u2016 \u2264 C\nI1 : \u2200\u1da0 (x : P \u00d7 G) in \ud835\udcdd q\u2080, AEStronglyMeasurable (fun a => \u2191(\u2191L (f a)) (g x.fst (x.snd - a))) \u03bc\nI2 : Integrable fun a => \u2191(\u2191L (f a)) (g q\u2080.fst (q\u2080.snd - a))\nI3 : AEStronglyMeasurable (fun a => ContinuousLinearMap.comp (\u2191L (f a)) (g' (q\u2080.fst, q\u2080.snd - a))) \u03bc\nK' : Set G := -k + {q\u2080.snd}\nK'_def : K' = -k + {q\u2080.snd}\nhK' : IsCompact K'\nU : Set G\nU_open : IsOpen U\nK'U : K' \u2286 U\nhU : IntegrableOn f U\n\u03b4 : \u211d\n\u03b4pos : 0 < \u03b4\n\u03b4\u03b5 : \u03b4 \u2264 \u03b5\nh\u03b4 : K' + ball 0 \u03b4 \u2286 U\nthis : Norm (E \u2192L[\ud835\udd5c] (P \u00d7 G \u2192L[\ud835\udd5c] E') \u2192L[\ud835\udd5c] P \u00d7 G \u2192L[\ud835\udd5c] F) := hasOpNorm\nbound : G \u2192 \u211d := indicator U fun t => \u2016precompR (P \u00d7 G) L\u2016 * \u2016f t\u2016 * C\nI4 : \u2200\u1d50 (a : G) \u2202\u03bc, \u2200 (x : P \u00d7 G), dist x q\u2080 < \u03b4 \u2192 \u2016\u2191(\u2191(precompR (P \u00d7 G) L) (f a)) (g' (x.fst, x.snd - a))\u2016 \u2264 bound a\nI5 : Integrable bound\nI6 :\n  \u2200\u1d50 (a : G) \u2202\u03bc,\n    \u2200 (x : P \u00d7 G),\n      dist x q\u2080 < \u03b4 \u2192\n        HasFDerivAt (fun x => \u2191(\u2191L (f a)) (g x.fst (x.snd - a)))\n          (ContinuousLinearMap.comp (\u2191L (f a)) (g' (x.fst, x.snd - a))) x\n\u22a2 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f \u22c6[precompR (P \u00d7 G) L, q\u2080.snd] fun x => fderiv \ud835\udd5c (\u21bfg) (q\u2080.fst, x)) q\u2080\n[PROOFSTEP]\nexact hasFDerivAt_integral_of_dominated_of_fderiv_le \u03b4pos I1 I2 I3 I4 I5 I6\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d : Type uE'\nE'' : Type uE''\nF\u271d : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b2\u2078 : NormedAddCommGroup E\ninst\u271d\u00b2\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b2\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b2\u2075 : NormedAddCommGroup F\u271d\nf\u271d f' : G\u271d \u2192 E\ng\u271d g' : G\u271d \u2192 E'\u271d\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b2\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u00b2 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b2\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2070 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u2079 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u2078 : CompleteSpace F\u271d\ninst\u271d\u00b9\u2077 : MeasurableSpace G\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup G\u271d\ninst\u271d\u00b9\u2075 : BorelSpace G\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\nG E' F P : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\ninduction' n using ENat.nat_induction with n ih ih generalizing g E' F\n[GOAL]\ncase h0\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContDiffOn \ud835\udd5c 0 (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c 0 (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrw [contDiffOn_zero] at hg \u22a2\n[GOAL]\ncase h0\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContinuousOn (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact continuousOn_convolution_right_with_param L hk hgs hf hg\n[GOAL]\ncase hsuc\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet f' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => (f \u22c6[L.precompR (P \u00d7 G), \u03bc] fun x : G => fderiv \ud835\udd5c (uncurry g) (p, x)) a\n[GOAL]\ncase hsuc\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (\u21bfg) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave A : \u2200 q\u2080 : P \u00d7 G, q\u2080.1 \u2208 s \u2192 HasFDerivAt (fun q : P \u00d7 G => (f \u22c6[L, \u03bc] g q.1) q.2) (f' q\u2080.1 q\u2080.2) q\u2080 :=\n  hasFDerivAt_convolution_right_with_param L hs hk hgs hf hg.one_of_succ\n[GOAL]\ncase hsuc\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (\u21bfg) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 ContDiffOn \ud835\udd5c (\u2191(Nat.succ n)) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrw [contDiffOn_succ_iff_fderiv_of_open (hs.prod (@isOpen_univ G _))] at hg \u22a2\n[GOAL]\ncase hsuc\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 DifferentiableOn \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) \u2227\n    ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) y) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase hsuc.left\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 DifferentiableOn \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase hsuc.left.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 DifferentiableWithinAt \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ) (p, x)\n[PROOFSTEP]\nexact (A (p, x) hp).differentiableAt.differentiableWithinAt\n[GOAL]\ncase hsuc.right\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) y) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nsuffices H : ContDiffOn \ud835\udd5c n (\u21bff') (s \u00d7\u02e2 univ)\n[GOAL]\ncase hsuc.right\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\nH : ContDiffOn \ud835\udd5c (\u2191n) (\u21bff') (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) y) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\napply H.congr\n[GOAL]\ncase hsuc.right\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\nH : ContDiffOn \ud835\udd5c (\u2191n) (\u21bff') (s \u00d7\u02e2 univ)\n\u22a2 \u2200 (x : P \u00d7 G), x \u2208 s \u00d7\u02e2 univ \u2192 fderiv \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) x = (\u21bff') x\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase hsuc.right.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\nH : ContDiffOn \ud835\udd5c (\u2191n) (\u21bff') (s \u00d7\u02e2 univ)\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 fderiv \ud835\udd5c (fun q => f \u22c6[L, q.snd] g q.fst) (p, x) = (\u21bff') (p, x)\n[PROOFSTEP]\nexact (A (p, x) hp).fderiv\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (\u21bff') (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave B : \u2200 (p : P) (x : G), p \u2208 s \u2192 x \u2209 k \u2192 fderiv \ud835\udd5c (uncurry g) (p, x) = 0 :=\n  by\n  intro p x hp hx\n  apply (hasFDerivAt_zero_of_eventually_const (0 : E') _).fderiv\n  have M2 : k\u1d9c \u2208 \ud835\udcdd x := IsOpen.mem_nhds hk.isClosed.isOpen_compl hx\n  have M1 : s \u2208 \ud835\udcdd p := hs.mem_nhds hp\n  rw [nhds_prod_eq]\n  filter_upwards [prod_mem_prod M1 M2]\n  rintro \u27e8p, y\u27e9 \u27e8hp, hy\u27e9\n  exact hgs p y hp hy\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\n\u22a2 \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 fderiv \ud835\udd5c (uncurry g) (p, x) = 0\n[PROOFSTEP]\nintro p x hp hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 fderiv \ud835\udd5c (uncurry g) (p, x) = 0\n[PROOFSTEP]\napply (hasFDerivAt_zero_of_eventually_const (0 : E') _).fderiv\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\n\u22a2 uncurry g =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nhave M2 : k\u1d9c \u2208 \ud835\udcdd x := IsOpen.mem_nhds hk.isClosed.isOpen_compl hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\n\u22a2 uncurry g =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nhave M1 : s \u2208 \ud835\udcdd p := hs.mem_nhds hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 uncurry g =\u1da0[\ud835\udcdd (p, x)] fun x => 0\n[PROOFSTEP]\nrw [nhds_prod_eq]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 uncurry g =\u1da0[\ud835\udcdd p \u00d7\u02e2 \ud835\udcdd x] fun x => 0\n[PROOFSTEP]\nfilter_upwards [prod_mem_prod M1 M2]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np : P\nx : G\nhp : p \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\n\u22a2 \u2200 (a : P \u00d7 G), a \u2208 s \u00d7\u02e2 k\u1d9c \u2192 uncurry g a = 0\n[PROOFSTEP]\nrintro \u27e8p, y\u27e9 \u27e8hp, hy\u27e9\n[GOAL]\ncase h.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx\u271d x' : G\u271d\ny\u271d y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\np\u271d : P\nx : G\nhp\u271d : p\u271d \u2208 s\nhx : \u00acx \u2208 k\nM2 : k\u1d9c \u2208 \ud835\udcdd x\nM1 : s \u2208 \ud835\udcdd p\u271d\np : P\ny : G\nhp : (p, y).fst \u2208 s\nhy : (p, y).snd \u2208 k\u1d9c\n\u22a2 uncurry g (p, y) = 0\n[PROOFSTEP]\nexact hgs p y hp hy\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\nB : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 fderiv \ud835\udd5c (uncurry g) (p, x) = 0\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (\u21bff') (s \u00d7\u02e2 univ)\n[PROOFSTEP]\napply ih (L.precompR (P \u00d7 G) : _) B\n[GOAL]\ncase H\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f'\u271d : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nn : \u2115\nih :\n  \u2200 {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : DifferentiableOn \ud835\udd5c (\u21bfg) (s \u00d7\u02e2 univ) \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c (\u21bfg) y) (s \u00d7\u02e2 univ)\nf' : P \u2192 G \u2192 P \u00d7 G \u2192L[\ud835\udd5c] F := fun p a => f \u22c6[precompR (P \u00d7 G) L, a] fun x => fderiv \ud835\udd5c (uncurry g) (p, x)\nA : \u2200 (q\u2080 : P \u00d7 G), q\u2080.fst \u2208 s \u2192 HasFDerivAt (fun q => f \u22c6[L, q.snd] g q.fst) (f' q\u2080.fst q\u2080.snd) q\u2080\nB : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 fderiv \ud835\udd5c (uncurry g) (p, x) = 0\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (\u21bffun p x => fderiv \ud835\udd5c (uncurry g) (p, x)) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nconvert hg.2\n[GOAL]\ncase htop\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nih :\n  \u2200 (n : \u2115) {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : ContDiffOn \ud835\udd5c \u22a4 (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nrw [contDiffOn_top] at hg \u22a2\n[GOAL]\ncase htop\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nih :\n  \u2200 (n : \u2115) {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase htop\n\ud835\udd5c : Type u\ud835\udd5c\nG\u271d : Type uG\nE : Type uE\nE'\u271d\u00b9 : Type uE'\nE'' : Type uE''\nF\u271d\u00b9 : Type uF\nF' : Type uF'\nF'' : Type uF''\nP\u271d : Type uP\ninst\u271d\u00b3\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3\u00b3 : NormedAddCommGroup E'\u271d\u00b9\ninst\u271d\u00b3\u00b2 : NormedAddCommGroup E''\ninst\u271d\u00b3\u00b9 : NormedAddCommGroup F\u271d\u00b9\nf\u271d f' : G\u271d \u2192 E\ng\u271d\u00b9 g' : G\u271d \u2192 E'\u271d\u00b9\nx x' : G\u271d\ny y' : E\ninst\u271d\u00b3\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2\u2078 : NormedSpace \ud835\udd5c E'\u271d\u00b9\ninst\u271d\u00b2\u2077 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b2\u2076 : NormedSpace \u211d F\u271d\u00b9\ninst\u271d\u00b2\u2075 : NormedSpace \ud835\udd5c F\u271d\u00b9\ninst\u271d\u00b2\u2074 : CompleteSpace F\u271d\u00b9\ninst\u271d\u00b2\u00b3 : MeasurableSpace G\u271d\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup G\u271d\ninst\u271d\u00b2\u00b9 : BorelSpace G\u271d\ninst\u271d\u00b2\u2070 : NormedSpace \ud835\udd5c G\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup P\u271d\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c P\u271d\n\u03bc\u271d : Measure G\u271d\nL\u271d\u00b9 : E \u2192L[\ud835\udd5c] E'\u271d\u00b9 \u2192L[\ud835\udd5c] F\u271d\u00b9\nG E'\u271d F\u271d P : Type uP\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\u271d\ninst\u271d\u00b9\u2076 : NormedAddCommGroup F\u271d\ninst\u271d\u00b9\u2075 : NormedSpace \ud835\udd5c E'\u271d\ninst\u271d\u00b9\u2074 : NormedSpace \u211d F\u271d\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c F\u271d\ninst\u271d\u00b9\u00b2 : CompleteSpace F\u271d\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\n\u03bc : Measure G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : BorelSpace G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\ninst\u271d\u2077 : NormedAddCommGroup P\ninst\u271d\u2076 : NormedSpace \ud835\udd5c P\nf : G \u2192 E\nn\u271d : \u2115\u221e\nL\u271d : E \u2192L[\ud835\udd5c] E'\u271d \u2192L[\ud835\udd5c] F\u271d\ng\u271d : P \u2192 G \u2192 E'\u271d\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs\u271d : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g\u271d p x = 0\nhf : LocallyIntegrable f\nhg\u271d : ContDiffOn \ud835\udd5c n\u271d (\u21bfg\u271d) (s \u00d7\u02e2 univ)\nih :\n  \u2200 (n : \u2115) {E' F : Type uP} [inst : NormedAddCommGroup E'] [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace \ud835\udd5c E']\n    [inst_3 : NormedSpace \u211d F] [inst_4 : NormedSpace \ud835\udd5c F] [inst_5 : CompleteSpace F] (L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F)\n    {g : P \u2192 G \u2192 E'},\n    (\u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0) \u2192\n      ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ) \u2192 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\nE' F : Type uP\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhg : \u2200 (n : \u2115), ContDiffOn \ud835\udd5c (\u2191n) (\u21bfg) (s \u00d7\u02e2 univ)\nn : \u2115\n\u22a2 ContDiffOn \ud835\udd5c (\u2191n) (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact ih n L hgs (hg n)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eG : Type max uG uE' uF uP := ULift.{max uE' uF uP} G\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nborelize eG\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eE' : Type max uE' uG uF uP := ULift.{max uG uF uP} E'\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eF : Type max uF uG uE' uP := ULift.{max uG uE' uP} F\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eP : Type max uP uG uE' uF := ULift.{max uG uE' uF} P\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave isoG : eG \u2243L[\ud835\udd5c] G := ContinuousLinearEquiv.ulift\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave isoE' : eE' \u2243L[\ud835\udd5c] E' := ContinuousLinearEquiv.ulift\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave isoF : eF \u2243L[\ud835\udd5c] F := ContinuousLinearEquiv.ulift\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave isoP : eP \u2243L[\ud835\udd5c] P := ContinuousLinearEquiv.ulift\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet ef := f \u2218 isoG\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet e\u03bc : MeasureTheory.Measure eG := Measure.map isoG.symm \u03bc\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eg : eP \u2192 eG \u2192 eE' := fun ep ex => isoE'.symm (g (isoP ep) (isoG ex))\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet eL :=\n  ContinuousLinearMap.comp ((ContinuousLinearEquiv.arrowCongr isoE' isoF).symm : (E' \u2192L[\ud835\udd5c] F) \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF) L\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nlet R := fun q : eP \u00d7 eG => (ef \u22c6[eL, e\u03bc] eg q.1) q.2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave R_contdiff : ContDiffOn \ud835\udd5c n R ((isoP \u207b\u00b9' s) \u00d7\u02e2 univ) :=\n  by\n  have hek : IsCompact (isoG \u207b\u00b9' k) := isoG.toHomeomorph.closedEmbedding.isCompact_preimage hk\n  have hes : IsOpen (isoP \u207b\u00b9' s) := isoP.continuous.isOpen_preimage _ hs\n  refine' contDiffOn_convolution_right_with_param_aux eL hes hek _ _ _\n  \u00b7 intro p x hp hx\n    simp only [(\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.prod_apply, LinearIsometryEquiv.coe_coe,\n      ContinuousLinearEquiv.map_eq_zero_iff]\n    exact hgs _ _ hp hx\n  \u00b7 apply (locallyIntegrable_map_homeomorph isoG.symm.toHomeomorph).2\n    convert hf\n    ext1 x\n    simp only [ContinuousLinearEquiv.coe_toHomeomorph, (\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.apply_symm_apply]\n  \u00b7 apply isoE'.symm.contDiff.comp_contDiffOn\n    apply hg.comp (isoP.prod isoG).contDiff.contDiffOn\n    rintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n    simpa only [mem_preimage, ContinuousLinearEquiv.prod_apply, prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using\n      hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\n\u22a2 ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave hek : IsCompact (isoG \u207b\u00b9' k) := isoG.toHomeomorph.closedEmbedding.isCompact_preimage hk\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\n\u22a2 ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave hes : IsOpen (isoP \u207b\u00b9' s) := isoP.continuous.isOpen_preimage _ hs\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' contDiffOn_convolution_right_with_param_aux eL hes hek _ _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 \u2200 (p : eP) (x : eG), p \u2208 \u2191isoP \u207b\u00b9' s \u2192 \u00acx \u2208 \u2191isoG \u207b\u00b9' k \u2192 eg p x = 0\n[PROOFSTEP]\nintro p x hp hx\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\np : eP\nx : eG\nhp : p \u2208 \u2191isoP \u207b\u00b9' s\nhx : \u00acx \u2208 \u2191isoG \u207b\u00b9' k\n\u22a2 eg p x = 0\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.prod_apply, LinearIsometryEquiv.coe_coe,\n  ContinuousLinearEquiv.map_eq_zero_iff]\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\np : eP\nx : eG\nhp : p \u2208 \u2191isoP \u207b\u00b9' s\nhx : \u00acx \u2208 \u2191isoG \u207b\u00b9' k\n\u22a2 g (\u2191isoP p) (\u2191isoG x) = 0\n[PROOFSTEP]\nexact hgs _ _ hp hx\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 LocallyIntegrable ef\n[PROOFSTEP]\napply (locallyIntegrable_map_homeomorph isoG.symm.toHomeomorph).2\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 LocallyIntegrable (ef \u2218 \u2191(ContinuousLinearEquiv.toHomeomorph (ContinuousLinearEquiv.symm isoG)))\n[PROOFSTEP]\nconvert hf\n[GOAL]\ncase h.e'_6\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 ef \u2218 \u2191(ContinuousLinearEquiv.toHomeomorph (ContinuousLinearEquiv.symm isoG)) = f\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_6.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\nx : G\n\u22a2 (ef \u2218 \u2191(ContinuousLinearEquiv.toHomeomorph (ContinuousLinearEquiv.symm isoG))) x = f x\n[PROOFSTEP]\nsimp only [ContinuousLinearEquiv.coe_toHomeomorph, (\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (\u21bfeg) ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n[PROOFSTEP]\napply isoE'.symm.contDiff.comp_contDiffOn\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 ContDiffOn \ud835\udd5c n (fun x => g (\u2191isoP x.fst) (\u2191isoG x.snd)) ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n[PROOFSTEP]\napply hg.comp (isoP.prod isoG).contDiff.contDiffOn\n[GOAL]\ncase refine'_3\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\n\u22a2 (\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ \u2286 \u2191(ContinuousLinearEquiv.prod isoP isoG) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase refine'_3.mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nhek : IsCompact (\u2191isoG \u207b\u00b9' k)\nhes : IsOpen (\u2191isoP \u207b\u00b9' s)\np : eP\nx : eG\nhp : (p, x).fst \u2208 \u2191isoP \u207b\u00b9' s\n\u22a2 (p, x) \u2208 \u2191(ContinuousLinearEquiv.prod isoP isoG) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [mem_preimage, ContinuousLinearEquiv.prod_apply, prod_mk_mem_set_prod_eq, mem_univ, and_true_iff] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave A : ContDiffOn \ud835\udd5c n (isoF \u2218 R \u2218 (isoP.prod isoG).symm) (s \u00d7\u02e2 univ) :=\n  by\n  apply isoF.contDiff.comp_contDiffOn\n  apply R_contdiff.comp (ContinuousLinearEquiv.contDiff _).contDiffOn\n  rintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n  simpa only [mem_preimage, mem_prod, mem_univ, and_true_iff, ContinuousLinearEquiv.prod_symm,\n    ContinuousLinearEquiv.prod_apply, ContinuousLinearEquiv.apply_symm_apply] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\napply isoF.contDiff.comp_contDiffOn\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\napply R_contdiff.comp (ContinuousLinearEquiv.contDiff _).contDiffOn\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\n\u22a2 s \u00d7\u02e2 univ \u2286 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG)) \u207b\u00b9' (\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9 \u27e8hp, -\u27e9\n[GOAL]\ncase mk.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\np : P\nx : G\nhp : (p, x).fst \u2208 s\n\u22a2 (p, x) \u2208 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG)) \u207b\u00b9' (\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [mem_preimage, mem_prod, mem_univ, and_true_iff, ContinuousLinearEquiv.prod_symm,\n  ContinuousLinearEquiv.prod_apply, ContinuousLinearEquiv.apply_symm_apply] using hp\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave : isoF \u2218 R \u2218 (isoP.prod isoG).symm = fun q : P \u00d7 G => (f \u22c6[L, \u03bc] g q.1) q.2 :=\n  by\n  apply funext\n  rintro \u27e8p, x\u27e9\n  simp only [LinearIsometryEquiv.coe_coe, (\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.prod_symm, ContinuousLinearEquiv.prod_apply]\n  simp only [convolution, coe_comp', ContinuousLinearEquiv.coe_coe, (\u00b7 \u2218 \u00b7)]\n  rw [ClosedEmbedding.integral_map, \u2190 isoF.integral_comp_comm]\n  swap; \u00b7 exact isoG.symm.toHomeomorph.closedEmbedding\n  congr 1\n  ext1 a\n  simp only [(\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.apply_symm_apply, coe_comp', ContinuousLinearEquiv.prod_apply,\n    ContinuousLinearEquiv.map_sub, ContinuousLinearEquiv.arrowCongr, ContinuousLinearEquiv.arrowCongrSL_symm_apply,\n    ContinuousLinearEquiv.coe_coe, Function.comp_apply, ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\n\u22a2 \u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG)) = fun q => f \u22c6[L, q.snd] g q.fst\n[PROOFSTEP]\napply funext\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\n\u22a2 \u2200 (x : P \u00d7 G),\n    (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) x = f \u22c6[L, x.snd] g x.fst\n[PROOFSTEP]\nrintro \u27e8p, x\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (p, x) =\n    f \u22c6[L, (p, x).snd] g (p, x).fst\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.coe_coe, (\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.prod_symm, ContinuousLinearEquiv.prod_apply]\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 \u2191isoF\n      ((fun x =>\n          f\n            (\u2191isoG\n              x)) \u22c6[ContinuousLinearMap.comp\n          (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L,\n        \u2191(ContinuousLinearEquiv.symm isoG) x] fun ex =>\n        \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p)) (\u2191isoG ex))) =\n    f \u22c6[L, x] g p\n[PROOFSTEP]\nsimp only [convolution, coe_comp', ContinuousLinearEquiv.coe_coe, (\u00b7 \u2218 \u00b7)]\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 \u2191isoF\n      (\u222b (t : ULift G),\n        \u2191(\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF)) (\u2191L (f (\u2191isoG t))))\n          (\u2191(ContinuousLinearEquiv.symm isoE')\n            (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p))\n              (\u2191isoG\n                (\u2191(ContinuousLinearEquiv.symm isoG) x - t)))) \u2202Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc) =\n    \u222b (t : G), \u2191(\u2191L (f t)) (g p (x - t)) \u2202\u03bc\n[PROOFSTEP]\nrw [ClosedEmbedding.integral_map, \u2190 isoF.integral_comp_comm]\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 \u222b (a : G),\n      \u2191isoF\n        (\u2191(\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))\n              (\u2191L (f (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) a)))))\n          (\u2191(ContinuousLinearEquiv.symm isoE')\n            (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p))\n              (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) x - \u2191(ContinuousLinearEquiv.symm isoG) a))))) \u2202\u03bc =\n    \u222b (t : G), \u2191(\u2191L (f t)) (g p (x - t)) \u2202\u03bc\ncase h.mk.h\u03c6\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 ClosedEmbedding \u2191(ContinuousLinearEquiv.symm isoG)\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.mk.h\u03c6\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 ClosedEmbedding \u2191(ContinuousLinearEquiv.symm isoG)\n[PROOFSTEP]\nexact isoG.symm.toHomeomorph.closedEmbedding\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 \u222b (a : G),\n      \u2191isoF\n        (\u2191(\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))\n              (\u2191L (f (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) a)))))\n          (\u2191(ContinuousLinearEquiv.symm isoE')\n            (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p))\n              (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) x - \u2191(ContinuousLinearEquiv.symm isoG) a))))) \u2202\u03bc =\n    \u222b (t : G), \u2191(\u2191L (f t)) (g p (x - t)) \u2202\u03bc\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.mk.e_f\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx : G\n\u22a2 (fun a =>\n      \u2191isoF\n        (\u2191(\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))\n              (\u2191L (f (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) a)))))\n          (\u2191(ContinuousLinearEquiv.symm isoE')\n            (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p))\n              (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) x - \u2191(ContinuousLinearEquiv.symm isoG) a)))))) =\n    fun t => \u2191(\u2191L (f t)) (g p (x - t))\n[PROOFSTEP]\next1 a\n[GOAL]\ncase h.mk.e_f.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\np : P\nx a : G\n\u22a2 \u2191isoF\n      (\u2191(\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))\n            (\u2191L (f (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) a)))))\n        (\u2191(ContinuousLinearEquiv.symm isoE')\n          (g (\u2191isoP (\u2191(ContinuousLinearEquiv.symm isoP) p))\n            (\u2191isoG (\u2191(ContinuousLinearEquiv.symm isoG) x - \u2191(ContinuousLinearEquiv.symm isoG) a))))) =\n    \u2191(\u2191L (f a)) (g p (x - a))\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), ContinuousLinearEquiv.apply_symm_apply, coe_comp', ContinuousLinearEquiv.prod_apply,\n  ContinuousLinearEquiv.map_sub, ContinuousLinearEquiv.arrowCongr, ContinuousLinearEquiv.arrowCongrSL_symm_apply,\n  ContinuousLinearEquiv.coe_coe, Function.comp_apply, ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nA : ContDiffOn \ud835\udd5c n (\u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s \u00d7\u02e2 univ)\nthis : \u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG)) = fun q => f \u22c6[L, q.snd] g q.fst\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nsimp_rw [this] at A \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\neG : Type (max uG uE' uF uP) := ULift G\nthis\u271d\u00b9 : MeasurableSpace eG := borel eG\nthis\u271d : BorelSpace eG\neE' : Type (max uE' uG uF uP) := ULift E'\neF : Type (max uF uG uE' uP) := ULift F\neP : Type (max uP uG uE' uF) := ULift P\nisoG : eG \u2243L[\ud835\udd5c] G\nisoE' : eE' \u2243L[\ud835\udd5c] E'\nisoF : eF \u2243L[\ud835\udd5c] F\nisoP : eP \u2243L[\ud835\udd5c] P\nef : eG \u2192 E := f \u2218 \u2191isoG\ne\u03bc : Measure eG := Measure.map (\u2191(ContinuousLinearEquiv.symm isoG)) \u03bc\neg : eP \u2192 eG \u2192 eE' := fun ep ex => \u2191(ContinuousLinearEquiv.symm isoE') (g (\u2191isoP ep) (\u2191isoG ex))\neL : E \u2192L[\ud835\udd5c] eE' \u2192L[\ud835\udd5c] eF :=\n  ContinuousLinearMap.comp (\u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L\nR : eP \u00d7 eG \u2192 eF := fun q => ef \u22c6[eL, q.snd] eg q.fst\nR_contdiff : ContDiffOn \ud835\udd5c n R ((\u2191isoP \u207b\u00b9' s) \u00d7\u02e2 univ)\nthis : \u2191isoF \u2218 R \u2218 \u2191(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG)) = fun q => f \u22c6[L, q.snd] g q.fst\nA : ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => f \u22c6[L, q.snd] g q.fst) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact A\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ns : Set P\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun x => f \u22c6[L, v x] g x) s\n[PROOFSTEP]\napply (contDiffOn_convolution_right_with_param L hs hk hgs hf hg).comp (contDiffOn_id.prod hv)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ns : Set P\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 s \u2286 (fun x => (_root_.id x, v x)) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ns : Set P\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\nx : P\nhx : x \u2208 s\n\u22a2 x \u2208 (fun x => (_root_.id x, v x)) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self_iff, id.def]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nL : E' \u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F\nf : G \u2192 E\nn : \u2115\u221e\ng : P \u2192 G \u2192 E'\ns : Set P\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun q => g q.fst \u22c6[L, q.snd] f) (s \u00d7\u02e2 univ)\n[PROOFSTEP]\nsimpa only [convolution_flip] using contDiffOn_convolution_right_with_param L.flip hs hk hgs hf hg\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nL : E' \u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F\ns : Set P\nn : \u2115\u221e\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \ud835\udd5c n (fun x => g x \u22c6[L, v x] f) s\n[PROOFSTEP]\napply (contDiffOn_convolution_left_with_param L hs hk hgs hf hg).comp (contDiffOn_id.prod hv)\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nL : E' \u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F\ns : Set P\nn : \u2115\u221e\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\n\u22a2 s \u2286 (fun x => (_root_.id x, v x)) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL\u271d : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nL : E' \u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] F\ns : Set P\nn : \u2115\u221e\nv : P \u2192 G\nhv : ContDiffOn \ud835\udd5c n v s\nf : G \u2192 E\ng : P \u2192 G \u2192 E'\nk : Set G\nhs : IsOpen s\nhk : IsCompact k\nhgs : \u2200 (p : P) (x : G), p \u2208 s \u2192 \u00acx \u2208 k \u2192 g p x = 0\nhf : LocallyIntegrable f\nhg : ContDiffOn \ud835\udd5c n (\u21bfg) (s \u00d7\u02e2 univ)\nx : P\nhx : x \u2208 s\n\u22a2 x \u2208 (fun x => (_root_.id x, v x)) \u207b\u00b9' s \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [hx, mem_preimage, prod_mk_mem_set_prod_eq, mem_univ, and_self_iff, id.def]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c n g\n\u22a2 ContDiff \ud835\udd5c n (convolution f g L)\n[PROOFSTEP]\nrcases exists_compact_iff_hasCompactSupport.2 hcg with \u27e8k, hk, h'k\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c n g\nk : Set G\nhk : IsCompact k\nh'k : \u2200 (x : G), \u00acx \u2208 k \u2192 g x = 0\n\u22a2 ContDiff \ud835\udd5c n (convolution f g L)\n[PROOFSTEP]\nrw [\u2190 contDiffOn_univ]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E'\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E''\ninst\u271d\u2078 : NormedSpace \u211d F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : MeasurableSpace G\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : BorelSpace G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : NormedAddCommGroup P\ninst\u271d : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\nn : \u2115\u221e\nhcg : HasCompactSupport g\nhf : LocallyIntegrable f\nhg : ContDiff \ud835\udd5c n g\nk : Set G\nhk : IsCompact k\nh'k : \u2200 (x : G), \u00acx \u2208 k \u2192 g x = 0\n\u22a2 ContDiffOn \ud835\udd5c n (convolution f g L) univ\n[PROOFSTEP]\nexact\n  contDiffOn_convolution_right_with_param_comp L contDiffOn_id isOpen_univ hk (fun p x _ hx => h'k x hx) hf\n    (hg.comp contDiff_snd).contDiffOn\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nn : \u2115\u221e\nhcf : HasCompactSupport f\nhf : ContDiff \ud835\udd5c n f\nhg : LocallyIntegrable g\n\u22a2 ContDiff \ud835\udd5c n (convolution f g L)\n[PROOFSTEP]\nrw [\u2190 convolution_flip]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\nf f' : G \u2192 E\ng g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u00b9\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E''\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : CompleteSpace F\ninst\u271d\u2077 : MeasurableSpace G\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b3 : NormedAddCommGroup P\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c P\n\u03bc : Measure G\nL : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : IsAddLeftInvariant \u03bc\ninst\u271d : IsNegInvariant \u03bc\nn : \u2115\u221e\nhcf : HasCompactSupport f\nhf : ContDiff \ud835\udd5c n f\nhg : LocallyIntegrable g\n\u22a2 ContDiff \ud835\udd5c n (convolution g f (ContinuousLinearMap.flip L))\n[PROOFSTEP]\nexact hcf.contDiff_convolution_right L.flip hg hf\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\n\u22a2 posConvolution f g L = convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\n\u22a2 posConvolution f g L x = indicator (Ioi 0) f \u22c6[L, x] indicator (Ioi 0) g\n[PROOFSTEP]\nunfold convolution posConvolution indicator\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\n\u22a2 (if x \u2208 Ioi 0 then (fun x => \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd) x else 0) =\n    \u222b (t : \u211d), \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) \u2202\u03bd\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\n\u22a2 (if x \u2208 Ioi 0 then \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd else 0) =\n    \u222b (t : \u211d), \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) \u2202\u03bd\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\n\u22a2 \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd =\n    \u222b (t : \u211d), \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) \u2202\u03bd\n[PROOFSTEP]\nrw [intervalIntegral.integral_of_le (le_of_lt h), integral_Ioc_eq_integral_Ioo, \u2190\n  integral_indicator (measurableSet_Ioo : MeasurableSet (Ioo 0 x))]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\n\u22a2 \u222b (x_1 : \u211d), indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) x_1 \u2202\u03bd =\n    \u222b (t : \u211d), \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) \u2202\u03bd\n[PROOFSTEP]\ncongr 1 with t : 1\n[GOAL]\ncase pos.e_f.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\n\u22a2 indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) t =\n    \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0)\n[PROOFSTEP]\nhave : t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t := by\n  rcases le_or_lt t 0 with (h | h)\n  \u00b7 exact Or.inl h\n  \u00b7 rcases lt_or_le t x with (h' | h')\n    exacts [Or.inr (Or.inl \u27e8h, h'\u27e9), Or.inr (Or.inr h')]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\n\u22a2 t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\n[PROOFSTEP]\nrcases le_or_lt t 0 with (h | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh\u271d : x \u2208 Ioi 0\nt : \u211d\nh : t \u2264 0\n\u22a2 t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh\u271d : x \u2208 Ioi 0\nt : \u211d\nh : 0 < t\n\u22a2 t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\n[PROOFSTEP]\nrcases lt_or_le t x with (h' | h')\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh\u271d : x \u2208 Ioi 0\nt : \u211d\nh : 0 < t\nh' : t < x\n\u22a2 t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\ncase inr.inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh\u271d : x \u2208 Ioi 0\nt : \u211d\nh : 0 < t\nh' : x \u2264 t\n\u22a2 t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\n[PROOFSTEP]\nexacts [Or.inr (Or.inl \u27e8h, h'\u27e9), Or.inr (Or.inr h')]\n[GOAL]\ncase pos.e_f.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\nthis : t \u2264 0 \u2228 t \u2208 Ioo 0 x \u2228 x \u2264 t\n\u22a2 indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) t =\n    \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0)\n[PROOFSTEP]\nrcases this with (ht | ht | ht)\n[GOAL]\ncase pos.e_f.h.inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\nht : t \u2264 0\n\u22a2 indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) t =\n    \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0)\n[PROOFSTEP]\nrw [indicator_of_not_mem (not_mem_Ioo_of_le ht), if_neg (not_mem_Ioi.mpr ht), ContinuousLinearMap.map_zero,\n  ContinuousLinearMap.zero_apply]\n[GOAL]\ncase pos.e_f.h.inr.inl\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\nht : t \u2208 Ioo 0 x\n\u22a2 indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) t =\n    \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0)\n[PROOFSTEP]\nrw [indicator_of_mem ht, if_pos (mem_Ioi.mpr ht.1), if_pos (mem_Ioi.mpr <| sub_pos.mpr ht.2)]\n[GOAL]\ncase pos.e_f.h.inr.inr\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2208 Ioi 0\nt : \u211d\nht : x \u2264 t\n\u22a2 indicator (Ioo 0 x) (fun t => \u2191(\u2191L (f t)) (g (x - t))) t =\n    \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0)\n[PROOFSTEP]\nrw [indicator_of_not_mem (not_mem_Ioo_of_ge ht), if_neg (not_mem_Ioi.mpr (sub_nonpos_of_le ht)),\n  ContinuousLinearMap.map_zero]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : \u00acx \u2208 Ioi 0\n\u22a2 0 = \u222b (t : \u211d), \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) \u2202\u03bd\n[PROOFSTEP]\nconvert (integral_zero \u211d F).symm with t\n[GOAL]\ncase h.e'_3.h.e'_7.h\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : \u00acx \u2208 Ioi 0\nt : \u211d\n\u22a2 \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) = 0\n[PROOFSTEP]\nby_cases ht : 0 < t\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : \u00acx \u2208 Ioi 0\nt : \u211d\nht : 0 < t\n\u22a2 \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) = 0\n[PROOFSTEP]\nrw [if_neg (_ : x - t \u2209 Ioi 0), ContinuousLinearMap.map_zero]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : \u00acx \u2208 Ioi 0\nt : \u211d\nht : 0 < t\n\u22a2 \u00acx - t \u2208 Ioi 0\n[PROOFSTEP]\nrw [not_mem_Ioi] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : x \u2264 0\nt : \u211d\nht : 0 < t\n\u22a2 x - t \u2264 0\n[PROOFSTEP]\nexact sub_nonpos.mpr (h.trans ht.le)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b2 : NormedSpace \u211d F\ninst\u271d\u00b9 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u03bd : autoParam (Measure \u211d) _auto\u271d\ninst\u271d : NoAtoms \u03bd\nx : \u211d\nh : \u00acx \u2208 Ioi 0\nt : \u211d\nht : \u00ac0 < t\n\u22a2 \u2191(\u2191L (if t \u2208 Ioi 0 then f t else 0)) (if x - t \u2208 Ioi 0 then g (x - t) else 0) = 0\n[PROOFSTEP]\nrw [if_neg (mem_Ioi.not.mpr ht), ContinuousLinearMap.map_zero, ContinuousLinearMap.zero_apply]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nhf : IntegrableOn f (Ioi 0)\nhg : IntegrableOn g (Ioi 0)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 Integrable (posConvolution f g L)\n[PROOFSTEP]\nrw [\u2190 integrable_indicator_iff (measurableSet_Ioi : MeasurableSet (Ioi (0 : \u211d)))] at hf hg \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nhf : Integrable (indicator (Ioi 0) f)\nhg : Integrable (indicator (Ioi 0) g)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 Integrable (posConvolution f g L)\n[PROOFSTEP]\nrw [posConvolution_eq_convolution_indicator f g L \u03bd]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E'\ninst\u271d\u2079 : NormedAddCommGroup E''\ninst\u271d\u2078 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nhf : Integrable (indicator (Ioi 0) f)\nhg : Integrable (indicator (Ioi 0) g)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 Integrable (convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L)\n[PROOFSTEP]\nexact (hf.convolution_integrand L hg).integral_prod_left\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : CompleteSpace E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nhf : IntegrableOn f (Ioi 0)\nhg : IntegrableOn g (Ioi 0)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd \u2202\u03bc =\n    \u2191(\u2191L (\u222b (x : \u211d) in Ioi 0, f x \u2202\u03bd)) (\u222b (x : \u211d) in Ioi 0, g x \u2202\u03bc)\n[PROOFSTEP]\nrw [\u2190 integrable_indicator_iff measurableSet_Ioi] at hf hg \n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : CompleteSpace E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nhf : Integrable (indicator (Ioi 0) f)\nhg : Integrable (indicator (Ioi 0) g)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 \u222b (x : \u211d) in Ioi 0, \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd \u2202\u03bc =\n    \u2191(\u2191L (\u222b (x : \u211d) in Ioi 0, f x \u2202\u03bd)) (\u222b (x : \u211d) in Ioi 0, g x \u2202\u03bc)\n[PROOFSTEP]\nsimp_rw [\u2190 integral_indicator measurableSet_Ioi]\n[GOAL]\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx x' : G\ny y' : E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : CompleteSpace E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nhf : Integrable (indicator (Ioi 0) f)\nhg : Integrable (indicator (Ioi 0) g)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\n\u22a2 \u222b (x : \u211d), indicator (Ioi 0) (fun x => \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd) x \u2202\u03bc =\n    \u2191(\u2191L (\u222b (x : \u211d), indicator (Ioi 0) (fun x => f x) x \u2202\u03bd)) (\u222b (x : \u211d), indicator (Ioi 0) (fun x => g x) x \u2202\u03bc)\n[PROOFSTEP]\nconvert integral_convolution L hf hg using 4 with x\n[GOAL]\ncase h.e'_2.h.e'_7.h.h.e\n\ud835\udd5c : Type u\ud835\udd5c\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E''\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\nf\u271d f' : G \u2192 E\ng\u271d g' : G \u2192 E'\nx\u271d x' : G\ny y' : E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedSpace \u211d E'\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : CompleteSpace F\ninst\u271d\u2075 : CompleteSpace E\ninst\u271d\u2074 : CompleteSpace E'\n\u03bc \u03bd : Measure \u211d\ninst\u271d\u00b3 : SigmaFinite \u03bc\ninst\u271d\u00b2 : SigmaFinite \u03bd\ninst\u271d\u00b9 : IsAddRightInvariant \u03bc\ninst\u271d : NoAtoms \u03bd\nf : \u211d \u2192 E\ng : \u211d \u2192 E'\nhf : Integrable (indicator (Ioi 0) f)\nhg : Integrable (indicator (Ioi 0) g)\nL : E \u2192L[\u211d] E' \u2192L[\u211d] F\nx : \u211d\n\u22a2 (indicator (Ioi 0) fun x => \u222b (t : \u211d) in 0 ..x, \u2191(\u2191L (f t)) (g (x - t)) \u2202\u03bd) =\n    convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L\n[PROOFSTEP]\napply posConvolution_eq_convolution_indicator\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convolution", "llama_tokens": 437973, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321936479701, "lm_q2_score": 0.6619228825191871, "lm_q1q2_score": 0.5332663838697203}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\n\u22a2 HasDerivAt (gronwallBound \u03b4 K \u03b5) (K * gronwallBound \u03b4 K \u03b5 x + \u03b5) x\n[PROOFSTEP]\nby_cases hK : K = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\nhK : K = 0\n\u22a2 HasDerivAt (gronwallBound \u03b4 K \u03b5) (K * gronwallBound \u03b4 K \u03b5 x + \u03b5) x\n[PROOFSTEP]\nsubst K\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 \u03b5 x : \u211d\n\u22a2 HasDerivAt (gronwallBound \u03b4 0 \u03b5) (0 * gronwallBound \u03b4 0 \u03b5 x + \u03b5) x\n[PROOFSTEP]\nsimp only [gronwallBound_K0, zero_mul, zero_add]\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 \u03b5 x : \u211d\n\u22a2 HasDerivAt (fun x => \u03b4 + \u03b5 * x) \u03b5 x\n[PROOFSTEP]\nconvert ((hasDerivAt_id x).const_mul \u03b5).const_add \u03b4\n[GOAL]\ncase h.e'_7\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 \u03b5 x : \u211d\n\u22a2 \u03b5 = \u03b5 * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\nhK : \u00acK = 0\n\u22a2 HasDerivAt (gronwallBound \u03b4 K \u03b5) (K * gronwallBound \u03b4 K \u03b5 x + \u03b5) x\n[PROOFSTEP]\nsimp only [gronwallBound_of_K_ne_0 hK]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\nhK : \u00acK = 0\n\u22a2 HasDerivAt (fun x => \u03b4 * exp (K * x) + \u03b5 / K * (exp (K * x) - 1))\n    (K * (\u03b4 * exp (K * x) + \u03b5 / K * (exp (K * x) - 1)) + \u03b5) x\n[PROOFSTEP]\nconvert\n  (((hasDerivAt_id x).const_mul K).exp.const_mul \u03b4).add\n    ((((hasDerivAt_id x).const_mul K).exp.sub_const 1).const_mul (\u03b5 / K)) using\n  1\n[GOAL]\ncase h.e'_7\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\nhK : \u00acK = 0\n\u22a2 K * (\u03b4 * exp (K * x) + \u03b5 / K * (exp (K * x) - 1)) + \u03b5 =\n    \u03b4 * (exp (K * id x) * (K * 1)) + \u03b5 / K * (exp (K * id x) * (K * 1))\n[PROOFSTEP]\nsimp only [id, mul_add, (mul_assoc _ _ _).symm, mul_comm _ K, mul_div_cancel' _ hK]\n[GOAL]\ncase h.e'_7\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x : \u211d\nhK : \u00acK = 0\n\u22a2 K * \u03b4 * exp (K * x) + \u03b5 * (exp (K * x) - 1) + \u03b5 = K * \u03b4 * exp (K * x) * 1 + \u03b5 * exp (K * x) * 1\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x a : \u211d\n\u22a2 HasDerivAt (fun y => gronwallBound \u03b4 K \u03b5 (y - a)) (K * gronwallBound \u03b4 K \u03b5 (x - a) + \u03b5) x\n[PROOFSTEP]\nconvert (hasDerivAt_gronwallBound \u03b4 K \u03b5 _).comp x ((hasDerivAt_id x).sub_const a) using 1\n[GOAL]\ncase h.e'_7\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 x a : \u211d\n\u22a2 K * gronwallBound \u03b4 K \u03b5 (x - a) + \u03b5 = (K * gronwallBound \u03b4 K \u03b5 (id x - a) + \u03b5) * 1\n[PROOFSTEP]\nrw [id, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 : \u211d\n\u22a2 gronwallBound \u03b4 K \u03b5 0 = \u03b4\n[PROOFSTEP]\nby_cases hK : K = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 : \u211d\nhK : K = 0\n\u22a2 gronwallBound \u03b4 K \u03b5 0 = \u03b4\n[PROOFSTEP]\nsimp only [gronwallBound, if_pos hK, mul_zero, add_zero]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K \u03b5 : \u211d\nhK : \u00acK = 0\n\u22a2 gronwallBound \u03b4 K \u03b5 0 = \u03b4\n[PROOFSTEP]\nsimp only [gronwallBound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one, add_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\n\u22a2 gronwallBound \u03b4 K 0 x = \u03b4 * exp (K * x)\n[PROOFSTEP]\nby_cases hK : K = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : K = 0\n\u22a2 gronwallBound \u03b4 K 0 x = \u03b4 * exp (K * x)\n[PROOFSTEP]\nsimp only [gronwallBound_K0, hK, zero_mul, exp_zero, add_zero, mul_one]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : \u00acK = 0\n\u22a2 gronwallBound \u03b4 K 0 x = \u03b4 * exp (K * x)\n[PROOFSTEP]\nsimp only [gronwallBound_of_K_ne_0 hK, zero_div, zero_mul, add_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nK x : \u211d\n\u22a2 gronwallBound 0 K 0 x = 0\n[PROOFSTEP]\nsimp only [gronwallBound_\u03b50, zero_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\n\u22a2 Continuous fun \u03b5 => gronwallBound \u03b4 K \u03b5 x\n[PROOFSTEP]\nby_cases hK : K = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : K = 0\n\u22a2 Continuous fun \u03b5 => gronwallBound \u03b4 K \u03b5 x\n[PROOFSTEP]\nsimp only [gronwallBound_K0, hK]\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : K = 0\n\u22a2 Continuous fun \u03b5 => \u03b4 + \u03b5 * x\n[PROOFSTEP]\nexact continuous_const.add (continuous_id.mul continuous_const)\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : \u00acK = 0\n\u22a2 Continuous fun \u03b5 => gronwallBound \u03b4 K \u03b5 x\n[PROOFSTEP]\nsimp only [gronwallBound_of_K_ne_0 hK]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u03b4 K x : \u211d\nhK : \u00acK = 0\n\u22a2 Continuous fun \u03b5 => \u03b4 * exp (K * x) + \u03b5 / K * (exp (K * x) - 1)\n[PROOFSTEP]\nexact continuous_const.add ((continuous_id.mul continuous_const).mul continuous_const)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5 (x - a)\n[PROOFSTEP]\nhave H : \u2200 x \u2208 Icc a b, \u2200 \u03b5' \u2208 Ioi \u03b5, f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a) :=\n  by\n  intro x hx \u03b5' h\u03b5'\n  apply image_le_of_liminf_slope_right_lt_deriv_boundary hf hf'\n  \u00b7 rwa [sub_self, gronwallBound_x0]\n  \u00b7 exact fun x => hasDerivAt_gronwallBound_shift \u03b4 K \u03b5' x a\n  \u00b7 intro x hx hfB\n    rw [\u2190 hfB]\n    apply lt_of_le_of_lt (bound x hx)\n    exact add_lt_add_left (mem_Ioi.1 h\u03b5') _\n  \u00b7 exact hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\n[PROOFSTEP]\nintro x hx \u03b5' h\u03b5'\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx : \u211d\nhx : x \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\n\u22a2 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\n[PROOFSTEP]\napply image_le_of_liminf_slope_right_lt_deriv_boundary hf hf'\n[GOAL]\ncase ha\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx : \u211d\nhx : x \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\n\u22a2 f a \u2264 gronwallBound \u03b4 K \u03b5' (a - a)\n[PROOFSTEP]\nrwa [sub_self, gronwallBound_x0]\n[GOAL]\ncase hB\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx : \u211d\nhx : x \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\n\u22a2 \u2200 (x : \u211d), HasDerivAt (fun x => gronwallBound \u03b4 K \u03b5' (x - a)) (?m.35213 x) x\n[PROOFSTEP]\nexact fun x => hasDerivAt_gronwallBound_shift \u03b4 K \u03b5' x a\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx : \u211d\nhx : x \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f x = gronwallBound \u03b4 K \u03b5' (x - a) \u2192 f' x < K * gronwallBound \u03b4 K \u03b5' (x - a) + \u03b5'\n[PROOFSTEP]\nintro x hx hfB\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\nx : \u211d\nhx : x \u2208 Ico a b\nhfB : f x = gronwallBound \u03b4 K \u03b5' (x - a)\n\u22a2 f' x < K * gronwallBound \u03b4 K \u03b5' (x - a) + \u03b5'\n[PROOFSTEP]\nrw [\u2190 hfB]\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\nx : \u211d\nhx : x \u2208 Ico a b\nhfB : f x = gronwallBound \u03b4 K \u03b5' (x - a)\n\u22a2 f' x < K * f x + \u03b5'\n[PROOFSTEP]\napply lt_of_le_of_lt (bound x hx)\n[GOAL]\ncase bound\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx\u271d : \u211d\nhx\u271d : x\u271d \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\nx : \u211d\nhx : x \u2208 Ico a b\nhfB : f x = gronwallBound \u03b4 K \u03b5' (x - a)\n\u22a2 K * f x + \u03b5 < K * f x + \u03b5'\n[PROOFSTEP]\nexact add_lt_add_left (mem_Ioi.1 h\u03b5') _\n[GOAL]\ncase a\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nx : \u211d\nhx : x \u2208 Icc a b\n\u03b5' : \u211d\nh\u03b5' : \u03b5' \u2208 Ioi \u03b5\n\u22a2 x \u2208 Icc a b\n[PROOFSTEP]\nexact hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nH : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\n\u22a2 \u2200 (x : \u211d), x \u2208 Icc a b \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5 (x - a)\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nH : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 f x \u2264 gronwallBound \u03b4 K \u03b5 (x - a)\n[PROOFSTEP]\nchange f x \u2264 (fun \u03b5' => gronwallBound \u03b4 K \u03b5' (x - a)) \u03b5\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nH : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 f x \u2264 (fun \u03b5' => gronwallBound \u03b4 K \u03b5' (x - a)) \u03b5\n[PROOFSTEP]\nconvert continuousWithinAt_const.closure_le _ _ (H x hx)\n[GOAL]\ncase convert_2\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nH : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 \u03b5 \u2208 closure (Ioi \u03b5)\n[PROOFSTEP]\nsimp only [closure_Ioi, left_mem_Ici]\n[GOAL]\ncase convert_3\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf f' : \u211d \u2192 \u211d\n\u03b4 K \u03b5 a b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2200 (r : \u211d), f' x < r \u2192 \u2203\u1da0 (z : \u211d) in \ud835\udcdd[Ioi x] x, (z - x)\u207b\u00b9 * (f z - f x) < r\nha : f a \u2264 \u03b4\nbound : \u2200 (x : \u211d), x \u2208 Ico a b \u2192 f' x \u2264 K * f x + \u03b5\nH : \u2200 (x : \u211d), x \u2208 Icc a b \u2192 \u2200 (\u03b5' : \u211d), \u03b5' \u2208 Ioi \u03b5 \u2192 f x \u2264 gronwallBound \u03b4 K \u03b5' (x - a)\nx : \u211d\nhx : x \u2208 Icc a b\n\u22a2 ContinuousWithinAt (fun y => gronwallBound \u03b4 K y (x - a)) (Ioi \u03b5) \u03b5\n[PROOFSTEP]\nexact (gronwallBound_continuous_\u03b5 \u03b4 K (x - a)).continuousWithinAt\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 dist (f t) (g t) \u2264 gronwallBound \u03b4 K (\u03b5f + \u03b5g) (t - a)\n[PROOFSTEP]\nsimp only [dist_eq_norm] at ha \u22a2\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 \u2016f t - g t\u2016 \u2264 gronwallBound \u03b4 K (\u03b5f + \u03b5g) (t - a)\n[PROOFSTEP]\nhave h_deriv : \u2200 t \u2208 Ico a b, HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t := fun t ht =>\n  (hf' t ht).sub (hg' t ht)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 \u2016f t - g t\u2016 \u2264 gronwallBound \u03b4 K (\u03b5f + \u03b5g) (t - a)\n[PROOFSTEP]\napply norm_le_gronwallBound_of_norm_deriv_right_le (hf.sub hg) h_deriv ha\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\n\u22a2 \u2200 (x : \u211d), x \u2208 Ico a b \u2192 \u2016f' x - g' x\u2016 \u2264 K * \u2016f x - g x\u2016 + (\u03b5f + \u03b5g)\n[PROOFSTEP]\nintro t ht\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\nt : \u211d\nht : t \u2208 Ico a b\n\u22a2 \u2016f' t - g' t\u2016 \u2264 K * \u2016f t - g t\u2016 + (\u03b5f + \u03b5g)\n[PROOFSTEP]\nhave := dist_triangle4_right (f' t) (g' t) (v t (f t)) (v t (g t))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\nt : \u211d\nht : t \u2208 Ico a b\nthis : dist (f' t) (g' t) \u2264 dist (f' t) (v t (f t)) + dist (g' t) (v t (g t)) + dist (v t (f t)) (v t (g t))\n\u22a2 \u2016f' t - g' t\u2016 \u2264 K * \u2016f t - g t\u2016 + (\u03b5f + \u03b5g)\n[PROOFSTEP]\nrw [dist_eq_norm] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\nt : \u211d\nht : t \u2208 Ico a b\nthis : \u2016f' t - g' t\u2016 \u2264 dist (f' t) (v t (f t)) + dist (g' t) (v t (g t)) + dist (v t (f t)) (v t (g t))\n\u22a2 \u2016f' t - g' t\u2016 \u2264 K * \u2016f t - g t\u2016 + (\u03b5f + \u03b5g)\n[PROOFSTEP]\nrefine'\n  this.trans ((add_le_add (add_le_add (f_bound t ht) (g_bound t ht)) (hv t (f t) (hfs t ht) (g t) (hgs t ht))).trans _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g f' g' : \u211d \u2192 E\na b \u03b5f \u03b5g \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (f' t) (v t (f t)) \u2264 \u03b5f\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (g' t) (Ici t) t\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (g' t) (v t (g t)) \u2264 \u03b5g\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : \u2016f a - g a\u2016 \u2264 \u03b4\nh_deriv : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t\nt : \u211d\nht : t \u2208 Ico a b\nthis : \u2016f' t - g' t\u2016 \u2264 dist (f' t) (v t (f t)) + dist (g' t) (v t (g t)) + dist (v t (f t)) (v t (g t))\n\u22a2 \u03b5f + \u03b5g + K * dist (f t) (g t) \u2264 K * \u2016f t - g t\u2016 + (\u03b5f + \u03b5g)\n[PROOFSTEP]\nrw [dist_eq_norm, add_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 dist (f t) (g t) \u2264 \u03b4 * exp (K * (t - a))\n[PROOFSTEP]\nhave f_bound : \u2200 t \u2208 Ico a b, dist (v t (f t)) (v t (f t)) \u2264 0 := by intros; rw [dist_self]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\n\u22a2 \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\n[PROOFSTEP]\nintros\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nt\u271d : \u211d\na\u271d : t\u271d \u2208 Ico a b\n\u22a2 dist (v t\u271d (f t\u271d)) (v t\u271d (f t\u271d)) \u2264 0\n[PROOFSTEP]\nrw [dist_self]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 dist (f t) (g t) \u2264 \u03b4 * exp (K * (t - a))\n[PROOFSTEP]\nhave g_bound : \u2200 t \u2208 Ico a b, dist (v t (g t)) (v t (g t)) \u2264 0 := by intros; rw [dist_self]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\n\u22a2 \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (g t)) (v t (g t)) \u2264 0\n[PROOFSTEP]\nintros\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\nt\u271d : \u211d\na\u271d : t\u271d \u2208 Ico a b\n\u22a2 dist (v t\u271d (g t\u271d)) (v t\u271d (g t\u271d)) \u2264 0\n[PROOFSTEP]\nrw [dist_self]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (g t)) (v t (g t)) \u2264 0\n\u22a2 \u2200 (t : \u211d), t \u2208 Icc a b \u2192 dist (f t) (g t) \u2264 \u03b4 * exp (K * (t - a))\n[PROOFSTEP]\nintro t ht\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (g t)) (v t (g t)) \u2264 0\nt : \u211d\nht : t \u2208 Icc a b\n\u22a2 dist (f t) (g t) \u2264 \u03b4 * exp (K * (t - a))\n[PROOFSTEP]\nhave := dist_le_of_approx_trajectories_ODE_of_mem_set hv hf hf' f_bound hfs hg hg' g_bound hgs ha t ht\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b \u03b4 : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : dist (f a) (g a) \u2264 \u03b4\nf_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (f t)) (v t (f t)) \u2264 0\ng_bound : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 dist (v t (g t)) (v t (g t)) \u2264 0\nt : \u211d\nht : t \u2208 Icc a b\nthis : dist (f t) (g t) \u2264 gronwallBound \u03b4 K (0 + 0) (t - a)\n\u22a2 dist (f t) (g t) \u2264 \u03b4 * exp (K * (t - a))\n[PROOFSTEP]\nrwa [zero_add, gronwallBound_\u03b50] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : f a = g a\nt : \u211d\nht : t \u2208 Icc a b\n\u22a2 f t = g t\n[PROOFSTEP]\nhave := dist_le_of_trajectories_ODE_of_mem_set hv hf hf' hfs hg hg' hgs (dist_le_zero.2 ha) t ht\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nv : \u211d \u2192 E \u2192 E\ns : \u211d \u2192 Set E\nK : \u211d\nhv : \u2200 (t : \u211d) (x : E), x \u2208 s t \u2192 \u2200 (y : E), y \u2208 s t \u2192 dist (v t x) (v t y) \u2264 K * dist x y\nf g : \u211d \u2192 E\na b : \u211d\nhf : ContinuousOn f (Icc a b)\nhf' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt f (v t (f t)) (Ici t) t\nhfs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 f t \u2208 s t\nhg : ContinuousOn g (Icc a b)\nhg' : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 HasDerivWithinAt g (v t (g t)) (Ici t) t\nhgs : \u2200 (t : \u211d), t \u2208 Ico a b \u2192 g t \u2208 s t\nha : f a = g a\nt : \u211d\nht : t \u2208 Icc a b\nthis : dist (f t) (g t) \u2264 0 * exp (K * (t - a))\n\u22a2 f t = g t\n[PROOFSTEP]\nrwa [zero_mul, dist_le_zero] at this \n", "meta": {"mathlib_filename": "Mathlib.Analysis.ODE.Gronwall", "llama_tokens": 17914, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321796478256, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.5332663692297879}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\n\u22a2 Saturated H \u2194 \u2200 (n : \u2124) (g : G), g ^ n \u2208 H \u2192 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\n\u22a2 Saturated H \u2192 \u2200 (n : \u2124) (g : G), g ^ n \u2208 H \u2192 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nintros hH n g hgn\n[GOAL]\ncase mp\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn : \u2124\ng : G\nhgn : g ^ n \u2208 H\n\u22a2 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\ninduction' n with n n\n[GOAL]\ncase mp.ofNat\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ Int.ofNat n \u2208 H\n\u22a2 Int.ofNat n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nsimp only [Int.coe_nat_eq_zero, Int.ofNat_eq_coe, zpow_ofNat] at hgn \u22a2\n[GOAL]\ncase mp.ofNat\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ n \u2208 H\n\u22a2 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nexact hH hgn\n[GOAL]\ncase mp.negSucc\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ Int.negSucc n \u2208 H\n\u22a2 Int.negSucc n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nsuffices g ^ (n + 1) \u2208 H by\n  refine' (hH this).imp _ id\n  simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ Int.negSucc n \u2208 H\nthis : g ^ (n + 1) \u2208 H\n\u22a2 Int.negSucc n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nrefine' (hH this).imp _ id\n[GOAL]\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ Int.negSucc n \u2208 H\nthis : g ^ (n + 1) \u2208 H\n\u22a2 n + 1 = 0 \u2192 Int.negSucc n = 0\n[PROOFSTEP]\nsimp only [IsEmpty.forall_iff, Nat.succ_ne_zero]\n[GOAL]\ncase mp.negSucc\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nhH : Saturated H\nn\u271d : \u2124\ng : G\nhgn\u271d : g ^ n\u271d \u2208 H\nn : \u2115\nhgn : g ^ Int.negSucc n \u2208 H\n\u22a2 g ^ (n + 1) \u2208 H\n[PROOFSTEP]\nsimpa only [inv_mem_iff, zpow_negSucc] using hgn\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\n\u22a2 (\u2200 (n : \u2124) (g : G), g ^ n \u2208 H \u2192 n = 0 \u2228 g \u2208 H) \u2192 Saturated H\n[PROOFSTEP]\nintro h n g hgn\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nh : \u2200 (n : \u2124) (g : G), g ^ n \u2208 H \u2192 n = 0 \u2228 g \u2208 H\nn : \u2115\ng : G\nhgn : g ^ n \u2208 H\n\u22a2 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nspecialize h n g\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nn : \u2115\ng : G\nhgn : g ^ n \u2208 H\nh : g ^ \u2191n \u2208 H \u2192 \u2191n = 0 \u2228 g \u2208 H\n\u22a2 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\nsimp only [Int.coe_nat_eq_zero, zpow_ofNat] at h \n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d : Group G\nH : Subgroup G\nn : \u2115\ng : G\nhgn : g ^ n \u2208 H\nh : g ^ n \u2208 H \u2192 n = 0 \u2228 g \u2208 H\n\u22a2 n = 0 \u2228 g \u2208 H\n[PROOFSTEP]\napply h hgn\n[GOAL]\nA\u2081 : Type u_1\nA\u2082 : Type u_2\ninst\u271d\u00b2 : AddCommGroup A\u2081\ninst\u271d\u00b9 : AddCommGroup A\u2082\ninst\u271d : NoZeroSMulDivisors \u2115 A\u2082\nf : A\u2081 \u2192+ A\u2082\n\u22a2 Saturated (AddMonoidHom.ker f)\n[PROOFSTEP]\nintro n g hg\n[GOAL]\nA\u2081 : Type u_1\nA\u2082 : Type u_2\ninst\u271d\u00b2 : AddCommGroup A\u2081\ninst\u271d\u00b9 : AddCommGroup A\u2082\ninst\u271d : NoZeroSMulDivisors \u2115 A\u2082\nf : A\u2081 \u2192+ A\u2082\nn : \u2115\ng : A\u2081\nhg : n \u2022 g \u2208 AddMonoidHom.ker f\n\u22a2 n = 0 \u2228 g \u2208 AddMonoidHom.ker f\n[PROOFSTEP]\nsimpa only [f.mem_ker, nsmul_eq_smul, f.map_nsmul, smul_eq_zero] using hg\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.Subgroup.Saturated", "llama_tokens": 1725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.7057850340255385, "lm_q1q2_score": 0.532807699692758}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA B\u271d : C\nf g : A \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nB : D\n\u22a2 F.map (NatTrans.app adj.unit (G.obj B)) \u226b F.map (G.map (NatTrans.app adj.counit B)) = \ud835\udfd9 (F.obj (G.obj ((\ud835\udfed D).obj B)))\n[PROOFSTEP]\nrw [\u2190 F.map_comp, adj.right_triangle_components]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nA B\u271d : C\nf g : A \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\nB : D\n\u22a2 F.map (\ud835\udfd9 (G.obj B)) = \ud835\udfd9 (F.obj (G.obj ((\ud835\udfed D).obj B)))\n[PROOFSTEP]\napply F.map_id\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasReflexiveCoequalizers C\nA B : C\nf g : A \u27f6 B\nr : B \u27f6 A\nrf : r \u226b f = \ud835\udfd9 B\nrg : r \u226b g = \ud835\udfd9 B\n\u22a2 HasCoequalizer f g\n[PROOFSTEP]\nletI := IsReflexivePair.mk' r rf rg\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasReflexiveCoequalizers C\nA B : C\nf g : A \u27f6 B\nr : B \u27f6 A\nrf : r \u226b f = \ud835\udfd9 B\nrg : r \u226b g = \ud835\udfd9 B\nthis : IsReflexivePair f g := IsReflexivePair.mk' r rf rg\n\u22a2 HasCoequalizer f g\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasCoreflexiveEqualizers C\nA B : C\nf g : A \u27f6 B\nr : B \u27f6 A\nfr : f \u226b r = \ud835\udfd9 A\ngr : g \u226b r = \ud835\udfd9 A\n\u22a2 HasEqualizer f g\n[PROOFSTEP]\nletI := IsCoreflexivePair.mk' r fr gr\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasCoreflexiveEqualizers C\nA B : C\nf g : A \u27f6 B\nr : B \u27f6 A\nfr : f \u226b r = \ud835\udfd9 A\ngr : g \u226b r = \ud835\udfd9 A\nthis : IsCoreflexivePair f g := IsCoreflexivePair.mk' r fr gr\n\u22a2 HasEqualizer f g\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasCoequalizers C\nA B : C\nf g : A \u27f6 B\nx\u271d : IsReflexivePair f g\n\u22a2 HasCoequalizer f g\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nA\u271d B\u271d : C\nf\u271d g\u271d : A\u271d \u27f6 B\u271d\nF : C \u2964 D\nG : D \u2964 C\nadj : F \u22a3 G\ninst\u271d : HasEqualizers C\nA B : C\nf g : A \u27f6 B\nx\u271d : IsCoreflexivePair f g\n\u22a2 HasEqualizer f g\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.Shapes.Reflexive", "llama_tokens": 1445, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8558511396138365, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.5327325279723737}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ng : \ud835\udce2(E, F)\ntoFun\u271d : E \u2192 F\nsmooth'\u271d : ContDiff \u211d \u22a4 toFun\u271d\ndecay'\u271d : \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n toFun\u271d x\u2016 \u2264 C\nh : (fun f => f.toFun) { toFun := toFun\u271d, smooth' := smooth'\u271d, decay' := decay'\u271d } = (fun f => f.toFun) g\n\u22a2 { toFun := toFun\u271d, smooth' := smooth'\u271d, decay' := decay'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\ntoFun\u271d\u00b9 : E \u2192 F\nsmooth'\u271d\u00b9 : ContDiff \u211d \u22a4 toFun\u271d\u00b9\ndecay'\u271d\u00b9 : \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n toFun\u271d\u00b9 x\u2016 \u2264 C\ntoFun\u271d : E \u2192 F\nsmooth'\u271d : ContDiff \u211d \u22a4 toFun\u271d\ndecay'\u271d : \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n toFun\u271d x\u2016 \u2264 C\nh :\n  (fun f => f.toFun) { toFun := toFun\u271d\u00b9, smooth' := smooth'\u271d\u00b9, decay' := decay'\u271d\u00b9 } =\n    (fun f => f.toFun) { toFun := toFun\u271d, smooth' := smooth'\u271d, decay' := decay'\u271d }\n\u22a2 { toFun := toFun\u271d\u00b9, smooth' := smooth'\u271d\u00b9, decay' := decay'\u271d\u00b9 } =\n    { toFun := toFun\u271d, smooth' := smooth'\u271d, decay' := decay'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk n : \u2115\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nrcases f.decay' k n with \u27e8C, hC\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk n : \u2115\nC : \u211d\nhC : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f.toFun x\u2016 \u2264 C\n\u22a2 \u2203 C, 0 < C \u2227 \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nexact \u27e8max C 1, by positivity, fun x => (hC x).trans (le_max_left _ _)\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk n : \u2115\nC : \u211d\nhC : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n f.toFun x\u2016 \u2264 C\n\u22a2 0 < max C 1\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)\n[PROOFSTEP]\nobtain \u27e8d, _, hd'\u27e9 := f.decay k 0\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d 0 (\u2191f) x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)\n[PROOFSTEP]\nsimp only [norm_iteratedFDeriv_zero] at hd' \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ (-\u2191k)\n[PROOFSTEP]\nsimp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2203 c, \u2200\u1da0 (x : E) in cocompact E, \u2016\u2191f x\u2016 \u2264 c * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016\n[PROOFSTEP]\nrefine' \u27e8d, Filter.Eventually.filter_mono Filter.cocompact_le_cofinite _\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\n\u22a2 \u2200\u1da0 (x : E) in cofinite, \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016\n[PROOFSTEP]\nrefine' (Filter.eventually_cofinite_ne 0).mono fun x hx => _\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016\u2191f x\u2016 \u2264 d * \u2016\u2016x\u2016 ^ (-\u2191k)\u2016\n[PROOFSTEP]\nrw [Real.norm_of_nonneg (zpow_nonneg (norm_nonneg _) _), zpow_neg, \u2190 div_eq_mul_inv, le_div_iff']\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 \u2016x\u2016 ^ \u2191k * \u2016\u2191f x\u2016 \u2264 d\ncase intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\nk : \u2115\nd : \u211d\nleft\u271d : 0 < d\nhd' : \u2200 (x : E), \u2016x\u2016 ^ k * \u2016\u2191f x\u2016 \u2264 d\nx : E\nhx : x \u2260 0\n\u22a2 0 < \u2016x\u2016 ^ \u2191k\n[PROOFSTEP]\nexacts [hd' x, zpow_pos_of_pos (norm_pos_iff.mpr hx) _]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ s\n[PROOFSTEP]\nlet k := \u2308-s\u2309\u208a\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ s\n[PROOFSTEP]\nhave hk : -(k : \u211d) \u2264 s := neg_le.mp (Nat.le_ceil (-s))\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ s\n[PROOFSTEP]\nrefine' (isBigO_cocompact_zpow_neg_nat f k).trans _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\n\u22a2 (fun x => \u2016x\u2016 ^ (-\u2191k)) =O[cocompact E] fun x => \u2016x\u2016 ^ s\n[PROOFSTEP]\nsuffices (fun x : \u211d => x ^ (-k : \u2124)) =O[atTop] fun x : \u211d => x ^ s from this.comp_tendsto tendsto_norm_cocompact_atTop\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\n\u22a2 (fun x => x ^ (-\u2191k)) =O[atTop] fun x => x ^ s\n[PROOFSTEP]\nsimp_rw [Asymptotics.IsBigO, Asymptotics.IsBigOWith]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\n\u22a2 \u2203 c, \u2200\u1da0 (x : \u211d) in atTop, \u2016x ^ (-\u2191\u2308-s\u2309\u208a)\u2016 \u2264 c * \u2016x ^ s\u2016\n[PROOFSTEP]\nrefine' \u27e81, (Filter.eventually_ge_atTop 1).mono fun x hx => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\nx : \u211d\nhx : 1 \u2264 x\n\u22a2 \u2016x ^ (-\u2191\u2308-s\u2309\u208a)\u2016 \u2264 1 * \u2016x ^ s\u2016\n[PROOFSTEP]\nrw [one_mul, Real.norm_of_nonneg (Real.rpow_nonneg_of_nonneg (zero_le_one.trans hx) _),\n  Real.norm_of_nonneg (zpow_nonneg (zero_le_one.trans hx) _), \u2190 Real.rpow_int_cast, Int.cast_neg, Int.cast_ofNat]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\ns : \u211d\nk : \u2115 := \u2308-s\u2309\u208a\nhk : -\u2191k \u2264 s\nx : \u211d\nhx : 1 \u2264 x\n\u22a2 x ^ (-\u2191\u2308-s\u2309\u208a) \u2264 x ^ s\n[PROOFSTEP]\nexact Real.rpow_le_rpow_of_exponent_le hx hk\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\nf : \ud835\udce2(E, F)\ninst\u271d : ProperSpace E\nk : \u2124\n\u22a2 \u2191f =O[cocompact E] fun x => \u2016x\u2016 ^ k\n[PROOFSTEP]\nsimpa only [Real.rpow_int_cast] using isBigO_cocompact_rpow f k\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf g : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f + \u2191g) x\u2016 \u2264\n    \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016\n[PROOFSTEP]\nrw [\u2190 mul_add]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf g : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f + \u2191g) x\u2016 \u2264 \u2016x\u2016 ^ k * (\u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf g : \ud835\udce2(E, F)\nx : E\n\u22a2 0 \u2264 \u2016x\u2016 ^ k\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf g : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d n (\u2191f + \u2191g) x\u2016 \u2264 \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_add_apply (f.smooth _) (g.smooth _)]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf g : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d n (\u2191f) x + iteratedFDeriv \u211d n (\u2191g) x\u2016 \u2264 \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016\n[PROOFSTEP]\nexact norm_add_le _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (-\u2191f) x\u2016 = \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_neg_apply, norm_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nk n : \u2115\nf : \ud835\udce2(E, F)\nc : \ud835\udd5c\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 = \u2016c\u2016 * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016\n[PROOFSTEP]\nrw [mul_comm \u2016c\u2016, mul_assoc, iteratedFDeriv_const_smul_apply (f.smooth _), norm_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\n\u22a2 \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nrefine' \u27e8f.seminormAux k n * (\u2016c\u2016 + 1), fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 SchwartzMap.seminormAux k n f * (\u2016c\u2016 + 1)\n[PROOFSTEP]\nhave hc : 0 \u2264 \u2016c\u2016 := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 0 \u2264 \u2016c\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 SchwartzMap.seminormAux k n f * (\u2016c\u2016 + 1)\n[PROOFSTEP]\nrefine' le_trans _ ((mul_le_mul_of_nonneg_right (f.le_seminormAux k n x) hc).trans _)\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 * \u2016c\u2016\n[PROOFSTEP]\napply Eq.le\n[GOAL]\ncase refine'_1.a\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 = \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 * \u2016c\u2016\n[PROOFSTEP]\nrw [mul_comm _ \u2016c\u2016, \u2190 mul_assoc]\n[GOAL]\ncase refine'_1.a\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 = \u2016c\u2016 * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016\n[PROOFSTEP]\nexact decay_smul_aux k n f c x\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 SchwartzMap.seminormAux k n f * \u2016c\u2016 \u2264 SchwartzMap.seminormAux k n f * (\u2016c\u2016 + 1)\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left _ (f.seminormAux_nonneg k n)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nk n : \u2115\nx : E\nhc : 0 \u2264 \u2016c\u2016\n\u22a2 \u2016c\u2016 \u2264 \u2016c\u2016 + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nk n : \u2115\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\n\u22a2 SchwartzMap.seminormAux k n (c \u2022 f) \u2264 \u2016c\u2016 * SchwartzMap.seminormAux k n f\n[PROOFSTEP]\nrefine'\n  (c \u2022 f).seminormAux_le_bound k n (mul_nonneg (norm_nonneg _) (seminormAux_nonneg _ _ _)) fun x =>\n    (decay_smul_aux k n f c x).le.trans _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nk n : \u2115\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016c\u2016 * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 \u2016c\u2016 * SchwartzMap.seminormAux k n f\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nk n : \u2115\nc : \ud835\udd5c\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016c\u2016 * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016) \u2264 \u2016c\u2016 * SchwartzMap.seminormAux k n f\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (f.le_seminormAux k n x) (norm_nonneg _)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nhave : c \u2022 (f : E \u2192 F) = (c : \u211d) \u2022 f := by\n  ext x\n  simp only [Pi.smul_apply, smul_apply]\n  exact nsmul_eq_smul_cast _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\n\u22a2 c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 (c \u2022 \u2191f) x = \u2191(\u2191c \u2022 f) x\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_apply]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 c \u2022 \u2191f x = \u2191c \u2022 \u2191f x\n[PROOFSTEP]\nexact nsmul_eq_smul_cast _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\nthis : c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2115\nf : \ud835\udce2(E, F)\nthis : c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191(\u2191c \u2022 f)) x\u2016 \u2264 C\n[PROOFSTEP]\nexact ((c : \u211d) \u2022 f).decay'\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nhave : c \u2022 (f : E \u2192 F) = (c : \u211d) \u2022 f := by\n  ext x\n  simp only [Pi.smul_apply, smul_apply]\n  exact zsmul_eq_smul_cast _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\n\u22a2 c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 (c \u2022 \u2191f) x = \u2191(\u2191c \u2022 f) x\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_apply]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 c \u2022 \u2191f x = \u2191c \u2022 \u2191f x\n[PROOFSTEP]\nexact zsmul_eq_smul_cast _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\nthis : c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (c \u2022 \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \u211d E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\ninst\u271d\u2075 : NormedField \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b2 : NormedField \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' F\nc : \u2124\nf : \ud835\udce2(E, F)\nthis : c \u2022 \u2191f = \u2191(\u2191c \u2022 f)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191(\u2191c \u2022 f)) x\u2016 \u2264 C\n[PROOFSTEP]\nexact ((c : \u211d) \u2022 f).decay'\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx\u271d\u00b2 x\u271d\u00b9 : \u2115\nx\u271d : E\n\u22a2 \u2016x\u271d\u2016 ^ x\u271d\u00b2 * \u2016iteratedFDeriv \u211d x\u271d\u00b9 (fun x => 0) x\u271d\u2016 \u2264 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nk n : \u2115\nx\u271d : E\n\u22a2 \u2016x\u271d\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u21910) x\u271d\u2016 \u2264 0\n[PROOFSTEP]\nsimp [Pi.zero_def]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f - \u2191g) x\u2016 \u2264 C\n[PROOFSTEP]\nintro k n\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nk n : \u2115\n\u22a2 \u2203 C, \u2200 (x : E), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f - \u2191g) x\u2016 \u2264 C\n[PROOFSTEP]\nrefine' \u27e8f.seminormAux k n + g.seminormAux k n, fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f - \u2191g) x\u2016 \u2264 SchwartzMap.seminormAux k n f + SchwartzMap.seminormAux k n g\n[PROOFSTEP]\nrefine' le_trans _ (add_le_add (f.le_seminormAux k n x) (g.le_seminormAux k n x))\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f - \u2191g) x\u2016 \u2264\n    \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f + -\u2191g) x\u2016 \u2264\n    \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191g) x\u2016\n[PROOFSTEP]\nrw [\u2190 decay_neg_aux k n g x]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf g : \ud835\udce2(E, F)\nk n : \u2115\nx : E\n\u22a2 \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f + -\u2191g) x\u2016 \u2264\n    \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 + \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (-\u2191g) x\u2016\n[PROOFSTEP]\nconvert decay_add_le_aux k n f (-g) x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u22a2 Function.Injective \u2191(coeHom E F)\n[PROOFSTEP]\nrw [coe_coeHom]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\n\u22a2 Function.Injective FunLike.coe\n[PROOFSTEP]\nexact FunLike.coe_injective\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nk n : \u2115\nf : \ud835\udce2(\u211d, F)\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (x : \u211d), |x| ^ k * \u2016iteratedDeriv n (\u2191f) x\u2016 \u2264 M\n\u22a2 \u2191(SchwartzMap.seminorm \ud835\udd5c k n) f \u2264 M\n[PROOFSTEP]\nrefine' seminorm_le_bound \ud835\udd5c k n f hMp _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nk n : \u2115\nf : \ud835\udce2(\u211d, F)\nM : \u211d\nhMp : 0 \u2264 M\nhM : \u2200 (x : \u211d), |x| ^ k * \u2016iteratedDeriv n (\u2191f) x\u2016 \u2264 M\n\u22a2 \u2200 (x : \u211d), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 M\n[PROOFSTEP]\nsimpa only [Real.norm_eq_abs, norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nk n : \u2115\nf : \ud835\udce2(\u211d, F)\nx : \u211d\n\u22a2 |x| ^ k * \u2016iteratedDeriv n (\u2191f) x\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k n) f\n[PROOFSTEP]\nhave := le_seminorm \ud835\udd5c k n f x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nk n : \u2115\nf : \ud835\udce2(\u211d, F)\nx : \u211d\nthis : \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k n) f\n\u22a2 |x| ^ k * \u2016iteratedDeriv n (\u2191f) x\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k n) f\n[PROOFSTEP]\nrwa [\u2190 Real.norm_eq_abs, \u2190 norm_iteratedFDeriv_eq_norm_iteratedDeriv]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nn : \u2115\nx\u2080 : E\n\u22a2 \u2016iteratedFDeriv \u211d n (\u2191f) x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 n) f\n[PROOFSTEP]\nhave := SchwartzMap.le_seminorm \ud835\udd5c 0 n f x\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nn : \u2115\nx\u2080 : E\nthis : \u2016x\u2080\u2016 ^ 0 * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 n) f\n\u22a2 \u2016iteratedFDeriv \u211d n (\u2191f) x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 n) f\n[PROOFSTEP]\nrwa [pow_zero, one_mul] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nk : \u2115\nx\u2080 : E\n\u22a2 \u2016x\u2080\u2016 ^ k * \u2016\u2191f x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k 0) f\n[PROOFSTEP]\nhave := SchwartzMap.le_seminorm \ud835\udd5c k 0 f x\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nk : \u2115\nx\u2080 : E\nthis : \u2016x\u2080\u2016 ^ k * \u2016iteratedFDeriv \u211d 0 (\u2191f) x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k 0) f\n\u22a2 \u2016x\u2080\u2016 ^ k * \u2016\u2191f x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c k 0) f\n[PROOFSTEP]\nrwa [norm_iteratedFDeriv_zero] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nx\u2080 : E\n\u22a2 \u2016\u2191f x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 0) f\n[PROOFSTEP]\nhave := norm_pow_mul_le_seminorm \ud835\udd5c f 0 x\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf : \ud835\udce2(E, F)\nx\u2080 : E\nthis : \u2016x\u2080\u2016 ^ 0 * \u2016\u2191f x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 0) f\n\u22a2 \u2016\u2191f x\u2080\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 0) f\n[PROOFSTEP]\nrwa [pow_zero, one_mul] at this \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 (1 + \u2016x\u2016) ^ k * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(2 ^ m.fst) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrw [add_comm, add_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 (\u2211 m in Finset.range (k + 1), \u2016x\u2016 ^ m * 1 ^ (k - m) * \u2191(Nat.choose k m)) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(2 ^ m.fst) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nsimp only [one_pow, mul_one, Finset.sum_congr, Finset.sum_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(2 ^ m.fst) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(2 ^ m.fst) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrw [\u2190 Nat.sum_range_choose m.1]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(\u2211 m_1 in Finset.range (m.fst + 1), Nat.choose m.fst m_1) *\n      \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    (\u2211 x in Finset.range (m.fst + 1), \u2191(Nat.choose m.fst x)) *\n      \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrw [Finset.sum_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2211 x in Finset.range (m.fst + 1),\n      \u2191(Nat.choose m.fst x) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nhave hk' : Finset.range (k + 1) \u2286 Finset.range (m.1 + 1) := by rwa [Finset.range_subset, add_le_add_iff_right]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\n[PROOFSTEP]\nrwa [Finset.range_subset, add_le_add_iff_right]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\n\u22a2 \u2211 x_1 in Finset.range (k + 1), \u2016x\u2016 ^ x_1 * \u2191(Nat.choose k x_1) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2211 x in Finset.range (m.fst + 1),\n      \u2191(Nat.choose m.fst x) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrefine' le_trans (Finset.sum_le_sum_of_subset_of_nonneg hk' fun _ _ _ => by positivity) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\nx\u271d\u00b2 : \u2115\nx\u271d\u00b9 : x\u271d\u00b2 \u2208 Finset.range (m.fst + 1)\nx\u271d : \u00acx\u271d\u00b2 \u2208 Finset.range (k + 1)\n\u22a2 0 \u2264 \u2016x\u2016 ^ x\u271d\u00b2 * \u2191(Nat.choose k x\u271d\u00b2) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\n\u22a2 \u2211 i in Finset.range (m.fst + 1), \u2016x\u2016 ^ i * \u2191(Nat.choose k i) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2211 x in Finset.range (m.fst + 1),\n      \u2191(Nat.choose m.fst x) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrefine' Finset.sum_le_sum fun i hi => _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2016x\u2016 ^ i * \u2191(Nat.choose k i) * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264\n    \u2191(Nat.choose m.fst i) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrw [mul_comm (\u2016x\u2016 ^ i), mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2191(Nat.choose k i) * (\u2016x\u2016 ^ i * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016) \u2264\n    \u2191(Nat.choose m.fst i) * \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\nrefine' mul_le_mul _ _ (by positivity) (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 0 \u2264 \u2016x\u2016 ^ i * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 0 \u2264 \u2191(Nat.choose m.fst i)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2191(Nat.choose k i) \u2264 \u2191(Nat.choose m.fst i)\n[PROOFSTEP]\nexact_mod_cast Nat.choose_le_choose i hk\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2016x\u2016 ^ i * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\ntrans\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2016x\u2016 ^ i * \u2016iteratedFDeriv \u211d n (\u2191f) x\u2016 \u2264 ?m.375189\n[PROOFSTEP]\nexact le_seminorm \ud835\udd5c i n f x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 \u2191(SchwartzMap.seminorm \ud835\udd5c i n) f \u2264 \u2191(Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd) f\n[PROOFSTEP]\napply Seminorm.le_def.1\n[GOAL]\ncase a\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : \u2115 \u00d7 \u2115\nk n : \u2115\nhk : k \u2264 m.fst\nhn : n \u2264 m.snd\nf : \ud835\udce2(E, F)\nx : E\nhk' : Finset.range (k + 1) \u2286 Finset.range (m.fst + 1)\ni : \u2115\nhi : i \u2208 Finset.range (m.fst + 1)\n\u22a2 SchwartzMap.seminorm \ud835\udd5c i n \u2264 Finset.sup (Finset.Iic m) fun m => SchwartzMap.seminorm \ud835\udd5c m.fst m.snd\n[PROOFSTEP]\nexact Finset.le_sup_of_le (Finset.mem_Iic.2 <| Prod.mk_le_mk.2 \u27e8Finset.mem_range_succ_iff.mp hi, hn\u27e9) le_rfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\n\u22a2 WithSeminorms (schwartzSeminormFamily \ud835\udd5c E F)\n[PROOFSTEP]\nhave A : WithSeminorms (schwartzSeminormFamily \u211d E F) := \u27e8rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nA : WithSeminorms (schwartzSeminormFamily \u211d E F)\n\u22a2 WithSeminorms (schwartzSeminormFamily \ud835\udd5c E F)\n[PROOFSTEP]\nrw [SeminormFamily.withSeminorms_iff_nhds_eq_iInf] at A \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nA : nhds 0 = \u2a05 (i : \u2115 \u00d7 \u2115), Filter.comap (\u2191(schwartzSeminormFamily \u211d E F i)) (nhds 0)\n\u22a2 nhds 0 = \u2a05 (i : \u2115 \u00d7 \u2115), Filter.comap (\u2191(schwartzSeminormFamily \ud835\udd5c E F i)) (nhds 0)\n[PROOFSTEP]\nrw [A]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nA : nhds 0 = \u2a05 (i : \u2115 \u00d7 \u2115), Filter.comap (\u2191(schwartzSeminormFamily \u211d E F i)) (nhds 0)\n\u22a2 \u2a05 (i : \u2115 \u00d7 \u2115), Filter.comap (\u2191(schwartzSeminormFamily \u211d E F i)) (nhds 0) =\n    \u2a05 (i : \u2115 \u00d7 \u2115), Filter.comap (\u2191(schwartzSeminormFamily \ud835\udd5c E F i)) (nhds 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\n\u22a2 ContinuousSMul \ud835\udd5c \ud835\udce2(E, F)\n[PROOFSTEP]\nrw [(schwartz_withSeminorms \ud835\udd5c E F).withSeminorms_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\n\u22a2 ContinuousSMul \ud835\udd5c \ud835\udce2(E, F)\n[PROOFSTEP]\nexact (schwartzSeminormFamily \ud835\udd5c E F).moduleFilterBasis.continuousSMul\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\nn : \u2115\n\u22a2 \u2203 k C x, \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ k\n[PROOFSTEP]\nchoose k C f using hf_temperate.2\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\n\u22a2 \u2203 k C x, \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ k\n[PROOFSTEP]\nuse(Finset.range (n + 1)).sup k\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\n\u22a2 \u2203 C x, \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nlet C' := max (0 : \u211d) ((Finset.range (n + 1)).sup' (by simp) C)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\n\u22a2 Finset.Nonempty (Finset.range (n + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\n\u22a2 \u2203 C x, \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nhave hC' : 0 \u2264 C' := by simp only [le_refl, Finset.le_sup'_iff, true_or_iff, le_max_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\n\u22a2 0 \u2264 C'\n[PROOFSTEP]\nsimp only [le_refl, Finset.le_sup'_iff, true_or_iff, le_max_iff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\n\u22a2 \u2203 C x, \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nuse C', hC'\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\n\u22a2 \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C' * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nintro N hN x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2264 n\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C' * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nrw [\u2190 Finset.mem_range_succ_iff] at hN \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d N f\u271d x\u2016 \u2264 C' * (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nrefine' le_trans (f N x) (mul_le_mul _ _ (by positivity) hC')\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 0 \u2264 (1 + \u2016x\u2016) ^ k N\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 C N \u2264 C'\n[PROOFSTEP]\nsimp only [Finset.le_sup'_iff, le_max_iff]\n[GOAL]\ncase h.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 C N \u2264 0 \u2228 \u2203 b, b \u2208 Finset.range (n + 1) \u2227 C N \u2264 C b\n[PROOFSTEP]\nright\n[GOAL]\ncase h.refine'_1.h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 \u2203 b, b \u2208 Finset.range (n + 1) \u2227 C N \u2264 C b\n[PROOFSTEP]\nexact \u27e8N, hN, rfl.le\u27e9\n[GOAL]\ncase h.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 (1 + \u2016x\u2016) ^ k N \u2264 (1 + \u2016x\u2016) ^ Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nrefine' pow_le_pow (by simp only [le_add_iff_nonneg_right, norm_nonneg]) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 1 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, norm_nonneg]\n[GOAL]\ncase h.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf\u271d : E \u2192 F\nhf_temperate : Function.HasTemperateGrowth f\u271d\nn : \u2115\nk : \u2115 \u2192 \u2115\nC : \u2115 \u2192 \u211d\nf : \u2200 (n : \u2115) (x : E), \u2016iteratedFDeriv \u211d n f\u271d x\u2016 \u2264 C n * (1 + \u2016x\u2016) ^ k n\nC' : \u211d := max 0 (Finset.sup' (Finset.range (n + 1)) (_ : Finset.Nonempty (Finset.range (n + 1))) C)\nhC' : 0 \u2264 C'\nN : \u2115\nhN : N \u2208 Finset.range (Nat.succ n)\nx : E\n\u22a2 k N \u2264 Finset.sup (Finset.range (n + 1)) k\n[PROOFSTEP]\nexact Finset.le_sup hN\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedField \ud835\udd5c\ninst\u271d\u2078 : NormedField \ud835\udd5c'\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\n\u22a2 \u2200 (k n : \u2115), \u2203 C, \u2200 (x : F), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (A \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nintro k n\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedField \ud835\udd5c\ninst\u271d\u2078 : NormedField \ud835\udd5c'\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\nk n : \u2115\n\u22a2 \u2203 C, \u2200 (x : F), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (A \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nrcases hbound \u27e8k, n\u27e9 with \u27e8s, C, _, h\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \u211d F\ninst\u271d\u2079 : NormedField \ud835\udd5c\ninst\u271d\u2078 : NormedField \ud835\udd5c'\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\ninst\u271d\u2074 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\nk n : \u2115\ns : Finset (\u2115 \u00d7 \u2115)\nC : \u211d\nleft\u271d : 0 \u2264 C\nh :\n  \u2200 (f : \ud835\udce2(D, E)) (x : F),\n    \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n\u22a2 \u2203 C, \u2200 (x : F), \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d n (A \u2191f) x\u2016 \u2264 C\n[PROOFSTEP]\nexact \u27e8C * (s.sup (schwartzSeminormFamily \ud835\udd5c D E)) f, h f\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n\u22a2 Continuous (mkLM A hadd hsmul hsmooth hbound).toAddHom.toFun\n[PROOFSTEP]\nchange Continuous (mkLM A hadd hsmul hsmooth hbound : \ud835\udce2(D, E) \u2192\u209b\u2097[\u03c3] \ud835\udce2(F, G))\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n\u22a2 Continuous \u2191(mkLM A hadd hsmul hsmooth hbound)\n[PROOFSTEP]\nrefine' Seminorm.continuous_from_bounded (schwartz_withSeminorms \ud835\udd5c D E) (schwartz_withSeminorms \ud835\udd5c' F G) _ fun n => _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nn : \u2115 \u00d7 \u2115\n\u22a2 \u2203 s C,\n    Seminorm.comp (schwartzSeminormFamily \ud835\udd5c' F G n) (mkLM A hadd hsmul hsmooth hbound) \u2264\n      C \u2022 Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)\n[PROOFSTEP]\nrcases hbound n with \u27e8s, C, hC, h\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nn : \u2115 \u00d7 \u2115\ns : Finset (\u2115 \u00d7 \u2115)\nC : \u211d\nhC : 0 \u2264 C\nh :\n  \u2200 (f : \ud835\udce2(D, E)) (x : F),\n    \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n\u22a2 \u2203 s C,\n    Seminorm.comp (schwartzSeminormFamily \ud835\udd5c' F G n) (mkLM A hadd hsmul hsmooth hbound) \u2264\n      C \u2022 Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)\n[PROOFSTEP]\nrefine' \u27e8s, \u27e8C, hC\u27e9, fun f => _\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nn : \u2115 \u00d7 \u2115\ns : Finset (\u2115 \u00d7 \u2115)\nC : \u211d\nhC : 0 \u2264 C\nh :\n  \u2200 (f : \ud835\udce2(D, E)) (x : F),\n    \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\n\u22a2 (fun f => \u2191f) (Seminorm.comp (schwartzSeminormFamily \ud835\udd5c' F G n) (mkLM A hadd hsmul hsmooth hbound)) f \u2264\n    (fun f => \u2191f) ({ val := C, property := hC } \u2022 Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n[PROOFSTEP]\nsimp only [Seminorm.comp_apply, Seminorm.smul_apply, NNReal.smul_def, Algebra.id.smul_eq_mul, Subtype.coe_mk]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nn : \u2115 \u00d7 \u2115\ns : Finset (\u2115 \u00d7 \u2115)\nC : \u211d\nhC : 0 \u2264 C\nh :\n  \u2200 (f : \ud835\udce2(D, E)) (x : F),\n    \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\n\u22a2 \u2191(schwartzSeminormFamily \ud835\udd5c' F G n) (\u2191(mkLM A hadd hsmul hsmooth hbound) f) \u2264\n    \u2191{ val := C, property := hC } * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n[PROOFSTEP]\nexact (mkLM A hadd hsmul hsmooth hbound f).seminorm_le_bound \ud835\udd5c' n.1 n.2 (by positivity) (h f)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d F\ninst\u271d\u00b9\u2070 : NormedField \ud835\udd5c\ninst\u271d\u2079 : NormedField \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup D\ninst\u271d\u2077 : NormedSpace \u211d D\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c' G\ninst\u271d\u00b9 : SMulCommClass \u211d \ud835\udd5c' G\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c'\ninst\u271d : RingHomIsometric \u03c3\nA : (D \u2192 E) \u2192 F \u2192 G\nhadd : \u2200 (f g : \ud835\udce2(D, E)) (x : F), A (\u2191f + \u2191g) x = A (\u2191f) x + A (\u2191g) x\nhsmul : \u2200 (a : \ud835\udd5c) (f : \ud835\udce2(D, E)) (x : F), A (\u2191(a \u2022 f)) x = \u2191\u03c3 a \u2022 A (\u2191f) x\nhsmooth : \u2200 (f : \ud835\udce2(D, E)), ContDiff \u211d \u22a4 (A \u2191f)\nhbound :\n  \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : F),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nn : \u2115 \u00d7 \u2115\ns : Finset (\u2115 \u00d7 \u2115)\nC : \u211d\nhC : 0 \u2264 C\nh :\n  \u2200 (f : \ud835\udce2(D, E)) (x : F),\n    \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd (A \u2191f) x\u2016 \u2264 C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\nf : \ud835\udce2(D, E)\n\u22a2 0 \u2264 \u2191{ val := C, property := hC } * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c D E)) f\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\n\u22a2 \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(E, E \u2192L[\u211d] F)) (x : E),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd ((fun f x => \u2191(f x) m) \u2191f) x\u2016 \u2264\n            C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nrintro \u27e8k, n\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, E \u2192L[\u211d] F)) (x : E),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(f x) m) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nuse{(k, n)}, \u2016m\u2016, norm_nonneg _\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\n\u22a2 \u2200 (f : \ud835\udce2(E, E \u2192L[\u211d] F)) (x : E),\n    \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(f x) m) \u2191f) x\u2016 \u2264\n      \u2016m\u2016 * \u2191(Finset.sup {(k, n)} (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nintro f x\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\nf : \ud835\udce2(E, E \u2192L[\u211d] F)\nx : E\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(f x) m) \u2191f) x\u2016 \u2264\n    \u2016m\u2016 * \u2191(Finset.sup {(k, n)} (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nrefine' le_trans (mul_le_mul_of_nonneg_left (norm_iteratedFDeriv_clm_apply_const f.2 le_top) (by positivity)) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\nf : \ud835\udce2(E, E \u2192L[\u211d] F)\nx : E\n\u22a2 0 \u2264 \u2016x\u2016 ^ (k, n).fst\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\nf : \ud835\udce2(E, E \u2192L[\u211d] F)\nx : E\n\u22a2 \u2016x\u2016 ^ (k, n).fst * (\u2016m\u2016 * \u2016iteratedFDeriv \u211d (k, n).snd (fun x => \u2191f x) x\u2016) \u2264\n    \u2016m\u2016 * \u2191(Finset.sup {(k, n)} (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 mul_comm \u2016m\u2016, mul_assoc]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\nf : \ud835\udce2(E, E \u2192L[\u211d] F)\nx : E\n\u22a2 \u2016m\u2016 * (\u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd (fun x => \u2191f x) x\u2016) \u2264\n    \u2016m\u2016 * \u2191(Finset.sup {(k, n)} (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (norm_nonneg _)\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nm : E\nk n : \u2115\nf : \ud835\udce2(E, E \u2192L[\u211d] F)\nx : E\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd (fun x => \u2191f x) x\u2016 \u2264\n    \u2191(Finset.sup {(k, n)} (schwartzSeminormFamily \ud835\udd5c E (E \u2192L[\u211d] F))) f\n[PROOFSTEP]\nsimp only [Finset.sup_singleton, schwartzSeminormFamily_apply, le_seminorm]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nx\u271d\u00b2 x\u271d\u00b9 : \ud835\udce2(D, E)\nx\u271d : D\n\u22a2 (fun f x => \u2191(\u2191B (f x)) (g x)) (\u2191x\u271d\u00b2 + \u2191x\u271d\u00b9) x\u271d =\n    (fun f x => \u2191(\u2191B (f x)) (g x)) (\u2191x\u271d\u00b2) x\u271d + (fun f x => \u2191(\u2191B (f x)) (g x)) (\u2191x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp only [map_add, add_left_inj, Pi.add_apply, eq_self_iff_true, ContinuousLinearMap.add_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nx\u271d\u00b2 : \u211d\nx\u271d\u00b9 : \ud835\udce2(D, E)\nx\u271d : D\n\u22a2 (fun f x => \u2191(\u2191B (f x)) (g x)) (\u2191(x\u271d\u00b2 \u2022 x\u271d\u00b9)) x\u271d = \u2191(RingHom.id \u211d) x\u271d\u00b2 \u2022 (fun f x => \u2191(\u2191B (f x)) (g x)) (\u2191x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp only [smul_apply, map_smul, ContinuousLinearMap.coe_smul', Pi.smul_apply, RingHom.id_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\n\u22a2 \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(D, E)) (x : D),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n            C * \u2191(Finset.sup s (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrintro \u27e8k, n\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n : \u2115\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(D, E)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrcases hg.norm_iteratedFDeriv_le_uniform_aux n with \u27e8l, C, hC, hgrowth\u27e9\n[GOAL]\ncase mk.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(D, E)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nuse Finset.Iic (l + k, n), \u2016B\u2016 * ((n : \u211d) + (1 : \u211d)) * n.choose (n / 2) * (C * 2 ^ (l + k)), by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 0 \u2264 \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k)))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2200 (f : \ud835\udce2(D, E)) (x : D),\n    \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n      \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n        \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nintro f x\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n    \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nhave hxk : 0 \u2264 \u2016x\u2016 ^ k := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\n\u22a2 0 \u2264 \u2016x\u2016 ^ k\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n    \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nhave hnorm_mul := ContinuousLinearMap.norm_iteratedFDeriv_le_of_bilinear B f.smooth' hg.1 x (n := n) le_top\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => \u2191(\u2191B (f x)) (g x)) \u2191f) x\u2016 \u2264\n    \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrefine' le_trans (mul_le_mul_of_nonneg_left hnorm_mul hxk) _\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2016x\u2016 ^ (k, n).fst *\n      (\u2016B\u2016 *\n        \u2211 i in Finset.range (n + 1),\n          \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016) \u2264\n    \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc (\u2016x\u2016 ^ k), mul_comm (\u2016x\u2016 ^ k)]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2016B\u2016 * \u2016x\u2016 ^ k *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264\n    \u2016B\u2016 * (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nsimp_rw [mul_assoc \u2016B\u2016]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2016B\u2016 *\n      (\u2016x\u2016 ^ k *\n        \u2211 x_1 in Finset.range (n + 1),\n          \u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    \u2016B\u2016 *\n      ((\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n        \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 0 \u2264 \u2016B\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2016x\u2016 ^ k *\n      \u2211 x_1 in Finset.range (n + 1),\n        \u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016 \u2264\n    (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [Finset.mul_sum]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nhave : (\u2211 _x : \u2115 in Finset.range (n + 1), (1 : \u211d)) = n + 1 := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\n\u22a2 \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrepeat rw [mul_assoc ((n : \u211d) + 1)]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) * \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [mul_assoc ((n : \u211d) + 1)]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) * (\u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k)))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [mul_assoc ((n : \u211d) + 1)]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) *\n      (\u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n        \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrw [mul_assoc ((n : \u211d) + 1)]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    (\u2191n + 1) *\n      (\u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n        \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrw [\u2190 this, Finset.sum_mul]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\n\u22a2 \u2211 x_1 in Finset.range (n + 1),\n      \u2016x\u2016 ^ k * (\u2191(Nat.choose n x_1) * \u2016iteratedFDeriv \u211d x_1 f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - x_1) g x\u2016) \u2264\n    \u2211 x in Finset.range (n + 1),\n      1 *\n        (\u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n          \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrefine' Finset.sum_le_sum fun i hi => _\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 \u2016x\u2016 ^ k * (\u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016) \u2264\n    1 *\n      (\u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n        \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nsimp only [one_mul]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 \u2016x\u2016 ^ k * (\u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016) \u2264\n    \u2191(Nat.choose n (n / 2)) * (C * \u2191(2 ^ (l + k))) *\n      \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm (\u2016x\u2016 ^ k), mul_assoc, mul_assoc, mul_assoc]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 \u2191(Nat.choose n i) * (\u2016iteratedFDeriv \u211d i f.toFun x\u2016 * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d (n - i) g x\u2016)) \u2264\n    \u2191(Nat.choose n (n / 2)) *\n      (C * \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrefine' mul_le_mul _ _ (by positivity) (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 0 \u2264 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d (n - i) g x\u2016)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 0 \u2264 \u2191(Nat.choose n (n / 2))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 \u2191(Nat.choose n i) \u2264 \u2191(Nat.choose n (n / 2))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase right.refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 Nat.choose n i \u2264 Nat.choose n (n / 2)\n[PROOFSTEP]\nexact i.choose_le_middle n\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d (n - i) g x\u2016) \u2264\n    C * \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nspecialize hgrowth (n - i) (by simp only [tsub_le_self]) x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\n\u22a2 n - i \u2264 n\n[PROOFSTEP]\nsimp only [tsub_le_self]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d (n - i) g x\u2016) \u2264\n    C * \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264\n    C * \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrefine' le_trans (mul_le_mul_of_nonneg_left hgrowth (by positivity)) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 0 \u2264 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016x\u2016 ^ k\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016x\u2016 ^ k * (C * (1 + \u2016x\u2016) ^ l) \u2264\n    C * \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [mul_comm _ (C * _), mul_assoc, mul_assoc C]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 C * ((1 + \u2016x\u2016) ^ l * (\u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016x\u2016 ^ k)) \u2264\n    C * (\u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ hC\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 (1 + \u2016x\u2016) ^ l * (\u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016x\u2016 ^ k) \u2264\n    \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [mul_comm _ (\u2016x\u2016 ^ k)]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 (1 + \u2016x\u2016) ^ l * (\u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d i f.toFun x\u2016) \u2264\n    \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2208 Finset.range (n + 1)\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 \u2264\n    \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrw [Finset.mem_range_succ_iff] at hi \n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhi : i \u2264 n\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 \u2264\n    \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nchange i \u2264 (l + k, n).snd at hi \n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 \u2264\n    \u2191(2 ^ (l + k)) * \u2191(Finset.sup (Finset.Iic (l + k, n)) (schwartzSeminormFamily \u211d D E)) f\n[PROOFSTEP]\nrefine' le_trans _ (one_add_le_sup_seminorm_apply le_rfl hi f x)\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 \u2264 (1 + \u2016x\u2016) ^ (l + k, n).fst * \u2016iteratedFDeriv \u211d i (\u2191f) x\u2016\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right _ (norm_nonneg _)\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ (l + k, n).fst\n[PROOFSTEP]\nrw [pow_add]\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 (1 + \u2016x\u2016) ^ l * \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ l * (1 + \u2016x\u2016) ^ k\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 0 \u2264 (1 + \u2016x\u2016) ^ l\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n[PROOFSTEP]\nrefine' pow_le_pow_of_le_left (norm_nonneg _) _ _\n[GOAL]\ncase right.refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ninst\u271d\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b2 : NormedSpace \u211d D\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nB : E \u2192L[\u211d] F \u2192L[\u211d] G\ng : D \u2192 F\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nf : \ud835\udce2(D, E)\nx : D\nhxk : 0 \u2264 \u2016x\u2016 ^ k\nhnorm_mul :\n  \u2016iteratedFDeriv \u211d n (fun y => \u2191(\u2191B (toFun f y)) (g y)) x\u2016 \u2264\n    \u2016B\u2016 *\n      \u2211 i in Finset.range (n + 1), \u2191(Nat.choose n i) * \u2016iteratedFDeriv \u211d i f.toFun x\u2016 * \u2016iteratedFDeriv \u211d (n - i) g x\u2016\nthis : \u2211 _x in Finset.range (n + 1), 1 = \u2191n + 1\ni : \u2115\nhgrowth : \u2016iteratedFDeriv \u211d (n - i) g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nhi : i \u2264 (l + k, n).snd\n\u22a2 \u2016x\u2016 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\nsimp only [zero_le_one, le_add_iff_nonneg_left]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nhg_upper : \u2203 k C, \u2200 (x : D), \u2016x\u2016 \u2264 C * (1 + \u2016g x\u2016) ^ k\nx\u271d\u00b2 x\u271d\u00b9 : \ud835\udce2(E, F)\nx\u271d : D\n\u22a2 (fun f x => f (g x)) (\u2191x\u271d\u00b2 + \u2191x\u271d\u00b9) x\u271d = (fun f x => f (g x)) (\u2191x\u271d\u00b2) x\u271d + (fun f x => f (g x)) (\u2191x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nsimp only [add_left_inj, Pi.add_apply, eq_self_iff_true]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nhg_upper : \u2203 k C, \u2200 (x : D), \u2016x\u2016 \u2264 C * (1 + \u2016g x\u2016) ^ k\n\u22a2 \u2200 (n : \u2115 \u00d7 \u2115),\n    \u2203 s C,\n      0 \u2264 C \u2227\n        \u2200 (f : \ud835\udce2(E, F)) (x : D),\n          \u2016x\u2016 ^ n.fst * \u2016iteratedFDeriv \u211d n.snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n            C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrintro \u27e8k, n\u27e9\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nhg_upper : \u2203 k C, \u2200 (x : D), \u2016x\u2016 \u2264 C * (1 + \u2016g x\u2016) ^ k\nk n : \u2115\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, F)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrcases hg.norm_iteratedFDeriv_le_uniform_aux n with \u27e8l, C, hC, hgrowth\u27e9\n[GOAL]\ncase mk.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nhg_upper : \u2203 k C, \u2200 (x : D), \u2016x\u2016 \u2264 C * (1 + \u2016g x\u2016) ^ k\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, F)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrcases hg_upper with \u27e8kg, Cg, hg_upper'\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, F)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hCg : 1 \u2264 1 + Cg := by\n  refine' le_add_of_nonneg_right _\n  specialize hg_upper' 0\n  rw [norm_zero] at hg_upper' \n  refine' nonneg_of_mul_nonneg_left hg_upper' (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\n\u22a2 1 \u2264 1 + Cg\n[PROOFSTEP]\nrefine' le_add_of_nonneg_right _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\n\u22a2 0 \u2264 Cg\n[PROOFSTEP]\nspecialize hg_upper' 0\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u20160\u2016 \u2264 Cg * (1 + \u2016g 0\u2016) ^ kg\n\u22a2 0 \u2264 Cg\n[PROOFSTEP]\nrw [norm_zero] at hg_upper' \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : 0 \u2264 Cg * (1 + \u2016g 0\u2016) ^ kg\n\u22a2 0 \u2264 Cg\n[PROOFSTEP]\nrefine' nonneg_of_mul_nonneg_left hg_upper' (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : 0 \u2264 Cg * (1 + \u2016g 0\u2016) ^ kg\n\u22a2 0 < (1 + \u2016g 0\u2016) ^ kg\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, F)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nlet k' := kg * (k + l * n)\n[GOAL]\ncase mk.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\n\u22a2 \u2203 s C,\n    0 \u2264 C \u2227\n      \u2200 (f : \ud835\udce2(E, F)) (x : D),\n        \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n          C * \u2191(Finset.sup s (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nuse Finset.Iic (k', n), (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * n ! * 2 ^ k'), by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\n\u22a2 0 \u2264 (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k'))\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\n\u22a2 \u2200 (f : \ud835\udce2(E, F)) (x : D),\n    \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n      (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n        \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nintro f x\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nlet seminorm_f := ((Finset.Iic (k', n)).sup (schwartzSeminormFamily \ud835\udd5c _ _)) f\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k' :=\n  by\n  rw [pow_mul, \u2190 mul_pow]\n  refine' pow_le_pow_of_le_left (by positivity) _ _\n  rw [add_mul]\n  refine' add_le_add _ (hg_upper' x)\n  nth_rw 1 [\u2190 one_mul (1 : \u211d)]\n  refine' mul_le_mul (le_refl _) (one_le_pow_of_one_le _ _) zero_le_one zero_le_one\n  simp only [le_add_iff_nonneg_right, norm_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\nrw [pow_mul, \u2190 mul_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) \u2264 ((1 + Cg) * (1 + \u2016g x\u2016) ^ kg) ^ (k + l * n)\n[PROOFSTEP]\nrefine' pow_le_pow_of_le_left (by positivity) _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 0 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 1 + \u2016x\u2016 \u2264 (1 + Cg) * (1 + \u2016g x\u2016) ^ kg\n[PROOFSTEP]\nrw [add_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 1 + \u2016x\u2016 \u2264 1 * (1 + \u2016g x\u2016) ^ kg + Cg * (1 + \u2016g x\u2016) ^ kg\n[PROOFSTEP]\nrefine' add_le_add _ (hg_upper' x)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 1 \u2264 1 * (1 + \u2016g x\u2016) ^ kg\n[PROOFSTEP]\nnth_rw 1 [\u2190 one_mul (1 : \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 1 * 1 \u2264 1 * (1 + \u2016g x\u2016) ^ kg\n[PROOFSTEP]\nrefine' mul_le_mul (le_refl _) (one_le_pow_of_one_le _ _) zero_le_one zero_le_one\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n\u22a2 1 \u2264 1 + \u2016g x\u2016\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, norm_nonneg]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hbound : \u2200 i, i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i f (g x)\u2016 \u2264 2 ^ k' * seminorm_f / (1 + \u2016g x\u2016) ^ k' :=\n  by\n  intro i hi\n  have hpos : 0 < (1 + \u2016g x\u2016) ^ k' := by positivity\n  rw [le_div_iff' hpos]\n  change i \u2264 (k', n).snd at hi \n  exact one_add_le_sup_seminorm_apply le_rfl hi _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\n\u22a2 \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\nintro i hi\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\ni : \u2115\nhi : i \u2264 n\n\u22a2 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\nhave hpos : 0 < (1 + \u2016g x\u2016) ^ k' := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\ni : \u2115\nhi : i \u2264 n\n\u22a2 0 < (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\ni : \u2115\nhi : i \u2264 n\nhpos : 0 < (1 + \u2016g x\u2016) ^ k'\n\u22a2 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\nrw [le_div_iff' hpos]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\ni : \u2115\nhi : i \u2264 n\nhpos : 0 < (1 + \u2016g x\u2016) ^ k'\n\u22a2 (1 + \u2016g x\u2016) ^ k' * \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f\n[PROOFSTEP]\nchange i \u2264 (k', n).snd at hi \n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\ni : \u2115\nhpos : 0 < (1 + \u2016g x\u2016) ^ k'\nhi : i \u2264 (k', n).snd\n\u22a2 (1 + \u2016g x\u2016) ^ k' * \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f\n[PROOFSTEP]\nexact one_add_le_sup_seminorm_apply le_rfl hi _ _\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hgrowth' : \u2200 N : \u2115, 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N :=\n  by\n  intro N hN\u2081 hN\u2082\n  refine' (hgrowth N hN\u2082 x).trans _\n  rw [mul_pow]\n  have hN\u2081' := (lt_of_lt_of_le zero_lt_one hN\u2081).ne'\n  refine' mul_le_mul _ _ (by positivity) (by positivity)\n  \u00b7 exact le_trans (by simp [hC]) (le_self_pow (by simp [hC]) hN\u2081')\n  \u00b7 refine' le_self_pow (one_le_pow_of_one_le _ l) hN\u2081'\n    simp only [le_add_iff_nonneg_right, norm_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\n\u22a2 \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nintro N hN\u2081 hN\u2082\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\n\u22a2 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nrefine' (hgrowth N hN\u2082 x).trans _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\n\u22a2 C * (1 + \u2016x\u2016) ^ l \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nrw [mul_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\n\u22a2 C * (1 + \u2016x\u2016) ^ l \u2264 (C + 1) ^ N * ((1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nhave hN\u2081' := (lt_of_lt_of_le zero_lt_one hN\u2081).ne'\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 C * (1 + \u2016x\u2016) ^ l \u2264 (C + 1) ^ N * ((1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nrefine' mul_le_mul _ _ (by positivity) (by positivity)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 0 \u2264 (1 + \u2016x\u2016) ^ l\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 0 \u2264 (C + 1) ^ N\n[PROOFSTEP]\npositivity\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 C \u2264 (C + 1) ^ N\n[PROOFSTEP]\nexact le_trans (by simp [hC]) (le_self_pow (by simp [hC]) hN\u2081')\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 C \u2264 C + 1\n[PROOFSTEP]\nsimp [hC]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 1 \u2264 C + 1\n[PROOFSTEP]\nsimp [hC]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 (1 + \u2016x\u2016) ^ l \u2264 ((1 + \u2016x\u2016) ^ l) ^ N\n[PROOFSTEP]\nrefine' le_self_pow (one_le_pow_of_one_le _ l) hN\u2081'\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nN : \u2115\nhN\u2081 : 1 \u2264 N\nhN\u2082 : N \u2264 n\nhN\u2081' : N \u2260 0\n\u22a2 1 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, norm_nonneg]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave := norm_iteratedFDeriv_comp_le f.smooth' hg.1 le_top x hbound hgrowth'\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k :=\n  pow_le_pow_of_le_left (norm_nonneg _) (by simp only [zero_le_one, le_add_iff_nonneg_left]) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\n\u22a2 \u2016x\u2016 \u2264 1 + \u2016x\u2016\n[PROOFSTEP]\nsimp only [zero_le_one, le_add_iff_nonneg_left]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016 \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrefine' le_trans (mul_le_mul hxk this (by positivity) (by positivity)) _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 0 \u2264 \u2016iteratedFDeriv \u211d (k, n).snd ((fun f x => f (g x)) \u2191f) x\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 0 \u2264 (1 + \u2016x\u2016) ^ k\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave rearrange :\n  (1 + \u2016x\u2016) ^ k * (n ! * (2 ^ k' * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * n ! * 2 ^ k' * seminorm_f) :=\n  by\n  rw [mul_pow, pow_add, \u2190 pow_mul]\n  ring\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\n[PROOFSTEP]\nrw [mul_pow, pow_add, \u2190 pow_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\n\u22a2 (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) ^ n * (1 + \u2016x\u2016) ^ (l * n))) =\n    (1 + \u2016x\u2016) ^ k * (1 + \u2016x\u2016) ^ (l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\n[PROOFSTEP]\nring\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\n\u22a2 (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrw [rearrange]\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hgxk' : 0 < (1 + \u2016g x\u2016) ^ k' := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\n\u22a2 0 < (1 + \u2016g x\u2016) ^ k'\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) \u2264 (1 + Cg) ^ (k + l * n) * (1 + \u2016g x\u2016) ^ k'\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrw [\u2190 div_le_iff hgxk'] at hg_upper'' \n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' \u2264 (1 + Cg) ^ (k + l * n)\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave hpos : (0 : \u211d) \u2264 (C + 1) ^ n * n ! * 2 ^ k' * seminorm_f :=\n  by\n  have : 0 \u2264 seminorm_f := map_nonneg _ _\n  positivity\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' \u2264 (1 + Cg) ^ (k + l * n)\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\n\u22a2 0 \u2264 (C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f\n[PROOFSTEP]\nhave : 0 \u2264 seminorm_f := map_nonneg _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' \u2264 (1 + Cg) ^ (k + l * n)\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis\u271d :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\nthis : 0 \u2264 seminorm_f\n\u22a2 0 \u2264 (C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' \u2264 (1 + Cg) ^ (k + l * n)\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\nhpos : 0 \u2264 (C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f\n\u22a2 (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrefine' le_trans (mul_le_mul_of_nonneg_right hg_upper'' hpos) _\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup D\ninst\u271d\u2076 : NormedSpace \u211d D\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : NormedSpace \u211d G\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b2 : SMulCommClass \u211d \ud835\udd5c F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : SMulCommClass \u211d \ud835\udd5c G\ng : D \u2192 E\nhg : Function.HasTemperateGrowth g\nk n l : \u2115\nC : \u211d\nhC : 0 \u2264 C\nhgrowth : \u2200 (N : \u2115), N \u2264 n \u2192 \u2200 (x : D), \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 C * (1 + \u2016x\u2016) ^ l\nkg : \u2115\nCg : \u211d\nhg_upper' : \u2200 (x : D), \u2016x\u2016 \u2264 Cg * (1 + \u2016g x\u2016) ^ kg\nhCg : 1 \u2264 1 + Cg\nk' : \u2115 := kg * (k + l * n)\nf : \ud835\udce2(E, F)\nx : D\nseminorm_f : (fun a => \u211d) f := \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\nhg_upper'' : (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' \u2264 (1 + Cg) ^ (k + l * n)\nhbound : \u2200 (i : \u2115), i \u2264 n \u2192 \u2016iteratedFDeriv \u211d i (\u2191f) (g x)\u2016 \u2264 \u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k'\nhgrowth' : \u2200 (N : \u2115), 1 \u2264 N \u2192 N \u2264 n \u2192 \u2016iteratedFDeriv \u211d N g x\u2016 \u2264 ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ N\nthis :\n  \u2016iteratedFDeriv \u211d n (f.toFun \u2218 g) x\u2016 \u2264\n    \u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n\nhxk : \u2016x\u2016 ^ k \u2264 (1 + \u2016x\u2016) ^ k\nrearrange :\n  (1 + \u2016x\u2016) ^ k * (\u2191n ! * (\u2191(2 ^ k') * seminorm_f / (1 + \u2016g x\u2016) ^ k') * ((C + 1) * (1 + \u2016x\u2016) ^ l) ^ n) =\n    (1 + \u2016x\u2016) ^ (k + l * n) / (1 + \u2016g x\u2016) ^ k' * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f)\nhgxk' : 0 < (1 + \u2016g x\u2016) ^ k'\nhpos : 0 \u2264 (C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f\n\u22a2 (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k') * seminorm_f) \u2264\n    (1 + Cg) ^ (k + l * n) * ((C + 1) ^ n * \u2191n ! * \u2191(2 ^ k')) *\n      \u2191(Finset.sup (Finset.Iic (k', n)) (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nx\u271d : \u2115 \u00d7 \u2115\nk n : \u2115\nf : \ud835\udce2(E, F)\nx : E\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd (fderiv \u211d \u2191f) x\u2016 \u2264\n    1 * \u2191(Finset.sup {(k, n + 1)} (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nsimpa only [schwartzSeminormFamily_apply, Seminorm.comp_apply, Finset.sup_singleton, one_smul,\n  norm_iteratedFDeriv_fderiv, one_mul] using f.le_seminorm \ud835\udd5c k (n + 1) x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nx\u271d : \u2115 \u00d7 \u2115\nk n : \u2115\nf : \ud835\udce2(\u211d, F)\nx : \u211d\n\u22a2 \u2016x\u2016 ^ (k, n).fst * \u2016iteratedFDeriv \u211d (k, n).snd ((fun f => deriv f) \u2191f) x\u2016 \u2264\n    1 * \u2191(Finset.sup {(k, n + 1)} (schwartzSeminormFamily \ud835\udd5c \u211d F)) f\n[PROOFSTEP]\nsimpa only [Real.norm_eq_abs, Finset.sup_singleton, schwartzSeminormFamily_apply, one_mul,\n  norm_iteratedFDeriv_eq_norm_iteratedDeriv, \u2190 iteratedDeriv_succ'] using f.le_seminorm' \ud835\udd5c k (n + 1) x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn : \u2115\nm : Fin (n + 1) \u2192 E\nf : \ud835\udce2(E, F)\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn : \u2115\nm\u271d : Fin (n + 1) \u2192 E\nf : \ud835\udce2(E, F)\nm : Fin (Nat.zero + 1) \u2192 E\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last Nat.zero))) f)\n[PROOFSTEP]\nrw [iteratedPDeriv_zero, iteratedPDeriv_one]\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn : \u2115\nm\u271d : Fin (n + 1) \u2192 E\nf : \ud835\udce2(E, F)\nm : Fin (Nat.zero + 1) \u2192 E\n\u22a2 \u2191(pderivCLM \ud835\udd5c (m 0)) f = \u2191(pderivCLM \ud835\udd5c (m (Fin.last Nat.zero))) f\n[PROOFSTEP]\nrfl\n  -- The proof is `\u2202^{n + 2} = \u2202 \u2202^{n + 1} = \u2202 \u2202^n \u2202 = \u2202^{n+1} \u2202`\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last (Nat.succ n)))) f)\n[PROOFSTEP]\nhave hmzero : Fin.init m 0 = m 0 := by simp only [Fin.init_def, Fin.castSucc_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\n\u22a2 Fin.init m 0 = m 0\n[PROOFSTEP]\nsimp only [Fin.init_def, Fin.castSucc_zero]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last (Nat.succ n)))) f)\n[PROOFSTEP]\nhave hmtail : Fin.tail m (Fin.last n) = m (Fin.last n.succ) := by\n  simp only [Fin.tail_def, Fin.succ_last]\n    -- Porting note: changed to `calc` proof\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\n\u22a2 Fin.tail m (Fin.last n) = m (Fin.last (Nat.succ n))\n[PROOFSTEP]\nsimp only [Fin.tail_def, Fin.succ_last]\n  -- Porting note: changed to `calc` proof\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\nhmtail : Fin.tail m (Fin.last n) = m (Fin.last (Nat.succ n))\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last (Nat.succ n)))) f)\n[PROOFSTEP]\ncalc\n  _ = pderivCLM \ud835\udd5c (m 0) (iteratedPDeriv \ud835\udd5c _ f) := iteratedPDeriv_succ_left _ _ _\n  _ = pderivCLM \ud835\udd5c (m 0) ((iteratedPDeriv \ud835\udd5c _) ((pderivCLM \ud835\udd5c _) f)) :=\n    by\n    congr 1\n    exact IH _\n  _ = _ := by simp only [hmtail, iteratedPDeriv_succ_left, hmzero, Fin.tail_init_eq_init_tail]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\nhmtail : Fin.tail m (Fin.last n) = m (Fin.last (Nat.succ n))\n\u22a2 \u2191(pderivCLM \ud835\udd5c (m 0)) (\u2191(iteratedPDeriv \ud835\udd5c (Fin.tail m)) f) =\n    \u2191(pderivCLM \ud835\udd5c (m 0)) (\u2191(iteratedPDeriv \ud835\udd5c ?m.961203) (\u2191(pderivCLM \ud835\udd5c ?m.962305) f))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_6.h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\nhmtail : Fin.tail m (Fin.last n) = m (Fin.last (Nat.succ n))\n\u22a2 \u2191(iteratedPDeriv \ud835\udd5c (Fin.tail m)) f = \u2191(iteratedPDeriv \ud835\udd5c ?m.961203) (\u2191(pderivCLM \ud835\udd5c ?m.962305) f)\n[PROOFSTEP]\nexact IH _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nn\u271d : \u2115\nm\u271d : Fin (n\u271d + 1) \u2192 E\nf : \ud835\udce2(E, F)\nn : \u2115\nIH :\n  \u2200 (m : Fin (n + 1) \u2192 E),\n    \u2191(iteratedPDeriv \ud835\udd5c m) f = \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last n))) f)\nm : Fin (Nat.succ n + 1) \u2192 E\nhmzero : Fin.init m 0 = m 0\nhmtail : Fin.tail m (Fin.last n) = m (Fin.last (Nat.succ n))\n\u22a2 \u2191(pderivCLM \ud835\udd5c (m 0)) (\u2191(iteratedPDeriv \ud835\udd5c (Fin.init (Fin.tail m))) (\u2191(pderivCLM \ud835\udd5c (Fin.tail m (Fin.last n))) f)) =\n    \u2191(iteratedPDeriv \ud835\udd5c (Fin.init m)) (\u2191(pderivCLM \ud835\udd5c (m (Fin.last (Nat.succ n)))) f)\n[PROOFSTEP]\nsimp only [hmtail, iteratedPDeriv_succ_left, hmzero, Fin.tail_init_eq_init_tail]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf g : \ud835\udce2(E, F)\n\u22a2 (fun f => toBoundedContinuousFunction f) (f + g) =\n    (fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nf g : \ud835\udce2(E, F)\nx\u271d : E\n\u22a2 \u2191((fun f => toBoundedContinuousFunction f) (f + g)) x\u271d =\n    \u2191((fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g) x\u271d\n[PROOFSTEP]\nexact add_apply\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\na : \ud835\udd5c\nf : \ud835\udce2(E, F)\n\u22a2 AddHom.toFun\n      { toFun := fun f => toBoundedContinuousFunction f,\n        map_add' :=\n          (_ :\n            \u2200 (f g : \ud835\udce2(E, F)),\n              (fun f => toBoundedContinuousFunction f) (f + g) =\n                (fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g) }\n      (a \u2022 f) =\n    \u2191(RingHom.id \ud835\udd5c) a \u2022\n      AddHom.toFun\n        { toFun := fun f => toBoundedContinuousFunction f,\n          map_add' :=\n            (_ :\n              \u2200 (f g : \ud835\udce2(E, F)),\n                (fun f => toBoundedContinuousFunction f) (f + g) =\n                  (fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g) }\n        f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\na : \ud835\udd5c\nf : \ud835\udce2(E, F)\nx\u271d : E\n\u22a2 \u2191(AddHom.toFun\n          { toFun := fun f => toBoundedContinuousFunction f,\n            map_add' :=\n              (_ :\n                \u2200 (f g : \ud835\udce2(E, F)),\n                  (fun f => toBoundedContinuousFunction f) (f + g) =\n                    (fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g) }\n          (a \u2022 f))\n      x\u271d =\n    \u2191(\u2191(RingHom.id \ud835\udd5c) a \u2022\n          AddHom.toFun\n            { toFun := fun f => toBoundedContinuousFunction f,\n              map_add' :=\n                (_ :\n                  \u2200 (f g : \ud835\udce2(E, F)),\n                    (fun f => toBoundedContinuousFunction f) (f + g) =\n                      (fun f => toBoundedContinuousFunction f) f + (fun f => toBoundedContinuousFunction f) g) }\n            f)\n      x\u271d\n[PROOFSTEP]\nexact smul_apply\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\n\u22a2 Continuous\n    { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : \ud835\udd5c) (x : \ud835\udce2(E, F)),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id \ud835\udd5c) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nchange Continuous (toBoundedContinuousFunctionLM \ud835\udd5c E F)\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\n\u22a2 Continuous \u2191(toBoundedContinuousFunctionLM \ud835\udd5c E F)\n[PROOFSTEP]\nrefine'\n  Seminorm.continuous_from_bounded (schwartz_withSeminorms \ud835\udd5c E F) (norm_withSeminorms \ud835\udd5c (E \u2192\u1d47 F)) _ fun _ =>\n    \u27e8{0}, 1, fun f => _\u27e9\n      -- Porting note: Lean failed to find this instance\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\nx\u271d : Fin 1\nf : \ud835\udce2(E, F)\n\u22a2 (fun f => \u2191f) (Seminorm.comp (normSeminorm \ud835\udd5c (E \u2192\u1d47 F)) (toBoundedContinuousFunctionLM \ud835\udd5c E F)) f \u2264\n    (fun f => \u2191f) (1 \u2022 Finset.sup {0} (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nhave : MulAction NNReal (Seminorm \ud835\udd5c \ud835\udce2(E, F)) := Seminorm.instDistribMulAction.toMulAction\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\nx\u271d : Fin 1\nf : \ud835\udce2(E, F)\nthis : MulAction NNReal (Seminorm \ud835\udd5c \ud835\udce2(E, F))\n\u22a2 (fun f => \u2191f) (Seminorm.comp (normSeminorm \ud835\udd5c (E \u2192\u1d47 F)) (toBoundedContinuousFunctionLM \ud835\udd5c E F)) f \u2264\n    (fun f => \u2191f) (1 \u2022 Finset.sup {0} (schwartzSeminormFamily \ud835\udd5c E F)) f\n[PROOFSTEP]\nsimp only [Seminorm.comp_apply, coe_normSeminorm, Finset.sup_singleton, schwartzSeminormFamily_apply_zero,\n  Seminorm.smul_apply, one_smul, ge_iff_le, BoundedContinuousFunction.norm_le (map_nonneg _ _)]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\nx\u271d : Fin 1\nf : \ud835\udce2(E, F)\nthis : MulAction NNReal (Seminorm \ud835\udd5c \ud835\udce2(E, F))\n\u22a2 \u2200 (x : E), \u2016\u2191(\u2191(toBoundedContinuousFunctionLM \ud835\udd5c E F) f) x\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 0) f\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nsrc\u271d : \ud835\udce2(E, F) \u2192\u2097[\ud835\udd5c] E \u2192\u1d47 F := toBoundedContinuousFunctionLM \ud835\udd5c E F\nx\u271d : Fin 1\nf : \ud835\udce2(E, F)\nthis : MulAction NNReal (Seminorm \ud835\udd5c \ud835\udce2(E, F))\nx : E\n\u22a2 \u2016\u2191(\u2191(toBoundedContinuousFunctionLM \ud835\udd5c E F) f) x\u2016 \u2264 \u2191(SchwartzMap.seminorm \ud835\udd5c 0 0) f\n[PROOFSTEP]\nexact norm_le_seminorm \ud835\udd5c _ _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Distribution.SchwartzSpace", "llama_tokens": 115155, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891479496521, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.5326126592708166}}
{"text": "[GOAL]\nJ\u271d : Type\ninst\u271d\u2074 : SmallCategory J\u271d\ninst\u271d\u00b3 : FinCategory J\u271d\nk : Type v\ninst\u271d\u00b2 : Field k\nJ : Type\ninst\u271d\u00b9 : Fintype J\nZ : J \u2192 ModuleCat k\ninst\u271d : \u2200 (j : J), FiniteDimensional k \u2191(Z j)\n\u22a2 FiniteDimensional k \u2191(ModuleCat.of k ((j : J) \u2192 \u2191(Z j)))\n[PROOFSTEP]\nunfold ModuleCat.of\n[GOAL]\nJ\u271d : Type\ninst\u271d\u2074 : SmallCategory J\u271d\ninst\u271d\u00b3 : FinCategory J\u271d\nk : Type v\ninst\u271d\u00b2 : Field k\nJ : Type\ninst\u271d\u00b9 : Fintype J\nZ : J \u2192 ModuleCat k\ninst\u271d : \u2200 (j : J), FiniteDimensional k \u2191(Z j)\n\u22a2 FiniteDimensional k \u2191(ModuleCat.mk ((j : J) \u2192 \u2191(Z j)))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ\u271d : Type\ninst\u271d\u2074 : SmallCategory J\u271d\ninst\u271d\u00b3 : FinCategory J\u271d\nk : Type v\ninst\u271d\u00b2 : Field k\nJ : Type\ninst\u271d\u00b9 : Fintype J\nZ : J \u2192 ModuleCat k\ninst\u271d : \u2200 (j : J), FiniteDimensional k \u2191(Z j)\nthis : FiniteDimensional k \u2191(ModuleCat.of k ((j : J) \u2192 \u2191(Z j)))\n\u22a2 Mono (ModuleCat.piIsoPi fun j => Z j).hom\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nJ : Type\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\nk : Type v\ninst\u271d : Field k\nF : J \u2964 FGModuleCat k\n\u22a2 \u2200 (j : J), FiniteDimensional k \u2191((F \u22d9 forget\u2082 (FGModuleCat k) (ModuleCat k)).obj j)\n[PROOFSTEP]\nintro j\n[GOAL]\nJ : Type\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\nk : Type v\ninst\u271d : Field k\nF : J \u2964 FGModuleCat k\nj : J\n\u22a2 FiniteDimensional k \u2191((F \u22d9 forget\u2082 (FGModuleCat k) (ModuleCat k)).obj j)\n[PROOFSTEP]\nchange FiniteDimensional k (F.obj j)\n[GOAL]\nJ : Type\ninst\u271d\u00b2 : SmallCategory J\ninst\u271d\u00b9 : FinCategory J\nk : Type v\ninst\u271d : Field k\nF : J \u2964 FGModuleCat k\nj : J\n\u22a2 FiniteDimensional k \u2191(F.obj j)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.FGModuleCat.Limits", "llama_tokens": 730, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744673038222, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.5325311175715618}}
{"text": "[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 a \u2264 c\n[PROOFSTEP]\nshow a \u2294 c = c\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b c : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 a \u2294 c = c\n[PROOFSTEP]\nrw [\u2190 hbc, \u2190 sup_assoc, hab]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b : \u03b1\nhab : a \u2264 b\nhba : b \u2264 a\n\u22a2 a = b\n[PROOFSTEP]\nrwa [\u2190 hba, sup_comm]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b : \u03b1\n\u22a2 a \u2294 (a \u2294 b) = a \u2294 b\n[PROOFSTEP]\nrw [\u2190 sup_assoc, sup_idem]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b : \u03b1\n\u22a2 b \u2294 (a \u2294 b) = a \u2294 b\n[PROOFSTEP]\nrw [sup_comm, sup_assoc, sup_idem]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b c : \u03b1\nhac : a \u2264 c\nhbc : b \u2264 c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nshow (a \u2294 b) \u2294 c = c\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Sup \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\na b c : \u03b1\nhac : a \u2264 c\nhbc : b \u2264 c\n\u22a2 a \u2294 b \u2294 c = c\n[PROOFSTEP]\nrwa [sup_assoc, hbc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b \u2264 a \u2227 a \u2264 a \u2294 b \u2194 b \u2264 a\n[PROOFSTEP]\nsimp [le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b \u2264 b \u2227 b \u2264 a \u2294 b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2264 b \u2194 \u2203 c, b = a \u2294 c\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2264 b \u2192 \u2203 c, b = a \u2294 c\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 \u2203 c, b = a \u2294 c\n[PROOFSTEP]\nexact \u27e8b, (sup_eq_right.mpr h).symm\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 (\u2203 c, b = a \u2294 c) \u2192 a \u2264 b\n[PROOFSTEP]\nrintro \u27e8c, rfl : _ = _ \u2294 _\u27e9\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na c\u271d d c : \u03b1\n\u22a2 a \u2264 a \u2294 c\n[PROOFSTEP]\nexact le_sup_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 a = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b = b \u2294 a\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b \u2264 b \u2294 a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 b \u2294 a \u2264 a \u2294 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d x : \u03b1\n\u22a2 a \u2294 b \u2294 c \u2264 x \u2194 a \u2294 (b \u2294 c) \u2264 x\n[PROOFSTEP]\nsimp only [sup_le_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d x : \u03b1\n\u22a2 (a \u2264 x \u2227 b \u2264 x) \u2227 c \u2264 x \u2194 a \u2264 x \u2227 b \u2264 x \u2227 c \u2264 x\n[PROOFSTEP]\nrw [and_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 b \u2294 c = c \u2294 b \u2294 a\n[PROOFSTEP]\nrw [sup_comm, @sup_comm _ _ a, sup_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 (a \u2294 b) = a \u2294 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b \u2294 b = a \u2294 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 (b \u2294 c) = b \u2294 (a \u2294 c)\n[PROOFSTEP]\nrw [\u2190 sup_assoc, \u2190 sup_assoc, @sup_comm \u03b1 _ a]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 b \u2294 c = a \u2294 c \u2294 b\n[PROOFSTEP]\nrw [sup_assoc, sup_assoc, @sup_comm _ _ b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d\u271d a b c d : \u03b1\n\u22a2 a \u2294 b \u2294 (c \u2294 d) = a \u2294 c \u2294 (b \u2294 d)\n[PROOFSTEP]\nrw [sup_assoc, sup_left_comm b, \u2190 sup_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 (b \u2294 c) = a \u2294 b \u2294 (a \u2294 c)\n[PROOFSTEP]\nrw [sup_sup_sup_comm, sup_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 b \u2294 c = a \u2294 c \u2294 (b \u2294 c)\n[PROOFSTEP]\nrw [sup_sup_sup_comm, sup_idem]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 x \u2294 y \u2264 c \u2194 x \u2294 y \u2264 c\n[PROOFSTEP]\nsimp only [sup_le_iff]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 x \u2294 y \u2264 c \u2194 x \u2264 c \u2227 y \u2264 c\n[PROOFSTEP]\nrw [\u2190 H, @sup_le_iff \u03b1 A, H, H]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\nhave ss : A.toSup = B.toSup := by ext; apply SemilatticeSup.ext_sup H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 toSup = toSup\n[PROOFSTEP]\next\n[GOAL]\ncase sup.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 x\u271d\u00b9 \u2294 x\u271d = x\u271d\u00b9 \u2294 x\u271d\n[PROOFSTEP]\napply SemilatticeSup.ext_sup H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeSup \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toSup = toSup\n\u22a2 A = B\n[PROOFSTEP]\ncases A\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nB : SemilatticeSup \u03b1\ntoSup\u271d : Sup \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\nle_sup_left\u271d : \u2200 (a b : \u03b1), a \u2264 a \u2294 b\nle_sup_right\u271d : \u2200 (a b : \u03b1), b \u2264 a \u2294 b\nsup_le\u271d : \u2200 (a b c : \u03b1), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toSup = toSup\n\u22a2 mk le_sup_left\u271d le_sup_right\u271d sup_le\u271d = B\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoSup\u271d\u00b9 : Sup \u03b1\ntoPartialOrder\u271d\u00b9 : PartialOrder \u03b1\nle_sup_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 a \u2294 b\nle_sup_right\u271d\u00b9 : \u2200 (a b : \u03b1), b \u2264 a \u2294 b\nsup_le\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\ntoSup\u271d : Sup \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\nle_sup_left\u271d : \u2200 (a b : \u03b1), a \u2264 a \u2294 b\nle_sup_right\u271d : \u2200 (a b : \u03b1), b \u2264 a \u2294 b\nsup_le\u271d : \u2200 (a b c : \u03b1), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toSup = toSup\n\u22a2 mk le_sup_left\u271d\u00b9 le_sup_right\u271d\u00b9 sup_le\u271d\u00b9 = mk le_sup_left\u271d le_sup_right\u271d sup_le\u271d\n[PROOFSTEP]\ncases PartialOrder.ext H\n[GOAL]\ncase mk.mk.refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoSup\u271d\u00b9 : Sup \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\nle_sup_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 a \u2294 b\nle_sup_right\u271d\u00b9 : \u2200 (a b : \u03b1), b \u2264 a \u2294 b\nsup_le\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\ntoSup\u271d : Sup \u03b1\nle_sup_left\u271d : \u2200 (a b : \u03b1), a \u2264 a \u2294 b\nle_sup_right\u271d : \u2200 (a b : \u03b1), b \u2264 a \u2294 b\nsup_le\u271d : \u2200 (a b c : \u03b1), a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toSup = toSup\n\u22a2 mk le_sup_left\u271d\u00b9 le_sup_right\u271d\u00b9 sup_le\u271d\u00b9 = mk le_sup_left\u271d le_sup_right\u271d sup_le\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 b \u2264 a \u2227 a \u2264 a \u2293 b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 b \u2264 b \u2227 b \u2264 a \u2293 b \u2194 b \u2264 a\n[PROOFSTEP]\nsimp [le_rfl]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 c \u2264 x \u2293 y \u2194 c \u2264 x \u2293 y\n[PROOFSTEP]\nsimp only [le_inf_iff]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c\u271d d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y c : \u03b1\n\u22a2 c \u2264 x \u2293 y \u2194 c \u2264 x \u2227 c \u2264 y\n[PROOFSTEP]\nrw [\u2190 H, @le_inf_iff \u03b1 A, H, H]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\nhave ss : A.toInf = B.toInf := by ext; apply SemilatticeInf.ext_inf H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 toInf = toInf\n[PROOFSTEP]\next\n[GOAL]\ncase inf.h.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx\u271d\u00b9 x\u271d : \u03b1\n\u22a2 x\u271d\u00b9 \u2293 x\u271d = x\u271d\u00b9 \u2293 x\u271d\n[PROOFSTEP]\napply SemilatticeInf.ext_inf H\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : SemilatticeInf \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toInf = toInf\n\u22a2 A = B\n[PROOFSTEP]\ncases A\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nB : SemilatticeInf \u03b1\ntoInf\u271d : Inf \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toInf = toInf\n\u22a2 mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d = B\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoInf\u271d\u00b9 : Inf \u03b1\ntoPartialOrder\u271d\u00b9 : PartialOrder \u03b1\ninf_le_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\ntoInf\u271d : Inf \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toInf = toInf\n\u22a2 mk inf_le_left\u271d\u00b9 inf_le_right\u271d\u00b9 le_inf\u271d\u00b9 = mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d\n[PROOFSTEP]\ncases PartialOrder.ext H\n[GOAL]\ncase mk.mk.refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeInf \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoInf\u271d\u00b9 : Inf \u03b1\ntoPartialOrder\u271d : PartialOrder \u03b1\ninf_le_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\ntoInf\u271d : Inf \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nss : toInf = toInf\n\u22a2 mk inf_le_left\u271d\u00b9 inf_le_right\u271d\u00b9 le_inf\u271d\u00b9 = mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Inf \u03b1\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\ninf_idem : \u2200 (a : \u03b1), a \u2293 a = a\n\u22a2 SemilatticeInf \u03b1\n[PROOFSTEP]\nhaveI : SemilatticeSup \u03b1\u1d52\u1d48 := SemilatticeSup.mk' inf_comm inf_assoc inf_idem\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Inf \u03b1\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\ninf_idem : \u2200 (a : \u03b1), a \u2293 a = a\nthis : SemilatticeSup \u03b1\u1d52\u1d48\n\u22a2 SemilatticeInf \u03b1\n[PROOFSTEP]\nhaveI i := OrderDual.semilatticeInf \u03b1\u1d52\u1d48\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d : Inf \u03b1\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\ninf_idem : \u2200 (a : \u03b1), a \u2293 a = a\nthis : SemilatticeSup \u03b1\u1d52\u1d48\ni : SemilatticeInf \u03b1\u1d52\u1d48\u1d52\u1d48\n\u22a2 SemilatticeInf \u03b1\n[PROOFSTEP]\nexact i\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\ninf_idem : \u2200 (a : \u03b1), a \u2293 a = a\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\na b : \u03b1\nh : a \u2294 b = b\n\u22a2 b \u2293 a = a\n[PROOFSTEP]\nrw [\u2190 h, inf_comm, inf_sup_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\nsup_idem : \u2200 (a : \u03b1), a \u2294 a = a\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\ninf_idem : \u2200 (a : \u03b1), a \u2293 a = a\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\na b : \u03b1\nh : b \u2293 a = a\n\u22a2 a \u2294 b = b\n[PROOFSTEP]\nrw [\u2190 h, sup_comm, sup_inf_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nb : \u03b1\n\u22a2 b \u2294 b = b \u2294 b \u2293 (b \u2294 b)\n[PROOFSTEP]\nrw [inf_sup_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nb : \u03b1\n\u22a2 b \u2294 b \u2293 (b \u2294 b) = b\n[PROOFSTEP]\nrw [sup_inf_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\nb : \u03b1\n\u22a2 b \u2293 b = b \u2293 (b \u2294 b \u2293 b)\n[PROOFSTEP]\nrw [sup_inf_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\nb : \u03b1\n\u22a2 b \u2293 (b \u2294 b \u2293 b) = b\n[PROOFSTEP]\nrw [inf_sup_self]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b : \u03b1\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\nrw [partial_order_eq]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b : \u03b1\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\napply inf_le_left\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b : \u03b1\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\nrw [partial_order_eq]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b : \u03b1\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\napply inf_le_right\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b c : \u03b1\n\u22a2 a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nrw [partial_order_eq]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nsup_comm : \u2200 (a b : \u03b1), a \u2294 b = b \u2294 a\nsup_assoc : \u2200 (a b c : \u03b1), a \u2294 b \u2294 c = a \u2294 (b \u2294 c)\ninf_comm : \u2200 (a b : \u03b1), a \u2293 b = b \u2293 a\ninf_assoc : \u2200 (a b c : \u03b1), a \u2293 b \u2293 c = a \u2293 (b \u2293 c)\nsup_inf_self : \u2200 (a b : \u03b1), a \u2294 a \u2293 b = a\ninf_sup_self : \u2200 (a b : \u03b1), a \u2293 (a \u2294 b) = a\nsup_idem : \u2200 (b : \u03b1), b \u2294 b = b\ninf_idem : \u2200 (b : \u03b1), b \u2293 b = b\nsemilatt_inf_inst : SemilatticeInf \u03b1 := SemilatticeInf.mk' inf_comm inf_assoc inf_idem\nsemilatt_sup_inst : SemilatticeSup \u03b1 := SemilatticeSup.mk' sup_comm sup_assoc sup_idem\npartial_order_eq : SemilatticeSup.toPartialOrder = SemilatticeInf.toPartialOrder\na b c : \u03b1\n\u22a2 a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\napply le_inf\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b \u2264 a \u2293 b \u2194 a = b\n[PROOFSTEP]\nsimp [le_antisymm_iff, and_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 b = a \u2294 b \u2194 a = b\n[PROOFSTEP]\nrw [\u2190 inf_le_sup.ge_iff_eq, sup_le_inf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 b < a \u2294 b \u2194 a \u2260 b\n[PROOFSTEP]\nrw [inf_le_sup.lt_iff_ne, Ne.def, inf_eq_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 b = c \u2227 a \u2294 b = c \u2194 a = c \u2227 b = c\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\nh : a \u2293 b = c \u2227 a \u2294 b = c\n\u22a2 a = c \u2227 b = c\ncase refine'_2 \u03b1 : Type u \u03b2 : Type v inst\u271d : Lattice \u03b1 a b c d : \u03b1 \u22a2 a = c \u2227 b = c \u2192 a \u2293 b = c \u2227 a \u2294 b = c\n[PROOFSTEP]\n{ obtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)\n  simpa using h\n}\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\nh : a \u2293 b = c \u2227 a \u2294 b = c\n\u22a2 a = c \u2227 b = c\n[PROOFSTEP]\nobtain rfl := sup_eq_inf.1 (h.2.trans h.1.symm)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na c d : \u03b1\nh : a \u2293 a = c \u2227 a \u2294 a = c\n\u22a2 a = c \u2227 a = c\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a = c \u2227 b = c \u2192 a \u2293 b = c \u2227 a \u2294 b = c\n[PROOFSTEP]\n{ rintro \u27e8rfl, rfl\u27e9\n  exact \u27e8inf_idem, sup_idem\u27e9\n}\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a = c \u2227 b = c \u2192 a \u2293 b = c \u2227 a \u2294 b = c\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\nb d : \u03b1\n\u22a2 b \u2293 b = b \u2227 b \u2294 b = b\n[PROOFSTEP]\nexact \u27e8inf_idem, sup_idem\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2293 (a \u2294 b) = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 a \u2293 b = a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\na b c d : \u03b1\n\u22a2 a \u2294 b = b \u2194 a \u2293 b = a\n[PROOFSTEP]\nrw [sup_eq_right, \u2190 inf_eq_left]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nA B : Lattice \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\ncases A\n[GOAL]\ncase mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\nB : Lattice \u03b1\ntoSemilatticeSup\u271d : SemilatticeSup \u03b1\ntoInf\u271d : Inf \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d = B\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoSemilatticeSup\u271d\u00b9 : SemilatticeSup \u03b1\ntoInf\u271d\u00b9 : Inf \u03b1\ninf_le_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\ntoSemilatticeSup\u271d : SemilatticeSup \u03b1\ntoInf\u271d : Inf \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 mk inf_le_left\u271d\u00b9 inf_le_right\u271d\u00b9 le_inf\u271d\u00b9 = mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d\n[PROOFSTEP]\ncases SemilatticeSup.ext H\n[GOAL]\ncase mk.mk.refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoSemilatticeSup\u271d : SemilatticeSup \u03b1\ntoInf\u271d\u00b9 : Inf \u03b1\ninf_le_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\ntoInf\u271d : Inf \u03b1\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 mk inf_le_left\u271d\u00b9 inf_le_right\u271d\u00b9 le_inf\u271d\u00b9 = mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d\n[PROOFSTEP]\ncases SemilatticeInf.ext H\n[GOAL]\ncase mk.mk.refl.refl\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d : Lattice \u03b1\u271d\na b c d : \u03b1\u271d\n\u03b1 : Type u_1\ntoSemilatticeSup\u271d : SemilatticeSup \u03b1\ntoInf\u271d : Inf \u03b1\ninf_le_left\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\ninf_le_left\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 a\ninf_le_right\u271d : \u2200 (a b : \u03b1), a \u2293 b \u2264 b\nle_inf\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 mk inf_le_left\u271d\u00b9 inf_le_right\u271d\u00b9 le_inf\u271d\u00b9 = mk inf_le_left\u271d inf_le_right\u271d le_inf\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 y \u2293 z \u2294 x = (y \u2294 x) \u2293 (z \u2294 x)\n[PROOFSTEP]\nsimp only [sup_inf_left, fun y : \u03b1 => @sup_comm \u03b1 _ y x, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 x \u2293 (y \u2294 z) = x \u2293 (x \u2294 z) \u2293 (y \u2294 z)\n[PROOFSTEP]\nrw [inf_sup_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 x \u2293 (x \u2294 z) \u2293 (y \u2294 z) = x \u2293 (x \u2293 y \u2294 z)\n[PROOFSTEP]\nsimp only [inf_assoc, sup_inf_right, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 x \u2293 (x \u2293 y \u2294 z) = (x \u2294 x \u2293 y) \u2293 (x \u2293 y \u2294 z)\n[PROOFSTEP]\nrw [sup_inf_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 (x \u2294 x \u2293 y) \u2293 (x \u2293 y \u2294 z) = (x \u2293 y \u2294 x) \u2293 (x \u2293 y \u2294 z)\n[PROOFSTEP]\nrw [sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 (x \u2293 y \u2294 x) \u2293 (x \u2293 y \u2294 z) = x \u2293 y \u2294 x \u2293 z\n[PROOFSTEP]\nrw [sup_inf_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\n\u22a2 (y \u2294 z) \u2293 x = y \u2293 x \u2294 z \u2293 x\n[PROOFSTEP]\nsimp only [inf_sup_left, fun y : \u03b1 => @inf_comm \u03b1 _ y x, eq_self_iff_true]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : DistribLattice \u03b1\nx y z : \u03b1\nh\u2081 : x \u2293 z \u2264 y \u2293 z\nh\u2082 : x \u2294 z \u2264 y \u2294 z\n\u22a2 y \u2293 z \u2294 x = (y \u2294 x) \u2293 (x \u2294 z)\n[PROOFSTEP]\nrw [sup_inf_right, @sup_comm _ _ x]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na\u271d b\u271d c d a b : \u03b1\np : \u03b1 \u2192 Prop\nha : p a\nhb : p b\nh : a \u2264 b\n\u22a2 p (a \u2294 b)\n[PROOFSTEP]\nrwa [sup_eq_right.2 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na\u271d b\u271d c d a b : \u03b1\np : \u03b1 \u2192 Prop\nha : p a\nhb : p b\nh : b \u2264 a\n\u22a2 p (a \u2294 b)\n[PROOFSTEP]\nrwa [sup_eq_left.2 h]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\n\u22a2 a \u2264 b \u2294 c \u2194 a \u2264 b \u2228 a \u2264 c\n[PROOFSTEP]\nexact\n  \u27e8fun h => (le_total c b).imp (fun bc => by rwa [sup_eq_left.2 bc] at h ) (fun bc => by rwa [sup_eq_right.2 bc] at h ),\n    fun h => h.elim le_sup_of_le_left le_sup_of_le_right\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh : a \u2264 b \u2294 c\nbc : c \u2264 b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrwa [sup_eq_left.2 bc] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh : a \u2264 b \u2294 c\nbc : b \u2264 c\n\u22a2 a \u2264 c\n[PROOFSTEP]\nrwa [sup_eq_right.2 bc] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\n\u22a2 a < b \u2294 c \u2194 a < b \u2228 a < c\n[PROOFSTEP]\nexact\n  \u27e8fun h => (le_total c b).imp (fun bc => by rwa [sup_eq_left.2 bc] at h ) (fun bc => by rwa [sup_eq_right.2 bc] at h ),\n    fun h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh : a < b \u2294 c\nbc : c \u2264 b\n\u22a2 a < b\n[PROOFSTEP]\nrwa [sup_eq_left.2 bc] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : LinearOrder \u03b1\na b c d : \u03b1\nh : a < b \u2294 c\nbc : b \u2264 c\n\u22a2 a < c\n[PROOFSTEP]\nrwa [sup_eq_right.2 bc] at h \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\n\u22a2 (fun x x_1 => x \u2294 x_1) = maxDefault\n[PROOFSTEP]\next x y\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\n\u22a2 x \u2294 y = maxDefault x y\n[PROOFSTEP]\nunfold maxDefault\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\n\u22a2 x \u2294 y = if x \u2264 y then y else x\n[PROOFSTEP]\nsplit_ifs with h'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\nh' : x \u2264 y\n\u22a2 x \u2294 y = y\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeSup \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\nh' : \u00acx \u2264 y\n\u22a2 x \u2294 y = x\n[PROOFSTEP]\nexacts [sup_of_le_right h', sup_of_le_left $ (total_of (\u00b7 \u2264 \u00b7) x y).resolve_left h']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\n\u22a2 (fun x x_1 => x \u2293 x_1) = minDefault\n[PROOFSTEP]\next x y\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\n\u22a2 x \u2293 y = minDefault x y\n[PROOFSTEP]\nunfold minDefault\n[GOAL]\ncase h.h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\n\u22a2 x \u2293 y = if x \u2264 y then x else y\n[PROOFSTEP]\nsplit_ifs with h'\n[GOAL]\ncase pos\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\nh' : x \u2264 y\n\u22a2 x \u2293 y = x\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nx y : \u03b1\nh' : \u00acx \u2264 y\n\u22a2 x \u2293 y = y\n[PROOFSTEP]\nexacts [inf_of_le_left h', inf_of_le_right $ (total_of (\u00b7 \u2264 \u00b7) x y).resolve_left h']\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nsrc\u271d : Lattice \u03b1 := inst\u271d\u2074\n\u22a2 \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nexact congr_fun\u2082 inf_eq_minDefault\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b1 : Type u\ninst\u271d\u2074 : Lattice \u03b1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x < x_1\ninst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1\nsrc\u271d : Lattice \u03b1 := inst\u271d\u2074\n\u22a2 \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nexact congr_fun\u2082 sup_eq_maxDefault\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeSup (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\nj : \u03b9\n\u22a2 update f i (a \u2294 b) j = (update f i a \u2294 update f i b) j\n[PROOFSTEP]\nobtain rfl | hji := eq_or_ne j i\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeSup (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\nj : \u03b9\na b : \u03c0 j\n\u22a2 update f j (a \u2294 b) j = (update f j a \u2294 update f j b) j\n[PROOFSTEP]\nsimp [update_noteq, *]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeSup (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\nj : \u03b9\nhji : j \u2260 i\n\u22a2 update f i (a \u2294 b) j = (update f i a \u2294 update f i b) j\n[PROOFSTEP]\nsimp [update_noteq, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeInf (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\nj : \u03b9\n\u22a2 update f i (a \u2293 b) j = (update f i a \u2293 update f i b) j\n[PROOFSTEP]\nobtain rfl | hji := eq_or_ne j i\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeInf (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\nj : \u03b9\na b : \u03c0 j\n\u22a2 update f j (a \u2293 b) j = (update f j a \u2293 update f j b) j\n[PROOFSTEP]\nsimp [update_noteq, *]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 SemilatticeInf (\u03c0 i)\nf : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\nj : \u03b9\nhji : j \u2260 i\n\u22a2 update f i (a \u2293 b) j = (update f i a \u2293 update f i b) j\n[PROOFSTEP]\nsimp [update_noteq, *]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nh : \u2200 (x y : \u03b1), f (x \u2293 y) = f x \u2293 f y\nx y : \u03b1\nhxy : x \u2264 y\n\u22a2 f x \u2293 f y = f x\n[PROOFSTEP]\nrw [\u2190 h, inf_eq_left.2 hxy]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\nx y : \u03b1\nh : x \u2264 y\n\u22a2 f (x \u2294 y) = f x \u2294 f y\n[PROOFSTEP]\nsimp only [h, hf h, sup_of_le_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\nx y : \u03b1\nh : y \u2264 x\n\u22a2 f (x \u2294 y) = f x \u2294 f y\n[PROOFSTEP]\nsimp only [h, hf h, sup_of_le_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 a \u2264 a \u2294 b\n[PROOFSTEP]\nchange f a \u2264 f (a \u2294 b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f a \u2264 f (a \u2294 b)\n[PROOFSTEP]\nrw [map_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f a \u2264 f a \u2294 f b\n[PROOFSTEP]\nexact le_sup_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 b \u2264 a \u2294 b\n[PROOFSTEP]\nchange f b \u2264 f (a \u2294 b)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f b \u2264 f (a \u2294 b)\n[PROOFSTEP]\nrw [map_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f b \u2264 f a \u2294 f b\n[PROOFSTEP]\nexact le_sup_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 c\nhb : b \u2264 c\n\u22a2 a \u2294 b \u2264 c\n[PROOFSTEP]\nchange f (a \u2294 b) \u2264 f c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 c\nhb : b \u2264 c\n\u22a2 f (a \u2294 b) \u2264 f c\n[PROOFSTEP]\nrw [map_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 c\nhb : b \u2264 c\n\u22a2 f a \u2294 f b \u2264 f c\n[PROOFSTEP]\nexact sup_le ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\nchange f (a \u2293 b) \u2264 f a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f (a \u2293 b) \u2264 f a\n[PROOFSTEP]\nrw [map_inf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f a \u2293 f b \u2264 f a\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\nchange f (a \u2293 b) \u2264 f b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f (a \u2293 b) \u2264 f b\n[PROOFSTEP]\nrw [map_inf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b : \u03b1\n\u22a2 f a \u2293 f b \u2264 f b\n[PROOFSTEP]\nexact inf_le_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 b\nhb : a \u2264 c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\nchange f a \u2264 f (b \u2293 c)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 b\nhb : a \u2264 c\n\u22a2 f a \u2264 f (b \u2293 c)\n[PROOFSTEP]\nrw [map_inf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : SemilatticeInf \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f hf_inj\na b c : \u03b1\nha : a \u2264 b\nhb : a \u2264 c\n\u22a2 f a \u2264 f b \u2293 f c\n[PROOFSTEP]\nexact le_inf ha hb\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : DistribLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : Lattice \u03b1 := Injective.lattice f hf_inj map_sup map_inf\na b c : \u03b1\n\u22a2 (a \u2294 b) \u2293 (a \u2294 c) \u2264 a \u2294 b \u2293 c\n[PROOFSTEP]\nchange f ((a \u2294 b) \u2293 (a \u2294 c)) \u2264 f (a \u2294 b \u2293 c)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : DistribLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : Lattice \u03b1 := Injective.lattice f hf_inj map_sup map_inf\na b c : \u03b1\n\u22a2 f ((a \u2294 b) \u2293 (a \u2294 c)) \u2264 f (a \u2294 b \u2293 c)\n[PROOFSTEP]\nrw [map_inf, map_sup, map_sup, map_sup, map_inf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : DistribLattice \u03b2\nf : \u03b1 \u2192 \u03b2\nhf_inj : Injective f\nmap_sup : \u2200 (a b : \u03b1), f (a \u2294 b) = f a \u2294 f b\nmap_inf : \u2200 (a b : \u03b1), f (a \u2293 b) = f a \u2293 f b\nsrc\u271d : Lattice \u03b1 := Injective.lattice f hf_inj map_sup map_inf\na b c : \u03b1\n\u22a2 (f a \u2294 f b) \u2293 (f a \u2294 f c) \u2264 f a \u2294 f b \u2293 f c\n[PROOFSTEP]\nexact le_sup_inf\n", "meta": {"mathlib_filename": "Mathlib.Order.Lattice", "llama_tokens": 21368, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430520409023, "lm_q2_score": 0.6406358548398982, "lm_q1q2_score": 0.5324600396384654}}
{"text": "[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\nf : M \u2192\u2097[R] M'\ng : N \u2192\u2097[R] N'\n\u22a2 \u2191(toMatrix (Basis.tensorProduct bM bN) (Basis.tensorProduct bM' bN')) (map f g) =\n    kroneckerMap (fun x x_1 => x * x_1) (\u2191(toMatrix bM bM') f) (\u2191(toMatrix bN bN') g)\n[PROOFSTEP]\next \u27e8i, j\u27e9 \u27e8i', j'\u27e9\n[GOAL]\ncase a.mk.h.mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\nf : M \u2192\u2097[R] M'\ng : N \u2192\u2097[R] N'\ni : \u03b9'\nj : \u03ba'\ni' : \u03b9\nj' : \u03ba\n\u22a2 \u2191(toMatrix (Basis.tensorProduct bM bN) (Basis.tensorProduct bM' bN')) (map f g) (i, j) (i', j') =\n    kroneckerMap (fun x x_1 => x * x_1) (\u2191(toMatrix bM bM') f) (\u2191(toMatrix bN bN') g) (i, j) (i', j')\n[PROOFSTEP]\nsimp_rw [Matrix.kroneckerMap_apply, toMatrix_apply, Basis.tensorProduct_apply, TensorProduct.map_tmul,\n  Basis.tensorProduct_repr_tmul_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\nA : Matrix \u03b9' \u03b9 R\nB : Matrix \u03ba' \u03ba R\n\u22a2 \u2191(toLin (Basis.tensorProduct bM bN) (Basis.tensorProduct bM' bN')) (kroneckerMap (fun x x_1 => x * x_1) A B) =\n    TensorProduct.map (\u2191(toLin bM bM') A) (\u2191(toLin bN bN') B)\n[PROOFSTEP]\nrw [\u2190 LinearEquiv.eq_symm_apply, toLin_symm, TensorProduct.toMatrix_map, toMatrix_toLin, toMatrix_toLin]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\n\u22a2 \u2191(toMatrix (Basis.tensorProduct bM bN) (Basis.tensorProduct bN bM)) \u2191(TensorProduct.comm R M N) =\n    submatrix 1 Prod.swap _root_.id\n[PROOFSTEP]\next \u27e8i, j\u27e9 \u27e8i', j'\u27e9\n[GOAL]\ncase a.mk.h.mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\n\u22a2 \u2191(toMatrix (Basis.tensorProduct bM bN) (Basis.tensorProduct bN bM)) \u2191(TensorProduct.comm R M N) (i, j) (i', j') =\n    submatrix 1 Prod.swap _root_.id (i, j) (i', j')\n[PROOFSTEP]\nsimp_rw [toMatrix_apply, Basis.tensorProduct_apply, LinearEquiv.coe_coe, TensorProduct.comm_tmul,\n  Basis.tensorProduct_repr_tmul_apply, Matrix.submatrix_apply, Prod.swap_prod_mk, id.def, Basis.repr_self_apply,\n  Matrix.one_apply, Prod.ext_iff, ite_and, @eq_comm _ i', @eq_comm _ j']\n[GOAL]\ncase a.mk.h.mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\n\u22a2 ((if i = j' then 1 else 0) * if j = i' then 1 else 0) = if j = i' then if i = j' then 1 else 0 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\nh\u271d\u00b9 : i = j'\nh\u271d : j = i'\n\u22a2 1 * 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\nh\u271d\u00b9 : i = j'\nh\u271d : \u00acj = i'\n\u22a2 1 * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\nh\u271d\u00b9 : \u00aci = j'\nh\u271d : j = i'\n\u22a2 0 * 1 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03ba\nj i' : \u03b9\nj' : \u03ba\nh\u271d\u00b9 : \u00aci = j'\nh\u271d : \u00acj = i'\n\u22a2 0 * 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\n\u22a2 \u2191(toMatrix (Basis.tensorProduct (Basis.tensorProduct bM bN) bP) (Basis.tensorProduct bM (Basis.tensorProduct bN bP)))\n      \u2191(TensorProduct.assoc R M N P) =\n    submatrix 1 _root_.id \u2191(Equiv.prodAssoc \u03b9 \u03ba \u03c4)\n[PROOFSTEP]\next \u27e8i, j, k\u27e9 \u27e8\u27e8i', j'\u27e9, k'\u27e9\n[GOAL]\ncase a.mk.mk.h.mk.mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\n\u22a2 \u2191(toMatrix (Basis.tensorProduct (Basis.tensorProduct bM bN) bP) (Basis.tensorProduct bM (Basis.tensorProduct bN bP)))\n      \u2191(TensorProduct.assoc R M N P) (i, j, k) ((i', j'), k') =\n    submatrix 1 _root_.id \u2191(Equiv.prodAssoc \u03b9 \u03ba \u03c4) (i, j, k) ((i', j'), k')\n[PROOFSTEP]\nsimp_rw [toMatrix_apply, Basis.tensorProduct_apply, LinearEquiv.coe_coe, TensorProduct.assoc_tmul,\n  Basis.tensorProduct_repr_tmul_apply, Matrix.submatrix_apply, Equiv.prodAssoc_apply, id.def, Basis.repr_self_apply,\n  Matrix.one_apply, Prod.ext_iff, ite_and, @eq_comm _ i', @eq_comm _ j', @eq_comm _ k']\n[GOAL]\ncase a.mk.mk.h.mk.mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\n\u22a2 (if i = i' then 1 else 0) * ((if j = j' then 1 else 0) * if k = k' then 1 else 0) =\n    if i = i' then if j = j' then if k = k' then 1 else 0 else 0 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : i = i'\nh\u271d\u00b9 : j = j'\nh\u271d : k = k'\n\u22a2 1 * (1 * 1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : i = i'\nh\u271d\u00b9 : j = j'\nh\u271d : \u00ack = k'\n\u22a2 1 * (1 * 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : i = i'\nh\u271d\u00b9 : \u00acj = j'\nh\u271d : k = k'\n\u22a2 1 * (0 * 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : i = i'\nh\u271d\u00b9 : \u00acj = j'\nh\u271d : \u00ack = k'\n\u22a2 1 * (0 * 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : \u00aci = i'\nh\u271d\u00b9 : j = j'\nh\u271d : k = k'\n\u22a2 0 * (1 * 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : \u00aci = i'\nh\u271d\u00b9 : j = j'\nh\u271d : \u00ack = k'\n\u22a2 0 * (1 * 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : \u00aci = i'\nh\u271d\u00b9 : \u00acj = j'\nh\u271d : k = k'\n\u22a2 0 * (0 * 1) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nM' : Type u_5\nN' : Type u_6\n\u03b9 : Type u_7\n\u03ba : Type u_8\n\u03c4 : Type u_9\n\u03b9' : Type u_10\n\u03ba' : Type u_11\ninst\u271d\u00b9\u2078 : DecidableEq \u03b9\ninst\u271d\u00b9\u2077 : DecidableEq \u03ba\ninst\u271d\u00b9\u2076 : DecidableEq \u03c4\ninst\u271d\u00b9\u2075 : Fintype \u03b9\ninst\u271d\u00b9\u2074 : Fintype \u03ba\ninst\u271d\u00b9\u00b3 : Fintype \u03c4\ninst\u271d\u00b9\u00b2 : Fintype \u03b9'\ninst\u271d\u00b9\u00b9 : Fintype \u03ba'\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup N\ninst\u271d\u2077 : AddCommGroup P\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup N'\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R N\ninst\u271d\u00b2 : Module R P\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R N'\nbM : Basis \u03b9 R M\nbN : Basis \u03ba R N\nbP : Basis \u03c4 R P\nbM' : Basis \u03b9' R M'\nbN' : Basis \u03ba' R N'\ni : \u03b9\nj : \u03ba\nk k' : \u03c4\ni' : \u03b9\nj' : \u03ba\nh\u271d\u00b2 : \u00aci = i'\nh\u271d\u00b9 : \u00acj = j'\nh\u271d : \u00ack = k'\n\u22a2 0 * (0 * 0) = 0\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.TensorProduct.Matrix", "llama_tokens": 10216, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672227971211, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.5320456011199011}}
{"text": "[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 -\u2045y, x\u2046 = \u2045x, y\u2046\n[PROOFSTEP]\nhave h : \u2045x + y, x\u2046 + \u2045x + y, y\u2046 = 0 := by rw [\u2190 lie_add]; apply lie_self\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x + y, x\u2046 + \u2045x + y, y\u2046 = 0\n[PROOFSTEP]\nrw [\u2190 lie_add]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x + y, x + y\u2046 = 0\n[PROOFSTEP]\napply lie_self\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\nh : \u2045x + y, x\u2046 + \u2045x + y, y\u2046 = 0\n\u22a2 -\u2045y, x\u2046 = \u2045x, y\u2046\n[PROOFSTEP]\nsimpa [neg_eq_iff_add_eq_zero] using h\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm\u271d n : M\nt : R\nx m : L\n\u22a2 \u2045t \u2022 x, m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\nrw [\u2190 lie_skew, \u2190 lie_skew x m, LieAlgebra.lie_smul, smul_neg]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2200 (t : R) (x m : L), \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\napply LieAlgebra.lie_smul\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045-x, m\u2046 = -\u2045x, m\u2046\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, sub_neg_eq_add, \u2190 add_lie]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045-x + x, m\u2046 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x, -m\u2046 = -\u2045x, m\u2046\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, sub_neg_eq_add, \u2190 lie_add]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x, -m + m\u2046 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x - y, m\u2046 = \u2045x, m\u2046 - \u2045y, m\u2046\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x, m - n\u2046 = \u2045x, m\u2046 - \u2045x, n\u2046\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045\u2045x, y\u2046, m\u2046 = \u2045x, \u2045y, m\u2046\u2046 - \u2045y, \u2045x, m\u2046\u2046\n[PROOFSTEP]\nrw [leibniz_lie, add_sub_cancel]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 \u2045x, \u2045y, z\u2046\u2046 + \u2045y, \u2045z, x\u2046\u2046 + \u2045z, \u2045x, y\u2046\u2046 = 0\n[PROOFSTEP]\nrw [\u2190 neg_neg \u2045x, y\u2046, lie_neg z, lie_skew y x, \u2190 lie_skew, lie_lie]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 -(\u2045y, \u2045z, x\u2046\u2046 - \u2045z, \u2045y, x\u2046\u2046) + \u2045y, \u2045z, x\u2046\u2046 + -\u2045z, \u2045y, x\u2046\u2046 = 0\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx y z : L\nm n : M\n\u22a2 -(\u2045y, \u2045z, x\u2046\u2046 - \u2045z, \u2045y, x\u2046\u2046) + \u2045y, \u2045z, x\u2046\u2046 + -\u2045z, \u2045y, x\u2046\u2046 = 0\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm\u271d n\u271d : M\nx : L\nf : M \u2192\u2097[R] N\nm n : M\n\u22a2 (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) (m + n) = (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) m + (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) n\n[PROOFSTEP]\nsimp only [lie_add, LinearMap.map_add]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm\u271d n\u271d : M\nx : L\nf : M \u2192\u2097[R] N\nm n : M\n\u22a2 \u2045x, \u2191f m\u2046 + \u2045x, \u2191f n\u2046 - (\u2191f \u2045x, m\u2046 + \u2191f \u2045x, n\u2046) = \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046 + (\u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm\u271d n\u271d : M\nx : L\nf : M \u2192\u2097[R] N\nm n : M\n\u22a2 \u2045x, \u2191f m\u2046 + \u2045x, \u2191f n\u2046 - (\u2191f \u2045x, m\u2046 + \u2191f \u2045x, n\u2046) = \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046 + (\u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm\u271d n : M\nx : L\nf : M \u2192\u2097[R] N\nt : R\nm : M\n\u22a2 AddHom.toFun\n      { toFun := fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046,\n        map_add' :=\n          (_ :\n            \u2200 (m n : M),\n              (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) (m + n) =\n                (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) m + (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) n) }\n      (t \u2022 m) =\n    \u2191(RingHom.id R) t \u2022\n      AddHom.toFun\n        { toFun := fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046,\n          map_add' :=\n            (_ :\n              \u2200 (m n : M),\n                (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) (m + n) =\n                  (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) m + (fun m => \u2045x, \u2191f m\u2046 - \u2191f \u2045x, m\u2046) n) }\n        m\n[PROOFSTEP]\nsimp only [smul_sub, LinearMap.map_smul, lie_smul, RingHom.id_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n : M\nx y : L\nf : M \u2192\u2097[R] N\n\u22a2 \u2045x + y, f\u2046 = \u2045x, f\u2046 + \u2045y, f\u2046\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2191\u2045x + y, f\u2046 n = \u2191(\u2045x, f\u2046 + \u2045y, f\u2046) n\n[PROOFSTEP]\nsimp only [add_lie, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.add_apply, LinearMap.map_add]\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2191f n\u2046 + \u2045y, \u2191f n\u2046 - (\u2191f \u2045x, n\u2046 + \u2191f \u2045y, n\u2046) = \u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046 + (\u2045y, \u2191f n\u2046 - \u2191f \u2045y, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2191f n\u2046 + \u2045y, \u2191f n\u2046 - (\u2191f \u2045x, n\u2046 + \u2191f \u2045y, n\u2046) = \u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046 + (\u2045y, \u2191f n\u2046 - \u2191f \u2045y, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm n : M\nx : L\nf g : M \u2192\u2097[R] N\n\u22a2 \u2045x, f + g\u2046 = \u2045x, f\u2046 + \u2045x, g\u2046\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm n\u271d : M\nx : L\nf g : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2191\u2045x, f + g\u2046 n = \u2191(\u2045x, f\u2046 + \u2045x, g\u2046) n\n[PROOFSTEP]\nsimp only [LinearMap.coe_mk, AddHom.coe_mk, lie_add, LinearMap.add_apply]\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm n\u271d : M\nx : L\nf g : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2191f n\u2046 + \u2045x, \u2191g n\u2046 - (\u2191f \u2045x, n\u2046 + \u2191g \u2045x, n\u2046) = \u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046 + (\u2045x, \u2191g n\u2046 - \u2191g \u2045x, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y z : L\nm n\u271d : M\nx : L\nf g : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2191f n\u2046 + \u2045x, \u2191g n\u2046 - (\u2191f \u2045x, n\u2046 + \u2191g \u2045x, n\u2046) = \u2045x, \u2191f n\u2046 - \u2191f \u2045x, n\u2046 + (\u2045x, \u2191g n\u2046 - \u2191g \u2045x, n\u2046)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n : M\nx y : L\nf : M \u2192\u2097[R] N\n\u22a2 \u2045x, \u2045y, f\u2046\u2046 = \u2045\u2045x, y\u2046, f\u2046 + \u2045y, \u2045x, f\u2046\u2046\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2191\u2045x, \u2045y, f\u2046\u2046 n = \u2191(\u2045\u2045x, y\u2046, f\u2046 + \u2045y, \u2045x, f\u2046\u2046) n\n[PROOFSTEP]\nsimp only [lie_lie, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.map_sub, LinearMap.add_apply, lie_sub]\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2045y, \u2191f n\u2046\u2046 - \u2045x, \u2191f \u2045y, n\u2046\u2046 - (\u2045y, \u2191f \u2045x, n\u2046\u2046 - \u2191f \u2045y, \u2045x, n\u2046\u2046) =\n    \u2045x, \u2045y, \u2191f n\u2046\u2046 - \u2045y, \u2045x, \u2191f n\u2046\u2046 - (\u2191f \u2045x, \u2045y, n\u2046\u2046 - \u2191f \u2045y, \u2045x, n\u2046\u2046) +\n      (\u2045y, \u2045x, \u2191f n\u2046\u2046 - \u2045y, \u2191f \u2045x, n\u2046\u2046 - (\u2045x, \u2191f \u2045y, n\u2046\u2046 - \u2191f \u2045x, \u2045y, n\u2046\u2046))\n[PROOFSTEP]\nabel\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt : R\nx\u271d y\u271d z : L\nm n\u271d : M\nx y : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2045x, \u2045y, \u2191f n\u2046\u2046 - \u2045x, \u2191f \u2045y, n\u2046\u2046 - (\u2045y, \u2191f \u2045x, n\u2046\u2046 - \u2191f \u2045y, \u2045x, n\u2046\u2046) =\n    \u2045x, \u2045y, \u2191f n\u2046\u2046 - \u2045y, \u2045x, \u2191f n\u2046\u2046 - (\u2191f \u2045x, \u2045y, n\u2046\u2046 - \u2191f \u2045y, \u2045x, n\u2046\u2046) +\n      (\u2045y, \u2045x, \u2191f n\u2046\u2046 - \u2045y, \u2191f \u2045x, n\u2046\u2046 - (\u2045x, \u2191f \u2045y, n\u2046\u2046 - \u2191f \u2045x, \u2045y, n\u2046\u2046))\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm n : M\nt : R\nx : L\nf : M \u2192\u2097[R] N\n\u22a2 \u2045t \u2022 x, f\u2046 = t \u2022 \u2045x, f\u2046\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm n\u271d : M\nt : R\nx : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2191\u2045t \u2022 x, f\u2046 n = \u2191(t \u2022 \u2045x, f\u2046) n\n[PROOFSTEP]\nsimp only [smul_sub, smul_lie, LinearMap.smul_apply, LieHom.lie_apply, LinearMap.map_smul]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm n : M\nt : R\nx : L\nf : M \u2192\u2097[R] N\n\u22a2 \u2045x, t \u2022 f\u2046 = t \u2022 \u2045x, f\u2046\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : LieRing L\ninst\u271d\u2078 : LieAlgebra R L\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup N\ninst\u271d\u00b2 : Module R N\ninst\u271d\u00b9 : LieRingModule L N\ninst\u271d : LieModule R L N\nt\u271d : R\nx\u271d y z : L\nm n\u271d : M\nt : R\nx : L\nf : M \u2192\u2097[R] N\nn : M\n\u22a2 \u2191\u2045x, t \u2022 f\u2046 n = \u2191(t \u2022 \u2045x, f\u2046) n\n[PROOFSTEP]\nsimp only [smul_sub, LinearMap.smul_apply, LieHom.lie_apply, lie_smul]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nx y : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh : (fun f => f.toFun) x = (fun f => f.toFun) y\n\u22a2 x = y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\ny : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ntoLinearMap\u271d : L\u2081 \u2192\u2097[R] L\u2082\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun toLinearMap\u271d.toAddHom x, AddHom.toFun toLinearMap\u271d.toAddHom y\u2046\nh : (fun f => f.toFun) { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d } = (fun f => f.toFun) y\n\u22a2 { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d } = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\ntoLinearMap\u271d\u00b9 : L\u2081 \u2192\u2097[R] L\u2082\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun toLinearMap\u271d\u00b9.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun toLinearMap\u271d\u00b9.toAddHom x, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom y\u2046\ntoLinearMap\u271d : L\u2081 \u2192\u2097[R] L\u2082\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun toLinearMap\u271d.toAddHom x, AddHom.toFun toLinearMap\u271d.toAddHom y\u2046\nh :\n  (fun f => f.toFun) { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } =\n    (fun f => f.toFun) { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n\u22a2 { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } = { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\ntoLinearMap\u271d\u00b9 : L\u2081 \u2192\u2097[R] L\u2082\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun toLinearMap\u271d\u00b9.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun toLinearMap\u271d\u00b9.toAddHom x, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom y\u2046\ntoLinearMap\u271d : L\u2081 \u2192\u2097[R] L\u2082\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun toLinearMap\u271d.toAddHom x, AddHom.toFun toLinearMap\u271d.toAddHom y\u2046\nh : toLinearMap\u271d\u00b9 = toLinearMap\u271d\n\u22a2 { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } = { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nsrc\u271d : L\u2081 \u2192\u2097[R] L\u2082 := 0\n\u22a2 \u2200 {x y : L\u2081},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, y\u2046 =\n      \u2045AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          y\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\n\u22a2 Injective FunLike.coe\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8f, _\u27e9, _\u27e9, _\u27e9 \u27e8\u27e8\u27e8g, _\u27e9, _\u27e9, _\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nh :\n  \u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n        map_lie' := map_lie'\u271d\u00b9 } =\n    \u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n        map_lie' := map_lie'\u271d }\n\u22a2 { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n      map_lie' := map_lie'\u271d\u00b9 } =\n    { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n      map_lie' := map_lie'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf g : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\n\u22a2 f = g \u2192 \u2200 (x : L\u2081), \u2191f x = \u2191g x\n[PROOFSTEP]\nrintro rfl x\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nx : L\u2081\n\u22a2 \u2191f x = \u2191f x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh\u2081 : \u2200 (x y : L\u2081), \u2191f (x + y) = \u2191f x + \u2191f y\nh\u2082 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := \u2191f, map_add' := h\u2081 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := \u2191f, map_add' := h\u2081 } x\nh\u2083 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom y\u2046\n\u22a2 { toLinearMap := { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }, map_lie' := h\u2083 } = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh\u2081 : \u2200 (x y : L\u2081), \u2191f (x + y) = \u2191f x + \u2191f y\nh\u2082 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := \u2191f, map_add' := h\u2081 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := \u2191f, map_add' := h\u2081 } x\nh\u2083 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }.toAddHom y\u2046\nx\u271d : L\u2081\n\u22a2 \u2191{ toLinearMap := { toAddHom := { toFun := \u2191f, map_add' := h\u2081 }, map_smul' := h\u2082 }, map_lie' := h\u2083 } x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2082 \u2192\u2097\u2045R\u2046 L\u2083\ng : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : L\u2081 \u2192\u2097[R] L\u2083 := LinearMap.comp \u2191f \u2191g\n\u22a2 \u2200 {x y : L\u2081},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, y\u2046 =\n      \u2045AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          y\u2046\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2082 \u2192\u2097\u2045R\u2046 L\u2083\ng : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : L\u2081 \u2192\u2097[R] L\u2083 := LinearMap.comp \u2191f \u2191g\nx y : L\u2081\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : L\u2081),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      \u2045x, y\u2046 =\n    \u2045AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x,\n      AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        y\u2046\n[PROOFSTEP]\nchange f (g \u2045x, y\u2046) = \u2045f (g x), f (g y)\u2046\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2082 \u2192\u2097\u2045R\u2046 L\u2083\ng : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nsrc\u271d : L\u2081 \u2192\u2097[R] L\u2083 := LinearMap.comp \u2191f \u2191g\nx y : L\u2081\n\u22a2 \u2191f (\u2191g \u2045x, y\u2046) = \u2045\u2191f (\u2191g x), \u2191f (\u2191g y)\u2046\n[PROOFSTEP]\nrw [map_lie, map_lie]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\n\u22a2 comp f id = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nx\u271d : L\u2081\n\u22a2 \u2191(comp f id) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\n\u22a2 comp id f = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nx\u271d : L\u2081\n\u22a2 \u2191(comp id f) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\n\u22a2 \u2200 {x y : L\u2082},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2082),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, y\u2046 =\n      \u2045AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2082),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2082),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          y\u2046\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : L\u2082),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      \u2045x, y\u2046 =\n    \u2045AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2082),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x,\n      AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2082),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        y\u2046\n[PROOFSTEP]\ncalc\n  g \u2045x, y\u2046 = g \u2045f (g x), f (g y)\u2046 := by conv_lhs => rw [\u2190 h\u2082 x, \u2190 h\u2082 y]\n  _ = g (f \u2045g x, g y\u2046) := by rw [map_lie]\n  _ = \u2045g x, g y\u2046 := h\u2081 _\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n\u22a2 g \u2045x, y\u2046 = g \u2045\u2191f (g x), \u2191f (g y)\u2046\n[PROOFSTEP]\nconv_lhs => rw [\u2190 h\u2082 x, \u2190 h\u2082 y]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n| g \u2045x, y\u2046\n[PROOFSTEP]\nrw [\u2190 h\u2082 x, \u2190 h\u2082 y]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n| g \u2045x, y\u2046\n[PROOFSTEP]\nrw [\u2190 h\u2082 x, \u2190 h\u2082 y]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n| g \u2045x, y\u2046\n[PROOFSTEP]\nrw [\u2190 h\u2082 x, \u2190 h\u2082 y]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : LieRing L\u2083\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ng : L\u2082 \u2192 L\u2081\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : L\u2082 \u2192\u2097[R] L\u2081 := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx y : L\u2082\n\u22a2 g \u2045\u2191f (g x), \u2191f (g y)\u2046 = g (\u2191f \u2045g x, g y\u2046)\n[PROOFSTEP]\nrw [map_lie]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nM : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : LieRingModule L\u2082 M\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nx y : L\u2081\nm : M\n\u22a2 \u2045x + y, m\u2046 = \u2045x, m\u2046 + \u2045y, m\u2046\n[PROOFSTEP]\nsimp only [LieHom.map_add, add_lie]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nM : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieAlgebra R L\u2081\ninst\u271d\u00b3 : LieRing L\u2082\ninst\u271d\u00b2 : LieAlgebra R L\u2082\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : LieRingModule L\u2082 M\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nx y : L\u2081\nm : M\n\u22a2 \u2045x, \u2045y, m\u2046\u2046 = \u2045\u2045x, y\u2046, m\u2046 + \u2045y, \u2045x, m\u2046\u2046\n[PROOFSTEP]\nsimp only [lie_lie, sub_add_cancel, LieHom.map_lie]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nM : Type w\u2081\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\u2081\ninst\u271d\u2076 : LieAlgebra R L\u2081\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : LieRingModule L\u2082 M\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : LieModule R L\u2082 M\nsrc\u271d : LieRingModule L\u2081 M := LieRingModule.compLieHom M f\nt : R\nx : L\u2081\nm : M\n\u22a2 \u2045t \u2022 x, m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [LieRingModule.compLieHom_apply, smul_lie, LieHom.map_smul]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nM : Type w\u2081\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : LieRing L\u2081\ninst\u271d\u2076 : LieAlgebra R L\u2081\ninst\u271d\u2075 : LieRing L\u2082\ninst\u271d\u2074 : LieAlgebra R L\u2082\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : LieRingModule L\u2082 M\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : LieModule R L\u2082 M\nsrc\u271d : LieRingModule L\u2081 M := LieRingModule.compLieHom M f\nt : R\nx : L\u2081\nm : M\n\u22a2 \u2045x, t \u2022 m\u2046 = t \u2022 \u2045x, m\u2046\n[PROOFSTEP]\nsimp only [LieRingModule.compLieHom_apply, lie_smul]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf g : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nh\u2081 : (fun f => f.toFun) f = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) f = (fun f => f.invFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\ng : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\ntoLieHom\u271d : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninvFun\u271d : L\u2082 \u2192 L\u2081\nleft_inv\u271d : LeftInverse invFun\u271d toLieHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieHom\u271d.toFun\nh\u2081 :\n  (fun f => f.toFun) { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } =\n    (fun f => f.toFun) g\nh\u2082 :\n  (fun f => f.invFun) { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } =\n    (fun f => f.invFun) g\n\u22a2 { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\ntoLieHom\u271d\u00b9 : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninvFun\u271d\u00b9 : L\u2082 \u2192 L\u2081\nleft_inv\u271d\u00b9 : LeftInverse invFun\u271d\u00b9 toLieHom\u271d\u00b9.toFun\nright_inv\u271d\u00b9 : Function.RightInverse invFun\u271d\u00b9 toLieHom\u271d\u00b9.toFun\ntoLieHom\u271d : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninvFun\u271d : L\u2082 \u2192 L\u2081\nleft_inv\u271d : LeftInverse invFun\u271d toLieHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieHom\u271d.toFun\nh\u2081 :\n  (fun f => f.toFun) { toLieHom := toLieHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    (fun f => f.toFun) { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\nh\u2082 :\n  (fun f => f.invFun) { toLieHom := toLieHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    (fun f => f.invFun) { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n\u22a2 { toLieHom := toLieHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp at h\u2081 h\u2082 \n[GOAL]\ncase mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\ntoLieHom\u271d\u00b9 : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninvFun\u271d\u00b9 : L\u2082 \u2192 L\u2081\nleft_inv\u271d\u00b9 : LeftInverse invFun\u271d\u00b9 toLieHom\u271d\u00b9.toFun\nright_inv\u271d\u00b9 : Function.RightInverse invFun\u271d\u00b9 toLieHom\u271d\u00b9.toFun\ntoLieHom\u271d : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\ninvFun\u271d : L\u2082 \u2192 L\u2081\nleft_inv\u271d : LeftInverse invFun\u271d toLieHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieHom\u271d.toFun\nh\u2082 : invFun\u271d\u00b9 = invFun\u271d\nh\u2081 : toLieHom\u271d\u00b9 = toLieHom\u271d\n\u22a2 { toLieHom := toLieHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    { toLieHom := toLieHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\n\u22a2 Injective toLinearEquiv\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8\u27e8f, -\u27e9, -\u27e9, -\u27e9, f_inv\u27e9 \u27e8\u27e8\u27e8\u27e8g, -\u27e9, -\u27e9, -\u27e9, g_inv\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf_inv : L\u2082 \u2192 L\u2081\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : L\u2082 \u2192 L\u2081\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\n\u22a2 toLinearEquiv\n        {\n          toLieHom :=\n            { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n              map_lie' := map_lie'\u271d\u00b9 },\n          invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n      toLinearEquiv\n        {\n          toLieHom :=\n            { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n              map_lie' := map_lie'\u271d },\n          invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d } \u2192\n    {\n        toLieHom :=\n          { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 },\n        invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n      {\n        toLieHom :=\n          { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d },\n        invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf_inv : L\u2082 \u2192 L\u2081\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : L\u2082 \u2192 L\u2081\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh :\n  toLinearEquiv\n      {\n        toLieHom :=\n          { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 },\n        invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    toLinearEquiv\n      {\n        toLieHom :=\n          { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d },\n        invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n\u22a2 {\n      toLieHom :=\n        { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n          map_lie' := map_lie'\u271d\u00b9 },\n      invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    {\n      toLieHom :=\n        { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n          map_lie' := map_lie'\u271d },\n      invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp only [to_linearEquiv_mk, LinearEquiv.mk.injEq, LinearMap.mk.injEq, AddHom.mk.injEq] at h \n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf_inv : L\u2082 \u2192 L\u2081\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : L\u2082 \u2192 L\u2081\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 {\n      toLieHom :=\n        { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n          map_lie' := map_lie'\u271d\u00b9 },\n      invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    {\n      toLieHom :=\n        { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n          map_lie' := map_lie'\u271d },\n      invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.e_toLieHom.e_toLinearMap.e_toAddHom.e_toFun\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf_inv : L\u2082 \u2192 L\u2081\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : L\u2082 \u2192 L\u2081\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 f = g\ncase mk.mk.mk.mk.mk.mk.mk.mk.e_invFun\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf_inv : L\u2082 \u2192 L\u2081\nf : L\u2081 \u2192 L\u2082\nmap_add'\u271d\u00b9 : \u2200 (x y : L\u2081), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom y\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : L\u2082 \u2192 L\u2081\ng : L\u2081 \u2192 L\u2082\nmap_add'\u271d : \u2200 (x y : L\u2081), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : L\u2081),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x y : L\u2081},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, y\u2046 =\n      \u2045AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom x,\n        AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom y\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 f_inv = g_inv\n[PROOFSTEP]\nexacts [h.1, h.2]\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\n\u22a2 symm (symm e) = e\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\ne : L\u2081 \u2243\u2097\u2045R\u2046 L\u2082\nx\u271d : L\u2081\n\u22a2 \u2191(symm (symm e)) x\u271d = \u2191e x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh : Bijective \u2191f\nsrc\u271d : L\u2081 \u2243\u2097[R] L\u2082 := LinearEquiv.ofBijective (\u2191f) h\n\u22a2 \u2200 {x y : L\u2081},\n    AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191f,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : L\u2081),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, y\u2046 =\n      \u2045AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := \u2191f,\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : L\u2081),\n                        AddHom.toFun src\u271d.toAddHom (x + y) =\n                          AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          x,\n        AddHom.toFun\n          {\n              toAddHom :=\n                { toFun := \u2191f,\n                  map_add' :=\n                    (_ :\n                      \u2200 (x y : L\u2081),\n                        AddHom.toFun src\u271d.toAddHom (x + y) =\n                          AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          y\u2046\n[PROOFSTEP]\nintros x y\n[GOAL]\nR : Type u\nL\u2081 : Type v\nL\u2082 : Type w\nL\u2083 : Type w\u2081\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\u2081\ninst\u271d\u2074 : LieRing L\u2082\ninst\u271d\u00b3 : LieRing L\u2083\ninst\u271d\u00b2 : LieAlgebra R L\u2081\ninst\u271d\u00b9 : LieAlgebra R L\u2082\ninst\u271d : LieAlgebra R L\u2083\nf : L\u2081 \u2192\u2097\u2045R\u2046 L\u2082\nh : Bijective \u2191f\nsrc\u271d : L\u2081 \u2243\u2097[R] L\u2082 := LinearEquiv.ofBijective (\u2191f) h\nx y : L\u2081\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191f,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : L\u2081),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : L\u2081),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      \u2045x, y\u2046 =\n    \u2045AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191f,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : L\u2081),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x,\n      AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191f,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : L\u2081),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : L\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        y\u2046\n[PROOFSTEP]\nexact f.map_lie x y\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nx y : M \u2192\u2097\u2045R,L\u2046 N\nh : (fun f => f.toFun) x = (fun f => f.toFun) y\n\u22a2 x = y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ny : M \u2192\u2097\u2045R,L\u2046 N\ntoLinearMap\u271d : M \u2192\u2097[R] N\nmap_lie'\u271d : \u2200 {x : L} {m : M}, AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun toLinearMap\u271d.toAddHom m\u2046\nh : (fun f => f.toFun) { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d } = (fun f => f.toFun) y\n\u22a2 { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d } = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ntoLinearMap\u271d\u00b9 : M \u2192\u2097[R] N\nmap_lie'\u271d\u00b9 : \u2200 {x : L} {m : M}, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom m\u2046\ntoLinearMap\u271d : M \u2192\u2097[R] N\nmap_lie'\u271d : \u2200 {x : L} {m : M}, AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun toLinearMap\u271d.toAddHom m\u2046\nh :\n  (fun f => f.toFun) { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } =\n    (fun f => f.toFun) { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n\u22a2 { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } = { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ntoLinearMap\u271d\u00b9 : M \u2192\u2097[R] N\nmap_lie'\u271d\u00b9 : \u2200 {x : L} {m : M}, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun toLinearMap\u271d\u00b9.toAddHom m\u2046\ntoLinearMap\u271d : M \u2192\u2097[R] N\nmap_lie'\u271d : \u2200 {x : L} {m : M}, AddHom.toFun toLinearMap\u271d.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun toLinearMap\u271d.toAddHom m\u2046\nh : toLinearMap\u271d\u00b9 = toLinearMap\u271d\n\u22a2 { toLinearMap := toLinearMap\u271d\u00b9, map_lie' := map_lie'\u271d\u00b9 } = { toLinearMap := toLinearMap\u271d, map_lie' := map_lie'\u271d }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N \u2192\u2097[R] P\nx : L\nm : M\nn : N\n\u22a2 \u2045x, \u2191(\u2191f m) n\u2046 = \u2191(\u2191f \u2045x, m\u2046) n + \u2191(\u2191f m) \u2045x, n\u2046\n[PROOFSTEP]\nsimp only [sub_add_cancel, map_lie, LieHom.lie_apply]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nsrc\u271d : M \u2192\u2097[R] N := 0\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\n\u22a2 Injective FunLike.coe\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8f, _\u27e9\u27e9\u27e9 \u27e8\u27e8\u27e8g, _\u27e9\u27e9\u27e9 h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nh :\n  \u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n        map_lie' := map_lie'\u271d\u00b9 } =\n    \u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n        map_lie' := map_lie'\u271d }\n\u22a2 { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n      map_lie' := map_lie'\u271d\u00b9 } =\n    { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n      map_lie' := map_lie'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf g : M \u2192\u2097\u2045R,L\u2046 N\n\u22a2 f = g \u2192 \u2200 (m : M), \u2191f m = \u2191g m\n[PROOFSTEP]\nrintro rfl m\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\nm : M\n\u22a2 \u2191f m = \u2191f m\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\nh : \u2200 {x : L} {m : M}, AddHom.toFun f.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun f.toAddHom m\u2046\n\u22a2 { toLinearMap := \u2191f, map_lie' := h } = f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097[R] N\nh : \u2200 {x : L} {m : M}, AddHom.toFun f.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun f.toAddHom m\u2046\n\u22a2 \u2191{ toLinearMap := f, map_lie' := h } = \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097[R] N\nh : \u2200 {x : L} {m : M}, AddHom.toFun f.toAddHom \u2045x, m\u2046 = \u2045x, AddHom.toFun f.toAddHom m\u2046\n\u22a2 \u2191{ toLinearMap := f, map_lie' := h } = f\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : N \u2192\u2097\u2045R,L\u2046 P\ng : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] P := LinearMap.comp \u2191f \u2191g\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nintros x m\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : N \u2192\u2097\u2045R,L\u2046 P\ng : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] P := LinearMap.comp \u2191f \u2191g\nx : L\nm : M\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : M),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      \u2045x, m\u2046 =\n    \u2045x,\n      AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        m\u2046\n[PROOFSTEP]\nchange f (g \u2045x, m\u2046) = \u2045x, f (g m)\u2046\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : N \u2192\u2097\u2045R,L\u2046 P\ng : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] P := LinearMap.comp \u2191f \u2191g\nx : L\nm : M\n\u22a2 \u2191f (\u2191g \u2045x, m\u2046) = \u2045x, \u2191f (\u2191g m)\u2046\n[PROOFSTEP]\nrw [map_lie, map_lie]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\ng : N \u2192 M\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : N \u2192\u2097[R] M := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\n\u22a2 \u2200 {x : L} {m : N},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : N),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : N),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nintros x n\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\ng : N \u2192 M\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : N \u2192\u2097[R] M := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx : L\nn : N\n\u22a2 AddHom.toFun\n      { toAddHom := src\u271d.toAddHom,\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : N),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      \u2045x, n\u2046 =\n    \u2045x,\n      AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : N),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        n\u2046\n[PROOFSTEP]\ncalc\n  g \u2045x, n\u2046 = g \u2045x, f (g n)\u2046 := by rw [h\u2082]\n  _ = g (f \u2045x, g n\u2046) := by rw [map_lie]\n  _ = \u2045x, g n\u2046 := h\u2081 _\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\ng : N \u2192 M\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : N \u2192\u2097[R] M := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx : L\nn : N\n\u22a2 g \u2045x, n\u2046 = g \u2045x, \u2191f (g n)\u2046\n[PROOFSTEP]\nrw [h\u2082]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\ng : N \u2192 M\nh\u2081 : LeftInverse g \u2191f\nh\u2082 : Function.RightInverse g \u2191f\nsrc\u271d : N \u2192\u2097[R] M := LinearMap.inverse (\u2191f) g h\u2081 h\u2082\nx : L\nn : N\n\u22a2 g \u2045x, \u2191f (g n)\u2046 = g (\u2191f \u2045x, g n\u2046)\n[PROOFSTEP]\nrw [map_lie]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf g : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := \u2191f + \u2191g\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf g : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := \u2191f - \u2191g\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := -\u2191f\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nn : \u2115\nf : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := n \u2022 \u2191f\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nz : \u2124\nf : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := z \u2022 \u2191f\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nt : R\nf : M \u2192\u2097\u2045R,L\u2046 N\nsrc\u271d : M \u2192\u2097[R] N := t \u2022 \u2191f\n\u22a2 \u2200 {x : L} {m : M},\n    AddHom.toFun\n        { toAddHom := src\u271d.toAddHom,\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        \u2045x, m\u2046 =\n      \u2045x,\n        AddHom.toFun\n          { toAddHom := src\u271d.toAddHom,\n              map_smul' :=\n                (_ :\n                  \u2200 (r : R) (x : M),\n                    AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n          m\u2046\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf g : M \u2243\u2097\u2045R,L\u2046 N\nh\u2081 : (fun f => f.toFun) f = (fun f => f.toFun) g\nh\u2082 : (fun f => f.invFun) f = (fun f => f.invFun) g\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ng : M \u2243\u2097\u2045R,L\u2046 N\ntoLieModuleHom\u271d : M \u2192\u2097\u2045R,L\u2046 N\ninvFun\u271d : N \u2192 M\nleft_inv\u271d : LeftInverse invFun\u271d toLieModuleHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieModuleHom\u271d.toFun\nh\u2081 :\n  (fun f => f.toFun)\n      { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } =\n    (fun f => f.toFun) g\nh\u2082 :\n  (fun f => f.invFun)\n      { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } =\n    (fun f => f.invFun) g\n\u22a2 { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ntoLieModuleHom\u271d\u00b9 : M \u2192\u2097\u2045R,L\u2046 N\ninvFun\u271d\u00b9 : N \u2192 M\nleft_inv\u271d\u00b9 : LeftInverse invFun\u271d\u00b9 toLieModuleHom\u271d\u00b9.toFun\nright_inv\u271d\u00b9 : Function.RightInverse invFun\u271d\u00b9 toLieModuleHom\u271d\u00b9.toFun\ntoLieModuleHom\u271d : M \u2192\u2097\u2045R,L\u2046 N\ninvFun\u271d : N \u2192 M\nleft_inv\u271d : LeftInverse invFun\u271d toLieModuleHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieModuleHom\u271d.toFun\nh\u2081 :\n  (fun f => f.toFun)\n      { toLieModuleHom := toLieModuleHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    (fun f => f.toFun)\n      { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\nh\u2082 :\n  (fun f => f.invFun)\n      { toLieModuleHom := toLieModuleHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    (fun f => f.invFun)\n      { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n\u22a2 { toLieModuleHom := toLieModuleHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp at h\u2081 h\u2082 \n[GOAL]\ncase mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ntoLieModuleHom\u271d\u00b9 : M \u2192\u2097\u2045R,L\u2046 N\ninvFun\u271d\u00b9 : N \u2192 M\nleft_inv\u271d\u00b9 : LeftInverse invFun\u271d\u00b9 toLieModuleHom\u271d\u00b9.toFun\nright_inv\u271d\u00b9 : Function.RightInverse invFun\u271d\u00b9 toLieModuleHom\u271d\u00b9.toFun\ntoLieModuleHom\u271d : M \u2192\u2097\u2045R,L\u2046 N\ninvFun\u271d : N \u2192 M\nleft_inv\u271d : LeftInverse invFun\u271d toLieModuleHom\u271d.toFun\nright_inv\u271d : Function.RightInverse invFun\u271d toLieModuleHom\u271d.toFun\nh\u2082 : invFun\u271d\u00b9 = invFun\u271d\nh\u2081 : toLieModuleHom\u271d\u00b9 = toLieModuleHom\u271d\n\u22a2 { toLieModuleHom := toLieModuleHom\u271d\u00b9, invFun := invFun\u271d\u00b9, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    { toLieModuleHom := toLieModuleHom\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\n\u22a2 Injective toEquiv\n[PROOFSTEP]\nrintro \u27e8\u27e8\u27e8\u27e8f, -\u27e9, -\u27e9, -\u27e9, f_inv\u27e9 \u27e8\u27e8\u27e8\u27e8g, -\u27e9, -\u27e9, -\u27e9, g_inv\u27e9\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf_inv : N \u2192 M\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : N \u2192 M\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\n\u22a2 toEquiv\n        {\n          toLieModuleHom :=\n            { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n              map_lie' := map_lie'\u271d\u00b9 },\n          invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n      toEquiv\n        {\n          toLieModuleHom :=\n            { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n              map_lie' := map_lie'\u271d },\n          invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d } \u2192\n    {\n        toLieModuleHom :=\n          { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 },\n        invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n      {\n        toLieModuleHom :=\n          { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d },\n        invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf_inv : N \u2192 M\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : N \u2192 M\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh :\n  toEquiv\n      {\n        toLieModuleHom :=\n          { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 },\n        invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    toEquiv\n      {\n        toLieModuleHom :=\n          { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d },\n        invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n\u22a2 {\n      toLieModuleHom :=\n        { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n          map_lie' := map_lie'\u271d\u00b9 },\n      invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    {\n      toLieModuleHom :=\n        { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n          map_lie' := map_lie'\u271d },\n      invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\nsimp only [toEquiv_mk, LieModuleHom.coe_mk, LinearMap.coe_mk, AddHom.coe_mk, Equiv.mk.injEq] at h \n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf_inv : N \u2192 M\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : N \u2192 M\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 {\n      toLieModuleHom :=\n        { toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n          map_lie' := map_lie'\u271d\u00b9 },\n      invFun := f_inv, left_inv := left_inv\u271d\u00b9, right_inv := right_inv\u271d\u00b9 } =\n    {\n      toLieModuleHom :=\n        { toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n          map_lie' := map_lie'\u271d },\n      invFun := g_inv, left_inv := left_inv\u271d, right_inv := right_inv\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.mk.mk.mk.mk.mk.mk.e_toLieModuleHom.e_toLinearMap.e_toAddHom.e_toFun\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf_inv : N \u2192 M\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : N \u2192 M\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 f = g\ncase mk.mk.mk.mk.mk.mk.mk.mk.e_invFun\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\nf_inv : N \u2192 M\nf : M \u2192 N\nmap_add'\u271d\u00b9 : \u2200 (x y : M), f (x + y) = f x + f y\nmap_smul'\u271d\u00b9 :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := f, map_add' := map_add'\u271d\u00b9 } x\nmap_lie'\u271d\u00b9 :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 }.toAddHom m\u2046\nleft_inv\u271d\u00b9 :\n  LeftInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\nright_inv\u271d\u00b9 :\n  Function.RightInverse f_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := f, map_add' := map_add'\u271d\u00b9 }, map_smul' := map_smul'\u271d\u00b9 },\n            map_lie' := map_lie'\u271d\u00b9 }).toAddHom.toFun\ng_inv : N \u2192 M\ng : M \u2192 N\nmap_add'\u271d : \u2200 (x y : M), g (x + y) = g x + g y\nmap_smul'\u271d :\n  \u2200 (r : R) (x : M),\n    AddHom.toFun { toFun := g, map_add' := map_add'\u271d } (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := g, map_add' := map_add'\u271d } x\nmap_lie'\u271d :\n  \u2200 {x : L} {m : M},\n    AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom \u2045x, m\u2046 =\n      \u2045x, AddHom.toFun { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d }.toAddHom m\u2046\nleft_inv\u271d :\n  LeftInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nright_inv\u271d :\n  Function.RightInverse g_inv\n    (\u2191{ toLinearMap := { toAddHom := { toFun := g, map_add' := map_add'\u271d }, map_smul' := map_smul'\u271d },\n            map_lie' := map_lie'\u271d }).toAddHom.toFun\nh : f = g \u2227 f_inv = g_inv\n\u22a2 f_inv = g_inv\n[PROOFSTEP]\nexacts [h.1, h.2]\n[GOAL]\nR : Type u\nL : Type v\nM : Type w\nN : Type w\u2081\nP : Type w\u2082\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : LieRing L\ninst\u271d\u00b9\u00b2 : LieAlgebra R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : AddCommGroup P\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieRingModule L N\ninst\u271d\u00b3 : LieRingModule L P\ninst\u271d\u00b2 : LieModule R L M\ninst\u271d\u00b9 : LieModule R L N\ninst\u271d : LieModule R L P\ne : M \u2243\u2097\u2045R,L\u2046 N\n\u22a2 symm (symm e) = e\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.Basic", "llama_tokens": 51595, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.6992544147913994, "lm_q1q2_score": 0.5318883269985532}}
{"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace (\u03b1 \u00d7 \u03b1)\n\u22a2 \ud835\udcdd\u02e2 (diagonal \u03b1) = \u2a06 (x : \u03b1), \ud835\udcdd (x, x)\n[PROOFSTEP]\nrw [nhdsSet, \u2190 range_diag, \u2190 range_comp]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b1\u271d\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\u271d\nx : \u03b1\u271d\n\u03b1 : Type u_3\ninst\u271d : TopologicalSpace (\u03b1 \u00d7 \u03b1)\n\u22a2 sSup (range (\ud835\udcdd \u2218 fun x => (x, x))) = \u2a06 (x : \u03b1), \ud835\udcdd (x, x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 s \u2208 \ud835\udcdd\u02e2 t \u2194 \u2200 (x : \u03b1), x \u2208 t \u2192 s \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimp_rw [nhdsSet, Filter.mem_sSup, ball_image_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 s \u2286 interior t \u2194 t \u2208 \ud835\udcdd\u02e2 s\n[PROOFSTEP]\nsimp_rw [mem_nhdsSet_iff_forall, subset_interior_iff_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 Disjoint (\ud835\udcdf s) (\ud835\udcdd\u02e2 t) \u2194 Disjoint (closure s) t\n[PROOFSTEP]\nrw [disjoint_principal_left, \u2190 subset_interior_iff_mem_nhdsSet, interior_compl, subset_compl_iff_disjoint_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdf t) \u2194 Disjoint s (closure t)\n[PROOFSTEP]\nrw [disjoint_comm, disjoint_principal_nhdsSet, disjoint_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 s \u2208 \ud835\udcdd\u02e2 t \u2194 \u2203 U, IsOpen U \u2227 t \u2286 U \u2227 U \u2286 s\n[PROOFSTEP]\nrw [\u2190 subset_interior_iff_mem_nhdsSet, subset_interior_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\n\u22a2 t \u2208 \ud835\udcdd\u02e2 s \u2194 \u2203 i, (IsOpen i \u2227 s \u2286 i) \u2227 i \u2286 t\n[PROOFSTEP]\nsimp [mem_nhdsSet_iff_exists, and_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\nhU : IsOpen s\n\u22a2 s \u2208 \ud835\udcdd\u02e2 t \u2194 t \u2286 s\n[PROOFSTEP]\nrw [\u2190 subset_interior_iff_mem_nhdsSet, hU.interior_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 \ud835\udcdd\u02e2 s = \ud835\udcdf s \u2194 IsOpen s\n[PROOFSTEP]\nrw [\u2190 principal_le_nhdsSet.le_iff_eq, le_principal_iff, mem_nhdsSet_iff_forall, isOpen_iff_mem_nhds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 \ud835\udcdd\u02e2 {x} = \ud835\udcdd x\n[PROOFSTEP]\nsimp [nhdsSet]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 \ud835\udcdd\u02e2 \u2205 = \u22a5\n[PROOFSTEP]\nrw [isOpen_empty.nhdsSet_eq, principal_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 s \u2208 \ud835\udcdd\u02e2 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\n\u22a2 \ud835\udcdd\u02e2 univ = \u22a4\n[PROOFSTEP]\nrw [isOpen_univ.nhdsSet_eq, principal_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t\u271d s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\ns t : Set \u03b1\n\u22a2 \ud835\udcdd\u02e2 (s \u222a t) = \ud835\udcdd\u02e2 s \u2294 \ud835\udcdd\u02e2 t\n[PROOFSTEP]\nsimp only [nhdsSet, image_union, sSup_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\nh\u2081 : s\u2081 \u2208 \ud835\udcdd\u02e2 t\u2081\nh\u2082 : s\u2082 \u2208 \ud835\udcdd\u02e2 t\u2082\n\u22a2 s\u2081 \u222a s\u2082 \u2208 \ud835\udcdd\u02e2 (t\u2081 \u222a t\u2082)\n[PROOFSTEP]\nrw [nhdsSet_union]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx : \u03b1\nh\u2081 : s\u2081 \u2208 \ud835\udcdd\u02e2 t\u2081\nh\u2082 : s\u2082 \u2208 \ud835\udcdd\u02e2 t\u2082\n\u22a2 s\u2081 \u222a s\u2082 \u2208 \ud835\udcdd\u02e2 t\u2081 \u2294 \ud835\udcdd\u02e2 t\u2082\n[PROOFSTEP]\nexact union_mem_sup h\u2081 h\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns\u271d t s\u2081 s\u2082 t\u2081 t\u2082 : Set \u03b1\nx\u271d x : \u03b1\ns : Set \u03b1\n\u22a2 \ud835\udcdd\u02e2 (insert x s) = \ud835\udcdd x \u2294 \ud835\udcdd\u02e2 s\n[PROOFSTEP]\nrw [insert_eq, nhdsSet_union, nhdsSet_singleton]\n", "meta": {"mathlib_filename": "Mathlib.Topology.NhdsSet", "llama_tokens": 2136, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389817407017, "lm_q2_score": 0.7122321720225278, "lm_q1q2_score": 0.5314241875958572}}
{"text": "[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX x\u271d\u00b9 x\u271d : Discrete M\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 (fun X Y => { as := X.as * Y.as }) X x\u271d\u00b9 = (fun X Y => { as := X.as * Y.as }) X x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX x\u271d\u00b9 x\u271d : Discrete M\nf : x\u271d\u00b9 \u27f6 x\u271d\n\u22a2 { as := X.as * x\u271d\u00b9.as } = { as := X.as * x\u271d.as }\n[PROOFSTEP]\nrw [eq_of_hom f]\n[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX\u2081\u271d X\u2082\u271d : Discrete M\nf : X\u2081\u271d \u27f6 X\u2082\u271d\nX : Discrete M\n\u22a2 (fun X Y => { as := X.as * Y.as }) X\u2081\u271d X = (fun X Y => { as := X.as * Y.as }) X\u2082\u271d X\n[PROOFSTEP]\ndsimp\n[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX\u2081\u271d X\u2082\u271d : Discrete M\nf : X\u2081\u271d \u27f6 X\u2082\u271d\nX : Discrete M\n\u22a2 { as := X\u2081\u271d.as * X.as } = { as := X\u2082\u271d.as * X.as }\n[PROOFSTEP]\nrw [eq_of_hom f]\n[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Discrete M\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 (fun X Y => { as := X.as * Y.as }) X\u2081\u271d X\u2082\u271d = (fun X Y => { as := X.as * Y.as }) Y\u2081\u271d Y\u2082\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\nM : Type u\ninst\u271d : Monoid M\nX\u2081\u271d Y\u2081\u271d X\u2082\u271d Y\u2082\u271d : Discrete M\nf : X\u2081\u271d \u27f6 Y\u2081\u271d\ng : X\u2082\u271d \u27f6 Y\u2082\u271d\n\u22a2 { as := X\u2081\u271d.as * X\u2082\u271d.as } = { as := Y\u2081\u271d.as * Y\u2082\u271d.as }\n[PROOFSTEP]\nrw [eq_of_hom f, eq_of_hom g]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Discrete", "llama_tokens": 696, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.6859494485880927, "lm_q1q2_score": 0.5313250523922459}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DivisionSemiring \u03b1\nm n : \u2115\nn_dvd : n \u2223 m\nn_nonzero : \u2191n \u2260 0\n\u22a2 \u2191(m / n) = \u2191m / \u2191n\n[PROOFSTEP]\nrcases n_dvd with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : DivisionSemiring \u03b1\nn : \u2115\nn_nonzero : \u2191n \u2260 0\nk : \u2115\n\u22a2 \u2191(n * k / n) = \u2191(n * k) / \u2191n\n[PROOFSTEP]\nhave : n \u2260 0 := by\n  rintro rfl\n  simp at n_nonzero \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DivisionSemiring \u03b1\nn : \u2115\nn_nonzero : \u2191n \u2260 0\nk : \u2115\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : DivisionSemiring \u03b1\nk : \u2115\nn_nonzero : \u21910 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at n_nonzero \n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : DivisionSemiring \u03b1\nn : \u2115\nn_nonzero : \u2191n \u2260 0\nk : \u2115\nthis : n \u2260 0\n\u22a2 \u2191(n * k / n) = \u2191(n * k) / \u2191n\n[PROOFSTEP]\nrw [Nat.mul_div_cancel_left _ this.bot_lt, mul_comm n k, cast_mul, mul_div_cancel _ n_nonzero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\n\u22a2 \u2191(m / d) / \u2191(n / d) = \u2191m / \u2191n\n[PROOFSTEP]\nrcases eq_or_ne d 0 with (rfl | hd)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n : \u2115\nhn : 0 \u2223 n\nhm : 0 \u2223 m\n\u22a2 \u2191(m / 0) / \u2191(n / 0) = \u2191m / \u2191n\n[PROOFSTEP]\nsimp [zero_dvd_iff.mp hm]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\nhd : d \u2260 0\n\u22a2 \u2191(m / d) / \u2191(n / d) = \u2191m / \u2191n\n[PROOFSTEP]\nreplace hd : (d : \u03b1) \u2260 0\n[GOAL]\ncase hd\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\nhd : d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\nhd : \u2191d \u2260 0\n\u22a2 \u2191(m / d) / \u2191(n / d) = \u2191m / \u2191n\n[PROOFSTEP]\nrw [cast_div hm, cast_div hn, div_div_div_cancel_right _ hd]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\nhd : \u2191d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nexact hd\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DivisionSemiring \u03b1\ninst\u271d : CharZero \u03b1\nm n d : \u2115\nhn : d \u2223 n\nhm : d \u2223 m\nhd : \u2191d \u2260 0\n\u22a2 \u2191d \u2260 0\n[PROOFSTEP]\nexact hd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nm n : \u2115\n\u22a2 \u2191(m / n) \u2264 \u2191m / \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nm : \u2115\n\u22a2 \u2191(m / zero) \u2264 \u2191m / \u2191zero\n[PROOFSTEP]\nrw [cast_zero, div_zero, Nat.div_zero, cast_zero]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nm n\u271d : \u2115\n\u22a2 \u2191(m / succ n\u271d) \u2264 \u2191m / \u2191(succ n\u271d)\n[PROOFSTEP]\nrw [le_div_iff, \u2190 Nat.cast_mul, @Nat.cast_le]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nm n\u271d : \u2115\n\u22a2 m / succ n\u271d * succ n\u271d \u2264 m\ncase succ \u03b1 : Type u_1 inst\u271d : LinearOrderedSemifield \u03b1 m n\u271d : \u2115 \u22a2 0 < \u2191(succ n\u271d)\n[PROOFSTEP]\nexact (Nat.div_mul_le_self m _)\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nm n\u271d : \u2115\n\u22a2 0 < \u2191(succ n\u271d)\n[PROOFSTEP]\nexact Nat.cast_pos.2 (Nat.succ_pos _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn : \u2115\n\u22a2 0 < 1 / (\u2191n + 1)\n[PROOFSTEP]\nrw [one_div]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn : \u2115\n\u22a2 0 < (\u2191n + 1)\u207b\u00b9\n[PROOFSTEP]\nexact inv_pos_of_nat\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n \u2264 m\n\u22a2 1 / (\u2191m + 1) \u2264 1 / (\u2191n + 1)\n[PROOFSTEP]\nrefine' one_div_le_one_div_of_le _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n \u2264 m\n\u22a2 0 < \u2191n + 1\ncase refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedSemifield \u03b1 n m : \u2115 h : n \u2264 m \u22a2 \u2191n + 1 \u2264 \u2191m + 1\n[PROOFSTEP]\nexact Nat.cast_add_one_pos _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n \u2264 m\n\u22a2 \u2191n + 1 \u2264 \u2191m + 1\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n < m\n\u22a2 1 / (\u2191m + 1) < 1 / (\u2191n + 1)\n[PROOFSTEP]\nrefine' one_div_lt_one_div_of_lt _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n < m\n\u22a2 0 < \u2191n + 1\ncase refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedSemifield \u03b1 n m : \u2115 h : n < m \u22a2 \u2191n + 1 < \u2191m + 1\n[PROOFSTEP]\nexact Nat.cast_add_one_pos _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\nn m : \u2115\nh : n < m\n\u22a2 \u2191n + 1 < \u2191m + 1\n[PROOFSTEP]\nsimpa\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Cast.Field", "llama_tokens": 2268, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577157, "lm_q2_score": 0.6859494485880927, "lm_q1q2_score": 0.5313250381146096}}
{"text": "[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n\u22a2 \u2200 {X X' Y Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y'), (f \u2297 g) \u226b (\u03b2 Y Y').hom = (\u03b2 X X').hom \u226b (g \u2297 f)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d X'\u271d Y\u271d Y'\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : X'\u271d \u27f6 Y'\u271d\n\u22a2 (f\u271d \u2297 g\u271d) \u226b (\u03b2 Y\u271d Y'\u271d).hom = (\u03b2 X\u271d X'\u271d).hom \u226b (g\u271d \u2297 f\u271d)\n[PROOFSTEP]\napply F.map_injective\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d X'\u271d Y\u271d Y'\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : X'\u271d \u27f6 Y'\u271d\n\u22a2 F.map ((f\u271d \u2297 g\u271d) \u226b (\u03b2 Y\u271d Y'\u271d).hom) = F.map ((\u03b2 X\u271d X'\u271d).hom \u226b (g\u271d \u2297 f\u271d))\n[PROOFSTEP]\nrefine (cancel_epi (F.\u03bc ?_ ?_)).1 ?_\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d X'\u271d Y\u271d Y'\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : X'\u271d \u27f6 Y'\u271d\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d X'\u271d \u226b F.map ((f\u271d \u2297 g\u271d) \u226b (\u03b2 Y\u271d Y'\u271d).hom) =\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d X'\u271d \u226b F.map ((\u03b2 X\u271d X'\u271d).hom \u226b (g\u271d \u2297 f\u271d))\n[PROOFSTEP]\nrw [Functor.map_comp, \u2190 LaxMonoidalFunctor.\u03bc_natural_assoc, w, Functor.map_comp, reassoc_of% w,\n  braiding_naturality_assoc, LaxMonoidalFunctor.\u03bc_natural]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n\u22a2 \u2200 (X Y Z : C),\n    (\u03b1_ X Y Z).hom \u226b (\u03b2 X (Y \u2297 Z)).hom \u226b (\u03b1_ Y Z X).hom = ((\u03b2 X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ Y X Z).hom \u226b (\ud835\udfd9 Y \u2297 (\u03b2 X Z).hom)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 (\u03b1_ X\u271d Y\u271d Z\u271d).hom \u226b (\u03b2 X\u271d (Y\u271d \u2297 Z\u271d)).hom \u226b (\u03b1_ Y\u271d Z\u271d X\u271d).hom =\n    ((\u03b2 X\u271d Y\u271d).hom \u2297 \ud835\udfd9 Z\u271d) \u226b (\u03b1_ Y\u271d X\u271d Z\u271d).hom \u226b (\ud835\udfd9 Y\u271d \u2297 (\u03b2 X\u271d Z\u271d).hom)\n[PROOFSTEP]\napply F.map_injective\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).hom \u226b (\u03b2 X\u271d (Y\u271d \u2297 Z\u271d)).hom \u226b (\u03b1_ Y\u271d Z\u271d X\u271d).hom) =\n    F.map (((\u03b2 X\u271d Y\u271d).hom \u2297 \ud835\udfd9 Z\u271d) \u226b (\u03b1_ Y\u271d X\u271d Z\u271d).hom \u226b (\ud835\udfd9 Y\u271d \u2297 (\u03b2 X\u271d Z\u271d).hom))\n[PROOFSTEP]\nrefine (cancel_epi (F.\u03bc _ _)).1 ?_\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (X\u271d \u2297 Y\u271d) Z\u271d \u226b\n      F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).hom \u226b (\u03b2 X\u271d (Y\u271d \u2297 Z\u271d)).hom \u226b (\u03b1_ Y\u271d Z\u271d X\u271d).hom) =\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (X\u271d \u2297 Y\u271d) Z\u271d \u226b\n      F.map (((\u03b2 X\u271d Y\u271d).hom \u2297 \ud835\udfd9 Z\u271d) \u226b (\u03b1_ Y\u271d X\u271d Z\u271d).hom \u226b (\ud835\udfd9 Y\u271d \u2297 (\u03b2 X\u271d Z\u271d).hom))\n[PROOFSTEP]\nrefine (cancel_epi (F.\u03bc _ _ \u2297 \ud835\udfd9 _)).1 ?_\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d Y\u271d \u2297 \ud835\udfd9 (F.obj Z\u271d)) \u226b\n      LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (X\u271d \u2297 Y\u271d) Z\u271d \u226b\n        F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).hom \u226b (\u03b2 X\u271d (Y\u271d \u2297 Z\u271d)).hom \u226b (\u03b1_ Y\u271d Z\u271d X\u271d).hom) =\n    (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d Y\u271d \u2297 \ud835\udfd9 (F.obj Z\u271d)) \u226b\n      LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor (X\u271d \u2297 Y\u271d) Z\u271d \u226b\n        F.map (((\u03b2 X\u271d Y\u271d).hom \u2297 \ud835\udfd9 Z\u271d) \u226b (\u03b1_ Y\u271d X\u271d Z\u271d).hom \u226b (\ud835\udfd9 Y\u271d \u2297 (\u03b2 X\u271d Z\u271d).hom))\n[PROOFSTEP]\nrw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, \u2190 LaxMonoidalFunctor.\u03bc_natural_assoc,\n  Functor.map_id, \u2190 comp_tensor_id_assoc, w, comp_tensor_id, assoc, LaxMonoidalFunctor.associativity_assoc,\n  LaxMonoidalFunctor.associativity_assoc, \u2190 LaxMonoidalFunctor.\u03bc_natural, Functor.map_id, \u2190 id_tensor_comp_assoc, w,\n  id_tensor_comp_assoc, reassoc_of% w, braiding_naturality_assoc, LaxMonoidalFunctor.associativity,\n  hexagon_forward_assoc]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n\u22a2 \u2200 (X Y Z : C),\n    (\u03b1_ X Y Z).inv \u226b (\u03b2 (X \u2297 Y) Z).hom \u226b (\u03b1_ Z X Y).inv = (\ud835\udfd9 X \u2297 (\u03b2 Y Z).hom) \u226b (\u03b1_ X Z Y).inv \u226b ((\u03b2 X Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 (\u03b1_ X\u271d Y\u271d Z\u271d).inv \u226b (\u03b2 (X\u271d \u2297 Y\u271d) Z\u271d).hom \u226b (\u03b1_ Z\u271d X\u271d Y\u271d).inv =\n    (\ud835\udfd9 X\u271d \u2297 (\u03b2 Y\u271d Z\u271d).hom) \u226b (\u03b1_ X\u271d Z\u271d Y\u271d).inv \u226b ((\u03b2 X\u271d Z\u271d).hom \u2297 \ud835\udfd9 Y\u271d)\n[PROOFSTEP]\napply F.toFunctor.map_injective\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).inv \u226b (\u03b2 (X\u271d \u2297 Y\u271d) Z\u271d).hom \u226b (\u03b1_ Z\u271d X\u271d Y\u271d).inv) =\n    F.map ((\ud835\udfd9 X\u271d \u2297 (\u03b2 Y\u271d Z\u271d).hom) \u226b (\u03b1_ X\u271d Z\u271d Y\u271d).inv \u226b ((\u03b2 X\u271d Z\u271d).hom \u2297 \ud835\udfd9 Y\u271d))\n[PROOFSTEP]\nrefine (cancel_epi (F.\u03bc _ _)).1 ?_\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d (Y\u271d \u2297 Z\u271d) \u226b\n      F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).inv \u226b (\u03b2 (X\u271d \u2297 Y\u271d) Z\u271d).hom \u226b (\u03b1_ Z\u271d X\u271d Y\u271d).inv) =\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d (Y\u271d \u2297 Z\u271d) \u226b\n      F.map ((\ud835\udfd9 X\u271d \u2297 (\u03b2 Y\u271d Z\u271d).hom) \u226b (\u03b1_ X\u271d Z\u271d Y\u271d).inv \u226b ((\u03b2 X\u271d Z\u271d).hom \u2297 \ud835\udfd9 Y\u271d))\n[PROOFSTEP]\nrefine (cancel_epi (\ud835\udfd9 _ \u2297 F.\u03bc _ _)).1 ?_\n[GOAL]\ncase a\nC : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{?u.40211, u_1} C\ninst\u271d\u2074 : Category.{?u.40215, u_2} D\ninst\u271d\u00b3 : MonoidalCategory C\ninst\u271d\u00b2 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u03b2 : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\nw :\n  \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2 X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\nX\u271d Y\u271d Z\u271d : C\n\u22a2 (\ud835\udfd9 (F.obj X\u271d) \u2297 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y\u271d Z\u271d) \u226b\n      LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d (Y\u271d \u2297 Z\u271d) \u226b\n        F.map ((\u03b1_ X\u271d Y\u271d Z\u271d).inv \u226b (\u03b2 (X\u271d \u2297 Y\u271d) Z\u271d).hom \u226b (\u03b1_ Z\u271d X\u271d Y\u271d).inv) =\n    (\ud835\udfd9 (F.obj X\u271d) \u2297 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y\u271d Z\u271d) \u226b\n      LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X\u271d (Y\u271d \u2297 Z\u271d) \u226b\n        F.map ((\ud835\udfd9 X\u271d \u2297 (\u03b2 Y\u271d Z\u271d).hom) \u226b (\u03b1_ X\u271d Z\u271d Y\u271d).inv \u226b ((\u03b2 X\u271d Z\u271d).hom \u2297 \ud835\udfd9 Y\u271d))\n[PROOFSTEP]\nrw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, \u2190 LaxMonoidalFunctor.\u03bc_natural_assoc,\n  Functor.map_id, \u2190 id_tensor_comp_assoc, w, id_tensor_comp_assoc, LaxMonoidalFunctor.associativity_inv_assoc,\n  LaxMonoidalFunctor.associativity_inv_assoc, \u2190 LaxMonoidalFunctor.\u03bc_natural, Functor.map_id, \u2190 comp_tensor_id_assoc, w,\n  comp_tensor_id_assoc, reassoc_of% w, braiding_naturality_assoc, LaxMonoidalFunctor.associativity_inv,\n  hexagon_reverse_assoc]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2076 : Category.{?u.78484, u_1} C\ninst\u271d\u2075 : Category.{?u.78488, u_2} D\ninst\u271d\u2074 : MonoidalCategory C\ninst\u271d\u00b3 : MonoidalCategory D\nF : MonoidalFunctor C D\ninst\u271d\u00b2 : Full F.toFunctor\ninst\u271d\u00b9 : Faithful F.toFunctor\ninst\u271d : BraidedCategory D\n\u22a2 \u2200 (X Y : C),\n    LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b\n        F.map\n          ((fun X Y =>\n                Functor.preimageIso F.toFunctor\n                  ((asIso (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)).symm \u226a\u226b\n                    \u03b2_ (F.obj X) (F.obj Y) \u226a\u226b asIso (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)))\n              X Y).hom =\n      (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n[PROOFSTEP]\naesop_cat\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) X).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 leftUnitor_tensor, leftUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) X).hom \u226b (\u03bb_ (\ud835\udfd9_ C \u2297 X)).hom \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv = ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\nslice_rhs 3 4 => rw [\u2190 id_tensor_comp, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 id_tensor_comp, Iso.hom_inv_id, tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C \u2297 X \u2297 \ud835\udfd9_ C) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv) \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b\n      (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\u03b1_ tensorUnit' (\ud835\udfd9_ C) X).hom \u226b\n          (\ud835\udfd9 tensorUnit' \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ tensorUnit' X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 hexagon_forward]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 hexagon_forward]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 hexagon_forward]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ X (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [\u2190 hexagon_forward]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((((\u03b1_ X (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) X).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv)) \u226b\n        (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv) \u226b\n      ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b\n      (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\u03b1_ tensorUnit' (\ud835\udfd9_ C) X).hom \u226b\n          (\ud835\udfd9 tensorUnit' \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ tensorUnit' X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b\n      (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\u03b1_ tensorUnit' (\ud835\udfd9_ C) X).hom \u226b\n          (\ud835\udfd9 tensorUnit' \u2297 (\u03b2_ X (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ tensorUnit' X (\ud835\udfd9_ C)).inv \u226b ((\u03bb_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [braiding_leftUnitor_aux\u2081]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (tensorUnit' \u2297 \ud835\udfd9_ C)).hom \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ X (\ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b ((\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom \u226b (\u03b2_ X (\ud835\udfd9_ C)).inv =\n    (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom)\n[PROOFSTEP]\nrw [Iso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X tensorUnit' (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) = (\u03c1_ X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\n[PROOFSTEP]\nrw [triangle]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b2_ X (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X).hom = (\u03c1_ X).hom\n[PROOFSTEP]\nrw [\u2190 tensor_right_iff, comp_tensor_id, braiding_leftUnitor_aux\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv \u226b ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv\n[PROOFSTEP]\nrw [\u2190 rightUnitor_tensor, rightUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ X (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv \u226b (\u03c1_ (X \u2297 \ud835\udfd9_ C)).hom \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv = (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom)\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b\n        ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n          ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom)\n[PROOFSTEP]\nslice_rhs 3 4 => rw [\u2190 comp_tensor_id, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, Iso.hom_inv_id, tensor_id]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 comp_tensor_id, Iso.hom_inv_id, tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b (\ud835\udfd9 ((\ud835\udfd9_ C \u2297 X) \u2297 \ud835\udfd9_ C) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom)\n[PROOFSTEP]\nrw [id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b\n      (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b\n        ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n          ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b\n      (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b\n        (\u03b1_ X (\ud835\udfd9_ C) tensorUnit').inv \u226b\n          ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 tensorUnit') \u226b (\u03b1_ (\ud835\udfd9_ C) X tensorUnit').hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom)\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 hexagon_reverse]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\n[PROOFSTEP]\nrw [\u2190 hexagon_reverse]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\n[PROOFSTEP]\nrw [\u2190 hexagon_reverse]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).inv \u226b ((\u03b2_ (\ud835\udfd9_ C) X).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom\n[PROOFSTEP]\nrw [\u2190 hexagon_reverse]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((((\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) X).inv \u226b (\u03b2_ (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C) X).hom \u226b (\u03b1_ X (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) \u226b ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b\n        (\u03b1_ (\ud835\udfd9_ C) X (\ud835\udfd9_ C)).hom) \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b\n      (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b\n        (\u03b1_ X (\ud835\udfd9_ C) tensorUnit').inv \u226b\n          ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 tensorUnit') \u226b (\u03b1_ (\ud835\udfd9_ C) X tensorUnit').hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom)\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b\n      (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b\n        (\u03b1_ X (\ud835\udfd9_ C) tensorUnit').inv \u226b\n          ((\u03b2_ (\ud835\udfd9_ C) X).inv \u2297 \ud835\udfd9 tensorUnit') \u226b (\u03b1_ (\ud835\udfd9_ C) X tensorUnit').hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X).hom) =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv\n[PROOFSTEP]\nrw [braiding_rightUnitor_aux\u2081]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom) \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b ((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C \u2297 tensorUnit') X).hom \u226b (\ud835\udfd9 X \u2297 (\u03c1_ (\ud835\udfd9_ C)).hom)\ncase a.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b2_ (\ud835\udfd9_ C) X).inv\ncase a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n| (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv\n[PROOFSTEP]\nrw [\u2190 braiding_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b (((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b ((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b ((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom \u226b (\u03b2_ (\ud835\udfd9_ C) X).inv =\n    (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b ((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X)\n[PROOFSTEP]\nrw [Iso.hom_inv_id, comp_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) tensorUnit' X).inv \u226b ((\u03c1_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X) = \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X).hom\n[PROOFSTEP]\nrw [triangle_assoc_comp_right]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03b2_ (\ud835\udfd9_ C) X).hom \u226b (\u03c1_ X).hom = (\u03bb_ X).hom\n[PROOFSTEP]\nrw [\u2190 tensor_left_iff, id_tensor_comp, braiding_rightUnitor_aux\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03bb_ X).inv \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom = (\u03c1_ X).inv\n[PROOFSTEP]\napply (cancel_mono (\u03c1_ X).hom).1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03bb_ X).inv \u226b (\u03b2_ (\ud835\udfd9_ C) X).hom) \u226b (\u03c1_ X).hom = (\u03c1_ X).inv \u226b (\u03c1_ X).hom\n[PROOFSTEP]\nsimp only [assoc, braiding_rightUnitor, Iso.inv_hom_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 (\u03c1_ X).inv \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom = (\u03bb_ X).inv\n[PROOFSTEP]\napply (cancel_mono (\u03bb_ X).hom).1\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX : C\n\u22a2 ((\u03c1_ X).inv \u226b (\u03b2_ X (\ud835\udfd9_ C)).hom) \u226b (\u03bb_ X).hom = (\u03bb_ X).inv \u226b (\u03bb_ X).hom\n[PROOFSTEP]\nsimp only [assoc, braiding_leftUnitor, Iso.inv_hom_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc (LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc) X Y \u226b\n      (LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc).toFunctor.map (\u03b2_ X Y).hom =\n    (\u03b2_ ((LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc).toFunctor.obj X)\n          ((LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc).toFunctor.obj Y)).hom \u226b\n      LaxMonoidalFunctor.\u03bc (LaxMonoidalFunctor.mk src\u271d.toFunctor src\u271d.\u03b5 src\u271d.\u03bc) Y X\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n\u22a2 (LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b\n        G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y)) \u226b\n      G.map (F.map (\u03b2_ X Y).hom) =\n    (\u03b2_ (G.obj (F.obj X)) (G.obj (F.obj Y))).hom \u226b\n      LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj Y) (F.obj X) \u226b\n        G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 CategoryTheory.Functor.map_comp, F.braided, CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y) \u226b G.map (F.map (\u03b2_ X Y).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y)\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.Functor.map_comp, F.braided, CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y) \u226b G.map (F.map (\u03b2_ X Y).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y)\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.Functor.map_comp, F.braided, CategoryTheory.Functor.map_comp]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y) \u226b G.map (F.map (\u03b2_ X Y).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y)\n[PROOFSTEP]\nrw [\u2190 CategoryTheory.Functor.map_comp, F.braided, CategoryTheory.Functor.map_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b\n      G.map (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) =\n    (\u03b2_ (G.obj (F.obj X)) (G.obj (F.obj Y))).hom \u226b\n      LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj Y) (F.obj X) \u226b\n        G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nslice_lhs 1 2 => rw [G.braided]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b G.map (\u03b2_ (F.obj X) (F.obj Y)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nrw [G.braided]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b G.map (\u03b2_ (F.obj X) (F.obj Y)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nrw [G.braided]\n[GOAL]\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj X) (F.obj Y) \u226b G.map (\u03b2_ (F.obj X) (F.obj Y)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n| G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nrw [G.braided]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : LaxBraidedFunctor C D\nG : LaxBraidedFunctor D E\nsrc\u271d : LaxMonoidalFunctor C E := F.toLaxMonoidalFunctor \u2297\u22d9 G.toLaxMonoidalFunctor\nX Y : C\n\u22a2 ((\u03b2_ (G.obj (F.obj X)) (G.obj (F.obj Y))).hom \u226b LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj Y) (F.obj X)) \u226b\n      G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X) =\n    (\u03b2_ (G.obj (F.obj X)) (G.obj (F.obj Y))).hom \u226b\n      LaxMonoidalFunctor.\u03bc G.toLaxMonoidalFunctor (F.obj Y) (F.obj X) \u226b\n        G.map (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X)\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nC\u271d : Type u\u2081\ninst\u271d\u00b9\u2075 : Category.{v\u2081, u\u2081} C\u271d\ninst\u271d\u00b9\u2074 : MonoidalCategory C\u271d\ninst\u271d\u00b9\u00b3 : BraidedCategory C\u271d\nD\u271d : Type u\u2082\ninst\u271d\u00b9\u00b2 : Category.{v\u2082, u\u2082} D\u271d\ninst\u271d\u00b9\u00b9 : MonoidalCategory D\u271d\ninst\u271d\u00b9\u2070 : BraidedCategory D\u271d\nE : Type u\u2083\ninst\u271d\u2079 : Category.{v\u2083, u\u2083} E\ninst\u271d\u2078 : MonoidalCategory E\ninst\u271d\u2077 : BraidedCategory E\nC : Type u_1\nD : Type u_2\ninst\u271d\u2076 : Category.{?u.308379, u_1} C\ninst\u271d\u2075 : Category.{?u.308383, u_2} D\ninst\u271d\u2074 : MonoidalCategory C\ninst\u271d\u00b3 : MonoidalCategory D\ninst\u271d\u00b2 : BraidedCategory C\ninst\u271d\u00b9 : SymmetricCategory D\nF : BraidedFunctor C D\ninst\u271d : Faithful F.toFunctor\nX Y : C\n\u22a2 F.map ((\u03b2_ X Y).hom \u226b (\u03b2_ Y X).hom) = F.map (\ud835\udfd9 (X \u2297 Y))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : BraidedFunctor C D\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b F.map (\u03b2_ X Y).hom =\n    (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n[PROOFSTEP]\nrw [F.braided]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nF : BraidedFunctor C D\nX Y : C\n\u22a2 LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y \u226b\n      inv (LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor X Y) \u226b\n        (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X =\n    (\u03b2_ (F.obj X) (F.obj Y)).hom \u226b LaxMonoidalFunctor.\u03bc F.toLaxMonoidalFunctor Y X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 : C\n\u22a2 tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) =\n    (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082)\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).hom) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).inv) \u226b\n      (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) =\n    (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 : C\n\u22a2 (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).hom) \u226b (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).inv \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) =\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).hom) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).inv\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 : C\n\u22a2 (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).hom) \u226b\n      (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).inv \u226b\n        (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).hom) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).inv =\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).hom) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b tensor_\u03bc C (Y\u2081, Y\u2082) (V\u2081, V\u2082) = tensor_\u03bc C (X\u2081, X\u2082) (U\u2081, U\u2082) \u226b ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b\n      (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nslice_lhs 1 2 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| ((f\u2081 \u2297 f\u2082) \u2297 g\u2081 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 Y\u2082 (V\u2081 \u2297 V\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (((((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n        (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n      (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nslice_lhs 2 3 => rw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 f\u2082 \u2297 g\u2081 \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 V\u2081 V\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_inv_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      ((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)) \u226b\n          (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n        (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nslice_lhs 3 4 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, comp_id g\u2082, \u2190 id_comp g\u2082, braiding_naturality,\n    tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, comp_id g\u2082, \u2190 id_comp g\u2082, braiding_naturality, tensor_comp,\n    tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, comp_id g\u2082, \u2190 id_comp g\u2082, braiding_naturality, tensor_comp,\n    tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (f\u2082 \u2297 g\u2081) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 V\u2081).hom \u2297 \ud835\udfd9 V\u2082)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, comp_id g\u2082, \u2190 id_comp g\u2082, braiding_naturality, tensor_comp,\n  tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082)) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)) \u226b\n          (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nslice_lhs 4 5 => rw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 (g\u2081 \u2297 f\u2082) \u2297 g\u2082) \u226b (\ud835\udfd9 Y\u2081 \u2297 (\u03b1_ V\u2081 Y\u2082 V\u2082).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, comp_id f\u2081, \u2190 id_comp f\u2081, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082)) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nslice_lhs 5 6 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (f\u2081 \u2297 g\u2081 \u2297 f\u2082 \u2297 g\u2082) \u226b (\u03b1_ Y\u2081 V\u2081 (Y\u2082 \u2297 V\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 U\u2081 U\u2082 V\u2081 V\u2082 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\ng\u2081 : U\u2081 \u27f6 V\u2081\ng\u2082 : U\u2082 \u27f6 V\u2082\n\u22a2 (\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv \u226b ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082) =\n    ((\u03b1_ X\u2081 X\u2082 (U\u2081 \u2297 U\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 U\u2081 U\u2082).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 U\u2081).hom \u2297 \ud835\udfd9 U\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ U\u2081 X\u2082 U\u2082).hom) \u226b (\u03b1_ X\u2081 U\u2081 (X\u2082 \u2297 U\u2082)).inv) \u226b\n      ((f\u2081 \u2297 g\u2081) \u2297 f\u2082 \u2297 g\u2082)\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom = ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b tensor_\u03bc C (\ud835\udfd9_ C, \ud835\udfd9_ C) (X\u2081, X\u2082) \u226b ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b\n      ((\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n              (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv) \u226b\n        ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\nhave :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082 :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b\n      ((\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n              (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv) \u226b\n        ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\nslice_rhs 1 3 => rw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  ((\u03bb_ (\ud835\udfd9_ C)).inv \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) =\n    \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    ((((\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082)) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom)) \u226b\n        (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    ((((\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082)) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom)) \u226b\n        (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, leftUnitor_inv_braiding]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, leftUnitor_inv_braiding]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, leftUnitor_inv_braiding]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, leftUnitor_inv_braiding]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    (((\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom)) \u226b (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ (X\u2081 \u2297 X\u2082)).hom =\n    (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03c1_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv \u226b ((\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom = (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b tensor_\u03bc C (X\u2081, X\u2082) (\ud835\udfd9_ C, \ud835\udfd9_ C) \u226b ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv) \u226b\n        ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\nhave :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv) \u226b\n        ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\nslice_rhs 1 3 => rw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).inv) =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (((((\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom)) \u226b\n        (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv) \u226b\n      ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (((((\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom)) \u226b\n        (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv) \u226b\n      ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, rightUnitor_inv_braiding]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, rightUnitor_inv_braiding]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, rightUnitor_inv_braiding]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C))\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, comp_id, comp_id, rightUnitor_inv_braiding]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b\n      (((\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom)) \u226b (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv) \u226b\n        ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\nsimp only [assoc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ (X\u2081 \u2297 X\u2082)).hom =\n    (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).inv \u226b ((\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom)\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        ((\u03b1_ W X Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b2_ W (X \u2297 Z)).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X Z W).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (Z \u2297 W) Y).hom\n[PROOFSTEP]\nslice_rhs 3 5 => rw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b2_ W (X \u2297 Z)).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X Z W).hom \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b2_ W (X \u2297 Z)).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X Z W).hom \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b2_ W (X \u2297 Z)).hom \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X Z W).hom \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        ((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (Z \u2297 W) Y).hom\n[PROOFSTEP]\nslice_rhs 5 6 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom) \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (Z \u2297 W) Y).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (\u03b1_ (W \u2297 X) Z Y).inv \u226b\n        (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv \u226b (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv \u226b (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 X) Z Y).inv \u226b (((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z) \u2297 \ud835\udfd9 Y)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (X \u2297 W) Z Y).inv) \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y)) \u226b (\u03b1_ X (W \u2297 Z) Y).hom) \u226b\n        (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nslice_rhs 3 5 => rw [\u2190 pentagon_hom_inv]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (X \u2297 W) Z Y).inv \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [\u2190 pentagon_hom_inv]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (X \u2297 W) Z Y).inv \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [\u2190 pentagon_hom_inv]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (X \u2297 W) Z Y).inv \u226b ((\u03b1_ X W Z).hom \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (W \u2297 Z) Y).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [\u2190 pentagon_hom_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b\n      ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ X W (Z \u2297 Y)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nslice_rhs 1 2 => rw [tensor_id, id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [tensor_id, id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [tensor_id, id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (W \u2297 X) \u2297 (\u03b2_ Y Z).hom) \u226b ((\u03b2_ W X).hom \u2297 \ud835\udfd9 Z \u2297 \ud835\udfd9 Y)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\n[PROOFSTEP]\nrw [tensor_id, id_tensor_comp_tensor_id, \u2190 tensor_id_comp_id_tensor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    (((((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom)) \u226b (\u03b1_ X W (Z \u2297 Y)).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b\n      (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom) \u226b (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom) \u226b (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 (X \u2297 W) \u2297 (\u03b2_ Y Z).hom) \u226b (\u03b1_ X W (Z \u2297 Y)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Y Z).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ (W \u2297 Y) Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ Z W Y).inv) =\n    ((\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)) \u226b\n      (((\u03b1_ X W (Y \u2297 Z)).hom \u226b (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom)) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv)) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\n[PROOFSTEP]\nslice_rhs 3 5 => rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Y \u2297 Z)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Y \u2297 Z)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 X \u2297 \ud835\udfd9 W \u2297 (\u03b2_ Y Z).hom) \u226b (\ud835\udfd9 X \u2297 (\u03b1_ W Z Y).inv) \u226b (\ud835\udfd9 X \u2297 (\u03b2_ W Z).hom \u2297 \ud835\udfd9 Y)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W X).hom \u2297 \ud835\udfd9 (Y \u2297 Z)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ X W (Y \u2297 Z)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b ((\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nhave :\n  (\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom =\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom =\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom =\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b ((\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ X\u2081 Y\u2081 Z\u2081).hom \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2082).hom =\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b\n        (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n                (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                        (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b\n        (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n            (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n                (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                        (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 2 4 => rw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081 \u2297 Y\u2081, X\u2082 \u2297 Y\u2082) (Z\u2081, Z\u2082) \u226b\n    (\u03b1_ (X\u2081 \u2297 Y\u2081) Z\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ Z\u2081 (X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      (((((((((((\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n                            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv)) \u226b\n                      (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n              (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n        (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 4 5 => rw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 Y\u2081 ((Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n          ((((((((((\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv)) \u226b\n                          (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 5 6 => rw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n          (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n            (((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n              (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 6 7 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082 \u2297 Y\u2082) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n          (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b\n              ((((((((\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nhave :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n          (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b\n              ((((((((\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 2 6 => rw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n      (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n      (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n      (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n        (\u03b1_ X\u2081 Y\u2081 (((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) \u2297 Z\u2082)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 ((X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 (X\u2082 \u2297 Y\u2082) \u2297 Z\u2081) Z\u2082).inv =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((((((((((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                          (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n                            ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                  ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n              (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n        (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((((((((((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                          (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n                            ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                  ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n              (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n        (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 1 3 => rw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082))\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082))\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082))\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 (((((((((((((((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n                                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082))) \u226b\n                              (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv) \u226b\n                            ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n            (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n      (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (((((((((((((\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                                ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n          (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 4 5 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          (((((((((((((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                  (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n            (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 5 6 => rw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((((((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                    (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n              (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 6 10 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190\n    tensor_comp, tensor_id, tensor_associativity_aux, \u2190 tensor_id, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082), tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190\n    tensor_comp, tensor_id, tensor_associativity_aux, \u2190 tensor_id, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082), tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190\n    tensor_comp, tensor_id, tensor_associativity_aux, \u2190 tensor_id, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082), tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2081) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 (Y\u2082 \u2297 Z\u2081)).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ X\u2082 Y\u2082 Z\u2081).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 Y\u2082) Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190\n  tensor_comp, tensor_id, tensor_associativity_aux, \u2190 tensor_id, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n  id_comp (\ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082), tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,\n  tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((((((((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                      (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 11 12 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (Z\u2081 \u2297 X\u2082) Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 ((Y\u2081 \u2297 Z\u2081 \u2297 X\u2082) \u2297 Y\u2082)) \u2297 \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                        ((((\ud835\udfd9 ((X\u2081 \u2297 (Y\u2081 \u2297 Z\u2081 \u2297 X\u2082) \u2297 Y\u2082) \u2297 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                                (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                          (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nsimp only [assoc, id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                          (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 10 11 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n  simp\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 Z\u2081 X\u2082).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, Iso.hom_inv_id]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2082 \u226b \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082 \u226b \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      (((\ud835\udfd9 ((X\u2081 \u2297 ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) \u2297 Z\u2082) \u226b (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                        (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nsimp only [assoc, id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                      (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 9 10 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 (((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    ((((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                      (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 10 11 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((Y\u2081 \u2297 Z\u2081) \u2297 X\u2082) Y\u2082 Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom)) \u226b\n                        (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 11 13 => rw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 (Y\u2081 \u2297 Z\u2081)).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 Z\u2081) X\u2082 (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 (Y\u2081 \u2297 Z\u2081) (X\u2082 \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b\n                          (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nhave :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b\n                          (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 7 12 => rw [this]\n[GOAL]\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Z\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Z\u2082) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Z\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Z\u2082) \u226b\n        (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082) Z\u2082).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081 \u2297 Z\u2081) Y\u2082 Z\u2082).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (Y\u2081 \u2297 Z\u2081) (Y\u2082 \u2297 Z\u2082)).hom) \u226b (\u03b1_ X\u2081 X\u2082 ((Y\u2081 \u2297 Z\u2081) \u2297 Y\u2082 \u2297 Z\u2082)).inv =\n    (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                  tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n                ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                  tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 6 7 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (((((((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b (\ud835\udfd9 X\u2081 \u2297 ((\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom)) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom)) \u226b\n                      (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv) \u226b\n                    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom)) \u226b\n                  (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 7 8 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Z\u2081 \u2297 Y\u2082) Z\u2082).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b\n                ((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom)) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv) \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom)) \u226b\n                    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                  tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 8 9 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081) \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom) \u226b\n                  (((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv) \u226b\n                        (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom)) \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                    tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 9 10 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Z\u2081 \u2297 Y\u2082) \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom) \u226b\n                    ((((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082)) \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom)) \u226b\n                        (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)) \u226b\n                      tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\nslice_lhs 10 12 => rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b2_ Y\u2082 Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Z\u2081 Y\u2082 Z\u2082).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Z\u2081 (Y\u2082 \u2297 Z\u2082)).inv)\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom) \u226b\n                    (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv \u226b\n                      (((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 Z\u2081 Z\u2082).hom) \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Y\u2082 (Z\u2081 \u2297 Z\u2082)).inv) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082))) \u226b\n                        tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 Y\u2081 Y\u2082 Z\u2081 Z\u2082 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (Z\u2081 \u2297 Z\u2082)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081 Z\u2082).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) Z\u2081).hom \u2297 \ud835\udfd9 Z\u2082) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 Z\u2081).hom) \u2297 \ud835\udfd9 Z\u2082) \u226b\n              (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082 \u2297 Z\u2081) Z\u2082).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) (Y\u2082 \u2297 Z\u2081) Z\u2082).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 ((Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).hom) \u226b\n                    (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 (Y\u2082 \u2297 Z\u2081) \u2297 Z\u2082)).inv \u226b\n                      (((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082) \u2297 \ud835\udfd9 Y\u2081 \u2297 (\u03b1_ Y\u2082 Z\u2081 Z\u2082).hom) \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ Y\u2081 Y\u2082 (Z\u2081 \u2297 Z\u2082)).inv) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082))) \u226b\n                        tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (Z\u2081 \u2297 Z\u2082)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 tensor_\u03bc C (Y\u2081, Y\u2082) (Z\u2081, Z\u2082)) \u226b tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081 \u2297 Z\u2081, Y\u2082 \u2297 Z\u2082)\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nsrc\u271d : C \u00d7 C \u2964 C := tensor C\n\u22a2 \u2200 (X Y : C \u00d7 C),\n    IsIso\n      (LaxMonoidalFunctor.\u03bc\n        (LaxMonoidalFunctor.mk (Functor.mk src\u271d.toPrefunctor) (\u03bb_ (\ud835\udfd9_ C)).inv fun X Y => tensor_\u03bc C X Y) X Y)\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nsrc\u271d : C \u00d7 C \u2964 C := tensor C\n\u22a2 \u2200 (X Y : C \u00d7 C),\n    IsIso\n      ((\u03b1_ X.fst X.snd (Y.fst \u2297 Y.snd)).hom \u226b\n        (\ud835\udfd9 X.fst \u2297 (\u03b1_ X.snd Y.fst Y.snd).inv) \u226b\n          (\ud835\udfd9 X.fst \u2297 (\u03b2_ X.snd Y.fst).hom \u2297 \ud835\udfd9 Y.snd) \u226b\n            (\ud835\udfd9 X.fst \u2297 (\u03b1_ Y.fst X.snd Y.snd).hom) \u226b (\u03b1_ X.fst Y.fst (X.snd \u2297 Y.snd)).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom = tensor_\u03bc C (\ud835\udfd9_ C, X\u2081) (\ud835\udfd9_ C, X\u2082) \u226b ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nhave :\n  (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\n\u22a2 (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\u03bb_ X\u2081).hom \u2297 (\u03bb_ X\u2082).hom =\n    (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03c1_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 braiding_leftUnitor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 id_comp (\ud835\udfd9 X\u2082), tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2082), tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2082), tensor_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u226b (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2082), tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n        (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b ((\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 leftUnitor_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 leftUnitor_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 leftUnitor_naturality]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ ((X\u2081 \u2297 \ud835\udfd9_ C) \u2297 X\u2082)).hom \u226b ((\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv\n[PROOFSTEP]\nrw [\u2190 leftUnitor_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n        ((\ud835\udfd9 tensorUnit' \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b (\u03bb_ ((\ud835\udfd9_ C \u2297 X\u2081) \u2297 X\u2082)).hom) \u226b ((\u03bb_ X\u2081).hom \u2297 \ud835\udfd9 X\u2082) =\n    ((\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).inv) \u226b\n          (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b2_ X\u2081 (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 X\u2082) \u226b\n            (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).hom) \u226b (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).inv) \u226b\n      ((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom = tensor_\u03bc C (X\u2081, \ud835\udfd9_ C) (X\u2082, \ud835\udfd9_ C) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\ndsimp [tensor_\u03bc]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nhave :\n  (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom)\n\u22a2 (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\nthis :\n  (\u03c1_ X\u2081).hom \u2297 (\u03c1_ X\u2082).hom =\n    (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom)\n\u22a2 (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom) =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom) =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nrw [\u2190 braiding_rightUnitor]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u226b (\u03c1_ X\u2082).hom) =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 id_comp (\ud835\udfd9 X\u2081), tensor_comp, id_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u226b (\u03c1_ X\u2082).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2081), tensor_comp, id_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u226b (\u03c1_ X\u2082).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2081), tensor_comp, id_comp]\n[GOAL]\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u226b (\u03c1_ X\u2082).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 id_comp (\ud835\udfd9 X\u2081), tensor_comp, id_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).hom) =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_comp, \u2190 rightUnitor_naturality, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 rightUnitor_naturality, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 rightUnitor_naturality, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (\ud835\udfd9_ C \u2297 X\u2082)).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 rightUnitor_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 : C\n\u22a2 (\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 tensorUnit') \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ (X\u2082 \u2297 \ud835\udfd9_ C)).hom)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03c1_ X\u2082).hom) =\n    ((\u03b1_ X\u2081 (\ud835\udfd9_ C) (X\u2082 \u2297 \ud835\udfd9_ C)).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (\ud835\udfd9_ C) X\u2082).hom \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).hom) \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).hom\n[PROOFSTEP]\ncoherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\u03b1_ W X (Y \u2297 Z)).inv \u226b\n      (\u03b1_ (W \u2297 X) Y Z).inv \u226b\n        ((\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (Y \u2297 W) X Z).hom \u226b (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\n[PROOFSTEP]\nslice_rhs 1 2 => rw [\u2190 pentagon_inv]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W X (Y \u2297 Z)).inv \u226b (\u03b1_ (W \u2297 X) Y Z).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\n[PROOFSTEP]\nrw [\u2190 pentagon_inv]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W X (Y \u2297 Z)).inv \u226b (\u03b1_ (W \u2297 X) Y Z).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\n[PROOFSTEP]\nrw [\u2190 pentagon_inv]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W X (Y \u2297 Z)).inv \u226b (\u03b1_ (W \u2297 X) Y Z).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\n[PROOFSTEP]\nrw [\u2190 pentagon_inv]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (((((\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z)) \u226b ((\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z)) \u226b\n          ((\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z)) \u226b\n        (\u03b1_ (Y \u2297 W) X Z).hom) \u226b\n      (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\n[PROOFSTEP]\nslice_rhs 3 5 => rw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z) \u226b ((\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z) \u226b ((\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z) \u226b ((\u03b2_ (W \u2297 X) Y).hom \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ Y W X).inv \u2297 \ud835\udfd9 Z)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (\u03b1_ W (X \u2297 Y) Z).inv \u226b\n        ((((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b (((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Z)) \u226b\n            (\u03b1_ (Y \u2297 W) X Z).hom) \u226b\n          (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\n[PROOFSTEP]\nslice_rhs 5 6 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (Y \u2297 W) X Z).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (\u03b1_ W (X \u2297 Y) Z).inv \u226b\n        ((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z) \u226b\n          ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b ((\u03b1_ (W \u2297 Y) X Z).hom \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Z)) \u226b (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\n[PROOFSTEP]\nslice_rhs 6 7 => rw [tensor_id, tensor_id_comp_id_tensor, \u2190 id_tensor_comp_tensor_id]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\n[PROOFSTEP]\nrw [tensor_id, tensor_id_comp_id_tensor, \u2190 id_tensor_comp_tensor_id]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\n[PROOFSTEP]\nrw [tensor_id, tensor_id_comp_id_tensor, \u2190 id_tensor_comp_tensor_id]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 (Y \u2297 W) \u2297 (\u03b2_ X Z).hom)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\n[PROOFSTEP]\nrw [tensor_id, tensor_id_comp_id_tensor, \u2190 id_tensor_comp_tensor_id]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (\u03b1_ W (X \u2297 Y) Z).inv \u226b\n        ((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z) \u226b\n          ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 Y) X Z).hom \u226b (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom) \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X))\n[PROOFSTEP]\nslice_rhs 2 3 => rw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z)\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z)\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom) \u2297 \ud835\udfd9 Z)\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (((((\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ W (Y \u2297 X) Z).inv) \u226b ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z)) \u226b (\u03b1_ (W \u2297 Y) X Z).hom) \u226b\n          (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom)) \u226b\n        ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X))\n[PROOFSTEP]\nslice_rhs 3 5 => rw [pentagon_inv_inv_hom]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (Y \u2297 X) Z).inv \u226b ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [pentagon_inv_inv_hom]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (Y \u2297 X) Z).inv \u226b ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [pentagon_inv_inv_hom]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W (Y \u2297 X) Z).inv \u226b ((\u03b1_ W Y X).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 Y) X Z).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\n[PROOFSTEP]\nrw [pentagon_inv_inv_hom]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b\n        (((\ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom) \u226b (\u03b1_ W Y (X \u2297 Z)).inv) \u226b (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom)) \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X))\n[PROOFSTEP]\nslice_rhs 4 5 => rw [\u2190 tensor_id, \u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (X \u2297 Z)).inv \u226b (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, \u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (X \u2297 Z)).inv \u226b (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, \u2190 associator_inv_naturality]\n[GOAL]\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (X \u2297 Z)).inv \u226b (\ud835\udfd9 (W \u2297 Y) \u2297 (\u03b2_ X Z).hom)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom\n[PROOFSTEP]\nrw [\u2190 tensor_id, \u2190 associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b\n        (\ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom) \u226b ((\ud835\udfd9 W \u2297 \ud835\udfd9 Y \u2297 (\u03b2_ X Z).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv) \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X))\n[PROOFSTEP]\nslice_rhs 2 4 => rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom) \u226b (\ud835\udfd9 W \u2297 \ud835\udfd9 Y \u2297 (\u03b2_ X Z).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (Z \u2297 X)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom) \u226b (\ud835\udfd9 W \u2297 \ud835\udfd9 Y \u2297 (\u03b2_ X Z).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (Z \u2297 X)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\ud835\udfd9 W \u2297 (\u03b2_ X Y).hom \u2297 \ud835\udfd9 Z) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y X Z).hom) \u226b (\ud835\udfd9 W \u2297 \ud835\udfd9 Y \u2297 (\u03b2_ X Z).hom)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b1_ W Y (Z \u2297 X)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| (\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n| \ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 hexagon_forward, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nW X Y Z : C\n\u22a2 (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom) \u226b (\u03b1_ W Y (Z \u2297 X)).inv \u226b ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X)) =\n    (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b\n      (((\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b2_ X (Y \u2297 Z)).hom) \u226b (\ud835\udfd9 W \u2297 (\u03b1_ Y Z X).hom)) \u226b (\u03b1_ W Y (Z \u2297 X)).inv) \u226b\n        ((\u03b2_ W Y).hom \u2297 \ud835\udfd9 (Z \u2297 X))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nhave :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n              (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n              (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n              (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  (\u03b1_ (X\u2081 \u2297 Y\u2081) (X\u2082 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom =\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n              (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n            (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n              ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                        (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n            (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n              ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                        (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 2 4 => rw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083))\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083))\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (tensor_\u03bc C (X\u2081, X\u2082) (Y\u2081, Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083))\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, tensor_\u03bc_def\u2081, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      (((((((((((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083))) \u226b\n                        (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n                      ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                  (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 4 5 => rw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b (\u03b1_ (X\u2081 \u2297 (Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          (((((((((\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                      (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 5 6 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n            (((((((((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                  (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 6 7 => rw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, associator_naturality, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n            ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n              ((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nhave :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083) :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n          (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n            ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n              ((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 2 6 => rw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n        ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n        ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n        ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\u03b1_ X\u2081 X\u2082 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 (X\u2083 \u2297 Y\u2083)) \u226b\n        (\u03b1_ (X\u2081 \u2297 (X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083 Y\u2083).inv \u226b\n          ((\u03b1_ X\u2081 ((X\u2082 \u2297 Y\u2081) \u2297 Y\u2082) X\u2083).hom \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 Y\u2081) Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) =\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((((((((\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n                            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                  (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n      ((((((((\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n                            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                  (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 1 3 => rw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| tensor_\u03bc C (X\u2081 \u2297 X\u2082, X\u2083) (Y\u2081 \u2297 Y\u2082, Y\u2083) \u226b\n    (\u03b1_ (X\u2081 \u2297 X\u2082) (Y\u2081 \u2297 Y\u2082) (X\u2083 \u2297 Y\u2083)).hom \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ (Y\u2081 \u2297 Y\u2082) X\u2083 Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_\u03bc_def\u2081]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (((((((((((((\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n                              (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                            (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv)) \u226b\n                        (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n          (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 3 4 => rw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 X\u2082 (((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_id, associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        ((((((((((((\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                              (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv)) \u226b\n                            (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 4 5 => rw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 ((Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_inv_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (((((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                              (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                        ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                      (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 5 6 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 (X\u2082 \u2297 (Y\u2081 \u2297 Y\u2082) \u2297 X\u2083) Y\u2083).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            ((((((((((\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                  (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 6 9 =>\n  rw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, tensor_id,\n    associator_monoidal_aux, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), \u2190 id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, tensor_id,\n    associator_monoidal_aux, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), \u2190 id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, tensor_id,\n    associator_monoidal_aux, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n    id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), \u2190 id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), tensor_comp, tensor_comp,\n    tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 (Y\u2081 \u2297 Y\u2082)).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2081 Y\u2082 X\u2083).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 Y\u2081 (Y\u2082 \u2297 X\u2083)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ X\u2082 Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 X\u2083) \u2297 \ud835\udfd9 Y\u2083)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_comp, tensor_id,\n  associator_monoidal_aux, \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190 id_comp (\ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081 \u226b \ud835\udfd9 X\u2081), \u2190\n  id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), \u2190 id_comp (\ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083 \u226b \ud835\udfd9 Y\u2083), tensor_comp, tensor_comp,\n  tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((((((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                            ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                              ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                                    ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                    (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 11 12 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        ((((((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)) \u226b\n                                (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                          (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 12 13 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (Y\u2082 \u2297 X\u2083) Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n                          (((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)) \u226b\n                                (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                            (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 13 14 => rw [\u2190 tensor_comp, \u2190 tensor_id, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_id, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_id, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (Y\u2081 \u2297 X\u2082) \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).hom)\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, \u2190 tensor_id, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n                            ((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n                                    (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                                  (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv) \u226b\n                                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 14 15 => rw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081 \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (Y\u2082 \u2297 X\u2083) \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\n[PROOFSTEP]\nrw [associator_inv_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n                              (((\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom)) \u226b\n                                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 15 17 => rw [tensor_id, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b2_ X\u2083 Y\u2082).hom \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ Y\u2082 X\u2083 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 Y\u2082 (X\u2083 \u2297 Y\u2083)).inv)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_comp, \u2190 tensor_comp, \u2190 tensor_\u03bc_def\u2082, tensor_comp, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n                              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b\n                                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                                    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nhave :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv :=\n  by pure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n[PROOFSTEP]\npure_coherence\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n                            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n                              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b\n                                  (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                                    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 9 16 => rw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv)\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv)\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n    ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n            (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n              (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv)\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [this]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nthis :\n  ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 X\u2083).inv \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) X\u2083 Y\u2082).hom) \u2297 \ud835\udfd9 Y\u2083) \u226b\n        (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082) \u2297 X\u2083 \u2297 Y\u2082) Y\u2083).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082) (X\u2083 \u2297 Y\u2082) Y\u2083).hom) \u226b\n            (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 X\u2082 ((X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).hom) \u226b\n              (\u03b1_ X\u2081 Y\u2081 (X\u2082 \u2297 (X\u2083 \u2297 Y\u2082) \u2297 Y\u2083)).inv \u226b\n                (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 \ud835\udfd9 X\u2082 \u2297 (\u03b1_ X\u2083 Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2082 \u2297 Y\u2083)).inv) =\n    (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nclear this\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                    ((\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 8 9 => rw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| ((\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\u03b1_ X\u2081 ((Y\u2081 \u2297 X\u2082 \u2297 X\u2083) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\n[PROOFSTEP]\nrw [associator_naturality]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (((((\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom \u226b (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom)) \u226b\n                        (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom)) \u226b\n                      (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                    (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 9 10 => rw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 ((\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082) \u2297 \ud835\udfd9 Y\u2083) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (Y\u2081 \u2297 X\u2082 \u2297 X\u2083) Y\u2082 Y\u2083).hom)\ncase a.a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\n[PROOFSTEP]\nrw [\u2190 tensor_comp, associator_naturality, tensor_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom \u226b\n                    ((((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom) \u226b (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Y\u2083)) \u226b\n                          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom)) \u226b\n                        (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv) \u226b\n                      (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\nslice_lhs 10 12 => rw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b2_ (X\u2082 \u2297 X\u2083) Y\u2081).hom \u2297 \ud835\udfd9 Y\u2082 \u2297 \ud835\udfd9 Y\u2083) \u226b\n    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ Y\u2081 (X\u2082 \u2297 X\u2083) (Y\u2082 \u2297 Y\u2083)).hom) \u226b (\u03b1_ X\u2081 Y\u2081 ((X\u2082 \u2297 X\u2083) \u2297 Y\u2082 \u2297 Y\u2083)).inv\ncase a.a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)\ncase a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom\ncase a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv\ncase a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083\ncase a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom\ncase a.a.a.a.a.a.a.a.a\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n| \ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom\n[PROOFSTEP]\nrw [tensor_id, \u2190 tensor_\u03bc_def\u2082]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 (Y\u2082 \u2297 Y\u2083)).hom) \u226b\n                          (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083) (Y\u2081 \u2297 Y\u2082 \u2297 Y\u2083)).inv \u226b tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083)) \u226b\n                        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2077 : MonoidalCategory C\ninst\u271d\u2076 : BraidedCategory C\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : MonoidalCategory D\ninst\u271d\u00b3 : BraidedCategory D\nE : Type u\u2083\ninst\u271d\u00b2 : Category.{v\u2083, u\u2083} E\ninst\u271d\u00b9 : MonoidalCategory E\ninst\u271d : BraidedCategory E\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\n\u22a2 (\u03b1_ (X\u2081 \u2297 X\u2082) X\u2083 ((Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n      (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03b1_ X\u2083 (Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n        (\u03b1_ X\u2081 X\u2082 ((X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) \u2297 Y\u2083)).hom \u226b\n          (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv) \u226b\n            (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083 \u2297 Y\u2081 \u2297 Y\u2082) Y\u2083).inv \u226b\n              ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 X\u2083 (Y\u2081 \u2297 Y\u2082)).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 Y\u2082).inv) \u2297 \ud835\udfd9 Y\u2083) \u226b\n                  (\u03b1_ X\u2081 (((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) \u2297 Y\u2082) Y\u2083).hom \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ ((X\u2082 \u2297 X\u2083) \u2297 Y\u2081) Y\u2082 Y\u2083).hom) \u226b\n                      ((\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (X\u2082 \u2297 X\u2083) Y\u2081 (Y\u2082 \u2297 Y\u2083)).hom) \u226b\n                          (\u03b1_ X\u2081 (X\u2082 \u2297 X\u2083) (Y\u2081 \u2297 Y\u2082 \u2297 Y\u2083)).inv \u226b tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083)) \u226b\n                        (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083)) =\n    ((\u03b1_ X\u2081 X\u2082 X\u2083).hom \u2297 (\u03b1_ Y\u2081 Y\u2082 Y\u2083).hom) \u226b\n      tensor_\u03bc C (X\u2081, X\u2082 \u2297 X\u2083) (Y\u2081, Y\u2082 \u2297 Y\u2083) \u226b (\ud835\udfd9 (X\u2081 \u2297 Y\u2081) \u2297 tensor_\u03bc C (X\u2082, X\u2083) (Y\u2082, Y\u2083))\n[PROOFSTEP]\ncoherence\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Monoidal.Braided", "llama_tokens": 566469, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245911726382, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.5308534321171604}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n[PROOFSTEP]\nhave h\u03b5' : 0 < \u03b5 / 3 := div_pos h\u03b5 zero_lt_three\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b4, h\u27e9 := exists_forall_sphere_dist_add_le_two_sub E h\u03b5'\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n[PROOFSTEP]\nset \u03b4' := min (1 / 2) (min (\u03b5 / 3) <| \u03b4 / 3)\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n[PROOFSTEP]\nrefine' \u27e8\u03b4', lt_min one_half_pos <| lt_min h\u03b5' (div_pos h\u03b4 zero_lt_three), fun x hx y hy hxy => _\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nobtain hx' | hx' := le_or_lt \u2016x\u2016 (1 - \u03b4')\n[GOAL]\ncase intro.intro.inl\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : \u2016x\u2016 \u2264 1 - \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nrw [\u2190 one_add_one_eq_two]\n[GOAL]\ncase intro.intro.inl\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : \u2016x\u2016 \u2264 1 - \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 1 + 1 - \u03b4'\n[PROOFSTEP]\nexact (norm_add_le_of_le hx' hy).trans (sub_add_eq_add_sub _ _ _).le\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nobtain hy' | hy' := le_or_lt \u2016y\u2016 (1 - \u03b4')\n[GOAL]\ncase intro.intro.inr.inl\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : \u2016y\u2016 \u2264 1 - \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nrw [\u2190 one_add_one_eq_two]\n[GOAL]\ncase intro.intro.inr.inl\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : \u2016y\u2016 \u2264 1 - \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 1 + 1 - \u03b4'\n[PROOFSTEP]\nexact (norm_add_le_of_le hx hy').trans (add_sub_assoc _ _ _).ge\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nhave h\u03b4' : 0 < 1 - \u03b4' := sub_pos_of_lt (min_lt_of_left_lt one_half_lt_one)\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nhave h\u2081 : \u2200 z : E, 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1 := by\n  rintro z hz\n  rw [norm_smul_of_nonneg (inv_nonneg.2 <| norm_nonneg _), inv_mul_cancel (h\u03b4'.trans hz).ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\n\u22a2 \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\n[PROOFSTEP]\nrintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nz : E\nhz : 1 - \u03b4' < \u2016z\u2016\n\u22a2 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\n[PROOFSTEP]\nrw [norm_smul_of_nonneg (inv_nonneg.2 <| norm_nonneg _), inv_mul_cancel (h\u03b4'.trans hz).ne']\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nhave h\u2082 : \u2200 z : E, \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4' :=\n  by\n  rintro z hz h\u03b4z\n  nth_rw 3 [\u2190 one_smul \u211d z]\n  rwa [\u2190 sub_smul, norm_smul_of_nonneg (sub_nonneg_of_le <| one_le_inv (h\u03b4'.trans_le h\u03b4z) hz), sub_mul,\n    inv_mul_cancel (h\u03b4'.trans_le h\u03b4z).ne', one_mul, sub_le_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\n\u22a2 \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\n[PROOFSTEP]\nrintro z hz h\u03b4z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nz : E\nhz : \u2016z\u2016 \u2264 1\nh\u03b4z : 1 - \u03b4' \u2264 \u2016z\u2016\n\u22a2 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\n[PROOFSTEP]\nnth_rw 3 [\u2190 one_smul \u211d z]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nz : E\nhz : \u2016z\u2016 \u2264 1\nh\u03b4z : 1 - \u03b4' \u2264 \u2016z\u2016\n\u22a2 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - 1 \u2022 z\u2016 \u2264 \u03b4'\n[PROOFSTEP]\nrwa [\u2190 sub_smul, norm_smul_of_nonneg (sub_nonneg_of_le <| one_le_inv (h\u03b4'.trans_le h\u03b4z) hz), sub_mul,\n  inv_mul_cancel (h\u03b4'.trans_le h\u03b4z).ne', one_mul, sub_le_comm]\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nset x' := \u2016x\u2016\u207b\u00b9 \u2022 x\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nset y' := \u2016y\u2016\u207b\u00b9 \u2022 y\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\nhave hxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016 :=\n  calc\n    \u03b5 / 3 = \u03b5 - (\u03b5 / 3 + \u03b5 / 3) := by ring\n    _ \u2264 \u2016x - y\u2016 - (\u2016x' - x\u2016 + \u2016y' - y\u2016) := by\n      gcongr\n      \u00b7 exact (h\u2082 _ hx hx'.le).trans <| min_le_of_right_le <| min_le_left _ _\n      \u00b7 exact (h\u2082 _ hy hy'.le).trans <| min_le_of_right_le <| min_le_left _ _\n    _ \u2264 _ := by\n      have : \u2200 x' y', x - y = x' - y' + (x - x') + (y' - y) := fun _ _ => by abel\n      rw [sub_le_iff_le_add, norm_sub_rev _ x, \u2190 add_assoc, this]\n      exact norm_add\u2083_le _ _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u03b5 / 3 = \u03b5 - (\u03b5 / 3 + \u03b5 / 3)\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u03b5 - (\u03b5 / 3 + \u03b5 / 3) \u2264 \u2016x - y\u2016 - (\u2016x' - x\u2016 + \u2016y' - y\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase hcd.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x' - x\u2016 \u2264 \u03b5 / 3\n[PROOFSTEP]\nexact (h\u2082 _ hx hx'.le).trans <| min_le_of_right_le <| min_le_left _ _\n[GOAL]\ncase hcd.h\u2082\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016y' - y\u2016 \u2264 \u03b5 / 3\n[PROOFSTEP]\nexact (h\u2082 _ hy hy'.le).trans <| min_le_of_right_le <| min_le_left _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\n\u22a2 \u2016x - y\u2016 - (\u2016x' - x\u2016 + \u2016y' - y\u2016) \u2264 \u2016x' - y'\u2016\n[PROOFSTEP]\nhave : \u2200 x' y', x - y = x' - y' + (x - x') + (y' - y) := fun _ _ => by abel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nx\u271d\u00b9 x\u271d : E\n\u22a2 x - y = x\u271d\u00b9 - x\u271d + (x - x\u271d\u00b9) + (x\u271d - y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nx\u271d\u00b9 x\u271d : E\n\u22a2 x - y = x\u271d\u00b9 - x\u271d + (x - x\u271d\u00b9) + (x\u271d - y)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nthis : \u2200 (x' y' : E), x - y = x' - y' + (x - x') + (y' - y)\n\u22a2 \u2016x - y\u2016 - (\u2016x' - x\u2016 + \u2016y' - y\u2016) \u2264 \u2016x' - y'\u2016\n[PROOFSTEP]\nrw [sub_le_iff_le_add, norm_sub_rev _ x, \u2190 add_assoc, this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nthis : \u2200 (x' y' : E), x - y = x' - y' + (x - x') + (y' - y)\n\u22a2 \u2016?x' - ?y' + (x - ?x') + (?y' - y)\u2016 \u2264 \u2016x' - y'\u2016 + \u2016x - x'\u2016 + \u2016y' - y\u2016\ncase x'\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nthis : \u2200 (x' y' : E), x - y = x' - y' + (x - x') + (y' - y)\n\u22a2 E\ncase y'\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nthis : \u2200 (x' y' : E), x - y = x' - y' + (x - x') + (y' - y)\n\u22a2 E\n[PROOFSTEP]\nexact norm_add\u2083_le _ _ _\n[GOAL]\ncase intro.intro.inr.inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 - \u03b4'\n[PROOFSTEP]\ncalc\n  \u2016x + y\u2016 \u2264 \u2016x' + y'\u2016 + \u2016x' - x\u2016 + \u2016y' - y\u2016 :=\n    by\n    have : \u2200 x' y', x + y = x' + y' + (x - x') + (y - y') := fun _ _ => by abel\n    rw [norm_sub_rev, norm_sub_rev y', this]\n    exact norm_add\u2083_le _ _ _\n  _ \u2264 2 - \u03b4 + \u03b4' + \u03b4' := (add_le_add_three (h (h\u2081 _ hx') (h\u2081 _ hy') hxy') (h\u2082 _ hx hx'.le) (h\u2082 _ hy hy'.le))\n  _ \u2264 2 - \u03b4' := by\n    rw [\u2190 le_sub_iff_add_le, \u2190 le_sub_iff_add_le, sub_sub, sub_sub]\n    refine' sub_le_sub_left _ _\n    ring_nf\n    rw [\u2190 mul_div_cancel' \u03b4 three_ne_zero]\n    norm_num\n      -- Porting note: these three extra lines needed to make `exact` work\n    have : 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3 := by linarith\n    rw [this, mul_comm]\n    gcongr\n    exact min_le_of_right_le <| min_le_right _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 \u2016x + y\u2016 \u2264 \u2016x' + y'\u2016 + \u2016x' - x\u2016 + \u2016y' - y\u2016\n[PROOFSTEP]\nhave : \u2200 x' y', x + y = x' + y' + (x - x') + (y - y') := fun _ _ => by abel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nx\u271d\u00b9 x\u271d : E\n\u22a2 x + y = x\u271d\u00b9 + x\u271d + (x - x\u271d\u00b9) + (y - x\u271d)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nx\u271d\u00b9 x\u271d : E\n\u22a2 x + y = x\u271d\u00b9 + x\u271d + (x - x\u271d\u00b9) + (y - x\u271d)\n[PROOFSTEP]\nabel\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : \u2200 (x' y' : E), x + y = x' + y' + (x - x') + (y - y')\n\u22a2 \u2016x + y\u2016 \u2264 \u2016x' + y'\u2016 + \u2016x' - x\u2016 + \u2016y' - y\u2016\n[PROOFSTEP]\nrw [norm_sub_rev, norm_sub_rev y', this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : \u2200 (x' y' : E), x + y = x' + y' + (x - x') + (y - y')\n\u22a2 \u2016?x' + ?y' + (x - ?x') + (y - ?y')\u2016 \u2264 \u2016x' + y'\u2016 + \u2016x - x'\u2016 + \u2016y - y'\u2016\ncase x'\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : \u2200 (x' y' : E), x + y = x' + y' + (x - x') + (y - y')\n\u22a2 E\ncase y'\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : \u2200 (x' y' : E), x + y = x' + y' + (x - x') + (y - y')\n\u22a2 E\n[PROOFSTEP]\nexact norm_add\u2083_le _ _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 2 - \u03b4 + \u03b4' + \u03b4' \u2264 2 - \u03b4'\n[PROOFSTEP]\nrw [\u2190 le_sub_iff_add_le, \u2190 le_sub_iff_add_le, sub_sub, sub_sub]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 2 - \u03b4 \u2264 2 - (\u03b4' + (\u03b4' + \u03b4'))\n[PROOFSTEP]\nrefine' sub_le_sub_left _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 \u03b4' + (\u03b4' + \u03b4') \u2264 \u03b4\n[PROOFSTEP]\nring_nf\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 min (\u2191(Int.ofNat 1) / \u21912) (min (\u03b5 * (\u2191(Int.ofNat 1) / \u21913)) (\u03b4 * (\u2191(Int.ofNat 1) / \u21913))) * 3 \u2264 \u03b4\n[PROOFSTEP]\nrw [\u2190 mul_div_cancel' \u03b4 three_ne_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 min (\u2191(Int.ofNat 1) / \u21912) (min (\u03b5 * (\u2191(Int.ofNat 1) / \u21913)) (3 * (\u03b4 / 3) * (\u2191(Int.ofNat 1) / \u21913))) * 3 \u2264 3 * (\u03b4 / 3)\n[PROOFSTEP]\nnorm_num\n  -- Porting note: these three extra lines needed to make `exact` work\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 min (1 / 2) (min (\u03b5 * (1 / 3)) (3 * (\u03b4 / 3) * (1 / 3))) * 3 \u2264 3 * (\u03b4 / 3)\n[PROOFSTEP]\nhave : 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3 := by linarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\n\u22a2 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3\n[PROOFSTEP]\nlinarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3\n\u22a2 min (1 / 2) (min (\u03b5 * (1 / 3)) (3 * (\u03b4 / 3) * (1 / 3))) * 3 \u2264 3 * (\u03b4 / 3)\n[PROOFSTEP]\nrw [this, mul_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3\n\u22a2 3 * min (1 / 2) (min (\u03b5 * (1 / 3)) (\u03b4 / 3)) \u2264 3 * (\u03b4 / 3)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nh\u03b5' : 0 < \u03b5 / 3\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 / 3 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u03b4' : \u211d := min (1 / 2) (min (\u03b5 / 3) (\u03b4 / 3))\nx : E\nhx : \u2016x\u2016 \u2264 1\ny : E\nhy : \u2016y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nhx' : 1 - \u03b4' < \u2016x\u2016\nhy' : 1 - \u03b4' < \u2016y\u2016\nh\u03b4' : 0 < 1 - \u03b4'\nh\u2081 : \u2200 (z : E), 1 - \u03b4' < \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z\u2016 = 1\nh\u2082 : \u2200 (z : E), \u2016z\u2016 \u2264 1 \u2192 1 - \u03b4' \u2264 \u2016z\u2016 \u2192 \u2016\u2016z\u2016\u207b\u00b9 \u2022 z - z\u2016 \u2264 \u03b4'\nx' : E := \u2016x\u2016\u207b\u00b9 \u2022 x\ny' : E := \u2016y\u2016\u207b\u00b9 \u2022 y\nhxy' : \u03b5 / 3 \u2264 \u2016x' - y'\u2016\nthis : 3 * (\u03b4 / 3) * (1 / 3) = \u03b4 / 3\n\u22a2 min (1 / 2) (min (\u03b5 * (1 / 3)) (\u03b4 / 3)) \u2264 \u03b4 / 3\n[PROOFSTEP]\nexact min_le_of_right_le <| min_le_right _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 r \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 r \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4\n[PROOFSTEP]\nobtain hr | hr := le_or_lt r 0\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : r \u2264 0\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 r \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 r \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4\n[PROOFSTEP]\nexact\n  \u27e81, one_pos, fun x hx y hy h =>\n    (h\u03b5.not_le <| h.trans <| (norm_sub_le _ _).trans <| add_nonpos (hx.trans hr) (hy.trans hr)).elim\u27e9\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 r \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 r \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4\n[PROOFSTEP]\nobtain \u27e8\u03b4, h\u03b4, h\u27e9 := exists_forall_closed_ball_dist_add_le_two_sub E (div_pos h\u03b5 hr)\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 r \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 r \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4\n[PROOFSTEP]\nrefine' \u27e8\u03b4 * r, mul_pos h\u03b4 hr, fun x hx y hy hxy => _\u27e9\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx : \u2016x\u2016 \u2264 r\ny : E\nhy : \u2016y\u2016 \u2264 r\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\nrw [\u2190 div_le_one hr, div_eq_inv_mul, \u2190 norm_smul_of_nonneg (inv_nonneg.2 hr.le)] at hx hy \n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx\u271d : r\u207b\u00b9 * \u2016x\u2016 \u2264 1\nhx : \u2016r\u207b\u00b9 \u2022 x\u2016 \u2264 1\ny : E\nhy\u271d : r\u207b\u00b9 * \u2016y\u2016 \u2264 1\nhy : \u2016r\u207b\u00b9 \u2022 y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\ntry infer_instance\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx\u271d : r\u207b\u00b9 * \u2016x\u2016 \u2264 1\nhx : \u2016r\u207b\u00b9 \u2022 x\u2016 \u2264 1\ny : E\nhy\u271d : r\u207b\u00b9 * \u2016y\u2016 \u2264 1\nhy : \u2016r\u207b\u00b9 \u2022 y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx\u271d : r\u207b\u00b9 * \u2016x\u2016 \u2264 1\nhx : \u2016r\u207b\u00b9 \u2022 x\u2016 \u2264 1\ny : E\nhy\u271d : r\u207b\u00b9 * \u2016y\u2016 \u2264 1\nhy : \u2016r\u207b\u00b9 \u2022 y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\nhave := h hx hy\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx\u271d : r\u207b\u00b9 * \u2016x\u2016 \u2264 1\nhx : \u2016r\u207b\u00b9 \u2022 x\u2016 \u2264 1\ny : E\nhy\u271d : r\u207b\u00b9 * \u2016y\u2016 \u2264 1\nhy : \u2016r\u207b\u00b9 \u2022 y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nthis : \u03b5 / r \u2264 \u2016r\u207b\u00b9 \u2022 x - r\u207b\u00b9 \u2022 y\u2016 \u2192 \u2016r\u207b\u00b9 \u2022 x + r\u207b\u00b9 \u2022 y\u2016 \u2264 2 - \u03b4\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\nsimp_rw [\u2190 smul_add, \u2190 smul_sub, norm_smul_of_nonneg (inv_nonneg.2 hr.le), \u2190 div_eq_inv_mul, div_le_div_right hr,\n  div_le_iff hr, sub_mul] at this \n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : UniformConvexSpace E\n\u03b5 : \u211d\ninst\u271d : NormedSpace \u211d E\nh\u03b5 : 0 < \u03b5\nr : \u211d\nhr : 0 < r\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nh : \u2200 \u2983x : E\u2984, \u2016x\u2016 \u2264 1 \u2192 \u2200 \u2983y : E\u2984, \u2016y\u2016 \u2264 1 \u2192 \u03b5 / r \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\nx : E\nhx\u271d : r\u207b\u00b9 * \u2016x\u2016 \u2264 1\nhx : \u2016r\u207b\u00b9 \u2022 x\u2016 \u2264 1\ny : E\nhy\u271d : r\u207b\u00b9 * \u2016y\u2016 \u2264 1\nhy : \u2016r\u207b\u00b9 \u2022 y\u2016 \u2264 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\nthis : \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n\u22a2 \u2016x + y\u2016 \u2264 2 * r - \u03b4 * r\n[PROOFSTEP]\nexact this hxy\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Uniform", "llama_tokens": 24574, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321796478255, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.5304423314956593}}
{"text": "[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (kernel.\u03b9 g \u226b cokernel.\u03c0 f) \u226b cokernel.desc f g w = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 kernel.lift g f w \u226b\n      kernel.lift (cokernel.desc f g w) (kernel.\u03b9 g \u226b cokernel.\u03c0 f)\n        (_ : (kernel.\u03b9 g \u226b cokernel.\u03c0 f) \u226b cokernel.desc f g w = 0) =\n    0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (kernel.lift g f w \u226b\n        kernel.lift (cokernel.desc f g w) (kernel.\u03b9 g \u226b cokernel.\u03c0 f)\n          (_ : (kernel.\u03b9 g \u226b cokernel.\u03c0 f) \u226b cokernel.desc f g w = 0)) \u226b\n      equalizer.\u03b9 (cokernel.desc f g w) 0 =\n    0 \u226b equalizer.\u03b9 (cokernel.desc f g w) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 Mono (homologyCToK f g w)\n[PROOFSTEP]\napply Pseudoelement.mono_of_zero_of_map_zero\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 \u2200 (a : Pseudoelement (homologyC f g w)), Pseudoelement.pseudoApply (homologyCToK f g w) a = 0 \u2192 a = 0\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyC f g w)\nha : Pseudoelement.pseudoApply (homologyCToK f g w) a = 0\n\u22a2 a = 0\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := Pseudoelement.pseudo_surjective_of_epi (cokernel.\u03c0 (kernel.lift g f w)) a\n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (homologyCToK f g w) (Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a) = 0\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a = 0\n[PROOFSTEP]\napply_fun kernel.\u03b9 (cokernel.desc f g w) at ha \n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha :\n  Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w))\n      (Pseudoelement.pseudoApply (homologyCToK f g w) (Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a)) =\n    Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) 0\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a = 0\n[PROOFSTEP]\nsimp only [\u2190 Pseudoelement.comp_apply, cokernel.\u03c0_desc, kernel.lift_\u03b9, Pseudoelement.apply_zero] at ha \n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (kernel.\u03b9 g \u226b cokernel.\u03c0 f) a = 0\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a = 0\n[PROOFSTEP]\nsimp only [Pseudoelement.comp_apply] at ha \n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a = 0\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 : \u2203 b, f b = _ := (Pseudoelement.pseudo_exact_of_exact (exact_cokernel f)).2 _ ha\n[GOAL]\ncase a.intro.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) a = 0\n[PROOFSTEP]\nrsuffices \u27e8c, rfl\u27e9 : \u2203 c, kernel.lift g f w c = a\n[GOAL]\ncase a.intro.intro.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nb c : Pseudoelement X\nha :\n  Pseudoelement.pseudoApply (cokernel.\u03c0 f)\n      (Pseudoelement.pseudoApply (kernel.\u03b9 g) (Pseudoelement.pseudoApply (kernel.lift g f w) c)) =\n    0\nhb :\n  Pseudoelement.pseudoApply f b =\n    Pseudoelement.pseudoApply (kernel.\u03b9 g) (Pseudoelement.pseudoApply (kernel.lift g f w) c)\n\u22a2 Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) (Pseudoelement.pseudoApply (kernel.lift g f w) c) = 0\n[PROOFSTEP]\nsimp [\u2190 Pseudoelement.comp_apply]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 \u2203 c, Pseudoelement.pseudoApply (kernel.lift g f w) c = a\n[PROOFSTEP]\nuse b\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Pseudoelement.pseudoApply (kernel.lift g f w) b = a\n[PROOFSTEP]\napply_fun kernel.\u03b9 g\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Pseudoelement.pseudoApply (kernel.\u03b9 g) (Pseudoelement.pseudoApply (kernel.lift g f w) b) =\n    Pseudoelement.pseudoApply (kernel.\u03b9 g) a\ncase h.inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Function.Injective (Pseudoelement.pseudoApply (kernel.\u03b9 g))\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Function.Injective (Pseudoelement.pseudoApply (kernel.\u03b9 g))\n[PROOFSTEP]\napply Pseudoelement.pseudo_injective_of_mono\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (kernel g)\nha : Pseudoelement.pseudoApply (cokernel.\u03c0 f) (Pseudoelement.pseudoApply (kernel.\u03b9 g) a) = 0\nb : Pseudoelement X\nhb : Pseudoelement.pseudoApply f b = Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n\u22a2 Pseudoelement.pseudoApply (kernel.\u03b9 g) (Pseudoelement.pseudoApply (kernel.lift g f w) b) =\n    Pseudoelement.pseudoApply (kernel.\u03b9 g) a\n[PROOFSTEP]\nsimpa [\u2190 Pseudoelement.comp_apply]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 Epi (homologyCToK f g w)\n[PROOFSTEP]\napply Pseudoelement.epi_of_pseudo_surjective\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 Function.Surjective (Pseudoelement.pseudoApply (homologyCToK f g w))\n[PROOFSTEP]\nintro a\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\nlet b := kernel.\u03b9 (cokernel.desc f g w) a\n[GOAL]\ncase a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 : \u2203 c, cokernel.\u03c0 f c = b\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\n\u22a2 \u2203 c, Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\napply Pseudoelement.pseudo_surjective_of_epi (cokernel.\u03c0 f)\n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\nhave : g c = 0 :=\n  by\n  rw [show g = cokernel.\u03c0 f \u226b cokernel.desc f g w by simp, Pseudoelement.comp_apply, hc]\n  simp [\u2190 Pseudoelement.comp_apply]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\n\u22a2 Pseudoelement.pseudoApply g c = 0\n[PROOFSTEP]\nrw [show g = cokernel.\u03c0 f \u226b cokernel.desc f g w by simp, Pseudoelement.comp_apply, hc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\n\u22a2 g = cokernel.\u03c0 f \u226b cokernel.desc f g w\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\n\u22a2 Pseudoelement.pseudoApply (cokernel.desc f g w) b = 0\n[PROOFSTEP]\nsimp [\u2190 Pseudoelement.comp_apply]\n[GOAL]\ncase a.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\nobtain \u27e8d, hd\u27e9 : \u2203 d, kernel.\u03b9 g d = c := by apply (Pseudoelement.pseudo_exact_of_exact exact_kernel_\u03b9).2 _ this\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\n\u22a2 \u2203 d, Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n[PROOFSTEP]\napply (Pseudoelement.pseudo_exact_of_exact exact_kernel_\u03b9).2 _ this\n[GOAL]\ncase a.intro.intro\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 \u2203 a_1, Pseudoelement.pseudoApply (homologyCToK f g w) a_1 = a\n[PROOFSTEP]\nuse cokernel.\u03c0 (kernel.lift g f w) d\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Pseudoelement.pseudoApply (homologyCToK f g w) (Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) d) = a\n[PROOFSTEP]\napply_fun kernel.\u03b9 (cokernel.desc f g w)\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w))\n      (Pseudoelement.pseudoApply (homologyCToK f g w) (Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) d)) =\n    Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\ncase h.inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Function.Injective (Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)))\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Function.Injective (Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)))\n[PROOFSTEP]\napply Pseudoelement.pseudo_injective_of_mono\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w))\n      (Pseudoelement.pseudoApply (homologyCToK f g w) (Pseudoelement.pseudoApply (cokernel.\u03c0 (kernel.lift g f w)) d)) =\n    Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\n[PROOFSTEP]\nsimp only [\u2190 Pseudoelement.comp_apply, cokernel.\u03c0_desc, kernel.lift_\u03b9]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\na : Pseudoelement (homologyK f g w)\nb : Pseudoelement (cokernel f) := Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\nc : Pseudoelement Y\nhc : Pseudoelement.pseudoApply (cokernel.\u03c0 f) c = b\nthis : Pseudoelement.pseudoApply g c = 0\nd : Pseudoelement (kernel g)\nhd : Pseudoelement.pseudoApply (kernel.\u03b9 g) d = c\n\u22a2 Pseudoelement.pseudoApply (kernel.\u03b9 g \u226b cokernel.\u03c0 f) d = Pseudoelement.pseudoApply (kernel.\u03b9 (cokernel.desc f g w)) a\n[PROOFSTEP]\nsimp only [Pseudoelement.comp_apply, hd, hc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\ne : kernel g \u27f6 W\nhe : kernel.lift g f w \u226b e = 0\n\u22a2 \u03c0' f g w \u226b desc' f g w e he = e\n[PROOFSTEP]\ndsimp [\u03c0', desc']\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\ne : kernel g \u27f6 W\nhe : kernel.lift g f w \u226b e = 0\n\u22a2 (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b\n      (homologyIsoCokernelLift f g w).hom \u226b cokernel.desc (kernel.lift g f w) e he =\n    e\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\ne : W \u27f6 cokernel f\nhe : e \u226b cokernel.desc f g w = 0\n\u22a2 lift f g w e he \u226b \u03b9 f g w = e\n[PROOFSTEP]\ndsimp [\u03b9, lift]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\ne : W \u27f6 cokernel f\nhe : e \u226b cokernel.desc f g w = 0\n\u22a2 (kernel.lift (cokernel.desc f g w) e he \u226b (homologyIsoKernelDesc f g w).inv) \u226b\n      (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    e\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 kernel.lift g f w \u226b \u03c0' f g w = 0\n[PROOFSTEP]\ndsimp [\u03c0']\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 kernel.lift g f w \u226b cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 \u03b9 f g w \u226b cokernel.desc f g w = 0\n[PROOFSTEP]\ndsimp [\u03b9]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 ((homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)) \u226b cokernel.desc f g w = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh : \u03c0' f g w \u226b a = \u03c0' f g w \u226b b\n\u22a2 a = b\n[PROOFSTEP]\ndsimp [\u03c0'] at h \n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\n\u22a2 a = b\n[PROOFSTEP]\napply_fun fun e => (homologyIsoCokernelLift f g w).inv \u226b e\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\n\u22a2 (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) a = (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) b\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\n\u22a2 Function.Injective fun e => (homologyIsoCokernelLift f g w).inv \u226b e\n[PROOFSTEP]\nswap\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\n\u22a2 Function.Injective fun e => (homologyIsoCokernelLift f g w).inv \u226b e\n[PROOFSTEP]\nintro i j hh\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\ni j : homology f g w \u27f6 W\nhh : (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) i = (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) j\n\u22a2 i = j\n[PROOFSTEP]\napply_fun fun e => (homologyIsoCokernelLift f g w).hom \u226b e at hh \n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\ni j : homology f g w \u27f6 W\nhh :\n  (homologyIsoCokernelLift f g w).hom \u226b (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) i =\n    (homologyIsoCokernelLift f g w).hom \u226b (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) j\n\u22a2 i = j\n[PROOFSTEP]\nsimpa using hh\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b a =\n    (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b b\n\u22a2 (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) a = (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) b\n[PROOFSTEP]\nsimp only [Category.assoc] at h \n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : homology f g w \u27f6 W\nh :\n  cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv \u226b a =\n    cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv \u226b b\n\u22a2 (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) a = (fun e => (homologyIsoCokernelLift f g w).inv \u226b e) b\n[PROOFSTEP]\nexact coequalizer.hom_ext h\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh : a \u226b \u03b9 f g w = b \u226b \u03b9 f g w\n\u22a2 a = b\n[PROOFSTEP]\ndsimp [\u03b9] at h \n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 a = b\n[PROOFSTEP]\napply_fun fun e => e \u226b (homologyIsoKernelDesc f g w).hom\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) a = (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) b\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 Function.Injective fun e => e \u226b (homologyIsoKernelDesc f g w).hom\n[PROOFSTEP]\nswap\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 Function.Injective fun e => e \u226b (homologyIsoKernelDesc f g w).hom\n[PROOFSTEP]\nintro i j hh\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\ni j : W \u27f6 homology f g w\nhh : (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) i = (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) j\n\u22a2 i = j\n[PROOFSTEP]\napply_fun fun e => e \u226b (homologyIsoKernelDesc f g w).inv at hh \n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\ni j : W \u27f6 homology f g w\nhh :\n  (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) i \u226b (homologyIsoKernelDesc f g w).inv =\n    (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) j \u226b (homologyIsoKernelDesc f g w).inv\n\u22a2 i = j\n[PROOFSTEP]\nsimpa using hh\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  a \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    b \u226b (homologyIsoKernelDesc f g w).hom \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) a = (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) b\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc] at h \n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nW : A\na b : W \u27f6 homology f g w\nh :\n  (a \u226b (homologyIsoKernelDesc f g w).hom) \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    (b \u226b (homologyIsoKernelDesc f g w).hom) \u226b kernel.\u03b9 (cokernel.desc f g w)\n\u22a2 (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) a = (fun e => e \u226b (homologyIsoKernelDesc f g w).hom) b\n[PROOFSTEP]\nexact equalizer.hom_ext h\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 \u03c0' f g w \u226b \u03b9 f g w = kernel.\u03b9 g \u226b cokernel.\u03c0 f\n[PROOFSTEP]\ndsimp [\u03c0', \u03b9, homologyIsoKernelDesc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (cokernel.\u03c0 (kernel.lift g f w) \u226b (homologyIsoCokernelLift f g w).inv) \u226b\n      ((homologyIsoCokernelLift f g w).hom \u226b Abelian.homologyCToK f g w) \u226b kernel.\u03b9 (cokernel.desc f g w) =\n    kernel.\u03b9 g \u226b cokernel.\u03c0 f\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (kernelSubobjectIso g).hom \u226b \u03c0' f g w = \u03c0 f g w\n[PROOFSTEP]\ndsimp [\u03c0', homologyIsoCokernelLift]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (kernelSubobjectIso g).hom \u226b\n      cokernel.\u03c0 (kernel.lift g f w) \u226b\n        ((cokernelIsoOfEq (_ : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w)).inv \u226b\n            cokernel.desc (factorThruImage f \u226b imageToKernel' f g w) (cokernel.\u03c0 (imageToKernel' f g w))\n              (_ : (factorThruImage f \u226b imageToKernel' f g w) \u226b cokernel.\u03c0 (imageToKernel' f g w) = 0)) \u226b\n          (homologyIsoCokernelImageToKernel' f g w).inv =\n    \u03c0 f g w\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 ((((kernelSubobjectIso g).hom \u226b cokernel.\u03c0 (kernel.lift g f w)) \u226b\n          (cokernelIsoOfEq (_ : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w)).inv) \u226b\n        cokernel.desc (factorThruImage f \u226b imageToKernel' f g w) (cokernel.\u03c0 (imageToKernel' f g w))\n          (_ : (factorThruImage f \u226b imageToKernel' f g w) \u226b cokernel.\u03c0 (imageToKernel' f g w) = 0)) \u226b\n      (homologyIsoCokernelImageToKernel' f g w).inv =\n    \u03c0 f g w\n[PROOFSTEP]\nrw [Iso.comp_inv_eq]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (((kernelSubobjectIso g).hom \u226b cokernel.\u03c0 (kernel.lift g f w)) \u226b\n        (cokernelIsoOfEq (_ : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w)).inv) \u226b\n      cokernel.desc (factorThruImage f \u226b imageToKernel' f g w) (cokernel.\u03c0 (imageToKernel' f g w))\n        (_ : (factorThruImage f \u226b imageToKernel' f g w) \u226b cokernel.\u03c0 (imageToKernel' f g w) = 0) =\n    \u03c0 f g w \u226b (homologyIsoCokernelImageToKernel' f g w).hom\n[PROOFSTEP]\ndsimp [\u03c0, homologyIsoCokernelImageToKernel']\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (((kernelSubobjectIso g).hom \u226b cokernel.\u03c0 (kernel.lift g f w)) \u226b\n        (cokernelIsoOfEq (_ : factorThruImage f \u226b imageToKernel' f g w = kernel.lift g f w)).inv) \u226b\n      cokernel.desc (factorThruImage f \u226b imageToKernel' f g w) (cokernel.\u03c0 (imageToKernel' f g w))\n        (_ : (factorThruImage f \u226b imageToKernel' f g w) \u226b cokernel.\u03c0 (imageToKernel' f g w) = 0) =\n    cokernel.\u03c0 (imageToKernel f g w) \u226b\n      cokernel.map (imageToKernel f g w) (imageToKernel' f g w) (imageSubobjectIso f).hom (kernelSubobjectIso g).hom\n        (_ : imageToKernel f g w \u226b (kernelSubobjectIso g).hom = (imageSubobjectIso f).hom \u226b imageToKernel' f g w)\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 g \u226b \u03b2.right = \u03b1.right \u226b g'\n[PROOFSTEP]\nsimp [h, \u03b2.w.symm]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b map w w' \u03b1 \u03b2 h = kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\napply_fun fun e => (kernelSubobjectIso _).hom \u226b e\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (fun e => (kernelSubobjectIso g).hom \u226b e) (\u03c0' f g w \u226b map w w' \u03b1 \u03b2 h) =\n    (fun e => (kernelSubobjectIso g).hom \u226b e)\n      (kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w')\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 Function.Injective fun e => (kernelSubobjectIso g).hom \u226b e\n[PROOFSTEP]\nswap\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 Function.Injective fun e => (kernelSubobjectIso g).hom \u226b e\n[PROOFSTEP]\nintro i j hh\n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\ni j : kernel g \u27f6 homology f' g' w'\nhh : (fun e => (kernelSubobjectIso g).hom \u226b e) i = (fun e => (kernelSubobjectIso g).hom \u226b e) j\n\u22a2 i = j\n[PROOFSTEP]\napply_fun fun e => (kernelSubobjectIso _).inv \u226b e at hh \n[GOAL]\ncase inj\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\ni j : kernel g \u27f6 homology f' g' w'\nhh :\n  (kernelSubobjectIso g).inv \u226b (fun e => (kernelSubobjectIso g).hom \u226b e) i =\n    (kernelSubobjectIso g).inv \u226b (fun e => (kernelSubobjectIso g).hom \u226b e) j\n\u22a2 i = j\n[PROOFSTEP]\nsimpa using hh\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (fun e => (kernelSubobjectIso g).hom \u226b e) (\u03c0' f g w \u226b map w w' \u03b1 \u03b2 h) =\n    (fun e => (kernelSubobjectIso g).hom \u226b e)\n      (kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w')\n[PROOFSTEP]\ndsimp [map]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernelSubobjectIso g).hom \u226b\n      \u03c0' f g w \u226b\n        cokernel.desc (imageToKernel f g w) (kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w'))\n          (_ : imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w') = 0) =\n    (kernelSubobjectIso g).hom \u226b kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nsimp only [\u03c0'_eq_\u03c0_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0 f g w \u226b\n      cokernel.desc (imageToKernel f g w) (kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w'))\n        (_ : imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w') = 0) =\n    (kernelSubobjectIso g).hom \u226b kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\ndsimp [\u03c0]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 cokernel.\u03c0 (imageToKernel f g w) \u226b\n      cokernel.desc (imageToKernel f g w) (kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w'))\n        (_ : imageToKernel f g w \u226b kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w') = 0) =\n    (kernelSubobjectIso g).hom \u226b kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nsimp only [cokernel.\u03c0_desc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernelSubobjectMap \u03b2 \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    (kernelSubobjectIso g).hom \u226b kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nrw [\u2190 Iso.inv_comp_eq, \u2190 Category.assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 ((kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2) \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nhave :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map _ _ \u03b2.left \u03b2.right \u03b2.w.symm \u226b (kernelSubobjectIso _).inv :=\n  by\n  rw [Iso.inv_comp_eq, \u2190 Category.assoc, Iso.eq_comp_inv]\n  ext\n  dsimp\n  simp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, \u2190 Category.assoc, Iso.eq_comp_inv]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernelSubobjectMap \u03b2 \u226b (kernelSubobjectIso (Arrow.mk g').hom).hom =\n    (kernelSubobjectIso g).hom \u226b\n      kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernelSubobjectMap \u03b2 \u226b (kernelSubobjectIso (Arrow.mk g').hom).hom) \u226b equalizer.\u03b9 (Arrow.mk g').hom 0 =\n    ((kernelSubobjectIso g).hom \u226b\n        kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n          (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom)) \u226b\n      equalizer.\u03b9 (Arrow.mk g').hom 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernelSubobjectMap \u03b2 \u226b (kernelSubobjectIso g').hom) \u226b kernel.\u03b9 g' =\n    ((kernelSubobjectIso g).hom \u226b kernel.map g g' \u03b2.left \u03b2.right (_ : g \u226b \u03b2.right = \u03b2.left \u226b g')) \u226b kernel.\u03b9 g'\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 ((kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2) \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nrw [this]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 (kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n          (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n        (kernelSubobjectIso (Arrow.mk g').hom).inv) \u226b\n      cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\nsimp only [Category.assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w'\n[PROOFSTEP]\ndsimp [\u03c0', homologyIsoCokernelLift]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 kernel.map g g' \u03b2.left \u03b2.right (_ : g \u226b \u03b2.right = \u03b2.left \u226b g') \u226b\n      (kernelSubobjectIso g').inv \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n      cokernel.\u03c0 (kernel.lift g' f' w') \u226b\n        ((cokernelIsoOfEq (_ : factorThruImage f' \u226b imageToKernel' f' g' w' = kernel.lift g' f' w')).inv \u226b\n            cokernel.desc (factorThruImage f' \u226b imageToKernel' f' g' w') (cokernel.\u03c0 (imageToKernel' f' g' w'))\n              (_ : (factorThruImage f' \u226b imageToKernel' f' g' w') \u226b cokernel.\u03c0 (imageToKernel' f' g' w') = 0)) \u226b\n          (homologyIsoCokernelImageToKernel' f' g' w').inv\n[PROOFSTEP]\nsimp only [cokernelIsoOfEq_inv_comp_desc, cokernel.\u03c0_desc_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 kernel.map g g' \u03b2.left \u03b2.right (_ : g \u226b \u03b2.right = \u03b2.left \u226b g') \u226b\n      (kernelSubobjectIso g').inv \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n      cokernel.\u03c0 (imageToKernel' f' g' w') \u226b (homologyIsoCokernelImageToKernel' f' g' w').inv\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 kernel.map g g' \u03b2.left \u03b2.right (_ : g \u226b \u03b2.right = \u03b2.left \u226b g') =\n    kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g')\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_p\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 \u03b2.left = \u03b1.right\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase e_a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 (kernelSubobjectIso g').inv \u226b cokernel.\u03c0 (imageToKernel f' g' w') =\n    cokernel.\u03c0 (imageToKernel' f' g' w') \u226b (homologyIsoCokernelImageToKernel' f' g' w').inv\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, \u2190 Category.assoc, Iso.eq_comp_inv]\n[GOAL]\ncase e_a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 cokernel.\u03c0 (imageToKernel f' g' w') \u226b (homologyIsoCokernelImageToKernel' f' g' w').hom =\n    (kernelSubobjectIso g').hom \u226b cokernel.\u03c0 (imageToKernel' f' g' w')\n[PROOFSTEP]\ndsimp [homologyIsoCokernelImageToKernel']\n[GOAL]\ncase e_a\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\nthis :\n  (kernelSubobjectIso g).inv \u226b kernelSubobjectMap \u03b2 =\n    kernel.map (Arrow.mk g).hom (Arrow.mk g').hom \u03b2.left \u03b2.right\n        (_ : (Arrow.mk g).hom \u226b (\ud835\udfed A).map \u03b2.right = (\ud835\udfed A).map \u03b2.left \u226b (Arrow.mk g').hom) \u226b\n      (kernelSubobjectIso (Arrow.mk g').hom).inv\n\u22a2 cokernel.\u03c0 (imageToKernel f' g' w') \u226b\n      cokernel.map (imageToKernel f' g' w') (imageToKernel' f' g' w') (imageSubobjectIso f').hom\n        (kernelSubobjectIso g').hom\n        (_ :\n          imageToKernel f' g' w' \u226b (kernelSubobjectIso g').hom = (imageSubobjectIso f').hom \u226b imageToKernel' f' g' w') =\n    (kernelSubobjectIso g').hom \u226b cokernel.\u03c0 (imageToKernel' f' g' w')\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.lift g f w \u226b\n      lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n    0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.lift g f w \u226b\n        lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n          (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0)) \u226b\n      \u03b9 f' g' w' =\n    0 \u226b \u03b9 f' g' w'\n[PROOFSTEP]\nsimp only [\u2190 h, Category.assoc, zero_comp, lift_\u03b9, kernel.lift_\u03b9_assoc]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nerw [\u2190 reassoc_of% \u03b1.w]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h =\n    desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0)\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b map w w' \u03b1 \u03b2 h =\n    \u03c0' f g w \u226b\n      desc' f g w\n        (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n          (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n        (_ :\n          kernel.lift g f w \u226b\n              lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n            0)\n[PROOFSTEP]\nsimp only [\u03c0'_map, \u03c0'_desc']\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b \u03c0' f' g' w' =\n    lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n      (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0)\n[PROOFSTEP]\ndsimp [\u03c0', lift]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n      cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homologyIsoCokernelLift f' g' w').inv =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) \u226b\n      (homologyIsoKernelDesc f' g' w').inv\n[PROOFSTEP]\nrw [Iso.eq_comp_inv]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n        cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homologyIsoCokernelLift f' g' w').inv) \u226b\n      (homologyIsoKernelDesc f' g' w').hom =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n      (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0)\n[PROOFSTEP]\ndsimp [homologyIsoKernelDesc]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n        cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homologyIsoCokernelLift f' g' w').inv) \u226b\n      (homologyIsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w' =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n      (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 ((kernel.map g g' \u03b1.right \u03b2.right (_ : g \u226b \u03b2.right = \u03b1.right \u226b g') \u226b\n          cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homologyIsoCokernelLift f' g' w').inv) \u226b\n        (homologyIsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w') \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0 =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp only [kernel.lift_\u03b9_assoc, \u2190 h]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nerw [\u2190 reassoc_of% \u03b1.w]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n      cokernel.desc f' g' w' =\n    0\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n          (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n        cokernel.desc f' g' w' =\n    \u03c0' f g w \u226b 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h =\n    lift f' g' w'\n      (desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0))\n      (_ :\n        desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n              (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n            cokernel.desc f' g' w' =\n          0)\n[PROOFSTEP]\nrw [map_eq_desc'_lift_left]\n  -- Porting note: once was known as ext\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0) =\n    lift f' g' w'\n      (desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0))\n      (_ :\n        desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n              (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n            cokernel.desc f' g' w' =\n          0)\n[PROOFSTEP]\napply homology.hom_to_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w\n        (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n          (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n        (_ :\n          kernel.lift g f w \u226b\n              lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n            0) \u226b\n      \u03b9 f' g' w' =\n    lift f' g' w'\n        (desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n          (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0))\n        (_ :\n          desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n              cokernel.desc f' g' w' =\n            0) \u226b\n      \u03b9 f' g' w'\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w\n          (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n            (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n          (_ :\n            kernel.lift g f w \u226b\n                lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                  (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n              0) \u226b\n        \u03b9 f' g' w' =\n    \u03c0' f g w \u226b\n      lift f' g' w'\n          (desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n            (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0))\n          (_ :\n            desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                  (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) \u226b\n                cokernel.desc f' g' w' =\n              0) \u226b\n        \u03b9 f' g' w'\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.lift g f w \u226b\n      lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n    0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.lift g f w \u226b\n        lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n          (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0)) \u226b\n      \u03b9 f' g' w' =\n    0 \u226b \u03b9 f' g' w'\n[PROOFSTEP]\nsimp only [Category.assoc, zero_comp, lift_\u03b9, kernel.lift_\u03b9_assoc]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nerw [\u2190 reassoc_of% \u03b1.w]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h =\n    desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0)\n[PROOFSTEP]\nrw [map_eq_desc'_lift_left]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0) =\n    desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (\u03c0' f g w \u226b\n        desc' f g w\n          (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n            (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n          (_ :\n            kernel.lift g f w \u226b\n                lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n                  (_ : (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n              0)) \u226b\n      \u03b9 f' g' w' =\n    (\u03c0' f g w \u226b\n        desc' f g w\n          (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n            (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n          (_ :\n            kernel.lift g f w \u226b\n                lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n                  (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n              0)) \u226b\n      \u03b9 f' g' w'\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp only [kernel.lift_\u03b9_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nerw [\u2190 reassoc_of% \u03b1.w]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n      cokernel.desc f' g' w' =\n    0\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n          (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n        cokernel.desc f' g' w' =\n    \u03c0' f g w \u226b 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h =\n    lift f' g' w'\n      (desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0))\n      (_ :\n        desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n              (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n            cokernel.desc f' g' w' =\n          0)\n[PROOFSTEP]\nrw [map_eq_desc'_lift_right]\n  -- Porting note: once was known as ext\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w\n      (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n      (_ :\n        kernel.lift g f w \u226b\n            lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n              (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n          0) =\n    lift f' g' w'\n      (desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0))\n      (_ :\n        desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n              (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n            cokernel.desc f' g' w' =\n          0)\n[PROOFSTEP]\napply homology.hom_to_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w\n        (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n          (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n        (_ :\n          kernel.lift g f w \u226b\n              lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n                (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n            0) \u226b\n      \u03b9 f' g' w' =\n    lift f' g' w'\n        (desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n          (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0))\n        (_ :\n          desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n                (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n              cokernel.desc f' g' w' =\n            0) \u226b\n      \u03b9 f' g' w'\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w\n          (lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n            (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0))\n          (_ :\n            kernel.lift g f w \u226b\n                lift f' g' w' (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n                  (_ : (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0) =\n              0) \u226b\n        \u03b9 f' g' w' =\n    \u03c0' f g w \u226b\n      lift f' g' w'\n          (desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n            (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0))\n          (_ :\n            desc' f g w (kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f')\n                  (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0) \u226b\n                cokernel.desc f' g' w' =\n              0) \u226b\n        \u03b9 f' g' w'\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b2.left = \u03b1.left \u226b f'\n[PROOFSTEP]\nsimp [h, \u03b2.w.symm]\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h \u226b \u03b9 f' g' w' = \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')\n[PROOFSTEP]\nrw [map_eq_lift_desc'_left, lift_\u03b9]\n  -- Porting note: once was known as ext\n[GOAL]\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')\n[PROOFSTEP]\napply homology.hom_from_ext\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f' = 0) =\n    \u03c0' f g w \u226b \u03b9 f g w \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b\n      desc' f g w ((kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f')\n        (_ : kernel.lift g f w \u226b (kernel.\u03b9 g \u226b \u03b2.left) \u226b cokernel.\u03c0 f' = 0) =\n    (\u03c0' f g w \u226b \u03b9 f g w) \u226b cokernel.map f f' \u03b1.left \u03b2.left (_ : f \u226b \u03b2.left = \u03b1.left \u226b f')\n[PROOFSTEP]\nrw [\u03c0'_\u03b9, \u03c0'_desc', Category.assoc, Category.assoc, cokernel.\u03c0_desc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.196788, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\nC : HomologicalComplex A c\nj : \u03b9\n\u22a2 F.map\n        (imageToKernel (HomologicalComplex.dTo C j) (HomologicalComplex.dFrom C j)\n          (_ : HomologicalComplex.dTo C j \u226b HomologicalComplex.dFrom C j = 0)) \u226b\n      (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom C j)) \u226a\u226b\n          PreservesKernel.iso F (HomologicalComplex.dFrom C j) \u226a\u226b\n            (kernelSubobjectIso (F.map (HomologicalComplex.dFrom C j))).symm).hom =\n    (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo C j)) \u226a\u226b\n          (PreservesImage.iso F (HomologicalComplex.dTo C j)).symm \u226a\u226b\n            (imageSubobjectIso (F.map (HomologicalComplex.dTo C j))).symm).hom \u226b\n      imageToKernel (HomologicalComplex.dTo ((mapHomologicalComplex F c).obj C) j)\n        (HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj C) j)\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj C) j \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj C) j =\n            0)\n[PROOFSTEP]\ndsimp\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.196788, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\nC : HomologicalComplex A c\nj : \u03b9\n\u22a2 F.map\n        (imageToKernel (HomologicalComplex.dTo C j) (HomologicalComplex.dFrom C j)\n          (_ : HomologicalComplex.dTo C j \u226b HomologicalComplex.dFrom C j = 0)) \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.dFrom C j)).hom \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom C j)).hom \u226b\n          (kernelSubobjectIso (F.map (HomologicalComplex.dFrom C j))).inv =\n    (F.map (imageSubobjectIso (HomologicalComplex.dTo C j)).hom \u226b\n        IsImage.lift\n            (StrongEpiMonoFactorisation.toMonoIsImage\n              (StrongEpiMonoFactorisation.mk\n                (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo C j)))\n                  (F.map (image.\u03b9 (HomologicalComplex.dTo C j)))\n                  (F.map (factorThruImage (HomologicalComplex.dTo C j))))))\n            (Image.monoFactorisation (F.map (HomologicalComplex.dTo C j))) \u226b\n          (imageSubobjectIso (F.map (HomologicalComplex.dTo C j))).inv) \u226b\n      imageToKernel (F.map (HomologicalComplex.d C (ComplexShape.prev c j) j))\n        (F.map (HomologicalComplex.d C j (ComplexShape.next c j)))\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj C) j \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj C) j =\n            0)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.196788, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\nC : HomologicalComplex A c\nj : \u03b9\n\u22a2 (F.map\n          (imageToKernel (HomologicalComplex.dTo C j) (HomologicalComplex.dFrom C j)\n            (_ : HomologicalComplex.dTo C j \u226b HomologicalComplex.dFrom C j = 0)) \u226b\n        F.map (kernelSubobjectIso (HomologicalComplex.dFrom C j)).hom \u226b\n          (PreservesKernel.iso F (HomologicalComplex.dFrom C j)).hom \u226b\n            (kernelSubobjectIso (F.map (HomologicalComplex.dFrom C j))).inv) \u226b\n      Subobject.arrow (kernelSubobject (F.map (HomologicalComplex.d C j (ComplexShape.next c j)))) =\n    ((F.map (imageSubobjectIso (HomologicalComplex.dTo C j)).hom \u226b\n          IsImage.lift\n              (StrongEpiMonoFactorisation.toMonoIsImage\n                (StrongEpiMonoFactorisation.mk\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo C j)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo C j)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo C j))))))\n              (Image.monoFactorisation (F.map (HomologicalComplex.dTo C j))) \u226b\n            (imageSubobjectIso (F.map (HomologicalComplex.dTo C j))).inv) \u226b\n        imageToKernel (F.map (HomologicalComplex.d C (ComplexShape.prev c j) j))\n          (F.map (HomologicalComplex.d C j (ComplexShape.next c j)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj C) j \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj C) j =\n              0)) \u226b\n      Subobject.arrow (kernelSubobject (F.map (HomologicalComplex.d C j (ComplexShape.next c j))))\n[PROOFSTEP]\nsimp only [Category.assoc, imageToKernel_arrow]\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.196788, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\nC : HomologicalComplex A c\nj : \u03b9\n\u22a2 F.map\n        (imageToKernel (HomologicalComplex.dTo C j) (HomologicalComplex.dFrom C j)\n          (_ : HomologicalComplex.dTo C j \u226b HomologicalComplex.dFrom C j = 0)) \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.dFrom C j)).hom \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom C j)).hom \u226b\n          (kernelSubobjectIso (F.map (HomologicalComplex.dFrom C j))).inv \u226b\n            Subobject.arrow (kernelSubobject (F.map (HomologicalComplex.d C j (ComplexShape.next c j)))) =\n    F.map (imageSubobjectIso (HomologicalComplex.dTo C j)).hom \u226b\n      IsImage.lift\n          (StrongEpiMonoFactorisation.toMonoIsImage\n            (StrongEpiMonoFactorisation.mk\n              (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo C j)))\n                (F.map (image.\u03b9 (HomologicalComplex.dTo C j))) (F.map (factorThruImage (HomologicalComplex.dTo C j))))))\n          (Image.monoFactorisation (F.map (HomologicalComplex.dTo C j))) \u226b\n        (imageSubobjectIso (F.map (HomologicalComplex.dTo C j))).inv \u226b\n          Subobject.arrow (imageSubobject (F.map (HomologicalComplex.d C (ComplexShape.prev c j) j)))\n[PROOFSTEP]\nerw [kernelSubobject_arrow', imageSubobject_arrow']\n[GOAL]\ncase h\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.196788, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\nC : HomologicalComplex A c\nj : \u03b9\n\u22a2 F.map\n        (imageToKernel (HomologicalComplex.dTo C j) (HomologicalComplex.dFrom C j)\n          (_ : HomologicalComplex.dTo C j \u226b HomologicalComplex.dFrom C j = 0)) \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.dFrom C j)).hom \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom C j)).hom \u226b kernel.\u03b9 (F.map (HomologicalComplex.dFrom C j)) =\n    F.map (imageSubobjectIso (HomologicalComplex.dTo C j)).hom \u226b\n      IsImage.lift\n          (StrongEpiMonoFactorisation.toMonoIsImage\n            (StrongEpiMonoFactorisation.mk\n              (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo C j)))\n                (F.map (image.\u03b9 (HomologicalComplex.dTo C j))) (F.map (factorThruImage (HomologicalComplex.dTo C j))))))\n          (Image.monoFactorisation (F.map (HomologicalComplex.dTo C j))) \u226b\n        image.\u03b9 (F.map (HomologicalComplex.dTo C j))\n[PROOFSTEP]\nsimp [\u2190 F.map_comp]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\n\u22a2 \u2200 {X Y : HomologicalComplex A c} (f : X \u27f6 Y),\n    (homologyFunctor A c i \u22d9 F).map f \u226b ((fun X => homologyIso F X i) Y).hom =\n      ((fun X => homologyIso F X i) X).hom \u226b (mapHomologicalComplex F c \u22d9 homologyFunctor B c i).map f\n[PROOFSTEP]\nintro X Y f\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (homologyFunctor A c i \u22d9 F).map f \u226b ((fun X => homologyIso F X i) Y).hom =\n    ((fun X => homologyIso F X i) X).hom \u226b (mapHomologicalComplex F c \u22d9 homologyFunctor B c i).map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 F.map\n        (homology.map (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)\n          (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0) (HomologicalComplex.Hom.sqTo f i)\n          (HomologicalComplex.Hom.sqFrom f i)\n          (_ : (HomologicalComplex.Hom.sqTo f i).right = (HomologicalComplex.Hom.sqTo f i).right)) \u226b\n      (homologyIso F Y i).hom =\n    (homologyIso F X i).hom \u226b\n      homology.map\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n            0)\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n            0)\n        (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i)\n        (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i)\n        (_ :\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right =\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right)\n[PROOFSTEP]\nrw [\u2190 Iso.inv_comp_eq, \u2190 Category.assoc, \u2190 Iso.eq_comp_inv]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (homologyIso F X i).inv \u226b\n      F.map\n        (homology.map (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)\n          (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0) (HomologicalComplex.Hom.sqTo f i)\n          (HomologicalComplex.Hom.sqFrom f i)\n          (_ : (HomologicalComplex.Hom.sqTo f i).right = (HomologicalComplex.Hom.sqTo f i).right)) =\n    homology.map\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n            0)\n        (_ :\n          HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n              HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n            0)\n        (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i)\n        (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i)\n        (_ :\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right =\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right) \u226b\n      (homologyIso F Y i).inv\n[PROOFSTEP]\nrefine' coequalizer.hom_ext _\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 coequalizer.\u03c0\n        (imageToKernel (HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i)\n          (HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0))\n        0 \u226b\n      (homologyIso F X i).inv \u226b\n        F.map\n          (homology.map (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)\n            (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0) (HomologicalComplex.Hom.sqTo f i)\n            (HomologicalComplex.Hom.sqFrom f i)\n            (_ : (HomologicalComplex.Hom.sqTo f i).right = (HomologicalComplex.Hom.sqTo f i).right)) =\n    coequalizer.\u03c0\n        (imageToKernel (HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i)\n          (HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0))\n        0 \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right) \u226b\n        (homologyIso F Y i).inv\n[PROOFSTEP]\ndsimp [homologyIso]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      (cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo X i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo X i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo X i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo X i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom X i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom X i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo X i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo X i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                      (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))) \u226b\n          (PreservesCokernel.iso F\n              (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))).inv) \u226b\n        F.map\n          (homology.map (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)\n            (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0) (HomologicalComplex.Hom.sqTo f i)\n            (HomologicalComplex.Hom.sqFrom f i)\n            (_ : (HomologicalComplex.Hom.sqTo f i).right = (HomologicalComplex.Hom.sqTo f i).right)) =\n    cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right) \u226b\n        cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo Y i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo Y i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo Y i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo Y i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom Y i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom Y i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo Y i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo Y i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                      (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) \u226b\n          (PreservesCokernel.iso F\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))).inv\n[PROOFSTEP]\nsimp only [PreservesCokernel.iso_inv]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      (cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo X i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo X i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo X i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo X i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom X i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom X i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo X i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo X i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                      (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))) \u226b\n          cokernelComparison\n            (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n              (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))\n            F) \u226b\n        F.map\n          (homology.map (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)\n            (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0) (HomologicalComplex.Hom.sqTo f i)\n            (HomologicalComplex.Hom.sqFrom f i)\n            (_ : (HomologicalComplex.Hom.sqTo f i).right = (HomologicalComplex.Hom.sqTo f i).right)) =\n    cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      homology.map\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n              0)\n          (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i)\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i)\n          (_ :\n            (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right =\n              (HomologicalComplex.Hom.sqTo ((mapHomologicalComplex F c).map f) i).right) \u226b\n        cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo Y i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo Y i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo Y i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo Y i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom Y i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom Y i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo Y i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo Y i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                      (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) \u226b\n          cokernelComparison\n            (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n              (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))\n            F\n[PROOFSTEP]\ndsimp [homology.map]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      (cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo X i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo X i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo X i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo X i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom X i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom X i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo X i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo X i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo X i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                      (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))) \u226b\n          cokernelComparison\n            (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n              (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))\n            F) \u226b\n        F.map\n          (cokernel.desc\n            (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n              (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))\n            (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n            (_ :\n              imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                    (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0) \u226b\n                  kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n                    cokernel.\u03c0\n                      (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                        (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)) =\n                0)) =\n    cokernel.\u03c0\n        (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n          (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (_ :\n            HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n              0)) \u226b\n      cokernel.desc\n          (imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n            (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n            (_ :\n              HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                  HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                0))\n          (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i) \u226b\n            cokernel.\u03c0\n              (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                (_ :\n                  HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                      HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                    0)))\n          (_ :\n            imageToKernel (F.map (HomologicalComplex.d X (ComplexShape.prev c i) i))\n                  (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n                  (_ :\n                    HomologicalComplex.dTo ((mapHomologicalComplex F c).obj X) i \u226b\n                        HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj X) i =\n                      0) \u226b\n                kernelSubobjectMap (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i) \u226b\n                  cokernel.\u03c0\n                    (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                      (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                      (_ :\n                        HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                            HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                          0)) =\n              0) \u226b\n        cokernel.map\n            (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                  0))\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n            (((imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).hom \u226b\n                image.lift\n                  (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo Y i)))\n                    (F.map (image.\u03b9 (HomologicalComplex.dTo Y i)))\n                    (F.map (factorThruImage (HomologicalComplex.dTo Y i))))) \u226b\n              F.map (imageSubobjectIso (HomologicalComplex.dTo Y i)).inv)\n            (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n                (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv) \u226b\n              F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv)\n            (_ :\n              imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                    (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                    (_ :\n                      HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                          HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                        0) \u226b\n                  (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom Y i)) \u226a\u226b\n                      PreservesKernel.iso F (HomologicalComplex.dFrom Y i) \u226a\u226b\n                        (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).symm).inv =\n                (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo Y i)) \u226a\u226b\n                      (PreservesImage.iso F (HomologicalComplex.dTo Y i)).symm \u226a\u226b\n                        (imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).symm).inv \u226b\n                  F.map\n                    (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                      (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) \u226b\n          cokernelComparison\n            (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n              (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))\n            F\n[PROOFSTEP]\nsimp only [\u2190 Category.assoc, cokernel.\u03c0_desc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (((((kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n              (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv) \u226b\n            F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv) \u226b\n          cokernel.\u03c0\n            (F.map\n              (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0)))) \u226b\n        cokernelComparison\n          (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n            (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))\n          F) \u226b\n      F.map\n        (cokernel.desc\n          (imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n            (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0))\n          (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n            cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n          (_ :\n            imageToKernel (HomologicalComplex.dTo X i) (HomologicalComplex.dFrom X i)\n                  (_ : HomologicalComplex.dTo X i \u226b HomologicalComplex.dFrom X i = 0) \u226b\n                kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n                  cokernel.\u03c0\n                    (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                      (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)) =\n              0)) =\n    ((kernelSubobjectMap (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i) \u226b\n          cokernel.\u03c0\n            (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n              (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n              (_ :\n                HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                    HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                  0))) \u226b\n        cokernel.map\n          (imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n            (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n            (_ :\n              HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                  HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                0))\n          (F.map\n            (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n              (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n          (((imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).hom \u226b\n              image.lift\n                (MonoFactorisation.mk (F.obj (image (HomologicalComplex.dTo Y i)))\n                  (F.map (image.\u03b9 (HomologicalComplex.dTo Y i)))\n                  (F.map (factorThruImage (HomologicalComplex.dTo Y i))))) \u226b\n            F.map (imageSubobjectIso (HomologicalComplex.dTo Y i)).inv)\n          (((kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n              (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv) \u226b\n            F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv)\n          (_ :\n            imageToKernel (F.map (HomologicalComplex.d Y (ComplexShape.prev c i) i))\n                  (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n                  (_ :\n                    HomologicalComplex.dTo ((mapHomologicalComplex F c).obj Y) i \u226b\n                        HomologicalComplex.dFrom ((mapHomologicalComplex F c).obj Y) i =\n                      0) \u226b\n                (F.mapIso (kernelSubobjectIso (HomologicalComplex.dFrom Y i)) \u226a\u226b\n                    PreservesKernel.iso F (HomologicalComplex.dFrom Y i) \u226a\u226b\n                      (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).symm).inv =\n              (F.mapIso (imageSubobjectIso (HomologicalComplex.dTo Y i)) \u226a\u226b\n                    (PreservesImage.iso F (HomologicalComplex.dTo Y i)).symm \u226a\u226b\n                      (imageSubobjectIso (F.map (HomologicalComplex.dTo Y i))).symm).inv \u226b\n                F.map\n                  (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                    (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))) \u226b\n      cokernelComparison\n        (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n          (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))\n        F\n[PROOFSTEP]\nsimp only [Category.assoc, cokernelComparison_map_desc, cokernel.\u03c0_desc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv \u226b\n        F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv \u226b\n          F.map\n            (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    kernelSubobjectMap (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i) \u226b\n      (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv \u226b\n            cokernel.\u03c0\n                (F.map\n                  (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                    (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) \u226b\n              cokernelComparison\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))\n                F\n[PROOFSTEP]\nsimp only [\u03c0_comp_cokernelComparison, \u2190 F.map_comp]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv \u226b\n        F.map\n          ((kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv \u226b\n            kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    kernelSubobjectMap (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i) \u226b\n      (kernelSubobjectIso (F.map (HomologicalComplex.dFrom Y i))).hom \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv \u226b\n          F.map\n            ((kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nerw [\u2190 kernelSubobjectIso_comp_kernel_map_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv \u226b\n        F.map\n          ((kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv \u226b\n            kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i) \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      kernel.map (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i).left\n          (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i).right\n          (_ :\n            (Arrow.mk (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n                (\ud835\udfed B).map (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i).right =\n              (\ud835\udfed B).map (HomologicalComplex.Hom.sqFrom ((mapHomologicalComplex F c).map f) i).left \u226b\n                (Arrow.mk (F.map (HomologicalComplex.d Y i (ComplexShape.next c i)))).hom) \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv \u226b\n          F.map\n            ((kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv \u226b\n              cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nsimp only [HomologicalComplex.Hom.sqFrom_right, HomologicalComplex.Hom.sqFrom_left, F.mapHomologicalComplex_map_f,\n  F.map_comp]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.dFrom X i))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.dFrom X i)).inv \u226b\n        F.map (kernelSubobjectIso (HomologicalComplex.dFrom X i)).inv \u226b\n          F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      kernel.map (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (F.map (HomologicalComplex.d Y i (ComplexShape.next c i))) (F.map (HomologicalComplex.Hom.f f i))\n          (F.map (HomologicalComplex.Hom.f f (ComplexShape.next c i)))\n          (_ :\n            F.map (HomologicalComplex.d X i (ComplexShape.next c i)) \u226b\n                F.map (HomologicalComplex.Hom.f f (ComplexShape.next c i)) =\n              F.map (HomologicalComplex.Hom.f f i) \u226b F.map (HomologicalComplex.d Y i (ComplexShape.next c i))) \u226b\n        (PreservesKernel.iso F (HomologicalComplex.dFrom Y i)).inv \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.dFrom Y i)).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.dFrom Y i)\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\ndsimp [HomologicalComplex.dFrom, HomologicalComplex.Hom.next]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map (kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n          F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      kernel.map (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))\n          (F.map (HomologicalComplex.d Y i (ComplexShape.next c i))) (F.map (HomologicalComplex.Hom.f f i))\n          (F.map (HomologicalComplex.Hom.f f (ComplexShape.next c i)))\n          (_ :\n            F.map (HomologicalComplex.d X i (ComplexShape.next c i)) \u226b\n                F.map (HomologicalComplex.Hom.f f (ComplexShape.next c i)) =\n              F.map (HomologicalComplex.Hom.f f i) \u226b F.map (HomologicalComplex.d Y i (ComplexShape.next c i))) \u226b\n        (PreservesKernel.iso F (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nrw [kernel_map_comp_preserves_kernel_iso_inv_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map (kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n          F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i)) ?hpq) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\ncase hpq\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n    HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i)\n[PROOFSTEP]\nconv_lhs => erw [\u2190 F.map_comp_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n| (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n    (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nerw [\u2190 F.map_comp_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n| (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n    (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nerw [\u2190 F.map_comp_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n| (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n    (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n      F.map (kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map (kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nerw [\u2190 F.map_comp_assoc]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            ((kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n              kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i)) ?hpq) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\ncase hpq\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n    HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i)\n[PROOFSTEP]\nrotate_right\n[GOAL]\ncase hpq\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n    HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i)\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            ((kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n              kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i)) ?hpq) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nsimp\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            ((kernelSubobjectIso (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n              kernelSubobjectMap (HomologicalComplex.Hom.sqFrom f i)) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i))\n              (_ :\n                HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n                  HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i))) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nrw [\u2190 kernel_map_comp_kernelSubobjectIso_inv]\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n                (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.sqFrom f i).left\n                (HomologicalComplex.Hom.sqFrom f i).right\n                (_ :\n                  (Arrow.mk (HomologicalComplex.d X i (ComplexShape.next c i))).hom \u226b\n                      (\ud835\udfed A).map (HomologicalComplex.Hom.sqFrom f i).right =\n                    (\ud835\udfed A).map (HomologicalComplex.Hom.sqFrom f i).left \u226b\n                      (Arrow.mk (HomologicalComplex.d Y i (ComplexShape.next c i))).hom) \u226b\n              (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i))\n              (_ :\n                HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n                  HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i))) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nany_goals simp\n[GOAL]\nA : Type u\ninst\u271d\u2076 : Category.{v, u} A\ninst\u271d\u2075 : Abelian A\nX\u271d Y\u271d Z : A\nf\u271d : X\u271d \u27f6 Y\u271d\ng : Y\u271d \u27f6 Z\nw : f\u271d \u226b g = 0\n\u03b9 : Type u_1\nc : ComplexShape \u03b9\nB : Type u_2\ninst\u271d\u2074 : Category.{?u.214307, u_2} B\ninst\u271d\u00b3 : Abelian B\nF : A \u2964 B\ninst\u271d\u00b2 : Additive F\ninst\u271d\u00b9 : PreservesFiniteLimits F\ninst\u271d : PreservesFiniteColimits F\ni : \u03b9\nX Y : HomologicalComplex A c\nf : X \u27f6 Y\n\u22a2 (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n                (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.sqFrom f i).left\n                (HomologicalComplex.Hom.sqFrom f i).right\n                (_ :\n                  (Arrow.mk (HomologicalComplex.d X i (ComplexShape.next c i))).hom \u226b\n                      (\ud835\udfed A).map (HomologicalComplex.Hom.sqFrom f i).right =\n                    (\ud835\udfed A).map (HomologicalComplex.Hom.sqFrom f i).left \u226b\n                      (Arrow.mk (HomologicalComplex.d Y i (ComplexShape.next c i))).hom) \u226b\n              (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv) \u226b\n          F.map\n            (cokernel.\u03c0\n              (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0))) =\n    (kernelSubobjectIso (F.map (HomologicalComplex.d X i (ComplexShape.next c i)))).hom \u226b\n      (PreservesKernel.iso F (HomologicalComplex.d X i (ComplexShape.next c i))).inv \u226b\n        F.map\n            (kernel.map (HomologicalComplex.d X i (ComplexShape.next c i))\n              (HomologicalComplex.d Y i (ComplexShape.next c i)) (HomologicalComplex.Hom.f f i)\n              (HomologicalComplex.Hom.f f (ComplexShape.next c i))\n              (_ :\n                HomologicalComplex.d X i (ComplexShape.next c i) \u226b HomologicalComplex.Hom.f f (ComplexShape.next c i) =\n                  HomologicalComplex.Hom.f f i \u226b HomologicalComplex.d Y i (ComplexShape.next c i))) \u226b\n          F.map (kernelSubobjectIso (HomologicalComplex.d Y i (ComplexShape.next c i))).inv \u226b\n            F.map\n              (cokernel.\u03c0\n                (imageToKernel (HomologicalComplex.dTo Y i) (HomologicalComplex.d Y i (ComplexShape.next c i))\n                  (_ : HomologicalComplex.dTo Y i \u226b HomologicalComplex.dFrom Y i = 0)))\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.Homology", "llama_tokens": 61383, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.5304423268411982}}
{"text": "[GOAL]\nM : Type ?u.4198\nN : Type ?u.4201\n\u03b1 : Type ?u.4204\n\u03b2 : Type ?u.4214\ninst\u271d\u2078 : MeasurableSpace M\ninst\u271d\u2077 : MeasurableSpace N\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : SMul M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : SMul N \u03b2\ninst\u271d : ContinuousConstSMul N \u03b2\n\u22a2 \u2200 (m : M\u1d48\u1d50\u1d43) (n : N) (a : \u03b1 \u2192\u2098[\u03bc] \u03b2), m \u2022 n \u2022 a = n \u2022 m \u2022 a\n[PROOFSTEP]\nrintro _ _ \u27e8_\u27e9\n[GOAL]\ncase mk\nM : Type ?u.4198\nN : Type ?u.4201\n\u03b1 : Type ?u.4204\n\u03b2 : Type ?u.4214\ninst\u271d\u2078 : MeasurableSpace M\ninst\u271d\u2077 : MeasurableSpace N\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : SMul M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : SMul N \u03b2\ninst\u271d : ContinuousConstSMul N \u03b2\nm\u271d : M\u1d48\u1d50\u1d43\nn\u271d : N\na\u271d\u00b9 : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : { f // AEStronglyMeasurable f \u03bc }\n\u22a2 m\u271d \u2022 n\u271d \u2022 Quot.mk Setoid.r a\u271d = n\u271d \u2022 m\u271d \u2022 Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type ?u.8732\nN : Type ?u.8735\n\u03b1 : Type ?u.8738\n\u03b2 : Type ?u.8748\ninst\u271d\u2078 : MeasurableSpace M\ninst\u271d\u2077 : MeasurableSpace N\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : SMul M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : AddMonoid \u03b2\ninst\u271d : ContinuousAdd \u03b2\n\u22a2 \u2200 (a : M\u1d48\u1d50\u1d43) (x y : \u03b1 \u2192\u2098[\u03bc] \u03b2), a \u2022 (x + y) = a \u2022 x + a \u2022 y\n[PROOFSTEP]\nrintro _ \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase mk.mk\nM : Type ?u.8732\nN : Type ?u.8735\n\u03b1 : Type ?u.8738\n\u03b2 : Type ?u.8748\ninst\u271d\u2078 : MeasurableSpace M\ninst\u271d\u2077 : MeasurableSpace N\ninst\u271d\u2076 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : SMul M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : AddMonoid \u03b2\ninst\u271d : ContinuousAdd \u03b2\na\u271d\u00b2 : M\u1d48\u1d50\u1d43\nx\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc }\ny\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : { f // AEStronglyMeasurable f \u03bc }\n\u22a2 a\u271d\u00b2 \u2022 (Quot.mk Setoid.r a\u271d\u00b9 + Quot.mk Setoid.r a\u271d) = a\u271d\u00b2 \u2022 Quot.mk Setoid.r a\u271d\u00b9 + a\u271d\u00b2 \u2022 Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type ?u.19747\nN : Type ?u.19750\n\u03b1 : Type ?u.19753\n\u03b2 : Type ?u.19763\ninst\u271d\u2079 : MeasurableSpace M\ninst\u271d\u2078 : MeasurableSpace N\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : MulAction M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : ContinuousMul \u03b2\n\u22a2 \u2200 (r : M\u1d48\u1d50\u1d43) (x y : \u03b1 \u2192\u2098[\u03bc] \u03b2), r \u2022 (x * y) = r \u2022 x * r \u2022 y\n[PROOFSTEP]\nrintro _ \u27e8\u27e9 \u27e8\u27e9\n[GOAL]\ncase mk.mk\nM : Type ?u.19747\nN : Type ?u.19750\n\u03b1 : Type ?u.19753\n\u03b2 : Type ?u.19763\ninst\u271d\u2079 : MeasurableSpace M\ninst\u271d\u2078 : MeasurableSpace N\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u2076 : TopologicalSpace \u03b2\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : MulAction M \u03b1\ninst\u271d\u00b3 : MeasurableSMul M \u03b1\ninst\u271d\u00b2 : SMulInvariantMeasure M \u03b1 \u03bc\ninst\u271d\u00b9 : Monoid \u03b2\ninst\u271d : ContinuousMul \u03b2\nr\u271d : M\u1d48\u1d50\u1d43\nx\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc }\ny\u271d : \u03b1 \u2192\u2098[\u03bc] \u03b2\na\u271d : { f // AEStronglyMeasurable f \u03bc }\n\u22a2 r\u271d \u2022 (Quot.mk Setoid.r a\u271d\u00b9 * Quot.mk Setoid.r a\u271d) = r\u271d \u2022 Quot.mk Setoid.r a\u271d\u00b9 * r\u271d \u2022 Quot.mk Setoid.r a\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.AEEqFun.DomAct", "llama_tokens": 1725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.851952809486198, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.5303059760083197}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : Bounded s\nht : Bounded t\n\u22a2 Bounded (s * t)\n[PROOFSTEP]\nobtain \u27e8Rs, hRs\u27e9 : \u2203 R, \u2200 x \u2208 s, \u2016x\u2016 \u2264 R := hs.exists_norm_le'\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : Bounded s\nht : Bounded t\nRs : \u211d\nhRs : \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 Rs\n\u22a2 Bounded (s * t)\n[PROOFSTEP]\nobtain \u27e8Rt, hRt\u27e9 : \u2203 R, \u2200 x \u2208 t, \u2016x\u2016 \u2264 R := ht.exists_norm_le'\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : Bounded s\nht : Bounded t\nRs : \u211d\nhRs : \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 Rs\nRt : \u211d\nhRt : \u2200 (x : E), x \u2208 t \u2192 \u2016x\u2016 \u2264 Rt\n\u22a2 Bounded (s * t)\n[PROOFSTEP]\nrefine' bounded_iff_forall_norm_le'.2 \u27e8Rs + Rt, _\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : Bounded s\nht : Bounded t\nRs : \u211d\nhRs : \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 Rs\nRt : \u211d\nhRt : \u2200 (x : E), x \u2208 t \u2192 \u2016x\u2016 \u2264 Rt\n\u22a2 \u2200 (x : E), x \u2208 s * t \u2192 \u2016x\u2016 \u2264 Rs + Rt\n[PROOFSTEP]\nrintro z \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y\u271d : E\nhs : Bounded s\nht : Bounded t\nRs : \u211d\nhRs : \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 Rs\nRt : \u211d\nhRt : \u2200 (x : E), x \u2208 t \u2192 \u2016x\u2016 \u2264 Rt\nx y : E\nhx : x \u2208 s\nhy : y \u2208 t\n\u22a2 \u2016(fun x x_1 => x * x_1) x y\u2016 \u2264 Rs + Rt\n[PROOFSTEP]\nexact norm_mul_le_of_le (hRs x hx) (hRt y hy)\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 Bounded s \u2192 Bounded s\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [bounded_iff_forall_norm_le', \u2190 image_inv, ball_image_iff, norm_inv']\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 (\u2203 C, \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 C) \u2192 \u2203 C, \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 C\n[PROOFSTEP]\nexact id\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns\u271d t : Set E\nx\u271d y x : E\ns : Set E\n\u22a2 infEdist x\u207b\u00b9 s\u207b\u00b9 = infEdist x s\n[PROOFSTEP]\nrw [\u2190 image_inv, infEdist_image isometry_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns\u271d t : Set E\nx\u271d y x : E\ns : Set E\n\u22a2 infEdist x\u207b\u00b9 s = infEdist x s\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 infEdist_inv_inv, inv_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 (thickening \u03b4 s)\u207b\u00b9 = thickening \u03b4 s\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [thickening, \u2190 infEdist_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x | EMetric.infEdist x s < ENNReal.ofReal \u03b4}\u207b\u00b9 = {x | EMetric.infEdist x\u207b\u00b9 s < ENNReal.ofReal \u03b4}\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 (cthickening \u03b4 s)\u207b\u00b9 = cthickening \u03b4 s\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [cthickening, \u2190 infEdist_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x | EMetric.infEdist x s \u2264 ENNReal.ofReal \u03b4}\u207b\u00b9 = {x | EMetric.infEdist x\u207b\u00b9 s \u2264 ENNReal.ofReal \u03b4}\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} * ball y \u03b4 = ball (x * y) \u03b4\n[PROOFSTEP]\nsimp only [preimage_mul_ball, image_mul_left, singleton_mul, div_inv_eq_mul, mul_comm y x]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} / ball y \u03b4 = ball (x / y) \u03b4\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, inv_ball, singleton_mul_ball]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball x \u03b4 * {y} = ball (x * y) \u03b4\n[PROOFSTEP]\nrw [mul_comm, singleton_mul_ball, mul_comm y]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball x \u03b4 / {y} = ball (x / y) \u03b4\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, inv_singleton, ball_mul_singleton]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} * ball 1 \u03b4 = ball x \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} / ball 1 \u03b4 = ball x \u03b4\n[PROOFSTEP]\nrw [singleton_div_ball, div_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball 1 \u03b4 * {x} = ball x \u03b4\n[PROOFSTEP]\nsimp [ball_mul_singleton]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball 1 \u03b4 / {x} = ball x\u207b\u00b9 \u03b4\n[PROOFSTEP]\nrw [ball_div_singleton, one_div]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 x \u2022 ball 1 \u03b4 = ball x \u03b4\n[PROOFSTEP]\nrw [smul_ball, smul_eq_mul, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} * closedBall y \u03b4 = closedBall (x * y) \u03b4\n[PROOFSTEP]\nsimp_rw [singleton_mul, \u2190 smul_eq_mul, image_smul, smul_closedBall]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} / closedBall y \u03b4 = closedBall (x / y) \u03b4\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, inv_closedBall, singleton_mul_closedBall]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 closedBall x \u03b4 * {y} = closedBall (x * y) \u03b4\n[PROOFSTEP]\nsimp [mul_comm _ { y }, mul_comm y]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 closedBall x \u03b4 / {y} = closedBall (x / y) \u03b4\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} * closedBall 1 \u03b4 = closedBall x \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 {x} / closedBall 1 \u03b4 = closedBall x \u03b4\n[PROOFSTEP]\nrw [singleton_div_closedBall, div_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 closedBall 1 \u03b4 * {x} = closedBall x \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 closedBall 1 \u03b4 / {x} = closedBall x\u207b\u00b9 \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 x \u2022 closedBall 1 \u03b4 = closedBall x \u03b4\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s * ball 1 \u03b4 = thickening \u03b4 s\n[PROOFSTEP]\nrw [thickening_eq_biUnion_ball]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s * ball 1 \u03b4 = \u22c3 (x : E) (_ : x \u2208 s), ball x \u03b4\n[PROOFSTEP]\nconvert iUnion\u2082_mul (fun x (_ : x \u2208 s) => { x }) (ball (1 : E) \u03b4)\n[GOAL]\ncase h.e'_2.h.e'_5\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s = \u22c3 (i : E) (_ : i \u2208 s), {i}\ncase h.e'_3.h.e'_3.h.f\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y x\u271d\u00b9 : E\nx\u271d : x\u271d\u00b9 \u2208 s\n\u22a2 ball x\u271d\u00b9 \u03b4 = {x\u271d\u00b9} * ball 1 \u03b4\n[PROOFSTEP]\nexact s.biUnion_of_singleton.symm\n[GOAL]\ncase h.e'_3.h.e'_3.h.f\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y x\u271d\u00b9 : E\nx\u271d : x\u271d\u00b9 \u2208 s\n\u22a2 ball x\u271d\u00b9 \u03b4 = {x\u271d\u00b9} * ball 1 \u03b4\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h.e'_3.h.f.h\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d\u00b2 y x\u271d\u00b9 : E\nx\u271d : x\u271d\u00b9 \u2208 s\nx : E\n\u22a2 x \u2208 ball x\u271d\u00b9 \u03b4 \u2194 x \u2208 {x\u271d\u00b9} * ball 1 \u03b4\n[PROOFSTEP]\nsimp_rw [singleton_mul_ball, mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s / ball 1 \u03b4 = thickening \u03b4 s\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_ball_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball 1 \u03b4 * s = thickening \u03b4 s\n[PROOFSTEP]\nrw [mul_comm, mul_ball_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball 1 \u03b4 / s = thickening \u03b4 s\u207b\u00b9\n[PROOFSTEP]\nsimp [div_eq_mul_inv, ball_mul_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s * ball x \u03b4 = x \u2022 thickening \u03b4 s\n[PROOFSTEP]\nrw [\u2190 smul_ball_one, mul_smul_comm, mul_ball_one]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s / ball x \u03b4 = x\u207b\u00b9 \u2022 thickening \u03b4 s\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball x \u03b4 * s = x \u2022 thickening \u03b4 s\n[PROOFSTEP]\nrw [mul_comm, mul_ball]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 ball x \u03b4 / s = x \u2022 thickening \u03b4 s\u207b\u00b9\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 s * closedBall 1 \u03b4 = cthickening \u03b4 s\n[PROOFSTEP]\nrw [hs.cthickening_eq_biUnion_closedBall h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 s * closedBall 1 \u03b4 = \u22c3 (x : E) (_ : x \u2208 s), closedBall x \u03b4\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 x \u2208 s * closedBall 1 \u03b4 \u2194 x \u2208 \u22c3 (x : E) (_ : x \u2208 s), closedBall x \u03b4\n[PROOFSTEP]\nsimp only [mem_mul, dist_eq_norm_div, exists_prop, mem_iUnion, mem_closedBall, exists_and_left, mem_closedBall_one_iff,\n  \u2190 eq_div_iff_mul_eq'', div_one, exists_eq_right]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 s / closedBall 1 \u03b4 = cthickening \u03b4 s\n[PROOFSTEP]\nsimp [div_eq_mul_inv, hs.mul_closedBall_one h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 closedBall 1 \u03b4 * s = cthickening \u03b4 s\n[PROOFSTEP]\nrw [mul_comm, hs.mul_closedBall_one h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\n\u22a2 closedBall 1 \u03b4 / s = cthickening \u03b4 s\u207b\u00b9\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_comm, hs.inv.mul_closedBall_one h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 s * closedBall x \u03b4 = x \u2022 cthickening \u03b4 s\n[PROOFSTEP]\nrw [\u2190 smul_closedBall_one, mul_smul_comm, hs.mul_closedBall_one h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 s / closedBall x \u03b4 = x\u207b\u00b9 \u2022 cthickening \u03b4 s\n[PROOFSTEP]\nsimp [div_eq_mul_inv, mul_comm, hs.mul_closedBall h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 closedBall x \u03b4 * s = x \u2022 cthickening \u03b4 s\n[PROOFSTEP]\nrw [mul_comm, hs.mul_closedBall h\u03b4]\n[GOAL]\nE : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx\u271d y : E\nhs : IsCompact s\nh\u03b4 : 0 \u2264 \u03b4\nx : E\n\u22a2 closedBall x \u03b4 * s = x \u2022 cthickening \u03b4 s\n[PROOFSTEP]\nsimp [div_eq_mul_inv, hs.closedBall_mul h\u03b4]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.Pointwise", "llama_tokens": 5510, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707283, "lm_q2_score": 0.6893056295505783, "lm_q1q2_score": 0.5301313369889463}}
{"text": "[GOAL]\n\u22a2 riemannZeta 0 = -1 / 2\n[PROOFSTEP]\nrw [riemannZeta_def]\n[GOAL]\n\u22a2 Function.update (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) 0 (-1 / 2) 0 = -1 / 2\n[PROOFSTEP]\nexact Function.update_same _ _ _\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 Summable fun n => rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\n[PROOFSTEP]\nhave : 0 < (\u2191t * I).im := by rwa [ofReal_mul_im, I_im, mul_one]\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 0 < (\u2191t * I).im\n[PROOFSTEP]\nrwa [ofReal_mul_im, I_im, mul_one]\n[GOAL]\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\n\u22a2 Summable fun n => rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\n[PROOFSTEP]\nconvert summable_norm_iff.mpr (hasSum_nat_jacobiTheta this).summable using 1\n[GOAL]\ncase h.e'_5\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\n\u22a2 (fun n => rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = fun x => \u2016cexp (\u2191\u03c0 * I * (\u2191x + 1) ^ 2 * (\u2191t * I))\u2016\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_5.h\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\nn : \u2115\n\u22a2 rexp (-\u03c0 * t * (\u2191n + 1) ^ 2) = \u2016cexp (\u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I))\u2016\n[PROOFSTEP]\nrw [Complex.norm_eq_abs, Complex.abs_exp]\n[GOAL]\ncase h.e'_5.h\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\nn : \u2115\n\u22a2 rexp (-\u03c0 * t * (\u2191n + 1) ^ 2) = rexp (\u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I)).re\n[PROOFSTEP]\nrw [show \u2191\u03c0 * I * ((n : \u2102) + 1) ^ 2 * (\u2191t * I) = ((\u03c0 * t * ((n : \u211d) + 1) ^ 2) : \u211d) * I ^ 2 by push_cast ; ring]\n[GOAL]\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\nn : \u2115\n\u22a2 \u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I) = \u2191(\u03c0 * t * (\u2191n + 1) ^ 2) * I ^ 2\n[PROOFSTEP]\npush_cast\n[GOAL]\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\nn : \u2115\n\u22a2 \u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I) = \u2191\u03c0 * \u2191t * (\u2191n + 1) ^ 2 * I ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase h.e'_5.h\nt : \u211d\nht : 0 < t\nthis : 0 < (\u2191t * I).im\nn : \u2115\n\u22a2 rexp (-\u03c0 * t * (\u2191n + 1) ^ 2) = rexp (\u2191(\u03c0 * t * (\u2191n + 1) ^ 2) * I ^ 2).re\n[PROOFSTEP]\nrw [I_sq, mul_neg_one, \u2190 ofReal_neg, ofReal_re, neg_mul, neg_mul]\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 zetaKernel\u2081 t = (jacobiTheta (\u2191t * I) - 1) / 2\n[PROOFSTEP]\nrw [jacobiTheta_eq_tsum_nat ((mul_I_im t).symm \u25b8 ht : 0 < (\u2191t * I).im), add_comm, add_sub_cancel,\n  mul_div_cancel_left _ (two_ne_zero' \u2102), zetaKernel\u2081]\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 \u2211' (n : \u2115), \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = \u2211' (n : \u2115), cexp (\u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I))\n[PROOFSTEP]\ncongr 1 with n : 1\n[GOAL]\ncase e_f.h\nt : \u211d\nht : 0 < t\nn : \u2115\n\u22a2 \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = cexp (\u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_f.h\nt : \u211d\nht : 0 < t\nn : \u2115\n\u22a2 cexp (-\u2191\u03c0 * \u2191t * (\u2191n + 1) ^ 2) = cexp (\u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I))\n[PROOFSTEP]\nrw [(by ring : \u2191\u03c0 * I * ((n : \u2102) + 1) ^ 2 * (t * I) = I ^ 2 * \u03c0 * t * ((n : \u2102) + 1) ^ 2), I_sq, neg_one_mul]\n[GOAL]\nt : \u211d\nht : 0 < t\nn : \u2115\n\u22a2 \u2191\u03c0 * I * (\u2191n + 1) ^ 2 * (\u2191t * I) = I ^ 2 * \u2191\u03c0 * \u2191t * (\u2191n + 1) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 ContinuousAt zetaKernel\u2081 t\n[PROOFSTEP]\nhave : ContinuousAt (fun u : \u211d => (jacobiTheta (u * I) - 1) / 2) t :=\n  by\n  refine' (ContinuousAt.sub _ continuousAt_const).div_const _\n  refine' (continuousAt_jacobiTheta _).comp (ContinuousAt.mul _ continuousAt_const)\n  \u00b7 rwa [mul_I_im, ofReal_re]\n  \u00b7 exact continuous_ofReal.continuousAt\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 ContinuousAt (fun u => (jacobiTheta (\u2191u * I) - 1) / 2) t\n[PROOFSTEP]\nrefine' (ContinuousAt.sub _ continuousAt_const).div_const _\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 ContinuousAt (fun u => jacobiTheta (\u2191u * I)) t\n[PROOFSTEP]\nrefine' (continuousAt_jacobiTheta _).comp (ContinuousAt.mul _ continuousAt_const)\n[GOAL]\ncase refine'_1\nt : \u211d\nht : 0 < t\n\u22a2 0 < (\u2191t * I).im\n[PROOFSTEP]\nrwa [mul_I_im, ofReal_re]\n[GOAL]\ncase refine'_2\nt : \u211d\nht : 0 < t\n\u22a2 ContinuousAt (fun u => \u2191u) t\n[PROOFSTEP]\nexact continuous_ofReal.continuousAt\n[GOAL]\nt : \u211d\nht : 0 < t\nthis : ContinuousAt (fun u => (jacobiTheta (\u2191u * I) - 1) / 2) t\n\u22a2 ContinuousAt zetaKernel\u2081 t\n[PROOFSTEP]\nrefine' this.congr (eventually_of_mem (Ioi_mem_nhds ht) fun u hu => _)\n[GOAL]\nt : \u211d\nht : 0 < t\nthis : ContinuousAt (fun u => (jacobiTheta (\u2191u * I) - 1) / 2) t\nu : \u211d\nhu : u \u2208 Ioi 0\n\u22a2 (fun u => (jacobiTheta (\u2191u * I) - 1) / 2) u = zetaKernel\u2081 u\n[PROOFSTEP]\nrw [zetaKernel\u2081_eq_jacobiTheta hu]\n[GOAL]\n\u22a2 LocallyIntegrableOn zetaKernel\u2082 (Ioi 0)\n[PROOFSTEP]\nrefine (locallyIntegrableOn_iff (Or.inr isOpen_Ioi)).mpr fun k hk hk' => Integrable.add ?_ ?_\n[GOAL]\ncase refine_1\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 Integrable zetaKernel\u2081\n[PROOFSTEP]\nrefine ContinuousOn.integrableOn_compact hk' ?_\n[GOAL]\ncase refine_1\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 ContinuousOn zetaKernel\u2081 k\n[PROOFSTEP]\nexact ContinuousAt.continuousOn fun x hx => continuousAt_zetaKernel\u2081 (hk hx)\n[GOAL]\ncase refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 Integrable (indicator (Ioc 0 1) fun t => (1 - 1 / \u2191(sqrt t)) / 2)\n[PROOFSTEP]\nrefine (integrable_indicator_iff measurableSet_Ioc).mpr ?_\n[GOAL]\ncase refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 IntegrableOn (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (Ioc 0 1)\n[PROOFSTEP]\nrw [IntegrableOn, Measure.restrict_restrict, \u2190 IntegrableOn]\n[GOAL]\ncase refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 IntegrableOn (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (Ioc 0 1 \u2229 k)\ncase refine_2 k : Set \u211d hk : k \u2286 Ioi 0 hk' : IsCompact k \u22a2 MeasurableSet (Ioc 0 1)\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 MeasurableSet (Ioc 0 1)\n[PROOFSTEP]\nexact measurableSet_Ioc\n[GOAL]\ncase refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 IntegrableOn (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (Ioc 0 1 \u2229 k)\n[PROOFSTEP]\napply ContinuousOn.integrableOn_compact\n[GOAL]\ncase refine_2.hK\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 IsCompact (Ioc 0 1 \u2229 k)\n[PROOFSTEP]\nconvert (isCompact_Icc : IsCompact <| Icc (0 : \u211d) 1).inter hk' using 1\n[GOAL]\ncase h.e'_3\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 Ioc 0 1 \u2229 k = Icc 0 1 \u2229 k\n[PROOFSTEP]\nexact Set.ext fun t => \u27e8fun h => \u27e8Ioc_subset_Icc_self h.1, h.2\u27e9, fun h => \u27e8\u27e8hk h.2, h.1.2\u27e9, h.2\u27e9\u27e9\n[GOAL]\ncase refine_2.hf\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 ContinuousOn (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (Ioc 0 1 \u2229 k)\n[PROOFSTEP]\nrefine ContinuousOn.mono ?_ ((inter_subset_right _ _).trans hk)\n[GOAL]\ncase refine_2.hf\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 ContinuousOn (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (Ioi 0)\n[PROOFSTEP]\nrefine (continuousOn_const.sub ?_).div_const _\n[GOAL]\ncase refine_2.hf\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 ContinuousOn (fun t => 1 / \u2191(sqrt t)) (Ioi 0)\n[PROOFSTEP]\nrefine ContinuousOn.div continuousOn_const ?_ fun x hx => ?_\n[GOAL]\ncase refine_2.hf.refine_1\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\n\u22a2 ContinuousOn (fun t => \u2191(sqrt t)) (Ioi 0)\n[PROOFSTEP]\nexact (continuous_ofReal.comp continuous_sqrt).continuousOn\n[GOAL]\ncase refine_2.hf.refine_2\nk : Set \u211d\nhk : k \u2286 Ioi 0\nhk' : IsCompact k\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(sqrt x) \u2260 0\n[PROOFSTEP]\nexact ofReal_ne_zero.mpr (sqrt_ne_zero'.mpr hx)\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nhave aux : \u2200 {u : \u211d} (_ : 1 < u), zetaKernel\u2082 (1 / u) = sqrt u * zetaKernel\u2082 u :=\n  by\n  intro u hu\n  simp_rw [zetaKernel\u2082, Pi.add_apply]\n  rw [indicator_of_mem, indicator_of_not_mem (not_mem_Ioc_of_gt hu), add_zero]\n  swap; \u00b7 exact \u27e8one_div_pos.mpr (zero_lt_one.trans hu), (one_div u).symm \u25b8 inv_le_one hu.le\u27e9\n  rw [zetaKernel\u2081_eq_jacobiTheta (one_div_pos.mpr <| zero_lt_one.trans hu),\n    zetaKernel\u2081_eq_jacobiTheta (zero_lt_one.trans hu), \u2190 add_div, \u2190 mul_div_assoc, add_sub, sub_add_cancel,\n    sqrt_div zero_le_one, sqrt_one, one_div (sqrt _), ofReal_inv, \u2190 one_div, one_div_one_div, mul_sub, mul_one]\n  congr 2\n  let \u03c4 : UpperHalfPlane := .mk (u * I) ((mul_I_im u).symm \u25b8 zero_lt_one.trans hu)\n  convert jacobiTheta_S_smul \u03c4 using 2\n  \u00b7\n    rw [UpperHalfPlane.modular_S_smul, UpperHalfPlane.coe_mk, UpperHalfPlane.coe_mk, \u2190 neg_inv, mul_inv, inv_I, mul_neg,\n      neg_neg, one_div, ofReal_inv]\n  \u00b7 rw [UpperHalfPlane.coe_mk, mul_comm, mul_assoc, mul_neg, I_mul_I, neg_neg, mul_one, sqrt_eq_rpow,\n      ofReal_cpow (zero_lt_one.trans hu).le]\n    push_cast\n    rfl\n[GOAL]\nt : \u211d\nht : 0 < t\n\u22a2 \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\n[PROOFSTEP]\nintro u hu\n[GOAL]\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\n[PROOFSTEP]\nsimp_rw [zetaKernel\u2082, Pi.add_apply]\n[GOAL]\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 zetaKernel\u2081 (1 / u) + indicator (Ioc 0 1) (fun t => (1 - 1 / \u2191(sqrt t)) / 2) (1 / u) =\n    \u2191(sqrt u) * (zetaKernel\u2081 u + indicator (Ioc 0 1) (fun t => (1 - 1 / \u2191(sqrt t)) / 2) u)\n[PROOFSTEP]\nrw [indicator_of_mem, indicator_of_not_mem (not_mem_Ioc_of_gt hu), add_zero]\n[GOAL]\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 zetaKernel\u2081 (1 / u) + (1 - 1 / \u2191(sqrt (1 / u))) / 2 = \u2191(sqrt u) * zetaKernel\u2081 u\ncase h t : \u211d ht : 0 < t u : \u211d hu : 1 < u \u22a2 1 / u \u2208 Ioc 0 1\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 1 / u \u2208 Ioc 0 1\n[PROOFSTEP]\nexact \u27e8one_div_pos.mpr (zero_lt_one.trans hu), (one_div u).symm \u25b8 inv_le_one hu.le\u27e9\n[GOAL]\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 zetaKernel\u2081 (1 / u) + (1 - 1 / \u2191(sqrt (1 / u))) / 2 = \u2191(sqrt u) * zetaKernel\u2081 u\n[PROOFSTEP]\nrw [zetaKernel\u2081_eq_jacobiTheta (one_div_pos.mpr <| zero_lt_one.trans hu),\n  zetaKernel\u2081_eq_jacobiTheta (zero_lt_one.trans hu), \u2190 add_div, \u2190 mul_div_assoc, add_sub, sub_add_cancel,\n  sqrt_div zero_le_one, sqrt_one, one_div (sqrt _), ofReal_inv, \u2190 one_div, one_div_one_div, mul_sub, mul_one]\n[GOAL]\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 (jacobiTheta (\u2191(1 / u) * I) - \u2191(sqrt u)) / 2 = (\u2191(sqrt u) * jacobiTheta (\u2191u * I) - \u2191(sqrt u)) / 2\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u22a2 jacobiTheta (\u2191(1 / u) * I) = \u2191(sqrt u) * jacobiTheta (\u2191u * I)\n[PROOFSTEP]\nlet \u03c4 : UpperHalfPlane := .mk (u * I) ((mul_I_im u).symm \u25b8 zero_lt_one.trans hu)\n[GOAL]\ncase e_a.e_a\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u03c4 : UpperHalfPlane := UpperHalfPlane.mk (\u2191u * I) (_ : 0 < (\u2191u * I).im)\n\u22a2 jacobiTheta (\u2191(1 / u) * I) = \u2191(sqrt u) * jacobiTheta (\u2191u * I)\n[PROOFSTEP]\nconvert jacobiTheta_S_smul \u03c4 using 2\n[GOAL]\ncase h.e'_2.h.e'_1\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u03c4 : UpperHalfPlane := UpperHalfPlane.mk (\u2191u * I) (_ : 0 < (\u2191u * I).im)\n\u22a2 \u2191(1 / u) * I = \u2191(ModularGroup.S \u2022 \u03c4)\n[PROOFSTEP]\nrw [UpperHalfPlane.modular_S_smul, UpperHalfPlane.coe_mk, UpperHalfPlane.coe_mk, \u2190 neg_inv, mul_inv, inv_I, mul_neg,\n  neg_neg, one_div, ofReal_inv]\n[GOAL]\ncase h.e'_3.h.e'_5\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u03c4 : UpperHalfPlane := UpperHalfPlane.mk (\u2191u * I) (_ : 0 < (\u2191u * I).im)\n\u22a2 \u2191(sqrt u) = (-I * \u2191\u03c4) ^ (1 / 2)\n[PROOFSTEP]\nrw [UpperHalfPlane.coe_mk, mul_comm, mul_assoc, mul_neg, I_mul_I, neg_neg, mul_one, sqrt_eq_rpow,\n  ofReal_cpow (zero_lt_one.trans hu).le]\n[GOAL]\ncase h.e'_3.h.e'_5\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u03c4 : UpperHalfPlane := UpperHalfPlane.mk (\u2191u * I) (_ : 0 < (\u2191u * I).im)\n\u22a2 \u2191u ^ \u2191(1 / 2) = \u2191u ^ (1 / 2)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_3.h.e'_5\nt : \u211d\nht : 0 < t\nu : \u211d\nhu : 1 < u\n\u03c4 : UpperHalfPlane := UpperHalfPlane.mk (\u2191u * I) (_ : 0 < (\u2191u * I).im)\n\u22a2 \u2191u ^ (1 / 2) = \u2191u ^ (1 / 2)\n[PROOFSTEP]\nrfl\n[GOAL]\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nrcases lt_trichotomy 1 t with (h | h | h)\n[GOAL]\ncase inl\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : 1 < t\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nexact aux h\n[GOAL]\ncase inr.inl\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : 1 = t\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nsimp only [\u2190 h, div_self, Ne.def, one_ne_zero, not_false_iff, sqrt_one, ofReal_one, one_mul]\n[GOAL]\ncase inr.inr\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : t < 1\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nhave := aux (show 1 < 1 / t by rwa [lt_one_div (zero_lt_one' \u211d) ht, div_one])\n[GOAL]\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : t < 1\n\u22a2 1 < 1 / t\n[PROOFSTEP]\nrwa [lt_one_div (zero_lt_one' \u211d) ht, div_one]\n[GOAL]\ncase inr.inr\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : t < 1\nthis : zetaKernel\u2082 (1 / (1 / t)) = \u2191(sqrt (1 / t)) * zetaKernel\u2082 (1 / t)\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nrw [one_div_one_div] at this \n[GOAL]\ncase inr.inr\nt : \u211d\nht : 0 < t\naux : \u2200 {u : \u211d}, 1 < u \u2192 zetaKernel\u2082 (1 / u) = \u2191(sqrt u) * zetaKernel\u2082 u\nh : t < 1\nthis : zetaKernel\u2082 t = \u2191(sqrt (1 / t)) * zetaKernel\u2082 (1 / t)\n\u22a2 zetaKernel\u2082 (1 / t) = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nrw [this, \u2190 mul_assoc, \u2190 ofReal_mul, \u2190 sqrt_mul ht.le, mul_one_div_cancel ht.ne', sqrt_one, ofReal_one, one_mul]\n[GOAL]\n\u22a2 zetaKernel\u2081 =O[atTop] fun t => rexp (-\u03c0 * t)\n[PROOFSTEP]\nhave h := isBigO_at_im_infty_jacobiTheta_sub_one.const_mul_left (1 / 2)\n[GOAL]\nh : (fun x => 1 / 2 * (jacobiTheta x - 1)) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\n\u22a2 zetaKernel\u2081 =O[atTop] fun t => rexp (-\u03c0 * t)\n[PROOFSTEP]\nsimp_rw [mul_comm (1 / 2 : \u2102) _, mul_one_div] at h \n[GOAL]\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\n\u22a2 zetaKernel\u2081 =O[atTop] fun t => rexp (-\u03c0 * t)\n[PROOFSTEP]\nhave h' : Tendsto (fun t : \u211d => \u2191t * I) atTop (comap im atTop) :=\n  by\n  rw [tendsto_comap_iff]\n  convert tendsto_id\n  ext1 t\n  rw [Function.comp_apply, mul_I_im, ofReal_re, id.def]\n[GOAL]\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\n\u22a2 Tendsto (fun t => \u2191t * I) atTop (comap im atTop)\n[PROOFSTEP]\nrw [tendsto_comap_iff]\n[GOAL]\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\n\u22a2 Tendsto (im \u2218 fun t => \u2191t * I) atTop atTop\n[PROOFSTEP]\nconvert tendsto_id\n[GOAL]\ncase h.e'_3\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\n\u22a2 (im \u2218 fun t => \u2191t * I) = id\n[PROOFSTEP]\next1 t\n[GOAL]\ncase h.e'_3.h\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\nt : \u211d\n\u22a2 (im \u2218 fun t => \u2191t * I) t = id t\n[PROOFSTEP]\nrw [Function.comp_apply, mul_I_im, ofReal_re, id.def]\n[GOAL]\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\nh' : Tendsto (fun t => \u2191t * I) atTop (comap im atTop)\n\u22a2 zetaKernel\u2081 =O[atTop] fun t => rexp (-\u03c0 * t)\n[PROOFSTEP]\nconvert\n  ((h.norm_left.comp_tendsto h').congr' (eventually_of_mem (Ioi_mem_atTop 0) fun t ht => _)\n        (eventually_of_mem (Ioi_mem_atTop 0) fun t _ => _)).of_norm_left\n    (E' := \u2102)\n[GOAL]\ncase convert_1\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\nh' : Tendsto (fun t => \u2191t * I) atTop (comap im atTop)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 ((fun x => \u2016(jacobiTheta x - 1) / 2\u2016) \u2218 fun t => \u2191t * I) t = \u2016zetaKernel\u2081 t\u2016\n[PROOFSTEP]\nrw [Function.comp_apply, \u2190 zetaKernel\u2081_eq_jacobiTheta ht]\n[GOAL]\ncase convert_2\nh : (fun x => (jacobiTheta x - 1) / 2) =O[comap im atTop] fun \u03c4 => rexp (-\u03c0 * \u03c4.im)\nh' : Tendsto (fun t => \u2191t * I) atTop (comap im atTop)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n\u22a2 ((fun \u03c4 => rexp (-\u03c0 * \u03c4.im)) \u2218 fun t => \u2191t * I) t = rexp (-\u03c0 * t)\n[PROOFSTEP]\nrw [Function.comp_apply, mul_I_im, ofReal_re]\n[GOAL]\n\u22a2 zetaKernel\u2082 =O[atTop] fun t => rexp (-\u03c0 * t)\n[PROOFSTEP]\nrefine' (eventuallyEq_of_mem (Ioi_mem_atTop (1 : \u211d)) fun t ht => _).trans_isBigO isBigO_atTop_zetaKernel\u2081\n[GOAL]\nt : \u211d\nht : t \u2208 Ioi 1\n\u22a2 zetaKernel\u2082 t = zetaKernel\u2081 t\n[PROOFSTEP]\nrw [zetaKernel\u2082, Pi.add_apply, indicator_of_not_mem (not_mem_Ioc_of_gt (Set.mem_Iio.mp ht)), add_zero]\n[GOAL]\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nhave h1 := isBigO_atTop_zetaKernel\u2082.comp_tendsto tendsto_inv_zero_atTop\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => x\u207b\u00b9) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => x\u207b\u00b9)\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nsimp_rw [\u2190 one_div] at h1 \n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nhave h2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[>] 0] fun t => sqrt t * zetaKernel\u2082 t :=\n  eventually_of_mem self_mem_nhdsWithin fun t ht => by dsimp only; rw [\u2190 zetaKernel\u2082_one_div ht]; rfl\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 (zetaKernel\u2082 \u2218 Div.div 1) t = (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) t\n[PROOFSTEP]\ndsimp only\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 (zetaKernel\u2082 \u2218 Div.div 1) t = \u2191(sqrt t) * zetaKernel\u2082 t\n[PROOFSTEP]\nrw [\u2190 zetaKernel\u2082_one_div ht]\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 (zetaKernel\u2082 \u2218 Div.div 1) t = zetaKernel\u2082 (1 / t)\n[PROOFSTEP]\nrfl\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nhave h3 := h1.congr' h2 (EventuallyEq.refl _ _)\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nhave h4 := h3.mul (isBigO_refl (fun t : \u211d => 1 / (sqrt t : \u2102)) (\ud835\udcdd[>] 0)).norm_right\n[GOAL]\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nrefine h4.congr' ?_ ?_\n[GOAL]\ncase refine_1\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\n\u22a2 (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082\n[PROOFSTEP]\nrefine eventually_of_mem self_mem_nhdsWithin fun x hx => ?_\n[GOAL]\ncase refine_1\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) x = zetaKernel\u2082 x\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine_1\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x)) = zetaKernel\u2082 x\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc, one_div_mul_cancel, one_mul]\n[GOAL]\ncase refine_1\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2191(sqrt x) \u2260 0\n[PROOFSTEP]\nexact ofReal_ne_zero.mpr ((sqrt_ne_zero <| le_of_lt hx).mpr (ne_of_gt hx))\n[GOAL]\ncase refine_2\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\n\u22a2 (fun x => ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t =>\n    rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nrefine eventually_of_mem self_mem_nhdsWithin fun x _ => ?_\n[GOAL]\ncase refine_2\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\nx : \u211d\nx\u271d : x \u2208 Ioi 0\n\u22a2 (fun x => ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016) x = (fun t => rexp (-\u03c0 / t) / sqrt t) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine_2\nh1 : (zetaKernel\u2082 \u2218 fun x => 1 / x) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh2 : zetaKernel\u2082 \u2218 Div.div 1 =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => \u2191(sqrt t) * zetaKernel\u2082 t\nh3 : (fun t => \u2191(sqrt t) * zetaKernel\u2082 t) =O[\ud835\udcdd[Ioi 0] 0] ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x)\nh4 :\n  (fun x => \u2191(sqrt x) * zetaKernel\u2082 x * (1 / \u2191(sqrt x))) =O[\ud835\udcdd[Ioi 0] 0] fun x =>\n    ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016\nx : \u211d\nx\u271d : x \u2208 Ioi 0\n\u22a2 ((fun t => rexp (-\u03c0 * t)) \u2218 fun x => 1 / x) x * \u20161 / \u2191(sqrt x)\u2016 = rexp (-\u03c0 / x) / sqrt x\n[PROOFSTEP]\nrw [Function.comp_apply, mul_one_div, one_div (sqrt x : \u2102), norm_inv, Complex.norm_eq_abs, abs_ofReal,\n  abs_of_nonneg (sqrt_nonneg _), \u2190 div_eq_mul_inv]\n[GOAL]\na : \u211d\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ a\n[PROOFSTEP]\nhave aux1 : IsBigO atTop (fun t => exp (-\u03c0 * t)) fun t => t ^ (-a - 1 / 2) :=\n  (isLittleO_exp_neg_mul_rpow_atTop pi_pos _).isBigO\n[GOAL]\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ a\n[PROOFSTEP]\nhave aux2 : IsBigO atTop (fun t => exp (-\u03c0 * t) * sqrt t) fun t => t ^ (-a) :=\n  by\n  refine (aux1.mul (isBigO_refl sqrt _)).congr' (EventuallyEq.refl _ _) ?_\n  refine (eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_)\n  simp_rw [sqrt_eq_rpow, \u2190 rpow_add ht, sub_add_cancel]\n[GOAL]\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\n\u22a2 (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\n[PROOFSTEP]\nrefine (aux1.mul (isBigO_refl sqrt _)).congr' (EventuallyEq.refl _ _) ?_\n[GOAL]\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\n\u22a2 (fun x => x ^ (-a - 1 / 2) * sqrt x) =\u1da0[atTop] fun t => t ^ (-a)\n[PROOFSTEP]\nrefine (eventually_gt_atTop 0).mp (eventually_of_forall fun t ht => ?_)\n[GOAL]\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\nt : \u211d\nht : 0 < t\n\u22a2 (fun x => x ^ (-a - 1 / 2) * sqrt x) t = (fun t => t ^ (-a)) t\n[PROOFSTEP]\nsimp_rw [sqrt_eq_rpow, \u2190 rpow_add ht, sub_add_cancel]\n[GOAL]\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\naux2 : (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\n\u22a2 zetaKernel\u2082 =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ a\n[PROOFSTEP]\nrefine isBigO_zero_zetaKernel\u2082.trans ((aux2.comp_tendsto tendsto_inv_zero_atTop).congr' ?_ ?_)\n[GOAL]\ncase refine_1\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\naux2 : (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\n\u22a2 ((fun t => rexp (-\u03c0 * t) * sqrt t) \u2218 fun x => x\u207b\u00b9) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => rexp (-\u03c0 / t) / sqrt t\n[PROOFSTEP]\nrefine eventually_of_mem self_mem_nhdsWithin fun x _ => ?_\n[GOAL]\ncase refine_1\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\naux2 : (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\nx : \u211d\nx\u271d : x \u2208 Ioi 0\n\u22a2 ((fun t => rexp (-\u03c0 * t) * sqrt t) \u2218 fun x => x\u207b\u00b9) x = (fun t => rexp (-\u03c0 / t) / sqrt t) x\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, sqrt_inv, \u2190 div_eq_mul_inv]\n[GOAL]\ncase refine_2\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\naux2 : (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\n\u22a2 ((fun t => t ^ (-a)) \u2218 fun x => x\u207b\u00b9) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun t => t ^ a\n[PROOFSTEP]\nrefine eventually_of_mem self_mem_nhdsWithin fun x hx => ?_\n[GOAL]\ncase refine_2\na : \u211d\naux1 : (fun t => rexp (-\u03c0 * t)) =O[atTop] fun t => t ^ (-a - 1 / 2)\naux2 : (fun t => rexp (-\u03c0 * t) * sqrt t) =O[atTop] fun t => t ^ (-a)\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ((fun t => t ^ (-a)) \u2218 fun x => x\u207b\u00b9) x = (fun t => t ^ a) x\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, inv_rpow (le_of_lt hx), rpow_neg (le_of_lt hx), inv_inv]\n[GOAL]\n\u22a2 zetaKernel\u2081 =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\nhave : zetaKernel\u2081 =\u1da0[\ud835\udcdd[>] 0] zetaKernel\u2082 + fun t => ((1 / sqrt t - 1) / 2 : \u2102) :=\n  by\n  refine eventuallyEq_of_mem (Ioc_mem_nhdsWithin_Ioi <| left_mem_Ico.mpr zero_lt_one) fun t h => ?_\n  rw [Pi.add_apply, zetaKernel\u2082, Pi.add_apply, indicator_of_mem h]\n  ring\n[GOAL]\n\u22a2 zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n[PROOFSTEP]\nrefine eventuallyEq_of_mem (Ioc_mem_nhdsWithin_Ioi <| left_mem_Ico.mpr zero_lt_one) fun t h => ?_\n[GOAL]\nt : \u211d\nh : t \u2208 Ioc 0 1\n\u22a2 zetaKernel\u2081 t = (zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2) t\n[PROOFSTEP]\nrw [Pi.add_apply, zetaKernel\u2082, Pi.add_apply, indicator_of_mem h]\n[GOAL]\nt : \u211d\nh : t \u2208 Ioc 0 1\n\u22a2 zetaKernel\u2081 t = zetaKernel\u2081 t + (1 - 1 / \u2191(sqrt t)) / 2 + (1 / \u2191(sqrt t) - 1) / 2\n[PROOFSTEP]\nring\n[GOAL]\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 zetaKernel\u2081 =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\nrefine ((isBigO_zero_zetaKernel\u2082_rpow _).add ?_).congr' this.symm (EventuallyEq.refl _ _)\n[GOAL]\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => (fun t => (1 / \u2191(sqrt t) - 1) / 2) x) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\nsimp_rw [sub_div]\n[GOAL]\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => 1 / \u2191(sqrt x) / 2 - 1 / 2) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\napply IsBigO.sub\n[GOAL]\ncase h\u2081\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => 1 / \u2191(sqrt x) / 2) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\napply IsBigO.of_norm_left\n[GOAL]\ncase h\u2081.a\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => \u20161 / \u2191(sqrt x) / 2\u2016) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\nsimp_rw [norm_div, norm_one, div_eq_mul_inv, one_mul, mul_comm _ \u2016(2 : \u2102)\u2016\u207b\u00b9]\n[GOAL]\ncase h\u2081.a\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => \u20162\u2016\u207b\u00b9 * \u2016\u2191(sqrt x)\u2016\u207b\u00b9) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-2\u207b\u00b9)\n[PROOFSTEP]\nrefine\n  ((isBigO_refl _ _).congr' (EventuallyEq.refl _ _)\n        (eventuallyEq_of_mem self_mem_nhdsWithin fun x hx => ?_)).const_mul_left\n    _\n[GOAL]\ncase h\u2081.a\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 \u2016\u2191(sqrt x)\u2016\u207b\u00b9 = x ^ (-2\u207b\u00b9)\n[PROOFSTEP]\nrw [Complex.norm_eq_abs, abs_ofReal, abs_of_nonneg (sqrt_nonneg _), sqrt_eq_rpow, rpow_neg (le_of_lt hx), one_div]\n[GOAL]\ncase h\u2082\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 (fun x => 1 / 2) =O[\ud835\udcdd[Ioi 0] 0] fun t => t ^ (-(1 / 2))\n[PROOFSTEP]\nrefine isBigO_iff.mpr \u27e8\u2016(1 / 2 : \u2102)\u2016, ?_\u27e9\n[GOAL]\ncase h\u2082\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi 0] 0, \u20161 / 2\u2016 \u2264 \u20161 / 2\u2016 * \u2016x ^ (-(1 / 2))\u2016\n[PROOFSTEP]\nrefine eventually_of_mem (Ioc_mem_nhdsWithin_Ioi <| left_mem_Ico.mpr zero_lt_one) fun t ht => ?_\n[GOAL]\ncase h\u2082\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 \u20161 / 2\u2016 \u2264 \u20161 / 2\u2016 * \u2016t ^ (-(1 / 2))\u2016\n[PROOFSTEP]\nrefine le_mul_of_one_le_right (norm_nonneg _) ?_\n[GOAL]\ncase h\u2082\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 1 \u2264 \u2016t ^ (-(1 / 2))\u2016\n[PROOFSTEP]\nrw [norm_of_nonneg (rpow_nonneg_of_nonneg ht.1.le _), rpow_neg ht.1.le]\n[GOAL]\ncase h\u2082\nthis : zetaKernel\u2081 =\u1da0[\ud835\udcdd[Ioi 0] 0] zetaKernel\u2082 + fun t => (1 / \u2191(sqrt t) - 1) / 2\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 1 \u2264 (t ^ (1 / 2))\u207b\u00b9\n[PROOFSTEP]\nexact one_le_inv (rpow_pos_of_pos ht.1 _) (rpow_le_one ht.1.le ht.2 one_half_pos.le)\n[GOAL]\ns : \u2102\nhs : s \u2260 0\nhs' : s \u2260 1\n\u22a2 DifferentiableAt \u2102 riemannCompletedZeta s\n[PROOFSTEP]\nrefine (differentiable_completed_zeta\u2080.differentiableAt.sub ?_).add ?_\n[GOAL]\ncase refine_1\ns : \u2102\nhs : s \u2260 0\nhs' : s \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => 1 / y) s\n[PROOFSTEP]\nexact (Differentiable.differentiableAt (differentiable_const _)).div differentiableAt_id hs\n[GOAL]\ncase refine_2\ns : \u2102\nhs : s \u2260 0\nhs' : s \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => 1 / (y - 1)) s\n[PROOFSTEP]\nrefine (differentiable_const _).differentiableAt.div ?_ (sub_ne_zero.mpr hs')\n[GOAL]\ncase refine_2\ns : \u2102\nhs : s \u2260 0\nhs' : s \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => y - 1) s\n[PROOFSTEP]\nexact differentiableAt_id.sub (differentiableAt_const _)\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\n\u22a2 DifferentiableAt \u2102 riemannZeta s\n[PROOFSTEP]\nhave c1 :\n  \u2200 (t : \u2102) (_ : t \u2260 0) (_ : t \u2260 1),\n    DifferentiableAt \u2102 (fun u : \u2102 => (\u03c0 : \u2102) ^ (u / 2) * riemannCompletedZeta u / Gamma (u / 2)) t :=\n  by\n  intro t ht ht'\n  apply DifferentiableAt.mul\n  \u00b7 refine (DifferentiableAt.const_cpow ?_ ?_).mul (differentiableAt_completed_zeta ht ht')\n    \u00b7 exact DifferentiableAt.div_const differentiableAt_id _\n    \u00b7 exact Or.inl (ofReal_ne_zero.mpr pi_pos.ne')\n  \u00b7 refine differentiable_one_div_Gamma.differentiableAt.comp t ?_\n    exact\n      DifferentiableAt.div_const differentiableAt_id\n        _\n          -- Second claim: the limit at `s = 0` exists and is equal to `-1 / 2`.\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\n\u22a2 \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n[PROOFSTEP]\nintro t ht ht'\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n[PROOFSTEP]\napply DifferentiableAt.mul\n[GOAL]\ncase ha\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => \u2191\u03c0 ^ (y / 2) * riemannCompletedZeta y) t\n[PROOFSTEP]\nrefine (DifferentiableAt.const_cpow ?_ ?_).mul (differentiableAt_completed_zeta ht ht')\n[GOAL]\ncase ha.refine_1\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => y / 2) t\n[PROOFSTEP]\nexact DifferentiableAt.div_const differentiableAt_id _\n[GOAL]\ncase ha.refine_2\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 \u2191\u03c0 \u2260 0 \u2228 t / 2 \u2260 0\n[PROOFSTEP]\nexact Or.inl (ofReal_ne_zero.mpr pi_pos.ne')\n[GOAL]\ncase hb\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => (Complex.Gamma (y / 2))\u207b\u00b9) t\n[PROOFSTEP]\nrefine differentiable_one_div_Gamma.differentiableAt.comp t ?_\n[GOAL]\ncase hb\ns : \u2102\nhs' : s \u2260 1\nt : \u2102\nht : t \u2260 0\nht' : t \u2260 1\n\u22a2 DifferentiableAt \u2102 (fun y => y / 2) t\n[PROOFSTEP]\nexact\n  DifferentiableAt.div_const differentiableAt_id\n    _\n      -- Second claim: the limit at `s = 0` exists and is equal to `-1 / 2`.\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n\u22a2 DifferentiableAt \u2102 riemannZeta s\n[PROOFSTEP]\nhave c2 : Tendsto (fun s : \u2102 => (\u03c0 : \u2102) ^ (s / 2) * riemannCompletedZeta s / Gamma (s / 2)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd <| -1 / 2) :=\n  by\n  have h1 : Tendsto (fun z : \u2102 => (\u03c0 : \u2102) ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1) :=\n    by\n    convert\n      (ContinuousAt.comp (f := fun z => z / 2) (continuousAt_const_cpow (ofReal_ne_zero.mpr pi_pos.ne'))\n          ?_).tendsto using\n      2\n    \u00b7 simp_rw [Function.comp_apply, zero_div, cpow_zero]\n    \u00b7 exact continuousAt_id.div continuousAt_const two_ne_zero\n  suffices h2 : Tendsto (fun z => riemannCompletedZeta z / Gamma (z / 2)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd <| -1 / 2)\n  \u00b7 convert (h1.mono_left nhdsWithin_le_nhds).mul h2 using 1\n    \u00b7 ext1 x; rw [mul_div]\n    \u00b7 simp only [one_mul]\n  suffices h3 : Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Gamma (z / 2))) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd <| -1 / 2)\n  \u00b7 refine Tendsto.congr' (eventuallyEq_of_mem self_mem_nhdsWithin fun z hz => ?_) h3\n    rw [\u2190 div_div, mul_div_cancel _ (div_ne_zero hz two_ne_zero)]\n  have h4 : Tendsto (fun z : \u2102 => z / 2 * Gamma (z / 2)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd 1) :=\n    by\n    refine tendsto_self_mul_Gamma_nhds_zero.comp ?_\n    rw [tendsto_nhdsWithin_iff, (by simp : \ud835\udcdd (0 : \u2102) = \ud835\udcdd (0 / 2))]\n    exact\n      \u27e8(tendsto_id.div_const _).mono_left nhdsWithin_le_nhds,\n        eventually_of_mem self_mem_nhdsWithin fun x hx => div_ne_zero hx two_ne_zero\u27e9\n  suffices Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd (-1 / 2 : \u2102))\n    by\n    have := this.div h4 one_ne_zero\n    simp_rw [div_one, mul_div_assoc] at this \n    exact this\n  refine Tendsto.div ?_ tendsto_const_nhds two_ne_zero\n  simp_rw [riemannCompletedZeta, add_mul, sub_mul]\n  rw [show \ud835\udcdd (-1 : \u2102) = \ud835\udcdd (0 - 1 + 0) by rw [zero_sub, add_zero]]\n  refine (Tendsto.sub ?_ ?_).add ?_\n  \u00b7 refine Tendsto.mono_left ?_ nhdsWithin_le_nhds\n    have : ContinuousAt riemannCompletedZeta\u2080 0 := differentiable_completed_zeta\u2080.continuous.continuousAt\n    simpa only [id.def, mul_zero] using Tendsto.mul this tendsto_id\n  \u00b7 refine tendsto_const_nhds.congr' (eventuallyEq_of_mem self_mem_nhdsWithin fun t ht => ?_)\n    simp_rw [one_div_mul_cancel ht]\n  \u00b7 refine Tendsto.mono_left ?_ nhdsWithin_le_nhds\n    suffices ContinuousAt (fun z : \u2102 => 1 / (z - 1)) 0 by\n      simpa only [id.def, mul_zero] using Tendsto.mul this tendsto_id\n    refine continuousAt_const.div (continuousAt_id.sub continuousAt_const) ?_\n    simpa only [zero_sub] using neg_ne_zero.mpr one_ne_zero\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n\u22a2 Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nhave h1 : Tendsto (fun z : \u2102 => (\u03c0 : \u2102) ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1) :=\n  by\n  convert\n    (ContinuousAt.comp (f := fun z => z / 2) (continuousAt_const_cpow (ofReal_ne_zero.mpr pi_pos.ne')) ?_).tendsto using\n    2\n  \u00b7 simp_rw [Function.comp_apply, zero_div, cpow_zero]\n  \u00b7 exact continuousAt_id.div continuousAt_const two_ne_zero\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n\u22a2 Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nconvert\n  (ContinuousAt.comp (f := fun z => z / 2) (continuousAt_const_cpow (ofReal_ne_zero.mpr pi_pos.ne')) ?_).tendsto using 2\n[GOAL]\ncase h.e'_5.h.e'_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n\u22a2 1 = ((fun x => \u2191\u03c0 ^ x) \u2218 fun z => z / 2) 0\n[PROOFSTEP]\nsimp_rw [Function.comp_apply, zero_div, cpow_zero]\n[GOAL]\ncase convert_1\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\n\u22a2 ContinuousAt (fun z => z / 2) 0\n[PROOFSTEP]\nexact continuousAt_id.div continuousAt_const two_ne_zero\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nsuffices h2 : Tendsto (fun z => riemannCompletedZeta z / Gamma (z / 2)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd <| -1 / 2)\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh2 : Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nconvert (h1.mono_left nhdsWithin_le_nhds).mul h2 using 1\n[GOAL]\ncase h.e'_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh2 : Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) = fun x =>\n    \u2191\u03c0 ^ (x / 2) * (riemannCompletedZeta x / Complex.Gamma (x / 2))\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_3.h\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh2 : Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nx : \u2102\n\u22a2 \u2191\u03c0 ^ (x / 2) * riemannCompletedZeta x / Complex.Gamma (x / 2) =\n    \u2191\u03c0 ^ (x / 2) * (riemannCompletedZeta x / Complex.Gamma (x / 2))\n[PROOFSTEP]\nrw [mul_div]\n[GOAL]\ncase h.e'_5\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh2 : Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 \ud835\udcdd (-1 / 2) = \ud835\udcdd (1 * (-1 / 2))\n[PROOFSTEP]\nsimp only [one_mul]\n[GOAL]\ncase h2\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nsuffices h3 : Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Gamma (z / 2))) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd <| -1 / 2)\n[GOAL]\ncase h2\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh3 : Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 Tendsto (fun z => riemannCompletedZeta z / Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nrefine Tendsto.congr' (eventuallyEq_of_mem self_mem_nhdsWithin fun z hz => ?_) h3\n[GOAL]\ncase h2\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh3 : Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nz : \u2102\nhz : z \u2208 {0}\u1d9c\n\u22a2 riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2)) = riemannCompletedZeta z / Complex.Gamma (z / 2)\n[PROOFSTEP]\nrw [\u2190 div_div, mul_div_cancel _ (div_ne_zero hz two_ne_zero)]\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nhave h4 : Tendsto (fun z : \u2102 => z / 2 * Gamma (z / 2)) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd 1) :=\n  by\n  refine tendsto_self_mul_Gamma_nhds_zero.comp ?_\n  rw [tendsto_nhdsWithin_iff, (by simp : \ud835\udcdd (0 : \u2102) = \ud835\udcdd (0 / 2))]\n  exact\n    \u27e8(tendsto_id.div_const _).mono_left nhdsWithin_le_nhds,\n      eventually_of_mem self_mem_nhdsWithin fun x hx => div_ne_zero hx two_ne_zero\u27e9\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine tendsto_self_mul_Gamma_nhds_zero.comp ?_\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd[{0}\u1d9c] 0)\n[PROOFSTEP]\nrw [tendsto_nhdsWithin_iff, (by simp : \ud835\udcdd (0 : \u2102) = \ud835\udcdd (0 / 2))]\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 \ud835\udcdd 0 = \ud835\udcdd (0 / 2)\n[PROOFSTEP]\nsimp\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (0 / 2)) \u2227 \u2200\u1da0 (n : \u2102) in \ud835\udcdd[{0}\u1d9c] 0, n / 2 \u2208 {0}\u1d9c\n[PROOFSTEP]\nexact\n  \u27e8(tendsto_id.div_const _).mono_left nhdsWithin_le_nhds,\n    eventually_of_mem self_mem_nhdsWithin fun x hx => div_ne_zero hx two_ne_zero\u27e9\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nsuffices Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd (-1 / 2 : \u2102))\n  by\n  have := this.div h4 one_ne_zero\n  simp_rw [div_one, mul_div_assoc] at this \n  exact this\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nthis : Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nhave := this.div h4 one_ne_zero\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nthis\u271d : Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nthis :\n  Tendsto ((fun z => riemannCompletedZeta z * z / 2) / fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0)\n    (\ud835\udcdd (-1 / 2 / 1))\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nsimp_rw [div_one, mul_div_assoc] at this \n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nthis\u271d : Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nthis :\n  Tendsto ((fun z => riemannCompletedZeta z * (z / 2)) / fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0)\n    (\ud835\udcdd (-1 / 2))\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * (z / 2) / (z / 2 * Complex.Gamma (z / 2))) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nexact this\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * z / 2) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n[PROOFSTEP]\nrefine Tendsto.div ?_ tendsto_const_nhds two_ne_zero\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta z * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1))\n[PROOFSTEP]\nsimp_rw [riemannCompletedZeta, add_mul, sub_mul]\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta\u2080 z * z - 1 / z * z + 1 / (z - 1) * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1))\n[PROOFSTEP]\nrw [show \ud835\udcdd (-1 : \u2102) = \ud835\udcdd (0 - 1 + 0) by rw [zero_sub, add_zero]]\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 \ud835\udcdd (-1) = \ud835\udcdd (0 - 1 + 0)\n[PROOFSTEP]\nrw [zero_sub, add_zero]\n[GOAL]\ncase h3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta\u2080 z * z - 1 / z * z + 1 / (z - 1) * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (0 - 1 + 0))\n[PROOFSTEP]\nrefine (Tendsto.sub ?_ ?_).add ?_\n[GOAL]\ncase h3.refine_1\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta\u2080 z * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine Tendsto.mono_left ?_ nhdsWithin_le_nhds\n[GOAL]\ncase h3.refine_1\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => riemannCompletedZeta\u2080 z * z) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : ContinuousAt riemannCompletedZeta\u2080 0 := differentiable_completed_zeta\u2080.continuous.continuousAt\n[GOAL]\ncase h3.refine_1\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nthis : ContinuousAt riemannCompletedZeta\u2080 0\n\u22a2 Tendsto (fun z => riemannCompletedZeta\u2080 z * z) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [id.def, mul_zero] using Tendsto.mul this tendsto_id\n[GOAL]\ncase h3.refine_2\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => 1 / z * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine tendsto_const_nhds.congr' (eventuallyEq_of_mem self_mem_nhdsWithin fun t ht => ?_)\n[GOAL]\ncase h3.refine_2\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nt : \u2102\nht : t \u2208 {0}\u1d9c\n\u22a2 1 = 1 / t * t\n[PROOFSTEP]\nsimp_rw [one_div_mul_cancel ht]\n[GOAL]\ncase h3.refine_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => 1 / (z - 1) * z) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine Tendsto.mono_left ?_ nhdsWithin_le_nhds\n[GOAL]\ncase h3.refine_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun z => 1 / (z - 1) * z) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices ContinuousAt (fun z : \u2102 => 1 / (z - 1)) 0 by simpa only [id.def, mul_zero] using Tendsto.mul this tendsto_id\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\nthis : ContinuousAt (fun z => 1 / (z - 1)) 0\n\u22a2 Tendsto (fun z => 1 / (z - 1) * z) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [id.def, mul_zero] using Tendsto.mul this tendsto_id\n[GOAL]\ncase h3.refine_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 ContinuousAt (fun z => 1 / (z - 1)) 0\n[PROOFSTEP]\nrefine continuousAt_const.div (continuousAt_id.sub continuousAt_const) ?_\n[GOAL]\ncase h3.refine_3\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nh1 : Tendsto (fun z => \u2191\u03c0 ^ (z / 2)) (\ud835\udcdd 0) (\ud835\udcdd 1)\nh4 : Tendsto (fun z => z / 2 * Complex.Gamma (z / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 1)\n\u22a2 0 - 1 \u2260 0\n[PROOFSTEP]\nsimpa only [zero_sub] using neg_ne_zero.mpr one_ne_zero\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\n\u22a2 DifferentiableAt \u2102 riemannZeta s\n[PROOFSTEP]\nrcases ne_or_eq s 0 with (hs | rfl)\n[GOAL]\ncase inl\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs : s \u2260 0\n\u22a2 DifferentiableAt \u2102 riemannZeta s\n[PROOFSTEP]\nhave : {(0 : \u2102)}\u1d9c \u2208 \ud835\udcdd s := isOpen_compl_singleton.mem_nhds hs\n[GOAL]\ncase inl\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs : s \u2260 0\nthis : {0}\u1d9c \u2208 \ud835\udcdd s\n\u22a2 DifferentiableAt \u2102 riemannZeta s\n[PROOFSTEP]\nrefine (c1 s hs hs').congr_of_eventuallyEq (eventuallyEq_of_mem this fun x hx => ?_)\n[GOAL]\ncase inl\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs : s \u2260 0\nthis : {0}\u1d9c \u2208 \ud835\udcdd s\nx : \u2102\nhx : x \u2208 {0}\u1d9c\n\u22a2 riemannZeta x = \u2191\u03c0 ^ (x / 2) * riemannCompletedZeta x / Complex.Gamma (x / 2)\n[PROOFSTEP]\nrw [riemannZeta_def]\n[GOAL]\ncase inl\ns : \u2102\nhs' : s \u2260 1\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs : s \u2260 0\nthis : {0}\u1d9c \u2208 \ud835\udcdd s\nx : \u2102\nhx : x \u2208 {0}\u1d9c\n\u22a2 Function.update (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) 0 (-1 / 2) x =\n    \u2191\u03c0 ^ (x / 2) * riemannCompletedZeta x / Complex.Gamma (x / 2)\n[PROOFSTEP]\napply Function.update_noteq hx\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\n\u22a2 DifferentiableAt \u2102 riemannZeta 0\n[PROOFSTEP]\nrw [riemannZeta, \u2190 (lim_eq_iff \u27e8-1 / 2, c2\u27e9).mpr c2]\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\n\u22a2 DifferentiableAt \u2102\n    (Function.update (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) 0\n      (lim (map (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0))))\n    0\n[PROOFSTEP]\nhave S_nhds : {(1 : \u2102)}\u1d9c \u2208 \ud835\udcdd (0 : \u2102) := isOpen_compl_singleton.mem_nhds hs'\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 DifferentiableAt \u2102\n    (Function.update (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) 0\n      (lim (map (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0))))\n    0\n[PROOFSTEP]\nrefine\n  ((Complex.differentiableOn_update_limUnder_of_isLittleO S_nhds (fun t ht => (c1 t ht.2 ht.1).differentiableWithinAt)\n          ?_)\n        0 hs').differentiableAt\n    S_nhds\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 (fun z =>\n      \u2191\u03c0 ^ (z / 2) * riemannCompletedZeta z / Complex.Gamma (z / 2) -\n        \u2191\u03c0 ^ (0 / 2) * riemannCompletedZeta 0 / Complex.Gamma (0 / 2)) =o[\ud835\udcdd[{0}\u1d9c] 0]\n    fun z => (z - 0)\u207b\u00b9\n[PROOFSTEP]\nsimp only [zero_div, div_zero, Complex.Gamma_zero, mul_zero, cpow_zero, sub_zero]\n  -- Remains to show completed zeta is `o (s ^ (-1))` near 0.\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 (fun z => \u2191\u03c0 ^ (z / 2) * riemannCompletedZeta z / Complex.Gamma (z / 2)) =o[\ud835\udcdd[{0}\u1d9c] 0] fun z => z\u207b\u00b9\n[PROOFSTEP]\nrefine (isBigO_const_of_tendsto c2 <| one_ne_zero' \u2102).trans_isLittleO ?_\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 (fun _x => 1) =o[\ud835\udcdd[{0}\u1d9c] 0] fun z => z\u207b\u00b9\n[PROOFSTEP]\nrw [isLittleO_iff_tendsto']\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 Tendsto (fun x => 1 / x\u207b\u00b9) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nexact Tendsto.congr (fun x => by rw [\u2190 one_div, one_div_one_div]) nhdsWithin_le_nhds\n[GOAL]\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\nx : \u2102\n\u22a2 x = 1 / x\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 one_div, one_div_one_div]\n[GOAL]\ncase inr\nc1 :\n  \u2200 (t : \u2102),\n    t \u2260 0 \u2192 t \u2260 1 \u2192 DifferentiableAt \u2102 (fun u => \u2191\u03c0 ^ (u / 2) * riemannCompletedZeta u / Complex.Gamma (u / 2)) t\nc2 : Tendsto (fun s => \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s / Complex.Gamma (s / 2)) (\ud835\udcdd[{0}\u1d9c] 0) (\ud835\udcdd (-1 / 2))\nhs' : 0 \u2260 1\nS_nhds : {1}\u1d9c \u2208 \ud835\udcdd 0\n\u22a2 \u2200\u1da0 (x : \u2102) in \ud835\udcdd[{0}\u1d9c] 0, x\u207b\u00b9 = 0 \u2192 1 = 0\n[PROOFSTEP]\nexact eventually_of_mem self_mem_nhdsWithin fun x hx hx' => (hx <| inv_eq_zero.mp hx').elim\n[GOAL]\nn : \u2115\n\u22a2 riemannZeta (-2 * (\u2191n + 1)) = 0\n[PROOFSTEP]\nhave : (-2 : \u2102) * (n + 1) \u2260 0 := mul_ne_zero (neg_ne_zero.mpr two_ne_zero) (Nat.cast_add_one_ne_zero n)\n[GOAL]\nn : \u2115\nthis : -2 * (\u2191n + 1) \u2260 0\n\u22a2 riemannZeta (-2 * (\u2191n + 1)) = 0\n[PROOFSTEP]\nrw [riemannZeta, Function.update_noteq this, show -2 * ((n : \u2102) + 1) / 2 = -\u2191(n + 1) by push_cast ; ring,\n  Complex.Gamma_neg_nat_eq_zero, div_zero]\n[GOAL]\nn : \u2115\nthis : -2 * (\u2191n + 1) \u2260 0\n\u22a2 -2 * (\u2191n + 1) / 2 = -\u2191(n + 1)\n[PROOFSTEP]\npush_cast\n[GOAL]\nn : \u2115\nthis : -2 * (\u2191n + 1) \u2260 0\n\u22a2 -2 * (\u2191n + 1) / 2 = -(\u2191n + 1)\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => 1 / \u2191(sqrt t)) s (1 / (s - 1 / 2))\n[PROOFSTEP]\nhave h1 : EqOn (fun t => 1 / \u2191(sqrt t) : \u211d \u2192 \u2102) (fun t => (t : \u2102) ^ (-1 / 2 : \u2102)) (Ioc 0 1) :=\n  by\n  intro t ht\n  simp_rw [neg_div, cpow_neg, \u2190 one_div, sqrt_eq_rpow, ofReal_cpow ht.1.le]\n  norm_num\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n[PROOFSTEP]\nintro t ht\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 (fun t => 1 / \u2191(sqrt t)) t = (fun t => \u2191t ^ (-1 / 2)) t\n[PROOFSTEP]\nsimp_rw [neg_div, cpow_neg, \u2190 one_div, sqrt_eq_rpow, ofReal_cpow ht.1.le]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nt : \u211d\nht : t \u2208 Ioc 0 1\n\u22a2 1 / \u2191t ^ \u2191(1 / 2) = 1 / \u2191t ^ (1 / 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh1 : EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => 1 / \u2191(sqrt t)) s (1 / (s - 1 / 2))\n[PROOFSTEP]\nsimp_rw [indicator_congr h1, (by ring : s - 1 / 2 = s + -1 / 2)]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh1 : EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n\u22a2 s - 1 / 2 = s + -1 / 2\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh1 : EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => \u2191t ^ (-1 / 2)) s (1 / (s + -1 / 2))\n[PROOFSTEP]\nconvert hasMellin_cpow_Ioc (-1 / 2) _\n[GOAL]\ncase convert_2\ns : \u2102\nhs : 1 / 2 < s.re\nh1 : EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n\u22a2 0 < s.re + (-1 / 2).re\n[PROOFSTEP]\nrwa [(by norm_num : (-1 / 2 : \u2102) = (-1 / 2 : \u211d)), ofReal_re, neg_div, \u2190 sub_eq_add_neg, sub_pos]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh1 : EqOn (fun t => 1 / \u2191(sqrt t)) (fun t => \u2191t ^ (-1 / 2)) (Ioc 0 1)\n\u22a2 -1 / 2 = \u2191(-1 / 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => (1 - 1 / \u2191(sqrt t)) / 2) s (1 / (2 * s) - 1 / (2 * s - 1))\n[PROOFSTEP]\nhave step1 : HasMellin (indicator (Ioc 0 1) (fun t => 1 - 1 / \u2191(sqrt t) : \u211d \u2192 \u2102)) s (1 / s - 1 / (s - 1 / 2)) :=\n  by\n  have a := hasMellin_one_Ioc (one_half_pos.trans hs)\n  have b := hasMellin_one_div_sqrt_Ioc hs\n  simpa only [a.2, b.2, \u2190 indicator_sub] using\n    hasMellin_sub a.1\n      b.1\n        -- todo: implement something like \"indicator.const_div\" (blocked by the port for now)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n[PROOFSTEP]\nhave a := hasMellin_one_Ioc (one_half_pos.trans hs)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\na : HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s)\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n[PROOFSTEP]\nhave b := hasMellin_one_div_sqrt_Ioc hs\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\na : HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s)\nb : HasMellin (indicator (Ioc 0 1) fun t => 1 / \u2191(sqrt t)) s (1 / (s - 1 / 2))\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n[PROOFSTEP]\nsimpa only [a.2, b.2, \u2190 indicator_sub] using\n  hasMellin_sub a.1\n    b.1\n      -- todo: implement something like \"indicator.const_div\" (blocked by the port for now)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n\u22a2 HasMellin (indicator (Ioc 0 1) fun t => (1 - 1 / \u2191(sqrt t)) / 2) s (1 / (2 * s) - 1 / (2 * s - 1))\n[PROOFSTEP]\nrw [show\n    ((Ioc 0 1).indicator fun t => (1 - 1 / (sqrt t : \u2102)) / 2) = fun t =>\n      (Ioc 0 1).indicator (fun t => 1 - 1 / (sqrt t : \u2102)) t / 2\n    by ext1 t; simp_rw [div_eq_inv_mul, indicator_mul_right]]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n\u22a2 (indicator (Ioc 0 1) fun t => (1 - 1 / \u2191(sqrt t)) / 2) = fun t =>\n    indicator (Ioc 0 1) (fun t => 1 - 1 / \u2191(sqrt t)) t / 2\n[PROOFSTEP]\next1 t\n[GOAL]\ncase h\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\nt : \u211d\n\u22a2 indicator (Ioc 0 1) (fun t => (1 - 1 / \u2191(sqrt t)) / 2) t = indicator (Ioc 0 1) (fun t => 1 - 1 / \u2191(sqrt t)) t / 2\n[PROOFSTEP]\nsimp_rw [div_eq_inv_mul, indicator_mul_right]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n\u22a2 HasMellin (fun t => indicator (Ioc 0 1) (fun t => 1 - 1 / \u2191(sqrt t)) t / 2) s (1 / (2 * s) - 1 / (2 * s - 1))\n[PROOFSTEP]\nsimp_rw [HasMellin, mellin_div_const, step1.2, sub_div, div_div]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n\u22a2 MellinConvergent (fun t => indicator (Ioc 0 1) (fun t => 1 - 1 / \u2191(sqrt t)) t / 2) s \u2227\n    1 / (s * 2) - 1 / ((s - 1 / 2) * 2) = 1 / (2 * s) - 1 / (2 * s - 1)\n[PROOFSTEP]\nrefine \u27e8step1.1.div_const _, ?_\u27e9\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nstep1 : HasMellin (indicator (Ioc 0 1) fun t => 1 - 1 / \u2191(sqrt t)) s (1 / s - 1 / (s - 1 / 2))\n\u22a2 1 / (s * 2) - 1 / ((s - 1 / 2) * 2) = 1 / (2 * s) - 1 / (2 * s - 1)\n[PROOFSTEP]\nrw [mul_comm, sub_mul, div_mul_cancel _ (two_ne_zero' \u2102), mul_comm s 2]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 mellin zetaKernel\u2082 s = mellin zetaKernel\u2081 s + 1 / (2 * s) - 1 / (2 * s - 1)\n[PROOFSTEP]\nhave h :=\n  mellinConvergent_of_isBigO_rpow_exp pi_pos locally_integrable_zetaKernel\u2081 isBigO_atTop_zetaKernel\u2081\n    isBigO_zero_zetaKernel\u2081 hs\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh : MellinConvergent zetaKernel\u2081 s\n\u22a2 mellin zetaKernel\u2082 s = mellin zetaKernel\u2081 s + 1 / (2 * s) - 1 / (2 * s - 1)\n[PROOFSTEP]\nhave h' := hasMellin_one_div_sqrt_sub_one_div_two_Ioc hs\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nh : MellinConvergent zetaKernel\u2081 s\nh' : HasMellin (indicator (Ioc 0 1) fun t => (1 - 1 / \u2191(sqrt t)) / 2) s (1 / (2 * s) - 1 / (2 * s - 1))\n\u22a2 mellin zetaKernel\u2082 s = mellin zetaKernel\u2081 s + 1 / (2 * s) - 1 / (2 * s - 1)\n[PROOFSTEP]\nsimp_rw [zetaKernel\u2082, Pi.add_def, add_sub_assoc, (hasMellin_add h h'.1).2, h'.2]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 riemannCompletedZeta s = mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nhave : 1 / 2 < (s / 2).re :=\n  by\n  rw [show s / 2 = \u2191(2\u207b\u00b9 : \u211d) * s by push_cast ; rw [mul_comm]; rfl]\n  rwa [ofReal_mul_re, \u2190 div_eq_inv_mul, div_lt_div_right (zero_lt_two' \u211d)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 1 / 2 < (s / 2).re\n[PROOFSTEP]\nrw [show s / 2 = \u2191(2\u207b\u00b9 : \u211d) * s by push_cast ; rw [mul_comm]; rfl]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = \u21912\u207b\u00b9 * s\n[PROOFSTEP]\npush_cast\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = 2\u207b\u00b9 * s\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = s * 2\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 1 / 2 < (\u21912\u207b\u00b9 * s).re\n[PROOFSTEP]\nrwa [ofReal_mul_re, \u2190 div_eq_inv_mul, div_lt_div_right (zero_lt_two' \u211d)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 riemannCompletedZeta s = mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nrw [riemannCompletedZeta, riemannCompletedZeta\u2080, mellin_zetaKernel\u2082_eq_of_lt_re this, sub_add, sub_sub, \u2190 add_sub]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 mellin zetaKernel\u2081 (s / 2) + (1 / (2 * (s / 2)) - (1 / (2 * (s / 2) - 1) + (1 / s - 1 / (s - 1)))) =\n    mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 add_zero (mellin zetaKernel\u2081 <| s / 2)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n| mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nrw [\u2190 add_zero (mellin zetaKernel\u2081 <| s / 2)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n| mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nrw [\u2190 add_zero (mellin zetaKernel\u2081 <| s / 2)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n| mellin zetaKernel\u2081 (s / 2)\n[PROOFSTEP]\nrw [\u2190 add_zero (mellin zetaKernel\u2081 <| s / 2)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 mellin zetaKernel\u2081 (s / 2) + (1 / (2 * (s / 2)) - (1 / (2 * (s / 2) - 1) + (1 / s - 1 / (s - 1)))) =\n    mellin zetaKernel\u2081 (s / 2) + 0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 1 / (2 * (s / 2)) - (1 / (2 * (s / 2) - 1) + (1 / s - 1 / (s - 1))) = 0\n[PROOFSTEP]\nrw [mul_div_cancel' _ (two_ne_zero' \u2102)]\n[GOAL]\ncase e_a\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 1 / s - (1 / (s - 1) + (1 / s - 1 / (s - 1))) = 0\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\ns : \u2102\nhs : 1 < s.re\nthis : 1 / 2 < (s / 2).re\n\u22a2 1 / s - (1 / (s - 1) + (1 / s - 1 / (s - 1))) = 0\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    \u2191\u03c0 ^ (-s) * Complex.Gamma s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nrw [Complex.Gamma_eq_integral hs, GammaIntegral_eq_mellin]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    \u2191\u03c0 ^ (-s) * mellin (fun x => \u2191(rexp (-x))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nconv_rhs =>\n  congr\n  rw [\u2190 smul_eq_mul, \u2190 mellin_comp_mul_left _ _ pi_pos]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n| \u2191\u03c0 ^ (-s) * mellin (fun x => \u2191(rexp (-x))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\n  congr\n  rw [\u2190 smul_eq_mul, \u2190 mellin_comp_mul_left _ _ pi_pos]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n| \u2191\u03c0 ^ (-s) * mellin (fun x => \u2191(rexp (-x))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\n  congr\n  rw [\u2190 smul_eq_mul, \u2190 mellin_comp_mul_left _ _ pi_pos]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n| \u2191\u03c0 ^ (-s) * mellin (fun x => \u2191(rexp (-x))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n| \u2191\u03c0 ^ (-s) * mellin (fun x => \u2191(rexp (-x))) s\ncase a s : \u2102 hs : 0 < s.re n : \u2115 | 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, \u2190 mellin_comp_mul_left _ _ pi_pos]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nhave : 1 / ((n : \u2102) + 1) ^ (2 * s) = (((n : \u211d) + 1) ^ (2 : \u211d) : \u2102) ^ (-s) :=\n  by\n  rw [(by norm_num : (n : \u2102) + 1 = \u2191((n : \u211d) + 1)), (by norm_num : 2 * s = \u2191(2 : \u211d) * s), cpow_mul_ofReal_nonneg,\n    one_div, cpow_neg]\n  rw [\u2190 Nat.cast_succ]\n  exact Nat.cast_nonneg _\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n[PROOFSTEP]\nrw [(by norm_num : (n : \u2102) + 1 = \u2191((n : \u211d) + 1)), (by norm_num : 2 * s = \u2191(2 : \u211d) * s), cpow_mul_ofReal_nonneg, one_div,\n  cpow_neg]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 \u2191n + 1 = \u2191(\u2191n + 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 2 * s = \u21912 * s\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase hx\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 0 \u2264 \u2191n + 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_succ]\n[GOAL]\ncase hx\ns : \u2102\nhs : 0 < s.re\nn : \u2115\n\u22a2 0 \u2264 \u2191(Nat.succ n)\n[PROOFSTEP]\nexact Nat.cast_nonneg _\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nconv_rhs => rw [this, mul_comm, \u2190 smul_eq_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n| mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nrw [this, mul_comm, \u2190 smul_eq_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n| mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nrw [this, mul_comm, \u2190 smul_eq_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n| mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s * (1 / (\u2191n + 1) ^ (2 * s))\n[PROOFSTEP]\nrw [this, mul_comm, \u2190 smul_eq_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    \u2191((\u2191n + 1) ^ 2) ^ (-s) \u2022 mellin (fun t => \u2191(rexp (-(\u03c0 * t)))) s\n[PROOFSTEP]\nrw [\u2190 mellin_comp_mul_right _ _ (show 0 < ((n : \u211d) + 1) ^ (2 : \u211d) by positivity)]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n\u22a2 0 < (\u2191n + 1) ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\n\u22a2 \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) =\n    mellin (fun t => \u2191(rexp (-(\u03c0 * (t * (\u2191n + 1) ^ 2))))) s\n[PROOFSTEP]\nrefine set_integral_congr measurableSet_Ioi fun t _ => ?_\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n\u22a2 \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = \u2191t ^ (s - 1) \u2022 (fun t => \u2191(rexp (-(\u03c0 * (t * (\u2191n + 1) ^ 2))))) t\n[PROOFSTEP]\nsimp_rw [smul_eq_mul]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n\u22a2 \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = \u2191t ^ (s - 1) * \u2191(rexp (-(\u03c0 * (t * (\u2191n + 1) ^ 2))))\n[PROOFSTEP]\ncongr 3\n[GOAL]\ncase e_a.e_r.e_x\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n\u22a2 -\u03c0 * t * (\u2191n + 1) ^ 2 = -(\u03c0 * (t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Nat.cast_two, rpow_nat_cast]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n| -(\u03c0 * (t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, rpow_nat_cast]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n| -(\u03c0 * (t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, rpow_nat_cast]\n[GOAL]\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n| -(\u03c0 * (t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, rpow_nat_cast]\n[GOAL]\ncase e_a.e_r.e_x\ns : \u2102\nhs : 0 < s.re\nn : \u2115\nthis : 1 / (\u2191n + 1) ^ (2 * s) = \u2191((\u2191n + 1) ^ 2) ^ (-s)\nt : \u211d\nx\u271d : t \u2208 Ioi 0\n\u22a2 -\u03c0 * t * (\u2191n + 1) ^ 2 = -(\u03c0 * (t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nlet bd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * exp (-\u03c0 * t * ((n : \u211d) + 1) ^ 2)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nlet f : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => (t : \u2102) ^ (s - 1) * exp (-\u03c0 * t * ((n : \u211d) + 1) ^ 2)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave hm : MeasurableSet (Ioi (0 : \u211d)) := measurableSet_Ioi\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_norm : \u2200 (n : \u2115) {t : \u211d} (_ : 0 < t), \u2016f n t\u2016 = bd n t :=\n  by\n  intro n t ht\n  rw [norm_mul, Complex.norm_eq_abs, Complex.norm_eq_abs, Complex.abs_of_nonneg (exp_pos _).le,\n    abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\n\u22a2 \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\n[PROOFSTEP]\nintro n t ht\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nn : \u2115\nt : \u211d\nht : 0 < t\n\u22a2 \u2016f n t\u2016 = bd n t\n[PROOFSTEP]\nrw [norm_mul, Complex.norm_eq_abs, Complex.norm_eq_abs, Complex.abs_of_nonneg (exp_pos _).le,\n  abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave hf_meas : \u2200 n : \u2115, AEStronglyMeasurable (f n) (volume.restrict <| Ioi 0) :=\n  by\n  intro n\n  refine (ContinuousOn.mul ?_ ?_).aestronglyMeasurable hm\n  \u00b7 exact ContinuousAt.continuousOn fun x hx => continuousAt_ofReal_cpow_const _ _ <| Or.inr <| ne_of_gt hx\n  \u00b7 apply Continuous.continuousOn\n    exact continuous_ofReal.comp (continuous_exp.comp ((continuous_const.mul continuous_id').mul continuous_const))\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\n\u22a2 \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\n[PROOFSTEP]\nintro n\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nn : \u2115\n\u22a2 AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\n[PROOFSTEP]\nrefine (ContinuousOn.mul ?_ ?_).aestronglyMeasurable hm\n[GOAL]\ncase refine_1\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nn : \u2115\n\u22a2 ContinuousOn (fun t => \u2191t ^ (s - 1)) (Ioi 0)\n[PROOFSTEP]\nexact ContinuousAt.continuousOn fun x hx => continuousAt_ofReal_cpow_const _ _ <| Or.inr <| ne_of_gt hx\n[GOAL]\ncase refine_2\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nn : \u2115\n\u22a2 ContinuousOn (fun t => \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))) (Ioi 0)\n[PROOFSTEP]\napply Continuous.continuousOn\n[GOAL]\ncase refine_2.h\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nn : \u2115\n\u22a2 Continuous fun t => \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\n[PROOFSTEP]\nexact continuous_ofReal.comp (continuous_exp.comp ((continuous_const.mul continuous_id').mul continuous_const))\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_le : \u2200 n : \u2115, \u2200\u1d50 t : \u211d \u2202volume.restrict (Ioi 0), \u2016f n t\u2016 \u2264 bd n t := fun n =>\n  (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => le_of_eq (h_norm n ht))\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_sum0 : \u2200 {t : \u211d} (_ : 0 < t), HasSum (fun n => f n t) ((t : \u2102) ^ (s - 1) * zetaKernel\u2081 t) :=\n  by\n  intro t ht\n  rw [zetaKernel\u2081]\n  convert (hasSum_ofReal.mpr (summable_exp_neg_pi_mul_nat_sq ht).hasSum).mul_left ((t : \u2102) ^ (s - 1))\n  simp only [neg_mul, ofReal_exp, ofReal_neg, ofReal_mul, ofReal_pow, ofReal_add, ofReal_nat_cast, ofReal_one,\n    ofReal_tsum]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\n\u22a2 \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\n[PROOFSTEP]\nintro t ht\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nt : \u211d\nht : 0 < t\n\u22a2 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\n[PROOFSTEP]\nrw [zetaKernel\u2081]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nt : \u211d\nht : 0 < t\n\u22a2 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * \u2211' (n : \u2115), \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)))\n[PROOFSTEP]\nconvert (hasSum_ofReal.mpr (summable_exp_neg_pi_mul_nat_sq ht).hasSum).mul_left ((t : \u2102) ^ (s - 1))\n[GOAL]\ncase h.e'_6.h.e'_6\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nt : \u211d\nht : 0 < t\n\u22a2 \u2211' (n : \u2115), \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)) = \u2191(\u2211' (b : \u2115), rexp (-\u03c0 * t * (\u2191b + 1) ^ 2))\n[PROOFSTEP]\nsimp only [neg_mul, ofReal_exp, ofReal_neg, ofReal_mul, ofReal_pow, ofReal_add, ofReal_nat_cast, ofReal_one,\n  ofReal_tsum]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_sum' : \u2200\u1d50 t : \u211d \u2202volume.restrict (Ioi 0), HasSum (fun n : \u2115 => f n t) ((t : \u2102) ^ (s - 1) * zetaKernel\u2081 t) :=\n  (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => h_sum0 ht)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_sum : \u2200\u1d50 t : \u211d \u2202volume.restrict (Ioi 0), Summable fun n : \u2115 => bd n t :=\n  by\n  refine (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => ?_)\n  simpa only [fun n => h_norm n ht] using summable_norm_iff.mpr (h_sum0 ht).summable\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\n\u22a2 \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\n[PROOFSTEP]\nrefine (ae_restrict_iff' hm).mpr (ae_of_all _ fun t ht => ?_)\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 Summable fun n => bd n t\n[PROOFSTEP]\nsimpa only [fun n => h_norm n ht] using summable_norm_iff.mpr (h_sum0 ht).summable\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nhave h_int : Integrable (fun t : \u211d => \u2211' n : \u2115, bd n t) (volume.restrict (Ioi 0)) :=\n  by\n  refine\n    IntegrableOn.congr_fun\n      (mellinConvergent_of_isBigO_rpow_exp pi_pos locally_integrable_zetaKernel\u2081 isBigO_atTop_zetaKernel\u2081\n          isBigO_zero_zetaKernel\u2081 hs).norm\n      (fun t ht => ?_) hm\n  rw [tsum_mul_left, norm_smul, Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re, zetaKernel\u2081, \u2190\n    ofReal_tsum, Complex.norm_eq_abs, Complex.abs_of_nonneg]\n  exact tsum_nonneg fun n => (exp_pos _).le\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\n\u22a2 Integrable fun t => \u2211' (n : \u2115), bd n t\n[PROOFSTEP]\nrefine\n  IntegrableOn.congr_fun\n    (mellinConvergent_of_isBigO_rpow_exp pi_pos locally_integrable_zetaKernel\u2081 isBigO_atTop_zetaKernel\u2081\n        isBigO_zero_zetaKernel\u2081 hs).norm\n    (fun t ht => ?_) hm\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 \u2016\u2191t ^ (s - 1) \u2022 zetaKernel\u2081 t\u2016 = \u2211' (n : \u2115), bd n t\n[PROOFSTEP]\nrw [tsum_mul_left, norm_smul, Complex.norm_eq_abs, abs_cpow_eq_rpow_re_of_pos ht, sub_re, one_re, zetaKernel\u2081, \u2190\n  ofReal_tsum, Complex.norm_eq_abs, Complex.abs_of_nonneg]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 0 \u2264 \u2211' (a : \u2115), rexp (-\u03c0 * t * (\u2191a + 1) ^ 2)\n[PROOFSTEP]\nexact tsum_nonneg fun n => (exp_pos _).le\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nh_int : Integrable fun t => \u2211' (n : \u2115), bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2191\u03c0 ^ (-s) * Complex.Gamma s * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ (2 * s)\n[PROOFSTEP]\nrw [\u2190 tsum_mul_left]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nh_int : Integrable fun t => \u2211' (n : \u2115), bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2211' (x : \u2115), \u2191\u03c0 ^ (-s) * Complex.Gamma s * (1 / (\u2191x + 1) ^ (2 * s))\n[PROOFSTEP]\nsimp_rw [\u2190 integral_cpow_mul_exp_neg_pi_mul_sq (one_half_pos.trans hs)]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nh_int : Integrable fun t => \u2211' (n : \u2115), bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u2211' (x : \u2115), \u222b (t : \u211d) in Ioi 0, \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191x + 1) ^ 2))\n[PROOFSTEP]\nrw [\u2190 (hasSum_integral_of_dominated_convergence bd hf_meas h_le h_sum h_int h_sum').tsum_eq.symm]\n[GOAL]\ns : \u2102\nhs : 1 / 2 < s.re\nbd : \u2115 \u2192 \u211d \u2192 \u211d := fun n t => t ^ (s.re - 1) * rexp (-\u03c0 * t * (\u2191n + 1) ^ 2)\nf : \u2115 \u2192 \u211d \u2192 \u2102 := fun n t => \u2191t ^ (s - 1) * \u2191(rexp (-\u03c0 * t * (\u2191n + 1) ^ 2))\nhm : MeasurableSet (Ioi 0)\nh_norm : \u2200 (n : \u2115) {t : \u211d}, 0 < t \u2192 \u2016f n t\u2016 = bd n t\nhf_meas : \u2200 (n : \u2115), AEStronglyMeasurable (f n) (Measure.restrict volume (Ioi 0))\nh_le : \u2200 (n : \u2115), \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), \u2016f n t\u2016 \u2264 bd n t\nh_sum0 : \u2200 {t : \u211d}, 0 < t \u2192 HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum' : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), HasSum (fun n => f n t) (\u2191t ^ (s - 1) * zetaKernel\u2081 t)\nh_sum : \u2200\u1d50 (t : \u211d) \u2202Measure.restrict volume (Ioi 0), Summable fun n => bd n t\nh_int : Integrable fun t => \u2211' (n : \u2115), bd n t\n\u22a2 mellin zetaKernel\u2081 s = \u222b (a : \u211d) in Ioi 0, \u2191a ^ (s - 1) * zetaKernel\u2081 a\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 riemannCompletedZeta s = \u2191\u03c0 ^ (-s / 2) * Complex.Gamma (s / 2) * \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ s\n[PROOFSTEP]\nrw [completed_zeta_eq_mellin_of_one_lt_re hs, mellin_zetaKernel\u2081_eq_tsum, neg_div, mul_div_cancel' _ (two_ne_zero' \u2102)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 1 / 2 < (s / 2).re\n[PROOFSTEP]\nrw [show s / 2 = \u2191(2\u207b\u00b9 : \u211d) * s by push_cast ; rw [mul_comm]; rfl]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = \u21912\u207b\u00b9 * s\n[PROOFSTEP]\npush_cast\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = 2\u207b\u00b9 * s\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s / 2 = s * 2\u207b\u00b9\n[PROOFSTEP]\nrfl\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 1 / 2 < (\u21912\u207b\u00b9 * s).re\n[PROOFSTEP]\nrwa [ofReal_mul_re, \u2190 div_eq_inv_mul, div_lt_div_right (zero_lt_two' \u211d)]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 riemannZeta s = \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ s\n[PROOFSTEP]\nhave : s \u2260 0 := by contrapose! hs; rw [hs, zero_re]; exact zero_le_one\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s \u2260 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ns : \u2102\nhs : s = 0\n\u22a2 s.re \u2264 1\n[PROOFSTEP]\nrw [hs, zero_re]\n[GOAL]\ns : \u2102\nhs : s = 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 riemannZeta s = \u2211' (n : \u2115), 1 / (\u2191n + 1) ^ s\n[PROOFSTEP]\nrw [riemannZeta, Function.update_noteq this, completed_zeta_eq_tsum_of_one_lt_re hs, \u2190 mul_assoc, neg_div, cpow_neg,\n  mul_inv_cancel_left\u2080, mul_div_cancel_left]\n[GOAL]\ncase ha\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 Complex.Gamma (s / 2) \u2260 0\n[PROOFSTEP]\napply Gamma_ne_zero_of_re_pos\n[GOAL]\ncase ha.hs\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 0 < (s / 2).re\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_comm, show (2\u207b\u00b9 : \u2102) = (2\u207b\u00b9 : \u211d) by norm_num, ofReal_mul_re]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 2\u207b\u00b9 = \u21912\u207b\u00b9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase ha.hs\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 0 < 2\u207b\u00b9 * s.re\n[PROOFSTEP]\nexact mul_pos (inv_pos_of_pos two_pos) (zero_lt_one.trans hs)\n[GOAL]\ncase h\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 \u2191\u03c0 ^ (s / 2) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, cpow_eq_zero_iff, not_and_or, \u2190 Ne.def, ofReal_ne_zero]\n[GOAL]\ncase h\ns : \u2102\nhs : 1 < s.re\nthis : s \u2260 0\n\u22a2 \u03c0 \u2260 0 \u2228 \u00acs / 2 \u2260 0\n[PROOFSTEP]\nexact Or.inl pi_pos.ne'\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 riemannZeta s = \u2211' (n : \u2115), 1 / \u2191n ^ s\n[PROOFSTEP]\nhave hs' : s \u2260 0 := by contrapose! hs; rw [hs, zero_re]; exact zero_le_one\n[GOAL]\ns : \u2102\nhs : 1 < s.re\n\u22a2 s \u2260 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ns : \u2102\nhs : s = 0\n\u22a2 s.re \u2264 1\n[PROOFSTEP]\nrw [hs, zero_re]\n[GOAL]\ns : \u2102\nhs : s = 0\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nhs' : s \u2260 0\n\u22a2 riemannZeta s = \u2211' (n : \u2115), 1 / \u2191n ^ s\n[PROOFSTEP]\nrw [tsum_eq_zero_add]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nhs' : s \u2260 0\n\u22a2 riemannZeta s = 1 / \u21910 ^ s + \u2211' (b : \u2115), 1 / \u2191(b + 1) ^ s\n[PROOFSTEP]\nsimp_rw [Nat.cast_zero, zero_cpow hs', div_zero, zero_add, zeta_eq_tsum_one_div_nat_add_one_cpow hs, Nat.cast_add,\n  Nat.cast_one]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nhs' : s \u2260 0\n\u22a2 Summable fun n => 1 / \u2191n ^ s\n[PROOFSTEP]\nrw [\u2190 summable_norm_iff]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nhs' : s \u2260 0\n\u22a2 Summable fun x => \u20161 / \u2191x ^ s\u2016\n[PROOFSTEP]\nsimp_rw [norm_div, norm_one, Complex.norm_eq_abs, \u2190 ofReal_nat_cast,\n  abs_cpow_eq_rpow_re_of_nonneg (Nat.cast_nonneg _) (zero_lt_one.trans hs).ne', summable_one_div_nat_rpow]\n[GOAL]\ns : \u2102\nhs : 1 < s.re\nhs' : s \u2260 0\n\u22a2 1 < s.re\n[PROOFSTEP]\nassumption\n[GOAL]\nk : \u2115\nhk : 1 < k\n\u22a2 riemannZeta \u2191k = \u2211' (n : \u2115), 1 / \u2191n ^ k\n[PROOFSTEP]\nsimp only [zeta_eq_tsum_one_div_nat_cpow\n    (by rwa [\u2190 ofReal_nat_cast, ofReal_re, \u2190 Nat.cast_one, Nat.cast_lt] : 1 < re k),\n  cpow_nat_cast]\n[GOAL]\nk : \u2115\nhk : 1 < k\n\u22a2 1 < (\u2191k).re\n[PROOFSTEP]\nrwa [\u2190 ofReal_nat_cast, ofReal_re, \u2190 Nat.cast_one, Nat.cast_lt]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n\u22a2 riemannZeta (2 * \u2191k) = (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u2191\u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)!\n[PROOFSTEP]\nconvert congr_arg ((\u2191) : \u211d \u2192 \u2102) (hasSum_zeta_nat hk).tsum_eq\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 riemannZeta (2 * \u2191k) = \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * k))\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, \u2190 Nat.cast_mul, zeta_nat_eq_tsum_of_gt_one]\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 \u2211' (n : \u2115), 1 / \u2191n ^ (2 * k) = \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * k))\n[PROOFSTEP]\nrw [ofReal_tsum]\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 \u2211' (n : \u2115), 1 / \u2191n ^ (2 * k) = \u2211' (a : \u2115), \u2191(1 / \u2191a ^ (2 * k))\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 1 < 2 * k\n[PROOFSTEP]\nrefine one_lt_two.trans_le ?_\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 2 \u2264 2 * k\n[PROOFSTEP]\nconv_lhs => rw [\u2190 mul_one 2]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 mul_one 2]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 mul_one 2]\n[GOAL]\nk : \u2115\nhk : k \u2260 0\n| 2\n[PROOFSTEP]\nrw [\u2190 mul_one 2]\n[GOAL]\ncase h.e'_2\nk : \u2115\nhk : k \u2260 0\n\u22a2 2 * 1 \u2264 2 * k\n[PROOFSTEP]\nrwa [mul_le_mul_left (zero_lt_two' \u2115), Nat.one_le_iff_ne_zero]\n[GOAL]\ncase h.e'_3\nk : \u2115\nhk : k \u2260 0\n\u22a2 (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u2191\u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)! =\n    \u2191((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * \u03c0 ^ (2 * k) * \u2191(bernoulli (2 * k)) / \u2191(2 * k)!)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 riemannZeta 2 = \u2191\u03c0 ^ 2 / 6\n[PROOFSTEP]\nconvert congr_arg ((\u2191) : \u211d \u2192 \u2102) hasSum_zeta_two.tsum_eq\n[GOAL]\ncase h.e'_2\n\u22a2 riemannZeta 2 = \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ 2)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two, ofReal_tsum]\n[GOAL]\ncase h.e'_2\n\u22a2 \u2211' (n : \u2115), 1 / \u2191n ^ 2 = \u2211' (a : \u2115), \u2191(1 / \u2191a ^ 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_3\n\u22a2 \u2191\u03c0 ^ 2 / 6 = \u2191(\u03c0 ^ 2 / 6)\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 riemannZeta 4 = \u2191(\u03c0 ^ 4) / 90\n[PROOFSTEP]\nconvert congr_arg ((\u2191) : \u211d \u2192 \u2102) hasSum_zeta_four.tsum_eq\n[GOAL]\ncase h.e'_2\n\u22a2 riemannZeta 4 = \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ 4)\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one, show (4 : \u2102) = (4 : \u2115) by norm_num, zeta_nat_eq_tsum_of_gt_one (by norm_num : 1 < 4), ofReal_tsum]\n[GOAL]\n\u22a2 4 = \u21914\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\u22a2 1 < 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_2\n\u22a2 \u2211' (n : \u2115), 1 / \u2191n ^ 4 = \u2211' (a : \u2115), \u2191(\u21911 / \u2191a ^ 4)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_3\n\u22a2 \u2191(\u03c0 ^ 4) / 90 = \u2191(\u03c0 ^ 4 / 90)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ns : \u2102\n\u22a2 riemannCompletedZeta\u2080 (1 - s) = riemannCompletedZeta\u2080 s\n[PROOFSTEP]\nhave := mellin_comp_rpow zetaKernel\u2082 (s / 2 - 1 / 2) neg_one_lt_zero.ne\n[GOAL]\ns : \u2102\nthis :\n  mellin (fun t => zetaKernel\u2082 (t ^ (-1))) (s / 2 - 1 / 2) = |(-1)|\u207b\u00b9 \u2022 mellin zetaKernel\u2082 ((s / 2 - 1 / 2) / \u2191(-1))\n\u22a2 riemannCompletedZeta\u2080 (1 - s) = riemannCompletedZeta\u2080 s\n[PROOFSTEP]\nsimp_rw [rpow_neg_one, \u2190 one_div, abs_neg, abs_one, div_one, one_smul, ofReal_neg, ofReal_one, div_neg, div_one,\n  neg_sub] at this \n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\n\u22a2 riemannCompletedZeta\u2080 (1 - s) = riemannCompletedZeta\u2080 s\n[PROOFSTEP]\nconv_lhs => rw [riemannCompletedZeta\u2080, sub_div, \u2190 this]\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\n| riemannCompletedZeta\u2080 (1 - s)\n[PROOFSTEP]\nrw [riemannCompletedZeta\u2080, sub_div, \u2190 this]\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\n| riemannCompletedZeta\u2080 (1 - s)\n[PROOFSTEP]\nrw [riemannCompletedZeta\u2080, sub_div, \u2190 this]\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\n| riemannCompletedZeta\u2080 (1 - s)\n[PROOFSTEP]\nrw [riemannCompletedZeta\u2080, sub_div, \u2190 this]\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\n\u22a2 mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = riemannCompletedZeta\u2080 s\n[PROOFSTEP]\nrefine set_integral_congr measurableSet_Ioi fun t ht => ?_\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 \u2191t ^ (s / 2 - 1 / 2 - 1) \u2022 (fun t => zetaKernel\u2082 (1 / t)) t = \u2191t ^ (s / 2 - 1) \u2022 zetaKernel\u2082 t\n[PROOFSTEP]\nsimp_rw [zetaKernel\u2082_one_div ht, smul_eq_mul, \u2190 mul_assoc, sqrt_eq_rpow, ofReal_cpow (le_of_lt ht), \u2190\n  cpow_add _ _ (ofReal_ne_zero.mpr <| ne_of_gt ht)]\n[GOAL]\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 \u2191t ^ (s / 2 - 1 / 2 - 1 + \u2191(1 / 2)) * zetaKernel\u2082 t = \u2191t ^ (s / 2 - 1) * zetaKernel\u2082 t\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 s / 2 - 1 / 2 - 1 + \u2191(1 / 2) = s / 2 - 1\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_a.e_a\ns : \u2102\nthis : mellin (fun t => zetaKernel\u2082 (1 / t)) (s / 2 - 1 / 2) = mellin zetaKernel\u2082 (1 / 2 - s / 2)\nt : \u211d\nht : t \u2208 Ioi 0\n\u22a2 s / 2 - 1 / 2 - 1 + 1 / 2 = s / 2 - 1\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\n\u22a2 riemannCompletedZeta (1 - s) = riemannCompletedZeta s\n[PROOFSTEP]\nsimp_rw [riemannCompletedZeta, riemannCompletedZeta\u2080_one_sub, sub_add, (by abel : 1 - s - 1 = -s),\n  (by abel : 1 - s = -(s - 1)), div_neg, neg_sub_neg]\n[GOAL]\ns : \u2102\n\u22a2 1 - s - 1 = -s\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u2102\n\u22a2 1 - s - 1 = -s\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u2102\n\u22a2 1 - s = -(s - 1)\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u2102\n\u22a2 1 - s = -(s - 1)\n[PROOFSTEP]\nabel\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave hs_ne : s \u2260 0 := by contrapose! hs; rw [hs]; exact \u27e80, by rw [Nat.cast_zero, neg_zero]\u27e9\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\n\u22a2 s \u2260 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nhs : s = 0\n\u22a2 \u2203 n, s = -\u2191n\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nhs : s = 0\n\u22a2 \u2203 n, 0 = -\u2191n\n[PROOFSTEP]\nexact \u27e80, by rw [Nat.cast_zero, neg_zero]\u27e9\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nhs : s = 0\n\u22a2 0 = -\u21910\n[PROOFSTEP]\nrw [Nat.cast_zero, neg_zero]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave h_sqrt : (sqrt \u03c0 : \u2102) \u2260 0 := ofReal_ne_zero.mpr (sqrt_ne_zero'.mpr pi_pos)\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave h_pow : (2 : \u2102) ^ (s - 1) \u2260 0 := by\n  rw [Ne.def, cpow_eq_zero_iff, not_and_or]\n  exact Or.inl two_ne_zero\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\n\u22a2 2 ^ (s - 1) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, cpow_eq_zero_iff, not_and_or]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\n\u22a2 \u00ac2 = 0 \u2228 \u00acs - 1 \u2260 0\n[PROOFSTEP]\nexact Or.inl two_ne_zero\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave h_Ga_ne1 : Gamma (s / 2) \u2260 0 := by\n  rw [Ne.def, Complex.Gamma_eq_zero_iff]\n  contrapose! hs\n  obtain \u27e8m, hm\u27e9 := hs\n  rw [div_eq_iff (two_ne_zero' \u2102), \u2190 Nat.cast_two, neg_mul, \u2190 Nat.cast_mul] at hm \n  exact \u27e8m * 2, by rw [hm]\u27e9\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\n\u22a2 Complex.Gamma (s / 2) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, Complex.Gamma_eq_zero_iff]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\n\u22a2 \u00ac\u2203 m, s / 2 = -\u2191m\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nhs : \u2203 m, s / 2 = -\u2191m\n\u22a2 \u2203 n, s = -\u2191n\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hs\n[GOAL]\ncase intro\ns : \u2102\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nm : \u2115\nhm : s / 2 = -\u2191m\n\u22a2 \u2203 n, s = -\u2191n\n[PROOFSTEP]\nrw [div_eq_iff (two_ne_zero' \u2102), \u2190 Nat.cast_two, neg_mul, \u2190 Nat.cast_mul] at hm \n[GOAL]\ncase intro\ns : \u2102\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nm : \u2115\nhm : s = -\u2191(m * 2)\n\u22a2 \u2203 n, s = -\u2191n\n[PROOFSTEP]\nexact \u27e8m * 2, by rw [hm]\u27e9\n[GOAL]\ns : \u2102\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nm : \u2115\nhm : s = -\u2191(m * 2)\n\u22a2 s = -\u2191(m * 2)\n[PROOFSTEP]\nrw [hm]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave h_Ga_eq : Gamma s = Gamma (s / 2) * Gamma ((s + 1) / 2) * (2 : \u2102) ^ (s - 1) / sqrt \u03c0 := by\n  rw [add_div, Complex.Gamma_mul_Gamma_add_half, mul_div_cancel' _ (two_ne_zero' \u2102), (by ring : 1 - s = -(s - 1)),\n    cpow_neg, \u2190 div_eq_mul_inv, eq_div_iff h_sqrt, div_mul_eq_mul_div\u2080, div_mul_cancel _ h_pow]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\n\u22a2 Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\n[PROOFSTEP]\nrw [add_div, Complex.Gamma_mul_Gamma_add_half, mul_div_cancel' _ (two_ne_zero' \u2102), (by ring : 1 - s = -(s - 1)),\n  cpow_neg, \u2190 div_eq_mul_inv, eq_div_iff h_sqrt, div_mul_eq_mul_div\u2080, div_mul_cancel _ h_pow]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\n\u22a2 1 - s = -(s - 1)\n[PROOFSTEP]\nring\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nhave h_Ga_ne3 : Gamma ((s + 1) / 2) \u2260 0 :=\n  by\n  have h_Ga_aux : Gamma s \u2260 0 := Complex.Gamma_ne_zero hs\n  contrapose! h_Ga_aux\n  rw [h_Ga_eq, h_Ga_aux, mul_zero, zero_mul, zero_div]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\n\u22a2 Complex.Gamma ((s + 1) / 2) \u2260 0\n[PROOFSTEP]\nhave h_Ga_aux : Gamma s \u2260 0 := Complex.Gamma_ne_zero hs\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_aux : Complex.Gamma s \u2260 0\n\u22a2 Complex.Gamma ((s + 1) / 2) \u2260 0\n[PROOFSTEP]\ncontrapose! h_Ga_aux\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_aux : Complex.Gamma ((s + 1) / 2) = 0\n\u22a2 Complex.Gamma s = 0\n[PROOFSTEP]\nrw [h_Ga_eq, h_Ga_aux, mul_zero, zero_mul, zero_div]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\n\u22a2 riemannZeta (1 - s) = 2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * riemannZeta s\n[PROOFSTEP]\nrw [riemannZeta, Function.update_noteq (by rwa [sub_ne_zero, ne_comm] : 1 - s \u2260 0), Function.update_noteq hs_ne,\n  riemannCompletedZeta_one_sub, mul_div, eq_div_iff h_Ga_ne1, mul_comm, \u2190 mul_div_assoc]\n  -- Now rule out case of s = positive odd integer & deduce further non-vanishing statements\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\n\u22a2 1 - s \u2260 0\n[PROOFSTEP]\nrwa [sub_ne_zero, ne_comm]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nby_cases hs_pos_odd : \u2203 n : \u2115, s = 1 + 2 * n\n[GOAL]\ncase pos\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u2203 n, s = 1 + 2 * \u2191n\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hs_pos_odd\n[GOAL]\ncase pos.intro\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\n\u22a2 Complex.Gamma ((1 + 2 * \u2191n) / 2) * (\u2191\u03c0 ^ ((1 - (1 + 2 * \u2191n)) / 2) * riemannCompletedZeta (1 + 2 * \u2191n)) /\n      Complex.Gamma ((1 - (1 + 2 * \u2191n)) / 2) =\n    2 ^ (1 - (1 + 2 * \u2191n)) * \u2191\u03c0 ^ (-(1 + 2 * \u2191n)) * Complex.Gamma (1 + 2 * \u2191n) *\n        Complex.sin (\u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2) *\n      (\u2191\u03c0 ^ ((1 + 2 * \u2191n) / 2) * riemannCompletedZeta (1 + 2 * \u2191n))\n[PROOFSTEP]\nhave : (1 - (1 + 2 * (n : \u2102))) / 2 = -\u2191n := by\n  rw [\u2190 sub_sub, sub_self, zero_sub, neg_div, mul_div_cancel_left _ (two_ne_zero' \u2102)]\n[GOAL]\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\n\u22a2 (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n[PROOFSTEP]\nrw [\u2190 sub_sub, sub_self, zero_sub, neg_div, mul_div_cancel_left _ (two_ne_zero' \u2102)]\n[GOAL]\ncase pos.intro\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n\u22a2 Complex.Gamma ((1 + 2 * \u2191n) / 2) * (\u2191\u03c0 ^ ((1 - (1 + 2 * \u2191n)) / 2) * riemannCompletedZeta (1 + 2 * \u2191n)) /\n      Complex.Gamma ((1 - (1 + 2 * \u2191n)) / 2) =\n    2 ^ (1 - (1 + 2 * \u2191n)) * \u2191\u03c0 ^ (-(1 + 2 * \u2191n)) * Complex.Gamma (1 + 2 * \u2191n) *\n        Complex.sin (\u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2) *\n      (\u2191\u03c0 ^ ((1 + 2 * \u2191n) / 2) * riemannCompletedZeta (1 + 2 * \u2191n))\n[PROOFSTEP]\nrw [this, Complex.Gamma_neg_nat_eq_zero, div_zero]\n[GOAL]\ncase pos.intro\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n\u22a2 0 =\n    2 ^ (1 - (1 + 2 * \u2191n)) * \u2191\u03c0 ^ (-(1 + 2 * \u2191n)) * Complex.Gamma (1 + 2 * \u2191n) *\n        Complex.sin (\u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2) *\n      (\u2191\u03c0 ^ ((1 + 2 * \u2191n) / 2) * riemannCompletedZeta (1 + 2 * \u2191n))\n[PROOFSTEP]\nhave : (\u03c0 : \u2102) * (1 - (1 + 2 * \u2191n)) / 2 = \u2191(-n : \u2124) * \u03c0 := by push_cast ; field_simp; ring\n[GOAL]\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n\u22a2 \u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2 = \u2191(-\u2191n) * \u2191\u03c0\n[PROOFSTEP]\npush_cast\n[GOAL]\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n\u22a2 \u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n * \u2191\u03c0\n[PROOFSTEP]\nfield_simp\n[GOAL]\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\n\u22a2 \u2191\u03c0 * (2 * \u2191n) = \u2191n * \u2191\u03c0 * 2\n[PROOFSTEP]\nring\n[GOAL]\ncase pos.intro\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nn : \u2115\nhs : \u2200 (n_1 : \u2115), 1 + 2 * \u2191n \u2260 -\u2191n_1\nhs' : 1 + 2 * \u2191n \u2260 1\nhs_ne : 1 + 2 * \u2191n \u2260 0\nh_pow : 2 ^ (1 + 2 * \u2191n - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma ((1 + 2 * \u2191n) / 2) \u2260 0\nh_Ga_eq :\n  Complex.Gamma (1 + 2 * \u2191n) =\n    Complex.Gamma ((1 + 2 * \u2191n) / 2) * Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) * 2 ^ (1 + 2 * \u2191n - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((1 + 2 * \u2191n + 1) / 2) \u2260 0\nthis\u271d : (1 - (1 + 2 * \u2191n)) / 2 = -\u2191n\nthis : \u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2 = \u2191(-\u2191n) * \u2191\u03c0\n\u22a2 0 =\n    2 ^ (1 - (1 + 2 * \u2191n)) * \u2191\u03c0 ^ (-(1 + 2 * \u2191n)) * Complex.Gamma (1 + 2 * \u2191n) *\n        Complex.sin (\u2191\u03c0 * (1 - (1 + 2 * \u2191n)) / 2) *\n      (\u2191\u03c0 ^ ((1 + 2 * \u2191n) / 2) * riemannCompletedZeta (1 + 2 * \u2191n))\n[PROOFSTEP]\nrw [this, Complex.sin_int_mul_pi, mul_zero, zero_mul]\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nhave h_Ga_ne4 : Gamma ((1 - s) / 2) \u2260 0 :=\n  by\n  rw [Ne.def, Complex.Gamma_eq_zero_iff]\n  contrapose! hs_pos_odd\n  obtain \u27e8m, hm\u27e9 := hs_pos_odd\n  rw [div_eq_iff (two_ne_zero' \u2102), sub_eq_iff_eq_add, neg_mul, \u2190 sub_eq_neg_add, eq_sub_iff_add_eq] at hm \n  exact\n    \u27e8m, by rw [\u2190 hm, mul_comm]\u27e9\n      -- At last the main proof\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\n\u22a2 Complex.Gamma ((1 - s) / 2) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, Complex.Gamma_eq_zero_iff]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\n\u22a2 \u00ac\u2203 m, (1 - s) / 2 = -\u2191m\n[PROOFSTEP]\ncontrapose! hs_pos_odd\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u2203 m, (1 - s) / 2 = -\u2191m\n\u22a2 \u2203 n, s = 1 + 2 * \u2191n\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := hs_pos_odd\n[GOAL]\ncase intro\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nm : \u2115\nhm : (1 - s) / 2 = -\u2191m\n\u22a2 \u2203 n, s = 1 + 2 * \u2191n\n[PROOFSTEP]\nrw [div_eq_iff (two_ne_zero' \u2102), sub_eq_iff_eq_add, neg_mul, \u2190 sub_eq_neg_add, eq_sub_iff_add_eq] at hm \n[GOAL]\ncase intro\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nm : \u2115\nhm : 1 + \u2191m * 2 = s\n\u22a2 \u2203 n, s = 1 + 2 * \u2191n\n[PROOFSTEP]\nexact\n  \u27e8m, by rw [\u2190 hm, mul_comm]\u27e9\n    -- At last the main proof\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nm : \u2115\nhm : 1 + \u2191m * 2 = s\n\u22a2 s = 1 + 2 * \u2191m\n[PROOFSTEP]\nrw [\u2190 hm, mul_comm]\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * Complex.sin (\u2191\u03c0 * (1 - s) / 2) * (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nrw [show sin (\u2191\u03c0 * (1 - s) / 2) = \u03c0 * (Gamma ((1 - s) / 2) * Gamma (s / 2 + 1 / 2))\u207b\u00b9\n    by\n    have := congr_arg Inv.inv (Complex.Gamma_mul_Gamma_one_sub ((1 - s) / 2)).symm\n    rwa [(by ring : 1 - (1 - s) / 2 = s / 2 + 1 / 2), inv_div, div_eq_iff (ofReal_ne_zero.mpr pi_pos.ne'),\n      mul_comm _ (\u03c0 : \u2102), mul_div_assoc'] at this ]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 Complex.sin (\u2191\u03c0 * (1 - s) / 2) = \u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9\n[PROOFSTEP]\nhave := congr_arg Inv.inv (Complex.Gamma_mul_Gamma_one_sub ((1 - s) / 2)).symm\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : (\u2191\u03c0 / Complex.sin (\u2191\u03c0 * ((1 - s) / 2)))\u207b\u00b9 = (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (1 - (1 - s) / 2))\u207b\u00b9\n\u22a2 Complex.sin (\u2191\u03c0 * (1 - s) / 2) = \u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9\n[PROOFSTEP]\nrwa [(by ring : 1 - (1 - s) / 2 = s / 2 + 1 / 2), inv_div, div_eq_iff (ofReal_ne_zero.mpr pi_pos.ne'),\n  mul_comm _ (\u03c0 : \u2102), mul_div_assoc'] at this \n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : (\u2191\u03c0 / Complex.sin (\u2191\u03c0 * ((1 - s) / 2)))\u207b\u00b9 = (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (1 - (1 - s) / 2))\u207b\u00b9\n\u22a2 1 - (1 - s) / 2 = s / 2 + 1 / 2\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    2 ^ (1 - s) * \u2191\u03c0 ^ (-s) * Complex.Gamma s * (\u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9) *\n      (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nrw [(by rw [\u2190 neg_sub] : (2 : \u2102) ^ (1 - s) = (2 : \u2102) ^ (-(s - 1))), cpow_neg, h_Ga_eq]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 2 ^ (1 - s) = 2 ^ (-(s - 1))\n[PROOFSTEP]\nrw [\u2190 neg_sub]\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    (2 ^ (s - 1))\u207b\u00b9 * \u2191\u03c0 ^ (-s) * (Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)) *\n        (\u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9) *\n      (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nsuffices (\u03c0 : \u2102) ^ ((1 - s) / 2) = (\u03c0 : \u2102) ^ (-s) * sqrt \u03c0 * (\u03c0 : \u2102) ^ (s / 2) by rw [this]; field_simp; ring_nf;\n  rw [\u2190 ofReal_pow, sq_sqrt pi_pos.le]; ring\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ ((1 - s) / 2) * riemannCompletedZeta s) / Complex.Gamma ((1 - s) / 2) =\n    (2 ^ (s - 1))\u207b\u00b9 * \u2191\u03c0 ^ (-s) * (Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)) *\n        (\u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9) *\n      (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s) /\n      Complex.Gamma ((1 - s) / 2) =\n    (2 ^ (s - 1))\u207b\u00b9 * \u2191\u03c0 ^ (-s) * (Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)) *\n        (\u2191\u03c0 * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma (s / 2 + 1 / 2))\u207b\u00b9) *\n      (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s)\n[PROOFSTEP]\nfield_simp\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n\u22a2 Complex.Gamma (s / 2) * (\u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s) *\n      (2 ^ (s - 1) * \u2191(sqrt \u03c0) * (Complex.Gamma ((1 - s) / 2) * Complex.Gamma ((s + 1) / 2))) =\n    \u2191\u03c0 ^ (-s) * (Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1)) * \u2191\u03c0 *\n        (\u2191\u03c0 ^ (s / 2) * riemannCompletedZeta s) *\n      Complex.Gamma ((1 - s) / 2)\n[PROOFSTEP]\nring_nf\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n\u22a2 Complex.Gamma (s * (\u2191(Int.ofNat 1) / \u21912)) * \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) ^ 2 * \u2191\u03c0 ^ (s * (\u2191(Int.ofNat 1) / \u21912)) *\n            riemannCompletedZeta s *\n          2 ^ (-1 + s) *\n        Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * \u2191(Int.negOfNat 1) * (\u2191(Int.ofNat 1) / \u21912)) *\n      Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * (\u2191(Int.ofNat 1) / \u21912)) =\n    Complex.Gamma (s * (\u2191(Int.ofNat 1) / \u21912)) * \u2191\u03c0 ^ (-s) * \u2191\u03c0 ^ (s * (\u2191(Int.ofNat 1) / \u21912)) * riemannCompletedZeta s *\n            2 ^ (-1 + s) *\n          Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * \u2191(Int.negOfNat 1) * (\u2191(Int.ofNat 1) / \u21912)) *\n        Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * (\u2191(Int.ofNat 1) / \u21912)) *\n      \u2191\u03c0\n[PROOFSTEP]\nrw [\u2190 ofReal_pow, sq_sqrt pi_pos.le]\n[GOAL]\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\nthis : \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n\u22a2 Complex.Gamma (s * (\u2191(Int.ofNat 1) / \u21912)) * \u2191\u03c0 ^ (-s) * \u2191\u03c0 * \u2191\u03c0 ^ (s * (\u2191(Int.ofNat 1) / \u21912)) *\n            riemannCompletedZeta s *\n          2 ^ (-1 + s) *\n        Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * \u2191(Int.negOfNat 1) * (\u2191(Int.ofNat 1) / \u21912)) *\n      Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * (\u2191(Int.ofNat 1) / \u21912)) =\n    Complex.Gamma (s * (\u2191(Int.ofNat 1) / \u21912)) * \u2191\u03c0 ^ (-s) * \u2191\u03c0 ^ (s * (\u2191(Int.ofNat 1) / \u21912)) * riemannCompletedZeta s *\n            2 ^ (-1 + s) *\n          Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * \u2191(Int.negOfNat 1) * (\u2191(Int.ofNat 1) / \u21912)) *\n        Complex.Gamma (\u2191(Int.ofNat 1) / \u21912 + s * (\u2191(Int.ofNat 1) / \u21912)) *\n      \u2191\u03c0\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s) * \u2191(sqrt \u03c0) * \u2191\u03c0 ^ (s / 2)\n[PROOFSTEP]\nsimp_rw [sqrt_eq_rpow, ofReal_cpow pi_pos.le, \u2190 cpow_add _ _ (ofReal_ne_zero.mpr pi_pos.ne')]\n[GOAL]\ncase neg\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 \u2191\u03c0 ^ ((1 - s) / 2) = \u2191\u03c0 ^ (-s + \u2191(1 / 2) + s / 2)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.e_a\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 (1 - s) / 2 = -s + \u2191(1 / 2) + s / 2\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase neg.e_a\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 (1 - s) / 2 = -s + 1 / 2 + s / 2\n[PROOFSTEP]\nfield_simp\n[GOAL]\ncase neg.e_a\ns : \u2102\nhs : \u2200 (n : \u2115), s \u2260 -\u2191n\nhs' : s \u2260 1\nhs_ne : s \u2260 0\nh_sqrt : \u2191(sqrt \u03c0) \u2260 0\nh_pow : 2 ^ (s - 1) \u2260 0\nh_Ga_ne1 : Complex.Gamma (s / 2) \u2260 0\nh_Ga_eq : Complex.Gamma s = Complex.Gamma (s / 2) * Complex.Gamma ((s + 1) / 2) * 2 ^ (s - 1) / \u2191(sqrt \u03c0)\nh_Ga_ne3 : Complex.Gamma ((s + 1) / 2) \u2260 0\nhs_pos_odd : \u00ac\u2203 n, s = 1 + 2 * \u2191n\nh_Ga_ne4 : Complex.Gamma ((1 - s) / 2) \u2260 0\n\u22a2 1 - s = -(s * 2) + 1 + s\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\n\u22a2 riemannZeta (-\u2191k) = (-1) ^ k * \u2191(bernoulli (k + 1)) / (\u2191k + 1)\n[PROOFSTEP]\nrcases Nat.even_or_odd' k with \u27e8m, rfl | rfl\u27e9\n[GOAL]\ncase intro.inl\nm : \u2115\n\u22a2 riemannZeta (-\u2191(2 * m)) = (-1) ^ (2 * m) * \u2191(bernoulli (2 * m + 1)) / (\u2191(2 * m) + 1)\n[PROOFSTEP]\ncases' m with m m\n[GOAL]\ncase intro.inl.zero\n\u22a2 riemannZeta (-\u2191(2 * Nat.zero)) = (-1) ^ (2 * Nat.zero) * \u2191(bernoulli (2 * Nat.zero + 1)) / (\u2191(2 * Nat.zero) + 1)\n[PROOFSTEP]\nrw [Nat.zero_eq, mul_zero, Nat.cast_zero, pow_zero, one_mul, zero_add, neg_zero, zero_add, div_one, bernoulli_one,\n  riemannZeta_zero]\n[GOAL]\ncase intro.inl.zero\n\u22a2 -1 / 2 = \u2191(-1 / 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inl.succ\nm : \u2115\n\u22a2 riemannZeta (-\u2191(2 * Nat.succ m)) =\n    (-1) ^ (2 * Nat.succ m) * \u2191(bernoulli (2 * Nat.succ m + 1)) / (\u2191(2 * Nat.succ m) + 1)\n[PROOFSTEP]\nrw [Nat.cast_mul, \u2190 neg_mul, Nat.cast_two, Nat.cast_succ, riemannZeta_neg_two_mul_nat_add_one,\n  bernoulli_eq_bernoulli'_of_ne_one]\n[GOAL]\ncase intro.inl.succ\nm : \u2115\n\u22a2 0 = (-1) ^ (2 * Nat.succ m) * \u2191(bernoulli' (2 * Nat.succ m + 1)) / (2 * (\u2191m + 1) + 1)\ncase intro.inl.succ m : \u2115 \u22a2 2 * Nat.succ m + 1 \u2260 1\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.inl.succ\nm : \u2115\n\u22a2 2 * Nat.succ m + 1 \u2260 1\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase intro.inl.succ.h\nm : \u2115\n\u22a2 1 < 2 * Nat.succ m + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inl.succ\nm : \u2115\n\u22a2 0 = (-1) ^ (2 * Nat.succ m) * \u2191(bernoulli' (2 * Nat.succ m + 1)) / (2 * (\u2191m + 1) + 1)\n[PROOFSTEP]\nrw [bernoulli'_odd_eq_zero \u27e8m + 1, rfl\u27e9 (by norm_num), Rat.cast_zero, mul_zero, zero_div]\n[GOAL]\nm : \u2115\n\u22a2 1 < 2 * (m + 1) + 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 riemannZeta (-\u2191(2 * m + 1)) = (-1) ^ (2 * m + 1) * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [Odd.neg_one_pow \u27e8m, rfl\u27e9]\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 riemannZeta (-\u2191(2 * m + 1)) = -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [show -(\u2191(2 * m + 1) : \u2102) = 1 - (2 * m + 2) by push_cast ; ring]\n[GOAL]\nm : \u2115\n\u22a2 -\u2191(2 * m + 1) = 1 - (2 * \u2191m + 2)\n[PROOFSTEP]\npush_cast\n[GOAL]\nm : \u2115\n\u22a2 -(2 * \u2191m + 1) = 1 - (2 * \u2191m + 2)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 riemannZeta (1 - (2 * \u2191m + 2)) = -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [riemannZeta_one_sub]\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) *\n        Complex.sin (\u2191\u03c0 * (1 - (2 * \u2191m + 2)) / 2) *\n      riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\ncase intro.inr.hs\nm : \u2115\n\u22a2 \u2200 (n : \u2115), 2 * \u2191m + 2 \u2260 -\u2191n\ncase intro.inr.hs' m : \u2115 \u22a2 2 * \u2191m + 2 \u2260 1\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase intro.inr.hs\nm : \u2115\n\u22a2 \u2200 (n : \u2115), 2 * \u2191m + 2 \u2260 -\u2191n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase intro.inr.hs\nm n : \u2115\n\u22a2 2 * \u2191m + 2 \u2260 -\u2191n\n[PROOFSTEP]\nrw [(by norm_cast : 2 * (m : \u2102) + 2 = \u2191(2 * m + 2)), \u2190 Int.cast_neg_natCast, \u2190 Int.cast_ofNat, Ne.def, Int.cast_inj]\n[GOAL]\nm n : \u2115\n\u22a2 2 * \u2191m + 2 = \u2191(2 * m + 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.inr.hs\nm n : \u2115\n\u22a2 \u00ac\u2191(2 * m + 2) = -\u2191n\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase intro.inr.hs.h\nm n : \u2115\n\u22a2 -\u2191n < \u2191(2 * m + 2)\n[PROOFSTEP]\nexact lt_of_le_of_lt (by norm_num : (-n : \u2124) \u2264 0) (by positivity)\n[GOAL]\nm n : \u2115\n\u22a2 -\u2191n \u2264 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm n : \u2115\n\u22a2 0 < \u2191(2 * m + 2)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.inr.hs'\nm : \u2115\n\u22a2 2 * \u2191m + 2 \u2260 1\n[PROOFSTEP]\nrw [(by norm_cast : 2 * (m : \u2102) + 2 = \u2191(2 * m + 2)), Ne.def, Nat.cast_eq_one]\n[GOAL]\nm : \u2115\n\u22a2 2 * \u2191m + 2 = \u2191(2 * m + 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.inr.hs'\nm : \u2115\n\u22a2 \u00ac2 * m + 2 = 1\n[PROOFSTEP]\nnorm_num\n  -- get rid of sine term\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) *\n        Complex.sin (\u2191\u03c0 * (1 - (2 * \u2191m + 2)) / 2) *\n      riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [show Complex.sin (\u2191\u03c0 * (1 - (2 * \u2191m + 2)) / 2) = -(-1 : \u2102) ^ m\n    by\n    rw [(by field_simp; ring : (\u03c0 : \u2102) * (1 - (2 * \u2191m + 2)) / 2 = \u03c0 / 2 - (\u03c0 * m + \u03c0))]\n    rw [Complex.sin_pi_div_two_sub, Complex.cos_add_pi, neg_inj]\n    rcases Nat.even_or_odd' m with \u27e8t, rfl | rfl\u27e9\n    \u00b7 rw [pow_mul, neg_one_sq, one_pow]\n      convert Complex.cos_nat_mul_two_pi t using 2\n      push_cast ; ring_nf\n    \u00b7 rw [pow_add, pow_one, pow_mul, neg_one_sq, one_pow, one_mul]\n      convert Complex.cos_nat_mul_two_pi_add_pi t using 2\n      push_cast ; ring_nf]\n  -- substitute in what we know about zeta values at positive integers\n[GOAL]\nm : \u2115\n\u22a2 Complex.sin (\u2191\u03c0 * (1 - (2 * \u2191m + 2)) / 2) = -(-1) ^ m\n[PROOFSTEP]\nrw [(by field_simp; ring : (\u03c0 : \u2102) * (1 - (2 * \u2191m + 2)) / 2 = \u03c0 / 2 - (\u03c0 * m + \u03c0))]\n[GOAL]\nm : \u2115\n\u22a2 \u2191\u03c0 * (1 - (2 * \u2191m + 2)) / 2 = \u2191\u03c0 / 2 - (\u2191\u03c0 * \u2191m + \u2191\u03c0)\n[PROOFSTEP]\nfield_simp\n[GOAL]\nm : \u2115\n\u22a2 \u2191\u03c0 * (1 - (2 * \u2191m + 2)) = \u2191\u03c0 - 2 * (\u2191\u03c0 * \u2191m + \u2191\u03c0)\n[PROOFSTEP]\nring\n[GOAL]\nm : \u2115\n\u22a2 Complex.sin (\u2191\u03c0 / 2 - (\u2191\u03c0 * \u2191m + \u2191\u03c0)) = -(-1) ^ m\n[PROOFSTEP]\nrw [Complex.sin_pi_div_two_sub, Complex.cos_add_pi, neg_inj]\n[GOAL]\nm : \u2115\n\u22a2 Complex.cos (\u2191\u03c0 * \u2191m) = (-1) ^ m\n[PROOFSTEP]\nrcases Nat.even_or_odd' m with \u27e8t, rfl | rfl\u27e9\n[GOAL]\ncase intro.inl\nt : \u2115\n\u22a2 Complex.cos (\u2191\u03c0 * \u2191(2 * t)) = (-1) ^ (2 * t)\n[PROOFSTEP]\nrw [pow_mul, neg_one_sq, one_pow]\n[GOAL]\ncase intro.inl\nt : \u2115\n\u22a2 Complex.cos (\u2191\u03c0 * \u2191(2 * t)) = 1\n[PROOFSTEP]\nconvert Complex.cos_nat_mul_two_pi t using 2\n[GOAL]\ncase h.e'_2.h.e'_1\nt : \u2115\n\u22a2 \u2191\u03c0 * \u2191(2 * t) = \u2191t * (2 * \u2191\u03c0)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1\nt : \u2115\n\u22a2 \u2191\u03c0 * (2 * \u2191t) = \u2191t * (2 * \u2191\u03c0)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase intro.inr\nt : \u2115\n\u22a2 Complex.cos (\u2191\u03c0 * \u2191(2 * t + 1)) = (-1) ^ (2 * t + 1)\n[PROOFSTEP]\nrw [pow_add, pow_one, pow_mul, neg_one_sq, one_pow, one_mul]\n[GOAL]\ncase intro.inr\nt : \u2115\n\u22a2 Complex.cos (\u2191\u03c0 * \u2191(2 * t + 1)) = -1\n[PROOFSTEP]\nconvert Complex.cos_nat_mul_two_pi_add_pi t using 2\n[GOAL]\ncase h.e'_2.h.e'_1\nt : \u2115\n\u22a2 \u2191\u03c0 * \u2191(2 * t + 1) = \u2191t * (2 * \u2191\u03c0) + \u2191\u03c0\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1\nt : \u2115\n\u22a2 \u2191\u03c0 * (2 * \u2191t + 1) = \u2191t * (2 * \u2191\u03c0) + \u2191\u03c0\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase intro.inr\nm : \u2115\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m * riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nhave step1 := congr_arg ((\u2191) : \u211d \u2192 \u2102) (hasSum_zeta_nat (by norm_num : m + 1 \u2260 0)).tsum_eq\n[GOAL]\nm : \u2115\n\u22a2 m + 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inr\nm : \u2115\nstep1 :\n  \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * (m + 1))) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1))) / \u2191(2 * (m + 1))!)\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m * riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nhave step2 := zeta_nat_eq_tsum_of_gt_one (by rw [mul_add]; norm_num : 1 < 2 * (m + 1))\n[GOAL]\nm : \u2115\nstep1 :\n  \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * (m + 1))) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1))) / \u2191(2 * (m + 1))!)\n\u22a2 1 < 2 * (m + 1)\n[PROOFSTEP]\nrw [mul_add]\n[GOAL]\nm : \u2115\nstep1 :\n  \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * (m + 1))) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1))) / \u2191(2 * (m + 1))!)\n\u22a2 1 < 2 * m + 2 * 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inr\nm : \u2115\nstep1 :\n  \u2191(\u2211' (b : \u2115), 1 / \u2191b ^ (2 * (m + 1))) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1))) / \u2191(2 * (m + 1))!)\nstep2 : riemannZeta \u2191(2 * (m + 1)) = \u2211' (n : \u2115), 1 / \u2191n ^ (2 * (m + 1))\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m * riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nsimp_rw [ofReal_tsum, ofReal_div, ofReal_one, ofReal_pow, ofReal_nat_cast] at step1 \n[GOAL]\ncase intro.inr\nm : \u2115\nstep2 : riemannZeta \u2191(2 * (m + 1)) = \u2211' (n : \u2115), 1 / \u2191n ^ (2 * (m + 1))\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m * riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [step1, (by norm_cast : (\u2191(2 * (m + 1)) : \u2102) = 2 * \u2191m + 2)] at step2 \n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta \u2191(2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u2191(2 * (m + 1)) = 2 * \u2191m + 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.inr\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m * riemannZeta (2 * \u2191m + 2) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [step2, mul_div]\n  -- now the rest is just a lengthy but elementary rearrangement\n[GOAL]\ncase intro.inr\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n        \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) /\n      \u2191(2 * (m + 1))! =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [show ((2 * (m + 1))! : \u2102) = Complex.Gamma (2 * m + 2) * (\u2191(2 * m + 1) + 1)\n    by\n    rw [(by push_cast ; ring : (2 * m + 2 : \u2102) = \u2191(2 * m + 1) + 1), Complex.Gamma_nat_eq_factorial,\n      (by ring : 2 * (m + 1) = 2 * m + 1 + 1), Nat.factorial_succ, Nat.cast_mul, mul_comm]\n    norm_num]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u2191(2 * (m + 1))! = Complex.Gamma (2 * \u2191m + 2) * (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [(by push_cast ; ring : (2 * m + 2 : \u2102) = \u2191(2 * m + 1) + 1), Complex.Gamma_nat_eq_factorial,\n  (by ring : 2 * (m + 1) = 2 * m + 1 + 1), Nat.factorial_succ, Nat.cast_mul, mul_comm]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 * \u2191m + 2 = \u2191(2 * m + 1) + 1\n[PROOFSTEP]\npush_cast\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 * \u2191m + 2 = 2 * \u2191m + 1 + 1\n[PROOFSTEP]\nring\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 * (m + 1) = 2 * m + 1 + 1\n[PROOFSTEP]\nring\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u2191(2 * m + 1)! * \u2191(2 * m + 1 + 1) = \u2191(2 * m + 1)! * (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.inr\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n        \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) /\n      (Complex.Gamma (2 * \u2191m + 2) * (\u2191(2 * m + 1) + 1)) =\n    -1 * \u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\nrw [\u2190 div_div, neg_one_mul]\n[GOAL]\ncase intro.inr\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n          \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) /\n        Complex.Gamma (2 * \u2191m + 2) /\n      (\u2191(2 * m + 1) + 1) =\n    -\u2191(bernoulli (2 * m + 1 + 1)) / (\u2191(2 * m + 1) + 1)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.inr.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n        \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) /\n      Complex.Gamma (2 * \u2191m + 2) =\n    -\u2191(bernoulli (2 * m + 1 + 1))\n[PROOFSTEP]\nrw [div_eq_iff (Gamma_ne_zero_of_re_pos _)]\n[GOAL]\ncase intro.inr.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n      \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) =\n    -\u2191(bernoulli (2 * m + 1 + 1)) * Complex.Gamma (2 * \u2191m + 2)\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 0 < (2 * \u2191m + 2).re\n[PROOFSTEP]\nswap\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 0 < (2 * \u2191m + 2).re\n[PROOFSTEP]\nrw [(by norm_num : 2 * (m : \u2102) + 2 = \u2191(2 * (m : \u211d) + 2)), ofReal_re]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 * \u2191m + 2 = \u2191(2 * \u2191m + 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 0 < 2 * \u2191m + 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.inr.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * -(-1) ^ m *\n      \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) =\n    -\u2191(bernoulli (2 * m + 1 + 1)) * Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nsimp_rw [ofReal_mul, \u2190 mul_assoc, ofReal_rat_cast, mul_add, Nat.add_assoc, mul_one, one_add_one_eq_two, mul_neg,\n  neg_mul, neg_inj]\n[GOAL]\ncase intro.inr.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * \u2191((-1) ^ (m + 2)) *\n          \u2191(2 ^ (2 * m + 2 - 1)) *\n        \u2191(\u03c0 ^ (2 * m + 2)) *\n      \u2191(bernoulli (2 * m + 2)) =\n    \u2191(bernoulli (2 * m + 2)) * Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nconv_rhs => rw [mul_comm]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(bernoulli (2 * m + 2)) * Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(bernoulli (2 * m + 2)) * Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(bernoulli (2 * m + 2)) * Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase intro.inr.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * \u2191((-1) ^ (m + 2)) *\n          \u2191(2 ^ (2 * m + 2 - 1)) *\n        \u2191(\u03c0 ^ (2 * m + 2)) *\n      \u2191(bernoulli (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2) * \u2191(bernoulli (2 * m + 2))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * \u2191((-1) ^ (m + 2)) *\n        \u2191(2 ^ (2 * m + 2 - 1)) *\n      \u2191(\u03c0 ^ (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [ofReal_pow, ofReal_neg, ofReal_one, pow_add, neg_one_sq, mul_one]\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m *\n        \u2191(2 ^ (2 * m + 2 - 1)) *\n      \u2191(\u03c0 ^ (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  congr\n  rw [mul_assoc, \u2190 pow_add, \u2190 two_mul, pow_mul, neg_one_sq, one_pow, mul_one]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m *\n      \u2191(2 ^ (2 * m + 2 - 1)) *\n    \u2191(\u03c0 ^ (2 * m + 2))\n[PROOFSTEP]\n  congr\n  congr\n  rw [mul_assoc, \u2190 pow_add, \u2190 two_mul, pow_mul, neg_one_sq, one_pow, mul_one]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m *\n      \u2191(2 ^ (2 * m + 2 - 1)) *\n    \u2191(\u03c0 ^ (2 * m + 2))\n[PROOFSTEP]\n  congr\n  congr\n  rw [mul_assoc, \u2190 pow_add, \u2190 two_mul, pow_mul, neg_one_sq, one_pow, mul_one]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m *\n      \u2191(2 ^ (2 * m + 2 - 1)) *\n    \u2191(\u03c0 ^ (2 * m + 2))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m *\n    \u2191(2 ^ (2 * m + 2 - 1))\ncase a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(\u03c0 ^ (2 * m + 2))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * (-1) ^ m * (-1) ^ m\ncase a.a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(2 ^ (2 * m + 2 - 1))\ncase a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n| \u2191(\u03c0 ^ (2 * m + 2))\n[PROOFSTEP]\nrw [mul_assoc, \u2190 pow_add, \u2190 two_mul, pow_mul, neg_one_sq, one_pow, mul_one]\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * \u2191(2 ^ (2 * m + 2 - 1)) *\n      \u2191(\u03c0 ^ (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [show (2 : \u2102) ^ (1 - (2 * (m : \u2102) + 2)) = (\u2191((2 : \u211d) ^ (2 * m + 2 - 1)))\u207b\u00b9\n    by\n    rw [ofReal_pow, \u2190 cpow_nat_cast, \u2190 cpow_neg, show (2 : \u211d) = (2 : \u2102) by norm_num]\n    congr 1\n    rw [Nat.add_sub_assoc one_le_two, Nat.cast_add, Nat.cast_mul, Nat.cast_two, (by norm_num : 2 - 1 = 1)]\n    push_cast ; ring]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) = (\u2191(2 ^ (2 * m + 2 - 1)))\u207b\u00b9\n[PROOFSTEP]\nrw [ofReal_pow, \u2190 cpow_nat_cast, \u2190 cpow_neg, show (2 : \u211d) = (2 : \u2102) by norm_num]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u21912 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 ^ (1 - (2 * \u2191m + 2)) = 2 ^ (-\u2191(2 * m + 2 - 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 1 - (2 * \u2191m + 2) = -\u2191(2 * m + 2 - 1)\n[PROOFSTEP]\nrw [Nat.add_sub_assoc one_le_two, Nat.cast_add, Nat.cast_mul, Nat.cast_two, (by norm_num : 2 - 1 = 1)]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 2 - 1 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 1 - (2 * \u2191m + 2) = -(2 * \u2191m + \u21911)\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 1 - (2 * \u2191m + 2) = -(2 * \u2191m + 1)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 (\u2191(2 ^ (2 * m + 2 - 1)))\u207b\u00b9 * \u2191\u03c0 ^ (-(2 * \u2191m + 2)) * Complex.Gamma (2 * \u2191m + 2) * \u2191(2 ^ (2 * m + 2 - 1)) *\n      \u2191(\u03c0 ^ (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [show (\u03c0 : \u2102) ^ (-(2 * (m : \u2102) + 2)) = (\u2191(\u03c0 ^ (2 * m + 2)))\u207b\u00b9 by\n    rw [ofReal_pow, \u2190 cpow_nat_cast, \u2190 cpow_neg, Nat.cast_add, Nat.cast_mul, Nat.cast_two]]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u2191\u03c0 ^ (-(2 * \u2191m + 2)) = (\u2191(\u03c0 ^ (2 * m + 2)))\u207b\u00b9\n[PROOFSTEP]\nrw [ofReal_pow, \u2190 cpow_nat_cast, \u2190 cpow_neg, Nat.cast_add, Nat.cast_mul, Nat.cast_two]\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 (\u2191(2 ^ (2 * m + 2 - 1)))\u207b\u00b9 * (\u2191(\u03c0 ^ (2 * m + 2)))\u207b\u00b9 * Complex.Gamma (2 * \u2191m + 2) * \u2191(2 ^ (2 * m + 2 - 1)) *\n      \u2191(\u03c0 ^ (2 * m + 2)) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [(by intros; ring : \u2200 a b c d e : \u2102, a * b * c * d * e = a * d * (b * e) * c)]\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 \u2200 (a b c d e : \u2102), a * b * c * d * e = a * d * (b * e) * c\n[PROOFSTEP]\nintros\n[GOAL]\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\na\u271d b\u271d c\u271d d\u271d e\u271d : \u2102\n\u22a2 a\u271d * b\u271d * c\u271d * d\u271d * e\u271d = a\u271d * d\u271d * (b\u271d * e\u271d) * c\u271d\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.inr.e_a.e_a\nm : \u2115\nstep2 :\n  riemannZeta (2 * \u2191m + 2) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\nstep1 :\n  \u2211' (a : \u2115), 1 / \u2191a ^ (2 * (m + 1)) =\n    \u2191((-1) ^ (m + 1 + 1) * 2 ^ (2 * (m + 1) - 1) * \u03c0 ^ (2 * (m + 1)) * \u2191(bernoulli (2 * (m + 1)))) / \u2191(2 * (m + 1))!\n\u22a2 (\u2191(2 ^ (2 * m + 2 - 1)))\u207b\u00b9 * \u2191(2 ^ (2 * m + 2 - 1)) * ((\u2191(\u03c0 ^ (2 * m + 2)))\u207b\u00b9 * \u2191(\u03c0 ^ (2 * m + 2))) *\n      Complex.Gamma (2 * \u2191m + 2) =\n    Complex.Gamma (2 * \u2191m + 2)\n[PROOFSTEP]\nrw [inv_mul_cancel (ofReal_ne_zero.mpr <| pow_ne_zero _ pi_pos.ne'),\n  inv_mul_cancel (ofReal_ne_zero.mpr <| pow_ne_zero _ two_ne_zero), one_mul, one_mul]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.ZetaFunction", "llama_tokens": 91325, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959545, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.5299963420175048}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\n\u22a2 (sum l).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 l\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\n\u22a2 (sum []).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH : (sum tl).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\n\u22a2 (sum (hd :: tl)).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 (hd :: tl)\n[PROOFSTEP]\nsimp only [List.sum_cons, Finset.union_comm]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH : (sum tl).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\n\u22a2 (hd + sum tl).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 (hd :: tl)\n[PROOFSTEP]\nrefine' Finsupp.support_add.trans (Finset.union_subset_union _ IH)\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH : (sum tl).support \u2286 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\n\u22a2 hd.support \u2286 hd.support\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\n\u22a2 (sum s).support \u2286 sup (map Finsupp.support s)\n[PROOFSTEP]\ninduction s using Quot.inductionOn\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\na\u271d : List (\u03b9 \u2192\u2080 M)\n\u22a2 (sum (Quot.mk Setoid.r a\u271d)).support \u2286 sup (map Finsupp.support (Quot.mk Setoid.r a\u271d))\n[PROOFSTEP]\nsimpa only [Multiset.quot_mk_to_coe'', Multiset.coe_sum, Multiset.coe_map, Multiset.sup_coe, List.foldr_map] using\n  List.support_sum_subset _\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\n\u22a2 (Finset.sum s id).support \u2286 sup s Finsupp.support\n[PROOFSTEP]\nclassical convert Multiset.support_sum_subset s.1; simp\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\n\u22a2 (Finset.sum s id).support \u2286 sup s Finsupp.support\n[PROOFSTEP]\nconvert Multiset.support_sum_subset s.1\n[GOAL]\ncase h.e'_3.h.e'_4\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\n\u22a2 Finset.sum s id = Multiset.sum s.val\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Zero M\nl : List (\u03b9 \u2192\u2080 M)\nx : \u03b9\n\u22a2 x \u2208 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 l \u2194 \u2203 f x_1, x \u2208 f.support\n[PROOFSTEP]\nsimp only [Finset.sup_eq_union, List.foldr_map, Finsupp.mem_support_iff, exists_prop]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Zero M\nl : List (\u03b9 \u2192\u2080 M)\nx : \u03b9\n\u22a2 x \u2208 foldr ((fun x x_1 => x \u222a x_1) \u2218 Finsupp.support) \u2205 l \u2194 \u2203 f, f \u2208 l \u2227 \u2191f x \u2260 0\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Zero M\nx : \u03b9\n\u22a2 x \u2208 foldr ((fun x x_1 => x \u222a x_1) \u2218 Finsupp.support) \u2205 [] \u2194 \u2203 f, f \u2208 [] \u2227 \u2191f x \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Zero M\nx : \u03b9\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH : x \u2208 foldr ((fun x x_1 => x \u222a x_1) \u2218 Finsupp.support) \u2205 tl \u2194 \u2203 f, f \u2208 tl \u2227 \u2191f x \u2260 0\n\u22a2 x \u2208 foldr ((fun x x_1 => x \u222a x_1) \u2218 Finsupp.support) \u2205 (hd :: tl) \u2194 \u2203 f, f \u2208 hd :: tl \u2227 \u2191f x \u2260 0\n[PROOFSTEP]\nsimp only [foldr, Function.comp_apply, Finset.mem_union, Finsupp.mem_support_iff, ne_eq, IH, find?, mem_cons,\n  exists_eq_or_imp]\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Zero M\ns : Multiset (\u03b9 \u2192\u2080 M)\nx : \u03b9\nx\u271d : List (\u03b9 \u2192\u2080 M)\n\u22a2 x \u2208 sup (map Finsupp.support (Quot.mk Setoid.r x\u271d)) \u2194 \u2203 f x_1, x \u2208 f.support\n[PROOFSTEP]\nsimpa only [Multiset.quot_mk_to_coe'', Multiset.coe_map, Multiset.sup_coe, List.foldr_map] using\n  List.mem_foldr_sup_support_iff\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl : Pairwise (_root_.Disjoint on Finsupp.support) l\n\u22a2 (sum l).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 l\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhl : Pairwise (_root_.Disjoint on Finsupp.support) []\n\u22a2 (sum []).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl : Pairwise (_root_.Disjoint on Finsupp.support) (hd :: tl)\n\u22a2 (sum (hd :: tl)).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 (hd :: tl)\n[PROOFSTEP]\nsimp only [List.pairwise_cons] at hl \n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\n\u22a2 (sum (hd :: tl)).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 (hd :: tl)\n[PROOFSTEP]\nsimp only [List.sum_cons, List.foldr_cons, Function.comp_apply]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\n\u22a2 (hd + sum tl).support = hd.support \u2294 foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\n[PROOFSTEP]\nrw [Finsupp.support_add_eq, IH hl.right, Finset.sup_eq_union]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\n\u22a2 _root_.Disjoint hd.support (sum tl).support\n[PROOFSTEP]\nsuffices _root_.Disjoint hd.support (tl.foldr (fun x y \u21a6 (Finsupp.support x \u2294 y)) \u2205) by\n  exact Finset.disjoint_of_subset_right (List.support_sum_subset _) this\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\nthis : _root_.Disjoint hd.support (foldr (fun x y => x.support \u2294 y) \u2205 tl)\n\u22a2 _root_.Disjoint hd.support (sum tl).support\n[PROOFSTEP]\nexact Finset.disjoint_of_subset_right (List.support_sum_subset _) this\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\n\u22a2 _root_.Disjoint hd.support (foldr (fun x y => x.support \u2294 y) \u2205 tl)\n[PROOFSTEP]\nrw [\u2190 List.foldr_map, \u2190 Finset.bot_eq_empty, List.foldr_sup_eq_sup_toFinset, Finset.disjoint_sup_right]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\n\u22a2 \u2200 \u2983i : Finset \u03b9\u2984, i \u2208 toFinset (map (fun x => x.support) tl) \u2192 _root_.Disjoint hd.support (id i)\n[PROOFSTEP]\nintro f hf\n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\nf : Finset \u03b9\nhf : f \u2208 toFinset (map (fun x => x.support) tl)\n\u22a2 _root_.Disjoint hd.support (id f)\n[PROOFSTEP]\nsimp only [List.mem_toFinset, List.mem_map] at hf \n[GOAL]\ncase cons\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\nf : Finset \u03b9\nhf : \u2203 a, a \u2208 tl \u2227 a.support = f\n\u22a2 _root_.Disjoint hd.support (id f)\n[PROOFSTEP]\nobtain \u27e8f, hf, rfl\u27e9 := hf\n[GOAL]\ncase cons.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhl\u271d : Pairwise (_root_.Disjoint on Finsupp.support) l\nhd : \u03b9 \u2192\u2080 M\ntl : List (\u03b9 \u2192\u2080 M)\nIH :\n  Pairwise (_root_.Disjoint on Finsupp.support) tl \u2192\n    (sum tl).support = foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 tl\nhl :\n  (\u2200 (a' : \u03b9 \u2192\u2080 M), a' \u2208 tl \u2192 (_root_.Disjoint on Finsupp.support) hd a') \u2227\n    Pairwise (_root_.Disjoint on Finsupp.support) tl\nf : \u03b9 \u2192\u2080 M\nhf : f \u2208 tl\n\u22a2 _root_.Disjoint hd.support (id f.support)\n[PROOFSTEP]\nexact hl.left _ hf\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\n\u22a2 (sum s).support = sup (map Finsupp.support s)\n[PROOFSTEP]\ninduction' s using Quot.inductionOn with a\n[GOAL]\ncase h\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs\u271d : Pairwise (_root_.Disjoint on Finsupp.support) s\na : List (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) (Quot.mk Setoid.r a)\n\u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))\n[PROOFSTEP]\nobtain \u27e8l, hl, hd\u27e9 := hs\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n\u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))\n[PROOFSTEP]\nsuffices : a.Pairwise (_root_.Disjoint on Finsupp.support)\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n\u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a))\n[PROOFSTEP]\nconvert List.support_sum_eq a this\n[GOAL]\ncase h.e'_2.h.e'_4\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n\u22a2 sum (Quot.mk Setoid.r a) = List.sum a\n[PROOFSTEP]\nsimp only [Multiset.quot_mk_to_coe'', Multiset.coe_sum]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n\u22a2 sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 a\n[PROOFSTEP]\ndsimp only [Function.comp]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nthis : List.Pairwise (_root_.Disjoint on Finsupp.support) a\n\u22a2 sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr (fun x x_1 => x.support \u2294 x_1) \u2205 a\n[PROOFSTEP]\nsimp only [quot_mk_to_coe'', coe_map, sup_coe, ge_iff_le, Finset.le_eq_subset, Finset.sup_eq_union, Finset.bot_eq_empty,\n  List.foldr_map]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhl : Quot.mk Setoid.r a = \u2191l\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\n\u22a2 List.Pairwise (_root_.Disjoint on Finsupp.support) a\n[PROOFSTEP]\nsimp only [Multiset.quot_mk_to_coe'', Multiset.coe_map, Multiset.coe_eq_coe] at hl \n[GOAL]\ncase this\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Multiset (\u03b9 \u2192\u2080 M)\nhs : Pairwise (_root_.Disjoint on Finsupp.support) s\na l : List (\u03b9 \u2192\u2080 M)\nhd : List.Pairwise (_root_.Disjoint on Finsupp.support) l\nhl : a ~ l\n\u22a2 List.Pairwise (_root_.Disjoint on Finsupp.support) a\n[PROOFSTEP]\nexact hl.symm.pairwise hd fun _ _ h \u21a6 _root_.Disjoint.symm h\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\n\u22a2 (Finset.sum s id).support = sup s Finsupp.support\n[PROOFSTEP]\nclassical\nsuffices : s.1.Pairwise (_root_.Disjoint on Finsupp.support)\n\u00b7 convert Multiset.support_sum_eq s.1 this\n  \u00b7 exact (Finset.sum_val _).symm\n\u00b7 obtain \u27e8l, hl, hn\u27e9 : \u2203 l : List (\u03b9 \u2192\u2080 M), l.toFinset = s \u2227 l.Nodup :=\n    by\n    refine' \u27e8s.toList, _, Finset.nodup_toList _\u27e9\n    simp\n  subst hl\n  rwa [List.toFinset_val, List.dedup_eq_self.mpr hn, Multiset.pairwise_coe_iff_pairwise, \u2190\n    List.pairwiseDisjoint_iff_coe_toFinset_pairwise_disjoint hn]\n  intro x y hxy\n  exact symmetric_disjoint hxy\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\n\u22a2 (Finset.sum s id).support = sup s Finsupp.support\n[PROOFSTEP]\nsuffices : s.1.Pairwise (_root_.Disjoint on Finsupp.support)\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\nthis : Multiset.Pairwise (Disjoint on Finsupp.support) s.val\n\u22a2 (Finset.sum s id).support = sup s Finsupp.support\n[PROOFSTEP]\nconvert Multiset.support_sum_eq s.1 this\n[GOAL]\ncase h.e'_2.h.e'_4\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\nthis : Multiset.Pairwise (Disjoint on Finsupp.support) s.val\n\u22a2 Finset.sum s id = Multiset.sum s.val\n[PROOFSTEP]\nexact (Finset.sum_val _).symm\n[GOAL]\ncase this\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\n\u22a2 Multiset.Pairwise (Disjoint on Finsupp.support) s.val\n[PROOFSTEP]\nobtain \u27e8l, hl, hn\u27e9 : \u2203 l : List (\u03b9 \u2192\u2080 M), l.toFinset = s \u2227 l.Nodup :=\n  by\n  refine' \u27e8s.toList, _, Finset.nodup_toList _\u27e9\n  simp\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\n\u22a2 \u2203 l, List.toFinset l = s \u2227 List.Nodup l\n[PROOFSTEP]\nrefine' \u27e8s.toList, _, Finset.nodup_toList _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\n\u22a2 List.toFinset (toList s) = s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\ns : Finset (\u03b9 \u2192\u2080 M)\nhs : Set.PairwiseDisjoint (\u2191s) Finsupp.support\nl : List (\u03b9 \u2192\u2080 M)\nhl : List.toFinset l = s\nhn : List.Nodup l\n\u22a2 Multiset.Pairwise (Disjoint on Finsupp.support) s.val\n[PROOFSTEP]\nsubst hl\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhn : List.Nodup l\nhs : Set.PairwiseDisjoint (\u2191(List.toFinset l)) Finsupp.support\n\u22a2 Multiset.Pairwise (Disjoint on Finsupp.support) (List.toFinset l).val\n[PROOFSTEP]\nrwa [List.toFinset_val, List.dedup_eq_self.mpr hn, Multiset.pairwise_coe_iff_pairwise, \u2190\n  List.pairwiseDisjoint_iff_coe_toFinset_pairwise_disjoint hn]\n[GOAL]\ncase this.intro.intro.hr\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhn : List.Nodup l\nhs : Set.PairwiseDisjoint (\u2191(List.toFinset l)) Finsupp.support\n\u22a2 Symmetric (Disjoint on Finsupp.support)\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase this.intro.intro.hr\n\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : AddCommMonoid M\nl : List (\u03b9 \u2192\u2080 M)\nhn : List.Nodup l\nhs : Set.PairwiseDisjoint (\u2191(List.toFinset l)) Finsupp.support\nx y : \u03b9 \u2192\u2080 M\nhxy : (Disjoint on Finsupp.support) x y\n\u22a2 (Disjoint on Finsupp.support) y x\n[PROOFSTEP]\nexact symmetric_disjoint hxy\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.BigOperators", "llama_tokens": 8706, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.5297511196029537}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ici a) (Ici (a + d))\n[PROOFSTEP]\nrefine' \u27e8fun x h => add_le_add_right (mem_Ici.mp h) _, (add_left_injective d).injOn _, fun _ h => _\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d x\u271d : M\nh : x\u271d \u2208 Ici (a + d)\n\u22a2 x\u271d \u2208 (fun x => x + d) '' Ici a\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := exists_add_of_le (mem_Ici.mp h)\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a + d + c \u2208 Ici (a + d)\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ici a\n[PROOFSTEP]\nrw [mem_Ici, add_right_comm, add_le_add_iff_right] at h \n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a \u2264 a + c\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ici a\n[PROOFSTEP]\nexact \u27e8a + c, h, by rw [add_right_comm]\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a \u2264 a + c\n\u22a2 (fun x => x + d) (a + c) = a + d + c\n[PROOFSTEP]\nrw [add_right_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a) (Ioi (a + d))\n[PROOFSTEP]\nrefine' \u27e8fun x h => add_lt_add_right (mem_Ioi.mp h) _, fun _ _ _ _ h => add_right_cancel h, fun _ h => _\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d x\u271d : M\nh : x\u271d \u2208 Ioi (a + d)\n\u22a2 x\u271d \u2208 (fun x => x + d) '' Ioi a\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := exists_add_of_le (mem_Ioi.mp h).le\n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a + d + c \u2208 Ioi (a + d)\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a\n[PROOFSTEP]\nrw [mem_Ioi, add_right_comm, add_lt_add_iff_right] at h \n[GOAL]\ncase intro\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a < a + c\n\u22a2 a + d + c \u2208 (fun x => x + d) '' Ioi a\n[PROOFSTEP]\nexact \u27e8a + c, h, by rw [add_right_comm]\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c\u271d d c : M\nh : a < a + c\n\u22a2 (fun x => x + d) (a + c) = a + d + c\n[PROOFSTEP]\nrw [add_right_comm]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Icc a b) (Icc (a + d) (b + d))\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iic, \u2190 Ici_inter_Iic]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ici a \u2229 Iic b) (Ici (a + d) \u2229 Iic (b + d))\n[PROOFSTEP]\nexact (Ici_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx => le_of_add_le_add_right hx.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioo a b) (Ioo (a + d) (b + d))\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iio, \u2190 Ioi_inter_Iio]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iio b) (Ioi (a + d) \u2229 Iio (b + d))\n[PROOFSTEP]\nexact (Ioi_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx => lt_of_add_lt_add_right hx.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioc a b) (Ioc (a + d) (b + d))\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, \u2190 Ioi_inter_Iic]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ioi a \u2229 Iic b) (Ioi (a + d) \u2229 Iic (b + d))\n[PROOFSTEP]\nexact (Ioi_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx => le_of_add_le_add_right hx.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ico a b) (Ico (a + d) (b + d))\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iio, \u2190 Ici_inter_Iio]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 BijOn (fun x => x + d) (Ici a \u2229 Iio b) (Ici (a + d) \u2229 Iio (b + d))\n[PROOFSTEP]\nexact (Ici_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx => lt_of_add_lt_add_right hx.2\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ici b = Ici (a + b)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Ici]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ioi b = Ioi (a + b)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Ioi]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Icc b c = Icc (a + b) (a + c)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Icc]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ico b c = Ico (a + b) (a + c)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Ico]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ioc b c = Ioc (a + b) (a + c)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Ioc]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : OrderedCancelAddCommMonoid M\ninst\u271d : ExistsAddOfLE M\na b c d : M\n\u22a2 (fun x => a + x) '' Ioo b c = Ioo (a + b) (a + c)\n[PROOFSTEP]\nsimp only [add_comm a, image_add_const_Ioo]\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.Monoid", "llama_tokens": 2766, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.5297511092566216}}
{"text": "[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nrla : IsLeftRegular a\n\u22a2 IsLeftRegular (a ^ n)\n[PROOFSTEP]\nsimp only [IsLeftRegular, \u2190 mul_left_iterate, rla.iterate n]\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nrra : IsRightRegular a\n\u22a2 IsRightRegular (a ^ n)\n[PROOFSTEP]\nrw [IsRightRegular, \u2190 mul_right_iterate]\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nrra : IsRightRegular a\n\u22a2 Function.Injective (fun x => x * a)^[n]\n[PROOFSTEP]\nexact rra.iterate n\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsLeftRegular (a ^ n) \u2194 IsLeftRegular a\n[PROOFSTEP]\nrefine' \u27e8_, IsLeftRegular.pow n\u27e9\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsLeftRegular (a ^ n) \u2192 IsLeftRegular a\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos n0, pow_succ']\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsLeftRegular (a ^ Nat.pred n * a) \u2192 IsLeftRegular a\n[PROOFSTEP]\nexact IsLeftRegular.of_mul\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsRightRegular (a ^ n) \u2194 IsRightRegular a\n[PROOFSTEP]\nrefine' \u27e8_, IsRightRegular.pow n\u27e9\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsRightRegular (a ^ n) \u2192 IsRightRegular a\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos n0, pow_succ]\n[GOAL]\nR : Type u_1\na b : R\ninst\u271d : Monoid R\nn : \u2115\nn0 : 0 < n\n\u22a2 IsRightRegular (a * a ^ Nat.pred n) \u2192 IsRightRegular a\n[PROOFSTEP]\nexact IsRightRegular.of_mul\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Regular.Pow", "llama_tokens": 721, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760727, "lm_q2_score": 0.7154239897159439, "lm_q1q2_score": 0.5295384869745418}}
{"text": "[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso f\nH : P (f \u226b g)\n\u22a2 P g\n[PROOFSTEP]\nconvert hP.2 (f \u226b g) (asIso f).symm.commRingCatIsoToRingEquiv H\n[GOAL]\ncase h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso f\nH : P (f \u226b g)\n\u22a2 g = comp (f \u226b g) (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso f).symm))\n[PROOFSTEP]\nexact (IsIso.inv_hom_id_assoc _ _).symm\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso g\nH : P (f \u226b g)\n\u22a2 P f\n[PROOFSTEP]\nconvert hP.1 (f \u226b g) (asIso g).symm.commRingCatIsoToRingEquiv H\n[GOAL]\ncase h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso g\nH : P (f \u226b g)\n\u22a2 f = comp (RingEquiv.toRingHom (Iso.commRingCatIsoToRingEquiv (asIso g).symm)) (f \u226b g)\n[PROOFSTEP]\nchange f = f \u226b g \u226b inv g\n[GOAL]\ncase h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : S \u27f6 T\ninst\u271d : IsIso g\nH : P (f \u226b g)\n\u22a2 f = f \u226b g \u226b inv g\n[PROOFSTEP]\nsimp\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\n\u22a2 P (Localization.awayMap f r) \u2194 P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nlet e\u2081 : R' \u2243+* Localization.Away r := (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\n\u22a2 P (Localization.awayMap f r) \u2194 P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nlet e\u2082 : Localization.Away (f r) \u2243+* S' := (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 P (Localization.awayMap f r) \u2194 P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nrefine' (hP.cancel_left_isIso e\u2081.toCommRingCatIso.hom (CommRingCat.ofHom _)).symm.trans _\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 P\n      ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n        CommRingCat.ofHom\n          (IsLocalization.map (Localization.Away (\u2191f r)) f\n            (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u2194\n    P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nrefine' (hP.cancel_right_isIso (CommRingCat.ofHom _) e\u2082.toCommRingCatIso.hom).symm.trans _\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 P\n      (CommRingCat.ofHom\n          ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n            CommRingCat.ofHom\n              (IsLocalization.map (Localization.Away (\u2191f r)) f\n                (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u226b\n        (RingEquiv.toCommRingCatIso e\u2082).hom) \u2194\n    P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\nrw [\u2190 eq_iff_iff]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 P\n      (CommRingCat.ofHom\n          ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n            CommRingCat.ofHom\n              (IsLocalization.map (Localization.Away (\u2191f r)) f\n                (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u226b\n        (RingEquiv.toCommRingCatIso e\u2082).hom) =\n    P (IsLocalization.Away.map R' S' f r)\n[PROOFSTEP]\ncongr 1\n  -- Porting Note : Here, the proof used to have a huge `simp` involving `[anonymous]`, which didn't\n    -- work out anymore. The issue seemed to be that it couldn't handle a term in which Ring\n    -- homomorphisms were repeatedly casted to the bundled category and back. Here we resolve the\n    -- problem by converting the goal to a more straightforward form.\n[GOAL]\ncase e_x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 CommRingCat.ofHom\n        ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n          CommRingCat.ofHom\n            (IsLocalization.map (Localization.Away (\u2191f r)) f\n              (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u226b\n      (RingEquiv.toCommRingCatIso e\u2082).hom =\n    IsLocalization.Away.map R' S' f r\n[PROOFSTEP]\nlet e :=\n  (e\u2082 : Localization.Away (f r) \u2192+* S').comp\n    (((IsLocalization.map (Localization.Away (f r)) f\n            (by rintro x \u27e8n, rfl\u27e9; use n; simp : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (f r)))) :\n          Localization.Away r \u2192+* Localization.Away (f r)).comp\n      (e\u2081 : R' \u2192+* Localization.Away r))\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\n\u22a2 Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))\n[PROOFSTEP]\nrintro x \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\nn : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) r n \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\nn : \u2115\n\u22a2 (fun x x_1 => x ^ x_1) (\u2191f r) n = \u2191f ((fun x x_1 => x ^ x_1) r n)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\n\u22a2 CommRingCat.ofHom\n        ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n          CommRingCat.ofHom\n            (IsLocalization.map (Localization.Away (\u2191f r)) f\n              (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u226b\n      (RingEquiv.toCommRingCatIso e\u2082).hom =\n    IsLocalization.Away.map R' S' f r\n[PROOFSTEP]\nsuffices e = IsLocalization.Away.map R' S' f r by convert this\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\nthis : e = IsLocalization.Away.map R' S' f r\n\u22a2 CommRingCat.ofHom\n        ((RingEquiv.toCommRingCatIso e\u2081).hom \u226b\n          CommRingCat.ofHom\n            (IsLocalization.map (Localization.Away (\u2191f r)) f\n              (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))) \u226b\n      (RingEquiv.toCommRingCatIso e\u2082).hom =\n    IsLocalization.Away.map R' S' f r\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase e_x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\n\u22a2 e = IsLocalization.Away.map R' S' f r\n[PROOFSTEP]\napply IsLocalization.ringHom_ext (Submonoid.powers r) _\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\n\u22a2 comp e (algebraMap R R') = comp (IsLocalization.Away.map R' S' f r) (algebraMap R R')\n[PROOFSTEP]\next1 x\n[GOAL]\ncase a\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\nx : R\n\u22a2 \u2191(comp e (algebraMap R R')) x = \u2191(comp (IsLocalization.Away.map R' S' f r) (algebraMap R R')) x\n[PROOFSTEP]\ndsimp [IsLocalization.Away.map]\n[GOAL]\ncase a\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : RespectsIso P\nR S R' S' : Type u\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : CommRing R'\ninst\u271d\u2074 : CommRing S'\ninst\u271d\u00b3 : Algebra R R'\ninst\u271d\u00b2 : Algebra S S'\nf : R \u2192+* S\nr : R\ninst\u271d\u00b9 : IsLocalization.Away r R'\ninst\u271d : IsLocalization.Away (\u2191f r) S'\ne\u2081 : R' \u2243+* Localization.Away r :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers r) R' (Localization.Away r))\ne\u2082 : Localization.Away (\u2191f r) \u2243+* S' :=\n  AlgEquiv.toRingEquiv (IsLocalization.algEquiv (Submonoid.powers (\u2191f r)) (Localization.Away (\u2191f r)) S')\ne : R' \u2192+* S' :=\n  comp (\u2191e\u2082)\n    (comp\n      (IsLocalization.map (Localization.Away (\u2191f r)) f\n        (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      \u2191e\u2081)\nx : R\n\u22a2 \u2191(IsLocalization.map S' (id S) (_ : Submonoid.powers (\u2191f r) \u2264 Submonoid.comap (id S) (Submonoid.powers (\u2191f r))))\n      (\u2191(IsLocalization.map (Localization.Away (\u2191f r)) f\n            (_ : \u2200 \u2983x : R\u2984, x \u2208 Submonoid.powers r \u2192 x \u2208 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n        (\u2191(IsLocalization.map (Localization.Away r) (id R)\n              (_ : Submonoid.powers r \u2264 Submonoid.comap (id R) (Submonoid.powers r)))\n          (\u2191(algebraMap R R') x))) =\n    \u2191(IsLocalization.map S' f (_ : Submonoid.powers r \u2264 Submonoid.comap f (Submonoid.powers (\u2191f r))))\n      (\u2191(algebraMap R R') x)\n[PROOFSTEP]\nsimp only [IsLocalization.map_eq, id_apply, RingHomCompTriple.comp_apply]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\n\u22a2 RespectsIso P\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\n\u22a2 \u2200 {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : R \u2192+* S) (e : S \u2243+* T),\n    P f \u2192 P (comp (RingEquiv.toRingHom e) f)\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase left\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ne : S \u2243+* T\nH : P f\n\u22a2 P (comp (RingEquiv.toRingHom e) f)\n[PROOFSTEP]\nskip\n[GOAL]\ncase left\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ne : S \u2243+* T\nH : P f\n\u22a2 P (comp (RingEquiv.toRingHom e) f)\n[PROOFSTEP]\napply hP\n[GOAL]\ncase left.x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ne : S \u2243+* T\nH : P f\n\u22a2 P f\ncase left.x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : R \u2192+* S\ne : S \u2243+* T\nH : P f\n\u22a2 P (RingEquiv.toRingHom e)\n[PROOFSTEP]\nexacts [H, hP' e]\n[GOAL]\ncase right\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\n\u22a2 \u2200 {R S T : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] (f : S \u2192+* T) (e : R \u2243+* S),\n    P f \u2192 P (comp f (RingEquiv.toRingHom e))\n[PROOFSTEP]\nintrov H\n[GOAL]\ncase right\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : S \u2192+* T\ne : R \u2243+* S\nH : P f\n\u22a2 P (comp f (RingEquiv.toRingHom e))\n[PROOFSTEP]\nskip\n[GOAL]\ncase right\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : S \u2192+* T\ne : R \u2243+* S\nH : P f\n\u22a2 P (comp f (RingEquiv.toRingHom e))\n[PROOFSTEP]\napply hP\n[GOAL]\ncase right.x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : S \u2192+* T\ne : R \u2243+* S\nH : P f\n\u22a2 P (RingEquiv.toRingHom e)\ncase right.x\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderComposition P\nhP' : \u2200 {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (e : R \u2243+* S), P (RingEquiv.toRingHom e)\nR S T : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\nf : S \u2192+* T\ne : R \u2243+* S\nH : P f\n\u22a2 P f\n[PROOFSTEP]\nexacts [hP' e, H]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\n\u22a2 StableUnderBaseChange P\n[PROOFSTEP]\nintrov R h H\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\n\u22a2 P (algebraMap R' S')\n[PROOFSTEP]\nlet e := h.symm.1.equiv\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\n\u22a2 P (algebraMap R' S')\n[PROOFSTEP]\nlet f' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\n\u22a2 P (algebraMap R' S')\n[PROOFSTEP]\nhave : \u2200 x, e x = f' x := by\n  intro x\n  change e.toLinearMap.restrictScalars R x = f'.toLinearMap x\n  congr 1\n  apply TensorProduct.ext'\n  intro x y\n  simp [IsBaseChange.equiv_tmul, Algebra.smul_def]\n    -- Porting Note: This had a lot of implicit inferences which didn't resolve anymore.\n      -- Added those in\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\n\u22a2 \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n[PROOFSTEP]\nintro x\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nx : TensorProduct R R' S\n\u22a2 \u2191e x = \u2191f' x\n[PROOFSTEP]\nchange e.toLinearMap.restrictScalars R x = f'.toLinearMap x\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nx : TensorProduct R R' S\n\u22a2 \u2191(\u2191R \u2191e) x = \u2191(AlgHom.toLinearMap f') x\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nx : TensorProduct R R' S\n\u22a2 \u2191R \u2191e = AlgHom.toLinearMap f'\n[PROOFSTEP]\napply TensorProduct.ext'\n[GOAL]\ncase e_a.H\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nx : TensorProduct R R' S\n\u22a2 \u2200 (x : R') (y : S), \u2191(\u2191R \u2191e) (x \u2297\u209c[R] y) = \u2191(AlgHom.toLinearMap f') (x \u2297\u209c[R] y)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase e_a.H\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nx\u271d : TensorProduct R R' S\nx : R'\ny : S\n\u22a2 \u2191(\u2191R \u2191e) (x \u2297\u209c[R] y) = \u2191(AlgHom.toLinearMap f') (x \u2297\u209c[R] y)\n[PROOFSTEP]\nsimp [IsBaseChange.equiv_tmul, Algebra.smul_def]\n  -- Porting Note: This had a lot of implicit inferences which didn't resolve anymore.\n    -- Added those in\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n\u22a2 P (algebraMap R' S')\n[PROOFSTEP]\nconvert\n  h\u2081.1 (_ : R' \u2192+* TensorProduct R R' S) (_ : TensorProduct R R' S \u2243+* S') (h\u2082 H : P (_ : R' \u2192+* TensorProduct R R' S))\n[GOAL]\ncase h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n\u22a2 algebraMap R' S' = comp (RingEquiv.toRingHom ?m.221851) Algebra.TensorProduct.includeLeftRingHom\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n\u22a2 TensorProduct R R' S \u2243+* S'\n[PROOFSTEP]\nswap\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n\u22a2 TensorProduct R R' S \u2243+* S'\n[PROOFSTEP]\nrefine' { e with map_mul' := fun x y => _ }\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\nx y : TensorProduct R R' S\n\u22a2 Equiv.toFun\n      { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n        right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n      (x * y) =\n    Equiv.toFun\n        { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n          right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n        x *\n      Equiv.toFun\n        { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n          right_inv := (_ : Function.RightInverse e.invFun e.toFun) }\n        y\n[PROOFSTEP]\nchange e (x * y) = e x * e y\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\nx y : TensorProduct R R' S\n\u22a2 \u2191e (x * y) = \u2191e x * \u2191e y\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\nx y : TensorProduct R R' S\n\u22a2 \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) (x * y) =\n    \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) x *\n      \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) y\n[PROOFSTEP]\nexact map_mul f' _ _\n[GOAL]\ncase h.e'_5\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\n\u22a2 algebraMap R' S' =\n    comp\n      (RingEquiv.toRingHom\n        {\n          toEquiv :=\n            { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n              right_inv := (_ : Function.RightInverse e.invFun e.toFun) },\n          map_mul' := (_ : \u2200 (x y : TensorProduct R R' S), \u2191e (x * y) = \u2191e x * \u2191e y),\n          map_add' :=\n            (_ :\n              \u2200 (x y : TensorProduct R R' S),\n                AddHom.toFun e.toAddHom (x + y) = AddHom.toFun e.toAddHom x + AddHom.toFun e.toAddHom y) })\n      Algebra.TensorProduct.includeLeftRingHom\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_5.a\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\nx : R'\n\u22a2 \u2191(algebraMap R' S') x =\n    \u2191(comp\n          (RingEquiv.toRingHom\n            {\n              toEquiv :=\n                { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun),\n                  right_inv := (_ : Function.RightInverse e.invFun e.toFun) },\n              map_mul' := (_ : \u2200 (x y : TensorProduct R R' S), \u2191e (x * y) = \u2191e x * \u2191e y),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : TensorProduct R R' S),\n                    AddHom.toFun e.toAddHom (x + y) = AddHom.toFun e.toAddHom x + AddHom.toFun e.toAddHom y) })\n          Algebra.TensorProduct.includeLeftRingHom)\n      x\n[PROOFSTEP]\nchange\n  _ =\n    e\n      (x \u2297\u209c[R] 1)\n        --Porting note: Had `dsimp only [e]` here, which didn't work anymore\n[GOAL]\ncase h.e'_5.a\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nh\u2081 : RespectsIso P\nh\u2082 :\n  \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]\n    [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom\nR S R' S' : Type u\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : CommRing R'\ninst\u271d\u2077 : CommRing S'\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R R'\ninst\u271d\u2074 : Algebra R S'\ninst\u271d\u00b3 : Algebra S S'\ninst\u271d\u00b2 : Algebra R' S'\ninst\u271d\u00b9 : IsScalarTower R S S'\ninst\u271d : IsScalarTower R R' S'\nh : Algebra.IsPushout R S R' S'\nH : P (algebraMap R S)\ne : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' :=\n  IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S')))\nf' : TensorProduct R R' S \u2192\u2090[R] S' :=\n  Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')\nthis : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x\nx : R'\n\u22a2 \u2191(algebraMap R' S') x = \u2191e (x \u2297\u209c[R] 1)\n[PROOFSTEP]\nrw [h.symm.1.equiv_tmul, Algebra.smul_def, AlgHom.toLinearMap_apply, map_one, mul_one]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\n\u22a2 P pushout.inl\n[PROOFSTEP]\nrw [\u2190\n  show _ = pushout.inl from colimit.isoColimitCocone_\u03b9_inv \u27e8_, CommRingCat.pushoutCoconeIsColimit f g\u27e9 WalkingSpan.left,\n  hP'.cancel_right_isIso]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\n\u22a2 P\n    (NatTrans.app\n      { cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9\n      WalkingSpan.left)\n[PROOFSTEP]\nletI := f.toAlgebra\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis : Algebra \u2191R \u2191S := toAlgebra f\n\u22a2 P\n    (NatTrans.app\n      { cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9\n      WalkingSpan.left)\n[PROOFSTEP]\nletI := g.toAlgebra\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P\n    (NatTrans.app\n      { cocone := CommRingCat.pushoutCocone f g, isColimit := CommRingCat.pushoutCoconeIsColimit f g }.cocone.\u03b9\n      WalkingSpan.left)\n[PROOFSTEP]\ndsimp only [CommRingCat.pushoutCocone_inl, PushoutCocone.\u03b9_app_left]\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P Algebra.TensorProduct.includeLeftRingHom\n[PROOFSTEP]\napply hP R T S (TensorProduct R S T)\n[GOAL]\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : StableUnderBaseChange P\nhP' : RespectsIso P\nR S T : CommRingCat\nf : R \u27f6 S\ng : R \u27f6 T\nH : P g\nthis\u271d : Algebra \u2191R \u2191S := toAlgebra f\nthis : Algebra \u2191R \u2191T := toAlgebra g\n\u22a2 P (algebraMap \u2191R \u2191T)\n[PROOFSTEP]\nexact H\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.RingHomProperties", "llama_tokens": 21794, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673087708699, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.5294650648931649}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave x\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5 := mem_ball_self \u03b5_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave nneg : \u2200 x, 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9 := fun x => inv_nonneg.mpr (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nset b : \u03b1 \u2192 \u211d := fun a => |bound a|\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave b_int : Integrable b \u03bc := bound_integrable.norm\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave b_nonneg : \u2200 a, 0 \u2264 b a := fun a => abs_nonneg _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nreplace h_lipsch : \u2200\u1d50 a \u2202\u03bc, \u2200 x \u2208 ball x\u2080 \u03b5, \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n[GOAL]\ncase h_lipsch\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 bound a * \u2016x - x\u2080\u2016\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact h_lipsch.mono fun a ha x hx => (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave hF_int' : \u2200 x \u2208 ball x\u2080 \u03b5, Integrable (F x) \u03bc := fun x x_in \u21a6\n  by\n  have : \u2200\u1d50 a \u2202\u03bc, \u2016F x\u2080 a - F x a\u2016 \u2264 \u03b5 * b a :=\n    by\n    simp only [norm_sub_rev (F x\u2080 _)]\n    refine' h_lipsch.mono fun a ha => (ha x x_in).trans _\n    rw [mul_comm \u03b5]\n    rw [mem_ball, dist_eq_norm] at x_in \n    exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)\n  exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int (bound_integrable.norm.const_mul \u03b5) this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable (F x)\n[PROOFSTEP]\nhave : \u2200\u1d50 a \u2202\u03bc, \u2016F x\u2080 a - F x a\u2016 \u2264 \u03b5 * b a :=\n  by\n  simp only [norm_sub_rev (F x\u2080 _)]\n  refine' h_lipsch.mono fun a ha => (ha x x_in).trans _\n  rw [mul_comm \u03b5]\n  rw [mem_ball, dist_eq_norm] at x_in \n  exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x\u2080 a - F x a\u2016 \u2264 \u03b5 * b a\n[PROOFSTEP]\nsimp only [norm_sub_rev (F x\u2080 _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x a - F x\u2080 a\u2016 \u2264 \u03b5 * |bound a|\n[PROOFSTEP]\nrefine' h_lipsch.mono fun a ha => (ha x x_in).trans _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 b a * \u2016x - x\u2080\u2016 \u2264 \u03b5 * |bound a|\n[PROOFSTEP]\nrw [mul_comm \u03b5]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 b a * \u2016x - x\u2080\u2016 \u2264 |bound a| * \u03b5\n[PROOFSTEP]\nrw [mem_ball, dist_eq_norm] at x_in \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : \u2016x - x\u2080\u2016 < \u03b5\na : \u03b1\nha : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 b a * \u2016x - x\u2080\u2016 \u2264 |bound a| * \u03b5\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F x\u2080 a - F x a\u2016 \u2264 \u03b5 * b a\n\u22a2 Integrable (F x)\n[PROOFSTEP]\nexact integrable_of_norm_sub_le (hF_meas x x_in) hF_int (bound_integrable.norm.const_mul \u03b5) this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave hF'_int : Integrable F' \u03bc :=\n  have : \u2200\u1d50 a \u2202\u03bc, \u2016F' a\u2016 \u2264 b a := by\n    apply (h_diff.and h_lipsch).mono\n    rintro a \u27e8ha_diff, ha_lip\u27e9\n    refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ \u03b5_pos) <| ha_lip)\n  b_int.mono' hF'_meas this\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016F' a\u2016 \u2264 b a\n[PROOFSTEP]\napply (h_diff.and h_lipsch).mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\n\u22a2 \u2200 (x : \u03b1),\n    (HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080 \u2227\n        \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x_1 x - F x\u2080 x\u2016 \u2264 b x * \u2016x_1 - x\u2080\u2016) \u2192\n      \u2016F' x\u2016 \u2264 b x\n[PROOFSTEP]\nrintro a \u27e8ha_diff, ha_lip\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\na : \u03b1\nha_diff : HasFDerivAt (fun x => F x a) (F' a) x\u2080\nha_lip : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016F' a\u2016 \u2264 b a\n[PROOFSTEP]\nrefine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ \u03b5_pos) <| ha_lip)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrefine' \u27e8hF'_int, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave h_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080 := ball_mem_nhds x\u2080 \u03b5_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave :\n  \u2200\u1da0 x in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016((\u222b a, F x a \u2202\u03bc) - \u222b a, F x\u2080 a \u2202\u03bc) - (\u222b a, F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b a, \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - F' a (x - x\u2080)) \u2202\u03bc\u2016 :=\n  by\n  apply mem_of_superset (ball_mem_nhds _ \u03b5_pos)\n  intro x x_in; simp only\n  rw [Set.mem_setOf_eq, \u2190 norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub, \u2190\n    ContinuousLinearMap.integral_apply hF'_int]\n  exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int, hF'_int.apply_continuousLinearMap _]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\n\u22a2 \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n[PROOFSTEP]\napply mem_of_superset (ball_mem_nhds _ \u03b5_pos)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\n\u22a2 ball x\u2080 \u03b5 \u2286\n    {x |\n      (fun x =>\n          \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n            \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016)\n        x}\n[PROOFSTEP]\nintro x x_in\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 x \u2208\n    {x |\n      (fun x =>\n          \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n            \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016)\n        x}\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 x \u2208\n    {x |\n      \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n        \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016}\n[PROOFSTEP]\nrw [Set.mem_setOf_eq, \u2190 norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub, \u2190\n  ContinuousLinearMap.integral_apply hF'_int]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable fun a => F x a\ncase hg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable fun a => F x\u2080 a\ncase hf\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable fun a => F x a - F x\u2080 a\ncase hg\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable fun a => \u2191(F' a) (x - x\u2080)\n[PROOFSTEP]\nexacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int, hF'_int.apply_continuousLinearMap _]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrw [hasFDerivAt_iff_tendsto, tendsto_congr' this, \u2190 tendsto_zero_iff_norm_tendsto_zero, \u2190\n  show (\u222b a : \u03b1, \u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - (F' a) (x\u2080 - x\u2080)) \u2202\u03bc) = 0 by simp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u222b (a : \u03b1), \u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - \u2191(F' a) (x\u2080 - x\u2080)) \u2202\u03bc = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 Tendsto (fun x => \u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc) (\ud835\udcdd x\u2080)\n    (\ud835\udcdd (\u222b (a : \u03b1), \u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - \u2191(F' a) (x\u2080 - x\u2080)) \u2202\u03bc))\n[PROOFSTEP]\napply tendsto_integral_filter_of_dominated_convergence\n[GOAL]\ncase hF_meas\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u2200\u1da0 (n : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (fun a => \u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))) \u03bc\n[PROOFSTEP]\nfilter_upwards [h_ball] with _ x_in\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na\u271d : H\nx_in : a\u271d \u2208 ball x\u2080 \u03b5\n\u22a2 AEStronglyMeasurable (fun a => \u2016a\u271d - x\u2080\u2016\u207b\u00b9 \u2022 (F a\u271d a - F x\u2080 a - \u2191(F' a) (a\u271d - x\u2080))) \u03bc\n[PROOFSTEP]\napply AEStronglyMeasurable.const_smul\n[GOAL]\ncase h.hf\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na\u271d : H\nx_in : a\u271d \u2208 ball x\u2080 \u03b5\n\u22a2 AEStronglyMeasurable (fun a => F a\u271d a - F x\u2080 a - \u2191(F' a) (a\u271d - x\u2080)) \u03bc\n[PROOFSTEP]\nexact ((hF_meas _ x_in).sub (hF_meas _ x\u2080_in)).sub (hF'_meas.apply_continuousLinearMap _)\n[GOAL]\ncase h_bound\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u2200\u1da0 (n : H) in \ud835\udcdd x\u2080, \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))\u2016 \u2264 ?bound a\n[PROOFSTEP]\nrefine mem_of_superset h_ball fun x hx \u21a6 ?_\n[GOAL]\ncase h_bound\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\n\u22a2 x \u2208 {x | (fun n => \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2016\u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))\u2016 \u2264 ?bound a) x}\n[PROOFSTEP]\napply (h_diff.and h_lipsch).mono\n[GOAL]\ncase h_bound\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2200 (x_1 : \u03b1),\n    (HasFDerivAt (fun x => F x x_1) (F' x_1) x\u2080 \u2227 \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x x_1 - F x\u2080 x_1\u2016 \u2264 b x_1 * \u2016x - x\u2080\u2016) \u2192\n      \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x x_1 - F x\u2080 x_1 - \u2191(F' x_1) (x - x\u2080))\u2016 \u2264 ?bound x_1\ncase bound\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u03b1 \u2192 \u211d\n[PROOFSTEP]\nrintro a \u27e8-, ha_bound\u27e9\n[GOAL]\ncase h_bound.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016 \u2264 ?bound a\ncase bound\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u03b1 \u2192 \u211d\n[PROOFSTEP]\nshow \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - F' a (x - x\u2080))\u2016 \u2264 b a + \u2016F' a\u2016\n[GOAL]\ncase h_bound.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016 \u2264 b a + \u2016F' a\u2016\n[PROOFSTEP]\nreplace ha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016 := ha_bound x hx\n[GOAL]\ncase h_bound.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016 \u2264 b a + \u2016F' a\u2016\n[PROOFSTEP]\ncalc\n  \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - F' a (x - x\u2080))\u2016 = \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a) - \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 F' a (x - x\u2080)\u2016 := by\n    rw [smul_sub]\n  _ \u2264 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a)\u2016 + \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 F' a (x - x\u2080)\u2016 := (norm_sub_le _ _)\n  _ = \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a\u2016 + \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F' a (x - x\u2080)\u2016 := by\n    rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _\n  _ \u2264 \u2016x - x\u2080\u2016\u207b\u00b9 * (b a * \u2016x - x\u2080\u2016) + \u2016x - x\u2080\u2016\u207b\u00b9 * (\u2016F' a\u2016 * \u2016x - x\u2080\u2016) := by gcongr; exact (F' a).le_op_norm _\n  _ \u2264 b a + \u2016F' a\u2016 := ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016 = \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a) - \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 \u2191(F' a) (x - x\u2080)\u2016\n[PROOFSTEP]\nrw [smul_sub]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a)\u2016 + \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 \u2191(F' a) (x - x\u2080)\u2016 =\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a\u2016 + \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u2191(F' a) (x - x\u2080)\u2016\n[PROOFSTEP]\nrw [norm_smul_of_nonneg, norm_smul_of_nonneg]\n[GOAL]\ncase ht\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\n[PROOFSTEP]\nexact nneg _\n[GOAL]\ncase ht\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\n[PROOFSTEP]\nexact nneg _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a\u2016 + \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u2191(F' a) (x - x\u2080)\u2016 \u2264\n    \u2016x - x\u2080\u2016\u207b\u00b9 * (b a * \u2016x - x\u2080\u2016) + \u2016x - x\u2080\u2016\u207b\u00b9 * (\u2016F' a\u2016 * \u2016x - x\u2080\u2016)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016\u2191(F' a) (x - x\u2080)\u2016 \u2264 \u2016F' a\u2016 * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nexact (F' a).le_op_norm _\n[GOAL]\ncase h_bound.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016x - x\u2080\u2016\u207b\u00b9 * (b a * \u2016x - x\u2080\u2016) + \u2016x - x\u2080\u2016\u207b\u00b9 * (\u2016F' a\u2016 * \u2016x - x\u2080\u2016) \u2264 b a + \u2016F' a\u2016\n[PROOFSTEP]\nsimp only [\u2190 div_eq_inv_mul]\n[GOAL]\ncase h_bound.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 |bound a| * \u2016x - x\u2080\u2016 / \u2016x - x\u2080\u2016 + \u2016F' a\u2016 * \u2016x - x\u2080\u2016 / \u2016x - x\u2080\u2016 \u2264 |bound a| + \u2016F' a\u2016\n[PROOFSTEP]\napply_rules [add_le_add, div_le_of_nonneg_of_le_mul]\n[GOAL]\ncase h_bound.intro.h\u2081.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\nfirst\n| rfl\n| positivity\n[GOAL]\ncase h_bound.intro.h\u2081.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bound.intro.h\u2081.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h_bound.intro.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 |bound a| * \u2016x - x\u2080\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nfirst\n| rfl\n| positivity\n[GOAL]\ncase h_bound.intro.h\u2081.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 |bound a| * \u2016x - x\u2080\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bound.intro.h\u2082.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\nfirst\n| rfl\n| positivity\n[GOAL]\ncase h_bound.intro.h\u2082.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bound.intro.h\u2082.hb\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016x - x\u2080\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h_bound.intro.h\u2082.hc\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016F' a\u2016\n[PROOFSTEP]\nfirst\n| rfl\n| positivity\n[GOAL]\ncase h_bound.intro.h\u2082.hc\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016F' a\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_bound.intro.h\u2082.hc\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 0 \u2264 \u2016F' a\u2016\n[PROOFSTEP]\npositivity\n[GOAL]\ncase h_bound.intro.h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016F' a\u2016 * \u2016x - x\u2080\u2016 \u2264 \u2016F' a\u2016 * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nfirst\n| rfl\n| positivity\n[GOAL]\ncase h_bound.intro.h\u2082.h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\nx : H\nhx : x \u2208 ball x\u2080 \u03b5\na : \u03b1\nha_bound : \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\n\u22a2 \u2016F' a\u2016 * \u2016x - x\u2080\u2016 \u2264 \u2016F' a\u2016 * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nrfl\n[GOAL]\ncase bound_integrable\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 Integrable fun a => b a + \u2016F' a\u2016\n[PROOFSTEP]\nexact b_int.add hF'_int.norm\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    Tendsto (fun n => \u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))) (\ud835\udcdd x\u2080)\n      (\ud835\udcdd (\u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - \u2191(F' a) (x\u2080 - x\u2080))))\n[PROOFSTEP]\napply h_diff.mono\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\n\u22a2 \u2200 (x : \u03b1),\n    HasFDerivAt (fun x_1 => F x_1 x) (F' x) x\u2080 \u2192\n      Tendsto (fun n => \u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n x - F x\u2080 x - \u2191(F' x) (n - x\u2080))) (\ud835\udcdd x\u2080)\n        (\ud835\udcdd (\u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 x - F x\u2080 x - \u2191(F' x) (x\u2080 - x\u2080))))\n[PROOFSTEP]\nintro a ha\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Tendsto (fun n => \u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))) (\ud835\udcdd x\u2080)\n    (\ud835\udcdd (\u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - \u2191(F' a) (x\u2080 - x\u2080))))\n[PROOFSTEP]\nsuffices Tendsto (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - F' a (x - x\u2080))) (\ud835\udcdd x\u2080) (\ud835\udcdd 0) by simpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis\u271d :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\nthis : Tendsto (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2016n - x\u2080\u2016\u207b\u00b9 \u2022 (F n a - F x\u2080 a - \u2191(F' a) (n - x\u2080))) (\ud835\udcdd x\u2080)\n    (\ud835\udcdd (\u2016x\u2080 - x\u2080\u2016\u207b\u00b9 \u2022 (F x\u2080 a - F x\u2080 a - \u2191(F' a) (x\u2080 - x\u2080))))\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Tendsto (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_zero_iff_norm_tendsto_zero]\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Tendsto (fun e => \u2016\u2016e - x\u2080\u2016\u207b\u00b9 \u2022 (F e a - F x\u2080 a - \u2191(F' a) (e - x\u2080))\u2016) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave :\n  (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a - F' a (x - x\u2080)\u2016) = fun x => \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - F' a (x - x\u2080))\u2016 :=\n  by\n  ext x\n  rw [norm_smul_of_nonneg (nneg _)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)\u2016) = fun x =>\n    \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx : H\n\u22a2 \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)\u2016 = \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016\n[PROOFSTEP]\nrw [norm_smul_of_nonneg (nneg _)]\n[GOAL]\ncase h_lim\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\nnneg : \u2200 (x : H), 0 \u2264 \u2016x - x\u2080\u2016\u207b\u00b9\nb : \u03b1 \u2192 \u211d := fun a => |bound a|\nb_int : Integrable b\nb_nonneg : \u2200 (a : \u03b1), 0 \u2264 b a\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 b a * \u2016x - x\u2080\u2016\nhF_int' : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x\u2080 \u03b5 \u2208 \ud835\udcdd x\u2080\nthis\u271d :\n  \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080,\n    \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016\u222b (a : \u03b1), F x a \u2202\u03bc - \u222b (a : \u03b1), F x\u2080 a \u2202\u03bc - \u2191(\u222b (a : \u03b1), F' a \u2202\u03bc) (x - x\u2080)\u2016 =\n      \u2016\u222b (a : \u03b1), \u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)) \u2202\u03bc\u2016\na : \u03b1\nha : HasFDerivAt (fun x => F x a) (F' a) x\u2080\nthis :\n  (fun x => \u2016x - x\u2080\u2016\u207b\u00b9 * \u2016F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080)\u2016) = fun x =>\n    \u2016\u2016x - x\u2080\u2016\u207b\u00b9 \u2022 (F x a - F x\u2080 a - \u2191(F' a) (x - x\u2080))\u2016\n\u22a2 Tendsto (fun e => \u2016\u2016e - x\u2080\u2016\u207b\u00b9 \u2022 (F e a - F x\u2080 a - \u2191(F' a) (e - x\u2080))\u2016) (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [hasFDerivAt_iff_tendsto, this] at ha \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4_pos, h\u03b4\u27e9 : \u2203 \u03b4 > 0, \u2200 x \u2208 ball x\u2080 \u03b4, AEStronglyMeasurable (F x) \u03bc \u2227 x \u2208 ball x\u2080 \u03b5\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc \u2227 x \u2208 ball x\u2080 \u03b5\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc \u2227 x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x\u2080 \u03b5_pos))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc \u2227 x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nchoose h\u03b4_meas h\u03b4\u03b5 using h\u03b4\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nreplace h_lip : \u2200\u1d50 a : \u03b1 \u2202\u03bc, \u2200 x \u2208 ball x\u2080 \u03b4, \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\n[GOAL]\ncase h_lip\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact h_lip.mono fun a lip x hx => lip.norm_sub_le (h\u03b4\u03b5 x hx) (mem_ball_self \u03b5_pos)\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nreplace bound_integrable := bound_integrable.norm\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 Integrable F' \u2227 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\napply hasFDerivAt_integral_of_dominated_loc_of_lip' \u03b4_pos\n[GOAL]\ncase intro.intro.hF_meas\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (fun a => F x a) \u03bc\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.hF_int\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 Integrable fun a => F x\u2080 a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.hF'_meas\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 AEStronglyMeasurable F' \u03bc\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.h_lipsch\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 ?m.181944 a * \u2016x - x\u2080\u2016\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.bound_integrable\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 Integrable fun a => |bound a|\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.h_diff\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n\u03b4 : \u211d\n\u03b4_pos : \u03b4 > 0\nh\u03b4_meas : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 AEStronglyMeasurable (F x) \u03bc\nh\u03b4\u03b5 : \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 x \u2208 ball x\u2080 \u03b5\nh_lip : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b4 \u2192 \u2016F x a - F x\u2080 a\u2016 \u2264 |bound a| * \u2016x - x\u2080\u2016\nbound_integrable : Integrable fun a => \u2016bound a\u2016\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' a) x\u2080\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nletI : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave x\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5 := mem_ball_self \u03b5_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave diff_x\u2080 : \u2200\u1d50 a \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080 := h_diff.mono fun a ha => ha x\u2080 x\u2080_in\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave : \u2200\u1d50 a \u2202\u03bc, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5) :=\n  by\n  apply (h_diff.and h_bound).mono\n  rintro a \u27e8ha_deriv, ha_bound\u27e9\n  refine'\n    (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasFDerivWithinAt)\n      fun x x_in => _\n  rw [\u2190 NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]\n  exact (ha_bound x x_in).trans (le_abs_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n[PROOFSTEP]\napply (h_diff.and h_bound).mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 \u2200 (x : \u03b1),\n    ((\u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) \u2227\n        \u2200 (x_1 : H), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x) \u2192\n      LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\n[PROOFSTEP]\nrintro a \u27e8ha_deriv, ha_bound\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\n\u22a2 LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n[PROOFSTEP]\nrefine'\n  (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasFDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasFDerivWithinAt)\n    fun x x_in => _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2016F' x a\u2016\u208a \u2264 \u2191Real.nnabs (bound a)\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nx : H\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2016F' x a\u2016 \u2264 |bound a|\n[PROOFSTEP]\nexact (ha_bound x x_in).trans (le_abs_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : H \u2192 \u03b1 \u2192 E\nF' : H \u2192 \u03b1 \u2192 H \u2192L[\ud835\udd5c] E\nx\u2080 : H\nbound : \u03b1 \u2192 \u211d\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : H) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : H), x \u2208 ball x\u2080 \u03b5 \u2192 HasFDerivAt (fun x => F x a) (F' x a) x\nthis\u271d : NormedSpace \u211d H := NormedSpace.restrictScalars \u211d \ud835\udd5c H\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact (hasFDerivAt_integral_of_dominated_loc_of_lip \u03b5_pos hF_meas hF_int hF'_meas this bound_integrable diff_x\u2080).2\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' a) x\u2080\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nset L : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E 1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' a) x\u2080\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nreplace h_diff : \u2200\u1d50 a \u2202\u03bc, HasFDerivAt (fun x => F x a) (L (F' a)) x\u2080 := h_diff.mono fun x hx => hx.hasFDerivAt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave hm : AEStronglyMeasurable (L \u2218 F') \u03bc := L.continuous.comp_aestronglyMeasurable hF'_meas\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\ncases' hasFDerivAt_integral_of_dominated_loc_of_lip \u03b5_pos hF_meas hF_int hm h_lipsch bound_integrable h_diff with\n  hF'_int key\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nhF'_int : Integrable (\u2191L \u2218 F')\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nreplace hF'_int : Integrable F' \u03bc\n[GOAL]\ncase hF'_int\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nhF'_int : Integrable (\u2191L \u2218 F')\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\n\u22a2 Integrable F'\n[PROOFSTEP]\nrw [\u2190 integrable_norm_iff hm] at hF'_int \n[GOAL]\ncase hF'_int\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nhF'_int : Integrable fun a => \u2016(\u2191L \u2218 F') a\u2016\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\n\u22a2 Integrable F'\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), integrable_norm_iff, hF'_meas, one_mul, norm_one, ContinuousLinearMap.comp_apply,\n  ContinuousLinearMap.coe_restrict_scalarsL', ContinuousLinearMap.norm_restrictScalars,\n  ContinuousLinearMap.norm_smulRightL_apply] using hF'_int\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\nhF'_int : Integrable F'\n\u22a2 Integrable F' \u2227 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nrefine' \u27e8hF'_int, _\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191L (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\nhF'_int : Integrable F'\n\u22a2 HasDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), F' a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nsimp_rw [hasDerivAt_iff_hasFDerivAt] at h_diff \u22a2\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF : \ud835\udd5c \u2192 \u03b1 \u2192 E\nF' : \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable F' \u03bc\nbound : \u03b1 \u2192 \u211d\nh_lipsch : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\nbound_integrable : Integrable bound\nL : E \u2192L[\ud835\udd5c] \ud835\udd5c \u2192L[\ud835\udd5c] E := \u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasFDerivAt (fun x => F x a) (\u2191(\u2191(ContinuousLinearMap.smulRightL \ud835\udd5c \ud835\udd5c E) 1) (F' a)) x\u2080\nhm : AEStronglyMeasurable (\u2191L \u2218 F') \u03bc\nkey : HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (\u222b (a : \u03b1), (\u2191L \u2218 F') a \u2202\u03bc) x\u2080\nhF'_int : Integrable F'\n\u22a2 HasFDerivAt (fun x => \u222b (a : \u03b1), F x a \u2202\u03bc) (ContinuousLinearMap.smulRight 1 (\u222b (a : \u03b1), F' a \u2202\u03bc)) x\u2080\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), ContinuousLinearMap.integral_comp_comm _ hF'_int] using key\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\n\u22a2 Integrable (F' x\u2080) \u2227 HasDerivAt (fun n => \u222b (a : \u03b1), F n a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave x\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5 := mem_ball_self \u03b5_pos\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\n\u22a2 Integrable (F' x\u2080) \u2227 HasDerivAt (fun n => \u222b (a : \u03b1), F n a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave diff_x\u2080 : \u2200\u1d50 a \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080 := h_diff.mono fun a ha => ha x\u2080 x\u2080_in\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 Integrable (F' x\u2080) \u2227 HasDerivAt (fun n => \u222b (a : \u03b1), F n a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nhave : \u2200\u1d50 a \u2202\u03bc, LipschitzOnWith (Real.nnabs (bound a)) (fun x : \ud835\udd5c => F x a) (ball x\u2080 \u03b5) :=\n  by\n  apply (h_diff.and h_bound).mono\n  rintro a \u27e8ha_deriv, ha_bound\u27e9\n  refine'\n    (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt)\n      fun x x_in => _\n  rw [\u2190 NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]\n  exact (ha_bound x x_in).trans (le_abs_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n[PROOFSTEP]\napply (h_diff.and h_bound).mono\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\n\u22a2 \u2200 (x : \u03b1),\n    ((\u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1) \u2227\n        \u2200 (x_1 : \ud835\udd5c), x_1 \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x_1 x\u2016 \u2264 bound x) \u2192\n      LipschitzOnWith (\u2191Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x\u2080 \u03b5)\n[PROOFSTEP]\nrintro a \u27e8ha_deriv, ha_bound\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\n\u22a2 LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n[PROOFSTEP]\nrefine'\n  (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le (fun x x_in => (ha_deriv x x_in).hasDerivWithinAt)\n    fun x x_in => _\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nx : \ud835\udd5c\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2016F' x a\u2016\u208a \u2264 \u2191Real.nnabs (bound a)\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\na : \u03b1\nha_deriv : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nha_bound : \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nx : \ud835\udd5c\nx_in : x \u2208 ball x\u2080 \u03b5\n\u22a2 \u2016F' x a\u2016 \u2264 |bound a|\n[PROOFSTEP]\nexact (ha_bound x x_in).trans (le_abs_self _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2077 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\n\ud835\udd5c : Type u_2\ninst\u271d\u2076 : IsROrC \ud835\udd5c\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b2 : CompleteSpace E\nH : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup H\ninst\u271d : NormedSpace \ud835\udd5c H\nF F' : \ud835\udd5c \u2192 \u03b1 \u2192 E\nx\u2080 : \ud835\udd5c\n\u03b5 : \u211d\n\u03b5_pos : 0 < \u03b5\nhF_meas : \u2200\u1da0 (x : \ud835\udd5c) in \ud835\udcdd x\u2080, AEStronglyMeasurable (F x) \u03bc\nhF_int : Integrable (F x\u2080)\nhF'_meas : AEStronglyMeasurable (F' x\u2080) \u03bc\nbound : \u03b1 \u2192 \u211d\nh_bound : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 \u2016F' x a\u2016 \u2264 bound a\nbound_integrable : Integrable bound\nh_diff : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (x : \ud835\udd5c), x \u2208 ball x\u2080 \u03b5 \u2192 HasDerivAt (fun x => F x a) (F' x a) x\nx\u2080_in : x\u2080 \u2208 ball x\u2080 \u03b5\ndiff_x\u2080 : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, HasDerivAt (fun x => F x a) (F' x\u2080 a) x\u2080\nthis : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, LipschitzOnWith (\u2191Real.nnabs (bound a)) (fun x => F x a) (ball x\u2080 \u03b5)\n\u22a2 Integrable (F' x\u2080) \u2227 HasDerivAt (fun n => \u222b (a : \u03b1), F n a \u2202\u03bc) (\u222b (a : \u03b1), F' x\u2080 a \u2202\u03bc) x\u2080\n[PROOFSTEP]\nexact hasDerivAt_integral_of_dominated_loc_of_lip \u03b5_pos hF_meas hF_int hF'_meas this bound_integrable diff_x\u2080\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.ParametricIntegral", "llama_tokens": 76183, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430394931456, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5294637222299213}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\nx : \u211d\n\u22a2 \u2016x\u22c6 * x\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimp only [star, id.def, norm_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\n\u22a2 \u2200 (x : E), \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nby_cases htriv : x = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : x = 0\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimp only [htriv, star_zero]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave hnt : 0 < \u2016x\u2016 := norm_pos_iff.mpr htriv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\nhnt : 0 < \u2016x\u2016\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave hnt_star : 0 < \u2016x\u22c6\u2016 := norm_pos_iff.mpr ((AddEquiv.map_ne_zero_iff starAddEquiv (M := E)).mpr htriv)\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\nhnt : 0 < \u2016x\u2016\nhnt_star : 0 < \u2016x\u22c6\u2016\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave h\u2081 :=\n  calc\n    \u2016x\u2016 * \u2016x\u2016 = \u2016x\u22c6 * x\u2016 := norm_star_mul_self.symm\n    _ \u2264 \u2016x\u22c6\u2016 * \u2016x\u2016 := norm_mul_le _ _\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\nhnt : 0 < \u2016x\u2016\nhnt_star : 0 < \u2016x\u22c6\u2016\nh\u2081 : \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016x\u22c6\u2016 * \u2016x\u2016\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave h\u2082 :=\n  calc\n    \u2016x\u22c6\u2016 * \u2016x\u22c6\u2016 = \u2016x * x\u22c6\u2016 := by rw [\u2190 norm_star_mul_self, star_star]\n    _ \u2264 \u2016x\u2016 * \u2016x\u22c6\u2016 := norm_mul_le _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\nhnt : 0 < \u2016x\u2016\nhnt_star : 0 < \u2016x\u22c6\u2016\nh\u2081 : \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016x\u22c6\u2016 * \u2016x\u2016\n\u22a2 \u2016x\u22c6\u2016 * \u2016x\u22c6\u2016 = \u2016x * x\u22c6\u2016\n[PROOFSTEP]\nrw [\u2190 norm_star_mul_self, star_star]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhtriv : \u00acx = 0\nhnt : 0 < \u2016x\u2016\nhnt_star : 0 < \u2016x\u22c6\u2016\nh\u2081 : \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016x\u22c6\u2016 * \u2016x\u2016\nh\u2082 : \u2016x\u22c6\u2016 * \u2016x\u22c6\u2016 \u2264 \u2016x\u2016 * \u2016x\u22c6\u2016\n\u22a2 \u2016x\u22c6\u2016 = \u2016x\u2016\n[PROOFSTEP]\nexact le_antisymm (le_of_mul_le_mul_right h\u2082 hnt_star) (le_of_mul_le_mul_right h\u2081 hnt)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x * x\u22c6\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nnth_rw 1 [\u2190 star_star x]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u22c6\u22c6 * x\u22c6\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimp only [norm_star_mul_self, norm_star]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u22c6 * x\u2016 = \u2016x\u22c6\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [norm_star_mul_self, norm_star]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 x\u22c6 * x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [\u2190 norm_eq_zero, norm_star_mul_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 \u2016x\u2016 * \u2016x\u2016 = 0 \u2194 x = 0\n[PROOFSTEP]\nexact mul_self_eq_zero.trans norm_eq_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 x\u22c6 * x \u2260 0 \u2194 x \u2260 0\n[PROOFSTEP]\nsimp only [Ne.def, star_mul_self_eq_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 x * x\u22c6 = 0 \u2194 x = 0\n[PROOFSTEP]\nsimpa only [star_eq_zero, star_star] using @star_mul_self_eq_zero_iff _ _ _ _ (star x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NonUnitalNormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\n\u22a2 x * x\u22c6 \u2260 0 \u2194 x \u2260 0\n[PROOFSTEP]\nsimp only [Ne.def, mul_star_self_eq_zero_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x\u22c6 * x\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\ndsimp only [norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016(x\u22c6 * x).fst\u2016 \u2294 \u2016(x\u22c6 * x).snd\u2016 = (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) * (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016)\n[PROOFSTEP]\nsimp only [Prod.fst_mul, Prod.fst_star, Prod.snd_mul, Prod.snd_star, norm_star_mul_self, \u2190 sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.fst\u2016 ^ 2 \u2294 \u2016x.snd\u2016 ^ 2 = (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.fst\u2016 ^ 2 \u2294 \u2016x.snd\u2016 ^ 2 \u2264 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2\n[PROOFSTEP]\nrefine' max_le _ _\n[GOAL]\ncase refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.fst\u2016 ^ 2 \u2264 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2\n[PROOFSTEP]\nrw [sq_le_sq, abs_of_nonneg (norm_nonneg _)]\n[GOAL]\ncase refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.snd\u2016 ^ 2 \u2264 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2\n[PROOFSTEP]\nrw [sq_le_sq, abs_of_nonneg (norm_nonneg _)]\n[GOAL]\ncase refine'_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.fst\u2016 \u2264 |\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016|\ncase refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.snd\u2016 \u2264 |\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016|\n[PROOFSTEP]\nexact (le_max_left _ _).trans (le_abs_self _)\n[GOAL]\ncase refine'_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 \u2016x.snd\u2016 \u2264 |\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016|\n[PROOFSTEP]\nexact (le_max_right _ _).trans (le_abs_self _)\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.fst\u2016 ^ 2 \u2294 \u2016x.snd\u2016 ^ 2\n[PROOFSTEP]\nrw [le_sup_iff]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\n\u22a2 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.fst\u2016 ^ 2 \u2228 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.snd\u2016 ^ 2\n[PROOFSTEP]\nrcases le_total \u2016x.fst\u2016 \u2016x.snd\u2016 with (h | h)\n[GOAL]\ncase refine'_2.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\nh : \u2016x.fst\u2016 \u2264 \u2016x.snd\u2016\n\u22a2 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.fst\u2016 ^ 2 \u2228 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.snd\u2016 ^ 2\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase refine'_2.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : R\u2081 \u00d7 R\u2082\nh : \u2016x.snd\u2016 \u2264 \u2016x.fst\u2016\n\u22a2 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.fst\u2016 ^ 2 \u2228 (\u2016x.fst\u2016 \u2294 \u2016x.snd\u2016) ^ 2 \u2264 \u2016x.snd\u2016 ^ 2\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : (i : \u03b9) \u2192 R i\n\u22a2 \u2016x\u22c6 * x\u2016 = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimp only [norm, Pi.mul_apply, Pi.star_apply, nnnorm_star_mul_self, \u2190 sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : (i : \u03b9) \u2192 R i\n\u22a2 \u2191(Finset.sup Finset.univ fun b => \u2016x b\u2016\u208a ^ 2) = \u2191(Finset.sup Finset.univ fun b => \u2016x b\u2016\u208a) ^ 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : (i : \u03b9) \u2192 R i\n\u22a2 (Finset.sup Finset.univ fun b => \u2016x b\u2016\u208a ^ 2) = (Finset.sup Finset.univ fun b => \u2016x b\u2016\u208a) ^ 2\n[PROOFSTEP]\nexact\n  (Finset.comp_sup_eq_sup_comp_of_is_total (fun x : NNReal => x ^ 2)\n      (fun x y h => by simpa only [sq] using mul_le_mul' h h) (by simp)).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx\u271d : (i : \u03b9) \u2192 R i\nx y : NNReal\nh : x \u2264 y\n\u22a2 (fun x => x ^ 2) x \u2264 (fun x => x ^ 2) y\n[PROOFSTEP]\nsimpa only [sq] using mul_le_mul' h h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\nR\u2081 : Type u_5\nR\u2082 : Type u_6\nR : \u03b9 \u2192 Type u_7\ninst\u271d\u2079 : NonUnitalNormedRing R\u2081\ninst\u271d\u2078 : StarRing R\u2081\ninst\u271d\u2077 : CstarRing R\u2081\ninst\u271d\u2076 : NonUnitalNormedRing R\u2082\ninst\u271d\u2075 : StarRing R\u2082\ninst\u271d\u2074 : CstarRing R\u2082\ninst\u271d\u00b3 : (i : \u03b9) \u2192 NonUnitalNormedRing (R i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 StarRing (R i)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : \u2200 (i : \u03b9), CstarRing (R i)\nx : (i : \u03b9) \u2192 R i\n\u22a2 (fun x => x ^ 2) \u22a5 = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : NormedRing E\ninst\u271d\u00b2 : StarRing E\ninst\u271d\u00b9 : CstarRing E\ninst\u271d : Nontrivial E\n\u22a2 \u20161\u2016 = 1\n[PROOFSTEP]\nhave : 0 < \u2016(1 : E)\u2016 := norm_pos_iff.mpr one_ne_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : NormedRing E\ninst\u271d\u00b2 : StarRing E\ninst\u271d\u00b9 : CstarRing E\ninst\u271d : Nontrivial E\nthis : 0 < \u20161\u2016\n\u22a2 \u20161\u2016 = 1\n[PROOFSTEP]\nrw [\u2190 mul_left_inj' this.ne', \u2190 norm_star_mul_self, mul_one, star_one, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b3 : NormedRing E\ninst\u271d\u00b2 : StarRing E\ninst\u271d\u00b9 : CstarRing E\ninst\u271d : Nontrivial E\nU : { x // x \u2208 unitary E }\n\u22a2 \u2016\u2191U\u2016 = 1\n[PROOFSTEP]\nrw [\u2190 sq_eq_sq (norm_nonneg _) zero_le_one, one_pow 2, sq, \u2190 CstarRing.norm_star_mul_self, unitary.coe_star_mul_self,\n  CstarRing.norm_one]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u22a2 \u2016\u2191U * A\u2016 = \u2016A\u2016\n[PROOFSTEP]\nnontriviality E\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016\u2191U * A\u2016 = \u2016A\u2016\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016\u2191U * A\u2016 \u2264 \u2016A\u2016\n[PROOFSTEP]\ncalc\n  _ \u2264 \u2016(U : E)\u2016 * \u2016A\u2016 := norm_mul_le _ _\n  _ = \u2016A\u2016 := by rw [norm_coe_unitary, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016\u2191U\u2016 * \u2016A\u2016 = \u2016A\u2016\n[PROOFSTEP]\nrw [norm_coe_unitary, one_mul]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016A\u2016 \u2264 \u2016\u2191U * A\u2016\n[PROOFSTEP]\ncalc\n  _ = \u2016(U : E)\u22c6 * U * A\u2016 := by rw [unitary.coe_star_mul_self U, one_mul]\n  _ \u2264 \u2016(U : E)\u22c6\u2016 * \u2016(U : E) * A\u2016 := by\n    rw [mul_assoc]\n    exact norm_mul_le _ _\n  _ = \u2016(U : E) * A\u2016 := by rw [norm_star, norm_coe_unitary, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016A\u2016 = \u2016(\u2191U)\u22c6 * \u2191U * A\u2016\n[PROOFSTEP]\nrw [unitary.coe_star_mul_self U, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016(\u2191U)\u22c6 * \u2191U * A\u2016 \u2264 \u2016(\u2191U)\u22c6\u2016 * \u2016\u2191U * A\u2016\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016(\u2191U)\u22c6 * (\u2191U * A)\u2016 \u2264 \u2016(\u2191U)\u22c6\u2016 * \u2016\u2191U * A\u2016\n[PROOFSTEP]\nexact norm_mul_le _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nU : { x // x \u2208 unitary E }\nA : E\n\u271d : Nontrivial E\n\u22a2 \u2016(\u2191U)\u22c6\u2016 * \u2016\u2191U * A\u2016 = \u2016\u2191U * A\u2016\n[PROOFSTEP]\nrw [norm_star, norm_coe_unitary, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nA : E\nU : { x // x \u2208 unitary E }\n\u22a2 \u2016A * \u2191U\u2016 = \u2016((\u2191U)\u22c6 * A\u22c6)\u22c6\u2016\n[PROOFSTEP]\nsimp only [star_star, star_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nA : E\nU : { x // x \u2208 unitary E }\n\u22a2 \u2016((\u2191U)\u22c6 * A\u22c6)\u22c6\u2016 = \u2016(\u2191U)\u22c6 * A\u22c6\u2016\n[PROOFSTEP]\nrw [norm_star]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhx : IsSelfAdjoint x\nn : \u2115\n\u22a2 \u2016x ^ 2 ^ n\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ n\n[PROOFSTEP]\ninduction' n with k hk\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhx : IsSelfAdjoint x\n\u22a2 \u2016x ^ 2 ^ Nat.zero\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ Nat.zero\n[PROOFSTEP]\nsimp only [pow_zero, pow_one, Nat.zero_eq]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhx : IsSelfAdjoint x\nk : \u2115\nhk : \u2016x ^ 2 ^ k\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ k\n\u22a2 \u2016x ^ 2 ^ Nat.succ k\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ Nat.succ k\n[PROOFSTEP]\nrw [pow_succ, pow_mul', sq]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhx : IsSelfAdjoint x\nk : \u2115\nhk : \u2016x ^ 2 ^ k\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ k\n\u22a2 \u2016x ^ 2 ^ k * x ^ 2 ^ k\u2016\u208a = \u2016x\u2016\u208a ^ (2 * 2 ^ k)\n[PROOFSTEP]\nnth_rw 1 [\u2190 selfAdjoint.mem_iff.mp hx]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\nE : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b2 : NormedRing E\ninst\u271d\u00b9 : StarRing E\ninst\u271d : CstarRing E\nx : E\nhx : IsSelfAdjoint x\nk : \u2115\nhk : \u2016x ^ 2 ^ k\u2016\u208a = \u2016x\u2016\u208a ^ 2 ^ k\n\u22a2 \u2016x\u22c6 ^ 2 ^ k * x ^ 2 ^ k\u2016\u208a = \u2016x\u2016\u208a ^ (2 * 2 ^ k)\n[PROOFSTEP]\nrw [\u2190 star_pow, CstarRing.nnnorm_star_mul_self, \u2190 sq, hk, pow_mul']\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Star.Basic", "llama_tokens": 10779, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430562234877, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.5294637099781395}}
{"text": "[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\nh : Nonempty \u03b1\ninst\u271d : IsEmpty \u03b2\n\u22a2 IsEmpty (\u03b1 \u2192 \u03b2)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np : Prop\n\u22a2 IsEmpty p \u2194 \u00acp\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_Prop]\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03c0 : \u03b1 \u2192 Sort u_4\n\u22a2 IsEmpty ((a : \u03b1) \u2192 \u03c0 a) \u2194 \u2203 a, IsEmpty (\u03c0 a)\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, Classical.nonempty_pi, not_forall]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Type u_5\nE : \u03b1 \u2192 Type u_4\n\u22a2 IsEmpty (Sigma E) \u2194 \u2200 (a : \u03b1), IsEmpty (E a)\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_sigma, not_exists]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Sort u_5\nE : \u03b1 \u2192 Sort u_4\n\u22a2 IsEmpty (PSigma E) \u2194 \u2200 (a : \u03b1), IsEmpty (E a)\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_psigma, not_exists]\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\np : \u03b1 \u2192 Prop\n\u22a2 IsEmpty (Subtype p) \u2194 \u2200 (x : \u03b1), \u00acp x\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_subtype, not_exists]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u22a2 IsEmpty (\u03b1 \u00d7 \u03b2) \u2194 IsEmpty \u03b1 \u2228 IsEmpty \u03b2\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_prod, not_and_or]\n[GOAL]\n\u03b1 : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u22a2 IsEmpty (PProd \u03b1 \u03b2) \u2194 IsEmpty \u03b1 \u2228 IsEmpty \u03b2\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_pprod, not_and_or]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u22a2 IsEmpty (\u03b1 \u2295 \u03b2) \u2194 IsEmpty \u03b1 \u2227 IsEmpty \u03b2\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_sum, not_or]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2\u271d : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u22a2 IsEmpty (\u03b1 \u2295' \u03b2) \u2194 IsEmpty \u03b1 \u2227 IsEmpty \u03b2\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_psum, not_or]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Type u_4\n\u22a2 IsEmpty (ULift \u03b1) \u2194 IsEmpty \u03b1\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_ulift]\n[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b3 : Sort u_3\n\u03b1 : Sort u_4\n\u22a2 IsEmpty (PLift \u03b1) \u2194 IsEmpty \u03b1\n[PROOFSTEP]\nsimp only [\u2190 not_nonempty_iff, nonempty_plift]\n", "meta": {"mathlib_filename": "Mathlib.Logic.IsEmpty", "llama_tokens": 1017, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.795658090372256, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.5294392944860259}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 a + (b - a) = b\n[PROOFSTEP]\nrefine' le_antisymm _ le_add_tsub\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 a + (b - a) \u2264 b\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := exists_add_of_le h\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c\u271d d c : \u03b1\nh : a \u2264 a + c\n\u22a2 a + (a + c - a) \u2264 a + c\n[PROOFSTEP]\nexact add_le_add_left add_tsub_le_left a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 b - a + a = b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 a + (b - a) = b\n[PROOFSTEP]\nexact add_tsub_cancel_of_le h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : c \u2264 b\n\u22a2 a - c \u2264 b - c \u2194 a \u2264 b\n[PROOFSTEP]\nrw [tsub_le_iff_right, tsub_add_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh1 : c \u2264 a\nh2 : c \u2264 b\n\u22a2 a - c = b - c \u2194 a = b\n[PROOFSTEP]\nsimp_rw [le_antisymm_iff, tsub_le_tsub_iff_right h1, tsub_le_tsub_iff_right h2]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : c \u2264 b\nh2 : a - c < b - c\n\u22a2 a < b\n[PROOFSTEP]\nrefine' ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : c \u2264 b\nh2 : a - c < b - c\n\u22a2 a \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nh : c \u2264 a\nh2 : a - c < a - c\n\u22a2 False\n[PROOFSTEP]\nexact h2.false\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhab : b \u2264 a\nhcb : c \u2264 b\n\u22a2 a - b + (b - c) = a - c\n[PROOFSTEP]\nconvert tsub_add_cancel_of_le (tsub_le_tsub_right hab c) using 2\n[GOAL]\ncase h.e'_2.h.e'_5\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhab : b \u2264 a\nhcb : c \u2264 b\n\u22a2 a - b = a - c - (b - c)\n[PROOFSTEP]\nrw [tsub_tsub, add_tsub_cancel_of_le hcb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : c \u2264 b\n\u22a2 a - c - (b - c) = a - b\n[PROOFSTEP]\nrw [tsub_tsub, add_tsub_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\n\u22a2 a = b - c \u2192 a + c = b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\n\u22a2 b - c + c = b\n[PROOFSTEP]\nexact tsub_add_cancel_of_le h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : b \u2264 a\n\u22a2 a - b = c \u2194 a = c + b\n[PROOFSTEP]\nrw [eq_comm, hb.eq_tsub_iff_add_eq_of_le h, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\na : \u03b1\n\u22a2 a + b - c = a + (b - c)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 add_tsub_cancel_of_le h, add_comm c, \u2190 add_assoc, hc.add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\na : \u03b1\n| a + b - c\n[PROOFSTEP]\nrw [\u2190 add_tsub_cancel_of_le h, add_comm c, \u2190 add_assoc, hc.add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\na : \u03b1\n| a + b - c\n[PROOFSTEP]\nrw [\u2190 add_tsub_cancel_of_le h, add_comm c, \u2190 add_assoc, hc.add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\na : \u03b1\n| a + b - c\n[PROOFSTEP]\nrw [\u2190 add_tsub_cancel_of_le h, add_comm c, \u2190 add_assoc, hc.add_tsub_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : b \u2264 a\n\u22a2 a - b + c = a + c - b\n[PROOFSTEP]\nrw [add_comm a, hb.add_tsub_assoc_of_le h, add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhbc : AddLECancellable (b - c)\nh\u2081 : b \u2264 a\nh\u2082 : c \u2264 b\n\u22a2 a = a - b + c + (b - c)\n[PROOFSTEP]\nrw [add_assoc, add_tsub_cancel_of_le h\u2082, tsub_add_cancel_of_le h\u2081]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhd : AddLECancellable d\nhba : b \u2264 a\nhdc : d \u2264 c\n\u22a2 a - b + (c - d) = a + c - (b + d)\n[PROOFSTEP]\nrw [hb.tsub_add_eq_add_tsub hba, \u2190 hd.add_tsub_assoc_of_le hdc, tsub_tsub, add_comm d]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a \u2264 c\n\u22a2 b \u2264 c - a \u2194 b + a \u2264 c\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a \u2264 c\n\u22a2 b \u2264 c - a \u2194 a + b \u2264 c\n[PROOFSTEP]\nexact ha.le_tsub_iff_left h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\n\u22a2 a - b < c \u2194 a < b + c\n[PROOFSTEP]\nrefine' \u27e8hb.lt_add_of_tsub_lt_left, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\n\u22a2 a < b + c \u2192 a - b < c\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\nh : a < b + c\n\u22a2 a - b < c\n[PROOFSTEP]\nrefine' (tsub_le_iff_left.mpr h.le).lt_of_ne _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\nh : a < b + c\n\u22a2 a - b \u2260 c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\nh : a < b + (a - b)\n\u22a2 False\n[PROOFSTEP]\nexact h.ne' (add_tsub_cancel_of_le hba)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\n\u22a2 a - b < c \u2194 a < c + b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhba : b \u2264 a\n\u22a2 a - b < c \u2194 a < b + c\n[PROOFSTEP]\nexact hb.tsub_lt_iff_left hba\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nhc : AddLECancellable c\nh\u2081 : b \u2264 a\nh\u2082 : c \u2264 a\n\u22a2 a - b < c \u2194 a - c < b\n[PROOFSTEP]\nrw [hb.tsub_lt_iff_left h\u2081, hc.tsub_lt_iff_right h\u2082]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nhc : AddLECancellable c\nh\u2081 : a \u2264 b\nh\u2082 : c \u2264 b\n\u22a2 a \u2264 b - c \u2194 c \u2264 b - a\n[PROOFSTEP]\nrw [ha.le_tsub_iff_left h\u2081, hc.le_tsub_iff_right h\u2082]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\n\u22a2 a < b - c \u2194 a + c < b\n[PROOFSTEP]\nrefine' \u27e8fun h' => (add_le_of_le_tsub_right_of_le h h'.le).lt_of_ne _, hc.lt_tsub_of_add_lt_right\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\nh' : a < b - c\n\u22a2 a + c \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 a + c\nh' : a < a + c - c\n\u22a2 False\n[PROOFSTEP]\nexact h'.ne' hc.add_tsub_cancel_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\n\u22a2 a < b - c \u2194 c + a < b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 b\n\u22a2 a < b - c \u2194 a + c < b\n[PROOFSTEP]\nexact hc.lt_tsub_iff_right_of_le h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhab : AddLECancellable (a - b)\nh\u2081 : b \u2264 a\nh\u2082 : c \u2264 a\nh\u2083 : a - b = a - c\n\u22a2 b = c\n[PROOFSTEP]\nrw [\u2190 hab.inj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhab : AddLECancellable (a - b)\nh\u2081 : b \u2264 a\nh\u2082 : c \u2264 a\nh\u2083 : a - b = a - c\n\u22a2 a - b + b = a - b + c\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le h\u2081, h\u2083, tsub_add_cancel_of_le h\u2082]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : AddCommSemigroup \u03b1\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : ExistsAddOfLE \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : AddLECancellable b\nhca : c \u2264 a\nh : a - b < a - c\n\u22a2 c < b\n[PROOFSTEP]\nconv_lhs at h => rw [\u2190 tsub_add_cancel_of_le hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : AddCommSemigroup \u03b1\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : ExistsAddOfLE \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : AddLECancellable b\nhca : c \u2264 a\nh : a - b < a - c\n| a - b\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : AddCommSemigroup \u03b1\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : ExistsAddOfLE \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : AddLECancellable b\nhca : c \u2264 a\nh : a - b < a - c\n| a - b\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : AddCommSemigroup \u03b1\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : ExistsAddOfLE \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : AddLECancellable b\nhca : c \u2264 a\nh : a - b < a - c\n| a - b\n[PROOFSTEP]\nrw [\u2190 tsub_add_cancel_of_le hca]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2076 : AddCommSemigroup \u03b1\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : ExistsAddOfLE \u03b1\ninst\u271d\u00b3 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x < x_1\nhb : AddLECancellable b\nhca : c \u2264 a\nh : a - c + c - b < a - c\n\u22a2 c < b\n[PROOFSTEP]\nexact lt_of_add_lt_add_left (hb.lt_add_of_tsub_lt_right h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 a\nh2 : a < b\n\u22a2 a - c < b - c\n[PROOFSTEP]\napply hc.lt_tsub_of_add_lt_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 a\nh2 : a < b\n\u22a2 c + (a - c) < b\n[PROOFSTEP]\nrwa [add_tsub_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhac : AddLECancellable (a - c)\nh : c \u2264 a\n\u22a2 a + b = b + c + (a - c)\n[PROOFSTEP]\nrw [add_assoc, add_tsub_cancel_of_le h, add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2075 : AddCommSemigroup \u03b1\ninst\u271d\u2074 : PartialOrder \u03b1\ninst\u271d\u00b3 : ExistsAddOfLE \u03b1\ninst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhab : AddLECancellable (a - b)\nh : b \u2264 a\n\u22a2 a - c - (a - b) = b - c\n[PROOFSTEP]\nrw [tsub_right_comm, hab.tsub_tsub_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 b - a + a = b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a + (b - a) = b \u2194 a \u2264 b\n[PROOFSTEP]\nexact add_tsub_cancel_iff_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b = 0 \u2194 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 nonpos_iff_eq_zero, tsub_le_iff_left, add_zero]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 0 < a - b \u2194 \u00aca \u2264 b\n[PROOFSTEP]\nrw [pos_iff_ne_zero, Ne.def, tsub_eq_zero_iff_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nhc : AddLECancellable c\nh : c \u2264 a\n\u22a2 a - b \u2264 a - c \u2194 c \u2264 b\n[PROOFSTEP]\nrefine' \u27e8_, fun h => tsub_le_tsub_left h a\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nhc : AddLECancellable c\nh : c \u2264 a\n\u22a2 a - b \u2264 a - c \u2192 c \u2264 b\n[PROOFSTEP]\nrw [tsub_le_iff_left, \u2190 hc.add_tsub_assoc_of_le h, hc.le_tsub_iff_right (h.trans le_add_self), add_comm b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nhc : AddLECancellable c\nh : c \u2264 a\n\u22a2 a + c \u2264 a + b \u2192 c \u2264 b\n[PROOFSTEP]\napply ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nhb : AddLECancellable b\nhc : AddLECancellable c\nhba : b \u2264 a\nhca : c \u2264 a\n\u22a2 a - b = a - c \u2194 b = c\n[PROOFSTEP]\nsimp_rw [le_antisymm_iff, ha.tsub_le_tsub_iff_left hb hba, ha.tsub_le_tsub_iff_left hc hca, and_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 AddCommMonoid \u03b1\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : CanonicallyOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nsrc\u271d : AddCommMonoid \u03b1 := inferInstance\na b c : \u03b1\nh : a + b = a + c\n\u22a2 b = c\n[PROOFSTEP]\nsimpa only [add_tsub_cancel_left] using congr_arg (fun x => x - a) h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 0 < a - b \u2194 b < a\n[PROOFSTEP]\nrw [tsub_pos_iff_not_le, not_le]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 a - b = a - min a b\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : a \u2264 b\n\u22a2 a - b = a - min a b\n[PROOFSTEP]\nrw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c d a b : \u03b1\nh : b \u2264 a\n\u22a2 a - b = a - min a b\n[PROOFSTEP]\nrw [min_eq_right h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : c \u2264 a\n\u22a2 a - c < b - c \u2194 a < b\n[PROOFSTEP]\nrw [hc.lt_tsub_iff_left, add_tsub_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh\u2081 : 0 < a\nh\u2082 : 0 < b\n\u22a2 a - b < a\n[PROOFSTEP]\nrefine' tsub_le_self.lt_of_ne fun h => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh\u2081 : 0 < a\nh\u2082 : 0 < b\nh : a - b = a\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 h, tsub_pos_iff_lt] at h\u2081 \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh\u2081 : b < a\nh\u2082 : 0 < b\nh : a - b = a\n\u22a2 False\n[PROOFSTEP]\nexact h\u2082.not_le (ha.add_le_iff_nonpos_left.1 <| add_le_of_le_tsub_left_of_le h\u2081.le h.ge)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\n\u22a2 a - b < a \u2194 0 < a \u2227 0 < b\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8(zero_le _).trans_lt h, (zero_le b).lt_of_ne _\u27e9, fun h => ha.tsub_lt_self h.1 h.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a - b < a\n\u22a2 0 \u2260 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nha : AddLECancellable a\nh : a - 0 < a\n\u22a2 False\n[PROOFSTEP]\nrw [tsub_zero] at h \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na c d : \u03b1\nha : AddLECancellable a\nh : a < a\n\u22a2 False\n[PROOFSTEP]\nexact h.false\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b + b = max a b\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 a - b + b = max a b\n[PROOFSTEP]\nrw [max_eq_right h, tsub_eq_zero_of_le h, zero_add]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : b \u2264 a\n\u22a2 a - b + b = max a b\n[PROOFSTEP]\nrw [max_eq_left h, tsub_add_cancel_of_le h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a + (b - a) = max a b\n[PROOFSTEP]\nrw [add_comm, max_comm, tsub_add_eq_max]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - min a b = a - b\n[PROOFSTEP]\ncases' le_total a b with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : a \u2264 b\n\u22a2 a - min a b = a - b\n[PROOFSTEP]\nrw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nh : b \u2264 a\n\u22a2 a - min a b = a - b\n[PROOFSTEP]\nrw [min_eq_right h]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b + min a b = a\n[PROOFSTEP]\nrw [\u2190 tsub_min, @tsub_add_cancel_of_le]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b2 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 min a b \u2264 a\n[PROOFSTEP]\napply min_le_left\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Sub.Canonical", "llama_tokens": 12336, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.7122321781307374, "lm_q1q2_score": 0.52930816949115}}
{"text": "[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn : \u2115\nx : \ud835\udd4e R\nk : \u2115\n\u22a2 coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\n[PROOFSTEP]\ninduction' n with n ih generalizing k\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d k : \u2115\n\u22a2 coeff (x * \u2191Nat.zero) k = \u2191(aeval x.coeff) (wittMulN p Nat.zero k)\n[PROOFSTEP]\nsimp only [Nat.zero_eq, Nat.cast_zero, mul_zero, zero_coeff, wittMulN, AlgHom.map_zero, Pi.zero_apply]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk : \u2115\n\u22a2 coeff (x * \u2191(Nat.succ n)) k = \u2191(aeval x.coeff) (wittMulN p (Nat.succ n) k)\n[PROOFSTEP]\nrw [wittMulN, Nat.succ_eq_add_one, Nat.cast_add, Nat.cast_one, mul_add, mul_one, aeval_bind\u2081, add_coeff]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk : \u2115\n\u22a2 peval (wittAdd p k) ![(x * \u2191n).coeff, x.coeff] =\n    \u2191(aeval fun i => \u2191(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)) (wittAdd p k)\n[PROOFSTEP]\napply eval\u2082Hom_congr (RingHom.ext_int _ _) _ rfl\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk : \u2115\n\u22a2 Function.uncurry ![(x * \u2191n).coeff, x.coeff] = fun i => \u2191(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] i)\n[PROOFSTEP]\next1 \u27e8b, i\u27e9\n[GOAL]\ncase h.mk\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk : \u2115\nb : Fin 2\ni : \u2115\n\u22a2 Function.uncurry ![(x * \u2191n).coeff, x.coeff] (b, i) = \u2191(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] (b, i))\n[PROOFSTEP]\nfin_cases b\n[GOAL]\ncase h.mk.head\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk i : \u2115\n\u22a2 Function.uncurry ![(x * \u2191n).coeff, x.coeff] ({ val := 0, isLt := (_ : 0 < 2) }, i) =\n    \u2191(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 0, isLt := (_ : 0 < 2) }, i))\n[PROOFSTEP]\nsimp [Function.uncurry, Matrix.cons_val_zero, ih]\n[GOAL]\ncase h.mk.tail.head\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx : \ud835\udd4e R\nk\u271d n : \u2115\nih : \u2200 (k : \u2115), coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\nk i : \u2115\n\u22a2 Function.uncurry ![(x * \u2191n).coeff, x.coeff] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i) =\n    \u2191(aeval x.coeff) (Function.uncurry ![wittMulN p n, X] ({ val := 1, isLt := (_ : (fun a => a < 2) 1) }, i))\n[PROOFSTEP]\nsimp [Function.uncurry, Matrix.cons_val_one, Matrix.head_cons, aeval_X]\n[GOAL]\np : \u2115\nR\u271d : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\u271d\nn : \u2115\nR : Type u_2\n_Rcr : CommRing R\nx : \ud835\udd4e R\n\u22a2 (x * \u2191n).coeff = fun n_1 => \u2191(aeval x.coeff) (wittMulN p n n_1)\n[PROOFSTEP]\nfunext k\n[GOAL]\ncase h\np : \u2115\nR\u271d : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\u271d\nn : \u2115\nR : Type u_2\n_Rcr : CommRing R\nx : \ud835\udd4e R\nk : \u2115\n\u22a2 coeff (x * \u2191n) k = \u2191(aeval x.coeff) (wittMulN p n k)\n[PROOFSTEP]\nexact mulN_coeff n x k\n[GOAL]\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nn k : \u2115\n\u22a2 \u2191(bind\u2081 (wittMulN p n)) (wittPolynomial p \u2124 k) = \u2191n * wittPolynomial p \u2124 k\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nk : \u2115\n\u22a2 \u2191(bind\u2081 (wittMulN p Nat.zero)) (wittPolynomial p \u2124 k) = \u2191Nat.zero * wittPolynomial p \u2124 k\n[PROOFSTEP]\nsimp [wittMulN, Nat.cast_zero, zero_mul, bind\u2081_zero_wittPolynomial]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nk n : \u2115\nih : \u2191(bind\u2081 (wittMulN p n)) (wittPolynomial p \u2124 k) = \u2191n * wittPolynomial p \u2124 k\n\u22a2 \u2191(bind\u2081 (wittMulN p (Nat.succ n))) (wittPolynomial p \u2124 k) = \u2191(Nat.succ n) * wittPolynomial p \u2124 k\n[PROOFSTEP]\nrw [wittMulN, \u2190 bind\u2081_bind\u2081, wittAdd, wittStructureInt_prop]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nk n : \u2115\nih : \u2191(bind\u2081 (wittMulN p n)) (wittPolynomial p \u2124 k) = \u2191n * wittPolynomial p \u2124 k\n\u22a2 \u2191(bind\u2081 (Function.uncurry ![wittMulN p n, X]))\n      (\u2191(bind\u2081 fun i => \u2191(rename (Prod.mk i)) (wittPolynomial p \u2124 k)) (X 0 + X 1)) =\n    \u2191(Nat.succ n) * wittPolynomial p \u2124 k\n[PROOFSTEP]\nsimp only [AlgHom.map_add, Nat.cast_succ, bind\u2081_X_right]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nk n : \u2115\nih : \u2191(bind\u2081 (wittMulN p n)) (wittPolynomial p \u2124 k) = \u2191n * wittPolynomial p \u2124 k\n\u22a2 \u2191(bind\u2081 (Function.uncurry ![wittMulN p n, X])) (\u2191(rename (Prod.mk 0)) (wittPolynomial p \u2124 k)) +\n      \u2191(bind\u2081 (Function.uncurry ![wittMulN p n, X])) (\u2191(rename (Prod.mk 1)) (wittPolynomial p \u2124 k)) =\n    (\u2191n + 1) * wittPolynomial p \u2124 k\n[PROOFSTEP]\nrw [add_mul, one_mul, bind\u2081_rename, bind\u2081_rename]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nk n : \u2115\nih : \u2191(bind\u2081 (wittMulN p n)) (wittPolynomial p \u2124 k) = \u2191n * wittPolynomial p \u2124 k\n\u22a2 \u2191(bind\u2081 (Function.uncurry ![wittMulN p n, X] \u2218 Prod.mk 0)) (wittPolynomial p \u2124 k) +\n      \u2191(bind\u2081 (Function.uncurry ![wittMulN p n, X] \u2218 Prod.mk 1)) (wittPolynomial p \u2124 k) =\n    \u2191n * wittPolynomial p \u2124 k + wittPolynomial p \u2124 k\n[PROOFSTEP]\nsimp only [ih, Function.uncurry, Function.comp, bind\u2081_X_left, AlgHom.id_apply, Matrix.cons_val_zero, Matrix.head_cons,\n  Matrix.cons_val_one]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.MulP", "llama_tokens": 2900, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835330070839, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5292038449405104}}
{"text": "[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 1 \u2208\n    { carrier := {x | \u2016x\u2016 \u2264 1},\n        mul_mem' := (_ : \u2200 {a b : \u211a_[p]}, a \u2208 {x | \u2016x\u2016 \u2264 1} \u2192 b \u2208 {x | \u2016x\u2016 \u2264 1} \u2192 \u2016a * b\u2016 \u2264 1) }.carrier\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 0 \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {x | \u2016x\u2016 \u2264 1},\n              mul_mem' := (_ : \u2200 {a b : \u211a_[p]}, a \u2208 {x | \u2016x\u2016 \u2264 1} \u2192 b \u2208 {x | \u2016x\u2016 \u2264 1} \u2192 \u2016a * b\u2016 \u2264 1) },\n          one_mem' := (_ : \u20161\u2016 \u2264 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Add { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Mul { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Neg { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Sub { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 Zero { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u20161\u2016 \u2264 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\n\u22a2 \u2191z = 0 \u2194 z = 0\n[PROOFSTEP]\nrw [\u2190 coe_zero, Subtype.coe_inj]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 AddCommGroup { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 CommRing { x // x \u2208 subring p }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016k\u2016 = 1\n\u22a2 \u2016k\u207b\u00b9\u2016 \u2264 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nm n : \u2115\nh : \u2191m = \u2191n\n\u22a2 \u2191m = \u2191n\n[PROOFSTEP]\nrw [Subtype.ext_iff] at h \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nm n : \u2115\nh\u271d : \u2191m = \u2191n\nh : \u2191\u2191m = \u2191\u2191n\n\u22a2 \u2191m = \u2191n\n[PROOFSTEP]\nnorm_cast at h \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124\n\u22a2 \u2191z1 = \u2191z2 \u2194 z1 = z2\n[PROOFSTEP]\nsuffices (z1 : \u211a_[p]) = z2 \u2194 z1 = z2 from Iff.trans (by norm_cast) this\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124\nthis : \u2191z1 = \u2191z2 \u2194 z1 = z2\n\u22a2 \u2191z1 = \u2191z2 \u2194 \u2191z1 = \u2191z2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124\n\u22a2 \u2191z1 = \u2191z2 \u2194 z1 = z2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nseq : \u2115 \u2192 \u2124\nh : IsCauSeq (padicNorm p) fun n => \u2191(seq n)\n\u22a2 \u2191(PadicSeq.norm { val := fun n => \u2191(seq n), property := h }) \u2264 1\n[PROOFSTEP]\nrw [PadicSeq.norm]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nseq : \u2115 \u2192 \u2124\nh : IsCauSeq (padicNorm p) fun n => \u2191(seq n)\n\u22a2 \u2191(if hf : { val := fun n => \u2191(seq n), property := h } \u2248 0 then 0\n      else padicNorm p (\u2191{ val := fun n => \u2191(seq n), property := h } (PadicSeq.stationaryPoint hf))) \u2264\n    1\n[PROOFSTEP]\nsplit_ifs with hne\n[GOAL]\ncase pos\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nseq : \u2115 \u2192 \u2124\nh : IsCauSeq (padicNorm p) fun n => \u2191(seq n)\nhne : { val := fun n => \u2191(seq n), property := h } \u2248 0\n\u22a2 \u21910 \u2264 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nseq : \u2115 \u2192 \u2124\nh : IsCauSeq (padicNorm p) fun n => \u2191(seq n)\nhne : \u00ac{ val := fun n => \u2191(seq n), property := h } \u2248 0\n\u22a2 \u2191(padicNorm p (\u2191{ val := fun n => \u2191(seq n), property := h } (PadicSeq.stationaryPoint hne))) \u2264 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase neg\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nseq : \u2115 \u2192 \u2124\nh : IsCauSeq (padicNorm p) fun n => \u2191(seq n)\nhne : \u00ac{ val := fun n => \u2191(seq n), property := h } \u2248 0\n\u22a2 padicNorm p (\u2191{ val := fun n => \u2191(seq n), property := h } (PadicSeq.stationaryPoint hne)) \u2264 1\n[PROOFSTEP]\napply padicNorm.of_int\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nsrc\u271d : CommRing \u2124_[p] := instCommRing\n\u22a2 \u2200 (a b : \u2124_[p]), \u2016a * b\u2016 \u2264 \u2016a\u2016 * \u2016b\u2016\n[PROOFSTEP]\nsimp [norm_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2200 {x : \u2124_[p]}, \u2016x\u2016 = 0 \u2194 x = 0\n[PROOFSTEP]\nsimp [norm_eq_zero]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nx\u271d\u00b9 x\u271d : \u2124_[p]\n\u22a2 \u2016x\u271d\u00b9 * x\u271d\u2016 = \u2016x\u271d\u00b9\u2016 * \u2016x\u271d\u2016\n[PROOFSTEP]\nsimp only [norm_def, padicNormE.mul, PadicInt.coe_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\n\u22a2 \u2016z1 * z2\u2016 = \u2016z1\u2016 * \u2016z2\u2016\n[PROOFSTEP]\nsimp [norm_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\n\u22a2 \u2016z ^ 0\u2016 = \u2016z\u2016 ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\nk : \u2115\n\u22a2 \u2016z ^ (k + 1)\u2016 = \u2016z\u2016 ^ (k + 1)\n[PROOFSTEP]\nrw [pow_succ, pow_succ, norm_mul]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\nk : \u2115\n\u22a2 \u2016z\u2016 * \u2016z ^ k\u2016 = \u2016z\u2016 * \u2016z\u2016 ^ k\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\nk : \u2115\n\u22a2 \u2016z ^ k\u2016 = \u2016z\u2016 ^ k\n[PROOFSTEP]\napply norm_pow\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\nh : \u2016z1 + z2\u2016 < \u2016z2\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 \u2016z1 + z2\u2016 \u2265 \u2016z2\u2016\n[PROOFSTEP]\nrw [norm_add_eq_max_of_ne hne]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\nh : \u2016z1 + z2\u2016 < \u2016z2\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z2\u2016\n[PROOFSTEP]\napply le_max_right\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 \u2016z1 + z2\u2016 \u2265 \u2016z1\u2016\n[PROOFSTEP]\nrw [norm_add_eq_max_of_ne hne]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\nh : \u2016z1 + z2\u2016 < \u2016z1\u2016\nhne : \u00ac\u2016z1\u2016 = \u2016z2\u2016\n\u22a2 max \u2016z1\u2016 \u2016z2\u2016 \u2265 \u2016z1\u2016\n[PROOFSTEP]\napply le_max_left\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124_[p]\n\u22a2 \u2016\u2191z\u2016 = \u2016z\u2016\n[PROOFSTEP]\nsimp [norm_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nz : \u2124\n\u22a2 \u2016\u2191z\u2016 = \u2016\u2191z\u2016\n[PROOFSTEP]\nsimp [norm_def]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u2124_[p] norm\nx\u271d : \u211d\nh\u03b5 : x\u271d > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 (fun a => \u2016a\u2016) ((fun n => \u2191(\u2191f n)) j - (fun n => \u2191(\u2191f n)) i) < x\u271d\n[PROOFSTEP]\nsimpa [norm, norm_def] using f.cauchy h\u03b5\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : CauSeq \u2124_[p] norm\nhqn : \u2016CauSeq.lim (PadicInt.cauSeq_to_rat_cauSeq f)\u2016 \u2264 1\n\u03b5 : \u211d\n\u22a2 \u03b5 > 0 \u2192\n    \u2203 i,\n      \u2200 (j : \u2115),\n        j \u2265 i \u2192\n          \u2016\u2191(f - CauSeq.const norm { val := CauSeq.lim (PadicInt.cauSeq_to_rat_cauSeq f), property := hqn }) j\u2016 < \u03b5\n[PROOFSTEP]\nsimpa [norm, norm_def] using CauSeq.equiv_lim (cauSeq_to_rat_cauSeq f) \u03b5\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 k, \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := exists_nat_gt \u03b5\u207b\u00b9\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 \u2203 k, \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nrw [\u2190 inv_lt_inv h\u03b5 (_root_.zpow_pos_of_pos _ _)]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 \u03b5\u207b\u00b9 < (\u2191p ^ (-\u2191k))\u207b\u00b9\n[PROOFSTEP]\nrw [zpow_neg, inv_inv, zpow_ofNat]\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 \u03b5\u207b\u00b9 < \u2191p ^ k\n[PROOFSTEP]\napply lt_of_lt_of_le hk\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 \u2191k \u2264 \u2191p ^ k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 k \u2264 p ^ k\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase h.a\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 k < p ^ k\n[PROOFSTEP]\nconvert Nat.lt_pow_self _ _ using 1\n[GOAL]\ncase h.a.convert_2\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 1 < p\n[PROOFSTEP]\nexact hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u03b5\u207b\u00b9 < \u2191k\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 k, \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := @exists_pow_neg_lt p _ \u03b5 (by exact_mod_cast h\u03b5)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\n\u22a2 0 < \u2191\u03b5\n[PROOFSTEP]\nexact_mod_cast h\u03b5\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u2191\u03b5\n\u22a2 \u2203 k, \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nuse k\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u2191\u03b5\n\u22a2 \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nrw [show (p : \u211d) = (p : \u211a) by simp] at hk \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u2191p ^ (-\u2191k) < \u2191\u03b5\n\u22a2 \u2191p = \u2191\u2191p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\n\u03b5 : \u211a\nh\u03b5 : 0 < \u03b5\nk : \u2115\nhk : \u2191\u2191p ^ (-\u2191k) < \u2191\u03b5\n\u22a2 \u2191p ^ (-\u2191k) < \u03b5\n[PROOFSTEP]\nexact_mod_cast hk\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nthis : \u2016\u2191k\u2016 < 1 \u2194 \u2191p \u2223 k\n\u22a2 \u2016\u2191k\u2016 < 1 \u2194 \u2191p \u2223 k\n[PROOFSTEP]\nrwa [norm_int_cast_eq_padic_norm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u2124\nn : \u2115\nthis : \u2016\u2191k\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191(p ^ n) \u2223 k\n\u22a2 \u2016\u2191k\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191(p ^ n) \u2223 k\n[PROOFSTEP]\nsimpa [norm_int_cast_eq_padic_norm]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2016x\u2016 = \u2191p ^ (-valuation x)\n[PROOFSTEP]\nrefine @Padic.norm_eq_pow_val p hp x ?_\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191x \u2260 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u2191x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nexact Subtype.val_injective hx\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 valuation \u2191p = 1\n[PROOFSTEP]\nsimp [valuation]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\n\u22a2 0 \u2264 valuation x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x = 0\n\u22a2 0 \u2264 valuation x\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\n\u22a2 0 \u2264 valuation x\n[PROOFSTEP]\nhave h : (1 : \u211d) < p := by exact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\nh : 1 < \u2191p\n\u22a2 0 \u2264 valuation x\n[PROOFSTEP]\nrw [\u2190 neg_nonpos, \u2190 (zpow_strictMono h).le_iff_le]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\nh : 1 < \u2191p\n\u22a2 (fun x x_1 => x ^ x_1) (\u2191p) (-valuation x) \u2264 (fun x x_1 => x ^ x_1) (\u2191p) 0\n[PROOFSTEP]\nshow (p : \u211d) ^ (-valuation x) \u2264 (p : \u211d) ^ (0 : \u2124)\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\nh : 1 < \u2191p\n\u22a2 \u2191p ^ (-valuation x) \u2264 \u2191p ^ 0\n[PROOFSTEP]\nrw [\u2190 norm_eq_pow_val hx]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u00acx = 0\nh : 1 < \u2191p\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ 0\n[PROOFSTEP]\nsimpa using x.property\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\n\u22a2 valuation (\u2191p ^ n * c) = \u2191n + valuation c\n[PROOFSTEP]\nhave : \u2016(p : \u2124_[p]) ^ n * c\u2016 = \u2016(p : \u2124_[p]) ^ n\u2016 * \u2016c\u2016 := norm_mul _ _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\n\u22a2 valuation (\u2191p ^ n * c) = \u2191n + valuation c\n[PROOFSTEP]\nhave aux : (p : \u2124_[p]) ^ n * c \u2260 0 := by\n  contrapose! hc\n  rw [mul_eq_zero] at hc \n  cases' hc with hc hc\n  \u00b7 refine (hp.1.ne_zero ?_).elim\n    exact_mod_cast pow_eq_zero hc\n  \u00b7 exact hc\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\n\u22a2 \u2191p ^ n * c \u2260 0\n[PROOFSTEP]\ncontrapose! hc\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\nhc : \u2191p ^ n * c = 0\n\u22a2 c = 0\n[PROOFSTEP]\nrw [mul_eq_zero] at hc \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\nhc : \u2191p ^ n = 0 \u2228 c = 0\n\u22a2 c = 0\n[PROOFSTEP]\ncases' hc with hc hc\n[GOAL]\ncase inl\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\nhc : \u2191p ^ n = 0\n\u22a2 c = 0\n[PROOFSTEP]\nrefine (hp.1.ne_zero ?_).elim\n[GOAL]\ncase inl\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\nhc : \u2191p ^ n = 0\n\u22a2 p = 0\n[PROOFSTEP]\nexact_mod_cast pow_eq_zero hc\n[GOAL]\ncase inr\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\nhc : c = 0\n\u22a2 c = 0\n[PROOFSTEP]\nexact hc\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2016\u2191p ^ n * c\u2016 = \u2016\u2191p ^ n\u2016 * \u2016c\u2016\naux : \u2191p ^ n * c \u2260 0\n\u22a2 valuation (\u2191p ^ n * c) = \u2191n + valuation c\n[PROOFSTEP]\nrwa [norm_eq_pow_val aux, norm_p_pow, norm_eq_pow_val hc, \u2190 zpow_add\u2080, \u2190 neg_add, zpow_inj, neg_inj] at this \n[GOAL]\ncase h\u2080\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2191p ^ (-valuation (\u2191p ^ n * c)) = \u2191p ^ (-(\u2191n + valuation c))\naux : \u2191p ^ n * c \u2260 0\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.pos\n[GOAL]\ncase h\u2081\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2191p ^ (-valuation (\u2191p ^ n * c)) = \u2191p ^ (-(\u2191n + valuation c))\naux : \u2191p ^ n * c \u2260 0\n\u22a2 \u2191p \u2260 1\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_one\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhc : c \u2260 0\nthis : \u2191p ^ (-valuation (\u2191p ^ n * c)) = \u2191p ^ (-\u2191n) * \u2191p ^ (-valuation c)\naux : \u2191p ^ n * c \u2260 0\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016{ val := k, property := property\u271d }\u2016 = 1\n\u22a2 { val := k, property := property\u271d } * inv { val := k, property := property\u271d } = 1\n[PROOFSTEP]\nhave hk : k \u2260 0 := fun h' => zero_ne_one' \u211a_[p] (by simp [h'] at h )\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016{ val := k, property := property\u271d }\u2016 = 1\nh' : k = 0\n\u22a2 0 = 1\n[PROOFSTEP]\nsimp [h'] at h \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016{ val := k, property := property\u271d }\u2016 = 1\nhk : k \u2260 0\n\u22a2 { val := k, property := property\u271d } * inv { val := k, property := property\u271d } = 1\n[PROOFSTEP]\nunfold PadicInt.inv\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016{ val := k, property := property\u271d }\u2016 = 1\nhk : k \u2260 0\n\u22a2 ({ val := k, property := property\u271d } *\n      match { val := k, property := property\u271d } with\n      | { val := k, property := property } => if h : \u2016k\u2016 = 1 then { val := k\u207b\u00b9, property := (_ : \u2016k\u207b\u00b9\u2016 \u2264 1) } else 0) =\n    1\n[PROOFSTEP]\nrw [norm_eq_padic_norm] at h \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016k\u2016 = 1\nhk : k \u2260 0\n\u22a2 ({ val := k, property := property\u271d } *\n      match { val := k, property := property\u271d } with\n      | { val := k, property := property } => if h : \u2016k\u2016 = 1 then { val := k\u207b\u00b9, property := (_ : \u2016k\u207b\u00b9\u2016 \u2264 1) } else 0) =\n    1\n[PROOFSTEP]\ndsimp only\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016k\u2016 = 1\nhk : k \u2260 0\n\u22a2 ({ val := k, property := property\u271d } * if h : \u2016k\u2016 = 1 then { val := k\u207b\u00b9, property := (_ : \u2016k\u207b\u00b9\u2016 \u2264 1) } else 0) = 1\n[PROOFSTEP]\nrw [dif_pos h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016k\u2016 = 1\nhk : k \u2260 0\n\u22a2 { val := k, property := property\u271d } * { val := k\u207b\u00b9, property := (_ : \u2016k\u207b\u00b9\u2016 \u2264 1) } = 1\n[PROOFSTEP]\napply Subtype.ext_iff_val.2\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nk : \u211a_[p]\nproperty\u271d : \u2016k\u2016 \u2264 1\nh : \u2016k\u2016 = 1\nhk : k \u2260 0\n\u22a2 \u2191({ val := k, property := property\u271d } * { val := k\u207b\u00b9, property := (_ : \u2016k\u207b\u00b9\u2016 \u2264 1) }) = \u21911\n[PROOFSTEP]\nsimp [mul_inv_cancel hk]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\nhz : \u2016z\u2016 = 1\n\u22a2 inv z * z = 1\n[PROOFSTEP]\nrw [mul_comm, mul_inv hz]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\nh : IsUnit z\n\u22a2 \u2016z\u2016 = 1\n[PROOFSTEP]\nrcases isUnit_iff_dvd_one.1 h with \u27e8w, eq\u27e9\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\nh : IsUnit z\nw : \u2124_[p]\neq : 1 = z * w\n\u22a2 \u2016z\u2016 = 1\n[PROOFSTEP]\nrefine' le_antisymm (norm_le_one _) _\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\nh : IsUnit z\nw : \u2124_[p]\neq : 1 = z * w\n\u22a2 1 \u2264 \u2016z\u2016\n[PROOFSTEP]\nhave := mul_le_mul_of_nonneg_left (norm_le_one w) (norm_nonneg z)\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\nh : IsUnit z\nw : \u2124_[p]\neq : 1 = z * w\nthis : \u2016z\u2016 * \u2016w\u2016 \u2264 \u2016z\u2016 * 1\n\u22a2 1 \u2264 \u2016z\u2016\n[PROOFSTEP]\nrwa [mul_one, \u2190 norm_mul, \u2190 eq, norm_one] at this \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz1 z2 : \u2124_[p]\nhz2 : \u2016z2\u2016 < 1\n\u22a2 \u2016z1 * z2\u2016 = \u2016z1\u2016 * \u2016z2\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\n\u22a2 z \u2208 nonunits \u2124_[p] \u2194 \u2016z\u2016 < 1\n[PROOFSTEP]\nrw [lt_iff_le_and_ne]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nz : \u2124_[p]\n\u22a2 z \u2208 nonunits \u2124_[p] \u2194 \u2016z\u2016 \u2264 1 \u2227 \u2016z\u2016 \u2260 1\n[PROOFSTEP]\nsimp [norm_le_one z, nonunits, isUnit_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nu : \u2124_[p]\u02e3\n\u22a2 IsUnit \u2191u\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nu : \u211a_[p] := \u2191x * \u2191p ^ (-valuation x)\n\u22a2 \u2016u\u2016 = 1\n[PROOFSTEP]\nsimp [hx, Nat.zpow_ne_zero_of_pos (by exact_mod_cast hp.1.pos) x.valuation, norm_eq_pow_val, zpow_neg, inv_mul_cancel]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nu : \u211a_[p] := \u2191x * \u2191p ^ (-valuation x)\n\u22a2 0 < ?m.493673\n[PROOFSTEP]\nexact_mod_cast hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 x = \u2191(unitCoeff hx) * \u2191p ^ Int.natAbs (valuation x)\n[PROOFSTEP]\napply Subtype.coe_injective\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 (fun a => \u2191a) x = (fun a => \u2191a) (\u2191(unitCoeff hx) * \u2191p ^ Int.natAbs (valuation x))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191x = \u2191\u2191(unitCoeff hx) * \u2191p ^ Int.natAbs (valuation x)\n[PROOFSTEP]\nhave repr : (x : \u211a_[p]) = unitCoeff hx * (p : \u211a_[p]) ^ x.valuation :=\n  by\n  rw [unitCoeff_coe, mul_assoc, \u2190 zpow_add\u2080]\n  \u00b7 simp\n  \u00b7 exact_mod_cast hp.1.ne_zero\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191x = \u2191\u2191(unitCoeff hx) * \u2191p ^ valuation x\n[PROOFSTEP]\nrw [unitCoeff_coe, mul_assoc, \u2190 zpow_add\u2080]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191x = \u2191x * \u2191p ^ (-valuation x + valuation x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nrepr : \u2191x = \u2191\u2191(unitCoeff hx) * \u2191p ^ valuation x\n\u22a2 \u2191x = \u2191\u2191(unitCoeff hx) * \u2191p ^ Int.natAbs (valuation x)\n[PROOFSTEP]\nconvert repr using 2\n[GOAL]\ncase h.e'_3.h.e'_6\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nrepr : \u2191x = \u2191\u2191(unitCoeff hx) * \u2191p ^ valuation x\n\u22a2 \u2191p ^ Int.natAbs (valuation x) = \u2191p ^ valuation x\n[PROOFSTEP]\nrw [\u2190 zpow_ofNat, Int.natAbs_of_nonneg (valuation_nonneg x)]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191n \u2264 valuation x\n[PROOFSTEP]\nrw [norm_eq_pow_val hx]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2191p ^ (-valuation x) \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191n \u2264 valuation x\n[PROOFSTEP]\nlift x.valuation to \u2115 using x.valuation_nonneg with k\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\n\u22a2 \u2191p ^ (-\u2191k) \u2264 \u2191p ^ (-\u2191n) \u2194 \u2191n \u2264 \u2191k\n[PROOFSTEP]\nsimp only [Int.ofNat_le, zpow_neg, zpow_ofNat]\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\n\u22a2 (\u2191p ^ k)\u207b\u00b9 \u2264 (\u2191p ^ n)\u207b\u00b9 \u2194 n \u2264 k\n[PROOFSTEP]\nhave aux : \u2200 m : \u2115, 0 < (p : \u211d) ^ m := by\n  intro m\n  refine pow_pos ?_ m\n  exact_mod_cast hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\n\u22a2 \u2200 (m : \u2115), 0 < \u2191p ^ m\n[PROOFSTEP]\nintro m\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k m : \u2115\n\u22a2 0 < \u2191p ^ m\n[PROOFSTEP]\nrefine pow_pos ?_ m\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k m : \u2115\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.pos\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\naux : \u2200 (m : \u2115), 0 < \u2191p ^ m\n\u22a2 (\u2191p ^ k)\u207b\u00b9 \u2264 (\u2191p ^ n)\u207b\u00b9 \u2194 n \u2264 k\n[PROOFSTEP]\nrw [inv_le_inv (aux _) (aux _)]\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\naux : \u2200 (m : \u2115), 0 < \u2191p ^ m\n\u22a2 \u2191p ^ n \u2264 \u2191p ^ k \u2194 n \u2264 k\n[PROOFSTEP]\nhave : p ^ n \u2264 p ^ k \u2194 n \u2264 k := (pow_strictMono_right hp.1.one_lt).le_iff_le\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\naux : \u2200 (m : \u2115), 0 < \u2191p ^ m\nthis : p ^ n \u2264 p ^ k \u2194 n \u2264 k\n\u22a2 \u2191p ^ n \u2264 \u2191p ^ k \u2194 n \u2264 k\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\naux : \u2200 (m : \u2115), 0 < \u2191p ^ m\nthis : p ^ n \u2264 p ^ k \u2194 n \u2264 k\n\u22a2 \u2191p ^ n \u2264 \u2191p ^ k \u2194 p ^ n \u2264 p ^ k\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 x \u2208 Ideal.span {\u2191p ^ n} \u2194 \u2191n \u2264 valuation x\n[PROOFSTEP]\nrw [Ideal.mem_span_singleton]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2191p ^ n \u2223 x \u2194 \u2191n \u2264 valuation x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2191p ^ n \u2223 x \u2192 \u2191n \u2264 valuation x\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase mp.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhx : \u2191p ^ n * c \u2260 0\n\u22a2 \u2191n \u2264 valuation (\u2191p ^ n * c)\n[PROOFSTEP]\nsuffices c \u2260 0 by\n  rw [valuation_p_pow_mul _ _ this, le_add_iff_nonneg_right]\n  apply valuation_nonneg\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhx : \u2191p ^ n * c \u2260 0\nthis : c \u2260 0\n\u22a2 \u2191n \u2264 valuation (\u2191p ^ n * c)\n[PROOFSTEP]\nrw [valuation_p_pow_mul _ _ this, le_add_iff_nonneg_right]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhx : \u2191p ^ n * c \u2260 0\nthis : c \u2260 0\n\u22a2 0 \u2264 valuation c\n[PROOFSTEP]\napply valuation_nonneg\n[GOAL]\ncase mp.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhx : \u2191p ^ n * c \u2260 0\n\u22a2 c \u2260 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase mp.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nc : \u2124_[p]\nhx : c = 0\n\u22a2 \u2191p ^ n * c = 0\n[PROOFSTEP]\nrw [hx, mul_zero]\n[GOAL]\ncase mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2191n \u2264 valuation x \u2192 \u2191p ^ n \u2223 x\n[PROOFSTEP]\nnth_rewrite 2 [unitCoeff_spec hx]\n[GOAL]\ncase mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn : \u2115\n\u22a2 \u2191n \u2264 valuation x \u2192 \u2191p ^ n \u2223 \u2191(unitCoeff hx) * \u2191p ^ Int.natAbs (valuation x)\n[PROOFSTEP]\nlift x.valuation to \u2115 using x.valuation_nonneg with k\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\n\u22a2 \u2191n \u2264 \u2191k \u2192 \u2191p ^ n \u2223 \u2191(unitCoeff hx) * \u2191p ^ Int.natAbs \u2191k\n[PROOFSTEP]\nsimp only [Int.natAbs_ofNat, Units.isUnit, IsUnit.dvd_mul_left, Int.ofNat_le]\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\n\u22a2 n \u2264 k \u2192 \u2191p ^ n \u2223 \u2191p ^ k\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\nH : n \u2264 k\n\u22a2 \u2191p ^ n \u2223 \u2191p ^ k\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le H\n[GOAL]\ncase mpr.intro.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\nn k : \u2115\nH : n \u2264 n + k\n\u22a2 \u2191p ^ n \u2223 \u2191p ^ (n + k)\n[PROOFSTEP]\nsimp only [pow_add, dvd_mul_right]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2115\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 x \u2208 Ideal.span {\u2191p ^ n}\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2115\nhx : x = 0\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 x \u2208 Ideal.span {\u2191p ^ n}\n[PROOFSTEP]\nsubst hx\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 \u20160\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 0 \u2208 Ideal.span {\u2191p ^ n}\n[PROOFSTEP]\nsimp only [norm_zero, zpow_neg, zpow_ofNat, inv_nonneg, iff_true_iff, Submodule.zero_mem]\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\n\u22a2 0 \u2264 \u2191p ^ n\n[PROOFSTEP]\nexact_mod_cast Nat.zero_le _\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2115\nhx : \u00acx = 0\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ (-\u2191n) \u2194 x \u2208 Ideal.span {\u2191p ^ n}\n[PROOFSTEP]\nrw [norm_le_pow_iff_le_valuation x hx, mem_span_pow_iff_le_valuation x hx]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2124\n\u22a2 \u2016x\u2016 \u2264 \u2191p ^ n \u2194 \u2016x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nrw [norm_def]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2124\n\u22a2 \u2016\u2191x\u2016 \u2264 \u2191p ^ n \u2194 \u2016\u2191x\u2016 < \u2191p ^ (n + 1)\n[PROOFSTEP]\nexact Padic.norm_le_pow_iff_norm_lt_pow_add_one _ _\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nn : \u2124\n\u22a2 \u2016x\u2016 < \u2191p ^ n \u2194 \u2016x\u2016 \u2264 \u2191p ^ (n - 1)\n[PROOFSTEP]\nrw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x\n[PROOFSTEP]\nhave := norm_le_pow_iff_mem_span_pow x 1\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 x \u2208 Ideal.span {\u2191p ^ 1}\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x\n[PROOFSTEP]\nrw [Ideal.mem_span_singleton, pow_one] at this \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x\n[PROOFSTEP]\nrw [\u2190 this, norm_le_pow_iff_norm_lt_pow_add_one]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2016x\u2016 < \u2191p ^ (-\u21911 + 1)\n[PROOFSTEP]\nsimp only [zpow_zero, Int.ofNat_zero, Int.ofNat_succ, add_left_neg, zero_add]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\na : \u2124\n\u22a2 \u2191p ^ n \u2223 \u2191a \u2194 \u2191(p ^ n) \u2223 a\n[PROOFSTEP]\nrw [\u2190 Nat.cast_pow, \u2190 norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow, Ideal.mem_span_singleton, Nat.cast_pow]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (a b : \u2124_[p]), a \u2208 nonunits \u2124_[p] \u2192 b \u2208 nonunits \u2124_[p] \u2192 a + b \u2208 nonunits \u2124_[p]\n[PROOFSTEP]\nsimp only [mem_nonunits]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 (a b : \u2124_[p]), \u2016a\u2016 < 1 \u2192 \u2016b\u2016 < 1 \u2192 \u2016a + b\u2016 < 1\n[PROOFSTEP]\nexact fun x y => norm_lt_one_add\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2208 nonunits \u2124_[p]\n[PROOFSTEP]\nhave : (p : \u211d)\u207b\u00b9 < 1 := inv_lt_one <| by exact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 1 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.one_lt\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nthis : (\u2191p)\u207b\u00b9 < 1\n\u22a2 \u2191p \u2208 nonunits \u2124_[p]\n[PROOFSTEP]\nrwa [\u2190 norm_p, \u2190 mem_nonunits] at this \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 maximalIdeal \u2124_[p] = Ideal.span {\u2191p}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 maximalIdeal \u2124_[p] \u2264 Ideal.span {\u2191p}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2208 maximalIdeal \u2124_[p]\n\u22a2 x \u2208 Ideal.span {\u2191p}\n[PROOFSTEP]\nsimp only [LocalRing.mem_maximalIdeal, mem_nonunits] at hx \n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : \u2016x\u2016 < 1\n\u22a2 x \u2208 Ideal.span {\u2191p}\n[PROOFSTEP]\nrwa [Ideal.mem_span_singleton, \u2190 norm_lt_one_iff_dvd]\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Ideal.span {\u2191p} \u2264 maximalIdeal \u2124_[p]\n[PROOFSTEP]\nrw [Ideal.span_le, Set.singleton_subset_iff]\n[GOAL]\ncase a\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2208 \u2191(maximalIdeal \u2124_[p])\n[PROOFSTEP]\nexact p_nonnunit\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Prime \u2191p\n[PROOFSTEP]\nrw [\u2190 Ideal.span_singleton_prime, \u2190 maximalIdeal_eq_span_p]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 Ideal.IsPrime (maximalIdeal \u2124_[p])\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2191p \u2260 0\n[PROOFSTEP]\nexact_mod_cast hp.1.ne_zero\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nhx : x \u2260 0\n\u22a2 \u2191p ^ Int.natAbs (valuation x) * \u2191(unitCoeff hx) = x\n[PROOFSTEP]\nrw [mul_comm, \u2190 unitCoeff_spec hx]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 x m \u2261 x n [SMOD maximalIdeal \u2124_[p] ^ m \u2022 \u22a4]\n\u22a2 \u2203 L, \u2200 (n : \u2115), x n \u2261 L [SMOD maximalIdeal \u2124_[p] ^ n \u2022 \u22a4]\n[PROOFSTEP]\nsimp only [\u2190 Ideal.one_eq_top, smul_eq_mul, mul_one, SModEq.sub_mem, maximalIdeal_eq_span_p, Ideal.span_singleton_pow, \u2190\n  norm_le_pow_iff_mem_span_pow] at hx \u22a2\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u22a2 \u2203 L, \u2200 (n : \u2115), \u2016x n - L\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nlet x' : CauSeq \u2124_[p] norm := \u27e8x, ?_\u27e9\n[GOAL]\ncase refine_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := ?refine_1 }\n\u22a2 \u2203 L, \u2200 (n : \u2115), \u2016x n - L\u2016 \u2264 \u2191p ^ (-\u2191n)\ncase refine_1\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u22a2 IsCauSeq norm x\n[PROOFSTEP]\nswap\n[GOAL]\ncase refine_1\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u22a2 IsCauSeq norm x\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase refine_1\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := exists_pow_neg_lt p h\u03b5\n[GOAL]\ncase refine_1.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nm : \u2115\nhm : \u2191p ^ (-\u2191m) < \u03b5\n\u22a2 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5\n[PROOFSTEP]\nrefine \u27e8m, fun n hn => lt_of_le_of_lt ?_ hm\u27e9\n[GOAL]\ncase refine_1.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nm : \u2115\nhm : \u2191p ^ (-\u2191m) < \u03b5\nn : \u2115\nhn : n \u2265 m\n\u22a2 \u2016x n - x m\u2016 \u2264 \u2191p ^ (-\u2191m)\n[PROOFSTEP]\nrw [\u2190 neg_sub, norm_neg]\n[GOAL]\ncase refine_1.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nm : \u2115\nhm : \u2191p ^ (-\u2191m) < \u03b5\nn : \u2115\nhn : n \u2265 m\n\u22a2 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\n[PROOFSTEP]\nexact hx hn\n[GOAL]\ncase refine_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\n\u22a2 \u2203 L, \u2200 (n : \u2115), \u2016x n - L\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nrefine \u27e8x'.lim, fun n => ?_\u27e9\n[GOAL]\ncase refine_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nhave : (0 : \u211d) < (p : \u211d) ^ (-n : \u2124) := by\n  apply zpow_pos_of_pos\n  exact_mod_cast hp.1.pos\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\n\u22a2 0 < \u2191p ^ (-\u2191n)\n[PROOFSTEP]\napply zpow_pos_of_pos\n[GOAL]\ncase ha\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\n\u22a2 0 < \u2191p\n[PROOFSTEP]\nexact_mod_cast hp.1.pos\n[GOAL]\ncase refine_2\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := equiv_def\u2083 (equiv_lim x') this\n[GOAL]\ncase refine_2.intro\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 \u2016\u2191x' k - \u2191(const norm (CauSeq.lim x')) j\u2016 < \u2191p ^ (-\u2191n)\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nby_cases hin : i \u2264 n\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 \u2016\u2191x' k - \u2191(const norm (CauSeq.lim x')) j\u2016 < \u2191p ^ (-\u2191n)\nhin : i \u2264 n\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nexact (hi i le_rfl n hin).le\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 \u2016\u2191x' k - \u2191(const norm (CauSeq.lim x')) j\u2016 < \u2191p ^ (-\u2191n)\nhin : \u00aci \u2264 n\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\npush_neg at hin \n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhi : \u2200 (j : \u2115), j \u2265 i \u2192 \u2200 (k : \u2115), k \u2265 j \u2192 \u2016\u2191x' k - \u2191(const norm (CauSeq.lim x')) j\u2016 < \u2191p ^ (-\u2191n)\nhin : n < i\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nspecialize hi i le_rfl i le_rfl\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhin : n < i\nhi : \u2016\u2191x' i - \u2191(const norm (CauSeq.lim x')) i\u2016 < \u2191p ^ (-\u2191n)\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nspecialize hx hin.le\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx\u271d : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhin : n < i\nhi : \u2016\u2191x' i - \u2191(const norm (CauSeq.lim x')) i\u2016 < \u2191p ^ (-\u2191n)\nhx : \u2016x n - x i\u2016 \u2264 \u2191p ^ (-\u2191n)\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nhave := nonarchimedean (x n - x i : \u2124_[p]) (x i - x'.lim)\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx\u271d : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis\u271d : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhin : n < i\nhi : \u2016\u2191x' i - \u2191(const norm (CauSeq.lim x')) i\u2016 < \u2191p ^ (-\u2191n)\nhx : \u2016x n - x i\u2016 \u2264 \u2191p ^ (-\u2191n)\nthis : \u2016x n - x i + (x i - CauSeq.lim x')\u2016 \u2264 max \u2016x n - x i\u2016 \u2016x i - CauSeq.lim x'\u2016\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nrw [sub_add_sub_cancel] at this \n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2115 \u2192 \u2124_[p]\nhx\u271d : \u2200 {m n : \u2115}, m \u2264 n \u2192 \u2016x m - x n\u2016 \u2264 \u2191p ^ (-\u2191m)\nx' : CauSeq \u2124_[p] norm := { val := x, property := (_ : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 i, \u2200 (j : \u2115), j \u2265 i \u2192 \u2016x j - x i\u2016 < \u03b5) }\nn : \u2115\nthis\u271d : 0 < \u2191p ^ (-\u2191n)\ni : \u2115\nhin : n < i\nhi : \u2016\u2191x' i - \u2191(const norm (CauSeq.lim x')) i\u2016 < \u2191p ^ (-\u2191n)\nhx : \u2016x n - x i\u2016 \u2264 \u2191p ^ (-\u2191n)\nthis : \u2016x n - CauSeq.lim x'\u2016 \u2264 max \u2016x n - x i\u2016 \u2016x i - CauSeq.lim x'\u2016\n\u22a2 \u2016x n - CauSeq.lim x'\u2016 \u2264 \u2191p ^ (-\u2191n)\n[PROOFSTEP]\nrefine' this.trans (max_le_iff.mpr \u27e8hx, hi.le\u27e9)\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d : { x // x \u2208 nonZeroDivisors \u2124_[p] }\nx : \u2124_[p]\nhx : x \u2208 nonZeroDivisors \u2124_[p]\n\u22a2 IsUnit (\u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191{ val := x, property := hx })\n[PROOFSTEP]\nrwa [algebraMap_apply, isUnit_iff_ne_zero, PadicInt.coe_ne_zero, \u2190 mem_nonZeroDivisors_iff_ne_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nby_cases hx : \u2016x\u2016 \u2264 1\n[GOAL]\ncase pos\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u2016x\u2016 \u2264 1\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nuse(\u27e8x, hx\u27e9, 1)\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u2016x\u2016 \u2264 1\n\u22a2 x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191({ val := x, property := hx }, 1).snd =\n    \u2191(algebraMap \u2124_[p] \u211a_[p]) ({ val := x, property := hx }, 1).fst\n[PROOFSTEP]\nrw [Submonoid.coe_one, map_one, mul_one, PadicInt.algebraMap_apply, Subtype.coe_mk]\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nset n := Int.toNat (-x.valuation) with hn\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nhave hn_coe : (n : \u2124) = -x.valuation := by\n  rw [hn, Int.toNat_of_nonneg]\n  rw [Right.nonneg_neg_iff]\n  rw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx \n  exact hx.le\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\n\u22a2 \u2191n = -Padic.valuation x\n[PROOFSTEP]\nrw [hn, Int.toNat_of_nonneg]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\n\u22a2 0 \u2264 -Padic.valuation x\n[PROOFSTEP]\nrw [Right.nonneg_neg_iff]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\n\u22a2 Padic.valuation x \u2264 0\n[PROOFSTEP]\nrw [Padic.norm_le_one_iff_val_nonneg, not_le] at hx \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : Padic.valuation x < 0\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\n\u22a2 Padic.valuation x \u2264 0\n[PROOFSTEP]\nexact hx.le\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nset a := x * (p : \u211a_[p]) ^ n with ha\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nhave ha_norm : \u2016a\u2016 = 1 :=\n  by\n  have hx : x \u2260 0 := by\n    intro h0\n    rw [h0, norm_zero] at hx \n    exact hx zero_le_one\n  rw [ha, padicNormE.mul, padicNormE.norm_p_pow, Padic.norm_eq_pow_val hx, \u2190 zpow_add', hn_coe, neg_neg, add_left_neg,\n    zpow_zero]\n  exact Or.inl (Nat.cast_ne_zero.mpr (NeZero.ne p))\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\n\u22a2 \u2016a\u2016 = 1\n[PROOFSTEP]\nhave hx : x \u2260 0 := by\n  intro h0\n  rw [h0, norm_zero] at hx \n  exact hx zero_le_one\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\n\u22a2 x \u2260 0\n[PROOFSTEP]\nintro h0\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nh0 : x = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h0, norm_zero] at hx \n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac0 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nh0 : x = 0\n\u22a2 False\n[PROOFSTEP]\nexact hx zero_le_one\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx\u271d : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nhx : x \u2260 0\n\u22a2 \u2016a\u2016 = 1\n[PROOFSTEP]\nrw [ha, padicNormE.mul, padicNormE.norm_p_pow, Padic.norm_eq_pow_val hx, \u2190 zpow_add', hn_coe, neg_neg, add_left_neg,\n  zpow_zero]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx\u271d : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nhx : x \u2260 0\n\u22a2 \u2191p \u2260 0 \u2228 -Padic.valuation x + -\u2191n \u2260 0 \u2228 -Padic.valuation x = 0 \u2227 -\u2191n = 0\n[PROOFSTEP]\nexact Or.inl (Nat.cast_ne_zero.mpr (NeZero.ne p))\n[GOAL]\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nha_norm : \u2016a\u2016 = 1\n\u22a2 \u2203 x_1, x * \u2191(algebraMap \u2124_[p] \u211a_[p]) \u2191x_1.snd = \u2191(algebraMap \u2124_[p] \u211a_[p]) x_1.fst\n[PROOFSTEP]\nuse(\u27e8a, le_of_eq ha_norm\u27e9, \u27e8(p ^ n : \u2124_[p]), mem_nonZeroDivisors_iff_ne_zero.mpr (NeZero.ne _)\u27e9)\n[GOAL]\ncase h\np : \u2115\nhp : Fact (Nat.Prime p)\nx : \u211a_[p]\nhx : \u00ac\u2016x\u2016 \u2264 1\nn : \u2115 := Int.toNat (-Padic.valuation x)\nhn : n = Int.toNat (-Padic.valuation x)\nhn_coe : \u2191n = -Padic.valuation x\na : \u211a_[p] := x * \u2191p ^ n\nha : a = x * \u2191p ^ n\nha_norm : \u2016a\u2016 = 1\n\u22a2 x *\n      \u2191(algebraMap \u2124_[p] \u211a_[p])\n        \u2191({ val := a, property := (_ : \u2016a\u2016 \u2264 1) },\n              { val := \u2191(p ^ n), property := (_ : \u2191(p ^ n) \u2208 nonZeroDivisors \u2124_[p]) }).snd =\n    \u2191(algebraMap \u2124_[p] \u211a_[p])\n      ({ val := a, property := (_ : \u2016a\u2016 \u2264 1) },\n          { val := \u2191(p ^ n), property := (_ : \u2191(p ^ n) \u2208 nonZeroDivisors \u2124_[p]) }).fst\n[PROOFSTEP]\nsimp only [map_pow, map_natCast, algebraMap_apply, PadicInt.coe_pow, PadicInt.coe_nat_cast, Subtype.coe_mk,\n  Nat.cast_pow]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 {x y : \u2124_[p]}, \u2191(algebraMap \u2124_[p] \u211a_[p]) x = \u2191(algebraMap \u2124_[p] \u211a_[p]) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nsimp_rw [algebraMap_apply, Subtype.coe_inj]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\n\u22a2 \u2200 {x y : \u2124_[p]}, x = y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrefine \u27e8fun h => \u27e81, by rw [h]\u27e9, ?_\u27e9\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d y\u271d : \u2124_[p]\nh : x\u271d = y\u271d\n\u22a2 \u21911 * x\u271d = \u21911 * y\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\np : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d y\u271d : \u2124_[p]\n\u22a2 (\u2203 c, \u2191c * x\u271d = \u2191c * y\u271d) \u2192 x\u271d = y\u271d\n[PROOFSTEP]\nrintro \u27e8\u27e8c, hc\u27e9, h\u27e9\n[GOAL]\ncase intro.mk\np : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d y\u271d c : \u2124_[p]\nhc : c \u2208 nonZeroDivisors \u2124_[p]\nh : \u2191{ val := c, property := hc } * x\u271d = \u2191{ val := c, property := hc } * y\u271d\n\u22a2 x\u271d = y\u271d\n[PROOFSTEP]\nexact (mul_eq_mul_left_iff.mp h).resolve_right (mem_nonZeroDivisors_iff_ne_zero.mp hc)\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Padics.PadicIntegers", "llama_tokens": 23922, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672135527632, "lm_q2_score": 0.6548947425132315, "lm_q1q2_score": 0.5291989697530214}}
{"text": "[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\n\u22a2 discr \u211a \u2191(IsPrimitiveRoot.powerBasis \u211a h\u03b6).basis = discr \u211a \u2191(subOnePowerBasis \u211a h\u03b6).basis\n[PROOFSTEP]\nhaveI : NumberField K := @NumberField.mk _ _ _ (IsCyclotomicExtension.finiteDimensional { n } \u211a K)\n[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\n\u22a2 discr \u211a \u2191(IsPrimitiveRoot.powerBasis \u211a h\u03b6).basis = discr \u211a \u2191(subOnePowerBasis \u211a h\u03b6).basis\n[PROOFSTEP]\nhave H\u2081 : (aeval (h\u03b6.powerBasis \u211a).gen) (X - 1 : \u2124[X]) = (h\u03b6.subOnePowerBasis \u211a).gen := by simp\n[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\n\u22a2 \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\n\u22a2 discr \u211a \u2191(IsPrimitiveRoot.powerBasis \u211a h\u03b6).basis = discr \u211a \u2191(subOnePowerBasis \u211a h\u03b6).basis\n[PROOFSTEP]\nhave H\u2082 : (aeval (h\u03b6.subOnePowerBasis \u211a).gen) (X + 1 : \u2124[X]) = (h\u03b6.powerBasis \u211a).gen := by simp\n[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\n\u22a2 \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\n\u22a2 discr \u211a \u2191(IsPrimitiveRoot.powerBasis \u211a h\u03b6).basis = discr \u211a \u2191(subOnePowerBasis \u211a h\u03b6).basis\n[PROOFSTEP]\nrefine'\n  discr_eq_discr_of_toMatrix_coeff_isIntegral _ (fun i j => toMatrix_isIntegral H\u2081 _ _ _ _) fun i j =>\n    toMatrix_isIntegral H\u2082 _ _ _ _\n[GOAL]\ncase refine'_1\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\ni : Fin (IsPrimitiveRoot.powerBasis \u211a h\u03b6).dim\nj : Fin (subOnePowerBasis \u211a h\u03b6).dim\n\u22a2 IsIntegral \u2124 (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\n[PROOFSTEP]\nexact h\u03b6.isIntegral n.pos\n[GOAL]\ncase refine'_2\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\ni : Fin (IsPrimitiveRoot.powerBasis \u211a h\u03b6).dim\nj : Fin (subOnePowerBasis \u211a h\u03b6).dim\n\u22a2 minpoly \u211a (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen =\n    Polynomial.map (algebraMap \u2124 \u211a) (minpoly \u2124 (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen)\n[PROOFSTEP]\nrefine' minpoly.isIntegrallyClosed_eq_field_fractions' (K := \u211a) (h\u03b6.isIntegral n.pos)\n[GOAL]\ncase refine'_3\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\ni : Fin (subOnePowerBasis \u211a h\u03b6).dim\nj : Fin (IsPrimitiveRoot.powerBasis \u211a h\u03b6).dim\n\u22a2 IsIntegral \u2124 (subOnePowerBasis \u211a h\u03b6).gen\n[PROOFSTEP]\nexact isIntegral_sub (h\u03b6.isIntegral n.pos) isIntegral_one\n[GOAL]\ncase refine'_4\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\ni : Fin (subOnePowerBasis \u211a h\u03b6).dim\nj : Fin (IsPrimitiveRoot.powerBasis \u211a h\u03b6).dim\n\u22a2 minpoly \u211a (subOnePowerBasis \u211a h\u03b6).gen = Polynomial.map (algebraMap \u2124 \u211a) (minpoly \u2124 (subOnePowerBasis \u211a h\u03b6).gen)\n[PROOFSTEP]\nrefine' minpoly.isIntegrallyClosed_eq_field_fractions' (K := \u211a) _\n[GOAL]\ncase refine'_4\nn : \u2115+\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nce : IsCyclotomicExtension {n} \u211a K\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191n\nthis : NumberField K\nH\u2081 : \u2191(aeval (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen) (X - 1) = (subOnePowerBasis \u211a h\u03b6).gen\nH\u2082 : \u2191(aeval (subOnePowerBasis \u211a h\u03b6).gen) (X + 1) = (IsPrimitiveRoot.powerBasis \u211a h\u03b6).gen\ni : Fin (subOnePowerBasis \u211a h\u03b6).dim\nj : Fin (IsPrimitiveRoot.powerBasis \u211a h\u03b6).dim\n\u22a2 IsIntegral \u2124 (subOnePowerBasis \u211a h\u03b6).gen\n[PROOFSTEP]\nexact isIntegral_sub (h\u03b6.isIntegral n.pos) isIntegral_one\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ (k + 1)) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhaveI hne := IsCyclotomicExtension.neZero' (p ^ (k + 1)) K L\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ (k + 1)) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhaveI mf : Module.Finite K L := finiteDimensional {p ^ (k + 1)} K L\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ (k + 1)) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhaveI se : IsSeparable K L := (isGalois (p ^ (k + 1)) K L).to_isSeparable\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ (k + 1)) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nrw [discr_powerBasis_eq_norm, finrank L hirr, h\u03b6.powerBasis_gen _, \u2190 h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr,\n  PNat.pow_coe, totient_prime_pow hp.out (succ_pos k), succ_sub_one]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) *\n      \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhave coe_two : ((2 : \u2115+) : \u2115) = 2 := rfl\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) *\n      \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhave hp2 : p = 2 \u2192 k \u2260 0 := by\n  rintro rfl rfl\n  exact absurd rfl hk\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\n\u22a2 p = 2 \u2192 k \u2260 0\n[PROOFSTEP]\nrintro rfl rfl\n[GOAL]\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\ninst\u271d : IsCyclotomicExtension {2 ^ (0 + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (0 + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (0 + 1))) K)\nhk : 2 ^ (0 + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (0 + 1))\n\u22a2 False\n[PROOFSTEP]\nexact absurd rfl hk\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) *\n      \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) = \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2))\n[PROOFSTEP]\nrcases eq_or_ne p 2 with (rfl | hp2)\n[GOAL]\ncase e_a.inl\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\ninst\u271d : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u21912)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (k + 1))) K)\nhk : 2 ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (k + 1))\nhp2 : 2 = 2 \u2192 k \u2260 0\n\u22a2 (-1) ^ (\u21912 ^ k * (\u21912 - 1) * (\u21912 ^ k * (\u21912 - 1) - 1) / 2) = \u2191((-1) ^ (\u21912 ^ k * (\u21912 - 1) / 2))\n[PROOFSTEP]\nrcases Nat.exists_eq_succ_of_ne_zero (hp2 rfl) with \u27e8k, rfl\u27e9\n[GOAL]\ncase e_a.inl.intro\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nk : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ k + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ k + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ k + 1))) K)\nhk : 2 ^ (succ k + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ k + 1))\nhp2 : 2 = 2 \u2192 succ k \u2260 0\n\u22a2 (-1) ^ (\u21912 ^ succ k * (\u21912 - 1) * (\u21912 ^ succ k * (\u21912 - 1) - 1) / 2) = \u2191((-1) ^ (\u21912 ^ succ k * (\u21912 - 1) / 2))\n[PROOFSTEP]\nrw [coe_two, succ_sub_succ_eq_sub, tsub_zero, mul_one]\n[GOAL]\ncase e_a.inl.intro\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nk : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ k + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ k + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ k + 1))) K)\nhk : 2 ^ (succ k + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ k + 1))\nhp2 : 2 = 2 \u2192 succ k \u2260 0\n\u22a2 (-1) ^ (2 ^ succ k * (2 ^ succ k - 1) / 2) = \u2191((-1) ^ (2 ^ succ k / 2))\n[PROOFSTEP]\nsimp only [_root_.pow_succ]\n[GOAL]\ncase e_a.inl.intro\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nk : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ k + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ k + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ k + 1))) K)\nhk : 2 ^ (succ k + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ k + 1))\nhp2 : 2 = 2 \u2192 succ k \u2260 0\n\u22a2 (-1) ^ (2 * 2 ^ k * (2 * 2 ^ k - 1) / 2) = \u2191((-1) ^ (2 * 2 ^ k / 2))\n[PROOFSTEP]\nrw [mul_assoc, Nat.mul_div_cancel_left _ zero_lt_two, Nat.mul_div_cancel_left _ zero_lt_two]\n[GOAL]\ncase e_a.inl.intro\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nk : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ k + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ k + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ k + 1))) K)\nhk : 2 ^ (succ k + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ k + 1))\nhp2 : 2 = 2 \u2192 succ k \u2260 0\n\u22a2 (-1) ^ (2 ^ k * (2 * 2 ^ k - 1)) = \u2191((-1) ^ 2 ^ k)\n[PROOFSTEP]\ncases k\n[GOAL]\ncase e_a.inl.intro.zero\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\ninst\u271d : IsCyclotomicExtension {2 ^ (succ Nat.zero + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ Nat.zero + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ Nat.zero + 1))) K)\nhk : 2 ^ (succ Nat.zero + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ Nat.zero + 1))\nhp2 : 2 = 2 \u2192 succ Nat.zero \u2260 0\n\u22a2 (-1) ^ (2 ^ Nat.zero * (2 * 2 ^ Nat.zero - 1)) = \u2191((-1) ^ 2 ^ Nat.zero)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.inl.intro.succ\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nn\u271d : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ (succ n\u271d) + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ (succ n\u271d) + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ (succ n\u271d) + 1))) K)\nhk : 2 ^ (succ (succ n\u271d) + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ (succ n\u271d) + 1))\nhp2 : 2 = 2 \u2192 succ (succ n\u271d) \u2260 0\n\u22a2 (-1) ^ (2 ^ succ n\u271d * (2 * 2 ^ succ n\u271d - 1)) = \u2191((-1) ^ 2 ^ succ n\u271d)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_a.inl.intro.succ\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp : Fact (Nat.Prime \u21912)\nn\u271d : \u2115\ninst\u271d : IsCyclotomicExtension {2 ^ (succ (succ n\u271d) + 1)} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ (succ (succ n\u271d) + 1))\nhirr : Irreducible (cyclotomic (\u2191(2 ^ (succ (succ n\u271d) + 1))) K)\nhk : 2 ^ (succ (succ n\u271d) + 1) \u2260 2\nhne : NeZero \u2191\u2191(2 ^ (succ (succ n\u271d) + 1))\nhp2 : 2 = 2 \u2192 succ (succ n\u271d) \u2260 0\n\u22a2 (-1) ^ (2 ^ succ n\u271d * (2 * 2 ^ succ n\u271d - 1)) = (-1) ^ 2 ^ succ n\u271d\n[PROOFSTEP]\nsimp_rw [_root_.pow_succ, (even_two.mul_right _).neg_one_pow, ((even_two.mul_right _).mul_right _).neg_one_pow]\n[GOAL]\ncase e_a.inr\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : p \u2260 2\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) = \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2))\n[PROOFSTEP]\nreplace hp2 : (p : \u2115) \u2260 2\n[GOAL]\ncase hp2\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : p \u2260 2\n\u22a2 \u2191p \u2260 2\n[PROOFSTEP]\nrwa [Ne.def, \u2190 coe_two, PNat.coe_inj]\n[GOAL]\ncase e_a.inr\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) = \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2))\n[PROOFSTEP]\nhave hpo : Odd (p : \u2115) := hp.out.odd_of_ne_two hp2\n[GOAL]\ncase e_a.inr\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) = \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2))\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := (hp.out.even_sub_one hp2).two_dvd\n[GOAL]\ncase e_a.inr.intro\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\na : \u2115\nha : \u2191p - 1 = 2 * a\n\u22a2 (-1) ^ (\u2191p ^ k * (\u2191p - 1) * (\u2191p ^ k * (\u2191p - 1) - 1) / 2) = \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2))\n[PROOFSTEP]\nrw [ha, mul_left_comm, mul_assoc, Nat.mul_div_cancel_left _ two_pos, Nat.mul_div_cancel_left _ two_pos, mul_right_comm,\n  pow_mul, (hpo.pow.mul _).neg_one_pow, pow_mul, hpo.pow.neg_one_pow]\n[GOAL]\ncase e_a.inr.intro\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\na : \u2115\nha : \u2191p - 1 = 2 * a\n\u22a2 (-1) ^ a = \u2191((-1) ^ a)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\na : \u2115\nha : \u2191p - 1 = 2 * a\n\u22a2 Odd (2 * (\u2191p ^ k * a) - 1)\n[PROOFSTEP]\nrefine' Nat.Even.sub_odd _ (even_two_mul _) odd_one\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\na : \u2115\nha : \u2191p - 1 = 2 * a\n\u22a2 1 \u2264 2 * (\u2191p ^ k * a)\n[PROOFSTEP]\nrw [mul_left_comm, \u2190 ha]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2\u271d : p = 2 \u2192 k \u2260 0\nhp2 : \u2191p \u2260 2\nhpo : Odd \u2191p\na : \u2115\nha : \u2191p - 1 = 2 * a\n\u22a2 1 \u2264 \u2191p ^ k * (\u2191p - 1)\n[PROOFSTEP]\nexact one_le_mul (one_le_pow _ _ hp.1.pos) (succ_le_iff.2 <| tsub_pos_of_lt hp.1.one_lt)\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhave H := congr_arg (@derivative K _) (cyclotomic_prime_pow_mul_X_pow_sub_one K p k)\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH : \u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K * (X ^ \u2191p ^ k - 1)) = \u2191derivative (X ^ \u2191p ^ (k + 1) - 1)\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nrw [derivative_mul, derivative_sub, derivative_one, sub_zero, derivative_X_pow, C_eq_nat_cast, derivative_sub,\n  derivative_one, sub_zero, derivative_X_pow, C_eq_nat_cast, \u2190 PNat.pow_coe,\n  h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr] at H \n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191derivative (minpoly K \u03b6) * (X ^ \u2191p ^ k - 1) + minpoly K \u03b6 * (\u2191(\u2191p ^ k) * X ^ (\u2191p ^ k - 1)) =\n    \u2191\u2191(p ^ (k + 1)) * X ^ (\u2191(p ^ (k + 1)) - 1)\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nreplace H := congr_arg (fun P => aeval \u03b6 P) H\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  (fun P => \u2191(aeval \u03b6) P)\n      (\u2191derivative (minpoly K \u03b6) * (X ^ \u2191p ^ k - 1) + minpoly K \u03b6 * (\u2191(\u2191p ^ k) * X ^ (\u2191p ^ k - 1))) =\n    (fun P => \u2191(aeval \u03b6) P) (\u2191\u2191(p ^ (k + 1)) * X ^ (\u2191(p ^ (k + 1)) - 1))\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nsimp only [aeval_add, aeval_mul, minpoly.aeval, zero_mul, add_zero, aeval_nat_cast, _root_.map_sub, aeval_one,\n  aeval_X_pow] at H \n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH : \u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1)\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nreplace H := congr_arg (Algebra.norm K) H\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1)) =\n    \u2191(Algebra.norm K) (\u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhave hnorm : (norm K) (\u03b6 ^ (p : \u2115) ^ k - 1) = (p : K) ^ (p : \u2115) ^ k :=\n  by\n  by_cases hp : p = 2\n  \u00b7 exact_mod_cast h\u03b6.pow_sub_one_norm_prime_pow_of_ne_zero hirr le_rfl (hp2 hp)\n  \u00b7 exact_mod_cast h\u03b6.pow_sub_one_norm_prime_ne_two hirr le_rfl hp\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1)) =\n    \u2191(Algebra.norm K) (\u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n[PROOFSTEP]\nby_cases hp : p = 2\n[GOAL]\ncase pos\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1)) =\n    \u2191(Algebra.norm K) (\u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhp : p = 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n[PROOFSTEP]\nexact_mod_cast h\u03b6.pow_sub_one_norm_prime_pow_of_ne_zero hirr le_rfl (hp2 hp)\n[GOAL]\ncase neg\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1)) =\n    \u2191(Algebra.norm K) (\u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhp : \u00acp = 2\n\u22a2 \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n[PROOFSTEP]\nexact_mod_cast h\u03b6.pow_sub_one_norm_prime_ne_two hirr le_rfl hp\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6)) * (\u03b6 ^ \u2191p ^ k - 1)) =\n    \u2191(Algebra.norm K) (\u2191\u2191(p ^ (k + 1)) * \u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nrw [MonoidHom.map_mul, hnorm, MonoidHom.map_mul, \u2190 map_natCast (algebraMap K L), Algebra.norm_algebraMap,\n  finrank L hirr] at H \n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nconv_rhs at H =>\n  -- Porting note: need to drill down to successfully rewrite the totient\n  enter [1, 2]\n  rw [PNat.pow_coe, \u2190 succ_eq_add_one, totient_prime_pow hp.out (succ_pos k), Nat.sub_one, Nat.pred_succ]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n| \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\n[PROOFSTEP]\n  enter [1, 2]\n  rw [PNat.pow_coe, \u2190 succ_eq_add_one, totient_prime_pow hp.out (succ_pos k), Nat.sub_one, Nat.pred_succ]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n| \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\n[PROOFSTEP]\n  enter [1, 2]\n  rw [PNat.pow_coe, \u2190 succ_eq_add_one, totient_prime_pow hp.out (succ_pos k), Nat.sub_one, Nat.pred_succ]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n| \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\n[PROOFSTEP]\nenter [1, 2]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ \u03c6 \u2191(p ^ (k + 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n| \u03c6 \u2191(p ^ (k + 1))\n[PROOFSTEP]\nrw [PNat.pow_coe, \u2190 succ_eq_add_one, totient_prime_pow hp.out (succ_pos k), Nat.sub_one, Nat.pred_succ]\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (minpoly K \u03b6))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191(p ^ (k + 1)) ^ (\u2191p ^ k * (\u2191p - 1)) * \u2191(Algebra.norm K) (\u03b6 ^ (\u2191(p ^ (k + 1)) - 1))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nrw [\u2190 h\u03b6.minpoly_eq_cyclotomic_of_irreducible hirr, map_pow, h\u03b6.norm_eq_one hk hirr, one_pow, mul_one, PNat.pow_coe,\n  cast_pow, \u2190 pow_mul, \u2190 mul_assoc, mul_comm (k + 1), mul_assoc] at H \n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * ((k + 1) * (\u2191p - 1)))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nhave := mul_pos (succ_pos k) (tsub_pos_of_lt hp.out.one_lt)\n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * ((k + 1) * (\u2191p - 1)))\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis : 0 < succ k * (\u2191p - 1)\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nrw [\u2190 succ_pred_eq_of_pos this, mul_succ, pow_add _ _ ((p : \u2115) ^ k)] at H \n[GOAL]\ncase e_a\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) * \u2191\u2191p ^ \u2191p ^ k\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis : 0 < succ k * (\u2191p - 1)\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nreplace H := (mul_left_inj' fun h => ?_).1 H\n[GOAL]\ncase e_a.refine_2\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis : 0 < succ k * (\u2191p - 1)\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) = \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1)))\n\u22a2 \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) =\n    \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nsimp only [H, mul_comm _ (k + 1)]\n[GOAL]\ncase e_a.refine_2\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis : 0 < succ k * (\u2191p - 1)\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) = \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1)))\n\u22a2 \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) = \u2191\u2191(p ^ (\u2191p ^ k * ((k + 1) * (\u2191p - 1) - 1)))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase e_a.refine_1\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) * \u2191\u2191p ^ \u2191p ^ k\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis : 0 < succ k * (\u2191p - 1)\nh : \u2191\u2191p ^ \u2191p ^ k = 0\n\u22a2 False\n[PROOFSTEP]\nhave := hne.1\n[GOAL]\ncase e_a.refine_1\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) * \u2191\u2191p ^ \u2191p ^ k\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis\u271d : 0 < succ k * (\u2191p - 1)\nh : \u2191\u2191p ^ \u2191p ^ k = 0\nthis : \u2191\u2191(p ^ (k + 1)) \u2260 0\n\u22a2 False\n[PROOFSTEP]\nrw [PNat.pow_coe, Nat.cast_pow, Ne.def, pow_eq_zero_iff (by linarith)] at this \n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) * \u2191\u2191p ^ \u2191p ^ k\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis\u271d : 0 < succ k * (\u2191p - 1)\nh : \u2191\u2191p ^ \u2191p ^ k = 0\nthis : \u00ac\u2191\u2191p ^ (k + 1) = 0\n\u22a2 0 < k + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase e_a.refine_1\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\nhne : NeZero \u2191\u2191(p ^ (k + 1))\nmf : Module.Finite K L\nse : IsSeparable K L\ncoe_two : \u21912 = 2\nhp2 : p = 2 \u2192 k \u2260 0\nH :\n  \u2191(Algebra.norm K) (\u2191(aeval \u03b6) (\u2191derivative (cyclotomic (\u2191p ^ (k + 1)) K))) * \u2191\u2191p ^ \u2191p ^ k =\n    \u2191\u2191p ^ (\u2191p ^ k * pred (succ k * (\u2191p - 1))) * \u2191\u2191p ^ \u2191p ^ k\nhnorm : \u2191(Algebra.norm K) (\u03b6 ^ \u2191p ^ k - 1) = \u2191\u2191p ^ \u2191p ^ k\nthis\u271d : 0 < succ k * (\u2191p - 1)\nh : \u2191\u2191p ^ \u2191p ^ k = 0\nthis : \u00ac\u2191\u2191p = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd (pow_eq_zero h) this\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ (k + 1))\nhirr : Irreducible (cyclotomic (\u2191(p ^ (k + 1))) K)\nhk : p ^ (k + 1) \u2260 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u2191p ^ k * (\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p ^ k * ((\u2191p - 1) * (k + 1) - 1)))\n[PROOFSTEP]\nsimpa [totient_prime_pow hp.out (succ_pos k)] using discr_prime_pow_ne_two h\u03b6 hirr hk\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhcycl : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))\n[PROOFSTEP]\ncases' k with k k\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ Nat.zero) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nsimp only [coe_basis, _root_.pow_zero, powerBasis_gen _ h\u03b6, totient_one, mul_zero, mul_one, show 1 / 2 = 0 by rfl,\n  discr, traceMatrix]\n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 1 / 2 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nhave h\u03b6one : \u03b6 = 1 := by simpa using h\u03b6\n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\n\u22a2 \u03b6 = 1\n[PROOFSTEP]\nsimpa using h\u03b6\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u03b6 ^ \u2191i) (\u03b6 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nrw [h\u03b6.powerBasis_dim _, h\u03b6one, \u2190 (algebraMap K L).map_one,\n  minpoly.eq_X_sub_C_of_algebraMap_inj _ (algebraMap K L).injective, natDegree_X_sub_C]\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 Matrix.det\n      (\u2191Matrix.of fun i j => BilinForm.bilin (traceForm K L) (\u2191(algebraMap K L) 1 ^ \u2191i) (\u2191(algebraMap K L) 1 ^ \u2191j)) =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nsimp only [traceMatrix, map_one, one_pow, Matrix.det_unique, traceForm_apply, mul_one]\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u2191(trace K L) 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nrw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr]\n[GOAL]\ncase zero\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nhcycl : IsCyclotomicExtension {p ^ Nat.zero} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ Nat.zero)\nhirr : Irreducible (cyclotomic (\u2191(p ^ Nat.zero)) K)\nh\u03b6one : \u03b6 = 1\n\u22a2 \u2191Matrix.of (fun i j => \u03c6 \u2191(p ^ Nat.zero) \u2022 1) default default =\n    \u2191((-1) ^ (\u03c6 \u21911 / 2)) * \u2191\u2191(p ^ (\u2191p ^ (Nat.zero - 1) * ((\u2191p - 1) * Nat.zero - 1)))\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nby_cases hk : p ^ (k + 1) = 2\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nhave coe_two : 2 = ((2 : \u2115+) : \u2115) := rfl\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nhave hp : p = 2 := by\n  rw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at hk \n  replace hk := eq_of_prime_pow_eq (prime_iff.1 hp.out) (prime_iff.1 Nat.prime_two) (succ_pos _) hk\n  rwa [coe_two, PNat.coe_inj] at hk \n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\n\u22a2 p = 2\n[PROOFSTEP]\nrw [\u2190 PNat.coe_inj, PNat.pow_coe, \u2190 pow_one 2] at hk \n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\nhk : \u2191p ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\n\u22a2 p = 2\n[PROOFSTEP]\nreplace hk := eq_of_prime_pow_eq (prime_iff.1 hp.out) (prime_iff.1 Nat.prime_two) (succ_pos _) hk\n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u2191p ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhk : \u2191p = 2\n\u22a2 p = 2\n[PROOFSTEP]\nrwa [coe_two, PNat.coe_inj] at hk \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : p ^ (k + 1) = 2\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nrw [hp, \u2190 PNat.coe_inj, PNat.pow_coe] at hk \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nnth_rw 2 [\u2190 pow_one 2] at hk \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\nhk : \u21912 ^ (k + 1) = \u2191(2 ^ 1)\ncoe_two : 2 = \u21912\nhp : p = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nreplace hk := Nat.pow_right_injective rfl.le hk\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k + 1 = 1\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nrw [add_left_eq_self] at hk \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nrw [hp, hk] at h\u03b6 \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(2 ^ succ 0)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nnorm_num at h\u03b6 \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 \u21912\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nrw [\u2190 coe_two] at h\u03b6 \n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6\u271d).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nrw [coe_basis, powerBasis_gen]\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nsimp only [hp, hk]\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = \u2191((-1) ^ (\u03c6 \u2191(2 ^ succ 0) / 2)) * \u2191\u2191(2 ^ (\u21912 ^ (succ 0 - 1) * ((\u21912 - 1) * succ 0 - 1)))\n[PROOFSTEP]\nnorm_num\n  -- Porting note: the goal at this point is `(discr K fun i \u21a6 \u03b6 ^ \u2191i) = 1`.\n        -- This `simp_rw` is needed so the next `rw` can rewrite the type of `i` from\n        -- `Fin (natDegree (minpoly K \u03b6))` to `Fin 1`\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u03b6 ^ \u2191i) = 1\n[PROOFSTEP]\nsimp_rw [h\u03b6.eq_neg_one_of_two_right, show (-1 : L) = algebraMap K L (-1) by simp]\n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 -1 = \u2191(algebraMap K L) (-1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u2191(algebraMap K L) (-1) ^ \u2191i) = 1\n[PROOFSTEP]\nrw [h\u03b6.eq_neg_one_of_two_right, show (-1 : L) = algebraMap K L (-1) by simp,\n  minpoly.eq_X_sub_C_of_algebraMap_inj _ (algebraMap K L).injective, natDegree_X_sub_C]\n[GOAL]\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 -1 = \u2191(algebraMap K L) (-1)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 (discr K fun i => \u2191(algebraMap K L) (-1) ^ \u2191i) = 1\n[PROOFSTEP]\nsimp only [discr, traceMatrix_apply, Matrix.det_unique, Fin.default_eq_zero, Fin.val_zero, _root_.pow_zero,\n  traceForm_apply, mul_one]\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(trace K L) 1 = 1\n[PROOFSTEP]\nrw [\u2190 (algebraMap K L).map_one, trace_algebraMap, finrank _ hirr, hp, hk]\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u03c6 \u2191(2 ^ succ 0) \u2022 1 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp\u271d : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk\u271d : \u21912 ^ (k + 1) = \u21912\ncoe_two : 2 = \u21912\nhp : p = 2\nhk : k = 0\nh\u03b6 : IsPrimitiveRoot \u03b6 2\n\u22a2 \u2191(\u03c6 \u21912) = 1\n[PROOFSTEP]\nsimp [\u2190 coe_two]\n[GOAL]\ncase neg\np : \u2115+\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Field L\ninst\u271d : Algebra K L\nhp : Fact (Nat.Prime \u2191p)\nk : \u2115\nhcycl : IsCyclotomicExtension {p ^ succ k} K L\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ succ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ succ k)) K)\nhk : \u00acp ^ (k + 1) = 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis =\n    \u2191((-1) ^ (\u03c6 \u2191(p ^ succ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (succ k - 1) * ((\u2191p - 1) * succ k - 1)))\n[PROOFSTEP]\nexact discr_prime_pow_ne_two h\u03b6 hirr hk\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\n\u22a2 \u2203 u n, discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191\u2191u * \u2191\u2191(p ^ n)\n[PROOFSTEP]\nrw [discr_prime_pow h\u03b6 hirr]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\n\u22a2 \u2203 u n, \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) = \u2191\u2191u * \u2191\u2191(p ^ n)\n[PROOFSTEP]\nby_cases heven : Even ((p ^ k : \u2115).totient / 2)\n[GOAL]\ncase pos\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : Even (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u2203 u n, \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) = \u2191\u2191u * \u2191\u2191(p ^ n)\n[PROOFSTEP]\nrefine' \u27e81, (p : \u2115) ^ (k - 1) * ((p - 1) * k - 1), by rw [heven.neg_one_pow]; norm_num\u27e9\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : Even (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) =\n    \u2191\u21911 * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))\n[PROOFSTEP]\nrw [heven.neg_one_pow]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : Even (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u21911 * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) = \u2191\u21911 * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : \u00acEven (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u2203 u n, \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) = \u2191\u2191u * \u2191\u2191(p ^ n)\n[PROOFSTEP]\nexact \u27e8-1, (p : \u2115) ^ (k - 1) * ((p - 1) * k - 1), by rw [(odd_iff_not_even.2 heven).neg_one_pow]; norm_num\u27e9\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : \u00acEven (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u2191((-1) ^ (\u03c6 \u2191(p ^ k) / 2)) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) =\n    \u2191\u2191(-1) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))\n[PROOFSTEP]\nrw [(odd_iff_not_even.2 heven).neg_one_pow]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p ^ k} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191(p ^ k)\nhirr : Irreducible (cyclotomic (\u2191(p ^ k)) K)\nheven : \u00acEven (\u03c6 \u2191(p ^ k) / 2)\n\u22a2 \u2191(-1) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1))) = \u2191\u2191(-1) * \u2191\u2191(p ^ (\u2191p ^ (k - 1) * ((\u2191p - 1) * k - 1)))\n[PROOFSTEP]\nnorm_num\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ ((\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p - 2))\n[PROOFSTEP]\nhave : IsCyclotomicExtension {p ^ (0 + 1)} K L :=\n  by\n  rw [zero_add, pow_one]\n  infer_instance\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\n\u22a2 IsCyclotomicExtension {p ^ (0 + 1)} K L\n[PROOFSTEP]\nrw [zero_add, pow_one]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\n\u22a2 IsCyclotomicExtension {p} K L\n[PROOFSTEP]\ninfer_instance\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ ((\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p - 2))\n[PROOFSTEP]\nhave h\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1)) := by simpa using h\u03b6\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n\u22a2 IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n[PROOFSTEP]\nsimpa using h\u03b6\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\nh\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 discr K \u2191(IsPrimitiveRoot.powerBasis K h\u03b6).basis = \u2191((-1) ^ ((\u2191p - 1) / 2)) * \u2191\u2191(p ^ (\u2191p - 2))\n[PROOFSTEP]\nconvert discr_prime_pow_ne_two h\u03b6' (by simpa [hirr]) (by simp [hodd]) using 2\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\nh\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 Irreducible (cyclotomic \u2191(p ^ (0 + 1)) ?m.954125)\n[PROOFSTEP]\nsimpa [hirr]\n[GOAL]\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\nh\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 p ^ (0 + 1) \u2260 2\n[PROOFSTEP]\nsimp [hodd]\n[GOAL]\ncase h.e'_3.h.e'_5\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\nh\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 \u2191((-1) ^ ((\u2191p - 1) / 2)) = \u2191((-1) ^ (\u03c6 \u2191(p ^ (0 + 1)) / 2))\n[PROOFSTEP]\nrw [zero_add, pow_one, totient_prime hp.out]\n[GOAL]\ncase h.e'_3.h.e'_6\np : \u2115+\nk : \u2115\nK : Type u\nL : Type v\n\u03b6 : L\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {p} K L\nhp : Fact (Nat.Prime \u2191p)\nh\u03b6 : IsPrimitiveRoot \u03b6 \u2191p\nhirr : Irreducible (cyclotomic (\u2191p) K)\nhodd : p \u2260 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\nh\u03b6' : IsPrimitiveRoot \u03b6 \u2191(p ^ (0 + 1))\n\u22a2 \u2191\u2191(p ^ (\u2191p - 2)) = \u2191\u2191(p ^ (\u2191p ^ 0 * ((\u2191p - 1) * (0 + 1) - 1)))\n[PROOFSTEP]\nrw [_root_.pow_zero, one_mul, zero_add, mul_one, Nat.sub_sub]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Cyclotomic.Discriminant", "llama_tokens": 35697, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672227971211, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.5291989594787663}}
{"text": "[GOAL]\nx y : \u210d\n\u22a2 \u2191(starRingEnd \u211d) (inner y x) = inner x y\n[PROOFSTEP]\nsimp [inner_def, mul_comm]\n[GOAL]\nx y z : \u210d\n\u22a2 inner (x + y) z = inner x z + inner y z\n[PROOFSTEP]\nsimp only [inner_def, add_mul, add_re]\n[GOAL]\nx y : \u210d\nr : \u211d\n\u22a2 inner (r \u2022 x) y = \u2191(starRingEnd \u211d) r * inner x y\n[PROOFSTEP]\nsimp [inner_def]\n[GOAL]\na : \u210d\n\u22a2 \u2191normSq a = \u2016a\u2016 * \u2016a\u2016\n[PROOFSTEP]\nrw [\u2190 inner_self, real_inner_self_eq_norm_mul_norm]\n[GOAL]\n\u22a2 \u20161\u2016 = 1\n[PROOFSTEP]\nrw [norm_eq_sqrt_real_inner, inner_self, normSq.map_one, Real.sqrt_one]\n[GOAL]\na : \u211d\n\u22a2 \u2016\u2191a\u2016 = \u2016a\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]\n[GOAL]\na : \u210d\n\u22a2 \u2016star a\u2016 = \u2016a\u2016\n[PROOFSTEP]\nsimp_rw [norm_eq_sqrt_real_inner, inner_self, normSq_star]\n[GOAL]\na b : \u210d\n\u22a2 \u2016a * b\u2016 = \u2016a\u2016 * \u2016b\u2016\n[PROOFSTEP]\nsimp only [norm_eq_sqrt_real_inner, inner_self, normSq.map_mul]\n[GOAL]\na b : \u210d\n\u22a2 Real.sqrt (\u2191normSq a * \u2191normSq b) = Real.sqrt (\u2191normSq a) * Real.sqrt (\u2191normSq b)\n[PROOFSTEP]\nexact\n  Real.sqrt_mul normSq_nonneg\n    _\n      -- porting note: added `noncomputable`\n[GOAL]\nz w : \u2102\n\u22a2 \u2191(z + w) = \u2191z + \u2191w\n[PROOFSTEP]\next\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z + w)).re = (\u2191z + \u2191w).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z + w)).imI = (\u2191z + \u2191w).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z + w)).imJ = (\u2191z + \u2191w).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z + w)).imK = (\u2191z + \u2191w).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nz w : \u2102\n\u22a2 \u2191(z * w) = \u2191z * \u2191w\n[PROOFSTEP]\next\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z * w)).re = (\u2191z * \u2191w).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z * w)).imI = (\u2191z * \u2191w).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z * w)).imJ = (\u2191z * \u2191w).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nz w : \u2102\n\u22a2 (\u2191(z * w)).imK = (\u2191z * \u2191w).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nr : \u211d\nz : \u2102\n\u22a2 \u2191(r \u2022 z) = \u2191r * \u2191z\n[PROOFSTEP]\next\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\n\u22a2 (\u2191(r \u2022 z)).re = (\u2191r * \u2191z).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\n\u22a2 (\u2191(r \u2022 z)).imI = (\u2191r * \u2191z).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\n\u22a2 (\u2191(r \u2022 z)).imJ = (\u2191r * \u2191z).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a\nr : \u211d\nz : \u2102\n\u22a2 (\u2191(r \u2022 z)).imK = (\u2191r * \u2191z).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nx : \u210d\n\u22a2 \u2016\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x)\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, normSq_def', PiLp.inner_apply, Fin.sum_univ_four]\n[GOAL]\nx : \u210d\n\u22a2 Real.sqrt\n      (inner (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 0)\n              (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 0) +\n            inner (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 1)\n              (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 1) +\n          inner (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 2)\n            (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 2) +\n        inner (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 3)\n          (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 3)) =\n    Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)\n[PROOFSTEP]\nsimp_rw [IsROrC.inner_apply, starRingEnd_apply, star_trivial, \u2190 sq]\n[GOAL]\nx : \u210d\n\u22a2 Real.sqrt\n      (\u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 0 ^ 2 +\n            \u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 1 ^ 2 +\n          \u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 2 ^ 2 +\n        \u2191(PiLp.equiv 2 fun x => \u211d).symm (\u2191(equivTuple \u211d) x) 3 ^ 2) =\n    Real.sqrt (x.re ^ 2 + x.imI ^ 2 + x.imJ ^ 2 + x.imK ^ 2)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 Continuous \u2191normSq\n[PROOFSTEP]\nsimpa [\u2190 normSq_eq_norm_mul_self] using (continuous_norm.mul continuous_norm : Continuous fun q : \u210d => \u2016q\u2016 * \u2016q\u2016)\n[GOAL]\n\u22a2 Continuous fun q => im q\n[PROOFSTEP]\nsimpa only [\u2190 sub_self_re] using continuous_id.sub (continuous_coe.comp continuous_re)\n[GOAL]\nthis : UniformEmbedding \u2191(LinearEquiv.toEquiv linearIsometryEquivTuple.toLinearEquiv).symm\n\u22a2 CompleteSpace (EuclideanSpace \u211d (Fin 4))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nr : \u211d\nh : HasSum (fun a => \u2191(f a)) \u2191r\n\u22a2 HasSum f r\n[PROOFSTEP]\nsimpa only using h.map (show \u210d \u2192\u2097[\u211d] \u211d from QuaternionAlgebra.re\u2097 _ _) continuous_re\n[GOAL]\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nr : \u211d\nh : HasSum f r\n\u22a2 HasSum (fun a => \u2191(f a)) \u2191r\n[PROOFSTEP]\nsimpa only using h.map (algebraMap \u211d \u210d) (continuous_algebraMap _ _)\n[GOAL]\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\n\u22a2 (Summable fun a => \u2191(f a)) \u2194 Summable f\n[PROOFSTEP]\nsimpa only using\n  Summable.map_iff_of_leftInverse (algebraMap \u211d \u210d) (show \u210d \u2192\u2097[\u211d] \u211d from QuaternionAlgebra.re\u2097 _ _)\n    (continuous_algebraMap _ _) continuous_re coe_re\n[GOAL]\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\n\u22a2 \u2211' (a : \u03b1), \u2191(f a) = \u2191(\u2211' (a : \u03b1), f a)\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nhf : Summable f\n\u22a2 \u2211' (a : \u03b1), \u2191(f a) = \u2191(\u2211' (a : \u03b1), f a)\n[PROOFSTEP]\nexact (hasSum_coe.mpr hf.hasSum).tsum_eq\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nf : \u03b1 \u2192 \u211d\nhf : \u00acSummable f\n\u22a2 \u2211' (a : \u03b1), \u2191(f a) = \u2191(\u2211' (a : \u03b1), f a)\n[PROOFSTEP]\nsimp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (summable_coe.not.mpr hf)]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Quaternion", "llama_tokens": 2665, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339676722393, "lm_q2_score": 0.6297745935070808, "lm_q1q2_score": 0.5288431181448726}}
{"text": "[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u00b2 : MulZeroClass M\u2080\na\u271d b : M\u2080\ninst\u271d\u00b9 : Mul M\u2080'\ninst\u271d : Zero M\u2080'\nf : M\u2080' \u2192 M\u2080\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (a b : M\u2080'), f (a * b) = f a * f b\na : M\u2080'\n\u22a2 f (0 * a) = f 0\n[PROOFSTEP]\nsimp only [mul, zero, zero_mul]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u00b2 : MulZeroClass M\u2080\na\u271d b : M\u2080\ninst\u271d\u00b9 : Mul M\u2080'\ninst\u271d : Zero M\u2080'\nf : M\u2080' \u2192 M\u2080\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (a b : M\u2080'), f (a * b) = f a * f b\na : M\u2080'\n\u22a2 f (a * 0) = f 0\n[PROOFSTEP]\nsimp only [mul, zero, mul_zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u00b2 : MulZeroClass M\u2080\na b : M\u2080\ninst\u271d\u00b9 : Mul M\u2080'\ninst\u271d : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Surjective f\nzero : f 0 = 0\nmul : \u2200 (a b : M\u2080), f (a * b) = f a * f b\nx : M\u2080\n\u22a2 0 * f x = 0\n[PROOFSTEP]\nsimp only [\u2190 zero, \u2190 mul, zero_mul]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u00b2 : MulZeroClass M\u2080\na b : M\u2080\ninst\u271d\u00b9 : Mul M\u2080'\ninst\u271d : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Surjective f\nzero : f 0 = 0\nmul : \u2200 (a b : M\u2080), f (a * b) = f a * f b\nx : M\u2080\n\u22a2 f x * 0 = 0\n[PROOFSTEP]\nsimp only [\u2190 zero, \u2190 mul, mul_zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : NoZeroDivisors M\u2080'\na\u271d b\u271d : M\u2080\nH : a\u271d * b\u271d = 0\n\u22a2 f (?m.2223 H) * f (?m.2224 H) = 0\n[PROOFSTEP]\nrw [\u2190 mul, H, zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : NoZeroDivisors M\u2080'\na\u271d b\u271d : M\u2080\nH\u271d : a\u271d * b\u271d = 0\nthis : f a\u271d * f b\u271d = 0\nH : f a\u271d = 0\n\u22a2 f a\u271d = f 0\n[PROOFSTEP]\nrwa [zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : NoZeroDivisors M\u2080'\na\u271d b\u271d : M\u2080\nH\u271d : a\u271d * b\u271d = 0\nthis : f a\u271d * f b\u271d = 0\nH : f b\u271d = 0\n\u22a2 f b\u271d = f 0\n[PROOFSTEP]\nrwa [zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsLeftCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : a\u271d \u2260 0\nHe : a\u271d * b\u271d = a\u271d * c\u271d\n\u22a2 b\u271d = c\u271d\n[PROOFSTEP]\nhave := congr_arg f He\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsLeftCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : a\u271d \u2260 0\nHe : a\u271d * b\u271d = a\u271d * c\u271d\nthis : f (a\u271d * b\u271d) = f (a\u271d * c\u271d)\n\u22a2 b\u271d = c\u271d\n[PROOFSTEP]\nrw [mul, mul] at this \n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsLeftCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : a\u271d \u2260 0\nHe : a\u271d * b\u271d = a\u271d * c\u271d\nthis : f a\u271d * f b\u271d = f a\u271d * f c\u271d\n\u22a2 b\u271d = c\u271d\n[PROOFSTEP]\nexact hf (mul_left_cancel\u2080 (fun Hfa => Hne <| hf <| by rw [Hfa, zero]) this)\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsLeftCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : a\u271d \u2260 0\nHe : a\u271d * b\u271d = a\u271d * c\u271d\nthis : f a\u271d * f b\u271d = f a\u271d * f c\u271d\nHfa : f a\u271d = 0\n\u22a2 f a\u271d = f 0\n[PROOFSTEP]\nrw [Hfa, zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsRightCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : b\u271d \u2260 0\nHe : a\u271d * b\u271d = c\u271d * b\u271d\n\u22a2 a\u271d = c\u271d\n[PROOFSTEP]\nhave := congr_arg f He\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsRightCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : b\u271d \u2260 0\nHe : a\u271d * b\u271d = c\u271d * b\u271d\nthis : f (a\u271d * b\u271d) = f (c\u271d * b\u271d)\n\u22a2 a\u271d = c\u271d\n[PROOFSTEP]\nrw [mul, mul] at this \n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsRightCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : b\u271d \u2260 0\nHe : a\u271d * b\u271d = c\u271d * b\u271d\nthis : f a\u271d * f b\u271d = f c\u271d * f b\u271d\n\u22a2 a\u271d = c\u271d\n[PROOFSTEP]\nexact hf (mul_right_cancel\u2080 (fun Hfa => Hne <| hf <| by rw [Hfa, zero]) this)\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : Mul M\u2080\ninst\u271d\u00b3 : Zero M\u2080\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : Zero M\u2080'\nf : M\u2080 \u2192 M\u2080'\nhf : Injective f\nzero : f 0 = 0\nmul : \u2200 (x y : M\u2080), f (x * y) = f x * f y\ninst\u271d : IsRightCancelMulZero M\u2080'\na\u271d b\u271d c\u271d : M\u2080\nHne : b\u271d \u2260 0\nHe : a\u271d * b\u271d = c\u271d * b\u271d\nthis : f a\u271d * f b\u271d = f c\u271d * f b\u271d\nHfa : f b\u271d = 0\n\u22a2 f b\u271d = f 0\n[PROOFSTEP]\nrw [Hfa, zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : CancelMonoidWithZero M\u2080\na b c : M\u2080\ninst\u271d\u00b3 : Zero M\u2080'\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : One M\u2080'\ninst\u271d : Pow M\u2080' \u2115\nf : M\u2080' \u2192 M\u2080\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : M\u2080'), f (x * y) = f x * f y\nnpow : \u2200 (x : M\u2080') (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : Monoid M\u2080' := Injective.monoid f hf one mul npow\nsrc\u271d : MulZeroClass M\u2080' := Injective.mulZeroClass f hf zero mul\na\u271d b\u271d c\u271d : M\u2080'\nhx : a\u271d \u2260 0\nH : a\u271d * b\u271d = a\u271d * c\u271d\n\u22a2 f a\u271d * f b\u271d = f a\u271d * f c\u271d\n[PROOFSTEP]\nerw [\u2190 mul, \u2190 mul, H]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2074 : CancelMonoidWithZero M\u2080\na b c : M\u2080\ninst\u271d\u00b3 : Zero M\u2080'\ninst\u271d\u00b2 : Mul M\u2080'\ninst\u271d\u00b9 : One M\u2080'\ninst\u271d : Pow M\u2080' \u2115\nf : M\u2080' \u2192 M\u2080\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : M\u2080'), f (x * y) = f x * f y\nnpow : \u2200 (x : M\u2080') (n : \u2115), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : Monoid M\u2080' := Injective.monoid f hf one mul npow\nsrc\u271d : MulZeroClass M\u2080' := Injective.mulZeroClass f hf zero mul\na\u271d b\u271d c\u271d : M\u2080'\nhx : b\u271d \u2260 0\nH : a\u271d * b\u271d = c\u271d * b\u271d\n\u22a2 f a\u271d * f b\u271d = f c\u271d * f b\u271d\n[PROOFSTEP]\nerw [\u2190 mul, \u2190 mul, H]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2077 : GroupWithZero G\u2080\na b c g h x : G\u2080\ninst\u271d\u2076 : Zero G\u2080'\ninst\u271d\u2075 : Mul G\u2080'\ninst\u271d\u2074 : One G\u2080'\ninst\u271d\u00b3 : Inv G\u2080'\ninst\u271d\u00b2 : Div G\u2080'\ninst\u271d\u00b9 : Pow G\u2080' \u2115\ninst\u271d : Pow G\u2080' \u2124\nf : G\u2080' \u2192 G\u2080\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : G\u2080'), f (x * y) = f x * f y\ninv : \u2200 (x : G\u2080'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : G\u2080'), f (x / y) = f x / f y\nnpow : \u2200 (x : G\u2080') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : G\u2080') (n : \u2124), f (x ^ n) = f x ^ n\nsrc\u271d\u00b2 : MonoidWithZero G\u2080' := Injective.monoidWithZero f hf zero one mul npow\nsrc\u271d\u00b9 : DivInvMonoid G\u2080' := Injective.divInvMonoid f hf one mul inv div npow zpow\nsrc\u271d : Nontrivial G\u2080' := pullback_nonzero f zero one\n\u22a2 f 0\u207b\u00b9 = f 0\n[PROOFSTEP]\nerw [inv, zero, inv_zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2077 : GroupWithZero G\u2080\na b c g h x\u271d : G\u2080\ninst\u271d\u2076 : Zero G\u2080'\ninst\u271d\u2075 : Mul G\u2080'\ninst\u271d\u2074 : One G\u2080'\ninst\u271d\u00b3 : Inv G\u2080'\ninst\u271d\u00b2 : Div G\u2080'\ninst\u271d\u00b9 : Pow G\u2080' \u2115\ninst\u271d : Pow G\u2080' \u2124\nf : G\u2080' \u2192 G\u2080\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : G\u2080'), f (x * y) = f x * f y\ninv : \u2200 (x : G\u2080'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : G\u2080'), f (x / y) = f x / f y\nnpow : \u2200 (x : G\u2080') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : G\u2080') (n : \u2124), f (x ^ n) = f x ^ n\nsrc\u271d\u00b2 : MonoidWithZero G\u2080' := Injective.monoidWithZero f hf zero one mul npow\nsrc\u271d\u00b9 : DivInvMonoid G\u2080' := Injective.divInvMonoid f hf one mul inv div npow zpow\nsrc\u271d : Nontrivial G\u2080' := pullback_nonzero f zero one\nx : G\u2080'\nhx : x \u2260 0\n\u22a2 f (x * x\u207b\u00b9) = f 1\n[PROOFSTEP]\nerw [one, mul, inv, mul_inv_cancel ((hf.ne_iff' zero).2 hx)]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2077 : GroupWithZero G\u2080\na b c g h x : G\u2080\ninst\u271d\u2076 : Zero G\u2080'\ninst\u271d\u2075 : Mul G\u2080'\ninst\u271d\u2074 : One G\u2080'\ninst\u271d\u00b3 : Inv G\u2080'\ninst\u271d\u00b2 : Div G\u2080'\ninst\u271d\u00b9 : Pow G\u2080' \u2115\ninst\u271d : Pow G\u2080' \u2124\nh01 : 0 \u2260 1\nf : G\u2080 \u2192 G\u2080'\nhf : Surjective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : G\u2080), f (x * y) = f x * f y\ninv : \u2200 (x : G\u2080), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : G\u2080), f (x / y) = f x / f y\nnpow : \u2200 (x : G\u2080) (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : G\u2080) (n : \u2124), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : MonoidWithZero G\u2080' := Surjective.monoidWithZero f hf zero one mul npow\nsrc\u271d : DivInvMonoid G\u2080' := Surjective.divInvMonoid f hf one mul inv div npow zpow\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nerw [\u2190 zero, \u2190 inv, inv_zero]\n[GOAL]\nM\u2080 : Type u_1\nG\u2080 : Type u_2\nM\u2080' : Type u_3\nG\u2080' : Type u_4\ninst\u271d\u2077 : GroupWithZero G\u2080\na b c g h x\u271d : G\u2080\ninst\u271d\u2076 : Zero G\u2080'\ninst\u271d\u2075 : Mul G\u2080'\ninst\u271d\u2074 : One G\u2080'\ninst\u271d\u00b3 : Inv G\u2080'\ninst\u271d\u00b2 : Div G\u2080'\ninst\u271d\u00b9 : Pow G\u2080' \u2115\ninst\u271d : Pow G\u2080' \u2124\nh01 : 0 \u2260 1\nf : G\u2080 \u2192 G\u2080'\nhf : Surjective f\nzero : f 0 = 0\none : f 1 = 1\nmul : \u2200 (x y : G\u2080), f (x * y) = f x * f y\ninv : \u2200 (x : G\u2080), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : G\u2080), f (x / y) = f x / f y\nnpow : \u2200 (x : G\u2080) (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : G\u2080) (n : \u2124), f (x ^ n) = f x ^ n\nsrc\u271d\u00b9 : MonoidWithZero G\u2080' := Surjective.monoidWithZero f hf zero one mul npow\nsrc\u271d : DivInvMonoid G\u2080' := Surjective.divInvMonoid f hf one mul inv div npow zpow\nx : G\u2080\nhx : f x \u2260 0\n\u22a2 f x * (f x)\u207b\u00b9 = 1\n[PROOFSTEP]\nerw [\u2190 inv, \u2190 mul, mul_inv_cancel (mt (congr_arg f) <| fun h \u21a6 hx (h.trans zero)), one]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GroupWithZero.InjSurj", "llama_tokens": 5772, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.5286945395605271}}
{"text": "[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave sub : \u2200 x \u2208 Ioo a b, Ioo a x \u2286 Ioo a b := fun x hx => Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hg : \u2200 x \u2208 Ioo a b, g x \u2260 0 := by\n  intro x hx h\n  have : Tendsto g (\ud835\udcdd[<] x) (\ud835\udcdd 0) :=\n    by\n    rw [\u2190 h, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]\n    exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto\n  obtain \u27e8y, hyx, hy\u27e9 : \u2203 c \u2208 Ioo a x, g' c = 0\n  exact exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy\n  exact hg' y (sub x hx hyx) hy\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\n[PROOFSTEP]\nintro x hx h\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\n\u22a2 False\n[PROOFSTEP]\nhave : Tendsto g (\ud835\udcdd[<] x) (\ud835\udcdd 0) := by\n  rw [\u2190 h, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]\n  exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\n\u22a2 Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 h, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\n\u22a2 Tendsto g (\ud835\udcdd[Ioo a x] x) (\ud835\udcdd (g x))\n[PROOFSTEP]\nexact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd 0)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y, hyx, hy\u27e9 : \u2203 c \u2208 Ioo a x, g' c = 0\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd 0)\n\u22a2 \u2203 c, c \u2208 Ioo a x \u2227 g' c = 0\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd 0)\ny : \u211d\nhyx : y \u2208 Ioo a x\nhy : g' y = 0\n\u22a2 False\n[PROOFSTEP]\nexact exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nx : \u211d\nhx : x \u2208 Ioo a b\nh : g x = 0\nthis : Tendsto g (\ud835\udcdd[Iio x] x) (\ud835\udcdd 0)\ny : \u211d\nhyx : y \u2208 Ioo a x\nhy : g' y = 0\n\u22a2 False\n[PROOFSTEP]\nexact hg' y (sub x hx hyx) hy\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave : \u2200 x \u2208 Ioo a b, \u2203 c \u2208 Ioo a x, f x * g' c = g x * f' c :=\n  by\n  intro x hx\n  rw [\u2190 sub_zero (f x), \u2190 sub_zero (g x)]\n  exact\n    exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)\n      (fun y hy => hff' y <| sub x hx hy) hga hfa (tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto)\n      (tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 \u2203 c, c \u2208 Ioo a x \u2227 f x * g' c = g x * f' c\n[PROOFSTEP]\nintro x hx\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 \u2203 c, c \u2208 Ioo a x \u2227 f x * g' c = g x * f' c\n[PROOFSTEP]\nrw [\u2190 sub_zero (f x), \u2190 sub_zero (g x)]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 \u2203 c, c \u2208 Ioo a x \u2227 (f x - 0) * g' c = (g x - 0) * f' c\n[PROOFSTEP]\nexact\n  exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)\n    (fun y hy => hff' y <| sub x hx hy) hga hfa (tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto)\n    (tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 \u2203 c, c \u2208 Ioo a x \u2227 f x * g' c = g x * f' c\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nchoose! c hc using this\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave : \u2200 x \u2208 Ioo a b, ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x :=\n  by\n  intro x hx\n  rcases hc x hx with \u27e8h\u2081, h\u2082\u27e9\n  field_simp [hg x hx, hg' (c x) ((sub x hx) h\u2081)]\n  simp only [h\u2082]\n  rw [mul_comm]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\n[PROOFSTEP]\nintro x hx\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\n[PROOFSTEP]\nrcases hc x hx with \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nx : \u211d\nhx : x \u2208 Ioo a b\nh\u2081 : c x \u2208 Ioo a x\nh\u2082 : f x * g' (c x) = g x * f' (c x)\n\u22a2 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\n[PROOFSTEP]\nfield_simp [hg x hx, hg' (c x) ((sub x hx) h\u2081)]\n[GOAL]\ncase intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nx : \u211d\nhx : x \u2208 Ioo a b\nh\u2081 : c x \u2208 Ioo a x\nh\u2082 : f x * g' (c x) = g x * f' (c x)\n\u22a2 f' (c x) * g x = f x * g' (c x)\n[PROOFSTEP]\nsimp only [h\u2082]\n[GOAL]\ncase intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nx : \u211d\nhx : x \u2208 Ioo a b\nh\u2081 : c x \u2208 Ioo a x\nh\u2082 : f x * g' (c x) = g x * f' (c x)\n\u22a2 f' (c x) * g x = g x * f' (c x)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave cmp : \u2200 x \u2208 Ioo a b, a < c x \u2227 c x < x := fun x hx => (hc x hx).1\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrw [\u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioo a b] a) l\n[PROOFSTEP]\napply tendsto_nhdsWithin_congr this\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 Tendsto (fun x => ((fun x' => f' x' / g' x') \u2218 c) x) (\ud835\udcdd[Ioo a b] a) l\n[PROOFSTEP]\napply hdiv.comp\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 Tendsto c (\ud835\udcdd[Ioo a b] a) (\ud835\udcdd[Ioi a] a)\n[PROOFSTEP]\nrefine'\n  tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _\n    (tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) _ _)\n    _\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioo a b] a, a \u2264 c b\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioo a b] a, c b \u2264 id b\ncase refine'_3\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioo a b] a, c x \u2208 Ioi a\n[PROOFSTEP]\nall_goals\n  apply eventually_nhdsWithin_of_forall\n  intro x hx\n  have := cmp x hx\n  try simp\n  linarith [this]\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioo a b] a, a \u2264 c b\n[PROOFSTEP]\napply eventually_nhdsWithin_of_forall\n[GOAL]\ncase refine'_1.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a \u2264 c x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_1.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 a \u2264 c x\n[PROOFSTEP]\nhave := cmp x hx\n[GOAL]\ncase refine'_1.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 a \u2264 c x\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_1.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 a \u2264 c x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 a \u2264 c x\n[PROOFSTEP]\nlinarith [this]\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioo a b] a, c b \u2264 id b\n[PROOFSTEP]\napply eventually_nhdsWithin_of_forall\n[GOAL]\ncase refine'_2.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2264 id x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_2.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 c x \u2264 id x\n[PROOFSTEP]\nhave := cmp x hx\n[GOAL]\ncase refine'_2.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 c x \u2264 id x\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_2.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 c x \u2264 id x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 c x \u2264 x\n[PROOFSTEP]\nlinarith [this]\n[GOAL]\ncase refine'_3\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioo a b] a, c x \u2208 Ioi a\n[PROOFSTEP]\napply eventually_nhdsWithin_of_forall\n[GOAL]\ncase refine'_3.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioi a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_3.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\n\u22a2 c x \u2208 Ioi a\n[PROOFSTEP]\nhave := cmp x hx\n[GOAL]\ncase refine'_3.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 c x \u2208 Ioi a\n[PROOFSTEP]\ntry simp\n[GOAL]\ncase refine'_3.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 c x \u2208 Ioi a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\nsub : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 Ioo a x \u2286 Ioo a b\nhg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g x \u2260 0\nc : \u211d \u2192 \u211d\nhc : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 c x \u2208 Ioo a x \u2227 f x * g' (c x) = g x * f' (c x)\nthis\u271d : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x\ncmp : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 a < c x \u2227 c x < x\nx : \u211d\nhx : x \u2208 Ioo a b\nthis : a < c x \u2227 c x < x\n\u22a2 a < c x\n[PROOFSTEP]\nlinarith [this]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrefine' lhopital_zero_right_on_Ioo hab hff' hgg' hg' _ _ hdiv\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hfa, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Ioo a b] a) (\ud835\udcdd (f a))\n[PROOFSTEP]\nexact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hga, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Ioo a b] a) (\ud835\udcdd (g a))\n[PROOFSTEP]\nexact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave hdnf : \u2200 x \u2208 -Ioo a b, HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x := fun x hx =>\n  comp x (hff' (-x) hx) (hasDerivAt_neg x)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 -Ioo a b \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave hdng : \u2200 x \u2208 -Ioo a b, HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x := fun x hx =>\n  comp x (hgg' (-x) hx) (hasDerivAt_neg x)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 -Ioo a b \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 -Ioo a b \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nrw [preimage_neg_Ioo] at hdnf \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 -Ioo a b \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nrw [preimage_neg_Ioo] at hdng \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave :=\n  lhopital_zero_right_on_Ioo (neg_lt_neg hab) hdnf hdng\n    (by\n      intro x hx h\n      apply hg' _ (by rw [\u2190 preimage_neg_Ioo] at hx ; exact hx)\n      rwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h )\n    (hfb.comp tendsto_neg_nhdsWithin_Ioi_neg) (hgb.comp tendsto_neg_nhdsWithin_Ioi_neg)\n    (by\n      simp only [neg_div_neg_eq, mul_one, mul_neg]\n      exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_nhdsWithin_Ioi_neg))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 g' (-x) * -1 \u2260 0\n[PROOFSTEP]\nintro x hx h\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioo (-b) (-a)\nh : g' (-x) * -1 = 0\n\u22a2 False\n[PROOFSTEP]\napply hg' _ (by rw [\u2190 preimage_neg_Ioo] at hx ; exact hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioo (-b) (-a)\nh : g' (-x) * -1 = 0\n\u22a2 ?m.13460 \u2208 Ioo a b\n[PROOFSTEP]\nrw [\u2190 preimage_neg_Ioo] at hx \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx\u271d : x \u2208 Ioo (-b) (-a)\nhx : x \u2208 -Ioo a b\nh : g' (-x) * -1 = 0\n\u22a2 ?m.13535 h \u2208 Ioo a b\n[PROOFSTEP]\nexact hx\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioo (-b) (-a)\nh : g' (-x) * -1 = 0\n\u22a2 g' (-x) = 0\n[PROOFSTEP]\nrwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f' (-x) * -1 / (g' (-x) * -1)) (\ud835\udcdd[Ioi (-b)] (-b)) ?m.13087\n[PROOFSTEP]\nsimp only [neg_div_neg_eq, mul_one, mul_neg]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f' (-x) / g' (-x)) (\ud835\udcdd[Ioi (-b)] (-b)) ?m.13087\n[PROOFSTEP]\nexact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_nhdsWithin_Ioi_neg)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) (\ud835\udcdd[Ioi (-b)] (-b)) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave := this.comp tendsto_neg_nhdsWithin_Iio\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis\u271d : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) (\ud835\udcdd[Ioi (-b)] (-b)) l\nthis : Tendsto ((fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) \u2218 Neg.neg) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nunfold Function.comp at this \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo (-b) (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis\u271d : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) (\ud835\udcdd[Ioi (-b)] (-b)) l\nthis : Tendsto (fun x => (fun x => f (-x) / g (-x)) (-x)) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nsimpa only [neg_neg]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nrefine' lhopital_zero_left_on_Ioo hab hff' hgg' hg' _ _ hdiv\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hfb, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Ioo a b] b) (\ud835\udcdd (f b))\n[PROOFSTEP]\nexact ((hcf b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hgb, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2260 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Ioo a b] b) (\ud835\udcdd (g b))\n[PROOFSTEP]\nexact ((hcg b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nobtain \u27e8a', haa', ha'\u27e9 : \u2203 a', a < a' \u2227 0 < a' :=\n  \u27e81 + max a 0, \u27e8lt_of_le_of_lt (le_max_left a 0) (lt_one_add _), lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)\u27e9\u27e9\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave fact1 : \u2200 x : \u211d, x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0 := fun _ hx => (ne_of_lt hx.1).symm\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave fact2 : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, a < x\u207b\u00b9 := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdnf : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x := fun x hx =>\n  comp x (hff' x\u207b\u00b9 <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdng : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x := fun x hx =>\n  comp x (hgg' x\u207b\u00b9 <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave :=\n  lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng\n    (by\n      intro x hx\n      refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)\n      exact hg' _ (fact2 x hx))\n    (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)\n    (by\n      refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)\n      rw [eventuallyEq_iff_exists_mem]\n      use Ioi 0, self_mem_nhdsWithin\n      intro x hx\n      unfold Function.comp\n      simp only\n      erw [mul_div_mul_right]\n      refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioo 0 a'\u207b\u00b9\n\u22a2 g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrefine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioo 0 a'\u207b\u00b9\n\u22a2 g' x\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nexact hg' _ (fact2 x hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 Tendsto (fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)) (\ud835\udcdd[Ioi 0] 0) ?m.21930\n[PROOFSTEP]\nrefine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 ((fun x => f' x / g' x) \u2218 fun x => x\u207b\u00b9) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)\n[PROOFSTEP]\nrw [eventuallyEq_iff_exists_mem]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 \u2203 s,\n    s \u2208 \ud835\udcdd[Ioi 0] 0 \u2227\n      EqOn ((fun x => f' x / g' x) \u2218 fun x => x\u207b\u00b9) (fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)) s\n[PROOFSTEP]\nuse Ioi 0, self_mem_nhdsWithin\n[GOAL]\ncase right\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\n\u22a2 EqOn ((fun x => f' x / g' x) \u2218 fun x => x\u207b\u00b9) (fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)) (Ioi 0)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase right\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 ((fun x => f' x / g' x) \u2218 fun x => x\u207b\u00b9) x = (fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)) x\n[PROOFSTEP]\nunfold Function.comp\n[GOAL]\ncase right\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 (fun x => f' x / g' x) ((fun x => x\u207b\u00b9) x) = (fun x => f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)) x\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase right\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 f' x\u207b\u00b9 / g' x\u207b\u00b9 = f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9 / (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9)\n[PROOFSTEP]\nerw [mul_div_mul_right]\n[GOAL]\ncase right.hc\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 -(x ^ 2)\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nrefine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx)\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nthis : Tendsto (fun x => (f \u2218 Inv.inv) x / (g \u2218 Inv.inv) x) (\ud835\udcdd[Ioi 0] 0) l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave := this.comp tendsto_inv_atTop_zero'\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nthis\u271d : Tendsto (fun x => (f \u2218 Inv.inv) x / (g \u2218 Inv.inv) x) (\ud835\udcdd[Ioi 0] 0) l\nthis : Tendsto ((fun x => (f \u2218 Inv.inv) x / (g \u2218 Inv.inv) x) \u2218 fun r => r\u207b\u00b9) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nunfold Function.comp at this \n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\na' : \u211d\nhaa' : a < a'\nha' : 0 < a'\nfact1 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0\nfact2 : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 a < x\u207b\u00b9\nhdnf : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x\nthis\u271d : Tendsto (fun x => (f \u2218 Inv.inv) x / (g \u2218 Inv.inv) x) (\ud835\udcdd[Ioi 0] 0) l\nthis : Tendsto (fun x => (fun x => f x\u207b\u00b9 / g x\u207b\u00b9) ((fun r => r\u207b\u00b9) x)) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nsimpa only [inv_inv]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdnf : \u2200 x \u2208 -Iio a, HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x := fun x hx =>\n  comp x (hff' (-x) hx) (hasDerivAt_neg x)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 -Iio a \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdng : \u2200 x \u2208 -Iio a, HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x := fun x hx =>\n  comp x (hgg' (-x) hx) (hasDerivAt_neg x)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 -Iio a \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 -Iio a \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrw [preimage_neg_Iio] at hdnf \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 -Iio a \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrw [preimage_neg_Iio] at hdng \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave :=\n  lhopital_zero_atTop_on_Ioi hdnf hdng\n    (by\n      intro x hx h\n      apply hg' _ (by rw [\u2190 preimage_neg_Iio] at hx ; exact hx)\n      rwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h )\n    (hfbot.comp tendsto_neg_atTop_atBot) (hgbot.comp tendsto_neg_atTop_atBot)\n    (by\n      simp only [mul_one, mul_neg, neg_div_neg_eq]\n      exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_atTop_atBot))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 g' (-x) * -1 \u2260 0\n[PROOFSTEP]\nintro x hx h\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioi (-a)\nh : g' (-x) * -1 = 0\n\u22a2 False\n[PROOFSTEP]\napply hg' _ (by rw [\u2190 preimage_neg_Iio] at hx ; exact hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioi (-a)\nh : g' (-x) * -1 = 0\n\u22a2 ?m.26788 \u2208 Iio a\n[PROOFSTEP]\nrw [\u2190 preimage_neg_Iio] at hx \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx\u271d : x \u2208 Ioi (-a)\nhx : x \u2208 -Iio a\nh : g' (-x) * -1 = 0\n\u22a2 ?m.26876 h \u2208 Iio a\n[PROOFSTEP]\nexact hx\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nx : \u211d\nhx : x \u2208 Ioi (-a)\nh : g' (-x) * -1 = 0\n\u22a2 g' (-x) = 0\n[PROOFSTEP]\nrwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f' (-x) * -1 / (g' (-x) * -1)) atTop ?m.26621\n[PROOFSTEP]\nsimp only [mul_one, mul_neg, neg_div_neg_eq]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\n\u22a2 Tendsto (fun x => f' (-x) / g' (-x)) atTop ?m.26621\n[PROOFSTEP]\nexact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_atTop_atBot)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave := this.comp tendsto_neg_atBot_atTop\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis\u271d : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) atTop l\nthis : Tendsto ((fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) \u2218 Neg.neg) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nunfold Function.comp at this \n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt f (f' x) x\nhgg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 HasDerivAt g (g' x) x\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\nhdnf : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x\nhdng : \u2200 (x : \u211d), x \u2208 Ioi (-a) \u2192 HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x\nthis\u271d : Tendsto (fun x => (f \u2218 Neg.neg) x / (g \u2218 Neg.neg) x) atTop l\nthis : Tendsto (fun x => (fun x => f (-x) / g (-x)) (-x)) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nsimpa only [neg_neg]\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hdf : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d f x := fun x hx => (hdf x hx).differentiableAt (Ioo_mem_nhds hx.1 hx.2)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\nhdf : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d f x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hdg : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d g x := fun x hx =>\n  by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\nhdf : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d f x\nhdg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nexact\n  HasDerivAt.lhopital_zero_right_on_Ioo hab (fun x hx => (hdf x hx).hasDerivAt) (fun x hx => (hdg x hx).hasDerivAt) hg'\n    hfa hga hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrefine' lhopital_zero_right_on_Ioo hab hdf hg' _ _ hdiv\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hfa, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n[GOAL]\ncase refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[Ioo a b] a) (\ud835\udcdd (f a))\n[PROOFSTEP]\nexact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 hga, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n[GOAL]\ncase refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[Ioo a b] a) (\ud835\udcdd (g a))\n[PROOFSTEP]\nexact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio b] b) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave hdf : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d f x := fun x hx => (hdf x hx).differentiableAt (Ioo_mem_nhds hx.1 hx.2)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio b] b) l\nhdf : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d f x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nhave hdg : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d g x := fun x hx =>\n  by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioo a b)\nhg' : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 deriv g x \u2260 0\nhfb : Tendsto f (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhgb : Tendsto g (\ud835\udcdd[Iio b] b) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio b] b) l\nhdf : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d f x\nhdg : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio b] b) l\n[PROOFSTEP]\nexact\n  HasDerivAt.lhopital_zero_left_on_Ioo hab (fun x hx => (hdf x hx).hasDerivAt) (fun x hx => (hdg x hx).hasDerivAt) hg'\n    hfb hgb hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Ioi a)\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdf : \u2200 x \u2208 Ioi a, DifferentiableAt \u211d f x := fun x hx => (hdf x hx).differentiableAt (Ioi_mem_nhds hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioi a)\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 DifferentiableAt \u211d f x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdg : \u2200 x \u2208 Ioi a, DifferentiableAt \u211d g x := fun x hx =>\n  by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Ioi a)\nhg' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 DifferentiableAt \u211d f x\nhdg : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nexact\n  HasDerivAt.lhopital_zero_atTop_on_Ioi (fun x hx => (hdf x hx).hasDerivAt) (fun x hx => (hdg x hx).hasDerivAt) hg'\n    hftop hgtop hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : DifferentiableOn \u211d f (Iio a)\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdf : \u2200 x \u2208 Iio a, DifferentiableAt \u211d f x := fun x hx => (hdf x hx).differentiableAt (Iio_mem_nhds hx)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Iio a)\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : \u2200 (x : \u211d), x \u2208 Iio a \u2192 DifferentiableAt \u211d f x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdg : \u2200 x \u2208 Iio a, DifferentiableAt \u211d g x := fun x hx =>\n  by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf\u271d : DifferentiableOn \u211d f (Iio a)\nhg' : \u2200 (x : \u211d), x \u2208 Iio a \u2192 deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdf : \u2200 (x : \u211d), x \u2208 Iio a \u2192 DifferentiableAt \u211d f x\nhdg : \u2200 (x : \u211d), x \u2208 Iio a \u2192 DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nexact\n  HasDerivAt.lhopital_zero_atBot_on_Iio (fun x hx => (hdf x hx).hasDerivAt) (fun x hx => (hdg x hx).hasDerivAt) hg'\n    hfbot hgbot hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrw [eventually_iff_exists_mem] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt f (f' y) y\nhgg' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhgg' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Ioi a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nlet s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hs : s \u2208 \ud835\udcdd[>] a := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : s \u2208 \ud835\udcdd[Ioi a] a\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset] at hs \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : \u2203 u, u \u2208 Ioi a \u2227 Ioo a u \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrcases hs with \u27e8u, hau, hu\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nrefine' lhopital_zero_right_on_Ioo hau _ _ _ hfa hga hdiv\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a u \u2192 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a u \u2192 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo a u \u2192 g' x \u2260 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 g' x \u2260 0\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nfirst\n| exact (hu hx).1.1\n| exact (hu hx).1.2\n| exact (hu hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nexact (hu hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nfirst\n| exact (hu hx).1.1\n| exact (hu hx).1.2\n| exact (hu hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hu hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hu hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nfirst\n| exact (hu hx).1.1\n| exact (hu hx).1.2\n| exact (hu hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hu hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hu hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Ioi a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Ioi a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Ioi a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nu : \u211d\nhau : u \u2208 Ioi a\nhu : Ioo a u \u2286 s\nx : \u211d\nhx : x \u2208 Ioo a u\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hu hx).2\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrw [eventually_iff_exists_mem] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt f (f' y) y\nhgg' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhgg' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhg' : \u2203 v, v \u2208 \ud835\udcdd[Iio a] a \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nlet s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nhave hs : s \u2208 \ud835\udcdd[<] a := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : s \u2208 \ud835\udcdd[Iio a] a\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrw [mem_nhdsWithin_Iio_iff_exists_Ioo_subset] at hs \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : \u2203 l, l \u2208 Iio a \u2227 Ioo l a \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nrcases hs with \u27e8l, hal, hl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\u271d\n[PROOFSTEP]\nrefine' lhopital_zero_left_on_Ioo hal _ _ _ hfa hga hdiv\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo l a \u2192 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo l a \u2192 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioo l a \u2192 g' x \u2260 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 g' x \u2260 0\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nfirst\n| exact (hl hx).1.1\n| exact (hl hx).1.2\n| exact (hl hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nexact (hl hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nfirst\n| exact (hl hx).1.1\n| exact (hl hx).1.2\n| exact (hl hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nfirst\n| exact (hl hx).1.1\n| exact (hl hx).1.2\n| exact (hl hx).2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hl hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hl hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.refine'_3.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 \ud835\udcdd[Iio a] a\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 \ud835\udcdd[Iio a] a\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 \ud835\udcdd[Iio a] a\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhal : l \u2208 Iio a\nhl : Ioo l a \u2286 s\nx : \u211d\nhx : x \u2208 Ioo l a\n\u22a2 x \u2208 s\u2083\n[PROOFSTEP]\nexact (hl hx).2\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[{a}\u1d9c] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\nsimp only [\u2190 Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : (\u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt f (f' x) x) \u2227 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt f (f' x) x\nhgg' : (\u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt g (g' x) x) \u2227 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt g (g' x) x\nhg' : (\u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, g' x \u2260 0) \u2227 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0) \u2227 Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0) \u2227 Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Iio a] a) l \u2227 Tendsto (fun x => f' x / g' x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l \u2227 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nexact\n  \u27e8lhopital_zero_nhds_left hff'.1 hgg'.1 hg'.1 hfa.1 hga.1 hdiv.1,\n    lhopital_zero_nhds_right hff'.2 hgg'.2 hg'.2 hfa.2 hga.2 hdiv.2\u27e9\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply @lhopital_zero_nhds' _ _ _ f' _ g'\n[GOAL]\ncase hff'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hff'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hgg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hgg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, g' x \u2260 0\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, g' x \u2260 0\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f' x / g' x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f' x / g' x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f' x / g' x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hff'.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hgg'.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hg'.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hfa.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd a) (\ud835\udcdd 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hga.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd a) (\ud835\udcdd 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hdiv.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, g' x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l\n[PROOFSTEP]\nassumption\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in atTop, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in atTop, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in atTop, g' x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrw [eventually_iff_exists_mem] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt f (f' y) y\nhgg' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhgg' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhg' : \u2203 v, v \u2208 atTop \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nlet s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hs : s \u2208 atTop := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : s \u2208 atTop\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrw [mem_atTop_sets] at hs \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : \u2203 a, \u2200 (b : \u211d), b \u2265 a \u2192 b \u2208 s\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nrcases hs with \u27e8l, hl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\n\u22a2 Tendsto (fun x => f x / g x) atTop l\u271d\n[PROOFSTEP]\nhave hl' : Ioi l \u2286 s := fun x hx => hl x (le_of_lt hx)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) atTop l\u271d\n[PROOFSTEP]\nrefine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi l \u2192 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Ioi l \u2192 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nfirst\n| exact (hl' hx).1.1\n| exact (hl' hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nexact (hl' hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nfirst\n| exact (hl' hx).1.1\n| exact (hl' hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl' hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atTop\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atTop\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atTop\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2265 l \u2192 b \u2208 s\nhl' : Ioi l \u2286 s\nx : \u211d\nhx : x \u2208 Ioi l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl' hx).1.2\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2200\u1da0 (x : \u211d) in atBot, HasDerivAt f (f' x) x\nhgg' : \u2200\u1da0 (x : \u211d) in atBot, HasDerivAt g (g' x) x\nhg' : \u2200\u1da0 (x : \u211d) in atBot, g' x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrw [eventually_iff_exists_mem] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhff' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt f (f' y) y\nhgg' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n[GOAL]\ncase intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhgg' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 HasDerivAt g (g' y) y\nhg' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhg' : \u2203 v, v \u2208 atBot \u2227 \u2200 (y : \u211d), y \u2208 v \u2192 g' y \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nlet s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hs : s \u2208 atBot := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : s \u2208 atBot\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrw [mem_atBot_sets] at hs \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nhs : \u2203 a, \u2200 (b : \u211d), b \u2264 a \u2192 b \u2208 s\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nrcases hs with \u27e8l, hl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\n\u22a2 Tendsto (fun x => f x / g x) atBot l\u271d\n[PROOFSTEP]\nhave hl' : Iio l \u2286 s := fun x hx => hl x (le_of_lt hx)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\n\u22a2 Tendsto (fun x => f x / g x) atBot l\u271d\n[PROOFSTEP]\nrefine' lhopital_zero_atBot_on_Iio _ _ (fun x hx => hg' x <| (hl' hx).2) hfbot hgbot hdiv\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Iio l \u2192 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\n\u22a2 \u2200 (x : \u211d), x \u2208 Iio l \u2192 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 HasDerivAt (fun x => f x) (f' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 HasDerivAt (fun x => g x) (g' x) x\n[PROOFSTEP]\napply_assumption\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nfirst\n| exact (hl' hx).1.1\n| exact (hl' hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_1.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 x \u2208 s\u2081\n[PROOFSTEP]\nexact (hl' hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nfirst\n| exact (hl' hx).1.1\n| exact (hl' hx).1.2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl' hx).1.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : \u211d\nhab : a < b\nl\u271d : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => f' x / g' x) atBot l\u271d\ns\u2081 : Set \u211d\nhs\u2081 : s\u2081 \u2208 atBot\nhff' : \u2200 (y : \u211d), y \u2208 s\u2081 \u2192 HasDerivAt f (f' y) y\ns\u2082 : Set \u211d\nhs\u2082 : s\u2082 \u2208 atBot\nhgg' : \u2200 (y : \u211d), y \u2208 s\u2082 \u2192 HasDerivAt g (g' y) y\ns\u2083 : Set \u211d\nhs\u2083 : s\u2083 \u2208 atBot\nhg' : \u2200 (y : \u211d), y \u2208 s\u2083 \u2192 g' y \u2260 0\ns : Set \u211d := s\u2081 \u2229 s\u2082 \u2229 s\u2083\nl : \u211d\nhl : \u2200 (b : \u211d), b \u2264 l \u2192 b \u2208 s\nhl' : Iio l \u2286 s\nx : \u211d\nhx : x \u2208 Iio l\n\u22a2 x \u2208 s\u2082\n[PROOFSTEP]\nexact (hl' hx).1.2\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hdg : \u2200\u1da0 x in \ud835\udcdd[>] a, DifferentiableAt \u211d g x :=\n  hg'.mp (eventually_of_forall fun _ hg' => by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hdf' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt f (deriv f x) x :=\n  hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt f (deriv f x) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nhave hdg' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt g (deriv g x) x :=\n  hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt f (deriv f x) x\nhdg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, HasDerivAt g (deriv g x) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nexact HasDerivAt.lhopital_zero_nhds_right hdf' hdg' hg' hfa hga hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nhave hdg : \u2200\u1da0 x in \ud835\udcdd[<] a, DifferentiableAt \u211d g x :=\n  hg'.mp (eventually_of_forall fun _ hg' => by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nhave hdf' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt f (deriv f x) x :=\n  hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt f (deriv f x) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nhave hdg' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt g (deriv g x) x :=\n  hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio a] a) l\nhdg : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt f (deriv f x) x\nhdg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, HasDerivAt g (deriv g x) x\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l\n[PROOFSTEP]\nexact HasDerivAt.lhopital_zero_nhds_left hdf' hdg' hg' hfa hga hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[{a}\u1d9c] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\nsimp only [\u2190 Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : (\u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, DifferentiableAt \u211d f x) \u2227 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, DifferentiableAt \u211d f x\nhg' : (\u2200\u1da0 (x : \u211d) in \ud835\udcdd[Iio a] a, deriv g x \u2260 0) \u2227 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[Ioi a] a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0) \u2227 Tendsto f (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd[Iio a] a) (\ud835\udcdd 0) \u2227 Tendsto g (\ud835\udcdd[Ioi a] a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Iio a] a) l \u2227 Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd[Ioi a] a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[Iio a] a) l \u2227 Tendsto (fun x => f x / g x) (\ud835\udcdd[Ioi a] a) l\n[PROOFSTEP]\nexact \u27e8lhopital_zero_nhds_left hdf.1 hg'.1 hfa.1 hga.1 hdiv.1, lhopital_zero_nhds_right hdf.2 hg'.2 hfa.2 hga.2 hdiv.2\u27e9\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x / g x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply lhopital_zero_nhds'\n[GOAL]\ncase hdf\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, DifferentiableAt \u211d (fun x => f x) x\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdf\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, DifferentiableAt \u211d (fun x => f x) x\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, deriv (fun x => g x) x \u2260 0\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hg'\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd[{a}\u1d9c] a, deriv (fun x => g x) x \u2260 0\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hfa\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hga\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd[{a}\u1d9c] a) (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\nfirst\n| apply eventually_nhdsWithin_of_eventually_nhds\n| apply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply eventually_nhdsWithin_of_eventually_nhds\n[GOAL]\ncase hdiv\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (\ud835\udcdd[{a}\u1d9c] a) l\n[PROOFSTEP]\napply tendsto_nhdsWithin_of_tendsto_nhds\n[GOAL]\ncase hdf.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d (fun x => f x) x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hg'.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv (fun x => g x) x \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hfa.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => f x) (\ud835\udcdd a) (\ud835\udcdd 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hga.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => g x) (\ud835\udcdd a) (\ud835\udcdd 0)\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hdiv.h\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in \ud835\udcdd a, deriv g x \u2260 0\nhfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)\nhga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (\ud835\udcdd a) l\n\u22a2 Tendsto (fun x => deriv (fun x => f x) x / deriv (fun x => g x) x) (\ud835\udcdd a) l\n[PROOFSTEP]\nassumption\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atTop, deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdg : \u2200\u1da0 x in atTop, DifferentiableAt \u211d g x :=\n  hg'.mp (eventually_of_forall fun _ hg' => by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atTop, deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdf' : \u2200\u1da0 x in atTop, HasDerivAt f (deriv f x) x :=\n  hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atTop, deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in atTop, HasDerivAt f (deriv f x) x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nhave hdg' : \u2200\u1da0 x in atTop, HasDerivAt g (deriv g x) x :=\n  hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atTop, deriv g x \u2260 0\nhftop : Tendsto f atTop (\ud835\udcdd 0)\nhgtop : Tendsto g atTop (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdg : \u2200\u1da0 (x : \u211d) in atTop, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in atTop, HasDerivAt f (deriv f x) x\nhdg' : \u2200\u1da0 (x : \u211d) in atTop, HasDerivAt g (deriv g x) x\n\u22a2 Tendsto (fun x => f x / g x) atTop l\n[PROOFSTEP]\nexact HasDerivAt.lhopital_zero_atTop hdf' hdg' hg' hftop hgtop hdiv\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atBot, deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdg : \u2200\u1da0 x in atBot, DifferentiableAt \u211d g x :=\n  hg'.mp (eventually_of_forall fun _ hg' => by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atBot, deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdg : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d g x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdf' : \u2200\u1da0 x in atBot, HasDerivAt f (deriv f x) x :=\n  hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atBot, deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdg : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in atBot, HasDerivAt f (deriv f x) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nhave hdg' : \u2200\u1da0 x in atBot, HasDerivAt g (deriv g x) x :=\n  hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n[GOAL]\na b : \u211d\nhab : a < b\nl : Filter \u211d\nf f' g g' : \u211d \u2192 \u211d\nhdf : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d f x\nhg' : \u2200\u1da0 (x : \u211d) in atBot, deriv g x \u2260 0\nhfbot : Tendsto f atBot (\ud835\udcdd 0)\nhgbot : Tendsto g atBot (\ud835\udcdd 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atBot l\nhdg : \u2200\u1da0 (x : \u211d) in atBot, DifferentiableAt \u211d g x\nhdf' : \u2200\u1da0 (x : \u211d) in atBot, HasDerivAt f (deriv f x) x\nhdg' : \u2200\u1da0 (x : \u211d) in atBot, HasDerivAt g (deriv g x) x\n\u22a2 Tendsto (fun x => f x / g x) atBot l\n[PROOFSTEP]\nexact HasDerivAt.lhopital_zero_atBot hdf' hdg' hg' hfbot hgbot hdiv\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.LHopital", "llama_tokens": 91399, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.885631476836816, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.5282199681607689}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nhn : IsPrimePow n\n\u22a2 Nat.minFac n ^ \u2191(Nat.factorization n) (Nat.minFac n) = n\n[PROOFSTEP]\nobtain \u27e8p, k, hp, hk, rfl\u27e9 := hn\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhp : Prime p\nhk : 0 < k\n\u22a2 Nat.minFac (p ^ k) ^ \u2191(Nat.factorization (p ^ k)) (Nat.minFac (p ^ k)) = p ^ k\n[PROOFSTEP]\nrw [\u2190 Nat.prime_iff] at hp \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\n\u22a2 Nat.minFac (p ^ k) ^ \u2191(Nat.factorization (p ^ k)) (Nat.minFac (p ^ k)) = p ^ k\n[PROOFSTEP]\nrw [hp.pow_minFac hk.ne', hp.factorization_pow, Finsupp.single_eq_same]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : Nat.minFac n ^ \u2191(Nat.factorization n) (Nat.minFac n) = n\nhn : n \u2260 1\n\u22a2 IsPrimePow n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn')\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk : \u2115\nh : Nat.minFac 0 ^ \u2191(Nat.factorization 0) (Nat.minFac 0) = 0\nhn : 0 \u2260 1\n\u22a2 IsPrimePow 0\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : Nat.minFac n ^ \u2191(Nat.factorization n) (Nat.minFac n) = n\nhn : n \u2260 1\nhn' : n \u2260 0\n\u22a2 IsPrimePow n\n[PROOFSTEP]\nrefine' \u27e8_, _, (Nat.minFac_prime hn).prime, _, h\u27e9\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : Nat.minFac n ^ \u2191(Nat.factorization n) (Nat.minFac n) = n\nhn : n \u2260 1\nhn' : n \u2260 0\n\u22a2 0 < \u2191(Nat.factorization n) (Nat.minFac n)\n[PROOFSTEP]\nrw [pos_iff_ne_zero, \u2190 Finsupp.mem_support_iff, Nat.factor_iff_mem_factorization,\n  Nat.mem_factors_iff_dvd hn' (Nat.minFac_prime hn)]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : Nat.minFac n ^ \u2191(Nat.factorization n) (Nat.minFac n) = n\nhn : n \u2260 1\nhn' : n \u2260 0\n\u22a2 Nat.minFac n \u2223 n\n[PROOFSTEP]\napply Nat.minFac_dvd\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 IsPrimePow n \u2194 \u2203 p k, 0 < k \u2227 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nrw [isPrimePow_nat_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 (\u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n) \u2194 \u2203 p k, 0 < k \u2227 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nrefine' exists\u2082_congr fun p k => _\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\n\u22a2 Nat.Prime p \u2227 0 < k \u2227 p ^ k = n \u2194 0 < k \u2227 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\n\u22a2 Nat.Prime p \u2227 0 < k \u2227 p ^ k = n \u2192 0 < k \u2227 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nrintro \u27e8hp, hk, hn\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\nhn : p ^ k = n\n\u22a2 0 < k \u2227 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nexact \u27e8hk, by rw [\u2190 hn, Nat.Prime.factorization_pow hp]\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\nhn : p ^ k = n\n\u22a2 Nat.factorization n = Finsupp.single p k\n[PROOFSTEP]\nrw [\u2190 hn, Nat.Prime.factorization_pow hp]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\n\u22a2 0 < k \u2227 Nat.factorization n = Finsupp.single p k \u2192 Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrintro \u27e8hk, hn\u27e9\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhk : 0 < k\nhn : Nat.factorization n = Finsupp.single p k\n\u22a2 Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nhave hn0 : n \u2260 0 := by\n  rintro rfl\n  simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhk : 0 < k\nhn : Nat.factorization n = Finsupp.single p k\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhk : 0 < k\nhn : Nat.factorization 0 = Finsupp.single p k\n\u22a2 False\n[PROOFSTEP]\nsimp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhk : 0 < k\nhn : Nat.factorization n = Finsupp.single p k\nhn0 : n \u2260 0\n\u22a2 Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrw [Nat.eq_pow_of_factorization_eq_single hn0 hn]\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhk : 0 < k\nhn : Nat.factorization n = Finsupp.single p k\nhn0 : n \u2260 0\n\u22a2 Nat.Prime p \u2227 0 < k \u2227 p ^ k = p ^ k\n[PROOFSTEP]\nexact \u27e8Nat.prime_of_mem_factorization (by simp [hn, hk.ne'] : p \u2208 n.factorization.support), hk, rfl\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n p k : \u2115\nhk : 0 < k\nhn : Nat.factorization n = Finsupp.single p k\nhn0 : n \u2260 0\n\u22a2 p \u2208 (Nat.factorization n).support\n[PROOFSTEP]\nsimp [hn, hk.ne']\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 IsPrimePow n \u2194 Finset.card (Nat.factorization n).support = 1\n[PROOFSTEP]\nsimp_rw [isPrimePow_iff_factorization_eq_single, Finsupp.card_support_eq_one', exists_prop, pos_iff_ne_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : IsPrimePow n\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn0)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk : \u2115\nh : IsPrimePow 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 0 / p ^ \u2191(Nat.factorization 0) p = 1\n[PROOFSTEP]\ncases not_isPrimePow_zero h\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrcases isPrimePow_iff_factorization_eq_single.mp h with \u27e8p, k, hk0, h1\u27e9\n[GOAL]\ncase inr.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrcases em' p.Prime with (pp | pp)\n[GOAL]\ncase inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : \u00acNat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrefine' absurd _ hk0.ne'\n[GOAL]\ncase inr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : \u00acNat.Prime p\n\u22a2 k = 0\n[PROOFSTEP]\nsimp [\u2190 Nat.factorization_eq_zero_of_non_prime n pp, h1]\n[GOAL]\ncase inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : Nat.Prime p\n\u22a2 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrefine' \u27e8p, pp, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : Nat.Prime p\n\u22a2 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrefine' Nat.eq_of_factorization_eq (Nat.ord_compl_pos p hn0).ne' (by simp) fun q => _\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : Nat.Prime p\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Nat.factorization (n / p ^ \u2191(Nat.factorization n) p)) q = \u2191(Nat.factorization 1) q\n[PROOFSTEP]\nrw [Nat.factorization_ord_compl n p, h1]\n[GOAL]\ncase inr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n : \u2115\nh : IsPrimePow n\nhn0 : n \u2260 0\np k : \u2115\nhk0 : 0 < k\nh1 : Nat.factorization n = Finsupp.single p k\npp : Nat.Prime p\nq : \u2115\n\u22a2 \u2191(Finsupp.erase p (Finsupp.single p k)) q = \u2191(Nat.factorization 1) q\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nhn : n \u2260 1\n\u22a2 IsPrimePow n \u2194 \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nrefine' \u27e8fun h => IsPrimePow.exists_ord_compl_eq_one h, fun h => _\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\nhn : n \u2260 1\nh : \u2203 p, Nat.Prime p \u2227 n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 IsPrimePow n\n[PROOFSTEP]\nrcases h with \u27e8p, pp, h\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n : \u2115\nhn : n \u2260 1\np : \u2115\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 IsPrimePow n\n[PROOFSTEP]\nrw [isPrimePow_nat_iff]\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n : \u2115\nhn : n \u2260 1\np : \u2115\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrw [\u2190 Nat.eq_of_dvd_of_div_eq_one (Nat.ord_proj_dvd n p) h] at hn \u22a2\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhn : p ^ \u2191(Nat.factorization n) p \u2260 1\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 \u2203 p_1 k, Nat.Prime p_1 \u2227 0 < k \u2227 p_1 ^ k = p ^ \u2191(Nat.factorization n) p\n[PROOFSTEP]\nrefine' \u27e8p, n.factorization p, pp, _, by simp\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhn : p ^ \u2191(Nat.factorization n) p \u2260 1\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 p ^ \u2191(Nat.factorization n) p = p ^ \u2191(Nat.factorization n) p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhn : p ^ \u2191(Nat.factorization n) p \u2260 1\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\n\u22a2 0 < \u2191(Nat.factorization n) p\n[PROOFSTEP]\ncontrapose! hn\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\npp : Nat.Prime p\nh : n / p ^ \u2191(Nat.factorization n) p = 1\nhn : \u2191(Nat.factorization n) p \u2264 0\n\u22a2 p ^ \u2191(Nat.factorization n) p = 1\n[PROOFSTEP]\nsimp [le_zero_iff.1 hn]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 IsPrimePow n \u2194 \u2203! p, Nat.Prime p \u2227 p \u2223 n\n[PROOFSTEP]\nrw [isPrimePow_nat_iff]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 (\u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n) \u2194 \u2203! p, Nat.Prime p \u2227 p \u2223 n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 (\u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n) \u2192 \u2203! p, Nat.Prime p \u2227 p \u2223 n\n[PROOFSTEP]\nrintro \u27e8p, k, hp, hk, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\n\u22a2 \u2203! p_1, Nat.Prime p_1 \u2227 p_1 \u2223 p ^ k\n[PROOFSTEP]\nrefine' \u27e8p, \u27e8hp, dvd_pow_self _ hk.ne'\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\n\u22a2 \u2200 (y : \u2115), (fun p_1 => Nat.Prime p_1 \u2227 p_1 \u2223 p ^ k) y \u2192 y = p\n[PROOFSTEP]\nrintro q \u27e8hq, hq'\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d p k : \u2115\nhp : Nat.Prime p\nhk : 0 < k\nq : \u2115\nhq : Nat.Prime q\nhq' : q \u2223 p ^ k\n\u22a2 q = p\n[PROOFSTEP]\nexact (Nat.prime_dvd_prime_iff_eq hq hp).1 (hq.dvd_of_dvd_pow hq')\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n : \u2115\n\u22a2 (\u2203! p, Nat.Prime p \u2227 p \u2223 n) \u2192 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrintro \u27e8p, \u27e8hp, hn\u27e9, hq\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhq : \u2200 (y : \u2115), (fun p => Nat.Prime p \u2227 p \u2223 n) y \u2192 y = p\nhp : Nat.Prime p\nhn : p \u2223 n\n\u22a2 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn\u2080)\n[GOAL]\ncase mpr.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk p : \u2115\nhp : Nat.Prime p\nhq : \u2200 (y : \u2115), (fun p => Nat.Prime p \u2227 p \u2223 0) y \u2192 y = p\nhn : p \u2223 0\n\u22a2 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = 0\n[PROOFSTEP]\ncases (hq 2 \u27e8Nat.prime_two, dvd_zero 2\u27e9).trans (hq 3 \u27e8Nat.prime_three, dvd_zero 3\u27e9).symm\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhq : \u2200 (y : \u2115), (fun p => Nat.Prime p \u2227 p \u2223 n) y \u2192 y = p\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\n\u22a2 \u2203 p k, Nat.Prime p \u2227 0 < k \u2227 p ^ k = n\n[PROOFSTEP]\nrefine' \u27e8p, n.factorization p, hp, hp.factorization_pos_of_dvd hn\u2080 hn, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhq : \u2200 (y : \u2115), (fun p => Nat.Prime p \u2227 p \u2223 n) y \u2192 y = p\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\n\u22a2 p ^ \u2191(Nat.factorization n) p = n\n[PROOFSTEP]\nsimp only [and_imp] at hq \n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\n\u22a2 p ^ \u2191(Nat.factorization n) p = n\n[PROOFSTEP]\napply\n  Nat.dvd_antisymm\n    (Nat.ord_proj_dvd _ _)\n      -- We need to show n \u2223 p ^ n.factorization p\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\n\u22a2 n \u2223 p ^ \u2191(Nat.factorization n) p\n[PROOFSTEP]\napply Nat.dvd_of_factors_subperm hn\u2080\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\n\u22a2 Nat.factors n <+~ Nat.factors (p ^ \u2191(Nat.factorization n) p)\n[PROOFSTEP]\nrw [hp.factors_pow, List.subperm_ext_iff]\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 Nat.factors n \u2192 List.count x (Nat.factors n) \u2264 List.count x (List.replicate (\u2191(Nat.factorization n) p) p)\n[PROOFSTEP]\nintro q hq'\n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\nq : \u2115\nhq' : q \u2208 Nat.factors n\n\u22a2 List.count q (Nat.factors n) \u2264 List.count q (List.replicate (\u2191(Nat.factorization n) p) p)\n[PROOFSTEP]\nrw [Nat.mem_factors hn\u2080] at hq' \n[GOAL]\ncase mpr.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\nq : \u2115\nhq' : Nat.Prime q \u2227 q \u2223 n\n\u22a2 List.count q (Nat.factors n) \u2264 List.count q (List.replicate (\u2191(Nat.factorization n) p) p)\n[PROOFSTEP]\ncases hq _ hq'.1 hq'.2\n[GOAL]\ncase mpr.intro.intro.intro.inr.refl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk n p : \u2115\nhp : Nat.Prime p\nhn : p \u2223 n\nhn\u2080 : n \u2260 0\nhq : \u2200 (y : \u2115), Nat.Prime y \u2192 y \u2223 n \u2192 y = p\nhq' : Nat.Prime p \u2227 p \u2223 n\n\u22a2 List.count p (Nat.factors n) \u2264 List.count p (List.replicate (\u2191(Nat.factorization n) p) p)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk\u271d n k : \u2115\nhk : k \u2260 0\n\u22a2 IsPrimePow (n ^ k) \u2194 IsPrimePow n\n[PROOFSTEP]\nsimp only [isPrimePow_iff_unique_prime_dvd]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk\u271d n k : \u2115\nhk : k \u2260 0\n\u22a2 (\u2203! p, Nat.Prime p \u2227 p \u2223 n ^ k) \u2194 \u2203! p, Nat.Prime p \u2227 p \u2223 n\n[PROOFSTEP]\napply exists_unique_congr\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk\u271d n k : \u2115\nhk : k \u2260 0\n\u22a2 \u2200 (a : \u2115), Nat.Prime a \u2227 a \u2223 n ^ k \u2194 Nat.Prime a \u2227 a \u2223 n\n[PROOFSTEP]\nsimp only [and_congr_right_iff]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk\u271d n k : \u2115\nhk : k \u2260 0\n\u22a2 \u2200 (a : \u2115), Nat.Prime a \u2192 (a \u2223 n ^ k \u2194 a \u2223 n)\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p\u271d : R\nk\u271d n k : \u2115\nhk : k \u2260 0\np : \u2115\nhp : Nat.Prime p\n\u22a2 p \u2223 n ^ k \u2194 p \u2223 n\n[PROOFSTEP]\nexact \u27e8hp.dvd_of_dvd_pow, fun t => t.trans (dvd_pow_self _ hk)\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a b : \u2115\nhab : coprime a b\nhn : IsPrimePow n\n\u22a2 n \u2223 a * b \u2194 n \u2223 a \u2228 n \u2223 b\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n b : \u2115\nhn : IsPrimePow n\nhab : coprime 0 b\n\u22a2 n \u2223 0 * b \u2194 n \u2223 0 \u2228 n \u2223 b\n[PROOFSTEP]\nsimp only [Nat.coprime_zero_left] at hab \n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n b : \u2115\nhn : IsPrimePow n\nhab : b = 1\n\u22a2 n \u2223 0 * b \u2194 n \u2223 0 \u2228 n \u2223 b\n[PROOFSTEP]\nsimp [hab, Finset.filter_singleton, not_isPrimePow_one]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a b : \u2115\nhab : coprime a b\nhn : IsPrimePow n\nha : a \u2260 0\n\u22a2 n \u2223 a * b \u2194 n \u2223 a \u2228 n \u2223 b\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb)\n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a : \u2115\nhn : IsPrimePow n\nha : a \u2260 0\nhab : coprime a 0\n\u22a2 n \u2223 a * 0 \u2194 n \u2223 a \u2228 n \u2223 0\n[PROOFSTEP]\nsimp only [Nat.coprime_zero_right] at hab \n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a : \u2115\nhn : IsPrimePow n\nha : a \u2260 0\nhab : a = 1\n\u22a2 n \u2223 a * 0 \u2194 n \u2223 a \u2228 n \u2223 0\n[PROOFSTEP]\nsimp [hab, Finset.filter_singleton, not_isPrimePow_one]\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a b : \u2115\nhab : coprime a b\nhn : IsPrimePow n\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 n \u2223 a * b \u2194 n \u2223 a \u2228 n \u2223 b\n[PROOFSTEP]\nrefine'\n  \u27e8_, fun h =>\n    Or.elim h (fun i => i.trans ((@dvd_mul_right a b a hab).mpr (dvd_refl a))) fun i =>\n      i.trans ((@dvd_mul_left a b b hab.symm).mpr (dvd_refl b))\u27e9\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk n a b : \u2115\nhab : coprime a b\nhn : IsPrimePow n\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 n \u2223 a * b \u2192 n \u2223 a \u2228 n \u2223 b\n[PROOFSTEP]\nobtain \u27e8p, k, hp, _, rfl\u27e9 := (isPrimePow_nat_iff _).1 hn\n[GOAL]\ncase inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\n\u22a2 p ^ k \u2223 a * b \u2192 p ^ k \u2223 a \u2228 p ^ k \u2223 b\n[PROOFSTEP]\nsimp only [hp.pow_dvd_iff_le_factorization (mul_ne_zero ha hb), Nat.factorization_mul ha hb,\n  hp.pow_dvd_iff_le_factorization ha, hp.pow_dvd_iff_le_factorization hb, Pi.add_apply, Finsupp.coe_add]\n[GOAL]\ncase inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\n\u22a2 k \u2264 \u2191(factorization a) p + \u2191(factorization b) p \u2192 k \u2264 \u2191(factorization a) p \u2228 k \u2264 \u2191(factorization b) p\n[PROOFSTEP]\nhave : a.factorization p = 0 \u2228 b.factorization p = 0 :=\n  by\n  rw [\u2190 Finsupp.not_mem_support_iff, \u2190 Finsupp.not_mem_support_iff, \u2190 not_and_or, \u2190 Finset.mem_inter]\n  intro t\n  simpa using (Nat.factorization_disjoint_of_coprime hab).le_bot t\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\n\u22a2 \u2191(factorization a) p = 0 \u2228 \u2191(factorization b) p = 0\n[PROOFSTEP]\nrw [\u2190 Finsupp.not_mem_support_iff, \u2190 Finsupp.not_mem_support_iff, \u2190 not_and_or, \u2190 Finset.mem_inter]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\n\u22a2 \u00acp \u2208 (factorization a).support \u2229 (factorization b).support\n[PROOFSTEP]\nintro t\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\nt : p \u2208 (factorization a).support \u2229 (factorization b).support\n\u22a2 False\n[PROOFSTEP]\nsimpa using (Nat.factorization_disjoint_of_coprime hab).le_bot t\n[GOAL]\ncase inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : \u2191(factorization a) p = 0 \u2228 \u2191(factorization b) p = 0\n\u22a2 k \u2264 \u2191(factorization a) p + \u2191(factorization b) p \u2192 k \u2264 \u2191(factorization a) p \u2228 k \u2264 \u2191(factorization b) p\n[PROOFSTEP]\ncases' this with h h\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\nh : \u2191(factorization a) p = 0\n\u22a2 k \u2264 \u2191(factorization a) p + \u2191(factorization b) p \u2192 k \u2264 \u2191(factorization a) p \u2228 k \u2264 \u2191(factorization b) p\n[PROOFSTEP]\nsimp [h, imp_or]\n[GOAL]\ncase inr.inr.intro.intro.intro.intro.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p\u271d : R\nk\u271d a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\np k : \u2115\nhp : Prime p\nleft\u271d : 0 < k\nhn : IsPrimePow (p ^ k)\nh : \u2191(factorization b) p = 0\n\u22a2 k \u2264 \u2191(factorization a) p + \u2191(factorization b) p \u2192 k \u2264 \u2191(factorization a) p \u2228 k \u2264 \u2191(factorization b) p\n[PROOFSTEP]\nsimp [h, imp_or]\n[GOAL]\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk a b : \u2115\nhab : coprime a b\n\u22a2 Finset.filter IsPrimePow (divisors (a * b)) = Finset.filter IsPrimePow (divisors a \u222a divisors b)\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk b : \u2115\nhab : coprime 0 b\n\u22a2 Finset.filter IsPrimePow (divisors (0 * b)) = Finset.filter IsPrimePow (divisors 0 \u222a divisors b)\n[PROOFSTEP]\nsimp only [Nat.coprime_zero_left] at hab \n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk b : \u2115\nhab : b = 1\n\u22a2 Finset.filter IsPrimePow (divisors (0 * b)) = Finset.filter IsPrimePow (divisors 0 \u222a divisors b)\n[PROOFSTEP]\nsimp [hab, Finset.filter_singleton, not_isPrimePow_one]\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk a b : \u2115\nhab : coprime a b\nha : a \u2260 0\n\u22a2 Finset.filter IsPrimePow (divisors (a * b)) = Finset.filter IsPrimePow (divisors a \u222a divisors b)\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb)\n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk a : \u2115\nha : a \u2260 0\nhab : coprime a 0\n\u22a2 Finset.filter IsPrimePow (divisors (a * 0)) = Finset.filter IsPrimePow (divisors a \u222a divisors 0)\n[PROOFSTEP]\nsimp only [Nat.coprime_zero_right] at hab \n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk a : \u2115\nha : a \u2260 0\nhab : a = 1\n\u22a2 Finset.filter IsPrimePow (divisors (a * 0)) = Finset.filter IsPrimePow (divisors a \u222a divisors 0)\n[PROOFSTEP]\nsimp [hab, Finset.filter_singleton, not_isPrimePow_one]\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn p : R\nk a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 Finset.filter IsPrimePow (divisors (a * b)) = Finset.filter IsPrimePow (divisors a \u222a divisors b)\n[PROOFSTEP]\next n\n[GOAL]\ncase inr.inr.a\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\nn : \u2115\n\u22a2 n \u2208 Finset.filter IsPrimePow (divisors (a * b)) \u2194 n \u2208 Finset.filter IsPrimePow (divisors a \u222a divisors b)\n[PROOFSTEP]\nsimp only [ha, hb, Finset.mem_union, Finset.mem_filter, Nat.mul_eq_zero, and_true_iff, Ne.def, and_congr_left_iff,\n  not_false_iff, Nat.mem_divisors, or_self_iff]\n[GOAL]\ncase inr.inr.a\nR : Type u_1\ninst\u271d : CommMonoidWithZero R\nn\u271d p : R\nk a b : \u2115\nhab : coprime a b\nha : a \u2260 0\nhb : b \u2260 0\nn : \u2115\n\u22a2 IsPrimePow n \u2192 (n \u2223 a * b \u2194 n \u2223 a \u2228 n \u2223 b)\n[PROOFSTEP]\napply hab.isPrimePow_dvd_mul\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factorization.PrimePow", "llama_tokens": 12394, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936324115011, "lm_q2_score": 0.6893056104028799, "lm_q1q2_score": 0.5282105000372498}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2078 : CommSemiring R\ninst\u271d\u2077 : StarRing R\ninst\u271d\u2076 : TrivialStar R\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : StarAddMonoid A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarModule R A\ninst\u271d\u00b9 : TopologicalSpace A\ninst\u271d : ContinuousStar A\nsrc\u271d : A \u2243+ A := AddEquiv.refl A\nr : R\na : A\n\u22a2 AddHom.toFun\n      { toFun := src\u271d.toFun,\n        map_add' :=\n          (_ :\n            \u2200 (x y : A), Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n      (r \u2022 a) =\n    \u2191(starRingEnd R) r \u2022\n      AddHom.toFun\n        { toFun := src\u271d.toFun,\n          map_add' :=\n            (_ :\n              \u2200 (x y : A), Equiv.toFun src\u271d.toEquiv (x + y) = Equiv.toFun src\u271d.toEquiv x + Equiv.toFun src\u271d.toEquiv y) }\n        a\n[PROOFSTEP]\nsimp [starRingEnd_apply]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Module.Star", "llama_tokens": 370, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8289388083214156, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.5280595476406136}}
{"text": "[GOAL]\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\nf g : SlashInvariantForm \u0393 k\nh : f.toFun = g.toFun\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase mk\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\ng : SlashInvariantForm \u0393 k\ntoFun\u271d : \u210d \u2192 \u2102\nslash_action_eq'\u271d : \u2200 (\u03b3 : { x // x \u2208 \u0393 }), toFun\u271d \u2223[k] \u03b3 = toFun\u271d\nh : { toFun := toFun\u271d, slash_action_eq' := slash_action_eq'\u271d }.toFun = g.toFun\n\u22a2 { toFun := toFun\u271d, slash_action_eq' := slash_action_eq'\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\ntoFun\u271d\u00b9 : \u210d \u2192 \u2102\nslash_action_eq'\u271d\u00b9 : \u2200 (\u03b3 : { x // x \u2208 \u0393 }), toFun\u271d\u00b9 \u2223[k] \u03b3 = toFun\u271d\u00b9\ntoFun\u271d : \u210d \u2192 \u2102\nslash_action_eq'\u271d : \u2200 (\u03b3 : { x // x \u2208 \u0393 }), toFun\u271d \u2223[k] \u03b3 = toFun\u271d\nh :\n  { toFun := toFun\u271d\u00b9, slash_action_eq' := slash_action_eq'\u271d\u00b9 }.toFun =\n    { toFun := toFun\u271d, slash_action_eq' := slash_action_eq'\u271d }.toFun\n\u22a2 { toFun := toFun\u271d\u00b9, slash_action_eq' := slash_action_eq'\u271d\u00b9 } =\n    { toFun := toFun\u271d, slash_action_eq' := slash_action_eq'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u0393\u271d : outParam (Subgroup SL(2, \u2124))\nk\u271d : outParam \u2124\nk : \u2124\n\u0393 : Subgroup SL(2, \u2124)\ninst\u271d : SlashInvariantFormClass F \u0393 k\nf : F\n\u03b3 : { x // x \u2208 \u0393 }\nz : \u210d\n\u22a2 \u2191f (\u03b3 \u2022 z) = (\u2191(\u2191\u2191\u2191\u03b3 1 0) * \u2191z + \u2191(\u2191\u2191\u2191\u03b3 1 1)) ^ k * \u2191f z\n[PROOFSTEP]\nrw [\u2190 ModularForm.slash_action_eq'_iff, slash_action_eqn]\n[GOAL]\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\nf g : SlashInvariantForm \u0393 k\n\u03b3 : { x // x \u2208 \u0393 }\n\u22a2 (\u2191f + \u2191g) \u2223[k] \u03b3 = \u2191f + \u2191g\n[PROOFSTEP]\nrw [SlashAction.add_slash, slash_action_eqn, slash_action_eqn]\n[GOAL]\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\n\u03b1 : Type u_2\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nc : \u03b1\nf : SlashInvariantForm \u0393 k\n\u03b3 : { x // x \u2208 \u0393 }\n\u22a2 (c \u2022 \u2191f) \u2223[k] \u03b3 = c \u2022 \u2191f\n[PROOFSTEP]\nrw [SlashAction.smul_slash_of_tower, slash_action_eqn]\n[GOAL]\nF : Type u_1\n\u0393 : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\nf : SlashInvariantForm \u0393 k\n\u03b3 : { x // x \u2208 \u0393 }\n\u22a2 (-\u2191f) \u2223[k] \u03b3 = -\u2191f\n[PROOFSTEP]\nrw [SlashAction.neg_slash, slash_action_eqn]\n[GOAL]\nF : Type u_1\n\u0393\u271d : outParam (Subgroup SL(2, \u2124))\nk : outParam \u2124\nk\u2081 k\u2082 : \u2124\n\u0393 : Subgroup SL(2, \u2124)\nf : SlashInvariantForm \u0393 k\u2081\ng : SlashInvariantForm \u0393 k\u2082\nA : { x // x \u2208 \u0393 }\n\u22a2 (\u2191f * \u2191g) \u2223[k\u2081 + k\u2082] A = \u2191f * \u2191g\n[PROOFSTEP]\nsimp_rw [ModularForm.mul_slash_subgroup, SlashInvariantFormClass.slash_action_eq]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.ModularForms.SlashInvariantForms", "llama_tokens": 1246, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.5279878474123929}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a\n[PROOFSTEP]\nhave := edist_approxOn_le hf h\u2080 x n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a\n[PROOFSTEP]\nrw [edist_comm y\u2080] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) (f x) \u2264 edist (f x) y\u2080\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a\n[PROOFSTEP]\nsimp only [edist_nndist, nndist_eq_nnnorm] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2191\u2016f x - y\u2080\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2264 \u2016f x - y\u2080\u2016\u208a\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nhave := edist_approxOn_y0_le hf h\u2080 x n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist y\u2080 (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) \u2264 edist y\u2080 (f x) + edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nrepeat' rw [edist_comm y\u2080, edist_eq_coe_nnnorm_sub] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist y\u2080 (\u2191(approxOn f hf s y\u2080 h\u2080 n) x) \u2264 edist y\u2080 (f x) + edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nrw [edist_comm y\u2080, edist_eq_coe_nnnorm_sub] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016\u208a \u2264 edist y\u2080 (f x) + edist y\u2080 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nrw [edist_comm y\u2080, edist_eq_coe_nnnorm_sub] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016\u208a \u2264 \u2191\u2016f x - y\u2080\u2016\u208a + \u2191\u2016f x - y\u2080\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nrw [edist_comm y\u2080, edist_eq_coe_nnnorm_sub] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016\u208a \u2264 \u2191\u2016f x - y\u2080\u2016\u208a + \u2191\u2016f x - y\u2080\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\nh\u2080 : 0 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\n\u22a2 \u2016\u2191(approxOn f hf s 0 h\u2080 n) x\u2016 \u2264 \u2016f x\u2016 + \u2016f x\u2016\n[PROOFSTEP]\nhave := edist_approxOn_y0_le hf h\u2080 x n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\nh\u2080 : 0 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : edist 0 (\u2191(approxOn f hf s 0 h\u2080 n) x) \u2264 edist 0 (f x) + edist 0 (f x)\n\u22a2 \u2016\u2191(approxOn f hf s 0 h\u2080 n) x\u2016 \u2264 \u2016f x\u2016 + \u2016f x\u2016\n[PROOFSTEP]\nsimp [edist_comm (0 : E), edist_eq_coe_nnnorm] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\nh\u2080 : 0 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nx : \u03b2\nn : \u2115\nthis : \u2191\u2016\u2191(approxOn f hf s 0 h\u2080 n) x\u2016\u208a \u2264 \u2191\u2016f x\u2016\u208a + \u2191\u2016f x\u2016\u208a\n\u22a2 \u2016\u2191(approxOn f hf s 0 h\u2080 n) x\u2016 \u2264 \u2016f x\u2016 + \u2016f x\u2016\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nby_cases hp_zero : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : p = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices Tendsto (fun n => \u222b\u207b x, (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) atTop (\ud835\udcdd 0)\n  by\n  simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]\n  convert continuous_rpow_const.continuousAt.tendsto.comp this\n  simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]\n    -- We simply check the conditions of the Dominated Convergence Theorem:\n      -- (1) The function \"`p`-th power of distance between `f` and the approximation\" is measurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => (\u222b\u207b (x : \u03b2), \u2191\u2016(\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p))\n    atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert continuous_rpow_const.continuousAt.tendsto.comp this\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nthis : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n\u22a2 0 = 0 ^ (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)]\n  -- We simply check the conditions of the Dominated Convergence Theorem:\n    -- (1) The function \"`p`-th power of distance between `f` and the approximation\" is measurable\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hF_meas : \u2200 n, Measurable fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal := by\n  simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n    (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n      (measurable_edist_right.comp hf).pow_const p.toReal\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\n\u22a2 \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nsimpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n  (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n    (measurable_edist_right.comp hf).pow_const p.toReal\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_bound :\n  \u2200 n,\n    (fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) \u2264\u1d50[\u03bc] fun x => (\u2016f x - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal :=\n  fun n => eventually_of_forall fun x => rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h\u2080 x n)) toReal_nonneg\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_fin : (\u222b\u207b a : \u03b2, (\u2016f a - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) \u2260 \u22a4 :=\n  (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_zero hp_ne_top hi).ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h_lim : \u2200\u1d50 a : \u03b2 \u2202\u03bc, Tendsto (fun n => (\u2016approxOn f hf s y\u2080 h\u2080 n a - f a\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop (\ud835\udcdd 0) :=\n  by\n  filter_upwards [h\u03bc] with a ha\n  have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n    (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds\n  convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n  simp [zero_rpow_of_pos hp]\n    -- Then we apply the Dominated Convergence Theorem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\n\u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nfilter_upwards [h\u03bc] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n  (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\na : \u03b2\nha : f a \u2208 closure s\nthis : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a))\n\u22a2 0 = \u2191\u2016f a - f a\u2016\u208a ^ ENNReal.toReal p\n[PROOFSTEP]\nsimp [zero_rpow_of_pos hp]\n  -- Then we apply the Dominated Convergence Theorem\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\nhp_zero : \u00acp = 0\nhp : 0 < ENNReal.toReal p\nhF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p\nh_bound :\n  \u2200 (n : \u2115),\n    (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p\nh_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4\nh_lim : \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Mem\u2112p (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p\n[PROOFSTEP]\nrefine' \u27e8(approxOn f fmeas s y\u2080 h\u2080 n).aestronglyMeasurable, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4\n[PROOFSTEP]\nsuffices snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc < \u22a4\n  by\n  have : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n    \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\u27e9\n  convert snorm_add_lt_top this hi\u2080\n  ext x\n  simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4\n[PROOFSTEP]\nhave : Mem\u2112p (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc :=\n  \u27e8(approxOn f fmeas s y\u2080 h\u2080 n - const \u03b2 y\u2080).aestronglyMeasurable, this\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 snorm (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) p \u03bc < \u22a4\n[PROOFSTEP]\nconvert snorm_add_lt_top this hi\u2080\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) = (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nthis\u271d : snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\nthis : Mem\u2112p (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p\nx : \u03b2\n\u22a2 \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x = ((fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) + fun x => y\u2080) x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\n[PROOFSTEP]\nhave hf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p \u03bc :=\n  by\n  have h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 :=\n    by\n    simp only [\u2190 dist_eq_norm]\n    exact (continuous_id.dist continuous_const).measurable.comp fmeas\n  refine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9\n  rw [snorm_norm]\n  convert snorm_add_lt_top hf hi\u2080.neg with x\n  simp [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n[PROOFSTEP]\nhave h_meas : Measurable fun x => \u2016f x - y\u2080\u2016 :=\n  by\n  simp only [\u2190 dist_eq_norm]\n  exact (continuous_id.dist continuous_const).measurable.comp fmeas\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Measurable fun x => \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nsimp only [\u2190 dist_eq_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\n\u22a2 Measurable fun x => dist (f x) y\u2080\n[PROOFSTEP]\nexact (continuous_id.dist continuous_const).measurable.comp fmeas\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n[PROOFSTEP]\nrefine' \u27e8h_meas.aemeasurable.aestronglyMeasurable, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => \u2016f x - y\u2080\u2016) p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_norm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\n\u22a2 snorm (fun x => f x - y\u2080) p \u03bc < \u22a4\n[PROOFSTEP]\nconvert snorm_add_lt_top hf hi\u2080.neg with x\n[GOAL]\ncase h.e'_3.h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nh_meas : Measurable fun x => \u2016f x - y\u2080\u2016\nx : \u03b2\n\u22a2 f x - y\u2080 = (f + -fun x => y\u2080) x\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\n[PROOFSTEP]\nhave : \u2200\u1d50 x \u2202\u03bc, \u2016approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 :=\n  by\n  refine' eventually_of_forall _\n  intro x\n  convert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1\n  rw [Real.norm_eq_abs, abs_of_nonneg]\n  exact add_nonneg (norm_nonneg _) (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n[PROOFSTEP]\nrefine' eventually_of_forall _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\n\u22a2 \u2200 (x : \u03b2), \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n[PROOFSTEP]\nconvert norm_approxOn_y\u2080_le fmeas h\u2080 x n using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016 = \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nrw [Real.norm_eq_abs, abs_of_nonneg]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nx : \u03b2\n\u22a2 0 \u2264 \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\n[PROOFSTEP]\nexact add_nonneg (norm_nonneg _) (norm_nonneg _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f p\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) p\nn : \u2115\nhf' : Mem\u2112p (fun x => \u2016f x - y\u2080\u2016) p\nthis : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, \u2016\u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080\u2016 \u2264 \u2016\u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016\u2016\n\u22a2 snorm (fun x => \u2191(approxOn f fmeas s y\u2080 h\u2080 n) x - y\u2080) p \u03bc < \u22a4\n[PROOFSTEP]\ncalc\n  snorm (fun x => approxOn f fmeas s y\u2080 h\u2080 n x - y\u2080) p \u03bc \u2264 snorm (fun x => \u2016f x - y\u2080\u2016 + \u2016f x - y\u2080\u2016) p \u03bc :=\n    snorm_mono_ae this\n  _ < \u22a4 := snorm_add_lt_top hf' hf'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\n\u22a2 Tendsto (fun n => snorm (\u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n) - f) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' tendsto_approxOn_Lp_snorm fmeas _ hp_ne_top _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase refine'_1.hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\n\u22a2 \u2200 (x : \u03b2), f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_1.hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\nx : \u03b2\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\napply subset_closure\n[GOAL]\ncase refine'_1.hp.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\nx : \u03b2\n\u22a2 f x \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : snorm f p \u03bc < \u22a4\n\u22a2 snorm (fun x => f x - 0) p \u03bc < \u22a4\n[PROOFSTEP]\nsimpa using hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace E\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Mem\u2112p f p\n\u22a2 Tendsto\n    (fun n =>\n      Mem\u2112p.toLp \u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)\n        (_ : Mem\u2112p (\u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n)) p))\n    atTop (\ud835\udcdd (Mem\u2112p.toLp f hf))\n[PROOFSTEP]\nsimpa only [Lp.tendsto_Lp_iff_tendsto_\u2112p''] using tendsto_approxOn_range_Lp_snorm hp_ne_top fmeas hf.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nlet f' := hf.1.mk f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nrsuffices \u27e8g, hg, g_mem\u27e9 : \u2203 g : \u03b2 \u2192\u209b E, snorm (f' - \u21d1g) p \u03bc < \u03b5 \u2227 Mem\u2112p g p \u03bc\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nrefine' \u27e8g, _, g_mem\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 snorm (f - \u2191g) p \u03bc < \u03b5\n[PROOFSTEP]\nsuffices snorm (f - \u21d1g) p \u03bc = snorm (f' - \u21d1g) p \u03bc by rwa [this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nthis : snorm (f - \u2191g) p \u03bc = snorm (f' - \u2191g) p \u03bc\n\u22a2 snorm (f - \u2191g) p \u03bc < \u03b5\n[PROOFSTEP]\nrwa [this]\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 snorm (f - \u2191g) p \u03bc = snorm (f' - \u2191g) p \u03bc\n[PROOFSTEP]\napply snorm_congr_ae\n[GOAL]\ncase intro.intro.hfg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\n\u22a2 f - \u2191g =\u1d50[\u03bc] f' - \u2191g\n[PROOFSTEP]\nfilter_upwards [hf.1.ae_eq_mk] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\ng : \u03b2 \u2192\u209b E\nhg : snorm (f' - \u2191g) p \u03bc < \u03b5\ng_mem : Mem\u2112p (\u2191g) p\nx : \u03b2\nhx : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) x\n\u22a2 (f - \u2191g) x = (AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) - \u2191g) x\n[PROOFSTEP]\nsimpa only [Pi.sub_apply, sub_left_inj] using hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nhave hf' : Mem\u2112p f' p \u03bc := hf.ae_eq hf.1.ae_eq_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhf' : Mem\u2112p f' p\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nhave f'meas : Measurable f' := hf.1.measurable_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhf' : Mem\u2112p f' p\nf'meas : Measurable f'\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nhave : SeparableSpace (range f' \u222a {0} : Set E) :=\n  StronglyMeasurable.separableSpace_range_union_singleton hf.1.stronglyMeasurable_mk\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhf' : Mem\u2112p f' p\nf'meas : Measurable f'\nthis : SeparableSpace \u2191(Set.range f' \u222a {0})\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nrcases((tendsto_approxOn_range_Lp_snorm hp_ne_top f'meas hf'.2).eventually <| gt_mem_nhds h\u03b5.bot_lt).exists with \u27e8n, hn\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhf' : Mem\u2112p f' p\nf'meas : Measurable f'\nthis : SeparableSpace \u2191(Set.range f' \u222a {0})\nn : \u2115\nhn : snorm (\u2191(approxOn f' f'meas (Set.range f' \u222a {0}) 0 (_ : 0 \u2208 Set.range f' \u222a {0}) n) - f') p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nrw [\u2190 snorm_neg, neg_sub] at hn \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE\u271d : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9 : NormedAddCommGroup F\nq : \u211d\np : \u211d\u22650\u221e\nE : Type u_7\ninst\u271d : NormedAddCommGroup E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nhf : Mem\u2112p f p\nhp_ne_top : p \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc)\nhf' : Mem\u2112p f' p\nf'meas : Measurable f'\nthis : SeparableSpace \u2191(Set.range f' \u222a {0})\nn : \u2115\nhn : snorm (f' - \u2191(approxOn f' f'meas (Set.range f' \u222a {0}) 0 (_ : 0 \u2208 Set.range f' \u222a {0}) n)) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p\n[PROOFSTEP]\nexact \u27e8_, hn, mem\u2112p_approxOn_range f'meas hf' _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : HasFiniteIntegral fun x => f x - y\u2080\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a \u2202\u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa [snorm_one_eq_lintegral_nnnorm] using\n  tendsto_approxOn_Lp_snorm hf h\u2080 one_ne_top h\u03bc (by simpa [snorm_one_eq_lintegral_nnnorm] using hi)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\nhf : Measurable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\n\u03bc : Measure \u03b2\nh\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s\nhi : HasFiniteIntegral fun x => f x - y\u2080\n\u22a2 snorm (fun x => f x - y\u2080) 1 \u03bc < \u22a4\n[PROOFSTEP]\nsimpa [snorm_one_eq_lintegral_nnnorm] using hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Integrable f\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Integrable fun x => y\u2080\nn : \u2115\n\u22a2 Integrable \u2191(approxOn f fmeas s y\u2080 h\u2080 n)\n[PROOFSTEP]\nrw [\u2190 mem\u2112p_one_iff_integrable] at hf hi\u2080 \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\nhf : Mem\u2112p f 1\ns : Set E\ny\u2080 : E\nh\u2080 : y\u2080 \u2208 s\ninst\u271d : SeparableSpace \u2191s\nhi\u2080 : Mem\u2112p (fun x => y\u2080) 1\nn : \u2115\n\u22a2 Mem\u2112p (\u2191(approxOn f fmeas s y\u2080 h\u2080 n)) 1\n[PROOFSTEP]\nexact mem\u2112p_approxOn fmeas hf h\u2080 hi\u2080 n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\nn : \u2115\nx : \u03b2\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\n\u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f fmeas (Set.range f \u222a {0}) 0 (_ : 0 \u2208 Set.range f \u222a {0}) n) x - f x\u2016\u208a \u2202\u03bc)\n    atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_approxOn_L1_nnnorm fmeas\n[GOAL]\ncase h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase h\u03bc.hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\n\u22a2 \u2200 (x : \u03b2), f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\u03bc.hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\nx : \u03b2\n\u22a2 f x \u2208 closure (Set.range f \u222a {0})\n[PROOFSTEP]\napply subset_closure\n[GOAL]\ncase h\u03bc.hp.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\nx : \u03b2\n\u22a2 f x \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : OpensMeasurableSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nfmeas : Measurable f\nhf : Integrable f\n\u22a2 HasFiniteIntegral fun x => f x - 0\n[PROOFSTEP]\nsimpa using hf.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2074 : MeasurableSpace \u03b2\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : BorelSpace E\nf : \u03b2 \u2192 E\n\u03bc : Measure \u03b2\nfmeas : Measurable f\ninst\u271d : SeparableSpace \u2191(Set.range f \u222a {0})\nhf : Integrable f\nn : \u2115\n\u22a2 0 \u2208 Set.range f \u222a {0}\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc\u271d : Measure \u03b1\np\u271d : \u211d\u22650\u221e\np : \u211d\nf : \u03b1 \u2192\u209b F\n\u03bc : Measure \u03b1\n\u22a2 snorm' (\u2191f) p \u03bc = (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / p)\n[PROOFSTEP]\nhave h_map : (fun a => (\u2016f a\u2016\u208a : \u211d\u22650\u221e) ^ p) = f.map fun a : F => (\u2016a\u2016\u208a : \u211d\u22650\u221e) ^ p := by simp; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc\u271d : Measure \u03b1\np\u271d : \u211d\u22650\u221e\np : \u211d\nf : \u03b1 \u2192\u209b F\n\u03bc : Measure \u03b1\n\u22a2 (fun a => \u2191\u2016\u2191f a\u2016\u208a ^ p) = \u2191(map (fun a => \u2191\u2016a\u2016\u208a ^ p) f)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc\u271d : Measure \u03b1\np\u271d : \u211d\u22650\u221e\np : \u211d\nf : \u03b1 \u2192\u209b F\n\u03bc : Measure \u03b1\n\u22a2 (fun a => \u2191\u2016\u2191f a\u2016\u208a ^ p) = (fun a => \u2191\u2016a\u2016\u208a ^ p) \u2218 \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc\u271d : Measure \u03b1\np\u271d : \u211d\u22650\u221e\np : \u211d\nf : \u03b1 \u2192\u209b F\n\u03bc : Measure \u03b1\nh_map : (fun a => \u2191\u2016\u2191f a\u2016\u208a ^ p) = \u2191(map (fun a => \u2191\u2016a\u2016\u208a ^ p) f)\n\u22a2 snorm' (\u2191f) p \u03bc = (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / p)\n[PROOFSTEP]\nrw [snorm', h_map, lintegral_eq_lintegral, map_lintegral]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nhave hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nhave hf_snorm := Mem\u2112p.snorm_lt_top hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : snorm (\u2191f) p \u03bc < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, \u2190\n  @ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),\n  @ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real]), ENNReal.sum_lt_top_iff] at hf_snorm \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4\n\u22a2 0 < 1 / ENNReal.toReal p\n[PROOFSTEP]\nsimp [hp_pos_real]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 ^ (1 / (1 / ENNReal.toReal p))\n\u22a2 0 < 1 / (1 / ENNReal.toReal p)\n[PROOFSTEP]\nsimp [hp_pos_real]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nby_cases hyf : y \u2208 f.range\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nsuffices h_empty : f \u207b\u00b9' { y } = \u2205\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrw [h_empty, measure_empty]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nh_empty : \u2191f \u207b\u00b9' {y} = \u2205\n\u22a2 0 < \u22a4\n[PROOFSTEP]\nexact ENNReal.coe_lt_top\n[GOAL]\ncase h_empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\n\u22a2 \u2191f \u207b\u00b9' {y} = \u2205\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 x \u2208 \u2191f \u207b\u00b9' {y} \u2194 x \u2208 \u2205\n[PROOFSTEP]\nrw [Set.mem_preimage, Set.mem_singleton_iff, mem_empty_iff_false, iff_false_iff]\n[GOAL]\ncase h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\n\u22a2 \u00ac\u2191f x = y\n[PROOFSTEP]\nrefine' fun hxy => hyf _\n[GOAL]\ncase h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 y \u2208 SimpleFunc.range f\n[PROOFSTEP]\nrw [mem_range, Set.mem_range]\n[GOAL]\ncase h_empty.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : \u00acy \u2208 SimpleFunc.range f\nx : \u03b1\nhxy : \u2191f x = y\n\u22a2 \u2203 y_1, \u2191f y_1 = y\n[PROOFSTEP]\nexact \u27e8x, hxy\u27e9\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhf_snorm : \u2200 (a : E), a \u2208 SimpleFunc.range f \u2192 \u2191\u2016a\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {a}) < \u22a4\nhyf : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nspecialize hf_snorm y hyf\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrw [ENNReal.mul_lt_top_iff] at hf_snorm \n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u2228 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\ncases hf_snorm with\n| inl hf_snorm => exact hf_snorm.2\n| inr hf_snorm =>\n  cases hf_snorm with\n  | inl hf_snorm =>\n    refine' absurd _ hy_ne\n    simpa [hp_pos_real] using hf_snorm\n  | inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4 \u2228 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\ncases hf_snorm with\n| inl hf_snorm => exact hf_snorm.2\n| inr hf_snorm =>\n  cases hf_snorm with\n  | inl hf_snorm =>\n    refine' absurd _ hy_ne\n    simpa [hp_pos_real] using hf_snorm\n  | inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\n\n| inl hf_snorm => exact hf_snorm.2\n[GOAL]\ncase pos.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p < \u22a4 \u2227 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nexact hf_snorm.2\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\n\n| inr hf_snorm =>\n  cases hf_snorm with\n  | inl hf_snorm =>\n    refine' absurd _ hy_ne\n    simpa [hp_pos_real] using hf_snorm\n  | inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\ncases hf_snorm with\n| inl hf_snorm =>\n  refine' absurd _ hy_ne\n  simpa [hp_pos_real] using hf_snorm\n| inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0 \u2228 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\ncases hf_snorm with\n| inl hf_snorm =>\n  refine' absurd _ hy_ne\n  simpa [hp_pos_real] using hf_snorm\n| inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\n\n| inl hf_snorm =>\n  refine' absurd _ hy_ne\n  simpa [hp_pos_real] using hf_snorm\n[GOAL]\ncase pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrefine' absurd _ hy_ne\n[GOAL]\ncase pos.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p = 0\n\u22a2 y = 0\n[PROOFSTEP]\nsimpa [hp_pos_real] using hf_snorm\n[GOAL]\ncase pos.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\n\n| inr hf_snorm => simp [hf_snorm]\n[GOAL]\ncase pos.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nf : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\ny : E\nhy_ne : y \u2260 0\nhp_pos_real : 0 < ENNReal.toReal p\nhyf : y \u2208 SimpleFunc.range f\nhf_snorm : \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) = 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nsimp [hf_snorm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 Mem\u2112p (\u2191f) p\n[PROOFSTEP]\nby_cases hp0 : p = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 Mem\u2112p (\u2191f) p\n[PROOFSTEP]\nrw [hp0, mem\u2112p_zero_iff_aestronglyMeasurable]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : p = 0\n\u22a2 AEStronglyMeasurable (\u2191f) \u03bc\n[PROOFSTEP]\nexact f.aestronglyMeasurable\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\n\u22a2 Mem\u2112p (\u2191f) p\n[PROOFSTEP]\nby_cases hp_top : p = \u221e\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p\n[PROOFSTEP]\nrw [hp_top]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : p = \u22a4\n\u22a2 Mem\u2112p \u2191f \u22a4\n[PROOFSTEP]\nexact mem\u2112p_top f \u03bc\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 Mem\u2112p (\u2191f) p\n[PROOFSTEP]\nrefine' \u27e8f.aestronglyMeasurable, _\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm (\u2191f) p \u03bc < \u22a4\n[PROOFSTEP]\nrw [snorm_eq_snorm' hp0 hp_top, f.snorm'_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 (\u2211 y in SimpleFunc.range f, \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y})) ^ (1 / ENNReal.toReal p) < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.rpow_lt_top_of_nonneg (by simp) (ENNReal.sum_lt_top_iff.mpr fun y _ => _).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\n\u22a2 0 \u2264 1 / ENNReal.toReal p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nby_cases hy0 : y = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : y = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nsimp [hy0, ENNReal.toReal_pos hp0 hp_top]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p * \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrefine' ENNReal.mul_lt_top _ (hf y hy0).ne\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np\u271d p : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\nhf : \u2200 (y : E), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\nhp0 : \u00acp = 0\nhp_top : \u00acp = \u22a4\ny : E\nx\u271d : y \u2208 SimpleFunc.range f\nhy0 : \u00acy = 0\n\u22a2 \u2191\u2016y\u2016\u208a ^ ENNReal.toReal p \u2260 \u22a4\n[PROOFSTEP]\nexact (ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg ENNReal.coe_ne_top).ne\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nf : \u03b1 \u2192\u209b E\ng : \u03b1 \u2192\u209b F\n\u22a2 Integrable \u2191f \u2192 Integrable \u2191g \u2192 Integrable \u2191(pair f g)\n[PROOFSTEP]\nsimpa only [integrable_iff_finMeasSupp] using FinMeasSupp.pair\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192\u209b \u03b2\nhf : \u2200 (y : \u03b2), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (support \u2191f) < \u22a4\n[PROOFSTEP]\nrw [support_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192\u209b \u03b2\nhf : \u2200 (y : \u03b2), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n\u22a2 \u2191\u2191\u03bc (\u22c3 (y : \u03b2) (_ : y \u2208 filter (fun y => y \u2260 0) (SimpleFunc.range f)), \u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrefine' (measure_biUnion_finset_le _ _).trans_lt (ENNReal.sum_lt_top_iff.mpr fun y hy => _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192\u209b \u03b2\nhf : \u2200 (y : \u03b2), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\ny : \u03b2\nhy : y \u2208 filter (fun y => y \u2260 0) (SimpleFunc.range f)\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nrw [Finset.mem_filter] at hy \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d : Zero \u03b2\nf : \u03b1 \u2192\u209b \u03b2\nhf : \u2200 (y : \u03b2), y \u2260 0 \u2192 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\ny : \u03b2\nhy : y \u2208 SimpleFunc.range f \u2227 y \u2260 0\n\u22a2 \u2191\u2191\u03bc (\u2191f \u207b\u00b9' {y}) < \u22a4\n[PROOFSTEP]\nexact hf y hy.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nc : E\nhc : c \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\nhcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nhave : Function.support (const \u03b1 c) = Set.univ := Function.support_const hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nc : E\nhc : c \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\nhcs : Mem\u2112p (\u2191(piecewise s hs (const \u03b1 c) (const \u03b1 0))) p\nthis : support \u2191(const \u03b1 c) = Set.univ\n\u22a2 \u2191\u2191\u03bc s < \u22a4\n[PROOFSTEP]\nsimpa only [mem\u2112p_iff_finMeasSupp hp_pos hp_ne_top, finMeasSupp_iff_support, support_indicator, Set.inter_univ,\n  this] using hcs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2200 {a b : { x // x \u2208 Lp E p }},\n    a \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n      b \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n        a + b \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f}\n[PROOFSTEP]\nrintro f g \u27e8s, hs\u27e9 \u27e8t, ht\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\nt : \u03b1 \u2192\u209b E\nht : AEEqFun.mk \u2191t (_ : AEStronglyMeasurable (\u2191t) \u03bc) = \u2191g\n\u22a2 f + g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f}\n[PROOFSTEP]\nuse s + t\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\nt : \u03b1 \u2192\u209b E\nht : AEEqFun.mk \u2191t (_ : AEStronglyMeasurable (\u2191t) \u03bc) = \u2191g\n\u22a2 AEEqFun.mk \u2191(s + t) (_ : AEStronglyMeasurable (\u2191(s + t)) \u03bc) = \u2191(f + g)\n[PROOFSTEP]\nsimp only [\u2190 hs, \u2190 ht, AEEqFun.mk_add_mk, AddSubgroup.coe_add, AEEqFun.mk_eq_mk, SimpleFunc.coe_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2200 {x : { x // x \u2208 Lp E p }},\n    x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f},\n                  add_mem' :=\n                    (_ :\n                      \u2200 {f g : { x // x \u2208 Lp E p }},\n                        f \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                          g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                            f + g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f}) },\n              zero_mem' := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u21910) }.toAddSubsemigroup.carrier \u2192\n      -x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f},\n                  add_mem' :=\n                    (_ :\n                      \u2200 {f g : { x // x \u2208 Lp E p }},\n                        f \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                          g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                            f + g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f}) },\n              zero_mem' := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u21910) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrintro f \u27e8s, hs\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 -f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f},\n              add_mem' :=\n                (_ :\n                  \u2200 {f g : { x // x \u2208 Lp E p }},\n                    f \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                      g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f} \u2192\n                        f + g \u2208 {f | \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f}) },\n          zero_mem' := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u21910) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nuse-s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 AEEqFun.mk \u2191(-s) (_ : AEStronglyMeasurable (\u2191(-s)) \u03bc) = \u2191(-f)\n[PROOFSTEP]\nsimp only [\u2190 hs, AEEqFun.neg_mk, SimpleFunc.coe_neg, AEEqFun.mk_eq_mk, AddSubgroup.coe_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 k \u2022 \u2191f \u2208 simpleFunc E p \u03bc\n[PROOFSTEP]\nrcases f with \u27e8f, \u27e8s, hs\u27e9\u27e9\n[GOAL]\ncase mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 k \u2022 \u2191{ val := f, property := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f) } \u2208 simpleFunc E p \u03bc\n[PROOFSTEP]\nuse k \u2022 s\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 AEEqFun.mk \u2191(k \u2022 s) (_ : AEStronglyMeasurable (\u2191(k \u2022 s)) \u03bc) =\n    \u2191(k \u2022 \u2191{ val := f, property := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f) })\n[PROOFSTEP]\napply Eq.trans (AEEqFun.smul_mk k s s.aestronglyMeasurable).symm _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 k \u2022 AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) =\n    \u2191(k \u2022 \u2191{ val := f, property := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f) })\n[PROOFSTEP]\nrw [hs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 Lp E p }\ns : \u03b1 \u2192\u209b E\nhs : AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f\n\u22a2 k \u2022 \u2191f = \u2191(k \u2022 \u2191{ val := f, property := (_ : \u2203 s, AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) = \u2191f) })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 1 \u2022 f = f\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(1 \u2022 f) = \u2191f\n[PROOFSTEP]\nexact one_smul _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx y : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 (x * y) \u2022 f = x \u2022 y \u2022 f\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx y : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191((x * y) \u2022 f) = \u2191(x \u2022 y \u2022 f)\n[PROOFSTEP]\nexact mul_smul _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : \ud835\udd5c\n\u22a2 x \u2022 0 = 0\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : \ud835\udd5c\n\u22a2 \u2191(x \u2022 0) = \u21910\n[PROOFSTEP]\nexact smul_zero _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : \ud835\udd5c\nf g : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 x \u2022 (f + g) = x \u2022 f + x \u2022 g\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : \ud835\udd5c\nf g : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(x \u2022 (f + g)) = \u2191(x \u2022 f + x \u2022 g)\n[PROOFSTEP]\nexact smul_add _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx y : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 (x + y) \u2022 f = x \u2022 f + y \u2022 f\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx y : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191((x + y) \u2022 f) = \u2191(x \u2022 f + y \u2022 f)\n[PROOFSTEP]\nexact add_smul _ _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 0 \u2022 f = 0\n[PROOFSTEP]\next1\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(0 \u2022 f) = \u21910\n[PROOFSTEP]\nexact zero_smul _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192\u209b E\nhf : Mem\u2112p (\u2191f) p\nhg : Mem\u2112p (\u2191g) p\n\u22a2 toLp (f - g) (_ : Mem\u2112p (\u2191f - fun a => \u2191g a) p) = toLp f hf - toLp g hg\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, \u2190 toLp_neg, \u2190 toLp_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc f) =\u1d50[\u03bc] \u2191\u2191\u2191f\n[PROOFSTEP]\nconvert (AEEqFun.coeFn_mk (toSimpleFunc f) (toSimpleFunc f).aestronglyMeasurable).symm using 2\n[GOAL]\ncase h.e'_5.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191\u2191f = AEEqFun.mk \u2191(toSimpleFunc f) (_ : AEStronglyMeasurable (\u2191(toSimpleFunc f)) \u03bc)\n[PROOFSTEP]\nexact (Classical.choose_spec f.2).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhfi : Mem\u2112p (\u2191f) p\n\u22a2 \u2191(toSimpleFunc (toLp f hfi)) =\u1d50[\u03bc] \u2191f\n[PROOFSTEP]\nrw [\u2190 AEEqFun.mk_eq_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhfi : Mem\u2112p (\u2191f) p\n\u22a2 AEEqFun.mk \u2191(toSimpleFunc (toLp f hfi)) ?m.1694293 = AEEqFun.mk \u2191f ?m.1694294\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhfi : Mem\u2112p (\u2191f) p\n\u22a2 AEStronglyMeasurable (\u2191(toSimpleFunc (toLp f hfi))) \u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : \u03b1 \u2192\u209b E\nhfi : Mem\u2112p (\u2191f) p\n\u22a2 AEStronglyMeasurable (\u2191f) \u03bc\n[PROOFSTEP]\nexact Classical.choose_spec (toLp f hfi).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n\u22a2 \u2191(toSimpleFunc 0) =\u1d50[\u03bc] 0\n[PROOFSTEP]\nfilter_upwards [toSimpleFunc_eq_toFun (0 : Lp.simpleFunc E p \u03bc), Lp.coeFn_zero E 1 \u03bc] with _ h\u2081 _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\na\u271d\u00b9 : \u03b1\nh\u2081 : \u2191(toSimpleFunc 0) a\u271d\u00b9 = \u2191\u2191\u21910 a\u271d\u00b9\na\u271d : \u2191\u21910 a\u271d\u00b9 = OfNat.ofNat 0 a\u271d\u00b9\n\u22a2 \u2191(toSimpleFunc 0) a\u271d\u00b9 = OfNat.ofNat 0 a\u271d\u00b9\n[PROOFSTEP]\nrwa [h\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (f + g)) =\u1d50[\u03bc] \u2191(toSimpleFunc f) + \u2191(toSimpleFunc g)\n[PROOFSTEP]\nfilter_upwards [toSimpleFunc_eq_toFun (f + g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g,\n  Lp.coeFn_add (f : Lp E p \u03bc) g] with _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191\u2191\u2191(f + g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191\u2191(\u2191f + \u2191g) a\u271d = (\u2191\u2191\u2191f + \u2191\u2191\u2191g) a\u271d \u2192 \u2191(toSimpleFunc (f + g)) a\u271d = (\u2191(toSimpleFunc f) + \u2191(toSimpleFunc g)) a\u271d\n[PROOFSTEP]\nsimp only [AddSubgroup.coe_add, Pi.add_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(toSimpleFunc f) a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\niterate 4 intro h; rw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(toSimpleFunc f) a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n    \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n      \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(toSimpleFunc f) a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n    \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n      \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191(toSimpleFunc f) a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n    \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191(toSimpleFunc f) a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n    \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b2 : \u2191(toSimpleFunc (f + g)) a\u271d = \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d\nh\u271d\u00b9 : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh\u271d : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\nh : \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f + \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (-f)) =\u1d50[\u03bc] -\u2191(toSimpleFunc f)\n[PROOFSTEP]\nfilter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f, Lp.coeFn_neg (f : Lp E p \u03bc)] with _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191\u2191\u2191(-f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(-\u2191f) a\u271d = (-\u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = (-\u2191(toSimpleFunc f)) a\u271d\n[PROOFSTEP]\nsimp only [Pi.neg_apply, AddSubgroup.coe_neg]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrepeat intro h; rw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (-f)) a\u271d = -\u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d \u2192 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (-f)) a\u271d = \u2191(-\u2191\u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(-\u2191\u2191f) a\u271d = -\u2191\u2191\u2191f a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (f - g)) =\u1d50[\u03bc] \u2191(toSimpleFunc f) - \u2191(toSimpleFunc g)\n[PROOFSTEP]\nfilter_upwards [toSimpleFunc_eq_toFun (f - g), toSimpleFunc_eq_toFun f, toSimpleFunc_eq_toFun g,\n  Lp.coeFn_sub (f : Lp E p \u03bc) g] with _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191\u2191\u2191(f - g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191\u2191(\u2191f - \u2191g) a\u271d = (\u2191\u2191\u2191f - \u2191\u2191\u2191g) a\u271d \u2192 \u2191(toSimpleFunc (f - g)) a\u271d = (\u2191(toSimpleFunc f) - \u2191(toSimpleFunc g)) a\u271d\n[PROOFSTEP]\nsimp only [AddSubgroup.coe_sub, Pi.sub_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(toSimpleFunc f) a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrepeat' intro h; rw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n        \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(toSimpleFunc f) a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n    \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n      \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(toSimpleFunc f) a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n    \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n      \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191(toSimpleFunc f) a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n    \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191(toSimpleFunc f) a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d \u2192\n    \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191(toSimpleFunc g) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d \u2192 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf g : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b2 : \u2191(toSimpleFunc (f - g)) a\u271d = \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d\nh\u271d\u00b9 : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh\u271d : \u2191(toSimpleFunc g) a\u271d = \u2191\u2191\u2191g a\u271d\nh : \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d\n\u22a2 \u2191(\u2191\u2191f - \u2191\u2191g) a\u271d = \u2191\u2191\u2191f a\u271d - \u2191\u2191\u2191g a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) =\u1d50[\u03bc] k \u2022 \u2191(toSimpleFunc f)\n[PROOFSTEP]\nfilter_upwards [toSimpleFunc_eq_toFun (k \u2022 f), toSimpleFunc_eq_toFun f, Lp.coeFn_smul k (f : Lp E p \u03bc)] with _\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191\u2191(k \u2022 f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192\n      \u2191\u2191(k \u2022 \u2191f) a\u271d = (k \u2022 \u2191\u2191\u2191f) a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = (k \u2022 \u2191(toSimpleFunc f)) a\u271d\n[PROOFSTEP]\nsimp only [Pi.smul_apply, coe_smul]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrepeat intro h; rw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\n\u22a2 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d \u2192\n    \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191(toSimpleFunc (k \u2022 f)) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\n\u22a2 \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191(toSimpleFunc f) a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d \u2192 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2075 : MeasurableSpace \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : NormedRing \ud835\udd5c\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nk : \ud835\udd5c\nf : { x // x \u2208 simpleFunc E p \u03bc }\na\u271d : \u03b1\nh\u271d\u00b9 : \u2191(toSimpleFunc (k \u2022 f)) a\u271d = \u2191\u2191(k \u2022 \u2191f) a\u271d\nh\u271d : \u2191(toSimpleFunc f) a\u271d = \u2191\u2191\u2191f a\u271d\nh : \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d\n\u22a2 \u2191\u2191(k \u2022 \u2191f) a\u271d = k \u2022 \u2191\u2191\u2191f a\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NormedRing \ud835\udd5c\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : BoundedSMul \ud835\udd5c E\ninst\u271d : Fact (1 \u2264 p)\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2016f\u2016 = ENNReal.toReal (snorm (\u2191(toSimpleFunc f)) p \u03bc)\n[PROOFSTEP]\nsimpa [toLp_toSimpleFunc] using norm_toLp (toSimpleFunc f) (simpleFunc.mem\u2112p f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 P f\n[PROOFSTEP]\nsuffices \u2200 f : \u03b1 \u2192\u209b E, \u2200 hf : Mem\u2112p f p \u03bc, P (toLp f hf)\n  by\n  rw [\u2190 toLp_toSimpleFunc f]\n  apply this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf : { x // x \u2208 simpleFunc E p \u03bc }\nthis : \u2200 (f : \u03b1 \u2192\u209b E) (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\n\u22a2 P f\n[PROOFSTEP]\nrw [\u2190 toLp_toSimpleFunc f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf : { x // x \u2208 simpleFunc E p \u03bc }\nthis : \u2200 (f : \u03b1 \u2192\u209b E) (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\n\u22a2 P (toLp (toSimpleFunc f) (_ : Mem\u2112p (\u2191(toSimpleFunc f)) p))\n[PROOFSTEP]\napply this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf : { x // x \u2208 simpleFunc E p \u03bc }\n\u22a2 \u2200 (f : \u03b1 \u2192\u209b E) (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\n[PROOFSTEP]\nclear f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\n\u22a2 \u2200 (f : \u03b1 \u2192\u209b E) (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\n[PROOFSTEP]\napply SimpleFunc.induction\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s)\n    (hf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p),\n    P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf)\n[PROOFSTEP]\nintro c s hs hf\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\n\u22a2 P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf)\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c = 0\n\u22a2 P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf)\n[PROOFSTEP]\nconvert h_ind 0 MeasurableSet.empty (by simp) using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c = 0\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c = 0\n\u22a2 toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf =\n    indicatorConst p (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) 0\n[PROOFSTEP]\next1\n[GOAL]\ncase h.e'_1.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c = 0\n\u22a2 \u2191(toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf) =\n    \u2191(indicatorConst p (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) 0)\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nhf : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : \u00acc = 0\n\u22a2 P (toLp (SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) hf)\n[PROOFSTEP]\nexact h_ind c hs (SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs hf)\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (\u2200 (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)) \u2192\n        (\u2200 (hf : Mem\u2112p (\u2191g) p), P (toLp g hf)) \u2192 \u2200 (hf : Mem\u2112p (\u2191(f + g)) p), P (toLp (f + g) hf)\n[PROOFSTEP]\nintro f g hfg hf hg hfg'\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf g : \u03b1 \u2192\u209b E\nhfg : Disjoint (support \u2191f) (support \u2191g)\nhf : \u2200 (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\nhg : \u2200 (hf : Mem\u2112p (\u2191g) p), P (toLp g hf)\nhfg' : Mem\u2112p (\u2191(f + g)) p\n\u22a2 P (toLp (f + g) hfg')\n[PROOFSTEP]\nobtain \u27e8hf', hg'\u27e9 : Mem\u2112p f p \u03bc \u2227 Mem\u2112p g p \u03bc :=\n  (mem\u2112p_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable).mp hfg'\n[GOAL]\ncase h_add.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_pos : p \u2260 0\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 simpleFunc E p \u03bc } \u2192 Prop\nh_ind : \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P (indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192 P (toLp f hf) \u2192 P (toLp g hg) \u2192 P (toLp f hf + toLp g hg)\nf g : \u03b1 \u2192\u209b E\nhfg : Disjoint (support \u2191f) (support \u2191g)\nhf : \u2200 (hf : Mem\u2112p (\u2191f) p), P (toLp f hf)\nhg : \u2200 (hf : Mem\u2112p (\u2191g) p), P (toLp g hf)\nhfg' : Mem\u2112p (\u2191(f + g)) p\nhf' : Mem\u2112p (\u2191f) p\nhg' : Mem\u2112p (\u2191g) p\n\u22a2 P (toLp (f + g) hfg')\n[PROOFSTEP]\nexact h_add hf' hg' hfg (hf hf') (hg hg')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\n\u22a2 DenseEmbedding Subtype.val\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseEmbedding Subtype.val\n[PROOFSTEP]\napply simpleFunc.uniformEmbedding.denseEmbedding\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 DenseRange Subtype.val\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 f \u2208 closure (Set.range Subtype.val)\n[PROOFSTEP]\nrw [mem_closure_iff_seq_limit]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)\n[PROOFSTEP]\nhave hfi' : Mem\u2112p f p \u03bc := Lp.mem\u2112p f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)\n[PROOFSTEP]\nhaveI : SeparableSpace (range f \u222a {0} : Set E) := (Lp.stronglyMeasurable f).separableSpace_range_union_singleton\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range Subtype.val) \u2227 Tendsto x atTop (\ud835\udcdd f)\n[PROOFSTEP]\nrefine'\n  \u27e8fun n =>\n    toLp (SimpleFunc.approxOn f (Lp.stronglyMeasurable f).measurable (range f \u222a {0}) 0 _ n)\n      (SimpleFunc.mem\u2112p_approxOn_range (Lp.stronglyMeasurable f).measurable hfi' n),\n    fun n => mem_range_self _, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 Tendsto\n    (fun n =>\n      \u2191(toLp (SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n)\n          (_ :\n            Mem\u2112p\n              (\u2191(SimpleFunc.approxOn \u2191\u2191f (_ : Measurable \u2191\u2191f) (Set.range \u2191\u2191f \u222a {0}) 0 (_ : 0 \u2208 Set.range \u2191\u2191f \u222a {0}) n))\n              p)))\n    atTop (\ud835\udcdd f)\n[PROOFSTEP]\nconvert SimpleFunc.tendsto_approxOn_range_Lp hp_ne_top (Lp.stronglyMeasurable f).measurable hfi'\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\ninst\u271d : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nf : { x // x \u2208 Lp E p }\nhfi' : Mem\u2112p (\u2191\u2191f) p\nthis : SeparableSpace \u2191(Set.range \u2191\u2191f \u222a {0})\n\u22a2 f = Mem\u2112p.toLp (\u2191\u2191f) hfi'\n[PROOFSTEP]\nrw [toLp_coeFn f (Lp.mem\u2112p f)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g : { x // x \u2208 simpleFunc G p \u03bc }\n\u22a2 \u2191\u2191\u2191f \u2264\u1d50[\u03bc] \u2191\u2191\u2191g \u2194 f \u2264 g\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_le_coe, \u2190 Lp.coeFn_le]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\n\u22a2 CovariantClass { x // x \u2208 simpleFunc G p \u03bc } { x // x \u2208 simpleFunc G p \u03bc } (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nrefine' \u27e8fun f g\u2081 g\u2082 hg\u2081\u2082 => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : g\u2081 \u2264 g\u2082\n\u22a2 f + g\u2081 \u2264 f + g\u2082\n[PROOFSTEP]\nrw [\u2190 Lp.simpleFunc.coeFn_le] at hg\u2081\u2082 \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : \u2191\u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\u2082\n\u22a2 \u2191\u2191\u2191(f + g\u2081) \u2264\u1d50[\u03bc] \u2191\u2191\u2191(f + g\u2082)\n[PROOFSTEP]\nhave h_add_1 : ((f + g\u2081 : Lp.simpleFunc G p \u03bc) : \u03b1 \u2192 G) =\u1d50[\u03bc] (f : \u03b1 \u2192 G) + g\u2081 := Lp.coeFn_add _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : \u2191\u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\u2082\nh_add_1 : \u2191\u2191\u2191(f + g\u2081) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081\n\u22a2 \u2191\u2191\u2191(f + g\u2081) \u2264\u1d50[\u03bc] \u2191\u2191\u2191(f + g\u2082)\n[PROOFSTEP]\nhave h_add_2 : ((f + g\u2082 : Lp.simpleFunc G p \u03bc) : \u03b1 \u2192 G) =\u1d50[\u03bc] (f : \u03b1 \u2192 G) + g\u2082 := Lp.coeFn_add _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : \u2191\u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\u2082\nh_add_1 : \u2191\u2191\u2191(f + g\u2081) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081\nh_add_2 : \u2191\u2191\u2191(f + g\u2082) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2082\n\u22a2 \u2191\u2191\u2191(f + g\u2081) \u2264\u1d50[\u03bc] \u2191\u2191\u2191(f + g\u2082)\n[PROOFSTEP]\nfilter_upwards [h_add_1, h_add_2, hg\u2081\u2082] with _ h1 h2 h3\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : \u2191\u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\u2082\nh_add_1 : \u2191\u2191\u2191(f + g\u2081) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081\nh_add_2 : \u2191\u2191\u2191(f + g\u2082) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2082\na\u271d : \u03b1\nh1 : \u2191\u2191\u2191(f + g\u2081) a\u271d = (\u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081) a\u271d\nh2 : \u2191\u2191\u2191(f + g\u2082) a\u271d = (\u2191\u2191\u2191f + \u2191\u2191\u2191g\u2082) a\u271d\nh3 : \u2191\u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191\u2191g\u2082 a\u271d\n\u22a2 \u2191\u2191\u2191(f + g\u2081) a\u271d \u2264 \u2191\u2191\u2191(f + g\u2082) a\u271d\n[PROOFSTEP]\nrw [h1, h2, Pi.add_apply, Pi.add_apply]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf g\u2081 g\u2082 : { x // x \u2208 simpleFunc G p \u03bc }\nhg\u2081\u2082 : \u2191\u2191\u2191g\u2081 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\u2082\nh_add_1 : \u2191\u2191\u2191(f + g\u2081) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081\nh_add_2 : \u2191\u2191\u2191(f + g\u2082) =\u1d50[\u03bc] \u2191\u2191\u2191f + \u2191\u2191\u2191g\u2082\na\u271d : \u03b1\nh1 : \u2191\u2191\u2191(f + g\u2081) a\u271d = (\u2191\u2191\u2191f + \u2191\u2191\u2191g\u2081) a\u271d\nh2 : \u2191\u2191\u2191(f + g\u2082) a\u271d = (\u2191\u2191\u2191f + \u2191\u2191\u2191g\u2082) a\u271d\nh3 : \u2191\u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191\u2191g\u2082 a\u271d\n\u22a2 \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g\u2081 a\u271d \u2264 \u2191\u2191\u2191f a\u271d + \u2191\u2191\u2191g\u2082 a\u271d\n[PROOFSTEP]\nexact add_le_add le_rfl h3\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf : { x // x \u2208 simpleFunc G p \u03bc }\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191f \u2194 0 \u2264 f\n[PROOFSTEP]\nrw [\u2190 Subtype.coe_le_coe, Lp.coeFn_nonneg, AddSubmonoid.coe_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nf : { x // x \u2208 simpleFunc G p \u03bc }\nhf : 0 \u2264 f\n\u22a2 \u2203 f', 0 \u2264 f' \u2227 \u2191\u2191\u2191f =\u1d50[\u03bc] \u2191f'\n[PROOFSTEP]\nrcases f with \u27e8\u27e8f, hp\u27e9, g, (rfl : _ = f)\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\ng : \u03b1 \u2192\u209b G\nhp : AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc) \u2208 Lp G p\nhf :\n  0 \u2264\n    { val := { val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp },\n      property :=\n        (_ :\n          \u2203 s,\n            AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) =\n              \u2191{ val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp }) }\n\u22a2 \u2203 f',\n    0 \u2264 f' \u2227\n      \u2191\u2191\u2191{ val := { val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp },\n                property :=\n                  (_ :\n                    \u2203 s,\n                      AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) =\n                        \u2191{ val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp }) } =\u1d50[\u03bc]\n        \u2191f'\n[PROOFSTEP]\nchange 0 \u2264\u1d50[\u03bc] g at hf \n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\ng : \u03b1 \u2192\u209b G\nhp : AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc) \u2208 Lp G p\nhf : 0 \u2264\u1d50[\u03bc] \u2191g\n\u22a2 \u2203 f',\n    0 \u2264 f' \u2227\n      \u2191\u2191\u2191{ val := { val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp },\n                property :=\n                  (_ :\n                    \u2203 s,\n                      AEEqFun.mk \u2191s (_ : AEStronglyMeasurable (\u2191s) \u03bc) =\n                        \u2191{ val := AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc), property := hp }) } =\u1d50[\u03bc]\n        \u2191f'\n[PROOFSTEP]\nrefine \u27e8g \u2294 0, le_sup_right, (AEEqFun.coeFn_mk _ _).trans ?_\u27e9\n[GOAL]\ncase mk.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\ng : \u03b1 \u2192\u209b G\nhp : AEEqFun.mk \u2191g (_ : AEStronglyMeasurable (\u2191g) \u03bc) \u2208 Lp G p\nhf : 0 \u2264\u1d50[\u03bc] \u2191g\n\u22a2 \u2191g =\u1d50[\u03bc] \u2191(g \u2294 0)\n[PROOFSTEP]\nexact hf.mono fun x hx \u21a6 (sup_of_le_left hx).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\n\u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G))\n[PROOFSTEP]\nborelize G\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 g \u2208 closure (Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G))\n[PROOFSTEP]\nrw [mem_closure_iff_seq_limit]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave hg_mem\u2112p : Mem\u2112p (g : \u03b1 \u2192 G) p \u03bc := Lp.mem\u2112p (g : Lp G p \u03bc)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave zero_mem : (0 : G) \u2208 (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 {y | 0 \u2264 y} := by\n  simp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff, mem_setOf_eq, le_refl,\n    and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\n\u22a2 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n[PROOFSTEP]\nsimp only [union_singleton, mem_inter_iff, mem_insert_iff, eq_self_iff_true, true_or_iff, mem_setOf_eq, le_refl,\n  and_self_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave : SeparableSpace ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 {y | 0 \u2264 y} : Set G) :=\n  by\n  apply IsSeparable.separableSpace\n  apply IsSeparable.mono _ (Set.inter_subset_left _ _)\n  exact (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union (finite_singleton _).isSeparable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n[PROOFSTEP]\napply IsSeparable.separableSpace\n[GOAL]\ncase hs\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n[PROOFSTEP]\napply IsSeparable.mono _ (Set.inter_subset_left _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n\u22a2 IsSeparable (Set.range \u2191\u2191\u2191g \u222a {0})\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable (g : Lp G p \u03bc)).isSeparable_range.union (finite_singleton _).isSeparable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave g_meas : Measurable (g : \u03b1 \u2192 G) := (Lp.stronglyMeasurable (g : Lp G p \u03bc)).measurable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nlet x n := SimpleFunc.approxOn g g_meas ((range (g : \u03b1 \u2192 G) \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave hx_nonneg : \u2200 n, 0 \u2264 x n := by\n  intro n a\n  change x n a \u2208 {y : G | 0 \u2264 y}\n  have A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y} := inter_subset_right _ _\n  apply A\n  exact SimpleFunc.approxOn_mem g_meas _ n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\n\u22a2 \u2200 (n : \u2115), 0 \u2264 x n\n[PROOFSTEP]\nintro n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u21910 a \u2264 \u2191(x n) a\n[PROOFSTEP]\nchange x n a \u2208 {y : G | 0 \u2264 y}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}\n[PROOFSTEP]\nhave A : (range (g : \u03b1 \u2192 G) \u222a {0} : Set G) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y} := inter_subset_right _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 {y | 0 \u2264 y}\n[PROOFSTEP]\napply A\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nn : \u2115\na : \u03b1\nA : (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y} \u2286 {y | 0 \u2264 y}\n\u22a2 \u2191(x n) a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n[PROOFSTEP]\nexact SimpleFunc.approxOn_mem g_meas _ n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave hx_mem\u2112p : \u2200 n, Mem\u2112p (x n) p \u03bc := SimpleFunc.mem\u2112p_approxOn _ hg_mem\u2112p _ \u27e8aestronglyMeasurable_const, by simp\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\n\u22a2 snorm (fun x => 0) p \u03bc < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave h_toLp := fun n => Mem\u2112p.coeFn_toLp (hx_mem\u2112p n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave hx_nonneg_Lp : \u2200 n, 0 \u2264 toLp (x n) (hx_mem\u2112p n) := by\n  intro n\n  rw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]\n  filter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp\n  rw [ha0, ha_toLp]\n  exact hx_nonneg n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\n\u22a2 \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n[PROOFSTEP]\nrw [\u2190 Lp.simpleFunc.coeFn_le, Lp.simpleFunc.toLp_eq_toLp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\n\u22a2 \u2191\u2191\u21910 \u2264\u1d50[\u03bc] \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p))\n[PROOFSTEP]\nfilter_upwards [Lp.simpleFunc.coeFn_zero p \u03bc G, h_toLp n] with a ha0 ha_toLp\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp :\n  \u2191\u2191(Mem\u2112p.toLp \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n)\n            (_ : Mem\u2112p (\u2191(x n)) p))\n      a =\n    \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n) a\n\u22a2 \u2191\u2191\u21910 a \u2264\n    \u2191\u2191(Mem\u2112p.toLp \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n)\n            (_ : Mem\u2112p (\u2191(x n)) p))\n      a\n[PROOFSTEP]\nrw [ha0, ha_toLp]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nn : \u2115\na : \u03b1\nha0 : \u2191\u2191\u21910 a = OfNat.ofNat 0 a\nha_toLp :\n  \u2191\u2191(Mem\u2112p.toLp \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n)\n            (_ : Mem\u2112p (\u2191(x n)) p))\n      a =\n    \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191(SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n) a\n[PROOFSTEP]\nexact hx_nonneg n a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nhave hx_tendsto : Tendsto (fun n : \u2115 => snorm ((x n : \u03b1 \u2192 G) - (g : \u03b1 \u2192 G)) p \u03bc) atTop (\ud835\udcdd 0) :=\n  by\n  apply SimpleFunc.tendsto_approxOn_Lp_snorm g_meas zero_mem hp_ne_top\n  \u00b7 have hg_nonneg : (0 : \u03b1 \u2192 G) \u2264\u1d50[\u03bc] g := (Lp.coeFn_nonneg _).mpr g.2\n    refine' hg_nonneg.mono fun a ha => subset_closure _\n    simpa using ha\n  \u00b7 simp_rw [sub_zero]; exact hg_mem\u2112p.snorm_lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\napply SimpleFunc.tendsto_approxOn_Lp_snorm g_meas zero_mem hp_ne_top\n[GOAL]\ncase h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n[PROOFSTEP]\nhave hg_nonneg : (0 : \u03b1 \u2192 G) \u2264\u1d50[\u03bc] g := (Lp.coeFn_nonneg _).mpr g.2\n[GOAL]\ncase h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2191\u2191\u2191g x \u2208 closure ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\n[PROOFSTEP]\nrefine' hg_nonneg.mono fun a ha => subset_closure _\n[GOAL]\ncase h\u03bc\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhg_nonneg : 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191g\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191g a\n\u22a2 \u2191\u2191\u2191g a \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\n[PROOFSTEP]\nsimpa using ha\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x - 0) p \u03bc < \u22a4\n[PROOFSTEP]\nsimp_rw [sub_zero]\n[GOAL]\ncase hi\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\n\u22a2 snorm (fun x => \u2191\u2191\u2191g x) p \u03bc < \u22a4\n[PROOFSTEP]\nexact hg_mem\u2112p.snorm_lt_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 \u2203 x, (\u2200 (n : \u2115), x n \u2208 Set.range (coeSimpleFuncNonnegToLpNonneg p \u03bc G)) \u2227 Tendsto x atTop (\ud835\udcdd g)\n[PROOFSTEP]\nrefine'\n  \u27e8fun n => (coeSimpleFuncNonnegToLpNonneg p \u03bc G) \u27e8toLp (x n) (hx_mem\u2112p n), hx_nonneg_Lp n\u27e9, fun n => mem_range_self _,\n    _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun n =>\n      coeSimpleFuncNonnegToLpNonneg p \u03bc G\n        { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })\n    atTop (\ud835\udcdd g)\n[PROOFSTEP]\nsuffices Tendsto (fun n : \u2115 => (toLp (x n) (hx_mem\u2112p n) : Lp G p \u03bc)) atTop (\ud835\udcdd (g : Lp G p \u03bc))\n  by\n  rw [tendsto_iff_dist_tendsto_zero] at this \u22a2\n  simp_rw [Subtype.dist_eq]\n  exact this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g)\n\u22a2 Tendsto\n    (fun n =>\n      coeSimpleFuncNonnegToLpNonneg p \u03bc G\n        { val := toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p), property := (_ : 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) })\n    atTop (\ud835\udcdd g)\n[PROOFSTEP]\nrw [tendsto_iff_dist_tendsto_zero] at this \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        (coeSimpleFuncNonnegToLpNonneg p \u03bc G\n          { val := toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p), property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        g)\n    atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [Subtype.dist_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b2 : MeasurableSpace G := borel G\nthis\u271d\u00b9 : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis\u271d : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nthis : Tendsto (fun b => dist \u2191(toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) \u2191g) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto\n    (fun b =>\n      dist\n        \u2191(coeSimpleFuncNonnegToLpNonneg p \u03bc G\n            {\n              val :=\n                toLp (SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem b)\n                  (_ : Mem\u2112p (\u2191(x b)) p),\n              property := (_ : 0 \u2264 toLp (x b) (_ : Mem\u2112p (\u2191(x b)) p)) })\n        \u2191g)\n    atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => \u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p))) atTop (\ud835\udcdd \u2191g)\n[PROOFSTEP]\nrw [Lp.tendsto_Lp_iff_tendsto_\u2112p']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun n => snorm (\u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine Filter.Tendsto.congr (fun n => snorm_congr_ae (EventuallyEq.sub ?_ ?_)) hx_tendsto\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191(x n) x_1) =\u1d50[\u03bc] fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1\n[PROOFSTEP]\nsymm\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191\u2191(toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1\n[PROOFSTEP]\nrw [Lp.simpleFunc.toLp_eq_toLp]\n[GOAL]\ncase refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x_1 => \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) x_1) =\u1d50[\u03bc] fun x_1 => \u2191(x n) x_1\n[PROOFSTEP]\nexact h_toLp n\n[GOAL]\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b3 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedAddCommGroup F\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nG : Type u_7\ninst\u271d : NormedLatticeAddCommGroup G\nhp : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\ng : { g // 0 \u2264 g }\nthis\u271d\u00b9 : MeasurableSpace G := borel G\nthis\u271d : BorelSpace G\nhg_mem\u2112p : Mem\u2112p (\u2191\u2191\u2191g) p\nzero_mem : 0 \u2208 (Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}\nthis : SeparableSpace \u2191((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y})\ng_meas : Measurable \u2191\u2191\u2191g\nx : \u2115 \u2192 \u03b1 \u2192\u209b G := fun n => SimpleFunc.approxOn (\u2191\u2191\u2191g) g_meas ((Set.range \u2191\u2191\u2191g \u222a {0}) \u2229 {y | 0 \u2264 y}) 0 zero_mem n\nhx_nonneg : \u2200 (n : \u2115), 0 \u2264 x n\nhx_mem\u2112p : \u2200 (n : \u2115), Mem\u2112p (\u2191(x n)) p\nh_toLp : \u2200 (n : \u2115), \u2191\u2191(Mem\u2112p.toLp \u2191(x n) (_ : Mem\u2112p (\u2191(x n)) p)) =\u1d50[\u03bc] \u2191(x n)\nhx_nonneg_Lp : \u2200 (n : \u2115), 0 \u2264 toLp (x n) (_ : Mem\u2112p (\u2191(x n)) p)\nhx_tendsto : Tendsto (fun n => snorm (\u2191(x n) - \u2191\u2191\u2191g) p \u03bc) atTop (\ud835\udcdd 0)\nn : \u2115\n\u22a2 (fun x => \u2191\u2191\u2191g x) =\u1d50[\u03bc] fun x => \u2191\u2191\u2191g x\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P f}\n\u22a2 \u2200 (f : { x // x \u2208 Lp E p }), P f\n[PROOFSTEP]\nrefine' fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (a : { x // x \u2208 \u2191(simpleFunc E p \u03bc) }), P \u2191a\n[PROOFSTEP]\nrefine' Lp.simpleFunc.induction (\u03b1 := \u03b1) (E := E) (lt_of_lt_of_le zero_lt_one _i.elim).ne' hp_ne_top _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\n[PROOFSTEP]\nexact fun c s => h_ind c\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : { x // x \u2208 Lp E p } \u2192 Prop\nh_ind :\n  \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : \u2191\u2191\u03bc s < \u22a4), P \u2191(simpleFunc.indicatorConst p hs (_ : \u2191\u2191\u03bc s \u2260 \u22a4) c)\nh_add :\n  \u2200 \u2983f g : \u03b1 \u2192 E\u2984 (hf : Mem\u2112p f p) (hg : Mem\u2112p g p),\n    Disjoint (support f) (support g) \u2192 P (Mem\u2112p.toLp f hf) \u2192 P (Mem\u2112p.toLp g hg) \u2192 P (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)\nh_closed : IsClosed {f | P f}\nf : { x // x \u2208 Lp E p }\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984 (hf : Mem\u2112p (\u2191f) p) (hg : Mem\u2112p (\u2191g) p),\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      P \u2191(simpleFunc.toLp f hf) \u2192 P \u2191(simpleFunc.toLp g hg) \u2192 P \u2191(simpleFunc.toLp f hf + simpleFunc.toLp g hg)\n[PROOFSTEP]\nexact fun f g hf hg => h_add hf hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f p \u2192 P f\n[PROOFSTEP]\nhave : \u2200 f : SimpleFunc \u03b1 E, Mem\u2112p f p \u03bc \u2192 P f :=\n  by\n  apply SimpleFunc.induction\n  \u00b7 intro c s hs h\n    by_cases hc : c = 0\n    \u00b7 subst hc; convert h_ind 0 MeasurableSet.empty (by simp) using 1; ext; simp [const]\n    have hp_pos : p \u2260 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne'\n    exact h_ind c hs (SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs h)\n  \u00b7 intro f g hfg hf hg int_fg\n    rw [SimpleFunc.coe_add, mem\u2112p_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg \n    refine' h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\n[PROOFSTEP]\napply SimpleFunc.induction\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s),\n    Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u2192\n      P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nintro c s hs h\n[GOAL]\ncase h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\n\u22a2 P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c = 0\n\u22a2 P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nsubst hc\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nconvert h_ind 0 MeasurableSet.empty (by simp) using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) = Set.indicator \u2205 fun x => 0\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_1.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\nx\u271d : \u03b1\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) x\u271d = Set.indicator \u2205 (fun x => 0) x\u271d\n[PROOFSTEP]\nsimp [const]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : \u00acc = 0\n\u22a2 P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nhave hp_pos : p \u2260 0 := (lt_of_lt_of_le zero_lt_one _i.elim).ne'\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\nh : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : \u00acc = 0\nhp_pos : p \u2260 0\n\u22a2 P \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))\n[PROOFSTEP]\nexact h_ind c hs (SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs h)\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192 (Mem\u2112p (\u2191f) p \u2192 P \u2191f) \u2192 (Mem\u2112p (\u2191g) p \u2192 P \u2191g) \u2192 Mem\u2112p (\u2191(f + g)) p \u2192 P \u2191(f + g)\n[PROOFSTEP]\nintro f g hfg hf hg int_fg\n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf g : \u03b1 \u2192\u209b E\nhfg : Disjoint (support \u2191f) (support \u2191g)\nhf : Mem\u2112p (\u2191f) p \u2192 P \u2191f\nhg : Mem\u2112p (\u2191g) p \u2192 P \u2191g\nint_fg : Mem\u2112p (\u2191(f + g)) p\n\u22a2 P \u2191(f + g)\n[PROOFSTEP]\nrw [SimpleFunc.coe_add, mem\u2112p_add_of_disjoint hfg f.stronglyMeasurable g.stronglyMeasurable] at int_fg \n[GOAL]\ncase h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nf g : \u03b1 \u2192\u209b E\nhfg : Disjoint (support \u2191f) (support \u2191g)\nhf : Mem\u2112p (\u2191f) p \u2192 P \u2191f\nhg : Mem\u2112p (\u2191g) p \u2192 P \u2191g\nint_fg : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191g) p\n\u22a2 P \u2191(f + g)\n[PROOFSTEP]\nrefine' h_add hfg int_fg.1 int_fg.2 (hf int_fg.1) (hg int_fg.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nthis : \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f p \u2192 P f\n[PROOFSTEP]\nhave : \u2200 f : Lp.simpleFunc E p \u03bc, P f := by\n  intro f\n  exact\n    h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.mem\u2112p f)\n      (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.mem\u2112p f))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nthis : \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\n\u22a2 \u2200 (f : { x // x \u2208 Lp.simpleFunc E p \u03bc }), P \u2191\u2191\u2191f\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nthis : \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\nf : { x // x \u2208 Lp.simpleFunc E p \u03bc }\n\u22a2 P \u2191\u2191\u2191f\n[PROOFSTEP]\nexact\n  h_ae (Lp.simpleFunc.toSimpleFunc_eq_toFun f) (Lp.simpleFunc.mem\u2112p f)\n    (this (Lp.simpleFunc.toSimpleFunc f) (Lp.simpleFunc.mem\u2112p f))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nthis\u271d : \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\nthis : \u2200 (f : { x // x \u2208 Lp.simpleFunc E p \u03bc }), P \u2191\u2191\u2191f\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f p \u2192 P f\n[PROOFSTEP]\nhave : \u2200 f : Lp E p \u03bc, P f := fun f => (Lp.simpleFunc.denseRange hp_ne_top).induction_on f h_closed this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\n_i : Fact (1 \u2264 p)\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (Set.indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f p \u2192 Mem\u2112p g p \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f p \u2192 P f \u2192 P g\nthis\u271d\u00b9 : \u2200 (f : \u03b1 \u2192\u209b E), Mem\u2112p (\u2191f) p \u2192 P \u2191f\nthis\u271d : \u2200 (f : { x // x \u2208 Lp.simpleFunc E p \u03bc }), P \u2191\u2191\u2191f\nthis : \u2200 (f : { x // x \u2208 Lp E p }), P \u2191\u2191f\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f p \u2192 P f\n[PROOFSTEP]\nexact fun f hf => h_ae hf.coeFn_toLp (Lp.mem\u2112p _) (this (hf.toLp f))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp_pos)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top]) h\u03b5 with \u27e8g, _, Pg\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp only [measure_empty, WithTop.zero_lt_top]\n[GOAL]\ncase inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\ng : \u03b1 \u2192 E\nleft\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) 0 \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nexact \u27e8g, by simp only [snorm_exponent_zero, zero_le'], Pg\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_ne_top : 0 \u2260 \u22a4\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) 0 \u03bc \u2264 \u03b5 \u2227 P g\nhf : Mem\u2112p f 0\ng : \u03b1 \u2192 E\nleft\u271d : snorm (g - Set.indicator \u2205 fun x => 0) 0 \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 snorm (f - g) 0 \u03bc \u2264 \u03b5\n[PROOFSTEP]\nsimp only [snorm_exponent_zero, zero_le']\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nsuffices H : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e) (h\u03b4 : \u03b4 \u2260 0), Mem\u2112p f' p \u03bc \u2192 \u2203 g, snorm (\u21d1f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nobtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p h\u03b5\n[GOAL]\ncase inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases hf.exists_simpleFunc_snorm_sub_lt hp_ne_top \u03b7pos.ne' with \u27e8f', hf', f'_mem\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases H f' \u03b7 \u03b7pos.ne' f'_mem with \u27e8g, hg, Pg\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (f - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrefine' \u27e8g, _, Pg\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 snorm (f - g) p \u03bc \u2264 \u03b5\n[PROOFSTEP]\nconvert\n  (h\u03b7 _ _ (hf.aestronglyMeasurable.sub f'.aestronglyMeasurable) (f'.aestronglyMeasurable.sub (h2P g Pg)) hf'.le\n      hg).le using\n  2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nH : \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b5\nf' : \u03b1 \u2192\u209b E\nhf' : snorm (f - \u2191f') p \u03bc < \u03b7\nf'_mem : Mem\u2112p (\u2191f') p\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f' - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 f - g = f - \u2191f' + (\u2191f' - g)\n[PROOFSTEP]\nsimp only [sub_add_sub_cancel]\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (f' : \u03b1 \u2192\u209b E) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\napply SimpleFunc.induction\n[GOAL]\ncase H.h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 (c : E) {s : Set \u03b1} (hs : MeasurableSet s) (\u03b4 : \u211d\u22650\u221e),\n    \u03b4 \u2260 0 \u2192\n      Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p \u2192\n        \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nintro c s hs \u03b5 \u03b5pos Hs\n[GOAL]\ncase H.h_ind\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases eq_or_ne c 0 with (rfl | hc)\n[GOAL]\ncase H.h_ind.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases h0P (0 : E) MeasurableSet.empty (by simp only [measure_empty, WithTop.zero_lt_top]) \u03b5pos with \u27e8g, hg, Pg\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\n\u22a2 \u2191\u2191\u03bc \u2205 < \u22a4\n[PROOFSTEP]\nsimp only [measure_empty, WithTop.zero_lt_top]\n[GOAL]\ncase H.h_ind.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator \u2205 fun x => 0) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrw [\u2190 snorm_neg, neg_sub] at hg \n[GOAL]\ncase H.h_ind.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrefine' \u27e8g, _, Pg\u27e9\n[GOAL]\ncase H.h_ind.inl.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5\n[PROOFSTEP]\nconvert hg\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) = Set.indicator \u2205 fun x => 0\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_3.h.e'_5.h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0))) p\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator \u2205 fun x => 0) - g) p \u03bc \u2264 \u03b5\nPg : P g\nx : \u03b1\n\u22a2 \u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 0) (SimpleFunc.const \u03b1 0)) x = Set.indicator \u2205 (fun x => 0) x\n[PROOFSTEP]\nsimp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_zero, piecewise_eq_indicator,\n  indicator_zero', Pi.zero_apply, indicator_zero]\n[GOAL]\ncase H.h_ind.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nhave : \u03bc s < \u221e := SimpleFunc.measure_lt_top_of_mem\u2112p_indicator hp_pos hp_ne_top hc hs Hs\n[GOAL]\ncase H.h_ind.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrcases h0P c hs this \u03b5pos with \u27e8g, hg, Pg\u27e9\n[GOAL]\ncase H.h_ind.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nrw [\u2190 snorm_neg, neg_sub] at hg \n[GOAL]\ncase H.h_ind.inr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5\u271d : \u211d\u22650\u221e\nh\u03b5 : \u03b5\u271d \u2260 0\nhp_pos : p \u2260 0\nc : E\ns : Set \u03b1\nhs : MeasurableSet s\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : \u03b5 \u2260 0\nHs : Mem\u2112p (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0))) p\nhc : c \u2260 0\nthis : \u2191\u2191\u03bc s < \u22a4\ng : \u03b1 \u2192 E\nhg : snorm ((Set.indicator s fun x => c) - g) p \u03bc \u2264 \u03b5\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(SimpleFunc.piecewise s hs (SimpleFunc.const \u03b1 c) (SimpleFunc.const \u03b1 0)) - g) p \u03bc \u2264 \u03b5 \u2227 P g\n[PROOFSTEP]\nexact \u27e8g, hg, Pg\u27e9\n[GOAL]\ncase H.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf : \u03b1 \u2192 E\nhf : Mem\u2112p f p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192\u209b E\u2984,\n    Disjoint (support \u2191f) (support \u2191g) \u2192\n      (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g) \u2192\n        (\u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191g) p \u2192 \u2203 g_1, snorm (\u2191g - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1) \u2192\n          \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191(f + g)) p \u2192 \u2203 g_1, snorm (\u2191(f + g) - g_1) p \u03bc \u2264 \u03b4 \u2227 P g_1\n[PROOFSTEP]\nintro f f' hff' hf hf' \u03b4 \u03b4pos int_ff'\n[GOAL]\ncase H.h_add\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nobtain \u27e8\u03b7, \u03b7pos, h\u03b7\u27e9 := exists_Lp_half E \u03bc p \u03b4pos\n[GOAL]\ncase H.h_add.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191(f + f')) p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nrw [SimpleFunc.coe_add, mem\u2112p_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable] at int_ff' \n[GOAL]\ncase H.h_add.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nrcases hf \u03b7 \u03b7pos.ne' int_ff'.1 with \u27e8g, hg, Pg\u27e9\n[GOAL]\ncase H.h_add.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nrcases hf' \u03b7 \u03b7pos.ne' int_ff'.2 with \u27e8g', hg', Pg'\u27e9\n[GOAL]\ncase H.h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2203 g, snorm (\u2191(f + f') - g) p \u03bc \u2264 \u03b4 \u2227 P g\n[PROOFSTEP]\nrefine' \u27e8g + g', _, h1P g g' Pg Pg'\u27e9\n[GOAL]\ncase H.h_add.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 snorm (\u2191(f + f') - (g + g')) p \u03bc \u2264 \u03b4\n[PROOFSTEP]\nconvert (h\u03b7 _ _ (f.aestronglyMeasurable.sub (h2P g Pg)) (f'.aestronglyMeasurable.sub (h2P g' Pg')) hg hg').le using 2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191(f + f') - (g + g') = \u2191f - g + (\u2191f' - g')\n[PROOFSTEP]\nrw [SimpleFunc.coe_add]\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191f + \u2191f' - (g + g') = \u2191f - g + (\u2191f' - g')\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d\u00b9 : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nhp_ne_top : p \u2260 \u22a4\nP : (\u03b1 \u2192 E) \u2192 Prop\nh0P :\n  \u2200 (c : E) \u2983s : Set \u03b1\u2984,\n    MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 \u2200 {\u03b5 : \u211d\u22650\u221e}, \u03b5 \u2260 0 \u2192 \u2203 g, snorm (g - Set.indicator s fun x => c) p \u03bc \u2264 \u03b5 \u2227 P g\nh1P : \u2200 (f g : \u03b1 \u2192 E), P f \u2192 P g \u2192 P (f + g)\nh2P : \u2200 (f : \u03b1 \u2192 E), P f \u2192 AEStronglyMeasurable f \u03bc\nf\u271d : \u03b1 \u2192 E\nhf\u271d : Mem\u2112p f\u271d p\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhp_pos : p \u2260 0\nf f' : \u03b1 \u2192\u209b E\nhff' : Disjoint (support \u2191f) (support \u2191f')\nhf : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f) p \u2192 \u2203 g, snorm (\u2191f - g) p \u03bc \u2264 \u03b4 \u2227 P g\nhf' : \u2200 (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 Mem\u2112p (\u2191f') p \u2192 \u2203 g, snorm (\u2191f' - g) p \u03bc \u2264 \u03b4 \u2227 P g\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nint_ff' : Mem\u2112p (\u2191f) p \u2227 Mem\u2112p (\u2191f') p\n\u03b7 : \u211d\u22650\u221e\n\u03b7pos : 0 < \u03b7\nh\u03b7 :\n  \u2200 (f g : \u03b1 \u2192 E),\n    AEStronglyMeasurable f \u03bc \u2192 AEStronglyMeasurable g \u03bc \u2192 snorm f p \u03bc \u2264 \u03b7 \u2192 snorm g p \u03bc \u2264 \u03b7 \u2192 snorm (f + g) p \u03bc < \u03b4\ng : \u03b1 \u2192 E\nhg : snorm (\u2191f - g) p \u03bc \u2264 \u03b7\nPg : P g\ng' : \u03b1 \u2192 E\nhg' : snorm (\u2191f' - g') p \u03bc \u2264 \u03b7\nPg' : P g'\n\u22a2 \u2191f + \u2191f' - (g + g') = \u2191f - g + (\u2191f' - g')\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp.simpleFunc E 1 \u03bc }\n\u22a2 Integrable \u2191(Lp.simpleFunc.toSimpleFunc f)\n[PROOFSTEP]\nrw [\u2190 mem\u2112p_one_iff_integrable]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf\u271d : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nf : { x // x \u2208 Lp.simpleFunc E 1 \u03bc }\n\u22a2 Mem\u2112p (\u2191(Lp.simpleFunc.toSimpleFunc f)) 1\n[PROOFSTEP]\nexact Lp.simpleFunc.mem\u2112p f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c)\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Integrable f \u2192 Integrable g \u2192 P f \u2192 P g \u2192 P (f + g)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Integrable f \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Integrable f \u2192 P f\n[PROOFSTEP]\nsimp only [\u2190 mem\u2112p_one_iff_integrable] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nE : Type u_4\nF : Type u_5\n\ud835\udd5c : Type u_6\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : NormedAddCommGroup E\nf : \u03b1 \u2192 E\np : \u211d\u22650\u221e\n\u03bc : Measure \u03b1\nP : (\u03b1 \u2192 E) \u2192 Prop\nh_ind : \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191\u03bc s < \u22a4 \u2192 P (indicator s fun x => c)\nh_closed : IsClosed {f | P \u2191\u2191f}\nh_add : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Mem\u2112p f 1 \u2192 Mem\u2112p g 1 \u2192 P f \u2192 P g \u2192 P (f + g)\nh_ae : \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Mem\u2112p f 1 \u2192 P f \u2192 P g\n\u22a2 \u2200 \u2983f : \u03b1 \u2192 E\u2984, Mem\u2112p f 1 \u2192 P f\n[PROOFSTEP]\nexact Mem\u2112p.induction one_ne_top (P := P) h_ind h_add h_closed h_ae\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.SimpleFuncDenseLp", "llama_tokens": 127915, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149978955811, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.527877645070725}}
{"text": "[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\n\u22a2 Basis (Fin (natDegree (minpoly K x))) K { x_1 // x_1 \u2208 adjoin K {x} }\n[PROOFSTEP]\nhave hST : Function.Injective (algebraMap (adjoin K ({ x } : Set S)) S) := Subtype.coe_injective\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\n\u22a2 Basis (Fin (natDegree (minpoly K x))) K { x_1 // x_1 \u2208 adjoin K {x} }\n[PROOFSTEP]\nhave hx' : IsIntegral K (\u27e8x, subset_adjoin (Set.mem_singleton x)\u27e9 : adjoin K ({ x } : Set S)) :=\n  by\n  apply (isIntegral_algebraMap_iff hST).mp\n  convert hx\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\n\u22a2 IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\n[PROOFSTEP]\napply (isIntegral_algebraMap_iff hST).mp\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\n\u22a2 IsIntegral K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\n[PROOFSTEP]\nconvert hx\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\n\u22a2 Basis (Fin (natDegree (minpoly K x))) K { x_1 // x_1 \u2208 adjoin K {x} }\n[PROOFSTEP]\nhave minpoly_eq := minpoly.eq_of_algebraMap_eq hST hx' rfl\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\n\u22a2 Basis (Fin (natDegree (minpoly K x))) K { x_1 // x_1 \u2208 adjoin K {x} }\n[PROOFSTEP]\napply\n  @Basis.mk (Fin (minpoly K x).natDegree) _ (adjoin K { x }) fun i => \u27e8x, subset_adjoin (Set.mem_singleton x)\u27e9 ^ (i : \u2115)\n[GOAL]\ncase hli\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\n\u22a2 LinearIndependent K fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i\n[PROOFSTEP]\nhave : LinearIndependent K _ := linearIndependent_pow (\u27e8x, self_mem_adjoin_singleton _ _\u27e9 : adjoin K { x })\n[GOAL]\ncase hli\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\nthis : LinearIndependent K fun i => { val := x, property := (_ : x \u2208 adjoin K {x}) } ^ \u2191i\n\u22a2 LinearIndependent K fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i\n[PROOFSTEP]\nrwa [minpoly_eq] at this \n[GOAL]\ncase hsp\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\n\u22a2 \u22a4 \u2264 Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n[PROOFSTEP]\nrintro \u27e8y, hy\u27e9 _\n[GOAL]\ncase hsp.mk\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\ny : S\nhy : y \u2208 adjoin K {x}\na\u271d : { val := y, property := hy } \u2208 \u22a4\n\u22a2 { val := y, property := hy } \u2208\n    Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n[PROOFSTEP]\nhave := hx'.mem_span_pow (y := \u27e8y, hy\u27e9)\n[GOAL]\ncase hsp.mk\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\ny : S\nhy : y \u2208 adjoin K {x}\na\u271d : { val := y, property := hy } \u2208 \u22a4\nthis :\n  (\u2203 f, { val := y, property := hy } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f) \u2192\n    { val := y, property := hy } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 { val := y, property := hy } \u2208\n    Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n[PROOFSTEP]\nrw [minpoly_eq] at this \n[GOAL]\ncase hsp.mk\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\ny : S\nhy : y \u2208 adjoin K {x}\na\u271d : { val := y, property := hy } \u2208 \u22a4\nthis :\n  (\u2203 f, { val := y, property := hy } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f) \u2192\n    { val := y, property := hy } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 { val := y, property := hy } \u2208\n    Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n[PROOFSTEP]\napply this\n[GOAL]\ncase hsp.mk\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\ny : S\nhy : y \u2208 adjoin K {x}\na\u271d : { val := y, property := hy } \u2208 \u22a4\nthis :\n  (\u2203 f, { val := y, property := hy } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f) \u2192\n    { val := y, property := hy } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 \u2203 f, { val := y, property := hy } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f\n[PROOFSTEP]\nrw [adjoin_singleton_eq_range_aeval] at hy \n[GOAL]\ncase hsp.mk\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\ny : S\nhy\u271d : y \u2208 adjoin K {x}\nhy : y \u2208 AlgHom.range (aeval x)\na\u271d : { val := y, property := hy\u271d } \u2208 \u22a4\nthis :\n  (\u2203 f, { val := y, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f) \u2192\n    { val := y, property := hy\u271d } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 \u2203 f, { val := y, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f\n[PROOFSTEP]\nobtain \u27e8f, rfl\u27e9 := (aeval x).mem_range.mp hy\n[GOAL]\ncase hsp.mk.intro\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\nf : K[X]\nhy\u271d : \u2191(aeval x) f \u2208 adjoin K {x}\nhy : \u2191(aeval x) f \u2208 AlgHom.range (aeval x)\na\u271d : { val := \u2191(aeval x) f, property := hy\u271d } \u2208 \u22a4\nthis :\n  (\u2203 f_1, { val := \u2191(aeval x) f, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f_1) \u2192\n    { val := \u2191(aeval x) f, property := hy\u271d } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 \u2203 f_1, { val := \u2191(aeval x) f, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f_1\n[PROOFSTEP]\nuse f\n[GOAL]\ncase h\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\nf : K[X]\nhy\u271d : \u2191(aeval x) f \u2208 adjoin K {x}\nhy : \u2191(aeval x) f \u2208 AlgHom.range (aeval x)\na\u271d : { val := \u2191(aeval x) f, property := hy\u271d } \u2208 \u22a4\nthis :\n  (\u2203 f_1, { val := \u2191(aeval x) f, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f_1) \u2192\n    { val := \u2191(aeval x) f, property := hy\u271d } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 { val := \u2191(aeval x) f, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f\n[PROOFSTEP]\next\n[GOAL]\ncase h.a\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S)\nhx' : IsIntegral K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }\nminpoly_eq :\n  minpoly K { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } =\n    minpoly K (\u2191(algebraMap { x_1 // x_1 \u2208 adjoin K {x} } S) { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) })\nf : K[X]\nhy\u271d : \u2191(aeval x) f \u2208 adjoin K {x}\nhy : \u2191(aeval x) f \u2208 AlgHom.range (aeval x)\na\u271d : { val := \u2191(aeval x) f, property := hy\u271d } \u2208 \u22a4\nthis :\n  (\u2203 f_1, { val := \u2191(aeval x) f, property := hy\u271d } = \u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f_1) \u2192\n    { val := \u2191(aeval x) f, property := hy\u271d } \u2208\n      Submodule.span K (Set.range fun i => { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i)\n\u22a2 \u2191{ val := \u2191(aeval x) f, property := hy\u271d } = \u2191(\u2191(aeval { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) }) f)\n[PROOFSTEP]\nexact aeval_algebraMap_apply S (\u27e8x, _\u27e9 : adjoin K { x }) _\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Algebra K S\nx : S\nhx : IsIntegral K x\ni : Fin (natDegree (minpoly K x))\n\u22a2 \u2191(powerBasisAux hx) i = { val := x, property := (_ : x \u2208 \u2191(adjoin K {x})) } ^ \u2191i\n[PROOFSTEP]\nrw [adjoin.powerBasisAux, Basis.mk_apply]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nintro i\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nlet Q := X ^ n %\u2098 minpoly R B.gen\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nhave : B.gen ^ n = aeval B.gen Q :=\n  by\n  rw [\u2190 @aeval_X_pow R _ _ _ _ B.gen, \u2190 modByMonic_add_div (X ^ n) (minpoly.monic hB)]\n  simp\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\n\u22a2 B.gen ^ n = \u2191(aeval B.gen) Q\n[PROOFSTEP]\nrw [\u2190 @aeval_X_pow R _ _ _ _ B.gen, \u2190 modByMonic_add_div (X ^ n) (minpoly.monic hB)]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\n\u22a2 \u2191(aeval B.gen) (X ^ n %\u2098 minpoly R B.gen + minpoly R B.gen * (X ^ n /\u2098 minpoly R B.gen)) = \u2191(aeval B.gen) Q\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nby_cases hQ : Q = 0\n[GOAL]\ncase pos\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : Q = 0\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nsimp [this, hQ, isIntegral_zero]\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nhave hlt : Q.natDegree < B.dim :=\n  by\n  rw [\u2190 B.natDegree_minpoly, hmin, (minpoly.monic hB).natDegree_map, natDegree_lt_natDegree_iff hQ]\n  letI : Nontrivial R := Nontrivial.of_polynomial_ne hQ\n  exact degree_modByMonic_lt _ (minpoly.monic hB)\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\n\u22a2 natDegree Q < B.dim\n[PROOFSTEP]\nrw [\u2190 B.natDegree_minpoly, hmin, (minpoly.monic hB).natDegree_map, natDegree_lt_natDegree_iff hQ]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\n\u22a2 degree Q < degree (minpoly R B.gen)\n[PROOFSTEP]\nletI : Nontrivial R := Nontrivial.of_polynomial_ne hQ\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis\u271d : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nthis : Nontrivial R := Nontrivial.of_polynomial_ne hQ\n\u22a2 degree Q < degree (minpoly R B.gen)\n[PROOFSTEP]\nexact degree_modByMonic_lt _ (minpoly.monic hB)\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nrw [this, aeval_eq_sum_range' hlt]\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (Finset.sum (Finset.range B.dim) fun i => coeff Q i \u2022 B.gen ^ i)) i)\n[PROOFSTEP]\nsimp only [LinearEquiv.map_sum, LinearEquiv.map_smul\u209b\u2097, RingHom.id_apply, Finset.sum_apply']\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\n\u22a2 IsIntegral R\n    (Finset.sum (Finset.range B.dim) fun x => \u2191(\u2191B.basis.repr (coeff (X ^ n %\u2098 minpoly R B.gen) x \u2022 B.gen ^ x)) i)\n[PROOFSTEP]\nrefine' IsIntegral.sum _ fun j hj => _\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j \u2208 Finset.range B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 B.gen ^ j)) i)\n[PROOFSTEP]\nreplace hj := Finset.mem_range.1 hj\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 B.gen ^ j)) i)\n[PROOFSTEP]\nrw [\u2190 Fin.val_mk hj, \u2190 B.basis_eq_pow, Algebra.smul_def, IsScalarTower.algebraMap_apply R S A, \u2190 Algebra.smul_def,\n  LinearEquiv.map_smul]\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\n\u22a2 IsIntegral R\n    (\u2191(\u2191(algebraMap R S) (coeff (X ^ n %\u2098 minpoly R B.gen) \u2191{ val := j, isLt := hj }) \u2022\n          \u2191B.basis.repr (\u2191B.basis { val := j, isLt := hj }))\n      i)\n[PROOFSTEP]\nsimp only [algebraMap_smul, Finsupp.coe_smul, Pi.smul_apply, B.basis.repr_self_apply]\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\n\u22a2 IsIntegral R (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 if { val := j, isLt := hj } = i then 1 else 0)\n[PROOFSTEP]\nby_cases hij : (\u27e8j, hj\u27e9 : Fin _) = i\n[GOAL]\ncase pos\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\nhij : { val := j, isLt := hj } = i\n\u22a2 IsIntegral R (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 if { val := j, isLt := hj } = i then 1 else 0)\n[PROOFSTEP]\nsimp only [hij, eq_self_iff_true, if_true]\n[GOAL]\ncase pos\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\nhij : { val := j, isLt := hj } = i\n\u22a2 IsIntegral R (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 1)\n[PROOFSTEP]\nrw [Algebra.smul_def, mul_one]\n[GOAL]\ncase pos\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\nhij : { val := j, isLt := hj } = i\n\u22a2 IsIntegral R (\u2191(algebraMap R S) (coeff (X ^ n %\u2098 minpoly R B.gen) j))\n[PROOFSTEP]\nexact isIntegral_algebraMap\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\ni : Fin B.dim\nQ : R[X] := X ^ n %\u2098 minpoly R B.gen\nthis : B.gen ^ n = \u2191(aeval B.gen) Q\nhQ : \u00acQ = 0\nhlt : natDegree Q < B.dim\nj : \u2115\nhj : j < B.dim\nhij : \u00ac{ val := j, isLt := hj } = i\n\u22a2 IsIntegral R (coeff (X ^ n %\u2098 minpoly R B.gen) j \u2022 if { val := j, isLt := hj } = i then 1 else 0)\n[PROOFSTEP]\nsimp [hij, isIntegral_zero]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x * y)) i)\n[PROOFSTEP]\nintro i\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (x * y)) i)\n[PROOFSTEP]\nrw [\u2190 B.basis.sum_repr x, \u2190 B.basis.sum_repr y, Finset.sum_mul_sum, LinearEquiv.map_sum, Finset.sum_apply']\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\n\u22a2 IsIntegral R\n    (Finset.sum (Finset.univ \u00d7\u02e2 Finset.univ) fun k =>\n      \u2191(\u2191B.basis.repr (\u2191(\u2191B.basis.repr x) k.fst \u2022 \u2191B.basis k.fst * \u2191(\u2191B.basis.repr y) k.snd \u2022 \u2191B.basis k.snd)) i)\n[PROOFSTEP]\nrefine' IsIntegral.sum _ fun I _ => _\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim \u00d7 Fin B.dim\nx\u271d : I \u2208 Finset.univ \u00d7\u02e2 Finset.univ\n\u22a2 IsIntegral R\n    (\u2191(\u2191B.basis.repr (\u2191(\u2191B.basis.repr x) I.fst \u2022 \u2191B.basis I.fst * \u2191(\u2191B.basis.repr y) I.snd \u2022 \u2191B.basis I.snd)) i)\n[PROOFSTEP]\nsimp only [Algebra.smul_mul_assoc, Algebra.mul_smul_comm, LinearEquiv.map_smul\u209b\u2097, RingHom.id_apply, Finsupp.coe_smul,\n  Pi.smul_apply, id.smul_eq_mul]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim \u00d7 Fin B.dim\nx\u271d : I \u2208 Finset.univ \u00d7\u02e2 Finset.univ\n\u22a2 IsIntegral R\n    ((\u2191(\u2191B.basis.repr y) I.snd \u2022 \u2191(\u2191B.basis.repr x) I.fst \u2022 \u2191(\u2191B.basis.repr (\u2191B.basis I.fst * \u2191B.basis I.snd))) i)\n[PROOFSTEP]\nrefine' isIntegral_mul (hy _) (isIntegral_mul (hx _) _)\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim \u00d7 Fin B.dim\nx\u271d : I \u2208 Finset.univ \u00d7\u02e2 Finset.univ\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (\u2191B.basis I.fst * \u2191B.basis I.snd)) i)\n[PROOFSTEP]\nsimp only [coe_basis, \u2190 pow_add]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx y : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhy : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr y) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\ni : Fin B.dim\nI : Fin B.dim \u00d7 Fin B.dim\nx\u271d : I \u2208 Finset.univ \u00d7\u02e2 Finset.univ\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (B.gen ^ (\u2191I.fst + \u2191I.snd))) i)\n[PROOFSTEP]\nrefine' repr_gen_pow_isIntegral hB hmin _ _\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i)\n[PROOFSTEP]\nnontriviality A using Subsingleton.elim (x ^ n) 0, isIntegral_zero\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i)\n[PROOFSTEP]\nrevert hx\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\n\u22a2 (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n    \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i)\n[PROOFSTEP]\nrefine'\n  Nat.case_strong_induction_on (p := fun n \u21a6 _ \u2192 \u2200 (i : Fin B.dim), IsIntegral R (B.basis.repr (x ^ n) i)) n _\n    fun n hn => _\n[GOAL]\ncase refine'_1\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\n\u22a2 (fun n =>\n      (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n        \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i))\n    0\n[PROOFSTEP]\nintro _ i\n[GOAL]\ncase refine'_1\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\na\u271d : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\ni : Fin B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (x ^ 0)) i)\n[PROOFSTEP]\nrw [pow_zero, \u2190 pow_zero B.gen, \u2190 Fin.val_mk B.dim_pos, \u2190 B.basis_eq_pow, B.basis.repr_self_apply]\n[GOAL]\ncase refine'_1\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\na\u271d : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\ni : Fin B.dim\n\u22a2 IsIntegral R (if { val := 0, isLt := (_ : 0 < B.dim) } = i then 1 else 0)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\na\u271d : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\ni : Fin B.dim\nh\u271d : { val := 0, isLt := (_ : 0 < B.dim) } = i\n\u22a2 IsIntegral R 1\n[PROOFSTEP]\nexact isIntegral_one\n[GOAL]\ncase neg\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn : \u2115\n\u271d : Nontrivial A\na\u271d : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\ni : Fin B.dim\nh\u271d : \u00ac{ val := 0, isLt := (_ : 0 < B.dim) } = i\n\u22a2 IsIntegral R 0\n[PROOFSTEP]\nexact isIntegral_zero\n[GOAL]\ncase refine'_2\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn\u271d : \u2115\n\u271d : Nontrivial A\nn : \u2115\nhn :\n  \u2200 (m : \u2115),\n    m \u2264 n \u2192\n      (fun n =>\n          (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n            \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i))\n        m\n\u22a2 (fun n =>\n      (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n        \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i))\n    (Nat.succ n)\n[PROOFSTEP]\nintro hx\n[GOAL]\ncase refine'_2\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn\u271d : \u2115\n\u271d : Nontrivial A\nn : \u2115\nhn :\n  \u2200 (m : \u2115),\n    m \u2264 n \u2192\n      (fun n =>\n          (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n            \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i))\n        m\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ Nat.succ n)) i)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\ncase refine'_2\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : Algebra K S\nR : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra R K\ninst\u271d\u2075 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\ninst\u271d : IsDomain S\nx : A\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (minpoly R B.gen)\nn\u271d : \u2115\n\u271d : Nontrivial A\nn : \u2115\nhn :\n  \u2200 (m : \u2115),\n    m \u2264 n \u2192\n      (fun n =>\n          (\u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)) \u2192\n            \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x ^ n)) i))\n        m\nhx : \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr x) i)\n\u22a2 \u2200 (i : Fin B.dim), IsIntegral R (\u2191(\u2191B.basis.repr (x * x ^ n)) i)\n[PROOFSTEP]\nexact repr_mul_isIntegral hB hx (fun _ => hn _ le_rfl (fun _ => hx _) _) hmin\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\n\u22a2 \u2200 (i : Fin B.dim) (j : Fin B'.dim), IsIntegral R (Basis.toMatrix B.basis (\u2191B'.basis) i j)\n[PROOFSTEP]\nintro i j\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni : Fin B.dim\nj : Fin B'.dim\n\u22a2 IsIntegral R (Basis.toMatrix B.basis (\u2191B'.basis) i j)\n[PROOFSTEP]\nrw [B.basis.toMatrix_apply, B'.coe_basis]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni : Fin B.dim\nj : Fin B'.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr ((fun i => B'.gen ^ \u2191i) j)) i)\n[PROOFSTEP]\nrefine' repr_pow_isIntegral hB (fun i => _) hmin _ _\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni\u271d : Fin B.dim\nj : Fin B'.dim\ni : Fin B.dim\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr B'.gen) i)\n[PROOFSTEP]\nrw [\u2190 h, aeval_eq_sum_range, LinearEquiv.map_sum, Finset.sum_apply']\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni\u271d : Fin B.dim\nj : Fin B'.dim\ni : Fin B.dim\n\u22a2 IsIntegral R (Finset.sum (Finset.range (natDegree P + 1)) fun k => \u2191(\u2191B.basis.repr (coeff P k \u2022 B.gen ^ k)) i)\n[PROOFSTEP]\nrefine' IsIntegral.sum _ fun n _ => _\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni\u271d : Fin B.dim\nj : Fin B'.dim\ni : Fin B.dim\nn : \u2115\nx\u271d : n \u2208 Finset.range (natDegree P + 1)\n\u22a2 IsIntegral R (\u2191(\u2191B.basis.repr (coeff P n \u2022 B.gen ^ n)) i)\n[PROOFSTEP]\nrw [Algebra.smul_def, IsScalarTower.algebraMap_apply R K S, \u2190 Algebra.smul_def, LinearEquiv.map_smul, algebraMap_smul]\n[GOAL]\nK : Type u_1\nS : Type u_2\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra K S\nR : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K S\nA : Type u_4\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra S A\ninst\u271d : IsScalarTower R S A\nB\u271d : PowerBasis S A\nhB\u271d : IsIntegral R B\u271d.gen\nB B' : PowerBasis K S\nP : R[X]\nh : \u2191(aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)\ni\u271d : Fin B.dim\nj : Fin B'.dim\ni : Fin B.dim\nn : \u2115\nx\u271d : n \u2208 Finset.range (natDegree P + 1)\n\u22a2 IsIntegral R (\u2191(coeff P n \u2022 \u2191B.basis.repr (B.gen ^ n)) i)\n[PROOFSTEP]\nexact isIntegral_smul _ (repr_gen_pow_isIntegral hB hmin _ _)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Adjoin.PowerBasis", "llama_tokens": 23014, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059511841119, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5277410688604507}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g\u271d : G \u2192 E\nf'\u271d g'\u271d : G \u2192L[\u211d] E\ns\u271d : Set G\nx\u271d : G\nn : \u2115\u221e\nf g : \u211d \u2192 E\nf' g' : E\ns : Set \u211d\nx : \u211d\nhf : HasDerivWithinAt f f' s x\nhg : HasDerivWithinAt g g' s x\n\u22a2 HasDerivWithinAt (fun t => Inner.inner (f t) (g t)) (Inner.inner (f x) g' + Inner.inner f' (g x)) s x\n[PROOFSTEP]\nsimpa using (hf.hasFDerivWithinAt.inner \ud835\udd5c hg.hasFDerivWithinAt).hasDerivWithinAt\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf\u271d g\u271d : G \u2192 E\nf'\u271d g'\u271d : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nf g : \u211d \u2192 E\nf' g' : E\nx : \u211d\n\u22a2 HasDerivAt f f' x \u2192\n    HasDerivAt g g' x \u2192 HasDerivAt (fun t => Inner.inner (f t) (g t)) (Inner.inner (f x) g' + Inner.inner f' (g x)) x\n[PROOFSTEP]\nsimpa only [\u2190 hasDerivWithinAt_univ] using HasDerivWithinAt.inner \ud835\udd5c\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableAt \u211d f x\nhg : DifferentiableAt \u211d g x\ny : G\n\u22a2 \u2191(fderiv \u211d (fun t => inner (f t) (g t)) x) y = inner (f x) (\u2191(fderiv \u211d g x) y) + inner (\u2191(fderiv \u211d f x) y) (g x)\n[PROOFSTEP]\nrw [(hf.hasFDerivAt.inner \ud835\udd5c hg.hasFDerivAt).fderiv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableAt \u211d f x\nhg : DifferentiableAt \u211d g x\ny : G\n\u22a2 \u2191(ContinuousLinearMap.comp (fderivInnerClm \ud835\udd5c (f x, g x)) (ContinuousLinearMap.prod (fderiv \u211d f x) (fderiv \u211d g x))) y =\n    inner (f x) (\u2191(fderiv \u211d g x) y) + inner (\u2191(fderiv \u211d f x) y) (g x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\n\u22a2 ContDiff \u211d n fun x => \u2016x\u2016 ^ 2\n[PROOFSTEP]\nconvert (reClm : \ud835\udd5c \u2192L[\u211d] \u211d).contDiff.comp ((contDiff_id (E := E)).inner \ud835\udd5c (contDiff_id (E := E)))\n[GOAL]\ncase h.e'_10.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nx\u271d : E\n\u22a2 \u2016x\u271d\u2016 ^ 2 = (\u2191reClm \u2218 fun x => inner (id x) (id x)) x\u271d\n[PROOFSTEP]\nexact (inner_self_eq_norm_sq _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : E\nhx : x \u2260 0\n\u22a2 ContDiffAt \u211d n Norm.norm x\n[PROOFSTEP]\nhave : \u2016id x\u2016 ^ 2 \u2260 0 := pow_ne_zero 2 (norm_pos_iff.2 hx).ne'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : E\nhx : x \u2260 0\nthis : \u2016id x\u2016 ^ 2 \u2260 0\n\u22a2 ContDiffAt \u211d n Norm.norm x\n[PROOFSTEP]\nsimpa only [id, sqrt_sq, norm_nonneg] using (contDiffAt_id.norm_sq \ud835\udd5c).sqrt this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffAt \u211d n f x\nhg : ContDiffAt \u211d n g x\nhne : f x \u2260 g x\n\u22a2 ContDiffAt \u211d n (fun y => Dist.dist (f y) (g y)) x\n[PROOFSTEP]\nsimp only [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffAt \u211d n f x\nhg : ContDiffAt \u211d n g x\nhne : f x \u2260 g x\n\u22a2 ContDiffAt \u211d n (fun y => \u2016f y - g y\u2016) x\n[PROOFSTEP]\nexact (hf.sub hg).norm \ud835\udd5c (sub_ne_zero.2 hne)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffWithinAt \u211d n f s x\nhg : ContDiffWithinAt \u211d n g s x\nhne : f x \u2260 g x\n\u22a2 ContDiffWithinAt \u211d n (fun y => Dist.dist (f y) (g y)) s x\n[PROOFSTEP]\nsimp only [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : ContDiffWithinAt \u211d n f s x\nhg : ContDiffWithinAt \u211d n g s x\nhne : f x \u2260 g x\n\u22a2 ContDiffWithinAt \u211d n (fun y => \u2016f y - g y\u2016) s x\n[PROOFSTEP]\nexact (hf.sub hg).norm \ud835\udd5c (sub_ne_zero.2 hne)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : F\n\u22a2 HasStrictFDerivAt (fun x => \u2016x\u2016 ^ 2) (2 \u2022 \u2191(innerSL \u211d) x) x\n[PROOFSTEP]\nsimp only [sq, \u2190 @inner_self_eq_norm_mul_norm \u211d]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : F\n\u22a2 HasStrictFDerivAt (fun x => \u2191re (inner x x)) (2 \u2022 \u2191(innerSL \u211d) x) x\n[PROOFSTEP]\nconvert (hasStrictFDerivAt_id x).inner \u211d (hasStrictFDerivAt_id x)\n[GOAL]\ncase h.e'_10.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : F\ne_7\u271d : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe\u271d : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\n\u22a2 2 \u2022 \u2191(innerSL \u211d) x =\n    ContinuousLinearMap.comp (fderivInnerClm \u211d (id x, id x))\n      (ContinuousLinearMap.prod (ContinuousLinearMap.id \u211d F) (ContinuousLinearMap.id \u211d F))\n[PROOFSTEP]\next y\n[GOAL]\ncase h.e'_10.h.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx\u271d : G\nn : \u2115\u221e\nx : F\ne_7\u271d : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup\nhe\u271d : InnerProductSpace.toNormedSpace = NormedAlgebra.toNormedSpace'\ny : F\n\u22a2 \u2191(2 \u2022 \u2191(innerSL \u211d) x) y =\n    \u2191(ContinuousLinearMap.comp (fderivInnerClm \u211d (id x, id x))\n          (ContinuousLinearMap.prod (ContinuousLinearMap.id \u211d F) (ContinuousLinearMap.id \u211d F)))\n      y\n[PROOFSTEP]\nsimp [two_smul, real_inner_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableAt \u211d f x\nhg : DifferentiableAt \u211d g x\nhne : f x \u2260 g x\n\u22a2 DifferentiableAt \u211d (fun y => Dist.dist (f y) (g y)) x\n[PROOFSTEP]\nsimp only [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableAt \u211d f x\nhg : DifferentiableAt \u211d g x\nhne : f x \u2260 g x\n\u22a2 DifferentiableAt \u211d (fun y => \u2016f y - g y\u2016) x\n[PROOFSTEP]\nexact (hf.sub hg).norm \ud835\udd5c (sub_ne_zero.2 hne)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableWithinAt \u211d f s x\nhg : DifferentiableWithinAt \u211d g s x\nhne : f x \u2260 g x\n\u22a2 DifferentiableWithinAt \u211d (fun y => Dist.dist (f y) (g y)) s x\n[PROOFSTEP]\nsimp only [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ninst\u271d\u00b2 : NormedSpace \u211d E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nf g : G \u2192 E\nf' g' : G \u2192L[\u211d] E\ns : Set G\nx : G\nn : \u2115\u221e\nhf : DifferentiableWithinAt \u211d f s x\nhg : DifferentiableWithinAt \u211d g s x\nhne : f x \u2260 g x\n\u22a2 DifferentiableWithinAt \u211d (fun y => \u2016f y - g y\u2016) s x\n[PROOFSTEP]\nexact (hf.sub hg).norm \ud835\udd5c (sub_ne_zero.2 hne)\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 DifferentiableWithinAt \ud835\udd5c f t y \u2194 \u2200 (i : \u03b9), DifferentiableWithinAt \ud835\udd5c (fun x => f x i) t y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_differentiableWithinAt_iff, differentiableWithinAt_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9), DifferentiableWithinAt \ud835\udd5c (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) t y) \u2194\n    \u2200 (i : \u03b9), DifferentiableWithinAt \ud835\udd5c (fun x => f x i) t y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 DifferentiableAt \ud835\udd5c f y \u2194 \u2200 (i : \u03b9), DifferentiableAt \ud835\udd5c (fun x => f x i) y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_differentiableAt_iff, differentiableAt_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9), DifferentiableAt \ud835\udd5c (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) y) \u2194\n    \u2200 (i : \u03b9), DifferentiableAt \ud835\udd5c (fun x => f x i) y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 DifferentiableOn \ud835\udd5c f t \u2194 \u2200 (i : \u03b9), DifferentiableOn \ud835\udd5c (fun x => f x i) t\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_differentiableOn_iff, differentiableOn_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9), DifferentiableOn \ud835\udd5c (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) t) \u2194\n    \u2200 (i : \u03b9), DifferentiableOn \ud835\udd5c (fun x => f x i) t\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 Differentiable \ud835\udd5c f \u2194 \u2200 (i : \u03b9), Differentiable \ud835\udd5c fun x => f x i\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_differentiable_iff, differentiable_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9), Differentiable \ud835\udd5c fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) \u2194\n    \u2200 (i : \u03b9), Differentiable \ud835\udd5c fun x => f x i\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 HasStrictFDerivAt f f' y \u2194 \u2200 (i : \u03b9), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9),\n      HasStrictFDerivAt (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i)\n        (comp (proj i) (comp (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c)) f')) y) \u2194\n    \u2200 (i : \u03b9), HasStrictFDerivAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 HasFDerivWithinAt f f' t y \u2194 \u2200 (i : \u03b9), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') t y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\n\u22a2 (\u2200 (i : \u03b9),\n      HasFDerivWithinAt (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i)\n        (comp (proj i) (comp (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c)) f')) t y) \u2194\n    \u2200 (i : \u03b9), HasFDerivWithinAt (fun x => f x i) (comp (EuclideanSpace.proj i) f') t y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 ContDiffWithinAt \ud835\udd5c n f t y \u2194 \u2200 (i : \u03b9), ContDiffWithinAt \ud835\udd5c n (fun x => f x i) t y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_contDiffWithinAt_iff, contDiffWithinAt_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiffWithinAt \ud835\udd5c n (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) t y) \u2194\n    \u2200 (i : \u03b9), ContDiffWithinAt \ud835\udd5c n (fun x => f x i) t y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 ContDiffAt \ud835\udd5c n f y \u2194 \u2200 (i : \u03b9), ContDiffAt \ud835\udd5c n (fun x => f x i) y\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_contDiffAt_iff, contDiffAt_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiffAt \ud835\udd5c n (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) y) \u2194\n    \u2200 (i : \u03b9), ContDiffAt \ud835\udd5c n (fun x => f x i) y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 ContDiffOn \ud835\udd5c n f t \u2194 \u2200 (i : \u03b9), ContDiffOn \ud835\udd5c n (fun x => f x i) t\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_contDiffOn_iff, contDiffOn_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiffOn \ud835\udd5c n (fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) t) \u2194\n    \u2200 (i : \u03b9), ContDiffOn \ud835\udd5c n (fun x => f x i) t\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => f x i\n[PROOFSTEP]\nrw [\u2190 (EuclideanSpace.equiv \u03b9 \ud835\udd5c).comp_contDiff_iff, contDiff_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nH : Type u_3\ninst\u271d\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup H\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c H\ninst\u271d : Fintype \u03b9\nf : H \u2192 EuclideanSpace \ud835\udd5c \u03b9\nf' : H \u2192L[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nt : Set H\ny : H\nn : \u2115\u221e\n\u22a2 (\u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => (\u2191(EuclideanSpace.equiv \u03b9 \ud835\udd5c) \u2218 f) x i) \u2194 \u2200 (i : \u03b9), ContDiff \ud835\udd5c n fun x => f x i\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u22a2 ContDiff \u211d n \u2191univUnitBall\n[PROOFSTEP]\nsuffices ContDiff \u211d n fun x : E => ((1 : \u211d) + \u2016x\u2016 ^ 2).sqrt\u207b\u00b9 from this.smul contDiff_id\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\n\u22a2 ContDiff \u211d n fun x => (sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9\n[PROOFSTEP]\nhave h : \u2200 x : E, (0 : \u211d) < (1 : \u211d) + \u2016x\u2016 ^ 2 := fun x => by positivity\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nx : E\n\u22a2 0 < 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nh : \u2200 (x : E), 0 < 1 + \u2016x\u2016 ^ 2\n\u22a2 ContDiff \u211d n fun x => (sqrt (1 + \u2016x\u2016 ^ 2))\u207b\u00b9\n[PROOFSTEP]\nrefine' ContDiff.inv _ fun x => Real.sqrt_ne_zero'.mpr (h x)\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nh : \u2200 (x : E), 0 < 1 + \u2016x\u2016 ^ 2\n\u22a2 ContDiff \u211d n fun x => sqrt (1 + \u2016x\u2016 ^ 2)\n[PROOFSTEP]\nexact (contDiff_const.add <| contDiff_norm_sq \u211d).sqrt fun x => (h x).ne'\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 ContDiffWithinAt \u211d n (\u2191(LocalHomeomorph.symm univUnitBall)) (ball 0 1) y\n[PROOFSTEP]\napply ContDiffAt.contDiffWithinAt\n[GOAL]\ncase h\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 ContDiffAt \u211d n (\u2191(LocalHomeomorph.symm univUnitBall)) y\n[PROOFSTEP]\nsuffices ContDiffAt \u211d n (fun y : E => ((1 : \u211d) - \u2016y\u2016 ^ 2).sqrt\u207b\u00b9) y from this.smul contDiffAt_id\n[GOAL]\ncase h\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 ContDiffAt \u211d n (fun y => (sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9) y\n[PROOFSTEP]\nhave h : (0 : \u211d) < (1 : \u211d) - \u2016(y : E)\u2016 ^ 2 := by\n  rwa [mem_ball_zero_iff, \u2190 _root_.abs_one, \u2190 abs_norm, \u2190 sq_lt_sq, one_pow, \u2190 sub_pos] at hy \n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\n\u22a2 0 < 1 - \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrwa [mem_ball_zero_iff, \u2190 _root_.abs_one, \u2190 abs_norm, \u2190 sq_lt_sq, one_pow, \u2190 sub_pos] at hy \n[GOAL]\ncase h\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\nh : 0 < 1 - \u2016y\u2016 ^ 2\n\u22a2 ContDiffAt \u211d n (fun y => (sqrt (1 - \u2016y\u2016 ^ 2))\u207b\u00b9) y\n[PROOFSTEP]\nrefine' ContDiffAt.inv _ (Real.sqrt_ne_zero'.mpr h)\n[GOAL]\ncase h\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\nh : 0 < 1 - \u2016y\u2016 ^ 2\n\u22a2 ContDiffAt \u211d n (fun y => sqrt (1 - \u2016y\u2016 ^ 2)) y\n[PROOFSTEP]\nrefine' (contDiffAt_sqrt h.ne').comp y _\n[GOAL]\ncase h\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\ny : E\nhy : y \u2208 ball 0 1\nh : 0 < 1 - \u2016y\u2016 ^ 2\n\u22a2 ContDiffAt \u211d n (fun y => 1 - \u2016y\u2016 ^ 2) y\n[PROOFSTEP]\nexact contDiffAt_const.sub (contDiff_norm_sq \u211d).contDiffAt\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\n\u22a2 ContDiff \u211d n \u2191(univBall c r)\n[PROOFSTEP]\nunfold univBall\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\n\u22a2 ContDiff \u211d n\n    \u2191(if h : 0 < r then\n        LocalHomeomorph.trans' univUnitBall (unitBallBall c r h)\n          (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target)\n      else Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c)))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : 0 < r\n\u22a2 ContDiff \u211d n\n    \u2191(LocalHomeomorph.trans' univUnitBall (unitBallBall c r h)\n        (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target))\n[PROOFSTEP]\nexact (contDiff_unitBallBall h).comp contDiff_univUnitBall\n[GOAL]\ncase neg\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : \u00ac0 < r\n\u22a2 ContDiff \u211d n \u2191(Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c)))\n[PROOFSTEP]\nexact contDiff_id.add contDiff_const\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\n\u22a2 ContDiffOn \u211d n (\u2191(LocalHomeomorph.symm (univBall c r))) (ball c r)\n[PROOFSTEP]\nunfold univBall\n[GOAL]\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\n\u22a2 ContDiffOn \u211d n\n    (\u2191(LocalHomeomorph.symm\n        (if h : 0 < r then\n          LocalHomeomorph.trans' univUnitBall (unitBallBall c r h)\n            (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target)\n        else Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c)))))\n    (ball c r)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : 0 < r\n\u22a2 ContDiffOn \u211d n\n    (\u2191(LocalHomeomorph.symm\n        (LocalHomeomorph.trans' univUnitBall (unitBallBall c r h)\n          (_ : univUnitBall.toLocalEquiv.target = univUnitBall.toLocalEquiv.target))))\n    (ball c r)\n[PROOFSTEP]\nrefine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_\n[GOAL]\ncase pos\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : 0 < r\n\u22a2 ball c r \u2286 (fun x => \u2191(LocalEquiv.symm (unitBallBall c r h).toLocalEquiv) x) \u207b\u00b9' ball 0 1\n[PROOFSTEP]\nrw [\u2190 unitBallBall_source c r h, \u2190 unitBallBall_target c r h]\n[GOAL]\ncase pos\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : 0 < r\n\u22a2 (unitBallBall c r h).toLocalEquiv.target \u2286\n    (fun x => \u2191(LocalEquiv.symm (unitBallBall c r h).toLocalEquiv) x) \u207b\u00b9' (unitBallBall c r h).toLocalEquiv.source\n[PROOFSTEP]\napply LocalHomeomorph.symm_mapsTo\n[GOAL]\ncase neg\nn : \u2115\u221e\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nc : E\nr : \u211d\nh : \u00ac0 < r\n\u22a2 ContDiffOn \u211d n\n    (\u2191(LocalHomeomorph.symm (Homeomorph.toLocalHomeomorph (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst c)))))\n    (ball c r)\n[PROOFSTEP]\nexact contDiffOn_id.sub contDiffOn_const\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Calculus", "llama_tokens": 12361, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059511841119, "lm_q2_score": 0.6654105454764746, "lm_q1q2_score": 0.5277410635980582}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsPreorder \u03b1 r\ninst\u271d : Inhabited \u03b1\n\u22a2 Inhabited (Antisymmetrization \u03b1 r)\n[PROOFSTEP]\nunfold Antisymmetrization\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : IsPreorder \u03b1 r\ninst\u271d : Inhabited \u03b1\n\u22a2 Inhabited (Quotient (AntisymmRel.setoid \u03b1 r))\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na b : \u03b1\n\u22a2 Relation.Fibration (fun x x_1 => x < x_1) (fun x x_1 => x < x_1) (toAntisymmetrization fun x x_1 => x \u2264 x_1)\n[PROOFSTEP]\nrintro a \u27e8b\u27e9 h\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na\u271d b\u271d\u00b9 a : \u03b1\nb\u271d : Antisymmetrization \u03b1 fun x x_1 => x \u2264 x_1\nb : \u03b1\nh : Quot.mk Setoid.r b < toAntisymmetrization (fun x x_1 => x \u2264 x_1) a\n\u22a2 \u2203 a', (fun x x_1 => x < x_1) a' a \u2227 toAntisymmetrization (fun x x_1 => x \u2264 x_1) a' = Quot.mk Setoid.r b\n[PROOFSTEP]\nexact \u27e8b, h, rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na\u271d\u00b9 b : \u03b1\na\u271d : (Antisymmetrization \u03b1 fun x x_1 => x \u2264 x_1)\u1d52\u1d48\na : \u03b1\n\u22a2 Quotient.map' id\n      (_ :\n        \u2200 (x x_1 : \u03b1\u1d52\u1d48),\n          (fun x x_2 => x \u2264 x_2) x x_1 \u2227 (fun x x_2 => x \u2264 x_2) x_1 x \u2192\n            (fun x x_2 => x \u2264 x_2) x_1 x \u2227 (fun x x_2 => x \u2264 x_2) x x_1)\n      (Quotient.map' id\n        (_ :\n          \u2200 (x x_1 : \u03b1),\n            (fun x x_2 => x \u2264 x_2) x x_1 \u2227 (fun x x_2 => x \u2264 x_2) x_1 x \u2192\n              (fun x x_2 => x \u2264 x_2) x_1 x \u2227 (fun x x_2 => x \u2264 x_2) x x_1)\n        (Quotient.mk'' a)) =\n    Quotient.mk'' a\n[PROOFSTEP]\nsimp_rw [Quotient.map'_mk'', id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na\u271d\u00b9 b : \u03b1\na\u271d : Antisymmetrization \u03b1\u1d52\u1d48 fun x x_1 => x \u2264 x_1\na : \u03b1\u1d52\u1d48\n\u22a2 Quotient.map' id\n      (_ :\n        \u2200 (x x_1 : \u03b1),\n          (fun x x_2 => x \u2264 x_2) x x_1 \u2227 (fun x x_2 => x \u2264 x_2) x_1 x \u2192\n            (fun x x_2 => x \u2264 x_2) x_1 x \u2227 (fun x x_2 => x \u2264 x_2) x x_1)\n      (Quotient.map' id\n        (_ :\n          \u2200 (x x_1 : \u03b1\u1d52\u1d48),\n            (fun x x_2 => x \u2264 x_2) x x_1 \u2227 (fun x x_2 => x \u2264 x_2) x_1 x \u2192\n              (fun x x_2 => x \u2264 x_2) x_1 x \u2227 (fun x x_2 => x \u2264 x_2) x x_1)\n        (Quotient.mk'' a)) =\n    Quotient.mk'' a\n[PROOFSTEP]\nsimp_rw [Quotient.map'_mk'', id]\n", "meta": {"mathlib_filename": "Mathlib.Order.Antisymmetrization", "llama_tokens": 1171, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812554, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.5276042741104954}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d : RingHomInvPair \u03c3' \u03c3\nf : \u03b9 \u2192 M\n\u03c6 : M \u2192SL[\u03c3] M\u2082\nx : M\nhf : HasSum f x\n\u22a2 HasSum (fun b => \u2191\u03c6 (f b)) (\u2191\u03c6 x)\n[PROOFSTEP]\nsimpa only using hf.map \u03c6.toLinearMap.toAddMonoidHom \u03c6.continuous\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d : RingHomInvPair \u03c3' \u03c3\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nh : HasSum (fun b => \u2191e (f b)) y\n\u22a2 HasSum f (\u2191(ContinuousLinearEquiv.symm e) y)\n[PROOFSTEP]\nsimpa only [e.symm.coe_coe, e.symm_apply_apply] using h.mapL (e.symm : M\u2082 \u2192SL[\u03c3'] M)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d : RingHomInvPair \u03c3' \u03c3\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nh : HasSum f (\u2191(ContinuousLinearEquiv.symm e) y)\n\u22a2 HasSum (fun b => \u2191e (f b)) y\n[PROOFSTEP]\nsimpa only [e.coe_coe, e.apply_symm_apply] using (e : M \u2192SL[\u03c3] M\u2082).hasSum h\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalSpace M\ninst\u271d\u00b2 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d : RingHomInvPair \u03c3' \u03c3\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\nx : M\n\u22a2 HasSum (fun b => \u2191e (f b)) (\u2191e x) \u2194 HasSum f x\n[PROOFSTEP]\nrw [e.hasSum, ContinuousLinearEquiv.symm_apply_apply]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\n\u22a2 \u2211' (z : \u03b9), \u2191e (f z) = y \u2194 \u2211' (z : \u03b9), f z = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nby_cases hf : Summable f\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : Summable f\n\u22a2 \u2211' (z : \u03b9), \u2191e (f z) = y \u2194 \u2211' (z : \u03b9), f z = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nexact\n  \u27e8fun h => (e.hasSum.mp ((e.summable.mpr hf).hasSum_iff.mpr h)).tsum_eq, fun h =>\n    (e.hasSum.mpr (hf.hasSum_iff.mpr h)).tsum_eq\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : \u00acSummable f\n\u22a2 \u2211' (z : \u03b9), \u2191e (f z) = y \u2194 \u2211' (z : \u03b9), f z = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nhave hf' : \u00acSummable fun z => e (f z) := fun h => hf (e.summable.mp h)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : \u00acSummable f\nhf' : \u00acSummable fun z => \u2191e (f z)\n\u22a2 \u2211' (z : \u03b9), \u2191e (f z) = y \u2194 \u2211' (z : \u03b9), f z = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nrw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable hf']\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : \u00acSummable f\nhf' : \u00acSummable fun z => \u2191e (f z)\n\u22a2 0 = y \u2194 0 = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nrefine \u27e8?_, fun H => ?_\u27e9\n[GOAL]\ncase neg.refine_1\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : \u00acSummable f\nhf' : \u00acSummable fun z => \u2191e (f z)\n\u22a2 0 = y \u2192 0 = \u2191(ContinuousLinearEquiv.symm e) y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase neg.refine_1\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\nhf : \u00acSummable f\nhf' : \u00acSummable fun z => \u2191e (f z)\n\u22a2 0 = \u2191(ContinuousLinearEquiv.symm e) 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.refine_2\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\ny : M\u2082\nhf : \u00acSummable f\nhf' : \u00acSummable fun z => \u2191e (f z)\nH : 0 = \u2191(ContinuousLinearEquiv.symm e) y\n\u22a2 0 = y\n[PROOFSTEP]\nsimpa using congr_arg (fun z => e z) H\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\n\u22a2 \u2191e (\u2211' (z : \u03b9), f z) = \u2211' (z : \u03b9), \u2191e (f z)\n[PROOFSTEP]\nrefine' symm (e.tsum_eq_iff.mpr _)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : Module R\u2082 M\u2082\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : TopologicalSpace M\u2082\n\u03c3 : R \u2192+* R\u2082\n\u03c3' : R\u2082 \u2192+* R\ninst\u271d\u00b3 : RingHomInvPair \u03c3 \u03c3'\ninst\u271d\u00b2 : RingHomInvPair \u03c3' \u03c3\ninst\u271d\u00b9 : T2Space M\ninst\u271d : T2Space M\u2082\nf : \u03b9 \u2192 M\ne : M \u2243SL[\u03c3] M\u2082\n\u22a2 \u2211' (z : \u03b9), f z = \u2191(ContinuousLinearEquiv.symm e) (\u2191e (\u2211' (z : \u03b9), f z))\n[PROOFSTEP]\nrw [e.symm_apply_apply _]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.InfiniteSum.Module", "llama_tokens": 4318, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.6548947223065754, "lm_q1q2_score": 0.5276042686841177}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 (s.pt.map f \u226b (fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) }) Y) \u226b\n      NatTrans.app (((evaluation K C).obj Y).mapCone c).\u03c0 j =\n    ((fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) }) X \u226b\n        c.pt.map f) \u226b\n      NatTrans.app (((evaluation K C).obj Y).mapCone c).\u03c0 j\n[PROOFSTEP]\nrw [assoc, (t Y).fac _ j]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 s.pt.map f \u226b NatTrans.app { pt := s.pt.obj Y, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj Y) }.\u03c0 j =\n    ((fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) }) X \u226b\n        c.pt.map f) \u226b\n      NatTrans.app (((evaluation K C).obj Y).mapCone c).\u03c0 j\n[PROOFSTEP]\nsimpa using ((t X).fac_assoc \u27e8s.pt.obj X, whiskerRight s.\u03c0 ((evaluation K C).obj X)\u27e9 j _).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nj : J\n\u22a2 (fun s =>\n          NatTrans.mk fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) })\n        s \u226b\n      NatTrans.app c.\u03c0 j =\n    NatTrans.app s.\u03c0 j\n[PROOFSTEP]\next k\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nj : J\nk : K\n\u22a2 NatTrans.app\n      ((fun s =>\n            NatTrans.mk fun k =>\n              IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) })\n          s \u226b\n        NatTrans.app c.\u03c0 j)\n      k =\n    NatTrans.app (NatTrans.app s.\u03c0 j) k\n[PROOFSTEP]\nexact (t k).fac _ j\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nm : s.pt \u27f6 c.pt\nw : \u2200 (j : J), m \u226b NatTrans.app c.\u03c0 j = NatTrans.app s.\u03c0 j\n\u22a2 m =\n    (fun s =>\n        NatTrans.mk fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) })\n      s\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cone F\nt : (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone c)\ns : Cone F\nm : s.pt \u27f6 c.pt\nw : \u2200 (j : J), m \u226b NatTrans.app c.\u03c0 j = NatTrans.app s.\u03c0 j\nx : K\n\u22a2 NatTrans.app m x =\n    NatTrans.app\n      ((fun s =>\n          NatTrans.mk fun k => IsLimit.lift (t k) { pt := s.pt.obj k, \u03c0 := whiskerRight s.\u03c0 ((evaluation K C).obj k) })\n        s)\n      x\n[PROOFSTEP]\nexact\n  (t x).hom_ext fun j =>\n    (congr_app (w j) x).trans ((t x).fac \u27e8s.pt.obj _, whiskerRight s.\u03c0 ((evaluation K C).obj _)\u27e9 j).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 LimitCone ((Functor.flip F).obj k)\nk : K\nj : J\n\u22a2 { obj := fun k => (c k).cone.pt,\n            map := fun {k\u2081 k\u2082} f =>\n              IsLimit.lift (c k\u2082).isLimit { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } }.map\n        (\ud835\udfd9 k) \u226b\n      NatTrans.app (c k).cone.\u03c0 j =\n    \ud835\udfd9\n        ({ obj := fun k => (c k).cone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsLimit.lift (c k\u2082).isLimit { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } }.obj\n          k) \u226b\n      NatTrans.app (c k).cone.\u03c0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 LimitCone ((Functor.flip F).obj k)\nk : K\nj : J\n\u22a2 IsLimit.lift (c k).isLimit { pt := (c k).cone.pt, \u03c0 := (c k).cone.\u03c0 \u226b (Functor.flip F).map (\ud835\udfd9 k) } \u226b\n      NatTrans.app (c k).cone.\u03c0 j =\n    \ud835\udfd9 (c k).cone.pt \u226b NatTrans.app (c k).cone.\u03c0 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 LimitCone ((Functor.flip F).obj k)\nk\u2081 k\u2082 k\u2083 : K\nf\u2081 : k\u2081 \u27f6 k\u2082\nf\u2082 : k\u2082 \u27f6 k\u2083\nj : J\n\u22a2 { obj := fun k => (c k).cone.pt,\n            map := fun {k\u2081 k\u2082} f =>\n              IsLimit.lift (c k\u2082).isLimit { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } }.map\n        (f\u2081 \u226b f\u2082) \u226b\n      NatTrans.app (c k\u2083).cone.\u03c0 j =\n    ({ obj := fun k => (c k).cone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsLimit.lift (c k\u2082).isLimit { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } }.map\n          f\u2081 \u226b\n        { obj := fun k => (c k).cone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsLimit.lift (c k\u2082).isLimit { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } }.map\n          f\u2082) \u226b\n      NatTrans.app (c k\u2083).cone.\u03c0 j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 LimitCone ((Functor.flip F).obj k)\nj\u2081 j\u2082 : J\ng : j\u2081 \u27f6 j\u2082\n\u22a2 ((const J).obj\n            (Functor.mk\n              { obj := fun k => (c k).cone.pt,\n                map := fun {k\u2081 k\u2082} f =>\n                  IsLimit.lift (c k\u2082).isLimit\n                    { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } })).map\n        g \u226b\n      (fun j => NatTrans.mk fun k => NatTrans.app (c k).cone.\u03c0 j) j\u2082 =\n    (fun j => NatTrans.mk fun k => NatTrans.app (c k).cone.\u03c0 j) j\u2081 \u226b F.map g\n[PROOFSTEP]\next k\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 LimitCone ((Functor.flip F).obj k)\nj\u2081 j\u2082 : J\ng : j\u2081 \u27f6 j\u2082\nk : K\n\u22a2 NatTrans.app\n      (((const J).obj\n              (Functor.mk\n                { obj := fun k => (c k).cone.pt,\n                  map := fun {k\u2081 k\u2082} f =>\n                    IsLimit.lift (c k\u2082).isLimit\n                      { pt := (c k\u2081).cone.pt, \u03c0 := (c k\u2081).cone.\u03c0 \u226b (Functor.flip F).map f } })).map\n          g \u226b\n        (fun j => NatTrans.mk fun k => NatTrans.app (c k).cone.\u03c0 j) j\u2082)\n      k =\n    NatTrans.app ((fun j => NatTrans.mk fun k => NatTrans.app (c k).cone.\u03c0 j) j\u2081 \u226b F.map g) k\n[PROOFSTEP]\nexact (c k).cone.\u03c0.naturality g\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 NatTrans.app (((evaluation K C).obj X).mapCocone c).\u03b9 j \u226b\n      c.pt.map f \u226b\n        (fun k => IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) }) Y =\n    NatTrans.app (((evaluation K C).obj X).mapCocone c).\u03b9 j \u226b\n      (fun k => IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) }) X \u226b\n        s.pt.map f\n[PROOFSTEP]\nrw [(t X).fac_assoc _ j]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 NatTrans.app (((evaluation K C).obj X).mapCocone c).\u03b9 j \u226b\n      c.pt.map f \u226b\n        (fun k => IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) }) Y =\n    NatTrans.app { pt := s.pt.obj X, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj X) }.\u03b9 j \u226b s.pt.map f\n[PROOFSTEP]\nerw [\u2190 (c.\u03b9.app j).naturality_assoc f]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 (F.obj j).map f \u226b\n      NatTrans.app (NatTrans.app c.\u03b9 j) Y \u226b\n        (fun k => IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) }) Y =\n    NatTrans.app { pt := s.pt.obj X, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj X) }.\u03b9 j \u226b s.pt.map f\n[PROOFSTEP]\nerw [(t Y).fac \u27e8s.pt.obj _, whiskerRight s.\u03b9 _\u27e9 j]\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 (F.obj j).map f \u226b NatTrans.app { pt := s.pt.obj Y, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj Y) }.\u03b9 j =\n    NatTrans.app { pt := s.pt.obj X, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj X) }.\u03b9 j \u226b s.pt.map f\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nX Y : K\nf : X \u27f6 Y\nj : J\n\u22a2 (F.obj j).map f \u226b NatTrans.app (NatTrans.app s.\u03b9 j) Y = NatTrans.app (NatTrans.app s.\u03b9 j) X \u226b s.pt.map f\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nj : J\n\u22a2 NatTrans.app c.\u03b9 j \u226b\n      (fun s =>\n          NatTrans.mk fun k =>\n            IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) })\n        s =\n    NatTrans.app s.\u03b9 j\n[PROOFSTEP]\next k\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nj : J\nk : K\n\u22a2 NatTrans.app\n      (NatTrans.app c.\u03b9 j \u226b\n        (fun s =>\n            NatTrans.mk fun k =>\n              IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) })\n          s)\n      k =\n    NatTrans.app (NatTrans.app s.\u03b9 j) k\n[PROOFSTEP]\nexact (t k).fac _ j\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nm : c.pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app c.\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\n\u22a2 m =\n    (fun s =>\n        NatTrans.mk fun k => IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) })\n      s\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : Cocone F\nt : (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone c)\ns : Cocone F\nm : c.pt \u27f6 s.pt\nw : \u2200 (j : J), NatTrans.app c.\u03b9 j \u226b m = NatTrans.app s.\u03b9 j\nx : K\n\u22a2 NatTrans.app m x =\n    NatTrans.app\n      ((fun s =>\n          NatTrans.mk fun k =>\n            IsColimit.desc (t k) { pt := s.pt.obj k, \u03b9 := whiskerRight s.\u03b9 ((evaluation K C).obj k) })\n        s)\n      x\n[PROOFSTEP]\nexact\n  (t x).hom_ext fun j =>\n    (congr_app (w j) x).trans ((t x).fac \u27e8s.pt.obj _, whiskerRight s.\u03b9 ((evaluation K C).obj _)\u27e9 j).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k)\nk : K\nj : J\n\u22a2 NatTrans.app (c k).cocone.\u03b9 j \u226b\n      { obj := fun k => (c k).cocone.pt,\n            map := fun {k\u2081 k\u2082} f =>\n              IsColimit.desc (c k\u2081).isColimit\n                { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } }.map\n        (\ud835\udfd9 k) =\n    NatTrans.app (c k).cocone.\u03b9 j \u226b\n      \ud835\udfd9\n        ({ obj := fun k => (c k).cocone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsColimit.desc (c k\u2081).isColimit\n                  { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } }.obj\n          k)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k)\nk : K\nj : J\n\u22a2 NatTrans.app (c k).cocone.\u03b9 j \u226b\n      IsColimit.desc (c k).isColimit { pt := (c k).cocone.pt, \u03b9 := (Functor.flip F).map (\ud835\udfd9 k) \u226b (c k).cocone.\u03b9 } =\n    NatTrans.app (c k).cocone.\u03b9 j \u226b \ud835\udfd9 (c k).cocone.pt\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k)\nk\u2081 k\u2082 k\u2083 : K\nf\u2081 : k\u2081 \u27f6 k\u2082\nf\u2082 : k\u2082 \u27f6 k\u2083\nj : J\n\u22a2 NatTrans.app (c k\u2081).cocone.\u03b9 j \u226b\n      { obj := fun k => (c k).cocone.pt,\n            map := fun {k\u2081 k\u2082} f =>\n              IsColimit.desc (c k\u2081).isColimit\n                { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } }.map\n        (f\u2081 \u226b f\u2082) =\n    NatTrans.app (c k\u2081).cocone.\u03b9 j \u226b\n      { obj := fun k => (c k).cocone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsColimit.desc (c k\u2081).isColimit\n                  { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } }.map\n          f\u2081 \u226b\n        { obj := fun k => (c k).cocone.pt,\n              map := fun {k\u2081 k\u2082} f =>\n                IsColimit.desc (c k\u2081).isColimit\n                  { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } }.map\n          f\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k)\nj\u2081 j\u2082 : J\ng : j\u2081 \u27f6 j\u2082\n\u22a2 F.map g \u226b (fun j => NatTrans.mk fun k => NatTrans.app (c k).cocone.\u03b9 j) j\u2082 =\n    (fun j => NatTrans.mk fun k => NatTrans.app (c k).cocone.\u03b9 j) j\u2081 \u226b\n      ((const J).obj\n            (Functor.mk\n              { obj := fun k => (c k).cocone.pt,\n                map := fun {k\u2081 k\u2082} f =>\n                  IsColimit.desc (c k\u2081).isColimit\n                    { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } })).map\n        g\n[PROOFSTEP]\next k\n[GOAL]\ncase w.h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : J \u2964 K \u2964 C\nc : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k)\nj\u2081 j\u2082 : J\ng : j\u2081 \u27f6 j\u2082\nk : K\n\u22a2 NatTrans.app (F.map g \u226b (fun j => NatTrans.mk fun k => NatTrans.app (c k).cocone.\u03b9 j) j\u2082) k =\n    NatTrans.app\n      ((fun j => NatTrans.mk fun k => NatTrans.app (c k).cocone.\u03b9 j) j\u2081 \u226b\n        ((const J).obj\n              (Functor.mk\n                { obj := fun k => (c k).cocone.pt,\n                  map := fun {k\u2081 k\u2082} f =>\n                    IsColimit.desc (c k\u2081).isColimit\n                      { pt := (c k\u2082).cocone.pt, \u03b9 := (Functor.flip F).map f \u226b (c k\u2082).cocone.\u03b9 } })).map\n          g)\n      k\n[PROOFSTEP]\nexact (c k).cocone.\u03b9.naturality g\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nk : K\nF : J \u2964 K \u2964 C\n\u22a2 PreservesLimit F ((evaluation K C).obj k)\n[PROOFSTEP]\nlet X : (k : K) \u2192 LimitCone (Prefunctor.obj (Functor.flip F).toPrefunctor k) := fun k =>\n  getLimitCone (Prefunctor.obj (Functor.flip F).toPrefunctor k)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nk : K\nF : J \u2964 K \u2964 C\nX : (k : K) \u2192 LimitCone ((Functor.flip F).obj k) := fun k => getLimitCone ((Functor.flip F).obj k)\n\u22a2 PreservesLimit F ((evaluation K C).obj k)\n[PROOFSTEP]\nexact\n  preservesLimitOfPreservesLimitCone (combinedIsLimit _ _) <|\n    IsLimit.ofIsoLimit (limit.isLimit _) (evaluateCombinedCones F X k).symm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (limitObjIsoLimitCompEvaluation F k).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j = NatTrans.app (limit.\u03c0 F j) k\n[PROOFSTEP]\ndsimp [limitObjIsoLimitCompEvaluation]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (preservesLimitIso ((evaluation K C).obj k) F).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j =\n    NatTrans.app (limit.\u03c0 F j) k\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (limitObjIsoLimitCompEvaluation F k).inv \u226b NatTrans.app (limit.\u03c0 F j) k = limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\ndsimp [limitObjIsoLimitCompEvaluation]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 (preservesLimitIso ((evaluation K C).obj k) F).inv \u226b NatTrans.app (limit.\u03c0 F j) k =\n    limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\nrw [Iso.inv_comp_eq]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 NatTrans.app (limit.\u03c0 F j) k =\n    (preservesLimitIso ((evaluation K C).obj k) F).hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\n\u22a2 (limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom =\n    (limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\nj\u271d : J\n\u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d =\n    ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\nj\u271d : J\n\u22a2 ((limit F).map f \u226b (limitObjIsoLimitCompEvaluation F j).hom) \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d =\n    ((limitObjIsoLimitCompEvaluation F i).hom \u226b limMap (whiskerLeft F ((evaluation K C).map f))) \u226b\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj j) j\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimitsOfShape J C\ni j : K\nF : J \u2964 K \u2964 C\nf : i \u27f6 j\n\u22a2 (limitObjIsoLimitCompEvaluation F i).inv \u226b (limit F).map f =\n    limMap (whiskerLeft F ((evaluation K C).map f)) \u226b (limitObjIsoLimitCompEvaluation F j).inv\n[PROOFSTEP]\nrw [Iso.inv_comp_eq, \u2190 Category.assoc, Iso.eq_comp_inv, limit_map_limitObjIsoLimitCompEvaluation_hom]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasLimitsOfShape J C\nk : K\nW : C\nf g : W \u27f6 (limit H).obj k\nw : \u2200 (j : J), f \u226b NatTrans.app (limit.\u03c0 H j) k = g \u226b NatTrans.app (limit.\u03c0 H j) k\n\u22a2 f = g\n[PROOFSTEP]\napply (cancel_mono (limitObjIsoLimitCompEvaluation H k).hom).1\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasLimitsOfShape J C\nk : K\nW : C\nf g : W \u27f6 (limit H).obj k\nw : \u2200 (j : J), f \u226b NatTrans.app (limit.\u03c0 H j) k = g \u226b NatTrans.app (limit.\u03c0 H j) k\n\u22a2 f \u226b (limitObjIsoLimitCompEvaluation H k).hom = g \u226b (limitObjIsoLimitCompEvaluation H k).hom\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasLimitsOfShape J C\nk : K\nW : C\nf g : W \u27f6 (limit H).obj k\nw : \u2200 (j : J), f \u226b NatTrans.app (limit.\u03c0 H j) k = g \u226b NatTrans.app (limit.\u03c0 H j) k\nj : J\n\u22a2 (f \u226b (limitObjIsoLimitCompEvaluation H k).hom) \u226b limit.\u03c0 (H \u22d9 (evaluation K C).obj k) j =\n    (g \u226b (limitObjIsoLimitCompEvaluation H k).hom) \u226b limit.\u03c0 (H \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\nsimpa using w j\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nk : K\nF : J \u2964 K \u2964 C\n\u22a2 PreservesColimit F ((evaluation K C).obj k)\n[PROOFSTEP]\nlet X : (k : K) \u2192 ColimitCocone (Prefunctor.obj (Functor.flip F).toPrefunctor k) := fun k =>\n  getColimitCocone (Prefunctor.obj (Functor.flip F).toPrefunctor k)\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nk : K\nF : J \u2964 K \u2964 C\nX : (k : K) \u2192 ColimitCocone ((Functor.flip F).obj k) := fun k => getColimitCocone ((Functor.flip F).obj k)\n\u22a2 PreservesColimit F ((evaluation K C).obj k)\n[PROOFSTEP]\nrefine\n  preservesColimitOfPreservesColimitCocone (combinedIsColimit _ _) <|\n    IsColimit.ofIsoColimit (colimit.isColimit _) (evaluateCombinedCocones F X k).symm\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation F k).inv =\n    NatTrans.app (colimit.\u03b9 F j) k\n[PROOFSTEP]\ndsimp [colimitObjIsoColimitCompEvaluation]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j \u226b (preservesColimitIso ((evaluation K C).obj k) F).inv =\n    NatTrans.app (colimit.\u03b9 F j) k\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 NatTrans.app (colimit.\u03b9 F j) k \u226b (colimitObjIsoColimitCompEvaluation F k).hom =\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\ndsimp [colimitObjIsoColimitCompEvaluation]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 NatTrans.app (colimit.\u03b9 F j) k \u226b (preservesColimitIso ((evaluation K C).obj k) F).hom =\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j\n[PROOFSTEP]\nrw [\u2190 Iso.eq_comp_inv]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\nj : J\nk : K\n\u22a2 NatTrans.app (colimit.\u03b9 F j) k =\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j \u226b (preservesColimitIso ((evaluation K C).obj k) F).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\ni j : K\nf : i \u27f6 j\n\u22a2 (colimitObjIsoColimitCompEvaluation F i).inv \u226b (colimit F).map f =\n    colimMap (whiskerLeft F ((evaluation K C).map f)) \u226b (colimitObjIsoColimitCompEvaluation F j).inv\n[PROOFSTEP]\next\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\ni j : K\nf : i \u27f6 j\nj\u271d : J\n\u22a2 colimit.\u03b9 (F \u22d9 (evaluation K C).obj i) j\u271d \u226b (colimitObjIsoColimitCompEvaluation F i).inv \u226b (colimit F).map f =\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj i) j\u271d \u226b\n      colimMap (whiskerLeft F ((evaluation K C).map f)) \u226b (colimitObjIsoColimitCompEvaluation F j).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\ni j : K\nf : i \u27f6 j\nj\u271d : J\n\u22a2 colimit.\u03b9 (F \u22d9 (evaluation K C).obj i) j\u271d \u226b (colimitObjIsoColimitCompEvaluation F i).inv \u226b (colimit F).map f =\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj i) j\u271d \u226b\n      colimMap (whiskerLeft F ((evaluation K C).map f)) \u226b (colimitObjIsoColimitCompEvaluation F j).inv\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimitsOfShape J C\nF : J \u2964 K \u2964 C\ni j : K\nf : i \u27f6 j\n\u22a2 (colimit F).map f \u226b (colimitObjIsoColimitCompEvaluation F j).hom =\n    (colimitObjIsoColimitCompEvaluation F i).hom \u226b colimMap (whiskerLeft F ((evaluation K C).map f))\n[PROOFSTEP]\nrw [\u2190 Iso.inv_comp_eq, \u2190 Category.assoc, \u2190 Iso.eq_comp_inv, colimitObjIsoColimitCompEvaluation_inv_colimit_map]\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), NatTrans.app (colimit.\u03b9 H j) k \u226b f = NatTrans.app (colimit.\u03b9 H j) k \u226b g\n\u22a2 f = g\n[PROOFSTEP]\napply (cancel_epi (colimitObjIsoColimitCompEvaluation H k).inv).1\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), NatTrans.app (colimit.\u03b9 H j) k \u226b f = NatTrans.app (colimit.\u03b9 H j) k \u226b g\n\u22a2 (colimitObjIsoColimitCompEvaluation H k).inv \u226b f = (colimitObjIsoColimitCompEvaluation H k).inv \u226b g\n[PROOFSTEP]\next j\n[GOAL]\ncase w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), NatTrans.app (colimit.\u03b9 H j) k \u226b f = NatTrans.app (colimit.\u03b9 H j) k \u226b g\nj : J\n\u22a2 colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b f =\n    colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b g\n[PROOFSTEP]\nsimpa using w j\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ\u271d : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\u271d\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimits C\nk : K\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\n\u22a2 PreservesLimitsOfShape J ((evaluation K C).obj k)\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ\u271d : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\u271d\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasLimits C\nk : K\nJ : Type v\n\ud835\udca5 : Category.{v, v} J\n\u22a2 PreservesLimitsOfShape J ((evaluation K C).obj k)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesLimit G (F \u22d9 (evaluation K C).obj k)\nc : Cone G\nhc : IsLimit c\n\u22a2 IsLimit (F.mapCone c)\n[PROOFSTEP]\napply evaluationJointlyReflectsLimits\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesLimit G (F \u22d9 (evaluation K C).obj k)\nc : Cone G\nhc : IsLimit c\n\u22a2 (k : K) \u2192 IsLimit (((evaluation K C).obj k).mapCone (F.mapCone c))\n[PROOFSTEP]\nintro X\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesLimit G (F \u22d9 (evaluation K C).obj k)\nc : Cone G\nhc : IsLimit c\nX : K\n\u22a2 IsLimit (((evaluation K C).obj X).mapCone (F.mapCone c))\n[PROOFSTEP]\nhaveI := H X\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesLimit G (F \u22d9 (evaluation K C).obj k)\nc : Cone G\nhc : IsLimit c\nX : K\nthis : PreservesLimit G (F \u22d9 (evaluation K C).obj X)\n\u22a2 IsLimit (((evaluation K C).obj X).mapCone (F.mapCone c))\n[PROOFSTEP]\nchange IsLimit ((F \u22d9 (evaluation K C).obj X).mapCone c)\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesLimit G (F \u22d9 (evaluation K C).obj k)\nc : Cone G\nhc : IsLimit c\nX : K\nthis : PreservesLimit G (F \u22d9 (evaluation K C).obj X)\n\u22a2 IsLimit ((F \u22d9 (evaluation K C).obj X).mapCone c)\n[PROOFSTEP]\nexact PreservesLimit.preserves hc\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimits C\nk : K\n\u22a2 {J : Type v} \u2192 [inst : Category.{v, v} J] \u2192 PreservesColimitsOfShape J ((evaluation K C).obj k)\n[PROOFSTEP]\nskip\n[GOAL]\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : HasColimits C\nk : K\n\u22a2 {J : Type v} \u2192 [inst : Category.{v, v} J] \u2192 PreservesColimitsOfShape J ((evaluation K C).obj k)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesColimit G (F \u22d9 (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\n\u22a2 IsColimit (F.mapCocone c)\n[PROOFSTEP]\napply evaluationJointlyReflectsColimits\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesColimit G (F \u22d9 (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\n\u22a2 (k : K) \u2192 IsColimit (((evaluation K C).obj k).mapCocone (F.mapCocone c))\n[PROOFSTEP]\nintro X\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesColimit G (F \u22d9 (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\nX : K\n\u22a2 IsColimit (((evaluation K C).obj X).mapCocone (F.mapCocone c))\n[PROOFSTEP]\nhaveI := H X\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesColimit G (F \u22d9 (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\nX : K\nthis : PreservesColimit G (F \u22d9 (evaluation K C).obj X)\n\u22a2 IsColimit (((evaluation K C).obj X).mapCocone (F.mapCocone c))\n[PROOFSTEP]\nchange IsColimit ((F \u22d9 (evaluation K C).obj X).mapCocone c)\n[GOAL]\ncase t\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b2 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} K\nF : D \u2964 K \u2964 C\nG : J \u2964 D\nH : (k : K) \u2192 PreservesColimit G (F \u22d9 (evaluation K C).obj k)\nc : Cocone G\nhc : IsColimit c\nX : K\nthis : PreservesColimit G (F \u22d9 (evaluation K C).obj X)\n\u22a2 IsColimit ((F \u22d9 (evaluation K C).obj X).mapCocone c)\n[PROOFSTEP]\nexact PreservesColimit.preserves hc\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Limits.FunctorCategory", "llama_tokens": 16930, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.6959583187272712, "lm_q1q2_score": 0.5273932965732884}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nobtain \u27e8d : \u211d, d_pos : 0 < d, hd : Euclidean.closedBall x d \u2286 s\u27e9 := Euclidean.nhds_basis_closedBall.mem_iff.1 hs\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nlet c : ContDiffBump (toEuclidean x) :=\n  { rIn := d / 2\n    rOut := d\n    rIn_pos := half_pos d_pos\n    rIn_lt_rOut := half_lt_self d_pos }\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nlet f : E \u2192 \u211d := c \u2218 toEuclidean\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nhave f_supp : f.support \u2286 Euclidean.ball x d := by\n  intro y hy\n  have : toEuclidean y \u2208 Function.support c := by\n    simpa only [Function.mem_support, Function.comp_apply, Ne.def] using hy\n  rwa [c.support_eq] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\n\u22a2 support f \u2286 Euclidean.ball x d\n[PROOFSTEP]\nintro y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\ny : E\nhy : y \u2208 support f\n\u22a2 y \u2208 Euclidean.ball x d\n[PROOFSTEP]\nhave : toEuclidean y \u2208 Function.support c := by simpa only [Function.mem_support, Function.comp_apply, Ne.def] using hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\ny : E\nhy : y \u2208 support f\n\u22a2 \u2191toEuclidean y \u2208 support \u2191c\n[PROOFSTEP]\nsimpa only [Function.mem_support, Function.comp_apply, Ne.def] using hy\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\ny : E\nhy : y \u2208 support f\nthis : \u2191toEuclidean y \u2208 support \u2191c\n\u22a2 y \u2208 Euclidean.ball x d\n[PROOFSTEP]\nrwa [c.support_eq] at this \n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nhave f_tsupp : tsupport f \u2286 Euclidean.closedBall x d :=\n  by\n  rw [tsupport, \u2190 Euclidean.closure_ball _ d_pos.ne']\n  exact closure_mono f_supp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\n\u22a2 tsupport f \u2286 Euclidean.closedBall x d\n[PROOFSTEP]\nrw [tsupport, \u2190 Euclidean.closure_ball _ d_pos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\n\u22a2 closure (support f) \u2286 closure (Euclidean.ball x d)\n[PROOFSTEP]\nexact closure_mono f_supp\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 \u2203 f, tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n[PROOFSTEP]\nrefine' \u27e8f, f_tsupp.trans hd, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 HasCompactSupport f\n[PROOFSTEP]\nrefine' isCompact_of_isClosed_bounded isClosed_closure _\n[GOAL]\ncase intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 Metric.Bounded (tsupport f)\n[PROOFSTEP]\nhave : Bounded (Euclidean.closedBall x d) := Euclidean.isCompact_closedBall.bounded\n[GOAL]\ncase intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\nthis : Metric.Bounded (Euclidean.closedBall x d)\n\u22a2 Metric.Bounded (tsupport f)\n[PROOFSTEP]\napply this.mono _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\nthis : Metric.Bounded (Euclidean.closedBall x d)\n\u22a2 tsupport f \u2286 Euclidean.closedBall x d\n[PROOFSTEP]\nrefine' (IsClosed.closure_subset_iff Euclidean.isClosed_closedBall).2 _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\nthis : Metric.Bounded (Euclidean.closedBall x d)\n\u22a2 support f \u2286 Euclidean.closedBall x d\n[PROOFSTEP]\nexact f_supp.trans Euclidean.ball_subset_closedBall\n[GOAL]\ncase intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 ContDiff \u211d \u22a4 f\n[PROOFSTEP]\napply c.contDiff.comp\n[GOAL]\ncase intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 ContDiff \u211d \u22a4 \u2191toEuclidean\n[PROOFSTEP]\nexact ContinuousLinearEquiv.contDiff _\n[GOAL]\ncase intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 range f \u2286 Icc 0 1\n[PROOFSTEP]\nrintro t \u27e8y, rfl\u27e9\n[GOAL]\ncase intro.intro.refine'_3.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\ny : E\n\u22a2 f y \u2208 Icc 0 1\n[PROOFSTEP]\nexact \u27e8c.nonneg, c.le_one\u27e9\n[GOAL]\ncase intro.intro.refine'_4\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 f x = 1\n[PROOFSTEP]\napply c.one_of_mem_closedBall\n[GOAL]\ncase intro.intro.refine'_4\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 \u2191toEuclidean x \u2208 closedBall (\u2191toEuclidean x) c.rIn\n[PROOFSTEP]\napply mem_closedBall_self\n[GOAL]\ncase intro.intro.refine'_4.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nx : E\nhs : s \u2208 \ud835\udcdd x\nd : \u211d\nd_pos : 0 < d\nhd : Euclidean.closedBall x d \u2286 s\nc : ContDiffBump (\u2191toEuclidean x) :=\n  { rIn := d / 2, rOut := d, rIn_pos := (_ : 0 < d / 2), rIn_lt_rOut := (_ : d / 2 < d) }\nf : E \u2192 \u211d := \u2191c \u2218 \u2191toEuclidean\nf_supp : support f \u2286 Euclidean.ball x d\nf_tsupp : tsupport f \u2286 Euclidean.closedBall x d\n\u22a2 0 \u2264 c.rIn\n[PROOFSTEP]\nexact (half_pos d_pos).le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | h's)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nhs : IsOpen \u2205\n\u22a2 \u2203 f, support f = \u2205 \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nexact\n  \u27e8fun _ => 0, Function.support_zero, contDiff_const, by\n    simp only [range_const, singleton_subset_iff, left_mem_Icc, zero_le_one]\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nhs : IsOpen \u2205\n\u22a2 (range fun x => 0) \u2286 Icc 0 1\n[PROOFSTEP]\nsimp only [range_const, singleton_subset_iff, left_mem_Icc, zero_le_one]\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nlet \u03b9 := { f : E \u2192 \u211d // f.support \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nobtain \u27e8T, T_count, hT\u27e9 : \u2203 T : Set \u03b9, T.Countable \u2227 \u22c3 f \u2208 T, support (f : E \u2192 \u211d) = s :=\n  by\n  have : \u22c3 f : \u03b9, (f : E \u2192 \u211d).support = s :=\n    by\n    refine' Subset.antisymm (iUnion_subset fun f => f.2.1) _\n    intro x hx\n    rcases exists_smooth_tsupport_subset (hs.mem_nhds hx) with \u27e8f, hf\u27e9\n    let g : \u03b9 := \u27e8f, (subset_tsupport f).trans hf.1, hf.2.1, hf.2.2.1, hf.2.2.2.1\u27e9\n    have : x \u2208 support (g : E \u2192 \u211d) := by\n      simp only [hf.2.2.2.2, Subtype.coe_mk, mem_support, Ne.def, one_ne_zero, not_false_iff]\n    exact mem_iUnion_of_mem _ this\n  simp_rw [\u2190 this]\n  apply isOpen_iUnion_countable\n  rintro \u27e8f, hf\u27e9\n  exact hf.2.2.1.continuous.isOpen_support\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\n[PROOFSTEP]\nhave : \u22c3 f : \u03b9, (f : E \u2192 \u211d).support = s :=\n  by\n  refine' Subset.antisymm (iUnion_subset fun f => f.2.1) _\n  intro x hx\n  rcases exists_smooth_tsupport_subset (hs.mem_nhds hx) with \u27e8f, hf\u27e9\n  let g : \u03b9 := \u27e8f, (subset_tsupport f).trans hf.1, hf.2.1, hf.2.2.1, hf.2.2.2.1\u27e9\n  have : x \u2208 support (g : E \u2192 \u211d) := by\n    simp only [hf.2.2.2.2, Subtype.coe_mk, mem_support, Ne.def, one_ne_zero, not_false_iff]\n  exact mem_iUnion_of_mem _ this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\n\u22a2 \u22c3 (f : \u03b9), support \u2191f = s\n[PROOFSTEP]\nrefine' Subset.antisymm (iUnion_subset fun f => f.2.1) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\n\u22a2 s \u2286 \u22c3 (f : \u03b9), support \u2191f\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nx : E\nhx : x \u2208 s\n\u22a2 x \u2208 \u22c3 (f : \u03b9), support \u2191f\n[PROOFSTEP]\nrcases exists_smooth_tsupport_subset (hs.mem_nhds hx) with \u27e8f, hf\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nx : E\nhx : x \u2208 s\nf : E \u2192 \u211d\nhf : tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\n\u22a2 x \u2208 \u22c3 (f : \u03b9), support \u2191f\n[PROOFSTEP]\nlet g : \u03b9 := \u27e8f, (subset_tsupport f).trans hf.1, hf.2.1, hf.2.2.1, hf.2.2.2.1\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nx : E\nhx : x \u2208 s\nf : E \u2192 \u211d\nhf : tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\ng : \u03b9 := { val := f, property := (_ : support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1) }\n\u22a2 x \u2208 \u22c3 (f : \u03b9), support \u2191f\n[PROOFSTEP]\nhave : x \u2208 support (g : E \u2192 \u211d) := by\n  simp only [hf.2.2.2.2, Subtype.coe_mk, mem_support, Ne.def, one_ne_zero, not_false_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nx : E\nhx : x \u2208 s\nf : E \u2192 \u211d\nhf : tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\ng : \u03b9 := { val := f, property := (_ : support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1) }\n\u22a2 x \u2208 support \u2191g\n[PROOFSTEP]\nsimp only [hf.2.2.2.2, Subtype.coe_mk, mem_support, Ne.def, one_ne_zero, not_false_iff]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nx : E\nhx : x \u2208 s\nf : E \u2192 \u211d\nhf : tsupport f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 \u2227 f x = 1\ng : \u03b9 := { val := f, property := (_ : support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1) }\nthis : x \u2208 support \u2191g\n\u22a2 x \u2208 \u22c3 (f : \u03b9), support \u2191f\n[PROOFSTEP]\nexact mem_iUnion_of_mem _ this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nthis : \u22c3 (f : \u03b9), support \u2191f = s\n\u22a2 \u2203 T, Set.Countable T \u2227 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\n[PROOFSTEP]\nsimp_rw [\u2190 this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nthis : \u22c3 (f : \u03b9), support \u2191f = s\n\u22a2 \u2203 T,\n    Set.Countable T \u2227\n      \u22c3 (f : { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }) (_ : f \u2208 T),\n          support \u2191f =\n        \u22c3 (f : { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }), support \u2191f\n[PROOFSTEP]\napply isOpen_iUnion_countable\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nthis : \u22c3 (f : \u03b9), support \u2191f = s\n\u22a2 \u2200 (i : { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }), IsOpen (support \u2191i)\n[PROOFSTEP]\nrintro \u27e8f, hf\u27e9\n[GOAL]\ncase H.mk\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nthis : \u22c3 (f : \u03b9), support \u2191f = s\nf : E \u2192 \u211d\nhf : support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n\u22a2 IsOpen (support \u2191{ val := f, property := hf })\n[PROOFSTEP]\nexact hf.2.2.1.continuous.isOpen_support\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nobtain \u27e8g0, hg\u27e9 : \u2203 g0 : \u2115 \u2192 \u03b9, T = range g0 :=\n  by\n  apply Countable.exists_eq_range T_count\n  rcases eq_empty_or_nonempty T with (rfl | hT)\n  \u00b7 simp only [iUnion_false, iUnion_empty] at hT \n    simp only [\u2190 hT, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.not_nonempty_empty] at h's \n  \u00b7 exact hT\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\n\u22a2 \u2203 g0, T = range g0\n[PROOFSTEP]\napply Countable.exists_eq_range T_count\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\n\u22a2 Set.Nonempty T\n[PROOFSTEP]\nrcases eq_empty_or_nonempty T with (rfl | hT)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT_count : Set.Countable \u2205\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 \u2205), support \u2191f = s\n\u22a2 Set.Nonempty \u2205\n[PROOFSTEP]\nsimp only [iUnion_false, iUnion_empty] at hT \n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT_count : Set.Countable \u2205\nhT :\n  \u22c3 (f : { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }) (_ : f \u2208 \u2205), support \u2191f = s\n\u22a2 Set.Nonempty \u2205\n[PROOFSTEP]\nsimp only [\u2190 hT, mem_empty_iff_false, iUnion_of_empty, iUnion_empty, Set.not_nonempty_empty] at h's \n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT\u271d : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\nhT : Set.Nonempty T\n\u22a2 Set.Nonempty T\n[PROOFSTEP]\nexact hT\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nlet g : \u2115 \u2192 E \u2192 \u211d := fun n => (g0 n).1\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave g_s : \u2200 n, support (g n) \u2286 s := fun n => (g0 n).2.1\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave s_g : \u2200 x \u2208 s, \u2203 n, x \u2208 support (g n) := by\n  intro x hx\n  rw [\u2190 hT] at hx \n  obtain \u27e8i, iT, hi\u27e9 : \u2203 (i : \u03b9) (_ : i \u2208 T), x \u2208 support (i : E \u2192 \u211d) := by simpa only [mem_iUnion] using hx\n  rw [hg, mem_range] at iT \n  rcases iT with \u27e8n, hn\u27e9\n  rw [\u2190 hn] at hi \n  exact \u27e8n, hi\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\n\u22a2 \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nintro x hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nrw [\u2190 hT] at hx \n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nobtain \u27e8i, iT, hi\u27e9 : \u2203 (i : \u03b9) (_ : i \u2208 T), x \u2208 support (i : E \u2192 \u211d) := by simpa only [mem_iUnion] using hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\n\u22a2 \u2203 i x_1, x \u2208 support \u2191i\n[PROOFSTEP]\nsimpa only [mem_iUnion] using hx\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\ni : \u03b9\niT : i \u2208 T\nhi : x \u2208 support \u2191i\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nrw [hg, mem_range] at iT \n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\ni : \u03b9\niT : \u2203 y, g0 y = i\nhi : x \u2208 support \u2191i\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nrcases iT with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\ni : \u03b9\nhi : x \u2208 support \u2191i\nn : \u2115\nhn : g0 n = i\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nrw [\u2190 hn] at hi \n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\nx : E\nhx : x \u2208 \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f\ni : \u03b9\nn : \u2115\nhi : x \u2208 support \u2191(g0 n)\nhn : g0 n = i\n\u22a2 \u2203 n, x \u2208 support (g n)\n[PROOFSTEP]\nexact \u27e8n, hi\u27e9\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave g_smooth : \u2200 n, ContDiff \u211d \u22a4 (g n) := fun n => (g0 n).2.2.2.1\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave g_comp_supp : \u2200 n, HasCompactSupport (g n) := fun n => (g0 n).2.2.1\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave g_nonneg : \u2200 n x, 0 \u2264 g n x := fun n x => ((g0 n).2.2.2.2 (mem_range_self x)).1\n[GOAL]\ncase inr.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4pos, c, \u03b4c, c_lt\u27e9 : \u2203 \u03b4 : \u2115 \u2192 \u211d\u22650, (\u2200 i : \u2115, 0 < \u03b4 i) \u2227 \u2203 c : NNReal, HasSum \u03b4 c \u2227 c < 1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u22a2 \u2203 \u03b4, (\u2200 (i : \u2115), 0 < \u03b4 i) \u2227 \u2203 c, HasSum \u03b4 c \u2227 c < 1\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nexact NNReal.exists_pos_sum_of_countable one_ne_zero \u2115\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave : \u2200 n : \u2115, \u2203 r : \u211d, 0 < r \u2227 \u2200 i \u2264 n, \u2200 x, \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u03b4 n :=\n  by\n  intro n\n  have : \u2200 i, \u2203 R, \u2200 x, \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R :=\n    by\n    intro i\n    have : BddAbove (range fun x => \u2016iteratedFDeriv \u211d i (fun x : E => g n x) x\u2016) :=\n      by\n      apply ((g_smooth n).continuous_iteratedFDeriv le_top).norm.bddAbove_range_of_hasCompactSupport\n      apply HasCompactSupport.comp_left _ norm_zero\n      apply (g_comp_supp n).iteratedFDeriv\n    rcases this with \u27e8R, hR\u27e9\n    exact \u27e8R, fun x => hR (mem_range_self _)\u27e9\n  choose R hR using this\n  let M := max (((Finset.range (n + 1)).image R).max' (by simp)) 1\n  have M_pos : 0 < M := zero_lt_one.trans_le (le_max_right _ _)\n  have \u03b4npos : 0 < \u03b4 n := \u03b4pos n\n  have IR : \u2200 i \u2264 n, R i \u2264 M := by\n    intro i hi\n    refine' le_trans _ (le_max_left _ _)\n    apply Finset.le_max'\n    apply Finset.mem_image_of_mem\n    simpa only [Finset.mem_range, Nat.lt_add_one_iff]\n  refine' \u27e8M\u207b\u00b9 * \u03b4 n, by positivity, fun i hi x => _\u27e9\n  calc\n    \u2016iteratedFDeriv \u211d i ((M\u207b\u00b9 * \u03b4 n) \u2022 g n) x\u2016 = \u2016(M\u207b\u00b9 * \u03b4 n) \u2022 iteratedFDeriv \u211d i (g n) x\u2016 := by\n      rw [iteratedFDeriv_const_smul_apply]; exact (g_smooth n).of_le le_top\n    _ = M\u207b\u00b9 * \u03b4 n * \u2016iteratedFDeriv \u211d i (g n) x\u2016 := by rw [norm_smul, Real.norm_of_nonneg]; positivity\n    _ \u2264 M\u207b\u00b9 * \u03b4 n * M := (mul_le_mul_of_nonneg_left ((hR i x).trans (IR i hi)) (by positivity))\n    _ = \u03b4 n := by field_simp [M_pos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\n\u22a2 \u2200 (n : \u2115), \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nhave : \u2200 i, \u2203 R, \u2200 x, \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R :=\n  by\n  intro i\n  have : BddAbove (range fun x => \u2016iteratedFDeriv \u211d i (fun x : E => g n x) x\u2016) :=\n    by\n    apply ((g_smooth n).continuous_iteratedFDeriv le_top).norm.bddAbove_range_of_hasCompactSupport\n    apply HasCompactSupport.comp_left _ norm_zero\n    apply (g_comp_supp n).iteratedFDeriv\n  rcases this with \u27e8R, hR\u27e9\n  exact \u27e8R, fun x => hR (mem_range_self _)\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\n\u22a2 \u2200 (i : \u2115), \u2203 R, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R\n[PROOFSTEP]\nintro i\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\n\u22a2 \u2203 R, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R\n[PROOFSTEP]\nhave : BddAbove (range fun x => \u2016iteratedFDeriv \u211d i (fun x : E => g n x) x\u2016) :=\n  by\n  apply ((g_smooth n).continuous_iteratedFDeriv le_top).norm.bddAbove_range_of_hasCompactSupport\n  apply HasCompactSupport.comp_left _ norm_zero\n  apply (g_comp_supp n).iteratedFDeriv\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\n\u22a2 BddAbove (range fun x => \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016)\n[PROOFSTEP]\napply ((g_smooth n).continuous_iteratedFDeriv le_top).norm.bddAbove_range_of_hasCompactSupport\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\n\u22a2 HasCompactSupport fun x => \u2016iteratedFDeriv \u211d i (g n) x\u2016\n[PROOFSTEP]\napply HasCompactSupport.comp_left _ norm_zero\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\n\u22a2 HasCompactSupport fun x => iteratedFDeriv \u211d i (g n) x\n[PROOFSTEP]\napply (g_comp_supp n).iteratedFDeriv\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\nthis : BddAbove (range fun x => \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016)\n\u22a2 \u2203 R, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R\n[PROOFSTEP]\nrcases this with \u27e8R, hR\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn i : \u2115\nR : \u211d\nhR : R \u2208 upperBounds (range fun x => \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016)\n\u22a2 \u2203 R, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R\n[PROOFSTEP]\nexact \u27e8R, fun x => hR (mem_range_self _)\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nthis : \u2200 (i : \u2115), \u2203 R, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nchoose R hR using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nlet M := max (((Finset.range (n + 1)).image R).max' (by simp)) 1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\n\u22a2 Finset.Nonempty (Finset.image R (Finset.range (n + 1)))\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nhave M_pos : 0 < M := zero_lt_one.trans_le (le_max_right _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nhave \u03b4npos : 0 < \u03b4 n := \u03b4pos n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nhave IR : \u2200 i \u2264 n, R i \u2264 M := by\n  intro i hi\n  refine' le_trans _ (le_max_left _ _)\n  apply Finset.le_max'\n  apply Finset.mem_image_of_mem\n  simpa only [Finset.mem_range, Nat.lt_add_one_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\n\u22a2 \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\n[PROOFSTEP]\nintro i hi\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\ni : \u2115\nhi : i \u2264 n\n\u22a2 R i \u2264 M\n[PROOFSTEP]\nrefine' le_trans _ (le_max_left _ _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\ni : \u2115\nhi : i \u2264 n\n\u22a2 R i \u2264\n    Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1))))\n[PROOFSTEP]\napply Finset.le_max'\n[GOAL]\ncase H2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\ni : \u2115\nhi : i \u2264 n\n\u22a2 R i \u2208 Finset.image R (Finset.range (n + 1))\n[PROOFSTEP]\napply Finset.mem_image_of_mem\n[GOAL]\ncase H2.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\ni : \u2115\nhi : i \u2264 n\n\u22a2 i \u2208 Finset.range (n + 1)\n[PROOFSTEP]\nsimpa only [Finset.mem_range, Nat.lt_add_one_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\n\u22a2 \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nrefine' \u27e8M\u207b\u00b9 * \u03b4 n, by positivity, fun i hi x => _\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\n\u22a2 0 < M\u207b\u00b9 * \u2191(\u03b4 n)\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d i ((M\u207b\u00b9 * \u2191(\u03b4 n)) \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\ncalc\n  \u2016iteratedFDeriv \u211d i ((M\u207b\u00b9 * \u03b4 n) \u2022 g n) x\u2016 = \u2016(M\u207b\u00b9 * \u03b4 n) \u2022 iteratedFDeriv \u211d i (g n) x\u2016 := by\n    rw [iteratedFDeriv_const_smul_apply]; exact (g_smooth n).of_le le_top\n  _ = M\u207b\u00b9 * \u03b4 n * \u2016iteratedFDeriv \u211d i (g n) x\u2016 := by rw [norm_smul, Real.norm_of_nonneg]; positivity\n  _ \u2264 M\u207b\u00b9 * \u03b4 n * M := (mul_le_mul_of_nonneg_left ((hR i x).trans (IR i hi)) (by positivity))\n  _ = \u03b4 n := by field_simp [M_pos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 \u2016iteratedFDeriv \u211d i ((M\u207b\u00b9 * \u2191(\u03b4 n)) \u2022 g n) x\u2016 = \u2016(M\u207b\u00b9 * \u2191(\u03b4 n)) \u2022 iteratedFDeriv \u211d i (g n) x\u2016\n[PROOFSTEP]\nrw [iteratedFDeriv_const_smul_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 ContDiff \u211d (\u2191i) (g n)\n[PROOFSTEP]\nexact (g_smooth n).of_le le_top\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 \u2016(M\u207b\u00b9 * \u2191(\u03b4 n)) \u2022 iteratedFDeriv \u211d i (g n) x\u2016 = M\u207b\u00b9 * \u2191(\u03b4 n) * \u2016iteratedFDeriv \u211d i (g n) x\u2016\n[PROOFSTEP]\nrw [norm_smul, Real.norm_of_nonneg]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 0 \u2264 M\u207b\u00b9 * \u2191(\u03b4 n)\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 0 \u2264 M\u207b\u00b9 * \u2191(\u03b4 n)\n[PROOFSTEP]\npositivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nn : \u2115\nR : \u2115 \u2192 \u211d\nhR : \u2200 (i : \u2115) (x : E), \u2016iteratedFDeriv \u211d i (fun x => g n x) x\u2016 \u2264 R i\nM : \u211d :=\n  max\n    (Finset.max' (Finset.image R (Finset.range (n + 1))) (_ : Finset.Nonempty (Finset.image R (Finset.range (n + 1)))))\n    1\nM_pos : 0 < M\n\u03b4npos : 0 < \u03b4 n\nIR : \u2200 (i : \u2115), i \u2264 n \u2192 R i \u2264 M\ni : \u2115\nhi : i \u2264 n\nx : E\n\u22a2 M\u207b\u00b9 * \u2191(\u03b4 n) * M = \u2191(\u03b4 n)\n[PROOFSTEP]\nfield_simp [M_pos.ne']\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nthis : \u2200 (n : \u2115), \u2203 r, 0 < r \u2227 \u2200 (i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nchoose r rpos hr using this\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nhave S : \u2200 x, Summable fun n => (r n \u2022 g n) x := by\n  intro x\n  refine' summable_of_nnnorm_bounded _ \u03b4c.summable fun n => _\n  rw [\u2190 NNReal.coe_le_coe, coe_nnnorm]\n  simpa only [norm_iteratedFDeriv_zero] using hr n 0 (zero_le n) x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n\u22a2 \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n[PROOFSTEP]\nintro x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nx : E\n\u22a2 Summable fun n => (r n \u2022 g n) x\n[PROOFSTEP]\nrefine' summable_of_nnnorm_bounded _ \u03b4c.summable fun n => _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nx : E\nn : \u2115\n\u22a2 \u2016(r n \u2022 g n) x\u2016\u208a \u2264 \u03b4 n\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_le_coe, coe_nnnorm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nx : E\nn : \u2115\n\u22a2 \u2016(r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nsimpa only [norm_iteratedFDeriv_zero] using hr n 0 (zero_le n) x\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 \u2203 f, support f = s \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\n[PROOFSTEP]\nrefine' \u27e8fun x => \u2211' n, (r n \u2022 g n) x, _, _, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 (support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) = s\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 (support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) \u2286 s\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nsimp only [Pi.smul_apply, Algebra.id.smul_eq_mul, mem_support, Ne.def] at hx \n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : \u00ac\u2211' (n : \u2115), r n * \u2191(g0 n) x = 0\n\u22a2 x \u2208 s\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : \u00acx \u2208 s\n\u22a2 \u2211' (n : \u2115), r n * \u2191(g0 n) x = 0\n[PROOFSTEP]\nhave : \u2200 n, g n x = 0 := by\n  intro n\n  contrapose! hx\n  exact g_s n hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : \u00acx \u2208 s\n\u22a2 \u2200 (n : \u2115), g n x = 0\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : \u00acx \u2208 s\nn : \u2115\n\u22a2 g n x = 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nn : \u2115\nhx : (fun n => \u2191(g0 n)) n x \u2260 0\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact g_s n hx\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : \u00acx \u2208 s\nthis : \u2200 (n : \u2115), g n x = 0\n\u22a2 \u2211' (n : \u2115), r n * \u2191(g0 n) x = 0\n[PROOFSTEP]\nsimp only [this, mul_zero, tsum_zero]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 s \u2286 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 s\n\u22a2 x \u2208 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, x \u2208 support (g n)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 s\n\u22a2 \u2203 n, x \u2208 support (g n)\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2082.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 support (g n)\n\u22a2 x \u2208 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nexact s_g x hx\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2082.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 support (g n)\n\u22a2 x \u2208 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nhave I : 0 < r n * g n x := mul_pos (rpos n) (lt_of_le_of_ne (g_nonneg n x) (Ne.symm hn))\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\u2082.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\nx : E\nhx : x \u2208 s\nn : \u2115\nhn : x \u2208 support (g n)\nI : 0 < r n * g n x\n\u22a2 x \u2208 support fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nexact ne_of_gt (tsum_pos (S x) (fun i => mul_nonneg (rpos i).le (g_nonneg i x)) n I)\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 ContDiff \u211d \u22a4 fun x => \u2211' (n : \u2115), (r n \u2022 g n) x\n[PROOFSTEP]\nrefine'\n  contDiff_tsum_of_eventually (fun n => (g_smooth n).const_smul (r n)) (fun k _ => (NNReal.hasSum_coe.2 \u03b4c).summable) _\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 \u2200 (k : \u2115), \u2191k \u2264 \u22a4 \u2192 \u2200\u1da0 (i : \u2115) in Filter.cofinite, \u2200 (x : E), \u2016iteratedFDeriv \u211d k (fun x => (r i \u2022 g i) x) x\u2016 \u2264 \u2191(\u03b4 i)\n[PROOFSTEP]\nintro i _\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ni : \u2115\na\u271d : \u2191i \u2264 \u22a4\n\u22a2 \u2200\u1da0 (i_1 : \u2115) in Filter.cofinite, \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => (r i_1 \u2022 g i_1) x) x\u2016 \u2264 \u2191(\u03b4 i_1)\n[PROOFSTEP]\nsimp only [Nat.cofinite_eq_atTop, Pi.smul_apply, Algebra.id.smul_eq_mul, Filter.eventually_atTop, ge_iff_le]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ni : \u2115\na\u271d : \u2191i \u2264 \u22a4\n\u22a2 \u2203 a, \u2200 (b : \u2115), a \u2264 b \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (fun x => r b * \u2191(g0 b) x) x\u2016 \u2264 \u2191(\u03b4 b)\n[PROOFSTEP]\nexact \u27e8i, fun n hn x => hr _ _ hn _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\n\u22a2 (range fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) \u2286 Icc 0 1\n[PROOFSTEP]\nrintro - \u27e8y, rfl\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\n\u22a2 (fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) y \u2208 Icc 0 1\n[PROOFSTEP]\nrefine' \u27e8tsum_nonneg fun n => mul_nonneg (rpos n).le (g_nonneg n y), le_trans _ c_lt.le\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\n\u22a2 (fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) y \u2264 (fun a => \u2191a) c\n[PROOFSTEP]\nhave A : HasSum (fun n => (\u03b4 n : \u211d)) c := NNReal.hasSum_coe.2 \u03b4c\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\nA : HasSum (fun n => \u2191(\u03b4 n)) \u2191c\n\u22a2 (fun x => \u2211' (n : \u2115), (r n \u2022 g n) x) y \u2264 (fun a => \u2191a) c\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_eq_mul, NNReal.val_eq_coe, \u2190 A.tsum_eq, ge_iff_le]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.refine'_3.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\nA : HasSum (fun n => \u2191(\u03b4 n)) \u2191c\n\u22a2 \u2211' (n : \u2115), r n * \u2191(g0 n) y \u2264 \u2211' (b : \u2115), \u2191(\u03b4 b)\n[PROOFSTEP]\napply tsum_le_tsum _ (S y) A.summable\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\nA : HasSum (fun n => \u2191(\u03b4 n)) \u2191c\n\u22a2 \u2200 (i : \u2115), (r i \u2022 g i) y \u2264 \u2191(\u03b4 i)\n[PROOFSTEP]\nintro n\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\nA : HasSum (fun n => \u2191(\u03b4 n)) \u2191c\nn : \u2115\n\u22a2 (r n \u2022 g n) y \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\napply (le_abs_self _).trans\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Set E\nhs : IsOpen s\nh's : Set.Nonempty s\n\u03b9 : Type u_1 := { f // support f \u2286 s \u2227 HasCompactSupport f \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1 }\nT : Set \u03b9\nT_count : Set.Countable T\nhT : \u22c3 (f : \u03b9) (_ : f \u2208 T), support \u2191f = s\ng0 : \u2115 \u2192 \u03b9\nhg : T = range g0\ng : \u2115 \u2192 E \u2192 \u211d := fun n => \u2191(g0 n)\ng_s : \u2200 (n : \u2115), support (g n) \u2286 s\ns_g : \u2200 (x : E), x \u2208 s \u2192 \u2203 n, x \u2208 support (g n)\ng_smooth : \u2200 (n : \u2115), ContDiff \u211d \u22a4 (g n)\ng_comp_supp : \u2200 (n : \u2115), HasCompactSupport (g n)\ng_nonneg : \u2200 (n : \u2115) (x : E), 0 \u2264 g n x\n\u03b4 : \u2115 \u2192 \u211d\u22650\n\u03b4pos : \u2200 (i : \u2115), 0 < \u03b4 i\nc : \u211d\u22650\n\u03b4c : HasSum \u03b4 c\nc_lt : c < 1\nr : \u2115 \u2192 \u211d\nrpos : \u2200 (n : \u2115), 0 < r n\nhr : \u2200 (n i : \u2115), i \u2264 n \u2192 \u2200 (x : E), \u2016iteratedFDeriv \u211d i (r n \u2022 g n) x\u2016 \u2264 \u2191(\u03b4 n)\nS : \u2200 (x : E), Summable fun n => (r n \u2022 g n) x\ny : E\nA : HasSum (fun n => \u2191(\u03b4 n)) \u2191c\nn : \u2115\n\u22a2 |(r n \u2022 g n) y| \u2264 \u2191(\u03b4 n)\n[PROOFSTEP]\nsimpa only [norm_iteratedFDeriv_zero] using hr n 0 (zero_le n) y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 \u2203 u, ContDiff \u211d \u22a4 u \u2227 (\u2200 (x : E), u x \u2208 Icc 0 1) \u2227 support u = ball 0 1 \u2227 \u2200 (x : E), u (-x) = u x\n[PROOFSTEP]\nhave A : IsOpen (ball (0 : E) 1) := isOpen_ball\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\n\u22a2 \u2203 u, ContDiff \u211d \u22a4 u \u2227 (\u2200 (x : E), u x \u2208 Icc 0 1) \u2227 support u = ball 0 1 \u2227 \u2200 (x : E), u (-x) = u x\n[PROOFSTEP]\nobtain \u27e8f, f_support, f_smooth, f_range\u27e9 :\n  \u2203 f : E \u2192 \u211d, f.support = ball (0 : E) 1 \u2227 ContDiff \u211d \u22a4 f \u2227 Set.range f \u2286 Set.Icc 0 1\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\n\u22a2 \u2203 f, support f = ball 0 1 \u2227 ContDiff \u211d \u22a4 f \u2227 range f \u2286 Icc 0 1\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\n\u22a2 \u2203 u, ContDiff \u211d \u22a4 u \u2227 (\u2200 (x : E), u x \u2208 Icc 0 1) \u2227 support u = ball 0 1 \u2227 \u2200 (x : E), u (-x) = u x\n[PROOFSTEP]\nexact A.exists_smooth_support_eq\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\n\u22a2 \u2203 u, ContDiff \u211d \u22a4 u \u2227 (\u2200 (x : E), u x \u2208 Icc 0 1) \u2227 support u = ball 0 1 \u2227 \u2200 (x : E), u (-x) = u x\n[PROOFSTEP]\nhave B : \u2200 x, f x \u2208 Icc (0 : \u211d) 1 := fun x => f_range (mem_range_self x)\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\n\u22a2 \u2203 u, ContDiff \u211d \u22a4 u \u2227 (\u2200 (x : E), u x \u2208 Icc 0 1) \u2227 support u = ball 0 1 \u2227 \u2200 (x : E), u (-x) = u x\n[PROOFSTEP]\nrefine' \u27e8fun x => (f x + f (-x)) / 2, _, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\n\u22a2 ContDiff \u211d \u22a4 fun x => (f x + f (-x)) / 2\n[PROOFSTEP]\nexact (f_smooth.add (f_smooth.comp contDiff_neg)).div_const _\n[GOAL]\ncase intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\n\u22a2 \u2200 (x : E), (fun x => (f x + f (-x)) / 2) x \u2208 Icc 0 1\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\n\u22a2 (fun x => (f x + f (-x)) / 2) x \u2208 Icc 0 1\n[PROOFSTEP]\nsimp only [mem_Icc]\n[GOAL]\ncase intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\n\u22a2 0 \u2264 (f x + f (-x)) / 2 \u2227 (f x + f (-x)) / 2 \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.refine'_2.left\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\n\u22a2 0 \u2264 (f x + f (-x)) / 2\n[PROOFSTEP]\nlinarith [(B x).1, (B (-x)).1]\n[GOAL]\ncase intro.intro.intro.refine'_2.right\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\n\u22a2 (f x + f (-x)) / 2 \u2264 1\n[PROOFSTEP]\nlinarith [(B x).2, (B (-x)).2]\n[GOAL]\ncase intro.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\n\u22a2 (support fun x => (f x + f (-x)) / 2) = ball 0 1\n[PROOFSTEP]\nrefine' support_eq_iff.2 \u27e8fun x hx => _, fun x hx => _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_1\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : x \u2208 ball 0 1\n\u22a2 (f x + f (-x)) / 2 \u2260 0\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_1.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : x \u2208 ball 0 1\n\u22a2 0 < (f x + f (-x)) / 2\n[PROOFSTEP]\nhave : 0 < f x := by\n  apply lt_of_le_of_ne (B x).1 (Ne.symm _)\n  rwa [\u2190 f_support] at hx \n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : x \u2208 ball 0 1\n\u22a2 0 < f x\n[PROOFSTEP]\napply lt_of_le_of_ne (B x).1 (Ne.symm _)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : x \u2208 ball 0 1\n\u22a2 f x \u2260 0\n[PROOFSTEP]\nrwa [\u2190 f_support] at hx \n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_1.h\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : x \u2208 ball 0 1\nthis : 0 < f x\n\u22a2 0 < (f x + f (-x)) / 2\n[PROOFSTEP]\nlinarith [(B (-x)).1]\n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\n\u22a2 (f x + f (-x)) / 2 = 0\n[PROOFSTEP]\nhave I1 : x \u2209 support f := by rwa [f_support]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\n\u22a2 \u00acx \u2208 support f\n[PROOFSTEP]\nrwa [f_support]\n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\nI1 : \u00acx \u2208 support f\n\u22a2 (f x + f (-x)) / 2 = 0\n[PROOFSTEP]\nhave I2 : -x \u2209 support f := by\n  rw [f_support]\n  simpa using hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\nI1 : \u00acx \u2208 support f\n\u22a2 \u00ac-x \u2208 support f\n[PROOFSTEP]\nrw [f_support]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\nI1 : \u00acx \u2208 support f\n\u22a2 \u00ac-x \u2208 ball 0 1\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\nI1 : \u00acx \u2208 support f\nI2 : \u00ac-x \u2208 support f\n\u22a2 (f x + f (-x)) / 2 = 0\n[PROOFSTEP]\nsimp only [mem_support, Classical.not_not] at I1 I2 \n[GOAL]\ncase intro.intro.intro.refine'_3.refine'_2\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\nhx : \u00acx \u2208 ball 0 1\nI1 : f x = 0\nI2 : f (-x) = 0\n\u22a2 (f x + f (-x)) / 2 = 0\n[PROOFSTEP]\nsimp only [I1, I2, add_zero, zero_div]\n[GOAL]\ncase intro.intro.intro.refine'_4\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\n\u22a2 \u2200 (x : E), (fun x => (f x + f (-x)) / 2) (-x) = (fun x => (f x + f (-x)) / 2) x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase intro.intro.intro.refine'_4\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nA : IsOpen (ball 0 1)\nf : E \u2192 \u211d\nf_support : support f = ball 0 1\nf_smooth : ContDiff \u211d \u22a4 f\nf_range : range f \u2286 Icc 0 1\nB : \u2200 (x : E), f x \u2208 Icc 0 1\nx : E\n\u22a2 (fun x => (f x + f (-x)) / 2) (-x) = (fun x => (f x + f (-x)) / 2) x\n[PROOFSTEP]\nsimp only [add_comm, neg_neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 HasCompactSupport u\n[PROOFSTEP]\nrw [hasCompactSupport_def, u_support, closure_ball (0 : E) one_ne_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 IsCompact (closedBall 0 1)\n[PROOFSTEP]\nexact isCompact_closedBall _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\n\u22a2 0 < \u222b (x : E), u x \u2202\u03bc\n[PROOFSTEP]\nrefine' (integral_pos_iff_support_of_nonneg u_nonneg _).mpr _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\n\u22a2 Integrable fun i => u i\n[PROOFSTEP]\nexact (u_continuous E).integrable_of_hasCompactSupport (u_compact_support E)\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => u i)\n[PROOFSTEP]\nrw [u_support]\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\n\u22a2 0 < \u2191\u2191\u03bc (ball 0 1)\n[PROOFSTEP]\nexact measure_ball_pos _ _ zero_lt_one\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\n\u22a2 w D = fun x => ((\u222b (x : E), u x \u2202\u03bc) * |D| ^ \u2191(finrank \u211d E))\u207b\u00b9 \u2022 u (D\u207b\u00b9 \u2022 x)\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 w D x = ((\u222b (x : E), u x \u2202\u03bc) * |D| ^ \u2191(finrank \u211d E))\u207b\u00b9 \u2022 u (D\u207b\u00b9 \u2022 x)\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 0 \u2264 w D x\n[PROOFSTEP]\napply mul_nonneg _ (u_nonneg _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 0 \u2264 ((\u222b (x : E), u x \u2202\u03bc) * |D| ^ \u2191(finrank \u211d E))\u207b\u00b9\n[PROOFSTEP]\napply inv_nonneg.2\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 0 \u2264 (\u222b (x : E), u x \u2202\u03bc) * |D| ^ \u2191(finrank \u211d E)\n[PROOFSTEP]\napply mul_nonneg (u_int_pos E).le\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 0 \u2264 |D| ^ \u2191(finrank \u211d E)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 0 \u2264 |D| ^ finrank \u211d E\n[PROOFSTEP]\napply pow_nonneg (abs_nonneg D)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx y : E\n\u22a2 \u2200 (a : E), a \u2208 closedBall 0 1 \u2192 0 \u2264 1\n[PROOFSTEP]\nsimp only [zero_le_one, imp_true_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 \u222b (x : E), w D x \u2202\u03bc = 1\n[PROOFSTEP]\nsimp_rw [w, integral_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 ((\u222b (x : E), u x \u2202\u03bc) * |D| ^ \u2191(finrank \u211d E))\u207b\u00b9 \u2022 \u222b (a : E), u (D\u207b\u00b9 \u2022 a) \u2202\u03bc = 1\n[PROOFSTEP]\nrw [integral_comp_inv_smul_of_nonneg \u03bc (u : E \u2192 \u211d) Dpos.le, abs_of_nonneg Dpos.le, mul_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 (D ^ \u2191(finrank \u211d E) * \u222b (x : E), u x \u2202\u03bc)\u207b\u00b9 \u2022 D ^ finrank \u211d E \u2022 \u222b (x : E), u x \u2202\u03bc = 1\n[PROOFSTEP]\nfield_simp [Dpos.ne', (u_int_pos E).ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 support (w D) = ball 0 D\n[PROOFSTEP]\nhave B : D \u2022 ball (0 : E) 1 = ball 0 D := by rw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 D \u2022 ball 0 1 = ball 0 D\n[PROOFSTEP]\nrw [smul_unitBall Dpos.ne', Real.norm_of_nonneg Dpos.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\nB : D \u2022 ball 0 1 = ball 0 D\n\u22a2 support (w D) = ball 0 D\n[PROOFSTEP]\nhave C : D ^ finrank \u211d E \u2260 0 := by\n  norm_cast\n  exact pow_ne_zero _ Dpos.ne'\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\nB : D \u2022 ball 0 1 = ball 0 D\n\u22a2 D ^ \u2191(finrank \u211d E) \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\nB : D \u2022 ball 0 1 = ball 0 D\n\u22a2 \u00acD ^ finrank \u211d E = 0\n[PROOFSTEP]\nexact pow_ne_zero _ Dpos.ne'\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\nB : D \u2022 ball 0 1 = ball 0 D\nC : D ^ \u2191(finrank \u211d E) \u2260 0\n\u22a2 support (w D) = ball 0 D\n[PROOFSTEP]\nsimp only [w_def, Algebra.id.smul_eq_mul, support_mul, support_inv, univ_inter, support_comp_inv_smul\u2080 Dpos.ne',\n  u_support, B, support_const (u_int_pos E).ne', support_const C, abs_of_nonneg Dpos.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 HasCompactSupport (w D)\n[PROOFSTEP]\nrw [hasCompactSupport_def, w_support E Dpos, closure_ball (0 : E) Dpos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nDpos : 0 < D\n\u22a2 IsCompact (closedBall 0 D)\n[PROOFSTEP]\nexact isCompact_closedBall _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 y D (-x) = y D x\n[PROOFSTEP]\napply convolution_neg_of_neg_eq\n[GOAL]\ncase h1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, w D (-x) = w D x\n[PROOFSTEP]\napply eventually_of_forall fun x => _\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 \u2200 (x : E), w D (-x) = w D x\n[PROOFSTEP]\nsimp only [w_def, Real.rpow_nat_cast, mul_inv_rev, smul_neg, u_neg, smul_eq_mul, forall_const]\n[GOAL]\ncase h2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 \u2200\u1d50 (x : E) \u2202\u03bc, \u03c6 (-x) = \u03c6 x\n[PROOFSTEP]\napply eventually_of_forall fun x => _\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\n\u22a2 \u2200 (x : E), \u03c6 (-x) = \u03c6 x\n[PROOFSTEP]\nsimp only [\u03c6, indicator, mem_closedBall, dist_zero_right, norm_neg, forall_const]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\n\u22a2 y D x = 1\n[PROOFSTEP]\nchange (w D \u22c6[lsmul \u211d \u211d, \u03bc] \u03c6) x = 1\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 1\n[PROOFSTEP]\nhave B : \u2200 y : E, y \u2208 ball x D \u2192 \u03c6 y = 1 :=\n  by\n  have C : ball x D \u2286 ball 0 1 := by\n    apply ball_subset_ball'\n    simp only [mem_closedBall] at hx \n    linarith only [hx]\n  intro y hy\n  simp only [\u03c6, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]\n  intro h'y\n  linarith only [mem_ball.1 (C hy), h'y]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\n[PROOFSTEP]\nhave C : ball x D \u2286 ball 0 1 := by\n  apply ball_subset_ball'\n  simp only [mem_closedBall] at hx \n  linarith only [hx]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\n\u22a2 ball x D \u2286 ball 0 1\n[PROOFSTEP]\napply ball_subset_ball'\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\n\u22a2 D + dist x 0 \u2264 1\n[PROOFSTEP]\nsimp only [mem_closedBall] at hx \n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : dist x 0 \u2264 1 - D\n\u22a2 D + dist x 0 \u2264 1\n[PROOFSTEP]\nlinarith only [hx]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nC : ball x D \u2286 ball 0 1\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\n[PROOFSTEP]\nintro y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nC : ball x D \u2286 ball 0 1\ny : E\nhy : y \u2208 ball x D\n\u22a2 \u03c6 y = 1\n[PROOFSTEP]\nsimp only [\u03c6, indicator, mem_closedBall, ite_eq_left_iff, not_le, zero_ne_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nC : ball x D \u2286 ball 0 1\ny : E\nhy : y \u2208 ball x D\n\u22a2 1 < dist y 0 \u2192 False\n[PROOFSTEP]\nintro h'y\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nC : ball x D \u2286 ball 0 1\ny : E\nhy : y \u2208 ball x D\nh'y : 1 < dist y 0\n\u22a2 False\n[PROOFSTEP]\nlinarith only [mem_ball.1 (C hy), h'y]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 1\n[PROOFSTEP]\nhave Bx : \u03c6 x = 1 := B _ (mem_ball_self Dpos)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\nBx : \u03c6 x = 1\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 1\n[PROOFSTEP]\nhave B' : \u2200 y, y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x := by rw [Bx]; exact B\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\nBx : \u03c6 x = 1\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n[PROOFSTEP]\nrw [Bx]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\nBx : \u03c6 x = 1\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\n[PROOFSTEP]\nexact B\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\nBx : \u03c6 x = 1\nB' : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 1\n[PROOFSTEP]\nrw [convolution_eq_right' _ (le_of_eq (w_support E Dpos)) B']\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : x \u2208 closedBall 0 (1 - D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 1\nBx : \u03c6 x = 1\nB' : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n\u22a2 \u222b (t : E), \u2191(\u2191(lsmul \u211d \u211d) (w D t)) (\u03c6 x) \u2202\u03bc = 1\n[PROOFSTEP]\nsimp only [lsmul_apply, Algebra.id.smul_eq_mul, integral_mul_right, w_integral E Dpos, Bx, one_mul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\n\u22a2 y D x = 0\n[PROOFSTEP]\nchange (w D \u22c6[lsmul \u211d \u211d, \u03bc] \u03c6) x = 0\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 0\n[PROOFSTEP]\nhave B : \u2200 y, y \u2208 ball x D \u2192 \u03c6 y = 0 := by\n  intro y hy\n  simp only [\u03c6, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]\n  intro h'y\n  have C : ball y D \u2286 ball 0 (1 + D) := by\n    apply ball_subset_ball'\n    rw [\u2190 dist_zero_right] at h'y \n    linarith only [h'y]\n  exact hx (C (mem_ball_comm.1 hy))\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\n[PROOFSTEP]\nintro y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\n\u22a2 \u03c6 y = 0\n[PROOFSTEP]\nsimp only [\u03c6, indicator, mem_closedBall_zero_iff, ite_eq_right_iff, one_ne_zero]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\n\u22a2 \u2016y\u2016 \u2264 1 \u2192 False\n[PROOFSTEP]\nintro h'y\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\nh'y : \u2016y\u2016 \u2264 1\n\u22a2 False\n[PROOFSTEP]\nhave C : ball y D \u2286 ball 0 (1 + D) := by\n  apply ball_subset_ball'\n  rw [\u2190 dist_zero_right] at h'y \n  linarith only [h'y]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\nh'y : \u2016y\u2016 \u2264 1\n\u22a2 ball y D \u2286 ball 0 (1 + D)\n[PROOFSTEP]\napply ball_subset_ball'\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\nh'y : \u2016y\u2016 \u2264 1\n\u22a2 D + dist y 0 \u2264 1 + D\n[PROOFSTEP]\nrw [\u2190 dist_zero_right] at h'y \n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\nh'y : dist y 0 \u2264 1\n\u22a2 D + dist y 0 \u2264 1 + D\n[PROOFSTEP]\nlinarith only [h'y]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\ny : E\nhy : y \u2208 ball x D\nh'y : \u2016y\u2016 \u2264 1\nC : ball y D \u2286 ball 0 (1 + D)\n\u22a2 False\n[PROOFSTEP]\nexact hx (C (mem_ball_comm.1 hy))\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 0\n[PROOFSTEP]\nhave Bx : \u03c6 x = 0 := B _ (mem_ball_self Dpos)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\nBx : \u03c6 x = 0\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 0\n[PROOFSTEP]\nhave B' : \u2200 y, y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x := by rw [Bx]; exact B\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\nBx : \u03c6 x = 0\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n[PROOFSTEP]\nrw [Bx]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\nBx : \u03c6 x = 0\n\u22a2 \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\n[PROOFSTEP]\nexact B\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\nBx : \u03c6 x = 0\nB' : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 = 0\n[PROOFSTEP]\nrw [convolution_eq_right' _ (le_of_eq (w_support E Dpos)) B']\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nhx : \u00acx \u2208 ball 0 (1 + D)\nB : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = 0\nBx : \u03c6 x = 0\nB' : \u2200 (y : E), y \u2208 ball x D \u2192 \u03c6 y = \u03c6 x\n\u22a2 \u222b (t : E), \u2191(\u2191(lsmul \u211d \u211d) (w D t)) (\u03c6 x) \u2202\u03bc = 0\n[PROOFSTEP]\nsimp only [lsmul_apply, Algebra.id.smul_eq_mul, Bx, mul_zero, integral_const]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\n\u22a2 y D x \u2264 1\n[PROOFSTEP]\nhave A : (w D \u22c6[lsmul \u211d \u211d, \u03bc] \u03c6) x \u2264 (w D \u22c6[lsmul \u211d \u211d, \u03bc] 1) x :=\n  by\n  apply convolution_mono_right_of_nonneg _ (w_nonneg D) (indicator_le_self' fun x _ => zero_le_one) fun _ => zero_le_one\n  refine'\n    (HasCompactSupport.convolutionExistsLeft _ (w_compact_support E Dpos) _ (locallyIntegrable_const (1 : \u211d))\n        x).integrable\n  exact continuous_const.mul ((u_continuous E).comp (continuous_id.const_smul _))\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\n\u22a2 w D \u22c6[lsmul \u211d \u211d, x] \u03c6 \u2264 w D \u22c6[lsmul \u211d \u211d, x] 1\n[PROOFSTEP]\napply convolution_mono_right_of_nonneg _ (w_nonneg D) (indicator_le_self' fun x _ => zero_le_one) fun _ => zero_le_one\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\n\u22a2 ConvolutionExistsAt (fun x => w D x) (fun x => 1) x (lsmul \u211d \u211d)\n[PROOFSTEP]\nrefine'\n  (HasCompactSupport.convolutionExistsLeft _ (w_compact_support E Dpos) _ (locallyIntegrable_const (1 : \u211d))\n      x).integrable\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\n\u22a2 Continuous (w D)\n[PROOFSTEP]\nexact continuous_const.mul ((u_continuous E).comp (continuous_id.const_smul _))\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nA : w D \u22c6[lsmul \u211d \u211d, x] \u03c6 \u2264 w D \u22c6[lsmul \u211d \u211d, x] 1\n\u22a2 y D x \u2264 1\n[PROOFSTEP]\nhave B : (w D \u22c6[lsmul \u211d \u211d, \u03bc] fun _ => (1 : \u211d)) x = 1 := by\n  simp only [convolution, ContinuousLinearMap.map_smul, mul_inv_rev, coe_smul', mul_one, lsmul_apply,\n    Algebra.id.smul_eq_mul, integral_mul_left, w_integral E Dpos, Pi.smul_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nA : w D \u22c6[lsmul \u211d \u211d, x] \u03c6 \u2264 w D \u22c6[lsmul \u211d \u211d, x] 1\n\u22a2 (w D \u22c6[lsmul \u211d \u211d, x] fun x => 1) = 1\n[PROOFSTEP]\nsimp only [convolution, ContinuousLinearMap.map_smul, mul_inv_rev, coe_smul', mul_one, lsmul_apply,\n  Algebra.id.smul_eq_mul, integral_mul_left, w_integral E Dpos, Pi.smul_apply]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nA : w D \u22c6[lsmul \u211d \u211d, x] \u03c6 \u2264 w D \u22c6[lsmul \u211d \u211d, x] 1\nB : (w D \u22c6[lsmul \u211d \u211d, x] fun x => 1) = 1\n\u22a2 y D x \u2264 1\n[PROOFSTEP]\nexact A.trans (le_of_eq B)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : x \u2208 ball 0 (1 + D)\n\u22a2 0 < y D x\n[PROOFSTEP]\nsimp only [mem_ball_zero_iff] at hx \n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\n\u22a2 0 < y D x\n[PROOFSTEP]\nrefine' (integral_pos_iff_support_of_nonneg (w_mul_\u03c6_nonneg D x) _).2 _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\n\u22a2 Integrable fun i => w D i * \u03c6 (x - i)\n[PROOFSTEP]\nhave F_comp : HasCompactSupport (w D) := w_compact_support E Dpos\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nF_comp : HasCompactSupport (w D)\n\u22a2 Integrable fun i => w D i * \u03c6 (x - i)\n[PROOFSTEP]\nhave B : LocallyIntegrable (\u03c6 : E \u2192 \u211d) \u03bc := (locallyIntegrable_const _).indicator measurableSet_closedBall\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nF_comp : HasCompactSupport (w D)\nB : LocallyIntegrable \u03c6\n\u22a2 Integrable fun i => w D i * \u03c6 (x - i)\n[PROOFSTEP]\nhave C : Continuous (w D : E \u2192 \u211d) := continuous_const.mul ((u_continuous E).comp (continuous_id.const_smul _))\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nF_comp : HasCompactSupport (w D)\nB : LocallyIntegrable \u03c6\nC : Continuous (w D)\n\u22a2 Integrable fun i => w D i * \u03c6 (x - i)\n[PROOFSTEP]\nexact (HasCompactSupport.convolutionExistsLeft (lsmul \u211d \u211d : \u211d \u2192L[\u211d] \u211d \u2192L[\u211d] \u211d) F_comp C B x).integrable\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => w D i * \u03c6 (x - i))\n[PROOFSTEP]\nset z := (D / (1 + D)) \u2022 x with hz\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => w D i * \u03c6 (x - i))\n[PROOFSTEP]\nhave B : 0 < 1 + D := by linarith\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\n\u22a2 0 < 1 + D\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => w D i * \u03c6 (x - i))\n[PROOFSTEP]\nhave C : ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y : E => w D y * \u03c6 (x - y) :=\n  by\n  intro y hy\n  simp only [support_mul, w_support E Dpos]\n  simp only [\u03c6, mem_inter_iff, mem_support, Ne.def, indicator_apply_eq_zero, mem_closedBall_zero_iff, one_ne_zero,\n    not_forall, not_false_iff, exists_prop, and_true_iff]\n  constructor\n  \u00b7 apply ball_subset_ball' _ hy\n    simp only [hz, norm_smul, abs_of_nonneg Dpos.le, abs_of_nonneg B.le, dist_zero_right, Real.norm_eq_abs, abs_div]\n    simp only [div_le_iff B, field_simps]\n    ring_nf\n    rfl\n  \u00b7 have ID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D) := by\n      rw [Real.norm_of_nonpos]\n      \u00b7 simp only [B.ne', Ne.def, not_false_iff, mul_one, neg_sub, add_tsub_cancel_right, field_simps]\n      \u00b7 simp only [B.ne', Ne.def, not_false_iff, mul_one, field_simps]\n        apply div_nonpos_of_nonpos_of_nonneg _ B.le\n        linarith only\n    rw [\u2190 mem_closedBall_iff_norm']\n    apply closedBall_subset_closedBall' _ (ball_subset_closedBall hy)\n    rw [\u2190 one_smul \u211d x, dist_eq_norm, hz, \u2190 sub_smul, one_smul, norm_smul, ID]\n    simp only [B.ne', div_le_iff B, field_simps]\n    nlinarith only [hx, D_lt_one]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\n\u22a2 ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y => w D y * \u03c6 (x - y)\n[PROOFSTEP]\nintro y hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 y \u2208 support fun y => w D y * \u03c6 (x - y)\n[PROOFSTEP]\nsimp only [support_mul, w_support E Dpos]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 y \u2208 ball 0 D \u2229 support fun x_1 => \u03c6 (x - x_1)\n[PROOFSTEP]\nsimp only [\u03c6, mem_inter_iff, mem_support, Ne.def, indicator_apply_eq_zero, mem_closedBall_zero_iff, one_ne_zero,\n  not_forall, not_false_iff, exists_prop, and_true_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 y \u2208 ball 0 D \u2227 \u2016x - y\u2016 \u2264 1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 y \u2208 ball 0 D\n[PROOFSTEP]\napply ball_subset_ball' _ hy\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D * (1 + D - \u2016x\u2016) / (1 + D) + dist z 0 \u2264 D\n[PROOFSTEP]\nsimp only [hz, norm_smul, abs_of_nonneg Dpos.le, abs_of_nonneg B.le, dist_zero_right, Real.norm_eq_abs, abs_div]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D * (1 + D - \u2016x\u2016) / (1 + D) + D / (1 + D) * \u2016x\u2016 \u2264 D\n[PROOFSTEP]\nsimp only [div_le_iff B, field_simps]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D * (1 + D - \u2016x\u2016) + D * \u2016x\u2016 \u2264 D * (1 + D)\n[PROOFSTEP]\nring_nf\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D + D ^ 2 \u2264 D + D ^ 2\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 \u2016x - y\u2016 \u2264 1\n[PROOFSTEP]\nhave ID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D) := by\n  rw [Real.norm_of_nonpos]\n  \u00b7 simp only [B.ne', Ne.def, not_false_iff, mul_one, neg_sub, add_tsub_cancel_right, field_simps]\n  \u00b7 simp only [B.ne', Ne.def, not_false_iff, mul_one, field_simps]\n    apply div_nonpos_of_nonpos_of_nonneg _ B.le\n    linarith only\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n[PROOFSTEP]\nrw [Real.norm_of_nonpos]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 -(D / (1 + D) - 1) = 1 / (1 + D)\n[PROOFSTEP]\nsimp only [B.ne', Ne.def, not_false_iff, mul_one, neg_sub, add_tsub_cancel_right, field_simps]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D / (1 + D) - 1 \u2264 0\n[PROOFSTEP]\nsimp only [B.ne', Ne.def, not_false_iff, mul_one, field_simps]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 (D - (1 + D)) / (1 + D) \u2264 0\n[PROOFSTEP]\napply div_nonpos_of_nonpos_of_nonneg _ B.le\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\n\u22a2 D - (1 + D) \u2264 0\n[PROOFSTEP]\nlinarith only\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\nID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n\u22a2 \u2016x - y\u2016 \u2264 1\n[PROOFSTEP]\nrw [\u2190 mem_closedBall_iff_norm']\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\nID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n\u22a2 y \u2208 closedBall x 1\n[PROOFSTEP]\napply closedBall_subset_closedBall' _ (ball_subset_closedBall hy)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\nID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n\u22a2 D * (1 + D - \u2016x\u2016) / (1 + D) + dist z x \u2264 1\n[PROOFSTEP]\nrw [\u2190 one_smul \u211d x, dist_eq_norm, hz, \u2190 sub_smul, one_smul, norm_smul, ID]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\nID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n\u22a2 D * (1 + D - \u2016x\u2016) / (1 + D) + 1 / (1 + D) * \u2016x\u2016 \u2264 1\n[PROOFSTEP]\nsimp only [B.ne', div_le_iff B, field_simps]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\ny : E\nhy : y \u2208 ball z (D * (1 + D - \u2016x\u2016) / (1 + D))\nID : \u2016D / (1 + D) - 1\u2016 = 1 / (1 + D)\n\u22a2 D * (1 + D - \u2016x\u2016) + 1 * \u2016x\u2016 \u2264 1 * (1 + D)\n[PROOFSTEP]\nnlinarith only [hx, D_lt_one]\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\nC : ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y => w D y * \u03c6 (x - y)\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => w D i * \u03c6 (x - i))\n[PROOFSTEP]\napply lt_of_lt_of_le _ (measure_mono C)\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\nC : ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y => w D y * \u03c6 (x - y)\n\u22a2 0 < \u2191\u2191\u03bc (ball z (D * (1 + D - \u2016x\u2016) / (1 + D)))\n[PROOFSTEP]\napply measure_ball_pos\n[GOAL]\ncase hr\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\nC : ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y => w D y * \u03c6 (x - y)\n\u22a2 0 < D * (1 + D - \u2016x\u2016) / (1 + D)\n[PROOFSTEP]\nexact div_pos (mul_pos Dpos (by linarith only [hx])) B\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nD : \u211d\nx : E\nDpos : 0 < D\nD_lt_one : D < 1\nhx : \u2016x\u2016 < 1 + D\nz : E := (D / (1 + D)) \u2022 x\nhz : z = (D / (1 + D)) \u2022 x\nB : 0 < 1 + D\nC : ball z (D * (1 + D - \u2016x\u2016) / (1 + D)) \u2286 support fun y => w D y * \u03c6 (x - y)\n\u22a2 0 < 1 + D - \u2016x\u2016\n[PROOFSTEP]\nlinarith only [hx]\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry y) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave hs : IsOpen (Ioo (0 : \u211d) (1 : \u211d)) := isOpen_Ioo\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry y) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave hk : IsCompact (closedBall (0 : E) 1) := ProperSpace.isCompact_closedBall _ _\n[GOAL]\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry y) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' contDiffOn_convolution_left_with_param (lsmul \u211d \u211d) hs hk _ _ _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 \u2200 (p : \u211d) (x : E), p \u2208 Ioo 0 1 \u2192 \u00acx \u2208 closedBall 0 1 \u2192 w p x = 0\n[PROOFSTEP]\nrintro p x hp hx\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : \u00acx \u2208 closedBall 0 1\n\u22a2 w p x = 0\n[PROOFSTEP]\nsimp only [w, mul_inv_rev, Algebra.id.smul_eq_mul, mul_eq_zero, inv_eq_zero]\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : \u00acx \u2208 closedBall 0 1\n\u22a2 (|p| ^ \u2191(finrank \u211d E) = 0 \u2228 \u222b (x : E), u x \u2202\u03bc = 0) \u2228 u (p\u207b\u00b9 \u2022 x) = 0\n[PROOFSTEP]\nright\n[GOAL]\ncase refine'_1.h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : \u00acx \u2208 closedBall 0 1\n\u22a2 u (p\u207b\u00b9 \u2022 x) = 0\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase refine'_1.h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : u (p\u207b\u00b9 \u2022 x) \u2260 0\n\u22a2 x \u2208 closedBall 0 1\n[PROOFSTEP]\nhave : p\u207b\u00b9 \u2022 x \u2208 support u := mem_support.2 hx\n[GOAL]\ncase refine'_1.h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : u (p\u207b\u00b9 \u2022 x) \u2260 0\nthis : p\u207b\u00b9 \u2022 x \u2208 support u\n\u22a2 x \u2208 closedBall 0 1\n[PROOFSTEP]\nsimp only [u_support, norm_smul, mem_ball_zero_iff, Real.norm_eq_abs, abs_inv, abs_of_nonneg hp.1.le, \u2190 div_eq_inv_mul,\n  div_lt_one hp.1] at this \n[GOAL]\ncase refine'_1.h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : u (p\u207b\u00b9 \u2022 x) \u2260 0\nthis : \u2016x\u2016 < p\n\u22a2 x \u2208 closedBall 0 1\n[PROOFSTEP]\nrw [mem_closedBall_zero_iff]\n[GOAL]\ncase refine'_1.h\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\np : \u211d\nx : E\nhp : p \u2208 Ioo 0 1\nhx : u (p\u207b\u00b9 \u2022 x) \u2260 0\nthis : \u2016x\u2016 < p\n\u22a2 \u2016x\u2016 \u2264 1\n[PROOFSTEP]\nexact this.le.trans hp.2.le\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 LocallyIntegrable \u03c6\n[PROOFSTEP]\nexact (locallyIntegrable_const _).indicator measurableSet_closedBall\n[GOAL]\ncase refine'_3\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (\u21bfw) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply ContDiffOn.mul\n[GOAL]\ncase refine'_3.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => ((\u222b (x : E), u x \u2202\u03bc) * |x.fst| ^ \u2191(finrank \u211d E))\u207b\u00b9) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_3.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => ((\u222b (x : E), u x \u2202\u03bc) * |x.fst| ^ finrank \u211d E)\u207b\u00b9) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine'\n  (contDiffOn_const.mul _).inv fun x hx =>\n    ne_of_gt (mul_pos (u_int_pos E) (pow_pos (abs_pos_of_pos hx.1.1) (finrank \u211d E)))\n[GOAL]\ncase refine'_3.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => |x.fst| ^ finrank \u211d E) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply ContDiffOn.pow\n[GOAL]\ncase refine'_3.hf.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun y => |y.fst|) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nsimp_rw [\u2190 Real.norm_eq_abs]\n[GOAL]\ncase refine'_3.hf.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun y => \u2016y.fst\u2016) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply @ContDiffOn.norm \u211d\n[GOAL]\ncase refine'_3.hf.hf.hf\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun y => y.fst) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact contDiffOn_fst\n[GOAL]\ncase refine'_3.hf.hf.h0\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 \u2200 (x : \u211d \u00d7 E), x \u2208 Ioo 0 1 \u00d7\u02e2 univ \u2192 x.fst \u2260 0\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_3.hf.hf.h0\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\nx : \u211d \u00d7 E\nhx : x \u2208 Ioo 0 1 \u00d7\u02e2 univ\n\u22a2 x.fst \u2260 0\n[PROOFSTEP]\nexact ne_of_gt hx.1.1\n[GOAL]\ncase refine'_3.hg\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => u (x.fst\u207b\u00b9 \u2022 x.snd)) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply (u_smooth E).comp_contDiffOn\n[GOAL]\ncase refine'_3.hg\nE : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \u211d E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nhs : IsOpen (Ioo 0 1)\nhk : IsCompact (closedBall 0 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => x.fst\u207b\u00b9 \u2022 x.snd) (Ioo 0 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nexact ContDiffOn.smul (contDiffOn_fst.inv fun x hx => ne_of_gt hx.1.1) contDiffOn_snd\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 HasContDiffBump E\n[PROOFSTEP]\nrefine' \u27e8\u27e8_\u27e9\u27e9\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\n\u22a2 ContDiffBumpBase E\n[PROOFSTEP]\nborelize E\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 ContDiffBumpBase E\n[PROOFSTEP]\nhave IR : \u2200 R : \u211d, 1 < R \u2192 0 < (R - 1) / (R + 1) := by intro R hR; apply div_pos <;> linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n[PROOFSTEP]\nintro R hR\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nR : \u211d\nhR : 1 < R\n\u22a2 0 < (R - 1) / (R + 1)\n[PROOFSTEP]\napply div_pos\n[GOAL]\ncase ha\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nR : \u211d\nhR : 1 < R\n\u22a2 0 < R - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hb\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nR : \u211d\nhR : 1 < R\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffBumpBase E\n[PROOFSTEP]\nexact\n  { toFun := fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0\n    mem_Icc := fun R x => by\n      simp only [mem_Icc]\n      split_ifs with h\n      \u00b7 refine' \u27e8y_nonneg _ _, y_le_one _ (IR R h)\u27e9\n      \u00b7 simp only [le_refl, zero_le_one, and_self]\n    symmetric := fun R x => by\n      simp only\n      split_ifs\n      \u00b7 simp only [y_neg, smul_neg]\n      \u00b7 rfl\n    smooth :=\n      by\n      suffices\n        ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p : \u211d \u00d7 E => ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)\u207b\u00b9 \u2022 p.2)) (Ioi 1 \u00d7\u02e2 univ)\n        by\n        apply this.congr\n        rintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n        simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply]\n      apply (y_smooth E).comp\n      \u00b7 apply ContDiffOn.prod\n        \u00b7 refine' (contDiffOn_fst.sub contDiffOn_const).div (contDiffOn_fst.add contDiffOn_const) _\n          rintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n          apply ne_of_gt\n          dsimp only\n          linarith\n        \u00b7 apply ContDiffOn.smul _ contDiffOn_snd\n          refine' ((contDiffOn_fst.add contDiffOn_const).div_const _).inv _\n          rintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n          apply ne_of_gt\n          dsimp only\n          linarith\n      \u00b7 rintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n        have A : 0 < (R - 1) / (R + 1) := by apply div_pos <;> linarith\n        have B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith\n        simp only [mem_preimage, prod_mk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true_iff, A, B]\n    eq_one := fun R hR x hx => by\n      have A : 0 < R + 1 := by linarith\n      simp only [hR, if_true]\n      apply y_eq_one_of_mem_closedBall (IR R hR)\n      simp only [norm_smul, inv_div, mem_closedBall_zero_iff, Real.norm_eq_abs, abs_div, abs_two, abs_of_nonneg A.le]\n      calc\n        2 / (R + 1) * \u2016x\u2016 \u2264 2 / (R + 1) * 1 := mul_le_mul_of_nonneg_left hx (div_nonneg zero_le_two A.le)\n        _ = 1 - (R - 1) / (R + 1) := by field_simp [A.ne']; ring\n    support := fun R hR => by\n      have A : 0 < (R + 1) / 2 := by linarith\n      have A' : 0 < R + 1 := by linarith\n      have C : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith\n      simp only [hR, if_true, support_comp_inv_smul\u2080 A.ne', y_support _ (IR R hR) C, _root_.smul_ball A.ne',\n        Real.norm_of_nonneg A.le, smul_zero]\n      refine' congr (congr_arg ball (Eq.refl 0)) _\n      field_simp [A'.ne']\n      ring }\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\n\u22a2 (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R x \u2208 Icc 0 1\n[PROOFSTEP]\nsimp only [mem_Icc]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\n\u22a2 (0 \u2264 if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) \u2227\n    (if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) \u2264 1\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nh : 1 < R\n\u22a2 0 \u2264 y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) \u2227 y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) \u2264 1\n[PROOFSTEP]\nrefine' \u27e8y_nonneg _ _, y_le_one _ (IR R h)\u27e9\n[GOAL]\ncase neg\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nh : \u00ac1 < R\n\u22a2 0 \u2264 0 \u2227 0 \u2264 1\n[PROOFSTEP]\nsimp only [le_refl, zero_le_one, and_self]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\n\u22a2 (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R (-x) =\n    (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R x\n[PROOFSTEP]\nsimp only\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\n\u22a2 (if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 -x) else 0) =\n    if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nh\u271d : 1 < R\n\u22a2 y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 -x) = y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x)\n[PROOFSTEP]\nsimp only [y_neg, smul_neg]\n[GOAL]\ncase neg\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nh\u271d : \u00ac1 < R\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nsuffices ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p : \u211d \u00d7 E => ((p.1 - 1) / (p.1 + 1), ((p.1 + 1) / 2)\u207b\u00b9 \u2022 p.2)) (Ioi 1 \u00d7\u02e2 univ)\n  by\n  apply this.congr\n  rintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n  simp only [hR, uncurry_apply_pair, if_true, Function.comp_apply]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nthis : ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (Ioi 1 \u00d7\u02e2 univ)\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply this.congr\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nthis : ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (Ioi 1 \u00d7\u02e2 univ)\n\u22a2 \u2200 (x : \u211d \u00d7 E),\n    x \u2208 Ioi 1 \u00d7\u02e2 univ \u2192\n      uncurry (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) x =\n        (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) x\n[PROOFSTEP]\nrintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n[GOAL]\ncase mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nthis : ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (Ioi 1 \u00d7\u02e2 univ)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 uncurry (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) (R, x) =\n    (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (R, x)\n[PROOFSTEP]\nsimp only [hR, uncurry_apply_pair, if_true, Function.comp_apply]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (uncurry y \u2218 fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply (y_smooth E).comp\n[GOAL]\ncase hf\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply ContDiffOn.prod\n[GOAL]\ncase hf.hf\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => (x.fst - 1) / (x.fst + 1)) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' (contDiffOn_fst.sub contDiffOn_const).div (contDiffOn_fst.add contDiffOn_const) _\n[GOAL]\ncase hf.hf\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 \u2200 (x : \u211d \u00d7 E), x \u2208 Ioi 1 \u00d7\u02e2 univ \u2192 x.fst + 1 \u2260 0\n[PROOFSTEP]\nrintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n[GOAL]\ncase hf.hf.mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 (R, x).fst + 1 \u2260 0\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase hf.hf.mk.intro.h\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < (R, x).fst + 1\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase hf.hf.mk.intro.h\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hf.hg\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => ((x.fst + 1) / 2)\u207b\u00b9 \u2022 x.snd) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\napply ContDiffOn.smul _ contDiffOn_snd\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 ContDiffOn \u211d \u22a4 (fun x => ((x.fst + 1) / 2)\u207b\u00b9) (Ioi 1 \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' ((contDiffOn_fst.add contDiffOn_const).div_const _).inv _\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 \u2200 (x : \u211d \u00d7 E), x \u2208 Ioi 1 \u00d7\u02e2 univ \u2192 (x.fst + 1) / 2 \u2260 0\n[PROOFSTEP]\nrintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n[GOAL]\ncase mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 ((R, x).fst + 1) / 2 \u2260 0\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase mk.intro.h\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < ((R, x).fst + 1) / 2\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase mk.intro.h\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < (R + 1) / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase st\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\n\u22a2 Ioi 1 \u00d7\u02e2 univ \u2286 (fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) \u207b\u00b9' Ioo 0 1 \u00d7\u02e2 univ\n[PROOFSTEP]\nrintro \u27e8R, x\u27e9 \u27e8hR : 1 < R, _\u27e9\n[GOAL]\ncase st.mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 (R, x) \u2208 (fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) \u207b\u00b9' Ioo 0 1 \u00d7\u02e2 univ\n[PROOFSTEP]\nhave A : 0 < (R - 1) / (R + 1) := by apply div_pos <;> linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < (R - 1) / (R + 1)\n[PROOFSTEP]\napply div_pos\n[GOAL]\ncase ha\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < R - 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hb\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase st.mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\nA : 0 < (R - 1) / (R + 1)\n\u22a2 (R, x) \u2208 (fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) \u207b\u00b9' Ioo 0 1 \u00d7\u02e2 univ\n[PROOFSTEP]\nhave B : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\nA : 0 < (R - 1) / (R + 1)\n\u22a2 (R - 1) / (R + 1) < 1\n[PROOFSTEP]\napply (div_lt_one _).2\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\nA : 0 < (R - 1) / (R + 1)\n\u22a2 R - 1 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\nA : 0 < (R - 1) / (R + 1)\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase st.mk.intro\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nx : E\nhR : 1 < R\nright\u271d : (R, x).snd \u2208 univ\nA : 0 < (R - 1) / (R + 1)\nB : (R - 1) / (R + 1) < 1\n\u22a2 (R, x) \u2208 (fun p => ((p.fst - 1) / (p.fst + 1), ((p.fst + 1) / 2)\u207b\u00b9 \u2022 p.snd)) \u207b\u00b9' Ioo 0 1 \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [mem_preimage, prod_mk_mem_set_prod_eq, mem_Ioo, mem_univ, and_true_iff, A, B]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\n\u22a2 (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R x = 1\n[PROOFSTEP]\nhave A : 0 < R + 1 := by linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 (fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R x = 1\n[PROOFSTEP]\nsimp only [hR, if_true]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) = 1\n[PROOFSTEP]\napply y_eq_one_of_mem_closedBall (IR R hR)\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 ((R + 1) / 2)\u207b\u00b9 \u2022 x \u2208 closedBall 0 (1 - (R - 1) / (R + 1))\n[PROOFSTEP]\nsimp only [norm_smul, inv_div, mem_closedBall_zero_iff, Real.norm_eq_abs, abs_div, abs_two, abs_of_nonneg A.le]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 2 / (R + 1) * \u2016x\u2016 \u2264 1 - (R - 1) / (R + 1)\n[PROOFSTEP]\ncalc\n  2 / (R + 1) * \u2016x\u2016 \u2264 2 / (R + 1) * 1 := mul_le_mul_of_nonneg_left hx (div_nonneg zero_le_two A.le)\n  _ = 1 - (R - 1) / (R + 1) := by field_simp [A.ne']; ring\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 2 / (R + 1) * 1 = 1 - (R - 1) / (R + 1)\n[PROOFSTEP]\nfield_simp [A.ne']\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nx : E\nhx : \u2016x\u2016 \u2264 1\nA : 0 < R + 1\n\u22a2 2 = 1 + 1\n[PROOFSTEP]\nring\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\n\u22a2 support ((fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R) = ball 0 R\n[PROOFSTEP]\nhave A : 0 < (R + 1) / 2 := by linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\n\u22a2 0 < (R + 1) / 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\n\u22a2 support ((fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R) = ball 0 R\n[PROOFSTEP]\nhave A' : 0 < R + 1 := by linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\n\u22a2 support ((fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R) = ball 0 R\n[PROOFSTEP]\nhave C : (R - 1) / (R + 1) < 1 := by apply (div_lt_one _).2 <;> linarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\n\u22a2 (R - 1) / (R + 1) < 1\n[PROOFSTEP]\napply (div_lt_one _).2\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\n\u22a2 R - 1 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\n\u22a2 0 < R + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\nC : (R - 1) / (R + 1) < 1\n\u22a2 support ((fun R x => if 1 < R then y ((R - 1) / (R + 1)) (((R + 1) / 2)\u207b\u00b9 \u2022 x) else 0) R) = ball 0 R\n[PROOFSTEP]\nsimp only [hR, if_true, support_comp_inv_smul\u2080 A.ne', y_support _ (IR R hR) C, _root_.smul_ball A.ne',\n  Real.norm_of_nonneg A.le, smul_zero]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\nC : (R - 1) / (R + 1) < 1\n\u22a2 ball 0 ((R + 1) / 2 * (1 + (R - 1) / (R + 1))) = ball 0 R\n[PROOFSTEP]\nrefine' congr (congr_arg ball (Eq.refl 0)) _\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\nC : (R - 1) / (R + 1) < 1\n\u22a2 (R + 1) / 2 * (1 + (R - 1) / (R + 1)) = R\n[PROOFSTEP]\nfield_simp [A'.ne']\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\u271d\ninst\u271d\u2074 : NormedSpace \u211d E\u271d\ninst\u271d\u00b3 : FiniteDimensional \u211d E\u271d\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\nIR : \u2200 (R : \u211d), 1 < R \u2192 0 < (R - 1) / (R + 1)\nR : \u211d\nhR : 1 < R\nA : 0 < (R + 1) / 2\nA' : 0 < R + 1\nC : (R - 1) / (R + 1) < 1\n\u22a2 (R + 1) * (R + R) = R * (2 * (R + 1))\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension", "llama_tokens": 99482, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.7122321720225278, "lm_q1q2_score": 0.5271759935504374}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq\u2081 q\u2082 : Basis A c\u2081 c\u2082\nhi : q\u2081.i = q\u2082.i\nhj : q\u2081.j = q\u2082.j\n\u22a2 q\u2081 = q\u2082\n[PROOFSTEP]\ncases q\u2081\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq\u2082 : Basis A c\u2081 c\u2082\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\ni_mul_j\u271d : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d }.i =\n    q\u2082.i\nhj :\n  { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d }.j =\n    q\u2082.j\n\u22a2 { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d } = q\u2082\n[PROOFSTEP]\nrename_i q\u2081_i_mul_j _\n[GOAL]\ncase mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq\u2082 : Basis A c\u2081 c\u2082\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\nq\u2081_i_mul_j : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.i =\n    q\u2082.i\nhj :\n  { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.j =\n    q\u2082.j\n\u22a2 { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2081_i_mul_j, j_mul_i := j_mul_i\u271d } =\n    q\u2082\n[PROOFSTEP]\ncases q\u2082\n[GOAL]\ncase mk.mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\ni\u271d\u00b9 j\u271d\u00b9 k\u271d\u00b9 : A\ni_mul_i\u271d\u00b9 : i\u271d\u00b9 * i\u271d\u00b9 = c\u2081 \u2022 1\nj_mul_j\u271d\u00b9 : j\u271d\u00b9 * j\u271d\u00b9 = c\u2082 \u2022 1\nq\u2081_i_mul_j : i\u271d\u00b9 * j\u271d\u00b9 = k\u271d\u00b9\nj_mul_i\u271d\u00b9 : j\u271d\u00b9 * i\u271d\u00b9 = -k\u271d\u00b9\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\ni_mul_j\u271d : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.i =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d }.i\nhj :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.j =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d }.j\n\u22a2 { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n      j_mul_i := j_mul_i\u271d\u00b9 } =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := i_mul_j\u271d, j_mul_i := j_mul_i\u271d }\n[PROOFSTEP]\nrename_i q\u2082_i_mul_j _\n[GOAL]\ncase mk.mk\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\ni\u271d\u00b9 j\u271d\u00b9 k\u271d\u00b9 : A\ni_mul_i\u271d\u00b9 : i\u271d\u00b9 * i\u271d\u00b9 = c\u2081 \u2022 1\nj_mul_j\u271d\u00b9 : j\u271d\u00b9 * j\u271d\u00b9 = c\u2082 \u2022 1\nq\u2081_i_mul_j : i\u271d\u00b9 * j\u271d\u00b9 = k\u271d\u00b9\nj_mul_i\u271d\u00b9 : j\u271d\u00b9 * i\u271d\u00b9 = -k\u271d\u00b9\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\nq\u2082_i_mul_j : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.i =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.i\nhj :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.j =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.j\n\u22a2 { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n      j_mul_i := j_mul_i\u271d\u00b9 } =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j, j_mul_i := j_mul_i\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.mk.e_k\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\ni\u271d\u00b9 j\u271d\u00b9 k\u271d\u00b9 : A\ni_mul_i\u271d\u00b9 : i\u271d\u00b9 * i\u271d\u00b9 = c\u2081 \u2022 1\nj_mul_j\u271d\u00b9 : j\u271d\u00b9 * j\u271d\u00b9 = c\u2082 \u2022 1\nq\u2081_i_mul_j : i\u271d\u00b9 * j\u271d\u00b9 = k\u271d\u00b9\nj_mul_i\u271d\u00b9 : j\u271d\u00b9 * i\u271d\u00b9 = -k\u271d\u00b9\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\nq\u2082_i_mul_j : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.i =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.i\nhj :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.j =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.j\n\u22a2 k\u271d\u00b9 = k\u271d\n[PROOFSTEP]\nrw [\u2190 q\u2081_i_mul_j, \u2190 q\u2082_i_mul_j]\n[GOAL]\ncase mk.mk.e_k\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\ni\u271d\u00b9 j\u271d\u00b9 k\u271d\u00b9 : A\ni_mul_i\u271d\u00b9 : i\u271d\u00b9 * i\u271d\u00b9 = c\u2081 \u2022 1\nj_mul_j\u271d\u00b9 : j\u271d\u00b9 * j\u271d\u00b9 = c\u2082 \u2022 1\nq\u2081_i_mul_j : i\u271d\u00b9 * j\u271d\u00b9 = k\u271d\u00b9\nj_mul_i\u271d\u00b9 : j\u271d\u00b9 * i\u271d\u00b9 = -k\u271d\u00b9\ni\u271d j\u271d k\u271d : A\ni_mul_i\u271d : i\u271d * i\u271d = c\u2081 \u2022 1\nj_mul_j\u271d : j\u271d * j\u271d = c\u2082 \u2022 1\nq\u2082_i_mul_j : i\u271d * j\u271d = k\u271d\nj_mul_i\u271d : j\u271d * i\u271d = -k\u271d\nhi :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.i =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.i\nhj :\n  { i := i\u271d\u00b9, j := j\u271d\u00b9, k := k\u271d\u00b9, i_mul_i := i_mul_i\u271d\u00b9, j_mul_j := j_mul_j\u271d\u00b9, i_mul_j := q\u2081_i_mul_j,\n        j_mul_i := j_mul_i\u271d\u00b9 }.j =\n    { i := i\u271d, j := j\u271d, k := k\u271d, i_mul_i := i_mul_i\u271d, j_mul_j := j_mul_j\u271d, i_mul_j := q\u2082_i_mul_j,\n        j_mul_i := j_mul_i\u271d }.j\n\u22a2 i\u271d\u00b9 * j\u271d\u00b9 = i\u271d * j\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } = c\u2081 \u2022 1\n[PROOFSTEP]\next\n[GOAL]\ncase re\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).re = (c\u2081 \u2022 1).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imI = (c\u2081 \u2022 1).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imJ = (c\u2081 \u2022 1).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imK = (c\u2081 \u2022 1).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } = c\u2082 \u2022 1\n[PROOFSTEP]\next\n[GOAL]\ncase re\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).re = (c\u2082 \u2022 1).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imI = (c\u2082 \u2022 1).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imJ = (c\u2082 \u2022 1).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imK = (c\u2082 \u2022 1).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 { re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 } =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }\n[PROOFSTEP]\next\n[GOAL]\ncase re\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).re =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imI =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imJ =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 1, imJ := 0, imK := 0 } * { re := 0, imI := 0, imJ := 1, imK := 0 }).imK =\n    { re := 0, imI := 0, imJ := 0, imK := 1 }.imK\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 { re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 } =\n    -{ re := 0, imI := 0, imJ := 0, imK := 1 }\n[PROOFSTEP]\next\n[GOAL]\ncase re\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).re =\n    (-{ re := 0, imI := 0, imJ := 0, imK := 1 }).re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imI\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imI =\n    (-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imJ\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imJ =\n    (-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imK\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\n\u22a2 ({ re := 0, imI := 0, imJ := 1, imK := 0 } * { re := 0, imI := 1, imJ := 0, imK := 0 }).imK =\n    (-{ re := 0, imI := 0, imJ := 0, imK := 1 }).imK\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 q.i * q.k = c\u2081 \u2022 q.j\n[PROOFSTEP]\nrw [\u2190 i_mul_j, \u2190 mul_assoc, i_mul_i, smul_mul_assoc, one_mul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 q.k * q.i = -c\u2081 \u2022 q.j\n[PROOFSTEP]\nrw [\u2190 i_mul_j, mul_assoc, j_mul_i, mul_neg, i_mul_k, neg_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 q.k * q.j = c\u2082 \u2022 q.i\n[PROOFSTEP]\nrw [\u2190 i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 q.j * q.k = -c\u2082 \u2022 q.i\n[PROOFSTEP]\nrw [\u2190 i_mul_j, \u2190 mul_assoc, j_mul_i, neg_mul, k_mul_j, neg_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 q.k * q.k = -((c\u2081 * c\u2082) \u2022 1)\n[PROOFSTEP]\nrw [\u2190 i_mul_j, mul_assoc, \u2190 mul_assoc q.j _ _, j_mul_i, \u2190 i_mul_j, \u2190 mul_assoc, mul_neg, \u2190 mul_assoc, i_mul_i,\n  smul_mul_assoc, one_mul, neg_mul, smul_mul_assoc, j_mul_j, smul_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 lift q 0 = 0\n[PROOFSTEP]\nsimp [lift]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 lift q 1 = 1\n[PROOFSTEP]\nsimp [lift]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 lift q (x + y) = lift q x + lift q y\n[PROOFSTEP]\nsimp [lift, add_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(algebraMap R A) x.re + \u2191(algebraMap R A) y.re + (x.imI \u2022 q.i + y.imI \u2022 q.i) + (x.imJ \u2022 q.j + y.imJ \u2022 q.j) +\n      (x.imK \u2022 q.k + y.imK \u2022 q.k) =\n    \u2191(algebraMap R A) x.re + x.imI \u2022 q.i + x.imJ \u2022 q.j + x.imK \u2022 q.k +\n      (\u2191(algebraMap R A) y.re + y.imI \u2022 q.i + y.imJ \u2022 q.j + y.imK \u2022 q.k)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(algebraMap R A) x.re + \u2191(algebraMap R A) y.re + (x.imI \u2022 q.i + y.imI \u2022 q.i) + (x.imJ \u2022 q.j + y.imJ \u2022 q.j) +\n      (x.imK \u2022 q.k + y.imK \u2022 q.k) =\n    \u2191(algebraMap R A) x.re + x.imI \u2022 q.i + x.imJ \u2022 q.j + x.imK \u2022 q.k +\n      (\u2191(algebraMap R A) y.re + y.imI \u2022 q.i + y.imJ \u2022 q.j + y.imK \u2022 q.k)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 lift q (x * y) = lift q x * lift q y\n[PROOFSTEP]\nsimp only [lift, Algebra.algebraMap_eq_smul_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re \u2022 1 + x.imI \u2022 q.i + x.imJ \u2022 q.j + x.imK \u2022 q.k) * (y.re \u2022 1 + y.imI \u2022 q.i + y.imJ \u2022 q.j + y.imK \u2022 q.k)\n[PROOFSTEP]\nsimp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k +\n          ((x.imI * y.re) \u2022 q.i + (x.imI * y.imI) \u2022 (q.i * q.i) + (x.imI * y.imJ) \u2022 (q.i * q.j) +\n            (x.imI * y.imK) \u2022 (q.i * q.k)) +\n        ((x.imJ * y.re) \u2022 q.j + (x.imJ * y.imI) \u2022 (q.j * q.i) + (x.imJ * y.imJ) \u2022 (q.j * q.j) +\n          (x.imJ * y.imK) \u2022 (q.j * q.k)) +\n      ((x.imK * y.re) \u2022 q.k + (x.imK * y.imI) \u2022 (q.k * q.i) + (x.imK * y.imJ) \u2022 (q.k * q.j) +\n        (x.imK * y.imK) \u2022 (q.k * q.k))\n[PROOFSTEP]\nsimp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k +\n          ((x.imI * y.re) \u2022 q.i + (x.imI * y.imI) \u2022 c\u2081 \u2022 1 + (x.imI * y.imJ) \u2022 q.k + (x.imI * y.imK) \u2022 c\u2081 \u2022 q.j) +\n        ((x.imJ * y.re) \u2022 q.j + (x.imJ * y.imI) \u2022 -q.k + (x.imJ * y.imJ) \u2022 c\u2082 \u2022 1 + (x.imJ * y.imK) \u2022 -c\u2082 \u2022 q.i) +\n      ((x.imK * y.re) \u2022 q.k + (x.imK * y.imI) \u2022 -c\u2081 \u2022 q.j + (x.imK * y.imJ) \u2022 c\u2082 \u2022 q.i +\n        (x.imK * y.imK) \u2022 -((c\u2081 * c\u2082) \u2022 1))\n[PROOFSTEP]\nsimp only [smul_smul, smul_neg, sub_eq_add_neg, add_smul, \u2190 add_assoc, mul_neg, neg_smul]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (x.imI * y.imI * c\u2081) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (x.imI * y.imK * c\u2081) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (x.imJ * y.imJ * c\u2082) \u2022 1 +\n              -((x.imJ * y.imK * c\u2082) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((x.imK * y.imI * c\u2081) \u2022 q.j) +\n        (x.imK * y.imJ * c\u2082) \u2022 q.i +\n      -((x.imK * y.imK * (c\u2081 * c\u2082)) \u2022 1)\n[PROOFSTEP]\nsimp only [mul_right_comm _ _ (c\u2081 * c\u2082), mul_comm _ (c\u2081 * c\u2082)]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (x.imI * y.imI * c\u2081) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (x.imI * y.imK * c\u2081) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (x.imJ * y.imJ * c\u2082) \u2022 1 +\n              -((x.imJ * y.imK * c\u2082) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((x.imK * y.imI * c\u2081) \u2022 q.j) +\n        (x.imK * y.imJ * c\u2082) \u2022 q.i +\n      -((c\u2081 * c\u2082 * (x.imK * y.imK)) \u2022 1)\n[PROOFSTEP]\nsimp only [mul_comm _ c\u2081, mul_right_comm _ _ c\u2081]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (c\u2081 * (x.imI * y.imI)) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (c\u2081 * (x.imI * y.imK)) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (x.imJ * y.imJ * c\u2082) \u2022 1 +\n              -((x.imJ * y.imK * c\u2082) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((c\u2081 * (x.imK * y.imI)) \u2022 q.j) +\n        (x.imK * y.imJ * c\u2082) \u2022 q.i +\n      -((c\u2081 * c\u2082 * (x.imK * y.imK)) \u2022 1)\n[PROOFSTEP]\nsimp only [mul_comm _ c\u2082, mul_right_comm _ _ c\u2082]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (c\u2081 * (x.imI * y.imI)) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (c\u2081 * (x.imI * y.imK)) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (c\u2082 * (x.imJ * y.imJ)) \u2022 1 +\n              -((c\u2082 * (x.imJ * y.imK)) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((c\u2081 * (x.imK * y.imI)) \u2022 q.j) +\n        (c\u2082 * (x.imK * y.imJ)) \u2022 q.i +\n      -((c\u2082 * c\u2081 * (x.imK * y.imK)) \u2022 1)\n[PROOFSTEP]\nsimp only [\u2190 mul_comm c\u2081 c\u2082, \u2190 mul_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x * y).re \u2022 1 + (x * y).imI \u2022 q.i + (x * y).imJ \u2022 q.j + (x * y).imK \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (c\u2081 * x.imI * y.imI) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (c\u2081 * x.imI * y.imK) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (c\u2082 * x.imJ * y.imJ) \u2022 1 +\n              -((c\u2082 * x.imJ * y.imK) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((c\u2081 * x.imK * y.imI) \u2022 q.j) +\n        (c\u2082 * x.imK * y.imJ) \u2022 q.i +\n      -((c\u2081 * c\u2082 * x.imK * y.imK) \u2022 1)\n[PROOFSTEP]\nsimp [sub_eq_add_neg, add_smul, \u2190 add_assoc]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x.re * y.re) \u2022 1 + (c\u2081 * x.imI * y.imI) \u2022 1 + (c\u2082 * x.imJ * y.imJ) \u2022 1 + -((c\u2081 * c\u2082 * x.imK * y.imK) \u2022 1) +\n                            (x.re * y.imI) \u2022 q.i +\n                          (x.imI * y.re) \u2022 q.i +\n                        -((c\u2082 * x.imJ * y.imK) \u2022 q.i) +\n                      (c\u2082 * x.imK * y.imJ) \u2022 q.i +\n                    (x.re * y.imJ) \u2022 q.j +\n                  (c\u2081 * x.imI * y.imK) \u2022 q.j +\n                (x.imJ * y.re) \u2022 q.j +\n              -((c\u2081 * x.imK * y.imI) \u2022 q.j) +\n            (x.re * y.imK) \u2022 q.k +\n          (x.imI * y.imJ) \u2022 q.k +\n        -((x.imJ * y.imI) \u2022 q.k) +\n      (x.imK * y.re) \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (c\u2081 * x.imI * y.imI) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (c\u2081 * x.imI * y.imK) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (c\u2082 * x.imJ * y.imJ) \u2022 1 +\n              -((c\u2082 * x.imJ * y.imK) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((c\u2081 * x.imK * y.imI) \u2022 q.j) +\n        (c\u2082 * x.imK * y.imJ) \u2022 q.i +\n      -((c\u2081 * c\u2082 * x.imK * y.imK) \u2022 1)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nx y : \u210d[R,c\u2081,c\u2082]\n\u22a2 (x.re * y.re) \u2022 1 + (c\u2081 * x.imI * y.imI) \u2022 1 + (c\u2082 * x.imJ * y.imJ) \u2022 1 + -((c\u2081 * c\u2082 * x.imK * y.imK) \u2022 1) +\n                            (x.re * y.imI) \u2022 q.i +\n                          (x.imI * y.re) \u2022 q.i +\n                        -((c\u2082 * x.imJ * y.imK) \u2022 q.i) +\n                      (c\u2082 * x.imK * y.imJ) \u2022 q.i +\n                    (x.re * y.imJ) \u2022 q.j +\n                  (c\u2081 * x.imI * y.imK) \u2022 q.j +\n                (x.imJ * y.re) \u2022 q.j +\n              -((c\u2081 * x.imK * y.imI) \u2022 q.j) +\n            (x.re * y.imK) \u2022 q.k +\n          (x.imI * y.imJ) \u2022 q.k +\n        -((x.imJ * y.imI) \u2022 q.k) +\n      (x.imK * y.re) \u2022 q.k =\n    (x.re * y.re) \u2022 1 + (x.re * y.imI) \u2022 q.i + (x.re * y.imJ) \u2022 q.j + (x.re * y.imK) \u2022 q.k + (x.imI * y.re) \u2022 q.i +\n                          (c\u2081 * x.imI * y.imI) \u2022 1 +\n                        (x.imI * y.imJ) \u2022 q.k +\n                      (c\u2081 * x.imI * y.imK) \u2022 q.j +\n                    (x.imJ * y.re) \u2022 q.j +\n                  -((x.imJ * y.imI) \u2022 q.k) +\n                (c\u2082 * x.imJ * y.imJ) \u2022 1 +\n              -((c\u2082 * x.imJ * y.imK) \u2022 q.i) +\n            (x.imK * y.re) \u2022 q.k +\n          -((c\u2081 * x.imK * y.imI) \u2022 q.j) +\n        (c\u2082 * x.imK * y.imJ) \u2022 q.i +\n      -((c\u2081 * c\u2082 * x.imK * y.imK) \u2022 1)\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nr : R\nx : \u210d[R,c\u2081,c\u2082]\n\u22a2 lift q (r \u2022 x) = r \u2022 lift q x\n[PROOFSTEP]\nsimp [lift, mul_smul, \u2190 Algebra.smul_def]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nF : A \u2192\u2090[R] B\n\u22a2 \u2191F q.i * \u2191F q.i = c\u2081 \u2022 1\n[PROOFSTEP]\nrw [\u2190 F.map_mul, q.i_mul_i, F.map_smul, F.map_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nF : A \u2192\u2090[R] B\n\u22a2 \u2191F q.j * \u2191F q.j = c\u2082 \u2022 1\n[PROOFSTEP]\nrw [\u2190 F.map_mul, q.j_mul_j, F.map_smul, F.map_one]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nF : A \u2192\u2090[R] B\n\u22a2 \u2191F q.i * \u2191F q.j = \u2191F q.k\n[PROOFSTEP]\nrw [\u2190 F.map_mul, q.i_mul_j]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\nF : A \u2192\u2090[R] B\n\u22a2 \u2191F q.j * \u2191F q.i = -\u2191F q.k\n[PROOFSTEP]\nrw [\u2190 F.map_mul, q.j_mul_i, F.map_neg]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 Basis.compHom (Basis.self R) (Basis.liftHom q) = q\n[PROOFSTEP]\next\n[GOAL]\ncase hi\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 (Basis.compHom (Basis.self R) (Basis.liftHom q)).i = q.i\n[PROOFSTEP]\nsimp [Basis.lift]\n[GOAL]\ncase hj\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nq : Basis A c\u2081 c\u2082\n\u22a2 (Basis.compHom (Basis.self R) (Basis.liftHom q)).j = q.j\n[PROOFSTEP]\nsimp [Basis.lift]\n[GOAL]\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\n\u22a2 Basis.liftHom (Basis.compHom (Basis.self R) F) = F\n[PROOFSTEP]\next\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(Basis.liftHom (Basis.compHom (Basis.self R) F)) x\u271d = \u2191F x\u271d\n[PROOFSTEP]\ndsimp [Basis.lift]\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191(algebraMap R A) x\u271d.re + x\u271d.imI \u2022 \u2191F { re := 0, imI := 1, imJ := 0, imK := 0 } +\n        x\u271d.imJ \u2022 \u2191F { re := 0, imI := 0, imJ := 1, imK := 0 } +\n      x\u271d.imK \u2022 \u2191F { re := 0, imI := 0, imJ := 0, imK := 1 } =\n    \u2191F x\u271d\n[PROOFSTEP]\nrw [\u2190 F.commutes]\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191F (\u2191(algebraMap R \u210d[R,c\u2081,c\u2082]) x\u271d.re) + x\u271d.imI \u2022 \u2191F { re := 0, imI := 1, imJ := 0, imK := 0 } +\n        x\u271d.imJ \u2022 \u2191F { re := 0, imI := 0, imJ := 1, imK := 0 } +\n      x\u271d.imK \u2022 \u2191F { re := 0, imI := 0, imJ := 0, imK := 1 } =\n    \u2191F x\u271d\n[PROOFSTEP]\nsimp only [\u2190 F.commutes, \u2190 F.map_smul, \u2190 F.map_add, mk_add_mk, smul_mk, smul_zero, algebraMap_eq]\n[GOAL]\ncase H\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 \u2191F\n      { re := x\u271d.re + 0 + 0 + 0, imI := 0 + x\u271d.imI \u2022 1 + 0 + 0, imJ := 0 + 0 + x\u271d.imJ \u2022 1 + 0,\n        imK := 0 + 0 + 0 + x\u271d.imK \u2022 1 } =\n    \u2191F x\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\ncase H.h.e_6.h.e_re\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 x\u271d.re + 0 + 0 + 0 = x\u271d.re\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.h.e_6.h.e_imI\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 0 + x\u271d.imI \u2022 1 + 0 + 0 = x\u271d.imI\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.h.e_6.h.e_imJ\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 0 + 0 + x\u271d.imJ \u2022 1 + 0 = x\u271d.imJ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase H.h.e_6.h.e_imK\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Algebra R B\nc\u2081 c\u2082 : R\nF : \u210d[R,c\u2081,c\u2082] \u2192\u2090[R] A\nx\u271d : \u210d[R,c\u2081,c\u2082]\n\u22a2 0 + 0 + 0 + x\u271d.imK \u2022 1 = x\u271d.imK\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Algebra.QuaternionBasis", "llama_tokens": 18048, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8723473846343393, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.5268383556482603}}
{"text": "[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V1\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V2\ninst\u271d : AffineSpace V2 P2\nx\u271d\u00b9 x\u271d : P1 \u2192\u1d43[k] P2\nf : P1 \u2192 P2\nf_linear : V1 \u2192\u2097[k] V2\nf_add : \u2200 (p : P1) (v : V1), f (v +\u1d65 p) = \u2191f_linear v +\u1d65 f p\ng : P1 \u2192 P2\ng_linear : V1 \u2192\u2097[k] V2\ng_add : \u2200 (p : P1) (v : V1), g (v +\u1d65 p) = \u2191g_linear v +\u1d65 g p\nh : f = g\n\u22a2 { toFun := f, linear := f_linear, map_vadd' := f_add } = { toFun := g, linear := g_linear, map_vadd' := g_add }\n[PROOFSTEP]\ncases' (AddTorsor.Nonempty : Nonempty P1) with p\n[GOAL]\ncase intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V1\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V2\ninst\u271d : AffineSpace V2 P2\nx\u271d\u00b9 x\u271d : P1 \u2192\u1d43[k] P2\nf : P1 \u2192 P2\nf_linear : V1 \u2192\u2097[k] V2\nf_add : \u2200 (p : P1) (v : V1), f (v +\u1d65 p) = \u2191f_linear v +\u1d65 f p\ng : P1 \u2192 P2\ng_linear : V1 \u2192\u2097[k] V2\ng_add : \u2200 (p : P1) (v : V1), g (v +\u1d65 p) = \u2191g_linear v +\u1d65 g p\nh : f = g\np : P1\n\u22a2 { toFun := f, linear := f_linear, map_vadd' := f_add } = { toFun := g, linear := g_linear, map_vadd' := g_add }\n[PROOFSTEP]\ncongr with v\n[GOAL]\ncase intro.e_linear.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V1\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V2\ninst\u271d : AffineSpace V2 P2\nx\u271d\u00b9 x\u271d : P1 \u2192\u1d43[k] P2\nf : P1 \u2192 P2\nf_linear : V1 \u2192\u2097[k] V2\nf_add : \u2200 (p : P1) (v : V1), f (v +\u1d65 p) = \u2191f_linear v +\u1d65 f p\ng : P1 \u2192 P2\ng_linear : V1 \u2192\u2097[k] V2\ng_add : \u2200 (p : P1) (v : V1), g (v +\u1d65 p) = \u2191g_linear v +\u1d65 g p\nh : f = g\np : P1\nv : V1\n\u22a2 \u2191f_linear v = \u2191g_linear v\n[PROOFSTEP]\napply vadd_right_cancel (f p)\n[GOAL]\ncase intro.e_linear.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V1\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V2\ninst\u271d : AffineSpace V2 P2\nx\u271d\u00b9 x\u271d : P1 \u2192\u1d43[k] P2\nf : P1 \u2192 P2\nf_linear : V1 \u2192\u2097[k] V2\nf_add : \u2200 (p : P1) (v : V1), f (v +\u1d65 p) = \u2191f_linear v +\u1d65 f p\ng : P1 \u2192 P2\ng_linear : V1 \u2192\u2097[k] V2\ng_add : \u2200 (p : P1) (v : V1), g (v +\u1d65 p) = \u2191g_linear v +\u1d65 g p\nh : f = g\np : P1\nv : V1\n\u22a2 \u2191f_linear v +\u1d65 f p = \u2191g_linear v +\u1d65 f p\n[PROOFSTEP]\nerw [\u2190 f_add, h, \u2190 g_add]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np1 p2 : P1\n\u22a2 \u2191f.linear (p1 -\u1d65 p2) = \u2191f p1 -\u1d65 \u2191f p2\n[PROOFSTEP]\nconv_rhs => rw [\u2190 vsub_vadd p1 p2, map_vadd, vadd_vsub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np1 p2 : P1\n| \u2191f p1 -\u1d65 \u2191f p2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p1 p2, map_vadd, vadd_vsub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np1 p2 : P1\n| \u2191f p1 -\u1d65 \u2191f p2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p1 p2, map_vadd, vadd_vsub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np1 p2 : P1\n| \u2191f p1 -\u1d65 \u2191f p2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p1 p2, map_vadd, vadd_vsub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np : P2\nx\u271d\u00b9 : P1\nx\u271d : V1\nthis : AddAction V2 P2 := inferInstance\n\u22a2 Function.const P1 p (x\u271d +\u1d65 x\u271d\u00b9) = \u21910 x\u271d +\u1d65 Function.const P1 p x\u271d\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\n\u22a2 f.linear = 0 \u2194 \u2203 q, f = const k P1 q\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 \u2203 q, f = const k P1 q\n[PROOFSTEP]\nuse f (Classical.arbitrary P1)\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\n\u22a2 f = const k P1 (\u2191f (Classical.arbitrary P1))\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : f.linear = 0\np\u271d : P1\n\u22a2 \u2191f p\u271d = \u2191(const k P1 (\u2191f (Classical.arbitrary P1))) p\u271d\n[PROOFSTEP]\nrw [coe_const, Function.const_apply, \u2190 @vsub_eq_zero_iff_eq V2, \u2190 f.linearMap_vsub, h, LinearMap.zero_apply]\n[GOAL]\ncase refine'_2\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\nh : \u2203 q, f = const k P1 q\n\u22a2 f.linear = 0\n[PROOFSTEP]\nrcases h with \u27e8q, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nq : P2\n\u22a2 (const k P1 q).linear = 0\n[PROOFSTEP]\nexact const_linear k P1 q\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192 P2\nf' : V1 \u2192\u2097[k] V2\np : P1\nh : \u2200 (p' : P1), f p' = \u2191f' (p' -\u1d65 p) +\u1d65 f p\np' : P1\nv : V1\n\u22a2 f (v +\u1d65 p') = \u2191f' v +\u1d65 f p'\n[PROOFSTEP]\nrw [h, h p', vadd_vsub_assoc, f'.map_add, vadd_vadd]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u2075 : Ring k\ninst\u271d\u00b9\u2074 : AddCommGroup V1\ninst\u271d\u00b9\u00b3 : Module k V1\ninst\u271d\u00b9\u00b2 : AffineSpace V1 P1\ninst\u271d\u00b9\u00b9 : AddCommGroup V2\ninst\u271d\u00b9\u2070 : Module k V2\ninst\u271d\u2079 : AffineSpace V2 P2\ninst\u271d\u2078 : AddCommGroup V3\ninst\u271d\u2077 : Module k V3\ninst\u271d\u2076 : AffineSpace V3 P3\ninst\u271d\u2075 : AddCommGroup V4\ninst\u271d\u2074 : Module k V4\ninst\u271d\u00b3 : AffineSpace V4 P4\nR : Type u_10\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R V2\ninst\u271d : SMulCommClass k R V2\nc : R\nf : P1 \u2192\u1d43[k] V2\np : P1\nv : V1\n\u22a2 (c \u2022 \u2191f) (v +\u1d65 p) = \u2191(c \u2022 f.linear) v +\u1d65 (c \u2022 \u2191f) p\n[PROOFSTEP]\nsimp [smul_add, map_vadd f]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf g : P1 \u2192\u1d43[k] V2\np : P1\nv : V1\n\u22a2 (\u2191f + \u2191g) (v +\u1d65 p) = \u2191(f.linear + g.linear) v +\u1d65 (\u2191f + \u2191g) p\n[PROOFSTEP]\nsimp [add_add_add_comm, (map_vadd)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf g : P1 \u2192\u1d43[k] V2\np : P1\nv : V1\n\u22a2 (\u2191f - \u2191g) (v +\u1d65 p) = \u2191(f.linear - g.linear) v +\u1d65 (\u2191f - \u2191g) p\n[PROOFSTEP]\nsimp [sub_add_sub_comm, (map_vadd)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] V2\np : P1\nv : V1\n\u22a2 (-\u2191f) (v +\u1d65 p) = \u2191(-f.linear) v +\u1d65 (-\u2191f) p\n[PROOFSTEP]\nsimp [add_comm, map_vadd f]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] V2\ng : P1 \u2192\u1d43[k] P2\nthis : AddAction V2 P2 := inferInstance\np : P1\nv : V1\n\u22a2 (fun p => \u2191f p +\u1d65 \u2191g p) (v +\u1d65 p) = \u2191(f.linear + g.linear) v +\u1d65 (fun p => \u2191f p +\u1d65 \u2191g p) p\n[PROOFSTEP]\nsimp [vadd_vadd, add_right_comm, (map_vadd)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf g : P1 \u2192\u1d43[k] P2\np : P1\nv : V1\n\u22a2 (fun p => \u2191f p -\u1d65 \u2191g p) (v +\u1d65 p) = \u2191(f.linear - g.linear) v +\u1d65 (fun p => \u2191f p -\u1d65 \u2191g p) p\n[PROOFSTEP]\nsimp [(map_vadd), (vsub_vadd_eq_vsub_sub), (vadd_vsub_assoc), add_sub, sub_add_eq_add_sub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P2 \u2192\u1d43[k] P3\ng : P1 \u2192\u1d43[k] P2\n\u22a2 \u2200 (p : P1) (v : V1), (\u2191f \u2218 \u2191g) (v +\u1d65 p) = \u2191(LinearMap.comp f.linear g.linear) v +\u1d65 (\u2191f \u2218 \u2191g) p\n[PROOFSTEP]\nintro p v\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P2 \u2192\u1d43[k] P3\ng : P1 \u2192\u1d43[k] P2\np : P1\nv : V1\n\u22a2 (\u2191f \u2218 \u2191g) (v +\u1d65 p) = \u2191(LinearMap.comp f.linear g.linear) v +\u1d65 (\u2191f \u2218 \u2191g) p\n[PROOFSTEP]\nrw [Function.comp_apply, g.map_vadd, f.map_vadd]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P2 \u2192\u1d43[k] P3\ng : P1 \u2192\u1d43[k] P2\np : P1\nv : V1\n\u22a2 \u2191f.linear (\u2191g.linear v) +\u1d65 \u2191f (\u2191g p) = \u2191(LinearMap.comp f.linear g.linear) v +\u1d65 (\u2191f \u2218 \u2191g) p\n[PROOFSTEP]\nrfl\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\n\u22a2 Function.Injective \u2191f.linear \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nobtain \u27e8p\u27e9 := (inferInstance : Nonempty P1)\n[GOAL]\ncase intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\n\u22a2 Function.Injective \u2191f.linear \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nhave h : \u21d1f.linear = (Equiv.vaddConst (f p)).symm \u2218 f \u2218 Equiv.vaddConst p :=\n  by\n  ext v\n  simp [f.map_vadd, vadd_vsub_assoc]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\n\u22a2 \u2191f.linear = \u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\nv : V1\n\u22a2 \u2191f.linear v = (\u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)) v\n[PROOFSTEP]\nsimp [f.map_vadd, vadd_vsub_assoc]\n[GOAL]\ncase intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\nh : \u2191f.linear = \u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)\n\u22a2 Function.Injective \u2191f.linear \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nrw [h, Equiv.comp_injective, Equiv.injective_comp]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\n\u22a2 Function.Surjective \u2191f.linear \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nobtain \u27e8p\u27e9 := (inferInstance : Nonempty P1)\n[GOAL]\ncase intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\n\u22a2 Function.Surjective \u2191f.linear \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nhave h : \u21d1f.linear = (Equiv.vaddConst (f p)).symm \u2218 f \u2218 Equiv.vaddConst p :=\n  by\n  ext v\n  simp [f.map_vadd, vadd_vsub_assoc]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\n\u22a2 \u2191f.linear = \u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\nv : V1\n\u22a2 \u2191f.linear v = (\u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)) v\n[PROOFSTEP]\nsimp [f.map_vadd, vadd_vsub_assoc]\n[GOAL]\ncase intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np : P1\nh : \u2191f.linear = \u2191(Equiv.vaddConst (\u2191f p)).symm \u2218 \u2191f \u2218 \u2191(Equiv.vaddConst p)\n\u22a2 Function.Surjective \u2191f.linear \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nrw [h, Equiv.comp_surjective, Equiv.surjective_comp]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\n\u22a2 \u2191f '' s -\u1d65 \u2191f '' t = \u2191f.linear '' (s -\u1d65 t)\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 v \u2208 \u2191f '' s -\u1d65 \u2191f '' t \u2194 v \u2208 \u2191f.linear '' (s -\u1d65 t)\n[PROOFSTEP]\nsimp only [(Set.mem_vsub), Set.mem_image, exists_exists_and_eq_and, exists_and_left, \u2190 f.linearMap_vsub]\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 a_1, a_1 \u2208 t \u2227 \u2191f.linear (a -\u1d65 a_1) = v) \u2194\n    \u2203 x, (\u2203 x_1, x_1 \u2208 s \u2227 \u2203 x_2, x_2 \u2208 t \u2227 x_1 -\u1d65 x_2 = x) \u2227 \u2191f.linear x = v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a, a \u2208 s \u2227 \u2203 a_1, a_1 \u2208 t \u2227 \u2191f.linear (a -\u1d65 a_1) = v) \u2192\n    \u2203 x, (\u2203 x_1, x_1 \u2208 s \u2227 \u2203 x_2, x_2 \u2208 t \u2227 x_1 -\u1d65 x_2 = x) \u2227 \u2191f.linear x = v\n[PROOFSTEP]\nrintro \u27e8x, hx, y, hy, hv\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\nhv : \u2191f.linear (x -\u1d65 y) = v\n\u22a2 \u2203 x, (\u2203 x_1, x_1 \u2208 s \u2227 \u2203 x_2, x_2 \u2208 t \u2227 x_1 -\u1d65 x_2 = x) \u2227 \u2191f.linear x = v\n[PROOFSTEP]\nexact \u27e8x -\u1d65 y, \u27e8x, hx, y, hy, rfl\u27e9, hv\u27e9\n[GOAL]\ncase h.mpr\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 x, (\u2203 x_1, x_1 \u2208 s \u2227 \u2203 x_2, x_2 \u2208 t \u2227 x_1 -\u1d65 x_2 = x) \u2227 \u2191f.linear x = v) \u2192\n    \u2203 a, a \u2208 s \u2227 \u2203 a_1, a_1 \u2208 t \u2227 \u2191f.linear (a -\u1d65 a_1) = v\n[PROOFSTEP]\nrintro \u27e8-, \u27e8x, hx, y, hy, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\n\u22a2 \u2203 a, a \u2208 s \u2227 \u2203 a_1, a_1 \u2208 t \u2227 \u2191f.linear (a -\u1d65 a_1) = \u2191f.linear (x -\u1d65 y)\n[PROOFSTEP]\nexact \u27e8x, hx, y, hy, rfl\u27e9\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : V1\nc : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c = (1 - c) \u2022 p\u2080 + c \u2022 p\u2081\n[PROOFSTEP]\nsimp [lineMap_apply_module', smul_sub, sub_smul]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : V1\nc : k\n\u22a2 c \u2022 p\u2081 - c \u2022 p\u2080 + p\u2080 = p\u2080 - c \u2022 p\u2080 + c \u2022 p\u2081\n[PROOFSTEP]\nabel\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : V1\nc : k\n\u22a2 c \u2022 p\u2081 - c \u2022 p\u2080 + p\u2080 = p\u2080 - c \u2022 p\u2080 + c \u2022 p\u2081\n[PROOFSTEP]\nabel\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np : P1\nv : V1\nc : k\n\u22a2 \u2191(lineMap p (v +\u1d65 p)) c = c \u2022 v +\u1d65 p\n[PROOFSTEP]\nrw [lineMap_apply, vadd_vsub]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np : P1\nc : k\n\u22a2 \u2191(lineMap p p) c = p\n[PROOFSTEP]\nletI : AddAction V1 P1 := inferInstance\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np : P1\nc : k\nthis : AddAction V1 P1 := inferInstance\n\u22a2 \u2191(lineMap p p) c = p\n[PROOFSTEP]\nsimp [(lineMap_apply), (vsub_self)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\n\u22a2 \u2191(lineMap p\u2080 p\u2081) 0 = p\u2080\n[PROOFSTEP]\nletI : AddAction V1 P1 := inferInstance\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nthis : AddAction V1 P1 := inferInstance\n\u22a2 \u2191(lineMap p\u2080 p\u2081) 0 = p\u2080\n[PROOFSTEP]\nsimp [(lineMap_apply)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\n\u22a2 \u2191(lineMap p\u2080 p\u2081) 1 = p\u2081\n[PROOFSTEP]\nsimp [(lineMap_apply), (vsub_vadd)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k V4\ninst\u271d\u00b9 : AffineSpace V4 P4\ninst\u271d : NoZeroSMulDivisors k V1\np\u2080 p\u2081 : P1\nc\u2081 c\u2082 : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c\u2081 = \u2191(lineMap p\u2080 p\u2081) c\u2082 \u2194 p\u2080 = p\u2081 \u2228 c\u2081 = c\u2082\n[PROOFSTEP]\nrw [lineMap_apply, lineMap_apply, \u2190 @vsub_eq_zero_iff_eq V1, vadd_vsub_vadd_cancel_right, \u2190 sub_smul, smul_eq_zero,\n  sub_eq_zero, vsub_eq_zero_iff_eq, or_comm, eq_comm]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k V4\ninst\u271d\u00b9 : AffineSpace V4 P4\ninst\u271d : NoZeroSMulDivisors k V1\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c = p\u2080 \u2194 p\u2080 = p\u2081 \u2228 c = 0\n[PROOFSTEP]\nrw [\u2190 @lineMap_eq_lineMap_iff k V1, lineMap_apply_zero]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k V4\ninst\u271d\u00b9 : AffineSpace V4 P4\ninst\u271d : NoZeroSMulDivisors k V1\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c = p\u2081 \u2194 p\u2080 = p\u2081 \u2228 c = 1\n[PROOFSTEP]\nrw [\u2190 @lineMap_eq_lineMap_iff k V1, lineMap_apply_one]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : P1 \u2192\u1d43[k] P2\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191f (\u2191(lineMap p\u2080 p\u2081) c) = \u2191(lineMap (\u2191f p\u2080) (\u2191f p\u2081)) c\n[PROOFSTEP]\nsimp [(lineMap_apply), (map_vadd), (linearMap_vsub)]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\n\u22a2 lineMap p\u2080 p\u2081 = comp (lineMap p\u2081 p\u2080) (lineMap 1 0)\n[PROOFSTEP]\nrw [comp_lineMap]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\n\u22a2 lineMap p\u2080 p\u2081 = lineMap (\u2191(lineMap p\u2081 p\u2080) 1) (\u2191(lineMap p\u2081 p\u2080) 0)\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) (1 - c) = \u2191(lineMap p\u2081 p\u2080) c\n[PROOFSTEP]\nrw [lineMap_symm p\u2080, comp_apply]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap p\u2081 p\u2080) (\u2191(lineMap 1 0) (1 - c)) = \u2191(lineMap p\u2081 p\u2080) c\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_6.h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap 1 0) (1 - c) = c\n[PROOFSTEP]\nsimp [lineMap_apply]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 p\u2080 -\u1d65 \u2191(lineMap p\u2080 p\u2081) c = c \u2022 (p\u2080 -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev, lineMap_vsub_left, \u2190 smul_neg, neg_vsub_eq_vsub_rev]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c -\u1d65 p\u2081 = (1 - c) \u2022 (p\u2080 -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [\u2190 lineMap_apply_one_sub, lineMap_vsub_left]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\np\u2080 p\u2081 : P1\nc : k\n\u22a2 p\u2081 -\u1d65 \u2191(lineMap p\u2080 p\u2081) c = (1 - c) \u2022 (p\u2081 -\u1d65 p\u2080)\n[PROOFSTEP]\nrw [\u2190 lineMap_apply_one_sub, left_vsub_lineMap]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : V1 \u2192\u1d43[k] V2\n\u22a2 \u2191f = \u2191f.linear + fun x => \u2191f 0\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : V1 \u2192\u1d43[k] V2\nx : V1\n\u22a2 \u2191f x = (\u2191f.linear + fun x => \u2191f 0) x\n[PROOFSTEP]\ncalc\n  f x = f.linear x +\u1d65 f 0 := by rw [\u2190 f.map_vadd, vadd_eq_add, add_zero]\n  _ = (f.linear + fun _ : V1 => f 0) x := rfl\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : V1 \u2192\u1d43[k] V2\nx : V1\n\u22a2 \u2191f x = \u2191f.linear x +\u1d65 \u2191f 0\n[PROOFSTEP]\nrw [\u2190 f.map_vadd, vadd_eq_add, add_zero]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : V1 \u2192\u1d43[k] V2\n\u22a2 \u2191f.linear = \u2191f - fun x => \u2191f 0\n[PROOFSTEP]\nrw [decomp]\n[GOAL]\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\nf : V1 \u2192\u1d43[k] V2\n\u22a2 \u2191f.linear = (\u2191f.linear + fun x => \u2191f 0) - fun x => (\u2191f.linear + fun x => \u2191f 0) 0\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, Pi.add_apply, add_sub_cancel, zero_add]\n[GOAL]\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b : k\n\u22a2 \u2191f '' Set.uIcc a b = Set.uIcc (\u2191f a) (\u2191f b)\n[PROOFSTEP]\nhave : \u21d1f = (fun x => x + f 0) \u2218 fun x => x * (f 1 - f 0) :=\n  by\n  ext x\n  change f x = x \u2022 (f 1 -\u1d65 f 0) +\u1d65 f 0\n  rw [\u2190 f.linearMap_vsub, \u2190 f.linear.map_smul, \u2190 f.map_vadd]\n  simp only [vsub_eq_sub, add_zero, mul_one, vadd_eq_add, sub_zero, smul_eq_mul]\n[GOAL]\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b : k\n\u22a2 \u2191f = (fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b x : k\n\u22a2 \u2191f x = ((fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)) x\n[PROOFSTEP]\nchange f x = x \u2022 (f 1 -\u1d65 f 0) +\u1d65 f 0\n[GOAL]\ncase h\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b x : k\n\u22a2 \u2191f x = x \u2022 (\u2191f 1 -\u1d65 \u2191f 0) +\u1d65 \u2191f 0\n[PROOFSTEP]\nrw [\u2190 f.linearMap_vsub, \u2190 f.linear.map_smul, \u2190 f.map_vadd]\n[GOAL]\ncase h\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b x : k\n\u22a2 \u2191f x = \u2191f (x \u2022 (1 -\u1d65 0) +\u1d65 0)\n[PROOFSTEP]\nsimp only [vsub_eq_sub, add_zero, mul_one, vadd_eq_add, sub_zero, smul_eq_mul]\n[GOAL]\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b : k\nthis : \u2191f = (fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)\n\u22a2 \u2191f '' Set.uIcc a b = Set.uIcc (\u2191f a) (\u2191f b)\n[PROOFSTEP]\nrw [this, Set.image_comp]\n[GOAL]\nk\u271d : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b3 : Ring k\u271d\ninst\u271d\u00b9\u00b2 : AddCommGroup V1\ninst\u271d\u00b9\u00b9 : Module k\u271d V1\ninst\u271d\u00b9\u2070 : AffineSpace V1 P1\ninst\u271d\u2079 : AddCommGroup V2\ninst\u271d\u2078 : Module k\u271d V2\ninst\u271d\u2077 : AffineSpace V2 P2\ninst\u271d\u2076 : AddCommGroup V3\ninst\u271d\u2075 : Module k\u271d V3\ninst\u271d\u2074 : AffineSpace V3 P3\ninst\u271d\u00b3 : AddCommGroup V4\ninst\u271d\u00b2 : Module k\u271d V4\ninst\u271d\u00b9 : AffineSpace V4 P4\nk : Type u_10\ninst\u271d : LinearOrderedField k\nf : k \u2192\u1d43[k] k\na b : k\nthis : \u2191f = (fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)\n\u22a2 (fun x => x + \u2191f 0) '' ((fun x => x * (\u2191f 1 - \u2191f 0)) '' Set.uIcc a b) =\n    Set.uIcc (((fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)) a)\n      (((fun x => x + \u2191f 0) \u2218 fun x => x * (\u2191f 1 - \u2191f 0)) b)\n[PROOFSTEP]\nsimp only [Set.image_add_const_uIcc, Set.image_mul_const_uIcc, Function.comp_apply]\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\n\u22a2 \u2200 (x y : V1 \u2192\u1d43[k] V2),\n    (fun f => (\u2191f 0, f.linear)) (x + y) = (fun f => (\u2191f 0, f.linear)) x + (fun f => (\u2191f 0, f.linear)) y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\n\u22a2 \u2200 (r : R) (x : V1 \u2192\u1d43[k] V2),\n    AddHom.toFun\n        { toFun := fun f => (\u2191f 0, f.linear),\n          map_add' :=\n            (_ :\n              \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) }\n        (r \u2022 x) =\n      \u2191(RingHom.id R) r \u2022\n        AddHom.toFun\n          { toFun := fun f => (\u2191f 0, f.linear),\n            map_add' :=\n              (_ :\n                \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                  (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) }\n          x\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nf : V1 \u2192\u1d43[k] V2\n\u22a2 (fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f => (\u2191f 0, f.linear),\n                map_add' :=\n                  (_ :\n                    \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                      (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n            map_smul' :=\n              (_ :\n                \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2), (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nf : V1 \u2192\u1d43[k] V2\np\u271d : V1\n\u22a2 \u2191((fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst)\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => (\u2191f 0, f.linear),\n                    map_add' :=\n                      (_ :\n                        \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                          (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2),\n                      (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n            f))\n      p\u271d =\n    \u2191f p\u271d\n[PROOFSTEP]\nrw [f.decomp]\n[GOAL]\ncase h\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nf : V1 \u2192\u1d43[k] V2\np\u271d : V1\n\u22a2 \u2191((fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst)\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => (\u2191f 0, f.linear),\n                    map_add' :=\n                      (_ :\n                        \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                          (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2),\n                      (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n            f))\n      p\u271d =\n    (\u2191f.linear + fun x => \u2191f 0) p\u271d\n[PROOFSTEP]\nsimp [const_apply _ _]\n  -- porting note: `simp` needs `_`s to use this lemma\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\n\u22a2 Function.RightInverse (fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst)\n    {\n          toAddHom :=\n            { toFun := fun f => (\u2191f 0, f.linear),\n              map_add' :=\n                (_ :\n                  \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                    (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n          map_smul' :=\n            (_ :\n              \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2),\n                (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom.toFun\n[PROOFSTEP]\nrintro \u27e8v, f\u27e9\n[GOAL]\ncase mk\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nv : V2\nf : V1 \u2192\u2097[k] V2\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f => (\u2191f 0, f.linear),\n              map_add' :=\n                (_ :\n                  \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                    (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n          map_smul' :=\n            (_ : \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2), (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n      ((fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst) (v, f)) =\n    (v, f)\n[PROOFSTEP]\next\n[GOAL]\ncase mk.h\u2081\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nv : V2\nf : V1 \u2192\u2097[k] V2\n\u22a2 (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f => (\u2191f 0, f.linear),\n                map_add' :=\n                  (_ :\n                    \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                      (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n            map_smul' :=\n              (_ :\n                \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2), (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n        ((fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst) (v, f))).fst =\n    (v, f).fst\n[PROOFSTEP]\nsimp [const_apply _ _, const_linear _ _]\n  -- porting note: `simp` needs `_`s\n[GOAL]\ncase mk.h\u2082.h\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V1\ninst\u271d\u2076 : AffineSpace V1 P1\ninst\u271d\u2075 : AddCommGroup V2\ninst\u271d\u2074 : Module k V1\ninst\u271d\u00b3 : Module k V2\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : Module R V2\ninst\u271d : SMulCommClass k R V2\nv : V2\nf : V1 \u2192\u2097[k] V2\nx\u271d : V1\n\u22a2 \u2191(AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => (\u2191f 0, f.linear),\n                    map_add' :=\n                      (_ :\n                        \u2200 (a a_1 : V1 \u2192\u1d43[k] V2),\n                          (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear) = (\u2191a 0 + \u2191a_1 0, a.linear + a_1.linear)) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (a : R) (a_1 : V1 \u2192\u1d43[k] V2),\n                      (a \u2022 \u2191a_1 0, a \u2022 a_1.linear) = (a \u2022 \u2191a_1 0, a \u2022 a_1.linear)) }.toAddHom\n            ((fun p => LinearMap.toAffineMap p.snd + const k V1 p.fst) (v, f))).snd\n      x\u271d =\n    \u2191(v, f).snd x\u271d\n[PROOFSTEP]\nsimp [const_apply _ _, const_linear _ _]\n  -- porting note: `simp` needs `_`s\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc : P1\n\u22a2 homothety c 1 = id k P1\n[PROOFSTEP]\next p\n[GOAL]\ncase h\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc p : P1\n\u22a2 \u2191(homothety c 1) p = \u2191(id k P1) p\n[PROOFSTEP]\nsimp [homothety_apply]\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc : P1\nr\u2081 r\u2082 : k\np : P1\n\u22a2 \u2191(homothety c (r\u2081 * r\u2082)) p = \u2191(homothety c r\u2081) (\u2191(homothety c r\u2082) p)\n[PROOFSTEP]\nsimp only [homothety_apply, mul_smul, vadd_vsub]\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc : P1\n\u22a2 homothety c 0 = const k P1 c\n[PROOFSTEP]\next p\n[GOAL]\ncase h\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc p : P1\n\u22a2 \u2191(homothety c 0) p = \u2191(const k P1 c) p\n[PROOFSTEP]\nsimp [homothety_apply]\n[GOAL]\nR : Type u_1\nk : Type u_2\nV1 : Type u_3\nP1 : Type u_4\nV2 : Type u_5\ninst\u271d\u2075 : CommRing k\ninst\u271d\u2074 : AddCommGroup V1\ninst\u271d\u00b3 : AffineSpace V1 P1\ninst\u271d\u00b2 : AddCommGroup V2\ninst\u271d\u00b9 : Module k V1\ninst\u271d : Module k V2\nc : P1\nr\u2081 r\u2082 : k\n\u22a2 homothety c (r\u2081 + r\u2082) = r\u2081 \u2022 (id k P1 -\u1d65 const k P1 c) +\u1d65 homothety c r\u2082\n[PROOFSTEP]\nsimp only [homothety_def, add_smul, vadd_vadd]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Ring \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na b : \ud835\udd5c\nf : E \u2192\u1d43[\ud835\udd5c] F\nh : a + b = 1\n\u22a2 \u2191f (a \u2022 x + b \u2022 y) = a \u2022 \u2191f x + b \u2022 \u2191f y\n[PROOFSTEP]\nsimp only [Convex.combo_eq_smul_sub_add h, \u2190 vsub_eq_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : Ring \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na b : \ud835\udd5c\nf : E \u2192\u1d43[\ud835\udd5c] F\nh : a + b = 1\n\u22a2 \u2191f (b \u2022 (y -\u1d65 x) + x) = b \u2022 (\u2191f y -\u1d65 \u2191f x) + \u2191f x\n[PROOFSTEP]\nexact f.apply_lineMap _ _ _\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.AffineMap", "llama_tokens": 29871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744761936437, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.5267019984018098}}
{"text": "[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\n\u22a2 rootMultiplicity t (\u2191derivative p) = rootMultiplicity t p - 1\n[PROOFSTEP]\nrcases eq_or_ne p 0 with (rfl | hp)\n[GOAL]\ncase inl\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\nt : R\nhpt : IsRoot 0 t\n\u22a2 rootMultiplicity t (\u2191derivative 0) = rootMultiplicity t 0 - 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\n\u22a2 rootMultiplicity t (\u2191derivative p) = rootMultiplicity t p - 1\n[PROOFSTEP]\nnth_rw 1 [\u2190 p.divByMonic_mul_pow_rootMultiplicity_eq t]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\n\u22a2 rootMultiplicity t (\u2191derivative (p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p * (X - \u2191C t) ^ rootMultiplicity t p)) =\n    rootMultiplicity t p - 1\n[PROOFSTEP]\nsimp only [derivative_pow, derivative_mul, derivative_sub, derivative_X, derivative_C, sub_zero, mul_one]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\n\u22a2 rootMultiplicity t\n      (\u2191derivative (p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p) * (X - \u2191C t) ^ rootMultiplicity t p +\n        p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p *\n          (\u2191C \u2191(rootMultiplicity t p) * (X - \u2191C t) ^ (rootMultiplicity t p - 1))) =\n    rootMultiplicity t p - 1\n[PROOFSTEP]\nset n := p.rootMultiplicity t - 1\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\n\u22a2 rootMultiplicity t\n      (\u2191derivative (p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p) * (X - \u2191C t) ^ rootMultiplicity t p +\n        p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p * (\u2191C \u2191(rootMultiplicity t p) * (X - \u2191C t) ^ n)) =\n    n\n[PROOFSTEP]\nhave hn : n + 1 = _ := tsub_add_cancel_of_le ((rootMultiplicity_pos hp).mpr hpt)\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\n\u22a2 rootMultiplicity t\n      (\u2191derivative (p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p) * (X - \u2191C t) ^ rootMultiplicity t p +\n        p /\u2098 (X - \u2191C t) ^ rootMultiplicity t p * (\u2191C \u2191(rootMultiplicity t p) * (X - \u2191C t) ^ n)) =\n    n\n[PROOFSTEP]\nrw [\u2190 hn]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\n\u22a2 rootMultiplicity t\n      (\u2191derivative (p /\u2098 (X - \u2191C t) ^ (n + 1)) * (X - \u2191C t) ^ (n + 1) +\n        p /\u2098 (X - \u2191C t) ^ (n + 1) * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n)) =\n    n\n[PROOFSTEP]\nset q := p /\u2098 (X - C t) ^ (n + 1) with _hq\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 rootMultiplicity t (\u2191derivative q * (X - \u2191C t) ^ (n + 1) + q * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n)) = n\n[PROOFSTEP]\nconvert_to rootMultiplicity t ((X - C t) ^ n * (derivative q * (X - C t) + q * C \u2191(n + 1))) = n\n[GOAL]\ncase h.e'_2\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 rootMultiplicity t (\u2191derivative q * (X - \u2191C t) ^ (n + 1) + q * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n)) =\n    rootMultiplicity t ((X - \u2191C t) ^ n * (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e'_2.e_p\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 \u2191derivative q * (X - \u2191C t) ^ (n + 1) + q * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n) =\n    (X - \u2191C t) ^ n * (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1))\n[PROOFSTEP]\nrw [mul_add, mul_left_comm <| (X - C t) ^ n, \u2190 pow_succ']\n[GOAL]\ncase h.e'_2.e_p\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 \u2191derivative q * (X - \u2191C t) ^ (n + 1) + q * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n) =\n    \u2191derivative q * (X - \u2191C t) ^ (n + 1) + (X - \u2191C t) ^ n * (q * \u2191C \u2191(n + 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_2.e_p.e_a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 q * (\u2191C \u2191(n + 1) * (X - \u2191C t) ^ n) = (X - \u2191C t) ^ n * (q * \u2191C \u2191(n + 1))\n[PROOFSTEP]\nrw [mul_left_comm <| (X - C t) ^ n, mul_comm <| (X - C t) ^ n]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 rootMultiplicity t ((X - \u2191C t) ^ n * (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1))) = n\n[PROOFSTEP]\nhave h : eval t (derivative q * (X - C t) + q * C (R := R) \u2191(n + 1)) \u2260 0 :=\n  by\n  suffices eval t q * \u2191(n + 1) \u2260 0 by simpa\n  refine' mul_ne_zero _ (Nat.cast_ne_zero.mpr n.succ_ne_zero)\n  convert eval_divByMonic_pow_rootMultiplicity_ne_zero t hp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 eval t (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n[PROOFSTEP]\nsuffices eval t q * \u2191(n + 1) \u2260 0 by simpa\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\nthis : eval t q * \u2191(n + 1) \u2260 0\n\u22a2 eval t (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n[PROOFSTEP]\nsimpa\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 eval t q * \u2191(n + 1) \u2260 0\n[PROOFSTEP]\nrefine' mul_ne_zero _ (Nat.cast_ne_zero.mpr n.succ_ne_zero)\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\n\u22a2 eval t q \u2260 0\n[PROOFSTEP]\nconvert eval_divByMonic_pow_rootMultiplicity_ne_zero t hp\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\nh : eval t (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n\u22a2 rootMultiplicity t ((X - \u2191C t) ^ n * (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1))) = n\n[PROOFSTEP]\nrw [rootMultiplicity_mul, rootMultiplicity_X_sub_C_pow, rootMultiplicity_eq_zero h, add_zero]\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\nh : eval t (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n\u22a2 (X - \u2191C t) ^ n * (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n[PROOFSTEP]\nrefine' mul_ne_zero (pow_ne_zero n <| X_sub_C_ne_zero t) _\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\nh : eval t (\u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1)) \u2260 0\n\u22a2 \u2191derivative q * (X - \u2191C t) + q * \u2191C \u2191(n + 1) \u2260 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : IsRoot p t\nhp : p \u2260 0\nn : \u2115 := rootMultiplicity t p - 1\nhn : n + 1 = rootMultiplicity t p\nq : R[X] := p /\u2098 (X - \u2191C t) ^ (n + 1)\n_hq : q = p /\u2098 (X - \u2191C t) ^ (n + 1)\nh :\n  \u2191derivative (p /\u2098 (X - \u2191C t) ^ (n + 1)) * (X - \u2191C t) +\n      p /\u2098 (X - \u2191C t) ^ (n + 1) * \u2191C \u2191(rootMultiplicity t p - 1 + 1) =\n    0\n\u22a2 eval t\n      (\u2191derivative (p /\u2098 (X - \u2191C t) ^ (n + 1)) * (X - \u2191C t) +\n        p /\u2098 (X - \u2191C t) ^ (n + 1) * \u2191C \u2191(rootMultiplicity t p - 1 + 1)) =\n    0\n[PROOFSTEP]\nrw [h, eval_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\n\u22a2 rootMultiplicity t p - 1 \u2264 rootMultiplicity t (\u2191derivative p)\n[PROOFSTEP]\nby_cases p.IsRoot t\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\n\u22a2 rootMultiplicity t p - 1 \u2264 rootMultiplicity t (\u2191derivative p)\n[PROOFSTEP]\nby_cases p.IsRoot t\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nh : IsRoot p t\n\u22a2 rootMultiplicity t p - 1 \u2264 rootMultiplicity t (\u2191derivative p)\n[PROOFSTEP]\nexact (derivative_rootMultiplicity_of_root h).symm.le\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nh : \u00acIsRoot p t\n\u22a2 rootMultiplicity t p - 1 \u2264 rootMultiplicity t (\u2191derivative p)\n[PROOFSTEP]\nrw [rootMultiplicity_eq_zero h, zero_tsub]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nh : \u00acIsRoot p t\n\u22a2 0 \u2264 rootMultiplicity t (\u2191derivative p)\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\n\u22a2 \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 RingHom.map_mul, Units.mul_inv, C_1]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\n\u22a2 \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1\n[PROOFSTEP]\nrw [\u2190 RingHom.map_mul, Units.inv_mul, C_1]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\n\u22a2 \u2191((fun p =>\n          { val := \u2191C \u2191(normUnit (leadingCoeff p)), inv := \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9,\n            val_inv := (_ : \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1),\n            inv_val := (_ : \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1) })\n        0) =\n    \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\na\u271d b\u271d : R[X]\nhp0 : a\u271d \u2260 0\nhq0 : b\u271d \u2260 0\n\u22a2 \u2191((fun p =>\n          { val := \u2191C \u2191(normUnit (leadingCoeff p)), inv := \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9,\n            val_inv := (_ : \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1),\n            inv_val := (_ : \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1) })\n        (a\u271d * b\u271d)) =\n    \u2191((fun p =>\n            { val := \u2191C \u2191(normUnit (leadingCoeff p)), inv := \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9,\n              val_inv := (_ : \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1),\n              inv_val := (_ : \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1) })\n          a\u271d *\n        (fun p =>\n            { val := \u2191C \u2191(normUnit (leadingCoeff p)), inv := \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9,\n              val_inv := (_ : \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1),\n              inv_val := (_ : \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1) })\n          b\u271d)\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\na\u271d b\u271d : R[X]\nhp0 : a\u271d \u2260 0\nhq0 : b\u271d \u2260 0\n\u22a2 \u2191C \u2191(normUnit (leadingCoeff (a\u271d * b\u271d))) = \u2191C \u2191(normUnit (leadingCoeff a\u271d)) * \u2191C \u2191(normUnit (leadingCoeff b\u271d))\n[PROOFSTEP]\nrw [Ne.def, \u2190 leadingCoeff_eq_zero] at *\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\na\u271d b\u271d : R[X]\nhp0 : \u00acleadingCoeff a\u271d = 0\nhq0 : \u00acleadingCoeff b\u271d = 0\n\u22a2 \u2191C \u2191(normUnit (leadingCoeff (a\u271d * b\u271d))) = \u2191C \u2191(normUnit (leadingCoeff a\u271d)) * \u2191C \u2191(normUnit (leadingCoeff b\u271d))\n[PROOFSTEP]\nrw [leadingCoeff_mul, normUnit_mul hp0 hq0, Units.val_mul, C_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\nu : R[X]\u02e3\n\u22a2 \u2191((fun p =>\n          { val := \u2191C \u2191(normUnit (leadingCoeff p)), inv := \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9,\n            val_inv := (_ : \u2191C \u2191(normUnit (leadingCoeff p)) * \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 = 1),\n            inv_val := (_ : \u2191C \u2191(normUnit (leadingCoeff p))\u207b\u00b9 * \u2191C \u2191(normUnit (leadingCoeff p)) = 1) })\n        \u2191u) =\n    \u2191u\u207b\u00b9\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\nu : R[X]\u02e3\n\u22a2 \u2191C \u2191(normUnit (leadingCoeff \u2191u)) = \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 mul_one u\u207b\u00b9, Units.val_mul, Units.eq_inv_mul_iff_mul_eq]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\nu : R[X]\u02e3\n\u22a2 \u2191u * \u2191C \u2191(normUnit (leadingCoeff \u2191u)) = \u21911\n[PROOFSTEP]\nrcases Polynomial.isUnit_iff.1 \u27e8u, rfl\u27e9 with \u27e8_, \u27e8w, rfl\u27e9, h2\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\nu : R[X]\u02e3\nw : R\u02e3\nh2 : \u2191C \u2191w = \u2191u\n\u22a2 \u2191u * \u2191C \u2191(normUnit (leadingCoeff \u2191u)) = \u21911\n[PROOFSTEP]\nrw [\u2190 h2, leadingCoeff_C, normUnit_coe_units, \u2190 C_mul, Units.mul_inv, C_1]\n[GOAL]\ncase intro.intro.intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\nu : R[X]\u02e3\nw : R\u02e3\nh2 : \u2191C \u2191w = \u2191u\n\u22a2 1 = \u21911\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\n\u22a2 \u2191(normUnit p) = \u2191C \u2191(normUnit (leadingCoeff p))\n[PROOFSTEP]\nsimp [normUnit]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\n\u22a2 leadingCoeff (\u2191normalize p) = \u2191normalize (leadingCoeff p)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\nhp : Monic p\n\u22a2 \u2191normalize p = p\n[PROOFSTEP]\nsimp only [Polynomial.coe_normUnit, normalize_apply, hp.leadingCoeff, normUnit_one, Units.val_one, Polynomial.C.map_one,\n  mul_one]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizationMonoid R\np : R[X]\n\u22a2 roots (\u2191normalize p) = roots p\n[PROOFSTEP]\nrw [normalize_apply, mul_comm, coe_normUnit, roots_C_mul _ (normUnit (leadingCoeff p)).ne_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np q : R[X]\nhp0 : p \u2260 0\nhp : \u00acIsUnit p\nh : 0 \u2265 degree p\n\u22a2 False\n[PROOFSTEP]\nrw [eq_C_of_degree_le_zero h] at hp0 hp \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np q : R[X]\nhp0 : \u2191C (coeff p 0) \u2260 0\nhp : \u00acIsUnit (\u2191C (coeff p 0))\nh : 0 \u2265 degree p\n\u22a2 False\n[PROOFSTEP]\nexact hp (IsUnit.map C (IsUnit.mk0 (coeff p 0) (mt C_inj.2 (by simpa using hp0))))\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np q : R[X]\nhp0 : \u2191C (coeff p 0) \u2260 0\nhp : \u00acIsUnit (\u2191C (coeff p 0))\nh : 0 \u2265 degree p\n\u22a2 \u00ac\u2191C (coeff p 0) = \u2191C 0\n[PROOFSTEP]\nsimpa using hp0\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np q : R[X]\nh : p \u2260 0\n\u22a2 Monic (p * \u2191C (leadingCoeff p)\u207b\u00b9)\n[PROOFSTEP]\nrw [Monic, leadingCoeff_mul, leadingCoeff_C, mul_inv_cancel (show leadingCoeff p \u2260 0 from mt leadingCoeff_eq_zero.1 h)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np\u271d q p : R[X]\nh : q \u2260 0\n\u22a2 degree (p * \u2191C (leadingCoeff q)\u207b\u00b9) = degree p\n[PROOFSTEP]\nhave h\u2081 : (leadingCoeff q)\u207b\u00b9 \u2260 0 := inv_ne_zero (mt leadingCoeff_eq_zero.1 h)\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : DivisionRing R\np\u271d q p : R[X]\nh : q \u2260 0\nh\u2081 : (leadingCoeff q)\u207b\u00b9 \u2260 0\n\u22a2 degree (p * \u2191C (leadingCoeff q)\u207b\u00b9) = degree p\n[PROOFSTEP]\nrw [degree_mul, degree_C h\u2081, add_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : DivisionRing R\np q : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\n\u22a2 map f p = 0 \u2194 p = 0\n[PROOFSTEP]\nsimp only [Polynomial.ext_iff]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : DivisionRing R\np q : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\n\u22a2 (\u2200 (n : \u2115), coeff (map f p) n = coeff 0 n) \u2194 \u2200 (n : \u2115), coeff p n = coeff 0 n\n[PROOFSTEP]\ncongr!\n[GOAL]\ncase a.h.a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : DivisionRing R\np q : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\na\u271d : \u2115\n\u22a2 coeff (map f p) a\u271d = coeff 0 a\u271d \u2194 coeff p a\u271d = coeff 0 a\u271d\n[PROOFSTEP]\nsimp [map_eq_zero, coeff_map, coeff_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : DivisionRing R\np q : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\n\u22a2 leadingCoeff (map f p) = \u2191f (leadingCoeff p)\n[PROOFSTEP]\nsimp only [\u2190 coeff_natDegree, coeff_map f, natDegree_map]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : DivisionRing R\np\u271d q : R[X]\ninst\u271d\u00b9 : Semiring S\ninst\u271d : Nontrivial S\nf : R \u2192+* S\np : R[X]\n\u22a2 Monic (map f p) \u2194 Monic p\n[PROOFSTEP]\nrw [Monic, leadingCoeff_map, \u2190 f.map_one, Function.Injective.eq_iff f.injective, Monic]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\n\u22a2 degree p \u2264 0\n[PROOFSTEP]\nsimp [*, le_refl]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [eq_C_of_degree_le_zero this, hc] at h \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree (\u2191C 0) = 0\nthis : degree p \u2264 0\nhc : coeff p 0 = 0\n\u22a2 False\n[PROOFSTEP]\nsimp at h \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 \u2260 0\n\u22a2 1 = p * \u2191C (coeff p 0)\u207b\u00b9\n[PROOFSTEP]\nconv in p => rw [eq_C_of_degree_le_zero this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_C_of_degree_le_zero this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_C_of_degree_le_zero this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_C_of_degree_le_zero this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 0\nthis : degree p \u2264 0\nhc : coeff p 0 \u2260 0\n\u22a2 1 = \u2191C (coeff p 0) * \u2191C (coeff p 0)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 C_mul, _root_.mul_inv_cancel hc, C_1]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\n\u22a2 q * div p q + mod p q = p\n[PROOFSTEP]\nby_cases h : q = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : q = 0\n\u22a2 q * div p q + mod p q = p\n[PROOFSTEP]\nsimp only [h, zero_mul, mod, modByMonic_zero, zero_add]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n\u22a2 q * div p q + mod p q = p\n[PROOFSTEP]\nconv =>\n  rhs\n  rw [\u2190 modByMonic_add_div p (monic_mul_leadingCoeff_inv h)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n| q * div p q + mod p q = p\n[PROOFSTEP]\n  rhs\n  rw [\u2190 modByMonic_add_div p (monic_mul_leadingCoeff_inv h)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n| q * div p q + mod p q = p\n[PROOFSTEP]\n  rhs\n  rw [\u2190 modByMonic_add_div p (monic_mul_leadingCoeff_inv h)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n| q * div p q + mod p q = p\n[PROOFSTEP]\nrhs\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n| p\n[PROOFSTEP]\nrw [\u2190 modByMonic_add_div p (monic_mul_leadingCoeff_inv h)]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nh : \u00acq = 0\n\u22a2 q * div p q + mod p q =\n    p %\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9) + q * \u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9))\n[PROOFSTEP]\nrw [div, mod, add_comm, mul_assoc]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q p : R[X]\nhq : q \u2260 0\n\u22a2 degree (mod p q) < degree q\n[PROOFSTEP]\nrw [\u2190 degree_mul_leadingCoeff_inv q hq]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q p : R[X]\nhq : q \u2260 0\n\u22a2 degree (mod p q) < degree (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n[PROOFSTEP]\nexact degree_modByMonic_lt p (monic_mul_leadingCoeff_inv hq)\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q p : R[X]\nhq : Monic q\n\u22a2 p %\u2098 q = p %\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n[PROOFSTEP]\nsimp only [Monic.def.1 hq, inv_one, mul_one, C_1]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q p : R[X]\nhq : Monic q\n\u22a2 p /\u2098 q = \u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9))\n[PROOFSTEP]\nsimp only [Monic.def.1 hq, inv_one, C_1, one_mul, mul_one]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nsrc\u271d\u00b9 : CommRing R[X] := commRing\nsrc\u271d : Nontrivial R[X] := nontrivial\n\u22a2 \u2200 (a : R[X]), (fun x x_1 => x / x_1) a 0 = 0\n[PROOFSTEP]\nsimp [div_def]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\n\u22a2 p % q = p\n[PROOFSTEP]\nclassical\nhave : \u00acdegree (q * C (leadingCoeff q)\u207b\u00b9) \u2264 degree p := not_le_of_gt <| by rwa [degree_mul_leadingCoeff_inv q hq0]\nrw [mod_def, modByMonic, dif_pos (monic_mul_leadingCoeff_inv hq0)]\nunfold divModByMonicAux\ndsimp\nsimp only [this, false_and_iff, if_false]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\n\u22a2 p % q = p\n[PROOFSTEP]\nhave : \u00acdegree (q * C (leadingCoeff q)\u207b\u00b9) \u2264 degree p := not_le_of_gt <| by rwa [degree_mul_leadingCoeff_inv q hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\n\u22a2 degree (q * \u2191C (leadingCoeff q)\u207b\u00b9) > degree p\n[PROOFSTEP]\nrwa [degree_mul_leadingCoeff_inv q hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nthis : \u00acdegree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p\n\u22a2 p % q = p\n[PROOFSTEP]\nrw [mod_def, modByMonic, dif_pos (monic_mul_leadingCoeff_inv hq0)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nthis : \u00acdegree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p\n\u22a2 (divModByMonicAux p (_ : Monic (q * \u2191C (leadingCoeff q)\u207b\u00b9))).snd = p\n[PROOFSTEP]\nunfold divModByMonicAux\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nthis : \u00acdegree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p\n\u22a2 (if h : degree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p \u2227 p \u2260 0 then\n        let z := \u2191C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * \u2191C (leadingCoeff q)\u207b\u00b9));\n        let_fun _wf :=\n          (_ :\n            degree\n                (p -\n                  \u2191C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * \u2191C (leadingCoeff q)\u207b\u00b9)) *\n                    (q * \u2191C (leadingCoeff q)\u207b\u00b9)) <\n              degree p);\n        let dm := divModByMonicAux (p - z * (q * \u2191C (leadingCoeff q)\u207b\u00b9)) (_ : Monic (q * \u2191C (leadingCoeff q)\u207b\u00b9));\n        (z + dm.fst, dm.snd)\n      else (0, p)).snd =\n    p\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nthis : \u00acdegree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p\n\u22a2 (if degree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p \u2227 \u00acp = 0 then\n        (\u2191C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * \u2191C (leadingCoeff q)\u207b\u00b9)) +\n            (divModByMonicAux\n                (p -\n                  \u2191C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * \u2191C (leadingCoeff q)\u207b\u00b9)) *\n                    (q * \u2191C (leadingCoeff q)\u207b\u00b9))\n                (_ : Monic (q * \u2191C (leadingCoeff q)\u207b\u00b9))).fst,\n          (divModByMonicAux\n              (p -\n                \u2191C (leadingCoeff p) * X ^ (natDegree p - natDegree (q * \u2191C (leadingCoeff q)\u207b\u00b9)) *\n                  (q * \u2191C (leadingCoeff q)\u207b\u00b9))\n              (_ : Monic (q * \u2191C (leadingCoeff q)\u207b\u00b9))).snd)\n      else (0, p)).snd =\n    p\n[PROOFSTEP]\nsimp only [this, false_and_iff, if_false]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : p / q = 0\n\u22a2 degree p < degree q\n[PROOFSTEP]\nhave := EuclideanDomain.div_add_mod p q\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : p / q = 0\nthis : q * (p / q) + p % q = p\n\u22a2 degree p < degree q\n[PROOFSTEP]\nrwa [h, mul_zero, zero_add, mod_eq_self_iff hq0] at this \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\n\u22a2 p / q = 0\n[PROOFSTEP]\nhave hlt : degree p < degree (q * C (leadingCoeff q)\u207b\u00b9) := by rwa [degree_mul_leadingCoeff_inv q hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\n\u22a2 degree p < degree (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n[PROOFSTEP]\nrwa [degree_mul_leadingCoeff_inv q hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nhlt : degree p < degree (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n\u22a2 p / q = 0\n[PROOFSTEP]\nhave hm : Monic (q * C (leadingCoeff q)\u207b\u00b9) := monic_mul_leadingCoeff_inv hq0\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nh : degree p < degree q\nhlt : degree p < degree (q * \u2191C (leadingCoeff q)\u207b\u00b9)\nhm : Monic (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n\u22a2 p / q = 0\n[PROOFSTEP]\nrw [div_def, (divByMonic_eq_zero_iff hm).2 hlt, mul_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nhpq : degree q \u2264 degree p\n\u22a2 degree q + degree (p / q) = degree p\n[PROOFSTEP]\nhave : degree (p % q) < degree (q * (p / q)) :=\n  calc\n    degree (p % q) < degree q := EuclideanDomain.mod_lt _ hq0\n    _ \u2264 _ := degree_le_mul_left _ (mt (div_eq_zero_iff hq0).1 (not_lt_of_ge hpq))\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nhpq : degree q \u2264 degree p\nthis : degree (p % q) < degree (q * (p / q))\n\u22a2 degree q + degree (p / q) = degree p\n[PROOFSTEP]\nconv_rhs => rw [\u2190 EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nhpq : degree q \u2264 degree p\nthis : degree (p % q) < degree (q * (p / q))\n| degree p\n[PROOFSTEP]\nrw [\u2190 EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nhpq : degree q \u2264 degree p\nthis : degree (p % q) < degree (q * (p / q))\n| degree p\n[PROOFSTEP]\nrw [\u2190 EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhq0 : q \u2260 0\nhpq : degree q \u2264 degree p\nthis : degree (p % q) < degree (q * (p / q))\n| degree p\n[PROOFSTEP]\nrw [\u2190 EuclideanDomain.div_add_mod p q, degree_add_eq_left_of_degree_lt this, degree_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\n\u22a2 degree (p / q) \u2264 degree p\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nhq : q = 0\n\u22a2 degree (p / q) \u2264 degree p\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nhq : \u00acq = 0\n\u22a2 degree (p / q) \u2264 degree p\n[PROOFSTEP]\nrw [div_def, mul_comm, degree_mul_leadingCoeff_inv _ hq]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np\u271d q\u271d p q : R[X]\nhq : \u00acq = 0\n\u22a2 degree (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9)) \u2264 degree p\n[PROOFSTEP]\nexact degree_divByMonic_le _ _\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\n\u22a2 degree (p / q) < degree p\n[PROOFSTEP]\nhave hq0 : q \u2260 0 := fun hq0 => by simp [hq0] at hq \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\nhq0 : q = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [hq0] at hq \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\nhq0 : q \u2260 0\n\u22a2 degree (p / q) < degree p\n[PROOFSTEP]\nrw [div_def, mul_comm, degree_mul_leadingCoeff_inv _ hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\nhq0 : q \u2260 0\n\u22a2 degree (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9)) < degree p\n[PROOFSTEP]\nexact degree_divByMonic_lt _ (monic_mul_leadingCoeff_inv hq0) hp (by rw [degree_mul_leadingCoeff_inv _ hq0]; exact hq)\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\nhq0 : q \u2260 0\n\u22a2 0 < degree (q * \u2191C (leadingCoeff q)\u207b\u00b9)\n[PROOFSTEP]\nrw [degree_mul_leadingCoeff_inv _ hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp : p \u2260 0\nhq : 0 < degree q\nhq0 : q \u2260 0\n\u22a2 0 < degree q\n[PROOFSTEP]\nexact hq\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 IsUnit (map f p) \u2194 IsUnit p\n[PROOFSTEP]\nsimp_rw [isUnit_iff_degree_eq_zero, degree_map]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 map f (p / q) = map f p / map f q\n[PROOFSTEP]\nif hq0 : q = 0 then simp [hq0]\nelse\n  rw [div_def, div_def, Polynomial.map_mul, map_divByMonic f (monic_mul_leadingCoeff_inv hq0), Polynomial.map_mul,\n    map_C, leadingCoeff_map, map_inv\u2080]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nhq0 : q = 0\n\u22a2 map f (p / q) = map f p / map f q\n[PROOFSTEP]\nsimp [hq0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nhq0 : \u00acq = 0\n\u22a2 map f (p / q) = map f p / map f q\n[PROOFSTEP]\nrw [div_def, div_def, Polynomial.map_mul, map_divByMonic f (monic_mul_leadingCoeff_inv hq0), Polynomial.map_mul, map_C,\n  leadingCoeff_map, map_inv\u2080]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 map f (p % q) = map f p % map f q\n[PROOFSTEP]\nby_cases hq0 : q = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nhq0 : q = 0\n\u22a2 map f (p % q) = map f p % map f q\n[PROOFSTEP]\nsimp [hq0]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nhq0 : \u00acq = 0\n\u22a2 map f (p % q) = map f p % map f q\n[PROOFSTEP]\nrw [mod_def, mod_def, leadingCoeff_map f, \u2190 map_inv\u2080 f, \u2190 map_C f, \u2190 Polynomial.map_mul f,\n  map_modByMonic f (monic_mul_leadingCoeff_inv hq0)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : Field k\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : DecidableEq k\nf : R \u2192+* k\nx : R[X]\n\u22a2 EuclideanDomain.gcd (map f 0) (map f x) = map f (EuclideanDomain.gcd 0 x)\n[PROOFSTEP]\nsimp_rw [Polynomial.map_zero, EuclideanDomain.gcd_zero_left]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : Field k\ninst\u271d\u00b9 : DecidableEq R\ninst\u271d : DecidableEq k\nf : R \u2192+* k\nx y : R[X]\nx\u271d : x \u2260 0\nih : EuclideanDomain.gcd (map f (y % x)) (map f x) = map f (EuclideanDomain.gcd (y % x) x)\n\u22a2 EuclideanDomain.gcd (map f x) (map f y) = map f (EuclideanDomain.gcd x y)\n[PROOFSTEP]\nrw [gcd_val, \u2190 map_mod, ih, \u2190 gcd_val]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np q : R[X]\ninst\u271d\u00b9 : CommSemiring k\ninst\u271d : DecidableEq R\n\u03d5 : R \u2192+* k\nf g : R[X]\n\u03b1 : k\nhf : eval\u2082 \u03d5 \u03b1 f = 0\nhg : eval\u2082 \u03d5 \u03b1 g = 0\n\u22a2 eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0\n[PROOFSTEP]\nrw [EuclideanDomain.gcd_eq_gcd_ab f g, Polynomial.eval\u2082_add, Polynomial.eval\u2082_mul, Polynomial.eval\u2082_mul, hf, hg,\n  zero_mul, zero_mul, zero_add]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np q : R[X]\ninst\u271d\u00b9 : CommSemiring k\ninst\u271d : DecidableEq R\n\u03d5 : R \u2192+* k\nf g : R[X]\n\u03b1 : k\nh\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0\n\u22a2 eval\u2082 \u03d5 \u03b1 f = 0\n[PROOFSTEP]\ncases' EuclideanDomain.gcd_dvd_left f g with p hp\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np\u271d q : R[X]\ninst\u271d\u00b9 : CommSemiring k\ninst\u271d : DecidableEq R\n\u03d5 : R \u2192+* k\nf g : R[X]\n\u03b1 : k\nh\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0\np : R[X]\nhp : f = EuclideanDomain.gcd f g * p\n\u22a2 eval\u2082 \u03d5 \u03b1 f = 0\n[PROOFSTEP]\nrw [hp, Polynomial.eval\u2082_mul, h\u03b1, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np q : R[X]\ninst\u271d\u00b9 : CommSemiring k\ninst\u271d : DecidableEq R\n\u03d5 : R \u2192+* k\nf g : R[X]\n\u03b1 : k\nh\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0\n\u22a2 eval\u2082 \u03d5 \u03b1 g = 0\n[PROOFSTEP]\ncases' EuclideanDomain.gcd_dvd_right f g with p hp\n[GOAL]\ncase intro\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np\u271d q : R[X]\ninst\u271d\u00b9 : CommSemiring k\ninst\u271d : DecidableEq R\n\u03d5 : R \u2192+* k\nf g : R[X]\n\u03b1 : k\nh\u03b1 : eval\u2082 \u03d5 \u03b1 (EuclideanDomain.gcd f g) = 0\np : R[X]\nhp : g = EuclideanDomain.gcd f g * p\n\u22a2 eval\u2082 \u03d5 \u03b1 g = 0\n[PROOFSTEP]\nrw [hp, Polynomial.eval\u2082_mul, h\u03b1, zero_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 IsCoprime (map f p) (map f q) \u2194 IsCoprime p q\n[PROOFSTEP]\nclassical rw [\u2190 EuclideanDomain.gcd_isUnit_iff, \u2190 EuclideanDomain.gcd_isUnit_iff, gcd_map, isUnit_map]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\n\u22a2 IsCoprime (map f p) (map f q) \u2194 IsCoprime p q\n[PROOFSTEP]\nrw [\u2190 EuclideanDomain.gcd_isUnit_iff, \u2190 EuclideanDomain.gcd_isUnit_iff, gcd_map, isUnit_map]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : Field R\np q : R[X]\ninst\u271d\u00b9 : CommRing k\ninst\u271d : IsDomain k\nf : R \u2192+* k\nx : k\nhp : p \u2260 0\n\u22a2 x \u2208 roots (map f p) \u2194 eval\u2082 f x p = 0\n[PROOFSTEP]\nrw [mem_roots (map_ne_zero hp), IsRoot, Polynomial.eval_map]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\na : R\nha : a \u2260 0\n\u22a2 rootSet (\u2191(monomial n) a) S = {0}\n[PROOFSTEP]\nclassical rw [rootSet, map_monomial, roots_monomial ((_root_.map_ne_zero (algebraMap R S)).2 ha),\n  Multiset.toFinset_nsmul _ _ hn, Multiset.toFinset_singleton, Finset.coe_singleton]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\na : R\nha : a \u2260 0\n\u22a2 rootSet (\u2191(monomial n) a) S = {0}\n[PROOFSTEP]\nrw [rootSet, map_monomial, roots_monomial ((_root_.map_ne_zero (algebraMap R S)).2 ha), Multiset.toFinset_nsmul _ _ hn,\n  Multiset.toFinset_singleton, Finset.coe_singleton]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\na : R\nha : a \u2260 0\n\u22a2 rootSet (\u2191C a * X ^ n) S = {0}\n[PROOFSTEP]\nrw [C_mul_X_pow_eq_monomial, rootSet_monomial hn ha]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\n\u22a2 rootSet (X ^ n) S = {0}\n[PROOFSTEP]\nrw [\u2190 one_mul (X ^ n : R[X]), \u2190 C_1, rootSet_C_mul_X_pow hn]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\nn : \u2115\nhn : n \u2260 0\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns : Finset \u03b9\nh : Finset.prod s f \u2260 0\n\u22a2 rootSet (Finset.prod s f) S = \u22c3 (i : \u03b9) (_ : i \u2208 s), rootSet (f i) S\n[PROOFSTEP]\nclassical\nsimp only [rootSet, \u2190 Finset.mem_coe]\nrw [Polynomial.map_prod, roots_prod, Finset.bind_toFinset, s.val_toFinset, Finset.coe_biUnion]\nrwa [\u2190 Polynomial.map_prod, Ne, map_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns : Finset \u03b9\nh : Finset.prod s f \u2260 0\n\u22a2 rootSet (Finset.prod s f) S = \u22c3 (i : \u03b9) (_ : i \u2208 s), rootSet (f i) S\n[PROOFSTEP]\nsimp only [rootSet, \u2190 Finset.mem_coe]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns : Finset \u03b9\nh : Finset.prod s f \u2260 0\n\u22a2 \u2191(Multiset.toFinset (roots (map (algebraMap R S) (Finset.prod s f)))) =\n    \u22c3 (i : \u03b9) (_ : i \u2208 \u2191s), \u2191(Multiset.toFinset (roots (map (algebraMap R S) (f i))))\n[PROOFSTEP]\nrw [Polynomial.map_prod, roots_prod, Finset.bind_toFinset, s.val_toFinset, Finset.coe_biUnion]\n[GOAL]\ncase a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b3 : Field R\np q : R[X]\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Algebra R S\n\u03b9 : Type u_1\nf : \u03b9 \u2192 R[X]\ns : Finset \u03b9\nh : Finset.prod s f \u2260 0\n\u22a2 \u220f i in s, map (algebraMap R S) (f i) \u2260 0\n[PROOFSTEP]\nrwa [\u2190 Polynomial.map_prod, Ne, map_eq_zero]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\n\u22a2 IsRoot p (-(coeff p 0 / coeff p 1))\n[PROOFSTEP]\nhave : p.coeff 1 \u2260 0 := by\n  have h' := natDegree_eq_of_degree_eq_some h\n  change natDegree p = 1 at h' ; rw [\u2190 h']\n  exact mt leadingCoeff_eq_zero.1 fun h0 => by simp [h0] at h \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\n\u22a2 coeff p 1 \u2260 0\n[PROOFSTEP]\nhave h' := natDegree_eq_of_degree_eq_some h\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nh' : natDegree p = One.one\n\u22a2 coeff p 1 \u2260 0\n[PROOFSTEP]\nchange natDegree p = 1 at h' \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nh' : natDegree p = 1\n\u22a2 coeff p 1 \u2260 0\n[PROOFSTEP]\nrw [\u2190 h']\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nh' : natDegree p = 1\n\u22a2 coeff p (natDegree p) \u2260 0\n[PROOFSTEP]\nexact mt leadingCoeff_eq_zero.1 fun h0 => by simp [h0] at h \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nh' : natDegree p = 1\nh0 : p = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h0] at h \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n\u22a2 IsRoot p (-(coeff p 0 / coeff p 1))\n[PROOFSTEP]\nconv in p => rw [eq_X_add_C_of_degree_le_one (show degree p \u2264 1 by rw [h])]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_X_add_C_of_degree_le_one (show degree p \u2264 1 by rw [h])]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_X_add_C_of_degree_le_one (show degree p \u2264 1 by rw [h])]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n| p\n[PROOFSTEP]\nrw [eq_X_add_C_of_degree_le_one (show degree p \u2264 1 by rw [h])]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n\u22a2 degree p \u2264 1\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nh : degree p = 1\nthis : coeff p 1 \u2260 0\n\u22a2 IsRoot (\u2191C (coeff p 1) * X + \u2191C (coeff p 0)) (-(coeff p 0 / coeff p 1))\n[PROOFSTEP]\nsimp [IsRoot, mul_div_cancel' _ this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Field R\np q : R[X]\nu : R[X]\u02e3\nn : \u2115\n\u22a2 (coeff (\u2191u) n)\u207b\u00b9 = coeff (\u2191u\u207b\u00b9) n\n[PROOFSTEP]\nrw [eq_C_of_degree_eq_zero (degree_coe_units u), eq_C_of_degree_eq_zero (degree_coe_units u\u207b\u00b9), coeff_C, coeff_C,\n  inv_eq_one_div]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Field R\np q : R[X]\nu : R[X]\u02e3\nn : \u2115\n\u22a2 (1 / if n = 0 then coeff (\u2191u) 0 else 0) = if n = 0 then coeff (\u2191u\u207b\u00b9) 0 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Field R\np q : R[X]\nu : R[X]\u02e3\nn : \u2115\nh\u271d : n = 0\n\u22a2 1 / coeff (\u2191u) 0 = coeff (\u2191u\u207b\u00b9) 0\n[PROOFSTEP]\nrw [div_eq_iff_mul_eq (coeff_coe_units_zero_ne_zero u), coeff_zero_eq_eval_zero, coeff_zero_eq_eval_zero, \u2190 eval_mul, \u2190\n  Units.val_mul, inv_mul_self]\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Field R\np q : R[X]\nu : R[X]\u02e3\nn : \u2115\nh\u271d : n = 0\n\u22a2 eval 0 \u21911 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Field R\np q : R[X]\nu : R[X]\u02e3\nn : \u2115\nh\u271d : \u00acn = 0\n\u22a2 1 / 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nhp0 : p \u2260 0\n\u22a2 Monic (\u2191normalize p)\n[PROOFSTEP]\nrw [Ne.def, \u2190 leadingCoeff_eq_zero, \u2190 Ne.def, \u2190 isUnit_iff_ne_zero] at hp0 \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nhp0 : IsUnit (leadingCoeff p)\n\u22a2 Monic (\u2191normalize p)\n[PROOFSTEP]\nrw [Monic, leadingCoeff_normalize, normalize_eq_one]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nhp0 : IsUnit (leadingCoeff p)\n\u22a2 IsUnit (leadingCoeff p)\n[PROOFSTEP]\napply hp0\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhpq : degree q \u2264 degree p\n\u22a2 leadingCoeff (p / q) = leadingCoeff p / leadingCoeff q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhpq : degree q \u2264 degree p\nhq : q = 0\n\u22a2 leadingCoeff (p / q) = leadingCoeff p / leadingCoeff q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhpq : degree q \u2264 degree p\nhq : \u00acq = 0\n\u22a2 leadingCoeff (p / q) = leadingCoeff p / leadingCoeff q\n[PROOFSTEP]\nrw [div_def, leadingCoeff_mul, leadingCoeff_C, leadingCoeff_divByMonic_of_monic (monic_mul_leadingCoeff_inv hq) _,\n  mul_comm, div_eq_mul_inv]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhpq : degree q \u2264 degree p\nhq : \u00acq = 0\n\u22a2 degree (q * \u2191C (leadingCoeff q)\u207b\u00b9) \u2264 degree p\n[PROOFSTEP]\nrwa [degree_mul_leadingCoeff_inv q hq]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\n\u22a2 p / (\u2191C a * q) = \u2191C a\u207b\u00b9 * (p / q)\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : a = 0\n\u22a2 p / (\u2191C a * q) = \u2191C a\u207b\u00b9 * (p / q)\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : \u00aca = 0\n\u22a2 p / (\u2191C a * q) = \u2191C a\u207b\u00b9 * (p / q)\n[PROOFSTEP]\nsimp only [div_def, leadingCoeff_mul, mul_inv, leadingCoeff_C, C.map_mul, mul_assoc]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : \u00aca = 0\n\u22a2 \u2191C a\u207b\u00b9 * (\u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (\u2191C a * (q * (\u2191C a\u207b\u00b9 * \u2191C (leadingCoeff q)\u207b\u00b9))))) =\n    \u2191C a\u207b\u00b9 * (\u2191C (leadingCoeff q)\u207b\u00b9 * (p /\u2098 (q * \u2191C (leadingCoeff q)\u207b\u00b9)))\n[PROOFSTEP]\ncongr 3\n[GOAL]\ncase neg.e_a.e_a.e_q\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : \u00aca = 0\n\u22a2 \u2191C a * (q * (\u2191C a\u207b\u00b9 * \u2191C (leadingCoeff q)\u207b\u00b9)) = q * \u2191C (leadingCoeff q)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_left_comm q, \u2190 mul_assoc, \u2190 C.map_mul, mul_inv_cancel ha, C.map_one, one_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : a \u2260 0\nx\u271d : p \u2223 q\nr : R[X]\nhr : q = p * r\n\u22a2 q = \u2191C a * p * (\u2191C a\u207b\u00b9 * r)\n[PROOFSTEP]\nrw [mul_assoc, mul_left_comm p, \u2190 mul_assoc, \u2190 C.map_mul, _root_.mul_inv_cancel ha, C.map_one, one_mul, hr]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nha : a \u2260 0\nx\u271d : p \u2223 \u2191C a * q\nr : R[X]\nhr : \u2191C a * q = p * r\n\u22a2 q = p * (\u2191C a\u207b\u00b9 * r)\n[PROOFSTEP]\nrw [mul_left_comm p, \u2190 hr, \u2190 mul_assoc, \u2190 C.map_mul, _root_.inv_mul_cancel ha, C.map_one, one_mul]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nhp : p \u2260 0\n\u22a2 \u2191(normUnit p) = \u2191C (leadingCoeff p)\u207b\u00b9\n[PROOFSTEP]\nhave : p.leadingCoeff \u2260 0 := mt leadingCoeff_eq_zero.mp hp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nhp : p \u2260 0\nthis : leadingCoeff p \u2260 0\n\u22a2 \u2191(normUnit p) = \u2191C (leadingCoeff p)\u207b\u00b9\n[PROOFSTEP]\nsimp [CommGroupWithZero.coe_normUnit _ this]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\nh : Monic p\n\u22a2 \u2191normalize p = p\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nx y : R[X]\n\u22a2 map f x \u2223 map f y \u2194 x \u2223 y\n[PROOFSTEP]\nby_cases H : x = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nx y : R[X]\nH : x = 0\n\u22a2 map f x \u2223 map f y \u2194 x \u2223 y\n[PROOFSTEP]\nrw [H, Polynomial.map_zero, zero_dvd_iff, zero_dvd_iff, map_eq_zero]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nx y : R[X]\nH : \u00acx = 0\n\u22a2 map f x \u2223 map f y \u2194 x \u2223 y\n[PROOFSTEP]\nclassical rw [\u2190 normalize_dvd_iff, \u2190 @normalize_dvd_iff R[X], normalize_apply, normalize_apply,\n  coe_normUnit_of_ne_zero H, coe_normUnit_of_ne_zero (mt (map_eq_zero f).1 H), leadingCoeff_map, \u2190 map_inv\u2080 f, \u2190 map_C,\n  \u2190 Polynomial.map_mul, map_dvd_map _ f.injective (monic_mul_leadingCoeff_inv H)]\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : Field k\nf : R \u2192+* k\nx y : R[X]\nH : \u00acx = 0\n\u22a2 map f x \u2223 map f y \u2194 x \u2223 y\n[PROOFSTEP]\nrw [\u2190 normalize_dvd_iff, \u2190 @normalize_dvd_iff R[X], normalize_apply, normalize_apply, coe_normUnit_of_ne_zero H,\n  coe_normUnit_of_ne_zero (mt (map_eq_zero f).1 H), leadingCoeff_map, \u2190 map_inv\u2080 f, \u2190 map_C, \u2190 Polynomial.map_mul,\n  map_dvd_map _ f.injective (monic_mul_leadingCoeff_inv H)]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\ninst\u271d : DecidableEq R\n\u22a2 degree (\u2191normalize p) = degree p\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp1 : degree p = 1\n\u22a2 Prime p\n[PROOFSTEP]\nclassical\nhave : Prime (normalize p) :=\n  Monic.prime_of_degree_eq_one (hp1 \u25b8 degree_normalize) (monic_normalize fun hp0 => absurd hp1 (hp0.symm \u25b8 by simp))\nexact (normalize_associated _).prime this\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp1 : degree p = 1\n\u22a2 Prime p\n[PROOFSTEP]\nhave : Prime (normalize p) :=\n  Monic.prime_of_degree_eq_one (hp1 \u25b8 degree_normalize) (monic_normalize fun hp0 => absurd hp1 (hp0.symm \u25b8 by simp))\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp1 : degree p = 1\nhp0 : p = 0\n\u22a2 \u00acdegree 0 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nhp1 : degree p = 1\nthis : Prime (\u2191normalize p)\n\u22a2 Prime p\n[PROOFSTEP]\nexact (normalize_associated _).prime this\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\n\u22a2 \u00acIrreducible (\u2191C x)\n[PROOFSTEP]\nby_cases H : x = 0\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible (\u2191C x)\n[PROOFSTEP]\nrw [H, C_0]\n[GOAL]\ncase pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : x = 0\n\u22a2 \u00acIrreducible 0\n[PROOFSTEP]\nexact not_irreducible_zero\n[GOAL]\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Field R\np q : R[X]\nx : R\nH : \u00acx = 0\n\u22a2 \u00acIrreducible (\u2191C x)\n[PROOFSTEP]\nexact fun hx => Irreducible.not_unit hx <| isUnit_C.2 <| isUnit_iff_ne_zero.2 H\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\n\u22a2 (X - \u2191C a) * (f /\u2098 (X - \u2191C a)) = f - f %\u2098 (X - \u2191C a)\n[PROOFSTEP]\nrw [eq_sub_iff_add_eq, \u2190 eq_sub_iff_add_eq', modByMonic_eq_sub_mul_div]\n[GOAL]\ncase _hq\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\n\u22a2 Monic (X - \u2191C a)\n[PROOFSTEP]\nexact monic_X_sub_C a\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nhave key := by apply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\n\u22a2 ?m.1574688\n[PROOFSTEP]\napply congrArg derivative <| X_sub_C_mul_divByMonic_eq_sub_modByMonic f a\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nkey : \u2191derivative ((X - \u2191C a) * (f /\u2098 (X - \u2191C a))) = \u2191derivative (f - f %\u2098 (X - \u2191C a))\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nrw [modByMonic_X_sub_C_eq_C_eval] at key \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nkey : \u2191derivative ((X - \u2191C a) * (f /\u2098 (X - \u2191C a))) = \u2191derivative (f - \u2191C (eval a f))\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nrw [derivative_mul, derivative_sub, derivative_X, derivative_sub] at key \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nkey :\n  (1 - \u2191derivative (\u2191C a)) * (f /\u2098 (X - \u2191C a)) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) =\n    \u2191derivative f - \u2191derivative (\u2191C (eval a f))\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nrw [derivative_C, sub_zero, one_mul] at key \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nkey : f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f - \u2191derivative (\u2191C (eval a f))\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nrw [derivative_C, sub_zero] at key \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nkey : f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n\u22a2 f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n[PROOFSTEP]\nassumption\n  /- Porting note: factored out another have statement from\n  isCoprime_of_is_root_of_eval_derivative_ne_zero because the original proof was timing out -/\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\n\u22a2 X - \u2191C a \u2223 \u2191derivative f\n[PROOFSTEP]\nhave key := divByMonic_add_X_sub_C_mul_derivate_divByMonic_eq_derivative f a\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\nkey : f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\n\u22a2 X - \u2191C a \u2223 \u2191derivative f\n[PROOFSTEP]\nhave \u27e8u, hu\u27e9 := hf\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\nkey : f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\nu : K[X]\nhu : f /\u2098 (X - \u2191C a) = (X - \u2191C a) * u\n\u22a2 X - \u2191C a \u2223 \u2191derivative f\n[PROOFSTEP]\nrw [\u2190 key, hu, \u2190 mul_add (X - C a) u _]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\nkey : f /\u2098 (X - \u2191C a) + (X - \u2191C a) * \u2191derivative (f /\u2098 (X - \u2191C a)) = \u2191derivative f\nu : K[X]\nhu : f /\u2098 (X - \u2191C a) = (X - \u2191C a) * u\n\u22a2 X - \u2191C a \u2223 (X - \u2191C a) * (u + \u2191derivative ((X - \u2191C a) * u))\n[PROOFSTEP]\nuse(u + derivative ((X - C a) * u))\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf' : eval a (\u2191derivative f) \u2260 0\n\u22a2 IsCoprime (X - \u2191C a) (f /\u2098 (X - \u2191C a))\n[PROOFSTEP]\nclassical\nrefine\n  Or.resolve_left\n    (EuclideanDomain.dvd_or_coprime (X - C a) (f /\u2098 (X - C a))\n      (irreducible_of_degree_eq_one (Polynomial.degree_X_sub_C a)))\n    ?_\ncontrapose! hf' with h\nhave : X - C a \u2223 derivative f := X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic f h\nrw [\u2190 dvd_iff_modByMonic_eq_zero (monic_X_sub_C _), modByMonic_X_sub_C_eq_C_eval] at this \nrwa [\u2190 C_inj, C_0]\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf' : eval a (\u2191derivative f) \u2260 0\n\u22a2 IsCoprime (X - \u2191C a) (f /\u2098 (X - \u2191C a))\n[PROOFSTEP]\nrefine\n  Or.resolve_left\n    (EuclideanDomain.dvd_or_coprime (X - C a) (f /\u2098 (X - C a))\n      (irreducible_of_degree_eq_one (Polynomial.degree_X_sub_C a)))\n    ?_\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nhf' : eval a (\u2191derivative f) \u2260 0\n\u22a2 \u00acX - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\n[PROOFSTEP]\ncontrapose! hf' with h\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nh : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\n\u22a2 eval a (\u2191derivative f) = 0\n[PROOFSTEP]\nhave : X - C a \u2223 derivative f := X_sub_C_dvd_derivative_of_X_sub_C_dvd_divByMonic f h\n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nh : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\nthis : X - \u2191C a \u2223 \u2191derivative f\n\u22a2 eval a (\u2191derivative f) = 0\n[PROOFSTEP]\nrw [\u2190 dvd_iff_modByMonic_eq_zero (monic_X_sub_C _), modByMonic_X_sub_C_eq_C_eval] at this \n[GOAL]\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : Field R\np q : R[X]\nK : Type u_1\ninst\u271d : Field K\nf : K[X]\na : K\nh : X - \u2191C a \u2223 f /\u2098 (X - \u2191C a)\nthis : \u2191C (eval a (\u2191derivative f)) = 0\n\u22a2 eval a (\u2191derivative f) = 0\n[PROOFSTEP]\nrwa [\u2190 C_inj, C_0]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.FieldDivision", "llama_tokens": 31271, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.5266642924807479}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\n\u22a2 Associated (\u2191(Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)\n[PROOFSTEP]\nhave hI := span_singleton_eq_bot.not.2 hf\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\n\u22a2 Associated (\u2191(Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)\n[PROOFSTEP]\nlet b' := ringBasis b (span { f }) hI\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated (\u2191(Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)\n[PROOFSTEP]\nclassical\nrw [\u2190 Matrix.det_diagonal, \u2190 LinearMap.det_toLin b']\nlet e := (b'.equiv ((span { f }).selfBasis b hI) <| Equiv.refl _).trans ((LinearEquiv.coord S S f hf).restrictScalars R)\nrefine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm\ndsimp only [LinearEquiv.trans_apply]\nsimp_rw [\u2190 LinearEquiv.coe_toLinearMap, \u2190 LinearMap.comp_apply, \u2190 LinearMap.ext_iff]\nrefine b'.ext fun i => ?_\nsimp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,\n  Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul, zero_smul, Finset.sum_ite_eq',\n  mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]\nchange _ = f * _\nrw [mul_comm, \u2190 smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul, Ideal.selfBasis_def]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated (\u2191(Algebra.norm R) f) (\u220f i : \u03b9, smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)\n[PROOFSTEP]\nrw [\u2190 Matrix.det_diagonal, \u2190 LinearMap.det_toLin b']\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\n\u22a2 Associated (\u2191(Algebra.norm R) f)\n    (\u2191LinearMap.det (\u2191(Matrix.toLin b' b') (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)))\n[PROOFSTEP]\nlet e := (b'.equiv ((span { f }).selfBasis b hI) <| Equiv.refl _).trans ((LinearEquiv.coord S S f hf).restrictScalars R)\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n\u22a2 Associated (\u2191(Algebra.norm R) f)\n    (\u2191LinearMap.det (\u2191(Matrix.toLin b' b') (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)))\n[PROOFSTEP]\nrefine (LinearMap.associated_det_of_eq_comp e _ _ ?_).symm\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n\u22a2 \u2200 (x : S),\n    \u2191(\u2191(Matrix.toLin b' b') (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)) x =\n      \u2191(\u2191\u2191\u2191(Algebra.lmul R S) f) (\u2191e x)\n[PROOFSTEP]\ndsimp only [LinearEquiv.trans_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n\u22a2 \u2200 (x : S),\n    \u2191(\u2191(Matrix.toLin (ringBasis b (span {f}) hI) (ringBasis b (span {f}) hI))\n            (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i))\n        x =\n      \u2191(\u2191\u2191\u2191(Algebra.lmul R S) f)\n        (\u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n          (\u2191(Basis.equiv (ringBasis b (span {f}) hI) (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)) x))\n[PROOFSTEP]\nsimp_rw [\u2190 LinearEquiv.coe_toLinearMap, \u2190 LinearMap.comp_apply, \u2190 LinearMap.ext_iff]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n\u22a2 \u2191\u2191(Matrix.toLin (ringBasis b (span {f}) hI) (ringBasis b (span {f}) hI))\n      (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i) =\n    LinearMap.comp (\u2191\u2191\u2191(Algebra.lmul R S) f)\n      (LinearMap.comp \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n        \u2191(Basis.equiv (ringBasis b (span {f}) hI) (selfBasis b (span {f}) hI) (Equiv.refl \u03b9)))\n[PROOFSTEP]\nrefine b'.ext fun i => ?_\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\ni : \u03b9\n\u22a2 \u2191(\u2191\u2191(Matrix.toLin (ringBasis b (span {f}) hI) (ringBasis b (span {f}) hI))\n          (Matrix.diagonal fun i => smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i))\n      (\u2191b' i) =\n    \u2191(LinearMap.comp (\u2191\u2191\u2191(Algebra.lmul R S) f)\n          (LinearMap.comp \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\n            \u2191(Basis.equiv (ringBasis b (span {f}) hI) (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))))\n      (\u2191b' i)\n[PROOFSTEP]\nsimp_rw [LinearMap.comp_apply, LinearEquiv.coe_toLinearMap, Matrix.toLin_apply, Basis.repr_self,\n  Finsupp.single_eq_pi_single, Matrix.diagonal_mulVec_single, Pi.single_apply, ite_smul, zero_smul, Finset.sum_ite_eq',\n  mul_one, if_pos (Finset.mem_univ _), b'.equiv_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i \u2022 \u2191(ringBasis b (span {f}) hI) i =\n    \u2191(\u2191\u2191\u2191(Algebra.lmul R S) f)\n      (\u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) (\u2191(selfBasis b (span {f}) hI) (\u2191(Equiv.refl \u03b9) i)))\n[PROOFSTEP]\nchange _ = f * _\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i \u2022 \u2191(ringBasis b (span {f}) hI) i =\n    f * \u2191(LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf)) (\u2191(selfBasis b (span {f}) hI) (\u2191(Equiv.refl \u03b9) i))\n[PROOFSTEP]\nrw [mul_comm, \u2190 smul_eq_mul, LinearEquiv.restrictScalars_apply, LinearEquiv.coord_apply_smul, Ideal.selfBasis_def]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : CommRing F\ninst\u271d\u00b3 : Algebra F R\ninst\u271d\u00b2 : Algebra F S\ninst\u271d\u00b9 : IsScalarTower F R S\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nf : S\nhf : f \u2260 0\nhI : \u00acspan {f} = \u22a5\nb' : Basis \u03b9 R S := ringBasis b (span {f}) hI\ne : S \u2243\u2097[R] S :=\n  LinearEquiv.trans (Basis.equiv b' (selfBasis b (span {f}) hI) (Equiv.refl \u03b9))\n    (LinearEquiv.restrictScalars R (LinearEquiv.coord S S f hf))\ni : \u03b9\n\u22a2 smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i \u2022 \u2191(ringBasis b (span {f}) hI) i =\n    smithCoeffs b (span {f}) hI (\u2191(Equiv.refl \u03b9) i) \u2022 \u2191(ringBasis b (span {f}) hI) (\u2191(Equiv.refl \u03b9) i)\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R S\nF : Type u_4\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra F[X] S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 F[X] S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\n\u22a2 FiniteDimensional F (F[X] \u29f8 span {smithCoeffs b I hI i})\n[PROOFSTEP]\nrefine PowerBasis.finiteDimensional ?_\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R S\nF : Type u_4\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra F[X] S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 F[X] S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\n\u22a2 PowerBasis F (F[X] \u29f8 span {smithCoeffs b I hI i})\n[PROOFSTEP]\nrefine AdjoinRoot.powerBasis ?_\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R S\nF : Type u_4\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra F[X] S\ninst\u271d : Finite \u03b9\nb : Basis \u03b9 F[X] S\nI : Ideal S\nhI : I \u2260 \u22a5\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2260 0\n[PROOFSTEP]\nrefine I.smithCoeffs_ne_zero b hI i\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\n\u22a2 FiniteDimensional.finrank F (S \u29f8 span {f}) = natDegree (\u2191(Algebra.norm F[X]) f)\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b9\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\nthis : Fintype \u03b9\n\u22a2 FiniteDimensional.finrank F (S \u29f8 span {f}) = natDegree (\u2191(Algebra.norm F[X]) f)\n[PROOFSTEP]\nhave h := span_singleton_eq_bot.not.2 hf\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\nthis : Fintype \u03b9\nh : \u00acspan {f} = \u22a5\n\u22a2 FiniteDimensional.finrank F (S \u29f8 span {f}) = natDegree (\u2191(Algebra.norm F[X]) f)\n[PROOFSTEP]\nrw [natDegree_eq_of_degree_eq (degree_eq_degree_of_associated <| associated_norm_prod_smith b hf)]\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\nthis : Fintype \u03b9\nh : \u00acspan {f} = \u22a5\n\u22a2 FiniteDimensional.finrank F (S \u29f8 span {f}) = natDegree (\u220f i : \u03b9, smithCoeffs b (span {f}) (_ : \u00acspan {f} = \u22a5) i)\n[PROOFSTEP]\nrw [natDegree_prod _ _ fun i _ => smithCoeffs_ne_zero b _ h i, finrank_quotient_eq_sum F h b]\n  -- finrank_quotient_eq_sum slow\n[GOAL]\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\nthis : Fintype \u03b9\nh : \u00acspan {f} = \u22a5\n\u22a2 \u2211 i : \u03b9, FiniteDimensional.finrank F (F[X] \u29f8 span {smithCoeffs b (span {f}) h i}) =\n    \u2211 i : \u03b9, natDegree (smithCoeffs b (span {f}) h i)\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h\nR : Type u_1\nS : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : IsDomain R\ninst\u271d\u2078 : IsPrincipalIdealRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : IsDomain S\ninst\u271d\u2075 : Algebra R S\nF : Type u_4\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra F[X] S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F F[X] S\nb : Basis \u03b9 F[X] S\nf : S\nhf : f \u2260 0\nthis : Fintype \u03b9\nh : \u00acspan {f} = \u22a5\ni : \u03b9\n\u22a2 FiniteDimensional.finrank F (F[X] \u29f8 span {smithCoeffs b (span {f}) h i}) = natDegree (smithCoeffs b (span {f}) h i)\n[PROOFSTEP]\nexact (AdjoinRoot.powerBasis <| smithCoeffs_ne_zero b _ h i).finrank\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.FreeModule.Norm", "llama_tokens": 7851, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430353105599, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.526454540653331}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeSup \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : \u03b9 \u2192 \u03b1\n\u22a2 a \u2264 iSup s \u2194 \u2200 (b : \u03b1), (\u2200 (i : \u03b9), s i \u2264 b) \u2192 a \u2264 b\n[PROOFSTEP]\nsimp [iSup, le_sSup_iff, upperBounds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeInf \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : \u03b9 \u2192 \u03b1\n\u22a2 iInf s \u2264 a \u2194 \u2200 (b : \u03b1), (\u2200 (i : \u03b9), b \u2264 s i) \u2192 b \u2264 a\n[PROOFSTEP]\nsimp [iInf, sInf_le_iff, lowerBounds]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 y \u2264 x\n\u22a2 \u2200 (c : \u03b1), c \u2264 sInf t \u2192 c \u2264 sInf s\n[PROOFSTEP]\nsimp only [le_sInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeInf \u03b1\ns t : Set \u03b1\na b : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 y \u2264 x\n\u22a2 \u2200 (c : \u03b1), (\u2200 (b : \u03b1), b \u2208 t \u2192 c \u2264 b) \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 c \u2264 b\n[PROOFSTEP]\nintrov h\u2080 h\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeInf \u03b1\ns t : Set \u03b1\na b\u271d : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 y \u2264 x\nc : \u03b1\nh\u2080 : \u2200 (b : \u03b1), b \u2208 t \u2192 c \u2264 b\nb : \u03b1\nh\u2081 : b \u2208 s\n\u22a2 c \u2264 b\n[PROOFSTEP]\nrcases h _ h\u2081 with \u27e8y, hy, hy'\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteSemilatticeInf \u03b1\ns t : Set \u03b1\na b\u271d : \u03b1\nh : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 y, y \u2208 t \u2227 y \u2264 x\nc : \u03b1\nh\u2080 : \u2200 (b : \u03b1), b \u2208 t \u2192 c \u2264 b\nb : \u03b1\nh\u2081 : b \u2208 s\ny : \u03b1\nhy : y \u2208 t\nhy' : y \u2264 b\n\u22a2 c \u2264 b\n[PROOFSTEP]\nsolve_by_elim [le_trans _ hy']\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : InfSet \u03b1\nisGLB_sInf : \u2200 (s : Set \u03b1), IsGLB s (sInf s)\na b c : \u03b1\nhac : a \u2264 c\nhbc : b \u2264 c\n\u22a2 c \u2208 {x | a \u2264 x \u2227 b \u2264 x}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : InfSet \u03b1\nisGLB_sInf : \u2200 (s : Set \u03b1), IsGLB s (sInf s)\na b c : \u03b1\nhab : a \u2264 b\nhac : a \u2264 c\n\u22a2 a \u2264 b \u2293 c\n[PROOFSTEP]\napply (isGLB_sInf _).2\n[GOAL]\ncase a\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : InfSet \u03b1\nisGLB_sInf : \u2200 (s : Set \u03b1), IsGLB s (sInf s)\na b c : \u03b1\nhab : a \u2264 b\nhac : a \u2264 c\n\u22a2 a \u2208 lowerBounds {b, c}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : InfSet \u03b1\nisGLB_sInf : \u2200 (s : Set \u03b1), IsGLB s (sInf s)\na : \u03b1\n\u22a2 a \u2208 lowerBounds \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : SupSet \u03b1\nisLUB_sSup : \u2200 (s : Set \u03b1), IsLUB s (sSup s)\na b c : \u03b1\nhac : a \u2264 c\nhbc : b \u2264 c\n\u22a2 c \u2208 upperBounds {a, b}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : SupSet \u03b1\nisLUB_sSup : \u2200 (s : Set \u03b1), IsLUB s (sSup s)\na b c : \u03b1\nhab : a \u2264 b\nhac : a \u2264 c\n\u22a2 a \u2208 {x | x \u2264 b \u2227 x \u2264 c}\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\nH1 : PartialOrder \u03b1\nH2 : SupSet \u03b1\nisLUB_sSup : \u2200 (s : Set \u03b1), IsLUB s (sSup s)\nx : \u03b1\n\u22a2 x \u2208 upperBounds \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 min a b = a\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 min a b = b\n[PROOFSTEP]\nsimp [(CompleteLinearOrder.le_total a b).resolve_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 max a b = b\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ni : CompleteLinearOrder \u03b1\na b : \u03b1\nh : \u00aca \u2264 b\n\u22a2 max a b = a\n[PROOFSTEP]\nsimp [(CompleteLinearOrder.le_total a b).resolve_left h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nh_sup : sSup s = \u22a5\nhne : Set.Nonempty s\n\u22a2 s = {\u22a5}\n[PROOFSTEP]\nrw [Set.eq_singleton_iff_nonempty_unique_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nh_sup : sSup s = \u22a5\nhne : Set.Nonempty s\n\u22a2 Set.Nonempty s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = \u22a5\n[PROOFSTEP]\nrw [sSup_eq_bot] at h_sup \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Set \u03b1\nh_sup : \u2200 (a : \u03b1), a \u2208 s \u2192 a = \u22a5\nhne : Set.Nonempty s\n\u22a2 Set.Nonempty s \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 x = \u22a5\n[PROOFSTEP]\nexact \u27e8hne, h_sup\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf g : \u03b9 \u2192 \u03b1\ns : Set \u03b1\n\u22a2 sSup s = \u2a06 (a : \u2191s), \u2191a\n[PROOFSTEP]\nrw [iSup, Subtype.range_coe]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b9'\nhf : Surjective f\ng : \u03b9' \u2192 \u03b1\n\u22a2 \u2a06 (x : \u03b9), g (f x) = \u2a06 (y : \u03b9'), g y\n[PROOFSTEP]\nsimp [iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b9'\nhf : Surjective f\ng : \u03b9' \u2192 \u03b1\n\u22a2 sSup (range fun x => g (f x)) = sSup (range fun y => g y)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b9'\nhf : Surjective f\ng : \u03b9' \u2192 \u03b1\n\u22a2 (range fun x => g (f x)) = range fun y => g y\n[PROOFSTEP]\nexact hf.range_comp g\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\ng : \u03b9' \u2192 \u03b1\nh : \u03b9 \u2192 \u03b9'\nh1 : Surjective h\nh2 : \u2200 (x : \u03b9), g (h x) = f x\n\u22a2 \u2a06 (x : \u03b9), f x = \u2a06 (y : \u03b9'), g y\n[PROOFSTEP]\nconvert h1.iSup_comp g\n[GOAL]\ncase h.e'_2.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf g\u271d : \u03b9 \u2192 \u03b1\ng : \u03b9' \u2192 \u03b1\nh : \u03b9 \u2192 \u03b9'\nh1 : Surjective h\nh2 : \u2200 (x : \u03b9), g (h x) = f x\nx\u271d : \u03b9\n\u22a2 f x\u271d = g (h x\u271d)\n[PROOFSTEP]\nexact (h2 _).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np q : Prop\nf\u2081 : p \u2192 \u03b1\nf\u2082 : q \u2192 \u03b1\npq : p \u2194 q\nf : \u2200 (x : q), f\u2081 (_ : p) = f\u2082 x\n\u22a2 iSup f\u2081 = iSup f\u2082\n[PROOFSTEP]\nobtain rfl := propext pq\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\n\u22a2 iSup f\u2081 = iSup f\u2082\n[PROOFSTEP]\ncongr with x\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\np : Prop\nf\u2081 f\u2082 : p \u2192 \u03b1\npq : p \u2194 p\nf : \u2200 (x : p), f\u2081 (_ : p) = f\u2082 x\nx : p\n\u22a2 f\u2081 x = f\u2082 x\n[PROOFSTEP]\napply f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b2\n\u22a2 \u2a06 (b : \u2191(range f)), g \u2191b = \u2a06 (i : \u03b9), g (f i)\n[PROOFSTEP]\nrw [iSup, iSup, \u2190 image_eq_range, \u2190 range_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g\u271d : \u03b9 \u2192 \u03b1\ng : \u03b2 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b2\n\u22a2 sSup (range (g \u2218 f)) = sSup (range fun i => g (f i))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : SupSet \u03b1\nf\u271d g : \u03b9 \u2192 \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 sSup (f '' s) = \u2a06 (a : \u2191s), f \u2191a\n[PROOFSTEP]\nrw [iSup, image_eq_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 \u2a06 (i : \u03b9) (j : \u03ba i), f i j \u2264 a \u2194 \u2200 (i : \u03b9) (j : \u03ba i), f i j \u2264 a\n[PROOFSTEP]\nsimp_rw [iSup_le_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 a \u2264 \u2a05 (i : \u03b9) (j : \u03ba i), f i j \u2194 \u2200 (i : \u03b9) (j : \u03ba i), a \u2264 f i j\n[PROOFSTEP]\nsimp_rw [le_iInf_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 \u2a06 (a : \u03b1) (_ : a \u2208 s), f a \u2264 f (sSup s)\n[PROOFSTEP]\nrw [sSup_eq_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Monotone f\n\u22a2 \u2a06 (a : \u03b1) (_ : a \u2208 s), f a \u2264 f (\u2a06 (a : \u03b1) (_ : a \u2208 s), a)\n[PROOFSTEP]\nexact hf.le_map_iSup\u2082 _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2243o \u03b2\nx\u271d : \u03b9 \u2192 \u03b1\nx : \u03b1\n\u22a2 \u2191f (\u2a06 (i : \u03b9), x\u271d i) \u2264 \u2191f x \u2194 \u2a06 (i : \u03b9), \u2191f (x\u271d i) \u2264 \u2191f x\n[PROOFSTEP]\nsimp only [f.le_iff_le, iSup_le_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\nf : \u03b1 \u2243o \u03b2\ns : Set \u03b1\n\u22a2 \u2191f (sSup s) = \u2a06 (a : \u03b1) (_ : a \u2208 s), \u2191f a\n[PROOFSTEP]\nsimp only [sSup_eq_iSup, OrderIso.map_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Antitone f\n\u22a2 f (sSup s) \u2264 \u2a05 (a : \u03b1) (_ : a \u2208 s), f a\n[PROOFSTEP]\nrw [sSup_eq_iSup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : CompleteLattice \u03b2\ns : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : Antitone f\n\u22a2 f (\u2a06 (a : \u03b1) (_ : a \u2208 s), a) \u2264 \u2a05 (a : \u03b1) (_ : a \u2208 s), f a\n[PROOFSTEP]\nexact hf.map_iSup\u2082_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ninst\u271d : Nonempty \u03b9\n\u22a2 \u2a06 (x : \u03b9), a = a\n[PROOFSTEP]\nrw [iSup, range_const, sSup_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 \u2a06 (i : \u03b9) (j : \u03ba i), f i j = \u22a5 \u2194 \u2200 (i : \u03b9) (j : \u03ba i), f i j = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 \u2a05 (i : \u03b9) (j : \u03ba i), f i j = \u22a4 \u2194 \u2200 (i : \u03b9) (j : \u03ba i), f i j = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : Prop\ninst\u271d : Decidable p\na : p \u2192 \u03b1\n\u22a2 \u2a06 (h : p), a h = if h : p then a h else \u22a5\n[PROOFSTEP]\nby_cases h : p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : Prop\ninst\u271d : Decidable p\na : p \u2192 \u03b1\nh : p\n\u22a2 \u2a06 (h : p), a h = if h : p then a h else \u22a5\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : Prop\ninst\u271d : Decidable p\na : p \u2192 \u03b1\nh : \u00acp\n\u22a2 \u2a06 (h : p), a h = if h : p then a h else \u22a5\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u03b9\u2081 : Sort u_9\n\u03b9\u2082 : Sort u_10\n\u03ba\u2081 : \u03b9\u2081 \u2192 Sort u_11\n\u03ba\u2082 : \u03b9\u2082 \u2192 Sort u_12\nf : (i\u2081 : \u03b9\u2081) \u2192 \u03ba\u2081 i\u2081 \u2192 (i\u2082 : \u03b9\u2082) \u2192 \u03ba\u2082 i\u2082 \u2192 \u03b1\n\u22a2 \u2a06 (i\u2081 : \u03b9\u2081) (j\u2081 : \u03ba\u2081 i\u2081) (i\u2082 : \u03b9\u2082) (j\u2082 : \u03ba\u2082 i\u2082), f i\u2081 j\u2081 i\u2082 j\u2082 =\n    \u2a06 (i\u2082 : \u03b9\u2082) (j\u2082 : \u03ba\u2082 i\u2082) (i\u2081 : \u03b9\u2081) (j\u2081 : \u03ba\u2081 i\u2081), f i\u2081 j\u2081 i\u2082 j\u2082\n[PROOFSTEP]\nsimp only [@iSup_comm _ (\u03ba\u2081 _), @iSup_comm _ \u03b9\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u03b9\u2081 : Sort u_9\n\u03b9\u2082 : Sort u_10\n\u03ba\u2081 : \u03b9\u2081 \u2192 Sort u_11\n\u03ba\u2082 : \u03b9\u2082 \u2192 Sort u_12\nf : (i\u2081 : \u03b9\u2081) \u2192 \u03ba\u2081 i\u2081 \u2192 (i\u2082 : \u03b9\u2082) \u2192 \u03ba\u2082 i\u2082 \u2192 \u03b1\n\u22a2 \u2a05 (i\u2081 : \u03b9\u2081) (j\u2081 : \u03ba\u2081 i\u2081) (i\u2082 : \u03b9\u2082) (j\u2082 : \u03ba\u2082 i\u2082), f i\u2081 j\u2081 i\u2082 j\u2082 =\n    \u2a05 (i\u2082 : \u03b9\u2082) (j\u2082 : \u03ba\u2082 i\u2082) (i\u2081 : \u03b9\u2081) (j\u2081 : \u03ba\u2081 i\u2081), f i\u2081 j\u2081 i\u2082 j\u2082\n[PROOFSTEP]\nsimp only [@iInf_comm _ (\u03ba\u2081 _), @iInf_comm _ \u03b9\u2081]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b\u271d : \u03b1\nb : \u03b2\nf : (x : \u03b2) \u2192 x = b \u2192 \u03b1\nc : \u03b2\n\u22a2 \u2200 (i : c = b), f c i \u2264 f b (_ : b = b)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nc : \u03b2\nf : (x : \u03b2) \u2192 x = c \u2192 \u03b1\n\u22a2 f c (_ : c = c) \u2264 f c (_ : c = c)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b\u271d : \u03b1\nb : \u03b2\nf : (x : \u03b2) \u2192 b = x \u2192 \u03b1\nc : \u03b2\n\u22a2 \u2200 (j : b = c), f c j \u2264 f b (_ : b = b)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b\u271d : \u03b1\nb : \u03b2\nf : (x : \u03b2) \u2192 b = x \u2192 \u03b1\n\u22a2 f b (_ : b = b) \u2264 f b (_ : b = b)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9\u271d : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s\u271d t : \u03b9\u271d \u2192 \u03b1\na\u271d b : \u03b1\n\u03b9 : Type u_9\na : \u03b1\ns : Set \u03b9\nhs : Set.Nonempty s\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 s), a = a\n[PROOFSTEP]\nhaveI : Nonempty s := Set.nonempty_coe_sort.mpr hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9\u271d : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s\u271d t : \u03b9\u271d \u2192 \u03b1\na\u271d b : \u03b1\n\u03b9 : Type u_9\na : \u03b1\ns : Set \u03b9\nhs : Set.Nonempty s\nthis : Nonempty \u2191s\n\u22a2 \u2a06 (i : \u03b9) (_ : i \u2208 s), a = a\n[PROOFSTEP]\nrw [\u2190 iSup_subtype'', iSup_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2a06 (x : \u03b9), f x) \u2294 a = \u2a06 (x : \u03b9), f x \u2294 a\n[PROOFSTEP]\nrw [iSup_sup_eq, iSup_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 (\u2a05 (x : \u03b9), f x) \u2293 a = \u2a05 (x : \u03b9), f x \u2293 a\n[PROOFSTEP]\nrw [iInf_inf_eq, iInf_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2294 \u2a06 (x : \u03b9), f x = \u2a06 (x : \u03b9), a \u2294 f x\n[PROOFSTEP]\nrw [iSup_sup_eq, iSup_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\ninst\u271d : Nonempty \u03b9\nf : \u03b9 \u2192 \u03b1\na : \u03b1\n\u22a2 a \u2293 \u2a05 (x : \u03b9), f x = \u2a05 (x : \u03b9), a \u2293 f x\n[PROOFSTEP]\nrw [iInf_inf_eq, iInf_const]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\na : \u03b1\nh : \u2203 i, p i\n\u22a2 (\u2a06 (i : \u03b9) (h : p i), f i h) \u2294 a = \u2a06 (i : \u03b9) (h : p i), f i h \u2294 a\n[PROOFSTEP]\nhaveI : Nonempty { i // p i } :=\n  let \u27e8i, hi\u27e9 := h\n  \u27e8\u27e8i, hi\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\na : \u03b1\nh : \u2203 i, p i\nthis : Nonempty { i // p i }\n\u22a2 (\u2a06 (i : \u03b9) (h : p i), f i h) \u2294 a = \u2a06 (i : \u03b9) (h : p i), f i h \u2294 a\n[PROOFSTEP]\nrw [iSup_subtype', iSup_subtype', iSup_sup]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b : \u03b1\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\na : \u03b1\nh : \u2203 i, p i\n\u22a2 a \u2294 \u2a06 (i : \u03b9) (h : p i), f i h = \u2a06 (i : \u03b9) (h : p i), a \u2294 f i h\n[PROOFSTEP]\nsimpa only [sup_comm] using @biSup_sup \u03b1 _ _ p _ _ h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ns : False \u2192 \u03b1\n\u22a2 iSup s = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ns : False \u2192 \u03b1\n\u22a2 iInf s = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\ng : (i : \u03b9) \u2192 \u00acp i \u2192 \u03b1\n\u22a2 (\u2a06 (i : \u03b9), if h : p i then f i h else g i h) = (\u2a06 (i : \u03b9) (h : p i), f i h) \u2294 \u2a06 (i : \u03b9) (h : \u00acp i), g i h\n[PROOFSTEP]\nrw [\u2190 iSup_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\ng : (i : \u03b9) \u2192 \u00acp i \u2192 \u03b1\n\u22a2 (\u2a06 (i : \u03b9), if h : p i then f i h else g i h) = \u2a06 (x : \u03b9), (\u2a06 (h : p x), f x h) \u2294 \u2a06 (h : \u00acp x), g x h\n[PROOFSTEP]\ncongr 1 with i\n[GOAL]\ncase e_s.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\ng : (i : \u03b9) \u2192 \u00acp i \u2192 \u03b1\ni : \u03b9\n\u22a2 (if h : p i then f i h else g i h) = (\u2a06 (h : p i), f i h) \u2294 \u2a06 (h : \u00acp i), g i h\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\ng : (i : \u03b9) \u2192 \u00acp i \u2192 \u03b1\ni : \u03b9\nh : p i\n\u22a2 f i h = (\u2a06 (h : p i), f i h) \u2294 \u2a06 (h : \u00acp i), g i h\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b9 \u2192 Prop\ninst\u271d : DecidablePred p\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\ng : (i : \u03b9) \u2192 \u00acp i \u2192 \u03b1\ni : \u03b9\nh : \u00acp i\n\u22a2 g i h = (\u2a06 (h : p i), f i h) \u2294 \u2a06 (h : \u00acp i), g i h\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g\u271d s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ng : \u03b2 \u2192 \u03b1\nf : \u03b9 \u2192 \u03b2\n\u22a2 \u2a06 (b : \u03b2) (_ : b \u2208 range f), g b = \u2a06 (i : \u03b9), g (f i)\n[PROOFSTEP]\nrw [\u2190 iSup_subtype'', iSup_range']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 sSup (f '' s) = \u2a06 (a : \u03b2) (_ : a \u2208 s), f a\n[PROOFSTEP]\nrw [\u2190 iSup_subtype'', sSup_image']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 (x : \u03b2) (_ : x \u2208 \u2205), f x = \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a05 (x : \u03b2) (_ : x \u2208 \u2205), f x = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 (x : \u03b2) (_ : x \u2208 univ), f x = \u2a06 (x : \u03b2), f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a05 (x : \u03b2) (_ : x \u2208 univ), f x = \u2a05 (x : \u03b2), f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\ns t : Set \u03b2\n\u22a2 \u2a06 (x : \u03b2) (_ : x \u2208 s \u222a t), f x = (\u2a06 (x : \u03b2) (_ : x \u2208 s), f x) \u2294 \u2a06 (x : \u03b2) (_ : x \u2208 t), f x\n[PROOFSTEP]\nsimp_rw [mem_union, iSup_or, iSup_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\np : \u03b2 \u2192 Prop\n\u22a2 \u2a06 (i : \u03b2), f i = (\u2a06 (i : \u03b2) (_ : p i), f i) \u2294 \u2a06 (i : \u03b2) (_ : \u00acp i), f i\n[PROOFSTEP]\nsimpa [Classical.em] using @iSup_union _ _ _ f {i | p i} {i | \u00acp i}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\ni\u2080 : \u03b2\n\u22a2 \u2a06 (i : \u03b2), f i = f i\u2080 \u2294 \u2a06 (i : \u03b2) (_ : i \u2260 i\u2080), f i\n[PROOFSTEP]\nconvert iSup_split f (fun i => i = i\u2080)\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b1\ni\u2080 : \u03b2\n\u22a2 f i\u2080 = \u2a06 (i : \u03b2) (_ : i = i\u2080), f i\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\n\u22a2 \u2a06 (x : \u03b2) (_ : x \u2208 {b}), f x = f b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\n\u22a2 \u2a05 (x : \u03b2) (_ : x \u2208 {b}), f x = f b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\na b : \u03b2\n\u22a2 \u2a06 (x : \u03b2) (_ : x \u2208 {a, b}), f x = f a \u2294 f b\n[PROOFSTEP]\nrw [iSup_insert, iSup_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b\u271d : \u03b1\nf : \u03b2 \u2192 \u03b1\na b : \u03b2\n\u22a2 \u2a05 (x : \u03b2) (_ : x \u2208 {a, b}), f x = f a \u2293 f b\n[PROOFSTEP]\nrw [iInf_insert, iInf_singleton]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3\u271d : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g\u271d s t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u03b3 : Type u_9\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\nt : Set \u03b2\n\u22a2 \u2a06 (c : \u03b3) (_ : c \u2208 f '' t), g c = \u2a06 (b : \u03b2) (_ : b \u2208 t), g (f b)\n[PROOFSTEP]\nrw [\u2190 sSup_image, \u2190 sSup_image, \u2190 image_comp]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3\u271d : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g\u271d s t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\n\u03b3 : Type u_9\nf : \u03b2 \u2192 \u03b3\ng : \u03b3 \u2192 \u03b1\nt : Set \u03b2\n\u22a2 sSup ((fun c => g c) \u2218 f '' t) = sSup ((fun b => g (f b)) '' t)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ne : \u03b9 \u2192 \u03b2\nhe : Injective e\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2a06 (j : \u03b2), extend e f \u22a5 j = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nrw [iSup_split _ fun j => \u2203 i, e i = j]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\ne : \u03b9 \u2192 \u03b2\nhe : Injective e\nf : \u03b9 \u2192 \u03b1\n\u22a2 (\u2a06 (i : \u03b2) (_ : \u2203 i_1, e i_1 = i), extend e f \u22a5 i) \u2294 \u2a06 (i : \u03b2) (_ : \u00ac\u2203 i_1, e i_1 = i), extend e f \u22a5 i =\n    \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nsimp (config := { contextual := true }) [he.extend_apply, extend_apply', @iSup_comm _ \u03b2 \u03b9]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : Bool \u2192 \u03b1\n\u22a2 \u2a06 (b : Bool), f b = f true \u2294 f false\n[PROOFSTEP]\nrw [iSup, Bool.range_eq, sSup_pair, sup_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b x y : \u03b1\n\u22a2 x \u2294 y = \u2a06 (b : Bool), bif b then x else y\n[PROOFSTEP]\nrw [iSup_bool_eq, Bool.cond_true, Bool.cond_false]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 IsGLB (f '' s) (\u2a05 (x : \u03b2) (_ : x \u2208 s), f x)\n[PROOFSTEP]\nsimpa only [range_comp, Subtype.range_coe, iInf_subtype'] using @isGLB_iInf \u03b1 s _ (f \u2218 fun x => (x : \u03b2))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t : \u03b9 \u2192 \u03b1\na b : \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\n\u22a2 IsLUB (f '' s) (\u2a06 (x : \u03b2) (_ : x \u2208 s), f x)\n[PROOFSTEP]\nsimpa only [range_comp, Subtype.range_coe, iSup_subtype'] using @isLUB_iSup \u03b1 s _ (f \u2218 fun x => (x : \u03b2))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\np : \u03b2 \u2192 Type u_9\nf : Sigma p \u2192 \u03b1\nc : \u03b1\n\u22a2 \u2a06 (x : Sigma p), f x \u2264 c \u2194 \u2a06 (i : \u03b2) (j : p i), f { fst := i, snd := j } \u2264 c\n[PROOFSTEP]\nsimp only [iSup_le_iff, Sigma.forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 \u03b1\nc : \u03b1\n\u22a2 \u2a06 (x : \u03b2 \u00d7 \u03b3), f x \u2264 c \u2194 \u2a06 (i : \u03b2) (j : \u03b3), f (i, j) \u2264 c\n[PROOFSTEP]\nsimp only [iSup_le_iff, Prod.forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 \u03b1\ns : Set \u03b2\nt : Set \u03b3\n\u22a2 \u2a06 (x : \u03b2 \u00d7 \u03b3) (_ : x \u2208 s \u00d7\u02e2 t), f x = \u2a06 (a : \u03b2) (_ : a \u2208 s) (b : \u03b3) (_ : b \u2208 t), f (a, b)\n[PROOFSTEP]\nsimp_rw [iSup_prod, mem_prod, iSup_and]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 \u03b1\ns : Set \u03b2\nt : Set \u03b3\n\u22a2 \u2a06 (i : \u03b2) (j : \u03b3) (_ : i \u2208 s) (_ : j \u2208 t), f (i, j) = \u2a06 (a : \u03b2) (_ : a \u2208 s) (b : \u03b3) (_ : b \u2208 t), f (a, b)\n[PROOFSTEP]\nexact iSup_congr fun _ => iSup_comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2295 \u03b3 \u2192 \u03b1\nc : \u03b1\n\u22a2 \u2a06 (x : \u03b2 \u2295 \u03b3), f x \u2264 c \u2194 (\u2a06 (i : \u03b2), f (Sum.inl i)) \u2294 \u2a06 (j : \u03b3), f (Sum.inr j) \u2264 c\n[PROOFSTEP]\nsimp only [sup_le_iff, iSup_le_iff, Sum.forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : Option \u03b2 \u2192 \u03b1\nc : \u03b1\n\u22a2 \u2a06 (o : Option \u03b2), f o \u2264 c \u2194 f none \u2294 \u2a06 (b : \u03b2), f (some b) \u2264 c\n[PROOFSTEP]\nsimp only [iSup_le_iff, sup_le_iff, Option.forall]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na\u271d b a : \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 \u2a06 (o : Option \u03b2), Option.elim o a f = a \u2294 \u2a06 (b : \u03b2), f b\n[PROOFSTEP]\nsimp [iSup_option]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 \u2a06 (i : { i // f i \u2260 \u22a5 }), f \u2191i = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nby_cases htriv : \u2200 i, f i = \u22a5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u2200 (i : \u03b9), f i = \u22a5\n\u22a2 \u2a06 (i : { i // f i \u2260 \u22a5 }), f \u2191i = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nsimp only [iSup_bot, (funext htriv : f = _)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\n\u22a2 \u2a06 (i : { i // f i \u2260 \u22a5 }), f \u2191i = \u2a06 (i : \u03b9), f i\n[PROOFSTEP]\nrefine' (iSup_comp_le f _).antisymm (iSup_mono' fun i => _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\ni : \u03b9\n\u22a2 \u2203 i', f i \u2264 f \u2191i'\n[PROOFSTEP]\nby_cases hi : f i = \u22a5\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\ni : \u03b9\nhi : f i = \u22a5\n\u22a2 \u2203 i', f i \u2264 f \u2191i'\n[PROOFSTEP]\nrw [hi]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\ni : \u03b9\nhi : f i = \u22a5\n\u22a2 \u2203 i', \u22a5 \u2264 f \u2191i'\n[PROOFSTEP]\nobtain \u27e8i\u2080, hi\u2080\u27e9 := not_forall.mp htriv\n[GOAL]\ncase pos.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\ni : \u03b9\nhi : f i = \u22a5\ni\u2080 : \u03b9\nhi\u2080 : \u00acf i\u2080 = \u22a5\n\u22a2 \u2203 i', \u22a5 \u2264 f \u2191i'\n[PROOFSTEP]\nexact \u27e8\u27e8i\u2080, hi\u2080\u27e9, bot_le\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b9 \u2192 \u03b1\nhtriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5\ni : \u03b9\nhi : \u00acf i = \u22a5\n\u22a2 \u2203 i', f i \u2264 f \u2191i'\n[PROOFSTEP]\nexact \u27e8\u27e8i, hi\u27e9, rfl.le\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b2\nt : Set \u03b3\n\u22a2 sSup (image2 f s t) = \u2a06 (a : \u03b2) (_ : a \u2208 s) (b : \u03b3) (_ : b \u2208 t), f a b\n[PROOFSTEP]\nrw [\u2190 image_prod, sSup_image, biSup_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s\u271d t\u271d : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u03b2 \u2192 \u03b3 \u2192 \u03b1\ns : Set \u03b2\nt : Set \u03b3\n\u22a2 sInf (image2 f s t) = \u2a05 (a : \u03b2) (_ : a \u2208 s) (b : \u03b3) (_ : b \u2208 t), f a b\n[PROOFSTEP]\nrw [\u2190 image_prod, sInf_image, biInf_prod]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2a06 (i : \u2115) (_ : i \u2265 n), u i = \u2a06 (i : \u2115), u (i + n)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2a06 (i : \u2115) (_ : i \u2265 n), u i \u2264 \u2a06 (i : \u2115), u (i + n)\n[PROOFSTEP]\nsimp only [iSup_le_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2a06 (i : \u2115), u (i + n) \u2264 \u2a06 (i : \u2115) (_ : i \u2265 n), u i\n[PROOFSTEP]\nsimp only [iSup_le_iff]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2200 (i : \u2115), i \u2265 n \u2192 u i \u2264 \u2a06 (i : \u2115), u (i + n)\n[PROOFSTEP]\nrefine fun i hi => le_sSup \u27e8i - n, ?_\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn i : \u2115\nhi : i \u2265 n\n\u22a2 (fun i => u (i + n)) (i - n) = u i\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn i : \u2115\nhi : i \u2265 n\n\u22a2 u (i - n + n) = u i\n[PROOFSTEP]\nrw [Nat.sub_add_cancel hi]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\nn : \u2115\n\u22a2 \u2200 (i : \u2115), u (i + n) \u2264 \u2a06 (i : \u2115) (_ : i \u2265 n), u i\n[PROOFSTEP]\nexact fun i => le_sSup \u27e8i + n, iSup_pos (Nat.le_add_left _ _)\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\n\u22a2 \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n[PROOFSTEP]\nhave hf : Monotone fun n => \u2a05 i \u2265 n, f i := fun n m h => biInf_mono fun i => h.trans\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n[PROOFSTEP]\nrw [\u2190 Monotone.iSup_nat_add hf k]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\nk : \u2115\nhf : Monotone fun n => \u2a05 (i : \u2115) (_ : i \u2265 n), f i\n\u22a2 \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n), f (i + k) = \u2a06 (n : \u2115), \u2a05 (i : \u2115) (_ : i \u2265 n + k), f i\n[PROOFSTEP]\nsimp_rw [iInf_ge_eq_iInf_nat_add, \u2190 Nat.add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\n\u22a2 u 0 \u2294 \u2a06 (i : \u2115), u (i + 1) = \u2a06 (x : \u2115) (_ : x \u2208 {0} \u222a range Nat.succ), u x\n[PROOFSTEP]\n{rw [iSup_union, iSup_singleton, iSup_range]\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\n\u22a2 u 0 \u2294 \u2a06 (i : \u2115), u (i + 1) = \u2a06 (x : \u2115) (_ : x \u2208 {0} \u222a range Nat.succ), u x\n[PROOFSTEP]\nrw [iSup_union, iSup_singleton, iSup_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nu : \u2115 \u2192 \u03b1\n\u22a2 \u2a06 (x : \u2115) (_ : x \u2208 {0} \u222a range Nat.succ), u x = \u2a06 (i : \u2115), u i\n[PROOFSTEP]\nrw [Nat.zero_union_range_succ, iSup_univ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\n\u22a2 \u2a05 (i : \u2115) (_ : i > 0), f i = \u2a05 (i : \u2115), f (i + 1)\n[PROOFSTEP]\nrw [\u2190 iInf_range, Nat.range_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\nf\u271d g s t : \u03b9 \u2192 \u03b1\na b : \u03b1\nf : \u2115 \u2192 \u03b1\n\u22a2 \u2a05 (i : \u2115) (_ : i > 0), f i = \u2a05 (b : \u2115) (_ : b \u2208 {i | 0 < i}), f b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLinearOrder \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 iSup f = \u22a4 \u2194 \u2200 (b : \u03b1), b < \u22a4 \u2192 \u2203 i, b < f i\n[PROOFSTEP]\nsimp only [\u2190 sSup_range, sSup_eq_top, Set.exists_range_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLinearOrder \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 iInf f = \u22a5 \u2194 \u2200 (b : \u03b1), b > \u22a5 \u2192 \u2203 i, f i < b\n[PROOFSTEP]\nsimp only [\u2190 sInf_range, sInf_eq_bot, Set.exists_range_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9\u271d : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : \u03b1 \u2192 Type u_10\n\u03b9 : Sort u_11\ninst\u271d : (i : \u03b1) \u2192 SupSet (\u03b2 i)\nf : \u03b9 \u2192 (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\n\u22a2 iSup (fun i => f i) a = \u2a06 (i : \u03b9), f i a\n[PROOFSTEP]\nrw [iSup, sSup_apply, iSup, iSup, \u2190 image_eq_range (fun f : \u2200 i, \u03b2 i => f a) (range f), \u2190 range_comp]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9\u271d : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9\u271d \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : \u03b1 \u2192 Type u_10\n\u03b9 : Sort u_11\ninst\u271d : (i : \u03b1) \u2192 SupSet (\u03b2 i)\nf : \u03b9 \u2192 (a : \u03b1) \u2192 \u03b2 a\na : \u03b1\n\u22a2 sSup (range ((fun f => f a) \u2218 f)) = sSup (range fun i => f i a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\ns : Set (\u03b1 \u2192 Prop)\na : \u03b1\n\u22a2 sSup s a \u2194 \u2203 r, r \u2208 s \u2227 r a\n[PROOFSTEP]\nrw [sSup_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\ns : Set (\u03b1 \u2192 Prop)\na : \u03b1\n\u22a2 \u2a06 (f : \u2191s), \u2191f a \u2194 \u2203 r, r \u2208 s \u2227 r a\n[PROOFSTEP]\nsimp [\u2190 eq_iff_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\ns : Set (\u03b1 \u2192 Prop)\na : \u03b1\n\u22a2 sInf s a \u2194 \u2200 (r : \u03b1 \u2192 Prop), r \u2208 s \u2192 r a\n[PROOFSTEP]\nrw [sInf_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\ns : Set (\u03b1 \u2192 Prop)\na : \u03b1\n\u22a2 \u2a05 (f : \u2191s), \u2191f a \u2194 \u2200 (r : \u03b1 \u2192 Prop), r \u2208 s \u2192 r a\n[PROOFSTEP]\nsimp [\u2190 eq_iff_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : Type u_10\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 sSup s a b \u2194 \u2203 r, r \u2208 s \u2227 r a b\n[PROOFSTEP]\nrw [sSup_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : Type u_10\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 iSup (fun f => \u2191f a) b \u2194 \u2203 r, r \u2208 s \u2227 r a b\n[PROOFSTEP]\nsimp [\u2190 eq_iff_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : Type u_10\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 sInf s a b \u2194 \u2200 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2192 r a b\n[PROOFSTEP]\nrw [sInf_apply]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\n\u03b1 : Type u_9\n\u03b2 : Type u_10\ns : Set (\u03b1 \u2192 \u03b2 \u2192 Prop)\na : \u03b1\nb : \u03b2\n\u22a2 iInf (fun f => \u2191f a) b \u2194 \u2200 (r : \u03b1 \u2192 \u03b2 \u2192 Prop), r \u2208 s \u2192 r a b\n[PROOFSTEP]\nsimp [\u2190 eq_iff_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : InfSet \u03b1\ninst\u271d : InfSet \u03b2\nf : \u03b9 \u2192 \u03b1 \u00d7 \u03b2\n\u22a2 swap (iInf f) = \u2a05 (i : \u03b9), swap (f i)\n[PROOFSTEP]\nsimp_rw [iInf, swap_sInf, \u2190 range_comp, Function.comp]\n  -- Porting note: need to unfold `\u2218`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d\u00b9 : SupSet \u03b1\ninst\u271d : SupSet \u03b2\nf : \u03b9 \u2192 \u03b1 \u00d7 \u03b2\n\u22a2 swap (iSup f) = \u2a06 (i : \u03b9), swap (f i)\n[PROOFSTEP]\nsimp_rw [iSup, swap_sSup, \u2190 range_comp, Function.comp]\n  -- Porting note: need to unfold `\u2218`\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\ns : Set (ULift \u03b1)\n\u22a2 (sSup s).down = \u2a06 (a : ULift \u03b1) (_ : a \u2208 s), a.down\n[PROOFSTEP]\nrw [sSup_eq_iSup', down_iSup, iSup_subtype'']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b2\u2082 : Type u_3\n\u03b3 : Type u_4\n\u03b9 : Sort u_5\n\u03b9' : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba' : \u03b9' \u2192 Sort u_8\ninst\u271d : CompleteLattice \u03b1\ns : Set (ULift \u03b1)\n\u22a2 (sInf s).down = \u2a05 (a : ULift \u03b1) (_ : a \u2208 s), a.down\n[PROOFSTEP]\nrw [sInf_eq_iInf', down_iInf, iInf_subtype'']\n", "meta": {"mathlib_filename": "Mathlib.Order.CompleteLattice", "llama_tokens": 24681, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672089305841, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5263385129829428}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : IsROrC \ud835\udd5c\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (nhds 0)\n[PROOFSTEP]\nconvert tendsto_algebraMap_inverse_atTop_nhds_0_nat \ud835\udd5c\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\ninst\u271d : IsROrC \ud835\udd5c\nx\u271d : \u2115\n\u22a2 (\u2191x\u271d)\u207b\u00b9 = (\u2191(algebraMap \u211d \ud835\udd5c) \u2218 fun n => (\u2191n)\u207b\u00b9) x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecificLimits.IsROrC", "llama_tokens": 167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.782662489091802, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.526208806313555}}
{"text": "[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave hyp1 : v (x - y) < \u03b3 * (v y * v y) := lt_of_lt_of_le h (min_le_left _ _)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave hyp1' : v (x - y) * (v y * v y)\u207b\u00b9 < \u03b3 := mul_inv_lt_of_lt_mul\u2080 hyp1\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave hyp2 : v (x - y) < v y := lt_of_lt_of_le h (min_le_right _ _)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave key : v x = v y := Valuation.map_eq_of_sub_lt v hyp2\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave x_ne : x \u2260 0 := by\n  intro h\n  apply y_ne\n  rw [h, v.map_zero] at key \n  exact v.zero_iff.1 key.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh\u271d : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nh : x = 0\n\u22a2 False\n[PROOFSTEP]\napply y_ne\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh\u271d : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nh : x = 0\n\u22a2 y = 0\n[PROOFSTEP]\nrw [h, v.map_zero] at key \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh\u271d : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : 0 = \u2191v y\nh : x = 0\n\u22a2 y = 0\n[PROOFSTEP]\nexact v.zero_iff.1 key.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nhave decomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9 := by\n  rw [mul_sub_left_distrib, sub_mul, mul_assoc, show y * y\u207b\u00b9 = 1 from mul_inv_cancel y_ne,\n    show x\u207b\u00b9 * x = 1 from inv_mul_cancel x_ne, mul_one, one_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\n\u22a2 x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n[PROOFSTEP]\nrw [mul_sub_left_distrib, sub_mul, mul_assoc, show y * y\u207b\u00b9 = 1 from mul_inv_cancel y_ne,\n  show x\u207b\u00b9 * x = 1 from inv_mul_cancel x_ne, mul_one, one_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\ncalc\n  v (x\u207b\u00b9 - y\u207b\u00b9) = v (x\u207b\u00b9 * (y - x) * y\u207b\u00b9) := by rw [decomp]\n  _ = v x\u207b\u00b9 * (v <| y - x) * v y\u207b\u00b9 := by repeat' rw [Valuation.map_mul]\n  _ = (v x)\u207b\u00b9 * (v <| y - x) * (v y)\u207b\u00b9 := by rw [map_inv\u2080, map_inv\u2080]\n  _ = (v <| y - x) * (v y * v y)\u207b\u00b9 := by rw [mul_assoc, mul_comm, key, mul_assoc, mul_inv_rev]\n  _ = (v <| y - x) * (v y * v y)\u207b\u00b9 := rfl\n  _ = (v <| x - y) * (v y * v y)\u207b\u00b9 := by rw [Valuation.map_sub_swap]\n  _ < \u03b3 := hyp1'\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (x\u207b\u00b9 - y\u207b\u00b9) = \u2191v (x\u207b\u00b9 * (y - x) * y\u207b\u00b9)\n[PROOFSTEP]\nrw [decomp]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (x\u207b\u00b9 * (y - x) * y\u207b\u00b9) = \u2191v x\u207b\u00b9 * \u2191v (y - x) * \u2191v y\u207b\u00b9\n[PROOFSTEP]\nrepeat' rw [Valuation.map_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (x\u207b\u00b9 * (y - x) * y\u207b\u00b9) = \u2191v x\u207b\u00b9 * \u2191v (y - x) * \u2191v y\u207b\u00b9\n[PROOFSTEP]\nrw [Valuation.map_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (x\u207b\u00b9 * (y - x)) * \u2191v y\u207b\u00b9 = \u2191v x\u207b\u00b9 * \u2191v (y - x) * \u2191v y\u207b\u00b9\n[PROOFSTEP]\nrw [Valuation.map_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v x\u207b\u00b9 * \u2191v (y - x) * \u2191v y\u207b\u00b9 = (\u2191v x)\u207b\u00b9 * \u2191v (y - x) * (\u2191v y)\u207b\u00b9\n[PROOFSTEP]\nrw [map_inv\u2080, map_inv\u2080]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 (\u2191v x)\u207b\u00b9 * \u2191v (y - x) * (\u2191v y)\u207b\u00b9 = \u2191v (y - x) * (\u2191v y * \u2191v y)\u207b\u00b9\n[PROOFSTEP]\nrw [mul_assoc, mul_comm, key, mul_assoc, mul_inv_rev]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation K \u0393\u2080\nx y : K\n\u03b3 : \u0393\u2080\u02e3\ny_ne : y \u2260 0\nh : \u2191v (x - y) < min (\u2191\u03b3 * (\u2191v y * \u2191v y)) (\u2191v y)\nhyp1 : \u2191v (x - y) < \u2191\u03b3 * (\u2191v y * \u2191v y)\nhyp1' : \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9 < \u2191\u03b3\nhyp2 : \u2191v (x - y) < \u2191v y\nkey : \u2191v x = \u2191v y\nx_ne : x \u2260 0\ndecomp : x\u207b\u00b9 - y\u207b\u00b9 = x\u207b\u00b9 * (y - x) * y\u207b\u00b9\n\u22a2 \u2191v (y - x) * (\u2191v y * \u2191v y)\u207b\u00b9 = \u2191v (x - y) * (\u2191v y * \u2191v y)\u207b\u00b9\n[PROOFSTEP]\nrw [Valuation.map_sub_swap]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 TopologicalRing K\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\n\u22a2 \u2200 \u2983x : K\u2984, x \u2260 0 \u2192 ContinuousAt Inv.inv x\n[PROOFSTEP]\nintro x x_ne s s_in\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\ns_in : s \u2208 \ud835\udcdd x\u207b\u00b9\n\u22a2 s \u2208 map Inv.inv (\ud835\udcdd x)\n[PROOFSTEP]\ncases' Valued.mem_nhds.mp s_in with \u03b3 hs\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\ns_in : s \u2208 \ud835\udcdd x\u207b\u00b9\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\n\u22a2 s \u2208 map Inv.inv (\ud835\udcdd x)\n[PROOFSTEP]\nclear s_in\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\n\u22a2 s \u2208 map Inv.inv (\ud835\udcdd x)\n[PROOFSTEP]\nrw [mem_map, Valued.mem_nhds]\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - x) < \u2191\u03b3} \u2286 Inv.inv \u207b\u00b9' s\n[PROOFSTEP]\nchange \u2203 \u03b3 : \u0393\u2080\u02e3, {y : K | (v (y - x) : \u0393\u2080) < \u03b3} \u2286 {x : K | x\u207b\u00b9 \u2208 s}\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - x) < \u2191\u03b3} \u2286 {x | x\u207b\u00b9 \u2208 s}\n[PROOFSTEP]\nhave vx_ne := (Valuation.ne_zero_iff <| v).mpr x_ne\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - x) < \u2191\u03b3} \u2286 {x | x\u207b\u00b9 \u2208 s}\n[PROOFSTEP]\nlet \u03b3' := Units.mk0 _ vx_ne\n[GOAL]\ncase intro\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - x) < \u2191\u03b3} \u2286 {x | x\u207b\u00b9 \u2208 s}\n[PROOFSTEP]\nuse min (\u03b3 * (\u03b3' * \u03b3')) \u03b3'\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\n\u22a2 {y | \u2191v (y - x) < \u2191(min (\u03b3 * (\u03b3' * \u03b3')) \u03b3')} \u2286 {x | x\u207b\u00b9 \u2208 s}\n[PROOFSTEP]\nintro y y_in\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\ny : K\ny_in : y \u2208 {y | \u2191v (y - x) < \u2191(min (\u03b3 * (\u03b3' * \u03b3')) \u03b3')}\n\u22a2 y \u2208 {x | x\u207b\u00b9 \u2208 s}\n[PROOFSTEP]\napply hs\n[GOAL]\ncase h.a\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\ny : K\ny_in : y \u2208 {y | \u2191v (y - x) < \u2191(min (\u03b3 * (\u03b3' * \u03b3')) \u03b3')}\n\u22a2 y\u207b\u00b9 \u2208 {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3}\n[PROOFSTEP]\nsimp only [mem_setOf_eq] at y_in \n[GOAL]\ncase h.a\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\ny : K\ny_in : \u2191v (y - x) < \u2191(min (\u03b3 * (Units.mk0 (\u2191v x) vx_ne * Units.mk0 (\u2191v x) vx_ne)) (Units.mk0 (\u2191v x) vx_ne))\n\u22a2 y\u207b\u00b9 \u2208 {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3}\n[PROOFSTEP]\nrw [Units.min_val, Units.val_mul, Units.val_mul] at y_in \n[GOAL]\ncase h.a\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nsrc\u271d : TopologicalRing K := inferInstance\nx : K\nx_ne : x \u2260 0\ns : Set K\n\u03b3 : \u0393\u2080\u02e3\nhs : {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3} \u2286 s\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\ny : K\ny_in : \u2191v (y - x) < min (\u2191\u03b3 * (\u2191(Units.mk0 (\u2191v x) vx_ne) * \u2191(Units.mk0 (\u2191v x) vx_ne))) \u2191(Units.mk0 (\u2191v x) vx_ne)\n\u22a2 y\u207b\u00b9 \u2208 {y | \u2191v (y - x\u207b\u00b9) < \u2191\u03b3}\n[PROOFSTEP]\nexact Valuation.inversion_estimate _ x_ne y_in\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 SeparatedSpace K\n[PROOFSTEP]\nrw [separated_iff_t2]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 T2Space K\n[PROOFSTEP]\napply TopologicalAddGroup.t2Space_of_zero_sep\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 \u2200 (x : K), x \u2260 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd 0 \u2227 \u00acx \u2208 U\n[PROOFSTEP]\nintro x x_ne\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\n\u22a2 \u2203 U, U \u2208 \ud835\udcdd 0 \u2227 \u00acx \u2208 U\n[PROOFSTEP]\nrefine' \u27e8{k | v k < v x}, _, fun h => lt_irrefl _ h\u27e9\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\n\u22a2 {k | \u2191v k < \u2191v x} \u2208 \ud835\udcdd 0\n[PROOFSTEP]\nrw [Valued.mem_nhds]\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - 0) < \u2191\u03b3} \u2286 {k | \u2191v k < \u2191v x}\n[PROOFSTEP]\nhave vx_ne := (Valuation.ne_zero_iff <| v).mpr x_ne\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\nvx_ne : \u2191v x \u2260 0\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - 0) < \u2191\u03b3} \u2286 {k | \u2191v k < \u2191v x}\n[PROOFSTEP]\nlet \u03b3' := Units.mk0 _ vx_ne\n[GOAL]\ncase H\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\n\u22a2 \u2203 \u03b3, {y | \u2191v (y - 0) < \u2191\u03b3} \u2286 {k | \u2191v k < \u2191v x}\n[PROOFSTEP]\nexact \u27e8\u03b3', fun y hy => by simpa using hy\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nx_ne : x \u2260 0\nvx_ne : \u2191v x \u2260 0\n\u03b3' : ((fun x => \u0393\u2080) x)\u02e3 := Units.mk0 (\u2191v x) vx_ne\ny : K\nhy : y \u2208 {y | \u2191v (y - 0) < \u2191\u03b3'}\n\u22a2 y \u2208 {k | \u2191v k < \u2191v x}\n[PROOFSTEP]\nsimpa using hy\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 Continuous \u2191v\n[PROOFSTEP]\nrw [continuous_iff_continuousAt]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 \u2200 (x : K), ContinuousAt (\u2191v) x\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\n\u22a2 ContinuousAt (\u2191v) x\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h)\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 ContinuousAt (\u2191v) 0\n[PROOFSTEP]\nrw [ContinuousAt, map_zero, WithZeroTopology.tendsto_zero]\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u22a2 \u2200 (\u03b3\u2080 : \u0393\u2080), \u03b3\u2080 \u2260 0 \u2192 \u2200\u1da0 (x : K) in \ud835\udcdd 0, \u2191v x < \u03b3\u2080\n[PROOFSTEP]\nintro \u03b3 h\u03b3\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\n\u22a2 \u2200\u1da0 (x : K) in \ud835\udcdd 0, \u2191v x < \u03b3\n[PROOFSTEP]\nrw [Filter.Eventually, Valued.mem_nhds_zero]\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\n\u22a2 \u2203 \u03b3_1, {x | \u2191v x < \u2191\u03b3_1} \u2286 {x | \u2191v x < \u03b3}\n[PROOFSTEP]\nuse Units.mk0 \u03b3 h\u03b3\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\nh\u03b3 : \u03b3 \u2260 0\n\u22a2 {x | \u2191v x < \u2191(Units.mk0 \u03b3 h\u03b3)} \u2286 {x | \u2191v x < \u03b3}\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nh : x \u2260 0\n\u22a2 ContinuousAt (\u2191v) x\n[PROOFSTEP]\nhave v_ne : (v x : \u0393\u2080) \u2260 0 := (Valuation.ne_zero_iff _).mpr h\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nh : x \u2260 0\nv_ne : \u2191v x \u2260 0\n\u22a2 ContinuousAt (\u2191v) x\n[PROOFSTEP]\nrw [ContinuousAt, WithZeroTopology.tendsto_of_ne_zero v_ne]\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b2 : DivisionRing K\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\ninst\u271d : Valued K \u0393\u2080\nx : K\nh : x \u2260 0\nv_ne : \u2191v x \u2260 0\n\u22a2 \u2200\u1da0 (x_1 : K) in \ud835\udcdd x, \u2191v x_1 = \u2191v x\n[PROOFSTEP]\napply Valued.loc_const v_ne\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\n\u22a2 \u2200 (F : Filter K), Cauchy F \u2192 \ud835\udcdd 0 \u2293 F = \u22a5 \u2192 Cauchy (map (fun x => x\u207b\u00b9) F)\n[PROOFSTEP]\nrintro F hF h0\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u22a2 Cauchy (map (fun x => x\u207b\u00b9) F)\n[PROOFSTEP]\nhave : \u2203 \u03b3\u2080 : \u0393\u2080\u02e3, \u2203 M \u2208 F, \u2200 x \u2208 M, (\u03b3\u2080 : \u0393\u2080) \u2264 v x :=\n  by\n  rcases Filter.inf_eq_bot_iff.mp h0 with \u27e8U, U_in, M, M_in, H\u27e9\n  rcases Valued.mem_nhds_zero.mp U_in with \u27e8\u03b3\u2080, hU\u27e9\n  exists \u03b3\u2080, M, M_in\n  intro x xM\n  apply le_of_not_lt _\n  intro hyp\n  have : x \u2208 U \u2229 M := \u27e8hU hyp, xM\u27e9\n  rwa [H] at this \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u22a2 \u2203 \u03b3\u2080 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n[PROOFSTEP]\nrcases Filter.inf_eq_bot_iff.mp h0 with \u27e8U, U_in, M, M_in, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u22a2 \u2203 \u03b3\u2080 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n[PROOFSTEP]\nrcases Valued.mem_nhds_zero.mp U_in with \u27e8\u03b3\u2080, hU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\n\u22a2 \u2203 \u03b3\u2080 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n[PROOFSTEP]\nexists \u03b3\u2080, M, M_in\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\n\u22a2 \u2200 (x : K), x \u2208 M \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n[PROOFSTEP]\nintro x xM\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\nx : K\nxM : x \u2208 M\n\u22a2 \u2191\u03b3\u2080 \u2264 \u2191v x\n[PROOFSTEP]\napply le_of_not_lt _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\nx : K\nxM : x \u2208 M\n\u22a2 \u00ac\u2191v x < \u2191\u03b3\u2080\n[PROOFSTEP]\nintro hyp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\nx : K\nxM : x \u2208 M\nhyp : \u2191v x < \u2191\u03b3\u2080\n\u22a2 False\n[PROOFSTEP]\nhave : x \u2208 U \u2229 M := \u27e8hU hyp, xM\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nU : Set K\nU_in : U \u2208 \ud835\udcdd 0\nM : Set K\nM_in : M \u2208 F\nH : U \u2229 M = \u2205\n\u03b3\u2080 : \u0393\u2080\u02e3\nhU : {x | \u2191v x < \u2191\u03b3\u2080} \u2286 U\nx : K\nxM : x \u2208 M\nhyp : \u2191v x < \u2191\u03b3\u2080\nthis : x \u2208 U \u2229 M\n\u22a2 False\n[PROOFSTEP]\nrwa [H] at this \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\nthis : \u2203 \u03b3\u2080 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u22a2 Cauchy (map (fun x => x\u207b\u00b9) F)\n[PROOFSTEP]\nrcases this with \u27e8\u03b3\u2080, M\u2080, M\u2080_in, H\u2080\u27e9\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : Cauchy F\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u22a2 Cauchy (map (fun x => x\u207b\u00b9) F)\n[PROOFSTEP]\nrw [Valued.cauchy_iff] at hF \u22a2\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : NeBot F \u2227 \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u22a2 NeBot (map (fun x => x\u207b\u00b9) F) \u2227\n    \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nrefine' \u27e8hF.1.map _, _\u27e9\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nhF : NeBot F \u2227 \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u22a2 \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nreplace hF := hF.2\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\nhF : \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n\u22a2 \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nintro \u03b3\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\nhF : \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nrcases hF (min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080) with \u27e8M\u2081, M\u2081_in, H\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\nhF : \u2200 (\u03b3 : \u0393\u2080\u02e3), \u2203 M, M \u2208 F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nclear hF\n[GOAL]\ncase intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 \u2203 M, M \u2208 map (fun x => x\u207b\u00b9) F \u2227 \u2200 (x : K), x \u2208 M \u2192 \u2200 (y : K), y \u2208 M \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nuse(fun x : K => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081)\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2208 map (fun x => x\u207b\u00b9) F \u2227\n    \u2200 (x : K), x \u2208 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2192 \u2200 (y : K), y \u2208 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2208 map (fun x => x\u207b\u00b9) F\n[PROOFSTEP]\nrw [mem_map]\n[GOAL]\ncase h.left\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 (fun x => x\u207b\u00b9) \u207b\u00b9' ((fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081)) \u2208 F\n[PROOFSTEP]\napply mem_of_superset (Filter.inter_mem M\u2080_in M\u2081_in)\n[GOAL]\ncase h.left\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 M\u2080 \u2229 M\u2081 \u2286 (fun x => x\u207b\u00b9) \u207b\u00b9' ((fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081))\n[PROOFSTEP]\nexact subset_preimage_image _ _\n[GOAL]\ncase h.right\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 \u2200 (x : K), x \u2208 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2192 \u2200 (y : K), y \u2208 (fun x => x\u207b\u00b9) '' (M\u2080 \u2229 M\u2081) \u2192 \u2191v (y - x) < \u2191\u03b3\n[PROOFSTEP]\nrintro _ \u27e8x, \u27e8x_in\u2080, x_in\u2081\u27e9, rfl\u27e9 _ \u27e8y, \u27e8y_in\u2080, y_in\u2081\u27e9, rfl\u27e9\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx : K\nx_in\u2080 : x \u2208 M\u2080\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\n\u22a2 \u2191v ((fun x => x\u207b\u00b9) y - (fun x => x\u207b\u00b9) x) < \u2191\u03b3\n[PROOFSTEP]\nsimp only [mem_setOf_eq]\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nH\u2081 : \u2200 (x : K), x \u2208 M\u2081 \u2192 \u2200 (y : K), y \u2208 M\u2081 \u2192 \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx : K\nx_in\u2080 : x \u2208 M\u2080\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\n\u22a2 \u2191v (y\u207b\u00b9 - x\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nspecialize H\u2081 x x_in\u2081 y y_in\u2081\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2080 : x \u2208 M\u2080\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\n\u22a2 \u2191v (y\u207b\u00b9 - x\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nreplace x_in\u2080 := H\u2080 x x_in\u2080\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\n\u22a2 \u2191v (y\u207b\u00b9 - x\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nreplace := H\u2080 y y_in\u2080\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\nH\u2080 : \u2200 (x : K), x \u2208 M\u2080 \u2192 \u2191\u03b3\u2080 \u2264 \u2191v x\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191v (y\u207b\u00b9 - x\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\nclear H\u2080\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191v (y\u207b\u00b9 - x\u207b\u00b9) < \u2191\u03b3\n[PROOFSTEP]\napply Valuation.inversion_estimate\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro.y_ne\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 x \u2260 0\n[PROOFSTEP]\nhave : (v x : \u0393\u2080) \u2260 0 := by\n  intro h\n  rw [h] at x_in\u2080 \n  simp at x_in\u2080 \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191v x \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\nh : \u2191v x = 0\n\u22a2 False\n[PROOFSTEP]\nrw [h] at x_in\u2080 \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 0\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\nh : \u2191v x = 0\n\u22a2 False\n[PROOFSTEP]\nsimp at x_in\u2080 \n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro.y_ne\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis\u271d : \u2191\u03b3\u2080 \u2264 \u2191v y\nthis : \u2191v x \u2260 0\n\u22a2 x \u2260 0\n[PROOFSTEP]\nexact (Valuation.ne_zero_iff _).mp this\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191v (y - x) < min (\u2191\u03b3 * (\u2191v x * \u2191v x)) (\u2191v x)\n[PROOFSTEP]\nrefine' lt_of_lt_of_le H\u2081 _\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080) \u2264 min (\u2191\u03b3 * (\u2191v x * \u2191v x)) (\u2191v x)\n[PROOFSTEP]\nrw [Units.min_val]\n[GOAL]\ncase h.right.intro.intro.intro.intro.intro.intro.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 min \u2191(\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u2191\u03b3\u2080 \u2264 min (\u2191\u03b3 * (\u2191v x * \u2191v x)) (\u2191v x)\n[PROOFSTEP]\napply min_le_min _ x_in\u2080\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191(\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u2264 \u2191\u03b3 * (\u2191v x * \u2191v x)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis : \u2191\u03b3\u2080 \u2264 \u2191v y\n\u22a2 \u2191(\u03b3 * (\u03b3\u2080 * \u03b3\u2080)) \u2264 \u2191\u03b3 * (\u2191v x * \u2191v x)\n[PROOFSTEP]\nhave : ((\u03b3\u2080 * \u03b3\u2080 : \u0393\u2080\u02e3) : \u0393\u2080) \u2264 v x * v x :=\n  calc\n    \u2191\u03b3\u2080 * \u2191\u03b3\u2080 \u2264 \u2191\u03b3\u2080 * v x := mul_le_mul_left' x_in\u2080 \u2191\u03b3\u2080\n    _ \u2264 _ := mul_le_mul_right' x_in\u2080 (v x)\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis\u271d : \u2191\u03b3\u2080 \u2264 \u2191v y\nthis : \u2191(\u03b3\u2080 * \u03b3\u2080) \u2264 \u2191v x * \u2191v x\n\u22a2 \u2191(\u03b3 * (\u03b3\u2080 * \u03b3\u2080)) \u2264 \u2191\u03b3 * (\u2191v x * \u2191v x)\n[PROOFSTEP]\nrw [Units.val_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nsrc\u271d : SeparatedSpace K := ValuedRing.separated\nF : Filter K\nh0 : \ud835\udcdd 0 \u2293 F = \u22a5\n\u03b3\u2080 : \u0393\u2080\u02e3\nM\u2080 : Set K\nM\u2080_in : M\u2080 \u2208 F\n\u03b3 : \u0393\u2080\u02e3\nM\u2081 : Set K\nM\u2081_in : M\u2081 \u2208 F\nx : K\nx_in\u2081 : x \u2208 M\u2081\ny : K\ny_in\u2080 : y \u2208 M\u2080\ny_in\u2081 : y \u2208 M\u2081\nH\u2081 : \u2191v (y - x) < \u2191(min (\u03b3 * \u03b3\u2080 * \u03b3\u2080) \u03b3\u2080)\nx_in\u2080 : \u2191\u03b3\u2080 \u2264 \u2191v x\nthis\u271d : \u2191\u03b3\u2080 \u2264 \u2191v y\nthis : \u2191(\u03b3\u2080 * \u03b3\u2080) \u2264 \u2191v x * \u2191v x\n\u22a2 \u2191\u03b3 * \u2191(\u03b3\u2080 * \u03b3\u2080) \u2264 \u2191\u03b3 * (\u2191v x * \u2191v x)\n[PROOFSTEP]\nexact mul_le_mul_left' this \u03b3\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 Continuous extension\n[PROOFSTEP]\nrefine' Completion.denseInducing_coe.continuous_extend _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 \u2200 (b : hat K), \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd b)) (\ud835\udcdd c)\n[PROOFSTEP]\nintro x\u2080\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nrcases eq_or_ne x\u2080 0 with (rfl | h)\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd 0)) (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e80, _\u27e9\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd 0)) (\ud835\udcdd 0)\n[PROOFSTEP]\nerw [\u2190 Completion.denseInducing_coe.toInducing.nhds_eq_comap]\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 Tendsto (\u2191v) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nexact Valued.continuous_valuation.tendsto' 0 0 (map_zero v)\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave preimage_one : v \u207b\u00b9' {(1 : \u0393\u2080)} \u2208 \ud835\udcdd (1 : K) :=\n  by\n  have : (v (1 : K) : \u0393\u2080) \u2260 0 := by\n    rw [Valuation.map_one]\n    exact zero_ne_one.symm\n  convert Valued.loc_const this\n  ext x\n  rw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\n\u22a2 \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nhave : (v (1 : K) : \u0393\u2080) \u2260 0 := by\n  rw [Valuation.map_one]\n  exact zero_ne_one.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\n\u22a2 \u2191v 1 \u2260 0\n[PROOFSTEP]\nrw [Valuation.map_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact zero_ne_one.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\nthis : \u2191v 1 \u2260 0\n\u22a2 \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\n[PROOFSTEP]\nconvert Valued.loc_const this\n[GOAL]\ncase h.e'_4\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\nthis : \u2191v 1 \u2260 0\n\u22a2 \u2191v \u207b\u00b9' {1} = {y | \u2191v y = \u2191v 1}\n[PROOFSTEP]\next x\n[GOAL]\ncase h.e'_4.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\nthis : \u2191v 1 \u2260 0\nx : K\n\u22a2 x \u2208 \u2191v \u207b\u00b9' {1} \u2194 x \u2208 {y | \u2191v y = \u2191v 1}\n[PROOFSTEP]\nrw [Valuation.map_one, mem_preimage, mem_singleton_iff, mem_setOf_eq]\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nobtain \u27e8V, V_in, hV\u27e9 : \u2203 V \u2208 \ud835\udcdd (1 : hat K), \u2200 x : K, (x : hat K) \u2208 V \u2192 (v x : \u0393\u2080) = 1 := by\n  rwa [Completion.denseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd 1 \u2227 \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n[PROOFSTEP]\nrwa [Completion.denseInducing_coe.nhds_eq_comap, mem_comap] at preimage_one \n[GOAL]\ncase inr.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave : \u2203 V' \u2208 \ud835\udcdd (1 : hat K), (0 : hat K) \u2209 V' \u2227 \u2200 (x) (_ : x \u2208 V') (y) (_ : y \u2208 V'), x * y\u207b\u00b9 \u2208 V :=\n  by\n  have : Tendsto (fun p : hat K \u00d7 hat K => p.1 * p.2\u207b\u00b9) ((\ud835\udcdd 1) \u00d7\u02e2 (\ud835\udcdd 1)) (\ud835\udcdd 1) :=\n    by\n    rw [\u2190 nhds_prod_eq]\n    conv =>\n      congr\n      rfl\n      rfl\n      rw [\u2190 one_mul (1 : hat K)]\n    refine'\n      Tendsto.mul continuous_fst.continuousAt\n        (Tendsto.comp _ continuous_snd.continuousAt)\n          -- Porting note: Added `ContinuousAt.tendsto`\n    convert (continuousAt_inv\u2080 (zero_ne_one.symm : 1 \u2260 (0 : hat K))).tendsto\n    exact inv_one.symm\n  rcases tendsto_prod_self_iff.mp this V V_in with \u27e8U, U_in, hU\u27e9\n  let hatKstar := ({0}\u1d9c : Set <| hat K)\n  have : hatKstar \u2208 \ud835\udcdd (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm\n  use U \u2229 hatKstar, Filter.inter_mem U_in this\n  constructor\n  \u00b7 rintro \u27e8_, h'\u27e9\n    rw [mem_compl_singleton_iff] at h' \n    exact h' rfl\n  \u00b7 rintro x \u27e8hx, _\u27e9 y \u27e8hy, _\u27e9\n    apply hU <;> assumption\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nhave : Tendsto (fun p : hat K \u00d7 hat K => p.1 * p.2\u207b\u00b9) ((\ud835\udcdd 1) \u00d7\u02e2 (\ud835\udcdd 1)) (\ud835\udcdd 1) :=\n  by\n  rw [\u2190 nhds_prod_eq]\n  conv =>\n    congr\n    rfl\n    rfl\n    rw [\u2190 one_mul (1 : hat K)]\n  refine'\n    Tendsto.mul continuous_fst.continuousAt\n      (Tendsto.comp _ continuous_snd.continuousAt)\n        -- Porting note: Added `ContinuousAt.tendsto`\n  convert (continuousAt_inv\u2080 (zero_ne_one.symm : 1 \u2260 (0 : hat K))).tendsto\n  exact inv_one.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 nhds_prod_eq]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd (1, 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nconv =>\n  congr\n  rfl\n  rfl\n  rw [\u2190 one_mul (1 : hat K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd (1, 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\n  congr\n  rfl\n  rfl\n  rw [\u2190 one_mul (1 : hat K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd (1, 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\n  congr\n  rfl\n  rfl\n  rw [\u2190 one_mul (1 : hat K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd (1, 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| fun p => p.fst * p.snd\u207b\u00b9\ncase l\u2081\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| \ud835\udcdd (1, 1)\ncase l\u2082\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| \ud835\udcdd 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase l\u2081\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| \ud835\udcdd (1, 1)\ncase l\u2082\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| \ud835\udcdd 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase l\u2082\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n| \ud835\udcdd 1\n[PROOFSTEP]\nrw [\u2190 one_mul (1 : hat K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd (1, 1)) (\ud835\udcdd (1 * 1))\n[PROOFSTEP]\nrefine'\n  Tendsto.mul continuous_fst.continuousAt\n    (Tendsto.comp _ continuous_snd.continuousAt)\n      -- Porting note: Added `ContinuousAt.tendsto`\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 Tendsto Inv.inv (\ud835\udcdd (1, 1).snd) (\ud835\udcdd 1)\n[PROOFSTEP]\nconvert (continuousAt_inv\u2080 (zero_ne_one.symm : 1 \u2260 (0 : hat K))).tendsto\n[GOAL]\ncase h.e'_5.h.e'_3\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\n\u22a2 1 = 1\u207b\u00b9\n[PROOFSTEP]\nexact inv_one.symm\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\n\u22a2 \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nrcases tendsto_prod_self_iff.mp this V V_in with \u27e8U, U_in, hU\u27e9\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\n\u22a2 \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nlet hatKstar := ({0}\u1d9c : Set <| hat K)\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\n\u22a2 \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nhave : hatKstar \u2208 \ud835\udcdd (1 : hat K) := compl_singleton_mem_nhds zero_ne_one.symm\n[GOAL]\ncase intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\n\u22a2 \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nuse U \u2229 hatKstar, Filter.inter_mem U_in this\n[GOAL]\ncase right\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\n\u22a2 \u00ac0 \u2208 U \u2229 hatKstar \u2227 \u2200 (x : hat K), x \u2208 U \u2229 hatKstar \u2192 \u2200 (y : hat K), y \u2208 U \u2229 hatKstar \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\n\u22a2 \u00ac0 \u2208 U \u2229 hatKstar\n[PROOFSTEP]\nrintro \u27e8_, h'\u27e9\n[GOAL]\ncase right.left.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\nleft\u271d : 0 \u2208 U\nh' : 0 \u2208 hatKstar\n\u22a2 False\n[PROOFSTEP]\nrw [mem_compl_singleton_iff] at h' \n[GOAL]\ncase right.left.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\nleft\u271d : 0 \u2208 U\nh' : 0 \u2260 0\n\u22a2 False\n[PROOFSTEP]\nexact h' rfl\n[GOAL]\ncase right.right\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\n\u22a2 \u2200 (x : hat K), x \u2208 U \u2229 hatKstar \u2192 \u2200 (y : hat K), y \u2208 U \u2229 hatKstar \u2192 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nrintro x \u27e8hx, _\u27e9 y \u27e8hy, _\u27e9\n[GOAL]\ncase right.right.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\nx : hat K\nhx : x \u2208 U\nright\u271d\u00b9 : x \u2208 hatKstar\ny : hat K\nhy : y \u2208 U\nright\u271d : y \u2208 hatKstar\n\u22a2 x * y\u207b\u00b9 \u2208 V\n[PROOFSTEP]\napply hU\n[GOAL]\ncase right.right.intro.intro.a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\nx : hat K\nhx : x \u2208 U\nright\u271d\u00b9 : x \u2208 hatKstar\ny : hat K\nhy : y \u2208 U\nright\u271d : y \u2208 hatKstar\n\u22a2 x \u2208 U\n[PROOFSTEP]\nassumption\n[GOAL]\ncase right.right.intro.intro.a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis\u271d : Tendsto (fun p => p.fst * p.snd\u207b\u00b9) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1)\nU : Set (hat K)\nU_in : U \u2208 \ud835\udcdd 1\nhU : \u2200 (x x' : hat K), x \u2208 U \u2192 x' \u2208 U \u2192 (x, x').fst * (x, x').snd\u207b\u00b9 \u2208 V\nhatKstar : Set (hat K) := {0}\u1d9c\nthis : hatKstar \u2208 \ud835\udcdd 1\nx : hat K\nhx : x \u2208 U\nright\u271d\u00b9 : x \u2208 hatKstar\ny : hat K\nhy : y \u2208 U\nright\u271d : y \u2208 hatKstar\n\u22a2 y \u2208 U\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nthis : \u2203 V', V' \u2208 \ud835\udcdd 1 \u2227 \u00ac0 \u2208 V' \u2227 \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nrcases this with \u27e8V', V'_in, zeroV', hV'\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave nhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080 :=\n  by\n  have l : Function.LeftInverse (fun x : hat K => x * x\u2080\u207b\u00b9) fun x : hat K => x * x\u2080 :=\n    by\n    intro x\n    simp only [mul_assoc, mul_inv_cancel h, mul_one]\n  have r : Function.RightInverse (fun x : hat K => x * x\u2080\u207b\u00b9) fun x : hat K => x * x\u2080 :=\n    by\n    intro x\n    simp only [mul_assoc, inv_mul_cancel h, mul_one]\n  have c : Continuous fun x : hat K => x * x\u2080\u207b\u00b9 := continuous_id.mul continuous_const\n  rw [image_eq_preimage_of_inverse l r]\n  rw [\u2190 mul_inv_cancel h] at V'_in \n  exact c.continuousAt V'_in\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n\u22a2 (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nhave l : Function.LeftInverse (fun x : hat K => x * x\u2080\u207b\u00b9) fun x : hat K => x * x\u2080 :=\n  by\n  intro x\n  simp only [mul_assoc, mul_inv_cancel h, mul_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\n\u22a2 Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nx : hat K\n\u22a2 (fun x => x * x\u2080\u207b\u00b9) ((fun x => x * x\u2080) x) = x\n[PROOFSTEP]\nsimp only [mul_assoc, mul_inv_cancel h, mul_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\n\u22a2 (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nhave r : Function.RightInverse (fun x : hat K => x * x\u2080\u207b\u00b9) fun x : hat K => x * x\u2080 :=\n  by\n  intro x\n  simp only [mul_assoc, inv_mul_cancel h, mul_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\n\u22a2 Function.RightInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nx : hat K\n\u22a2 (fun x => x * x\u2080) ((fun x => x * x\u2080\u207b\u00b9) x) = x\n[PROOFSTEP]\nsimp only [mul_assoc, inv_mul_cancel h, mul_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nr : Function.RightInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\n\u22a2 (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nhave c : Continuous fun x : hat K => x * x\u2080\u207b\u00b9 := continuous_id.mul continuous_const\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nr : Function.RightInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nc : Continuous fun x => x * x\u2080\u207b\u00b9\n\u22a2 (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nrw [image_eq_preimage_of_inverse l r]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nr : Function.RightInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nc : Continuous fun x => x * x\u2080\u207b\u00b9\n\u22a2 (fun x => x * x\u2080\u207b\u00b9) \u207b\u00b9' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nrw [\u2190 mul_inv_cancel h] at V'_in \n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd (x\u2080 * x\u2080\u207b\u00b9)\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nl : Function.LeftInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nr : Function.RightInverse (fun x => x * x\u2080\u207b\u00b9) fun x => x * x\u2080\nc : Continuous fun x => x * x\u2080\u207b\u00b9\n\u22a2 (fun x => x * x\u2080\u207b\u00b9) \u207b\u00b9' V' \u2208 \ud835\udcdd x\u2080\n[PROOFSTEP]\nexact c.continuousAt V'_in\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave : \u2203 z\u2080 : K, \u2203 y\u2080 \u2208 V', \u2191z\u2080 = y\u2080 * x\u2080 \u2227 z\u2080 \u2260 0 :=\n  by\n  rcases Completion.denseRange_coe.mem_nhds nhds_right with \u27e8z\u2080, y\u2080, y\u2080_in, H : y\u2080 * x\u2080 = z\u2080\u27e9\n  refine' \u27e8z\u2080, y\u2080, y\u2080_in, \u27e8H.symm, _\u27e9\u27e9\n  rintro rfl\n  exact mul_ne_zero (ne_of_mem_of_not_mem y\u2080_in zeroV') h H\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\n\u22a2 \u2203 z\u2080 y\u2080, y\u2080 \u2208 V' \u2227 \u2191K z\u2080 = y\u2080 * x\u2080 \u2227 z\u2080 \u2260 0\n[PROOFSTEP]\nrcases Completion.denseRange_coe.mem_nhds nhds_right with \u27e8z\u2080, y\u2080, y\u2080_in, H : y\u2080 * x\u2080 = z\u2080\u27e9\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nH : y\u2080 * x\u2080 = \u2191K z\u2080\n\u22a2 \u2203 z\u2080 y\u2080, y\u2080 \u2208 V' \u2227 \u2191K z\u2080 = y\u2080 * x\u2080 \u2227 z\u2080 \u2260 0\n[PROOFSTEP]\nrefine' \u27e8z\u2080, y\u2080, y\u2080_in, \u27e8H.symm, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nH : y\u2080 * x\u2080 = \u2191K z\u2080\n\u22a2 z\u2080 \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nH : y\u2080 * x\u2080 = \u2191K 0\n\u22a2 False\n[PROOFSTEP]\nexact mul_ne_zero (ne_of_mem_of_not_mem y\u2080_in zeroV') h H\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nthis : \u2203 z\u2080 y\u2080, y\u2080 \u2208 V' \u2227 \u2191K z\u2080 = y\u2080 * x\u2080 \u2227 z\u2080 \u2260 0\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nrcases this with \u27e8z\u2080, y\u2080, y\u2080_in, hz\u2080, z\u2080_ne\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nhave vz\u2080_ne : (v z\u2080 : \u0393\u2080) \u2260 0 := by rwa [Valuation.ne_zero_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\n\u22a2 \u2191v z\u2080 \u2260 0\n[PROOFSTEP]\nrwa [Valuation.ne_zero_iff]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\n\u22a2 \u2203 c, Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd c)\n[PROOFSTEP]\nrefine' \u27e8v z\u2080, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\n\u22a2 Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd x\u2080)) (\ud835\udcdd (\u2191v z\u2080))\n[PROOFSTEP]\nrw [WithZeroTopology.tendsto_of_ne_zero vz\u2080_ne, eventually_comap]\n[GOAL]\ncase inr.intro.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\n\u22a2 \u2200\u1da0 (b : hat K) in \ud835\udcdd x\u2080, \u2200 (a : K), \u2191K a = b \u2192 \u2191v a = \u2191v z\u2080\n[PROOFSTEP]\nfilter_upwards [nhds_right] with x x_in a ha\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\nx : hat K\nx_in : x \u2208 (fun x => x * x\u2080) '' V'\na : K\nha : \u2191K a = x\n\u22a2 \u2191v a = \u2191v z\u2080\n[PROOFSTEP]\nrcases x_in with \u27e8y, y_in, rfl\u27e9\n[GOAL]\ncase h.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\n\u22a2 \u2191v a = \u2191v z\u2080\n[PROOFSTEP]\nhave : (v (a * z\u2080\u207b\u00b9) : \u0393\u2080) = 1 := by\n  apply hV\n  have : (z\u2080\u207b\u00b9 : K) = (z\u2080 : hat K)\u207b\u00b9 := map_inv\u2080 (Completion.coeRingHom : K \u2192+* hat K) z\u2080\n  rw [Completion.coe_mul, this, ha, hz\u2080, mul_inv, mul_comm y\u2080\u207b\u00b9, \u2190 mul_assoc, mul_assoc y, mul_inv_cancel h, mul_one]\n  solve_by_elim\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\n\u22a2 \u2191v (a * z\u2080\u207b\u00b9) = 1\n[PROOFSTEP]\napply hV\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\n\u22a2 \u2191K (a * z\u2080\u207b\u00b9) \u2208 V\n[PROOFSTEP]\nhave : (z\u2080\u207b\u00b9 : K) = (z\u2080 : hat K)\u207b\u00b9 := map_inv\u2080 (Completion.coeRingHom : K \u2192+* hat K) z\u2080\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\nthis : \u2191K z\u2080\u207b\u00b9 = (\u2191K z\u2080)\u207b\u00b9\n\u22a2 \u2191K (a * z\u2080\u207b\u00b9) \u2208 V\n[PROOFSTEP]\nrw [Completion.coe_mul, this, ha, hz\u2080, mul_inv, mul_comm y\u2080\u207b\u00b9, \u2190 mul_assoc, mul_assoc y, mul_inv_cancel h, mul_one]\n[GOAL]\ncase a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\nthis : \u2191K z\u2080\u207b\u00b9 = (\u2191K z\u2080)\u207b\u00b9\n\u22a2 y * y\u2080\u207b\u00b9 \u2208 V\n[PROOFSTEP]\nsolve_by_elim\n[GOAL]\ncase h.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\nthis : \u2191v (a * z\u2080\u207b\u00b9) = 1\n\u22a2 \u2191v a = \u2191v z\u2080\n[PROOFSTEP]\ncalc\n  v a = v (a * z\u2080\u207b\u00b9 * z\u2080) := by rw [mul_assoc, inv_mul_cancel z\u2080_ne, mul_one]\n  _ = v (a * z\u2080\u207b\u00b9) * v z\u2080 := (Valuation.map_mul _ _ _)\n  _ = v z\u2080 := by rw [this, one_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\nthis : \u2191v (a * z\u2080\u207b\u00b9) = 1\n\u22a2 \u2191v a = \u2191v (a * z\u2080\u207b\u00b9 * z\u2080)\n[PROOFSTEP]\nrw [mul_assoc, inv_mul_cancel z\u2080_ne, mul_one]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u2080 : hat K\nh : x\u2080 \u2260 0\npreimage_one : \u2191v \u207b\u00b9' {1} \u2208 \ud835\udcdd 1\nV : Set (hat K)\nV_in : V \u2208 \ud835\udcdd 1\nhV : \u2200 (x : K), \u2191K x \u2208 V \u2192 \u2191v x = 1\nV' : Set (hat K)\nV'_in : V' \u2208 \ud835\udcdd 1\nzeroV' : \u00ac0 \u2208 V'\nhV' : \u2200 (x : hat K), x \u2208 V' \u2192 \u2200 (y : hat K), y \u2208 V' \u2192 x * y\u207b\u00b9 \u2208 V\nnhds_right : (fun x => x * x\u2080) '' V' \u2208 \ud835\udcdd x\u2080\nz\u2080 : K\ny\u2080 : hat K\ny\u2080_in : y\u2080 \u2208 V'\nhz\u2080 : \u2191K z\u2080 = y\u2080 * x\u2080\nz\u2080_ne : z\u2080 \u2260 0\nvz\u2080_ne : \u2191v z\u2080 \u2260 0\na : K\ny : hat K\ny_in : y \u2208 V'\nha : \u2191K a = (fun x => x * x\u2080) y\nthis : \u2191v (a * z\u2080\u207b\u00b9) = 1\n\u22a2 \u2191v (a * z\u2080\u207b\u00b9) * \u2191v z\u2080 = \u2191v z\u2080\n[PROOFSTEP]\nrw [this, one_mul]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx : K\n\u22a2 extension (\u2191K x) = \u2191v x\n[PROOFSTEP]\nrefine' Completion.denseInducing_coe.extend_eq_of_tendsto _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx : K\n\u22a2 Tendsto (\u2191v) (Filter.comap (\u2191K) (\ud835\udcdd (\u2191K x))) (\ud835\udcdd (\u2191v x))\n[PROOFSTEP]\nrw [\u2190 Completion.denseInducing_coe.nhds_eq_comap]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx : K\n\u22a2 Tendsto (\u2191v) (\ud835\udcdd x) (\ud835\udcdd (\u2191v x))\n[PROOFSTEP]\nexact Valued.continuous_valuation.continuousAt\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 extension 0 = 0\n[PROOFSTEP]\nrw [\u2190 v.map_zero (R := K), \u2190 Valued.extension_extends (0 : K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 extension 0 = extension (\u2191K 0)\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 extension 1 = 1\n[PROOFSTEP]\nrw [\u2190 Completion.coe_one, Valued.extension_extends (1 : K)]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u22a2 \u2191v 1 = 1\n[PROOFSTEP]\nexact Valuation.map_one _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } (x * y) =\n    ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } x *\n      ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } y\n[PROOFSTEP]\napply Completion.induction_on\u2082 x y (p := fun x y => extension (x * y) = extension x * extension y)\n[GOAL]\ncase hp\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 IsClosed {x | extension (x.fst * x.snd) = extension x.fst * extension x.snd}\n[PROOFSTEP]\nhave c1 : Continuous fun x : hat K \u00d7 hat K => Valued.extension (x.1 * x.2) :=\n  Valued.continuous_extension.comp (continuous_fst.mul continuous_snd)\n[GOAL]\ncase hp\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\nc1 : Continuous fun x => extension (x.fst * x.snd)\n\u22a2 IsClosed {x | extension (x.fst * x.snd) = extension x.fst * extension x.snd}\n[PROOFSTEP]\nhave c2 : Continuous fun x : hat K \u00d7 hat K => Valued.extension x.1 * Valued.extension x.2 :=\n  (Valued.continuous_extension.comp continuous_fst).mul (Valued.continuous_extension.comp continuous_snd)\n[GOAL]\ncase hp\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\nc1 : Continuous fun x => extension (x.fst * x.snd)\nc2 : Continuous fun x => extension x.fst * extension x.snd\n\u22a2 IsClosed {x | extension (x.fst * x.snd) = extension x.fst * extension x.snd}\n[PROOFSTEP]\nexact isClosed_eq c1 c2\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 \u2200 (a b : K), extension (\u2191K a * \u2191K b) = extension (\u2191K a) * extension (\u2191K b)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u271d y\u271d : hat K\nx y : K\n\u22a2 extension (\u2191K x * \u2191K y) = extension (\u2191K x) * extension (\u2191K y)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u271d y\u271d : hat K\nx y : K\n\u22a2 \u2191v (x * y) = \u2191v x * \u2191v y\n[PROOFSTEP]\nexact Valuation.map_mul _ _ _\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 ZeroHom.toFun\n      (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n          map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n          map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n      (x + y) \u2264\n    max\n      (ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        x)\n      (ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        y)\n[PROOFSTEP]\nrw [le_max_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        (x + y) \u2264\n      ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        x \u2228\n    ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        (x + y) \u2264\n      ZeroHom.toFun\n        (\u2191{ toZeroHom := { toFun := extension, map_zero' := (_ : extension 0 = 0) },\n            map_one' := (_ : ZeroHom.toFun { toFun := extension, map_zero' := (_ : extension 0 = 0) } 1 = 1),\n            map_mul' := (_ : \u2200 (x y : hat K), extension (x * y) = extension x * extension y) })\n        y\n[PROOFSTEP]\napply Completion.induction_on\u2082 x y (p := fun x y => extension (x + y) \u2264 extension x \u2228 extension (x + y) \u2264 extension y)\n[GOAL]\ncase hp\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 IsClosed {x | extension (x.fst + x.snd) \u2264 extension x.fst \u2228 extension (x.fst + x.snd) \u2264 extension x.snd}\n[PROOFSTEP]\nhave cont : Continuous (Valued.extension : hat K \u2192 \u0393\u2080) := Valued.continuous_extension\n[GOAL]\ncase hp\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\ncont : Continuous extension\n\u22a2 IsClosed {x | extension (x.fst + x.snd) \u2264 extension x.fst \u2228 extension (x.fst + x.snd) \u2264 extension x.snd}\n[PROOFSTEP]\nexact\n  (isClosed_le (cont.comp continuous_add) <| cont.comp continuous_fst).union\n    (isClosed_le (cont.comp continuous_add) <| cont.comp continuous_snd)\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx y : hat K\n\u22a2 \u2200 (a b : K), extension (\u2191K a + \u2191K b) \u2264 extension (\u2191K a) \u2228 extension (\u2191K a + \u2191K b) \u2264 extension (\u2191K b)\n[PROOFSTEP]\nintro x y\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u271d y\u271d : hat K\nx y : K\n\u22a2 extension (\u2191K x + \u2191K y) \u2264 extension (\u2191K x) \u2228 extension (\u2191K x + \u2191K y) \u2264 extension (\u2191K y)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u271d y\u271d : hat K\nx y : K\n\u22a2 \u2191v (x + y) \u2264 \u2191v x \u2228 \u2191v (x + y) \u2264 \u2191v y\n[PROOFSTEP]\nrw [\u2190 le_max_iff]\n[GOAL]\ncase ih\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\nx\u271d y\u271d : hat K\nx y : K\n\u22a2 \u2191v (x + y) \u2264 max (\u2191v x) (\u2191v y)\n[PROOFSTEP]\nexact v.map_add x y\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) = {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\nlet \u03b3\u2080 := extensionValuation x\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\nsuffices \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure ((\u2191) '' {x : K | v x < (\u03b3 : \u0393\u2080)}) \u2194 \u03b3\u2080 < (\u03b3 : \u0393\u2080))\n  by\n  cases' eq_or_ne \u03b3\u2080 0 with h h\n  \u00b7 simp only [h, (Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt, iff_true_iff]\n    apply subset_closure\n    exact \u27e80, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]; rfl\u27e9\n  \u00b7 exact this h\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\ncases' eq_or_ne \u03b3\u2080 0 with h h\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\nsimp only [h, (Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt, iff_true_iff]\n[GOAL]\ncase inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3})\n[PROOFSTEP]\napply subset_closure\n[GOAL]\ncase inl.a\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 \u2191K '' {x | \u2191v x < \u2191\u03b3}\n[PROOFSTEP]\nexact \u27e80, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]; rfl\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 0 \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K 0 = 0\n[PROOFSTEP]\nsimp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 = 0\n\u22a2 \u2191K 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nthis : \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 x \u2208 {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\n\u22a2 \u03b3\u2080 \u2260 0 \u2192 (x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nhave h\u03b3\u2080 : extension \u207b\u00b9' { \u03b3\u2080 } \u2208 \ud835\udcdd x :=\n  continuous_extension.continuousAt.preimage_mem_nhds (WithZeroTopology.singleton_mem_nhds_of_ne_zero h)\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 x \u2208 closure (\u2191K '' {x | \u2191v x < \u2191\u03b3}) \u2194 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nrw [mem_closure_iff_nhds']\n[GOAL]\ncase h\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\n\u22a2 (\u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t) \u2194 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nrefine' \u27e8fun hx => _, fun hx s hs => _\u27e9\n[GOAL]\ncase h.refine'_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\n\u22a2 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nobtain \u27e8\u27e8-, y, hy\u2081 : v y < (\u03b3 : \u0393\u2080), rfl\u27e9, hy\u2082\u27e9 := hx _ h\u03b3\u2080\n[GOAL]\ncase h.refine'_1.intro.mk.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nreplace hy\u2082 : v y = \u03b3\u2080\n[GOAL]\ncase hy\u2082\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191{ val := \u2191K y, property := (_ : \u2203 a, a \u2208 {x | \u2191v x < \u2191\u03b3} \u2227 \u2191K a = \u2191K y) } \u2208 extension \u207b\u00b9' {\u03b3\u2080}\n\u22a2 \u2191v y = \u03b3\u2080\n[PROOFSTEP]\nsimpa using hy\u2082\n[GOAL]\ncase h.refine'_1.intro.mk.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u2200 (t : Set (hat K)), t \u2208 \ud835\udcdd x \u2192 \u2203 y, \u2191y \u2208 t\ny : K\nhy\u2081 : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191v y = \u03b3\u2080\n\u22a2 \u03b3\u2080 < \u2191\u03b3\n[PROOFSTEP]\nrwa [\u2190 hy\u2082]\n[GOAL]\ncase h.refine'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\n\u22a2 \u2203 y, \u2191y \u2208 s\n[PROOFSTEP]\nobtain \u27e8y, hy\u2081, hy\u2082\u27e9 := Completion.denseRange_coe.mem_nhds (inter_mem h\u03b3\u2080 hs)\n[GOAL]\ncase h.refine'_2.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2203 y, \u2191y \u2208 s\n[PROOFSTEP]\nreplace hy\u2081 : v y = \u03b3\u2080\n[GOAL]\ncase hy\u2081\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2081 : \u2191K y \u2208 extension \u207b\u00b9' {\u03b3\u2080}\nhy\u2082 : \u2191K y \u2208 s\n\u22a2 \u2191v y = \u03b3\u2080\n[PROOFSTEP]\nsimpa using hy\u2081\n[GOAL]\ncase h.refine'_2.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\nhx : \u03b3\u2080 < \u2191\u03b3\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s\n[PROOFSTEP]\nrw [\u2190 hy\u2081] at hx \n[GOAL]\ncase h.refine'_2.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\n\u03b3 : \u0393\u2080\u02e3\nx : hat K\n\u03b3\u2080 : (fun x => \u0393\u2080) x := \u2191extensionValuation x\nh : \u03b3\u2080 \u2260 0\nh\u03b3\u2080 : extension \u207b\u00b9' {\u03b3\u2080} \u2208 \ud835\udcdd x\ns : Set (hat K)\nhs : s \u2208 \ud835\udcdd x\ny : K\nhx : \u2191v y < \u2191\u03b3\nhy\u2082 : \u2191K y \u2208 s\nhy\u2081 : \u2191v y = \u03b3\u2080\n\u22a2 \u2203 y, \u2191y \u2208 s\n[PROOFSTEP]\nexact \u27e8\u27e8y, \u27e8y, hx, rfl\u27e9\u27e9, hy\u2082\u27e9\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 \u03b3, {x | \u2191extensionValuation x < \u2191\u03b3} \u2286 s\n[PROOFSTEP]\nsuffices HasBasis (\ud835\udcdd (0 : hat K)) (fun _ => True) fun \u03b3 : \u0393\u2080\u02e3 => {x | extensionValuation x < \u03b3}\n  by\n  rw [this.mem_iff]\n  exact exists_congr fun \u03b3 => by simp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\nthis : HasBasis (\ud835\udcdd 0) (fun x => True) fun \u03b3 => {x | \u2191extensionValuation x < \u2191\u03b3}\n\u22a2 s \u2208 \ud835\udcdd 0 \u2194 \u2203 \u03b3, {x | \u2191extensionValuation x < \u2191\u03b3} \u2286 s\n[PROOFSTEP]\nrw [this.mem_iff]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\nthis : HasBasis (\ud835\udcdd 0) (fun x => True) fun \u03b3 => {x | \u2191extensionValuation x < \u2191\u03b3}\n\u22a2 (\u2203 i, True \u2227 {x | \u2191extensionValuation x < \u2191i} \u2286 s) \u2194 \u2203 \u03b3, {x | \u2191extensionValuation x < \u2191\u03b3} \u2286 s\n[PROOFSTEP]\nexact exists_congr fun \u03b3 => by simp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\nthis : HasBasis (\ud835\udcdd 0) (fun x => True) fun \u03b3 => {x | \u2191extensionValuation x < \u2191\u03b3}\n\u03b3 : \u0393\u2080\u02e3\n\u22a2 True \u2227 {x | \u2191extensionValuation x < \u2191\u03b3} \u2286 s \u2194 {x | \u2191extensionValuation x < \u2191\u03b3} \u2286 s\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\n\u22a2 HasBasis (\ud835\udcdd 0) (fun x => True) fun \u03b3 => {x | \u2191extensionValuation x < \u2191\u03b3}\n[PROOFSTEP]\nsimp_rw [\u2190 closure_coe_completion_v_lt]\n[GOAL]\nK : Type u_1\ninst\u271d\u00b9 : Field K\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nhv : Valued K \u0393\u2080\ns : Set (hat K)\n\u22a2 HasBasis (\ud835\udcdd 0) (fun x => True) fun \u03b3 => closure (\u2191K '' {x | \u2191v x < \u2191\u03b3})\n[PROOFSTEP]\nexact (hasBasis_nhds_zero K \u0393\u2080).hasBasis_of_denseInducing Completion.denseInducing_coe\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.ValuedField", "llama_tokens": 52847, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835452961425, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.5261663333427182}}
{"text": "[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nP : Matrix (Fin (m + 1)) (Fin n) \u03b1 \u2192 Prop\n\u22a2 Forall P \u2194 \u2200 (x : Matrix (Fin (m + 1)) (Fin n) \u03b1), P x\n[PROOFSTEP]\nsimp only [Forall, FinVec.forall_iff, forall_iff]\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nP : Matrix (Fin (m + 1)) (Fin n) \u03b1 \u2192 Prop\n\u22a2 (\u2200 (x : Fin n \u2192 \u03b1) (x_1 : Matrix (Fin (Nat.add m 0)) (Fin n) \u03b1), P (\u2191of (vecCons x x_1))) \u2194\n    \u2200 (x : Matrix (Fin (m + 1)) (Fin n) \u03b1), P x\n[PROOFSTEP]\nexact Iff.symm Fin.forall_fin_succ_pi\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nP : Matrix (Fin (m + 1)) (Fin n) \u03b1 \u2192 Prop\n\u22a2 Exists P \u2194 \u2203 x, P x\n[PROOFSTEP]\nsimp only [Exists, FinVec.exists_iff, exists_iff]\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nP : Matrix (Fin (m + 1)) (Fin n) \u03b1 \u2192 Prop\n\u22a2 (\u2203 x x_1, P (\u2191of (vecCons x x_1))) \u2194 \u2203 x, P x\n[PROOFSTEP]\nexact Iff.symm Fin.exists_fin_succ_pi\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni : Fin (n + 1)\nj : Fin m\n\u22a2 transpose\u1d63 A i j = A\u1d40 i j\n[PROOFSTEP]\nsimp_rw [transpose\u1d63, transpose\u1d63_eq]\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni : Fin (n + 1)\nj : Fin m\n\u22a2 \u2191of (vecCons (FinVec.map (fun v => v 0) A) (submatrix A id Fin.succ)\u1d40) i j = A\u1d40 i j\n[PROOFSTEP]\nrefine' i.cases _ fun i => _\n[GOAL]\ncase refine'_1\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni : Fin (n + 1)\nj : Fin m\n\u22a2 \u2191of (vecCons (FinVec.map (fun v => v 0) A) (submatrix A id Fin.succ)\u1d40) 0 j = A\u1d40 0 j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase refine'_1\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni : Fin (n + 1)\nj : Fin m\n\u22a2 FinVec.map (fun v => v 0) A j = A j 0\n[PROOFSTEP]\nrw [FinVec.map_eq, Function.comp_apply]\n[GOAL]\ncase refine'_2\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni\u271d : Fin (n + 1)\nj : Fin m\ni : Fin n\n\u22a2 \u2191of (vecCons (FinVec.map (fun v => v 0) A) (submatrix A id Fin.succ)\u1d40) (Fin.succ i) j = A\u1d40 (Fin.succ i) j\n[PROOFSTEP]\nsimp only [of_apply, Matrix.cons_val_succ]\n[GOAL]\ncase refine'_2\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin (n + 1)) \u03b1\ni\u271d : Fin (n + 1)\nj : Fin m\ni : Fin n\n\u22a2 (submatrix A id Fin.succ)\u1d40 i j = A\u1d40 (Fin.succ i) j\n[PROOFSTEP]\nrfl\n[GOAL]\nl m\u271d n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : AddCommMonoid \u03b1\nm : \u2115\na b : Fin m \u2192 \u03b1\n\u22a2 dotProduct\u1d63 a b = a \u2b1d\u1d65 b\n[PROOFSTEP]\nsimp_rw [dotProduct\u1d63, dotProduct, FinVec.sum_eq, FinVec.seq_eq, FinVec.map_eq, Function.comp_apply]\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : AddCommMonoid \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\nB : Matrix (Fin m) (Fin n) \u03b1\n\u22a2 mul\u1d63 A B = A * B\n[PROOFSTEP]\nsimp [mul\u1d63, Function.comp, Matrix.transpose]\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ninst\u271d : AddCommMonoid \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\nB : Matrix (Fin m) (Fin n) \u03b1\n\u22a2 (\u2191of fun x x_1 => A x \u2b1d\u1d65 \u2191of (fun x y => B y x) x_1) = A * B\n[PROOFSTEP]\nrfl\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\nv : Fin m \u2192 \u03b1\n\u22a2 mulVec\u1d63 A v = mulVec A v\n[PROOFSTEP]\nsimp [mulVec\u1d63, Function.comp]\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\nv : Fin m \u2192 \u03b1\n\u22a2 (fun x => A x \u2b1d\u1d65 v) = mulVec A v\n[PROOFSTEP]\nrfl\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nv : Fin l \u2192 \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\n\u22a2 vecMul\u1d63 v A = vecMul v A\n[PROOFSTEP]\nsimp [vecMul\u1d63, Function.comp]\n[GOAL]\nl m n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalNonAssocSemiring \u03b1\nv : Fin l \u2192 \u03b1\nA : Matrix (Fin l) (Fin m) \u03b1\n\u22a2 (fun x => v \u2b1d\u1d65 A\u1d40 x) = vecMul v A\n[PROOFSTEP]\nrfl\n[GOAL]\nl m\u271d n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nA : Matrix (Fin m) (Fin n) \u03b1\n\u22a2 etaExpand A = A\n[PROOFSTEP]\nsimp_rw [etaExpand, FinVec.etaExpand_eq, Matrix.of, Equiv.refl_apply]\n", "meta": {"mathlib_filename": "Mathlib.Data.Matrix.Reflection", "llama_tokens": 2101, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649232, "lm_q2_score": 0.6791787056691698, "lm_q1q2_score": 0.5260805402899779}}
{"text": "[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V = \u2191n\n\u22a2 finrank K V = n\n[PROOFSTEP]\napply_fun toNat at h \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2191toNat (Module.rank K V) = \u2191toNat \u2191n\n\u22a2 finrank K V = n\n[PROOFSTEP]\nrw [toNat_cast] at h \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2191toNat (Module.rank K V) = n\n\u22a2 finrank K V = n\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V \u2264 \u2191n\n\u22a2 finrank K V \u2264 n\n[PROOFSTEP]\nrwa [\u2190 Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_cast] at h \n[GOAL]\ncase hc\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V \u2264 \u2191n\n\u22a2 Module.rank K V < \u2135\u2080\n[PROOFSTEP]\nexact h.trans_lt (nat_lt_aleph0 n)\n[GOAL]\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V \u2264 \u2191n\n\u22a2 \u2191n < \u2135\u2080\n[PROOFSTEP]\nexact nat_lt_aleph0 n\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V < \u2191n\n\u22a2 finrank K V < n\n[PROOFSTEP]\nrwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_cast] at h \n[GOAL]\ncase hc\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V < \u2191n\n\u22a2 Module.rank K V < \u2135\u2080\n[PROOFSTEP]\nexact h.trans (nat_lt_aleph0 n)\n[GOAL]\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : Module.rank K V < \u2191n\n\u22a2 \u2191n < \u2135\u2080\n[PROOFSTEP]\nexact nat_lt_aleph0 n\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 \u2191n < Module.rank K V\n[PROOFSTEP]\nrwa [\u2190 Cardinal.toNat_lt_iff_lt_of_lt_aleph0, toNat_cast]\n[GOAL]\ncase hc\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 \u2191n < \u2135\u2080\n[PROOFSTEP]\nexact nat_lt_aleph0 n\n[GOAL]\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : n < finrank K V\n\u22a2 Module.rank K V < \u2135\u2080\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 finrank K V \u2264 n\n[PROOFSTEP]\nrw [finrank, Cardinal.toNat_apply_of_aleph0_le h]\n[GOAL]\ncase hd\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nn : \u2115\nh : \u2135\u2080 \u2264 Module.rank K V\n\u22a2 0 \u2264 n\n[PROOFSTEP]\nexact n.zero_le\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nh : 1 < finrank K V\n\u22a2 1 < Module.rank K V\n[PROOFSTEP]\nsimpa using lt_rank_of_lt_finrank h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nh : lift (Module.rank K V) \u2264 lift (Module.rank K V\u2082)\nh' : Module.rank K V\u2082 < \u2135\u2080\n\u22a2 finrank K V \u2264 finrank K V\u2082\n[PROOFSTEP]\nsimpa only [toNat_lift] using toNat_le_of_le_of_lt_aleph0 (lift_lt_aleph0.mpr h') h\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : Ring K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : Nontrivial K\ninst\u271d : NoZeroSMulDivisors K V\nn : \u2115\nhn : finrank K V = Nat.succ n\n\u22a2 0 < finrank (?m.108424 hn) V\n[PROOFSTEP]\nrw [hn]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : Ring K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : Nontrivial K\ninst\u271d : NoZeroSMulDivisors K V\nn : \u2115\nhn : finrank K V = Nat.succ n\n\u22a2 0 < Nat.succ n\n[PROOFSTEP]\nexact n.succ_pos\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : Ring K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : Nontrivial K\ninst\u271d : NoZeroSMulDivisors K V\nh : Subsingleton V\n\u22a2 finrank K V = 0\n[PROOFSTEP]\nby_contra h0\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2076 : Ring K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : Nontrivial K\ninst\u271d : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : \u00acfinrank K V = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8x, y, hxy\u27e9 := nontrivial_of_finrank_pos (Nat.pos_of_ne_zero h0)\n[GOAL]\ncase mk.intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2076 : Ring K\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b3 : AddCommGroup V\u2082\ninst\u271d\u00b2 : Module K V\u2082\ninst\u271d\u00b9 : Nontrivial K\ninst\u271d : NoZeroSMulDivisors K V\nh : Subsingleton V\nh0 : \u00acfinrank K V = 0\nx y : V\nhxy : x \u2260 y\n\u22a2 False\n[PROOFSTEP]\nexact hxy (Subsingleton.elim _ _)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : Ring K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : StrongRankCondition K\n\u03b9 : Type w\nb : Finset \u03b9\nh : Basis { x // x \u2208 b } K V\n\u22a2 finrank K V = Finset.card b\n[PROOFSTEP]\nrw [finrank_eq_card_basis h, Fintype.card_coe]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : Ring K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : StrongRankCondition K\n\u22a2 Module.rank K K = \u21911\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2075 : Ring K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : StrongRankCondition K\nn : \u2115\n\u22a2 finrank K (Fin n \u2192 K) = n\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : StrongRankCondition K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Free K V\nh : \u2200 (s : Set V), Basis (\u2191s) K V \u2192 \u00acSet.Finite s\n\u22a2 finrank K V = 0\n[PROOFSTEP]\nobtain \u27e8_, \u27e8b\u27e9\u27e9 := (Module.free_iff_set K V).mp \u2039_\u203a\n[GOAL]\ncase intro.intro\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : StrongRankCondition K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Free K V\nh : \u2200 (s : Set V), Basis (\u2191s) K V \u2192 \u00acSet.Finite s\nw\u271d : Set V\nb : Basis (\u2191w\u271d) K V\n\u22a2 finrank K V = 0\n[PROOFSTEP]\nexact dif_neg fun rank_lt => h _ b (b.finite_index_of_rank_lt_aleph0 rank_lt)\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : StrongRankCondition K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Free K V\nh : \u2200 (s : Finset V), Basis (\u2191\u2191s) K V \u2192 False\ns : Set V\nb : Basis (\u2191s) K V\nhs : Set.Finite s\n\u22a2 Basis (\u2191\u2191(Set.Finite.toFinset hs)) K V\n[PROOFSTEP]\nconvert b\n[GOAL]\ncase h.e'_1.h.e'_2\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : StrongRankCondition K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : Free K V\nh : \u2200 (s : Finset V), Basis (\u2191\u2191s) K V \u2192 False\ns : Set V\nb : Basis (\u2191s) K V\nhs : Set.Finite s\n\u22a2 \u2191(Set.Finite.toFinset hs) = s\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nV\u2082 : Type v'\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module K V\u2082\nR : Type u_1\nM : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\nf : M \u2243\u2097[R] M\u2082\n\u22a2 finrank R M = finrank R M\u2082\n[PROOFSTEP]\nunfold finrank\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2079 : Ring K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nV\u2082 : Type v'\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module K V\u2082\nR : Type u_1\nM : Type u_2\nM\u2082 : Type u_3\ninst\u271d\u2074 : Ring R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup M\u2082\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R M\u2082\nf : M \u2243\u2097[R] M\u2082\n\u22a2 \u2191toNat (Module.rank R M) = \u2191toNat (Module.rank R M\u2082)\n[PROOFSTEP]\nrw [\u2190 Cardinal.toNat_lift, f.lift_rank_eq, Cardinal.toNat_lift]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\nf : V \u2192\u2097[K] V\u2082\nhf : Injective \u2191f\n\u22a2 finrank K { x // x \u2208 range f } = finrank K V\n[PROOFSTEP]\nrw [(LinearEquiv.ofInjective f hf).finrank_eq]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : Ring K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 finrank K { x // x \u2208 \u22a4 } = finrank K V\n[PROOFSTEP]\nunfold finrank\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : Ring K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 \u2191toNat (Module.rank K { x // x \u2208 \u22a4 }) = \u2191toNat (Module.rank K V)\n[PROOFSTEP]\nsimp [rank_top]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\ns t : Submodule K V\nle : s \u2264 t\nlt : finrank K { x // x \u2208 s } < finrank K { x // x \u2208 t }\nh : s = t\n\u22a2 finrank K { x // x \u2208 s } = finrank K { x // x \u2208 t }\n[PROOFSTEP]\nrw [h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\ns : Submodule K V\nlt : finrank K { x // x \u2208 s } < finrank K V\n\u22a2 s < \u22a4\n[PROOFSTEP]\nrw [\u2190 finrank_top K V] at lt \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b9 : AddCommGroup V\u2082\ninst\u271d : Module K V\u2082\ns : Submodule K V\nlt : finrank K { x // x \u2208 s } < finrank K { x // x \u2208 \u22a4 }\n\u22a2 s < \u22a4\n[PROOFSTEP]\nexact lt_of_le_of_finrank_lt_finrank le_top lt\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ns : Set V\ninst\u271d : Fintype \u2191s\n\u22a2 Module.rank K { x // x \u2208 span K s } \u2264 \u2191(Finset.card (Set.toFinset s))\n[PROOFSTEP]\nsimpa using rank_span_le (K := K) s\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\n\u22a2 Finset.card (Set.toFinset \u2191s) = Finset.card s\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 Set.finrank K (Set.range b) \u2264 Fintype.card \u03b9\n[PROOFSTEP]\nclassical\nrefine (finrank_span_le_card _).trans ?_\nrw [Set.toFinset_range]\nexact Finset.card_image_le\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 Set.finrank K (Set.range b) \u2264 Fintype.card \u03b9\n[PROOFSTEP]\nrefine (finrank_span_le_card _).trans ?_\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 Finset.card (Set.toFinset (Set.range b)) \u2264 Fintype.card \u03b9\n[PROOFSTEP]\nrw [Set.toFinset_range]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 Finset.card (Finset.image b Finset.univ) \u2264 Fintype.card \u03b9\n[PROOFSTEP]\nexact Finset.card_image_le\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhb : LinearIndependent K b\n\u22a2 Module.rank K { x // x \u2208 span K (Set.range b) } = \u2191(Fintype.card \u03b9)\n[PROOFSTEP]\nhave : Module.rank K (span K (Set.range b)) = #(Set.range b) := rank_span hb\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhb : LinearIndependent K b\nthis : Module.rank K { x // x \u2208 span K (Set.range b) } = #\u2191(Set.range b)\n\u22a2 Module.rank K { x // x \u2208 span K (Set.range b) } = \u2191(Fintype.card \u03b9)\n[PROOFSTEP]\nrwa [\u2190 lift_inj, mk_range_eq_of_injective hb.injective, Cardinal.mk_fintype, lift_natCast, lift_eq_nat_iff] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ns : Set V\ninst\u271d : Fintype \u2191s\nhs : LinearIndependent K Subtype.val\n\u22a2 Module.rank K { x // x \u2208 span K s } = \u2191(Finset.card (Set.toFinset s))\n[PROOFSTEP]\nhave : Module.rank K (span K s) = #s := rank_span_set hs\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ns : Set V\ninst\u271d : Fintype \u2191s\nhs : LinearIndependent K Subtype.val\nthis : Module.rank K { x // x \u2208 span K s } = #\u2191s\n\u22a2 Module.rank K { x // x \u2208 span K s } = \u2191(Finset.card (Set.toFinset s))\n[PROOFSTEP]\nrwa [Cardinal.mk_fintype, \u2190 Set.toFinset_card] at this \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : LinearIndependent K Subtype.val\n\u22a2 finrank K { x // x \u2208 span K \u2191s } = Finset.card s\n[PROOFSTEP]\nconvert finrank_span_set_eq_card (s : Set V) hs\n[GOAL]\ncase h.e'_3.h.e'_2\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : LinearIndependent K Subtype.val\n\u22a2 s = Set.toFinset \u2191s\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_3.h.e'_2.a\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : LinearIndependent K Subtype.val\na\u271d : V\n\u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 Set.toFinset \u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nclassical\nby_contra gx_ne_zero\nrefine'\n  not_le_of_gt (span_lt_top_of_card_lt_finrank (show (b '' (Set.univ \\ { i })).toFinset.card < finrank K V from _)) _\n\u00b7\n  calc\n    (b '' (Set.univ \\ { i })).toFinset.card = ((Set.univ \\ { i }).toFinset.image b).card := by\n      rw [Set.toFinset_card, Fintype.card_ofFinset]\n    _ \u2264 (Set.univ \\ { i }).toFinset.card := Finset.card_image_le\n    _ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))\n    _ < Finset.univ.card := (Finset.card_erase_lt_of_mem (Finset.mem_univ i))\n    _ = finrank K V := card_eq\nrefine' spans.trans (span_le.mpr _)\nrintro _\n  \u27e8j, rfl, rfl\u27e9\n      -- The case that `j \u2260 i` is easy because `b j \u2208 b '' (univ \\ {i})`.\nby_cases j_eq : j = i\nswap\n\u00b7 refine' subset_span \u27e8j, (Set.mem_diff _).mpr \u27e8Set.mem_univ _, _\u27e9, rfl\u27e9\n  exact mt Set.mem_singleton_iff.mp j_eq\nrw [j_eq, SetLike.mem_coe, show b i = -((g i)\u207b\u00b9 \u2022 (s.erase i).sum fun j => g j \u2022 b j) from _]\n\u00b7 refine' neg_mem (smul_mem _ _ (sum_mem fun k hk => _))\n  obtain \u27e8k_ne_i, _\u27e9 := Finset.mem_erase.mp hk\n  refine' smul_mem _ _ (subset_span \u27e8k, _, rfl\u27e9)\n  simp_all only [Set.mem_univ, Set.mem_diff, Set.mem_singleton_iff]\n    -- To show `b i` is a weighted sum of the other `b j`s, we'll rewrite this sum\n        -- to have the form of the assumption `dependent`.\napply eq_neg_of_add_eq_zero_left\ncalc\n  (b i + (g i)\u207b\u00b9 \u2022 (s.erase i).sum fun j => g j \u2022 b j) = (g i)\u207b\u00b9 \u2022 (g i \u2022 b i + (s.erase i).sum fun j => g j \u2022 b j) :=\n    by rw [smul_add, \u2190 mul_smul, inv_mul_cancel gx_ne_zero, one_smul]\n  _ = (g i)\u207b\u00b9 \u2022 (0 : V) := (congr_arg _ ?_)\n  _ = 0 :=\n    smul_zero\n      _\n        -- And then it's just a bit of manipulation with finite sums.\nrwa [\u2190 Finset.insert_erase i_mem_s, Finset.sum_insert (Finset.not_mem_erase _ _)] at dependent \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nby_contra gx_ne_zero\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 False\n[PROOFSTEP]\nrefine'\n  not_le_of_gt (span_lt_top_of_card_lt_finrank (show (b '' (Set.univ \\ { i })).toFinset.card < finrank K V from _)) _\n[GOAL]\ncase refine'_1\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 Finset.card (Set.toFinset (b '' (Set.univ \\ {i}))) < finrank K V\n[PROOFSTEP]\ncalc\n  (b '' (Set.univ \\ { i })).toFinset.card = ((Set.univ \\ { i }).toFinset.image b).card := by\n    rw [Set.toFinset_card, Fintype.card_ofFinset]\n  _ \u2264 (Set.univ \\ { i }).toFinset.card := Finset.card_image_le\n  _ = (Finset.univ.erase i).card := (congr_arg Finset.card (Finset.ext (by simp [and_comm])))\n  _ < Finset.univ.card := (Finset.card_erase_lt_of_mem (Finset.mem_univ i))\n  _ = finrank K V := card_eq\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 Finset.card (Set.toFinset (b '' (Set.univ \\ {i}))) = Finset.card (Finset.image b (Set.toFinset (Set.univ \\ {i})))\n[PROOFSTEP]\nrw [Set.toFinset_card, Fintype.card_ofFinset]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 \u2200 (a : \u03b9), a \u2208 Set.toFinset (Set.univ \\ {i}) \u2194 a \u2208 Finset.erase Finset.univ i\n[PROOFSTEP]\nsimp [and_comm]\n[GOAL]\ncase refine'_2\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 \u22a4 \u2264 span K (b '' (Set.univ \\ {i}))\n[PROOFSTEP]\nrefine' spans.trans (span_le.mpr _)\n[GOAL]\ncase refine'_2\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\n\u22a2 Set.range b \u2286 \u2191(span K (b '' (Set.univ \\ {i})))\n[PROOFSTEP]\nrintro _\n  \u27e8j, rfl, rfl\u27e9\n      -- The case that `j \u2260 i` is easy because `b j \u2208 b '' (univ \\ {i})`.\n[GOAL]\ncase refine'_2.intro.refl\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\n\u22a2 b j \u2208 \u2191(span K (b '' (Set.univ \\ {i})))\n[PROOFSTEP]\nby_cases j_eq : j = i\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 b j \u2208 \u2191(span K (b '' (Set.univ \\ {i})))\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : \u00acj = i\n\u22a2 b j \u2208 \u2191(span K (b '' (Set.univ \\ {i})))\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : \u00acj = i\n\u22a2 b j \u2208 \u2191(span K (b '' (Set.univ \\ {i})))\n[PROOFSTEP]\nrefine' subset_span \u27e8j, (Set.mem_diff _).mpr \u27e8Set.mem_univ _, _\u27e9, rfl\u27e9\n[GOAL]\ncase neg\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : \u00acj = i\n\u22a2 \u00acj \u2208 {i}\n[PROOFSTEP]\nexact mt Set.mem_singleton_iff.mp j_eq\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 b j \u2208 \u2191(span K (b '' (Set.univ \\ {i})))\n[PROOFSTEP]\nrw [j_eq, SetLike.mem_coe, show b i = -((g i)\u207b\u00b9 \u2022 (s.erase i).sum fun j => g j \u2022 b j) from _]\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 -((g i)\u207b\u00b9 \u2022 Finset.sum (Finset.erase s i) fun j => g j \u2022 b j) \u2208 span K (b '' (Set.univ \\ {i}))\n[PROOFSTEP]\nrefine' neg_mem (smul_mem _ _ (sum_mem fun k hk => _))\n[GOAL]\ncase pos\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\nk : \u03b9\nhk : k \u2208 Finset.erase s i\n\u22a2 g k \u2022 b k \u2208 span K (b '' (Set.univ \\ {i}))\n[PROOFSTEP]\nobtain \u27e8k_ne_i, _\u27e9 := Finset.mem_erase.mp hk\n[GOAL]\ncase pos.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\nk : \u03b9\nhk : k \u2208 Finset.erase s i\nk_ne_i : k \u2260 i\nright\u271d : k \u2208 s\n\u22a2 g k \u2022 b k \u2208 span K (b '' (Set.univ \\ {i}))\n[PROOFSTEP]\nrefine' smul_mem _ _ (subset_span \u27e8k, _, rfl\u27e9)\n[GOAL]\ncase pos.intro\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\nk : \u03b9\nhk : k \u2208 Finset.erase s i\nk_ne_i : k \u2260 i\nright\u271d : k \u2208 s\n\u22a2 k \u2208 Set.univ \\ {i}\n[PROOFSTEP]\nsimp_all only [Set.mem_univ, Set.mem_diff, Set.mem_singleton_iff]\n  -- To show `b i` is a weighted sum of the other `b j`s, we'll rewrite this sum\n      -- to have the form of the assumption `dependent`.\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 b i = -((g i)\u207b\u00b9 \u2022 Finset.sum (Finset.erase s i) fun j => g j \u2022 b j)\n[PROOFSTEP]\napply eq_neg_of_add_eq_zero_left\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 (b i + (g i)\u207b\u00b9 \u2022 Finset.sum (Finset.erase s i) fun j => g j \u2022 b j) = 0\n[PROOFSTEP]\ncalc\n  (b i + (g i)\u207b\u00b9 \u2022 (s.erase i).sum fun j => g j \u2022 b j) = (g i)\u207b\u00b9 \u2022 (g i \u2022 b i + (s.erase i).sum fun j => g j \u2022 b j) :=\n    by rw [smul_add, \u2190 mul_smul, inv_mul_cancel gx_ne_zero, one_smul]\n  _ = (g i)\u207b\u00b9 \u2022 (0 : V) := (congr_arg _ ?_)\n  _ = 0 :=\n    smul_zero\n      _\n        -- And then it's just a bit of manipulation with finite sums.\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 (b i + (g i)\u207b\u00b9 \u2022 Finset.sum (Finset.erase s i) fun j => g j \u2022 b j) =\n    (g i)\u207b\u00b9 \u2022 (g i \u2022 b i + Finset.sum (Finset.erase s i) fun j => g j \u2022 b j)\n[PROOFSTEP]\nrw [smul_add, \u2190 mul_smul, inv_mul_cancel gx_ne_zero, one_smul]\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nspans : \u22a4 \u2264 span K (Set.range b)\ncard_eq : Fintype.card \u03b9 = finrank K V\ns : Finset \u03b9\ng : \u03b9 \u2192 K\ndependent : (Finset.sum s fun i => g i \u2022 b i) = 0\ni : \u03b9\ni_mem_s : i \u2208 s\ngx_ne_zero : \u00acg i = 0\nj : \u03b9\nj_eq : j = i\n\u22a2 (g i \u2022 b i + Finset.sum (Finset.erase s i) fun j => g j \u2022 b j) = 0\n[PROOFSTEP]\nrwa [\u2190 Finset.insert_erase i_mem_s, Finset.sum_insert (Finset.not_mem_erase _ _)] at dependent \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 LinearIndependent K b \u2194 Fintype.card \u03b9 = Set.finrank K (Set.range b)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 LinearIndependent K b \u2192 Fintype.card \u03b9 = Set.finrank K (Set.range b)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nh : LinearIndependent K b\n\u22a2 Fintype.card \u03b9 = Set.finrank K (Set.range b)\n[PROOFSTEP]\nexact (finrank_span_eq_card h).symm\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 Fintype.card \u03b9 = Set.finrank K (Set.range b) \u2192 LinearIndependent K b\n[PROOFSTEP]\nintro hc\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\n\u22a2 LinearIndependent K b\n[PROOFSTEP]\nlet f := Submodule.subtype (span K (Set.range b))\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\n\u22a2 LinearIndependent K b\n[PROOFSTEP]\nlet b' : \u03b9 \u2192 span K (Set.range b) := fun i => \u27e8b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)\u27e9\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\n\u22a2 LinearIndependent K b\n[PROOFSTEP]\nhave hs : \u22a4 \u2264 span K (Set.range b') := by\n  intro x\n  have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f\n  have hf : f '' Set.range b' = Set.range b := by\n    ext x\n    simp [Set.mem_image, Set.mem_range]\n  rw [hf] at h \n  have hx : (x : V) \u2208 span K (Set.range b) := x.property\n  conv at hx =>\n    arg 2\n    rw [h]\n  simpa [mem_map] using hx\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\n\u22a2 \u22a4 \u2264 span K (Set.range b')\n[PROOFSTEP]\nintro x\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nhave h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b'))\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nhave hf : f '' Set.range b' = Set.range b := by\n  ext x\n  simp [Set.mem_image, Set.mem_range]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b'))\n\u22a2 \u2191f '' Set.range b' = Set.range b\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx\u271d : { x // x \u2208 span K (Set.range b) }\nh : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nx : V\n\u22a2 x \u2208 \u2191f '' Set.range b' \u2194 x \u2208 Set.range b\n[PROOFSTEP]\nsimp [Set.mem_image, Set.mem_range]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nrw [hf] at h \n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nhave hx : (x : V) \u2208 span K (Set.range b) := x.property\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 span K (Set.range b)\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nconv at hx =>\n  arg 2\n  rw [h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 span K (Set.range b)\n| \u2191x \u2208 span K (Set.range b)\n[PROOFSTEP]\n  arg 2\n  rw [h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 span K (Set.range b)\n| \u2191x \u2208 span K (Set.range b)\n[PROOFSTEP]\n  arg 2\n  rw [h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 span K (Set.range b)\n| \u2191x \u2208 span K (Set.range b)\n[PROOFSTEP]\narg 2\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 span K (Set.range b)\n| span K (Set.range b)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nx : { x // x \u2208 span K (Set.range b) }\nh : span K (Set.range b) = Submodule.map f (span K (Set.range b'))\nhf : \u2191f '' Set.range b' = Set.range b\nhx : \u2191x \u2208 Submodule.map f (span K (Set.range b'))\n\u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b')\n[PROOFSTEP]\nsimpa [mem_map] using hx\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nhs : \u22a4 \u2264 span K (Set.range b')\n\u22a2 LinearIndependent K b\n[PROOFSTEP]\nhave hi : LinearMap.ker f = \u22a5 := ker_subtype _\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\nhc : Fintype.card \u03b9 = Set.finrank K (Set.range b)\nf : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b))\nb' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) }\nhs : \u22a4 \u2264 span K (Set.range b')\nhi : LinearMap.ker f = \u22a5\n\u22a2 LinearIndependent K b\n[PROOFSTEP]\nconvert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nb : \u03b9 \u2192 V\n\u22a2 LinearIndependent K b \u2194 Fintype.card \u03b9 \u2264 Set.finrank K (Set.range b)\n[PROOFSTEP]\nrw [linearIndependent_iff_card_eq_finrank_span, finrank_range_le_card.le_iff_eq]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nn : v \u2260 0\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\n\u22a2 finrank K V = 1\n[PROOFSTEP]\nhaveI := nontrivial_of_invariantBasisNumber K\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nn : v \u2260 0\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\nthis : Nontrivial K\n\u22a2 finrank K V = 1\n[PROOFSTEP]\nobtain \u27e8b\u27e9 := (Basis.basis_singleton_iff.{u} PUnit).mpr \u27e8v, n, h\u27e9\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nn : v \u2260 0\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\nthis : Nontrivial K\nb : Basis PUnit K V\n\u22a2 finrank K V = 1\n[PROOFSTEP]\nrw [finrank_eq_card_basis b, Fintype.card_punit]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nhaveI := nontrivial_of_invariantBasisNumber K\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\nthis : Nontrivial K\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nrcases eq_or_ne v 0 with (rfl | hn)\n[GOAL]\ncase inl\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nthis : Nontrivial K\nh : \u2200 (w : V), \u2203 c, c \u2022 0 = w\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nhaveI :=\n  subsingleton_of_forall_eq (0 : V) fun w => by\n    obtain \u27e8c, rfl\u27e9 := h w\n    simp\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nthis : Nontrivial K\nh : \u2200 (w : V), \u2203 c, c \u2022 0 = w\nw : V\n\u22a2 w = 0\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := h w\n[GOAL]\ncase intro\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nthis : Nontrivial K\nh : \u2200 (w : V), \u2203 c, c \u2022 0 = w\nc : K\n\u22a2 c \u2022 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nthis\u271d : Nontrivial K\nh : \u2200 (w : V), \u2203 c, c \u2022 0 = w\nthis : Subsingleton V\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nrw [finrank_zero_of_subsingleton]\n[GOAL]\ncase inl\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nthis\u271d : Nontrivial K\nh : \u2200 (w : V), \u2203 c, c \u2022 0 = w\nthis : Subsingleton V\n\u22a2 0 \u2264 1\n[PROOFSTEP]\nexact zero_le_one\n[GOAL]\ncase inr\nK : Type u\nV : Type v\ninst\u271d\u2074 : Ring K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module K V\ninst\u271d\u00b9 : NoZeroSMulDivisors K V\ninst\u271d : StrongRankCondition K\nv : V\nh : \u2200 (w : V), \u2203 c, c \u2022 v = w\nthis : Nontrivial K\nhn : v \u2260 0\n\u22a2 finrank K V \u2264 1\n[PROOFSTEP]\nexact (finrank_eq_one v hn h).le\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nrw [\u2190 Algebra.top_toSubmodule]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F { x // x \u2208 \u2191Subalgebra.toSubmodule \u22a4 }\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 finrank F { x // x \u2208 \u22a4 } = finrank F { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nrw [\u2190 Algebra.top_toSubmodule]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 finrank F { x // x \u2208 \u22a4 } = finrank F { x // x \u2208 \u2191Subalgebra.toSubmodule \u22a4 }\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F E\n[PROOFSTEP]\nrw [subalgebra_top_rank_eq_submodule_top_rank]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : CommRing F\ninst\u271d\u00b9 : Ring E\ninst\u271d : Algebra F E\n\u22a2 Module.rank F { x // x \u2208 \u22a4 } = Module.rank F E\n[PROOFSTEP]\nexact _root_.rank_top F E\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u2075 : CommRing F\ninst\u271d\u2074 : Ring E\ninst\u271d\u00b3 : Algebra F E\ninst\u271d\u00b2 : StrongRankCondition F\ninst\u271d\u00b9 : NoZeroSMulDivisors F E\ninst\u271d : Nontrivial E\n\u22a2 Module.rank F { x // x \u2208 span F {1} } = 1\n[PROOFSTEP]\nletI := Module.nontrivial F E\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u2075 : CommRing F\ninst\u271d\u2074 : Ring E\ninst\u271d\u00b3 : Algebra F E\ninst\u271d\u00b2 : StrongRankCondition F\ninst\u271d\u00b9 : NoZeroSMulDivisors F E\ninst\u271d : Nontrivial E\nthis : Nontrivial F := Module.nontrivial F E\n\u22a2 Module.rank F { x // x \u2208 span F {1} } = 1\n[PROOFSTEP]\nrw [rank_span_set]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u2075 : CommRing F\ninst\u271d\u2074 : Ring E\ninst\u271d\u00b3 : Algebra F E\ninst\u271d\u00b2 : StrongRankCondition F\ninst\u271d\u00b9 : NoZeroSMulDivisors F E\ninst\u271d : Nontrivial E\nthis : Nontrivial F := Module.nontrivial F E\n\u22a2 #\u2191{1} = 1\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u2075 : CommRing F\ninst\u271d\u2074 : Ring E\ninst\u271d\u00b3 : Algebra F E\ninst\u271d\u00b2 : StrongRankCondition F\ninst\u271d\u00b9 : NoZeroSMulDivisors F E\ninst\u271d : Nontrivial E\nthis : Nontrivial F := Module.nontrivial F E\n\u22a2 LinearIndependent F fun x => \u2191x\n[PROOFSTEP]\nexacts [mk_singleton _, linearIndependent_singleton one_ne_zero]\n[GOAL]\nK : Type u\nV : Type v\nF : Type u_1\nE : Type u_2\ninst\u271d\u2075 : CommRing F\ninst\u271d\u2074 : Ring E\ninst\u271d\u00b3 : Algebra F E\ninst\u271d\u00b2 : StrongRankCondition F\ninst\u271d\u00b9 : NoZeroSMulDivisors F E\ninst\u271d : Nontrivial E\n\u22a2 Module.rank F { x // x \u2208 \u22a5 } = \u21911\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Finrank", "llama_tokens": 21476, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.6992544273261175, "lm_q1q2_score": 0.5258481216064468}}
{"text": "[GOAL]\n\u22a2 StrictConvexOn \u211d univ exp\n[PROOFSTEP]\napply strictConvexOn_of_slope_strict_mono_adjacent convex_univ\n[GOAL]\n\u22a2 \u2200 {x y z : \u211d}, x \u2208 univ \u2192 z \u2208 univ \u2192 x < y \u2192 y < z \u2192 (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)\n[PROOFSTEP]\nrintro x y z - - hxy hyz\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\n\u22a2 (exp y - exp x) / (y - x) < (exp z - exp y) / (z - y)\n[PROOFSTEP]\ntrans exp y\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\n\u22a2 (exp y - exp x) / (y - x) < exp y\n[PROOFSTEP]\nhave h1 : 0 < y - x := by linarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\n\u22a2 (exp y - exp x) / (y - x) < exp y\n[PROOFSTEP]\nhave h2 : x - y < 0 := by linarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\n\u22a2 x - y < 0\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 (exp y - exp x) / (y - x) < exp y\n[PROOFSTEP]\nrw [div_lt_iff h1]\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y - exp x < exp y * (y - x)\n[PROOFSTEP]\ncalc\n  exp y - exp x = exp y - exp y * exp (x - y) := by rw [\u2190 exp_add]; ring_nf\n  _ = exp y * (1 - exp (x - y)) := by ring\n  _ < exp y * -(x - y) := by gcongr; linarith [add_one_lt_exp_of_nonzero h2.ne]\n  _ = exp y * (y - x) := by ring\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y - exp x = exp y - exp y * exp (x - y)\n[PROOFSTEP]\nrw [\u2190 exp_add]\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y - exp x = exp y - exp (y + (x - y))\n[PROOFSTEP]\nring_nf\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y - exp y * exp (x - y) = exp y * (1 - exp (x - y))\n[PROOFSTEP]\nring\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y * (1 - exp (x - y)) < exp y * -(x - y)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 1 - exp (x - y) < -(x - y)\n[PROOFSTEP]\nlinarith [add_one_lt_exp_of_nonzero h2.ne]\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n\u22a2 exp y * -(x - y) = exp y * (y - x)\n[PROOFSTEP]\nring\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\n\u22a2 exp y < (exp z - exp y) / (z - y)\n[PROOFSTEP]\nhave h1 : 0 < z - y := by linarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp y < (exp z - exp y) / (z - y)\n[PROOFSTEP]\nrw [lt_div_iff h1]\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp y * (z - y) < exp z - exp y\n[PROOFSTEP]\ncalc\n  exp y * (z - y) < exp y * (exp (z - y) - 1) := by\n    gcongr _ * ?_\n    linarith [add_one_lt_exp_of_nonzero h1.ne']\n  _ = exp (z - y) * exp y - exp y := by ring\n  _ \u2264 exp z - exp y := by rw [\u2190 exp_add]; ring_nf; rfl\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp y * (z - y) < exp y * (exp (z - y) - 1)\n[PROOFSTEP]\ngcongr _ * ?_\n[GOAL]\ncase bc\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 z - y < exp (z - y) - 1\n[PROOFSTEP]\nlinarith [add_one_lt_exp_of_nonzero h1.ne']\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp y * (exp (z - y) - 1) = exp (z - y) * exp y - exp y\n[PROOFSTEP]\nring\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp (z - y) * exp y - exp y \u2264 exp z - exp y\n[PROOFSTEP]\nrw [\u2190 exp_add]\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp (z - y + y) - exp y \u2264 exp z - exp y\n[PROOFSTEP]\nring_nf\n[GOAL]\nx y z : \u211d\nhxy : x < y\nhyz : y < z\nh1 : 0 < z - y\n\u22a2 exp z - exp y \u2264 exp z - exp y\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ n\n[PROOFSTEP]\ninduction' n with k IH\n[GOAL]\ncase zero\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ Nat.zero\n[PROOFSTEP]\nexact convexOn_const (1 : \u211d) (convex_Ici _)\n[GOAL]\ncase succ\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ Nat.succ k\n[PROOFSTEP]\nrefine' \u27e8convex_Ici _, _\u27e9\n[GOAL]\ncase succ\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\n\u22a2 \u2200 \u2983x : \u211d\u2984,\n    x \u2208 Ici 0 \u2192\n      \u2200 \u2983y : \u211d\u2984,\n        y \u2208 Ici 0 \u2192\n          \u2200 \u2983a b : \u211d\u2984,\n            0 \u2264 a \u2192\n              0 \u2264 b \u2192\n                a + b = 1 \u2192\n                  (fun x => x ^ Nat.succ k) (a \u2022 x + b \u2022 y) \u2264\n                    a \u2022 (fun x => x ^ Nat.succ k) x + b \u2022 (fun x => x ^ Nat.succ k) y\n[PROOFSTEP]\nrintro a (ha : 0 \u2264 a) b (hb : 0 \u2264 b) \u03bc \u03bd h\u03bc h\u03bd h\n[GOAL]\ncase succ\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (fun x => x ^ Nat.succ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ Nat.succ k) a + \u03bd \u2022 (fun x => x ^ Nat.succ k) b\n[PROOFSTEP]\nhave H := IH.2 ha hb h\u03bc h\u03bd h\n[GOAL]\ncase succ\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\n\u22a2 (fun x => x ^ Nat.succ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ Nat.succ k) a + \u03bd \u2022 (fun x => x ^ Nat.succ k) b\n[PROOFSTEP]\nhave : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd :=\n  by\n  cases' le_or_lt a b with hab hab\n  \u00b7 have : a ^ k \u2264 b ^ k := by gcongr\n    have : 0 \u2264 (b ^ k - a ^ k) * (b - a) := by nlinarith\n    positivity\n  \u00b7 have : b ^ k \u2264 a ^ k := by gcongr\n    have : 0 \u2264 (b ^ k - a ^ k) * (b - a) := by nlinarith\n    positivity\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\ncases' le_or_lt a b with hab hab\n[GOAL]\ncase inl\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : a \u2264 b\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\nhave : a ^ k \u2264 b ^ k := by gcongr\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : a \u2264 b\n\u22a2 a ^ k \u2264 b ^ k\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase inl\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : a \u2264 b\nthis : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\nhave : 0 \u2264 (b ^ k - a ^ k) * (b - a) := by nlinarith\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : a \u2264 b\nthis : a ^ k \u2264 b ^ k\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a)\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inl\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : a \u2264 b\nthis\u271d : a ^ k \u2264 b ^ k\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a)\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\npositivity\n[GOAL]\ncase inr\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : b < a\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\nhave : b ^ k \u2264 a ^ k := by gcongr\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : b < a\n\u22a2 b ^ k \u2264 a ^ k\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase inr\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : b < a\nthis : b ^ k \u2264 a ^ k\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\nhave : 0 \u2264 (b ^ k - a ^ k) * (b - a) := by nlinarith\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : b < a\nthis : b ^ k \u2264 a ^ k\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a)\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nhab : b < a\nthis\u271d : b ^ k \u2264 a ^ k\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a)\n\u22a2 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\npositivity\n[GOAL]\ncase succ\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (fun x => x ^ Nat.succ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ Nat.succ k) a + \u03bd \u2022 (fun x => x ^ Nat.succ k) b\n[PROOFSTEP]\ncalc\n  (\u03bc * a + \u03bd * b) ^ k.succ = (\u03bc * a + \u03bd * b) * (\u03bc * a + \u03bd * b) ^ k := pow_succ _ _\n  _ \u2264 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) := by gcongr; exact H\n  _ \u2264 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) + (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd := by linarith\n  _ = (\u03bc + \u03bd) * (\u03bc * a ^ k.succ + \u03bd * b ^ k.succ) := by rw [Nat.succ_eq_add_one]; ring\n  _ = \u03bc * a ^ k.succ + \u03bd * b ^ k.succ := by rw [h]; ring\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) * (\u03bc * a + \u03bd * b) ^ k \u2264 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) ^ k \u2264 \u03bc * a ^ k + \u03bd * b ^ k\n[PROOFSTEP]\nexact H\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) \u2264\n    (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) + (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n[PROOFSTEP]\nlinarith\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) + (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd =\n    (\u03bc + \u03bd) * (\u03bc * a ^ Nat.succ k + \u03bd * b ^ Nat.succ k)\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one]\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) * (\u03bc * a ^ k + \u03bd * b ^ k) + (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd =\n    (\u03bc + \u03bd) * (\u03bc * a ^ (k + 1) + \u03bd * b ^ (k + 1))\n[PROOFSTEP]\nring\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 (\u03bc + \u03bd) * (\u03bc * a ^ Nat.succ k + \u03bd * b ^ Nat.succ k) = \u03bc * a ^ Nat.succ k + \u03bd * b ^ Nat.succ k\n[PROOFSTEP]\nrw [h]\n[GOAL]\nk : \u2115\nIH : ConvexOn \u211d (Ici 0) fun x => x ^ k\na : \u211d\nha : 0 \u2264 a\nb : \u211d\nhb : 0 \u2264 b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nH : (fun x => x ^ k) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ k) a + \u03bd \u2022 (fun x => x ^ k) b\nthis : 0 \u2264 (b ^ k - a ^ k) * (b - a) * \u03bc * \u03bd\n\u22a2 1 * (\u03bc * a ^ Nat.succ k + \u03bd * b ^ Nat.succ k) = \u03bc * a ^ Nat.succ k + \u03bd * b ^ Nat.succ k\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\nhn : Even n\n\u22a2 ConvexOn \u211d univ fun x => x ^ n\n[PROOFSTEP]\nrefine' \u27e8convex_univ, _\u27e9\n[GOAL]\nn : \u2115\nhn : Even n\n\u22a2 \u2200 \u2983x : \u211d\u2984,\n    x \u2208 univ \u2192\n      \u2200 \u2983y : \u211d\u2984,\n        y \u2208 univ \u2192\n          \u2200 \u2983a b : \u211d\u2984,\n            0 \u2264 a \u2192\n              0 \u2264 b \u2192 a + b = 1 \u2192 (fun x => x ^ n) (a \u2022 x + b \u2022 y) \u2264 a \u2022 (fun x => x ^ n) x + b \u2022 (fun x => x ^ n) y\n[PROOFSTEP]\nrintro a - b - \u03bc \u03bd h\u03bc h\u03bd h\n[GOAL]\nn : \u2115\nhn : Even n\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (fun x => x ^ n) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ n) a + \u03bd \u2022 (fun x => x ^ n) b\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 :=\n  hn.exists_two_nsmul\n    _\n      -- Porting note: added type ascription to LHS\n[GOAL]\ncase intro\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\n\u22a2 (fun x => x ^ (2 \u2022 k)) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ (2 \u2022 k)) a + \u03bd \u2022 (fun x => x ^ (2 \u2022 k)) b\n[PROOFSTEP]\nhave : (0 : \u211d) \u2264 (a - b) ^ 2 * \u03bc * \u03bd := by positivity\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\n\u22a2 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 (fun x => x ^ (2 \u2022 k)) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => x ^ (2 \u2022 k)) a + \u03bd \u2022 (fun x => x ^ (2 \u2022 k)) b\n[PROOFSTEP]\ncalc\n  (\u03bc * a + \u03bd * b) ^ (2 * k) = ((\u03bc * a + \u03bd * b) ^ 2) ^ k := by rw [pow_mul]\n  _ \u2264 ((\u03bc + \u03bd) * (\u03bc * a ^ 2 + \u03bd * b ^ 2)) ^ k := by gcongr; linarith\n  _ = (\u03bc * a ^ 2 + \u03bd * b ^ 2) ^ k := by rw [h]; ring\n  _ \u2264 \u03bc * (a ^ 2) ^ k + \u03bd * (b ^ 2) ^ k := ?_\n  _ \u2264 \u03bc * a ^ (2 * k) + \u03bd * b ^ (2 * k) := by ring_nf;\n    rfl\n      -- Porting note: `rw [mem_Ici]` was `dsimp`\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) ^ (2 * k) = ((\u03bc * a + \u03bd * b) ^ 2) ^ k\n[PROOFSTEP]\nrw [pow_mul]\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 ((\u03bc * a + \u03bd * b) ^ 2) ^ k \u2264 ((\u03bc + \u03bd) * (\u03bc * a ^ 2 + \u03bd * b ^ 2)) ^ k\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase hab\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 (\u03bc * a + \u03bd * b) ^ 2 \u2264 (\u03bc + \u03bd) * (\u03bc * a ^ 2 + \u03bd * b ^ 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 ((\u03bc + \u03bd) * (\u03bc * a ^ 2 + \u03bd * b ^ 2)) ^ k = (\u03bc * a ^ 2 + \u03bd * b ^ 2) ^ k\n[PROOFSTEP]\nrw [h]\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 (1 * (\u03bc * a ^ 2 + \u03bd * b ^ 2)) ^ k = (\u03bc * a ^ 2 + \u03bd * b ^ 2) ^ k\n[PROOFSTEP]\nring\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 \u03bc * (a ^ 2) ^ k + \u03bd * (b ^ 2) ^ k \u2264 \u03bc * a ^ (2 * k) + \u03bd * b ^ (2 * k)\n[PROOFSTEP]\nring_nf\n[GOAL]\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 \u03bc * a ^ (k * 2) + \u03bd * b ^ (k * 2) \u2264 \u03bc * a ^ (k * 2) + \u03bd * b ^ (k * 2)\n[PROOFSTEP]\nrfl\n  -- Porting note: `rw [mem_Ici]` was `dsimp`\n[GOAL]\ncase intro\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 (\u03bc * a ^ 2 + \u03bd * b ^ 2) ^ k \u2264 \u03bc * (a ^ 2) ^ k + \u03bd * (b ^ 2) ^ k\n[PROOFSTEP]\nrefine' (convexOn_pow k).2 _ _ h\u03bc h\u03bd h\n[GOAL]\ncase intro.refine'_1\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 a ^ 2 \u2208 Ici 0\n[PROOFSTEP]\nrw [mem_Ici]\n[GOAL]\ncase intro.refine'_2\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 b ^ 2 \u2208 Ici 0\n[PROOFSTEP]\nrw [mem_Ici]\n[GOAL]\ncase intro.refine'_1\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 0 \u2264 a ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.refine'_2\na b \u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nk : \u2115\nhn : Even (2 \u2022 k)\nthis : 0 \u2264 (a - b) ^ 2 * \u03bc * \u03bd\n\u22a2 0 \u2264 b ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\nn : \u2115\n\u22a2 ConvexOn \u211d (Ioi 0) fun x => x ^ \u2191n\n[PROOFSTEP]\nsimp_rw [zpow_ofNat]\n[GOAL]\nn : \u2115\n\u22a2 ConvexOn \u211d (Ioi 0) fun x => x ^ n\n[PROOFSTEP]\nexact (convexOn_pow n).subset Ioi_subset_Ici_self (convex_Ioi _)\n[GOAL]\nn : \u2115\n\u22a2 ConvexOn \u211d (Ioi 0) fun x => x ^ -[n+1]\n[PROOFSTEP]\nsimp_rw [zpow_negSucc]\n[GOAL]\nn : \u2115\n\u22a2 ConvexOn \u211d (Ioi 0) fun x => (x ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nrefine' \u27e8convex_Ioi _, _\u27e9\n[GOAL]\nn : \u2115\n\u22a2 \u2200 \u2983x : \u211d\u2984,\n    x \u2208 Ioi 0 \u2192\n      \u2200 \u2983y : \u211d\u2984,\n        y \u2208 Ioi 0 \u2192\n          \u2200 \u2983a b : \u211d\u2984,\n            0 \u2264 a \u2192\n              0 \u2264 b \u2192\n                a + b = 1 \u2192\n                  (fun x => (x ^ (n + 1))\u207b\u00b9) (a \u2022 x + b \u2022 y) \u2264\n                    a \u2022 (fun x => (x ^ (n + 1))\u207b\u00b9) x + b \u2022 (fun x => (x ^ (n + 1))\u207b\u00b9) y\n[PROOFSTEP]\nrintro a (ha : 0 < a) b (hb : 0 < b) \u03bc \u03bd h\u03bc h\u03bd h\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (fun x => (x ^ (n + 1))\u207b\u00b9) (\u03bc \u2022 a + \u03bd \u2022 b) \u2264 \u03bc \u2022 (fun x => (x ^ (n + 1))\u207b\u00b9) a + \u03bd \u2022 (fun x => (x ^ (n + 1))\u207b\u00b9) b\n[PROOFSTEP]\nfield_simp [ha.ne', hb.ne']\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 1 / (\u03bc * a + \u03bd * b) ^ (n + 1) \u2264 (\u03bc * b ^ (n + 1) + \u03bd * a ^ (n + 1)) / (a ^ (n + 1) * b ^ (n + 1))\n[PROOFSTEP]\nrw [div_le_div_iff]\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 1 * (a ^ (n + 1) * b ^ (n + 1)) \u2264 (\u03bc * b ^ (n + 1) + \u03bd * a ^ (n + 1)) * (\u03bc * a + \u03bd * b) ^ (n + 1)\n[PROOFSTEP]\ncalc\n  (1 : \u211d) * (a ^ (n + 1) * b ^ (n + 1)) = ((\u03bc + \u03bd) ^ 2 * (a * b)) ^ (n + 1) := by rw [h]; ring\n  _ \u2264 ((\u03bc * b + \u03bd * a) * (\u03bc * a + \u03bd * b)) ^ (n + 1) := ?_\n  _ = (\u03bc * b + \u03bd * a) ^ (n + 1) * (\u03bc * a + \u03bd * b) ^ (n + 1) := by rw [mul_pow]\n  _ \u2264 (\u03bc * b ^ (n + 1) + \u03bd * a ^ (n + 1)) * (\u03bc * a + \u03bd * b) ^ (n + 1) := ?_\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 1 * (a ^ (n + 1) * b ^ (n + 1)) = ((\u03bc + \u03bd) ^ 2 * (a * b)) ^ (n + 1)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 1 * (a ^ (n + 1) * b ^ (n + 1)) = (1 ^ 2 * (a * b)) ^ (n + 1)\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 ((\u03bc * b + \u03bd * a) * (\u03bc * a + \u03bd * b)) ^ (n + 1) = (\u03bc * b + \u03bd * a) ^ (n + 1) * (\u03bc * a + \u03bd * b) ^ (n + 1)\n[PROOFSTEP]\nrw [mul_pow]\n[GOAL]\ncase calc_1\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 ((\u03bc + \u03bd) ^ 2 * (a * b)) ^ (n + 1) \u2264 ((\u03bc * b + \u03bd * a) * (\u03bc * a + \u03bd * b)) ^ (n + 1)\n[PROOFSTEP]\ngcongr(?_ : \u211d) ^ _\n[GOAL]\ncase calc_1.hab\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (\u03bc + \u03bd) ^ 2 * (a * b) \u2264 (\u03bc * b + \u03bd * a) * (\u03bc * a + \u03bd * b)\n[PROOFSTEP]\nhave : (0 : \u211d) \u2264 \u03bc * \u03bd * (a - b) ^ 2 := by positivity\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 0 \u2264 \u03bc * \u03bd * (a - b) ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\ncase calc_1.hab\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nthis : 0 \u2264 \u03bc * \u03bd * (a - b) ^ 2\n\u22a2 (\u03bc + \u03bd) ^ 2 * (a * b) \u2264 (\u03bc * b + \u03bd * a) * (\u03bc * a + \u03bd * b)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase calc_2\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (\u03bc * b + \u03bd * a) ^ (n + 1) * (\u03bc * a + \u03bd * b) ^ (n + 1) \u2264\n    (\u03bc * b ^ (n + 1) + \u03bd * a ^ (n + 1)) * (\u03bc * a + \u03bd * b) ^ (n + 1)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase calc_2.h\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 (\u03bc * b + \u03bd * a) ^ (n + 1) \u2264 \u03bc * b ^ (n + 1) + \u03bd * a ^ (n + 1)\n[PROOFSTEP]\napply (convexOn_pow (n + 1)).2 hb.le ha.le h\u03bc h\u03bd h\n[GOAL]\ncase b0\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 0 < (\u03bc * a + \u03bd * b) ^ (n + 1)\n[PROOFSTEP]\nhave : 0 < \u03bc * a + \u03bd * b := by cases le_or_lt a b <;> nlinarith\n[GOAL]\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 0 < \u03bc * a + \u03bd * b\n[PROOFSTEP]\ncases le_or_lt a b\n[GOAL]\ncase inl\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nh\u271d : a \u2264 b\n\u22a2 0 < \u03bc * a + \u03bd * b\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nh\u271d : b < a\n\u22a2 0 < \u03bc * a + \u03bd * b\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase b0\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\nthis : 0 < \u03bc * a + \u03bd * b\n\u22a2 0 < (\u03bc * a + \u03bd * b) ^ (n + 1)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase d0\nn : \u2115\na : \u211d\nha : 0 < a\nb : \u211d\nhb : 0 < b\n\u03bc \u03bd : \u211d\nh\u03bc : 0 \u2264 \u03bc\nh\u03bd : 0 \u2264 \u03bd\nh : \u03bc + \u03bd = 1\n\u22a2 0 < a ^ (n + 1) * b ^ (n + 1)\n[PROOFSTEP]\npositivity\n[GOAL]\n\u22a2 StrictConcaveOn \u211d (Ioi 0) log\n[PROOFSTEP]\napply strictConcaveOn_of_slope_strict_anti_adjacent (convex_Ioi (0 : \u211d))\n[GOAL]\n\u22a2 \u2200 {x y z : \u211d}, x \u2208 Ioi 0 \u2192 z \u2208 Ioi 0 \u2192 x < y \u2192 y < z \u2192 (log z - log y) / (z - y) < (log y - log x) / (y - x)\n[PROOFSTEP]\nrintro x y z (hx : 0 < x) (hz : 0 < z) hxy hyz\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\n\u22a2 (log z - log y) / (z - y) < (log y - log x) / (y - x)\n[PROOFSTEP]\nhave hy : 0 < y := hx.trans hxy\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 (log z - log y) / (z - y) < (log y - log x) / (y - x)\n[PROOFSTEP]\ntrans y\u207b\u00b9\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 (log z - log y) / (z - y) < y\u207b\u00b9\n[PROOFSTEP]\nhave h : 0 < z - y := by linarith\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\n\u22a2 (log z - log y) / (z - y) < y\u207b\u00b9\n[PROOFSTEP]\nrw [div_lt_iff h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\n\u22a2 log z - log y < y\u207b\u00b9 * (z - y)\n[PROOFSTEP]\nhave hyz' : 0 < z / y := by positivity\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\n\u22a2 0 < z / y\n[PROOFSTEP]\npositivity\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\nhyz' : 0 < z / y\n\u22a2 log z - log y < y\u207b\u00b9 * (z - y)\n[PROOFSTEP]\nhave hyz'' : z / y \u2260 1 := by\n  contrapose! h\n  rw [div_eq_one_iff_eq hy.ne'] at h \n  simp [h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\nhyz' : 0 < z / y\n\u22a2 z / y \u2260 1\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhyz' : 0 < z / y\nh : z / y = 1\n\u22a2 z - y \u2264 0\n[PROOFSTEP]\nrw [div_eq_one_iff_eq hy.ne'] at h \n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhyz' : 0 < z / y\nh : z = y\n\u22a2 z - y \u2264 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\nhyz' : 0 < z / y\nhyz'' : z / y \u2260 1\n\u22a2 log z - log y < y\u207b\u00b9 * (z - y)\n[PROOFSTEP]\ncalc\n  log z - log y = log (z / y) := by rw [\u2190 log_div hz.ne' hy.ne']\n  _ < z / y - 1 := (log_lt_sub_one_of_pos hyz' hyz'')\n  _ = y\u207b\u00b9 * (z - y) := by field_simp [hy.ne']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\nhyz' : 0 < z / y\nhyz'' : z / y \u2260 1\n\u22a2 log z - log y = log (z / y)\n[PROOFSTEP]\nrw [\u2190 log_div hz.ne' hy.ne']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < z - y\nhyz' : 0 < z / y\nhyz'' : z / y \u2260 1\n\u22a2 z / y - 1 = y\u207b\u00b9 * (z - y)\n[PROOFSTEP]\nfield_simp [hy.ne']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 y\u207b\u00b9 < (log y - log x) / (y - x)\n[PROOFSTEP]\nhave h : 0 < y - x := by linarith\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\n\u22a2 y\u207b\u00b9 < (log y - log x) / (y - x)\n[PROOFSTEP]\nrw [lt_div_iff h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\n\u22a2 y\u207b\u00b9 * (y - x) < log y - log x\n[PROOFSTEP]\nhave hxy' : 0 < x / y := by positivity\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\n\u22a2 0 < x / y\n[PROOFSTEP]\npositivity\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\n\u22a2 y\u207b\u00b9 * (y - x) < log y - log x\n[PROOFSTEP]\nhave hxy'' : x / y \u2260 1 := by\n  contrapose! h\n  rw [div_eq_one_iff_eq hy.ne'] at h \n  simp [h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\n\u22a2 x / y \u2260 1\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhxy' : 0 < x / y\nh : x / y = 1\n\u22a2 y - x \u2264 0\n[PROOFSTEP]\nrw [div_eq_one_iff_eq hy.ne'] at h \n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhxy' : 0 < x / y\nh : x = y\n\u22a2 y - x \u2264 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y \u2260 1\n\u22a2 y\u207b\u00b9 * (y - x) < log y - log x\n[PROOFSTEP]\ncalc\n  y\u207b\u00b9 * (y - x) = 1 - x / y := by field_simp [hy.ne']\n  _ < -log (x / y) := by linarith [log_lt_sub_one_of_pos hxy' hxy'']\n  _ = -(log x - log y) := by rw [log_div hx.ne' hy.ne']\n  _ = log y - log x := by ring\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y \u2260 1\n\u22a2 y\u207b\u00b9 * (y - x) = 1 - x / y\n[PROOFSTEP]\nfield_simp [hy.ne']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y \u2260 1\n\u22a2 1 - x / y < -log (x / y)\n[PROOFSTEP]\nlinarith [log_lt_sub_one_of_pos hxy' hxy'']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y \u2260 1\n\u22a2 -log (x / y) = -(log x - log y)\n[PROOFSTEP]\nrw [log_div hx.ne' hy.ne']\n[GOAL]\nx y z : \u211d\nhx : 0 < x\nhz : 0 < z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nh : 0 < y - x\nhxy' : 0 < x / y\nhxy'' : x / y \u2260 1\n\u22a2 -(log x - log y) = log y - log x\n[PROOFSTEP]\nring\n[GOAL]\ns : \u211d\nhs : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\n\u22a2 1 + p * s < (1 + s) ^ p\n[PROOFSTEP]\nrcases eq_or_lt_of_le hs with (rfl | hs)\n[GOAL]\ncase inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p\n[PROOFSTEP]\nhave : p \u2260 0 := by positivity\n[GOAL]\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\n\u22a2 p \u2260 0\n[PROOFSTEP]\npositivity\n[GOAL]\ncase inl\np : \u211d\nhp : 1 < p\nhs : -1 \u2264 -1\nhs' : -1 \u2260 0\nthis : p \u2260 0\n\u22a2 1 + p * -1 < (1 + -1) ^ p\n[PROOFSTEP]\nsimpa [zero_rpow this]\n[GOAL]\ncase inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\n\u22a2 1 + p * s < (1 + s) ^ p\n[PROOFSTEP]\nhave hs1 : 0 < 1 + s := by linarith\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\n\u22a2 0 < 1 + s\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\n\u22a2 1 + p * s < (1 + s) ^ p\n[PROOFSTEP]\ncases' le_or_lt (1 + p * s) 0 with hs2 hs2\n[GOAL]\ncase inr.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 1 + p * s \u2264 0\n\u22a2 1 + p * s < (1 + s) ^ p\n[PROOFSTEP]\nexact hs2.trans_lt (rpow_pos_of_pos hs1 _)\n[GOAL]\ncase inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 1 + p * s < (1 + s) ^ p\n[PROOFSTEP]\nrw [rpow_def_of_pos hs1, \u2190 exp_log hs2]\n[GOAL]\ncase inr.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 exp (log (1 + p * s)) < exp (log (1 + s) * p)\n[PROOFSTEP]\napply exp_strictMono\n[GOAL]\ncase inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\nhave hp : 0 < p := by positivity\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\n\u22a2 0 < p\n[PROOFSTEP]\npositivity\n[GOAL]\ncase inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\nhave hs3 : 1 + s \u2260 1 := by contrapose! hs'; linarith\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\n\u22a2 1 + s \u2260 1\n[PROOFSTEP]\ncontrapose! hs'\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs' : 1 + s = 1\n\u22a2 s = 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\nhave hs4 : 1 + p * s \u2260 1 := by contrapose! hs'; nlinarith\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\n\u22a2 1 + p * s \u2260 1\n[PROOFSTEP]\ncontrapose! hs'\n[GOAL]\ns : \u211d\nhs\u271d : -1 \u2264 s\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs' : 1 + p * s = 1\n\u22a2 s = 0\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.a\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs' : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\ncases' lt_or_gt_of_ne hs' with hs' hs'\n[GOAL]\ncase inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\nrw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right_of_neg hs']\n  -- Porting note: previously we could write `zero_lt_one` inline,\n      -- but now Lean doesn't guess we are talking about `1` fast enough.\n[GOAL]\ncase inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s\n[PROOFSTEP]\nhaveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one\n[GOAL]\ncase inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s < log (1 + p * s) / p / s\n[PROOFSTEP]\nconvert strictConcaveOn_log_Ioi.secant_strict_mono this hs2 hs1 hs4 hs3 _ using 1\n[GOAL]\ncase h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)\n[PROOFSTEP]\nfield_simp [log_one]\n[GOAL]\ncase h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)\n[PROOFSTEP]\nfield_simp [log_one]\n[GOAL]\ncase inr.inr.a.inl\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s < 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + p * s < 1 + s\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) < log (1 + s) * p\n[PROOFSTEP]\nrw [\u2190 div_lt_iff hp, \u2190 div_lt_div_right hs']\n  -- Porting note: previously we could write `zero_lt_one` inline,\n      -- but now Lean doesn't guess we are talking about `1` fast enough.\n[GOAL]\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s\n[PROOFSTEP]\nhaveI : (1 : \u211d) \u2208 Ioi 0 := zero_lt_one\n[GOAL]\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s < log (1 + s) / s\n[PROOFSTEP]\nconvert strictConcaveOn_log_Ioi.secant_strict_mono this hs1 hs2 hs3 hs4 _ using 1\n[GOAL]\ncase h.e'_3\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + p * s) / p / s = (log (1 + p * s) - log 1) / (1 + p * s - 1)\n[PROOFSTEP]\nfield_simp [log_one, hp.ne']\n[GOAL]\ncase h.e'_4\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 log (1 + s) / s = (log (1 + s) - log 1) / (1 + s - 1)\n[PROOFSTEP]\nfield_simp [log_one]\n[GOAL]\ncase inr.inr.a.inr\ns : \u211d\nhs\u271d : -1 \u2264 s\nhs'\u271d : s \u2260 0\np : \u211d\nhp\u271d : 1 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhp : 0 < p\nhs3 : 1 + s \u2260 1\nhs4 : 1 + p * s \u2260 1\nhs' : s > 0\nthis : 1 \u2208 Ioi 0\n\u22a2 1 + s < 1 + p * s\n[PROOFSTEP]\nnlinarith\n[GOAL]\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp : 1 \u2264 p\n\u22a2 1 + p * s \u2264 (1 + s) ^ p\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp with (rfl | hp)\n[GOAL]\ncase inl\ns : \u211d\nhs : -1 \u2264 s\nhp : 1 \u2264 1\n\u22a2 1 + 1 * s \u2264 (1 + s) ^ 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 1 + p * s \u2264 (1 + s) ^ p\n[PROOFSTEP]\nby_cases hs' : s = 0\n[GOAL]\ncase pos\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : s = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p\n[PROOFSTEP]\nsimp [hs']\n[GOAL]\ncase neg\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : \u00acs = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p\n[PROOFSTEP]\nexact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le\n[GOAL]\np : \u211d\nhp : 1 < p\n\u22a2 StrictConvexOn \u211d (Ici 0) fun x => x ^ p\n[PROOFSTEP]\napply strictConvexOn_of_slope_strict_mono_adjacent (convex_Ici (0 : \u211d))\n[GOAL]\np : \u211d\nhp : 1 < p\n\u22a2 \u2200 {x y z : \u211d}, x \u2208 Ici 0 \u2192 z \u2208 Ici 0 \u2192 x < y \u2192 y < z \u2192 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nrintro x y z (hx : 0 \u2264 x) (hz : 0 \u2264 z) hxy hyz\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\n\u22a2 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hy : 0 < y := by linarith\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\n\u22a2 0 < y\n[PROOFSTEP]\nlinarith\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\n\u22a2 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hy' : 0 < y ^ p := rpow_pos_of_pos hy _\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\n\u22a2 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave H1 : y ^ (p - 1 + 1) = y ^ (p - 1) * y := rpow_add_one hy.ne' _\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ (p - 1 + 1) = y ^ (p - 1) * y\n\u22a2 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nring_nf at H1 \n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\n\u22a2 (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\ntrans p * y ^ (p - 1)\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave h3 : 0 < y - x := by linarith only [hxy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\n\u22a2 0 < y - x\n[PROOFSTEP]\nlinarith only [hxy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave hyx'' : x / y < 1 := by rwa [div_lt_one hy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\n\u22a2 x / y < 1\n[PROOFSTEP]\nrwa [div_lt_one hy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave hyx''' : x / y - 1 < 0 := by linarith only [hyx'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\n\u22a2 x / y - 1 < 0\n[PROOFSTEP]\nlinarith only [hyx'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave hyx'''' : 0 \u2264 x / y := by positivity\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\n\u22a2 0 \u2264 x / y\n[PROOFSTEP]\npositivity\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave hyx''''' : -1 \u2264 x / y - 1 := by linarith only [hyx'''']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\n\u22a2 -1 \u2264 x / y - 1\n[PROOFSTEP]\nlinarith only [hyx'''']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nhave : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1) := by\n  linarith [one_add_mul_self_lt_rpow_one_add hyx''''' hyx'''.ne hp]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\n\u22a2 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n[PROOFSTEP]\nlinarith [one_add_mul_self_lt_rpow_one_add hyx''''' hyx'''.ne hp]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 (y ^ p - x ^ p) / (y - x) < p * y ^ (p - 1)\n[PROOFSTEP]\nrw [div_lt_iff h3, \u2190 div_lt_div_right hy']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 (y ^ p - x ^ p) / y ^ p < p * y ^ (p - 1) * (y - x) / y ^ p\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 (y ^ p - x ^ p) / y ^ p = 1 - (1 + (x / y - 1)) ^ p\n[PROOFSTEP]\nhave H : (x / y) ^ p = x ^ p / y ^ p := div_rpow hx hy.le _\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\nH : (x / y) ^ p = x ^ p / y ^ p\n\u22a2 (y ^ p - x ^ p) / y ^ p = 1 - (1 + (x / y - 1)) ^ p\n[PROOFSTEP]\nring_nf at H \u22a2\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\nH : (x * y\u207b\u00b9) ^ p = x ^ p * (y ^ p)\u207b\u00b9\n\u22a2 -(x ^ p * (y ^ p)\u207b\u00b9) + y ^ p * (y ^ p)\u207b\u00b9 = 1 - (x * y\u207b\u00b9) ^ p\n[PROOFSTEP]\nfield_simp [hy.ne', hy'.ne'] at H \u22a2\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\nH : (x / y) ^ p * y ^ p = x ^ p\n\u22a2 -x ^ p + y ^ p = (1 - (x / y) ^ p) * y ^ p\n[PROOFSTEP]\nlinear_combination H\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 p * y ^ (p - 1) * (y - x) / y ^ p = -p * (x / y - 1)\n[PROOFSTEP]\nring_nf at H1 \u22a2\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 p * y ^ (-1 + p) * y * (y ^ p)\u207b\u00b9 - p * y ^ (-1 + p) * x * (y ^ p)\u207b\u00b9 = p - p * x * y\u207b\u00b9\n[PROOFSTEP]\nfield_simp [hy.ne', hy'.ne']\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nh3 : 0 < y - x\nhyx'' : x / y < 1\nhyx''' : x / y - 1 < 0\nhyx'''' : 0 \u2264 x / y\nhyx''''' : -1 \u2264 x / y - 1\nthis : 1 - (1 + (x / y - 1)) ^ p < -p * (x / y - 1)\n\u22a2 (p * y ^ (-1 + p) * y - p * y ^ (-1 + p) * x) * y = (p * y - p * x) * y ^ p\n[PROOFSTEP]\nlinear_combination p * (-y + x) * H1\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hyz' : 0 < z - y := by linarith only [hyz]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\n\u22a2 0 < z - y\n[PROOFSTEP]\nlinarith only [hyz]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hyz'' : 1 < z / y := by rwa [one_lt_div hy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\n\u22a2 1 < z / y\n[PROOFSTEP]\nrwa [one_lt_div hy]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hyz''' : 0 < z / y - 1 := by linarith only [hyz'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\n\u22a2 0 < z / y - 1\n[PROOFSTEP]\nlinarith only [hyz'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave hyz'''' : -1 \u2264 z / y - 1 := by linarith only [hyz'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\n\u22a2 -1 \u2264 z / y - 1\n[PROOFSTEP]\nlinarith only [hyz'']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nhave : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1 := by\n  linarith [one_add_mul_self_lt_rpow_one_add hyz'''' hyz'''.ne' hp]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\n\u22a2 p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n[PROOFSTEP]\nlinarith [one_add_mul_self_lt_rpow_one_add hyz'''' hyz'''.ne' hp]\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 p * y ^ (p - 1) < (z ^ p - y ^ p) / (z - y)\n[PROOFSTEP]\nrw [lt_div_iff hyz', \u2190 div_lt_div_right hy']\n[GOAL]\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 p * y ^ (p - 1) * (z - y) / y ^ p < (z ^ p - y ^ p) / y ^ p\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 p * y ^ (p - 1) * (z - y) / y ^ p = p * (z / y - 1)\n[PROOFSTEP]\nring_nf at H1 \u22a2\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 -(p * y ^ (-1 + p) * y * (y ^ p)\u207b\u00b9) + p * y ^ (-1 + p) * z * (y ^ p)\u207b\u00b9 = -p + p * z * y\u207b\u00b9\n[PROOFSTEP]\nfield_simp [hy.ne', hy'.ne'] at H1 \u22a2\n[GOAL]\ncase h.e'_3\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 (-(p * y ^ (-1 + p) * y * y ^ p) + p * y ^ (-1 + p) * z * y ^ p) * y = (-(p * y) + p * z) * (y ^ p * y ^ p)\n[PROOFSTEP]\nlinear_combination p * (y - z) * y ^ p * H1\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\n\u22a2 (z ^ p - y ^ p) / y ^ p = (1 + (z / y - 1)) ^ p - 1\n[PROOFSTEP]\nhave H : (z / y) ^ p = z ^ p / y ^ p := div_rpow hz hy.le _\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\nH : (z / y) ^ p = z ^ p / y ^ p\n\u22a2 (z ^ p - y ^ p) / y ^ p = (1 + (z / y - 1)) ^ p - 1\n[PROOFSTEP]\nring_nf at H \u22a2\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\nH : (z * y\u207b\u00b9) ^ p = z ^ p * (y ^ p)\u207b\u00b9\n\u22a2 z ^ p * (y ^ p)\u207b\u00b9 - y ^ p * (y ^ p)\u207b\u00b9 = -1 + (z * y\u207b\u00b9) ^ p\n[PROOFSTEP]\nfield_simp [hy.ne', hy'.ne'] at H \u22a2\n[GOAL]\ncase h.e'_4\np : \u211d\nhp : 1 < p\nx y z : \u211d\nhx : 0 \u2264 x\nhz : 0 \u2264 z\nhxy : x < y\nhyz : y < z\nhy : 0 < y\nhy' : 0 < y ^ p\nH1 : y ^ p = y ^ (-1 + p) * y\nhyz' : 0 < z - y\nhyz'' : 1 < z / y\nhyz''' : 0 < z / y - 1\nhyz'''' : -1 \u2264 z / y - 1\nthis : p * (z / y - 1) < (1 + (z / y - 1)) ^ p - 1\nH : (z / y) ^ p * y ^ p = z ^ p\n\u22a2 z ^ p - y ^ p = (-1 + (z / y) ^ p) * y ^ p\n[PROOFSTEP]\nlinear_combination -H\n[GOAL]\np : \u211d\nhp : 1 \u2264 p\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ p\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp with (rfl | hp)\n[GOAL]\ncase inl\nhp : 1 \u2264 1\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ 1\n[PROOFSTEP]\nsimpa using convexOn_id (convex_Ici _)\n[GOAL]\ncase inr\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 ConvexOn \u211d (Ici 0) fun x => x ^ p\n[PROOFSTEP]\nexact (strictConvexOn_rpow hp).convexOn\n[GOAL]\n\u22a2 StrictConcaveOn \u211d (Iio 0) log\n[PROOFSTEP]\nrefine' \u27e8convex_Iio _, _\u27e9\n[GOAL]\n\u22a2 \u2200 \u2983x : \u211d\u2984,\n    x \u2208 Iio 0 \u2192\n      \u2200 \u2983y : \u211d\u2984,\n        y \u2208 Iio 0 \u2192 x \u2260 y \u2192 \u2200 \u2983a b : \u211d\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 log x + b \u2022 log y < log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nrintro x (hx : x < 0) y (hy : y < 0) hxy a b ha hb hab\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 log x + b \u2022 log y < log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nhave hx' : 0 < -x := by linarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 0 < -x\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\n\u22a2 a \u2022 log x + b \u2022 log y < log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nhave hy' : 0 < -y := by linarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\n\u22a2 0 < -y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\n\u22a2 a \u2022 log x + b \u2022 log y < log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nhave hxy' : -x \u2260 -y := by contrapose! hxy; linarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\n\u22a2 -x \u2260 -y\n[PROOFSTEP]\ncontrapose! hxy\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy : -x = -y\n\u22a2 x = y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 a \u2022 log x + b \u2022 log y < log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\ncalc\n  a \u2022 log x + b \u2022 log y = a \u2022 log (-x) + b \u2022 log (-y) := by simp_rw [log_neg_eq_log]\n  _ < log (a \u2022 -x + b \u2022 -y) := (strictConcaveOn_log_Ioi.2 hx' hy' hxy' ha hb hab)\n  _ = log (-(a \u2022 x + b \u2022 y)) := by congr 1; simp only [Algebra.id.smul_eq_mul]; ring\n  _ = _ := by rw [log_neg_eq_log]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 a \u2022 log x + b \u2022 log y = a \u2022 log (-x) + b \u2022 log (-y)\n[PROOFSTEP]\nsimp_rw [log_neg_eq_log]\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 log (a \u2022 -x + b \u2022 -y) = log (-(a \u2022 x + b \u2022 y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_x\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 a \u2022 -x + b \u2022 -y = -(a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nsimp only [Algebra.id.smul_eq_mul]\n[GOAL]\ncase e_x\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 a * -x + b * -y = -(a * x + b * y)\n[PROOFSTEP]\nring\n[GOAL]\nx : \u211d\nhx : x < 0\ny : \u211d\nhy : y < 0\nhxy : x \u2260 y\na b : \u211d\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx' : 0 < -x\nhy' : 0 < -y\nhxy' : -x \u2260 -y\n\u22a2 log (-(a \u2022 x + b \u2022 y)) = log (a \u2022 x + b \u2022 y)\n[PROOFSTEP]\nrw [log_neg_eq_log]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.SpecificFunctions.Basic", "llama_tokens": 32142, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8397339556397749, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5257777320656227}}
{"text": "[GOAL]\n\u22a2 MeasurablePow \u211d\u22650\u221e \u211d\n[PROOFSTEP]\nrefine' \u27e8ENNReal.measurable_of_measurable_nnreal_prod _ _\u27e9\n[GOAL]\ncase refine'_1\n\u22a2 Measurable fun p => (\u2191p.fst, p.snd).fst ^ (\u2191p.fst, p.snd).snd\n[PROOFSTEP]\nsimp_rw [ENNReal.coe_rpow_def]\n[GOAL]\ncase refine'_1\n\u22a2 Measurable fun p => if p.fst = 0 \u2227 p.snd < 0 then \u22a4 else \u2191(p.fst ^ p.snd)\n[PROOFSTEP]\nrefine' Measurable.ite _ measurable_const (measurable_fst.pow measurable_snd).coe_nnreal_ennreal\n[GOAL]\ncase refine'_1\n\u22a2 MeasurableSet {a | a.fst = 0 \u2227 a.snd < 0}\n[PROOFSTEP]\nexact MeasurableSet.inter (measurable_fst (measurableSet_singleton 0)) (measurable_snd measurableSet_Iio)\n[GOAL]\ncase refine'_2\n\u22a2 Measurable fun x => (\u22a4, x).fst ^ (\u22a4, x).snd\n[PROOFSTEP]\nsimp_rw [ENNReal.top_rpow_def]\n[GOAL]\ncase refine'_2\n\u22a2 Measurable fun x => if 0 < x then \u22a4 else if x = 0 then 1 else 0\n[PROOFSTEP]\nrefine' Measurable.ite measurableSet_Ioi measurable_const _\n[GOAL]\ncase refine'_2\n\u22a2 Measurable fun x => if x = 0 then 1 else 0\n[PROOFSTEP]\nexact Measurable.ite (measurableSet_singleton 0) measurable_const measurable_const\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.SpecialFunctions.Basic", "llama_tokens": 494, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998714925403, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.5252717324556295}}
{"text": "[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 \u2200 \u2983X Y : C \u2964 D\u2984 (f : X \u27f6 Y),\n    (\ud835\udfed (C \u2964 D)).map f \u226b\n        (fun X =>\n            (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n          Y =\n      (fun X => (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n          X \u226b\n        ((whiskeringRight C D E).obj F \u22d9 (whiskeringRight C E D).obj G).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 D\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (C \u2964 D)).map f\u271d \u226b\n      (fun X => (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n        Y\u271d =\n    (fun X => (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n        X\u271d \u226b\n      ((whiskeringRight C D E).obj F \u22d9 (whiskeringRight C E D).obj G).map f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 D\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : C\n\u22a2 NatTrans.app\n      ((\ud835\udfed (C \u2964 D)).map f\u271d \u226b\n        (fun X =>\n            (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n          Y\u271d)\n      x\u271d =\n    NatTrans.app\n      ((fun X =>\n            (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n          X\u271d \u226b\n        ((whiskeringRight C D E).obj F \u22d9 (whiskeringRight C E D).obj G).map f\u271d)\n      x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 D\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : C\n\u22a2 NatTrans.app f\u271d x\u271d \u226b \ud835\udfd9 (Y\u271d.obj x\u271d) \u226b NatTrans.app adj.unit (Y\u271d.obj x\u271d) \u226b \ud835\udfd9 (G.obj (F.obj (Y\u271d.obj x\u271d))) =\n    (\ud835\udfd9 (X\u271d.obj x\u271d) \u226b NatTrans.app adj.unit (X\u271d.obj x\u271d) \u226b \ud835\udfd9 (G.obj (F.obj (X\u271d.obj x\u271d)))) \u226b\n      G.map (F.map (NatTrans.app f\u271d x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 \u2200 \u2983X Y : C \u2964 E\u2984 (f : X \u27f6 Y),\n    ((whiskeringRight C E D).obj G \u22d9 (whiskeringRight C D E).obj F).map f \u226b\n        (fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) Y =\n      (fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) X \u226b\n        (\ud835\udfed (C \u2964 E)).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 E\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((whiskeringRight C E D).obj G \u22d9 (whiskeringRight C D E).obj F).map f\u271d \u226b\n      (fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) Y\u271d =\n    (fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) X\u271d \u226b\n      (\ud835\udfed (C \u2964 E)).map f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 E\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : C\n\u22a2 NatTrans.app\n      (((whiskeringRight C E D).obj G \u22d9 (whiskeringRight C D E).obj F).map f\u271d \u226b\n        (fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) Y\u271d)\n      x\u271d =\n    NatTrans.app\n      ((fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) X\u271d \u226b\n        (\ud835\udfed (C \u2964 E)).map f\u271d)\n      x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : C \u2964 E\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : C\n\u22a2 F.map (G.map (NatTrans.app f\u271d x\u271d)) \u226b\n      \ud835\udfd9 (F.obj (G.obj (Y\u271d.obj x\u271d))) \u226b NatTrans.app adj.counit (Y\u271d.obj x\u271d) \u226b \ud835\udfd9 (Y\u271d.obj x\u271d) =\n    (\ud835\udfd9 (F.obj (G.obj (X\u271d.obj x\u271d))) \u226b NatTrans.app adj.counit (X\u271d.obj x\u271d) \u226b \ud835\udfd9 (X\u271d.obj x\u271d)) \u226b NatTrans.app f\u271d x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 whiskerRight\n        (NatTrans.mk fun X =>\n          (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n        ((whiskeringRight C D E).obj F) \u226b\n      (Functor.associator ((whiskeringRight C D E).obj F) ((whiskeringRight C E D).obj G)\n            ((whiskeringRight C D E).obj F)).hom \u226b\n        whiskerLeft ((whiskeringRight C D E).obj F)\n          (NatTrans.mk fun X =>\n            (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom) =\n    NatTrans.id (\ud835\udfed (C \u2964 D) \u22d9 (whiskeringRight C D E).obj F)\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx\u271d\u00b9 : C \u2964 D\nx\u271d : C\n\u22a2 NatTrans.app\n      (NatTrans.app\n        (whiskerRight\n            (NatTrans.mk fun X =>\n              (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv)\n            ((whiskeringRight C D E).obj F) \u226b\n          (Functor.associator ((whiskeringRight C D E).obj F) ((whiskeringRight C E D).obj G)\n                ((whiskeringRight C D E).obj F)).hom \u226b\n            whiskerLeft ((whiskeringRight C D E).obj F)\n              (NatTrans.mk fun X =>\n                (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom))\n        x\u271d\u00b9)\n      x\u271d =\n    NatTrans.app (NatTrans.app (NatTrans.id (\ud835\udfed (C \u2964 D) \u22d9 (whiskeringRight C D E).obj F)) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx\u271d\u00b9 : C \u2964 D\nx\u271d : C\n\u22a2 F.map (\ud835\udfd9 (x\u271d\u00b9.obj x\u271d) \u226b NatTrans.app adj.unit (x\u271d\u00b9.obj x\u271d) \u226b \ud835\udfd9 (G.obj (F.obj (x\u271d\u00b9.obj x\u271d)))) \u226b\n      \ud835\udfd9 (F.obj (G.obj (F.obj (x\u271d\u00b9.obj x\u271d)))) \u226b\n        \ud835\udfd9 (F.obj (G.obj (F.obj (x\u271d\u00b9.obj x\u271d)))) \u226b NatTrans.app adj.counit (F.obj (x\u271d\u00b9.obj x\u271d)) \u226b \ud835\udfd9 (F.obj (x\u271d\u00b9.obj x\u271d)) =\n    \ud835\udfd9 (F.obj (x\u271d\u00b9.obj x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 whiskerLeft ((whiskeringRight C E D).obj G)\n        (NatTrans.mk fun X =>\n          (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv) \u226b\n      (Functor.associator ((whiskeringRight C E D).obj G) ((whiskeringRight C D E).obj F)\n            ((whiskeringRight C E D).obj G)).inv \u226b\n        whiskerRight\n          (NatTrans.mk fun X => (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom)\n          ((whiskeringRight C E D).obj G) =\n    NatTrans.id ((whiskeringRight C E D).obj G \u22d9 \ud835\udfed (C \u2964 D))\n[PROOFSTEP]\next\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx\u271d\u00b9 : C \u2964 E\nx\u271d : C\n\u22a2 NatTrans.app\n      (NatTrans.app\n        (whiskerLeft ((whiskeringRight C E D).obj G)\n            (NatTrans.mk fun X =>\n              (Functor.rightUnitor ((\ud835\udfed (C \u2964 D)).obj X)).inv \u226b whiskerLeft X adj.unit \u226b (Functor.associator X F G).inv) \u226b\n          (Functor.associator ((whiskeringRight C E D).obj G) ((whiskeringRight C D E).obj F)\n                ((whiskeringRight C E D).obj G)).inv \u226b\n            whiskerRight\n              (NatTrans.mk fun X =>\n                (Functor.associator X G F).hom \u226b whiskerLeft X adj.counit \u226b (Functor.rightUnitor X).hom)\n              ((whiskeringRight C E D).obj G))\n        x\u271d\u00b9)\n      x\u271d =\n    NatTrans.app (NatTrans.app (NatTrans.id ((whiskeringRight C E D).obj G \u22d9 \ud835\udfed (C \u2964 D))) x\u271d\u00b9) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.63, u_1} C\ninst\u271d\u00b9 : Category.{?u.67, u_2} D\ninst\u271d : Category.{?u.71, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx\u271d\u00b9 : C \u2964 E\nx\u271d : C\n\u22a2 (\ud835\udfd9 (G.obj (x\u271d\u00b9.obj x\u271d)) \u226b NatTrans.app adj.unit (G.obj (x\u271d\u00b9.obj x\u271d)) \u226b \ud835\udfd9 (G.obj (F.obj (G.obj (x\u271d\u00b9.obj x\u271d))))) \u226b\n      \ud835\udfd9 (G.obj (F.obj (G.obj (x\u271d\u00b9.obj x\u271d)))) \u226b\n        G.map (\ud835\udfd9 (F.obj (G.obj (x\u271d\u00b9.obj x\u271d))) \u226b NatTrans.app adj.counit (x\u271d\u00b9.obj x\u271d) \u226b \ud835\udfd9 (x\u271d\u00b9.obj x\u271d)) =\n    \ud835\udfd9 (G.obj (x\u271d\u00b9.obj x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 \u2200 \u2983X Y : D \u2964 C\u2984 (f : X \u27f6 Y),\n    (\ud835\udfed (D \u2964 C)).map f \u226b\n        (fun X =>\n            (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n          Y =\n      (fun X => (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n          X \u226b\n        ((whiskeringLeft E D C).obj G \u22d9 (whiskeringLeft D E C).obj F).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : D \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 (\ud835\udfed (D \u2964 C)).map f\u271d \u226b\n      (fun X => (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n        Y\u271d =\n    (fun X => (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n        X\u271d \u226b\n      ((whiskeringLeft E D C).obj G \u22d9 (whiskeringLeft D E C).obj F).map f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : D \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : D\n\u22a2 NatTrans.app\n      ((\ud835\udfed (D \u2964 C)).map f\u271d \u226b\n        (fun X =>\n            (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n          Y\u271d)\n      x\u271d =\n    NatTrans.app\n      ((fun X =>\n            (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n          X\u271d \u226b\n        ((whiskeringLeft E D C).obj G \u22d9 (whiskeringLeft D E C).obj F).map f\u271d)\n      x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : D \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : D\n\u22a2 NatTrans.app f\u271d x\u271d \u226b \ud835\udfd9 (Y\u271d.obj x\u271d) \u226b Y\u271d.map (NatTrans.app adj.unit x\u271d) \u226b \ud835\udfd9 (Y\u271d.obj (G.obj (F.obj x\u271d))) =\n    (\ud835\udfd9 (X\u271d.obj x\u271d) \u226b X\u271d.map (NatTrans.app adj.unit x\u271d) \u226b \ud835\udfd9 (X\u271d.obj (G.obj (F.obj x\u271d)))) \u226b\n      NatTrans.app f\u271d (G.obj (F.obj x\u271d))\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 \u2200 \u2983X Y : E \u2964 C\u2984 (f : X \u27f6 Y),\n    ((whiskeringLeft D E C).obj F \u22d9 (whiskeringLeft E D C).obj G).map f \u226b\n        (fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) Y =\n      (fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) X \u226b\n        (\ud835\udfed (E \u2964 C)).map f\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : E \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 ((whiskeringLeft D E C).obj F \u22d9 (whiskeringLeft E D C).obj G).map f\u271d \u226b\n      (fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) Y\u271d =\n    (fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) X\u271d \u226b\n      (\ud835\udfed (E \u2964 C)).map f\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : E \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : E\n\u22a2 NatTrans.app\n      (((whiskeringLeft D E C).obj F \u22d9 (whiskeringLeft E D C).obj G).map f\u271d \u226b\n        (fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) Y\u271d)\n      x\u271d =\n    NatTrans.app\n      ((fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) X\u271d \u226b\n        (\ud835\udfed (E \u2964 C)).map f\u271d)\n      x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nX\u271d Y\u271d : E \u2964 C\nf\u271d : X\u271d \u27f6 Y\u271d\nx\u271d : E\n\u22a2 NatTrans.app f\u271d (F.obj (G.obj x\u271d)) \u226b\n      \ud835\udfd9 (Y\u271d.obj (F.obj (G.obj x\u271d))) \u226b Y\u271d.map (NatTrans.app adj.counit x\u271d) \u226b \ud835\udfd9 (Y\u271d.obj x\u271d) =\n    (\ud835\udfd9 (X\u271d.obj (F.obj (G.obj x\u271d))) \u226b X\u271d.map (NatTrans.app adj.counit x\u271d) \u226b \ud835\udfd9 (X\u271d.obj x\u271d)) \u226b NatTrans.app f\u271d x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 whiskerRight\n        (NatTrans.mk fun X =>\n          (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n        ((whiskeringLeft E D C).obj G) \u226b\n      (Functor.associator ((whiskeringLeft E D C).obj G) ((whiskeringLeft D E C).obj F)\n            ((whiskeringLeft E D C).obj G)).hom \u226b\n        whiskerLeft ((whiskeringLeft E D C).obj G)\n          (NatTrans.mk fun X =>\n            (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom) =\n    NatTrans.id (\ud835\udfed (D \u2964 C) \u22d9 (whiskeringLeft E D C).obj G)\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx : D \u2964 C\nx\u271d : E\n\u22a2 NatTrans.app\n      (NatTrans.app\n        (whiskerRight\n            (NatTrans.mk fun X =>\n              (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom)\n            ((whiskeringLeft E D C).obj G) \u226b\n          (Functor.associator ((whiskeringLeft E D C).obj G) ((whiskeringLeft D E C).obj F)\n                ((whiskeringLeft E D C).obj G)).hom \u226b\n            whiskerLeft ((whiskeringLeft E D C).obj G)\n              (NatTrans.mk fun X =>\n                (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom))\n        x)\n      x\u271d =\n    NatTrans.app (NatTrans.app (NatTrans.id (\ud835\udfed (D \u2964 C) \u22d9 (whiskeringLeft E D C).obj G)) x) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx : D \u2964 C\nx\u271d : E\n\u22a2 (\ud835\udfd9 (x.obj (G.obj x\u271d)) \u226b x.map (NatTrans.app adj.unit (G.obj x\u271d)) \u226b \ud835\udfd9 (x.obj (G.obj (F.obj (G.obj x\u271d))))) \u226b\n      \ud835\udfd9 (x.obj (G.obj (F.obj (G.obj x\u271d)))) \u226b\n        \ud835\udfd9 (x.obj (G.obj (F.obj (G.obj x\u271d)))) \u226b x.map (G.map (NatTrans.app adj.counit x\u271d)) \u226b \ud835\udfd9 (x.obj (G.obj x\u271d)) =\n    \ud835\udfd9 (x.obj (G.obj x\u271d))\n[PROOFSTEP]\nsimp [Category.id_comp, Category.comp_id, \u2190 x.map_comp]\n[GOAL]\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\n\u22a2 whiskerLeft ((whiskeringLeft D E C).obj F)\n        (NatTrans.mk fun X =>\n          (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom) \u226b\n      (Functor.associator ((whiskeringLeft D E C).obj F) ((whiskeringLeft E D C).obj G)\n            ((whiskeringLeft D E C).obj F)).inv \u226b\n        whiskerRight\n          (NatTrans.mk fun X => (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom)\n          ((whiskeringLeft D E C).obj F) =\n    NatTrans.id ((whiskeringLeft D E C).obj F \u22d9 \ud835\udfed (D \u2964 C))\n[PROOFSTEP]\next x\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx : E \u2964 C\nx\u271d : D\n\u22a2 NatTrans.app\n      (NatTrans.app\n        (whiskerLeft ((whiskeringLeft D E C).obj F)\n            (NatTrans.mk fun X =>\n              (Functor.leftUnitor ((\ud835\udfed (D \u2964 C)).obj X)).inv \u226b whiskerRight adj.unit X \u226b (Functor.associator F G X).hom) \u226b\n          (Functor.associator ((whiskeringLeft D E C).obj F) ((whiskeringLeft E D C).obj G)\n                ((whiskeringLeft D E C).obj F)).inv \u226b\n            whiskerRight\n              (NatTrans.mk fun X =>\n                (Functor.associator G F X).inv \u226b whiskerRight adj.counit X \u226b (Functor.leftUnitor X).hom)\n              ((whiskeringLeft D E C).obj F))\n        x)\n      x\u271d =\n    NatTrans.app (NatTrans.app (NatTrans.id ((whiskeringLeft D E C).obj F \u22d9 \ud835\udfed (D \u2964 C))) x) x\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase w.h.w.h\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b2 : Category.{?u.21134, u_1} C\ninst\u271d\u00b9 : Category.{?u.21138, u_2} D\ninst\u271d : Category.{?u.21142, u_3} E\nF : D \u2964 E\nG : E \u2964 D\nadj : F \u22a3 G\nx : E \u2964 C\nx\u271d : D\n\u22a2 (\ud835\udfd9 (x.obj (F.obj x\u271d)) \u226b x.map (F.map (NatTrans.app adj.unit x\u271d)) \u226b \ud835\udfd9 (x.obj (F.obj (G.obj (F.obj x\u271d))))) \u226b\n      \ud835\udfd9 (x.obj (F.obj (G.obj (F.obj x\u271d)))) \u226b\n        \ud835\udfd9 (x.obj (F.obj (G.obj (F.obj x\u271d)))) \u226b x.map (NatTrans.app adj.counit (F.obj x\u271d)) \u226b \ud835\udfd9 (x.obj (F.obj x\u271d)) =\n    \ud835\udfd9 (x.obj (F.obj x\u271d))\n[PROOFSTEP]\nsimp [Category.id_comp, Category.comp_id, \u2190 x.map_comp]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Adjunction.Whiskering", "llama_tokens": 9862, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.82893881677331, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5250583545051791}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : Fin n \u2192 \u03b2\n\u22a2 \u220f i : Fin n, f i = List.prod (List.map f (List.finRange n))\n[PROOFSTEP]\nsimp [univ_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : Fin n \u2192 \u03b2\n\u22a2 List.prod (List.ofFn f) = \u220f i : Fin n, f i\n[PROOFSTEP]\nrw [List.ofFn_eq_map, prod_univ_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b2\nx : Fin (n + 1)\n\u22a2 \u220f i : Fin (n + 1), f i = f x * \u220f i : Fin n, f (succAbove x i)\n[PROOFSTEP]\nrw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb.toEmbedding, RelEmbedding.coe_toEmbedding]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b2\nx : Fin (n + 1)\n\u22a2 f x * \u220f x_1 : Fin n, f (\u2191(succAboveEmb x) x_1) = f x * \u220f i : Fin n, f (succAbove x i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b2\n\u22a2 \u220f i : Fin (n + 1), f i = (\u220f i : Fin n, f (castSucc i)) * f (last n)\n[PROOFSTEP]\nsimpa [mul_comm] using prod_univ_succAbove f (last n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nx : \u03b2\nf : Fin n \u2192 \u03b2\n\u22a2 \u220f i : Fin (Nat.succ n), cons x f i = x * \u220f i : Fin n, f i\n[PROOFSTEP]\nsimp_rw [prod_univ_succ, cons_zero, cons_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 1 \u2192 \u03b2\n\u22a2 \u220f i : Fin 1, f i = f 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 2 \u2192 \u03b2\n\u22a2 \u220f i : Fin 2, f i = f 0 * f 1\n[PROOFSTEP]\nsimp [prod_univ_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 3 \u2192 \u03b2\n\u22a2 \u220f i : Fin 3, f i = f 0 * f 1 * f 2\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_two]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 3 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (last 2) = f 0 * f 1 * f 2\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 4 \u2192 \u03b2\n\u22a2 \u220f i : Fin 4, f i = f 0 * f 1 * f 2 * f 3\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_three]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 4 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (last 3) = f 0 * f 1 * f 2 * f 3\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 5 \u2192 \u03b2\n\u22a2 \u220f i : Fin 5, f i = f 0 * f 1 * f 2 * f 3 * f 4\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_four]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 5 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (last 4) = f 0 * f 1 * f 2 * f 3 * f 4\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 6 \u2192 \u03b2\n\u22a2 \u220f i : Fin 6, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_five]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 6 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (last 5) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 7 \u2192 \u03b2\n\u22a2 \u220f i : Fin 7, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_six]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 7 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (last 6) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 8 \u2192 \u03b2\n\u22a2 \u220f i : Fin 8, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7\n[PROOFSTEP]\nrw [prod_univ_castSucc, prod_univ_seven]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b2\nf : Fin 8 \u2192 \u03b2\n\u22a2 f (castSucc 0) * f (castSucc 1) * f (castSucc 2) * f (castSucc 3) * f (castSucc 4) * f (castSucc 5) * f (castSucc 6) *\n      f (last 7) =\n    f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nR : Type u_3\ninst\u271d : CommSemiring R\na b : R\n\u22a2 \u2211 s : Finset (Fin n), a ^ card s * b ^ (n - card s) = (a + b) ^ n\n[PROOFSTEP]\nsimpa using Fintype.sum_pow_mul_eq_add_pow (Fin n) a b\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 \u220f _i : Fin n, x = x ^ n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : AddCommMonoid \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 \u2211 _i : Fin n, x = n \u2022 x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\nn : \u2115\nv : Fin (Nat.succ n) \u2192 M\n\u22a2 \u220f i in Ioi 0, v i = \u220f j : Fin n, v (succ j)\n[PROOFSTEP]\nrw [Ioi_zero_eq_map, Finset.prod_map, RelEmbedding.coe_toEmbedding, val_succEmbedding]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\nn : \u2115\ni : Fin n\nv : Fin (Nat.succ n) \u2192 M\n\u22a2 \u220f j in Ioi (succ i), v j = \u220f j in Ioi i, v (succ j)\n[PROOFSTEP]\nrw [Ioi_succ, Finset.prod_map, RelEmbedding.coe_toEmbedding, val_succEmbedding]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin b \u2192 M\nh : a = b\n\u22a2 \u220f i : Fin a, f (\u2191(castIso h) i) = \u220f i : Fin b, f i\n[PROOFSTEP]\nsubst h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na : \u2115\nf : Fin a \u2192 M\n\u22a2 \u220f i : Fin a, f (\u2191(castIso (_ : a = a)) i) = \u220f i : Fin a, f i\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\n\u22a2 \u220f i : Fin (a + b), f i = (\u220f i : Fin a, f (castAdd b i)) * \u220f i : Fin b, f (natAdd a i)\n[PROOFSTEP]\nrw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\n\u22a2 \u220f x : Fin a \u2295 Fin b, f (Equiv.toFun finSumFinEquiv x) = (\u220f i : Fin a, f (castAdd b i)) * \u220f i : Fin b, f (natAdd a i)\n[PROOFSTEP]\napply Fintype.prod_sum_type\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\n\u22a2 \u2200 (x : Fin (a + b)), f x = f (Equiv.toFun finSumFinEquiv (\u2191finSumFinEquiv.symm x))\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\nx : Fin (a + b)\n\u22a2 f x = f (Equiv.toFun finSumFinEquiv (\u2191finSumFinEquiv.symm x))\n[PROOFSTEP]\nsimp only [Equiv.toFun_as_coe, Equiv.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\nhf : \u2200 (j : Fin b), f (natAdd a j) = 1\n\u22a2 \u220f i : Fin (a + b), f i = \u220f i : Fin a, f (castLE (_ : a \u2264 a + b) i)\n[PROOFSTEP]\nrw [prod_univ_add, Fintype.prod_eq_one _ hf, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nM : Type u_3\ninst\u271d : CommMonoid M\na b : \u2115\nf : Fin (a + b) \u2192 M\nhf : \u2200 (j : Fin b), f (natAdd a j) = 1\n\u22a2 \u220f i : Fin a, f (castAdd b i) = \u220f i : Fin a, f (castLE (_ : a \u2264 a + b) i)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Monoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 partialProd f 0 = 1\n[PROOFSTEP]\nsimp [partialProd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Monoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\nj : Fin n\n\u22a2 partialProd f (succ j) = partialProd f (castSucc j) * f j\n[PROOFSTEP]\nsimp [partialProd, List.take_succ, List.ofFnNthVal, dif_pos j.is_lt, \u2190 Option.coe_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Monoid \u03b1\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\nj : Fin (n + 1)\n\u22a2 partialProd f (succ j) = f 0 * partialProd (tail f) j\n[PROOFSTEP]\nsimp [partialProd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Monoid \u03b1\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1\nj : Fin (n + 1)\n\u22a2 f 0 * List.prod (List.take (\u2191j) (List.ofFn fun i => f (succ i))) =\n    f 0 * List.prod (List.take (\u2191j) (List.ofFn (tail f)))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin (n + 1) \u2192 G\nx : Fin (n + 1)\n\u22a2 (f 0 \u2022 partialProd fun i => (f \u2191\u2191i)\u207b\u00b9 * f (succ i)) 0 = f 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin (n + 1) \u2192 G\nx\u271d : Fin (n + 1)\nx : Fin n\nhx : (f 0 \u2022 partialProd fun i => (f \u2191\u2191i)\u207b\u00b9 * f (succ i)) (castSucc x) = f (castSucc x)\n\u22a2 (f 0 \u2022 partialProd fun i => (f \u2191\u2191i)\u207b\u00b9 * f (succ i)) (succ x) = f (succ x)\n[PROOFSTEP]\nsimp only [coe_eq_castSucc, Pi.smul_apply, smul_eq_mul] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin (n + 1) \u2192 G\nx\u271d : Fin (n + 1)\nx : Fin n\nhx : f 0 * partialProd (fun i => (f (castSucc i))\u207b\u00b9 * f (succ i)) (castSucc x) = f (castSucc x)\n\u22a2 f 0 * partialProd (fun i => (f (castSucc i))\u207b\u00b9 * f (succ i)) (succ x) = f (succ x)\n[PROOFSTEP]\nrw [partialProd_succ, \u2190 mul_assoc, hx, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : Fin n\n\u22a2 (partialProd f (castSucc i))\u207b\u00b9 * partialProd f (succ i) = f i\n[PROOFSTEP]\ncases' i with i hn\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : i < n\n\u22a2 (partialProd f (castSucc { val := i, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := i, isLt := hn }) =\n    f { val := i, isLt := hn }\n[PROOFSTEP]\ninduction i with\n| zero => simp [-Fin.succ_mk, partialProd_succ]\n| succ i hi =>\n  specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n  simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi \u22a2\n  rw [\u2190 Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), \u2190 Fin.succ_mk]\n  rw [Nat.succ_eq_add_one] at hn \n  simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n        -- assoc_rw [hi, inv_mul_cancel_left]\n  rw [\u2190 mul_assoc, mul_left_eq_self, mul_assoc, hi, mul_left_inv]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : i < n\n\u22a2 (partialProd f (castSucc { val := i, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := i, isLt := hn }) =\n    f { val := i, isLt := hn }\n[PROOFSTEP]\ninduction i with\n| zero => simp [-Fin.succ_mk, partialProd_succ]\n| succ i hi =>\n  specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n  simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi \u22a2\n  rw [\u2190 Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), \u2190 Fin.succ_mk]\n  rw [Nat.succ_eq_add_one] at hn \n  simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n        -- assoc_rw [hi, inv_mul_cancel_left]\n  rw [\u2190 mul_assoc, mul_left_eq_self, mul_assoc, hi, mul_left_inv]\n[GOAL]\ncase mk.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\nhn : Nat.zero < n\n\u22a2 (partialProd f (castSucc { val := Nat.zero, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := Nat.zero, isLt := hn }) =\n    f { val := Nat.zero, isLt := hn }\n[PROOFSTEP]\n\n| zero => simp [-Fin.succ_mk, partialProd_succ]\n[GOAL]\ncase mk.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\nhn : Nat.zero < n\n\u22a2 (partialProd f (castSucc { val := Nat.zero, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := Nat.zero, isLt := hn }) =\n    f { val := Nat.zero, isLt := hn }\n[PROOFSTEP]\nsimp [-Fin.succ_mk, partialProd_succ]\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhi :\n  \u2200 (hn : i < n),\n    (partialProd f (castSucc { val := i, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := i, isLt := hn }) =\n      f { val := i, isLt := hn }\nhn : Nat.succ i < n\n\u22a2 (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))\u207b\u00b9 *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }\n[PROOFSTEP]\n\n| succ i hi =>\n  specialize hi (lt_trans (Nat.lt_succ_self i) hn)\n  simp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi \u22a2\n  rw [\u2190 Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), \u2190 Fin.succ_mk]\n  rw [Nat.succ_eq_add_one] at hn \n  simp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n    -- Porting note: was\n        -- assoc_rw [hi, inv_mul_cancel_left]\n  rw [\u2190 mul_assoc, mul_left_eq_self, mul_assoc, hi, mul_left_inv]\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhi :\n  \u2200 (hn : i < n),\n    (partialProd f (castSucc { val := i, isLt := hn }))\u207b\u00b9 * partialProd f (succ { val := i, isLt := hn }) =\n      f { val := i, isLt := hn }\nhn : Nat.succ i < n\n\u22a2 (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))\u207b\u00b9 *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }\n[PROOFSTEP]\nspecialize hi (lt_trans (Nat.lt_succ_self i) hn)\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : Nat.succ i < n\nhi :\n  (partialProd f (castSucc { val := i, isLt := (_ : i < n) }))\u207b\u00b9 *\n      partialProd f (succ { val := i, isLt := (_ : i < n) }) =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 (partialProd f (castSucc { val := Nat.succ i, isLt := hn }))\u207b\u00b9 *\n      partialProd f (succ { val := Nat.succ i, isLt := hn }) =\n    f { val := Nat.succ i, isLt := hn }\n[PROOFSTEP]\nsimp only [Fin.coe_eq_castSucc, Fin.succ_mk, Fin.castSucc_mk] at hi \u22a2\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 (partialProd f { val := Nat.succ i, isLt := (_ : Nat.succ i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := Nat.succ i + 1, isLt := (_ : Nat.succ (Nat.succ i) < Nat.succ n) } =\n    f { val := Nat.succ i, isLt := hn }\n[PROOFSTEP]\nrw [\u2190 Fin.succ_mk _ _ (lt_trans (Nat.lt_succ_self _) hn), \u2190 Fin.succ_mk]\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 (partialProd f (succ { val := i, isLt := (_ : i < n) }))\u207b\u00b9 *\n      partialProd f (succ { val := i + 1, isLt := ?mk.succ.h }) =\n    f { val := Nat.succ i, isLt := hn }\ncase mk.succ.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn : Nat.succ i < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 i + 1 < n\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one] at hn \n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn\u271d : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 (partialProd f (succ { val := i, isLt := (_ : i < n) }))\u207b\u00b9 *\n      partialProd f (succ { val := i + 1, isLt := (_ : i + 1 < n) }) =\n    f { val := Nat.succ i, isLt := hn\u271d }\n[PROOFSTEP]\nsimp only [partialProd_succ, mul_inv_rev, Fin.castSucc_mk]\n  -- Porting note: was\n      -- assoc_rw [hi, inv_mul_cancel_left]\n[GOAL]\ncase mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\nf : Fin n \u2192 G\ni : \u2115\nhn\u271d : Nat.succ i < n\nhn : i + 1 < n\nhi :\n  (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      partialProd f { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ n) } =\n    f { val := i, isLt := (_ : i < n) }\n\u22a2 (f { val := i, isLt := (_ : i < n) })\u207b\u00b9 * (partialProd f { val := i, isLt := (_ : i < Nat.succ n) })\u207b\u00b9 *\n      (partialProd f { val := i + 1, isLt := (_ : i + 1 < Nat.succ n) } * f { val := i + 1, isLt := (_ : i + 1 < n) }) =\n    f { val := Nat.succ i, isLt := hn\u271d }\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_left_eq_self, mul_assoc, hi, mul_left_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\n\u22a2 (partialProd g (succAbove (succ j) (castSucc k)))\u207b\u00b9 * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k\n[PROOFSTEP]\nrcases lt_trichotomy (k : \u2115) j with (h | h | h)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k < \u2191j\n\u22a2 (partialProd g (succAbove (succ j) (castSucc k)))\u207b\u00b9 * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k\n[PROOFSTEP]\nrwa [succAbove_below, succAbove_below, partialProd_right_inv, contractNth_apply_of_lt]\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k < \u2191j\n\u22a2 castSucc k < j\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k < \u2191j\n\u22a2 castSucc (castSucc k) < succ j\n[PROOFSTEP]\nrw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n[GOAL]\ncase inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k < \u2191j\n\u22a2 \u2191(castSucc k) \u2264 \u2191j\n[PROOFSTEP]\nexact le_of_lt h\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k = \u2191j\n\u22a2 (partialProd g (succAbove (succ j) (castSucc k)))\u207b\u00b9 * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k\n[PROOFSTEP]\nrwa [succAbove_below, succAbove_above, partialProd_succ, castSucc_fin_succ, \u2190 mul_assoc, partialProd_right_inv,\n  contractNth_apply_of_eq]\n[GOAL]\ncase inr.inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k = \u2191j\n\u22a2 j \u2264 castSucc k\n[PROOFSTEP]\nsimp [le_iff_val_le_val, \u2190 h]\n[GOAL]\ncase inr.inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k = \u2191j\n\u22a2 castSucc (castSucc k) < succ j\n[PROOFSTEP]\nrw [castSucc_lt_iff_succ_le, succ_le_succ_iff, le_iff_val_le_val]\n[GOAL]\ncase inr.inl.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191k = \u2191j\n\u22a2 \u2191(castSucc k) \u2264 \u2191j\n[PROOFSTEP]\nexact le_of_eq h\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191j < \u2191k\n\u22a2 (partialProd g (succAbove (succ j) (castSucc k)))\u207b\u00b9 * partialProd g (succ (succAbove j k)) =\n    contractNth j (fun x x_1 => x * x_1) g k\n[PROOFSTEP]\nrwa [succAbove_above, succAbove_above, partialProd_succ, partialProd_succ, castSucc_fin_succ, partialProd_succ,\n  inv_mul_cancel_left, contractNth_apply_of_gt]\n[GOAL]\ncase inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191j < \u2191k\n\u22a2 j \u2264 castSucc k\n[PROOFSTEP]\nexact le_iff_val_le_val.2 (le_of_lt h)\n[GOAL]\ncase inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191j < \u2191k\n\u22a2 succ j \u2264 castSucc (castSucc k)\n[PROOFSTEP]\nrw [le_iff_val_le_val, val_succ]\n[GOAL]\ncase inr.inr.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\nG : Type u_3\ninst\u271d : Group G\ng : Fin (n + 1) \u2192 G\nj : Fin (n + 1)\nk : Fin n\nh : \u2191j < \u2191k\n\u22a2 \u2191j + 1 \u2264 \u2191(castSucc (castSucc k))\n[PROOFSTEP]\nexact Nat.succ_le_of_lt h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\n\u22a2 Fintype.card (Fin n \u2192 Fin m) = Fintype.card (Fin (m ^ n))\n[PROOFSTEP]\nsimp_rw [Fintype.card_fun, Fintype.card_fin]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nf : Fin n \u2192 Fin m\n\u22a2 \u2211 i : Fin n, \u2191(f i) * m ^ \u2191i < m ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\nf\u271d : Fin n \u2192 Fin m\nf : Fin Nat.zero \u2192 Fin m\n\u22a2 \u2211 i : Fin Nat.zero, \u2191(f i) * m ^ \u2191i < m ^ Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\nf\u271d : Fin n\u271d \u2192 Fin m\nn : \u2115\nih : \u2200 (f : Fin n \u2192 Fin m), \u2211 i : Fin n, \u2191(f i) * m ^ \u2191i < m ^ n\nf : Fin (Nat.succ n) \u2192 Fin m\n\u22a2 \u2211 i : Fin (Nat.succ n), \u2191(f i) * m ^ \u2191i < m ^ Nat.succ n\n[PROOFSTEP]\ncases m\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n : \u2115\nf\u271d : Fin n\u271d \u2192 Fin Nat.zero\nih : \u2200 (f : Fin n \u2192 Fin Nat.zero), \u2211 i : Fin n, \u2191(f i) * Nat.zero ^ \u2191i < Nat.zero ^ n\nf : Fin (Nat.succ n) \u2192 Fin Nat.zero\n\u22a2 \u2211 i : Fin (Nat.succ n), \u2191(f i) * Nat.zero ^ \u2191i < Nat.zero ^ Nat.succ n\n[PROOFSTEP]\ndsimp only [Nat.zero_eq] at f \n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n : \u2115\nf\u271d : Fin n\u271d \u2192 Fin Nat.zero\nih : \u2200 (f : Fin n \u2192 Fin Nat.zero), \u2211 i : Fin n, \u2191(f i) * Nat.zero ^ \u2191i < Nat.zero ^ n\nf : Fin (Nat.succ n) \u2192 Fin 0\n\u22a2 \u2211 i : Fin (Nat.succ n), \u2191(f i) * Nat.zero ^ \u2191i < Nat.zero ^ Nat.succ n\n[PROOFSTEP]\nexact isEmptyElim (f <| Fin.last _)\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d\u00b9 n n\u271d : \u2115\nf\u271d : Fin n\u271d\u00b9 \u2192 Fin (Nat.succ n\u271d)\nih : \u2200 (f : Fin n \u2192 Fin (Nat.succ n\u271d)), \u2211 i : Fin n, \u2191(f i) * Nat.succ n\u271d ^ \u2191i < Nat.succ n\u271d ^ n\nf : Fin (Nat.succ n) \u2192 Fin (Nat.succ n\u271d)\n\u22a2 \u2211 i : Fin (Nat.succ n), \u2191(f i) * Nat.succ n\u271d ^ \u2191i < Nat.succ n\u271d ^ Nat.succ n\n[PROOFSTEP]\nsimp_rw [Fin.sum_univ_castSucc, Fin.coe_castSucc, Fin.val_last]\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d\u00b9 n n\u271d : \u2115\nf\u271d : Fin n\u271d\u00b9 \u2192 Fin (Nat.succ n\u271d)\nih : \u2200 (f : Fin n \u2192 Fin (Nat.succ n\u271d)), \u2211 i : Fin n, \u2191(f i) * Nat.succ n\u271d ^ \u2191i < Nat.succ n\u271d ^ n\nf : Fin (Nat.succ n) \u2192 Fin (Nat.succ n\u271d)\n\u22a2 \u2211 x : Fin n, \u2191(f (Fin.castSucc x)) * Nat.succ n\u271d ^ \u2191x + \u2191(f (Fin.last n)) * Nat.succ n\u271d ^ n < Nat.succ n\u271d ^ Nat.succ n\n[PROOFSTEP]\nrefine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d\u00b9 n n\u271d : \u2115\nf\u271d : Fin n\u271d\u00b9 \u2192 Fin (Nat.succ n\u271d)\nih : \u2200 (f : Fin n \u2192 Fin (Nat.succ n\u271d)), \u2211 i : Fin n, \u2191(f i) * Nat.succ n\u271d ^ \u2191i < Nat.succ n\u271d ^ n\nf : Fin (Nat.succ n) \u2192 Fin (Nat.succ n\u271d)\n\u22a2 Nat.succ n\u271d ^ n + n\u271d * Nat.succ n\u271d ^ n = Nat.succ n\u271d ^ Nat.succ n\n[PROOFSTEP]\nrw [\u2190 one_add_mul (_ : \u2115), add_comm, pow_succ]\n  -- porting note: added, wrong `succ`\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d\u00b9 n n\u271d : \u2115\nf\u271d : Fin n\u271d\u00b9 \u2192 Fin (Nat.succ n\u271d)\nih : \u2200 (f : Fin n \u2192 Fin (Nat.succ n\u271d)), \u2211 i : Fin n, \u2191(f i) * Nat.succ n\u271d ^ \u2191i < Nat.succ n\u271d ^ n\nf : Fin (Nat.succ n) \u2192 Fin (Nat.succ n\u271d)\n\u22a2 (n\u271d + 1) * Nat.succ n\u271d ^ n = Nat.succ n\u271d * Nat.succ n\u271d ^ n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na : Fin (m ^ n)\nb : Fin n\n\u22a2 \u2191a / m ^ \u2191b % m < m\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\na : Fin (m ^ Nat.zero)\nb : Fin Nat.zero\n\u22a2 \u2191a / m ^ \u2191b % m < m\n[PROOFSTEP]\nexact b.elim0\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na : Fin (m ^ Nat.succ n)\nb : Fin (Nat.succ n)\n\u22a2 \u2191a / m ^ \u2191b % m < m\n[PROOFSTEP]\ncases' m with m\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nb : Fin (Nat.succ n)\na : Fin (Nat.zero ^ Nat.succ n)\n\u22a2 \u2191a / Nat.zero ^ \u2191b % Nat.zero < Nat.zero\n[PROOFSTEP]\ndsimp only [Nat.zero_eq] at a \n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nb : Fin (Nat.succ n)\na : Fin (0 ^ Nat.succ n)\n\u22a2 \u2191a / Nat.zero ^ \u2191b % Nat.zero < Nat.zero\n[PROOFSTEP]\nrw [zero_pow n.succ_pos] at a \n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nb : Fin (Nat.succ n)\na\u271d : Fin (0 ^ Nat.succ n)\na : Fin 0\n\u22a2 \u2191a\u271d / Nat.zero ^ \u2191b % Nat.zero < Nat.zero\n[PROOFSTEP]\nexact a.elim0\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\nb : Fin (Nat.succ n)\nm : \u2115\na : Fin (Nat.succ m ^ Nat.succ n)\n\u22a2 \u2191a / Nat.succ m ^ \u2191b % Nat.succ m < Nat.succ m\n[PROOFSTEP]\nexact Nat.mod_lt _ m.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na : Fin (m ^ n)\n\u22a2 (fun f => { val := \u2211 i : Fin n, \u2191(f i) * m ^ \u2191i, isLt := (_ : \u2211 i : Fin n, \u2191(f i) * m ^ \u2191i < m ^ n) })\n      ((fun a b => { val := \u2191a / m ^ \u2191b % m, isLt := (_ : \u2191a / m ^ \u2191b % m < m) }) a) =\n    a\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na : Fin (m ^ n)\n\u22a2 { val := \u2211 i : Fin n, \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n      isLt := (_ : \u2211 i : Fin n, \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i < m ^ n) } =\n    a\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na\u271d : Fin (m ^ n)\na : Fin (m ^ Nat.zero)\n\u22a2 { val := \u2211 i : Fin Nat.zero, \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n      isLt :=\n        (_ :\n          \u2211 i : Fin Nat.zero, \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i <\n            m ^ Nat.zero) } =\n    a\n[PROOFSTEP]\nhaveI : Subsingleton (Fin (m ^ 0)) := (Fin.castIso <| pow_zero _).toEquiv.subsingleton\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\na\u271d : Fin (m ^ n)\na : Fin (m ^ Nat.zero)\nthis : Subsingleton (Fin (m ^ 0))\n\u22a2 { val := \u2211 i : Fin Nat.zero, \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n      isLt :=\n        (_ :\n          \u2211 i : Fin Nat.zero, \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i <\n            m ^ Nat.zero) } =\n    a\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\na\u271d : Fin (m ^ n\u271d)\nn : \u2115\nih :\n  \u2200 (a : Fin (m ^ n)),\n    { val := \u2211 i : Fin n, \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n        isLt := (_ : \u2211 i : Fin n, \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i < m ^ n) } =\n      a\na : Fin (m ^ Nat.succ n)\n\u22a2 { val := \u2211 i : Fin (Nat.succ n), \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n      isLt :=\n        (_ :\n          \u2211 i : Fin (Nat.succ n), \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i <\n            m ^ Nat.succ n) } =\n    a\n[PROOFSTEP]\nsimp_rw [Fin.forall_iff, Fin.ext_iff] at ih \n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\na\u271d : Fin (m ^ n\u271d)\nn : \u2115\na : Fin (m ^ Nat.succ n)\nih : \u2200 (i : \u2115), i < m ^ n \u2192 \u2211 x : Fin n, i / m ^ \u2191x % m * m ^ \u2191x = i\n\u22a2 { val := \u2211 i : Fin (Nat.succ n), \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n      isLt :=\n        (_ :\n          \u2211 i : Fin (Nat.succ n), \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i <\n            m ^ Nat.succ n) } =\n    a\n[PROOFSTEP]\next\n[GOAL]\ncase succ.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\na\u271d : Fin (m ^ n\u271d)\nn : \u2115\na : Fin (m ^ Nat.succ n)\nih : \u2200 (i : \u2115), i < m ^ n \u2192 \u2211 x : Fin n, i / m ^ \u2191x % m * m ^ \u2191x = i\n\u22a2 \u2191{ val := \u2211 i : Fin (Nat.succ n), \u2191a / m ^ \u2191i % m * m ^ \u2191i,\n        isLt :=\n          (_ :\n            \u2211 i : Fin (Nat.succ n), \u2191{ val := \u2191a / m ^ \u2191i % m, isLt := (_ : \u2191a / m ^ \u2191i % m < m) } * m ^ \u2191i <\n              m ^ Nat.succ n) } =\n    \u2191a\n[PROOFSTEP]\nsimp_rw [Fin.sum_univ_succ, Fin.val_zero, Fin.val_succ, pow_zero, Nat.div_one, mul_one, pow_succ, \u2190\n  Nat.div_div_eq_div_mul, mul_left_comm _ m, \u2190 mul_sum]\n[GOAL]\ncase succ.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\na\u271d : Fin (m ^ n\u271d)\nn : \u2115\na : Fin (m ^ Nat.succ n)\nih : \u2200 (i : \u2115), i < m ^ n \u2192 \u2211 x : Fin n, i / m ^ \u2191x % m * m ^ \u2191x = i\n\u22a2 \u2191a % m + m * \u2211 x : Fin n, \u2191a / m / m ^ \u2191x % m * m ^ \u2191x = \u2191a\n[PROOFSTEP]\nrw [ih _ (Nat.div_lt_of_lt_mul ?_), Nat.mod_add_div]\n  -- porting note: replaces `a.is_lt` in the wildcard above. Caused by a refactor of the `npow`\n        -- instance for `Fin`.\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n\u271d : \u2115\na\u271d : Fin (m ^ n\u271d)\nn : \u2115\na : Fin (m ^ Nat.succ n)\nih : \u2200 (i : \u2115), i < m ^ n \u2192 \u2211 x : Fin n, i / m ^ \u2191x % m * m ^ \u2191x = i\n\u22a2 \u2191a < m * m ^ n\n[PROOFSTEP]\nexact a.is_lt.trans_eq (pow_succ _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\ninst\u271d : NeZero m\ni : Fin n\nj : Fin m\n\u22a2 \u2191(\u2191finFunctionFinEquiv (Pi.single i j)) = \u2191j * m ^ \u2191i\n[PROOFSTEP]\nrw [finFunctionFinEquiv_apply, Fintype.sum_eq_single i, Pi.single_eq_same]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\ninst\u271d : NeZero m\ni : Fin n\nj : Fin m\n\u22a2 \u2200 (x : Fin n), x \u2260 i \u2192 \u2191(Pi.single i j x) * m ^ \u2191x = 0\n[PROOFSTEP]\nrintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm n : \u2115\ninst\u271d : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x \u2260 i\n\u22a2 \u2191(Pi.single i j x) * m ^ \u2191x = 0\n[PROOFSTEP]\nrw [Pi.single_eq_of_ne hx, Fin.val_zero', zero_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 Fintype.card ((i : Fin m) \u2192 Fin (n i)) = Fintype.card (Fin (\u220f i : Fin m, n i))\n[PROOFSTEP]\nsimp_rw [Fintype.card_pi, Fintype.card_fin]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\n\u22a2 \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\n[PROOFSTEP]\ninduction' m with m ih\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nf\u271d : (i : Fin m) \u2192 Fin (n\u271d i)\nn : Fin Nat.zero \u2192 \u2115\nf : (i : Fin Nat.zero) \u2192 Fin (n i)\n\u22a2 \u2211 i : Fin Nat.zero, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 Nat.zero) j) < \u220f i : Fin Nat.zero, n i\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\n\u22a2 \u2211 i : Fin (Nat.succ m), \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 Nat.succ m) j) < \u220f i : Fin (Nat.succ m), n i\n[PROOFSTEP]\nrw [Fin.prod_univ_castSucc, Fin.sum_univ_castSucc]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\n\u22a2 \u2211 i : Fin m,\n        \u2191(f (Fin.castSucc i)) * \u220f j : Fin \u2191(Fin.castSucc i), n (Fin.castLE (_ : \u2191(Fin.castSucc i) \u2264 Nat.succ m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin \u2191(Fin.last m), n (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) j) <\n    (\u220f i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)\n[PROOFSTEP]\nsuffices\n  \u2200 (n : Fin m \u2192 \u2115) (nn : \u2115) (f : \u2200 i : Fin m, Fin (n i)) (fn : Fin nn),\n    ((\u2211 i : Fin m, \u2191(f i) * \u220f j : Fin i, n (Fin.castLE i.prop.le j)) + \u2191fn * \u220f j, n j) < (\u220f i : Fin m, n i) * nn\n  by\n  replace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n  rw [\u2190 Fin.snoc_init_self f]\n  simp (config := { singlePass := true }) only [\u2190 Fin.snoc_init_self n]\n  simp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n  exact this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\nthis :\n  \u2200 (n : Fin m \u2192 \u2115) (nn : \u2115) (f : (i : Fin m) \u2192 Fin (n i)) (fn : Fin nn),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j < (\u220f i : Fin m, n i) * nn\n\u22a2 \u2211 i : Fin m,\n        \u2191(f (Fin.castSucc i)) * \u220f j : Fin \u2191(Fin.castSucc i), n (Fin.castLE (_ : \u2191(Fin.castSucc i) \u2264 Nat.succ m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin \u2191(Fin.last m), n (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) j) <\n    (\u220f i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)\n[PROOFSTEP]\nreplace := this (Fin.init n) (n (Fin.last _)) (Fin.init f) (f (Fin.last _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\nthis :\n  \u2211 i : Fin m, \u2191(Fin.init f i) * \u220f j : Fin \u2191i, Fin.init n (Fin.castLE (_ : \u2191i \u2264 m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin m, Fin.init n j <\n    (\u220f i : Fin m, Fin.init n i) * n (Fin.last m)\n\u22a2 \u2211 i : Fin m,\n        \u2191(f (Fin.castSucc i)) * \u220f j : Fin \u2191(Fin.castSucc i), n (Fin.castLE (_ : \u2191(Fin.castSucc i) \u2264 Nat.succ m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin \u2191(Fin.last m), n (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) j) <\n    (\u220f i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)\n[PROOFSTEP]\nrw [\u2190 Fin.snoc_init_self f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\nthis :\n  \u2211 i : Fin m, \u2191(Fin.init f i) * \u220f j : Fin \u2191i, Fin.init n (Fin.castLE (_ : \u2191i \u2264 m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin m, Fin.init n j <\n    (\u220f i : Fin m, Fin.init n i) * n (Fin.last m)\n\u22a2 \u2211 i : Fin m,\n        \u2191(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.castSucc i)) *\n          \u220f j : Fin \u2191(Fin.castSucc i), n (Fin.castLE (_ : \u2191(Fin.castSucc i) \u2264 Nat.succ m) j) +\n      \u2191(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.last m)) *\n        \u220f j : Fin \u2191(Fin.last m), n (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) j) <\n    (\u220f i : Fin m, n (Fin.castSucc i)) * n (Fin.last m)\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [\u2190 Fin.snoc_init_self n]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\nthis :\n  \u2211 i : Fin m, \u2191(Fin.init f i) * \u220f j : Fin \u2191i, Fin.init n (Fin.castLE (_ : \u2191i \u2264 m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin m, Fin.init n j <\n    (\u220f i : Fin m, Fin.init n i) * n (Fin.last m)\n\u22a2 \u2211 x : Fin m,\n        \u2191(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.castSucc x)) *\n          \u220f x_1 : Fin \u2191(Fin.castSucc x),\n            Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castLE (_ : \u2191(Fin.castSucc x) \u2264 Nat.succ m) x_1) +\n      \u2191(Fin.snoc (Fin.init f) (f (Fin.last m)) (Fin.last m)) *\n        \u220f x : Fin \u2191(Fin.last m),\n          Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) x) <\n    (\u220f x : Fin m, Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.castSucc x)) *\n      Fin.snoc (Fin.init n) (n (Fin.last m)) (Fin.last m)\n[PROOFSTEP]\nsimp_rw [Fin.snoc_castSucc, Fin.snoc_last, Fin.snoc_init_self n]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\nthis :\n  \u2211 i : Fin m, \u2191(Fin.init f i) * \u220f j : Fin \u2191i, Fin.init n (Fin.castLE (_ : \u2191i \u2264 m) j) +\n      \u2191(f (Fin.last m)) * \u220f j : Fin m, Fin.init n j <\n    (\u220f i : Fin m, Fin.init n i) * n (Fin.last m)\n\u22a2 \u2211 x : Fin m,\n        \u2191(Fin.init f x) * \u220f x_1 : Fin \u2191(Fin.castSucc x), n (Fin.castLE (_ : \u2191(Fin.castSucc x) \u2264 Nat.succ m) x_1) +\n      \u2191(f (Fin.last m)) * \u220f x : Fin \u2191(Fin.last m), n (Fin.castLE (_ : \u2191(Fin.last m) \u2264 Nat.succ m) x) <\n    (\u220f x : Fin m, Fin.init n x) * n (Fin.last m)\n[PROOFSTEP]\nexact this\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d : Fin m\u271d \u2192 \u2115\nf\u271d : (i : Fin m\u271d) \u2192 Fin (n\u271d i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn : Fin (Nat.succ m) \u2192 \u2115\nf : (i : Fin (Nat.succ m)) \u2192 Fin (n i)\n\u22a2 \u2200 (n : Fin m \u2192 \u2115) (nn : \u2115) (f : (i : Fin m) \u2192 Fin (n i)) (fn : Fin nn),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j < (\u220f i : Fin m, n i) * nn\n[PROOFSTEP]\nintro n nn f fn\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b9 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b9 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d i)\nn : Fin m \u2192 \u2115\nnn : \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nfn : Fin nn\n\u22a2 \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j < (\u220f i : Fin m, n i) * nn\n[PROOFSTEP]\ncases nn\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b9 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b9 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d i)\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nfn : Fin Nat.zero\n\u22a2 \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j <\n    (\u220f i : Fin m, n i) * Nat.zero\n[PROOFSTEP]\ndsimp only [Nat.zero_eq] at fn \n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b9 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b9 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d i)\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nfn : Fin 0\n\u22a2 \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j <\n    (\u220f i : Fin m, n i) * Nat.zero\n[PROOFSTEP]\nexact isEmptyElim fn\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b2 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b2 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d\u00b9 : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d\u00b9 i)\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nn\u271d : \u2115\nfn : Fin (Nat.succ n\u271d)\n\u22a2 \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) + \u2191fn * \u220f j : Fin m, n j <\n    (\u220f i : Fin m, n i) * Nat.succ n\u271d\n[PROOFSTEP]\nrefine' (add_lt_add_of_lt_of_le (ih _) <| mul_le_mul_right' (Fin.is_le _) _).trans_eq _\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b2 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b2 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d\u00b9 : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d\u00b9 i)\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nn\u271d : \u2115\nfn : Fin (Nat.succ n\u271d)\n\u22a2 \u220f i : Fin m, n i + n\u271d * \u220f j : Fin m, n j = (\u220f i : Fin m, n i) * Nat.succ n\u271d\n[PROOFSTEP]\nrw [\u2190 one_add_mul (_ : \u2115), mul_comm, add_comm]\n  -- porting note: added, wrong `succ`\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : \u2115\nn\u271d\u00b2 : Fin m\u271d \u2192 \u2115\nf\u271d\u00b9 : (i : Fin m\u271d) \u2192 Fin (n\u271d\u00b2 i)\nm : \u2115\nih :\n  \u2200 {n : Fin m \u2192 \u2115} (f : (i : Fin m) \u2192 Fin (n i)),\n    \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i\nn\u271d\u00b9 : Fin (Nat.succ m) \u2192 \u2115\nf\u271d : (i : Fin (Nat.succ m)) \u2192 Fin (n\u271d\u00b9 i)\nn : Fin m \u2192 \u2115\nf : (i : Fin m) \u2192 Fin (n i)\nn\u271d : \u2115\nfn : Fin (Nat.succ n\u271d)\n\u22a2 (\u220f i : Fin m, n i) * (n\u271d + 1) = (\u220f i : Fin m, n i) * Nat.succ n\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\na : Fin (\u220f i : Fin m, n i)\nb : Fin m\n\u22a2 (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b < n b\n[PROOFSTEP]\ncases m\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : Fin Nat.zero \u2192 \u2115\na : Fin (\u220f i : Fin Nat.zero, n i)\nb : Fin Nat.zero\n\u22a2 (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 Nat.zero) j)) % n b < n b\n[PROOFSTEP]\nexact b.elim0\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\nn : Fin (Nat.succ n\u271d) \u2192 \u2115\na : Fin (\u220f i : Fin (Nat.succ n\u271d), n i)\nb : Fin (Nat.succ n\u271d)\n\u22a2 (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 Nat.succ n\u271d) j)) % n b < n b\n[PROOFSTEP]\ncases' h : n b with nb\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\nn : Fin (Nat.succ n\u271d) \u2192 \u2115\na : Fin (\u220f i : Fin (Nat.succ n\u271d), n i)\nb : Fin (Nat.succ n\u271d)\nh : n b = Nat.zero\n\u22a2 (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 Nat.succ n\u271d) j)) % Nat.zero < Nat.zero\n[PROOFSTEP]\nrw [prod_eq_zero (Finset.mem_univ _) h] at a \n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\nn : Fin (Nat.succ n\u271d) \u2192 \u2115\na\u271d : Fin (\u220f i : Fin (Nat.succ n\u271d), n i)\na : Fin 0\nb : Fin (Nat.succ n\u271d)\nh : n b = Nat.zero\n\u22a2 (\u2191a\u271d / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 Nat.succ n\u271d) j)) % Nat.zero < Nat.zero\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d : \u2115\nn : Fin (Nat.succ n\u271d) \u2192 \u2115\na : Fin (\u220f i : Fin (Nat.succ n\u271d), n i)\nb : Fin (Nat.succ n\u271d)\nnb : \u2115\nh : n b = Nat.succ nb\n\u22a2 (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 Nat.succ n\u271d) j)) % Nat.succ nb < Nat.succ nb\n[PROOFSTEP]\nexact Nat.mod_lt _ nb.succ_pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 Function.RightInverse\n    (fun a b =>\n      { val := (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b,\n        isLt := (_ : (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b < n b) })\n    fun f =>\n    { val := \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j),\n      isLt := (_ : \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i) }\n[PROOFSTEP]\nintro a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\na : Fin (\u220f i : Fin m, n i)\n\u22a2 (fun f =>\n        { val := \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j),\n          isLt := (_ : \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i) })\n      ((fun a b =>\n          { val := (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b,\n            isLt := (_ : (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b < n b) })\n        a) =\n    a\n[PROOFSTEP]\nrevert a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 \u2200 (a : Fin (\u220f i : Fin m, n i)),\n    (fun f =>\n          { val := \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j),\n            isLt := (_ : \u2211 i : Fin m, \u2191(f i) * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) < \u220f i : Fin m, n i) })\n        ((fun a b =>\n            { val := (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b,\n              isLt := (_ : (\u2191a / \u220f j : Fin \u2191b, n (Fin.castLE (_ : \u2191b \u2264 m) j)) % n b < n b) })\n          a) =\n      a\n[PROOFSTEP]\ndsimp only [Fin.val_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 \u2200 (a : Fin (\u220f i : Fin m, n i)),\n    {\n        val :=\n          \u2211 i : Fin m,\n            (\u2191a / \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j)) % n i * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j),\n        isLt :=\n          (_ :\n            \u2211 i : Fin m,\n                \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j)) % n i,\n                      isLt := (_ : (\u2191a / \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j)) % n i < n i) } *\n                  \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j) <\n              \u220f i : Fin m, n i) } =\n      a\n[PROOFSTEP]\nrefine' Fin.consInduction _ _ n\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 \u2200 (a : Fin (\u220f i : Fin 0, Fin.elim0 i)),\n    {\n        val :=\n          \u2211 i : Fin 0,\n            (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i *\n              \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j),\n        isLt :=\n          (_ :\n            \u2211 i : Fin 0,\n                \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i,\n                      isLt :=\n                        (_ : (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                  \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j) <\n              \u220f i : Fin 0, Fin.elim0 i) } =\n      a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\na : Fin (\u220f i : Fin 0, Fin.elim0 i)\n\u22a2 {\n      val :=\n        \u2211 i : Fin 0,\n          (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i *\n            \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j),\n      isLt :=\n        (_ :\n          \u2211 i : Fin 0,\n              \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i,\n                    isLt :=\n                      (_ : (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j) <\n            \u220f i : Fin 0, Fin.elim0 i) } =\n    a\n[PROOFSTEP]\nhaveI : Subsingleton (Fin (\u220f i : Fin 0, i.elim0)) := (Fin.castIso <| prod_empty).toEquiv.subsingleton\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\na : Fin (\u220f i : Fin 0, Fin.elim0 i)\nthis : Subsingleton (Fin (\u220f i : Fin 0, Fin.elim0 i))\n\u22a2 {\n      val :=\n        \u2211 i : Fin 0,\n          (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i *\n            \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j),\n      isLt :=\n        (_ :\n          \u2211 i : Fin 0,\n              \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i,\n                    isLt :=\n                      (_ : (\u2191a / \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j)) % Fin.elim0 i < Fin.elim0 i) } *\n                \u220f j : Fin \u2191i, Fin.elim0 (Fin.castLE (_ : \u2191i \u2264 0) j) <\n            \u220f i : Fin 0, Fin.elim0 i) } =\n    a\n[PROOFSTEP]\nexact Subsingleton.elim _ _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\n\u22a2 \u2200 {n : \u2115} (x\u2080 : \u2115) (x : Fin n \u2192 \u2115),\n    (\u2200 (a : Fin (\u220f i : Fin n, x i)),\n        {\n            val :=\n              \u2211 i : Fin n,\n                (\u2191a / \u220f j : Fin \u2191i, x (Fin.castLE (_ : \u2191i \u2264 n) j)) % x i * \u220f j : Fin \u2191i, x (Fin.castLE (_ : \u2191i \u2264 n) j),\n            isLt :=\n              (_ :\n                \u2211 i : Fin n,\n                    \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, x (Fin.castLE (_ : \u2191i \u2264 n) j)) % x i,\n                          isLt := (_ : (\u2191a / \u220f j : Fin \u2191i, x (Fin.castLE (_ : \u2191i \u2264 n) j)) % x i < x i) } *\n                      \u220f j : Fin \u2191i, x (Fin.castLE (_ : \u2191i \u2264 n) j) <\n                  \u220f i : Fin n, x i) } =\n          a) \u2192\n      \u2200 (a : Fin (\u220f i : Fin (n + 1), Fin.cons x\u2080 x i)),\n        {\n            val :=\n              \u2211 i : Fin (n + 1),\n                (\u2191a / \u220f j : Fin \u2191i, Fin.cons x\u2080 x (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x\u2080 x i *\n                  \u220f j : Fin \u2191i, Fin.cons x\u2080 x (Fin.castLE (_ : \u2191i \u2264 n + 1) j),\n            isLt :=\n              (_ :\n                \u2211 i : Fin (n + 1),\n                    \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.cons x\u2080 x (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x\u2080 x i,\n                          isLt :=\n                            (_ :\n                              (\u2191a / \u220f j : Fin \u2191i, Fin.cons x\u2080 x (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x\u2080 x i <\n                                Fin.cons x\u2080 x i) } *\n                      \u220f j : Fin \u2191i, Fin.cons x\u2080 x (Fin.castLE (_ : \u2191i \u2264 n + 1) j) <\n                  \u220f i : Fin (n + 1), Fin.cons x\u2080 x i) } =\n          a\n[PROOFSTEP]\nintro n x xs ih a\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\nih :\n  \u2200 (a : Fin (\u220f i : Fin n, xs i)),\n    {\n        val :=\n          \u2211 i : Fin n,\n            (\u2191a / \u220f j : Fin \u2191i, xs (Fin.castLE (_ : \u2191i \u2264 n) j)) % xs i * \u220f j : Fin \u2191i, xs (Fin.castLE (_ : \u2191i \u2264 n) j),\n        isLt :=\n          (_ :\n            \u2211 i : Fin n,\n                \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, xs (Fin.castLE (_ : \u2191i \u2264 n) j)) % xs i,\n                      isLt := (_ : (\u2191a / \u220f j : Fin \u2191i, xs (Fin.castLE (_ : \u2191i \u2264 n) j)) % xs i < xs i) } *\n                  \u220f j : Fin \u2191i, xs (Fin.castLE (_ : \u2191i \u2264 n) j) <\n              \u220f i : Fin n, xs i) } =\n      a\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\n\u22a2 {\n      val :=\n        \u2211 i : Fin (n + 1),\n          (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i *\n            \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j),\n      isLt :=\n        (_ :\n          \u2211 i : Fin (n + 1),\n              \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i,\n                    isLt :=\n                      (_ :\n                        (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i <\n                          Fin.cons x xs i) } *\n                \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j) <\n            \u220f i : Fin (n + 1), Fin.cons x xs i) } =\n    a\n[PROOFSTEP]\nsimp_rw [Fin.forall_iff, Fin.ext_iff] at ih \n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 {\n      val :=\n        \u2211 i : Fin (n + 1),\n          (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i *\n            \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j),\n      isLt :=\n        (_ :\n          \u2211 i : Fin (n + 1),\n              \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i,\n                    isLt :=\n                      (_ :\n                        (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i <\n                          Fin.cons x xs i) } *\n                \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j) <\n            \u220f i : Fin (n + 1), Fin.cons x xs i) } =\n    a\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u2191{\n        val :=\n          \u2211 i : Fin (n + 1),\n            (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i *\n              \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j),\n        isLt :=\n          (_ :\n            \u2211 i : Fin (n + 1),\n                \u2191{ val := (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i,\n                      isLt :=\n                        (_ :\n                          (\u2191a / \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j)) % Fin.cons x xs i <\n                            Fin.cons x xs i) } *\n                  \u220f j : Fin \u2191i, Fin.cons x xs (Fin.castLE (_ : \u2191i \u2264 n + 1) j) <\n              \u220f i : Fin (n + 1), Fin.cons x xs i) } =\n    \u2191a\n[PROOFSTEP]\nsimp_rw [Fin.sum_univ_succ, Fin.cons_succ]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 (\u2191a / \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j)) % Fin.cons x xs 0 *\n        \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j) +\n      \u2211 x_1 : Fin n,\n        (\u2191a / \u220f j : Fin \u2191(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ x_1) \u2264 n + 1) j)) % xs x_1 *\n          \u220f j : Fin \u2191(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ x_1) \u2264 n + 1) j) =\n    \u2191a\n[PROOFSTEP]\nhave := fun i : Fin n =>\n  Fintype.prod_equiv (Fin.castIso <| Fin.val_succ i).toEquiv\n    (fun j => (Fin.cons x xs : _ \u2192 \u2115) (Fin.castLE (Fin.is_lt _).le j))\n    (fun j => (Fin.cons x xs : _ \u2192 \u2115) (Fin.castLE (Nat.succ_le_succ (Fin.is_lt _).le) j)) fun j => rfl\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\nthis :\n  \u2200 (i : Fin n),\n    \u220f x_1 : Fin \u2191(Fin.succ i), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ i) \u2264 n + 1) x_1) =\n      \u220f x_1 : Fin (\u2191i + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191i \u2264 Nat.succ n) x_1)\n\u22a2 (\u2191a / \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j)) % Fin.cons x xs 0 *\n        \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j) +\n      \u2211 x_1 : Fin n,\n        (\u2191a / \u220f j : Fin \u2191(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ x_1) \u2264 n + 1) j)) % xs x_1 *\n          \u220f j : Fin \u2191(Fin.succ x_1), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ x_1) \u2264 n + 1) j) =\n    \u2191a\n[PROOFSTEP]\nsimp_rw [this]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\nthis :\n  \u2200 (i : Fin n),\n    \u220f x_1 : Fin \u2191(Fin.succ i), Fin.cons x xs (Fin.castLE (_ : \u2191(Fin.succ i) \u2264 n + 1) x_1) =\n      \u220f x_1 : Fin (\u2191i + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191i \u2264 Nat.succ n) x_1)\n\u22a2 (\u2191a / \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j)) % Fin.cons x xs 0 *\n        \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j) +\n      \u2211 x_1 : Fin n,\n        (\u2191a / \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2)) % xs x_1 *\n          \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2) =\n    \u2191a\n[PROOFSTEP]\nclear this\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 (\u2191a / \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j)) % Fin.cons x xs 0 *\n        \u220f j : Fin \u21910, Fin.cons x xs (Fin.castLE (_ : \u21910 \u2264 n + 1) j) +\n      \u2211 x_1 : Fin n,\n        (\u2191a / \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2)) % xs x_1 *\n          \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2) =\n    \u2191a\n[PROOFSTEP]\ndsimp only [Fin.val_zero]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 (\u2191a / \u220f j : Fin 0, Fin.cons x xs (Fin.castLE (_ : 0 \u2264 n + 1) j)) % Fin.cons x xs 0 *\n        \u220f j : Fin 0, Fin.cons x xs (Fin.castLE (_ : 0 \u2264 n + 1) j) +\n      \u2211 x_1 : Fin n,\n        (\u2191a / \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2)) % xs x_1 *\n          \u220f x_2 : Fin (\u2191x_1 + 1), Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) x_2) =\n    \u2191a\n[PROOFSTEP]\nsimp_rw [Fintype.prod_empty, Nat.div_one, mul_one, Fin.cons_zero, Fin.prod_univ_succ]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u2191a % x +\n      \u2211 x_1 : Fin n,\n        \u2191a /\n              (Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) 0) *\n                \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i))) %\n            xs x_1 *\n          (Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) 0) *\n            \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i))) =\n    \u2191a\n[PROOFSTEP]\nchange (_ + \u2211 y : _, _ / (x * _) % _ * (x * _)) = _\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u2191a % x +\n      \u2211 y : Fin n,\n        \u2191a / (x * \u220f i : Fin \u2191y, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191y \u2264 Nat.succ n) (Fin.succ i))) % xs y *\n          (x * \u220f i : Fin \u2191y, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191y \u2264 Nat.succ n) (Fin.succ i))) =\n    \u2191a\n[PROOFSTEP]\nsimp_rw [\u2190 Nat.div_div_eq_div_mul, mul_left_comm (_ % _ : \u2115), \u2190 mul_sum]\n[GOAL]\ncase refine'_2.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u2191a % x +\n      x *\n        \u2211 x_1 : Fin n,\n          (\u2191a / x / \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i))) % xs x_1 *\n            \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i)) =\n    \u2191a\n[PROOFSTEP]\nconvert\n  Nat.mod_add_div _\n    _\n      -- porting note: new\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u2211 x_1 : Fin n,\n      (\u2191a / x / \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i))) % xs x_1 *\n        \u220f i : Fin \u2191x_1, Fin.cons x xs (Fin.castLE (_ : Nat.succ \u2191x_1 \u2264 Nat.succ n) (Fin.succ i)) =\n    \u2191a / x\n[PROOFSTEP]\nrefine (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq ?_))\n[GOAL]\ncase h.e'_2.h.e'_6.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn\u271d : Fin m \u2192 \u2115\nn x : \u2115\nxs : Fin n \u2192 \u2115\na : Fin (\u220f i : Fin (n + 1), Fin.cons x xs i)\nih :\n  \u2200 (i : \u2115),\n    i < \u220f i : Fin n, xs i \u2192\n      \u2211 x : Fin n,\n          (i / \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j)) % xs x * \u220f j : Fin \u2191x, xs (Fin.castLE (_ : \u2191x \u2264 n) j) =\n        i\n\u22a2 \u220f i : Fin (n + 1), Fin.cons x xs i = x * \u220f i : Fin n, xs i\n[PROOFSTEP]\nexact\n  Fin.prod_univ_succ\n    _  -- porting note: was:\n              /-\n              refine' Eq.trans _ (ih (a / x) (Nat.div_lt_of_lt_mul <| a.is_lt.trans_eq _))\n              swap\n              \u00b7 convert Fin.prod_univ_succ (Fin.cons x xs : \u2200 _, \u2115)\n                simp_rw [Fin.cons_succ]\n              congr with i\n              congr with j\n              \u00b7 cases j\n                rfl\n              \u00b7 cases j\n                rfl-/\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\ninst\u271d : \u2200 (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n\u22a2 \u2191(\u2191finPiFinEquiv (Pi.single i j)) = \u2191j * \u220f j : Fin \u2191i, n (Fin.castLE (_ : \u2191i \u2264 m) j)\n[PROOFSTEP]\nrw [finPiFinEquiv_apply, Fintype.sum_eq_single i, Pi.single_eq_same]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\ninst\u271d : \u2200 (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\n\u22a2 \u2200 (x : Fin m), x \u2260 i \u2192 \u2191(Pi.single i j x) * \u220f j : Fin \u2191x, n (Fin.castLE (_ : \u2191x \u2264 m) j) = 0\n[PROOFSTEP]\nrintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : \u2115\nn : Fin m \u2192 \u2115\ninst\u271d : \u2200 (i : Fin m), NeZero (n i)\ni : Fin m\nj : Fin (n i)\nx : Fin m\nhx : x \u2260 i\n\u22a2 \u2191(Pi.single i j x) * \u220f j : Fin \u2191x, n (Fin.castLE (_ : \u2191x \u2264 m) j) = 0\n[PROOFSTEP]\nrw [Pi.single_eq_of_ne hx, Fin.val_zero', zero_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\n\u22a2 prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\n[PROOFSTEP]\ninduction i with\n| zero => simp\n| succ i IH =>\n  by_cases h : i < n\n  \u00b7 have : i < length (ofFn f) := by rwa [length_ofFn f]\n    rw [prod_take_succ _ _ this]\n    have A :\n      ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n        ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) \u222a {(\u27e8i, h\u27e9 : Fin n)} :=\n      by\n      ext \u27e8_, _\u27e9\n      simp [Nat.lt_succ_iff_lt_or_eq]\n    have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ) (singleton (\u27e8i, h\u27e9 : Fin n)) := by\n      simp\n    rw [A, Finset.prod_union B, IH]\n    simp\n  \u00b7 have A : (ofFn f).take i = (ofFn f).take i.succ :=\n      by\n      rw [\u2190 length_ofFn f] at h \n      have : length (ofFn f) \u2264 i := not_lt.mp h\n      rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]\n    have B : \u2200 j : Fin n, ((j : \u2115) < i.succ) = ((j : \u2115) < i) :=\n      by\n      intro j\n      have : (j : \u2115) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n      simp [this, lt_trans this (Nat.lt_succ_self _)]\n    simp [\u2190 A, B, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\n\u22a2 prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\n[PROOFSTEP]\ninduction i with\n| zero => simp\n| succ i IH =>\n  by_cases h : i < n\n  \u00b7 have : i < length (ofFn f) := by rwa [length_ofFn f]\n    rw [prod_take_succ _ _ this]\n    have A :\n      ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n        ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) \u222a {(\u27e8i, h\u27e9 : Fin n)} :=\n      by\n      ext \u27e8_, _\u27e9\n      simp [Nat.lt_succ_iff_lt_or_eq]\n    have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ) (singleton (\u27e8i, h\u27e9 : Fin n)) := by\n      simp\n    rw [A, Finset.prod_union B, IH]\n    simp\n  \u00b7 have A : (ofFn f).take i = (ofFn f).take i.succ :=\n      by\n      rw [\u2190 length_ofFn f] at h \n      have : length (ofFn f) \u2264 i := not_lt.mp h\n      rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]\n    have B : \u2200 j : Fin n, ((j : \u2115) < i.succ) = ((j : \u2115) < i) :=\n      by\n      intro j\n      have : (j : \u2115) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n      simp [this, lt_trans this (Nat.lt_succ_self _)]\n    simp [\u2190 A, B, IH]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 prod (take Nat.zero (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.zero) univ, f j\n[PROOFSTEP]\n\n| zero => simp\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 prod (take Nat.zero (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.zero) univ, f j\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\n\n| succ i IH =>\n  by_cases h : i < n\n  \u00b7 have : i < length (ofFn f) := by rwa [length_ofFn f]\n    rw [prod_take_succ _ _ this]\n    have A :\n      ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n        ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) \u222a {(\u27e8i, h\u27e9 : Fin n)} :=\n      by\n      ext \u27e8_, _\u27e9\n      simp [Nat.lt_succ_iff_lt_or_eq]\n    have B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ) (singleton (\u27e8i, h\u27e9 : Fin n)) := by\n      simp\n    rw [A, Finset.prod_union B, IH]\n    simp\n  \u00b7 have A : (ofFn f).take i = (ofFn f).take i.succ :=\n      by\n      rw [\u2190 length_ofFn f] at h \n      have : length (ofFn f) \u2264 i := not_lt.mp h\n      rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]\n    have B : \u2200 j : Fin n, ((j : \u2115) < i.succ) = ((j : \u2115) < i) :=\n      by\n      intro j\n      have : (j : \u2115) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n      simp [this, lt_trans this (Nat.lt_succ_self _)]\n    simp [\u2190 A, B, IH]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nby_cases h : i < n\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nhave : i < length (ofFn f) := by rwa [length_ofFn f]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\n\u22a2 i < length (ofFn f)\n[PROOFSTEP]\nrwa [length_ofFn f]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nrw [prod_take_succ _ _ this]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n\u22a2 prod (take i (ofFn f)) * nthLe (ofFn f) i this = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nhave A :\n  ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i + 1) =\n    ((Finset.univ : Finset (Fin n)).filter fun j => j.val < i) \u222a {(\u27e8i, h\u27e9 : Fin n)} :=\n  by\n  ext \u27e8_, _\u27e9\n  simp [Nat.lt_succ_iff_lt_or_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\n\u22a2 Finset.filter (fun j => \u2191j < i + 1) univ = Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\n[PROOFSTEP]\next \u27e8_, _\u27e9\n[GOAL]\ncase a.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\n\u22a2 { val := val\u271d, isLt := isLt\u271d } \u2208 Finset.filter (fun j => \u2191j < i + 1) univ \u2194\n    { val := val\u271d, isLt := isLt\u271d } \u2208 Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\n[PROOFSTEP]\nsimp [Nat.lt_succ_iff_lt_or_eq]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => \u2191j < i + 1) univ = Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\n\u22a2 prod (take i (ofFn f)) * nthLe (ofFn f) i this = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nhave B : _root_.Disjoint (Finset.filter (fun j : Fin n => j.val < i) Finset.univ) (singleton (\u27e8i, h\u27e9 : Fin n)) := by\n  simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => \u2191j < i + 1) univ = Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\n\u22a2 _root_.Disjoint (Finset.filter (fun j => \u2191j < i) univ) {{ val := i, isLt := h }}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => \u2191j < i + 1) univ = Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => \u2191j < i) univ) {{ val := i, isLt := h }}\n\u22a2 prod (take i (ofFn f)) * nthLe (ofFn f) i this = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nrw [A, Finset.prod_union B, IH]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : i < n\nthis : i < length (ofFn f)\nA : Finset.filter (fun j => \u2191j < i + 1) univ = Finset.filter (fun j => \u2191j < i) univ \u222a {{ val := i, isLt := h }}\nB : _root_.Disjoint (Finset.filter (fun j => \u2191j < i) univ) {{ val := i, isLt := h }}\n\u22a2 (\u220f j in Finset.filter (fun j => \u2191j < i) univ, f j) * nthLe (ofFn f) i this =\n    (\u220f x in Finset.filter (fun j => \u2191j < i) univ, f x) * \u220f x in {{ val := i, isLt := h }}, f x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nhave A : (ofFn f).take i = (ofFn f).take i.succ :=\n  by\n  rw [\u2190 length_ofFn f] at h \n  have : length (ofFn f) \u2264 i := not_lt.mp h\n  rw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\n\u22a2 take i (ofFn f) = take (Nat.succ i) (ofFn f)\n[PROOFSTEP]\nrw [\u2190 length_ofFn f] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < length (ofFn f)\n\u22a2 take i (ofFn f) = take (Nat.succ i) (ofFn f)\n[PROOFSTEP]\nhave : length (ofFn f) \u2264 i := not_lt.mp h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < length (ofFn f)\nthis : length (ofFn f) \u2264 i\n\u22a2 take i (ofFn f) = take (Nat.succ i) (ofFn f)\n[PROOFSTEP]\nrw [take_all_of_le this, take_all_of_le (le_trans this (Nat.le_succ _))]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nhave B : \u2200 j : Fin n, ((j : \u2115) < i.succ) = ((j : \u2115) < i) :=\n  by\n  intro j\n  have : (j : \u2115) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n  simp [this, lt_trans this (Nat.lt_succ_self _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\n\u22a2 \u2200 (j : Fin n), (\u2191j < Nat.succ i) = (\u2191j < i)\n[PROOFSTEP]\nintro j\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nj : Fin n\n\u22a2 (\u2191j < Nat.succ i) = (\u2191j < i)\n[PROOFSTEP]\nhave : (j : \u2115) < i := lt_of_lt_of_le j.2 (not_lt.mp h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nj : Fin n\nthis : \u2191j < i\n\u22a2 (\u2191j < Nat.succ i) = (\u2191j < i)\n[PROOFSTEP]\nsimp [this, lt_trans this (Nat.lt_succ_self _)]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\ni : \u2115\nIH : prod (take i (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < i) univ, f j\nh : \u00aci < n\nA : take i (ofFn f) = take (Nat.succ i) (ofFn f)\nB : \u2200 (j : Fin n), (\u2191j < Nat.succ i) = (\u2191j < i)\n\u22a2 prod (take (Nat.succ i) (ofFn f)) = \u220f j in Finset.filter (fun j => \u2191j < Nat.succ i) univ, f j\n[PROOFSTEP]\nsimp [\u2190 A, B, IH]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 prod (ofFn f) = \u220f i : Fin n, f i\n[PROOFSTEP]\nconvert prod_take_ofFn f n\n[GOAL]\ncase h.e'_2.h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 ofFn f = take n (ofFn f)\n[PROOFSTEP]\nrw [take_all_of_le (le_of_eq (length_ofFn f))]\n[GOAL]\ncase h.e'_3.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : CommMonoid \u03b1\nn : \u2115\nf : Fin n \u2192 \u03b1\n\u22a2 univ = Finset.filter (fun j => \u2191j < n) univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\n\u22a2 alternatingProd [] = \u220f i : Fin (length []), get [] i ^ (-1) ^ \u2191i\n[PROOFSTEP]\nrw [alternatingProd, Finset.prod_eq_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\n\u22a2 \u2200 (x : Fin (length [])), x \u2208 univ \u2192 get [] x ^ (-1) ^ \u2191x = 1\n[PROOFSTEP]\nrintro \u27e8i, \u27e8\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng : G\n\u22a2 alternatingProd [g] = \u220f i : Fin (length [g]), get [g] i ^ (-1) ^ \u2191i\n[PROOFSTEP]\nshow g = \u220f i : Fin 1, [g].get i ^ (-1 : \u2124) ^ (i : \u2115)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng : G\n\u22a2 g = \u220f i : Fin 1, get [g] i ^ (-1) ^ \u2191i\n[PROOFSTEP]\nrw [Fin.prod_univ_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng : G\n\u22a2 g = get [g] 0 ^ (-1) ^ \u21910 * \u220f i : Fin 0, get [g] (Fin.succ i) ^ (-1) ^ \u2191(Fin.succ i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng h : G\nL : List G\n\u22a2 g * h\u207b\u00b9 * \u220f i : Fin (length L), get L i ^ (-1) ^ \u2191i = \u220f i : Fin (length L + 2), get (g :: h :: L) i ^ (-1) ^ \u2191i\n[PROOFSTEP]\n{ rw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]\n  simp [Nat.succ_eq_add_one, pow_add]\n}\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng h : G\nL : List G\n\u22a2 g * h\u207b\u00b9 * \u220f i : Fin (length L), get L i ^ (-1) ^ \u2191i = \u220f i : Fin (length L + 2), get (g :: h :: L) i ^ (-1) ^ \u2191i\n[PROOFSTEP]\nrw [Fin.prod_univ_succ, Fin.prod_univ_succ, mul_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : CommGroup G\ng h : G\nL : List G\n\u22a2 g * (h\u207b\u00b9 * \u220f i : Fin (length L), get L i ^ (-1) ^ \u2191i) =\n    get (g :: h :: L) 0 ^ (-1) ^ \u21910 *\n      (get (g :: h :: L) (Fin.succ 0) ^ (-1) ^ \u2191(Fin.succ 0) *\n        \u220f i : Fin (length L), get (g :: h :: L) (Fin.succ (Fin.succ i)) ^ (-1) ^ \u2191(Fin.succ (Fin.succ i)))\n[PROOFSTEP]\nsimp [Nat.succ_eq_add_one, pow_add]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.Fin", "llama_tokens": 38484, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.6825737473266736, "lm_q1q2_score": 0.5249539794122534}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Add \u03b1\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : LeftDistribClass \u03b1\na b c : \u03b1\nh\u2081 : a \u2223 b\nh\u2082 : a \u2223 c\nd : \u03b1\nhd : b = a * d\ne : \u03b1\nhe : c = a * e\n\u22a2 a * (d + e) = b + c\n[PROOFSTEP]\nsimp [left_distrib, hd, he]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b c : \u03b1\n\u22a2 (\u2203 b_1, -b = a * \u2191(Equiv.neg \u03b1).symm b_1) \u2194 a \u2223 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b c : \u03b1\n\u22a2 (\u2203 b_1, b = a * b_1) \u2194 a \u2223 b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b c : \u03b1\n\u22a2 (\u2203 b_1, b = -a * \u2191(Equiv.neg \u03b1).symm b_1) \u2194 a \u2223 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Semigroup \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b c : \u03b1\n\u22a2 (\u2203 b_1, b = a * b_1) \u2194 a \u2223 b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh\u2081 : a \u2223 b\nh\u2082 : a \u2223 c\n\u22a2 a \u2223 b - c\n[PROOFSTEP]\nsimpa only [\u2190 sub_eq_add_neg] using h\u2081.add h\u2082.neg_right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 c\nH : a \u2223 b + c\n\u22a2 a \u2223 b\n[PROOFSTEP]\nsimpa only [add_sub_cancel] using dvd_sub H h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 b\n\u22a2 a \u2223 b + c \u2194 a \u2223 c\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 b\n\u22a2 a \u2223 c + b \u2194 a \u2223 c\n[PROOFSTEP]\nexact dvd_add_left h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 c\n\u22a2 a \u2223 b - c \u2194 a \u2223 b\n[PROOFSTEP]\nsimpa only [\u2190 sub_eq_add_neg] using dvd_add_left ((dvd_neg (\u03b1 := \u03b1)).2 h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 b\n\u22a2 a \u2223 b - c \u2194 a \u2223 c\n[PROOFSTEP]\nrw [sub_eq_add_neg, dvd_add_right h, dvd_neg (\u03b1 := \u03b1)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\nh : a \u2223 b - c\n\u22a2 a \u2223 b \u2194 a \u2223 c\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel b c, dvd_add_right h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalRing \u03b1\na b c : \u03b1\n\u22a2 a \u2223 b - c \u2194 a \u2223 c - b\n[PROOFSTEP]\nrw [\u2190 dvd_neg (\u03b1 := \u03b1), neg_sub]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalCommRing \u03b1\na\u271d b\u271d c k a b x y : \u03b1\nhab : k \u2223 a - b\nhxy : k \u2223 x - y\n\u22a2 k \u2223 a * x - b * y\n[PROOFSTEP]\nconvert dvd_add (hxy.mul_left a) (hab.mul_right y) using 1\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalCommRing \u03b1\na\u271d b\u271d c k a b x y : \u03b1\nhab : k \u2223 a - b\nhxy : k \u2223 x - y\n\u22a2 a * x - b * y = a * (x - y) + (a - b) * y\n[PROOFSTEP]\nrw [mul_sub_left_distrib, mul_sub_right_distrib]\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NonUnitalCommRing \u03b1\na\u271d b\u271d c k a b x y : \u03b1\nhab : k \u2223 a - b\nhxy : k \u2223 x - y\n\u22a2 a * x - b * y = a * x - a * y + (a * y - b * y)\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, add_assoc, neg_add_cancel_left]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Divisibility", "llama_tokens": 1523, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943822145998, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.524880198897589}}
{"text": "[GOAL]\nn iv : \u2115\nilt\u2081 : iv < n\njv : \u2115\njlt\u2081 : jv < n\nh : { val := iv, isLt := ilt\u2081 }.val = { val := jv, isLt := jlt\u2081 }.val\n\u22a2 { val := iv, isLt := ilt\u2081 } = { val := jv, isLt := jlt\u2081 }\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\nn iv : \u2115\nilt\u2081 jlt\u2081 : iv < n\n\u22a2 { val := iv, isLt := ilt\u2081 } = { val := iv, isLt := jlt\u2081 }\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.Fin.Basic", "llama_tokens": 172, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8333245787544825, "lm_q2_score": 0.629774621301746, "lm_q1q2_score": 0.5248066710065412}}
{"text": "[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\nm n : \u2115\nh : m \u2264 n\n\u22a2 \u2191(natLERec f' m n h) = fun a => Nat.leRecOn h (fun k => \u2191(f' k)) a\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le h\n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\nm k : \u2115\nh : m \u2264 m + k\n\u22a2 \u2191(natLERec f' m (m + k) h) = fun a => Nat.leRecOn h (fun k => \u2191(f' k)) a\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\nm k : \u2115\nh : m \u2264 m + k\nx : G' m\n\u22a2 \u2191(natLERec f' m (m + k) h) x = Nat.leRecOn h (fun k => \u2191(f' k)) x\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase intro.h.zero\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\nm k : \u2115\nh\u271d : m \u2264 m + k\nx : G' m\nh : m \u2264 m + Nat.zero\n\u22a2 \u2191(natLERec f' m (m + Nat.zero) h) x = Nat.leRecOn h (fun k => \u2191(f' k)) x\n[PROOFSTEP]\nrw [natLERec, Nat.leRecOn_self, Embedding.refl_apply, Nat.leRecOn_self]\n[GOAL]\ncase intro.h.succ\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\nm k\u271d : \u2115\nh\u271d : m \u2264 m + k\u271d\nx : G' m\nk : \u2115\nih : \u2200 (h : m \u2264 m + k), \u2191(natLERec f' m (m + k) h) x = Nat.leRecOn h (fun k => \u2191(f' k)) x\nh : m \u2264 m + Nat.succ k\n\u22a2 \u2191(natLERec f' m (m + Nat.succ k) h) x = Nat.leRecOn h (fun k => \u2191(f' k)) x\n[PROOFSTEP]\nrw [Nat.leRecOn_succ le_self_add, natLERec, Nat.leRecOn_succ le_self_add, \u2190 natLERec, Embedding.comp_apply, ih]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nG' : \u2115 \u2192 Type w\ninst\u271d : (i : \u2115) \u2192 Structure L (G' i)\nf' : (n : \u2115) \u2192 G' n \u21aa[L] G' (n + 1)\ni\u271d j\u271d k\u271d : \u2115\nhij : i\u271d \u2264 j\u271d\nhjk : j\u271d \u2264 k\u271d\n\u22a2 \u2200 (x : G' i\u271d), \u2191(natLERec f' j\u271d k\u271d hjk) (\u2191(natLERec f' i\u271d j\u271d hij) x) = \u2191(natLERec f' i\u271d k\u271d (_ : i\u271d \u2264 k\u271d)) x\n[PROOFSTEP]\nsimp [Nat.leRecOn_trans hij hjk]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u03b1 : Type u_1\ni : \u03b9\nx : \u03b1 \u2192 G i\n\u22a2 unify f (fun a => Structure.Sigma.mk f i (x a)) i\n      (_ : \u2200 (j : \u03b9), j \u2208 range (Sigma.fst \u2218 fun a => Structure.Sigma.mk f i (x a)) \u2192 j \u2264 i) =\n    x\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u03b1 : Type u_1\ni : \u03b9\nx : \u03b1 \u2192 G i\na : \u03b1\n\u22a2 unify f (fun a => Structure.Sigma.mk f i (x a)) i\n      (_ : \u2200 (j : \u03b9), j \u2208 range (Sigma.fst \u2218 fun a => Structure.Sigma.mk f i (x a)) \u2192 j \u2264 i) a =\n    x a\n[PROOFSTEP]\nrw [unify]\n[GOAL]\ncase h\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u03b1 : Type u_1\ni : \u03b9\nx : \u03b1 \u2192 G i\na : \u03b1\n\u22a2 \u2191(f (Structure.Sigma.mk f i (x a)).fst i (_ : (Structure.Sigma.mk f i (x a)).fst \u2264 i))\n      (Structure.Sigma.mk f i (x a)).snd =\n    x a\n[PROOFSTEP]\napply DirectedSystem.map_self\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u03b1 : Type u_1\nx : \u03b1 \u2192 \u03a3\u02e3 f\ni j : \u03b9\nij : i \u2264 j\nh : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\n\u22a2 \u2191(f i j ij) \u2218 unify f x i h = unify f x j (_ : \u2200 (k : \u03b9), k \u2208 range (Sigma.fst \u2218 x) \u2192 k \u2264 j)\n[PROOFSTEP]\next a\n[GOAL]\ncase h\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b2 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u03b1 : Type u_1\nx : \u03b1 \u2192 \u03a3\u02e3 f\ni j : \u03b9\nij : i \u2264 j\nh : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\na : \u03b1\n\u22a2 (\u2191(f i j ij) \u2218 unify f x i h) a = unify f x j (_ : \u2200 (k : \u03b9), k \u2208 range (Sigma.fst \u2218 x) \u2192 k \u2264 j) a\n[PROOFSTEP]\nsimp [unify, DirectedSystem.map_map]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\n\u22a2 match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\n[PROOFSTEP]\nobtain \u27e8ijk, hijijk, hjkijk\u27e9 := directed_of (\u00b7 \u2264 \u00b7) ij jk\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\n[PROOFSTEP]\nrefine' \u27e8ijk, le_trans hiij hijijk, le_trans hkjk hjkijk, _\u27e9\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 \u2191(f i ijk (_ : i \u2264 ijk)) x = \u2191(f k ijk (_ : k \u2264 ijk)) z\n[PROOFSTEP]\nrw [\u2190 DirectedSystem.map_map, hij, DirectedSystem.map_map]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 \u2191(f j ijk (_ : j \u2264 ijk)) y = \u2191(f k ijk (_ : k \u2264 ijk)) z\ncase intro.intro.hjk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 ij \u2264 ijk\ncase intro.intro.hjk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 ij \u2264 ijk\ncase intro.intro.hjk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 ij \u2264 ijk\n[PROOFSTEP]\nsymm\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 \u2191(f k ijk (_ : k \u2264 ijk)) z = \u2191(f j ijk (_ : j \u2264 ijk)) y\n[PROOFSTEP]\nrw [\u2190 DirectedSystem.map_map, \u2190 hjk, DirectedSystem.map_map]\n[GOAL]\ncase intro.intro.hjk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 jk \u2264 ijk\n[PROOFSTEP]\nassumption\n[GOAL]\ncase intro.intro.hjk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\nx\u271d\u2074 x\u271d\u00b3 x\u271d\u00b2 : \u03a3\u02e3 f\ni : \u03b9\nx : (fun i => G i) i\nj : \u03b9\ny : (fun i => G i) j\nx\u271d\u00b9 :\n  match { fst := i, snd := x } with\n  | { fst := i, snd := x } =>\n    match { fst := j, snd := y } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nk : \u03b9\nz : (fun i => G i) k\nx\u271d :\n  match { fst := j, snd := y } with\n  | { fst := i, snd := x } =>\n    match { fst := k, snd := z } with\n    | { fst := j, snd := y } => \u2203 k ik jk, \u2191(f i k ik) x = \u2191(f j k jk) y\nij : \u03b9\nhiij : i \u2264 ij\nhjij : j \u2264 ij\nhij : \u2191(f i ij hiij) x = \u2191(f j ij hjij) y\njk : \u03b9\nhjjk : j \u2264 jk\nhkjk : k \u2264 jk\nhjk : \u2191(f j jk hjjk) y = \u2191(f k jk hkjk) z\nijk : \u03b9\nhijijk : ij \u2264 ijk\nhjkijk : jk \u2264 ijk\n\u22a2 ij \u2264 ijk\n[PROOFSTEP]\nassumption\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nx y : \u03a3\u02e3 f\ni : \u03b9\nhx : x.fst \u2264 i\nhy : y.fst \u2264 i\n\u22a2 x \u2248 y \u2194 \u2191(f x.fst i hx) x.snd = \u2191(f y.fst i hy) y.snd\n[PROOFSTEP]\ncases x\n[GOAL]\ncase mk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ny : \u03a3\u02e3 f\ni : \u03b9\nhy : y.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhx : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\n\u22a2 { fst := fst\u271d, snd := snd\u271d } \u2248 y \u2194\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hx) { fst := fst\u271d, snd := snd\u271d }.snd = \u2191(f y.fst i hy) y.snd\n[PROOFSTEP]\ncases y\n[GOAL]\ncase mk.mk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\n\u22a2 { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } \u2248 { fst := fst\u271d, snd := snd\u271d } \u2194\n    \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n      \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nrefine' \u27e8fun xy => _, fun xy => \u27e8i, hx, hy, xy\u27e9\u27e9\n[GOAL]\ncase mk.mk\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nxy : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 } \u2248 { fst := fst\u271d, snd := snd\u271d }\n\u22a2 \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nobtain \u27e8j, _, _, h\u27e9 := xy\n[GOAL]\ncase mk.mk.intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nj : \u03b9\nw\u271d\u00b9 : fst\u271d\u00b9 \u2264 j\nw\u271d : fst\u271d \u2264 j\nh : \u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9 = \u2191(f fst\u271d j w\u271d) snd\u271d\n\u22a2 \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nobtain \u27e8k, ik, jk\u27e9 := directed_of (\u00b7 \u2264 \u00b7) i j\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nj : \u03b9\nw\u271d\u00b9 : fst\u271d\u00b9 \u2264 j\nw\u271d : fst\u271d \u2264 j\nh : \u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9 = \u2191(f fst\u271d j w\u271d) snd\u271d\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\n\u22a2 \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nhave h := congr_arg (f j k jk) h\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nj : \u03b9\nw\u271d\u00b9 : fst\u271d\u00b9 \u2264 j\nw\u271d : fst\u271d \u2264 j\nh\u271d : \u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9 = \u2191(f fst\u271d j w\u271d) snd\u271d\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\nh : \u2191(f j k jk) (\u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9) = \u2191(f j k jk) (\u2191(f fst\u271d j w\u271d) snd\u271d)\n\u22a2 \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\napply (f i k ik).injective\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.a\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nj : \u03b9\nw\u271d\u00b9 : fst\u271d\u00b9 \u2264 j\nw\u271d : fst\u271d \u2264 j\nh\u271d : \u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9 = \u2191(f fst\u271d j w\u271d) snd\u271d\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\nh : \u2191(f j k jk) (\u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9) = \u2191(f j k jk) (\u2191(f fst\u271d j w\u271d) snd\u271d)\n\u22a2 \u2191(f i k ik) (\u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst i hx) { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd) =\n    \u2191(f i k ik) (\u2191(f { fst := fst\u271d, snd := snd\u271d }.fst i hy) { fst := fst\u271d, snd := snd\u271d }.snd)\n[PROOFSTEP]\nrw [DirectedSystem.map_map, DirectedSystem.map_map] at *\n[GOAL]\ncase mk.mk.intro.intro.intro.intro.intro.a\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\ni fst\u271d\u00b9 : \u03b9\nsnd\u271d\u00b9 : (fun i => G i) fst\u271d\u00b9\nhx : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 i\nfst\u271d : \u03b9\nsnd\u271d : (fun i => G i) fst\u271d\nhy : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 i\nj : \u03b9\nw\u271d\u00b9 : fst\u271d\u00b9 \u2264 j\nw\u271d : fst\u271d \u2264 j\nh\u271d : \u2191(f fst\u271d\u00b9 j w\u271d\u00b9) snd\u271d\u00b9 = \u2191(f fst\u271d j w\u271d) snd\u271d\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\nh : \u2191(f fst\u271d\u00b9 k (_ : fst\u271d\u00b9 \u2264 k)) snd\u271d\u00b9 = \u2191(f fst\u271d k (_ : fst\u271d \u2264 k)) snd\u271d\n\u22a2 \u2191(f { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst k (_ : { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.fst \u2264 k))\n      { fst := fst\u271d\u00b9, snd := snd\u271d\u00b9 }.snd =\n    \u2191(f { fst := fst\u271d, snd := snd\u271d }.fst k (_ : { fst := fst\u271d, snd := snd\u271d }.fst \u2264 k)) { fst := fst\u271d, snd := snd\u271d }.snd\n[PROOFSTEP]\nexact h\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nn : \u2115\nF : Functions L n\nx : Fin n \u2192 \u03a3\u02e3 f\ni j : \u03b9\nhi : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhj : j \u2208 upperBounds (range (Sigma.fst \u2218 x))\n\u22a2 Structure.Sigma.mk f i (funMap F (unify f x i hi)) \u2248 Structure.Sigma.mk f j (funMap F (unify f x j hj))\n[PROOFSTEP]\nobtain \u27e8k, ik, jk\u27e9 := directed_of (\u00b7 \u2264 \u00b7) i j\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nn : \u2115\nF : Functions L n\nx : Fin n \u2192 \u03a3\u02e3 f\ni j : \u03b9\nhi : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhj : j \u2208 upperBounds (range (Sigma.fst \u2218 x))\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\n\u22a2 Structure.Sigma.mk f i (funMap F (unify f x i hi)) \u2248 Structure.Sigma.mk f j (funMap F (unify f x j hj))\n[PROOFSTEP]\nrefine' \u27e8k, ik, jk, _\u27e9\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nn : \u2115\nF : Functions L n\nx : Fin n \u2192 \u03a3\u02e3 f\ni j : \u03b9\nhi : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhj : j \u2208 upperBounds (range (Sigma.fst \u2218 x))\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\n\u22a2 \u2191(f i k ik) (funMap F (unify f x i hi)) = \u2191(f j k jk) (funMap F (unify f x j hj))\n[PROOFSTEP]\nrw [(f i k ik).map_fun, (f j k jk).map_fun, comp_unify, comp_unify]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nn : \u2115\nR : Relations L n\nx : Fin n \u2192 \u03a3\u02e3 f\ni j : \u03b9\nhi : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhj : j \u2208 upperBounds (range (Sigma.fst \u2218 x))\n\u22a2 RelMap R (unify f x i hi) = RelMap R (unify f x j hj)\n[PROOFSTEP]\nobtain \u27e8k, ik, jk\u27e9 := directed_of (\u00b7 \u2264 \u00b7) i j\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u00b3 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b9 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nn : \u2115\nR : Relations L n\nx : Fin n \u2192 \u03a3\u02e3 f\ni j : \u03b9\nhi : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhj : j \u2208 upperBounds (range (Sigma.fst \u2218 x))\nk : \u03b9\nik : i \u2264 k\njk : j \u2264 k\n\u22a2 RelMap R (unify f x i hi) = RelMap R (unify f x j hj)\n[PROOFSTEP]\nrw [\u2190 (f i k ik).map_rel, comp_unify, \u2190 (f j k jk).map_rel, comp_unify]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx y : \u03b1 \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\n\u22a2 \u2203 i hx hy, unify f x i hx = unify f y i hy\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Fintype.bddAbove_range (Sum.elim (fun a => (x a).1) fun a => (y a).1)\n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx y : \u03b1 \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhi : i \u2208 upperBounds (range (Sum.elim (fun a => (x a).fst) fun a => (y a).fst))\n\u22a2 \u2203 i hx hy, unify f x i hx = unify f y i hy\n[PROOFSTEP]\nrw [Sum.elim_range, upperBounds_union] at hi \n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx y : \u03b1 \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (x a).fst) \u2229 upperBounds (range fun a => (y a).fst)\n\u22a2 \u2203 i hx hy, unify f x i hx = unify f y i hy\n[PROOFSTEP]\nsimp_rw [\u2190 Function.comp_apply (f := Sigma.fst)] at hi \n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx y : \u03b1 \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Sigma.fst \u2218 x) a) \u2229 upperBounds (range fun a => (Sigma.fst \u2218 y) a)\n\u22a2 \u2203 i hx hy, unify f x i hx = unify f y i hy\n[PROOFSTEP]\nexact \u27e8i, hi.1, hi.2, funext fun a => (equiv_iff G f _ _).1 (xy a)\u27e9\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\n\u22a2 funMap F x \u2248 funMap F y\n[PROOFSTEP]\nobtain \u27e8i, hx, hy, h\u27e9 := exists_unify_eq G f xy\n[GOAL]\ncase intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhy : i \u2208 upperBounds (range (Sigma.fst \u2218 y))\nh : unify f x i hx = unify f y i hy\n\u22a2 funMap F x \u2248 funMap F y\n[PROOFSTEP]\nrefine' Setoid.trans (funMap_equiv_unify G f F x i hx) (Setoid.trans _ (Setoid.symm (funMap_equiv_unify G f F y i hy)))\n[GOAL]\ncase intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhy : i \u2208 upperBounds (range (Sigma.fst \u2218 y))\nh : unify f x i hx = unify f y i hy\n\u22a2 Structure.Sigma.mk f i (funMap F (unify f x i hx)) \u2248 Structure.Sigma.mk f i (funMap F (unify f y i hy))\n[PROOFSTEP]\nrw [h]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\n\u22a2 RelMap R x = RelMap R y\n[PROOFSTEP]\nobtain \u27e8i, hx, hy, h\u27e9 := exists_unify_eq G f xy\n[GOAL]\ncase intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhy : i \u2208 upperBounds (range (Sigma.fst \u2218 y))\nh : unify f x i hx = unify f y i hy\n\u22a2 RelMap R x = RelMap R y\n[PROOFSTEP]\nrefine' _root_.trans (relMap_equiv_unify G f R x i hx) (_root_.trans _ (symm (relMap_equiv_unify G f R y i hy)))\n[GOAL]\ncase intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\nx y : Fin n \u2192 \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : i \u2208 upperBounds (range (Sigma.fst \u2218 x))\nhy : i \u2208 upperBounds (range (Sigma.fst \u2218 y))\nh : unify f x i hx = unify f y i hy\n\u22a2 RelMap R (unify f x i hx) = RelMap R (unify f y i hy)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\ni : \u03b9\nx : Fin n \u2192 G i\n\u22a2 (funMap F fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (x a))) =\n    Quotient.mk (setoid G f) (Structure.Sigma.mk f i (funMap F x))\n[PROOFSTEP]\nsimp [Function.comp_apply, funMap_quotient_mk', Quotient.eq']\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\ni : \u03b9\nx : Fin n \u2192 G i\n\u22a2 (funMap F fun i_1 => Structure.Sigma.mk f i (x i_1)) \u2248 Structure.Sigma.mk f i (funMap F x)\n[PROOFSTEP]\nobtain \u27e8k, ik, jk\u27e9 := directed_of (\u00b7 \u2264 \u00b7) i (Classical.choose (Fintype.bddAbove_range fun _ : Fin n => i))\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nik : i \u2264 k\njk : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 (funMap F fun i_1 => Structure.Sigma.mk f i (x i_1)) \u2248 Structure.Sigma.mk f i (funMap F x)\n[PROOFSTEP]\nrefine' \u27e8k, jk, ik, _\u27e9\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nik : i \u2264 k\njk : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 \u2191(f (Classical.choose (_ : BddAbove (range fun a => (Structure.Sigma.mk f i (x a)).fst))) k jk)\n      (funMap F\n        (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1))\n          (Classical.choose (_ : BddAbove (range fun a => (Structure.Sigma.mk f i (x a)).fst)))\n          (_ :\n            Classical.choose (_ : BddAbove (range fun a => (Structure.Sigma.mk f i (x a)).fst)) \u2208\n              upperBounds (range fun a => (Structure.Sigma.mk f i (x a)).fst)))) =\n    \u2191(f i k ik) (funMap F x)\n[PROOFSTEP]\nsimp only [Embedding.map_fun, comp_unify]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nF : Functions L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nik : i \u2264 k\njk : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 funMap F\n      (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) k\n        (_ : \u2200 (k_1 : \u03b9), k_1 \u2208 range (Sigma.fst \u2218 fun i_1 => Structure.Sigma.mk f i (x i_1)) \u2192 k_1 \u2264 k)) =\n    funMap F (\u2191(f i k ik) \u2218 x)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\ni : \u03b9\nx : Fin n \u2192 G i\n\u22a2 (RelMap R fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (x a))) = RelMap R x\n[PROOFSTEP]\nrw [relMap_quotient_mk']\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\ni : \u03b9\nx : Fin n \u2192 G i\n\u22a2 (RelMap R fun a => Structure.Sigma.mk f i (x a)) = RelMap R x\n[PROOFSTEP]\nobtain \u27e8k, _, _\u27e9 := directed_of (\u00b7 \u2264 \u00b7) i (Classical.choose (Fintype.bddAbove_range fun _ : Fin n => i))\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nleft\u271d : i \u2264 k\nright\u271d : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 (RelMap R fun a => Structure.Sigma.mk f i (x a)) = RelMap R x\n[PROOFSTEP]\nrw [relMap_equiv_unify G f R (fun a => .mk f i (x a)) i]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nleft\u271d : i \u2264 k\nright\u271d : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 RelMap R (unify f (fun a => Structure.Sigma.mk f i (x a)) i ?intro.intro) = RelMap R x\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nn : \u2115\nR : Relations L n\ni : \u03b9\nx : Fin n \u2192 G i\nk : \u03b9\nleft\u271d : i \u2264 k\nright\u271d : Classical.choose (_ : BddAbove (range fun x => i)) \u2264 k\n\u22a2 i \u2208 upperBounds (range (Sigma.fst \u2218 fun a => Structure.Sigma.mk f i (x a)))\n[PROOFSTEP]\nrw [unify_sigma_mk_self]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\n\u22a2 \u2203 i y, x = fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))\n[PROOFSTEP]\nobtain \u27e8i, hi\u27e9 := Fintype.bddAbove_range fun a => (x a).out.1\n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\n\u22a2 \u2203 i y, x = fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))\n[PROOFSTEP]\nrefine' \u27e8i, unify f (Quotient.out \u2218 x) i hi, _\u27e9\n[GOAL]\ncase intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\n\u22a2 x = fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (unify f (Quotient.out \u2218 x) i hi a))\n[PROOFSTEP]\next a\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\n\u22a2 x a = Quotient.mk (setoid G f) (Structure.Sigma.mk f i (unify f (Quotient.out \u2218 x) i hi a))\n[PROOFSTEP]\nrw [Quotient.eq_mk_iff_out, unify]\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\n\u22a2 Quotient.out (x a) \u2248\n    Structure.Sigma.mk f i\n      (\u2191(f ((Quotient.out \u2218 x) a).fst i (_ : ((Quotient.out \u2218 x) a).fst \u2264 i)) ((Quotient.out \u2218 x) a).snd)\n[PROOFSTEP]\ngeneralize_proofs r\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\nr : ((Quotient.out \u2218 x) a).fst \u2264 i\n\u22a2 Quotient.out (x a) \u2248 Structure.Sigma.mk f i (\u2191(f ((Quotient.out \u2218 x) a).fst i r) ((Quotient.out \u2218 x) a).snd)\n[PROOFSTEP]\nchange _ \u2248 .mk f i (f (Quotient.out (x a)).fst i r (Quotient.out (x a)).snd)\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\nr : ((Quotient.out \u2218 x) a).fst \u2264 i\n\u22a2 Quotient.out (x a) \u2248 Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)\n[PROOFSTEP]\nhave : (.mk f i (f (Quotient.out (x a)).fst i r (Quotient.out (x a)).snd) : \u03a3\u02e3 f).fst \u2264 i := le_rfl\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\nr : ((Quotient.out \u2218 x) a).fst \u2264 i\nthis : (Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)).fst \u2264 i\n\u22a2 Quotient.out (x a) \u2248 Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)\n[PROOFSTEP]\nrw [equiv_iff G f (i := i) (hi _) this]\n[GOAL]\ncase intro.h\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\nr : ((Quotient.out \u2218 x) a).fst \u2264 i\nthis : (Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)).fst \u2264 i\n\u22a2 \u2191(f (Quotient.out (x a)).fst i (_ : (Quotient.out (x a)).fst \u2264 i)) (Quotient.out (x a)).snd =\n    \u2191(f (Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)).fst i this)\n      (Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)).snd\n[PROOFSTEP]\nsimp only [DirectedSystem.map_self]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\n\u03b1 : Type u_1\ninst\u271d : Fintype \u03b1\nx : \u03b1 \u2192 DirectLimit G f\ni : \u03b9\nhi : i \u2208 upperBounds (range fun a => (Quotient.out (x a)).fst)\na : \u03b1\nr : ((Quotient.out \u2218 x) a).fst \u2264 i\nthis : (Structure.Sigma.mk f i (\u2191(f (Quotient.out (x a)).fst i r) (Quotient.out (x a)).snd)).fst \u2264 i\n\u22a2 (Quotient.out (x a)).fst \u2208 range fun a => (Quotient.out (x a)).fst\n[PROOFSTEP]\nexact \u27e8a, rfl\u27e9\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nx y : G i\nh :\n  (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) x =\n    (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [Quotient.eq] at h \n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nx y : G i\nh : Structure.Sigma.mk f i x \u2248 Structure.Sigma.mk f i y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8j, h1, _, h3\u27e9 := h\n[GOAL]\ncase intro.intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nx y : G i\nj : \u03b9\nh1 w\u271d : i \u2264 j\nh3 : \u2191(f i j h1) x = \u2191(f i j w\u271d) y\n\u22a2 x = y\n[PROOFSTEP]\nexact (f i j h1).injective h3\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nn\u271d : \u2115\nF : Functions L n\u271d\nx : Fin n\u271d \u2192 G i\n\u22a2 Function.Embedding.toFun\n      { toFun := fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a),\n        inj' :=\n          (_ :\n            \u2200 (x y : G i),\n              (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) x =\n                  (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) y \u2192\n                x = y) }\n      (funMap F x) =\n    funMap F\n      ({ toFun := fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a),\n            inj' :=\n              (_ :\n                \u2200 (x y : G i),\n                  (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) x =\n                      (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) y \u2192\n                    x = y) }.toFun \u2218\n        x)\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nn\u271d : \u2115\nF : Functions L n\u271d\nx : Fin n\u271d \u2192 G i\n\u22a2 Quotient.mk (setoid G f) (Structure.Sigma.mk f i (funMap F x)) =\n    funMap F ((fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) \u2218 x)\n[PROOFSTEP]\nrw [\u2190 funMap_quotient_mk'_sigma_mk']\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nn\u271d : \u2115\nF : Functions L n\u271d\nx : Fin n\u271d \u2192 G i\n\u22a2 (funMap F fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (x a))) =\n    funMap F ((fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) \u2218 x)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\n\u22a2 \u2200 {n : \u2115} (r : Relations L n) (x : Fin n \u2192 G i),\n    RelMap r\n        ({ toFun := fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a),\n              inj' :=\n                (_ :\n                  \u2200 (x y : G i),\n                    (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) x =\n                        (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) y \u2192\n                      x = y) }.toFun \u2218\n          x) \u2194\n      RelMap r x\n[PROOFSTEP]\nintro n R x\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nn : \u2115\nR : Relations L n\nx : Fin n \u2192 G i\n\u22a2 RelMap R\n      ({ toFun := fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a),\n            inj' :=\n              (_ :\n                \u2200 (x y : G i),\n                  (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) x =\n                      (fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i a)) y \u2192\n                    x = y) }.toFun \u2218\n        x) \u2194\n    RelMap R x\n[PROOFSTEP]\nchange RelMap R (fun a => (\u27e6.mk f i (x a)\u27e7 : DirectLimit G f)) \u2194 _\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni : \u03b9\nn : \u2115\nR : Relations L n\nx : Fin n \u2192 G i\n\u22a2 (RelMap R fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (x a))) \u2194 RelMap R x\n[PROOFSTEP]\nsimp only [relMap_quotient_mk'_sigma_mk']\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni j : \u03b9\nhij : i \u2264 j\nx : G i\n\u22a2 \u2191(of L \u03b9 G f j) (\u2191(f i j hij) x) = \u2191(of L \u03b9 G f i) x\n[PROOFSTEP]\nrw [of_apply, of_apply, Quotient.eq]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni j : \u03b9\nhij : i \u2264 j\nx : G i\n\u22a2 Structure.Sigma.mk f j (\u2191(f i j hij) x) \u2248 Structure.Sigma.mk f i x\n[PROOFSTEP]\nrefine' Setoid.symm \u27e8j, hij, refl j, _\u27e9\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\ni j : \u03b9\nhij : i \u2264 j\nx : G i\n\u22a2 \u2191(f i j hij) x = \u2191(f j j (_ : j \u2264 j)) (\u2191(f i j hij) x)\n[PROOFSTEP]\nsimp only [DirectedSystem.map_self]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2074 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b2 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d : Nonempty \u03b9\nz : DirectLimit G f\n\u22a2 \u2191(of L \u03b9 G f (Quotient.out z).fst) (Quotient.out z).snd = z\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : \u03a3\u02e3 f\nxy : x \u2248 y\n\u22a2 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : \u03a3\u02e3 f\nxy : x \u2248 y\n\u22a2 \u2191(g x.fst) x.snd = \u2191(g y.fst) y.snd\n[PROOFSTEP]\nobtain \u27e8i, hx, hy\u27e9 := directed_of (\u00b7 \u2264 \u00b7) x.1 y.1\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : x.fst \u2264 i\nhy : y.fst \u2264 i\n\u22a2 \u2191(g x.fst) x.snd = \u2191(g y.fst) y.snd\n[PROOFSTEP]\nrw [\u2190 Hg x.1 i hx, \u2190 Hg y.1 i hy]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : \u03a3\u02e3 f\nxy : x \u2248 y\ni : \u03b9\nhx : x.fst \u2264 i\nhy : y.fst \u2264 i\n\u22a2 \u2191(g i) (\u2191(f x.fst i hx) x.snd) = \u2191(g i) (\u2191(f y.fst i hy) y.snd)\n[PROOFSTEP]\nexact congr_arg _ ((equiv_iff ..).1 xy)\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : DirectLimit G f\nxy :\n  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n      (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n    Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n      (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq x, \u2190 Quotient.out_eq y, Quotient.lift_mk, Quotient.lift_mk] at xy \n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : DirectLimit G f\nxy : \u2191(g (Quotient.out x).fst) (Quotient.out x).snd = \u2191(g (Quotient.out y).fst) (Quotient.out y).snd\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8i, hx, hy\u27e9 := directed_of (\u00b7 \u2264 \u00b7) x.out.1 y.out.1\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : DirectLimit G f\nxy : \u2191(g (Quotient.out x).fst) (Quotient.out x).snd = \u2191(g (Quotient.out y).fst) (Quotient.out y).snd\ni : \u03b9\nhx : (Quotient.out x).fst \u2264 i\nhy : (Quotient.out y).fst \u2264 i\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Hg x.out.1 i hx, \u2190 Hg y.out.1 i hy] at xy \n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : DirectLimit G f\ni : \u03b9\nhx : (Quotient.out x).fst \u2264 i\nhy : (Quotient.out y).fst \u2264 i\nxy :\n  \u2191(g i) (\u2191(f (Quotient.out x).fst i hx) (Quotient.out x).snd) =\n    \u2191(g i) (\u2191(f (Quotient.out y).fst i hy) (Quotient.out y).snd)\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq x, \u2190 Quotient.out_eq y, Quotient.eq, equiv_iff G f hx hy]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nx y : DirectLimit G f\ni : \u03b9\nhx : (Quotient.out x).fst \u2264 i\nhy : (Quotient.out y).fst \u2264 i\nxy :\n  \u2191(g i) (\u2191(f (Quotient.out x).fst i hx) (Quotient.out x).snd) =\n    \u2191(g i) (\u2191(f (Quotient.out y).fst i hy) (Quotient.out y).snd)\n\u22a2 \u2191(f (Quotient.out x).fst i hx) (Quotient.out x).snd = \u2191(f (Quotient.out y).fst i hy) (Quotient.out y).snd\n[PROOFSTEP]\nexact (g i).injective xy\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nF : Functions L n\u271d\nx : Fin n\u271d \u2192 DirectLimit G f\n\u22a2 Function.Embedding.toFun\n      {\n        toFun :=\n          Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n            (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n        inj' :=\n          (_ :\n            \u2200 (x y : DirectLimit G f),\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                x = y) }\n      (funMap F x) =\n    funMap F\n      ({\n            toFun :=\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n            inj' :=\n              (_ :\n                \u2200 (x y : DirectLimit G f),\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                      Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                    x = y) }.toFun \u2218\n        x)\n[PROOFSTEP]\nobtain \u27e8i, y, rfl\u27e9 := exists_quotient_mk'_sigma_mk'_eq G f x\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nF : Functions L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 Function.Embedding.toFun\n      {\n        toFun :=\n          Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n            (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n        inj' :=\n          (_ :\n            \u2200 (x y : DirectLimit G f),\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                x = y) }\n      (funMap F fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))) =\n    funMap F\n      ({\n            toFun :=\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n            inj' :=\n              (_ :\n                \u2200 (x y : DirectLimit G f),\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                      Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                    x = y) }.toFun \u2218\n        fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a)))\n[PROOFSTEP]\nchange _ = funMap F (Quotient.lift _ _ \u2218 Quotient.mk _ \u2218 Structure.Sigma.mk f i \u2218 y)\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nF : Functions L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 Function.Embedding.toFun\n      {\n        toFun :=\n          Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n            (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n        inj' :=\n          (_ :\n            \u2200 (x y : DirectLimit G f),\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                x = y) }\n      (funMap F fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))) =\n    funMap F\n      (Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n          (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) \u2218\n        Quotient.mk (setoid G f) \u2218 Structure.Sigma.mk f i \u2218 y)\n[PROOFSTEP]\nrw [funMap_quotient_mk'_sigma_mk', \u2190 Function.comp.assoc, Quotient.lift_comp_mk]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nF : Functions L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 Function.Embedding.toFun\n      {\n        toFun :=\n          Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n            (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n        inj' :=\n          (_ :\n            \u2200 (x y : DirectLimit G f),\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                    (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                x = y) }\n      (Quotient.mk (setoid G f) (Structure.Sigma.mk f i (funMap F fun a => y a))) =\n    funMap F ((fun x => \u2191(g x.fst) x.snd) \u2218 Structure.Sigma.mk f i \u2218 y)\n[PROOFSTEP]\nsimp only [Quotient.lift_mk, Embedding.map_fun]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nF : Functions L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 funMap F (\u2191(g i) \u2218 fun a => y a) = funMap F ((fun x => \u2191(g x.fst) x.snd) \u2218 Structure.Sigma.mk f i \u2218 y)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nR : Relations L n\u271d\nx : Fin n\u271d \u2192 DirectLimit G f\n\u22a2 RelMap R\n      ({\n            toFun :=\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n            inj' :=\n              (_ :\n                \u2200 (x y : DirectLimit G f),\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                      Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                    x = y) }.toFun \u2218\n        x) \u2194\n    RelMap R x\n[PROOFSTEP]\nobtain \u27e8i, y, rfl\u27e9 := exists_quotient_mk'_sigma_mk'_eq G f x\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nR : Relations L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 RelMap R\n      ({\n            toFun :=\n              Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y),\n            inj' :=\n              (_ :\n                \u2200 (x y : DirectLimit G f),\n                  Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) x =\n                      Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n                        (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) y \u2192\n                    x = y) }.toFun \u2218\n        fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))) \u2194\n    RelMap R fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))\n[PROOFSTEP]\nchange RelMap R (Quotient.lift _ _ \u2218 Quotient.mk _ \u2218 Structure.Sigma.mk f i \u2218 y) \u2194 _\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nR : Relations L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 RelMap R\n      (Quotient.lift (fun x => \u2191(g x.fst) x.snd)\n          (_ : \u2200 (x y : \u03a3\u02e3 f), x \u2248 y \u2192 (fun x => \u2191(g x.fst) x.snd) x = (fun x => \u2191(g x.fst) x.snd) y) \u2218\n        Quotient.mk (setoid G f) \u2218 Structure.Sigma.mk f i \u2218 y) \u2194\n    RelMap R fun a => Quotient.mk (setoid G f) (Structure.Sigma.mk f i (y a))\n[PROOFSTEP]\nrw [relMap_quotient_mk'_sigma_mk' G f, \u2190 (g i).map_rel R y, \u2190 Function.comp.assoc, Quotient.lift_comp_mk]\n[GOAL]\ncase intro.intro\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nn\u271d : \u2115\nR : Relations L n\u271d\ni : \u03b9\ny : Fin n\u271d \u2192 G i\n\u22a2 RelMap R ((fun x => \u2191(g x.fst) x.snd) \u2218 Structure.Sigma.mk f i \u2218 y) \u2194 RelMap R (\u2191(g i) \u2218 y)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\ni : \u03b9\nx : G i\n\u22a2 \u2191(lift L \u03b9 G f g Hg) (Quotient.mk (setoid G f) (Structure.Sigma.mk f i x)) = \u2191(g i) x\n[PROOFSTEP]\nchange (lift L \u03b9 G f g Hg).toFun \u27e6.mk f i x\u27e7 = _\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\ni : \u03b9\nx : G i\n\u22a2 Function.Embedding.toFun (lift L \u03b9 G f g Hg).toEmbedding (Quotient.mk (setoid G f) (Structure.Sigma.mk f i x)) =\n    \u2191(g i) x\n[PROOFSTEP]\nsimp only [lift, Quotient.lift_mk]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\ni : \u03b9\nx : G i\n\u22a2 \u2191(lift L \u03b9 G f g Hg) (\u2191(of L \u03b9 G f i) x) = \u2191(g i) x\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nF : DirectLimit G f \u21aa[L] P\nx\u271d : DirectLimit G f\ni j : \u03b9\nhij : i \u2264 j\nx : G i\n\u22a2 \u2191((fun i => Embedding.comp F (of L \u03b9 G f i)) j) (\u2191(f i j hij) x) = \u2191((fun i => Embedding.comp F (of L \u03b9 G f i)) i) x\n[PROOFSTEP]\nrw [F.comp_apply, F.comp_apply, of_f]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nF : DirectLimit G f \u21aa[L] P\nx\u271d : DirectLimit G f\ni : \u03b9\nx : G i\n\u22a2 \u2191F (\u2191(of L \u03b9 G f i) x) =\n    \u2191(lift L \u03b9 G f (fun i => Embedding.comp F (of L \u03b9 G f i))\n          (_ :\n            \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i),\n              \u2191((fun i => Embedding.comp F (of L \u03b9 G f i)) j) (\u2191(f i j hij) x) =\n                \u2191((fun i => Embedding.comp F (of L \u03b9 G f i)) i) x))\n      (\u2191(of L \u03b9 G f i) x)\n[PROOFSTEP]\nrw [lift_of]\n[GOAL]\nL : Language\n\u03b9 : Type v\ninst\u271d\u2075 : Preorder \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u2074 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : DirectedSystem G fun i j h => \u2191(f i j h)\ninst\u271d\u00b9 : Nonempty \u03b9\nP : Type u\u2081\ninst\u271d : Structure L P\ng : (i : \u03b9) \u2192 G i \u21aa[L] P\nHg : \u2200 (i j : \u03b9) (hij : i \u2264 j) (x : G i), \u2191(g j) (\u2191(f i j hij) x) = \u2191(g i) x\nF : DirectLimit G f \u21aa[L] P\nx\u271d : DirectLimit G f\ni : \u03b9\nx : G i\n\u22a2 \u2191F (\u2191(of L \u03b9 G f i) x) = \u2191(Embedding.comp F (of L \u03b9 G f i)) x\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 CG L (DirectLimit G f)\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u22c3 i, DirectLimit.of L \u03b9 G f i '' Classical.choose (h i).out, _, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 Set.Countable (\u22c3 (i : \u03b9), \u2191(of L \u03b9 G f i) '' Classical.choose (_ : Substructure.CG \u22a4))\n[PROOFSTEP]\nexact Set.countable_iUnion fun i => Set.Countable.image (Classical.choose_spec (h i).out).1 _\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 LowerAdjoint.toFun (Substructure.closure L) (\u22c3 (i : \u03b9), \u2191(of L \u03b9 G f i) '' Classical.choose (_ : Substructure.CG \u22a4)) =\n    \u22a4\n[PROOFSTEP]\nrw [eq_top_iff, Substructure.closure_union\u1d62]\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 \u22a4 \u2264\n    \u2a06 (i : \u03b9), LowerAdjoint.toFun (Substructure.closure L) (\u2191(of L \u03b9 G f i) '' Classical.choose (_ : Substructure.CG \u22a4))\n[PROOFSTEP]\nsimp_rw [\u2190 Embedding.coe_toHom, Substructure.closure_image]\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 \u22a4 \u2264\n    \u2a06 (i : \u03b9),\n      Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4)))\n[PROOFSTEP]\nrw [le_iSup_iff]\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\n\u22a2 \u2200 (b : Substructure L (DirectLimit G f)),\n    (\u2200 (i : \u03b9),\n        Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n            (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n          b) \u2192\n      \u22a4 \u2264 b\n[PROOFSTEP]\nintro S hS x _\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nS : Substructure L (DirectLimit G f)\nhS :\n  \u2200 (i : \u03b9),\n    Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n      S\nx : DirectLimit G f\na\u271d : x \u2208 \u22a4\n\u22a2 x \u2208 S\n[PROOFSTEP]\nlet out := Quotient.out (s := DirectLimit.setoid G f)\n[GOAL]\ncase refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nS : Substructure L (DirectLimit G f)\nhS :\n  \u2200 (i : \u03b9),\n    Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n      S\nx : DirectLimit G f\na\u271d : x \u2208 \u22a4\nout : Quotient (setoid G f) \u2192 \u03a3\u02e3 f := Quotient.out\n\u22a2 x \u2208 S\n[PROOFSTEP]\nrefine' hS (out x).1 \u27e8(out x).2, _, _\u27e9\n[GOAL]\ncase refine'_2.refine'_1\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nS : Substructure L (DirectLimit G f)\nhS :\n  \u2200 (i : \u03b9),\n    Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n      S\nx : DirectLimit G f\na\u271d : x \u2208 \u22a4\nout : Quotient (setoid G f) \u2192 \u03a3\u02e3 f := Quotient.out\n\u22a2 (out x).snd \u2208 \u2191(LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4)))\n[PROOFSTEP]\nrw [(Classical.choose_spec (h (out x).1).out).2]\n[GOAL]\ncase refine'_2.refine'_1\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nS : Substructure L (DirectLimit G f)\nhS :\n  \u2200 (i : \u03b9),\n    Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n      S\nx : DirectLimit G f\na\u271d : x \u2208 \u22a4\nout : Quotient (setoid G f) \u2192 \u03a3\u02e3 f := Quotient.out\n\u22a2 (out x).snd \u2208 \u2191\u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase refine'_2.refine'_2\nL : Language\n\u03b9\u271d : Type v\ninst\u271d\u00b9\u00b9 : Preorder \u03b9\u271d\nG\u271d : \u03b9\u271d \u2192 Type w\ninst\u271d\u00b9\u2070 : (i : \u03b9\u271d) \u2192 Structure L (G\u271d i)\nf\u271d : (i j : \u03b9\u271d) \u2192 i \u2264 j \u2192 G\u271d i \u21aa[L] G\u271d j\ninst\u271d\u2079 : IsDirected \u03b9\u271d fun x x_1 => x \u2264 x_1\ninst\u271d\u2078 : DirectedSystem G\u271d fun i j h => \u2191(f\u271d i j h)\ninst\u271d\u2077 : Nonempty \u03b9\u271d\nP : Type u\u2081\ninst\u271d\u2076 : Structure L P\ng : (i : \u03b9\u271d) \u2192 G\u271d i \u21aa[L] P\nHg : \u2200 (i j : \u03b9\u271d) (hij : i \u2264 j) (x : G\u271d i), \u2191(g j) (\u2191(f\u271d i j hij) x) = \u2191(g i) x\n\u03b9 : Type u_1\ninst\u271d\u2075 : Encodable \u03b9\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : IsDirected \u03b9 fun x x_1 => x \u2264 x_1\ninst\u271d\u00b2 : Nonempty \u03b9\nG : \u03b9 \u2192 Type w\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Structure L (G i)\nf : (i j : \u03b9) \u2192 i \u2264 j \u2192 G i \u21aa[L] G j\nh : \u2200 (i : \u03b9), CG L (G i)\ninst\u271d : DirectedSystem G fun i j h => \u2191(f i j h)\nS : Substructure L (DirectLimit G f)\nhS :\n  \u2200 (i : \u03b9),\n    Substructure.map (Embedding.toHom (of L \u03b9 G f i))\n        (LowerAdjoint.toFun (Substructure.closure L) (Classical.choose (_ : Substructure.CG \u22a4))) \u2264\n      S\nx : DirectLimit G f\na\u271d : x \u2208 \u22a4\nout : Quotient (setoid G f) \u2192 \u03a3\u02e3 f := Quotient.out\n\u22a2 \u2191(Embedding.toHom (of L \u03b9 G f (out x).fst)) (out x).snd = x\n[PROOFSTEP]\nsimp only [Embedding.coe_toHom, DirectLimit.of_apply, Sigma.eta, Quotient.out_eq]\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.DirectLimit", "llama_tokens": 43299, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.5247524454245026}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[l] fun z => expR (B * expR (c * |u z|))\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[l] fun z => expR (B * expR (c * |u z|))\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))\n[PROOFSTEP]\nhave :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 z, \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016 :=\n  fun hc hB\u2080 hB z \u21a6\n  by\n  rw [Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]\n  exact mul_le_mul hB (Real.exp_le_exp.2 <| mul_le_mul_of_nonneg_right hc <| abs_nonneg _) (Real.exp_pos _).le hB\u2080\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[l] fun z => expR (B * expR (c * |u z|))\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[l] fun z => expR (B * expR (c * |u z|))\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\nz : \u2102\n\u22a2 \u2016expR (B\u2081\u271d * expR (c\u2081\u271d * |u z|))\u2016 \u2264 \u2016expR (B\u2082\u271d * expR (c\u2082\u271d * |u z|))\u2016\n[PROOFSTEP]\nrw [Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[l] fun z => expR (B * expR (c * |u z|))\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[l] fun z => expR (B * expR (c * |u z|))\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\nz : \u2102\n\u22a2 B\u2081\u271d * expR (c\u2081\u271d * |u z|) \u2264 B\u2082\u271d * expR (c\u2082\u271d * |u z|)\n[PROOFSTEP]\nexact mul_le_mul hB (Real.exp_le_exp.2 <| mul_le_mul_of_nonneg_right hc <| abs_nonneg _) (Real.exp_pos _).le hB\u2080\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[l] fun z => expR (B * expR (c * |u z|))\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[l] fun z => expR (B * expR (c * |u z|))\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))\n[PROOFSTEP]\nrcases hBf with \u27e8cf, hcf, Bf, hOf\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[l] fun z => expR (B * expR (c * |u z|))\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))\n[PROOFSTEP]\nrcases hBg with \u27e8cg, hcg, Bg, hOg\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))\n[PROOFSTEP]\nrefine' \u27e8max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 (f - g) =O[l] fun z => expR (max 0 (max Bf Bg) * expR (max cf cg * |u z|))\n[PROOFSTEP]\nrefine' (hOf.trans_le <| this _ _ _).sub (hOg.trans_le <| this _ _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 cf \u2264 max cf cg\ncase intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 0 \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 Bf \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_4\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 cg \u2264 max cf cg\ncase intro.intro.intro.intro.intro.intro.refine'_5\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 0 \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_6\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nu : \u2102 \u2192 \u211d\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 0 \u2264 B\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 \u2200 (z : \u2102), \u2016expR (B\u2081 * expR (c\u2081 * |u z|))\u2016 \u2264 \u2016expR (B\u2082 * expR (c\u2082 * |u z|))\u2016\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))\n\u22a2 Bg \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nexacts [le_max_left _ _, le_max_left _ _, (le_max_left _ _).trans (le_max_right _ _), le_max_right _ _, le_max_left _ _,\n  (le_max_right _ _).trans (le_max_right _ _)]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nhave :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z : \u2102 => expR (B\u2081 * abs z ^ c\u2081)) =O[comap Complex.abs atTop \u2293 l] fun z => expR (B\u2082 * abs z ^ c\u2082) :=\n  fun hc hB\u2080 hB \u21a6\n  .of_bound 1 <|\n    by\n    have : \u2200\u1da0 z : \u2102 in comap Complex.abs atTop \u2293 l, 1 \u2264 abs z :=\n      ((eventually_ge_atTop 1).comap _).filter_mono inf_le_left\n    refine this.mono fun z hz => ?_\n    rw [one_mul, Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]\n    exact\n      mul_le_mul hB (Real.rpow_le_rpow_of_exponent_le hz hc) (Real.rpow_nonneg_of_nonneg (Complex.abs.nonneg _) _) hB\u2080\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\n\u22a2 \u2200\u1da0 (x : \u2102) in comap (\u2191Complex.abs) atTop \u2293 l,\n    \u2016expR (B\u2081\u271d * \u2191Complex.abs x ^ c\u2081\u271d)\u2016 \u2264 1 * \u2016expR (B\u2082\u271d * \u2191Complex.abs x ^ c\u2082\u271d)\u2016\n[PROOFSTEP]\nhave : \u2200\u1da0 z : \u2102 in comap Complex.abs atTop \u2293 l, 1 \u2264 abs z := ((eventually_ge_atTop 1).comap _).filter_mono inf_le_left\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\nthis : \u2200\u1da0 (z : \u2102) in comap (\u2191Complex.abs) atTop \u2293 l, 1 \u2264 \u2191Complex.abs z\n\u22a2 \u2200\u1da0 (x : \u2102) in comap (\u2191Complex.abs) atTop \u2293 l,\n    \u2016expR (B\u2081\u271d * \u2191Complex.abs x ^ c\u2081\u271d)\u2016 \u2264 1 * \u2016expR (B\u2082\u271d * \u2191Complex.abs x ^ c\u2082\u271d)\u2016\n[PROOFSTEP]\nrefine this.mono fun z hz => ?_\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\nthis : \u2200\u1da0 (z : \u2102) in comap (\u2191Complex.abs) atTop \u2293 l, 1 \u2264 \u2191Complex.abs z\nz : \u2102\nhz : 1 \u2264 \u2191Complex.abs z\n\u22a2 \u2016expR (B\u2081\u271d * \u2191Complex.abs z ^ c\u2081\u271d)\u2016 \u2264 1 * \u2016expR (B\u2082\u271d * \u2191Complex.abs z ^ c\u2082\u271d)\u2016\n[PROOFSTEP]\nrw [one_mul, Real.norm_eq_abs, Real.norm_eq_abs, Real.abs_exp, Real.abs_exp, Real.exp_le_exp]\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081\u271d c\u2082\u271d B\u2081\u271d B\u2082\u271d : \u211d\nhc : c\u2081\u271d \u2264 c\u2082\u271d\nhB\u2080 : 0 \u2264 B\u2082\u271d\nhB : B\u2081\u271d \u2264 B\u2082\u271d\nthis : \u2200\u1da0 (z : \u2102) in comap (\u2191Complex.abs) atTop \u2293 l, 1 \u2264 \u2191Complex.abs z\nz : \u2102\nhz : 1 \u2264 \u2191Complex.abs z\n\u22a2 B\u2081\u271d * \u2191Complex.abs z ^ c\u2081\u271d \u2264 B\u2082\u271d * \u2191Complex.abs z ^ c\u2082\u271d\n[PROOFSTEP]\nexact mul_le_mul hB (Real.rpow_le_rpow_of_exponent_le hz hc) (Real.rpow_nonneg_of_nonneg (Complex.abs.nonneg _) _) hB\u2080\n[GOAL]\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBf : \u2203 c, c < a \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hBf with \u27e8cf, hcf, Bf, hOf\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nhBg : \u2203 c, c < a \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hBg with \u27e8cg, hcg, Bg, hOg\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 \u2203 c, c < a \u2227 \u2203 B, (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine' \u27e8max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (max 0 (max Bf Bg) * \u2191Complex.abs z ^ max cf cg)\n[PROOFSTEP]\nrefine' (hOf.trans <| this _ _ _).sub (hOg.trans <| this _ _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 cf \u2264 max cf cg\ncase intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 0 \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 Bf \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_4\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 cg \u2264 max cf cg\ncase intro.intro.intro.intro.intro.intro.refine'_5\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 0 \u2264 max 0 (max Bf Bg)\ncase intro.intro.intro.intro.intro.intro.refine'_6\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\na : \u211d\nf g : \u2102 \u2192 E\nl : Filter \u2102\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      0 \u2264 B\u2082 \u2192\n        B\u2081 \u2264 B\u2082 \u2192\n          (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z =>\n            expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < a\nBf : \u211d\nhOf : f =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < a\nBg : \u211d\nhOg : g =O[comap (\u2191Complex.abs) atTop \u2293 l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 Bg \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nexacts [le_max_left _ _, le_max_left _ _, (le_max_left _ _).trans (le_max_right _ _), le_max_right _ _, le_max_left _ _,\n  (le_max_right _ _).trans (le_max_right _ _)]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.im\nhzb : z.im \u2264 b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrw [le_iff_eq_or_lt] at hza hzb \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a = z.im \u2228 a < z.im\nhzb : z.im = b \u2228 z.im < b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\ncases' hza with hza hza\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im = b \u2228 z.im < b\nhza : a = z.im\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nexact hle_a _ hza.symm\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im = b \u2228 z.im < b\nhza : a < z.im\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\ncases' hzb with hzb hzb\n[GOAL]\ncase inr.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im = b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nexact hle_b _ hzb\n[GOAL]\ncase inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nwlog hC\u2080 : 0 < C generalizing C\n[GOAL]\ncase inr.inr.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\nthis : \u2200 {C : \u211d}, (\u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C) \u2192 (\u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C) \u2192 0 < C \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : \u00ac0 < C\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' le_of_forall_le_of_dense fun C' hC' => this (fun w hw => _) (fun w hw => _) _\n[GOAL]\ncase inr.inr.inr.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\nthis : \u2200 {C : \u211d}, (\u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C) \u2192 (\u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C) \u2192 0 < C \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : \u00ac0 < C\nC' : \u211d\nhC' : C < C'\nw : \u2102\nhw : w.im = a\n\u22a2 \u2016f w\u2016 \u2264 C'\n[PROOFSTEP]\nexact (hle_a _ hw).trans hC'.le\n[GOAL]\ncase inr.inr.inr.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\nthis : \u2200 {C : \u211d}, (\u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C) \u2192 (\u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C) \u2192 0 < C \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : \u00ac0 < C\nC' : \u211d\nhC' : C < C'\nw : \u2102\nhw : w.im = b\n\u22a2 \u2016f w\u2016 \u2264 C'\n[PROOFSTEP]\nexact (hle_b _ hw).trans hC'.le\n[GOAL]\ncase inr.inr.inr.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\nthis : \u2200 {C : \u211d}, (\u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C) \u2192 (\u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C) \u2192 0 < C \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : \u00ac0 < C\nC' : \u211d\nhC' : C < C'\n\u22a2 0 < C'\n[PROOFSTEP]\nrefine' ((norm_nonneg (f (a * I))).trans (hle_a _ _)).trans_lt hC'\n[GOAL]\ncase inr.inr.inr.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a < z.im\nhzb : z.im < b\nthis : \u2200 {C : \u211d}, (\u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C) \u2192 (\u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C) \u2192 0 < C \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : \u00ac0 < C\nC' : \u211d\nhC' : C < C'\n\u22a2 (\u2191a * I).im = a\n[PROOFSTEP]\nrw [mul_I_im, ofReal_re]\n  -- After a change of variables, we deal with the strip `a - b < im z < a + b` instead\n    -- of `a < im z < b`\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhza : a < z.im\nhzb : z.im < b\nC : \u211d\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : 0 < C\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8a, b, rfl, rfl\u27e9 : \u2203 a' b', a = a' - b' \u2227 b = a' + b' := \u27e8(a + b) / 2, (b - a) / 2, by ring, by ring\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhza : a < z.im\nhzb : z.im < b\nC : \u211d\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : 0 < C\n\u22a2 a = (a + b) / 2 - (b - a) / 2\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\nhza : a < z.im\nhzb : z.im < b\nC : \u211d\nhle_a : \u2200 (z : \u2102), z.im = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.im = b \u2192 \u2016f z\u2016 \u2264 C\nhC\u2080 : 0 < C\n\u22a2 b = (a + b) / 2 + (b - a) / 2\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / (a + b - (a - b)) \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hab : a - b < a + b := hza.trans hzb\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / (a + b - (a - b)) \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nhab : a - b < a + b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hb : 0 < b := by simpa only [sub_eq_add_neg, add_lt_add_iff_left, neg_lt_self_iff] using hab\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / (a + b - (a - b)) \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nhab : a - b < a + b\n\u22a2 0 < b\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, add_lt_add_iff_left, neg_lt_self_iff] using hab\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / (a + b - (a - b)) \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nhab : a - b < a + b\nhb : 0 < b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrw [add_sub_sub_cancel, \u2190 two_mul, div_mul_eq_div_div] at hB \n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / 2 / b \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nhab : a - b < a + b\nhb : 0 < b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave h\u03c0b : 0 < \u03c0 / 2 / b := div_pos Real.pi_div_two_pos hb\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhB :\n  \u2203 c,\n    c < \u03c0 / 2 / b \u2227\n      \u2203 B, f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8d, \u27e8hcd, hd\u2080\u27e9, hd\u27e9 : \u2203 d, (c < d \u2227 0 < d) \u2227 d < \u03c0 / 2 / b := by\n  simpa only [max_lt_iff] using exists_between (max_lt hc h\u03c0b)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\n\u22a2 \u2203 d, (c < d \u2227 0 < d) \u2227 d < \u03c0 / 2 / b\n[PROOFSTEP]\nsimpa only [max_lt_iff] using exists_between (max_lt hc h\u03c0b)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hb' : d * b < \u03c0 / 2 := (lt_div_iff hb).1 hd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nset aff := (fun w => d * (w - a * I) : \u2102 \u2192 \u2102)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nset g := fun (\u03b5 : \u211d) (w : \u2102) =>\n  exp\n    (\u03b5 * (exp (aff w) + exp (-aff w)))\n      /- Since `g \u03b5 z \u2192 1` as `\u03b5 \u2192 0\u207b`, it suffices to prove that `\u2016g \u03b5 z \u2022 f z\u2016 \u2264 C`\n          for all negative `\u03b5`. -/\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nsuffices : \u2200\u1da0 \u03b5 : \u211d in \ud835\udcdd[<] (0 : \u211d), \u2016g \u03b5 z \u2022 f z\u2016 \u2264 C\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016g \u03b5 z \u2022 f z\u2016 \u2264 C\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' le_of_tendsto (Tendsto.mono_left _ nhdsWithin_le_nhds) this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016g \u03b5 z \u2022 f z\u2016 \u2264 C\n\u22a2 Tendsto (fun c => \u2016g c z \u2022 f z\u2016) (\ud835\udcdd 0) (\ud835\udcdd \u2016f z\u2016)\n[PROOFSTEP]\napply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016g \u03b5 z \u2022 f z\u2016 \u2264 C\n\u22a2 \u2016exp (\u21910 * (exp (aff z) + exp (-aff z))) \u2022 f z\u2016 = \u2016f z\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016g \u03b5 z \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with \u03b5 \u03b5\u2080\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 \u2208 Iio 0\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nchange \u03b5 < 0 at \u03b5\u2080 \n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b4\u2080, h\u03b4\u27e9 : \u2203 \u03b4 : \u211d, \u03b4 < 0 \u2227 \u2200 \u2983w\u2984, im w \u2208 Icc (a - b) (a + b) \u2192 abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |re w|)) :=\n  by\n  refine'\n    \u27e8\u03b5 * Real.cos (d * b),\n      mul_neg_of_neg_of_pos \u03b5\u2080 (Real.cos_pos_of_mem_Ioo <| abs_lt.1 <| (abs_of_pos (mul_pos hd\u2080 hb)).symm \u25b8 hb'),\n      fun w hw => _\u27e9\n  replace hw : |im (aff w)| \u2264 d * b\n  \u00b7 rw [\u2190 Real.closedBall_eq_Icc] at hw \n    rwa [ofReal_mul_im, sub_im, mul_I_im, ofReal_re, _root_.abs_mul, abs_of_pos hd\u2080, mul_le_mul_left hd\u2080]\n  simpa only [ofReal_mul_re, _root_.abs_mul, abs_of_pos hd\u2080, sub_re, mul_I_re, ofReal_im, zero_mul, neg_zero,\n    sub_zero] using abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le \u03b5\u2080.le hw hb'.le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u22a2 \u2203 \u03b4, \u03b4 < 0 \u2227 \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\n[PROOFSTEP]\nrefine'\n  \u27e8\u03b5 * Real.cos (d * b),\n    mul_neg_of_neg_of_pos \u03b5\u2080 (Real.cos_pos_of_mem_Ioo <| abs_lt.1 <| (abs_of_pos (mul_pos hd\u2080 hb)).symm \u25b8 hb'),\n    fun w hw => _\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\nw : \u2102\nhw : w.im \u2208 Icc (a - b) (a + b)\n\u22a2 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b5 * Real.cos (d * b) * expR (d * |w.re|))\n[PROOFSTEP]\nreplace hw : |im (aff w)| \u2264 d * b\n[GOAL]\ncase hw\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\nw : \u2102\nhw : w.im \u2208 Icc (a - b) (a + b)\n\u22a2 |(aff w).im| \u2264 d * b\n[PROOFSTEP]\nrw [\u2190 Real.closedBall_eq_Icc] at hw \n[GOAL]\ncase hw\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\nw : \u2102\nhw : w.im \u2208 closedBall a b\n\u22a2 |(aff w).im| \u2264 d * b\n[PROOFSTEP]\nrwa [ofReal_mul_im, sub_im, mul_I_im, ofReal_re, _root_.abs_mul, abs_of_pos hd\u2080, mul_le_mul_left hd\u2080]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\nw : \u2102\nhw : |(aff w).im| \u2264 d * b\n\u22a2 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b5 * Real.cos (d * b) * expR (d * |w.re|))\n[PROOFSTEP]\nsimpa only [ofReal_mul_re, _root_.abs_mul, abs_of_pos hd\u2080, sub_re, mul_I_re, ofReal_im, zero_mul, neg_zero,\n  sub_zero] using abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le \u03b5\u2080.le hw hb'.le\n[GOAL]\ncase h.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hg\u2081 : \u2200 w, im w = a - b \u2228 im w = a + b \u2192 abs (g \u03b5 w) \u2264 1 :=\n  by\n  refine' fun w hw => (h\u03b4 <| hw.by_cases _ _).trans (Real.exp_le_one_iff.2 _)\n  exacts [fun h => h.symm \u25b8 left_mem_Icc.2 hab.le, fun h => h.symm \u25b8 right_mem_Icc.2 hab.le,\n    mul_nonpos_of_nonpos_of_nonneg \u03b4\u2080.le (Real.exp_pos _).le]\n    /- Our apriori estimate on `f` implies that `g \u03b5 w \u2022 f w \u2192 0` as `|w.re| \u2192 \u221e` along the strip. In\n        particular, its norm is less than or equal to `C` for sufficiently large `|w.re|`. -/\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\n\u22a2 \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\n[PROOFSTEP]\nrefine' fun w hw => (h\u03b4 <| hw.by_cases _ _).trans (Real.exp_le_one_iff.2 _)\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nw : \u2102\nhw : w.im = a - b \u2228 w.im = a + b\n\u22a2 w.im = a - b \u2192 w.im \u2208 Icc (a - b) (a + b)\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nw : \u2102\nhw : w.im = a - b \u2228 w.im = a + b\n\u22a2 w.im = a + b \u2192 w.im \u2208 Icc (a - b) (a + b)\ncase refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nw : \u2102\nhw : w.im = a - b \u2228 w.im = a + b\n\u22a2 \u03b4 * expR (d * |w.re|) \u2264 0\n[PROOFSTEP]\nexacts [fun h => h.symm \u25b8 left_mem_Icc.2 hab.le, fun h => h.symm \u25b8 right_mem_Icc.2 hab.le,\n  mul_nonpos_of_nonpos_of_nonneg \u03b4\u2080.le (Real.exp_pos _).le]\n  /- Our apriori estimate on `f` implies that `g \u03b5 w \u2022 f w \u2192 0` as `|w.re| \u2192 \u221e` along the strip. In\n      particular, its norm is less than or equal to `C` for sufficiently large `|w.re|`. -/\n[GOAL]\ncase h.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8R, hzR, hR\u27e9 : \u2203 R : \u211d, |z.re| < R \u2227 \u2200 w, |re w| = R \u2192 im w \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C :=\n  by\n  refine' ((eventually_gt_atTop _).and _).exists\n  rcases hO.exists_pos with \u27e8A, hA\u2080, hA\u27e9\n  simp only [isBigOWith_iff, eventually_inf_principal, eventually_comap, mem_Ioo, \u2190 abs_lt, mem_preimage, (\u00b7 \u2218 \u00b7),\n    Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] at hA \n  suffices : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\n  \u00b7 filter_upwards [this.eventually (ge_mem_nhds hC\u2080), hA] with R hR Hle w hre him\n    calc\n      \u2016g \u03b5 w \u2022 f w\u2016 \u2264 expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) := ?_\n      _ \u2264 C := hR\n    rw [norm_smul, Real.exp_add, \u2190 hre, Real.exp_add, Real.exp_log hA\u2080, mul_assoc, mul_comm _ A]\n    exact mul_le_mul (h\u03b4 <| Ioo_subset_Icc_self him) (Hle _ hre him) (norm_nonneg _) (Real.exp_pos _).le\n  refine' Real.tendsto_exp_atBot.comp _\n  suffices H : Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd (\u03b4 + B * 0))\n  \u00b7 rw [mul_zero, add_zero] at H \n    refine' Tendsto.atBot_add _ tendsto_const_nhds\n    simpa only [id, (\u00b7 \u2218 \u00b7), add_mul, mul_assoc, \u2190 div_eq_inv_mul, \u2190 Real.exp_sub, \u2190 sub_mul, sub_sub_cancel] using\n      H.neg_mul_atTop \u03b4\u2080 <| Real.tendsto_exp_atTop.comp <| tendsto_const_nhds.mul_atTop hd\u2080 tendsto_id\n  refine' tendsto_const_nhds.add (tendsto_const_nhds.mul _)\n  exact\n    tendsto_inv_atTop_zero.comp <|\n      Real.tendsto_exp_atTop.comp <| tendsto_const_nhds.mul_atTop (sub_pos.2 hcd) tendsto_id\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\n\u22a2 \u2203 R, |z.re| < R \u2227 \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nrefine' ((eventually_gt_atTop _).and _).exists\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\n\u22a2 \u2200\u1da0 (R : \u211d) in atTop, \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nrcases hO.exists_pos with \u27e8A, hA\u2080, hA\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA : IsBigOWith A (comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))) f fun z => expR (B * expR (c * |z.re|))\n\u22a2 \u2200\u1da0 (R : \u211d) in atTop, \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [isBigOWith_iff, eventually_inf_principal, eventually_comap, mem_Ioo, \u2190 abs_lt, mem_preimage, (\u00b7 \u2218 \u00b7),\n  Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] at hA \n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\n\u22a2 \u2200\u1da0 (R : \u211d) in atTop, \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nsuffices : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nthis : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (R : \u211d) in atTop, \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nfilter_upwards [this.eventually (ge_mem_nhds hC\u2080), hA] with R hR Hle w hre him\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nthis : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\nR : \u211d\nhR : expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) \u2264 C\nHle : \u2200 (a_1 : \u2102), |a_1.re| = R \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nw : \u2102\nhre : |w.re| = R\nhim : w.im \u2208 Ioo (a - b) (a + b)\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (w - \u2191a * I)) + exp (-(\u2191d * (w - \u2191a * I))))) \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\ncalc\n  \u2016g \u03b5 w \u2022 f w\u2016 \u2264 expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) := ?_\n  _ \u2264 C := hR\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nthis : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\nR : \u211d\nhR : expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) \u2264 C\nHle : \u2200 (a_1 : \u2102), |a_1.re| = R \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nw : \u2102\nhre : |w.re| = R\nhim : w.im \u2208 Ioo (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)\n[PROOFSTEP]\nrw [norm_smul, Real.exp_add, \u2190 hre, Real.exp_add, Real.exp_log hA\u2080, mul_assoc, mul_comm _ A]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nthis : Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\nR : \u211d\nhR : expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) \u2264 C\nHle : \u2200 (a_1 : \u2102), |a_1.re| = R \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nw : \u2102\nhre : |w.re| = R\nhim : w.im \u2208 Ioo (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w\u2016 * \u2016f w\u2016 \u2264 expR (\u03b4 * expR (d * |w.re|)) * (A * expR (B * expR (c * |w.re|)))\n[PROOFSTEP]\nexact mul_le_mul (h\u03b4 <| Ioo_subset_Icc_self him) (Hle _ hre him) (norm_nonneg _) (Real.exp_pos _).le\n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\n\u22a2 Tendsto (fun R => expR (\u03b4 * expR (d * R) + B * expR (c * R) + Real.log A)) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' Real.tendsto_exp_atBot.comp _\n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\n\u22a2 Tendsto (fun R => \u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) atTop atBot\n[PROOFSTEP]\nsuffices H : Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd (\u03b4 + B * 0))\n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nH : Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd (\u03b4 + B * 0))\n\u22a2 Tendsto (fun R => \u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) atTop atBot\n[PROOFSTEP]\nrw [mul_zero, add_zero] at H \n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nH : Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd \u03b4)\n\u22a2 Tendsto (fun R => \u03b4 * expR (d * R) + B * expR (c * R) + Real.log A) atTop atBot\n[PROOFSTEP]\nrefine' Tendsto.atBot_add _ tendsto_const_nhds\n[GOAL]\ncase this\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\nH : Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd \u03b4)\n\u22a2 Tendsto (fun R => \u03b4 * expR (d * R) + B * expR (c * R)) atTop atBot\n[PROOFSTEP]\nsimpa only [id, (\u00b7 \u2218 \u00b7), add_mul, mul_assoc, \u2190 div_eq_inv_mul, \u2190 Real.exp_sub, \u2190 sub_mul, sub_sub_cancel] using\n  H.neg_mul_atTop \u03b4\u2080 <| Real.tendsto_exp_atTop.comp <| tendsto_const_nhds.mul_atTop hd\u2080 tendsto_id\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\n\u22a2 Tendsto (fun R => \u03b4 + B * (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd (\u03b4 + B * 0))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.add (tendsto_const_nhds.mul _)\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nA : \u211d\nhA\u2080 : 0 < A\nhA :\n  \u2200\u1da0 (b_1 : \u211d) in atTop,\n    \u2200 (a_1 : \u2102), |a_1.re| = b_1 \u2192 a - b < a_1.im \u2227 a_1.im < a + b \u2192 \u2016f a_1\u2016 \u2264 A * expR (B * expR (c * |a_1.re|))\n\u22a2 Tendsto (fun R => (expR ((d - c) * R))\u207b\u00b9) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact\n  tendsto_inv_atTop_zero.comp <| Real.tendsto_exp_atTop.comp <| tendsto_const_nhds.mul_atTop (sub_pos.2 hcd) tendsto_id\n[GOAL]\ncase h.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hR\u2080 : 0 < R := (_root_.abs_nonneg _).trans_lt hzR\n[GOAL]\ncase h.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hgd : Differentiable \u2102 (g \u03b5) :=\n  ((((differentiable_id.sub_const _).const_mul _).cexp.add\n          ((differentiable_id.sub_const _).const_mul _).neg.cexp).const_mul\n      _).cexp\n[GOAL]\ncase h.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nreplace hd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhd : d < \u03c0 / 2 / b\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\n\u22a2 DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\ncase h.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nexact (hgd.diffContOnCl.smul hfd).mono (inter_subset_right _ _)\n[GOAL]\ncase h.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n\u22a2 \u2016exp (\u2191\u03b5 * (exp (\u2191d * (z - \u2191a * I)) + exp (-(\u2191d * (z - \u2191a * I))))) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nconvert norm_le_of_forall_mem_frontier_norm_le ((bounded_Ioo _ _).reProdIm (bounded_Ioo _ _)) hd (fun w hw => _) _\n[GOAL]\ncase h.intro.intro.intro.intro.convert_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 frontier (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nrw [frontier_reProdIm, closure_Ioo (neg_lt_self hR\u2080).ne, frontier_Ioo hab, closure_Ioo hab.ne,\n  frontier_Ioo (neg_lt_self hR\u2080)] at hw \n[GOAL]\ncase h.intro.intro.intro.intro.convert_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 Icc (-R) R \u00d7\u2102 {a - b, a + b} \u222a {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nby_cases him : w.im = a - b \u2228 w.im = a + b\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 Icc (-R) R \u00d7\u2102 {a - b, a + b} \u222a {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhim : w.im = a - b \u2228 w.im = a + b\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nrw [norm_smul, \u2190 one_mul C]\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 Icc (-R) R \u00d7\u2102 {a - b, a + b} \u222a {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhim : w.im = a - b \u2228 w.im = a + b\n\u22a2 \u2016g \u03b5 w\u2016 * \u2016f w\u2016 \u2264 1 * C\n[PROOFSTEP]\nexact mul_le_mul (hg\u2081 _ him) (him.by_cases (hle_a _) (hle_b _)) (norm_nonneg _) zero_le_one\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 Icc (-R) R \u00d7\u2102 {a - b, a + b} \u222a {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nreplace hw : w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\n[GOAL]\ncase hw\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhw : w \u2208 Icc (-R) R \u00d7\u2102 {a - b, a + b} \u222a {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\n\u22a2 w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\nhw : w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nexact hw.resolve_left fun h => him h.2\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\nhw : w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nhave hw' := eq_endpoints_or_mem_Ioo_of_mem_Icc hw.2\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\nhw : w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhw' : w.im = a - b \u2228 w.im = a + b \u2228 w.im \u2208 Ioo (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nrw [\u2190 or_assoc] at hw' \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\nw : \u2102\nhim : \u00ac(w.im = a - b \u2228 w.im = a + b)\nhw : w \u2208 {-R, R} \u00d7\u2102 Icc (a - b) (a + b)\nhw' : (w.im = a - b \u2228 w.im = a + b) \u2228 w.im \u2208 Ioo (a - b) (a + b)\n\u22a2 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\n[PROOFSTEP]\nexact hR _ ((abs_eq hR\u2080.le).2 hw.1.symm) (hw'.resolve_left him)\n[GOAL]\ncase h.intro.intro.intro.intro.convert_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n\u22a2 z \u2208 closure (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n[PROOFSTEP]\nrw [closure_reProdIm, closure_Ioo hab.ne, closure_Ioo (neg_lt_self hR\u2080).ne]\n[GOAL]\ncase h.intro.intro.intro.intro.convert_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nC\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nC : \u211d\nhC\u2080 : 0 < C\na b : \u211d\nhza : a - b < z.im\nhle_a : \u2200 (z : \u2102), z.im = a - b \u2192 \u2016f z\u2016 \u2264 C\nhzb : z.im < a + b\nhle_b : \u2200 (z : \u2102), z.im = a + b \u2192 \u2016f z\u2016 \u2264 C\nhfd : DiffContOnCl \u2102 f (im \u207b\u00b9' Ioo (a - b) (a + b))\nhab : a - b < a + b\nhb : 0 < b\nh\u03c0b : 0 < \u03c0 / 2 / b\nc : \u211d\nhc : c < \u03c0 / 2 / b\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))\nd : \u211d\nhcd : c < d\nhd\u2080 : 0 < d\nhb' : d * b < \u03c0 / 2\naff : \u2102 \u2192 \u2102 := fun w => \u2191d * (w - \u2191a * I)\ng : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b5 w => exp (\u2191\u03b5 * (exp (aff w) + exp (-aff w)))\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u03b4 : \u211d\n\u03b4\u2080 : \u03b4 < 0\nh\u03b4 : \u2200 \u2983w : \u2102\u2984, w.im \u2208 Icc (a - b) (a + b) \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 expR (\u03b4 * expR (d * |w.re|))\nhg\u2081 : \u2200 (w : \u2102), w.im = a - b \u2228 w.im = a + b \u2192 \u2191Complex.abs (g \u03b5 w) \u2264 1\nR : \u211d\nhzR : |z.re| < R\nhR : \u2200 (w : \u2102), |w.re| = R \u2192 w.im \u2208 Ioo (a - b) (a + b) \u2192 \u2016g \u03b5 w \u2022 f w\u2016 \u2264 C\nhR\u2080 : 0 < R\nhgd : Differentiable \u2102 (g \u03b5)\nhd : DiffContOnCl \u2102 (fun w => g \u03b5 w \u2022 f w) (Ioo (-R) R \u00d7\u2102 Ioo (a - b) (a + b))\n\u22a2 z \u2208 Icc (-R) R \u00d7\u2102 Icc (a - b) (a + b)\n[PROOFSTEP]\nexact \u27e8abs_le.1 hzR.le, \u27e8hza.le, hzb.le\u27e9\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nsuffices \u2016f (z * I * -I)\u2016 \u2264 C by simpa [mul_assoc] using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nthis : \u2016f (z * I * -I)\u2016 \u2264 C\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nsimpa [mul_assoc] using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\n\u22a2 \u2016f (z * I * -I)\u2016 \u2264 C\n[PROOFSTEP]\nhave H : MapsTo (\u00b7 * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b) := fun z hz \u21a6 by simpa using hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz\u271d : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z\u271d.re\nhzb : z\u271d.re \u2264 b\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo a b\n\u22a2 (fun x => x * -I) z \u2208 re \u207b\u00b9' Ioo a b\n[PROOFSTEP]\nsimpa using hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 \u2016f (z * I * -I)\u2016 \u2264 C\n[PROOFSTEP]\nrefine'\n  horizontal_strip (f := fun z \u21a6 f (z * -I)) (hfd.comp (differentiable_id.mul_const _).diffContOnCl H) _\n    (fun z hz => hle_a _ _) (fun z hz => hle_b _ _) _ _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 \u2203 c,\n    c < \u03c0 / (b - a) \u2227\n      \u2203 B,\n        (fun z => f (z * -I)) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nc : \u211d\nhc : c < \u03c0 / (b - a)\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\n\u22a2 \u2203 c,\n    c < \u03c0 / (b - a) \u2227\n      \u2203 B,\n        (fun z => f (z * -I)) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nrefine \u27e8c, hc, B, ?_\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nc : \u211d\nhc : c < \u03c0 / (b - a)\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\n\u22a2 (fun z => f (z * -I)) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nhave : Tendsto (\u00b7 * -I) (comap (|re \u00b7|) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)) (comap (|im \u00b7|) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)) :=\n  by\n  refine' (tendsto_comap_iff.2 _).inf H.tendsto\n  simpa [(\u00b7 \u2218 \u00b7)] using tendsto_comap\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nc : \u211d\nhc : c < \u03c0 / (b - a)\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\n\u22a2 Tendsto (fun x => x * -I) (comap (fun x => |x.re|) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b))\n    (comap (fun x => |x.im|) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b))\n[PROOFSTEP]\nrefine' (tendsto_comap_iff.2 _).inf H.tendsto\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nc : \u211d\nhc : c < \u03c0 / (b - a)\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\n\u22a2 Tendsto ((fun x => |x.im|) \u2218 fun x => x * -I) (comap (fun x => |x.re|) atTop) atTop\n[PROOFSTEP]\nsimpa [(\u00b7 \u2218 \u00b7)] using tendsto_comap\n[GOAL]\ncase refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nc : \u211d\nhc : c < \u03c0 / (b - a)\nB : \u211d\nhO : f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nthis :\n  Tendsto (fun x => x * -I) (comap (fun x => |x.re|) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b))\n    (comap (fun x => |x.im|) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b))\n\u22a2 (fun z => f (z * -I)) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nsimpa [(\u00b7 \u2218 \u00b7)] using hO.comp_tendsto this\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz\u271d : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z\u271d.re\nhzb : z\u271d.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nz : \u2102\nhz : z.im = a\n\u22a2 (z * -I).re = a\ncase refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz\u271d : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z\u271d.re\nhzb : z\u271d.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nz : \u2102\nhz : z.im = b\n\u22a2 (z * -I).re = b\ncase refine'_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 a \u2264 (z * I).im\ncase refine'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 (z * I).im \u2264 b\n[PROOFSTEP]\nall_goals simpa\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz\u271d : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z\u271d.re\nhzb : z\u271d.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nz : \u2102\nhz : z.im = a\n\u22a2 (z * -I).re = a\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz\u271d : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z\u271d.re\nhzb : z\u271d.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\nz : \u2102\nhz : z.im = b\n\u22a2 (z * -I).re = b\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 a \u2264 (z * I).im\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhfd : DiffContOnCl \u2102 f (re \u207b\u00b9' Ioo a b)\nhB :\n  \u2203 c,\n    c < \u03c0 / (b - a) \u2227 \u2203 B, f =O[comap (Abs.abs \u2218 im) atTop \u2293 \ud835\udcdf (re \u207b\u00b9' Ioo a b)] fun z => expR (B * expR (c * |z.im|))\nhle_a : \u2200 (z : \u2102), z.re = a \u2192 \u2016f z\u2016 \u2264 C\nhle_b : \u2200 (z : \u2102), z.re = b \u2192 \u2016f z\u2016 \u2264 C\nhza : a \u2264 z.re\nhzb : z.re \u2264 b\nH : MapsTo (fun x => x * -I) (im \u207b\u00b9' Ioo a b) (re \u207b\u00b9' Ioo a b)\n\u22a2 (z * I).im \u2264 b\n[PROOFSTEP]\nsimpa\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrcases eq_or_ne z 0 with (rfl | hzne)\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 0.re\nhz_im : 0 \u2264 0.im\n\u22a2 \u2016f 0\u2016 \u2264 C\n[PROOFSTEP]\nexact hre 0 le_rfl\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nhzne : z \u2260 0\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8\u03b6, h\u03b6, rfl\u27e9 : \u2203 \u03b6 : \u2102, \u03b6.im \u2208 Icc 0 (\u03c0 / 2) \u2227 exp \u03b6 = z :=\n  by\n  refine' \u27e8log z, _, exp_log hzne\u27e9\n  rw [log_im]\n  exact\n    \u27e8arg_nonneg_iff.2 hz_im, arg_le_pi_div_two_iff.2 (Or.inl hz_re)\u27e9\n      -- porting note: failed to clear `clear hz_re hz_im hzne`\n        -- We are going to apply `PhragmenLindelof.horizontal_strip` to `f \u2218 Complex.exp` and `\u03b6`.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nhzne : z \u2260 0\n\u22a2 \u2203 \u03b6, \u03b6.im \u2208 Icc 0 (\u03c0 / 2) \u2227 exp \u03b6 = z\n[PROOFSTEP]\nrefine' \u27e8log z, _, exp_log hzne\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nhzne : z \u2260 0\n\u22a2 (log z).im \u2208 Icc 0 (\u03c0 / 2)\n[PROOFSTEP]\nrw [log_im]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nhzne : z \u2260 0\n\u22a2 arg z \u2208 Icc 0 (\u03c0 / 2)\n[PROOFSTEP]\nexact\n  \u27e8arg_nonneg_iff.2 hz_im, arg_le_pi_div_two_iff.2 (Or.inl hz_re)\u27e9\n    -- porting note: failed to clear `clear hz_re hz_im hzne`\n      -- We are going to apply `PhragmenLindelof.horizontal_strip` to `f \u2218 Complex.exp` and `\u03b6`.\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\n\u22a2 \u2016f (exp \u03b6)\u2016 \u2264 C\n[PROOFSTEP]\nchange \u2016(f \u2218 exp) \u03b6\u2016 \u2264 C\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\n\u22a2 \u2016(f \u2218 exp) \u03b6\u2016 \u2264 C\n[PROOFSTEP]\nhave H : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0) := fun z hz \u21a6\n  by\n  rw [mem_reProdIm, exp_re, exp_im, mem_Ioi, mem_Ioi]\n  have : 0 < Real.cos z.im := Real.cos_pos_of_mem_Ioo \u27e8by linarith [hz.1, hz.2], hz.2\u27e9\n  have : 0 < Real.sin z.im := Real.sin_pos_of_mem_Ioo \u27e8hz.1, hz.2.trans (half_lt_self Real.pi_pos)\u27e9\n  constructor <;> positivity\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\n\u22a2 exp z \u2208 Ioi 0 \u00d7\u2102 Ioi 0\n[PROOFSTEP]\nrw [mem_reProdIm, exp_re, exp_im, mem_Ioi, mem_Ioi]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\n\u22a2 0 < expR z.re * Real.cos z.im \u2227 0 < expR z.re * Real.sin z.im\n[PROOFSTEP]\nhave : 0 < Real.cos z.im := Real.cos_pos_of_mem_Ioo \u27e8by linarith [hz.1, hz.2], hz.2\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\n\u22a2 -(\u03c0 / 2) < z.im\n[PROOFSTEP]\nlinarith [hz.1, hz.2]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\nthis : 0 < Real.cos z.im\n\u22a2 0 < expR z.re * Real.cos z.im \u2227 0 < expR z.re * Real.sin z.im\n[PROOFSTEP]\nhave : 0 < Real.sin z.im := Real.sin_pos_of_mem_Ioo \u27e8hz.1, hz.2.trans (half_lt_self Real.pi_pos)\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\nthis\u271d : 0 < Real.cos z.im\nthis : 0 < Real.sin z.im\n\u22a2 0 < expR z.re * Real.cos z.im \u2227 0 < expR z.re * Real.sin z.im\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\nthis\u271d : 0 < Real.cos z.im\nthis : 0 < Real.sin z.im\n\u22a2 0 < expR z.re * Real.cos z.im\n[PROOFSTEP]\npositivity\n[GOAL]\ncase right\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nz : \u2102\nhz : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\nthis\u271d : 0 < Real.cos z.im\nthis : 0 < Real.sin z.im\n\u22a2 0 < expR z.re * Real.sin z.im\n[PROOFSTEP]\npositivity\n[GOAL]\ncase inr.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2016(f \u2218 exp) \u03b6\u2016 \u2264 C\n[PROOFSTEP]\nrefine'\n  horizontal_strip (hd.comp differentiable_exp.diffContOnCl H) _ _ _ h\u03b6.1\n    h\u03b6.2\n      -- porting note: failed to clear h\u03b6 \u03b6\n[GOAL]\ncase inr.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2203 c,\n    c < \u03c0 / (\u03c0 / 2 - 0) \u2227\n      \u2203 B, (f \u2218 exp) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nrw [sub_zero, div_div_cancel' Real.pi_pos.ne']\n[GOAL]\ncase inr.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 exp) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 exp) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (B * expR (c * |z.re|))\n[PROOFSTEP]\nrefine' \u27e8c, hc, max B 0, _\u27e9\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 exp) =O[comap (Abs.abs \u2218 re) atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (max B 0 * expR (c * |z.re|))\n[PROOFSTEP]\nrw [\u2190 comap_comap, comap_abs_atTop, comap_sup, inf_sup_right]\n  -- We prove separately the estimates as `\u03b6.re \u2192 \u221e` and as `\u03b6.re \u2192 -\u221e`\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 exp) =O[comap re atBot \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) \u2294 comap re atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z =>\n    expR (max B 0 * expR (c * |z.re|))\n[PROOFSTEP]\nrefine' IsBigO.sup _ ((hO.comp_tendsto <| tendsto_exp_comap_re_atTop.inf H.tendsto).trans <| .of_bound 1 _)\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 exp) =O[comap re atBot \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (max B 0 * expR (c * |z.re|))\n[PROOFSTEP]\nhave hc : ContinuousWithinAt f (Ioi 0 \u00d7\u2102 Ioi 0) 0 :=\n  by\n  refine' (hd.continuousOn _ _).mono subset_closure\n  simp [closure_reProdIm, mem_reProdIm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 ContinuousWithinAt f (Ioi 0 \u00d7\u2102 Ioi 0) 0\n[PROOFSTEP]\nrefine' (hd.continuousOn _ _).mono subset_closure\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 0 \u2208 closure (Ioi 0 \u00d7\u2102 Ioi 0)\n[PROOFSTEP]\nsimp [closure_reProdIm, mem_reProdIm]\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc\u271d : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhc : ContinuousWithinAt f (Ioi 0 \u00d7\u2102 Ioi 0) 0\n\u22a2 (f \u2218 exp) =O[comap re atBot \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2))] fun z => expR (max B 0 * expR (c * |z.re|))\n[PROOFSTEP]\nrefine' ((hc.tendsto.comp <| tendsto_exp_comap_re_atBot.inf H.tendsto).isBigO_one \u211d).trans (isBigO_of_le _ fun w => _)\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc\u271d : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhc : ContinuousWithinAt f (Ioi 0 \u00d7\u2102 Ioi 0) 0\nw : \u2102\n\u22a2 \u20161\u2016 \u2264 \u2016expR (max B 0 * expR (c * |w.re|))\u2016\n[PROOFSTEP]\nrw [norm_one, Real.norm_of_nonneg (Real.exp_pos _).le, Real.one_le_exp_iff]\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc\u271d : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhc : ContinuousWithinAt f (Ioi 0 \u00d7\u2102 Ioi 0) 0\nw : \u2102\n\u22a2 0 \u2264 max B 0 * expR (c * |w.re|)\n[PROOFSTEP]\nexact mul_nonneg (le_max_right _ _) (Real.exp_pos _).le\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200\u1da0 (x : \u2102) in comap re atTop \u2293 \ud835\udcdf (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)),\n    \u2016((fun z => expR (B * \u2191Complex.abs z ^ c)) \u2218 exp) x\u2016 \u2264 1 * \u2016expR (max B 0 * expR (c * |x.re|))\u2016\n[PROOFSTEP]\nsimp only [eventually_inf_principal, eventually_comap, comp_apply, one_mul, Real.norm_of_nonneg (Real.exp_pos _).le,\n  abs_exp, \u2190 Real.exp_mul, Real.exp_le_exp]\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200\u1da0 (b : \u211d) in atTop,\n    \u2200 (a : \u2102), a.re = b \u2192 a \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2) \u2192 B * expR (a.re * c) \u2264 max B 0 * expR (c * |a.re|)\n[PROOFSTEP]\nrefine' (eventually_ge_atTop 0).mono fun x hx z hz _ => _\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nx : \u211d\nhx : 0 \u2264 x\nz : \u2102\nhz : z.re = x\nx\u271d : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\n\u22a2 B * expR (z.re * c) \u2264 max B 0 * expR (c * |z.re|)\n[PROOFSTEP]\nrw [hz, _root_.abs_of_nonneg hx, mul_comm _ c]\n[GOAL]\ncase inr.intro.intro.refine'_1.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nx : \u211d\nhx : 0 \u2264 x\nz : \u2102\nhz : z.re = x\nx\u271d : z \u2208 im \u207b\u00b9' Ioo 0 (\u03c0 / 2)\n\u22a2 B * expR (c * x) \u2264 max B 0 * expR (c * x)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.exp_pos _).le\n[GOAL]\ncase inr.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2200 (z : \u2102), z.im = 0 \u2192 \u2016(f \u2218 exp) z\u2016 \u2264 C\n[PROOFSTEP]\nintro \u03b6 h\u03b6\n[GOAL]\ncase inr.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6\u271d : \u2102\nh\u03b6\u271d : \u03b6\u271d.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6\u271d).re\nhz_im : 0 \u2264 (exp \u03b6\u271d).im\nhzne : exp \u03b6\u271d \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im = 0\n\u22a2 \u2016(f \u2218 exp) \u03b6\u2016 \u2264 C\n[PROOFSTEP]\nlift \u03b6 to \u211d using h\u03b6\n[GOAL]\ncase inr.intro.intro.refine'_2.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6\u271d : \u2102\nhz_re : 0 \u2264 (exp \u03b6\u271d).re\nhz_im : 0 \u2264 (exp \u03b6\u271d).im\nhzne : exp \u03b6\u271d \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u03b6 : \u211d\n\u22a2 \u2016(f \u2218 exp) \u2191\u03b6\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, \u2190 ofReal_exp]\n[GOAL]\ncase inr.intro.intro.refine'_2.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6\u271d : \u2102\nhz_re : 0 \u2264 (exp \u03b6\u271d).re\nhz_im : 0 \u2264 (exp \u03b6\u271d).im\nhzne : exp \u03b6\u271d \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u03b6 : \u211d\n\u22a2 \u2016f \u2191(expR \u03b6)\u2016 \u2264 C\n[PROOFSTEP]\nexact hre _ (Real.exp_pos _).le\n[GOAL]\ncase inr.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6).re\nhz_im : 0 \u2264 (exp \u03b6).im\nhzne : exp \u03b6 \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2200 (z : \u2102), z.im = \u03c0 / 2 \u2192 \u2016(f \u2218 exp) z\u2016 \u2264 C\n[PROOFSTEP]\nintro \u03b6 h\u03b6\n[GOAL]\ncase inr.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6\u271d : \u2102\nh\u03b6\u271d : \u03b6\u271d.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6\u271d).re\nhz_im : 0 \u2264 (exp \u03b6\u271d).im\nhzne : exp \u03b6\u271d \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im = \u03c0 / 2\n\u22a2 \u2016(f \u2218 exp) \u03b6\u2016 \u2264 C\n[PROOFSTEP]\nrw [\u2190 re_add_im \u03b6, h\u03b6, comp_apply, exp_add_mul_I, \u2190 ofReal_cos, \u2190 ofReal_sin, Real.cos_pi_div_two, Real.sin_pi_div_two,\n  ofReal_zero, ofReal_one, one_mul, zero_add, \u2190 ofReal_exp]\n[GOAL]\ncase inr.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\n\u03b6\u271d : \u2102\nh\u03b6\u271d : \u03b6\u271d.im \u2208 Icc 0 (\u03c0 / 2)\nhz_re : 0 \u2264 (exp \u03b6\u271d).re\nhz_im : 0 \u2264 (exp \u03b6\u271d).im\nhzne : exp \u03b6\u271d \u2260 0\nH : MapsTo exp (im \u207b\u00b9' Ioo 0 (\u03c0 / 2)) (Ioi 0 \u00d7\u2102 Ioi 0)\n\u03b6 : \u2102\nh\u03b6 : \u03b6.im = \u03c0 / 2\n\u22a2 \u2016f (\u2191(expR \u03b6.re) * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact him _ (Real.exp_pos _).le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 : \u2203 z', z' * I = z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2203 z', z' * I = z\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : (z * I).re \u2264 0\nhz_im : 0 \u2264 (z * I).im\n\u22a2 \u2016f (z * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact \u27e8z / I, div_mul_cancel _ I_ne_zero\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : (z * I).re \u2264 0\nhz_im : 0 \u2264 (z * I).im\n\u22a2 \u2016f (z * I)\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [mul_I_re, mul_I_im, neg_nonpos] at hz_re hz_im \n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\n\u22a2 \u2016f (z * I)\u2016 \u2264 C\n[PROOFSTEP]\nchange \u2016(f \u2218 (\u00b7 * I)) z\u2016 \u2264 C\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\n\u22a2 \u2016(f \u2218 fun x => x * I) z\u2016 \u2264 C\n[PROOFSTEP]\nhave H : MapsTo (\u00b7 * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0) := fun w hw \u21a6 by\n  simpa only [mem_reProdIm, mul_I_re, mul_I_im, neg_lt_zero, mem_Iio] using hw.symm\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nw : \u2102\nhw : w \u2208 Ioi 0 \u00d7\u2102 Ioi 0\n\u22a2 (fun x => x * I) w \u2208 Iio 0 \u00d7\u2102 Ioi 0\n[PROOFSTEP]\nsimpa only [mem_reProdIm, mul_I_re, mul_I_im, neg_lt_zero, mem_Iio] using hw.symm\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nH : MapsTo (fun x => x * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0)\n\u22a2 \u2016(f \u2218 fun x => x * I) z\u2016 \u2264 C\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nH : MapsTo (fun x => x * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2016(f \u2218 fun x => x * I) z\u2016 \u2264 C\n[PROOFSTEP]\nrefine'\n  quadrant_I (hd.comp (differentiable_id.mul_const _).diffContOnCl H) \u27e8c, hc, B, ?_\u27e9 him (fun x hx => _) hz_im hz_re\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nH : MapsTo (fun x => x * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 fun x => x * I) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), map_mul, abs_I, mul_one] using\n  hO.comp_tendsto ((tendsto_mul_right_cobounded I_ne_zero).inf H.tendsto)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nH : MapsTo (fun x => x * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 fun x => x * I) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, mul_assoc, I_mul_I, mul_neg_one, \u2190 ofReal_neg]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Ioi 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.im\nhz_im : 0 \u2264 z.re\nH : MapsTo (fun x => x * I) (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Ioi 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f \u2191(-x)\u2016 \u2264 C\n[PROOFSTEP]\nexact hre _ (neg_nonpos.2 hx)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : z.re \u2264 0\nhz_im : z.im \u2264 0\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 : \u2203 z', -z' = z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : z.re \u2264 0\nhz_im : z.im \u2264 0\n\u22a2 \u2203 z', -z' = z\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : (-z).re \u2264 0\nhz_im : (-z).im \u2264 0\n\u22a2 \u2016f (-z)\u2016 \u2264 C\n[PROOFSTEP]\nexact \u27e8-z, neg_neg z\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : (-z).re \u2264 0\nhz_im : (-z).im \u2264 0\n\u22a2 \u2016f (-z)\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [neg_re, neg_im, neg_nonpos] at hz_re hz_im \n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f (-z)\u2016 \u2264 C\n[PROOFSTEP]\nchange \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[PROOFSTEP]\nhave H : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0) :=\n  by\n  intro w hw\n  simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\n\u22a2 MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\n[PROOFSTEP]\nintro w hw\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nw : \u2102\nhw : w \u2208 Ioi 0 \u00d7\u2102 Ioi 0\n\u22a2 -w \u2208 Iio 0 \u00d7\u2102 Iio 0\n[PROOFSTEP]\nsimpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, mem_Iio] using hw\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' quadrant_I (hd.comp differentiable_neg.diffContOnCl H) _ (fun x hx => _) (fun x hx => _) hz_re hz_im\n[GOAL]\ncase intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine \u27e8c, hc, B, ?_\u27e9\n[GOAL]\ncase intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), Complex.abs.map_neg] using hO.comp_tendsto (tendsto_neg_cobounded.inf H.tendsto)\n[GOAL]\ncase intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 Neg.neg) \u2191x\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, \u2190 ofReal_neg]\n[GOAL]\ncase intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f \u2191(-x)\u2016 \u2264 C\n[PROOFSTEP]\nexact hre (-x) (neg_nonpos.2 hx)\n[GOAL]\ncase intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 Neg.neg) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, \u2190 neg_mul, \u2190 ofReal_neg]\n[GOAL]\ncase intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Iio 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 z.re\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Ioi 0 \u00d7\u2102 Ioi 0) (Iio 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f (\u2191(-x) * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact him (-x) (neg_nonpos.2 hx)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nhz_re : 0 \u2264 z.re\nhz_im : z.im \u2264 0\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8z, rfl\u27e9 : \u2203 z', -z' = z := \u27e8-z, neg_neg z\u27e9\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : 0 \u2264 (-z).re\nhz_im : (-z).im \u2264 0\n\u22a2 \u2016f (-z)\u2016 \u2264 C\n[PROOFSTEP]\nsimp only [neg_re, neg_im, neg_nonpos, neg_nonneg] at hz_re hz_im \n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016f (-z)\u2016 \u2264 C\n[PROOFSTEP]\nchange \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[PROOFSTEP]\nhave H : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0) := fun w hw \u21a6 by\n  simpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, neg_pos, mem_Ioi, mem_Iio] using hw\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nw : \u2102\nhw : w \u2208 Iio 0 \u00d7\u2102 Ioi 0\n\u22a2 -w \u2208 Ioi 0 \u00d7\u2102 Iio 0\n[PROOFSTEP]\nsimpa only [mem_reProdIm, neg_re, neg_im, neg_lt_zero, neg_pos, mem_Ioi, mem_Iio] using hw\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2016(f \u2218 Neg.neg) z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' quadrant_II (hd.comp differentiable_neg.diffContOnCl H) _ (fun x hx => _) (fun x hx => _) hz_re hz_im\n[GOAL]\ncase intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hB with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c,\n    c < 2 \u2227\n      \u2203 B, (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine \u27e8c, hc, B, ?_\u27e9\n[GOAL]\ncase intro.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (f \u2218 Neg.neg) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Iio 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nsimpa only [(\u00b7 \u2218 \u00b7), Complex.abs.map_neg] using hO.comp_tendsto (tendsto_neg_cobounded.inf H.tendsto)\n[GOAL]\ncase intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016(f \u2218 Neg.neg) \u2191x\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, \u2190 ofReal_neg]\n[GOAL]\ncase intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : x \u2264 0\n\u22a2 \u2016f \u2191(-x)\u2016 \u2264 C\n[PROOFSTEP]\nexact hre (-x) (neg_nonneg.2 hx)\n[GOAL]\ncase intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016(f \u2218 Neg.neg) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nrw [comp_apply, \u2190 neg_mul, \u2190 ofReal_neg]\n[GOAL]\ncase intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f (Ioi 0 \u00d7\u2102 Iio 0)\nhB : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C\nhim : \u2200 (x : \u211d), x \u2264 0 \u2192 \u2016f (\u2191x * I)\u2016 \u2264 C\nz : \u2102\nhz_re : z.re \u2264 0\nhz_im : 0 \u2264 z.im\nH : MapsTo Neg.neg (Iio 0 \u00d7\u2102 Ioi 0) (Ioi 0 \u00d7\u2102 Iio 0)\nx : \u211d\nhx : 0 \u2264 x\n\u22a2 \u2016f (\u2191(-x) * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact him (-x) (neg_nonpos.2 hx)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrevert z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave hle : \u2200 C', (\u2200 x : \u211d, 0 \u2264 x \u2192 \u2016f x\u2016 \u2264 C') \u2192 \u2200 z : \u2102, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C' := fun C' hC' z hz \u21a6\n  by\n  rcases hexp with \u27e8c, hc, B, hO\u27e9\n  cases' le_total z.im 0 with h h\n  \u00b7 refine\n      quadrant_IV (hd.mono fun _ => And.left) \u27e8c, hc, B, ?_\u27e9 (fun x hx => (hC' x hx).trans <| le_max_right _ _)\n        (fun x _ => (him x).trans (le_max_left _ _)) hz h\n    exact hO.mono (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)\n  \u00b7 refine'\n      quadrant_I (hd.mono fun _ => And.left) \u27e8c, hc, B, ?_\u27e9 (fun x hx => (hC' x hx).trans <| le_max_right _ _)\n        (fun x _ => (him x).trans (le_max_left _ _)) hz h\n    exact\n      hO.mono\n        (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)\n          -- Since `f` is continuous on `Ici 0` and `\u2016f x\u2016` tends to zero as `x \u2192 \u221e`,\n            -- the norm `\u2016f x\u2016` takes its maximum value at some `x\u2080 : \u211d`.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\n\u22a2 \u2016f z\u2016 \u2264 max C C'\n[PROOFSTEP]\nrcases hexp with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2016f z\u2016 \u2264 max C C'\n[PROOFSTEP]\ncases' le_total z.im 0 with h h\n[GOAL]\ncase intro.intro.intro.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nh : z.im \u2264 0\n\u22a2 \u2016f z\u2016 \u2264 max C C'\n[PROOFSTEP]\nrefine\n  quadrant_IV (hd.mono fun _ => And.left) \u27e8c, hc, B, ?_\u27e9 (fun x hx => (hC' x hx).trans <| le_max_right _ _)\n    (fun x _ => (him x).trans (le_max_left _ _)) hz h\n[GOAL]\ncase intro.intro.intro.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nh : z.im \u2264 0\n\u22a2 f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Iio 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nexact hO.mono (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)\n[GOAL]\ncase intro.intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nh : 0 \u2264 z.im\n\u22a2 \u2016f z\u2016 \u2264 max C C'\n[PROOFSTEP]\nrefine'\n  quadrant_I (hd.mono fun _ => And.left) \u27e8c, hc, B, ?_\u27e9 (fun x hx => (hC' x hx).trans <| le_max_right _ _)\n    (fun x _ => (him x).trans (le_max_left _ _)) hz h\n[GOAL]\ncase intro.intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nC' : \u211d\nhC' : \u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C'\nz : \u2102\nhz : 0 \u2264 z.re\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nh : 0 \u2264 z.im\n\u22a2 f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf (Ioi 0 \u00d7\u2102 Ioi 0)] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nexact\n  hO.mono\n    (inf_le_inf_left _ <| principal_mono.2 fun _ => And.left)\n      -- Since `f` is continuous on `Ici 0` and `\u2016f x\u2016` tends to zero as `x \u2192 \u221e`,\n        -- the norm `\u2016f x\u2016` takes its maximum value at some `x\u2080 : \u211d`.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nobtain \u27e8x\u2080, hx\u2080, hmax\u27e9 : \u2203 x : \u211d, 0 \u2264 x \u2227 \u2200 y : \u211d, 0 \u2264 y \u2192 \u2016f y\u2016 \u2264 \u2016f x\u2016 :=\n  by\n  have hfc : ContinuousOn (fun x : \u211d => f x) (Ici 0) :=\n    by\n    refine' hd.continuousOn.comp continuous_ofReal.continuousOn fun x hx => _\n    rwa [closure_setOf_lt_re]\n  by_cases h\u2080 : \u2200 x : \u211d, 0 \u2264 x \u2192 f x = 0\n  \u00b7 refine' \u27e80, le_rfl, fun y hy => _\u27e9; rw [h\u2080 y hy, h\u2080 0 le_rfl]\n  push_neg at h\u2080 \n  rcases h\u2080 with \u27e8x\u2080, hx\u2080, hne\u27e9\n  have hlt : \u2016(0 : E)\u2016 < \u2016f x\u2080\u2016 := by rwa [norm_zero, norm_pos_iff]\n  suffices \u2200\u1da0 x : \u211d in cocompact \u211d \u2293 \ud835\udcdf (Ici 0), \u2016f x\u2016 \u2264 \u2016f x\u2080\u2016 by\n    simpa only [exists_prop] using hfc.norm.exists_forall_ge' isClosed_Ici hx\u2080 this\n  rw [Real.cocompact_eq, inf_sup_right, (disjoint_atBot_principal_Ici (0 : \u211d)).eq_bot, bot_sup_eq]\n  exact (hre.norm.eventually <| ge_mem_nhds hlt).filter_mono inf_le_left\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nhave hfc : ContinuousOn (fun x : \u211d => f x) (Ici 0) :=\n  by\n  refine' hd.continuousOn.comp continuous_ofReal.continuousOn fun x hx => _\n  rwa [closure_setOf_lt_re]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\n\u22a2 ContinuousOn (fun x => f \u2191x) (Ici 0)\n[PROOFSTEP]\nrefine' hd.continuousOn.comp continuous_ofReal.continuousOn fun x hx => _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx : \u211d\nhx : x \u2208 Ici 0\n\u22a2 \u2191x \u2208 closure {z | 0 < z.re}\n[PROOFSTEP]\nrwa [closure_setOf_lt_re]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nby_cases h\u2080 : \u2200 x : \u211d, 0 \u2264 x \u2192 f x = 0\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nh\u2080 : \u2200 (x : \u211d), 0 \u2264 x \u2192 f \u2191x = 0\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nrefine' \u27e80, le_rfl, fun y hy => _\u27e9\n[GOAL]\ncase pos\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nh\u2080 : \u2200 (x : \u211d), 0 \u2264 x \u2192 f \u2191x = 0\ny : \u211d\nhy : 0 \u2264 y\n\u22a2 \u2016f \u2191y\u2016 \u2264 \u2016f \u21910\u2016\n[PROOFSTEP]\nrw [h\u2080 y hy, h\u2080 0 le_rfl]\n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nh\u2080 : \u00ac\u2200 (x : \u211d), 0 \u2264 x \u2192 f \u2191x = 0\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\npush_neg at h\u2080 \n[GOAL]\ncase neg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nh\u2080 : \u2203 x, 0 \u2264 x \u2227 f \u2191x \u2260 0\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nrcases h\u2080 with \u27e8x\u2080, hx\u2080, hne\u27e9\n[GOAL]\ncase neg.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nhave hlt : \u2016(0 : E)\u2016 < \u2016f x\u2080\u2016 := by rwa [norm_zero, norm_pos_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\n\u22a2 \u20160\u2016 < \u2016f \u2191x\u2080\u2016\n[PROOFSTEP]\nrwa [norm_zero, norm_pos_iff]\n[GOAL]\ncase neg.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\nhlt : \u20160\u2016 < \u2016f \u2191x\u2080\u2016\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nsuffices \u2200\u1da0 x : \u211d in cocompact \u211d \u2293 \ud835\udcdf (Ici 0), \u2016f x\u2016 \u2264 \u2016f x\u2080\u2016 by\n  simpa only [exists_prop] using hfc.norm.exists_forall_ge' isClosed_Ici hx\u2080 this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\nhlt : \u20160\u2016 < \u2016f \u2191x\u2080\u2016\nthis : \u2200\u1da0 (x : \u211d) in cocompact \u211d \u2293 \ud835\udcdf (Ici 0), \u2016f \u2191x\u2016 \u2264 \u2016f \u2191x\u2080\u2016\n\u22a2 \u2203 x, 0 \u2264 x \u2227 \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2016\n[PROOFSTEP]\nsimpa only [exists_prop] using hfc.norm.exists_forall_ge' isClosed_Ici hx\u2080 this\n[GOAL]\ncase neg.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\nhlt : \u20160\u2016 < \u2016f \u2191x\u2080\u2016\n\u22a2 \u2200\u1da0 (x : \u211d) in cocompact \u211d \u2293 \ud835\udcdf (Ici 0), \u2016f \u2191x\u2016 \u2264 \u2016f \u2191x\u2080\u2016\n[PROOFSTEP]\nrw [Real.cocompact_eq, inf_sup_right, (disjoint_atBot_principal_Ici (0 : \u211d)).eq_bot, bot_sup_eq]\n[GOAL]\ncase neg.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nhfc : ContinuousOn (fun x => f \u2191x) (Ici 0)\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhne : f \u2191x\u2080 \u2260 0\nhlt : \u20160\u2016 < \u2016f \u2191x\u2080\u2016\n\u22a2 \u2200\u1da0 (x : \u211d) in atTop \u2293 \ud835\udcdf (Ici 0), \u2016f \u2191x\u2016 \u2264 \u2016f \u2191x\u2080\u2016\n[PROOFSTEP]\nexact (hre.norm.eventually <| ge_mem_nhds hlt).filter_mono inf_le_left\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhmax : \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2080\u2016\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\ncases' le_or_lt \u2016f x\u2080\u2016 C with h h\n[GOAL]\ncase intro.intro.inl\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhmax : \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2080\u2016\nh : \u2016f \u2191x\u2080\u2016 \u2264 C\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [max_eq_left h] using hle _ hmax\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhmax : \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2080\u2016\nh : C < \u2016f \u2191x\u2080\u2016\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nreplace hmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} x\u2080\n[GOAL]\ncase hmax\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhmax : \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2080\u2016\nh : C < \u2016f \u2191x\u2080\u2016\n\u22a2 IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\n[PROOFSTEP]\nrintro z (hz : 0 < z.re)\n[GOAL]\ncase hmax\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nhmax : \u2200 (y : \u211d), 0 \u2264 y \u2192 \u2016f \u2191y\u2016 \u2264 \u2016f \u2191x\u2080\u2016\nh : C < \u2016f \u2191x\u2080\u2016\nz : \u2102\nhz : 0 < z.re\n\u22a2 z \u2208 {x | (fun x => (norm \u2218 f) x \u2264 (norm \u2218 f) \u2191x\u2080) x}\n[PROOFSTEP]\nsimpa [max_eq_right h.le] using hle _ hmax _ hz.le\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nhave : \u2016f 0\u2016 = \u2016f x\u2080\u2016 := by\n  apply norm_eq_norm_of_isMaxOn_of_ball_subset hd hmax\n  intro z hz\n  rw [mem_ball, dist_zero_left, dist_eq, norm_eq_abs, Complex.abs_of_nonneg hx\u2080] at hz \n  rw [mem_setOf_eq]\n  contrapose! hz\n  calc\n    x\u2080 \u2264 x\u2080 - z.re := (le_sub_self_iff _).2 hz\n    _ \u2264 |x\u2080 - z.re| := (le_abs_self _)\n    _ = |(z - x\u2080).re| := by rw [sub_re, ofReal_re, _root_.abs_sub_comm]\n    _ \u2264 abs (z - x\u2080) :=\n      abs_re_le_abs\n        _\n          -- Thus we have `C < \u2016f x\u2080\u2016 = \u2016f 0\u2016 \u2264 C`. Contradiction completes the proof.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\n\u22a2 \u2016f 0\u2016 = \u2016f \u2191x\u2080\u2016\n[PROOFSTEP]\napply norm_eq_norm_of_isMaxOn_of_ball_subset hd hmax\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\n\u22a2 ball (\u2191x\u2080) (dist 0 \u2191x\u2080) \u2286 {z | 0 < z.re}\n[PROOFSTEP]\nintro z hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nz : \u2102\nhz : z \u2208 ball (\u2191x\u2080) (dist 0 \u2191x\u2080)\n\u22a2 z \u2208 {z | 0 < z.re}\n[PROOFSTEP]\nrw [mem_ball, dist_zero_left, dist_eq, norm_eq_abs, Complex.abs_of_nonneg hx\u2080] at hz \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nz : \u2102\nhz : \u2191Complex.abs (z - \u2191x\u2080) < x\u2080\n\u22a2 z \u2208 {z | 0 < z.re}\n[PROOFSTEP]\nrw [mem_setOf_eq]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nz : \u2102\nhz : \u2191Complex.abs (z - \u2191x\u2080) < x\u2080\n\u22a2 0 < z.re\n[PROOFSTEP]\ncontrapose! hz\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nz : \u2102\nhz : z.re \u2264 0\n\u22a2 x\u2080 \u2264 \u2191Complex.abs (z - \u2191x\u2080)\n[PROOFSTEP]\ncalc\n  x\u2080 \u2264 x\u2080 - z.re := (le_sub_self_iff _).2 hz\n  _ \u2264 |x\u2080 - z.re| := (le_abs_self _)\n  _ = |(z - x\u2080).re| := by rw [sub_re, ofReal_re, _root_.abs_sub_comm]\n  _ \u2264 abs (z - x\u2080) :=\n    abs_re_le_abs\n      _\n        -- Thus we have `C < \u2016f x\u2080\u2016 = \u2016f 0\u2016 \u2264 C`. Contradiction completes the proof.\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nz : \u2102\nhz : z.re \u2264 0\n\u22a2 |x\u2080 - z.re| = |(z - \u2191x\u2080).re|\n[PROOFSTEP]\nrw [sub_re, ofReal_re, _root_.abs_sub_comm]\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nthis : \u2016f 0\u2016 = \u2016f \u2191x\u2080\u2016\n\u22a2 \u2200 {z : \u2102}, 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' (h.not_le <| this \u25b8 _).elim\n[GOAL]\ncase intro.intro.inr\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : Tendsto (fun x => f \u2191x) atTop (\ud835\udcdd 0)\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhle : \u2200 (C' : \u211d), (\u2200 (x : \u211d), 0 \u2264 x \u2192 \u2016f \u2191x\u2016 \u2264 C') \u2192 \u2200 (z : \u2102), 0 \u2264 z.re \u2192 \u2016f z\u2016 \u2264 max C C'\nx\u2080 : \u211d\nhx\u2080 : 0 \u2264 x\u2080\nh : C < \u2016f \u2191x\u2080\u2016\nhmax : IsMaxOn (norm \u2218 f) {z | 0 < z.re} \u2191x\u2080\nthis : \u2016f 0\u2016 = \u2016f \u2191x\u2080\u2016\n\u22a2 \u2016f 0\u2016 \u2264 C\n[PROOFSTEP]\nsimpa using him 0\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nsuffices \u2200\u1da0 \u03b5 : \u211d in \ud835\udcdd[<] 0, \u2016exp (\u03b5 * z) \u2022 f z\u2016 \u2264 C\n  by\n  refine' le_of_tendsto (Tendsto.mono_left _ nhdsWithin_le_nhds) this\n  apply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'\n  simp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n\u22a2 \u2016f z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' le_of_tendsto (Tendsto.mono_left _ nhdsWithin_le_nhds) this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n\u22a2 Tendsto (fun c => \u2016exp (\u2191c * z) \u2022 f z\u2016) (\ud835\udcdd 0) (\ud835\udcdd \u2016f z\u2016)\n[PROOFSTEP]\napply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\nthis : \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n\u22a2 \u2016exp (\u21910 * z) \u2022 f z\u2016 = \u2016f z\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u22a2 \u2200\u1da0 (\u03b5 : \u211d) in \ud835\udcdd[Iio 0] 0, \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nfilter_upwards [self_mem_nhdsWithin] with \u03b5 \u03b5\u2080\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 \u2208 Iio 0\n\u22a2 \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nchange \u03b5 < 0 at \u03b5\u2080 \n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\n\u22a2 \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nset g : \u2102 \u2192 E := fun z => exp (\u03b5 * z) \u2022 f z\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\n\u22a2 \u2016exp (\u2191\u03b5 * z) \u2022 f z\u2016 \u2264 C\n[PROOFSTEP]\nchange \u2016g z\u2016 \u2264 C\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\n\u22a2 \u2016g z\u2016 \u2264 C\n[PROOFSTEP]\nreplace hd : DiffContOnCl \u2102 g {z : \u2102 | 0 < z.re}\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\n\u22a2 DiffContOnCl \u2102 g {z | 0 < z.re}\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\n\u22a2 \u2016g z\u2016 \u2264 C\n[PROOFSTEP]\nexact (differentiable_id.const_mul _).cexp.diffContOnCl.smul hd\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\n\u22a2 \u2016g z\u2016 \u2264 C\n[PROOFSTEP]\nhave hgn : \u2200 z, \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016 := fun z \u21a6 by rw [norm_smul, norm_eq_abs, abs_exp, ofReal_mul_re]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z\u271d.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nz : \u2102\n\u22a2 \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\n[PROOFSTEP]\nrw [norm_smul, norm_eq_abs, abs_exp, ofReal_mul_re]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\n\u22a2 \u2016g z\u2016 \u2264 C\n[PROOFSTEP]\nrefine' right_half_plane_of_tendsto_zero_on_real hd _ _ (fun y => _) hz\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hexp with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine \u27e8c, hc, B, (IsBigO.of_bound 1 ?_).trans hO\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200\u1da0 (x : \u2102) in comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}, \u2016g x\u2016 \u2264 1 * \u2016f x\u2016\n[PROOFSTEP]\nrefine' eventually_inf_principal.2 <| eventually_of_forall fun z hz => _\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz\u271d : 0 \u2264 z\u271d.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\nhz : z \u2208 {z | 0 < z.re}\n\u22a2 \u2016g z\u2016 \u2264 1 * \u2016f z\u2016\n[PROOFSTEP]\nrw [hgn, one_mul]\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz\u271d : 0 \u2264 z\u271d.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\nhz : z \u2208 {z | 0 < z.re}\n\u22a2 expR (\u03b5 * z.re) * \u2016f z\u2016 \u2264 \u2016f z\u2016\n[PROOFSTEP]\nrefine' mul_le_of_le_one_left (norm_nonneg _) (Real.exp_le_one_iff.2 _)\n[GOAL]\ncase h.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz\u271d : 0 \u2264 z\u271d.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\nhz : z \u2208 {z | 0 < z.re}\n\u22a2 \u03b5 * z.re \u2264 0\n[PROOFSTEP]\nexact mul_nonpos_of_nonpos_of_nonneg \u03b5\u2080.le (le_of_lt hz)\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\n\u22a2 Tendsto (fun x => g \u2191x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [\u2190 ofReal_mul, \u2190 ofReal_exp, coe_smul]\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\n\u22a2 Tendsto (fun x => expR (\u03b5 * x) \u2022 f \u2191x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nhave h\u2080 : Tendsto (fun x : \u211d => expR (\u03b5 * x)) atTop (\ud835\udcdd 0) :=\n  Real.tendsto_exp_atBot.comp (tendsto_const_nhds.neg_mul_atTop \u03b5\u2080 tendsto_id)\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\nh\u2080 : Tendsto (fun x => expR (\u03b5 * x)) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun x => expR (\u03b5 * x) \u2022 f \u2191x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact h\u2080.zero_smul_isBoundedUnder_le hre\n[GOAL]\ncase h.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\ny : \u211d\n\u22a2 \u2016g (\u2191y * I)\u2016 \u2264 C\n[PROOFSTEP]\nrw [hgn, ofReal_mul_re, I_re, mul_zero, mul_zero, Real.exp_zero, one_mul]\n[GOAL]\ncase h.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : IsBoundedUnder (fun x x_1 => x \u2264 x_1) atTop fun x => \u2016f \u2191x\u2016\nhim : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhz : 0 \u2264 z.re\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 < 0\ng : \u2102 \u2192 E := fun z => exp (\u2191\u03b5 * z) \u2022 f z\nhd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhgn : \u2200 (z : \u2102), \u2016g z\u2016 = expR (\u03b5 * z.re) * \u2016f z\u2016\ny : \u211d\n\u22a2 \u2016f (\u2191y * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact him y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nhim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\n\u22a2 EqOn f 0 {z | 0 \u2264 z.re}\n[PROOFSTEP]\nrcases him with\n  \u27e8C, hC\u27e9\n    -- Due to continuity, it suffices to prove the equality on the open right half-plane.\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\n\u22a2 EqOn f 0 {z | 0 \u2264 z.re}\n[PROOFSTEP]\nsuffices \u2200 z : \u2102, 0 < z.re \u2192 f z = 0 by\n  simpa only [closure_setOf_lt_re] using\n    EqOn.of_subset_closure this hd.continuousOn continuousOn_const subset_closure Subset.rfl\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nthis : \u2200 (z : \u2102), 0 < z.re \u2192 f z = 0\n\u22a2 EqOn f 0 {z | 0 \u2264 z.re}\n[PROOFSTEP]\nsimpa only [closure_setOf_lt_re] using\n  EqOn.of_subset_closure this hd.continuousOn continuousOn_const subset_closure Subset.rfl\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\n\u22a2 \u2200 (z : \u2102), 0 < z.re \u2192 f z = 0\n[PROOFSTEP]\nset g : \u2115 \u2192 \u2102 \u2192 E := fun (n : \u2115) (z : \u2102) => exp z ^ n \u2022 f z\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\n\u22a2 \u2200 (z : \u2102), 0 < z.re \u2192 f z = 0\n[PROOFSTEP]\nhave hg : \u2200 n z, \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016 := fun n z \u21a6 by\n  simp only [norm_smul, norm_eq_abs, Complex.abs_pow, abs_exp]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nn : \u2115\nz : \u2102\n\u22a2 \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\n[PROOFSTEP]\nsimp only [norm_smul, norm_eq_abs, Complex.abs_pow, abs_exp]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\n\u22a2 \u2200 (z : \u2102), 0 < z.re \u2192 f z = 0\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\n\u22a2 f z = 0\n[PROOFSTEP]\nsuffices H : \u2200 n : \u2115, \u2016g n z\u2016 \u2264 C\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nH : \u2200 (n : \u2115), \u2016g n z\u2016 \u2264 C\n\u22a2 f z = 0\n[PROOFSTEP]\ncontrapose! H\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nH : f z \u2260 0\n\u22a2 \u2203 n, C < \u2016(fun n z => exp z ^ n \u2022 f z) n z\u2016\n[PROOFSTEP]\nsimp only [hg]\n[GOAL]\ncase intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nH : f z \u2260 0\n\u22a2 \u2203 n, C < expR z.re ^ n * \u2016f z\u2016\n[PROOFSTEP]\nexact\n  (((tendsto_pow_atTop_atTop_of_one_lt (Real.one_lt_exp_iff.2 hz)).atTop_mul (norm_pos_iff.2 H)\n          tendsto_const_nhds).eventually\n      (eventually_gt_atTop C)).exists\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\n\u22a2 \u2200 (n : \u2115), \u2016g n z\u2016 \u2264 C\n[PROOFSTEP]\nintro n\n[GOAL]\ncase H\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\n\u22a2 \u2016g n z\u2016 \u2264 C\n[PROOFSTEP]\nrefine'\n  right_half_plane_of_tendsto_zero_on_real ((differentiable_exp.pow n).diffContOnCl.smul hd) _ _ (fun y => _) hz.le\n[GOAL]\ncase H.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, g n =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hexp with \u27e8c, hc, B, hO\u27e9\n[GOAL]\ncase H.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, g n =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine' \u27e8max c 1, max_lt hc one_lt_two, n + max B 0, .of_norm_left _\u27e9\n[GOAL]\ncase H.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (fun x => \u2016g n x\u2016) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z =>\n    expR ((\u2191n + max B 0) * \u2191Complex.abs z ^ max c 1)\n[PROOFSTEP]\nsimp only [hg]\n[GOAL]\ncase H.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 (fun x => expR x.re ^ n * \u2016f x\u2016) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z =>\n    expR ((\u2191n + max B 0) * \u2191Complex.abs z ^ max c 1)\n[PROOFSTEP]\nrefine' ((isBigO_refl (fun z : \u2102 => expR z.re ^ n) _).mul hO.norm_left).trans (.of_bound 1 _)\n[GOAL]\ncase H.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200\u1da0 (x : \u2102) in comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re},\n    \u2016expR x.re ^ n * expR (B * \u2191Complex.abs x ^ c)\u2016 \u2264 1 * \u2016expR ((\u2191n + max B 0) * \u2191Complex.abs x ^ max c 1)\u2016\n[PROOFSTEP]\nsimp only [\u2190 Real.exp_nat_mul, \u2190 Real.exp_add, Real.norm_of_nonneg (Real.exp_pos _).le, Real.exp_le_exp, add_mul,\n  eventually_inf_principal, eventually_comap, one_mul]\n  -- porting note: todo: `0 < z.re` is not used; where do we use it?\n[GOAL]\ncase H.refine'_1.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200\u1da0 (b : \u211d) in atTop,\n    \u2200 (a : \u2102),\n      \u2191Complex.abs a = b \u2192\n        a \u2208 {z | 0 < z.re} \u2192\n          \u2191n * a.re + B * \u2191Complex.abs a ^ c \u2264 \u2191n * \u2191Complex.abs a ^ max c 1 + max B 0 * \u2191Complex.abs a ^ max c 1\n[PROOFSTEP]\nfilter_upwards [eventually_ge_atTop (1 : \u211d)] with r hr z hzr _\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d\u00b9 : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz\u271d : \u2102\nhz : 0 < z\u271d.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nr : \u211d\nhr : 1 \u2264 r\nz : \u2102\nhzr : \u2191Complex.abs z = r\na\u271d : 0 < z.re\n\u22a2 \u2191n * z.re + B * \u2191Complex.abs z ^ c \u2264 \u2191n * \u2191Complex.abs z ^ max c 1 + max B 0 * \u2191Complex.abs z ^ max c 1\n[PROOFSTEP]\nsubst r\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d\u00b9 : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz\u271d : \u2102\nhz : 0 < z\u271d.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\na\u271d : 0 < z.re\nhr : 1 \u2264 \u2191Complex.abs z\n\u22a2 \u2191n * z.re + B * \u2191Complex.abs z ^ c \u2264 \u2191n * \u2191Complex.abs z ^ max c 1 + max B 0 * \u2191Complex.abs z ^ max c 1\n[PROOFSTEP]\nrefine' add_le_add (mul_le_mul_of_nonneg_left _ n.cast_nonneg) _\n[GOAL]\ncase h.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d\u00b9 : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz\u271d : \u2102\nhz : 0 < z\u271d.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\na\u271d : 0 < z.re\nhr : 1 \u2264 \u2191Complex.abs z\n\u22a2 z.re \u2264 \u2191Complex.abs z ^ max c 1\n[PROOFSTEP]\ncalc\n  z.re \u2264 abs z := re_le_abs _\n  _ = abs z ^ (1 : \u211d) := (Real.rpow_one _).symm\n  _ \u2264 abs z ^ max c 1 := Real.rpow_le_rpow_of_exponent_le hr (le_max_right _ _)\n[GOAL]\ncase h.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d\u00b9 : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz\u271d : \u2102\nhz : 0 < z\u271d.re\nn : \u2115\nc : \u211d\nhc : c < 2\nB : \u211d\nhO : f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nz : \u2102\na\u271d : 0 < z.re\nhr : 1 \u2264 \u2191Complex.abs z\n\u22a2 B * \u2191Complex.abs z ^ c \u2264 max B 0 * \u2191Complex.abs z ^ max c 1\n[PROOFSTEP]\nexact\n  mul_le_mul (le_max_left _ _) (Real.rpow_le_rpow_of_exponent_le hr (le_max_left _ _))\n    (Real.rpow_nonneg_of_nonneg (Complex.abs.nonneg _) _) (le_max_right _ _)\n[GOAL]\ncase H.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\n\u22a2 Tendsto (fun x => g n \u2191x) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_zero_iff_norm_tendsto_zero]\n[GOAL]\ncase H.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\n\u22a2 Tendsto (fun e => \u2016g n \u2191e\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [hg]\n[GOAL]\ncase H.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\n\u22a2 Tendsto (fun e => expR (\u2191e).re ^ n * \u2016f \u2191e\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact hre n\n[GOAL]\ncase H.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\ny : \u211d\n\u22a2 \u2016g n (\u2191y * I)\u2016 \u2264 C\n[PROOFSTEP]\nrw [hg, ofReal_mul_re, I_re, mul_zero, Real.exp_zero, one_pow, one_mul]\n[GOAL]\ncase H.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C\u271d : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\nhd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x\u2016\nC : \u211d\nhC : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\ng : \u2115 \u2192 \u2102 \u2192 E := fun n z => exp z ^ n \u2022 f z\nhg : \u2200 (n : \u2115) (z : \u2102), \u2016g n z\u2016 = expR z.re ^ n * \u2016f z\u2016\nz : \u2102\nhz : 0 < z.re\nn : \u2115\ny : \u211d\n\u22a2 \u2016f (\u2191y * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact hC y\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\n\u22a2 EqOn f g {z | 0 \u2264 z.re}\n[PROOFSTEP]\nsuffices EqOn (f - g) 0 {z : \u2102 | 0 \u2264 z.re} by simpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nthis : EqOn (f - g) 0 {z | 0 \u2264 z.re}\n\u22a2 EqOn f g {z | 0 \u2264 z.re}\n[PROOFSTEP]\nsimpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\n\u22a2 EqOn (f - g) 0 {z | 0 \u2264 z.re}\n[PROOFSTEP]\nrefine' eq_zero_on_right_half_plane_of_superexponential_decay (hfd.sub hgd) _ hre _\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, (f - g) =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nset l : Filter \u2102 := comap Complex.abs atTop \u2293 \ud835\udcdf {z : \u2102 | 0 < z.re}\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nsuffices\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192 B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * abs z ^ c\u2082)\n  by\n  rcases hfexp with \u27e8cf, hcf, Bf, hOf\u27e9; rcases hgexp with \u27e8cg, hcg, Bg, hOg\u27e9\n  refine' \u27e8max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _\u27e9\n  refine' .sub (hOf.trans <| this _ _ _) (hOg.trans <| this _ _ _) <;> simp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hfexp with \u27e8cf, hcf, Bf, hOf\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrcases hgexp with \u27e8cg, hcg, Bg, hOg\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 \u2203 c, c < 2 \u2227 \u2203 B, (f - g) =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n[PROOFSTEP]\nrefine' \u27e8max cf cg, max_lt hcf hcg, max 0 (max Bf Bg), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 (f - g) =O[l] fun z => expR (max 0 (max Bf Bg) * \u2191Complex.abs z ^ max cf cg)\n[PROOFSTEP]\nrefine' .sub (hOf.trans <| this _ _ _) (hOg.trans <| this _ _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 cf \u2264 max cf cg\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 Bf \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_3\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 0 \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_4\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 cg \u2264 max cf cg\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 Bg \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_6\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nthis :\n  \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\ncf : \u211d\nhcf : cf < 2\nBf : \u211d\nhOf : f =O[l] fun z => expR (Bf * \u2191Complex.abs z ^ cf)\ncg : \u211d\nhcg : cg < 2\nBg : \u211d\nhOg : g =O[l] fun z => expR (Bg * \u2191Complex.abs z ^ cg)\n\u22a2 0 \u2264 max 0 (max Bf Bg)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\n\u22a2 \u2200 {c\u2081 c\u2082 B\u2081 B\u2082 : \u211d},\n    c\u2081 \u2264 c\u2082 \u2192\n      B\u2081 \u2264 B\u2082 \u2192 0 \u2264 B\u2082 \u2192 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\n[PROOFSTEP]\nintro c\u2081 c\u2082 B\u2081 B\u2082 hc hB hB\u2082\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081 c\u2082 B\u2081 B\u2082 : \u211d\nhc : c\u2081 \u2264 c\u2082\nhB : B\u2081 \u2264 B\u2082\nhB\u2082 : 0 \u2264 B\u2082\n\u22a2 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\n[PROOFSTEP]\nhave : \u2200\u1da0 z : \u2102 in l, 1 \u2264 abs z := ((eventually_ge_atTop 1).comap _).filter_mono inf_le_left\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081 c\u2082 B\u2081 B\u2082 : \u211d\nhc : c\u2081 \u2264 c\u2082\nhB : B\u2081 \u2264 B\u2082\nhB\u2082 : 0 \u2264 B\u2082\nthis : \u2200\u1da0 (z : \u2102) in l, 1 \u2264 \u2191Complex.abs z\n\u22a2 (fun z => expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)) =O[l] fun z => expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\n[PROOFSTEP]\nrefine' .of_bound 1 (this.mono fun z hz => _)\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081 c\u2082 B\u2081 B\u2082 : \u211d\nhc : c\u2081 \u2264 c\u2082\nhB : B\u2081 \u2264 B\u2082\nhB\u2082 : 0 \u2264 B\u2082\nthis : \u2200\u1da0 (z : \u2102) in l, 1 \u2264 \u2191Complex.abs z\nz : \u2102\nhz : 1 \u2264 \u2191Complex.abs z\n\u22a2 \u2016expR (B\u2081 * \u2191Complex.abs z ^ c\u2081)\u2016 \u2264 1 * \u2016expR (B\u2082 * \u2191Complex.abs z ^ c\u2082)\u2016\n[PROOFSTEP]\nsimp only [Real.norm_of_nonneg (Real.exp_pos _).le, Real.exp_le_exp, one_mul]\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081 c\u2082 B\u2081 B\u2082 : \u211d\nhc : c\u2081 \u2264 c\u2082\nhB : B\u2081 \u2264 B\u2082\nhB\u2082 : 0 \u2264 B\u2082\nthis : \u2200\u1da0 (z : \u2102) in l, 1 \u2264 \u2191Complex.abs z\nz : \u2102\nhz : 1 \u2264 \u2191Complex.abs z\n\u22a2 B\u2081 * \u2191Complex.abs z ^ c\u2081 \u2264 B\u2082 * \u2191Complex.abs z ^ c\u2082\n[PROOFSTEP]\nhave := Real.rpow_le_rpow_of_exponent_le hz hc\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz\u271d : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nl : Filter \u2102 := comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[l] fun z => expR (B * \u2191Complex.abs z ^ c)\nc\u2081 c\u2082 B\u2081 B\u2082 : \u211d\nhc : c\u2081 \u2264 c\u2082\nhB : B\u2081 \u2264 B\u2082\nhB\u2082 : 0 \u2264 B\u2082\nthis\u271d : \u2200\u1da0 (z : \u2102) in l, 1 \u2264 \u2191Complex.abs z\nz : \u2102\nhz : 1 \u2264 \u2191Complex.abs z\nthis : \u2191Complex.abs z ^ c\u2081 \u2264 \u2191Complex.abs z ^ c\u2082\n\u22a2 B\u2081 * \u2191Complex.abs z ^ c\u2081 \u2264 B\u2082 * \u2191Complex.abs z ^ c\u2082\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhfim : \u2203 C, \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 C\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\n\u22a2 \u2203 C, \u2200 (x : \u211d), \u2016(f - g) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nrcases hfim with \u27e8Cf, hCf\u27e9\n[GOAL]\ncase refine'_2.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nhgim : \u2203 C, \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 C\nCf : \u211d\nhCf : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 Cf\n\u22a2 \u2203 C, \u2200 (x : \u211d), \u2016(f - g) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nrcases hgim with \u27e8Cg, hCg\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\na b C : \u211d\nf g\u271d : \u2102 \u2192 E\nz : \u2102\ng : \u2102 \u2192 E\nhfd : DiffContOnCl \u2102 f {z | 0 < z.re}\nhgd : DiffContOnCl \u2102 g {z | 0 < z.re}\nhfexp : \u2203 c, c < 2 \u2227 \u2203 B, f =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhgexp : \u2203 c, c < 2 \u2227 \u2203 B, g =O[comap (\u2191Complex.abs) atTop \u2293 \ud835\udcdf {z | 0 < z.re}] fun z => expR (B * \u2191Complex.abs z ^ c)\nhre : SuperpolynomialDecay atTop expR fun x => \u2016f \u2191x - g \u2191x\u2016\nCf : \u211d\nhCf : \u2200 (x : \u211d), \u2016f (\u2191x * I)\u2016 \u2264 Cf\nCg : \u211d\nhCg : \u2200 (x : \u211d), \u2016g (\u2191x * I)\u2016 \u2264 Cg\n\u22a2 \u2203 C, \u2200 (x : \u211d), \u2016(f - g) (\u2191x * I)\u2016 \u2264 C\n[PROOFSTEP]\nexact \u27e8Cf + Cg, fun x => norm_sub_le_of_le (hCf x) (hCg x)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.PhragmenLindelof", "llama_tokens": 135635, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.863391595913457, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.5246512739798503}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ny : Y\n\u22a2 Homotopic (const X y) (const X y)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(X, Y)\nhf : Nullhomotopic f\ng : C(Y, Z)\n\u22a2 Nullhomotopic (comp g f)\n[PROOFSTEP]\ncases' hf with y hy\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n\u22a2 Nullhomotopic (comp g f)\n[PROOFSTEP]\nuse g y\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n\u22a2 Homotopic (comp g f) (const X (\u2191g y))\n[PROOFSTEP]\nexact Homotopic.hcomp hy (Homotopic.refl g)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(Y, Z)\nhf : Nullhomotopic f\ng : C(X, Y)\n\u22a2 Nullhomotopic (comp f g)\n[PROOFSTEP]\ncases' hf with y hy\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(Y, Z)\ng : C(X, Y)\ny : Z\nhy : Homotopic f (const Y y)\n\u22a2 Nullhomotopic (comp f g)\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : C(Y, Z)\ng : C(X, Y)\ny : Z\nhy : Homotopic f (const Y y)\n\u22a2 Homotopic (comp f g) (const X y)\n[PROOFSTEP]\nexact Homotopic.hcomp (Homotopic.refl g) hy\n[GOAL]\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : ContractibleSpace X\n\u22a2 Nullhomotopic (ContinuousMap.id X)\n[PROOFSTEP]\nobtain \u27e8hv\u27e9 := ContractibleSpace.hequiv_unit X\n[GOAL]\ncase intro\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : ContractibleSpace X\nhv : X \u2243\u2095 Unit\n\u22a2 Nullhomotopic (ContinuousMap.id X)\n[PROOFSTEP]\nuse hv.invFun ()\n[GOAL]\ncase h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : ContractibleSpace X\nhv : X \u2243\u2095 Unit\n\u22a2 Homotopic (ContinuousMap.id X) (const X (\u2191hv.invFun ()))\n[PROOFSTEP]\nconvert hv.left_inv.symm\n[GOAL]\nY : Type u_1\ninst\u271d : TopologicalSpace Y\n\u22a2 ContractibleSpace Y \u2194 Nullhomotopic (ContinuousMap.id Y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nY : Type u_1\ninst\u271d : TopologicalSpace Y\n\u22a2 ContractibleSpace Y \u2192 Nullhomotopic (ContinuousMap.id Y)\n[PROOFSTEP]\nintro\n[GOAL]\ncase mp\nY : Type u_1\ninst\u271d : TopologicalSpace Y\na\u271d : ContractibleSpace Y\n\u22a2 Nullhomotopic (ContinuousMap.id Y)\n[PROOFSTEP]\napply id_nullhomotopic\n[GOAL]\ncase mpr\nY : Type u_1\ninst\u271d : TopologicalSpace Y\n\u22a2 Nullhomotopic (ContinuousMap.id Y) \u2192 ContractibleSpace Y\n[PROOFSTEP]\nrintro \u27e8p, h\u27e9\n[GOAL]\ncase mpr.intro\nY : Type u_1\ninst\u271d : TopologicalSpace Y\np : Y\nh : Homotopic (ContinuousMap.id Y) (const Y p)\n\u22a2 ContractibleSpace Y\n[PROOFSTEP]\nrefine\n  {\n    hequiv_unit' :=\n      \u27e8{  toFun := ContinuousMap.const _ ()\n          invFun := ContinuousMap.const _ p\n          left_inv := ?_\n          right_inv := ?_ }\u27e9 }\n[GOAL]\ncase mpr.intro.refine_1\nY : Type u_1\ninst\u271d : TopologicalSpace Y\np : Y\nh : Homotopic (ContinuousMap.id Y) (const Y p)\n\u22a2 Homotopic (comp (const Unit p) (const Y ())) (ContinuousMap.id Y)\n[PROOFSTEP]\nexact h.symm\n[GOAL]\ncase mpr.intro.refine_2\nY : Type u_1\ninst\u271d : TopologicalSpace Y\np : Y\nh : Homotopic (ContinuousMap.id Y) (const Y p)\n\u22a2 Homotopic (comp (const Y ()) (const Unit p)) (ContinuousMap.id Unit)\n[PROOFSTEP]\nconvert Homotopic.refl (ContinuousMap.id Unit)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\n\u22a2 PathConnectedSpace X\n[PROOFSTEP]\nobtain \u27e8p, \u27e8h\u27e9\u27e9 := id_nullhomotopic X\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\np : X\nh : Homotopy (ContinuousMap.id X) (const X p)\n\u22a2 PathConnectedSpace X\n[PROOFSTEP]\nhave : \u2200 x, Joined p x := fun x => \u27e8(h.evalAt x).symm\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\np : X\nh : Homotopy (ContinuousMap.id X) (const X p)\nthis : \u2200 (x : X), Joined p x\n\u22a2 PathConnectedSpace X\n[PROOFSTEP]\nrw [pathConnectedSpace_iff_eq]\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\np : X\nh : Homotopy (ContinuousMap.id X) (const X p)\nthis : \u2200 (x : X), Joined p x\n\u22a2 \u2203 x, pathComponent x = Set.univ\n[PROOFSTEP]\nuse p\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\np : X\nh : Homotopy (ContinuousMap.id X) (const X p)\nthis : \u2200 (x : X), Joined p x\n\u22a2 pathComponent p = Set.univ\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : ContractibleSpace X\np : X\nh : Homotopy (ContinuousMap.id X) (const X p)\nthis : \u2200 (x : X), Joined p x\nx\u271d : X\n\u22a2 x\u271d \u2208 pathComponent p \u2194 x\u271d \u2208 Set.univ\n[PROOFSTEP]\ntauto\n", "meta": {"mathlib_filename": "Mathlib.Topology.Homotopy.Contractible", "llama_tokens": 2360, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257126, "lm_q2_score": 0.6723317123102955, "lm_q1q2_score": 0.5244139553408433}}
{"text": "[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na b c : N \u22ca[\u03c6] G\n\u22a2 (a * b * c).left = (a * (b * c)).left\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na b c : N \u22ca[\u03c6] G\n\u22a2 (a * b * c).right = (a * (b * c)).right\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na : N \u22ca[\u03c6] G\n\u22a2 (1 * a).left = a.left\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na : N \u22ca[\u03c6] G\n\u22a2 (a * 1).left = a.left\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na : N \u22ca[\u03c6] G\n\u22a2 (a\u207b\u00b9 * a).left = 1.left\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\na : N \u22ca[\u03c6] G\n\u22a2 (a\u207b\u00b9 * a).right = 1.right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\n\u22a2 \u2200 (x y : N),\n    OneHom.toFun\n        { toFun := fun n => { left := n, right := 1 },\n          map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n        (x * y) =\n      OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          x *\n        OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          y\n[PROOFSTEP]\nintros\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : N\n\u22a2 OneHom.toFun\n      { toFun := fun n => { left := n, right := 1 },\n        map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n      (x\u271d * y\u271d) =\n    OneHom.toFun\n        { toFun := fun n => { left := n, right := 1 },\n          map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n        x\u271d *\n      OneHom.toFun\n        { toFun := fun n => { left := n, right := 1 },\n          map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n        y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : N\n\u22a2 (OneHom.toFun\n        { toFun := fun n => { left := n, right := 1 },\n          map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n        (x\u271d * y\u271d)).left =\n    (OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          x\u271d *\n        OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          y\u271d).left\n[PROOFSTEP]\nsimp only [mul_left, map_one, MulAut.one_apply, mul_right, mul_one]\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : N\n\u22a2 (OneHom.toFun\n        { toFun := fun n => { left := n, right := 1 },\n          map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n        (x\u271d * y\u271d)).right =\n    (OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          x\u271d *\n        OneHom.toFun\n          { toFun := fun n => { left := n, right := 1 },\n            map_one' := (_ : (fun n => { left := n, right := 1 }) 1 = (fun n => { left := n, right := 1 }) 1) }\n          y\u271d).right\n[PROOFSTEP]\nsimp only [mul_left, map_one, MulAut.one_apply, mul_right, mul_one]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\n\u22a2 \u2200 (x y : G),\n    OneHom.toFun\n        { toFun := fun g => { left := 1, right := g },\n          map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n        (x * y) =\n      OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          x *\n        OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          y\n[PROOFSTEP]\nintros\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : G\n\u22a2 OneHom.toFun\n      { toFun := fun g => { left := 1, right := g },\n        map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n      (x\u271d * y\u271d) =\n    OneHom.toFun\n        { toFun := fun g => { left := 1, right := g },\n          map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n        x\u271d *\n      OneHom.toFun\n        { toFun := fun g => { left := 1, right := g },\n          map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n        y\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : G\n\u22a2 (OneHom.toFun\n        { toFun := fun g => { left := 1, right := g },\n          map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n        (x\u271d * y\u271d)).left =\n    (OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          x\u271d *\n        OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          y\u271d).left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d y\u271d : G\n\u22a2 (OneHom.toFun\n        { toFun := fun g => { left := 1, right := g },\n          map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n        (x\u271d * y\u271d)).right =\n    (OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          x\u271d *\n        OneHom.toFun\n          { toFun := fun g => { left := 1, right := g },\n            map_one' := (_ : (fun g => { left := 1, right := g }) 1 = (fun g => { left := 1, right := g }) 1) }\n          y\u271d).right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 \u2191inl (\u2191(\u2191\u03c6 g) n) = \u2191inr g * \u2191inl n * \u2191inr g\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 (\u2191inl (\u2191(\u2191\u03c6 g) n)).left = (\u2191inr g * \u2191inl n * \u2191inr g\u207b\u00b9).left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 (\u2191inl (\u2191(\u2191\u03c6 g) n)).right = (\u2191inr g * \u2191inl n * \u2191inr g\u207b\u00b9).right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 \u2191inl (\u2191(\u2191\u03c6 g)\u207b\u00b9 n) = \u2191inr g\u207b\u00b9 * \u2191inl n * \u2191inr g\n[PROOFSTEP]\nrw [\u2190 MonoidHom.map_inv, inl_aut, inv_inv]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 { left := n, right := g } = \u2191inl n * \u2191inr g\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 { left := n, right := g }.left = (\u2191inl n * \u2191inr g).left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\nn : N\n\u22a2 { left := n, right := g }.right = (\u2191inl n * \u2191inr g).right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\n\u22a2 \u2191inl x.left * \u2191inr x.right = x\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\n\u22a2 (\u2191inl x.left * \u2191inr x.right).left = x.left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\n\u22a2 (\u2191inl x.left * \u2191inr x.right).right = x.right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\n\u22a2 MonoidHom.comp rightHom inl = 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d : N\n\u22a2 \u2191(MonoidHom.comp rightHom inl) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nsimp [rightHom]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\n\u22a2 MonoidHom.comp rightHom inr = MonoidHom.id G\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp rightHom inr) x\u271d = \u2191(MonoidHom.id G) x\u271d\n[PROOFSTEP]\nsimp [rightHom]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nn : N\n\u22a2 \u2191rightHom (\u2191inl n) = 1\n[PROOFSTEP]\nsimp [rightHom]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\ng : G\n\u22a2 \u2191rightHom (\u2191inr g) = g\n[PROOFSTEP]\nsimp [rightHom]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx\u271d : N \u22ca[\u03c6] G\n\u22a2 x\u271d \u2208 MonoidHom.range inl \u2192 x\u271d \u2208 MonoidHom.ker rightHom\n[PROOFSTEP]\nsimp (config := { contextual := true }) [MonoidHom.mem_ker, eq_comm]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\nhx : x \u2208 MonoidHom.ker rightHom\n\u22a2 \u2191inl x.left = x\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\nhx : x \u2208 MonoidHom.ker rightHom\n\u22a2 (\u2191inl x.left).left = x.left\n[PROOFSTEP]\nsimp_all [MonoidHom.mem_ker]\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nx : N \u22ca[\u03c6] G\nhx : x \u2208 MonoidHom.ker rightHom\n\u22a2 (\u2191inl x.left).right = x.right\n[PROOFSTEP]\nsimp_all [MonoidHom.mem_ker]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081\u271d : N \u2192* H\nf\u2082\u271d : G \u2192* H\nh\u271d :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081\u271d (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082\u271d g))) f\u2081\u271d\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\n\u22a2 (fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081\u271d : N \u2192* H\nf\u2082\u271d : G \u2192* H\nh\u271d :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081\u271d (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082\u271d g))) f\u2081\u271d\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\na b : N \u22ca[\u03c6] G\n\u22a2 OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } (a * b) =\n    OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } a *\n      OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } b\n[PROOFSTEP]\nhave := fun n g \u21a6 FunLike.ext_iff.1 (h n) g\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081\u271d : N \u2192* H\nf\u2082\u271d : G \u2192* H\nh\u271d :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081\u271d (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082\u271d g))) f\u2081\u271d\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\na b : N \u22ca[\u03c6] G\nthis :\n  \u2200 (n : G) (g : N),\n    \u2191(MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 n))) g =\n      \u2191(MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 n))) f\u2081) g\n\u22a2 OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } (a * b) =\n    OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } a *\n      OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } b\n[PROOFSTEP]\nsimp only [MulAut.conj_apply, MonoidHom.comp_apply, MulEquiv.coe_toMonoidHom] at this \n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081\u271d : N \u2192* H\nf\u2082\u271d : G \u2192* H\nh\u271d :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081\u271d (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082\u271d g))) f\u2081\u271d\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\na b : N \u22ca[\u03c6] G\nthis : \u2200 (n : G) (g : N), \u2191f\u2081 (\u2191(\u2191\u03c6 n) g) = \u2191f\u2082 n * \u2191f\u2081 g * (\u2191f\u2082 n)\u207b\u00b9\n\u22a2 OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } (a * b) =\n    OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } a *\n      OneHom.toFun { toFun := fun a => \u2191f\u2081 a.left * \u2191f\u2082 a.right, map_one' := (_ : \u2191f\u2081 1 * \u2191f\u2082 1 = 1) } b\n[PROOFSTEP]\nsimp only [mul_left, mul_right, map_mul, this, mul_assoc, inv_mul_cancel_left]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nn : N\n\u22a2 \u2191(lift f\u2081 f\u2082 h) (\u2191inl n) = \u2191f\u2081 n\n[PROOFSTEP]\nsimp [lift]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\n\u22a2 MonoidHom.comp (lift f\u2081 f\u2082 h) inl = f\u2081\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nx\u271d : N\n\u22a2 \u2191(MonoidHom.comp (lift f\u2081 f\u2082 h) inl) x\u271d = \u2191f\u2081 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\ng : G\n\u22a2 \u2191(lift f\u2081 f\u2082 h) (\u2191inr g) = \u2191f\u2082 g\n[PROOFSTEP]\nsimp [lift]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\n\u22a2 MonoidHom.comp (lift f\u2081 f\u2082 h) inr = f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp (lift f\u2081 f\u2082 h) inr) x\u271d = \u2191f\u2082 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nF : N \u22ca[\u03c6] G \u2192* H\nx\u271d : G\n\u22a2 MonoidHom.comp (MonoidHom.comp F inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x\u271d)) =\n    MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp F inr) x\u271d))) (MonoidHom.comp F inl)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nF : N \u22ca[\u03c6] G \u2192* H\nx\u271d\u00b9 : G\nx\u271d : N\n\u22a2 \u2191(MonoidHom.comp (MonoidHom.comp F inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x\u271d\u00b9))) x\u271d =\n    \u2191(MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp F inr) x\u271d\u00b9))) (MonoidHom.comp F inl)) x\u271d\n[PROOFSTEP]\nsimp [inl_aut]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nF : N \u22ca[\u03c6] G \u2192* H\n\u22a2 F =\n    lift (MonoidHom.comp F inl) (MonoidHom.comp F inr)\n      (_ :\n        \u2200 (x : G),\n          MonoidHom.comp (MonoidHom.comp F inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x)) =\n            MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp F inr) x))) (MonoidHom.comp F inl))\n[PROOFSTEP]\nrw [FunLike.ext_iff]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nF : N \u22ca[\u03c6] G \u2192* H\n\u22a2 \u2200 (x : N \u22ca[\u03c6] G),\n    \u2191F x =\n      \u2191(lift (MonoidHom.comp F inl) (MonoidHom.comp F inr)\n            (_ :\n              \u2200 (x : G),\n                MonoidHom.comp (MonoidHom.comp F inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x)) =\n                  MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp F inr) x)))\n                    (MonoidHom.comp F inl)))\n        x\n[PROOFSTEP]\nsimp only [lift, MonoidHom.comp_apply, MonoidHom.coe_mk, OneHom.coe_mk, \u2190 map_mul, inl_left_mul_inr_right, forall_const]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nf g : N \u22ca[\u03c6] G \u2192* H\nhl : MonoidHom.comp f inl = MonoidHom.comp g inl\nhr : MonoidHom.comp f inr = MonoidHom.comp g inr\n\u22a2 f = g\n[PROOFSTEP]\nrw [lift_unique f, lift_unique g]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nf\u2081 : N \u2192* H\nf\u2082 : G \u2192* H\nh :\n  \u2200 (g : G),\n    MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191f\u2082 g))) f\u2081\nf g : N \u22ca[\u03c6] G \u2192* H\nhl : MonoidHom.comp f inl = MonoidHom.comp g inl\nhr : MonoidHom.comp f inr = MonoidHom.comp g inr\n\u22a2 lift (MonoidHom.comp f inl) (MonoidHom.comp f inr)\n      (_ :\n        \u2200 (x : G),\n          MonoidHom.comp (MonoidHom.comp f inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x)) =\n            MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp f inr) x))) (MonoidHom.comp f inl)) =\n    lift (MonoidHom.comp g inl) (MonoidHom.comp g inr)\n      (_ :\n        \u2200 (x : G),\n          MonoidHom.comp (MonoidHom.comp g inl) (MulEquiv.toMonoidHom (\u2191\u03c6 x)) =\n            MonoidHom.comp (MulEquiv.toMonoidHom (\u2191MulAut.conj (\u2191(MonoidHom.comp g inr) x))) (MonoidHom.comp g inl))\n[PROOFSTEP]\nsimp only [*]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\n\u22a2 (fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right }) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nx y : N \u22ca[\u03c6] G\n\u22a2 OneHom.toFun\n      { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n        map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        y\n[PROOFSTEP]\nreplace h := FunLike.ext_iff.1 (h x.right) y.left\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nx y : N \u22ca[\u03c6] G\nh :\n  \u2191(MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 x.right))) y.left =\n    \u2191(MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 x.right))) f\u2081) y.left\n\u22a2 OneHom.toFun\n      { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n        map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        y\n[PROOFSTEP]\next\n[GOAL]\ncase left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nx y : N \u22ca[\u03c6] G\nh :\n  \u2191(MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 x.right))) y.left =\n    \u2191(MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 x.right))) f\u2081) y.left\n\u22a2 (OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        (x * y)).left =\n    (OneHom.toFun\n          { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n            map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n          x *\n        OneHom.toFun\n          { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n            map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n          y).left\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nx y : N \u22ca[\u03c6] G\nh :\n  \u2191(MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 x.right))) y.left =\n    \u2191(MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 x.right))) f\u2081) y.left\n\u22a2 (OneHom.toFun\n        { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n          map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n        (x * y)).right =\n    (OneHom.toFun\n          { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n            map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n          x *\n        OneHom.toFun\n          { toFun := fun x => { left := \u2191f\u2081 x.left, right := \u2191f\u2082 x.right },\n            map_one' := (_ : { left := \u2191f\u2081 1, right := \u2191f\u2082 1 } = 1) }\n          y).right\n[PROOFSTEP]\nsimp_all\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nn : N\n\u22a2 \u2191(map f\u2081 f\u2082 h) (\u2191inl n) = \u2191inl (\u2191f\u2081 n)\n[PROOFSTEP]\nsimp [map]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\n\u22a2 MonoidHom.comp (map f\u2081 f\u2082 h) inl = MonoidHom.comp inl f\u2081\n[PROOFSTEP]\next\n[GOAL]\ncase h.left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nx\u271d : N\n\u22a2 (\u2191(MonoidHom.comp (map f\u2081 f\u2082 h) inl) x\u271d).left = (\u2191(MonoidHom.comp inl f\u2081) x\u271d).left\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nx\u271d : N\n\u22a2 (\u2191(MonoidHom.comp (map f\u2081 f\u2082 h) inl) x\u271d).right = (\u2191(MonoidHom.comp inl f\u2081) x\u271d).right\n[PROOFSTEP]\nsimp\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\ng : G\n\u22a2 \u2191(map f\u2081 f\u2082 h) (\u2191inr g) = \u2191inr (\u2191f\u2082 g)\n[PROOFSTEP]\nsimp [map]\n[GOAL]\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\n\u22a2 MonoidHom.comp (map f\u2081 f\u2082 h) inr = MonoidHom.comp inr f\u2082\n[PROOFSTEP]\next\n[GOAL]\ncase h.left\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nx\u271d : G\n\u22a2 (\u2191(MonoidHom.comp (map f\u2081 f\u2082 h) inr) x\u271d).left = (\u2191(MonoidHom.comp inr f\u2082) x\u271d).left\n[PROOFSTEP]\nsimp [map]\n[GOAL]\ncase h.right\nN : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Group N\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* MulAut N\nN\u2081 : Type u_4\nG\u2081 : Type u_5\ninst\u271d\u00b9 : Group N\u2081\ninst\u271d : Group G\u2081\n\u03c6\u2081 : G\u2081 \u2192* MulAut N\u2081\nf\u2081 : N \u2192* N\u2081\nf\u2082 : G \u2192* G\u2081\nh : \u2200 (g : G), MonoidHom.comp f\u2081 (MulEquiv.toMonoidHom (\u2191\u03c6 g)) = MonoidHom.comp (MulEquiv.toMonoidHom (\u2191\u03c6\u2081 (\u2191f\u2082 g))) f\u2081\nx\u271d : G\n\u22a2 (\u2191(MonoidHom.comp (map f\u2081 f\u2082 h) inr) x\u271d).right = (\u2191(MonoidHom.comp inr f\u2082) x\u271d).right\n[PROOFSTEP]\nsimp [map]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.SemidirectProduct", "llama_tokens": 14341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959545, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.524368966365726}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\n\u22a2 coeff (trinomial k m n u v w) n = w\n[PROOFSTEP]\nrw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,\n  if_neg (hkm.trans hmn).ne', if_neg hmn.ne', if_pos rfl, zero_add, zero_add]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\n\u22a2 coeff (trinomial k m n u v w) m = v\n[PROOFSTEP]\nrw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_neg hkm.ne',\n  if_pos rfl, if_neg hmn.ne, zero_add, add_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\n\u22a2 coeff (trinomial k m n u v w) k = u\n[PROOFSTEP]\nrw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow, if_pos rfl,\n  if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\n\u22a2 natDegree (trinomial k m n u v w) = n\n[PROOFSTEP]\nrefine'\n  natDegree_eq_of_degree_eq_some\n    ((Finset.sup_le fun i h => _).antisymm <| le_degree_of_ne_zero <| by rwa [trinomial_leading_coeff' hkm hmn])\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\n\u22a2 coeff (trinomial k m n u v w) n \u2260 0\n[PROOFSTEP]\nrwa [trinomial_leading_coeff' hkm hmn]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\ni : \u2115\nh : i \u2208 support (trinomial k m n u v w)\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nreplace h := support_trinomial' k m n u v w h\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\ni : \u2115\nh : i \u2208 {k, m, n}\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nrw [mem_insert, mem_insert, mem_singleton] at h \n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\ni : \u2115\nh : i = k \u2228 i = m \u2228 i = n\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nrcases h with (rfl | rfl | rfl)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : Semiring R\nm n : \u2115\nu v w : R\nhmn : m < n\nhw : w \u2260 0\ni : \u2115\nhkm : i < m\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nexact WithBot.coe_le_coe.mpr (hkm.trans hmn).le\n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : Semiring R\nk n : \u2115\nu v w : R\nhw : w \u2260 0\ni : \u2115\nhkm : k < i\nhmn : i < n\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nexact WithBot.coe_le_coe.mpr hmn.le\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d : Semiring R\nk m : \u2115\nu v w : R\nhkm : k < m\nhw : w \u2260 0\ni : \u2115\nhmn : m < i\n\u22a2 \u2191i \u2264 \u2191i\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\n\u22a2 natTrailingDegree (trinomial k m n u v w) = k\n[PROOFSTEP]\nrefine'\n  natTrailingDegree_eq_of_trailingDegree_eq_some\n    ((Finset.le_inf fun i h => _).antisymm <|\n        le_trailingDegree_of_ne_zero <| by rwa [trinomial_trailing_coeff' hkm hmn]).symm\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\n\u22a2 coeff (trinomial k m n u v w) k \u2260 0\n[PROOFSTEP]\nrwa [trinomial_trailing_coeff' hkm hmn]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\ni : \u2115\nh : i \u2208 support (trinomial k m n u v w)\n\u22a2 \u2191k \u2264 \u2191i\n[PROOFSTEP]\nreplace h := support_trinomial' k m n u v w h\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\ni : \u2115\nh : i \u2208 {k, m, n}\n\u22a2 \u2191k \u2264 \u2191i\n[PROOFSTEP]\nrw [mem_insert, mem_insert, mem_singleton] at h \n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\ni : \u2115\nh : i = k \u2228 i = m \u2228 i = n\n\u22a2 \u2191k \u2264 \u2191i\n[PROOFSTEP]\nrcases h with (rfl | rfl | rfl)\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : Semiring R\nm n : \u2115\nu v w : R\nhmn : m < n\nhu : u \u2260 0\ni : \u2115\nhkm : i < m\n\u22a2 \u2191i \u2264 \u2191i\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase inr.inl\nR : Type u_1\ninst\u271d : Semiring R\nk n : \u2115\nu v w : R\nhu : u \u2260 0\ni : \u2115\nhkm : k < i\nhmn : i < n\n\u22a2 \u2191k \u2264 \u2191i\n[PROOFSTEP]\nexact WithTop.coe_le_coe.mpr hkm.le\n[GOAL]\ncase inr.inr\nR : Type u_1\ninst\u271d : Semiring R\nk m : \u2115\nu v w : R\nhkm : k < m\nhu : u \u2260 0\ni : \u2115\nhmn : m < i\n\u22a2 \u2191k \u2264 \u2191i\n[PROOFSTEP]\nexact WithTop.coe_le_coe.mpr (hkm.trans hmn).le\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhw : w \u2260 0\n\u22a2 leadingCoeff (trinomial k m n u v w) = w\n[PROOFSTEP]\nrw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\n\u22a2 trailingCoeff (trinomial k m n u v w) = u\n[PROOFSTEP]\nrw [trailingCoeff, trinomial_natTrailingDegree hkm hmn hu, trinomial_trailing_coeff' hkm hmn]\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\n\u22a2 Monic (trinomial k m n u v 1)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\n\u271d : Nontrivial R\n\u22a2 Monic (trinomial k m n u v 1)\n[PROOFSTEP]\nexact trinomial_leadingCoeff hkm hmn one_ne_zero\n[GOAL]\nR : Type u_1\ninst\u271d : Semiring R\nk m n : \u2115\nu v w : R\nhkm : k < m\nhmn : m < n\nhu : u \u2260 0\nhw : w \u2260 0\n\u22a2 mirror (trinomial k m n u v w) = trinomial k (n - m + k) n w v u\n[PROOFSTEP]\nrw [mirror, trinomial_natTrailingDegree hkm hmn hu, reverse, trinomial_natDegree hkm hmn hw, trinomial_def, reflect_add,\n  reflect_add, reflect_C_mul_X_pow, reflect_C_mul_X_pow, reflect_C_mul_X_pow, revAt_le (hkm.trans hmn).le,\n  revAt_le hmn.le, revAt_le le_rfl, add_mul, add_mul, mul_assoc, mul_assoc, mul_assoc, \u2190 pow_add, \u2190 pow_add, \u2190 pow_add,\n  Nat.sub_add_cancel (hkm.trans hmn).le, Nat.sub_self, zero_add, add_comm, add_comm (C u * X ^ n), \u2190 add_assoc, \u2190\n  trinomial_def]\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\n\u22a2 \u00acIsUnit p\n[PROOFSTEP]\nobtain \u27e8k, m, n, hkm, hmn, u, v, w, rfl\u27e9 := hp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\n\u22a2 \u00acIsUnit (trinomial k m n \u2191u \u2191v \u2191w)\n[PROOFSTEP]\nexact fun h =>\n  ne_zero_of_lt hmn\n    ((trinomial_natDegree hkm hmn w.ne_zero).symm.trans (natDegree_eq_of_degree_eq_some (degree_eq_zero_of_isUnit h)))\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\n\u22a2 card (support p) = 3\n[PROOFSTEP]\nobtain \u27e8k, m, n, hkm, hmn, u, v, w, rfl\u27e9 := hp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\n\u22a2 card (support (trinomial k m n \u2191u \u2191v \u2191w)) = 3\n[PROOFSTEP]\nexact card_support_trinomial hkm hmn u.ne_zero v.ne_zero w.ne_zero\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\n\u22a2 p \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nq : \u2124[X]\nhp : IsUnitTrinomial 0\n\u22a2 False\n[PROOFSTEP]\nexact Nat.zero_ne_bit1 1 hp.card_support_eq_three\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\nk : \u2115\nhk : k \u2208 support p\n\u22a2 IsUnit (coeff p k)\n[PROOFSTEP]\nobtain \u27e8k, m, n, hkm, hmn, u, v, w, rfl\u27e9 := hp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk\u271d k m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhk : k\u271d \u2208 support (trinomial k m n \u2191u \u2191v \u2191w)\n\u22a2 IsUnit (coeff (trinomial k m n \u2191u \u2191v \u2191w) k\u271d)\n[PROOFSTEP]\nhave := support_trinomial' k m n (u : \u2124) v w hk\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk\u271d k m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhk : k\u271d \u2208 support (trinomial k m n \u2191u \u2191v \u2191w)\nthis : k\u271d \u2208 {k, m, n}\n\u22a2 IsUnit (coeff (trinomial k m n \u2191u \u2191v \u2191w) k\u271d)\n[PROOFSTEP]\nrw [mem_insert, mem_insert, mem_singleton] at this \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk\u271d k m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhk : k\u271d \u2208 support (trinomial k m n \u2191u \u2191v \u2191w)\nthis : k\u271d = k \u2228 k\u271d = m \u2228 k\u271d = n\n\u22a2 IsUnit (coeff (trinomial k m n \u2191u \u2191v \u2191w) k\u271d)\n[PROOFSTEP]\nrcases this with (rfl | rfl | rfl)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.inl\nq : \u2124[X]\nk m n : \u2115\nhmn : m < n\nu v w : \u2124\u02e3\nhkm : k < m\nhk : k \u2208 support (trinomial k m n \u2191u \u2191v \u2191w)\n\u22a2 IsUnit (coeff (trinomial k m n \u2191u \u2191v \u2191w) k)\n[PROOFSTEP]\nrefine' \u27e8u, by rw [trinomial_trailing_coeff' hkm hmn]\u27e9\n[GOAL]\nq : \u2124[X]\nk m n : \u2115\nhmn : m < n\nu v w : \u2124\u02e3\nhkm : k < m\nhk : k \u2208 support (trinomial k m n \u2191u \u2191v \u2191w)\n\u22a2 \u2191u = coeff (trinomial k m n \u2191u \u2191v \u2191w) k\n[PROOFSTEP]\nrw [trinomial_trailing_coeff' hkm hmn]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.inr.inl\nq : \u2124[X]\nk\u271d k n : \u2115\nu v w : \u2124\u02e3\nhkm : k < k\u271d\nhmn : k\u271d < n\nhk : k\u271d \u2208 support (trinomial k k\u271d n \u2191u \u2191v \u2191w)\n\u22a2 IsUnit (coeff (trinomial k k\u271d n \u2191u \u2191v \u2191w) k\u271d)\n[PROOFSTEP]\nrefine' \u27e8v, by rw [trinomial_middle_coeff hkm hmn]\u27e9\n[GOAL]\nq : \u2124[X]\nk\u271d k n : \u2115\nu v w : \u2124\u02e3\nhkm : k < k\u271d\nhmn : k\u271d < n\nhk : k\u271d \u2208 support (trinomial k k\u271d n \u2191u \u2191v \u2191w)\n\u22a2 \u2191v = coeff (trinomial k k\u271d n \u2191u \u2191v \u2191w) k\u271d\n[PROOFSTEP]\nrw [trinomial_middle_coeff hkm hmn]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.inr.inr\nq : \u2124[X]\nk\u271d k m : \u2115\nhkm : k < m\nu v w : \u2124\u02e3\nhmn : m < k\u271d\nhk : k\u271d \u2208 support (trinomial k m k\u271d \u2191u \u2191v \u2191w)\n\u22a2 IsUnit (coeff (trinomial k m k\u271d \u2191u \u2191v \u2191w) k\u271d)\n[PROOFSTEP]\nrefine' \u27e8w, by rw [trinomial_leading_coeff' hkm hmn]\u27e9\n[GOAL]\nq : \u2124[X]\nk\u271d k m : \u2115\nhkm : k < m\nu v w : \u2124\u02e3\nhmn : m < k\u271d\nhk : k\u271d \u2208 support (trinomial k m k\u271d \u2191u \u2191v \u2191w)\n\u22a2 \u2191w = coeff (trinomial k m k\u271d \u2191u \u2191v \u2191w) k\u271d\n[PROOFSTEP]\nrw [trinomial_leading_coeff' hkm hmn]\n[GOAL]\np q : \u2124[X]\n\u22a2 IsUnitTrinomial p \u2194 card (support p) = 3 \u2227 \u2200 (k : \u2115), k \u2208 support p \u2192 IsUnit (coeff p k)\n[PROOFSTEP]\nrefine' \u27e8fun hp => \u27e8hp.card_support_eq_three, fun k => hp.coeff_isUnit\u27e9, fun hp => _\u27e9\n[GOAL]\np q : \u2124[X]\nhp : card (support p) = 3 \u2227 \u2200 (k : \u2115), k \u2208 support p \u2192 IsUnit (coeff p k)\n\u22a2 IsUnitTrinomial p\n[PROOFSTEP]\nobtain \u27e8k, m, n, hkm, hmn, x, y, z, hx, hy, hz, rfl\u27e9 := card_support_eq_three.mp hp.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhx : x \u2260 0\nhy : y \u2260 0\nhz : z \u2260 0\nhp :\n  card (support (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)) = 3 \u2227\n    \u2200 (k_1 : \u2115),\n      k_1 \u2208 support (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) \u2192\n        IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nrw [support_trinomial hkm hmn hx hy hz] at hp \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhx : x \u2260 0\nhy : y \u2260 0\nhz : z \u2260 0\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nreplace hx := hp.2 k (mem_insert_self k { m, n })\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhy : y \u2260 0\nhz : z \u2260 0\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nreplace hy := hp.2 m (mem_insert_of_mem (mem_insert_self m { n }))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhz : z \u2260 0\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k)\nhy : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) m)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nreplace hz := hp.2 n (mem_insert_of_mem (mem_insert_of_mem (mem_singleton_self n)))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k)\nhy : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) m)\nhz : IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) n)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nsimp_rw [coeff_add, coeff_C_mul, coeff_X_pow_self, mul_one, coeff_X_pow] at hx hy hz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit ((x + y * if k = m then 1 else 0) + z * if k = n then 1 else 0)\nhy : IsUnit ((x * if m = k then 1 else 0) + y + z * if m = n then 1 else 0)\nhz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nrw [if_neg hkm.ne, if_neg (hkm.trans hmn).ne] at hx \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (x + y * 0 + z * 0)\nhy : IsUnit ((x * if m = k then 1 else 0) + y + z * if m = n then 1 else 0)\nhz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nrw [if_neg hkm.ne', if_neg hmn.ne] at hy \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (x + y * 0 + z * 0)\nhy : IsUnit (x * 0 + y + z * 0)\nhz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nrw [if_neg (hkm.trans hmn).ne', if_neg hmn.ne'] at hz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhx : IsUnit (x + y * 0 + z * 0)\nhy : IsUnit (x * 0 + y + z * 0)\nhz : IsUnit (x * 0 + y * 0 + z)\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nsimp_rw [mul_zero, zero_add, add_zero] at hx hy hz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nx y z : \u2124\nhp : card {k, m, n} = 3 \u2227 \u2200 (k_1 : \u2115), k_1 \u2208 {k, m, n} \u2192 IsUnit (coeff (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n) k_1)\nhz : IsUnit z\nhx : IsUnit x\nhy : IsUnit y\n\u22a2 IsUnitTrinomial (\u2191C x * X ^ k + \u2191C y * X ^ m + \u2191C z * X ^ n)\n[PROOFSTEP]\nexact \u27e8k, m, n, hkm, hmn, hx.unit, hy.unit, hz.unit, rfl\u27e9\n[GOAL]\np q : \u2124[X]\n\u22a2 IsUnitTrinomial p \u2194 coeff (p * mirror p) ((natDegree (p * mirror p) + natTrailingDegree (p * mirror p)) / 2) = 3\n[PROOFSTEP]\nrw [natDegree_mul_mirror, natTrailingDegree_mul_mirror, \u2190 mul_add, Nat.mul_div_right _ zero_lt_two, coeff_mul_mirror]\n[GOAL]\np q : \u2124[X]\n\u22a2 IsUnitTrinomial p \u2194 (sum p fun n x => x ^ 2) = 3\n[PROOFSTEP]\nrefine' \u27e8_, fun hp => _\u27e9\n[GOAL]\ncase refine'_1\np q : \u2124[X]\n\u22a2 IsUnitTrinomial p \u2192 (sum p fun n x => x ^ 2) = 3\n[PROOFSTEP]\nrintro \u27e8k, m, n, hkm, hmn, u, v, w, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\n\u22a2 (sum (trinomial k m n \u2191u \u2191v \u2191w) fun n x => x ^ 2) = 3\n[PROOFSTEP]\nrw [sum_def, trinomial_support hkm hmn u.ne_zero v.ne_zero w.ne_zero,\n  sum_insert (mt mem_insert.mp (not_or_of_not hkm.ne (mt mem_singleton.mp (hkm.trans hmn).ne))),\n  sum_insert (mt mem_singleton.mp hmn.ne), sum_singleton, trinomial_leading_coeff' hkm hmn,\n  trinomial_middle_coeff hkm hmn, trinomial_trailing_coeff' hkm hmn]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.intro.intro.intro\nq : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\n\u22a2 \u2191u ^ 2 + (\u2191v ^ 2 + \u2191w ^ 2) = 3\n[PROOFSTEP]\nsimp_rw [\u2190 Units.val_pow_eq_pow_val, Int.units_sq]\n[GOAL]\ncase refine'_2\np q : \u2124[X]\nhp : (sum p fun n x => x ^ 2) = 3\n\u22a2 IsUnitTrinomial p\n[PROOFSTEP]\nhave key : \u2200 k \u2208 p.support, p.coeff k ^ 2 = 1 := fun k hk =>\n  Int.sq_eq_one_of_sq_le_three ((single_le_sum (fun k _ => sq_nonneg (p.coeff k)) hk).trans hp.le)\n    (mem_support_iff.mp hk)\n[GOAL]\ncase refine'_2\np q : \u2124[X]\nhp : (sum p fun n x => x ^ 2) = 3\nkey : \u2200 (k : \u2115), k \u2208 support p \u2192 coeff p k ^ 2 = 1\n\u22a2 IsUnitTrinomial p\n[PROOFSTEP]\nrefine' isUnitTrinomial_iff.mpr \u27e8_, fun k hk => isUnit_ofPowEqOne (key k hk) two_ne_zero\u27e9\n[GOAL]\ncase refine'_2\np q : \u2124[X]\nhp : (sum p fun n x => x ^ 2) = 3\nkey : \u2200 (k : \u2115), k \u2208 support p \u2192 coeff p k ^ 2 = 1\n\u22a2 card (support p) = 3\n[PROOFSTEP]\nrw [sum_def, sum_congr rfl key, sum_const, Nat.smul_one_eq_coe] at hp \n[GOAL]\ncase refine'_2\np q : \u2124[X]\nhp : \u2191(card (support p)) = 3\nkey : \u2200 (k : \u2115), k \u2208 support p \u2192 coeff p k ^ 2 = 1\n\u22a2 card (support p) = 3\n[PROOFSTEP]\nexact Nat.cast_injective hp\n[GOAL]\np q : \u2124[X]\nh : p * mirror p = q * mirror q\n\u22a2 IsUnitTrinomial p \u2194 IsUnitTrinomial q\n[PROOFSTEP]\nrw [isUnitTrinomial_iff', isUnitTrinomial_iff', h]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\n\u22a2 \u2191C \u2191v * (\u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n)) =\n    { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) (p * mirror p).toFinsupp }\n[PROOFSTEP]\nhave key : n - m + k < n := by rwa [\u2190 lt_tsub_iff_right, tsub_lt_tsub_iff_left_of_le hmn.le]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\n\u22a2 n - m + k < n\n[PROOFSTEP]\nrwa [\u2190 lt_tsub_iff_right, tsub_lt_tsub_iff_left_of_le hmn.le]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * (\u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n)) =\n    { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) (p * mirror p).toFinsupp }\n[PROOFSTEP]\nrw [hp, trinomial_mirror hkm hmn u.ne_zero w.ne_zero]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * (\u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n)) =\n    {\n      toFinsupp :=\n        Finsupp.filter (Set.Ioo (k + n) (n + n))\n          (trinomial k m n \u2191u \u2191v \u2191w * trinomial k (n - m + k) n \u2191w \u2191v \u2191u).toFinsupp }\n[PROOFSTEP]\nsimp_rw [trinomial_def, C_mul_X_pow_eq_monomial, add_mul, mul_add, monomial_mul_monomial, toFinsupp_add,\n  toFinsupp_monomial]\n  -- Porting note: added next line (less powerful `simp`).\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    {\n      toFinsupp :=\n        Finsupp.filter (Set.Ioo (k + n) (n + n))\n          (Finsupp.single (k + k) (\u2191u * \u2191w) + Finsupp.single (k + (n - m + k)) (\u2191u * \u2191v) +\n                Finsupp.single (k + n) (\u2191u * \u2191u) +\n              (Finsupp.single (m + k) (\u2191v * \u2191w) + Finsupp.single (m + (n - m + k)) (\u2191v * \u2191v) +\n                Finsupp.single (m + n) (\u2191v * \u2191u)) +\n            (Finsupp.single (n + k) (\u2191w * \u2191w) + Finsupp.single (n + (n - m + k)) (\u2191w * \u2191v) +\n              Finsupp.single (n + n) (\u2191w * \u2191u))) }\n[PROOFSTEP]\nrw [Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add,\n  Finsupp.filter_add, Finsupp.filter_add, Finsupp.filter_add]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    {\n      toFinsupp :=\n        Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + k) (\u2191u * \u2191w)) +\n                Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + (n - m + k)) (\u2191u * \u2191v)) +\n              Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (k + n) (\u2191u * \u2191u)) +\n            (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + k) (\u2191v * \u2191w)) +\n                Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + (n - m + k)) (\u2191v * \u2191v)) +\n              Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (m + n) (\u2191v * \u2191u))) +\n          (Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + k) (\u2191w * \u2191w)) +\n              Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + (n - m + k)) (\u2191w * \u2191v)) +\n            Finsupp.filter (Set.Ioo (k + n) (n + n)) (Finsupp.single (n + n) (\u2191w * \u2191u))) }\n[PROOFSTEP]\nrw [Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg,\n  Finsupp.filter_single_of_neg, Finsupp.filter_single_of_neg, Finsupp.filter_single_of_pos,\n  Finsupp.filter_single_of_neg, Finsupp.filter_single_of_pos, Finsupp.filter_single_of_neg]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    {\n      toFinsupp :=\n        0 + 0 + 0 + (0 + 0 + Finsupp.single (m + n) (\u2191v * \u2191u)) + (0 + Finsupp.single (n + (n - m + k)) (\u2191w * \u2191v) + 0) }\n[PROOFSTEP]\nsimp only [add_zero, zero_add, ofFinsupp_add, ofFinsupp_single]\n  -- Porting note: added next two lines (less powerful `simp`).\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    { toFinsupp := Finsupp.single (m + n) (\u2191v * \u2191u) + Finsupp.single (n + (n - m + k)) (\u2191w * \u2191v) }\n[PROOFSTEP]\nrw [ofFinsupp_add]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    { toFinsupp := Finsupp.single (m + n) (\u2191v * \u2191u) } + { toFinsupp := Finsupp.single (n + (n - m + k)) (\u2191w * \u2191v) }\n[PROOFSTEP]\nsimp only [ofFinsupp_single]\n[GOAL]\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u2191C \u2191v * \u2191(monomial (m + n)) \u2191u + \u2191C \u2191v * \u2191(monomial (n - m + k + n)) \u2191w =\n    \u2191(monomial (m + n)) (\u2191v * \u2191u) + \u2191(monomial (n + (n - m + k))) (\u2191w * \u2191v)\n[PROOFSTEP]\nrw [C_mul_monomial, C_mul_monomial, mul_comm (v : \u2124) w, add_comm (n - m + k) n]\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (n + n)\n[PROOFSTEP]\nexact fun h => h.2.ne rfl\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 Set.Ioo (k + n) (n + n) (n + (n - m + k))\n[PROOFSTEP]\nrefine' \u27e8_, add_lt_add_left key n\u27e9\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 k + n < n + (n - m + k)\n[PROOFSTEP]\nrwa [add_comm, add_lt_add_iff_left, lt_add_iff_pos_left, tsub_pos_iff_lt]\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (n + k)\n[PROOFSTEP]\nexact fun h => h.1.ne (add_comm k n)\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 Set.Ioo (k + n) (n + n) (m + n)\n[PROOFSTEP]\nexact \u27e8add_lt_add_right hkm n, add_lt_add_right hmn n\u27e9\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (m + (n - m + k))\n[PROOFSTEP]\nrw [\u2190 add_assoc, add_tsub_cancel_of_le hmn.le, add_comm]\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (n + k) (n + n) (n + k)\n[PROOFSTEP]\nexact fun h => h.1.ne rfl\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (m + k)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\n\u22a2 False\n[PROOFSTEP]\nhave := h.1\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : k + n < m + k\n\u22a2 False\n[PROOFSTEP]\nrw [add_comm, add_lt_add_iff_right] at this \n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\nh : Set.Ioo (k + n) (n + n) (m + k)\nthis : n < m\n\u22a2 False\n[PROOFSTEP]\nexact asymm this hmn\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (k + n)\n[PROOFSTEP]\nexact fun h => h.1.ne rfl\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (k + (n - m + k))\n[PROOFSTEP]\nexact fun h => asymm ((add_lt_add_iff_left k).mp h.1) key\n[GOAL]\ncase h\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nkey : n - m + k < n\n\u22a2 \u00acSet.Ioo (k + n) (n + n) (k + k)\n[PROOFSTEP]\nexact fun h => asymm ((add_lt_add_iff_left k).mp h.1) (hkm.trans hmn)\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nh : p * mirror p = q * mirror q\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nlet f : \u2124[X] \u2192 \u2124[X] := fun p => \u27e8Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp\u27e9\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nh : p * mirror p = q * mirror q\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nreplace h := congr_arg f h\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh : f (p * mirror p) = f (q * mirror q)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nreplace h := (irreducible_aux1 hkm hmn u v w hp).trans h\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh : \u2191C \u2191v * (\u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n)) = f (q * mirror q)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nreplace h := h.trans (irreducible_aux1 hkm' hmn' u v w hq).symm\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh :\n  \u2191C \u2191v * (\u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n)) =\n    \u2191C \u2191v * (\u2191C \u2191u * X ^ (m' + n) + \u2191C \u2191w * X ^ (n - m' + k + n))\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [(isUnit_C.mpr v.isUnit).mul_right_inj] at h \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh : \u2191C \u2191u * X ^ (m + n) + \u2191C \u2191w * X ^ (n - m + k + n) = \u2191C \u2191u * X ^ (m' + n) + \u2191C \u2191w * X ^ (n - m' + k + n)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [binomial_eq_binomial u.ne_zero w.ne_zero] at h \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh :\n  m + n = m' + n \u2227 n - m + k + n = n - m' + k + n \u2228\n    \u2191u = \u2191w \u2227 m + n = n - m' + k + n \u2227 n - m + k + n = m' + n \u2228\n      \u2191u + \u2191w = 0 \u2227 m + n = n - m + k + n \u2227 m' + n = n - m' + k + n\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nsimp only [add_left_inj, Units.eq_iff] at h \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nh : m = m' \u2227 n - m = n - m' \u2228 u = w \u2227 m = n - m' + k \u2227 n - m + k = m' \u2228 \u2191u + \u2191w = 0 \u2227 m = n - m + k \u2227 m' = n - m' + k\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrcases h with (\u27e8rfl, -\u27e9 | \u27e8rfl, rfl, h\u27e9 | \u27e8-, hm, hm'\u27e9)\n[GOAL]\ncase inl.intro\np q : \u2124[X]\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhkm' : k < m\nhmn' : m < n\nhq : q = trinomial k m n \u2191u \u2191v \u2191w\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nexact Or.inl (hq.trans hp.symm)\n[GOAL]\ncase inr.inl.intro.intro\np q : \u2124[X]\nk m' n : \u2115\nhkm' : k < m'\nhmn' : m' < n\nu v : \u2124\u02e3\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhq : q = trinomial k m' n \u2191u \u2191v \u2191u\nhkm : k < n - m' + k\nhmn : n - m' + k < n\nhp : p = trinomial k (n - m' + k) n \u2191u \u2191v \u2191u\nh : n - (n - m' + k) + k = m'\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrefine' Or.inr _\n[GOAL]\ncase inr.inl.intro.intro\np q : \u2124[X]\nk m' n : \u2115\nhkm' : k < m'\nhmn' : m' < n\nu v : \u2124\u02e3\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhq : q = trinomial k m' n \u2191u \u2191v \u2191u\nhkm : k < n - m' + k\nhmn : n - m' + k < n\nhp : p = trinomial k (n - m' + k) n \u2191u \u2191v \u2191u\nh : n - (n - m' + k) + k = m'\n\u22a2 q = mirror p\n[PROOFSTEP]\nrw [\u2190 trinomial_mirror hkm' hmn' u.ne_zero u.ne_zero, eq_comm, mirror_eq_iff] at hp \n[GOAL]\ncase inr.inl.intro.intro\np q : \u2124[X]\nk m' n : \u2115\nhkm' : k < m'\nhmn' : m' < n\nu v : \u2124\u02e3\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhq : q = trinomial k m' n \u2191u \u2191v \u2191u\nhkm : k < n - m' + k\nhmn : n - m' + k < n\nhp : trinomial k m' n \u2191u \u2191v \u2191u = mirror p\nh : n - (n - m' + k) + k = m'\n\u22a2 q = mirror p\n[PROOFSTEP]\nexact hq.trans hp\n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n - m + k\nhm' : m' = n - m' + k\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nsuffices m = m' by\n  rw [this] at hp \n  exact Or.inl (hq.trans hp.symm)\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n - m + k\nhm' : m' = n - m' + k\nthis : m = m'\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [this] at hp \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m' n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n - m + k\nhm' : m' = n - m' + k\nthis : m = m'\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nexact Or.inl (hq.trans hp.symm)\n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n - m + k\nhm' : m' = n - m' + k\n\u22a2 m = m'\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub hmn.le, eq_tsub_iff_add_eq_of_le, \u2190 two_mul] at hm \n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm\u271d : m = n + k - m\nhm : 2 * m = n + k\nhm' : m' = n - m' + k\n\u22a2 m = m'\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n + k - m\nhm' : m' = n - m' + k\n\u22a2 m \u2264 n + k\n[PROOFSTEP]\nrw [tsub_add_eq_add_tsub hmn'.le, eq_tsub_iff_add_eq_of_le, \u2190 two_mul] at hm' \n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm\u271d : m = n + k - m\nhm : 2 * m = n + k\nhm'\u271d : m' = n + k - m'\nhm' : 2 * m' = n + k\n\u22a2 m = m'\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm\u271d : m = n + k - m\nhm : 2 * m = n + k\nhm' : m' = n + k - m'\n\u22a2 m' \u2264 n + k\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n + k - m\nhm' : m' = n - m' + k\n\u22a2 m \u2264 n + k\n[PROOFSTEP]\nexact mul_left_cancel\u2080 two_ne_zero (hm.trans hm'.symm)\n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm\u271d : m = n + k - m\nhm : 2 * m = n + k\nhm' : m' = n + k - m'\n\u22a2 m' \u2264 n + k\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n + k - m\nhm' : m' = n - m' + k\n\u22a2 m \u2264 n + k\n[PROOFSTEP]\nexact hmn'.le.trans (Nat.le_add_right n k)\n[GOAL]\ncase inr.inr.intro.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nf : \u2124[X] \u2192 \u2124[X] := fun p => { toFinsupp := Finsupp.filter (Set.Ioo (k + n) (n + n)) p.toFinsupp }\nhm : m = n + k - m\nhm' : m' = n - m' + k\n\u22a2 m \u2264 n + k\n[PROOFSTEP]\nexact hmn.le.trans (Nat.le_add_right n k)\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nhave hmul := congr_arg leadingCoeff h\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : leadingCoeff (p * mirror p) = leadingCoeff (q * mirror q)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [leadingCoeff_mul, leadingCoeff_mul, mirror_leadingCoeff, mirror_leadingCoeff, hp, hq,\n  trinomial_leadingCoeff hkm hmn w.ne_zero, trinomial_leadingCoeff hkm' hmn' z.ne_zero,\n  trinomial_trailingCoeff hkm hmn u.ne_zero, trinomial_trailingCoeff hkm' hmn' x.ne_zero] at hmul \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nhave hadd := congr_arg (eval 1) h\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : eval 1 (p * mirror p) = eval 1 (q * mirror q)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [eval_mul, eval_mul, mirror_eval_one, mirror_eval_one, \u2190 sq, \u2190 sq, hp, hq] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : eval 1 (trinomial k m n \u2191u \u2191v \u2191w) ^ 2 = eval 1 (trinomial k m' n \u2191x \u2191v \u2191z) ^ 2\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nsimp only [eval_add, eval_C_mul, eval_pow, eval_X, one_pow, mul_one, trinomial_def] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : (\u2191u + \u2191v + \u2191w) ^ 2 = (\u2191x + \u2191v + \u2191z) ^ 2\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [add_assoc, add_assoc, add_comm (u : \u2124), add_comm (x : \u2124), add_assoc, add_assoc] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : (\u2191v + (\u2191w + \u2191u)) ^ 2 = (\u2191v + (\u2191z + \u2191x)) ^ 2\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nsimp only [add_sq', add_assoc, add_right_inj, \u2190 Units.val_pow_eq_pow_val, Int.units_sq] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : 2 * \u2191w * \u2191u + 2 * \u2191v * (\u2191w + \u2191u) = 2 * \u2191z * \u2191x + 2 * \u2191v * (\u2191z + \u2191x)\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [mul_assoc, hmul, \u2190 mul_assoc, add_right_inj,\n  mul_right_inj' (show 2 * (v : \u2124) \u2260 0 from mul_ne_zero two_ne_zero v.ne_zero)] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : \u2191w + \u2191u = \u2191z + \u2191x\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nreplace hadd := (Int.isUnit_add_isUnit_eq_isUnit_add_isUnit w.isUnit u.isUnit z.isUnit x.isUnit).mp hadd\n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : \u2191w = \u2191z \u2227 \u2191u = \u2191x \u2228 \u2191w = \u2191x \u2227 \u2191u = \u2191z\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nsimp only [Units.eq_iff] at hadd \n[GOAL]\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w x z : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh : p * mirror p = q * mirror q\nhmul : \u2191w * \u2191u = \u2191z * \u2191x\nhadd : w = z \u2227 u = x \u2228 w = x \u2227 u = z\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrcases hadd with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)\n[GOAL]\ncase inl.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nh : p * mirror p = q * mirror q\nhq : q = trinomial k m' n \u2191u \u2191v \u2191w\nhmul : \u2191w * \u2191u = \u2191w * \u2191u\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nexact irreducible_aux2 hkm hmn hkm' hmn' u v w hp hq h\n[GOAL]\ncase inr.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nh : p * mirror p = q * mirror q\nhq : q = trinomial k m' n \u2191w \u2191v \u2191u\nhmul : \u2191w * \u2191u = \u2191u * \u2191w\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [\u2190 mirror_inj, trinomial_mirror hkm' hmn' w.ne_zero u.ne_zero] at hq \n[GOAL]\ncase inr.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nh : p * mirror p = q * mirror q\nhq : mirror q = trinomial k (n - m' + k) n \u2191u \u2191v \u2191w\nhmul : \u2191w * \u2191u = \u2191u * \u2191w\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [mul_comm q, \u2190 q.mirror_mirror, q.mirror.mirror_mirror] at h \n[GOAL]\ncase inr.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nh : p * mirror p = mirror q * mirror (mirror q)\nhq : mirror q = trinomial k (n - m' + k) n \u2191u \u2191v \u2191w\nhmul : \u2191w * \u2191u = \u2191u * \u2191w\n\u22a2 q = p \u2228 q = mirror p\n[PROOFSTEP]\nrw [\u2190 mirror_inj, or_comm, \u2190 mirror_eq_iff]\n[GOAL]\ncase inr.intro\np q : \u2124[X]\nk m m' n : \u2115\nhkm : k < m\nhmn : m < n\nhkm' : k < m'\nhmn' : m' < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nh : p * mirror p = mirror q * mirror (mirror q)\nhq : mirror q = trinomial k (n - m' + k) n \u2191u \u2191v \u2191w\nhmul : \u2191w * \u2191u = \u2191u * \u2191w\n\u22a2 mirror q = p \u2228 mirror q = mirror p\n[PROOFSTEP]\nexact\n  irreducible_aux2 hkm hmn (lt_add_of_pos_left k (tsub_pos_of_lt hmn'))\n    (lt_tsub_iff_right.mp ((tsub_lt_tsub_iff_left_of_le hmn'.le).mpr hkm')) u v w hp hq h\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\n\u22a2 Irreducible p\n[PROOFSTEP]\nrefine' irreducible_of_mirror hp.not_isUnit (fun q hpq => _) h\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nhave hq : IsUnitTrinomial q := (isUnitTrinomial_iff'' hpq).mp hp\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nhq : IsUnitTrinomial q\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nobtain \u27e8k, m, n, hkm, hmn, u, v, w, hp\u27e9 := hp\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nhq : IsUnitTrinomial q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nobtain \u27e8k', m', n', hkm', hmn', x, y, z, hq\u27e9 := hq\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nk' m' n' : \u2115\nhkm' : k' < m'\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhq : q = trinomial k' m' n' \u2191x \u2191y \u2191z\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nhave hk : k = k' := by\n  rw [\u2190 mul_right_inj' (show 2 \u2260 0 from two_ne_zero), \u2190 trinomial_natTrailingDegree hkm hmn u.ne_zero, \u2190 hp, \u2190\n    natTrailingDegree_mul_mirror, hpq, natTrailingDegree_mul_mirror, hq,\n    trinomial_natTrailingDegree hkm' hmn' x.ne_zero]\n[GOAL]\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nk' m' n' : \u2115\nhkm' : k' < m'\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhq : q = trinomial k' m' n' \u2191x \u2191y \u2191z\n\u22a2 k = k'\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' (show 2 \u2260 0 from two_ne_zero), \u2190 trinomial_natTrailingDegree hkm hmn u.ne_zero, \u2190 hp, \u2190\n  natTrailingDegree_mul_mirror, hpq, natTrailingDegree_mul_mirror, hq, trinomial_natTrailingDegree hkm' hmn' x.ne_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nk' m' n' : \u2115\nhkm' : k' < m'\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhq : q = trinomial k' m' n' \u2191x \u2191y \u2191z\nhk : k = k'\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nhave hn : n = n' := by\n  rw [\u2190 mul_right_inj' (show 2 \u2260 0 from two_ne_zero), \u2190 trinomial_natDegree hkm hmn w.ne_zero, \u2190 hp, \u2190\n    natDegree_mul_mirror, hpq, natDegree_mul_mirror, hq, trinomial_natDegree hkm' hmn' z.ne_zero]\n[GOAL]\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nk' m' n' : \u2115\nhkm' : k' < m'\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhq : q = trinomial k' m' n' \u2191x \u2191y \u2191z\nhk : k = k'\n\u22a2 n = n'\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' (show 2 \u2260 0 from two_ne_zero), \u2190 trinomial_natDegree hkm hmn w.ne_zero, \u2190 hp, \u2190\n  natDegree_mul_mirror, hpq, natDegree_mul_mirror, hq, trinomial_natDegree hkm' hmn' z.ne_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nk' m' n' : \u2115\nhkm' : k' < m'\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhq : q = trinomial k' m' n' \u2191x \u2191y \u2191z\nhk : k = k'\nhn : n = n'\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nsubst hk\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' n' : \u2115\nhmn' : m' < n'\nx y z : \u2124\u02e3\nhn : n = n'\nhkm' : k < m'\nhq : q = trinomial k m' n' \u2191x \u2191y \u2191z\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191y \u2191z\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrcases eq_or_eq_neg_of_sq_eq_sq (y : \u2124) (v : \u2124) ((Int.isUnit_sq y.isUnit).trans (Int.isUnit_sq v.isUnit).symm) with\n  (h1 | h1)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191y \u2191z\nh1 : \u2191y = \u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrw [h1] at hq \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh1 : \u2191y = \u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrcases irreducible_aux3 hkm hmn hkm' hmn' u v w x z hp hq hpq with (h2 | h2)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl.inl\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh1 : \u2191y = \u2191v\nh2 : q = p\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nexact Or.inl h2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191v \u2191z\nh1 : \u2191y = \u2191v\nh2 : q = mirror p\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nexact Or.inr (Or.inr (Or.inl h2))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n \u2191x \u2191y \u2191z\nh1 : \u2191y = -\u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrw [h1] at hq \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = trinomial k m n \u2191u \u2191v \u2191w\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\nh1 : \u2191y = -\u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrw [trinomial_def] at hp \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp : p = \u2191C \u2191u * X ^ k + \u2191C \u2191v * X ^ m + \u2191C \u2191w * X ^ n\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\nh1 : \u2191y = -\u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrw [\u2190 neg_inj, neg_add, neg_add, \u2190 neg_mul, \u2190 neg_mul, \u2190 neg_mul, \u2190 C_neg, \u2190 C_neg, \u2190 C_neg] at hp \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq : p * mirror p = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp\u271d : p = \u2191C \u2191u * X ^ k + \u2191C \u2191v * X ^ m + \u2191C \u2191w * X ^ n\nhp : -p = \u2191C (-\u2191u) * X ^ k + \u2191C (-\u2191v) * X ^ m + \u2191C (-\u2191w) * X ^ n\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\nh1 : \u2191y = -\u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrw [\u2190 neg_mul_neg, \u2190 mirror_neg] at hpq \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\np q\u271d : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nq : \u2124[X]\nhpq\u271d : p * mirror p = q * mirror q\nhpq : -p * mirror (-p) = q * mirror q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp\u271d : p = \u2191C \u2191u * X ^ k + \u2191C \u2191v * X ^ m + \u2191C \u2191w * X ^ n\nhp : -p = \u2191C (-\u2191u) * X ^ k + \u2191C (-\u2191v) * X ^ m + \u2191C (-\u2191w) * X ^ n\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nhq : q = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\nh1 : \u2191y = -\u2191v\n\u22a2 q = p \u2228 q = -p \u2228 q = mirror p \u2228 q = -mirror p\n[PROOFSTEP]\nrcases irreducible_aux3 hkm hmn hkm' hmn' (-u) (-v) (-w) x z hp hq hpq with (rfl | rfl)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.inl\np q : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp\u271d : p = \u2191C \u2191u * X ^ k + \u2191C \u2191v * X ^ m + \u2191C \u2191w * X ^ n\nhp : -p = \u2191C (-\u2191u) * X ^ k + \u2191C (-\u2191v) * X ^ m + \u2191C (-\u2191w) * X ^ n\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nh1 : \u2191y = -\u2191v\nhpq\u271d : p * mirror p = -p * mirror (-p)\nhpq : -p * mirror (-p) = -p * mirror (-p)\nhq : -p = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\n\u22a2 -p = p \u2228 -p = -p \u2228 -p = mirror p \u2228 -p = -mirror p\n[PROOFSTEP]\nexact Or.inr (Or.inl rfl)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.inr\np q : \u2124[X]\nh : \u2200 (q : \u2124[X]), q \u2223 p \u2192 q \u2223 mirror p \u2192 IsUnit q\nk m n : \u2115\nhkm : k < m\nhmn : m < n\nu v w : \u2124\u02e3\nhp\u271d : p = \u2191C \u2191u * X ^ k + \u2191C \u2191v * X ^ m + \u2191C \u2191w * X ^ n\nhp : -p = \u2191C (-\u2191u) * X ^ k + \u2191C (-\u2191v) * X ^ m + \u2191C (-\u2191w) * X ^ n\nm' : \u2115\nx y z : \u2124\u02e3\nhkm' : k < m'\nhmn' : m' < n\nh1 : \u2191y = -\u2191v\nhpq\u271d : p * mirror p = mirror (-p) * mirror (mirror (-p))\nhpq : -p * mirror (-p) = mirror (-p) * mirror (mirror (-p))\nhq : mirror (-p) = trinomial k m' n (\u2191x) (-\u2191v) \u2191z\n\u22a2 mirror (-p) = p \u2228 mirror (-p) = -p \u2228 mirror (-p) = mirror p \u2228 mirror (-p) = -mirror p\n[PROOFSTEP]\nexact Or.inr (Or.inr (Or.inr p.mirror_neg))\n[GOAL]\np q : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\n\u22a2 Irreducible p\n[PROOFSTEP]\nrefine' hp.irreducible_of_coprime fun q hq hq' => _\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\n\u22a2 IsUnit q\n[PROOFSTEP]\nsuffices \u00ac0 < q.natDegree by\n  rcases hq with \u27e8p, rfl\u27e9\n  replace hp := hp.leadingCoeff_isUnit\n  rw [leadingCoeff_mul] at hp \n  replace hp := isUnit_of_mul_isUnit_left hp\n  rw [not_lt, le_zero_iff] at this \n  rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, \u2190 this]\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nthis : \u00ac0 < natDegree q\n\u22a2 IsUnit q\n[PROOFSTEP]\nrcases hq with \u27e8p, rfl\u27e9\n[GOAL]\ncase intro\nq\u271d q : \u2124[X]\nthis : \u00ac0 < natDegree q\np : \u2124[X]\nhp : IsUnitTrinomial (q * p)\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0)\nhq' : q \u2223 mirror (q * p)\n\u22a2 IsUnit q\n[PROOFSTEP]\nreplace hp := hp.leadingCoeff_isUnit\n[GOAL]\ncase intro\nq\u271d q : \u2124[X]\nthis : \u00ac0 < natDegree q\np : \u2124[X]\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0)\nhq' : q \u2223 mirror (q * p)\nhp : IsUnit (leadingCoeff (q * p))\n\u22a2 IsUnit q\n[PROOFSTEP]\nrw [leadingCoeff_mul] at hp \n[GOAL]\ncase intro\nq\u271d q : \u2124[X]\nthis : \u00ac0 < natDegree q\np : \u2124[X]\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0)\nhq' : q \u2223 mirror (q * p)\nhp : IsUnit (leadingCoeff q * leadingCoeff p)\n\u22a2 IsUnit q\n[PROOFSTEP]\nreplace hp := isUnit_of_mul_isUnit_left hp\n[GOAL]\ncase intro\nq\u271d q : \u2124[X]\nthis : \u00ac0 < natDegree q\np : \u2124[X]\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0)\nhq' : q \u2223 mirror (q * p)\nhp : IsUnit (leadingCoeff q)\n\u22a2 IsUnit q\n[PROOFSTEP]\nrw [not_lt, le_zero_iff] at this \n[GOAL]\ncase intro\nq\u271d q : \u2124[X]\nthis : natDegree q = 0\np : \u2124[X]\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0)\nhq' : q \u2223 mirror (q * p)\nhp : IsUnit (leadingCoeff q)\n\u22a2 IsUnit q\n[PROOFSTEP]\nrwa [eq_C_of_natDegree_eq_zero this, isUnit_C, \u2190 this]\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\n\u22a2 \u00ac0 < natDegree q\n[PROOFSTEP]\nintro hq''\n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < natDegree q\n\u22a2 False\n[PROOFSTEP]\nrw [natDegree_pos_iff_degree_pos] at hq'' \n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree q\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 degree_map_eq_of_injective (algebraMap \u2124 \u2102).injective_int] at hq'' \n[GOAL]\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\n\u22a2 False\n[PROOFSTEP]\ncases' Complex.exists_root hq'' with z hz\n[GOAL]\ncase intro\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : IsRoot (map (algebraMap \u2124 \u2102) q) z\n\u22a2 False\n[PROOFSTEP]\nrw [IsRoot, eval_map, \u2190 aeval_def] at hz \n[GOAL]\ncase intro\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : \u2191(aeval z) q = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' h z \u27e8_, _\u27e9\n[GOAL]\ncase intro.refine'_1\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : \u2191(aeval z) q = 0\n\u22a2 \u2191(aeval z) p = 0\n[PROOFSTEP]\ncases' hq with g' hg'\n[GOAL]\ncase intro.refine'_1.intro\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : \u2191(aeval z) q = 0\ng' : \u2124[X]\nhg' : p = q * g'\n\u22a2 \u2191(aeval z) p = 0\n[PROOFSTEP]\nrw [hg', aeval_mul, hz, zero_mul]\n[GOAL]\ncase intro.refine'_2\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq' : q \u2223 mirror p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : \u2191(aeval z) q = 0\n\u22a2 \u2191(aeval z) (mirror p) = 0\n[PROOFSTEP]\ncases' hq' with g' hg'\n[GOAL]\ncase intro.refine'_2.intro\np q\u271d : \u2124[X]\nhp : IsUnitTrinomial p\nh : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0)\nq : \u2124[X]\nhq : q \u2223 p\nhq'' : 0 < degree (map (algebraMap \u2124 \u2102) q)\nz : \u2102\nhz : \u2191(aeval z) q = 0\ng' : \u2124[X]\nhg' : mirror p = q * g'\n\u22a2 \u2191(aeval z) (mirror p) = 0\n[PROOFSTEP]\nrw [hg', aeval_mul, hz, zero_mul]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.UnitTrinomial", "llama_tokens": 31675, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5243207575921639}}
{"text": "[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring R\u2081\ninst\u271d\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\nx\u271d\u00b2 x\u271d\u00b9 : n \u2192 R\u2081\nx\u271d : m \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) (x\u271d\u00b2 + x\u271d\u00b9) x\u271d =\n    (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b2 x\u271d +\n      (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [Pi.add_apply, map_add, add_mul, sum_add_distrib]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring R\u2081\ninst\u271d\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\nx\u271d\u00b2 : R\u2081\nx\u271d\u00b9 : n \u2192 R\u2081\nx\u271d : m \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) (x\u271d\u00b2 \u2022 x\u271d\u00b9) x\u271d =\n    \u2191\u03c3\u2081 x\u271d\u00b2 \u2022 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_eq_mul, RingHom.map_mul, mul_assoc, mul_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring R\u2081\ninst\u271d\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\nx\u271d\u00b2 : n \u2192 R\u2081\nx\u271d\u00b9 x\u271d : m \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b2 (x\u271d\u00b9 + x\u271d) =\n    (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b2 x\u271d\u00b9 +\n      (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b2 x\u271d\n[PROOFSTEP]\nsimp only [Pi.add_apply, map_add, mul_add, sum_add_distrib]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : CommSemiring R\u2081\ninst\u271d\u00b2 : CommSemiring R\u2082\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\nx\u271d\u00b2 : R\u2082\nx\u271d\u00b9 : n \u2192 R\u2081\nx\u271d : m \u2192 R\u2082\n\u22a2 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b9 (x\u271d\u00b2 \u2022 x\u271d) =\n    \u2191\u03c3\u2082 x\u271d\u00b2 \u2022 (fun v w => \u2211 i : n, \u2211 j : m, \u2191\u03c3\u2081 (v i) * f i j * \u2191\u03c3\u2082 (w j)) x\u271d\u00b9 x\u271d\n[PROOFSTEP]\nsimp only [Pi.smul_apply, smul_eq_mul, RingHom.map_mul, mul_assoc, mul_left_comm, mul_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\n\u22a2 \u2191(\u2191(toLinearMap\u2082'Aux \u03c3\u2081 \u03c3\u2082 f) (\u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1))\n      (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1) =\n    f i j\n[PROOFSTEP]\nrw [Matrix.toLinearMap\u2082'Aux, mk\u2082'\u209b\u2097_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\n\u22a2 \u2211 i_1 : n,\n      \u2211 j_1 : m,\n        \u2191\u03c3\u2081 (\u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1 i_1) * f i_1 j_1 *\n          \u2191\u03c3\u2082 (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1) =\n    f i j\n[PROOFSTEP]\nhave : (\u2211 i', \u2211 j', (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j :=\n  by\n  simp_rw [mul_assoc, \u2190 Finset.mul_sum]\n  simp only [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true, mul_comm (f _ _)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\n\u22a2 (\u2211 i' : n, \u2211 j' : m, (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j\n[PROOFSTEP]\nsimp_rw [mul_assoc, \u2190 Finset.mul_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\n\u22a2 (\u2211 x : n, (if i = x then 1 else 0) * \u2211 x_1 : m, f x x_1 * if j = x_1 then 1 else 0) = f i j\n[PROOFSTEP]\nsimp only [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true, mul_comm (f _ _)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\nthis : (\u2211 i' : n, \u2211 j' : m, (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j\n\u22a2 \u2211 i_1 : n,\n      \u2211 j_1 : m,\n        \u2191\u03c3\u2081 (\u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1 i_1) * f i_1 j_1 *\n          \u2191\u03c3\u2082 (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1) =\n    f i j\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\nthis : (\u2211 i' : n, \u2211 j' : m, (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j\n\u22a2 \u2211 i_1 : n,\n      \u2211 j_1 : m,\n        \u2191\u03c3\u2081 (\u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1 i_1) * f i_1 j_1 *\n          \u2191\u03c3\u2082 (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 j_1) =\n    \u2211 i' : n, \u2211 j' : m, (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0\n[PROOFSTEP]\nexact Finset.sum_congr rfl fun _ _ => Finset.sum_congr rfl fun _ _ => by simp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : CommSemiring R\u2081\ninst\u271d\u2074 : CommSemiring R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\nf : Matrix n m R\ni : n\nj : m\nthis : (\u2211 i' : n, \u2211 j' : m, (if i = i' then 1 else 0) * f i' j' * if j = j' then 1 else 0) = f i j\nx\u271d\u00b3 : n\nx\u271d\u00b2 : x\u271d\u00b3 \u2208 univ\nx\u271d\u00b9 : m\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 \u2191\u03c3\u2081 (\u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1 x\u271d\u00b3) * f x\u271d\u00b3 x\u271d\u00b9 *\n      \u2191\u03c3\u2082 (\u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1 x\u271d\u00b9) =\n    (if i = x\u271d\u00b3 then 1 else 0) * f x\u271d\u00b3 x\u271d\u00b9 * if j = x\u271d\u00b9 then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : (n \u2192 R\u2081) \u2192\u209b\u2097[\u03c3\u2081] (m \u2192 R\u2082) \u2192\u209b\u2097[\u03c3\u2082] R\n\u22a2 toLinearMap\u2082'Aux \u03c3\u2081 \u03c3\u2082\n      (\u2191(toMatrix\u2082Aux (fun i => \u2191(stdBasis R\u2081 (fun x => R\u2081) i) 1) fun j => \u2191(stdBasis R\u2082 (fun x => R\u2082) j) 1) f) =\n    f\n[PROOFSTEP]\nrefine' ext_basis (Pi.basisFun R\u2081 n) (Pi.basisFun R\u2082 m) fun i j => _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : (n \u2192 R\u2081) \u2192\u209b\u2097[\u03c3\u2081] (m \u2192 R\u2082) \u2192\u209b\u2097[\u03c3\u2082] R\ni : n\nj : m\n\u22a2 \u2191(\u2191(toLinearMap\u2082'Aux \u03c3\u2081 \u03c3\u2082\n              (\u2191(toMatrix\u2082Aux (fun i => \u2191(stdBasis R\u2081 (fun x => R\u2081) i) 1) fun j => \u2191(stdBasis R\u2082 (fun x => R\u2082) j) 1) f))\n          (\u2191(Pi.basisFun R\u2081 n) i))\n      (\u2191(Pi.basisFun R\u2082 m) j) =\n    \u2191(\u2191f (\u2191(Pi.basisFun R\u2081 n) i)) (\u2191(Pi.basisFun R\u2082 m) j)\n[PROOFSTEP]\nsimp_rw [Pi.basisFun_apply, Matrix.toLinearMap\u2082'Aux_stdBasis, LinearMap.toMatrix\u2082Aux_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\n\u22a2 \u2191(toMatrix\u2082Aux (fun i => \u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1) fun j =>\n          \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1)\n      (toLinearMap\u2082'Aux \u03c3\u2081 \u03c3\u2082 f) =\n    f\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R\u2081 M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nf : Matrix n m R\ni : n\nj : m\n\u22a2 \u2191(toMatrix\u2082Aux (fun i => \u2191(LinearMap.stdBasis R\u2081 (fun x => R\u2081) i) 1) fun j =>\n          \u2191(LinearMap.stdBasis R\u2082 (fun x => R\u2082) j) 1)\n      (toLinearMap\u2082'Aux \u03c3\u2081 \u03c3\u2082 f) i j =\n    f i j\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrix\u2082Aux_apply, Matrix.toLinearMap\u2082'Aux_stdBasis]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : CommRing R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nM : Matrix n m R\nv : n \u2192 R\nw : m \u2192 R\n\u22a2 \u2191(\u2191(\u2191toLinearMap\u2082' M) v) w = v \u2b1d\u1d65 mulVec M w\n[PROOFSTEP]\nsimp_rw [Matrix.toLinearMap\u2082'_apply, Matrix.dotProduct, Matrix.mulVec, Matrix.dotProduct]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : CommRing R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nM : Matrix n m R\nv : n \u2192 R\nw : m \u2192 R\n\u22a2 \u2211 i : n, \u2211 j : m, v i * M i j * w j = \u2211 x : n, v x * \u2211 x_1 : m, M x x_1 * w x_1\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun _ _ => _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : CommRing R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nM : Matrix n m R\nv : n \u2192 R\nw : m \u2192 R\nx\u271d\u00b9 : n\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 \u2211 j : m, v x\u271d\u00b9 * M x\u271d\u00b9 j * w j = v x\u271d\u00b9 * \u2211 x : m, M x\u271d\u00b9 x * w x\n[PROOFSTEP]\nrw [Finset.mul_sum]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : CommRing R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nM : Matrix n m R\nv : n \u2192 R\nw : m \u2192 R\nx\u271d\u00b9 : n\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 \u2211 j : m, v x\u271d\u00b9 * M x\u271d\u00b9 j * w j = \u2211 x : m, v x\u271d\u00b9 * (M x\u271d\u00b9 x * w x)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun _ _ => _\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : CommRing R\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype m\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\nM : Matrix n m R\nv : n \u2192 R\nw : m \u2192 R\nx\u271d\u00b3 : n\nx\u271d\u00b2 : x\u271d\u00b3 \u2208 univ\nx\u271d\u00b9 : m\nx\u271d : x\u271d\u00b9 \u2208 univ\n\u22a2 v x\u271d\u00b3 * M x\u271d\u00b3 x\u271d\u00b9 * w x\u271d\u00b9 = v x\u271d\u00b3 * (M x\u271d\u00b3 x\u271d\u00b9 * w x\u271d\u00b9)\n[PROOFSTEP]\nrw [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 \u2191toMatrix\u2082' (compl\u2081\u2082 B l r) = (\u2191toMatrix' l)\u1d40 * \u2191toMatrix\u2082' B * \u2191toMatrix' r\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 \u2191toMatrix\u2082' (compl\u2081\u2082 B l r) i j = ((\u2191toMatrix' l)\u1d40 * \u2191toMatrix\u2082' B * \u2191toMatrix' r) i j\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix\u2082'_apply, LinearMap.compl\u2081\u2082_apply, transpose_apply, Matrix.mul_apply, LinearMap.toMatrix',\n  LinearEquiv.coe_mk, sum_mul]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 \u2191(\u2191B (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) (\u2191r (\u2191(stdBasis R (fun x => R) j) 1)) =\n    \u2211 x : m,\n      \u2211 x_1 : n,\n        \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) x_1 i *\n            \u2191(\u2191B (\u2191(stdBasis R (fun x => R) x_1) 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\nrw [sum_comm]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 \u2191(\u2191B (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) (\u2191r (\u2191(stdBasis R (fun x => R) j) 1)) =\n    \u2211 y : n,\n      \u2211 x : m,\n        \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) y i *\n            \u2191(\u2191B (\u2191(stdBasis R (fun x => R) y) 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\nconv_lhs => rw [\u2190 LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul (Pi.basisFun R n) (Pi.basisFun R m) (l _) (r _)]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) fun i xi =>\n      Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n        xi \u2022 yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i)) (\u2191(Pi.basisFun R m) j)) =\n    \u2211 y : n,\n      \u2211 x : m,\n        \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) y i *\n            \u2191(\u2191B (\u2191(stdBasis R (fun x => R) y) 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 (\u2211 i_1 : n,\n      Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n        \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i_1 \u2022\n          yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i_1)) (\u2191(Pi.basisFun R m) j)) =\n    \u2211 y : n,\n      \u2211 x : m,\n        \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) y i *\n            \u2191(\u2191B (\u2191(stdBasis R (fun x => R) y) 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      (Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n          \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) x \u2022\n            yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) x)) (\u2191(Pi.basisFun R m) j)) =\n        \u2211 x_1 : m,\n          \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) x i *\n              \u2191(\u2191B (\u2191(stdBasis R (fun x => R) x) 1)) (\u2191(stdBasis R (fun x => R) x_1) 1) *\n            \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x_1 j\n[PROOFSTEP]\nrintro i' -\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n      \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n        yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) j)) =\n    \u2211 x : m,\n      \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) i' i *\n          \u2191(\u2191B (\u2191(stdBasis R (fun x => R) i') 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n        \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\n\u22a2 \u2211 i_1 : m,\n      \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n        \u2191(\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) i_1 \u2022\n          \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) i_1) =\n    \u2211 x : m,\n      \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) i' i *\n          \u2191(\u2191B (\u2191(stdBasis R (fun x => R) i') 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n        \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\n\u22a2 \u2200 (x : m),\n    x \u2208 univ \u2192\n      \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n          \u2191(\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) x \u2022\n            \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) x) =\n        \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) i' i *\n            \u2191(\u2191B (\u2191(stdBasis R (fun x => R) i') 1)) (\u2191(stdBasis R (fun x => R) x) 1) *\n          \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) x j\n[PROOFSTEP]\nrintro j' -\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\nj' : m\n\u22a2 \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n      \u2191(\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) j' \u2022\n        \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) j') =\n    \u2191of (fun i j => \u2191l (\u2191(stdBasis R (fun x => R) j) 1) i) i' i *\n        \u2191(\u2191B (\u2191(stdBasis R (fun x => R) i') 1)) (\u2191(stdBasis R (fun x => R) j') 1) *\n      \u2191of (fun i j => \u2191r (\u2191(stdBasis R (fun x => R) j) 1) i) j' j\n[PROOFSTEP]\nsimp only [smul_eq_mul, Pi.basisFun_repr, mul_assoc, mul_comm, mul_left_comm, Pi.basisFun_apply, of_apply]\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\n\u22a2 \u2200 (i_1 : m),\n    \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n        0 \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) i_1) =\n      0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni' : n\ni\u271d : m\n\u22a2 \u2191(\u2191(Pi.basisFun R n).repr (\u2191l (\u2191(stdBasis R (fun x => R) i) 1))) i' \u2022\n      0 \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i')) (\u2191(Pi.basisFun R m) i\u271d) =\n    0\n[PROOFSTEP]\nsimp only [zero_smul, smul_zero]\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\n\u22a2 \u2200 (i : n),\n    (Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n        0 \u2022 yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i)) (\u2191(Pi.basisFun R m) j)) =\n      0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nl : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\nr : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\ni : n'\nj : m'\ni\u271d : n\n\u22a2 (Finsupp.sum (\u2191(Pi.basisFun R m).repr (\u2191r (\u2191(stdBasis R (fun x => R) j) 1))) fun j yj =>\n      0 \u2022 yj \u2022 \u2191(\u2191B (\u2191(Pi.basisFun R n) i\u271d)) (\u2191(Pi.basisFun R m) j)) =\n    0\n[PROOFSTEP]\nsimp only [zero_smul, Finsupp.sum_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nf : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 \u2191toMatrix\u2082' (comp B f) = (\u2191toMatrix' f)\u1d40 * \u2191toMatrix\u2082' B\n[PROOFSTEP]\nrw [\u2190 LinearMap.compl\u2082_id (B.comp f), \u2190 LinearMap.compl\u2081\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nf : (n' \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 \u2191toMatrix\u2082' (compl\u2081\u2082 B f id) = (\u2191toMatrix' f)\u1d40 * \u2191toMatrix\u2082' B\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nf : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 \u2191toMatrix\u2082' (compl\u2082 B f) = \u2191toMatrix\u2082' B * \u2191toMatrix' f\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_id B, \u2190 LinearMap.compl\u2081\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nf : (m' \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 \u2191toMatrix\u2082' (compl\u2081\u2082 B id f) = \u2191toMatrix\u2082' (comp B id) * \u2191toMatrix' f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nM : Matrix n' n R\nN : Matrix m m' R\n\u22a2 M * \u2191toMatrix\u2082' B * N = \u2191toMatrix\u2082' (compl\u2081\u2082 B (\u2191toLin' M\u1d40) (\u2191toLin' N))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nM : Matrix n' n R\n\u22a2 M * \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' (comp B (\u2191toLin' M\u1d40))\n[PROOFSTEP]\nsimp only [B.toMatrix\u2082'_comp, transpose_transpose, toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\nM : Matrix m m' R\n\u22a2 \u2191toMatrix\u2082' B * M = \u2191toMatrix\u2082' (compl\u2082 B (\u2191toLin' M))\n[PROOFSTEP]\nsimp only [B.toMatrix\u2082'_compl\u2082, toMatrix'_toLin']\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing R\u2081\ninst\u271d\u2078 : CommRing R\u2082\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : Fintype m\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : DecidableEq m\n\u03c3\u2081 : R\u2081 \u2192+* R\n\u03c3\u2082 : R\u2082 \u2192+* R\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nM : Matrix n m R\nP : Matrix n n' R\nQ : Matrix m m' R\n\u22a2 \u2191toMatrix\u2082' (compl\u2081\u2082 (\u2191toLinearMap\u2082' M) (\u2191toLin' P) (\u2191toLin' Q)) = \u2191toMatrix\u2082' (\u2191toLinearMap\u2082' (P\u1d40 * M * Q))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\ni : n\nj : m\n\u22a2 \u2191(toMatrix\u2082 b\u2081 b\u2082) B i j = \u2191(\u2191B (\u2191b\u2081 i)) (\u2191b\u2082 j)\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix\u2082, LinearEquiv.trans_apply, LinearMap.toMatrix\u2082'_apply, LinearEquiv.trans_apply,\n  LinearMap.toMatrix\u2082'_apply, LinearEquiv.arrowCongr_apply, Basis.equivFun_symm_stdBasis, LinearEquiv.refl_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\ni : n\nj : m\n\u22a2 \u2191(toMatrix\u2082Aux \u2191b\u2081 \u2191b\u2082) B i j = \u2191(toMatrix\u2082 b\u2081 b\u2082) B i j\n[PROOFSTEP]\nrw [LinearMap.toMatrix\u2082_apply, LinearMap.toMatrix\u2082Aux_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\n\u22a2 toLinearMap\u2082 (Pi.basisFun R n) (Pi.basisFun R m) = toLinearMap\u2082'\n[PROOFSTEP]\next M\n[GOAL]\ncase h.h.h.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\nM : Matrix n m R\ni\u271d\u00b9 : n\ni\u271d : m\n\u22a2 \u2191(comp (\u2191(comp (\u2191(toLinearMap\u2082 (Pi.basisFun R n) (Pi.basisFun R m)) M) (single i\u271d\u00b9)) 1) (single i\u271d)) 1 =\n    \u2191(comp (\u2191(comp (\u2191toLinearMap\u2082' M) (single i\u271d\u00b9)) 1) (single i\u271d)) 1\n[PROOFSTEP]\nsimp only [Matrix.toLinearMap\u2082_apply, Matrix.toLinearMap\u2082'_apply, Pi.basisFun_repr, coe_comp, Function.comp_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\n\u22a2 toMatrix\u2082 (Pi.basisFun R n) (Pi.basisFun R m) = toMatrix\u2082'\n[PROOFSTEP]\next B\n[GOAL]\ncase h.a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\nB : (n \u2192 R) \u2192\u2097[R] (m \u2192 R) \u2192\u2097[R] R\ni\u271d : n\nx\u271d : m\n\u22a2 \u2191(toMatrix\u2082 (Pi.basisFun R n) (Pi.basisFun R m)) B i\u271d x\u271d = \u2191toMatrix\u2082' B i\u271d x\u271d\n[PROOFSTEP]\nrw [LinearMap.toMatrix\u2082_apply, LinearMap.toMatrix\u2082'_apply, Pi.basisFun_apply, Pi.basisFun_apply]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\n\u22a2 \u2191(toMatrix\u2082 b\u2081' b\u2082') (compl\u2081\u2082 B l r) = (\u2191(toMatrix b\u2081' b\u2081) l)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix b\u2082' b\u2082) r\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 \u2191(toMatrix\u2082 b\u2081' b\u2082') (compl\u2081\u2082 B l r) i j = ((\u2191(toMatrix b\u2081' b\u2081) l)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix b\u2082' b\u2082) r) i j\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix\u2082_apply, compl\u2081\u2082_apply, transpose_apply, Matrix.mul_apply, LinearMap.toMatrix_apply,\n  LinearEquiv.coe_mk, sum_mul]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 \u2191(\u2191B (\u2191l (\u2191b\u2081' i))) (\u2191r (\u2191b\u2082' j)) =\n    \u2211 x : m, \u2211 x_1 : n, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) x_1 * \u2191(\u2191B (\u2191b\u2081 x_1)) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\nrw [sum_comm]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 \u2191(\u2191B (\u2191l (\u2191b\u2081' i))) (\u2191r (\u2191b\u2082' j)) =\n    \u2211 y : n, \u2211 x : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) y * \u2191(\u2191B (\u2191b\u2081 y)) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\nconv_lhs => rw [\u2190 LinearMap.sum_repr_mul_repr_mul b\u2081 b\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191b\u2081' i))) (\u2191r (\u2191b\u2082' j))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul b\u2081 b\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191b\u2081' i))) (\u2191r (\u2191b\u2082' j))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul b\u2081 b\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n| \u2191(\u2191B (\u2191l (\u2191b\u2081' i))) (\u2191r (\u2191b\u2082' j))\n[PROOFSTEP]\nrw [\u2190 LinearMap.sum_repr_mul_repr_mul b\u2081 b\u2082]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 (Finsupp.sum (\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) fun i xi =>\n      Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => xi \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 i)) (\u2191b\u2082 j)) =\n    \u2211 y : n, \u2211 x : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) y * \u2191(\u2191B (\u2191b\u2081 y)) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 (\u2211 i_1 : n,\n      Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i_1 \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 i_1)) (\u2191b\u2082 j)) =\n    \u2211 y : n, \u2211 x : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) y * \u2191(\u2191B (\u2191b\u2081 y)) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 \u2200 (x : n),\n    x \u2208 univ \u2192\n      (Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) x \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 x)) (\u2191b\u2082 j)) =\n        \u2211 x_1 : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) x * \u2191(\u2191B (\u2191b\u2081 x)) (\u2191b\u2082 x_1) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x_1\n[PROOFSTEP]\nrintro i' -\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\n\u22a2 (Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 j)) =\n    \u2211 x : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' * \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\nrw [Finsupp.sum_fintype]\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\n\u22a2 \u2211 i_1 : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) i_1 \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 i_1) =\n    \u2211 x : m, \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' * \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\n\u22a2 \u2200 (x : m),\n    x \u2208 univ \u2192\n      \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 x) =\n        \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' * \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 x) * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) x\n[PROOFSTEP]\nrintro j' -\n[GOAL]\ncase a.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\nj' : m\n\u22a2 \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) j' \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 j') =\n    \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' * \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 j') * \u2191(\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) j'\n[PROOFSTEP]\nsimp only [smul_eq_mul, LinearMap.toMatrix_apply, Basis.equivFun_apply, mul_assoc, mul_comm, mul_left_comm]\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\n\u22a2 \u2200 (i_1 : m), \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 0 \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 i_1) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni' : n\ni\u271d : m\n\u22a2 \u2191(\u2191b\u2081.repr (\u2191l (\u2191b\u2081' i))) i' \u2022 0 \u2022 \u2191(\u2191B (\u2191b\u2081 i')) (\u2191b\u2082 i\u271d) = 0\n[PROOFSTEP]\nsimp only [zero_smul, smul_zero]\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\n\u22a2 \u2200 (i : n), (Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => 0 \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 i)) (\u2191b\u2082 j)) = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase a.h.h\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nl : M\u2081' \u2192\u2097[R] M\u2081\nr : M\u2082' \u2192\u2097[R] M\u2082\ni : n'\nj : m'\ni\u271d : n\n\u22a2 (Finsupp.sum (\u2191b\u2082.repr (\u2191r (\u2191b\u2082' j))) fun j yj => 0 \u2022 yj \u2022 \u2191(\u2191B (\u2191b\u2081 i\u271d)) (\u2191b\u2082 j)) = 0\n[PROOFSTEP]\nsimp only [zero_smul, Finsupp.sum_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nf : M\u2081' \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toMatrix\u2082 b\u2081' b\u2082) (comp B f) = (\u2191(toMatrix b\u2081' b\u2081) f)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B\n[PROOFSTEP]\nrw [\u2190 LinearMap.compl\u2082_id (B.comp f), \u2190 LinearMap.compl\u2081\u2082, LinearMap.toMatrix\u2082_compl\u2081\u2082 b\u2081 b\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nf : M\u2081' \u2192\u2097[R] M\u2081\n\u22a2 (\u2191(toMatrix b\u2081' b\u2081) f)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix b\u2082 b\u2082) id = (\u2191(toMatrix b\u2081' b\u2081) f)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nf : M\u2082' \u2192\u2097[R] M\u2082\n\u22a2 \u2191(toMatrix\u2082 b\u2081 b\u2082') (compl\u2082 B f) = \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix b\u2082' b\u2082) f\n[PROOFSTEP]\nrw [\u2190 LinearMap.comp_id B, \u2190 LinearMap.compl\u2081\u2082, LinearMap.toMatrix\u2082_compl\u2081\u2082 b\u2081 b\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nf : M\u2082' \u2192\u2097[R] M\u2082\n\u22a2 (\u2191(toMatrix b\u2081 b\u2081) id)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix b\u2082' b\u2082) f =\n    \u2191(toMatrix\u2082 b\u2081 b\u2082) (comp B id) * \u2191(toMatrix b\u2082' b\u2082) f\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nc\u2081 : Basis n' R M\u2081\nc\u2082 : Basis m' R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 (Basis.toMatrix b\u2081 \u2191c\u2081)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * Basis.toMatrix b\u2082 \u2191c\u2082 = \u2191(toMatrix\u2082 c\u2081 c\u2082) B\n[PROOFSTEP]\nsimp_rw [\u2190 LinearMap.toMatrix_id_eq_basis_toMatrix]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nc\u2081 : Basis n' R M\u2081\nc\u2082 : Basis m' R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 (\u2191(toMatrix c\u2081 b\u2081) id)\u1d40 * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * \u2191(toMatrix c\u2082 b\u2082) id = \u2191(toMatrix\u2082 c\u2081 c\u2082) B\n[PROOFSTEP]\nrw [\u2190 LinearMap.toMatrix\u2082_compl\u2081\u2082, LinearMap.compl\u2081\u2082_id_id]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nM : Matrix n' n R\nN : Matrix m m' R\n\u22a2 M * \u2191(toMatrix\u2082 b\u2081 b\u2082) B * N = \u2191(toMatrix\u2082 b\u2081' b\u2082') (compl\u2081\u2082 B (\u2191(toLin b\u2081' b\u2081) M\u1d40) (\u2191(toLin b\u2082' b\u2082) N))\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrix\u2082_compl\u2081\u2082 b\u2081 b\u2082, toMatrix_toLin, transpose_transpose]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nM : Matrix n' n R\n\u22a2 M * \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081' b\u2082) (comp B (\u2191(toLin b\u2081' b\u2081) M\u1d40))\n[PROOFSTEP]\nrw [LinearMap.toMatrix\u2082_comp b\u2081, toMatrix_toLin, transpose_transpose]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nM : Matrix m m' R\n\u22a2 \u2191(toMatrix\u2082 b\u2081 b\u2082) B * M = \u2191(toMatrix\u2082 b\u2081 b\u2082') (compl\u2082 B (\u2191(toLin b\u2082' b\u2082) M))\n[PROOFSTEP]\nrw [LinearMap.toMatrix\u2082_compl\u2082 b\u2081 b\u2082, toMatrix_toLin]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommRing R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nM : Matrix n m R\nP : Matrix n n' R\nQ : Matrix m m' R\n\u22a2 \u2191(toMatrix\u2082 b\u2081' b\u2082') (compl\u2081\u2082 (\u2191(toLinearMap\u2082 b\u2081 b\u2082) M) (\u2191(toLin b\u2081' b\u2081) P) (\u2191(toLin b\u2082' b\u2082) Q)) =\n    \u2191(toMatrix\u2082 b\u2081' b\u2082') (\u2191(toLinearMap\u2082 b\u2081' b\u2082') (P\u1d40 * M * Q))\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix\u2082_compl\u2081\u2082 b\u2081 b\u2082, LinearMap.toMatrix\u2082_toLinearMap\u2082, toMatrix_toLin]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 LinearMap.IsAdjointPair (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J') (\u2191toLin' A) (\u2191toLin' A') \u2194\n    Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nrw [isAdjointPair_iff_comp_eq_compl\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 comp (\u2191toLinearMap\u2082' J') (\u2191toLin' A) = compl\u2082 (\u2191toLinearMap\u2082' J) (\u2191toLin' A') \u2194 Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nhave h : \u2200 B B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R, B = B' \u2194 LinearMap.toMatrix\u2082' B = LinearMap.toMatrix\u2082' B' :=\n  by\n  intro B B'\n  constructor <;> intro h\n  \u00b7 rw [h]\n  \u00b7 exact LinearMap.toMatrix\u2082'.injective h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 \u2200 (B B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R), B = B' \u2194 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n[PROOFSTEP]\nintro B B'\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R\n\u22a2 B = B' \u2194 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R\n\u22a2 B = B' \u2192 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R\n\u22a2 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B' \u2192 B = B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R\nh : B = B'\n\u22a2 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R\nh : \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n\u22a2 B = B'\n[PROOFSTEP]\nexact LinearMap.toMatrix\u2082'.injective h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : \u2200 (B B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R), B = B' \u2194 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n\u22a2 comp (\u2191toLinearMap\u2082' J') (\u2191toLin' A) = compl\u2082 (\u2191toLinearMap\u2082' J) (\u2191toLin' A') \u2194 Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nsimp_rw [h, LinearMap.toMatrix\u2082'_comp, LinearMap.toMatrix\u2082'_compl\u2082, LinearMap.toMatrix'_toLin',\n  LinearMap.toMatrix'_toLinearMap\u2082']\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : \u2200 (B B' : (n \u2192 R) \u2192\u2097[R] (n' \u2192 R) \u2192\u2097[R] R), B = B' \u2194 \u2191toMatrix\u2082' B = \u2191toMatrix\u2082' B'\n\u22a2 A\u1d40 * J' = J * A' \u2194 Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 LinearMap.IsAdjointPair (\u2191(toLinearMap\u2082 b\u2081 b\u2081) J) (\u2191(toLinearMap\u2082 b\u2082 b\u2082) J') (\u2191(toLin b\u2081 b\u2082) A) (\u2191(toLin b\u2082 b\u2081) A') \u2194\n    Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nrw [isAdjointPair_iff_comp_eq_compl\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 comp (\u2191(toLinearMap\u2082 b\u2082 b\u2082) J') (\u2191(toLin b\u2081 b\u2082) A) = compl\u2082 (\u2191(toLinearMap\u2082 b\u2081 b\u2081) J) (\u2191(toLin b\u2082 b\u2081) A') \u2194\n    Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nhave h : \u2200 B B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R, B = B' \u2194 LinearMap.toMatrix\u2082 b\u2081 b\u2082 B = LinearMap.toMatrix\u2082 b\u2081 b\u2082 B' :=\n  by\n  intro B B'\n  constructor <;> intro h\n  \u00b7 rw [h]\n  \u00b7 exact (LinearMap.toMatrix\u2082 b\u2081 b\u2082).injective h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 \u2200 (B B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R), B = B' \u2194 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n[PROOFSTEP]\nintro B B'\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 B = B' \u2194 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 B = B' \u2192 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B' \u2192 B = B'\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nh : B = B'\n\u22a2 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nB B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\nh : \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n\u22a2 B = B'\n[PROOFSTEP]\nexact (LinearMap.toMatrix\u2082 b\u2081 b\u2082).injective h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : \u2200 (B B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R), B = B' \u2194 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n\u22a2 comp (\u2191(toLinearMap\u2082 b\u2082 b\u2082) J') (\u2191(toLin b\u2081 b\u2082) A) = compl\u2082 (\u2191(toLinearMap\u2082 b\u2081 b\u2081) J) (\u2191(toLin b\u2082 b\u2081) A') \u2194\n    Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nsimp_rw [h, LinearMap.toMatrix\u2082_comp b\u2082 b\u2082, LinearMap.toMatrix\u2082_compl\u2082 b\u2081 b\u2081, LinearMap.toMatrix_toLin,\n  LinearMap.toMatrix\u2082_toLinearMap\u2082]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : \u2200 (B B' : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R), B = B' \u2194 \u2191(toMatrix\u2082 b\u2081 b\u2082) B = \u2191(toMatrix\u2082 b\u2081 b\u2082) B'\n\u22a2 A\u1d40 * J' = J * A' \u2194 Matrix.IsAdjointPair J J' A A'\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nhave h' : IsUnit P.det := P.isUnit_iff_isUnit_det.mp h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nlet u := P.nonsingInvUnit h'\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nlet v := P\u1d40.nonsingInvUnit (P.isUnit_det_transpose h')\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nlet x := A\u2081\u1d40 * P\u1d40 * J\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nlet y := J * P * A\u2081\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nsuffices x * u.val = v.val * y \u2194 (v\u207b\u00b9).val * x = y * (u\u207b\u00b9).val\n  by\n  dsimp only [Matrix.IsAdjointPair]\n  simp only [Matrix.transpose_mul]\n  simp only [\u2190 mul_assoc, P.transpose_nonsing_inv]\n    -- porting note: the previous proof used `conv` and was causing timeouts, so we use `convert`\n  convert this using 2\n  \u00b7 rw [mul_assoc, mul_assoc, \u2190 mul_assoc J]\n    rfl\n  \u00b7 rw [mul_assoc, mul_assoc, \u2190 mul_assoc _ _ J]\n    rfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 IsAdjointPair (P\u1d40 * J * P) (P\u1d40 * J * P) A\u2081 A\u2081 \u2194 IsAdjointPair J J (P * A\u2081 * P\u207b\u00b9) (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\ndsimp only [Matrix.IsAdjointPair]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 A\u2081\u1d40 * (P\u1d40 * J * P) = P\u1d40 * J * P * A\u2081 \u2194 (P * A\u2081 * P\u207b\u00b9)\u1d40 * J = J * (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nsimp only [Matrix.transpose_mul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 A\u2081\u1d40 * (P\u1d40 * J * P) = P\u1d40 * J * P * A\u2081 \u2194 P\u207b\u00b9\u1d40 * (A\u2081\u1d40 * P\u1d40) * J = J * (P * A\u2081 * P\u207b\u00b9)\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc, P.transpose_nonsing_inv]\n  -- porting note: the previous proof used `conv` and was causing timeouts, so we use `convert`\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 A\u2081\u1d40 * P\u1d40 * J * P = P\u1d40 * J * P * A\u2081 \u2194 P\u1d40\u207b\u00b9 * A\u2081\u1d40 * P\u1d40 * J = J * P * A\u2081 * P\u207b\u00b9\n[PROOFSTEP]\nconvert this using 2\n[GOAL]\ncase h.e'_1.h.e'_3\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 P\u1d40 * J * P * A\u2081 = \u2191v * y\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, \u2190 mul_assoc J]\n[GOAL]\ncase h.e'_1.h.e'_3\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 P\u1d40 * (J * P * A\u2081) = \u2191v * y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.e'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 P\u1d40\u207b\u00b9 * A\u2081\u1d40 * P\u1d40 * J = \u2191v\u207b\u00b9 * x\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, \u2190 mul_assoc _ _ J]\n[GOAL]\ncase h.e'_2.h.e'_2\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\nthis : x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n\u22a2 P\u1d40\u207b\u00b9 * (A\u2081\u1d40 * P\u1d40 * J) = \u2191v\u207b\u00b9 * x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n\u22a2 x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x = y * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [Units.eq_mul_inv_iff_mul_eq]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n\u22a2 x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * x * \u2191u = y\n[PROOFSTEP]\nconv_rhs => rw [mul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n| \u2191v\u207b\u00b9 * x * \u2191u = y\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n| \u2191v\u207b\u00b9 * x * \u2191u = y\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n| \u2191v\u207b\u00b9 * x * \u2191u = y\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nP : Matrix n n R\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R)\u02e3 := nonsingInvUnit P h'\nv : (Matrix n n R)\u02e3 := nonsingInvUnit P\u1d40 (_ : IsUnit (det P\u1d40))\nx : Matrix n n R := A\u2081\u1d40 * P\u1d40 * J\ny : Matrix n n R := J * P * A\u2081\n\u22a2 x * \u2191u = \u2191v * y \u2194 \u2191v\u207b\u00b9 * (x * \u2191u) = y\n[PROOFSTEP]\nrw [v.inv_mul_eq_iff_eq_mul]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 A\u2081 \u2208 pairSelfAdjointMatricesSubmodule J J\u2082 \u2194 Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nsimp only [pairSelfAdjointMatricesSubmodule, LinearEquiv.coe_coe, LinearMap.toMatrix'_apply, Submodule.mem_map,\n  mem_isPairSelfAdjointSubmodule]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 (\u2203 y, IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) y \u2227 \u2191toMatrix' y = A\u2081) \u2194\n    Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 (\u2203 y, IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) y \u2227 \u2191toMatrix' y = A\u2081) \u2192\n    Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nrintro \u27e8f, hf, hA\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nf : Module.End R (n \u2192 R)\nhf : IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) f\nhA : \u2191toMatrix' f = A\u2081\n\u22a2 Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nhave hf' : f = toLin' A\u2081 := by rw [\u2190 hA, Matrix.toLin'_toMatrix']\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nf : Module.End R (n \u2192 R)\nhf : IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) f\nhA : \u2191toMatrix' f = A\u2081\n\u22a2 f = \u2191toLin' A\u2081\n[PROOFSTEP]\nrw [\u2190 hA, Matrix.toLin'_toMatrix']\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nf : Module.End R (n \u2192 R)\nhf : IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) f\nhA : \u2191toMatrix' f = A\u2081\nhf' : f = \u2191toLin' A\u2081\n\u22a2 Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nrw [hf'] at hf \n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nf : Module.End R (n \u2192 R)\nhf : IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) (\u2191toLin' A\u2081)\nhA : \u2191toMatrix' f = A\u2081\nhf' : f = \u2191toLin' A\u2081\n\u22a2 Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n[PROOFSTEP]\nrw [\u2190 isAdjointPair_toLinearMap\u2082']\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nf : Module.End R (n \u2192 R)\nhf : IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) (\u2191toLin' A\u2081)\nhA : \u2191toMatrix' f = A\u2081\nhf' : f = \u2191toLin' A\u2081\n\u22a2 LinearMap.IsAdjointPair (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) (\u2191toLin' A\u2081) (\u2191toLin' A\u2081)\n[PROOFSTEP]\nexact hf\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081 \u2192 \u2203 y, IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) y \u2227 \u2191toMatrix' y = A\u2081\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n\u22a2 \u2203 y, IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) y \u2227 \u2191toMatrix' y = A\u2081\n[PROOFSTEP]\nrefine' \u27e8toLin' A\u2081, _, LinearMap.toMatrix'_toLin' _\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\nh : Matrix.IsAdjointPair J J\u2082 A\u2081 A\u2081\n\u22a2 IsPairSelfAdjoint (\u2191toLinearMap\u2082' J) (\u2191toLinearMap\u2082' J\u2082) (\u2191toLin' A\u2081)\n[PROOFSTEP]\nexact (isAdjointPair_toLinearMap\u2082' _ _ _ _).mpr h\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 A\u2081 \u2208 selfAdjointMatricesSubmodule J \u2194 Matrix.IsSelfAdjoint J A\u2081\n[PROOFSTEP]\nerw [mem_pairSelfAdjointMatricesSubmodule]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 Matrix.IsAdjointPair J J A\u2081 A\u2081 \u2194 Matrix.IsSelfAdjoint J A\u2081\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 A\u2081 \u2208 skewAdjointMatricesSubmodule J \u2194 Matrix.IsSkewAdjoint J A\u2081\n[PROOFSTEP]\nerw [mem_pairSelfAdjointMatricesSubmodule]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : Module R M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2082\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : Fintype n'\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis n' R M\u2082\nJ J\u2082 : Matrix n n R\nJ' : Matrix n' n' R\nA : Matrix n' n R\nA' : Matrix n n' R\nA\u2081 : Matrix n n R\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : DecidableEq n'\n\u22a2 Matrix.IsAdjointPair (-J) J A\u2081 A\u2081 \u2194 Matrix.IsSkewAdjoint J A\u2081\n[PROOFSTEP]\nsimp [Matrix.IsSkewAdjoint, Matrix.IsAdjointPair]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommRing R\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u2192\u2097[R\u2081] R\u2081\nM : Matrix \u03b9 \u03b9 R\u2081\nh : Matrix.Nondegenerate M\nx : \u03b9 \u2192 R\u2081\nhx : \u2200 (y : \u03b9 \u2192 R\u2081), \u2191(\u2191(\u2191Matrix.toLinearMap\u2082' M) x) y = 0\ny : \u03b9 \u2192 R\u2081\n\u22a2 x \u2b1d\u1d65 mulVec M y = 0\n[PROOFSTEP]\nsimpa only [toLinearMap\u2082'_apply'] using hx y\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2074 : CommRing R\u2081\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\nB : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u2192\u2097[R\u2081] R\u2081\nM : Matrix \u03b9 \u03b9 R\u2081\nb : Basis \u03b9 R\u2081 M\u2081\n\u22a2 SeparatingLeft (\u2191(toLinearMap\u2082 b b) M) \u2194 Matrix.Nondegenerate M\n[PROOFSTEP]\nrw [\u2190 Matrix.separatingLeft_toLinearMap\u2082'_iff_separatingLeft_toLinearMap\u2082, Matrix.separatingLeft_toLinearMap\u2082'_iff]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM\u271d : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nB : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u2192\u2097[R\u2081] R\u2081\ninst\u271d : IsDomain R\u2081\nM : Matrix \u03b9 \u03b9 R\u2081\n\u22a2 SeparatingLeft (\u2191toLinearMap\u2082' M) \u2194 det M \u2260 0\n[PROOFSTEP]\nrw [Matrix.separatingLeft_toLinearMap\u2082'_iff, Matrix.nondegenerate_iff_det_ne_zero]\n[GOAL]\nR : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u2075 : CommRing R\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2081\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : Fintype \u03b9\nB\u271d : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u2192\u2097[R\u2081] R\u2081\ninst\u271d : IsDomain R\u2081\nB : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u2192\u2097[R\u2081] R\u2081\nb : Basis \u03b9 R\u2081 M\u2081\n\u22a2 SeparatingLeft B \u2194 det (\u2191(toMatrix\u2082 b b) B) \u2260 0\n[PROOFSTEP]\nrw [\u2190 Matrix.nondegenerate_iff_det_ne_zero, nondegenerate_toMatrix_iff]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.SesquilinearForm", "llama_tokens": 60072, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.665410572017153, "lm_q1q2_score": 0.5242977409756244}}
{"text": "[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx y : FractionRing K[X]\nxy : { toFractionRing := x }.toFractionRing = { toFractionRing := y }.toFractionRing\n\u22a2 { toFractionRing := x } = { toFractionRing := y }\n[PROOFSTEP]\nsubst xy\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx : FractionRing K[X]\n\u22a2 { toFractionRing := x } = { toFractionRing := { toFractionRing := x }.toFractionRing }\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nx : RatFunc K\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n\u22a2 P\n[PROOFSTEP]\nrefine Localization.liftOn (toFractionRing x) (fun p q => f p q) ?_\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nx : RatFunc K\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n\u22a2 \u2200 {a c : K[X]} {b d : { x // x \u2208 K[X]\u2070 }},\n    \u2191(Localization.r K[X]\u2070) (a, b) (c, d) \u2192 (fun p q => f p \u2191q) a b = (fun p q => f p \u2191q) c d\n[PROOFSTEP]\nintros p p' q q' h\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nx : RatFunc K\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\np p' : K[X]\nq q' : { x // x \u2208 K[X]\u2070 }\nh : \u2191(Localization.r K[X]\u2070) (p, q) (p', q')\n\u22a2 (fun p q => f p \u2191q) p q = (fun p q => f p \u2191q) p' q'\n[PROOFSTEP]\nexact\n  H q.2 q'.2\n    (let \u27e8\u27e8c, hc\u27e9, mul_eq\u27e9 := Localization.r_iff_exists.mp h\n    mul_cancel_left_coe_nonZeroDivisors.mp mul_eq)\n      -- porting note: the definition above was as follows\n      --    (-- Fix timeout by manipulating elaboration order\n      --    fun p q => f p q)\n      --    fun p p' q q' h => by\n      --    exact H q.2 q'.2\n      --      (let \u27e8\u27e8c, hc\u27e9, mul_eq\u27e9 := Localization.r_iff_exists.mp h\n      --      mul_cancel_left_coe_nonZeroDivisors.mp mul_eq)\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x \u2208 K[X]\u2070 }\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n\u22a2 RatFunc.liftOn { toFractionRing := Localization.mk n d } f H = f n \u2191d\n[PROOFSTEP]\nrw [RatFunc.liftOn]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nn : K[X]\nd : { x // x \u2208 K[X]\u2070 }\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n\u22a2 Localization.liftOn { toFractionRing := Localization.mk n d }.toFractionRing (fun p q => f p \u2191q)\n      (_ : \u2200 {p p' : K[X]} {q q' : { x // x \u2208 K[X]\u2070 }}, \u2191(Localization.r K[X]\u2070) (p, q) (p', q') \u2192 f p \u2191q = f p' \u2191q') =\n    f n \u2191d\n[PROOFSTEP]\nexact Localization.liftOn_mk _ _ _ _\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nP : Sort v\nf : K[X] \u2192 K[X] \u2192 P\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\np q p' q' : K[X]\nhq : q \u2260 0\nhq' : q' \u2260 0\nh : q' * p = q * p'\n\u22a2 f (q' * p) (q' * q) = f (q * p') (q * q')\n[PROOFSTEP]\nrw [h, mul_comm q']\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : K[X]\n\u22a2 RatFunc.mk p q =\n    { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) p / \u2191(algebraMap K[X] (FractionRing K[X])) q }\n[PROOFSTEP]\nrw [RatFunc.mk]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : K[X]\n\u22a2 RatFunc.mk p 0 = { toFractionRing := 0 }\n[PROOFSTEP]\nrw [mk_eq_div', RingHom.map_zero, div_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : K[X]\nq : { x // x \u2208 K[X]\u2070 }\n\u22a2 RatFunc.mk p \u2191q = { toFractionRing := IsLocalization.mk' (FractionRing K[X]) p q }\n[PROOFSTEP]\nsimp only [mk_eq_div', \u2190 Localization.mk_eq_mk', FractionRing.mk_eq_div]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : K[X]\nhq : q \u2208 K[X]\u2070\n\u22a2 RatFunc.mk p q = { toFractionRing := IsLocalization.mk' (FractionRing K[X]) p { val := q, property := hq } }\n[PROOFSTEP]\nsimp only [\u2190 mk_coe_def]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : K[X]\nhq : q \u2260 0\n\u22a2 RatFunc.mk p q = { toFractionRing := Localization.mk p { val := q, property := (_ : q \u2208 K[X]\u2070) } }\n[PROOFSTEP]\nrw [mk_def_of_ne, Localization.mk_eq_mk']\n[GOAL]\ncase hq\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : K[X]\nhq : q \u2260 0\n\u22a2 q \u2260 0\n[PROOFSTEP]\nexact hq\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : K[X]\n\u22a2 RatFunc.mk p 1 = { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) p }\n[PROOFSTEP]\nrw [\u2190 IsLocalization.mk'_one (M := K[X]\u2070) (FractionRing K[X]) p, \u2190 mk_coe_def, Submonoid.coe_one]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q p' q' : K[X]\nhq : q \u2260 0\nhq' : q' \u2260 0\n\u22a2 RatFunc.mk p q = RatFunc.mk p' q' \u2194 p * q' = p' * q\n[PROOFSTEP]\nrw [mk_def_of_ne _ hq, mk_def_of_ne _ hq', ofFractionRing_injective.eq_iff, IsLocalization.mk'_eq_iff_eq',\n  -- porting note: removed `[anonymous], [anonymous]`(IsFractionRing.injective K[X] (FractionRing K[X])).eq_iff]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH' : \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\nH :\n  optParam (\u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n    (_ : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n\u22a2 RatFunc.liftOn (RatFunc.mk p q) f H = f p q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH' : \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\nH :\n  optParam (\u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n    (_ : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\nhq : q = 0\n\u22a2 RatFunc.liftOn (RatFunc.mk p q) f H = f p q\n[PROOFSTEP]\nsubst hq\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH' : \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\nH :\n  optParam (\u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n    (_ : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n\u22a2 RatFunc.liftOn (RatFunc.mk p 0) f H = f p 0\n[PROOFSTEP]\nsimp only [mk_zero, f0, \u2190 Localization.mk_zero 1, Localization.liftOn_mk, liftOn_ofFractionRing_mk, Submonoid.coe_one]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH' : \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\nH :\n  optParam (\u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n    (_ : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\nhq : \u00acq = 0\n\u22a2 RatFunc.liftOn (RatFunc.mk p q) f H = f p q\n[PROOFSTEP]\nsimp only [mk_eq_localization_mk _ hq, Localization.liftOn_mk, liftOn_ofFractionRing_mk]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 RatFunc.liftOn' (RatFunc.mk p q) f H = f p q\n[PROOFSTEP]\nrw [RatFunc.liftOn', RatFunc.liftOn_mk _ _ _ f0]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n[PROOFSTEP]\napply lift_on_condition_of_lift_on'_condition H\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : RatFunc K \u2192 Prop\nx : FractionRing K[X]\nf : \u2200 (p q : K[X]), q \u2260 0 \u2192 P (RatFunc.mk p q)\nx\u271d : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\np : K[X]\nq : { x // x \u2208 K[X]\u2070 }\n\u22a2 P { toFractionRing := Localization.mk (p, q).fst (p, q).snd }\n[PROOFSTEP]\nsimpa only [mk_coe_def, Localization.mk_eq_mk'] using f p q (mem_nonZeroDivisors_iff_ne_zero.mp q.2)\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 { toFractionRing := 0 } = 0\n[PROOFSTEP]\nsimp only [Zero.zero, OfNat.ofNat, RatFunc.zero]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\np q : FractionRing K[X]\n\u22a2 { toFractionRing := p + q } = { toFractionRing := p } + { toFractionRing := q }\n[PROOFSTEP]\nsimp only [HAdd.hAdd, Add.add, RatFunc.add]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\np q : FractionRing K[X]\n\u22a2 { toFractionRing := p - q } = { toFractionRing := p } - { toFractionRing := q }\n[PROOFSTEP]\nsimp only [Sub.sub, HSub.hSub, RatFunc.sub]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\np : FractionRing K[X]\n\u22a2 { toFractionRing := -p } = -{ toFractionRing := p }\n[PROOFSTEP]\nsimp only [Neg.neg, RatFunc.neg]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 { toFractionRing := 1 } = 1\n[PROOFSTEP]\nsimp only [One.one, OfNat.ofNat, RatFunc.one]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\np q : FractionRing K[X]\n\u22a2 { toFractionRing := p * q } = { toFractionRing := p } * { toFractionRing := q }\n[PROOFSTEP]\nsimp only [Mul.mul, HMul.hMul, RatFunc.mul]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : FractionRing K[X]\n\u22a2 { toFractionRing := p / q } = { toFractionRing := p } / { toFractionRing := q }\n[PROOFSTEP]\nsimp only [Div.div, HDiv.hDiv, RatFunc.div]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : FractionRing K[X]\n\u22a2 { toFractionRing := p\u207b\u00b9 } = { toFractionRing := p }\u207b\u00b9\n[PROOFSTEP]\nsimp only [Inv.inv, RatFunc.inv]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } \u2260 0\n\u22a2 { toFractionRing := p } * { toFractionRing := p }\u207b\u00b9 = 1\n[PROOFSTEP]\nhave : p \u2260 0 := fun hp => h <| by rw [hp, ofFractionRing_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } \u2260 0\nhp : p = 0\n\u22a2 { toFractionRing := p } = 0\n[PROOFSTEP]\nrw [hp, ofFractionRing_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } \u2260 0\nthis : p \u2260 0\n\u22a2 { toFractionRing := p } * { toFractionRing := p }\u207b\u00b9 = 1\n[PROOFSTEP]\nsimpa only [\u2190 ofFractionRing_inv, \u2190 ofFractionRing_mul, \u2190 ofFractionRing_one, ofFractionRing.injEq] using\n  -- porting note: `ofFractionRing.injEq` was not present_root_.mul_inv_cancel this\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\nR : Type u_1\ninst\u271d : SMul R (FractionRing K[X])\nc : R\np : FractionRing K[X]\n\u22a2 { toFractionRing := c \u2022 p } = c \u2022 { toFractionRing := p }\n[PROOFSTEP]\nsimp only [SMul.smul, HSMul.hSMul, RatFunc.smul]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\nR : Type u_1\ninst\u271d : SMul R (FractionRing K[X])\nc : R\np : RatFunc K\n\u22a2 (c \u2022 p).toFractionRing = c \u2022 p.toFractionRing\n[PROOFSTEP]\ncases p\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d\u00b9 : CommRing K\nR : Type u_1\ninst\u271d : SMul R (FractionRing K[X])\nc : R\ntoFractionRing\u271d : FractionRing K[X]\n\u22a2 (c \u2022 { toFractionRing := toFractionRing\u271d }).toFractionRing = c \u2022 { toFractionRing := toFractionRing\u271d }.toFractionRing\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_smul]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nR : Type u_1\nx : RatFunc K\nr : K\n\u22a2 r \u2022 x = \u2191Polynomial.C r \u2022 x\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d : CommRing K\nR : Type u_1\nr : K\nx : FractionRing K[X]\n\u22a2 r \u2022 { toFractionRing := x } = \u2191Polynomial.C r \u2022 { toFractionRing := x }\n[PROOFSTEP]\ninduction x using Localization.induction_on\n[GOAL]\ncase ofFractionRing.H\nK : Type u\ninst\u271d : CommRing K\nR : Type u_1\nr : K\ny\u271d : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 r \u2022 { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd } =\n    \u2191Polynomial.C r \u2022 { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd }\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_smul, \u2190 ofFractionRing_smul, Localization.smul_mk, Localization.smul_mk, smul_eq_mul,\n  Polynomial.smul_eq_C_mul]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\n\u22a2 RatFunc.mk (c \u2022 p) q = c \u2022 RatFunc.mk p q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : q = 0\n\u22a2 RatFunc.mk (c \u2022 p) q = c \u2022 RatFunc.mk p q\n[PROOFSTEP]\nrw [hq, mk_zero, mk_zero, \u2190 ofFractionRing_smul, smul_zero]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nhq : \u00acq = 0\n\u22a2 RatFunc.mk (c \u2022 p) q = c \u2022 RatFunc.mk p q\n[PROOFSTEP]\nrw [mk_eq_localization_mk _ hq, mk_eq_localization_mk _ hq, \u2190 Localization.smul_mk, \u2190 ofFractionRing_smul]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\n\u22a2 \u2200 (x : R) (y : K[X]) (z : RatFunc K), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n[PROOFSTEP]\nintros c p q\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np : K[X]\nq : RatFunc K\n\u22a2 (c \u2022 p) \u2022 q = c \u2022 p \u2022 q\n[PROOFSTEP]\napply q.induction_on' fun q r _ => by rw [\u2190 mk_smul, smul_assoc, mk_smul, mk_smul]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nR : Type u_1\ninst\u271d\u00b3 : IsDomain K\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np : K[X]\nq\u271d : RatFunc K\nq r : K[X]\nx\u271d : r \u2260 0\n\u22a2 (c \u2022 p) \u2022 RatFunc.mk q r = c \u2022 p \u2022 RatFunc.mk q r\n[PROOFSTEP]\nrw [\u2190 mk_smul, smul_assoc, mk_smul, mk_smul]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx\u271d\u00b9 x\u271d : RatFunc K\ntoFractionRing\u271d\u00b9 toFractionRing\u271d : FractionRing K[X]\n\u22a2 Equiv.toFun\n      { toFun := toFractionRing, invFun := ofFractionRing,\n        left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n        right_inv :=\n          (_ :\n            \u2200 (x : FractionRing K[X]),\n              { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n      ({ toFractionRing := toFractionRing\u271d\u00b9 } * { toFractionRing := toFractionRing\u271d }) =\n    Equiv.toFun\n        { toFun := toFractionRing, invFun := ofFractionRing,\n          left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n          right_inv :=\n            (_ :\n              \u2200 (x : FractionRing K[X]),\n                { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n        { toFractionRing := toFractionRing\u271d\u00b9 } *\n      Equiv.toFun\n        { toFun := toFractionRing, invFun := ofFractionRing,\n          left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n          right_inv :=\n            (_ :\n              \u2200 (x : FractionRing K[X]),\n                { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n        { toFractionRing := toFractionRing\u271d }\n[PROOFSTEP]\nsimp [\u2190 ofFractionRing_mul]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx\u271d\u00b9 x\u271d : RatFunc K\ntoFractionRing\u271d\u00b9 toFractionRing\u271d : FractionRing K[X]\n\u22a2 Equiv.toFun\n      { toFun := toFractionRing, invFun := ofFractionRing,\n        left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n        right_inv :=\n          (_ :\n            \u2200 (x : FractionRing K[X]),\n              { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n      ({ toFractionRing := toFractionRing\u271d\u00b9 } + { toFractionRing := toFractionRing\u271d }) =\n    Equiv.toFun\n        { toFun := toFractionRing, invFun := ofFractionRing,\n          left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n          right_inv :=\n            (_ :\n              \u2200 (x : FractionRing K[X]),\n                { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n        { toFractionRing := toFractionRing\u271d\u00b9 } +\n      Equiv.toFun\n        { toFun := toFractionRing, invFun := ofFractionRing,\n          left_inv := (_ : \u2200 (x : RatFunc K), { toFractionRing := x.toFractionRing } = x),\n          right_inv :=\n            (_ :\n              \u2200 (x : FractionRing K[X]),\n                { toFractionRing := x }.toFractionRing = { toFractionRing := x }.toFractionRing) }\n        { toFractionRing := toFractionRing\u271d }\n[PROOFSTEP]\nsimp [\u2190 ofFractionRing_add]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b c : RatFunc K), a * b * c = a * (b * c)\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), 1 * a = a\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), a * 1 = a\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b : RatFunc K), a * b = b * a\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b c : RatFunc K), a + b + c = a + (b + c)\n[PROOFSTEP]\nfrac_tac\n  -- porting note: `by frac_tac` didn't work:\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), 0 + a = a\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), a + 0 = a\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (x : RatFunc K), (fun x x_1 => x \u2022 x_1) 0 x = 0\n[PROOFSTEP]\nsmul_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx\u271d : \u2115\n\u22a2 \u2200 (x : RatFunc K), (fun x x_1 => x \u2022 x_1) (x\u271d + 1) x = x + (fun x x_1 => x \u2022 x_1) x\u271d x\n[PROOFSTEP]\nsmul_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b : RatFunc K), a - b = a + -b\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), (fun x x_1 => x \u2022 x_1) 0 a = 0\n[PROOFSTEP]\nsmul_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx\u271d : \u2115\n\u22a2 \u2200 (a : RatFunc K), (fun x x_1 => x \u2022 x_1) (Int.ofNat (Nat.succ x\u271d)) a = a + (fun x x_1 => x \u2022 x_1) (Int.ofNat x\u271d) a\n[PROOFSTEP]\nsmul_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nx\u271d : \u2115\n\u22a2 \u2200 (a : RatFunc K), (fun x x_1 => x \u2022 x_1) (Int.negSucc x\u271d) a = -(fun x x_1 => x \u2022 x_1) (\u2191(Nat.succ x\u271d)) a\n[PROOFSTEP]\nsmul_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a : RatFunc K), -a + a = 0\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b : RatFunc K), a + b = b + a\n[PROOFSTEP]\nrepeat rintro (\u27e8\u27e9 : RatFunc _) <;> simp only [\u2190 ofFractionRing_add, add_comm]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\n\u22a2 \u2200 (a b : RatFunc K), a + b = b + a\n[PROOFSTEP]\nrintro (\u27e8\u27e9 : RatFunc _)\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d : CommRing K\ntoFractionRing\u271d : FractionRing K[X]\n\u22a2 \u2200 (b : RatFunc K), { toFractionRing := toFractionRing\u271d } + b = b + { toFractionRing := toFractionRing\u271d }\n[PROOFSTEP]\nsimp only [\u2190 ofFractionRing_add, add_comm]\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d : CommRing K\ntoFractionRing\u271d : FractionRing K[X]\n\u22a2 \u2200 (b : RatFunc K), { toFractionRing := toFractionRing\u271d } + b = b + { toFractionRing := toFractionRing\u271d }\n[PROOFSTEP]\nrintro (\u27e8\u27e9 : RatFunc _)\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d : CommRing K\ntoFractionRing\u271d\u00b9 toFractionRing\u271d : FractionRing K[X]\n\u22a2 { toFractionRing := toFractionRing\u271d\u00b9 } + { toFractionRing := toFractionRing\u271d } =\n    { toFractionRing := toFractionRing\u271d } + { toFractionRing := toFractionRing\u271d\u00b9 }\n[PROOFSTEP]\nsimp only [\u2190 ofFractionRing_add, add_comm]\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nsrc\u271d\u00b9 : CommMonoid (RatFunc K) := instCommMonoid K\nsrc\u271d : AddCommGroup (RatFunc K) := instAddCommGroup K\n\u22a2 \u2200 (a b c : RatFunc K), a * (b + c) = a * b + a * c\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nsrc\u271d\u00b9 : CommMonoid (RatFunc K) := instCommMonoid K\nsrc\u271d : AddCommGroup (RatFunc K) := instAddCommGroup K\n\u22a2 \u2200 (a b c : RatFunc K), (a + b) * c = a * c + b * c\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nsrc\u271d\u00b9 : CommMonoid (RatFunc K) := instCommMonoid K\nsrc\u271d : AddCommGroup (RatFunc K) := instAddCommGroup K\n\u22a2 \u2200 (a : RatFunc K), 0 * a = 0\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d : CommRing K\nsrc\u271d\u00b9 : CommMonoid (RatFunc K) := instCommMonoid K\nsrc\u271d : AddCommGroup (RatFunc K) := instAddCommGroup K\n\u22a2 \u2200 (a : RatFunc K), a * 0 = 0\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      p q =\n    (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      p' q'\n[PROOFSTEP]\ndsimp only\n  -- porting note: force the function to be applied\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } } else 0) =\n    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } } else 0\n[PROOFSTEP]\nrw [dif_pos, dif_pos]\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := ?hc } } =\n    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := ?hc } }\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q' \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\n[PROOFSTEP]\ncongr 1\n  -- porting note: this was a `rw [ofFractionRing.inj_eq]` which was overkill anyway\n[GOAL]\ncase e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := ?hc } = Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := ?hc }\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q' \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\n[PROOFSTEP]\nrw [Localization.mk_eq_mk_iff]\n[GOAL]\ncase e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191(Localization.r S[X]\u2070) (\u2191\u03c6 p, { val := \u2191\u03c6 q, property := ?hc }) (\u2191\u03c6 p', { val := \u2191\u03c6 q', property := ?hc })\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q' \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q' \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q \u2208 S[X]\u2070\n[PROOFSTEP]\nexact h\u03c6 hq\n[GOAL]\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191\u03c6 q' \u2208 S[X]\u2070\n[PROOFSTEP]\nexact h\u03c6 hq'\n[GOAL]\ncase e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191(Localization.r S[X]\u2070) (\u2191\u03c6 p, { val := \u2191\u03c6 q, property := (_ : q \u2208 Submonoid.comap \u03c6 S[X]\u2070) })\n    (\u2191\u03c6 p', { val := \u2191\u03c6 q', property := (_ : q' \u2208 Submonoid.comap \u03c6 S[X]\u2070) })\n[PROOFSTEP]\nrefine' Localization.r_of_eq _\n[GOAL]\ncase e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 \u2191(\u2191\u03c6 p', { val := \u2191\u03c6 q', property := (_ : q' \u2208 Submonoid.comap \u03c6 S[X]\u2070) }).snd *\n      (\u2191\u03c6 p, { val := \u2191\u03c6 q, property := (_ : q \u2208 Submonoid.comap \u03c6 S[X]\u2070) }).fst =\n    \u2191(\u2191\u03c6 p, { val := \u2191\u03c6 q, property := (_ : q \u2208 Submonoid.comap \u03c6 S[X]\u2070) }).snd *\n      (\u2191\u03c6 p', { val := \u2191\u03c6 q', property := (_ : q' \u2208 Submonoid.comap \u03c6 S[X]\u2070) }).fst\n[PROOFSTEP]\nsimpa only [map_mul] using congr_arg \u03c6 h\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\n\u22a2 (fun f =>\n        RatFunc.liftOn f\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (fun n d =>\n                          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                            { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                          else 0)\n                        p q =\n                      (fun n d =>\n                          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                            { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                          else 0)\n                        p' q'))\n      1 =\n    1\n[PROOFSTEP]\ndsimp only\n  -- porting note: force the function to be applied\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\n\u22a2 RatFunc.liftOn 1\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    1\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_one, \u2190 Localization.mk_one, liftOn_ofFractionRing_mk, dif_pos]\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\n\u22a2 { toFractionRing := Localization.mk (\u2191\u03c6 1) { val := \u2191\u03c6 \u21911, property := ?hc } } = 1\n[PROOFSTEP]\nsimpa using ofFractionRing_one\n[GOAL]\ncase hc\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\n\u22a2 \u2191\u03c6 \u21911 \u2208 S[X]\u2070\n[PROOFSTEP]\nsimpa using Submonoid.one_mem _\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nx y : RatFunc R\n\u22a2 OneHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (fun n d =>\n                            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                              { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                            else 0)\n                          p q =\n                        (fun n d =>\n                            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                              { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                            else 0)\n                          p' q'),\n        map_one' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f\n                    (fun n d =>\n                      if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                      else 0)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192\n                            q' * p = q * p' \u2192\n                              (fun n d =>\n                                    if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                      { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                    else 0)\n                                  p q =\n                                (fun n d =>\n                                    if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                      { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                    else 0)\n                                  p' q'))\n                1 =\n              1) }\n      (x * y) =\n    OneHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f\n              (fun n d =>\n                if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                else 0)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192\n                      q' * p = q * p' \u2192\n                        (fun n d =>\n                              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                              else 0)\n                            p q =\n                          (fun n d =>\n                              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                              else 0)\n                            p' q'),\n          map_one' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f\n                      (fun n d =>\n                        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                        else 0)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192\n                                (fun n d =>\n                                      if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                        { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                      else 0)\n                                    p q =\n                                  (fun n d =>\n                                      if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                        { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                      else 0)\n                                    p' q'))\n                  1 =\n                1) }\n        x *\n      OneHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f\n              (fun n d =>\n                if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                else 0)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192\n                      q' * p = q * p' \u2192\n                        (fun n d =>\n                              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                              else 0)\n                            p q =\n                          (fun n d =>\n                              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                              else 0)\n                            p' q'),\n          map_one' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f\n                      (fun n d =>\n                        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                        else 0)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192\n                                (fun n d =>\n                                      if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                        { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                      else 0)\n                                    p q =\n                                  (fun n d =>\n                                      if h : \u2191\u03c6 d \u2208 S[X]\u2070 then\n                                        { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n                                      else 0)\n                                    p' q'))\n                  1 =\n                1) }\n        y\n[PROOFSTEP]\ndsimp only\n  -- porting note: force the function to be applied\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nx y : RatFunc R\n\u22a2 RatFunc.liftOn (x * y)\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn x\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn y\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\ny : RatFunc R\nx : FractionRing R[X]\n\u22a2 RatFunc.liftOn ({ toFractionRing := x } * y)\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := x }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn y\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\ncases' y with y\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nx y : FractionRing R[X]\n\u22a2 RatFunc.liftOn ({ toFractionRing := x } * { toFractionRing := y })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := x }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := y }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\ninduction' x using Localization.rec with p q\n[GOAL]\ncase ofFractionRing.ofFractionRing.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\n\u22a2 RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := y })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := Localization.mk p q }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := y }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := y }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0)) =\n    (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := y }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0))\n[PROOFSTEP]\ninduction' y using Localization.rec with p' q'\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk p' q' })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := Localization.mk p q }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := Localization.mk p' q' }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\nhave hq : \u03c6 q \u2208 S[X]\u2070 := h\u03c6 q.prop\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\nhq : \u2191\u03c6 \u2191q \u2208 S[X]\u2070\n\u22a2 RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk p' q' })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := Localization.mk p q }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := Localization.mk p' q' }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\nhave hq' : \u03c6 q' \u2208 S[X]\u2070 := h\u03c6 q'.prop\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\nhq : \u2191\u03c6 \u2191q \u2208 S[X]\u2070\nhq' : \u2191\u03c6 \u2191q' \u2208 S[X]\u2070\n\u22a2 RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk p' q' })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := Localization.mk p q }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := Localization.mk p' q' }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\nhave hqq' : \u03c6 \u2191(q * q') \u2208 S[X]\u2070 := by simpa using Submonoid.mul_mem _ hq hq'\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\nhq : \u2191\u03c6 \u2191q \u2208 S[X]\u2070\nhq' : \u2191\u03c6 \u2191q' \u2208 S[X]\u2070\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2208 S[X]\u2070\n[PROOFSTEP]\nsimpa using Submonoid.mul_mem _ hq hq'\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\nhq : \u2191\u03c6 \u2191q \u2208 S[X]\u2070\nhq' : \u2191\u03c6 \u2191q' \u2208 S[X]\u2070\nhqq' : \u2191\u03c6 \u2191(q * q') \u2208 S[X]\u2070\n\u22a2 RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk p' q' })\n      (fun n d =>\n        if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n      (_ :\n        \u2200 {p q p' q' : R[X]},\n          q \u2208 R[X]\u2070 \u2192\n            q' \u2208 R[X]\u2070 \u2192\n              q' * p = q * p' \u2192\n                (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                  else 0) =\n                  if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                    { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                  else 0) =\n    RatFunc.liftOn { toFractionRing := Localization.mk p q }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0) *\n      RatFunc.liftOn { toFractionRing := Localization.mk p' q' }\n        (fun n d =>\n          if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n        (_ :\n          \u2200 {p q p' q' : R[X]},\n            q \u2208 R[X]\u2070 \u2192\n              q' \u2208 R[X]\u2070 \u2192\n                q' * p = q * p' \u2192\n                  (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                    else 0) =\n                    if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                      { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                    else 0)\n[PROOFSTEP]\nsimp_rw [\u2190 ofFractionRing_mul, Localization.mk_mul, liftOn_ofFractionRing_mk, dif_pos hq, dif_pos hq', dif_pos hqq', \u2190\n  ofFractionRing_mul, Submonoid.coe_mul, map_mul, Localization.mk_mul, Submonoid.mk_mul_mk]\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk c\u271d d\u271d })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk p q }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0)) =\n    (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk c\u271d d\u271d })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk p q }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := y }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0)) =\n    (_ :\n      RatFunc.liftOn ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y })\n          (fun n d =>\n            if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } } else 0)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192\n                q' \u2208 R[X]\u2070 \u2192\n                  q' * p = q * p' \u2192\n                    (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                      else 0) =\n                      if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                        { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                      else 0) =\n        RatFunc.liftOn { toFractionRing := Localization.mk c\u271d d\u271d }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0) *\n          RatFunc.liftOn { toFractionRing := y }\n            (fun n d =>\n              if h : \u2191\u03c6 d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 d, property := h } }\n              else 0)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192\n                    q' * p = q * p' \u2192\n                      (if h : \u2191\u03c6 q \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p) { val := \u2191\u03c6 q, property := h } }\n                        else 0) =\n                        if h : \u2191\u03c6 q' \u2208 S[X]\u2070 then\n                          { toFractionRing := Localization.mk (\u2191\u03c6 p') { val := \u2191\u03c6 q', property := h } }\n                        else 0))\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nn : R[X]\nd : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(map \u03c6 h\u03c6) { toFractionRing := Localization.mk n d } =\n    { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 \u2191d, property := (_ : \u2191d \u2208 Submonoid.comap \u03c6 S[X]\u2070) } }\n[PROOFSTEP]\nrefine (liftOn_ofFractionRing_mk n _ _ _).trans ?_\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nn : R[X]\nd : { x // x \u2208 R[X]\u2070 }\n\u22a2 (if h : \u2191\u03c6 \u2191d \u2208 S[X]\u2070 then { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 \u2191d, property := h } } else 0) =\n    { toFractionRing := Localization.mk (\u2191\u03c6 n) { val := \u2191\u03c6 \u2191d, property := (_ : \u2191d \u2208 Submonoid.comap \u03c6 S[X]\u2070) } }\n[PROOFSTEP]\nrw [dif_pos]\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nhf : Function.Injective \u2191\u03c6\n\u22a2 Function.Injective \u2191(map \u03c6 h\u03c6)\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9 h\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nhf : Function.Injective \u2191\u03c6\nx y : FractionRing R[X]\nh : \u2191(map \u03c6 h\u03c6) { toFractionRing := x } = \u2191(map \u03c6 h\u03c6) { toFractionRing := y }\n\u22a2 { toFractionRing := x } = { toFractionRing := y }\n[PROOFSTEP]\ninduction x using Localization.induction_on\n[GOAL]\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nhf : Function.Injective \u2191\u03c6\ny : FractionRing R[X]\ny\u271d : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191(map \u03c6 h\u03c6) { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd } = \u2191(map \u03c6 h\u03c6) { toFractionRing := y }\n\u22a2 { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd } = { toFractionRing := y }\n[PROOFSTEP]\ninduction y using Localization.induction_on\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : MonoidHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nhf : Function.Injective \u2191\u03c6\ny\u271d\u00b9 y\u271d : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh :\n  \u2191(map \u03c6 h\u03c6) { toFractionRing := Localization.mk y\u271d\u00b9.fst y\u271d\u00b9.snd } =\n    \u2191(map \u03c6 h\u03c6) { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd }\n\u22a2 { toFractionRing := Localization.mk y\u271d\u00b9.fst y\u271d\u00b9.snd } = { toFractionRing := Localization.mk y\u271d.fst y\u271d.snd }\n[PROOFSTEP]\nsimpa only [map_apply_ofFractionRing_mk, ofFractionRing_injective.eq_iff, Localization.mk_eq_mk_iff,\n  Localization.r_iff_exists, mul_cancel_left_coe_nonZeroDivisors, exists_const, \u2190 map_mul, hf.eq_iff] using h\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := \u2191src\u271d,\n          map_mul' :=\n            (_ : \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n      0 =\n    0\n[PROOFSTEP]\nsimp_rw [MonoidHom.toFun_eq_coe, \u2190 ofFractionRing_zero, \u2190 Localization.mk_zero (1 : R[X]\u2070), \u2190\n  Localization.mk_zero (1 : S[X]\u2070), map_apply_ofFractionRing_mk, map_zero, Localization.mk_eq_mk',\n  IsLocalization.mk'_zero]\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\n\u22a2 \u2200 (x y : RatFunc R),\n    OneHom.toFun\n        (\u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n        (x + y) =\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          x +\n        OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          y\n[PROOFSTEP]\nrintro \u27e8x\u27e9\n  \u27e8y\u27e9\n      -- porting note: had to hint `induction` which induction principle to use\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\nx y : FractionRing R[X]\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := \u2191src\u271d,\n          map_mul' :=\n            (_ : \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n      ({ toFractionRing := x } + { toFractionRing := y }) =\n    OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := x } +\n      OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := y }\n[PROOFSTEP]\ninduction x using Localization.rec\n[GOAL]\ncase ofFractionRing.ofFractionRing.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\ny : FractionRing R[X]\na\u271d : R[X]\nb\u271d : { x // x \u2208 R[X]\u2070 }\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := \u2191src\u271d,\n          map_mul' :=\n            (_ : \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n      ({ toFractionRing := Localization.mk a\u271d b\u271d } + { toFractionRing := y }) =\n    OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := Localization.mk a\u271d b\u271d } +\n      OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := y }\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := y }) =\n    (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := y })\n[PROOFSTEP]\ninduction y using Localization.rec\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\na\u271d\u00b9 : R[X]\nb\u271d\u00b9 : { x // x \u2208 R[X]\u2070 }\na\u271d : R[X]\nb\u271d : { x // x \u2208 R[X]\u2070 }\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := \u2191src\u271d,\n          map_mul' :=\n            (_ : \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n      ({ toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } + { toFractionRing := Localization.mk a\u271d b\u271d }) =\n    OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } +\n      OneHom.toFun\n        \u2191{ toOneHom := \u2191src\u271d,\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R), OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n        { toFractionRing := Localization.mk a\u271d b\u271d }\n[PROOFSTEP]\nsimp only [\u2190 ofFractionRing_add, Localization.add_mk, map_add, map_mul, MonoidHom.toFun_eq_coe,\n  map_apply_ofFractionRing_mk, Submonoid.coe_mul]\n  -- Porting note: `Submonoid.mk_mul_mk` couldn't be applied: motive incorrect,\n          -- even though it is a rfl lemma.\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\na\u271d\u00b9 : R[X]\nb\u271d\u00b9 : { x // x \u2208 R[X]\u2070 }\na\u271d : R[X]\nb\u271d : { x // x \u2208 R[X]\u2070 }\n\u22a2 {\n      toFractionRing :=\n        Localization.mk (\u2191\u03c6 \u2191b\u271d\u00b9 * \u2191\u03c6 a\u271d + \u2191\u03c6 \u2191b\u271d * \u2191\u03c6 a\u271d\u00b9)\n          { val := \u2191\u03c6 \u2191b\u271d\u00b9 * \u2191\u03c6 \u2191b\u271d, property := (_ : (fun x => x \u2208 S[X]\u2070) (\u2191\u03c6 \u2191b\u271d\u00b9 * \u2191\u03c6 \u2191b\u271d)) } } =\n    {\n      toFractionRing :=\n        Localization.mk (\u2191\u03c6 \u2191b\u271d\u00b9 * \u2191\u03c6 a\u271d + \u2191\u03c6 \u2191b\u271d * \u2191\u03c6 a\u271d\u00b9)\n          ({ val := \u2191\u03c6 \u2191b\u271d\u00b9, property := (_ : \u2191b\u271d\u00b9 \u2208 Submonoid.comap \u03c6 S[X]\u2070) } *\n            { val := \u2191\u03c6 \u2191b\u271d, property := (_ : \u2191b\u271d \u2208 Submonoid.comap \u03c6 S[X]\u2070) }) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\ny : FractionRing R[X]\na\u271d\u00b9 : R[X]\nb\u271d\u00b9 : { x // x \u2208 R[X]\u2070 }\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } + { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d }) =\n    (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } + { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk a\u271d\u00b9 b\u271d\u00b9 } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : RingHomClass F R[X] S[X]\n\u03c6 : F\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 S[X]\u2070\nsrc\u271d : RatFunc R \u2192* RatFunc S := map \u03c6 h\u03c6\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := y }) =\n    (_ :\n      OneHom.toFun\n          (\u2191{ toOneHom := \u2191src\u271d,\n              map_mul' :=\n                (_ :\n                  \u2200 (x y : RatFunc R),\n                    OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) })\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } +\n          OneHom.toFun\n            \u2191{ toOneHom := \u2191src\u271d,\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : RatFunc R),\n                      OneHom.toFun (\u2191src\u271d) (x * y) = OneHom.toFun (\u2191src\u271d) x * OneHom.toFun (\u2191src\u271d) y) }\n            { toFractionRing := y })\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\n\u22a2 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\nh\u271d : Subsingleton R\n\u22a2 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'\n[PROOFSTEP]\nrw [Subsingleton.elim p q, Subsingleton.elim p' q, Subsingleton.elim q' q]\n[GOAL]\ncase inr\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\nh\u271d : Nontrivial R\n\u22a2 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'\n[PROOFSTEP]\nrw [div_eq_div_iff, \u2190 map_mul, mul_comm p, h, map_mul, mul_comm]\n[GOAL]\ncase inr.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\nh\u271d : Nontrivial R\n\u22a2 \u2191\u03c6 q \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 \u2039_\u203a)\n[GOAL]\ncase inr.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nf : RatFunc R\np q p' q' : R[X]\nhq : q \u2208 R[X]\u2070\nhq' : q' \u2208 R[X]\u2070\nh : q' * p = q * p'\nh\u271d : Nontrivial R\n\u22a2 \u2191\u03c6 q' \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 \u2039_\u203a)\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 (fun f =>\n        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n          (_ :\n            \u2200 {p q p' q' : R[X]},\n              q \u2208 R[X]\u2070 \u2192 q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n      0 =\n    0\n[PROOFSTEP]\ndsimp only\n  -- porting note: force the function to be applied\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 RatFunc.liftOn 0 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n      (_ : \u2200 {p q p' q' : R[X]}, q \u2208 R[X]\u2070 \u2192 q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 \u2191\u03c6 p / \u2191\u03c6 q = \u2191\u03c6 p' / \u2191\u03c6 q') =\n    0\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_zero, \u2190 Localization.mk_zero (1 : R[X]\u2070), liftOn_ofFractionRing_mk]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 \u2191\u03c6 0 / \u2191\u03c6 \u21911 = 0\n[PROOFSTEP]\nsimp only [map_zero, zero_div]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      1 =\n    1\n[PROOFSTEP]\ndsimp only\n  -- porting note: force the function to be applied\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 RatFunc.liftOn 1 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n      (_ : \u2200 {p q p' q' : R[X]}, q \u2208 R[X]\u2070 \u2192 q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 \u2191\u03c6 p / \u2191\u03c6 q = \u2191\u03c6 p' / \u2191\u03c6 q') =\n    1\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_one, \u2190 Localization.mk_one, liftOn_ofFractionRing_mk]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\n\u22a2 \u2191\u03c6 1 / \u2191\u03c6 \u21911 = 1\n[PROOFSTEP]\nsimp only [map_one, Submonoid.coe_one, div_one]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nx y : RatFunc R\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      (x * y) =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        x *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        y\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\ny : RatFunc R\nx : FractionRing R[X]\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      ({ toFractionRing := x } * y) =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := x } *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        y\n[PROOFSTEP]\ncases' y with y\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nx y : FractionRing R[X]\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      ({ toFractionRing := x } * { toFractionRing := y }) =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := x } *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := y }\n[PROOFSTEP]\ninduction' x using Localization.rec with p q\n[GOAL]\ncase ofFractionRing.ofFractionRing.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      ({ toFractionRing := Localization.mk p q } * { toFractionRing := y }) =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := Localization.mk p q } *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := y }\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := y }) =\n    (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := y })\n[PROOFSTEP]\ninduction' y using Localization.rec with p' q'\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk p' q' }) =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := Localization.mk p q } *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := Localization.mk p' q' }\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_mul, Localization.mk_mul]\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun f =>\n          RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n            (_ :\n              \u2200 {p q p' q' : R[X]},\n                q \u2208 R[X]\u2070 \u2192\n                  q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n        map_zero' :=\n          (_ :\n            (fun f =>\n                  RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                    (_ :\n                      \u2200 {p q p' q' : R[X]},\n                        q \u2208 R[X]\u2070 \u2192\n                          q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                0 =\n              0) }\n      { toFractionRing := Localization.mk (p * p') (q * q') } =\n    ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := Localization.mk p q } *\n      ZeroHom.toFun\n        {\n          toFun := fun f =>\n            RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n              (_ :\n                \u2200 {p q p' q' : R[X]},\n                  q \u2208 R[X]\u2070 \u2192\n                    q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n          map_zero' :=\n            (_ :\n              (fun f =>\n                    RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                      (_ :\n                        \u2200 {p q p' q' : R[X]},\n                          q \u2208 R[X]\u2070 \u2192\n                            q' \u2208 R[X]\u2070 \u2192\n                              q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                  0 =\n                0) }\n        { toFractionRing := Localization.mk p' q' }\n[PROOFSTEP]\nsimp only [liftOn_ofFractionRing_mk, div_mul_div_comm, map_mul, Submonoid.coe_mul]\n[GOAL]\ncase ofFractionRing.ofFractionRing.f.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk p q } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d }) =\n    (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk p q } * { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk p q } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := y }) =\n    (_ :\n      ZeroHom.toFun\n          {\n            toFun := fun f =>\n              RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                (_ :\n                  \u2200 {p q p' q' : R[X]},\n                    q \u2208 R[X]\u2070 \u2192\n                      q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n            map_zero' :=\n              (_ :\n                (fun f =>\n                      RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                        (_ :\n                          \u2200 {p q p' q' : R[X]},\n                            q \u2208 R[X]\u2070 \u2192\n                              q' \u2208 R[X]\u2070 \u2192\n                                q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                    0 =\n                  0) }\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } * { toFractionRing := y }) =\n        ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := Localization.mk c\u271d d\u271d } *\n          ZeroHom.toFun\n            {\n              toFun := fun f =>\n                RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                  (_ :\n                    \u2200 {p q p' q' : R[X]},\n                      q \u2208 R[X]\u2070 \u2192\n                        q' \u2208 R[X]\u2070 \u2192 q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'),\n              map_zero' :=\n                (_ :\n                  (fun f =>\n                        RatFunc.liftOn f (fun p q => \u2191\u03c6 p / \u2191\u03c6 q)\n                          (_ :\n                            \u2200 {p q p' q' : R[X]},\n                              q \u2208 R[X]\u2070 \u2192\n                                q' \u2208 R[X]\u2070 \u2192\n                                  q' * p = q * p' \u2192 (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p q = (fun p q => \u2191\u03c6 p / \u2191\u03c6 q) p' q'))\n                      0 =\n                    0) }\n            { toFractionRing := y })\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\n\u22a2 Function.Injective \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6')\n[PROOFSTEP]\nrintro \u27e8x\u27e9 \u27e8y\u27e9\n[GOAL]\ncase ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\nx y : FractionRing R[X]\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := x } = \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := y } \u2192\n    { toFractionRing := x } = { toFractionRing := y }\n[PROOFSTEP]\ninduction' x using Localization.induction_on with a\n[GOAL]\ncase ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\ny : FractionRing R[X]\na : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := Localization.mk a.fst a.snd } =\n      \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := y } \u2192\n    { toFractionRing := Localization.mk a.fst a.snd } = { toFractionRing := y }\n[PROOFSTEP]\ninduction' y using Localization.induction_on with a'\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := Localization.mk a.fst a.snd } =\n      \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6') { toFractionRing := Localization.mk a'.fst a'.snd } \u2192\n    { toFractionRing := Localization.mk a.fst a.snd } = { toFractionRing := Localization.mk a'.fst a'.snd }\n[PROOFSTEP]\nsimp_rw [liftMonoidWithZeroHom_apply_ofFractionRing_mk]\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd \u2192\n    { toFractionRing := Localization.mk a.fst a.snd } = { toFractionRing := Localization.mk a'.fst a'.snd }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 { toFractionRing := Localization.mk a.fst a.snd } = { toFractionRing := Localization.mk a'.fst a'.snd }\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 Localization.mk a.fst a.snd = Localization.mk a'.fst a'.snd\n[PROOFSTEP]\nrefine Localization.mk_eq_mk_iff.mpr (Localization.r_of_eq (M := R[X]) ?_)\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191(a'.fst, a'.snd).snd * (a.fst, a.snd).fst = \u2191(a.fst, a.snd).snd * (a'.fst, a'.snd).fst\n[PROOFSTEP]\nhave := mul_eq_mul_of_div_eq_div _ _ ?_ ?_ h\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_3\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\nthis : \u2191\u03c6 a.fst * \u2191\u03c6 \u2191a'.snd = \u2191\u03c6 a'.fst * \u2191\u03c6 \u2191a.snd\n\u22a2 \u2191(a'.fst, a'.snd).snd * (a.fst, a.snd).fst = \u2191(a.fst, a.snd).snd * (a'.fst, a'.snd).fst\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_1\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a.snd \u2260 0\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_2\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a'.snd \u2260 0\n[PROOFSTEP]\nrwa [\u2190 map_mul, \u2190 map_mul, h\u03c6.eq_iff, mul_comm, mul_comm a'.fst] at this \n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_1\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a.snd \u2260 0\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_2\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a'.snd \u2260 0\n[PROOFSTEP]\nall_goals exact map_ne_zero_of_mem_nonZeroDivisors _ h\u03c6 (SetLike.coe_mem _)\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_1\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a.snd \u2260 0\n[PROOFSTEP]\nexact map_ne_zero_of_mem_nonZeroDivisors _ h\u03c6 (SetLike.coe_mem _)\n[GOAL]\ncase ofFractionRing.ofFractionRing.H.H.e_toFractionRing.refine_2\nK : Type u\ninst\u271d\u2075 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u2074 : CommGroupWithZero G\u2080\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : Nontrivial R\n\u03c6 : R[X] \u2192*\u2080 G\u2080\nh\u03c6 : Function.Injective \u2191\u03c6\nh\u03c6' : optParam (R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070) (_ : R[X]\u2070 \u2264 Submonoid.comap \u03c6 G\u2080\u2070)\na a' : R[X] \u00d7 { x // x \u2208 R[X]\u2070 }\nh : \u2191\u03c6 a.fst / \u2191\u03c6 \u2191a.snd = \u2191\u03c6 a'.fst / \u2191\u03c6 \u2191a'.snd\n\u22a2 \u2191\u03c6 \u2191a'.snd \u2260 0\n[PROOFSTEP]\nexact map_ne_zero_of_mem_nonZeroDivisors _ h\u03c6 (SetLike.coe_mem _)\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nx y : RatFunc R\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : RatFunc R), ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n      (x + y) =\n    OneHom.toFun\n        (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R),\n                  ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n        x +\n      OneHom.toFun\n        (\u2191{ toOneHom := { toFun := src\u271d.toFun, map_one' := (_ : ZeroHom.toFun (\u2191src\u271d) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : RatFunc R),\n                  ZeroHom.toFun (\u2191src\u271d) (x * y) = ZeroHom.toFun (\u2191src\u271d) x * ZeroHom.toFun (\u2191src\u271d) y) })\n        y\n[PROOFSTEP]\nsimp only [ZeroHom.toFun_eq_coe, MonoidWithZeroHom.toZeroHom_coe]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nx y : RatFunc R\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) (x + y) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) x +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) y\n[PROOFSTEP]\ncases subsingleton_or_nontrivial R\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nx y : RatFunc R\nh\u271d : Subsingleton R\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) (x + y) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) x +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) y\n[PROOFSTEP]\nrw [Subsingleton.elim (x + y) y, Subsingleton.elim x 0, map_zero, zero_add]\n[GOAL]\ncase inr\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nx y : RatFunc R\nh\u271d : Nontrivial R\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) (x + y) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) x +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) y\n[PROOFSTEP]\ncases' x with x\n[GOAL]\ncase inr.ofFractionRing\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\ny : RatFunc R\nh\u271d : Nontrivial R\nx : FractionRing R[X]\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) ({ toFractionRing := x } + y) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := x } +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) y\n[PROOFSTEP]\ncases' y with y\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\nx y : FractionRing R[X]\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) ({ toFractionRing := x } + { toFractionRing := y }) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := x } +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y }\n[PROOFSTEP]\ninduction' x using Localization.rec with p q\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n      ({ toFractionRing := Localization.mk p q } + { toFractionRing := y }) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p q } +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y }\ncase inr.ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d\u00b9 : Nontrivial R\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y }) =\n    (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y })\n[PROOFSTEP]\ninduction' y using Localization.rec with p' q'\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n      ({ toFractionRing := Localization.mk p q } + { toFractionRing := Localization.mk p' q' }) =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p q } +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p' q' }\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_add, Localization.add_mk]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n      { toFractionRing := Localization.mk (\u2191q * p' + \u2191q' * p) (q * q') } =\n    \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p q } +\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p' q' }\n[PROOFSTEP]\nsimp only [RingHom.toMonoidWithZeroHom_eq_coe, liftMonoidWithZeroHom_apply_ofFractionRing_mk]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 (\u2191q * p' + \u2191q' * p) / \u2191\u03c6 \u2191(q * q') = \u2191\u03c6 p / \u2191\u03c6 \u2191q + \u2191\u03c6 p' / \u2191\u03c6 \u2191q'\n[PROOFSTEP]\nrw [div_add_div, div_eq_div_iff]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 (\u2191q * p' + \u2191q' * p) * (\u2191\u03c6 \u2191q * \u2191\u03c6 \u2191q') = (\u2191\u03c6 p * \u2191\u03c6 \u2191q' + \u2191\u03c6 \u2191q * \u2191\u03c6 p') * \u2191\u03c6 \u2191(q * q')\n[PROOFSTEP]\nrw [mul_comm _ p, mul_comm _ p', mul_comm _ (\u03c6 p'), add_comm]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 (p * \u2191q' + p' * \u2191q) * (\u2191\u03c6 \u2191q * \u2191\u03c6 \u2191q') = (\u2191\u03c6 p * \u2191\u03c6 \u2191q' + \u2191\u03c6 p' * \u2191\u03c6 \u2191q) * \u2191\u03c6 \u2191(q * q')\n[PROOFSTEP]\nsimp only [map_add, map_mul, Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2260 0\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q * \u2191\u03c6 \u2191q' \u2260 0\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q \u2260 0\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\nall_goals\n  try simp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n  exact nonZeroDivisors.ne_zero (h\u03c6 (SetLike.coe_mem _))\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2260 0\n[PROOFSTEP]\ntry simp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2260 0\n[PROOFSTEP]\nsimp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 (SetLike.coe_mem _))\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q * \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\ntry simp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q * \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\nsimp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191(q * q') \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 (SetLike.coe_mem _))\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q \u2260 0\n[PROOFSTEP]\ntry simp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q \u2260 0\n[PROOFSTEP]\nsimp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hb\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 (SetLike.coe_mem _))\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\ntry simp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\nsimp only [\u2190 map_mul, \u2190 Submonoid.coe_mul]\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.f.hd\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d : Nontrivial R\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\np' : R[X]\nq' : { x // x \u2208 R[X]\u2070 }\n\u22a2 \u2191\u03c6 \u2191q' \u2260 0\n[PROOFSTEP]\nexact nonZeroDivisors.ne_zero (h\u03c6 (SetLike.coe_mem _))\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.f.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d\u00b9 : Nontrivial R\ny : FractionRing R[X]\np : R[X]\nq : { x // x \u2208 R[X]\u2070 }\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk p q } + { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p q } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d }) =\n    (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk p q } + { toFractionRing := Localization.mk c\u271d d\u271d }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk p q } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.ofFractionRing.ofFractionRing.H\nK : Type u\ninst\u271d\u2074 : CommRing K\nG\u2080 : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst\u271d\u00b3 : CommGroupWithZero G\u2080\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u03c6 : R[X] \u2192+* L\nh\u03c6 : R[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc R \u2192*\u2080 L := liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6\nh\u271d\u00b9 : Nontrivial R\nx y : FractionRing R[X]\na\u271d c\u271d : R[X]\nb\u271d d\u271d : { x // x \u2208 R[X]\u2070 }\nh\u271d : \u2191(Localization.r R[X]\u2070) (a\u271d, b\u271d) (c\u271d, d\u271d)\n\u22a2 (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y }) =\n    (_ :\n      \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6)\n          ({ toFractionRing := Localization.mk c\u271d d\u271d } + { toFractionRing := y }) =\n        \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := Localization.mk c\u271d d\u271d } +\n          \u2191(liftMonoidWithZeroHom (RingHom.toMonoidWithZeroHom \u03c6) h\u03c6) { toFractionRing := y })\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nsrc\u271d\u00b9 : CommRing (RatFunc K) := instCommRing K\nsrc\u271d : Nontrivial (RatFunc K) := instNontrivial K\n\u22a2 \u2200 (a b : RatFunc K), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nfrac_tac\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nsrc\u271d\u00b9 : CommRing (RatFunc K) := instCommRing K\nsrc\u271d : Nontrivial (RatFunc K) := instNontrivial K\n\u22a2 0\u207b\u00b9 = 0\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_zero, \u2190 ofFractionRing_inv, inv_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\n\u22a2 (fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1) 1 = 1\n[PROOFSTEP]\nsimp only [mk_one', RingHom.map_one, ofFractionRing_one]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nx y : R\n\u22a2 OneHom.toFun\n      { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n        map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n          map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n          map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) }\n        y\n[PROOFSTEP]\nsimp only [mk_one', RingHom.map_mul, ofFractionRing_mul]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n              map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : R),\n                RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                  RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) })\n      0 =\n    0\n[PROOFSTEP]\nsimp only [mk_one', RingHom.map_zero, ofFractionRing_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nx y : R\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n              map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : R),\n                RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                  RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) })\n      (x + y) =\n    OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n                map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : R),\n                  RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                    RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) })\n        x +\n      OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n                map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : R),\n                  RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                    RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) })\n        y\n[PROOFSTEP]\nsimp only [mk_one', RingHom.map_add, ofFractionRing_add]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nc : R\nx : (fun x => RatFunc K) c\n\u22a2 c \u2022 x =\n    \u2191{\n            toMonoidHom :=\n              {\n                toOneHom :=\n                  { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n                    map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : R),\n                      RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                        RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) },\n            map_zero' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 0) 1 = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : R),\n                  RatFunc.mk (\u2191(algebraMap R K[X]) (x + y)) 1 =\n                    RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 + RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) }\n        c *\n      x\n[PROOFSTEP]\ninduction' x using RatFunc.induction_on' with p q hq\n[GOAL]\ncase _pq\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nc : R\np q : K[X]\nhq : q \u2260 0\n\u22a2 c \u2022 RatFunc.mk p q =\n    \u2191{\n            toMonoidHom :=\n              {\n                toOneHom :=\n                  { toFun := fun x => RatFunc.mk (\u2191(algebraMap R K[X]) x) 1,\n                    map_one' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 1) 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : R),\n                      RatFunc.mk (\u2191(algebraMap R K[X]) (x * y)) 1 =\n                        RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 * RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) },\n            map_zero' := (_ : RatFunc.mk (\u2191(algebraMap R K[X]) 0) 1 = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : R),\n                  RatFunc.mk (\u2191(algebraMap R K[X]) (x + y)) 1 =\n                    RatFunc.mk (\u2191(algebraMap R K[X]) x) 1 + RatFunc.mk (\u2191(algebraMap R K[X]) y) 1) }\n        c *\n      RatFunc.mk p q\n[PROOFSTEP]\nrw [RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk]\n[GOAL]\ncase _pq\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nc : R\np q : K[X]\nhq : q \u2260 0\n\u22a2 c \u2022 RatFunc.mk p q = RatFunc.mk (\u2191(algebraMap R K[X]) c) 1 * RatFunc.mk p q\n[PROOFSTEP]\nrw [mk_one', \u2190 mk_smul, mk_def_of_ne (c \u2022 p) hq, mk_def_of_ne p hq, \u2190 ofFractionRing_mul,\n  IsLocalization.mul_mk'_eq_mk'_of_mul, Algebra.smul_def]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx : K[X]\n\u22a2 { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) x } = \u2191(algebraMap K[X] (RatFunc K)) x\n[PROOFSTEP]\nrw [\u2190 mk_one, mk_one']\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\np q : K[X]\n\u22a2 RatFunc.mk p q = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nsimp only [mk_eq_div', ofFractionRing_div, ofFractionRing_algebraMap]\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (c \u2022 p) / \u2191(algebraMap K[X] (RatFunc K)) q =\n    c \u2022 (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)\n[PROOFSTEP]\nrw [\u2190 mk_eq_div, mk_smul, mk_eq_div]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nx : R\n\u22a2 \u2191(algebraMap R (RatFunc K)) x =\n    \u2191(algebraMap ((fun x => K[X]) x) (RatFunc K)) (\u2191(algebraMap R K[X]) x) / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nrw [\u2190 mk_eq_div]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : CommRing K\ninst\u271d\u00b2 : IsDomain K\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : Algebra R K[X]\nx : R\n\u22a2 \u2191(algebraMap R (RatFunc K)) x = RatFunc.mk (\u2191(algebraMap R K[X]) x) 1\n[PROOFSTEP]\nrfl\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) q) (RatFunc R)) (\u2191\u03c6 q)\n[PROOFSTEP]\nhave hq' : \u03c6 q \u2260 0 := nonZeroDivisors.ne_zero (h\u03c6 (mem_nonZeroDivisors_iff_ne_zero.mpr hq))\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np q : K[X]\nhq : q \u2260 0\nhq' : \u2191\u03c6 q \u2260 0\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) q) (RatFunc R)) (\u2191\u03c6 q)\n[PROOFSTEP]\nsimp only [\u2190 mk_eq_div, mk_eq_localization_mk _ hq, map_apply_ofFractionRing_mk, mk_eq_localization_mk _ hq']\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np q : K[X]\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) q) (RatFunc R)) (\u2191\u03c6 q)\n[PROOFSTEP]\nrcases eq_or_ne q 0 with (rfl | hq)\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 0) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) 0) (RatFunc R)) (\u2191\u03c6 0)\n[PROOFSTEP]\nhave : (0 : RatFunc K) = algebraMap K[X] _ 0 / algebraMap K[X] _ 1 := by simp\n[GOAL]\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\n\u22a2 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\nthis : 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 0) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) 0) (RatFunc R)) (\u2191\u03c6 0)\n[PROOFSTEP]\nrw [map_zero, map_zero, map_zero, div_zero, div_zero, this, map_apply_div_ne_zero, map_one, map_one, div_one, map_zero,\n  map_zero]\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\nthis : 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n\u22a2 \u2191(algebraMap ((fun x => R[X]) 0) (RatFunc R)) 0 / \u2191(algebraMap ((fun x => R[X]) 1) (RatFunc R)) 1 = 0\ncase inl.hq\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\nthis : 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nsimp only [map_zero, map_one, div_one]\n  -- porting note: this `simp` was not needed\n[GOAL]\ncase inl.hq\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np : K[X]\nthis : 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase inr\nK : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidWithZeroHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(map \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191(algebraMap ((fun x => R[X]) p) (RatFunc R)) (\u2191\u03c6 p) / \u2191(algebraMap ((fun x => R[X]) q) (RatFunc R)) (\u2191\u03c6 q)\n[PROOFSTEP]\nexact map_apply_div_ne_zero _ _ _ _ hq\n[GOAL]\nK : Type u\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\nL : Type u_1\ninst\u271d : CommGroupWithZero L\n\u03c6 : K[X] \u2192*\u2080 L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\np q : K[X]\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) = \u2191\u03c6 p / \u2191\u03c6 q\n[PROOFSTEP]\nrcases eq_or_ne q 0 with (rfl | hq)\n[GOAL]\ncase inl\nK : Type u\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\nL : Type u_1\ninst\u271d : CommGroupWithZero L\n\u03c6 : K[X] \u2192*\u2080 L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\np : K[X]\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 0) = \u2191\u03c6 p / \u2191\u03c6 0\n[PROOFSTEP]\nsimp only [div_zero, map_zero]\n[GOAL]\ncase inr\nK : Type u\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\nL : Type u_1\ninst\u271d : CommGroupWithZero L\n\u03c6 : K[X] \u2192*\u2080 L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) = \u2191\u03c6 p / \u2191\u03c6 q\n[PROOFSTEP]\nsimp only [\u2190 mk_eq_div, mk_eq_localization_mk _ hq, liftMonoidWithZeroHom_apply_ofFractionRing_mk]\n[GOAL]\nK : Type u\ninst\u271d\u00b2 : CommRing K\ninst\u271d\u00b9 : IsDomain K\nL : Type u_1\ninst\u271d : CommGroupWithZero L\n\u03c6 : K[X] \u2192*\u2080 L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\np q : K[X]\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p) /\n      \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191\u03c6 p / \u2191\u03c6 q\n[PROOFSTEP]\nrw [\u2190 map_div\u2080, liftMonoidWithZeroHom_apply_div]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u22a2 Function.Injective \u2191(algebraMap K[X] (RatFunc K))\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_comp_algebraMap]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u22a2 Function.Injective (ofFractionRing \u2218 \u2191(algebraMap K[X] (FractionRing K[X])))\n[PROOFSTEP]\nexact ofFractionRing_injective.comp (IsFractionRing.injective _ _)\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx : K[X]\nhx : x = 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) x = 0\n[PROOFSTEP]\nrw [hx, RingHom.map_zero]\n[GOAL]\nK : Type u\ninst\u271d\u2078 : CommRing K\ninst\u271d\u2077 : IsDomain K\nL : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra S K[X]\ninst\u271d\u00b9 : Algebra S L\ninst\u271d : Algebra S R[X]\n\u03c6\u271d : K[X] \u2192\u2090[S] L\nh\u03c6\u271d : K[X]\u2070 \u2264 Submonoid.comap \u03c6\u271d L\u2070\n\u03c6 : K[X] \u2192\u2090[S] R[X]\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\nsrc\u271d : RatFunc K \u2192+* RatFunc R := mapRingHom \u03c6 h\u03c6\nr : S\n\u22a2 OneHom.toFun\n      (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : RatFunc K),\n                  OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n      (\u2191(algebraMap S (RatFunc K)) r) =\n    \u2191(algebraMap S (RatFunc R)) r\n[PROOFSTEP]\nsimp_rw [RingHom.toFun_eq_coe, coe_mapRingHom_eq_coe_map, algebraMap_apply r, map_apply_div, map_one, AlgHom.commutes]\n[GOAL]\nK : Type u\ninst\u271d\u2078 : CommRing K\ninst\u271d\u2077 : IsDomain K\nL : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : CommSemiring S\ninst\u271d\u00b2 : Algebra S K[X]\ninst\u271d\u00b9 : Algebra S L\ninst\u271d : Algebra S R[X]\n\u03c6 : K[X] \u2192\u2090[S] L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nsrc\u271d : RatFunc K \u2192+* L := liftRingHom (\u2191\u03c6) h\u03c6\nr : S\n\u22a2 OneHom.toFun\n      (\u2191\u2191{ toMonoidHom := \u2191src\u271d, map_zero' := (_ : OneHom.toFun (\u2191\u2191src\u271d) 0 = 0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : RatFunc K),\n                  OneHom.toFun (\u2191\u2191src\u271d) (x + y) = OneHom.toFun (\u2191\u2191src\u271d) x + OneHom.toFun (\u2191\u2191src\u271d) y) })\n      (\u2191(algebraMap S (RatFunc K)) r) =\n    \u2191(algebraMap S L) r\n[PROOFSTEP]\nsimp_rw [RingHom.toFun_eq_coe, AlgHom.toRingHom_eq_coe, algebraMap_apply r, liftRingHom_apply_div, AlgHom.coe_toRingHom,\n  map_one, div_one, AlgHom.commutes]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\ny : { x // x \u2208 K[X]\u2070 }\n\u22a2 IsUnit (\u2191(algebraMap K[X] (RatFunc K)) \u2191y)\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_algebraMap]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\ny : { x // x \u2208 K[X]\u2070 }\n\u22a2 IsUnit { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) \u2191y }\n[PROOFSTEP]\nexact (toFractionRingRingEquiv K).symm.toRingHom.isUnit_map (IsLocalization.map_units _ y)\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u22a2 \u2200 (z : RatFunc K), \u2203 x, z * \u2191(algebraMap K[X] (RatFunc K)) \u2191x.snd = \u2191(algebraMap K[X] (RatFunc K)) x.fst\n[PROOFSTEP]\nrintro \u27e8z\u27e9\n[GOAL]\ncase ofFractionRing\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nz : FractionRing K[X]\n\u22a2 \u2203 x, { toFractionRing := z } * \u2191(algebraMap K[X] (RatFunc K)) \u2191x.snd = \u2191(algebraMap K[X] (RatFunc K)) x.fst\n[PROOFSTEP]\nconvert IsLocalization.surj K[X]\u2070 z\n[GOAL]\ncase h.e'_2.h.a\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nz : FractionRing K[X]\nx\u271d : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 { toFractionRing := z } * \u2191(algebraMap K[X] (RatFunc K)) \u2191x\u271d.snd = \u2191(algebraMap K[X] (RatFunc K)) x\u271d.fst \u2194\n    z * \u2191(algebraMap K[X] (FractionRing K[X])) \u2191x\u271d.snd = \u2191(algebraMap K[X] (FractionRing K[X])) x\u271d.fst\n[PROOFSTEP]\nsimp only [\u2190 ofFractionRing_algebraMap, Function.comp_apply, \u2190 ofFractionRing_mul]\n[GOAL]\ncase h.e'_2.h.a\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nz : FractionRing K[X]\nx\u271d : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 { toFractionRing := z * \u2191(algebraMap K[X] (FractionRing K[X])) \u2191x\u271d.snd } =\n      { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) x\u271d.fst } \u2194\n    z * \u2191(algebraMap K[X] (FractionRing K[X])) \u2191x\u271d.snd = \u2191(algebraMap K[X] (FractionRing K[X])) x\u271d.fst\n[PROOFSTEP]\nrw [ofFractionRing.injEq]\n  -- porting note: added\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx y : K[X]\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) x = \u2191(algebraMap K[X] (RatFunc K)) y \u2194 \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nrw [\u2190 ofFractionRing_algebraMap, \u2190 ofFractionRing_algebraMap]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx y : K[X]\n\u22a2 { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) x } =\n      { toFractionRing := \u2191(algebraMap K[X] (FractionRing K[X])) y } \u2194\n    \u2203 c, \u2191c * x = \u2191c * y\n[PROOFSTEP]\nexact (toFractionRingRingEquiv K).symm.injective.eq_iff.trans (IsLocalization.eq_iff_exists _ _)\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH' : \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\nH :\n  optParam (\u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n    (_ : \u2200 {p q p' q' : K[X]}, q \u2208 K[X]\u2070 \u2192 q' \u2208 K[X]\u2070 \u2192 q' * p = q * p' \u2192 f p q = f p' q')\n\u22a2 RatFunc.liftOn (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) f H = f p q\n[PROOFSTEP]\nrw [\u2190 mk_eq_div, liftOn_mk _ _ f f0 @H']\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 RatFunc.liftOn' (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) f H = f p q\n[PROOFSTEP]\nrw [RatFunc.liftOn', liftOn_div _ _ _ f0]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] \u2192 K[X] \u2192 P\nf0 : \u2200 (p : K[X]), f p 0 = f 0 1\nH : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q\n\u22a2 \u2200 {p q p' q' : K[X]}, q \u2260 0 \u2192 q' \u2260 0 \u2192 q' * p = q * p' \u2192 f p q = f p' q'\n[PROOFSTEP]\napply lift_on_condition_of_lift_on'_condition H\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nP : RatFunc K \u2192 Prop\nx : RatFunc K\nf : \u2200 (p q : K[X]), q \u2260 0 \u2192 P (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)\np q : K[X]\nhq : q \u2260 0\n\u22a2 P (RatFunc.mk p q)\n[PROOFSTEP]\nsimpa using f p q hq\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx : K[X]\ny : { x // x \u2208 K[X]\u2070 }\n\u22a2 { toFractionRing := IsLocalization.mk' (FractionRing K[X]) x y } = IsLocalization.mk' (RatFunc K) x y\n[PROOFSTEP]\nrw [IsFractionRing.mk'_eq_div, IsFractionRing.mk'_eq_div, \u2190 mk_eq_div', \u2190 mk_eq_div]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx\u271d : FractionRing K[X]\nx : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 { toFractionRing := Localization.mk x.fst x.snd } =\n    \u2191(IsLocalization.algEquiv K[X]\u2070 (FractionRing K[X]) (RatFunc K)) (Localization.mk x.fst x.snd)\n[PROOFSTEP]\nsimp only [IsLocalization.algEquiv_apply, IsLocalization.ringEquivOfRingEquiv_apply, Localization.mk_eq_mk'_apply,\n  IsLocalization.map_mk', ofFractionRing_mk', RingEquiv.coe_toRingHom, RingEquiv.refl_apply, SetLike.eta]\n  -- porting note: added following `simp`.  The previous one can be squeezed.\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx\u271d : FractionRing K[X]\nx : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 IsLocalization.mk' (RatFunc K) x.fst x.snd =\n    IsLocalization.mk' (RatFunc K) (\u2191(RingHom.id K[X]) x.fst)\n      { val := \u2191(RingHom.id K[X]) \u2191x.snd, property := (_ : \u2191x.snd \u2208 Submonoid.comap (RingHom.id K[X]) K[X]\u2070) }\n[PROOFSTEP]\nsimp only [IsFractionRing.mk'_eq_div, RingHom.id_apply, Subtype.coe_eta]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx\u271d\u00b9 : RatFunc K\nx\u271d : FractionRing K[X]\nx : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 { toFractionRing := Localization.mk x.fst x.snd }.toFractionRing =\n    \u2191(IsLocalization.algEquiv K[X]\u2070 (RatFunc K) (FractionRing K[X])) { toFractionRing := Localization.mk x.fst x.snd }\n[PROOFSTEP]\nsimp only [Localization.mk_eq_mk'_apply, ofFractionRing_mk', IsLocalization.algEquiv_apply,\n  IsLocalization.ringEquivOfRingEquiv_apply, IsLocalization.map_mk', RingEquiv.coe_toRingHom, RingEquiv.refl_apply,\n  SetLike.eta]\n  -- porting note: added following `simp`.  The previous one can be squeezed.\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx\u271d\u00b9 : RatFunc K\nx\u271d : FractionRing K[X]\nx : K[X] \u00d7 { x // x \u2208 K[X]\u2070 }\n\u22a2 IsLocalization.mk' (Localization K[X]\u2070) x.fst x.snd =\n    IsLocalization.mk' (FractionRing K[X]) (\u2191(RingHom.id K[X]) x.fst)\n      { val := \u2191(RingHom.id K[X]) \u2191x.snd, property := (_ : \u2191x.snd \u2208 Submonoid.comap (RingHom.id K[X]) K[X]\u2070) }\n[PROOFSTEP]\nsimp only [IsFractionRing.mk'_eq_div, RingHom.id_apply, Subtype.coe_eta]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\n\u22a2 RingEquiv.symm (toFractionRingRingEquiv K) =\n    AlgEquiv.toRingEquiv (IsLocalization.algEquiv K[X]\u2070 (FractionRing K[X]) (RatFunc K))\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nx : FractionRing K[X]\n\u22a2 \u2191(RingEquiv.symm (toFractionRingRingEquiv K)) x =\n    \u2191(AlgEquiv.toRingEquiv (IsLocalization.algEquiv K[X]\u2070 (FractionRing K[X]) (RatFunc K))) x\n[PROOFSTEP]\nsimp [toFractionRingRingEquiv, ofFractionRing_eq, AlgEquiv.coe_ringEquiv']\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 \u2200 {p q a : K[X]},\n    q \u2260 0 \u2192\n      a \u2260 0 \u2192\n        (fun p q =>\n              if q = 0 then (0, 1)\n              else\n                let r := gcd p q;\n                (\u2191C (leadingCoeff (q / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (q / r))\u207b\u00b9 * (q / r)))\n            (a * p) (a * q) =\n          (fun p q =>\n              if q = 0 then (0, 1)\n              else\n                let r := gcd p q;\n                (\u2191C (leadingCoeff (q / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (q / r))\u207b\u00b9 * (q / r)))\n            p q\n[PROOFSTEP]\nintros p q a hq ha\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\n\u22a2 (fun p q =>\n        if q = 0 then (0, 1)\n        else\n          let r := gcd p q;\n          (\u2191C (leadingCoeff (q / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (q / r))\u207b\u00b9 * (q / r)))\n      (a * p) (a * q) =\n    (fun p q =>\n        if q = 0 then (0, 1)\n        else\n          let r := gcd p q;\n          (\u2191C (leadingCoeff (q / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (q / r))\u207b\u00b9 * (q / r)))\n      p q\n[PROOFSTEP]\ndsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\n\u22a2 (if a * q = 0 then (0, 1)\n    else\n      (\u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * p / gcd (a * p) (a * q)),\n        \u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * q / gcd (a * p) (a * q)))) =\n    if q = 0 then (0, 1)\n    else (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q), \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nrw [if_neg hq, if_neg (mul_ne_zero ha hq)]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\n\u22a2 (\u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * p / gcd (a * p) (a * q)),\n      \u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * q / gcd (a * p) (a * q))) =\n    (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q), \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nhave ha' : a.leadingCoeff \u2260 0 := Polynomial.leadingCoeff_ne_zero.mpr ha\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\n\u22a2 (\u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * p / gcd (a * p) (a * q)),\n      \u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * q / gcd (a * p) (a * q))) =\n    (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q), \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nhave hainv : a.leadingCoeff\u207b\u00b9 \u2260 0 := inv_ne_zero ha'\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\n\u22a2 (\u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * p / gcd (a * p) (a * q)),\n      \u2191C (leadingCoeff (a * q / gcd (a * p) (a * q)))\u207b\u00b9 * (a * q / gcd (a * p) (a * q))) =\n    (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q), \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nsimp only [Prod.ext_iff, gcd_mul_left, normalize_apply, Polynomial.coe_normUnit, mul_assoc,\n  CommGroupWithZero.coe_normUnit _ ha']\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\n\u22a2 \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * p / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * q / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nhave hdeg : (gcd p q).degree \u2264 q.degree := degree_gcd_le_right _ hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\n\u22a2 \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * p / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * q / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nhave hdeg' : (Polynomial.C a.leadingCoeff\u207b\u00b9 * gcd p q).degree \u2264 q.degree :=\n  by\n  rw [Polynomial.degree_mul, Polynomial.degree_C hainv, zero_add]\n  exact hdeg\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\n\u22a2 degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\n[PROOFSTEP]\nrw [Polynomial.degree_mul, Polynomial.degree_C hainv, zero_add]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\n\u22a2 degree (gcd p q) \u2264 degree q\n[PROOFSTEP]\nexact hdeg\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\n\u22a2 \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * p / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * q / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nhave hdivp : Polynomial.C a.leadingCoeff\u207b\u00b9 * gcd p q \u2223 p := (C_mul_dvd hainv).mpr (gcd_dvd_left p q)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\n\u22a2 \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * p / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * q / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nhave hdivq : Polynomial.C a.leadingCoeff\u207b\u00b9 * gcd p q \u2223 q :=\n  (C_mul_dvd hainv).mpr\n    (gcd_dvd_right p q)\n      -- porting note: added `simp only [...]` and `rw [mul_assoc]`\n            -- porting note: note the unfolding of `normalize` and `normUnit`!\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * p / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (\u2191normalize a * gcd p q)))\u207b\u00b9 * (a * q / (\u2191normalize a * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nsimp only [normalize, normUnit, coe_normUnit, leadingCoeff_eq_zero, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, ha,\n  dite_false, Units.val_inv_eq_inv_val, Units.val_mk0]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C (leadingCoeff (a * q / (a * \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q)))\u207b\u00b9 *\n        (a * p / (a * \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (a * \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q)))\u207b\u00b9 *\n        (a * q / (a * \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q)) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C (leadingCoeff (a * q / (a * (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q))))\u207b\u00b9 *\n        (a * p / (a * (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q))) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C (leadingCoeff (a * q / (a * (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q))))\u207b\u00b9 *\n        (a * q / (a * (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q))) =\n      \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nrw [EuclideanDomain.mul_div_mul_cancel ha hdivp, EuclideanDomain.mul_div_mul_cancel ha hdivq, leadingCoeff_div hdeg,\n  leadingCoeff_div hdeg', Polynomial.leadingCoeff_mul, Polynomial.leadingCoeff_C, div_C_mul, div_C_mul, \u2190 mul_assoc, \u2190\n  Polynomial.C_mul, \u2190 mul_assoc, \u2190 Polynomial.C_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C ((leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9) * (p / gcd p q) =\n      \u2191C (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9 * (p / gcd p q) \u2227\n    \u2191C ((leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9) * (q / gcd p q) =\n      \u2191C (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C ((leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9) * (p / gcd p q) =\n    \u2191C (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9 * (p / gcd p q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase right\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 \u2191C ((leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9) * (q / gcd p q) =\n    \u2191C (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase left.e_a.h.e_6.h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 (leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9 =\n    (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9\n[PROOFSTEP]\nrw [inv_div, mul_comm, mul_div_assoc, \u2190 mul_assoc, inv_inv, _root_.mul_inv_cancel ha', one_mul, inv_div]\n[GOAL]\ncase right.e_a.h.e_6.h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q a : K[X]\nhq : q \u2260 0\nha : a \u2260 0\nha' : leadingCoeff a \u2260 0\nhainv : (leadingCoeff a)\u207b\u00b9 \u2260 0\nhdeg : degree (gcd p q) \u2264 degree q\nhdeg' : degree (\u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q) \u2264 degree q\nhdivp : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 p\nhdivq : \u2191C (leadingCoeff a)\u207b\u00b9 * gcd p q \u2223 q\n\u22a2 (leadingCoeff q / ((leadingCoeff a)\u207b\u00b9 * leadingCoeff (gcd p q)))\u207b\u00b9 * (leadingCoeff a)\u207b\u00b9\u207b\u00b9 =\n    (leadingCoeff q / leadingCoeff (gcd p q))\u207b\u00b9\n[PROOFSTEP]\nrw [inv_div, mul_comm, mul_div_assoc, \u2190 mul_assoc, inv_inv, _root_.mul_inv_cancel ha', one_mul, inv_div]\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 numDenom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q), \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nrw [numDenom, liftOn'_div, if_neg hq]\n[GOAL]\ncase f0\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2200 (p : K[X]),\n    (if 0 = 0 then (0, 1)\n      else\n        let r := gcd p 0;\n        (\u2191C (leadingCoeff (0 / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (0 / r))\u207b\u00b9 * (0 / r))) =\n      if 1 = 0 then (0, 1)\n      else\n        let r := gcd 0 1;\n        (\u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (0 / r), \u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (1 / r))\n[PROOFSTEP]\nintro p\n[GOAL]\ncase f0\nK : Type u\ninst\u271d : Field K\np\u271d q : K[X]\nhq : q \u2260 0\np : K[X]\n\u22a2 (if 0 = 0 then (0, 1)\n    else\n      let r := gcd p 0;\n      (\u2191C (leadingCoeff (0 / r))\u207b\u00b9 * (p / r), \u2191C (leadingCoeff (0 / r))\u207b\u00b9 * (0 / r))) =\n    if 1 = 0 then (0, 1)\n    else\n      let r := gcd 0 1;\n      (\u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (0 / r), \u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (1 / r))\n[PROOFSTEP]\nrw [if_pos rfl, if_neg (one_ne_zero' K[X])]\n[GOAL]\ncase f0\nK : Type u\ninst\u271d : Field K\np\u271d q : K[X]\nhq : q \u2260 0\np : K[X]\n\u22a2 (0, 1) =\n    let r := gcd 0 1;\n    (\u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (0 / r), \u2191C (leadingCoeff (1 / r))\u207b\u00b9 * (1 / r))\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q)\n[PROOFSTEP]\nrw [num, numDenom_div _ hq]\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 num 0 = 0\n[PROOFSTEP]\nconvert num_div' (0 : K[X]) one_ne_zero\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 0 = \u2191C (leadingCoeff (1 / gcd 0 1))\u207b\u00b9 * (0 / gcd 0 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q)\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q = 0\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q)\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : \u00acq = 0\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q)\n[PROOFSTEP]\nexact num_div' p hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 num 1 = 1\n[PROOFSTEP]\nconvert num_div (1 : K[X]) 1\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 1 = \u2191(algebraMap K[X] (RatFunc K)) 1 / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 1 = \u2191C (leadingCoeff (1 / gcd 1 1))\u207b\u00b9 * (1 / gcd 1 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p) = p\n[PROOFSTEP]\nconvert num_div p 1\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) p = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 p = \u2191C (leadingCoeff (1 / gcd p 1))\u207b\u00b9 * (p / gcd p 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 p\n[PROOFSTEP]\nrw [num_div _ q, C_mul_dvd]\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 p / gcd p q \u2223 p\n[PROOFSTEP]\nexact EuclideanDomain.div_dvd_of_dvd (gcd_dvd_left p q)\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 (leadingCoeff (q / gcd p q))\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, inv_eq_zero, Polynomial.leadingCoeff_eq_zero] using right_div_gcd_ne_zero hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q) \u2223 p\n[PROOFSTEP]\nsimpa using num_div_dvd p hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) =\n    \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q)\n[PROOFSTEP]\nrw [denom, numDenom_div _ hq]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 Monic (denom x)\n[PROOFSTEP]\ninduction x using RatFunc.induction_on with\n| f p q hq =>\n  rw [denom_div p hq, mul_comm]\n  exact Polynomial.monic_mul_leadingCoeff_inv (right_div_gcd_ne_zero hq)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 Monic (denom x)\n[PROOFSTEP]\ninduction x using RatFunc.induction_on with\n| f p q hq =>\n  rw [denom_div p hq, mul_comm]\n  exact Polynomial.monic_mul_leadingCoeff_inv (right_div_gcd_ne_zero hq)\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 Monic (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q))\n[PROOFSTEP]\n\n| f p q hq =>\n  rw [denom_div p hq, mul_comm]\n  exact Polynomial.monic_mul_leadingCoeff_inv (right_div_gcd_ne_zero hq)\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 Monic (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q))\n[PROOFSTEP]\nrw [denom_div p hq, mul_comm]\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 Monic (q / gcd p q * \u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9)\n[PROOFSTEP]\nexact Polynomial.monic_mul_leadingCoeff_inv (right_div_gcd_ne_zero hq)\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 denom 0 = 1\n[PROOFSTEP]\nconvert denom_div (0 : K[X]) one_ne_zero\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 0 = \u2191(algebraMap K[X] (RatFunc K)) 0 / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 1 = \u2191C (leadingCoeff (1 / gcd 0 1))\u207b\u00b9 * (1 / gcd 0 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 denom 1 = 1\n[PROOFSTEP]\nconvert denom_div (1 : K[X]) one_ne_zero\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 1 = \u2191(algebraMap K[X] (RatFunc K)) 1 / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\n\u22a2 1 = \u2191C (leadingCoeff (1 / gcd 1 1))\u207b\u00b9 * (1 / gcd 1 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p) = 1\n[PROOFSTEP]\nconvert denom_div p one_ne_zero\n[GOAL]\ncase h.e'_2.h.e'_3\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) p = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 1 = \u2191C (leadingCoeff (1 / gcd p 1))\u207b\u00b9 * (1 / gcd p 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nK : Type u\ninst\u271d : Field K\np q : K[X]\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 q\n[PROOFSTEP]\nby_cases hq : q = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q = 0\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 q\n[PROOFSTEP]\nsimp [hq]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : \u00acq = 0\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 q\n[PROOFSTEP]\nrw [denom_div _ hq, C_mul_dvd]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : \u00acq = 0\n\u22a2 q / gcd p q \u2223 q\n[PROOFSTEP]\nexact EuclideanDomain.div_dvd_of_dvd (gcd_dvd_right p q)\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : \u00acq = 0\n\u22a2 (leadingCoeff (q / gcd p q))\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nsimpa only [Ne.def, inv_eq_zero, Polynomial.leadingCoeff_eq_zero] using right_div_gcd_ne_zero hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) = x\n[PROOFSTEP]\ninduction' x using RatFunc.induction_on with p q hq\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) =\n    \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nhave q_div_ne_zero : q / gcd p q \u2260 0 := right_div_gcd_ne_zero hq\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) =\n    \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q)) =\n    \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nrw [num_div p q, denom_div p hq, RingHom.map_mul, RingHom.map_mul, mul_div_mul_left, div_eq_div_iff, \u2190 RingHom.map_mul,\n  \u2190 RingHom.map_mul, mul_comm _ q, \u2190 EuclideanDomain.mul_div_assoc, \u2190 EuclideanDomain.mul_div_assoc, mul_comm]\n[GOAL]\ncase f.h\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 gcd p q \u2223 q\n[PROOFSTEP]\napply gcd_dvd_right\n[GOAL]\ncase f.h\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 gcd p q \u2223 p\n[PROOFSTEP]\napply gcd_dvd_left\n[GOAL]\ncase f.hb\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (q / gcd p q) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero q_div_ne_zero\n[GOAL]\ncase f.hd\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) q \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero hq\n[GOAL]\ncase f.hc\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9) \u2260 0\n[PROOFSTEP]\nrefine' algebraMap_ne_zero (mt Polynomial.C_eq_zero.mp _)\n[GOAL]\ncase f.hc\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\nq_div_ne_zero : q / gcd p q \u2260 0\n\u22a2 \u00ac(leadingCoeff (q / gcd p q))\u207b\u00b9 = 0\n[PROOFSTEP]\nexact inv_ne_zero (Polynomial.leadingCoeff_ne_zero.mpr q_div_ne_zero)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 IsCoprime (num x) (denom x)\n[PROOFSTEP]\ninduction' x using RatFunc.induction_on with p q hq\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 IsCoprime (num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q))\n    (denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q))\n[PROOFSTEP]\nrw [num_div, denom_div _ hq]\n[GOAL]\ncase f\nK : Type u\ninst\u271d : Field K\np q : K[X]\nhq : q \u2260 0\n\u22a2 IsCoprime (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (p / gcd p q)) (\u2191C (leadingCoeff (q / gcd p q))\u207b\u00b9 * (q / gcd p q))\n[PROOFSTEP]\nexact\n  (isCoprime_mul_unit_left ((leadingCoeff_ne_zero.2 <| right_div_gcd_ne_zero hq).isUnit.inv.map C) _ _).2\n    (isCoprime_div_gcd_div_gcd hq)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nh : num x = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, h, RingHom.map_zero, zero_div]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 num x * q = p * denom x \u2194 x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nrw [\u2190 (algebraMap_injective K).eq_iff, eq_div_iff (algebraMap_ne_zero hq)]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p\n[PROOFSTEP]\nconv_rhs => rw [\u2190 num_div_denom x]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n| x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p\n[PROOFSTEP]\nrw [\u2190 num_div_denom x]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n| x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p\n[PROOFSTEP]\nrw [\u2190 num_div_denom x]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n| x * \u2191(algebraMap K[X] (RatFunc K)) q = \u2191(algebraMap K[X] (RatFunc K)) p\n[PROOFSTEP]\nrw [\u2190 num_div_denom x]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x * q) = \u2191(algebraMap K[X] (RatFunc K)) (p * denom x) \u2194\n    \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) *\n        \u2191(algebraMap K[X] (RatFunc K)) q =\n      \u2191(algebraMap K[X] (RatFunc K)) p\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, div_eq_mul_inv, mul_assoc, mul_comm (Inv.inv _), \u2190 mul_assoc, \u2190 div_eq_mul_inv,\n  div_eq_iff]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np q : K[X]\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero (denom_ne_zero x)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 x + y =\n    \u2191(algebraMap K[X] (RatFunc K)) (num x * denom y + denom x * num y) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 num_div_denom x, \u2190 num_div_denom y]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x + y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x + y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x + y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (num x) / \u2191(algebraMap K[X] (RatFunc K)) (denom x) +\n      \u2191(algebraMap K[X] (RatFunc K)) (num y) / \u2191(algebraMap K[X] (RatFunc K)) (denom y) =\n    \u2191(algebraMap K[X] (RatFunc K)) (num x * denom y + denom x * num y) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrw [div_add_div, RingHom.map_mul, RingHom.map_add, RingHom.map_mul, RingHom.map_mul]\n[GOAL]\ncase hb\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero (denom_ne_zero x)\n[GOAL]\ncase hd\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom y) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero (denom_ne_zero y)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 num (-x) * denom x = -num x * denom (-x)\n[PROOFSTEP]\nrw [num_mul_eq_mul_denom_iff (denom_ne_zero x), _root_.map_neg, neg_div, num_div_denom]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 x * y = \u2191(algebraMap K[X] (RatFunc K)) (num x * num y) / \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 num_div_denom x, \u2190 num_div_denom y, div_mul_div_comm, \u2190 RingHom.map_mul, \u2190 RingHom.map_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x * y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y, div_mul_div_comm, \u2190 RingHom.map_mul, \u2190 RingHom.map_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x * y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y, div_mul_div_comm, \u2190 RingHom.map_mul, \u2190 RingHom.map_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n| x * y\n[PROOFSTEP]\nrw [\u2190 num_div_denom x, \u2190 num_div_denom y, div_mul_div_comm, \u2190 RingHom.map_mul, \u2190 RingHom.map_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhp : p \u2260 0\n\u22a2 num x \u2223 p \u2194 \u2203 q hq, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhp : p \u2260 0\n\u22a2 num x \u2223 p \u2192 \u2203 q hq, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nrintro \u27e8q, rfl\u27e9\n[GOAL]\ncase mp.intro\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhp : num x * q \u2260 0\n\u22a2 \u2203 q_1 hq, x = \u2191(algebraMap K[X] (RatFunc K)) (num x * q) / \u2191(algebraMap K[X] (RatFunc K)) q_1\n[PROOFSTEP]\nobtain \u27e8_hx, hq\u27e9 := mul_ne_zero_iff.mp hp\n[GOAL]\ncase mp.intro.intro\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhp : num x * q \u2260 0\n_hx : num x \u2260 0\nhq : q \u2260 0\n\u22a2 \u2203 q_1 hq, x = \u2191(algebraMap K[X] (RatFunc K)) (num x * q) / \u2191(algebraMap K[X] (RatFunc K)) q_1\n[PROOFSTEP]\nuse denom x * q\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhp : num x * q \u2260 0\n_hx : num x \u2260 0\nhq : q \u2260 0\n\u22a2 \u2203 hq, x = \u2191(algebraMap K[X] (RatFunc K)) (num x * q) / \u2191(algebraMap K[X] (RatFunc K)) (denom x * q)\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, \u2190 div_mul_div_comm, div_self, mul_one, num_div_denom]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhp : num x * q \u2260 0\n_hx : num x \u2260 0\nhq : q \u2260 0\n\u22a2 \u2203 hq, x = x\n[PROOFSTEP]\nexact \u27e8mul_ne_zero (denom_ne_zero x) hq, rfl\u27e9\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhp : num x * q \u2260 0\n_hx : num x \u2260 0\nhq : q \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) q \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero hq\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhp : p \u2260 0\n\u22a2 (\u2203 q hq, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2192 num x \u2223 p\n[PROOFSTEP]\nrintro \u27e8q, hq, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nK : Type u\ninst\u271d : Field K\np : K[X]\nhp : p \u2260 0\nq : K[X]\nhq : q \u2260 0\n\u22a2 num (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 p\n[PROOFSTEP]\nexact num_div_dvd p hq\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhq : q \u2260 0\n\u22a2 denom x \u2223 q \u2194 \u2203 p, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhq : q \u2260 0\n\u22a2 denom x \u2223 q \u2192 \u2203 p, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nrintro \u27e8p, rfl\u27e9\n[GOAL]\ncase mp.intro\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhq : denom x * p \u2260 0\n\u22a2 \u2203 p_1, x = \u2191(algebraMap K[X] (RatFunc K)) p_1 / \u2191(algebraMap K[X] (RatFunc K)) (denom x * p)\n[PROOFSTEP]\nobtain \u27e8_hx, hp\u27e9 := mul_ne_zero_iff.mp hq\n[GOAL]\ncase mp.intro.intro\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhq : denom x * p \u2260 0\n_hx : denom x \u2260 0\nhp : p \u2260 0\n\u22a2 \u2203 p_1, x = \u2191(algebraMap K[X] (RatFunc K)) p_1 / \u2191(algebraMap K[X] (RatFunc K)) (denom x * p)\n[PROOFSTEP]\nuse num x * p\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhq : denom x * p \u2260 0\n_hx : denom x \u2260 0\nhp : p \u2260 0\n\u22a2 x = \u2191(algebraMap K[X] (RatFunc K)) (num x * p) / \u2191(algebraMap K[X] (RatFunc K)) (denom x * p)\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, \u2190 div_mul_div_comm, div_self, mul_one, num_div_denom]\n[GOAL]\ncase h\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\np : K[X]\nhq : denom x * p \u2260 0\n_hx : denom x \u2260 0\nhp : p \u2260 0\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) p \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero hp\n[GOAL]\ncase mpr\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nq : K[X]\nhq : q \u2260 0\n\u22a2 (\u2203 p, x = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2192 denom x \u2223 q\n[PROOFSTEP]\nrintro \u27e8p, rfl\u27e9\n[GOAL]\ncase mpr.intro\nK : Type u\ninst\u271d : Field K\nq : K[X]\nhq : q \u2260 0\np : K[X]\n\u22a2 denom (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) \u2223 q\n[PROOFSTEP]\nexact denom_div_dvd p q\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 num (x * y) \u2223 num x * num y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : x = 0\n\u22a2 num (x * y) \u2223 num x * num y\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : \u00acx = 0\n\u22a2 num (x * y) \u2223 num x * num y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 num (x * y) \u2223 num x * num y\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 num (x * y) \u2223 num x * num y\n[PROOFSTEP]\nrw [num_dvd (mul_ne_zero (num_ne_zero hx) (num_ne_zero hy))]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 \u2203 q hq, x * y = \u2191(algebraMap K[X] (RatFunc K)) (num x * num y) / \u2191(algebraMap K[X] (RatFunc K)) q\n[PROOFSTEP]\nrefine' \u27e8x.denom * y.denom, mul_ne_zero (denom_ne_zero x) (denom_ne_zero y), _\u27e9\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 x * y = \u2191(algebraMap K[X] (RatFunc K)) (num x * num y) / \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, \u2190 div_mul_div_comm, num_div_denom, num_div_denom]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 denom (x * y) \u2223 denom x * denom y\n[PROOFSTEP]\nrw [denom_dvd (mul_ne_zero (denom_ne_zero x) (denom_ne_zero y))]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2203 p, x * y = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrefine' \u27e8x.num * y.num, _\u27e9\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 x * y = \u2191(algebraMap K[X] (RatFunc K)) (num x * num y) / \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_mul, \u2190 div_mul_div_comm, num_div_denom, num_div_denom]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 denom (x + y) \u2223 denom x * denom y\n[PROOFSTEP]\nrw [denom_dvd (mul_ne_zero (denom_ne_zero x) (denom_ne_zero y))]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2203 p, x + y = \u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrefine' \u27e8x.num * y.denom + x.denom * y.num, _\u27e9\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 x + y =\n    \u2191(algebraMap K[X] (RatFunc K)) (num x * denom y + denom x * num y) /\n      \u2191(algebraMap K[X] (RatFunc K)) (denom x * denom y)\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_add, RingHom.map_mul, RingHom.map_mul, \u2190 div_add_div, num_div_denom, num_div_denom]\n[GOAL]\ncase hb\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom x) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero (denom_ne_zero x)\n[GOAL]\ncase hd\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\n\u22a2 \u2191(algebraMap K[X] (RatFunc K)) (denom y) \u2260 0\n[PROOFSTEP]\nexact algebraMap_ne_zero (denom_ne_zero y)\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\nf : RatFunc K\n\u22a2 \u2191(map \u03c6 h\u03c6) f =\n    \u2191(algebraMap ((fun x => R[X]) (num f)) (RatFunc R)) (\u2191\u03c6 (num f)) /\n      \u2191(algebraMap ((fun x => R[X]) (denom f)) (RatFunc R)) (\u2191\u03c6 (denom f))\n[PROOFSTEP]\nrw [\u2190 num_div_denom f, map_apply_div_ne_zero, num_div_denom f]\n[GOAL]\ncase hq\nK : Type u\ninst\u271d\u00b3 : Field K\nR : Type u_1\nF : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : MonoidHomClass F K[X] R[X]\n\u03c6 : F\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 R[X]\u2070\nf : RatFunc K\n\u22a2 denom f \u2260 0\n[PROOFSTEP]\nexact denom_ne_zero _\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : CommGroupWithZero L\n\u03c6 : K[X] \u2192*\u2080 L\nh\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070\nf : RatFunc K\n\u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) f = \u2191\u03c6 (num f) / \u2191\u03c6 (denom f)\n[PROOFSTEP]\nrw [\u2190 num_div_denom f, liftMonoidWithZeroHom_apply_div, num_div_denom]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhxy : x + y \u2260 0\n\u22a2 num x * denom y + denom x * num y \u2260 0\n[PROOFSTEP]\nintro h_zero\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhxy : x + y \u2260 0\nh_zero : num x * denom y + denom x * num y = 0\n\u22a2 False\n[PROOFSTEP]\nhave h := num_denom_add x y\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhxy : x + y \u2260 0\nh_zero : num x * denom y + denom x * num y = 0\nh : num (x + y) * (denom x * denom y) = (num x * denom y + denom x * num y) * denom (x + y)\n\u22a2 False\n[PROOFSTEP]\nrw [h_zero, zero_mul] at h \n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhxy : x + y \u2260 0\nh_zero : num x * denom y + denom x * num y = 0\nh : num (x + y) * (denom x * denom y) = 0\n\u22a2 False\n[PROOFSTEP]\nexact (mul_ne_zero (num_ne_zero hxy) (mul_ne_zero x.denom_ne_zero y.denom_ne_zero)) h\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : CommRing K\ninst\u271d : IsDomain K\nr : K\nx : RatFunc K\n\u22a2 r \u2022 x = \u2191C r * x\n[PROOFSTEP]\nrw [Algebra.smul_def, algebraMap_eq_C]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx : RatFunc K\nh : Polynomial.eval\u2082 f a (denom x) = 0\n\u22a2 eval f a x = 0\n[PROOFSTEP]\nrw [eval, h, div_zero]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nc : K\n\u22a2 eval f a (\u2191C c) = \u2191f c\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\n\u22a2 eval f a X = a\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\n\u22a2 eval f a 0 = 0\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\n\u22a2 eval f a 1 = 1\n[PROOFSTEP]\nsimp [eval]\n[GOAL]\nK : Type u\ninst\u271d\u00b3 : Field K\nL : Type u_1\ninst\u271d\u00b2 : Field L\nf : K \u2192+* L\na : L\nS : Type u_2\ninst\u271d\u00b9 : CommSemiring S\ninst\u271d : Algebra S K[X]\np : S\n\u22a2 eval f a (\u2191(algebraMap S (RatFunc K)) p) = Polynomial.eval\u2082 f a (\u2191(algebraMap S K[X]) p)\n[PROOFSTEP]\nsimp [eval, IsScalarTower.algebraMap_apply S K[X] (RatFunc K)]\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\n\u22a2 eval f a (x + y) = eval f a x + eval f a y\n[PROOFSTEP]\nunfold eval\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\nby_cases hxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\nhave := Polynomial.eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero f a (denom_add_dvd x y) hxy\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x * denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\nrw [Polynomial.eval\u2082_mul] at this \n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\ncases mul_eq_zero.mp this\n[GOAL]\ncase pos.inl\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\nh\u271d : Polynomial.eval\u2082 f a (denom x) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos.inr\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x + y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\nh\u271d : Polynomial.eval\u2082 f a (denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x + y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) / Polynomial.eval\u2082 f a (denom (x + y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) +\n      Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y)\n[PROOFSTEP]\nrw [div_add_div _ _ hx hy, eq_div_iff (mul_ne_zero hx hy), div_eq_mul_inv, mul_right_comm, \u2190 div_eq_mul_inv,\n  div_eq_iff hxy]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x + y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y)) * (Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y)) =\n    (Polynomial.eval\u2082 f a (num x) * Polynomial.eval\u2082 f a (denom y) +\n        Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (num y)) *\n      Polynomial.eval\u2082 f a (denom (x + y))\n[PROOFSTEP]\nsimp only [\u2190 Polynomial.eval\u2082_mul, \u2190 Polynomial.eval\u2082_add]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x + y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x + y) * (denom x * denom y)) =\n    Polynomial.eval\u2082 f a ((num x * denom y + denom x * num y) * denom (x + y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.e_p\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x + y)) = 0\n\u22a2 num (x + y) * (denom x * denom y) = (num x * denom y + denom x * num y) * denom (x + y)\n[PROOFSTEP]\napply num_denom_add\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\n\u22a2 eval f a (x * y) = eval f a x * eval f a y\n[PROOFSTEP]\nunfold eval\n[GOAL]\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\nby_cases hxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\nhave := Polynomial.eval\u2082_eq_zero_of_dvd_of_eval\u2082_eq_zero f a (denom_mul_dvd x y) hxy\n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x * denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\nrw [Polynomial.eval\u2082_mul] at this \n[GOAL]\ncase pos\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\ncases mul_eq_zero.mp this\n[GOAL]\ncase pos.inl\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\nh\u271d : Polynomial.eval\u2082 f a (denom x) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos.inr\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : Polynomial.eval\u2082 f a (denom (x * y)) = 0\nthis : Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y) = 0\nh\u271d : Polynomial.eval\u2082 f a (denom y) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x * y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) / Polynomial.eval\u2082 f a (denom (x * y)) =\n    Polynomial.eval\u2082 f a (num x) / Polynomial.eval\u2082 f a (denom x) *\n      (Polynomial.eval\u2082 f a (num y) / Polynomial.eval\u2082 f a (denom y))\n[PROOFSTEP]\nrw [div_mul_div_comm, eq_div_iff (mul_ne_zero hx hy), div_eq_mul_inv, mul_right_comm, \u2190 div_eq_mul_inv, div_eq_iff hxy]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x * y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y)) * (Polynomial.eval\u2082 f a (denom x) * Polynomial.eval\u2082 f a (denom y)) =\n    Polynomial.eval\u2082 f a (num x) * Polynomial.eval\u2082 f a (num y) * Polynomial.eval\u2082 f a (denom (x * y))\n[PROOFSTEP]\nsimp only [\u2190 Polynomial.eval\u2082_mul]\n  -- porting note: was `repeat' rw [\u2190 Polynomial.eval\u2082_mul]`\n[GOAL]\ncase neg\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x * y)) = 0\n\u22a2 Polynomial.eval\u2082 f a (num (x * y) * (denom x * denom y)) = Polynomial.eval\u2082 f a (num x * num y * denom (x * y))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase neg.e_p\nK : Type u\ninst\u271d\u00b9 : Field K\nL : Type u_1\ninst\u271d : Field L\nf : K \u2192+* L\na : L\nx y : RatFunc K\nhx : Polynomial.eval\u2082 f a (denom x) \u2260 0\nhy : Polynomial.eval\u2082 f a (denom y) \u2260 0\nhxy : \u00acPolynomial.eval\u2082 f a (denom (x * y)) = 0\n\u22a2 num (x * y) * (denom x * denom y) = num x * num y * denom (x * y)\n[PROOFSTEP]\napply num_denom_mul\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 intDegree 0 = 0\n[PROOFSTEP]\nrw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 intDegree 1 = 0\n[PROOFSTEP]\nrw [intDegree, num_one, denom_one, sub_self]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nk : K\n\u22a2 intDegree (\u2191C k) = 0\n[PROOFSTEP]\nrw [intDegree, num_C, natDegree_C, denom_C, natDegree_one, sub_self]\n[GOAL]\nK : Type u\ninst\u271d : Field K\n\u22a2 intDegree X = 1\n[PROOFSTEP]\nrw [intDegree, RatFunc.num_X, Polynomial.natDegree_X, RatFunc.denom_X, Polynomial.natDegree_one, Int.ofNat_one,\n  Int.ofNat_zero, sub_zero]\n[GOAL]\nK : Type u\ninst\u271d : Field K\np : K[X]\n\u22a2 intDegree (\u2191(algebraMap K[X] (RatFunc K)) p) = \u2191(natDegree p)\n[PROOFSTEP]\nrw [intDegree, RatFunc.num_algebraMap, RatFunc.denom_algebraMap, Polynomial.natDegree_one, Int.ofNat_zero, sub_zero]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 intDegree (x * y) = intDegree x + intDegree y\n[PROOFSTEP]\nsimp only [intDegree, add_sub, sub_add, sub_sub_eq_add_sub, sub_sub, sub_eq_sub_iff_add_eq_add]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 \u2191(natDegree (num (x * y))) + (\u2191(natDegree (denom x)) + \u2191(natDegree (denom y))) =\n    \u2191(natDegree (num x)) + \u2191(natDegree (num y)) + \u2191(natDegree (denom (x * y)))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhx : x \u2260 0\nhy : y \u2260 0\n\u22a2 natDegree (num (x * y)) + (natDegree (denom x) + natDegree (denom y)) =\n    natDegree (num x) + natDegree (num y) + natDegree (denom (x * y))\n[PROOFSTEP]\nrw [\u2190 Polynomial.natDegree_mul x.denom_ne_zero y.denom_ne_zero, \u2190\n  Polynomial.natDegree_mul (RatFunc.num_ne_zero (mul_ne_zero hx hy)) (mul_ne_zero x.denom_ne_zero y.denom_ne_zero), \u2190\n  Polynomial.natDegree_mul (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy), \u2190\n  Polynomial.natDegree_mul (mul_ne_zero (RatFunc.num_ne_zero hx) (RatFunc.num_ne_zero hy)) (x * y).denom_ne_zero,\n  RatFunc.num_denom_mul]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\n\u22a2 intDegree (-x) = intDegree x\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nhx : x = 0\n\u22a2 intDegree (-x) = intDegree x\n[PROOFSTEP]\nrw [hx, neg_zero]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nhx : \u00acx = 0\n\u22a2 intDegree (-x) = intDegree x\n[PROOFSTEP]\nrw [intDegree, intDegree, \u2190 natDegree_neg x.num]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nhx : \u00acx = 0\n\u22a2 \u2191(natDegree (num (-x))) - \u2191(natDegree (denom (-x))) = \u2191(natDegree (-num x)) - \u2191(natDegree (denom x))\n[PROOFSTEP]\nexact\n  natDegree_sub_eq_of_prod_eq (num_ne_zero (neg_ne_zero.mpr hx)) (denom_ne_zero (-x)) (neg_ne_zero.mpr (num_ne_zero hx))\n    (denom_ne_zero x) (num_denom_neg x)\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nhx : x \u2260 0\ns : K[X]\nhs : s \u2260 0\n\u22a2 \u2191(natDegree (num x * s)) - \u2191(natDegree (s * denom x)) = intDegree x\n[PROOFSTEP]\napply\n  natDegree_sub_eq_of_prod_eq (mul_ne_zero (num_ne_zero hx) hs) (mul_ne_zero hs x.denom_ne_zero) (num_ne_zero hx)\n    x.denom_ne_zero\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx : RatFunc K\nhx : x \u2260 0\ns : K[X]\nhs : s \u2260 0\n\u22a2 num x * s * denom x = num x * (s * denom x)\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhy : y \u2260 0\nhxy : x + y \u2260 0\n\u22a2 intDegree (x + y) \u2264 max (intDegree x) (intDegree y)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhy : y \u2260 0\nhxy : x + y \u2260 0\nhx : x = 0\n\u22a2 intDegree (x + y) \u2264 max (intDegree x) (intDegree y)\n[PROOFSTEP]\nsimp [hx] at hxy \n[GOAL]\ncase pos\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhy : y \u2260 0\nhx : x = 0\nhxy : \u00acy = 0\n\u22a2 intDegree (x + y) \u2264 max (intDegree x) (intDegree y)\n[PROOFSTEP]\nsimp [hx, hxy]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhy : y \u2260 0\nhxy : x + y \u2260 0\nhx : \u00acx = 0\n\u22a2 intDegree (x + y) \u2264 max (intDegree x) (intDegree y)\n[PROOFSTEP]\nrw [intDegree_add hxy, \u2190 natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hx y.denom_ne_zero,\n  mul_comm y.denom, \u2190 natDegree_num_mul_right_sub_natDegree_denom_mul_left_eq_intDegree hy x.denom_ne_zero, le_max_iff,\n  sub_le_sub_iff_right, Int.ofNat_le, sub_le_sub_iff_right, Int.ofNat_le, \u2190 le_max_iff, mul_comm y.num]\n[GOAL]\ncase neg\nK : Type u\ninst\u271d : Field K\nx y : RatFunc K\nhy : y \u2260 0\nhxy : x + y \u2260 0\nhx : \u00acx = 0\n\u22a2 natDegree (num x * denom y + denom x * num y) \u2264 max (natDegree (num x * denom y)) (natDegree (denom x * num y))\n[PROOFSTEP]\nexact natDegree_add_le _ _\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\nr : F\n\u22a2 \u2191(\u2191C r) = \u2191HahnSeries.C r\n[PROOFSTEP]\nrw [coe_num_denom, num_C, denom_C, Polynomial.coe_C,\n  -- porting note: removed `coe_C`Polynomial.coe_one, PowerSeries.coe_one, div_one]\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\nr : F\n\u22a2 \u2191(ofPowerSeries \u2124 F) (\u2191(PowerSeries.C F) r) = \u2191HahnSeries.C r\n[PROOFSTEP]\nsimp only [algebraMap_eq_C, ofPowerSeries_C, C_apply]\n  -- porting note: added\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\nr : F\n\u22a2 \u2191(r \u2022 f) = r \u2022 \u2191f\n[PROOFSTEP]\nrw [smul_eq_C_mul, \u2190 C_mul_eq_smul, coe_mul, coe_C]\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\n\u22a2 \u2191X = \u2191(single 1) 1\n[PROOFSTEP]\nrw [coe_num_denom, num_X, denom_X, Polynomial.coe_X,\n  -- porting note: removed `coe_C`Polynomial.coe_one, PowerSeries.coe_one, div_one]\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\n\u22a2 \u2191(ofPowerSeries \u2124 F) PowerSeries.X = \u2191(single 1) 1\n[PROOFSTEP]\nsimp only [ofPowerSeries_X]\n  -- porting note: added\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\n\u22a2 \u2191(algebraMap (RatFunc F) (LaurentSeries F)) (\u2191(algebraMap F[X] (RatFunc F)) p / \u2191(algebraMap F[X] (RatFunc F)) q) =\n    \u2191(algebraMap F[X] (LaurentSeries F)) p / \u2191(algebraMap F[X] (LaurentSeries F)) q\n[PROOFSTEP]\nconvert coe_div (algebraMap F[X] (RatFunc F) p) (algebraMap F[X] (RatFunc F) q)\n[GOAL]\ncase h.e'_3.h.e'_5\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\n\u22a2 \u2191(algebraMap F[X] (LaurentSeries F)) p = \u2191(\u2191(algebraMap F[X] (RatFunc F)) p)\n[PROOFSTEP]\nrw [\u2190 mk_one, coe_def, coeAlgHom, mk_eq_div, liftAlgHom_apply_div, map_one, div_one, Algebra.ofId_apply]\n[GOAL]\ncase h.e'_3.h.e'_6\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\n\u22a2 \u2191(algebraMap F[X] (LaurentSeries F)) q = \u2191(\u2191(algebraMap F[X] (RatFunc F)) q)\n[PROOFSTEP]\nrw [\u2190 mk_one, coe_def, coeAlgHom, mk_eq_div, liftAlgHom_apply_div, map_one, div_one, Algebra.ofId_apply]\n[GOAL]\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\nx : F[X]\ny : RatFunc F\nz : LaurentSeries F\n\u22a2 (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n[PROOFSTEP]\next\n[GOAL]\ncase coeff.h\nK F : Type u\ninst\u271d : Field F\np q : F[X]\nf g : RatFunc F\nx : F[X]\ny : RatFunc F\nz : LaurentSeries F\nx\u271d : \u2124\n\u22a2 HahnSeries.coeff ((x \u2022 y) \u2022 z) x\u271d = HahnSeries.coeff (x \u2022 y \u2022 z) x\u271d\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.RatFunc", "llama_tokens": 110234, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434873426302, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.5242196456760077}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\nM' : Type u_2\ninst\u271d : Mul M'\n\u22a2 commProb (M \u00d7 M') = commProb M * commProb M'\n[PROOFSTEP]\nsimp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, \u2190 Nat.cast_mul, \u2190 Nat.card_prod, Commute,\n  SemiconjBy, Prod.ext_iff]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\nM' : Type u_2\ninst\u271d : Mul M'\n\u22a2 \u2191(Nat.card { p // (p.fst * p.snd).fst = (p.snd * p.fst).fst \u2227 (p.fst * p.snd).snd = (p.snd * p.fst).snd }) /\n      (\u2191(Nat.card M) ^ 2 * \u2191(Nat.card M') ^ 2) =\n    \u2191(Nat.card ({ p // p.fst * p.snd = p.snd * p.fst } \u00d7 { p // p.fst * p.snd = p.snd * p.fst })) /\n      (\u2191(Nat.card M) ^ 2 * \u2191(Nat.card M') ^ 2)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nM : Type u_1\ninst\u271d\u00b9 : Mul M\nM' : Type u_2\ninst\u271d : Mul M'\n\u22a2 Nat.card { p // (p.fst * p.snd).fst = (p.snd * p.fst).fst \u2227 (p.fst * p.snd).snd = (p.snd * p.fst).snd } =\n    Nat.card ({ p // p.fst * p.snd = p.snd * p.fst } \u00d7 { p // p.fst * p.snd = p.snd * p.fst })\n[PROOFSTEP]\nexact\n  Nat.card_congr\n    \u27e8fun x => \u27e8\u27e8\u27e8x.1.1.1, x.1.2.1\u27e9, x.2.1\u27e9, \u27e8\u27e8x.1.1.2, x.1.2.2\u27e9, x.2.2\u27e9\u27e9, fun x =>\n      \u27e8\u27e8\u27e8x.1.1.1, x.2.1.1\u27e9, \u27e8x.1.1.2, x.2.1.2\u27e9\u27e9, \u27e8x.1.2, x.2.2\u27e9\u27e9, fun x => rfl, fun x => rfl\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Mul M\n\u03b1 : Type u_3\ni : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Mul (i a)\n\u22a2 commProb ((a : \u03b1) \u2192 i a) = \u220f a : \u03b1, commProb (i a)\n[PROOFSTEP]\nsimp_rw [commProb_def, Finset.prod_div_distrib, Finset.prod_pow, \u2190 Nat.cast_prod, \u2190 Nat.card_pi, Commute, SemiconjBy,\n  Function.funext_iff]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Mul M\n\u03b1 : Type u_3\ni : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Mul (i a)\n\u22a2 \u2191(Nat.card { p // \u2200 (a : \u03b1), (p.fst * p.snd) a = (p.snd * p.fst) a }) / \u2191(Nat.card ((a : \u03b1) \u2192 i a)) ^ 2 =\n    \u2191(Nat.card ((a : \u03b1) \u2192 { p // p.fst * p.snd = p.snd * p.fst })) / \u2191(Nat.card ((a : \u03b1) \u2192 i a)) ^ 2\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nM : Type u_1\ninst\u271d\u00b2 : Mul M\n\u03b1 : Type u_3\ni : \u03b1 \u2192 Type u_2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : (a : \u03b1) \u2192 Mul (i a)\n\u22a2 Nat.card { p // \u2200 (a : \u03b1), (p.fst * p.snd) a = (p.snd * p.fst) a } =\n    Nat.card ((a : \u03b1) \u2192 { p // p.fst * p.snd = p.snd * p.fst })\n[PROOFSTEP]\nexact\n  Nat.card_congr\n    \u27e8fun x a => \u27e8\u27e8x.1.1 a, x.1.2 a\u27e9, x.2 a\u27e9, fun x => \u27e8\u27e8fun a => (x a).1.1, fun a => (x a).1.2\u27e9, fun a => (x a).2\u27e9,\n      fun x => rfl, fun x => rfl\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Mul M\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Mul \u03b2\n\u22a2 commProb (\u03b1 \u2192 \u03b2) = commProb \u03b2 ^ card \u03b1\n[PROOFSTEP]\nrw [commProb_pi, Finset.prod_const, Finset.card_univ]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\n\u22a2 commProb M \u2264 1\n[PROOFSTEP]\nrefine' div_le_one_of_le _ (sq_nonneg (Nat.card M : \u211a))\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\n\u22a2 \u2191(Nat.card { p // Commute p.fst p.snd }) \u2264 \u2191(Nat.card M) ^ 2\n[PROOFSTEP]\nrw [\u2190 Nat.cast_pow, Nat.cast_le, sq, \u2190 Nat.card_prod]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\n\u22a2 Nat.card { p // Commute p.fst p.snd } \u2264 Nat.card (M \u00d7 M)\n[PROOFSTEP]\napply Finite.card_subtype_le\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\nh : Nonempty M\n\u22a2 commProb M = 1 \u2194 Commutative fun x x_1 => x * x_1\n[PROOFSTEP]\nhaveI := Fintype.ofFinite M\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\nh : Nonempty M\nthis : Fintype M\n\u22a2 commProb M = 1 \u2194 Commutative fun x x_1 => x * x_1\n[PROOFSTEP]\nrw [commProb, \u2190 Set.coe_setOf, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\nh : Nonempty M\nthis : Fintype M\n\u22a2 \u2191(card \u2191{x | Commute x.fst x.snd}) / \u2191(card M) ^ 2 = 1 \u2194 Commutative fun x x_1 => x * x_1\n[PROOFSTEP]\nrw [div_eq_one_iff_eq, \u2190 Nat.cast_pow, Nat.cast_inj, sq, \u2190 card_prod, set_fintype_card_eq_univ_iff,\n  Set.eq_univ_iff_forall]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\nh : Nonempty M\nthis : Fintype M\n\u22a2 (\u2200 (x : M \u00d7 M), x \u2208 {x | Commute x.fst x.snd}) \u2194 Commutative fun x x_1 => x * x_1\n[PROOFSTEP]\nexact \u27e8fun h x y \u21a6 h (x, y), fun h x \u21a6 h x.1 x.2\u27e9\n[GOAL]\nM : Type u_1\ninst\u271d\u00b9 : Mul M\ninst\u271d : Finite M\nh : Nonempty M\nthis : Fintype M\n\u22a2 \u2191(card M) ^ 2 \u2260 0\n[PROOFSTEP]\nexact pow_ne_zero 2 (Nat.cast_ne_zero.mpr card_ne_zero)\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Finite M\nG : Type u_2\ninst\u271d : Group G\n\u22a2 commProb G = \u2191(Nat.card (ConjClasses G)) / \u2191(Nat.card G)\n[PROOFSTEP]\nrw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Finite M\nG : Type u_2\ninst\u271d : Group G\n\u22a2 \u2191(Nat.card (ConjClasses G)) * \u2191(Nat.card G) / (\u2191(Nat.card G) * \u2191(Nat.card G)) =\n    \u2191(Nat.card (ConjClasses G)) / \u2191(Nat.card G)\n[PROOFSTEP]\nby_cases h : (Nat.card G : \u211a) = 0\n[GOAL]\ncase pos\nM : Type u_1\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Finite M\nG : Type u_2\ninst\u271d : Group G\nh : \u2191(Nat.card G) = 0\n\u22a2 \u2191(Nat.card (ConjClasses G)) * \u2191(Nat.card G) / (\u2191(Nat.card G) * \u2191(Nat.card G)) =\n    \u2191(Nat.card (ConjClasses G)) / \u2191(Nat.card G)\n[PROOFSTEP]\nrw [h, zero_mul, div_zero, div_zero]\n[GOAL]\ncase neg\nM : Type u_1\ninst\u271d\u00b2 : Mul M\ninst\u271d\u00b9 : Finite M\nG : Type u_2\ninst\u271d : Group G\nh : \u00ac\u2191(Nat.card G) = 0\n\u22a2 \u2191(Nat.card (ConjClasses G)) * \u2191(Nat.card G) / (\u2191(Nat.card G) * \u2191(Nat.card G)) =\n    \u2191(Nat.card (ConjClasses G)) / \u2191(Nat.card G)\n[PROOFSTEP]\nexact mul_div_mul_right _ _ h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\n\u22a2 commProb { x // x \u2208 H } \u2264 commProb G * \u2191(index H) ^ 2\n[PROOFSTEP]\nrw [commProb_def, commProb_def, div_le_iff, mul_assoc, \u2190 mul_pow, \u2190 Nat.cast_mul, mul_comm H.index, H.card_mul_index,\n  div_mul_cancel, Nat.cast_le]\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\n\u22a2 Nat.card { p // Commute p.fst p.snd } \u2264 Nat.card { p // Commute p.fst p.snd }\n[PROOFSTEP]\nrefine' Finite.card_le_of_injective (fun p \u21a6 \u27e8\u27e8p.1.1, p.1.2\u27e9, Subtype.ext_iff.mp p.2\u27e9) _\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\n\u22a2 Function.Injective fun p =>\n    { val := (\u2191(\u2191p).fst, \u2191(\u2191p).snd), property := (_ : \u2191((\u2191p).fst * (\u2191p).snd) = \u2191((\u2191p).snd * (\u2191p).fst)) }\n[PROOFSTEP]\nexact fun p q h \u21a6 by simpa only [Subtype.ext_iff, Prod.ext_iff] using h\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\np q : { p // Commute p.fst p.snd }\nh :\n  (fun p => { val := (\u2191(\u2191p).fst, \u2191(\u2191p).snd), property := (_ : \u2191((\u2191p).fst * (\u2191p).snd) = \u2191((\u2191p).snd * (\u2191p).fst)) }) p =\n    (fun p => { val := (\u2191(\u2191p).fst, \u2191(\u2191p).snd), property := (_ : \u2191((\u2191p).fst * (\u2191p).snd) = \u2191((\u2191p).snd * (\u2191p).fst)) }) q\n\u22a2 p = q\n[PROOFSTEP]\nsimpa only [Subtype.ext_iff, Prod.ext_iff] using h\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\n\u22a2 \u2191(Nat.card G) ^ 2 \u2260 0\n[PROOFSTEP]\nexact pow_ne_zero 2 (Nat.cast_ne_zero.mpr Finite.card_pos.ne')\n[GOAL]\nM : Type u_1\ninst\u271d\u00b3 : Mul M\ninst\u271d\u00b2 : Finite M\nG : Type u_2\ninst\u271d\u00b9 : Group G\ninst\u271d : Finite G\nH : Subgroup G\n\u22a2 0 < \u2191(Nat.card { x // x \u2208 H }) ^ 2\n[PROOFSTEP]\nexact pow_pos (Nat.cast_pos.mpr Finite.card_pos) 2\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 commProb (G \u29f8 H) \u2264 commProb G * \u2191(Nat.card { x // x \u2208 H })\n[PROOFSTEP]\nrw [commProb_def', commProb_def', div_le_iff, mul_assoc, \u2190 Nat.cast_mul, \u2190 Subgroup.index, H.card_mul_index,\n  div_mul_cancel, Nat.cast_le]\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 Nat.card (ConjClasses (G \u29f8 H)) \u2264 Nat.card (ConjClasses G)\n[PROOFSTEP]\napply Finite.card_le_of_surjective\n[GOAL]\ncase hf\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 Function.Surjective ?f\ncase f\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 ConjClasses G \u2192 ConjClasses (G \u29f8 H)\n[PROOFSTEP]\nshow Function.Surjective (ConjClasses.map (QuotientGroup.mk' H))\n[GOAL]\ncase hf\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 Function.Surjective (ConjClasses.map (QuotientGroup.mk' H))\n[PROOFSTEP]\nexact ConjClasses.map_surjective Quotient.surjective_Quotient_mk''\n[GOAL]\ncase h\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 \u2191(Nat.card G) \u2260 0\n[PROOFSTEP]\nexact Nat.cast_ne_zero.mpr Finite.card_pos.ne'\n[GOAL]\nM : Type u_1\ninst\u271d\u2074 : Mul M\ninst\u271d\u00b3 : Finite M\nG : Type u_2\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : Finite G\nH : Subgroup G\ninst\u271d : Normal H\n\u22a2 0 < \u2191(Nat.card (G \u29f8 H))\n[PROOFSTEP]\nexact Nat.cast_pos.mpr Finite.card_pos\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.CommutingProbability", "llama_tokens": 4650, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673269042767, "lm_q2_score": 0.6442251201477016, "lm_q1q2_score": 0.5236695513390487}}
{"text": "[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns\u2081 s\u2082 : Set \u03a9\nh : s\u2081 \u2286 s\u2082\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\u2081 \u2264 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\u2082\n[PROOFSTEP]\nchange ((\u03bc : Measure \u03a9) s\u2081).toNNReal \u2264 ((\u03bc : Measure \u03a9) s\u2082).toNNReal\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns\u2081 s\u2082 : Set \u03a9\nh : s\u2081 \u2286 s\u2082\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s\u2081) \u2264 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s\u2082)\n[PROOFSTEP]\nhave key : (\u03bc : Measure \u03a9) s\u2081 \u2264 (\u03bc : Measure \u03a9) s\u2082 := (\u03bc : Measure \u03a9).mono h\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\ns\u2081 s\u2082 : Set \u03a9\nh : s\u2081 \u2286 s\u2082\nkey : \u2191\u2191\u2191\u03bc s\u2081 \u2264 \u2191\u2191\u2191\u03bc s\u2082\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s\u2081) \u2264 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s\u2082)\n[PROOFSTEP]\napply (ENNReal.toNNReal_le_toNNReal (measure_ne_top _ s\u2081) (measure_ne_top _ s\u2082)).mpr key\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc = 0 \u2194 \u03bc = 0\n[PROOFSTEP]\nrefine' \u27e8fun \u03bc_mass => _, fun h\u03bc => by simp only [h\u03bc, zero_mass]\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nh\u03bc : \u03bc = 0\n\u22a2 mass \u03bc = 0\n[PROOFSTEP]\nsimp only [h\u03bc, zero_mass]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u03bc = 0\n[PROOFSTEP]\napply toMeasure_injective\n[GOAL]\ncase a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u03bc = \u21910\n[PROOFSTEP]\napply Measure.measure_univ_eq_zero.mp\n[GOAL]\ncase a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u03bc_mass : mass \u03bc = 0\n\u22a2 \u2191\u2191\u2191\u03bc univ = 0\n[PROOFSTEP]\nrwa [\u2190 ennreal_mass, ENNReal.coe_eq_zero]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc \u2260 0 \u2194 \u03bc \u2260 0\n[PROOFSTEP]\nrw [not_iff_not]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 mass \u03bc = 0 \u2194 \u03bc = 0\n[PROOFSTEP]\nexact FiniteMeasure.mass_zero_iff \u03bc\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\napply Subtype.ext\n[GOAL]\ncase a\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n\u22a2 \u2191\u03bc = \u2191\u03bd\n[PROOFSTEP]\next1 s s_mble\n[GOAL]\ncase a.h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n[PROOFSTEP]\nexact h s s_mble\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\n\u22a2 \u03bc = \u03bd\n[PROOFSTEP]\next1 s s_mble\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d : MeasurableSpace \u03a9\n\u03bc \u03bd : FiniteMeasure \u03a9\nh : \u2200 (s : Set \u03a9), MeasurableSet s \u2192 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) s\ns : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191\u03bc s = \u2191\u2191\u2191\u03bd s\n[PROOFSTEP]\nsimpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d\u22650\u221e) (h s s_mble)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u21910 s)) = 0\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\nx\u271d : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u21910 x\u271d) = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc \u03bd : FiniteMeasure \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(\u03bc + \u03bd) s)) = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) + fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc \u03bd : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(\u03bc + \u03bd) x\u271d) = ((fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) + fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bd s)) x\u271d\n[PROOFSTEP]\nsimp [\u2190 ENNReal.coe_eq_coe]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc \u03bd : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 \u2191\u2191\u2191(\u03bc + \u03bd) x\u271d = \u2191\u2191\u2191\u03bc x\u271d + \u2191\u2191\u2191\u03bd x\u271d\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) = c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) x\u271d) = (c \u2022 fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) x\u271d\n[PROOFSTEP]\nsimp [\u2190 ENNReal.coe_eq_coe, ENNReal.coe_smul]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\nx\u271d : Set \u03a9\n\u22a2 \u2191\u2191\u2191(c \u2022 \u03bc) x\u271d = c \u2022 \u2191\u2191\u2191\u03bc x\u271d\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650 \u211d\u22650\nc : R\n\u03bc : FiniteMeasure \u03a9\ns : Set \u03a9\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(c \u2022 \u03bc) s)) s = c \u2022 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) s\n[PROOFSTEP]\nrw [coeFn_smul, Pi.smul_apply]\n  -- porting note: why doesn't `simp only` work in place of `rw` here?\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA s : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191(restrict \u03bc A) s)) s = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) (s \u2229 A)\n[PROOFSTEP]\napply congr_arg ENNReal.toNNReal\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA s : Set \u03a9\ns_mble : MeasurableSet s\n\u22a2 \u2191\u2191\u2191(restrict \u03bc A) s = \u2191\u2191\u2191\u03bc (s \u2229 A)\n[PROOFSTEP]\nexact Measure.restrict_apply s_mble\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA : Set \u03a9\n\u22a2 mass (restrict \u03bc A) = (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) A\n[PROOFSTEP]\nsimp only [mass, restrict_apply \u03bc A MeasurableSet.univ, univ_inter]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA : Set \u03a9\n\u22a2 restrict \u03bc A = 0 \u2194 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) A = 0\n[PROOFSTEP]\nrw [\u2190 mass_zero_iff, restrict_mass]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u00b3 : SMul R \u211d\u22650\ninst\u271d\u00b2 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\n\u03bc : FiniteMeasure \u03a9\nA : Set \u03a9\n\u22a2 restrict \u03bc A \u2260 0 \u2194 (fun s => ENNReal.toNNReal (\u2191\u2191\u2191\u03bc s)) A \u2260 0\n[PROOFSTEP]\nrw [\u2190 mass_nonzero_iff, restrict_mass]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u03bc < \u22a4\n[PROOFSTEP]\napply IsFiniteMeasure.lintegral_lt_top_of_bounded_to_eNNReal\n[GOAL]\ncase f_bdd\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u2203 c, \u2200 (x : \u03a9), \u2191(\u2191f x) \u2264 \u2191c\n[PROOFSTEP]\nuse nndist f 0\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u2200 (x : \u03a9), \u2191(\u2191f x) \u2264 \u2191(nndist f 0)\n[PROOFSTEP]\nintro x\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)\n[PROOFSTEP]\nhave key := BoundedContinuousFunction.Nnreal.upper_bound f x\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 \u2191(\u2191f x) \u2264 \u2191(nndist f 0)\n[PROOFSTEP]\nrw [ENNReal.coe_le_coe]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 \u2191f x \u2264 nndist f 0\n[PROOFSTEP]\nhave eq : nndist f 0 = \u27e8dist f 0, dist_nonneg\u27e9 := by\n  ext\n  simp only [Real.coe_toNNReal', max_eq_left_iff, NNReal.coe_mk, coe_nndist]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 nndist f 0 = { val := dist f 0, property := (_ : 0 \u2264 dist f 0) }\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\nkey : \u2191f x \u2264 nndist f 0\n\u22a2 \u2191(nndist f 0) = \u2191{ val := dist f 0, property := (_ : 0 \u2264 dist f 0) }\n[PROOFSTEP]\nsimp only [Real.coe_toNNReal', max_eq_left_iff, NNReal.coe_mk, coe_nndist]\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nx : \u03a9\nkey : \u2191f x \u2264 nndist f 0\neq : nndist f 0 = { val := dist f 0, property := (_ : 0 \u2264 dist f 0) }\n\u22a2 \u2191f x \u2264 nndist f 0\n[PROOFSTEP]\nrwa [eq] at key \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nc : \u211d\u22650\n\u22a2 testAgainstNN \u03bc (const \u03a9 c) = c * mass \u03bc\n[PROOFSTEP]\nsimp [\u2190 ENNReal.coe_eq_coe]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nf_le_g : \u2191f \u2264 \u2191g\n\u22a2 testAgainstNN \u03bc f \u2264 testAgainstNN \u03bc g\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.coe_le_coe, testAgainstNN_coe_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nf_le_g : \u2191f \u2264 \u2191g\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191g \u03c9) \u2202\u2191\u03bc\n[PROOFSTEP]\nexact lintegral_mono fun \u03c9 => ENNReal.coe_mono (f_le_g \u03c9)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 testAgainstNN \u03bc 0 = 0\n[PROOFSTEP]\nsimpa only [zero_mul] using \u03bc.testAgainstNN_const 0\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 testAgainstNN \u03bc 1 = mass \u03bc\n[PROOFSTEP]\nsimp only [testAgainstNN, coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 ENNReal.toNNReal (\u2191\u2191\u2191\u03bc univ) = mass \u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN 0 f = 0\n[PROOFSTEP]\nsimp only [testAgainstNN, toMeasure_zero, lintegral_zero_measure, ENNReal.zero_toNNReal]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\n\u22a2 testAgainstNN 0 = 0\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\nx\u271d : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN 0 x\u271d = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nsimp only [zero_testAgainstNN_apply, Pi.zero_apply]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2075 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2074 : SMul R \u211d\u22650\ninst\u271d\u00b3 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b9 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d : TopologicalSpace \u03a9\nc : \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN (c \u2022 \u03bc) f = c \u2022 testAgainstNN \u03bc f\n[PROOFSTEP]\nsimp only [testAgainstNN, toMeasure_smul, smul_eq_mul, \u2190 ENNReal.smul_toNNReal, ENNReal.smul_def,\n  lintegral_smul_measure]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc (f\u2081 + f\u2082) = testAgainstNN \u03bc f\u2081 + testAgainstNN \u03bc f\u2082\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_add, ENNReal.coe_add, Pi.add_apply, testAgainstNN_coe_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) + \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc = \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2081 \u03c9) \u2202\u2191\u03bc + \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f\u2082 \u03c9) \u2202\u2191\u03bc\n[PROOFSTEP]\nexact lintegral_add_left (BoundedContinuousFunction.NNReal.coe_ennreal_comp_measurable _) _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc (c \u2022 f) = c \u2022 testAgainstNN \u03bc f\n[PROOFSTEP]\nsimp only [\u2190 ENNReal.coe_eq_coe, BoundedContinuousFunction.coe_smul, testAgainstNN_coe_eq, ENNReal.coe_smul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 smul_one_smul \u211d\u22650\u221e c (f _ : \u211d\u22650\u221e), \u2190 smul_one_smul \u211d\u22650\u221e c (lintegral _ _ : \u211d\u22650\u221e), smul_eq_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b9\u2070 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2079 : SMul R \u211d\u22650\ninst\u271d\u2078 : SMul R \u211d\u22650\u221e\ninst\u271d\u2077 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u2076 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u2075 : TopologicalSpace \u03a9\ninst\u271d\u2074 : OpensMeasurableSpace \u03a9\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\ninst\u271d\u00b2 : PseudoMetricSpace R\ninst\u271d\u00b9 : Zero R\ninst\u271d : BoundedSMul R \u211d\u22650\n\u03bc : FiniteMeasure \u03a9\nc : R\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), c \u2022 1 * \u2191(\u2191f \u03c9) \u2202\u2191\u03bc = c \u2022 1 * \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc\n[PROOFSTEP]\nexact\n  @lintegral_const_mul _ _ (\u03bc : Measure \u03a9) (c \u2022 (1 : \u211d\u22650\u221e)) _\n    (BoundedContinuousFunction.NNReal.coe_ennreal_comp_measurable f)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN \u03bc f \u2264 testAgainstNN \u03bc g + nndist f g * mass \u03bc\n[PROOFSTEP]\nsimp only [\u2190 \u03bc.testAgainstNN_const (nndist f g), \u2190 testAgainstNN_add, \u2190 ENNReal.coe_le_coe,\n  BoundedContinuousFunction.coe_add, const_apply, ENNReal.coe_add, Pi.add_apply, coe_nnreal_ennreal_nndist,\n  testAgainstNN_coe_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc \u2264 \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191g \u03c9) + edist f g \u2202\u2191\u03bc\n[PROOFSTEP]\napply lintegral_mono\n[GOAL]\ncase hfg\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g\n[PROOFSTEP]\nhave le_dist : \u2200 \u03c9, dist (f \u03c9) (g \u03c9) \u2264 nndist f g := BoundedContinuousFunction.dist_coe_le_dist\n[GOAL]\ncase hfg\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u2264 fun a => \u2191(\u2191g a) + edist f g\n[PROOFSTEP]\nintro \u03c9\n[GOAL]\ncase hfg\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9\n[PROOFSTEP]\nhave le' : f \u03c9 \u2264 g \u03c9 + nndist f g :=\n  by\n  apply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans\n  rw [add_le_add_iff_left]\n  exact dist_le_coe.mp (le_dist \u03c9)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n[PROOFSTEP]\napply (NNReal.le_add_nndist (f \u03c9) (g \u03c9)).trans\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 \u2191g \u03c9 + nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191g \u03c9 + nndist f g\n[PROOFSTEP]\nrw [add_le_add_iff_left]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\n\u22a2 nndist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 nndist f g\n[PROOFSTEP]\nexact dist_le_coe.mp (le_dist \u03c9)\n[GOAL]\ncase hfg\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9\n[PROOFSTEP]\nhave le : (f \u03c9 : \u211d\u22650\u221e) \u2264 (g \u03c9 : \u211d\u22650\u221e) + nndist f g := by rw [\u2190 ENNReal.coe_add]; exact ENNReal.coe_mono le'\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g)\n[PROOFSTEP]\nrw [\u2190 ENNReal.coe_add]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\n\u22a2 \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9 + nndist f g)\n[PROOFSTEP]\nexact ENNReal.coe_mono le'\n[GOAL]\ncase hfg\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf g : \u03a9 \u2192\u1d47 \u211d\u22650\nle_dist : \u2200 (\u03c9 : \u03a9), dist (\u2191f \u03c9) (\u2191g \u03c9) \u2264 \u2191(nndist f g)\n\u03c9 : \u03a9\nle' : \u2191f \u03c9 \u2264 \u2191g \u03c9 + nndist f g\nle : \u2191(\u2191f \u03c9) \u2264 \u2191(\u2191g \u03c9) + \u2191(nndist f g)\n\u22a2 (fun a => \u2191(\u2191f a)) \u03c9 \u2264 (fun a => \u2191(\u2191g a) + edist f g) \u03c9\n[PROOFSTEP]\nrwa [coe_nnreal_ennreal_nndist] at le \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 LipschitzWith (mass \u03bc) fun f => testAgainstNN \u03bc f\n[PROOFSTEP]\nrw [lipschitzWith_iff_dist_le_mul]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 \u2200 (x y : \u03a9 \u2192\u1d47 \u211d\u22650), dist (testAgainstNN \u03bc x) (testAgainstNN \u03bc y) \u2264 \u2191(mass \u03bc) * dist x y\n[PROOFSTEP]\nintro f\u2081 f\u2082\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 dist (testAgainstNN \u03bc f\u2081) (testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nsuffices abs (\u03bc.testAgainstNN f\u2081 - \u03bc.testAgainstNN f\u2082 : \u211d) \u2264 \u03bc.mass * dist f\u2081 f\u2082 by rwa [NNReal.dist_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nthis : |\u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)| \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n\u22a2 dist (testAgainstNN \u03bc f\u2081) (testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nrwa [NNReal.dist_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 |\u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)| \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\napply abs_le.mpr\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 -(\u2191(mass \u03bc) * dist f\u2081 f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2227\n    \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 -(\u2191(mass \u03bc) * dist f\u2081 f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)\n[PROOFSTEP]\nhave key' := \u03bc.testAgainstNN_lipschitz_estimate f\u2082 f\u2081\n[GOAL]\ncase left\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2082 \u2264 testAgainstNN \u03bc f\u2081 + nndist f\u2082 f\u2081 * mass \u03bc\n\u22a2 -(\u2191(mass \u03bc) * dist f\u2081 f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)\n[PROOFSTEP]\nrw [mul_comm] at key' \n[GOAL]\ncase left\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2082 \u2264 testAgainstNN \u03bc f\u2081 + mass \u03bc * nndist f\u2082 f\u2081\n\u22a2 -(\u2191(mass \u03bc) * dist f\u2081 f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)\n[PROOFSTEP]\nsuffices \u2191(\u03bc.testAgainstNN f\u2082) \u2264 \u2191(\u03bc.testAgainstNN f\u2081) + \u2191\u03bc.mass * dist f\u2081 f\u2082 by linarith\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2082 \u2264 testAgainstNN \u03bc f\u2081 + mass \u03bc * nndist f\u2082 f\u2081\nthis : \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n\u22a2 -(\u2191(mass \u03bc) * dist f\u2081 f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082)\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase left\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2082 \u2264 testAgainstNN \u03bc f\u2081 + mass \u03bc * nndist f\u2082 f\u2081\n\u22a2 \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nhave key := NNReal.coe_mono key'\n[GOAL]\ncase left\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2082 \u2264 testAgainstNN \u03bc f\u2081 + mass \u03bc * nndist f\u2082 f\u2081\nkey : \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081 + mass \u03bc * nndist f\u2082 f\u2081)\n\u22a2 \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(testAgainstNN \u03bc f\u2081) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nrwa [NNReal.coe_add, NNReal.coe_mul, nndist_comm] at key \n[GOAL]\ncase right\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nhave key' := \u03bc.testAgainstNN_lipschitz_estimate f\u2081 f\u2082\n[GOAL]\ncase right\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2081 \u2264 testAgainstNN \u03bc f\u2082 + nndist f\u2081 f\u2082 * mass \u03bc\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nrw [mul_comm] at key' \n[GOAL]\ncase right\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2081 \u2264 testAgainstNN \u03bc f\u2082 + mass \u03bc * nndist f\u2081 f\u2082\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nsuffices \u2191(\u03bc.testAgainstNN f\u2081) \u2264 \u2191(\u03bc.testAgainstNN f\u2082) + \u2191\u03bc.mass * dist f\u2081 f\u2082 by linarith\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2081 \u2264 testAgainstNN \u03bc f\u2082 + mass \u03bc * nndist f\u2081 f\u2082\nthis : \u2191(testAgainstNN \u03bc f\u2081) \u2264 \u2191(testAgainstNN \u03bc f\u2082) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) - \u2191(testAgainstNN \u03bc f\u2082) \u2264 \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase right\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2081 \u2264 testAgainstNN \u03bc f\u2082 + mass \u03bc * nndist f\u2081 f\u2082\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) \u2264 \u2191(testAgainstNN \u03bc f\u2082) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nhave key := NNReal.coe_mono key'\n[GOAL]\ncase right\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03bc : FiniteMeasure \u03a9\nf\u2081 f\u2082 : \u03a9 \u2192\u1d47 \u211d\u22650\nkey' : testAgainstNN \u03bc f\u2081 \u2264 testAgainstNN \u03bc f\u2082 + mass \u03bc * nndist f\u2081 f\u2082\nkey : \u2191(testAgainstNN \u03bc f\u2081) \u2264 \u2191(testAgainstNN \u03bc f\u2082 + mass \u03bc * nndist f\u2081 f\u2082)\n\u22a2 \u2191(testAgainstNN \u03bc f\u2081) \u2264 \u2191(testAgainstNN \u03bc f\u2082) + \u2191(mass \u03bc) * dist f\u2081 f\u2082\n[PROOFSTEP]\nrwa [NNReal.coe_add, NNReal.coe_mul] at key \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous fun \u03bc => testAgainstNN \u03bc f\n[PROOFSTEP]\nshow Continuous ((fun \u03c6 : WeakDual \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650) => \u03c6 f) \u2218 toWeakDualBCNN)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous ((fun \u03c6 => \u2191\u03c6 f) \u2218 toWeakDualBCNN)\n[PROOFSTEP]\nrefine Continuous.comp ?_ (toWeakDualBCNN_continuous (\u03a9 := \u03a9))\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Continuous fun \u03c6 => \u2191\u03c6 f\n[PROOFSTEP]\nexact\n  @WeakBilin.eval_continuous _ _ _ _ _ _ ContinuousLinearMap.module _ _ _\n    _\n      /- porting note: without explicitly providing `ContinuousLinearMap.module`, TC synthesis times\n        out trying to find `Module \u211d\u22650 ((\u03a9 \u2192\u1d47 \u211d\u22650) \u2192L[\u211d\u22650] \u211d\u22650)`, but it can find it with enough time:\n        `set_option synthInstance.maxHeartbeats 47000` was sufficient. -/\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u22a2 Continuous fun \u03bc => mass \u03bc\n[PROOFSTEP]\nsimp_rw [\u2190 testAgainstNN_one]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u22a2 Continuous fun \u03bc => testAgainstNN \u03bc 1\n[PROOFSTEP]\nexact continuous_testAgainstNN_eval 1\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))\n[PROOFSTEP]\nrw [tendsto_iff_weak_star_tendsto, tendsto_iff_forall_eval_tendsto_topDualPairing]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (y : \u03a9 \u2192\u1d47 \u211d\u22650),\n      Tendsto (fun i => \u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN (\u03bcs i))) y) F\n        (\ud835\udcdd (\u2191(\u2191(topDualPairing \u211d\u22650 (\u03a9 \u2192\u1d47 \u211d\u22650)) (toWeakDualBCNN \u03bc)) y))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nrw [FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd 0)\n[PROOFSTEP]\napply tendsto_iff_dist_tendsto_zero.mpr\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)\n[PROOFSTEP]\nhave obs := fun i => (\u03bcs i).testAgainstNN_lipschitz_estimate f 0\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 testAgainstNN (\u03bcs i) 0 + nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [testAgainstNN_zero, zero_add] at obs \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => dist (testAgainstNN (\u03bcs b) f) 0) F (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp_rw [show \u2200 i, dist ((\u03bcs i).testAgainstNN f) 0 = (\u03bcs i).testAgainstNN f by\n    simp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 \u2200 (i : \u03b3), dist (testAgainstNN (\u03bcs i) f) 0 = \u2191(testAgainstNN (\u03bcs i) f)\n[PROOFSTEP]\nsimp only [dist_nndist, NNReal.nndist_zero_eq_val', eq_self_iff_true, imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun b => \u2191(testAgainstNN (\u03bcs b) f)) F (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero (fun i => NNReal.coe_nonneg _) obs _\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)\n[PROOFSTEP]\nhave lim_pair : Tendsto (fun i => (\u27e8nndist f 0, (\u03bcs i).mass\u27e9 : \u211d \u00d7 \u211d)) F (\ud835\udcdd \u27e8nndist f 0, 0\u27e9) :=\n  by\n  refine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9\n  exact (NNReal.continuous_coe.tendsto 0).comp mass_lim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\n[PROOFSTEP]\nrefine' (Prod.tendsto_iff _ _).mpr \u27e8tendsto_const_nhds, _\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\n\u22a2 Tendsto (fun n => (\u2191(nndist f 0), \u2191(mass (\u03bcs n))).snd) F (\ud835\udcdd (\u2191(nndist f 0), 0).snd)\n[PROOFSTEP]\nexact (NNReal.continuous_coe.tendsto 0).comp mass_lim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)\n[PROOFSTEP]\nhave key := tendsto_mul.comp lim_pair\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nobs : \u2200 (i : \u03b3), testAgainstNN (\u03bcs i) f \u2264 nndist f 0 * mass (\u03bcs i)\nlim_pair : Tendsto (fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0), 0))\nkey : Tendsto ((fun p => p.fst * p.snd) \u2218 fun i => (\u2191(nndist f 0), \u2191(mass (\u03bcs i)))) F (\ud835\udcdd (\u2191(nndist f 0) * 0))\n\u22a2 Tendsto (fun t => (fun a => \u2191a) (nndist f 0 * mass (\u03bcs t))) F (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [mul_zero] at key \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_iff_forall_testAgainstNN_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN 0 f))\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => testAgainstNN (\u03bcs i) f) F (\ud835\udcdd (testAgainstNN 0 f))\n[PROOFSTEP]\nconvert tendsto_zero_testAgainstNN_of_tendsto_zero_mass mass_lim f\n[GOAL]\ncase h.e'_5.h.e'_3\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\nmass_lim : Tendsto (fun i => mass (\u03bcs i)) F (\ud835\udcdd 0)\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 testAgainstNN 0 f = 0\n[PROOFSTEP]\nrw [zero_testAgainstNN_apply]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\nrw [tendsto_iff_forall_toWeakDualBCNN_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\nsimp_rw [toWeakDualBCNN_apply _ _, \u2190 testAgainstNN_coe_eq, ENNReal.tendsto_coe, ENNReal.toNNReal_coe]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs i) \u03c9) \u2202\u03bc) L (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(f \u03c9) \u2202\u03bc))\n[PROOFSTEP]\nrefine\n  tendsto_lintegral_filter_of_dominated_convergence (fun _ => c)\n    (eventually_of_forall fun i => (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_\n    (@lintegral_const_lt_top _ _ \u03bc _ _ (@ENNReal.coe_ne_top c)).ne ?_\n[GOAL]\ncase refine_1\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1da0 (n : \u03b9) in L, \u2200\u1d50 (a : \u03a9) \u2202\u03bc, \u2191(\u2191(fs n) a) \u2264 (fun x => \u2191c) a\n[PROOFSTEP]\nsimpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const\n[GOAL]\ncase refine_2\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d\u00b9 : IsCountablyGenerated L\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (f \u03c9))\n\u22a2 \u2200\u1d50 (a : \u03a9) \u2202\u03bc, Tendsto (fun n => \u2191(\u2191(fs n) a)) L (\ud835\udcdd \u2191(f a))\n[PROOFSTEP]\nsimpa only [Function.comp_apply, ENNReal.tendsto_coe] using fs_lim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\n\u03bc : FiniteMeasure \u03a9\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u2191\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u2191\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (\u2191f \u03c9))\n\u22a2 Tendsto (fun i => testAgainstNN \u03bc (fs i)) L (\ud835\udcdd (testAgainstNN \u03bc f))\n[PROOFSTEP]\napply (ENNReal.tendsto_toNNReal (lintegral_lt_top_of_boundedContinuous_to_nnreal (\u03bc : Measure \u03a9) f).ne).comp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03b9 : Type u_2\nL : Filter \u03b9\ninst\u271d : IsCountablyGenerated L\n\u03bc : FiniteMeasure \u03a9\nfs : \u03b9 \u2192 \u03a9 \u2192\u1d47 \u211d\u22650\nc : \u211d\u22650\nfs_le_const : \u2200\u1da0 (i : \u03b9) in L, \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u2191\u03bc, \u2191(fs i) \u03c9 \u2264 c\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nfs_lim : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u2191\u03bc, Tendsto (fun i => \u2191(fs i) \u03c9) L (\ud835\udcdd (\u2191f \u03c9))\n\u22a2 Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191(fs i) \u03c9) \u2202\u2191\u03bc) L (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nexact tendsto_lintegral_nn_filter_of_le_const \u03bc fs_le_const fs_lim\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Integrable (NNReal.toReal \u2218 \u2191f)\n[PROOFSTEP]\nrefine' \u27e8(NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable, _\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 HasFiniteIntegral (NNReal.toReal \u2218 \u2191f)\n[PROOFSTEP]\nsimp only [HasFiniteIntegral, Function.comp_apply, NNReal.nnnorm_eq]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 \u222b\u207b (a : \u03a9), \u2191(\u2191f a) \u2202\u03bc < \u22a4\n[PROOFSTEP]\nexact lintegral_lt_top_of_boundedContinuous_to_nnreal _ f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Integrable \u2191f\n[PROOFSTEP]\nrefine' \u27e8f.continuous.measurable.aestronglyMeasurable, _\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 HasFiniteIntegral \u2191f\n[PROOFSTEP]\nhave aux : ((\u2191) : \u211d\u22650 \u2192 \u211d) \u2218 \u21d1f.nnnorm = fun x => \u2016f x\u2016 :=\n  by\n  ext \u03c9\n  simp only [Function.comp_apply, BoundedContinuousFunction.nnnorm_coeFn_eq, coe_nnnorm]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n[PROOFSTEP]\next \u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u03c9 : \u03a9\n\u22a2 (NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f)) \u03c9 = \u2016\u2191f \u03c9\u2016\n[PROOFSTEP]\nsimp only [Function.comp_apply, BoundedContinuousFunction.nnnorm_coeFn_eq, coe_nnnorm]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\naux : NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n\u22a2 HasFiniteIntegral \u2191f\n[PROOFSTEP]\napply (hasFiniteIntegral_iff_norm f).mpr\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\naux : NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n\u22a2 \u222b\u207b (a : \u03a9), ENNReal.ofReal \u2016\u2191f a\u2016 \u2202\u03bc < \u22a4\n[PROOFSTEP]\nrw [\u2190 ofReal_integral_eq_lintegral_ofReal]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\naux : NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n\u22a2 ENNReal.ofReal (\u222b (x : \u03a9), \u2016\u2191f x\u2016 \u2202\u03bc) < \u22a4\n[PROOFSTEP]\nexact ENNReal.ofReal_lt_top\n[GOAL]\ncase hfi\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\naux : NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n\u22a2 Integrable fun a => \u2016\u2191f a\u2016\n[PROOFSTEP]\nexact aux \u25b8 integrable_of_boundedContinuous_to_nnreal \u03bc f.nnnorm\n[GOAL]\ncase f_nn\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\naux : NNReal.toReal \u2218 \u2191(BoundedContinuousFunction.nnnorm f) = fun x => \u2016\u2191f x\u2016\n\u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2016\u2191f a\u2016\n[PROOFSTEP]\nexact eventually_of_forall fun \u03c9 => norm_nonneg (f \u03c9)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 \u222b (\u03c9 : \u03a9), \u2191f \u03c9 \u2202\u03bc = \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart f) \u03c9) \u2202\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u03bc\n[PROOFSTEP]\nsimp only [f.self_eq_nnrealPart_sub_nnrealPart_neg, Pi.sub_apply, integral_sub,\n  integrable_of_boundedContinuous_to_nnreal]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b3 : MeasurableSpace \u03a9\ninst\u271d\u00b2 : TopologicalSpace \u03a9\ninst\u271d\u00b9 : OpensMeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 \u222b (a : \u03a9), (NNReal.toReal \u2218 \u2191(nnrealPart f)) a \u2202\u03bc - \u222b (a : \u03a9), (NNReal.toReal \u2218 \u2191(nnrealPart (-f))) a \u2202\u03bc =\n    \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart f) \u03c9) \u2202\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u03bc\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc)\n[PROOFSTEP]\napply (@tendsto_iff_forall_lintegral_tendsto \u03a9 _ _ _ \u03b3 F \u03bcs \u03bc).mpr\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n\u22a2 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\nhave key :=\n  @ENNReal.tendsto_toReal_iff _ F _ (fun i => (lintegral_lt_top_of_boundedContinuous_to_nnreal (\u03bcs i : Measure \u03a9) f).ne)\n    _ (lintegral_lt_top_of_boundedContinuous_to_nnreal (\u03bc : Measure \u03a9) f).ne\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_coe_nnreal] at key \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\n[PROOFSTEP]\napply key.mp\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave lip : LipschitzWith 1 ((\u2191) : \u211d\u22650 \u2192 \u211d) := isometry_subtype_coe.lipschitz\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nset f\u2080 := BoundedContinuousFunction.comp _ lip f with _def_f\u2080\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave f\u2080_eq : \u21d1f\u2080 = ((\u2191) : \u211d\u22650 \u2192 \u211d) \u2218 \u21d1f := by rfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\n\u22a2 \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave f\u2080_nn : 0 \u2264 \u21d1f\u2080 := fun _ => by simp only [f\u2080_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nx\u271d : \u03a9\n\u22a2 OfNat.ofNat 0 x\u271d \u2264 \u2191f\u2080 x\u271d\n[PROOFSTEP]\nsimp only [f\u2080_eq, Pi.zero_apply, Function.comp_apply, NNReal.zero_le_coe]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave f\u2080_ae_nn : 0 \u2264\u1d50[(\u03bc : Measure \u03a9)] \u21d1f\u2080 := eventually_of_forall f\u2080_nn\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\nf\u2080_ae_nn : 0 \u2264\u1d50[\u2191\u03bc] \u2191f\u2080\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave f\u2080_ae_nns : \u2200 i, 0 \u2264\u1d50[(\u03bcs i : Measure \u03a9)] \u21d1f\u2080 := fun i => eventually_of_forall f\u2080_nn\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\nf\u2080_ae_nn : 0 \u2264\u1d50[\u2191\u03bc] \u2191f\u2080\nf\u2080_ae_nns : \u2200 (i : \u03b3), 0 \u2264\u1d50[\u2191(\u03bcs i)] \u2191f\u2080\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave aux := integral_eq_lintegral_of_nonneg_ae f\u2080_ae_nn f\u2080.continuous.measurable.aestronglyMeasurable\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\nf\u2080_ae_nn : 0 \u2264\u1d50[\u2191\u03bc] \u2191f\u2080\nf\u2080_ae_nns : \u2200 (i : \u03b3), 0 \u2264\u1d50[\u2191(\u03bcs i)] \u2191f\u2080\naux : \u222b (a : \u03a9), \u2191f\u2080 a \u2202\u2191\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03a9), ENNReal.ofReal (\u2191f\u2080 a) \u2202\u2191\u03bc)\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nhave auxs := fun i => integral_eq_lintegral_of_nonneg_ae (f\u2080_ae_nns i) f\u2080.continuous.measurable.aestronglyMeasurable\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\nf\u2080_ae_nn : 0 \u2264\u1d50[\u2191\u03bc] \u2191f\u2080\nf\u2080_ae_nns : \u2200 (i : \u03b3), 0 \u2264\u1d50[\u2191(\u03bcs i)] \u2191f\u2080\naux : \u222b (a : \u03a9), \u2191f\u2080 a \u2202\u2191\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03a9), ENNReal.ofReal (\u2191f\u2080 a) \u2202\u2191\u03bc)\nauxs : \u2200 (i : \u03b3), \u222b (a : \u03a9), \u2191f\u2080 a \u2202\u2191(\u03bcs i) = ENNReal.toReal (\u222b\u207b (a : \u03a9), ENNReal.ofReal (\u2191f\u2080 a) \u2202\u2191(\u03bcs i))\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nsimp_rw [f\u2080_eq, Function.comp_apply, ENNReal.ofReal_coe_nnreal] at aux auxs \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\u22650\nkey :\n  Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))) \u2194\n    Tendsto (fun i => \u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc))\nlip : LipschitzWith 1 NNReal.toReal\nf\u2080 : \u03a9 \u2192\u1d47 \u211d := comp NNReal.toReal lip f\n_def_f\u2080 : f\u2080 = comp NNReal.toReal lip f\nf\u2080_eq : \u2191f\u2080 = NNReal.toReal \u2218 \u2191f\nf\u2080_nn : 0 \u2264 \u2191f\u2080\nf\u2080_ae_nn : 0 \u2264\u1d50[\u2191\u03bc] \u2191f\u2080\nf\u2080_ae_nns : \u2200 (i : \u03b3), 0 \u2264\u1d50[\u2191(\u03bcs i)] \u2191f\u2080\naux : \u222b (a : \u03a9), \u2191(\u2191f a) \u2202\u2191\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03a9), \u2191(\u2191f a) \u2202\u2191\u03bc)\nauxs : \u2200 (i : \u03b3), \u222b (a : \u03a9), \u2191(\u2191f a) \u2202\u2191(\u03bcs i) = ENNReal.toReal (\u222b\u207b (a : \u03a9), \u2191(\u2191f a) \u2202\u2191(\u03bcs i))\n\u22a2 Tendsto (fun n => ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191(\u03bcs n))) F (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f \u03c9) \u2202\u2191\u03bc)))\n[PROOFSTEP]\nsimpa only [\u2190 aux, \u2190 auxs] using h f\u2080\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bc : Measure \u03a9\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u03bc) = \u222b (x : \u03a9), \u2191(\u2191f x) \u2202\u03bc\n[PROOFSTEP]\nrw [integral_eq_lintegral_of_nonneg_ae _\n    (by simpa [Function.comp_apply] using (NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable)]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bc : Measure \u03a9\n\u22a2 AEStronglyMeasurable (fun x => \u2191(\u2191f x)) \u03bc\n[PROOFSTEP]\nsimpa [Function.comp_apply] using (NNReal.continuous_coe.comp f.continuous).measurable.aestronglyMeasurable\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bc : Measure \u03a9\n\u22a2 ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03a9), ENNReal.ofReal \u2191(\u2191f a) \u2202\u03bc)\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_coe_nnreal]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bc : Measure \u03a9\n\u22a2 0 \u2264\u1d50[\u03bc] fun x => \u2191(\u2191f x)\n[PROOFSTEP]\napply eventually_of_forall\n[GOAL]\ncase hp\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\nf : \u03a9 \u2192\u1d47 \u211d\u22650\n\u03bc : Measure \u03a9\n\u22a2 \u2200 (x : \u03a9), OfNat.ofNat 0 x \u2264 (fun x => \u2191(\u2191f x)) x\n[PROOFSTEP]\nsimp only [Pi.zero_apply, NNReal.zero_le_coe, imp_true_iff]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n[PROOFSTEP]\nrefine' \u27e8_, tendsto_of_forall_integral_tendsto\u27e9\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2192 \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n[PROOFSTEP]\nrw [tendsto_iff_forall_lintegral_tendsto]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))) \u2192\n    \u2200 (f : \u03a9 \u2192\u1d47 \u211d), Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n[PROOFSTEP]\nintro h f\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (x : \u03a9), \u2191f x \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191f x \u2202\u2191\u03bc))\n[PROOFSTEP]\nsimp_rw [BoundedContinuousFunction.integral_eq_integral_nnrealPart_sub]\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart f) \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart f) \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nset f_pos := f.nnrealPart with _def_f_pos\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191(nnrealPart (-f)) \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nset f_neg := (-f).nnrealPart with _def_f_neg\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nhave tends_pos :=\n  (ENNReal.tendsto_toReal (lintegral_lt_top_of_boundedContinuous_to_nnreal (\u03bc : Measure \u03a9) f_pos).ne).comp (h f_pos)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nhave tends_neg :=\n  (ENNReal.tendsto_toReal (lintegral_lt_top_of_boundedContinuous_to_nnreal (\u03bc : Measure \u03a9) f_neg).ne).comp (h f_neg)\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc)))\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nhave aux :\n  \u2200 g : \u03a9 \u2192\u1d47 \u211d\u22650,\n    (ENNReal.toReal \u2218 fun i : \u03b3 => \u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)) = fun i : \u03b3 =>\n      (\u222b\u207b x : \u03a9, \u2191(g x) \u2202(\u03bcs i : Measure \u03a9)).toReal :=\n  fun _ => rfl\n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\ntends_pos :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_pos x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc)))\ntends_neg :\n  Tendsto (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f_neg x) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (ENNReal.toReal (\u222b\u207b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc)))\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nsimp_rw [aux, BoundedContinuousFunction.NNReal.toReal_lintegral_eq_integral] at tends_pos tends_neg \n[GOAL]\n\u03a9 : Type u_1\ninst\u271d\u00b2 : MeasurableSpace \u03a9\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_2\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))\nf : \u03a9 \u2192\u1d47 \u211d\nf_pos : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart f\n_def_f_pos : f_pos = nnrealPart f\nf_neg : \u03a9 \u2192\u1d47 \u211d\u22650 := nnrealPart (-f)\n_def_f_neg : f_neg = nnrealPart (-f)\naux :\n  \u2200 (g : \u03a9 \u2192\u1d47 \u211d\u22650),\n    (ENNReal.toReal \u2218 fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i)) = fun i => ENNReal.toReal (\u222b\u207b (x : \u03a9), \u2191(\u2191g x) \u2202\u2191(\u03bcs i))\ntends_pos : Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart f) x) \u2202\u2191\u03bc))\ntends_neg :\n  Tendsto (fun i => \u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b (x : \u03a9), \u2191(\u2191(nnrealPart (-f)) x) \u2202\u2191\u03bc))\n\u22a2 Tendsto (fun i => \u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191(\u03bcs i) - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191(\u03bcs i)) F\n    (\ud835\udcdd (\u222b (\u03c9 : \u03a9), \u2191(\u2191f_pos \u03c9) \u2202\u2191\u03bc - \u222b (\u03c9 : \u03a9), \u2191(\u2191f_neg \u03c9) \u2202\u2191\u03bc))\n[PROOFSTEP]\nexact Tendsto.sub tends_pos tends_neg\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.FiniteMeasure", "llama_tokens": 41684, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5235074827046599}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), Quotient.mk (isSetoid \u03b1) a \u2208 powersetAux l \u2194 Quotient.mk (isSetoid \u03b1) a \u2264 \u2191l\n[PROOFSTEP]\nsimp [powersetAux_eq_map_coe, Subperm, and_comm]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 powersetAux l ~ powersetAux' l\n[PROOFSTEP]\nrw [powersetAux_eq_map_coe]\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 List.map ofList (sublists l) ~ powersetAux' l\n[PROOFSTEP]\nexact (sublists_perm_sublists' _).map _\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\n\u22a2 powersetAux' (a :: l) = powersetAux' l ++ List.map (cons a) (powersetAux' l)\n[PROOFSTEP]\nsimp [powersetAux']\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\nl : List \u03b1\n\u22a2 List.map (ofList \u2218 List.cons a) (sublists' l) = List.map (cons a \u2218 ofList) (sublists' l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 powersetAux' l\u2081 ~ powersetAux' l\u2082\n[PROOFSTEP]\ninduction' p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 _ _ IH\u2081 IH\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 powersetAux' [] ~ powersetAux' []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : powersetAux' l\u2081 ~ powersetAux' l\u2082\n\u22a2 powersetAux' (a :: l\u2081) ~ powersetAux' (a :: l\u2082)\n[PROOFSTEP]\nsimp only [powersetAux'_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : powersetAux' l\u2081 ~ powersetAux' l\u2082\n\u22a2 powersetAux' l\u2081 ++ List.map (cons a) (powersetAux' l\u2081) ~ powersetAux' l\u2082 ++ List.map (cons a) (powersetAux' l\u2082)\n[PROOFSTEP]\nexact IH.append (IH.map _)\n[GOAL]\ncase swap\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 powersetAux' (b :: a :: l) ~ powersetAux' (a :: b :: l)\n[PROOFSTEP]\nsimp only [powersetAux'_cons, map_append, List.map_map, append_assoc]\n[GOAL]\ncase swap\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 powersetAux' l ++\n      (List.map (cons a) (powersetAux' l) ++\n        (List.map (cons b) (powersetAux' l) ++ List.map (cons b \u2218 cons a) (powersetAux' l))) ~\n    powersetAux' l ++\n      (List.map (cons b) (powersetAux' l) ++\n        (List.map (cons a) (powersetAux' l) ++ List.map (cons a \u2218 cons b) (powersetAux' l)))\n[PROOFSTEP]\napply Perm.append_left\n[GOAL]\ncase swap.a\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.map (cons a) (powersetAux' l) ++\n      (List.map (cons b) (powersetAux' l) ++ List.map (cons b \u2218 cons a) (powersetAux' l)) ~\n    List.map (cons b) (powersetAux' l) ++\n      (List.map (cons a) (powersetAux' l) ++ List.map (cons a \u2218 cons b) (powersetAux' l))\n[PROOFSTEP]\nrw [\u2190 append_assoc, \u2190 append_assoc, (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)]\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 cons b \u2218 cons a = cons a \u2218 cons b\n[PROOFSTEP]\nfunext s\n[GOAL]\ncase h\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\ns : Multiset \u03b1\n\u22a2 (cons b \u2218 cons a) s = (cons a \u2218 cons b) s\n[PROOFSTEP]\nsimp [cons_swap]\n[GOAL]\ncase swap.a\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.map (cons a) (powersetAux' l) ++ List.map (cons b) (powersetAux' l) ++\n      List.map (cons a \u2218 cons b) (powersetAux' l) ~\n    List.map (cons b) (powersetAux' l) ++ List.map (cons a) (powersetAux' l) ++\n      List.map (cons a \u2218 cons b) (powersetAux' l)\n[PROOFSTEP]\nexact perm_append_comm.append_right _\n[GOAL]\ncase trans\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l\u2083 : List \u03b1\na\u271d\u00b9 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : powersetAux' l\u2081 ~ powersetAux' l\u2082\nIH\u2082 : powersetAux' l\u2082 ~ powersetAux' l\u2083\n\u22a2 powersetAux' l\u2081 ~ powersetAux' l\u2083\n[PROOFSTEP]\nexact IH\u2081.trans IH\u2082\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 powerset (a ::\u2098 Quotient.mk (isSetoid \u03b1) l) =\n    powerset (Quotient.mk (isSetoid \u03b1) l) + map (cons a) (powerset (Quotient.mk (isSetoid \u03b1) l))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.map ofList (sublists' l) ++ List.map (ofList \u2218 List.cons a) (sublists' l) ~\n    List.map ofList (sublists' l) ++ List.map ((fun x => a ::\u2098 x) \u2218 ofList) (sublists' l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ns t : Multiset \u03b1\n\u22a2 \u2200 (a b : List \u03b1),\n    Quotient.mk (isSetoid \u03b1) a \u2208 powerset (Quotient.mk (isSetoid \u03b1) b) \u2194\n      Quotient.mk (isSetoid \u03b1) a \u2264 Quotient.mk (isSetoid \u03b1) b\n[PROOFSTEP]\nsimp [Subperm, and_comm]\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 map singleton (Quotient.mk (isSetoid \u03b1) l) \u2264 powerset (Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nsimp only [powerset_coe, quot_mk_to_coe, coe_le, coe_map]\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.map singleton l <+~ List.map ofList (sublists l)\n[PROOFSTEP]\nshow l.map (((\u2191) : List \u03b1 \u2192 Multiset \u03b1) \u2218 List.ret) <+~ (sublists l).map (\u2191)\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.map (ofList \u2218 List.ret) l <+~ List.map ofList (sublists l)\n[PROOFSTEP]\nrw [\u2190 List.map_map]\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.map ofList (List.map List.ret l) <+~ List.map ofList (sublists l)\n[PROOFSTEP]\nexact ((map_ret_sublist_sublists _).map _).subperm\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), \u2191card (powerset (Quotient.mk (isSetoid \u03b1) a)) = 2 ^ \u2191card (Quotient.mk (isSetoid \u03b1) a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : x \u2208 revzip (powersetAux l)\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nrw [revzip, powersetAux_eq_map_coe, \u2190 map_reverse, zip_map, \u2190 revzip, List.mem_map] at h \n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : \u2203 a, a \u2208 revzip (sublists l) \u2227 Prod.map ofList ofList a = x\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nsimp only [Prod_map, Prod.exists] at h \n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : \u2203 a b, (a, b) \u2208 revzip (sublists l) \u2227 (\u2191a, \u2191b) = x\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nrcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refl\n\u03b1 : Type u_1\nl l\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 revzip (sublists l)\n\u22a2 (\u2191l\u2081, \u2191l\u2082).fst + (\u2191l\u2081, \u2191l\u2082).snd = \u2191l\n[PROOFSTEP]\nexact Quot.sound (revzip_sublists _ _ _ h)\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : x \u2208 revzip (powersetAux' l)\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nrw [revzip, powersetAux', \u2190 map_reverse, zip_map, \u2190 revzip, List.mem_map] at h \n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : \u2203 a, a \u2208 revzip (sublists' l) \u2227 Prod.map ofList ofList a = x\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nsimp only [Prod_map, Prod.exists] at h \n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nx : Multiset \u03b1 \u00d7 Multiset \u03b1\nh : \u2203 a b, (a, b) \u2208 revzip (sublists' l) \u2227 (\u2191a, \u2191b) = x\n\u22a2 x.fst + x.snd = \u2191l\n[PROOFSTEP]\nrcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.refl\n\u03b1 : Type u_1\nl l\u2081 l\u2082 : List \u03b1\nh : (l\u2081, l\u2082) \u2208 revzip (sublists' l)\n\u22a2 (\u2191l\u2081, \u2191l\u2082).fst + (\u2191l\u2081, \u2191l\u2082).snd = \u2191l\n[PROOFSTEP]\nexact Quot.sound (revzip_sublists' _ _ _ h)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\n\u22a2 revzip l' = List.map (fun x => (x, \u2191l - x)) l'\n[PROOFSTEP]\nhave :\n  Forall\u2082 (fun (p : Multiset \u03b1 \u00d7 Multiset \u03b1) (s : Multiset \u03b1) => p = (s, \u2191l - s)) (revzip l')\n    ((revzip l').map Prod.fst) :=\n  by\n  rw [forall\u2082_map_right_iff, forall\u2082_same]\n  rintro \u27e8s, t\u27e9 h\n  dsimp\n  rw [\u2190 H h, add_tsub_cancel_left]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\n\u22a2 Forall\u2082 (fun p s => p = (s, \u2191l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n[PROOFSTEP]\nrw [forall\u2082_map_right_iff, forall\u2082_same]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\n\u22a2 \u2200 (x : Multiset \u03b1 \u00d7 Multiset \u03b1), x \u2208 revzip l' \u2192 x = (x.fst, \u2191l - x.fst)\n[PROOFSTEP]\nrintro \u27e8s, t\u27e9 h\n[GOAL]\ncase mk\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\ns t : Multiset \u03b1\nh : (s, t) \u2208 revzip l'\n\u22a2 (s, t) = ((s, t).fst, \u2191l - (s, t).fst)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mk\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\ns t : Multiset \u03b1\nh : (s, t) \u2208 revzip l'\n\u22a2 (s, t) = (s, \u2191l - s)\n[PROOFSTEP]\nrw [\u2190 H h, add_tsub_cancel_left]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\nthis : Forall\u2082 (fun p s => p = (s, \u2191l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n\u22a2 revzip l' = List.map (fun x => (x, \u2191l - x)) l'\n[PROOFSTEP]\nrw [\u2190 forall\u2082_eq_eq_eq, forall\u2082_map_right_iff]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nl' : List (Multiset \u03b1)\nH : \u2200 \u2983x : Multiset \u03b1 \u00d7 Multiset \u03b1\u2984, x \u2208 revzip l' \u2192 x.fst + x.snd = \u2191l\nthis : Forall\u2082 (fun p s => p = (s, \u2191l - s)) (revzip l') (List.map Prod.fst (revzip l'))\n\u22a2 Forall\u2082 (fun a c => a = (c, \u2191l - c)) (revzip l') l'\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 revzip (powersetAux l) ~ revzip (powersetAux' l)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nthis : DecidableEq \u03b1\n\u22a2 revzip (powersetAux l) ~ revzip (powersetAux' l)\n[PROOFSTEP]\nrw [revzip_powersetAux_lemma l revzip_powersetAux, revzip_powersetAux_lemma l revzip_powersetAux']\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\nthis : DecidableEq \u03b1\n\u22a2 List.map (fun x => (x, \u2191l - x)) (powersetAux l) ~ List.map (fun x => (x, \u2191l - x)) (powersetAux' l)\n[PROOFSTEP]\nexact powersetAux_perm_powersetAux'.map _\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 revzip (powersetAux l\u2081) ~ revzip (powersetAux l\u2082)\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b1\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nthis : DecidableEq \u03b1\n\u22a2 revzip (powersetAux l\u2081) ~ revzip (powersetAux l\u2082)\n[PROOFSTEP]\nsimp [fun l : List \u03b1 => revzip_powersetAux_lemma l revzip_powersetAux, coe_eq_coe.2 p]\n[GOAL]\n\u03b1 : Type u_1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nthis : DecidableEq \u03b1\n\u22a2 List.map (fun x => (x, \u2191l\u2082 - x)) (powersetAux l\u2081) ~ List.map (fun x => (x, \u2191l\u2082 - x)) (powersetAux l\u2082)\n[PROOFSTEP]\nexact (powersetAux_perm p).map _\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : List \u03b1\n\u22a2 powersetLenAux n l = List.map ofList (sublistsLen n l)\n[PROOFSTEP]\nrw [powersetLenAux, sublistsLenAux_eq, append_nil]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : List \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1),\n    Quotient.mk (isSetoid \u03b1) a \u2208 powersetLenAux n l \u2194\n      Quotient.mk (isSetoid \u03b1) a \u2264 \u2191l \u2227 \u2191card (Quotient.mk (isSetoid \u03b1) a) = n\n[PROOFSTEP]\nsimp [powersetLenAux_eq_map_coe, Subperm]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl : List \u03b1\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), (\u2203 a_1, (a_1 <+ l \u2227 length a_1 = n) \u2227 a_1 ~ a) \u2194 (\u2203 l_1, l_1 ~ a \u2227 l_1 <+ l) \u2227 length a = n\n[PROOFSTEP]\nexact fun l\u2081 =>\n  \u27e8fun \u27e8l\u2082, \u27e8s, e\u27e9, p\u27e9 => \u27e8\u27e8_, p, s\u27e9, p.symm.length_eq.trans e\u27e9, fun \u27e8\u27e8l\u2082, p, s\u27e9, e\u27e9 =>\n    \u27e8_, \u27e8s, p.length_eq.trans e\u27e9, p\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\nl : List \u03b1\n\u22a2 powersetLenAux 0 l = [0]\n[PROOFSTEP]\nsimp [powersetLenAux_eq_map_coe]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 powersetLenAux (n + 1) (a :: l) = powersetLenAux (n + 1) l ++ List.map (cons a) (powersetLenAux n l)\n[PROOFSTEP]\nsimp [powersetLenAux_eq_map_coe]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 List.map (ofList \u2218 List.cons a) (sublistsLen n l) = List.map (cons a \u2218 ofList) (sublistsLen n l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\n[PROOFSTEP]\ninduction' n with n IHn generalizing l\u2081 l\u2082\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 powersetLenAux Nat.zero l\u2081 ~ powersetLenAux Nat.zero l\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np\u271d : l\u2081\u271d ~ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\n\u22a2 powersetLenAux (Nat.succ n) l\u2081 ~ powersetLenAux (Nat.succ n) l\u2082\n[PROOFSTEP]\ninduction' p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 _ _ IH\u2081 IH\u2082\n[GOAL]\ncase succ.nil\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081 l\u2082 : List \u03b1\n\u22a2 powersetLenAux (Nat.succ n) [] ~ powersetLenAux (Nat.succ n) []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.cons\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np\u271d : l\u2081\u271d\u00b9 ~ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : powersetLenAux (Nat.succ n) l\u2081 ~ powersetLenAux (Nat.succ n) l\u2082\n\u22a2 powersetLenAux (Nat.succ n) (a :: l\u2081) ~ powersetLenAux (Nat.succ n) (a :: l\u2082)\n[PROOFSTEP]\nsimp only [powersetLenAux_cons]\n[GOAL]\ncase succ.cons\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np\u271d : l\u2081\u271d\u00b9 ~ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081\u271d l\u2082\u271d : List \u03b1\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\np : l\u2081 ~ l\u2082\nIH : powersetLenAux (Nat.succ n) l\u2081 ~ powersetLenAux (Nat.succ n) l\u2082\n\u22a2 powersetLenAux (n + 1) l\u2081 ++ List.map (cons a) (powersetLenAux n l\u2081) ~\n    powersetLenAux (n + 1) l\u2082 ++ List.map (cons a) (powersetLenAux n l\u2082)\n[PROOFSTEP]\nexact IH.append ((IHn p).map _)\n[GOAL]\ncase succ.swap\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 powersetLenAux (Nat.succ n) (b :: a :: l) ~ powersetLenAux (Nat.succ n) (a :: b :: l)\n[PROOFSTEP]\nsimp only [powersetLenAux_cons, append_assoc]\n[GOAL]\ncase succ.swap\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 powersetLenAux (n + 1) l ++\n      (List.map (cons a) (powersetLenAux n l) ++ List.map (cons b) (powersetLenAux n (a :: l))) ~\n    powersetLenAux (n + 1) l ++\n      (List.map (cons b) (powersetLenAux n l) ++ List.map (cons a) (powersetLenAux n (b :: l)))\n[PROOFSTEP]\napply Perm.append_left\n[GOAL]\ncase succ.swap.a\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 List.map (cons a) (powersetLenAux n l) ++ List.map (cons b) (powersetLenAux n (a :: l)) ~\n    List.map (cons b) (powersetLenAux n l) ++ List.map (cons a) (powersetLenAux n (b :: l))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase succ.swap.a.zero\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux Nat.zero l\u2081 ~ powersetLenAux Nat.zero l\u2082\n\u22a2 List.map (cons a) (powersetLenAux Nat.zero l) ++ List.map (cons b) (powersetLenAux Nat.zero (a :: l)) ~\n    List.map (cons b) (powersetLenAux Nat.zero l) ++ List.map (cons a) (powersetLenAux Nat.zero (b :: l))\n[PROOFSTEP]\nsimp [Perm.swap]\n[GOAL]\ncase succ.swap.a.succ\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux (Nat.succ n\u271d) l\u2081 ~ powersetLenAux (Nat.succ n\u271d) l\u2082\n\u22a2 List.map (cons a) (powersetLenAux (Nat.succ n\u271d) l) ++ List.map (cons b) (powersetLenAux (Nat.succ n\u271d) (a :: l)) ~\n    List.map (cons b) (powersetLenAux (Nat.succ n\u271d) l) ++ List.map (cons a) (powersetLenAux (Nat.succ n\u271d) (b :: l))\n[PROOFSTEP]\nsimp only [powersetLenAux_cons, map_append, List.map_map]\n[GOAL]\ncase succ.swap.a.succ\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux (Nat.succ n\u271d) l\u2081 ~ powersetLenAux (Nat.succ n\u271d) l\u2082\n\u22a2 List.map (cons a) (powersetLenAux (Nat.succ n\u271d) l) ++\n      (List.map (cons b) (powersetLenAux (n\u271d + 1) l) ++ List.map (cons b \u2218 cons a) (powersetLenAux n\u271d l)) ~\n    List.map (cons b) (powersetLenAux (Nat.succ n\u271d) l) ++\n      (List.map (cons a) (powersetLenAux (n\u271d + 1) l) ++ List.map (cons a \u2218 cons b) (powersetLenAux n\u271d l))\n[PROOFSTEP]\nrw [\u2190 append_assoc, \u2190 append_assoc, (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)]\n[GOAL]\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux (Nat.succ n\u271d) l\u2081 ~ powersetLenAux (Nat.succ n\u271d) l\u2082\n\u22a2 cons b \u2218 cons a = cons a \u2218 cons b\n[PROOFSTEP]\nfunext s\n[GOAL]\ncase h\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux (Nat.succ n\u271d) l\u2081 ~ powersetLenAux (Nat.succ n\u271d) l\u2082\ns : Multiset \u03b1\n\u22a2 (cons b \u2218 cons a) s = (cons a \u2218 cons b) s\n[PROOFSTEP]\nsimp [cons_swap]\n[GOAL]\ncase succ.swap.a.succ\n\u03b1 : Type u_1\nl\u2081\u271d l\u2082\u271d : List \u03b1\np : l\u2081\u271d ~ l\u2082\u271d\nl\u2081 l\u2082 : List \u03b1\na b : \u03b1\nl : List \u03b1\nn\u271d : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux (Nat.succ n\u271d) l\u2081 ~ powersetLenAux (Nat.succ n\u271d) l\u2082\n\u22a2 List.map (cons a) (powersetLenAux (Nat.succ n\u271d) l) ++ List.map (cons b) (powersetLenAux (n\u271d + 1) l) ++\n      List.map (cons a \u2218 cons b) (powersetLenAux n\u271d l) ~\n    List.map (cons b) (powersetLenAux (Nat.succ n\u271d) l) ++ List.map (cons a) (powersetLenAux (n\u271d + 1) l) ++\n      List.map (cons a \u2218 cons b) (powersetLenAux n\u271d l)\n[PROOFSTEP]\nexact perm_append_comm.append_right _\n[GOAL]\ncase succ.trans\n\u03b1 : Type u_1\nl\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List \u03b1\np : l\u2081\u271d\u00b9 ~ l\u2082\u271d\u00b9\nn : \u2115\nIHn : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 ~ l\u2082 \u2192 powersetLenAux n l\u2081 ~ powersetLenAux n l\u2082\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 l\u2083 : List \u03b1\na\u271d\u00b9 : l\u2081 ~ l\u2082\na\u271d : l\u2082 ~ l\u2083\nIH\u2081 : powersetLenAux (Nat.succ n) l\u2081 ~ powersetLenAux (Nat.succ n) l\u2082\nIH\u2082 : powersetLenAux (Nat.succ n) l\u2082 ~ powersetLenAux (Nat.succ n) l\u2083\n\u22a2 powersetLenAux (Nat.succ n) l\u2081 ~ powersetLenAux (Nat.succ n) l\u2083\n[PROOFSTEP]\nexact IH\u2081.trans IH\u2082\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 powersetLen 0 (Quotient.mk (isSetoid \u03b1) l) = {0}\n[PROOFSTEP]\nsimp [powersetLen_coe']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\na : \u03b1\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 powersetLen (n + 1) (a ::\u2098 Quotient.mk (isSetoid \u03b1) l) =\n    powersetLen (n + 1) (Quotient.mk (isSetoid \u03b1) l) + map (cons a) (powersetLen n (Quotient.mk (isSetoid \u03b1) l))\n[PROOFSTEP]\nsimp [powersetLen_coe']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns t : Multiset \u03b1\nl : List \u03b1\n\u22a2 s \u2208 powersetLen n (Quotient.mk (isSetoid \u03b1) l) \u2194 s \u2264 Quotient.mk (isSetoid \u03b1) l \u2227 \u2191card s = n\n[PROOFSTEP]\nsimp [powersetLen_coe']\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns : Multiset \u03b1\n\u22a2 \u2200 (a : List \u03b1), \u2191card (powersetLen n (Quotient.mk (isSetoid \u03b1) a)) = Nat.choose (\u2191card (Quotient.mk (isSetoid \u03b1) a)) n\n[PROOFSTEP]\nsimp [powersetLen_coe]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 powersetLen n (Quotient.mk (isSetoid \u03b1) l) \u2264 powerset (Quotient.mk (isSetoid \u03b1) l)\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, powersetLen_coe, powerset_coe', coe_le]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns : Multiset \u03b1\nl : List \u03b1\n\u22a2 List.map ofList (sublistsLen n l) <+~ List.map ofList (sublists' l)\n[PROOFSTEP]\nexact ((sublistsLen_sublist_sublists' _ _).map _).subperm\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns t : Multiset \u03b1\nh\u271d : s \u2264 t\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 powersetLen n \u2191l\u2081 \u2264 powersetLen n \u2191l\u2082\n[PROOFSTEP]\nsimp only [powersetLen_coe, coe_le]\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ns t : Multiset \u03b1\nh\u271d : s \u2264 t\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <+ l\u2082\n\u22a2 List.map ofList (sublistsLen n l\u2081) <+~ List.map ofList (sublistsLen n l\u2082)\n[PROOFSTEP]\nexact ((sublistsLen_sublist_of_sublist _ h).map _).subperm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\ns : Multiset \u03b1\n\u22a2 powersetLen n (map f s) = map (map f) (powersetLen n s)\n[PROOFSTEP]\ninduction' s using Multiset.induction with t s ih generalizing n\n[GOAL]\ncase empty\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn\u271d n : \u2115\n\u22a2 powersetLen n (map f 0) = map (map f) (powersetLen n 0)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase empty.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 powersetLen Nat.zero (map f 0) = map (map f) (powersetLen Nat.zero 0)\n[PROOFSTEP]\nsimp [powersetLen_zero_left, powersetLen_zero_right]\n[GOAL]\ncase empty.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn n\u271d : \u2115\n\u22a2 powersetLen (Nat.succ n\u271d) (map f 0) = map (map f) (powersetLen (Nat.succ n\u271d) 0)\n[PROOFSTEP]\nsimp [powersetLen_zero_left, powersetLen_zero_right]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nt : \u03b1\ns : Multiset \u03b1\nih : \u2200 (n : \u2115), powersetLen n (map f s) = map (map f) (powersetLen n s)\nn : \u2115\n\u22a2 powersetLen n (map f (t ::\u2098 s)) = map (map f) (powersetLen n (t ::\u2098 s))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase cons.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nt : \u03b1\ns : Multiset \u03b1\nih : \u2200 (n : \u2115), powersetLen n (map f s) = map (map f) (powersetLen n s)\n\u22a2 powersetLen Nat.zero (map f (t ::\u2098 s)) = map (map f) (powersetLen Nat.zero (t ::\u2098 s))\n[PROOFSTEP]\nsimp [ih, map_comp_cons]\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\nn : \u2115\nt : \u03b1\ns : Multiset \u03b1\nih : \u2200 (n : \u2115), powersetLen n (map f s) = map (map f) (powersetLen n s)\nn\u271d : \u2115\n\u22a2 powersetLen (Nat.succ n\u271d) (map f (t ::\u2098 s)) = map (map f) (powersetLen (Nat.succ n\u271d) (t ::\u2098 s))\n[PROOFSTEP]\nsimp [ih, map_comp_cons]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\nS : Multiset \u03b1\n\u22a2 (bind (range (\u2191card S + 1)) fun k => powersetLen k S) = powerset S\n[PROOFSTEP]\ninduction S using Quotient.inductionOn\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\na\u271d : List \u03b1\n\u22a2 (bind (range (\u2191card (Quotient.mk (isSetoid \u03b1) a\u271d) + 1)) fun k => powersetLen k (Quotient.mk (isSetoid \u03b1) a\u271d)) =\n    powerset (Quotient.mk (isSetoid \u03b1) a\u271d)\n[PROOFSTEP]\nsimp_rw [quot_mk_to_coe, powerset_coe', powersetLen_coe, \u2190 coe_range, coe_bind, \u2190 List.bind_map, coe_card]\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\na\u271d : List \u03b1\n\u22a2 \u2191(List.map ofList (List.bind (List.range (length a\u271d + 1)) fun a => sublistsLen a a\u271d)) =\n    \u2191(List.map ofList (sublists' a\u271d))\n[PROOFSTEP]\nexact coe_eq_coe.mpr ((List.range_bind_sublistsLen_perm _).map _)\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 Nodup (powerset (Quotient.mk (isSetoid \u03b1) l))\n[PROOFSTEP]\nsimp only [quot_mk_to_coe, powerset_coe', coe_nodup]\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 List.Nodup (List.map ofList (sublists' l))\n[PROOFSTEP]\nrefine' (nodup_sublists'.2 h).map_on _\n[GOAL]\n\u03b1 : Type u_1\ns : Multiset \u03b1\nl : List \u03b1\nh : Nodup (Quotient.mk (isSetoid \u03b1) l)\n\u22a2 \u2200 (x : List \u03b1), x \u2208 sublists' l \u2192 \u2200 (y : List \u03b1), y \u2208 sublists' l \u2192 \u2191x = \u2191y \u2192 x = y\n[PROOFSTEP]\nexact fun x sx y sy e => (h.sublist_ext (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1 (Quotient.exact e)\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Powerset", "llama_tokens": 11147, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.5234360194654252}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n\u22a2 \u2016\u2191f x\u2016 \u2264 \u2016\u2191f 1\u2016 * \u2016x\u2016\n[PROOFSTEP]\nconv_lhs => rw [\u2190 mul_one x]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n| \u2016\u2191f x\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n| \u2016\u2191f x\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n| \u2016\u2191f x\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one x]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n\u22a2 \u2016\u2191f (x * 1)\u2016 \u2264 \u2016\u2191f 1\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, f.map_smul, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2074 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b3 : Ring \ud835\udd5c\u2082\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nf : \ud835\udd5c \u2192\u2097[\ud835\udd5c] E\nx : \ud835\udd5c\n\u22a2 \u2016x \u2022 \u2191f 1\u2016 \u2264 \u2016x\u2016 * \u2016\u2191f 1\u2016\n[PROOFSTEP]\nexact norm_smul_le _ _\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\n\u22a2 (\u2200 (x : E), \u2016\u2191f x\u2016 = a * \u2016x\u2016) \u2192 \u2200 (y : F), \u2016\u2191(LinearEquiv.symm f) y\u2016 = a\u207b\u00b9 * \u2016y\u2016\n[PROOFSTEP]\nintro hf y\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016\u2191f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 \u2016\u2191(LinearEquiv.symm f) y\u2016 = a\u207b\u00b9 * \u2016y\u2016\n[PROOFSTEP]\ncalc\n  \u2016f.symm y\u2016 = a\u207b\u00b9 * (a * \u2016f.symm y\u2016) := by rw [\u2190 mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]\n  _ = a\u207b\u00b9 * \u2016f (f.symm y)\u2016 := by rw [hf]\n  _ = a\u207b\u00b9 * \u2016y\u2016 := by simp\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016\u2191f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 \u2016\u2191(LinearEquiv.symm f) y\u2016 = a\u207b\u00b9 * (a * \u2016\u2191(LinearEquiv.symm f) y\u2016)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016\u2191f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 a\u207b\u00b9 * (a * \u2016\u2191(LinearEquiv.symm f) y\u2016) = a\u207b\u00b9 * \u2016\u2191f (\u2191(LinearEquiv.symm f) y)\u2016\n[PROOFSTEP]\nrw [hf]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u2077 : Ring \ud835\udd5c\ninst\u271d\u2076 : Ring \ud835\udd5c\u2082\ninst\u271d\u2075 : SeminormedAddCommGroup E\ninst\u271d\u2074 : SeminormedAddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 F\n\u03c3 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082\nf\u271d : E \u2192\u209b\u2097[\u03c3] F\n\u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\ninst\u271d\u00b9 : RingHomInvPair \u03c3 \u03c3\u2082\u2081\ninst\u271d : RingHomInvPair \u03c3\u2082\u2081 \u03c3\na : \u211d\nha : 0 < a\nf : E \u2243\u209b\u2097[\u03c3] F\nhf : \u2200 (x : E), \u2016\u2191f x\u2016 = a * \u2016x\u2016\ny : F\n\u22a2 a\u207b\u00b9 * \u2016\u2191f (\u2191(LinearEquiv.symm f) y)\u2016 = a\u207b\u00b9 * \u2016y\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : E\nc : \ud835\udd5c\n\u22a2 \u2016\u2191(LinearMap.toSpanSingleton \ud835\udd5c E x) c\u2016 = \u2016x\u2016 * \u2016c\u2016\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : SeminormedAddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : BoundedSMul \ud835\udd5c E\nx : E\nc : \ud835\udd5c\n\u22a2 \u2016\u2191(LinearMap.toSpanSingleton \ud835\udd5c E x) c\u2016 = \u2016c\u2016 * \u2016x\u2016\n[PROOFSTEP]\nexact norm_smul _ _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.ContinuousLinearMap", "llama_tokens": 2802, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430645886584, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.5234327971982234}}
{"text": "[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 natDegree (det (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C)) \u2264 Fintype.card n\n[PROOFSTEP]\nrw [det_apply]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 natDegree (\u2211 \u03c3 : Equiv.Perm n, \u2191sign \u03c3 \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191\u03c3 i) i) \u2264 Fintype.card n\n[PROOFSTEP]\nrefine' (natDegree_sum_le _ _).trans _\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 Finset.fold max 0 (natDegree \u2218 fun \u03c3 => \u2191sign \u03c3 \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191\u03c3 i) i)\n      Finset.univ \u2264\n    Fintype.card n\n[PROOFSTEP]\nrefine' Multiset.max_nat_le_of_forall_le _ _ _\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2200 (x : \u2115),\n    x \u2208\n        Multiset.map (natDegree \u2218 fun \u03c3 => \u2191sign \u03c3 \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191\u03c3 i) i)\n          Finset.univ.val \u2192\n      x \u2264 Fintype.card n\n[PROOFSTEP]\nsimp only [forall_apply_eq_imp_iff', true_and_iff, Function.comp_apply, Multiset.map_map, Multiset.mem_map, exists_imp,\n  Finset.mem_univ_val]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2200 (a : Equiv.Perm n), natDegree (\u2191sign a \u2022 \u220f x : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191a x) x) \u2264 Fintype.card n\n[PROOFSTEP]\nintro g\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 natDegree (\u2191sign g \u2022 \u220f x : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x) x) \u2264 Fintype.card n\n[PROOFSTEP]\ncalc\n  natDegree (sign g \u2022 \u220f i : n, (X \u2022 A.map C + B.map C) (g i) i) \u2264\n      natDegree (\u220f i : n, (X \u2022 A.map C + B.map C) (g i) i) :=\n    by\n    cases' Int.units_eq_one_or (sign g) with sg sg\n    \u00b7 rw [sg, one_smul]\n    \u00b7 rw [sg, Units.neg_smul, one_smul, natDegree_neg]\n  _ \u2264 \u2211 i : n, natDegree (((X : \u03b1[X]) \u2022 A.map C + B.map C) (g i) i) :=\n    (natDegree_prod_le (Finset.univ : Finset n) fun i : n => (X \u2022 A.map C + B.map C) (g i) i)\n  _ \u2264 Finset.univ.card \u2022 1 := (Finset.sum_le_card_nsmul _ _ 1 fun (i : n) _ => ?_)\n  _ \u2264 Fintype.card n := by simp [mul_one, Algebra.id.smul_eq_mul, Finset.card_univ]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 natDegree (\u2191sign g \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) \u2264\n    natDegree (\u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i)\n[PROOFSTEP]\ncases' Int.units_eq_one_or (sign g) with sg sg\n[GOAL]\ncase inl\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\nsg : \u2191sign g = 1\n\u22a2 natDegree (\u2191sign g \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) \u2264\n    natDegree (\u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i)\n[PROOFSTEP]\nrw [sg, one_smul]\n[GOAL]\ncase inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\nsg : \u2191sign g = -1\n\u22a2 natDegree (\u2191sign g \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) \u2264\n    natDegree (\u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i)\n[PROOFSTEP]\nrw [sg, Units.neg_smul, one_smul, natDegree_neg]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 Finset.card Finset.univ \u2022 1 \u2264 Fintype.card n\n[PROOFSTEP]\nsimp [mul_one, Algebra.id.smul_eq_mul, Finset.card_univ]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\ni : n\nx\u271d : i \u2208 Finset.univ\n\u22a2 natDegree ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) \u2264 1\n[PROOFSTEP]\ncalc\n  natDegree (((X : \u03b1[X]) \u2022 A.map C + B.map C) (g i) i) = natDegree ((X : \u03b1[X]) * C (A (g i) i) + C (B (g i) i)) := by\n    simp\n  _ \u2264 max (natDegree ((X : \u03b1[X]) * C (A (g i) i))) (natDegree (C (B (g i) i))) := (natDegree_add_le _ _)\n  _ = natDegree ((X : \u03b1[X]) * C (A (g i) i)) := (max_eq_left ((natDegree_C _).le.trans (zero_le _)))\n  _ \u2264 natDegree (X : \u03b1[X]) := (natDegree_mul_C_le _ _)\n  _ \u2264 1 := natDegree_X_le\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\ni : n\nx\u271d : i \u2208 Finset.univ\n\u22a2 natDegree ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) = natDegree (X * \u2191C (A (\u2191g i) i) + \u2191C (B (\u2191g i) i))\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 coeff (det (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C)) 0 = det B\n[PROOFSTEP]\nrw [det_apply, finset_sum_coeff, det_apply]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2211 b : Equiv.Perm n, coeff (\u2191sign b \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191b i) i) 0 =\n    \u2211 \u03c3 : Equiv.Perm n, \u2191sign \u03c3 \u2022 \u220f i : n, B (\u2191\u03c3 i) i\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl _\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2200 (x : Equiv.Perm n),\n    x \u2208 Finset.univ \u2192\n      coeff (\u2191sign x \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191x i) i) 0 = \u2191sign x \u2022 \u220f i : n, B (\u2191x i) i\n[PROOFSTEP]\nrintro g -\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 coeff (\u2191sign g \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) 0 = \u2191sign g \u2022 \u220f i : n, B (\u2191g i) i\n[PROOFSTEP]\nconvert coeff_smul (R := \u03b1) (sign g) _ 0\n[GOAL]\ncase h.e'_3.h.e'_6\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u220f i : n, B (\u2191g i) i = coeff (\u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) 0\n[PROOFSTEP]\nrw [coeff_zero_prod]\n[GOAL]\ncase h.e'_3.h.e'_6\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u220f i : n, B (\u2191g i) i = \u220f i : n, coeff ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g i) i) 0\n[PROOFSTEP]\nrefine' Finset.prod_congr rfl _\n[GOAL]\ncase h.e'_3.h.e'_6\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u2200 (x : n), x \u2208 Finset.univ \u2192 B (\u2191g x) x = coeff ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x) x) 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 coeff (det (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C)) (Fintype.card n) = det A\n[PROOFSTEP]\nrw [det_apply, det_apply, finset_sum_coeff]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2211 b : Equiv.Perm n, coeff (\u2191sign b \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191b i) i) (Fintype.card n) =\n    \u2211 \u03c3 : Equiv.Perm n, \u2191sign \u03c3 \u2022 \u220f i : n, A (\u2191\u03c3 i) i\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl _\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2200 (x : Equiv.Perm n),\n    x \u2208 Finset.univ \u2192\n      coeff (\u2191sign x \u2022 \u220f i : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191x i) i) (Fintype.card n) =\n        \u2191sign x \u2022 \u220f i : n, A (\u2191x i) i\n[PROOFSTEP]\nsimp only [Algebra.id.smul_eq_mul, Finset.mem_univ, RingHom.mapMatrix_apply, forall_true_left, map_apply, Pi.smul_apply]\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\n\u22a2 \u2200 (x : Equiv.Perm n),\n    coeff (\u2191sign x \u2022 \u220f x_1 : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191x x_1) x_1) (Fintype.card n) =\n      \u2191sign x \u2022 \u220f x_1 : n, A (\u2191x x_1) x_1\n[PROOFSTEP]\nintro g\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 coeff (\u2191sign g \u2022 \u220f x : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x) x) (Fintype.card n) =\n    \u2191sign g \u2022 \u220f x : n, A (\u2191g x) x\n[PROOFSTEP]\nconvert coeff_smul (R := \u03b1) (sign g) _ _\n[GOAL]\ncase h.e'_3.h.e'_6\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u220f x : n, A (\u2191g x) x = coeff (\u220f x : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x) x) (Fintype.card n)\n[PROOFSTEP]\nrw [\u2190 mul_one (Fintype.card n)]\n[GOAL]\ncase h.e'_3.h.e'_6\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u220f x : n, A (\u2191g x) x = coeff (\u220f x : n, (X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x) x) (Fintype.card n * 1)\n[PROOFSTEP]\nconvert (coeff_prod_of_natDegree_le (R := \u03b1) _ _ _ _).symm\n[GOAL]\ncase h.e'_2.a\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\nx\u271d : n\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 A (\u2191g x\u271d) x\u271d = coeff ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g x\u271d) x\u271d) 1\n[PROOFSTEP]\nsimp [coeff_C]\n[GOAL]\ncase h.e'_3.h.e'_6.convert_5\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\n\u22a2 \u2200 (p : n), p \u2208 Finset.univ \u2192 natDegree ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g p) p) \u2264 1\n[PROOFSTEP]\nrintro p -\n[GOAL]\ncase h.e'_3.h.e'_6.convert_5\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\np : n\n\u22a2 natDegree ((X \u2022 Matrix.map A \u2191C + Matrix.map B \u2191C) (\u2191g p) p) \u2264 1\n[PROOFSTEP]\nrefine' (natDegree_add_le _ _).trans _\n[GOAL]\ncase h.e'_3.h.e'_6.convert_5\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA B : Matrix n n \u03b1\ng : Equiv.Perm n\np : n\n\u22a2 max (natDegree ((X \u2022 Matrix.map A \u2191C) (\u2191g p) p)) (natDegree (Matrix.map B (\u2191C) (\u2191g p) p)) \u2264 1\n[PROOFSTEP]\nsimpa [Pi.smul_apply, map_apply, Algebra.id.smul_eq_mul, X_mul_C, natDegree_C, max_eq_left, zero_le'] using\n  (natDegree_C_mul_le _ _).trans (natDegree_X_le (R := \u03b1))\n[GOAL]\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 leadingCoeff (det (X \u2022 1 + Matrix.map A \u2191C)) = 1\n[PROOFSTEP]\ncases subsingleton_or_nontrivial \u03b1\n[GOAL]\ncase inl\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Subsingleton \u03b1\n\u22a2 leadingCoeff (det (X \u2022 1 + Matrix.map A \u2191C)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 leadingCoeff (det (X \u2022 1 + Matrix.map A \u2191C)) = 1\n[PROOFSTEP]\nrw [\u2190 @det_one n, \u2190 coeff_det_X_add_C_card _ A, leadingCoeff]\n[GOAL]\ncase inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nsimp only [Matrix.map_one, C_eq_zero, RingHom.map_one]\n[GOAL]\ncase inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\ncases' (natDegree_det_X_add_C_le 1 A).eq_or_lt with h h\n[GOAL]\ncase inr.inl\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\nh : natDegree (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) = Fintype.card n\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nsimp only [RingHom.map_one, Matrix.map_one, C_eq_zero] at h \n[GOAL]\ncase inr.inl\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\nh : natDegree (det (X \u2022 1 + Matrix.map A \u2191C)) = Fintype.card n\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase inr.inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\nh : natDegree (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) < Fintype.card n\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nhave H := coeff_eq_zero_of_natDegree_lt h\n[GOAL]\ncase inr.inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\nh : natDegree (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) < Fintype.card n\nH : coeff (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) (Fintype.card n) = 0\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nrw [coeff_det_X_add_C_card] at H \n[GOAL]\ncase inr.inr\nn : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : CommRing \u03b1\nA : Matrix n n \u03b1\nh\u271d : Nontrivial \u03b1\nh : natDegree (det (X \u2022 Matrix.map 1 \u2191C + Matrix.map A \u2191C)) < Fintype.card n\nH : det 1 = 0\n\u22a2 coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (natDegree (det (X \u2022 1 + Matrix.map A \u2191C))) =\n    coeff (det (X \u2022 1 + Matrix.map A \u2191C)) (Fintype.card n)\n[PROOFSTEP]\nsimp at H \n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Polynomial", "llama_tokens": 7140, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430478583168, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.5234327866618791}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\n\u22a2 BaireSpace \u03b1\n[PROOFSTEP]\nrefine' \u27e8fun f ho hd => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nlet B : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave Bpos : \u2200 n, 0 < B n := by\n  intro n\n  simp only [one_div, one_mul, ENNReal.inv_pos]\n  exact pow_ne_top two_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\n\u22a2 \u2200 (n : \u2115), 0 < B n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nn : \u2115\n\u22a2 0 < B n\n[PROOFSTEP]\nsimp only [one_div, one_mul, ENNReal.inv_pos]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nn : \u2115\n\u22a2 2 ^ n \u2260 \u22a4\n[PROOFSTEP]\nexact pow_ne_top two_ne_top\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave : \u2200 n x \u03b4, \u03b4 \u2260 0 \u2192 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n :=\n  by\n  intro n x \u03b4 \u03b4pos\n  have : x \u2208 closure (f n) := hd n x\n  rcases EMetric.mem_closure_iff.1 this (\u03b4 / 2) (ENNReal.half_pos \u03b4pos) with \u27e8y, ys, xy\u27e9\n  rw [edist_comm] at xy \n  obtain \u27e8r, rpos, hr\u27e9 : \u2203 r > 0, closedBall y r \u2286 f n :=\n    nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n  refine' \u27e8y, min (min (\u03b4 / 2) r) (B (n + 1)), _, _, fun z hz => \u27e8_, _\u27e9\u27e9\n  show 0 < min (min (\u03b4 / 2) r) (B (n + 1))\n  exact lt_min (lt_min (ENNReal.half_pos \u03b4pos) rpos) (Bpos (n + 1))\n  show min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\n  exact min_le_right _ _\n  show z \u2208 closedBall x \u03b4\n  exact\n    calc\n      edist z x \u2264 edist z y + edist y x := edist_triangle _ _ _\n      _ \u2264 min (min (\u03b4 / 2) r) (B (n + 1)) + \u03b4 / 2 := (add_le_add hz (le_of_lt xy))\n      _ \u2264 \u03b4 / 2 + \u03b4 / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)\n      _ = \u03b4 := ENNReal.add_halves \u03b4\n  show z \u2208 f n\n  exact\n    hr\n      (calc\n        edist z y \u2264 min (min (\u03b4 / 2) r) (B (n + 1)) := hz\n        _ \u2264 r := le_trans (min_le_left _ _) (min_le_right _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\n\u22a2 \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nintro n x \u03b4 \u03b4pos\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\n\u22a2 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nhave : x \u2208 closure (f n) := hd n x\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\n\u22a2 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nrcases EMetric.mem_closure_iff.1 this (\u03b4 / 2) (ENNReal.half_pos \u03b4pos) with \u27e8y, ys, xy\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist x y < \u03b4 / 2\n\u22a2 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nrw [edist_comm] at xy \n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\n\u22a2 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nobtain \u27e8r, rpos, hr\u27e9 : \u2203 r > 0, closedBall y r \u2286 f n :=\n  nhds_basis_closed_eball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n[PROOFSTEP]\nrefine' \u27e8y, min (min (\u03b4 / 2) r) (B (n + 1)), _, _, fun z hz => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 0 < min (min (\u03b4 / 2) r) (B (n + 1))\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nshow 0 < min (min (\u03b4 / 2) r) (B (n + 1))\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 0 < min (min (\u03b4 / 2) r) (B (n + 1))\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nexact lt_min (lt_min (ENNReal.half_pos \u03b4pos) rpos) (Bpos (n + 1))\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nshow min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\n\u22a2 min (min (\u03b4 / 2) r) (B (n + 1)) \u2264 B (n + 1)\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nexact min_le_right _ _\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nshow z \u2208 closedBall x \u03b4\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 closedBall x \u03b4\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nexact\n  calc\n    edist z x \u2264 edist z y + edist y x := edist_triangle _ _ _\n    _ \u2264 min (min (\u03b4 / 2) r) (B (n + 1)) + \u03b4 / 2 := (add_le_add hz (le_of_lt xy))\n    _ \u2264 \u03b4 / 2 + \u03b4 / 2 := (add_le_add (le_trans (min_le_left _ _) (min_le_left _ _)) le_rfl)\n    _ = \u03b4 := ENNReal.add_halves \u03b4\n[GOAL]\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nshow z \u2208 f n\n[GOAL]\ncase intro.intro.intro.intro.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nn : \u2115\nx : \u03b1\n\u03b4 : \u211d\u22650\u221e\n\u03b4pos : \u03b4 \u2260 0\nthis : x \u2208 closure (f n)\ny : \u03b1\nys : y \u2208 f n\nxy : edist y x < \u03b4 / 2\nr : \u211d\u22650\u221e\nrpos : r > 0\nhr : closedBall y r \u2286 f n\nz : \u03b1\nhz : z \u2208 closedBall y (min (min (\u03b4 / 2) r) (B (n + 1)))\n\u22a2 z \u2208 f n\n[PROOFSTEP]\nexact\n  hr\n    (calc\n      edist z y \u2264 min (min (\u03b4 / 2) r) (B (n + 1)) := hz\n      _ \u2264 r := le_trans (min_le_left _ _) (min_le_right _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\nthis : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 \u2203 y r, 0 < r \u2227 r \u2264 B (n + 1) \u2227 closedBall y r \u2286 closedBall x \u03b4 \u2229 f n\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nchoose! center radius Hpos HB Hball using this\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nrefine' fun x =>\n  (mem_closure_iff_nhds_basis nhds_basis_closed_eball).2 fun \u03b5 \u03b5pos =>\n    _\n      /- `\u03b5` is positive. We have to find a point in the ball of radius `\u03b5` around `x` belonging to all\n          `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed\n          ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that\n          `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a\n          limit which belongs to all the `f n`. -/\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nlet F : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n =>\n  Nat.recOn n (Prod.mk x (min \u03b5 (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nlet c : \u2115 \u2192 \u03b1 := fun n => (F n).1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nlet r : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave rpos : \u2200 n, 0 < r n := by\n  intro n\n  induction' n with n hn\n  exact lt_min \u03b5pos (Bpos 0)\n  exact Hpos n (c n) (r n) hn.ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\n\u22a2 \u2200 (n : \u2115), 0 < r n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nn : \u2115\n\u22a2 0 < r n\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\n\u22a2 0 < r Nat.zero\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nn : \u2115\nhn : 0 < r n\n\u22a2 0 < r (Nat.succ n)\n[PROOFSTEP]\nexact lt_min \u03b5pos (Bpos 0)\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nn : \u2115\nhn : 0 < r n\n\u22a2 0 < r (Nat.succ n)\n[PROOFSTEP]\nexact Hpos n (c n) (r n) hn.ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave r0 : \u2200 n, r n \u2260 0 := fun n => (rpos n).ne'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave rB : \u2200 n, r n \u2264 B n := by\n  intro n\n  induction' n with n _\n  exact min_le_right _ _\n  exact HB n (c n) (r n) (r0 n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\n\u22a2 \u2200 (n : \u2115), r n \u2264 B n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nn : \u2115\n\u22a2 r n \u2264 B n\n[PROOFSTEP]\ninduction' n with n _\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\n\u22a2 r Nat.zero \u2264 B Nat.zero\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nn : \u2115\nn_ih\u271d : r n \u2264 B n\n\u22a2 r (Nat.succ n) \u2264 B (Nat.succ n)\n[PROOFSTEP]\nexact min_le_right _ _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nn : \u2115\nn_ih\u271d : r n \u2264 B n\n\u22a2 r (Nat.succ n) \u2264 B (Nat.succ n)\n[PROOFSTEP]\nexact HB n (c n) (r n) (r0 n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave incl : \u2200 n, closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n := fun n =>\n  Hball n (c n) (r n) (r0 n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave cdist : \u2200 n, edist (c n) (c (n + 1)) \u2264 B n := by\n  intro n\n  rw [edist_comm]\n  have A : c (n + 1) \u2208 closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self\n  have I :=\n    calc\n      closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) := Subset.trans (incl n) (inter_subset_left _ _)\n      _ \u2286 closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n  exact I A\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\n\u22a2 \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\nn : \u2115\n\u22a2 edist (c n) (c (n + 1)) \u2264 B n\n[PROOFSTEP]\nrw [edist_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\nn : \u2115\n\u22a2 edist (c (n + 1)) (c n) \u2264 B n\n[PROOFSTEP]\nhave A : c (n + 1) \u2208 closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\nn : \u2115\nA : c (n + 1) \u2208 closedBall (c (n + 1)) (r (n + 1))\n\u22a2 edist (c (n + 1)) (c n) \u2264 B n\n[PROOFSTEP]\nhave I :=\n  calc\n    closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) := Subset.trans (incl n) (inter_subset_left _ _)\n    _ \u2286 closedBall (c n) (B n) := closedBall_subset_closedBall (rB n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\nn : \u2115\nA : c (n + 1) \u2208 closedBall (c (n + 1)) (r (n + 1))\nI : closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (B n)\n\u22a2 edist (c (n + 1)) (c n) \u2264 B n\n[PROOFSTEP]\nexact I A\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave : CauchySeq c := cauchySeq_of_edist_le_geometric_two _ one_ne_top cdist\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nrcases cauchySeq_tendsto_of_complete this with\n  \u27e8y, ylim\u27e9\n    -- this point `y` will be the desired point. We will check that it belongs to all\n      -- `f n` and to `ball x \u03b5`.\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\n\u22a2 \u2203 y, y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nuse y\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\n\u22a2 y \u2208 \u22c2 (n : \u2115), f n \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iInter]\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\n\u22a2 (\u2200 (i : \u2115), y \u2208 f i) \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave I : \u2200 n, \u2200 m \u2265 n, closedBall (c m) (r m) \u2286 closedBall (c n) (r n) :=\n  by\n  intro n\n  refine' Nat.le_induction _ fun m _ h => _\n  \u00b7 exact Subset.refl _\n  \u00b7 exact Subset.trans (incl m) (Subset.trans (inter_subset_left _ _) h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\n\u22a2 \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nn : \u2115\n\u22a2 \u2200 (m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\n[PROOFSTEP]\nrefine' Nat.le_induction _ fun m _ h => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nn : \u2115\n\u22a2 closedBall (c n) (r n) \u2286 closedBall (c n) (r n)\n[PROOFSTEP]\nexact Subset.refl _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nn m : \u2115\nx\u271d : n \u2264 m\nh : closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\n\u22a2 closedBall (c (m + 1)) (r (m + 1)) \u2286 closedBall (c n) (r n)\n[PROOFSTEP]\nexact Subset.trans (incl m) (Subset.trans (inter_subset_left _ _) h)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\n\u22a2 (\u2200 (i : \u2115), y \u2208 f i) \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nhave yball : \u2200 n, y \u2208 closedBall (c n) (r n) := by\n  intro n\n  refine' isClosed_ball.mem_of_tendsto ylim _\n  refine' (Filter.eventually_ge_atTop n).mono fun m hm => _\n  exact I n m hm mem_closedBall_self\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\n\u22a2 \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n[PROOFSTEP]\nintro n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nn : \u2115\n\u22a2 y \u2208 closedBall (c n) (r n)\n[PROOFSTEP]\nrefine' isClosed_ball.mem_of_tendsto ylim _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nn : \u2115\n\u22a2 \u2200\u1da0 (x : \u2115) in atTop, c x \u2208 closedBall (c n) (r n)\n[PROOFSTEP]\nrefine' (Filter.eventually_ge_atTop n).mono fun m hm => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nn m : \u2115\nhm : n \u2264 m\n\u22a2 c m \u2208 closedBall (c n) (r n)\n[PROOFSTEP]\nexact I n m hm mem_closedBall_self\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 (\u2200 (i : \u2115), y \u2208 f i) \u2227 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 \u2200 (i : \u2115), y \u2208 f i\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nshow \u2200 n, y \u2208 f n\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 \u2200 (n : \u2115), y \u2208 f n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\nn : \u2115\n\u22a2 y \u2208 f n\n[PROOFSTEP]\nhave : closedBall (c (n + 1)) (r (n + 1)) \u2286 f n := Subset.trans (incl n) (inter_subset_right _ _)\n[GOAL]\ncase h.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis\u271d : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\nn : \u2115\nthis : closedBall (c (n + 1)) (r (n + 1)) \u2286 f n\n\u22a2 y \u2208 f n\n[PROOFSTEP]\nexact this (yball (n + 1))\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 y \u2208 closedBall x \u03b5\n[PROOFSTEP]\nshow edist y x \u2264 \u03b5\n[GOAL]\ncase h.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : CompleteSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nB : \u2115 \u2192 \u211d\u22650\u221e := fun n => 1 / 2 ^ n\nBpos : \u2200 (n : \u2115), 0 < B n\ncenter : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u03b1\nradius : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nHpos : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 0 < radius n x \u03b4\nHB : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 radius n x \u03b4 \u2264 B (n + 1)\nHball : \u2200 (n : \u2115) (x : \u03b1) (\u03b4 : \u211d\u22650\u221e), \u03b4 \u2260 0 \u2192 closedBall (center n x \u03b4) (radius n x \u03b4) \u2286 closedBall x \u03b4 \u2229 f n\nx : \u03b1\n\u03b5 : \u211d\u22650\u221e\n\u03b5pos : 0 < \u03b5\nF : \u2115 \u2192 \u03b1 \u00d7 \u211d\u22650\u221e := fun n => Nat.recOn n (x, min \u03b5 (B 0)) fun n p => (center n p.fst p.snd, radius n p.fst p.snd)\nc : \u2115 \u2192 \u03b1 := fun n => (F n).fst\nr : \u2115 \u2192 \u211d\u22650\u221e := fun n => (F n).snd\nrpos : \u2200 (n : \u2115), 0 < r n\nr0 : \u2200 (n : \u2115), r n \u2260 0\nrB : \u2200 (n : \u2115), r n \u2264 B n\nincl : \u2200 (n : \u2115), closedBall (c (n + 1)) (r (n + 1)) \u2286 closedBall (c n) (r n) \u2229 f n\ncdist : \u2200 (n : \u2115), edist (c n) (c (n + 1)) \u2264 B n\nthis : CauchySeq c\ny : \u03b1\nylim : Tendsto c atTop (\ud835\udcdd y)\nI : \u2200 (n m : \u2115), m \u2265 n \u2192 closedBall (c m) (r m) \u2286 closedBall (c n) (r n)\nyball : \u2200 (n : \u2115), y \u2208 closedBall (c n) (r n)\n\u22a2 edist y x \u2264 \u03b5\n[PROOFSTEP]\nexact le_trans (yball 0) (min_le_left _ _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\n\u22a2 BaireSpace \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase baire_property\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\n\u22a2 \u2200 (f : \u2115 \u2192 Set \u03b1), (\u2200 (n : \u2115), IsOpen (f n)) \u2192 (\u2200 (n : \u2115), Dense (f n)) \u2192 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nintro f ho hd\n[GOAL]\ncase baire_property\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\n\u22a2 Dense (\u22c2 (n : \u2115), f n)\n[PROOFSTEP]\napply dense_iff_inter_open.2\n[GOAL]\ncase baire_property\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\n\u22a2 \u2200 (U : Set \u03b1), IsOpen U \u2192 Set.Nonempty U \u2192 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nintro U U_open U_nonempty\n[GOAL]\ncase baire_property\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nrcases exists_positiveCompacts_subset U_open U_nonempty with \u27e8K\u2080, hK\u2080\u27e9\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave : \u2200 (n) (K : PositiveCompacts \u03b1), \u2203 K' : PositiveCompacts \u03b1, \u2191K' \u2286 f n \u2229 interior K :=\n  by\n  refine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _\n  rw [inter_comm]\n  exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\n\u22a2 \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2203 K', \u2191K' \u2286 f n \u2229 interior \u2191K\n[PROOFSTEP]\nrefine' fun n K => exists_positiveCompacts_subset ((ho n).inter isOpen_interior) _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nn : \u2115\nK : PositiveCompacts \u03b1\n\u22a2 Set.Nonempty (f n \u2229 interior \u2191K)\n[PROOFSTEP]\nrw [inter_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nn : \u2115\nK : PositiveCompacts \u03b1\n\u22a2 Set.Nonempty (interior \u2191K \u2229 f n)\n[PROOFSTEP]\nexact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nthis : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2203 K', \u2191K' \u2286 f n \u2229 interior \u2191K\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nchoose K_next hK_next using this\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nlet K : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave hK_decreasing : \u2200 n : \u2115, ((K (n + 1)).carrier) \u2286 (f n \u2229 (K n).carrier) := fun n =>\n  (hK_next n (K n)).trans <| inter_subset_inter_right _ interior_subset\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave hK_subset : (\u22c2 n, (K n).carrier : Set \u03b1) \u2286 U \u2229 \u22c2 n, f n :=\n  by\n  intro x hx\n  simp only [mem_iInter] at hx \n  simp only [mem_inter_iff, mem_inter] at hx \u22a2\n  refine' \u27e8hK\u2080 <| hx 0, _\u27e9\n  simp only [mem_iInter]\n  exact fun n =>\n    (hK_decreasing n (hx (n + 1))).1\n      /- Prove that `\u22c2 n : \u2115, K n` is not empty, as an intersection of a decreasing sequence\n          of nonempty compact subsets. -/\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\n\u22a2 \u22c2 (n : \u2115), (K n).carrier \u2286 U \u2229 \u22c2 (n : \u2115), f n\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nx : \u03b1\nhx : x \u2208 \u22c2 (n : \u2115), (K n).carrier\n\u22a2 x \u2208 U \u2229 \u22c2 (n : \u2115), f n\n[PROOFSTEP]\nsimp only [mem_iInter] at hx \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nx : \u03b1\nhx : \u2200 (i : \u2115), x \u2208 (Nat.rec K\u2080 K_next i).toCompacts.carrier\n\u22a2 x \u2208 U \u2229 \u22c2 (n : \u2115), f n\n[PROOFSTEP]\nsimp only [mem_inter_iff, mem_inter] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nx : \u03b1\nhx : \u2200 (i : \u2115), x \u2208 (Nat.rec K\u2080 K_next i).toCompacts.carrier\n\u22a2 x \u2208 U \u2227 x \u2208 \u22c2 (n : \u2115), f n\n[PROOFSTEP]\nrefine' \u27e8hK\u2080 <| hx 0, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nx : \u03b1\nhx : \u2200 (i : \u2115), x \u2208 (Nat.rec K\u2080 K_next i).toCompacts.carrier\n\u22a2 x \u2208 \u22c2 (n : \u2115), f n\n[PROOFSTEP]\nsimp only [mem_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nx : \u03b1\nhx : \u2200 (i : \u2115), x \u2208 (Nat.rec K\u2080 K_next i).toCompacts.carrier\n\u22a2 \u2200 (i : \u2115), x \u2208 f i\n[PROOFSTEP]\nexact fun n =>\n  (hK_decreasing n (hx (n + 1))).1\n    /- Prove that `\u22c2 n : \u2115, K n` is not empty, as an intersection of a decreasing sequence\n        of nonempty compact subsets. -/\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nhK_subset : \u22c2 (n : \u2115), (K n).carrier \u2286 U \u2229 \u22c2 (n : \u2115), f n\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nhave hK_nonempty : (\u22c2 n, (K n).carrier : Set \u03b1).Nonempty :=\n  IsCompact.nonempty_iInter_of_sequence_nonempty_compact_closed _\n    (fun n => (hK_decreasing n).trans (inter_subset_right _ _)) (fun n => (K n).nonempty) (K 0).isCompact fun n =>\n    (K n).isCompact.isClosed\n[GOAL]\ncase baire_property.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : T2Space \u03b1\ninst\u271d : LocallyCompactSpace \u03b1\nf : \u2115 \u2192 Set \u03b1\nho : \u2200 (n : \u2115), IsOpen (f n)\nhd : \u2200 (n : \u2115), Dense (f n)\nU : Set \u03b1\nU_open : IsOpen U\nU_nonempty : Set.Nonempty U\nK\u2080 : PositiveCompacts \u03b1\nhK\u2080 : \u2191K\u2080 \u2286 U\nK_next : \u2115 \u2192 PositiveCompacts \u03b1 \u2192 PositiveCompacts \u03b1\nhK_next : \u2200 (n : \u2115) (K : PositiveCompacts \u03b1), \u2191(K_next n K) \u2286 f n \u2229 interior \u2191K\nK : \u2115 \u2192 PositiveCompacts \u03b1 := fun n => Nat.recOn n K\u2080 K_next\nhK_decreasing : \u2200 (n : \u2115), (K (n + 1)).carrier \u2286 f n \u2229 (K n).carrier\nhK_subset : \u22c2 (n : \u2115), (K n).carrier \u2286 U \u2229 \u22c2 (n : \u2115), f n\nhK_nonempty : Set.Nonempty (\u22c2 (n : \u2115), (K n).carrier)\n\u22a2 Set.Nonempty (U \u2229 \u22c2 (n : \u2115), f n)\n[PROOFSTEP]\nexact hK_nonempty.mono hK_subset\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\ncases' S.eq_empty_or_nonempty with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : S = \u2205\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\nrcases hS.exists_eq_range h with \u27e8f, hf\u27e9\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\nf : \u2115 \u2192 Set \u03b1\nhf : S = range f\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\nhave F : \u2200 n, f n \u2208 S := fun n => by rw [hf]; exact mem_range_self _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\nf : \u2115 \u2192 Set \u03b1\nhf : S = range f\nn : \u2115\n\u22a2 f n \u2208 S\n[PROOFSTEP]\nrw [hf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\nf : \u2115 \u2192 Set \u03b1\nhf : S = range f\nn : \u2115\n\u22a2 f n \u2208 range f\n[PROOFSTEP]\nexact mem_range_self _\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\nf : \u2115 \u2192 Set \u03b1\nhf : S = range f\nF : \u2200 (n : \u2115), f n \u2208 S\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\nrw [hf, sInter_range]\n[GOAL]\ncase inr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set (Set \u03b1)\nho : \u2200 (s : Set \u03b1), s \u2208 S \u2192 IsOpen s\nhS : Set.Countable S\nhd : \u2200 (s : Set \u03b1), s \u2208 S \u2192 Dense s\nh : Set.Nonempty S\nf : \u2115 \u2192 Set \u03b1\nhf : S = range f\nF : \u2200 (n : \u2115), f n \u2208 S\n\u22a2 Dense (\u22c2 (x : \u2115), f x)\n[PROOFSTEP]\nexact dense_iInter_of_open_nat (fun n => ho _ (F n)) fun n => hd _ (F n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), s \u2208 S \u2192 IsOpen (f s)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2), s \u2208 S \u2192 Dense (f s)\n\u22a2 Dense (\u22c2 (s : \u03b2) (_ : s \u2208 S), f s)\n[PROOFSTEP]\nrw [\u2190 sInter_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), s \u2208 S \u2192 IsOpen (f s)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2), s \u2208 S \u2192 Dense (f s)\n\u22a2 Dense (\u22c2\u2080 ((fun s => f s) '' S))\n[PROOFSTEP]\napply dense_sInter_of_open\n[GOAL]\ncase ho\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), s \u2208 S \u2192 IsOpen (f s)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2), s \u2208 S \u2192 Dense (f s)\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 (fun s => f s) '' S \u2192 IsOpen s\n[PROOFSTEP]\nrwa [ball_image_iff]\n[GOAL]\ncase hS\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), s \u2208 S \u2192 IsOpen (f s)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2), s \u2208 S \u2192 Dense (f s)\n\u22a2 Set.Countable ((fun s => f s) '' S)\n[PROOFSTEP]\nexact hS.image _\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), s \u2208 S \u2192 IsOpen (f s)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2), s \u2208 S \u2192 Dense (f s)\n\u22a2 \u2200 (s : Set \u03b1), s \u2208 (fun s => f s) '' S \u2192 Dense s\n[PROOFSTEP]\nrwa [ball_image_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsOpen (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 Dense (\u22c2 (s : \u03b2), f s)\n[PROOFSTEP]\nrw [\u2190 sInter_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsOpen (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 Dense (\u22c2\u2080 range fun s => f s)\n[PROOFSTEP]\napply dense_sInter_of_open\n[GOAL]\ncase ho\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsOpen (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 \u2200 (s : Set \u03b1), (s \u2208 range fun s => f s) \u2192 IsOpen s\n[PROOFSTEP]\nrwa [forall_range_iff]\n[GOAL]\ncase hS\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsOpen (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 Set.Countable (range fun s => f s)\n[PROOFSTEP]\nexact countable_range _\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsOpen (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 \u2200 (s : Set \u03b1), (s \u2208 range fun s => f s) \u2192 Dense s\n[PROOFSTEP]\nrwa [forall_range_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 residual \u03b1 \u2194 \u2203 t x, IsG\u03b4 t \u2227 Dense t\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\n\u22a2 s \u2208 residual \u03b1 \u2192 \u2203 t x, IsG\u03b4 t \u2227 Dense t\n[PROOFSTEP]\nrw [mem_residual_iff]\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 S, (\u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t) \u2227 (\u2200 (t : Set \u03b1), t \u2208 S \u2192 Dense t) \u2227 Set.Countable S \u2227 \u22c2\u2080 S \u2286 s) \u2192\n    \u2203 t x, IsG\u03b4 t \u2227 Dense t\n[PROOFSTEP]\nrintro \u27e8S, hSo, hSd, Sct, Ss\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\nS : Set (Set \u03b1)\nhSo : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nhSd : \u2200 (t : Set \u03b1), t \u2208 S \u2192 Dense t\nSct : Set.Countable S\nSs : \u22c2\u2080 S \u2286 s\n\u22a2 \u2203 t x, IsG\u03b4 t \u2227 Dense t\n[PROOFSTEP]\nrefine' \u27e8_, Ss, \u27e8_, fun t ht => hSo _ ht, Sct, rfl\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\nS : Set (Set \u03b1)\nhSo : \u2200 (t : Set \u03b1), t \u2208 S \u2192 IsOpen t\nhSd : \u2200 (t : Set \u03b1), t \u2208 S \u2192 Dense t\nSct : Set.Countable S\nSs : \u22c2\u2080 S \u2286 s\n\u22a2 Dense (\u22c2\u2080 S)\n[PROOFSTEP]\nexact dense_sInter_of_open hSo Sct hSd\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns : Set \u03b1\n\u22a2 (\u2203 t x, IsG\u03b4 t \u2227 Dense t) \u2192 s \u2208 residual \u03b1\n[PROOFSTEP]\nrintro \u27e8t, ts, ho, hd\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns t : Set \u03b1\nts : t \u2286 s\nho : IsG\u03b4 t\nhd : Dense t\n\u22a2 s \u2208 residual \u03b1\n[PROOFSTEP]\nexact mem_of_superset (residual_of_dense_G\u03b4 ho hd) ts\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1) in residual \u03b1, p x) \u2194 \u2203 t, IsG\u03b4 t \u2227 Dense t \u2227 \u2200 (x : \u03b1), x \u2208 t \u2192 p x\n[PROOFSTEP]\nconvert @mem_residual _ _ _ p\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\np : \u03b1 \u2192 Prop\nx\u271d : Set \u03b1\n\u22a2 (IsG\u03b4 x\u271d \u2227 Dense x\u271d \u2227 \u2200 (x : \u03b1), x \u2208 x\u271d \u2192 p x) \u2194 \u2203 x, IsG\u03b4 x\u271d \u2227 Dense x\u271d\n[PROOFSTEP]\nsimp_rw [exists_prop, @and_comm ((_ : Set \u03b1) \u2286 p), and_assoc]\n[GOAL]\ncase h.e'_2.h.e'_2.h.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\np : \u03b1 \u2192 Prop\nx\u271d : Set \u03b1\n\u22a2 (IsG\u03b4 x\u271d \u2227 Dense x\u271d \u2227 \u2200 (x : \u03b1), x \u2208 x\u271d \u2192 p x) \u2194 IsG\u03b4 x\u271d \u2227 Dense x\u271d \u2227 x\u271d \u2286 p\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsG\u03b4 (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 Dense (\u22c2 (s : \u03b2), f s)\n[PROOFSTEP]\nrw [\u2190 sInter_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nho : \u2200 (s : \u03b2), IsG\u03b4 (f s)\nhd : \u2200 (s : \u03b2), Dense (f s)\n\u22a2 Dense (\u22c2\u2080 range fun s => f s)\n[PROOFSTEP]\nexact dense_sInter_of_G\u03b4 (forall_range_iff.2 \u2039_\u203a) (countable_range _) (forall_range_iff.2 \u2039_\u203a)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : (x : \u03b2) \u2192 x \u2208 S \u2192 Set \u03b1\nho : \u2200 (s : \u03b2) (H : s \u2208 S), IsG\u03b4 (f s H)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2) (H : s \u2208 S), Dense (f s H)\n\u22a2 Dense (\u22c2 (s : \u03b2) (h : s \u2208 S), f s h)\n[PROOFSTEP]\nrw [biInter_eq_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : (x : \u03b2) \u2192 x \u2208 S \u2192 Set \u03b1\nho : \u2200 (s : \u03b2) (H : s \u2208 S), IsG\u03b4 (f s H)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2) (H : s \u2208 S), Dense (f s H)\n\u22a2 Dense (\u22c2 (x : \u2191S), f \u2191x (_ : \u2191x \u2208 S))\n[PROOFSTEP]\nhaveI := hS.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nS : Set \u03b2\nf : (x : \u03b2) \u2192 x \u2208 S \u2192 Set \u03b1\nho : \u2200 (s : \u03b2) (H : s \u2208 S), IsG\u03b4 (f s H)\nhS : Set.Countable S\nhd : \u2200 (s : \u03b2) (H : s \u2208 S), Dense (f s H)\nthis : Encodable \u2191S\n\u22a2 Dense (\u22c2 (x : \u2191S), f \u2191x (_ : \u2191x \u2208 S))\n[PROOFSTEP]\nexact dense_iInter_of_G\u03b4 (fun s => ho s s.2) fun s => hd s s.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns t : Set \u03b1\nhs : IsG\u03b4 s\nht : IsG\u03b4 t\nhsc : Dense s\nhtc : Dense t\n\u22a2 Dense (s \u2229 t)\n[PROOFSTEP]\nrw [inter_eq_iInter]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns t : Set \u03b1\nhs : IsG\u03b4 s\nht : IsG\u03b4 t\nhsc : Dense s\nhtc : Dense t\n\u22a2 Dense (\u22c2 (b : Bool), bif b then s else t)\n[PROOFSTEP]\napply dense_iInter_of_G\u03b4\n[GOAL]\ncase ho\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns t : Set \u03b1\nhs : IsG\u03b4 s\nht : IsG\u03b4 t\nhsc : Dense s\nhtc : Dense t\n\u22a2 \u2200 (s_1 : Bool), IsG\u03b4 (bif s_1 then s else t)\n[PROOFSTEP]\nsimp [Bool.forall_bool, *]\n[GOAL]\ncase hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\ns t : Set \u03b1\nhs : IsG\u03b4 s\nht : IsG\u03b4 t\nhsc : Dense s\nhtc : Dense t\n\u22a2 \u2200 (s_1 : Bool), Dense (bif s_1 then s else t)\n[PROOFSTEP]\nsimp [Bool.forall_bool, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\n\u22a2 Dense (\u22c3 (i : \u03b9), interior (f i))\n[PROOFSTEP]\nlet g i := (frontier (f i))\u1d9c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\n\u22a2 Dense (\u22c3 (i : \u03b9), interior (f i))\n[PROOFSTEP]\nhave hgo : \u2200 i, IsOpen (g i) := fun i => isClosed_frontier.isOpen_compl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\n\u22a2 Dense (\u22c3 (i : \u03b9), interior (f i))\n[PROOFSTEP]\nhave hgd : Dense (\u22c2 i, g i) := by\n  refine' dense_iInter_of_open hgo fun i x => _\n  rw [closure_compl, interior_frontier (hc _)]\n  exact id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\n\u22a2 Dense (\u22c2 (i : \u03b9), g i)\n[PROOFSTEP]\nrefine' dense_iInter_of_open hgo fun i x => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 closure (g i)\n[PROOFSTEP]\nrw [closure_compl, interior_frontier (hc _)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\ni : \u03b9\nx : \u03b1\n\u22a2 x \u2208 \u2205\u1d9c\n[PROOFSTEP]\nexact id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\nhgd : Dense (\u22c2 (i : \u03b9), g i)\n\u22a2 Dense (\u22c3 (i : \u03b9), interior (f i))\n[PROOFSTEP]\nrefine' (hd.inter_of_G\u03b4 hs (isG\u03b4_iInter_of_open fun i => (hgo i)) hgd).mono _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\nhgd : Dense (\u22c2 (i : \u03b9), g i)\n\u22a2 s \u2229 \u22c2 (i : \u03b9), g i \u2286 \u22c3 (i : \u03b9), interior (f i)\n[PROOFSTEP]\nrintro x \u27e8hxs, hxg\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\nhgd : Dense (\u22c2 (i : \u03b9), g i)\nx : \u03b1\nhxs : x \u2208 s\nhxg : x \u2208 \u22c2 (i : \u03b9), g i\n\u22a2 x \u2208 \u22c3 (i : \u03b9), interior (f i)\n[PROOFSTEP]\nrw [mem_iInter] at hxg \n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\nhgd : Dense (\u22c2 (i : \u03b9), g i)\nx : \u03b1\nhxs : x \u2208 s\nhxg : \u2200 (i : \u03b9), x \u2208 g i\n\u22a2 x \u2208 \u22c3 (i : \u03b9), interior (f i)\n[PROOFSTEP]\nrcases mem_iUnion.1 (hU hxs) with \u27e8i, hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : BaireSpace \u03b1\ninst\u271d : Encodable \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9), f i\ng : \u03b9 \u2192 Set \u03b1 := fun i => (frontier (f i))\u1d9c\nhgo : \u2200 (i : \u03b9), IsOpen (g i)\nhgd : Dense (\u22c2 (i : \u03b9), g i)\nx : \u03b1\nhxs : x \u2208 s\nhxg : \u2200 (i : \u03b9), x \u2208 g i\ni : \u03b9\nhi : x \u2208 f i\n\u22a2 x \u2208 \u22c3 (i : \u03b9), interior (f i)\n[PROOFSTEP]\nexact mem_iUnion.2 \u27e8i, self_diff_frontier (f i) \u25b8 \u27e8hi, hxg _\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nt : Set \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nht : Set.Countable t\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), i \u2208 t \u2192 IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\n\u22a2 Dense (\u22c3 (i : \u03b9) (_ : i \u2208 t), interior (f i))\n[PROOFSTEP]\nhaveI := ht.toEncodable\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nt : Set \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nht : Set.Countable t\nf : \u03b9 \u2192 Set \u03b1\nhc : \u2200 (i : \u03b9), i \u2208 t \u2192 IsClosed (f i)\nhU : s \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 t), f i\nthis : Encodable \u2191t\n\u22a2 Dense (\u22c3 (i : \u03b9) (_ : i \u2208 t), interior (f i))\n[PROOFSTEP]\nsimp only [biUnion_eq_iUnion, SetCoe.forall'] at *\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nt : Set \u03b9\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nht : Set.Countable t\nf : \u03b9 \u2192 Set \u03b1\nthis : Encodable \u2191t\nhc : \u2200 (x : \u2191t), IsClosed (f \u2191x)\nhU : s \u2286 \u22c3 (x : \u2191t), f \u2191x\n\u22a2 Dense (\u22c3 (x : \u2191t), interior (f \u2191x))\n[PROOFSTEP]\nexact hs.dense_iUnion_interior_of_closed hd hc hU\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : BaireSpace \u03b1\nT : Set (Set \u03b1)\ns : Set \u03b1\nhs : IsG\u03b4 s\nhd : Dense s\nhc : Set.Countable T\nhc' : \u2200 (t : Set \u03b1), t \u2208 T \u2192 IsClosed t\nhU : s \u2286 \u22c3\u2080 T\n\u22a2 s \u2286 \u22c3 (i : Set \u03b1) (_ : i \u2208 T), i\n[PROOFSTEP]\nrwa [\u2190 sUnion_eq_biUnion]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : BaireSpace \u03b1\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : Encodable \u03b2\nf : \u03b2 \u2192 Set \u03b1\nhc : \u2200 (s : \u03b2), IsClosed (f s)\nhU : \u22c3 (s : \u03b2), f s = univ\n\u22a2 \u2203 s, Set.Nonempty (interior (f s))\n[PROOFSTEP]\nsimpa using (dense_iUnion_interior_of_closed hc hU).nonempty\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Baire", "llama_tokens": 53012, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.831143031127974, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.5234327645748628}}
{"text": "[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nhs : IsSubgroup s\nx y : G\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 x / y \u2208 s\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\n\u22a2 IsAddSubgroup s \u2192 IsSubgroup s\n[PROOFSTEP]\nrintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9\n[GOAL]\ncase mk.mk\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\nh\u2083 : \u2200 {a : Additive G}, a \u2208 s \u2192 -a \u2208 s\nh\u2081 : 0 \u2208 s\nh\u2082 : \u2200 {a b : Additive G}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s\n\u22a2 IsSubgroup s\n[PROOFSTEP]\nexact @IsSubgroup.mk G _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set A\n\u22a2 IsSubgroup s \u2192 IsAddSubgroup s\n[PROOFSTEP]\nrintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9\n[GOAL]\ncase mk.mk\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set A\nh\u2083 : \u2200 {a : Multiplicative A}, a \u2208 s \u2192 a\u207b\u00b9 \u2208 s\nh\u2081 : 1 \u2208 s\nh\u2082 : \u2200 {a b : Multiplicative A}, a \u2208 s \u2192 b \u2208 s \u2192 a * b \u2208 s\n\u22a2 IsAddSubgroup s\n[PROOFSTEP]\nexact @IsAddSubgroup.mk A _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\na : G\nha : a \u2208 s\n\u22a2 a\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nhave : 1 * a\u207b\u00b9 \u2208 s := div_mem one_mem ha\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\na : G\nha : a \u2208 s\nthis : 1 * a\u207b\u00b9 \u2208 s\n\u22a2 a\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_4\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\na : G\nha : a \u2208 s\nthis : 1 * a\u207b\u00b9 \u2208 s\n\u22a2 a\u207b\u00b9 = 1 * a\u207b\u00b9\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\ninv_mem : \u2200 (a : G), a \u2208 s \u2192 a\u207b\u00b9 \u2208 s\na b : G\nha : a \u2208 s\nhb : b \u2208 s\n\u22a2 a * b \u2208 s\n[PROOFSTEP]\nhave : a * b\u207b\u00b9\u207b\u00b9 \u2208 s := div_mem ha (inv_mem b hb)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\ninv_mem : \u2200 (a : G), a \u2208 s \u2192 a\u207b\u00b9 \u2208 s\na b : G\nha : a \u2208 s\nhb : b \u2208 s\nthis : a * b\u207b\u00b9\u207b\u00b9 \u2208 s\n\u22a2 a * b \u2208 s\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4.h.e'_6\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set G\none_mem : 1 \u2208 s\ndiv_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s\ninv_mem : \u2200 (a : G), a \u2208 s \u2192 a\u207b\u00b9 \u2208 s\na b : G\nha : a \u2208 s\nhb : b \u2208 s\nthis : a * b\u207b\u00b9\u207b\u00b9 \u2208 s\n\u22a2 b = b\u207b\u00b9\u207b\u00b9\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns : Set A\nzero_mem : 0 \u2208 s\nsub_mem : \u2200 {a b : A}, a \u2208 s \u2192 b \u2208 s \u2192 a - b \u2208 s\nx y : A\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 x + -y \u2208 s\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg] using sub_mem hx hy\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\nh : a\u207b\u00b9 \u2208 s\n\u22a2 a \u2208 s\n[PROOFSTEP]\nsimpa using hs.inv_mem h\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\nh : a \u2208 s\nhba : b * a \u2208 s\n\u22a2 b \u2208 s\n[PROOFSTEP]\nsimpa using hs.mul_mem hba (hs.inv_mem h)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\nh : a \u2208 s\nhab : a * b \u2208 s\n\u22a2 b \u2208 s\n[PROOFSTEP]\nsimpa using hs.mul_mem (hs.inv_mem h) hab\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : CommGroup G\ns : Set G\nhs : IsSubgroup s\nn : G\nhn : n \u2208 s\ng : G\n\u22a2 g * n * g\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrwa [mul_right_comm, mul_right_inv, one_mul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\n\u22a2 IsNormalAddSubgroup s \u2192 IsNormalSubgroup s\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mk\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nh\u2081 : IsAddSubgroup s\nh\u2082 : \u2200 (n : Additive G), n \u2208 s \u2192 \u2200 (g : Additive G), g + n + -g \u2208 s\n\u22a2 IsNormalSubgroup s\n[PROOFSTEP]\nexact @IsNormalSubgroup.mk G _ _ (Additive.isAddSubgroup_iff.1 h\u2081) @h\u2082\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : AddGroup A\ns : Set A\n\u22a2 IsNormalSubgroup s \u2192 IsNormalAddSubgroup s\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase mk\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : AddGroup A\ns : Set A\nh\u2081 : IsSubgroup s\nh\u2082 : \u2200 (n : Multiplicative A), n \u2208 s \u2192 \u2200 (g : Multiplicative A), g * n * g\u207b\u00b9 \u2208 s\n\u22a2 IsNormalAddSubgroup s\n[PROOFSTEP]\nexact @IsNormalAddSubgroup.mk A _ _ (Multiplicative.isSubgroup_iff.1 h\u2081) @h\u2082\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d : Group G\ns : Set G\nhs : IsNormalSubgroup s\na b : G\nhab : a * b \u2208 s\n\u22a2 b * a \u2208 s\n[PROOFSTEP]\nhave h : a\u207b\u00b9 * (a * b) * a\u207b\u00b9\u207b\u00b9 \u2208 s := hs.normal (a * b) hab a\u207b\u00b9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d : Group G\ns : Set G\nhs : IsNormalSubgroup s\na b : G\nhab : a * b \u2208 s\nh : a\u207b\u00b9 * (a * b) * a\u207b\u00b9\u207b\u00b9 \u2208 s\n\u22a2 b * a \u2208 s\n[PROOFSTEP]\nsimp at h \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d : Group G\ns : Set G\nhs : IsNormalSubgroup s\na b : G\nhab : a * b \u2208 s\nh : b * a \u2208 s\n\u22a2 b * a \u2208 s\n[PROOFSTEP]\nexact h\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 IsNormalSubgroup (trivial G)\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 1 \u2208 trivial G\n[PROOFSTEP]\nsimp (config := { contextual := true }) [trivial]\n[GOAL]\ncase refine'_2\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 {a b : G}, a \u2208 trivial G \u2192 b \u2208 trivial G \u2192 a * b \u2208 trivial G\n[PROOFSTEP]\nsimp (config := { contextual := true }) [trivial]\n[GOAL]\ncase refine'_3\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 {a : G}, a \u2208 trivial G \u2192 a\u207b\u00b9 \u2208 trivial G\n[PROOFSTEP]\nsimp (config := { contextual := true }) [trivial]\n[GOAL]\ncase refine'_4\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 (n : G), n \u2208 trivial G \u2192 \u2200 (g : G), g * n * g\u207b\u00b9 \u2208 trivial G\n[PROOFSTEP]\nsimp (config := { contextual := true }) [trivial]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\n\u22a2 s = trivial G \u2194 \u2200 (x : G), x \u2208 s \u2192 x = 1\n[PROOFSTEP]\nsimp only [Set.ext_iff, IsSubgroup.mem_trivial]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\n\u22a2 (\u2200 (x : G), x \u2208 s \u2194 x = 1) \u2194 \u2200 (x : G), x \u2208 s \u2192 x = 1\n[PROOFSTEP]\nexact \u27e8fun h x => (h x).1, fun h x => \u27e8h x, fun hx => hx.symm \u25b8 hs.toIsSubmonoid.one_mem\u27e9\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 IsNormalSubgroup univ\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 1 \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 {a b : G}, a \u2208 univ \u2192 b \u2208 univ \u2192 a * b \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 {a : G}, a \u2208 univ \u2192 a\u207b\u00b9 \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_4\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2200 (n : G), n \u2208 univ \u2192 \u2200 (g : G), g * n * g\u207b\u00b9 \u2208 univ\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 1 \u2208 center G\n[PROOFSTEP]\nsimp [center]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\na\u271d b\u271d : G\nha : a\u271d \u2208 center G\nhb : b\u271d \u2208 center G\ng : G\n\u22a2 g * (a\u271d * b\u271d) = a\u271d * b\u271d * g\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, \u2190 mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\na : G\nha : a \u2208 center G\ng : G\n\u22a2 g * a\u207b\u00b9 = a\u207b\u00b9 * (g * a) * a\u207b\u00b9\n[PROOFSTEP]\nsimp [ha g]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\na : G\nha : a \u2208 center G\ng : G\n\u22a2 a\u207b\u00b9 * (g * a) * a\u207b\u00b9 = a\u207b\u00b9 * g\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\na : G\nha : a \u2208 center G\ng : G\n\u22a2 a\u207b\u00b9 * g * (a * a\u207b\u00b9) = a\u207b\u00b9 * g\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\nn : G\nha : n \u2208 center G\ng h : G\n\u22a2 h * (g * n * g\u207b\u00b9) = h * n\n[PROOFSTEP]\nsimp [ha g, mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\nn : G\nha : n \u2208 center G\ng h : G\n\u22a2 h * n = g * g\u207b\u00b9 * n * h\n[PROOFSTEP]\nrw [ha h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\nn : G\nha : n \u2208 center G\ng h : G\n\u22a2 n * h = g * g\u207b\u00b9 * n * h\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\nn : G\nha : n \u2208 center G\ng h : G\n\u22a2 g * g\u207b\u00b9 * n * h = g * n * g\u207b\u00b9 * h\n[PROOFSTEP]\nrw [mul_assoc g, ha g\u207b\u00b9, \u2190 mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\n\u22a2 1 \u2208 normalizer s\n[PROOFSTEP]\nsimp [normalizer]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d : Group G\ns : Set G\na b : G\nha : \u2200 (n : G), n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s\nhb : \u2200 (n : G), n \u2208 s \u2194 b * n * b\u207b\u00b9 \u2208 s\nn : G\n\u22a2 n \u2208 s \u2194 a * b * n * (a * b)\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrw [mul_inv_rev, \u2190 mul_assoc, mul_assoc a, mul_assoc a, \u2190 ha, \u2190 hb]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\na : G\nha : \u2200 (n : G), n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s\nn : G\n\u22a2 n \u2208 s \u2194 a\u207b\u00b9 * n * a\u207b\u00b9\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrw [ha (a\u207b\u00b9 * n * a\u207b\u00b9\u207b\u00b9)]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\na : G\nha : \u2200 (n : G), n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s\nn : G\n\u22a2 n \u2208 s \u2194 a * (a\u207b\u00b9 * n * a\u207b\u00b9\u207b\u00b9) * a\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nsimp [mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\ng : G\nhg : g \u2208 s\nn : G\n\u22a2 n \u2208 s \u2194 g * n * g\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrw [IsSubgroup.mul_mem_cancel_right hs ((IsSubgroup.inv_mem_iff hs).2 hg), IsSubgroup.mul_mem_cancel_left hs hg]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f (a * b\u207b\u00b9) = 1\n\u22a2 f a = f b\n[PROOFSTEP]\nrw [hf.map_mul, hf.map_inv] at h \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a * (f b)\u207b\u00b9 = 1\n\u22a2 f a = f b\n[PROOFSTEP]\nrw [\u2190 inv_inv (f b), eq_inv_of_mul_eq_one_left h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f (a\u207b\u00b9 * b) = 1\n\u22a2 f a = f b\n[PROOFSTEP]\nrw [hf.map_mul, hf.map_inv] at h \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : (f a)\u207b\u00b9 * f b = 1\n\u22a2 f a = f b\n[PROOFSTEP]\napply inv_injective\n[GOAL]\ncase a\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : (f a)\u207b\u00b9 * f b = 1\n\u22a2 (f a)\u207b\u00b9 = (f b)\u207b\u00b9\n[PROOFSTEP]\nrw [eq_inv_of_mul_eq_one_left h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 f (a * b\u207b\u00b9) = 1\n[PROOFSTEP]\nhave : f a * (f b)\u207b\u00b9 = 1 := by rw [h, mul_right_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 f a * (f b)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [h, mul_right_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\nthis : f a * (f b)\u207b\u00b9 = 1\n\u22a2 f (a * b\u207b\u00b9) = 1\n[PROOFSTEP]\nrwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 f (a\u207b\u00b9 * b) = 1\n[PROOFSTEP]\nhave : (f a)\u207b\u00b9 * f b = 1 := by rw [h, mul_left_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\n\u22a2 (f a)\u207b\u00b9 * f b = 1\n[PROOFSTEP]\nrw [h, mul_left_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\nh : f a = f b\nthis : (f a)\u207b\u00b9 * f b = 1\n\u22a2 f (a\u207b\u00b9 * b) = 1\n[PROOFSTEP]\nrwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\n\u22a2 f a = f b \u2194 a * b\u207b\u00b9 \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\n\u22a2 f a = f b \u2194 f (a * b\u207b\u00b9) = 1\n[PROOFSTEP]\nexact one_iff_ker_inv hf _ _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\n\u22a2 f a = f b \u2194 a\u207b\u00b9 * b \u2208 ker f\n[PROOFSTEP]\nrw [mem_ker]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na b : G\n\u22a2 f a = f b \u2194 f (a\u207b\u00b9 * b) = 1\n[PROOFSTEP]\nexact one_iff_ker_inv' hf _ _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na\u2081 a\u2082 : H\nx\u271d\u00b9 : a\u2081 \u2208 f '' s\nx\u271d : a\u2082 \u2208 f '' s\nb\u2081 : G\nhb\u2081 : b\u2081 \u2208 s\neq\u2081 : f b\u2081 = a\u2081\nb\u2082 : G\nhb\u2082 : b\u2082 \u2208 s\neq\u2082 : f b\u2082 = a\u2082\n\u22a2 f (b\u2081 * b\u2082) = a\u2081 * a\u2082\n[PROOFSTEP]\nsimp [eq\u2081, eq\u2082, hf.map_mul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na : H\nx\u271d : a \u2208 f '' s\nb : G\nhb : b \u2208 s\nEq : f b = a\n\u22a2 f b\u207b\u00b9 = a\u207b\u00b9\n[PROOFSTEP]\nrw [hf.map_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b\u271d c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nhs : IsSubgroup s\na : H\nx\u271d : a \u2208 f '' s\nb : G\nhb : b \u2208 s\nEq : f b = a\n\u22a2 (f b)\u207b\u00b9 = a\u207b\u00b9\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsSubgroup s\n\u22a2 IsSubgroup (f \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' { .. }\n[GOAL]\ncase refine'_1\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsSubgroup s\n\u22a2 1 \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul, hf.map_one, hf.map_inv,\n  InvMemClass.inv_mem]\n[GOAL]\ncase refine'_2\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsSubgroup s\n\u22a2 \u2200 {a b : G}, a \u2208 f \u207b\u00b9' s \u2192 b \u2208 f \u207b\u00b9' s \u2192 a * b \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul, hf.map_one, hf.map_inv,\n  InvMemClass.inv_mem]\n[GOAL]\ncase refine'_3\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsSubgroup s\n\u22a2 \u2200 {a : G}, a \u2208 f \u207b\u00b9' s \u2192 a\u207b\u00b9 \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul, hf.map_one, hf.map_inv,\n  InvMemClass.inv_mem]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsNormalSubgroup s\n\u22a2 1 \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp [hf.map_one, hs.toIsSubgroup.one_mem]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsNormalSubgroup s\n\u22a2 \u2200 {a b : G}, a \u2208 f \u207b\u00b9' s \u2192 b \u2208 f \u207b\u00b9' s \u2192 a * b \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hf.map_mul, hs.toIsSubgroup.mul_mem]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsNormalSubgroup s\n\u22a2 \u2200 {a : G}, a \u2208 f \u207b\u00b9' s \u2192 a\u207b\u00b9 \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hf.map_inv, hs.toIsSubgroup.inv_mem]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set H\nhs : IsNormalSubgroup s\n\u22a2 \u2200 (n : G), n \u2208 f \u207b\u00b9' s \u2192 \u2200 (g : G), g * n * g\u207b\u00b9 \u2208 f \u207b\u00b9' s\n[PROOFSTEP]\nsimp (config := { contextual := true }) [hs.normal, hf.map_mul, hf.map_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : ker f = IsSubgroup.trivial G\n\u22a2 Injective f\n[PROOFSTEP]\nintro a\u2081 a\u2082 hfa\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : ker f = IsSubgroup.trivial G\na\u2081 a\u2082 : G\nhfa : f a\u2081 = f a\u2082\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nsimp [ext_iff, ker, IsSubgroup.trivial] at h \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na\u2081 a\u2082 : G\nhfa : f a\u2081 = f a\u2082\nh : \u2200 (x : G), f x = 1 \u2194 x = 1\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nhave ha : a\u2081 * a\u2082\u207b\u00b9 = 1 := by rw [\u2190 h]; exact hf.inv_ker_one hfa\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na\u2081 a\u2082 : G\nhfa : f a\u2081 = f a\u2082\nh : \u2200 (x : G), f x = 1 \u2194 x = 1\n\u22a2 a\u2081 * a\u2082\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na\u2081 a\u2082 : G\nhfa : f a\u2081 = f a\u2082\nh : \u2200 (x : G), f x = 1 \u2194 x = 1\n\u22a2 f (a\u2081 * a\u2082\u207b\u00b9) = 1\n[PROOFSTEP]\nexact hf.inv_ker_one hfa\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081\u271d a\u2082\u271d b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\na\u2081 a\u2082 : G\nhfa : f a\u2081 = f a\u2082\nh : \u2200 (x : G), f x = 1 \u2194 x = 1\nha : a\u2081 * a\u2082\u207b\u00b9 = 1\n\u22a2 a\u2081 = a\u2082\n[PROOFSTEP]\nrw [eq_inv_of_mul_eq_one_left ha, inv_inv a\u2082]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : Injective f\nx : G\nhx : x \u2208 ker f\n\u22a2 x \u2208 IsSubgroup.trivial G\n[PROOFSTEP]\nsuffices f x = f 1 by simpa using h this\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : Injective f\nx : G\nhx : x \u2208 ker f\nthis : f x = f 1\n\u22a2 x \u2208 IsSubgroup.trivial G\n[PROOFSTEP]\nsimpa using h this\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : Injective f\nx : G\nhx : x \u2208 ker f\n\u22a2 f x = f 1\n[PROOFSTEP]\nsimp [hf.map_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : Injective f\nx : G\nhx : x \u2208 ker f\n\u22a2 f x = 1\n[PROOFSTEP]\nrwa [mem_ker] at hx \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : Injective f\nx : G\n\u22a2 x \u2208 IsSubgroup.trivial G \u2192 x \u2208 ker f\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mem_ker, hf.map_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\n\u22a2 ker f = IsSubgroup.trivial G \u2194 \u2200 (x : G), f x = 1 \u2192 x = 1\n[PROOFSTEP]\nrw [Set.ext_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\n\u22a2 (\u2200 (x : G), x \u2208 ker f \u2194 x \u2208 IsSubgroup.trivial G) \u2194 \u2200 (x : G), f x = 1 \u2192 x = 1\n[PROOFSTEP]\nsimp [ker]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\n\u22a2 (\u2200 (x : G), f x = 1 \u2194 x = 1) \u2194 \u2200 (x : G), f x = 1 \u2192 x = 1\n[PROOFSTEP]\nexact \u27e8fun h x hx => (h x).1 hx, fun h x => \u27e8h x, fun hx => by rw [hx, hf.map_one]\u27e9\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\nh : \u2200 (x : G), f x = 1 \u2192 x = 1\nx : G\nhx : x = 1\n\u22a2 f x = 1\n[PROOFSTEP]\nrw [hx, hf.map_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s t : Set G\nht : IsSubgroup t\nh : s \u2286 t\na : G\nha : a \u2208 closure s\n\u22a2 a \u2208 t\n[PROOFSTEP]\ninduction ha\n[GOAL]\ncase basic\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d\u00b2 a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s t : Set G\nht : IsSubgroup t\nh : s \u2286 t\na a\u271d\u00b9 : G\na\u271d : a\u271d\u00b9 \u2208 s\n\u22a2 a\u271d\u00b9 \u2208 t\n[PROOFSTEP]\nsimp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n[GOAL]\ncase one\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s t : Set G\nht : IsSubgroup t\nh : s \u2286 t\na : G\n\u22a2 1 \u2208 t\n[PROOFSTEP]\nsimp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n[GOAL]\ncase inv\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d\u00b2 a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s t : Set G\nht : IsSubgroup t\nh : s \u2286 t\na a\u271d\u00b9 : G\na\u271d : InClosure s a\u271d\u00b9\na_ih\u271d : a\u271d\u00b9 \u2208 t\n\u22a2 a\u271d\u00b9\u207b\u00b9 \u2208 t\n[PROOFSTEP]\nsimp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n[GOAL]\ncase mul\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d\u00b3 a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s t : Set G\nht : IsSubgroup t\nh : s \u2286 t\na a\u271d\u00b2 b\u271d : G\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\na_ih\u271d\u00b9 : a\u271d\u00b2 \u2208 t\na_ih\u271d : b\u271d \u2208 t\n\u22a2 a\u271d\u00b2 * b\u271d \u2208 t\n[PROOFSTEP]\nsimp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s : Set G\na : G\nh : a \u2208 closure s\nx\u271d\u00b2 : G\nx\u271d\u00b9 : InClosure s x\u271d\u00b2\nx\u271d : \u2203 l, (\u2200 (x : G), x \u2208 l \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 List.prod l = x\u271d\u00b2\nL : List G\nHL1 : \u2200 (x : G), x \u2208 L \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s\nHL2 : List.prod L = x\u271d\u00b2\nx : G\nhx : x \u2208 List.map Inv.inv (List.reverse L)\ny : G\nhy1 : y \u2208 List.reverse L\nhy2 : y\u207b\u00b9 = x\n\u22a2 y \u2208 s \u2192 y\u207b\u00b9\u207b\u00b9 \u2208 s\n[PROOFSTEP]\nrw [inv_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s : Set G\na : G\nh : a \u2208 closure s\nx\u271d\u00b2 : G\nx\u271d\u00b9 : InClosure s x\u271d\u00b2\nx\u271d : \u2203 l, (\u2200 (x : G), x \u2208 l \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 List.prod l = x\u271d\u00b2\nL : List G\nHL1 : \u2200 (x : G), x \u2208 L \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s\nHL2 : List.prod L = x\u271d\u00b2\nx : G\nhx : x \u2208 List.map Inv.inv (List.reverse L)\ny : G\nhy1 : y \u2208 List.reverse L\nhy2 : y\u207b\u00b9 = x\n\u22a2 y \u2208 s \u2192 y \u2208 s\n[PROOFSTEP]\nexact id\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s : Set G\na : G\nh : a \u2208 closure s\nx : G\nx\u271d\u00b9 : InClosure s x\nx\u271d : \u2203 l, (\u2200 (x : G), x \u2208 l \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 List.prod l = x\nL : List G\nHL1 : \u2200 (x : G), x \u2208 L \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s\nHL2 : List.prod L = x\nhd : G\ntl : List G\nih : List.prod (List.map Inv.inv (List.reverse tl)) = (List.prod tl)\u207b\u00b9\n\u22a2 List.prod (List.map Inv.inv (List.reverse (hd :: tl))) = (List.prod (hd :: tl))\u207b\u00b9\n[PROOFSTEP]\nrw [List.reverse_cons, List.map_append, List.prod_append, ih, List.map_singleton, List.prod_cons, List.prod_nil,\n  mul_one, List.prod_cons, mul_inv_rev]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns\u271d s : Set G\na : G\nh : a \u2208 closure s\nx y : G\nx\u271d\u00b3 : InClosure s x\nx\u271d\u00b2 : InClosure s y\nx\u271d\u00b9 : \u2203 l, (\u2200 (x : G), x \u2208 l \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 List.prod l = x\nx\u271d : \u2203 l, (\u2200 (x : G), x \u2208 l \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 List.prod l = y\nL1 : List G\nHL1 : \u2200 (x : G), x \u2208 L1 \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s\nHL2 : List.prod L1 = x\nL2 : List G\nHL3 : \u2200 (x : G), x \u2208 L2 \u2192 x \u2208 s \u2228 x\u207b\u00b9 \u2208 s\nHL4 : List.prod L2 = y\n\u22a2 List.prod (L1 ++ L2) = x * y\n[PROOFSTEP]\nrw [List.prod_append, HL2, HL4]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\n\u22a2 f '' closure s \u2264 closure (f '' s)\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 f x \u2208 closure (f '' s)\n[PROOFSTEP]\nexact\n  InClosure.recOn hx (by intros _ ha; exact subset_closure (mem_image_of_mem f ha))\n    (by\n      rw [hf.map_one]\n      apply IsSubmonoid.one_mem (closure.isSubgroup _).toIsSubmonoid)\n    (by\n      intros _ _\n      rw [hf.map_inv]\n      apply IsSubgroup.inv_mem (closure.isSubgroup _))\n    (by\n      intros _ _ _ _ ha hb\n      rw [hf.map_mul]\n      exact (closure.isSubgroup (f '' s)).toIsSubmonoid.mul_mem ha hb)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 \u2200 {a : G}, a \u2208 s \u2192 f a \u2208 closure (f '' s)\n[PROOFSTEP]\nintros _ ha\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\na\u271d : G\nha : a\u271d \u2208 s\n\u22a2 f a\u271d \u2208 closure (f '' s)\n[PROOFSTEP]\nexact subset_closure (mem_image_of_mem f ha)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 f 1 \u2208 closure (f '' s)\n[PROOFSTEP]\nrw [hf.map_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 1 \u2208 closure (f '' s)\n[PROOFSTEP]\napply IsSubmonoid.one_mem (closure.isSubgroup _).toIsSubmonoid\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 \u2200 {a : G}, InClosure s a \u2192 f a \u2208 closure (f '' s) \u2192 f a\u207b\u00b9 \u2208 closure (f '' s)\n[PROOFSTEP]\nintros _ _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\na\u271d\u00b9 : G\na\u271d : InClosure s a\u271d\u00b9\n\u22a2 f a\u271d\u00b9 \u2208 closure (f '' s) \u2192 f a\u271d\u00b9\u207b\u00b9 \u2208 closure (f '' s)\n[PROOFSTEP]\nrw [hf.map_inv]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\na\u271d\u00b9 : G\na\u271d : InClosure s a\u271d\u00b9\n\u22a2 f a\u271d\u00b9 \u2208 closure (f '' s) \u2192 (f a\u271d\u00b9)\u207b\u00b9 \u2208 closure (f '' s)\n[PROOFSTEP]\napply IsSubgroup.inv_mem (closure.isSubgroup _)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\n\u22a2 \u2200 {a b : G},\n    InClosure s a \u2192 InClosure s b \u2192 f a \u2208 closure (f '' s) \u2192 f b \u2208 closure (f '' s) \u2192 f (a * b) \u2208 closure (f '' s)\n[PROOFSTEP]\nintros _ _ _ _ ha hb\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : G\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\nha : f a\u271d\u00b2 \u2208 closure (f '' s)\nhb : f b\u271d \u2208 closure (f '' s)\n\u22a2 f (a\u271d\u00b2 * b\u271d) \u2208 closure (f '' s)\n[PROOFSTEP]\nrw [hf.map_mul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d\u00b9 : Group G\ns\u271d : Set G\ninst\u271d : Group H\nf : G \u2192 H\nhf : IsGroupHom f\ns : Set G\nx : G\nhx : x \u2208 closure s\na\u271d\u00b2 b\u271d : G\na\u271d\u00b9 : InClosure s a\u271d\u00b2\na\u271d : InClosure s b\u271d\nha : f a\u271d\u00b2 \u2208 closure (f '' s)\nhb : f b\u271d \u2208 closure (f '' s)\n\u22a2 f a\u271d\u00b2 * f b\u271d \u2208 closure (f '' s)\n[PROOFSTEP]\nexact (closure.isSubgroup (f '' s)).toIsSubmonoid.mul_mem ha hb\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 x \u2208 closure (s \u222a t) \u2194 \u2203 y, y \u2208 closure s \u2227 \u2203 z, z \u2208 closure t \u2227 y * z = x\n[PROOFSTEP]\nsimp only [closure_eq_mclosure, Monoid.mem_closure_union_iff, exists_prop, preimage_union]\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2194\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x\n[PROOFSTEP]\nrintro \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * yt * (zs * zt)\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8_, hys, _, hzs, rfl\u27e9, _, \u27e8_, hyt, _, hzt, rfl\u27e9, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * zs * (yt * zt) = ys * yt * (zs * zt)\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, mul_left_comm zs]\n[GOAL]\ncase mpr\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nx : G\n\u22a2 (\u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 y_1 * z = y) \u2227\n        \u2203 z, (\u2203 y, y \u2208 Monoid.Closure t \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227 y * z = x) \u2192\n    \u2203 y,\n      (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n        \u2203 z,\n          (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n            y * z = x\n[PROOFSTEP]\nrintro \u27e8_, \u27e8ys, hys, zs, hzs, rfl\u27e9, _, \u27e8yt, hyt, zt, hzt, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 \u2203 y,\n    (\u2203 y_1, y_1 \u2208 Monoid.Closure s \u2227 \u2203 z, z \u2208 Monoid.Closure t \u2227 y_1 * z = y) \u2227\n      \u2203 z,\n        (\u2203 y, y \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2227 \u2203 z_1, z_1 \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t) \u2227 y * z_1 = z) \u2227\n          y * z = ys * zs * (yt * zt)\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\u271d\ninst\u271d\u00b9 : Group G\u271d\ns\u271d : Set G\u271d\nG : Type u_4\ninst\u271d : CommGroup G\ns t : Set G\nys : G\nhys : ys \u2208 Monoid.Closure s\nzs : G\nhzs : zs \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' s)\nyt : G\nhyt : yt \u2208 Monoid.Closure t\nzt : G\nhzt : zt \u2208 Monoid.Closure (Inv.inv \u207b\u00b9' t)\n\u22a2 ys * yt * (zs * zt) = ys * zs * (yt * zt)\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc, mul_left_comm yt]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 trivial G \u2286 Group.closure \u2205\n[PROOFSTEP]\nsimp [Set.subset_def, (Group.closure.isSubgroup _).one_mem]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\n\u22a2 \u2205 \u2286 trivial G\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nt : Set G\nht : IsNormalSubgroup t\na : G\nh : a \u2208 t\nx : G\nhc : x \u2208 conjugatesOf a\n\u22a2 x \u2208 t\n[PROOFSTEP]\nobtain \u27e8c, w\u27e9 := isConj_iff.1 hc\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c\u271d : G\ns : Set G\ninst\u271d : Group G\nt : Set G\nht : IsNormalSubgroup t\na : G\nh : a \u2208 t\nx : G\nhc : x \u2208 conjugatesOf a\nc : G\nw : c * a * c\u207b\u00b9 = x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nhave H := IsNormalSubgroup.normal ht a h c\n[GOAL]\ncase intro\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c\u271d : G\ns : Set G\ninst\u271d : Group G\nt : Set G\nht : IsNormalSubgroup t\na : G\nh : a \u2208 t\nx : G\nhc : x \u2208 conjugatesOf a\nc : G\nw : c * a * c\u207b\u00b9 = x\nH : c * a * c\u207b\u00b9 \u2208 t\n\u22a2 x \u2208 t\n[PROOFSTEP]\nrwa [\u2190 w]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn : G\nh : n \u2208 normalClosure s\ng : G\n\u22a2 g * n * g\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\ninduction' h with x hx x hx ihx x y hx hy ihx ihy\n[GOAL]\ncase basic\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g x : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 g * x * g\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\nexact conjugatesOfSet_subset_normalClosure (conj_mem_conjugatesOfSet hx)\n[GOAL]\ncase one\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g : G\n\u22a2 g * 1 * g\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\nsimpa using (normalClosure.isSubgroup s).one_mem\n[GOAL]\ncase inv\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g x : G\nhx : InClosure (conjugatesOfSet s) x\nihx : g * x * g\u207b\u00b9 \u2208 normalClosure s\n\u22a2 g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\nrw [\u2190 conj_inv]\n[GOAL]\ncase inv\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g x : G\nhx : InClosure (conjugatesOfSet s) x\nihx : g * x * g\u207b\u00b9 \u2208 normalClosure s\n\u22a2 (g * x * g\u207b\u00b9)\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\nexact (normalClosure.isSubgroup _).inv_mem ihx\n[GOAL]\ncase mul\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g x y : G\nhx : InClosure (conjugatesOfSet s) x\nhy : InClosure (conjugatesOfSet s) y\nihx : g * x * g\u207b\u00b9 \u2208 normalClosure s\nihy : g * y * g\u207b\u00b9 \u2208 normalClosure s\n\u22a2 g * (x * y) * g\u207b\u00b9 \u2208 normalClosure s\n[PROOFSTEP]\nrw [\u2190 conj_mul]\n[GOAL]\ncase mul\nG : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ns : Set G\ninst\u271d : Group G\nsrc\u271d : IsSubgroup (normalClosure s) := isSubgroup s\nn g x y : G\nhx : InClosure (conjugatesOfSet s) x\nhy : InClosure (conjugatesOfSet s) y\nihx : g * x * g\u207b\u00b9 \u2208 normalClosure s\nihy : g * y * g\u207b\u00b9 \u2208 normalClosure s\n\u22a2 g * x * g\u207b\u00b9 * (g * y * g\u207b\u00b9) \u2208 normalClosure s\n[PROOFSTEP]\nexact (normalClosure.isSubgroup _).toIsSubmonoid.mul_mem ihx ihy\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ns\u271d : Set G\ninst\u271d : Group G\ns t : Set G\nht : IsNormalSubgroup t\nh : s \u2286 t\na : G\nw : a \u2208 normalClosure s\n\u22a2 a \u2208 t\n[PROOFSTEP]\ninduction' w with x hx x _ ihx x y _ _ ihx ihy\n[GOAL]\ncase basic\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ns\u271d : Set G\ninst\u271d : Group G\ns t : Set G\nht : IsNormalSubgroup t\nh : s \u2286 t\na x : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact conjugatesOfSet_subset' ht h <| hx\n[GOAL]\ncase one\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d a\u2081 a\u2082 b c : G\ns\u271d : Set G\ninst\u271d : Group G\ns t : Set G\nht : IsNormalSubgroup t\nh : s \u2286 t\na : G\n\u22a2 1 \u2208 t\n[PROOFSTEP]\nexact ht.toIsSubgroup.toIsSubmonoid.one_mem\n[GOAL]\ncase inv\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d\u00b9 a\u2081 a\u2082 b c : G\ns\u271d : Set G\ninst\u271d : Group G\ns t : Set G\nht : IsNormalSubgroup t\nh : s \u2286 t\na x : G\na\u271d : InClosure (conjugatesOfSet s) x\nihx : x \u2208 t\n\u22a2 x\u207b\u00b9 \u2208 t\n[PROOFSTEP]\nexact ht.toIsSubgroup.inv_mem ihx\n[GOAL]\ncase mul\nG : Type u_1\nH : Type u_2\nA : Type u_3\na\u271d\u00b2 a\u2081 a\u2082 b c : G\ns\u271d : Set G\ninst\u271d : Group G\ns t : Set G\nht : IsNormalSubgroup t\nh : s \u2286 t\na x y : G\na\u271d\u00b9 : InClosure (conjugatesOfSet s) x\na\u271d : InClosure (conjugatesOfSet s) y\nihx : x \u2208 t\nihy : y \u2208 t\n\u22a2 x * y \u2208 t\n[PROOFSTEP]\nexact ht.toIsSubgroup.toIsSubmonoid.mul_mem ihx ihy\n", "meta": {"mathlib_filename": "Mathlib.Deprecated.Subgroup", "llama_tokens": 20229, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.7279754607093178, "lm_q1q2_score": 0.5231190839949014}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 -\u2191u * -\u2191u\u207b\u00b9 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 -\u2191u\u207b\u00b9 * -\u2191u = 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na\u271d b a : \u03b1\nu : \u03b1\u02e3\n\u22a2 -(a /\u209a u) = -a /\u209a u\n[PROOFSTEP]\nsimp only [divp, neg_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u + b /\u209a u = (a + b) /\u209a u\n[PROOFSTEP]\nsimp only [divp, add_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u - b /\u209a u = (a - b) /\u209a u\n[PROOFSTEP]\nrw [sub_eq_add_neg, sub_eq_add_neg, neg_divp, divp_add_divp_same]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a + b /\u209a u = (a * \u2191u + b) /\u209a u\n[PROOFSTEP]\nsimp only [divp, add_mul, Units.mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a - b /\u209a u = (a * \u2191u - b) /\u209a u\n[PROOFSTEP]\nsimp only [divp, sub_mul, Units.mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u + b = (a + b * \u2191u) /\u209a u\n[PROOFSTEP]\nsimp only [divp, add_mul, Units.mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a /\u209a u - b = (a - b * \u2191u) /\u209a u\n[PROOFSTEP]\nsimp only [divp, sub_mul, sub_right_inj]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Ring \u03b1\na\u271d b\u271d a b : \u03b1\nu : \u03b1\u02e3\n\u22a2 b = b * \u2191u * \u2191u\u207b\u00b9\n[PROOFSTEP]\nrw [mul_assoc, Units.mul_inv, mul_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : CommRing \u03b1\na b : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 a /\u209a u\u2081 + b /\u209a u\u2082 = (a * \u2191u\u2082 + \u2191u\u2081 * b) /\u209a (u\u2081 * u\u2082)\n[PROOFSTEP]\nsimp only [divp, add_mul, mul_inv_rev, val_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : CommRing \u03b1\na b : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 a * \u2191u\u2081\u207b\u00b9 + b * \u2191u\u2082\u207b\u00b9 = a * \u2191u\u2082 * (\u2191u\u2082\u207b\u00b9 * \u2191u\u2081\u207b\u00b9) + \u2191u\u2081 * b * (\u2191u\u2082\u207b\u00b9 * \u2191u\u2081\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_comm (\u2191u\u2081 * b), mul_comm b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : CommRing \u03b1\na b : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 a * \u2191u\u2081\u207b\u00b9 + \u2191u\u2082\u207b\u00b9 * b = a * \u2191u\u2082 * (\u2191u\u2082\u207b\u00b9 * \u2191u\u2081\u207b\u00b9) + \u2191u\u2082\u207b\u00b9 * \u2191u\u2081\u207b\u00b9 * (\u2191u\u2081 * b)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 mul_assoc, mul_assoc a, mul_assoc (\u2191u\u2082\u207b\u00b9 : \u03b1), mul_inv, inv_mul, mul_one, mul_one]\n  -- porting note: `assoc_rw` not ported: `assoc_rw [mul_inv, mul_inv, mul_one, mul_one]`\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : CommRing \u03b1\na b : \u03b1\nu\u2081 u\u2082 : \u03b1\u02e3\n\u22a2 a /\u209a u\u2081 - b /\u209a u\u2082 = (a * \u2191u\u2082 - \u2191u\u2081 * b) /\u209a (u\u2081 * u\u2082)\n[PROOFSTEP]\nsimp only [sub_eq_add_neg, neg_divp, divp_add_divp, mul_neg]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nR : Type x\ninst\u271d : Semiring R\na : R\u02e3\nb : R\n\u22a2 \u2191a + b = \u2191a * (1 + \u2191a\u207b\u00b9 * b)\n[PROOFSTEP]\nrw [mul_add, mul_one, \u2190 mul_assoc, Units.mul_inv, one_mul]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Ring.Units", "llama_tokens": 1674, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998508568416, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.52260332439502}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nc : R\nh\u2081 : a \u2264 b\nh\u2082 : 0 \u2264 c\n\u22a2 c \u2022 a \u2264 c \u2022 b\n[PROOFSTEP]\nrcases h\u2081.eq_or_lt with (rfl | hab)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na : M\nc : R\nh\u2082 : 0 \u2264 c\nh\u2081 : a \u2264 a\n\u22a2 c \u2022 a \u2264 c \u2022 a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nc : R\nh\u2081 : a \u2264 b\nh\u2082 : 0 \u2264 c\nhab : a < b\n\u22a2 c \u2022 a \u2264 c \u2022 b\n[PROOFSTEP]\nrcases h\u2082.eq_or_lt with (rfl | hc)\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nh\u2081 : a \u2264 b\nhab : a < b\nh\u2082 : 0 \u2264 0\n\u22a2 0 \u2022 a \u2264 0 \u2022 b\n[PROOFSTEP]\nrw [zero_smul, zero_smul]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nc : R\nh\u2081 : a \u2264 b\nh\u2082 : 0 \u2264 c\nhab : a < b\nhc : 0 < c\n\u22a2 c \u2022 a \u2264 c \u2022 b\n[PROOFSTEP]\nexact (smul_lt_smul_of_pos hab hc).le\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nc : R\nh : c \u2022 a < c \u2022 b\nhc\u271d : 0 \u2264 c\nhc : 0 = c\n\u22a2 0 < 0\n[PROOFSTEP]\nrwa [\u2190 hc, zero_smul, zero_smul] at h \n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : OrderedSemiring R\ninst\u271d\u00b2 : OrderedAddCommMonoid M\ninst\u271d\u00b9 : SMulWithZero R M\ninst\u271d : OrderedSMul R M\ns : Set M\na b : M\nc : R\nhc : 0 < c\n\u22a2 0 < c \u2022 a \u2194 c \u2022 0 < c \u2022 a\n[PROOFSTEP]\nrw [smul_zero]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nn : \u2115\nhn : 0 < n\na b : M\nhab : a < b\n\u22a2 (fun a => n \u2022 a) a < (fun a => n \u2022 a) b\n[PROOFSTEP]\ncases n with\n| zero => cases hn\n| succ n =>\n  induction n with\n  | zero => dsimp; rwa [one_nsmul, one_nsmul]\n  | succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\nn : \u2115\nhn : 0 < n\na b : M\nhab : a < b\n\u22a2 (fun a => n \u2022 a) a < (fun a => n \u2022 a) b\n[PROOFSTEP]\ncases n with\n| zero => cases hn\n| succ n =>\n  induction n with\n  | zero => dsimp; rwa [one_nsmul, one_nsmul]\n  | succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nhn : 0 < zero\n\u22a2 (fun a => zero \u2022 a) a < (fun a => zero \u2022 a) b\n[PROOFSTEP]\n\n| zero => cases hn\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nhn : 0 < zero\n\u22a2 (fun a => zero \u2022 a) a < (fun a => zero \u2022 a) b\n[PROOFSTEP]\ncases hn\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nn : \u2115\nhn : 0 < succ n\n\u22a2 (fun a => succ n \u2022 a) a < (fun a => succ n \u2022 a) b\n[PROOFSTEP]\n\n| succ n =>\n  induction n with\n  | zero => dsimp; rwa [one_nsmul, one_nsmul]\n  | succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nn : \u2115\nhn : 0 < succ n\n\u22a2 (fun a => succ n \u2022 a) a < (fun a => succ n \u2022 a) b\n[PROOFSTEP]\ninduction n with\n| zero => dsimp; rwa [one_nsmul, one_nsmul]\n| succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nn : \u2115\nhn : 0 < succ n\n\u22a2 (fun a => succ n \u2022 a) a < (fun a => succ n \u2022 a) b\n[PROOFSTEP]\ninduction n with\n| zero => dsimp; rwa [one_nsmul, one_nsmul]\n| succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\ncase succ.zero\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nhn : 0 < succ zero\n\u22a2 (fun a => succ zero \u2022 a) a < (fun a => succ zero \u2022 a) b\n[PROOFSTEP]\n\n| zero => dsimp; rwa [one_nsmul, one_nsmul]\n[GOAL]\ncase succ.zero\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nhn : 0 < succ zero\n\u22a2 (fun a => succ zero \u2022 a) a < (fun a => succ zero \u2022 a) b\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ.zero\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nhn : 0 < succ zero\n\u22a2 succ 0 \u2022 a < succ 0 \u2022 b\n[PROOFSTEP]\nrwa [one_nsmul, one_nsmul]\n[GOAL]\ncase succ.succ\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nn : \u2115\nih : 0 < succ n \u2192 (fun a => succ n \u2022 a) a < (fun a => succ n \u2022 a) b\nhn : 0 < succ (succ n)\n\u22a2 (fun a => succ (succ n) \u2022 a) a < (fun a => succ (succ n) \u2022 a) b\n[PROOFSTEP]\n\n| succ n ih => simp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\ncase succ.succ\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedCancelAddCommMonoid M\na b : M\nhab : a < b\nn : \u2115\nih : 0 < succ n \u2192 (fun a => succ n \u2022 a) a < (fun a => succ n \u2022 a) b\nhn : 0 < succ (succ n)\n\u22a2 (fun a => succ (succ n) \u2022 a) a < (fun a => succ (succ n) \u2022 a) b\n[PROOFSTEP]\nsimp only [succ_nsmul _ n.succ, _root_.add_lt_add hab (ih n.succ_pos)]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedAddCommGroup M\nn : \u2124\nhn : 0 < n\n\u22a2 StrictMono fun a => n \u2022 a\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedAddCommGroup M\na\u271d : \u2115\nhn : 0 < ofNat a\u271d\n\u22a2 StrictMono fun a => ofNat a\u271d \u2022 a\n[PROOFSTEP]\nsimp only [Int.ofNat_eq_coe, Int.coe_nat_pos, coe_nat_zsmul] at hn \u22a2\n[GOAL]\ncase ofNat\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedAddCommGroup M\na\u271d : \u2115\nhn : 0 < a\u271d\n\u22a2 StrictMono fun a => a\u271d \u2022 a\n[PROOFSTEP]\nexact strictMono_smul_left hn\n[GOAL]\ncase negSucc\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d : LinearOrderedAddCommGroup M\na\u271d : \u2115\nhn : 0 < -[a\u271d+1]\n\u22a2 StrictMono fun a => -[a\u271d+1] \u2022 a\n[PROOFSTEP]\ncases (Int.negSucc_not_pos _).1 hn\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\n\u22a2 OrderedSMul \ud835\udd5c M\n[PROOFSTEP]\nhave hlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b :=\n  by\n  refine' fun a b c hab hc => (h hab hc).lt_of_ne _\n  rw [Ne.def, hc.ne'.isUnit.smul_left_cancel]\n  exact hab.ne\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\n\u22a2 \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\n[PROOFSTEP]\nrefine' fun a b c hab hc => (h hab hc).lt_of_ne _\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\na b : M\nc : \ud835\udd5c\nhab : a < b\nhc : 0 < c\n\u22a2 c \u2022 a \u2260 c \u2022 b\n[PROOFSTEP]\nrw [Ne.def, hc.ne'.isUnit.smul_left_cancel]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\na b : M\nc : \ud835\udd5c\nhab : a < b\nhc : 0 < c\n\u22a2 \u00aca = b\n[PROOFSTEP]\nexact hab.ne\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\n\u22a2 OrderedSMul \ud835\udd5c M\n[PROOFSTEP]\nrefine' { smul_lt_smul_of_pos := fun {a b c} => hlt' a b c .. }\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\n\u22a2 \u2200 {a b : M} {c : \ud835\udd5c}, c \u2022 a < c \u2022 b \u2192 0 < c \u2192 a < b\n[PROOFSTEP]\nintro a b c hab hc\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\na b : M\nc : \ud835\udd5c\nhab : c \u2022 a < c \u2022 b\nhc : 0 < c\n\u22a2 a < b\n[PROOFSTEP]\nobtain \u27e8c, rfl\u27e9 := hc.ne'.isUnit\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\na b : M\nc : \ud835\udd5c\u02e3\nhab : \u2191c \u2022 a < \u2191c \u2022 b\nhc : 0 < \u2191c\n\u22a2 a < b\n[PROOFSTEP]\nrw [\u2190 inv_smul_smul c a, \u2190 inv_smul_smul c b]\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\na b : M\nc : \ud835\udd5c\u02e3\nhab : \u2191c \u2022 a < \u2191c \u2022 b\nhc : 0 < \u2191c\n\u22a2 c\u207b\u00b9 \u2022 c \u2022 a < c\u207b\u00b9 \u2022 c \u2022 b\n[PROOFSTEP]\nrefine' hlt' _ _ _ hab (pos_of_mul_pos_right _ hc.le)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2074 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b3 : OrderedAddCommMonoid M\ninst\u271d\u00b2 : OrderedAddCommMonoid N\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c M\ninst\u271d : MulActionWithZero \ud835\udd5c N\nh : \u2200 \u2983a b : M\u2984 \u2983c : \ud835\udd5c\u2984, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b\nhlt' : \u2200 (a b : M) (c : \ud835\udd5c), a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\na b : M\nc : \ud835\udd5c\u02e3\nhab : \u2191c \u2022 a < \u2191c \u2022 b\nhc : 0 < \u2191c\n\u22a2 0 < \u2191c * \u2191c\u207b\u00b9\n[PROOFSTEP]\nsimp only [c.mul_inv, zero_lt_one]\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u2074 : OrderedAddCommMonoid M\ninst\u271d\u00b3 : OrderedAddCommMonoid N\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c M\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c N\ninst\u271d : OrderedSMul \ud835\udd5c M\ns : Set M\na b : M\nc : \ud835\udd5c\nh : 0 < c\n\u22a2 c\u207b\u00b9 \u2022 a \u2264 b \u2194 a \u2264 c \u2022 b\n[PROOFSTEP]\nrw [\u2190 smul_le_smul_iff_of_pos h, smul_inv_smul\u2080 h.ne']\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u2074 : OrderedAddCommMonoid M\ninst\u271d\u00b3 : OrderedAddCommMonoid N\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c M\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c N\ninst\u271d : OrderedSMul \ud835\udd5c M\ns : Set M\na b : M\nc : \ud835\udd5c\nh : 0 < c\n\u22a2 c\u207b\u00b9 \u2022 a < b \u2194 a < c \u2022 b\n[PROOFSTEP]\nrw [\u2190 smul_lt_smul_iff_of_pos h, smul_inv_smul\u2080 h.ne']\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u2074 : OrderedAddCommMonoid M\ninst\u271d\u00b3 : OrderedAddCommMonoid N\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c M\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c N\ninst\u271d : OrderedSMul \ud835\udd5c M\ns : Set M\na b : M\nc : \ud835\udd5c\nh : 0 < c\n\u22a2 a \u2264 c\u207b\u00b9 \u2022 b \u2194 c \u2022 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 smul_le_smul_iff_of_pos h, smul_inv_smul\u2080 h.ne']\n[GOAL]\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2075 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u2074 : OrderedAddCommMonoid M\ninst\u271d\u00b3 : OrderedAddCommMonoid N\ninst\u271d\u00b2 : MulActionWithZero \ud835\udd5c M\ninst\u271d\u00b9 : MulActionWithZero \ud835\udd5c N\ninst\u271d : OrderedSMul \ud835\udd5c M\ns : Set M\na b : M\nc : \ud835\udd5c\nh : 0 < c\n\u22a2 a < c\u207b\u00b9 \u2022 b \u2194 c \u2022 a < b\n[PROOFSTEP]\nrw [\u2190 smul_lt_smul_iff_of_pos h, smul_inv_smul\u2080 h.ne']\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.SMul", "llama_tokens": 6751, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.5225526215971906}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 (a + b) / c = a / c + b / c\n[PROOFSTEP]\nsimp_rw [div_eq_mul_inv, add_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nh : b \u2260 0\n\u22a2 (b + a) / b = 1 + a / b\n[PROOFSTEP]\nrw [\u2190 div_self h, add_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nh : b \u2260 0\n\u22a2 (a + b) / b = a / b + 1\n[PROOFSTEP]\nrw [\u2190 div_self h, add_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 1 / a * (a + b) * (1 / b) = 1 / a + 1 / b\n[PROOFSTEP]\nrw [mul_add, one_div_mul_cancel ha, add_mul, one_mul, mul_assoc, mul_one_div_cancel hb, mul_one, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na\u271d b\u271d c d a b : \u03b1\nhc : c \u2260 0\n\u22a2 (a + b / c) * c = a * c + b\n[PROOFSTEP]\nrw [right_distrib, div_mul_cancel _ hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\nhc : c \u2260 0\n\u22a2 b + a / c = (b * c + a) / c\n[PROOFSTEP]\nrw [add_div, mul_div_cancel _ hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\nhc : c \u2260 0\n\u22a2 a / c + b = (a + b * c) / c\n[PROOFSTEP]\nrwa [add_comm, add_div', add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nhbc : Commute b c\nhbd : Commute b d\nhb : b \u2260 0\nhd : d \u2260 0\n\u22a2 a / b + c / d = (a * d + b * c) / (b * d)\n[PROOFSTEP]\nrw [add_div, mul_div_mul_right _ b hd, hbc.eq, hbd.eq, mul_div_mul_right c d hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nhab : Commute a b\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 1 / a + 1 / b = (a + b) / (a * b)\n[PROOFSTEP]\nrw [(Commute.one_right a).div_add_div hab ha hb, one_mul, mul_one, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nhab : Commute a b\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a\u207b\u00b9 + b\u207b\u00b9 = (a + b) / (a * b)\n[PROOFSTEP]\nrw [inv_eq_one_div, inv_eq_one_div, hab.one_div_add_one_div ha hb]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na b : K\n\u22a2 -1 * -1 = 1\n[PROOFSTEP]\nrw [neg_mul_neg, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b a : K\n\u22a2 1 / -a = 1 / (-1 * a)\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b a : K\n\u22a2 1 / (-1 * a) = 1 / a * (1 / -1)\n[PROOFSTEP]\nrw [one_div_mul_one_div_rev]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b a : K\n\u22a2 1 / a * (1 / -1) = 1 / a * -1\n[PROOFSTEP]\nrw [one_div_neg_one_eq_neg_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b a : K\n\u22a2 1 / a * -1 = -(1 / a)\n[PROOFSTEP]\nrw [mul_neg, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 b / -a = b * (1 / -a)\n[PROOFSTEP]\nrw [\u2190 inv_eq_one_div, division_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 b * (1 / -a) = b * -(1 / a)\n[PROOFSTEP]\nrw [one_div_neg_eq_neg_one_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 b * -(1 / a) = -(b * (1 / a))\n[PROOFSTEP]\nrw [neg_mul_eq_mul_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 -(b * (1 / a)) = -(b / a)\n[PROOFSTEP]\nrw [mul_one_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 -b / a = -(b / a)\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul, mul_div_assoc, \u2190 neg_eq_neg_one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 -(b / a) = -b / a\n[PROOFSTEP]\nsimp [neg_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b\u271d a b : K\n\u22a2 -a / -b = a / b\n[PROOFSTEP]\nrw [div_neg_eq_neg_div, neg_div, neg_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na b : K\n\u22a2 -a\u207b\u00b9 = (-a)\u207b\u00b9\n[PROOFSTEP]\nrw [inv_eq_one_div, inv_eq_one_div, div_neg_eq_neg_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na\u271d b a : K\n\u22a2 a / -b = -(a / b)\n[PROOFSTEP]\nrw [\u2190 div_neg_eq_neg_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na b : K\n\u22a2 (-a)\u207b\u00b9 = -a\u207b\u00b9\n[PROOFSTEP]\nrw [neg_inv]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9 : DivisionMonoid K\ninst\u271d : HasDistribNeg K\na b : K\n\u22a2 (-1)\u207b\u00b9 = -1\n[PROOFSTEP]\nrw [\u2190 neg_inv, inv_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b c d a : K\nh : a \u2260 0\n\u22a2 a / -a = -1\n[PROOFSTEP]\nrw [div_neg_eq_neg_div, div_self h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b c d a : K\nh : a \u2260 0\n\u22a2 -a / a = -1\n[PROOFSTEP]\nrw [neg_div, div_self h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b\u271d c\u271d d a b c : K\n\u22a2 a / c - b / c = (a - b) / c\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 neg_div, div_add_div_same, sub_eq_add_neg]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b\u271d c d a b : K\nh : b \u2260 0\n\u22a2 (b - a) / b = 1 - a / b\n[PROOFSTEP]\nsimpa only [\u2190 @div_self _ _ b h] using (div_sub_div_same b a b).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b\u271d c d a b : K\nh : b \u2260 0\n\u22a2 (a - b) / b = a / b - 1\n[PROOFSTEP]\nsimpa only [\u2190 @div_self _ _ b h] using (div_sub_div_same a b b).symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na\u271d b\u271d c d a b : K\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a\u207b\u00b9 - b\u207b\u00b9 = a\u207b\u00b9 * (b - a) * b\u207b\u00b9\n[PROOFSTEP]\nrw [mul_sub, sub_mul, mul_inv_cancel_right\u2080 hb, inv_mul_cancel ha, one_mul]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na b c d : K\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 1 / a * (b - a) * (1 / b) = 1 / a - 1 / b\n[PROOFSTEP]\nrw [mul_sub_left_distrib (1 / a), one_div_mul_cancel ha, mul_sub_right_distrib, one_mul, mul_assoc,\n  mul_one_div_cancel hb, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na b c d : K\nhbc : Commute b c\nhbd : Commute b d\nhb : b \u2260 0\nhd : d \u2260 0\n\u22a2 a / b - c / d = (a * d - b * c) / (b * d)\n[PROOFSTEP]\nsimpa only [mul_neg, neg_div, \u2190 sub_eq_add_neg] using hbc.neg_right.div_add_div hbd hb hd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionRing K\na b c d : K\nhab : Commute a b\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a\u207b\u00b9 - b\u207b\u00b9 = (b - a) / (a * b)\n[PROOFSTEP]\nsimp only [inv_eq_one_div, (Commute.one_right a).div_sub_div hab ha hb, one_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : Field K\na b : K\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a\u207b\u00b9 - b\u207b\u00b9 = (b - a) / (a * b)\n[PROOFSTEP]\nrw [inv_eq_one_div, inv_eq_one_div, div_sub_div _ _ ha hb, one_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : Field K\na b c : K\nhc : c \u2260 0\n\u22a2 b - a / c = (b * c - a) / c\n[PROOFSTEP]\nsimpa using div_sub_div b a one_ne_zero hc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : Field K\na b c : K\nhc : c \u2260 0\n\u22a2 a / c - b = (a - c * b) / c\n[PROOFSTEP]\nsimpa using div_sub_div a b hc one_ne_zero\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : DivisionRing K\nK' : Type ?u.52132\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : One K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Mul K'\ninst\u271d\u00b9\u00b9 : Neg K'\ninst\u271d\u00b9\u2070 : Sub K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : GroupWithZero K' := Injective.groupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : Ring K' := Injective.ring f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 f \u2191(Rat.mk' a b) = f (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nerw [rat_cast, mul, inv, int_cast, nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : DivisionRing K\nK' : Type ?u.52132\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : One K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Mul K'\ninst\u271d\u00b9\u00b9 : Neg K'\ninst\u271d\u00b9\u2070 : Sub K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : GroupWithZero K' := Injective.groupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : Ring K' := Injective.ring f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nexact DivisionRing.ratCast_mk a b h1 h2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : DivisionRing K\nK' : Type ?u.52132\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : One K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Mul K'\ninst\u271d\u00b9\u00b9 : Neg K'\ninst\u271d\u00b9\u2070 : Sub K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : GroupWithZero K' := Injective.groupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : Ring K' := Injective.ring f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u211a\nx : K'\n\u22a2 f ((fun x x_1 => x \u2022 x_1) a x) = f (\u2191a * x)\n[PROOFSTEP]\nerw [qsmul, mul, Rat.smul_def, rat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : Field K\nK' : Type ?u.64255\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : Mul K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Neg K'\ninst\u271d\u00b9\u00b9 : Sub K'\ninst\u271d\u00b9\u2070 : One K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : CommGroupWithZero K' := Injective.commGroupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : CommRing K' := Injective.commRing f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 f \u2191(Rat.mk' a b) = f (\u2191a * (\u2191b)\u207b\u00b9)\n[PROOFSTEP]\nerw [rat_cast, mul, inv, int_cast, nat_cast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : Field K\nK' : Type ?u.64255\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : Mul K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Neg K'\ninst\u271d\u00b9\u00b9 : Sub K'\ninst\u271d\u00b9\u2070 : One K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : CommGroupWithZero K' := Injective.commGroupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : CommRing K' := Injective.commRing f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u2124\nb : \u2115\nh1 : b \u2260 0\nh2 : Nat.coprime (Int.natAbs a) b\n\u22a2 \u2191(Rat.mk' a b) = \u2191a * (\u2191b)\u207b\u00b9\n[PROOFSTEP]\nexact DivisionRing.ratCast_mk a b h1 h2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d\u00b9\u2076 : Field K\nK' : Type ?u.64255\ninst\u271d\u00b9\u2075 : Zero K'\ninst\u271d\u00b9\u2074 : Mul K'\ninst\u271d\u00b9\u00b3 : Add K'\ninst\u271d\u00b9\u00b2 : Neg K'\ninst\u271d\u00b9\u00b9 : Sub K'\ninst\u271d\u00b9\u2070 : One K'\ninst\u271d\u2079 : Inv K'\ninst\u271d\u2078 : Div K'\ninst\u271d\u2077 : SMul \u2115 K'\ninst\u271d\u2076 : SMul \u2124 K'\ninst\u271d\u2075 : SMul \u211a K'\ninst\u271d\u2074 : Pow K' \u2115\ninst\u271d\u00b3 : Pow K' \u2124\ninst\u271d\u00b2 : NatCast K'\ninst\u271d\u00b9 : IntCast K'\ninst\u271d : RatCast K'\nf : K' \u2192 K\nhf : Injective f\nzero : f 0 = 0\none : f 1 = 1\nadd : \u2200 (x y : K'), f (x + y) = f x + f y\nmul : \u2200 (x y : K'), f (x * y) = f x * f y\nneg : \u2200 (x : K'), f (-x) = -f x\nsub : \u2200 (x y : K'), f (x - y) = f x - f y\ninv : \u2200 (x : K'), f x\u207b\u00b9 = (f x)\u207b\u00b9\ndiv : \u2200 (x y : K'), f (x / y) = f x / f y\nnsmul : \u2200 (x : K') (n : \u2115), f (n \u2022 x) = n \u2022 f x\nzsmul : \u2200 (x : K') (n : \u2124), f (n \u2022 x) = n \u2022 f x\nqsmul : \u2200 (x : K') (n : \u211a), f (n \u2022 x) = n \u2022 f x\nnpow : \u2200 (x : K') (n : \u2115), f (x ^ n) = f x ^ n\nzpow : \u2200 (x : K') (n : \u2124), f (x ^ n) = f x ^ n\nnat_cast : \u2200 (n : \u2115), f \u2191n = \u2191n\nint_cast : \u2200 (n : \u2124), f \u2191n = \u2191n\nrat_cast : \u2200 (n : \u211a), f \u2191n = \u2191n\nsrc\u271d\u00b9 : CommGroupWithZero K' := Injective.commGroupWithZero f hf zero one mul inv div npow zpow\nsrc\u271d : CommRing K' := Injective.commRing f hf zero one add mul neg sub nsmul zsmul npow nat_cast int_cast\na : \u211a\nx : K'\n\u22a2 f ((fun x x_1 => x \u2022 x_1) a x) = f (\u2191a * x)\n[PROOFSTEP]\nerw [qsmul, mul, Rat.smul_def, rat_cast]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Field.Basic", "llama_tokens": 8555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943822145998, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.522351914355635}}
{"text": "[GOAL]\n\u03b1 : Sort ?u.352\np q : Prop\ns : \u03b1 \u2192 Prop\n\u22a2 \u00acp \u2227 q \u2228 \u00ac\u00acp \u2227 \u00acq \u2194 p \u2227 \u00acq \u2228 \u00acp \u2227 q\n[PROOFSTEP]\nrw [not_not, or_comm]\n", "meta": {"mathlib_filename": "Mathlib.Tactic.PushNeg", "llama_tokens": 79, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8198933447152497, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5222973200728539}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\n\u22a2 IsLocallyHomeomorphOn f s\n[PROOFSTEP]\nintro x hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 e, x \u2208 e.source \u2227 f = \u2191e\n[PROOFSTEP]\nobtain \u27e8e, hx, he\u27e9 := h x hx\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx\u271d : x \u2208 s\ne : LocalHomeomorph X Y\nhx : x \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\n\u22a2 \u2203 e, x \u2208 e.source \u2227 f = \u2191e\n[PROOFSTEP]\nexact\n  \u27e8{ e with\n      toFun := f\n      map_source' := fun x hx => by rw [he x hx]; exact e.map_source' hx\n      left_inv' := fun x hx => by rw [he x hx]; exact e.left_inv' hx\n      right_inv' := fun y hy => by rw [he _ (e.map_target' hy)]; exact e.right_inv' hy\n      continuous_toFun := (continuousOn_congr he).mpr e.continuous_toFun },\n    hx, rfl\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx\u271d : X\nhx\u271d\u00b9 : x\u271d \u2208 s\ne : LocalHomeomorph X Y\nhx\u271d : x\u271d \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx : x \u2208 e.source\n\u22a2 f x \u2208 e.target\n[PROOFSTEP]\nrw [he x hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx\u271d : X\nhx\u271d\u00b9 : x\u271d \u2208 s\ne : LocalHomeomorph X Y\nhx\u271d : x\u271d \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx : x \u2208 e.source\n\u22a2 \u2191e x \u2208 e.target\n[PROOFSTEP]\nexact e.map_source' hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx\u271d : X\nhx\u271d\u00b9 : x\u271d \u2208 s\ne : LocalHomeomorph X Y\nhx\u271d : x\u271d \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx : x \u2208 e.source\n\u22a2 LocalEquiv.invFun e.toLocalEquiv (f x) = x\n[PROOFSTEP]\nrw [he x hx]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx\u271d : X\nhx\u271d\u00b9 : x\u271d \u2208 s\ne : LocalHomeomorph X Y\nhx\u271d : x\u271d \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx : x \u2208 e.source\n\u22a2 LocalEquiv.invFun e.toLocalEquiv (\u2191e x) = x\n[PROOFSTEP]\nexact e.left_inv' hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx\u271d : x \u2208 s\ne : LocalHomeomorph X Y\nhx : x \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\ny : Y\nhy : y \u2208 e.target\n\u22a2 f (LocalEquiv.invFun e.toLocalEquiv y) = y\n[PROOFSTEP]\nrw [he _ (e.map_target' hy)]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nh : \u2200 (x : X), x \u2208 s \u2192 \u2203 e, x \u2208 e.source \u2227 \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\nx : X\nhx\u271d : x \u2208 s\ne : LocalHomeomorph X Y\nhx : x \u2208 e.source\nhe : \u2200 (y : X), y \u2208 e.source \u2192 f y = \u2191e y\ny : Y\nhy : y \u2208 e.target\n\u22a2 \u2191e (LocalEquiv.invFun e.toLocalEquiv y) = y\n[PROOFSTEP]\nexact e.right_inv' hy\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nhg : IsLocallyHomeomorphOn g t\nhf : IsLocallyHomeomorphOn f s\nh : Set.MapsTo f s t\n\u22a2 IsLocallyHomeomorphOn (g \u2218 f) s\n[PROOFSTEP]\nintro x hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nhg : IsLocallyHomeomorphOn g t\nhf : IsLocallyHomeomorphOn f s\nh : Set.MapsTo f s t\nx : X\nhx : x \u2208 s\n\u22a2 \u2203 e, x \u2208 e.source \u2227 g \u2218 f = \u2191e\n[PROOFSTEP]\nobtain \u27e8eg, hxg, rfl\u27e9 := hg (f x) (h hx)\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\nhf : IsLocallyHomeomorphOn f s\nh : Set.MapsTo f s t\nx : X\nhx : x \u2208 s\neg : LocalHomeomorph Y Z\nhxg : f x \u2208 eg.source\nhg : IsLocallyHomeomorphOn (\u2191eg) t\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2191eg \u2218 f = \u2191e\n[PROOFSTEP]\nobtain \u27e8ef, hxf, rfl\u27e9 := hf x hx\n[GOAL]\ncase intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ns : Set X\nt : Set Y\nx : X\nhx : x \u2208 s\neg : LocalHomeomorph Y Z\nhg : IsLocallyHomeomorphOn (\u2191eg) t\nef : LocalHomeomorph X Y\nhxf : x \u2208 ef.source\nhf : IsLocallyHomeomorphOn (\u2191ef) s\nh : Set.MapsTo (\u2191ef) s t\nhxg : \u2191ef x \u2208 eg.source\n\u22a2 \u2203 e, x \u2208 e.source \u2227 \u2191eg \u2218 \u2191ef = \u2191e\n[PROOFSTEP]\nexact \u27e8ef.trans eg, \u27e8hxf, hxg\u27e9, rfl\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\ng : Y \u2192 Z\nf : X \u2192 Y\ns : Set X\nt : Set Y\n\u22a2 IsLocallyHomeomorph f \u2194 IsLocallyHomeomorphOn f Set.univ\n[PROOFSTEP]\nsimp only [IsLocallyHomeomorph, IsLocallyHomeomorphOn, Set.mem_univ, forall_true_left]\n", "meta": {"mathlib_filename": "Mathlib.Topology.IsLocallyHomeomorph", "llama_tokens": 3034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324893519999, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.5222671627092765}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nb : \u2124\n\u22a2 b \u2264 \u230a\u2191(b + 1)\u230b\n[PROOFSTEP]\nrw [floor_intCast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nb : \u2124\n\u22a2 b \u2264 b + 1\n[PROOFSTEP]\nexact (lt_add_one _).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nb : \u2124\n\u22a2 \u2308\u2191(b - 1)\u2309 \u2264 b\n[PROOFSTEP]\nrw [ceil_intCast]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nb : \u2124\n\u22a2 b - 1 \u2264 b\n[PROOFSTEP]\nexact (sub_one_lt _).le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 Tendsto floor (\ud835\udcdd[Ici \u2191n] \u2191n) (pure n)\n[PROOFSTEP]\nsimpa only [floor_intCast] using\n  tendsto_floor_right_pure_floor\n    (n : \u03b1)\n      -- porting note: new theorem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 Tendsto ceil (\ud835\udcdd[Iic \u2191n] \u2191n) (pure n)\n[PROOFSTEP]\nsimpa only [ceil_intCast] using\n  tendsto_ceil_left_pure_ceil\n    (n : \u03b1)\n      -- porting note: new theorem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\n\u22a2 \u2191(\u2308x\u2309 - 1) < x\n[PROOFSTEP]\nrw [cast_sub, cast_one, sub_lt_iff_lt_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\n\u22a2 \u2191\u2308x\u2309 < x + 1\n[PROOFSTEP]\nexact ceil_lt_add_one _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\nh\u2081 : \u2191(\u2308x\u2309 - 1) < x\n\u22a2 x \u2264 \u2191(\u2308x\u2309 - 1) + 1\n[PROOFSTEP]\nrw [cast_sub, cast_one, sub_add_cancel]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\nh\u2081 : \u2191(\u2308x\u2309 - 1) < x\n\u22a2 x \u2264 \u2191\u2308x\u2309\n[PROOFSTEP]\nexact le_ceil _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 Tendsto floor (\ud835\udcdd[Iio \u2191n] \u2191n) (pure (n - 1))\n[PROOFSTEP]\nsimpa only [ceil_intCast] using\n  tendsto_floor_left_pure_ceil_sub_one\n    (n : \u03b1)\n      -- porting note: new theorem\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\n\u22a2 \u2191(\u230ax\u230b + 1) - 1 \u2264 x\n[PROOFSTEP]\nrw [cast_add, cast_one, add_sub_cancel]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nx : \u03b1\n\u22a2 \u2191\u230ax\u230b \u2264 x\n[PROOFSTEP]\nexact floor_le _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 Tendsto ceil (\ud835\udcdd[Ioi \u2191n] \u2191n) (pure (n + 1))\n[PROOFSTEP]\nsimpa only [floor_intCast] using tendsto_ceil_right_pure_floor_add_one (n : \u03b1)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 pure (IntCast.intCast (n - 1)) \u2264 \ud835\udcdd[Iic (\u2191n - 1)] (\u2191n - 1)\n[PROOFSTEP]\nrw [\u2190 @cast_one \u03b1, \u2190 cast_sub]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 pure (IntCast.intCast (n - 1)) \u2264 \ud835\udcdd[Iic \u2191(n - 1)] \u2191(n - 1)\n[PROOFSTEP]\nexact pure_le_nhdsWithin le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 pure (IntCast.intCast (n + 1)) \u2264 \ud835\udcdd[Ici (\u2191n + 1)] (\u2191n + 1)\n[PROOFSTEP]\nrw [\u2190 @cast_one \u03b1, \u2190 cast_add]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : FloorRing \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\nn : \u2124\n\u22a2 pure (IntCast.intCast (n + 1)) \u2264 \ud835\udcdd[Ici \u2191(n + 1)] \u2191(n + 1)\n[PROOFSTEP]\nexact pure_le_nhdsWithin le_rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : FloorRing \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderClosedTopology \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nn : \u2124\n\u22a2 Tendsto fract (\ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 sub_sub_cancel (n : \u03b1) 1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : FloorRing \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderClosedTopology \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nn : \u2124\n\u22a2 Tendsto fract (\ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd (\u2191n - (\u2191n - 1)))\n[PROOFSTEP]\nrefine (tendsto_id.mono_left nhdsWithin_le_nhds).sub ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : FloorRing \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : OrderClosedTopology \u03b1\ninst\u271d : TopologicalAddGroup \u03b1\nn : \u2124\n\u22a2 Tendsto (fun x => \u2191\u230ax\u230b) (\ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd (\u2191n - 1))\n[PROOFSTEP]\nexact tendsto_floor_left' n\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 Continuous fun st => f st.fst (fract st.snd)\n[PROOFSTEP]\nchange Continuous (uncurry f \u2218 Prod.map id fract)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 Continuous (uncurry f \u2218 Prod.map id fract)\n[PROOFSTEP]\nrw [continuous_iff_continuousAt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\n\u22a2 \u2200 (x : \u03b2 \u00d7 \u03b1), ContinuousAt (uncurry f \u2218 Prod.map id fract) x\n[PROOFSTEP]\nrintro \u27e8s, t\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)\n[PROOFSTEP]\nrcases em (\u2203 n : \u2124, t = n) with (\u27e8n, rfl\u27e9 | ht)\n[GOAL]\ncase mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, \u2191n)\n[PROOFSTEP]\nrw [ContinuousAt, nhds_prod_eq, \u2190 nhds_left'_sup_nhds_right (n : \u03b1), prod_sup, tendsto_sup]\n[GOAL]\ncase mk.inl.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))) \u2227\n    Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.inl.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))\n[PROOFSTEP]\nrefine\n  (((h (s, 1) \u27e8trivial, zero_le_one, le_rfl\u27e9).tendsto.mono_left ?_).comp\n        (tendsto_id.prod_map (tendsto_fract_left _))).mono_right\n    (le_of_eq ?_)\n[GOAL]\ncase mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Iio 1] 1 \u2264 \ud835\udcdd[univ \u00d7\u02e2 Icc 0 1] (s, 1)\n[PROOFSTEP]\nrw [nhdsWithin_prod_eq, nhdsWithin_univ, \u2190 nhdsWithin_Ico_eq_nhdsWithin_Iio one_pos]\n[GOAL]\ncase mk.inl.intro.left.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ico 0 1] 1 \u2264 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Icc 0 1] 1\n[PROOFSTEP]\nexact Filter.prod_mono le_rfl (nhdsWithin_mono _ Ico_subset_Icc_self)\n[GOAL]\ncase mk.inl.intro.left.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd (uncurry f (s, 1)) = \ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\ncase mk.inl.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 Tendsto (uncurry f \u2218 Prod.map id fract) (\ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici \u2191n] \u2191n) (\ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n)))\n[PROOFSTEP]\nrefine\n  (((h (s, 0) \u27e8trivial, le_rfl, zero_le_one\u27e9).tendsto.mono_left <| le_of_eq ?_).comp\n        (tendsto_id.prod_map (tendsto_fract_right _))).mono_right\n    (le_of_eq ?_)\n[GOAL]\ncase mk.inl.intro.right.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd s \u00d7\u02e2 \ud835\udcdd[Ici 0] 0 = \ud835\udcdd[univ \u00d7\u02e2 Icc 0 1] (s, 0)\n[PROOFSTEP]\nsimp [nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\ncase mk.inl.intro.right.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nn : \u2124\n\u22a2 \ud835\udcdd (uncurry f (s, 0)) = \ud835\udcdd ((uncurry f \u2218 Prod.map id fract) (s, \u2191n))\n[PROOFSTEP]\nsimp [nhdsWithin_prod_eq, nhdsWithin_univ]\n[GOAL]\ncase mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : \u00ac\u2203 n, t = \u2191n\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)\n[PROOFSTEP]\nreplace ht : t \u2260 \u230at\u230b := fun ht' => ht \u27e8_, ht'\u27e9\n[GOAL]\ncase mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 ContinuousAt (uncurry f \u2218 Prod.map id fract) (s, t)\n[PROOFSTEP]\nrefine (h.continuousAt ?_).comp (continuousAt_id.prod_map (continuousAt_fract ht))\n[GOAL]\ncase mk.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2075 : LinearOrderedRing \u03b1\ninst\u271d\u2074 : FloorRing \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : TopologicalSpace \u03b3\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b3\nh : ContinuousOn (uncurry f) (univ \u00d7\u02e2 Icc 0 1)\nhf : \u2200 (s : \u03b2), f s 0 = f s 1\ns : \u03b2\nt : \u03b1\nht : t \u2260 \u2191\u230at\u230b\n\u22a2 univ \u00d7\u02e2 Icc 0 1 \u2208 \ud835\udcdd (Prod.map id fract (s, t))\n[PROOFSTEP]\nexact prod_mem_nhds univ_mem (Icc_mem_nhds (fract_pos.2 ht) (fract_lt_one _))\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.Order.Floor", "llama_tokens": 6737, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.743167997235783, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.5220978557523345}}
{"text": "[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 IsNoetherian K V \u2194 Module.rank K V < \u2135\u2080\n[PROOFSTEP]\nlet b := Basis.ofVectorSpace K V\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 IsNoetherian K V \u2194 Module.rank K V < \u2135\u2080\n[PROOFSTEP]\nrw [\u2190 b.mk_eq_rank'', lt_aleph0_iff_set_finite]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 IsNoetherian K V \u2194 Set.Finite (Basis.ofVectorSpaceIndex K V)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 IsNoetherian K V \u2192 Set.Finite (Basis.ofVectorSpaceIndex K V)\n[PROOFSTEP]\nintro\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\na\u271d : IsNoetherian K V\n\u22a2 Set.Finite (Basis.ofVectorSpaceIndex K V)\n[PROOFSTEP]\nexact finite_of_linearIndependent (Basis.ofVectorSpaceIndex.linearIndependent K V)\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\n\u22a2 Set.Finite (Basis.ofVectorSpaceIndex K V) \u2192 IsNoetherian K V\n[PROOFSTEP]\nintro hbfinite\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhbfinite : Set.Finite (Basis.ofVectorSpaceIndex K V)\n\u22a2 IsNoetherian K V\n[PROOFSTEP]\nrefine' @isNoetherian_of_linearEquiv K (\u22a4 : Submodule K V) V _ _ _ _ _ (LinearEquiv.ofTop _ rfl) (id _)\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhbfinite : Set.Finite (Basis.ofVectorSpaceIndex K V)\n\u22a2 IsNoetherian K { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nrefine' isNoetherian_of_fg_of_noetherian _ \u27e8Set.Finite.toFinset hbfinite, _\u27e9\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nb : Basis (\u2191(Basis.ofVectorSpaceIndex K V)) K V := Basis.ofVectorSpace K V\nhbfinite : Set.Finite (Basis.ofVectorSpaceIndex K V)\n\u22a2 span K \u2191(Set.Finite.toFinset hbfinite) = \u22a4\n[PROOFSTEP]\nrw [Set.Finite.coe_toFinset, \u2190 b.span_eq, Basis.coe_ofVectorSpace, Subtype.range_coe]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : IsNoetherian K V\n\u22a2 \u2191(Basis.ofVectorSpaceIndex K V) \u2243 { x // x \u2208 finsetBasisIndex K V }\n[PROOFSTEP]\nrw [coeSort_finsetBasisIndex]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b3 : DivisionRing K\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module K V\ninst\u271d : IsNoetherian K V\n\u22a2 Set.range \u2191(finsetBasis K V) = Basis.ofVectorSpaceIndex K V\n[PROOFSTEP]\nrw [finsetBasis, Basis.range_reindex, Basis.range_ofVectorSpace]\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 IsNoetherian K V \u2194 Module.Finite K V\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 IsNoetherian K V \u2192 Module.Finite K V\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : IsNoetherian K V\n\u22a2 Module.Finite K V\n[PROOFSTEP]\nexact\n  \u27e8\u27e8finsetBasisIndex K V, by\n      convert (finsetBasis K V).span_eq\n      simp\u27e9\u27e9\n[GOAL]\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : IsNoetherian K V\n\u22a2 span K \u2191(finsetBasisIndex K V) = \u22a4\n[PROOFSTEP]\nconvert (finsetBasis K V).span_eq\n[GOAL]\ncase h.e'_2.h.e'_6\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nh : IsNoetherian K V\n\u22a2 \u2191(finsetBasisIndex K V) = Set.range \u2191(finsetBasis K V)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\n\u22a2 Module.Finite K V \u2192 IsNoetherian K V\n[PROOFSTEP]\nrintro \u27e8s, hs\u27e9\n[GOAL]\ncase mpr.mk.intro\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : span K \u2191s = \u22a4\n\u22a2 IsNoetherian K V\n[PROOFSTEP]\nrw [IsNoetherian.iff_rank_lt_aleph0, \u2190 rank_top, \u2190 hs]\n[GOAL]\ncase mpr.mk.intro\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\ns : Finset V\nhs : span K \u2191s = \u22a4\n\u22a2 Module.rank K { x // x \u2208 span K \u2191s } < \u2135\u2080\n[PROOFSTEP]\nexact lt_of_le_of_lt (rank_span_le _) s.finite_toSet.lt_aleph0\n", "meta": {"mathlib_filename": "Mathlib.FieldTheory.Finiteness", "llama_tokens": 2188, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321983146848, "lm_q2_score": 0.6477982179521105, "lm_q1q2_score": 0.5218871023930941}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nh : PseudoMetrizableSpace X\n\u22a2 FirstCountableTopology X\n[PROOFSTEP]\nrcases h with \u27e8_, hm\u27e9\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nw\u271d : PseudoMetricSpace X\nhm : UniformSpace.toTopologicalSpace = inst\u271d\u00b3\n\u22a2 FirstCountableTopology X\n[PROOFSTEP]\nrw [\u2190 hm]\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nw\u271d : PseudoMetricSpace X\nhm : UniformSpace.toTopologicalSpace = inst\u271d\u00b3\n\u22a2 FirstCountableTopology X\n[PROOFSTEP]\nexact @UniformSpace.firstCountableTopology X PseudoMetricSpace.toUniformSpace EMetric.instIsCountablyGeneratedUniformity\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), PseudoMetrizableSpace (\u03c0 i)\n\u22a2 PseudoMetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), PseudoMetrizableSpace (\u03c0 i)\nval\u271d : Fintype \u03b9\n\u22a2 PseudoMetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\nletI := fun i => pseudoMetrizableSpacePseudoMetric (\u03c0 i)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), PseudoMetrizableSpace (\u03c0 i)\nval\u271d : Fintype \u03b9\nthis : (i : \u03b9) \u2192 PseudoMetricSpace (\u03c0 i) := fun i => pseudoMetrizableSpacePseudoMetric (\u03c0 i)\n\u22a2 PseudoMetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), MetrizableSpace (\u03c0 i)\n\u22a2 MetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), MetrizableSpace (\u03c0 i)\nval\u271d : Fintype \u03b9\n\u22a2 MetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\nletI := fun i => metrizableSpaceMetric (\u03c0 i)\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : \u2200 (i : \u03b9), MetrizableSpace (\u03c0 i)\nval\u271d : Fintype \u03b9\nthis : (i : \u03b9) \u2192 MetricSpace (\u03c0 i) := fun i => metrizableSpaceMetric (\u03c0 i)\n\u22a2 MetrizableSpace ((i : \u03b9) \u2192 \u03c0 i)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\n\u22a2 SecondCountableTopology \u2191s\n[PROOFSTEP]\nletI := pseudoMetrizableSpacePseudoMetric X\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\nthis : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X\n\u22a2 SecondCountableTopology \u2191s\n[PROOFSTEP]\nhave := hs.separableSpace\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : TopologicalSpace Y\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : PseudoMetrizableSpace X\ns : Set X\nhs : IsSeparable s\nthis\u271d : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X\nthis : SeparableSpace \u2191s\n\u22a2 SecondCountableTopology \u2191s\n[PROOFSTEP]\nexact UniformSpace.secondCountable_of_separable s\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhaveI : NormalSpace X := normalSpaceOfT3SecondCountable X\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis : NormalSpace X\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nrcases exists_countable_basis X with \u27e8B, hBc, -, hB\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nlet s : Set (Set X \u00d7 Set X) :=\n  {UV \u2208 B \u00d7\u02e2 B | closure UV.1 \u2286 UV.2}\n    -- `s` is a countable set.\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhaveI : Encodable s := ((hBc.prod hBc).mono (inter_subset_left _ _)).toEncodable\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis : Encodable \u2191s\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nletI : TopologicalSpace s := \u22a5\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b9 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d : Encodable \u2191s\nthis : TopologicalSpace \u2191s := \u22a5\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhaveI : DiscreteTopology s := \u27e8rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nrsuffices \u27e8f, hf\u27e9 : \u2203 f : X \u2192 s \u2192\u1d47 \u211d, Embedding f\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nf : X \u2192 \u2191s \u2192\u1d47 \u211d\nhf : Embedding f\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nexact\n  \u27e8fun x => (f x).extend (Encodable.encode' s) 0,\n    (BoundedContinuousFunction.isometry_extend (Encodable.encode' s) (0 : \u2115 \u2192\u1d47 \u211d)).embedding.comp hf\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhave hd : \u2200 UV : s, Disjoint (closure UV.1.1) UV.1.2\u1d9c := fun UV =>\n  disjoint_compl_right.mono_right\n    (compl_subset_compl.2 UV.2.2)\n      -- Choose a sequence of `\u03b5\u2099 > 0`, `n : s`, that is bounded above by `1` and tends to zero\n        -- along the `cofinite` filter.\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nobtain \u27e8\u03b5, \u03b501, h\u03b5\u27e9 : \u2203 \u03b5 : s \u2192 \u211d, (\u2200 UV, \u03b5 UV \u2208 Ioc (0 : \u211d) 1) \u2227 Tendsto \u03b5 cofinite (\ud835\udcdd 0) :=\n  by\n  rcases posSumOfEncodable zero_lt_one s with \u27e8\u03b5, \u03b50, c, h\u03b5c, hc1\u27e9\n  refine' \u27e8\u03b5, fun UV => \u27e8\u03b50 UV, _\u27e9, h\u03b5c.summable.tendsto_cofinite_zero\u27e9\n  exact (le_hasSum h\u03b5c UV fun _ _ => (\u03b50 _).le).trans hc1\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u22a2 \u2203 \u03b5, (\u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1) \u2227 Tendsto \u03b5 cofinite (\ud835\udcdd 0)\n[PROOFSTEP]\nrcases posSumOfEncodable zero_lt_one s with \u27e8\u03b5, \u03b50, c, h\u03b5c, hc1\u27e9\n[GOAL]\ncase mk.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b50 : \u2200 (i : \u2191s), 0 < \u03b5 i\nc : \u211d\nh\u03b5c : HasSum \u03b5 c\nhc1 : c \u2264 1\n\u22a2 \u2203 \u03b5, (\u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1) \u2227 Tendsto \u03b5 cofinite (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' \u27e8\u03b5, fun UV => \u27e8\u03b50 UV, _\u27e9, h\u03b5c.summable.tendsto_cofinite_zero\u27e9\n[GOAL]\ncase mk.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b50 : \u2200 (i : \u2191s), 0 < \u03b5 i\nc : \u211d\nh\u03b5c : HasSum \u03b5 c\nhc1 : c \u2264 1\nUV : \u2191s\n\u22a2 \u03b5 UV \u2264 1\n[PROOFSTEP]\nexact (le_hasSum h\u03b5c UV fun _ _ => (\u03b50 _).le).trans hc1\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhave : \u2200 UV : s, \u2203 f : C(X, \u211d), EqOn f 0 UV.1.1 \u2227 EqOn f (fun _ => \u03b5 UV) UV.1.2\u1d9c \u2227 \u2200 x, f x \u2208 Icc 0 (\u03b5 UV) :=\n  by\n  intro UV\n  rcases exists_continuous_zero_one_of_closed isClosed_closure (hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with\n    \u27e8f, hf\u2080, hf\u2081, hf01\u27e9\n  exact\n    \u27e8\u03b5 UV \u2022 f, fun x hx => by simp [hf\u2080 (subset_closure hx)], fun x hx => by simp [hf\u2081 hx], fun x =>\n      \u27e8mul_nonneg (\u03b501 _).1.le (hf01 _).1, mul_le_of_le_one_right (\u03b501 _).1.le (hf01 _).2\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\n\u22a2 \u2200 (UV : \u2191s), \u2203 f, EqOn (\u2191f) 0 (\u2191UV).fst \u2227 EqOn (\u2191f) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 (\u03b5 UV)\n[PROOFSTEP]\nintro UV\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nUV : \u2191s\n\u22a2 \u2203 f, EqOn (\u2191f) 0 (\u2191UV).fst \u2227 EqOn (\u2191f) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 (\u03b5 UV)\n[PROOFSTEP]\nrcases exists_continuous_zero_one_of_closed isClosed_closure (hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with\n  \u27e8f, hf\u2080, hf\u2081, hf01\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nUV : \u2191s\nf : C(X, \u211d)\nhf\u2080 : EqOn (\u2191f) 0 (closure (\u2191UV).fst)\nhf\u2081 : EqOn (\u2191f) 1 (\u2191UV).snd\u1d9c\nhf01 : \u2200 (x : X), \u2191f x \u2208 Icc 0 1\n\u22a2 \u2203 f, EqOn (\u2191f) 0 (\u2191UV).fst \u2227 EqOn (\u2191f) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 (\u03b5 UV)\n[PROOFSTEP]\nexact\n  \u27e8\u03b5 UV \u2022 f, fun x hx => by simp [hf\u2080 (subset_closure hx)], fun x hx => by simp [hf\u2081 hx], fun x =>\n    \u27e8mul_nonneg (\u03b501 _).1.le (hf01 _).1, mul_le_of_le_one_right (\u03b501 _).1.le (hf01 _).2\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nUV : \u2191s\nf : C(X, \u211d)\nhf\u2080 : EqOn (\u2191f) 0 (closure (\u2191UV).fst)\nhf\u2081 : EqOn (\u2191f) 1 (\u2191UV).snd\u1d9c\nhf01 : \u2200 (x : X), \u2191f x \u2208 Icc 0 1\nx : X\nhx : x \u2208 (\u2191UV).fst\n\u22a2 \u2191(\u03b5 UV \u2022 f) x = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp [hf\u2080 (subset_closure hx)]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nUV : \u2191s\nf : C(X, \u211d)\nhf\u2080 : EqOn (\u2191f) 0 (closure (\u2191UV).fst)\nhf\u2081 : EqOn (\u2191f) 1 (\u2191UV).snd\u1d9c\nhf01 : \u2200 (x : X), \u2191f x \u2208 Icc 0 1\nx : X\nhx : x \u2208 (\u2191UV).snd\u1d9c\n\u22a2 \u2191(\u03b5 UV \u2022 f) x = (fun x => \u03b5 UV) x\n[PROOFSTEP]\nsimp [hf\u2081 hx]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nthis : \u2200 (UV : \u2191s), \u2203 f, EqOn (\u2191f) 0 (\u2191UV).fst \u2227 EqOn (\u2191f) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c \u2227 \u2200 (x : X), \u2191f x \u2208 Icc 0 (\u03b5 UV)\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nchoose f hf0 hf\u03b5 hf0\u03b5 using this\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhave hf01 : \u2200 UV x, f UV x \u2208 Icc (0 : \u211d) 1 := fun UV x =>\n  Icc_subset_Icc_right (\u03b501 _).2\n    (hf0\u03b5 _ _)\n      -- The embedding is given by `F x UV = f UV x`.\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nset F : X \u2192 s \u2192\u1d47 \u211d := fun x =>\n  \u27e8\u27e8fun UV => f UV x, continuous_of_discreteTopology\u27e9, 1, fun UV\u2081 UV\u2082 =>\n    Real.dist_le_of_mem_Icc_01 (hf01 _ _) (hf01 _ _)\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nhave hF : \u2200 x UV, F x UV = f UV x := fun _ _ => rfl\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\n\u22a2 \u2203 f, Embedding f\n[PROOFSTEP]\nrefine' \u27e8F, Embedding.mk' _ (fun x y hxy => _) fun x => le_antisymm _ _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\n\u22a2 x = y\n[PROOFSTEP]\nby_contra Hne\n[GOAL]\ncase intro.intro.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\n\u22a2 False\n[PROOFSTEP]\nrcases hB.mem_nhds_iff.1 (isOpen_ne.mem_nhds Hne) with \u27e8V, hVB, hxV, hVy\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nhVy : V \u2286 {y_1 | y_1 \u2260 y}\n\u22a2 False\n[PROOFSTEP]\nrcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with \u27e8U, hUB, hxU, hUV\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nhVy : V \u2286 {y_1 | y_1 \u2260 y}\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\n\u22a2 False\n[PROOFSTEP]\nset UV : \u21a5s := \u27e8(U, V), \u27e8hUB, hVB\u27e9, hUV\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nhVy : V \u2286 {y_1 | y_1 \u2260 y}\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\n\u22a2 False\n[PROOFSTEP]\napply (\u03b501 UV).1.ne\n[GOAL]\ncase intro.intro.refine'_1.intro.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nhVy : V \u2286 {y_1 | y_1 \u2260 y}\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\n\u22a2 0 = \u03b5 UV\n[PROOFSTEP]\ncalc\n  (0 : \u211d) = F x UV := (hf0 UV hxU).symm\n  _ = F y UV := by rw [hxy]\n  _ = \u03b5 UV := hf\u03b5 UV fun h : y \u2208 V => hVy h rfl\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx y : X\nhxy : F x = F y\nHne : \u00acx = y\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nhVy : V \u2286 {y_1 | y_1 \u2260 y}\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\n\u22a2 \u2191(F x) UV = \u2191(F y) UV\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u22a2 comap F (\ud835\udcdd (F x)) \u2264 \ud835\udcdd x\n[PROOFSTEP]\nrefine' ((nhds_basis_ball.comap _).le_basis_iff hB.nhds_hasBasis).2 _\n[GOAL]\ncase intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u22a2 \u2200 (i' : Set X), i' \u2208 B \u2227 x \u2208 i' \u2192 \u2203 i, 0 < i \u2227 F \u207b\u00b9' ball (F x) i \u2286 i'\n[PROOFSTEP]\nrintro V \u27e8hVB, hxV\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\n\u22a2 \u2203 i, 0 < i \u2227 F \u207b\u00b9' ball (F x) i \u2286 V\n[PROOFSTEP]\nrcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with \u27e8U, hUB, hxU, hUV\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\n\u22a2 \u2203 i, 0 < i \u2227 F \u207b\u00b9' ball (F x) i \u2286 V\n[PROOFSTEP]\nset UV : \u21a5s := \u27e8(U, V), \u27e8hUB, hVB\u27e9, hUV\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\n\u22a2 \u2203 i, 0 < i \u2227 F \u207b\u00b9' ball (F x) i \u2286 V\n[PROOFSTEP]\nrefine' \u27e8\u03b5 UV, (\u03b501 UV).1, fun y (hy : dist (F y) (F x) < \u03b5 UV) => _\u27e9\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : dist (F y) (F x) < \u03b5 UV\n\u22a2 y \u2208 V\n[PROOFSTEP]\nreplace hy : dist (F y UV) (F x UV) < \u03b5 UV\n[GOAL]\ncase hy\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : dist (F y) (F x) < \u03b5 UV\n\u22a2 dist (\u2191(F y) UV) (\u2191(F x) UV) < \u03b5 UV\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : dist (\u2191(F y) UV) (\u2191(F x) UV) < \u03b5 UV\n\u22a2 y \u2208 V\n[PROOFSTEP]\nexact (BoundedContinuousFunction.dist_coe_le_dist _).trans_lt hy\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : dist (\u2191(F y) UV) (\u2191(F x) UV) < \u03b5 UV\n\u22a2 y \u2208 V\n[PROOFSTEP]\ncontrapose! hy\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : \u00acy \u2208 V\n\u22a2 \u03b5 { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) } \u2264\n    dist\n      (\u2191((fun x =>\n              { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n                map_bounded' :=\n                  (_ :\n                    \u2203 C,\n                      \u2200 (x_1 y : \u2191s),\n                        dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                            (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                          C) })\n            y)\n        { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) })\n      (\u2191((fun x =>\n              { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n                map_bounded' :=\n                  (_ :\n                    \u2203 C,\n                      \u2200 (x_1 y : \u2191s),\n                        dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                            (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                          C) })\n            x)\n        { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) })\n[PROOFSTEP]\nrw [hF, hF, hf\u03b5 UV hy, hf0 UV hxU, Pi.zero_apply, dist_zero_right]\n[GOAL]\ncase intro.intro.refine'_2.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\nV : Set X\nhVB : V \u2208 B\nhxV : x \u2208 V\nU : Set X\nhUB : U \u2208 B\nhxU : x \u2208 U\nhUV : closure U \u2286 V\nUV : \u2191s := { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) }\ny : X\nhy : \u00acy \u2208 V\n\u22a2 \u03b5 { val := (U, V), property := (_ : (U, V) \u2208 B \u00d7\u02e2 B \u2227 closure (U, V).fst \u2286 (U, V).snd) } \u2264 \u2016(fun x => \u03b5 UV) y\u2016\n[PROOFSTEP]\nexact le_abs_self _\n[GOAL]\ncase intro.intro.refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u22a2 \ud835\udcdd x \u2264 comap F (\ud835\udcdd (F x))\n[PROOFSTEP]\nrefine' (nhds_basis_closedBall.comap _).ge_iff.2 fun \u03b4 \u03b40 => _\n[GOAL]\ncase intro.intro.refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u22a2 F \u207b\u00b9' closedBall (F x) \u03b4 \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave h_fin : {UV : s | \u03b4 \u2264 \u03b5 UV}.Finite := by simpa only [\u2190 not_lt] using h\u03b5 (gt_mem_nhds \u03b40)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\n\u22a2 Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\n[PROOFSTEP]\nsimpa only [\u2190 not_lt] using h\u03b5 (gt_mem_nhds \u03b40)\n[GOAL]\ncase intro.intro.refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\n\u22a2 F \u207b\u00b9' closedBall (F x) \u03b4 \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave : \u2200\u1da0 y in \ud835\udcdd x, \u2200 UV, \u03b4 \u2264 \u03b5 UV \u2192 dist (F y UV) (F x UV) \u2264 \u03b4 :=\n  by\n  refine' (eventually_all_finite h_fin).2 fun UV _ => _\n  exact (f UV).continuous.tendsto x (closedBall_mem_nhds _ \u03b40)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\n\u22a2 \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\n[PROOFSTEP]\nrefine' (eventually_all_finite h_fin).2 fun UV _ => _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b2 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b9 : Encodable \u2191s\nthis\u271d : TopologicalSpace \u2191s := \u22a5\nthis : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nUV : \u2191s\nx\u271d : UV \u2208 {UV | \u03b4 \u2264 \u03b5 UV}\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (\u2191(F x_1) UV) (\u2191(F x) UV) \u2264 \u03b4\n[PROOFSTEP]\nexact (f UV).continuous.tendsto x (closedBall_mem_nhds _ \u03b40)\n[GOAL]\ncase intro.intro.refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nthis : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\n\u22a2 F \u207b\u00b9' closedBall (F x) \u03b4 \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrefine' this.mono fun y hy => (BoundedContinuousFunction.dist_le \u03b40.le).2 fun UV => _\n[GOAL]\ncase intro.intro.refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nthis : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\ny : X\nhy : \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\nUV : \u2191s\n\u22a2 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\n[PROOFSTEP]\ncases' le_total \u03b4 (\u03b5 UV) with hle hle\n[GOAL]\ncase intro.intro.refine'_3.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nthis : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\ny : X\nhy : \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\nUV : \u2191s\nhle : \u03b4 \u2264 \u03b5 UV\n\u22a2 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\ncase intro.intro.refine'_3.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nthis : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\ny : X\nhy : \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\nUV : \u2191s\nhle : \u03b5 UV \u2264 \u03b4\n\u22a2 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\n[PROOFSTEP]\nexacts [hy _ hle, (Real.dist_le_of_mem_Icc (hf0\u03b5 _ _) (hf0\u03b5 _ _)).trans (by rwa [sub_zero])]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : T3Space X\ninst\u271d : SecondCountableTopology X\nthis\u271d\u00b3 : NormalSpace X\nB : Set (Set X)\nhBc : Set.Countable B\nhB : IsTopologicalBasis B\ns : Set (Set X \u00d7 Set X) := {UV | UV \u2208 B \u00d7\u02e2 B \u2227 closure UV.fst \u2286 UV.snd}\nthis\u271d\u00b2 : Encodable \u2191s\nthis\u271d\u00b9 : TopologicalSpace \u2191s := \u22a5\nthis\u271d : DiscreteTopology \u2191s\nhd : \u2200 (UV : \u2191s), Disjoint (closure (\u2191UV).fst) (\u2191UV).snd\u1d9c\n\u03b5 : \u2191s \u2192 \u211d\n\u03b501 : \u2200 (UV : \u2191s), \u03b5 UV \u2208 Ioc 0 1\nh\u03b5 : Tendsto \u03b5 cofinite (\ud835\udcdd 0)\nf : \u2191s \u2192 C(X, \u211d)\nhf0 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) 0 (\u2191UV).fst\nhf\u03b5 : \u2200 (UV : \u2191s), EqOn (\u2191(f UV)) (fun x => \u03b5 UV) (\u2191UV).snd\u1d9c\nhf0\u03b5 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 (\u03b5 UV)\nhf01 : \u2200 (UV : \u2191s) (x : X), \u2191(f UV) x \u2208 Icc 0 1\nF : X \u2192 \u2191s \u2192\u1d47 \u211d :=\n  fun x =>\n    { toContinuousMap := ContinuousMap.mk fun UV => \u2191(f UV) x,\n      map_bounded' :=\n        (_ :\n          \u2203 C,\n            \u2200 (x_1 y : \u2191s),\n              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) x_1)\n                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => \u2191(f UV) x) y) \u2264\n                C) }\nhF : \u2200 (x : X) (UV : \u2191s), \u2191(F x) UV = \u2191(f UV) x\nx : X\n\u03b4 : \u211d\n\u03b40 : 0 < \u03b4\nh_fin : Set.Finite {UV | \u03b4 \u2264 \u03b5 UV}\nthis : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\ny : X\nhy : \u2200 (UV : \u2191s), \u03b4 \u2264 \u03b5 UV \u2192 dist (\u2191(F y) UV) (\u2191(F x) UV) \u2264 \u03b4\nUV : \u2191s\nhle : \u03b5 UV \u2264 \u03b4\n\u22a2 \u03b5 UV - 0 \u2264 \u03b4\n[PROOFSTEP]\nrwa [sub_zero]\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Metrizable", "llama_tokens": 36399, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321983146848, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.521887102393094}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b1\nb : \u03b2\n\u22a2 b \u2208 upperBounds (l '' s) \u2194 b \u2208 u \u207b\u00b9' upperBounds s\n[PROOFSTEP]\nsimp [upperBounds, gc _ _]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b1\nx\u271d : BddAbove (l '' s)\nx : \u03b2\nhx : x \u2208 upperBounds (l '' s)\n\u22a2 u x \u2208 upperBounds s\n[PROOFSTEP]\nrwa [gc.upperBounds_l_image] at hx \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b1\na : \u03b1\nh : IsLUB s a\nb : \u03b2\nhb : b \u2208 upperBounds (l '' s)\n\u22a2 u b \u2208 upperBounds s\n[PROOFSTEP]\nrwa [gc.upperBounds_l_image] at hb \n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nz : \u03b1\ny : \u03b2\n\u22a2 u y = z \u2194 \u2200 (x : \u03b1), x \u2264 z \u2194 l x \u2264 y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nz : \u03b1\ny : \u03b2\n\u22a2 u y = z \u2192 \u2200 (x : \u03b1), x \u2264 z \u2194 l x \u2264 y\n[PROOFSTEP]\nrintro rfl x\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ny : \u03b2\nx : \u03b1\n\u22a2 x \u2264 u y \u2194 l x \u2264 y\n[PROOFSTEP]\nexact (gc x y).symm\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nz : \u03b1\ny : \u03b2\n\u22a2 (\u2200 (x : \u03b1), x \u2264 z \u2194 l x \u2264 y) \u2192 u y = z\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : Preorder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nz : \u03b1\ny : \u03b2\nH : \u2200 (x : \u03b1), x \u2264 z \u2194 l x \u2264 y\n\u22a2 u y = z\n[PROOFSTEP]\nexact ((H <| u y).mpr (gc.l_u_le y)).antisymm ((gc _ _).mp <| (H z).mp le_rfl)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PartialOrder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nx : \u03b1\nz : \u03b2\n\u22a2 l x = z \u2194 \u2200 (y : \u03b2), z \u2264 y \u2194 x \u2264 u y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PartialOrder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nx : \u03b1\nz : \u03b2\n\u22a2 l x = z \u2192 \u2200 (y : \u03b2), z \u2264 y \u2194 x \u2264 u y\n[PROOFSTEP]\nrintro rfl y\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PartialOrder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nx : \u03b1\ny : \u03b2\n\u22a2 l x \u2264 y \u2194 x \u2264 u y\n[PROOFSTEP]\nexact gc x y\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PartialOrder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nx : \u03b1\nz : \u03b2\n\u22a2 (\u2200 (y : \u03b2), z \u2264 y \u2194 x \u2264 u y) \u2192 l x = z\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PartialOrder \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nx : \u03b1\nz : \u03b2\nH : \u2200 (y : \u03b2), z \u2264 y \u2194 x \u2264 u y\n\u22a2 l x = z\n[PROOFSTEP]\nexact ((gc _ _).mpr <| (H z).mp le_rfl).antisymm ((H <| l x).mpr (gc.le_u_l x))\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\n\u22a2 IsLUB (l '' {a\u2081, a\u2082}) (l a\u2081 \u2294 l a\u2082)\n[PROOFSTEP]\nsimp only [image_pair, isLUB_pair]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\n\u22a2 IsLUB (range (l \u2218 f)) (l (iSup f))\n[PROOFSTEP]\nrw [range_comp, \u2190 sSup_range]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : \u03b9 \u2192 \u03b1\n\u22a2 IsLUB (l '' range f) (l (sSup (range f)))\n[PROOFSTEP]\nexact gc.isLUB_l_image (isLUB_sSup _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\nf : (i : \u03b9) \u2192 \u03ba i \u2192 \u03b1\n\u22a2 l (\u2a06 (i : \u03b9) (j : \u03ba i), f i j) = \u2a06 (i : \u03b9) (j : \u03ba i), l (f i j)\n[PROOFSTEP]\nsimp_rw [gc.l_iSup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\ns : Set \u03b1\n\u22a2 l (sSup s) = \u2a06 (a : \u03b1) (_ : a \u2208 s), l a\n[PROOFSTEP]\nsimp only [sSup_eq_iSup, gc.l_iSup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nl1 : \u03b1 \u2192 \u03b2\nu1 : \u03b2 \u2192 \u03b1\nl2 : \u03b2 \u2192 \u03b3\nu2 : \u03b3 \u2192 \u03b2\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\n\u22a2 GaloisConnection (l2 \u2218 l1) (u1 \u2218 u2)\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nl1 : \u03b1 \u2192 \u03b2\nu1 : \u03b2 \u2192 \u03b1\nl2 : \u03b2 \u2192 \u03b3\nu2 : \u03b3 \u2192 \u03b2\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : \u03b1\nb : \u03b3\n\u22a2 (l2 \u2218 l1) a \u2264 b \u2194 a \u2264 (u1 \u2218 u2) b\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nl1 : \u03b1 \u2192 \u03b2\nu1 : \u03b2 \u2192 \u03b1\nl2 : \u03b2 \u2192 \u03b3\nu2 : \u03b3 \u2192 \u03b2\ngc1 : GaloisConnection l1 u1\ngc2 : GaloisConnection l2 u2\na : \u03b1\nb : \u03b3\n\u22a2 l2 (l1 a) \u2264 b \u2194 a \u2264 u1 (u2 b)\n[PROOFSTEP]\nrw [gc2, gc1]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\n\u22a2 GaloisConnection (compl \u2218 u \u2218 compl) (compl \u2218 l \u2218 compl)\n[PROOFSTEP]\nintro a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\na : \u03b2\nb : \u03b1\n\u22a2 (compl \u2218 u \u2218 compl) a \u2264 b \u2194 a \u2264 (compl \u2218 l \u2218 compl) b\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : BooleanAlgebra \u03b1\ninst\u271d : BooleanAlgebra \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ngc : GaloisConnection l u\na : \u03b2\nb : \u03b1\n\u22a2 (u a\u1d9c)\u1d9c \u2264 b \u2194 a \u2264 (l b\u1d9c)\u1d9c\n[PROOFSTEP]\nrw [le_compl_iff_le_compl, gc, compl_le_iff_compl_le]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b2 : CompleteLattice \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nl u : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 u\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03b1\nl\u2082 u\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b2\nh\u2081 : \u2200 (b : \u03b2), GaloisConnection (swap l b) (u\u2081 b)\nh\u2082 : \u2200 (a : \u03b1), GaloisConnection (l a) (u\u2082 a)\n\u22a2 sSup (image2 l s t) = l (sSup s) (sSup t)\n[PROOFSTEP]\nsimp_rw [sSup_image2, \u2190 (h\u2082 _).l_sSup, \u2190 (h\u2081 _).l_sSup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b2 : CompleteLattice \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b2\ninst\u271d : CompleteLattice \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nl u : \u03b1 \u2192 \u03b2 \u2192 \u03b3\nl\u2081 u\u2081 : \u03b2 \u2192 \u03b3 \u2192 \u03b1\nl\u2082 u\u2082 : \u03b1 \u2192 \u03b3 \u2192 \u03b2\nh\u2081 : \u2200 (b : \u03b2), GaloisConnection (l\u2081 b) (swap u b)\nh\u2082 : \u2200 (a : \u03b1), GaloisConnection (l\u2082 a) (u a)\n\u22a2 sInf (image2 u s t) = u (sInf s) (sInf t)\n[PROOFSTEP]\nsimp_rw [sInf_image2, \u2190 (h\u2082 _).u_sInf, \u2190 (h\u2081 _).u_sInf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\ns : Set \u03b2\n\u22a2 BddAbove (\u2191e \u207b\u00b9' s) \u2194 BddAbove s\n[PROOFSTEP]\nrw [\u2190 e.bddAbove_image, e.image_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\ns : Set \u03b2\n\u22a2 BddBelow (\u2191e \u207b\u00b9' s) \u2194 BddBelow s\n[PROOFSTEP]\nrw [\u2190 e.bddBelow_image, e.image_preimage]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : SemilatticeSup \u03b1\ninst\u271d : SemilatticeSup \u03b2\ngi : GaloisInsertion l u\na b : \u03b2\n\u22a2 l (u a) \u2294 l (u b) = a \u2294 b\n[PROOFSTEP]\nsimp only [gi.l_u_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9\u271d : Sort x\n\u03ba : \u03b9\u271d \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\n\u03b9 : Sort x\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b2\n\u22a2 l (\u2a06 (i : \u03b9) (hi : p i), u (f i hi)) = \u2a06 (i : \u03b9) (hi : p i), f i hi\n[PROOFSTEP]\nsimp only [iSup_subtype', gi.l_iSup_u]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\ns : Set \u03b2\n\u22a2 l (sSup (u '' s)) = sSup s\n[PROOFSTEP]\nrw [sSup_image, gi.l_biSup_u, sSup_eq_iSup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : SemilatticeInf \u03b2\ngi : GaloisInsertion l u\na b : \u03b2\n\u22a2 l (u (a \u2293 b)) = a \u2293 b\n[PROOFSTEP]\nsimp only [gi.l_u_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9\u271d : Sort x\n\u03ba : \u03b9\u271d \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\n\u03b9 : Sort x\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b2\n\u22a2 l (\u2a05 (i : \u03b9) (hi : p i), u (f i hi)) = \u2a05 (i : \u03b9) (hi : p i), f i hi\n[PROOFSTEP]\nsimp only [iInf_subtype', gi.l_iInf_u]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\ns : Set \u03b2\n\u22a2 l (sInf (u '' s)) = sInf s\n[PROOFSTEP]\nrw [sInf_image, gi.l_biInf_u, sInf_eq_iInf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9\u271d : Sort x\n\u03ba : \u03b9\u271d \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\n\u03b9 : Sort x\nf : \u03b9 \u2192 \u03b1\nhf : \u2200 (i : \u03b9), u (l (f i)) = f i\n\u22a2 l (\u2a05 (i : \u03b9), f i) = l (\u2a05 (i : \u03b9), u (l (f i)))\n[PROOFSTEP]\nsimp [hf]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9\u271d : Sort x\n\u03ba : \u03b9\u271d \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\n\u03b9 : Sort x\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\nhf : \u2200 (i : \u03b9) (hi : p i), u (l (f i hi)) = f i hi\n\u22a2 l (\u2a05 (i : \u03b9) (hi : p i), f i hi) = \u2a05 (i : \u03b9) (hi : p i), l (f i hi)\n[PROOFSTEP]\nrw [iInf_subtype', iInf_subtype']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9\u271d : Sort x\n\u03ba : \u03b9\u271d \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : CompleteLattice \u03b2\ngi : GaloisInsertion l u\n\u03b9 : Sort x\np : \u03b9 \u2192 Prop\nf : (i : \u03b9) \u2192 p i \u2192 \u03b1\nhf : \u2200 (i : \u03b9) (hi : p i), u (l (f i hi)) = f i hi\n\u22a2 l (\u2a05 (x : { i // p i }), f \u2191x (_ : p \u2191x)) = \u2a05 (x : { i // p i }), l (f \u2191x (_ : p \u2191x))\n[PROOFSTEP]\nexact gi.l_iInf_of_ul_eq_self _ fun _ => hf _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b : \u03b2), a \u2293 b \u2264 a\n[PROOFSTEP]\nsimp only [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b : \u03b2), l (u a \u2293 u b) \u2264 a\n[PROOFSTEP]\nexact fun a b => gi.gc.l_le inf_le_left\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b : \u03b2), a \u2293 b \u2264 b\n[PROOFSTEP]\nsimp only [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b : \u03b2), l (u a \u2293 u b) \u2264 b\n[PROOFSTEP]\nexact fun a b => gi.gc.l_le inf_le_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b c : \u03b2), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nsimp only [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : SemilatticeInf \u03b1\ngi : GaloisInsertion l u\nsrc\u271d : PartialOrder \u03b2 := inst\u271d\u00b9\n\u22a2 \u2200 (a b c : \u03b2), a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 l (u b \u2293 u c)\n[PROOFSTEP]\nexact fun a b c hac hbc =>\n  (gi.le_l_u a).trans <| gi.gc.monotone_l <| le_inf (gi.gc.monotone_u hac) (gi.gc.monotone_u hbc)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderTop \u03b1\ngi : GaloisInsertion l u\n\u22a2 \u2200 (a : \u03b2), a \u2264 \u22a4\n[PROOFSTEP]\nsimp only [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : OrderTop \u03b1\ngi : GaloisInsertion l u\n\u22a2 \u2200 (a : \u03b2), a \u2264 l \u22a4\n[PROOFSTEP]\nexact fun b => (gi.le_l_u b).trans (gi.gc.monotone_l le_top)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), a \u2208 s \u2192 sInf s \u2264 a\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), a \u2208 s \u2192 choice gi (sInf (u '' s)) (_ : u (l (sInf (u '' s))) \u2264 sInf (u '' s)) \u2264 a\n[PROOFSTEP]\nrw [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), a \u2208 s \u2192 l (sInf (u '' s)) \u2264 a\n[PROOFSTEP]\nexact (gi.isGLB_of_u_image (isGLB_sInf _)).1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), (\u2200 (b : \u03b2), b \u2208 s \u2192 a \u2264 b) \u2192 a \u2264 sInf s\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), (\u2200 (b : \u03b2), b \u2208 s \u2192 a \u2264 b) \u2192 a \u2264 choice gi (sInf (u '' s)) (_ : u (l (sInf (u '' s))) \u2264 sInf (u '' s))\n[PROOFSTEP]\nrw [gi.choice_eq]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b2\ninst\u271d : CompleteLattice \u03b1\ngi : GaloisInsertion l u\nsrc\u271d\u00b9 : BoundedOrder \u03b2 := liftBoundedOrder gi\nsrc\u271d : Lattice \u03b2 := liftLattice gi\ns : Set \u03b2\n\u22a2 \u2200 (a : \u03b2), (\u2200 (b : \u03b2), b \u2208 s \u2192 a \u2264 b) \u2192 a \u2264 l (sInf (u '' s))\n[PROOFSTEP]\nexact (gi.isGLB_of_u_image (isGLB_sInf _)).2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeInf \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b : \u03b1\n\u22a2 a \u2293 b \u2264 a\n[PROOFSTEP]\nexact (@OrderDual.semilatticeInf \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeSup).inf_le_left a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeInf \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b : \u03b1\n\u22a2 a \u2293 b \u2264 b\n[PROOFSTEP]\nexact (@OrderDual.semilatticeInf \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeSup).inf_le_right a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeInf \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b c : \u03b1\n\u22a2 a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c\n[PROOFSTEP]\nexact (@OrderDual.semilatticeInf \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeSup).le_inf a b c\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeSup \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b : \u03b1\n\u22a2 a \u2264 a \u2294 b\n[PROOFSTEP]\nexact (@OrderDual.semilatticeSup \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeInf).le_sup_left a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeSup \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b : \u03b1\n\u22a2 b \u2264 a \u2294 b\n[PROOFSTEP]\nexact (@OrderDual.semilatticeSup \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeInf).le_sup_right a b\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na\u271d a\u2081 a\u2082 : \u03b1\nb\u271d b\u2081 b\u2082 : \u03b2\nl : \u03b1 \u2192 \u03b2\nu : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SemilatticeSup \u03b2\ngi : GaloisCoinsertion l u\nsrc\u271d : PartialOrder \u03b1 := inst\u271d\u00b9\na b c : \u03b1\n\u22a2 a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c\n[PROOFSTEP]\nexact (@OrderDual.semilatticeSup \u03b1\u1d52\u1d48 gi.dual.liftSemilatticeInf).sup_le a b c\n", "meta": {"mathlib_filename": "Mathlib.Order.GaloisConnection", "llama_tokens": 9875, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8056321889812553, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.521887090868963}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns t : Set \u03b1\nhs : \u2191\u2191\u03bc s = 0\nht : \u2191\u2191\u03bd t = 0\nhst : univ \u2286 s \u222a t\n\u22a2 \u03bc \u27c2\u2098 \u03bd\n[PROOFSTEP]\nuse toMeasurable \u03bc s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs\n[GOAL]\ncase right\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns t : Set \u03b1\nhs : \u2191\u2191\u03bc s = 0\nht : \u2191\u2191\u03bd t = 0\nhst : univ \u2286 s \u222a t\n\u22a2 \u2191\u2191\u03bd (toMeasurable \u03bc s)\u1d9c = 0\n[PROOFSTEP]\nrefine' measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx _) ht\n[GOAL]\ncase right\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns t : Set \u03b1\nhs : \u2191\u2191\u03bc s = 0\nht : \u2191\u2191\u03bd t = 0\nhst : univ \u2286 s \u222a t\nx : \u03b1\nhx : x \u2208 (toMeasurable \u03bc s)\u1d9c\nhxs : x \u2208 s\n\u22a2 x \u2208 toMeasurable \u03bc s\n[PROOFSTEP]\nexact subset_toMeasurable _ _ hxs\n[GOAL]\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd \u2194 \u2200 (i : \u03b9), \u03bc i \u27c2\u2098 \u03bd\n[PROOFSTEP]\nrefine' \u27e8fun h i => h.mono (le_sum _ _) le_rfl, fun H => _\u27e9\n[GOAL]\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\nH : \u2200 (i : \u03b9), \u03bc i \u27c2\u2098 \u03bd\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd\n[PROOFSTEP]\nchoose s hsm hs\u03bc hs\u03bd using H\n[GOAL]\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u2191\u2191\u03bd (s i)\u1d9c = 0\n\u22a2 sum \u03bc \u27c2\u2098 \u03bd\n[PROOFSTEP]\nrefine' \u27e8\u22c2 i, s i, MeasurableSet.iInter hsm, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u2191\u2191\u03bd (s i)\u1d9c = 0\n\u22a2 \u2191\u2191(sum \u03bc) (\u22c2 (i : \u03b9), s i) = 0\n[PROOFSTEP]\nrw [sum_apply _ (MeasurableSet.iInter hsm), ENNReal.tsum_eq_zero]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u2191\u2191\u03bd (s i)\u1d9c = 0\n\u22a2 \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (\u22c2 (b : \u03b9), s b) = 0\n[PROOFSTEP]\nexact fun i => measure_mono_null (iInter_subset _ _) (hs\u03bc i)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u03b9 : Type u_2\ninst\u271d : Countable \u03b9\n\u03bc : \u03b9 \u2192 Measure \u03b1\ns : \u03b9 \u2192 Set \u03b1\nhsm : \u2200 (i : \u03b9), MeasurableSet (s i)\nhs\u03bc : \u2200 (i : \u03b9), \u2191\u2191(\u03bc i) (s i) = 0\nhs\u03bd : \u2200 (i : \u03b9), \u2191\u2191\u03bd (s i)\u1d9c = 0\n\u22a2 \u2191\u2191\u03bd (\u22c2 (i : \u03b9), s i)\u1d9c = 0\n[PROOFSTEP]\nrwa [compl_iInter, measure_iUnion_null_iff]\n[GOAL]\n\u03b1 : Type u_1\nm0 : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\n\u22a2 \u03bc\u2081 + \u03bc\u2082 \u27c2\u2098 \u03bd \u2194 \u03bc\u2081 \u27c2\u2098 \u03bd \u2227 \u03bc\u2082 \u27c2\u2098 \u03bd\n[PROOFSTEP]\nrw [\u2190 sum_cond, sum_left, Bool.forall_bool, cond, cond, and_comm]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.MutuallySingular", "llama_tokens": 1658, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506418255927, "lm_q2_score": 0.6859494614282923, "lm_q1q2_score": 0.5217678980953502}}
{"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u03b1 \u03b2 : F \u27f6 G\n\u22a2 T.map (\u03b1.f + \u03b2.f) \u226b G.a = F.a \u226b (\u03b1.f + \u03b2.f)\n[PROOFSTEP]\nsimp only [Functor.map_add, add_comp, Monad.Algebra.Hom.h, comp_add]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a b c : F \u27f6 G), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d c\u271d : F \u27f6 G\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d c\u271d : F \u27f6 G\n\u22a2 (a\u271d + b\u271d + c\u271d).f = (a\u271d + (b\u271d + c\u271d)).f\n[PROOFSTEP]\napply add_assoc\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 T.map 0 \u226b G.a = F.a \u226b 0\n[PROOFSTEP]\nsimp only [Functor.map_zero, zero_comp, comp_zero]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a : F \u27f6 G), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 (0 + a\u271d).f = a\u271d.f\n[PROOFSTEP]\napply zero_add\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a : F \u27f6 G), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 (a\u271d + 0).f = a\u271d.f\n[PROOFSTEP]\napply add_zero\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn : \u2115\n\u03b1 : F \u27f6 G\n\u22a2 T.map (n \u2022 \u03b1.f) \u226b G.a = F.a \u226b (n \u2022 \u03b1.f)\n[PROOFSTEP]\nrw [Functor.map_nsmul, nsmul_comp, Monad.Algebra.Hom.h, comp_nsmul]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (x : F \u27f6 G), (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) 0 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nx\u271d : F \u27f6 G\n\u22a2 (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nx\u271d : F \u27f6 G\n\u22a2 ((fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d).f = 0.f\n[PROOFSTEP]\napply zero_smul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (n : \u2115) (x : F \u27f6 G), (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) (n + 1) x = x + (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) n x\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\nx\u271d : F \u27f6 G\n\u22a2 (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) (n\u271d + 1) x\u271d = x\u271d + (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\nx\u271d : F \u27f6 G\n\u22a2 ((fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) (n\u271d + 1) x\u271d).f = (x\u271d + (fun n \u03b1 => Algebra.Hom.mk (n \u2022 \u03b1.f)) n\u271d x\u271d).f\n[PROOFSTEP]\napply succ_nsmul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u03b1 : F \u27f6 G\n\u22a2 T.map (-\u03b1.f) \u226b G.a = F.a \u226b (-\u03b1.f)\n[PROOFSTEP]\nsimp only [Functor.map_neg, neg_comp, Monad.Algebra.Hom.h, comp_neg]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u03b1 \u03b2 : F \u27f6 G\n\u22a2 T.map (\u03b1.f - \u03b2.f) \u226b G.a = F.a \u226b (\u03b1.f - \u03b2.f)\n[PROOFSTEP]\nsimp only [Functor.map_sub, sub_comp, Monad.Algebra.Hom.h, comp_sub]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a b : F \u27f6 G), a - b = a + -b\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d : F \u27f6 G\n\u22a2 (a\u271d - b\u271d).f = (a\u271d + -b\u271d).f\n[PROOFSTEP]\napply sub_eq_add_neg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nr : \u2124\n\u03b1 : F \u27f6 G\n\u22a2 T.map (r \u2022 \u03b1.f) \u226b G.a = F.a \u226b (r \u2022 \u03b1.f)\n[PROOFSTEP]\nrw [Functor.map_zsmul, zsmul_comp, Monad.Algebra.Hom.h, comp_zsmul]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a : F \u27f6 G), (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) 0 a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d).f = 0.f\n[PROOFSTEP]\napply zero_smul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (n : \u2115) (a : F \u27f6 G),\n    (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n)) a =\n      a + (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n\u271d)) a\u271d =\n    a\u271d + (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n\u271d)) a\u271d).f =\n    (a\u271d + (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n\u271d) a\u271d).f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 \u2191(Nat.succ n\u271d) \u2022 a\u271d.f = (a\u271d + Algebra.Hom.mk (\u2191n\u271d \u2022 a\u271d.f)).f\n[PROOFSTEP]\nsimp only [coe_nat_zsmul, succ_nsmul]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 a\u271d.f + n\u271d \u2022 a\u271d.f = (a\u271d + Algebra.Hom.mk (n\u271d \u2022 a\u271d.f)).f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (n : \u2115) (a : F \u27f6 G),\n    (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n) a = -(fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n)) a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n\u271d) a\u271d = -(fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n\u271d)) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n\u271d) a\u271d).f =\n    (-(fun r \u03b1 => Algebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n\u271d)) a\u271d).f\n[PROOFSTEP]\nsimp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast \u2124]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a : F \u27f6 G), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d : F \u27f6 G\n\u22a2 (-a\u271d + a\u271d).f = 0.f\n[PROOFSTEP]\napply add_left_neg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\n\u22a2 \u2200 (a b : F \u27f6 G), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nF G : Algebra T\na\u271d b\u271d : F \u27f6 G\n\u22a2 (a\u271d + b\u271d).f = (b\u271d + a\u271d).f\n[PROOFSTEP]\napply add_comm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\n\u22a2 \u2200 (P Q R : Algebra T) (f f' : P \u27f6 Q) (g : Q \u27f6 R), (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nP\u271d Q\u271d R\u271d : Algebra T\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d + f'\u271d) \u226b g\u271d = f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nP\u271d Q\u271d R\u271d : Algebra T\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 ((f\u271d + f'\u271d) \u226b g\u271d).f = (f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d).f\n[PROOFSTEP]\napply add_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\n\u22a2 \u2200 (P Q R : Algebra T) (f : P \u27f6 Q) (g g' : Q \u27f6 R), f \u226b (g + g') = f \u226b g + f \u226b g'\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nP\u271d Q\u271d R\u271d : Algebra T\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 f\u271d \u226b (g\u271d + g'\u271d) = f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b9 : Preadditive C\nT : Monad C\ninst\u271d : Functor.Additive T.toFunctor\nP\u271d Q\u271d R\u271d : Algebra T\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d \u226b (g\u271d + g'\u271d)).f = (f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d).f\n[PROOFSTEP]\napply comp_add\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u03b1 \u03b2 : F \u27f6 G\n\u22a2 F.a \u226b U.map (\u03b1.f + \u03b2.f) = (\u03b1.f + \u03b2.f) \u226b G.a\n[PROOFSTEP]\nsimp only [Functor.map_add, comp_add, Comonad.Coalgebra.Hom.h, add_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a b c : F \u27f6 G), a + b + c = a + (b + c)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d c\u271d : F \u27f6 G\n\u22a2 a\u271d + b\u271d + c\u271d = a\u271d + (b\u271d + c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d c\u271d : F \u27f6 G\n\u22a2 (a\u271d + b\u271d + c\u271d).f = (a\u271d + (b\u271d + c\u271d)).f\n[PROOFSTEP]\napply add_assoc\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 F.a \u226b U.map 0 = 0 \u226b G.a\n[PROOFSTEP]\nsimp only [Functor.map_zero, comp_zero, zero_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a : F \u27f6 G), 0 + a = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 0 + a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 (0 + a\u271d).f = a\u271d.f\n[PROOFSTEP]\napply zero_add\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a : F \u27f6 G), a + 0 = a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 a\u271d + 0 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 (a\u271d + 0).f = a\u271d.f\n[PROOFSTEP]\napply add_zero\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn : \u2115\n\u03b1 : F \u27f6 G\n\u22a2 F.a \u226b U.map (n \u2022 \u03b1.f) = (n \u2022 \u03b1.f) \u226b G.a\n[PROOFSTEP]\nrw [Functor.map_nsmul, comp_nsmul, Comonad.Coalgebra.Hom.h, nsmul_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (x : F \u27f6 G), (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) 0 x = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nx\u271d : F \u27f6 G\n\u22a2 (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nx\u271d : F \u27f6 G\n\u22a2 ((fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) 0 x\u271d).f = 0.f\n[PROOFSTEP]\napply zero_smul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (n : \u2115) (x : F \u27f6 G),\n    (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) (n + 1) x = x + (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) n x\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\nx\u271d : F \u27f6 G\n\u22a2 (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) (n\u271d + 1) x\u271d = x\u271d + (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\nx\u271d : F \u27f6 G\n\u22a2 ((fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) (n\u271d + 1) x\u271d).f = (x\u271d + (fun n \u03b1 => Coalgebra.Hom.mk (n \u2022 \u03b1.f)) n\u271d x\u271d).f\n[PROOFSTEP]\napply succ_nsmul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u03b1 : F \u27f6 G\n\u22a2 F.a \u226b U.map (-\u03b1.f) = (-\u03b1.f) \u226b G.a\n[PROOFSTEP]\nsimp only [Functor.map_neg, comp_neg, Comonad.Coalgebra.Hom.h, neg_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u03b1 \u03b2 : F \u27f6 G\n\u22a2 F.a \u226b U.map (\u03b1.f - \u03b2.f) = (\u03b1.f - \u03b2.f) \u226b G.a\n[PROOFSTEP]\nsimp only [Functor.map_sub, comp_sub, Comonad.Coalgebra.Hom.h, sub_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a b : F \u27f6 G), a - b = a + -b\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d - b\u271d = a\u271d + -b\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d : F \u27f6 G\n\u22a2 (a\u271d - b\u271d).f = (a\u271d + -b\u271d).f\n[PROOFSTEP]\napply sub_eq_add_neg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nr : \u2124\n\u03b1 : F \u27f6 G\n\u22a2 F.a \u226b U.map (r \u2022 \u03b1.f) = (r \u2022 \u03b1.f) \u226b G.a\n[PROOFSTEP]\nrw [Functor.map_zsmul, comp_zsmul, Comonad.Coalgebra.Hom.h, zsmul_comp]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a : F \u27f6 G), (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) 0 a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) 0 a\u271d).f = 0.f\n[PROOFSTEP]\napply zero_smul\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (n : \u2115) (a : F \u27f6 G),\n    (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n)) a =\n      a + (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n\u271d)) a\u271d =\n    a\u271d + (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat (Nat.succ n\u271d)) a\u271d).f =\n    (a\u271d + (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.ofNat n\u271d) a\u271d).f\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 \u2191(Nat.succ n\u271d) \u2022 a\u271d.f = (a\u271d + Coalgebra.Hom.mk (\u2191n\u271d \u2022 a\u271d.f)).f\n[PROOFSTEP]\nsimp only [coe_nat_zsmul, succ_nsmul]\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 a\u271d.f + n\u271d \u2022 a\u271d.f = (a\u271d + Coalgebra.Hom.mk (n\u271d \u2022 a\u271d.f)).f\n[PROOFSTEP]\nrfl\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (n : \u2115) (a : F \u27f6 G),\n    (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n) a =\n      -(fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n)) a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 (fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n\u271d) a\u271d =\n    -(fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n\u271d)) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\nn\u271d : \u2115\na\u271d : F \u27f6 G\n\u22a2 ((fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (Int.negSucc n\u271d) a\u271d).f =\n    (-(fun r \u03b1 => Coalgebra.Hom.mk (r \u2022 \u03b1.f)) (\u2191(Nat.succ n\u271d)) a\u271d).f\n[PROOFSTEP]\nsimp only [negSucc_zsmul, neg_inj, nsmul_eq_smul_cast \u2124]\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a : F \u27f6 G), -a + a = 0\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 -a\u271d + a\u271d = 0\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d : F \u27f6 G\n\u22a2 (-a\u271d + a\u271d).f = 0.f\n[PROOFSTEP]\napply add_left_neg\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\n\u22a2 \u2200 (a b : F \u27f6 G), a + b = b + a\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d : F \u27f6 G\n\u22a2 a\u271d + b\u271d = b\u271d + a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nF G : Coalgebra U\na\u271d b\u271d : F \u27f6 G\n\u22a2 (a\u271d + b\u271d).f = (b\u271d + a\u271d).f\n[PROOFSTEP]\napply add_comm\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\n\u22a2 \u2200 (P Q R : Coalgebra U) (f f' : P \u27f6 Q) (g : Q \u27f6 R), (f + f') \u226b g = f \u226b g + f' \u226b g\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nP\u271d Q\u271d R\u271d : Coalgebra U\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d + f'\u271d) \u226b g\u271d = f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nP\u271d Q\u271d R\u271d : Coalgebra U\nf\u271d f'\u271d : P\u271d \u27f6 Q\u271d\ng\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 ((f\u271d + f'\u271d) \u226b g\u271d).f = (f\u271d \u226b g\u271d + f'\u271d \u226b g\u271d).f\n[PROOFSTEP]\napply add_comp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\n\u22a2 \u2200 (P Q R : Coalgebra U) (f : P \u27f6 Q) (g g' : Q \u27f6 R), f \u226b (g + g') = f \u226b g + f \u226b g'\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nP\u271d Q\u271d R\u271d : Coalgebra U\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 f\u271d \u226b (g\u271d + g'\u271d) = f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase h\nC : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : Preadditive C\nT : Monad C\ninst\u271d\u00b9 : Functor.Additive T.toFunctor\nU : Comonad C\ninst\u271d : Functor.Additive U.toFunctor\nP\u271d Q\u271d R\u271d : Coalgebra U\nf\u271d : P\u271d \u27f6 Q\u271d\ng\u271d g'\u271d : Q\u271d \u27f6 R\u271d\n\u22a2 (f\u271d \u226b (g\u271d + g'\u271d)).f = (f\u271d \u226b g\u271d + f\u271d \u226b g'\u271d).f\n[PROOFSTEP]\napply comp_add\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.EilenbergMoore", "llama_tokens": 13851, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257127, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.5217218370710884}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 _root_.Disjoint (map Embedding.inl s) (map Embedding.inr t)\n[PROOFSTEP]\nsimp_rw [disjoint_left, mem_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns : Finset \u03b1\nt : Finset \u03b2\n\u22a2 \u2200 \u2983a : \u03b1 \u2295 \u03b2\u2984, (\u2203 a_1, a_1 \u2208 s \u2227 \u2191Embedding.inl a_1 = a) \u2192 \u00ac\u2203 a_2, a_2 \u2208 t \u2227 \u2191Embedding.inr a_2 = a\n[PROOFSTEP]\nrintro x \u27e8a, _, rfl\u27e9 \u27e8b, _, \u27e8\u27e9\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.Sum", "llama_tokens": 216, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7718434873426303, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.5215845113132972}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\n\u22a2 1\u207b = 1\n[PROOFSTEP]\nrw [m_neg_part_def, inv_one, sup_idem]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a\u207b = (a \u2293 1)\u207b\u00b9\n[PROOFSTEP]\nrw [m_neg_part_def, \u2190 inv_inj, inv_sup_eq_inv_inf_inv, inv_inv, inv_inv, inv_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 a\u207b\u00b9 \u2294 a\u207b\u00b9\u207b\u00b9 = a \u2294 a\u207b\u00b9\n[PROOFSTEP]\nrw [inv_inv, sup_comm]\n  -- 0 \u2264 a\u207a\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 a\u207a \u2264 1 \u2194 a \u2264 1\n[PROOFSTEP]\nrw [m_pos_part_def, sup_le_iff, and_iff_left le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 a\u207b \u2264 1 \u2194 a\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrw [m_neg_part_def, sup_le_iff, and_iff_left le_rfl]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 Mul.mul LE.le\na : \u03b1\n\u22a2 a\u207b = 1 \u2194 1 \u2264 a\n[PROOFSTEP]\nrw [le_antisymm_iff, neg_le_one_iff, inv_le_one', and_iff_left (one_le_neg _)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\n\u22a2 a\u207a = a\u207b\u00b9\u207b\n[PROOFSTEP]\nrw [neg_eq_pos_inv, inv_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * (a \u2293 b) = c * a \u2293 c * b\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * (a \u2293 b) \u2264 c * a \u2293 c * b\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * a \u2293 c * b \u2264 c * (a \u2293 b)\n[PROOFSTEP]\nrw [le_inf_iff, mul_le_mul_iff_left, mul_le_mul_iff_left]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 a \u2293 b \u2264 a \u2227 a \u2293 b \u2264 b\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * a \u2293 c * b \u2264 c * (a \u2293 b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * a \u2293 c * b \u2264 c * (a \u2293 b)\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_iff_left c\u207b\u00b9, \u2190 mul_assoc, inv_mul_self, one_mul, le_inf_iff, inv_mul_le_iff_le_mul,\n  inv_mul_le_iff_le_mul]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 c * a \u2293 c * b \u2264 c * a \u2227 c * a \u2293 c * b \u2264 c * b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a\u207a / a\u207b = a\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a = a\u207a / a\u207b\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a = a\u207a * a\u207b\u207b\u00b9\n[PROOFSTEP]\napply eq_mul_inv_of_mul_eq\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a * a\u207b = a\u207a\n[PROOFSTEP]\nrw [m_neg_part_def, mul_sup, mul_one, mul_right_inv, sup_comm, m_pos_part_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\nh : 1 \u2264 a\n\u22a2 a\u207a = a\n[PROOFSTEP]\nrw [m_pos_part_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\nh : 1 \u2264 a\n\u22a2 a \u2294 1 = a\n[PROOFSTEP]\nexact sup_of_le_left h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\nh : 1 \u2264 a\u207b\u00b9\n\u22a2 a\u207b = a\u207b\u00b9\n[PROOFSTEP]\nrw [neg_eq_pos_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na : \u03b1\nh : 1 \u2264 a\u207b\u00b9\n\u22a2 a\u207b\u00b9\u207a = a\u207b\u00b9\n[PROOFSTEP]\nexact pos_of_one_le _ h\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : Group \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommGroup \u03b1\nx : \u03b1\nhx : 1 < x\u207a\n\u22a2 x\u207a = x\n[PROOFSTEP]\nrw [m_pos_part_def, right_lt_sup, not_le] at hx \n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : Group \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommGroup \u03b1\nx : \u03b1\nhx : 1 < x\n\u22a2 x\u207a = x\n[PROOFSTEP]\nrw [m_pos_part_def, sup_eq_left]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : Group \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CommGroup \u03b1\nx : \u03b1\nhx : 1 < x\n\u22a2 1 \u2264 x\n[PROOFSTEP]\nexact hx.le\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na b : \u03b1\n\u22a2 |a / b| = |b / a|\n[PROOFSTEP]\ndsimp only [Abs.abs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Group \u03b1\na b : \u03b1\n\u22a2 a / b \u2294 (a / b)\u207b\u00b9 = b / a \u2294 (b / a)\u207b\u00b9\n[PROOFSTEP]\nrw [inv_div a b, \u2190 inv_inv (a / b), inv_div, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 a ^ 2 \u2192 1 \u2264 a\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\nh : 1 \u2264 a ^ 2\n\u22a2 1 \u2264 a\n[PROOFSTEP]\nhave e1 : (a \u2293 1) * (a \u2293 1) = a \u2293 1 := by\n  rw [mul_inf, inf_mul, \u2190 pow_two, mul_one, one_mul, inf_assoc, inf_left_idem, inf_comm, inf_assoc, (inf_of_le_left h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\nh : 1 \u2264 a ^ 2\n\u22a2 (a \u2293 1) * (a \u2293 1) = a \u2293 1\n[PROOFSTEP]\nrw [mul_inf, inf_mul, \u2190 pow_two, mul_one, one_mul, inf_assoc, inf_left_idem, inf_comm, inf_assoc, (inf_of_le_left h)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\nh : 1 \u2264 a ^ 2\ne1 : (a \u2293 1) * (a \u2293 1) = a \u2293 1\n\u22a2 1 \u2264 a\n[PROOFSTEP]\nrw [\u2190 inf_eq_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\nh : 1 \u2264 a ^ 2\ne1 : (a \u2293 1) * (a \u2293 1) = a \u2293 1\n\u22a2 a \u2293 1 = 1\n[PROOFSTEP]\nexact mul_right_eq_self.mp e1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 |a|\n[PROOFSTEP]\napply pow_two_semiclosed _ _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 |a| ^ 2\n[PROOFSTEP]\nrw [abs_eq_sup_inv, pow_two, mul_sup, sup_mul, \u2190 pow_two, mul_left_inv, sup_comm, \u2190 sup_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 (a \u2294 a\u207b\u00b9) * a\u207b\u00b9 \u2294 a ^ 2 \u2294 1\n[PROOFSTEP]\napply le_sup_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a| = a\u207a * a\u207b\n[PROOFSTEP]\nrw [m_pos_part_def, sup_mul, one_mul, m_neg_part_def, mul_sup, mul_one, mul_inv_self, sup_assoc, \u2190 @sup_assoc _ _ a,\n  sup_eq_right.2 le_sup_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a| = a \u2294 a\u207b\u00b9 \u2294 1\n[PROOFSTEP]\nexact (sup_eq_left.2 <| one_le_abs a).symm\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a|\u207a = |a|\n[PROOFSTEP]\nrw [m_pos_part_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a| \u2294 1 = |a|\n[PROOFSTEP]\napply sup_of_le_left\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 |a|\n[PROOFSTEP]\napply one_le_abs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a|\u207b = 1\n[PROOFSTEP]\nrw [m_neg_part_def]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a|\u207b\u00b9 \u2294 1 = 1\n[PROOFSTEP]\napply sup_of_le_right\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 |a|\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrw [Left.inv_le_one_iff]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 1 \u2264 |a|\n[PROOFSTEP]\napply one_le_abs\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2294 b) / (a \u2293 b) = (a \u2294 b) * (a\u207b\u00b9 \u2294 b\u207b\u00b9)\n[PROOFSTEP]\nrw [div_eq_mul_inv, \u2190 inv_inf_eq_sup_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2294 b) * (a\u207b\u00b9 \u2294 b\u207b\u00b9) = a * a\u207b\u00b9 \u2294 b * a\u207b\u00b9 \u2294 (a * b\u207b\u00b9 \u2294 b * b\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_sup, sup_mul, sup_mul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * a\u207b\u00b9 \u2294 b * a\u207b\u00b9 \u2294 (a * b\u207b\u00b9 \u2294 b * b\u207b\u00b9) = 1 \u2294 b / a \u2294 (a / b \u2294 1)\n[PROOFSTEP]\nrw [mul_right_inv, mul_right_inv, \u2190 div_eq_mul_inv, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 (a / b \u2294 1) = 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1)\n[PROOFSTEP]\nrw [one_div_div]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 (1 / (b / a) \u2294 1) = 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1)\n[PROOFSTEP]\nrw [inv_eq_one_div]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 b / a \u2294 ((b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1)\n[PROOFSTEP]\nrw [sup_assoc, sup_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 (b / a \u2294 (b / a)\u207b\u00b9 \u2294 1) = 1 \u2294 (|b / a| \u2294 1)\n[PROOFSTEP]\nrw [abs_eq_sup_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 (|b / a| \u2294 1) = 1 \u2294 |b / a|\n[PROOFSTEP]\nrw [\u2190 m_pos_part_def, m_pos_abs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 1 \u2294 |b / a| = |b / a| \u2294 1\n[PROOFSTEP]\nrw [sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |b / a| \u2294 1 = |b / a|\n[PROOFSTEP]\nrw [\u2190 m_pos_part_def, m_pos_abs]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na : \u03b1\n\u22a2 a\u207a \u2293 a\u207b = 1\n[PROOFSTEP]\nrw [\u2190 mul_left_inj (a\u207b)\u207b\u00b9, inf_mul, one_mul, mul_right_inv, \u2190 div_eq_mul_inv, pos_div_neg, neg_eq_inv_inf_one, inv_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2293 b) * (a \u2294 b) = (a \u2293 b) * (a * b * (b\u207b\u00b9 \u2294 a\u207b\u00b9))\n[PROOFSTEP]\nrw [mul_sup b\u207b\u00b9 a\u207b\u00b9 (a * b), mul_inv_cancel_right, mul_inv_cancel_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2293 b) * (a * b * (b\u207b\u00b9 \u2294 a\u207b\u00b9)) = (a \u2293 b) * (a * b * (a \u2293 b)\u207b\u00b9)\n[PROOFSTEP]\nrw [inv_inf_eq_sup_inv, sup_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2293 b) * (a * b * (a \u2293 b)\u207b\u00b9) = a * b\n[PROOFSTEP]\nrw [mul_comm, inv_mul_cancel_right]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2294 b = b * (a / b) \u2294 b * 1\n[PROOFSTEP]\nrw [mul_one b, div_eq_mul_inv, mul_comm a, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 b * (a / b) \u2294 b * 1 = b * (a / b \u2294 1)\n[PROOFSTEP]\nrw [\u2190 mul_sup (a / b) 1 b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2293 b = a * 1 \u2293 a * (b / a)\n[PROOFSTEP]\nrw [mul_one a, div_eq_mul_inv, mul_comm b, mul_inv_cancel_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * 1 \u2293 a * (b / a) = a * (1 \u2293 b / a)\n[PROOFSTEP]\nrw [\u2190 mul_inf 1 (b / a) a]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * (1 \u2293 b / a) = a * (b / a \u2293 1)\n[PROOFSTEP]\nrw [inf_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * (b / a \u2293 1) = a * ((a / b)\u207b\u00b9 \u2293 1)\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * (b * a\u207b\u00b9 \u2293 1) = a * ((a / b)\u207b\u00b9 \u2293 1)\n[PROOFSTEP]\nnth_rw 1 [\u2190 inv_inv b]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * (b\u207b\u00b9\u207b\u00b9 * a\u207b\u00b9 \u2293 1) = a * ((a / b)\u207b\u00b9 \u2293 1)\n[PROOFSTEP]\nrw [\u2190 mul_inv, mul_comm b\u207b\u00b9, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * ((a / b)\u207b\u00b9 \u2293 1) = a * ((a / b)\u207b\u00b9 \u2293 1\u207b\u00b9)\n[PROOFSTEP]\nrw [inv_one]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * ((a / b)\u207b\u00b9 \u2293 1\u207b\u00b9) = a / (a / b \u2294 1)\n[PROOFSTEP]\nrw [\u2190 inv_sup_eq_inv_inf_inv, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2264 b \u2194 a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a \u2264 b \u2192 a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b \u2192 a \u2264 b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a\u207a \u2264 b\u207a\n[PROOFSTEP]\nexact sup_le (h.trans (m_le_pos b)) (one_le_pos b)\n[GOAL]\ncase mp.right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a \u2264 b\n\u22a2 b\u207b \u2264 a\u207b\n[PROOFSTEP]\nrw [\u2190 inv_le_inv_iff] at h \n[GOAL]\ncase mp.right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh\u271d : a \u2264 b\nh : b\u207b\u00b9 \u2264 a\u207b\u00b9\n\u22a2 b\u207b \u2264 a\u207b\n[PROOFSTEP]\nexact sup_le (h.trans (inv_le_neg a)) (one_le_neg a)\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 pos_div_neg a, \u2190 pos_div_neg b]\n[GOAL]\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a\u207a \u2264 b\u207a \u2227 b\u207b \u2264 a\u207b\n\u22a2 a\u207a / a\u207b \u2264 b\u207a / b\u207b\n[PROOFSTEP]\nexact div_le_div'' h.1 h.2\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2294 b) ^ 2 = a * b * |b / a|\n[PROOFSTEP]\nrw [\u2190 inf_mul_sup a b, \u2190 sup_div_inf_eq_abs_div, div_eq_mul_inv, \u2190 mul_assoc, mul_comm, mul_assoc, \u2190 pow_two,\n  inv_mul_cancel_left]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a \u2293 b) ^ 2 = a * b / |b / a|\n[PROOFSTEP]\nrw [\u2190 inf_mul_sup a b, \u2190 sup_div_inf_eq_abs_div, div_eq_mul_inv, div_eq_mul_inv, mul_inv_rev, inv_inv, mul_assoc,\n  mul_inv_cancel_comm_assoc, \u2190 pow_two]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\n\u22a2 \u2200 (x y z : \u03b1), (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y \u2293 z\n[PROOFSTEP]\nintros x y z\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y \u2293 z\n[PROOFSTEP]\nrw [\u2190 mul_le_mul_iff_left (x \u2293 (y \u2293 z)), inf_mul_sup x (y \u2293 z), \u2190 inv_mul_le_iff_le_mul, le_inf_iff]\n[GOAL]\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 x\u207b\u00b9 * ((x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z))) \u2264 y \u2227 x\u207b\u00b9 * ((x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z))) \u2264 z\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 x\u207b\u00b9 * ((x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z))) \u2264 y\n[PROOFSTEP]\nrw [inv_mul_le_iff_le_mul, \u2190 inf_mul_sup x y]\n[GOAL]\ncase left\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 (x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z)) \u2264 (x \u2293 y) * (x \u2294 y)\n[PROOFSTEP]\napply mul_le_mul'\n[GOAL]\ncase left.h\u2081\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 x \u2293 (y \u2293 z) \u2264 x \u2293 y\n[PROOFSTEP]\napply inf_le_inf_left\n[GOAL]\ncase left.h\u2081.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 y \u2293 z \u2264 y\n[PROOFSTEP]\napply inf_le_left\n[GOAL]\ncase left.h\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 y\n[PROOFSTEP]\napply inf_le_left\n[GOAL]\ncase right\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 x\u207b\u00b9 * ((x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z))) \u2264 z\n[PROOFSTEP]\nrw [inv_mul_le_iff_le_mul, \u2190 inf_mul_sup x z]\n[GOAL]\ncase right\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 (x \u2293 (y \u2293 z)) * ((x \u2294 y) \u2293 (x \u2294 z)) \u2264 (x \u2293 z) * (x \u2294 z)\n[PROOFSTEP]\napply mul_le_mul'\n[GOAL]\ncase right.h\u2081\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 x \u2293 (y \u2293 z) \u2264 x \u2293 z\n[PROOFSTEP]\napply inf_le_inf_left\n[GOAL]\ncase right.h\u2081.h\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 y \u2293 z \u2264 z\n[PROOFSTEP]\napply inf_le_right\n[GOAL]\ncase right.h\u2082\n\u03b1\u271d : Type u\n\u03b2 : Type v\ninst\u271d\u00b3 : Lattice \u03b1\u271d\ninst\u271d\u00b2 : CommGroup \u03b1\u271d\n\u03b1 : Type u\ns : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx y z : \u03b1\n\u22a2 (x \u2294 y) \u2293 (x \u2294 z) \u2264 x \u2294 z\n[PROOFSTEP]\napply inf_le_right\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 |(a \u2294 c) / (b \u2294 c)| * |(a \u2293 c) / (b \u2293 c)| = |a / b|\n[PROOFSTEP]\nletI : DistribLattice \u03b1 := LatticeOrderedCommGroup.latticeOrderedCommGroupToDistribLattice \u03b1\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 |(a \u2294 c) / (b \u2294 c)| * |(a \u2293 c) / (b \u2293 c)| = |a / b|\n[PROOFSTEP]\ncalc\n  |(a \u2294 c) / (b \u2294 c)| * |(a \u2293 c) / (b \u2293 c)| = (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * |(a \u2293 c) / (b \u2293 c)| := by\n    rw [sup_div_inf_eq_abs_div]\n  _ = (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * ((b \u2293 c \u2294 a \u2293 c) / (b \u2293 c \u2293 (a \u2293 c))) := by\n    rw [sup_div_inf_eq_abs_div (b \u2293 c) (a \u2293 c)]\n  _ = (b \u2294 a \u2294 c) / (b \u2293 a \u2294 c) * (((b \u2294 a) \u2293 c) / (b \u2293 a \u2293 c)) := by\n    rw [\u2190 sup_inf_right, \u2190 inf_sup_right, sup_assoc, @sup_comm _ _ c (a \u2294 c), sup_right_idem, sup_assoc, inf_assoc,\n      @inf_comm _ _ c (a \u2293 c), inf_right_idem, inf_assoc]\n  _ = (b \u2294 a \u2294 c) * ((b \u2294 a) \u2293 c) / ((b \u2293 a \u2294 c) * (b \u2293 a \u2293 c)) := by rw [div_mul_div_comm]\n  _ = (b \u2294 a) * c / ((b \u2293 a) * c) := by rw [mul_comm, inf_mul_sup, mul_comm (b \u2293 a \u2294 c), inf_mul_sup]\n  _ = (b \u2294 a) / (b \u2293 a) := by rw [div_eq_mul_inv, mul_inv_rev, mul_assoc, mul_inv_cancel_left, \u2190 div_eq_mul_inv]\n  _ = |a / b| := by rw [sup_div_inf_eq_abs_div]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 |(a \u2294 c) / (b \u2294 c)| * |(a \u2293 c) / (b \u2293 c)| = (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * |(a \u2293 c) / (b \u2293 c)|\n[PROOFSTEP]\nrw [sup_div_inf_eq_abs_div]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * |(a \u2293 c) / (b \u2293 c)| =\n    (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * ((b \u2293 c \u2294 a \u2293 c) / (b \u2293 c \u2293 (a \u2293 c)))\n[PROOFSTEP]\nrw [sup_div_inf_eq_abs_div (b \u2293 c) (a \u2293 c)]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 c \u2294 (a \u2294 c)) / ((b \u2294 c) \u2293 (a \u2294 c)) * ((b \u2293 c \u2294 a \u2293 c) / (b \u2293 c \u2293 (a \u2293 c))) =\n    (b \u2294 a \u2294 c) / (b \u2293 a \u2294 c) * (((b \u2294 a) \u2293 c) / (b \u2293 a \u2293 c))\n[PROOFSTEP]\nrw [\u2190 sup_inf_right, \u2190 inf_sup_right, sup_assoc, @sup_comm _ _ c (a \u2294 c), sup_right_idem, sup_assoc, inf_assoc,\n  @inf_comm _ _ c (a \u2293 c), inf_right_idem, inf_assoc]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 a \u2294 c) / (b \u2293 a \u2294 c) * (((b \u2294 a) \u2293 c) / (b \u2293 a \u2293 c)) = (b \u2294 a \u2294 c) * ((b \u2294 a) \u2293 c) / ((b \u2293 a \u2294 c) * (b \u2293 a \u2293 c))\n[PROOFSTEP]\nrw [div_mul_div_comm]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 a \u2294 c) * ((b \u2294 a) \u2293 c) / ((b \u2293 a \u2294 c) * (b \u2293 a \u2293 c)) = (b \u2294 a) * c / ((b \u2293 a) * c)\n[PROOFSTEP]\nrw [mul_comm, inf_mul_sup, mul_comm (b \u2293 a \u2294 c), inf_mul_sup]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 a) * c / ((b \u2293 a) * c) = (b \u2294 a) / (b \u2293 a)\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_inv_rev, mul_assoc, mul_inv_cancel_left, \u2190 div_eq_mul_inv]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nthis : DistribLattice \u03b1 := latticeOrderedCommGroupToDistribLattice \u03b1\n\u22a2 (b \u2294 a) / (b \u2293 a) = |a / b|\n[PROOFSTEP]\nrw [sup_div_inf_eq_abs_div]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 |(a \u2294 c) / (b \u2294 c)| \u2264 |a / b|\n[PROOFSTEP]\napply le_of_mul_le_of_one_le_left\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 |(a \u2294 c) / (b \u2294 c)| * ?b \u2264 |a / b|\n[PROOFSTEP]\nrw [abs_div_sup_mul_abs_div_inf]\n[GOAL]\ncase hle\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 1 \u2264 |(a \u2293 c) / (b \u2293 c)|\n[PROOFSTEP]\nexact one_le_abs _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 |(a \u2293 c) / (b \u2293 c)| \u2264 |a / b|\n[PROOFSTEP]\napply le_of_mul_le_of_one_le_right\n[GOAL]\ncase h\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 ?a * |(a \u2293 c) / (b \u2293 c)| \u2264 |a / b|\n[PROOFSTEP]\nrw [abs_div_sup_mul_abs_div_inf]\n[GOAL]\ncase hle\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 1 \u2264 |(a \u2294 c) / (b \u2294 c)|\n[PROOFSTEP]\nexact one_le_abs _\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a * b| \u2264 |a| * |b|\n[PROOFSTEP]\napply sup_le\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a * b \u2264 |a| * |b|\n[PROOFSTEP]\nexact mul_le_mul' (le_mabs a) (le_mabs b)\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (a * b)\u207b\u00b9 \u2264 |a| * |b|\n[PROOFSTEP]\nrw [mul_inv]\n[GOAL]\ncase a\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a\u207b\u00b9 * b\u207b\u00b9 \u2264 |a| * |b|\n[PROOFSTEP]\nexact mul_le_mul' (inv_le_abs _) (inv_le_abs _)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 ||a| / |b|| \u2264 |a / b|\n[PROOFSTEP]\nrw [abs_eq_sup_inv, sup_le_iff]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b| \u2227 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| / |b| \u2264 |a / b|\n[PROOFSTEP]\napply div_le_iff_le_mul.2\n[GOAL]\ncase left\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |a| \u2264 |a / b| * |b|\n[PROOFSTEP]\nconvert mabs_mul_le (a / b) b\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a = a / b * b\n[PROOFSTEP]\nrw [div_mul_cancel']\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 (|a| / |b|)\u207b\u00b9 \u2264 |a / b|\n[PROOFSTEP]\nrw [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_le_iff_le_mul, abs_div_comm]\n[GOAL]\ncase right\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 |b| \u2264 |b / a| * |a|\n[PROOFSTEP]\nconvert mabs_mul_le (b / a) a\n[GOAL]\ncase h.e'_3.h.e'_3\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b2 : Lattice \u03b1\ninst\u271d\u00b9 : CommGroup \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 b = b / a * a\n[PROOFSTEP]\nrw [div_mul_cancel']\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : Semiring \u03b1\ninst\u271d\u2074 : Invertible 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\n\u22a2 x \u2293 y = \u215f2 \u2022 (x + y - |y - x|)\n[PROOFSTEP]\nrw [\u2190 LatticeOrderedCommGroup.two_inf_eq_add_sub_abs_sub x y, two_smul, \u2190 two_smul \u03b1, smul_smul, invOf_mul_self,\n  one_smul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : Semiring \u03b1\ninst\u271d\u2074 : Invertible 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\n\u22a2 x \u2294 y = \u215f2 \u2022 (x + y + |y - x|)\n[PROOFSTEP]\nrw [\u2190 LatticeOrderedCommGroup.two_sup_eq_add_add_abs_sub x y, two_smul, \u2190 two_smul \u03b1, smul_smul, invOf_mul_self,\n  one_smul]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : DivisionSemiring \u03b1\ninst\u271d\u2074 : NeZero 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\n\u22a2 x \u2293 y = 2\u207b\u00b9 \u2022 (x + y - |y - x|)\n[PROOFSTEP]\nletI := invertibleOfNonzero (two_ne_zero' \u03b1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : DivisionSemiring \u03b1\ninst\u271d\u2074 : NeZero 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\nthis : Invertible 2 := invertibleOfNonzero (_ : 2 \u2260 0)\n\u22a2 x \u2293 y = 2\u207b\u00b9 \u2022 (x + y - |y - x|)\n[PROOFSTEP]\nexact inf_eq_half_smul_add_sub_abs_sub \u03b1 x y\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : DivisionSemiring \u03b1\ninst\u271d\u2074 : NeZero 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\n\u22a2 x \u2294 y = 2\u207b\u00b9 \u2022 (x + y + |y - x|)\n[PROOFSTEP]\nletI := invertibleOfNonzero (two_ne_zero' \u03b1)\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u2077 : Lattice \u03b1\ninst\u271d\u2076 : CommGroup \u03b1\ninst\u271d\u2075 : DivisionSemiring \u03b1\ninst\u271d\u2074 : NeZero 2\ninst\u271d\u00b3 : Lattice \u03b2\ninst\u271d\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : CovariantClass \u03b2 \u03b2 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nx y : \u03b2\nthis : Invertible 2 := invertibleOfNonzero (_ : 2 \u2260 0)\n\u22a2 x \u2294 y = 2\u207b\u00b9 \u2022 (x + y + |y - x|)\n[PROOFSTEP]\nexact sup_eq_half_smul_add_add_abs_sub \u03b1 x y\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.LatticeGroup", "llama_tokens": 19134, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.5210722958510333}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\n\u22a2 \u2200 (a b c : \u03a3\u2097' (i : \u03b9), \u03b1 i), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 \u27e8a\u2083, b\u2083\u27e9 \u27e8h\u2081r\u27e9 \u27e8h\u2082r\u27e9\n[GOAL]\ncase mk.mk.mk.left.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 { fst := a\u2081, snd := b\u2081 } \u2264 { fst := a\u2083, snd := b\u2083 }\n[PROOFSTEP]\nleft\n[GOAL]\ncase mk.mk.mk.left.left.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 a\u2081 < a\u2083\n[PROOFSTEP]\napply lt_trans\n[GOAL]\ncase mk.mk.mk.left.left.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 a\u2081 < ?mk.mk.mk.left.left.a.b\ncase mk.mk.mk.left.left.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 ?mk.mk.mk.left.left.a.b < a\u2083\ncase mk.mk.mk.left.left.a.b\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 \u03b9\n[PROOFSTEP]\nrepeat' assumption\n[GOAL]\ncase mk.mk.mk.left.left.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 a\u2081 < ?mk.mk.mk.left.left.a.b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.mk.left.left.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\na\u271d\u00b9 : a\u2081 < a\u2082\na\u271d : a\u2082 < a\u2083\n\u22a2 a\u2082 < a\u2083\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.mk.left.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u271d\u00b9 : a\u2081 < a\u2082\nb\u2082\u271d : \u03b1 a\u2082\na\u271d : b\u2082 \u2264 b\u2082\u271d\n\u22a2 { fst := a\u2081, snd := b\u2081 } \u2264 { fst := a\u2082, snd := b\u2082\u271d }\n[PROOFSTEP]\nleft\n[GOAL]\ncase mk.mk.mk.left.right.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\na\u271d\u00b9 : a\u2081 < a\u2082\nb\u2082\u271d : \u03b1 a\u2082\na\u271d : b\u2082 \u2264 b\u2082\u271d\n\u22a2 a\u2081 < a\u2082\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.mk.right.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\nb\u2082\u271d : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\na\u271d : a\u2081 < a\u2083\n\u22a2 { fst := a\u2081, snd := b\u2081 } \u2264 { fst := a\u2083, snd := b\u2083 }\n[PROOFSTEP]\nleft\n[GOAL]\ncase mk.mk.mk.right.left.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2083 : \u03b9\nb\u2083 : \u03b1 a\u2083\nb\u2082\u271d : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\na\u271d : a\u2081 < a\u2083\n\u22a2 a\u2081 < a\u2083\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.mk.right.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 { fst := a\u2081, snd := b\u2081 } \u2264 { fst := a\u2081, snd := b\u2082\u271d }\n[PROOFSTEP]\nright\n[GOAL]\ncase mk.mk.mk.right.right.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 b\u2081 \u2264 b\u2082\u271d\n[PROOFSTEP]\napply le_trans\n[GOAL]\ncase mk.mk.mk.right.right.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 b\u2081 \u2264 ?mk.mk.mk.right.right.a.b\ncase mk.mk.mk.right.right.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 ?mk.mk.mk.right.right.a.b \u2264 b\u2082\u271d\ncase mk.mk.mk.right.right.a.b\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 (fun i => \u03b1 i) a\u2081\n[PROOFSTEP]\nrepeat' assumption\n[GOAL]\ncase mk.mk.mk.right.right.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 b\u2081 \u2264 ?mk.mk.mk.right.right.a.b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mk.mk.mk.right.right.a.a\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d\u00b9 : \u03b1 a\u2081\na\u271d\u00b9 : b\u2081 \u2264 b\u2082\u271d\u00b9\nb\u2082\u271d : \u03b1 a\u2081\na\u271d : b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n\u22a2 b\u2082\u271d\u00b9 \u2264 b\u2082\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\n\u22a2 \u2200 (a b : \u03a3\u2097' (i : \u03b9), \u03b1 i), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nrefine' fun a b => \u27e8fun hab => \u27e8hab.mono_right fun i a b => le_of_lt, _\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na b : \u03a3\u2097' (i : \u03b9), \u03b1 i\nhab : a < b\n\u22a2 \u00acb \u2264 a\n[PROOFSTEP]\nrintro (\u27e8i, a, hji\u27e9 | \u27e8i, hba\u27e9)\n[GOAL]\ncase refine'_1.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081\u271d : \u03b9\ni : \u03b1 a\u2081\u271d\na\u2082\u271d : \u03b9\na : \u03b1 a\u2082\u271d\nhji : a\u2081\u271d < a\u2082\u271d\nhab : { fst := a\u2082\u271d, snd := a } < { fst := a\u2081\u271d, snd := i }\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8_, _, hij\u27e9 | \u27e8_, hab\u27e9 := hab\n[GOAL]\ncase refine'_1.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhba : b\u2081\u271d \u2264 b\u2082\u271d\nhab : { fst := i, snd := b\u2082\u271d } < { fst := i, snd := b\u2081\u271d }\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8_, _, hij\u27e9 | \u27e8_, hab\u27e9 := hab\n[GOAL]\ncase refine'_1.left.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081\u271d : \u03b9\ni : \u03b1 a\u2081\u271d\na\u2082\u271d : \u03b9\na : \u03b1 a\u2082\u271d\nhji : a\u2081\u271d < a\u2082\u271d\nhij : a\u2082\u271d < a\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hij.not_lt hji\n[GOAL]\ncase refine'_1.left.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081\u271d : \u03b9\ni a : \u03b1 a\u2081\u271d\nhji : a\u2081\u271d < a\u2081\u271d\nhab : a < i\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ hji\n[GOAL]\ncase refine'_1.right.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhba : b\u2081\u271d \u2264 b\u2082\u271d\nhij : i < i\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ hij\n[GOAL]\ncase refine'_1.right.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhba : b\u2081\u271d \u2264 b\u2082\u271d\nhab : b\u2082\u271d < b\u2081\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hab.not_le hba\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na b : \u03a3\u2097' (i : \u03b9), \u03b1 i\n\u22a2 a \u2264 b \u2227 \u00acb \u2264 a \u2192 a < b\n[PROOFSTEP]\nrintro \u27e8\u27e8j, b, hij\u27e9 | \u27e8i, hab\u27e9, hba\u27e9\n[GOAL]\ncase refine'_2.intro.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\na\u2081\u271d : \u03b9\nj : \u03b1 a\u2081\u271d\na\u2082\u271d : \u03b9\nb : \u03b1 a\u2082\u271d\nhij : a\u2081\u271d < a\u2082\u271d\nhba : \u00ac{ fst := a\u2082\u271d, snd := b } \u2264 { fst := a\u2081\u271d, snd := j }\n\u22a2 { fst := a\u2081\u271d, snd := j } < { fst := a\u2082\u271d, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ hij\n[GOAL]\ncase refine'_2.intro.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : Preorder \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nsrc\u271d\u00b9 : LE (\u03a3\u2097' (i : \u03b9), \u03b1 i) := le\nsrc\u271d : LT (\u03a3\u2097' (i : \u03b9), \u03b1 i) := lt\ni : \u03b9\nb\u2081\u271d b\u2082\u271d : \u03b1 i\nhab : b\u2081\u271d \u2264 b\u2082\u271d\nhba : \u00ac{ fst := i, snd := b\u2082\u271d } \u2264 { fst := i, snd := b\u2081\u271d }\n\u22a2 { fst := i, snd := b\u2081\u271d } < { fst := i, snd := b\u2082\u271d }\n[PROOFSTEP]\nexact Lex.right _ (hab.lt_of_not_le fun h => hba <| Lex.right _ h)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nsrc\u271d : Preorder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := preorder\n\u22a2 \u2200 (a b : \u03a3\u2097' (i : \u03b9), \u03b1 i), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nrintro \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 (\u27e8_, _, hlt\u2081\u27e9 | \u27e8_, hlt\u2081\u27e9) (\u27e8_, _, hlt\u2082\u27e9 | \u27e8_, hlt\u2082\u27e9)\n[GOAL]\ncase mk.mk.left.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nsrc\u271d : Preorder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := preorder\na\u2081 : \u03b9\nb\u2081 : \u03b1 a\u2081\na\u2082 : \u03b9\nb\u2082 : \u03b1 a\u2082\nhlt\u2081 : a\u2081 < a\u2082\nhlt\u2082 : a\u2082 < a\u2081\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2082, snd := b\u2082 }\n[PROOFSTEP]\nexact (lt_irrefl a\u2081 <| hlt\u2081.trans hlt\u2082).elim\n[GOAL]\ncase mk.mk.left.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nsrc\u271d : Preorder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := preorder\na\u2081 : \u03b9\nb\u2081 b\u2082 : \u03b1 a\u2081\nhlt\u2081 : a\u2081 < a\u2081\nhlt\u2082 : b\u2082 \u2264 b\u2081\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082 }\n[PROOFSTEP]\nexact (lt_irrefl a\u2081 hlt\u2081).elim\n[GOAL]\ncase mk.mk.right.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nsrc\u271d : Preorder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := preorder\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d : \u03b1 a\u2081\nhlt\u2081 : b\u2081 \u2264 b\u2082\u271d\nhlt\u2082 : a\u2081 < a\u2081\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082\u271d }\n[PROOFSTEP]\nexact (lt_irrefl a\u2081 hlt\u2082).elim\n[GOAL]\ncase mk.mk.right.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : PartialOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 PartialOrder (\u03b1 i)\nsrc\u271d : Preorder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := preorder\na\u2081 : \u03b9\nb\u2081 b\u2082\u271d : \u03b1 a\u2081\nhlt\u2081 : b\u2081 \u2264 b\u2082\u271d\nhlt\u2082 : b\u2082\u271d \u2264 b\u2081\n\u22a2 { fst := a\u2081, snd := b\u2081 } = { fst := a\u2081, snd := b\u2082\u271d }\n[PROOFSTEP]\nrw [hlt\u2081.antisymm hlt\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\n\u22a2 \u2200 (a b : \u03a3\u2097' (i : \u03b9), \u03b1 i), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\n\u22a2 { fst := i, snd := a } \u2264 { fst := j, snd := b } \u2228 { fst := j, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nobtain hij | rfl | hji := lt_trichotomy i j\n[GOAL]\ncase mk.mk.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhij : i < j\n\u22a2 { fst := i, snd := a } \u2264 { fst := j, snd := b } \u2228 { fst := j, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.left _ _ hij)\n[GOAL]\ncase mk.mk.inr.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na b : \u03b1 i\n\u22a2 { fst := i, snd := a } \u2264 { fst := i, snd := b } \u2228 { fst := i, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nobtain hab | hba := le_total a b\n[GOAL]\ncase mk.mk.inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na b : \u03b1 i\nhab : a \u2264 b\n\u22a2 { fst := i, snd := a } \u2264 { fst := i, snd := b } \u2228 { fst := i, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inl (Lex.right _ hab)\n[GOAL]\ncase mk.mk.inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na b : \u03b1 i\nhba : b \u2264 a\n\u22a2 { fst := i, snd := a } \u2264 { fst := i, snd := b } \u2228 { fst := i, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Lex.right _ hba)\n[GOAL]\ncase mk.mk.inr.inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b9\ninst\u271d : (i : \u03b9) \u2192 LinearOrder (\u03b1 i)\nsrc\u271d : PartialOrder (\u03a3\u2097' (i : \u03b9), \u03b1 i) := partialOrder\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nhji : j < i\n\u22a2 { fst := i, snd := a } \u2264 { fst := j, snd := b } \u2228 { fst := j, snd := b } \u2264 { fst := i, snd := a }\n[PROOFSTEP]\nexact Or.inr (Lex.left _ _ hji)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderBot \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderBot (\u03b1 \u22a5)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\na : \u03b9\nb : \u03b1 a\n\u22a2 \u22a5 \u2264 { fst := a, snd := b }\n[PROOFSTEP]\nobtain rfl | ha := eq_bot_or_bot_lt a\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderBot \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderBot (\u03b1 \u22a5)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\nb : \u03b1 \u22a5\n\u22a2 \u22a5 \u2264 { fst := \u22a5, snd := b }\n[PROOFSTEP]\nexact Lex.right _ bot_le\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderBot \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderBot (\u03b1 \u22a5)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\na : \u03b9\nb : \u03b1 a\nha : \u22a5 < a\n\u22a2 \u22a5 \u2264 { fst := a, snd := b }\n[PROOFSTEP]\nexact Lex.left _ _ ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderTop \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderTop (\u03b1 \u22a4)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\na : \u03b9\nb : \u03b1 a\n\u22a2 { fst := a, snd := b } \u2264 \u22a4\n[PROOFSTEP]\nobtain rfl | ha := eq_top_or_lt_top a\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderTop \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderTop (\u03b1 \u22a4)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\nb : \u03b1 \u22a4\n\u22a2 { fst := \u22a4, snd := b } \u2264 \u22a4\n[PROOFSTEP]\nexact Lex.right _ le_top\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b9\ninst\u271d\u00b2 : OrderTop \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : OrderTop (\u03b1 \u22a4)\nx\u271d : \u03a3\u2097' (i : \u03b9), \u03b1 i\na : \u03b9\nb : \u03b1 a\nha : a < \u22a4\n\u22a2 { fst := a, snd := b } \u2264 \u22a4\n[PROOFSTEP]\nexact Lex.left _ _ ha\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03a3\u2097' (i : \u03b9), \u03b1 i), a\u2081 < a\u2082 \u2192 \u2203 a, a\u2081 < a \u2227 a < a\u2082\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9 (\u27e8_, _, h\u27e9 | @\u27e8_, _, b, h\u27e9)\n[GOAL]\ncase mk.mk.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nobtain \u27e8k, hi, hj\u27e9 := exists_between h\n[GOAL]\ncase mk.mk.left.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\nk : \u03b9\nhi : i < k\nhj : k < j\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nobtain \u27e8c\u27e9 : Nonempty (\u03b1 k) := inferInstance\n[GOAL]\ncase mk.mk.left.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\nk : \u03b9\nhi : i < k\nhj : k < j\nc : \u03b1 k\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8k, c\u27e9, left _ _ hi, left _ _ hj\u27e9\n[GOAL]\ncase mk.mk.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between h\n[GOAL]\ncase mk.mk.right.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Preorder \u03b9\ninst\u271d\u00b3 : DenselyOrdered \u03b9\ninst\u271d\u00b2 : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\nc : \u03b1 i\nha : a < c\nhb : c < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8i, c\u27e9, right _ ha, right _ hb\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03a3\u2097' (i : \u03b9), \u03b1 i), a\u2081 < a\u2082 \u2192 \u2203 a, a\u2081 < a \u2227 a < a\u2082\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9 (\u27e8_, _, h\u27e9 | @\u27e8_, _, b, h\u27e9)\n[GOAL]\ncase mk.mk.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nobtain \u27e8c, ha\u27e9 := exists_gt a\n[GOAL]\ncase mk.mk.left.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\nc : \u03b1 i\nha : a < c\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8i, c\u27e9, right _ ha, left _ _ h\u27e9\n[GOAL]\ncase mk.mk.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between h\n[GOAL]\ncase mk.mk.right.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\nc : \u03b1 i\nha : a < c\nhb : c < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8i, c\u27e9, right _ ha, right _ hb\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\n\u22a2 \u2200 (a\u2081 a\u2082 : \u03a3\u2097' (i : \u03b9), \u03b1 i), a\u2081 < a\u2082 \u2192 \u2203 a, a\u2081 < a \u2227 a < a\u2082\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9 (\u27e8_, _, h\u27e9 | @\u27e8_, _, b, h\u27e9)\n[GOAL]\ncase mk.mk.left\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nobtain \u27e8c, hb\u27e9 := exists_lt b\n[GOAL]\ncase mk.mk.left.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nb : \u03b1 j\nh : i < j\nc : \u03b1 j\nhb : c < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := j, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8j, c\u27e9, left _ _ h, right _ hb\u27e9\n[GOAL]\ncase mk.mk.right\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between h\n[GOAL]\ncase mk.mk.right.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\nc : \u03b1 i\nha : a < c\nhb : c < b\n\u22a2 \u2203 a_1, { fst := i, snd := a } < a_1 \u2227 a_1 < { fst := i, snd := b }\n[PROOFSTEP]\nexact \u27e8\u27e8i, c\u27e9, right _ ha, right _ hb\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 \u2200 (a : \u03a3\u2097' (i : \u03b9), \u03b1 i), \u2203 b, a < b\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\n\u22a2 \u2203 b, { fst := i, snd := a } < b\n[PROOFSTEP]\nobtain \u27e8j, h\u27e9 := exists_gt i\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nh : i < j\n\u22a2 \u2203 b, { fst := i, snd := a } < b\n[PROOFSTEP]\nobtain \u27e8b\u27e9 : Nonempty (\u03b1 j) := inferInstance\n[GOAL]\ncase mk.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMaxOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nh : i < j\nb : \u03b1 j\n\u22a2 \u2203 b, { fst := i, snd := a } < b\n[PROOFSTEP]\nexact \u27e8\u27e8j, b\u27e9, left _ _ h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\n\u22a2 \u2200 (a : \u03a3\u2097' (i : \u03b9), \u03b1 i), \u2203 b, b < a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\n\u22a2 \u2203 b, b < { fst := i, snd := a }\n[PROOFSTEP]\nobtain \u27e8j, h\u27e9 := exists_lt i\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nh : j < i\n\u22a2 \u2203 b, b < { fst := i, snd := a }\n[PROOFSTEP]\nobtain \u27e8b\u27e9 : Nonempty (\u03b1 j) := inferInstance\n[GOAL]\ncase mk.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b3 : Preorder \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d\u00b9 : NoMinOrder \u03b9\ninst\u271d : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\ni : \u03b9\na : \u03b1 i\nj : \u03b9\nh : j < i\nb : \u03b1 j\n\u22a2 \u2203 b, b < { fst := i, snd := a }\n[PROOFSTEP]\nexact \u27e8\u27e8j, b\u27e9, left _ _ h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\n\u22a2 \u2200 (a : \u03a3\u2097' (i : \u03b9), \u03b1 i), \u2203 b, a < b\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\n\u22a2 \u2203 b, { fst := i, snd := a } < b\n[PROOFSTEP]\nobtain \u27e8b, h\u27e9 := exists_gt a\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMaxOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : a < b\n\u22a2 \u2203 b, { fst := i, snd := a } < b\n[PROOFSTEP]\nexact \u27e8\u27e8i, b\u27e9, right _ h\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\n\u22a2 \u2200 (a : \u03a3\u2097' (i : \u03b9), \u03b1 i), \u2203 b, b < a\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na : \u03b1 i\n\u22a2 \u2203 b, b < { fst := i, snd := a }\n[PROOFSTEP]\nobtain \u27e8b, h\u27e9 := exists_lt a\n[GOAL]\ncase mk.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : Preorder \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), NoMinOrder (\u03b1 i)\ni : \u03b9\na b : \u03b1 i\nh : b < a\n\u22a2 \u2203 b, b < { fst := i, snd := a }\n[PROOFSTEP]\nexact \u27e8\u27e8i, b\u27e9, right _ h\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.PSigma.Order", "llama_tokens": 14140, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5209052415985508}}
{"text": "[GOAL]\na : \u2124\n\u22a2 J(a | 0) = 1\n[PROOFSTEP]\nsimp only [jacobiSym, factors_zero, List.prod_nil, List.pmap]\n[GOAL]\na : \u2124\n\u22a2 J(a | 1) = 1\n[PROOFSTEP]\nsimp only [jacobiSym, factors_one, List.prod_nil, List.pmap]\n[GOAL]\np : \u2115\nfp : Fact (Nat.Prime p)\na : \u2124\n\u22a2 legendreSym p a = J(a | p)\n[PROOFSTEP]\nsimp only [jacobiSym, factors_prime fp.1, List.prod_cons, List.prod_nil, mul_one, List.pmap]\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 J(a | b\u2081 * b\u2082) = J(a | b\u2081) * J(a | b\u2082)\n[PROOFSTEP]\nrw [jacobiSym, ((perm_factors_mul hb\u2081 hb\u2082).pmap _).prod_eq, List.pmap_append, List.prod_append]\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2081) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2081 \u2192 Nat.Prime a)) *\n      List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2082) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2082 \u2192 Nat.Prime a)) =\n    J(a | b\u2081) * J(a | b\u2082)\ncase h\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 \u2200 (a : \u2115), a \u2208 factors b\u2081 ++ factors b\u2082 \u2192 Nat.Prime a\ncase h a : \u2124 b\u2081 b\u2082 : \u2115 hb\u2081 : b\u2081 \u2260 0 hb\u2082 : b\u2082 \u2260 0 \u22a2 \u2200 (a : \u2115), a \u2208 factors b\u2081 ++ factors b\u2082 \u2192 Nat.Prime a\n[PROOFSTEP]\ncase h => exact fun p hp => (List.mem_append.mp hp).elim prime_of_mem_factors prime_of_mem_factors\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 \u2200 (a : \u2115), a \u2208 factors b\u2081 ++ factors b\u2082 \u2192 Nat.Prime a\n[PROOFSTEP]\ncase h => exact fun p hp => (List.mem_append.mp hp).elim prime_of_mem_factors prime_of_mem_factors\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 \u2200 (a : \u2115), a \u2208 factors b\u2081 ++ factors b\u2082 \u2192 Nat.Prime a\n[PROOFSTEP]\nexact fun p hp => (List.mem_append.mp hp).elim prime_of_mem_factors prime_of_mem_factors\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2081) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2081 \u2192 Nat.Prime a)) *\n      List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2082) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2082 \u2192 Nat.Prime a)) =\n    J(a | b\u2081) * J(a | b\u2082)\n[PROOFSTEP]\ncase _ => rfl\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2081) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2081 \u2192 Nat.Prime a)) *\n      List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2082) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2082 \u2192 Nat.Prime a)) =\n    J(a | b\u2081) * J(a | b\u2082)\n[PROOFSTEP]\ncase _ => rfl\n[GOAL]\na : \u2124\nb\u2081 b\u2082 : \u2115\nhb\u2081 : b\u2081 \u2260 0\nhb\u2082 : b\u2082 \u2260 0\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2081) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2081 \u2192 Nat.Prime a)) *\n      List.prod (List.pmap (fun p pp => legendreSym p a) (factors b\u2082) (_ : \u2200 (a : \u2115), a \u2208 factors b\u2082 \u2192 Nat.Prime a)) =\n    J(a | b\u2081) * J(a | b\u2082)\n[PROOFSTEP]\nrfl\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 {0, 1, -1} = \u2191(MonoidHom.mrange \u2191SignType.castHom)\n[PROOFSTEP]\nrw [Set.pair_comm]\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 {0, -1, 1} = \u2191(MonoidHom.mrange \u2191SignType.castHom)\n[PROOFSTEP]\nexact (SignType.range_eq SignType.castHom).symm\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 \u2200 (x : \u2124),\n    x \u2208 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x) \u2192\n      x \u2208\n        Submonoid.copy (MonoidHom.mrange \u2191SignType.castHom) {0, 1, -1}\n          (_ : {0, 1, -1} = \u2191(MonoidHom.mrange \u2191SignType.castHom))\n[PROOFSTEP]\nintro _ ha'\n[GOAL]\na : \u2124\nb : \u2115\nx\u271d : \u2124\nha' : x\u271d \u2208 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\n\u22a2 x\u271d \u2208\n    Submonoid.copy (MonoidHom.mrange \u2191SignType.castHom) {0, 1, -1}\n      (_ : {0, 1, -1} = \u2191(MonoidHom.mrange \u2191SignType.castHom))\n[PROOFSTEP]\nrcases List.mem_pmap.mp ha' with \u27e8p, hp, rfl\u27e9\n[GOAL]\ncase intro.intro\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nha' : legendreSym p a \u2208 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\n\u22a2 legendreSym p a \u2208\n    Submonoid.copy (MonoidHom.mrange \u2191SignType.castHom) {0, 1, -1}\n      (_ : {0, 1, -1} = \u2191(MonoidHom.mrange \u2191SignType.castHom))\n[PROOFSTEP]\nhaveI : Fact p.Prime := \u27e8prime_of_mem_factors hp\u27e9\n[GOAL]\ncase intro.intro\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nha' : legendreSym p a \u2208 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\nthis : Fact (Nat.Prime p)\n\u22a2 legendreSym p a \u2208\n    Submonoid.copy (MonoidHom.mrange \u2191SignType.castHom) {0, 1, -1}\n      (_ : {0, 1, -1} = \u2191(MonoidHom.mrange \u2191SignType.castHom))\n[PROOFSTEP]\nexact quadraticChar_isQuadratic (ZMod p) a\n[GOAL]\nb : \u2115\nz : \u2124\nhz : z \u2208 List.pmap (fun p pp => legendreSym p 1) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\n\u22a2 z = 1\n[PROOFSTEP]\nlet \u27e8p, hp, he\u27e9 := List.mem_pmap.1 hz\n[GOAL]\nb : \u2115\nz : \u2124\nhz : z \u2208 List.pmap (fun p pp => legendreSym p 1) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\np : \u2115\nhp : p \u2208 factors b\nhe : legendreSym p 1 = z\n\u22a2 z = 1\n[PROOFSTEP]\nletI : Fact p.Prime := \u27e8prime_of_mem_factors hp\u27e9\n[GOAL]\nb : \u2115\nz : \u2124\nhz : z \u2208 List.pmap (fun p pp => legendreSym p 1) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\np : \u2115\nhp : p \u2208 factors b\nhe : legendreSym p 1 = z\nthis : Fact (Nat.Prime p) := { out := prime_of_mem_factors hp }\n\u22a2 z = 1\n[PROOFSTEP]\nrw [\u2190 he, legendreSym.at_one]\n[GOAL]\na\u2081 a\u2082 : \u2124\nb : \u2115\n\u22a2 J(a\u2081 * a\u2082 | b) = J(a\u2081 | b) * J(a\u2082 | b)\n[PROOFSTEP]\nsimp_rw [jacobiSym, List.pmap_eq_map_attach, legendreSym.mul _ _ _]\n[GOAL]\na\u2081 a\u2082 : \u2124\nb : \u2115\n\u22a2 List.prod (List.map (fun x => legendreSym (\u2191x) a\u2081 * legendreSym (\u2191x) a\u2082) (List.attach (factors b))) =\n    List.prod (List.map (fun x => legendreSym (\u2191x) a\u2081) (List.attach (factors b))) *\n      List.prod (List.map (fun x => legendreSym (\u2191x) a\u2082) (List.attach (factors b)))\n[PROOFSTEP]\nexact\n  List.prod_map_mul (\u03b1 := \u2124) (l := (factors b).attach) (f := fun x \u21a6\n    @legendreSym x { out := prime_of_mem_factors x.2 } a\u2081) (g := fun x \u21a6\n    @legendreSym x { out := prime_of_mem_factors x.2 } a\u2082)\n[GOAL]\na : \u2124\nb : \u2115\ninst\u271d : NeZero b\n\u22a2 0 \u2208 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x) \u2194\n    Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\nrw [List.mem_pmap, Int.gcd_eq_natAbs, Ne, Prime.not_coprime_iff_dvd]\n  -- porting note: Initially, `and_assoc'` and `and_comm'` were used on line 164 but they have\n        -- been deprecated so we replace them with `and_assoc` and `and_comm`\n[GOAL]\na : \u2124\nb : \u2115\ninst\u271d : NeZero b\n\u22a2 (\u2203 a_1 h, legendreSym a_1 a = 0) \u2194 \u2203 p, Nat.Prime p \u2227 p \u2223 Int.natAbs a \u2227 p \u2223 Int.natAbs \u2191b\n[PROOFSTEP]\nsimp_rw [legendreSym.eq_zero_iff _ _, int_cast_zmod_eq_zero_iff_dvd, mem_factors (NeZero.ne b), \u2190 Int.coe_nat_dvd_left,\n  Int.coe_nat_dvd, exists_prop, and_assoc, and_comm]\n[GOAL]\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\n\u22a2 J(a | b) \u2260 0\n[PROOFSTEP]\ncases' eq_zero_or_neZero b with hb\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nhb : b = 0\n\u22a2 J(a | b) \u2260 0\n[PROOFSTEP]\nrw [hb, zero_right]\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nhb : b = 0\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nh\u271d : NeZero b\n\u22a2 J(a | b) \u2260 0\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nh\u271d : NeZero b\nh : J(a | b) = 0\n\u22a2 Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\nexact eq_zero_iff_not_coprime.1 h\n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = 0\n\u22a2 b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\ncases' eq_or_ne b 0 with hb hb\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : J(a | b) = 0\nhb : b = 0\n\u22a2 b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\nrw [hb, zero_right] at h \n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : 1 = 0\nhb : b = 0\n\u22a2 b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\ncases h\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nh : J(a | b) = 0\nhb : b \u2260 0\n\u22a2 b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\n[PROOFSTEP]\nexact \u27e8hb, mt jacobiSym.ne_zero <| Classical.not_not.2 h\u27e9\n[GOAL]\na : \u2124\nb : \u2115\nx\u271d : b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\nhb : b \u2260 0\nh : Int.gcd a \u2191b \u2260 1\n\u22a2 J(a | b) = 0\n[PROOFSTEP]\nrw [\u2190 neZero_iff] at hb \n[GOAL]\na : \u2124\nb : \u2115\nx\u271d : b \u2260 0 \u2227 Int.gcd a \u2191b \u2260 1\nhb : NeZero b\nh : Int.gcd a \u2191b \u2260 1\n\u22a2 J(a | b) = 0\n[PROOFSTEP]\nexact eq_zero_iff_not_coprime.2 h\n[GOAL]\nb : \u2115\nhb : 1 < b\n\u22a2 Int.gcd 0 \u2191b \u2260 1\n[PROOFSTEP]\nrw [Int.gcd_zero_left, Int.natAbs_ofNat]\n[GOAL]\nb : \u2115\nhb : 1 < b\n\u22a2 b \u2260 1\n[PROOFSTEP]\nexact hb.ne'\n[GOAL]\na : \u2124\ne b : \u2115\n\u22a2 J(a ^ zero | b) = J(a | b) ^ zero\n[PROOFSTEP]\nrw [_root_.pow_zero, _root_.pow_zero, one_left]\n[GOAL]\na : \u2124\ne b x\u271d : \u2115\nih : J(a ^ x\u271d | b) = J(a | b) ^ x\u271d\n\u22a2 J(a ^ succ x\u271d | b) = J(a | b) ^ succ x\u271d\n[PROOFSTEP]\nrw [_root_.pow_succ, _root_.pow_succ, mul_left, ih]\n[GOAL]\na : \u2124\nb e : \u2115\n\u22a2 J(a | b ^ e) = J(a | b) ^ e\n[PROOFSTEP]\ninduction' e with e ih\n[GOAL]\ncase zero\na : \u2124\nb : \u2115\n\u22a2 J(a | b ^ zero) = J(a | b) ^ zero\n[PROOFSTEP]\nrw [Nat.pow_zero, _root_.pow_zero, one_right]\n[GOAL]\ncase succ\na : \u2124\nb e : \u2115\nih : J(a | b ^ e) = J(a | b) ^ e\n\u22a2 J(a | b ^ succ e) = J(a | b) ^ succ e\n[PROOFSTEP]\ncases' eq_zero_or_neZero b with hb\n[GOAL]\ncase succ.inl\na : \u2124\nb e : \u2115\nih : J(a | b ^ e) = J(a | b) ^ e\nhb : b = 0\n\u22a2 J(a | b ^ succ e) = J(a | b) ^ succ e\n[PROOFSTEP]\nrw [hb, zero_pow (succ_pos e), zero_right, one_pow]\n[GOAL]\ncase succ.inr\na : \u2124\nb e : \u2115\nih : J(a | b ^ e) = J(a | b) ^ e\nh\u271d : NeZero b\n\u22a2 J(a | b ^ succ e) = J(a | b) ^ succ e\n[PROOFSTEP]\nrw [_root_.pow_succ, _root_.pow_succ, mul_right, ih]\n[GOAL]\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\n\u22a2 J(a | b) ^ 2 = 1\n[PROOFSTEP]\ncases' eq_one_or_neg_one h with h\u2081 h\u2081\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nh\u2081 : J(a | b) = 1\n\u22a2 J(a | b) ^ 2 = 1\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nh\u2081 : J(a | b) = -1\n\u22a2 J(a | b) ^ 2 = 1\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nh\u2081 : J(a | b) = 1\n\u22a2 1 ^ 2 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\nh\u2081 : J(a | b) = -1\n\u22a2 (-1) ^ 2 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\na : \u2124\nb : \u2115\nh : Int.gcd a \u2191b = 1\n\u22a2 J(a ^ 2 | b) = 1\n[PROOFSTEP]\nrw [pow_left, sq_one h]\n[GOAL]\na : \u2124\nb : \u2115\n\u22a2 \u2200 (a_1 : \u2115), a_1 \u2208 factors b \u2192 \u2200 (h\u2081 h\u2082 : Nat.Prime a_1), legendreSym a_1 a = legendreSym a_1 (a % \u2191b)\n[PROOFSTEP]\nrintro p hp _ h\u2082\n[GOAL]\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nh\u2081\u271d h\u2082 : Nat.Prime p\n\u22a2 legendreSym p a = legendreSym p (a % \u2191b)\n[PROOFSTEP]\nletI : Fact p.Prime := \u27e8h\u2082\u27e9\n[GOAL]\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nh\u2081\u271d h\u2082 : Nat.Prime p\nthis : Fact (Nat.Prime p) := { out := h\u2082 }\n\u22a2 legendreSym p a = legendreSym p (a % \u2191b)\n[PROOFSTEP]\nconv_rhs => rw [legendreSym.mod, Int.emod_emod_of_dvd _ (Int.coe_nat_dvd.2 <| dvd_of_mem_factors hp), \u2190 legendreSym.mod]\n[GOAL]\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nh\u2081\u271d h\u2082 : Nat.Prime p\nthis : Fact (Nat.Prime p) := { out := h\u2082 }\n| legendreSym p (a % \u2191b)\n[PROOFSTEP]\nrw [legendreSym.mod, Int.emod_emod_of_dvd _ (Int.coe_nat_dvd.2 <| dvd_of_mem_factors hp), \u2190 legendreSym.mod]\n[GOAL]\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nh\u2081\u271d h\u2082 : Nat.Prime p\nthis : Fact (Nat.Prime p) := { out := h\u2082 }\n| legendreSym p (a % \u2191b)\n[PROOFSTEP]\nrw [legendreSym.mod, Int.emod_emod_of_dvd _ (Int.coe_nat_dvd.2 <| dvd_of_mem_factors hp), \u2190 legendreSym.mod]\n[GOAL]\na : \u2124\nb p : \u2115\nhp : p \u2208 factors b\nh\u2081\u271d h\u2082 : Nat.Prime p\nthis : Fact (Nat.Prime p) := { out := h\u2082 }\n| legendreSym p (a % \u2191b)\n[PROOFSTEP]\nrw [legendreSym.mod, Int.emod_emod_of_dvd _ (Int.coe_nat_dvd.2 <| dvd_of_mem_factors hp), \u2190 legendreSym.mod]\n[GOAL]\na\u2081 a\u2082 : \u2124\nb : \u2115\nh : a\u2081 % \u2191b = a\u2082 % \u2191b\n\u22a2 J(a\u2081 | b) = J(a\u2082 | b)\n[PROOFSTEP]\nrw [mod_left, h, \u2190 mod_left]\n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : J(a | p) = -1\nx y : \u2124\nhxy : \u2191p \u2223 x ^ 2 - a * y ^ 2\n\u22a2 \u2191p \u2223 x \u2227 \u2191p \u2223 y\n[PROOFSTEP]\nrw [\u2190 legendreSym.to_jacobiSym] at h \n[GOAL]\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\na : \u2124\nh : legendreSym p a = -1\nx y : \u2124\nhxy : \u2191p \u2223 x ^ 2 - a * y ^ 2\n\u22a2 \u2191p \u2223 x \u2227 \u2191p \u2223 y\n[PROOFSTEP]\nexact legendreSym.prime_dvd_of_eq_neg_one h hxy\n[GOAL]\nl : List \u2124\nn : \u2115\n\u22a2 J(List.prod l | n) = List.prod (List.map (fun a => J(a | n)) l)\n[PROOFSTEP]\ninduction' l with n l' ih\n[GOAL]\ncase nil\nn : \u2115\n\u22a2 J(List.prod [] | n) = List.prod (List.map (fun a => J(a | n)) [])\n[PROOFSTEP]\nsimp only [List.prod_nil, List.map_nil, one_left]\n[GOAL]\ncase cons\nn\u271d : \u2115\nn : \u2124\nl' : List \u2124\nih : J(List.prod l' | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) l')\n\u22a2 J(List.prod (n :: l') | n\u271d) = List.prod (List.map (fun a => J(a | n\u271d)) (n :: l'))\n[PROOFSTEP]\nrw [List.map, List.prod_cons, List.prod_cons, mul_left, ih]\n[GOAL]\na : \u2124\nl : List \u2115\nhl : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\n\u22a2 J(a | List.prod l) = List.prod (List.map (fun n => J(a | n)) l)\n[PROOFSTEP]\ninduction' l with n l' ih\n[GOAL]\ncase nil\na : \u2124\nl : List \u2115\nhl\u271d : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\nhl : \u2200 (n : \u2115), n \u2208 [] \u2192 n \u2260 0\n\u22a2 J(a | List.prod []) = List.prod (List.map (fun n => J(a | n)) [])\n[PROOFSTEP]\nsimp only [List.prod_nil, one_right, List.map_nil]\n[GOAL]\ncase cons\na : \u2124\nl : List \u2115\nhl\u271d : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\nn : \u2115\nl' : List \u2115\nih : (\u2200 (n : \u2115), n \u2208 l' \u2192 n \u2260 0) \u2192 J(a | List.prod l') = List.prod (List.map (fun n => J(a | n)) l')\nhl : \u2200 (n_1 : \u2115), n_1 \u2208 n :: l' \u2192 n_1 \u2260 0\n\u22a2 J(a | List.prod (n :: l')) = List.prod (List.map (fun n => J(a | n)) (n :: l'))\n[PROOFSTEP]\nhave hn :=\n  hl n\n    (List.mem_cons_self n l')\n      -- `n \u2260 0`\n[GOAL]\ncase cons\na : \u2124\nl : List \u2115\nhl\u271d : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\nn : \u2115\nl' : List \u2115\nih : (\u2200 (n : \u2115), n \u2208 l' \u2192 n \u2260 0) \u2192 J(a | List.prod l') = List.prod (List.map (fun n => J(a | n)) l')\nhl : \u2200 (n_1 : \u2115), n_1 \u2208 n :: l' \u2192 n_1 \u2260 0\nhn : n \u2260 0\n\u22a2 J(a | List.prod (n :: l')) = List.prod (List.map (fun n => J(a | n)) (n :: l'))\n[PROOFSTEP]\nhave hl' := List.prod_ne_zero fun hf => hl 0 (List.mem_cons_of_mem _ hf) rfl\n[GOAL]\ncase cons\na : \u2124\nl : List \u2115\nhl\u271d : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\nn : \u2115\nl' : List \u2115\nih : (\u2200 (n : \u2115), n \u2208 l' \u2192 n \u2260 0) \u2192 J(a | List.prod l') = List.prod (List.map (fun n => J(a | n)) l')\nhl : \u2200 (n_1 : \u2115), n_1 \u2208 n :: l' \u2192 n_1 \u2260 0\nhn : n \u2260 0\nhl' : List.prod l' \u2260 0\n\u22a2 J(a | List.prod (n :: l')) = List.prod (List.map (fun n => J(a | n)) (n :: l'))\n[PROOFSTEP]\nhave h := fun m hm =>\n  hl m\n    (List.mem_cons_of_mem _ hm)\n      -- `\u2200 (m : \u2115), m \u2208 l' \u2192 m \u2260 0`\n[GOAL]\ncase cons\na : \u2124\nl : List \u2115\nhl\u271d : \u2200 (n : \u2115), n \u2208 l \u2192 n \u2260 0\nn : \u2115\nl' : List \u2115\nih : (\u2200 (n : \u2115), n \u2208 l' \u2192 n \u2260 0) \u2192 J(a | List.prod l') = List.prod (List.map (fun n => J(a | n)) l')\nhl : \u2200 (n_1 : \u2115), n_1 \u2208 n :: l' \u2192 n_1 \u2260 0\nhn : n \u2260 0\nhl' : List.prod l' \u2260 0\nh : \u2200 (m : \u2115), m \u2208 l' \u2192 m \u2260 0\n\u22a2 J(a | List.prod (n :: l')) = List.prod (List.map (fun n => J(a | n)) (n :: l'))\n[PROOFSTEP]\nrw [List.map, List.prod_cons, List.prod_cons, mul_right' a hn hl', ih h]\n[GOAL]\na : \u2124\nn : \u2115\nh : J(a | n) = -1\n\u22a2 \u2203 p x, p \u2223 n \u2227 J(a | p) = -1\n[PROOFSTEP]\nhave hn\u2080 : n \u2260 0 := by\n  rintro rfl\n  rw [zero_right, eq_neg_self_iff] at h \n  exact one_ne_zero h\n[GOAL]\na : \u2124\nn : \u2115\nh : J(a | n) = -1\n\u22a2 n \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\na : \u2124\nh : J(a | 0) = -1\n\u22a2 False\n[PROOFSTEP]\nrw [zero_right, eq_neg_self_iff] at h \n[GOAL]\na : \u2124\nh : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero h\n[GOAL]\na : \u2124\nn : \u2115\nh : J(a | n) = -1\nhn\u2080 : n \u2260 0\n\u22a2 \u2203 p x, p \u2223 n \u2227 J(a | p) = -1\n[PROOFSTEP]\nhave hf\u2080 : \u2200 p \u2208 n.factors, p \u2260 0 := fun p hp => (Nat.pos_of_mem_factors hp).ne.symm\n[GOAL]\na : \u2124\nn : \u2115\nh : J(a | n) = -1\nhn\u2080 : n \u2260 0\nhf\u2080 : \u2200 (p : \u2115), p \u2208 factors n \u2192 p \u2260 0\n\u22a2 \u2203 p x, p \u2223 n \u2227 J(a | p) = -1\n[PROOFSTEP]\nrw [\u2190 Nat.prod_factors hn\u2080, list_prod_right hf\u2080] at h \n[GOAL]\na : \u2124\nn : \u2115\nh : List.prod (List.map (fun n => J(a | n)) (factors n)) = -1\nhn\u2080 : n \u2260 0\nhf\u2080 : \u2200 (p : \u2115), p \u2208 factors n \u2192 p \u2260 0\n\u22a2 \u2203 p x, p \u2223 n \u2227 J(a | p) = -1\n[PROOFSTEP]\nobtain \u27e8p, hmem, hj\u27e9 := List.mem_map.mp (List.neg_one_mem_of_prod_eq_neg_one h)\n[GOAL]\ncase intro.intro\na : \u2124\nn : \u2115\nh : List.prod (List.map (fun n => J(a | n)) (factors n)) = -1\nhn\u2080 : n \u2260 0\nhf\u2080 : \u2200 (p : \u2115), p \u2208 factors n \u2192 p \u2260 0\np : \u2115\nhmem : p \u2208 factors n\nhj : J(a | p) = -1\n\u22a2 \u2203 p x, p \u2223 n \u2227 J(a | p) = -1\n[PROOFSTEP]\nexact \u27e8p, Nat.prime_of_mem_factors hmem, Nat.dvd_of_mem_factors hmem, hj\u27e9\n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = -1\nx\u271d : IsSquare \u2191a\nr : ZMod b\nha : \u2191a = r * r\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 r.coe_valMinAbs, \u2190 Int.cast_mul, int_cast_eq_int_cast_iff', \u2190 sq] at ha \n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = -1\nx\u271d : IsSquare \u2191a\nr : ZMod b\nha : a % \u2191b = valMinAbs r ^ 2 % \u2191b\n\u22a2 False\n[PROOFSTEP]\napply (by norm_num : \u00ac(0 : \u2124) \u2264 -1)\n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = -1\nx\u271d : IsSquare \u2191a\nr : ZMod b\nha : a % \u2191b = valMinAbs r ^ 2 % \u2191b\n\u22a2 \u00ac0 \u2264 -1\n[PROOFSTEP]\nnorm_num\n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = -1\nx\u271d : IsSquare \u2191a\nr : ZMod b\nha : a % \u2191b = valMinAbs r ^ 2 % \u2191b\n\u22a2 0 \u2264 -1\n[PROOFSTEP]\nrw [\u2190 h, mod_left, ha, \u2190 mod_left, pow_left]\n[GOAL]\na : \u2124\nb : \u2115\nh : J(a | b) = -1\nx\u271d : IsSquare \u2191a\nr : ZMod b\nha : a % \u2191b = valMinAbs r ^ 2 % \u2191b\n\u22a2 0 \u2264 J(valMinAbs r | b) ^ 2\n[PROOFSTEP]\napply sq_nonneg\n[GOAL]\na : \u2124\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 J(a | p) = -1 \u2194 \u00acIsSquare \u2191a\n[PROOFSTEP]\nrw [\u2190 legendreSym.to_jacobiSym]\n[GOAL]\na : \u2124\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 legendreSym p a = -1 \u2194 \u00acIsSquare \u2191a\n[PROOFSTEP]\nexact legendreSym.eq_neg_one_iff p\n[GOAL]\na : \u2124\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : J(a | p) = 1\n\u22a2 \u00ac\u00acIsSquare \u2191a\n[PROOFSTEP]\nrw [\u2190 nonsquare_iff_jacobiSym_eq_neg_one, h]\n[GOAL]\na : \u2124\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nh : J(a | p) = 1\n\u22a2 \u00ac1 = -1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = \u2191\u03c7 \u2191b\n[PROOFSTEP]\nconv_rhs => rw [\u2190 prod_factors hb.pos.ne', cast_list_prod, \u03c7.map_list_prod]\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n| \u2191\u03c7 \u2191b\n[PROOFSTEP]\nrw [\u2190 prod_factors hb.pos.ne', cast_list_prod, \u03c7.map_list_prod]\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n| \u2191\u03c7 \u2191b\n[PROOFSTEP]\nrw [\u2190 prod_factors hb.pos.ne', cast_list_prod, \u03c7.map_list_prod]\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n| \u2191\u03c7 \u2191b\n[PROOFSTEP]\nrw [\u2190 prod_factors hb.pos.ne', cast_list_prod, \u03c7.map_list_prod]\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = List.prod (List.map (\u2191\u03c7) (List.map Nat.cast (factors b)))\n[PROOFSTEP]\nrw [jacobiSym, List.map_map, \u2190 List.pmap_eq_map Nat.Prime _ _ fun _ => prime_of_mem_factors]\n[GOAL]\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.prod (List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)) =\n    List.prod (List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 List.pmap (fun p pp => legendreSym p a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x) =\n    List.pmap (fun a x => (\u2191\u03c7 \u2218 Nat.cast) a) (factors b) (_ : \u2200 (x : \u2115), x \u2208 factors b \u2192 Nat.Prime x)\n[PROOFSTEP]\napply List.pmap_congr\n[GOAL]\ncase e_a.h\na : \u2124\nR : Type u_1\ninst\u271d : CommSemiring R\n\u03c7 : R \u2192* \u2124\nhp : \u2200 (p : \u2115) (pp : Nat.Prime p), p \u2260 2 \u2192 legendreSym p a = \u2191\u03c7 \u2191p\nb : \u2115\nhb : Odd b\n\u22a2 \u2200 (a_1 : \u2115), a_1 \u2208 factors b \u2192 \u2200 (h\u2081 : Nat.Prime a_1), Nat.Prime a_1 \u2192 legendreSym a_1 a = (\u2191\u03c7 \u2218 Nat.cast) a_1\n[PROOFSTEP]\nexact fun p h pp _ => hp p pp (hb.ne_two_of_dvd_nat <| dvd_of_mem_factors h)\n[GOAL]\na : \u2124\nb : \u2115\nhb : Odd b\n\u22a2 J(-a | b) = \u2191\u03c7\u2084 \u2191b * J(a | b)\n[PROOFSTEP]\nrw [neg_eq_neg_one_mul, mul_left, at_neg_one hb]\n[GOAL]\nm n : \u2115\nhm : Odd m\nhn : Odd n\n\u22a2 qrSign m n = (-1) ^ (m / 2 * (n / 2))\n[PROOFSTEP]\nrw [qrSign, pow_mul, \u2190 \u03c7\u2084_eq_neg_one_pow (odd_iff.mp hm)]\n[GOAL]\nm n : \u2115\nhm : Odd m\nhn : Odd n\n\u22a2 J(\u2191\u03c7\u2084 \u2191m | n) = \u2191\u03c7\u2084 \u2191m ^ (n / 2)\n[PROOFSTEP]\ncases' odd_mod_four_iff.mp (odd_iff.mp hm) with h h\n[GOAL]\ncase inl\nm n : \u2115\nhm : Odd m\nhn : Odd n\nh : m % 4 = 1\n\u22a2 J(\u2191\u03c7\u2084 \u2191m | n) = \u2191\u03c7\u2084 \u2191m ^ (n / 2)\n[PROOFSTEP]\nrw [\u03c7\u2084_nat_one_mod_four h, jacobiSym.one_left, one_pow]\n[GOAL]\ncase inr\nm n : \u2115\nhm : Odd m\nhn : Odd n\nh : m % 4 = 3\n\u22a2 J(\u2191\u03c7\u2084 \u2191m | n) = \u2191\u03c7\u2084 \u2191m ^ (n / 2)\n[PROOFSTEP]\nrw [\u03c7\u2084_nat_three_mod_four h, \u2190 \u03c7\u2084_eq_neg_one_pow (odd_iff.mp hn), jacobiSym.at_neg_one hn]\n[GOAL]\nm n : \u2115\nhm : Odd m\nhn : Odd n\n\u22a2 qrSign m n ^ 2 = 1\n[PROOFSTEP]\nrw [neg_one_pow hm hn, \u2190 pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]\n[GOAL]\nm\u2081 m\u2082 n : \u2115\n\u22a2 qrSign (m\u2081 * m\u2082) n = qrSign m\u2081 n * qrSign m\u2082 n\n[PROOFSTEP]\nsimp_rw [qrSign, Nat.cast_mul, map_mul, jacobiSym.mul_left]\n[GOAL]\nm n : \u2115\nhm : Odd m\nhn : Odd n\n\u22a2 qrSign m n = qrSign n m\n[PROOFSTEP]\nrw [neg_one_pow hm hn, neg_one_pow hn hm, mul_comm (m / 2)]\n[GOAL]\nm n : \u2115\nhm : Odd m\nhn : Odd n\nx y : \u2124\n\u22a2 qrSign m n * x = y \u2194 x = qrSign m n * y\n[PROOFSTEP]\nrefine'\n  \u27e8fun h' =>\n    let h := h'.symm\n    _,\n    fun h => _\u27e9\n[GOAL]\ncase refine'_1\nm n : \u2115\nhm : Odd m\nhn : Odd n\nx y : \u2124\nh' : qrSign m n * x = y\nh : y = qrSign m n * x := Eq.symm h'\n\u22a2 x = qrSign m n * y\n[PROOFSTEP]\nrw [h, \u2190 mul_assoc, \u2190 pow_two, sq_eq_one hm hn, one_mul]\n[GOAL]\ncase refine'_2\nm n : \u2115\nhm : Odd m\nhn : Odd n\nx y : \u2124\nh : x = qrSign m n * y\n\u22a2 qrSign m n * x = y\n[PROOFSTEP]\nrw [h, \u2190 mul_assoc, \u2190 pow_two, sq_eq_one hm hn, one_mul]\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\n\u22a2 J(\u2191a | b) = qrSign b a * J(\u2191b | a)\n[PROOFSTEP]\nlet rhs : \u2115 \u2192 \u2115 \u2192* \u2124 := fun a =>\n  { toFun := fun x => qrSign x a * J(x | a)\n    map_one' := by convert \u2190 mul_one (M := \u2124) _; symm; all_goals apply one_left\n    map_mul' := fun x y => by\n      -- porting note: `simp_rw` on line 423 replaces `rw` to allow the rewrite rules to be\n              -- applied under the binder `fun \u21a6 ...`simp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left,\n        mul_mul_mul_comm] }\n[GOAL]\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na : \u2115\n\u22a2 (fun x => qrSign x a * J(\u2191x | a)) 1 = 1\n[PROOFSTEP]\nconvert \u2190 mul_one (M := \u2124) _\n[GOAL]\ncase h.e'_2.h.e'_6\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na : \u2115\n\u22a2 1 = J(\u21911 | a)\ncase h.e'_3 a\u271d b : \u2115 ha : Odd a\u271d hb : Odd b a : \u2115 \u22a2 qrSign 1 a = 1\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.e'_2.h.e'_6\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na : \u2115\n\u22a2 J(\u21911 | a) = 1\ncase h.e'_3 a\u271d b : \u2115 ha : Odd a\u271d hb : Odd b a : \u2115 \u22a2 qrSign 1 a = 1\n[PROOFSTEP]\nall_goals apply one_left\n[GOAL]\ncase h.e'_2.h.e'_6\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na : \u2115\n\u22a2 J(\u21911 | a) = 1\n[PROOFSTEP]\napply one_left\n[GOAL]\ncase h.e'_3\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na : \u2115\n\u22a2 qrSign 1 a = 1\n[PROOFSTEP]\napply one_left\n[GOAL]\na\u271d b : \u2115\nha : Odd a\u271d\nhb : Odd b\na x y : \u2115\n\u22a2 OneHom.toFun { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n      (x * y) =\n    OneHom.toFun { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) } y\n[PROOFSTEP]\nsimp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left, mul_mul_mul_comm]\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\n\u22a2 J(\u2191a | b) = qrSign b a * J(\u2191b | a)\n[PROOFSTEP]\nhave rhs_apply : \u2200 a b : \u2115, rhs a b = qrSign b a * J(b | a) := fun a b => rfl\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\n\u22a2 J(\u2191a | b) = qrSign b a * J(\u2191b | a)\n[PROOFSTEP]\nrefine' value_at a (rhs a) (fun p pp hp => Eq.symm _) hb\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\np : \u2115\npp : Nat.Prime p\nhp : p \u2260 2\n\u22a2 \u2191(rhs a) \u2191p = legendreSym p \u2191a\n[PROOFSTEP]\nhave hpo := pp.eq_two_or_odd'.resolve_left hp\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\np : \u2115\npp : Nat.Prime p\nhp : p \u2260 2\nhpo : Odd p\n\u22a2 \u2191(rhs a) \u2191p = legendreSym p \u2191a\n[PROOFSTEP]\nrw [@legendreSym.to_jacobiSym p \u27e8pp\u27e9, rhs_apply, Nat.cast_id, qrSign.eq_iff_eq hpo ha, qrSign.symm hpo ha]\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\np : \u2115\npp : Nat.Prime p\nhp : p \u2260 2\nhpo : Odd p\n\u22a2 J(\u2191p | a) = qrSign a p * J(\u2191a | p)\n[PROOFSTEP]\nrefine' value_at p (rhs p) (fun q pq hq => _) ha\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\np : \u2115\npp : Nat.Prime p\nhp : p \u2260 2\nhpo : Odd p\nq : \u2115\npq : Nat.Prime q\nhq : q \u2260 2\n\u22a2 legendreSym q \u2191p = \u2191(rhs p) \u2191q\n[PROOFSTEP]\nhave hqo := pq.eq_two_or_odd'.resolve_left hq\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\nrhs : \u2115 \u2192 \u2115 \u2192* \u2124 :=\n  fun a =>\n    {\n      toOneHom :=\n        { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) },\n      map_mul' :=\n        (_ :\n          \u2200 (x y : \u2115),\n            OneHom.toFun\n                { toFun := fun x => qrSign x a * J(\u2191x | a), map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                (x * y) =\n              OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  x *\n                OneHom.toFun\n                  { toFun := fun x => qrSign x a * J(\u2191x | a),\n                    map_one' := (_ : (fun x => qrSign x a * J(\u2191x | a)) 1 = 1) }\n                  y) }\nrhs_apply : \u2200 (a b : \u2115), \u2191(rhs a) b = qrSign b a * J(\u2191b | a)\np : \u2115\npp : Nat.Prime p\nhp : p \u2260 2\nhpo : Odd p\nq : \u2115\npq : Nat.Prime q\nhq : q \u2260 2\nhqo : Odd q\n\u22a2 legendreSym q \u2191p = \u2191(rhs p) \u2191q\n[PROOFSTEP]\nrw [rhs_apply, Nat.cast_id, \u2190 @legendreSym.to_jacobiSym p \u27e8pp\u27e9, qrSign.symm hqo hpo, qrSign.neg_one_pow hpo hqo,\n  @legendreSym.quadratic_reciprocity' p q \u27e8pp\u27e9 \u27e8pq\u27e9 hp hq]\n[GOAL]\na b : \u2115\nha : Odd a\nhb : Odd b\n\u22a2 J(\u2191a | b) = (-1) ^ (a / 2 * (b / 2)) * J(\u2191b | a)\n[PROOFSTEP]\nrw [\u2190 qrSign.neg_one_pow ha hb, qrSign.symm ha hb, quadratic_reciprocity' ha hb]\n[GOAL]\na b : \u2115\nha : a % 4 = 1\nhb : Odd b\n\u22a2 J(\u2191a | b) = J(\u2191b | a)\n[PROOFSTEP]\nrw [quadratic_reciprocity (odd_iff.mpr (odd_of_mod_four_eq_one ha)) hb, pow_mul, neg_one_pow_div_two_of_one_mod_four ha,\n  one_pow, one_mul]\n[GOAL]\na b : \u2115\nha : a % 4 = 3\nhb : b % 4 = 3\n\u22a2 J(\u2191a | b) = -J(\u2191b | a)\n[PROOFSTEP]\nlet nop := @neg_one_pow_div_two_of_three_mod_four\n[GOAL]\na b : \u2115\nha : a % 4 = 3\nhb : b % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 J(\u2191a | b) = -J(\u2191b | a)\n[PROOFSTEP]\nrw [quadratic_reciprocity, pow_mul, nop ha, nop hb, neg_one_mul]\n[GOAL]\ncase ha\na b : \u2115\nha : a % 4 = 3\nhb : b % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 Odd a\n[PROOFSTEP]\nrwa [odd_iff, odd_of_mod_four_eq_three]\n[GOAL]\ncase hb\na b : \u2115\nha : a % 4 = 3\nhb : b % 4 = 3\nnop : \u2200 {n : \u2115}, n % 4 = 3 \u2192 (-1) ^ (n / 2) = -1 := @neg_one_pow_div_two_of_three_mod_four\n\u22a2 Odd b\n[PROOFSTEP]\nrwa [odd_iff, odd_of_mod_four_eq_three]\n[GOAL]\na b : \u2115\nhb : Odd b\n\u22a2 J(\u2191a | b) = J(\u2191a | b % (4 * a))\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha\u2080)\n[GOAL]\ncase inl\nb : \u2115\nhb : Odd b\n\u22a2 J(\u21910 | b) = J(\u21910 | b % (4 * 0))\n[PROOFSTEP]\nrw [mul_zero, mod_zero]\n[GOAL]\ncase inr\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\n\u22a2 J(\u2191a | b) = J(\u2191a | b % (4 * a))\n[PROOFSTEP]\nhave hb' : Odd (b % (4 * a)) := hb.mod_even (Even.mul_right (by norm_num) _)\n[GOAL]\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\n\u22a2 Even 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\n\u22a2 J(\u2191a | b) = J(\u2191a | b % (4 * a))\n[PROOFSTEP]\nrcases exists_eq_pow_mul_and_not_dvd ha\u2080 2 (by norm_num) with \u27e8e, a', ha\u2081', ha\u2082\u27e9\n[GOAL]\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\n\u22a2 2 \u2260 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.intro.intro.intro\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\n\u22a2 J(\u2191a | b) = J(\u2191a | b % (4 * a))\n[PROOFSTEP]\nhave ha\u2081 := odd_iff.mpr (two_dvd_ne_zero.mp ha\u2081')\n[GOAL]\ncase inr.intro.intro.intro\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 J(\u2191a | b) = J(\u2191a | b % (4 * a))\n[PROOFSTEP]\nnth_rw 2 [ha\u2082]\n[GOAL]\ncase inr.intro.intro.intro\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 J(\u2191a | b) = J(\u2191(2 ^ e * a') | b % (4 * a))\n[PROOFSTEP]\nnth_rw 1 [ha\u2082]\n[GOAL]\ncase inr.intro.intro.intro\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 J(\u2191(2 ^ e * a') | b) = J(\u2191(2 ^ e * a') | b % (4 * a))\n[PROOFSTEP]\nrw [Nat.cast_mul, mul_left, mul_left, quadratic_reciprocity' ha\u2081 hb, quadratic_reciprocity' ha\u2081 hb', Nat.cast_pow,\n  pow_left, pow_left, Nat.cast_two, at_two hb, at_two hb']\n[GOAL]\ncase inr.intro.intro.intro\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ e * (qrSign b a' * J(\u2191b | a')) = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ e * (qrSign (b % (4 * a)) a' * J(\u2191(b % (4 * a)) | a'))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase inr.intro.intro.intro.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ e = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ e\ncase inr.intro.intro.intro.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 qrSign b a' * J(\u2191b | a') = qrSign (b % (4 * a)) a' * J(\u2191(b % (4 * a)) | a')\n[PROOFSTEP]\nswap\n[GOAL]\ncase inr.intro.intro.intro.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 qrSign b a' * J(\u2191b | a') = qrSign (b % (4 * a)) a' * J(\u2191(b % (4 * a)) | a')\ncase inr.intro.intro.intro.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ e = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ e\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase inr.intro.intro.intro.e_a.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 qrSign b a' = qrSign (b % (4 * a)) a'\n[PROOFSTEP]\nsimp_rw [qrSign]\n[GOAL]\ncase inr.intro.intro.intro.e_a.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 J(\u2191\u03c7\u2084 \u2191b | a') = J(\u2191\u03c7\u2084 \u2191(b % (4 * a)) | a')\n[PROOFSTEP]\nrw [\u03c7\u2084_nat_mod_four, \u03c7\u2084_nat_mod_four (b % (4 * a)), mod_mod_of_dvd b (dvd_mul_right 4 a)]\n[GOAL]\ncase inr.intro.intro.intro.e_a.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 J(\u2191b | a') = J(\u2191(b % (4 * a)) | a')\n[PROOFSTEP]\nrw [mod_left \u2191(b % _), mod_left b, Int.coe_nat_mod, Int.emod_emod_of_dvd b]\n[GOAL]\ncase inr.intro.intro.intro.e_a.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191a' \u2223 \u2191(4 * a)\n[PROOFSTEP]\nsimp only [ha\u2082, Nat.cast_mul, \u2190 mul_assoc]\n[GOAL]\ncase inr.intro.intro.intro.e_a.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191a' \u2223 \u21914 * \u2191(2 ^ e) * \u2191a'\n[PROOFSTEP]\nexact\n  dvd_mul_left (a' : \u2124)\n    (\u21914 * \u2191(2 ^ e))\n      -- porting note: In mathlib3, it was written `cases' e`. In Lean 4, this resulted in the choice\n        -- of a name other than e (for the case distinction of line 482) so we indicate the name\n        -- to use explicitly.\n[GOAL]\ncase inr.intro.intro.intro.e_a\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\ne a' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2082 : a = 2 ^ e * a'\nha\u2081 : Odd a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ e = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ e\n[PROOFSTEP]\ncases' e with e\n[GOAL]\ncase inr.intro.intro.intro.e_a.zero\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\na' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2081 : Odd a'\nha\u2082 : a = 2 ^ zero * a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ zero = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.intro.intro.intro.e_a.succ\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\na' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2081 : Odd a'\ne : \u2115\nha\u2082 : a = 2 ^ succ e * a'\n\u22a2 \u2191\u03c7\u2088 \u2191b ^ succ e = \u2191\u03c7\u2088 \u2191(b % (4 * a)) ^ succ e\n[PROOFSTEP]\nrw [\u03c7\u2088_nat_mod_eight, \u03c7\u2088_nat_mod_eight (b % (4 * a)), mod_mod_of_dvd b]\n[GOAL]\ncase inr.intro.intro.intro.e_a.succ\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\na' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2081 : Odd a'\ne : \u2115\nha\u2082 : a = 2 ^ succ e * a'\n\u22a2 8 \u2223 4 * a\n[PROOFSTEP]\nuse 2 ^ e * a'\n[GOAL]\ncase h\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\na' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2081 : Odd a'\ne : \u2115\nha\u2082 : a = 2 ^ succ e * a'\n\u22a2 4 * a = 8 * (2 ^ e * a')\n[PROOFSTEP]\nrw [ha\u2082, Nat.pow_succ]\n[GOAL]\ncase h\na b : \u2115\nhb : Odd b\nha\u2080 : a \u2260 0\nhb' : Odd (b % (4 * a))\na' : \u2115\nha\u2081' : \u00ac2 \u2223 a'\nha\u2081 : Odd a'\ne : \u2115\nha\u2082 : a = 2 ^ succ e * a'\n\u22a2 4 * (2 ^ e * 2 * a') = 8 * (2 ^ e * a')\n[PROOFSTEP]\nring\n[GOAL]\na : \u2124\nb : \u2115\nhb : Odd b\n\u22a2 J(a | b) = J(a | b % (4 * Int.natAbs a))\n[PROOFSTEP]\ncases' Int.natAbs_eq a with ha ha\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = \u2191(Int.natAbs a)\n\u22a2 J(a | b) = J(a | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nnth_rw 2 [ha]\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = -\u2191(Int.natAbs a)\n\u22a2 J(a | b) = J(a | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nnth_rw 2 [ha]\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = \u2191(Int.natAbs a)\n\u22a2 J(a | b) = J(\u2191(Int.natAbs a) | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nnth_rw 1 [ha]\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = -\u2191(Int.natAbs a)\n\u22a2 J(a | b) = J(-\u2191(Int.natAbs a) | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nnth_rw 1 [ha]\n[GOAL]\ncase inl\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = \u2191(Int.natAbs a)\n\u22a2 J(\u2191(Int.natAbs a) | b) = J(\u2191(Int.natAbs a) | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nexact mod_right' a.natAbs hb\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = -\u2191(Int.natAbs a)\n\u22a2 J(-\u2191(Int.natAbs a) | b) = J(-\u2191(Int.natAbs a) | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nhave hb' : Odd (b % (4 * a.natAbs)) := hb.mod_even (Even.mul_right (by norm_num) _)\n[GOAL]\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = -\u2191(Int.natAbs a)\n\u22a2 Even 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\na : \u2124\nb : \u2115\nhb : Odd b\nha : a = -\u2191(Int.natAbs a)\nhb' : Odd (b % (4 * Int.natAbs a))\n\u22a2 J(-\u2191(Int.natAbs a) | b) = J(-\u2191(Int.natAbs a) | b % (4 * Int.natAbs a))\n[PROOFSTEP]\nrw [jacobiSym.neg _ hb, jacobiSym.neg _ hb', mod_right' _ hb, \u03c7\u2084_nat_mod_four, \u03c7\u2084_nat_mod_four (b % (4 * _)),\n  mod_mod_of_dvd b (dvd_mul_right 4 _)]\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol", "llama_tokens": 20672, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8418256512199033, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5209052415985508}}
{"text": "[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = lt) = if c then a = lt else b = lt\n[PROOFSTEP]\nby_cases c\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = lt) = if c then a = lt else b = lt\n[PROOFSTEP]\nby_cases c\n[GOAL]\ncase pos\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : c\n\u22a2 ((if c then a else b) = lt) = if c then a = lt else b = lt\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : \u00acc\n\u22a2 ((if c then a else b) = lt) = if c then a = lt else b = lt\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = eq) = if c then a = eq else b = eq\n[PROOFSTEP]\nby_cases c\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = eq) = if c then a = eq else b = eq\n[PROOFSTEP]\nby_cases c\n[GOAL]\ncase pos\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : c\n\u22a2 ((if c then a else b) = eq) = if c then a = eq else b = eq\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : \u00acc\n\u22a2 ((if c then a else b) = eq) = if c then a = eq else b = eq\n[PROOFSTEP]\nsimp [*]\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = gt) = if c then a = gt else b = gt\n[PROOFSTEP]\nby_cases c\n[GOAL]\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\n\u22a2 ((if c then a else b) = gt) = if c then a = gt else b = gt\n[PROOFSTEP]\nby_cases c\n[GOAL]\ncase pos\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : c\n\u22a2 ((if c then a else b) = gt) = if c then a = gt else b = gt\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\nc : Prop\ninst\u271d : Decidable c\na b : Ordering\nh : \u00acc\n\u22a2 ((if c then a else b) = gt) = if c then a = gt else b = gt\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel lt\na b : \u03b1\n\u22a2 (cmpUsing lt a b = Ordering.lt) = lt a b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\n\u22a2 (cmpUsing lt a b = Ordering.gt) = lt b a\n[PROOFSTEP]\nsimp only [cmpUsing, Ordering.ite_eq_gt_distrib, if_false_right_eq_and, and_true, if_false_left_eq_and]\n[GOAL]\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\n\u22a2 (\u00aclt a b \u2227 lt b a) = lt b a\n[PROOFSTEP]\napply propext\n[GOAL]\ncase a\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\n\u22a2 \u00aclt a b \u2227 lt b a \u2194 lt b a\n[PROOFSTEP]\napply Iff.intro\n[GOAL]\ncase a.mp\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\n\u22a2 \u00aclt a b \u2227 lt b a \u2192 lt b a\n[PROOFSTEP]\nexact fun h => h.2\n[GOAL]\ncase a.mpr\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\n\u22a2 lt b a \u2192 \u00aclt a b \u2227 lt b a\n[PROOFSTEP]\nintro hba\n[GOAL]\ncase a.mpr\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\nhba : lt b a\n\u22a2 \u00aclt a b \u2227 lt b a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mpr.left\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\nhba : lt b a\n\u22a2 \u00aclt a b\n[PROOFSTEP]\nintro hab\n[GOAL]\ncase a.mpr.left\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\nhba : lt b a\nhab : lt a b\n\u22a2 False\n[PROOFSTEP]\nexact absurd (_root_.trans hab hba) (irrefl a)\n[GOAL]\ncase a.mpr.right\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableRel lt\ninst\u271d : IsStrictOrder \u03b1 lt\na b : \u03b1\nhba : lt b a\n\u22a2 lt b a\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u\nlt : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : DecidableRel lt\na b : \u03b1\n\u22a2 (cmpUsing lt a b = Ordering.eq) = (\u00aclt a b \u2227 \u00aclt b a)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Init.Data.Ordering.Lemmas", "llama_tokens": 1745, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.5208754852024533}}
{"text": "[GOAL]\n\u03b1 \u03b2 : PreordCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191e \u226b \u2191(OrderIso.symm e) = \ud835\udfd9 \u03b1\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : PreordCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget PreordCat).obj \u03b1\n\u22a2 \u2191(\u2191e \u226b \u2191(OrderIso.symm e)) x = \u2191(\ud835\udfd9 \u03b1) x\n[PROOFSTEP]\nexact e.symm_apply_apply x\n[GOAL]\n\u03b1 \u03b2 : PreordCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 \u2191(OrderIso.symm e) \u226b \u2191e = \ud835\udfd9 \u03b2\n[PROOFSTEP]\next x\n[GOAL]\ncase w\n\u03b1 \u03b2 : PreordCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx : (forget PreordCat).obj \u03b2\n\u22a2 \u2191(\u2191(OrderIso.symm e) \u226b \u2191e) x = \u2191(\ud835\udfd9 \u03b2) x\n[PROOFSTEP]\nexact e.apply_symm_apply x\n[GOAL]\nX\u271d Y\u271d : PreordCat\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : preordCatToCat.map a\u2081\u271d = preordCatToCat.map a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\next x\n[GOAL]\ncase w\nX\u271d Y\u271d : PreordCat\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : preordCatToCat.map a\u2081\u271d = preordCatToCat.map a\u2082\u271d\nx : (forget PreordCat).obj X\u271d\n\u22a2 \u2191a\u2081\u271d x = \u2191a\u2082\u271d x\n[PROOFSTEP]\nexact Functor.congr_obj h x\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.PreordCat", "llama_tokens": 521, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.5208754756600326}}
{"text": "[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\n\u22a2 Quiver.IsThin C \u2194 \u2200 (c : C), Subsingleton (c \u27f6 c)\n[PROOFSTEP]\nrefine' \u27e8fun h c => h c c, fun h c d => Subsingleton.intro fun f g => _\u27e9\n[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\n\u22a2 f = g\n[PROOFSTEP]\nhaveI := h d\n[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\nthis : Subsingleton (d \u27f6 d)\n\u22a2 f = g\n[PROOFSTEP]\ncalc\n  f = f \u226b inv g \u226b g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]\n  _ = f \u226b inv f \u226b g := by\n    congr 1\n    simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]\n  _ = g := by simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]\n[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\nthis : Subsingleton (d \u27f6 d)\n\u22a2 f = f \u226b inv g \u226b g\n[PROOFSTEP]\nsimp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]\n[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\nthis : Subsingleton (d \u27f6 d)\n\u22a2 f \u226b inv g \u226b g = f \u226b inv f \u226b g\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\nthis : Subsingleton (d \u27f6 d)\n\u22a2 inv g \u226b g = inv f \u226b g\n[PROOFSTEP]\nsimp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]\n[GOAL]\nC : Type u_1\ninst\u271d : Groupoid C\nh : \u2200 (c : C), Subsingleton (c \u27f6 c)\nc d : C\nf g : c \u27f6 d\nthis : Subsingleton (d \u27f6 d)\n\u22a2 f \u226b inv f \u226b g = g\n[PROOFSTEP]\nsimp only [inv_eq_inv, IsIso.hom_inv_id_assoc]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Groupoid.Basic", "llama_tokens": 803, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.7154239897159439, "lm_q1q2_score": 0.5208111000715696}}
{"text": "[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d : SeminormedGroup E\nh : \u2200 (x : E), \u2016x\u2016 = 0 \u2192 x = 1\nsrc\u271d : SeminormedGroup E := inst\u271d\nx\u271d y\u271d : E\nhxy : dist x\u271d y\u271d = 0\n\u22a2 \u2016x\u271d / y\u271d\u2016 = 0\n[PROOFSTEP]\nexact (\u2039SeminormedGroup E\u203a.dist_eq _ _).symm.trans hxy\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist x y \u2264 dist (x * z) (y * z)\nx y : E\n\u22a2 dist x y = \u2016x / y\u2016\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist x y \u2264 dist (x * z) (y * z)\nx y : E\n\u22a2 dist x y = dist (x / y) 1\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist x y \u2264 dist (x * z) (y * z)\nx y : E\n\u22a2 dist x y \u2264 dist (x / y) 1\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv, \u2190 mul_right_inv y] using h\u2082 _ _ _\n[GOAL]\ncase a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist x y \u2264 dist (x * z) (y * z)\nx y : E\n\u22a2 dist (x / y) 1 \u2264 dist x y\n[PROOFSTEP]\nsimpa only [div_mul_cancel', one_mul] using h\u2082 (x / y) 1 y\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist (x * z) (y * z) \u2264 dist x y\nx y : E\n\u22a2 dist x y = \u2016x / y\u2016\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist (x * z) (y * z) \u2264 dist x y\nx y : E\n\u22a2 dist x y = dist (x / y) 1\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist (x * z) (y * z) \u2264 dist x y\nx y : E\n\u22a2 dist x y \u2264 dist (x / y) 1\n[PROOFSTEP]\nsimpa only [div_mul_cancel', one_mul] using h\u2082 (x / y) 1 y\n[GOAL]\ncase a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Norm E\ninst\u271d\u00b9 : Group E\ninst\u271d : PseudoMetricSpace E\nh\u2081 : \u2200 (x : E), \u2016x\u2016 = dist x 1\nh\u2082 : \u2200 (x y z : E), dist (x * z) (y * z) \u2264 dist x y\nx y : E\n\u22a2 dist (x / y) 1 \u2264 dist x y\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv, \u2190 mul_right_inv y] using h\u2082 _ _ _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d : Group E\nf : GroupSeminorm E\nx : E\n\u22a2 dist x x = 0\n[PROOFSTEP]\nsimp only [div_self', map_one_eq_zero]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d : Group E\nf : GroupSeminorm E\nx y : E\n\u22a2 (fun x y => \u2191{ val := \u2191f (x / y), property := (_ : 0 \u2264 \u2191f (x / y)) }) x y = ENNReal.ofReal (dist x y)\n[PROOFSTEP]\nexact\n  ENNReal.coe_nnreal_eq\n    _\n      -- porting note: how did `mathlib3` solve this automatically?\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist a b = \u2016b / a\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\u1d50\u1d52\u1d56\nb c : E\n\u22a2 dist (a \u2022 b) (a \u2022 c) = dist b c\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 dist a 1 = \u2016a\u2016\n[PROOFSTEP]\nrw [dist_eq_norm_div, div_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 dist 1 a = \u2016a\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nhi : Isometry f\nh\u2081 : f 1 = 1\nx : E\n\u22a2 \u2016f x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 dist_one_right, \u2190 h\u2081, hi.dist_eq, dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : ProperSpace E\n\u22a2 Tendsto norm (cocompact E) atTop\n[PROOFSTEP]\nsimpa only [dist_one_right] using tendsto_dist_right_cocompact_atTop (1 : E)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 \u2016a / b\u2016 = \u2016b / a\u2016\n[PROOFSTEP]\nsimpa only [dist_eq_norm_div] using dist_comm a b\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 \u2016a\u207b\u00b9\u2016 = \u2016a\u2016\n[PROOFSTEP]\nsimpa using norm_div_rev 1 a\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist b (a * b) = \u2016a\u2016\n[PROOFSTEP]\nrw [\u2190 dist_one_left, \u2190 dist_mul_right 1 a b, one_mul]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist (a * b) b = \u2016a\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_mul_self_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b c : E\n\u22a2 dist (a / b) c = dist a (c * b)\n[PROOFSTEP]\nrw [\u2190 dist_mul_right _ _ b, div_mul_cancel']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b c : E\n\u22a2 dist a (b / c) = dist (a * c) b\n[PROOFSTEP]\nrw [\u2190 dist_mul_right _ _ c, div_mul_cancel']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 Tendsto Inv.inv (comap norm atTop) (comap norm atTop)\n[PROOFSTEP]\nsimpa only [norm_inv', tendsto_comap_iff, (\u00b7 \u2218 \u00b7)] using tendsto_comap\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 \u2016a * b\u2016 \u2264 \u2016a\u2016 + \u2016b\u2016\n[PROOFSTEP]\nsimpa [dist_eq_norm_div] using dist_triangle a 1 b\u207b\u00b9\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 0 \u2264 \u2016a\u2016\n[PROOFSTEP]\nrw [\u2190 dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 0 \u2264 dist a 1\n[PROOFSTEP]\nexact dist_nonneg\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 \u20161\u2016 = 0\n[PROOFSTEP]\nrw [\u2190 dist_one_right, dist_self]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a = 1 \u2192 \u2016a\u2016 = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 \u20161\u2016 = 0\n[PROOFSTEP]\nexact norm_one'\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : Subsingleton E\na : E\n\u22a2 \u2016a\u2016 = 0\n[PROOFSTEP]\nrw [Subsingleton.elim a 1, norm_one']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 0 < 1 + \u2016x\u2016 ^ 2\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 \u2016a / b\u2016 \u2264 \u2016a\u2016 + \u2016b\u2016\n[PROOFSTEP]\nsimpa [dist_eq_norm_div] using dist_triangle a 1 b\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist a b \u2264 \u2016a\u2016 + \u2016b\u2016\n[PROOFSTEP]\nrw [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 \u2016a / b\u2016 \u2264 \u2016a\u2016 + \u2016b\u2016\n[PROOFSTEP]\napply norm_div_le\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 |\u2016a\u2016 - \u2016b\u2016| \u2264 \u2016a / b\u2016\n[PROOFSTEP]\nsimpa [dist_eq_norm_div] using abs_dist_sub_le a b 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016u\u2016 \u2264 \u2016v\u2016 + \u2016u / v\u2016\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016u\u2016 \u2264 \u2016u / v\u2016 + \u2016v\u2016\n[PROOFSTEP]\nrefine' (norm_mul_le' _ _).trans_eq' _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016u\u2016 = \u2016u / v * v\u2016\n[PROOFSTEP]\nrw [div_mul_cancel']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016v\u2016 \u2264 \u2016u\u2016 + \u2016u / v\u2016\n[PROOFSTEP]\nrw [norm_div_rev]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016v\u2016 \u2264 \u2016u\u2016 + \u2016v / u\u2016\n[PROOFSTEP]\nexact norm_le_norm_add_norm_div' v u\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : E\n\u22a2 \u2016u\u2016 = \u2016u * v / v\u2016\n[PROOFSTEP]\nrw [mul_div_cancel'']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ny : E\n\u03b5 : \u211d\na : E\n\u22a2 a \u2208 ball y \u03b5 \u2194 a \u2208 {x | \u2016x / y\u2016 < \u03b5}\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\na : E\n\u22a2 a \u2208 ball 1 r \u2194 a \u2208 {x | \u2016x\u2016 < r}\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 b \u2208 ball a r \u2194 \u2016b / a\u2016 < r\n[PROOFSTEP]\nrw [mem_ball, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 b \u2208 ball a r \u2194 \u2016a / b\u2016 < r\n[PROOFSTEP]\nrw [mem_ball', dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2208 ball 1 r \u2194 \u2016a\u2016 < r\n[PROOFSTEP]\nrw [mem_ball, dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 b \u2208 closedBall a r \u2194 \u2016b / a\u2016 \u2264 r\n[PROOFSTEP]\nrw [mem_closedBall, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2208 closedBall 1 r \u2194 \u2016a\u2016 \u2264 r\n[PROOFSTEP]\nrw [mem_closedBall, dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 b \u2208 closedBall a r \u2194 \u2016a / b\u2016 \u2264 r\n[PROOFSTEP]\nrw [mem_closedBall', dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nh : b \u2208 closedBall a r\n\u22a2 \u2016b / a\u2016 \u2264 r\n[PROOFSTEP]\nrwa [\u2190 dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nh : b \u2208 ball a r\n\u22a2 \u2016b / a\u2016 < r\n[PROOFSTEP]\nrwa [\u2190 dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v w : E\n\u22a2 \u2016u / w\u2016 - \u2016v / w\u2016 \u2264 \u2016u / v\u2016\n[PROOFSTEP]\nsimpa only [div_div_div_cancel_right'] using norm_sub_norm_le' (u / w) (v / w)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 Bounded s \u2194 \u2203 C, \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 C\n[PROOFSTEP]\nsimpa only [Set.subset_def, mem_closedBall_one_iff] using bounded_iff_subset_ball (1 : E)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhs : Bounded s\nR\u2080 : \u211d\nhR\u2080 : \u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 R\u2080\n\u22a2 max R\u2080 1 > 0\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 b \u2208 sphere a r \u2194 \u2016b / a\u2016 = r\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2208 sphere 1 r \u2194 \u2016a\u2016 = r\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhr : r \u2260 0\nx : \u2191(sphere 1 r)\n\u22a2 \u2016\u2191x\u2016 \u2260 0\n[PROOFSTEP]\nrwa [norm_eq_of_mem_sphere' x]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\nl : Filter \u03b1\n\u22a2 (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, dist (f x) 1 < \u03b5) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\n[PROOFSTEP]\nsimp only [dist_one_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nx : E\ny : F\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd y) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x' : E), \u2016x' / x\u2016 < \u03b4 \u2192 \u2016f x' / y\u2016 < \u03b5\n[PROOFSTEP]\nsimp_rw [Metric.tendsto_nhds_nhds, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u2074 : SeminormedGroup E\ninst\u271d\u00b3 : SeminormedGroup F\ninst\u271d\u00b2 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d\u00b9 : Nonempty \u03b1\ninst\u271d : SemilatticeSup \u03b1\nu : \u03b1 \u2192 E\n\u22a2 CauchySeq u \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : \u03b1), N \u2264 m \u2192 \u2200 (n : \u03b1), N \u2264 n \u2192 \u2016u m / u n\u2016 < \u03b5\n[PROOFSTEP]\nsimp [Metric.cauchySeq_iff, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 HasBasis (\ud835\udcdd x) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {y | \u2016y / x\u2016 < \u03b5}\n[PROOFSTEP]\nsimp_rw [\u2190 ball_eq']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 HasBasis (\ud835\udcdd x) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => ball x \u03b5\n[PROOFSTEP]\nexact Metric.nhds_basis_ball\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 HasBasis (\ud835\udcdd 1) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {y | \u2016y\u2016 < \u03b5}\n[PROOFSTEP]\nconvert NormedCommGroup.nhds_basis_norm_lt (1 : E)\n[GOAL]\ncase h.e'_5.h.h.e'_2.h.h.e'_3.h.e'_3\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 x\u271d\u00b9 : \u211d\nx\u271d : E\n\u22a2 x\u271d = x\u271d / 1\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 HasBasis (\ud835\udce4 E) (fun \u03b5 => 0 < \u03b5) fun \u03b5 => {p | \u2016p.fst / p.snd\u2016 < \u03b5}\n[PROOFSTEP]\nconvert Metric.uniformity_basis_dist (\u03b1 := E) using 1\n[GOAL]\ncase h.e'_5\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 (fun \u03b5 => {p | \u2016p.fst / p.snd\u2016 < \u03b5}) = fun \u03b5 => {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nC : \u211d\nh : \u2200 (x : E), \u2016\u2191f x\u2016 \u2264 C * \u2016x\u2016\nx y : E\n\u22a2 dist (\u2191f x) (\u2191f y) \u2264 C * dist x y\n[PROOFSTEP]\nsimpa only [dist_eq_norm_div, map_div] using h (x / y)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nC : \u211d\u22650\n\u22a2 LipschitzOnWith C f s \u2194 \u2200 \u2983x : E\u2984, x \u2208 s \u2192 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2016f x / f y\u2016 \u2264 \u2191C * \u2016x / y\u2016\n[PROOFSTEP]\nsimp only [lipschitzOnWith_iff_dist_le_mul, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nC : \u211d\u22650\nh : LipschitzOnWith C f s\nha : a \u2208 s\nhb : b \u2208 s\nhr : \u2016a / b\u2016 \u2264 r\n\u22a2 \u2191C * \u2016a / b\u2016 \u2264 \u2191C * r\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nC : \u211d\u22650\n\u22a2 LipschitzWith C f \u2194 \u2200 (x y : E), \u2016f x / f y\u2016 \u2264 \u2191C * \u2016x / y\u2016\n[PROOFSTEP]\nsimp only [lipschitzWith_iff_dist_le_mul, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nC : \u211d\u22650\nh : LipschitzWith C f\nhr : \u2016a / b\u2016 \u2264 r\n\u22a2 \u2191C * \u2016a / b\u2016 \u2264 \u2191C * r\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\n\u22a2 Isometry \u2191f \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimp only [isometry_iff_dist_eq, dist_eq_norm_div, \u2190 map_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\n\u22a2 (\u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016) \u2194 \u2200 (x : E), \u2016\u2191f x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrefine' \u27e8fun h x => _, fun h x y => h _\u27e9\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nh : \u2200 (x y : E), \u2016\u2191f (x / y)\u2016 = \u2016x / y\u2016\nx : E\n\u22a2 \u2016\u2191f x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimpa using h x 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 a = 1 \u2192 \u2016a\u2016\u208a = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 \u20161\u2016\u208a = 0\n[PROOFSTEP]\nexact nnnorm_one'\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 edist a b = \u2191\u2016a / b\u2016\u208a\n[PROOFSTEP]\nrw [edist_dist, dist_eq_norm_div, ofReal_norm_eq_coe_nnnorm']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 edist x 1 = \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrw [edist_eq_coe_nnnorm_div, div_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\nr : \u211d\u22650\u221e\n\u22a2 a \u2208 EMetric.ball 1 r \u2194 \u2191\u2016a\u2016\u208a < r\n[PROOFSTEP]\nrw [EMetric.mem_ball, edist_eq_coe_nnnorm']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nK : \u211d\u22650\nh : \u2200 (x : E), \u2016x\u2016 \u2264 \u2191K * \u2016\u2191f x\u2016\nx y : E\n\u22a2 dist x y \u2264 \u2191K * dist (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nsimpa only [dist_eq_norm_div, map_div] using h (x / y)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nK : \u211d\u22650\nh : LipschitzWith K f\nhf : f 1 = 1\nx : E\n\u22a2 \u2016f x\u2016 \u2264 \u2191K * \u2016x\u2016\n[PROOFSTEP]\nsimpa only [dist_one_right, hf] using h.dist_le_mul x 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : E \u2192 F\nK : \u211d\u22650\nh : AntilipschitzWith K f\nhf : f 1 = 1\nx : E\n\u22a2 \u2016x\u2016 \u2264 \u2191K * \u2016f x\u2016\n[PROOFSTEP]\nsimpa only [dist_one_right, hf] using h.le_mul_dist x 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\na : Filter \u03b1\nb : E\n\u22a2 Tendsto f a (\ud835\udcdd b) \u2194 Tendsto (fun e => \u2016f e / b\u2016) a (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert tendsto_iff_dist_tendsto_zero (f := f) (x := a) (a := b) using 1\n[GOAL]\ncase h.e'_2.a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\na : Filter \u03b1\nb : E\n\u22a2 Tendsto (fun e => \u2016f e / b\u2016) a (\ud835\udcdd 0) \u2194 Tendsto (fun b_1 => dist (f b_1) b) a (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\na : Filter \u03b1\n\u22a2 Tendsto f a (\ud835\udcdd 1) \u2194 Tendsto (fun e => \u2016f e\u2016) a (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_iff_norm_tendsto_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\na : Filter \u03b1\n\u22a2 Tendsto (fun e => \u2016f e / 1\u2016) a (\ud835\udcdd 0) \u2194 Tendsto (fun e => \u2016f e\u2016) a (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [div_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 comap norm (\ud835\udcdd 0) = \ud835\udcdd 1\n[PROOFSTEP]\nsimpa only [dist_one_right] using nhds_comap_dist (1 : E)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 Tendsto (fun a => \u2016a / x\u2016) (\ud835\udcdd x) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa [dist_eq_norm_div] using tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (\ud835\udcdd x) _)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 Tendsto (fun a => \u2016a\u2016) (\ud835\udcdd x) (\ud835\udcdd \u2016x\u2016)\n[PROOFSTEP]\nsimpa using tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (1 : E)) _ _)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 Tendsto (fun a => \u2016a\u2016) (\ud835\udcdd 1) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using tendsto_norm_div_self (1 : E)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 Continuous fun a => \u2016a\u2016\n[PROOFSTEP]\nsimpa using continuous_id.dist (continuous_const : Continuous fun _a => (1 : E))\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 LipschitzWith 1 norm\n[PROOFSTEP]\nsimpa only [dist_one_left] using LipschitzWith.dist_right (1 : E)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx : E\n\u22a2 x \u2208 closure {1} \u2194 \u2016x\u2016 = 0\n[PROOFSTEP]\nrw [\u2190 closedBall_zero', mem_closedBall_one_iff, (norm_nonneg' x).le_iff_eq]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : Tendsto f l (\ud835\udcdd 1)\nhg : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 g)\nop : E \u2192 F \u2192 G\nh_op : \u2203 A, \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\n\u22a2 Tendsto (fun x => op (f x) (g x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\ncases' h_op with A h_op\n[GOAL]\ncase intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : Tendsto f l (\ud835\udcdd 1)\nhg : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 g)\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\n\u22a2 Tendsto (fun x => op (f x) (g x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\nrcases hg with \u27e8C, hC\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : Tendsto f l (\ud835\udcdd 1)\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (x : \u211d) in map (norm \u2218 g) l, (fun x x_1 => x \u2264 x_1) x C\n\u22a2 Tendsto (fun x => op (f x) (g x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [eventually_map] at hC \n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : Tendsto f l (\ud835\udcdd 1)\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u22a2 Tendsto (fun x => op (f x) (g x)) l (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [NormedCommGroup.tendsto_nhds_one] at hf \u22a2\n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016op (f x) (g x)\u2016 < \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5\u2080\n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016op (f x) (g x)\u2016 < \u03b5\n[PROOFSTEP]\nrcases exists_pos_mul_lt \u03b5\u2080 (A * C) with \u27e8\u03b4, \u03b4\u2080, h\u03b4\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016op (f x) (g x)\u2016 < \u03b5\n[PROOFSTEP]\nfilter_upwards [hf \u03b4 \u03b4\u2080, hC] with i hf hg\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\n\u22a2 \u2016op (f i) (g i)\u2016 < \u03b5\n[PROOFSTEP]\nrefine' (h_op _ _).trans_lt _\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\n\u22a2 A * \u2016f i\u2016 * \u2016g i\u2016 < \u03b5\n[PROOFSTEP]\ncases' le_total A 0 with hA hA\n[GOAL]\ncase h.inl\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\nhA : A \u2264 0\n\u22a2 A * \u2016f i\u2016 * \u2016g i\u2016 < \u03b5\n[PROOFSTEP]\nexact\n  (mul_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonpos_of_nonneg hA <| norm_nonneg' _) <| norm_nonneg' _).trans_lt \u03b5\u2080\n[GOAL]\ncase h.inr\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\nhA : 0 \u2264 A\n\u22a2 A * \u2016f i\u2016 * \u2016g i\u2016 < \u03b5\n[PROOFSTEP]\ncalc\n  A * \u2016f i\u2016 * \u2016g i\u2016 \u2264 A * \u03b4 * C := by gcongr; exact hg\n  _ = A * C * \u03b4 := (mul_right_comm _ _ _)\n  _ < \u03b5 := h\u03b4\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\nhA : 0 \u2264 A\n\u22a2 A * \u2016f i\u2016 * \u2016g i\u2016 \u2264 A * \u03b4 * C\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2082\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nl : Filter \u03b1\nhf\u271d : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b1) in l, \u2016f x\u2016 < \u03b5\nop : E \u2192 F \u2192 G\nA : \u211d\nh_op : \u2200 (x : E) (y : F), \u2016op x y\u2016 \u2264 A * \u2016x\u2016 * \u2016y\u2016\nC : \u211d\nhC : \u2200\u1da0 (a : \u03b1) in l, (fun x x_1 => x \u2264 x_1) ((norm \u2218 g) a) C\n\u03b5 : \u211d\n\u03b5\u2080 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b4\u2080 : 0 < \u03b4\nh\u03b4 : A * C * \u03b4 < \u03b5\ni : \u03b1\nhf : \u2016f i\u2016 < \u03b4\nhg : (norm \u2218 g) i \u2264 C\nhA : 0 \u2264 A\n\u22a2 \u2016g i\u2016 \u2264 C\n[PROOFSTEP]\nexact hg\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2208 closure s \u2194 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 b, b \u2208 s \u2227 \u2016a / b\u2016 < \u03b5\n[PROOFSTEP]\nsimp [Metric.mem_closure_iff, dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : T0Space E\na : E\n\u22a2 \u2016a\u2016 \u2264 0 \u2194 a = 1\n[PROOFSTEP]\nletI : NormedGroup E := { \u2039SeminormedGroup E\u203a with toMetricSpace := MetricSpace.ofT0PseudoMetricSpace E }\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : T0Space E\na : E\nthis : NormedGroup E :=\n  let src := inst\u271d\u00b3;\n  NormedGroup.mk\n\u22a2 \u2016a\u2016 \u2264 0 \u2194 a = 1\n[PROOFSTEP]\nrw [\u2190 dist_one_right, dist_le_zero]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b3 : SeminormedGroup E\ninst\u271d\u00b2 : SeminormedGroup F\ninst\u271d\u00b9 : SeminormedGroup G\ns : Set E\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : T0Space E\na : E\n\u22a2 0 < \u2016a\u2016 \u2194 a \u2260 1\n[PROOFSTEP]\nrw [\u2190 not_le, norm_le_zero_iff''']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns\u271d : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\ns : Set \u03ba\nl : Filter \u03b9\n\u22a2 TendstoUniformlyOn f 1 l s \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (i : \u03b9) in l, \u2200 (x : \u03ba), x \u2208 s \u2192 \u2016f i x\u2016 < \u03b5\n[PROOFSTEP]\nsimp_rw [tendstoUniformlyOn_iff, Pi.one_apply, dist_one_left]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\n\u22a2 UniformCauchySeqOnFilter f l l' \u2194 TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l \u00d7\u02e2 l) l'\n[PROOFSTEP]\nrefine' \u27e8fun hf u hu => _, fun hf u hu => _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) \u2208 u\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, H\u27e9 := uniformity_basis_dist.mem_uniformity_iff.mp hu\n[GOAL]\ncase refine'_1.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nH : \u2200 (a b : G), (a, b) \u2208 {p | dist p.fst p.snd < \u03b5} \u2192 (a, b) \u2208 u\n\u22a2 \u2200\u1da0 (n : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) \u2208 u\n[PROOFSTEP]\nrefine'\n  (hf {p : G \u00d7 G | dist p.fst p.snd < \u03b5} <| dist_mem_uniformity h\u03b5).mono fun x hx =>\n    H 1 (f x.fst.fst x.snd / f x.fst.snd x.snd) _\n[GOAL]\ncase refine'_1.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nH : \u2200 (a b : G), (a, b) \u2208 {p | dist p.fst p.snd < \u03b5} \u2192 (a, b) \u2208 u\nx : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba\nhx : (f x.fst.fst x.snd, f x.fst.snd x.snd) \u2208 {p | dist p.fst p.snd < \u03b5}\n\u22a2 (1, f x.fst.fst x.snd / f x.fst.snd x.snd) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa [dist_eq_norm_div, norm_div_rev] using hx\n[GOAL]\ncase refine'_2\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l \u00d7\u02e2 l) l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', (f m.fst.fst m.snd, f m.fst.snd m.snd) \u2208 u\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, H\u27e9 := uniformity_basis_dist.mem_uniformity_iff.mp hu\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l \u00d7\u02e2 l) l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nH : \u2200 (a b : G), (a, b) \u2208 {p | dist p.fst p.snd < \u03b5} \u2192 (a, b) \u2208 u\n\u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba) in (l \u00d7\u02e2 l) \u00d7\u02e2 l', (f m.fst.fst m.snd, f m.fst.snd m.snd) \u2208 u\n[PROOFSTEP]\nrefine'\n  (hf {p : G \u00d7 G | dist p.fst p.snd < \u03b5} <| dist_mem_uniformity h\u03b5).mono fun x hx =>\n    H (f x.fst.fst x.snd) (f x.fst.snd x.snd) _\n[GOAL]\ncase refine'_2.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\nl : Filter \u03b9\nl' : Filter \u03ba\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l \u00d7\u02e2 l) l'\nu : Set (G \u00d7 G)\nhu : u \u2208 \ud835\udce4 G\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nH : \u2200 (a b : G), (a, b) \u2208 {p | dist p.fst p.snd < \u03b5} \u2192 (a, b) \u2208 u\nx : (\u03b9 \u00d7 \u03b9) \u00d7 \u03ba\nhx : (OfNat.ofNat 1 x.snd, (fun n z => f n.fst z / f n.snd z) x.fst x.snd) \u2208 {p | dist p.fst p.snd < \u03b5}\n\u22a2 (f x.fst.fst x.snd, f x.fst.snd x.snd) \u2208 {p | dist p.fst p.snd < \u03b5}\n[PROOFSTEP]\nsimpa [dist_eq_norm_div, norm_div_rev] using hx\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedGroup E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : SeminormedGroup G\ns\u271d : Set E\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nf : \u03b9 \u2192 \u03ba \u2192 G\ns : Set \u03ba\nl : Filter \u03b9\n\u22a2 UniformCauchySeqOn f l s \u2194 TendstoUniformlyOn (fun n z => f n.fst z / f n.snd z) 1 (l \u00d7\u02e2 l) s\n[PROOFSTEP]\nrw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter, uniformCauchySeqOn_iff_uniformCauchySeqOnFilter,\n  SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Group E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nsrc\u271d : PseudoMetricSpace E := PseudoMetricSpace.induced (\u2191f) toPseudoMetricSpace\nx y : E\n\u22a2 dist x y = \u2016x / y\u2016\n[PROOFSTEP]\nsimp only [map_div, \u2190 dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : Group E\ninst\u271d\u00b9 : SeminormedGroup F\ninst\u271d : MonoidHomClass \ud835\udcd5 E F\nf : \ud835\udcd5\nsrc\u271d : PseudoMetricSpace E := PseudoMetricSpace.induced (\u2191f) toPseudoMetricSpace\nx y : E\n\u22a2 dist x y = dist (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b c : E\n\u22a2 dist (a \u2022 b) (a \u2022 c) = dist b c\n[PROOFSTEP]\nsimp [dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx y : E\n\u22a2 dist x\u207b\u00b9 y = dist x y\u207b\u00b9\n[PROOFSTEP]\nsimp_rw [dist_eq_norm_div, \u2190 norm_inv' (x\u207b\u00b9 / y), inv_div, div_inv_eq_mul, mul_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist a (a * b) = \u2016b\u2016\n[PROOFSTEP]\nrw [\u2190 dist_one_left, \u2190 dist_mul_left a 1 b, mul_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist (a * b) a = \u2016b\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_self_mul_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist a (a / b) = \u2016b\u2016\n[PROOFSTEP]\nrw [div_eq_mul_inv, dist_self_mul_right, norm_inv']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na b : E\n\u22a2 dist (a / b) a = \u2016b\u2016\n[PROOFSTEP]\nrw [dist_comm, dist_self_div_right]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 dist (a\u2081 * a\u2082) (b\u2081 * b\u2082) \u2264 dist a\u2081 b\u2081 + dist a\u2082 b\u2082\n[PROOFSTEP]\nsimpa only [dist_mul_left, dist_mul_right] using dist_triangle (a\u2081 * a\u2082) (b\u2081 * a\u2082) (b\u2081 * b\u2082)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 dist (a\u2081 / a\u2082) (b\u2081 / b\u2082) \u2264 dist a\u2081 b\u2081 + dist a\u2082 b\u2082\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv, dist_inv_inv] using dist_mul_mul_le a\u2081 a\u2082\u207b\u00b9 b\u2081 b\u2082\u207b\u00b9\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 |dist a\u2081 b\u2081 - dist a\u2082 b\u2082| \u2264 dist (a\u2081 * a\u2082) (b\u2081 * b\u2082)\n[PROOFSTEP]\nsimpa only [dist_mul_left, dist_mul_right, dist_comm b\u2082] using abs_dist_sub_le (a\u2081 * a\u2082) (b\u2081 * b\u2082) (b\u2081 * a\u2082)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 \u2016Multiset.prod m\u2016 \u2264 Multiset.sum (Multiset.map (fun x => \u2016x\u2016) m)\n[PROOFSTEP]\nrw [\u2190 Multiplicative.ofAdd_le, ofAdd_multiset_prod, Multiset.map_map]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 \u2191Multiplicative.ofAdd \u2016Multiset.prod m\u2016 \u2264 Multiset.prod (Multiset.map (\u2191Multiplicative.ofAdd \u2218 fun x => \u2016x\u2016) m)\n[PROOFSTEP]\nrefine' Multiset.le_prod_of_submultiplicative (Multiplicative.ofAdd \u2218 norm) _ (fun x y => _) _\n[GOAL]\ncase refine'_1\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 (\u2191Multiplicative.ofAdd \u2218 norm) 1 = 1\n[PROOFSTEP]\nsimp only [comp_apply, norm_one', ofAdd_zero]\n[GOAL]\ncase refine'_2\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\nx y : E\n\u22a2 (\u2191Multiplicative.ofAdd \u2218 norm) (x * y) \u2264 (\u2191Multiplicative.ofAdd \u2218 norm) x * (\u2191Multiplicative.ofAdd \u2218 norm) y\n[PROOFSTEP]\nexact norm_mul_le' x y\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\n\u22a2 \u2016\u220f i in s, f i\u2016 \u2264 \u2211 i in s, \u2016f i\u2016\n[PROOFSTEP]\nrw [\u2190 Multiplicative.ofAdd_le, ofAdd_sum]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\n\u22a2 \u2191Multiplicative.ofAdd \u2016\u220f i in s, f i\u2016 \u2264 \u220f i in s, \u2191Multiplicative.ofAdd \u2016f i\u2016\n[PROOFSTEP]\nrefine' Finset.le_prod_of_submultiplicative (Multiplicative.ofAdd \u2218 norm) _ (fun x y => _) _ _\n[GOAL]\ncase refine'_1\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\n\u22a2 (\u2191Multiplicative.ofAdd \u2218 norm) 1 = 1\n[PROOFSTEP]\nsimp only [comp_apply, norm_one', ofAdd_zero]\n[GOAL]\ncase refine'_2\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\nx y : E\n\u22a2 (\u2191Multiplicative.ofAdd \u2218 norm) (x * y) \u2264 (\u2191Multiplicative.ofAdd \u2218 norm) x * (\u2191Multiplicative.ofAdd \u2218 norm) y\n[PROOFSTEP]\nexact norm_mul_le' x y\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf a : \u03b9 \u2192 E\nd : \u03b9 \u2192 \u211d\nh : \u2200 (b : \u03b9), b \u2208 s \u2192 dist (f b) (a b) \u2264 d b\n\u22a2 dist (\u220f b in s, f b) (\u220f b in s, a b) \u2264 \u2211 b in s, d b\n[PROOFSTEP]\nsimp only [dist_eq_norm_div, \u2190 Finset.prod_div_distrib] at *\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf a : \u03b9 \u2192 E\nd : \u03b9 \u2192 \u211d\nh : \u2200 (b : \u03b9), b \u2208 s \u2192 \u2016f b / a b\u2016 \u2264 d b\n\u22a2 \u2016\u220f x in s, f x / a x\u2016 \u2264 \u2211 b in s, d b\n[PROOFSTEP]\nexact norm_prod_le_of_le s h\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a * b \u2208 ball a r \u2194 \u2016b\u2016 < r\n[PROOFSTEP]\nrw [mem_ball_iff_norm'', mul_div_cancel''']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a * b \u2208 closedBall a r \u2194 \u2016b\u2016 \u2264 r\n[PROOFSTEP]\nrw [mem_closedBall_iff_norm'', mul_div_cancel''']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\n\u22a2 (fun x x_1 => x * x_1) b \u207b\u00b9' ball a r = ball (a / b) r\n[PROOFSTEP]\next c\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\nc : E\n\u22a2 c \u2208 (fun x x_1 => x * x_1) b \u207b\u00b9' ball a r \u2194 c \u2208 ball (a / b) r\n[PROOFSTEP]\nsimp only [dist_eq_norm_div, Set.mem_preimage, mem_ball, div_div_eq_mul_div, mul_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\n\u22a2 (fun x x_1 => x * x_1) b \u207b\u00b9' closedBall a r = closedBall (a / b) r\n[PROOFSTEP]\next c\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\nc : E\n\u22a2 c \u2208 (fun x x_1 => x * x_1) b \u207b\u00b9' closedBall a r \u2194 c \u2208 closedBall (a / b) r\n[PROOFSTEP]\nsimp only [dist_eq_norm_div, Set.mem_preimage, mem_closedBall, div_div_eq_mul_div, mul_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\n\u22a2 (fun x x_1 => x * x_1) b \u207b\u00b9' sphere a r = sphere (a / b) r\n[PROOFSTEP]\next c\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 : \u211d\na b : E\nr : \u211d\nc : E\n\u22a2 c \u2208 (fun x x_1 => x * x_1) b \u207b\u00b9' sphere a r \u2194 c \u2208 sphere (a / b) r\n[PROOFSTEP]\nsimp only [Set.mem_preimage, mem_sphere_iff_norm', div_div_eq_mul_div, mul_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\na : E\n\u22a2 \u2016a ^ n\u2016 \u2264 \u2191n * \u2016a\u2016\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\n\u22a2 \u2016a ^ Nat.zero\u2016 \u2264 \u2191Nat.zero * \u2016a\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\na : E\nn : \u2115\nih : \u2016a ^ n\u2016 \u2264 \u2191n * \u2016a\u2016\n\u22a2 \u2016a ^ Nat.succ n\u2016 \u2264 \u2191(Nat.succ n) * \u2016a\u2016\n[PROOFSTEP]\nsimpa only [pow_succ', Nat.cast_succ, add_mul, one_mul] using norm_mul_le_of_le ih le_rfl\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\na : E\n\u22a2 \u2016a ^ n\u2016\u208a \u2264 \u2191n * \u2016a\u2016\u208a\n[PROOFSTEP]\nsimpa only [\u2190 NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_nat_cast] using norm_pow_le_mul_norm n a\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nh : a \u2208 closedBall b r\n\u22a2 a ^ n \u2208 closedBall (b ^ n) (n \u2022 r)\n[PROOFSTEP]\nsimp only [mem_closedBall, dist_eq_norm_div, \u2190 div_pow] at h \u22a2\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nh : \u2016a / b\u2016 \u2264 r\n\u22a2 \u2016(a / b) ^ n\u2016 \u2264 n \u2022 r\n[PROOFSTEP]\nrefine' (norm_pow_le_mul_norm n (a / b)).trans _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nh : \u2016a / b\u2016 \u2264 r\n\u22a2 \u2191n * \u2016a / b\u2016 \u2264 n \u2022 r\n[PROOFSTEP]\nsimpa only [nsmul_eq_mul] using mul_le_mul_of_nonneg_left h n.cast_nonneg\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nhn : 0 < n\nh : a \u2208 ball b r\n\u22a2 a ^ n \u2208 ball (b ^ n) (n \u2022 r)\n[PROOFSTEP]\nsimp only [mem_ball, dist_eq_norm_div, \u2190 div_pow] at h \u22a2\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nhn : 0 < n\nh : \u2016a / b\u2016 < r\n\u22a2 \u2016(a / b) ^ n\u2016 < n \u2022 r\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (norm_pow_le_mul_norm n (a / b)) _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nhn : 0 < n\nh : \u2016a / b\u2016 < r\n\u22a2 \u2191n * \u2016a / b\u2016 < n \u2022 r\n[PROOFSTEP]\nreplace hn : 0 < (n : \u211d)\n[GOAL]\ncase hn\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nhn : 0 < n\nh : \u2016a / b\u2016 < r\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nh : \u2016a / b\u2016 < r\nhn : 0 < \u2191n\n\u22a2 \u2191n * \u2016a / b\u2016 < n \u2022 r\n[PROOFSTEP]\nrw [nsmul_eq_mul]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\nh : \u2016a / b\u2016 < r\nhn : 0 < \u2191n\n\u22a2 \u2191n * \u2016a / b\u2016 < \u2191n * r\n[PROOFSTEP]\nnlinarith\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nc : E\n\u22a2 a * c \u2208 closedBall (b * c) r \u2194 a \u2208 closedBall b r\n[PROOFSTEP]\nsimp only [mem_closedBall, dist_eq_norm_div, mul_div_mul_right_eq_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nc : E\n\u22a2 a * c \u2208 ball (b * c) r \u2194 a \u2208 ball b r\n[PROOFSTEP]\nsimp only [mem_ball, dist_eq_norm_div, mul_div_mul_right_eq_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2022 closedBall b r = closedBall (a \u2022 b) r\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx\u271d : E\n\u22a2 x\u271d \u2208 a \u2022 closedBall b r \u2194 x\u271d \u2208 closedBall (a \u2022 b) r\n[PROOFSTEP]\nsimp [mem_closedBall, Set.mem_smul_set, dist_eq_norm_div, _root_.div_eq_inv_mul, \u2190 eq_inv_mul_iff_mul_eq, mul_assoc]\n  -- porting note: `ENNReal.div_eq_inv_mul` should be `protected`?\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 a \u2022 ball b r = ball (a \u2022 b) r\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nx\u271d : E\n\u22a2 x\u271d \u2208 a \u2022 ball b r \u2194 x\u271d \u2208 ball (a \u2022 b) r\n[PROOFSTEP]\nsimp [mem_ball, Set.mem_smul_set, dist_eq_norm_div, _root_.div_eq_inv_mul, \u2190 eq_inv_mul_iff_mul_eq, mul_assoc]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nobtain \u27e8u : \u2115 \u2192 E, u_in : \u2200 n, u n \u2208 s, lim_u : Tendsto u atTop (\ud835\udcdd a)\u27e9 := mem_closure_iff_seq_limit.mp hg\n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nobtain \u27e8n\u2080, hn\u2080\u27e9 : \u2203 n\u2080, \u2200 n \u2265 n\u2080, \u2016u n / a\u2016 < b 0 :=\n  haveI : {x | \u2016x / a\u2016 < b 0} \u2208 \ud835\udcdd a := by\n    simp_rw [\u2190 dist_eq_norm_div]\n    exact Metric.ball_mem_nhds _ (b_pos _)\n  Filter.tendsto_atTop'.mp lim_u _ this\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 {x | \u2016x / a\u2016 < b 0} \u2208 \ud835\udcdd a\n[PROOFSTEP]\nsimp_rw [\u2190 dist_eq_norm_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\n\u22a2 {x | dist x a < b 0} \u2208 \ud835\udcdd a\n[PROOFSTEP]\nexact Metric.ball_mem_nhds _ (b_pos _)\n[GOAL]\ncase intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nset z : \u2115 \u2192 E := fun n => u (n + n\u2080)\n[GOAL]\ncase intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nhave lim_z : Tendsto z atTop (\ud835\udcdd a) := lim_u.comp (tendsto_add_atTop_nat n\u2080)\n[GOAL]\ncase intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nhave mem_\ud835\udce4 : \u2200 n, {p : E \u00d7 E | \u2016p.1 / p.2\u2016 < b (n + 1)} \u2208 \ud835\udce4 E := fun n => by\n  simpa [\u2190 dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n[PROOFSTEP]\nsimpa [\u2190 dist_eq_norm_div] using Metric.dist_mem_uniformity (b_pos <| n + 1)\n[GOAL]\ncase intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nobtain \u27e8\u03c6 : \u2115 \u2192 \u2115, \u03c6_extr : StrictMono \u03c6, h\u03c6 : \u2200 n, \u2016z (\u03c6 <| n + 1) / z (\u03c6 n)\u2016 < b (n + 1)\u27e9 :=\n  lim_z.cauchySeq.subseq_mem mem_\ud835\udce4\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nset w : \u2115 \u2192 E := z \u2218 \u03c6\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nhave hw : Tendsto w atTop (\ud835\udcdd a) := lim_z.comp \u03c6_extr.tendsto_atTop\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nset v : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n\u22a2 \u2203 v,\n    Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd a) \u2227\n      (\u2200 (n : \u2115), v n \u2208 s) \u2227 \u2016v 0 / a\u2016 < b 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nrefine' \u27e8v, Tendsto.congr (Finset.eq_prod_range_div' w) hw, _, hn\u2080 _ (n\u2080.le_add_left _), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n\u22a2 \u2200 (n : \u2115), v n \u2208 s\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.zero\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n\u22a2 v Nat.zero \u2208 s\n[PROOFSTEP]\nchange w 0 \u2208 s\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.zero\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n\u22a2 w 0 \u2208 s\n[PROOFSTEP]\napply u_in\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.succ\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nn\u271d : \u2115\n\u22a2 v (Nat.succ n\u271d) \u2208 s\n[PROOFSTEP]\napply s.div_mem\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.succ.hx\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nn\u271d : \u2115\n\u22a2 w (Nat.succ n\u271d) \u2208 s\n[PROOFSTEP]\napply u_in\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.succ.hy\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nn\u271d : \u2115\n\u22a2 w (Nat.succ n\u271d - 1) \u2208 s\n[PROOFSTEP]\napply u_in\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < b n\n[PROOFSTEP]\nintro l hl\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nl : \u2115\nhl : 0 < l\n\u22a2 \u2016v l\u2016 < b l\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 : \u2203 k, l = k + 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nl : \u2115\nhl : 0 < l\n\u22a2 \u2203 k, l = k + 1\ncase intro.intro.intro.intro.intro.refine'_2.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nk : \u2115\nhl : 0 < k + 1\n\u22a2 \u2016v (k + 1)\u2016 < b (k + 1)\n[PROOFSTEP]\nexact Nat.exists_eq_succ_of_ne_zero hl.ne'\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Subgroup E\nhg : a \u2208 closure \u2191s\nb : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < b n\nu : \u2115 \u2192 E\nu_in : \u2200 (n : \u2115), u n \u2208 s\nlim_u : Tendsto u atTop (\ud835\udcdd a)\nn\u2080 : \u2115\nhn\u2080 : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 \u2016u n / a\u2016 < b 0\nz : \u2115 \u2192 E := fun n => u (n + n\u2080)\nlim_z : Tendsto z atTop (\ud835\udcdd a)\nmem_\ud835\udce4 : \u2200 (n : \u2115), {p | \u2016p.fst / p.snd\u2016 < b (n + 1)} \u2208 \ud835\udce4 E\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6_extr : StrictMono \u03c6\nh\u03c6 : \u2200 (n : \u2115), \u2016z (\u03c6 (n + 1)) / z (\u03c6 n)\u2016 < b (n + 1)\nw : \u2115 \u2192 E := z \u2218 \u03c6\nhw : Tendsto w atTop (\ud835\udcdd a)\nv : \u2115 \u2192 E := fun i => if i = 0 then w 0 else w i / w (i - 1)\nk : \u2115\nhl : 0 < k + 1\n\u22a2 \u2016v (k + 1)\u2016 < b (k + 1)\n[PROOFSTEP]\napply h\u03c6\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\n\u22a2 \u2203 a,\n    Tendsto (fun n => \u220f i in range (n + 1), \u2191j (a i)) atTop (\ud835\udcdd b) \u2227\n      \u2016\u2191j (a 0) / b\u2016 < f 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016\u2191j (a n)\u2016 < f n\n[PROOFSTEP]\nobtain \u27e8v, sum_v, v_in, hv\u2080, hv_pos\u27e9 := controlled_prod_of_mem_closure hb b_pos\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\nv : \u2115 \u2192 F\nsum_v : Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd b)\nv_in : \u2200 (n : \u2115), v n \u2208 MonoidHom.range j\nhv\u2080 : \u2016v 0 / b\u2016 < f 0\nhv_pos : \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < f n\n\u22a2 \u2203 a,\n    Tendsto (fun n => \u220f i in range (n + 1), \u2191j (a i)) atTop (\ud835\udcdd b) \u2227\n      \u2016\u2191j (a 0) / b\u2016 < f 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016\u2191j (a n)\u2016 < f n\n[PROOFSTEP]\nchoose g hg using v_in\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\nv : \u2115 \u2192 F\nsum_v : Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd b)\nhv\u2080 : \u2016v 0 / b\u2016 < f 0\nhv_pos : \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < f n\ng : \u2115 \u2192 E\nhg : \u2200 (n : \u2115), \u2191j (g n) = v n\n\u22a2 \u2203 a,\n    Tendsto (fun n => \u220f i in range (n + 1), \u2191j (a i)) atTop (\ud835\udcdd b) \u2227\n      \u2016\u2191j (a 0) / b\u2016 < f 0 \u2227 \u2200 (n : \u2115), 0 < n \u2192 \u2016\u2191j (a n)\u2016 < f n\n[PROOFSTEP]\nexact \u27e8g, by simpa [\u2190 hg] using sum_v, by simpa [hg 0] using hv\u2080, fun n hn => by simpa [hg] using hv_pos n hn\u27e9\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\nv : \u2115 \u2192 F\nsum_v : Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd b)\nhv\u2080 : \u2016v 0 / b\u2016 < f 0\nhv_pos : \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < f n\ng : \u2115 \u2192 E\nhg : \u2200 (n : \u2115), \u2191j (g n) = v n\n\u22a2 Tendsto (fun n => \u220f i in range (n + 1), \u2191j (g i)) atTop (\ud835\udcdd b)\n[PROOFSTEP]\nsimpa [\u2190 hg] using sum_v\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\nv : \u2115 \u2192 F\nsum_v : Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd b)\nhv\u2080 : \u2016v 0 / b\u2016 < f 0\nhv_pos : \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < f n\ng : \u2115 \u2192 E\nhg : \u2200 (n : \u2115), \u2191j (g n) = v n\n\u22a2 \u2016\u2191j (g 0) / b\u2016 < f 0\n[PROOFSTEP]\nsimpa [hg 0] using hv\u2080\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nj : E \u2192* F\nb : F\nhb : b \u2208 closure \u2191(MonoidHom.range j)\nf : \u2115 \u2192 \u211d\nb_pos : \u2200 (n : \u2115), 0 < f n\nv : \u2115 \u2192 F\nsum_v : Tendsto (fun n => \u220f i in range (n + 1), v i) atTop (\ud835\udcdd b)\nhv\u2080 : \u2016v 0 / b\u2016 < f 0\nhv_pos : \u2200 (n : \u2115), 0 < n \u2192 \u2016v n\u2016 < f n\ng : \u2115 \u2192 E\nhg : \u2200 (n : \u2115), \u2191j (g n) = v n\nn : \u2115\nhn : 0 < n\n\u22a2 \u2016\u2191j (g n)\u2016 < f n\n[PROOFSTEP]\nsimpa [hg] using hv_pos n hn\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 edist (a\u2081 * a\u2082) (b\u2081 * b\u2082) \u2264 edist a\u2081 b\u2081 + edist a\u2082 b\u2082\n[PROOFSTEP]\nsimp only [edist_nndist]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 \u2191(nndist (a\u2081 * a\u2082) (b\u2081 * b\u2082)) \u2264 \u2191(nndist a\u2081 b\u2081) + \u2191(nndist a\u2082 b\u2082)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081\u271d a\u2082\u271d b b\u2081\u271d b\u2082\u271d : E\nr r\u2081 r\u2082 : \u211d\na\u2081 a\u2082 b\u2081 b\u2082 : E\n\u22a2 nndist (a\u2081 * a\u2082) (b\u2081 * b\u2082) \u2264 nndist a\u2081 b\u2081 + nndist a\u2082 b\u2082\n[PROOFSTEP]\napply nndist_mul_mul_le\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 \u2191\u2016Multiset.prod m\u2016\u208a \u2264 \u2191(Multiset.sum (Multiset.map (fun x => \u2016x\u2016\u208a) m))\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 \u2016Multiset.prod m\u2016 \u2264 Multiset.sum (Multiset.map toReal (Multiset.map (fun x => \u2016x\u2016\u208a) m))\n[PROOFSTEP]\nrw [Multiset.map_map]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : Multiset E\n\u22a2 \u2016Multiset.prod m\u2016 \u2264 Multiset.sum (Multiset.map (toReal \u2218 fun x => \u2016x\u2016\u208a) m)\n[PROOFSTEP]\nexact norm_multiset_prod_le _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\n\u22a2 \u2191\u2016\u220f a in s, f a\u2016\u208a \u2264 \u2191(\u2211 a in s, \u2016f a\u2016\u208a)\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ns : Finset \u03b9\nf : \u03b9 \u2192 E\n\u22a2 \u2016\u220f a in s, f a\u2016 \u2264 \u2211 x in s, \u2016f x\u2016\n[PROOFSTEP]\nexact norm_prod_le _ _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\n\u22a2 \u2191\u20162\u2016\u208a = \u21912\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\n\u22a2 \u2016|r|\u2016\u208a = \u2016r\u2016\u208a\n[PROOFSTEP]\nsimp [nnnorm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191\u2016r\u2016\u208a = ENNReal.ofReal r\n[PROOFSTEP]\nrw [\u2190 ofReal_norm_eq_coe_nnnorm, norm_of_nonneg hr]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\n\u22a2 \u2191\u2016r\u2016\u208a = ENNReal.ofReal |r|\n[PROOFSTEP]\nrw [\u2190 Real.nnnorm_abs r, Real.ennnorm_eq_ofReal (abs_nonneg _)]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhr : 0 \u2264 r\n\u22a2 toNNReal r = \u2016r\u2016\u208a\n[PROOFSTEP]\nrw [Real.toNNReal_of_nonneg hr]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhr : 0 \u2264 r\n\u22a2 { val := r, property := hr } = \u2016r\u2016\u208a\n[PROOFSTEP]\next\n[GOAL]\ncase a\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2191{ val := r, property := hr } = \u2191\u2016r\u2016\u208a\n[PROOFSTEP]\nrw [coe_mk, coe_nnnorm r, Real.norm_eq_abs r, abs_of_nonneg hr]\n  -- porting note: this is due to the change from `Subtype.val` to `NNReal.toReal` for the coercion\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\n\u22a2 ENNReal.ofReal r \u2264 \u2191\u2016r\u2016\u208a\n[PROOFSTEP]\nobtain hr | hr := le_total 0 r\n[GOAL]\ncase inl\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\nhr : 0 \u2264 r\n\u22a2 ENNReal.ofReal r \u2264 \u2191\u2016r\u2016\u208a\n[PROOFSTEP]\nexact (Real.ennnorm_eq_ofReal hr).ge\n[GOAL]\ncase inr\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\nhr : r \u2264 0\n\u22a2 ENNReal.ofReal r \u2264 \u2191\u2016r\u2016\u208a\n[PROOFSTEP]\nrw [ENNReal.ofReal_eq_zero.2 hr]\n[GOAL]\ncase inr\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr\u271d r\u2081 r\u2082 r : \u211d\nhr : r \u2264 0\n\u22a2 0 \u2264 \u2191\u2016r\u2016\u208a\n[PROOFSTEP]\nexact bot_le\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm n : \u2124\n\u22a2 dist m n = \u2016m - n\u2016\n[PROOFSTEP]\nsimp only [Int.dist_eq, norm, Int.cast_sub]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2124\n\u22a2 \u2016\u2191n\u2016 = \u2191|n|\n[PROOFSTEP]\nrw [Real.norm_eq_abs, cast_abs]\n  -- porting note: I'm not sure why this isn't `rfl` anymore, but I suspect it's about coercions\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2115\n\u22a2 \u2016\u2191n\u2016 = \u2191n\n[PROOFSTEP]\nsimp [Int.norm_eq_abs]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2124\n\u22a2 \u2191\u2191(natAbs n) = \u2191\u2191(natAbs n)\n[PROOFSTEP]\nsimp only [Int.cast_ofNat, NNReal.coe_nat_cast]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nn : \u2124\n\u22a2 \u2191\u2191(natAbs n) = \u2191|n|\n[PROOFSTEP]\nsimp only [Int.coe_natAbs, Int.cast_abs]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nz : \u2124\nc : \u211d\u22650\n\u22a2 |z| \u2264 \u2191\u230ac\u230b\u208a \u2194 \u2016z\u2016\u208a \u2264 c\n[PROOFSTEP]\nrw [Int.abs_eq_natAbs, Int.ofNat_le, Nat.le_floor_iff (zero_le c), NNReal.coe_natAbs z]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081\u271d r\u2082\u271d : \u211d\nr\u2081 r\u2082 : \u211a\n\u22a2 dist r\u2081 r\u2082 = \u2016r\u2081 - r\u2082\u2016\n[PROOFSTEP]\nsimp only [Rat.dist_eq, norm, Rat.cast_sub]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : \u2124\n\u22a2 \u2016\u2191m\u2016 = \u2016m\u2016\n[PROOFSTEP]\nrw [\u2190 Rat.norm_cast_real, \u2190 Int.norm_cast_real]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nm : \u2124\n\u22a2 \u2016\u2191\u2191m\u2016 = \u2016\u2191m\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : SeminormedCommGroup \u03b1\nn : \u2124\na : \u03b1\n\u22a2 \u2016a ^ n\u2016 \u2264 \u2016n\u2016 * \u2016a\u2016\n[PROOFSTEP]\nrcases n.eq_nat_or_neg with \u27e8n, rfl | rfl\u27e9\n[GOAL]\ncase intro.inl\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : SeminormedCommGroup \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2016a ^ \u2191n\u2016 \u2264 \u2016\u2191n\u2016 * \u2016a\u2016\n[PROOFSTEP]\nsimpa using norm_pow_le_mul_norm n a\n[GOAL]\ncase intro.inr\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : SeminormedCommGroup \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u2016a ^ (-\u2191n)\u2016 \u2264 \u2016-\u2191n\u2016 * \u2016a\u2016\n[PROOFSTEP]\nsimpa using norm_pow_le_mul_norm n a\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : SeminormedCommGroup \u03b1\nn : \u2124\na : \u03b1\n\u22a2 \u2016a ^ n\u2016\u208a \u2264 \u2016n\u2016\u208a * \u2016a\u2016\u208a\n[PROOFSTEP]\nsimpa only [\u2190 NNReal.coe_le_coe, NNReal.coe_mul] using norm_zpow_le_mul_norm n a\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : LipschitzWith Kf f\nhg : LipschitzWith Kg g\n\u22a2 LipschitzWith (Kf + Kg) fun x => f x / g x\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using hf.mul' hg.inv\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\n\u22a2 AntilipschitzWith (Kf\u207b\u00b9 - Kg)\u207b\u00b9 fun x => f x * g x\n[PROOFSTEP]\nletI : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace hf.edist_ne_top\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\nthis : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace (_ : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\n\u22a2 AntilipschitzWith (Kf\u207b\u00b9 - Kg)\u207b\u00b9 fun x => f x * g x\n[PROOFSTEP]\nrefine' AntilipschitzWith.of_le_mul_dist fun x y => _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\nthis : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace (_ : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\nx y : \u03b1\n\u22a2 dist x y \u2264 \u2191(Kf\u207b\u00b9 - Kg)\u207b\u00b9 * dist (f x * g x) (f y * g y)\n[PROOFSTEP]\nrw [NNReal.coe_inv, \u2190 _root_.div_eq_inv_mul]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\nthis : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace (_ : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\nx y : \u03b1\n\u22a2 dist x y \u2264 dist (f x * g x) (f y * g y) / \u2191(Kf\u207b\u00b9 - Kg)\n[PROOFSTEP]\nrw [le_div_iff (NNReal.coe_pos.2 <| tsub_pos_iff_lt.2 hK)]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\nthis : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace (_ : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\nx y : \u03b1\n\u22a2 dist x y * \u2191(Kf\u207b\u00b9 - Kg) \u2264 dist (f x * g x) (f y * g y)\n[PROOFSTEP]\nrw [mul_comm, NNReal.coe_sub hK.le, _root_.sub_mul]\n  -- porting note: `ENNReal.sub_mul` should be `protected`?\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf\u207b\u00b9\nthis : PseudoMetricSpace \u03b1 := PseudoEMetricSpace.toPseudoMetricSpace (_ : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\nx y : \u03b1\n\u22a2 \u2191Kf\u207b\u00b9 * dist x y - \u2191Kg * dist x y \u2264 dist (f x * g x) (f y * g y)\n[PROOFSTEP]\ncalc\n  \u2191Kf\u207b\u00b9 * dist x y - Kg * dist x y \u2264 dist (f x) (f y) - dist (g x) (g y) :=\n    sub_le_sub (hf.mul_le_dist x y) (hg.dist_le_mul x y)\n  _ \u2264 _ := le_trans (le_abs_self _) (abs_dist_sub_le_dist_mul_mul _ _ _ _)\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf g : \u03b1 \u2192 E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg (g / f)\nhK : Kg < Kf\u207b\u00b9\n\u22a2 AntilipschitzWith (Kf\u207b\u00b9 - Kg)\u207b\u00b9 g\n[PROOFSTEP]\nsimpa only [Pi.div_apply, mul_div_cancel'_right] using hf.mul_lipschitzWith hg hK\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b2 : SeminormedCommGroup E\ninst\u271d\u00b9 : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\ninst\u271d : PseudoEMetricSpace \u03b1\nK Kf Kg : \u211d\u22650\nf\u271d g : \u03b1 \u2192 E\nf : E \u2192 F\nhf : AntilipschitzWith K f\nx y : E\n\u22a2 \u2016x / y\u2016 \u2264 \u2191K * \u2016f x / f y\u2016\n[PROOFSTEP]\nsimp [\u2190 dist_eq_norm_div, hf.le_mul_dist x y]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\n\u22a2 CauchySeq fun n => \u220f k in range (n + 1), u k\n[PROOFSTEP]\nlet d : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\n\u22a2 CauchySeq fun n => \u220f k in range (n + 1), u k\n[PROOFSTEP]\nrw [show (fun n => \u220f k in range (n + 1), u k) = d * fun n => \u220f k in range (n + 1), v k by ext n; simp]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\n\u22a2 (fun n => \u220f k in range (n + 1), u k) = d * fun n => \u220f k in range (n + 1), v k\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nn : \u2115\n\u22a2 \u220f k in range (n + 1), u k = (d * fun n => \u220f k in range (n + 1), v k) n\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\n\u22a2 CauchySeq (d * fun n => \u220f k in range (n + 1), v k)\n[PROOFSTEP]\nsuffices \u2200 n \u2265 N, d n = d N by exact (tendsto_atTop_of_eventually_const this).cauchySeq.mul hv\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nthis : \u2200 (n : \u2115), n \u2265 N \u2192 d n = d N\n\u22a2 CauchySeq (d * fun n => \u220f k in range (n + 1), v k)\n[PROOFSTEP]\nexact (tendsto_atTop_of_eventually_const this).cauchySeq.mul hv\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\n\u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 d n = d N\n[PROOFSTEP]\nintro n hn\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nn : \u2115\nhn : n \u2265 N\n\u22a2 d n = d N\n[PROOFSTEP]\ndsimp\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nn : \u2115\nhn : n \u2265 N\n\u22a2 \u220f k in range (n + 1), u k / v k = \u220f k in range (N + 1), u k / v k\n[PROOFSTEP]\nrw [eventually_constant_prod _ hn]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nn : \u2115\nhn : n \u2265 N\n\u22a2 \u2200 (n : \u2115), n \u2265 N \u2192 u n / v n = 1\n[PROOFSTEP]\nintro m hm\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedCommGroup E\ninst\u271d : SeminormedCommGroup F\na a\u2081 a\u2082 b b\u2081 b\u2082 : E\nr r\u2081 r\u2082 : \u211d\nu v : \u2115 \u2192 E\nN : \u2115\nhuv : \u2200 (n : \u2115), n \u2265 N \u2192 u n = v n\nhv : CauchySeq fun n => \u220f k in range (n + 1), v k\nd : \u2115 \u2192 E := fun n => \u220f k in range (n + 1), u k / v k\nn : \u2115\nhn : n \u2265 N\nm : \u2115\nhm : m \u2265 N\n\u22a2 u m / v m = 1\n[PROOFSTEP]\nsimp [huv m hm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\n\u22a2 \u2016a / b\u2016 = 0 \u2194 a = b\n[PROOFSTEP]\nrw [norm_eq_zero'', div_eq_one]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\n\u22a2 0 < \u2016a / b\u2016 \u2194 a \u2260 b\n[PROOFSTEP]\nrw [(norm_nonneg' _).lt_iff_ne, ne_comm]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\n\u22a2 \u2016a / b\u2016 \u2260 0 \u2194 a \u2260 b\n[PROOFSTEP]\nexact norm_div_eq_zero_iff.not\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\nh : \u2016a / b\u2016 \u2264 0\n\u22a2 a = b\n[PROOFSTEP]\nrwa [\u2190 div_eq_one, \u2190 norm_le_zero_iff'']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\n\u22a2 \u2016a\u2016\u208a = 0 \u2194 a = 1\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq_zero, coe_nnnorm', norm_eq_zero'']\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup E\ninst\u271d : TopologicalSpace \u03b1\nf : \u03b1 \u2192 E\nhf : Continuous f\nh : HasCompactSupport f\n\u22a2 \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C\n[PROOFSTEP]\nsimpa [bddAbove_def] using hf.norm.bddAbove_range_of_hasCompactSupport h.norm\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\n\u22a2 \u2203 R, 0 < R \u2227 \u2200 (x : \u03b1), R \u2264 \u2016x\u2016 \u2192 f x = 1\n[PROOFSTEP]\nobtain \u27e8K, \u27e8hK1, hK2\u27e9\u27e9 := exists_compact_iff_hasCompactMulSupport.mpr hf\n[GOAL]\ncase intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\nK : Set \u03b1\nhK1 : IsCompact K\nhK2 : \u2200 (x : \u03b1), \u00acx \u2208 K \u2192 f x = 1\n\u22a2 \u2203 R, 0 < R \u2227 \u2200 (x : \u03b1), R \u2264 \u2016x\u2016 \u2192 f x = 1\n[PROOFSTEP]\nobtain \u27e8S, hS, hS'\u27e9 := hK1.bounded.exists_pos_norm_le\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\nK : Set \u03b1\nhK1 : IsCompact K\nhK2 : \u2200 (x : \u03b1), \u00acx \u2208 K \u2192 f x = 1\nS : \u211d\nhS : S > 0\nhS' : \u2200 (x : \u03b1), x \u2208 K \u2192 \u2016x\u2016 \u2264 S\n\u22a2 \u2203 R, 0 < R \u2227 \u2200 (x : \u03b1), R \u2264 \u2016x\u2016 \u2192 f x = 1\n[PROOFSTEP]\nrefine'\n  \u27e8S + 1, by positivity, fun x hx => hK2 x ((mt <| hS' x) _)\u27e9\n    -- porting note: `ENNReal.add_lt_add` should be `protected`?\n      -- [context: we used `_root_.add_lt_add` in a previous version of this proof]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\nK : Set \u03b1\nhK1 : IsCompact K\nhK2 : \u2200 (x : \u03b1), \u00acx \u2208 K \u2192 f x = 1\nS : \u211d\nhS : S > 0\nhS' : \u2200 (x : \u03b1), x \u2208 K \u2192 \u2016x\u2016 \u2264 S\n\u22a2 0 < S + 1\n[PROOFSTEP]\npositivity\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\nK : Set \u03b1\nhK1 : IsCompact K\nhK2 : \u2200 (x : \u03b1), \u00acx \u2208 K \u2192 f x = 1\nS : \u211d\nhS : S > 0\nhS' : \u2200 (x : \u03b1), x \u2208 K \u2192 \u2016x\u2016 \u2264 S\nx : \u03b1\nhx : S + 1 \u2264 \u2016x\u2016\n\u22a2 \u00ac\u2016x\u2016 \u2264 S\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedAddGroup \u03b1\nf : \u03b1 \u2192 E\ninst\u271d : One E\nhf : HasCompactMulSupport f\nK : Set \u03b1\nhK1 : IsCompact K\nhK2 : \u2200 (x : \u03b1), \u00acx \u2208 K \u2192 f x = 1\nS : \u211d\nhS : S > 0\nhS' : \u2200 (x : \u03b1), x \u2208 K \u2192 \u2016x\u2016 \u2264 S\nx : \u03b1\nhx : \u2016x\u2016 \u2264 S\n\u22a2 \u2016x\u2016 < S + 1\n[PROOFSTEP]\nexact lt_add_of_le_of_pos hx zero_lt_one\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : SeminormedGroup E\ninst\u271d : SeminormedGroup F\nx y : E \u00d7 F\n\u22a2 dist x y = \u2016x / y\u2016\n[PROOFSTEP]\nsimp only [Prod.norm_def, Prod.dist_eq, dist_eq_norm_div, Prod.fst_div, Prod.snd_div]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 \u2016x\u2016 \u2264 r \u2194 \u2200 (i : \u03b9), \u2016x i\u2016 \u2264 r\n[PROOFSTEP]\nsimp only [\u2190 dist_one_right, dist_pi_le_iff hr, Pi.one_apply]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d\u00b9 : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\ninst\u271d : Nonempty \u03b9\n\u22a2 \u2016f\u2016 \u2264 r \u2194 \u2200 (b : \u03b9), \u2016f b\u2016 \u2264 r\n[PROOFSTEP]\nby_cases hr : 0 \u2264 r\n[GOAL]\ncase pos\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d\u00b9 : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\ninst\u271d : Nonempty \u03b9\nhr : 0 \u2264 r\n\u22a2 \u2016f\u2016 \u2264 r \u2194 \u2200 (b : \u03b9), \u2016f b\u2016 \u2264 r\n[PROOFSTEP]\nexact pi_norm_le_iff_of_nonneg' hr\n[GOAL]\ncase neg\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d\u00b9 : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\ninst\u271d : Nonempty \u03b9\nhr : \u00ac0 \u2264 r\n\u22a2 \u2016f\u2016 \u2264 r \u2194 \u2200 (b : \u03b9), \u2016f b\u2016 \u2264 r\n[PROOFSTEP]\nexact\n  iff_of_false (fun h => hr <| (norm_nonneg' _).trans h) fun h =>\n    hr <| (norm_nonneg' _).trans <| h <| Classical.arbitrary _\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\nhr : 0 < r\n\u22a2 \u2016x\u2016 < r \u2194 \u2200 (i : \u03b9), \u2016x i\u2016 < r\n[PROOFSTEP]\nsimp only [\u2190 dist_one_right, dist_pi_lt_iff hr, Pi.one_apply]\n[GOAL]\n\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\n\u03c0 : \u03b9 \u2192 Type u_9\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : (i : \u03b9) \u2192 SeminormedGroup (\u03c0 i)\ninst\u271d\u00b9 : SeminormedGroup E\nf x : (i : \u03b9) \u2192 \u03c0 i\nr : \u211d\ninst\u271d : Nonempty \u03b9\na : E\n\u22a2 \u2016fun _i => a\u2016 = \u2016a\u2016\n[PROOFSTEP]\nsimpa only [\u2190 dist_one_right] using dist_pi_const a 1\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.Basic", "llama_tokens": 62635, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.520791882623339}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 \u2200 (a : \u03b1\u1d52\u1d48), (\u2191toDual \u2218 SuccOrder.succ \u2218 \u2191ofDual) a \u2264 a\n[PROOFSTEP]\nsimp only [comp, OrderDual.forall, ofDual_toDual, toDual_le_toDual, SuccOrder.le_succ, implies_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na\u271d : \u03b1\u1d52\u1d48\nh : a\u271d \u2264 (\u2191toDual \u2218 SuccOrder.succ \u2218 \u2191ofDual) a\u271d\n\u22a2 IsMin a\u271d\n[PROOFSTEP]\napply SuccOrder.max_of_succ_le h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 \u2200 {a b : \u03b1\u1d52\u1d48}, a < b \u2192 a \u2264 (\u2191toDual \u2218 SuccOrder.succ \u2218 \u2191ofDual) b\n[PROOFSTEP]\nintro a b h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\u1d52\u1d48\nh : a < b\n\u22a2 a \u2264 (\u2191toDual \u2218 SuccOrder.succ \u2218 \u2191ofDual) b\n[PROOFSTEP]\nexact SuccOrder.succ_le_of_lt h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 \u2200 (a : \u03b1\u1d52\u1d48), a \u2264 (\u2191toDual \u2218 PredOrder.pred \u2218 \u2191ofDual) a\n[PROOFSTEP]\nsimp only [comp, OrderDual.forall, ofDual_toDual, toDual_le_toDual, PredOrder.pred_le, implies_true]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na\u271d : \u03b1\u1d52\u1d48\nh : (\u2191toDual \u2218 PredOrder.pred \u2218 \u2191ofDual) a\u271d \u2264 a\u271d\n\u22a2 IsMax a\u271d\n[PROOFSTEP]\napply PredOrder.min_of_le_pred h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 \u2200 {a b : \u03b1\u1d52\u1d48}, a < b \u2192 (\u2191toDual \u2218 PredOrder.pred \u2218 \u2191ofDual) a \u2264 b\n[PROOFSTEP]\nintro a b h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\u1d52\u1d48\nh : a < b\n\u22a2 (\u2191toDual \u2218 PredOrder.pred \u2218 \u2191ofDual) a \u2264 b\n[PROOFSTEP]\nexact PredOrder.le_pred_of_lt h\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsucc : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMax a \u2192 \u2200 (b : \u03b1), a < b \u2194 succ a \u2264 b\nhm : \u2200 (a : \u03b1), IsMax a \u2192 succ a = a\na : \u03b1\nh : \u00acIsMax a\n\u22a2 a < succ a\n[PROOFSTEP]\nsimpa using (hn h a).not\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsucc : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMax a \u2192 \u2200 (b : \u03b1), a < b \u2194 succ a \u2264 b\nhm : \u2200 (a : \u03b1), IsMax a \u2192 succ a = a\na : \u03b1\nh : \u00acIsMax a\n\u22a2 \u00acsucc a \u2264 a\n[PROOFSTEP]\nsimpa using (hn h a).not\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrder \u03b1\nsucc : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMax a \u2192 \u2200 (b : \u03b1), a < b \u2194 succ a \u2264 b\nhm : \u2200 (a : \u03b1), IsMax a \u2192 succ a = a\na b : \u03b1\nhab : a < succ b\nh : \u00acIsMax b\n\u22a2 a \u2264 b\n[PROOFSTEP]\nsimpa [hab] using (hn h a).not\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\n\u03b1 : Type ?u.4579\ninst\u271d : LinearOrder \u03b1\npred : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMin a \u2192 \u2200 (b : \u03b1), b \u2264 pred a \u2194 b < a\nhm : \u2200 (a : \u03b1), IsMin a \u2192 pred a = a\na : \u03b1\nh : \u00acIsMin a\n\u22a2 pred a < a\n[PROOFSTEP]\nsimpa using (hn h a).not\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\n\u03b1 : Type ?u.4579\ninst\u271d : LinearOrder \u03b1\npred : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMin a \u2192 \u2200 (b : \u03b1), b \u2264 pred a \u2194 b < a\nhm : \u2200 (a : \u03b1), IsMin a \u2192 pred a = a\na : \u03b1\nh : \u00acIsMin a\n\u22a2 \u00aca \u2264 pred a\n[PROOFSTEP]\nsimpa using (hn h a).not\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\u271d\n\u03b1 : Type ?u.4579\ninst\u271d : LinearOrder \u03b1\npred : \u03b1 \u2192 \u03b1\nhn : \u2200 {a : \u03b1}, \u00acIsMin a \u2192 \u2200 (b : \u03b1), b \u2264 pred a \u2194 b < a\nhm : \u2200 (a : \u03b1), IsMin a \u2192 pred a = a\na b : \u03b1\nhab : pred a < b\nh : \u00acIsMin a\n\u22a2 a \u2264 b\n[PROOFSTEP]\nsimpa [hab] using (hn h b).not\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nha : \u00acIsMax a\nhb : \u00acIsMax b\n\u22a2 succ a < succ b \u2194 a < b\n[PROOFSTEP]\nrw [lt_succ_iff_of_not_isMax hb, succ_le_iff_of_not_isMax ha]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nha : \u00acIsMax a\nhb : \u00acIsMax b\n\u22a2 succ a \u2264 succ b \u2194 a \u2264 b\n[PROOFSTEP]\nrw [succ_le_iff_of_not_isMax ha, lt_succ_iff_of_not_isMax hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 b\n\u22a2 succ a \u2264 succ b\n[PROOFSTEP]\nby_cases hb : IsMax b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nhb : IsMax b\n\u22a2 succ a \u2264 succ b\n[PROOFSTEP]\nby_cases hba : b \u2264 a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nhb : IsMax b\nhba : b \u2264 a\n\u22a2 succ a \u2264 succ b\n[PROOFSTEP]\nexact (hb <| hba.trans <| le_succ _).trans (le_succ _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nhb : IsMax b\nhba : \u00acb \u2264 a\n\u22a2 succ a \u2264 succ b\n[PROOFSTEP]\nexact succ_le_of_lt ((h.lt_of_not_le hba).trans_le <| le_succ b)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 b\nhb : \u00acIsMax b\n\u22a2 succ a \u2264 succ b\n[PROOFSTEP]\nrwa [succ_le_iff_of_not_isMax fun ha => hb <| ha.mono h, lt_succ_iff_of_not_isMax hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 x \u2264 succ^[k] x\n[PROOFSTEP]\nconv_lhs => rw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 x = id^[k] x\n[PROOFSTEP]\nsimp only [Function.iterate_id, id.def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 id^[k] x \u2264 succ^[k] x\n[PROOFSTEP]\nexact Monotone.le_iterate_of_le succ_mono le_succ k x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\n\u22a2 IsMax (succ^[n] a)\n[PROOFSTEP]\nrefine' max_of_succ_le (le_trans _ h_eq.symm.le)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\n\u22a2 succ (succ^[n] a) \u2264 succ^[m] a\n[PROOFSTEP]\nhave : succ (succ^[n] a) = succ^[n + 1] a := by rw [Function.iterate_succ', comp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\n\u22a2 succ (succ^[n] a) = succ^[n + 1] a\n[PROOFSTEP]\nrw [Function.iterate_succ', comp]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\nthis : succ (succ^[n] a) = succ^[n + 1] a\n\u22a2 succ (succ^[n] a) \u2264 succ^[m] a\n[PROOFSTEP]\nrw [this]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\nthis : succ (succ^[n] a) = succ^[n + 1] a\n\u22a2 succ^[n + 1] a \u2264 succ^[m] a\n[PROOFSTEP]\nhave h_le : n + 1 \u2264 m := Nat.succ_le_of_lt h_lt\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_lt : n < m\nthis : succ (succ^[n] a) = succ^[n + 1] a\nh_le : n + 1 \u2264 m\n\u22a2 succ^[n + 1] a \u2264 succ^[m] a\n[PROOFSTEP]\nexact Monotone.monotone_iterate_of_le_map succ_mono (le_succ a) h_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_ne : n \u2260 m\n\u22a2 IsMax (succ^[n] a)\n[PROOFSTEP]\ncases' le_total n m with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_ne : n \u2260 m\nh : n \u2264 m\n\u22a2 IsMax (succ^[n] a)\n[PROOFSTEP]\nexact isMax_iterate_succ_of_eq_of_lt h_eq (lt_of_le_of_ne h h_ne)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_ne : n \u2260 m\nh : m \u2264 n\n\u22a2 IsMax (succ^[n] a)\n[PROOFSTEP]\nrw [h_eq]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nn m : \u2115\nh_eq : succ^[n] a = succ^[m] a\nh_ne : n \u2260 m\nh : m \u2264 n\n\u22a2 IsMax (succ^[m] a)\n[PROOFSTEP]\nexact isMax_iterate_succ_of_eq_of_lt h_eq.symm (lt_of_le_of_ne h h_ne.symm)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nhb : \u00acIsMax b\n\u22a2 Ico a (succ b) = Icc a b\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iio, Iio_succ_of_not_isMax hb, Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nhb : \u00acIsMax b\n\u22a2 Ioo a (succ b) = Ioc a b\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iio, Iio_succ_of_not_isMax hb, Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nha : \u00acIsMax a\n\u22a2 Icc (succ a) b = Ioc a b\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iic, Ici_succ_of_not_isMax ha, Ioi_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nha : \u00acIsMax a\n\u22a2 Ico (succ a) b = Ioo a b\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iio, Ici_succ_of_not_isMax ha, Ioi_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 succ a \u2264 succ b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 succ a < succ b \u2194 a < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nha : \u00acIsMax a\nhb : \u00acIsMax b\n\u22a2 succ a = succ b \u2194 a = b\n[PROOFSTEP]\nrw [eq_iff_le_not_lt, eq_iff_le_not_lt, succ_le_succ_iff_of_not_isMax ha hb, succ_lt_succ_iff_of_not_isMax ha hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\n\u22a2 a \u2264 b \u2227 b \u2264 succ a \u2194 b = a \u2228 b = succ a\n[PROOFSTEP]\nrefine' \u27e8fun h => or_iff_not_imp_left.2 fun hba : b \u2260 a => h.2.antisymm (succ_le_of_lt <| h.1.lt_of_ne <| hba.symm), _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\n\u22a2 b = a \u2228 b = succ a \u2192 a \u2264 b \u2227 b \u2264 succ a\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb : \u03b1\n\u22a2 b \u2264 b \u2227 b \u2264 succ b\n[PROOFSTEP]\nexact \u27e8le_rfl, le_succ b\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 a \u2264 succ a \u2227 succ a \u2264 succ a\n[PROOFSTEP]\nexact \u27e8le_succ a, le_rfl\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2a7f b\n\u22a2 b \u2264 succ a\n[PROOFSTEP]\nobtain h | rfl := h.covby_or_eq\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh\u271d : a \u2a7f b\nh : a \u22d6 b\n\u22a2 b \u2264 succ a\n[PROOFSTEP]\nexact (Covby.succ_eq h).ge\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\nh : a \u2a7f a\n\u22a2 a \u2264 succ a\n[PROOFSTEP]\nexact le_succ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\n\u22a2 a \u2264 succ b \u2194 a = succ b \u2228 a \u2264 b\n[PROOFSTEP]\nby_cases hb : IsMax b\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nhb : IsMax b\n\u22a2 a \u2264 succ b \u2194 a = succ b \u2228 a \u2264 b\n[PROOFSTEP]\nrw [hb.succ_eq, or_iff_right_of_imp le_of_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nhb : \u00acIsMax b\n\u22a2 a \u2264 succ b \u2194 a = succ b \u2228 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 lt_succ_iff_of_not_isMax hb, le_iff_eq_or_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a \u2264 succ b\n\u22a2 Icc a (succ b) = insert (succ b) (Icc a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ici_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ici.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh : a < succ b\n\u22a2 Ioc a (succ b) = insert (succ b) (Ioc a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ioi_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ioi.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh\u2081 : a \u2264 b\nh\u2082 : \u00acIsMax b\n\u22a2 Ico a (succ b) = insert b (Ico a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Iio_inter_Ici, Iio_succ_eq_insert_of_not_isMax h\u2082, insert_inter_of_mem (mem_Ici.2 h\u2081)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : \u03b1\nh\u2081 : a < b\nh\u2082 : \u00acIsMax b\n\u22a2 Ioo a (succ b) = insert b (Ioo a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Iio_inter_Ioi, Iio_succ_eq_insert_of_not_isMax h\u2082, insert_inter_of_mem (mem_Ioi.2 h\u2081)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 succ a = b \u2192 a \u22d6 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na : \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 a \u22d6 succ a\n[PROOFSTEP]\nexact covby_succ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d : OrderTop \u03b1\n\u22a2 succ \u22a4 = \u22a4\n[PROOFSTEP]\nrw [succ_eq_iff_isMax, isMax_iff_eq_top]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : NoMaxOrder \u03b1\n\u22a2 a < succ \u22a5 \u2194 a = \u22a5\n[PROOFSTEP]\nrw [lt_succ_iff, le_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\na b : \u03b1\ninst\u271d : OrderBot \u03b1\n\u22a2 a \u2264 succ \u22a5 \u2194 a = \u22a5 \u2228 a = succ \u22a5\n[PROOFSTEP]\nrw [le_succ_iff_eq_or_le, le_bot_iff, or_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\n\u22a2 \u2200 (a b : SuccOrder \u03b1), a = b\n[PROOFSTEP]\nintro h\u2080 h\u2081\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : SuccOrder \u03b1\n\u22a2 h\u2080 = h\u2081\n[PROOFSTEP]\next a\n[GOAL]\ncase succ.h\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : SuccOrder \u03b1\na : \u03b1\n\u22a2 SuccOrder.succ a = SuccOrder.succ a\n[PROOFSTEP]\nby_cases ha : IsMax a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : SuccOrder \u03b1\na : \u03b1\nha : IsMax a\n\u22a2 SuccOrder.succ a = SuccOrder.succ a\n[PROOFSTEP]\nexact (@IsMax.succ_eq _ _ h\u2080 _ ha).trans ha.succ_eq.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : SuccOrder \u03b1\na : \u03b1\nha : \u00acIsMax a\n\u22a2 SuccOrder.succ a = SuccOrder.succ a\n[PROOFSTEP]\nexact @Covby.succ_eq _ _ h\u2080 _ _ (covby_succ_of_not_isMax ha)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 succ a = \u2a05 (b : \u03b1) (_ : a < b), b\n[PROOFSTEP]\nrefine' le_antisymm (le_iInf fun b => le_iInf succ_le_of_lt) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 \u2a05 (b : \u03b1) (_ : a < b), b \u2264 succ a\n[PROOFSTEP]\nobtain rfl | ha := eq_or_ne a \u22a4\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 \u2a05 (b : \u03b1) (_ : \u22a4 < b), b \u2264 succ \u22a4\n[PROOFSTEP]\nrw [succ_top]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 \u2a05 (b : \u03b1) (_ : \u22a4 < b), b \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\nha : a \u2260 \u22a4\n\u22a2 \u2a05 (b : \u03b1) (_ : a < b), b \u2264 succ a\n[PROOFSTEP]\nexact iInf\u2082_le _ (lt_succ_iff_ne_top.2 ha)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 pred^[k] x \u2264 x\n[PROOFSTEP]\nconv_rhs => rw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n| x\n[PROOFSTEP]\nrw [(by simp only [Function.iterate_id, id.def] : x = id^[k] x)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 x = id^[k] x\n[PROOFSTEP]\nsimp only [Function.iterate_id, id.def]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nk : \u2115\nx : \u03b1\n\u22a2 pred^[k] x \u2264 id^[k] x\n[PROOFSTEP]\nexact Monotone.iterate_le_of_le pred_mono pred_le k x\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 Ioc (pred a) b = Icc a b\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, Ioi_pred_of_not_isMin ha, Ici_inter_Iic]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 Ioo (pred a) b = Ico a b\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iio, Ioi_pred_of_not_isMin ha, Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin b\n\u22a2 Icc a (pred b) = Ico a b\n[PROOFSTEP]\nrw [\u2190 Ici_inter_Iic, Iic_pred_of_not_isMin ha, Ici_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin b\n\u22a2 Ioc a (pred b) = Ioo a b\n[PROOFSTEP]\nrw [\u2190 Ioi_inter_Iic, Iic_pred_of_not_isMin ha, Ioi_inter_Iio]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred a \u2264 pred b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred a < pred b \u2194 a < b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 pred a \u2264 b \u2227 b \u2264 a \u2194 b = a \u2228 b = pred a\n[PROOFSTEP]\nrefine' \u27e8fun h => or_iff_not_imp_left.2 fun hba : b \u2260 a => (le_pred_of_lt <| h.2.lt_of_ne hba).antisymm h.1, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na\u271d b\u271d a b : \u03b1\n\u22a2 b = a \u2228 b = pred a \u2192 pred a \u2264 b \u2227 b \u2264 a\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\n\u22a2 pred b \u2264 b \u2227 b \u2264 b\n[PROOFSTEP]\nexact \u27e8pred_le b, le_rfl\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na\u271d b a : \u03b1\n\u22a2 pred a \u2264 pred a \u2227 pred a \u2264 a\n[PROOFSTEP]\nexact \u27e8le_rfl, pred_le a\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nh : a \u2a7f b\n\u22a2 pred b \u2264 a\n[PROOFSTEP]\nobtain h | rfl := h.covby_or_eq\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nh\u271d : a \u2a7f b\nh : a \u22d6 b\n\u22a2 pred b \u2264 a\n[PROOFSTEP]\nexact (Covby.pred_eq h).le\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nh : a \u2a7f a\n\u22a2 pred a \u2264 a\n[PROOFSTEP]\nexact pred_le _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b\n[PROOFSTEP]\nby_cases ha : IsMin a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : IsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b\n[PROOFSTEP]\nrw [ha.pred_eq, or_iff_right_of_imp ge_of_eq]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 pred a \u2264 b \u2194 b = pred a \u2228 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 pred_lt_iff_of_not_isMin ha, le_iff_eq_or_lt, eq_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\n\u22a2 Ioi (pred a) = insert a (Ioi a)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\nx : \u03b1\n\u22a2 x \u2208 Ioi (pred a) \u2194 x \u2208 insert a (Ioi a)\n[PROOFSTEP]\nsimp only [insert, mem_setOf, @eq_comm _ x a, mem_Ioi, Set.insert]\n[GOAL]\ncase h\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nha : \u00acIsMin a\nx : \u03b1\n\u22a2 pred a < x \u2194 a = x \u2228 a < x\n[PROOFSTEP]\nexact pred_lt_iff_eq_or_lt_of_not_isMin ha\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nh : pred a \u2264 b\n\u22a2 Icc (pred a) b = insert (pred a) (Icc a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ici_inter_Iic, Ici_pred, insert_inter_of_mem (mem_Iic.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nh : pred a < b\n\u22a2 Ico (pred a) b = insert (pred a) (Ico a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ici_inter_Iio, Ici_pred, insert_inter_of_mem (mem_Iio.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred a = pred b \u2194 a = b\n[PROOFSTEP]\nsimp_rw [eq_iff_le_not_lt, pred_le_pred_iff, pred_lt_pred_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred b = a \u2192 a \u22d6 b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\nb : \u03b1\ninst\u271d : NoMinOrder \u03b1\n\u22a2 pred b \u22d6 b\n[PROOFSTEP]\nexact pred_covby _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\nh : a \u2264 b\n\u22a2 Ioc (pred a) b = insert a (Ioc a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ioi_inter_Iic, Ioi_pred_eq_insert, insert_inter_of_mem (mem_Iic.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\na b : \u03b1\ninst\u271d : NoMinOrder \u03b1\nh : a < b\n\u22a2 Ioo (pred a) b = insert a (Ioo a b)\n[PROOFSTEP]\nsimp_rw [\u2190 Ioi_inter_Iio, Ioi_pred_eq_insert, insert_inter_of_mem (mem_Iio.2 h)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\n\u22a2 \u2200 (a b : PredOrder \u03b1), a = b\n[PROOFSTEP]\nintro h\u2080 h\u2081\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : PredOrder \u03b1\n\u22a2 h\u2080 = h\u2081\n[PROOFSTEP]\next a\n[GOAL]\ncase pred.h\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : PredOrder \u03b1\na : \u03b1\n\u22a2 PredOrder.pred a = PredOrder.pred a\n[PROOFSTEP]\nby_cases ha : IsMin a\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : PredOrder \u03b1\na : \u03b1\nha : IsMin a\n\u22a2 PredOrder.pred a = PredOrder.pred a\n[PROOFSTEP]\nexact (@IsMin.pred_eq _ _ h\u2080 _ ha).trans ha.pred_eq.symm\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d : PartialOrder \u03b1\nh\u2080 h\u2081 : PredOrder \u03b1\na : \u03b1\nha : \u00acIsMin a\n\u22a2 PredOrder.pred a = PredOrder.pred a\n[PROOFSTEP]\nexact @Covby.pred_eq _ _ h\u2080 _ _ (pred_covby_of_not_isMin ha)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\n\u22a2 pred a = \u2a06 (b : \u03b1) (_ : b < a), b\n[PROOFSTEP]\nrefine' le_antisymm _ (iSup_le fun b => iSup_le le_pred_of_lt)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\n\u22a2 pred a \u2264 \u2a06 (b : \u03b1) (_ : b < a), b\n[PROOFSTEP]\nobtain rfl | ha := eq_or_ne a \u22a5\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 pred \u22a5 \u2264 \u2a06 (b : \u03b1) (_ : b < \u22a5), b\n[PROOFSTEP]\nrw [pred_bot]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 \u22a5 \u2264 \u2a06 (b : \u03b1) (_ : b < \u22a5), b\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha : a \u2260 \u22a5\n\u22a2 pred a \u2264 \u2a06 (b : \u03b1) (_ : b < a), b\n[PROOFSTEP]\nexact @le_iSup\u2082 _ _ (fun b => b < a) _ (fun a _ => a) (pred a) (pred_lt_iff_ne_bot.2 ha)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn : \u2115\nhin : \u00acIsMax (succ^[n - 1] i)\n\u22a2 pred^[n] (succ^[n] i) = i\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn : \u2115\nhin\u271d : \u00acIsMax (succ^[n - 1] i)\nhin : \u00acIsMax (succ^[Nat.zero - 1] i)\n\u22a2 pred^[Nat.zero] (succ^[Nat.zero] i) = i\n[PROOFSTEP]\nsimp only [Nat.zero_eq, Function.iterate_zero, id.def]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n - 1] i)\n\u22a2 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\n[PROOFSTEP]\nrw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hin \n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\n\u22a2 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\n[PROOFSTEP]\nhave h_not_max : \u00acIsMax (succ^[n - 1] i) := by\n  cases' n with n\n  \u00b7 simpa using hin\n  rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hn \u22a2\n  have h_sub_le : succ^[n] i \u2264 succ^[n.succ] i :=\n    by\n    rw [Function.iterate_succ']\n    exact le_succ _\n  refine' fun h_max => hin fun j hj => _\n  have hj_le : j \u2264 succ^[n] i := h_max (h_sub_le.trans hj)\n  exact hj_le.trans h_sub_le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\n\u22a2 \u00acIsMax (succ^[n - 1] i)\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn : \u2115\nhin\u271d : \u00acIsMax (succ^[n - 1] i)\nhn : \u00acIsMax (succ^[Nat.zero - 1] i) \u2192 pred^[Nat.zero] (succ^[Nat.zero] i) = i\nhin : \u00acIsMax (succ^[Nat.zero] i)\n\u22a2 \u00acIsMax (succ^[Nat.zero - 1] i)\n[PROOFSTEP]\nsimpa using hin\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[Nat.succ n - 1] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\n\u22a2 \u00acIsMax (succ^[Nat.succ n - 1] i)\n[PROOFSTEP]\nrw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hn \u22a2\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\n\u22a2 \u00acIsMax (succ^[n] i)\n[PROOFSTEP]\nhave h_sub_le : succ^[n] i \u2264 succ^[n.succ] i :=\n  by\n  rw [Function.iterate_succ']\n  exact le_succ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\n\u22a2 succ^[n] i \u2264 succ^[Nat.succ n] i\n[PROOFSTEP]\nrw [Function.iterate_succ']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\n\u22a2 succ^[n] i \u2264 (succ \u2218 succ^[n]) i\n[PROOFSTEP]\nexact le_succ _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\nh_sub_le : succ^[n] i \u2264 succ^[Nat.succ n] i\n\u22a2 \u00acIsMax (succ^[n] i)\n[PROOFSTEP]\nrefine' fun h_max => hin fun j hj => _\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\nh_sub_le : succ^[n] i \u2264 succ^[Nat.succ n] i\nh_max : IsMax (succ^[n] i)\nj : \u03b1\nhj : succ^[Nat.succ n] i \u2264 j\n\u22a2 j \u2264 succ^[Nat.succ n] i\n[PROOFSTEP]\nhave hj_le : j \u2264 succ^[n] i := h_max (h_sub_le.trans hj)\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n] i) \u2192 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\nhin : \u00acIsMax (succ^[Nat.succ n] i)\nh_sub_le : succ^[n] i \u2264 succ^[Nat.succ n] i\nh_max : IsMax (succ^[n] i)\nj : \u03b1\nhj : succ^[Nat.succ n] i \u2264 j\nhj_le : j \u2264 succ^[n] i\n\u22a2 j \u2264 succ^[Nat.succ n] i\n[PROOFSTEP]\nexact hj_le.trans h_sub_le\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\nh_not_max : \u00acIsMax (succ^[n - 1] i)\n\u22a2 pred^[Nat.succ n] (succ^[Nat.succ n] i) = i\n[PROOFSTEP]\nrw [Function.iterate_succ, Function.iterate_succ']\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\nh_not_max : \u00acIsMax (succ^[n - 1] i)\n\u22a2 (pred^[n] \u2218 pred) ((succ \u2218 succ^[n]) i) = i\n[PROOFSTEP]\nsimp only [Function.comp_apply]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\nh_not_max : \u00acIsMax (succ^[n - 1] i)\n\u22a2 pred^[n] (pred (succ (succ^[n] i))) = i\n[PROOFSTEP]\nrw [pred_succ_of_not_isMax hin]\n[GOAL]\ncase succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b i : \u03b1\nn\u271d : \u2115\nhin\u271d : \u00acIsMax (succ^[n\u271d - 1] i)\nn : \u2115\nhn : \u00acIsMax (succ^[n - 1] i) \u2192 pred^[n] (succ^[n] i) = i\nhin : \u00acIsMax (succ^[n] i)\nh_not_max : \u00acIsMax (succ^[n - 1] i)\n\u22a2 pred^[n] (succ^[n] i) = i\n[PROOFSTEP]\nexact hn h_not_max\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      a\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 Option.some a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      (Option.some a)\n[PROOFSTEP]\nchange _ \u2264 ite _ _ _\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 Option.some a \u2264 if a = \u22a4 then \u22a4 else \u2191(succ a)\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\nh\u271d : a = \u22a4\n\u22a2 Option.some a \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\nh\u271d : \u00aca = \u22a4\n\u22a2 Option.some a \u2264 \u2191(succ a)\n[PROOFSTEP]\nexact some_le_some.2 (le_succ a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      a \u2264\n    a\n\u22a2 IsMax a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      none \u2264\n    none\n\u22a2 IsMax none\n[PROOFSTEP]\nexact isMax_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\ndsimp only at ha \n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha : (if val\u271d = \u22a4 then \u22a4 else \u2191(succ val\u271d)) \u2264 Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nsplit_ifs at ha  with ha'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha' : val\u271d = \u22a4\nha : \u22a4 \u2264 Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nexact (not_top_le_coe _ ha).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha' : \u00acval\u271d = \u22a4\nha : \u2191(succ val\u271d) \u2264 Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nrw [some_eq_coe, coe_le_coe, succ_le_iff_eq_top] at ha \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha' : \u00acval\u271d = \u22a4\nha\u271d : succ val\u271d \u2264 val\u271d\nha : val\u271d = \u22a4\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nexact (ha' ha).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithTop \u03b1\nh : a < b\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      a \u2264\n    b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\nh : a < none\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      a \u2264\n    none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\nval\u271d : \u03b1\nh : a < Option.some val\u271d\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      a \u2264\n    Option.some val\u271d\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh : none < Option.some val\u271d\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      none \u2264\n    Option.some val\u271d\n[PROOFSTEP]\nexact (not_top_lt h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d < Option.some val\u271d\u00b9\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\u00b9\n[PROOFSTEP]\nrw [some_lt_some] at h \n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d < val\u271d\u00b9\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\u00b9\n[PROOFSTEP]\nchange ite _ _ _ \u2264 _\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d < val\u271d\u00b9\n\u22a2 (if val\u271d = \u22a4 then \u22a4 else \u2191(succ val\u271d)) \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nsplit_ifs with ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d < val\u271d\u00b9\nha : val\u271d = \u22a4\n\u22a2 \u22a4 \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nrw [ha] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : \u22a4 < val\u271d\u00b9\nha : val\u271d = \u22a4\n\u22a2 \u22a4 \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact (not_top_lt h).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d < val\u271d\u00b9\nha : \u00acval\u271d = \u22a4\n\u22a2 \u2191(succ val\u271d) \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact some_le_some.2 (succ_le_of_lt h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithTop \u03b1\nh :\n  a <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nh :\n  none <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      b\n\u22a2 none \u2264 b\n[PROOFSTEP]\nexact (not_top_lt h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      b\n\u22a2 Option.some val\u271d \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      none\n\u22a2 Option.some val\u271d \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  Option.some val\u271d\u00b9 <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => if a = \u22a4 then \u22a4 else \u2191(succ a))\n      (Option.some val\u271d)\n\u22a2 Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\ndsimp only at h \n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d\u00b9 < if val\u271d = \u22a4 then \u22a4 else \u2191(succ val\u271d)\n\u22a2 Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\nrw [some_le_some]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d\u00b9 < if val\u271d = \u22a4 then \u22a4 else \u2191(succ val\u271d)\n\u22a2 val\u271d\u00b9 \u2264 val\u271d\n[PROOFSTEP]\nsplit_ifs at h  with hb\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nhb : val\u271d = \u22a4\nh : Option.some val\u271d\u00b9 < \u22a4\n\u22a2 val\u271d\u00b9 \u2264 val\u271d\n[PROOFSTEP]\nrw [hb]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nhb : val\u271d = \u22a4\nh : Option.some val\u271d\u00b9 < \u22a4\n\u22a2 val\u271d\u00b9 \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nhb : \u00acval\u271d = \u22a4\nh : Option.some val\u271d\u00b9 < \u2191(succ val\u271d)\n\u22a2 val\u271d\u00b9 \u2264 val\u271d\n[PROOFSTEP]\nexact le_of_lt_succ (some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na : WithTop \u03b1\nha :\n  a \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      a\n\u22a2 IsMin a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nha :\n  none \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      none\n\u22a2 IsMin none\n[PROOFSTEP]\nexact ((coe_lt_top (\u22a4 : \u03b1)).not_le ha).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nha :\n  Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d)\n\u22a2 IsMin (Option.some val\u271d)\n[PROOFSTEP]\nexact (min_of_le_pred <| some_le_some.1 ha).withTop\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithTop \u03b1\nh : a < b\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nb : WithTop \u03b1\nh : none < b\n\u22a2 none \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      b\n[PROOFSTEP]\nexact (le_top.not_lt h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < b\n\u22a2 Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < none\n\u22a2 Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      none\n[PROOFSTEP]\nexact some_le_some.2 le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d\u00b9 < Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 \u2264\n    (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d)\n[PROOFSTEP]\nexact some_le_some.2 (le_pred_of_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithTop \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      a <\n    b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na : WithTop \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      a <\n    none\n\u22a2 a \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na : WithTop \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      a <\n    Option.some val\u271d\n\u22a2 a \u2264 Option.some val\u271d\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      none <\n    Option.some val\u271d\n\u22a2 none \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact (not_top_lt <| some_lt_some.1 h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a4\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d) <\n    Option.some val\u271d\u00b9\n\u22a2 Option.some val\u271d \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact some_le_some.2 (le_of_pred_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na : WithTop \u03b1\nha : a \u2260 \u22a4\n\u22a2 pred a \u2260 \u22a4\n[PROOFSTEP]\ninduction a using WithTop.recTopCoe\n[GOAL]\ncase top\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\nha : \u22a4 \u2260 \u22a4\n\u22a2 pred \u22a4 \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase coe\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderTop \u03b1\ninst\u271d : PredOrder \u03b1\na\u271d : \u03b1\nha : \u2191a\u271d \u2260 \u22a4\n\u22a2 pred \u2191a\u271d \u2260 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      a\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\n\u22a2 none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : \u03b1\n\u22a2 Option.some a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some a)\n[PROOFSTEP]\nexact some_le_some.2 (le_succ a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithTop \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    a\n\u22a2 IsMax a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      none \u2264\n    none\n\u22a2 IsMax none\n[PROOFSTEP]\nexact isMax_top\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nexact (not_isMax _ <| max_of_succ_le <| some_le_some.1 ha).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithTop \u03b1\nh : a < b\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nh : none < b\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      none \u2264\n    b\n[PROOFSTEP]\nexact (not_top_lt h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < b\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < none\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d\u00b9 < Option.some val\u271d\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d\u00b9) \u2264\n    Option.some val\u271d\n[PROOFSTEP]\nexact some_le_some.2 (succ_le_of_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithTop \u03b1\nh :\n  a <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nh :\n  none <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 none \u2264 b\n[PROOFSTEP]\nexact (not_top_lt h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithTop \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 Option.some val\u271d \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      none\n\u22a2 Option.some val\u271d \u2264 none\n[PROOFSTEP]\nexact le_top\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMaxOrder \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  Option.some val\u271d\u00b9 <\n    (fun a =>\n        match a with\n        | none => \u22a4\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d)\n\u22a2 Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact some_le_some.2 (le_of_lt_succ <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\n\u22a2 PredOrder (WithTop \u03b1) \u2192 False\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : PredOrder (WithTop \u03b1)\n\u22a2 False\n[PROOFSTEP]\ncases' h : pred (\u22a4 : WithTop \u03b1) with a ha\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : PredOrder (WithTop \u03b1)\nh : pred \u22a4 = none\n\u22a2 False\n[PROOFSTEP]\nexact h\u03b1.elim fun a => (min_of_le_pred h.ge).not_lt <| coe_lt_top a\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : PredOrder (WithTop \u03b1)\na : \u03b1\nh : pred \u22a4 = Option.some a\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := exists_gt a\n[GOAL]\ncase some.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : PredOrder (WithTop \u03b1)\na : \u03b1\nh : pred \u22a4 = Option.some a\nc : \u03b1\nhc : a < c\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 some_lt_some, \u2190 h] at hc \n[GOAL]\ncase some.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMaxOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : PredOrder (WithTop \u03b1)\na : \u03b1\nh : pred \u22a4 = Option.some a\nc : \u03b1\nhc : pred \u22a4 < Option.some c\n\u22a2 False\n[PROOFSTEP]\nexact (le_of_pred_lt hc).not_lt (some_lt_none _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithBot \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    a\n\u22a2 IsMax a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      none \u2264\n    none\n\u22a2 IsMax none\n[PROOFSTEP]\nexact ((none_lt_some (\u22a5 : \u03b1)).not_le ha).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nha :\n  (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\n\u22a2 IsMax (Option.some val\u271d)\n[PROOFSTEP]\nexact (max_of_succ_le <| some_le_some.1 ha).withBot\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithBot \u03b1\nh : a < b\n\u22a2 (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithBot \u03b1\nh : a < none\n\u22a2 (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    none\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithBot \u03b1\nval\u271d : \u03b1\nh : a < Option.some val\u271d\n\u22a2 (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      a \u2264\n    Option.some val\u271d\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh : none < Option.some val\u271d\n\u22a2 (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      none \u2264\n    Option.some val\u271d\n[PROOFSTEP]\nexact some_le_some.2 bot_le\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d < Option.some val\u271d\u00b9\n\u22a2 (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d) \u2264\n    Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact some_le_some.2 (succ_le_of_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na b : WithBot \u03b1\nh :\n  a <\n    (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithBot \u03b1\nh :\n  none <\n    (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 none \u2264 b\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nb : WithBot \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      b\n\u22a2 Option.some val\u271d \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d : \u03b1\nh :\n  Option.some val\u271d <\n    (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      none\n\u22a2 Option.some val\u271d \u2264 none\n[PROOFSTEP]\nexact (not_lt_bot <| some_lt_some.1 h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  Option.some val\u271d\u00b9 <\n    (fun a =>\n        match a with\n        | none => \u2191\u22a5\n        | Option.some a => \u2191(succ a))\n      (Option.some val\u271d)\n\u22a2 Option.some val\u271d\u00b9 \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact some_le_some.2 (le_of_lt_succ <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na : WithBot \u03b1\nha : a \u2260 \u22a5\n\u22a2 succ a \u2260 \u22a5\n[PROOFSTEP]\ninduction a using WithBot.recBotCoe\n[GOAL]\ncase bot\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\nha : \u22a5 \u2260 \u22a5\n\u22a2 succ \u22a5 \u2260 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\ncase coe\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : SuccOrder \u03b1\na\u271d : \u03b1\nha : \u2191a\u271d \u2260 \u22a5\n\u22a2 succ \u2191a\u271d \u2260 \u22a5\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      a \u2264\n    a\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      none \u2264\n    none\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      (Option.some a) \u2264\n    Option.some a\n[PROOFSTEP]\nchange ite _ _ _ \u2264 _\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\n\u22a2 (if a = \u22a5 then \u22a5 else \u2191(pred a)) \u2264 Option.some a\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nh\u271d : a = \u22a5\n\u22a2 \u22a5 \u2264 Option.some a\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nh\u271d : \u00aca = \u22a5\n\u22a2 \u2191(pred a) \u2264 Option.some a\n[PROOFSTEP]\nexact some_le_some.2 (pred_le a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nha :\n  a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      a\n\u22a2 IsMin a\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nha :\n  none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      none\n\u22a2 IsMin none\n[PROOFSTEP]\nexact isMin_bot\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha :\n  Option.some a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      (Option.some a)\n\u22a2 IsMin (Option.some a)\n[PROOFSTEP]\ndsimp only at ha \n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha : Option.some a \u2264 if a = \u22a5 then \u22a5 else \u2191(pred a)\n\u22a2 IsMin (Option.some a)\n[PROOFSTEP]\nsplit_ifs at ha  with ha'\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha' : a = \u22a5\nha : Option.some a \u2264 \u22a5\n\u22a2 IsMin (Option.some a)\n[PROOFSTEP]\nexact (not_coe_le_bot _ ha).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha' : \u00aca = \u22a5\nha : Option.some a \u2264 \u2191(pred a)\n\u22a2 IsMin (Option.some a)\n[PROOFSTEP]\nrw [some_eq_coe, coe_le_coe, le_pred_iff_eq_bot] at ha \n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\nha' : \u00aca = \u22a5\nha\u271d : a \u2264 pred a\nha : a = \u22a5\n\u22a2 IsMin (Option.some a)\n[PROOFSTEP]\nexact (ha' ha).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithBot \u03b1\nh : a < b\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nb : WithBot \u03b1\nh : none < b\n\u22a2 none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      b\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nb : WithBot \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < b\n\u22a2 Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh : Option.some val\u271d < none\n\u22a2 Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      none\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d\u00b9 < Option.some val\u271d\n\u22a2 Option.some val\u271d\u00b9 \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      (Option.some val\u271d)\n[PROOFSTEP]\nrw [some_lt_some] at h \n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d\u00b9 < val\u271d\n\u22a2 Option.some val\u271d\u00b9 \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      (Option.some val\u271d)\n[PROOFSTEP]\nchange _ \u2264 ite _ _ _\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d\u00b9 < val\u271d\n\u22a2 Option.some val\u271d\u00b9 \u2264 if val\u271d = \u22a5 then \u22a5 else \u2191(pred val\u271d)\n[PROOFSTEP]\nsplit_ifs with hb\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d\u00b9 < val\u271d\nhb : val\u271d = \u22a5\n\u22a2 Option.some val\u271d\u00b9 \u2264 \u22a5\n[PROOFSTEP]\nrw [hb] at h \n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d\u00b9 < \u22a5\nhb : val\u271d = \u22a5\n\u22a2 Option.some val\u271d\u00b9 \u2264 \u22a5\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : val\u271d\u00b9 < val\u271d\nhb : \u00acval\u271d = \u22a5\n\u22a2 Option.some val\u271d\u00b9 \u2264 \u2191(pred val\u271d)\n[PROOFSTEP]\nexact some_le_some.2 (le_pred_of_lt h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithBot \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      a <\n    b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      a <\n    none\n\u22a2 a \u2264 none\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      a <\n    Option.some val\u271d\n\u22a2 a \u2264 Option.some val\u271d\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      none <\n    Option.some val\u271d\n\u22a2 none \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => if a = \u22a5 then \u22a5 else \u2191(pred a))\n      (Option.some val\u271d) <\n    Option.some val\u271d\u00b9\n\u22a2 Option.some val\u271d \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\ndsimp only at h \n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : (if val\u271d = \u22a5 then \u22a5 else \u2191(pred val\u271d)) < Option.some val\u271d\u00b9\n\u22a2 Option.some val\u271d \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nrw [some_le_some]\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : (if val\u271d = \u22a5 then \u22a5 else \u2191(pred val\u271d)) < Option.some val\u271d\u00b9\n\u22a2 val\u271d \u2264 val\u271d\u00b9\n[PROOFSTEP]\nsplit_ifs at h  with ha\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nha : val\u271d = \u22a5\nh : \u22a5 < Option.some val\u271d\u00b9\n\u22a2 val\u271d \u2264 val\u271d\u00b9\n[PROOFSTEP]\nrw [ha]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nha : val\u271d = \u22a5\nh : \u22a5 < Option.some val\u271d\u00b9\n\u22a2 \u22a5 \u2264 val\u271d\u00b9\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b3 : DecidableEq \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nha : \u00acval\u271d = \u22a5\nh : \u2191(pred val\u271d) < Option.some val\u271d\u00b9\n\u22a2 val\u271d \u2264 val\u271d\u00b9\n[PROOFSTEP]\nexact le_of_pred_lt (some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\n\u22a2 SuccOrder (WithBot \u03b1) \u2192 False\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : SuccOrder (WithBot \u03b1)\n\u22a2 False\n[PROOFSTEP]\ncases' h : succ (\u22a5 : WithBot \u03b1) with a ha\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : SuccOrder (WithBot \u03b1)\nh : succ \u22a5 = none\n\u22a2 False\n[PROOFSTEP]\nexact h\u03b1.elim fun a => (max_of_succ_le h.le).not_lt <| bot_lt_coe a\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : SuccOrder (WithBot \u03b1)\na : \u03b1\nh : succ \u22a5 = Option.some a\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := exists_lt a\n[GOAL]\ncase some.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : SuccOrder (WithBot \u03b1)\na : \u03b1\nh : succ \u22a5 = Option.some a\nc : \u03b1\nhc : c < a\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 some_lt_some, \u2190 h] at hc \n[GOAL]\ncase some.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : NoMinOrder \u03b1\nh\u03b1 : Nonempty \u03b1\na\u271d : SuccOrder (WithBot \u03b1)\na : \u03b1\nh : succ \u22a5 = Option.some a\nc : \u03b1\nhc : Option.some c < succ \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact (le_of_lt_succ hc).not_lt (none_lt_some _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      a \u2264\n    a\n[PROOFSTEP]\ncases' a with a a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      none \u2264\n    none\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : \u03b1\n\u22a2 (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some a) \u2264\n    Option.some a\n[PROOFSTEP]\nexact some_le_some.2 (pred_le a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nha :\n  a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      a\n\u22a2 IsMin a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nha :\n  none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      none\n\u22a2 IsMin none\n[PROOFSTEP]\nexact isMin_bot\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nha :\n  Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d)\n\u22a2 IsMin (Option.some val\u271d)\n[PROOFSTEP]\nexact (not_isMin _ <| min_of_le_pred <| some_le_some.1 ha).elim\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithBot \u03b1\nh : a < b\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nh : a < none\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      none\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nval\u271d : \u03b1\nh : a < Option.some val\u271d\n\u22a2 a \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh : none < Option.some val\u271d\n\u22a2 none \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d)\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh : Option.some val\u271d < Option.some val\u271d\u00b9\n\u22a2 Option.some val\u271d \u2264\n    (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d\u00b9)\n[PROOFSTEP]\nexact some_le_some.2 (le_pred_of_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na b : WithBot \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      a <\n    b\n\u22a2 a \u2264 b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      a <\n    none\n\u22a2 a \u2264 none\n[PROOFSTEP]\nexact (not_lt_bot h).elim\n[GOAL]\ncase some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\na : WithBot \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      a <\n    Option.some val\u271d\n\u22a2 a \u2264 Option.some val\u271d\n[PROOFSTEP]\ncases a\n[GOAL]\ncase some.none\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      none <\n    Option.some val\u271d\n\u22a2 none \u2264 Option.some val\u271d\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase some.some\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : NoMinOrder \u03b1\ninst\u271d : PredOrder \u03b1\nval\u271d\u00b9 val\u271d : \u03b1\nh :\n  (fun a =>\n        match a with\n        | none => \u22a5\n        | Option.some a => \u2191(pred a))\n      (Option.some val\u271d) <\n    Option.some val\u271d\u00b9\n\u22a2 Option.some val\u271d \u2264 Option.some val\u271d\u00b9\n[PROOFSTEP]\nexact some_le_some.2 (le_of_pred_lt <| some_lt_some.1 h)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nh : a \u2264 b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nconvert exists_succ_iterate_of_le h.ofDual\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\n\u22a2 (\u2203 n, succ^[n] a = b) \u2194 a \u2264 b\n[PROOFSTEP]\nrefine' \u27e8_, exists_succ_iterate_of_le\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\n\u22a2 (\u2203 n, succ^[n] a = b) \u2192 a \u2264 b\n[PROOFSTEP]\nrintro \u27e8n, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na : \u03b1\nn : \u2115\n\u22a2 a \u2264 succ^[n] a\n[PROOFSTEP]\nexact id_le_iterate_of_id_le le_succ n a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\nn : \u03b1\nhmn : m \u2264 n\n\u22a2 P n\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := hmn.exists_succ_iterate\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\nn : \u2115\nhmn : m \u2264 succ^[n] m\n\u22a2 P (succ^[n] m)\n[PROOFSTEP]\nclear hmn\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\nn : \u2115\n\u22a2 P (succ^[n] m)\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase intro.zero\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\n\u22a2 P (succ^[Nat.zero] m)\n[PROOFSTEP]\nexact h0\n[GOAL]\ncase intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\nn : \u2115\nih : P (succ^[n] m)\n\u22a2 P (succ^[Nat.succ n] m)\n[PROOFSTEP]\nrw [Function.iterate_succ_apply']\n[GOAL]\ncase intro.succ\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\nm : \u03b1\nh0 : P m\nh1 : \u2200 (n : \u03b1), m \u2264 n \u2192 P n \u2192 P (succ n)\nn : \u2115\nih : P (succ^[n] m)\n\u22a2 P (succ (succ^[n] m))\n[PROOFSTEP]\nexact h1 _ (id_le_iterate_of_id_le le_succ n m) ih\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na\u271d b\u271d : \u03b1\np : \u03b1 \u2192 Prop\nhsucc : \u2200 (a : \u03b1), p a \u2194 p (succ a)\na b : \u03b1\nh : a \u2264 b\n\u22a2 p a \u2194 p b\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := h.exists_succ_iterate\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\na\u271d b : \u03b1\np : \u03b1 \u2192 Prop\nhsucc : \u2200 (a : \u03b1), p a \u2194 p (succ a)\na : \u03b1\nn : \u2115\nh : a \u2264 succ^[n] a\n\u22a2 p a \u2194 p (succ^[n] a)\n[PROOFSTEP]\nexact Iterate.rec (fun b => p a \u2194 p b) (fun c hc => hc.trans (hsucc _)) Iff.rfl n\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : PredOrder \u03b1\ninst\u271d : IsPredArchimedean \u03b1\na\u271d b\u271d : \u03b1\na b : \u03b1\u1d52\u1d48\nh : a \u2264 b\n\u22a2 \u2203 n, succ^[n] a = b\n[PROOFSTEP]\nconvert exists_pred_iterate_of_le h.ofDual\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\n\u22a2 a \u2264 b \u2192 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nrefine' WellFounded.fix (C := fun b => a \u2264 b \u2192 \u2203 n, Nat.iterate pred n b = a) h.wf _ b\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\n\u22a2 \u2200 (x : \u03b1), (\u2200 (y : \u03b1), y < x \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y) \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) x\n[PROOFSTEP]\nintros b ih hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\nhab : a \u2264 b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nreplace hab := eq_or_lt_of_le hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\nhab : a = b \u2228 a < b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nrcases hab with (rfl | hab)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b : \u03b1\nih : \u2200 (y : \u03b1), y < a \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\n\u22a2 \u2203 n, pred^[n] a = a\n[PROOFSTEP]\nexact \u27e80, rfl\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\nhab : a < b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\ncases' le_or_lt b (pred b) with hb hb\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\nhab : a < b\nhb : b \u2264 pred b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\ncases (min_of_le_pred hb).not_lt hab\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 (fun b => a \u2264 b \u2192 \u2203 n, pred^[n] b = a) y\nhab : a < b\nhb : pred b < b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\ndsimp at ih \n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 a \u2264 y \u2192 \u2203 n, pred^[n] y = a\nhab : a < b\nhb : pred b < b\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := ih (pred b) hb (le_pred_of_lt hab)\n[GOAL]\ncase inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 a \u2264 y \u2192 \u2203 n, pred^[n] y = a\nhab : a < b\nhb : pred b < b\nk : \u2115\nhk : pred^[k] (pred b) = a\n\u22a2 \u2203 n, pred^[n] b = a\n[PROOFSTEP]\nrefine' \u27e8k + 1, _\u27e9\n[GOAL]\ncase inr.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x < x_1\ninst\u271d : PredOrder \u03b1\na b\u271d b : \u03b1\nih : \u2200 (y : \u03b1), y < b \u2192 a \u2264 y \u2192 \u2203 n, pred^[n] y = a\nhab : a < b\nhb : pred b < b\nk : \u2115\nhk : pred^[k] (pred b) = a\n\u22a2 pred^[k + 1] b = a\n[PROOFSTEP]\nrw [iterate_add_apply, iterate_one, hk]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nh : IsWellOrder \u03b1 fun x x_1 => x > x_1\ninst\u271d : SuccOrder \u03b1\n\u22a2 IsPredArchimedean \u03b1\u1d52\u1d48\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\n\u22a2 (\u2200 (i : \u03b1), i \u2260 \u22a5 \u2192 P i) \u2194 \u2200 (i : \u03b1), P (succ i)\n[PROOFSTEP]\nrefine' \u27e8fun h i \u21a6 h _ (Order.succ_ne_bot i), fun h i hi \u21a6 _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\ni : \u03b1\nhi : i \u2260 \u22a5\n\u22a2 P i\n[PROOFSTEP]\nobtain \u27e8j, rfl\u27e9 := exists_succ_iterate_of_le (bot_le : \u22a5 \u2264 i)\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\n\u22a2 P (Order.succ^[j] \u22a5)\n[PROOFSTEP]\nhave hj : 0 < j := by apply Nat.pos_of_ne_zero; contrapose! hi; simp [hi]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\n\u22a2 0 < j\n[PROOFSTEP]\napply Nat.pos_of_ne_zero\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\n\u22a2 j \u2260 0\n[PROOFSTEP]\ncontrapose! hi\n[GOAL]\ncase a\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : j = 0\n\u22a2 Order.succ^[j] \u22a5 = \u22a5\n[PROOFSTEP]\nsimp [hi]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\nhj : 0 < j\n\u22a2 P (Order.succ^[j] \u22a5)\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos hj]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\nhj : 0 < j\n\u22a2 P (Order.succ^[Nat.succ (Nat.pred j)] \u22a5)\n[PROOFSTEP]\nsimp only [Function.iterate_succ', Function.comp_apply]\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u2074 : Nontrivial \u03b1\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : OrderBot \u03b1\ninst\u271d\u00b9 : SuccOrder \u03b1\ninst\u271d : IsSuccArchimedean \u03b1\nP : \u03b1 \u2192 Prop\nh : \u2200 (i : \u03b1), P (succ i)\nj : \u2115\nhi : Order.succ^[j] \u22a5 \u2260 \u22a5\nhj : 0 < j\n\u22a2 P (Order.succ (Order.succ^[Nat.pred j] \u22a5))\n[PROOFSTEP]\napply h\n", "meta": {"mathlib_filename": "Mathlib.Order.SuccPred.Basic", "llama_tokens": 37436, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.782662489091802, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5207918789836498}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\n\u22a2 Tendsto f cofinite (cocompact Y) \u2194 \u2200 (K : Set Y), IsCompact K \u2192 Set.Finite (f \u207b\u00b9' K)\n[PROOFSTEP]\nrw [hasBasis_cocompact.tendsto_right_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\n\u22a2 (\u2200 (i : Set Y), IsCompact i \u2192 \u2200\u1da0 (x : X) in cofinite, f x \u2208 i\u1d9c) \u2194 \u2200 (K : Set Y), IsCompact K \u2192 Set.Finite (f \u207b\u00b9' K)\n[PROOFSTEP]\nrefine' forall\u2082_congr (fun K _ \u21a6 _)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nf : X \u2192 Y\nK : Set Y\nx\u271d : IsCompact K\n\u22a2 (\u2200\u1da0 (x : X) in cofinite, f x \u2208 K\u1d9c) \u2194 Set.Finite (f \u207b\u00b9' K)\n[PROOFSTEP]\nsimp only [mem_compl_iff, eventually_cofinite, not_not, preimage]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : T1Space X\ninst\u271d : LocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\n\u22a2 DiscreteTopology X\n[PROOFSTEP]\nrefine' singletons_open_iff_discrete.mp (fun x \u21a6 _)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : T1Space X\ninst\u271d : LocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\n\u22a2 IsOpen {x}\n[PROOFSTEP]\nobtain \u27e8K : Set Y, hK : IsCompact K, hK' : K \u2208 \ud835\udcdd (f x)\u27e9 := exists_compact_mem_nhds (f x)\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : T1Space X\ninst\u271d : LocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K \u2208 \ud835\udcdd (f x)\n\u22a2 IsOpen {x}\n[PROOFSTEP]\nobtain \u27e8U : Set Y, hU\u2081 : U \u2286 K, hU\u2082 : IsOpen U, hU\u2083 : f x \u2208 U\u27e9 := mem_nhds_iff.mp hK'\n[GOAL]\ncase intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : T1Space X\ninst\u271d : LocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K \u2208 \ud835\udcdd (f x)\nU : Set Y\nhU\u2081 : U \u2286 K\nhU\u2082 : IsOpen U\nhU\u2083 : f x \u2208 U\n\u22a2 IsOpen {x}\n[PROOFSTEP]\nhave hU\u2084 : Set.Finite (f \u207b\u00b9' U) := Finite.subset (tendsto_cofinite_cocompact_iff.mp hf K hK) (preimage_mono hU\u2081)\n[GOAL]\ncase intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d\u00b9 : T1Space X\ninst\u271d : LocallyCompactSpace Y\nhf' : Continuous f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K \u2208 \ud835\udcdd (f x)\nU : Set Y\nhU\u2081 : U \u2286 K\nhU\u2082 : IsOpen U\nhU\u2083 : f x \u2208 U\nhU\u2084 : Set.Finite (f \u207b\u00b9' U)\n\u22a2 IsOpen {x}\n[PROOFSTEP]\nexact isOpen_singleton_of_finite_mem_nhds _ ((hU\u2082.preimage hf').mem_nhds hU\u2083) hU\u2084\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d : DiscreteTopology X\nhf : Tendsto f (cocompact X) (cocompact Y)\n\u22a2 Tendsto f cofinite (cocompact Y)\n[PROOFSTEP]\nconvert hf\n[GOAL]\ncase h.e'_4\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nf : X \u2192 Y\ninst\u271d : DiscreteTopology X\nhf : Tendsto f (cocompact X) (cocompact Y)\n\u22a2 cofinite = cocompact X\n[PROOFSTEP]\nrw [cocompact_eq_cofinite X]\n", "meta": {"mathlib_filename": "Mathlib.Topology.DiscreteSubset", "llama_tokens": 1588, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.519813719528277}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nlet a := I.smithCoeffs b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nlet b' := I.ringBasis b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nlet ab := I.selfBasis b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nhave ab_eq := I.selfBasis_def b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nhave mem_I_iff : \u2200 x, x \u2208 I \u2194 \u2200 i, a i \u2223 b'.repr x i := by\n  intro x\n  rw [ab.mem_ideal_iff']\n  simp_rw [ab_eq]\n  have : \u2200 (c : \u03b9 \u2192 R) (i), b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 b' j) i = a i * c i :=\n    by\n    intro c i\n    simp only [\u2190 MulAction.mul_smul, b'.repr_sum_self, mul_comm]\n  constructor\n  \u00b7 rintro \u27e8c, rfl\u27e9 i\n    exact \u27e8c i, this c i\u27e9\n  \u00b7 rintro ha\n    choose c hc using ha\n    exact\n      \u27e8c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm\u27e9\n        -- Now we map everything through the linear equiv `S \u2243\u2097 (\u03b9 \u2192 R)`,\n          -- which maps `I` to `I' := \u03a0 i, a i \u2124`.\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\n\u22a2 \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\n\u22a2 x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\n[PROOFSTEP]\nrw [ab.mem_ideal_iff']\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\n\u22a2 (\u2203 c, x = \u2211 i : \u03b9, c i \u2022 \u2191(\u2191ab i)) \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\n[PROOFSTEP]\nsimp_rw [ab_eq]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\n\u22a2 (\u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x) \u2194\n    \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr x) i\n[PROOFSTEP]\nhave : \u2200 (c : \u03b9 \u2192 R) (i), b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 b' j) i = a i * c i :=\n  by\n  intro c i\n  simp only [\u2190 MulAction.mul_smul, b'.repr_sum_self, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\n\u22a2 \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\n[PROOFSTEP]\nintro c i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nc : \u03b9 \u2192 R\ni : \u03b9\n\u22a2 \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\n[PROOFSTEP]\nsimp only [\u2190 MulAction.mul_smul, b'.repr_sum_self, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\n\u22a2 (\u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x) \u2194\n    \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr x) i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\n\u22a2 (\u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x) \u2192\n    \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr x) i\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9 i\n[GOAL]\ncase mp.intro\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\nc : \u03b9 \u2192 R\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr (\u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x)) i\n[PROOFSTEP]\nexact \u27e8c i, this c i\u27e9\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\n\u22a2 (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr x) i) \u2192\n    \u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x\n[PROOFSTEP]\nrintro ha\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\nha : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr x) i\n\u22a2 \u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x\n[PROOFSTEP]\nchoose c hc using ha\n[GOAL]\ncase mpr\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nx : S\nthis : \u2200 (c : \u03b9 \u2192 R) (i : \u03b9), \u2191(\u2191b'.repr (\u2211 j : \u03b9, c j \u2022 a j \u2022 \u2191b' j)) i = a i * c i\nc : \u03b9 \u2192 R\nhc : \u2200 (i : \u03b9), \u2191(\u2191(ringBasis b I hI).repr x) i = smithCoeffs b I hI i * c i\n\u22a2 \u2203 c, x = \u2211 x : \u03b9, c x \u2022 smithCoeffs b I hI x \u2022 \u2191(ringBasis b I hI) x\n[PROOFSTEP]\nexact\n  \u27e8c, b'.ext_elem fun i => Eq.trans (hc i) (this c i).symm\u27e9\n    -- Now we map everything through the linear equiv `S \u2243\u2097 (\u03b9 \u2192 R)`,\n      -- which maps `I` to `I' := \u03a0 i, a i \u2124`.\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nlet I' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span ({a i} : Set R)\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nhave : Submodule.map (b'.equivFun : S \u2192\u2097[R] \u03b9 \u2192 R) (I.restrictScalars R) = I' :=\n  by\n  ext x\n  simp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ, Submodule.restrictScalars_mem,\n    mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe, Basis.equivFun_apply]\n  constructor\n  \u00b7 rintro \u27e8y, hy, rfl\u27e9 i\n    exact hy i\n  \u00b7 rintro hdvd\n    refine' \u27e8\u2211 i, x i \u2022 b' i, fun i => _, _\u27e9 <;> rw [b'.repr_sum_self]\n    \u00b7 exact hdvd i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\n\u22a2 Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\n\u22a2 x \u2208 Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) \u2194 x \u2208 I'\n[PROOFSTEP]\nsimp only [Submodule.mem_map, Submodule.mem_pi, mem_span_singleton, Set.mem_univ, Submodule.restrictScalars_mem,\n  mem_I_iff, smul_eq_mul, forall_true_left, LinearEquiv.coe_coe, Basis.equivFun_apply]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\n\u22a2 (\u2203 y, (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i) \u2227 \u2191(\u2191(ringBasis b I hI).repr y) = x) \u2194\n    \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\n\u22a2 (\u2203 y, (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i) \u2227 \u2191(\u2191(ringBasis b I hI).repr y) = x) \u2192\n    \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\n[PROOFSTEP]\nrintro \u27e8y, hy, rfl\u27e9 i\n[GOAL]\ncase h.mp.intro.intro\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\ny : S\nhy : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i\n[PROOFSTEP]\nexact hy i\n[GOAL]\ncase h.mpr\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\n\u22a2 (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i) \u2192\n    \u2203 y, (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i) \u2227 \u2191(\u2191(ringBasis b I hI).repr y) = x\n[PROOFSTEP]\nrintro hdvd\n[GOAL]\ncase h.mpr\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\nhdvd : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\n\u22a2 \u2203 y, (\u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr y) i) \u2227 \u2191(\u2191(ringBasis b I hI).repr y) = x\n[PROOFSTEP]\nrefine' \u27e8\u2211 i, x i \u2022 b' i, fun i => _, _\u27e9\n[GOAL]\ncase h.mpr.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\nhdvd : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2223 \u2191(\u2191(ringBasis b I hI).repr (\u2211 i : \u03b9, x i \u2022 \u2191b' i)) i\n[PROOFSTEP]\nrw [b'.repr_sum_self]\n[GOAL]\ncase h.mpr.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\nhdvd : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\n\u22a2 \u2191(\u2191(ringBasis b I hI).repr (\u2211 i : \u03b9, x i \u2022 \u2191b' i)) = x\n[PROOFSTEP]\nrw [b'.repr_sum_self]\n[GOAL]\ncase h.mpr.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nx : \u03b9 \u2192 R\nhdvd : \u2200 (i : \u03b9), smithCoeffs b I hI i \u2223 x i\ni : \u03b9\n\u22a2 smithCoeffs b I hI i \u2223 x i\n[PROOFSTEP]\nexact hdvd i\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 (S \u29f8 I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nrefine'\n  ((Submodule.Quotient.restrictScalarsEquiv R I).restrictScalars R).symm.trans (\u03c3\u2081\u2082 := RingHom.id R) (\u03c3\u2083\u2082 :=\n    RingHom.id R) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 RingHomInvPair (RingHom.id R) (RingHom.id R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 RingHomInvPair (RingHom.id R) (RingHom.id R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 (S \u29f8 Submodule.restrictScalars R I) \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nrefine'\n  (Submodule.Quotient.equiv (I.restrictScalars R) I' b'.equivFun this).trans (\u03c3\u2081\u2082 := RingHom.id R) (\u03c3\u2083\u2082 := RingHom.id R)\n    _\n[GOAL]\ncase refine'_3.refine'_1\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 RingHomInvPair (RingHom.id R) (RingHom.id R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_3.refine'_2\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 RingHomInvPair (RingHom.id R) (RingHom.id R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_3.refine'_3\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 ((\u03b9 \u2192 R) \u29f8 I') \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nclassical\nlet this := Submodule.quotientPi (show \u2200 _, Submodule R R from fun i => span ({a i} : Set R))\nexact this\n[GOAL]\ncase refine'_3.refine'_3\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\n\u22a2 ((\u03b9 \u2192 R) \u29f8 I') \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nlet this := Submodule.quotientPi (show \u2200 _, Submodule R R from fun i => span ({a i} : Set R))\n[GOAL]\ncase refine'_3.refine'_3\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsPrincipalIdealRing R\ninst\u271d\u00b9 : IsDomain S\ninst\u271d : Finite \u03b9\nI : Ideal S\nb : Basis \u03b9 R S\nhI : I \u2260 \u22a5\nthis\u271d\u00b9 : Fintype \u03b9\na : \u03b9 \u2192 R := smithCoeffs b I hI\nb' : Basis \u03b9 R S := ringBasis b I hI\nab : Basis \u03b9 R { x // x \u2208 I } := selfBasis b I hI\nab_eq : \u2200 (i : \u03b9), \u2191(\u2191(selfBasis b I hI) i) = smithCoeffs b I hI i \u2022 \u2191(ringBasis b I hI) i\nmem_I_iff : \u2200 (x : S), x \u2208 I \u2194 \u2200 (i : \u03b9), a i \u2223 \u2191(\u2191b'.repr x) i\nI' : Submodule R (\u03b9 \u2192 R) := Submodule.pi Set.univ fun i => span {a i}\nthis\u271d : Submodule.map (\u2191(Basis.equivFun b')) (Submodule.restrictScalars R I) = I'\nthis : ((\u03b9 \u2192 R) \u29f8\n    Submodule.pi Set.univ\n      (let_fun this := fun i => span {a i};\n      this)) \u2243\u2097[R]\n  (i : \u03b9) \u2192 R \u29f8 (fun this => this) (fun i => span {a i}) i :=\n  Submodule.quotientPi\n    (let_fun this := fun i => span {a i};\n    this)\n\u22a2 ((\u03b9 \u2192 R) \u29f8 I') \u2243\u2097[R] (i : \u03b9) \u2192 R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nlet b := Module.Free.chooseBasis \u2124 S\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\nb : Basis (Module.Free.ChooseBasisIndex \u2124 S) \u2124 S := Module.Free.chooseBasis \u2124 S\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nlet a := I.smithCoeffs b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\nb : Basis (Module.Free.ChooseBasisIndex \u2124 S) \u2124 S := Module.Free.chooseBasis \u2124 S\na : Module.Free.ChooseBasisIndex \u2124 S \u2192 \u2124 := smithCoeffs b I hI\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nlet e := I.quotientEquivPiZMod b hI\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\nb : Basis (Module.Free.ChooseBasisIndex \u2124 S) \u2124 S := Module.Free.chooseBasis \u2124 S\na : Module.Free.ChooseBasisIndex \u2124 S \u2192 \u2124 := smithCoeffs b I hI\ne : S \u29f8 I \u2243+ ((i : Module.Free.ChooseBasisIndex \u2124 S) \u2192 ZMod (Int.natAbs (smithCoeffs b I hI i))) :=\n  quotientEquivPiZMod I b hI\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nhaveI : \u2200 i, NeZero (a i).natAbs := fun i => \u27e8Int.natAbs_ne_zero.mpr (smithCoeffs_ne_zero b I hI i)\u27e9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\nb : Basis (Module.Free.ChooseBasisIndex \u2124 S) \u2124 S := Module.Free.chooseBasis \u2124 S\na : Module.Free.ChooseBasisIndex \u2124 S \u2192 \u2124 := smithCoeffs b I hI\ne : S \u29f8 I \u2243+ ((i : Module.Free.ChooseBasisIndex \u2124 S) \u2192 ZMod (Int.natAbs (smithCoeffs b I hI i))) :=\n  quotientEquivPiZMod I b hI\nthis : \u2200 (i : Module.Free.ChooseBasisIndex \u2124 S), NeZero (Int.natAbs (a i))\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nclassical exact Fintype.ofEquiv (\u2200 i, ZMod (a i).natAbs) e.symm\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : IsDomain R\ninst\u271d\u2074 : IsPrincipalIdealRing R\ninst\u271d\u00b3 : IsDomain S\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : I \u2260 \u22a5\nb : Basis (Module.Free.ChooseBasisIndex \u2124 S) \u2124 S := Module.Free.chooseBasis \u2124 S\na : Module.Free.ChooseBasisIndex \u2124 S \u2192 \u2124 := smithCoeffs b I hI\ne : S \u29f8 I \u2243+ ((i : Module.Free.ChooseBasisIndex \u2124 S) \u2192 ZMod (Int.natAbs (smithCoeffs b I hI i))) :=\n  quotientEquivPiZMod I b hI\nthis : \u2200 (i : Module.Free.ChooseBasisIndex \u2124 S), NeZero (Int.natAbs (a i))\n\u22a2 Fintype (S \u29f8 I)\n[PROOFSTEP]\nexact Fintype.ofEquiv (\u2200 i, ZMod (a i).natAbs) e.symm\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\ninst\u271d\u2075 : IsDomain S\ninst\u271d\u2074 : Finite \u03b9\nF : Type u_4\ninst\u271d\u00b3 : CommRing F\ninst\u271d\u00b2 : Algebra F R\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F R S\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\n\u22a2 (S \u29f8 I) \u2243\u2097[F] \u2a01 (i : \u03b9), R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nhaveI := Fintype.ofFinite \u03b9\n[GOAL]\n\u03b9 : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : IsDomain R\ninst\u271d\u2076 : IsPrincipalIdealRing R\ninst\u271d\u2075 : IsDomain S\ninst\u271d\u2074 : Finite \u03b9\nF : Type u_4\ninst\u271d\u00b3 : CommRing F\ninst\u271d\u00b2 : Algebra F R\ninst\u271d\u00b9 : Algebra F S\ninst\u271d : IsScalarTower F R S\nb : Basis \u03b9 R S\nI : Ideal S\nhI : I \u2260 \u22a5\nthis : Fintype \u03b9\n\u22a2 (S \u29f8 I) \u2243\u2097[F] \u2a01 (i : \u03b9), R \u29f8 span {smithCoeffs b I hI i}\n[PROOFSTEP]\nexact ((I.quotientEquivPiSpan b _).restrictScalars F).trans (DirectSum.linearEquivFunOnFintype _ _ _).symm\n[GOAL]\n\u03b9\u271d : Type u_1\nR : Type u_2\nS : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : IsDomain R\ninst\u271d\u00b9\u2070 : IsPrincipalIdealRing R\ninst\u271d\u2079 : IsDomain S\ninst\u271d\u2078 : Finite \u03b9\u271d\nF : Type u_4\ninst\u271d\u2077 : CommRing F\ninst\u271d\u2076 : Algebra F R\ninst\u271d\u2075 : Algebra F S\ninst\u271d\u2074 : IsScalarTower F R S\nb\u271d : Basis \u03b9\u271d R S\nI : Ideal S\nhI : I \u2260 \u22a5\n\u03b9 : Type u_5\ninst\u271d\u00b3 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b2 : Nontrivial F\ninst\u271d\u00b9 : \u2200 (i : \u03b9), Module.Free F (R \u29f8 span {smithCoeffs b I hI i})\ninst\u271d : \u2200 (i : \u03b9), Module.Finite F (R \u29f8 span {smithCoeffs b I hI i})\n\u22a2 FiniteDimensional.finrank F (S \u29f8 I) = \u2211 i : \u03b9, FiniteDimensional.finrank F (R \u29f8 span {smithCoeffs b I hI i})\n[PROOFSTEP]\nrw [LinearEquiv.finrank_eq <| quotientEquivDirectSum F b hI, FiniteDimensional.finrank_directSum]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.FreeModule.IdealQuotient", "llama_tokens": 18066, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324983301568, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5193282173329716}}
{"text": "[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf : a\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\nhg : b\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\ni : \u2115\n\u22a2 coeff (a\u271d * b\u271d) i \u2208 I ^ i\n[PROOFSTEP]\nrw [coeff_mul]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf : a\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\nhg : b\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\ni : \u2115\n\u22a2 \u2211 x in Finset.Nat.antidiagonal i, coeff a\u271d x.fst * coeff b\u271d x.snd \u2208 I ^ i\n[PROOFSTEP]\napply Ideal.sum_mem\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf : a\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\nhg : b\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\ni : \u2115\n\u22a2 \u2200 (c : \u2115 \u00d7 \u2115), c \u2208 Finset.Nat.antidiagonal i \u2192 coeff a\u271d c.fst * coeff b\u271d c.snd \u2208 I ^ i\n[PROOFSTEP]\nrintro \u27e8j, k\u27e9 e\n[GOAL]\ncase a.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf : a\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\nhg : b\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\ni j k : \u2115\ne : (j, k) \u2208 Finset.Nat.antidiagonal i\n\u22a2 coeff a\u271d (j, k).fst * coeff b\u271d (j, k).snd \u2208 I ^ i\n[PROOFSTEP]\nrw [\u2190 Finset.Nat.mem_antidiagonal.mp e, pow_add]\n[GOAL]\ncase a.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf : a\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\nhg : b\u271d \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i}\ni j k : \u2115\ne : (j, k) \u2208 Finset.Nat.antidiagonal i\n\u22a2 coeff a\u271d (j, k).fst * coeff b\u271d (j, k).snd \u2208 I ^ (j, k).fst * I ^ (j, k).snd\n[PROOFSTEP]\nexact Ideal.mul_mem_mul (hf j) (hg k)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ni : \u2115\n\u22a2 coeff 1 i \u2208 I ^ i\n[PROOFSTEP]\nrw [coeff_one]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ni : \u2115\n\u22a2 (if 0 = i then 1 else 0) \u2208 I ^ i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ni : \u2115\nh : 0 = i\n\u22a2 1 \u2208 I ^ i\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\n\u22a2 1 \u2208 I ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ni : \u2115\nh : \u00ac0 = i\n\u22a2 0 \u2208 I ^ i\n[PROOFSTEP]\nsimp\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf :\n  a\u271d \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i},\n              mul_mem' :=\n                (_ :\n                  \u2200 {a b : R[X]},\n                    a \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192 \u2200 (i : \u2115), coeff (a * b) i \u2208 I ^ i) },\n          one_mem' := (_ : \u2200 (i : \u2115), coeff 1 i \u2208 I ^ i) }.toSubsemigroup.carrier\nhg :\n  b\u271d \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i},\n              mul_mem' :=\n                (_ :\n                  \u2200 {a b : R[X]},\n                    a \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192 \u2200 (i : \u2115), coeff (a * b) i \u2208 I ^ i) },\n          one_mem' := (_ : \u2200 (i : \u2115), coeff 1 i \u2208 I ^ i) }.toSubsemigroup.carrier\ni : \u2115\n\u22a2 coeff (a\u271d + b\u271d) i \u2208 I ^ i\n[PROOFSTEP]\nrw [coeff_add]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\na\u271d b\u271d : R[X]\nhf :\n  a\u271d \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i},\n              mul_mem' :=\n                (_ :\n                  \u2200 {a b : R[X]},\n                    a \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192 \u2200 (i : \u2115), coeff (a * b) i \u2208 I ^ i) },\n          one_mem' := (_ : \u2200 (i : \u2115), coeff 1 i \u2208 I ^ i) }.toSubsemigroup.carrier\nhg :\n  b\u271d \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i},\n              mul_mem' :=\n                (_ :\n                  \u2200 {a b : R[X]},\n                    a \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), coeff f i \u2208 I ^ i} \u2192 \u2200 (i : \u2115), coeff (a * b) i \u2208 I ^ i) },\n          one_mem' := (_ : \u2200 (i : \u2115), coeff 1 i \u2208 I ^ i) }.toSubsemigroup.carrier\ni : \u2115\n\u22a2 coeff a\u271d i + coeff b\u271d i \u2208 I ^ i\n[PROOFSTEP]\nexact Ideal.add_mem _ (hf i) (hg i)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\ni : \u2115\n\u22a2 coeff (\u2191(algebraMap R R[X]) r) i \u2208 I ^ i\n[PROOFSTEP]\nrw [algebraMap_apply, coeff_C]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\ni : \u2115\n\u22a2 (if i = 0 then \u2191(algebraMap R R) r else 0) \u2208 I ^ i\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\ni : \u2115\nh : i = 0\n\u22a2 \u2191(algebraMap R R) r \u2208 I ^ i\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\n\u22a2 \u2191(algebraMap R R) r \u2208 I ^ 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\ni : \u2115\nh : \u00aci = 0\n\u22a2 0 \u2208 I ^ i\n[PROOFSTEP]\nsimp\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nf : R[X]\n\u22a2 f \u2208 reesAlgebra I \u2194 \u2200 (i : \u2115), i \u2208 support f \u2192 coeff f i \u2208 I ^ i\n[PROOFSTEP]\napply forall_congr'\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nf : R[X]\n\u22a2 \u2200 (a : \u2115), coeff f a \u2208 I ^ a \u2194 a \u2208 support f \u2192 coeff f a \u2208 I ^ a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nf : R[X]\na : \u2115\n\u22a2 coeff f a \u2208 I ^ a \u2194 a \u2208 support f \u2192 coeff f a \u2208 I ^ a\n[PROOFSTEP]\nrw [mem_support_iff, Iff.comm, imp_iff_right_iff, Ne.def, \u2190 imp_iff_not_or]\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nf : R[X]\na : \u2115\n\u22a2 coeff f a = 0 \u2192 coeff f a \u2208 I ^ a\n[PROOFSTEP]\nexact fun e => e.symm \u25b8 (I ^ a).zero_mem\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\ni : \u2115\nr : R\n\u22a2 \u2191(monomial i) r \u2208 reesAlgebra I \u2194 r \u2208 I ^ i\n[PROOFSTEP]\nsimp (config := { contextual := true }) [mem_reesAlgebra_iff_support, coeff_monomial, \u2190 imp_iff_not_or]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn : \u2115\nr : R\nhr : r \u2208 I ^ n\n\u22a2 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\ninduction' n with n hn generalizing r\n[GOAL]\ncase zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\nr : R\nhr : r \u2208 I ^ Nat.zero\n\u22a2 \u2191(monomial Nat.zero) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nexact Subalgebra.algebraMap_mem _ _\n[GOAL]\ncase succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I ^ Nat.succ n\n\u22a2 \u2191(monomial (Nat.succ n)) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nrw [pow_succ] at hr \n[GOAL]\ncase succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I * I ^ n\n\u22a2 \u2191(monomial (Nat.succ n)) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\napply\n  Submodule.smul_induction_on (p := fun r =>\n    (monomial (Nat.succ n)) r \u2208 Algebra.adjoin R (Submodule.map (monomial 1) I)) hr\n[GOAL]\ncase succ.Hb\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I * I ^ n\n\u22a2 \u2200 (r : R),\n    r \u2208 I \u2192\n      \u2200 (n_1 : R), n_1 \u2208 I ^ n \u2192 \u2191(monomial (Nat.succ n)) (r \u2022 n_1) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nintro r hr s hs\n[GOAL]\ncase succ.Hb\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d\u00b9 : R\nhr\u271d\u00b9 : r\u271d\u00b9 \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr\u271d : R\nhr\u271d : r\u271d \u2208 I * I ^ n\nr : R\nhr : r \u2208 I\ns : R\nhs : s \u2208 I ^ n\n\u22a2 \u2191(monomial (Nat.succ n)) (r \u2022 s) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, smul_eq_mul, \u2190 monomial_mul_monomial]\n[GOAL]\ncase succ.Hb\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d\u00b9 : R\nhr\u271d\u00b9 : r\u271d\u00b9 \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr\u271d : R\nhr\u271d : r\u271d \u2208 I * I ^ n\nr : R\nhr : r \u2208 I\ns : R\nhs : s \u2208 I ^ n\n\u22a2 \u2191(monomial 1) r * \u2191(monomial n) s \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nexact Subalgebra.mul_mem _ (Algebra.subset_adjoin (Set.mem_image_of_mem _ hr)) (hn hs)\n[GOAL]\ncase succ.H1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I * I ^ n\n\u22a2 \u2200 (x y : R),\n    \u2191(monomial (Nat.succ n)) x \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I) \u2192\n      \u2191(monomial (Nat.succ n)) y \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I) \u2192\n        \u2191(monomial (Nat.succ n)) (x + y) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase succ.H1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I * I ^ n\nx y : R\nhx : \u2191(monomial (Nat.succ n)) x \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nhy : \u2191(monomial (Nat.succ n)) y \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n\u22a2 \u2191(monomial (Nat.succ n)) (x + y) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nrw [monomial_add]\n[GOAL]\ncase succ.H1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d I : Ideal R\nn\u271d : \u2115\nr\u271d : R\nhr\u271d : r\u271d \u2208 I ^ n\u271d\nn : \u2115\nhn : \u2200 {r : R}, r \u2208 I ^ n \u2192 \u2191(monomial n) r \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nr : R\nhr : r \u2208 I * I ^ n\nx y : R\nhx : \u2191(monomial (Nat.succ n)) x \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\nhy : \u2191(monomial (Nat.succ n)) y \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n\u22a2 \u2191(monomial (Nat.succ n)) x + \u2191(monomial (Nat.succ n)) y \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nexact Subalgebra.add_mem _ hx hy\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\n\u22a2 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I) = reesAlgebra I\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\n\u22a2 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I) \u2264 reesAlgebra I\n[PROOFSTEP]\napply Algebra.adjoin_le _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\n\u22a2 \u2191(Submodule.map (monomial 1) I) \u2286 \u2191(reesAlgebra I)\n[PROOFSTEP]\nrintro _ \u27e8r, hr, rfl\u27e9\n[GOAL]\ncase intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\nhr : r \u2208 \u2191I\n\u22a2 \u2191(monomial 1) r \u2208 \u2191(reesAlgebra I)\n[PROOFSTEP]\nexact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nr : R\nhr : r \u2208 \u2191I\n\u22a2 r \u2208 I ^ 1\n[PROOFSTEP]\nrwa [pow_one]\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\n\u22a2 reesAlgebra I \u2264 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\np : R[X]\nhp : p \u2208 reesAlgebra I\n\u22a2 p \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nrw [p.as_sum_support]\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\np : R[X]\nhp : p \u2208 reesAlgebra I\n\u22a2 \u2211 i in support p, \u2191(monomial i) (coeff p i) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\napply Subalgebra.sum_mem _ _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\np : R[X]\nhp : p \u2208 reesAlgebra I\n\u22a2 \u2200 (x : \u2115), x \u2208 support p \u2192 \u2191(monomial x) (coeff p x) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nrintro i -\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\np : R[X]\nhp : p \u2208 reesAlgebra I\ni : \u2115\n\u22a2 \u2191(monomial i) (coeff p i) \u2208 Algebra.adjoin R \u2191(Submodule.map (monomial 1) I)\n[PROOFSTEP]\nexact monomial_mem_adjoin_monomial (hp i)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhI : Ideal.FG I\n\u22a2 Subalgebra.FG (reesAlgebra I)\n[PROOFSTEP]\nclassical\nobtain \u27e8s, hs\u27e9 := hI\nrw [\u2190 adjoin_monomial_eq_reesAlgebra, \u2190 hs]\nuse s.image (monomial 1)\nrw [Finset.coe_image]\nchange _ = Algebra.adjoin R (Submodule.map (monomial 1 : R \u2192\u2097[R] R[X]) (Submodule.span R \u2191s) : Set R[X])\nrw [Submodule.map_span, Algebra.adjoin_span]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nhI : Ideal.FG I\n\u22a2 Subalgebra.FG (reesAlgebra I)\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := hI\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (reesAlgebra I)\n[PROOFSTEP]\nrw [\u2190 adjoin_monomial_eq_reesAlgebra, \u2190 hs]\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Subalgebra.FG (Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s)))\n[PROOFSTEP]\nuse s.image (monomial 1)\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R \u2191(Finset.image (\u2191(monomial 1)) s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))\n[PROOFSTEP]\nrw [Finset.coe_image]\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Ideal.span \u2191s))\n[PROOFSTEP]\nchange _ = Algebra.adjoin R (Submodule.map (monomial 1 : R \u2192\u2097[R] R[X]) (Submodule.span R \u2191s) : Set R[X])\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\ns : Finset R\nhs : Ideal.span \u2191s = I\n\u22a2 Algebra.adjoin R (\u2191(monomial 1) '' \u2191s) = Algebra.adjoin R \u2191(Submodule.map (monomial 1) (Submodule.span R \u2191s))\n[PROOFSTEP]\nrw [Submodule.map_span, Algebra.adjoin_span]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.ReesAlgebra", "llama_tokens": 7991, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152325073083132, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.5193282005813651}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\n\u22a2 IsCoatomic \u03b1\n[PROOFSTEP]\nrefine \u27e8fun x => le_top.eq_or_lt.imp_right fun hx => ?_\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a\n[PROOFSTEP]\nhave : \u2203 y \u2208 Ico x \u22a4, x \u2264 y \u2227 \u2200 z \u2208 Ico x \u22a4, y \u2264 z \u2192 z = y :=\n  by\n  refine zorn_nonempty_partialOrder\u2080 (Ico x \u22a4) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)\n  rcases h c hc \u27e8y, hy\u27e9 fun h => (hxc h).2.ne rfl with \u27e8z, hz, hcz\u27e9\n  exact \u27e8z, \u27e8le_trans (hxc hy).1 (hcz hy), hz.lt_top\u27e9, hcz\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\n\u22a2 \u2203 y, y \u2208 Ico x \u22a4 \u2227 x \u2264 y \u2227 \u2200 (z : \u03b1), z \u2208 Ico x \u22a4 \u2192 y \u2264 z \u2192 z = y\n[PROOFSTEP]\nrefine zorn_nonempty_partialOrder\u2080 (Ico x \u22a4) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\n\u22a2 \u2203 ub, ub \u2208 Ico x \u22a4 \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nrcases h c hc \u27e8y, hy\u27e9 fun h => (hxc h).2.ne rfl with \u27e8z, hz, hcz\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nc : Set \u03b1\nhxc : c \u2286 Ico x \u22a4\nhc : IsChain (fun x x_1 => x \u2264 x_1) c\ny : \u03b1\nhy : y \u2208 c\nz : \u03b1\nhz : z \u2260 \u22a4\nhcz : z \u2208 upperBounds c\n\u22a2 \u2203 ub, ub \u2208 Ico x \u22a4 \u2227 \u2200 (z : \u03b1), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nexact \u27e8z, \u27e8le_trans (hxc hy).1 (hcz hy), hz.lt_top\u27e9, hcz\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\nthis : \u2203 y, y \u2208 Ico x \u22a4 \u2227 x \u2264 y \u2227 \u2200 (z : \u03b1), z \u2208 Ico x \u22a4 \u2192 y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a\n[PROOFSTEP]\nrcases this with \u27e8y, \u27e8hxy, hy\u27e9, -, hy'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 (z : \u03b1), z \u2208 Ico x \u22a4 \u2192 y \u2264 z \u2192 z = y\n\u22a2 \u2203 a, IsCoatom a \u2227 x \u2264 a\n[PROOFSTEP]\nrefine \u27e8y, \u27e8hy.ne, fun z hyz => le_top.eq_or_lt.resolve_right fun hz => ?_\u27e9, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\nh : \u2200 (c : Set \u03b1), IsChain (fun x x_1 => x \u2264 x_1) c \u2192 Set.Nonempty c \u2192 \u00ac\u22a4 \u2208 c \u2192 \u2203 x x_1, x \u2208 upperBounds c\nx : \u03b1\nhx : x < \u22a4\ny : \u03b1\nhxy : x \u2264 y\nhy : y < \u22a4\nhy' : \u2200 (z : \u03b1), z \u2208 Ico x \u22a4 \u2192 y \u2264 z \u2192 z = y\nz : \u03b1\nhyz : y < z\nhz : z < \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact hyz.ne' (hy' z \u27e8hxy.trans hyz.le, hz\u27e9 hyz.le)\n", "meta": {"mathlib_filename": "Mathlib.Order.ZornAtoms", "llama_tokens": 1705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324938410784, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.5193281920023219}}
{"text": "[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\n\u22a2 IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n))) n\n[PROOFSTEP]\nrw [IsPrimitiveRoot.iff_def]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\n\u22a2 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) ^ n = 1 \u2227 \u2200 (l : \u2115), exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) ^ l = 1 \u2192 n \u2223 l\n[PROOFSTEP]\nsimp only [\u2190 exp_nat_mul, exp_eq_one_iff]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\n\u22a2 (\u2203 n_1, \u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)) \u2227\n    \u2200 (l : \u2115), (\u2203 n_1, \u2191l * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)) \u2192 n \u2223 l\n[PROOFSTEP]\nhave hn0 : (n : \u2102) \u2260 0 := by exact_mod_cast h0\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast h0\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 (\u2203 n_1, \u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)) \u2227\n    \u2200 (l : \u2115), (\u2203 n_1, \u2191l * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)) \u2192 n \u2223 l\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 \u2203 n_1, \u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 \u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191\u2191i * (2 * \u2191\u03c0 * I)\n[PROOFSTEP]\nfield_simp [hn0, mul_comm (i : \u2102), mul_comm (n : \u2102)]\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 \u2200 (l : \u2115), (\u2203 n_1, \u2191l * (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)) \u2192 n \u2223 l\n[PROOFSTEP]\nsimp only [hn0, mul_right_comm _ _ \u2191n, mul_left_inj' two_pi_I_ne_zero, Ne.def, not_false_iff, mul_comm _ (i : \u2102), \u2190\n  mul_assoc _ (i : \u2102), exists_imp, field_simps]\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 \u2200 (l : \u2115) (x : \u2124), \u2191i * \u2191l * (2 * \u2191\u03c0 * I) = \u2191x * (2 * \u2191\u03c0 * I) * \u2191n \u2192 n \u2223 l\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\n\u22a2 \u2200 (l : \u2115) (x : \u2124), \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191x * (\u2191(2 * \u03c0) * I) * \u2191n \u2192 n \u2223 l\n[PROOFSTEP]\nrintro l k hk\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n\u22a2 n \u2223 l\n[PROOFSTEP]\nconv_rhs at hk => rw [mul_comm, \u2190 mul_assoc]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n| \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n| \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n| \u2191k * (\u2191(2 * \u03c0) * I) * \u2191n\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc]\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191n * \u2191k * (\u2191(2 * \u03c0) * I)\n\u22a2 n \u2223 l\n[PROOFSTEP]\nhave hz : 2 * \u2191\u03c0 * I \u2260 0 := by simp [pi_pos.ne.symm, I_ne_zero]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191n * \u2191k * (\u2191(2 * \u03c0) * I)\n\u22a2 2 * \u2191\u03c0 * I \u2260 0\n[PROOFSTEP]\nsimp [pi_pos.ne.symm, I_ne_zero]\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhk : \u2191(i * l) * (\u2191(2 * \u03c0) * I) = \u2191n * \u2191k * (\u2191(2 * \u03c0) * I)\nhz : 2 * \u2191\u03c0 * I \u2260 0\n\u22a2 n \u2223 l\n[PROOFSTEP]\nfield_simp [hz] at hk \n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhz : 2 * \u2191\u03c0 * I \u2260 0\nhk : \u2191i * \u2191l = \u2191n * \u2191k\n\u22a2 n \u2223 l\n[PROOFSTEP]\nnorm_cast at hk \n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhz : 2 * \u2191\u03c0 * I \u2260 0\nhk : \u2191(i * l) = \u2191n * k\n\u22a2 n \u2223 l\n[PROOFSTEP]\nhave : n \u2223 i * l := by rw [\u2190 Int.coe_nat_dvd, hk, mul_comm]; apply dvd_mul_left\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhz : 2 * \u2191\u03c0 * I \u2260 0\nhk : \u2191(i * l) = \u2191n * k\n\u22a2 n \u2223 i * l\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd, hk, mul_comm]\n[GOAL]\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhz : 2 * \u2191\u03c0 * I \u2260 0\nhk : \u2191(i * l) = \u2191n * k\n\u22a2 \u2191n \u2223 k * \u2191n\n[PROOFSTEP]\napply dvd_mul_left\n[GOAL]\ncase right\ni n : \u2115\nh0 : n \u2260 0\nhi : Nat.coprime i n\nhn0 : \u2191n \u2260 0\nl : \u2115\nk : \u2124\nhz : 2 * \u2191\u03c0 * I \u2260 0\nhk : \u2191(i * l) = \u2191n * k\nthis : n \u2223 i * l\n\u22a2 n \u2223 l\n[PROOFSTEP]\nexact hi.symm.dvd_of_dvd_mul_left this\n[GOAL]\nn : \u2115\nh0 : n \u2260 0\n\u22a2 IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I / \u2191n)) n\n[PROOFSTEP]\nsimpa only [Nat.cast_one, one_div] using isPrimitiveRoot_exp_of_coprime 1 n h0 n.coprime_one_left\n[GOAL]\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\n\u22a2 IsPrimitiveRoot \u03b6 n \u2194 \u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6\n[PROOFSTEP]\nhave hn0 : (n : \u2102) \u2260 0 := by exact_mod_cast hn\n[GOAL]\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 IsPrimitiveRoot \u03b6 n \u2194 \u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 IsPrimitiveRoot \u03b6 n \u2192 \u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6\ncase mpr\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 (\u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6) \u2192 IsPrimitiveRoot \u03b6 n\n[PROOFSTEP]\nswap\n[GOAL]\ncase mpr\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 (\u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6) \u2192 IsPrimitiveRoot \u03b6 n\n[PROOFSTEP]\nrintro \u27e8i, -, hi, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\ni : \u2115\nhi : Nat.coprime i n\n\u22a2 IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n))) n\n[PROOFSTEP]\nexact isPrimitiveRoot_exp_of_coprime i n hn hi\n[GOAL]\ncase mp\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\n\u22a2 IsPrimitiveRoot \u03b6 n \u2192 \u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b6 : \u2102\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\nh : IsPrimitiveRoot \u03b6 n\n\u22a2 \u2203 i, i < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = \u03b6\n[PROOFSTEP]\nobtain \u27e8i, hi, rfl\u27e9 := (isPrimitiveRoot_exp n hn).eq_pow_of_pow_eq_one h.pow_eq_one (Nat.pos_of_ne_zero hn)\n[GOAL]\ncase mp.intro.intro\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\ni : \u2115\nhi : i < n\nh : IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I / \u2191n) ^ i) n\n\u22a2 \u2203 i_1, i_1 < n \u2227 \u2203 x, exp (2 * \u2191\u03c0 * I * (\u2191i_1 / \u2191n)) = exp (2 * \u2191\u03c0 * I / \u2191n) ^ i\n[PROOFSTEP]\nrefine' \u27e8i, hi, ((isPrimitiveRoot_exp n hn).pow_iff_coprime (Nat.pos_of_ne_zero hn) i).mp h, _\u27e9\n[GOAL]\ncase mp.intro.intro\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\ni : \u2115\nhi : i < n\nh : IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I / \u2191n) ^ i) n\n\u22a2 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = exp (2 * \u2191\u03c0 * I / \u2191n) ^ i\n[PROOFSTEP]\nrw [\u2190 exp_nat_mul]\n[GOAL]\ncase mp.intro.intro\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\ni : \u2115\nhi : i < n\nh : IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I / \u2191n) ^ i) n\n\u22a2 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191n)) = exp (\u2191i * (2 * \u2191\u03c0 * I / \u2191n))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mp.intro.intro.e_z\nn : \u2115\nhn : n \u2260 0\nhn0 : \u2191n \u2260 0\ni : \u2115\nhi : i < n\nh : IsPrimitiveRoot (exp (2 * \u2191\u03c0 * I / \u2191n) ^ i) n\n\u22a2 2 * \u2191\u03c0 * I * (\u2191i / \u2191n) = \u2191i * (2 * \u2191\u03c0 * I / \u2191n)\n[PROOFSTEP]\nfield_simp [hn0, mul_comm (i : \u2102)]\n[GOAL]\nn : \u2115+\nx : \u2102\u02e3\n\u22a2 x \u2208 rootsOfUnity n \u2102 \u2194 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nrw [mem_rootsOfUnity, Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one]\n[GOAL]\nn : \u2115+\nx : \u2102\u02e3\n\u22a2 \u2191x ^ \u2191n = 1 \u2194 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nhave hn0 : (n : \u2102) \u2260 0 := by exact_mod_cast n.ne_zero\n[GOAL]\nn : \u2115+\nx : \u2102\u02e3\n\u22a2 \u2191\u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast n.ne_zero\n[GOAL]\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\n\u22a2 \u2191x ^ \u2191n = 1 \u2194 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\n\u22a2 \u2191x ^ \u2191n = 1 \u2192 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\n\u22a2 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nobtain \u27e8i, hi, H\u27e9 : \u2203 i < (n : \u2115), exp (2 * \u03c0 * I / n) ^ i = x := by\n  simpa only using (isPrimitiveRoot_exp n n.ne_zero).eq_pow_of_pow_eq_one h n.pos\n[GOAL]\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\n\u22a2 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I / \u2191\u2191n) ^ i = \u2191x\n[PROOFSTEP]\nsimpa only using (isPrimitiveRoot_exp n n.ne_zero).eq_pow_of_pow_eq_one h n.pos\n[GOAL]\ncase mp.intro.intro\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\ni : \u2115\nhi : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I / \u2191\u2191n) ^ i = \u2191x\n\u22a2 \u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nrefine' \u27e8i, hi, _\u27e9\n[GOAL]\ncase mp.intro.intro\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\ni : \u2115\nhi : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I / \u2191\u2191n) ^ i = \u2191x\n\u22a2 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n[PROOFSTEP]\nrw [\u2190 H, \u2190 exp_nat_mul]\n[GOAL]\ncase mp.intro.intro\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\ni : \u2115\nhi : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I / \u2191\u2191n) ^ i = \u2191x\n\u22a2 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = exp (\u2191i * (2 * \u2191\u03c0 * I / \u2191\u2191n))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mp.intro.intro.e_z\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\nh : \u2191x ^ \u2191n = 1\ni : \u2115\nhi : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I / \u2191\u2191n) ^ i = \u2191x\n\u22a2 2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n) = \u2191i * (2 * \u2191\u03c0 * I / \u2191\u2191n)\n[PROOFSTEP]\nfield_simp [hn0, mul_comm (i : \u2102)]\n[GOAL]\ncase mpr\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\n\u22a2 (\u2203 i, i < \u2191n \u2227 exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x) \u2192 \u2191x ^ \u2191n = 1\n[PROOFSTEP]\nrintro \u27e8i, _, H\u27e9\n[GOAL]\ncase mpr.intro.intro\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\ni : \u2115\nleft\u271d : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n\u22a2 \u2191x ^ \u2191n = 1\n[PROOFSTEP]\nrw [\u2190 H, \u2190 exp_nat_mul, exp_eq_one_iff]\n[GOAL]\ncase mpr.intro.intro\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\ni : \u2115\nleft\u271d : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n\u22a2 \u2203 n_1, \u2191\u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191n_1 * (2 * \u2191\u03c0 * I)\n[PROOFSTEP]\nuse i\n[GOAL]\ncase h\nn : \u2115+\nx : \u2102\u02e3\nhn0 : \u2191\u2191n \u2260 0\ni : \u2115\nleft\u271d : i < \u2191n\nH : exp (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191x\n\u22a2 \u2191\u2191n * (2 * \u2191\u03c0 * I * (\u2191i / \u2191\u2191n)) = \u2191\u2191i * (2 * \u2191\u03c0 * I)\n[PROOFSTEP]\nfield_simp [hn0, mul_comm ((n : \u2115) : \u2102), mul_comm (i : \u2102)]\n[GOAL]\nk : \u2115\n\u22a2 Finset.card (primitiveRoots k \u2102) = \u03c6 k\n[PROOFSTEP]\nby_cases h : k = 0\n[GOAL]\ncase pos\nk : \u2115\nh : k = 0\n\u22a2 Finset.card (primitiveRoots k \u2102) = \u03c6 k\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nk : \u2115\nh : \u00ack = 0\n\u22a2 Finset.card (primitiveRoots k \u2102) = \u03c6 k\n[PROOFSTEP]\nexact (isPrimitiveRoot_exp k h).card_primitiveRoots\n[GOAL]\nn : \u2115\n\u03b6 : \u2102\nh : IsPrimitiveRoot \u03b6 n\nhn : n \u2260 0\n\u22a2 \u2203 i, Complex.arg \u03b6 = \u2191i / \u2191n * (2 * Real.pi) \u2227 IsCoprime i \u2191n \u2227 Int.natAbs i < n\n[PROOFSTEP]\nrw [Complex.isPrimitiveRoot_iff _ _ hn] at h \n[GOAL]\nn : \u2115\n\u03b6 : \u2102\nh : \u2203 i, i < n \u2227 \u2203 x, Complex.exp (2 * \u2191Real.pi * Complex.I * (\u2191i / \u2191n)) = \u03b6\nhn : n \u2260 0\n\u22a2 \u2203 i, Complex.arg \u03b6 = \u2191i / \u2191n * (2 * Real.pi) \u2227 IsCoprime i \u2191n \u2227 Int.natAbs i < n\n[PROOFSTEP]\nobtain \u27e8i, h, hin, rfl\u27e9 := h\n[GOAL]\ncase intro.intro.intro\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 \u2203 i_1,\n    Complex.arg (Complex.exp (2 * \u2191Real.pi * Complex.I * (\u2191i / \u2191n))) = \u2191i_1 / \u2191n * (2 * Real.pi) \u2227\n      IsCoprime i_1 \u2191n \u2227 Int.natAbs i_1 < n\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc, Complex.exp_mul_I]\n[GOAL]\ncase intro.intro.intro\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 \u2203 i_1,\n    Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n        \u2191i_1 / \u2191n * (2 * Real.pi) \u2227\n      IsCoprime i_1 \u2191n \u2227 Int.natAbs i_1 < n\n[PROOFSTEP]\nrefine' \u27e8if i * 2 \u2264 n then i else i - n, _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) / \u2191n * (2 * Real.pi)\ncase intro.intro.intro.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 IsCoprime (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) \u2191n\ncase intro.intro.intro.refine'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Int.natAbs (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) < n\n[PROOFSTEP]\non_goal 2 =>\n  replace hin := Nat.isCoprime_iff_coprime.mpr hin\n  split_ifs\n  \u00b7 exact hin\n  \u00b7 convert hin.add_mul_left_left (-1) using 1\n    rw [mul_neg_one, sub_eq_add_neg]\n[GOAL]\ncase intro.intro.intro.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) / \u2191n * (2 * Real.pi)\ncase intro.intro.intro.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 IsCoprime (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) \u2191n\ncase intro.intro.intro.refine'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Int.natAbs (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) < n\n[PROOFSTEP]\non_goal 2 =>\n  replace hin := Nat.isCoprime_iff_coprime.mpr hin\n  split_ifs\n  \u00b7 exact hin\n  \u00b7 convert hin.add_mul_left_left (-1) using 1\n    rw [mul_neg_one, sub_eq_add_neg]\n[GOAL]\ncase intro.intro.intro.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 IsCoprime (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) \u2191n\n[PROOFSTEP]\nreplace hin := Nat.isCoprime_iff_coprime.mpr hin\n[GOAL]\ncase intro.intro.intro.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : IsCoprime \u2191i \u2191n\n\u22a2 IsCoprime (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) \u2191n\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : IsCoprime \u2191i \u2191n\nh\u271d : i * 2 \u2264 n\n\u22a2 IsCoprime \u2191i \u2191n\n[PROOFSTEP]\nexact hin\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : IsCoprime \u2191i \u2191n\nh\u271d : \u00aci * 2 \u2264 n\n\u22a2 IsCoprime (\u2191i - \u2191n) \u2191n\n[PROOFSTEP]\nconvert hin.add_mul_left_left (-1) using 1\n[GOAL]\ncase h.e'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : IsCoprime \u2191i \u2191n\nh\u271d : \u00aci * 2 \u2264 n\n\u22a2 \u2191i - \u2191n = \u2191i + \u2191n * -1\n[PROOFSTEP]\nrw [mul_neg_one, sub_eq_add_neg]\n[GOAL]\ncase intro.intro.intro.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) / \u2191n * (2 * Real.pi)\ncase intro.intro.intro.refine'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Int.natAbs (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) < n\n[PROOFSTEP]\non_goal 2 =>\n  split_ifs with h\u2082\n  \u00b7 exact_mod_cast h\n  suffices (i - n : \u2124).natAbs = n - i by\n    rw [this]\n    apply tsub_lt_self hn.bot_lt\n    contrapose! h\u2082\n    rw [Nat.eq_zero_of_le_zero h\u2082, zero_mul]\n    exact zero_le _\n  rw [\u2190 Int.natAbs_neg, neg_sub, Int.natAbs_eq_iff]\n  exact Or.inl (Int.ofNat_sub h.le).symm\n[GOAL]\ncase intro.intro.intro.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) / \u2191n * (2 * Real.pi)\ncase intro.intro.intro.refine'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Int.natAbs (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) < n\n[PROOFSTEP]\non_goal 2 =>\n  split_ifs with h\u2082\n  \u00b7 exact_mod_cast h\n  suffices (i - n : \u2124).natAbs = n - i by\n    rw [this]\n    apply tsub_lt_self hn.bot_lt\n    contrapose! h\u2082\n    rw [Nat.eq_zero_of_le_zero h\u2082, zero_mul]\n    exact zero_le _\n  rw [\u2190 Int.natAbs_neg, neg_sub, Int.natAbs_eq_iff]\n  exact Or.inl (Int.ofNat_sub h.le).symm\n[GOAL]\ncase intro.intro.intro.refine'_3\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Int.natAbs (if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) < n\n[PROOFSTEP]\nsplit_ifs with h\u2082\n[GOAL]\ncase pos\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 Int.natAbs \u2191i < n\n[PROOFSTEP]\nexact_mod_cast h\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 Int.natAbs (\u2191i - \u2191n) < n\n[PROOFSTEP]\nsuffices (i - n : \u2124).natAbs = n - i by\n  rw [this]\n  apply tsub_lt_self hn.bot_lt\n  contrapose! h\u2082\n  rw [Nat.eq_zero_of_le_zero h\u2082, zero_mul]\n  exact zero_le _\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\nthis : Int.natAbs (\u2191i - \u2191n) = n - i\n\u22a2 Int.natAbs (\u2191i - \u2191n) < n\n[PROOFSTEP]\nrw [this]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\nthis : Int.natAbs (\u2191i - \u2191n) = n - i\n\u22a2 n - i < n\n[PROOFSTEP]\napply tsub_lt_self hn.bot_lt\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\nthis : Int.natAbs (\u2191i - \u2191n) = n - i\n\u22a2 0 < i\n[PROOFSTEP]\ncontrapose! h\u2082\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nthis : Int.natAbs (\u2191i - \u2191n) = n - i\nh\u2082 : i \u2264 0\n\u22a2 i * 2 \u2264 n\n[PROOFSTEP]\nrw [Nat.eq_zero_of_le_zero h\u2082, zero_mul]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nthis : Int.natAbs (\u2191i - \u2191n) = n - i\nh\u2082 : i \u2264 0\n\u22a2 0 \u2264 n\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 Int.natAbs (\u2191i - \u2191n) = n - i\n[PROOFSTEP]\nrw [\u2190 Int.natAbs_neg, neg_sub, Int.natAbs_eq_iff]\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191n - \u2191i = \u2191(n - i) \u2228 \u2191n - \u2191i = -\u2191(n - i)\n[PROOFSTEP]\nexact Or.inl (Int.ofNat_sub h.le).symm\n[GOAL]\ncase intro.intro.intro.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(if i * 2 \u2264 n then \u2191i else \u2191i - \u2191n) / \u2191n * (2 * Real.pi)\n[PROOFSTEP]\nsplit_ifs with h\u2082\n[GOAL]\ncase pos\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191\u2191i / \u2191n * (2 * Real.pi)\n[PROOFSTEP]\nconvert Complex.arg_cos_add_sin_mul_I _\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) = \u2191(\u2191\u2191i / \u2191n * (2 * Real.pi))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) = \u2191i / \u2191n * (2 * \u2191Real.pi)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_6.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) = \u2191(\u2191\u2191i / \u2191n * (2 * Real.pi))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_6.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) = \u2191i / \u2191n * (2 * \u2191Real.pi)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos.convert_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191\u2191i / \u2191n * (2 * Real.pi) \u2208 Set.Ioc (-Real.pi) Real.pi\n[PROOFSTEP]\nfield_simp [hn]\n[GOAL]\ncase pos.convert_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 -Real.pi < \u2191i * (2 * Real.pi) / \u2191n \u2227 \u2191i * (2 * Real.pi) / \u2191n \u2264 Real.pi\n[PROOFSTEP]\nrefine' \u27e8(neg_lt_neg Real.pi_pos).trans_le _, _\u27e9\n[GOAL]\ncase pos.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 -0 \u2264 \u2191i * (2 * Real.pi) / \u2191n\n[PROOFSTEP]\nrw [neg_zero]\n[GOAL]\ncase pos.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 0 \u2264 \u2191i * (2 * Real.pi) / \u2191n\n[PROOFSTEP]\nexact\n  mul_nonneg (mul_nonneg i.cast_nonneg <| by simp [Real.pi_pos.le]) (by rw [inv_nonneg]; simp only [Nat.cast_nonneg])\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 0 \u2264 2 * Real.pi\n[PROOFSTEP]\nsimp [Real.pi_pos.le]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 0 \u2264 (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrw [inv_nonneg]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nsimp only [Nat.cast_nonneg]\n[GOAL]\ncase pos.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 \u2191i * (2 * Real.pi) / \u2191n \u2264 Real.pi\n[PROOFSTEP]\nrw [\u2190 mul_rotate', mul_div_assoc]\n[GOAL]\ncase pos.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n\n\u22a2 Real.pi * (\u2191i * 2 / \u2191n) \u2264 Real.pi\n[PROOFSTEP]\nrw [\u2190 mul_one n] at h\u2082 \n[GOAL]\ncase pos.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n * 1\n\u22a2 Real.pi * (\u2191i * 2 / \u2191n) \u2264 Real.pi\n[PROOFSTEP]\nexact mul_le_of_le_one_right Real.pi_pos.le ((div_le_iff' <| by exact_mod_cast pos_of_gt h).mpr <| by exact_mod_cast h\u2082)\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n * 1\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact_mod_cast pos_of_gt h\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : i * 2 \u2264 n * 1\n\u22a2 \u2191i * 2 \u2264 \u2191n * 1\n[PROOFSTEP]\nexact_mod_cast h\u2082\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 Complex.arg (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi)) + Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi)) * Complex.I) =\n    \u2191(\u2191i - \u2191n) / \u2191n * (2 * Real.pi)\n[PROOFSTEP]\nrw [\u2190 Complex.cos_sub_two_pi, \u2190 Complex.sin_sub_two_pi]\n[GOAL]\ncase neg\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 Complex.arg\n      (Complex.cos (\u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi) +\n        Complex.sin (\u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi) * Complex.I) =\n    \u2191(\u2191i - \u2191n) / \u2191n * (2 * Real.pi)\n[PROOFSTEP]\nconvert Complex.arg_cos_add_sin_mul_I _\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi = \u2191(\u2191(\u2191i - \u2191n) / \u2191n * (2 * Real.pi))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi = (\u2191i - \u2191n) / \u2191n * (2 * \u2191Real.pi)\n[PROOFSTEP]\nrw [\u2190 sub_one_mul, sub_div, div_self]\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_6.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi = \u2191(\u2191(\u2191i - \u2191n) / \u2191n * (2 * Real.pi))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_6.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191i / \u2191n * (2 * \u2191Real.pi) - 2 * \u2191Real.pi = (\u2191i - \u2191n) / \u2191n * (2 * \u2191Real.pi)\n[PROOFSTEP]\nrw [\u2190 sub_one_mul, sub_div, div_self]\n[GOAL]\ncase h.e'_2.h.e'_1.h.e'_6.h.e'_5.h.e'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact_mod_cast hn\n[GOAL]\ncase neg.convert_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191(\u2191i - \u2191n) / \u2191n * (2 * Real.pi) \u2208 Set.Ioc (-Real.pi) Real.pi\n[PROOFSTEP]\nfield_simp [hn]\n[GOAL]\ncase neg.convert_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 -Real.pi < (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n \u2227 (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n \u2264 Real.pi\n[PROOFSTEP]\nrefine' \u27e8_, le_trans _ Real.pi_pos.le\u27e9\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 -Real.pi < (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n\ncase neg.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n \u2264 0\n[PROOFSTEP]\non_goal 2 =>\n  rw [mul_div_assoc]\n  exact\n    mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr <| by exact_mod_cast h.le)\n      (div_nonneg (by simp [Real.pi_pos.le]) <| by simp)\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 -Real.pi < (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n\ncase neg.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n \u2264 0\n[PROOFSTEP]\non_goal 2 =>\n  rw [mul_div_assoc]\n  exact\n    mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr <| by exact_mod_cast h.le)\n      (div_nonneg (by simp [Real.pi_pos.le]) <| by simp)\n[GOAL]\ncase neg.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n \u2264 0\n[PROOFSTEP]\nrw [mul_div_assoc]\n[GOAL]\ncase neg.convert_2.refine'_2\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 (\u2191i - \u2191n) * (2 * Real.pi / \u2191n) \u2264 0\n[PROOFSTEP]\nexact\n  mul_nonpos_of_nonpos_of_nonneg (sub_nonpos.mpr <| by exact_mod_cast h.le)\n    (div_nonneg (by simp [Real.pi_pos.le]) <| by simp)\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191i \u2264 \u2191n\n[PROOFSTEP]\nexact_mod_cast h.le\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 0 \u2264 2 * Real.pi\n[PROOFSTEP]\nsimp [Real.pi_pos.le]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 -Real.pi < (\u2191i - \u2191n) * (2 * Real.pi) / \u2191n\n[PROOFSTEP]\nrw [\u2190 mul_rotate', mul_div_assoc, neg_lt, \u2190 mul_neg, mul_lt_iff_lt_one_right Real.pi_pos, \u2190 neg_div, \u2190 neg_mul, neg_sub,\n  div_lt_iff, one_mul, sub_mul, sub_lt_comm, \u2190 mul_sub_one]\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191n * (2 - 1) < \u2191i * 2\ncase neg.convert_2.refine'_1 n : \u2115 hn : n \u2260 0 i : \u2115 h : i < n hin : Nat.coprime i n h\u2082 : \u00aci * 2 \u2264 n \u22a2 0 < \u2191n\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 \u2191n < \u2191i * 2\ncase neg.convert_2.refine'_1 n : \u2115 hn : n \u2260 0 i : \u2115 h : i < n hin : Nat.coprime i n h\u2082 : \u00aci * 2 \u2264 n \u22a2 0 < \u2191n\n[PROOFSTEP]\nexact_mod_cast not_le.mp h\u2082\n[GOAL]\ncase neg.convert_2.refine'_1\nn : \u2115\nhn : n \u2260 0\ni : \u2115\nh : i < n\nhin : Nat.coprime i n\nh\u2082 : \u00aci * 2 \u2264 n\n\u22a2 0 < \u2191n\n[PROOFSTEP]\nexact Nat.cast_pos.mpr hn.bot_lt\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.RootsOfUnity.Complex", "llama_tokens": 14978, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8438950947024555, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5190696240810612}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na b : \u03b1\nh : Monotone f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Icc a b\n\u22a2 f x\u271d\u00b9 \u2208 Icc (f a) (f b)\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na b : \u03b1\nh : StrictMono f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Ioo a b\n\u22a2 f x\u271d\u00b9 \u2208 Ioo (f a) (f b)\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nh : Monotone f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Ici a\n\u22a2 f x\u271d\u00b9 \u2208 Ici (f a)\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nh : Monotone f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Iic a\n\u22a2 f x\u271d\u00b9 \u2208 Iic (f a)\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nh : StrictMono f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Ioi a\n\u22a2 f x\u271d\u00b9 \u2208 Ioi (f a)\n[PROOFSTEP]\naesop\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nh : StrictMono f\nx\u271d\u00b9 : \u03b1\nx\u271d : x\u271d\u00b9 \u2208 Iio a\n\u22a2 f x\u271d\u00b9 \u2208 Iio (f a)\n[PROOFSTEP]\naesop\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Intervals.Image", "llama_tokens": 657, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5190155064012479}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave h3 : (0 : \u211d) < 3 := by norm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\n\u22a2 0 < 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave h23 : 0 < (2 / 3 : \u211d) := by\n  norm_num1\n    -- In the trivial case `f = 0`, we take `g = 0`\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\n\u22a2 0 < 2 / 3\n[PROOFSTEP]\nnorm_num1\n  -- In the trivial case `f = 0`, we take `g = 0`\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nrcases eq_or_ne f 0 with (rfl | hf)\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u20160\u2016 / 3 \u2227 dist (compContinuous g e) 0 \u2264 2 / 3 * \u20160\u2016\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\n\u22a2 \u20160\u2016 \u2264 \u20160\u2016 / 3 \u2227 dist (compContinuous 0 e) 0 \u2264 2 / 3 * \u20160\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : f \u2260 0\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nreplace hf : 0 < \u2016f\u2016 := norm_pos_iff.2 hf\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave hf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3 := (div_lt_div_right h3).2 (Left.neg_lt_self hf)\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave hc\u2081 : IsClosed (e '' (f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) := he.isClosedMap _ (isClosed_Iic.preimage f.continuous)\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave hc\u2082 : IsClosed (e '' (f \u207b\u00b9' Ici (\u2016f\u2016 / 3))) := he.isClosedMap _ (isClosed_Ici.preimage f.continuous)\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave hd : Disjoint (e '' (f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (e '' (f \u207b\u00b9' Ici (\u2016f\u2016 / 3))) :=\n  by\n  refine' disjoint_image_of_injective he.inj (Disjoint.preimage _ _)\n  rwa [Iic_disjoint_Ici, not_le]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\n\u22a2 Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\n[PROOFSTEP]\nrefine' disjoint_image_of_injective he.inj (Disjoint.preimage _ _)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\n\u22a2 Disjoint (Iic (-\u2016f\u2016 / 3)) (Ici (\u2016f\u2016 / 3))\n[PROOFSTEP]\nrwa [Iic_disjoint_Ici, not_le]\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nrcases exists_bounded_mem_Icc_of_closed_of_le hc\u2081 hc\u2082 hd hf3.le with \u27e8g, hg\u2081, hg\u2082, hgf\u27e9\n[GOAL]\ncase inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\n\u22a2 \u2203 g, \u2016g\u2016 \u2264 \u2016f\u2016 / 3 \u2227 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nrefine' \u27e8g, _, _\u27e9\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\n\u22a2 \u2016g\u2016 \u2264 \u2016f\u2016 / 3\n[PROOFSTEP]\nrefine' (norm_le <| div_nonneg hf.le h3.le).mpr fun y => _\n[GOAL]\ncase inr.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\ny : Y\n\u22a2 \u2016\u2191g y\u2016 \u2264 \u2016f\u2016 / 3\n[PROOFSTEP]\nsimpa [abs_le, neg_div] using hgf y\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\n\u22a2 dist (compContinuous g e) f \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nrefine' (dist_le <| mul_nonneg h23.le hf.le).mpr fun x => _\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nhave hfx : -\u2016f\u2016 \u2264 f x \u2227 f x \u2264 \u2016f\u2016 := by simpa only [Real.norm_eq_abs, abs_le] using f.norm_coe_le_norm x\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\n\u22a2 -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\n[PROOFSTEP]\nsimpa only [Real.norm_eq_abs, abs_le] using f.norm_coe_le_norm x\n[GOAL]\ncase inr.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\ncases' le_total (f x) (-\u2016f\u2016 / 3) with hle\u2081 hle\u2081\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : \u2191f x \u2264 -\u2016f\u2016 / 3\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\ncalc\n  |g (e x) - f x| = -\u2016f\u2016 / 3 - f x := by\n    rw [hg\u2081 (mem_image_of_mem _ hle\u2081), Function.const_apply, abs_of_nonneg (sub_nonneg.2 hle\u2081)]\n  _ \u2264 2 / 3 * \u2016f\u2016 := by linarith\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : \u2191f x \u2264 -\u2016f\u2016 / 3\n\u22a2 |\u2191g (\u2191e x) - \u2191f x| = -\u2016f\u2016 / 3 - \u2191f x\n[PROOFSTEP]\nrw [hg\u2081 (mem_image_of_mem _ hle\u2081), Function.const_apply, abs_of_nonneg (sub_nonneg.2 hle\u2081)]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : \u2191f x \u2264 -\u2016f\u2016 / 3\n\u22a2 -\u2016f\u2016 / 3 - \u2191f x \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : -\u2016f\u2016 / 3 \u2264 \u2191f x\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\ncases' le_total (f x) (\u2016f\u2016 / 3) with hle\u2082 hle\u2082\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.inr.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : -\u2016f\u2016 / 3 \u2264 \u2191f x\nhle\u2082 : \u2191f x \u2264 \u2016f\u2016 / 3\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nsimp only [neg_div] at *\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.inr.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\ng : Y \u2192\u1d47 \u211d\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2082 : \u2191f x \u2264 \u2016f\u2016 / 3\nhf3 : -(\u2016f\u2016 / 3) < \u2016f\u2016 / 3\nhc\u2081 : IsClosed ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3))))\nhc\u2082 : IsClosed ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3)))) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-(\u2016f\u2016 / 3))) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3))))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-(\u2016f\u2016 / 3)) (\u2016f\u2016 / 3)\nhle\u2081 : -(\u2016f\u2016 / 3) \u2264 \u2191f x\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\ncalc\n  dist (g (e x)) (f x) \u2264 |g (e x)| + |f x| := dist_le_norm_add_norm _ _\n  _ \u2264 \u2016f\u2016 / 3 + \u2016f\u2016 / 3 := (add_le_add (abs_le.2 <| hgf _) (abs_le.2 \u27e8hle\u2081, hle\u2082\u27e9))\n  _ = 2 / 3 * \u2016f\u2016 := by linarith\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\ng : Y \u2192\u1d47 \u211d\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2082 : \u2191f x \u2264 \u2016f\u2016 / 3\nhf3 : -(\u2016f\u2016 / 3) < \u2016f\u2016 / 3\nhc\u2081 : IsClosed ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3))))\nhc\u2082 : IsClosed ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3)))) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-(\u2016f\u2016 / 3))) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Iic (-(\u2016f\u2016 / 3))))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) ((fun a => \u2191e a) '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-(\u2016f\u2016 / 3)) (\u2016f\u2016 / 3)\nhle\u2081 : -(\u2016f\u2016 / 3) \u2264 \u2191f x\n\u22a2 \u2016f\u2016 / 3 + \u2016f\u2016 / 3 = 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase inr.intro.intro.intro.refine'_2.inr.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : -\u2016f\u2016 / 3 \u2264 \u2191f x\nhle\u2082 : \u2016f\u2016 / 3 \u2264 \u2191f x\n\u22a2 dist (\u2191(compContinuous g e) x) (\u2191f x) \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\ncalc\n  |g (e x) - f x| = f x - \u2016f\u2016 / 3 := by\n    rw [hg\u2082 (mem_image_of_mem _ hle\u2082), abs_sub_comm, Function.const_apply, abs_of_nonneg (sub_nonneg.2 hle\u2082)]\n  _ \u2264 2 / 3 * \u2016f\u2016 := by linarith\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : -\u2016f\u2016 / 3 \u2264 \u2191f x\nhle\u2082 : \u2016f\u2016 / 3 \u2264 \u2191f x\n\u22a2 |\u2191g (\u2191e x) - \u2191f x| = \u2191f x - \u2016f\u2016 / 3\n[PROOFSTEP]\nrw [hg\u2082 (mem_image_of_mem _ hle\u2082), abs_sub_comm, Function.const_apply, abs_of_nonneg (sub_nonneg.2 hle\u2082)]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < \u2016f\u2016\nhf3 : -\u2016f\u2016 / 3 < \u2016f\u2016 / 3\nhc\u2081 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhc\u2082 : IsClosed (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhd : Disjoint (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3))) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\ng : Y \u2192\u1d47 \u211d\nhg\u2081 : EqOn (\u2191g) (Function.const Y (-\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Iic (-\u2016f\u2016 / 3)))\nhg\u2082 : EqOn (\u2191g) (Function.const Y (\u2016f\u2016 / 3)) (\u2191e '' (\u2191f \u207b\u00b9' Ici (\u2016f\u2016 / 3)))\nhgf : \u2200 (x : Y), \u2191g x \u2208 Icc (-\u2016f\u2016 / 3) (\u2016f\u2016 / 3)\nx : X\nhfx : -\u2016f\u2016 \u2264 \u2191f x \u2227 \u2191f x \u2264 \u2016f\u2016\nhle\u2081 : -\u2016f\u2016 / 3 \u2264 \u2191f x\nhle\u2082 : \u2016f\u2016 / 3 \u2264 \u2191f x\n\u22a2 \u2191f x - \u2016f\u2016 / 3 \u2264 2 / 3 * \u2016f\u2016\n[PROOFSTEP]\nlinarith\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nchoose F hF_norm hF_dist using fun f : X \u2192\u1d47 \u211d => tietze_extension_step f e he\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nset g : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - g.compContinuous e))^[n] 0\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave g0 : g 0 = 0 := rfl\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave g_succ : \u2200 n, g (n + 1) = g n + F (f - (g n).compContinuous e) := fun n => Function.iterate_succ_apply' _ _ _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave hgf : \u2200 n, dist ((g n).compContinuous e) f \u2264 (2 / 3) ^ n * \u2016f\u2016 :=\n  by\n  intro n\n  induction' n with n ihn\n  \u00b7 simp [g0]\n  \u00b7 rw [g_succ n, add_compContinuous, \u2190 dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]\n    refine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))\n    rwa [\u2190 dist_eq_norm']\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\n\u22a2 \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n[PROOFSTEP]\nintro n\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nn : \u2115\n\u22a2 dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n[PROOFSTEP]\ninduction' n with n ihn\n[GOAL]\ncase zero\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\n\u22a2 dist (compContinuous (g Nat.zero) e) f \u2264 (2 / 3) ^ Nat.zero * \u2016f\u2016\n[PROOFSTEP]\nsimp [g0]\n[GOAL]\ncase succ\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nn : \u2115\nihn : dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 dist (compContinuous (g (Nat.succ n)) e) f \u2264 (2 / 3) ^ Nat.succ n * \u2016f\u2016\n[PROOFSTEP]\nrw [g_succ n, add_compContinuous, \u2190 dist_sub_right, add_sub_cancel', pow_succ, mul_assoc]\n[GOAL]\ncase succ\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nn : \u2115\nihn : dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 dist (compContinuous (F (f - compContinuous (g n) e)) e) (f - compContinuous (g n) e) \u2264 2 / 3 * ((2 / 3) ^ n * \u2016f\u2016)\n[PROOFSTEP]\nrefine' (hF_dist _).trans (mul_le_mul_of_nonneg_left _ (by norm_num1))\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nn : \u2115\nihn : dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 0 \u2264 2 / 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase succ\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nn : \u2115\nihn : dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 \u2016f - compContinuous (g n) e\u2016 \u2264 (2 / 3) ^ n * \u2016f\u2016\n[PROOFSTEP]\nrwa [\u2190 dist_eq_norm']\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave hg_dist : \u2200 n, dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n :=\n  by\n  intro n\n  calc\n    dist (g n) (g (n + 1)) = \u2016F (f - (g n).compContinuous e)\u2016 := by rw [g_succ, dist_eq_norm', add_sub_cancel']\n    _ \u2264 \u2016f - (g n).compContinuous e\u2016 / 3 := (hF_norm _)\n    _ = 1 / 3 * dist ((g n).compContinuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]\n    _ \u2264 1 / 3 * ((2 / 3) ^ n * \u2016f\u2016) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))\n    _ = 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n := by ac_rfl\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\n\u22a2 \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\n[PROOFSTEP]\nintro n\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nn : \u2115\n\u22a2 dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\n[PROOFSTEP]\ncalc\n  dist (g n) (g (n + 1)) = \u2016F (f - (g n).compContinuous e)\u2016 := by rw [g_succ, dist_eq_norm', add_sub_cancel']\n  _ \u2264 \u2016f - (g n).compContinuous e\u2016 / 3 := (hF_norm _)\n  _ = 1 / 3 * dist ((g n).compContinuous e) f := by rw [dist_eq_norm', one_div, div_eq_inv_mul]\n  _ \u2264 1 / 3 * ((2 / 3) ^ n * \u2016f\u2016) := (mul_le_mul_of_nonneg_left (hgf n) (by norm_num1))\n  _ = 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n := by ac_rfl\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nn : \u2115\n\u22a2 dist (g n) (g (n + 1)) = \u2016F (f - compContinuous (g n) e)\u2016\n[PROOFSTEP]\nrw [g_succ, dist_eq_norm', add_sub_cancel']\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nn : \u2115\n\u22a2 \u2016f - compContinuous (g n) e\u2016 / 3 = 1 / 3 * dist (compContinuous (g n) e) f\n[PROOFSTEP]\nrw [dist_eq_norm', one_div, div_eq_inv_mul]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nn : \u2115\n\u22a2 0 \u2264 1 / 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nn : \u2115\n\u22a2 1 / 3 * ((2 / 3) ^ n * \u2016f\u2016) = 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\n[PROOFSTEP]\nac_rfl\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave hg_cau : CauchySeq g := cauchySeq_of_le_geometric _ _ (by norm_num1) hg_dist\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\n\u22a2 2 / 3 < 1\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave : Tendsto (fun n => (g n).compContinuous e) atTop (\ud835\udcdd <| (limUnder atTop g).compContinuous e) :=\n  ((continuous_compContinuous e).tendsto _).comp hg_cau.tendsto_limUnder\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nhave hge : (limUnder atTop g).compContinuous e = f :=\n  by\n  refine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)\n  refine' squeeze_zero (fun _ => dist_nonneg) hgf _\n  rw [\u2190 zero_mul \u2016f\u2016]\n  refine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 compContinuous (limUnder atTop g) e = f\n[PROOFSTEP]\nrefine' tendsto_nhds_unique this (tendsto_iff_dist_tendsto_zero.2 _)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 Tendsto (fun b => dist (compContinuous (g b) e) f) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero (fun _ => dist_nonneg) hgf _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 Tendsto (fun t => (2 / 3) ^ t * \u2016f\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 zero_mul \u2016f\u2016]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 Tendsto (fun t => (2 / 3) ^ t * \u2016f\u2016) atTop (\ud835\udcdd (0 * \u2016f\u2016))\n[PROOFSTEP]\nrefine' (tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 0 \u2264 2 / 3\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\n\u22a2 2 / 3 < 1\n[PROOFSTEP]\nnorm_num1\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 compContinuous g e = f\n[PROOFSTEP]\nrefine' \u27e8limUnder atTop g, le_antisymm _ _, hge\u27e9\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 \u2016limUnder atTop g\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrw [\u2190 dist_zero_left, \u2190 g0]\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 dist (g 0) (limUnder atTop g) \u2264 \u2016f\u2016\n[PROOFSTEP]\nrefine' (dist_le_of_le_geometric_of_tendsto\u2080 _ _ (by norm_num1) hg_dist hg_cau.tendsto_limUnder).trans_eq _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 2 / 3 < 1\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 1 / 3 * \u2016f\u2016 / (1 - 2 / 3) = \u2016f\u2016\n[PROOFSTEP]\nfield_simp [show (3 - 2 : \u211d) = 1 by norm_num1]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 3 - 2 = 1\n[PROOFSTEP]\nnorm_num1\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 \u2016f\u2016 \u2264 \u2016limUnder atTop g\u2016\n[PROOFSTEP]\nrw [\u2190 hge]\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : C(X, Y)\nhe : ClosedEmbedding \u2191e\nF : (X \u2192\u1d47 \u211d) \u2192 Y \u2192\u1d47 \u211d\nhF_norm : \u2200 (f : X \u2192\u1d47 \u211d), \u2016F f\u2016 \u2264 \u2016f\u2016 / 3\nhF_dist : \u2200 (f : X \u2192\u1d47 \u211d), dist (compContinuous (F f) e) f \u2264 2 / 3 * \u2016f\u2016\ng : \u2115 \u2192 Y \u2192\u1d47 \u211d := fun n => (fun g => g + F (f - compContinuous g e))^[n] 0\ng0 : g 0 = 0\ng_succ : \u2200 (n : \u2115), g (n + 1) = g n + F (f - compContinuous (g n) e)\nhgf : \u2200 (n : \u2115), dist (compContinuous (g n) e) f \u2264 (2 / 3) ^ n * \u2016f\u2016\nhg_dist : \u2200 (n : \u2115), dist (g n) (g (n + 1)) \u2264 1 / 3 * \u2016f\u2016 * (2 / 3) ^ n\nhg_cau : CauchySeq g\nthis : Tendsto (fun n => compContinuous (g n) e) atTop (\ud835\udcdd (compContinuous (limUnder atTop g) e))\nhge : compContinuous (limUnder atTop g) e = f\n\u22a2 \u2016compContinuous (limUnder atTop g) e\u2016 \u2264 \u2016limUnder atTop g\u2016\n[PROOFSTEP]\nexact norm_compContinuous_le _ _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, \u2016g\u2016 = \u2016f\u2016 \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_extension_norm_eq_of_closedEmbedding' f \u27e8e, he.continuous\u27e9 he with \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\ne : X \u2192 Y\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhg : \u2016g\u2016 = \u2016compContinuous g (ContinuousMap.mk e)\u2016\n\u22a2 \u2203 g_1, \u2016g_1\u2016 = \u2016compContinuous g (ContinuousMap.mk e)\u2016 \u2227 \u2191g_1 \u2218 e = \u2191(compContinuous g (ContinuousMap.mk e))\n[PROOFSTEP]\nexact \u27e8g, hg, rfl\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 Icc a b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_extension_norm_eq_of_closedEmbedding (f - const X ((a + b) / 2)) he with \u27e8g, hgf, hge\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 Icc a b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8const Y ((a + b) / 2) + g, fun y => _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\ny : Y\n\u22a2 \u2191(const Y ((a + b) / 2) + g) y \u2208 Icc a b\n[PROOFSTEP]\nsuffices \u2016f - const X ((a + b) / 2)\u2016 \u2264 (b - a) / 2 by\n  simpa [Real.Icc_eq_closedBall, add_mem_closedBall_iff_norm] using (norm_coe_le_norm g y).trans (hgf.trans_le this)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\ny : Y\nthis : \u2016f - const X ((a + b) / 2)\u2016 \u2264 (b - a) / 2\n\u22a2 \u2191(const Y ((a + b) / 2) + g) y \u2208 Icc a b\n[PROOFSTEP]\nsimpa [Real.Icc_eq_closedBall, add_mem_closedBall_iff_norm] using (norm_coe_le_norm g y).trans (hgf.trans_le this)\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\ny : Y\n\u22a2 \u2016f - const X ((a + b) / 2)\u2016 \u2264 (b - a) / 2\n[PROOFSTEP]\nrefine' (norm_le <| div_nonneg (sub_nonneg.2 hle) zero_le_two).2 fun x => _\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\ny : Y\nx : X\n\u22a2 \u2016\u2191(f - const X ((a + b) / 2)) x\u2016 \u2264 (b - a) / 2\n[PROOFSTEP]\nsimpa only [Real.Icc_eq_closedBall] using hf x\n[GOAL]\ncase intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\n\u22a2 \u2191(const Y ((a + b) / 2) + g) \u2218 e = \u2191f\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.refine'_2.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\nx : X\n\u22a2 (\u2191(const Y ((a + b) / 2) + g) \u2218 e) x = \u2191f x\n[PROOFSTEP]\nhave : g (e x) = f x - (a + b) / 2 := congr_fun hge x\n[GOAL]\ncase intro.intro.refine'_2.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\na b : \u211d\ne : X \u2192 Y\nhf : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhe : ClosedEmbedding e\ng : Y \u2192\u1d47 \u211d\nhgf : \u2016g\u2016 = \u2016f - const X ((a + b) / 2)\u2016\nhge : \u2191g \u2218 e = \u2191(f - const X ((a + b) / 2))\nx : X\nthis : \u2191g (e x) = \u2191f x - (a + b) / 2\n\u22a2 (\u2191(const Y ((a + b) / 2) + g) \u2218 e) x = \u2191f x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\ninhabit X\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a, IsGLB (range f) a\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\n\u22a2 \u2203 a, IsGLB (range \u2191f) a\ncase intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nexact \u27e8_, isGLB_ciInf (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1\u27e9\n[GOAL]\ncase intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 : \u2203 b, IsLUB (range f) b\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\n\u22a2 \u2203 b, IsLUB (range \u2191f) b\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nexact\n  \u27e8_, isLUB_ciSup (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).2\u27e9\n    -- Then `f x \u2208 [a, b]` for all `x`\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hmem : \u2200 x, f x \u2208 Icc a b := fun x =>\n  \u27e8ha.1 \u27e8x, rfl\u27e9, hb.1 \u27e8x, rfl\u27e9\u27e9\n    -- Rule out the trivial case `a = b`\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hle : a \u2264 b := (hmem default).1.trans (hmem default).2\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases hle.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nhb : IsLUB (range \u2191f) a\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a a\nhle : a \u2264 a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave : \u2200 x, f x = a := by simpa using hmem\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nhb : IsLUB (range \u2191f) a\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a a\nhle : a \u2264 a\n\u22a2 \u2200 (x : X), \u2191f x = a\n[PROOFSTEP]\nsimpa using hmem\n[GOAL]\ncase intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nhb : IsLUB (range \u2191f) a\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), \u2191f x = a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nuse const Y a\n[GOAL]\ncase h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nhb : IsLUB (range \u2191f) a\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a a\nhle : a \u2264 a\nthis : \u2200 (x : X), \u2191f x = a\n\u22a2 (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191(const Y a) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191(const Y a) \u2218 e = \u2191f\n[PROOFSTEP]\nsimp [this, Function.funext_iff]\n  -- Put `c = (a + b) / 2`. Then `a < c < b` and `c - a = b - c`.\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nset c := (a + b) / 2\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hac : a < c := left_lt_add_div_two.2 hlt\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hcb : c < b := add_div_two_lt_right.2 hlt\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hsub : c - a = b - c := by\n  field_simp\n  ring\n    /- Due to `exists_extension_forall_mem_Icc_of_closedEmbedding`, there exists an extension `g`\n        such that `g y \u2208 [a, b]` for all `y`. However, if `a` and/or `b` do not belong to the range of\n        `f`, then we need to ensure that these points do not belong to the range of `g`. This is done\n        in two almost identical steps. First we deal with the case `\u2200 x, f x \u2260 a`. -/\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 c - a = b - c\n[PROOFSTEP]\nfield_simp\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\n\u22a2 a + b - 2 * a = b * 2 - (a + b)\n[PROOFSTEP]\nring\n  /- Due to `exists_extension_forall_mem_Icc_of_closedEmbedding`, there exists an extension `g`\n      such that `g y \u2208 [a, b]` for all `y`. However, if `a` and/or `b` do not belong to the range of\n      `f`, then we need to ensure that these points do not belong to the range of `g`. This is done\n      in two almost identical steps. First we deal with the case `\u2200 x, f x \u2260 a`. -/\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nobtain \u27e8g, hg_mem, hgf\u27e9 : \u2203 g : Y \u2192\u1d47 \u211d, (\u2200 y, \u2203 x, g y \u2208 Icc (f x) b) \u2227 g \u2218 e = f :=\n  by\n  rcases exists_extension_forall_mem_Icc_of_closedEmbedding f hmem hle he with\n    \u27e8g, hg_mem, hgf\u27e9\n      -- If `a \u2208 range f`, then we are done.\n  rcases em (\u2203 x, f x = a) with (\u27e8x, rfl\u27e9 | ha')\n  \u00b7\n    exact\n      \u27e8g, fun y => \u27e8x, hg_mem _\u27e9, hgf\u27e9\n        /- Otherwise, `g \u207b\u00b9' {a}` is disjoint with `range e \u222a g \u207b\u00b9' (Ici c)`, hence there exists a\n                function `dg : Y \u2192 \u211d` such that `dg \u2218 e = 0`, `dg y = 0` whenever `c \u2264 g y`, `dg y = c - a`\n                whenever `g y = a`, and `0 \u2264 dg y \u2264 c - a` for all `y`.  -/\n  have hd : Disjoint (range e \u222a g \u207b\u00b9' Ici c) (g \u207b\u00b9' { a }) :=\n    by\n    refine' disjoint_union_left.2 \u27e8_, Disjoint.preimage _ _\u27e9\n    \u00b7 rw [Set.disjoint_left]\n      rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = a)\n      exact ha' \u27e8x, (congr_fun hgf x).symm\u27e9\n    \u00b7 exact Set.disjoint_singleton_right.2 hac.not_le\n  rcases exists_bounded_mem_Icc_of_closed_of_le (he.closed_range.union <| isClosed_Ici.preimage g.continuous)\n      (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with\n    \u27e8dg, dg0, dga, dgmem\u27e9\n  replace hgf : \u2200 x, (g + dg) (e x) = f x\n  \u00b7 intro x\n    simp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]\n  refine' \u27e8g + dg, fun y => _, funext hgf\u27e9\n  \u00b7 have hay : a < (g + dg) y := by\n      rcases(hg_mem y).1.eq_or_lt with (rfl | hlt)\n      \u00b7 refine' (lt_add_iff_pos_right _).2 _\n        calc\n          0 < c - g y := sub_pos.2 hac\n          _ = dg y := (dga rfl).symm\n      \u00b7 exact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)\n    rcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9\n    refine' \u27e8x, hxy.le, _\u27e9\n    cases' le_total c (g y) with hc hc\n    \u00b7 simp [dg0 (Or.inr hc), (hg_mem y).2]\n    \u00b7\n      calc\n        g y + dg y \u2264 c + (c - a) := add_le_add hc (dgmem _).2\n        _ = b := by\n          rw [hsub, add_sub_cancel'_right]\n            /- Now we deal with the case `\u2200 x, f x \u2260 b`. The proof is the same as in the first case, with\n                minor modifications that make it hard to deduplicate code. -/\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_extension_forall_mem_Icc_of_closedEmbedding f hmem hle he with\n  \u27e8g, hg_mem, hgf\u27e9\n    -- If `a \u2208 range f`, then we are done.\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases em (\u2203 x, f x = a) with (\u27e8x, rfl\u27e9 | ha')\n[GOAL]\ncase intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nx : X\nha : IsGLB (range \u2191f) (\u2191f x)\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc (\u2191f x) b\nhle : \u2191f x \u2264 b\nhlt : \u2191f x < b\nc : \u211d := (\u2191f x + b) / 2\nhac : \u2191f x < c\nhcb : c < b\nhsub : c - \u2191f x = b - c\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc (\u2191f x) b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nexact\n  \u27e8g, fun y => \u27e8x, hg_mem _\u27e9, hgf\u27e9\n    /- Otherwise, `g \u207b\u00b9' {a}` is disjoint with `range e \u222a g \u207b\u00b9' (Ici c)`, hence there exists a\n            function `dg : Y \u2192 \u211d` such that `dg \u2218 e = 0`, `dg y = 0` whenever `c \u2264 g y`, `dg y = c - a`\n            whenever `g y = a`, and `0 \u2264 dg y \u2264 c - a` for all `y`.  -/\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hd : Disjoint (range e \u222a g \u207b\u00b9' Ici c) (g \u207b\u00b9' { a }) :=\n  by\n  refine' disjoint_union_left.2 \u27e8_, Disjoint.preimage _ _\u27e9\n  \u00b7 rw [Set.disjoint_left]\n    rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = a)\n    exact ha' \u27e8x, (congr_fun hgf x).symm\u27e9\n  \u00b7 exact Set.disjoint_singleton_right.2 hac.not_le\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\n\u22a2 Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\n[PROOFSTEP]\nrefine' disjoint_union_left.2 \u27e8_, Disjoint.preimage _ _\u27e9\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\n\u22a2 Disjoint (range e) (\u2191g \u207b\u00b9' {a})\n[PROOFSTEP]\nrw [Set.disjoint_left]\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\n\u22a2 \u2200 \u2983a_1 : Y\u2984, a_1 \u2208 range e \u2192 \u00aca_1 \u2208 \u2191g \u207b\u00b9' {a}\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = a)\n[GOAL]\ncase refine'_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nx : X\nha : IsGLB (range \u2191f) (\u2191g (e x))\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc (\u2191g (e x)) b\nhle : \u2191g (e x) \u2264 b\nhlt : \u2191g (e x) < b\nc : \u211d := (\u2191g (e x) + b) / 2\nhac : \u2191g (e x) < c\nhcb : c < b\nhsub : c - \u2191g (e x) = b - c\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc (\u2191g (e x)) b\nha' : \u00ac\u2203 x_1, \u2191f x_1 = \u2191g (e x)\n\u22a2 False\n[PROOFSTEP]\nexact ha' \u27e8x, (congr_fun hgf x).symm\u27e9\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\n\u22a2 Disjoint (Ici c) {a}\n[PROOFSTEP]\nexact Set.disjoint_singleton_right.2 hac.not_le\n[GOAL]\ncase intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_bounded_mem_Icc_of_closed_of_le (he.closed_range.union <| isClosed_Ici.preimage g.continuous)\n    (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hac.le) with\n  \u27e8dg, dg0, dga, dgmem\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nreplace hgf : \u2200 x, (g + dg) (e x) = f x\n[GOAL]\ncase hgf\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\n\u22a2 \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hgf\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nhgf : \u2191g \u2218 e = \u2191f\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nx : X\n\u22a2 \u2191(g + dg) (e x) = \u2191f x\n[PROOFSTEP]\nsimp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]\n[GOAL]\ncase intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8g + dg, fun y => _, funext hgf\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\n\u22a2 \u2203 x, \u2191(g + dg) y \u2208 Icc (\u2191f x) b\n[PROOFSTEP]\nhave hay : a < (g + dg) y := by\n  rcases(hg_mem y).1.eq_or_lt with (rfl | hlt)\n  \u00b7 refine' (lt_add_iff_pos_right _).2 _\n    calc\n      0 < c - g y := sub_pos.2 hac\n      _ = dg y := (dga rfl).symm\n  \u00b7 exact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\n\u22a2 a < \u2191(g + dg) y\n[PROOFSTEP]\nrcases(hg_mem y).1.eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nha : IsGLB (range \u2191f) (\u2191g y)\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc (\u2191g y) b\nhle : \u2191g y \u2264 b\nhlt : \u2191g y < b\nc : \u211d := (\u2191g y + b) / 2\nhac : \u2191g y < c\nhcb : c < b\nhsub : c - \u2191g y = b - c\nhg_mem : \u2200 (y_1 : Y), \u2191g y_1 \u2208 Icc (\u2191g y) b\nha' : \u00ac\u2203 x, \u2191f x = \u2191g y\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {\u2191g y})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - \u2191g y)) (\u2191g \u207b\u00b9' {\u2191g y})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - \u2191g y)\n\u22a2 \u2191g y < \u2191(g + dg) y\n[PROOFSTEP]\nrefine' (lt_add_iff_pos_right _).2 _\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng dg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nha : IsGLB (range \u2191f) (\u2191g y)\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc (\u2191g y) b\nhle : \u2191g y \u2264 b\nhlt : \u2191g y < b\nc : \u211d := (\u2191g y + b) / 2\nhac : \u2191g y < c\nhcb : c < b\nhsub : c - \u2191g y = b - c\nhg_mem : \u2200 (y_1 : Y), \u2191g y_1 \u2208 Icc (\u2191g y) b\nha' : \u00ac\u2203 x, \u2191f x = \u2191g y\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {\u2191g y})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - \u2191g y)) (\u2191g \u207b\u00b9' {\u2191g y})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - \u2191g y)\n\u22a2 0 < \u2191dg.toContinuousMap y\n[PROOFSTEP]\ncalc\n  0 < c - g y := sub_pos.2 hac\n  _ = dg y := (dga rfl).symm\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhlt : a < \u2191g y\n\u22a2 a < \u2191(g + dg) y\n[PROOFSTEP]\nexact hlt.trans_le ((le_add_iff_nonneg_right _).2 <| (dgmem y).1)\n[GOAL]\ncase intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\n\u22a2 \u2203 x, \u2191(g + dg) y \u2208 Icc (\u2191f x) b\n[PROOFSTEP]\nrcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g + dg) y\n\u22a2 \u2203 x, \u2191(g + dg) y \u2208 Icc (\u2191f x) b\n[PROOFSTEP]\nrefine' \u27e8x, hxy.le, _\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g + dg) y\n\u22a2 \u2191(g + dg) y \u2264 b\n[PROOFSTEP]\ncases' le_total c (g y) with hc hc\n[GOAL]\ncase intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g + dg) y\nhc : c \u2264 \u2191g y\n\u22a2 \u2191(g + dg) y \u2264 b\n[PROOFSTEP]\nsimp [dg0 (Or.inr hc), (hg_mem y).2]\n[GOAL]\ncase intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g + dg) y\nhc : \u2191g y \u2264 c\n\u22a2 \u2191(g + dg) y \u2264 b\n[PROOFSTEP]\ncalc\n  g y + dg y \u2264 c + (c - a) := add_le_add hc (dgmem _).2\n  _ = b := by\n    rw [hsub, add_sub_cancel'_right]\n      /- Now we deal with the case `\u2200 x, f x \u2260 b`. The proof is the same as in the first case, with\n          minor modifications that make it hard to deduplicate code. -/\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2191g y \u2208 Icc a b\nha' : \u00ac\u2203 x, \u2191f x = a\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Ici c) (\u2191g \u207b\u00b9' {a})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Ici c)\ndga : EqOn (\u2191dg) (Function.const Y (c - a)) (\u2191g \u207b\u00b9' {a})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (c - a)\nhgf : \u2200 (x : X), \u2191(g + dg) (e x) = \u2191f x\ny : Y\nhay : a < \u2191(g + dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g + dg) y\nhc : \u2191g y \u2264 c\n\u22a2 c + (c - a) = b\n[PROOFSTEP]\nrw [hsub, add_sub_cancel'_right]\n  /- Now we deal with the case `\u2200 x, f x \u2260 b`. The proof is the same as in the first case, with\n      minor modifications that make it hard to deduplicate code. -/\n[GOAL]\ncase intro.intro.inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhg_mem : \u2200 (y : Y), \u2203 x, \u2191g y \u2208 Icc (\u2191f x) b\nhgf : \u2191g \u2218 e = \u2191f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nchoose xl hxl hgb using hg_mem\n[GOAL]\ncase intro.intro.inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases em (\u2203 x, f x = b) with (\u27e8x, rfl\u27e9 | hb')\n[GOAL]\ncase intro.intro.inr.intro.intro.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nx : X\nhb : IsLUB (range \u2191f) (\u2191f x)\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc a (\u2191f x)\nhle : a \u2264 \u2191f x\nhlt : a < \u2191f x\nc : \u211d := (a + \u2191f x) / 2\nhac : a < c\nhcb : c < \u2191f x\nhsub : c - a = \u2191f x - c\nhgb : \u2200 (y : Y), \u2191g y \u2264 \u2191f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nexact \u27e8g, fun y => \u27e8xl y, x, hxl y, hgb y\u27e9, hgf\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hd : Disjoint (range e \u222a g \u207b\u00b9' Iic c) (g \u207b\u00b9' { b }) :=\n  by\n  refine' disjoint_union_left.2 \u27e8_, Disjoint.preimage _ _\u27e9\n  \u00b7 rw [Set.disjoint_left]\n    rintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = b)\n    exact hb' \u27e8x, (congr_fun hgf x).symm\u27e9\n  \u00b7 exact Set.disjoint_singleton_right.2 hcb.not_le\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\n\u22a2 Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\n[PROOFSTEP]\nrefine' disjoint_union_left.2 \u27e8_, Disjoint.preimage _ _\u27e9\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\n\u22a2 Disjoint (range e) (\u2191g \u207b\u00b9' {b})\n[PROOFSTEP]\nrw [Set.disjoint_left]\n[GOAL]\ncase refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\n\u22a2 \u2200 \u2983a : Y\u2984, a \u2208 range e \u2192 \u00aca \u2208 \u2191g \u207b\u00b9' {b}\n[PROOFSTEP]\nrintro _ \u27e8x, rfl\u27e9 (rfl : g (e x) = b)\n[GOAL]\ncase refine'_1.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nx : X\nhb : IsLUB (range \u2191f) (\u2191g (e x))\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc a (\u2191g (e x))\nhle : a \u2264 \u2191g (e x)\nhlt : a < \u2191g (e x)\nc : \u211d := (a + \u2191g (e x)) / 2\nhac : a < c\nhcb : c < \u2191g (e x)\nhsub : c - a = \u2191g (e x) - c\nhgb : \u2200 (y : Y), \u2191g y \u2264 \u2191g (e x)\nhb' : \u00ac\u2203 x_1, \u2191f x_1 = \u2191g (e x)\n\u22a2 False\n[PROOFSTEP]\nexact hb' \u27e8x, (congr_fun hgf x).symm\u27e9\n[GOAL]\ncase refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\n\u22a2 Disjoint (Iic c) {b}\n[PROOFSTEP]\nexact Set.disjoint_singleton_right.2 hcb.not_le\n[GOAL]\ncase intro.intro.inr.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_bounded_mem_Icc_of_closed_of_le (he.closed_range.union <| isClosed_Iic.preimage g.continuous)\n    (isClosed_singleton.preimage g.continuous) hd (sub_nonneg.2 hcb.le) with\n  \u27e8dg, dg0, dgb, dgmem\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nreplace hgf : \u2200 x, (g - dg) (e x) = f x\n[GOAL]\ncase hgf\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\n\u22a2 \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hgf\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nhgf : \u2191g \u2218 e = \u2191f\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nx : X\n\u22a2 \u2191(g - dg) (e x) = \u2191f x\n[PROOFSTEP]\nsimp [dg0 (Or.inl <| mem_range_self _), \u2190 hgf]\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8g - dg, fun y => _, funext hgf\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nhave hyb : (g - dg) y < b := by\n  rcases(hgb y).eq_or_lt with (rfl | hlt)\n  \u00b7 refine' (sub_lt_self_iff _).2 _\n    calc\n      0 < g y - c := sub_pos.2 hcb\n      _ = dg y := (dgb rfl).symm\n  \u00b7 exact ((sub_le_self_iff _).2 (dgmem _).1).trans_lt hlt\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\n\u22a2 \u2191(g - dg) y < b\n[PROOFSTEP]\nrcases(hgb y).eq_or_lt with (rfl | hlt)\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhb : IsLUB (range \u2191f) (\u2191g y)\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a (\u2191g y)\nhle : a \u2264 \u2191g y\nhlt : a < \u2191g y\nc : \u211d := (a + \u2191g y) / 2\nhac : a < c\nhcb : c < \u2191g y\nhsub : c - a = \u2191g y - c\nhgb : \u2200 (y_1 : Y), \u2191g y_1 \u2264 \u2191g y\nhb' : \u00ac\u2203 x, \u2191f x = \u2191g y\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {\u2191g y})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (\u2191g y - c)) (\u2191g \u207b\u00b9' {\u2191g y})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (\u2191g y - c)\n\u22a2 \u2191(g - dg) y < \u2191g y\n[PROOFSTEP]\nrefine' (sub_lt_self_iff _).2 _\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhb : IsLUB (range \u2191f) (\u2191g y)\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a (\u2191g y)\nhle : a \u2264 \u2191g y\nhlt : a < \u2191g y\nc : \u211d := (a + \u2191g y) / 2\nhac : a < c\nhcb : c < \u2191g y\nhsub : c - a = \u2191g y - c\nhgb : \u2200 (y_1 : Y), \u2191g y_1 \u2264 \u2191g y\nhb' : \u00ac\u2203 x, \u2191f x = \u2191g y\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {\u2191g y})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (\u2191g y - c)) (\u2191g \u207b\u00b9' {\u2191g y})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (\u2191g y - c)\n\u22a2 0 < \u2191dg y\n[PROOFSTEP]\ncalc\n  0 < g y - c := sub_pos.2 hcb\n  _ = dg y := (dgb rfl).symm\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt\u271d : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhlt : \u2191g y < b\n\u22a2 \u2191(g - dg) y < b\n[PROOFSTEP]\nexact ((sub_le_self_iff _).2 (dgmem _).1).trans_lt hlt\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nrcases hb.exists_between hyb with \u27e8_, \u27e8xu, rfl\u27e9, hyxu, _\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\ncases' lt_or_le c (g y) with hc hc\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nrcases em (a \u2208 range f) with (\u27e8x, rfl\u27e9 | _)\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nx : X\nha : IsGLB (range \u2191f) (\u2191f x)\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc (\u2191f x) b\nhle : \u2191f x \u2264 b\nhlt : \u2191f x < b\nc : \u211d := (\u2191f x + b) / 2\nhac : \u2191f x < c\nhcb : c < b\nhsub : c - \u2191f x = b - c\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhc : c < \u2191g y\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nrefine' \u27e8x, xu, _, hyxu.le\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nx : X\nha : IsGLB (range \u2191f) (\u2191f x)\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc (\u2191f x) b\nhle : \u2191f x \u2264 b\nhlt : \u2191f x < b\nc : \u211d := (\u2191f x + b) / 2\nhac : \u2191f x < c\nhcb : c < b\nhsub : c - \u2191f x = b - c\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhc : c < \u2191g y\n\u22a2 \u2191f x \u2264 \u2191(g - dg) y\n[PROOFSTEP]\ncalc\n  f x = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n  _ \u2264 g y - dg y := sub_le_sub hc.le (dgmem _).2\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\nb : \u211d\nhb : IsLUB (range \u2191f) b\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\ndg : Y \u2192\u1d47 \u211d\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nx : X\nha : IsGLB (range \u2191f) (\u2191f x)\nhmem : \u2200 (x_1 : X), \u2191f x_1 \u2208 Icc (\u2191f x) b\nhle : \u2191f x \u2264 b\nhlt : \u2191f x < b\nc : \u211d := (\u2191f x + b) / 2\nhac : \u2191f x < c\nhcb : c < b\nhsub : c - \u2191f x = b - c\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhc : c < \u2191g y\n\u22a2 \u2191f x = c - (b - c)\n[PROOFSTEP]\nrw [\u2190 hsub, sub_sub_cancel]\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\nh\u271d : \u00aca \u2208 range \u2191f\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nhave hay : a < (g - dg) y := by\n  calc\n    a = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n    _ < g y - (b - c) := (sub_lt_sub_right hc _)\n    _ \u2264 g y - dg y := sub_le_sub_left (dgmem _).2 _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\nh\u271d : \u00aca \u2208 range \u2191f\n\u22a2 a < \u2191(g - dg) y\n[PROOFSTEP]\ncalc\n  a = c - (b - c) := by rw [\u2190 hsub, sub_sub_cancel]\n  _ < g y - (b - c) := (sub_lt_sub_right hc _)\n  _ \u2264 g y - dg y := sub_le_sub_left (dgmem _).2 _\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\nh\u271d : \u00aca \u2208 range \u2191f\n\u22a2 a = c - (b - c)\n[PROOFSTEP]\nrw [\u2190 hsub, sub_sub_cancel]\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\nh\u271d : \u00aca \u2208 range \u2191f\nhay : a < \u2191(g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nrcases ha.exists_between hay with \u27e8_, \u27e8x, rfl\u27e9, _, hxy\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inl.inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : c < \u2191g y\nh\u271d : \u00aca \u2208 range \u2191f\nhay : a < \u2191(g - dg) y\nx : X\nleft\u271d : a \u2264 \u2191f x\nhxy : \u2191f x < \u2191(g - dg) y\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nexact \u27e8x, xu, hxy.le, hyxu.le\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : \u2191g y \u2264 c\n\u22a2 \u2203 x\u2081 x\u2082, \u2191(g - dg) y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\n[PROOFSTEP]\nrefine' \u27e8xl y, xu, _, hyxu.le\u27e9\n[GOAL]\ncase intro.intro.inr.intro.intro.inr.intro.intro.intro.intro.intro.intro.intro.inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ninst\u271d : Nonempty X\nf : X \u2192\u1d47 \u211d\ne : X \u2192 Y\nhe : ClosedEmbedding e\ninhabited_h : Inhabited X\na : \u211d\nha : IsGLB (range \u2191f) a\nb : \u211d\nhb : IsLUB (range \u2191f) b\nhmem : \u2200 (x : X), \u2191f x \u2208 Icc a b\nhle : a \u2264 b\nhlt : a < b\nc : \u211d := (a + b) / 2\nhac : a < c\nhcb : c < b\nhsub : c - a = b - c\ng : Y \u2192\u1d47 \u211d\nxl : Y \u2192 X\nhxl : \u2200 (y : Y), \u2191f (xl y) \u2264 \u2191g y\nhgb : \u2200 (y : Y), \u2191g y \u2264 b\nhb' : \u00ac\u2203 x, \u2191f x = b\nhd : Disjoint (range e \u222a \u2191g \u207b\u00b9' Iic c) (\u2191g \u207b\u00b9' {b})\ndg : Y \u2192\u1d47 \u211d\ndg0 : EqOn (\u2191dg) (Function.const Y 0) (range e \u222a \u2191g \u207b\u00b9' Iic c)\ndgb : EqOn (\u2191dg) (Function.const Y (b - c)) (\u2191g \u207b\u00b9' {b})\ndgmem : \u2200 (x : Y), \u2191dg x \u2208 Icc 0 (b - c)\nhgf : \u2200 (x : X), \u2191(g - dg) (e x) = \u2191f x\ny : Y\nhyb : \u2191(g - dg) y < b\nxu : X\nhyxu : \u2191(g - dg) y < \u2191f xu\nright\u271d : \u2191f xu \u2264 b\nhc : \u2191g y \u2264 c\n\u22a2 \u2191f (xl y) \u2264 \u2191(g - dg) y\n[PROOFSTEP]\nsimp [dg0 (Or.inr hc), hxl]\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\ncases isEmpty_or_nonempty X\n[GOAL]\ncase inl\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh\u271d : IsEmpty X\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases hne with \u27e8c, hc\u27e9\n[GOAL]\ncase inl.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhe : ClosedEmbedding e\nh\u271d : IsEmpty X\nc : \u211d\nhc : c \u2208 t\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8const Y c, fun _ => hc, funext fun x => isEmptyElim x\u27e9\n[GOAL]\ncase inr\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh\u271d : Nonempty X\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases exists_extension_forall_exists_le_ge_of_closedEmbedding f he with \u27e8g, hg, hgf\u27e9\n[GOAL]\ncase inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh\u271d : Nonempty X\ng : Y \u2192\u1d47 \u211d\nhg : \u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\nhgf : \u2191g \u2218 e = \u2191f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8g, fun y => _, hgf\u27e9\n[GOAL]\ncase inr.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh\u271d : Nonempty X\ng : Y \u2192\u1d47 \u211d\nhg : \u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\nhgf : \u2191g \u2218 e = \u2191f\ny : Y\n\u22a2 \u2191g y \u2208 t\n[PROOFSTEP]\nrcases hg y with \u27e8xl, xu, h\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : X \u2192\u1d47 \u211d\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh\u271d : Nonempty X\ng : Y \u2192\u1d47 \u211d\nhg : \u2200 (y : Y), \u2203 x\u2081 x\u2082, \u2191g y \u2208 Icc (\u2191f x\u2081) (\u2191f x\u2082)\nhgf : \u2191g \u2218 e = \u2191f\ny : Y\nxl xu : X\nh : \u2191g y \u2208 Icc (\u2191f xl) (\u2191f xu)\n\u22a2 \u2191g y \u2208 t\n[PROOFSTEP]\nexact hs.out (hf _) (hf _) h\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ns : Set Y\nf : \u2191s \u2192\u1d47 \u211d\nhs : IsClosed s\nt : Set \u211d\ninst\u271d : OrdConnected t\nhf : \u2200 (x : \u2191s), \u2191f x \u2208 t\nhne : Set.Nonempty t\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 restrict g s = f\n[PROOFSTEP]\nrcases exists_extension_forall_mem_of_closedEmbedding f hf hne (closedEmbedding_subtype_val hs) with \u27e8g, hg, hgf\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : NormalSpace Y\ns : Set Y\nf : \u2191s \u2192\u1d47 \u211d\nhs : IsClosed s\nt : Set \u211d\ninst\u271d : OrdConnected t\nhf : \u2200 (x : \u2191s), \u2191f x \u2208 t\nhne : Set.Nonempty t\ng : Y \u2192\u1d47 \u211d\nhg : \u2200 (y : Y), \u2191g y \u2208 t\nhgf : \u2191g \u2218 Subtype.val = \u2191f\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 restrict g s = f\n[PROOFSTEP]\nexact \u27e8g, hg, FunLike.coe_injective hgf\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave h : \u211d \u2243o Ioo (-1 : \u211d) 1 := orderIsoIooNegOneOne \u211d\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nlet F : X \u2192\u1d47 \u211d :=\n  { toFun := (\u2191) \u2218 h \u2218 f\n    continuous_toFun := continuous_subtype_val.comp (h.continuous.comp f.continuous)\n    map_bounded' := bounded_range_iff.1 ((bounded_Ioo (-1 : \u211d) 1).mono <| forall_range_iff.2 fun x => (h (f x)).2) }\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nlet t' : Set \u211d := (\u2191) \u2218 h '' t\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave ht_sub : t' \u2286 Ioo (-1 : \u211d) 1 := image_subset_iff.2 fun x _ => (h x).2\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave : OrdConnected t' := by\n  constructor\n  rintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9 z hz\n  lift z to Ioo (-1 : \u211d) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz\n  change z \u2208 Icc (h x) (h y) at hz \n  rw [\u2190 h.image_Icc] at hz \n  rcases hz with \u27e8z, hz, rfl\u27e9\n  exact \u27e8z, hs.out hx hy hz, rfl\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\n\u22a2 OrdConnected t'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase out'\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\n\u22a2 \u2200 \u2983x : \u211d\u2984, x \u2208 t' \u2192 \u2200 \u2983y : \u211d\u2984, y \u2208 t' \u2192 Icc x y \u2286 t'\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9 _ \u27e8y, hy, rfl\u27e9 z hz\n[GOAL]\ncase out'.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nx : \u211d\nhx : x \u2208 t\ny : \u211d\nhy : y \u2208 t\nz : \u211d\nhz : z \u2208 Icc ((Subtype.val \u2218 \u2191h) x) ((Subtype.val \u2218 \u2191h) y)\n\u22a2 z \u2208 t'\n[PROOFSTEP]\nlift z to Ioo (-1 : \u211d) 1 using Icc_subset_Ioo (h x).2.1 (h y).2.2 hz\n[GOAL]\ncase out'.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nx : \u211d\nhx : x \u2208 t\ny : \u211d\nhy : y \u2208 t\nz : { x // x \u2208 Ioo (-1) 1 }\nhz : \u2191z \u2208 Icc ((Subtype.val \u2218 \u2191h) x) ((Subtype.val \u2218 \u2191h) y)\n\u22a2 \u2191z \u2208 t'\n[PROOFSTEP]\nchange z \u2208 Icc (h x) (h y) at hz \n[GOAL]\ncase out'.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nx : \u211d\nhx : x \u2208 t\ny : \u211d\nhy : y \u2208 t\nz : { x // x \u2208 Ioo (-1) 1 }\nhz : z \u2208 Icc (\u2191h x) (\u2191h y)\n\u22a2 \u2191z \u2208 t'\n[PROOFSTEP]\nrw [\u2190 h.image_Icc] at hz \n[GOAL]\ncase out'.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nx : \u211d\nhx : x \u2208 t\ny : \u211d\nhy : y \u2208 t\nz : { x // x \u2208 Ioo (-1) 1 }\nhz : z \u2208 \u2191h '' Icc x y\n\u22a2 \u2191z \u2208 t'\n[PROOFSTEP]\nrcases hz with \u27e8z, hz, rfl\u27e9\n[GOAL]\ncase out'.intro.intro.intro.intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nx : \u211d\nhx : x \u2208 t\ny : \u211d\nhy : y \u2208 t\nz : \u211d\nhz : z \u2208 Icc x y\n\u22a2 \u2191(\u2191h z) \u2208 t'\n[PROOFSTEP]\nexact \u27e8z, hs.out hx hy hz, rfl\u27e9\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hFt : \u2200 x, F x \u2208 t' := fun x => mem_image_of_mem _ (hf x)\n[GOAL]\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrcases F.exists_extension_forall_mem_of_closedEmbedding hFt (hne.image _) he with \u27e8G, hG, hGF\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nlet g : C(Y, \u211d) :=\n  \u27e8h.symm \u2218 codRestrict G _ fun y => ht_sub (hG y), h.symm.continuous.comp <| G.continuous.subtype_mk _\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nhave hgG : \u2200 {y a}, g y = a \u2194 G y = h a := @fun y a => h.toEquiv.symm_apply_eq.trans Subtype.ext_iff\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\n\u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\nrefine' \u27e8g, fun y => _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\ny : Y\n\u22a2 \u2191g y \u2208 t\n[PROOFSTEP]\nrcases hG y with \u27e8a, ha, hay\u27e9\n[GOAL]\ncase intro.intro.refine'_1.intro.intro\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\ny : Y\na : \u211d\nha : a \u2208 t\nhay : (Subtype.val \u2218 \u2191h) a = \u2191G y\n\u22a2 \u2191g y \u2208 t\n[PROOFSTEP]\nconvert ha\n[GOAL]\ncase h.e'_4\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\ny : Y\na : \u211d\nha : a \u2208 t\nhay : (Subtype.val \u2218 \u2191h) a = \u2191G y\n\u22a2 \u2191g y = a\n[PROOFSTEP]\nexact hgG.2 hay.symm\n[GOAL]\ncase intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\n\u22a2 \u2191g \u2218 e = \u2191f\n[PROOFSTEP]\next x\n[GOAL]\ncase intro.intro.refine'_2.h\nX : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : NormalSpace Y\nf : C(X, \u211d)\nt : Set \u211d\ne : X \u2192 Y\nhs : OrdConnected t\nhf : \u2200 (x : X), \u2191f x \u2208 t\nhne : Set.Nonempty t\nhe : ClosedEmbedding e\nh : \u211d \u2243o \u2191(Ioo (-1) 1)\nF : X \u2192\u1d47 \u211d :=\n  { toContinuousMap := mk (Subtype.val \u2218 \u2191h \u2218 \u2191f),\n    map_bounded' :=\n      (_ :\n        \u2203 C,\n          \u2200 (x y : X),\n            dist (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) x)\n                (ContinuousMap.toFun (mk (Subtype.val \u2218 \u2191h \u2218 \u2191f)) y) \u2264\n              C) }\nt' : Set \u211d := Subtype.val \u2218 \u2191h '' t\nht_sub : t' \u2286 Ioo (-1) 1\nthis : OrdConnected t'\nhFt : \u2200 (x : X), \u2191F x \u2208 t'\nG : Y \u2192\u1d47 \u211d\nhG : \u2200 (y : Y), \u2191G y \u2208 t'\nhGF : \u2191G \u2218 e = \u2191F\ng : C(Y, \u211d) := mk (\u2191(OrderIso.symm h) \u2218 Set.codRestrict (\u2191G) (Ioo (-1) 1) (_ : \u2200 (y : Y), \u2191G y \u2208 Ioo (-1) 1))\nhgG : \u2200 {y : Y} {a : (fun x => \u211d) y}, \u2191g y = a \u2194 \u2191G y = \u2191(\u2191h a)\nx : X\n\u22a2 (\u2191g \u2218 e) x = \u2191f x\n[PROOFSTEP]\nexact hgG.2 (congr_fun hGF _)\n", "meta": {"mathlib_filename": "Mathlib.Topology.TietzeExtension", "llama_tokens": 74789, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5190155064012479}}
{"text": "[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nz : A\n\u22a2 \u2203 a b, IsUnit (gcd a b) \u2227 z * \u2191(algebraMap R A) b = \u2191(algebraMap R A) a\n[PROOFSTEP]\nobtain \u27e8x, \u27e8y, hy\u27e9, rfl\u27e9 := IsLocalization.mk'_surjective M z\n[GOAL]\ncase intro.intro.mk\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\n\u22a2 \u2203 a b, IsUnit (gcd a b) \u2227 mk' A x { val := y, property := hy } * \u2191(algebraMap R A) b = \u2191(algebraMap R A) a\n[PROOFSTEP]\nobtain \u27e8x', y', hx', hy', hu\u27e9 := extract_gcd x y\n[GOAL]\ncase intro.intro.mk.intro.intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n\u22a2 \u2203 a b, IsUnit (gcd a b) \u2227 mk' A x { val := y, property := hy } * \u2191(algebraMap R A) b = \u2191(algebraMap R A) a\n[PROOFSTEP]\nuse x', y', hu\n[GOAL]\ncase right\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n\u22a2 mk' A x { val := y, property := hy } * \u2191(algebraMap R A) y' = \u2191(algebraMap R A) x'\n[PROOFSTEP]\nrw [mul_comm, IsLocalization.mul_mk'_eq_mk'_of_mul]\n[GOAL]\ncase right\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n\u22a2 mk' A (y' * x) { val := y, property := hy } = \u2191(algebraMap R A) x'\n[PROOFSTEP]\nconvert IsLocalization.mk'_mul_cancel_left (M := M) (S := A) _ _ using 2\n[GOAL]\ncase h.e'_2.h.e'_8\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n\u22a2 y' * x = \u2191{ val := y, property := hy } * x'\n[PROOFSTEP]\nrw [Subtype.coe_mk, hy', \u2190 mul_comm y', mul_assoc]\n[GOAL]\ncase h.e'_2.h.e'_8\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n\u22a2 y' * x = y' * (gcd x y * x')\n[PROOFSTEP]\nconv_lhs => rw [hx']\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n| y' * x\n[PROOFSTEP]\nrw [hx']\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n| y' * x\n[PROOFSTEP]\nrw [hx']\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : IsDomain R\ninst\u271d\u00b3 : GCDMonoid R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\nM : Submonoid R\ninst\u271d : IsLocalization M A\nx y : R\nhy : y \u2208 M\nx' y' : R\nhx' : x = gcd x y * x'\nhy' : y = gcd x y * y'\nhu : IsUnit (gcd x' y')\n| y' * x\n[PROOFSTEP]\nrw [hx']\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = X\n[PROOFSTEP]\nobtain \u27e8x, y, hg, he\u27e9 := IsLocalization.surj_of_gcd_domain (nonZeroDivisors R) X\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = X\n[PROOFSTEP]\nhave :=\n  Polynomial.dvd_pow_natDegree_of_eval\u2082_eq_zero (IsFractionRing.injective R <| FractionRing R) hp\u2081 y x _ hp\u2082\n    (by rw [mul_comm, he])\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\n\u22a2 \u2191(algebraMap R (FractionRing R)) y * X = \u2191(algebraMap R (FractionRing R)) x\n[PROOFSTEP]\nrw [mul_comm, he]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis : y \u2223 x ^ Polynomial.natDegree p\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = X\n[PROOFSTEP]\nhave : IsUnit y := by\n  rw [isUnit_iff_dvd_one, \u2190 one_pow]\n  exact\n    (dvd_gcd this <| dvd_refl y).trans\n      (gcd_pow_left_dvd_pow_gcd.trans <| pow_dvd_pow_of_dvd (isUnit_iff_dvd_one.1 hg) _)\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis : y \u2223 x ^ Polynomial.natDegree p\n\u22a2 IsUnit y\n[PROOFSTEP]\nrw [isUnit_iff_dvd_one, \u2190 one_pow]\n[GOAL]\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis : y \u2223 x ^ Polynomial.natDegree p\n\u22a2 y \u2223 1 ^ ?m.56326\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis : y \u2223 x ^ Polynomial.natDegree p\n\u22a2 \u2115\n[PROOFSTEP]\nexact\n  (dvd_gcd this <| dvd_refl y).trans (gcd_pow_left_dvd_pow_gcd.trans <| pow_dvd_pow_of_dvd (isUnit_iff_dvd_one.1 hg) _)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis\u271d : y \u2223 x ^ Polynomial.natDegree p\nthis : IsUnit y\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = X\n[PROOFSTEP]\nuse x * (this.unit\u207b\u00b9 : _)\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis\u271d : y \u2223 x ^ Polynomial.natDegree p\nthis : IsUnit y\n\u22a2 \u2191(algebraMap R (FractionRing R)) (x * \u2191(IsUnit.unit this)\u207b\u00b9) = X\n[PROOFSTEP]\nerw [map_mul, \u2190 Units.coe_map_inv, eq_comm, Units.eq_mul_inv_iff_mul_eq]\n[GOAL]\ncase h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : GCDMonoid R\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra R A\nX : FractionRing R\nx\u271d : IsIntegral R X\np : R[X]\nhp\u2081 : Polynomial.Monic p\nhp\u2082 : Polynomial.eval\u2082 (algebraMap R (FractionRing R)) X p = 0\nx y : R\nhg : IsUnit (gcd x y)\nhe : X * \u2191(algebraMap R (FractionRing R)) y = \u2191(algebraMap R (FractionRing R)) x\nthis\u271d : y \u2223 x ^ Polynomial.natDegree p\nthis : IsUnit y\n\u22a2 X * \u2191(\u2191(Units.map \u2191(algebraMap R (FractionRing R))) (IsUnit.unit this)) = \u2191(algebraMap R (FractionRing R)) x\n[PROOFSTEP]\nexact he\n", "meta": {"mathlib_filename": "Mathlib.Algebra.GCDMonoid.IntegrallyClosed", "llama_tokens": 4580, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390747, "lm_q2_score": 0.6723317123102956, "lm_q1q2_score": 0.5189348605377131}}
{"text": "[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\nf : P \u2192\u1d43[R] P'\n\u22a2 WSameSide (AffineSubspace.map f s) (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrcases h with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WSameSide (AffineSubspace.map f s) (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrefine' \u27e8f p\u2081, mem_map_of_mem f hp\u2081, f p\u2082, mem_map_of_mem f hp\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f y -\u1d65 \u2191f p\u2082)\n[PROOFSTEP]\nsimp_rw [\u2190 linearMap_vsub]\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (\u2191f.linear (x -\u1d65 p\u2081)) (\u2191f.linear (y -\u1d65 p\u2082))\n[PROOFSTEP]\nexact h.map f.linear\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\n\u22a2 WSameSide (map f s) (\u2191f x) (\u2191f y) \u2194 WSameSide s x y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.map _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nh : WSameSide (map f s) (\u2191f x) (\u2191f y)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrcases h with \u27e8fp\u2081, hfp\u2081, fp\u2082, hfp\u2082, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2081 : P'\nhfp\u2081 : fp\u2081 \u2208 map f s\nfp\u2082 : P'\nhfp\u2082 : fp\u2082 \u2208 map f s\nh : SameRay R (\u2191f x -\u1d65 fp\u2081) (\u2191f y -\u1d65 fp\u2082)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrw [mem_map] at hfp\u2081 hfp\u2082 \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2081 : P'\nhfp\u2081 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2081\nfp\u2082 : P'\nhfp\u2082 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2082\nh : SameRay R (\u2191f x -\u1d65 fp\u2081) (\u2191f y -\u1d65 fp\u2082)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrcases hfp\u2081 with \u27e8p\u2081, hp\u2081, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2082 : P'\nhfp\u2082 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2082\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f y -\u1d65 fp\u2082)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrcases hfp\u2082 with \u27e8p\u2082, hp\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f y -\u1d65 \u2191f p\u2082)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f y -\u1d65 \u2191f p\u2082)\n\u22a2 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nsimp_rw [\u2190 linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\n\u22a2 SSameSide (map f s) (\u2191f x) (\u2191f y) \u2194 SSameSide s x y\n[PROOFSTEP]\nsimp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WOppSide s x y\nf : P \u2192\u1d43[R] P'\n\u22a2 WOppSide (AffineSubspace.map f s) (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrcases h with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WOppSide (AffineSubspace.map f s) (\u2191f x) (\u2191f y)\n[PROOFSTEP]\nrefine' \u27e8f p\u2081, mem_map_of_mem f hp\u2081, f p\u2082, mem_map_of_mem f hp\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f p\u2082 -\u1d65 \u2191f y)\n[PROOFSTEP]\nsimp_rw [\u2190 linearMap_vsub]\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (\u2191f.linear (x -\u1d65 p\u2081)) (\u2191f.linear (p\u2082 -\u1d65 y))\n[PROOFSTEP]\nexact h.map f.linear\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\n\u22a2 WOppSide (map f s) (\u2191f x) (\u2191f y) \u2194 WOppSide s x y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h.map _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nh : WOppSide (map f s) (\u2191f x) (\u2191f y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrcases h with \u27e8fp\u2081, hfp\u2081, fp\u2082, hfp\u2082, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2081 : P'\nhfp\u2081 : fp\u2081 \u2208 map f s\nfp\u2082 : P'\nhfp\u2082 : fp\u2082 \u2208 map f s\nh : SameRay R (\u2191f x -\u1d65 fp\u2081) (fp\u2082 -\u1d65 \u2191f y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrw [mem_map] at hfp\u2081 hfp\u2082 \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2081 : P'\nhfp\u2081 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2081\nfp\u2082 : P'\nhfp\u2082 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2082\nh : SameRay R (\u2191f x -\u1d65 fp\u2081) (fp\u2082 -\u1d65 \u2191f y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrcases hfp\u2081 with \u27e8p\u2081, hp\u2081, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\nfp\u2082 : P'\nhfp\u2082 : \u2203 y, y \u2208 s \u2227 \u2191f y = fp\u2082\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (fp\u2082 -\u1d65 \u2191f y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrcases hfp\u2082 with \u27e8p\u2082, hp\u2082, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f p\u2082 -\u1d65 \u2191f y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (\u2191f x -\u1d65 \u2191f p\u2081) (\u2191f p\u2082 -\u1d65 \u2191f y)\n\u22a2 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nsimp_rw [\u2190 linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P \u2192\u1d43[R] P'\nhf : Function.Injective \u2191f\n\u22a2 SOppSide (map f s) (\u2191f x) (\u2191f y) \u2194 SOppSide s x y\n[PROOFSTEP]\nsimp_rw [SOppSide, hf.wOppSide_map_iff, mem_map_iff_mem_of_injective hf]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 SSameSide s x y \u2194 SSameSide s y x\n[PROOFSTEP]\nrw [SSameSide, SSameSide, wSameSide_comm, and_comm (b := x \u2209 s)]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WOppSide s x y \u2194 WOppSide s y x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WOppSide s x y \u2192 WOppSide s y x\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WOppSide s y x\n[PROOFSTEP]\nrefine' \u27e8p\u2082, hp\u2082, p\u2081, hp\u2081, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (y -\u1d65 p\u2082) (p\u2081 -\u1d65 x)\n[PROOFSTEP]\nrwa [SameRay.sameRay_comm, \u2190 sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WOppSide s y x \u2192 WOppSide s x y\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (y -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrefine' \u27e8p\u2082, hp\u2082, p\u2081, hp\u2081, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (y -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\n\u22a2 SameRay R (x -\u1d65 p\u2082) (p\u2081 -\u1d65 y)\n[PROOFSTEP]\nrwa [SameRay.sameRay_comm, \u2190 sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 SOppSide s x y \u2194 SOppSide s y x\n[PROOFSTEP]\nrw [SOppSide, SOppSide, wOppSide_comm, and_comm (b := x \u2209 s)]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrefine' \u27e8x, hx, x, hx, _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 SameRay R (x -\u1d65 x) (y -\u1d65 x)\n[PROOFSTEP]\nrw [vsub_self]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 SameRay R 0 (y -\u1d65 x)\n[PROOFSTEP]\napply SameRay.zero_left\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrefine' \u27e8x, hx, x, hx, _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 SameRay R (x -\u1d65 x) (x -\u1d65 y)\n[PROOFSTEP]\nrw [vsub_self]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\n\u22a2 SameRay R 0 (x -\u1d65 y)\n[PROOFSTEP]\napply SameRay.zero_left\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WSameSide s (v +\u1d65 x) y \u2194 WSameSide s x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WSameSide s (v +\u1d65 x) y \u2192 WSameSide s x y\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nrefine' \u27e8-v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (x -\u1d65 (-v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, \u2190 vadd_vsub_assoc]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WSameSide s x y \u2192 WSameSide s (v +\u1d65 x) y\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WSameSide s (v +\u1d65 x) y\n[PROOFSTEP]\nrefine' \u27e8v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction hv hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 SameRay R (v +\u1d65 x -\u1d65 (v +\u1d65 p\u2081)) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrwa [vadd_vsub_vadd_cancel_left]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WSameSide s x (v +\u1d65 y) \u2194 WSameSide s x y\n[PROOFSTEP]\nrw [wSameSide_comm, wSameSide_vadd_left_iff hv, wSameSide_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 SSameSide s (v +\u1d65 x) y \u2194 SSameSide s x y\n[PROOFSTEP]\nrw [SSameSide, SSameSide, wSameSide_vadd_left_iff hv, vadd_mem_iff_mem_of_mem_direction hv]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 SSameSide s x (v +\u1d65 y) \u2194 SSameSide s x y\n[PROOFSTEP]\nrw [sSameSide_comm, sSameSide_vadd_left_iff hv, sSameSide_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WOppSide s (v +\u1d65 x) y \u2194 WOppSide s x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WOppSide s (v +\u1d65 x) y \u2192 WOppSide s x y\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nrefine' \u27e8-v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (v +\u1d65 x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (x -\u1d65 (-v +\u1d65 p\u2081)) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, \u2190 vadd_vsub_assoc]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WOppSide s x y \u2192 WOppSide s (v +\u1d65 x) y\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WOppSide s (v +\u1d65 x) y\n[PROOFSTEP]\nrefine' \u27e8v +\u1d65 p\u2081, AffineSubspace.vadd_mem_of_mem_direction hv hp\u2081, p\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (v +\u1d65 x -\u1d65 (v +\u1d65 p\u2081)) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrwa [vadd_vsub_vadd_cancel_left]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 WOppSide s x (v +\u1d65 y) \u2194 WOppSide s x y\n[PROOFSTEP]\nrw [wOppSide_comm, wOppSide_vadd_left_iff hv, wOppSide_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 SOppSide s (v +\u1d65 x) y \u2194 SOppSide s x y\n[PROOFSTEP]\nrw [SOppSide, SOppSide, wOppSide_vadd_left_iff hv, vadd_mem_iff_mem_of_mem_direction hv]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v \u2208 direction s\n\u22a2 SOppSide s x (v +\u1d65 y) \u2194 SOppSide s x y\n[PROOFSTEP]\nrw [sOppSide_comm, sOppSide_vadd_left_iff hv, sOppSide_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : 0 \u2264 t\n\u22a2 WSameSide s (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082) x\n[PROOFSTEP]\nrefine' \u27e8p\u2082, hp\u2082, p\u2081, hp\u2081, _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : 0 \u2264 t\n\u22a2 SameRay R (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082 -\u1d65 p\u2082) (x -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [vadd_vsub]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : 0 \u2264 t\n\u22a2 SameRay R (t \u2022 (x -\u1d65 p\u2081)) (x -\u1d65 p\u2081)\n[PROOFSTEP]\nexact SameRay.sameRay_nonneg_smul_left _ ht\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : t \u2264 0\n\u22a2 WOppSide s (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082) x\n[PROOFSTEP]\nrefine' \u27e8p\u2082, hp\u2082, p\u2081, hp\u2081, _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : t \u2264 0\n\u22a2 SameRay R (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082 -\u1d65 p\u2082) (p\u2081 -\u1d65 x)\n[PROOFSTEP]\nrw [vadd_vsub, \u2190 neg_neg t, neg_smul, \u2190 smul_neg, neg_vsub_eq_vsub_rev]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\np\u2081 p\u2082 x : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : t \u2264 0\n\u22a2 SameRay R (-t \u2022 (p\u2081 -\u1d65 x)) (p\u2081 -\u1d65 x)\n[PROOFSTEP]\nexact SameRay.sameRay_nonneg_smul_left _ (neg_nonneg.2 ht)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nh : Wbtw R x y z\nhx : x \u2208 s\n\u22a2 WSameSide s y z\n[PROOFSTEP]\nrcases h with \u27e8t, \u27e8ht0, -\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : x \u2208 s\nt : R\nht0 : 0 \u2264 t\n\u22a2 WSameSide s (\u2191(lineMap x z) t) z\n[PROOFSTEP]\nexact wSameSide_lineMap_left z hx ht0\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nh : Wbtw R x y z\nhy : y \u2208 s\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrcases h with \u27e8t, \u27e8ht0, ht1\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrefine' \u27e8_, hy, _, hy, _\u27e9\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) t) (\u2191(lineMap x z) t -\u1d65 z)\n[PROOFSTEP]\nrcases ht1.lt_or_eq with (ht1' | rfl)\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) t) (\u2191(lineMap x z) t -\u1d65 z)\ncase intro.intro.intro.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nhy : \u2191(lineMap x z) 1 \u2208 s\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) 1) (\u2191(lineMap x z) 1 -\u1d65 z)\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.intro.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nhy : \u2191(lineMap x z) 1 \u2208 s\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) 1) (\u2191(lineMap x z) 1 -\u1d65 z)\n[PROOFSTEP]\nrw [lineMap_apply_one]\n[GOAL]\ncase intro.intro.intro.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nhy : \u2191(lineMap x z) 1 \u2208 s\n\u22a2 SameRay R (x -\u1d65 z) (z -\u1d65 z)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) t) (\u2191(lineMap x z) t -\u1d65 z)\n[PROOFSTEP]\nrcases ht0.lt_or_eq with (ht0' | rfl)\n[GOAL]\ncase intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\nht0' : 0 < t\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) t) (\u2191(lineMap x z) t -\u1d65 z)\ncase intro.intro.intro.inl.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 0\nht1 : 0 \u2264 1\nhy : \u2191(lineMap x z) 0 \u2208 s\nht1' : 0 < 1\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) 0) (\u2191(lineMap x z) 0 -\u1d65 z)\n[PROOFSTEP]\nswap\n[GOAL]\ncase intro.intro.intro.inl.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 0\nht1 : 0 \u2264 1\nhy : \u2191(lineMap x z) 0 \u2208 s\nht1' : 0 < 1\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) 0) (\u2191(lineMap x z) 0 -\u1d65 z)\n[PROOFSTEP]\nrw [lineMap_apply_zero]\n[GOAL]\ncase intro.intro.intro.inl.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nht0 : 0 \u2264 0\nht1 : 0 \u2264 1\nhy : \u2191(lineMap x z) 0 \u2208 s\nht1' : 0 < 1\n\u22a2 SameRay R (x -\u1d65 x) (x -\u1d65 z)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\nht0' : 0 < t\n\u22a2 SameRay R (x -\u1d65 \u2191(lineMap x z) t) (\u2191(lineMap x z) t -\u1d65 z)\n[PROOFSTEP]\nrefine'\n  Or.inr\n    (Or.inr \u27e81 - t, t, sub_pos.2 ht1', ht0', _\u27e9)\n      -- TODO: after lean4#2336 \"simp made no progress feature\"\n        -- had to add `_` to several lemmas here. Not sure why!\n[GOAL]\ncase intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\nht0' : 0 < t\n\u22a2 (1 - t) \u2022 (x -\u1d65 \u2191(lineMap x z) t) = t \u2022 (\u2191(lineMap x z) t -\u1d65 z)\n[PROOFSTEP]\nsimp_rw [lineMap_apply _, vadd_vsub_assoc _, vsub_vadd_eq_vsub_sub _, \u2190 neg_vsub_eq_vsub_rev z x, vsub_self _, zero_sub,\n  \u2190 neg_one_smul R (z -\u1d65 x), \u2190 add_smul, smul_neg, \u2190 neg_smul, smul_smul]\n[GOAL]\ncase intro.intro.intro.inl.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : StrictOrderedCommRing R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nht1' : t < 1\nht0' : 0 < t\n\u22a2 (-(1 - t) * t) \u2022 (z -\u1d65 x) = (t * (t + -1)) \u2022 (z -\u1d65 x)\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx : P\n\u22a2 WOppSide s x x \u2194 x \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx : P\n\u22a2 WOppSide s x x \u2192 x \u2208 s\n[PROOFSTEP]\nrintro \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\n\u22a2 x \u2208 s\n[PROOFSTEP]\nobtain \u27e8a, -, -, -, -, h\u2081, -\u27e9 := h.exists_eq_smul_add\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\na : R\nh\u2081 : x -\u1d65 p\u2081 = a \u2022 (x -\u1d65 p\u2081 + (p\u2082 -\u1d65 x))\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [add_comm, vsub_add_vsub_cancel, \u2190 eq_vadd_iff_vsub_eq] at h\u2081 \n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\na : R\nh\u2081 : x = a \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 p\u2081\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 x)\na : R\nh\u2081 : x = a \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 p\u2081\n\u22a2 a \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 p\u2081 \u2208 s\n[PROOFSTEP]\nexact s.smul_vsub_vadd_mem a hp\u2082 hp\u2081 hp\u2081\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx : P\n\u22a2 x \u2208 s \u2192 WOppSide s x x\n[PROOFSTEP]\nexact fun h => \u27e8x, h, x, h, SameRay.rfl\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx : P\n\u22a2 \u00acSOppSide s x x\n[PROOFSTEP]\nrw [SOppSide]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx : P\n\u22a2 \u00ac(WOppSide s x x \u2227 \u00acx \u2208 s \u2227 \u00acx \u2208 s)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 WSameSide s x y \u2194 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 WSameSide s x y \u2192 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrintro \u27e8p\u2081', hp\u2081', p\u2082', hp\u2082', h0 | h0 | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, hr\u27e9\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x -\u1d65 p\u2081' = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h0 \n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [h0]\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 p\u2081' \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (p\u2081' -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nexact Or.inl hp\u2081'\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : y -\u1d65 p\u2082' = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrefine' Or.inr \u27e8p\u2082', hp\u2082', _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : y -\u1d65 p\u2082' = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082')\n[PROOFSTEP]\nrw [h0]\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : y -\u1d65 p\u2082' = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) 0\n[PROOFSTEP]\nexact SameRay.zero_right _\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (y -\u1d65 p\u2082')\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrefine'\n  Or.inr \u27e8(r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082', s.smul_vsub_vadd_mem _ h hp\u2081' hp\u2082', Or.inr (Or.inr \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, _\u27e9)\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (y -\u1d65 p\u2082')\n\u22a2 r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (y -\u1d65 ((r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082'))\n[PROOFSTEP]\nrw [vsub_vadd_eq_vsub_sub, smul_sub, \u2190 hr, smul_smul, mul_div_cancel' _ hr\u2082.ne.symm, \u2190 smul_sub,\n  vsub_sub_vsub_cancel_right]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)) \u2192 WSameSide s x y\n[PROOFSTEP]\nrintro (h' | \u27e8h\u2081, h\u2082, h\u2083\u27e9)\n[GOAL]\ncase mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh' : x \u2208 s\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nexact wSameSide_of_left_mem y h'\n[GOAL]\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh\u2081 : P\nh\u2082 : h\u2081 \u2208 s\nh\u2083 : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 h\u2081)\n\u22a2 WSameSide s x y\n[PROOFSTEP]\nexact \u27e8p\u2081, h, h\u2081, h\u2082, h\u2083\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 WSameSide s x y \u2194 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [wSameSide_comm, wSameSide_iff_exists_left h]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 (y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (x -\u1d65 p\u2082_1)) \u2194 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nsimp_rw [SameRay.sameRay_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 SSameSide s x y \u2194 \u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [SSameSide, and_comm, wSameSide_iff_exists_left h, and_assoc, and_congr_right_iff]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 \u00acx \u2208 s \u2192\n    (\u00acy \u2208 s \u2227 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)) \u2194\n      \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082))\n[PROOFSTEP]\nintro hx\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nhx : \u00acx \u2208 s\n\u22a2 \u00acy \u2208 s \u2227 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)) \u2194\n    \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [or_iff_right hx]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 SSameSide s x y \u2194 \u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [sSameSide_comm, sSameSide_iff_exists_left h, \u2190 and_assoc, and_comm (a := y \u2209 s), and_assoc]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 (\u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (x -\u1d65 p\u2082_1)) \u2194\n    \u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n[PROOFSTEP]\nsimp_rw [SameRay.sameRay_comm]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 WOppSide s x y \u2194 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 WOppSide s x y \u2192 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrintro \u27e8p\u2081', hp\u2081', p\u2082', hp\u2082', h0 | h0 | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, hr\u27e9\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x -\u1d65 p\u2081' = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h0 \n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [h0]\n[GOAL]\ncase mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : x = p\u2081'\n\u22a2 p\u2081' \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (p\u2081' -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nexact Or.inl hp\u2081'\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrefine' Or.inr \u27e8p\u2082', hp\u2082', _\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) (p\u2082' -\u1d65 y)\n[PROOFSTEP]\nrw [h0]\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nh0 : p\u2082' -\u1d65 y = 0\n\u22a2 SameRay R (x -\u1d65 p\u2081) 0\n[PROOFSTEP]\nexact SameRay.zero_right _\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrefine'\n  Or.inr \u27e8(-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082', s.smul_vsub_vadd_mem _ h hp\u2081' hp\u2082', Or.inr (Or.inr \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, _\u27e9)\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\np\u2081' : P\nhp\u2081' : p\u2081' \u2208 s\np\u2082' : P\nhp\u2082' : p\u2082' \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nhr : r\u2081 \u2022 (x -\u1d65 p\u2081') = r\u2082 \u2022 (p\u2082' -\u1d65 y)\n\u22a2 r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 ((-r\u2081 / r\u2082) \u2022 (p\u2081 -\u1d65 p\u2081') +\u1d65 p\u2082' -\u1d65 y)\n[PROOFSTEP]\nrw [vadd_vsub_assoc, smul_add, \u2190 hr, smul_smul, neg_div, mul_neg, mul_div_cancel' _ hr\u2082.ne.symm, neg_smul,\n  neg_add_eq_sub, \u2190 smul_sub, vsub_sub_vsub_cancel_right]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2192 WOppSide s x y\n[PROOFSTEP]\nrintro (h' | \u27e8h\u2081, h\u2082, h\u2083\u27e9)\n[GOAL]\ncase mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh' : x \u2208 s\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nexact wOppSide_of_left_mem y h'\n[GOAL]\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nh\u2081 : P\nh\u2082 : h\u2081 \u2208 s\nh\u2083 : SameRay R (x -\u1d65 p\u2081) (h\u2081 -\u1d65 y)\n\u22a2 WOppSide s x y\n[PROOFSTEP]\nexact \u27e8p\u2081, h, h\u2081, h\u2082, h\u2083\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 WOppSide s x y \u2194 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [wOppSide_comm, wOppSide_iff_exists_left h]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 (y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 x)) \u2194 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 (y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 x)) \u2192 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrintro (hy | \u27e8p, hp, hr\u27e9)\n[GOAL]\ncase mp.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\nhy : y \u2208 s\n\u22a2 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nexact Or.inl hy\n[GOAL]\ncase mp.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\np : P\nhp : p \u2208 s\nhr : SameRay R (y -\u1d65 p\u2082) (p -\u1d65 x)\n\u22a2 y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrefine' Or.inr \u27e8p, hp, _\u27e9\n[GOAL]\ncase mp.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\np : P\nhp : p \u2208 s\nhr : SameRay R (y -\u1d65 p\u2082) (p -\u1d65 x)\n\u22a2 SameRay R (x -\u1d65 p) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrwa [SameRay.sameRay_comm, \u2190 sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 (y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2192 y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 x)\n[PROOFSTEP]\nrintro (hy | \u27e8p, hp, hr\u27e9)\n[GOAL]\ncase mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\nhy : y \u2208 s\n\u22a2 y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 x)\n[PROOFSTEP]\nexact Or.inl hy\n[GOAL]\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\np : P\nhp : p \u2208 s\nhr : SameRay R (x -\u1d65 p) (p\u2082 -\u1d65 y)\n\u22a2 y \u2208 s \u2228 \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 x)\n[PROOFSTEP]\nrefine' Or.inr \u27e8p, hp, _\u27e9\n[GOAL]\ncase mpr.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\np : P\nhp : p \u2208 s\nhr : SameRay R (x -\u1d65 p) (p\u2082 -\u1d65 y)\n\u22a2 SameRay R (y -\u1d65 p\u2082) (p -\u1d65 x)\n[PROOFSTEP]\nrwa [SameRay.sameRay_comm, \u2190 sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 SOppSide s x y \u2194 \u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [SOppSide, and_comm, wOppSide_iff_exists_left h, and_assoc, and_congr_right_iff]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\n\u22a2 \u00acx \u2208 s \u2192\n    (\u00acy \u2208 s \u2227 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2194\n      \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y))\n[PROOFSTEP]\nintro hx\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nh : p\u2081 \u2208 s\nhx : \u00acx \u2208 s\n\u22a2 \u00acy \u2208 s \u2227 (x \u2208 s \u2228 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2194\n    \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [or_iff_right hx]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 SOppSide s x y \u2194 \u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [SOppSide, and_comm, wOppSide_iff_exists_right h, and_assoc, and_congr_right_iff, and_congr_right_iff]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\n\u22a2 \u00acx \u2208 s \u2192\n    \u00acy \u2208 s \u2192 ((y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2194 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y))\n[PROOFSTEP]\nrintro _ hy\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2082 : P\nh : p\u2082 \u2208 s\na\u271d : \u00acx \u2208 s\nhy : \u00acy \u2208 s\n\u22a2 (y \u2208 s \u2228 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)) \u2194 \u2203 p\u2081, p\u2081 \u2208 s \u2227 SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n[PROOFSTEP]\nrw [or_iff_right hy]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WSameSide s x y\nhyz : WSameSide s y z\nhy : \u00acy \u2208 s\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrcases hxy with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WSameSide s y z\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrw [wSameSide_iff_exists_left hp\u2082, or_iff_right hy] at hyz \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhyz : \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (z -\u1d65 p\u2082_1)\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrcases hyz with \u27e8p\u2083, hp\u2083, hyz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (z -\u1d65 p\u2083)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, p\u2083, hp\u2083, hxy.trans hyz _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (z -\u1d65 p\u2083)\n\u22a2 y -\u1d65 p\u2082 = 0 \u2192 x -\u1d65 p\u2081 = 0 \u2228 z -\u1d65 p\u2083 = 0\n[PROOFSTEP]\nrefine' fun h => False.elim _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (z -\u1d65 p\u2083)\nh : y -\u1d65 p\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (z -\u1d65 p\u2083)\nh : y = p\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hy (h.symm \u25b8 hp\u2082)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WSameSide s x y\nhyz : WOppSide s y z\nhy : \u00acy \u2208 s\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrcases hxy with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrw [wOppSide_iff_exists_left hp\u2082, or_iff_right hy] at hyz \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhyz : \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 z)\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrcases hyz with \u27e8p\u2083, hp\u2083, hyz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (p\u2083 -\u1d65 z)\n\u22a2 WOppSide s x z\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, p\u2083, hp\u2083, hxy.trans hyz _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (p\u2083 -\u1d65 z)\n\u22a2 y -\u1d65 p\u2082 = 0 \u2192 x -\u1d65 p\u2081 = 0 \u2228 p\u2083 -\u1d65 z = 0\n[PROOFSTEP]\nrefine' fun h => False.elim _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (p\u2083 -\u1d65 z)\nh : y -\u1d65 p\u2082 = 0\n\u22a2 False\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (y -\u1d65 p\u2082)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (p\u2083 -\u1d65 z)\nh : y = p\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hy (h.symm \u25b8 hp\u2082)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WOppSide s x y\nhyz : WOppSide s y z\nhy : \u00acy \u2208 s\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrcases hxy with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, hxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WOppSide s y z\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrw [wOppSide_iff_exists_left hp\u2082, or_iff_right hy] at hyz \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhyz : \u2203 p\u2082_1, p\u2082_1 \u2208 s \u2227 SameRay R (y -\u1d65 p\u2082) (p\u2082_1 -\u1d65 z)\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrcases hyz with \u27e8p\u2083, hp\u2083, hyz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (y -\u1d65 p\u2082) (p\u2083 -\u1d65 z)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrw [\u2190 sameRay_neg_iff, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev] at hyz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (p\u2082 -\u1d65 y) (z -\u1d65 p\u2083)\n\u22a2 WSameSide s x z\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, p\u2083, hp\u2083, hxy.trans hyz _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (p\u2082 -\u1d65 y) (z -\u1d65 p\u2083)\n\u22a2 p\u2082 -\u1d65 y = 0 \u2192 x -\u1d65 p\u2081 = 0 \u2228 z -\u1d65 p\u2083 = 0\n[PROOFSTEP]\nrefine' fun h => False.elim _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (p\u2082 -\u1d65 y) (z -\u1d65 p\u2083)\nh : p\u2082 -\u1d65 y = 0\n\u22a2 False\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : \u00acy \u2208 s\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nhxy : SameRay R (x -\u1d65 p\u2081) (p\u2082 -\u1d65 y)\np\u2083 : P\nhp\u2083 : p\u2083 \u2208 s\nhyz : SameRay R (p\u2082 -\u1d65 y) (z -\u1d65 p\u2083)\nh : p\u2082 = y\n\u22a2 False\n[PROOFSTEP]\nexact hy (h \u25b8 hp\u2082)\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WSameSide s x y \u2227 WOppSide s x y \u2194 x \u2208 s \u2228 y \u2208 s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WSameSide s x y \u2227 WOppSide s x y \u2192 x \u2208 s \u2228 y \u2208 s\n[PROOFSTEP]\nrintro \u27e8hs, ho\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhs : WSameSide s x y\nho : WOppSide s x y\n\u22a2 x \u2208 s \u2228 y \u2208 s\n[PROOFSTEP]\nrw [wOppSide_comm] at ho \n[GOAL]\ncase mp.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhs : WSameSide s x y\nho : WOppSide s y x\n\u22a2 x \u2208 s \u2228 y \u2208 s\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase mp.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhs : WSameSide s x y\nho : WOppSide s y x\nh : \u00ac(x \u2208 s \u2228 y \u2208 s)\n\u22a2 False\n[PROOFSTEP]\nrw [not_or] at h \n[GOAL]\ncase mp.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhs : WSameSide s x y\nho : WOppSide s y x\nh : \u00acx \u2208 s \u2227 \u00acy \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact h.1 (wOppSide_self_iff.1 (hs.trans_wOppSide ho h.2))\n[GOAL]\ncase mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 x \u2208 s \u2228 y \u2208 s \u2192 WSameSide s x y \u2227 WOppSide s x y\n[PROOFSTEP]\nrintro (h | h)\n[GOAL]\ncase mpr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : x \u2208 s\n\u22a2 WSameSide s x y \u2227 WOppSide s x y\n[PROOFSTEP]\nexact \u27e8wSameSide_of_left_mem y h, wOppSide_of_left_mem y h\u27e9\n[GOAL]\ncase mpr.inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : y \u2208 s\n\u22a2 WSameSide s x y \u2227 WOppSide s x y\n[PROOFSTEP]\nexact \u27e8wSameSide_of_right_mem x h, wOppSide_of_right_mem x h\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\n\u22a2 \u00acSOppSide s x y\n[PROOFSTEP]\nintro ho\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\nho : SOppSide s x y\n\u22a2 False\n[PROOFSTEP]\nhave hxy := wSameSide_and_wOppSide_iff.1 \u27e8h, ho.1\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\nho : SOppSide s x y\nhxy : x \u2208 s \u2228 y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrcases hxy with (hx | hy)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\nho : SOppSide s x y\nhx : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact ho.2.1 hx\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WSameSide s x y\nho : SOppSide s x y\nhy : y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact ho.2.2 hy\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SSameSide s x y\n\u22a2 \u00acWOppSide s x y\n[PROOFSTEP]\nintro ho\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SSameSide s x y\nho : WOppSide s x y\n\u22a2 False\n[PROOFSTEP]\nhave hxy := wSameSide_and_wOppSide_iff.1 \u27e8h.1, ho\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SSameSide s x y\nho : WOppSide s x y\nhxy : x \u2208 s \u2228 y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nrcases hxy with (hx | hy)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SSameSide s x y\nho : WOppSide s x y\nhx : x \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact h.2.1 hx\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SSameSide s x y\nho : WOppSide s x y\nhy : y \u2208 s\n\u22a2 False\n[PROOFSTEP]\nexact h.2.2 hy\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\n\u22a2 WOppSide s x y \u2194 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun \u27e8p, hp, h\u27e9 => h.wOppSide\u2081\u2083 hp\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : WOppSide s x y\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrcases h with \u27e8p\u2081, hp\u2081, p\u2082, hp\u2082, h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x -\u1d65 p\u2081 = 0\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\u2081\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\u2081\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R p\u2081 p y\n[PROOFSTEP]\nexact \u27e8p\u2081, hp\u2081, wbtw_self_left _ _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 -\u1d65 y = 0\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 = y\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\ncase intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 = y\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p p\u2082\n[PROOFSTEP]\nexact \u27e8p\u2082, hp\u2082, wbtw_self_right _ _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 \u2203 p, p \u2208 s \u2227 Wbtw R x p y\n[PROOFSTEP]\nrefine' \u27e8lineMap x y (r\u2082 / (r\u2081 + r\u2082)), _, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.inr.inr.intro.intro.intro.intro.refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 \u2191(lineMap x y) (r\u2082 / (r\u2081 + r\u2082)) \u2208 s\n[PROOFSTEP]\nhave : (r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 p\u2082 + (p\u2082 -\u1d65 p\u2081) - (x -\u1d65 p\u2081)) + (x -\u1d65 p\u2081) = (r\u2082 / (r\u2081 + r\u2082)) \u2022 (p\u2082 -\u1d65 p\u2081) := by\n  rw [add_comm (y -\u1d65 p\u2082), smul_sub, smul_add, add_sub_assoc, add_assoc, add_right_eq_self, div_eq_inv_mul, \u2190\n    neg_vsub_eq_vsub_rev, smul_neg, \u2190 smul_smul, \u2190 h, smul_smul, \u2190 neg_smul, \u2190 sub_smul, \u2190 div_eq_inv_mul, \u2190\n    div_eq_inv_mul, \u2190 neg_div, \u2190 sub_div, sub_eq_add_neg, \u2190 neg_add, neg_div, div_self (Left.add_pos hr\u2081 hr\u2082).ne.symm,\n    neg_one_smul, neg_add_self]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 (r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 p\u2082 + (p\u2082 -\u1d65 p\u2081) - (x -\u1d65 p\u2081)) + (x -\u1d65 p\u2081) = (r\u2082 / (r\u2081 + r\u2082)) \u2022 (p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nrw [add_comm (y -\u1d65 p\u2082), smul_sub, smul_add, add_sub_assoc, add_assoc, add_right_eq_self, div_eq_inv_mul, \u2190\n  neg_vsub_eq_vsub_rev, smul_neg, \u2190 smul_smul, \u2190 h, smul_smul, \u2190 neg_smul, \u2190 sub_smul, \u2190 div_eq_inv_mul, \u2190\n  div_eq_inv_mul, \u2190 neg_div, \u2190 sub_div, sub_eq_add_neg, \u2190 neg_add, neg_div, div_self (Left.add_pos hr\u2081 hr\u2082).ne.symm,\n  neg_one_smul, neg_add_self]\n[GOAL]\ncase intro.intro.intro.intro.inr.inr.intro.intro.intro.intro.refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\nthis : (r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 p\u2082 + (p\u2082 -\u1d65 p\u2081) - (x -\u1d65 p\u2081)) + (x -\u1d65 p\u2081) = (r\u2082 / (r\u2081 + r\u2082)) \u2022 (p\u2082 -\u1d65 p\u2081)\n\u22a2 \u2191(lineMap x y) (r\u2082 / (r\u2081 + r\u2082)) \u2208 s\n[PROOFSTEP]\nrw [lineMap_apply, \u2190 vsub_vadd x p\u2081, \u2190 vsub_vadd y p\u2082, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc, \u2190 vadd_assoc,\n  vadd_eq_add, this]\n[GOAL]\ncase intro.intro.intro.intro.inr.inr.intro.intro.intro.intro.refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\nthis : (r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 p\u2082 + (p\u2082 -\u1d65 p\u2081) - (x -\u1d65 p\u2081)) + (x -\u1d65 p\u2081) = (r\u2082 / (r\u2081 + r\u2082)) \u2022 (p\u2082 -\u1d65 p\u2081)\n\u22a2 (r\u2082 / (r\u2081 + r\u2082)) \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 p\u2081 \u2208 s\n[PROOFSTEP]\nexact s.smul_vsub_vadd_mem (r\u2082 / (r\u2081 + r\u2082)) hp\u2082 hp\u2081 hp\u2081\n[GOAL]\ncase intro.intro.intro.intro.inr.inr.intro.intro.intro.intro.refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y p\u2081 : P\nhp\u2081 : p\u2081 \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p\u2081) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 Wbtw R x (\u2191(lineMap x y) (r\u2082 / (r\u2081 + r\u2082))) y\n[PROOFSTEP]\nexact\n  Set.mem_image_of_mem _\n    \u27e8div_nonneg hr\u2082.le (Left.add_pos hr\u2081 hr\u2082).le,\n      div_le_one_of_le (le_add_of_nonneg_left hr\u2081.le) (Left.add_pos hr\u2081 hr\u2082).le\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SOppSide s x y\n\u22a2 \u2203 p, p \u2208 s \u2227 Sbtw R x p y\n[PROOFSTEP]\nobtain \u27e8p, hp, hw\u27e9 := wOppSide_iff_exists_wbtw.1 h.wOppSide\n[GOAL]\ncase intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SOppSide s x y\np : P\nhp : p \u2208 s\nhw : Wbtw R x p y\n\u22a2 \u2203 p, p \u2208 s \u2227 Sbtw R x p y\n[PROOFSTEP]\nrefine' \u27e8p, hp, hw, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SOppSide s x y\np : P\nhp : p \u2208 s\nhw : Wbtw R x p y\n\u22a2 p \u2260 x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\ny p : P\nhp : p \u2208 s\nh : SOppSide s p y\nhw : Wbtw R p p y\n\u22a2 False\n[PROOFSTEP]\nexact h.2.1 hp\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nh : SOppSide s x y\np : P\nhp : p \u2208 s\nhw : Wbtw R x p y\n\u22a2 p \u2260 y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhp : p \u2208 s\nh : SOppSide s x p\nhw : Wbtw R x p p\n\u22a2 False\n[PROOFSTEP]\nexact h.2.2 hp\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nh : Sbtw R x y z\nhx : \u00acx \u2208 s\nhy : y \u2208 s\n\u22a2 SOppSide s x z\n[PROOFSTEP]\nrefine' \u27e8h.wbtw.wOppSide\u2081\u2083 hy, hx, fun hz => hx _\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nh : Sbtw R x y z\nhx : \u00acx \u2208 s\nhy : y \u2208 s\nhz : z \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases h with \u27e8\u27e8t, \u27e8ht0, ht1\u27e9, rfl\u27e9, hyx, hyz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : \u2191(lineMap x z) t \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [lineMap_apply] at hy \n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : t \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\n\u22a2 x \u2208 s\n[PROOFSTEP]\nhave ht : t \u2260 1 := by\n  rintro rfl\n  simp [lineMap_apply] at hyz \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : t \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\n\u22a2 t \u2260 1\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nht0 : 0 \u2264 1\nht1 : 1 \u2264 1\nhy : 1 \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) 1 \u2260 x\nhyz : \u2191(lineMap x z) 1 \u2260 z\n\u22a2 False\n[PROOFSTEP]\nsimp [lineMap_apply] at hyz \n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : t \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\nht : t \u2260 1\n\u22a2 x \u2208 s\n[PROOFSTEP]\nhave hy' := vsub_mem_direction hy hz\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : t \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\nht : t \u2260 1\nhy' : t \u2022 (z -\u1d65 x) +\u1d65 x -\u1d65 z \u2208 direction s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrw [vadd_vsub_assoc, \u2190 neg_vsub_eq_vsub_rev z, \u2190 neg_one_smul R (z -\u1d65 x), \u2190 add_smul, \u2190 sub_eq_add_neg,\n  s.direction.smul_mem_iff (sub_ne_zero_of_ne ht)] at hy' \n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx z : P\nhx : \u00acx \u2208 s\nhz : z \u2208 s\nt : R\nht0 : 0 \u2264 t\nht1 : t \u2264 1\nhy : t \u2022 (z -\u1d65 x) +\u1d65 x \u2208 s\nhyx : \u2191(lineMap x z) t \u2260 x\nhyz : \u2191(lineMap x z) t \u2260 z\nht : t \u2260 1\nhy' : z -\u1d65 x \u2208 direction s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrwa [vadd_mem_iff_mem_of_mem_direction (Submodule.smul_mem _ _ hy')] at hy \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 p\u2082 : P\nhx : \u00acx \u2208 s\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : 0 < t\n\u22a2 SSameSide s (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082) x\n[PROOFSTEP]\nrefine' \u27e8wSameSide_smul_vsub_vadd_left x hp\u2081 hp\u2082 ht.le, fun h => hx _, hx\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 p\u2082 : P\nhx : \u00acx \u2208 s\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : 0 < t\nh : t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082 \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrwa [vadd_mem_iff_mem_direction _ hp\u2082, s.direction.smul_mem_iff ht.ne.symm, vsub_right_mem_direction_iff_mem hp\u2081] at h \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 p\u2082 : P\nhx : \u00acx \u2208 s\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : t < 0\n\u22a2 SOppSide s (t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082) x\n[PROOFSTEP]\nrefine' \u27e8wOppSide_smul_vsub_vadd_left x hp\u2081 hp\u2082 ht.le, fun h => hx _, hx\u27e9\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p\u2081 p\u2082 : P\nhx : \u00acx \u2208 s\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nt : R\nht : t < 0\nh : t \u2022 (x -\u1d65 p\u2081) +\u1d65 p\u2082 \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrwa [vadd_mem_iff_mem_direction _ hp\u2082, s.direction.smul_mem_iff ht.ne, vsub_right_mem_direction_iff_mem hp\u2081] at h \n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\n\u22a2 {y | WSameSide s x y} = Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Ici 0) \u2191s\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 y \u2208 {y | WSameSide s x y} \u2194 y \u2208 Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Ici 0) \u2191s\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Ici]\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 WSameSide s x y \u2194 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 WSameSide s x y \u2192 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [wSameSide_iff_exists_left hp, or_iff_right hx]\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p) (y -\u1d65 p\u2082)) \u2192 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrintro \u27e8p\u2082, hp\u2082, h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x -\u1d65 p = 0\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hx (h.symm \u25b8 hp))\n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : y -\u1d65 p\u2082 = 0\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : y = p\u2082\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e80, p\u2082, le_refl _, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : y = p\u2082\n\u22a2 0 \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.mp.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (y -\u1d65 p\u2082)\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e8r\u2081 / r\u2082, p\u2082, (div_pos hr\u2081 hr\u2082).le, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (y -\u1d65 p\u2082)\n\u22a2 (r\u2081 / r\u2082) \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nrw [div_eq_inv_mul, \u2190 smul_smul, h, smul_smul, inv_mul_cancel hr\u2082.ne.symm, one_smul, vsub_vadd]\n[GOAL]\ncase h.mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 a b, 0 \u2264 a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y) \u2192 WSameSide s x y\n[PROOFSTEP]\nrintro \u27e8t, p', ht, hp', rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\nt : R\np' : P\nht : 0 \u2264 t\nhp' : p' \u2208 \u2191s\n\u22a2 WSameSide s x (t \u2022 (x -\u1d65 p) +\u1d65 p')\n[PROOFSTEP]\nexact wSameSide_smul_vsub_vadd_right x hp hp' ht\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\n\u22a2 {y | SSameSide s x y} = Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Ioi 0) \u2191s\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 y \u2208 {y | SSameSide s x y} \u2194 y \u2208 Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Ioi 0) \u2191s\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Ioi]\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 SSameSide s x y \u2194 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 SSameSide s x y \u2192 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [sSameSide_iff_exists_left hp]\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p) (y -\u1d65 p\u2082)) \u2192 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrintro \u27e8-, hy, p\u2082, hp\u2082, h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x -\u1d65 p = 0\n\u22a2 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\n\u22a2 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hx (h.symm \u25b8 hp))\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : y -\u1d65 p\u2082 = 0\n\u22a2 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : y = p\u2082\n\u22a2 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hy (h.symm \u25b8 hp\u2082))\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (y -\u1d65 p\u2082)\n\u22a2 \u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e8r\u2081 / r\u2082, p\u2082, div_pos hr\u2081 hr\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (y -\u1d65 p\u2082)\n\u22a2 (r\u2081 / r\u2082) \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nrw [div_eq_inv_mul, \u2190 smul_smul, h, smul_smul, inv_mul_cancel hr\u2082.ne.symm, one_smul, vsub_vadd]\n[GOAL]\ncase h.mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 a b, 0 < a \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y) \u2192 SSameSide s x y\n[PROOFSTEP]\nrintro \u27e8t, p', ht, hp', rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\nt : R\np' : P\nht : 0 < t\nhp' : p' \u2208 \u2191s\n\u22a2 SSameSide s x (t \u2022 (x -\u1d65 p) +\u1d65 p')\n[PROOFSTEP]\nexact sSameSide_smul_vsub_vadd_right hx hp hp' ht\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\n\u22a2 {y | WOppSide s x y} = Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Iic 0) \u2191s\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 y \u2208 {y | WOppSide s x y} \u2194 y \u2208 Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Iic 0) \u2191s\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Iic]\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 WOppSide s x y \u2194 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 WOppSide s x y \u2192 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [wOppSide_iff_exists_left hp, or_iff_right hx]\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p) (p\u2082 -\u1d65 y)) \u2192 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrintro \u27e8p\u2082, hp\u2082, h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x -\u1d65 p = 0\n\u22a2 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\n\u22a2 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hx (h.symm \u25b8 hp))\n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 -\u1d65 y = 0\n\u22a2 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 = y\n\u22a2 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e80, p\u2082, le_refl _, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 = y\n\u22a2 0 \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase h.mp.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 \u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e8-r\u2081 / r\u2082, p\u2082, (div_neg_of_neg_of_pos (Left.neg_neg_iff.2 hr\u2081) hr\u2082).le, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny p\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 (-r\u2081 / r\u2082) \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nrw [div_eq_inv_mul, \u2190 smul_smul, neg_smul, h, smul_neg, smul_smul, inv_mul_cancel hr\u2082.ne.symm, one_smul,\n  neg_vsub_eq_vsub_rev, vsub_vadd]\n[GOAL]\ncase h.mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 a b, a \u2264 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y) \u2192 WOppSide s x y\n[PROOFSTEP]\nrintro \u27e8t, p', ht, hp', rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\nt : R\np' : P\nht : t \u2264 0\nhp' : p' \u2208 \u2191s\n\u22a2 WOppSide s x (t \u2022 (x -\u1d65 p) +\u1d65 p')\n[PROOFSTEP]\nexact wOppSide_smul_vsub_vadd_right x hp hp' ht\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\n\u22a2 {y | SOppSide s x y} = Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Iio 0) \u2191s\n[PROOFSTEP]\next y\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 y \u2208 {y | SOppSide s x y} \u2194 y \u2208 Set.image2 (fun t q => t \u2022 (x -\u1d65 p) +\u1d65 q) (Set.Iio 0) \u2191s\n[PROOFSTEP]\nsimp_rw [Set.mem_setOf, Set.mem_image2, Set.mem_Iio]\n[GOAL]\ncase h\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 SOppSide s x y \u2194 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 SOppSide s x y \u2192 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [sOppSide_iff_exists_left hp]\n[GOAL]\ncase h.mp\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u00acx \u2208 s \u2227 \u00acy \u2208 s \u2227 \u2203 p\u2082, p\u2082 \u2208 s \u2227 SameRay R (x -\u1d65 p) (p\u2082 -\u1d65 y)) \u2192 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrintro \u27e8-, hy, p\u2082, hp\u2082, h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x -\u1d65 p = 0\n\u22a2 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : x = p\n\u22a2 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hx (h.symm \u25b8 hp))\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 -\u1d65 y = 0\n\u22a2 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrw [vsub_eq_zero_iff_eq] at h \n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2082 = y\n\u22a2 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nexact False.elim (hy (h \u25b8 hp\u2082))\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 \u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y\n[PROOFSTEP]\nrefine' \u27e8-r\u2081 / r\u2082, p\u2082, div_neg_of_neg_of_pos (Left.neg_neg_iff.2 hr\u2081) hr\u2082, hp\u2082, _\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.inr.inr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\nhy : \u00acy \u2208 s\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 s\nr\u2081 r\u2082 : R\nhr\u2081 : 0 < r\u2081\nhr\u2082 : 0 < r\u2082\nh : r\u2081 \u2022 (x -\u1d65 p) = r\u2082 \u2022 (p\u2082 -\u1d65 y)\n\u22a2 (-r\u2081 / r\u2082) \u2022 (x -\u1d65 p) +\u1d65 p\u2082 = y\n[PROOFSTEP]\nrw [div_eq_inv_mul, \u2190 smul_smul, neg_smul, h, smul_neg, smul_smul, inv_mul_cancel hr\u2082.ne.symm, one_smul,\n  neg_vsub_eq_vsub_rev, vsub_vadd]\n[GOAL]\ncase h.mpr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\ny : P\n\u22a2 (\u2203 a b, a < 0 \u2227 b \u2208 \u2191s \u2227 a \u2022 (x -\u1d65 p) +\u1d65 b = y) \u2192 SOppSide s x y\n[PROOFSTEP]\nrintro \u27e8t, p', ht, hp', rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx p : P\nhx : \u00acx \u2208 s\nhp : p \u2208 s\nt : R\np' : P\nht : t < 0\nhp' : p' \u2208 \u2191s\n\u22a2 SOppSide s x (t \u2022 (x -\u1d65 p) +\u1d65 p')\n[PROOFSTEP]\nexact sOppSide_smul_vsub_vadd_right hx hp hp' ht\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\nhy : \u00acy \u2208 s\n\u22a2 SOppSide s y (\u2191(pointReflection R x) y)\n[PROOFSTEP]\nrefine' (sbtw_pointReflection_of_ne R fun h => hy _).sOppSide_of_not_mem_of_mem hy hx\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u2076 : LinearOrderedField R\ninst\u271d\u2075 : AddCommGroup V\ninst\u271d\u2074 : Module R V\ninst\u271d\u00b3 : AddTorsor V P\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module R V'\ninst\u271d : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nhx : x \u2208 s\nhy : \u00acy \u2208 s\nh : x = y\n\u22a2 y \u2208 s\n[PROOFSTEP]\nrwa [\u2190 h]\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsConnected {y | WSameSide s x y}\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := h\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\n\u22a2 IsConnected {y | WSameSide s x y}\n[PROOFSTEP]\nhaveI : Nonempty s := \u27e8\u27e8p, hp\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected {y | WSameSide s x y}\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\n\u22a2 IsConnected {y | WSameSide s x y}\n[PROOFSTEP]\nsimp only [wSameSide_of_left_mem, hx]\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\n\u22a2 IsConnected {y | True}\n[PROOFSTEP]\nhave := AddTorsor.connectedSpace V P\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis\u271d : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\nthis : ConnectedSpace P\n\u22a2 IsConnected {y | True}\n[PROOFSTEP]\nexact isConnected_univ\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 IsConnected {y | WSameSide s x y}\n[PROOFSTEP]\nrw [setOf_wSameSide_eq_image2 hx hp, \u2190 Set.image_prod]\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 IsConnected ((fun x_1 => x_1.fst \u2022 (x -\u1d65 p) +\u1d65 x_1.snd) '' Set.Ici 0 \u00d7\u02e2 \u2191s)\n[PROOFSTEP]\nrefine'\n  (isConnected_Ici.prod (isConnected_iff_connectedSpace.2 _)).image _\n    ((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 ConnectedSpace \u2191\u2191s\n[PROOFSTEP]\nconvert AddTorsor.connectedSpace s.direction s\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\n\u22a2 IsPreconnected {y | WSameSide s x y}\n[PROOFSTEP]\nrcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : \u2191s = \u2205\n\u22a2 IsPreconnected {y | WSameSide s x y}\n[PROOFSTEP]\nrw [coe_eq_bot_iff] at h \n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | WSameSide s x y}\n[PROOFSTEP]\nsimp only [h, not_wSameSide_bot]\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsPreconnected {y | WSameSide s x y}\n[PROOFSTEP]\nexact (isConnected_setOf_wSameSide x h).isPreconnected\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\nh : Set.Nonempty \u2191s\n\u22a2 IsConnected {y | SSameSide s x y}\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := h\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\n\u22a2 IsConnected {y | SSameSide s x y}\n[PROOFSTEP]\nhaveI : Nonempty s := \u27e8\u27e8p, hp\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected {y | SSameSide s x y}\n[PROOFSTEP]\nrw [setOf_sSameSide_eq_image2 hx hp, \u2190 Set.image_prod]\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected ((fun x_1 => x_1.fst \u2022 (x -\u1d65 p) +\u1d65 x_1.snd) '' Set.Ioi 0 \u00d7\u02e2 \u2191s)\n[PROOFSTEP]\nrefine'\n  (isConnected_Ioi.prod (isConnected_iff_connectedSpace.2 _)).image _\n    ((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 ConnectedSpace \u2191\u2191s\n[PROOFSTEP]\nconvert AddTorsor.connectedSpace s.direction s\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nrcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : \u2191s = \u2205\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nrw [coe_eq_bot_iff] at h \n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nsimp only [h, not_sSameSide_bot]\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : x \u2208 s\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nsimp only [hx, SSameSide, not_true, false_and_iff, and_false_iff]\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : x \u2208 s\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : \u00acx \u2208 s\n\u22a2 IsPreconnected {y | SSameSide s x y}\n[PROOFSTEP]\nexact (isConnected_setOf_sSameSide hx h).isPreconnected\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsConnected {y | WOppSide s x y}\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := h\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\n\u22a2 IsConnected {y | WOppSide s x y}\n[PROOFSTEP]\nhaveI : Nonempty s := \u27e8\u27e8p, hp\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected {y | WOppSide s x y}\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\n\u22a2 IsConnected {y | WOppSide s x y}\n[PROOFSTEP]\nsimp only [wOppSide_of_left_mem, hx]\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\n\u22a2 IsConnected {y | True}\n[PROOFSTEP]\nhave := AddTorsor.connectedSpace V P\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis\u271d : Nonempty { x // x \u2208 s }\nhx : x \u2208 s\nthis : ConnectedSpace P\n\u22a2 IsConnected {y | True}\n[PROOFSTEP]\nexact isConnected_univ\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 IsConnected {y | WOppSide s x y}\n[PROOFSTEP]\nrw [setOf_wOppSide_eq_image2 hx hp, \u2190 Set.image_prod]\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 IsConnected ((fun x_1 => x_1.fst \u2022 (x -\u1d65 p) +\u1d65 x_1.snd) '' Set.Iic 0 \u00d7\u02e2 \u2191s)\n[PROOFSTEP]\nrefine'\n  (isConnected_Iic.prod (isConnected_iff_connectedSpace.2 _)).image _\n    ((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx p : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\nhx : \u00acx \u2208 s\n\u22a2 ConnectedSpace \u2191\u2191s\n[PROOFSTEP]\nconvert AddTorsor.connectedSpace s.direction s\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\n\u22a2 IsPreconnected {y | WOppSide s x y}\n[PROOFSTEP]\nrcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : \u2191s = \u2205\n\u22a2 IsPreconnected {y | WOppSide s x y}\n[PROOFSTEP]\nrw [coe_eq_bot_iff] at h \n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | WOppSide s x y}\n[PROOFSTEP]\nsimp only [h, not_wOppSide_bot]\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsPreconnected {y | WOppSide s x y}\n[PROOFSTEP]\nexact (isConnected_setOf_wOppSide x h).isPreconnected\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\nh : Set.Nonempty \u2191s\n\u22a2 IsConnected {y | SOppSide s x y}\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := h\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\n\u22a2 IsConnected {y | SOppSide s x y}\n[PROOFSTEP]\nhaveI : Nonempty s := \u27e8\u27e8p, hp\u27e9\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected {y | SOppSide s x y}\n[PROOFSTEP]\nrw [setOf_sOppSide_eq_image2 hx hp, \u2190 Set.image_prod]\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 IsConnected ((fun x_1 => x_1.fst \u2022 (x -\u1d65 p) +\u1d65 x_1.snd) '' Set.Iio 0 \u00d7\u02e2 \u2191s)\n[PROOFSTEP]\nrefine'\n  (isConnected_Iio.prod (isConnected_iff_connectedSpace.2 _)).image _\n    ((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn\n[GOAL]\ncase intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nhx : \u00acx \u2208 s\np : P\nhp : p \u2208 \u2191s\nthis : Nonempty { x // x \u2208 s }\n\u22a2 ConnectedSpace \u2191\u2191s\n[PROOFSTEP]\nconvert AddTorsor.connectedSpace s.direction s\n[GOAL]\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nrcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : \u2191s = \u2205\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nrw [coe_eq_bot_iff] at h \n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nsimp only [h, not_sOppSide_bot]\n[GOAL]\ncase inl\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : s = \u22a5\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase inr\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nby_cases hx : x \u2208 s\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : x \u2208 s\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nsimp only [hx, SOppSide, not_true, false_and_iff, and_false_iff]\n[GOAL]\ncase pos\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : x \u2208 s\n\u22a2 IsPreconnected {y | False}\n[PROOFSTEP]\nexact isPreconnected_empty\n[GOAL]\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : SeminormedAddCommGroup V\ninst\u271d\u00b2 : NormedSpace \u211d V\ninst\u271d\u00b9 : PseudoMetricSpace P\ninst\u271d : NormedAddTorsor V P\ns : AffineSubspace \u211d P\nx : P\nh : Set.Nonempty \u2191s\nhx : \u00acx \u2208 s\n\u22a2 IsPreconnected {y | SOppSide s x y}\n[PROOFSTEP]\nexact (isConnected_setOf_sOppSide hx h).isPreconnected\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Side", "llama_tokens": 78467, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434873426302, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.5189348433455363}}
{"text": "[GOAL]\nn : \u2115\nF : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F\nq : MvQPF F\nG : TypeVec n \u2192 Type u\ninst\u271d : MvFunctor G\nFG_abs : {\u03b1 : TypeVec n} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : TypeVec n} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : TypeVec n} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (x : F \u03b1), FG_abs (f <$$> x) = f <$$> FG_abs x\n\u03b1\u271d : TypeVec n\nx : G \u03b1\u271d\n\u22a2 (fun {\u03b1} p => FG_abs (abs p)) ((fun {\u03b1} x => repr (FG_repr x)) x) = x\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F\nq : MvQPF F\nG : TypeVec n \u2192 Type u\ninst\u271d : MvFunctor G\nFG_abs : {\u03b1 : TypeVec n} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : TypeVec n} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : TypeVec n} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (x : F \u03b1), FG_abs (f <$$> x) = f <$$> FG_abs x\n\u03b1\u271d : TypeVec n\nx : G \u03b1\u271d\n\u22a2 FG_abs (abs (repr (FG_repr x))) = x\n[PROOFSTEP]\nrw [abs_repr, FG_abs_repr]\n[GOAL]\nn : \u2115\nF : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F\nq : MvQPF F\nG : TypeVec n \u2192 Type u\ninst\u271d : MvFunctor G\nFG_abs : {\u03b1 : TypeVec n} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : TypeVec n} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : TypeVec n} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (x : F \u03b1), FG_abs (f <$$> x) = f <$$> FG_abs x\n\u03b1\u271d \u03b2\u271d : TypeVec n\nf : \u03b1\u271d \u27f9 \u03b2\u271d\np : MvPFunctor.Obj (P F) \u03b1\u271d\n\u22a2 (fun {\u03b1} p => FG_abs (abs p)) (f <$$> p) = f <$$> (fun {\u03b1} p => FG_abs (abs p)) p\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\nF : TypeVec n \u2192 Type u\ninst\u271d\u00b9 : MvFunctor F\nq : MvQPF F\nG : TypeVec n \u2192 Type u\ninst\u271d : MvFunctor G\nFG_abs : {\u03b1 : TypeVec n} \u2192 F \u03b1 \u2192 G \u03b1\nFG_repr : {\u03b1 : TypeVec n} \u2192 G \u03b1 \u2192 F \u03b1\nFG_abs_repr : \u2200 {\u03b1 : TypeVec n} (x : G \u03b1), FG_abs (FG_repr x) = x\nFG_abs_map : \u2200 {\u03b1 \u03b2 : TypeVec n} (f : \u03b1 \u27f9 \u03b2) (x : F \u03b1), FG_abs (f <$$> x) = f <$$> FG_abs x\n\u03b1\u271d \u03b2\u271d : TypeVec n\nf : \u03b1\u271d \u27f9 \u03b2\u271d\np : MvPFunctor.Obj (P F) \u03b1\u271d\n\u22a2 FG_abs (abs (f <$$> p)) = f <$$> FG_abs (abs p)\n[PROOFSTEP]\nrw [abs_map, FG_abs_map]\n", "meta": {"mathlib_filename": "Mathlib.Data.QPF.Multivariate.Constructions.Quot", "llama_tokens": 1097, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333245911726382, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.5187106676953793}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : Set.Nonempty s\n\u22a2 ContractibleSpace \u2191s\n[PROOFSTEP]\nrefine'\n  (contractible_iff_id_nullhomotopic s).2\n    \u27e8\u27e8x, h.mem hne\u27e9, \u27e8\u27e8\u27e8fun p => \u27e8p.1.1 \u2022 x + (1 - p.1.1) \u2022 (p.2 : E), _\u27e9, _\u27e9, fun x => _, fun x => _\u27e9\u27e9\u27e9\n[GOAL]\ncase refine'_1\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : Set.Nonempty s\np : \u2191unitInterval \u00d7 \u2191s\n\u22a2 \u2191p.fst \u2022 x + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s\n[PROOFSTEP]\nexact h p.2.2 p.1.2.1 (sub_nonneg.2 p.1.2.2) (add_sub_cancel'_right _ _)\n[GOAL]\ncase refine'_2\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx : E\nh : StarConvex \u211d x s\nhne : Set.Nonempty s\n\u22a2 Continuous fun p =>\n    { val := \u2191p.fst \u2022 x + (1 - \u2191p.fst) \u2022 \u2191p.snd, property := (_ : \u2191p.fst \u2022 x + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s) }\n[PROOFSTEP]\nexact\n  ((continuous_subtype_val.fst'.smul continuous_const).add\n        ((continuous_const.sub continuous_subtype_val.fst').smul continuous_subtype_val.snd')).subtype_mk\n    _\n[GOAL]\ncase refine'_3\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : Set.Nonempty s\nx : \u2191s\n\u22a2 ContinuousMap.toFun\n      (ContinuousMap.mk fun p =>\n        { val := \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd, property := (_ : \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s) })\n      (0, x) =\n    \u2191(ContinuousMap.id \u2191s) x\n[PROOFSTEP]\next1\n[GOAL]\ncase refine'_3.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : Set.Nonempty s\nx : \u2191s\n\u22a2 \u2191(ContinuousMap.toFun\n        (ContinuousMap.mk fun p =>\n          { val := \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd, property := (_ : \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s) })\n        (0, x)) =\n    \u2191(\u2191(ContinuousMap.id \u2191s) x)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_4\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : Set.Nonempty s\nx : \u2191s\n\u22a2 ContinuousMap.toFun\n      (ContinuousMap.mk fun p =>\n        { val := \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd, property := (_ : \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s) })\n      (1, x) =\n    \u2191(ContinuousMap.const \u2191s { val := x\u271d, property := (_ : x\u271d \u2208 s) }) x\n[PROOFSTEP]\next1\n[GOAL]\ncase refine'_4.a\nE : Type u_1\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : Module \u211d E\ninst\u271d\u00b2 : TopologicalSpace E\ninst\u271d\u00b9 : ContinuousAdd E\ninst\u271d : ContinuousSMul \u211d E\ns : Set E\nx\u271d : E\nh : StarConvex \u211d x\u271d s\nhne : Set.Nonempty s\nx : \u2191s\n\u22a2 \u2191(ContinuousMap.toFun\n        (ContinuousMap.mk fun p =>\n          { val := \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd, property := (_ : \u2191p.fst \u2022 x\u271d + (1 - \u2191p.fst) \u2022 \u2191p.snd \u2208 s) })\n        (1, x)) =\n    \u2191(\u2191(ContinuousMap.const \u2191s { val := x\u271d, property := (_ : x\u271d \u2208 s) }) x)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Contractible", "llama_tokens": 1678, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649232, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.5181035459047243}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 Homeomorph.trans h (Homeomorph.symm h) = Homeomorph.refl \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nx\u271d : \u03b1\n\u22a2 \u2191(Homeomorph.trans h (Homeomorph.symm h)) x\u271d = \u2191(Homeomorph.refl \u03b1) x\u271d\n[PROOFSTEP]\napply symm_apply_apply\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 Homeomorph.trans (Homeomorph.symm h) h = Homeomorph.refl \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nx\u271d : \u03b2\n\u22a2 \u2191(Homeomorph.trans (Homeomorph.symm h) h) x\u271d = \u2191(Homeomorph.refl \u03b2) x\u271d\n[PROOFSTEP]\napply apply_symm_apply\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2243\u209c \u03b2\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g \u2191f\nthis : g = \u2191(Homeomorph.symm f)\n\u22a2 Function.LeftInverse g \u2191f\n[PROOFSTEP]\nconvert f.left_inv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2243\u209c \u03b2\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g \u2191f\nthis : g = \u2191(Homeomorph.symm f)\n\u22a2 Function.RightInverse g \u2191f\n[PROOFSTEP]\nconvert f.right_inv using 1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2243\u209c \u03b2\ng : \u03b2 \u2192 \u03b1\nhg : Function.RightInverse g \u2191f\nthis : g = \u2191(Homeomorph.symm f)\n\u22a2 Continuous\n    { toFun := \u2191f, invFun := g, left_inv := (_ : Function.LeftInverse g \u2191f),\n        right_inv := (_ : Function.RightInverse g \u2191f) }.invFun\n[PROOFSTEP]\nconvert f.symm.continuous\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 Inducing (\u2191(Homeomorph.symm h) \u2218 \u2191h)\n[PROOFSTEP]\nsimp only [symm_comp_self, inducing_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 QuotientMap (\u2191h \u2218 \u2191(Homeomorph.symm h))\n[PROOFSTEP]\nsimp only [self_comp_symm, QuotientMap.id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nf : \u03b1 \u2192 \u03b2\nhf : Embedding f\n\u22a2 Continuous (f \u2218 (Equiv.ofInjective f (_ : Function.Injective f)).invFun)\n[PROOFSTEP]\nsimp [continuous_subtype_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns : Set \u03b2\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 IsCompact (\u2191h \u207b\u00b9' s) \u2194 IsCompact s\n[PROOFSTEP]\nrw [\u2190 image_symm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns : Set \u03b2\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 IsCompact (\u2191(Homeomorph.symm h) '' s) \u2194 IsCompact s\n[PROOFSTEP]\nexact h.symm.isCompact_image\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns : Set \u03b1\nh : \u03b1 \u2243\u209c \u03b2\nhs : IsPreconnected (\u2191h '' s)\n\u22a2 IsPreconnected s\n[PROOFSTEP]\nsimpa only [image_symm, preimage_image] using hs.image _ h.symm.continuous.continuousOn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns : Set \u03b2\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 IsPreconnected (\u2191h \u207b\u00b9' s) \u2194 IsPreconnected s\n[PROOFSTEP]\nrw [\u2190 image_symm, isPreconnected_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ns : Set \u03b2\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 IsConnected (\u2191h \u207b\u00b9' s) \u2194 IsConnected s\n[PROOFSTEP]\nrw [\u2190 image_symm, isConnected_image]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 map (\u2191h) (cocompact \u03b1) = cocompact \u03b2\n[PROOFSTEP]\nrw [\u2190 h.comap_cocompact, map_comap_of_surjective h.surjective]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b1\n\u22a2 IsOpen (\u2191h '' s) \u2194 IsOpen s\n[PROOFSTEP]\nrw [\u2190 preimage_symm, isOpen_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b2\n\u22a2 IsClosed (\u2191h \u207b\u00b9' s) \u2194 IsClosed s\n[PROOFSTEP]\nsimp only [\u2190 isOpen_compl_iff, \u2190 preimage_compl, isOpen_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b1\n\u22a2 IsClosed (\u2191h '' s) \u2194 IsClosed s\n[PROOFSTEP]\nrw [\u2190 preimage_symm, isClosed_preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b1\n\u22a2 \u2191h '' closure s = closure (\u2191h '' s)\n[PROOFSTEP]\nrw [\u2190 preimage_symm, preimage_closure]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b1\n\u22a2 \u2191h '' interior s = interior (\u2191h '' s)\n[PROOFSTEP]\nrw [\u2190 preimage_symm, preimage_interior]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ns : Set \u03b1\n\u22a2 \u2191h '' frontier s = frontier (\u2191h '' s)\n[PROOFSTEP]\nrw [\u2190 preimage_symm, preimage_frontier]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nx : \u03b1\n\u22a2 range \u2191h \u2208 \ud835\udcdd (\u2191h x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nx : \u03b1\n\u22a2 map (\u2191(Homeomorph.symm h)) (\ud835\udcdd (\u2191h x)) = \ud835\udcdd x\n[PROOFSTEP]\nrw [h.symm.map_nhds_eq, h.symm_apply_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\ny : \u03b2\n\u22a2 comap (\u2191h) (\ud835\udcdd y) = \ud835\udcdd (\u2191(Homeomorph.symm h) y)\n[PROOFSTEP]\nrw [h.nhds_eq_comap, h.apply_symm_apply]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ni : LocallyConnectedSpace \u03b2\nh : \u03b1 \u2243\u209c \u03b2\n\u22a2 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nhave : \u2200 x, (\ud835\udcdd x).HasBasis (fun s \u21a6 IsOpen s \u2227 h x \u2208 s \u2227 IsConnected s) (h.symm '' \u00b7) := fun x \u21a6\n  by\n  rw [\u2190 h.symm_map_nhds_eq]\n  exact (i.1 _).map _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ni : LocallyConnectedSpace \u03b2\nh : \u03b1 \u2243\u209c \u03b2\nx : \u03b1\n\u22a2 HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 \u2191h x \u2208 s \u2227 IsConnected s) fun x => \u2191(Homeomorph.symm h) '' x\n[PROOFSTEP]\nrw [\u2190 h.symm_map_nhds_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ni : LocallyConnectedSpace \u03b2\nh : \u03b1 \u2243\u209c \u03b2\nx : \u03b1\n\u22a2 HasBasis (map (\u2191(Homeomorph.symm h)) (\ud835\udcdd (\u2191h x))) (fun s => IsOpen s \u2227 \u2191h x \u2208 s \u2227 IsConnected s) fun x =>\n    \u2191(Homeomorph.symm h) '' x\n[PROOFSTEP]\nexact (i.1 _).map _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ni : LocallyConnectedSpace \u03b2\nh : \u03b1 \u2243\u209c \u03b2\nthis : \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 \u2191h x \u2208 s \u2227 IsConnected s) fun x => \u2191(Homeomorph.symm h) '' x\n\u22a2 LocallyConnectedSpace \u03b1\n[PROOFSTEP]\nrefine locallyConnectedSpace_of_connected_bases _ _ this fun _ _ hs \u21a6 ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ni : LocallyConnectedSpace \u03b2\nh : \u03b1 \u2243\u209c \u03b2\nthis : \u2200 (x : \u03b1), HasBasis (\ud835\udcdd x) (fun s => IsOpen s \u2227 \u2191h x \u2208 s \u2227 IsConnected s) fun x => \u2191(Homeomorph.symm h) '' x\nx\u271d\u00b9 : \u03b1\nx\u271d : Set \u03b2\nhs : IsOpen x\u271d \u2227 \u2191h x\u271d\u00b9 \u2208 x\u271d \u2227 IsConnected x\u271d\n\u22a2 IsPreconnected (\u2191(Homeomorph.symm h) '' x\u271d)\n[PROOFSTEP]\nexact hs.2.2.2.image _ h.symm.continuous.continuousOn\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ne : \u03b1 \u2243 \u03b2\nh\u2081 : Continuous \u2191e\nh\u2082 : IsOpenMap \u2191e\n\u22a2 Continuous e.invFun\n[PROOFSTEP]\nrw [continuous_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ne : \u03b1 \u2243 \u03b2\nh\u2081 : Continuous \u2191e\nh\u2082 : IsOpenMap \u2191e\n\u22a2 \u2200 (s : Set \u03b1), IsOpen s \u2192 IsOpen (e.invFun \u207b\u00b9' s)\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ne : \u03b1 \u2243 \u03b2\nh\u2081 : Continuous \u2191e\nh\u2082 : IsOpenMap \u2191e\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 IsOpen (e.invFun \u207b\u00b9' s)\n[PROOFSTEP]\nconvert \u2190 h\u2082 s hs using 1\n[GOAL]\ncase h.e'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\ne : \u03b1 \u2243 \u03b2\nh\u2081 : Continuous \u2191e\nh\u2082 : IsOpenMap \u2191e\ns : Set \u03b1\nhs : IsOpen s\n\u22a2 \u2191e '' s = e.invFun \u207b\u00b9' s\n[PROOFSTEP]\napply e.image_eq_preimage\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b2 \u2192 \u03b3\nx : \u03b1\n\u22a2 range \u2191h \u2208 \ud835\udcdd (\u2191h x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b3 \u2192 \u03b1\n\u22a2 IsOpenMap (\u2191h \u2218 f) \u2194 IsOpenMap f\n[PROOFSTEP]\nrefine' \u27e8_, fun hf => h.isOpenMap.comp hf\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b3 \u2192 \u03b1\n\u22a2 IsOpenMap (\u2191h \u2218 f) \u2192 IsOpenMap f\n[PROOFSTEP]\nintro hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b3 \u2192 \u03b1\nhf : IsOpenMap (\u2191h \u2218 f)\n\u22a2 IsOpenMap f\n[PROOFSTEP]\nrw [\u2190 Function.comp.left_id f, \u2190 h.symm_comp_self, Function.comp.assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b3 \u2192 \u03b1\nhf : IsOpenMap (\u2191h \u2218 f)\n\u22a2 IsOpenMap (\u2191(Homeomorph.symm h) \u2218 \u2191h \u2218 f)\n[PROOFSTEP]\nexact h.symm.isOpenMap.comp hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b2 \u2192 \u03b3\n\u22a2 IsOpenMap (f \u2218 \u2191h) \u2194 IsOpenMap f\n[PROOFSTEP]\nrefine' \u27e8_, fun hf => hf.comp h.isOpenMap\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b2 \u2192 \u03b3\n\u22a2 IsOpenMap (f \u2218 \u2191h) \u2192 IsOpenMap f\n[PROOFSTEP]\nintro hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b2 \u2192 \u03b3\nhf : IsOpenMap (f \u2218 \u2191h)\n\u22a2 IsOpenMap f\n[PROOFSTEP]\nrw [\u2190 Function.comp.right_id f, \u2190 h.self_comp_symm, \u2190 Function.comp.assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : TopologicalSpace \u03b4\nh : \u03b1 \u2243\u209c \u03b2\nf : \u03b2 \u2192 \u03b3\nhf : IsOpenMap (f \u2218 \u2191h)\n\u22a2 IsOpenMap ((f \u2218 \u2191h) \u2218 \u2191(Homeomorph.symm h))\n[PROOFSTEP]\nexact hf.comp h.symm.isOpenMap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\n\u22a2 Continuous (Equiv.piEquivPiSubtypeProd p \u03b2).toFun\n[PROOFSTEP]\napply Continuous.prod_mk\n[GOAL]\ncase hf\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\n\u22a2 Continuous fun x x_1 => x \u2191x_1\n[PROOFSTEP]\nexact continuous_pi fun j => continuous_apply j.1\n[GOAL]\ncase hg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\n\u22a2 Continuous fun x x_1 => x \u2191x_1\n[PROOFSTEP]\nexact continuous_pi fun j => continuous_apply j.1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\nj : \u03b9\n\u22a2 Continuous fun a => Equiv.invFun (Equiv.piEquivPiSubtypeProd p \u03b2) a j\n[PROOFSTEP]\ndsimp only [Equiv.piEquivPiSubtypeProd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\nj : \u03b9\n\u22a2 Continuous fun a => if h : p j then Prod.fst a { val := j, property := h } else Prod.snd a { val := j, property := h }\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\nj : \u03b9\nh\u271d : p j\n\u22a2 Continuous fun a => Prod.fst a { val := j, property := h\u271d }\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\np : \u03b9 \u2192 Prop\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03b2 i)\ninst\u271d : DecidablePred p\nj : \u03b9\nh\u271d : \u00acp j\n\u22a2 Continuous fun a => Prod.snd a { val := j, property := h\u271d }\n[PROOFSTEP]\nexacts [(continuous_apply _).comp continuous_fst, (continuous_apply _).comp continuous_snd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\ni : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\n\u22a2 Continuous fun a => Equiv.invFun (Equiv.piSplitAt i \u03b2) a j\n[PROOFSTEP]\ndsimp only [Equiv.piSplitAt]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\ni : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\n\u22a2 Continuous fun a => if h : j = i then (_ : i = j) \u25b8 a.fst else Prod.snd a { val := j, property := h }\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\ni : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\nh : j = i\n\u22a2 Continuous fun a => (_ : i = j) \u25b8 a.fst\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\ni : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\nh : \u00acj = i\n\u22a2 Continuous fun a => Prod.snd a { val := j, property := h }\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\n\u22a2 Continuous fun a => (_ : j = j) \u25b8 a.fst\ncase neg\n\u03b1 : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\u271d\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : TopologicalSpace \u03b4\n\u03b9 : Type u_5\ninst\u271d\u00b9 : DecidableEq \u03b9\ni : \u03b9\n\u03b2 : \u03b9 \u2192 Type u_6\ninst\u271d : (j : \u03b9) \u2192 TopologicalSpace (\u03b2 j)\nj : \u03b9\nh : \u00acj = i\n\u22a2 Continuous fun a => Prod.snd a { val := j, property := h }\n[PROOFSTEP]\nexacts [continuous_fst, (continuous_apply _).comp continuous_snd]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2243 \u03b2\nhf : Inducing \u2191f\n\u22a2 Continuous\n    (\u2191f \u2218\n      { toFun := f.toFun, invFun := f.invFun, left_inv := (_ : Function.LeftInverse f.invFun f.toFun),\n          right_inv := (_ : Function.RightInverse f.invFun f.toFun) }.invFun)\n[PROOFSTEP]\nsimpa using continuous_id\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2243 \u03b2\nhf : Continuous \u2191f\n\u22a2 Continuous \u2191f.symm\n[PROOFSTEP]\nrw [continuous_iff_isClosed]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2243 \u03b2\nhf : Continuous \u2191f\n\u22a2 \u2200 (s : Set \u03b1), IsClosed s \u2192 IsClosed (\u2191f.symm \u207b\u00b9' s)\n[PROOFSTEP]\nintro C hC\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2243 \u03b2\nhf : Continuous \u2191f\nC : Set \u03b1\nhC : IsClosed C\n\u22a2 IsClosed (\u2191f.symm \u207b\u00b9' C)\n[PROOFSTEP]\nhave hC' : IsClosed (f '' C) := (hC.isCompact.image hf).isClosed\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : CompactSpace \u03b1\ninst\u271d : T2Space \u03b2\nf : \u03b1 \u2243 \u03b2\nhf : Continuous \u2191f\nC : Set \u03b1\nhC : IsClosed C\nhC' : IsClosed (\u2191f '' C)\n\u22a2 IsClosed (\u2191f.symm \u207b\u00b9' C)\n[PROOFSTEP]\nrwa [Equiv.image_eq_preimage] at hC' \n", "meta": {"mathlib_filename": "Mathlib.Topology.Homeomorph", "llama_tokens": 9687, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649233, "lm_q2_score": 0.6688802603710085, "lm_q1q2_score": 0.5181035356792908}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\nr : \u211d\n\u22a2 closedBall x r = \u2191(ContinuousLinearEquiv.symm toEuclidean) '' Metric.closedBall (\u2191toEuclidean x) r\n[PROOFSTEP]\nrw [toEuclidean.image_symm_eq_preimage, closedBall_eq_preimage]\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\nr : \u211d\n\u22a2 IsCompact (closedBall x r)\n[PROOFSTEP]\nrw [closedBall_eq_image]\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\nr : \u211d\n\u22a2 IsCompact (\u2191(ContinuousLinearEquiv.symm toEuclidean) '' Metric.closedBall (\u2191toEuclidean x) r)\n[PROOFSTEP]\nexact (isCompact_closedBall _ _).image toEuclidean.symm.continuous\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\nr : \u211d\nh : r \u2260 0\n\u22a2 closure (ball x r) = closedBall x r\n[PROOFSTEP]\nrw [ball_eq_preimage, \u2190 toEuclidean.preimage_closure, closure_ball (toEuclidean x) h, closedBall_eq_preimage]\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nR : \u211d\ns : Set E\nx : E\nhR : 0 < R\nhs : IsClosed s\nh : s \u2286 ball x R\n\u22a2 \u2203 r, r \u2208 Ioo 0 R \u2227 s \u2286 ball x r\n[PROOFSTEP]\nrw [ball_eq_preimage, \u2190 image_subset_iff] at h \n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nR : \u211d\ns : Set E\nx : E\nhR : 0 < R\nhs : IsClosed s\nh : \u2191toEuclidean '' s \u2286 Metric.ball (\u2191toEuclidean x) R\n\u22a2 \u2203 r, r \u2208 Ioo 0 R \u2227 s \u2286 ball x r\n[PROOFSTEP]\nrcases exists_pos_lt_subset_ball hR (toEuclidean.isClosed_image.2 hs) h with \u27e8r, hr, hsr\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nR : \u211d\ns : Set E\nx : E\nhR : 0 < R\nhs : IsClosed s\nh : \u2191toEuclidean '' s \u2286 Metric.ball (\u2191toEuclidean x) R\nr : \u211d\nhr : r \u2208 Ioo 0 R\nhsr : \u2191toEuclidean '' s \u2286 Metric.ball (\u2191toEuclidean x) r\n\u22a2 \u2203 r, r \u2208 Ioo 0 R \u2227 s \u2286 ball x r\n[PROOFSTEP]\nexact \u27e8r, hr, image_subset_iff.1 hsr\u27e9\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r) (closedBall x)\n[PROOFSTEP]\nrw [toEuclidean.toHomeomorph.nhds_eq_comap x]\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis\n    (Filter.comap (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean))\n      (\ud835\udcdd (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean) x)))\n    (fun r => 0 < r) (closedBall x)\n[PROOFSTEP]\nexact Metric.nhds_basis_closedBall.comap _\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r) (ball x)\n[PROOFSTEP]\nrw [toEuclidean.toHomeomorph.nhds_eq_comap x]\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalAddGroup E\ninst\u271d\u00b3 : T2Space E\ninst\u271d\u00b2 : Module \u211d E\ninst\u271d\u00b9 : ContinuousSMul \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\n\u22a2 Filter.HasBasis\n    (Filter.comap (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean))\n      (\ud835\udcdd (\u2191(ContinuousLinearEquiv.toHomeomorph toEuclidean) x)))\n    (fun r => 0 < r) (ball x)\n[PROOFSTEP]\nexact Metric.nhds_basis_ball.comap _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : TopologicalSpace E\ninst\u271d\u2079 : TopologicalAddGroup E\ninst\u271d\u2078 : T2Space E\ninst\u271d\u2077 : Module \u211d E\ninst\u271d\u2076 : ContinuousSMul \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nG : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\ninst\u271d : FiniteDimensional \u211d G\nf g : F \u2192 G\nn : \u2115\u221e\nhf : ContDiff \u211d n f\nhg : ContDiff \u211d n g\nh : \u2200 (x : F), f x \u2260 g x\n\u22a2 ContDiff \u211d n fun x => Euclidean.dist (f x) (g x)\n[PROOFSTEP]\nsimp only [Euclidean.dist]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : TopologicalSpace E\ninst\u271d\u2079 : TopologicalAddGroup E\ninst\u271d\u2078 : T2Space E\ninst\u271d\u2077 : Module \u211d E\ninst\u271d\u2076 : ContinuousSMul \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nG : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\ninst\u271d : FiniteDimensional \u211d G\nf g : F \u2192 G\nn : \u2115\u221e\nhf : ContDiff \u211d n f\nhg : ContDiff \u211d n g\nh : \u2200 (x : F), f x \u2260 g x\n\u22a2 ContDiff \u211d n fun x => Dist.dist (\u2191toEuclidean (f x)) (\u2191toEuclidean (g x))\n[PROOFSTEP]\napply @ContDiff.dist \u211d\n[GOAL]\ncase hf\nE : Type u_1\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : TopologicalSpace E\ninst\u271d\u2079 : TopologicalAddGroup E\ninst\u271d\u2078 : T2Space E\ninst\u271d\u2077 : Module \u211d E\ninst\u271d\u2076 : ContinuousSMul \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nG : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\ninst\u271d : FiniteDimensional \u211d G\nf g : F \u2192 G\nn : \u2115\u221e\nhf : ContDiff \u211d n f\nhg : ContDiff \u211d n g\nh : \u2200 (x : F), f x \u2260 g x\n\u22a2 ContDiff \u211d n fun y => \u2191toEuclidean (f y)\ncase hg\nE : Type u_1\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : TopologicalSpace E\ninst\u271d\u2079 : TopologicalAddGroup E\ninst\u271d\u2078 : T2Space E\ninst\u271d\u2077 : Module \u211d E\ninst\u271d\u2076 : ContinuousSMul \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nG : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\ninst\u271d : FiniteDimensional \u211d G\nf g : F \u2192 G\nn : \u2115\u221e\nhf : ContDiff \u211d n f\nhg : ContDiff \u211d n g\nh : \u2200 (x : F), f x \u2260 g x\n\u22a2 ContDiff \u211d n fun y => \u2191toEuclidean (g y)\ncase hne\nE : Type u_1\ninst\u271d\u00b9\u00b9 : AddCommGroup E\ninst\u271d\u00b9\u2070 : TopologicalSpace E\ninst\u271d\u2079 : TopologicalAddGroup E\ninst\u271d\u2078 : T2Space E\ninst\u271d\u2077 : Module \u211d E\ninst\u271d\u2076 : ContinuousSMul \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\nF : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\nG : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \u211d G\ninst\u271d : FiniteDimensional \u211d G\nf g : F \u2192 G\nn : \u2115\u221e\nhf : ContDiff \u211d n f\nhg : ContDiff \u211d n g\nh : \u2200 (x : F), f x \u2260 g x\n\u22a2 \u2200 (x : F), \u2191toEuclidean (f x) \u2260 \u2191toEuclidean (g x)\n[PROOFSTEP]\nexacts [(@toEuclidean G _ _ _ _ _ _ _).contDiff.comp hf, (@toEuclidean G _ _ _ _ _ _ _).contDiff.comp hg, fun x =>\n  toEuclidean.injective.ne (h x)]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.EuclideanDist", "llama_tokens": 3361, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.5181035287181152}}
{"text": "[GOAL]\nX : Type u_1\n\u03b9 : Type u_2\nY : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (Y i)\ninst\u271d : Nonempty X\n\u22a2 Function.Injective fun g => comp (sigmaMk g.fst) g.snd\n[PROOFSTEP]\nrintro \u27e8i, g\u27e9 \u27e8i', g'\u27e9 h\n[GOAL]\ncase mk.mk\nX : Type u_1\n\u03b9 : Type u_2\nY : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (Y i)\ninst\u271d : Nonempty X\ni : \u03b9\ng : C(X, Y i)\ni' : \u03b9\ng' : C(X, Y i')\nh :\n  (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g } =\n    (fun g => comp (sigmaMk g.fst) g.snd) { fst := i', snd := g' }\n\u22a2 { fst := i, snd := g } = { fst := i', snd := g' }\n[PROOFSTEP]\nobtain \u27e8rfl, hg\u27e9 : i = i' \u2227 HEq (\u21d1g) (\u21d1g') := Function.eq_of_sigmaMk_comp <| congr_arg FunLike.coe h\n[GOAL]\ncase mk.mk.intro\nX : Type u_1\n\u03b9 : Type u_2\nY : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (Y i)\ninst\u271d : Nonempty X\ni : \u03b9\ng g' : C(X, Y i)\nh :\n  (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g } =\n    (fun g => comp (sigmaMk g.fst) g.snd) { fst := i, snd := g' }\nhg : HEq \u2191g \u2191g'\n\u22a2 { fst := i, snd := g } = { fst := i, snd := g' }\n[PROOFSTEP]\nsimpa using hg\n", "meta": {"mathlib_filename": "Mathlib.Topology.ContinuousFunction.Sigma", "llama_tokens": 597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529376, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.5180622041081412}}
{"text": "[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\n\u22a2 \u2045x, x\u2046 = 0\n[PROOFSTEP]\nsimp only [bracket_def]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) x = 0\n[PROOFSTEP]\nrefine' x.induction_on _ _ _\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) 0 = 0\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, eq_self_iff_true, LinearMap.zero_apply]\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\n\u22a2 \u2200 (x : A) (y : L), \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y)) (x \u2297\u209c[R] y) = 0\n[PROOFSTEP]\nintro a l\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\na : A\nl : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a \u2297\u209c[R] l)) (a \u2297\u209c[R] l) = 0\n[PROOFSTEP]\nsimp only [bracket'_tmul, TensorProduct.tmul_zero, eq_self_iff_true, lie_self]\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx : A \u2297[R] L\n\u22a2 \u2200 (x y : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) x = 0 \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) y = 0 \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y)) (x + y) = 0\n[PROOFSTEP]\nintro z\u2081 z\u2082 h\u2081 h\u2082\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (z\u2081 + z\u2082)) (z\u2081 + z\u2082) = 0\n[PROOFSTEP]\nsuffices bracket' R A L z\u2081 z\u2082 + bracket' R A L z\u2082 z\u2081 = 0 by\n  rw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h\u2081, h\u2082, zero_add, add_zero,\n    add_comm, this]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\nthis : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2081 = 0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (z\u2081 + z\u2082)) (z\u2081 + z\u2082) = 0\n[PROOFSTEP]\nrw [LinearMap.map_add, LinearMap.map_add, LinearMap.add_apply, LinearMap.add_apply, h\u2081, h\u2082, zero_add, add_zero,\n  add_comm, this]\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2081 = 0\n[PROOFSTEP]\nrefine' z\u2081.induction_on _ _ _\n[GOAL]\ncase refine'_3.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) 0 = 0\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n[GOAL]\ncase refine'_3.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\n\u22a2 \u2200 (x : A) (y : L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y)) z\u2082 +\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) (x \u2297\u209c[R] y) =\n      0\n[PROOFSTEP]\nintro a\u2081 l\u2081\n[GOAL]\ncase refine'_3.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z\u2082 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\n[PROOFSTEP]\nrefine' z\u2082.induction_on _ _ _\n[GOAL]\ncase refine'_3.refine'_2.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) 0 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n[GOAL]\ncase refine'_3.refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2200 (x : A) (y : L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x \u2297\u209c[R] y) +\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y)) (a\u2081 \u2297\u209c[R] l\u2081) =\n      0\n[PROOFSTEP]\nintro a\u2082 l\u2082\n[GOAL]\ncase refine'_3.refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082) +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\n[PROOFSTEP]\nsimp only [\u2190 lie_skew l\u2082 l\u2081, mul_comm a\u2081 a\u2082, TensorProduct.tmul_neg, bracket'_tmul, add_right_neg]\n[GOAL]\ncase refine'_3.refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2200 (x y : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) x +\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) (a\u2081 \u2297\u209c[R] l\u2081) =\n        0 \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (a\u2081 \u2297\u209c[R] l\u2081) =\n          0 \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x + y) +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y)) (a\u2081 \u2297\u209c[R] l\u2081) =\n          0\n[PROOFSTEP]\nintro y\u2081 y\u2082 hy\u2081 hy\u2082\n[GOAL]\ncase refine'_3.refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\na\u2081 : A\nl\u2081 : L\ny\u2081 y\u2082 : A \u2297[R] L\nhy\u2081 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y\u2081 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y\u2081) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\nhy\u2082 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y\u2082 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y\u2082) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (y\u2081 + y\u2082) +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (y\u2081 + y\u2082)) (a\u2081 \u2297\u209c[R] l\u2081) =\n    0\n[PROOFSTEP]\nsimp only [hy\u2081, hy\u2082, add_add_add_comm, add_zero, LinearMap.add_apply, LinearMap.map_add]\n[GOAL]\ncase refine'_3.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\n\u22a2 \u2200 (x y : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) x = 0 \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) y = 0 \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y)) z\u2082 +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) (x + y) =\n          0\n[PROOFSTEP]\nintro y\u2081 y\u2082 hy\u2081 hy\u2082\n[GOAL]\ncase refine'_3.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx z\u2081 z\u2082 : A \u2297[R] L\nh\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2081) z\u2081 = 0\nh\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) z\u2082 = 0\ny\u2081 y\u2082 : A \u2297[R] L\nhy\u2081 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y\u2081) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) y\u2081 = 0\nhy\u2082 : \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y\u2082) z\u2082 + \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) y\u2082 = 0\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (y\u2081 + y\u2082)) z\u2082 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) z\u2082) (y\u2081 + y\u2082) =\n    0\n[PROOFSTEP]\nsimp only [add_add_add_comm, hy\u2081, hy\u2082, add_zero, LinearMap.add_apply, LinearMap.map_add]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2045x, \u2045y, z\u2046\u2046 = \u2045\u2045x, y\u2046, z\u2046 + \u2045y, \u2045x, z\u2046\u2046\n[PROOFSTEP]\nsimp only [bracket_def]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) z)\n[PROOFSTEP]\nrefine' x.induction_on _ _ _\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) z)\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2200 (x : A) (y_1 : L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y_1)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y_1)) y)) z +\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y_1)) z)\n[PROOFSTEP]\nintro a\u2081 l\u2081\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nrefine' y.induction_on _ _ _\n[GOAL]\ncase refine'_2.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) 0)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) 0) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nsimp only [LinearMap.map_zero, add_zero, LinearMap.zero_apply]\n[GOAL]\ncase refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2200 (x : A) (y : L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y)) z) =\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x \u2297\u209c[R] y)))\n          z +\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x \u2297\u209c[R] y))\n          (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nintro a\u2082 l\u2082\n[GOAL]\ncase refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nrefine' z.induction_on _ _ _\n[GOAL]\ncase refine'_2.refine'_2.refine'_1\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) 0) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        0 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) 0)\n[PROOFSTEP]\nrw [LinearMap.map_zero, LinearMap.map_zero, LinearMap.map_zero, LinearMap.map_zero, add_zero]\n[GOAL]\ncase refine'_2.refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\n\u22a2 \u2200 (x : A) (y : L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) (x \u2297\u209c[R] y)) =\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n          (x \u2297\u209c[R] y) +\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n          (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x \u2297\u209c[R] y))\n[PROOFSTEP]\nintro a\u2083 l\u2083\n[GOAL]\ncase refine'_2.refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\na\u2083 : A\nl\u2083 : L\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) (a\u2083 \u2297\u209c[R] l\u2083)) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        (a\u2083 \u2297\u209c[R] l\u2083) +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2083 \u2297\u209c[R] l\u2083))\n[PROOFSTEP]\nsimp only [bracket'_tmul]\n[GOAL]\ncase refine'_2.refine'_2.refine'_2\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\na\u2083 : A\nl\u2083 : L\n\u22a2 (a\u2081 * (a\u2082 * a\u2083)) \u2297\u209c[R] \u2045l\u2081, \u2045l\u2082, l\u2083\u2046\u2046 = (a\u2081 * a\u2082 * a\u2083) \u2297\u209c[R] \u2045\u2045l\u2081, l\u2082\u2046, l\u2083\u2046 + (a\u2082 * (a\u2081 * a\u2083)) \u2297\u209c[R] \u2045l\u2082, \u2045l\u2081, l\u2083\u2046\u2046\n[PROOFSTEP]\nrw [mul_left_comm a\u2082 a\u2081 a\u2083, mul_assoc, leibniz_lie, TensorProduct.tmul_add]\n[GOAL]\ncase refine'_2.refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\n\u22a2 \u2200 (x y : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n          (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) x) =\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n                (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n            x +\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) x) \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) y) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n                  (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n              y +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y) \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) (x + y)) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n                  (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n              (x + y) +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x + y))\n[PROOFSTEP]\nintro u\u2081 u\u2082 h\u2081 h\u2082\n[GOAL]\ncase refine'_2.refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\na\u2082 : A\nl\u2082 : L\nu\u2081 u\u2082 : A \u2297[R] L\nh\u2081 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) u\u2081) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        u\u2081 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) u\u2081)\nh\u2082 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) u\u2082) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        u\u2082 +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) u\u2082)\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082)) (u\u2081 + u\u2082)) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (a\u2082 \u2297\u209c[R] l\u2082)))\n        (u\u2081 + u\u2082) +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2082 \u2297\u209c[R] l\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (u\u2081 + u\u2082))\n[PROOFSTEP]\nrw [map_add, map_add, map_add, map_add, map_add, h\u2081, h\u2082, add_add_add_comm]\n[GOAL]\ncase refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\n\u22a2 \u2200 (x y : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) z) =\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) x))\n            z +\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z) \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) y))\n              z +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y)\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z) \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y)) z) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n                  (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (x + y)))\n              z +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y))\n              (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nintro u\u2081 u\u2082 h\u2081 h\u2082\n[GOAL]\ncase refine'_2.refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\na\u2081 : A\nl\u2081 : L\nu\u2081 u\u2082 : A \u2297[R] L\nh\u2081 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2081) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) u\u2081)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2081) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\nh\u2082 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2082) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) u\u2082)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2082) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081))\n      (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (u\u2081 + u\u2082)) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L)\n            (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) (u\u2081 + u\u2082)))\n        z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (u\u2081 + u\u2082))\n        (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (a\u2081 \u2297\u209c[R] l\u2081)) z)\n[PROOFSTEP]\nrw [map_add, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, map_add, LinearMap.add_apply, h\u2081, h\u2082,\n  add_add_add_comm]\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2200 (x y_1 : A \u2297[R] L),\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) y)) z +\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) x) z) \u2192\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y_1) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y_1) y)) z +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y_1) z) \u2192\n        \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n          \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) y)) z +\n            \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (x + y_1)) z)\n[PROOFSTEP]\nintro u\u2081 u\u2082 h\u2081 h\u2082\n[GOAL]\ncase refine'_3\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z u\u2081 u\u2082 : A \u2297[R] L\nh\u2081 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2081) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2081) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2081) z)\nh\u2082 :\n  \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2082) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2082) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) u\u2082) z)\n\u22a2 \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (u\u2081 + u\u2082)) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) z) =\n    \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (u\u2081 + u\u2082)) y)) z +\n      \u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) y) (\u2191(\u2191(LieAlgebra.ExtendScalars.bracket' R A L) (u\u2081 + u\u2082)) z)\n[PROOFSTEP]\nrw [map_add, LinearMap.add_apply, LinearMap.add_apply, LinearMap.add_apply, map_add, map_add, LinearMap.add_apply, h\u2081,\n  h\u2082, add_add_add_comm]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2045x + y, z\u2046 = \u2045x, z\u2046 + \u2045y, z\u2046\n[PROOFSTEP]\nsimp only [bracket_def, LinearMap.add_apply, LinearMap.map_add]\n[GOAL]\nR : Type u\nA : Type w\nL : Type v\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : CommRing A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : LieRing L\ninst\u271d : LieAlgebra R L\nx y z : A \u2297[R] L\n\u22a2 \u2045x, y + z\u2046 = \u2045x, y\u2046 + \u2045x, z\u2046\n[PROOFSTEP]\nsimp only [bracket_def, LinearMap.map_add]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Lie.BaseChange", "llama_tokens": 14217, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744806385542, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5178600673575399}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nx : E\n\u22a2 0 \u2264 \u2191re (inner x (\u2191T x))\n[PROOFSTEP]\nrw [inner_re_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nx : E\n\u22a2 0 \u2264 \u2191re (inner (\u2191T x) x)\n[PROOFSTEP]\nexact hT.inner_nonneg_left x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\n\u22a2 IsPositive 0\n[PROOFSTEP]\nrefine' \u27e8isSelfAdjoint_zero _, fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nx : E\n\u22a2 0 \u2264 reApplyInnerSelf 0 x\n[PROOFSTEP]\nchange 0 \u2264 re \u27ea_, _\u27eb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nx : E\n\u22a2 0 \u2264 \u2191re (inner (\u21910 x) x)\n[PROOFSTEP]\nrw [zero_apply, inner_zero_left, ZeroHomClass.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT S : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nhS : IsPositive S\n\u22a2 IsPositive (T + S)\n[PROOFSTEP]\nrefine' \u27e8hT.isSelfAdjoint.add hS.isSelfAdjoint, fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT S : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n\u22a2 0 \u2264 reApplyInnerSelf (T + S) x\n[PROOFSTEP]\nrw [reApplyInnerSelf, add_apply, inner_add_left, map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT S : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n\u22a2 0 \u2264 \u2191re (inner (\u2191T x) x) + \u2191re (inner (\u2191S x) x)\n[PROOFSTEP]\nexact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 IsPositive (comp S (comp T (\u2191adjoint S)))\n[PROOFSTEP]\nrefine' \u27e8hT.isSelfAdjoint.conj_adjoint S, fun x => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nS : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 0 \u2264 reApplyInnerSelf (comp S (comp T (\u2191adjoint S))) x\n[PROOFSTEP]\nrw [reApplyInnerSelf, comp_apply, \u2190 adjoint_inner_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nS : E \u2192L[\ud835\udd5c] F\nx : F\n\u22a2 0 \u2264 \u2191re (inner (\u2191(comp T (\u2191adjoint S)) x) (\u2191(\u2191adjoint S) x))\n[PROOFSTEP]\nexact hT.inner_nonneg_left _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nS : F \u2192L[\ud835\udd5c] E\n\u22a2 IsPositive (comp (\u2191adjoint S) (comp T S))\n[PROOFSTEP]\nconvert hT.conj_adjoint (S\u2020)\n[GOAL]\ncase h.e'_7.h.e'_24.h.e'_24\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nS : F \u2192L[\ud835\udd5c] E\n\u22a2 S = \u2191adjoint (\u2191adjoint S)\n[PROOFSTEP]\nrw [adjoint_adjoint]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nU : Submodule \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsPositive\n    (comp (Submodule.subtypeL U)\n      (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))\n[PROOFSTEP]\nhave := hT.conj_adjoint (U.subtypeL \u2218L orthogonalProjection U)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nU : Submodule \ud835\udd5c E\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\ninst\u271d : CompleteSpace { x // x \u2208 U }\nthis :\n  IsPositive\n    (comp (comp (Submodule.subtypeL U) (orthogonalProjection U))\n      (comp T (\u2191adjoint (comp (Submodule.subtypeL U) (orthogonalProjection U)))))\n\u22a2 IsPositive\n    (comp (Submodule.subtypeL U)\n      (comp (orthogonalProjection U) (comp T (comp (Submodule.subtypeL U) (orthogonalProjection U)))))\n[PROOFSTEP]\nrwa [(orthogonalProjection_isSelfAdjoint U).adjoint_eq] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\n\u22a2 IsPositive (comp (orthogonalProjection U) (comp T (Submodule.subtypeL U)))\n[PROOFSTEP]\nhave := hT.conj_adjoint (orthogonalProjection U : E \u2192L[\ud835\udd5c] U)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsPositive T\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\nthis : IsPositive (comp (orthogonalProjection U) (comp T (\u2191adjoint (orthogonalProjection U))))\n\u22a2 IsPositive (comp (orthogonalProjection U) (comp T (Submodule.subtypeL U)))\n[PROOFSTEP]\nrwa [U.adjoint_orthogonalProjection] at this \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : CompleteSpace F\nE' : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : InnerProductSpace \u2102 E'\ninst\u271d : CompleteSpace E'\nT : E' \u2192L[\u2102] E'\n\u22a2 IsPositive T \u2194 \u2200 (x : E'), \u2191(\u2191re (inner (\u2191T x) x)) = inner (\u2191T x) x \u2227 0 \u2264 \u2191re (inner (\u2191T x) x)\n[PROOFSTEP]\nsimp_rw [IsPositive, forall_and, isSelfAdjoint_iff_isSymmetric, LinearMap.isSymmetric_iff_inner_map_self_real,\n  conj_eq_iff_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u2074 : CompleteSpace E\ninst\u271d\u00b3 : CompleteSpace F\nE' : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : InnerProductSpace \u2102 E'\ninst\u271d : CompleteSpace E'\nT : E' \u2192L[\u2102] E'\n\u22a2 ((\u2200 (v : E'), \u2191(\u2191re (inner (\u2191\u2191T v) v)) = inner (\u2191\u2191T v) v) \u2227 \u2200 (x : E'), 0 \u2264 reApplyInnerSelf T x) \u2194\n    (\u2200 (x : E'), \u2191(\u2191re (inner (\u2191T x) x)) = inner (\u2191T x) x) \u2227 \u2200 (x : E'), 0 \u2264 \u2191re (inner (\u2191T x) x)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Positive", "llama_tokens": 3866, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.685949467848392, "lm_q1q2_score": 0.5178335335126254}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nH : Type u_2\ninst\u271d\u2076 : TopologicalSpace H\nE : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nI : ModelWithCorners \ud835\udd5c E H\nR : Type u_4\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : ChartedSpace H R\ninst\u271d : SmoothRing I R\n\u22a2 Smooth I I fun a => -a\n[PROOFSTEP]\nsimpa only [neg_one_mul] using @smooth_mul_left \ud835\udd5c _ H _ E _ _ I R _ _ _ _ (-1)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nsrc\u271d : LieAddGroup \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udd5c := normedSpaceLieAddGroup\n\u22a2 Smooth (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) fun p => p.fst * p.snd\n[PROOFSTEP]\nrw [smooth_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nsrc\u271d : LieAddGroup \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udd5c := normedSpaceLieAddGroup\n\u22a2 (Continuous fun p => p.fst * p.snd) \u2227\n    \u2200 (x : \ud835\udd5c \u00d7 \ud835\udd5c) (y : \ud835\udd5c),\n      ContDiffOn \ud835\udd5c \u22a4\n        (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) y) \u2218\n          (fun p => p.fst * p.snd) \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x)))\n        ((extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x).target \u2229\n          \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x)) \u207b\u00b9'\n            ((fun p => p.fst * p.snd) \u207b\u00b9' (extChartAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) y).source))\n[PROOFSTEP]\nrefine' \u27e8continuous_mul, fun x y => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nsrc\u271d : LieAddGroup \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udd5c := normedSpaceLieAddGroup\nx : \ud835\udd5c \u00d7 \ud835\udd5c\ny : \ud835\udd5c\n\u22a2 ContDiffOn \ud835\udd5c \u22a4\n    (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) y) \u2218\n      (fun p => p.fst * p.snd) \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x)))\n    ((extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x).target \u2229\n      \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udcd8(\ud835\udd5c, \ud835\udd5c)) x)) \u207b\u00b9'\n        ((fun p => p.fst * p.snd) \u207b\u00b9' (extChartAt \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) y).source))\n[PROOFSTEP]\nsimp only [Prod.mk.eta, mfld_simps]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nsrc\u271d : LieAddGroup \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udd5c := normedSpaceLieAddGroup\nx : \ud835\udd5c \u00d7 \ud835\udd5c\ny : \ud835\udd5c\n\u22a2 ContDiffOn \ud835\udd5c \u22a4 (fun p => p.fst * p.snd) Set.univ\n[PROOFSTEP]\nrw [contDiffOn_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d : NontriviallyNormedField \ud835\udd5c\nsrc\u271d : LieAddGroup \ud835\udcd8(\ud835\udd5c, \ud835\udd5c) \ud835\udd5c := normedSpaceLieAddGroup\nx : \ud835\udd5c \u00d7 \ud835\udd5c\ny : \ud835\udd5c\n\u22a2 ContDiff \ud835\udd5c \u22a4 fun p => p.fst * p.snd\n[PROOFSTEP]\nexact contDiff_mul\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Manifold.Algebra.Structures", "llama_tokens": 1214, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467770088162, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.5177199813300628}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nr : \u2102\n\u22a2 fourierIntegral e \u03bc L (r \u2022 f) = r \u2022 fourierIntegral e \u03bc L f\n[PROOFSTEP]\next1 w\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nr : \u2102\nw : W\n\u22a2 fourierIntegral e \u03bc L (r \u2022 f) w = (r \u2022 fourierIntegral e \u03bc L f) w\n[PROOFSTEP]\nsimp only [Pi.smul_apply, fourierIntegral, \u2190 integral_smul]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nr : \u2102\nw : W\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 r \u2022 f v \u2202\u03bc =\n    \u222b (a : V), r \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) w))) \u2022 f a \u2202\u03bc\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_f\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nr : \u2102\nw : W\n\u22a2 (fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 r \u2022 f v) = fun a =>\n    r \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) w))) \u2022 f a\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h.e_f.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nr : \u2102\nw : W\nx\u271d : V\n\u22a2 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L x\u271d) w))) \u2022 r \u2022 f x\u271d = r \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L x\u271d) w))) \u2022 f x\u271d\n[PROOFSTEP]\nrw [smul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nw : W\n\u22a2 \u2016fourierIntegral e \u03bc L f w\u2016 \u2264 \u222b (v : V), \u2016f v\u2016 \u2202\u03bc\n[PROOFSTEP]\nrefine' (norm_integral_le_integral_norm _).trans (le_of_eq _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2076 : AddCommGroup V\ninst\u271d\u2075 : Module \ud835\udd5c V\ninst\u271d\u2074 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b3 : AddCommGroup W\ninst\u271d\u00b2 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nw : W\n\u22a2 \u222b (a : V), \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) w))) \u2022 f a\u2016 \u2202\u03bc = \u222b (v : V), \u2016f v\u2016 \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [norm_smul, Complex.norm_eq_abs, abs_coe_circle, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\n\u22a2 fourierIntegral e \u03bc L (f \u2218 fun v => v + v\u2080) = fun w =>\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 fourierIntegral e \u03bc L f w\n[PROOFSTEP]\next1 w\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 fourierIntegral e \u03bc L (f \u2218 fun v => v + v\u2080) w = \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 fourierIntegral e \u03bc L f w\n[PROOFSTEP]\ndsimp only [fourierIntegral, Function.comp_apply]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f (v + v\u2080) \u2202\u03bc =\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc\n[PROOFSTEP]\nconv in L _ => rw [\u2190 add_sub_cancel v v\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n| \u2191L v\n[PROOFSTEP]\nrw [\u2190 add_sub_cancel v v\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n| \u2191L v\n[PROOFSTEP]\nrw [\u2190 add_sub_cancel v v\u2080]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n| \u2191L v\n[PROOFSTEP]\nrw [\u2190 add_sub_cancel v v\u2080]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L (v + v\u2080 - v\u2080)) w))) \u2022 f (v + v\u2080) \u2202\u03bc =\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc\n[PROOFSTEP]\nrw [integral_add_right_eq_self fun v : V => e[-L (v - v\u2080) w] \u2022 f v]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (x : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L (x - v\u2080)) w))) \u2022 f x \u2202\u03bc =\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (x : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L (x - v\u2080)) w))) \u2022 f x \u2202\u03bc =\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_smul]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\n\u22a2 \u222b (x : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L (x - v\u2080)) w))) \u2022 f x \u2202\u03bc =\n    \u222b (a : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) w))) \u2022 f a \u2202\u03bc\n[PROOFSTEP]\ncongr 1 with v\n[GOAL]\ncase h.e_f.h\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module \ud835\udd5c V\ninst\u271d\u2076 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2075 : AddCommGroup W\ninst\u271d\u2074 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u2102 E\ninst\u271d\u00b9 : MeasurableAdd V\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\ninst\u271d : Measure.IsAddRightInvariant \u03bc\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nf : V \u2192 E\nv\u2080 : V\nw : W\nv : V\n\u22a2 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L (v - v\u2080)) w))) \u2022 f v =\n    \u2191(\u2191e (\u2191Multiplicative.ofAdd (\u2191(\u2191L v\u2080) w))) \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\n[PROOFSTEP]\nrw [\u2190 smul_assoc, smul_eq_mul, \u2190 Submonoid.coe_mul, \u2190 e.map_mul, \u2190 ofAdd_add, \u2190 LinearMap.neg_apply, \u2190 sub_eq_add_neg, \u2190\n  LinearMap.sub_apply, LinearMap.map_sub, neg_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\n\u22a2 Integrable f \u2194 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\n[PROOFSTEP]\nhave aux : \u2200 {g : V \u2192 E} (_ : Integrable g \u03bc) (x : W), Integrable (fun v : V => e[-L v x] \u2022 g v) \u03bc :=\n  by\n  intro g hg x\n  have c : Continuous fun v => e[-L v x] :=\n    by\n    refine' (continuous_induced_rng.mp he).comp (continuous_ofAdd.comp (Continuous.neg _))\n    exact hL.comp (continuous_prod_mk.mpr \u27e8continuous_id, continuous_const\u27e9)\n  rw [\u2190 integrable_norm_iff (c.aestronglyMeasurable.smul hg.1)]\n  convert hg.norm using 2\n  rw [norm_smul, Complex.norm_eq_abs, abs_coe_circle, one_mul]\n    -- then use it for both directions\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\n\u22a2 \u2200 {g : V \u2192 E}, Integrable g \u2192 \u2200 (x : W), Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\n[PROOFSTEP]\nintro g hg x\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\n[PROOFSTEP]\nhave c : Continuous fun v => e[-L v x] :=\n  by\n  refine' (continuous_induced_rng.mp he).comp (continuous_ofAdd.comp (Continuous.neg _))\n  exact hL.comp (continuous_prod_mk.mpr \u27e8continuous_id, continuous_const\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\n\u22a2 Continuous fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x)))\n[PROOFSTEP]\nrefine' (continuous_induced_rng.mp he).comp (continuous_ofAdd.comp (Continuous.neg _))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\n\u22a2 Continuous fun v => \u2191(\u2191L v) x\n[PROOFSTEP]\nexact hL.comp (continuous_prod_mk.mpr \u27e8continuous_id, continuous_const\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\nc : Continuous fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x)))\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\n[PROOFSTEP]\nrw [\u2190 integrable_norm_iff (c.aestronglyMeasurable.smul hg.1)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\nc : Continuous fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x)))\n\u22a2 Integrable fun a => \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) x))) \u2022 g a\u2016\n[PROOFSTEP]\nconvert hg.norm using 2\n[GOAL]\ncase h.e'_5.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\ng : V \u2192 E\nhg : Integrable g\nx : W\nc : Continuous fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x)))\nx\u271d : V\n\u22a2 \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L x\u271d) x))) \u2022 g x\u271d\u2016 = \u2016g x\u271d\u2016\n[PROOFSTEP]\nrw [norm_smul, Complex.norm_eq_abs, abs_coe_circle, one_mul]\n  -- then use it for both directions\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\naux : \u2200 {g : V \u2192 E}, Integrable g \u2192 \u2200 (x : W), Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\n\u22a2 Integrable f \u2194 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\n[PROOFSTEP]\nrefine' \u27e8fun hf => aux hf w, fun hf => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\naux : \u2200 {g : V \u2192 E}, Integrable g \u2192 \u2200 (x : W), Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\nhf : Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\n\u22a2 Integrable f\n[PROOFSTEP]\nconvert aux hf (-w)\n[GOAL]\ncase h.e'_5.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b9 : AddCommGroup V\ninst\u271d\u00b9\u2070 : Module \ud835\udd5c V\ninst\u271d\u2079 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2078 : AddCommGroup W\ninst\u271d\u2077 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\ninst\u271d\u2074 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b3 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace V\ninst\u271d\u00b9 : BorelSpace V\ninst\u271d : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nw : W\naux : \u2200 {g : V \u2192 E}, Integrable g \u2192 \u2200 (x : W), Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) x))) \u2022 g v\nhf : Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\nx\u271d : V\n\u22a2 f x\u271d = \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L x\u271d) (-w)))) \u2022 \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L x\u271d) w))) \u2022 f x\u271d\n[PROOFSTEP]\nrw [\u2190 smul_assoc, smul_eq_mul, \u2190 Submonoid.coe_mul, \u2190 MonoidHom.map_mul, \u2190 ofAdd_add, LinearMap.map_neg, neg_neg, \u2190\n  sub_eq_add_neg, sub_self, ofAdd_zero, MonoidHom.map_one, Submonoid.coe_one, one_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\n\u22a2 fourierIntegral e \u03bc L f + fourierIntegral e \u03bc L g = fourierIntegral e \u03bc L (f + g)\n[PROOFSTEP]\next1 w\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\nw : W\n\u22a2 (fourierIntegral e \u03bc L f + fourierIntegral e \u03bc L g) w = fourierIntegral e \u03bc L (f + g) w\n[PROOFSTEP]\ndsimp only [Pi.add_apply, fourierIntegral]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\nw : W\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc +\n      \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 g v \u2202\u03bc =\n    \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 (f v + g v) \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [smul_add]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\nw : W\n\u22a2 \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v \u2202\u03bc +\n      \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 g v \u2202\u03bc =\n    \u222b (v : V), \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v + \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 g v \u2202\u03bc\n[PROOFSTEP]\nrw [integral_add]\n[GOAL]\ncase h.hf\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\nw : W\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\n[PROOFSTEP]\nexact (fourier_integral_convergent_iff he hL w).mp hf\n[GOAL]\ncase h.hg\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup V\ninst\u271d\u00b9\u00b9 : Module \ud835\udd5c V\ninst\u271d\u00b9\u2070 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u2079 : AddCommGroup W\ninst\u271d\u2078 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u2102 E\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalRing \ud835\udd5c\ninst\u271d\u00b3 : TopologicalSpace V\ninst\u271d\u00b2 : BorelSpace V\ninst\u271d\u00b9 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d : CompleteSpace E\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf g : V \u2192 E\nhf : Integrable f\nhg : Integrable g\nw : W\n\u22a2 Integrable fun v => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 g v\n[PROOFSTEP]\nexact (fourier_integral_convergent_iff he hL w).mp hg\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 Continuous (fourierIntegral e \u03bc L f)\n[PROOFSTEP]\napply continuous_of_dominated\n[GOAL]\ncase hF_meas\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 \u2200 (x : W), AEStronglyMeasurable (fun a => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) x))) \u2022 f a) \u03bc\n[PROOFSTEP]\nexact fun w => ((fourier_integral_convergent_iff he hL w).mp hf).1\n[GOAL]\ncase h_bound\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 \u2200 (x : W), \u2200\u1d50 (a : V) \u2202\u03bc, \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) x))) \u2022 f a\u2016 \u2264 ?bound a\n[PROOFSTEP]\nrefine' fun w => ae_of_all _ fun v => _\n[GOAL]\ncase bound\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 V \u2192 \u211d\n[PROOFSTEP]\nexact fun v => \u2016f v\u2016\n[GOAL]\ncase h_bound\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\nw : W\nv : V\n\u22a2 \u2016\u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L v) w))) \u2022 f v\u2016 \u2264 \u2016f v\u2016\n[PROOFSTEP]\nrw [norm_smul, Complex.norm_eq_abs, abs_coe_circle, one_mul]\n[GOAL]\ncase bound_integrable\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 Integrable fun v => \u2016f v\u2016\n[PROOFSTEP]\nexact hf.norm\n[GOAL]\ncase h_cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous \u2191e\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 \u2200\u1d50 (a : V) \u2202\u03bc, Continuous fun x => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) x))) \u2022 f a\n[PROOFSTEP]\nrw [continuous_induced_rng] at he \n[GOAL]\ncase h_cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous (Subtype.val \u2218 \u2191e)\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\n\u22a2 \u2200\u1d50 (a : V) \u2202\u03bc, Continuous fun x => \u2191(\u2191e (\u2191Multiplicative.ofAdd (-\u2191(\u2191L a) x))) \u2022 f a\n[PROOFSTEP]\nrefine' ae_of_all _ fun v => (he.comp (continuous_ofAdd.comp _)).smul continuous_const\n[GOAL]\ncase h_cont\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : CommRing \ud835\udd5c\nV : Type u_2\ninst\u271d\u00b9\u00b3 : AddCommGroup V\ninst\u271d\u00b9\u00b2 : Module \ud835\udd5c V\ninst\u271d\u00b9\u00b9 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9\u2070 : AddCommGroup W\ninst\u271d\u2079 : Module \ud835\udd5c W\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u2102 E\ninst\u271d\u2076 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2075 : TopologicalRing \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace V\ninst\u271d\u00b3 : BorelSpace V\ninst\u271d\u00b2 : TopologicalSpace W\ne : Multiplicative \ud835\udd5c \u2192* { x // x \u2208 \ud835\udd4a }\n\u03bc : Measure V\nL : V \u2192\u2097[\ud835\udd5c] W \u2192\u2097[\ud835\udd5c] \ud835\udd5c\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : TopologicalSpace.FirstCountableTopology W\nhe : Continuous (Subtype.val \u2218 \u2191e)\nhL : Continuous fun p => \u2191(\u2191L p.fst) p.snd\nf : V \u2192 E\nhf : Integrable f\nv : V\n\u22a2 Continuous fun x => -\u2191(\u2191L v) x\n[PROOFSTEP]\nrefine' (hL.comp (continuous_prod_mk.mpr \u27e8continuous_const, continuous_id\u27e9)).neg\n[GOAL]\n\u22a2 (fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z)) 1 = 1\n[PROOFSTEP]\nsimp only\n[GOAL]\n\u22a2 \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd 1) = 1\n[PROOFSTEP]\nrw [toAdd_one, mul_zero, expMapCircle_zero]\n[GOAL]\nx y : Multiplicative \u211d\n\u22a2 OneHom.toFun\n      { toFun := fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z),\n        map_one' := (_ : (fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z)) 1 = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z),\n          map_one' := (_ : (fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z)) 1 = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z),\n          map_one' := (_ : (fun z => \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd z)) 1 = 1) }\n        y\n[PROOFSTEP]\nsimp only\n[GOAL]\nx y : Multiplicative \u211d\n\u22a2 \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd (x * y)) =\n    \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd x) * \u2191expMapCircle (2 * \u03c0 * \u2191Multiplicative.toAdd y)\n[PROOFSTEP]\nrw [toAdd_mul, mul_add, expMapCircle_add]\n[GOAL]\nx : \u211d\n\u22a2 \u2191(\u2191fourierChar (\u2191Multiplicative.ofAdd x)) = Complex.exp (\u2191(2 * \u03c0 * x) * Complex.I)\n[PROOFSTEP]\nrfl\n[GOAL]\nE : Type u_1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nV : Type u_2\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module \u211d V\ninst\u271d\u00b2 : MeasurableSpace V\nW : Type u_3\ninst\u271d\u00b9 : AddCommGroup W\ninst\u271d : Module \u211d W\nL : V \u2192\u2097[\u211d] W \u2192\u2097[\u211d] \u211d\n\u03bc : Measure V\nf : V \u2192 E\nw : W\n\u22a2 VectorFourier.fourierIntegral fourierChar \u03bc L f w =\n    \u222b (v : V), Complex.exp (\u2191(-2 * \u03c0 * \u2191(\u2191L v) w) * Complex.I) \u2022 f v \u2202\u03bc\n[PROOFSTEP]\nsimp_rw [VectorFourier.fourierIntegral, Real.fourierChar_apply, mul_neg, neg_mul]\n[GOAL]\nE\u271d : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedSpace \u2102 E\u271d\nE : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nf : \u211d \u2192 E\nw : \u211d\n\u22a2 \ud835\udcd5 f w = \u222b (v : \u211d), Complex.exp (\u2191(-2 * \u03c0 * v * w) * Complex.I) \u2022 f v\n[PROOFSTEP]\nsimp_rw [fourierIntegral_def, Real.fourierChar_apply, mul_neg, neg_mul, mul_assoc]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Fourier.FourierTransform", "llama_tokens": 16666, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117940706733, "lm_q2_score": 0.6261241842048092, "lm_q1q2_score": 0.5176242476349945}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 ack 0 n = n + 1\n[PROOFSTEP]\nrw [ack]\n[GOAL]\nm : \u2115\n\u22a2 ack (m + 1) 0 = ack m 1\n[PROOFSTEP]\nrw [ack]\n[GOAL]\nm n : \u2115\n\u22a2 ack (m + 1) (n + 1) = ack m (ack (m + 1) n)\n[PROOFSTEP]\nrw [ack]\n[GOAL]\nn : \u2115\n\u22a2 ack 1 n = n + 2\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u22a2 ack 1 zero = zero + 2\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 1 n = n + 2\n\u22a2 ack 1 (succ n) = succ n + 2\n[PROOFSTEP]\nsimp [IH]\n[GOAL]\nn : \u2115\n\u22a2 ack 2 n = 2 * n + 3\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u22a2 ack 2 zero = 2 * zero + 3\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 2 n = 2 * n + 3\n\u22a2 ack 2 (succ n) = 2 * succ n + 3\n[PROOFSTEP]\nsimpa [mul_succ]\n[GOAL]\nn : \u2115\n\u22a2 ack 3 n = 2 ^ (n + 3) - 3\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\u22a2 ack 3 zero = 2 ^ (zero + 3) - 3\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 ack 3 (succ n) = 2 ^ (succ n + 3) - 3\n[PROOFSTEP]\nrw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2, Nat.mul_sub_left_distrib, \u2190\n  Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)\n[PROOFSTEP]\nhave H : 2 * 3 \u2264 2 * 2 ^ 3 := by norm_num\n[GOAL]\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 3 \u2264 2 * 2 ^ (n + 3)\n[PROOFSTEP]\napply H.trans\n[GOAL]\ncase succ\nn : \u2115\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 \u2264 2 * 2 ^ 3\n\u22a2 2 * 2 ^ 3 \u2264 2 * 2 ^ (n + 3)\n[PROOFSTEP]\nsimp [pow_le_pow]\n[GOAL]\nn : \u2115\n\u22a2 0 < ack 0 n\n[PROOFSTEP]\nsimp\n[GOAL]\nm : \u2115\n\u22a2 0 < ack (m + 1) 0\n[PROOFSTEP]\nrw [ack_succ_zero]\n[GOAL]\nm : \u2115\n\u22a2 0 < ack m 1\n[PROOFSTEP]\napply ack_pos\n[GOAL]\nm n : \u2115\n\u22a2 0 < ack (m + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_succ]\n[GOAL]\nm n : \u2115\n\u22a2 0 < ack m (ack (m + 1) n)\n[PROOFSTEP]\napply ack_pos\n[GOAL]\nn : \u2115\n\u22a2 1 < ack (0 + 1) n\n[PROOFSTEP]\nsimp\n[GOAL]\nm : \u2115\n\u22a2 1 < ack (m + 1 + 1) 0\n[PROOFSTEP]\nrw [ack_succ_zero]\n[GOAL]\nm : \u2115\n\u22a2 1 < ack (m + 1) 1\n[PROOFSTEP]\napply one_lt_ack_succ_left\n[GOAL]\nm n : \u2115\n\u22a2 1 < ack (m + 1 + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_succ]\n[GOAL]\nm n : \u2115\n\u22a2 1 < ack (m + 1) (ack (m + 1 + 1) n)\n[PROOFSTEP]\napply one_lt_ack_succ_left\n[GOAL]\nn : \u2115\n\u22a2 1 < ack 0 (n + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nm n : \u2115\n\u22a2 1 < ack (m + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_succ]\n[GOAL]\nm n : \u2115\n\u22a2 1 < ack m (ack (m + 1) n)\n[PROOFSTEP]\ncases' exists_eq_succ_of_ne_zero (ack_pos (m + 1) n).ne' with h h\n[GOAL]\ncase intro\nm n h\u271d : \u2115\nh : ack (m + 1) n = succ h\u271d\n\u22a2 1 < ack m (ack (m + 1) n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase intro\nm n h\u271d : \u2115\nh : ack (m + 1) n = succ h\u271d\n\u22a2 1 < ack m (succ h\u271d)\n[PROOFSTEP]\napply one_lt_ack_succ_right\n[GOAL]\nn\u2081 n\u2082 : \u2115\nh : n\u2081 < n\u2082\n\u22a2 ack 0 n\u2081 < ack 0 n\u2082\n[PROOFSTEP]\nsimpa using h\n[GOAL]\nm n : \u2115\n_h : 0 < n + 1\n\u22a2 ack (m + 1) 0 < ack (m + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_zero, ack_succ_succ]\n[GOAL]\nm n : \u2115\n_h : 0 < n + 1\n\u22a2 ack m 1 < ack m (ack (m + 1) n)\n[PROOFSTEP]\nexact ack_strictMono_right _ (one_lt_ack_succ_left m n)\n[GOAL]\nm n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 ack (m + 1) (n\u2081 + 1) < ack (m + 1) (n\u2082 + 1)\n[PROOFSTEP]\nrw [ack_succ_succ, ack_succ_succ]\n[GOAL]\nm n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 ack m (ack (m + 1) n\u2081) < ack m (ack (m + 1) n\u2082)\n[PROOFSTEP]\napply ack_strictMono_right _ (ack_strictMono_right _ _)\n[GOAL]\nm n\u2081 n\u2082 : \u2115\nh : n\u2081 + 1 < n\u2082 + 1\n\u22a2 n\u2081 < n\u2082\n[PROOFSTEP]\nrwa [add_lt_add_iff_right] at h \n[GOAL]\nn : \u2115\n\u22a2 0 + n < ack 0 n\n[PROOFSTEP]\nsimp\n[GOAL]\nm : \u2115\n\u22a2 m + 1 + 0 < ack (m + 1) 0\n[PROOFSTEP]\nsimpa using add_lt_ack m 1\n[GOAL]\nm n : \u2115\n\u22a2 m + 1 + n + 1 \u2264 m + (m + n + 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\nm n : \u2115\n\u22a2 m + n + 2 = succ (m + 1 + n)\n[PROOFSTEP]\nrw [succ_eq_add_one]\n[GOAL]\nm n : \u2115\n\u22a2 m + n + 2 = m + 1 + n + 1\n[PROOFSTEP]\nring_nf\n[GOAL]\nm : \u2115\n_h : 0 < m + 1\n\u22a2 ack 0 0 < ack (m + 1) 0\n[PROOFSTEP]\nsimpa using one_lt_ack_succ_right m 0\n[GOAL]\nm n : \u2115\nh : 0 < m + 1\n\u22a2 ack 0 (n + 1) < ack (m + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_zero, ack_succ_succ]\n[GOAL]\nm n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 < ack m (ack (m + 1) n)\n[PROOFSTEP]\napply lt_of_le_of_lt (le_trans _ <| add_le_add_left (add_add_one_le_ack _ _) m) (add_lt_ack _ _)\n[GOAL]\nm n : \u2115\nh : 0 < m + 1\n\u22a2 n + 1 + 1 \u2264 m + (m + 1 + n + 1)\n[PROOFSTEP]\nlinarith\n[GOAL]\nm\u2081 m\u2082 : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack (m\u2081 + 1) 0 < ack (m\u2082 + 1) 0\n[PROOFSTEP]\nsimpa using ack_strict_mono_left' 1 ((add_lt_add_iff_right 1).1 h)\n[GOAL]\nm\u2081 m\u2082 n : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack (m\u2081 + 1) (n + 1) < ack (m\u2082 + 1) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_succ, ack_succ_succ]\n[GOAL]\nm\u2081 m\u2082 n : \u2115\nh : m\u2081 + 1 < m\u2082 + 1\n\u22a2 ack m\u2081 (ack (m\u2081 + 1) n) < ack m\u2082 (ack (m\u2082 + 1) n)\n[PROOFSTEP]\nexact\n  (ack_strict_mono_left' _ <| (add_lt_add_iff_right 1).1 h).trans (ack_strictMono_right _ <| ack_strict_mono_left' n h)\n[GOAL]\nm n : \u2115\n\u22a2 ack m (n + 1) \u2264 ack (m + 1) n\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase zero\nm : \u2115\n\u22a2 ack m (zero + 1) \u2264 ack (m + 1) zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nm n : \u2115\n\u22a2 ack m (succ n + 1) \u2264 ack (m + 1) (succ n)\n[PROOFSTEP]\nrw [ack_succ_succ, succ_eq_add_one]\n[GOAL]\ncase succ\nm n : \u2115\n\u22a2 ack m (n + 1 + 1) \u2264 ack m (ack (m + 1) n)\n[PROOFSTEP]\napply ack_mono_right m (le_trans _ <| add_add_one_le_ack _ n)\n[GOAL]\nm n : \u2115\n\u22a2 n + 1 + 1 \u2264 m + 1 + n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nn : \u2115\n\u22a2 n ^ 2 \u2264 2 ^ (n + 1) - 3\n[PROOFSTEP]\ninduction' n with k hk\n[GOAL]\ncase zero\n\u22a2 zero ^ 2 \u2264 2 ^ (zero + 1) - 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\nk : \u2115\nhk : k ^ 2 \u2264 2 ^ (k + 1) - 3\n\u22a2 succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n[PROOFSTEP]\ncases' k with k k\n[GOAL]\ncase succ.zero\nhk : zero ^ 2 \u2264 2 ^ (zero + 1) - 3\n\u22a2 succ zero ^ 2 \u2264 2 ^ (succ zero + 1) - 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ.succ\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 succ (succ k) ^ 2 \u2264 2 ^ (succ (succ k) + 1) - 3\n[PROOFSTEP]\nrw [succ_eq_add_one, add_sq, Nat.pow_succ 2, mul_comm _ 2, two_mul (2 ^ _), add_tsub_assoc_of_le, add_comm (2 ^ _),\n  add_assoc]\n[GOAL]\ncase succ.succ\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 (k + 1) ^ 2 + (2 * (k + 1) * 1 + 1 ^ 2) \u2264 2 ^ (k + 2) - 3 + 2 ^ (k + 2)\n[PROOFSTEP]\napply Nat.add_le_add hk\n[GOAL]\ncase succ.succ\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 2 * (k + 1) * 1 + 1 ^ 2 \u2264 2 ^ (k + 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ.succ\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 2 * (k + 1) + 1 \u2264 2 ^ (k + 2)\n[PROOFSTEP]\napply succ_le_of_lt\n[GOAL]\ncase succ.succ.h\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 2 * (k + 1) < 2 ^ (k + 2)\n[PROOFSTEP]\nrw [Nat.pow_succ, mul_comm _ 2, mul_lt_mul_left (zero_lt_two' \u2115)]\n[GOAL]\ncase succ.succ.h\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 k + 1 < 2 ^ (k + 1)\n[PROOFSTEP]\napply lt_two_pow\n[GOAL]\ncase succ.succ.h\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 3 \u2264 2 ^ (k + 2)\n[PROOFSTEP]\nrw [Nat.pow_succ, Nat.pow_succ]\n[GOAL]\ncase succ.succ.h\nk : \u2115\nhk : succ k ^ 2 \u2264 2 ^ (succ k + 1) - 3\n\u22a2 3 \u2264 2 ^ k * 2 * 2\n[PROOFSTEP]\nlinarith [one_le_pow k 2 zero_lt_two]\n[GOAL]\nn : \u2115\n\u22a2 (ack 0 n + 1) ^ 2 \u2264 ack (0 + 3) n\n[PROOFSTEP]\nsimpa using sq_le_two_pow_add_one_minus_three (n + 2)\n[GOAL]\nm : \u2115\n\u22a2 (ack (m + 1) 0 + 1) ^ 2 \u2264 ack (m + 1 + 3) 0\n[PROOFSTEP]\nrw [ack_succ_zero, ack_succ_zero]\n[GOAL]\nm : \u2115\n\u22a2 (ack m 1 + 1) ^ 2 \u2264 ack (m + 3) 1\n[PROOFSTEP]\napply ack_add_one_sq_lt_ack_add_three\n[GOAL]\nm n : \u2115\n\u22a2 (ack (m + 1) (n + 1) + 1) ^ 2 \u2264 ack (m + 1 + 3) (n + 1)\n[PROOFSTEP]\nrw [ack_succ_succ, ack_succ_succ]\n[GOAL]\nm n : \u2115\n\u22a2 (ack m (ack (m + 1) n) + 1) ^ 2 \u2264 ack (m + 3) (ack (m + 3 + 1) n)\n[PROOFSTEP]\napply (ack_add_one_sq_lt_ack_add_three _ _).trans (ack_mono_right _ <| ack_mono_left _ _)\n[GOAL]\nm n : \u2115\n\u22a2 m + 1 \u2264 m + 3 + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\nm n : \u2115\n\u22a2 m \u2264 m + 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nf : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 \u2203 m, \u2200 (n : \u2115), f n < ack m n\n[PROOFSTEP]\ninduction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg\n[GOAL]\ncase zero\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun x => 0) n < ack m n\n[PROOFSTEP]\nexact\n  \u27e80, ack_pos 0\u27e9\n    -- Successor function:\n[GOAL]\ncase succ\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), succ n < ack m n\n[PROOFSTEP]\nrefine' \u27e81, fun n => _\u27e9\n[GOAL]\ncase succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 succ n < ack 1 n\n[PROOFSTEP]\nrw [succ_eq_one_add]\n[GOAL]\ncase succ\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 1 + n < ack 1 n\n[PROOFSTEP]\napply add_lt_ack\n[GOAL]\ncase left\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).fst) n < ack m n\n[PROOFSTEP]\nrefine' \u27e80, fun n => _\u27e9\n[GOAL]\ncase left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).fst) n < ack 0 n\n[PROOFSTEP]\nrw [ack_zero, lt_succ_iff]\n[GOAL]\ncase left\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).fst) n \u2264 n\n[PROOFSTEP]\nexact unpair_left_le n\n[GOAL]\ncase right\nf : \u2115 \u2192 \u2115\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => (unpair n).snd) n < ack m n\n[PROOFSTEP]\nrefine' \u27e80, fun n => _\u27e9\n[GOAL]\ncase right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).snd) n < ack 0 n\n[PROOFSTEP]\nrw [ack_zero, lt_succ_iff]\n[GOAL]\ncase right\nf : \u2115 \u2192 \u2115\nn : \u2115\n\u22a2 (fun n => (unpair n).snd) n \u2264 n\n[PROOFSTEP]\nexact unpair_right_le n\n[GOAL]\ncase pair\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\ncase comp\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\ncase prec\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n\n[PROOFSTEP]\nall_goals cases' IHf with a ha; cases' IHg with b hb\n[GOAL]\ncase pair\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n[PROOFSTEP]\ncases' IHf with a ha\n[GOAL]\ncase pair.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n[PROOFSTEP]\ncases' IHg with b hb\n[GOAL]\ncase comp\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n[PROOFSTEP]\ncases' IHf with a ha\n[GOAL]\ncase comp.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n[PROOFSTEP]\ncases' IHg with b hb\n[GOAL]\ncase prec\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : \u2203 m, \u2200 (n : \u2115), f n < ack m n\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n\n[PROOFSTEP]\ncases' IHf with a ha\n[GOAL]\ncase prec.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : \u2203 m, \u2200 (n : \u2115), g n < ack m n\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n\n[PROOFSTEP]\ncases' IHg with b hb\n[GOAL]\ncase pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => pair (f n) (g n)) n < ack m n\n[PROOFSTEP]\nrefine'\n  \u27e8max a b + 3, fun n =>\n    (pair_lt_max_add_one_sq _ _).trans_le <|\n      (pow_le_pow_of_le_left (add_le_add_right _ _) 2).trans <| ack_add_one_sq_lt_ack_add_three _ _\u27e9\n[GOAL]\ncase pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 ack (max a b) n\n[PROOFSTEP]\nrw [max_ack_left]\n[GOAL]\ncase pair.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nn : \u2115\n\u22a2 max (f n) (g n) \u2264 max (ack a n) (ack b n)\n[PROOFSTEP]\nexact max_le_max (ha n).le (hb n).le\n[GOAL]\ncase comp.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), (fun n => f (g n)) n < ack m n\n[PROOFSTEP]\nexact\n  \u27e8max a b + 2, fun n => (ha _).trans <| (ack_strictMono_right a <| hb n).trans <| ack_ack_lt_ack_max_add_two a b n\u27e9\n    -- Primitive recursion operator:\n[GOAL]\ncase prec.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n\n[PROOFSTEP]\nhave : \u2200 {m n}, rec (f m) (fun y IH => g <| pair m <| pair y IH) n < ack (max a b + 9) (m + n) :=\n  by\n  intro m n\n  induction' n with n IH\n  \u00b7 apply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _)\n    linarith\n  \u00b7\n    -- We get rid of the first `pair`.\n    simp only [ge_iff_le]\n    apply\n      (hb _).trans\n        ((ack_pair_lt _ _ _).trans_le _)\n          -- If m is the maximum, we get a very weak inequality.\n    cases' lt_or_le _ m with h\u2081 h\u2081\n    \u00b7 rw [max_eq_left h\u2081.le]\n      exact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num) (self_le_add_right m _)\n    rw [max_eq_right h\u2081]\n      -- We get rid of the second `pair`.\n    apply (ack_pair_lt _ _ _).le.trans\n    cases' lt_or_le _ n with h\u2082 h\u2082\n    \u00b7 rw [max_eq_left h\u2082.le, add_assoc]\n      exact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num) ((le_succ n).trans <| self_le_add_left _ _)\n    rw [max_eq_right h\u2082]\n      -- We now use the inductive hypothesis, and some simple algebraic manipulation.\n    apply (ack_strictMono_right _ IH).le.trans\n    rw [add_succ m, add_succ _ 8, succ_eq_add_one, succ_eq_add_one, ack_succ_succ (_ + 8), add_assoc]\n    exact\n      ack_mono_left _\n        (Nat.add_le_add (le_max_right a b) le_rfl)\n          -- The proof is now simple.\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\n\u22a2 \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n[PROOFSTEP]\nintro m n\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero)\n[PROOFSTEP]\napply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _)\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm : \u2115\n\u22a2 max a b < max a b + 9\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nsimp only [ge_iff_le]\n[GOAL]\ncase succ\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\napply\n  (hb _).trans\n    ((ack_pair_lt _ _ _).trans_le _)\n      -- If m is the maximum, we get a very weak inequality.\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\ncases' lt_or_le _ m with h\u2081 h\u2081\n[GOAL]\ncase inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : ?m.127608 < m\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nrw [max_eq_left h\u2081.le]\n[GOAL]\ncase inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n\u22a2 ack (b + 4) m \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nexact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num) (self_le_add_right m _)\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n\u22a2 4 \u2264 9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nrw [max_eq_right h\u2081]\n  -- We get rid of the second `pair`.\n[GOAL]\ncase inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\napply (ack_pair_lt _ _ _).le.trans\n[GOAL]\ncase inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\ncases' lt_or_le _ n with h\u2082 h\u2082\n[GOAL]\ncase inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : ?m.127977 < n\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nrw [max_eq_left h\u2082.le, add_assoc]\n[GOAL]\ncase inr.inl\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n\u22a2 ack (b + (4 + 4)) n \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nexact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num) ((le_succ n).trans <| self_le_add_left _ _)\n[GOAL]\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n\u22a2 4 + 4 \u2264 9\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nrw [max_eq_right h\u2082]\n  -- We now use the inductive hypothesis, and some simple algebraic manipulation.\n[GOAL]\ncase inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\napply (ack_strictMono_right _ IH).le.trans\n[GOAL]\ncase inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + 4 + 4) (ack (max a b + 9) (m + n)) \u2264 ack (max a b + 9) (m + succ n)\n[PROOFSTEP]\nrw [add_succ m, add_succ _ 8, succ_eq_add_one, succ_eq_add_one, ack_succ_succ (_ + 8), add_assoc]\n[GOAL]\ncase inr.inr\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nm n : \u2115\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh\u2081 : m \u2264 pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh\u2082 : n \u2264 rec (f m) (fun y IH => g (pair m (pair y IH))) n\n\u22a2 ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) \u2264 ack (max a b + 8) (ack (max a b + 8 + 1) (m + n))\n[PROOFSTEP]\nexact\n  ack_mono_left _\n    (Nat.add_le_add (le_max_right a b) le_rfl)\n      -- The proof is now simple.\n[GOAL]\ncase prec.intro.intro\nf\u271d f g : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : \u2115\nha : \u2200 (n : \u2115), f n < ack a n\nb : \u2115\nhb : \u2200 (n : \u2115), g n < ack b n\nthis : \u2200 {m n : \u2115}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n\u22a2 \u2203 m, \u2200 (n : \u2115), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n\n[PROOFSTEP]\nexact \u27e8max a b + 9, fun n => this.trans_le <| ack_mono_right _ <| unpair_add_le n\u27e9\n[GOAL]\nh : Nat.Primrec fun n => ack n n\n\u22a2 False\n[PROOFSTEP]\ncases' exists_lt_ack_of_nat_primrec h with m hm\n[GOAL]\ncase intro\nh : Nat.Primrec fun n => ack n n\nm : \u2115\nhm : \u2200 (n : \u2115), ack n n < ack m n\n\u22a2 False\n[PROOFSTEP]\nexact (hm m).false\n[GOAL]\n\u22a2 \u00acPrimrec fun n => ack n n\n[PROOFSTEP]\nrw [Primrec.nat_iff]\n[GOAL]\n\u22a2 \u00acNat.Primrec fun n => ack n n\n[PROOFSTEP]\nexact not_nat_primrec_ack_self\n", "meta": {"mathlib_filename": "Mathlib.Computability.Ackermann", "llama_tokens": 12949, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267118026095991, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5176242472138628}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nx : R\n\u22a2 MeasurableSet (floor \u207b\u00b9' {\u230ax\u230b})\n[PROOFSTEP]\nsimpa only [Int.preimage_floor_singleton] using measurableSet_Ico\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nx : R\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308x\u2309})\n[PROOFSTEP]\nsimpa only [Int.preimage_ceil_singleton] using measurableSet_Ioc\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\n\u22a2 Measurable Int.fract\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Int.fract \u207b\u00b9' s)\n[PROOFSTEP]\nrw [Int.preimage_fract]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 (m : \u2124), (fun x => x - \u2191m) \u207b\u00b9' (s \u2229 Ico 0 1))\n[PROOFSTEP]\nexact MeasurableSet.iUnion fun z => measurable_id.sub_const _ (hs.inter measurableSet_Ico)\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (Int.fract '' s)\n[PROOFSTEP]\nsimp only [Int.image_fract, sub_eq_add_neg, image_add_right']\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedRing R\ninst\u271d\u2074 : FloorRing R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n\u22a2 MeasurableSet (\u22c3 (m : \u2124), (fun x => x + \u2191m) \u207b\u00b9' s \u2229 Ico 0 1)\n[PROOFSTEP]\nexact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\n\u22a2 MeasurableSet (floor \u207b\u00b9' {\u230an\u230b\u208a})\n[PROOFSTEP]\ncases' eq_or_ne \u230an\u230b\u208a 0 with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\nh : \u230an\u230b\u208a = 0\n\u22a2 MeasurableSet (floor \u207b\u00b9' {\u230an\u230b\u208a})\n[PROOFSTEP]\nsimp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\nh : \u230an\u230b\u208a \u2260 0\n\u22a2 MeasurableSet (floor \u207b\u00b9' {\u230an\u230b\u208a})\n[PROOFSTEP]\nsimp_all [h, Nat.preimage_floor_of_ne_zero, -floor_eq_zero]\n[GOAL]\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308n\u2309\u208a})\n[PROOFSTEP]\ncases' eq_or_ne \u2308n\u2309\u208a 0 with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\nh : \u2308n\u2309\u208a = 0\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308n\u2309\u208a})\n[PROOFSTEP]\nsimp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : LinearOrderedSemiring R\ninst\u271d\u2074 : FloorSemiring R\ninst\u271d\u00b3 : TopologicalSpace R\ninst\u271d\u00b2 : OrderTopology R\ninst\u271d\u00b9 : MeasurableSpace R\ninst\u271d : OpensMeasurableSpace R\nf : \u03b1 \u2192 R\nn : R\nh : \u2308n\u2309\u208a \u2260 0\n\u22a2 MeasurableSet (ceil \u207b\u00b9' {\u2308n\u2309\u208a})\n[PROOFSTEP]\nsimp_all [h, Nat.preimage_ceil_of_ne_zero, -ceil_eq_zero]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.Floor", "llama_tokens": 2234, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.6442250928250375, "lm_q1q2_score": 0.5174247135126626}}
{"text": "[GOAL]\nf : \u2115 \u2192. \u2115\nhf : Partrec f\n\u22a2 Partrec\u2082 fun a m =>\n    Part.map (fun x => x + m) (Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + m)))\n[PROOFSTEP]\nrefine'\n  Partrec.map ((@Partrec\u2082.unpaired' fun a b : \u2115 => Nat.rfind fun n => (fun m => m = 0) <$> f (Nat.pair a (n + b))).1 _)\n    (Primrec.nat_add.comp Primrec.snd <| Primrec.snd.comp Primrec.fst).to_comp.to\u2082\n[GOAL]\nf : \u2115 \u2192. \u2115\nhf : Partrec f\n\u22a2 Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b)))\n[PROOFSTEP]\nhave :\n  Nat.Partrec\n    (fun a =>\n      Nat.rfind\n        (fun n =>\n          (fun m => decide (m = 0)) <$>\n            Nat.unpaired (fun a b => f (Nat.pair (Nat.unpair a).1 (b + (Nat.unpair a).2))) (Nat.pair a n))) :=\n  rfind\n    (Partrec\u2082.unpaired'.2\n      ((Partrec.nat_iff.2 hf).comp\n        (Primrec\u2082.pair.comp (Primrec.fst.comp <| Primrec.unpair.comp Primrec.fst)\n            (Primrec.nat_add.comp Primrec.snd (Primrec.snd.comp <| Primrec.unpair.comp Primrec.fst))).to_comp))\n[GOAL]\nf : \u2115 \u2192. \u2115\nhf : Partrec f\nthis :\n  Partrec fun a =>\n    Nat.rfind fun n =>\n      (fun m => decide (m = 0)) <$>\n        unpaired (fun a b => f (Nat.pair (unpair a).fst (b + (unpair a).snd))) (Nat.pair a n)\n\u22a2 Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b)))\n[PROOFSTEP]\nsimp at this \n[GOAL]\nf : \u2115 \u2192. \u2115\nhf : Partrec f\nthis :\n  Partrec fun a =>\n    Nat.rfind fun n => Part.map (fun m => decide (m = 0)) (f (Nat.pair (unpair a).fst (n + (unpair a).snd)))\n\u22a2 Partrec (unpaired fun a b => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a (n + b)))\n[PROOFSTEP]\nexact this\n[GOAL]\nx\u271d : Code.const 0 = Code.const 0\n\u22a2 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn\u2081 n\u2082 : \u2115\nh : Code.const (n\u2081 + 1) = Code.const (n\u2082 + 1)\n\u22a2 n\u2081 + 1 = n\u2082 + 1\n[PROOFSTEP]\ndsimp [Nat.add_one, Nat.Partrec.Code.const] at h \n[GOAL]\nn\u2081 n\u2082 : \u2115\nh : comp succ (Code.const n\u2081) = comp succ (Code.const n\u2082)\n\u22a2 n\u2081 + 1 = n\u2082 + 1\n[PROOFSTEP]\ninjection h with h\u2081 h\u2082\n[GOAL]\nn\u2081 n\u2082 : \u2115\nh\u2081 : succ = succ\nh\u2082 : Code.const n\u2081 = Code.const n\u2082\n\u22a2 n\u2081 + 1 = n\u2082 + 1\n[PROOFSTEP]\nsimp only [const_inj h\u2082]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\n\u22a2 encodeCode (ofNatCode 0) = 0\n[PROOFSTEP]\nsimp [ofNatCode, encodeCode]\n[GOAL]\n\u22a2 encodeCode (ofNatCode 1) = 1\n[PROOFSTEP]\nsimp [ofNatCode, encodeCode]\n[GOAL]\n\u22a2 encodeCode (ofNatCode 2) = 2\n[PROOFSTEP]\nsimp [ofNatCode, encodeCode]\n[GOAL]\n\u22a2 encodeCode (ofNatCode 3) = 3\n[PROOFSTEP]\nsimp [ofNatCode, encodeCode]\n[GOAL]\nn : \u2115\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nlet m := n.div2.div2\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave hm : m < n + 4 := by\n  simp [Nat.div2_val]\n  exact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave _m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave _m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave IH := encode_ofNatCode m\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave IH1 := encode_ofNatCode m.unpair.1\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nhave IH2 := encode_ofNatCode m.unpair.2\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode (ofNatCode (n + 4)) = n + 4\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n| n + 4\n[PROOFSTEP]\nrw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n| n + 4\n[PROOFSTEP]\nrw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n| n + 4\n[PROOFSTEP]\nrw [\u2190 Nat.bit_decomp n, \u2190 Nat.bit_decomp n.div2]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode (ofNatCode (n + 4)) = bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode]\n[GOAL]\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit (bodd n) (bit (bodd (div2 n)) (div2 (div2 n))) + 4\n[PROOFSTEP]\ncases n.bodd\n[GOAL]\ncase false\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match false, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit false (bit (bodd (div2 n)) (div2 (div2 n))) + 4\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase true\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match true, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit true (bit (bodd (div2 n)) (div2 (div2 n))) + 4\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase false.false\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match false, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit false (bit false (div2 (div2 n))) + 4\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]\n[GOAL]\ncase false.true\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match false, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit false (bit true (div2 (div2 n))) + 4\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]\n[GOAL]\ncase true.false\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match true, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit true (bit false (div2 (div2 n))) + 4\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]\n[GOAL]\ncase true.true\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n_m1 : (unpair m).fst < n + 4\n_m2 : (unpair m).snd < n + 4\nIH : encodeCode (ofNatCode m) = m\nIH1 : encodeCode (ofNatCode (unpair m).fst) = (unpair m).fst\nIH2 : encodeCode (ofNatCode (unpair m).snd) = (unpair m).snd\n\u22a2 encodeCode\n      (match true, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n)))) =\n    bit true (bit true (div2 (div2 n))) + 4\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, IH, IH1, IH2, Nat.bit_val]\n[GOAL]\nc : Code\n\u22a2 ofNatCode (encodeCode c) = c\n[PROOFSTEP]\ninduction c\n[GOAL]\ncase zero\n\u22a2 ofNatCode (encodeCode zero) = zero\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase zero\n\u22a2 ofNatCode (encodeCode zero) = zero\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase zero\n\u22a2 ofNatCode (encodeCode zero) = zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u22a2 ofNatCode (encodeCode succ) = succ\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase succ\n\u22a2 ofNatCode (encodeCode succ) = succ\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase succ\n\u22a2 ofNatCode (encodeCode succ) = succ\n[PROOFSTEP]\nrfl\n[GOAL]\ncase left\n\u22a2 ofNatCode (encodeCode left) = left\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase left\n\u22a2 ofNatCode (encodeCode left) = left\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase left\n\u22a2 ofNatCode (encodeCode left) = left\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\n\u22a2 ofNatCode (encodeCode right) = right\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase right\n\u22a2 ofNatCode (encodeCode right) = right\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase right\n\u22a2 ofNatCode (encodeCode right) = right\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (pair a\u271d\u00b9 a\u271d)) = pair a\u271d\u00b9 a\u271d\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (pair a\u271d\u00b9 a\u271d)) = pair a\u271d\u00b9 a\u271d\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (pair a\u271d\u00b9 a\u271d)) = pair a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (comp a\u271d\u00b9 a\u271d)) = comp a\u271d\u00b9 a\u271d\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (comp a\u271d\u00b9 a\u271d)) = comp a\u271d\u00b9 a\u271d\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (comp a\u271d\u00b9 a\u271d)) = comp a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (prec a\u271d\u00b9 a\u271d)) = prec a\u271d\u00b9 a\u271d\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (prec a\u271d\u00b9 a\u271d)) = prec a\u271d\u00b9 a\u271d\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (prec a\u271d\u00b9 a\u271d)) = prec a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (rfind' a\u271d)) = rfind' a\u271d\n[PROOFSTEP]\ntry {rfl\n}\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (rfind' a\u271d)) = rfind' a\u271d\n[PROOFSTEP]\n{rfl\n}\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (rfind' a\u271d)) = rfind' a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (pair a\u271d\u00b9 a\u271d)) = pair a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, Nat.div2_val, *]\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (comp a\u271d\u00b9 a\u271d)) = comp a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, Nat.div2_val, *]\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : ofNatCode (encodeCode a\u271d\u00b9) = a\u271d\u00b9\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (prec a\u271d\u00b9 a\u271d)) = prec a\u271d\u00b9 a\u271d\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, Nat.div2_val, *]\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : ofNatCode (encodeCode a\u271d) = a\u271d\n\u22a2 ofNatCode (encodeCode (rfind' a\u271d)) = rfind' a\u271d\n[PROOFSTEP]\nsimp [encodeCode, ofNatCode, Nat.div2_val, *]\n[GOAL]\ncf cg : Code\n\u22a2 encode cf < encode (pair cf cg) \u2227 encode cg < encode (pair cf cg)\n[PROOFSTEP]\nsimp [encodeCode_eq, encodeCode]\n[GOAL]\ncf cg : Code\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n[PROOFSTEP]\nhave := Nat.mul_le_mul_right (Nat.pair cf.encodeCode cg.encodeCode) (by decide : 1 \u2264 2 * 2)\n[GOAL]\ncf cg : Code\n\u22a2 1 \u2264 2 * 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncf cg : Code\nthis : 1 * Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * 2 * Nat.pair (encodeCode cf) (encodeCode cg)\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n[PROOFSTEP]\nrw [one_mul, mul_assoc] at this \n[GOAL]\ncf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n[PROOFSTEP]\nhave := lt_of_le_of_lt this (lt_add_of_pos_right _ (by decide : 0 < 4))\n[GOAL]\ncf cg : Code\nthis : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\n\u22a2 0 < 4\n[PROOFSTEP]\ndecide\n[GOAL]\ncf cg : Code\nthis\u271d : Nat.pair (encodeCode cf) (encodeCode cg) \u2264 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg))\nthis : Nat.pair (encodeCode cf) (encodeCode cg) < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n\u22a2 encodeCode cf < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4 \u2227\n    encodeCode cg < 2 * (2 * Nat.pair (encodeCode cf) (encodeCode cg)) + 4\n[PROOFSTEP]\nexact \u27e8lt_of_le_of_lt (Nat.left_le_pair _ _) this, lt_of_le_of_lt (Nat.right_le_pair _ _) this\u27e9\n[GOAL]\ncf cg : Code\n\u22a2 encode cf < encode (comp cf cg) \u2227 encode cg < encode (comp cf cg)\n[PROOFSTEP]\nsuffices\n[GOAL]\ncf cg : Code\nthis : ?m.354291\n\u22a2 encode cf < encode (comp cf cg) \u2227 encode cg < encode (comp cf cg)\ncase this cf cg : Code \u22a2 ?m.354291\n[PROOFSTEP]\nexact (encode_lt_pair cf cg).imp (fun h => lt_trans h this) fun h => lt_trans h this\n[GOAL]\ncase this\ncf cg : Code\n\u22a2 encode (pair cf cg) < encode (comp cf cg)\n[PROOFSTEP]\nchange _\n[GOAL]\ncase this\ncf cg : Code\n\u22a2 encode (pair cf cg) < encode (comp cf cg)\n[PROOFSTEP]\nsimp [encodeCode_eq, encodeCode]\n[GOAL]\ncf cg : Code\n\u22a2 encode cf < encode (prec cf cg) \u2227 encode cg < encode (prec cf cg)\n[PROOFSTEP]\nsuffices\n[GOAL]\ncf cg : Code\nthis : ?m.356899\n\u22a2 encode cf < encode (prec cf cg) \u2227 encode cg < encode (prec cf cg)\ncase this cf cg : Code \u22a2 ?m.356899\n[PROOFSTEP]\nexact (encode_lt_pair cf cg).imp (fun h => lt_trans h this) fun h => lt_trans h this\n[GOAL]\ncase this\ncf cg : Code\n\u22a2 encode (pair cf cg) < encode (prec cf cg)\n[PROOFSTEP]\nchange _\n[GOAL]\ncase this\ncf cg : Code\n\u22a2 encode (pair cf cg) < encode (prec cf cg)\n[PROOFSTEP]\nsimp [encodeCode_eq, encodeCode]\n[GOAL]\ncf : Code\n\u22a2 encode cf < encode (rfind' cf)\n[PROOFSTEP]\nsimp [encodeCode_eq, encodeCode]\n[GOAL]\ncf : Code\n\u22a2 encodeCode cf < 2 * (2 * encodeCode cf + 1) + 1 + 4\n[PROOFSTEP]\nhave := Nat.mul_le_mul_right cf.encodeCode (by decide : 1 \u2264 2 * 2)\n[GOAL]\ncf : Code\n\u22a2 1 \u2264 2 * 2\n[PROOFSTEP]\ndecide\n[GOAL]\ncf : Code\nthis : 1 * encodeCode cf \u2264 2 * 2 * encodeCode cf\n\u22a2 encodeCode cf < 2 * (2 * encodeCode cf + 1) + 1 + 4\n[PROOFSTEP]\nrw [one_mul, mul_assoc] at this \n[GOAL]\ncf : Code\nthis : encodeCode cf \u2264 2 * (2 * encodeCode cf)\n\u22a2 encodeCode cf < 2 * (2 * encodeCode cf + 1) + 1 + 4\n[PROOFSTEP]\nrefine' lt_of_le_of_lt (le_trans this _) (lt_add_of_pos_right _ (by decide : 0 < 4))\n[GOAL]\ncf : Code\nthis : encodeCode cf \u2264 2 * (2 * encodeCode cf)\n\u22a2 0 < 4\n[PROOFSTEP]\ndecide\n[GOAL]\ncf : Code\nthis : encodeCode cf \u2264 2 * (2 * encodeCode cf)\n\u22a2 2 * (2 * encodeCode cf) \u2264 2 * (2 * encodeCode cf + 1) + 1\n[PROOFSTEP]\nexact\n  le_of_lt\n    (Nat.lt_succ_of_le <| Nat.mul_le_mul_left _ <| le_of_lt <| Nat.lt_succ_of_le <| Nat.mul_le_mul_left _ <| le_rfl)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\n\u22a2 let PR := fun a cf cg hf hg => pr a (cf, cg, hf, hg);\n  let CO := fun a cf cg hf hg => co a (cf, cg, hf, hg);\n  let PC := fun a cf cg hf hg => pc a (cf, cg, hf, hg);\n  let RF := fun a cf hf => rf a (cf, hf);\n  let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a);\n  Primrec fun a => F a (c a)\n[PROOFSTEP]\nintros _ _ _ _ F\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nlet G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n  let a := p.1.1\n  let IH := p.1.2\n  let n := p.2.1\n  let m := p.2.2\n  (IH.get? m).bind fun s =>\n    (IH.get? m.unpair.1).bind fun s\u2081 =>\n      (IH.get? m.unpair.2).map fun s\u2082 =>\n        cond n.bodd\n          (cond n.div2.bodd (rf a (ofNat Code m, s)) (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n          (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))\n            (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nhave : Primrec G\u2081 := by\n  refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n  unfold Primrec\u2082\n  refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _\n  unfold Primrec\u2082\n  refine'\n    option_map ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n  have a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n    fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n  have n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n    fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n    snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m\u2081 := fst.comp (Primrec.unpair.comp m)\n  have m\u2082 := snd.comp (Primrec.unpair.comp m)\n  have s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n  have s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n  have s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n  unfold Primrec\u2082\n  exact\n    (nat_bodd.comp n).cond\n      ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Primrec.ofNat Code).comp m).pair s))\n        (hpc.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n      (Primrec.cond (nat_bodd.comp <| nat_div2.comp n)\n        (hco.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n        (hpr.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec G\u2081\n[PROOFSTEP]\nrefine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s =>\n    Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n      Option.map\n        (fun s\u2082 =>\n          bif bodd p.snd.fst then\n            bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd, s)\n            else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n          else\n            bif bodd (div2 p.snd.fst) then\n              co p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n            else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082))\n        (List.get? p.fst.snd (unpair p.snd.snd).snd)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun p =>\n    (fun p s =>\n        Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n          Option.map\n            (fun s\u2082 =>\n              bif bodd p.snd.fst then\n                bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd, s)\n                else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n              else\n                bif bodd (div2 p.snd.fst) then\n                  co p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n                else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082))\n            (List.get? p.fst.snd (unpair p.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s\u2081 =>\n    Option.map\n      (fun s\u2082 =>\n        bif bodd p.fst.snd.fst then\n          bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd, p.snd)\n          else pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n        else\n          bif bodd (div2 p.fst.snd.fst) then\n            co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n          else pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082))\n      (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun p =>\n    (fun p s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd p.fst.snd.fst then\n              bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd, p.snd)\n              else\n                pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 p.fst.snd.fst) then\n                co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n              else\n                pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082))\n          (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine'\n  option_map ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n  fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n  fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n  snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m\u2081 := fst.comp (Primrec.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m\u2082 := snd.comp (Primrec.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\n\u22a2 Primrec fun p =>\n    (fun p s\u2082 =>\n        bif bodd p.fst.fst.snd.fst then\n          bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n          else\n            pc p.fst.fst.fst.fst\n              (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n        else\n          bif bodd (div2 p.fst.fst.snd.fst) then\n            co p.fst.fst.fst.fst\n              (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n          else\n            pr p.fst.fst.fst.fst\n              (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082))\n      p.fst p.snd\n[PROOFSTEP]\nexact\n  (nat_bodd.comp n).cond\n    ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Primrec.ofNat Code).comp m).pair s))\n      (hpc.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n    (Primrec.cond (nat_bodd.comp <| nat_div2.comp n)\n      (hco.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n      (hpr.comp a (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nlet G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n  IH.length.casesOn (some (z a)) fun n =>\n    n.casesOn (some (s a)) fun n =>\n      n.casesOn (some (l a)) fun n => n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nhave : Primrec\u2082 G := by\n  unfold Primrec\u2082\n  refine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n  unfold Primrec\u2082\n  exact\n    this.comp <|\n      ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n        snd.pair <| nat_div2.comp <| nat_div2.comp snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 G\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p => G p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n =>\n        Nat.casesOn n (some (s p.fst)) fun n =>\n          Nat.casesOn n (some (l p.fst)) fun n =>\n            Nat.casesOn n (some (r p.fst)) fun n => G\u2081 ((p.fst, p.snd), n, div2 (div2 n)))\n      n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n =>\n            Nat.casesOn n (some (s p.fst)) fun n =>\n              Nat.casesOn n (some (l p.fst)) fun n =>\n                Nat.casesOn n (some (r p.fst)) fun n => G\u2081 ((p.fst, p.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n =>\n        Nat.casesOn n (some (l p.fst.fst)) fun n =>\n          Nat.casesOn n (some (r p.fst.fst)) fun n => G\u2081 ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n      n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n =>\n            Nat.casesOn n (some (l p.fst.fst)) fun n =>\n              Nat.casesOn n (some (r p.fst.fst)) fun n => G\u2081 ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G\u2081 ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n))) n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G\u2081 ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n => (fun n => G\u2081 ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p => (fun p n => (fun n => G\u2081 ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n) p.fst p.snd\n[PROOFSTEP]\nexact\n  this.comp <|\n    ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n      snd.pair <| nat_div2.comp <| nat_div2.comp snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nrefine'\n  ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp _root_.Primrec.id <|\n        encode_iff.2 hc).of_eq\n    fun a => by simp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).fst\n      (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).snd =\n    some ((fun a n => F a (ofNat Code n)) a n)\n[PROOFSTEP]\nsimp (config := { zeta := false })\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\niterate 4 cases' n with n; \u00b7 simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range Nat.zero)) = some (F a (ofNat Code Nat.zero))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [] = some (F a zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ n))) = some (F a (ofNat Code (Nat.succ n)))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ Nat.zero))) =\n    some (F a (ofNat Code (Nat.succ Nat.zero)))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero] = some (F a succ)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ n)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ n))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ Nat.zero)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ Nat.zero))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ] = some (F a left)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n)))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ, F a left] = some (F a right)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (List.length\n        (List.map\n          (fun n =>\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nrw [List.length_map, List.length_range]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nlet m := n.div2.div2\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nshow G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) = some (F a (ofNat Code (n + 4)))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave hm : m < n + 4 := by\n  simp [Nat.div2_val]\n  exact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nsimp [List.get?_map, List.get?_range, hm, m1, m2]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4))\n[PROOFSTEP]\nrw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4))\n[PROOFSTEP]\nsimp [ofNatCode]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.bodd\n[GOAL]\ncase succ.succ.succ.succ.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.false.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif false then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.false.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif true then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif false then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Primrec\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Primrec\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Primrec\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Primrec\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif true then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\n\u22a2 let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a);\n  Primrec fun a => F a (c a)\n[PROOFSTEP]\nintros F\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nlet G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n  let a := p.1.1\n  let IH := p.1.2\n  let n := p.2.1\n  let m := p.2.2\n  (IH.get? m).bind fun s =>\n    (IH.get? m.unpair.1).bind fun s\u2081 =>\n      (IH.get? m.unpair.2).map fun s\u2082 =>\n        cond n.bodd\n          (cond n.div2.bodd (rf a (ofNat Code m) s) (pc a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s\u2081 s\u2082))\n          (cond n.div2.bodd (co a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s\u2081 s\u2082)\n            (pr a (ofNat Code m.unpair.1) (ofNat Code m.unpair.2) s\u2081 s\u2082))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nhave : Primrec G\u2081 := by\n  refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n  unfold Primrec\u2082\n  refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _\n  unfold Primrec\u2082\n  refine'\n    option_map ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n  have a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n    fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n  have n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n    fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n    snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m\u2081 := fst.comp (Primrec.unpair.comp m)\n  have m\u2082 := snd.comp (Primrec.unpair.comp m)\n  have s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n  have s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n  have s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n  have h\u2081 := hrf.comp <| a.pair (((Primrec.ofNat Code).comp m).pair s)\n  have h\u2082 :=\n    hpc.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n  have h\u2083 :=\n    hco.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n  have h\u2084 :=\n    hpr.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n  unfold Primrec\u2082\n  exact\n    (nat_bodd.comp n).cond ((nat_bodd.comp <| nat_div2.comp n).cond h\u2081 h\u2082)\n      (cond (nat_bodd.comp <| nat_div2.comp n) h\u2083 h\u2084)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec G\u2081\n[PROOFSTEP]\nrefine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s =>\n    Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n      Option.map\n        (fun s\u2082 =>\n          bif bodd p.snd.fst then\n            bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n            else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082\n          else\n            bif bodd (div2 p.snd.fst) then\n              co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082\n            else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082)\n        (List.get? p.fst.snd (unpair p.snd.snd).snd)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun p =>\n    (fun p s =>\n        Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n          Option.map\n            (fun s\u2082 =>\n              bif bodd p.snd.fst then\n                bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd) s\n                else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082\n              else\n                bif bodd (div2 p.snd.fst) then\n                  co p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082\n                else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst) (ofNat Code (unpair p.snd.snd).snd) s\u2081 s\u2082)\n            (List.get? p.fst.snd (unpair p.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s\u2081 =>\n    Option.map\n      (fun s\u2082 =>\n        bif bodd p.fst.snd.fst then\n          bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n          else pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082\n        else\n          bif bodd (div2 p.fst.snd.fst) then\n            co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082\n          else pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082)\n      (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec fun p =>\n    (fun p s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd p.fst.snd.fst then\n              bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd) p.snd\n              else\n                pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 p.fst.snd.fst) then\n                co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082\n              else\n                pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst) (ofNat Code (unpair p.fst.snd.snd).snd) s\u2081 s\u2082)\n          (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine'\n  option_map ((list_get?.comp (snd.comp fst) (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n  fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n  fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n  snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave m\u2081 := fst.comp (Primrec.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave m\u2082 := snd.comp (Primrec.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave h\u2081 := hrf.comp <| a.pair (((Primrec.ofNat Code).comp m).pair s)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\nh\u2081 :\n  Primrec fun a =>\n    rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave h\u2082 := hpc.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\nh\u2081 :\n  Primrec fun a =>\n    rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh\u2082 :\n  Primrec fun a =>\n    pc\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave h\u2083 := hco.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\nh\u2081 :\n  Primrec fun a =>\n    rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh\u2082 :\n  Primrec fun a =>\n    pc\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh\u2083 :\n  Primrec fun a =>\n    co\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nhave h\u2084 := hpr.comp <| a.pair (((Primrec.ofNat Code).comp m\u2081).pair <| ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\nh\u2081 :\n  Primrec fun a =>\n    rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh\u2082 :\n  Primrec fun a =>\n    pc\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh\u2083 :\n  Primrec fun a =>\n    co\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh\u2084 :\n  Primrec fun a =>\n    pr\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n\u22a2 Primrec\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n      else\n        pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n      else\n        pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n          p.snd s\u2082\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Primrec s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\na : Primrec fun p => p.fst.fst.fst.fst.fst\nn : Primrec fun p => p.fst.fst.fst.snd.fst\nm : Primrec fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Primrec fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Primrec fun p => p.fst.fst.snd\ns\u2081 : Primrec fun p => p.fst.snd\ns\u2082 : Primrec fun p => p.snd\nh\u2081 :\n  Primrec fun a =>\n    rf (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code a.fst.fst.fst.snd.snd, a.fst.fst.snd).snd.snd\nh\u2082 :\n  Primrec fun a =>\n    pc\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh\u2083 :\n  Primrec fun a =>\n    co\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\nh\u2084 :\n  Primrec fun a =>\n    pr\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n          ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n            ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n              ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.fst\n      (a.fst.fst.fst.fst.fst, ofNat Code (unpair a.fst.fst.fst.snd.snd).fst,\n                ofNat Code (unpair a.fst.fst.fst.snd.snd).snd, a.fst.snd, a.snd).snd.snd.snd.snd\n\u22a2 Primrec fun p =>\n    (fun p s\u2082 =>\n        bif bodd p.fst.fst.snd.fst then\n          bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd) p.fst.snd\n          else\n            pc p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n              p.snd s\u2082\n        else\n          bif bodd (div2 p.fst.fst.snd.fst) then\n            co p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n              p.snd s\u2082\n          else\n            pr p.fst.fst.fst.fst (ofNat Code (unpair p.fst.fst.snd.snd).fst) (ofNat Code (unpair p.fst.fst.snd.snd).snd)\n              p.snd s\u2082)\n      p.fst p.snd\n[PROOFSTEP]\nexact\n  (nat_bodd.comp n).cond ((nat_bodd.comp <| nat_div2.comp n).cond h\u2081 h\u2082) (cond (nat_bodd.comp <| nat_div2.comp n) h\u2083 h\u2084)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nlet G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n  IH.length.casesOn (some (z a)) fun n =>\n    n.casesOn (some (s a)) fun n =>\n      n.casesOn (some (l a)) fun n => n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nhave : Primrec\u2082 G := by\n  unfold Primrec\u2082\n  refine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n  unfold Primrec\u2082\n  refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n  unfold Primrec\u2082\n  exact\n    this.comp <|\n      ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n        snd.pair <| nat_div2.comp <| nat_div2.comp snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 G\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p => G p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n =>\n        Nat.casesOn n (some (s p.fst)) fun n =>\n          Nat.casesOn n (some (l p.fst)) fun n =>\n            Nat.casesOn n (some (r p.fst)) fun n => G\u2081 ((p.fst, p.snd), n, div2 (div2 n)))\n      n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n =>\n            Nat.casesOn n (some (s p.fst)) fun n =>\n              Nat.casesOn n (some (l p.fst)) fun n =>\n                Nat.casesOn n (some (r p.fst)) fun n => G\u2081 ((p.fst, p.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n =>\n        Nat.casesOn n (some (l p.fst.fst)) fun n =>\n          Nat.casesOn n (some (r p.fst.fst)) fun n => G\u2081 ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n      n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n =>\n            Nat.casesOn n (some (l p.fst.fst)) fun n =>\n              Nat.casesOn n (some (r p.fst.fst)) fun n => G\u2081 ((p.fst.fst, p.fst.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n =>\n    (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G\u2081 ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n))) n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun n => Nat.casesOn n (some (r p.fst.fst.fst)) fun n => G\u2081 ((p.fst.fst.fst, p.fst.fst.snd), n, div2 (div2 n)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec\u2082 fun p n => (fun n => G\u2081 ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Primrec fun p => (fun p n => (fun n => G\u2081 ((p.fst.fst.fst.fst, p.fst.fst.fst.snd), n, div2 (div2 n))) n) p.fst p.snd\n[PROOFSTEP]\nexact\n  this.comp <|\n    ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n      snd.pair <| nat_div2.comp <| nat_div2.comp snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\n\u22a2 Primrec fun a => F a (c a)\n[PROOFSTEP]\nrefine'\n  ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp _root_.Primrec.id <|\n        encode_iff.2 hc).of_eq\n    fun a => by simp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).fst\n      (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).snd =\n    some ((fun a n => F a (ofNat Code n)) a n)\n[PROOFSTEP]\nsimp (config := { zeta := false })\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\niterate 4 cases' n with n; \u00b7 simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range Nat.zero)) = some (F a (ofNat Code Nat.zero))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [] = some (F a zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ n))) = some (F a (ofNat Code (Nat.succ n)))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ Nat.zero))) =\n    some (F a (ofNat Code (Nat.succ Nat.zero)))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero] = some (F a succ)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ n)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ n))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ Nat.zero)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ Nat.zero))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ] = some (F a left)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n)))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ, F a left] = some (F a right)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n                              else\n                                pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                            else\n                              bif bodd (div2 n_4) then\n                                co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                              else\n                                pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082)\n                          (List.get?\n                            (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (List.length\n        (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nrw [List.length_map, List.length_range]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n                              else\n                                pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                            else\n                              bif bodd (div2 n_4) then\n                                co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                              else\n                                pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082)\n                          (List.get?\n                            (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nlet m := n.div2.div2\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4))) s_1\n                              else\n                                pc a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                            else\n                              bif bodd (div2 n_4) then\n                                co a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082\n                              else\n                                pr a (ofNat Code (unpair (div2 (div2 n_4))).fst)\n                                  (ofNat Code (unpair (div2 (div2 n_4))).snd) s\u2081 s\u2082)\n                          (List.get?\n                            (List.map (fun n => rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nshow G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) = some (F a (ofNat Code (n + 4)))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave hm : m < n + 4 := by\n  simp [Nat.div2_val]\n  exact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nsimp [List.get?_map, List.get?_range, hm, m1, m2]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (n + 4))\n[PROOFSTEP]\nrw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNatCode (n + 4))\n[PROOFSTEP]\nsimp [ofNatCode]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.bodd\n[GOAL]\ncase succ.succ.succ.succ.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif bodd (div2 n) then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match false, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif bodd (div2 n) then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match true, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.false.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif false then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match false, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.false.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif true then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match false, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif false then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match true, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Primrec c\nz : \u03b1 \u2192 \u03c3\nhz : Primrec z\ns : \u03b1 \u2192 \u03c3\nhs : Primrec s\nl : \u03b1 \u2192 \u03c3\nhl : Primrec l\nr : \u03b1 \u2192 \u03c3\nhr : Primrec r\npr : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpr : Primrec fun a => pr a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nco : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhco : Primrec fun a => co a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\npc : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3\nhpc : Primrec fun a => pc a.fst a.snd.fst a.snd.snd.fst a.snd.snd.snd.fst a.snd.snd.snd.snd\nrf : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3\nhrf : Primrec fun a => rf a.fst a.snd.fst a.snd.snd\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m) s\n              else pc a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082\n              else pr a (ofNat Code (unpair m).fst) (ofNat Code (unpair m).snd) s\u2081 s\u2082)\n          (List.get? IH (unpair m).snd)\nthis\u271d : Primrec G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Primrec\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif true then\n        rf a (ofNat Code (div2 (div2 n)))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (div2 (div2 n))))\n      else\n        pc a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a (ofNat Code (unpair (div2 (div2 n))).fst) (ofNat Code (unpair (div2 (div2 n))).snd)\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).fst))\n          (rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (pr a) (co a) (pc a) (rf a)\n      (match true, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\n\u22a2 let PR := fun a cf cg hf hg => pr a (cf, cg, hf, hg);\n  let CO := fun a cf cg hf hg => co a (cf, cg, hf, hg);\n  let PC := fun a cf cg hf hg => pc a (cf, cg, hf, hg);\n  let RF := fun a cf hf => rf a (cf, hf);\n  let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a);\n  Computable fun a => F a (c a)\n[PROOFSTEP]\nintros _ _ _ _ F\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\n\u22a2 Computable fun a => F a (c a)\n[PROOFSTEP]\nlet G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n  let a := p.1.1\n  let IH := p.1.2\n  let n := p.2.1\n  let m := p.2.2\n  (IH.get? m).bind fun s =>\n    (IH.get? m.unpair.1).bind fun s\u2081 =>\n      (IH.get? m.unpair.2).map fun s\u2082 =>\n        cond n.bodd\n          (cond n.div2.bodd (rf a (ofNat Code m, s)) (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n          (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))\n            (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable fun a => F a (c a)\n[PROOFSTEP]\nhave : Computable G\u2081 := by\n  refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n  unfold Computable\u2082\n  refine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _\n  unfold Computable\u2082\n  refine'\n    option_map\n      ((list_get?.comp (snd.comp fst) (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n  have a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n    fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n  have n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n    fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n    snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n  have m\u2081 := fst.comp (Computable.unpair.comp m)\n  have m\u2082 := snd.comp (Computable.unpair.comp m)\n  have s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n  have s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n  have s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n  exact\n    (nat_bodd.comp n).cond\n      ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n        (hpc.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n      (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n        (hco.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n        (hpr.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable G\u2081\n[PROOFSTEP]\nrefine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable\u2082 fun p s =>\n    Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n      Option.map\n        (fun s\u2082 =>\n          bif bodd p.snd.fst then\n            bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd, s)\n            else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n          else\n            bif bodd (div2 p.snd.fst) then\n              co p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n            else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082))\n        (List.get? p.fst.snd (unpair p.snd.snd).snd)\n[PROOFSTEP]\nunfold Computable\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable fun p =>\n    (fun p s =>\n        Option.bind (List.get? p.fst.snd (unpair p.snd.snd).fst) fun s\u2081 =>\n          Option.map\n            (fun s\u2082 =>\n              bif bodd p.snd.fst then\n                bif bodd (div2 p.snd.fst) then rf p.fst.fst (ofNat Code p.snd.snd, s)\n                else pc p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n              else\n                bif bodd (div2 p.snd.fst) then\n                  co p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082)\n                else pr p.fst.fst (ofNat Code (unpair p.snd.snd).fst, ofNat Code (unpair p.snd.snd).snd, s\u2081, s\u2082))\n            (List.get? p.fst.snd (unpair p.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine' option_bind ((list_get?.comp (snd.comp fst) (fst.comp <| Computable.unpair.comp (snd.comp snd))).comp fst) _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable\u2082 fun p s\u2081 =>\n    Option.map\n      (fun s\u2082 =>\n        bif bodd p.fst.snd.fst then\n          bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd, p.snd)\n          else pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n        else\n          bif bodd (div2 p.fst.snd.fst) then\n            co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n          else pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082))\n      (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd)\n[PROOFSTEP]\nunfold Computable\u2082\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable fun p =>\n    (fun p s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd p.fst.snd.fst then\n              bif bodd (div2 p.fst.snd.fst) then rf p.fst.fst.fst (ofNat Code p.fst.snd.snd, p.snd)\n              else\n                pc p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 p.fst.snd.fst) then\n                co p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082)\n              else\n                pr p.fst.fst.fst (ofNat Code (unpair p.fst.snd.snd).fst, ofNat Code (unpair p.fst.snd.snd).snd, s\u2081, s\u2082))\n          (List.get? p.fst.fst.snd (unpair p.fst.snd.snd).snd))\n      p.fst p.snd\n[PROOFSTEP]\nrefine'\n  option_map ((list_get?.comp (snd.comp fst) (snd.comp <| Computable.unpair.comp (snd.comp snd))).comp <| fst.comp fst)\n    _\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave a : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n  fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave n : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n  fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n  snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m\u2081 := fst.comp (Computable.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).fst\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave m\u2082 := snd.comp (Computable.unpair.comp m)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).snd\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) := snd.comp (fst.comp fst)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Computable fun p => p.fst.fst.snd\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s\u2081 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) := snd.comp fst\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Computable fun p => p.fst.fst.snd\ns\u2081 : Computable fun p => p.fst.snd\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nhave s\u2082 : Computable (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) := snd\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns\u271d : \u03b1 \u2192 \u03c3\nhs : Computable s\u271d\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\na : Computable fun p => p.fst.fst.fst.fst.fst\nn : Computable fun p => p.fst.fst.fst.snd.fst\nm : Computable fun p => p.fst.fst.fst.snd.snd\nm\u2081 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).fst\nm\u2082 : Computable fun a => (unpair a.fst.fst.fst.snd.snd).snd\ns : Computable fun p => p.fst.fst.snd\ns\u2081 : Computable fun p => p.fst.snd\ns\u2082 : Computable fun p => p.snd\n\u22a2 Computable\u2082 fun p s\u2082 =>\n    bif bodd p.fst.fst.snd.fst then\n      bif bodd (div2 p.fst.fst.snd.fst) then rf p.fst.fst.fst.fst (ofNat Code p.fst.fst.snd.snd, p.fst.snd)\n      else\n        pc p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n    else\n      bif bodd (div2 p.fst.fst.snd.fst) then\n        co p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n      else\n        pr p.fst.fst.fst.fst\n          (ofNat Code (unpair p.fst.fst.snd.snd).fst, ofNat Code (unpair p.fst.fst.snd.snd).snd, p.snd, s\u2082)\n[PROOFSTEP]\nexact\n  (nat_bodd.comp n).cond\n    ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Computable.ofNat Code).comp m).pair s))\n      (hpc.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n    (Computable.cond (nat_bodd.comp <| nat_div2.comp n)\n      (hco.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n      (hpr.comp a (((Computable.ofNat Code).comp m\u2081).pair <| ((Computable.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Computable G\u2081\n\u22a2 Computable fun a => F a (c a)\n[PROOFSTEP]\nlet G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n  IH.length.casesOn (some (z a)) fun n =>\n    n.casesOn (some (s a)) fun n =>\n      n.casesOn (some (l a)) fun n => n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\n\u22a2 Computable fun a => F a (c a)\n[PROOFSTEP]\nhave : Computable\u2082 G :=\n  Computable.nat_casesOn (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) <|\n    Computable.nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) <|\n      Computable.nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) <|\n        Computable.nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst)))\n          (this.comp <|\n            ((Computable.fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n              snd.pair <| nat_div2.comp <| nat_div2.comp snd)\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\n\u22a2 Computable fun a => F a (c a)\n[PROOFSTEP]\nrefine'\n  ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp Computable.id <| encode_iff.2 hc).of_eq\n    fun a => by simp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).fst\n      (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).snd =\n    some ((fun a n => F a (ofNat Code n)) a n)\n[PROOFSTEP]\nsimp (config := { zeta := false })\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\niterate 4 cases' n with n; \u00b7 simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]; rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range n)) = some (F a (ofNat Code n))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range Nat.zero)) = some (F a (ofNat Code Nat.zero))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a [] = some (F a zero)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ n))) = some (F a (ofNat Code (Nat.succ n)))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ Nat.zero))) =\n    some (F a (ofNat Code (Nat.succ Nat.zero)))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a [F a zero] = some (F a succ)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ n)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ n))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ Nat.zero)))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ Nat.zero))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ] = some (F a left)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n)))))\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero)))))\n[PROOFSTEP]\nsimp (config := { zeta := false }) [ofNatCode_eq, ofNatCode]\n[GOAL]\ncase succ.succ.succ.zero\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\n\u22a2 G a [F a zero, F a succ, F a left] = some (F a right)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) =\n    some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nsimp only []\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (List.length\n        (List.map\n          (fun n =>\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nrw [List.length_map, List.length_range]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nlet m := n.div2.div2\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 Nat.rec (some (z a))\n      (fun n_1 n_ih =>\n        Nat.rec (some (s a))\n          (fun n_2 n_ih =>\n            Nat.rec (some (l a))\n              (fun n_3 n_ih =>\n                Nat.rec (some (r a))\n                  (fun n_4 n_ih =>\n                    Option.bind\n                      (List.get?\n                        (List.map\n                          (fun n =>\n                            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                              (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                          (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                        (div2 (div2 n_4)))\n                      fun s_1 =>\n                      Option.bind\n                        (List.get?\n                          (List.map\n                            (fun n =>\n                              rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                            (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                          (unpair (div2 (div2 n_4))).fst)\n                        fun s\u2081 =>\n                        Option.map\n                          (fun s\u2082 =>\n                            bif bodd n_4 then\n                              bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1)\n                              else\n                                pc a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                            else\n                              bif bodd (div2 n_4) then\n                                co a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082)\n                              else\n                                pr a\n                                  (ofNat Code (unpair (div2 (div2 n_4))).fst, ofNat Code (unpair (div2 (div2 n_4))).snd,\n                                    s\u2081, s\u2082))\n                          (List.get?\n                            (List.map\n                              (fun n =>\n                                rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n                                  (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n                                  (fun cf hf => rf a (cf, hf)) (ofNat Code n))\n                              (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n                            (unpair (div2 (div2 n_4))).snd))\n                  n_3)\n              n_2)\n          n_1)\n      (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) =\n    some\n      (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n        (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n        (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))\n[PROOFSTEP]\nshow G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) = some (F a (ofNat Code (n + 4)))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave hm : m < n + 4 := by\n  simp [Nat.div2_val]\n  exact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 m < n + 4\n[PROOFSTEP]\nsimp [Nat.div2_val]\n[GOAL]\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\n\u22a2 n / 2 / 2 < n + 4\n[PROOFSTEP]\nexact lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _)) (Nat.succ_le_succ (Nat.le_add_right _ _))\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nhave m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4)))\n[PROOFSTEP]\nsimp [List.get?_map, List.get?_range, hm, m1, m2]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4))\n[PROOFSTEP]\nrw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4))\n[PROOFSTEP]\nsimp [ofNatCode]\n[GOAL]\ncase succ.succ.succ.succ\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif bodd n then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match bodd n, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.bodd\n[GOAL]\ncase succ.succ.succ.succ.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif bodd (div2 n) then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif bodd (div2 n) then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, bodd (div2 n) with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\ncases n.div2.bodd\n[GOAL]\ncase succ.succ.succ.succ.false.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif false then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.false.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif false then\n      bif true then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match false, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.false\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif false then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif false then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, false with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.succ.succ.succ.true.true\n\u03b1 : Type u_1\n\u03c3 : Type u_2\ninst\u271d\u00b9 : Primcodable \u03b1\ninst\u271d : Primcodable \u03c3\nc : \u03b1 \u2192 Code\nhc : Computable c\nz : \u03b1 \u2192 \u03c3\nhz : Computable z\ns : \u03b1 \u2192 \u03c3\nhs : Computable s\nl : \u03b1 \u2192 \u03c3\nhl : Computable l\nr : \u03b1 \u2192 \u03c3\nhr : Computable r\npr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpr : Computable\u2082 pr\nco : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhco : Computable\u2082 co\npc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3\nhpc : Computable\u2082 pc\nrf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3\nhrf : Computable\u2082 rf\nPR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg)\nCO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg)\nPC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg)\nRF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf)\nF : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a)\nG\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 :=\n  fun p =>\n    let a := p.fst.fst;\n    let IH := p.fst.snd;\n    let n := p.snd.fst;\n    let m := p.snd.snd;\n    Option.bind (List.get? IH m) fun s =>\n      Option.bind (List.get? IH (unpair m).fst) fun s\u2081 =>\n        Option.map\n          (fun s\u2082 =>\n            bif bodd n then\n              bif bodd (div2 n) then rf a (ofNat Code m, s)\n              else pc a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n            else\n              bif bodd (div2 n) then co a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082)\n              else pr a (ofNat Code (unpair m).fst, ofNat Code (unpair m).snd, s\u2081, s\u2082))\n          (List.get? IH (unpair m).snd)\nthis\u271d : Computable G\u2081\nG : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 :=\n  fun a IH =>\n    Nat.casesOn (List.length IH) (some (z a)) fun n =>\n      Nat.casesOn n (some (s a)) fun n =>\n        Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n))\nthis : Computable\u2082 G\na : \u03b1\nn : \u2115\nm : \u2115 := div2 (div2 n)\nhm : m < n + 4\nm1 : (unpair m).fst < n + 4\nm2 : (unpair m).snd < n + 4\n\u22a2 (bif true then\n      bif true then\n        rf a\n          (ofNat Code (div2 (div2 n)),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n))))\n      else\n        pc a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n    else\n      bif true then\n        co a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))\n      else\n        pr a\n          (ofNat Code (unpair (div2 (div2 n))).fst, ofNat Code (unpair (div2 (div2 n))).snd,\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).fst),\n            rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg))\n              (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg))\n              (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).snd))) =\n    rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg))\n      (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf))\n      (match true, true with\n      | false, false => pair (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | false, true => comp (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, false => prec (ofNatCode (unpair (div2 (div2 n))).fst) (ofNatCode (unpair (div2 (div2 n))).snd)\n      | true, true => rfind' (ofNatCode (div2 (div2 n))))\n[PROOFSTEP]\nrfl\n[GOAL]\ncf cg : Code\na : \u2115\n\u22a2 eval (prec cf cg) (Nat.pair a 0) = eval cf a\n[PROOFSTEP]\nrw [eval, Nat.unpaired, Nat.unpair_pair]\n[GOAL]\ncf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf (a, 0).fst)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair (a, 0).fst (Nat.pair y i)))\n      (a, 0).snd =\n    eval cf a\n[PROOFSTEP]\nsimp (config := { Lean.Meta.Simp.neutralConfig with proj := true }) only []\n[GOAL]\ncf cg : Code\na : \u2115\n\u22a2 Nat.rec (eval cf a)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair a (Nat.pair y i)))\n      0 =\n    eval cf a\n[PROOFSTEP]\nrw [Nat.rec_zero]\n[GOAL]\ncf cg : Code\na k : \u2115\n\u22a2 eval (prec cf cg) (Nat.pair a (Nat.succ k)) = do\n    let ih \u2190 eval (prec cf cg) (Nat.pair a k)\n    eval cg (Nat.pair a (Nat.pair k ih))\n[PROOFSTEP]\nrw [eval, Nat.unpaired, Part.bind_eq_bind, Nat.unpair_pair]\n[GOAL]\ncf cg : Code\na k : \u2115\n\u22a2 Nat.rec (eval cf (a, Nat.succ k).fst)\n      (fun y IH => do\n        let i \u2190 IH\n        eval cg (Nat.pair (a, Nat.succ k).fst (Nat.pair y i)))\n      (a, Nat.succ k).snd =\n    Part.bind\n      (unpaired\n        (fun a n =>\n          Nat.rec (eval cf a)\n            (fun y IH => do\n              let i \u2190 IH\n              eval cg (Nat.pair a (Nat.pair y i)))\n            n)\n        (Nat.pair a k))\n      fun ih => eval cg (Nat.pair a (Nat.pair k ih))\n[PROOFSTEP]\nsimp\n[GOAL]\nn m : \u2115\n\u22a2 eval (Code.const (n + 1)) m = Part.some (n + 1)\n[PROOFSTEP]\nsimp! [eval_const n m]\n[GOAL]\nn : \u2115\n\u22a2 eval Code.id n = Part.some n\n[PROOFSTEP]\nsimp! [Seq.seq]\n[GOAL]\nc : Code\nn x : \u2115\n\u22a2 eval (curry c n) x = eval c (Nat.pair n x)\n[PROOFSTEP]\nsimp! [Seq.seq]\n[GOAL]\nn : \u2115\n\u22a2 (fun b => comp succ (n, b).snd)^[id n] zero = Code.const n\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 (fun b => comp succ b)^[n] zero = Code.const n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u22a2 (fun b => comp succ b)^[Nat.zero] zero = Code.const Nat.zero\n[PROOFSTEP]\nsimp [*, Code.const, Function.iterate_succ', -Function.iterate_succ]\n[GOAL]\ncase succ\nn\u271d : \u2115\nn_ih\u271d : (fun b => comp succ b)^[n\u271d] zero = Code.const n\u271d\n\u22a2 (fun b => comp succ b)^[Nat.succ n\u271d] zero = Code.const (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*, Code.const, Function.iterate_succ', -Function.iterate_succ]\n[GOAL]\nc\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh : curry c\u2081 n\u2081 = curry c\u2082 n\u2082\n\u22a2 c\u2081 = c\u2082\n[PROOFSTEP]\ninjection h\n[GOAL]\nc\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh : curry c\u2081 n\u2081 = curry c\u2082 n\u2082\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\ninjection h with h\u2081 h\u2082\n[GOAL]\nc\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2082 : pair (Code.const n\u2081) Code.id = pair (Code.const n\u2082) Code.id\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\ninjection h\u2082 with h\u2083 h\u2084\n[GOAL]\nc\u2081 c\u2082 : Code\nn\u2081 n\u2082 : \u2115\nh\u2081 : c\u2081 = c\u2082\nh\u2083 : Code.const n\u2081 = Code.const n\u2082\nh\u2084 : Code.id = Code.id\n\u22a2 n\u2081 = n\u2082\n[PROOFSTEP]\nexact const_inj h\u2083\n[GOAL]\nf : \u2115 \u2192. \u2115\nh : Partrec f\n\u22a2 \u2203 c, eval c = f\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase zero\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0\ncase succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).fst\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase zero => exact \u27e8zero, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0\n[PROOFSTEP]\ncase zero => exact \u27e8zero, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = pure 0\n[PROOFSTEP]\nexact \u27e8zero, rfl\u27e9\n[GOAL]\ncase succ\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).fst\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase succ => exact \u27e8succ, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n[PROOFSTEP]\ncase succ => exact \u27e8succ, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191Nat.succ\n[PROOFSTEP]\nexact \u27e8succ, rfl\u27e9\n[GOAL]\ncase left\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).fst\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase left => exact \u27e8left, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).fst\n[PROOFSTEP]\ncase left => exact \u27e8left, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).fst\n[PROOFSTEP]\nexact \u27e8left, rfl\u27e9\n[GOAL]\ncase right\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase right => exact \u27e8right, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\n[PROOFSTEP]\ncase right => exact \u27e8right, rfl\u27e9\n[GOAL]\nf : \u2115 \u2192. \u2115\n\u22a2 \u2203 c, eval c = \u2191fun n => (unpair n).snd\n[PROOFSTEP]\nexact \u27e8right, rfl\u27e9\n[GOAL]\ncase pair\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f\u271d n) fun x => g\u271d n\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase pair f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8pair cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f n) fun x => g n\n[PROOFSTEP]\ncase pair f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8pair cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> f n) fun x => g n\n[PROOFSTEP]\nrcases hf with \u27e8cf, rfl\u27e9\n[GOAL]\ncase intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => g n\n[PROOFSTEP]\nrcases hg with \u27e8cg, rfl\u27e9\n[GOAL]\ncase intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => Seq.seq (Nat.pair <$> eval cf n) fun x => eval cg n\n[PROOFSTEP]\nexact \u27e8pair cf cg, rfl\u27e9\n[GOAL]\ncase comp\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c, eval c = fun n => g\u271d n >>= f\u271d\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase comp f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8comp cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => g n >>= f\n[PROOFSTEP]\ncase comp f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8comp cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c, eval c = fun n => g n >>= f\n[PROOFSTEP]\nrcases hf with \u27e8cf, rfl\u27e9\n[GOAL]\ncase intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun n => g n >>= eval cf\n[PROOFSTEP]\nrcases hg with \u27e8cg, rfl\u27e9\n[GOAL]\ncase intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c, eval c = fun n => eval cg n >>= eval cf\n[PROOFSTEP]\nexact \u27e8comp cf cg, rfl\u27e9\n[GOAL]\ncase prec\nf f\u271d g\u271d : \u2115 \u2192. \u2115\na\u271d\u00b9 : Partrec f\u271d\na\u271d : Partrec g\u271d\na_ih\u271d\u00b9 : \u2203 c, eval c = f\u271d\na_ih\u271d : \u2203 c, eval c = g\u271d\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f\u271d a)\n          (fun y IH => do\n            let i \u2190 IH\n            g\u271d (Nat.pair a (Nat.pair y i)))\n          n\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase prec f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8prec cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n\n[PROOFSTEP]\ncase prec f g pf pg hf hg =>\n  rcases hf with \u27e8cf, rfl\u27e9; rcases hg with \u27e8cg, rfl\u27e9\n  exact \u27e8prec cf cg, rfl\u27e9\n[GOAL]\nf\u271d f g : \u2115 \u2192. \u2115\npf : Partrec f\npg : Partrec g\nhf : \u2203 c, eval c = f\nhg : \u2203 c, eval c = g\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (f a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n\n[PROOFSTEP]\nrcases hf with \u27e8cf, rfl\u27e9\n[GOAL]\ncase intro\nf g : \u2115 \u2192. \u2115\npg : Partrec g\nhg : \u2203 c, eval c = g\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            g (Nat.pair a (Nat.pair y i)))\n          n\n[PROOFSTEP]\nrcases hg with \u27e8cg, rfl\u27e9\n[GOAL]\ncase intro.intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\ncg : Code\npg : Partrec (eval cg)\n\u22a2 \u2203 c,\n    eval c =\n      unpaired fun a n =>\n        Nat.rec (eval cf a)\n          (fun y IH => do\n            let i \u2190 IH\n            eval cg (Nat.pair a (Nat.pair y i)))\n          n\n[PROOFSTEP]\nexact \u27e8prec cf cg, rfl\u27e9\n[GOAL]\ncase rfind\nf f\u271d : \u2115 \u2192. \u2115\na\u271d : Partrec f\u271d\na_ih\u271d : \u2203 c, eval c = f\u271d\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f\u271d (Nat.pair a n)\n[PROOFSTEP]\ncase rfind f pf hf =>\n  rcases hf with \u27e8cf, rfl\u27e9\n  refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9\n  simp [eval, Seq.seq, pure, PFun.pure, Part.map_id']\n[GOAL]\nf\u271d f : \u2115 \u2192. \u2115\npf : Partrec f\nhf : \u2203 c, eval c = f\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a n)\n[PROOFSTEP]\ncase rfind f pf hf =>\n  rcases hf with \u27e8cf, rfl\u27e9\n  refine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9\n  simp [eval, Seq.seq, pure, PFun.pure, Part.map_id']\n[GOAL]\nf\u271d f : \u2115 \u2192. \u2115\npf : Partrec f\nhf : \u2203 c, eval c = f\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> f (Nat.pair a n)\n[PROOFSTEP]\nrcases hf with \u27e8cf, rfl\u27e9\n[GOAL]\ncase intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 \u2203 c, eval c = fun a => Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)\n[PROOFSTEP]\nrefine' \u27e8comp (rfind' cf) (pair Code.id zero), _\u27e9\n[GOAL]\ncase intro\nf : \u2115 \u2192. \u2115\ncf : Code\npf : Partrec (eval cf)\n\u22a2 eval (comp (rfind' cf) (pair Code.id zero)) = fun a =>\n    Nat.rfind fun n => (fun m => decide (m = 0)) <$> eval cf (Nat.pair a n)\n[PROOFSTEP]\nsimp [eval, Seq.seq, pure, PFun.pure, Part.map_id']\n[GOAL]\nf : \u2115 \u2192. \u2115\nh : \u2203 c, eval c = f\n\u22a2 Partrec f\n[PROOFSTEP]\nrcases h with \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\nc : Code\n\u22a2 Partrec (eval c)\n[PROOFSTEP]\ninduction c\n[GOAL]\ncase intro.zero\n\u22a2 Partrec (eval zero)\ncase intro.succ\n\u22a2 Partrec (eval succ)\ncase intro.left\n\u22a2 Partrec (eval left)\ncase intro.right\n\u22a2 Partrec (eval right)\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase zero => exact Nat.Partrec.zero\n[GOAL]\n\u22a2 Partrec (eval zero)\n[PROOFSTEP]\ncase zero => exact Nat.Partrec.zero\n[GOAL]\n\u22a2 Partrec (eval zero)\n[PROOFSTEP]\nexact Nat.Partrec.zero\n[GOAL]\ncase intro.succ\n\u22a2 Partrec (eval succ)\ncase intro.left\n\u22a2 Partrec (eval left)\ncase intro.right\n\u22a2 Partrec (eval right)\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase succ => exact Nat.Partrec.succ\n[GOAL]\n\u22a2 Partrec (eval succ)\n[PROOFSTEP]\ncase succ => exact Nat.Partrec.succ\n[GOAL]\n\u22a2 Partrec (eval succ)\n[PROOFSTEP]\nexact Nat.Partrec.succ\n[GOAL]\ncase intro.left\n\u22a2 Partrec (eval left)\ncase intro.right\n\u22a2 Partrec (eval right)\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase left => exact Nat.Partrec.left\n[GOAL]\n\u22a2 Partrec (eval left)\n[PROOFSTEP]\ncase left => exact Nat.Partrec.left\n[GOAL]\n\u22a2 Partrec (eval left)\n[PROOFSTEP]\nexact Nat.Partrec.left\n[GOAL]\ncase intro.right\n\u22a2 Partrec (eval right)\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase right => exact Nat.Partrec.right\n[GOAL]\n\u22a2 Partrec (eval right)\n[PROOFSTEP]\ncase right => exact Nat.Partrec.right\n[GOAL]\n\u22a2 Partrec (eval right)\n[PROOFSTEP]\nexact Nat.Partrec.right\n[GOAL]\ncase intro.pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (pair a\u271d\u00b9 a\u271d))\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase pair cf cg pf pg => exact pf.pair pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (pair cf cg))\n[PROOFSTEP]\ncase pair cf cg pf pg => exact pf.pair pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (pair cf cg))\n[PROOFSTEP]\nexact pf.pair pg\n[GOAL]\ncase intro.comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (comp a\u271d\u00b9 a\u271d))\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase comp cf cg pf pg => exact pf.comp pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (comp cf cg))\n[PROOFSTEP]\ncase comp cf cg pf pg => exact pf.comp pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (comp cf cg))\n[PROOFSTEP]\nexact pf.comp pg\n[GOAL]\ncase intro.prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : Partrec (eval a\u271d\u00b9)\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (prec a\u271d\u00b9 a\u271d))\ncase intro.rfind' a\u271d : Code a_ih\u271d : Partrec (eval a\u271d) \u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase prec cf cg pf pg => exact pf.prec pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (prec cf cg))\n[PROOFSTEP]\ncase prec cf cg pf pg => exact pf.prec pg\n[GOAL]\ncf cg : Code\npf : Partrec (eval cf)\npg : Partrec (eval cg)\n\u22a2 Partrec (eval (prec cf cg))\n[PROOFSTEP]\nexact pf.prec pg\n[GOAL]\ncase intro.rfind'\na\u271d : Code\na_ih\u271d : Partrec (eval a\u271d)\n\u22a2 Partrec (eval (rfind' a\u271d))\n[PROOFSTEP]\ncase rfind' cf pf => exact pf.rfind'\n[GOAL]\ncf : Code\npf : Partrec (eval cf)\n\u22a2 Partrec (eval (rfind' cf))\n[PROOFSTEP]\ncase rfind' cf pf => exact pf.rfind'\n[GOAL]\ncf : Code\npf : Partrec (eval cf)\n\u22a2 Partrec (eval (rfind' cf))\n[PROOFSTEP]\nexact pf.rfind'\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := pair cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := pair cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cf < sizeOf (pair cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := pair cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := pair cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cg < sizeOf (pair cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := comp cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := comp cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cg < sizeOf (comp cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := comp cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := comp cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cf < sizeOf (comp cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cf < sizeOf (prec cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k, snd := prec cf cg } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k, snd := prec cf cg } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 k < Nat.succ k\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf cg : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cg } { fst := Nat.succ k, snd := prec cf cg }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf cg : Code\n\u22a2 sizeOf cg < sizeOf (prec cf cg)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := rfind' cf }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k + 1, snd := cf } { fst := Nat.succ k, snd := rfind' cf }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf : Code\n\u22a2 sizeOf cf < sizeOf (rfind' cf)\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nk : \u2115\ncf : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k, snd := rfind' cf } { fst := Nat.succ k, snd := rfind' cf }\n[PROOFSTEP]\n{decreasing_with simp (config := { arith := true }) [Zero.zero]; done\n}\n[GOAL]\nk : \u2115\ncf : Code\n\u22a2 (invImage (fun a => PSigma.casesOn a fun k snd => (k, snd)) Prod.instWellFoundedRelationProd).1\n    { fst := k, snd := rfind' cf } { fst := Nat.succ k, snd := rfind' cf }\n[PROOFSTEP]\ndecreasing_with simp (config := { arith := true }) [Zero.zero]; done\n[GOAL]\ncase h\nk : \u2115\ncf : Code\n\u22a2 k < Nat.succ k\n[PROOFSTEP]\nsimp (config := { arith := true }) [Zero.zero]\n[GOAL]\n\n[PROOFSTEP]\ndone\n[GOAL]\nc : Code\nn x : \u2115\nh : x \u2208 evaln 0 c n\n\u22a2 n < 0\n[PROOFSTEP]\nsimp [evaln] at h \n[GOAL]\nk : \u2115\nc : Code\nn x : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 n < k + 1\n[PROOFSTEP]\nsuffices\n  \u2200 {o : Option \u2115},\n    x \u2208 do {\n        guard (n \u2264 k);\n        o\n        } \u2192\n      n < k + 1\n  by cases c <;> rw [evaln] at h  <;> exact this h\n[GOAL]\nk : \u2115\nc : Code\nn x : \u2115\nh : x \u2208 evaln (k + 1) c n\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\n\u22a2 n < k + 1\n[PROOFSTEP]\ncases c\n[GOAL]\ncase zero\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh : x \u2208 evaln (k + 1) zero n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase succ\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh : x \u2208 evaln (k + 1) succ n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase left\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh : x \u2208 evaln (k + 1) left n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase right\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh : x \u2208 evaln (k + 1) right n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase pair\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh : x \u2208 evaln (k + 1) (pair a\u271d\u00b9 a\u271d) n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase comp\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh : x \u2208 evaln (k + 1) (comp a\u271d\u00b9 a\u271d) n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase prec\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh : x \u2208 evaln (k + 1) (prec a\u271d\u00b9 a\u271d) n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase rfind'\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d : Code\nh : x \u2208 evaln (k + 1) (rfind' a\u271d) n\n\u22a2 n < k + 1\n[PROOFSTEP]\nrw [evaln] at h \n[GOAL]\ncase zero\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        pure 0)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase succ\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        pure (Nat.succ n))\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase left\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase right\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase pair\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) a\u271d\u00b9 n) fun x => evaln (k + 1) a\u271d n)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase comp\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) a\u271d n\n        evaln (k + 1) a\u271d\u00b9 x)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase prec\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d\u00b9 a\u271d : Code\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) a\u271d\u00b9 a) fun y => do\n                let i \u2190 evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair a y)\n                evaln (k + 1) a\u271d (Nat.pair a (Nat.pair y i)))\n            n)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\ncase rfind'\nk n x : \u2115\nthis :\n  \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\na\u271d : Code\nh :\n  x \u2208\n    (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) a\u271d (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' a\u271d) (Nat.pair a (m + 1)))\n            n)\n      n\n\u22a2 n < k + 1\n[PROOFSTEP]\nexact this h\n[GOAL]\nk : \u2115\nc : Code\nn x : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 \u2200 {o : Option \u2115},\n    (x \u2208 do\n        guard (n \u2264 k)\n        o) \u2192\n      n < k + 1\n[PROOFSTEP]\nsimpa [Bind.bind] using Nat.lt_succ_of_le\n[GOAL]\nk\u2082 : \u2115\nc : Code\nn x : \u2115\nx\u271d : 0 \u2264 k\u2082\nh : x \u2208 evaln 0 c n\n\u22a2 x \u2208 evaln k\u2082 c n\n[PROOFSTEP]\nsimp [evaln] at h \n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nh : x \u2208 evaln (k + 1) c n\n\u22a2 x \u2208 evaln (k\u2082 + 1) c n\n[PROOFSTEP]\nhave hl' := Nat.le_of_succ_le_succ hl\n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nh : x \u2208 evaln (k + 1) c n\nhl' : k \u2264 k\u2082\n\u22a2 x \u2208 evaln (k\u2082 + 1) c n\n[PROOFSTEP]\nhave :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        x \u2208 do {\n            guard (n \u2264 k);\n            o\u2081\n            } \u2192\n          x \u2208 do {\n            guard (n \u2264 k\u2082);\n            o\u2082\n            } :=\n  by\n  simp [Bind.bind]\n  introv h h\u2081 h\u2082 h\u2083\n  exact \u27e8le_trans h\u2082 h, h\u2081 h\u2083\u27e9\n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nh : x \u2208 evaln (k + 1) c n\nhl' : k \u2264 k\u2082\n\u22a2 \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\n[PROOFSTEP]\nsimp [Bind.bind]\n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nh : x \u2208 evaln (k + 1) c n\nhl' : k \u2264 k\u2082\n\u22a2 \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115}, k \u2264 k\u2082 \u2192 (o\u2081 = some x \u2192 o\u2082 = some x) \u2192 n \u2264 k \u2192 o\u2081 = some x \u2192 n \u2264 k\u2082 \u2227 o\u2082 = some x\n[PROOFSTEP]\nintrov h h\u2081 h\u2082 h\u2083\n[GOAL]\nk\u271d k\u2082\u271d : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k\u271d + 1 \u2264 k\u2082\u271d + 1\nh\u271d : x\u271d \u2208 evaln (k\u271d + 1) c n\u271d\nhl' : k\u271d \u2264 k\u2082\u271d\nk k\u2082 n x : \u2115\no\u2081 o\u2082 : Option \u2115\nh : k \u2264 k\u2082\nh\u2081 : o\u2081 = some x \u2192 o\u2082 = some x\nh\u2082 : n \u2264 k\nh\u2083 : o\u2081 = some x\n\u22a2 n \u2264 k\u2082 \u2227 o\u2082 = some x\n[PROOFSTEP]\nexact \u27e8le_trans h\u2082 h, h\u2081 h\u2083\u27e9\n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nh : x \u2208 evaln (k + 1) c n\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\n\u22a2 x \u2208 evaln (k\u2082 + 1) c n\n[PROOFSTEP]\nsimp at h \u22a2\n[GOAL]\nk k\u2082 : \u2115\nc : Code\nn x : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh : evaln (k + 1) c n = some x\n\u22a2 evaln (k\u2082 + 1) c n = some x\n[PROOFSTEP]\ninduction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generalizing x n\n[GOAL]\ncase zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh : evaln (k + 1) zero n = some x\n\u22a2 evaln (k\u2082 + 1) zero n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase succ\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh : evaln (k + 1) succ n = some x\n\u22a2 evaln (k\u2082 + 1) succ n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh : evaln (k + 1) left n = some x\n\u22a2 evaln (k\u2082 + 1) left n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh : evaln (k + 1) right n = some x\n\u22a2 evaln (k\u2082 + 1) right n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh : evaln (k + 1) (pair cf cg) n = some x\n\u22a2 evaln (k\u2082 + 1) (pair cf cg) n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh : evaln (k + 1) (comp cf cg) n = some x\n\u22a2 evaln (k\u2082 + 1) (comp cf cg) n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh : evaln (k + 1) (prec cf cg) n = some x\n\u22a2 evaln (k\u2082 + 1) (prec cf cg) n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh : evaln (k + 1) (rfind' cf) n = some x\n\u22a2 evaln (k\u2082 + 1) (rfind' cf) n = some x\n[PROOFSTEP]\nrw [evaln] at h \u22a2\n[GOAL]\ncase zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        pure 0)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        pure 0)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase succ\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (Nat.succ n))\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        pure (Nat.succ n))\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        pure (unpair n).fst)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        pure (unpair n).snd)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        let x \u2190 evaln (k\u2082 + 1) cg n\n        evaln (k\u2082 + 1) cf x)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n                let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n                evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\n\u22a2 (fun n => do\n        guard (n \u2264 k\u2082)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\n[PROOFSTEP]\nrefine' this hl' (fun h => _) h\n[GOAL]\ncase zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure 0)\n      n =\n    some x\nh : x \u2208 pure 0\n\u22a2 x \u2208 pure 0\ncase succ\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (Nat.succ n))\n      n =\n    some x\nh : x \u2208 pure (Nat.succ n)\n\u22a2 x \u2208 pure (Nat.succ n)\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n =\n    some x\nh : x \u2208 pure (unpair n).fst\n\u22a2 x \u2208 pure (unpair n).fst\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\nh : x \u2208 pure (unpair n).snd\n\u22a2 x \u2208 pure (unpair n).snd\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\niterate 4 exact h\n[GOAL]\ncase zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure 0)\n      n =\n    some x\nh : x \u2208 pure 0\n\u22a2 x \u2208 pure 0\ncase succ\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (Nat.succ n))\n      n =\n    some x\nh : x \u2208 pure (Nat.succ n)\n\u22a2 x \u2208 pure (Nat.succ n)\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n =\n    some x\nh : x \u2208 pure (unpair n).fst\n\u22a2 x \u2208 pure (unpair n).fst\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\nh : x \u2208 pure (unpair n).snd\n\u22a2 x \u2208 pure (unpair n).snd\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\nexact h\n[GOAL]\ncase succ\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (Nat.succ n))\n      n =\n    some x\nh : x \u2208 pure (Nat.succ n)\n\u22a2 x \u2208 pure (Nat.succ n)\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n =\n    some x\nh : x \u2208 pure (unpair n).fst\n\u22a2 x \u2208 pure (unpair n).fst\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\nh : x \u2208 pure (unpair n).snd\n\u22a2 x \u2208 pure (unpair n).snd\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\nexact h\n[GOAL]\ncase left\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).fst)\n      n =\n    some x\nh : x \u2208 pure (unpair n).fst\n\u22a2 x \u2208 pure (unpair n).fst\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\nh : x \u2208 pure (unpair n).snd\n\u22a2 x \u2208 pure (unpair n).snd\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\nexact h\n[GOAL]\ncase right\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        pure (unpair n).snd)\n      n =\n    some x\nh : x \u2208 pure (unpair n).snd\n\u22a2 x \u2208 pure (unpair n).snd\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\nexact h\n[GOAL]\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : x \u2208 Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n\n\u22a2 x \u2208 Seq.seq (Nat.pair <$> evaln (k\u2082 + 1) cf n) fun x => evaln (k\u2082 + 1) cg n\n[PROOFSTEP]\nsimp [Seq.seq] at h \u22a2\n[GOAL]\ncase pair\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        Seq.seq (Nat.pair <$> evaln (k + 1) cf n) fun x => evaln (k + 1) cg n)\n      n =\n    some x\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, evaln (k\u2082 + 1) cf n = some a \u2227 \u2203 a_1, evaln (k\u2082 + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n[PROOFSTEP]\nexact h.imp fun a => And.imp (hf _ _) <| Exists.imp fun b => And.imp_left (hg _ _)\n[GOAL]\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh :\n  x \u2208 do\n    let x \u2190 evaln (k + 1) cg n\n    evaln (k + 1) cf x\n\u22a2 x \u2208 do\n    let x \u2190 evaln (k\u2082 + 1) cg n\n    evaln (k\u2082 + 1) cf x\n[PROOFSTEP]\nsimp [Bind.bind] at h \u22a2\n[GOAL]\ncase comp\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        let x \u2190 evaln (k + 1) cg n\n        evaln (k + 1) cf x)\n      n =\n    some x\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, evaln (k\u2082 + 1) cg n = some a \u2227 evaln (k\u2082 + 1) cf a = some x\n[PROOFSTEP]\nexact h.imp fun a => And.imp (hg _ _) (hf _ _)\n[GOAL]\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n          let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n          evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a n =>\n        Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n          let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n          evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n      n\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n\u22a2 x \u2208\n      unpaired\n        (fun a n =>\n          Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n            let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n            evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n        n \u2192\n    x \u2208\n      unpaired\n        (fun a n =>\n          Nat.casesOn n (evaln (k\u2082 + 1) cf a) fun y => do\n            let i \u2190 evaln k\u2082 (prec cf cg) (Nat.pair a y)\n            evaln (k\u2082 + 1) cg (Nat.pair a (Nat.pair y i)))\n        n\n[PROOFSTEP]\nsimp [Bind.bind]\n[GOAL]\ncase prec\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (unpair n).snd =\n      some x \u2192\n    Nat.rec (evaln (k\u2082 + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (unpair n).snd =\n      some x\n[PROOFSTEP]\ninduction n.unpair.2\n[GOAL]\ncase prec.zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        Nat.zero =\n      some x \u2192\n    Nat.rec (evaln (k\u2082 + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        Nat.zero =\n      some x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec.succ\nk k\u2082 : \u2115\nc : Code\nn\u271d\u00b9 x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d\u00b9 = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nn\u271d : \u2115\nn_ih\u271d :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        n\u271d =\n      some x \u2192\n    Nat.rec (evaln (k\u2082 + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        n\u271d =\n      some x\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (Nat.succ n\u271d) =\n      some x \u2192\n    Nat.rec (evaln (k\u2082 + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (Nat.succ n\u271d) =\n      some x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec.zero\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\n\u22a2 evaln (k + 1) cf (unpair n).fst = some x \u2192 evaln (k\u2082 + 1) cf (unpair n).fst = some x\n[PROOFSTEP]\napply hf\n[GOAL]\ncase prec.succ\nk k\u2082 : \u2115\nc : Code\nn\u271d\u00b9 x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d : evaln (k + 1) c n\u271d\u00b9 = some x\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nhg : \u2200 (n x : \u2115), evaln (k + 1) cg n = some x \u2192 evaln (k\u2082 + 1) cg n = some x\nn x : \u2115\nh :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a n =>\n              Nat.casesOn n (evaln (k + 1) cf a) fun y => do\n                let i \u2190 evaln k (prec cf cg) (Nat.pair a y)\n                evaln (k + 1) cg (Nat.pair a (Nat.pair y i)))\n            n)\n      n =\n    some x\nn\u271d : \u2115\nn_ih\u271d :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        n\u271d =\n      some x \u2192\n    Nat.rec (evaln (k\u2082 + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        n\u271d =\n      some x\n\u22a2 \u2200 (x_1 : \u2115),\n    evaln k (prec cf cg) (Nat.pair (unpair n).fst n\u271d) = some x_1 \u2192\n      evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d x_1)) = some x \u2192\n        \u2203 a,\n          evaln k\u2082 (prec cf cg) (Nat.pair (unpair n).fst n\u271d) = some a \u2227\n            evaln (k\u2082 + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d a)) = some x\n[PROOFSTEP]\nexact fun y h\u2081 h\u2082 => \u27e8y, evaln_mono hl' h\u2081, hg _ _ h\u2082\u27e9\n[GOAL]\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n      n\n\u22a2 x \u2208\n    unpaired\n      (fun a m => do\n        let x \u2190 evaln (k\u2082 + 1) cf (Nat.pair a m)\n        if x = 0 then pure m else evaln k\u2082 (rfind' cf) (Nat.pair a (m + 1)))\n      n\n[PROOFSTEP]\nsimp [Bind.bind] at h \u22a2\n[GOAL]\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    evaln (k\u2082 + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k\u2082 (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n[PROOFSTEP]\nrefine' h.imp fun x => And.imp (hf _ _) _\n[GOAL]\ncase rfind'\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\u00b9\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x\u271d : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\u271d\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\u271d\nx : \u2115\n\u22a2 (if x = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d \u2192\n    (if x = 0 then pure (unpair n).snd else evaln k\u2082 (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d\n[PROOFSTEP]\nby_cases x0 : x = 0\n[GOAL]\ncase pos\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\u00b9\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x\u271d : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\u271d\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\u271d\nx : \u2115\nx0 : x = 0\n\u22a2 (if x = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d \u2192\n    (if x = 0 then pure (unpair n).snd else evaln k\u2082 (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d\n[PROOFSTEP]\nsimp [x0]\n[GOAL]\ncase neg\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\u00b9\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x\u271d : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\u271d\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\u271d\nx : \u2115\nx0 : \u00acx = 0\n\u22a2 (if x = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d \u2192\n    (if x = 0 then pure (unpair n).snd else evaln k\u2082 (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n      some x\u271d\n[PROOFSTEP]\nsimp [x0]\n[GOAL]\ncase neg\nk k\u2082 : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nhl : k + 1 \u2264 k\u2082 + 1\nhl' : k \u2264 k\u2082\nthis :\n  \u2200 {k k\u2082 n x : \u2115} {o\u2081 o\u2082 : Option \u2115},\n    k \u2264 k\u2082 \u2192\n      (x \u2208 o\u2081 \u2192 x \u2208 o\u2082) \u2192\n        (x \u2208 do\n            guard (n \u2264 k)\n            o\u2081) \u2192\n          x \u2208 do\n            guard (n \u2264 k\u2082)\n            o\u2082\nh\u271d\u00b9 : evaln (k + 1) c n\u271d = some x\u271d\u00b9\ncf : Code\nhf : \u2200 (n x : \u2115), evaln (k + 1) cf n = some x \u2192 evaln (k\u2082 + 1) cf n = some x\nn x\u271d : \u2115\nh\u271d :\n  (fun n => do\n        guard (n \u2264 k)\n        unpaired\n            (fun a m => do\n              let x \u2190 evaln (k + 1) cf (Nat.pair a m)\n              if x = 0 then pure m else evaln k (rfind' cf) (Nat.pair a (m + 1)))\n            n)\n      n =\n    some x\u271d\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\u271d\nx : \u2115\nx0 : \u00acx = 0\n\u22a2 evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\u271d \u2192\n    evaln k\u2082 (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\u271d\n[PROOFSTEP]\nexact evaln_mono hl'\n[GOAL]\nx\u271d : Code\nn x : \u2115\nh : x \u2208 evaln 0 x\u271d n\n\u22a2 x \u2208 eval x\u271d n\n[PROOFSTEP]\nsimp [evaln] at h \n[GOAL]\nk : \u2115\nc : Code\nn x : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 x \u2208 eval c n\n[PROOFSTEP]\ninduction' c with cf cg hf hg cf cg hf hg cf cg hf hg cf hf generalizing x n\n[GOAL]\ncase zero\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : x \u2208 evaln (k + 1) zero n\n\u22a2 x \u2208 eval zero n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase succ\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : x \u2208 evaln (k + 1) succ n\n\u22a2 x \u2208 eval succ n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase left\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : x \u2208 evaln (k + 1) left n\n\u22a2 x \u2208 eval left n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase right\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : x \u2208 evaln (k + 1) right n\n\u22a2 x \u2208 eval right n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase pair\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh : x \u2208 evaln (k + 1) (pair cf cg) n\n\u22a2 x \u2208 eval (pair cf cg) n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase comp\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh : x \u2208 evaln (k + 1) (comp cf cg) n\n\u22a2 x \u2208 eval (comp cf cg) n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase prec\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh : x \u2208 evaln (k + 1) (prec cf cg) n\n\u22a2 x \u2208 eval (prec cf cg) n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase rfind'\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nh : x \u2208 evaln (k + 1) (rfind' cf) n\n\u22a2 x \u2208 eval (rfind' cf) n\n[PROOFSTEP]\nsimp [eval, evaln, Bind.bind, Seq.seq] at h \u22a2\n[GOAL]\ncase zero\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : n \u2264 k \u2227 pure 0 = some x\n\u22a2 x \u2208 pure 0 n\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase succ\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : n \u2264 k \u2227 pure (Nat.succ n) = some x\n\u22a2 x = Nat.succ n\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase left\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : n \u2264 k \u2227 pure (unpair n).fst = some x\n\u22a2 x = (unpair n).fst\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase right\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nh : n \u2264 k \u2227 pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase pair\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh : n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase comp\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh : n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase prec\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nh :\n  n \u2264 k \u2227\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (unpair n).snd =\n      some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase rfind'\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nh :\n  n \u2264 k \u2227\n    \u2203 a,\n      evaln (k + 1) cf n = some a \u2227\n        (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n          some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\ncases' h with _ h\n[GOAL]\ncase zero.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure 0 = some x\n\u22a2 x \u2208 pure 0 n\ncase succ.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (Nat.succ n) = some x\n\u22a2 x = Nat.succ n\ncase left.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).fst = some x\n\u22a2 x = (unpair n).fst\ncase right.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\niterate 4 simpa [pure, PFun.pure, eq_comm] using h\n[GOAL]\ncase zero.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure 0 = some x\n\u22a2 x \u2208 pure 0 n\ncase succ.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (Nat.succ n) = some x\n\u22a2 x = Nat.succ n\ncase left.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).fst = some x\n\u22a2 x = (unpair n).fst\ncase right.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimpa [pure, PFun.pure, eq_comm] using h\n[GOAL]\ncase succ.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (Nat.succ n) = some x\n\u22a2 x = Nat.succ n\ncase left.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).fst = some x\n\u22a2 x = (unpair n).fst\ncase right.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimpa [pure, PFun.pure, eq_comm] using h\n[GOAL]\ncase left.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).fst = some x\n\u22a2 x = (unpair n).fst\ncase right.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimpa [pure, PFun.pure, eq_comm] using h\n[GOAL]\ncase right.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\nn x : \u2115\nleft\u271d : n \u2264 k\nh : pure (unpair n).snd = some x\n\u22a2 x = (unpair n).snd\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimpa [pure, PFun.pure, eq_comm] using h\n[GOAL]\ncase pair.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\n[PROOFSTEP]\nrcases h with \u27e8y, ef, z, eg, rfl\u27e9\n[GOAL]\ncase pair.intro.intro.intro.intro.intro\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn : \u2115\nleft\u271d : n \u2264 k\ny : \u2115\nef : evaln (k + 1) cf n = some y\nz : \u2115\neg : evaln (k + 1) cg n = some z\n\u22a2 \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = Nat.pair y z\n[PROOFSTEP]\nexact \u27e8_, hf _ _ ef, _, hg _ _ eg, rfl\u27e9\n[GOAL]\ncase comp.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh : \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n[PROOFSTEP]\nrcases h with \u27e8y, eg, ef\u27e9\n[GOAL]\ncase comp.intro.intro.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\ny : \u2115\neg : evaln (k + 1) cg n = some y\nef : evaln (k + 1) cf y = some x\n\u22a2 \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n[PROOFSTEP]\nexact \u27e8_, hg _ _ eg, hf _ _ ef\u27e9\n[GOAL]\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    some x\n\u22a2 x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n[PROOFSTEP]\nrevert h\n[GOAL]\ncase prec.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x : \u2115\nleft\u271d : n \u2264 k\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (unpair n).snd =\n      some x \u2192\n    x \u2208\n      Nat.rec (eval cf (unpair n).fst)\n        (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n[PROOFSTEP]\ninduction' n.unpair.2 with m IH generalizing x\n[GOAL]\ncase prec.intro.zero\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nx : \u2115\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        Nat.zero =\n      some x \u2192\n    x \u2208\n      Nat.rec (eval cf (unpair n).fst)\n        (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) Nat.zero\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec.intro.succ\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx : \u2115\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        (Nat.succ m) =\n      some x \u2192\n    x \u2208\n      Nat.rec (eval cf (unpair n).fst)\n        (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (Nat.succ m)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase prec.intro.zero\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nx : \u2115\n\u22a2 evaln (k + 1) cf (unpair n).fst = some x \u2192 x \u2208 eval cf (unpair n).fst\n[PROOFSTEP]\napply hf\n[GOAL]\ncase prec.intro.succ\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx : \u2115\n\u22a2 \u2200 (x_1 : \u2115),\n    evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some x_1 \u2192\n      evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m x_1)) = some x \u2192\n        \u2203 a,\n          a \u2208\n              Nat.rec (eval cf (unpair n).fst)\n                (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m \u2227\n            x \u2208 eval cg (Nat.pair (unpair n).fst (Nat.pair m a))\n[PROOFSTEP]\nrefine' fun y h\u2081 h\u2082 => \u27e8y, IH _ _, _\u27e9\n[GOAL]\ncase prec.intro.succ.refine'_1\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : \u2115\nh\u2081 : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh\u2082 : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      m =\n    some y\n[PROOFSTEP]\nhave := evaln_mono k.le_succ h\u2081\n[GOAL]\ncase prec.intro.succ.refine'_1\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : \u2115\nh\u2081 : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh\u2082 : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis : y \u2208 evaln (Nat.succ k) (prec cf cg) (Nat.pair (unpair n).fst m)\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      m =\n    some y\n[PROOFSTEP]\nsimp [evaln, Bind.bind] at this \n[GOAL]\ncase prec.intro.succ.refine'_1\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : \u2115\nh\u2081 : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh\u2082 : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\nthis :\n  Nat.pair (unpair n).fst m \u2264 k \u2227\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n        (fun n_1 n_ih =>\n          Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n            evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n        m =\n      some y\n\u22a2 Nat.rec (evaln (k + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      m =\n    some y\n[PROOFSTEP]\nexact this.2\n[GOAL]\ncase prec.intro.succ.refine'_2\nk : \u2115\nc : Code\nn\u271d x\u271d\u00b9 : \u2115\nh : x\u271d\u00b9 \u2208 evaln (k + 1) c n\u271d\ncf cg : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nhg : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cg n \u2192 x \u2208 eval cg n\nn x\u271d : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nIH :\n  \u2200 (x : \u2115),\n    Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          m =\n        some x \u2192\n      x \u2208\n        Nat.rec (eval cf (unpair n).fst)\n          (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) m\nx y : \u2115\nh\u2081 : evaln k (prec cf cg) (Nat.pair (unpair n).fst m) = some y\nh\u2082 : evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair m y)) = some x\n\u22a2 x \u2208 eval cg (Nat.pair (unpair n).fst (Nat.pair m y))\n[PROOFSTEP]\nexact hg _ _ h\u2082\n[GOAL]\ncase rfind'.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh\u271d : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nh :\n  \u2203 a,\n    evaln (k + 1) cf n = some a \u2227\n      (if a = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n        some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nrcases h with \u27e8m, h\u2081, h\u2082\u27e9\n[GOAL]\ncase rfind'.intro.intro.intro\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nh\u2082 :\n  (if m = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) = some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nby_cases m0 : m = 0\n[GOAL]\ncase pos\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nh\u2082 :\n  (if m = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) = some x\nm0 : m = 0\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimp [m0] at h\u2082 \n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nh\u2082 :\n  (if m = 0 then pure (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) = some x\nm0 : \u00acm = 0\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimp [m0] at h\u2082 \n[GOAL]\ncase pos\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : m = 0\nh\u2082 : pure (unpair n).snd = some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nexact \u27e80, \u27e8by simpa [m0] using hf _ _ h\u2081, fun {m} => (Nat.not_lt_zero _).elim\u27e9, by injection h\u2082 with h\u2082; simp [h\u2082]\u27e9\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : m = 0\nh\u2082 : pure (unpair n).snd = some x\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).fst (0 + (unpair n).snd))\n[PROOFSTEP]\nsimpa [m0] using hf _ _ h\u2081\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : m = 0\nh\u2082 : pure (unpair n).snd = some x\n\u22a2 0 + (unpair n).snd = x\n[PROOFSTEP]\ninjection h\u2082 with h\u2082\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : m = 0\nh\u2082 : (unpair n).snd = x\n\u22a2 0 + (unpair n).snd = x\n[PROOFSTEP]\nsimp [h\u2082]\n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nhave := evaln_sound h\u2082\n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis : x \u2208 eval (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nsimp [eval] at this \n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn\u271d x\u271d : \u2115\nh : x\u271d \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn x : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some x\nthis :\n  \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + ((unpair n).snd + 1))) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0) \u2227\n      a + ((unpair n).snd + 1) = x\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n[PROOFSTEP]\nrcases this with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.intro\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n\u22a2 \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = y + ((unpair n).snd + 1)\n[PROOFSTEP]\nrefine' \u27e8y + 1, \u27e8by simpa [add_comm, add_left_comm] using hy\u2081, fun {i} im => _\u27e9, by simp [add_comm, add_left_comm]\u27e9\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + 1 + (unpair n).snd))\n[PROOFSTEP]\nsimpa [add_comm, add_left_comm] using hy\u2081\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\n\u22a2 y + 1 + (unpair n).snd = y + ((unpair n).snd + 1)\n[PROOFSTEP]\nsimp [add_comm, add_left_comm]\n[GOAL]\ncase neg.intro.intro.intro\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : \u2115\nim : i < y + 1\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (i + (unpair n).snd)) \u2227 \u00aca = 0\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase neg.intro.intro.intro.zero\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\nim : Nat.zero < y + 1\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (Nat.zero + (unpair n).snd)) \u2227 \u00aca = 0\n[PROOFSTEP]\nexact \u27e8m, by simpa using hf _ _ h\u2081, m0\u27e9\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\nim : Nat.zero < y + 1\n\u22a2 m \u2208 eval cf (Nat.pair (unpair n).fst (Nat.zero + (unpair n).snd))\n[PROOFSTEP]\nsimpa using hf _ _ h\u2081\n[GOAL]\ncase neg.intro.intro.intro.succ\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : \u2115\nim : Nat.succ i < y + 1\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) \u2227 \u00aca = 0\n[PROOFSTEP]\nrcases hy\u2082 (Nat.lt_of_succ_lt_succ im) with \u27e8z, hz, z0\u27e9\n[GOAL]\ncase neg.intro.intro.intro.succ.intro.intro\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : \u2115\nim : Nat.succ i < y + 1\nz : \u2115\nhz : z \u2208 eval cf (Nat.pair (unpair n).fst (i + ((unpair n).snd + 1)))\nz0 : \u00acz = 0\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd)) \u2227 \u00aca = 0\n[PROOFSTEP]\nexact \u27e8z, by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hz, z0\u27e9\n[GOAL]\nk : \u2115\nc : Code\nn\u271d x : \u2115\nh : x \u2208 evaln (k + 1) c n\u271d\ncf : Code\nhf : \u2200 (n x : \u2115), x \u2208 evaln (k + 1) cf n \u2192 x \u2208 eval cf n\nn : \u2115\nleft\u271d : n \u2264 k\nm : \u2115\nh\u2081 : evaln (k + 1) cf n = some m\nm0 : \u00acm = 0\ny : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + ((unpair n).snd + 1)))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + ((unpair n).snd + 1))) \u2227 \u00aca = 0\nh\u2082 : evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) = some (y + ((unpair n).snd + 1))\ni : \u2115\nim : Nat.succ i < y + 1\nz : \u2115\nhz : z \u2208 eval cf (Nat.pair (unpair n).fst (i + ((unpair n).snd + 1)))\nz0 : \u00acz = 0\n\u22a2 z \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ i + (unpair n).snd))\n[PROOFSTEP]\nsimpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hz\n[GOAL]\nc : Code\nn x : \u2115\nh : x \u2208 eval c n\n\u22a2 \u2203 k, x \u2208 evaln k c n\n[PROOFSTEP]\nrsuffices \u27e8k, h\u27e9 : \u2203 k, x \u2208 evaln (k + 1) c n\n[GOAL]\ncase intro\nc : Code\nn x : \u2115\nh\u271d : x \u2208 eval c n\nk : \u2115\nh : x \u2208 evaln (k + 1) c n\n\u22a2 \u2203 k, x \u2208 evaln k c n\n[PROOFSTEP]\nexact \u27e8k + 1, h\u27e9\n[GOAL]\nc : Code\nn x : \u2115\nh : x \u2208 eval c n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) c n\n[PROOFSTEP]\ninduction c generalizing n x\n[GOAL]\ncase zero\nn x : \u2115\nh : x \u2208 eval zero n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) zero n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase succ\nn x : \u2115\nh : x \u2208 eval succ n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) succ n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase left\nn x : \u2115\nh : x \u2208 eval left n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) left n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase right\nn x : \u2115\nh : x \u2208 eval right n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) right n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : x \u2208 eval (pair a\u271d\u00b9 a\u271d) n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) (pair a\u271d\u00b9 a\u271d) n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : x \u2208 eval (comp a\u271d\u00b9 a\u271d) n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) (comp a\u271d\u00b9 a\u271d) n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : x \u2208 eval (prec a\u271d\u00b9 a\u271d) n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) (prec a\u271d\u00b9 a\u271d) n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : x \u2208 eval (rfind' a\u271d) n\n\u22a2 \u2203 k, x \u2208 evaln (k + 1) (rfind' a\u271d) n\n[PROOFSTEP]\nsimp [eval, evaln, pure, PFun.pure, Seq.seq, Bind.bind] at h \u22a2\n[GOAL]\ncase zero\nn x : \u2115\nh : x = 0\n\u22a2 (\u2203 x, n \u2264 x) \u2227 0 = x\ncase succ\nn x : \u2115\nh : x = Nat.succ n\n\u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x\ncase left\nn x : \u2115\nh : x = (unpair n).fst\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).fst = x\ncase right\nn x : \u2115\nh : x = (unpair n).snd\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).snd = x\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\niterate 4 exact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9\n[GOAL]\ncase zero\nn x : \u2115\nh : x = 0\n\u22a2 (\u2203 x, n \u2264 x) \u2227 0 = x\ncase succ\nn x : \u2115\nh : x = Nat.succ n\n\u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x\ncase left\nn x : \u2115\nh : x = (unpair n).fst\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).fst = x\ncase right\nn x : \u2115\nh : x = (unpair n).snd\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).snd = x\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\nexact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9\n[GOAL]\ncase succ\nn x : \u2115\nh : x = Nat.succ n\n\u22a2 (\u2203 x, n \u2264 x) \u2227 Nat.succ n = x\ncase left\nn x : \u2115\nh : x = (unpair n).fst\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).fst = x\ncase right\nn x : \u2115\nh : x = (unpair n).snd\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).snd = x\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\nexact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9\n[GOAL]\ncase left\nn x : \u2115\nh : x = (unpair n).fst\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).fst = x\ncase right\nn x : \u2115\nh : x = (unpair n).snd\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).snd = x\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\nexact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9\n[GOAL]\ncase right\nn x : \u2115\nh : x = (unpair n).snd\n\u22a2 (\u2203 x, n \u2264 x) \u2227 (unpair n).snd = x\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\nexact \u27e8\u27e8_, le_rfl\u27e9, h.symm\u27e9\n[GOAL]\ncase pair\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d\u00b9 n \u2227 \u2203 a_1, a_1 \u2208 eval a\u271d n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d\u00b9 n = some a \u2227 \u2203 a_1, evaln (k + 1) a\u271d n = some a_1 \u2227 Nat.pair a a_1 = x\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\ncase pair cf cg hf hg =>\n  rcases h with \u27e8x, hx, y, hy, rfl\u27e9\n  rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9; rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  refine'\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n[PROOFSTEP]\ncase pair cf cg hf hg =>\n  rcases h with \u27e8x, hx, y, hy, rfl\u27e9\n  rcases hf hx with \u27e8k\u2081, hk\u2081\u27e9; rcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  refine'\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cf n \u2227 \u2203 a_1, a_1 \u2208 eval cg n \u2227 Nat.pair a a_1 = x\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = x\n[PROOFSTEP]\nrcases h with \u27e8x, hx, y, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y\n[PROOFSTEP]\nrcases hf hx with \u27e8k\u2081, hk\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y\n[PROOFSTEP]\nrcases hg hy with \u27e8k\u2082, hk\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cf n = some a \u2227 \u2203 a_1, evaln (k + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y\n[PROOFSTEP]\nrefine' \u27e8max k\u2081 k\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nhx : x \u2208 eval cf n\ny : \u2115\nhy : y \u2208 eval cg n\nk\u2081 : \u2115\nhk\u2081 : x \u2208 evaln (k\u2081 + 1) cf n\nk\u2082 : \u2115\nhk\u2082 : y \u2208 evaln (k\u2082 + 1) cg n\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227\n    \u2203 a,\n      evaln (max k\u2081 k\u2082 + 1) cf n = some a \u2227 \u2203 a_1, evaln (max k\u2081 k\u2082 + 1) cg n = some a_1 \u2227 Nat.pair a a_1 = Nat.pair x y\n[PROOFSTEP]\nrefine'\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n    _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082, rfl\u27e9\n[GOAL]\ncase comp\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh : \u2203 a, a \u2208 eval a\u271d n \u2227 x \u2208 eval a\u271d\u00b9 a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) a\u271d n = some a \u2227 evaln (k + 1) a\u271d\u00b9 a = some x\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\ncase comp cf cg hf hg =>\n  rcases h with \u27e8y, hy, hx\u27e9\n  rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9; rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  exact\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n[PROOFSTEP]\ncase comp cf cg hf hg =>\n  rcases h with \u27e8y, hy, hx\u27e9\n  rcases hg hy with \u27e8k\u2081, hk\u2081\u27e9; rcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n  refine' \u27e8max k\u2081 k\u2082, _\u27e9\n  exact\n    \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh : \u2203 a, a \u2208 eval cg n \u2227 x \u2208 eval cf a\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n[PROOFSTEP]\nrcases h with \u27e8y, hy, hx\u27e9\n[GOAL]\ncase intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n[PROOFSTEP]\nrcases hg hy with \u27e8k\u2081, hk\u2081\u27e9\n[GOAL]\ncase intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n[PROOFSTEP]\nrcases hf hx with \u27e8k\u2082, hk\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 \u2203 k, n \u2264 k \u2227 \u2203 a, evaln (k + 1) cg n = some a \u2227 evaln (k + 1) cf a = some x\n[PROOFSTEP]\nrefine' \u27e8max k\u2081 k\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x y : \u2115\nhy : y \u2208 eval cg n\nhx : x \u2208 eval cf y\nk\u2081 : \u2115\nhk\u2081 : y \u2208 evaln (k\u2081 + 1) cg n\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cf y\n\u22a2 n \u2264 max k\u2081 k\u2082 \u2227 \u2203 a, evaln (max k\u2081 k\u2082 + 1) cg n = some a \u2227 evaln (max k\u2081 k\u2082 + 1) cf a = some x\n[PROOFSTEP]\nexact\n  \u27e8le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081, _, evaln_mono (Nat.succ_le_succ <| le_max_left _ _) hk\u2081,\n    evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\u27e9\n[GOAL]\ncase prec\na\u271d\u00b9 a\u271d : Code\na_ih\u271d\u00b9 : \u2200 {n x : \u2115}, x \u2208 eval a\u271d\u00b9 n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d\u00b9 n\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval a\u271d\u00b9 (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval a\u271d (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) a\u271d\u00b9 (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec a\u271d\u00b9 a\u271d) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) a\u271d (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\ncase prec cf cg hf hg =>\n  revert h\n  generalize n.unpair.1 = n\u2081; generalize n.unpair.2 = n\u2082\n  induction' n\u2082 with m IH generalizing x n <;> simp\n  \u00b7 intro h\n    rcases hf h with \u27e8k, hk\u27e9\n    exact \u27e8_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u27e9\n  \u00b7 intro y hy hx\n    rcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9\n    rcases hg hx with \u27e8k\u2082, hk\u2082\u27e9\n    refine'\n      \u27e8(max k\u2081 k\u2082).succ, Nat.le_succ_of_le <| le_max_of_le_left <| le_trans (le_max_left _ (Nat.pair n\u2081 m)) nk\u2081, y,\n        evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,\n        evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk\u2082\u27e9\n    simp [evaln, Bind.bind]\n    exact \u27e8le_trans (le_max_right _ _) nk\u2081, hk\u2081\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\n[PROOFSTEP]\ncase prec cf cg hf hg =>\n  revert h\n  generalize n.unpair.1 = n\u2081; generalize n.unpair.2 = n\u2082\n  induction' n\u2082 with m IH generalizing x n <;> simp\n  \u00b7 intro h\n    rcases hf h with \u27e8k, hk\u27e9\n    exact \u27e8_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u27e9\n  \u00b7 intro y hy hx\n    rcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9\n    rcases hg hx with \u27e8k\u2082, hk\u2082\u27e9\n    refine'\n      \u27e8(max k\u2081 k\u2082).succ, Nat.le_succ_of_le <| le_max_of_le_left <| le_trans (le_max_left _ (Nat.pair n\u2081 m)) nk\u2081, y,\n        evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,\n        evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk\u2082\u27e9\n    simp [evaln, Bind.bind]\n    exact \u27e8le_trans (le_max_right _ _) nk\u2081, hk\u2081\u27e9\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\nh :\n  x \u2208\n    Nat.rec (eval cf (unpair n).fst)\n      (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      Nat.rec (evaln (k + 1) cf (unpair n).fst)\n          (fun n_1 n_ih =>\n            Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n              evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd =\n        some x\n[PROOFSTEP]\nrevert h\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x : \u2115\n\u22a2 x \u2208\n      Nat.rec (eval cf (unpair n).fst)\n        (fun y IH => Part.bind IH fun i => eval cg (Nat.pair (unpair n).fst (Nat.pair y i))) (unpair n).snd \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf (unpair n).fst)\n            (fun n_1 n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n                evaln (k + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n            (unpair n).snd =\n          some x\n[PROOFSTEP]\ngeneralize n.unpair.1 = n\u2081\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) (unpair n).snd \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            (unpair n).snd =\n          some x\n[PROOFSTEP]\ngeneralize n.unpair.2 = n\u2082\n[GOAL]\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn x n\u2081 n\u2082 : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) n\u2082 \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            n\u2082 =\n          some x\n[PROOFSTEP]\ninduction' n\u2082 with m IH generalizing x n\n[GOAL]\ncase zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) Nat.zero \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            Nat.zero =\n          some x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x : \u2115\n\u22a2 x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) (Nat.succ m) \u2192\n    \u2203 k,\n      n \u2264 k \u2227\n        Nat.rec (evaln (k + 1) cf n\u2081)\n            (fun n n_ih =>\n              Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n            (Nat.succ m) =\n          some x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\n\u22a2 x \u2208 eval cf n\u2081 \u2192 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase zero\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x\n[PROOFSTEP]\nrcases hf h with \u27e8k, hk\u27e9\n[GOAL]\ncase zero.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 n x : \u2115\nh : x \u2208 eval cf n\u2081\nk : \u2115\nhk : x \u2208 evaln (k + 1) cf n\u2081\n\u22a2 \u2203 k, n \u2264 k \u2227 evaln (k + 1) cf n\u2081 = some x\n[PROOFSTEP]\nexact \u27e8_, le_max_left _ _, evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u27e9\n[GOAL]\ncase succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x : \u2115\n\u22a2 \u2200 (x_1 : \u2115),\n    x_1 \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m x_1)) \u2192\n        \u2203 k,\n          n \u2264 k \u2227\n            \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x\n[PROOFSTEP]\nintro y hy hx\n[GOAL]\ncase succ\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x\n[PROOFSTEP]\nrcases IH hy with \u27e8k\u2081, nk\u2081, hk\u2081\u27e9\n[GOAL]\ncase succ.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1242894 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x\n[PROOFSTEP]\nrcases hg hx with \u27e8k\u2082, hk\u2082\u27e9\n[GOAL]\ncase succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : ?m.1242894 \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 \u2203 k,\n    n \u2264 k \u2227 \u2203 a, evaln k (prec cf cg) (Nat.pair n\u2081 m) = some a \u2227 evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair m a)) = some x\n[PROOFSTEP]\nrefine'\n  \u27e8(max k\u2081 k\u2082).succ, Nat.le_succ_of_le <| le_max_of_le_left <| le_trans (le_max_left _ (Nat.pair n\u2081 m)) nk\u2081, y,\n    evaln_mono (Nat.succ_le_succ <| le_max_left _ _) _,\n    evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_right _ _) hk\u2082\u27e9\n[GOAL]\ncase succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 y \u2208 evaln (Nat.succ k\u2081) (prec cf cg) (Nat.pair n\u2081 m)\n[PROOFSTEP]\nsimp [evaln, Bind.bind]\n[GOAL]\ncase succ.intro.intro.intro\ncf cg : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nhg : \u2200 {n x : \u2115}, x \u2208 eval cg n \u2192 \u2203 k, x \u2208 evaln (k + 1) cg n\nn\u271d x\u271d n\u2081 m : \u2115\nIH :\n  \u2200 {n x : \u2115},\n    x \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m \u2192\n      \u2203 k,\n        n \u2264 k \u2227\n          Nat.rec (evaln (k + 1) cf n\u2081)\n              (fun n n_ih =>\n                Option.bind (evaln k (prec cf cg) (Nat.pair n\u2081 n)) fun i =>\n                  evaln (k + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n              m =\n            some x\nn x y : \u2115\nhy : y \u2208 Nat.rec (eval cf n\u2081) (fun y IH => Part.bind IH fun i => eval cg (Nat.pair n\u2081 (Nat.pair y i))) m\nhx : x \u2208 eval cg (Nat.pair n\u2081 (Nat.pair m y))\nk\u2081 : \u2115\nnk\u2081 : max n (Nat.pair n\u2081 m) \u2264 k\u2081\nhk\u2081 :\n  Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n      (fun n n_ih =>\n        Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n      m =\n    some y\nk\u2082 : \u2115\nhk\u2082 : x \u2208 evaln (k\u2082 + 1) cg (Nat.pair n\u2081 (Nat.pair m y))\n\u22a2 Nat.pair n\u2081 m \u2264 k\u2081 \u2227\n    Nat.rec (evaln (k\u2081 + 1) cf n\u2081)\n        (fun n n_ih =>\n          Option.bind (evaln k\u2081 (prec cf cg) (Nat.pair n\u2081 n)) fun i => evaln (k\u2081 + 1) cg (Nat.pair n\u2081 (Nat.pair n i)))\n        m =\n      some y\n[PROOFSTEP]\nexact \u27e8le_trans (le_max_right _ _) nk\u2081, hk\u2081\u27e9\n[GOAL]\ncase rfind'\na\u271d : Code\na_ih\u271d : \u2200 {n x : \u2115}, x \u2208 eval a\u271d n \u2192 \u2203 k, x \u2208 evaln (k + 1) a\u271d n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval a\u271d (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval a\u271d (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) a\u271d n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' a\u271d) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\ncase rfind' cf hf =>\n  rcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9\n  suffices \u2203 k, y + n.unpair.2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by simpa [evaln, Bind.bind]\n  revert hy\u2081 hy\u2082\n  generalize n.unpair.2 = m\n  intro hy\u2081 hy\u2082\n  induction' y with y IH generalizing m <;> simp [evaln, Bind.bind]\n  \u00b7 simp at hy\u2081 \n    rcases hf hy\u2081 with \u27e8k, hk\u27e9\n    exact \u27e8_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl\u27e9\n  \u00b7 rcases hy\u2082 (Nat.succ_pos _) with \u27e8a, ha, a0\u27e9\n    rcases hf ha with \u27e8k\u2081, hk\u2081\u27e9\n    rcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081) fun {i} hi => by\n        simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2082 (Nat.succ_lt_succ hi) with\n      \u27e8k\u2082, hk\u2082\u27e9\n    use(max k\u2081 k\u2082).succ\n    rw [zero_add] at hk\u2081 \n    use Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081\n    use a\n    use evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk\u2081\n    simpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\ncase rfind' cf hf =>\n  rcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9\n  suffices \u2203 k, y + n.unpair.2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by simpa [evaln, Bind.bind]\n  revert hy\u2081 hy\u2082\n  generalize n.unpair.2 = m\n  intro hy\u2081 hy\u2082\n  induction' y with y IH generalizing m <;> simp [evaln, Bind.bind]\n  \u00b7 simp at hy\u2081 \n    rcases hf hy\u2081 with \u27e8k, hk\u27e9\n    exact \u27e8_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl\u27e9\n  \u00b7 rcases hy\u2082 (Nat.succ_pos _) with \u27e8a, ha, a0\u27e9\n    rcases hf ha with \u27e8k\u2081, hk\u2081\u27e9\n    rcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081) fun {i} hi => by\n        simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2082 (Nat.succ_lt_succ hi) with\n      \u27e8k\u2082, hk\u2082\u27e9\n    use(max k\u2081 k\u2082).succ\n    rw [zero_add] at hk\u2081 \n    use Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081\n    use a\n    use evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk\u2081\n    simpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n      evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn x : \u2115\nh :\n  \u2203 a,\n    (0 \u2208 eval cf (Nat.pair (unpair n).fst (a + (unpair n).snd)) \u2227\n        \u2200 {m : \u2115}, m < a \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2227\n      a + (unpair n).snd = x\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some x\n[PROOFSTEP]\nrcases h with \u27e8y, \u27e8hy\u2081, hy\u2082\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + (unpair n).snd))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some (y + (unpair n).snd)\n[PROOFSTEP]\nsuffices \u2203 k, y + n.unpair.2 \u2208 evaln (k + 1) (rfind' cf) (Nat.pair n.unpair.1 n.unpair.2) by simpa [evaln, Bind.bind]\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + (unpair n).snd))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0\nthis : \u2203 k, y + (unpair n).snd \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst (unpair n).snd)\n\u22a2 \u2203 k,\n    n \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf n = some a \u2227\n          (if a = 0 then some (unpair n).snd else evaln k (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))) =\n            some (y + (unpair n).snd)\n[PROOFSTEP]\nsimpa [evaln, Bind.bind]\n[GOAL]\ncase intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + (unpair n).snd))\nhy\u2082 : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + (unpair n).snd \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst (unpair n).snd)\n[PROOFSTEP]\nrevert hy\u2081 hy\u2082\n[GOAL]\ncase intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + (unpair n).snd)) \u2192\n    (\u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + (unpair n).snd)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + (unpair n).snd \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst (unpair n).snd)\n[PROOFSTEP]\ngeneralize n.unpair.2 = m\n[GOAL]\ncase intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n    (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n      \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\n[PROOFSTEP]\nintro hy\u2081 hy\u2082\n[GOAL]\ncase intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\n[PROOFSTEP]\ninduction' y with y IH generalizing m\n[GOAL]\ncase intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.zero + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, Nat.zero + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\n[PROOFSTEP]\nsimp [evaln, Bind.bind]\n[GOAL]\ncase intro.intro.intro.succ\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k, Nat.succ y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\n[PROOFSTEP]\nsimp [evaln, Bind.bind]\n[GOAL]\ncase intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.zero + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some m\n[PROOFSTEP]\nsimp at hy\u2081 \n[GOAL]\ncase intro.intro.intro.zero\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some m\n[PROOFSTEP]\nrcases hf hy\u2081 with \u27e8k, hk\u27e9\n[GOAL]\ncase intro.intro.intro.zero.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).fst m)\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some m\n[PROOFSTEP]\nexact \u27e8_, Nat.le_of_lt_succ <| evaln_bound hk, _, hk, by simp; rfl\u27e9\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).fst m)\n\u22a2 (if 0 = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some m\n[PROOFSTEP]\nsimp\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\nm : \u2115\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.zero \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst m)\nk : \u2115\nhk : 0 \u2208 evaln (k + 1) cf (Nat.pair (unpair n).fst m)\n\u22a2 pure m = some m\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.succ\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some (Nat.succ y + m)\n[PROOFSTEP]\nrcases hy\u2082 (Nat.succ_pos _) with \u27e8a, ha, a0\u27e9\n[GOAL]\ncase intro.intro.intro.succ.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some (Nat.succ y + m)\n[PROOFSTEP]\nrcases hf ha with \u27e8k\u2081, hk\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.succ.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst (0 + m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some (Nat.succ y + m)\n[PROOFSTEP]\nrcases IH m.succ (by simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081) fun {i} hi => by\n    simpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2082 (Nat.succ_lt_succ hi) with\n  \u27e8k\u2082, hk\u2082\u27e9\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst (0 + m))\n\u22a2 0 \u2208 eval cf (Nat.pair (unpair n).fst (y + Nat.succ m))\n[PROOFSTEP]\nsimpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2081\n[GOAL]\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst (0 + m))\ni : \u2115\nhi : i < y\n\u22a2 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (i + Nat.succ m)) \u2227 \u00aca = 0\n[PROOFSTEP]\nsimpa [Nat.succ_eq_add_one, add_comm, add_left_comm] using hy\u2082 (Nat.succ_lt_succ hi)\n[GOAL]\ncase intro.intro.intro.succ.intro.intro.intro.intro\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 \u2203 k,\n    Nat.pair (unpair n).fst m \u2264 k \u2227\n      \u2203 a,\n        evaln (k + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n          (if a = 0 then pure m else evaln k (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) = some (Nat.succ y + m)\n[PROOFSTEP]\nuse(max k\u2081 k\u2082).succ\n[GOAL]\ncase h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst (0 + m))\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 Nat.pair (unpair n).fst m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) =\n          some (Nat.succ y + m)\n[PROOFSTEP]\nrw [zero_add] at hk\u2081 \n[GOAL]\ncase h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 Nat.pair (unpair n).fst m \u2264 Nat.succ (max k\u2081 k\u2082) \u2227\n    \u2203 a,\n      evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n        (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) =\n          some (Nat.succ y + m)\n[PROOFSTEP]\nuse Nat.le_succ_of_le <| le_max_of_le_left <| Nat.le_of_lt_succ <| evaln_bound hk\u2081\n[GOAL]\ncase right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 \u2203 a,\n    evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n      (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) =\n        some (Nat.succ y + m)\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 evaln (Nat.succ (max k\u2081 k\u2082) + 1) cf (Nat.pair (unpair n).fst m) = some a \u2227\n    (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) =\n      some (Nat.succ y + m)\n[PROOFSTEP]\nuse evaln_mono (Nat.succ_le_succ <| Nat.le_succ_of_le <| le_max_left _ _) hk\u2081\n[GOAL]\ncase right\ncf : Code\nhf : \u2200 {n x : \u2115}, x \u2208 eval cf n \u2192 \u2203 k, x \u2208 evaln (k + 1) cf n\nn y\u271d m\u271d : \u2115\nhy\u2081\u271d : 0 \u2208 eval cf (Nat.pair (unpair n).fst (y\u271d + m\u271d))\nhy\u2082\u271d : \u2200 {m : \u2115}, m < y\u271d \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m + m\u271d)) \u2227 \u00aca = 0\ny : \u2115\nIH :\n  \u2200 (m : \u2115),\n    0 \u2208 eval cf (Nat.pair (unpair n).fst (y + m)) \u2192\n      (\u2200 {m_1 : \u2115}, m_1 < y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0) \u2192\n        \u2203 k, y + m \u2208 evaln (k + 1) (rfind' cf) (Nat.pair (unpair n).fst m)\nm : \u2115\nhy\u2081 : 0 \u2208 eval cf (Nat.pair (unpair n).fst (Nat.succ y + m))\nhy\u2082 : \u2200 {m_1 : \u2115}, m_1 < Nat.succ y \u2192 \u2203 a, a \u2208 eval cf (Nat.pair (unpair n).fst (m_1 + m)) \u2227 \u00aca = 0\na : \u2115\nha : a \u2208 eval cf (Nat.pair (unpair n).fst (0 + m))\na0 : \u00aca = 0\nk\u2081 : \u2115\nhk\u2081 : a \u2208 evaln (k\u2081 + 1) cf (Nat.pair (unpair n).fst m)\nk\u2082 : \u2115\nhk\u2082 : y + Nat.succ m \u2208 evaln (k\u2082 + 1) (rfind' cf) (Nat.pair (unpair n).fst (Nat.succ m))\n\u22a2 (if a = 0 then pure m else evaln (Nat.succ (max k\u2081 k\u2082)) (rfind' cf) (Nat.pair (unpair n).fst (m + 1))) =\n    some (Nat.succ y + m)\n[PROOFSTEP]\nsimpa [Nat.succ_eq_add_one, a0, -max_eq_left, -max_eq_right, add_comm, add_left_comm] using\n  evaln_mono (Nat.succ_le_succ <| le_max_right _ _) hk\u2082\n[GOAL]\n\u22a2 Primrec Nat.Partrec.Code.G\n[PROOFSTEP]\nhave a := (Primrec.ofNat (\u2115 \u00d7 Code)).comp (Primrec.list_length (\u03b1 := List (Option \u2115)))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\n\u22a2 Primrec Nat.Partrec.Code.G\n[PROOFSTEP]\nhave k := Primrec.fst.comp a\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).fst\n\u22a2 Primrec Nat.Partrec.Code.G\n[PROOFSTEP]\nrefine' Primrec.option_some.comp (Primrec.list_map (Primrec.list_range.comp k) (_ : Primrec _))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).fst\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).fst Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).snd (some 0) (some (Nat.succ n)) (some (unpair n).fst)\n            (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).fst;\n              Nat.casesOn (unpair n).snd (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) z)\n                fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).fst;\n            let m := (unpair n).snd;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z (m + 1)))\n      p.fst p.snd\n[PROOFSTEP]\nreplace k := k.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).fst Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).snd (some 0) (some (Nat.succ n)) (some (unpair n).fst)\n            (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).fst;\n              Nat.casesOn (unpair n).snd (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) z)\n                fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).fst;\n            let m := (unpair n).snd;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z (m + 1)))\n      p.fst p.snd\n[PROOFSTEP]\nhave n := Primrec.snd (\u03b1 := List (List (Option \u2115))) (\u03b2 := \u2115)\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).fst Option.none fun k' =>\n          Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).snd (some 0) (some (Nat.succ n)) (some (unpair n).fst)\n            (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) n\n              let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) n\n              Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) x)\n            (fun cf cg x x =>\n              let z := (unpair n).fst;\n              Nat.casesOn (unpair n).snd (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) z)\n                fun y => do\n                let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z y)\n                Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cg) (Nat.pair z (Nat.pair y i)))\n            fun cf x =>\n            let z := (unpair n).fst;\n            let m := (unpair n).snd;\n            do\n            let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).fst, cf) (Nat.pair z m)\n            Nat.casesOn x (some m) fun x =>\n                Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).snd) (Nat.pair z (m + 1)))\n      p.fst p.snd\n[PROOFSTEP]\nrefine' Primrec.nat_casesOn k (_root_.Primrec.const Option.none) (_ : Primrec _)\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd (some 0) (some (Nat.succ p.snd))\n              (some (unpair p.snd).fst) (some (unpair p.snd).snd)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) p.snd\n                let y \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.snd).fst;\n                Nat.casesOn (unpair p.snd).snd\n                  (Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) z) fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg)\n                      (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.snd).fst;\n              let m := (unpair p.snd).snd;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z (m + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave k := k.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd (some 0) (some (Nat.succ p.snd))\n              (some (unpair p.snd).fst) (some (unpair p.snd).snd)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) p.snd\n                let y \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.snd).fst;\n                Nat.casesOn (unpair p.snd).snd\n                  (Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) z) fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg)\n                      (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.snd).fst;\n              let m := (unpair p.snd).snd;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z (m + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave n := n.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd (some 0) (some (Nat.succ p.snd))\n              (some (unpair p.snd).fst) (some (unpair p.snd).snd)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) p.snd\n                let y \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.snd).fst;\n                Nat.casesOn (unpair p.snd).snd\n                  (Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) z) fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg)\n                      (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.snd).fst;\n              let m := (unpair p.snd).snd;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z (m + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave k' := Primrec.snd (\u03b1 := List (List (Option \u2115)) \u00d7 \u2115) (\u03b2 := \u2115)\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd (some 0) (some (Nat.succ p.snd))\n              (some (unpair p.snd).fst) (some (unpair p.snd).snd)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) p.snd\n                let y \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.snd).fst;\n                Nat.casesOn (unpair p.snd).snd\n                  (Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) z) fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg)\n                      (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.snd).fst;\n              let m := (unpair p.snd).snd;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z (m + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave c := Primrec.snd.comp (a.comp <| (Primrec.fst (\u03b2 := \u2115)).comp (Primrec.fst (\u03b2 := \u2115)))\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun k' =>\n            Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd (some 0) (some (Nat.succ p.snd))\n              (some (unpair p.snd).fst) (some (unpair p.snd).snd)\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) p.snd\n                let y \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg) p.snd\n                Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) x)\n              (fun cf cg x x =>\n                let z := (unpair p.snd).fst;\n                Nat.casesOn (unpair p.snd).snd\n                  (Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) z) fun y => do\n                  let i \u2190 Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z y)\n                  Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cg)\n                      (Nat.pair z (Nat.pair y i)))\n              fun cf x =>\n              let z := (unpair p.snd).fst;\n              let m := (unpair p.snd).snd;\n              do\n              let x \u2190 Nat.Partrec.Code.lup p.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst)).fst, cf) (Nat.pair z m)\n              Nat.casesOn x (some m) fun x =>\n                  Nat.Partrec.Code.lup p.fst (k', (ofNat (\u2115 \u00d7 Code) (List.length p.fst)).snd) (Nat.pair z (m + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\napply\n  Nat.Partrec.Code.rec_prim c (_root_.Primrec.const (some 0)) (Primrec.option_some.comp (_root_.Primrec.succ.comp n))\n    (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))\n    (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nhave L :=\n  (Primrec.fst.comp Primrec.fst).comp\n    (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nhave k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nhave n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nhave cf :=\n  Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nhave cg :=\n  (Primrec.fst.comp Primrec.snd).comp\n    (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) a.fst.fst.snd\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nrefine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cf).pair n) ?_\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec\u2082 fun a x => do\n    let y \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    some (Nat.pair x y)\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\nconv =>\n  congr\n  \u00b7 ext p\n    dsimp only []\n    erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| Primrec fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\n  congr\n  \u00b7 ext p\n    dsimp only []\n    erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| Primrec fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\n  congr\n  \u00b7 ext p\n    dsimp only []\n    erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| Primrec fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\ncongr\n[GOAL]\ncase f\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\n\u00b7 ext p\n  dsimp only []\n  erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\ncase f\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\n  ext p\n  dsimp only []\n  erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\ncase f\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\n  ext p\n  dsimp only []\n  erw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\ncase f\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n| fun p =>\n    (fun a x => do\n        let y \u2190\n          Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n              a.fst.fst.snd\n        some (Nat.pair x y))\n      p.fst p.snd\n[PROOFSTEP]\next p\n[GOAL]\ncase f.h\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\np : (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115\n| (fun a x => do\n      let y \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n            a.fst.fst.snd\n      some (Nat.pair x y))\n    p.fst p.snd\n[PROOFSTEP]\ndsimp only []\n[GOAL]\ncase f.h\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\np : (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115\n| do\n    let y \u2190\n      Nat.Partrec.Code.lup p.fst.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          p.fst.fst.fst.snd\n    some (Nat.pair p.snd y)\n[PROOFSTEP]\nerw [Option.bind_eq_bind, \u2190 Option.map_eq_bind]\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun p =>\n    Option.map (fun y => Nat.pair p.snd y)\n      (Nat.Partrec.Code.lup p.fst.fst.fst.fst\n        ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst) p.fst.fst.fst.snd)\n[PROOFSTEP]\nrefine Primrec.option_map ((hlup.comp <| L.pair <| (k.pair cg).pair n).comp Primrec.fst) ?_\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec\u2082 fun p y => Nat.pair p.snd y\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\ncase hpr\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun p => (fun p y => Nat.pair p.snd y) p.fst p.snd\n[PROOFSTEP]\nexact Primrec\u2082.natPair.comp (Primrec.snd.comp Primrec.fst) Primrec.snd\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nhave L :=\n  (Primrec.fst.comp Primrec.fst).comp\n    (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nhave k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nhave n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nhave cf :=\n  Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nhave cg :=\n  (Primrec.fst.comp Primrec.snd).comp\n    (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun a => do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          a.fst.fst.snd\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nrefine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec\u2082 fun a x =>\n    Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x)\n      p.fst p.snd\n[PROOFSTEP]\nhave h := hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd)\n[GOAL]\ncase hco\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nh :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).fst, a.fst.snd.fst), a.snd).fst\n      (a.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).fst, a.fst.snd.fst), a.snd).snd.fst\n      (a.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).fst, a.fst.snd.fst), a.snd).snd.snd\n\u22a2 Primrec fun p =>\n    (fun a x => Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) x)\n      p.fst p.snd\n[PROOFSTEP]\nexact h\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave L :=\n  (Primrec.fst.comp Primrec.fst).comp\n    (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave cf :=\n  Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave cg :=\n  (Primrec.fst.comp Primrec.snd).comp\n    (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nhave z := Primrec.fst.comp (Primrec.unpair.comp n)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    Nat.casesOn (unpair a.fst.fst.snd).snd\n      (Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) z) fun y => do\n      let i \u2190\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair z y)\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n          (Nat.pair z (Nat.pair y i))\n[PROOFSTEP]\nrefine'\n  Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n)) (hlup.comp <| L.pair <| (k.pair cf).pair z)\n    (_ : Primrec _)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n                  (Nat.pair (unpair a.fst.fst.snd).fst y)\n            Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n                (Nat.pair (unpair a.fst.fst.snd).fst (Nat.pair y i)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave L := L.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n                  (Nat.pair (unpair a.fst.fst.snd).fst y)\n            Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n                (Nat.pair (unpair a.fst.fst.snd).fst (Nat.pair y i)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave z := z.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n                  (Nat.pair (unpair a.fst.fst.snd).fst y)\n            Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n                (Nat.pair (unpair a.fst.fst.snd).fst (Nat.pair y i)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave y := Primrec.snd (\u03b1 := ((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) (\u03b2 := \u2115)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n                  (Nat.pair (unpair a.fst.fst.snd).fst y)\n            Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n                (Nat.pair (unpair a.fst.fst.snd).fst (Nat.pair y i)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nhave h\u2081 := hlup.comp <| L.pair <| (((k'.pair c).comp Primrec.fst).comp Primrec.fst).pair (Primrec\u2082.natPair.comp z y)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\n\u22a2 Primrec fun p =>\n    (fun a n =>\n        (fun y => do\n            let i \u2190\n              Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n                  (Nat.pair (unpair a.fst.fst.snd).fst y)\n            Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.snd.fst)\n                (Nat.pair (unpair a.fst.fst.snd).fst (Nat.pair y i)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nrefine' Primrec.option_bind h\u2081 (_ : Primrec _)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst\n          ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst (Nat.pair p.snd i)))\n      p.fst p.snd\n[PROOFSTEP]\nhave z := z.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d\u00b9 : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\nz : Primrec fun a => (unpair a.fst.fst.fst.fst.snd).fst\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst\n          ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst (Nat.pair p.snd i)))\n      p.fst p.snd\n[PROOFSTEP]\nhave y := y.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d\u00b9 : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\nz : Primrec fun a => (unpair a.fst.fst.fst.fst.snd).fst\ny : Primrec fun a => a.fst.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst\n          ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst (Nat.pair p.snd i)))\n      p.fst p.snd\n[PROOFSTEP]\nhave i := Primrec.snd (\u03b1 := (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115) (\u03b2 := \u2115)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d\u00b9 : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\nz : Primrec fun a => (unpair a.fst.fst.fst.fst.snd).fst\ny : Primrec fun a => a.fst.snd\ni : Primrec Prod.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst\n          ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst (Nat.pair p.snd i)))\n      p.fst p.snd\n[PROOFSTEP]\nhave h\u2082 :=\n  hlup.comp\n    ((L.comp Primrec.fst).pair <|\n      ((k.pair cg).comp <| Primrec.fst.comp Primrec.fst).pair <| Primrec\u2082.natPair.comp z <| Primrec\u2082.natPair.comp y i)\n[GOAL]\ncase hpc\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL\u271d : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\ncg : Primrec fun a => a.snd.snd.fst\nz\u271d\u00b9 : Primrec fun a => (unpair a.fst.fst.snd).fst\nL : Primrec fun a => a.fst.fst.fst.fst\nz\u271d : Primrec fun a => (unpair a.fst.fst.fst.snd).fst\ny\u271d : Primrec Prod.snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n          Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.fst\n      (a.fst.fst.fst.fst, (a.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst)).snd),\n            Nat.pair (unpair a.fst.fst.fst.snd).fst a.snd).snd.snd\nz : Primrec fun a => (unpair a.fst.fst.fst.fst.snd).fst\ny : Primrec fun a => a.fst.snd\ni : Primrec Prod.snd\nh\u2082 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst.fst)).fst, a.fst.fst.snd.snd.fst),\n          Nat.pair (unpair a.fst.fst.fst.fst.snd).fst (Nat.pair a.fst.snd a.snd)).fst\n      (a.fst.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst.fst)).fst, a.fst.fst.snd.snd.fst),\n            Nat.pair (unpair a.fst.fst.fst.fst.snd).fst (Nat.pair a.fst.snd a.snd)).snd.fst\n      (a.fst.fst.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst.fst.fst)).fst, a.fst.fst.snd.snd.fst),\n            Nat.pair (unpair a.fst.fst.fst.fst.snd).fst (Nat.pair a.fst.snd a.snd)).snd.snd\n\u22a2 Primrec fun p =>\n    (fun p i =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst\n          ((ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).fst, p.fst.snd.snd.fst)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst (Nat.pair p.snd i)))\n      p.fst p.snd\n[PROOFSTEP]\nexact h\u2082\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave L :=\n  (Primrec.fst.comp Primrec.fst).comp (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Option \u2115))\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115))\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115))\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) (\u03b2 := Code \u00d7 Option \u2115))\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave z := Primrec.fst.comp (Primrec.unpair.comp n)\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave m := Primrec.snd.comp (Primrec.unpair.comp n)\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm : Primrec fun a => (unpair a.fst.fst.snd).snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nhave h\u2081 := hlup.comp <| L.pair <| (k.pair cf).pair (Primrec\u2082.natPair.comp z m)\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm : Primrec fun a => (unpair a.fst.fst.snd).snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n          Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.snd\n\u22a2 Primrec fun a =>\n    let z := (unpair a.fst.fst.snd).fst;\n    let m := (unpair a.fst.fst.snd).snd;\n    do\n    let x \u2190\n      Nat.Partrec.Code.lup a.fst.fst.fst ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst) (Nat.pair z m)\n    Nat.casesOn x (some m) fun x =>\n        Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n          (Nat.pair z (m + 1))\n[PROOFSTEP]\nrefine' Primrec.option_bind h\u2081 (_ : Primrec _)\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm : Primrec fun a => (unpair a.fst.fst.snd).snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n          Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.snd\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.fst.fst.snd).snd) fun x =>\n          Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair (unpair a.fst.fst.snd).fst ((unpair a.fst.fst.snd).snd + 1)))\n      p.fst p.snd\n[PROOFSTEP]\nhave m := m.comp (Primrec.fst (\u03b2 := \u2115))\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm\u271d : Primrec fun a => (unpair a.fst.fst.snd).snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n          Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.snd\nm : Primrec fun a => (unpair a.fst.fst.fst.snd).snd\n\u22a2 Primrec fun p =>\n    (fun a x =>\n        Nat.casesOn x (some (unpair a.fst.fst.snd).snd) fun x =>\n          Nat.Partrec.Code.lup a.fst.fst.fst (a.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).snd)\n            (Nat.pair (unpair a.fst.fst.snd).fst ((unpair a.fst.fst.snd).snd + 1)))\n      p.fst p.snd\n[PROOFSTEP]\nrefine Primrec.nat_casesOn Primrec.snd (Primrec.option_some.comp m) ?_\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm\u271d : Primrec fun a => (unpair a.fst.fst.snd).snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n          Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.snd\nm : Primrec fun a => (unpair a.fst.fst.fst.snd).snd\n\u22a2 Primrec\u2082 fun p n =>\n    (fun x =>\n        Nat.Partrec.Code.lup p.fst.fst.fst.fst (p.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).snd)\n          (Nat.pair (unpair p.fst.fst.fst.snd).fst ((unpair p.fst.fst.fst.snd).snd + 1)))\n      n\n[PROOFSTEP]\nunfold Primrec\u2082\n[GOAL]\ncase hrf\na : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a)\nk\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst)).fst\nn\u271d\u00b9 : Primrec Prod.snd\nk\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).fst\nn\u271d : Primrec fun a => a.fst.snd\nk' : Primrec Prod.snd\nc : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst)).snd\nL : Primrec fun a => a.fst.fst.fst\nk : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst\nn : Primrec fun a => a.fst.fst.snd\ncf : Primrec fun a => a.snd.fst\nz : Primrec fun a => (unpair a.fst.fst.snd).fst\nm\u271d : Primrec fun a => (unpair a.fst.fst.snd).snd\nh\u2081 :\n  Primrec fun a =>\n    Nat.Partrec.Code.lup\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n          Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.fst\n      (a.fst.fst.fst, ((ofNat (\u2115 \u00d7 Code) (List.length a.fst.fst.fst)).fst, a.snd.fst),\n            Nat.pair (unpair a.fst.fst.snd).fst (unpair a.fst.fst.snd).snd).snd.snd\nm : Primrec fun a => (unpair a.fst.fst.fst.snd).snd\n\u22a2 Primrec fun p =>\n    (fun p n =>\n        (fun x =>\n            Nat.Partrec.Code.lup p.fst.fst.fst.fst\n              (p.fst.fst.snd, (ofNat (\u2115 \u00d7 Code) (List.length p.fst.fst.fst.fst)).snd)\n              (Nat.pair (unpair p.fst.fst.fst.snd).fst ((unpair p.fst.fst.fst.snd).snd + 1)))\n          n)\n      p.fst p.snd\n[PROOFSTEP]\nexact\n  (hlup.comp\n        ((L.comp Primrec.fst).pair <|\n          ((k'.pair c).comp <| Primrec.fst.comp Primrec.fst).pair\n            (Primrec\u2082.natPair.comp (z.comp Primrec.fst) (_root_.Primrec.succ.comp m)))).comp\n    Primrec.fst\n[GOAL]\nk : \u2115\nc : Code\nn : \u2115\n\u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n\n[PROOFSTEP]\nby_cases kn : n < k\n[GOAL]\ncase pos\nk : \u2115\nc : Code\nn : \u2115\nkn : n < k\n\u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n\n[PROOFSTEP]\nsimp [List.get?_range kn]\n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn : \u2115\nkn : \u00acn < k\n\u22a2 (Option.bind (Option.map (evaln k c) (List.get? (List.range k) n)) fun b => b) = evaln k c n\n[PROOFSTEP]\nrw [List.get?_len_le]\n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn : \u2115\nkn : \u00acn < k\n\u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = evaln k c n\n[PROOFSTEP]\ncases e : evaln k c n\n[GOAL]\ncase neg.none\nk : \u2115\nc : Code\nn : \u2115\nkn : \u00acn < k\ne : evaln k c n = Option.none\n\u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = Option.none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg.some\nk : \u2115\nc : Code\nn : \u2115\nkn : \u00acn < k\nval\u271d : \u2115\ne : evaln k c n = some val\u271d\n\u22a2 (Option.bind (Option.map (evaln k c) Option.none) fun b => b) = some val\u271d\n[PROOFSTEP]\nexact kn.elim (evaln_bound e)\n[GOAL]\ncase neg\nk : \u2115\nc : Code\nn : \u2115\nkn : \u00acn < k\n\u22a2 List.length (List.range k) \u2264 n\n[PROOFSTEP]\nsimpa using kn\n[GOAL]\nx\u271d : Unit\np : \u2115\n\u22a2 Nat.Partrec.Code.G\n      (x\u271d,\n          List.map\n            (fun n =>\n              let a := ofNat (\u2115 \u00d7 Code) n;\n              List.map (evaln a.fst a.snd) (List.range a.fst))\n            (List.range p)).snd =\n    some\n      (let a := ofNat (\u2115 \u00d7 Code) p;\n      List.map (evaln a.fst a.snd) (List.range a.fst))\n[PROOFSTEP]\nsimp only [G, prod_ofNat_val, ofNat_nat, List.length_map, List.length_range, Nat.pair_unpair, Option.some_inj]\n[GOAL]\nx\u271d : Unit\np : \u2115\n\u22a2 List.map\n      (fun n =>\n        Nat.rec Option.none\n          (fun n_1 n_ih =>\n            rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cf) n\n                let y \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cg) n\n                some (Nat.pair x y))\n              (fun cf cg x x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cg) n\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) x)\n              (fun cf cg x x =>\n                Nat.rec\n                  (Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) (unpair n).fst)\n                  (fun n_2 n_ih => do\n                    let i \u2190\n                      Nat.Partrec.Code.lup\n                          (List.map\n                            (fun n =>\n                              List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                            (List.range p))\n                          (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range p))\n                        ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                  (unpair n).snd)\n              (fun cf x => do\n                let x \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cf) n\n                Nat.rec (some (unpair n).snd)\n                    (fun n_2 n_ih =>\n                      Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                    x)\n              (ofNat Code (unpair p).snd))\n          (unpair p).fst)\n      (List.range (unpair p).fst) =\n    List.map (evaln (unpair p).fst (ofNat Code (unpair p).snd)) (List.range (unpair p).fst)\n[PROOFSTEP]\nrefine List.map_congr fun n => ?_\n[GOAL]\nx\u271d : Unit\np n : \u2115\n\u22a2 n \u2208 List.range (unpair p).fst \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range p))\n                  ((unpair p).fst, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range p))\n                  ((unpair p).fst, cf) (unpair n).fst)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                (unpair n).snd)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) n\n              Nat.rec (some (unpair n).snd)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                  x)\n            (ofNat Code (unpair p).snd))\n        (unpair p).fst =\n      evaln (unpair p).fst (ofNat Code (unpair p).snd) n\n[PROOFSTEP]\nhave : List.range p = List.range (Nat.pair p.unpair.1 (encode (ofNat Code p.unpair.2))) := by simp\n[GOAL]\nx\u271d : Unit\np n : \u2115\n\u22a2 List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n[PROOFSTEP]\nsimp\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n\u22a2 n \u2208 List.range (unpair p).fst \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range p))\n                  ((unpair p).fst, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range p))\n                  ((unpair p).fst, cf) (unpair n).fst)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range p))\n                        (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                (unpair n).snd)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range p))\n                    ((unpair p).fst, cf) n\n              Nat.rec (some (unpair n).snd)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range p))\n                      (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                  x)\n            (ofNat Code (unpair p).snd))\n        (unpair p).fst =\n      evaln (unpair p).fst (ofNat Code (unpair p).snd) n\n[PROOFSTEP]\nrw [this]\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\n\u22a2 n \u2208 List.range (unpair p).fst \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                    ((unpair p).fst, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                    ((unpair p).fst, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                    ((unpair p).fst, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                  ((unpair p).fst, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                  ((unpair p).fst, cf) (unpair n).fst)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                        (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                      ((unpair p).fst, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                (unpair n).snd)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                    ((unpair p).fst, cf) n\n              Nat.rec (some (unpair n).snd)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))))\n                      (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                  x)\n            (ofNat Code (unpair p).snd))\n        (unpair p).fst =\n      evaln (unpair p).fst (ofNat Code (unpair p).snd) n\n[PROOFSTEP]\ngeneralize p.unpair.1 = k\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : \u2115\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                  (k, cf) (unpair n).fst)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                        (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                      (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                (unpair n).snd)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).snd)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair k (encode (ofNat Code (unpair p).snd)))))\n                      (n_1, ofNat Code (unpair p).snd) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                  x)\n            (ofNat Code (unpair p).snd))\n        k =\n      evaln k (ofNat Code (unpair p).snd) n\n[PROOFSTEP]\ngeneralize ofNat Code p.unpair.2 = c\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : \u2115\nc : Code\n\u22a2 n \u2208 List.range k \u2192\n    Nat.rec Option.none\n        (fun n_1 n_ih =>\n          rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              let y \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              some (Nat.pair x y))\n            (fun cf cg x x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) n\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) x)\n            (fun cf cg x x =>\n              Nat.rec\n                (Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) (unpair n).fst)\n                (fun n_2 n_ih => do\n                  let i \u2190\n                    Nat.Partrec.Code.lup\n                        (List.map\n                          (fun n =>\n                            List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                          (List.range (Nat.pair k (encode c))))\n                        (n_1, c) (Nat.pair (unpair n).fst n_2)\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair k (encode c))))\n                      (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n                (unpair n).snd)\n            (fun cf x => do\n              let x \u2190\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cf) n\n              Nat.rec (some (unpair n).snd)\n                  (fun n_2 n_ih =>\n                    Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                  x)\n            c)\n        k =\n      evaln k c n\n[PROOFSTEP]\nintro nk\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk : \u2115\nc : Code\nnk : n \u2208 List.range k\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cg) n\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair k (encode c))))\n                (k, cf) (unpair n).fst)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair k (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).fst n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (k, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n              (unpair n).snd)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair k (encode c))))\n                  (k, cf) n\n            Nat.rec (some (unpair n).snd)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair k (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                x)\n          c)\n      k =\n    evaln k c n\n[PROOFSTEP]\ncases' k with k'\n[GOAL]\ncase zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nnk : n \u2208 List.range Nat.zero\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cg) n\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair Nat.zero (encode c))))\n                (Nat.zero, cf) (unpair n).fst)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair Nat.zero (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).fst n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (Nat.zero, cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n              (unpair n).snd)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair Nat.zero (encode c))))\n                  (Nat.zero, cf) n\n            Nat.rec (some (unpair n).snd)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair Nat.zero (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                x)\n          c)\n      Nat.zero =\n    evaln Nat.zero c n\n[PROOFSTEP]\nsimp [evaln]\n[GOAL]\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).fst)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).fst n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n              (unpair n).snd)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).snd)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n\n[PROOFSTEP]\nlet k := k' + 1\n[GOAL]\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 Nat.rec Option.none\n      (fun n_1 n_ih =>\n        rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            let y \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            some (Nat.pair x y))\n          (fun cf cg x x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cg) n\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) x)\n          (fun cf cg x x =>\n            Nat.rec\n              (Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (Nat.succ k') (encode c))))\n                (Nat.succ k', cf) (unpair n).fst)\n              (fun n_2 n_ih => do\n                let i \u2190\n                  Nat.Partrec.Code.lup\n                      (List.map\n                        (fun n =>\n                          List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                        (List.range (Nat.pair (Nat.succ k') (encode c))))\n                      (n_1, c) (Nat.pair (unpair n).fst n_2)\n                Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (Nat.succ k', cg) (Nat.pair (unpair n).fst (Nat.pair n_2 i)))\n              (unpair n).snd)\n          (fun cf x => do\n            let x \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (Nat.succ k') (encode c))))\n                  (Nat.succ k', cf) n\n            Nat.rec (some (unpair n).snd)\n                (fun n_2 n_ih =>\n                  Nat.Partrec.Code.lup\n                    (List.map\n                      (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                      (List.range (Nat.pair (Nat.succ k') (encode c))))\n                    (n_1, c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n                x)\n          c)\n      (Nat.succ k') =\n    evaln (Nat.succ k') c n\n[PROOFSTEP]\nsimp only [show k'.succ = k from rfl]\n[GOAL]\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nnk : n \u2208 List.range (Nat.succ k')\nk : \u2115 := k' + 1\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n\n[PROOFSTEP]\nsimp [Nat.lt_succ_iff] at nk \n[GOAL]\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n\n[PROOFSTEP]\nhave hg :\n  \u2200 {k' c' n},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      lup\n          ((List.range (Nat.pair k (encode c))).map fun n =>\n            (List.range n.unpair.1).map (evaln n.unpair.1 (ofNat Code n.unpair.2)))\n          (k', c') n =\n        evaln k' c' n :=\n  by\n  intro k\u2081 c\u2081 n\u2081 hl\n  simp [lup, List.get?_range hl, evaln_map, Bind.bind]\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\n\u22a2 \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode c))))\n          (k', c') n =\n        evaln k' c' n\n[PROOFSTEP]\nintro k\u2081 c\u2081 n\u2081 hl\n[GOAL]\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nk\u2081 : \u2115\nc\u2081 : Code\nn\u2081 : \u2115\nhl : Nat.pair k\u2081 (encode c\u2081) < Nat.pair k (encode c)\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n        (List.range (Nat.pair k (encode c))))\n      (k\u2081, c\u2081) n\u2081 =\n    evaln k\u2081 c\u2081 n\u2081\n[PROOFSTEP]\nsimp [lup, List.get?_range hl, evaln_map, Bind.bind]\n[GOAL]\ncase succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nc : Code\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode c) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode c))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode c))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode c))))\n                  (k', c) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode c))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode c))))\n                (k', c) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      c =\n    evaln (k' + 1) c n\n[PROOFSTEP]\ncases' c with cf cg cf cg cf cg cf\n[GOAL]\ncase succ.zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode zero) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode zero))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode zero))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode zero))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode zero))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode zero))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode zero))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode zero))))\n                  (k', zero) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode zero))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode zero))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode zero))))\n                (k', zero) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      zero =\n    evaln (k' + 1) zero n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode succ) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode succ))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode succ))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode succ))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode succ))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode succ))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode succ))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode succ))))\n                  (k', succ) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode succ))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode succ))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode succ))))\n                (k', succ) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      succ =\n    evaln (k' + 1) succ n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.left\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode left) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode left))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode left))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode left))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode left))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode left))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode left))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode left))))\n                  (k', left) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode left))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode left))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode left))))\n                (k', left) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      left =\n    evaln (k' + 1) left n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.right\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode right) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode right))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode right))))\n              (k' + 1, cf) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode right))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode right))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode right))))\n            (k' + 1, cf) x)\n      (fun cf cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode right))))\n            (k' + 1, cf) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode right))))\n                  (k', right) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode right))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode right))))\n              (k' + 1, cf) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode right))))\n                (k', right) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      right =\n    evaln (k' + 1) right n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.pair\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n              (k' + 1, cf_1) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n              (k' + 1, cg_1) n\n        some (Nat.pair x y))\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n              (k' + 1, cg_1) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n            (k' + 1, cf_1) x)\n      (fun cf_1 cg_1 x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n            (k' + 1, cf_1) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n                  (k', pair cf cg) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n                (k' + 1, cg_1) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf_1 x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n              (k' + 1, cf_1) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n                (k', pair cf cg) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      (pair cf cg) =\n    evaln (k' + 1) (pair cf cg) n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.comp\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n              (k' + 1, cf_1) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n              (k' + 1, cg_1) n\n        some (Nat.pair x y))\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n              (k' + 1, cg_1) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n            (k' + 1, cf_1) x)\n      (fun cf_1 cg_1 x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n            (k' + 1, cf_1) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n                  (k', comp cf cg) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n                (k' + 1, cg_1) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf_1 x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n              (k' + 1, cf_1) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n                (k', comp cf cg) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      (comp cf cg) =\n    evaln (k' + 1) (comp cf cg) n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.prec\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n              (k' + 1, cf_1) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n              (k' + 1, cg_1) n\n        some (Nat.pair x y))\n      (fun cf_1 cg_1 x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n              (k' + 1, cg_1) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cf_1) x)\n      (fun cf_1 cg_1 x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cf_1) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n                  (k', prec cf cg) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n                (k' + 1, cg_1) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf_1 x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n              (k' + 1, cf_1) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n                (k', prec cf cg) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      (prec cf cg) =\n    evaln (k' + 1) (prec cf cg) n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 rec (some 0) (some (Nat.succ n)) (some (unpair n).fst) (some (unpair n).snd)\n      (fun cf_1 cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n              (k' + 1, cf_1) n\n        let y \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n              (k' + 1, cg) n\n        some (Nat.pair x y))\n      (fun cf_1 cg x x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n              (k' + 1, cg) n\n        Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k' + 1, cf_1) x)\n      (fun cf_1 cg x x =>\n        Nat.rec\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k' + 1, cf_1) (unpair n).fst)\n          (fun n_1 n_ih => do\n            let i \u2190\n              Nat.Partrec.Code.lup\n                  (List.map\n                    (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                    (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n                  (k', rfind' cf) (Nat.pair (unpair n).fst n_1)\n            Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n                (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n          (unpair n).snd)\n      (fun cf_1 x => do\n        let x \u2190\n          Nat.Partrec.Code.lup\n              (List.map\n                (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n              (k' + 1, cf_1) n\n        Nat.rec (some (unpair n).snd)\n            (fun n_1 n_ih =>\n              Nat.Partrec.Code.lup\n                (List.map\n                  (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n                  (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n                (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n            x)\n      (rfind' cf) =\n    evaln (k' + 1) (rfind' cf) n\n[PROOFSTEP]\nsimp [evaln, nk, Bind.bind, Functor.map, Seq.seq, pure]\n[GOAL]\ncase succ.pair\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)\n[PROOFSTEP]\ncases' encode_lt_pair cf cg with lf lg\n[GOAL]\ncase succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Option.bind\n        (Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair (k' + 1) (encode (pair cf cg)))))\n          (k' + 1, cg) n)\n        fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_right _ lf), hg (Nat.pair_lt_pair_right _ lg)]\n[GOAL]\ncase succ.pair.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind (evaln k cf n) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (evaln (k' + 1) cf n)) fun y => Option.map y (evaln (k' + 1) cg n)\n[PROOFSTEP]\ncases evaln k cf n\n[GOAL]\ncase succ.pair.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\n\u22a2 (Option.bind Option.none fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair Option.none) fun y => Option.map y (evaln (k' + 1) cg n)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.pair.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x => Option.bind (evaln k cg n) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (some val\u271d)) fun y => Option.map y (evaln (k' + 1) cg n)\n[PROOFSTEP]\ncases evaln k cg n\n[GOAL]\ncase succ.pair.intro.some.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x => Option.bind Option.none fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (some val\u271d)) fun y => Option.map y Option.none\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.pair.intro.some.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (pair cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (pair cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (pair cf cg)\nlg : encode cg < encode (pair cf cg)\nval\u271d\u00b9 val\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d\u00b9) fun x => Option.bind (some val\u271d) fun y => some (Nat.pair x y)) =\n    Option.bind (Option.map Nat.pair (some val\u271d\u00b9)) fun y => Option.map y (some val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.comp\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x\n[PROOFSTEP]\ncases' encode_lt_comp cf cg with lf lg\n[GOAL]\ncase succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cg) n)\n      fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_right _ lg)]\n[GOAL]\ncase succ.comp.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind (evaln k cg n) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (evaln (k' + 1) cg n) fun x => evaln (k' + 1) cf x\n[PROOFSTEP]\ncases evaln k cg n\n[GOAL]\ncase succ.comp.intro.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind Option.none fun x => evaln (k' + 1) cf x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.comp.intro.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (comp cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (comp cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (comp cf cg)\nlg : encode cg < encode (comp cf cg)\nval\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun x =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (comp cf cg)))))\n        (k' + 1, cf) x) =\n    Option.bind (some val\u271d) fun x => evaln (k' + 1) cf x\n[PROOFSTEP]\nsimp [hg (Nat.pair_lt_pair_right _ lf)]\n[GOAL]\ncase succ.prec\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd\n[PROOFSTEP]\ncases' encode_lt_prec cf cg with lf lg\n[GOAL]\ncase succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cf) (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_right _ lf)]\n[GOAL]\ncase succ.prec.intro\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd =\n    Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (unpair n).snd\n[PROOFSTEP]\ncases n.unpair.2\n[GOAL]\ncase succ.prec.intro.zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\n\u22a2 Nat.rec (evaln k cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      Nat.zero =\n    Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      Nat.zero\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 Nat.rec (evaln k cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind\n          (Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k', prec cf cg) (Nat.pair (unpair n).fst n_1))\n          fun i =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n            (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (Nat.succ n\u271d) =\n    Nat.rec (evaln (k' + 1) cf (unpair n).fst)\n      (fun n_1 n_ih =>\n        Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n_1)) fun i =>\n          evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n_1 i)))\n      (Nat.succ n\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k', prec cf cg) (Nat.pair (unpair n).fst n\u271d))\n      fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d i))\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]\n[GOAL]\ncase succ.prec.intro.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n\u271d)) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n\u271d i))) =\n    Option.bind (evaln k' (prec cf cg) (Nat.pair (unpair n).fst n\u271d)) fun i =>\n      evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d i))\n[PROOFSTEP]\ncases evaln k' _ _\n[GOAL]\ncase succ.prec.intro.succ.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d : \u2115\n\u22a2 (Option.bind Option.none fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n\u271d i))) =\n    Option.bind Option.none fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d i))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.prec.intro.succ.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf cg : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (prec cf cg)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (prec cf cg)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (prec cf cg)\nlg : encode cg < encode (prec cf cg)\nn\u271d val\u271d : \u2115\n\u22a2 (Option.bind (some val\u271d) fun i =>\n      Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (prec cf cg)))))\n        (k' + 1, cg) (Nat.pair (unpair n).fst (Nat.pair n\u271d i))) =\n    Option.bind (some val\u271d) fun i => evaln (k' + 1) cg (Nat.pair (unpair n).fst (Nat.pair n\u271d i))\n[PROOFSTEP]\nsimp [hg (Nat.pair_lt_pair_right _ lg)]\n[GOAL]\ncase succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).snd)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nhave lf := encode_lt_rfind' cf\n[GOAL]\ncase succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind\n      (Nat.Partrec.Code.lup\n        (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n          (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n        (k' + 1, cf) n)\n      fun x =>\n      Nat.rec (some (unpair n).snd)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_right _ lf)]\n[GOAL]\ncase succ.rfind'\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind (evaln k cf n) fun x =>\n      Nat.rec (some (unpair n).snd)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n        x) =\n    Option.bind (evaln (k' + 1) cf n) fun x =>\n      if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\ncases' evaln k cf n with x\n[GOAL]\ncase succ.rfind'.none\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 (Option.bind Option.none fun x =>\n      Nat.rec (some (unpair n).snd)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n        x) =\n    Option.bind Option.none fun x =>\n      if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 (Option.bind (some x) fun x =>\n      Nat.rec (some (unpair n).snd)\n        (fun n_1 n_ih =>\n          Nat.Partrec.Code.lup\n            (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n              (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n            (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n        x) =\n    Option.bind (some x) fun x =>\n      if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.rfind'.some\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nx : \u2115\n\u22a2 Nat.rec (some (unpair n).snd)\n      (fun n_1 n_ih =>\n        Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n          (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n      x =\n    if x = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\ncases x\n[GOAL]\ncase succ.rfind'.some.zero\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\n\u22a2 Nat.rec (some (unpair n).snd)\n      (fun n_1 n_ih =>\n        Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n          (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n      Nat.zero =\n    if Nat.zero = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nsimp [Nat.succ_ne_zero]\n[GOAL]\ncase succ.rfind'.some.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn\u271d : \u2115\n\u22a2 Nat.rec (some (unpair n).snd)\n      (fun n_1 n_ih =>\n        Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n          (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)))\n      (Nat.succ n\u271d) =\n    if Nat.succ n\u271d = 0 then some (unpair n).snd else evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nsimp [Nat.succ_ne_zero]\n[GOAL]\ncase succ.rfind'.some.succ\nx\u271d : Unit\np n : \u2115\nthis : List.range p = List.range (Nat.pair (unpair p).fst (encode (ofNat Code (unpair p).snd)))\nk' : \u2115\nk : \u2115 := k' + 1\nnk : n \u2264 k'\ncf : Code\nhg :\n  \u2200 {k' : \u2115} {c' : Code} {n : \u2115},\n    Nat.pair k' (encode c') < Nat.pair k (encode (rfind' cf)) \u2192\n      Nat.Partrec.Code.lup\n          (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n            (List.range (Nat.pair k (encode (rfind' cf)))))\n          (k', c') n =\n        evaln k' c' n\nlf : encode cf < encode (rfind' cf)\nn\u271d : \u2115\n\u22a2 Nat.Partrec.Code.lup\n      (List.map (fun n => List.map (evaln (unpair n).fst (ofNat Code (unpair n).snd)) (List.range (unpair n).fst))\n        (List.range (Nat.pair (k' + 1) (encode (rfind' cf)))))\n      (k', rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1)) =\n    evaln k' (rfind' cf) (Nat.pair (unpair n).fst ((unpair n).snd + 1))\n[PROOFSTEP]\nrw [hg (Nat.pair_lt_pair_left _ k'.lt_succ_self)]\n[GOAL]\nthis :\n  Primrec\u2082 fun x n =>\n    let a := ofNat (\u2115 \u00d7 Code) n;\n    List.map (evaln a.fst a.snd) (List.range a.fst)\nx\u271d : (\u2115 \u00d7 Code) \u00d7 \u2115\nk : \u2115\nc : Code\nn : \u2115\n\u22a2 (Option.bind\n      (List.get?\n        (let a := ofNat (\u2115 \u00d7 Code) (encode ((k, c), n).fst);\n        List.map (evaln a.fst a.snd) (List.range a.fst))\n        ((k, c), n).snd)\n      fun b => (((k, c), n), b).snd) =\n    evaln ((k, c), n).fst.fst ((k, c), n).fst.snd ((k, c), n).snd\n[PROOFSTEP]\nsimp [evaln_map]\n[GOAL]\nc : Code\nn x : \u2115\n\u22a2 x \u2208 eval c n \u2194 x \u2208 rfindOpt fun k => evaln k c n\n[PROOFSTEP]\nrefine' evaln_complete.trans (Nat.rfindOpt_mono _).symm\n[GOAL]\nc : Code\nn x : \u2115\n\u22a2 \u2200 {a m n_1 : \u2115}, m \u2264 n_1 \u2192 a \u2208 evaln m c n \u2192 a \u2208 evaln n_1 c n\n[PROOFSTEP]\nintro a m n hl\n[GOAL]\nc : Code\nn\u271d x a m n : \u2115\nhl : m \u2264 n\n\u22a2 a \u2208 evaln m c n\u271d \u2192 a \u2208 evaln n c n\u271d\n[PROOFSTEP]\napply evaln_mono hl\n[GOAL]\na : Code \u00d7 \u2115\n\u22a2 (rfindOpt fun b =>\n      evaln (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).fst.fst\n        (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).fst.snd (((a, b).snd, (a, b).fst.fst), (a, b).fst.snd).snd) =\n    eval a.fst a.snd\n[PROOFSTEP]\nsimp [eval_eq_rfindOpt]\n[GOAL]\nf : Code \u2192 Code\nhf : Computable f\ng : \u2115 \u2192 \u2115 \u2192 Part \u2115 :=\n  fun x y => do\n    let b \u2190 eval (ofNat Code x) x\n    eval (ofNat Code b) y\nthis : Partrec\u2082 g\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\n\u22a2 \u2200 (a n : \u2115), eval cg (Nat.pair a n) = Part.map encode (g a n)\n[PROOFSTEP]\nsimp [eg]\n[GOAL]\nf : Code \u2192 Code\nhf : Computable f\ng : \u2115 \u2192 \u2115 \u2192 Part \u2115 :=\n  fun x y => do\n    let b \u2190 eval (ofNat Code x) x\n    eval (ofNat Code b) y\nthis\u271d : Partrec\u2082 g\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\neg' : \u2200 (a n : \u2115), eval cg (Nat.pair a n) = Part.map encode (g a n)\nF : \u2115 \u2192 Code := fun x => f (curry cg x)\nthis : Computable F\ncF : Code\neF : eval cF = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (\u2191F a)\n\u22a2 eval cF (encode cF) = Part.some (encode (F (encode cF)))\n[PROOFSTEP]\nsimp [eF]\n[GOAL]\nf : Code \u2192 Code\nhf : Computable f\ng : \u2115 \u2192 \u2115 \u2192 Part \u2115 :=\n  fun x y => do\n    let b \u2190 eval (ofNat Code x) x\n    eval (ofNat Code b) y\nthis\u271d : Partrec\u2082 g\ncg : Code\neg : eval cg = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => g p.fst p.snd) a)\neg' : \u2200 (a n : \u2115), eval cg (Nat.pair a n) = Part.map encode (g a n)\nF : \u2115 \u2192 Code := fun x => f (curry cg x)\nthis : Computable F\ncF : Code\neF : eval cF = fun n => Part.bind \u2191(decode n) fun a => Part.map encode (\u2191F a)\neF' : eval cF (encode cF) = Part.some (encode (F (encode cF)))\nn : \u2115\n\u22a2 eval (f (curry cg (encode cF))) n = eval (curry cg (encode cF)) n\n[PROOFSTEP]\nsimp [eg', eF', Part.map_id']\n[GOAL]\nf : Code \u2192 \u2115 \u2192. \u2115\nhf : Partrec\u2082 f\ncf : Code\nef : eval cf = fun n => Part.bind \u2191(decode n) fun a => Part.map encode ((fun p => f p.fst p.snd) a)\nc : Code\ne : eval (curry cf (encode c)) = eval c\nn : \u2115\n\u22a2 eval c n = f c n\n[PROOFSTEP]\nsimp [e.symm, ef, Part.map_id']\n", "meta": {"mathlib_filename": "Mathlib.Computability.PartrecCode", "llama_tokens": 431280, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707283, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.5170770175455709}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Closeds \u03b1\nh : s.carrier = t.carrier\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Closeds \u03b1\ncarrier\u271d : Set \u03b1\nclosed'\u271d : IsClosed carrier\u271d\nh : { carrier := carrier\u271d, closed' := closed'\u271d }.carrier = t.carrier\n\u22a2 { carrier := carrier\u271d, closed' := closed'\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ncarrier\u271d\u00b9 : Set \u03b1\nclosed'\u271d\u00b9 : IsClosed carrier\u271d\u00b9\ncarrier\u271d : Set \u03b1\nclosed'\u271d : IsClosed carrier\u271d\nh : { carrier := carrier\u271d\u00b9, closed' := closed'\u271d\u00b9 }.carrier = { carrier := carrier\u271d, closed' := closed'\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, closed' := closed'\u271d\u00b9 } = { carrier := carrier\u271d, closed' := closed'\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u03b9 : Sort u_4\nx : \u03b1\ns : \u03b9 \u2192 Closeds \u03b1\n\u22a2 x \u2208 iInf s \u2194 \u2200 (i : \u03b9), x \u2208 s i\n[PROOFSTEP]\nsimp [iInf]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u03b9 : Sort u_4\ns : \u03b9 \u2192 Closeds \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), s i) = \u22c2 (i : \u03b9), \u2191(s i)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u03b9 : Sort u_4\ns : \u03b9 \u2192 Closeds \u03b1\nx\u271d : \u03b1\n\u22a2 x\u271d \u2208 \u2191(\u2a05 (i : \u03b9), s i) \u2194 x\u271d \u2208 \u22c2 (i : \u03b9), \u2191(s i)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u03b9 : Sort u_4\ns : \u03b9 \u2192 Closeds \u03b1\n\u22a2 \u2a05 (i : \u03b9), s i = { carrier := \u22c2 (i : \u03b9), \u2191(s i), closed' := (_ : IsClosed (\u22c2 (i : \u03b9), \u2191(s i))) }\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\n\u03b9 : Sort u_4\ns : \u03b9 \u2192 Closeds \u03b1\n\u22a2 \u2191(\u2a05 (i : \u03b9), s i) = \u2191{ carrier := \u22c2 (i : \u03b9), \u2191(s i), closed' := (_ : IsClosed (\u22c2 (i : \u03b9), \u2191(s i))) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nsrc\u271d : CompleteLattice (Closeds \u03b1) := inferInstanceAs (CompleteLattice (Closeds \u03b1))\na : Closeds \u03b1\ns : Set (Closeds \u03b1)\n\u22a2 \u2191(\u2a05 (b : Closeds \u03b1) (_ : b \u2208 s), a \u2294 b) = \u2191(a \u2294 sInf s)\n[PROOFSTEP]\nsimp only [coe_sup, coe_iInf, coe_sInf, Set.union_iInter\u2082]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Closeds \u03b1\n\u22a2 (Opens.compl \u2218 \u2191ofDual) ((\u2191toDual \u2218 compl) s) = s\n[PROOFSTEP]\nsimp [Closeds.compl_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : (Opens \u03b1)\u1d52\u1d48\n\u22a2 (\u2191toDual \u2218 compl) ((Opens.compl \u2218 \u2191ofDual) s) = s\n[PROOFSTEP]\nsimp [Opens.compl_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : Opens \u03b1\n\u22a2 (Closeds.compl \u2218 \u2191ofDual) ((\u2191toDual \u2218 compl) s) = s\n[PROOFSTEP]\nsimp [Opens.compl_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns : (Closeds \u03b1)\u1d52\u1d48\n\u22a2 (\u2191toDual \u2218 compl) ((Closeds.compl \u2218 \u2191ofDual) s) = s\n[PROOFSTEP]\nsimp [Closeds.compl_compl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Closeds \u03b1\n\u22a2 IsAtom s \u2194 \u2203 x, s = singleton x\n[PROOFSTEP]\nhave : IsAtom (s : Set \u03b1) \u2194 IsAtom s :=\n  by\n  refine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s\n  obtain \u27e8x, rfl\u27e9 := t.isAtom_iff.mp ht\n  exact closure_singleton\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Closeds \u03b1\n\u22a2 IsAtom \u2191s \u2194 IsAtom s\n[PROOFSTEP]\nrefine' Closeds.gi.isAtom_iff' rfl (fun t ht => _) s\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Closeds \u03b1\nt : Set \u03b1\nht : IsAtom t\n\u22a2 \u2191(Closeds.closure t) = t\n[PROOFSTEP]\nobtain \u27e8x, rfl\u27e9 := t.isAtom_iff.mp ht\n[GOAL]\ncase intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Closeds \u03b1\nx : \u03b1\nht : IsAtom {x}\n\u22a2 \u2191(Closeds.closure {x}) = {x}\n[PROOFSTEP]\nexact closure_singleton\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Closeds \u03b1\nthis : IsAtom \u2191s \u2194 IsAtom s\n\u22a2 IsAtom s \u2194 \u2203 x, s = singleton x\n[PROOFSTEP]\nsimp only [\u2190 this, (s : Set \u03b1).isAtom_iff, SetLike.ext'_iff, Closeds.singleton_coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Opens \u03b1\n\u22a2 IsCoatom s \u2194 \u2203 x, s = Closeds.compl (Closeds.singleton x)\n[PROOFSTEP]\nrw [\u2190 s.compl_compl, \u2190 isAtom_dual_iff_isCoatom]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Opens \u03b1\n\u22a2 IsAtom (\u2191toDual (Closeds.compl (compl s))) \u2194 \u2203 x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)\n[PROOFSTEP]\nchange IsAtom (Closeds.complOrderIso \u03b1 s.compl) \u2194 _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b2\ninst\u271d : T1Space \u03b1\ns : Opens \u03b1\n\u22a2 IsAtom (\u2191(Closeds.complOrderIso \u03b1) (compl s)) \u2194 \u2203 x, Closeds.compl (compl s) = Closeds.compl (Closeds.singleton x)\n[PROOFSTEP]\nsimp only [(Closeds.complOrderIso \u03b1).isAtom_iff, Closeds.isAtom_iff, Closeds.compl_bijective.injective.eq_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ns t : Clopens \u03b1\nh : (fun s => s.carrier) s = (fun s => s.carrier) t\n\u22a2 s = t\n[PROOFSTEP]\ncases s\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nt : Clopens \u03b1\ncarrier\u271d : Set \u03b1\nclopen'\u271d : IsClopen carrier\u271d\nh : (fun s => s.carrier) { carrier := carrier\u271d, clopen' := clopen'\u271d } = (fun s => s.carrier) t\n\u22a2 { carrier := carrier\u271d, clopen' := clopen'\u271d } = t\n[PROOFSTEP]\ncases t\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\ncarrier\u271d\u00b9 : Set \u03b1\nclopen'\u271d\u00b9 : IsClopen carrier\u271d\u00b9\ncarrier\u271d : Set \u03b1\nclopen'\u271d : IsClopen carrier\u271d\nh :\n  (fun s => s.carrier) { carrier := carrier\u271d\u00b9, clopen' := clopen'\u271d\u00b9 } =\n    (fun s => s.carrier) { carrier := carrier\u271d, clopen' := clopen'\u271d }\n\u22a2 { carrier := carrier\u271d\u00b9, clopen' := clopen'\u271d\u00b9 } = { carrier := carrier\u271d, clopen' := clopen'\u271d }\n[PROOFSTEP]\ncongr\n", "meta": {"mathlib_filename": "Mathlib.Topology.Sets.Closeds", "llama_tokens": 3277, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.5170770175455708}}
{"text": "[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx : PiLp 2 f\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner x x)\n[PROOFSTEP]\nsimp only [PiLp.norm_sq_eq_of_L2, map_sum, \u2190 norm_sq_eq_inner, one_div]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\n\u22a2 \u2200 (x y : PiLp 2 f), \u2191(starRingEnd \ud835\udd5c) (inner y x) = inner x y\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) = inner x y\n[PROOFSTEP]\nunfold inner\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\n\u22a2 \u2191(starRingEnd \ud835\udd5c) ({ inner := fun x y => \u2211 i : \u03b9, InnerProductSpace.toInner.1 (x i) (y i) }.1 y x) =\n    { inner := fun x y => \u2211 i : \u03b9, InnerProductSpace.toInner.1 (x i) (y i) }.1 x y\n[PROOFSTEP]\nrw [map_sum]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\n\u22a2 \u2211 x_1 : \u03b9, \u2191(starRingEnd \ud835\udd5c) (InnerProductSpace.toInner.1 (y x_1) (x x_1)) =\n    { inner := fun x y => \u2211 i : \u03b9, InnerProductSpace.toInner.1 (x i) (y i) }.1 x y\n[PROOFSTEP]\napply Finset.sum_congr rfl\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\n\u22a2 \u2200 (x_1 : \u03b9),\n    x_1 \u2208 Finset.univ \u2192\n      \u2191(starRingEnd \ud835\udd5c) (InnerProductSpace.toInner.1 (y x_1) (x x_1)) = InnerProductSpace.toInner.1 (x x_1) (y x_1)\n[PROOFSTEP]\nrintro z -\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\nz : \u03b9\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (InnerProductSpace.toInner.1 (y z) (x z)) = InnerProductSpace.toInner.1 (x z) (y z)\n[PROOFSTEP]\napply inner_conj_symm\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y z : PiLp 2 f\n\u22a2 \u2211 i : \u03b9, inner (x i + y i) (z i) = \u2211 i : \u03b9, inner (x i) (z i) + \u2211 i : \u03b9, inner (y i) (z i)\n[PROOFSTEP]\nsimp only [inner_add_left, Finset.sum_add_distrib]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\n\u03b9 : Type u_8\ninst\u271d\u00b2 : Fintype \u03b9\nf : \u03b9 \u2192 Type u_9\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (f i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (f i)\nx y : PiLp 2 f\nr : \ud835\udd5c\n\u22a2 \u2211 i : \u03b9, inner (r \u2022 x i) (y i) = \u2191(starRingEnd \ud835\udd5c) r * \u2211 i : \u03b9, inner (x i) (y i)\n[PROOFSTEP]\nsimp only [Finset.mul_sum, inner_smul_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c\u271d : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\u271d\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c\u271d E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c\u271d E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\n\ud835\udd5c : Type u_8\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\nn : Type u_9\ninst\u271d : Fintype n\nx : EuclideanSpace \ud835\udd5c n\n\u22a2 \u2016x\u2016 = Real.sqrt (\u2211 i : n, \u2016x i\u2016 ^ 2)\n[PROOFSTEP]\nsimpa only [Real.coe_sqrt, NNReal.coe_sum] using congr_arg ((\u2191) : \u211d\u22650 \u2192 \u211d) x.nnnorm_eq\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 FiniteDimensional \ud835\udd5c (EuclideanSpace \ud835\udd5c \u03b9)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 InnerProductSpace \ud835\udd5c (EuclideanSpace \ud835\udd5c \u03b9)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 FiniteDimensional.finrank \ud835\udd5c (EuclideanSpace \ud835\udd5c \u03b9) = Fintype.card \u03b9\n[PROOFSTEP]\nsimp [EuclideanSpace, PiLp, WithLp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nn : \u2115\n\u22a2 FiniteDimensional.finrank \ud835\udd5c (EuclideanSpace \ud835\udd5c (Fin n)) = n\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\n\u22a2 E \u2243\u2097\u1d62[\ud835\udd5c] PiLp 2 fun i => { x // x \u2208 V i }\n[PROOFSTEP]\nlet e\u2081 := DirectSum.linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => V i\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\n\u22a2 E \u2243\u2097\u1d62[\ud835\udd5c] PiLp 2 fun i => { x // x \u2208 V i }\n[PROOFSTEP]\nlet e\u2082 := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\n\u22a2 E \u2243\u2097\u1d62[\ud835\udd5c] PiLp 2 fun i => { x // x \u2208 V i }\n[PROOFSTEP]\nrefine' LinearEquiv.isometryOfInner (e\u2082.symm.trans e\u2081) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\n\u22a2 \u2200 (x y : E),\n    inner (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) x) (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) y) =\n      inner x y\n[PROOFSTEP]\nsuffices \u2200 (v w : PiLp 2 fun i => V i), \u27eav, w\u27eb = \u27eae\u2082 (e\u2081.symm v), e\u2082 (e\u2081.symm w)\u27eb\n  by\n  intro v\u2080 w\u2080\n  convert this (e\u2081 (e\u2082.symm v\u2080)) (e\u2081 (e\u2082.symm w\u2080)) <;>\n    simp only [LinearEquiv.symm_apply_apply, LinearEquiv.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nthis :\n  \u2200 (v w : PiLp 2 fun i => { x // x \u2208 V i }),\n    inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\n\u22a2 \u2200 (x y : E),\n    inner (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) x) (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) y) =\n      inner x y\n[PROOFSTEP]\nintro v\u2080 w\u2080\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nthis :\n  \u2200 (v w : PiLp 2 fun i => { x // x \u2208 V i }),\n    inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\nv\u2080 w\u2080 : E\n\u22a2 inner (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) v\u2080) (\u2191(LinearEquiv.trans (LinearEquiv.symm e\u2082) e\u2081) w\u2080) =\n    inner v\u2080 w\u2080\n[PROOFSTEP]\nconvert this (e\u2081 (e\u2082.symm v\u2080)) (e\u2081 (e\u2082.symm w\u2080))\n[GOAL]\ncase h.e'_3.h.e'_4\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nthis :\n  \u2200 (v w : PiLp 2 fun i => { x // x \u2208 V i }),\n    inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\nv\u2080 w\u2080 : E\n\u22a2 v\u2080 = \u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) (\u2191e\u2081 (\u2191(LinearEquiv.symm e\u2082) v\u2080)))\n[PROOFSTEP]\nsimp only [LinearEquiv.symm_apply_apply, LinearEquiv.apply_symm_apply]\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nthis :\n  \u2200 (v w : PiLp 2 fun i => { x // x \u2208 V i }),\n    inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\nv\u2080 w\u2080 : E\n\u22a2 w\u2080 = \u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) (\u2191e\u2081 (\u2191(LinearEquiv.symm e\u2082) w\u2080)))\n[PROOFSTEP]\nsimp only [LinearEquiv.symm_apply_apply, LinearEquiv.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\n\u22a2 \u2200 (v w : PiLp 2 fun i => { x // x \u2208 V i }),\n    inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\n[PROOFSTEP]\nintro v w\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv w : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 inner v w = inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\n[PROOFSTEP]\ntrans \u27ea\u2211 i, (V i).subtype\u2097\u1d62 (v i), \u2211 i, (V i).subtype\u2097\u1d62 (w i)\u27eb\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv w : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 inner v w = inner (\u2211 i : \u03b9, \u2191(subtype\u2097\u1d62 (V i)) (v i)) (\u2211 i : \u03b9, \u2191(subtype\u2097\u1d62 (V i)) (w i))\n[PROOFSTEP]\nsimp only [sum_inner, hV'.inner_right_fintype, PiLp.inner_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv w : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 inner (\u2211 i : \u03b9, \u2191(subtype\u2097\u1d62 (V i)) (v i)) (\u2211 i : \u03b9, \u2191(subtype\u2097\u1d62 (V i)) (w i)) =\n    inner (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) v)) (\u2191e\u2082 (\u2191(LinearEquiv.symm e\u2081) w))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_s\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv w : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 Finset.univ =\n    Multiset.toFinset \u2191{ val := Finset.univ.val, property := (_ : \u2200 (x : \u03b9), x \u2208 Finset.univ.val \u2228 v x = 0) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase e_a.e_s\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv w : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 Finset.univ =\n    Multiset.toFinset \u2191{ val := Finset.univ.val, property := (_ : \u2200 (x : \u03b9), x \u2208 Finset.univ.val \u2228 w x = 0) }\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryL2OfOrthogonalFamily hV hV')) w = \u2211 i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nclassical\nlet e\u2081 := DirectSum.linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => V i\nlet e\u2082 := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV\nsuffices \u2200 v : \u2a01 i, V i, e\u2082 v = \u2211 i, e\u2081 v i by exact this (e\u2081.symm w)\nintro v\nsimp [DirectSum.coeLinearMap, DirectSum.toModule, DFinsupp.lsum, DFinsupp.sumAddHom_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryL2OfOrthogonalFamily hV hV')) w = \u2211 i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nlet e\u2081 := DirectSum.linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => V i\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryL2OfOrthogonalFamily hV hV')) w = \u2211 i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nlet e\u2082 := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryL2OfOrthogonalFamily hV hV')) w = \u2211 i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nsuffices \u2200 v : \u2a01 i, V i, e\u2082 v = \u2211 i, e\u2081 v i by exact this (e\u2081.symm w)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nthis : \u2200 (v : \u2a01 (i : \u03b9), { x // x \u2208 V i }), \u2191e\u2082 v = \u2211 i : \u03b9, \u2191(\u2191e\u2081 v i)\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryL2OfOrthogonalFamily hV hV')) w = \u2211 i : \u03b9, \u2191(w i)\n[PROOFSTEP]\nexact this (e\u2081.symm w)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\n\u22a2 \u2200 (v : \u2a01 (i : \u03b9), { x // x \u2208 V i }), \u2191e\u2082 v = \u2211 i : \u03b9, \u2191(\u2191e\u2081 v i)\n[PROOFSTEP]\nintro v\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\nw : PiLp 2 fun i => { x // x \u2208 V i }\ne\u2081 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] (i : \u03b9) \u2192 { x // x \u2208 V i } :=\n  linearEquivFunOnFintype \ud835\udd5c \u03b9 fun i => { x // x \u2208 V i }\ne\u2082 : (\u2a01 (i : \u03b9), { x // x \u2208 V i }) \u2243\u2097[\ud835\udd5c] E := LinearEquiv.ofBijective (coeLinearMap V) hV\nv : \u2a01 (i : \u03b9), { x // x \u2208 V i }\n\u22a2 \u2191e\u2082 v = \u2211 i : \u03b9, \u2191(\u2191e\u2081 v i)\n[PROOFSTEP]\nsimp [DirectSum.coeLinearMap, DirectSum.toModule, DFinsupp.lsum, DFinsupp.sumAddHom_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\na : \ud835\udd5c\nj : \u03b9\n\u22a2 single i a j = if j = i then a else 0\n[PROOFSTEP]\nrw [EuclideanSpace.single, PiLp.equiv_symm_apply, \u2190 Pi.single_apply i a j]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\na : \ud835\udd5c\nv : EuclideanSpace \ud835\udd5c \u03b9\n\u22a2 inner (single i a) v = \u2191(starRingEnd \ud835\udd5c) a * v i\n[PROOFSTEP]\nsimp [apply_ite conj]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\na : \ud835\udd5c\nv : EuclideanSpace \ud835\udd5c \u03b9\n\u22a2 inner v (single i a) = a * \u2191(starRingEnd ((fun x => \ud835\udd5c) i)) (v i)\n[PROOFSTEP]\nsimp [apply_ite conj, mul_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 Orthonormal \ud835\udd5c fun i => single i 1\n[PROOFSTEP]\nsimp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul, EuclideanSpace.single_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u03b9 \u2192 \u03b9 \u2192 True\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ni\u271d j\u271d : \u03b9\n\u22a2 True\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nf g : OrthonormalBasis \u03b9 \ud835\udd5c E\nh : f.repr = g.repr\n\u22a2 f = g\n[PROOFSTEP]\ncases f\n[GOAL]\ncase ofRepr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\ng : OrthonormalBasis \u03b9 \ud835\udd5c E\nrepr\u271d : E \u2243\u2097\u1d62[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nh : { repr := repr\u271d }.repr = g.repr\n\u22a2 { repr := repr\u271d } = g\n[PROOFSTEP]\ncases g\n[GOAL]\ncase ofRepr.ofRepr\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nrepr\u271d\u00b9 repr\u271d : E \u2243\u2097\u1d62[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nh : { repr := repr\u271d\u00b9 }.repr = { repr := repr\u271d }.repr\n\u22a2 { repr := repr\u271d\u00b9 } = { repr := repr\u271d }\n[PROOFSTEP]\ncongr\n  -- Porting note: `CoeFun` \u2192 `FunLike`\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 E\n[PROOFSTEP]\nclassical exact b.repr.symm (EuclideanSpace.single i (1 : \ud835\udd5c))\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 E\n[PROOFSTEP]\nexact b.repr.symm (EuclideanSpace.single i (1 : \ud835\udd5c))\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\n\u22a2 \u2191(LinearEquiv.symm b.repr.toLinearEquiv) = \u2191(LinearEquiv.symm b'.repr.toLinearEquiv)\n[PROOFSTEP]\nclassical\nrw [\u2190 LinearMap.cancel_right (PiLp.linearEquiv 2 \ud835\udd5c (fun _ => \ud835\udd5c)).symm.surjective]\nsimp only [LinearIsometryEquiv.toLinearEquiv_symm]\nrefine LinearMap.pi_ext fun i k => ?_\nhave : k = k \u2022 (1 : \ud835\udd5c) := by rw [smul_eq_mul, mul_one]\nrw [this, Pi.single_smul]\nreplace h := congr_fun h i\nsimp only [LinearEquiv.comp_coe, SMulHomClass.map_smul, LinearEquiv.coe_coe, LinearEquiv.trans_apply,\n  WithLp.linearEquiv_symm_apply, PiLp.equiv_symm_single, LinearIsometryEquiv.coe_toLinearEquiv] at h \u22a2\nrw [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\n\u22a2 \u2191(LinearEquiv.symm b.repr.toLinearEquiv) = \u2191(LinearEquiv.symm b'.repr.toLinearEquiv)\n[PROOFSTEP]\nrw [\u2190 LinearMap.cancel_right (PiLp.linearEquiv 2 \ud835\udd5c (fun _ => \ud835\udd5c)).symm.surjective]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\n\u22a2 LinearMap.comp \u2191(LinearEquiv.symm b.repr.toLinearEquiv) \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)) =\n    LinearMap.comp \u2191(LinearEquiv.symm b'.repr.toLinearEquiv) \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c))\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.toLinearEquiv_symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\n\u22a2 LinearMap.comp \u2191(LinearIsometryEquiv.symm b.repr).toLinearEquiv\n      \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)) =\n    LinearMap.comp \u2191(LinearIsometryEquiv.symm b'.repr).toLinearEquiv\n      \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c))\n[PROOFSTEP]\nrefine LinearMap.pi_ext fun i k => ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\ni : \u03b9\nk : \ud835\udd5c\n\u22a2 \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (Pi.single i k) =\n    \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b'.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (Pi.single i k)\n[PROOFSTEP]\nhave : k = k \u2022 (1 : \ud835\udd5c) := by rw [smul_eq_mul, mul_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\ni : \u03b9\nk : \ud835\udd5c\n\u22a2 k = k \u2022 1\n[PROOFSTEP]\nrw [smul_eq_mul, mul_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\ni : \u03b9\nk : \ud835\udd5c\nthis : k = k \u2022 1\n\u22a2 \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (Pi.single i k) =\n    \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b'.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (Pi.single i k)\n[PROOFSTEP]\nrw [this, Pi.single_smul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b'\ni : \u03b9\nk : \ud835\udd5c\nthis : k = k \u2022 1\n\u22a2 \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (k \u2022 Pi.single i 1) =\n    \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b'.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (k \u2022 Pi.single i 1)\n[PROOFSTEP]\nreplace h := congr_fun h i\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\nk : \ud835\udd5c\nthis : k = k \u2022 1\nh :\n  (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b i =\n    (fun b i => \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)) b' i\n\u22a2 \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (k \u2022 Pi.single i 1) =\n    \u2191(LinearMap.comp \u2191(LinearIsometryEquiv.symm b'.repr).toLinearEquiv\n          \u2191(LinearEquiv.symm (PiLp.linearEquiv 2 \ud835\udd5c fun x => \ud835\udd5c)))\n      (k \u2022 Pi.single i 1)\n[PROOFSTEP]\nsimp only [LinearEquiv.comp_coe, SMulHomClass.map_smul, LinearEquiv.coe_coe, LinearEquiv.trans_apply,\n  WithLp.linearEquiv_symm_apply, PiLp.equiv_symm_single, LinearIsometryEquiv.coe_toLinearEquiv] at h \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb b' : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\nk : \ud835\udd5c\nthis : k = k \u2022 1\nh :\n  \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1) =\n    \u2191(LinearIsometryEquiv.symm b'.repr) (EuclideanSpace.single i 1)\n\u22a2 k \u2022 \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1) =\n    k \u2022 \u2191(LinearIsometryEquiv.symm b'.repr) (EuclideanSpace.single i 1)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ne : E \u2243\u2097\u1d62[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\n\u22a2 \u2191{ repr := e } = fun i => \u2191(LinearIsometryEquiv.symm e) (EuclideanSpace.single i 1)\n[PROOFSTEP]\ndsimp only [FunLike.coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ne : E \u2243\u2097\u1d62[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\n\u22a2 (fun i => EquivLike.coe (LinearIsometryEquiv.symm e) (EuclideanSpace.single i 1)) = fun i =>\n    EquivLike.coe (LinearIsometryEquiv.symm e) (EuclideanSpace.single i 1)\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\ne : E \u2243\u2097\u1d62[\ud835\udd5c] EuclideanSpace \ud835\udd5c \u03b9\nx\u271d : \u03b9\n\u22a2 EquivLike.coe (LinearIsometryEquiv.symm e) (EuclideanSpace.single x\u271d 1) =\n    EquivLike.coe (LinearIsometryEquiv.symm e) (EuclideanSpace.single x\u271d 1)\n[PROOFSTEP]\ncongr!\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 \u2191(LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1) = \u2191b i\n[PROOFSTEP]\ndsimp only [FunLike.coe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 EquivLike.coe (LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1) =\n    EquivLike.coe (LinearIsometryEquiv.symm b.repr) (EuclideanSpace.single i 1)\n[PROOFSTEP]\ncongr!\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni : \u03b9\n\u22a2 \u2191b.repr (\u2191b i) = EuclideanSpace.single i 1\n[PROOFSTEP]\nrw [\u2190 b.repr_symm_single i, LinearIsometryEquiv.apply_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nv : E\ni : \u03b9\n\u22a2 \u2191b.repr v i = inner (\u2191b i) v\n[PROOFSTEP]\nclassical\nrw [\u2190 b.repr.inner_map_map (b i) v, b.repr_self i, EuclideanSpace.inner_single_left]\nsimp only [one_mul, eq_self_iff_true, map_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nv : E\ni : \u03b9\n\u22a2 \u2191b.repr v i = inner (\u2191b i) v\n[PROOFSTEP]\nrw [\u2190 b.repr.inner_map_map (b i) v, b.repr_self i, EuclideanSpace.inner_single_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nv : E\ni : \u03b9\n\u22a2 \u2191b.repr v i = \u2191(starRingEnd \ud835\udd5c) 1 * \u2191b.repr v i\n[PROOFSTEP]\nsimp only [one_mul, eq_self_iff_true, map_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 Orthonormal \ud835\udd5c \u2191b\n[PROOFSTEP]\nclassical\nrw [orthonormal_iff_ite]\nintro i j\nrw [\u2190 b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j, EuclideanSpace.inner_single_left,\n  EuclideanSpace.single_apply, map_one, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 Orthonormal \ud835\udd5c \u2191b\n[PROOFSTEP]\nrw [orthonormal_iff_ite]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2200 (i j : \u03b9), inner (\u2191b i) (\u2191b j) = if i = j then 1 else 0\n[PROOFSTEP]\nintro i j\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 inner (\u2191b i) (\u2191b j) = if i = j then 1 else 0\n[PROOFSTEP]\nrw [\u2190 b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j, EuclideanSpace.inner_single_left,\n  EuclideanSpace.single_apply, map_one, one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2191(OrthonormalBasis.toBasis b) = \u2191b\n[PROOFSTEP]\nrw [OrthonormalBasis.toBasis]\n  -- Porting note: was `change`\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2191(Basis.ofEquivFun b.repr.toLinearEquiv) = \u2191b\n[PROOFSTEP]\next j\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 \u2191(Basis.ofEquivFun b.repr.toLinearEquiv) j = \u2191b j\n[PROOFSTEP]\nclassical\nrw [Basis.coe_ofEquivFun]\ncongr\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 \u2191(Basis.ofEquivFun b.repr.toLinearEquiv) j = \u2191b j\n[PROOFSTEP]\nrw [Basis.coe_ofEquivFun]\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nj : \u03b9\n\u22a2 (fun i => \u2191(LinearEquiv.symm b.repr.toLinearEquiv) (update 0 i 1)) j = \u2191b j\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx : E\ni : \u03b9\n\u22a2 \u2191(\u2191(OrthonormalBasis.toBasis b).repr x) i = \u2191b.repr x i\n[PROOFSTEP]\nrw [\u2190 Basis.equivFun_apply, OrthonormalBasis.coe_toBasis_repr, LinearIsometryEquiv.coe_toLinearEquiv]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx : E\n\u22a2 \u2211 i : \u03b9, \u2191b.repr x i \u2022 \u2191b i = x\n[PROOFSTEP]\nsimp_rw [\u2190 b.coe_toBasis_repr_apply, \u2190 b.coe_toBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx : E\n\u22a2 \u2211 x_1 : \u03b9, \u2191(\u2191(OrthonormalBasis.toBasis b).repr x) x_1 \u2022 \u2191(OrthonormalBasis.toBasis b) x_1 = x\n[PROOFSTEP]\nexact b.toBasis.sum_repr x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nv : EuclideanSpace \ud835\udd5c \u03b9\n\u22a2 \u2211 i : \u03b9, v i \u2022 \u2191b i = \u2191(LinearIsometryEquiv.symm b.repr) v\n[PROOFSTEP]\nsimpa using (b.toBasis.equivFun_symm_apply v).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx y : E\n\u22a2 \u2211 i : \u03b9, inner x (\u2191b i) * inner (\u2191b i) y = inner x y\n[PROOFSTEP]\nhave := congr_arg (innerSL \ud835\udd5c x) (b.sum_repr y)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx y : E\nthis : \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2211 i : \u03b9, \u2191b.repr y i \u2022 \u2191b i) = \u2191(\u2191(innerSL \ud835\udd5c) x) y\n\u22a2 \u2211 i : \u03b9, inner x (\u2191b i) * inner (\u2191b i) y = inner x y\n[PROOFSTEP]\nrw [map_sum] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx y : E\nthis : \u2211 x_1 : \u03b9, \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191b.repr y x_1 \u2022 \u2191b x_1) = \u2191(\u2191(innerSL \ud835\udd5c) x) y\n\u22a2 \u2211 i : \u03b9, inner x (\u2191b i) * inner (\u2191b i) y = inner x y\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx y : E\nthis : \u2211 x_1 : \u03b9, \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191b.repr y x_1 \u2022 \u2191b x_1) = \u2191(\u2191(innerSL \ud835\udd5c) x) y\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 inner x (\u2191b x\u271d) * inner (\u2191b x\u271d) y = \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191b.repr y x\u271d \u2022 \u2191b x\u271d)\n[PROOFSTEP]\nrw [SMulHomClass.map_smul, b.repr_apply_apply, mul_comm]\n[GOAL]\ncase h.e'_2.a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\nx y : E\nthis : \u2211 x_1 : \u03b9, \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191b.repr y x_1 \u2022 \u2191b x_1) = \u2191(\u2191(innerSL \ud835\udd5c) x) y\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 inner (\u2191b x\u271d) y * inner x (\u2191b x\u271d) = inner (\u2191b x\u271d) y \u2022 \u2191(\u2191(innerSL \ud835\udd5c) x) (\u2191b x\u271d)\n[PROOFSTEP]\nsimp only [innerSL_apply, smul_eq_mul]\n  -- Porting note: was `rfl`\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nU : Submodule \ud835\udd5c E\ninst\u271d : CompleteSpace { x // x \u2208 U }\nb : OrthonormalBasis \u03b9 \ud835\udd5c { x // x \u2208 U }\nx : E\n\u22a2 \u2191(orthogonalProjection U) x = \u2211 i : \u03b9, inner (\u2191(\u2191b i)) x \u2022 \u2191b i\n[PROOFSTEP]\nsimpa only [b.repr_apply_apply, inner_orthogonalProjection_eq_of_mem_left] using\n  (b.sum_repr (orthogonalProjection U x)).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\n\u22a2 \u2200 (x y : E), inner (\u2191(Basis.equivFun v) x) (\u2191(Basis.equivFun v) y) = inner x y\n[PROOFSTEP]\nintro x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\n\u22a2 inner (\u2191(Basis.equivFun v) x) (\u2191(Basis.equivFun v) y) = inner x y\n[PROOFSTEP]\nlet p : EuclideanSpace \ud835\udd5c \u03b9 := v.equivFun x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\n\u22a2 inner (\u2191(Basis.equivFun v) x) (\u2191(Basis.equivFun v) y) = inner x y\n[PROOFSTEP]\nlet q : EuclideanSpace \ud835\udd5c \u03b9 := v.equivFun y\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\nq : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) y\n\u22a2 inner (\u2191(Basis.equivFun v) x) (\u2191(Basis.equivFun v) y) = inner x y\n[PROOFSTEP]\nhave key : \u27eap, q\u27eb = \u27ea\u2211 i, p i \u2022 v i, \u2211 i, q i \u2022 v i\u27eb := by simp [sum_inner, inner_smul_left, hv.inner_right_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\nq : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) y\n\u22a2 inner p q = inner (\u2211 i : \u03b9, p i \u2022 \u2191v i) (\u2211 i : \u03b9, q i \u2022 \u2191v i)\n[PROOFSTEP]\nsimp [sum_inner, inner_smul_left, hv.inner_right_fintype]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\nq : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) y\nkey : inner p q = inner (\u2211 i : \u03b9, p i \u2022 \u2191v i) (\u2211 i : \u03b9, q i \u2022 \u2191v i)\n\u22a2 inner (\u2191(Basis.equivFun v) x) (\u2191(Basis.equivFun v) y) = inner x y\n[PROOFSTEP]\nconvert key\n[GOAL]\ncase h.e'_3.h.e'_4\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\nq : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) y\nkey : inner p q = inner (\u2211 i : \u03b9, p i \u2022 \u2191v i) (\u2211 i : \u03b9, q i \u2022 \u2191v i)\n\u22a2 x = \u2211 i : \u03b9, p i \u2022 \u2191v i\n[PROOFSTEP]\nrw [\u2190 v.equivFun.symm_apply_apply x, v.equivFun_symm_apply]\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nx y : E\np : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) x\nq : EuclideanSpace \ud835\udd5c \u03b9 := \u2191(Basis.equivFun v) y\nkey : inner p q = inner (\u2211 i : \u03b9, p i \u2022 \u2191v i) (\u2211 i : \u03b9, q i \u2022 \u2191v i)\n\u22a2 y = \u2211 i : \u03b9, q i \u2022 \u2191v i\n[PROOFSTEP]\nrw [\u2190 v.equivFun.symm_apply_apply y, v.equivFun_symm_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\n\u22a2 OrthonormalBasis.toBasis (Basis.toOrthonormalBasis v hv) = v\n[PROOFSTEP]\nsimp [Basis.toOrthonormalBasis, OrthonormalBasis.toBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\n\u22a2 \u2191(Basis.toOrthonormalBasis v hv) = \u2191(OrthonormalBasis.toBasis (Basis.toOrthonormalBasis v hv))\n[PROOFSTEP]\nclassical rw [OrthonormalBasis.coe_toBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\n\u22a2 \u2191(Basis.toOrthonormalBasis v hv) = \u2191(OrthonormalBasis.toBasis (Basis.toOrthonormalBasis v hv))\n[PROOFSTEP]\nrw [OrthonormalBasis.coe_toBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\n\u22a2 \u2191(OrthonormalBasis.toBasis (Basis.toOrthonormalBasis v hv)) = \u2191v\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 span \ud835\udd5c (range v)\n\u22a2 Orthonormal \ud835\udd5c \u2191(Basis.mk (_ : LinearIndependent \ud835\udd5c v) hsp)\n[PROOFSTEP]\nrwa [Basis.coe_mk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 span \ud835\udd5c (range v)\n\u22a2 \u2191(OrthonormalBasis.mk hon hsp) = v\n[PROOFSTEP]\nclassical rw [OrthonormalBasis.mk, _root_.Basis.coe_toOrthonormalBasis, Basis.coe_mk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : \u22a4 \u2264 span \ud835\udd5c (range v)\n\u22a2 \u2191(OrthonormalBasis.mk hon hsp) = v\n[PROOFSTEP]\nrw [OrthonormalBasis.mk, _root_.Basis.coe_toOrthonormalBasis, Basis.coe_mk]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : DecidableEq E\nv' : \u03b9' \u2192 E\nh : Orthonormal \ud835\udd5c v'\ns : Finset \u03b9'\ne\u2080' : Basis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  Basis.span (_ : LinearIndependent \ud835\udd5c (v' \u2218 Subtype.val))\n\u22a2 Orthonormal \ud835\udd5c \u2191e\u2080'\n[PROOFSTEP]\nconvert orthonormal_span (h.comp ((\u2191) : s \u2192 \u03b9') Subtype.val_injective)\n[GOAL]\ncase h.e'_7.h.h.e'_3\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : DecidableEq E\nv' : \u03b9' \u2192 E\nh : Orthonormal \ud835\udd5c v'\ns : Finset \u03b9'\ne\u2080' : Basis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  Basis.span (_ : LinearIndependent \ud835\udd5c (v' \u2218 Subtype.val))\nx\u271d : { x // x \u2208 s }\n\u22a2 \u2191(\u2191e\u2080' x\u271d) = (v' \u2218 Subtype.val) x\u271d\n[PROOFSTEP]\nsimp [Basis.span_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : DecidableEq E\nv' : \u03b9' \u2192 E\nh : Orthonormal \ud835\udd5c v'\ns : Finset \u03b9'\ne\u2080' : Basis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  Basis.span (_ : LinearIndependent \ud835\udd5c (v' \u2218 Subtype.val))\ne\u2080 : OrthonormalBasis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  OrthonormalBasis.mk (_ : Orthonormal \ud835\udd5c \u2191e\u2080') (_ : \u22a4 \u2264 span \ud835\udd5c (range \u2191e\u2080'))\n\u22a2 span \ud835\udd5c \u2191(Finset.image v' s) = span \ud835\udd5c (range (v' \u2218 Subtype.val))\n[PROOFSTEP]\nrw [Finset.coe_image, image_eq_range]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : DecidableEq E\nv' : \u03b9' \u2192 E\nh : Orthonormal \ud835\udd5c v'\ns : Finset \u03b9'\ne\u2080' : Basis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  Basis.span (_ : LinearIndependent \ud835\udd5c (v' \u2218 Subtype.val))\ne\u2080 : OrthonormalBasis { x // x \u2208 s } \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range (v' \u2218 Subtype.val)) } :=\n  OrthonormalBasis.mk (_ : Orthonormal \ud835\udd5c \u2191e\u2080') (_ : \u22a4 \u2264 span \ud835\udd5c (range \u2191e\u2080'))\n\u22a2 span \ud835\udd5c (range fun x => v' \u2191x) = span \ud835\udd5c (range (v' \u2218 Subtype.val))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : DecidableEq E\nv' : \u03b9' \u2192 E\nh : Orthonormal \ud835\udd5c v'\ns : Finset \u03b9'\ni : { x // x \u2208 s }\n\u22a2 \u2191(\u2191(OrthonormalBasis.span h s) i) = v' \u2191i\n[PROOFSTEP]\nsimp only [OrthonormalBasis.span, Basis.span_apply, LinearIsometryEquiv.ofEq_symm, OrthonormalBasis.map_apply,\n  OrthonormalBasis.coe_mk, LinearIsometryEquiv.coe_ofEq_apply, comp_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : (span \ud835\udd5c (range v))\u15ee = \u22a5\n\u22a2 \u22a4 \u2264 span \ud835\udd5c (range v)\n[PROOFSTEP]\nrefine' Eq.ge _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : (span \ud835\udd5c (range v))\u15ee = \u22a5\n\u22a2 span \ud835\udd5c (range v) = \u22a4\n[PROOFSTEP]\nhaveI : FiniteDimensional \ud835\udd5c (span \ud835\udd5c (range v)) := FiniteDimensional.span_of_finite \ud835\udd5c (finite_range v)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : (span \ud835\udd5c (range v))\u15ee = \u22a5\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range v) }\n\u22a2 span \ud835\udd5c (range v) = \u22a4\n[PROOFSTEP]\nhaveI : CompleteSpace (span \ud835\udd5c (range v)) := FiniteDimensional.complete \ud835\udd5c _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : \u03b9 \u2192 E\nhon : Orthonormal \ud835\udd5c v\nhsp : (span \ud835\udd5c (range v))\u15ee = \u22a5\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 span \ud835\udd5c (range v) }\nthis : CompleteSpace { x // x \u2208 span \ud835\udd5c (range v) }\n\u22a2 span \ud835\udd5c (range v) = \u22a4\n[PROOFSTEP]\nrwa [orthogonal_eq_bot_iff] at hsp \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ni' : \u03b9'\n\u22a2 \u2191(reindex b e) i' = \u2191b (\u2191e.symm i')\n[PROOFSTEP]\nclassical\ndsimp [reindex]\nrw [coe_ofRepr]\ndsimp\nrw [\u2190 b.repr_symm_single, LinearIsometryEquiv.piLpCongrLeft_symm, EuclideanSpace.piLpCongrLeft_single]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ni' : \u03b9'\n\u22a2 \u2191(reindex b e) i' = \u2191b (\u2191e.symm i')\n[PROOFSTEP]\ndsimp [reindex]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ni' : \u03b9'\n\u22a2 \u2191{ repr := LinearIsometryEquiv.trans b.repr (LinearIsometryEquiv.piLpCongrLeft 2 \ud835\udd5c \ud835\udd5c e) } i' = \u2191b (\u2191e.symm i')\n[PROOFSTEP]\nrw [coe_ofRepr]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ni' : \u03b9'\n\u22a2 (fun i =>\n        \u2191(LinearIsometryEquiv.symm (LinearIsometryEquiv.trans b.repr (LinearIsometryEquiv.piLpCongrLeft 2 \ud835\udd5c \ud835\udd5c e)))\n          (EuclideanSpace.single i 1))\n      i' =\n    \u2191b (\u2191e.symm i')\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\ni' : \u03b9'\n\u22a2 \u2191(LinearIsometryEquiv.symm b.repr)\n      (\u2191(LinearIsometryEquiv.symm (LinearIsometryEquiv.piLpCongrLeft 2 \ud835\udd5c \ud835\udd5c e)) (EuclideanSpace.single i' 1)) =\n    \u2191b (\u2191e.symm i')\n[PROOFSTEP]\nrw [\u2190 b.repr_symm_single, LinearIsometryEquiv.piLpCongrLeft_symm, EuclideanSpace.piLpCongrLeft_single]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx : E\ni' : \u03b9'\n\u22a2 \u2191(reindex b e).repr x i' = \u2191b.repr x (\u2191e.symm i')\n[PROOFSTEP]\nclassical rw [OrthonormalBasis.repr_apply_apply, b.repr_apply_apply, OrthonormalBasis.coe_reindex, comp_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : \u03b9 \u2192 E\ninst\u271d : Fintype \u03b9'\nb : OrthonormalBasis \u03b9 \ud835\udd5c E\ne : \u03b9 \u2243 \u03b9'\nx : E\ni' : \u03b9'\n\u22a2 \u2191(reindex b e).repr x i' = \u2191b.repr x (\u2191e.symm i')\n[PROOFSTEP]\nrw [OrthonormalBasis.repr_apply_apply, b.repr_apply_apply, OrthonormalBasis.coe_reindex, comp_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 Orthonormal \u211d \u2191basisOneI\n[PROOFSTEP]\nrw [orthonormal_iff_ite]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (i j : Fin 2), inner (\u2191basisOneI i) (\u2191basisOneI j) = if i = j then 1 else 0\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\ni : Fin 2\n\u22a2 \u2200 (j : Fin 2), inner (\u2191basisOneI i) (\u2191basisOneI j) = if i = j then 1 else 0\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (j : Fin 2),\n    inner (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) (\u2191basisOneI j) =\n      if { val := 0, isLt := (_ : 0 < 2) } = j then 1 else 0\n[PROOFSTEP]\nintro j\n[GOAL]\ncase tail.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (j : Fin 2),\n    inner (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (\u2191basisOneI j) =\n      if { val := 1, isLt := (_ : (fun a => a < 2) 1) } = j then 1 else 0\n[PROOFSTEP]\nintro j\n[GOAL]\ncase head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nj : Fin 2\n\u22a2 inner (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) (\u2191basisOneI j) =\n    if { val := 0, isLt := (_ : 0 < 2) } = j then 1 else 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase tail.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nj : Fin 2\n\u22a2 inner (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (\u2191basisOneI j) =\n    if { val := 1, isLt := (_ : (fun a => a < 2) 1) } = j then 1 else 0\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase head.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 inner (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) =\n    if { val := 0, isLt := (_ : 0 < 2) } = { val := 0, isLt := (_ : 0 < 2) } then 1 else 0\n[PROOFSTEP]\nsimp [real_inner_eq_re_inner]\n[GOAL]\ncase head.tail.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 inner (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    if { val := 0, isLt := (_ : 0 < 2) } = { val := 1, isLt := (_ : (fun a => a < 2) 1) } then 1 else 0\n[PROOFSTEP]\nsimp [real_inner_eq_re_inner]\n[GOAL]\ncase tail.head.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 inner (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) =\n    if { val := 1, isLt := (_ : (fun a => a < 2) 1) } = { val := 0, isLt := (_ : 0 < 2) } then 1 else 0\n[PROOFSTEP]\nsimp [real_inner_eq_re_inner]\n[GOAL]\ncase tail.head.tail.head\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 inner (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    if { val := 1, isLt := (_ : (fun a => a < 2) 1) } = { val := 1, isLt := (_ : (fun a => a < 2) 1) } then 1 else 0\n[PROOFSTEP]\nsimp [real_inner_eq_re_inner]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\n\u22a2 \u2191orthonormalBasisOneI = ![1, I]\n[PROOFSTEP]\nsimp [Complex.orthonormalBasisOneI]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : OrthonormalBasis (Fin 2) \u211d F\nf : F \u2243\u2097\u1d62[\u211d] F'\n\u22a2 isometryOfOrthonormal (OrthonormalBasis.map v f) = LinearIsometryEquiv.trans (isometryOfOrthonormal v) f\n[PROOFSTEP]\nsimp [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_assoc, OrthonormalBasis.map]\n  -- Porting note: `LinearIsometryEquiv.trans_assoc` doesn't trigger in the `simp` above\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : OrthonormalBasis (Fin 2) \u211d F\nf : F \u2243\u2097\u1d62[\u211d] F'\n\u22a2 LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.trans (LinearIsometryEquiv.symm v.repr) f) =\n    LinearIsometryEquiv.trans (LinearIsometryEquiv.trans orthonormalBasisOneI.repr (LinearIsometryEquiv.symm v.repr)) f\n[PROOFSTEP]\nrw [LinearIsometryEquiv.trans_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : OrthonormalBasis (Fin 2) \u211d F\nf : F\n\u22a2 \u2191(LinearIsometryEquiv.symm (isometryOfOrthonormal v)) f =\n    \u2191(\u2191(Basis.coord (OrthonormalBasis.toBasis v) 0) f) + \u2191(\u2191(Basis.coord (OrthonormalBasis.toBasis v) 1) f) * I\n[PROOFSTEP]\nsimp [Complex.isometryOfOrthonormal]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : OrthonormalBasis (Fin 2) \u211d F\nz : \u2102\n\u22a2 \u2191(isometryOfOrthonormal v) z = z.re \u2022 \u2191v 0 + z.im \u2022 \u2191v 1\n[PROOFSTEP]\nrw [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_apply]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2079 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2076 : NormedAddCommGroup E'\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : Fintype \u03b9\nv : OrthonormalBasis (Fin 2) \u211d F\nz : \u2102\n\u22a2 \u2191(LinearIsometryEquiv.symm v.repr) (\u2191orthonormalBasisOneI.repr z) = z.re \u2022 \u2191v 0 + z.im \u2022 \u2191v 1\n[PROOFSTEP]\nsimp [\u2190 v.sum_repr_symm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 Basis.toMatrix (OrthonormalBasis.toBasis a) \u2191b \u2208 Matrix.unitaryGroup \u03b9 \ud835\udd5c\n[PROOFSTEP]\nrw [Matrix.mem_unitaryGroup_iff']\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 star (Basis.toMatrix (OrthonormalBasis.toBasis a) \u2191b) * Basis.toMatrix (OrthonormalBasis.toBasis a) \u2191b = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 (star (Basis.toMatrix (OrthonormalBasis.toBasis a) \u2191b) * Basis.toMatrix (OrthonormalBasis.toBasis a) \u2191b) i j =\n    OfNat.ofNat 1 i j\n[PROOFSTEP]\nconvert a.repr.inner_map_map (b i) (b j)\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 OfNat.ofNat 1 i j = inner (\u2191b i) (\u2191b j)\n[PROOFSTEP]\nrw [orthonormal_iff_ite.mp b.orthonormal i j]\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\ni j : \u03b9\n\u22a2 OfNat.ofNat 1 i j = if i = j then 1 else 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2016\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b\u2016 = 1\n[PROOFSTEP]\nhave : (normSq (a.toBasis.det b) : \ud835\udd5c) = 1 := by\n  simpa [IsROrC.mul_conj] using (Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)).2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\n\u22a2 \u2191(\u2191normSq (\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b)) = 1\n[PROOFSTEP]\nsimpa [IsROrC.mul_conj] using (Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)).2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\nthis : \u2191(\u2191normSq (\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b)) = 1\n\u22a2 \u2016\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b\u2016 = 1\n[PROOFSTEP]\nnorm_cast at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \ud835\udd5c E\nthis : \u2191normSq (\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b) = 1\n\u22a2 \u2016\u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b\u2016 = 1\n[PROOFSTEP]\nrwa [\u2190 sqrt_normSq_eq_norm, sqrt_eq_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b = 1 \u2228 \u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b = -1\n[PROOFSTEP]\nrw [\u2190 sq_eq_one_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\na b : OrthonormalBasis \u03b9 \u211d F\n\u22a2 \u2191(Basis.det (OrthonormalBasis.toBasis a)) \u2191b ^ 2 = 1\n[PROOFSTEP]\nsimpa [unitary, sq] using Matrix.det_of_mem_unitary (a.toMatrix_orthonormalBasis_mem_unitary b)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 A i }) fun i => subtype\u2097\u1d62 (A i)\ninst\u271d\u00b9 : DecidableEq \u03b9\nhV_sum : IsInternal fun i => A i\n\u03b1 : \u03b9 \u2192 Type u_8\ninst\u271d : (i : \u03b9) \u2192 Fintype (\u03b1 i)\nv_family : (i : \u03b9) \u2192 OrthonormalBasis (\u03b1 i) \ud835\udd5c { x // x \u2208 A i }\n\u22a2 Orthonormal \ud835\udd5c \u2191(collectedBasis hV_sum fun i => OrthonormalBasis.toBasis (v_family i))\n[PROOFSTEP]\nsimpa using hV.orthonormal_sigma_orthonormal (show \u2200 i, Orthonormal \ud835\udd5c (v_family i).toBasis by simp)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 A i }) fun i => subtype\u2097\u1d62 (A i)\ninst\u271d\u00b9 : DecidableEq \u03b9\nhV_sum : IsInternal fun i => A i\n\u03b1 : \u03b9 \u2192 Type u_8\ninst\u271d : (i : \u03b9) \u2192 Fintype (\u03b1 i)\nv_family : (i : \u03b9) \u2192 OrthonormalBasis (\u03b1 i) \ud835\udd5c { x // x \u2208 A i }\n\u22a2 \u2200 (i : \u03b9), Orthonormal \ud835\udd5c \u2191(OrthonormalBasis.toBasis (v_family i))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\nv\u271d : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : DecidableEq \u03b9\nh : IsInternal A\n\u03b1 : \u03b9 \u2192 Type u_8\ninst\u271d : (i : \u03b9) \u2192 Fintype (\u03b1 i)\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 A i }) fun i => subtype\u2097\u1d62 (A i)\nv : (i : \u03b9) \u2192 OrthonormalBasis (\u03b1 i) \ud835\udd5c { x // x \u2208 A i }\na : (i : \u03b9) \u00d7 \u03b1 i\n\u22a2 \u2191(collectedOrthonormalBasis hV h v) a \u2208 A a.fst\n[PROOFSTEP]\nsimp [DirectSum.IsInternal.collectedOrthonormalBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nobtain \u27e8u\u2080, hu\u2080s, hu\u2080, hu\u2080_max\u27e9 := exists_maximal_orthonormal hv\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : \u2200 (u : Set E), u \u2287 u\u2080 \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = u\u2080\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nrw [maximal_orthonormal_iff_orthogonalComplement_eq_bot hu\u2080] at hu\u2080_max \n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nhave hu\u2080_finite : u\u2080.Finite := hu\u2080.linearIndependent.finite\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nlet u : Finset E := hu\u2080_finite.toFinset\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nlet fu : \u21a5u \u2243 \u21a5u\u2080 := Equiv.cast (congr_arg (\u21a5) hu\u2080_finite.coe_toFinset)\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nhave hfu : ((\u2191) : u \u2192 E) = ((\u2191) : u\u2080 \u2192 E) \u2218 fu := by ext; simp\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\n\u22a2 Subtype.val = Subtype.val \u2218 \u2191fu\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nx\u271d : { x // x \u2208 u }\n\u22a2 \u2191x\u271d = (Subtype.val \u2218 \u2191fu) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nhave hu : Orthonormal \ud835\udd5c ((\u2191) : u \u2192 E) := by simpa [hfu] using hu\u2080.comp _ fu.injective\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\n\u22a2 Orthonormal \ud835\udd5c Subtype.val\n[PROOFSTEP]\nsimpa [hfu] using hu\u2080.comp _ fu.injective\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2203 u b, v \u2286 \u2191u \u2227 \u2191b = Subtype.val\n[PROOFSTEP]\nrefine' \u27e8u, OrthonormalBasis.mkOfOrthogonalEqBot hu _, _, _\u27e9\n[GOAL]\ncase intro.intro.intro.refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 (span \ud835\udd5c (range Subtype.val))\u15ee = \u22a5\n[PROOFSTEP]\nsimpa using hu\u2080_max\n[GOAL]\ncase intro.intro.intro.refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 v \u2286 \u2191u\n[PROOFSTEP]\nsimpa using hu\u2080s\n[GOAL]\ncase intro.intro.intro.refine'_3\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c Subtype.val\nu\u2080 : Set E\nhu\u2080s : u\u2080 \u2287 v\nhu\u2080 : Orthonormal \ud835\udd5c Subtype.val\nhu\u2080_max : (span \ud835\udd5c u\u2080)\u15ee = \u22a5\nhu\u2080_finite : Set.Finite u\u2080\nu : Finset E := Finite.toFinset hu\u2080_finite\nfu : { x // x \u2208 u } \u2243 \u2191u\u2080 := Equiv.cast (_ : \u2191\u2191(Finite.toFinset hu\u2080_finite) = \u2191u\u2080)\nhfu : Subtype.val = Subtype.val \u2218 \u2191fu\nhu : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2191(OrthonormalBasis.mkOfOrthogonalEqBot hu (_ : (span \ud835\udd5c (range Subtype.val))\u15ee = \u22a5)) = Subtype.val\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nhave hsv : Injective (s.restrict v) := hv.linearIndependent.injective\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nhave hX : Orthonormal \ud835\udd5c ((\u2191) : Set.range (s.restrict v) \u2192 E) := by rwa [orthonormal_subtype_range hsv]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\n\u22a2 Orthonormal \ud835\udd5c Subtype.val\n[PROOFSTEP]\nrwa [orthonormal_subtype_range hsv]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nobtain \u27e8Y, b\u2080, hX, hb\u2080\u27e9 := hX.exists_orthonormalBasis_extension\n[GOAL]\ncase intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nhave h\u03b9Y : Fintype.card \u03b9 = Y.card := by\n  refine' card_\u03b9.symm.trans _\n  exact FiniteDimensional.finrank_eq_card_finset_basis b\u2080.toBasis\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\n\u22a2 Fintype.card \u03b9 = Finset.card Y\n[PROOFSTEP]\nrefine' card_\u03b9.symm.trans _\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\n\u22a2 finrank \ud835\udd5c E = Finset.card Y\n[PROOFSTEP]\nexact FiniteDimensional.finrank_eq_card_finset_basis b\u2080.toBasis\n[GOAL]\ncase intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nhave hvsY : s.MapsTo v Y := (s.mapsTo_image v).mono_right (by rwa [\u2190 range_restrict])\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\n\u22a2 v '' s \u2286 \u2191Y\n[PROOFSTEP]\nrwa [\u2190 range_restrict]\n[GOAL]\ncase intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nhave hsv' : Set.InjOn v s := by\n  rw [Set.injOn_iff_injective]\n  exact hsv\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\n\u22a2 InjOn v s\n[PROOFSTEP]\nrw [Set.injOn_iff_injective]\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\n\u22a2 Injective (restrict s v)\n[PROOFSTEP]\nexact hsv\n[GOAL]\ncase intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\nhsv' : InjOn v s\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nobtain \u27e8g, hg\u27e9 := hvsY.exists_equiv_extend_of_card_eq h\u03b9Y hsv'\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\nhsv' : InjOn v s\ng : \u03b9 \u2243 { x // x \u2208 Y }\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191g i) = v i\n\u22a2 \u2203 b, \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191b i = v i\n[PROOFSTEP]\nuse b\u2080.reindex g.symm\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\nhsv' : InjOn v s\ng : \u03b9 \u2243 { x // x \u2208 Y }\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191g i) = v i\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(OrthonormalBasis.reindex b\u2080 g.symm) i = v i\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\u271d\nv\u271d : Set E\nA : \u03b9\u271d \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\n\u03b9 : Type u_8\ninst\u271d : Fintype \u03b9\ncard_\u03b9 : finrank \ud835\udd5c E = Fintype.card \u03b9\nv : \u03b9 \u2192 E\ns : Set \u03b9\nhv : Orthonormal \ud835\udd5c (restrict s v)\nhsv : Injective (restrict s v)\nhX\u271d : Orthonormal \ud835\udd5c Subtype.val\nY : Finset E\nb\u2080 : OrthonormalBasis { x // x \u2208 Y } \ud835\udd5c E\nhX : range (restrict s v) \u2286 \u2191Y\nhb\u2080 : \u2191b\u2080 = Subtype.val\nh\u03b9Y : Fintype.card \u03b9 = Finset.card Y\nhvsY : MapsTo v s \u2191Y\nhsv' : InjOn v s\ng : \u03b9 \u2243 { x // x \u2208 Y }\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191g i) = v i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(OrthonormalBasis.reindex b\u2080 g.symm) i = v i\n[PROOFSTEP]\nsimp [hb\u2080, hg i hi]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\n\u22a2 OrthonormalBasis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E\n[PROOFSTEP]\nlet b := Classical.choose (Classical.choose_spec <| exists_orthonormalBasis \ud835\udd5c E)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis { x // x \u2208 Classical.choose (_ : \u2203 w b, \u2191b = Subtype.val) } \ud835\udd5c E :=\n  Classical.choose (_ : \u2203 b, \u2191b = Subtype.val)\n\u22a2 OrthonormalBasis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E\n[PROOFSTEP]\nrw [finrank_eq_card_basis b.toBasis]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis { x // x \u2208 Classical.choose (_ : \u2203 w b, \u2191b = Subtype.val) } \ud835\udd5c E :=\n  Classical.choose (_ : \u2203 b, \u2191b = Subtype.val)\n\u22a2 OrthonormalBasis (Fin (Fintype.card { x // x \u2208 Classical.choose (_ : \u2203 w b, \u2191b = Subtype.val) })) \ud835\udd5c E\n[PROOFSTEP]\nexact b.reindex (Fintype.equivFinOfCardEq rfl)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\n\u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1\n[PROOFSTEP]\nhave : Unique \u03b9 := b.toBasis.unique\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis : Unique \u03b9\n\u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1\n[PROOFSTEP]\nhave : b default = 1 \u2228 b default = -1 :=\n  by\n  have : \u2016b default\u2016 = 1 := b.orthonormal.1 _\n  rwa [Real.norm_eq_abs, abs_eq (zero_le_one' \u211d)] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis : Unique \u03b9\n\u22a2 \u2191b default = 1 \u2228 \u2191b default = -1\n[PROOFSTEP]\nhave : \u2016b default\u2016 = 1 := b.orthonormal.1 _\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2016\u2191b default\u2016 = 1\n\u22a2 \u2191b default = 1 \u2228 \u2191b default = -1\n[PROOFSTEP]\nrwa [Real.norm_eq_abs, abs_eq (zero_le_one' \u211d)] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\n\u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1\n[PROOFSTEP]\nrw [eq_const_of_unique b]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\n\u22a2 (const \u03b9 (\u2191b default) = fun x => 1) \u2228 const \u03b9 (\u2191b default) = fun x => -1\n[PROOFSTEP]\nrefine' this.imp _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\n\u22a2 \u2191b default = 1 \u2192 const \u03b9 (\u2191b default) = fun x => 1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\na\u271d : \u2191b default = 1\n\u22a2 const \u03b9 (\u2191b default) = fun x => 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\na\u271d : \u2191b default = 1\nx\u271d : \u03b9\n\u22a2 const \u03b9 (\u2191b default) x\u271d = 1\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\n\u22a2 \u2191b default = -1 \u2192 const \u03b9 (\u2191b default) = fun x => -1\n[PROOFSTEP]\nintro\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\na\u271d : \u2191b default = -1\n\u22a2 const \u03b9 (\u2191b default) = fun x => -1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2.h\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d : FiniteDimensional \ud835\udd5c E\nb : OrthonormalBasis \u03b9 \u211d \u211d\nthis\u271d : Unique \u03b9\nthis : \u2191b default = 1 \u2228 \u2191b default = -1\na\u271d : \u2191b default = -1\nx\u271d : \u03b9\n\u22a2 const \u03b9 (\u2191b default) x\u271d = -1\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2078 : NormedAddCommGroup E'\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2074 : NormedAddCommGroup F'\ninst\u271d\u00b3 : InnerProductSpace \u211d F'\ninst\u271d\u00b2 : Fintype \u03b9\nv : Set E\nA : \u03b9 \u2192 Submodule \ud835\udd5c E\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\nn : \u2115\nhn : finrank \ud835\udd5c E = n\ninst\u271d : DecidableEq \u03b9\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : IsInternal V\na : Fin n\nhV' : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => subtype\u2097\u1d62 (V i)\n\u22a2 \u2191(subordinateOrthonormalBasis hn hV hV') a \u2208 V (subordinateOrthonormalBasisIndex hn hV a hV')\n[PROOFSTEP]\nsimpa only [DirectSum.IsInternal.subordinateOrthonormalBasis, OrthonormalBasis.coe_reindex,\n  DirectSum.IsInternal.subordinateOrthonormalBasisIndex] using\n  hV.collectedOrthonormalBasis_mem hV' (fun i => stdOrthonormalBasis \ud835\udd5c (V i))\n    ((hV.sigmaOrthonormalBasisIndexEquiv hn hV').symm a)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet d := finrank \ud835\udd5c S\u15ee\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet LS := LinearMap.range L.toLinearMap\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nhave E : S\u15ee \u2243\u2097\u1d62[\ud835\udd5c] LS\u15ee :=\n  by\n  have dim_LS_perp : finrank \ud835\udd5c LS\u15ee = d :=\n    calc\n      finrank \ud835\udd5c LS\u15ee = finrank \ud835\udd5c V - finrank \ud835\udd5c LS := by\n        simp only [\u2190 LS.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n      _ = finrank \ud835\udd5c V - finrank \ud835\udd5c S := by simp only [LinearMap.finrank_range_of_inj L.injective]\n      _ = finrank \ud835\udd5c S\u15ee := by simp only [\u2190 S.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n  exact (stdOrthonormalBasis \ud835\udd5c S\u15ee).repr.trans ((stdOrthonormalBasis \ud835\udd5c LS\u15ee).reindex <| finCongr dim_LS_perp).repr.symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\n\u22a2 { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\n[PROOFSTEP]\nhave dim_LS_perp : finrank \ud835\udd5c LS\u15ee = d :=\n  calc\n    finrank \ud835\udd5c LS\u15ee = finrank \ud835\udd5c V - finrank \ud835\udd5c LS := by\n      simp only [\u2190 LS.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n    _ = finrank \ud835\udd5c V - finrank \ud835\udd5c S := by simp only [LinearMap.finrank_range_of_inj L.injective]\n    _ = finrank \ud835\udd5c S\u15ee := by simp only [\u2190 S.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\n\u22a2 finrank \ud835\udd5c { x // x \u2208 LS\u15ee } = finrank \ud835\udd5c V - finrank \ud835\udd5c { x // x \u2208 LS }\n[PROOFSTEP]\nsimp only [\u2190 LS.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\n\u22a2 finrank \ud835\udd5c V - finrank \ud835\udd5c { x // x \u2208 LS } = finrank \ud835\udd5c V - finrank \ud835\udd5c { x // x \u2208 S }\n[PROOFSTEP]\nsimp only [LinearMap.finrank_range_of_inj L.injective]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\n\u22a2 finrank \ud835\udd5c V - finrank \ud835\udd5c { x // x \u2208 S } = finrank \ud835\udd5c { x // x \u2208 S\u15ee }\n[PROOFSTEP]\nsimp only [\u2190 S.finrank_add_finrank_orthogonal, add_tsub_cancel_left]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\ndim_LS_perp : finrank \ud835\udd5c { x // x \u2208 LS\u15ee } = d\n\u22a2 { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\n[PROOFSTEP]\nexact (stdOrthonormalBasis \ud835\udd5c S\u15ee).repr.trans ((stdOrthonormalBasis \ud835\udd5c LS\u15ee).reindex <| finCongr dim_LS_perp).repr.symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet L3 := LS\u15ee.subtype\u2097\u1d62.comp E.toLinearIsometry\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nhaveI : CompleteSpace S := FiniteDimensional.complete \ud835\udd5c S\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis : CompleteSpace { x // x \u2208 S }\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nhaveI : CompleteSpace V := FiniteDimensional.complete \ud835\udd5c V\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet p1 := (orthogonalProjection S).toLinearMap\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet p2 := (orthogonalProjection S\u15ee).toLinearMap\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nlet M := L.toLinearMap.comp p1 + L3.toLinearMap.comp p2\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nhave M_norm_map : \u2200 x : V, \u2016M x\u2016 = \u2016x\u2016 := by\n  intro x\n  have Mx_decomp : M x = L (p1 x) + L3 (p2 x) := by\n    simp only [LinearMap.add_apply, LinearMap.comp_apply, LinearMap.comp_apply, LinearIsometry.coe_toLinearMap]\n      -- Mx_decomp is the orthogonal decomposition of M x\n  have Mx_orth : \u27eaL (p1 x), L3 (p2 x)\u27eb = 0 :=\n    by\n    have Lp1x : L (p1 x) \u2208 LinearMap.range L.toLinearMap := LinearMap.mem_range_self L.toLinearMap (p1 x)\n    have Lp2x : L3 (p2 x) \u2208 (LinearMap.range L.toLinearMap)\u15ee :=\n      by\n      simp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtype\u2097\u1d62, \u2190 Submodule.range_subtype LS\u15ee]\n      apply LinearMap.mem_range_self\n    apply Submodule.inner_right_of_mem_orthogonal Lp1x Lp2x\n  rw [\u2190 sq_eq_sq (norm_nonneg _) (norm_nonneg _), norm_sq_eq_add_norm_sq_projection x S]\n  simp only [sq, Mx_decomp]\n  rw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth]\n  simp only [LinearIsometry.norm_map, _root_.add_left_inj, mul_eq_mul_left_iff, norm_eq_zero, true_or_iff,\n    eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm, Submodule.coe_eq_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\n\u22a2 \u2200 (x : V), \u2016\u2191M x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\n\u22a2 \u2016\u2191M x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave Mx_decomp : M x = L (p1 x) + L3 (p2 x) := by\n  simp only [LinearMap.add_apply, LinearMap.comp_apply, LinearMap.comp_apply, LinearIsometry.coe_toLinearMap]\n    -- Mx_decomp is the orthogonal decomposition of M x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\n\u22a2 \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\n[PROOFSTEP]\nsimp only [LinearMap.add_apply, LinearMap.comp_apply, LinearMap.comp_apply, LinearIsometry.coe_toLinearMap]\n  -- Mx_decomp is the orthogonal decomposition of M x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\n\u22a2 \u2016\u2191M x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nhave Mx_orth : \u27eaL (p1 x), L3 (p2 x)\u27eb = 0 :=\n  by\n  have Lp1x : L (p1 x) \u2208 LinearMap.range L.toLinearMap := LinearMap.mem_range_self L.toLinearMap (p1 x)\n  have Lp2x : L3 (p2 x) \u2208 (LinearMap.range L.toLinearMap)\u15ee :=\n    by\n    simp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtype\u2097\u1d62, \u2190 Submodule.range_subtype LS\u15ee]\n    apply LinearMap.mem_range_self\n  apply Submodule.inner_right_of_mem_orthogonal Lp1x Lp2x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\n\u22a2 inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n[PROOFSTEP]\nhave Lp1x : L (p1 x) \u2208 LinearMap.range L.toLinearMap := LinearMap.mem_range_self L.toLinearMap (p1 x)\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nLp1x : \u2191L (\u2191p1 x) \u2208 LinearMap.range L.toLinearMap\n\u22a2 inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n[PROOFSTEP]\nhave Lp2x : L3 (p2 x) \u2208 (LinearMap.range L.toLinearMap)\u15ee :=\n  by\n  simp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtype\u2097\u1d62, \u2190 Submodule.range_subtype LS\u15ee]\n  apply LinearMap.mem_range_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nLp1x : \u2191L (\u2191p1 x) \u2208 LinearMap.range L.toLinearMap\n\u22a2 \u2191L3 (\u2191p2 x) \u2208 (LinearMap.range L.toLinearMap)\u15ee\n[PROOFSTEP]\nsimp only [LinearIsometry.coe_comp, Function.comp_apply, Submodule.coe_subtype\u2097\u1d62, \u2190 Submodule.range_subtype LS\u15ee]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nLp1x : \u2191L (\u2191p1 x) \u2208 LinearMap.range L.toLinearMap\n\u22a2 \u2191(Submodule.subtype (LinearMap.range L.toLinearMap)\u15ee)\n      (\u2191(LinearIsometryEquiv.toLinearIsometry E) (\u2191\u2191(orthogonalProjection S\u15ee) x)) \u2208\n    LinearMap.range (Submodule.subtype (LinearMap.range L.toLinearMap)\u15ee)\n[PROOFSTEP]\napply LinearMap.mem_range_self\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nLp1x : \u2191L (\u2191p1 x) \u2208 LinearMap.range L.toLinearMap\nLp2x : \u2191L3 (\u2191p2 x) \u2208 (LinearMap.range L.toLinearMap)\u15ee\n\u22a2 inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n[PROOFSTEP]\napply Submodule.inner_right_of_mem_orthogonal Lp1x Lp2x\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nMx_orth : inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n\u22a2 \u2016\u2191M x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrw [\u2190 sq_eq_sq (norm_nonneg _) (norm_nonneg _), norm_sq_eq_add_norm_sq_projection x S]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nMx_orth : inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n\u22a2 \u2016\u2191M x\u2016 ^ 2 = \u2016\u2191(orthogonalProjection S) x\u2016 ^ 2 + \u2016\u2191(orthogonalProjection S\u15ee) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [sq, Mx_decomp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nMx_orth : inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n\u22a2 \u2016\u2191L (\u2191\u2191(orthogonalProjection S) x) +\n          \u2191(comp (subtype\u2097\u1d62 (LinearMap.range L.toLinearMap)\u15ee) (LinearIsometryEquiv.toLinearIsometry E))\n            (\u2191\u2191(orthogonalProjection S\u15ee) x)\u2016 *\n      \u2016\u2191L (\u2191\u2191(orthogonalProjection S) x) +\n          \u2191(comp (subtype\u2097\u1d62 (LinearMap.range L.toLinearMap)\u15ee) (LinearIsometryEquiv.toLinearIsometry E))\n            (\u2191\u2191(orthogonalProjection S\u15ee) x)\u2016 =\n    \u2016\u2191(orthogonalProjection S) x\u2016 * \u2016\u2191(orthogonalProjection S) x\u2016 +\n      \u2016\u2191(orthogonalProjection S\u15ee) x\u2016 * \u2016\u2191(orthogonalProjection S\u15ee) x\u2016\n[PROOFSTEP]\nrw [norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nx : V\nMx_decomp : \u2191M x = \u2191L (\u2191p1 x) + \u2191L3 (\u2191p2 x)\nMx_orth : inner (\u2191L (\u2191p1 x)) (\u2191L3 (\u2191p2 x)) = 0\n\u22a2 \u2016\u2191L (\u2191p1 x)\u2016 * \u2016\u2191L (\u2191p1 x)\u2016 + \u2016\u2191L3 (\u2191p2 x)\u2016 * \u2016\u2191L3 (\u2191p2 x)\u2016 =\n    \u2016\u2191(orthogonalProjection S) x\u2016 * \u2016\u2191(orthogonalProjection S) x\u2016 +\n      \u2016\u2191(orthogonalProjection S\u15ee) x\u2016 * \u2016\u2191(orthogonalProjection S\u15ee) x\u2016\n[PROOFSTEP]\nsimp only [LinearIsometry.norm_map, _root_.add_left_inj, mul_eq_mul_left_iff, norm_eq_zero, true_or_iff,\n  eq_self_iff_true, ContinuousLinearMap.coe_coe, Submodule.coe_norm, Submodule.coe_eq_zero]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE\u271d : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\u271d\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\u271d\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\nd : \u2115 := finrank \ud835\udd5c { x // x \u2208 S\u15ee }\nLS : Submodule \ud835\udd5c V := LinearMap.range L.toLinearMap\nE : { x // x \u2208 S\u15ee } \u2243\u2097\u1d62[\ud835\udd5c] { x // x \u2208 LS\u15ee }\nL3 : { x // x \u2208 S\u15ee } \u2192\u2097\u1d62[\ud835\udd5c] V := comp (subtype\u2097\u1d62 LS\u15ee) (LinearIsometryEquiv.toLinearIsometry E)\nthis\u271d : CompleteSpace { x // x \u2208 S }\nthis : CompleteSpace V\np1 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S } := \u2191(orthogonalProjection S)\np2 : V \u2192\u2097[\ud835\udd5c] { x // x \u2208 S\u15ee } := \u2191(orthogonalProjection S\u15ee)\nM : V \u2192\u2097[\ud835\udd5c] V := LinearMap.comp L.toLinearMap p1 + LinearMap.comp L3.toLinearMap p2\nM_norm_map : \u2200 (x : V), \u2016\u2191M x\u2016 = \u2016x\u2016\n\u22a2 V \u2192\u2097\u1d62[\ud835\udd5c] V\n[PROOFSTEP]\nexact\n  { toLinearMap := M\n    norm_map' := M_norm_map }\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\ns : { x // x \u2208 S }\n\u22a2 \u2191(extend L) \u2191s = \u2191L s\n[PROOFSTEP]\nhaveI : CompleteSpace S := FiniteDimensional.complete \ud835\udd5c S\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\ns : { x // x \u2208 S }\nthis : CompleteSpace { x // x \u2208 S }\n\u22a2 \u2191(extend L) \u2191s = \u2191L s\n[PROOFSTEP]\nsimp only [LinearIsometry.extend, \u2190 LinearIsometry.coe_toLinearMap]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u00b2 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u2075 : NormedAddCommGroup F'\ninst\u271d\u2074 : InnerProductSpace \u211d F'\ninst\u271d\u00b3 : Fintype \u03b9\nV : Type u_8\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \ud835\udd5c V\ninst\u271d : FiniteDimensional \ud835\udd5c V\nS : Submodule \ud835\udd5c V\nL\u271d L : { x // x \u2208 S } \u2192\u2097\u1d62[\ud835\udd5c] V\ns : { x // x \u2208 S }\nthis : CompleteSpace { x // x \u2208 S }\n\u22a2 \u2191(LinearMap.comp L.toLinearMap \u2191(orthogonalProjection S) +\n          LinearMap.comp\n            (comp (subtype\u2097\u1d62 (LinearMap.range L.toLinearMap)\u15ee)\n                (LinearIsometryEquiv.toLinearIsometry\n                  (LinearIsometryEquiv.trans (stdOrthonormalBasis \ud835\udd5c { x // x \u2208 S\u15ee }).repr\n                    (LinearIsometryEquiv.symm\n                      (OrthonormalBasis.reindex (stdOrthonormalBasis \ud835\udd5c { x // x \u2208 (LinearMap.range L.toLinearMap)\u15ee })\n                          (finCongr\n                            (_ :\n                              finrank \ud835\udd5c { x // x \u2208 (LinearMap.range L.toLinearMap)\u15ee } =\n                                finrank \ud835\udd5c { x // x \u2208 S\u15ee }))).repr)))).toLinearMap\n            \u2191(orthogonalProjection S\u15ee))\n      \u2191s =\n    \u2191L.toLinearMap s\n[PROOFSTEP]\nsimp only [add_right_eq_self, LinearIsometry.coe_toLinearMap, LinearIsometryEquiv.coe_toLinearIsometry,\n  LinearIsometry.coe_comp, Function.comp_apply, orthogonalProjection_mem_subspace_eq_self, LinearMap.coe_comp,\n  ContinuousLinearMap.coe_coe, Submodule.coeSubtype, LinearMap.add_apply, Submodule.coe_eq_zero,\n  LinearIsometryEquiv.map_eq_zero_iff, Submodule.coe_subtype\u2097\u1d62,\n  orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.orthogonal_orthogonal, Submodule.coe_mem]\n[GOAL]\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c\nE : Type u_4\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E\nE' : Type u_5\ninst\u271d\u2077 : NormedAddCommGroup E'\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E'\nF : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\nF' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b2 : InnerProductSpace \u211d F'\ninst\u271d\u00b9 : Fintype \u03b9\nm : Type u_8\nn : Type u_9\ninst\u271d : Fintype n\nA B : Matrix m n \ud835\udd5c\ni j : m\n\u22a2 inner (\u2191(PiLp.equiv 2 fun i => \ud835\udd5c).symm (A i)) (\u2191(PiLp.equiv 2 fun i => \ud835\udd5c).symm (B j)) = (B * A\u1d34) j i\n[PROOFSTEP]\nsimp_rw [EuclideanSpace.inner_piLp_equiv_symm, Matrix.mul_apply', Matrix.dotProduct_comm, Matrix.conjTranspose_apply,\n  Pi.star_def]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.PiL2", "llama_tokens": 84606, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835330070839, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5169808629399676}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 TerminatedAt g n \u2194 Stream'.Seq.TerminatedAt g.s n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 TerminatedAt g n \u2194 Stream'.Seq.get? g.s n = none\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 Stream'.Seq.get? (partialNumerators g) n = none \u2194 Stream'.Seq.get? g.s n = none\n[PROOFSTEP]\ncases s_nth_eq : g.s.get? n\n[GOAL]\ncase none\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\ns_nth_eq : Stream'.Seq.get? g.s n = none\n\u22a2 Stream'.Seq.get? (partialNumerators g) n = none \u2194 none = none\n[PROOFSTEP]\nsimp [partialNumerators, s_nth_eq]\n[GOAL]\ncase some\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\nval\u271d : Pair \u03b1\ns_nth_eq : Stream'.Seq.get? g.s n = some val\u271d\n\u22a2 Stream'.Seq.get? (partialNumerators g) n = none \u2194 some val\u271d = none\n[PROOFSTEP]\nsimp [partialNumerators, s_nth_eq]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 TerminatedAt g n \u2194 Stream'.Seq.get? (partialNumerators g) n = none\n[PROOFSTEP]\nrw [terminatedAt_iff_s_none, part_num_none_iff_s_none]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 Stream'.Seq.get? (partialDenominators g) n = none \u2194 Stream'.Seq.get? g.s n = none\n[PROOFSTEP]\ncases s_nth_eq : g.s.get? n\n[GOAL]\ncase none\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\ns_nth_eq : Stream'.Seq.get? g.s n = none\n\u22a2 Stream'.Seq.get? (partialDenominators g) n = none \u2194 none = none\n[PROOFSTEP]\nsimp [partialDenominators, s_nth_eq]\n[GOAL]\ncase some\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\nval\u271d : Pair \u03b1\ns_nth_eq : Stream'.Seq.get? g.s n = some val\u271d\n\u22a2 Stream'.Seq.get? (partialDenominators g) n = none \u2194 some val\u271d = none\n[PROOFSTEP]\nsimp [partialDenominators, s_nth_eq]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\n\u22a2 TerminatedAt g n \u2194 Stream'.Seq.get? (partialDenominators g) n = none\n[PROOFSTEP]\nrw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\ngp : Pair \u03b1\ns_nth_eq : Stream'.Seq.get? g.s n = some gp\n\u22a2 Stream'.Seq.get? (partialNumerators g) n = some gp.a\n[PROOFSTEP]\nsimp [partialNumerators, s_nth_eq]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\ngp : Pair \u03b1\ns_nth_eq : Stream'.Seq.get? g.s n = some gp\n\u22a2 Stream'.Seq.get? (partialDenominators g) n = some gp.b\n[PROOFSTEP]\nsimp [partialDenominators, s_nth_eq]\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\na : \u03b1\nnth_part_num_eq : Stream'.Seq.get? (partialNumerators g) n = some a\n\u22a2 \u2203 gp, Stream'.Seq.get? g.s n = some gp \u2227 gp.a = a\n[PROOFSTEP]\nsimpa [partialNumerators, Stream'.Seq.map_get?] using nth_part_num_eq\n[GOAL]\n\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\nb : \u03b1\nnth_part_denom_eq : Stream'.Seq.get? (partialDenominators g) n = some b\n\u22a2 \u2203 gp, Stream'.Seq.get? g.s n = some gp \u2227 gp.b = b\n[PROOFSTEP]\nsimpa [partialDenominators, Stream'.Seq.map_get?] using nth_part_denom_eq\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\nA : K\nnth_num_eq : numerators g n = A\n\u22a2 \u2203 conts, continuants g n = conts \u2227 conts.a = A\n[PROOFSTEP]\nsimpa\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\nB : K\nnth_denom_eq : denominators g n = B\n\u22a2 \u2203 conts, continuants g n = conts \u2227 conts.b = B\n[PROOFSTEP]\nsimpa\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\n\u22a2 convergents g 0 = g.h\n[PROOFSTEP]\nsimp [convergent_eq_num_div_denom, num_eq_conts_a, denom_eq_conts_b, div_one]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nzeroth_s_eq : Stream'.Seq.get? g.s 0 = some gp\n\u22a2 continuantsAux g 2 = { a := gp.b * g.h + gp.a, b := gp.b }\n[PROOFSTEP]\nsimp [zeroth_s_eq, continuantsAux, nextContinuants, nextDenominator, nextNumerator]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nzeroth_s_eq : Stream'.Seq.get? g.s 0 = some gp\n\u22a2 continuants g 1 = { a := gp.b * g.h + gp.a, b := gp.b }\n[PROOFSTEP]\nsimp [nth_cont_eq_succ_nth_cont_aux]\n  -- porting note: simp used to work here, but now it can't figure out that 1 + 1 = 2\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nzeroth_s_eq : Stream'.Seq.get? g.s 0 = some gp\n\u22a2 continuantsAux g (1 + 1) = { a := gp.b * g.h + gp.a, b := gp.b }\n[PROOFSTEP]\nconvert second_continuant_aux_eq zeroth_s_eq\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nzeroth_s_eq : Stream'.Seq.get? g.s 0 = some gp\n\u22a2 numerators g 1 = gp.b * g.h + gp.a\n[PROOFSTEP]\nsimp [num_eq_conts_a, first_continuant_eq zeroth_s_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\ngp : Pair K\nzeroth_s_eq : Stream'.Seq.get? g.s 0 = some gp\n\u22a2 denominators g 1 = gp.b\n[PROOFSTEP]\nsimp [denom_eq_conts_b, first_continuant_eq zeroth_s_eq]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn : \u2115\ninst\u271d : DivisionRing K\n\u22a2 convergents' g 0 = g.h\n[PROOFSTEP]\nsimp [convergents']\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\nh : Stream'.Seq.head s = none\nn : \u2115\n\u22a2 convergents'Aux s (n + 1) = 0\n[PROOFSTEP]\nsimp [convergents'Aux, h, convergents'Aux.match_1]\n[GOAL]\nK : Type u_1\ng : GeneralizedContinuedFraction K\nn\u271d : \u2115\ninst\u271d : DivisionRing K\ns : Stream'.Seq (Pair K)\np : Pair K\nh : Stream'.Seq.head s = some p\nn : \u2115\n\u22a2 convergents'Aux s (n + 1) = p.a / (p.b + convergents'Aux (Stream'.Seq.tail s) n)\n[PROOFSTEP]\nsimp [convergents'Aux, h, convergents'Aux.match_1]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.ContinuedFractions.Translations", "llama_tokens": 2568, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.6893056231680121, "lm_q1q2_score": 0.5163500282432367}}
{"text": "[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\n\u22a2 Injective (mapFun f)\n[PROOFSTEP]\nintros _ _ h\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\na\u2081\u271d a\u2082\u271d : \ud835\udd4e \u03b1\nh : mapFun f a\u2081\u271d = mapFun f a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\next p\n[GOAL]\ncase h\np\u271d : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p\u271d)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Injective f\na\u2081\u271d a\u2082\u271d : WittVector p\u271d \u03b1\nh : mapFun f a\u2081\u271d = mapFun f a\u2082\u271d\np : \u2115\n\u22a2 coeff a\u2081\u271d p = coeff a\u2082\u271d p\n[PROOFSTEP]\nexact hf (congr_arg (fun x => coeff x p) h : _)\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx : \ud835\udd4e \u03b2\n\u22a2 mapFun f (mk p fun n => Classical.choose (_ : \u2203 a, f a = coeff x n)) = x\n[PROOFSTEP]\next n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx : \ud835\udd4e \u03b2\nn : \u2115\n\u22a2 coeff (mapFun f (mk p fun n => Classical.choose (_ : \u2203 a, f a = coeff x n))) n = coeff x n\n[PROOFSTEP]\nsimp only [mapFun, coeff_mk, comp_apply, Classical.choose_spec (hf (x.coeff n))]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) 0 = 0\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) 1 = 1\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) (x + y) = mapFun (\u2191f) x + mapFun (\u2191f) y\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) (x - y) = mapFun (\u2191f) x - mapFun (\u2191f) y\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) (x * y) = mapFun (\u2191f) x * mapFun (\u2191f) y\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) (-x) = -mapFun (\u2191f) x\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 mapFun (\u2191f) (n \u2022 x) = n \u2022 mapFun (\u2191f) x\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nz : \u2124\n\u22a2 mapFun (\u2191f) (z \u2022 x) = z \u2022 mapFun (\u2191f) x\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 mapFun (\u2191f) (x ^ n) = mapFun (\u2191f) x ^ n\n[PROOFSTEP]\nmap_fun_tac\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn : \u2115\n\u22a2 mapFun (\u2191f) (Nat.unaryCast n) = \u2191n\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 mapFun (\u2191f) (Nat.unaryCast Nat.zero) = \u2191Nat.zero\n[PROOFSTEP]\nsimp [*, Nat.unaryCast, add, one, zero]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn\u271d : \u2115\nn_ih\u271d : mapFun (\u2191f) (Nat.unaryCast n\u271d) = \u2191n\u271d\n\u22a2 mapFun (\u2191f) (Nat.unaryCast (Nat.succ n\u271d)) = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*, Nat.unaryCast, add, one, zero]\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\n\u22a2 0 = \u21910\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn\u271d : \u2115\nn_ih\u271d : mapFun (\u2191f) (Nat.unaryCast n\u271d) = \u2191n\u271d\n\u22a2 \u2191n\u271d + 1 = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\nn : \u2124\n\u22a2 mapFun (\u2191f) (Int.castDef n) = \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase ofNat\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 mapFun (\u2191f) (Int.castDef (Int.ofNat a\u271d)) = \u2191(Int.ofNat a\u271d)\n[PROOFSTEP]\nsimp [*, Int.castDef, add, one, neg, zero, nat_cast]\n[GOAL]\ncase negSucc\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 mapFun (\u2191f) (Int.castDef (Int.negSucc a\u271d)) = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nsimp [*, Int.castDef, add, one, neg, zero, nat_cast]\n[GOAL]\ncase ofNat\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 \u2191a\u271d = \u2191\u2191a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase negSucc\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : R \u2192+* S\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 -\u2191(a\u271d + 1) = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nR\u271d : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\u271d\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\u271d\nR : Type u_6\ni : Fin 0\nj : \u2115\n\u22a2 coeff ![] j = ![]\n[PROOFSTEP]\nrcases i with \u27e8_ | _ | _ | _ | i_val, \u27e8\u27e9\u27e9\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun 0 = 0\n[PROOFSTEP]\nghost_fun_tac 0,![]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun 1 = 1\n[PROOFSTEP]\nghost_fun_tac 1,![]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun (x + y) = WittVector.ghostFun x + WittVector.ghostFun y\n[PROOFSTEP]\nghost_fun_tac X 0 + X 1,![x.coeff, y.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\ni : \u2115\n\u22a2 WittVector.ghostFun (Nat.unaryCast i) = \u2191i\n[PROOFSTEP]\ninduction i\n[GOAL]\ncase zero\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun (Nat.unaryCast Nat.zero) = \u2191Nat.zero\n[PROOFSTEP]\nsimp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.coe_nat]\n[GOAL]\ncase succ\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\nn\u271d : \u2115\nn_ih\u271d : WittVector.ghostFun (Nat.unaryCast n\u271d) = \u2191n\u271d\n\u22a2 WittVector.ghostFun (Nat.unaryCast (Nat.succ n\u271d)) = \u2191(Nat.succ n\u271d)\n[PROOFSTEP]\nsimp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.coe_nat]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun (x - y) = WittVector.ghostFun x - WittVector.ghostFun y\n[PROOFSTEP]\nghost_fun_tac X 0 - X 1,![x.coeff, y.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun (x * y) = WittVector.ghostFun x * WittVector.ghostFun y\n[PROOFSTEP]\nghost_fun_tac X 0 * X 1,![x.coeff, y.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\n\u22a2 WittVector.ghostFun (-x) = -WittVector.ghostFun x\n[PROOFSTEP]\nghost_fun_tac-X 0,![x.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\ni : \u2124\n\u22a2 WittVector.ghostFun (Int.castDef i) = \u2191i\n[PROOFSTEP]\ncases i\n[GOAL]\ncase ofNat\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 WittVector.ghostFun (Int.castDef (Int.ofNat a\u271d)) = \u2191(Int.ofNat a\u271d)\n[PROOFSTEP]\nsimp [*, Int.castDef, ghostFun_nat_cast, ghostFun_neg, -Pi.coe_nat, -Pi.coe_int]\n[GOAL]\ncase negSucc\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\na\u271d : \u2115\n\u22a2 WittVector.ghostFun (Int.castDef (Int.negSucc a\u271d)) = \u2191(Int.negSucc a\u271d)\n[PROOFSTEP]\nsimp [*, Int.castDef, ghostFun_nat_cast, ghostFun_neg, -Pi.coe_nat, -Pi.coe_int]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\nm : \u2115\n\u22a2 WittVector.ghostFun (m \u2022 x) = m \u2022 WittVector.ghostFun x\n[PROOFSTEP]\nghost_fun_tac m \u2022 (X 0 : MvPolynomial _ \u2124),![x.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\nm : \u2124\n\u22a2 WittVector.ghostFun (m \u2022 x) = m \u2022 WittVector.ghostFun x\n[PROOFSTEP]\nghost_fun_tac m \u2022 (X 0 : MvPolynomial _ \u2124),![x.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\nm : \u2115\n\u22a2 WittVector.ghostFun (x ^ m) = WittVector.ghostFun x ^ m\n[PROOFSTEP]\nghost_fun_tac X 0 ^ m,![x.coeff]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\n\u22a2 LeftInverse (fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) WittVector.ghostFun\n[PROOFSTEP]\nintro x\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \ud835\udd4e R\n\u22a2 (fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) (WittVector.ghostFun x) = x\n[PROOFSTEP]\next n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \ud835\udd4e R\nn : \u2115\n\u22a2 coeff ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) (WittVector.ghostFun x)) n = coeff x n\n[PROOFSTEP]\nhave := bind\u2081_wittPolynomial_xInTermsOfW p R n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \ud835\udd4e R\nn : \u2115\nthis : \u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n) = X n\n\u22a2 coeff ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) (WittVector.ghostFun x)) n = coeff x n\n[PROOFSTEP]\napply_fun aeval x.coeff at this \n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \ud835\udd4e R\nn : \u2115\nthis : \u2191(aeval x.coeff) (\u2191(bind\u2081 (W_ R)) (xInTermsOfW p R n)) = \u2191(aeval x.coeff) (X n)\n\u22a2 coeff ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) (WittVector.ghostFun x)) n = coeff x n\n[PROOFSTEP]\nsimpa only [aeval_bind\u2081, aeval_X, ghostFun, aeval_wittPolynomial]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\n\u22a2 Function.RightInverse (fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) WittVector.ghostFun\n[PROOFSTEP]\nintro x\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \u2115 \u2192 R\n\u22a2 WittVector.ghostFun ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) x) = x\n[PROOFSTEP]\next n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \u2115 \u2192 R\nn : \u2115\n\u22a2 WittVector.ghostFun ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) x) n = x n\n[PROOFSTEP]\nhave := bind\u2081_xInTermsOfW_wittPolynomial p R n\n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \u2115 \u2192 R\nn : \u2115\nthis : \u2191(bind\u2081 (xInTermsOfW p R)) (W_ R n) = X n\n\u22a2 WittVector.ghostFun ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) x) n = x n\n[PROOFSTEP]\napply_fun aeval x at this \n[GOAL]\ncase h\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Invertible \u2191p\nx : \u2115 \u2192 R\nn : \u2115\nthis : \u2191(aeval x) (\u2191(bind\u2081 (xInTermsOfW p R)) (W_ R n)) = \u2191(aeval x) (X n)\n\u22a2 WittVector.ghostFun ((fun x => mk p fun n => \u2191(aeval x) (xInTermsOfW p R n)) x) n = x n\n[PROOFSTEP]\nsimpa only [aeval_bind\u2081, aeval_X, ghostFun, aeval_wittPolynomial]\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u22a2 (fun x => coeff x 0) 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\np : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\n\u22a2 OneHom.toFun\n      (\u2191{ toOneHom := { toFun := fun x => coeff x 0, map_one' := (_ : coeff 1 0 = 1) },\n          map_mul' := (_ : \u2200 (x y : \ud835\udd4e R), coeff (x * y) 0 = coeff x 0 * coeff y 0) })\n      0 =\n    0\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.WittVector.Basic", "llama_tokens": 8625, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789086703224, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.5163000286848346}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddZeroClass \u03b2\ninst\u271d : ContinuousAdd \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : ZeroAtFilter l f\nhg : ZeroAtFilter l g\n\u22a2 ZeroAtFilter l (f + g)\n[PROOFSTEP]\nsimpa using hf.add hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : AddGroup \u03b2\ninst\u271d : ContinuousNeg \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : ZeroAtFilter l f\n\u22a2 ZeroAtFilter l (-f)\n[PROOFSTEP]\nsimpa using hf.neg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : TopologicalSpace \ud835\udd5c\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : Zero \ud835\udd5c\ninst\u271d\u00b2 : Zero \u03b2\ninst\u271d\u00b9 : SMulWithZero \ud835\udd5c \u03b2\ninst\u271d : ContinuousSMul \ud835\udd5c \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nc : \ud835\udd5c\nhf : ZeroAtFilter l f\n\u22a2 ZeroAtFilter l (c \u2022 f)\n[PROOFSTEP]\nsimpa using hf.const_smul c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : ZeroAtFilter l f\n\u22a2 BoundedAtFilter l f\n[PROOFSTEP]\nrw [ZeroAtFilter, \u2190 Asymptotics.isLittleO_const_iff (one_ne_zero' \u211d)] at hf \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nl : Filter \u03b1\nf : \u03b1 \u2192 \u03b2\nhf : f =o[l] fun _x => 1\n\u22a2 BoundedAtFilter l f\n[PROOFSTEP]\nexact hf.isBigO\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedAddCommGroup \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : BoundedAtFilter l f\nhg : BoundedAtFilter l g\n\u22a2 BoundedAtFilter l (f + g)\n[PROOFSTEP]\nsimpa using hf.add hg\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : BoundedAtFilter l f\nhg : BoundedAtFilter l g\n\u22a2 BoundedAtFilter l (f * g)\n[PROOFSTEP]\nrefine' (hf.mul hg).trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : BoundedAtFilter l f\nhg : BoundedAtFilter l g\n\u22a2 (fun x => OfNat.ofNat 1 x * OfNat.ofNat 1 x) =O[l] 1\n[PROOFSTEP]\nconvert Asymptotics.isBigO_refl (E := \u211d) _ l\n[GOAL]\ncase h.e'_8.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : BoundedAtFilter l f\nhg : BoundedAtFilter l g\nx\u271d : \u03b1\n\u22a2 OfNat.ofNat 1 x\u271d = OfNat.ofNat 1 x\u271d * OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\n\u22a2 Subalgebra \u03b2 (\u03b1 \u2192 \u03b2)\n[PROOFSTEP]\nrefine' Submodule.toSubalgebra (boundedFilterSubmodule l) _ fun f g hf hg \u21a6 _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\n\u22a2 1 \u2208 boundedFilterSubmodule l\n[PROOFSTEP]\nexact const_boundedAtFilter l (1 : \u03b2)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : NormedField \u03b2\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u03b2\nhf : f \u2208 boundedFilterSubmodule l\nhg : g \u2208 boundedFilterSubmodule l\n\u22a2 f * g \u2208 boundedFilterSubmodule l\n[PROOFSTEP]\nsimpa only [Pi.one_apply, mul_one, norm_mul] using hf.mul hg\n", "meta": {"mathlib_filename": "Mathlib.Order.Filter.ZeroAndBoundedAtFilter", "llama_tokens": 1283, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.7122321720225278, "lm_q1q2_score": 0.5162759312359896}}
{"text": "[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d x y p : R\nh : p \u2223 x - y\n\u22a2 p \u2223 \u2211 i in range n, x ^ i * y ^ (n - 1 - i) \u2194 p \u2223 \u2191n * y ^ (n - 1)\n[PROOFSTEP]\nrw [\u2190 mem_span_singleton, \u2190 Ideal.Quotient.eq] at h \n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d x y p : R\nh : \u2191(Ideal.Quotient.mk (span {p})) x = \u2191(Ideal.Quotient.mk (span {p})) y\n\u22a2 p \u2223 \u2211 i in range n, x ^ i * y ^ (n - 1 - i) \u2194 p \u2223 \u2191n * y ^ (n - 1)\n[PROOFSTEP]\nsimp only [\u2190 mem_span_singleton, \u2190 eq_zero_iff_mem, RingHom.map_geom_sum\u2082, h, geom_sum\u2082_self, _root_.map_mul, map_pow,\n  map_natCast]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d x y p : R\nh : p \u2223 x - y\n\u22a2 p \u2223 \u2211 i in range n, x ^ i * y ^ (n - 1 - i) \u2194 p \u2223 \u2191n * x ^ (n - 1)\n[PROOFSTEP]\nrw [geom_sum\u2082_comm, dvd_geom_sum\u2082_iff_of_dvd_sub]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d x y p : R\nh : p \u2223 x - y\n\u22a2 p \u2223 y - x\n[PROOFSTEP]\nsimpa using h.neg_right\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\n\u22a2 p ^ 2 \u2223 (x + p) ^ n - x ^ (n - 1) * p * \u2191n - x ^ n\n[PROOFSTEP]\ncases' n with n n\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\n\u22a2 p ^ 2 \u2223 (x + p) ^ Nat.zero - x ^ (Nat.zero - 1) * p * \u2191Nat.zero - x ^ Nat.zero\n[PROOFSTEP]\nsimp only [pow_zero, Nat.cast_zero, sub_zero, sub_self, dvd_zero, Nat.zero_eq, mul_zero]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\n\u22a2 p ^ 2 \u2223 (x + p) ^ Nat.succ n - x ^ (Nat.succ n - 1) * p * \u2191(Nat.succ n) - x ^ Nat.succ n\n[PROOFSTEP]\nsimp only [Nat.succ_sub_succ_eq_sub, tsub_zero, Nat.cast_succ, add_pow, Finset.sum_range_succ, Nat.choose_self,\n  Nat.succ_sub _, tsub_self, pow_one, Nat.choose_succ_self_right, pow_zero, mul_one, Nat.cast_zero, zero_add,\n  Nat.succ_eq_add_one, add_tsub_cancel_left]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\n\u22a2 p ^ 2 \u2223\n    \u2211 m in range n, x ^ m * p ^ (n + 1 - m) * \u2191(Nat.choose (n + 1) m) + x ^ n * p * (\u2191n + 1) + x ^ (n + 1) -\n        x ^ n * p * (\u2191n + 1) -\n      x ^ (n + 1)\n[PROOFSTEP]\nsuffices p ^ 2 \u2223 \u2211 i : \u2115 in range n, x ^ i * p ^ (n + 1 - i) * \u2191((n + 1).choose i) by convert this; abel\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\nthis : p ^ 2 \u2223 \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n\u22a2 p ^ 2 \u2223\n    \u2211 m in range n, x ^ m * p ^ (n + 1 - m) * \u2191(Nat.choose (n + 1) m) + x ^ n * p * (\u2191n + 1) + x ^ (n + 1) -\n        x ^ n * p * (\u2191n + 1) -\n      x ^ (n + 1)\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_4\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\nthis : p ^ 2 \u2223 \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n\u22a2 \u2211 m in range n, x ^ m * p ^ (n + 1 - m) * \u2191(Nat.choose (n + 1) m) + x ^ n * p * (\u2191n + 1) + x ^ (n + 1) -\n        x ^ n * p * (\u2191n + 1) -\n      x ^ (n + 1) =\n    \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\nthis : p ^ 2 \u2223 \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n\u22a2 \u2211 m in range n, x ^ m * p ^ (n + 1 - m) * \u2191(Nat.choose (n + 1) m) + x ^ n * p * (\u2191n + 1) + x ^ (n + 1) -\n        x ^ n * p * (\u2191n + 1) -\n      x ^ (n + 1) =\n    \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n[PROOFSTEP]\nabel\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\n\u22a2 p ^ 2 \u2223 \u2211 i in range n, x ^ i * p ^ (n + 1 - i) * \u2191(Nat.choose (n + 1) i)\n[PROOFSTEP]\napply Finset.dvd_sum\n[GOAL]\ncase succ.h\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y p x : R\nn : \u2115\n\u22a2 \u2200 (x_1 : \u2115), x_1 \u2208 range n \u2192 p ^ 2 \u2223 x ^ x_1 * p ^ (n + 1 - x_1) * \u2191(Nat.choose (n + 1) x_1)\n[PROOFSTEP]\nintro y hy\n[GOAL]\ncase succ.h\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d p x : R\nn y : \u2115\nhy : y \u2208 range n\n\u22a2 p ^ 2 \u2223 x ^ y * p ^ (n + 1 - y) * \u2191(Nat.choose (n + 1) y)\n[PROOFSTEP]\ncalc\n  p ^ 2 \u2223 p ^ (n + 1 - y) := pow_dvd_pow p (le_tsub_of_add_le_left (by linarith [Finset.mem_range.mp hy]))\n  _ \u2223 x ^ y * p ^ (n + 1 - y) * \u2191((n + 1).choose y) := dvd_mul_of_dvd_left (dvd_mul_left _ _) _\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d p x : R\nn y : \u2115\nhy : y \u2208 range n\n\u22a2 y + 2 \u2264 n + 1\n[PROOFSTEP]\nlinarith [Finset.mem_range.mp hy]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\n\u22a2 \u2191p ^ 2 \u2223 \u2211 i in range p, (a + \u2191p * b) ^ i * a ^ (p - 1 - i) - \u2191p * a ^ (p - 1)\n[PROOFSTEP]\nhave h1 : \u2200 (i : \u2115), (p : R) ^ 2 \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * i + a ^ i) :=\n  by\n  intro i\n  calc\n    \u2191p ^ 2 \u2223 (\u2191p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]\n    _ \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i) := by\n      simp only [sq_dvd_add_pow_sub_sub (\u2191p * b) a i, \u2190 sub_sub]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\n\u22a2 \u2200 (i : \u2115), \u2191p ^ 2 \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\n[PROOFSTEP]\nintro i\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\ni : \u2115\n\u22a2 \u2191p ^ 2 \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\n[PROOFSTEP]\ncalc\n  \u2191p ^ 2 \u2223 (\u2191p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]\n  _ \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i) := by\n    simp only [sq_dvd_add_pow_sub_sub (\u2191p * b) a i, \u2190 sub_sub]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\ni : \u2115\n\u22a2 \u2191p ^ 2 \u2223 (\u2191p * b) ^ 2\n[PROOFSTEP]\nsimp only [mul_pow, dvd_mul_right]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\ni : \u2115\n\u22a2 (\u2191p * b) ^ 2 \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\n[PROOFSTEP]\nsimp only [sq_dvd_add_pow_sub_sub (\u2191p * b) a i, \u2190 sub_sub]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 : \u2200 (i : \u2115), \u2191p ^ 2 \u2223 (a + \u2191p * b) ^ i - (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\n\u22a2 \u2191p ^ 2 \u2223 \u2211 i in range p, (a + \u2191p * b) ^ i * a ^ (p - 1 - i) - \u2191p * a ^ (p - 1)\n[PROOFSTEP]\nsimp_rw [\u2190 mem_span_singleton, \u2190 Ideal.Quotient.eq] at *\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 i in range p, (a + \u2191p * b) ^ i * a ^ (p - 1 - i)) =\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nlet s : R := (p : R) ^ 2\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 i in range p, (a + \u2191p * b) ^ i * a ^ (p - 1 - i)) =\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\ncalc\n  (Ideal.Quotient.mk (span { s })) (\u2211 i in range p, (a + (p : R) * b) ^ i * a ^ (p - 1 - i)) =\n      \u2211 i : \u2115 in Finset.range p, mk (span { s }) ((a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i) * a ^ (p - 1 - i)) :=\n    by simp_rw [RingHom.map_geom_sum\u2082, \u2190 map_pow, h1, \u2190 _root_.map_mul]\n  _ =\n      mk (span { s }) (\u2211 x : \u2115 in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n        mk (span { s }) (\u2211 x : \u2115 in Finset.range p, a ^ (x + (p - 1 - x))) :=\n    by\n    ring_nf\n    simp only [\u2190 pow_add, map_add, Finset.sum_add_distrib, \u2190 map_sum]\n    congr\n    simp [pow_add a, mul_assoc]\n  _ =\n      mk (span { s }) (\u2211 x : \u2115 in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n        mk (span { s }) (\u2211 x : \u2115 in Finset.range p, a ^ (p - 1)) :=\n    by\n    rw [add_right_inj]\n    have : \u2200 (x : \u2115), (hx : x \u2208 range p) \u2192 a ^ (x + (p - 1 - x)) = a ^ (p - 1) :=\n      by\n      intro x hx\n      rw [\u2190 Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]\n      exact Nat.le_pred_of_lt (Finset.mem_range.mp hx)\n    rw [Finset.sum_congr rfl this]\n  _ =\n      mk (span { s }) (\u2211 x : \u2115 in Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n        mk (span { s }) (\u2191p * a ^ (p - 1)) :=\n    by simp only [add_right_inj, Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n  _ = mk (span { s }) (\u2191p * b * \u2211 x : \u2115 in Finset.range p, a ^ (p - 2) * x) + mk (span { s }) (\u2191p * a ^ (p - 1)) :=\n    by\n    simp only [Finset.mul_sum, \u2190 mul_assoc, \u2190 pow_add]\n    rw [Finset.sum_congr rfl]\n    rintro (\u27e8\u27e9 | \u27e8x\u27e9) hx\n    \u00b7 rw [Nat.cast_zero, mul_zero, mul_zero]\n    \u00b7 have : x.succ - 1 + (p - 1 - x.succ) = p - 2 :=\n        by\n        rw [\u2190 Nat.add_sub_assoc (Nat.le_pred_of_lt (Finset.mem_range.mp hx))]\n        exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)\n      rw [this]\n      ring1\n  _ = mk (span { s }) (\u2191p * a ^ (p - 1)) :=\n    by\n    have : Finset.sum (range p) (fun (x : \u2115) \u21a6 (x : R)) = ((Finset.sum (range p) (fun (x : \u2115) \u21a6 (x : \u2115)))) := by\n      simp only [Nat.cast_sum]\n    simp only [add_left_eq_self, \u2190 Finset.mul_sum, this]\n    norm_cast\n    simp only [Finset.sum_range_id]\n    norm_cast\n    simp only [Nat.cast_mul, _root_.map_mul, Nat.mul_div_assoc p (even_iff_two_dvd.mp (Nat.Odd.sub_odd hp odd_one))]\n    ring_nf\n    rw [mul_assoc, mul_assoc]\n    refine' mul_eq_zero_of_left _ _\n    refine' Ideal.Quotient.eq_zero_iff_mem.mpr _\n    simp [mem_span_singleton]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 i in range p, (a + \u2191p * b) ^ i * a ^ (p - 1 - i)) =\n    \u2211 i in range p, \u2191(Ideal.Quotient.mk (span {s})) ((a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i) * a ^ (p - 1 - i))\n[PROOFSTEP]\nsimp_rw [RingHom.map_geom_sum\u2082, \u2190 map_pow, h1, \u2190 _root_.map_mul]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2211 i in range p, \u2191(Ideal.Quotient.mk (span {s})) ((a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i) * a ^ (p - 1 - i)) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x + (p - 1 - x)))\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2211 x in range p,\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (x - 1) * a ^ (p - 1 - x) * \u2191p * b * \u2191x + a ^ x * a ^ (p - 1 - x)) =\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x + (p - 1 - x)))\n[PROOFSTEP]\nsimp only [\u2190 pow_add, map_add, Finset.sum_add_distrib, \u2190 map_sum]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x - 1 + (p - 1 - x)) * \u2191p * b * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x + (p - 1 - x))) =\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x + (p - 1 - x)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.h.e_6.h.e_f\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 (fun x => a ^ (x - 1 + (p - 1 - x)) * \u2191p * b * \u2191x) = fun x => a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))\n[PROOFSTEP]\nsimp [pow_add a, mul_assoc]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x + (p - 1 - x))) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (p - 1))\n[PROOFSTEP]\nrw [add_right_inj]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x + (p - 1 - x))) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (p - 1))\n[PROOFSTEP]\nhave : \u2200 (x : \u2115), (hx : x \u2208 range p) \u2192 a ^ (x + (p - 1 - x)) = a ^ (p - 1) :=\n  by\n  intro x hx\n  rw [\u2190 Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]\n  exact Nat.le_pred_of_lt (Finset.mem_range.mp hx)\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2200 (x : \u2115), x \u2208 range p \u2192 a ^ (x + (p - 1 - x)) = a ^ (p - 1)\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : x \u2208 range p\n\u22a2 a ^ (x + (p - 1 - x)) = a ^ (p - 1)\n[PROOFSTEP]\nrw [\u2190 Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : x \u2208 range p\n\u22a2 x \u2264 p - 1\n[PROOFSTEP]\nexact Nat.le_pred_of_lt (Finset.mem_range.mp hx)\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2200 (x : \u2115), x \u2208 range p \u2192 a ^ (x + (p - 1 - x)) = a ^ (p - 1)\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x + (p - 1 - x))) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (p - 1))\n[PROOFSTEP]\nrw [Finset.sum_congr rfl this]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (p - 1)) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nsimp only [add_right_inj, Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2211 x in range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (\u2191p * (b * \u2191x)))) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1)) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * b * \u2211 x in range p, a ^ (p - 2) * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nsimp only [Finset.mul_sum, \u2190 mul_assoc, \u2190 pow_add]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, a ^ (x - 1 + (p - 1 - x)) * \u2191p * b * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * a ^ (p - 1)) =\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2211 x in range p, \u2191p * b * a ^ (p - 2) * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nrw [Finset.sum_congr rfl]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2200 (x : \u2115), x \u2208 range p \u2192 a ^ (x - 1 + (p - 1 - x)) * \u2191p * b * \u2191x = \u2191p * b * a ^ (p - 2) * \u2191x\n[PROOFSTEP]\nrintro (\u27e8\u27e9 | \u27e8x\u27e9) hx\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nhx : Nat.zero \u2208 range p\n\u22a2 a ^ (Nat.zero - 1 + (p - 1 - Nat.zero)) * \u2191p * b * \u2191Nat.zero = \u2191p * b * a ^ (p - 2) * \u2191Nat.zero\n[PROOFSTEP]\nrw [Nat.cast_zero, mul_zero, mul_zero]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : Nat.succ x \u2208 range p\n\u22a2 a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * \u2191p * b * \u2191(Nat.succ x) = \u2191p * b * a ^ (p - 2) * \u2191(Nat.succ x)\n[PROOFSTEP]\nhave : x.succ - 1 + (p - 1 - x.succ) = p - 2 :=\n  by\n  rw [\u2190 Nat.add_sub_assoc (Nat.le_pred_of_lt (Finset.mem_range.mp hx))]\n  exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : Nat.succ x \u2208 range p\n\u22a2 Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n[PROOFSTEP]\nrw [\u2190 Nat.add_sub_assoc (Nat.le_pred_of_lt (Finset.mem_range.mp hx))]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : Nat.succ x \u2208 range p\n\u22a2 Nat.succ x - 1 + (p - 1) - Nat.succ x = p - 2\n[PROOFSTEP]\nexact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : Nat.succ x \u2208 range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n\u22a2 a ^ (Nat.succ x - 1 + (p - 1 - Nat.succ x)) * \u2191p * b * \u2191(Nat.succ x) = \u2191p * b * a ^ (p - 2) * \u2191(Nat.succ x)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nx : \u2115\nhx : Nat.succ x \u2208 range p\nthis : Nat.succ x - 1 + (p - 1 - Nat.succ x) = p - 2\n\u22a2 a ^ (p - 2) * \u2191p * b * \u2191(Nat.succ x) = \u2191p * b * a ^ (p - 2) * \u2191(Nat.succ x)\n[PROOFSTEP]\nring1\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * b * \u2211 x in range p, a ^ (p - 2) * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1)) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nhave : Finset.sum (range p) (fun (x : \u2115) \u21a6 (x : R)) = ((Finset.sum (range p) (fun (x : \u2115) \u21a6 (x : \u2115)))) := by\n  simp only [Nat.cast_sum]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\n\u22a2 \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n[PROOFSTEP]\nsimp only [Nat.cast_sum]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * b * \u2211 x in range p, a ^ (p - 2) * \u2191x) +\n      \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1)) =\n    \u2191(Ideal.Quotient.mk (span {s})) (\u2191p * a ^ (p - 1))\n[PROOFSTEP]\nsimp only [add_left_eq_self, \u2190 Finset.mul_sum, this]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * b * (a ^ (p - 2) * \u2191(\u2211 x in range p, x))) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * b * (a ^ (p - 2) * \u2191(\u2211 x in range p, x))) = 0\n[PROOFSTEP]\nsimp only [Finset.sum_range_id]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * b * (a ^ (p - 2) * \u2191(p * (p - 1) / 2))) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (\u2191p * b * (a ^ (p - 2) * \u2191(p * (p - 1) / 2))) = 0\n[PROOFSTEP]\nsimp only [Nat.cast_mul, _root_.map_mul, Nat.mul_div_assoc p (even_iff_two_dvd.mp (Nat.Odd.sub_odd hp odd_one))]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191p * \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) b *\n      (\u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (p - 2)) *\n        (\u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191p * \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191((p - 1) / 2))) =\n    0\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191p ^ 2 * \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) b *\n        \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (p - 2)) *\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191((p - 1) / 2) =\n    0\n[PROOFSTEP]\nrw [mul_assoc, mul_assoc]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191p ^ 2 *\n      (\u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) b *\n        (\u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (p - 2)) * \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191((p - 1) / 2))) =\n    0\n[PROOFSTEP]\nrefine' mul_eq_zero_of_left _ _\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) \u2191p ^ 2 = 0\n[PROOFSTEP]\nrefine' Ideal.Quotient.eq_zero_iff_mem.mpr _\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x y : R\np : \u2115\nhp : Odd p\nh1 :\n  \u2200 (i : \u2115),\n    \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) ((a + \u2191p * b) ^ i) =\n      \u2191(Ideal.Quotient.mk (span {\u2191p ^ 2})) (a ^ (i - 1) * (\u2191p * b) * \u2191i + a ^ i)\ns : R := \u2191p ^ 2\nthis : \u2211 x in range p, \u2191x = \u2191(\u2211 x in range p, x)\n\u22a2 (fun x => x ^ 2) \u2191p \u2208 span {\u2191p ^ 2}\n[PROOFSTEP]\nsimp [mem_span_singleton]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d\u00b2 : CommRing R\na b x\u271d y\u271d : R\np\u271d : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\np : R\nhp : Prime p\nx y : R\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : \u00acp \u2223 \u2191n\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y)\n[PROOFSTEP]\nrw [\u2190 geom_sum\u2082_mul, multiplicity.mul hp, multiplicity_eq_zero.2 (not_dvd_geom_sum\u2082 hp hxy hx hn), zero_add]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 multiplicity (\u2191p) (\u2211 i in range p, x ^ i * y ^ (p - 1 - i)) = 1\n[PROOFSTEP]\nrw [\u2190 Nat.cast_one]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 multiplicity (\u2191p) (\u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)) = 1\n[PROOFSTEP]\nrefine' multiplicity.eq_coe_iff.2 \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 \u2191p ^ 1 \u2223 \u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\nrw [pow_one]\n[GOAL]\ncase refine'_1\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 \u2191p \u2223 \u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\nexact dvd_geom_sum\u2082_self hxy\n[GOAL]\ncase refine'_2\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 \u00ac\u2191p ^ (1 + 1) \u2223 \u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\nrw [dvd_iff_dvd_of_dvd_sub hxy] at hx \n[GOAL]\ncase refine'_2\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 y\n\u22a2 \u00ac\u2191p ^ (1 + 1) \u2223 \u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\ncases' hxy with k hk\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhx : \u00ac\u2191p \u2223 y\nk : R\nhk : x - y = \u2191p * k\n\u22a2 \u00ac\u2191p ^ (1 + 1) \u2223 \u2211 i in range p, x ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\nrw [one_add_one_eq_two, eq_add_of_sub_eq' hk]\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhx : \u00ac\u2191p \u2223 y\nk : R\nhk : x - y = \u2191p * k\n\u22a2 \u00ac\u2191p ^ 2 \u2223 \u2211 i in range p, (y + \u2191p * k) ^ i * y ^ (p - \u21911 - i)\n[PROOFSTEP]\nrefine' mt (dvd_iff_dvd_of_dvd_sub (@odd_sq_dvd_geom_sum\u2082_sub _ _ y k _ hp1)).mp _\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhx : \u00ac\u2191p \u2223 y\nk : R\nhk : x - y = \u2191p * k\n\u22a2 \u00ac\u2191p ^ 2 \u2223 \u2191p * y ^ (p - 1)\n[PROOFSTEP]\nrw [pow_two, mul_dvd_mul_iff_left hp.ne_zero]\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhx : \u00ac\u2191p \u2223 y\nk : R\nhk : x - y = \u2191p * k\n\u22a2 \u00ac\u2191p \u2223 y ^ (p - 1)\n[PROOFSTEP]\nexact mt hp.dvd_of_dvd_pow hx\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 multiplicity (\u2191p) (x ^ p - y ^ p) = multiplicity (\u2191p) (x - y) + 1\n[PROOFSTEP]\nrw [\u2190 geom_sum\u2082_mul, multiplicity.mul hp, geom_sum\u2082_eq_one hp hp1 hxy hx, add_comm]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\n\u22a2 multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n[PROOFSTEP]\ninduction' a with a h_ind\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 multiplicity (\u2191p) (x ^ p ^ Nat.zero - y ^ p ^ Nat.zero) = multiplicity (\u2191p) (x - y) + \u2191Nat.zero\n[PROOFSTEP]\nrw [Nat.cast_zero, add_zero, pow_zero, pow_one, pow_one]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\nh_ind : multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n\u22a2 multiplicity (\u2191p) (x ^ p ^ Nat.succ a - y ^ p ^ Nat.succ a) = multiplicity (\u2191p) (x - y) + \u2191(Nat.succ a)\n[PROOFSTEP]\nrw [\u2190 Nat.add_one, Nat.cast_add, Nat.cast_one, \u2190 add_assoc, \u2190 h_ind, pow_succ', pow_mul, pow_mul]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\nh_ind : multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n\u22a2 multiplicity (\u2191p) ((x ^ p ^ a) ^ p - (y ^ p ^ a) ^ p) = multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) + 1\n[PROOFSTEP]\napply pow_prime_sub_pow_prime hp hp1\n[GOAL]\ncase succ.hxy\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\nh_ind : multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n\u22a2 \u2191p \u2223 x ^ p ^ a - y ^ p ^ a\n[PROOFSTEP]\nrw [\u2190 geom_sum\u2082_mul]\n[GOAL]\ncase succ.hxy\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\nh_ind : multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n\u22a2 \u2191p \u2223 (\u2211 i in range (p ^ a), x ^ i * y ^ (p ^ a - 1 - i)) * (x - y)\n[PROOFSTEP]\nexact dvd_mul_of_dvd_right hxy _\n[GOAL]\ncase succ.hx\nR : Type u_1\nn : \u2115\ninst\u271d\u00b2 : CommRing R\na\u271d b x y : R\np : \u2115\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\nhp : Prime \u2191p\nhp1 : Odd p\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\na : \u2115\nh_ind : multiplicity (\u2191p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (\u2191p) (x - y) + \u2191a\n\u22a2 \u00ac\u2191p \u2223 x ^ p ^ a\n[PROOFSTEP]\nexact fun h => hx (hp.dvd_of_dvd_pow h)\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\n\u22a2 multiplicity (\u2191p) (x ^ n - y ^ n) = multiplicity (\u2191p) (x - y) + multiplicity p n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\n\u22a2 multiplicity (\u2191p) (x ^ Nat.zero - y ^ Nat.zero) = multiplicity (\u2191p) (x - y) + multiplicity p Nat.zero\n[PROOFSTEP]\nsimp only [multiplicity.zero, add_top, pow_zero, sub_self, Nat.zero_eq]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\n\u22a2 multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity (\u2191p) (x - y) + multiplicity p (Nat.succ n)\n[PROOFSTEP]\nhave h : (multiplicity _ _).Dom := finite_nat_iff.mpr \u27e8hp.ne_one, n.succ_pos\u27e9\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\n\u22a2 multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity (\u2191p) (x - y) + multiplicity p (Nat.succ n)\n[PROOFSTEP]\nrcases eq_coe_iff.mp (PartENat.natCast_get h).symm with \u27e8\u27e8k, hk\u27e9, hpn\u27e9\n[GOAL]\ncase succ.intro.intro\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity (\u2191p) (x - y) + multiplicity p (Nat.succ n)\n[PROOFSTEP]\nconv_lhs => rw [hk, pow_mul, pow_mul]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n| multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n)\n[PROOFSTEP]\nrw [hk, pow_mul, pow_mul]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n| multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n)\n[PROOFSTEP]\nrw [hk, pow_mul, pow_mul]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n| multiplicity (\u2191p) (x ^ Nat.succ n - y ^ Nat.succ n)\n[PROOFSTEP]\nrw [hk, pow_mul, pow_mul]\n[GOAL]\ncase succ.intro.intro\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 multiplicity (\u2191p)\n      ((x ^ p ^ Part.get (multiplicity p (Nat.succ n)) h) ^ k -\n        (y ^ p ^ Part.get (multiplicity p (Nat.succ n)) h) ^ k) =\n    multiplicity (\u2191p) (x - y) + multiplicity p (Nat.succ n)\n[PROOFSTEP]\nrw [Nat.prime_iff_prime_int] at hp \n[GOAL]\ncase succ.intro.intro\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 multiplicity (\u2191p)\n      ((x ^ p ^ Part.get (multiplicity p (Nat.succ n)) h) ^ k -\n        (y ^ p ^ Part.get (multiplicity p (Nat.succ n)) h) ^ k) =\n    multiplicity (\u2191p) (x - y) + multiplicity p (Nat.succ n)\n[PROOFSTEP]\nrw [pow_sub_pow_of_prime hp, pow_prime_pow_sub_pow_prime_pow hp hp1 hxy hx, PartENat.natCast_get]\n[GOAL]\ncase succ.intro.intro.hxy\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 \u2191p \u2223 x ^ p ^ Part.get (multiplicity p (Nat.succ n)) h - y ^ p ^ Part.get (multiplicity p (Nat.succ n)) h\n[PROOFSTEP]\nrw [\u2190 geom_sum\u2082_mul]\n[GOAL]\ncase succ.intro.intro.hxy\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 \u2191p \u2223\n    (\u2211 i in range (p ^ Part.get (multiplicity p (Nat.succ n)) h),\n        x ^ i * y ^ (p ^ Part.get (multiplicity p (Nat.succ n)) h - 1 - i)) *\n      (x - y)\n[PROOFSTEP]\nexact dvd_mul_of_dvd_right hxy _\n[GOAL]\ncase succ.intro.intro.hx\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 \u00ac\u2191p \u2223 x ^ p ^ Part.get (multiplicity p (Nat.succ n)) h\n[PROOFSTEP]\nexact fun h => hx (hp.dvd_of_dvd_pow h)\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 \u00ac\u2191p \u2223 \u2191k\n[PROOFSTEP]\nrw [Int.coe_nat_dvd]\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * k\n\u22a2 \u00acp \u2223 k\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase succ.intro.intro.hn.intro\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nc : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * (p * c)\n\u22a2 False\n[PROOFSTEP]\nrefine' hpn \u27e8c, _\u27e9\n[GOAL]\ncase succ.intro.intro.hn.intro\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Prime \u2191p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nh : (multiplicity p (Nat.succ n)).Dom\nhpn : \u00acp ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) \u2223 Nat.succ n\nc : \u2115\nhk : Nat.succ n = p ^ Part.get (multiplicity p (Nat.succ n)) h * (p * c)\n\u22a2 Nat.succ n = p ^ (Part.get (multiplicity p (Nat.succ n)) h + 1) * c\n[PROOFSTEP]\nrwa [pow_succ', mul_assoc]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x + y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity (\u2191p) (x ^ n + y ^ n) = multiplicity (\u2191p) (x + y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 sub_neg_eq_add] at hxy \n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - -y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity (\u2191p) (x ^ n + y ^ n) = multiplicity (\u2191p) (x + y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 sub_neg_eq_add, \u2190 sub_neg_eq_add, \u2190 Odd.neg_pow hn]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2124\nhxy : \u2191p \u2223 x - -y\nhx : \u00ac\u2191p \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity (\u2191p) (x ^ n - (-y) ^ n) = multiplicity (\u2191p) (x - -y) + multiplicity p n\n[PROOFSTEP]\nexact Int.pow_sub_pow hp hp1 hxy hx n\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\nobtain hyx | hyx := le_total y x\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\niterate 2 rw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity \u2191p \u2191(x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity \u2191p \u2191(x ^ n - y ^ n) = multiplicity \u2191p \u2191(x - y) + multiplicity p n\n[PROOFSTEP]\nrw [Int.ofNat_sub (Nat.pow_le_pow_of_le_left hyx n)]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity (\u2191p) (\u2191(x ^ n) - \u2191(y ^ n)) = multiplicity \u2191p \u2191(x - y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd] at hxy hx \n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : \u2191p \u2223 \u2191(x - y)\nhx : \u00ac\u2191p \u2223 \u2191x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity (\u2191p) (\u2191(x ^ n) - \u2191(y ^ n)) = multiplicity \u2191p \u2191(x - y) + multiplicity p n\n[PROOFSTEP]\nrw [Int.coe_nat_sub hyx] at *\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : \u2191p \u2223 \u2191x - \u2191y\nhx : \u00ac\u2191p \u2223 \u2191x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity (\u2191p) (\u2191(x ^ n) - \u2191(y ^ n)) = multiplicity (\u2191p) (\u2191x - \u2191y) + multiplicity p n\n[PROOFSTEP]\npush_cast at *\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : \u2191p \u2223 \u2191x - \u2191y\nhx : \u00ac\u2191p \u2223 \u2191x\nn : \u2115\nhyx : y \u2264 x\n\u22a2 multiplicity (\u2191p) (\u2191x ^ n - \u2191y ^ n) = multiplicity (\u2191p) (\u2191x - \u2191y) + multiplicity p n\n[PROOFSTEP]\nexact Int.pow_sub_pow hp hp1 hxy hx n\n[GOAL]\ncase inr\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhyx : x \u2264 y\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\nsimp only [Nat.sub_eq_zero_iff_le.mpr hyx, Nat.sub_eq_zero_iff_le.mpr (Nat.pow_le_pow_of_le_left hyx n),\n  multiplicity.zero, PartENat.top_add]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x + y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity p (x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n\n[PROOFSTEP]\niterate 2 rw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x + y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity p (x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x + y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity \u2191p \u2191(x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_multiplicity]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : p \u2223 x + y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity \u2191p \u2191(x ^ n + y ^ n) = multiplicity \u2191p \u2191(x + y) + multiplicity p n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd] at hxy hx \n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhxy : \u2191p \u2223 \u2191(x + y)\nhx : \u00ac\u2191p \u2223 \u2191x\nn : \u2115\nhn : Odd n\n\u22a2 multiplicity \u2191p \u2191(x ^ n + y ^ n) = multiplicity \u2191p \u2191(x + y) + multiplicity p n\n[PROOFSTEP]\npush_cast at *\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\na b x\u271d y\u271d : R\np : \u2115\nhp : Nat.Prime p\nhp1 : Odd p\nx y : \u2115\nhx : \u00ac\u2191p \u2223 \u2191x\nn : \u2115\nhn : Odd n\nhxy : \u2191p \u2223 \u2191x + \u2191y\n\u22a2 multiplicity (\u2191p) (\u2191x ^ n + \u2191y ^ n) = multiplicity (\u2191p) (\u2191x + \u2191y) + multiplicity p n\n[PROOFSTEP]\nexact Int.pow_add_pow hp hp1 hxy hx hn\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\ninst\u271d : CommRing R\nx y : R\nn : \u2115\n\u22a2 x ^ 2 ^ n - y ^ 2 ^ n = (\u220f i in range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n[PROOFSTEP]\ninduction' n with d hd\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\nx y : R\n\u22a2 x ^ 2 ^ Nat.zero - y ^ 2 ^ Nat.zero = (\u220f i in range Nat.zero, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n[PROOFSTEP]\nsimp only [pow_zero, pow_one, range_zero, prod_empty, one_mul, Nat.zero_eq]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\nx y : R\nd : \u2115\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (\u220f i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n\u22a2 x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (\u220f i in range (Nat.succ d), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n[PROOFSTEP]\nsuffices x ^ 2 ^ d.succ - y ^ 2 ^ d.succ = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d) by\n  rw [this, hd, Finset.prod_range_succ, \u2190 mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]\n[GOAL]\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\nx y : R\nd : \u2115\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (\u220f i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\nthis : x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)\n\u22a2 x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (\u220f i in range (Nat.succ d), (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n[PROOFSTEP]\nrw [this, hd, Finset.prod_range_succ, \u2190 mul_assoc, mul_comm (x ^ 2 ^ d + y ^ 2 ^ d)]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\nx y : R\nd : \u2115\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (\u220f i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n\u22a2 x ^ 2 ^ Nat.succ d - y ^ 2 ^ Nat.succ d = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one]\n[GOAL]\ncase succ\nR : Type u_1\nn : \u2115\ninst\u271d : CommRing R\nx y : R\nd : \u2115\nhd : x ^ 2 ^ d - y ^ 2 ^ d = (\u220f i in range d, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)\n\u22a2 x ^ 2 ^ (d + 1) - y ^ 2 ^ (d + 1) = (x ^ 2 ^ d + y ^ 2 ^ d) * (x ^ 2 ^ d - y ^ 2 ^ d)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nn : \u2115\nx : \u2124\n\u22a2 Odd x \u2192 x ^ 2 % 4 = 1\n[PROOFSTEP]\nintro hx\n[GOAL]\nR : Type u_1\nn : \u2115\nx : \u2124\nhx : Odd x\n\u22a2 x ^ 2 % 4 = 1\n[PROOFSTEP]\nunfold Odd at hx \n[GOAL]\nR : Type u_1\nn : \u2115\nx : \u2124\nhx : \u2203 k, x = 2 * k + 1\n\u22a2 x ^ 2 % 4 = 1\n[PROOFSTEP]\nrcases hx with \u27e8_, rfl\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nw\u271d : \u2124\n\u22a2 (2 * w\u271d + 1) ^ 2 % 4 = 1\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nw\u271d : \u2124\n\u22a2 (1 + w\u271d * 4 + w\u271d ^ 2 * 4) % 4 = 1\n[PROOFSTEP]\nrw [add_assoc, \u2190 add_mul, Int.add_mul_emod_self]\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nw\u271d : \u2124\n\u22a2 1 % 4 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\n\u22a2 multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = \u21911\n[PROOFSTEP]\nhave hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\n\u22a2 Odd x\n[PROOFSTEP]\nrwa [Int.odd_iff_not_even, even_iff_two_dvd]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\n\u22a2 multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = \u21911\n[PROOFSTEP]\nhave hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by norm_num) hxy)\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\n\u22a2 2 \u2223 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\n\u22a2 multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = \u21911\n[PROOFSTEP]\nhave hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\n\u22a2 Odd y\n[PROOFSTEP]\nsimpa using hx_odd.sub_even hxy_even\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = \u21911\n[PROOFSTEP]\nrefine' multiplicity.eq_coe_iff.mpr \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 2 ^ 1 \u2223 x ^ 2 ^ i + y ^ 2 ^ i\n[PROOFSTEP]\nrw [pow_one, \u2190 even_iff_two_dvd]\n[GOAL]\ncase refine'_1\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 Even (x ^ 2 ^ i + y ^ 2 ^ i)\n[PROOFSTEP]\nexact hx_odd.pow.add_odd hy_odd.pow\n[GOAL]\ncase refine'_2\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\ni : \u2115\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 \u00ac2 ^ (1 + 1) \u2223 x ^ 2 ^ i + y ^ 2 ^ i\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase refine'_2.zero\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 \u00ac2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\n[PROOFSTEP]\nintro hxy'\n[GOAL]\ncase refine'_2.zero\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\n\u22a2 False\n[PROOFSTEP]\nhave : 2 * 2 \u2223 2 * x := by\n  have := dvd_add hxy hxy'\n  norm_num at *\n  rw [two_mul]\n  exact this\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\n\u22a2 2 * 2 \u2223 2 * x\n[PROOFSTEP]\nhave := dvd_add hxy hxy'\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\nthis : 4 \u2223 x - y + (x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero)\n\u22a2 2 * 2 \u2223 2 * x\n[PROOFSTEP]\nnorm_num at *\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhxy_even : Even (x - y)\nhx : x % 2 = 1\nhx_odd : \u00acEven x\nhy_odd : \u00acEven y\nhxy' : 4 \u2223 x + y\nthis : 4 \u2223 x + x\n\u22a2 4 \u2223 2 * x\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhxy_even : Even (x - y)\nhx : x % 2 = 1\nhx_odd : \u00acEven x\nhy_odd : \u00acEven y\nhxy' : 4 \u2223 x + y\nthis : 4 \u2223 x + x\n\u22a2 4 \u2223 x + x\n[PROOFSTEP]\nexact this\n[GOAL]\ncase refine'_2.zero\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\nthis : 2 * 2 \u2223 2 * x\n\u22a2 False\n[PROOFSTEP]\nhave : 2 \u2223 x := (mul_dvd_mul_iff_left (by norm_num)).mp this\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\nthis : 2 * 2 \u2223 2 * x\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2.zero\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nhxy' : 2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.zero + y ^ 2 ^ Nat.zero\nthis\u271d : 2 * 2 \u2223 2 * x\nthis : 2 \u2223 x\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase refine'_2.succ\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : \u2115\n\u22a2 \u00ac2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.succ i + y ^ 2 ^ Nat.succ i\n[PROOFSTEP]\nsuffices \u2200 x : \u2124, Odd x \u2192 x ^ 2 ^ (i + 1) % 4 = 1\n  by\n  rw [show (2 ^ (1 + 1) : \u2124) = 4 by norm_num, Int.dvd_iff_emod_eq_zero, Int.add_emod, this _ hx_odd, this _ hy_odd]\n  norm_num\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : \u2115\nthis : \u2200 (x : \u2124), Odd x \u2192 x ^ 2 ^ (i + 1) % 4 = 1\n\u22a2 \u00ac2 ^ (1 + 1) \u2223 x ^ 2 ^ Nat.succ i + y ^ 2 ^ Nat.succ i\n[PROOFSTEP]\nrw [show (2 ^ (1 + 1) : \u2124) = 4 by norm_num, Int.dvd_iff_emod_eq_zero, Int.add_emod, this _ hx_odd, this _ hy_odd]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : \u2115\nthis : \u2200 (x : \u2124), Odd x \u2192 x ^ 2 ^ (i + 1) % 4 = 1\n\u22a2 2 ^ (1 + 1) = 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : \u2115\nthis : \u2200 (x : \u2124), Odd x \u2192 x ^ 2 ^ (i + 1) % 4 = 1\n\u22a2 \u00ac(1 + 1) % 4 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2.succ\nR : Type u_1\nn : \u2115\nx y : \u2124\nhx : \u00ac2 \u2223 x\nhxy : 4 \u2223 x - y\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\ni : \u2115\n\u22a2 \u2200 (x : \u2124), Odd x \u2192 x ^ 2 ^ (i + 1) % 4 = 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase refine'_2.succ\nR : Type u_1\nn : \u2115\nx\u271d y : \u2124\nhx\u271d : \u00ac2 \u2223 x\u271d\nhxy : 4 \u2223 x\u271d - y\nhx_odd : Odd x\u271d\nhxy_even : Even (x\u271d - y)\nhy_odd : Odd y\ni : \u2115\nx : \u2124\nhx : Odd x\n\u22a2 x ^ 2 ^ (i + 1) % 4 = 1\n[PROOFSTEP]\nrw [pow_succ, mul_comm, pow_mul, Int.sq_mod_four_eq_one_of_odd hx.pow]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\n\u22a2 multiplicity 2 (x ^ 2 ^ n - y ^ 2 ^ n) = multiplicity 2 (x - y) + \u2191n\n[PROOFSTEP]\nsimp only [pow_two_pow_sub_pow_two_pow n, multiplicity.mul Int.prime_two, multiplicity.Finset.prod Int.prime_two,\n  add_comm, Nat.cast_one, Finset.sum_const, Finset.card_range, nsmul_one,\n  Int.two_pow_two_pow_add_two_pow_two_pow hx hxy]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\n\u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\nhave hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\n\u22a2 Odd x\n[PROOFSTEP]\nrwa [Int.odd_iff_not_even, even_iff_two_dvd]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\n\u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\nhave hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by norm_num) hxy)\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\n\u22a2 2 \u2223 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n\u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\nhave hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n\u22a2 Odd y\n[PROOFSTEP]\nsimpa using hx_odd.sub_even hxy_even\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\ncases' n with n\n[GOAL]\ncase zero\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\n\u22a2 multiplicity 2 (x ^ Nat.zero - y ^ Nat.zero) = multiplicity 2 (x - y) + multiplicity 2 \u2191Nat.zero\n[PROOFSTEP]\nsimp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\n\u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n)\n[PROOFSTEP]\nhave h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr \u27e8by norm_num, n.succ_pos\u27e9\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\n\u22a2 2 \u2260 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\n\u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n)\n[PROOFSTEP]\nrcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with \u27e8\u27e8k, hk\u27e9, hpn\u27e9\n[GOAL]\ncase succ.intro.intro\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n)\n[PROOFSTEP]\nrw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime, Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, \u2190 hk,\n  PartENat.natCast_get]\n[GOAL]\ncase succ.intro.intro\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 multiplicity 2 (x - y) + multiplicity 2 (Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase succ.intro.intro.hp\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 Prime 2\n[PROOFSTEP]\nexact Int.prime_two\n[GOAL]\ncase succ.intro.intro.hxy\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 2 \u2223 x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h - y ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h\n[PROOFSTEP]\nsimpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow\n[GOAL]\ncase succ.intro.intro.hx\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 \u00ac2 \u2223 x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h\n[PROOFSTEP]\nsimpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 \u00ac2 \u2223 \u2191k\n[PROOFSTEP]\nerw [Int.coe_nat_dvd]\n  -- `erw` to deal with `2 : \u2124` vs `(2 : \u2115) : \u2124`\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nhpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n\u22a2 \u00ac2 \u2223 k\n[PROOFSTEP]\ncontrapose! hpn\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n\u22a2 2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n\u22a2 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * 2 \u2223 Nat.succ n\n[PROOFSTEP]\nconv_rhs => rw [hk]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n| Nat.succ n\n[PROOFSTEP]\nrw [hk]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n| Nat.succ n\n[PROOFSTEP]\nrw [hk]\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n| Nat.succ n\n[PROOFSTEP]\nrw [hk]\n[GOAL]\ncase succ.intro.intro.hn\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nhxy : 4 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : \u2115\nh : (multiplicity 2 (Nat.succ n)).Dom\nk : \u2115\nhk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\nhpn : 2 \u2223 k\n\u22a2 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * 2 \u2223 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k\n[PROOFSTEP]\nexact mul_dvd_mul_left _ hpn\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhn : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\nhave hy : Odd y := by\n  rw [\u2190 even_iff_two_dvd, \u2190 Int.odd_iff_not_even] at hx \n  replace hxy := (@even_neg _ _ (x - y)).mpr (even_iff_two_dvd.mpr hxy)\n  convert Even.add_odd hxy hx\n  abel\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhn : Even n\n\u22a2 Odd y\n[PROOFSTEP]\nrw [\u2190 even_iff_two_dvd, \u2190 Int.odd_iff_not_even] at hx \n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 2 \u2223 x - y\nhx : Odd x\nhn : Even n\n\u22a2 Odd y\n[PROOFSTEP]\nreplace hxy := (@even_neg _ _ (x - y)).mpr (even_iff_two_dvd.mpr hxy)\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n\u22a2 Odd y\n[PROOFSTEP]\nconvert Even.add_odd hxy hx\n[GOAL]\ncase h.e'_3\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n\u22a2 y = -(x - y) + x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n\u22a2 y = -(x - y) + x\n[PROOFSTEP]\nabel\n[GOAL]\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhn : Even n\nhy : Odd y\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\ncases' hn with d hd\n[GOAL]\ncase intro\nR : Type u_1\nn\u271d : \u2115\nx y : \u2124\nn : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhd : n = d + d\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191n\n[PROOFSTEP]\nsubst hd\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\n\u22a2 multiplicity 2 (x ^ (d + d) - y ^ (d + d)) + 1 =\n    multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191(d + d)\n[PROOFSTEP]\nsimp only [\u2190 two_mul, pow_mul]\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\n\u22a2 multiplicity 2 ((x ^ 2) ^ d - (y ^ 2) ^ d) + 1 =\n    multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191(2 * d)\n[PROOFSTEP]\nhave hxy4 : 4 \u2223 x ^ 2 - y ^ 2 :=\n  by\n  rw [Int.dvd_iff_emod_eq_zero, Int.sub_emod, Int.sq_mod_four_eq_one_of_odd _, Int.sq_mod_four_eq_one_of_odd hy]\n  \u00b7 norm_num\n  \u00b7 simp only [Int.odd_iff_not_even, even_iff_two_dvd, hx, not_false_iff]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\n\u22a2 4 \u2223 x ^ 2 - y ^ 2\n[PROOFSTEP]\nrw [Int.dvd_iff_emod_eq_zero, Int.sub_emod, Int.sq_mod_four_eq_one_of_odd _, Int.sq_mod_four_eq_one_of_odd hy]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\n\u22a2 (1 - 1) % 4 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\n\u22a2 Odd x\n[PROOFSTEP]\nsimp only [Int.odd_iff_not_even, even_iff_two_dvd, hx, not_false_iff]\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 multiplicity 2 ((x ^ 2) ^ d - (y ^ 2) ^ d) + 1 =\n    multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191(2 * d)\n[PROOFSTEP]\nrw [Int.two_pow_sub_pow' d hxy4 _, sq_sub_sq, \u2190 Int.ofNat_mul_out, multiplicity.mul Int.prime_two,\n  multiplicity.mul Int.prime_two]\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191d + 1 =\n    multiplicity 2 (x + y) + multiplicity 2 (x - y) + (multiplicity 2 \u21912 + multiplicity 2 \u2191d)\nR : Type u_1 n : \u2115 x y : \u2124 hxy : 2 \u2223 x - y hx : \u00ac2 \u2223 x hy : Odd y d : \u2115 hxy4 : 4 \u2223 x ^ 2 - y ^ 2 \u22a2 \u00ac2 \u2223 x ^ 2\n[PROOFSTEP]\nsuffices multiplicity (2 : \u2124) \u2191(2 : \u2115) = 1 by rw [this, add_comm (1 : PartENat), \u2190 add_assoc]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\nthis : multiplicity 2 \u21912 = 1\n\u22a2 multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 \u2191d + 1 =\n    multiplicity 2 (x + y) + multiplicity 2 (x - y) + (multiplicity 2 \u21912 + multiplicity 2 \u2191d)\n[PROOFSTEP]\nrw [this, add_comm (1 : PartENat), \u2190 add_assoc]\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 multiplicity 2 \u21912 = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 multiplicity 2 2 = 1\n[PROOFSTEP]\nrw [multiplicity.multiplicity_self _ _]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 \u00acIsUnit 2\n[PROOFSTEP]\napply Prime.not_unit\n[GOAL]\ncase hp\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 Prime 2\n[PROOFSTEP]\nsimp only [\u2190 Nat.prime_iff, Nat.prime_two]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 \u00ac2 \u2223 x ^ 2\n[PROOFSTEP]\nrw [\u2190 even_iff_two_dvd, \u2190 Int.odd_iff_not_even]\n[GOAL]\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 Odd (x ^ 2)\n[PROOFSTEP]\napply Odd.pow\n[GOAL]\ncase hm\nR : Type u_1\nn : \u2115\nx y : \u2124\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nhy : Odd y\nd : \u2115\nhxy4 : 4 \u2223 x ^ 2 - y ^ 2\n\u22a2 Odd x\n[PROOFSTEP]\nsimp only [Int.odd_iff_not_even, even_iff_two_dvd, hx, not_false_iff]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nobtain hyx | hyx := le_total y x\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\niterate 3 rw [\u2190 multiplicity.Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nrw [\u2190 multiplicity.Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity \u21912 \u2191(x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nrw [\u2190 multiplicity.Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity \u21912 \u2191(x ^ n - y ^ n) + 1 = multiplicity \u21912 \u2191(x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nrw [\u2190 multiplicity.Int.coe_nat_multiplicity]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity \u21912 \u2191(x ^ n - y ^ n) + 1 = multiplicity \u21912 \u2191(x + y) + multiplicity \u21912 \u2191(x - y) + multiplicity 2 n\n[PROOFSTEP]\nsimp only [Int.ofNat_sub hyx, Int.ofNat_sub (pow_le_pow_of_le_left' hyx _), Int.ofNat_add, Int.coe_nat_pow]\n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity (\u21912) (\u2191x ^ n - \u2191y ^ n) + 1 = multiplicity (\u21912) (\u2191x + \u2191y) + multiplicity (\u21912) (\u2191x - \u2191y) + multiplicity 2 n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd] at hx \n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac\u21912 \u2223 \u2191x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity (\u21912) (\u2191x ^ n - \u2191y ^ n) + 1 = multiplicity (\u21912) (\u2191x + \u2191y) + multiplicity (\u21912) (\u2191x - \u2191y) + multiplicity 2 n\n[PROOFSTEP]\nrw [\u2190 Int.coe_nat_dvd, Int.ofNat_sub hyx] at hxy \n[GOAL]\ncase inl\nR : Type u_1\nn\u271d x y : \u2115\nhxy : \u21912 \u2223 \u2191x - \u2191y\nhx : \u00ac\u21912 \u2223 \u2191x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity (\u21912) (\u2191x ^ n - \u2191y ^ n) + 1 = multiplicity (\u21912) (\u2191x + \u2191y) + multiplicity (\u21912) (\u2191x - \u2191y) + multiplicity 2 n\n[PROOFSTEP]\nconvert Int.two_pow_sub_pow hxy hx hn using 2\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nn\u271d x y : \u2115\nhxy : \u21912 \u2223 \u2191x - \u2191y\nhx : \u00ac\u21912 \u2223 \u2191x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity 2 n = multiplicity 2 \u2191n\n[PROOFSTEP]\nrw [\u2190 multiplicity.Int.coe_nat_multiplicity]\n[GOAL]\ncase h.e'_3.h.e'_6\nR : Type u_1\nn\u271d x y : \u2115\nhxy : \u21912 \u2223 \u2191x - \u2191y\nhx : \u00ac\u21912 \u2223 \u2191x\nn : \u2115\nhn : Even n\nhyx : y \u2264 x\n\u22a2 multiplicity \u21912 \u2191n = multiplicity 2 \u2191n\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nR : Type u_1\nn\u271d x y : \u2115\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : Even n\nhyx : x \u2264 y\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nsimp only [Nat.sub_eq_zero_iff_le.mpr hyx, Nat.sub_eq_zero_iff_le.mpr (pow_le_pow_of_le_left' hyx n), multiplicity.zero,\n  PartENat.top_add, PartENat.add_top]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 padicValNat 2 (x ^ n - y ^ n) + 1 = padicValNat 2 (x + y) + padicValNat 2 (x - y) + padicValNat 2 n\n[PROOFSTEP]\nsimp only [\u2190 PartENat.natCast_inj, Nat.cast_add]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 \u2191(padicValNat 2 (x ^ n - y ^ n)) + \u21911 = \u2191(padicValNat 2 (x + y)) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)\n[PROOFSTEP]\niterate 4 rw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 \u2191(padicValNat 2 (x ^ n - y ^ n)) + \u21911 = \u2191(padicValNat 2 (x + y)) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + \u21911 = \u2191(padicValNat 2 (x + y)) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x ^ n - y ^ n\nR : Type u_1 n\u271d x y : \u2115 hyx : y < x hxy : 2 \u2223 x - y hx : \u00ac2 \u2223 x n : \u2115 hn : 0 < n hneven : Even n \u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + \u21911 = multiplicity 2 (x + y) + \u2191(padicValNat 2 (x - y)) + \u2191(padicValNat 2 n)\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x + y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x + y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x ^ n - y ^ n\nR : Type u_1 n\u271d x y : \u2115 hyx : y < x hxy : 2 \u2223 x - y hx : \u00ac2 \u2223 x n : \u2115 hn : 0 < n hneven : Even n \u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + \u21911 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + \u2191(padicValNat 2 n)\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x - y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x - y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x + y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x + y\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x ^ n - y ^ n\nR : Type u_1 n\u271d x y : \u2115 hyx : y < x hxy : 2 \u2223 x - y hx : \u00ac2 \u2223 x n : \u2115 hn : 0 < n hneven : Even n \u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 multiplicity 2 (x ^ n - y ^ n) + \u21911 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n\n[PROOFSTEP]\nconvert Nat.two_pow_sub_pow hxy hx hneven using 2\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < n\n[PROOFSTEP]\nexact hn\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x - y\n[PROOFSTEP]\nexact Nat.sub_pos_of_lt hyx\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x + y\n[PROOFSTEP]\nlinarith\n[GOAL]\nR : Type u_1\nn\u271d x y : \u2115\nhyx : y < x\nhxy : 2 \u2223 x - y\nhx : \u00ac2 \u2223 x\nn : \u2115\nhn : 0 < n\nhneven : Even n\n\u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nsimp only [tsub_pos_iff_lt, pow_lt_pow_of_lt_left hyx (@zero_le' _ y _) hn]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 padicValNat p (x ^ n - y ^ n) = padicValNat p (x - y) + padicValNat p n\n[PROOFSTEP]\nrw [\u2190 PartENat.natCast_inj, Nat.cast_add]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191(padicValNat p (x ^ n - y ^ n)) = \u2191(padicValNat p (x - y)) + \u2191(padicValNat p n)\n[PROOFSTEP]\niterate 3 rw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 \u2191(padicValNat p (x ^ n - y ^ n)) = \u2191(padicValNat p (x - y)) + \u2191(padicValNat p n)\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 multiplicity p (x ^ n - y ^ n) = \u2191(padicValNat p (x - y)) + \u2191(padicValNat p n)\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x ^ n - y ^ n\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + \u2191(padicValNat p n)\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x - y\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x - y\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x ^ n - y ^ n\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) + multiplicity p n\n[PROOFSTEP]\nexact multiplicity.Nat.pow_sub_pow hp.out hp1 hxy hx n\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < n\n[PROOFSTEP]\nexact hn\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x - y\n[PROOFSTEP]\nexact Nat.sub_pos_of_lt hyx\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhyx : y < x\nhxy : p \u2223 x - y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : 0 < n\n\u22a2 0 < x ^ n - y ^ n\n[PROOFSTEP]\nexact Nat.sub_pos_of_lt (Nat.pow_lt_pow_of_lt_left hyx hn)\n[GOAL]\nR : Type u_1\nn\u271d x y p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhxy : p \u2223 x + y\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\n\u22a2 padicValNat p (x ^ n + y ^ n) = padicValNat p (x + y) + padicValNat p n\n[PROOFSTEP]\ncases' y with y\n[GOAL]\ncase zero\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\nhxy : p \u2223 x + Nat.zero\n\u22a2 padicValNat p (x ^ n + Nat.zero ^ n) = padicValNat p (x + Nat.zero) + padicValNat p n\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 padicValNat p (x ^ n + Nat.succ y ^ n) = padicValNat p (x + Nat.succ y) + padicValNat p n\n[PROOFSTEP]\nrw [\u2190 PartENat.natCast_inj, Nat.cast_add]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 \u2191(padicValNat p (x ^ n + Nat.succ y ^ n)) = \u2191(padicValNat p (x + Nat.succ y)) + \u2191(padicValNat p n)\n[PROOFSTEP]\niterate 3 rw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 \u2191(padicValNat p (x ^ n + Nat.succ y ^ n)) = \u2191(padicValNat p (x + Nat.succ y)) + \u2191(padicValNat p n)\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 multiplicity p (x ^ n + Nat.succ y ^ n) = \u2191(padicValNat p (x + Nat.succ y)) + \u2191(padicValNat p n)\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x ^ n + Nat.succ y ^ n\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x ^ n + Nat.succ y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 multiplicity p (x ^ n + Nat.succ y ^ n) = multiplicity p (x + Nat.succ y) + \u2191(padicValNat p n)\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x + Nat.succ y\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x + Nat.succ y\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x ^ n + Nat.succ y ^ n\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x ^ n + Nat.succ y ^ n\n[PROOFSTEP]\nrw [padicValNat_def, PartENat.natCast_get]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 multiplicity p (x ^ n + Nat.succ y ^ n) = multiplicity p (x + Nat.succ y) + multiplicity p n\n[PROOFSTEP]\nexact multiplicity.Nat.pow_add_pow hp.out hp1 hxy hx hn\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < n\n[PROOFSTEP]\nexact Odd.pos hn\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x + Nat.succ y\n[PROOFSTEP]\nsimp only [add_pos_iff, Nat.succ_pos', or_true_iff]\n[GOAL]\ncase succ\nR : Type u_1\nn\u271d x p : \u2115\nhp : Fact (Nat.Prime p)\nhp1 : Odd p\nhx : \u00acp \u2223 x\nn : \u2115\nhn : Odd n\ny : \u2115\nhxy : p \u2223 x + Nat.succ y\n\u22a2 0 < x ^ n + Nat.succ y ^ n\n[PROOFSTEP]\nexact Nat.lt_add_left _ _ _ (pow_pos y.succ_pos _)\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Multiplicity", "llama_tokens": 45745, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718435083355188, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5162708817265025}}
{"text": "[GOAL]\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\n\u22a2 coprime n b\n[PROOFSTEP]\nby_cases h\u2083 : 2 \u2264 n\n[GOAL]\ncase pos\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\nh\u2083 : 2 \u2264 n\n\u22a2 coprime n b\n[PROOFSTEP]\napply Nat.coprime_of_dvd\n[GOAL]\ncase pos.H\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\nh\u2083 : 2 \u2264 n\n\u22a2 \u2200 (k : \u2115), Prime k \u2192 k \u2223 n \u2192 \u00ack \u2223 b\n[PROOFSTEP]\nrintro k hk \u27e8m, rfl\u27e9\n  \u27e8j, rfl\u27e9\n      -- Because prime numbers do not divide 1, it suffices to show that `k \u2223 1` to prove a\n          -- contradiction\n[GOAL]\ncase pos.H.intro.intro\nk : \u2115\nhk : Prime k\nm : \u2115\nh\u2081 : 1 \u2264 k * m\nh\u2083 : 2 \u2264 k * m\nj : \u2115\nh\u2082 : 1 \u2264 k * j\nh : ProbablePrime (k * m) (k * j)\n\u22a2 False\n[PROOFSTEP]\napply Nat.Prime.not_dvd_one hk\n[GOAL]\ncase pos.H.intro.intro\nk : \u2115\nhk : Prime k\nm : \u2115\nh\u2081 : 1 \u2264 k * m\nh\u2083 : 2 \u2264 k * m\nj : \u2115\nh\u2082 : 1 \u2264 k * j\nh : ProbablePrime (k * m) (k * j)\n\u22a2 k \u2223 1\n[PROOFSTEP]\nreplace h := dvd_of_mul_right_dvd h\n[GOAL]\ncase pos.H.intro.intro\nk : \u2115\nhk : Prime k\nm : \u2115\nh\u2081 : 1 \u2264 k * m\nh\u2083 : 2 \u2264 k * m\nj : \u2115\nh\u2082 : 1 \u2264 k * j\nh : k \u2223 (k * j) ^ (k * m - 1) - 1\n\u22a2 k \u2223 1\n[PROOFSTEP]\nrw [Nat.dvd_add_iff_right h, Nat.sub_add_cancel (Nat.one_le_pow _ _ h\u2082)]\n  -- Since `k` divides `b`, `k` also divides any power of `b` except `b ^ 0`. Therefore, it\n      -- suffices to show that `n - 1` isn't zero. However, we know that `n - 1` isn't zero because we\n      -- assumed `2 \u2264 n` when doing `by_cases`.\n[GOAL]\ncase pos.H.intro.intro\nk : \u2115\nhk : Prime k\nm : \u2115\nh\u2081 : 1 \u2264 k * m\nh\u2083 : 2 \u2264 k * m\nj : \u2115\nh\u2082 : 1 \u2264 k * j\nh : k \u2223 (k * j) ^ (k * m - 1) - 1\n\u22a2 k \u2223 (k * j) ^ (k * m - 1)\n[PROOFSTEP]\nrefine' dvd_of_mul_right_dvd (dvd_pow_self (k * j) _)\n[GOAL]\ncase pos.H.intro.intro\nk : \u2115\nhk : Prime k\nm : \u2115\nh\u2081 : 1 \u2264 k * m\nh\u2083 : 2 \u2264 k * m\nj : \u2115\nh\u2082 : 1 \u2264 k * j\nh : k \u2223 (k * j) ^ (k * m - 1) - 1\n\u22a2 k * m - 1 \u2260 0\n[PROOFSTEP]\nlinarith [tsub_pos_of_lt (one_lt_two.trans_le h\u2083)]\n  -- If `n = 1`, then it follows trivially that `n` is coprime with `b`.\n[GOAL]\ncase neg\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\nh\u2083 : \u00ac2 \u2264 n\n\u22a2 coprime n b\n[PROOFSTEP]\nrw [show n = 1 by linarith]\n[GOAL]\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\nh\u2083 : \u00ac2 \u2264 n\n\u22a2 n = 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nn b : \u2115\nh : ProbablePrime n b\nh\u2081 : 1 \u2264 n\nh\u2082 : 1 \u2264 b\nh\u2083 : \u00ac2 \u2264 n\n\u22a2 coprime 1 b\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn b : \u2115\nh : 1 \u2264 b\n\u22a2 ProbablePrime n b \u2194 b ^ (n - 1) \u2261 1 [MOD n]\n[PROOFSTEP]\nhave : 1 \u2264 b ^ (n - 1) :=\n  one_le_pow_of_one_le h\n    (n - 1)\n      -- For exact_mod_cast\n[GOAL]\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\n\u22a2 ProbablePrime n b \u2194 b ^ (n - 1) \u2261 1 [MOD n]\n[PROOFSTEP]\nrw [Nat.ModEq.comm]\n[GOAL]\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\n\u22a2 ProbablePrime n b \u2194 1 \u2261 b ^ (n - 1) [MOD n]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\n\u22a2 ProbablePrime n b \u2192 1 \u2261 b ^ (n - 1) [MOD n]\n[PROOFSTEP]\nintro h\u2081\n[GOAL]\ncase mp\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\nh\u2081 : ProbablePrime n b\n\u22a2 1 \u2261 b ^ (n - 1) [MOD n]\n[PROOFSTEP]\napply Nat.modEq_of_dvd\n[GOAL]\ncase mp.a\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\nh\u2081 : ProbablePrime n b\n\u22a2 \u2191n \u2223 \u2191(b ^ (n - 1)) - \u21911\n[PROOFSTEP]\nexact_mod_cast h\u2081\n[GOAL]\ncase mpr\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\n\u22a2 1 \u2261 b ^ (n - 1) [MOD n] \u2192 ProbablePrime n b\n[PROOFSTEP]\nintro h\u2081\n[GOAL]\ncase mpr\nn b : \u2115\nh : 1 \u2264 b\nthis : 1 \u2264 b ^ (n - 1)\nh\u2081 : 1 \u2261 b ^ (n - 1) [MOD n]\n\u22a2 ProbablePrime n b\n[PROOFSTEP]\nexact_mod_cast Nat.ModEq.dvd h\u2081\n[GOAL]\nn b : \u2115\nh : FermatPsp n b\nh\u2081 : 1 \u2264 b\n\u22a2 coprime n b\n[PROOFSTEP]\nrcases h with \u27e8hp, _, hn\u2082\u27e9\n[GOAL]\ncase intro.intro\nn b : \u2115\nh\u2081 : 1 \u2264 b\nhp : ProbablePrime n b\nleft\u271d : \u00acPrime n\nhn\u2082 : 1 < n\n\u22a2 coprime n b\n[PROOFSTEP]\nexact coprime_of_probablePrime hp (by linarith) h\u2081\n[GOAL]\nn b : \u2115\nh\u2081 : 1 \u2264 b\nhp : ProbablePrime n b\nleft\u271d : \u00acPrime n\nhn\u2082 : 1 < n\n\u22a2 1 \u2264 n\n[PROOFSTEP]\nlinarith\n[GOAL]\nn : \u2115\nh\u2081 : 1 < n\nh\u2082 : \u00acPrime n\n\u22a2 FermatPsp n 1\n[PROOFSTEP]\nrefine' \u27e8show n \u2223 1 ^ (n - 1) - 1 from _, h\u2082, h\u2081\u27e9\n[GOAL]\nn : \u2115\nh\u2081 : 1 < n\nh\u2082 : \u00acPrime n\n\u22a2 n \u2223 1 ^ (n - 1) - 1\n[PROOFSTEP]\nexact show 0 = 1 ^ (n - 1) - 1 by norm_num \u25b8 dvd_zero n\n[GOAL]\nn : \u2115\nh\u2081 : 1 < n\nh\u2082 : \u00acPrime n\n\u22a2 0 = 1 ^ (n - 1) - 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\n\u22a2 2 \u2264 (a ^ b - 1) / (a - 1)\n[PROOFSTEP]\nchange 1 < _\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\n\u22a2 1 < (a ^ b - 1) / (a - 1)\n[PROOFSTEP]\nhave h\u2081 : a - 1 \u2223 a ^ b - 1 := by simpa only [one_pow] using nat_sub_dvd_pow_sub_pow a 1 b\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\n\u22a2 a - 1 \u2223 a ^ b - 1\n[PROOFSTEP]\nsimpa only [one_pow] using nat_sub_dvd_pow_sub_pow a 1 b\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\nh\u2081 : a - 1 \u2223 a ^ b - 1\n\u22a2 1 < (a ^ b - 1) / (a - 1)\n[PROOFSTEP]\nrw [Nat.lt_div_iff_mul_lt h\u2081, mul_one, tsub_lt_tsub_iff_right (Nat.le_of_succ_le ha)]\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\nh\u2081 : a - 1 \u2223 a ^ b - 1\n\u22a2 a < a ^ b\n[PROOFSTEP]\nconvert pow_lt_pow (Nat.lt_of_succ_le ha) hb\n[GOAL]\ncase h.e'_3\na b : \u2115\nha : 2 \u2264 a\nhb : 2 \u2264 b\nh\u2081 : a - 1 \u2223 a ^ b - 1\n\u22a2 a = a ^ 1\n[PROOFSTEP]\nrw [pow_one]\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 < b\n\u22a2 2 \u2264 (a ^ b + 1) / (a + 1)\n[PROOFSTEP]\nrw [Nat.le_div_iff_mul_le (Nat.zero_lt_succ _)]\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 < b\n\u22a2 2 * succ a \u2264 a ^ b + 1\n[PROOFSTEP]\napply Nat.succ_le_succ\n[GOAL]\ncase a\na b : \u2115\nha : 2 \u2264 a\nhb : 2 < b\n\u22a2 Nat.mul 2 a + 1 \u2264 a ^ b\n[PROOFSTEP]\ncalc\n  2 * a + 1 \u2264 a ^ 2 * a := by nlinarith\n  _ = a ^ 3 := by rw [pow_succ a 2]\n  _ \u2264 a ^ b := pow_le_pow (Nat.le_of_succ_le ha) hb\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 < b\n\u22a2 2 * a + 1 \u2264 a ^ 2 * a\n[PROOFSTEP]\nnlinarith\n[GOAL]\na b : \u2115\nha : 2 \u2264 a\nhb : 2 < b\n\u22a2 a ^ 2 * a = a ^ 3\n[PROOFSTEP]\nrw [pow_succ a 2]\n[GOAL]\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\n\u22a2 (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nhave q\u2081 : b - 1 \u2223 b ^ p - 1 := by simpa only [one_pow] using nat_sub_dvd_pow_sub_pow b 1 p\n[GOAL]\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\n\u22a2 b - 1 \u2223 b ^ p - 1\n[PROOFSTEP]\nsimpa only [one_pow] using nat_sub_dvd_pow_sub_pow b 1 p\n[GOAL]\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\n\u22a2 (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nhave q\u2082 : b + 1 \u2223 b ^ p + 1 := by simpa only [one_pow] using hp.nat_add_dvd_pow_add_pow b 1\n[GOAL]\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\n\u22a2 b + 1 \u2223 b ^ p + 1\n[PROOFSTEP]\nsimpa only [one_pow] using hp.nat_add_dvd_pow_add_pow b 1\n[GOAL]\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\nq\u2082 : b + 1 \u2223 b ^ p + 1\n\u22a2 (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nconvert Nat.div_mul_div_comm q\u2081 q\u2082 using 2\n[GOAL]\ncase h.e'_3.h.e'_5\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\nq\u2082 : b + 1 \u2223 b ^ p + 1\n\u22a2 b ^ (2 * p) - 1 = (b ^ p - 1) * (b ^ p + 1)\n[PROOFSTEP]\nrw [mul_comm (_ - 1), \u2190 Nat.sq_sub_sq]\n[GOAL]\ncase h.e'_3.h.e'_6\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\nq\u2082 : b + 1 \u2223 b ^ p + 1\n\u22a2 b ^ 2 - 1 = (b - 1) * (b + 1)\n[PROOFSTEP]\nrw [mul_comm (_ - 1), \u2190 Nat.sq_sub_sq]\n[GOAL]\ncase h.e'_3.h.e'_5\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\nq\u2082 : b + 1 \u2223 b ^ p + 1\n\u22a2 b ^ (2 * p) - 1 = (b ^ p) ^ 2 - 1 ^ 2\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase h.e'_3.h.e'_6\nb p : \u2115\nx\u271d : 2 \u2264 b\nhp : Odd p\nq\u2081 : b - 1 \u2223 b ^ p - 1\nq\u2082 : b + 1 \u2223 b ^ p + 1\n\u22a2 b ^ 2 - 1 = b ^ 2 - 1 ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 b ^ (2 * p) - 1 - (b ^ 2 - 1) = b ^ (2 * p) - (1 + (b ^ 2 - 1))\n[PROOFSTEP]\nrw [Nat.sub_sub]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 b ^ (2 * p) - (1 + (b ^ 2 - 1)) = b ^ (2 * p) - (1 + b ^ 2 - 1)\n[PROOFSTEP]\nrw [Nat.add_sub_assoc hi_bsquared]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 b ^ (2 * p) - (1 + b ^ 2 - 1) = b ^ (2 * p) - b ^ 2\n[PROOFSTEP]\nrw [Nat.add_sub_cancel_left]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 b ^ (2 * p) - b ^ 2 = b ^ (p * 2) - b ^ 2\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 b ^ (p * 2) - b ^ 2 = (b ^ p) ^ 2 - b ^ 2\n[PROOFSTEP]\nrw [pow_mul]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b ^ p) ^ 2 - b ^ 2 = (b ^ p + b) * (b ^ p - b)\n[PROOFSTEP]\nrw [Nat.sq_sub_sq]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b ^ p + b) * (b ^ p - b) = (b ^ p - b) * (b ^ p + b)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b ^ p - b) * (b ^ p + b) = (b ^ (p - 1 + 1) - b) * (b ^ p + b)\n[PROOFSTEP]\nrw [Nat.sub_add_cancel hp]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b ^ (p - 1 + 1) - b) * (b ^ p + b) = (b * b ^ (p - 1) - b) * (b ^ p + b)\n[PROOFSTEP]\nrw [pow_succ']\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b * b ^ (p - 1) - b) * (b ^ p + b) = (b * b ^ (p - 1) - b * 1) * (b ^ p + b)\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\nb p : \u2115\nhb : 0 < b\nhp : 1 \u2264 p\nhi_bsquared : 1 \u2264 b ^ 2\n\u22a2 (b * b ^ (p - 1) - b * 1) * (b ^ p + b) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nrw [Nat.mul_sub_left_distrib]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\n\u22a2 FermatPsp (Nat.psp_from_prime b p) b\n[PROOFSTEP]\nunfold psp_from_prime\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\n\u22a2 FermatPsp ((b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1))) b\n[PROOFSTEP]\nset A := (b ^ p - 1) / (b - 1)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\n\u22a2 FermatPsp (A * ((b ^ p + 1) / (b + 1))) b\n[PROOFSTEP]\nset B :=\n  (b ^ p + 1) /\n    (b + 1)\n      -- Inequalities\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_A : 1 < A := a_id_helper (Nat.succ_le_iff.mp b_ge_two) (Nat.Prime.one_lt p_prime)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_B : 1 < B := b_id_helper (Nat.succ_le_iff.mp b_ge_two) p_gt_two\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_AB : 1 < A * B := one_lt_mul'' hi_A hi_B\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_b : 0 < b := by linarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\n\u22a2 0 < b\n[PROOFSTEP]\nlinarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_p : 1 \u2264 p := Nat.one_le_of_lt p_gt_two\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_bsquared : 0 < b ^ 2 - 1 := by\n  -- Porting note: was `by nlinarith [Nat.one_le_pow 2 b hi_b]`\n  have h0 := mul_le_mul b_ge_two b_ge_two zero_le_two hi_b.le\n  have h1 : 1 < 2 * 2 := by linarith\n  have := tsub_pos_of_lt (h1.trans_le h0)\n  rwa [pow_two]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\n\u22a2 0 < b ^ 2 - 1\n[PROOFSTEP]\nhave h0 := mul_le_mul b_ge_two b_ge_two zero_le_two hi_b.le\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nh0 : 2 * 2 \u2264 b * b\n\u22a2 0 < b ^ 2 - 1\n[PROOFSTEP]\nhave h1 : 1 < 2 * 2 := by linarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nh0 : 2 * 2 \u2264 b * b\n\u22a2 1 < 2 * 2\n[PROOFSTEP]\nlinarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nh0 : 2 * 2 \u2264 b * b\nh1 : 1 < 2 * 2\n\u22a2 0 < b ^ 2 - 1\n[PROOFSTEP]\nhave := tsub_pos_of_lt (h1.trans_le h0)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nh0 : 2 * 2 \u2264 b * b\nh1 : 1 < 2 * 2\nthis : 0 < b * b - 1\n\u22a2 0 < b ^ 2 - 1\n[PROOFSTEP]\nrwa [pow_two]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_bpowtwop : 1 \u2264 b ^ (2 * p) := Nat.one_le_pow (2 * p) b hi_b\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hi_bpowpsubone : 1 \u2264 b ^ (p - 1) := Nat.one_le_pow (p - 1) b hi_b\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave p_odd : Odd p := p_prime.odd_of_ne_two p_gt_two.ne.symm\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave AB_not_prime : \u00acNat.Prime (A * B) := Nat.not_prime_mul hi_A hi_B\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) := AB_id_helper _ _ b_ge_two p_odd\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nhave hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 := by simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\u22a2 b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n[PROOFSTEP]\nsimpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 FermatPsp (A * B) b\n[PROOFSTEP]\nrefine'\n  \u27e8_, AB_not_prime, hi_AB\u27e9\n    -- Used to prove that `2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)`.\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) :=\n  by\n  apply_fun fun x => x * (b ^ 2 - 1) at AB_id \n  rw [Nat.div_mul_cancel hd] at AB_id \n  apply_fun fun x => x - (b ^ 2 - 1) at AB_id \n  nth_rw 2 [\u2190 one_mul (b ^ 2 - 1)] at AB_id \n  rw [\u2190 Nat.mul_sub_right_distrib, mul_comm] at AB_id \n  rw [AB_id]\n  exact bp_helper hi_b hi_p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\napply_fun fun x => x * (b ^ 2 - 1) at AB_id \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1)\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nrw [Nat.div_mul_cancel hd] at AB_id \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : A * B * (b ^ 2 - 1) = b ^ (2 * p) - 1\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\napply_fun fun x => x - (b ^ 2 - 1) at AB_id \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : A * B * (b ^ 2 - 1) - (b ^ 2 - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1)\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_mul (b ^ 2 - 1)] at AB_id \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : A * B * (b ^ 2 - 1) - 1 * (b ^ 2 - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1)\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nrw [\u2190 Nat.mul_sub_right_distrib, mul_comm] at AB_id \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : (b ^ 2 - 1) * (A * B - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1)\n\u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nrw [AB_id]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nAB_id : (b ^ 2 - 1) * (A * B - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1)\n\u22a2 b ^ (2 * p) - 1 - (b ^ 2 - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nexact bp_helper hi_b hi_p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2082 : 2 \u2223 b ^ p + b := by\n  -- Porting note: golfedrw [\u2190 even_iff_two_dvd, Nat.even_add, Nat.even_pow' p_prime.ne_zero]\n    -- Since `b` isn't divisible by `p`, `b` is coprime with `p`. we can use Fermat's Little Theorem\n      -- to prove this.\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\n\u22a2 2 \u2223 b ^ p + b\n[PROOFSTEP]\nrw [\u2190 even_iff_two_dvd, Nat.even_add, Nat.even_pow' p_prime.ne_zero]\n  -- Since `b` isn't divisible by `p`, `b` is coprime with `p`. we can use Fermat's Little Theorem\n    -- to prove this.\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2083 : p \u2223 b ^ (p - 1) - 1 :=\n  by\n  have : \u00acp \u2223 b := mt (fun h : p \u2223 b => dvd_mul_of_dvd_left h _) not_dvd\n  have : p.coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this\n  have : IsCoprime (b : \u2124) \u2191p := this.symm.isCoprime\n  have : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this\n  have : \u2191p \u2223 \u2191b ^ (p - 1) - \u21911 := by exact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this)\n  exact_mod_cast this\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : \u00acp \u2223 b := mt (fun h : p \u2223 b => dvd_mul_of_dvd_left h _) not_dvd\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis : \u00acp \u2223 b\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : p.coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis\u271d : \u00acp \u2223 b\nthis : coprime p b\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : IsCoprime (b : \u2124) \u2191p := this.symm.isCoprime\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis\u271d\u00b9 : \u00acp \u2223 b\nthis\u271d : coprime p b\nthis : IsCoprime \u2191b \u2191p\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis\u271d\u00b2 : \u00acp \u2223 b\nthis\u271d\u00b9 : coprime p b\nthis\u271d : IsCoprime \u2191b \u2191p\nthis : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p]\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : \u2191p \u2223 \u2191b ^ (p - 1) - \u21911 := by exact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis\u271d\u00b2 : \u00acp \u2223 b\nthis\u271d\u00b9 : coprime p b\nthis\u271d : IsCoprime \u2191b \u2191p\nthis : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p]\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nexact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nthis\u271d\u00b3 : \u00acp \u2223 b\nthis\u271d\u00b2 : coprime p b\nthis\u271d\u00b9 : IsCoprime \u2191b \u2191p\nthis\u271d : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p]\nthis : p \u2223 b ^ (p - 1) - 1\n\u22a2 p \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 := by\n  cases' p_odd with k hk\n  have : 2 \u2223 p - 1 := \u27e8k, by simp [hk]\u27e9\n  cases' this with c hc\n  have : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 := by simpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c\n  have : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 := by rwa [\u2190 pow_mul] at this \n  rwa [\u2190 hc] at this \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\ncases' p_odd with k hk\n[GOAL]\ncase intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : 2 \u2223 p - 1 := \u27e8k, by simp [hk]\u27e9\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\n\u22a2 p - 1 = 2 * k\n[PROOFSTEP]\nsimp [hk]\n[GOAL]\ncase intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nthis : 2 \u2223 p - 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\ncases' this with c hc\n[GOAL]\ncase intro.intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nc : \u2115\nhc : p - 1 = 2 * c\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 := by simpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nc : \u2115\nhc : p - 1 = 2 * c\n\u22a2 b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1\n[PROOFSTEP]\nsimpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c\n[GOAL]\ncase intro.intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nc : \u2115\nhc : p - 1 = 2 * c\nthis : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nhave : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 := by rwa [\u2190 pow_mul] at this \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nc : \u2115\nhc : p - 1 = 2 * c\nthis : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (2 * c) - 1\n[PROOFSTEP]\nrwa [\u2190 pow_mul] at this \n[GOAL]\ncase intro.intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nk : \u2115\nhk : p = 2 * k + 1\nc : \u2115\nhc : p - 1 = 2 * c\nthis\u271d : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1\nthis : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1\n\u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n[PROOFSTEP]\nrwa [\u2190 hc] at this \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) :=\n  by\n  suffices q : 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n  \u00b7\n    rwa [ha\u2081]\n      -- We already proved that `b ^ 2 - 1 \u2223 b ^ (p - 1) - 1`.\n          -- Since `2 \u2223 b ^ p + b` and `p \u2223 b ^ p + b`, if we show that 2 and p are coprime, then we\n          -- know that `2 * p \u2223 b ^ p + b`\n  have q\u2081 : Nat.coprime p (b ^ 2 - 1) :=\n    haveI q\u2082 : \u00acp \u2223 b ^ 2 - 1 := by\n      rw [mul_comm] at not_dvd \n      exact mt (fun h : p \u2223 b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd\n    (Nat.Prime.coprime_iff_not_dvd p_prime).mpr q\u2082\n  have q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 := Nat.coprime.mul_dvd_of_dvd_of_dvd q\u2081 ha\u2083 ha\u2084\n  have q\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b) := mul_dvd_mul q\u2082 ha\u2082\n  have q\u2084 : p * (b ^ 2 - 1) * 2 \u2223 b * ((b ^ (p - 1) - 1) * (b ^ p + b)) := dvd_mul_of_dvd_right q\u2083 _\n  rwa [mul_assoc, mul_comm, mul_assoc b]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\n[PROOFSTEP]\nsuffices q : 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nq : 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\n[PROOFSTEP]\nrwa [ha\u2081]\n  -- We already proved that `b ^ 2 - 1 \u2223 b ^ (p - 1) - 1`.\n      -- Since `2 \u2223 b ^ p + b` and `p \u2223 b ^ p + b`, if we show that 2 and p are coprime, then we\n      -- know that `2 * p \u2223 b ^ p + b`\n[GOAL]\ncase q\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nhave q\u2081 : Nat.coprime p (b ^ 2 - 1) :=\n  haveI q\u2082 : \u00acp \u2223 b ^ 2 - 1 := by\n    rw [mul_comm] at not_dvd \n    exact mt (fun h : p \u2223 b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd\n  (Nat.Prime.coprime_iff_not_dvd p_prime).mpr q\u2082\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n\u22a2 \u00acp \u2223 b ^ 2 - 1\n[PROOFSTEP]\nrw [mul_comm] at not_dvd \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 (b ^ 2 - 1) * b\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\n\u22a2 \u00acp \u2223 b ^ 2 - 1\n[PROOFSTEP]\nexact mt (fun h : p \u2223 b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd\n[GOAL]\ncase q\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nq\u2081 : coprime p (b ^ 2 - 1)\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nhave q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 := Nat.coprime.mul_dvd_of_dvd_of_dvd q\u2081 ha\u2083 ha\u2084\n[GOAL]\ncase q\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nq\u2081 : coprime p (b ^ 2 - 1)\nq\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nhave q\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b) := mul_dvd_mul q\u2082 ha\u2082\n[GOAL]\ncase q\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nq\u2081 : coprime p (b ^ 2 - 1)\nq\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1\nq\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b)\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nhave q\u2084 : p * (b ^ 2 - 1) * 2 \u2223 b * ((b ^ (p - 1) - 1) * (b ^ p + b)) := dvd_mul_of_dvd_right q\u2083 _\n[GOAL]\ncase q\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nq\u2081 : coprime p (b ^ 2 - 1)\nq\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1\nq\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b)\nq\u2084 : p * (b ^ 2 - 1) * 2 \u2223 b * ((b ^ (p - 1) - 1) * (b ^ p + b))\n\u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b)\n[PROOFSTEP]\nrwa [mul_assoc, mul_comm, mul_assoc b]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2086 : 2 * p \u2223 A * B - 1 := by\n  rw [mul_comm] at ha\u2085 \n  exact Nat.dvd_of_mul_dvd_mul_left hi_bsquared ha\u2085\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\n\u22a2 2 * p \u2223 A * B - 1\n[PROOFSTEP]\nrw [mul_comm] at ha\u2085 \n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : (b ^ 2 - 1) * (2 * p) \u2223 (b ^ 2 - 1) * (A * B - 1)\n\u22a2 2 * p \u2223 A * B - 1\n[PROOFSTEP]\nexact Nat.dvd_of_mul_dvd_mul_left hi_bsquared ha\u2085\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2086 : 2 * p \u2223 A * B - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 := by\n  use b ^ 2 - 1\n  have : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) :=\n    congr_arg (fun x : \u2115 => x * (b ^ 2 - 1)) AB_id\n  simpa only [add_comm, Nat.div_mul_cancel hd, Nat.sub_add_cancel hi_bpowtwop] using this.symm\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2086 : 2 * p \u2223 A * B - 1\n\u22a2 A * B \u2223 b ^ (2 * p) - 1\n[PROOFSTEP]\nuse b ^ 2 - 1\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2086 : 2 * p \u2223 A * B - 1\n\u22a2 b ^ (2 * p) - 1 = A * B * (b ^ 2 - 1)\n[PROOFSTEP]\nhave : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) :=\n  congr_arg (fun x : \u2115 => x * (b ^ 2 - 1)) AB_id\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2086 : 2 * p \u2223 A * B - 1\nthis : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1)\n\u22a2 b ^ (2 * p) - 1 = A * B * (b ^ 2 - 1)\n[PROOFSTEP]\nsimpa only [add_comm, Nat.div_mul_cancel hd, Nat.sub_add_cancel hi_bpowtwop] using this.symm\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2086 : 2 * p \u2223 A * B - 1\nha\u2087 : A * B \u2223 b ^ (2 * p) - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\ncases' ha\u2086 with q hq\n[GOAL]\ncase intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2087 : A * B \u2223 b ^ (2 * p) - 1\nq : \u2115\nhq : A * B - 1 = 2 * p * q\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nhave ha\u2088 : b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1 := by\n  simpa only [one_pow, pow_mul, hq] using nat_sub_dvd_pow_sub_pow _ 1 q\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2087 : A * B \u2223 b ^ (2 * p) - 1\nq : \u2115\nhq : A * B - 1 = 2 * p * q\n\u22a2 b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1\n[PROOFSTEP]\nsimpa only [one_pow, pow_mul, hq] using nat_sub_dvd_pow_sub_pow _ 1 q\n[GOAL]\ncase intro\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : \u00acp \u2223 b * (b ^ 2 - 1)\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nhi_A : 1 < A\nhi_B : 1 < B\nhi_AB : 1 < A * B\nhi_b : 0 < b\nhi_p : 1 \u2264 p\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 \u2264 b ^ (2 * p)\nhi_bpowpsubone : 1 \u2264 b ^ (p - 1)\np_odd : Odd p\nAB_not_prime : \u00acPrime (A * B)\nAB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nhd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)\nha\u2082 : 2 \u2223 b ^ p + b\nha\u2083 : p \u2223 b ^ (p - 1) - 1\nha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1\nha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1)\nha\u2087 : A * B \u2223 b ^ (2 * p) - 1\nq : \u2115\nhq : A * B - 1 = 2 * p * q\nha\u2088 : b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1\n\u22a2 ProbablePrime (A * B) b\n[PROOFSTEP]\nexact dvd_trans ha\u2087 ha\u2088\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\n\u22a2 p < Nat.psp_from_prime b p\n[PROOFSTEP]\nunfold psp_from_prime\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\n\u22a2 p < (b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1))\n[PROOFSTEP]\nset A := (b ^ p - 1) / (b - 1)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\n\u22a2 p < A * ((b ^ p + 1) / (b + 1))\n[PROOFSTEP]\nset B := (b ^ p + 1) / (b + 1)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\n\u22a2 p < A * B\n[PROOFSTEP]\nrw [show A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) from\n    AB_id_helper _ _ b_ge_two (p_prime.odd_of_ne_two p_gt_two.ne.symm)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\n\u22a2 p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nhave AB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 := by simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\n\u22a2 b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n[PROOFSTEP]\nsimpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nsuffices h : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\n\u22a2 p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nhave h\u2081 : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1) := Nat.div_lt_div_of_lt_of_dvd AB_dvd h\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh\u2081 : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\u22a2 p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nhave h\u2082 : 0 < b ^ 2 - 1 := by linarith [show 3 \u2264 b ^ 2 - 1 from le_tsub_of_add_le_left (show 4 \u2264 b ^ 2 by nlinarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh\u2081 : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\u22a2 0 < b ^ 2 - 1\n[PROOFSTEP]\nlinarith [show 3 \u2264 b ^ 2 - 1 from le_tsub_of_add_le_left (show 4 \u2264 b ^ 2 by nlinarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh\u2081 : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n\u22a2 4 \u2264 b ^ 2\n[PROOFSTEP]\nnlinarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * (b ^ 2 - 1) < b ^ (2 * p) - 1\nh\u2081 : p * (b ^ 2 - 1) / (b ^ 2 - 1) < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\nh\u2082 : 0 < b ^ 2 - 1\n\u22a2 p < (b ^ (2 * p) - 1) / (b ^ 2 - 1)\n[PROOFSTEP]\nrwa [Nat.mul_div_cancel _ h\u2082] at h\u2081 \n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p * (b ^ 2 - 1) < b ^ (2 * p) - 1\n[PROOFSTEP]\nrw [Nat.mul_sub_left_distrib, mul_one, pow_mul]\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p * b ^ 2 - p < (b ^ 2) ^ p - 1\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Nat.sub_add_cancel (show 1 \u2264 p by linarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n| (b ^ 2) ^ p - 1\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel (show 1 \u2264 p by linarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n| (b ^ 2) ^ p - 1\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel (show 1 \u2264 p by linarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n| (b ^ 2) ^ p - 1\n[PROOFSTEP]\nrw [\u2190 Nat.sub_add_cancel (show 1 \u2264 p by linarith)]\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 1 \u2264 p\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p * b ^ 2 - p < (b ^ 2) ^ (p - 1 + 1) - 1\n[PROOFSTEP]\nrw [pow_succ (b ^ 2)]\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p * b ^ 2 - p < (b ^ 2) ^ (p - 1) * b ^ 2 - 1\n[PROOFSTEP]\nsuffices h : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n\u22a2 p * b ^ 2 - p < (b ^ 2) ^ (p - 1) * b ^ 2 - 1\n[PROOFSTEP]\napply gt_of_ge_of_gt\n[GOAL]\ncase h.h\u2081\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n\u22a2 (b ^ 2) ^ (p - 1) * b ^ 2 - 1 \u2265 ?h.b\n[PROOFSTEP]\nexact tsub_le_tsub_left (one_le_of_lt p_gt_two) ((b ^ 2) ^ (p - 1) * b ^ 2)\n[GOAL]\ncase h.h\u2082\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n\u22a2 (b ^ 2) ^ (p - 1) * b ^ 2 - p > p * b ^ 2 - p\n[PROOFSTEP]\nhave : p \u2264 p * b ^ 2 := Nat.le_mul_of_pos_right (show 0 < b ^ 2 by nlinarith)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n\u22a2 0 < b ^ 2\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase h.h\u2082\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\nthis : p \u2264 p * b ^ 2\n\u22a2 (b ^ 2) ^ (p - 1) * b ^ 2 - p > p * b ^ 2 - p\n[PROOFSTEP]\nexact tsub_lt_tsub_right_of_le this h\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n[PROOFSTEP]\nsuffices h : p < (b ^ 2) ^ (p - 1)\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n\u22a2 p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n[PROOFSTEP]\nhave : 4 \u2264 b ^ 2 := by nlinarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\n\u22a2 4 \u2264 b ^ 2\n[PROOFSTEP]\nnlinarith\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis : 4 \u2264 b ^ 2\n\u22a2 p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n[PROOFSTEP]\nhave : 0 < b ^ 2 := by linarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis : 4 \u2264 b ^ 2\n\u22a2 0 < b ^ 2\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nh : p < (b ^ 2) ^ (p - 1)\nthis\u271d : 4 \u2264 b ^ 2\nthis : 0 < b ^ 2\n\u22a2 p * b ^ 2 < (b ^ 2) ^ (p - 1) * b ^ 2\n[PROOFSTEP]\nexact mul_lt_mul_of_pos_right h this\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p < (b ^ 2) ^ (p - 1)\n[PROOFSTEP]\nrw [\u2190 pow_mul, Nat.mul_sub_left_distrib, mul_one]\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 p < b ^ (2 * p - 2)\n[PROOFSTEP]\nhave : 2 \u2264 2 * p - 2 := le_tsub_of_add_le_left (show 4 \u2264 2 * p by linarith)\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\n\u22a2 4 \u2264 2 * p\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nthis : 2 \u2264 2 * p - 2\n\u22a2 p < b ^ (2 * p - 2)\n[PROOFSTEP]\nhave : 2 + p \u2264 2 * p := by linarith\n[GOAL]\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nthis : 2 \u2264 2 * p - 2\n\u22a2 2 + p \u2264 2 * p\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nthis\u271d : 2 \u2264 2 * p - 2\nthis : 2 + p \u2264 2 * p\n\u22a2 p < b ^ (2 * p - 2)\n[PROOFSTEP]\nhave : p \u2264 2 * p - 2 := le_tsub_of_add_le_left this\n[GOAL]\ncase h\nb : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\np_prime : Prime p\np_gt_two : 2 < p\nA : \u2115 := (b ^ p - 1) / (b - 1)\nB : \u2115 := (b ^ p + 1) / (b + 1)\nAB_dvd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1\nthis\u271d\u00b9 : 2 \u2264 2 * p - 2\nthis\u271d : 2 + p \u2264 2 * p\nthis : p \u2264 2 * p - 2\n\u22a2 p < b ^ (2 * p - 2)\n[PROOFSTEP]\nexact Nat.lt_of_le_of_lt this (pow_gt_exponent _ b_ge_two)\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nby_cases b_ge_two : 2 \u2264 b\n[GOAL]\ncase pos\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h := Nat.exists_infinite_primes (b * (b ^ 2 - 1) + 1 + m)\n[GOAL]\ncase pos\nb : \u2115\nh\u271d : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\nh : \u2203 p, b * (b ^ 2 - 1) + 1 + m \u2264 p \u2227 Prime p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\ncases' h with p hp\n[GOAL]\ncase pos.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp : b * (b ^ 2 - 1) + 1 + m \u2264 p \u2227 Prime p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\ncases' hp with hp\u2081 hp\u2082\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2081 : 0 < b := pos_of_gt (Nat.succ_le_iff.mp b_ge_two)\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2082 : 4 \u2264 b ^ 2 := pow_le_pow_of_le_left' b_ge_two 2\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2083 : 0 < b ^ 2 - 1 := tsub_pos_of_lt (gt_of_ge_of_gt h\u2082 (by norm_num))\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\n\u22a2 4 > 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2084 : 0 < b * (b ^ 2 - 1) := mul_pos h\u2081 h\u2083\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2085 : b * (b ^ 2 - 1) < p := by linarith\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\n\u22a2 b * (b ^ 2 - 1) < p\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2086 : \u00acp \u2223 b * (b ^ 2 - 1) := Nat.not_dvd_of_pos_of_lt h\u2084 h\u2085\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2087 : b \u2264 b * (b ^ 2 - 1) := Nat.le_mul_of_pos_right h\u2083\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2088 : 2 \u2264 b * (b ^ 2 - 1) := le_trans b_ge_two h\u2087\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2089 : 2 < p := gt_of_gt_of_ge h\u2085 h\u2088\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2081\u2080 := psp_from_prime_gt_p b_ge_two hp\u2082 h\u2089\n[GOAL]\ncase pos.intro.intro\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\nh\u2081\u2080 : p < Nat.psp_from_prime b p\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nuse psp_from_prime b p\n[GOAL]\ncase h\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\nh\u2081\u2080 : p < Nat.psp_from_prime b p\n\u22a2 FermatPsp (Nat.psp_from_prime b p) b \u2227 m \u2264 Nat.psp_from_prime b p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\nh\u2081\u2080 : p < Nat.psp_from_prime b p\n\u22a2 FermatPsp (Nat.psp_from_prime b p) b\n[PROOFSTEP]\nexact psp_from_prime_psp b_ge_two hp\u2082 h\u2089 h\u2086\n[GOAL]\ncase h.right\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\nh\u2081\u2080 : p < Nat.psp_from_prime b p\n\u22a2 m \u2264 Nat.psp_from_prime b p\n[PROOFSTEP]\nexact\n  le_trans (show m \u2264 p by linarith)\n    (le_of_lt h\u2081\u2080)\n      -- If `\u00ac2 \u2264 b`, then `b = 1`. Since all composite numbers are pseudoprimes to base 1, we can pick\n        -- any composite number greater than m. We choose `2 * (m + 2)` because it is greater than `m` and\n        -- is composite for all natural numbers `m`.\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : 2 \u2264 b\np : \u2115\nhp\u2081 : b * (b ^ 2 - 1) + 1 + m \u2264 p\nhp\u2082 : Prime p\nh\u2081 : 0 < b\nh\u2082 : 4 \u2264 b ^ 2\nh\u2083 : 0 < b ^ 2 - 1\nh\u2084 : 0 < b * (b ^ 2 - 1)\nh\u2085 : b * (b ^ 2 - 1) < p\nh\u2086 : \u00acp \u2223 b * (b ^ 2 - 1)\nh\u2087 : b \u2264 b * (b ^ 2 - 1)\nh\u2088 : 2 \u2264 b * (b ^ 2 - 1)\nh\u2089 : 2 < p\nh\u2081\u2080 : p < Nat.psp_from_prime b p\n\u22a2 m \u2264 p\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nhave h\u2081 : b = 1 := by linarith\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\n\u22a2 b = 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase neg\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\n\u22a2 \u2203 n, FermatPsp n b \u2227 m \u2264 n\n[PROOFSTEP]\nrw [h\u2081]\n[GOAL]\ncase neg\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\n\u22a2 \u2203 n, FermatPsp n 1 \u2227 m \u2264 n\n[PROOFSTEP]\nuse 2 * (m + 2)\n[GOAL]\ncase h\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\n\u22a2 FermatPsp (2 * (m + 2)) 1 \u2227 m \u2264 2 * (m + 2)\n[PROOFSTEP]\nhave : \u00acNat.Prime (2 * (m + 2)) := Nat.not_prime_mul (by norm_num) (by norm_num)\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\n\u22a2 1 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\n\u22a2 1 < m + 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\nthis : \u00acPrime (2 * (m + 2))\n\u22a2 FermatPsp (2 * (m + 2)) 1 \u2227 m \u2264 2 * (m + 2)\n[PROOFSTEP]\nexact \u27e8fermatPsp_base_one (by linarith) this, by linarith\u27e9\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\nthis : \u00acPrime (2 * (m + 2))\n\u22a2 1 < 2 * (m + 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nm : \u2115\nb_ge_two : \u00ac2 \u2264 b\nh\u2081 : b = 1\nthis : \u00acPrime (2 * (m + 2))\n\u22a2 m \u2264 2 * (m + 2)\n[PROOFSTEP]\nlinarith\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\n\u22a2 \u2203\u1da0 (n : \u2115) in Filter.atTop, FermatPsp n b\n[PROOFSTEP]\nrefine' Filter.frequently_atTop.2 fun n => _\n[GOAL]\nb : \u2115\nh : 1 \u2264 b\nn : \u2115\n\u22a2 \u2203 b_1, b_1 \u2265 n \u2227 FermatPsp b_1 b\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := exists_infinite_pseudoprimes h n\n[GOAL]\ncase intro\nb : \u2115\nh : 1 \u2264 b\nn p : \u2115\nhp : FermatPsp p b \u2227 n \u2264 p\n\u22a2 \u2203 b_1, b_1 \u2265 n \u2227 FermatPsp b_1 b\n[PROOFSTEP]\nexact \u27e8p, hp.2, hp.1\u27e9\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.FermatPsp", "llama_tokens": 46117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8244619091240701, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5162155345063847}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\n\u22a2 join [l] = l\n[PROOFSTEP]\nrw [join, join, append_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nL : List (List \u03b1)\n\u22a2 join (l :: L) = [] \u2194 \u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 l_1 = []\n[PROOFSTEP]\nsimp only [join, append_eq_nil, join_eq_nil, forall_mem_cons]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL\u2081 L\u2082 : List (List \u03b1)\n\u22a2 join (L\u2081 ++ L\u2082) = join L\u2081 ++ join L\u2082\n[PROOFSTEP]\ninduction L\u2081\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL\u2082 : List (List \u03b1)\n\u22a2 join ([] ++ L\u2082) = join [] ++ join L\u2082\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL\u2082 : List (List \u03b1)\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : join (tail\u271d ++ L\u2082) = join tail\u271d ++ join L\u2082\n\u22a2 join (head\u271d :: tail\u271d ++ L\u2082) = join (head\u271d :: tail\u271d) ++ join L\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nl : List \u03b1\n\u22a2 join (concat L l) = join L ++ l\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidablePred fun l => isEmpty l = false\nL : List (List \u03b1)\n\u22a2 join (filter (fun l => decide (isEmpty l = false)) ([] :: L)) = join ([] :: L)\n[PROOFSTEP]\nsimp [join_filter_isEmpty_eq_false (L := L), isEmpty_iff_eq_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidablePred fun l => isEmpty l = false\na : \u03b1\nl : List \u03b1\nL : List (List \u03b1)\n\u22a2 join (filter (fun l => decide (isEmpty l = false)) ((a :: l) :: L)) = join ((a :: l) :: L)\n[PROOFSTEP]\nhave cons_not_empty : isEmpty (a :: l) = false := rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidablePred fun l => isEmpty l = false\na : \u03b1\nl : List \u03b1\nL : List (List \u03b1)\ncons_not_empty : isEmpty (a :: l) = false\n\u22a2 join (filter (fun l => decide (isEmpty l = false)) ((a :: l) :: L)) = join ((a :: l) :: L)\n[PROOFSTEP]\nsimp [join_filter_isEmpty_eq_false (L := L), cons_not_empty]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : DecidablePred fun l => l \u2260 []\nL : List (List \u03b1)\n\u22a2 join (filter (fun l => decide (l \u2260 [])) L) = join L\n[PROOFSTEP]\nsimp [join_filter_isEmpty_eq_false, \u2190 isEmpty_iff_eq_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List (List (List \u03b1))\n\u22a2 join (join l) = join (map join l)\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 join (join []) = join (map join [])\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List (List \u03b1)\ntail\u271d : List (List (List \u03b1))\ntail_ih\u271d : join (join tail\u271d) = join (map join tail\u271d)\n\u22a2 join (join (head\u271d :: tail\u271d)) = join (map join (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\n\u22a2 length (join L) = sum (map length L)\n[PROOFSTEP]\ninduction L <;> [rfl; simp only [*, join, map, sum_cons, length_append]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\n\u22a2 length (join L) = sum (map length L)\n[PROOFSTEP]\ninduction L\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 length (join []) = sum (map length [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : length (join tail\u271d) = sum (map length tail\u271d)\n\u22a2 length (join (head\u271d :: tail\u271d)) = sum (map length (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp only [*, join, map, sum_cons, length_append]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 length (List.bind l f) = sum (map (length \u2218 f) l)\n[PROOFSTEP]\nrw [List.bind, length_join, map_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 (\u2200 (l_1 : List \u03b2), l_1 \u2208 map f l \u2192 l_1 = []) \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 f x = []\n[PROOFSTEP]\nsimp only [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : \u2115\n\u22a2 take (sum (take i (map length L))) (join L) = join (take i L)\n[PROOFSTEP]\ninduction L generalizing i\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ni : \u2115\n\u22a2 take (sum (take i (map length []))) (join []) = join (take i [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), take (sum (take i (map length tail\u271d))) (join tail\u271d) = join (take i tail\u271d)\ni : \u2115\n\u22a2 take (sum (take i (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) = join (take i (head\u271d :: tail\u271d))\n[PROOFSTEP]\ncases i\n[GOAL]\ncase cons.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), take (sum (take i (map length tail\u271d))) (join tail\u271d) = join (take i tail\u271d)\n\u22a2 take (sum (take Nat.zero (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) =\n    join (take Nat.zero (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [take_append, *]\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), take (sum (take i (map length tail\u271d))) (join tail\u271d) = join (take i tail\u271d)\nn\u271d : \u2115\n\u22a2 take (sum (take (Nat.succ n\u271d) (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) =\n    join (take (Nat.succ n\u271d) (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [take_append, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : \u2115\n\u22a2 drop (sum (take i (map length L))) (join L) = join (drop i L)\n[PROOFSTEP]\ninduction L generalizing i\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ni : \u2115\n\u22a2 drop (sum (take i (map length []))) (join []) = join (drop i [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), drop (sum (take i (map length tail\u271d))) (join tail\u271d) = join (drop i tail\u271d)\ni : \u2115\n\u22a2 drop (sum (take i (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) = join (drop i (head\u271d :: tail\u271d))\n[PROOFSTEP]\ncases i\n[GOAL]\ncase cons.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), drop (sum (take i (map length tail\u271d))) (join tail\u271d) = join (drop i tail\u271d)\n\u22a2 drop (sum (take Nat.zero (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) =\n    join (drop Nat.zero (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [drop_append, *]\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\ntail_ih\u271d : \u2200 (i : \u2115), drop (sum (take i (map length tail\u271d))) (join tail\u271d) = join (drop i tail\u271d)\nn\u271d : \u2115\n\u22a2 drop (sum (take (Nat.succ n\u271d) (map length (head\u271d :: tail\u271d)))) (join (head\u271d :: tail\u271d)) =\n    join (drop (Nat.succ n\u271d) (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp [drop_append, *]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni : Fin (length L)\n\u22a2 drop (\u2191i) (take (\u2191i + 1) L) = [get L i]\n[PROOFSTEP]\ninduction' L with head tail ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : Fin (length L)\ni : Fin (length [])\n\u22a2 drop (\u2191i) (take (\u2191i + 1) []) = [get [] i]\n[PROOFSTEP]\nexact (Nat.not_succ_le_zero i i.isLt).elim\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : Fin (length L)\nhead : \u03b1\ntail : List \u03b1\nih : \u2200 (i : Fin (length tail)), drop (\u2191i) (take (\u2191i + 1) tail) = [get tail i]\ni : Fin (length (head :: tail))\n\u22a2 drop (\u2191i) (take (\u2191i + 1) (head :: tail)) = [get (head :: tail) i]\n[PROOFSTEP]\nrcases i with \u27e8_ | i, hi\u27e9\n[GOAL]\ncase cons.mk.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni : Fin (length L)\nhead : \u03b1\ntail : List \u03b1\nih : \u2200 (i : Fin (length tail)), drop (\u2191i) (take (\u2191i + 1) tail) = [get tail i]\nhi : Nat.zero < length (head :: tail)\n\u22a2 drop (\u2191{ val := Nat.zero, isLt := hi }) (take (\u2191{ val := Nat.zero, isLt := hi } + 1) (head :: tail)) =\n    [get (head :: tail) { val := Nat.zero, isLt := hi }]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : Fin (length L)\nhead : \u03b1\ntail : List \u03b1\nih : \u2200 (i : Fin (length tail)), drop (\u2191i) (take (\u2191i + 1) tail) = [get tail i]\ni : \u2115\nhi : Nat.succ i < length (head :: tail)\n\u22a2 drop (\u2191{ val := Nat.succ i, isLt := hi }) (take (\u2191{ val := Nat.succ i, isLt := hi } + 1) (head :: tail)) =\n    [get (head :: tail) { val := Nat.succ i, isLt := hi }]\n[PROOFSTEP]\nsimpa using ih \u27e8i, Nat.lt_of_succ_lt_succ hi\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni : \u2115\nhi : i < length L\n\u22a2 drop i (take (i + 1) L) = [nthLe L i hi]\n[PROOFSTEP]\ninduction' L with head tail generalizing i\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\ni : \u2115\nhi : i < length []\n\u22a2 drop i (take (i + 1) []) = [nthLe [] i hi]\n[PROOFSTEP]\nsimp only [length] at hi \n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\ni : \u2115\nhi : i < 0\n\u22a2 drop i (take (i + 1) []) = [nthLe [] i hi]\n[PROOFSTEP]\nexact (Nat.not_succ_le_zero i hi).elim\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : i < length (head :: tail)\n\u22a2 drop i (take (i + 1) (head :: tail)) = [nthLe (head :: tail) i hi]\n[PROOFSTEP]\ncases' i with i hi\n[GOAL]\ncase cons.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni : \u2115\nhi\u271d : i < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\nhi : Nat.zero < length (head :: tail)\n\u22a2 drop Nat.zero (take (Nat.zero + 1) (head :: tail)) = [nthLe (head :: tail) Nat.zero hi]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni : \u2115\nhi\u271d : i < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\nhi : Nat.zero < length (head :: tail)\n\u22a2 head = nthLe (head :: tail) 0 hi\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : Nat.succ i < length (head :: tail)\n\u22a2 drop (Nat.succ i) (take (Nat.succ i + 1) (head :: tail)) = [nthLe (head :: tail) (Nat.succ i) hi]\n[PROOFSTEP]\nhave : i < tail.length := by\n  simp at hi \n  exact Nat.lt_of_succ_lt_succ hi\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : Nat.succ i < length (head :: tail)\n\u22a2 i < length tail\n[PROOFSTEP]\nsimp at hi \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : Nat.succ i < Nat.succ (length tail)\n\u22a2 i < length tail\n[PROOFSTEP]\nexact Nat.lt_of_succ_lt_succ hi\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : Nat.succ i < length (head :: tail)\nthis : i < length tail\n\u22a2 drop (Nat.succ i) (take (Nat.succ i + 1) (head :: tail)) = [nthLe (head :: tail) (Nat.succ i) hi]\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase cons.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List \u03b1\ni\u271d : \u2115\nhi\u271d : i\u271d < length L\nhead : \u03b1\ntail : List \u03b1\ntail_ih\u271d : \u2200 {i : \u2115} (hi : i < length tail), drop i (take (i + 1) tail) = [nthLe tail i hi]\ni : \u2115\nhi : Nat.succ i < length (head :: tail)\nthis : i < length tail\n\u22a2 nthLe tail i (_ : i < length tail) = nthLe (head :: tail) (Nat.succ i) hi\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : Fin (length L)\n\u22a2 drop (sum (take (\u2191i) (map length L))) (take (sum (take (\u2191i + 1) (map length L))) (join L)) = get L i\n[PROOFSTEP]\nhave : (L.map length).take i = ((L.take (i + 1)).map length).take i := by simp [map_take, take_take]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : Fin (length L)\n\u22a2 take (\u2191i) (map length L) = take (\u2191i) (map length (take (\u2191i + 1) L))\n[PROOFSTEP]\nsimp [map_take, take_take]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : Fin (length L)\nthis : take (\u2191i) (map length L) = take (\u2191i) (map length (take (\u2191i + 1) L))\n\u22a2 drop (sum (take (\u2191i) (map length L))) (take (sum (take (\u2191i + 1) (map length L))) (join L)) = get L i\n[PROOFSTEP]\nsimp only [this, length_map, take_sum_join, drop_sum_join, drop_take_succ_eq_cons_get, join, append_nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : \u2115\nhi : i < length L\n\u22a2 drop (sum (take i (map length L))) (take (sum (take (i + 1) (map length L))) (join L)) = nthLe L i hi\n[PROOFSTEP]\nhave : (L.map length).take i = ((L.take (i + 1)).map length).take i := by simp [map_take, take_take]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : \u2115\nhi : i < length L\n\u22a2 take i (map length L) = take i (map length (take (i + 1) L))\n[PROOFSTEP]\nsimp [map_take, take_take]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni : \u2115\nhi : i < length L\nthis : take i (map length L) = take i (map length (take (i + 1) L))\n\u22a2 drop (sum (take i (map length L))) (take (sum (take (i + 1) (map length L))) (join L)) = nthLe L i hi\n[PROOFSTEP]\nsimp [take_sum_join, this, drop_sum_join, drop_take_succ_eq_cons_nthLe _ hi]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\n\u22a2 sum (take i (map length L)) + j < sum (take (i + 1) (map length L))\n[PROOFSTEP]\nsimp [hi, sum_take_succ, hj]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\n\u22a2 sum (take i (map length L)) + j < length (join L)\n[PROOFSTEP]\nconvert lt_of_lt_of_le (sum_take_map_length_lt1 L hi hj) (monotone_sum_take _ hi)\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\n\u22a2 length (join L) = (fun i => sum (take i (map length L))) (length L)\n[PROOFSTEP]\nhave : L.length = (L.map length).length := by simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\n\u22a2 length L = length (map length L)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\nthis : length L = length (map length L)\n\u22a2 length (join L) = (fun i => sum (take i (map length L))) (length L)\n[PROOFSTEP]\nsimp [this, -length_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\n\u22a2 nthLe (join L) (sum (take i (map length L)) + j) (_ : sum (take i (map length L)) + j < length (join L)) =\n    nthLe (nthLe L i hi) j hj\n[PROOFSTEP]\nhave := nthLe_take L.join (sum_take_map_length_lt2 L hi hj) (sum_take_map_length_lt1 L hi hj)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\ni j : \u2115\nhi : i < length L\nhj : j < length (nthLe L i hi)\nthis :\n  nthLe (join L) (sum (take i (map length L)) + j) (_ : sum (take i (map length L)) + j < length (join L)) =\n    nthLe (take (sum (take (i + 1) (map length L))) (join L)) (sum (take i (map length L)) + j)\n      (_ : sum (take i (map length L)) + j < length (take (sum (take (i + 1) (map length L))) (join L)))\n\u22a2 nthLe (join L) (sum (take i (map length L)) + j) (_ : sum (take i (map length L)) + j < length (join L)) =\n    nthLe (nthLe L i hi) j hj\n[PROOFSTEP]\nrw [this, nthLe_drop, nthLe_of_eq (drop_take_succ_join_eq_nthLe L hi)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\n\u22a2 L = L' \u2194 join L = join L' \u2227 map length L = map length L'\n[PROOFSTEP]\nrefine' \u27e8fun H => by simp [H], _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\nH : L = L'\n\u22a2 join L = join L' \u2227 map length L = map length L'\n[PROOFSTEP]\nsimp [H]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\n\u22a2 join L = join L' \u2227 map length L = map length L' \u2192 L = L'\n[PROOFSTEP]\nrintro \u27e8join_eq, length_eq\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\n\u22a2 L = L'\n[PROOFSTEP]\napply ext_get\n[GOAL]\ncase intro.hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\n\u22a2 length L = length L'\n[PROOFSTEP]\nhave : length (map length L) = length (map length L') := by rw [length_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\n\u22a2 length (map length L) = length (map length L')\n[PROOFSTEP]\nrw [length_eq]\n[GOAL]\ncase intro.hl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\nthis : length (map length L) = length (map length L')\n\u22a2 length L = length L'\n[PROOFSTEP]\nsimpa using this\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\n\u22a2 \u2200 (n : \u2115) (h\u2081 : n < length L) (h\u2082 : n < length L'), get L { val := n, isLt := h\u2081 } = get L' { val := n, isLt := h\u2082 }\n[PROOFSTEP]\nintro n h\u2081 h\u2082\n[GOAL]\ncase intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL L' : List (List \u03b1)\njoin_eq : join L = join L'\nlength_eq : map length L = map length L'\nn : \u2115\nh\u2081 : n < length L\nh\u2082 : n < length L'\n\u22a2 get L { val := n, isLt := h\u2081 } = get L' { val := n, isLt := h\u2082 }\n[PROOFSTEP]\nrw [\u2190 drop_take_succ_join_eq_get, \u2190 drop_take_succ_join_eq_get, join_eq, length_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nh : L \u2260 []\n\u22a2 join (drop (length L - 1) L) = getLast L h\n[PROOFSTEP]\ninduction L using List.reverseRecOn\n[GOAL]\ncase H0\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nh : [] \u2260 []\n\u22a2 join (drop (length [] - 1) []) = getLast [] h\n[PROOFSTEP]\ncases h rfl\n[GOAL]\ncase H1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d : List (List \u03b1)\na\u271d\u00b9 : List \u03b1\na\u271d : \u2200 (h : l\u271d \u2260 []), join (drop (length l\u271d - 1) l\u271d) = getLast l\u271d h\nh : l\u271d ++ [a\u271d\u00b9] \u2260 []\n\u22a2 join (drop (length (l\u271d ++ [a\u271d\u00b9]) - 1) (l\u271d ++ [a\u271d\u00b9])) = getLast (l\u271d ++ [a\u271d\u00b9]) h\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\nx : List \u03b1\n\u22a2 x ++ join (map (fun l => l ++ x) L) = join (map (fun l => x ++ l) L) ++ x\n[PROOFSTEP]\ninduction' L with _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : List \u03b1\n\u22a2 x ++ join (map (fun l => l ++ x) []) = join (map (fun l => x ++ l) []) ++ x\n[PROOFSTEP]\nrw [map_nil, join, append_nil, map_nil, join, nil_append]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx head\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\nih : x ++ join (map (fun l => l ++ x) tail\u271d) = join (map (fun l => x ++ l) tail\u271d) ++ x\n\u22a2 x ++ join (map (fun l => l ++ x) (head\u271d :: tail\u271d)) = join (map (fun l => x ++ l) (head\u271d :: tail\u271d)) ++ x\n[PROOFSTEP]\nrw [map_cons, join, map_cons, join, append_assoc, ih, append_assoc, append_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\n\u22a2 reverse (join L) = join (reverse (map reverse L))\n[PROOFSTEP]\ninduction' L with _ _ ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u22a2 reverse (join []) = join (reverse (map reverse []))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhead\u271d : List \u03b1\ntail\u271d : List (List \u03b1)\nih : reverse (join tail\u271d) = join (reverse (map reverse tail\u271d))\n\u22a2 reverse (join (head\u271d :: tail\u271d)) = join (reverse (map reverse (head\u271d :: tail\u271d)))\n[PROOFSTEP]\nrw [join, reverse_append, ih, map_cons, reverse_cons', join_concat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nL : List (List \u03b1)\n\u22a2 join (reverse L) = reverse (join (map reverse L))\n[PROOFSTEP]\nsimpa [reverse_reverse, map_reverse] using congr_arg List.reverse (reverse_join L.reverse)\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Join", "llama_tokens": 8648, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879991, "lm_q2_score": 0.7279754489059774, "lm_q1q2_score": 0.5161485132481478}}
{"text": "[GOAL]\n\u22a2 SuccOrder (Fin 0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase le_succ\n\u22a2 \u2200 (a : Fin 0), a \u2264 ?succ a\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase le_succ\n\u22a2 \u2200 (a : Fin 0), a \u2264 ?succ a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase le_succ\n\u22a2 \u2200 (a : Fin 0), a \u2264 ?succ a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase le_succ\na : Fin 0\n\u22a2 a \u2264 ?succ a\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase max_of_succ_le\n\u22a2 \u2200 {a : Fin 0}, ?succ a \u2264 a \u2192 IsMax a\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase max_of_succ_le\n\u22a2 \u2200 {a : Fin 0}, ?succ a \u2264 a \u2192 IsMax a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase max_of_succ_le\n\u22a2 \u2200 {a : Fin 0}, ?succ a \u2264 a \u2192 IsMax a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase max_of_succ_le\na : Fin 0\n\u22a2 ?succ a \u2264 a \u2192 IsMax a\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase succ_le_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 ?succ a \u2264 b\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase succ_le_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 ?succ a \u2264 b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ_le_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 ?succ a \u2264 b\n[PROOFSTEP]\nintro a\n[GOAL]\ncase succ_le_of_lt\na : Fin 0\n\u22a2 \u2200 {b : Fin 0}, a < b \u2192 ?succ a \u2264 b\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase le_of_lt_succ\n\u22a2 \u2200 {a b : Fin 0}, a < ?succ b \u2192 a \u2264 b\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase le_of_lt_succ\n\u22a2 \u2200 {a b : Fin 0}, a < ?succ b \u2192 a \u2264 b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase le_of_lt_succ\n\u22a2 \u2200 {a b : Fin 0}, a < ?succ b \u2192 a \u2264 b\n[PROOFSTEP]\nintro a\n[GOAL]\ncase le_of_lt_succ\na : Fin 0\n\u22a2 \u2200 {b : Fin 0}, a < ?succ b \u2192 a \u2264 b\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase succ\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase succ\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase succ\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nintro a\n[GOAL]\ncase succ\na : Fin 0\n\u22a2 Fin 0\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {a : Fin (n + 1)}, \u00acIsMax a \u2192 \u2200 (b : Fin (n + 1)), a < b \u2194 (fun i => if i < last n then i + 1 else i) a \u2264 b\n[PROOFSTEP]\nintro a ha b\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha : \u00acIsMax a\nb : Fin (n + 1)\n\u22a2 a < b \u2194 (fun i => if i < last n then i + 1 else i) a \u2264 b\n[PROOFSTEP]\nrw [isMax_iff_eq_top, eq_top_iff, not_le, top_eq_last] at ha \n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMax a\nha : a < last n\nb : Fin (n + 1)\n\u22a2 a < b \u2194 (fun i => if i < last n then i + 1 else i) a \u2264 b\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMax a\nha : a < last n\nb : Fin (n + 1)\n\u22a2 a < b \u2194 (if a < last n then a + 1 else a) \u2264 b\n[PROOFSTEP]\nrw [if_pos ha, lt_iff_val_lt_val, le_iff_val_le_val, val_add_one_of_lt ha]\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMax a\nha : a < last n\nb : Fin (n + 1)\n\u22a2 \u2191a < \u2191b \u2194 \u2191a + 1 \u2264 \u2191b\n[PROOFSTEP]\nexact Nat.lt_iff_add_one_le\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : Fin (n + 1)), IsMax a \u2192 (fun i => if i < last n then i + 1 else i) a = a\n[PROOFSTEP]\nintro a ha\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha : IsMax a\n\u22a2 (fun i => if i < last n then i + 1 else i) a = a\n[PROOFSTEP]\nrw [isMax_iff_eq_top, top_eq_last] at ha \n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : IsMax a\nha : a = last n\n\u22a2 (fun i => if i < last n then i + 1 else i) a = a\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : IsMax a\nha : a = last n\n\u22a2 (if a < last n then a + 1 else a) = a\n[PROOFSTEP]\nrw [if_neg ha.not_lt]\n[GOAL]\n\u22a2 PredOrder (Fin 0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase pred_le\n\u22a2 \u2200 (a : Fin 0), ?pred a \u2264 a\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase pred_le\n\u22a2 \u2200 (a : Fin 0), ?pred a \u2264 a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pred_le\n\u22a2 \u2200 (a : Fin 0), ?pred a \u2264 a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase pred_le\na : Fin 0\n\u22a2 ?pred a \u2264 a\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase min_of_le_pred\n\u22a2 \u2200 {a : Fin 0}, a \u2264 ?pred a \u2192 IsMin a\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase min_of_le_pred\n\u22a2 \u2200 {a : Fin 0}, a \u2264 ?pred a \u2192 IsMin a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase min_of_le_pred\n\u22a2 \u2200 {a : Fin 0}, a \u2264 ?pred a \u2192 IsMin a\n[PROOFSTEP]\nintro a\n[GOAL]\ncase min_of_le_pred\na : Fin 0\n\u22a2 a \u2264 ?pred a \u2192 IsMin a\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase le_pred_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 a \u2264 ?pred b\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase le_pred_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 a \u2264 ?pred b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase le_pred_of_lt\n\u22a2 \u2200 {a b : Fin 0}, a < b \u2192 a \u2264 ?pred b\n[PROOFSTEP]\nintro a\n[GOAL]\ncase le_pred_of_lt\na : Fin 0\n\u22a2 \u2200 {b : Fin 0}, a < b \u2192 a \u2264 ?pred b\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase le_of_pred_lt\n\u22a2 \u2200 {a b : Fin 0}, ?pred a < b \u2192 a \u2264 b\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase le_of_pred_lt\n\u22a2 \u2200 {a b : Fin 0}, ?pred a < b \u2192 a \u2264 b\n[PROOFSTEP]\nassumption\n[GOAL]\ncase le_of_pred_lt\n\u22a2 \u2200 {a b : Fin 0}, ?pred a < b \u2192 a \u2264 b\n[PROOFSTEP]\nintro a\n[GOAL]\ncase le_of_pred_lt\na : Fin 0\n\u22a2 \u2200 {b : Fin 0}, ?pred a < b \u2192 a \u2264 b\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\ncase pred\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nfirst\n| assumption\n| intro a; exact elim0 a\n[GOAL]\ncase pred\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase pred\n\u22a2 Fin 0 \u2192 Fin 0\n[PROOFSTEP]\nintro a\n[GOAL]\ncase pred\na : Fin 0\n\u22a2 Fin 0\n[PROOFSTEP]\nexact elim0 a\n[GOAL]\nn : \u2115\n\u22a2 \u2200 {a : Fin (n + 1)}, \u00acIsMin a \u2192 \u2200 (b : Fin (n + 1)), b \u2264 (fun x => if x = 0 then 0 else x - 1) a \u2194 b < a\n[PROOFSTEP]\nintro a ha b\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha : \u00acIsMin a\nb : Fin (n + 1)\n\u22a2 b \u2264 (fun x => if x = 0 then 0 else x - 1) a \u2194 b < a\n[PROOFSTEP]\nrw [isMin_iff_eq_bot, eq_bot_iff, not_le, bot_eq_zero] at ha \n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMin a\nha : 0 < a\nb : Fin (n + 1)\n\u22a2 b \u2264 (fun x => if x = 0 then 0 else x - 1) a \u2194 b < a\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMin a\nha : 0 < a\nb : Fin (n + 1)\n\u22a2 (b \u2264 if a = 0 then 0 else a - 1) \u2194 b < a\n[PROOFSTEP]\nrw [if_neg ha.ne', lt_iff_val_lt_val, le_iff_val_le_val, coe_sub_one, if_neg ha.ne', le_tsub_iff_right, Iff.comm]\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMin a\nha : 0 < a\nb : Fin (n + 1)\n\u22a2 \u2191b < \u2191a \u2194 \u2191b + 1 \u2264 \u2191a\nn : \u2115 a : Fin (n + 1) ha\u271d : \u00acIsMin a ha : 0 < a b : Fin (n + 1) \u22a2 1 \u2264 \u2191a\n[PROOFSTEP]\nexact Nat.lt_iff_add_one_le\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : \u00acIsMin a\nha : 0 < a\nb : Fin (n + 1)\n\u22a2 1 \u2264 \u2191a\n[PROOFSTEP]\nexact ha\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : Fin (n + 1)), IsMin a \u2192 (fun x => if x = 0 then 0 else x - 1) a = a\n[PROOFSTEP]\nintro a ha\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha : IsMin a\n\u22a2 (fun x => if x = 0 then 0 else x - 1) a = a\n[PROOFSTEP]\nrw [isMin_iff_eq_bot, bot_eq_zero] at ha \n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : IsMin a\nha : a = 0\n\u22a2 (fun x => if x = 0 then 0 else x - 1) a = a\n[PROOFSTEP]\ndsimp\n[GOAL]\nn : \u2115\na : Fin (n + 1)\nha\u271d : IsMin a\nha : a = 0\n\u22a2 (if a = 0 then 0 else a - 1) = a\n[PROOFSTEP]\nrwa [if_pos ha, eq_comm]\n", "meta": {"mathlib_filename": "Mathlib.Data.Fin.SuccPred", "llama_tokens": 3491, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.5160119856590055}}
{"text": "[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\n\u22a2 IsometryEquiv (weightedSumSquares \u211d w) (weightedSumSquares \u211d (sign \u2218 w))\n[PROOFSTEP]\nlet u i := if h : w i = 0 then (1 : \u211d\u02e3) else Units.mk0 (w i) h\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\n\u22a2 IsometryEquiv (weightedSumSquares \u211d w) (weightedSumSquares \u211d (sign \u2218 w))\n[PROOFSTEP]\nhave hu' : \u2200 i : \u03b9, (Real.sign (u i) * u i) ^ (-(1 / 2 : \u211d)) \u2260 0 :=\n  by\n  intro i\n  refine' (ne_of_lt (Real.rpow_pos_of_pos (sign_mul_pos_of_ne_zero _ <| Units.ne_zero _) _)).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\n\u22a2 \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\ni : \u03b9\n\u22a2 (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\n[PROOFSTEP]\nrefine' (ne_of_lt (Real.rpow_pos_of_pos (sign_mul_pos_of_ne_zero _ <| Units.ne_zero _) _)).symm\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\n\u22a2 IsometryEquiv (weightedSumSquares \u211d w) (weightedSumSquares \u211d (sign \u2218 w))\n[PROOFSTEP]\nconvert\n  (weightedSumSquares \u211d w).isometryEquivBasisRepr\n    ((Pi.basisFun \u211d \u03b9).unitsSMul fun i => (isUnit_iff_ne_zero.2 <| hu' i).unit)\n[GOAL]\ncase h.e'_10\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\n\u22a2 weightedSumSquares \u211d (sign \u2218 w) =\n    basisRepr (weightedSumSquares \u211d w)\n      (Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i => IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\n\u22a2 \u2191(weightedSumSquares \u211d (sign \u2218 w)) v =\n    \u2191(basisRepr (weightedSumSquares \u211d w)\n          (Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i => IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))))\n      v\n[PROOFSTEP]\nrw [basisRepr_apply, weightedSumSquares_apply, weightedSumSquares_apply]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\n\u22a2 \u2211 i : \u03b9, (sign \u2218 w) i \u2022 (v i * v i) =\n    \u2211 i : \u03b9,\n      w i \u2022\n        (Finset.sum univ\n            (fun i =>\n              v i \u2022\n                \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                      IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                  i)\n            i *\n          Finset.sum univ\n            (fun i =>\n              v i \u2022\n                \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                      IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                  i)\n            i)\n[PROOFSTEP]\nrefine' sum_congr rfl fun j hj => _\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 (sign \u2218 w) j \u2022 (v j * v j) =\n    w j \u2022\n      (Finset.sum univ\n          (fun i =>\n            v i \u2022\n              \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                    IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                i)\n          j *\n        Finset.sum univ\n          (fun i =>\n            v i \u2022\n              \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                    IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                i)\n          j)\n[PROOFSTEP]\nhave hsum :\n  (\u2211 i : \u03b9, v i \u2022 ((isUnit_iff_ne_zero.2 <| hu' i).unit : \u211d) \u2022 (Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (Real.sign (u j) * u j) ^ (-(1 / 2 : \u211d)) :=\n  by\n  rw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec, LinearMap.stdBasis_apply, Pi.smul_apply,\n    Pi.smul_apply, Function.update_same, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_one]\n  intro i _ hij\n  rw [Pi.basisFun_apply, LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_noteq hij.symm,\n    Pi.zero_apply, smul_eq_mul, smul_eq_mul, mul_zero, mul_zero]\n  intro hj'; exact False.elim (hj' hj)\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\n[PROOFSTEP]\nrw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec, LinearMap.stdBasis_apply, Pi.smul_apply,\n  Pi.smul_apply, Function.update_same, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_one]\n[GOAL]\ncase h\u2080\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u2200 (b : \u03b9),\n    b \u2208 univ \u2192\n      b \u2260 j \u2192 (v b \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u b) * \u2191(u b)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) b) j = 0\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) j) j = 0\n[PROOFSTEP]\nintro i _ hij\n[GOAL]\ncase h\u2080\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\ni : \u03b9\na\u271d : i \u2208 univ\nhij : i \u2260 j\n\u22a2 (v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j = 0\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) j) j = 0\n[PROOFSTEP]\nrw [Pi.basisFun_apply, LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_noteq hij.symm,\n  Pi.zero_apply, smul_eq_mul, smul_eq_mul, mul_zero, mul_zero]\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\n\u22a2 \u00acj \u2208 univ \u2192 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) j) j = 0\n[PROOFSTEP]\nintro hj'\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhj' : \u00acj \u2208 univ\n\u22a2 (v j \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) j) j = 0\n[PROOFSTEP]\nexact False.elim (hj' hj)\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\n\u22a2 (sign \u2218 w) j \u2022 (v j * v j) =\n    w j \u2022\n      (Finset.sum univ\n          (fun i =>\n            v i \u2022\n              \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                    IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                i)\n          j *\n        Finset.sum univ\n          (fun i =>\n            v i \u2022\n              \u2191(Basis.unitsSMul (Pi.basisFun \u211d \u03b9) fun i =>\n                    IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)))))\n                i)\n          j)\n[PROOFSTEP]\nsimp_rw [Basis.unitsSMul_apply]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\n\u22a2 (sign \u2218 w) j \u2022 (v j * v j) =\n    w j \u2022\n      (Finset.sum univ\n          (fun x => v x \u2022 IsUnit.unit (_ : IsUnit ((sign \u2191(u x) * \u2191(u x)) ^ (-(1 / 2)))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) x) j *\n        Finset.sum univ\n          (fun x => v x \u2022 IsUnit.unit (_ : IsUnit ((sign \u2191(u x) * \u2191(u x)) ^ (-(1 / 2)))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) x) j)\n[PROOFSTEP]\nerw [hsum]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\n\u22a2 (sign \u2218 w) j \u2022 (v j * v j) =\n    w j \u2022 (v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)))\n[PROOFSTEP]\nsimp only [Function.comp, smul_eq_mul]\n[GOAL]\ncase h.e'_10.H\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\n\u22a2 sign (w j) * (v j * v j) =\n    w j *\n      (v j *\n          (sign \u2191(if h : w j = 0 then 1 else Units.mk0 (w j) h) * \u2191(if h : w j = 0 then 1 else Units.mk0 (w j) h)) ^\n            (-(1 / 2)) *\n        (v j *\n          (sign \u2191(if h : w j = 0 then 1 else Units.mk0 (w j) h) * \u2191(if h : w j = 0 then 1 else Units.mk0 (w j) h)) ^\n            (-(1 / 2))))\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : w j = 0\n\u22a2 sign (w j) * (v j * v j) = w j * (v j * (sign \u21911 * \u21911) ^ (-(1 / 2)) * (v j * (sign \u21911 * \u21911) ^ (-(1 / 2))))\n[PROOFSTEP]\nsimp only [h, zero_smul, zero_mul, Real.sign_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\n\u22a2 sign (w j) * (v j * v j) =\n    w j *\n      (v j * (sign \u2191(Units.mk0 (w j) h) * \u2191(Units.mk0 (w j) h)) ^ (-(1 / 2)) *\n        (v j * (sign \u2191(Units.mk0 (w j) h) * \u2191(Units.mk0 (w j) h)) ^ (-(1 / 2))))\n[PROOFSTEP]\nhave hwu : w j = u j := by simp only [dif_neg h, Units.val_mk0]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\n\u22a2 w j = \u2191(u j)\n[PROOFSTEP]\nsimp only [dif_neg h, Units.val_mk0]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\n\u22a2 sign (w j) * (v j * v j) =\n    w j *\n      (v j * (sign \u2191(Units.mk0 (w j) h) * \u2191(Units.mk0 (w j) h)) ^ (-(1 / 2)) *\n        (v j * (sign \u2191(Units.mk0 (w j) h) * \u2191(Units.mk0 (w j) h)) ^ (-(1 / 2))))\n[PROOFSTEP]\nsimp only [Units.val_mk0]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\n\u22a2 sign (w j) * (v j * v j) = w j * (v j * (sign (w j) * w j) ^ (-(1 / 2)) * (v j * (sign (w j) * w j) ^ (-(1 / 2))))\n[PROOFSTEP]\nrw [hwu]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\n\u22a2 sign \u2191(u j) * (v j * v j) =\n    \u2191(u j) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))\n[PROOFSTEP]\nsuffices\n  (u j : \u211d).sign * v j * v j =\n    (Real.sign (u j) * u j) ^ (-(1 / 2 : \u211d)) * (Real.sign (u j) * u j) ^ (-(1 / 2 : \u211d)) * u j * v j * v j\n  by erw [\u2190 mul_assoc, this]; ring\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\nthis :\n  sign \u2191(u j) * v j * v j =\n    (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * \u2191(u j) * v j * v j\n\u22a2 sign \u2191(u j) * (v j * v j) =\n    \u2191(u j) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))\n[PROOFSTEP]\nerw [\u2190 mul_assoc, this]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\nthis :\n  sign \u2191(u j) * v j * v j =\n    (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * \u2191(u j) * v j * v j\n\u22a2 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * \u2191(u j) * v j * v j =\n    \u2191(u j) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (v j * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))))\n[PROOFSTEP]\nring\n[GOAL]\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\n\u22a2 sign \u2191(u j) * v j * v j =\n    (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2)) * \u2191(u j) * v j * v j\n[PROOFSTEP]\nrw [\u2190 Real.rpow_add (sign_mul_pos_of_ne_zero _ <| Units.ne_zero _), show -(1 / 2 : \u211d) + -(1 / 2) = -1 by ring,\n  Real.rpow_neg_one, mul_inv, inv_sign, mul_assoc (Real.sign (u j)) (u j)\u207b\u00b9, inv_mul_cancel (Units.ne_zero _), mul_one]\n[GOAL]\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nw : \u03b9 \u2192 \u211d\nu : \u03b9 \u2192 \u211d\u02e3 := fun i => if h : w i = 0 then 1 else Units.mk0 (w i) h\nhu' : \u2200 (i : \u03b9), (sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2)) \u2260 0\nv : \u03b9 \u2192 \u211d\nj : \u03b9\nhj : j \u2208 univ\nhsum :\n  Finset.sum univ\n      (fun i => v i \u2022 \u2191(IsUnit.unit (_ : IsUnit ((sign \u2191(u i) * \u2191(u i)) ^ (-(1 / 2))))) \u2022 \u2191(Pi.basisFun \u211d \u03b9) i) j =\n    v j \u2022 (sign \u2191(u j) * \u2191(u j)) ^ (-(1 / 2))\nh : \u00acw j = 0\nhwu : w j = \u2191(u j)\n\u22a2 -(1 / 2) + -(1 / 2) = -1\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.QuadraticForm.Real", "llama_tokens": 9686, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.787931190663057, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.5160119836131236}}
{"text": "[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\n\u22a2 F \u2192\u2097[R] E b\n[PROOFSTEP]\nrefine' IsLinearMap.mk' (e.symm b) _\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\n\u22a2 IsLinearMap R (Pretrivialization.symm e b)\n[PROOFSTEP]\nby_cases hb : b \u2208 e.baseSet\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\n\u22a2 IsLinearMap R (Pretrivialization.symm e b)\n[PROOFSTEP]\nexact\n  (((e.linear R hb).mk' _).inverse (e.symm b) (e.symm_apply_apply_mk hb) fun v \u21a6\n      congr_arg Prod.snd <| e.apply_mk_symm hb v).isLinear\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : \u00acb \u2208 e.baseSet\n\u22a2 IsLinearMap R (Pretrivialization.symm e b)\n[PROOFSTEP]\nrw [e.coe_symm_of_not_mem hb]\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : \u00acb \u2208 e.baseSet\n\u22a2 IsLinearMap R 0\n[PROOFSTEP]\nexact (0 : F \u2192\u2097[R] E b).isLinear\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nv : F\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun y => (\u2191e { proj := b, snd := y }).snd,\n              map_add' :=\n                (_ :\n                  \u2200 (v w : E b),\n                    (\u2191e { proj := b, snd := v + w }).snd =\n                      (\u2191e { proj := b, snd := v }).snd + (\u2191e { proj := b, snd := w }).snd) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (v : E b),\n                (\u2191e { proj := b, snd := c \u2022 v }).snd = c \u2022 (\u2191e { proj := b, snd := v }).snd) }.toAddHom\n      (Pretrivialization.symm e b v) =\n    v\n[PROOFSTEP]\nsimp_rw [e.apply_mk_symm hb v]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\n\u22a2 \u2191(Pretrivialization.linearMapAt R e b) = fun y => if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).snd else 0\n[PROOFSTEP]\nrw [Pretrivialization.linearMapAt]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\n\u22a2 \u2191(if hb : b \u2208 e.baseSet then \u2191(linearEquivAt R e b hb) else 0) = fun y =>\n    if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).snd else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nh\u271d : b \u2208 e.baseSet\n\u22a2 \u2191\u2191(linearEquivAt R e b h\u271d) = fun y => (\u2191e { proj := b, snd := y }).snd\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nh\u271d : \u00acb \u2208 e.baseSet\n\u22a2 \u21910 = fun y => 0\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\n\u22a2 \u2191(Pretrivialization.linearMapAt R e b) = fun y => (\u2191e { proj := b, snd := y }).snd\n[PROOFSTEP]\nsimp_rw [coe_linearMapAt, if_pos hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\ny : E b\n\u22a2 \u2191(Pretrivialization.linearMapAt R e b) y = if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).snd else 0\n[PROOFSTEP]\nrw [coe_linearMapAt]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\ny : E b\n\u22a2 \u2191(Pretrivialization.symm\u2097 R e b) (\u2191(Pretrivialization.linearMapAt R e b) y) = y\n[PROOFSTEP]\nrw [e.linearMapAt_def_of_mem hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\ny : E b\n\u22a2 \u2191(Pretrivialization.symm\u2097 R e b) (\u2191\u2191(linearEquivAt R e b hb) y) = y\n[PROOFSTEP]\nexact (e.linearEquivAt R b hb).left_inv y\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\ny : F\n\u22a2 \u2191(Pretrivialization.linearMapAt R e b) (\u2191(Pretrivialization.symm\u2097 R e b) y) = y\n[PROOFSTEP]\nrw [e.linearMapAt_def_of_mem hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : TopologicalSpace F\ninst\u271d\u2075 : TopologicalSpace B\ne\u271d : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ninst\u271d : Pretrivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\ny : F\n\u22a2 \u2191\u2191(linearEquivAt R e b hb) (\u2191(Pretrivialization.symm\u2097 R e b) y) = y\n[PROOFSTEP]\nexact (e.linearEquivAt R b hb).right_inv y\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\n\u22a2 \u2191(Trivialization.linearMapAt R e b) = fun y => (\u2191e { proj := b, snd := y }).snd\n[PROOFSTEP]\nsimp_rw [coe_linearMapAt, if_pos hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : TopologicalSpace F\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2074 : AddCommMonoid F\ninst\u271d\u00b3 : Module R F\ninst\u271d\u00b2 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b9 : (x : B) \u2192 Module R (E x)\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\ny : E b\n\u22a2 \u2191(Trivialization.linearMapAt R e b) y = if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).snd else 0\n[PROOFSTEP]\nrw [coe_linearMapAt]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\n\u22a2 Continuous\n    (\u2191(if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n            LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n              (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n          else LinearEquiv.refl R F)).toAddHom.toFun\n[PROOFSTEP]\nby_cases hb : b \u2208 e.baseSet \u2229 e'.baseSet\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (\u2191(if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n            LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n              (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n          else LinearEquiv.refl R F)).toAddHom.toFun\n[PROOFSTEP]\nrw [dif_pos hb]\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (\u2191(LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n            (linearEquivAt R e' b (_ : b \u2208 e'.baseSet)))).toAddHom.toFun\n[PROOFSTEP]\nrefine' (e'.continuousOn.comp_continuous _ _).snd\n[GOAL]\ncase pos.refine'_1\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous fun a => { proj := b, snd := \u2191\u2191(LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet))) a }\n[PROOFSTEP]\nexact e.continuousOn_symm.comp_continuous (Continuous.Prod.mk b) fun y => mk_mem_prod hb.1 (mem_univ y)\n[GOAL]\ncase pos.refine'_2\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2200 (x : F), { proj := b, snd := \u2191\u2191(LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet))) x } \u2208 e'.source\n[PROOFSTEP]\nexact fun y => e'.mem_source.mpr hb.2\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : \u00acb \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (\u2191(if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n            LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n              (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n          else LinearEquiv.refl R F)).toAddHom.toFun\n[PROOFSTEP]\nrw [dif_neg hb]\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : \u00acb \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous (\u2191(LinearEquiv.refl R F)).toAddHom.toFun\n[PROOFSTEP]\nexact continuous_id\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\n\u22a2 Continuous\n    (if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n        LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n          (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n      else LinearEquiv.refl R F).invFun\n[PROOFSTEP]\nby_cases hb : b \u2208 e.baseSet \u2229 e'.baseSet\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n        LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n          (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n      else LinearEquiv.refl R F).invFun\n[PROOFSTEP]\nrw [dif_pos hb]\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n        (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))).invFun\n[PROOFSTEP]\nrefine' (e.continuousOn.comp_continuous _ _).snd\n[GOAL]\ncase pos.refine'_1\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous fun a => { proj := b, snd := \u2191(LinearEquiv.toEquiv (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))).symm a }\ncase pos.refine'_2\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2200 (x : F), { proj := b, snd := \u2191(LinearEquiv.toEquiv (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))).symm x } \u2208 e.source\n[PROOFSTEP]\nexact e'.continuousOn_symm.comp_continuous (Continuous.Prod.mk b) fun y => mk_mem_prod hb.2 (mem_univ y)\n[GOAL]\ncase pos.refine'_2\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2200 (x : F), { proj := b, snd := \u2191(LinearEquiv.toEquiv (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))).symm x } \u2208 e.source\n[PROOFSTEP]\nexact fun y => e.mem_source.mpr hb.1\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : \u00acb \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous\n    (if hb : b \u2208 e.baseSet \u2229 e'.baseSet then\n        LinearEquiv.trans (LinearEquiv.symm (linearEquivAt R e b (_ : b \u2208 e.baseSet)))\n          (linearEquivAt R e' b (_ : b \u2208 e'.baseSet))\n      else LinearEquiv.refl R F).invFun\n[PROOFSTEP]\nrw [dif_neg hb]\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : \u00acb \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 Continuous (LinearEquiv.refl R F).invFun\n[PROOFSTEP]\nexact continuous_id\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e'.baseSet \u2229 e.baseSet\n\u22a2 ContinuousLinearEquiv.symm (coordChangeL R e e' b) = coordChangeL R e' e b\n[PROOFSTEP]\napply ContinuousLinearEquiv.toLinearEquiv_injective\n[GOAL]\ncase a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e'.baseSet \u2229 e.baseSet\n\u22a2 (ContinuousLinearEquiv.symm (coordChangeL R e e' b)).toLinearEquiv = (coordChangeL R e' e b).toLinearEquiv\n[PROOFSTEP]\nrw [coe_coordChangeL' e' e hb, (coordChangeL R e e' b).symm_toLinearEquiv, coe_coordChangeL' e e' hb.symm,\n  LinearEquiv.trans_symm, LinearEquiv.symm_symm]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 (b, \u2191(coordChangeL R e e' b) y) = \u2191e' { proj := b, snd := Trivialization.symm e b y }\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 (b, \u2191(coordChangeL R e e' b) y).fst = (\u2191e' { proj := b, snd := Trivialization.symm e b y }).fst\n[PROOFSTEP]\nrw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y)).proj \u2208 e'.baseSet\n[PROOFSTEP]\nrw [e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 b \u2208 e'.baseSet\n[PROOFSTEP]\nexact hb.2\n[GOAL]\ncase h\u2082\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 (b, \u2191(coordChangeL R e e' b) y).snd = (\u2191e' { proj := b, snd := Trivialization.symm e b y }).snd\n[PROOFSTEP]\nexact e.coordChangeL_apply e' hb y\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191e' (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, v)) = (b, \u2191(coordChangeL R e e' b) v)\n[PROOFSTEP]\nrw [e.mk_coordChangeL e' hb, e.mk_symm hb.1]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : Semiring R\ninst\u271d\u2078 : TopologicalSpace F\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F TotalSpace.proj\nx : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d\u2075 : AddCommMonoid F\ninst\u271d\u2074 : Module R F\ninst\u271d\u00b3 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2 : (x : B) \u2192 Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\ny : F\n\u22a2 \u2191(coordChangeL R e e' b) y = (\u2191e' (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y))).snd\n[PROOFSTEP]\nrw [e.coordChangeL_apply e' hb, e.mk_symm hb.1]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : E b \u2192\u2097[R] F := Trivialization.linearMapAt R e b\n\u22a2 Continuous\n    {\n          toAddHom :=\n            { toFun := \u2191(Trivialization.linearMapAt R e b),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : E b),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : E b),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : E b \u2192\u2097[R] F := Trivialization.linearMapAt R e b\n\u22a2 Continuous \u2191(Trivialization.linearMapAt R e b)\n[PROOFSTEP]\nrw [e.coe_linearMapAt b]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : E b \u2192\u2097[R] F := Trivialization.linearMapAt R e b\n\u22a2 Continuous fun y => if b \u2208 e.baseSet then (\u2191e { proj := b, snd := y }).snd else 0\n[PROOFSTEP]\nrefine' continuous_if_const _ (fun hb => _) fun _ => continuous_zero\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : E b \u2192\u2097[R] F := Trivialization.linearMapAt R e b\nhb : b \u2208 e.baseSet\n\u22a2 Continuous fun y => (\u2191e { proj := b, snd := y }).snd\n[PROOFSTEP]\nexact\n  (e.continuousOn.comp_continuous (FiberBundle.totalSpaceMk_inducing F E b).continuous fun x => e.mem_source.mpr hb).snd\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : F \u2192\u2097[R] E b := Trivialization.symm\u2097 R e b\n\u22a2 Continuous\n    {\n          toAddHom :=\n            { toFun := Trivialization.symm e b,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : F),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : F),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nby_cases hb : b \u2208 e.baseSet\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : F \u2192\u2097[R] E b := Trivialization.symm\u2097 R e b\nhb : b \u2208 e.baseSet\n\u22a2 Continuous\n    {\n          toAddHom :=\n            { toFun := Trivialization.symm e b,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : F),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : F),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nrw [(FiberBundle.totalSpaceMk_inducing F E b).continuous_iff]\n[GOAL]\ncase pos\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : F \u2192\u2097[R] E b := Trivialization.symm\u2097 R e b\nhb : b \u2208 e.baseSet\n\u22a2 Continuous\n    (TotalSpace.mk b \u2218\n      {\n            toAddHom :=\n              { toFun := Trivialization.symm e b,\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : F),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : F),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) =\n                    \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun)\n[PROOFSTEP]\nexact e.continuousOn_symm.comp_continuous (continuous_const.prod_mk continuous_id) fun x \u21a6 mk_mem_prod hb (mem_univ x)\n[GOAL]\ncase neg\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nsrc\u271d : F \u2192\u2097[R] E b := Trivialization.symm\u2097 R e b\nhb : \u00acb \u2208 e.baseSet\n\u22a2 Continuous\n    {\n          toAddHom :=\n            { toFun := Trivialization.symm e b,\n              map_add' :=\n                (_ :\n                  \u2200 (x y : F),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : F),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom.toFun\n[PROOFSTEP]\nrefine' continuous_zero.congr fun x => (e.symm_apply_of_not_mem hb x).symm\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : E b\n\u22a2 \u2191e { proj := b, snd := z } = (b, \u2191(continuousLinearEquivAt R e b hb) z)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : E b\n\u22a2 (\u2191e { proj := b, snd := z }).fst = (b, \u2191(continuousLinearEquivAt R e b hb) z).fst\n[PROOFSTEP]\nrefine' e.coe_fst _\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : E b\n\u22a2 { proj := b, snd := z } \u2208 e.source\n[PROOFSTEP]\nrw [e.source_eq]\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : E b\n\u22a2 { proj := b, snd := z } \u2208 TotalSpace.proj \u207b\u00b9' e.baseSet\n[PROOFSTEP]\nexact hb\n[GOAL]\ncase h\u2082\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : E b\n\u22a2 (\u2191e { proj := b, snd := z }).snd = (b, \u2191(continuousLinearEquivAt R e b hb) z).snd\n[PROOFSTEP]\nsimp only [coe_coe, continuousLinearEquivAt_apply]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nx : B\nhx : x \u2208 e.baseSet\n\u22a2 \u2191e (zeroSection F E x) = (x, 0)\n[PROOFSTEP]\nsimp_rw [zeroSection, e.apply_eq_prod_continuousLinearEquivAt R x hx 0, map_zero]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\n\u22a2 \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) =\n    { proj := b, snd := \u2191(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z }\n[PROOFSTEP]\nhave h : (b, z) \u2208 e.target\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\n\u22a2 (b, z) \u2208 e.target\n[PROOFSTEP]\nrw [e.target_eq]\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\n\u22a2 (b, z) \u2208 e.baseSet \u00d7\u02e2 univ\n[PROOFSTEP]\nexact \u27e8hb, mem_univ _\u27e9\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\nh : (b, z) \u2208 e.target\n\u22a2 \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) =\n    { proj := b, snd := \u2191(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z }\n[PROOFSTEP]\napply e.injOn (e.map_target h)\n[GOAL]\ncase a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\nh : (b, z) \u2208 e.target\n\u22a2 { proj := b, snd := \u2191(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z } \u2208 e.source\n[PROOFSTEP]\nsimpa only [e.source_eq, mem_preimage]\n[GOAL]\ncase a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2079 : NontriviallyNormedField R\ninst\u271d\u2078 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2077 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace R F\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b2 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b9 : FiberBundle F E\ne : Trivialization F TotalSpace.proj\ninst\u271d : Trivialization.IsLinear R e\nb : B\nhb : b \u2208 e.baseSet\nz : F\nh : (b, z) \u2208 e.target\n\u22a2 \u2191e.toLocalHomeomorph (\u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z)) =\n    \u2191e.toLocalHomeomorph { proj := b, snd := \u2191(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z }\n[PROOFSTEP]\nsimp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuousLinearEquivAt R b hb, ContinuousLinearEquiv.apply_symm_apply]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b9\u2070 : NontriviallyNormedField R\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace R F\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b \u2208 e.baseSet)))\n      (continuousLinearEquivAt R e' b (_ : b \u2208 e'.baseSet)) =\n    coordChangeL R e e' b\n[PROOFSTEP]\next v\n[GOAL]\ncase h.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b9\u2070 : NontriviallyNormedField R\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace R F\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b \u2208 e.baseSet)))\n          (continuousLinearEquivAt R e' b (_ : b \u2208 e'.baseSet)))\n      v =\n    \u2191(coordChangeL R e e' b) v\n[PROOFSTEP]\nrw [coordChangeL_apply e e' hb]\n[GOAL]\ncase h.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b9\u2070 : NontriviallyNormedField R\ninst\u271d\u2079 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2078 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace R F\ninst\u271d\u2075 : TopologicalSpace B\ninst\u271d\u2074 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b3 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d\u00b2 : FiberBundle F E\ne e' : Trivialization F TotalSpace.proj\ninst\u271d\u00b9 : Trivialization.IsLinear R e\ninst\u271d : Trivialization.IsLinear R e'\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191(ContinuousLinearEquiv.trans (ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b (_ : b \u2208 e.baseSet)))\n          (continuousLinearEquivAt R e' b (_ : b \u2208 e'.baseSet)))\n      v =\n    (\u2191e' { proj := b, snd := Trivialization.symm e b v }).snd\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\ni j k : \u03b9\nx : B\nhx : x \u2208 baseSet Z i \u2229 baseSet Z j \u2229 baseSet Z k\n\u22a2 ContinuousLinearMap.comp (coordChange Z j k x) (coordChange Z i j x) = coordChange Z i k x\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\ni j k : \u03b9\nx : B\nhx : x \u2208 baseSet Z i \u2229 baseSet Z j \u2229 baseSet Z k\nv : F\n\u22a2 \u2191(ContinuousLinearMap.comp (coordChange Z j k x) (coordChange Z i j x)) v = \u2191(coordChange Z i k x) v\n[PROOFSTEP]\nexact Z.coordChange_comp i j k x hx v\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni : \u03b9\nx : B\nx\u271d\u00b2 : x \u2208 (localTriv Z i).baseSet\nx\u271d\u00b9 x\u271d : Fiber Z x\n\u22a2 (\u2191(localTriv Z i) { proj := x, snd := x\u271d\u00b9 + x\u271d }).snd =\n    (\u2191(localTriv Z i) { proj := x, snd := x\u271d\u00b9 }).snd + (\u2191(localTriv Z i) { proj := x, snd := x\u271d }).snd\n[PROOFSTEP]\nsimp only [map_add, localTriv_apply, mfld_simps]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni : \u03b9\nx : B\nx\u271d\u00b2 : x \u2208 (localTriv Z i).baseSet\nx\u271d\u00b9 : R\nx\u271d : Fiber Z x\n\u22a2 (\u2191(localTriv Z i) { proj := x, snd := x\u271d\u00b9 \u2022 x\u271d }).snd = x\u271d\u00b9 \u2022 (\u2191(localTriv Z i) { proj := x, snd := x\u271d }).snd\n[PROOFSTEP]\nsimp only [map_smul, localTriv_apply, mfld_simps]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : F\n\u22a2 Trivialization.symm (localTriv Z i) b v = \u2191(coordChange Z i (indexAt Z b) b) v\n[PROOFSTEP]\napply (Z.localTriv i).symm_apply hb v\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z j\nv : F\n\u22a2 \u2191(Trivialization.coordChangeL R (localTriv Z i) (localTriv Z j) b) v = \u2191(coordChange Z i j b) v\n[PROOFSTEP]\nrw [Trivialization.coordChangeL_apply', localTriv_symm_fst, localTriv_apply, coordChange_comp]\n[GOAL]\ncase a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z j\nv : F\n\u22a2 { proj := (b, v).fst, snd := \u2191(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.proj \u2208\n    baseSet Z i \u2229\n        baseSet Z\n          (indexAt Z\n            { proj := (b, v).fst, snd := \u2191(coordChange Z i (indexAt Z (b, v).fst) (b, v).fst) (b, v).snd }.proj) \u2229\n      baseSet Z j\ncase hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z j\nv : F\n\u22a2 b \u2208 (localTriv Z i).baseSet \u2229 (localTriv Z j).baseSet\n[PROOFSTEP]\nexacts [\u27e8\u27e8hb.1, Z.mem_baseSet_at b\u27e9, hb.2\u27e9, hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni j : \u03b9\n\u22a2 { proj := b, snd := a } \u2208 (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source\n[PROOFSTEP]\nrw [localTrivAt, mem_localTriv_source]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni j : \u03b9\n\u22a2 { proj := b, snd := a }.proj \u2208 baseSet Z (indexAt Z b)\n[PROOFSTEP]\nexact Z.mem_baseSet_at b\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni j : \u03b9\n\u22a2 \u2200 (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear R e\n[PROOFSTEP]\nrintro _ \u27e8i, rfl\u27e9\n[GOAL]\ncase mk.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni\u271d j i : \u03b9\n\u22a2 Trivialization.IsLinear R (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n[PROOFSTEP]\napply localTriv.isLinear\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni j : \u03b9\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n    ContinuousOn (fun b => \u2191(Trivialization.coordChangeL R e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8i, rfl\u27e9 \u27e8i', rfl\u27e9\n[GOAL]\ncase mk.intro.mk.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb : B\na : F\ni\u271d j i i' : \u03b9\n\u22a2 ContinuousOn\n    (fun b =>\n      \u2191(Trivialization.coordChangeL R (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n          (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b))\n    ((FiberBundleCore.localTriv (toFiberBundleCore Z) i).baseSet \u2229\n      (FiberBundleCore.localTriv (toFiberBundleCore Z) i').baseSet)\n[PROOFSTEP]\nrefine' (Z.continuousOn_coordChange i i').congr fun b hb => _\n[GOAL]\ncase mk.intro.mk.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni\u271d j i i' : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z i'\n\u22a2 \u2191(Trivialization.coordChangeL R (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n        (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b) =\n    coordChange Z i i' b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.mk.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni\u271d j i i' : \u03b9\nb : B\nhb : b \u2208 baseSet Z i \u2229 baseSet Z i'\nv : F\n\u22a2 \u2191\u2191(Trivialization.coordChangeL R (FiberBundleCore.localTriv (toFiberBundleCore Z) i)\n            (FiberBundleCore.localTriv (toFiberBundleCore Z) i') b)\n      v =\n    \u2191(coordChange Z i i' b) v\n[PROOFSTEP]\nexact Z.localTriv_coordChange_eq i i' hb v\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\n\u22a2 Trivialization.continuousLinearMapAt R (localTriv Z i) b = coordChange Z (indexAt Z b) i b\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : Fiber Z b\n\u22a2 \u2191(Trivialization.continuousLinearMapAt R (localTriv Z i) b) v = \u2191(coordChange Z (indexAt Z b) i b) v\n[PROOFSTEP]\nrw [(Z.localTriv i).continuousLinearMapAt_apply R, (Z.localTriv i).coe_linearMapAt_of_mem]\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : Fiber Z b\n\u22a2 (fun y => (\u2191(localTriv Z i) { proj := b, snd := y }).snd) v = \u2191(coordChange Z (indexAt Z b) i b) v\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : Fiber Z b\n\u22a2 b \u2208 (localTriv Z i).baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\n\u22a2 Trivialization.symmL R (localTriv Z i) b = coordChange Z i (indexAt Z b) b\n[PROOFSTEP]\next1 v\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : F\n\u22a2 \u2191(Trivialization.symmL R (localTriv Z i) b) v = \u2191(coordChange Z i (indexAt Z b) b) v\n[PROOFSTEP]\nrw [(Z.localTriv i).symmL_apply R, (Z.localTriv i).symm_apply]\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : F\n\u22a2 cast (_ : Fiber Z (\u2191(LocalHomeomorph.symm (localTriv Z i).toLocalHomeomorph) (b, v)).proj = Fiber Z b)\n      (\u2191(LocalHomeomorph.symm (localTriv Z i).toLocalHomeomorph) (b, v)).snd =\n    \u2191(coordChange Z i (indexAt Z b) b) v\ncase h.hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2078 : NontriviallyNormedField R\ninst\u271d\u2077 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2076 : (x : B) \u2192 Module R (E x)\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace R F\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F E)\ninst\u271d\u00b9 : (x : B) \u2192 TopologicalSpace (E x)\ninst\u271d : FiberBundle F E\n\u03b9 : Type u_5\nZ : VectorBundleCore R B F \u03b9\nb\u271d : B\na : F\ni j : \u03b9\nb : B\nhb : b \u2208 baseSet Z i\nv : F\n\u22a2 b \u2208 (localTriv Z i).baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(coordChange a he he' b) v) = \u2191e' { proj := b, snd := Pretrivialization.symm e b v }\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(coordChange a he he' b) v).fst = (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).fst\n[PROOFSTEP]\nrw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (\u2191(LocalEquiv.symm e.toLocalEquiv) (b, v)).proj \u2208 e'.baseSet\n[PROOFSTEP]\nrw [e.proj_symm_apply' hb.1]\n[GOAL]\ncase h\u2081\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 b \u2208 e'.baseSet\n[PROOFSTEP]\nexact hb.2\n[GOAL]\ncase h\u2082\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne e' : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\nhe' : e' \u2208 a.pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 (b, \u2191(coordChange a he he' b) v).snd = (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).snd\n[PROOFSTEP]\nexact a.coordChange_apply he he' hb v\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\n\u22a2 ContinuousOn (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv)) (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nhave : ContinuousOn (fun x : B \u00d7 F \u21a6 a.coordChange he' he x.1 x.2) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ) :=\n  isBoundedBilinearMapApply.continuous.comp_continuousOn ((a.continuousOn_coordChange he' he).prod_map continuousOn_id)\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : ContinuousOn (fun x => \u2191(coordChange a he' he x.fst) x.snd) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv)) (e'.target \u2229 \u2191(LocalEquiv.symm e'.toLocalEquiv) \u207b\u00b9' e.source)\n[PROOFSTEP]\nrw [e.target_inter_preimage_symm_source_eq e', inter_comm]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : ContinuousOn (fun x => \u2191(coordChange a he' he x.fst) x.snd) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\n\u22a2 ContinuousOn (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv)) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' (continuousOn_fst.prod this).congr _\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : ContinuousOn (fun x => \u2191(coordChange a he' he x.fst) x.snd) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\n\u22a2 EqOn (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv)) (fun x => (x.fst, \u2191(coordChange a he' he x.fst) x.snd))\n    ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\n[PROOFSTEP]\nrintro \u27e8b, f\u27e9 \u27e8hb, -\u27e9\n[GOAL]\ncase mk.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : ContinuousOn (fun x => \u2191(coordChange a he' he x.fst) x.snd) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\nb : B\nf : F\nhb : (b, f).fst \u2208 e'.baseSet \u2229 e.baseSet\n\u22a2 (\u2191e \u2218 \u2191(LocalEquiv.symm e'.toLocalEquiv)) (b, f) = (fun x => (x.fst, \u2191(coordChange a he' he x.fst) x.snd)) (b, f)\n[PROOFSTEP]\ndsimp only [Function.comp, Prod.map]\n[GOAL]\ncase mk.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 a.pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 a.pretrivializationAtlas\nthis : ContinuousOn (fun x => \u2191(coordChange a he' he x.fst) x.snd) ((e'.baseSet \u2229 e.baseSet) \u00d7\u02e2 univ)\nb : B\nf : F\nhb : (b, f).fst \u2208 e'.baseSet \u2229 e.baseSet\n\u22a2 \u2191e (\u2191(LocalEquiv.symm e'.toLocalEquiv) (b, f)) = (b, \u2191(coordChange a he' he b) f)\n[PROOFSTEP]\nrw [a.mk_coordChange _ _ hb, e'.mk_symm hb.1]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n\u22a2 \u2200 (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear R e\n[PROOFSTEP]\nrintro _ \u27e8e, he, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\n\u22a2 Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n[PROOFSTEP]\napply linear_trivializationOfMemPretrivializationAtlas\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n\u22a2 \u2200 (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n    ContinuousOn (fun b => \u2191(Trivialization.coordChangeL R e e' b)) (e.baseSet \u2229 e'.baseSet)\n[PROOFSTEP]\nrintro _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\n\u22a2 ContinuousOn\n    (fun b =>\n      \u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n          (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b))\n    ((FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet \u2229\n      (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet)\n[PROOFSTEP]\nrefine (a.continuousOn_coordChange he he').congr fun b hb \u21a6 ?_\n[GOAL]\ncase mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\n\u22a2 \u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n        (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) =\n    coordChange a he he' b\n[PROOFSTEP]\next v\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\n\u22a2 \u2191\u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n            (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n      v =\n    \u2191(coordChange a he he' b) v\n[PROOFSTEP]\nhaveI h\u2081 := a.linear_trivializationOfMemPretrivializationAtlas he\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\nh\u2081 : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\n\u22a2 \u2191\u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n            (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n      v =\n    \u2191(coordChange a he he' b) v\n[PROOFSTEP]\nhaveI h\u2082 := a.linear_trivializationOfMemPretrivializationAtlas he'\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\nh\u2081 : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\nh\u2082 : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he')\n\u22a2 \u2191\u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n            (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n      v =\n    \u2191(coordChange a he he' b) v\n[PROOFSTEP]\nrw [trivializationOfMemPretrivializationAtlas] at h\u2081 h\u2082 \n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\nh\u2081 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh\u2082 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n\u22a2 \u2191\u2191(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n            (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n      v =\n    \u2191(coordChange a he he' b) v\n[PROOFSTEP]\nrw [a.coordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe, Trivialization.coordChangeL_apply]\n[GOAL]\ncase mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\nh\u2081 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh\u2082 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n\u22a2 (\u2191(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n        { proj := b,\n          snd :=\n            Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b\n              v }).snd =\n    (\u2191e' { proj := b, snd := Pretrivialization.symm e b v }).snd\ncase mk.intro.intro.mk.intro.intro.h.hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u2076 : NontriviallyNormedField R\ninst\u271d\u2075 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u2074 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace R F\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : (x : B) \u2192 TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis\u271d : TopologicalSpace (TotalSpace F E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e \u2208 (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' \u2208 (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b \u2208 e.baseSet \u2229 e'.baseSet\nv : F\nh\u2081 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh\u2082 : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n\u22a2 b \u2208\n    (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet \u2229\n      (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet\n[PROOFSTEP]\nexacts [rfl, hb]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField R\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2070 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace R F\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : (x : B) \u2192 TopologicalSpace (E x)\n\ud835\udd5c\u2081 : Type u_5\n\ud835\udd5c\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nB' : Type u_7\ninst\u271d\u00b9\u00b3 : TopologicalSpace B'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2081 F\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E x)\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\nF' : Type u_8\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082 F'\nE' : B' \u2192 Type u_9\ninst\u271d\u2077 : (x : B') \u2192 AddCommMonoid (E' x)\ninst\u271d\u2076 : (x : B') \u2192 Module \ud835\udd5c\u2082 (E' x)\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F' E')\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c\u2081 F E\ninst\u271d\u00b2 : (x : B') \u2192 TopologicalSpace (E' x)\ninst\u271d\u00b9 : FiberBundle F' E'\ninst\u271d : VectorBundle \ud835\udd5c\u2082 F' E'\nx\u2080 x : B\ny\u2080 y : B'\n\u03d5 : E x \u2192SL[\u03c3] E' y\nhx : x \u2208 (trivializationAt F E x\u2080).baseSet\nhy : y \u2208 (trivializationAt F' E' y\u2080).baseSet\n\u22a2 inCoordinates F E F' E' x\u2080 x y\u2080 y \u03d5 =\n    comp (\u2191(Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F' E' y\u2080) y hy))\n      (comp \u03d5 \u2191(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F E x\u2080) x hx)))\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField R\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2070 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace R F\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : (x : B) \u2192 TopologicalSpace (E x)\n\ud835\udd5c\u2081 : Type u_5\n\ud835\udd5c\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nB' : Type u_7\ninst\u271d\u00b9\u00b3 : TopologicalSpace B'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2081 F\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E x)\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\nF' : Type u_8\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082 F'\nE' : B' \u2192 Type u_9\ninst\u271d\u2077 : (x : B') \u2192 AddCommMonoid (E' x)\ninst\u271d\u2076 : (x : B') \u2192 Module \ud835\udd5c\u2082 (E' x)\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F' E')\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c\u2081 F E\ninst\u271d\u00b2 : (x : B') \u2192 TopologicalSpace (E' x)\ninst\u271d\u00b9 : FiberBundle F' E'\ninst\u271d : VectorBundle \ud835\udd5c\u2082 F' E'\nx\u2080 x : B\ny\u2080 y : B'\n\u03d5 : E x \u2192SL[\u03c3] E' y\nhx : x \u2208 (trivializationAt F E x\u2080).baseSet\nhy : y \u2208 (trivializationAt F' E' y\u2080).baseSet\nx\u271d : F\n\u22a2 \u2191(inCoordinates F E F' E' x\u2080 x y\u2080 y \u03d5) x\u271d =\n    \u2191(comp (\u2191(Trivialization.continuousLinearEquivAt \ud835\udd5c\u2082 (trivializationAt F' E' y\u2080) y hy))\n          (comp \u03d5\n            \u2191(ContinuousLinearEquiv.symm (Trivialization.continuousLinearEquivAt \ud835\udd5c\u2081 (trivializationAt F E x\u2080) x hx))))\n      x\u271d\n[PROOFSTEP]\nsimp_rw [inCoordinates, ContinuousLinearMap.coe_comp', ContinuousLinearEquiv.coe_coe,\n  Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq]\n[GOAL]\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\ninst\u271d\u00b2\u00b2 : NontriviallyNormedField R\ninst\u271d\u00b2\u00b9 : (x : B) \u2192 AddCommMonoid (E x)\ninst\u271d\u00b2\u2070 : (x : B) \u2192 Module R (E x)\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace R F\ninst\u271d\u00b9\u2077 : TopologicalSpace B\ninst\u271d\u00b9\u2076 : (x : B) \u2192 TopologicalSpace (E x)\n\ud835\udd5c\u2081 : Type u_5\n\ud835\udd5c\u2082 : Type u_6\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nB' : Type u_7\ninst\u271d\u00b9\u00b3 : TopologicalSpace B'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c\u2081 F\ninst\u271d\u00b9\u00b9 : (x : B) \u2192 Module \ud835\udd5c\u2081 (E x)\ninst\u271d\u00b9\u2070 : TopologicalSpace (TotalSpace F E)\nF' : Type u_8\ninst\u271d\u2079 : NormedAddCommGroup F'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c\u2082 F'\nE' : B' \u2192 Type u_9\ninst\u271d\u2077 : (x : B') \u2192 AddCommMonoid (E' x)\ninst\u271d\u2076 : (x : B') \u2192 Module \ud835\udd5c\u2082 (E' x)\ninst\u271d\u2075 : TopologicalSpace (TotalSpace F' E')\ninst\u271d\u2074 : FiberBundle F E\ninst\u271d\u00b3 : VectorBundle \ud835\udd5c\u2081 F E\ninst\u271d\u00b2 : (x : B') \u2192 TopologicalSpace (E' x)\ninst\u271d\u00b9 : FiberBundle F' E'\ninst\u271d : VectorBundle \ud835\udd5c\u2082 F' E'\n\u03b9 : Type u_10\n\u03b9' : Type u_11\nZ : VectorBundleCore \ud835\udd5c\u2081 B F \u03b9\nZ' : VectorBundleCore \ud835\udd5c\u2082 B' F' \u03b9'\nx\u2080 x : B\ny\u2080 y : B'\n\u03d5 : F \u2192SL[\u03c3] F'\nhx : x \u2208 VectorBundleCore.baseSet Z (VectorBundleCore.indexAt Z x\u2080)\nhy : y \u2208 VectorBundleCore.baseSet Z' (VectorBundleCore.indexAt Z' y\u2080)\n\u22a2 inCoordinates F (VectorBundleCore.Fiber Z) F' (VectorBundleCore.Fiber Z') x\u2080 x y\u2080 y \u03d5 =\n    comp (VectorBundleCore.coordChange Z' (VectorBundleCore.indexAt Z' y) (VectorBundleCore.indexAt Z' y\u2080) y)\n      (comp \u03d5 (VectorBundleCore.coordChange Z (VectorBundleCore.indexAt Z x\u2080) (VectorBundleCore.indexAt Z x) x))\n[PROOFSTEP]\nsimp_rw [inCoordinates, Z'.trivializationAt_continuousLinearMapAt hy, Z.trivializationAt_symmL hx]\n", "meta": {"mathlib_filename": "Mathlib.Topology.VectorBundle.Basic", "llama_tokens": 38059, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837527911056, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.5159352985748727}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 (mulDysonEtransform e x).fst * (mulDysonEtransform e x).snd \u2286 x.fst * x.snd\n[PROOFSTEP]\nrefine' union_mul_inter_subset_union.trans (union_subset Subset.rfl _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 e \u2022 x.snd * e\u207b\u00b9 \u2022 x.fst \u2286 x.fst * x.snd\n[PROOFSTEP]\nrw [mul_smul_comm, smul_mul_assoc, inv_smul_smul, mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 Finset.card (mulDysonEtransform e x).fst + Finset.card (mulDysonEtransform e x).snd =\n    Finset.card x.fst + Finset.card x.snd\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 Finset.card (x.fst \u222a e \u2022 x.snd) + Finset.card (x.snd \u2229 e\u207b\u00b9 \u2022 x.fst) = Finset.card x.fst + Finset.card x.snd\n[PROOFSTEP]\nrw [\u2190 card_smul_finset e (_ \u2229 _), smul_finset_inter, smul_inv_smul, inter_comm, card_union_add_card_inter,\n  card_smul_finset]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulDysonEtransform e (mulDysonEtransform e x) = mulDysonEtransform e x\n[PROOFSTEP]\next : 1\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 (mulDysonEtransform e (mulDysonEtransform e x)).fst = (mulDysonEtransform e x).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 (mulDysonEtransform e (mulDysonEtransform e x)).snd = (mulDysonEtransform e x).snd\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 x.fst \u222a e \u2022 x.snd \u222a e \u2022 (x.snd \u2229 e\u207b\u00b9 \u2022 x.fst) = x.fst \u222a e \u2022 x.snd\n[PROOFSTEP]\nrw [smul_finset_inter, smul_inv_smul, inter_comm, union_eq_left_iff_subset]\n[GOAL]\ncase h\u2081\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 x.fst \u2229 e \u2022 x.snd \u2286 x.fst \u222a e \u2022 x.snd\n[PROOFSTEP]\nexact inter_subset_union\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 x.snd \u2229 e\u207b\u00b9 \u2022 x.fst \u2229 e\u207b\u00b9 \u2022 (x.fst \u222a e \u2022 x.snd) = x.snd \u2229 e\u207b\u00b9 \u2022 x.fst\n[PROOFSTEP]\nrw [smul_finset_union, inv_smul_smul, union_comm, inter_eq_left_iff_subset]\n[GOAL]\ncase h\u2082\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 x.snd \u2229 e\u207b\u00b9 \u2022 x.fst \u2286 x.snd \u222a e\u207b\u00b9 \u2022 x.fst\n[PROOFSTEP]\nexact inter_subset_union\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 e \u2022 (mulDysonEtransform e x).snd \u2286 (mulDysonEtransform e x).fst\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 e \u2022 (x.snd \u2229 e\u207b\u00b9 \u2022 x.fst) \u2286 x.fst \u222a e \u2022 x.snd\n[PROOFSTEP]\nrw [smul_finset_inter, smul_inv_smul, inter_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 x.fst \u2229 e \u2022 x.snd \u2286 x.fst \u222a e \u2022 x.snd\n[PROOFSTEP]\nexact inter_subset_union\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulEtransformLeft 1 x = x\n[PROOFSTEP]\nsimp [mulEtransformLeft]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulEtransformRight 1 x = x\n[PROOFSTEP]\nsimp [mulEtransformRight]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 (mulEtransformLeft e x).fst * (mulEtransformLeft e x).snd \u2286 x.fst * x.snd\n[PROOFSTEP]\nrefine' inter_mul_union_subset_union.trans (union_subset Subset.rfl _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 op e \u2022 x.fst * e\u207b\u00b9 \u2022 x.snd \u2286 x.fst * x.snd\n[PROOFSTEP]\nrw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 (mulEtransformRight e x).fst * (mulEtransformRight e x).snd \u2286 x.fst * x.snd\n[PROOFSTEP]\nrefine' union_mul_inter_subset_union.trans (union_subset Subset.rfl _)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 op e \u2022 x.fst * e\u207b\u00b9 \u2022 x.snd \u2286 x.fst * x.snd\n[PROOFSTEP]\nrw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 Finset.card x.fst + Finset.card (op e \u2022 x.fst) = 2 * Finset.card x.fst\n[PROOFSTEP]\nrw [card_smul_finset, two_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 Finset.card x.snd + Finset.card (e\u207b\u00b9 \u2022 x.snd) = 2 * Finset.card x.snd\n[PROOFSTEP]\nrw [card_smul_finset, two_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Group \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 card (mulEtransformLeft e x).fst + card (mulEtransformLeft e x).snd +\n      (card (mulEtransformRight e x).fst + card (mulEtransformRight e x).snd) =\n    card x.fst + card x.snd + (card x.fst + card x.snd)\n[PROOFSTEP]\nrw [add_add_add_comm, mulEtransformLeft.card, mulEtransformRight.card, \u2190 mul_add, two_mul]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulEtransformLeft e\u207b\u00b9 x = Prod.swap (mulEtransformRight e (Prod.swap x))\n[PROOFSTEP]\nsimp [-op_inv, op_smul_eq_smul, mulEtransformLeft, mulEtransformRight]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : CommGroup \u03b1\ne : \u03b1\nx : Finset \u03b1 \u00d7 Finset \u03b1\n\u22a2 mulEtransformRight e\u207b\u00b9 x = Prod.swap (mulEtransformLeft e (Prod.swap x))\n[PROOFSTEP]\nsimp [-op_inv, op_smul_eq_smul, mulEtransformLeft, mulEtransformRight]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Additive.Etransform", "llama_tokens": 2739, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754472, "lm_q2_score": 0.6859494678483918, "lm_q1q2_score": 0.5158426324106066}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nV\u2081 : Type u_2\nV\u2082 : Type u_3\nP\u2081 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u00b9\u00b9 : NormedField \ud835\udd5c\ninst\u271d\u00b9\u2070 : NormedAddCommGroup V\u2081\ninst\u271d\u2079 : SeminormedAddCommGroup V\u2082\ninst\u271d\u2078 : NormedSpace \ud835\udd5c V\u2081\ninst\u271d\u2077 : NormedSpace \ud835\udd5c V\u2082\ninst\u271d\u2076 : MetricSpace P\u2081\ninst\u271d\u2075 : PseudoMetricSpace P\u2082\ninst\u271d\u2074 : NormedAddTorsor V\u2081 P\u2081\ninst\u271d\u00b3 : NormedAddTorsor V\u2082 P\u2082\ninst\u271d\u00b2 : FiniteDimensional \ud835\udd5c V\u2081\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c V\u2082\ninst\u271d : Inhabited P\u2081\nli : P\u2081 \u2192\u1d43\u2071[\ud835\udd5c] P\u2082\nh : finrank \ud835\udd5c V\u2081 = finrank \ud835\udd5c V\u2082\np : P\u2081\n\u22a2 \u2191li p = \u2191(LinearIsometry.toLinearIsometryEquiv (AffineIsometry.linearIsometry li) h) (p -\u1d65 default) +\u1d65 \u2191li default\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\n\u22a2 Continuous fun f => det f\n[PROOFSTEP]\nchange\n  Continuous fun f : E \u2192L[\ud835\udd5c] E =>\n    LinearMap.det\n      (f : E \u2192\u2097[\ud835\udd5c] E)\n        -- Porting note: this could be easier with `det_cases`\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\n\u22a2 Continuous fun f => \u2191LinearMap.det \u2191f\n[PROOFSTEP]\nby_cases h : \u2203 s : Finset E, Nonempty (Basis (\u21a5s) \ud835\udd5c E)\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c E)\n\u22a2 Continuous fun f => \u2191LinearMap.det \u2191f\n[PROOFSTEP]\nrcases h with \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\ns : Finset E\nb : Basis { x // x \u2208 s } \ud835\udd5c E\n\u22a2 Continuous fun f => \u2191LinearMap.det \u2191f\n[PROOFSTEP]\nhaveI : FiniteDimensional \ud835\udd5c E := FiniteDimensional.of_fintype_basis b\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\ns : Finset E\nb : Basis { x // x \u2208 s } \ud835\udd5c E\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 Continuous fun f => \u2191LinearMap.det \u2191f\n[PROOFSTEP]\nsimp_rw [LinearMap.det_eq_det_toMatrix_of_finset b]\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\ns : Finset E\nb : Basis { x // x \u2208 s } \ud835\udd5c E\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 Continuous fun f => Matrix.det (\u2191(LinearMap.toMatrix b b) \u2191f)\n[PROOFSTEP]\nrefine' Continuous.matrix_det _\n[GOAL]\ncase pos.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\ns : Finset E\nb : Basis { x // x \u2208 s } \ud835\udd5c E\nthis : FiniteDimensional \ud835\udd5c E\n\u22a2 Continuous fun f => \u2191(LinearMap.toMatrix b b) \u2191f\n[PROOFSTEP]\nexact ((LinearMap.toMatrix b b).toLinearMap.comp (ContinuousLinearMap.coeLM \ud835\udd5c)).continuous_of_finiteDimensional\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c E)\n\u22a2 Continuous fun f => \u2191LinearMap.det \u2191f\n[PROOFSTEP]\nrw [LinearMap.det_def]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c E)\n\u22a2 Continuous fun f =>\n    \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } \ud835\udd5c E) then\n          LinearMap.detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } \ud835\udd5c E))))\n        else 1)\n      \u2191f\n[PROOFSTEP]\nsimpa only [h, MonoidHom.one_apply, dif_neg, not_false_iff] using continuous_const\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nE' : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\n\u22a2 0 < lipschitzExtensionConstant E'\n[PROOFSTEP]\nrw [lipschitzExtensionConstant]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nE' : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\n\u22a2 0 <\n    let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'));\n    max (\u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a * \u2016\u2191A\u2016\u208a) 1\n[PROOFSTEP]\nexact zero_lt_one.trans_le (le_max_right _ _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nlet \u03b9 : Type _ := Basis.ofVectorSpaceIndex \u211d E'\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nlet A := (Basis.ofVectorSpace \u211d E').equivFun.toContinuousLinearEquiv\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nhave LA : LipschitzWith \u2016A.toContinuousLinearMap\u2016\u208a A := by apply A.lipschitz\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\n\u22a2 LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\n[PROOFSTEP]\napply A.lipschitz\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nhave L : LipschitzOnWith (\u2016A.toContinuousLinearMap\u2016\u208a * K) (A \u2218 f) s := LA.comp_lipschitzOnWith hf\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nobtain \u27e8g, hg, gs\u27e9 : \u2203 g : \u03b1 \u2192 \u03b9 \u2192 \u211d, LipschitzWith (\u2016A.toContinuousLinearMap\u2016\u208a * K) g \u2227 EqOn (A \u2218 f) g s := L.extend_pi\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\n\u22a2 \u2203 g, LipschitzWith (lipschitzExtensionConstant E' * K) g \u2227 EqOn f g s\n[PROOFSTEP]\nrefine' \u27e8A.symm \u2218 g, _, _\u27e9\n[GOAL]\ncase intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\n\u22a2 LipschitzWith (lipschitzExtensionConstant E' * K) (\u2191(ContinuousLinearEquiv.symm A) \u2218 g)\n[PROOFSTEP]\nhave LAsymm : LipschitzWith \u2016A.symm.toContinuousLinearMap\u2016\u208a A.symm := by apply A.symm.lipschitz\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\n\u22a2 LipschitzWith \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a \u2191(ContinuousLinearEquiv.symm A)\n[PROOFSTEP]\napply A.symm.lipschitz\n[GOAL]\ncase intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\nLAsymm : LipschitzWith \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a \u2191(ContinuousLinearEquiv.symm A)\n\u22a2 LipschitzWith (lipschitzExtensionConstant E' * K) (\u2191(ContinuousLinearEquiv.symm A) \u2218 g)\n[PROOFSTEP]\napply (LAsymm.comp hg).weaken\n[GOAL]\ncase intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\nLAsymm : LipschitzWith \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a \u2191(ContinuousLinearEquiv.symm A)\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a * (\u2016\u2191A\u2016\u208a * K) \u2264 lipschitzExtensionConstant E' * K\n[PROOFSTEP]\nrw [lipschitzExtensionConstant, \u2190 mul_assoc]\n[GOAL]\ncase intro.intro.refine'_1\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\nLAsymm : LipschitzWith \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a \u2191(ContinuousLinearEquiv.symm A)\n\u22a2 \u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a * \u2016\u2191A\u2016\u208a * K \u2264\n    (let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'));\n      max (\u2016\u2191(ContinuousLinearEquiv.symm A)\u2016\u208a * \u2016\u2191A\u2016\u208a) 1) *\n      K\n[PROOFSTEP]\nrefine' mul_le_mul' (le_max_left _ _) le_rfl\n[GOAL]\ncase intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\n\u22a2 EqOn f (\u2191(ContinuousLinearEquiv.symm A) \u2218 g) s\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 f x = (\u2191(ContinuousLinearEquiv.symm A) \u2218 g) x\n[PROOFSTEP]\nhave : A (f x) = g x := gs hx\n[GOAL]\ncase intro.intro.refine'_2\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2079 : AddCommGroup F'\ninst\u271d\u2078 : Module \ud835\udd5c F'\ninst\u271d\u2077 : TopologicalSpace F'\ninst\u271d\u2076 : TopologicalAddGroup F'\ninst\u271d\u2075 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u2074 : CompleteSpace \ud835\udd5c\n\u03b1 : Type u_1\ninst\u271d\u00b3 : PseudoMetricSpace \u03b1\nE' : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E'\ninst\u271d\u00b9 : NormedSpace \u211d E'\ninst\u271d : FiniteDimensional \u211d E'\ns : Set \u03b1\nf : \u03b1 \u2192 E'\nK : \u211d\u22650\nhf : LipschitzOnWith K f s\n\u03b9 : Type u_2 := \u2191(Basis.ofVectorSpaceIndex \u211d E')\nA : E' \u2243L[\u211d] \u2191(Basis.ofVectorSpaceIndex \u211d E') \u2192 \u211d :=\n  LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace \u211d E'))\nLA : LipschitzWith \u2016\u2191A\u2016\u208a \u2191A\nL : LipschitzOnWith (\u2016\u2191A\u2016\u208a * K) (\u2191A \u2218 f) s\ng : \u03b1 \u2192 \u03b9 \u2192 \u211d\nhg : LipschitzWith (\u2016\u2191A\u2016\u208a * K) g\ngs : EqOn (\u2191A \u2218 f) g s\nx : \u03b1\nhx : x \u2208 s\nthis : \u2191A (f x) = g x\n\u22a2 f x = (\u2191(ContinuousLinearEquiv.symm A) \u2218 g) x\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7), \u2190 this, A.symm_apply_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : ker f = \u22a5\n\u22a2 \u2203 K, K > 0 \u2227 AntilipschitzWith K \u2191f\n[PROOFSTEP]\ncases subsingleton_or_nontrivial E\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : ker f = \u22a5\nh\u271d : Subsingleton E\n\u22a2 \u2203 K, K > 0 \u2227 AntilipschitzWith K \u2191f\n[PROOFSTEP]\nexact \u27e81, zero_lt_one, AntilipschitzWith.of_subsingleton\u27e9\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : ker f = \u22a5\nh\u271d : Nontrivial E\n\u22a2 \u2203 K, K > 0 \u2227 AntilipschitzWith K \u2191f\n[PROOFSTEP]\nrw [LinearMap.ker_eq_bot] at hf \n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : Function.Injective \u2191f\nh\u271d : Nontrivial E\n\u22a2 \u2203 K, K > 0 \u2227 AntilipschitzWith K \u2191f\n[PROOFSTEP]\nlet e : E \u2243L[\ud835\udd5c] LinearMap.range f := (LinearEquiv.ofInjective f hf).toContinuousLinearEquiv\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : Function.Injective \u2191f\nh\u271d : Nontrivial E\ne : E \u2243L[\ud835\udd5c] { x // x \u2208 range f } := LinearEquiv.toContinuousLinearEquiv (LinearEquiv.ofInjective f hf)\n\u22a2 \u2203 K, K > 0 \u2227 AntilipschitzWith K \u2191f\n[PROOFSTEP]\nexact \u27e8_, e.nnnorm_symm_pos, e.antilipschitz\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nhf : LinearIndependent \ud835\udd5c f\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f, LinearIndependent \ud835\udd5c g\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nhf : LinearIndependent \ud835\udd5c f\nval\u271d : Fintype \u03b9\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f, LinearIndependent \ud835\udd5c g\n[PROOFSTEP]\nsimp only [Fintype.linearIndependent_iff'] at hf \u22a2\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f,\n    LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nrcases LinearMap.exists_antilipschitzWith _ hf with \u27e8K, K0, hK\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f,\n    LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nhave : Tendsto (fun g : \u03b9 \u2192 E => \u2211 i, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd <| \u2211 i, \u2016f i - f i\u2016) :=\n  tendsto_finset_sum _ fun i _ => Tendsto.norm <| ((continuous_apply i).tendsto _).sub tendsto_const_nhds\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd (\u2211 i : \u03b9, \u2016f i - f i\u2016))\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f,\n    LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nsimp only [sub_self, norm_zero, Finset.sum_const_zero] at this \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (g : \u03b9 \u2192 E) in \ud835\udcdd f,\n    LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nrefine' (this.eventually (gt_mem_nhds <| inv_pos.2 K0)).mono fun g hg => _\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016 < ((fun a => \u2191a) K)\u207b\u00b9\n\u22a2 LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nreplace hg : \u2211 i, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\n[GOAL]\ncase hg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016 < ((fun a => \u2191a) K)\u207b\u00b9\n\u22a2 \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_lt_coe]\n[GOAL]\ncase hg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016 < ((fun a => \u2191a) K)\u207b\u00b9\n\u22a2 \u2191(\u2211 i : \u03b9, \u2016g i - f i\u2016\u208a) < \u2191K\u207b\u00b9\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase hg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016 < ((fun a => \u2191a) K)\u207b\u00b9\n\u22a2 \u2211 x : \u03b9, \u2016g x - f x\u2016 < (\u2191K)\u207b\u00b9\n[PROOFSTEP]\nexact hg\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\n\u22a2 LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) = \u22a5\n[PROOFSTEP]\nrw [LinearMap.ker_eq_bot]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\n\u22a2 Function.Injective \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i))\n[PROOFSTEP]\nrefine' (hK.add_sub_lipschitzWith (LipschitzWith.of_dist_le_mul fun v u => _) hg).injective\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\nv u : \u03b9 \u2192 \ud835\udd5c\n\u22a2 dist\n      ((\u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n          \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)))\n        v)\n      ((\u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (g i)) -\n          \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)))\n        u) \u2264\n    \u2191(\u2211 i : \u03b9, \u2016g i - f i\u2016\u208a) * dist v u\n[PROOFSTEP]\nsimp only [dist_eq_norm, LinearMap.lsum_apply, Pi.sub_apply, LinearMap.sum_apply, LinearMap.comp_apply,\n  LinearMap.proj_apply, LinearMap.smulRight_apply, LinearMap.id_apply, \u2190 Finset.sum_sub_distrib, \u2190 smul_sub, \u2190 sub_smul,\n  NNReal.coe_sum, coe_nnnorm, Finset.sum_mul]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\nv u : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2016\u2211 x : \u03b9, (v x - u x) \u2022 (g x - f x)\u2016 \u2264 \u2211 x : \u03b9, \u2016g x - f x\u2016 * \u2016v - u\u2016\n[PROOFSTEP]\nrefine' norm_sum_le_of_le _ fun i _ => _\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\nv u : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016(v i - u i) \u2022 (g i - f i)\u2016 \u2264 \u2016g i - f i\u2016 * \u2016v - u\u2016\n[PROOFSTEP]\nrw [norm_smul, mul_comm]\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\nv u : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016g i - f i\u2016 * \u2016v i - u i\u2016 \u2264 \u2016g i - f i\u2016 * \u2016v - u\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 E\nval\u271d : Fintype \u03b9\nhf : LinearMap.ker (\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i)) = \u22a5\nK : \u211d\u22650\nK0 : K > 0\nhK : AntilipschitzWith K \u2191(\u2191(LinearMap.lsum \ud835\udd5c (fun x => \ud835\udd5c) \u2115) fun i => LinearMap.smulRight LinearMap.id (f i))\nthis : Tendsto (fun g => \u2211 i : \u03b9, \u2016g i - f i\u2016) (\ud835\udcdd f) (\ud835\udcdd 0)\ng : \u03b9 \u2192 E\nhg : \u2211 i : \u03b9, \u2016g i - f i\u2016\u208a < K\u207b\u00b9\nv u : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016v i - u i\u2016 \u2264 \u2016v - u\u2016\n[PROOFSTEP]\nexact norm_le_pi_norm (v - u) i\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nn : \u2115\n\u22a2 IsOpen {f | \u2191n \u2264 LinearMap.rank \u2191f}\n[PROOFSTEP]\nsimp only [LinearMap.le_rank_iff_exists_linearIndependent_finset, setOf_exists, \u2190 exists_prop]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nn : \u2115\n\u22a2 IsOpen (\u22c3 (i : Finset E) (_ : Finset.card i = n), {x | LinearIndependent \ud835\udd5c fun x_1 => \u2191\u2191x \u2191x_1})\n[PROOFSTEP]\nrefine' isOpen_biUnion fun t _ => _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nn : \u2115\nt : Finset E\nx\u271d : t \u2208 fun i => Finset.card i = n\n\u22a2 IsOpen {x | LinearIndependent \ud835\udd5c fun x_1 => \u2191\u2191x \u2191x_1}\n[PROOFSTEP]\nhave : Continuous fun f : E \u2192L[\ud835\udd5c] F => fun x : (t : Set E) => f x :=\n  continuous_pi fun x => (ContinuousLinearMap.apply \ud835\udd5c F (x : E)).continuous\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nn : \u2115\nt : Finset E\nx\u271d : t \u2208 fun i => Finset.card i = n\nthis : Continuous fun f x => \u2191f \u2191x\n\u22a2 IsOpen {x | LinearIndependent \ud835\udd5c fun x_1 => \u2191\u2191x \u2191x_1}\n[PROOFSTEP]\nexact isOpen_setOf_linearIndependent.preimage this\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u22a2 \u2016\u2191u e\u2016\u208a \u2264 Fintype.card \u03b9 \u2022 \u2016\u2191(equivFunL v)\u2016\u208a * M * \u2016e\u2016\u208a\n[PROOFSTEP]\nset \u03c6 := v.equivFunL.toContinuousLinearMap\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2016\u2191u e\u2016\u208a \u2264 Fintype.card \u03b9 \u2022 \u2016\u03c6\u2016\u208a * M * \u2016e\u2016\u208a\n[PROOFSTEP]\ncalc\n  \u2016u e\u2016\u208a = \u2016u (\u2211 i, v.equivFun e i \u2022 v i)\u2016\u208a := by rw [v.sum_equivFun]\n  _ = \u2016\u2211 i, v.equivFun e i \u2022 (u <| v i)\u2016\u208a := by simp [u.map_sum, LinearMap.map_smul]\n  _ \u2264 \u2211 i, \u2016v.equivFun e i \u2022 (u <| v i)\u2016\u208a := (nnnorm_sum_le _ _)\n  _ = \u2211 i, \u2016v.equivFun e i\u2016\u208a * \u2016u (v i)\u2016\u208a := by simp only [nnnorm_smul]\n  _ \u2264 \u2211 i, \u2016v.equivFun e i\u2016\u208a * M := by gcongr; apply hu\n  _ = (\u2211 i, \u2016v.equivFun e i\u2016\u208a) * M := Finset.sum_mul.symm\n  _ \u2264 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a) * M := by\n    gcongr\n    calc\n      \u2211 i, \u2016v.equivFun e i\u2016\u208a \u2264 Fintype.card \u03b9 \u2022 \u2016\u03c6 e\u2016\u208a := Pi.sum_nnnorm_apply_le_nnnorm _\n      _ \u2264 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a) := nsmul_le_nsmul_of_le_right (\u03c6.le_op_nnnorm e) _\n  _ = Fintype.card \u03b9 \u2022 \u2016\u03c6\u2016\u208a * M * \u2016e\u2016\u208a := by simp only [smul_mul_assoc, mul_right_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2016\u2191u e\u2016\u208a = \u2016\u2191u (\u2211 i : \u03b9, \u2191(equivFun v) e i \u2022 \u2191v i)\u2016\u208a\n[PROOFSTEP]\nrw [v.sum_equivFun]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2016\u2191u (\u2211 i : \u03b9, \u2191(equivFun v) e i \u2022 \u2191v i)\u2016\u208a = \u2016\u2211 i : \u03b9, \u2191(equivFun v) e i \u2022 \u2191u (\u2191v i)\u2016\u208a\n[PROOFSTEP]\nsimp [u.map_sum, LinearMap.map_smul]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2211 i : \u03b9, \u2016\u2191(equivFun v) e i \u2022 \u2191u (\u2191v i)\u2016\u208a = \u2211 i : \u03b9, \u2016\u2191(equivFun v) e i\u2016\u208a * \u2016\u2191u (\u2191v i)\u2016\u208a\n[PROOFSTEP]\nsimp only [nnnorm_smul]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2211 i : \u03b9, \u2016\u2191(equivFun v) e i\u2016\u208a * \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 \u2211 i : \u03b9, \u2016\u2191(equivFun v) e i\u2016\u208a * M\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.bc\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\ni\u271d : \u03b9\na\u271d : i\u271d \u2208 Finset.univ\n\u22a2 \u2016\u2191u (\u2191v i\u271d)\u2016\u208a \u2264 M\n[PROOFSTEP]\napply hu\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 (\u2211 i : \u03b9, \u2016\u2191(equivFun v) e i\u2016\u208a) * M \u2264 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a) * M\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 \u2211 i : \u03b9, \u2016\u2191(equivFun v) e i\u2016\u208a \u2264 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a)\n[PROOFSTEP]\ncalc\n  \u2211 i, \u2016v.equivFun e i\u2016\u208a \u2264 Fintype.card \u03b9 \u2022 \u2016\u03c6 e\u2016\u208a := Pi.sum_nnnorm_apply_le_nnnorm _\n  _ \u2264 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a) := nsmul_le_nsmul_of_le_right (\u03c6.le_op_nnnorm e) _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\u22650\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M\ne : E\n\u03c6 : E \u2192L[\ud835\udd5c] \u03b9 \u2192 \ud835\udd5c := \u2191(equivFunL v)\n\u22a2 Fintype.card \u03b9 \u2022 (\u2016\u03c6\u2016\u208a * \u2016e\u2016\u208a) * M = Fintype.card \u03b9 \u2022 \u2016\u03c6\u2016\u208a * M * \u2016e\u2016\u208a\n[PROOFSTEP]\nsimp only [smul_mul_assoc, mul_right_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\nhM : 0 \u2264 M\nhu : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016 \u2264 M\n\u22a2 \u2016u\u2016 \u2264 Fintype.card \u03b9 \u2022 \u2016\u2191(equivFunL v)\u2016 * M\n[PROOFSTEP]\nsimpa using NNReal.coe_le_coe.mpr (v.op_nnnorm_le \u27e8M, hM\u27e9 hu)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nval\u271d : Fintype \u03b9\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M\n[PROOFSTEP]\nexact\n  \u27e8max (Fintype.card \u03b9 \u2022 \u2016v.equivFunL.toContinuousLinearMap\u2016\u208a) 1, zero_lt_one.trans_le (le_max_right _ _),\n    fun {u} M hu => (v.op_nnnorm_le M hu).trans <| mul_le_mul_of_nonneg_right (le_max_left _ _) (zero_le M)\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 {u : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016 \u2264 M) \u2192 \u2016u\u2016 \u2264 C * M\n[PROOFSTEP]\nobtain \u27e8C, hC, h\u27e9 :=\n  v.exists_op_nnnorm_le (F := F)\n    -- Porting note: used `Subtype.forall'` below\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nC : \u211d\u22650\nhC : C > 0\nh : \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 {u : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016 \u2264 M) \u2192 \u2016u\u2016 \u2264 C * M\n[PROOFSTEP]\nrefine \u27e8C, hC, ?_\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nC : \u211d\u22650\nhC : C > 0\nh : \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M\n\u22a2 \u2200 {u : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016 \u2264 M) \u2192 \u2016u\u2016 \u2264 \u2191C * M\n[PROOFSTEP]\nintro u M hM H\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d : Finite \u03b9\nv : Basis \u03b9 \ud835\udd5c E\nC : \u211d\u22650\nhC : C > 0\nh : \u2200 {u : E \u2192L[\ud835\udd5c] F} (M : \u211d\u22650), (\u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016\u208a \u2264 M) \u2192 \u2016u\u2016\u208a \u2264 C * M\nu : E \u2192L[\ud835\udd5c] F\nM : \u211d\nhM : 0 \u2264 M\nH : \u2200 (i : \u03b9), \u2016\u2191u (\u2191v i)\u2016 \u2264 M\n\u22a2 \u2016u\u2016 \u2264 \u2191C * M\n[PROOFSTEP]\nsimpa using h \u27e8M, hM\u27e9 H\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\n\u22a2 SecondCountableTopology (E \u2192L[\ud835\udd5c] F)\n[PROOFSTEP]\nset d := FiniteDimensional.finrank \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u22a2 SecondCountableTopology (E \u2192L[\ud835\udd5c] F)\n[PROOFSTEP]\nsuffices \u2200 \u03b5 > (0 : \u211d), \u2203 n : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115, \u2200 f g : E \u2192L[\ud835\udd5c] F, n f = n g \u2192 dist f g \u2264 \u03b5 from\n  Metric.secondCountable_of_countable_discretization fun \u03b5 \u03b5_pos => \u27e8Fin d \u2192 \u2115, by infer_instance, this \u03b5 \u03b5_pos\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\nthis : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\n\u22a2 Encodable (Fin d \u2192 \u2115)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b5_pos\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8u : \u2115 \u2192 F, hu : DenseRange u\u27e9 := exists_dense_seq F\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nlet v := FiniteDimensional.finBasis \ud835\udd5c E\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8C : \u211d, C_pos : 0 < C, hC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 i, \u2016\u03c6 (v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\u27e9 :=\n  v.exists_op_norm_le (E := E) (F := F)\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nhave h_2C : 0 < 2 * C := mul_pos zero_lt_two C_pos\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nhave h\u03b52C : 0 < \u03b5 / (2 * C) := div_pos \u03b5_pos h_2C\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nhave : \u2200 \u03c6 : E \u2192L[\ud835\udd5c] F, \u2203 n : Fin d \u2192 \u2115, \u2016\u03c6 - (v.constrL <| u \u2218 n)\u2016 \u2264 \u03b5 / 2 :=\n  by\n  intro \u03c6\n  have : \u2200 i, \u2203 n, \u2016\u03c6 (v i) - u n\u2016 \u2264 \u03b5 / (2 * C) :=\n    by\n    simp only [norm_sub_rev]\n    intro i\n    have : \u03c6 (v i) \u2208 closure (range u) := hu _\n    obtain \u27e8n, hn\u27e9 : \u2203 n, \u2016u n - \u03c6 (v i)\u2016 < \u03b5 / (2 * C) :=\n      by\n      rw [mem_closure_iff_nhds_basis Metric.nhds_basis_ball] at this \n      specialize this (\u03b5 / (2 * C)) h\u03b52C\n      simpa [dist_eq_norm]\n    exact \u27e8n, le_of_lt hn\u27e9\n  choose n hn using this\n  use n\n  replace hn : \u2200 i : Fin d, \u2016(\u03c6 - (v.constrL <| u \u2218 n)) (v i)\u2016 \u2264 \u03b5 / (2 * C)\n  \u00b7 simp [hn]\n  have : C * (\u03b5 / (2 * C)) = \u03b5 / 2 := by\n    rw [eq_div_iff (two_ne_zero : (2 : \u211d) \u2260 0), mul_comm, \u2190 mul_assoc, mul_div_cancel' _ (ne_of_gt h_2C)]\n  specialize hC (le_of_lt h\u03b52C) hn\n  rwa [this] at hC \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u22a2 \u2200 (\u03c6 : E \u2192L[\ud835\udd5c] F), \u2203 n, \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nintro \u03c6\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2203 n, \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nhave : \u2200 i, \u2203 n, \u2016\u03c6 (v i) - u n\u2016 \u2264 \u03b5 / (2 * C) :=\n  by\n  simp only [norm_sub_rev]\n  intro i\n  have : \u03c6 (v i) \u2208 closure (range u) := hu _\n  obtain \u27e8n, hn\u27e9 : \u2203 n, \u2016u n - \u03c6 (v i)\u2016 < \u03b5 / (2 * C) :=\n    by\n    rw [mem_closure_iff_nhds_basis Metric.nhds_basis_ball] at this \n    specialize this (\u03b5 / (2 * C)) h\u03b52C\n    simpa [dist_eq_norm]\n  exact \u27e8n, le_of_lt hn\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2203 n, \u2016\u2191\u03c6 (\u2191v i) - u n\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nsimp only [norm_sub_rev]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2203 n, \u2016u n - \u2191\u03c6 (\u2191(finBasis \ud835\udd5c E) i)\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nintro i\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191(finBasis \ud835\udd5c E) i)\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nhave : \u03c6 (v i) \u2208 closure (range u) := hu _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\nthis : \u2191\u03c6 (\u2191v i) \u2208 closure (range u)\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191(finBasis \ud835\udd5c E) i)\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 : \u2203 n, \u2016u n - \u03c6 (v i)\u2016 < \u03b5 / (2 * C) :=\n  by\n  rw [mem_closure_iff_nhds_basis Metric.nhds_basis_ball] at this \n  specialize this (\u03b5 / (2 * C)) h\u03b52C\n  simpa [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\nthis : \u2191\u03c6 (\u2191v i) \u2208 closure (range u)\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191v i)\u2016 < \u03b5 / (2 * C)\n[PROOFSTEP]\nrw [mem_closure_iff_nhds_basis Metric.nhds_basis_ball] at this \n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\nthis : \u2200 (i_1 : \u211d), 0 < i_1 \u2192 \u2203 y, y \u2208 range u \u2227 y \u2208 Metric.ball (\u2191\u03c6 (\u2191v i)) i_1\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191v i)\u2016 < \u03b5 / (2 * C)\n[PROOFSTEP]\nspecialize this (\u03b5 / (2 * C)) h\u03b52C\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\nthis : \u2203 y, y \u2208 range u \u2227 y \u2208 Metric.ball (\u2191\u03c6 (\u2191v i)) (\u03b5 / (2 * C))\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191v i)\u2016 < \u03b5 / (2 * C)\n[PROOFSTEP]\nsimpa [dist_eq_norm]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\ni : Fin (finrank \ud835\udd5c E)\nthis : \u2191\u03c6 (\u2191v i) \u2208 closure (range u)\nn : \u2115\nhn : \u2016u n - \u2191\u03c6 (\u2191v i)\u2016 < \u03b5 / (2 * C)\n\u22a2 \u2203 n, \u2016u n - \u2191\u03c6 (\u2191(finBasis \ud835\udd5c E) i)\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nexact \u27e8n, le_of_lt hn\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nthis : \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2203 n, \u2016\u2191\u03c6 (\u2191v i) - u n\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 \u2203 n, \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nchoose n hn using this\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i) - u (n i)\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 \u2203 n, \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i) - u (n i)\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nreplace hn : \u2200 i : Fin d, \u2016(\u03c6 - (v.constrL <| u \u2218 n)) (v i)\u2016 \u2264 \u03b5 / (2 * C)\n[GOAL]\ncase hn\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i) - u (n i)\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 \u2200 (i : Fin d), \u2016\u2191(\u03c6 - Basis.constrL v (u \u2218 n)) (\u2191v i)\u2016 \u2264 \u03b5 / (2 * C)\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin d), \u2016\u2191(\u03c6 - Basis.constrL v (u \u2218 n)) (\u2191v i)\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nhave : C * (\u03b5 / (2 * C)) = \u03b5 / 2 := by\n  rw [eq_div_iff (two_ne_zero : (2 : \u211d) \u2260 0), mul_comm, \u2190 mul_assoc, mul_div_cancel' _ (ne_of_gt h_2C)]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin d), \u2016\u2191(\u03c6 - Basis.constrL v (u \u2218 n)) (\u2191v i)\u2016 \u2264 \u03b5 / (2 * C)\n\u22a2 C * (\u03b5 / (2 * C)) = \u03b5 / 2\n[PROOFSTEP]\nrw [eq_div_iff (two_ne_zero : (2 : \u211d) \u2260 0), mul_comm, \u2190 mul_assoc, mul_div_cancel' _ (ne_of_gt h_2C)]\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin d), \u2016\u2191(\u03c6 - Basis.constrL v (u \u2218 n)) (\u2191v i)\u2016 \u2264 \u03b5 / (2 * C)\nthis : C * (\u03b5 / (2 * C)) = \u03b5 / 2\n\u22a2 \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nspecialize hC (le_of_lt h\u03b52C) hn\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\n\u03c6 : E \u2192L[\ud835\udd5c] F\nn : Fin (finrank \ud835\udd5c E) \u2192 \u2115\nhn : \u2200 (i : Fin d), \u2016\u2191(\u03c6 - Basis.constrL v (u \u2218 n)) (\u2191v i)\u2016 \u2264 \u03b5 / (2 * C)\nthis : C * (\u03b5 / (2 * C)) = \u03b5 / 2\nhC : \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 C * (\u03b5 / (2 * C))\n\u22a2 \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n[PROOFSTEP]\nrwa [this] at hC \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nthis : \u2200 (\u03c6 : E \u2192L[\ud835\udd5c] F), \u2203 n, \u2016\u03c6 - Basis.constrL v (u \u2218 n)\u2016 \u2264 \u03b5 / 2\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nchoose n hn using this\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\nhn : \u2200 (\u03c6 : E \u2192L[\ud835\udd5c] F), \u2016\u03c6 - Basis.constrL v (u \u2218 n \u03c6)\u2016 \u2264 \u03b5 / 2\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nset \u03a6 := fun \u03c6 : E \u2192L[\ud835\udd5c] F => v.constrL <| u \u2218 n \u03c6\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\nhn : \u2200 (\u03c6 : E \u2192L[\ud835\udd5c] F), \u2016\u03c6 - Basis.constrL v (u \u2218 n \u03c6)\u2016 \u2264 \u03b5 / 2\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nchange \u2200 z, dist z (\u03a6 z) \u2264 \u03b5 / 2 at hn \n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\nhn : \u2200 (z : E \u2192L[\ud835\udd5c] F), dist z (\u03a6 z) \u2264 \u03b5 / 2\n\u22a2 \u2203 n, \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\nhn : \u2200 (z : E \u2192L[\ud835\udd5c] F), dist z (\u03a6 z) \u2264 \u03b5 / 2\n\u22a2 \u2200 (f g : E \u2192L[\ud835\udd5c] F), n f = n g \u2192 dist f g \u2264 \u03b5\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\nhn : \u2200 (z : E \u2192L[\ud835\udd5c] F), dist z (\u03a6 z) \u2264 \u03b5 / 2\nx y : E \u2192L[\ud835\udd5c] F\nhxy : n x = n y\n\u22a2 dist x y \u2264 \u03b5\n[PROOFSTEP]\ncalc\n  dist x y \u2264 dist x (\u03a6 x) + dist (\u03a6 x) y := dist_triangle _ _ _\n  _ = dist x (\u03a6 x) + dist y (\u03a6 y) := by simp [hxy, dist_comm]\n  _ \u2264 \u03b5 := by linarith [hn x, hn y]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\nhn : \u2200 (z : E \u2192L[\ud835\udd5c] F), dist z (\u03a6 z) \u2264 \u03b5 / 2\nx y : E \u2192L[\ud835\udd5c] F\nhxy : n x = n y\n\u22a2 dist x (\u03a6 x) + dist (\u03a6 x) y = dist x (\u03a6 x) + dist y (\u03a6 y)\n[PROOFSTEP]\nsimp [hxy, dist_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\ninst\u271d\u00b9 : FiniteDimensional \ud835\udd5c E\ninst\u271d : SecondCountableTopology F\nd : \u2115 := finrank \ud835\udd5c E\n\u03b5 : \u211d\n\u03b5_pos : \u03b5 > 0\nu : \u2115 \u2192 F\nhu : DenseRange u\nv : Basis (Fin (finrank \ud835\udd5c E)) \ud835\udd5c E := finBasis \ud835\udd5c E\nC : \u211d\nC_pos : 0 < C\nhC : \u2200 {\u03c6 : E \u2192L[\ud835\udd5c] F} {M : \u211d}, 0 \u2264 M \u2192 (\u2200 (i : Fin (finrank \ud835\udd5c E)), \u2016\u2191\u03c6 (\u2191v i)\u2016 \u2264 M) \u2192 \u2016\u03c6\u2016 \u2264 C * M\nh_2C : 0 < 2 * C\nh\u03b52C : 0 < \u03b5 / (2 * C)\nn : (E \u2192L[\ud835\udd5c] F) \u2192 Fin d \u2192 \u2115\n\u03a6 : (E \u2192L[\ud835\udd5c] F) \u2192 E \u2192L[\ud835\udd5c] F := fun \u03c6 => Basis.constrL v (u \u2218 n \u03c6)\nhn : \u2200 (z : E \u2192L[\ud835\udd5c] F), dist z (\u03a6 z) \u2264 \u03b5 / 2\nx y : E \u2192L[\ud835\udd5c] F\nhxy : n x = n y\n\u22a2 dist x (\u03a6 x) + dist y (\u03a6 y) \u2264 \u03b5\n[PROOFSTEP]\nlinarith [hn x, hn y]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\n\u22a2 CompleteSpace E\n[PROOFSTEP]\nset e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun \ud835\udd5c _ _ (finrank \ud835\udd5c E)).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\ne : E \u2243L[\ud835\udd5c] Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c :=\n  ContinuousLinearEquiv.ofFinrankEq (_ : finrank \ud835\udd5c E = finrank \ud835\udd5c (Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c))\n\u22a2 CompleteSpace E\n[PROOFSTEP]\nhave : UniformEmbedding e.toLinearEquiv.toEquiv.symm := e.symm.uniformEmbedding\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\ne : E \u2243L[\ud835\udd5c] Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c :=\n  ContinuousLinearEquiv.ofFinrankEq (_ : finrank \ud835\udd5c E = finrank \ud835\udd5c (Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c))\nthis : UniformEmbedding \u2191(LinearEquiv.toEquiv e.toLinearEquiv).symm\n\u22a2 CompleteSpace E\n[PROOFSTEP]\nexact (completeSpace_congr this).1 (by infer_instance)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\ne : E \u2243L[\ud835\udd5c] Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c :=\n  ContinuousLinearEquiv.ofFinrankEq (_ : finrank \ud835\udd5c E = finrank \ud835\udd5c (Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c))\nthis : UniformEmbedding \u2191(LinearEquiv.toEquiv e.toLinearEquiv).symm\n\u22a2 CompleteSpace (Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nlet F := Submodule.span \ud835\udd5c (s : Set E)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nhaveI : FiniteDimensional \ud835\udd5c F :=\n  Module.finite_def.2 ((Submodule.fg_top _).2 (Submodule.fg_def.2 \u27e8s, Finset.finite_toSet _, rfl\u27e9))\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nhave Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nhave : \u2203 x, x \u2209 F := by\n  contrapose! h\n  have : (\u22a4 : Submodule \ud835\udd5c E) = F := by\n    ext x\n    simp [h]\n  have : FiniteDimensional \ud835\udd5c (\u22a4 : Submodule \ud835\udd5c E) := by rwa [this]\n  refine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\n\u22a2 \u2203 x, \u00acx \u2208 F\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nhave : (\u22a4 : Submodule \ud835\udd5c E) = F := by\n  ext x\n  simp [h]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\n\u22a2 \u22a4 = F\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nx : E\n\u22a2 x \u2208 \u22a4 \u2194 x \u2208 F\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis : \u22a4 = F\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nhave : FiniteDimensional \ud835\udd5c (\u22a4 : Submodule \ud835\udd5c E) := by rwa [this]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis : \u22a4 = F\n\u22a2 FiniteDimensional \ud835\udd5c { x // x \u2208 \u22a4 }\n[PROOFSTEP]\nrwa [this]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d\u00b9 : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nh : \u2200 (x : E), x \u2208 Submodule.span \ud835\udd5c \u2191s\nthis\u271d : \u22a4 = F\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 \u22a4 }\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nrefine' Module.finite_def.2 ((Submodule.fg_top _).1 (Module.finite_def.1 this))\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nobtain \u27e8x, xR, hx\u27e9 : \u2203 x : E, \u2016x\u2016 \u2264 R \u2227 \u2200 y : E, y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016 := riesz_lemma_of_norm_lt hc hR Fclosed this\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nhave hx' : \u2200 y : E, y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016 := by\n  intro y hy\n  rw [\u2190 norm_neg]\n  simpa using hx y hy\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nrw [\u2190 norm_neg]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\ny : E\nhy : y \u2208 F\n\u22a2 1 \u2264 \u2016-(y - x)\u2016\n[PROOFSTEP]\nsimpa using hx y hy\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF\u271d : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\u271d\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u271d\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\ns : Finset E\nF : Submodule \ud835\udd5c E := Submodule.span \ud835\udd5c \u2191s\nthis\u271d : FiniteDimensional \ud835\udd5c { x // x \u2208 F }\nFclosed : IsClosed \u2191F\nthis : \u2203 x, \u00acx \u2208 F\nx : E\nxR : \u2016x\u2016 \u2264 R\nhx : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016x - y\u2016\nhx' : \u2200 (y : E), y \u2208 F \u2192 1 \u2264 \u2016y - x\u2016\n\u22a2 \u2203 x, \u2016x\u2016 \u2264 R \u2227 \u2200 (y : E), y \u2208 s \u2192 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nexact \u27e8x, xR, fun y hy => hx' _ (Submodule.subset_span hy)\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 \u2203 f, (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\nhave : IsSymm E fun x y : E => 1 \u2264 \u2016x - y\u2016 := by\n  constructor\n  intro x y hxy\n  rw [\u2190 norm_neg]\n  simpa\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 IsSymm E fun x y => 1 \u2264 \u2016x - y\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase symm\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 \u2200 (a b : E), 1 \u2264 \u2016a - b\u2016 \u2192 1 \u2264 \u2016b - a\u2016\n[PROOFSTEP]\nintro x y hxy\n[GOAL]\ncase symm\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\nx y : E\nhxy : 1 \u2264 \u2016x - y\u2016\n\u22a2 1 \u2264 \u2016y - x\u2016\n[PROOFSTEP]\nrw [\u2190 norm_neg]\n[GOAL]\ncase symm\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\nx y : E\nhxy : 1 \u2264 \u2016x - y\u2016\n\u22a2 1 \u2264 \u2016-(y - x)\u2016\n[PROOFSTEP]\nsimpa\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\nthis : IsSymm E fun x y => 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2203 f, (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\napply exists_seq_of_forall_finset_exists' (fun x : E => \u2016x\u2016 \u2264 R) fun (x : E) (y : E) => 1 \u2264 \u2016x - y\u2016\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\nthis : IsSymm E fun x y => 1 \u2264 \u2016x - y\u2016\n\u22a2 \u2200 (s : Finset E), (\u2200 (x : E), x \u2208 s \u2192 \u2016x\u2016 \u2264 R) \u2192 \u2203 y, \u2016y\u2016 \u2264 R \u2227 \u2200 (x : E), x \u2208 s \u2192 1 \u2264 \u2016x - y\u2016\n[PROOFSTEP]\nrintro s -\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nR : \u211d\nhR : \u2016c\u2016 < R\nh : \u00acFiniteDimensional \ud835\udd5c E\nthis : IsSymm E fun x y => 1 \u2264 \u2016x - y\u2016\ns : Finset E\n\u22a2 \u2203 y, \u2016y\u2016 \u2264 R \u2227 \u2200 (x : E), x \u2208 s \u2192 1 \u2264 \u2016x - y\u2016\n[PROOFSTEP]\nexact exists_norm_le_le_norm_sub_of_finset hc hR h s\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 : \u2203 c : \ud835\udd5c, 1 < \u2016c\u2016 := NormedField.exists_one_lt_norm \ud835\udd5c\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\nhave A : \u2016c\u2016 < \u2016c\u2016 + 1 := by linarith\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2016c\u2016 < \u2016c\u2016 + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\nrcases exists_seq_norm_le_one_le_norm_sub' hc A h with \u27e8f, hf\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nh : \u00acFiniteDimensional \ud835\udd5c E\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\nA : \u2016c\u2016 < \u2016c\u2016 + 1\nf : \u2115 \u2192 E\nhf : (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 \u2016c\u2016 + 1) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n\u22a2 \u2203 R f, 1 < R \u2227 (\u2200 (n : \u2115), \u2016f n\u2016 \u2264 R) \u2227 \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n[PROOFSTEP]\nexact \u27e8\u2016c\u2016 + 1, f, hc.trans A, hf.1, hf.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nby_contra hfin\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8R, f, Rgt, fle, lef\u27e9 : \u2203 (R : \u211d) (f : \u2115 \u2192 E), 1 < R \u2227 (\u2200 n, \u2016f n\u2016 \u2264 R) \u2227 \u2200 m n, m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016 :=\n  exists_seq_norm_le_one_le_norm_sub hfin\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\n\u22a2 False\n[PROOFSTEP]\nhave rRpos : 0 < r / R := div_pos rpos (zero_lt_one.trans Rgt)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 : \u2203 c : \ud835\udd5c, 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R := NormedField.exists_norm_lt _ rRpos\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\n\u22a2 False\n[PROOFSTEP]\nlet g := fun n : \u2115 => c \u2022 f n\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\n\u22a2 False\n[PROOFSTEP]\nhave A : \u2200 n, g n \u2208 Metric.closedBall (0 : E) r := by\n  intro n\n  simp only [norm_smul, dist_zero_right, Metric.mem_closedBall]\n  calc\n    \u2016c\u2016 * \u2016f n\u2016 \u2264 r / R * R := by gcongr; exact hc.2.le; apply fle\n    _ = r := by\n      field_simp [(zero_lt_one.trans Rgt).ne']\n        -- Porting note: moved type ascriptions because of exists_prop changes\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\n\u22a2 \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 g n \u2208 Metric.closedBall 0 r\n[PROOFSTEP]\nsimp only [norm_smul, dist_zero_right, Metric.mem_closedBall]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 \u2016c\u2016 * \u2016f n\u2016 \u2264 r\n[PROOFSTEP]\ncalc\n  \u2016c\u2016 * \u2016f n\u2016 \u2264 r / R * R := by gcongr; exact hc.2.le; apply fle\n  _ = r := by\n    field_simp [(zero_lt_one.trans Rgt).ne']\n      -- Porting note: moved type ascriptions because of exists_prop changes\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 \u2016c\u2016 * \u2016f n\u2016 \u2264 r / R * R\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 \u2016c\u2016 \u2264 r / R\ncase h\u2082\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 \u2016f n\u2016 \u2264 R\n[PROOFSTEP]\nexact hc.2.le\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 \u2016f n\u2016 \u2264 R\n[PROOFSTEP]\napply fle\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nn : \u2115\n\u22a2 r / R * R = r\n[PROOFSTEP]\nfield_simp [(zero_lt_one.trans Rgt).ne']\n  -- Porting note: moved type ascriptions because of exists_prop changes\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\n\u22a2 False\n[PROOFSTEP]\nobtain\n  \u27e8x : E, _ : x \u2208 Metric.closedBall (0 : E) r, \u03c6 : \u2115 \u2192 \u2115, \u03c6mono : StrictMono \u03c6, \u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\u27e9 :=\n  h.tendsto_subseq A\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\n\u22a2 False\n[PROOFSTEP]\nhave B : CauchySeq (g \u2218 \u03c6) := \u03c6lim.cauchySeq\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8N, hN\u27e9 : \u2203 N : \u2115, \u2200 n : \u2115, N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016 := Metric.cauchySeq_iff'.1 B \u2016c\u2016 hc.1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 False\n[PROOFSTEP]\napply lt_irrefl \u2016c\u2016\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 \u2016c\u2016 < \u2016c\u2016\n[PROOFSTEP]\ncalc\n  \u2016c\u2016 \u2264 dist (g (\u03c6 (N + 1))) (g (\u03c6 N)) := by\n    conv_lhs => rw [\u2190 mul_one \u2016c\u2016]\n    simp only [dist_eq_norm, \u2190 smul_sub, norm_smul]\n    gcongr\n    apply lef _ _ (ne_of_gt _)\n    exact \u03c6mono (Nat.lt_succ_self N)\n  _ < \u2016c\u2016 := hN (N + 1) (Nat.le_succ N)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 \u2016c\u2016 \u2264 dist (g (\u03c6 (N + 1))) (g (\u03c6 N))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 mul_one \u2016c\u2016]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n| \u2016c\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one \u2016c\u2016]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n| \u2016c\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one \u2016c\u2016]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n| \u2016c\u2016\n[PROOFSTEP]\nrw [\u2190 mul_one \u2016c\u2016]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 \u2016c\u2016 * 1 \u2264 dist (g (\u03c6 (N + 1))) (g (\u03c6 N))\n[PROOFSTEP]\nsimp only [dist_eq_norm, \u2190 smul_sub, norm_smul]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 \u2016c\u2016 * 1 \u2264 \u2016c\u2016 * \u2016f (\u03c6 (N + 1)) - f (\u03c6 N)\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 1 \u2264 \u2016f (\u03c6 (N + 1)) - f (\u03c6 N)\u2016\n[PROOFSTEP]\napply lef _ _ (ne_of_gt _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nh : IsCompact (Metric.closedBall 0 r)\nhfin : \u00acFiniteDimensional \ud835\udd5c E\nR : \u211d\nf : \u2115 \u2192 E\nRgt : 1 < R\nfle : \u2200 (n : \u2115), \u2016f n\u2016 \u2264 R\nlef : \u2200 (m n : \u2115), m \u2260 n \u2192 1 \u2264 \u2016f m - f n\u2016\nrRpos : 0 < r / R\nc : \ud835\udd5c\nhc : 0 < \u2016c\u2016 \u2227 \u2016c\u2016 < r / R\ng : \u2115 \u2192 E := fun n => c \u2022 f n\nA : \u2200 (n : \u2115), g n \u2208 Metric.closedBall 0 r\nx : E\nleft\u271d : x \u2208 Metric.closedBall 0 r\n\u03c6 : \u2115 \u2192 \u2115\n\u03c6mono : StrictMono \u03c6\n\u03c6lim : Tendsto (g \u2218 \u03c6) atTop (\ud835\udcdd x)\nB : CauchySeq (g \u2218 \u03c6)\nN : \u2115\nhN : \u2200 (n : \u2115), N \u2264 n \u2192 dist ((g \u2218 \u03c6) n) ((g \u2218 \u03c6) N) < \u2016c\u2016\n\u22a2 \u03c6 N < \u03c6 (N + 1)\n[PROOFSTEP]\nexact \u03c6mono (Nat.lt_succ_self N)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nc : E\nh : IsCompact (Metric.closedBall c r)\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\napply finiteDimensional_of_isCompact_closed_ball\u2080 \ud835\udd5c rpos\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nc : E\nh : IsCompact (Metric.closedBall c r)\n\u22a2 IsCompact (Metric.closedBall 0 r)\n[PROOFSTEP]\nhave : Continuous fun x => -c + x := continuous_const.add continuous_id\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nr : \u211d\nrpos : 0 < r\nc : E\nh : IsCompact (Metric.closedBall c r)\nthis : Continuous fun x => -c + x\n\u22a2 IsCompact (Metric.closedBall 0 r)\n[PROOFSTEP]\nsimpa using h.image this\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\n\u22a2 f = 1 \u2228 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nby_cases h : \u2200 x, f x = 1\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u2200 (x : E), f x = 1\n\u22a2 f = 1 \u2228 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\napply Or.inl\n[GOAL]\ncase pos.h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u2200 (x : E), f x = 1\n\u22a2 f = 1\n[PROOFSTEP]\next x\n[GOAL]\ncase pos.h.h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u2200 (x : E), f x = 1\nx : E\n\u22a2 f x = OfNat.ofNat 1 x\n[PROOFSTEP]\nexact h x\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u00ac\u2200 (x : E), f x = 1\n\u22a2 f = 1 \u2228 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\napply Or.inr\n[GOAL]\ncase neg.h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u00ac\u2200 (x : E), f x = 1\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg.h\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : \u2203 x, f x \u2260 1\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x, f x \u2260 1 := h\n[GOAL]\ncase neg.h.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x \u2260 1\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nhave : Function.mulSupport f \u2208 \ud835\udcdd x := h'f.isOpen_mulSupport.mem_nhds hx\n[GOAL]\ncase neg.h.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x \u2260 1\nthis : Function.mulSupport f \u2208 \ud835\udcdd x\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nobtain \u27e8r : \u211d, rpos : 0 < r, hr : Metric.closedBall x r \u2286 Function.mulSupport f\u27e9 :=\n  Metric.nhds_basis_closedBall.mem_iff.1 this\n[GOAL]\ncase neg.h.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x \u2260 1\nthis : Function.mulSupport f \u2208 \ud835\udcdd x\nr : \u211d\nrpos : 0 < r\nhr : Metric.closedBall x r \u2286 Function.mulSupport f\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nhave : IsCompact (Metric.closedBall x r) :=\n  isCompact_of_isClosed_subset hf Metric.isClosed_ball (hr.trans (subset_mulTSupport _))\n[GOAL]\ncase neg.h.intro.intro.intro\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b3 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2078 : AddCommGroup F'\ninst\u271d\u2077 : Module \ud835\udd5c F'\ninst\u271d\u2076 : TopologicalSpace F'\ninst\u271d\u2075 : TopologicalAddGroup F'\ninst\u271d\u2074 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b3 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : One X\ninst\u271d : T2Space X\nf : E \u2192 X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x \u2260 1\nthis\u271d : Function.mulSupport f \u2208 \ud835\udcdd x\nr : \u211d\nrpos : 0 < r\nhr : Metric.closedBall x r \u2286 Function.mulSupport f\nthis : IsCompact (Metric.closedBall x r)\n\u22a2 FiniteDimensional \ud835\udd5c E\n[PROOFSTEP]\nexact finiteDimensional_of_isCompact_closedBall \ud835\udd5c rpos this\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : LinearMap.ker f = \u22a5\ninst\u271d : FiniteDimensional \ud835\udd5c E\ng : E \u2243\u2097[\ud835\udd5c] { x // x \u2208 LinearMap.range f } := ofInjective f (_ : Function.Injective \u2191f)\nsrc\u271d : Embedding (Subtype.val \u2218 \u2191(ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g))) :=\n  Embedding.comp embedding_subtype_val\n    (Homeomorph.embedding (ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g)))\n\u22a2 IsClosed (range \u2191f)\n[PROOFSTEP]\nhaveI := f.finiteDimensional_range\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2076 : AddCommGroup F'\ninst\u271d\u2075 : Module \ud835\udd5c F'\ninst\u271d\u2074 : TopologicalSpace F'\ninst\u271d\u00b3 : TopologicalAddGroup F'\ninst\u271d\u00b2 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\nf : E \u2192\u2097[\ud835\udd5c] F\nhf : LinearMap.ker f = \u22a5\ninst\u271d : FiniteDimensional \ud835\udd5c E\ng : E \u2243\u2097[\ud835\udd5c] { x // x \u2208 LinearMap.range f } := ofInjective f (_ : Function.Injective \u2191f)\nsrc\u271d : Embedding (Subtype.val \u2218 \u2191(ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g))) :=\n  Embedding.comp embedding_subtype_val\n    (Homeomorph.embedding (ContinuousLinearEquiv.toHomeomorph (toContinuousLinearEquiv g)))\nthis : FiniteDimensional \ud835\udd5c { x // x \u2208 LinearMap.range f }\n\u22a2 IsClosed (range \u2191f)\n[PROOFSTEP]\nsimpa [LinearMap.range_coe f] using f.range.closed_of_finiteDimensional\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : E\n\u22a2 IsClosedMap fun x => x \u2022 c\n[PROOFSTEP]\nby_cases hc : c = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : E\nhc : c = 0\n\u22a2 IsClosedMap fun x => x \u2022 c\n[PROOFSTEP]\nsimp_rw [hc, smul_zero]\n[GOAL]\ncase pos\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : E\nhc : c = 0\n\u22a2 IsClosedMap fun x => 0\n[PROOFSTEP]\nexact isClosedMap_const\n[GOAL]\ncase neg\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2075 : AddCommGroup F'\ninst\u271d\u2074 : Module \ud835\udd5c F'\ninst\u271d\u00b3 : TopologicalSpace F'\ninst\u271d\u00b2 : TopologicalAddGroup F'\ninst\u271d\u00b9 : ContinuousSMul \ud835\udd5c F'\ninst\u271d : CompleteSpace \ud835\udd5c\nc : E\nhc : \u00acc = 0\n\u22a2 IsClosedMap fun x => x \u2022 c\n[PROOFSTEP]\nexact (closedEmbedding_smul_left hc).isClosedMap\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\n\u22a2 Continuous\n    (\u2191{ toLinearMap := \u2191src\u271d, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }).toAddHom.toFun\n[PROOFSTEP]\nrefine' continuous_pi fun i => _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ni : \u03b9\n\u22a2 Continuous fun a =>\n    AddHom.toFun\n      (\u2191{ toLinearMap := \u2191src\u271d, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n            right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }).toAddHom\n      a i\n[PROOFSTEP]\nexact (ContinuousLinearMap.apply \ud835\udd5c E (Pi.single i 1)).continuous\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\n\u22a2 Continuous\n    { toLinearMap := \u2191src\u271d, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n        right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }.invFun\n[PROOFSTEP]\nsimp_rw [LinearEquiv.invFun_eq_symm, LinearEquiv.trans_symm, LinearEquiv.symm_symm]\n  -- Note: added explicit type and removed `change` that tried to achieve the same\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\n\u22a2 Continuous \u2191(LinearEquiv.trans (LinearEquiv.symm (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)) LinearMap.toContinuousLinearMap)\n[PROOFSTEP]\nrefine\n  AddMonoidHomClass.continuous_of_bound\n    (LinearMap.toContinuousLinearMap.toLinearMap.comp (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c).symm.toLinearMap)\n    (Fintype.card \u03b9 : \u211d) fun g => ?_\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\n\u22a2 \u2016\u2191(LinearMap.comp \u2191LinearMap.toContinuousLinearMap \u2191(LinearEquiv.symm (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c))) g\u2016 \u2264\n    \u2191(Fintype.card \u03b9) * \u2016g\u2016\n[PROOFSTEP]\nrw [\u2190 nsmul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\n\u22a2 \u2016\u2191(LinearMap.comp \u2191LinearMap.toContinuousLinearMap \u2191(LinearEquiv.symm (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c))) g\u2016 \u2264\n    Fintype.card \u03b9 \u2022 \u2016g\u2016\n[PROOFSTEP]\nrefine op_norm_le_bound _ (nsmul_nonneg (norm_nonneg g) (Fintype.card \u03b9)) fun t => ?_\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2016\u2191(\u2191(LinearMap.comp \u2191LinearMap.toContinuousLinearMap \u2191(LinearEquiv.symm (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c))) g) t\u2016 \u2264\n    Fintype.card \u03b9 \u2022 \u2016g\u2016 * \u2016t\u2016\n[PROOFSTEP]\nsimp_rw [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, Function.comp_apply, LinearMap.coe_toContinuousLinearMap',\n  LinearEquiv.piRing_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2016\u2211 i : \u03b9, t i \u2022 g i\u2016 \u2264 Fintype.card \u03b9 \u2022 \u2016g\u2016 * \u2016t\u2016\n[PROOFSTEP]\napply le_trans (norm_sum_le _ _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2211 i : \u03b9, \u2016t i \u2022 g i\u2016 \u2264 Fintype.card \u03b9 \u2022 \u2016g\u2016 * \u2016t\u2016\n[PROOFSTEP]\nrw [smul_mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2211 i : \u03b9, \u2016t i \u2022 g i\u2016 \u2264 Fintype.card \u03b9 \u2022 (\u2016g\u2016 * \u2016t\u2016)\n[PROOFSTEP]\nrefine' Finset.sum_le_card_nsmul _ _ _ fun i _ => _\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016t i \u2022 g i\u2016 \u2264 \u2016g\u2016 * \u2016t\u2016\n[PROOFSTEP]\nrw [norm_smul, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016g i\u2016 * \u2016t i\u2016 \u2264 \u2016g\u2016 * \u2016t\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016g i\u2016 \u2264 \u2016g\u2016\n[PROOFSTEP]\napply norm_le_pi_norm\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\n\u03b9 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nsrc\u271d : ((\u03b9 \u2192 \ud835\udd5c) \u2192L[\ud835\udd5c] E) \u2243\u2097[\ud835\udd5c] \u03b9 \u2192 E :=\n  LinearEquiv.trans (LinearEquiv.symm LinearMap.toContinuousLinearMap) (LinearEquiv.piRing \ud835\udd5c E \u03b9 \ud835\udd5c)\ng : \u03b9 \u2192 E\nt : \u03b9 \u2192 \ud835\udd5c\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2016t i\u2016 \u2264 \u2016t\u2016\n[PROOFSTEP]\napply norm_le_pi_norm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\n\u22a2 ContinuousOn f s \u2194 \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\n[PROOFSTEP]\nrefine' \u27e8fun h y => (ContinuousLinearMap.apply \ud835\udd5c F y).continuous.comp_continuousOn h, fun h => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nlet d := finrank \ud835\udd5c E\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\nd : \u2115 := finrank \ud835\udd5c E\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nhave hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) := (finrank_fin_fun \ud835\udd5c).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\nd : \u2115 := finrank \ud835\udd5c E\nhd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c)\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nlet e\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\nd : \u2115 := finrank \ud835\udd5c E\nhd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c)\ne\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nlet e\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F := (e\u2081.arrowCongr (1 : F \u2243L[\ud835\udd5c] F)).trans (ContinuousLinearEquiv.piRing (Fin d))\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\nd : \u2115 := finrank \ud835\udd5c E\nhd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c)\ne\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd\ne\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F :=\n  ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e\u2081 1) (ContinuousLinearEquiv.piRing (Fin d))\n\u22a2 ContinuousOn f s\n[PROOFSTEP]\nrw [\u2190 Function.comp.left_id f, \u2190 e\u2082.symm_comp_self]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\ns : Set X\nh : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s\nd : \u2115 := finrank \ud835\udd5c E\nhd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c)\ne\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd\ne\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F :=\n  ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e\u2081 1) (ContinuousLinearEquiv.piRing (Fin d))\n\u22a2 ContinuousOn ((\u2191(ContinuousLinearEquiv.symm e\u2082) \u2218 \u2191e\u2082) \u2218 f) s\n[PROOFSTEP]\nexact e\u2082.symm.continuous.comp_continuousOn (continuousOn_pi.mpr fun i => h _)\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type w\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nF' : Type x\ninst\u271d\u2077 : AddCommGroup F'\ninst\u271d\u2076 : Module \ud835\udd5c F'\ninst\u271d\u2075 : TopologicalSpace F'\ninst\u271d\u2074 : TopologicalAddGroup F'\ninst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F'\ninst\u271d\u00b2 : CompleteSpace \ud835\udd5c\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FiniteDimensional \ud835\udd5c E\nf : X \u2192 E \u2192L[\ud835\udd5c] F\n\u22a2 Continuous f \u2194 \u2200 (y : E), Continuous fun x => \u2191(f x) y\n[PROOFSTEP]\nsimp_rw [continuous_iff_continuousOn_univ, continuousOn_clm_apply]\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : ProperSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\n\u22a2 ProperSpace E\n[PROOFSTEP]\nset e := ContinuousLinearEquiv.ofFinrankEq (@finrank_fin_fun \ud835\udd5c _ _ (finrank \ud835\udd5c E)).symm\n[GOAL]\n\ud835\udd5c : Type u\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : Type v\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : ProperSpace \ud835\udd5c\ninst\u271d : FiniteDimensional \ud835\udd5c E\ne : E \u2243L[\ud835\udd5c] Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c :=\n  ContinuousLinearEquiv.ofFinrankEq (_ : finrank \ud835\udd5c E = finrank \ud835\udd5c (Fin (finrank \ud835\udd5c E) \u2192 \ud835\udd5c))\n\u22a2 ProperSpace E\n[PROOFSTEP]\nexact e.symm.antilipschitz.properSpace e.symm.continuous e.symm.surjective\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\n\u22a2 \u2203 y, y \u2208 frontier s \u2227 Metric.infDist x s\u1d9c = dist x y\n[PROOFSTEP]\nrcases Metric.exists_mem_closure_infDist_eq_dist (nonempty_compl.2 hs) x with \u27e8y, hys, hyd\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 closure s\u1d9c\nhyd : Metric.infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y, y \u2208 frontier s \u2227 Metric.infDist x s\u1d9c = dist x y\n[PROOFSTEP]\nrw [closure_compl] at hys \n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : Metric.infDist x s\u1d9c = dist x y\n\u22a2 \u2203 y, y \u2208 frontier s \u2227 Metric.infDist x s\u1d9c = dist x y\n[PROOFSTEP]\nrefine' \u27e8y, \u27e8Metric.closedBall_infDist_compl_subset_closure hx <| Metric.mem_closedBall.2 <| ge_of_eq _, hys\u27e9, hyd\u27e9\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nx : E\ns : Set E\nhx : x \u2208 s\nhs : s \u2260 univ\ny : E\nhys : y \u2208 (interior s)\u1d9c\nhyd : Metric.infDist x s\u1d9c = dist x y\n\u22a2 Metric.infDist x s\u1d9c = dist y x\n[PROOFSTEP]\nrwa [dist_comm]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nobtain hx' | hx' : x \u2208 interior K \u222a frontier K :=\n  by\n  rw [\u2190 closure_eq_interior_union_frontier]\n  exact subset_closure hx\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\n\u22a2 x \u2208 interior K \u222a frontier K\n[PROOFSTEP]\nrw [\u2190 closure_eq_interior_union_frontier]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\n\u22a2 x \u2208 closure K\n[PROOFSTEP]\nexact subset_closure hx\n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nhx' : x \u2208 interior K\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nrw [mem_interior_iff_mem_nhds, Metric.nhds_basis_closedBall.mem_iff] at hx' \n[GOAL]\ncase inl\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nhx' : \u2203 i, 0 < i \u2227 Metric.closedBall x i \u2286 K\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nrcases hx' with \u27e8r, hr\u2080, hrK\u27e9\n[GOAL]\ncase inl.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nr : \u211d\nhr\u2080 : 0 < r\nhrK : Metric.closedBall x r \u2286 K\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nhave : FiniteDimensional \u211d E :=\n  finiteDimensional_of_isCompact_closedBall \u211d hr\u2080 (isCompact_of_isClosed_subset hK Metric.isClosed_ball hrK)\n[GOAL]\ncase inl.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nr : \u211d\nhr\u2080 : 0 < r\nhrK : Metric.closedBall x r \u2286 K\nthis : FiniteDimensional \u211d E\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nexact exists_mem_frontier_infDist_compl_eq_dist hx hK.ne_univ\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nhx' : x \u2208 frontier K\n\u22a2 \u2203 y, y \u2208 frontier K \u2227 Metric.infDist x K\u1d9c = dist x y\n[PROOFSTEP]\nrefine' \u27e8x, hx', _\u27e9\n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nhx' : x \u2208 frontier K\n\u22a2 Metric.infDist x K\u1d9c = dist x x\n[PROOFSTEP]\nrw [frontier_eq_closure_inter_closure] at hx' \n[GOAL]\ncase inr\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nK : Set E\nhK : IsCompact K\nhx : x \u2208 K\nhx' : x \u2208 closure K \u2229 closure K\u1d9c\n\u22a2 Metric.infDist x K\u1d9c = dist x x\n[PROOFSTEP]\nrw [Metric.infDist_zero_of_mem_closure hx'.2, dist_self]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\n\u22a2 (Summable fun x => \u2016f x\u2016) \u2194 Summable f\n[PROOFSTEP]\nrefine'\n  \u27e8summable_of_summable_norm, fun hf => _\u27e9\n    -- First we use a finite basis to reduce the problem to the case `E = Fin N \u2192 \u211d`\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\n\u22a2 Summable fun x => \u2016f x\u2016\n[PROOFSTEP]\nsuffices \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\n  by\n  obtain v := finBasis \u211d E\n  set e := v.equivFunL\n  have : Summable fun x => \u2016e (f x)\u2016 := this (e.summable.2 hf)\n  refine' summable_of_norm_bounded _ (this.mul_left \u2191\u2016(e.symm : (Fin (finrank \u211d E) \u2192 \u211d) \u2192L[\u211d] E)\u2016\u208a) fun i => _\n  simpa using (e.symm : (Fin (finrank \u211d E) \u2192 \u211d) \u2192L[\u211d] E).le_op_norm (e <| f i)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\nthis : \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\n\u22a2 Summable fun x => \u2016f x\u2016\n[PROOFSTEP]\nobtain v := finBasis \u211d E\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\nthis : \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\nv : Basis (Fin (finrank \u211d E)) \u211d E\n\u22a2 Summable fun x => \u2016f x\u2016\n[PROOFSTEP]\nset e := v.equivFunL\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\nthis : \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\nv : Basis (Fin (finrank \u211d E)) \u211d E\ne : E \u2243L[\u211d] Fin (finrank \u211d E) \u2192 \u211d := Basis.equivFunL v\n\u22a2 Summable fun x => \u2016f x\u2016\n[PROOFSTEP]\nhave : Summable fun x => \u2016e (f x)\u2016 := this (e.summable.2 hf)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\nthis\u271d : \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\nv : Basis (Fin (finrank \u211d E)) \u211d E\ne : E \u2243L[\u211d] Fin (finrank \u211d E) \u2192 \u211d := Basis.equivFunL v\nthis : Summable fun x => \u2016\u2191e (f x)\u2016\n\u22a2 Summable fun x => \u2016f x\u2016\n[PROOFSTEP]\nrefine' summable_of_norm_bounded _ (this.mul_left \u2191\u2016(e.symm : (Fin (finrank \u211d E) \u2192 \u211d) \u2192L[\u211d] E)\u2016\u208a) fun i => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\nthis\u271d : \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\nv : Basis (Fin (finrank \u211d E)) \u211d E\ne : E \u2243L[\u211d] Fin (finrank \u211d E) \u2192 \u211d := Basis.equivFunL v\nthis : Summable fun x => \u2016\u2191e (f x)\u2016\ni : \u03b1\n\u22a2 \u2016\u2016f i\u2016\u2016 \u2264 \u2191\u2016\u2191(ContinuousLinearEquiv.symm e)\u2016\u208a * \u2016\u2191e (f i)\u2016\n[PROOFSTEP]\nsimpa using (e.symm : (Fin (finrank \u211d E) \u2192 \u211d) \u2192L[\u211d] E).le_op_norm (e <| f i)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\nf : \u03b1 \u2192 E\nhf : Summable f\n\u22a2 \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\n[PROOFSTEP]\nclear! E\n[GOAL]\n\u03b1 : Type u_1\n\u22a2 \u2200 {N : \u2115} {g : \u03b1 \u2192 Fin N \u2192 \u211d}, Summable g \u2192 Summable fun x => \u2016g x\u2016\n[PROOFSTEP]\nintro N g hg\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\n\u22a2 Summable fun x => \u2016g x\u2016\n[PROOFSTEP]\nhave : \u2200 i, Summable fun x => \u2016g x i\u2016 := fun i => (Pi.summable.1 hg i).abs\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\nthis : \u2200 (i : Fin N), Summable fun x => \u2016g x i\u2016\n\u22a2 Summable fun x => \u2016g x\u2016\n[PROOFSTEP]\nrefine' summable_of_norm_bounded _ (summable_sum fun i (_ : i \u2208 Finset.univ) => this i) fun x => _\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\nthis : \u2200 (i : Fin N), Summable fun x => \u2016g x i\u2016\nx : \u03b1\n\u22a2 \u2016\u2016g x\u2016\u2016 \u2264 \u2211 i : Fin N, \u2016g x i\u2016\n[PROOFSTEP]\nrw [norm_norm, pi_norm_le_iff_of_nonneg]\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\nthis : \u2200 (i : Fin N), Summable fun x => \u2016g x i\u2016\nx : \u03b1\n\u22a2 \u2200 (i : Fin N), \u2016g x i\u2016 \u2264 \u2211 i : Fin N, \u2016g x i\u2016\n[PROOFSTEP]\nrefine' fun i => Finset.single_le_sum (f := fun i => \u2016g x i\u2016) (fun i _ => _) (Finset.mem_univ i)\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\nthis : \u2200 (i : Fin N), Summable fun x => \u2016g x i\u2016\nx : \u03b1\ni\u271d i : Fin N\nx\u271d : i \u2208 Finset.univ\n\u22a2 0 \u2264 (fun i => \u2016g x i\u2016) i\n[PROOFSTEP]\nexact norm_nonneg (g x i)\n[GOAL]\n\u03b1 : Type u_1\nN : \u2115\ng : \u03b1 \u2192 Fin N \u2192 \u211d\nhg : Summable g\nthis : \u2200 (i : Fin N), Summable fun x => \u2016g x i\u2016\nx : \u03b1\n\u22a2 0 \u2264 \u2211 i : Fin N, \u2016g x i\u2016\n[PROOFSTEP]\nexact Finset.sum_nonneg fun _ _ => norm_nonneg _\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.FiniteDimension", "llama_tokens": 85376, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920116079208, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5158258964408983}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nf : Bool \u2192 \u03b1\n\u22a2 range f = {f false, f true}\n[PROOFSTEP]\nrw [\u2190 image_univ, univ_eq, image_pair]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bool.Set", "llama_tokens": 55, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5158258903130507}}
{"text": "[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n\u22a2 \u220f p in antidiagonal (n + 1), f p = f (0, n + 1) * \u220f p in antidiagonal n, f (p.fst + 1, p.snd)\n[PROOFSTEP]\nrw [antidiagonal_succ, prod_cons, prod_map]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n\u22a2 f (0, n + 1) *\n      \u220f x in antidiagonal n,\n        f\n          (\u2191(Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl \u2115))\n            x) =\n    f (0, n + 1) * \u220f p in antidiagonal n, f (p.fst + 1, p.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n\u22a2 \u220f p in antidiagonal n, f (Prod.swap p) = \u220f p in antidiagonal n, f p\n[PROOFSTEP]\nconv_lhs => rw [\u2190 map_swap_antidiagonal, Finset.prod_map]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n| \u220f p in antidiagonal n, f (Prod.swap p)\n[PROOFSTEP]\nrw [\u2190 map_swap_antidiagonal, Finset.prod_map]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n| \u220f p in antidiagonal n, f (Prod.swap p)\n[PROOFSTEP]\nrw [\u2190 map_swap_antidiagonal, Finset.prod_map]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n| \u220f p in antidiagonal n, f (Prod.swap p)\n[PROOFSTEP]\nrw [\u2190 map_swap_antidiagonal, Finset.prod_map]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n\u22a2 \u220f p in antidiagonal (n + 1), f p = f (n + 1, 0) * \u220f p in antidiagonal n, f (p.fst, p.snd + 1)\n[PROOFSTEP]\nrw [\u2190 prod_antidiagonal_swap, prod_antidiagonal_succ, \u2190 prod_antidiagonal_swap]\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 M\n\u22a2 f (Prod.swap (0, n + 1)) * \u220f p in antidiagonal n, f (Prod.swap ((Prod.swap p).fst + 1, (Prod.swap p).snd)) =\n    f (n + 1, 0) * \u220f p in antidiagonal n, f (p.fst, p.snd + 1)\n[PROOFSTEP]\nrfl\n[GOAL]\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : AddCommMonoid N\nn : \u2115\nf : \u2115 \u00d7 \u2115 \u2192 \u2115 \u2192 M\np : \u2115 \u00d7 \u2115\nhp : p \u2208 antidiagonal n\n\u22a2 f p n = f p (p.fst + p.snd)\n[PROOFSTEP]\nrw [Nat.mem_antidiagonal.1 hp]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.BigOperators.NatAntidiagonal", "llama_tokens": 1224, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581510799253, "lm_q2_score": 0.6992544273261176, "lm_q1q2_score": 0.5154611007821729}}
{"text": "[GOAL]\nn : Type ?u.20\nm : Type ?u.8\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nR : Type ?u.35\ninst\u271d : Fintype R\n\u22a2 Fintype (Matrix m n R)\n[PROOFSTEP]\nunfold Matrix\n[GOAL]\nn : Type ?u.20\nm : Type ?u.8\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq n\nR : Type ?u.35\ninst\u271d : Fintype R\n\u22a2 Fintype (m \u2192 n \u2192 R)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\n\u22a2 vecMul (\u2191(LinearMap.stdBasis R (fun x => R) i) 1) M j = M i j\n[PROOFSTEP]\nhave : (\u2211 i', (if i = i' then 1 else 0) * M i' j) = M i j := by\n  simp_rw [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\n\u22a2 \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\n[PROOFSTEP]\nsimp_rw [boole_mul, Finset.sum_ite_eq, Finset.mem_univ, if_true]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\n\u22a2 vecMul (\u2191(LinearMap.stdBasis R (fun x => R) i) 1) M j = M i j\n[PROOFSTEP]\nsimp only [vecMul, dotProduct]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\n\u22a2 \u2211 x : m, \u2191(LinearMap.stdBasis R (fun x => R) i) 1 x * M x j = M i j\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.a.h.e'_5\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\nx\u271d : m\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 \u2191(LinearMap.stdBasis R (fun x => R) i) 1 x\u271d = if i = x\u271d then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\nx\u271d : m\na\u271d : x\u271d \u2208 Finset.univ\nh : i = x\u271d\n\u22a2 \u2191(LinearMap.stdBasis R (fun x => R) i) 1 x\u271d = 1\n[PROOFSTEP]\nsimp only [stdBasis_apply]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\nx\u271d : m\na\u271d : x\u271d \u2208 Finset.univ\nh : \u00aci = x\u271d\n\u22a2 \u2191(LinearMap.stdBasis R (fun x => R) i) 1 x\u271d = 0\n[PROOFSTEP]\nsimp only [stdBasis_apply]\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\nx\u271d : m\na\u271d : x\u271d \u2208 Finset.univ\nh : i = x\u271d\n\u22a2 Function.update 0 i 1 x\u271d = 1\n[PROOFSTEP]\nrw [h, Function.update_same]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\nthis : \u2211 i' : m, (if i = i' then 1 else 0) * M i' j = M i j\nx\u271d : m\na\u271d : x\u271d \u2208 Finset.univ\nh : \u00aci = x\u271d\n\u22a2 Function.update 0 i 1 x\u271d = 0\n[PROOFSTEP]\nrw [Function.update_noteq (Ne.symm h), Pi.zero_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf g : (m \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f + (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf g : (m \u2192 R) \u2192\u2097[R] n \u2192 R\ni : m\nj : n\n\u22a2 (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) i j =\n    ((fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f + (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) i\n      j\n[PROOFSTEP]\nsimp only [Pi.add_apply, LinearMap.add_apply, Matrix.add_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nc : R\u1d50\u1d52\u1d56\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 AddHom.toFun\n      { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n        map_add' :=\n          (_ :\n            \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n      (c \u2022 f) =\n    \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n      AddHom.toFun\n        { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n          map_add' :=\n            (_ :\n              \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n        f\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nc : R\u1d50\u1d52\u1d56\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\ni : m\nj : n\n\u22a2 AddHom.toFun\n      { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n        map_add' :=\n          (_ :\n            \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n      (c \u2022 f) i j =\n    (\u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n        AddHom.toFun\n          { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n            map_add' :=\n              (_ :\n                \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n          f)\n      i j\n[PROOFSTEP]\nsimp only [Pi.smul_apply, LinearMap.smul_apply, RingHom.id_apply, Matrix.smul_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 vecMulLinear\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                          (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                  AddHom.toFun\n                      { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                        f) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\napply (Pi.basisFun R m).ext\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 \u2200 (i : m),\n    \u2191(vecMulLinear\n            (AddHom.toFun\n              {\n                  toAddHom :=\n                    { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                            (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                        AddHom.toFun\n                            { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                            (c \u2022 f) =\n                          \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                            AddHom.toFun\n                              { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                          (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                              f) }.toAddHom\n              f))\n        (\u2191(Pi.basisFun R m) i) =\n      \u2191f (\u2191(Pi.basisFun R m) i)\n[PROOFSTEP]\nintro j\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\nj : m\n\u22a2 \u2191(vecMulLinear\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                    map_add' :=\n                      (_ :\n                        \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                          (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                            (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                      AddHom.toFun\n                          { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                          (c \u2022 f) =\n                        \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                          AddHom.toFun\n                            { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                            f) }.toAddHom\n            f))\n      (\u2191(Pi.basisFun R m) j) =\n    \u2191f (\u2191(Pi.basisFun R m) j)\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nf : (m \u2192 R) \u2192\u2097[R] n \u2192 R\nj : m\ni : n\n\u22a2 \u2191(vecMulLinear\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                    map_add' :=\n                      (_ :\n                        \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                          (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                            (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                      AddHom.toFun\n                          { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                          (c \u2022 f) =\n                        \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                          AddHom.toFun\n                            { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                            f) }.toAddHom\n            f))\n      (\u2191(Pi.basisFun R m) j) i =\n    \u2191f (\u2191(Pi.basisFun R m) j) i\n[PROOFSTEP]\nsimp only [Pi.basisFun_apply, Matrix.vecMul_stdBasis, Matrix.vecMulLinear_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n              map_add' :=\n                (_ :\n                  \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                AddHom.toFun\n                    { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                            (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                    (c \u2022 f) =\n                  \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                    AddHom.toFun\n                      { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                      f) }.toAddHom\n      (vecMulLinear M) =\n    M\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\ni : m\nj : n\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n              map_add' :=\n                (_ :\n                  \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                    (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                      (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                        (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R\u1d50\u1d52\u1d56) (f : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                AddHom.toFun\n                    { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                            (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                    (c \u2022 f) =\n                  \u2191(RingHom.id R\u1d50\u1d52\u1d56) c \u2022\n                    AddHom.toFun\n                      { toFun := fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (m \u2192 R) \u2192\u2097[R] n \u2192 R),\n                              (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) (f + g) =\n                                (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) f +\n                                  (fun f i j => \u2191f (\u2191(stdBasis R (fun x => R) i) 1) j) g) }\n                      f) }.toAddHom\n      (vecMulLinear M) i j =\n    M i j\n[PROOFSTEP]\nsimp only [Matrix.vecMul_stdBasis, Matrix.vecMulLinear_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\n\u22a2 \u2191toLinearMapRight' 1 = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.h\nR : Type u_1\ninst\u271d\u00b2 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\ni\u271d x\u271d : m\n\u22a2 \u2191(comp (\u2191toLinearMapRight' 1) (single i\u271d)) 1 x\u271d = \u2191(comp LinearMap.id (single i\u271d)) 1 x\u271d\n[PROOFSTEP]\nsimp [LinearMap.one_apply, stdBasis_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (n \u2192 R) \u2192\u2097[R] m \u2192 R) M' := \u2191(LinearEquiv.symm toMatrixRight') M'\nx : n \u2192 R\n\u22a2 \u2191(\u2191toLinearMapRight' M)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191toLinearMapRight' M'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : n \u2192 R),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : n \u2192 R),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\ndsimp only\n  -- porting note: needed due to non-flat structures\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (n \u2192 R) \u2192\u2097[R] m \u2192 R) M' := \u2191(LinearEquiv.symm toMatrixRight') M'\nx : n \u2192 R\n\u22a2 \u2191(\u2191toLinearMapRight' M) (\u2191(\u2191toLinearMapRight' M') x) = x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLinearMapRight'_mul_apply, hM'M, Matrix.toLinearMapRight'_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (n \u2192 R) \u2192\u2097[R] m \u2192 R) M' := \u2191(LinearEquiv.symm toMatrixRight') M'\nx : m \u2192 R\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191toLinearMapRight' M'),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : n \u2192 R),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : n \u2192 R),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (\u2191(\u2191toLinearMapRight' M) x) =\n    x\n[PROOFSTEP]\ndsimp only\n  -- porting note: needed due to non-flat structures\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : Semiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (n \u2192 R) \u2192\u2097[R] m \u2192 R) M' := \u2191(LinearEquiv.symm toMatrixRight') M'\nx : m \u2192 R\n\u22a2 \u2191(\u2191toLinearMapRight' M') (\u2191(\u2191toLinearMapRight' M) x) = x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLinearMapRight'_mul_apply, hMM', Matrix.toLinearMapRight'_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\n\u22a2 mulVecLin 1 = LinearMap.id\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\ni\u271d x\u271d : n\n\u22a2 \u2191(comp (mulVecLin 1) (single i\u271d)) 1 x\u271d = \u2191(comp LinearMap.id (single i\u271d)) 1 x\u271d\n[PROOFSTEP]\nsimp [Matrix.one_apply, Pi.single_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d : Fintype n\nM : Matrix n n R\n\u22a2 ker (mulVecLin M) = \u22a5 \u2194 \u2200 (v : n \u2192 R), mulVec M v = 0 \u2192 v = 0\n[PROOFSTEP]\nsimp only [Submodule.eq_bot_iff, LinearMap.mem_ker, Matrix.mulVecLin_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d : Fintype n\nM : Matrix m n R\n\u22a2 LinearMap.range (mulVecLin M) = span R (Set.range M\u1d40)\n[PROOFSTEP]\nletI := Classical.decEq n\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d : Fintype n\nM : Matrix m n R\nthis : DecidableEq n := Classical.decEq n\n\u22a2 LinearMap.range (mulVecLin M) = span R (Set.range M\u1d40)\n[PROOFSTEP]\nsimp_rw [range_eq_map, \u2190 iSup_range_stdBasis, Submodule.map_iSup, range_eq_map, \u2190 Ideal.span_singleton_one, Ideal.span,\n  Submodule.map_span, image_image, image_singleton, Matrix.mulVecLin_apply, M.mulVec_stdBasis_apply, iSup_span,\n  range_eq_iUnion]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf g : (n \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf g : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) i j =\n    ((fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g)\n      i j\n[PROOFSTEP]\nsimp only [Pi.add_apply, LinearMap.add_apply, of_apply, Matrix.add_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nc : R\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 AddHom.toFun\n      { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n        map_add' :=\n          (_ :\n            \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n      (c \u2022 f) =\n    \u2191(RingHom.id R) c \u2022\n      AddHom.toFun\n        { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n          map_add' :=\n            (_ :\n              \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n        f\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nc : R\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 AddHom.toFun\n      { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n        map_add' :=\n          (_ :\n            \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n      (c \u2022 f) i j =\n    (\u2191(RingHom.id R) c \u2022\n        AddHom.toFun\n          { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n            map_add' :=\n              (_ :\n                \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n          f)\n      i j\n[PROOFSTEP]\nsimp only [Pi.smul_apply, LinearMap.smul_apply, RingHom.id_apply, of_apply, Matrix.smul_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 mulVecLin\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                map_add' :=\n                  (_ :\n                    \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                          (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n            map_smul' :=\n              (_ :\n                \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                  AddHom.toFun\n                      { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                      (c \u2022 f) =\n                    \u2191(RingHom.id R) c \u2022\n                      AddHom.toFun\n                        { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                          map_add' :=\n                            (_ :\n                              \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                        f) }.toAddHom\n        f) =\n    f\n[PROOFSTEP]\napply (Pi.basisFun R n).ext\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\n\u22a2 \u2200 (i : n),\n    \u2191(mulVecLin\n            (AddHom.toFun\n              {\n                  toAddHom :=\n                    { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                            (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n                  map_smul' :=\n                    (_ :\n                      \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                        AddHom.toFun\n                            { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                            (c \u2022 f) =\n                          \u2191(RingHom.id R) c \u2022\n                            AddHom.toFun\n                              { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                                map_add' :=\n                                  (_ :\n                                    \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                          (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                              f) }.toAddHom\n              f))\n        (\u2191(Pi.basisFun R n) i) =\n      \u2191f (\u2191(Pi.basisFun R n) i)\n[PROOFSTEP]\nintro j\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\nj : n\n\u22a2 \u2191(mulVecLin\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                    map_add' :=\n                      (_ :\n                        \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                          (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                            (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                      AddHom.toFun\n                          { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                          (c \u2022 f) =\n                        \u2191(RingHom.id R) c \u2022\n                          AddHom.toFun\n                            { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                            f) }.toAddHom\n            f))\n      (\u2191(Pi.basisFun R n) j) =\n    \u2191f (\u2191(Pi.basisFun R n) j)\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\nj : n\ni : m\n\u22a2 \u2191(mulVecLin\n          (AddHom.toFun\n            {\n                toAddHom :=\n                  { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                    map_add' :=\n                      (_ :\n                        \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                          (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                            (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n                map_smul' :=\n                  (_ :\n                    \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                      AddHom.toFun\n                          { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                            map_add' :=\n                              (_ :\n                                \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                          (c \u2022 f) =\n                        \u2191(RingHom.id R) c \u2022\n                          AddHom.toFun\n                            { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                              map_add' :=\n                                (_ :\n                                  \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                            f) }.toAddHom\n            f))\n      (\u2191(Pi.basisFun R n) j) i =\n    \u2191f (\u2191(Pi.basisFun R n) j) i\n[PROOFSTEP]\nsimp only [Pi.basisFun_apply, Matrix.mulVec_stdBasis, Matrix.mulVecLin_apply, of_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n              map_add' :=\n                (_ :\n                  \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                AddHom.toFun\n                    { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                            (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                    (c \u2022 f) =\n                  \u2191(RingHom.id R) c \u2022\n                    AddHom.toFun\n                      { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                      f) }.toAddHom\n      (mulVecLin M) =\n    M\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix m n R\ni : m\nj : n\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n              map_add' :=\n                (_ :\n                  \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                    (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                      (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                        (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) },\n          map_smul' :=\n            (_ :\n              \u2200 (c : R) (f : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                AddHom.toFun\n                    { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                      map_add' :=\n                        (_ :\n                          \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                            (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                    (c \u2022 f) =\n                  \u2191(RingHom.id R) c \u2022\n                    AddHom.toFun\n                      { toFun := fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i,\n                        map_add' :=\n                          (_ :\n                            \u2200 (f g : (n \u2192 R) \u2192\u2097[R] m \u2192 R),\n                              (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) (f + g) =\n                                (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) f +\n                                  (fun f => \u2191of fun i j => \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i) g) }\n                      f) }.toAddHom\n      (mulVecLin M) i j =\n    M i j\n[PROOFSTEP]\nsimp only [Matrix.mulVec_stdBasis, Matrix.mulVecLin_apply, of_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191toMatrix' f i j = \u2191f (fun j' => if j' = j then 1 else 0) i\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix', LinearEquiv.coe_mk, of_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) i = \u2191f (fun j' => if j' = j then 1 else 0) i\n[PROOFSTEP]\nrefine\n  congr_fun ?_\n    _\n      -- porting note: `congr` didn't do this\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191f (\u2191(stdBasis R (fun x => R) j) 1) = \u2191f fun j' => if j' = j then 1 else 0\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_6.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 = fun j' => if j' = j then 1 else 0\n[PROOFSTEP]\next j'\n[GOAL]\ncase h.e_6.h.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = if j' = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : j' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 1\n[PROOFSTEP]\nrw [h, stdBasis_same]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ni : m\nj j' : n\nh : \u00acj' = j\n\u22a2 \u2191(stdBasis R (fun x => R) j) 1 j' = 0\n[PROOFSTEP]\napply stdBasis_ne _ _ _ _ h\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\n\u22a2 \u2191toMatrix' id = 1\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\ni\u271d x\u271d : n\n\u22a2 \u2191toMatrix' id i\u271d x\u271d = OfNat.ofNat 1 i\u271d x\u271d\n[PROOFSTEP]\nrw [Matrix.one_apply, LinearMap.toMatrix'_apply, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix l m R\nN : Matrix m n R\nx : n \u2192 R\n\u22a2 \u2191(\u2191toLin' (M * N)) x = \u2191(\u2191toLin' M) (\u2191(\u2191toLin' N) x)\n[PROOFSTEP]\nrw [Matrix.toLin'_mul, LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq l\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ng : (l \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 \u2191toMatrix' (comp f g) = \u2191toMatrix' f * \u2191toMatrix' g\n[PROOFSTEP]\nsuffices f.comp g = Matrix.toLin' (LinearMap.toMatrix' f * LinearMap.toMatrix' g) by\n  rw [this, LinearMap.toMatrix'_toLin']\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq l\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ng : (l \u2192 R) \u2192\u2097[R] n \u2192 R\nthis : comp f g = \u2191toLin' (\u2191toMatrix' f * \u2191toMatrix' g)\n\u22a2 \u2191toMatrix' (comp f g) = \u2191toMatrix' f * \u2191toMatrix' g\n[PROOFSTEP]\nrw [this, LinearMap.toMatrix'_toLin']\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq l\nf : (n \u2192 R) \u2192\u2097[R] m \u2192 R\ng : (l \u2192 R) \u2192\u2097[R] n \u2192 R\n\u22a2 comp f g = \u2191toLin' (\u2191toMatrix' f * \u2191toMatrix' g)\n[PROOFSTEP]\nrw [Matrix.toLin'_mul, Matrix.toLin'_toMatrix', Matrix.toLin'_toMatrix']\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nx : R\n\u22a2 \u2191toMatrix' (\u2191(algebraMap R (Module.End R (n \u2192 R))) x) = \u2191(scalar n) x\n[PROOFSTEP]\nsimp [Module.algebraMap_end_eq_smul_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (m \u2192 R) \u2192\u2097[R] n \u2192 R) M' := \u2191toLin' M'\nx : m \u2192 R\n\u22a2 \u2191(\u2191toLin' M)\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191toLin' M'),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : m \u2192 R),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : m \u2192 R),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin'_mul_apply, hMM', Matrix.toLin'_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (m \u2192 R) \u2192\u2097[R] n \u2192 R) M' := \u2191toLin' M'\nx : n \u2192 R\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191toLin' M'),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : m \u2192 R),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : m \u2192 R),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (\u2191(\u2191toLin' M) x) =\n    x\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => (m \u2192 R) \u2192\u2097[R] n \u2192 R) M' := \u2191toLin' M'\nx : n \u2192 R\n\u22a2 \u2191(\u2191toLin' M') (\u2191(\u2191toLin' M) x) = x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin'_mul_apply, hM'M, Matrix.toLin'_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nf : (n \u2192 R) \u2192\u2097[R] n \u2192 R\ni j : n\n\u22a2 \u2191toMatrixAlgEquiv' f i j = \u2191f (fun j' => if j' = j then 1 else 0) i\n[PROOFSTEP]\nsimp [LinearMap.toMatrixAlgEquiv']\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\n\u22a2 \u2191(toLin v\u2081 v\u2082) (\u2191(toMatrix v\u2081 v\u2082) f) = f\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin_symm, LinearEquiv.apply_symm_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\n\u22a2 \u2191(toMatrix v\u2081 v\u2082) (\u2191(toLin v\u2081 v\u2082) M) = M\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin_symm, LinearEquiv.symm_apply_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj : n\n\u22a2 \u2191(toMatrix v\u2081 v\u2082) f i j = \u2191(\u2191v\u2082.repr (\u2191f (\u2191v\u2081 j))) i\n[PROOFSTEP]\nrw [LinearMap.toMatrix, LinearEquiv.trans_apply, LinearMap.toMatrix'_apply, LinearEquiv.arrowCongr_apply,\n  Basis.equivFun_symm_apply, Finset.sum_eq_single j, if_pos rfl, one_smul, Basis.equivFun_apply]\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj : n\n\u22a2 \u2200 (b : n), b \u2208 Finset.univ \u2192 b \u2260 j \u2192 (if b = j then 1 else 0) \u2022 \u2191v\u2081 b = 0\n[PROOFSTEP]\nintro j' _ hj'\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj j' : n\na\u271d : j' \u2208 Finset.univ\nhj' : j' \u2260 j\n\u22a2 (if j' = j then 1 else 0) \u2022 \u2191v\u2081 j' = 0\n[PROOFSTEP]\nrw [if_neg hj', zero_smul]\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj : n\n\u22a2 \u00acj \u2208 Finset.univ \u2192 (if j = j then 1 else 0) \u2022 \u2191v\u2081 j = 0\n[PROOFSTEP]\nintro hj\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj : n\nhj : \u00acj \u2208 Finset.univ\n\u22a2 (if j = j then 1 else 0) \u2022 \u2191v\u2081 j = 0\n[PROOFSTEP]\nhave := Finset.mem_univ j\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\ni : m\nj : n\nhj : \u00acj \u2208 Finset.univ\nthis : j \u2208 Finset.univ\n\u22a2 (if j = j then 1 else 0) \u2022 \u2191v\u2081 j = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\nv : M\u2081\n\u22a2 \u2191(LinearEquiv.symm (Basis.equivFun v\u2082)) (\u2191(\u2191toLin' M) \u2191(\u2191v\u2081.repr v)) = \u2211 j : m, mulVec M (\u2191(\u2191v\u2081.repr v)) j \u2022 \u2191v\u2082 j\n[PROOFSTEP]\nrw [Matrix.toLin'_apply, v\u2082.equivFun_symm_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\n\u22a2 \u2191(\u2191(toLin v\u2081 v\u2082) M) (\u2191v\u2081 i) = \u2211 j : m, M j i \u2022 \u2191v\u2082 j\n[PROOFSTEP]\nrw [Matrix.toLin_apply, Finset.sum_congr rfl fun j _hj => ?_]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\n\u22a2 mulVec M (\u2191(\u2191v\u2081.repr (\u2191v\u2081 i))) j \u2022 \u2191v\u2082 j = M j i \u2022 \u2191v\u2082 j\n[PROOFSTEP]\nrw [Basis.repr_self, Matrix.mulVec, dotProduct, Finset.sum_eq_single i, Finsupp.single_eq_same, mul_one]\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\n\u22a2 \u2200 (b : n), b \u2208 Finset.univ \u2192 b \u2260 i \u2192 M j b * \u2191(Finsupp.single i 1) b = 0\n[PROOFSTEP]\nintro i' _ i'_ne\n[GOAL]\ncase h\u2080\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\ni' : n\na\u271d : i' \u2208 Finset.univ\ni'_ne : i' \u2260 i\n\u22a2 M j i' * \u2191(Finsupp.single i 1) i' = 0\n[PROOFSTEP]\nrw [Finsupp.single_eq_of_ne i'_ne.symm, mul_zero]\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\n\u22a2 \u00aci \u2208 Finset.univ \u2192 M j i * \u2191(Finsupp.single i 1) i = 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\na\u271d : \u00aci \u2208 Finset.univ\n\u22a2 M j i * \u2191(Finsupp.single i 1) i = 0\n[PROOFSTEP]\nhave := Finset.mem_univ i\n[GOAL]\ncase h\u2081\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM : Matrix m n R\ni : n\nj : m\n_hj : j \u2208 Finset.univ\na\u271d : \u00aci \u2208 Finset.univ\nthis : i \u2208 Finset.univ\n\u22a2 M j i * \u2191(Finsupp.single i 1) i = 0\n[PROOFSTEP]\ncontradiction\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) id = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\ni j : n\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) id i j = OfNat.ofNat 1 i j\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply, Matrix.one_apply, Finsupp.single_apply, eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2077 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2076 : Fintype n\ninst\u271d\u2075 : Fintype m\ninst\u271d\u2074 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2082\ninst\u271d\u00b9 : Module R M\u2081\ninst\u271d : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\n\u22a2 \u2191(toLin v\u2081 v\u2081) 1 = LinearMap.id\n[PROOFSTEP]\nrw [\u2190 LinearMap.toMatrix_id v\u2081, Matrix.toLin_toMatrix]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\ninst\u271d\u00b9 : DecidableEq M\u2081\ninst\u271d : DecidableEq M\u2082\nf : M\u2081 \u2192\u2097[R] M\u2082\nk : m\ni : n\n\u22a2 \u2191(toMatrix (Basis.reindexRange v\u2081) (Basis.reindexRange v\u2082)) f\n      { val := \u2191v\u2082 k, property := (_ : \u2191v\u2082 k \u2208 Set.range \u2191v\u2082) }\n      { val := \u2191v\u2081 i, property := (_ : \u2191v\u2081 i \u2208 Set.range \u2191v\u2081) } =\n    \u2191(toMatrix v\u2081 v\u2082) f k i\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrix_apply, Basis.reindexRange_self, Basis.reindexRange_repr]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nf : M\u2082 \u2192\u2097[R] M\u2083\ng : M\u2081 \u2192\u2097[R] M\u2082\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (comp f g) = \u2191(toMatrix v\u2082 v\u2083) f * \u2191(toMatrix v\u2081 v\u2082) g\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrix, LinearEquiv.trans_apply, LinearEquiv.arrowCongr_comp _ v\u2082.equivFun,\n  LinearMap.toMatrix'_comp]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf g : M\u2081 \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) (f * g) = \u2191(toMatrix v\u2081 v\u2081) f * \u2191(toMatrix v\u2081 v\u2081) g\n[PROOFSTEP]\nrw [LinearMap.mul_eq_comp, LinearMap.toMatrix_comp v\u2081 v\u2081 v\u2081 f g]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nx : R\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) (\u2191(algebraMap R (Module.End R M\u2081)) x) = \u2191(scalar n) x\n[PROOFSTEP]\nsimp [Module.algebraMap_end_eq_smul_id, LinearMap.toMatrix_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2082\nx : M\u2081\n\u22a2 mulVec (\u2191(toMatrix v\u2081 v\u2082) f) \u2191(\u2191v\u2081.repr x) = \u2191(\u2191v\u2082.repr (\u2191f x))\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2082\nx : M\u2081\ni : m\n\u22a2 mulVec (\u2191(toMatrix v\u2081 v\u2082) f) (\u2191(\u2191v\u2081.repr x)) i = \u2191(\u2191v\u2082.repr (\u2191f x)) i\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin'_apply, LinearMap.toMatrix, LinearEquiv.trans_apply, Matrix.toLin'_toMatrix',\n  LinearEquiv.arrowCongr_apply, v\u2082.equivFun_apply]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2082\nx : M\u2081\ni : m\n\u22a2 \u2191(\u2191v\u2082.repr (\u2191f (\u2191(LinearEquiv.symm (Basis.equivFun v\u2081)) \u2191(\u2191v\u2081.repr x)))) i = \u2191(\u2191v\u2082.repr (\u2191f x)) i\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_a.h.e_6.h.h.e_6.h\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2082\nx : M\u2081\ni : m\n\u22a2 \u2191(LinearEquiv.symm (Basis.equivFun v\u2081)) \u2191(\u2191v\u2081.repr x) = x\n[PROOFSTEP]\nexact v\u2081.equivFun.symm_apply_apply x\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq l\nb : Basis l R M\u2081\nb' : Basis l R M\u2082\n\u22a2 \u2191(toMatrix b' b) \u2191(Basis.equiv b' b (Equiv.refl l)) = 1\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq l\nb : Basis l R M\u2081\nb' : Basis l R M\u2082\ni j : l\n\u22a2 \u2191(toMatrix b' b) (\u2191(Basis.equiv b' b (Equiv.refl l))) i j = OfNat.ofNat 1 i j\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply, Matrix.one_apply, Finsupp.single_apply, eq_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\n\u22a2 \u2191(toLin v\u2081 v\u2083) (A * B) = comp (\u2191(toLin v\u2082 v\u2083) A) (\u2191(toLin v\u2081 v\u2082) B)\n[PROOFSTEP]\napply (LinearMap.toMatrix v\u2081 v\u2083).injective\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (\u2191(toLin v\u2081 v\u2083) (A * B)) = \u2191(toMatrix v\u2081 v\u2083) (comp (\u2191(toLin v\u2082 v\u2083) A) (\u2191(toLin v\u2081 v\u2082) B))\n[PROOFSTEP]\nhaveI : DecidableEq l := fun _ _ => Classical.propDecidable _\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (\u2191(toLin v\u2081 v\u2083) (A * B)) = \u2191(toMatrix v\u2081 v\u2083) (comp (\u2191(toLin v\u2082 v\u2083) A) (\u2191(toLin v\u2081 v\u2082) B))\n[PROOFSTEP]\nrw [LinearMap.toMatrix_comp v\u2081 v\u2082 v\u2083]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (\u2191(toLin v\u2081 v\u2083) (A * B)) =\n    \u2191(toMatrix v\u2082 v\u2083) (\u2191(toLin v\u2082 v\u2083) A) * \u2191(toMatrix v\u2081 v\u2082) (\u2191(toLin v\u2081 v\u2082) B)\n[PROOFSTEP]\nrepeat' rw [LinearMap.toMatrix_toLin]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n\u22a2 \u2191(toMatrix v\u2081 v\u2083) (\u2191(toLin v\u2081 v\u2083) (A * B)) =\n    \u2191(toMatrix v\u2082 v\u2083) (\u2191(toLin v\u2082 v\u2083) A) * \u2191(toMatrix v\u2081 v\u2082) (\u2191(toLin v\u2081 v\u2082) B)\n[PROOFSTEP]\nrw [LinearMap.toMatrix_toLin]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n\u22a2 A * B = \u2191(toMatrix v\u2082 v\u2083) (\u2191(toLin v\u2082 v\u2083) A) * \u2191(toMatrix v\u2081 v\u2082) (\u2191(toLin v\u2081 v\u2082) B)\n[PROOFSTEP]\nrw [LinearMap.toMatrix_toLin]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nthis : DecidableEq l\n\u22a2 A * B = A * \u2191(toMatrix v\u2081 v\u2082) (\u2191(toLin v\u2081 v\u2082) B)\n[PROOFSTEP]\nrw [LinearMap.toMatrix_toLin]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u00b9 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : Fintype m\ninst\u271d\u2078 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : Module R M\u2081\ninst\u271d\u2074 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b2 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d\u00b9 : Fintype l\ninst\u271d : DecidableEq m\nA : Matrix l m R\nB : Matrix m n R\nx : M\u2081\n\u22a2 \u2191(\u2191(toLin v\u2081 v\u2083) (A * B)) x = \u2191(\u2191(toLin v\u2082 v\u2083) A) (\u2191(\u2191(toLin v\u2081 v\u2082) B) x)\n[PROOFSTEP]\nrw [Matrix.toLin_mul v\u2081 v\u2082, LinearMap.comp_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2076 : AddCommMonoid M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2081\ninst\u271d\u00b3 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => M\u2081 \u2192\u2097[R] M\u2082) M := \u2191(toLin v\u2081 v\u2082) M\nx : M\u2081\n\u22a2 \u2191(\u2191(toLin v\u2082 v\u2081) M')\n      (AddHom.toFun\n        {\n            toAddHom :=\n              { toFun := \u2191(\u2191(toLin v\u2081 v\u2082) M),\n                map_add' :=\n                  (_ :\n                    \u2200 (x y : M\u2081),\n                      AddHom.toFun src\u271d.toAddHom (x + y) =\n                        AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n            map_smul' :=\n              (_ :\n                \u2200 (r : R) (x : M\u2081),\n                  AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n        x) =\n    x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin_mul_apply, hM'M, Matrix.toLin_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2076 : AddCommMonoid M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2081\ninst\u271d\u00b3 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => M\u2081 \u2192\u2097[R] M\u2082) M := \u2191(toLin v\u2081 v\u2082) M\nx : M\u2082\n\u22a2 AddHom.toFun\n      {\n          toAddHom :=\n            { toFun := \u2191(\u2191(toLin v\u2081 v\u2082) M),\n              map_add' :=\n                (_ :\n                  \u2200 (x y : M\u2081),\n                    AddHom.toFun src\u271d.toAddHom (x + y) = AddHom.toFun src\u271d.toAddHom x + AddHom.toFun src\u271d.toAddHom y) },\n          map_smul' :=\n            (_ :\n              \u2200 (r : R) (x : M\u2081),\n                AddHom.toFun src\u271d.toAddHom (r \u2022 x) = \u2191(RingHom.id R) r \u2022 AddHom.toFun src\u271d.toAddHom x) }.toAddHom\n      (\u2191(\u2191(toLin v\u2082 v\u2081) M') x) =\n    x\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2076 : AddCommMonoid M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2081\ninst\u271d\u00b3 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nsrc\u271d : (fun x => M\u2081 \u2192\u2097[R] M\u2082) M := \u2191(toLin v\u2081 v\u2082) M\nx : M\u2082\n\u22a2 \u2191(\u2191(toLin v\u2081 v\u2082) M) (\u2191(\u2191(toLin v\u2082 v\u2081) M') x) = x\n[PROOFSTEP]\nrw [\u2190 Matrix.toLin_mul_apply, hMM', Matrix.toLin_one, id_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toLinAlgEquiv v\u2081) (\u2191(toMatrixAlgEquiv v\u2081) f) = f\n[PROOFSTEP]\nrw [\u2190 Matrix.toLinAlgEquiv_symm, AlgEquiv.apply_symm_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nM : Matrix n n R\n\u22a2 \u2191(toMatrixAlgEquiv v\u2081) (\u2191(toLinAlgEquiv v\u2081) M) = M\n[PROOFSTEP]\nrw [\u2190 Matrix.toLinAlgEquiv_symm, AlgEquiv.symm_apply_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf : M\u2081 \u2192\u2097[R] M\u2081\ni j : n\n\u22a2 \u2191(toMatrixAlgEquiv v\u2081) f i j = \u2191(\u2191v\u2081.repr (\u2191f (\u2191v\u2081 j))) i\n[PROOFSTEP]\nsimp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nM : Matrix n n R\nv : M\u2081\n\u22a2 \u2191(LinearEquiv.symm (Basis.equivFun v\u2081)) (\u2191(\u2191toLinAlgEquiv' M) \u2191(\u2191v\u2081.repr v)) =\n    \u2211 j : n, mulVec M (\u2191(\u2191v\u2081.repr v)) j \u2022 \u2191v\u2081 j\n[PROOFSTEP]\nrw [Matrix.toLinAlgEquiv'_apply, v\u2081.equivFun_symm_apply]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\n\u22a2 \u2191(toMatrixAlgEquiv v\u2081) id = 1\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrixAlgEquiv, AlgEquiv.ofLinearEquiv_apply, LinearMap.toMatrix_id]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\n\u22a2 \u2191(toLinAlgEquiv v\u2081) 1 = LinearMap.id\n[PROOFSTEP]\nrw [\u2190 LinearMap.toMatrixAlgEquiv_id v\u2081, Matrix.toLinAlgEquiv_toMatrixAlgEquiv]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9\u2070 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2079 : Fintype n\ninst\u271d\u2078 : Fintype m\ninst\u271d\u2077 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2076 : AddCommMonoid M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : Module R M\u2081\ninst\u271d\u00b3 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b2 : AddCommMonoid M\u2083\ninst\u271d\u00b9 : Module R M\u2083\nv\u2083 : Basis l R M\u2083\ninst\u271d : DecidableEq M\u2081\nf : M\u2081 \u2192\u2097[R] M\u2081\nk i : n\n\u22a2 \u2191(toMatrixAlgEquiv (Basis.reindexRange v\u2081)) f { val := \u2191v\u2081 k, property := (_ : \u2191v\u2081 k \u2208 Set.range \u2191v\u2081) }\n      { val := \u2191v\u2081 i, property := (_ : \u2191v\u2081 i \u2208 Set.range \u2191v\u2081) } =\n    \u2191(toMatrixAlgEquiv v\u2081) f k i\n[PROOFSTEP]\nsimp_rw [LinearMap.toMatrixAlgEquiv_apply, Basis.reindexRange_self, Basis.reindexRange_repr]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf g : M\u2081 \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toMatrixAlgEquiv v\u2081) (comp f g) = \u2191(toMatrixAlgEquiv v\u2081) f * \u2191(toMatrixAlgEquiv v\u2081) g\n[PROOFSTEP]\nsimp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_comp v\u2081 v\u2081 v\u2081 f g]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nf g : M\u2081 \u2192\u2097[R] M\u2081\n\u22a2 \u2191(toMatrixAlgEquiv v\u2081) (f * g) = \u2191(toMatrixAlgEquiv v\u2081) f * \u2191(toMatrixAlgEquiv v\u2081) g\n[PROOFSTEP]\nrw [LinearMap.mul_eq_comp, LinearMap.toMatrixAlgEquiv_comp v\u2081 f g]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nA B : Matrix n n R\n\u22a2 \u2191(toLinAlgEquiv v\u2081) (A * B) = comp (\u2191(toLinAlgEquiv v\u2081) A) (\u2191(toLinAlgEquiv v\u2081) B)\n[PROOFSTEP]\nconvert Matrix.toLin_mul v\u2081 v\u2081 v\u2081 A B\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\na b c d : R\nx : R \u00d7 R\n\u22a2 \u2191(\u2191(toLin (Basis.finTwoProd R) (Basis.finTwoProd R)) (\u2191of ![![a, b], ![c, d]])) x =\n    (a * x.fst + b * x.snd, c * x.fst + d * x.snd)\n[PROOFSTEP]\nsimp [Matrix.toLin_apply, Matrix.mulVec, Matrix.dotProduct, (Prod.smul_mk)]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nx : R\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) (DistribMulAction.toLinearMap R M\u2081 x) = Matrix.diagonal fun x_1 => x\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\ninst\u271d\u2079 : CommSemiring R\nl : Type u_2\nm : Type u_3\nn : Type u_4\ninst\u271d\u2078 : Fintype n\ninst\u271d\u2077 : Fintype m\ninst\u271d\u2076 : DecidableEq n\nM\u2081 : Type u_5\nM\u2082 : Type u_6\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\u2081\ninst\u271d\u00b2 : Module R M\u2082\nv\u2081 : Basis n R M\u2081\nv\u2082 : Basis m R M\u2082\nM\u2083 : Type u_7\ninst\u271d\u00b9 : AddCommMonoid M\u2083\ninst\u271d : Module R M\u2083\nv\u2083 : Basis l R M\u2083\nx : R\ni\u271d x\u271d : n\n\u22a2 \u2191(toMatrix v\u2081 v\u2081) (DistribMulAction.toLinearMap R M\u2081 x) i\u271d x\u271d = Matrix.diagonal (fun x_1 => x) i\u271d x\u271d\n[PROOFSTEP]\nrw [LinearMap.toMatrix_apply, DistribMulAction.toLinearMap_apply, LinearEquiv.map_smul, Basis.repr_self,\n  Finsupp.smul_single_one, Finsupp.single_eq_pi_single, Matrix.diagonal_apply, Pi.single_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx : S\ni j : m\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) x) i j = \u2191(\u2191b.repr (x * \u2191b j)) i\n[PROOFSTEP]\nsimp only [LinearMap.toMatrix_apply', coe_lmul_eq_mul, LinearMap.mul_apply']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\n\u22a2 (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1\n[PROOFSTEP]\ndsimp only\n  -- porting node: needed due to new-style structures\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) 1) = 1\n[PROOFSTEP]\nrw [AlgHom.map_one, LinearMap.toMatrix_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx y : S\n\u22a2 OneHom.toFun\n      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n      (x * y) =\n    OneHom.toFun\n        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n        x *\n      OneHom.toFun\n        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n        y\n[PROOFSTEP]\ndsimp only\n  -- porting node: needed due to new-style structures\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx y : S\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) (x * y)) = \u2191(toMatrix b b) (\u2191(lmul R S) x) * \u2191(toMatrix b b) (\u2191(lmul R S) y)\n[PROOFSTEP]\nrw [AlgHom.map_mul, LinearMap.toMatrix_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n              map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : S),\n                OneHom.toFun\n                    { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                      map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                    (x * y) =\n                  OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      x *\n                    OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      y) })\n      0 =\n    0\n[PROOFSTEP]\ndsimp only\n  -- porting node: needed due to new-style structures\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) 0) = 0\n[PROOFSTEP]\nrw [AlgHom.map_zero, LinearEquiv.map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx y : S\n\u22a2 OneHom.toFun\n      (\u2191{\n          toOneHom :=\n            { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n              map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : S),\n                OneHom.toFun\n                    { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                      map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                    (x * y) =\n                  OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      x *\n                    OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      y) })\n      (x + y) =\n    OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      (x * y) =\n                    OneHom.toFun\n                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                        x *\n                      OneHom.toFun\n                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                        y) })\n        x +\n      OneHom.toFun\n        (\u2191{\n            toOneHom :=\n              { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n            map_mul' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                      (x * y) =\n                    OneHom.toFun\n                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                        x *\n                      OneHom.toFun\n                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                        y) })\n        y\n[PROOFSTEP]\ndsimp only\n  -- porting node: needed due to new-style structures\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx y : S\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) (x + y)) = \u2191(toMatrix b b) (\u2191(lmul R S) x) + \u2191(toMatrix b b) (\u2191(lmul R S) y)\n[PROOFSTEP]\nrw [AlgHom.map_add, LinearEquiv.map_add]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nr : R\n\u22a2 OneHom.toFun\n      (\u2191\u2191{\n            toMonoidHom :=\n              {\n                toOneHom :=\n                  { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                    map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n                map_mul' :=\n                  (_ :\n                    \u2200 (x y : S),\n                      OneHom.toFun\n                          { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                            map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                          (x * y) =\n                        OneHom.toFun\n                            { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                              map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                            x *\n                          OneHom.toFun\n                            { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                              map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                            y) },\n            map_zero' :=\n              (_ :\n                OneHom.toFun\n                    (\u2191{\n                        toOneHom :=\n                          { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                            map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n                        map_mul' :=\n                          (_ :\n                            \u2200 (x y : S),\n                              OneHom.toFun\n                                  { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                    map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                  (x * y) =\n                                OneHom.toFun\n                                    { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                      map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                    x *\n                                  OneHom.toFun\n                                    { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                      map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                    y) })\n                    0 =\n                  0),\n            map_add' :=\n              (_ :\n                \u2200 (x y : S),\n                  OneHom.toFun\n                      (\u2191{\n                          toOneHom :=\n                            { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                              map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n                          map_mul' :=\n                            (_ :\n                              \u2200 (x y : S),\n                                OneHom.toFun\n                                    { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                      map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                    (x * y) =\n                                  OneHom.toFun\n                                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                      x *\n                                    OneHom.toFun\n                                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                      y) })\n                      (x + y) =\n                    OneHom.toFun\n                        (\u2191{\n                            toOneHom :=\n                              { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n                            map_mul' :=\n                              (_ :\n                                \u2200 (x y : S),\n                                  OneHom.toFun\n                                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                      (x * y) =\n                                    OneHom.toFun\n                                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                        x *\n                                      OneHom.toFun\n                                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                        y) })\n                        x +\n                      OneHom.toFun\n                        (\u2191{\n                            toOneHom :=\n                              { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) },\n                            map_mul' :=\n                              (_ :\n                                \u2200 (x y : S),\n                                  OneHom.toFun\n                                      { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                        map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                      (x * y) =\n                                    OneHom.toFun\n                                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                        x *\n                                      OneHom.toFun\n                                        { toFun := fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x),\n                                          map_one' := (_ : (fun x => \u2191(toMatrix b b) (\u2191(lmul R S) x)) 1 = 1) }\n                                        y) })\n                        y) })\n      (\u2191(algebraMap R S) r) =\n    \u2191(algebraMap R (Matrix m m R)) r\n[PROOFSTEP]\ndsimp only\n  -- porting node: needed due to new-style structures\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nr : R\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) (\u2191(algebraMap R S) r)) = \u2191(algebraMap R (Matrix m m R)) r\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nr : R\ni\u271d x\u271d : m\n\u22a2 \u2191(toMatrix b b) (\u2191(lmul R S) (\u2191(algebraMap R S) r)) i\u271d x\u271d = \u2191(algebraMap R (Matrix m m R)) r i\u271d x\u271d\n[PROOFSTEP]\nrw [lmul_algebraMap, toMatrix_lsmul, algebraMap_eq_diagonal, Pi.algebraMap_def, Algebra.id.map_eq_self]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx : S\ni j : m\n\u22a2 \u2191(leftMulMatrix b) x i j = \u2191(\u2191b.repr (x * \u2191b j)) i\n[PROOFSTEP]\nrw [leftMulMatrix_apply, toMatrix_lmul' b x i j]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : Ring S\ninst\u271d\u00b2 : Algebra R S\nm : Type u_3\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nb : Basis m R S\nx x' : S\nh : \u2191(leftMulMatrix b) x = \u2191(leftMulMatrix b) x'\n\u22a2 \u2191(\u2191(lmul R S) x) 1 = \u2191(\u2191(lmul R S) x') 1\n[PROOFSTEP]\nrw [(LinearMap.toMatrix b b).injective h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : T\nik jk : m \u00d7 n\n\u22a2 \u2191(leftMulMatrix (Basis.smul b c)) x ik jk = \u2191(leftMulMatrix b) (\u2191(leftMulMatrix c) x ik.snd jk.snd) ik.fst jk.fst\n[PROOFSTEP]\nsimp only [leftMulMatrix_apply, LinearMap.toMatrix_apply, mul_comm, Basis.smul_apply, Basis.smul_repr,\n  Finsupp.smul_apply, id.smul_eq_mul, LinearEquiv.map_smul, mul_smul_comm, coe_lmul_eq_mul, LinearMap.mul_apply']\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\n\u22a2 \u2191(leftMulMatrix (Basis.smul b c)) (\u2191(algebraMap S T) x) = blockDiagonal fun x_1 => \u2191(leftMulMatrix b) x\n[PROOFSTEP]\next \u27e8i, k\u27e9 \u27e8j, k'\u27e9\n[GOAL]\ncase a.mk.h.mk\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\n\u22a2 \u2191(leftMulMatrix (Basis.smul b c)) (\u2191(algebraMap S T) x) (i, k) (j, k') =\n    blockDiagonal (fun x_1 => \u2191(leftMulMatrix b) x) (i, k) (j, k')\n[PROOFSTEP]\nrw [smul_leftMulMatrix, AlgHom.commutes, blockDiagonal_apply, algebraMap_matrix_apply]\n[GOAL]\ncase a.mk.h.mk\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\n\u22a2 \u2191(leftMulMatrix b) (if (i, k).snd = (j, k').snd then \u2191(algebraMap S S) x else 0) (i, k).fst (j, k').fst =\n    if (i, k).snd = (j, k').snd then \u2191(leftMulMatrix b) x (i, k).fst (j, k').fst else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\nh : (i, k).snd = (j, k').snd\n\u22a2 \u2191(leftMulMatrix b) (if (i, k).snd = (j, k').snd then \u2191(algebraMap S S) x else 0) (i, k).fst (j, k').fst =\n    \u2191(leftMulMatrix b) x (i, k).fst (j, k').fst\n[PROOFSTEP]\nsimp only at h \n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\nh : \u00ac(i, k).snd = (j, k').snd\n\u22a2 \u2191(leftMulMatrix b) (if (i, k).snd = (j, k').snd then \u2191(algebraMap S S) x else 0) (i, k).fst (j, k').fst = 0\n[PROOFSTEP]\nsimp only at h \n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\nh : k = k'\n\u22a2 \u2191(leftMulMatrix b) (if (i, k).snd = (j, k').snd then \u2191(algebraMap S S) x else 0) (i, k).fst (j, k').fst =\n    \u2191(leftMulMatrix b) x (i, k).fst (j, k').fst\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni : m\nk : n\nj : m\nk' : n\nh : \u00ack = k'\n\u22a2 \u2191(leftMulMatrix b) (if (i, k).snd = (j, k').snd then \u2191(algebraMap S S) x else 0) (i, k).fst (j, k').fst = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni j : m\nk : n\n\u22a2 \u2191(leftMulMatrix (Basis.smul b c)) (\u2191(algebraMap S T) x) (i, k) (j, k) = \u2191(leftMulMatrix b) x i j\n[PROOFSTEP]\nrw [smul_leftMulMatrix_algebraMap, blockDiagonal_apply_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Ring T\ninst\u271d\u2077 : Algebra R S\ninst\u271d\u2076 : Algebra S T\ninst\u271d\u2075 : Algebra R T\ninst\u271d\u2074 : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nb : Basis m R S\nc : Basis n S T\nx : S\ni j : m\nk k' : n\nh : k \u2260 k'\n\u22a2 \u2191(leftMulMatrix (Basis.smul b c)) (\u2191(algebraMap S T) x) (i, k) (j, k') = 0\n[PROOFSTEP]\nrw [smul_leftMulMatrix_algebraMap, blockDiagonal_apply_ne _ _ _ h]\n[GOAL]\nR : Type v\ninst\u271d\u2077 : CommRing R\nn : Type u_1\ninst\u271d\u2076 : DecidableEq n\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : AddCommGroup M\u2082\ninst\u271d : Module R M\u2082\ne : M\u2081 \u2243\u2097[R] M\u2082\nsrc\u271d : Module.End R M\u2081 \u2243\u2097[R] Module.End R M\u2082 := conj e\nf g : Module.End R M\u2081\n\u22a2 Equiv.toFun\n      { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n        right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n      (f * g) =\n    Equiv.toFun\n        { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n          right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        f *\n      Equiv.toFun\n        { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n          right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n        g\n[PROOFSTEP]\napply e.arrowCongr_comp\n[GOAL]\nR : Type v\ninst\u271d\u2077 : CommRing R\nn : Type u_1\ninst\u271d\u2076 : DecidableEq n\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : AddCommGroup M\u2082\ninst\u271d : Module R M\u2082\ne : M\u2081 \u2243\u2097[R] M\u2082\nsrc\u271d : Module.End R M\u2081 \u2243\u2097[R] Module.End R M\u2082 := conj e\nr : R\n\u22a2 Equiv.toFun\n      { toFun := src\u271d.toFun, invFun := src\u271d.invFun, left_inv := (_ : Function.LeftInverse src\u271d.invFun src\u271d.toFun),\n        right_inv := (_ : Function.RightInverse src\u271d.invFun src\u271d.toFun) }\n      (\u2191(algebraMap R (Module.End R M\u2081)) r) =\n    \u2191(algebraMap R (Module.End R M\u2082)) r\n[PROOFSTEP]\nchange e.conj (r \u2022 LinearMap.id) = r \u2022 LinearMap.id\n[GOAL]\nR : Type v\ninst\u271d\u2077 : CommRing R\nn : Type u_1\ninst\u271d\u2076 : DecidableEq n\nM : Type u_2\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommGroup M\u2081\ninst\u271d\u00b2 : Module R M\u2081\ninst\u271d\u00b9 : AddCommGroup M\u2082\ninst\u271d : Module R M\u2082\ne : M\u2081 \u2243\u2097[R] M\u2082\nsrc\u271d : Module.End R M\u2081 \u2243\u2097[R] Module.End R M\u2082 := conj e\nr : R\n\u22a2 \u2191(conj e) (r \u2022 LinearMap.id) = r \u2022 LinearMap.id\n[PROOFSTEP]\nrw [LinearEquiv.map_smul, LinearEquiv.conj_id]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.ToLin", "llama_tokens": 45910, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303087996142, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.5149157623025867}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\nF' : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\n\u03c6' : (i : \u03b9) \u2192 E \u2192L[\ud835\udd5c] F' i\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\n\u03a6' : E \u2192L[\ud835\udd5c] (i : \u03b9) \u2192 F' i\n\u22a2 HasStrictFDerivAt \u03a6 \u03a6' x \u2194 \u2200 (i : \u03b9), HasStrictFDerivAt (fun x => \u03a6 x i) (comp (proj i) \u03a6') x\n[PROOFSTEP]\nsimp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\nF' : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\n\u03c6' : (i : \u03b9) \u2192 E \u2192L[\ud835\udd5c] F' i\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\n\u03a6' : E \u2192L[\ud835\udd5c] (i : \u03b9) \u2192 F' i\n\u22a2 ((fun p => \u03a6 p.fst - \u03a6 p.snd - \u2191\u03a6' (p.fst - p.snd)) =o[\ud835\udcdd (x, x)] fun p => p.fst - p.snd) \u2194\n    \u2200 (i : \u03b9),\n      (fun p => \u03a6 p.fst i - \u03a6 p.snd i - \u2191(comp (proj i) \u03a6') (p.fst - p.snd)) =o[\ud835\udcdd (x, x)] fun p => p.fst - p.snd\n[PROOFSTEP]\nexact isLittleO_pi\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\nF' : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\n\u03c6' : (i : \u03b9) \u2192 E \u2192L[\ud835\udd5c] F' i\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\n\u03a6' : E \u2192L[\ud835\udd5c] (i : \u03b9) \u2192 F' i\n\u22a2 HasFDerivAtFilter \u03a6 \u03a6' x L \u2194 \u2200 (i : \u03b9), HasFDerivAtFilter (fun x => \u03a6 x i) (comp (proj i) \u03a6') x L\n[PROOFSTEP]\nsimp only [HasFDerivAtFilter, ContinuousLinearMap.coe_pi]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup G'\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c G'\nf f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\n\u03b9 : Type u_6\ninst\u271d\u00b2 : Fintype \u03b9\nF' : \u03b9 \u2192 Type u_7\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\n\u03c6' : (i : \u03b9) \u2192 E \u2192L[\ud835\udd5c] F' i\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\n\u03a6' : E \u2192L[\ud835\udd5c] (i : \u03b9) \u2192 F' i\n\u22a2 ((fun x' => \u03a6 x' - \u03a6 x - \u2191\u03a6' (x' - x)) =o[L] fun x' => x' - x) \u2194\n    \u2200 (i : \u03b9), (fun x' => \u03a6 x' i - \u03a6 x i - \u2191(comp (proj i) \u03a6') (x' - x)) =o[L] fun x' => x' - x\n[PROOFSTEP]\nexact isLittleO_pi\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.FDeriv.Prod", "llama_tokens": 1875, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673223709251, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.514878516248395}}
{"text": "[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\n\u22a2 FG (\u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)))\n[PROOFSTEP]\nrcases fg_def.1 h1 with \u27e8p, hp, hp'\u27e9\n[GOAL]\ncase intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 FG (\u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)))\n[PROOFSTEP]\nrcases fg_def.1 h2 with \u27e8q, hq, hq'\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 FG (\u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)))\n[PROOFSTEP]\nrefine' fg_def.2 \u27e8p * q, hp.mul hq, le_antisymm _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 span R (p * q) \u2264 \u2191Subalgebra.toSubmodule (adjoin R (s \u222a t))\n[PROOFSTEP]\nrw [span_le]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 p * q \u2286 \u2191(\u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)))\n[PROOFSTEP]\nrintro _ \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 (fun x x_1 => x * x_1) x y \u2208 \u2191(\u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)))\n[PROOFSTEP]\nchange x * y \u2208 adjoin R (s \u222a t)\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 x * y \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nrefine' Subalgebra.mul_mem _ _ _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 x \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nhave : x \u2208 Subalgebra.toSubmodule (adjoin R s) := by\n  rw [\u2190 hp']\n  exact subset_span hx\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 x \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n[PROOFSTEP]\nrw [\u2190 hp']\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 x \u2208 span R p\n[PROOFSTEP]\nexact subset_span hx\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\nthis : x \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 x \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nexact adjoin_mono (Set.subset_union_left _ _) this\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 y \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nhave : y \u2208 Subalgebra.toSubmodule (adjoin (adjoin R s) t) :=\n  by\n  rw [\u2190 hq']\n  exact subset_span hy\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 y \u2208 \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n[PROOFSTEP]\nrw [\u2190 hq']\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\n\u22a2 y \u2208 span { x // x \u2208 adjoin R s } q\n[PROOFSTEP]\nexact subset_span hy\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\nthis : y \u2208 \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 y \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nchange y \u2208 adjoin R (s \u222a t)\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nx y : A\nhx : x \u2208 p\nhy : y \u2208 q\nthis : y \u2208 \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 y \u2208 adjoin R (s \u222a t)\n[PROOFSTEP]\nrwa [adjoin_union_eq_adjoin_adjoin]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 \u2191Subalgebra.toSubmodule (adjoin R (s \u222a t)) \u2264 span R (p * q)\n[PROOFSTEP]\nintro r hr\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nr : A\nhr : r \u2208 \u2191Subalgebra.toSubmodule (adjoin R (s \u222a t))\n\u22a2 r \u2208 span R (p * q)\n[PROOFSTEP]\nchange r \u2208 adjoin R (s \u222a t) at hr \n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nr : A\nhr : r \u2208 adjoin R (s \u222a t)\n\u22a2 r \u2208 span R (p * q)\n[PROOFSTEP]\nrw [adjoin_union_eq_adjoin_adjoin] at hr \n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nr : A\nhr : r \u2208 Subalgebra.restrictScalars R (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 r \u2208 span R (p * q)\n[PROOFSTEP]\nchange r \u2208 Subalgebra.toSubmodule (adjoin (adjoin R s) t) at hr \n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nr : A\nhr : r \u2208 \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\n\u22a2 r \u2208 span R (p * q)\n[PROOFSTEP]\nrw [\u2190 hq', \u2190 Set.image_id q, Finsupp.mem_span_image_iff_total (adjoin R s)] at hr \n[GOAL]\ncase intro.intro.intro.intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nr : A\nhr :\n  \u2203 l,\n    l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q \u2227\n      \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } _root_.id) l = r\n\u22a2 r \u2208 span R (p * q)\n[PROOFSTEP]\nrcases hr with \u27e8l, hlq, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\n\u22a2 \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } _root_.id) l \u2208 span R (p * q)\n[PROOFSTEP]\nhave := @Finsupp.total_apply A A (adjoin R s)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } _root_.id) l \u2208 span R (p * q)\n[PROOFSTEP]\nrw [this, Finsupp.sum]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 (Finset.sum l.support fun a => \u2191l a \u2022 _root_.id a) \u2208 span R (p * q)\n[PROOFSTEP]\nrefine' sum_mem _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 \u2200 (c : A), c \u2208 l.support \u2192 \u2191l c \u2022 _root_.id c \u2208 span R (p * q)\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\n\u22a2 \u2191l z \u2022 _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nchange (l z).1 * _ \u2208 _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nhave : (l z).1 \u2208 Subalgebra.toSubmodule (adjoin R s) := (l z).2\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nthis : \u2191(\u2191l z) \u2208 \u2191Subalgebra.toSubmodule (adjoin R s)\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nrw [\u2190 hp', \u2190 Set.image_id p, Finsupp.mem_span_image_iff_total R] at this \n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nthis : \u2203 l_1, l_1 \u2208 Finsupp.supported R R p \u2227 \u2191(Finsupp.total A A R _root_.id) l_1 = \u2191(\u2191l z)\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nrcases this with \u27e8l2, hlp, hl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : \u2191(Finsupp.total A A R _root_.id) l2 = \u2191(\u2191l z)\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nhave := @Finsupp.total_apply A A R\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : \u2191(Finsupp.total A A R _root_.id) l2 = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nrw [this] at hl \n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 \u2191(\u2191l z) * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nrw [\u2190 hl, Finsupp.sum_mul]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 (Finsupp.sum l2 fun a c => c \u2022 _root_.id a * _root_.id z) \u2208 span R (p * q)\n[PROOFSTEP]\nrefine' sum_mem _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\n\u22a2 \u2200 (c : A), c \u2208 l2.support \u2192 (fun a c => c \u2022 _root_.id a * _root_.id z) c (\u2191l2 c) \u2208 span R (p * q)\n[PROOFSTEP]\nintro t ht\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t\u271d : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\nt : A\nht : t \u2208 l2.support\n\u22a2 (fun a c => c \u2022 _root_.id a * _root_.id z) t (\u2191l2 t) \u2208 span R (p * q)\n[PROOFSTEP]\nchange _ * _ \u2208 _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t\u271d : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\nt : A\nht : t \u2208 l2.support\n\u22a2 \u2191l2 t \u2022 _root_.id t * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nrw [smul_mul_assoc]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t\u271d : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\nt : A\nht : t \u2208 l2.support\n\u22a2 \u2191l2 t \u2022 (_root_.id t * _root_.id z) \u2208 span R (p * q)\n[PROOFSTEP]\nrefine' smul_mem _ _ _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.intro.intro.intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : CommSemiring A\ninst\u271d : Algebra R A\ns t\u271d : Set A\nh1 : FG (\u2191Subalgebra.toSubmodule (adjoin R s))\nh2 : FG (\u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d))\np : Set A\nhp : Set.Finite p\nhp' : span R p = \u2191Subalgebra.toSubmodule (adjoin R s)\nq : Set A\nhq : Set.Finite q\nhq' : span { x // x \u2208 adjoin R s } q = \u2191Subalgebra.toSubmodule (adjoin { x // x \u2208 adjoin R s } t\u271d)\nl : A \u2192\u2080 { x // x \u2208 adjoin R s }\nhlq : l \u2208 Finsupp.supported { x // x \u2208 adjoin R s } { x // x \u2208 adjoin R s } q\nthis\u271d :\n  \u2200 [inst : Semiring { x // x \u2208 adjoin R s }] [inst_1 : AddCommMonoid A] [inst_2 : Module { x // x \u2208 adjoin R s } A]\n    {v : A \u2192 A} (l : A \u2192\u2080 { x // x \u2208 adjoin R s }),\n    \u2191(Finsupp.total A A { x // x \u2208 adjoin R s } v) l = Finsupp.sum l fun i a => a \u2022 v i\nz : A\nhz : z \u2208 l.support\nl2 : A \u2192\u2080 R\nhlp : l2 \u2208 Finsupp.supported R R p\nhl : (Finsupp.sum l2 fun i a => a \u2022 _root_.id i) = \u2191(\u2191l z)\nthis :\n  \u2200 [inst : Semiring R] [inst_1 : AddCommMonoid A] [inst_2 : Module R A] {v : A \u2192 A} (l : A \u2192\u2080 R),\n    \u2191(Finsupp.total A A R v) l = Finsupp.sum l fun i a => a \u2022 v i\nt : A\nht : t \u2208 l2.support\n\u22a2 _root_.id t * _root_.id z \u2208 span R (p * q)\n[PROOFSTEP]\nexact subset_span \u27e8t, z, hlp ht, hlq hz, rfl\u27e9\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nx\u271d : Submodule.FG (\u2191toSubmodule S)\nt : Finset A\nht : span R \u2191t = \u2191toSubmodule S\nx : A\nhx : x \u2208 \u2191toSubmodule S\n\u22a2 x \u2208 span R \u2191t\n[PROOFSTEP]\nrw [ht]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nx\u271d : Submodule.FG (\u2191toSubmodule S)\nt : Finset A\nht : span R \u2191t = \u2191toSubmodule S\nx : A\nhx : x \u2208 \u2191toSubmodule S\n\u22a2 x \u2208 \u2191toSubmodule S\n[PROOFSTEP]\nexact hx\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nh : Submodule.FG \u22a4\ns : Finset A\nhs : span R \u2191s = \u22a4\n\u22a2 \u2191toSubmodule (Algebra.adjoin R \u2191s) = \u2191toSubmodule \u22a4\n[PROOFSTEP]\nrw [Algebra.top_toSubmodule, eq_top_iff, \u2190 hs, span_le]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nh : Submodule.FG \u22a4\ns : Finset A\nhs : span R \u2191s = \u22a4\n\u22a2 \u2191s \u2286 \u2191(\u2191toSubmodule (Algebra.adjoin R \u2191s))\n[PROOFSTEP]\nexact Algebra.subset_adjoin\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nT : Subalgebra R B\nhS : FG S\nhT : FG T\n\u22a2 FG (Subalgebra.prod S T)\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := fg_def.1 hS\n[GOAL]\ncase intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nT : Subalgebra R B\nhS : FG S\nhT : FG T\ns : Set A\nhs : Set.Finite s \u2227 Algebra.adjoin R s = S\n\u22a2 FG (Subalgebra.prod S T)\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := fg_def.1 hT\n[GOAL]\ncase intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nT : Subalgebra R B\nhS : FG S\nhT : FG T\ns : Set A\nhs : Set.Finite s \u2227 Algebra.adjoin R s = S\nt : Set B\nht : Set.Finite t \u2227 Algebra.adjoin R t = T\n\u22a2 FG (Subalgebra.prod S T)\n[PROOFSTEP]\nrw [\u2190 hs.2, \u2190 ht.2]\n[GOAL]\ncase intro.intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nT : Subalgebra R B\nhS : FG S\nhT : FG T\ns : Set A\nhs : Set.Finite s \u2227 Algebra.adjoin R s = S\nt : Set B\nht : Set.Finite t \u2227 Algebra.adjoin R t = T\n\u22a2 FG (Subalgebra.prod (Algebra.adjoin R s) (Algebra.adjoin R t))\n[PROOFSTEP]\nexact\n  fg_def.2\n    \u27e8LinearMap.inl R A B '' (s \u222a {1}) \u222a LinearMap.inr R A B '' (t \u222a {1}),\n      Set.Finite.union (Set.Finite.image _ (Set.Finite.union hs.1 (Set.finite_singleton _)))\n        (Set.Finite.image _ (Set.Finite.union ht.1 (Set.finite_singleton _))),\n      Algebra.adjoin_inl_union_inr_eq_prod R s t\u27e9\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nf : A \u2192\u2090[R] B\nhs\u271d : FG S\ns : Finset A\nhs : Algebra.adjoin R \u2191s = S\n\u22a2 Algebra.adjoin R \u2191(Finset.image (\u2191f) s) = Subalgebra.map f S\n[PROOFSTEP]\nrw [Finset.coe_image, Algebra.adjoin_image, hs]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nhs\u271d : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R \u2191s = map f S\n\u22a2 map f\n      (Algebra.adjoin R\n        \u2191(Finset.preimage s \u2191f\n            (_ : \u2200 (x : A), x \u2208 \u2191f \u207b\u00b9' \u2191s \u2192 \u2200 (x_2 : A), x_2 \u2208 \u2191f \u207b\u00b9' \u2191s \u2192 \u2191f x = \u2191f x_2 \u2192 x = x_2))) =\n    map f S\n[PROOFSTEP]\nrw [\u2190 Algebra.adjoin_image, Finset.coe_preimage, Set.image_preimage_eq_of_subset, hs]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nhs\u271d : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R \u2191s = map f S\n\u22a2 \u2191s \u2286 Set.range \u2191f\n[PROOFSTEP]\nrw [\u2190 AlgHom.coe_range, \u2190 Algebra.adjoin_le_iff, hs, \u2190 Algebra.map_top]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nf : A \u2192\u2090[R] B\nhf : Function.Injective \u2191f\nhs\u271d : FG (map f S)\ns : Finset B\nhs : Algebra.adjoin R \u2191s = map f S\n\u22a2 map f S \u2264 map f \u22a4\n[PROOFSTEP]\nexact map_mono le_top\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nh : FG \u22a4\n\u22a2 FG S\n[PROOFSTEP]\nrw [\u2190 S.range_val, \u2190 Algebra.map_top]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nh : FG \u22a4\n\u22a2 FG (map (val S) \u22a4)\n[PROOFSTEP]\nexact FG.map _ h\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nh : FG S\n\u22a2 FG (map (val S) \u22a4)\n[PROOFSTEP]\nrw [Algebra.map_top, range_val]\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : Semiring A\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Semiring B\ninst\u271d : Algebra R B\nS : Subalgebra R A\nh : FG S\n\u22a2 FG S\n[PROOFSTEP]\nexact h\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nS : Subalgebra R A\n\u22a2 P S\n[PROOFSTEP]\nclassical\nobtain \u27e8t, rfl\u27e9 := S.fg_of_noetherian\nrefine' Finset.induction_on t _ _\n\u00b7 simpa using base\nintro x t _ h\nrw [Finset.coe_insert]\nsimpa only [Algebra.adjoin_insert_adjoin] using ih _ x h\n[GOAL]\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nS : Subalgebra R A\n\u22a2 P S\n[PROOFSTEP]\nobtain \u27e8t, rfl\u27e9 := S.fg_of_noetherian\n[GOAL]\ncase intro\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nt : Finset A\n\u22a2 P (Algebra.adjoin R \u2191t)\n[PROOFSTEP]\nrefine' Finset.induction_on t _ _\n[GOAL]\ncase intro.refine'_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nt : Finset A\n\u22a2 P (Algebra.adjoin R \u2191\u2205)\n[PROOFSTEP]\nsimpa using base\n[GOAL]\ncase intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nt : Finset A\n\u22a2 \u2200 \u2983a : A\u2984 {s : Finset A}, \u00aca \u2208 s \u2192 P (Algebra.adjoin R \u2191s) \u2192 P (Algebra.adjoin R \u2191(insert a s))\n[PROOFSTEP]\nintro x t _ h\n[GOAL]\ncase intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nt\u271d : Finset A\nx : A\nt : Finset A\na\u271d : \u00acx \u2208 t\nh : P (Algebra.adjoin R \u2191t)\n\u22a2 P (Algebra.adjoin R \u2191(insert x t))\n[PROOFSTEP]\nrw [Finset.coe_insert]\n[GOAL]\ncase intro.refine'_2\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Semiring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Semiring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : IsNoetherian R A\nP : Subalgebra R A \u2192 Prop\nbase : P \u22a5\nih : \u2200 (S : Subalgebra R A) (x : A), P S \u2192 P (Algebra.adjoin R (insert x \u2191S))\nt\u271d : Finset A\nx : A\nt : Finset A\na\u271d : \u00acx \u2208 t\nh : P (Algebra.adjoin R \u2191t)\n\u22a2 P (Algebra.adjoin R (insert x \u2191t))\n[PROOFSTEP]\nsimpa only [Algebra.adjoin_insert_adjoin] using ih _ x h\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Adjoin.FG", "llama_tokens": 19103, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8128673178375734, "lm_q2_score": 0.6334102705979902, "lm_q1q2_score": 0.5148785077517598}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i in s, f i) \u2022 \u2211 i in s, g i \u2264 card s \u2022 \u2211 i in s, f i \u2022 g i\n[PROOFSTEP]\nclassical\nobtain \u27e8\u03c3, h\u03c3, hs\u27e9 := s.countable_toSet.exists_cycleOn\nrw [\u2190 card_range s.card, sum_smul_sum_eq_sum_perm h\u03c3]\nexact\n  sum_le_card_nsmul _ _ _ fun n _ =>\n    hfg.sum_smul_comp_perm_le_sum_smul fun x hx => hs fun h => hx <| IsFixedPt.perm_pow h _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i in s, f i) \u2022 \u2211 i in s, g i \u2264 card s \u2022 \u2211 i in s, f i \u2022 g i\n[PROOFSTEP]\nobtain \u27e8\u03c3, h\u03c3, hs\u27e9 := s.countable_toSet.exists_cycleOn\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3\u271d : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\n\u03c3 : Perm \u03b9\nh\u03c3 : IsCycleOn \u03c3 \u2191s\nhs : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\n\u22a2 (\u2211 i in s, f i) \u2022 \u2211 i in s, g i \u2264 card s \u2022 \u2211 i in s, f i \u2022 g i\n[PROOFSTEP]\nrw [\u2190 card_range s.card, sum_smul_sum_eq_sum_perm h\u03c3]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3\u271d : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : MonovaryOn f g \u2191s\n\u03c3 : Perm \u03b9\nh\u03c3 : IsCycleOn \u03c3 \u2191s\nhs : {x | \u2191\u03c3 x \u2260 x} \u2286 \u2191s\n\u22a2 \u2211 k in range (card s), \u2211 i in s, f i \u2022 g (\u2191(\u03c3 ^ k) i) \u2264 card (range (card s)) \u2022 \u2211 i in s, f i \u2022 g i\n[PROOFSTEP]\nexact\n  sum_le_card_nsmul _ _ _ fun n _ =>\n    hfg.sum_smul_comp_perm_le_sum_smul fun x hx => hs fun h => hx <| IsFixedPt.perm_pow h _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\nhfg : AntivaryOn f g \u2191s\n\u22a2 card s \u2022 \u2211 i in s, f i \u2022 g i \u2264 (\u2211 i in s, f i) \u2022 \u2211 i in s, g i\n[PROOFSTEP]\nrefine hfg.dual_right.sum_smul_sum_le_card_smul_sum\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u2074 : LinearOrderedRing \u03b1\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b2 : Module \u03b1 \u03b2\ninst\u271d\u00b9 : OrderedSMul \u03b1 \u03b2\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\ninst\u271d : Fintype \u03b9\nhfg : Antivary f g\n\u22a2 Fintype.card \u03b9 \u2022 \u2211 i : \u03b9, f i \u2022 g i \u2264 (\u2211 i : \u03b9, f i) \u2022 \u2211 i : \u03b9, g i\n[PROOFSTEP]\nrefine (hfg.dual_right.monovaryOn _).sum_smul_sum_le_card_smul_sum\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i in s, f i) * \u2211 i in s, g i \u2264 \u2191(card s) * \u2211 i in s, f i * g i\n[PROOFSTEP]\nrw [\u2190 nsmul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i in s, f i) * \u2211 i in s, g i \u2264 card s \u2022 \u2211 i in s, f i * g i\n[PROOFSTEP]\nexact hfg.sum_smul_sum_le_card_smul_sum\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : AntivaryOn f g \u2191s\n\u22a2 \u2191(card s) * \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i) * \u2211 i in s, g i\n[PROOFSTEP]\nrw [\u2190 nsmul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : AntivaryOn f g \u2191s\n\u22a2 card s \u2022 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i) * \u2211 i in s, g i\n[PROOFSTEP]\nexact hfg.card_smul_sum_le_sum_smul_sum\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\n\u22a2 (\u2211 i in s, f i) ^ 2 \u2264 \u2191(card s) * \u2211 i in s, f i ^ 2\n[PROOFSTEP]\nsimp_rw [sq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\n\u22a2 (\u2211 i in s, f i) * \u2211 i in s, f i \u2264 \u2191(card s) * \u2211 x in s, f x * f x\n[PROOFSTEP]\nexact (monovaryOn_self _ _).sum_mul_sum_le_card_mul_sum\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\n\u22a2 ((\u2211 i in s, f i) / \u2191(card s)) ^ 2 \u2264 (\u2211 i in s, f i ^ 2) / \u2191(card s)\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\nf : \u03b9 \u2192 \u03b1\n\u22a2 ((\u2211 i in \u2205, f i) / \u2191(card \u2205)) ^ 2 \u2264 (\u2211 i in \u2205, f i ^ 2) / \u2191(card \u2205)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : Finset.Nonempty s\n\u22a2 ((\u2211 i in s, f i) / \u2191(card s)) ^ 2 \u2264 (\u2211 i in s, f i ^ 2) / \u2191(card s)\n[PROOFSTEP]\nrw [\u2190 card_pos, \u2190 @Nat.cast_pos \u03b1] at hs \n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : 0 < \u2191(card s)\n\u22a2 ((\u2211 i in s, f i) / \u2191(card s)) ^ 2 \u2264 (\u2211 i in s, f i ^ 2) / \u2191(card s)\n[PROOFSTEP]\nrw [div_pow, div_le_div_iff (sq_pos_of_ne_zero _ hs.ne') hs, sq (s.card : \u03b1), mul_left_comm, \u2190 mul_assoc]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedField \u03b1\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : 0 < \u2191(card s)\n\u22a2 (\u2211 i in s, f i) ^ 2 * \u2191(card s) \u2264 (\u2191(card s) * \u2211 i in s, f i ^ 2) * \u2191(card s)\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right sq_sum_le_card_mul_sum_sq hs.le\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Chebyshev", "llama_tokens": 2845, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5147924928741103}}
{"text": "[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Preadditive D\nF : C \u2964 D\ninst\u271d\u00b9 : Additive F\nJ\u271d : Type\ninst\u271d : Fintype J\u271d\nf\u271d : J\u271d \u2192 C\nb\u271d : Bicone f\u271d\nhb : Bicone.IsBilimit b\u271d\n\u22a2 (Finset.sum Finset.univ fun j => Bicone.\u03c0 (mapBicone F b\u271d) j \u226b Bicone.\u03b9 (mapBicone F b\u271d) j) = \ud835\udfd9 (mapBicone F b\u271d).pt\n[PROOFSTEP]\nsimp_rw [F.mapBicone_\u03c0, F.mapBicone_\u03b9, \u2190 F.map_comp]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2074 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Preadditive D\nF : C \u2964 D\ninst\u271d\u00b9 : Additive F\nJ\u271d : Type\ninst\u271d : Fintype J\u271d\nf\u271d : J\u271d \u2192 C\nb\u271d : Bicone f\u271d\nhb : Bicone.IsBilimit b\u271d\n\u22a2 (Finset.sum Finset.univ fun x => F.map (Bicone.\u03c0 b\u271d x \u226b Bicone.\u03b9 b\u271d x)) = \ud835\udfd9 (mapBicone F b\u271d).pt\n[PROOFSTEP]\nerw [\u2190 F.map_sum, \u2190 F.map_id, IsBilimit.total hb]\n[GOAL]\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} D\ninst\u271d\u2074 : Preadditive C\ninst\u271d\u00b3 : Preadditive D\nF : C \u2964 D\ninst\u271d\u00b2 : HasBinaryBiproducts C\ninst\u271d\u00b9 : PreservesZeroMorphisms F\ninst\u271d : PreservesBinaryBiproducts F\nX Y : C\nf g : X \u27f6 Y\n\u22a2 F.map (f + g) = F.map f + F.map g\n[PROOFSTEP]\nrw [biprod.add_eq_lift_id_desc, F.map_comp, \u2190 biprod.lift_mapBiprod, \u2190 biprod.mapBiprod_hom_desc, Category.assoc,\n  Iso.inv_hom_id_assoc, F.map_id, biprod.add_eq_lift_id_desc]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u2074 : Category.{?u.46335, u_1} C\ninst\u271d\u00b3 : Category.{?u.46339, u_2} D\ninst\u271d\u00b2 : Preadditive C\ninst\u271d\u00b9 : Preadditive D\ne : C \u224c D\ninst\u271d : Functor.Additive e.functor\nf g : D\nf\u271d g\u271d : f \u27f6 g\n\u22a2 e.functor.map (e.inverse.map (f\u271d + g\u271d)) = e.functor.map (e.inverse.map f\u271d + e.inverse.map g\u271d)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Preadditive.AdditiveFunctor", "llama_tokens": 965, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8221891261650247, "lm_q2_score": 0.6261241702517975, "lm_q1q2_score": 0.5147924844101265}}
{"text": "[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nclassical\n  -- The limit cone for `F` whose topology is defined as an infimum.\nlet D := limitConeInfi F\nlet E : C.pt \u2245 D.pt := hC.conePointUniqueUpToIso (limitConeInfiIsLimit _)\nhave hE : Inducing E.hom := (TopCat.homeoOfIso E).inducing\nsuffices IsTopologicalBasis {U : Set D.pt | \u2203 (j : _) (V : Set (F.obj j)), V \u2208 T j \u2227 U = D.\u03c0.app j \u207b\u00b9' V}\n  by\n  convert this.inducing hE\n  ext U0\n  constructor\n  \u00b7 rintro \u27e8j, V, hV, rfl\u27e9\n    refine' \u27e8D.\u03c0.app j \u207b\u00b9' V, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n  \u00b7 rintro \u27e8W, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n    refine'\n      \u27e8j, V, hV, rfl\u27e9\n        -- Using `D`, we can apply the characterization of the topological basis of a\n          -- topology defined as an infimum...\nconvert isTopologicalBasis_iInf hT fun j (x : D.pt) => D.\u03c0.app j x using 1\next U0\nconstructor\n\u00b7 rintro \u27e8j, V, hV, rfl\u27e9\n  let U : \u2200 i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ\n  refine' \u27e8U, { j }, _, _\u27e9\n  \u00b7 rintro i h\n    rw [Finset.mem_singleton] at h \n    dsimp\n    rw [dif_pos h]\n    subst h\n    exact hV\n  \u00b7 dsimp\n    simp\n\u00b7 rintro \u27e8U, G, h1, h2\u27e9\n  obtain \u27e8j, hj\u27e9 := IsCofiltered.inf_objs_exists G\n  let g : \u2200 (e) (_he : e \u2208 G), j \u27f6 e := fun _ he => (hj he).some\n  let Vs : J \u2192 Set (F.obj j) := fun e => if h : e \u2208 G then F.map (g e h) \u207b\u00b9' U e else Set.univ\n  let V : Set (F.obj j) := \u22c2 (e : J) (_he : e \u2208 G), Vs e\n  refine' \u27e8j, V, _, _\u27e9\n  \u00b7\n    -- An intermediate claim used to apply induction along `G : Finset J` later on.\n    have :\n      \u2200 (S : Set (Set (F.obj j))) (E : Finset J) (P : J \u2192 Set (F.obj j)) (_univ : Set.univ \u2208 S)\n        (_inter : \u2200 A B : Set (F.obj j), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) (_cond : \u2200 (e : J) (_he : e \u2208 E), P e \u2208 S),\n        (\u22c2 (e) (_he : e \u2208 E), P e) \u2208 S :=\n      by\n      intro S E\n      induction E using Finset.induction_on with\n      | empty =>\n        intro P he _hh\n        simpa\n      | @insert a E _ha hh1 =>\n        intro hh2 hh3 hh4 hh5\n        rw [Finset.set_biInter_insert]\n        refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n        intro e he\n        exact\n          hh5 e\n            (Finset.mem_insert_of_mem he)\n              -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n    refine' this _ _ _ (univ _) (inter _) _\n    intro e he\n    dsimp\n    rw [dif_pos he]\n    exact compat j e (g e he) (U e) (h1 e he)\n  \u00b7\n    -- conclude...\n    rw [h2]\n    change _ = (D.\u03c0.app j) \u207b\u00b9' \u22c2 (e : J) (_ : e \u2208 G), Vs e\n    rw [Set.preimage_iInter]\n    apply congrArg\n    ext1 e\n    erw [Set.preimage_iInter]\n    apply congrArg\n    ext1 he\n    change (D.\u03c0.app e) \u207b\u00b9' U e = (D.\u03c0.app j) \u207b\u00b9' if h : e \u2208 G then F.map (g e h) \u207b\u00b9' U e else Set.univ\n    rw [dif_pos he, \u2190 Set.preimage_comp]\n    apply congrFun\n    apply congrArg\n    rw [\u2190 coe_comp, D.w]\n    rfl\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nlet D := limitConeInfi F\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nlet E : C.pt \u2245 D.pt := hC.conePointUniqueUpToIso (limitConeInfiIsLimit _)\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nhave hE : Inducing E.hom := (TopCat.homeoOfIso E).inducing\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nsuffices IsTopologicalBasis {U : Set D.pt | \u2203 (j : _) (V : Set (F.obj j)), V \u2208 T j \u2227 U = D.\u03c0.app j \u207b\u00b9' V}\n  by\n  convert this.inducing hE\n  ext U0\n  constructor\n  \u00b7 rintro \u27e8j, V, hV, rfl\u27e9\n    refine' \u27e8D.\u03c0.app j \u207b\u00b9' V, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n  \u00b7 rintro \u27e8W, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n    refine'\n      \u27e8j, V, hV, rfl\u27e9\n        -- Using `D`, we can apply the characterization of the topological basis of a\n          -- topology defined as an infimum...\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nconvert this.inducing hE\n[GOAL]\ncase h.e'_3.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\n\u22a2 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V} =\n    Set.preimage \u2191E.hom '' {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\next U0\n[GOAL]\ncase h.e'_3.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\nU0 : Set \u2191C.pt\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V} \u2194\n    U0 \u2208 Set.preimage \u2191E.hom '' {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_3.h.h.mp\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\nU0 : Set \u2191C.pt\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V} \u2192\n    U0 \u2208 Set.preimage \u2191E.hom '' {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrintro \u27e8j, V, hV, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.h.mp.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\n\u22a2 \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V \u2208 Set.preimage \u2191E.hom '' {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine' \u27e8D.\u03c0.app j \u207b\u00b9' V, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.h.mpr\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\nU0 : Set \u2191C.pt\n\u22a2 U0 \u2208 Set.preimage \u2191E.hom '' {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V} \u2192\n    U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrintro \u27e8W, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.h.mpr.intro.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nthis : IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\ne_1\u271d : \u2191C.pt = (forget TopCat).obj C.pt\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\n\u22a2 \u2191E.hom \u207b\u00b9' (\u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V) \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app C.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine'\n  \u27e8j, V, hV, rfl\u27e9\n    -- Using `D`, we can apply the characterization of the topological basis of a\n      -- topology defined as an infimum...\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\n\u22a2 IsTopologicalBasis {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nconvert isTopologicalBasis_iInf hT fun j (x : D.pt) => D.\u03c0.app j x using 1\n[GOAL]\ncase h.e'_3\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\n\u22a2 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V} =\n    {S |\n      \u2203 U F_1,\n        (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i}\n[PROOFSTEP]\next U0\n[GOAL]\ncase h.e'_3.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V} \u2194\n    U0 \u2208\n      {S |\n        \u2203 U F_1,\n          (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.e'_3.h.mp\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V} \u2192\n    U0 \u2208\n      {S |\n        \u2203 U F_1,\n          (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i}\n[PROOFSTEP]\nrintro \u27e8j, V, hV, rfl\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\n\u22a2 \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V \u2208\n    {S |\n      \u2203 U F_1,\n        (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i}\n[PROOFSTEP]\nlet U : \u2200 i, Set (F.obj i) := fun i => if h : i = j then by rw [h]; exact V else Set.univ\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\ni : J\nh : i = j\n\u22a2 Set \u2191(F.obj i)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\ni : J\nh : i = j\n\u22a2 Set \u2191(F.obj j)\n[PROOFSTEP]\nexact V\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\n\u22a2 \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V \u2208\n    {S |\n      \u2203 U F_1,\n        (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i}\n[PROOFSTEP]\nrefine' \u27e8U, { j }, _, _\u27e9\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\n\u22a2 \u2200 (i : J), i \u2208 {j} \u2192 U i \u2208 T i\n[PROOFSTEP]\nrintro i h\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\ni : J\nh : i \u2208 {j}\n\u22a2 U i \u2208 T i\n[PROOFSTEP]\nrw [Finset.mem_singleton] at h \n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\ni : J\nh : i = j\n\u22a2 U i \u2208 T i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\ni : J\nh : i = j\n\u22a2 (if h : i = j then cast (_ : Set \u2191(F.obj j) = Set \u2191(F.obj i)) V else Set.univ) \u2208 T i\n[PROOFSTEP]\nrw [dif_pos h]\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\ni : J\nh : i = j\n\u22a2 cast (_ : Set \u2191(F.obj j) = Set \u2191(F.obj i)) V \u2208 T i\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\ni : J\nV : Set \u2191(F.obj i)\nhV : V \u2208 T i\nU : (i : J) \u2192 Set \u2191(F.obj i) :=\n  fun i_1 => if h : i_1 = i then Eq.mpr (_ : Set \u2191(F.obj i_1) = Set \u2191(F.obj i)) V else Set.univ\n\u22a2 cast (_ : Set \u2191(F.obj i) = Set \u2191(F.obj i)) V \u2208 T i\n[PROOFSTEP]\nexact hV\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\n\u22a2 \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V = \u22c2 (i : J) (_ : i \u2208 {j}), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.mp.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nj : J\nV : Set \u2191(F.obj j)\nhV : V \u2208 T j\nU : (i : J) \u2192 Set \u2191(F.obj i) := fun i => if h : i = j then Eq.mpr (_ : Set \u2191(F.obj i) = Set \u2191(F.obj j)) V else Set.univ\n\u22a2 \u2191(NatTrans.app (limitConeInfi F).\u03c0 j) \u207b\u00b9' V =\n    \u22c2 (i : J) (_ : i \u2208 {j}),\n      (fun x => \u2191(NatTrans.app (limitConeInfi F).\u03c0 i) x) \u207b\u00b9'\n        if h : i = j then cast (_ : Set \u2191(F.obj j) = Set \u2191(F.obj i)) V else Set.univ\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3.h.mpr\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\n\u22a2 U0 \u2208\n      {S |\n        \u2203 U F_1,\n          (\u2200 (i : J), i \u2208 F_1 \u2192 U i \u2208 T i) \u2227 S = \u22c2 (i : J) (_ : i \u2208 F_1), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i} \u2192\n    U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrintro \u27e8U, G, h1, h2\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nobtain \u27e8j, hj\u27e9 := IsCofiltered.inf_objs_exists G\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nlet g : \u2200 (e) (_he : e \u2208 G), j \u27f6 e := fun _ he => (hj he).some\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nlet Vs : J \u2192 Set (F.obj j) := fun e => if h : e \u2208 G then F.map (g e h) \u207b\u00b9' U e else Set.univ\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nlet V : Set (F.obj j) := \u22c2 (e : J) (_he : e \u2208 G), Vs e\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 U0 \u2208 {U | \u2203 j V, V \u2208 T j \u2227 U = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V}\n[PROOFSTEP]\nrefine' \u27e8j, V, _, _\u27e9\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 V \u2208 T j\n[PROOFSTEP]\nhave :\n  \u2200 (S : Set (Set (F.obj j))) (E : Finset J) (P : J \u2192 Set (F.obj j)) (_univ : Set.univ \u2208 S)\n    (_inter : \u2200 A B : Set (F.obj j), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) (_cond : \u2200 (e : J) (_he : e \u2208 E), P e \u2208 S),\n    (\u22c2 (e) (_he : e \u2208 E), P e) \u2208 S :=\n  by\n  intro S E\n  induction E using Finset.induction_on with\n  | empty =>\n    intro P he _hh\n    simpa\n  | @insert a E _ha hh1 =>\n    intro hh2 hh3 hh4 hh5\n    rw [Finset.set_biInter_insert]\n    refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n    intro e he\n    exact\n      hh5 e\n        (Finset.mem_insert_of_mem he)\n          -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n[PROOFSTEP]\nintro S E\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\nE : Finset J\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n[PROOFSTEP]\ninduction E using Finset.induction_on with\n| empty =>\n  intro P he _hh\n  simpa\n| @insert a E _ha hh1 =>\n  intro hh2 hh3 hh4 hh5\n  rw [Finset.set_biInter_insert]\n  refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n  intro e he\n  exact\n    hh5 e\n      (Finset.mem_insert_of_mem he)\n        -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n[GOAL]\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\nE : Finset J\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n[PROOFSTEP]\ninduction E using Finset.induction_on with\n| empty =>\n  intro P he _hh\n  simpa\n| @insert a E _ha hh1 =>\n  intro hh2 hh3 hh4 hh5\n  rw [Finset.set_biInter_insert]\n  refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n  intro e he\n  exact\n    hh5 e\n      (Finset.mem_insert_of_mem he)\n        -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n[GOAL]\ncase empty\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 \u2205 \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 \u2205), P e \u2208 S\n[PROOFSTEP]\n\n| empty =>\n  intro P he _hh\n  simpa\n[GOAL]\ncase empty\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 \u2205 \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 \u2205), P e \u2208 S\n[PROOFSTEP]\nintro P he _hh\n[GOAL]\ncase empty\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\nP : J \u2192 Set \u2191(F.obj j)\nhe : Set.univ \u2208 S\n_hh : \u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S\n\u22a2 (\u2200 (e : J), e \u2208 \u2205 \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 \u2205), P e \u2208 S\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 insert a E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 insert a E), P e \u2208 S\n[PROOFSTEP]\n\n| @insert a E _ha hh1 =>\n  intro hh2 hh3 hh4 hh5\n  rw [Finset.set_biInter_insert]\n  refine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n  intro e he\n  exact\n    hh5 e\n      (Finset.mem_insert_of_mem he)\n        -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n\u22a2 \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 insert a E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 insert a E), P e \u2208 S\n[PROOFSTEP]\nintro hh2 hh3 hh4 hh5\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\nhh2 : J \u2192 Set \u2191(F.obj j)\nhh3 : Set.univ \u2208 S\nhh4 : \u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S\nhh5 : \u2200 (e : J), e \u2208 insert a E \u2192 hh2 e \u2208 S\n\u22a2 \u22c2 (e : J) (_ : e \u2208 insert a E), hh2 e \u2208 S\n[PROOFSTEP]\nrw [Finset.set_biInter_insert]\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\nhh2 : J \u2192 Set \u2191(F.obj j)\nhh3 : Set.univ \u2208 S\nhh4 : \u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S\nhh5 : \u2200 (e : J), e \u2208 insert a E \u2192 hh2 e \u2208 S\n\u22a2 hh2 a \u2229 \u22c2 (x : J) (_ : x \u2208 E), hh2 x \u2208 S\n[PROOFSTEP]\nrefine' hh4 _ _ (hh5 _ (Finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _)\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\nhh2 : J \u2192 Set \u2191(F.obj j)\nhh3 : Set.univ \u2208 S\nhh4 : \u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S\nhh5 : \u2200 (e : J), e \u2208 insert a E \u2192 hh2 e \u2208 S\n\u22a2 \u2200 (e : J), e \u2208 E \u2192 hh2 e \u2208 S\n[PROOFSTEP]\nintro e he\n[GOAL]\ncase insert\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE\u271d : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E\u271d.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nS : Set (Set \u2191(F.obj j))\na : J\nE : Finset J\n_ha : \u00aca \u2208 E\nhh1 :\n  \u2200 (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\nhh2 : J \u2192 Set \u2191(F.obj j)\nhh3 : Set.univ \u2208 S\nhh4 : \u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S\nhh5 : \u2200 (e : J), e \u2208 insert a E \u2192 hh2 e \u2208 S\ne : J\nhe : e \u2208 E\n\u22a2 hh2 e \u2208 S\n[PROOFSTEP]\nexact\n  hh5 e\n    (Finset.mem_insert_of_mem he)\n      -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nthis :\n  \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n\u22a2 V \u2208 T j\n[PROOFSTEP]\nrefine' this _ _ _ (univ _) (inter _) _\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nthis :\n  \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\n\u22a2 \u2200 (e : J), e \u2208 G \u2192 Vs e \u2208 T j\n[PROOFSTEP]\nintro e he\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nthis :\n  \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\ne : J\nhe : e \u2208 G\n\u22a2 Vs e \u2208 T j\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nthis :\n  \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\ne : J\nhe : e \u2208 G\n\u22a2 (if h : e \u2208 G then \u2191(F.map (Nonempty.some (_ : Nonempty (j \u27f6 e)))) \u207b\u00b9' U e else Set.univ) \u2208 T j\n[PROOFSTEP]\nrw [dif_pos he]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_1\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\nthis :\n  \u2200 (S : Set (Set \u2191(F.obj j))) (E : Finset J) (P : J \u2192 Set \u2191(F.obj j)),\n    Set.univ \u2208 S \u2192\n      (\u2200 (A B : Set \u2191(F.obj j)), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S) \u2192\n        (\u2200 (e : J), e \u2208 E \u2192 P e \u2208 S) \u2192 \u22c2 (e : J) (_ : e \u2208 E), P e \u2208 S\ne : J\nhe : e \u2208 G\n\u22a2 \u2191(F.map (Nonempty.some (_ : Nonempty (j \u27f6 e)))) \u207b\u00b9' U e \u2208 T j\n[PROOFSTEP]\nexact compat j e (g e he) (U e) (h1 e he)\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 U0 = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V\n[PROOFSTEP]\nrw [h2]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' V\n[PROOFSTEP]\nchange _ = (D.\u03c0.app j) \u207b\u00b9' \u22c2 (e : J) (_ : e \u2208 G), Vs e\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i =\n    \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' \u22c2 (e : J) (_ : e \u2208 G), Vs e\n[PROOFSTEP]\nrw [Set.preimage_iInter]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i =\n    \u22c2 (i : J), \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' \u22c2 (_ : i \u2208 G), Vs i\n[PROOFSTEP]\napply congrArg\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\n\u22a2 (fun i => \u22c2 (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i) = fun i =>\n    \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' \u22c2 (_ : i \u2208 G), Vs i\n[PROOFSTEP]\next1 e\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\n\u22a2 \u22c2 (_ : e \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 e) x) \u207b\u00b9' U e = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' \u22c2 (_ : e \u2208 G), Vs e\n[PROOFSTEP]\nerw [Set.preimage_iInter]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\n\u22a2 \u22c2 (_ : e \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 e) x) \u207b\u00b9' U e = \u22c2 (_ : e \u2208 G), \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' Vs e\n[PROOFSTEP]\napply congrArg\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\n\u22a2 (fun x => (fun x => \u2191(NatTrans.app D.\u03c0 e) x) \u207b\u00b9' U e) = fun i => \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' Vs e\n[PROOFSTEP]\next1 he\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 (fun x => \u2191(NatTrans.app D.\u03c0 e) x) \u207b\u00b9' U e = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' Vs e\n[PROOFSTEP]\nchange (D.\u03c0.app e) \u207b\u00b9' U e = (D.\u03c0.app j) \u207b\u00b9' if h : e \u2208 G then F.map (g e h) \u207b\u00b9' U e else Set.univ\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 \u2191(NatTrans.app D.\u03c0 e) \u207b\u00b9' U e = \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\n[PROOFSTEP]\nrw [dif_pos he, \u2190 Set.preimage_comp]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 \u2191(NatTrans.app D.\u03c0 e) \u207b\u00b9' U e = \u2191(F.map (g e he)) \u2218 \u2191(NatTrans.app D.\u03c0 j) \u207b\u00b9' U e\n[PROOFSTEP]\napply congrFun\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 Set.preimage \u2191(NatTrans.app D.\u03c0 e) = Set.preimage (\u2191(F.map (g e he)) \u2218 \u2191(NatTrans.app D.\u03c0 j))\n[PROOFSTEP]\napply congrArg\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 \u2191(NatTrans.app D.\u03c0 e) = \u2191(F.map (g e he)) \u2218 \u2191(NatTrans.app D.\u03c0 j)\n[PROOFSTEP]\nrw [\u2190 coe_comp, D.w]\n[GOAL]\ncase h.e'_3.h.mpr.intro.intro.intro.intro.refine'_2.h.h.h.h.h.h\nJ : Type v\ninst\u271d\u00b9 : SmallCategory J\ninst\u271d : IsCofiltered J\nF : J \u2964 TopCatMax\nC : Cone F\nhC : IsLimit C\nT : (j : J) \u2192 Set (Set \u2191(F.obj j))\nhT : \u2200 (j : J), IsTopologicalBasis (T j)\nuniv : \u2200 (i : J), Set.univ \u2208 T i\ninter : \u2200 (i : J) (U1 U2 : Set \u2191(F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i\ncompat : \u2200 (i j : J) (f : i \u27f6 j) (V : Set \u2191(F.obj j)), V \u2208 T j \u2192 \u2191(F.map f) \u207b\u00b9' V \u2208 T i\nD : Cone F := limitConeInfi F\nE : C.pt \u2245 D.pt := IsLimit.conePointUniqueUpToIso hC (limitConeInfiIsLimit F)\nhE : Inducing \u2191E.hom\nU0 : Set \u2191D.pt\nU : (i : J) \u2192 Set \u2191(F.obj i)\nG : Finset J\nh1 : \u2200 (i : J), i \u2208 G \u2192 U i \u2208 T i\nh2 : U0 = \u22c2 (i : J) (_ : i \u2208 G), (fun x => \u2191(NatTrans.app D.\u03c0 i) x) \u207b\u00b9' U i\nj : J\nhj : \u2200 {X : J}, X \u2208 G \u2192 Nonempty (j \u27f6 X)\ng : (e : J) \u2192 e \u2208 G \u2192 (j \u27f6 e) := fun x he => Nonempty.some (_ : Nonempty (j \u27f6 x))\nVs : J \u2192 Set \u2191(F.obj j) := fun e => if h : e \u2208 G then \u2191(F.map (g e h)) \u207b\u00b9' U e else Set.univ\nV : Set \u2191(F.obj j) := \u22c2 (e : J) (_ : e \u2208 G), Vs e\ne : J\nhe : e \u2208 G\n\u22a2 \u2191(NatTrans.app D.\u03c0 e) = \u2191(NatTrans.app D.\u03c0 e)\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Topology.Category.TopCat.Limits.Cofiltered", "llama_tokens": 39302, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8080672181749421, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.5147637019875364}}
{"text": "[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nl : List M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\n\u22a2 \u2191(\u2191(foldr Q f hf) n) (List.prod (List.map (\u2191(\u03b9 Q)) l)) = List.foldr (fun m n => \u2191(\u2191f m) n) n l\n[PROOFSTEP]\ninduction' l with hd tl ih\n[GOAL]\ncase nil\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\n\u22a2 \u2191(\u2191(foldr Q f hf) n) (List.prod (List.map \u2191(\u03b9 Q) [])) = List.foldr (fun m n => \u2191(\u2191f m) n) n []\n[PROOFSTEP]\nrw [List.map_nil, List.prod_nil, List.foldr_nil, foldr_one]\n[GOAL]\ncase cons\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\nhd : M\ntl : List M\nih : \u2191(\u2191(foldr Q f hf) n) (List.prod (List.map (\u2191(\u03b9 Q)) tl)) = List.foldr (fun m n => \u2191(\u2191f m) n) n tl\n\u22a2 \u2191(\u2191(foldr Q f hf) n) (List.prod (List.map (\u2191(\u03b9 Q)) (hd :: tl))) = List.foldr (fun m n => \u2191(\u2191f m) n) n (hd :: tl)\n[PROOFSTEP]\nrw [List.map_cons, List.prod_cons, List.foldr_cons, foldr_mul, foldr_\u03b9, ih]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\nm : M\n\u22a2 \u2191(\u2191(foldl Q f hf) n) (\u2191(\u03b9 Q) m) = \u2191(\u2191f m) n\n[PROOFSTEP]\nrw [\u2190 foldr_reverse, reverse_\u03b9, foldr_\u03b9]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\nr : R\n\u22a2 \u2191(\u2191(foldl Q f hf) n) (\u2191(algebraMap R (CliffordAlgebra Q)) r) = r \u2022 n\n[PROOFSTEP]\nrw [\u2190 foldr_reverse, reverse.commutes, foldr_algebraMap]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\n\u22a2 \u2191(\u2191(foldl Q f hf) n) 1 = n\n[PROOFSTEP]\nrw [\u2190 foldr_reverse, reverse.map_one, foldr_one]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\na b : CliffordAlgebra Q\n\u22a2 \u2191(\u2191(foldl Q f hf) n) (a * b) = \u2191(\u2191(foldl Q f hf) (\u2191(\u2191(foldl Q f hf) n) a)) b\n[PROOFSTEP]\nrw [\u2190 foldr_reverse, \u2190 foldr_reverse, \u2190 foldr_reverse, reverse.map_mul, foldr_mul]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nl : List M\nf : M \u2192\u2097[R] N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : N), \u2191(\u2191f m) (\u2191(\u2191f m) x) = \u2191Q m \u2022 x\nn : N\n\u22a2 \u2191(\u2191(foldl Q f hf) n) (List.prod (List.map (\u2191(\u03b9 Q)) l)) = List.foldl (fun m n => \u2191(\u2191f n) m) n l\n[PROOFSTEP]\nrw [\u2190 foldr_reverse, reverse_prod_map_\u03b9, \u2190 List.map_reverse, foldr_prod_map_\u03b9, List.foldr_reverse]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\n\u22a2 \u2200 (x : CliffordAlgebra Q), P x\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\n\u22a2 P x\n[PROOFSTEP]\nhave : x \u2208 \u22a4 := Submodule.mem_top (R := R)\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\nthis : x \u2208 \u22a4\n\u22a2 P x\n[PROOFSTEP]\nrw [\u2190 iSup_\u03b9_range_eq_top] at this \n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\nthis : x \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\n\u22a2 P x\n[PROOFSTEP]\ninduction this using Submodule.iSup_induction' with\n  -- _ this (fun i x hx => ?_) _ h_add\n| hp i x hx =>\n  induction hx using Submodule.pow_induction_on_right' with\n  | hr r => exact hr r\n  | hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n  | hmul _i x _hx px m hm =>\n    obtain \u27e8m, rfl\u27e9 := hm\n    exact h_\u03b9_mul _ _ px\n| h0 => simpa only [map_zero] using hr 0\n| hadd _x _y _ _ ihx ihy => exact h_add _ _ ihx ihy\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\nthis : x \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\n\u22a2 P x\n[PROOFSTEP]\ninduction this using Submodule.iSup_induction' with\n  -- _ this (fun i x hx => ?_) _ h_add\n| hp i x hx =>\n  induction hx using Submodule.pow_induction_on_right' with\n  | hr r => exact hr r\n  | hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n  | hmul _i x _hx px m hm =>\n    obtain \u27e8m, rfl\u27e9 := hm\n    exact h_\u03b9_mul _ _ px\n| h0 => simpa only [map_zero] using hr 0\n| hadd _x _y _ _ ihx ihy => exact h_add _ _ ihx ihy\n[GOAL]\ncase hp\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 LinearMap.range (\u03b9 Q) ^ i\n\u22a2 P x\n[PROOFSTEP]\n\n| hp i x hx =>\n  induction hx using Submodule.pow_induction_on_right' with\n  | hr r => exact hr r\n  | hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n  | hmul _i x _hx px m hm =>\n    obtain \u27e8m, rfl\u27e9 := hm\n    exact h_\u03b9_mul _ _ px\n[GOAL]\ncase hp\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 LinearMap.range (\u03b9 Q) ^ i\n\u22a2 P x\n[PROOFSTEP]\ninduction hx using Submodule.pow_induction_on_right' with\n| hr r => exact hr r\n| hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n| hmul _i x _hx px m hm =>\n  obtain \u27e8m, rfl\u27e9 := hm\n  exact h_\u03b9_mul _ _ px\n[GOAL]\ncase hp\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 LinearMap.range (\u03b9 Q) ^ i\n\u22a2 P x\n[PROOFSTEP]\ninduction hx using Submodule.pow_induction_on_right' with\n| hr r => exact hr r\n| hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n| hmul _i x _hx px m hm =>\n  obtain \u27e8m, rfl\u27e9 := hm\n  exact h_\u03b9_mul _ _ px\n[GOAL]\ncase hp.hr\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx : CliffordAlgebra Q\nr : R\n\u22a2 P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\n[PROOFSTEP]\n\n| hr r => exact hr r\n[GOAL]\ncase hp.hr\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx : CliffordAlgebra Q\nr : R\n\u22a2 P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\n[PROOFSTEP]\nexact hr r\n[GOAL]\ncase hp.hadd\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx _x _y : CliffordAlgebra Q\n_i : \u2115\nhx\u271d : _x \u2208 LinearMap.range (\u03b9 Q) ^ _i\nhy\u271d : _y \u2208 LinearMap.range (\u03b9 Q) ^ _i\nihx : P _x\nihy : P _y\n\u22a2 P (_x + _y)\n[PROOFSTEP]\n\n| hadd _x _y _i _ _ ihx ihy => exact h_add _ _ ihx ihy\n[GOAL]\ncase hp.hadd\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d : CliffordAlgebra Q\ni : \u2115\nx _x _y : CliffordAlgebra Q\n_i : \u2115\nhx\u271d : _x \u2208 LinearMap.range (\u03b9 Q) ^ _i\nhy\u271d : _y \u2208 LinearMap.range (\u03b9 Q) ^ _i\nihx : P _x\nihy : P _y\n\u22a2 P (_x + _y)\n[PROOFSTEP]\nexact h_add _ _ ihx ihy\n[GOAL]\ncase hp.hmul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d\u00b9 : CliffordAlgebra Q\ni : \u2115\nx\u271d : CliffordAlgebra Q\n_i : \u2115\nx : CliffordAlgebra Q\n_hx : x \u2208 LinearMap.range (\u03b9 Q) ^ _i\npx : P x\nm : CliffordAlgebra Q\nhm : m \u2208 LinearMap.range (\u03b9 Q)\n\u22a2 P (x * m)\n[PROOFSTEP]\n\n| hmul _i x _hx px m hm =>\n  obtain \u27e8m, rfl\u27e9 := hm\n  exact h_\u03b9_mul _ _ px\n[GOAL]\ncase hp.hmul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d\u00b9 : CliffordAlgebra Q\ni : \u2115\nx\u271d : CliffordAlgebra Q\n_i : \u2115\nx : CliffordAlgebra Q\n_hx : x \u2208 LinearMap.range (\u03b9 Q) ^ _i\npx : P x\nm : CliffordAlgebra Q\nhm : m \u2208 LinearMap.range (\u03b9 Q)\n\u22a2 P (x * m)\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := hm\n[GOAL]\ncase hp.hmul.intro\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx\u271d\u00b9 : CliffordAlgebra Q\ni : \u2115\nx\u271d : CliffordAlgebra Q\n_i : \u2115\nx : CliffordAlgebra Q\n_hx : x \u2208 LinearMap.range (\u03b9 Q) ^ _i\npx : P x\nm : M\n\u22a2 P (x * \u2191(\u03b9 Q) m)\n[PROOFSTEP]\nexact h_\u03b9_mul _ _ px\n[GOAL]\ncase h0\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\n\u22a2 P 0\n[PROOFSTEP]\n\n| h0 => simpa only [map_zero] using hr 0\n[GOAL]\ncase h0\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx : CliffordAlgebra Q\n\u22a2 P 0\n[PROOFSTEP]\nsimpa only [map_zero] using hr 0\n[GOAL]\ncase hadd\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx _x _y : CliffordAlgebra Q\nhx\u271d : _x \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\nhy\u271d : _y \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\nihx : P _x\nihy : P _y\n\u22a2 P (_x + _y)\n[PROOFSTEP]\n\n| hadd _x _y _ _ ihx ihy => exact h_add _ _ ihx ihy\n[GOAL]\ncase hadd\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_\u03b9_mul : \u2200 (m : M) (x : CliffordAlgebra Q), P x \u2192 P (x * \u2191(\u03b9 Q) m)\nx _x _y : CliffordAlgebra Q\nhx\u271d : _x \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\nhy\u271d : _y \u2208 \u2a06 (i : \u2115), LinearMap.range (\u03b9 Q) ^ i\nihx : P _x\nihy : P _y\n\u22a2 P (_x + _y)\n[PROOFSTEP]\nexact h_add _ _ ihx ihy\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\n\u22a2 \u2200 (x : CliffordAlgebra Q), P x\n[PROOFSTEP]\nrefine' reverse_involutive.surjective.forall.2 _\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\n\u22a2 \u2200 (x : CliffordAlgebra Q), P (\u2191reverse x)\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\nx : CliffordAlgebra Q\n\u22a2 P (\u2191reverse x)\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.right_induction with r x y hx hy m x hx\n[GOAL]\ncase hr\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\nr : R\n\u22a2 P (\u2191reverse (\u2191(algebraMap R (CliffordAlgebra Q)) r))\n[PROOFSTEP]\nsimpa only [reverse.commutes] using hr r\n[GOAL]\ncase h_add\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\nx y : CliffordAlgebra Q\nhx : P (\u2191reverse x)\nhy : P (\u2191reverse y)\n\u22a2 P (\u2191reverse (x + y))\n[PROOFSTEP]\nsimpa only [map_add] using h_add _ _ hx hy\n[GOAL]\ncase h_\u03b9_mul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nP : CliffordAlgebra Q \u2192 Prop\nhr : \u2200 (r : R), P (\u2191(algebraMap R (CliffordAlgebra Q)) r)\nh_add : \u2200 (x y : CliffordAlgebra Q), P x \u2192 P y \u2192 P (x + y)\nh_mul_\u03b9 : \u2200 (x : CliffordAlgebra Q) (m : M), P x \u2192 P (\u2191(\u03b9 Q) m * x)\nm : M\nx : CliffordAlgebra Q\nhx : P (\u2191reverse x)\n\u22a2 P (\u2191reverse (x * \u2191(\u03b9 Q) m))\n[PROOFSTEP]\nsimpa only [reverse.map_mul, reverse_\u03b9] using h_mul_\u03b9 _ _ hx\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\n\u22a2 M \u2192\u2097[R] Module.End R (CliffordAlgebra Q \u00d7 N)\n[PROOFSTEP]\nhave v_mul := (Algebra.lmul R (CliffordAlgebra Q)).toLinearMap \u2218\u2097 \u03b9 Q\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nv_mul : M \u2192\u2097[R] Module.End R (CliffordAlgebra Q)\n\u22a2 M \u2192\u2097[R] Module.End R (CliffordAlgebra Q \u00d7 N)\n[PROOFSTEP]\nhave l := v_mul.compl\u2082 (LinearMap.fst _ _ N)\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nv_mul : M \u2192\u2097[R] Module.End R (CliffordAlgebra Q)\nl : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] CliffordAlgebra Q\n\u22a2 M \u2192\u2097[R] Module.End R (CliffordAlgebra Q \u00d7 N)\n[PROOFSTEP]\nexact\n  { toFun := fun m => (l m).prod (f m)\n    map_add' := fun v\u2082 v\u2082 =>\n      LinearMap.ext fun x => Prod.ext (LinearMap.congr_fun (l.map_add _ _) x) (LinearMap.congr_fun (f.map_add _ _) x)\n    map_smul' := fun c v =>\n      LinearMap.ext fun x =>\n        Prod.ext (LinearMap.congr_fun (l.map_smul _ _) x) (LinearMap.congr_fun (f.map_smul _ _) x) }\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nv : M\nx_fx : CliffordAlgebra Q \u00d7 N\n\u22a2 \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx\n[PROOFSTEP]\ncases' x_fx with x fx\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nv : M\nx : CliffordAlgebra Q\nfx : N\n\u22a2 \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) (x, fx)) = \u2191Q v \u2022 (x, fx)\n[PROOFSTEP]\nsimp only [foldr'Aux_apply_apply]\n[GOAL]\ncase mk\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nv : M\nx : CliffordAlgebra Q\nfx : N\n\u22a2 (\u2191(\u03b9 Q) v * (\u2191(\u03b9 Q) v * x), \u2191(\u2191f v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191f v) (x, fx))) = \u2191Q v \u2022 (x, fx)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u03b9_sq_scalar, \u2190 Algebra.smul_def, hf, Prod.smul_mk]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx : CliffordAlgebra Q\n\u22a2 \u2191(foldr' Q f hf n) (\u2191(\u03b9 Q) m * x) = \u2191(\u2191f m) (x, \u2191(foldr' Q f hf n) x)\n[PROOFSTEP]\ndsimp [foldr']\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx : CliffordAlgebra Q\n\u22a2 (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        (\u2191(\u03b9 Q) m * x)).snd =\n    \u2191(\u2191f m)\n      (x,\n        (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                    (_ :\n                      \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                        \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n                (1, n))\n            x).snd)\n[PROOFSTEP]\nrw [foldr_mul, foldr_\u03b9, foldr'Aux_apply_apply]\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx : CliffordAlgebra Q\n\u22a2 (\u2191(\u03b9 Q) m *\n          (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                      (_ :\n                        \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                          \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n                  (1, n))\n              x).fst,\n        \u2191(\u2191f m)\n          (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                    (_ :\n                      \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                        \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n                (1, n))\n            x)).snd =\n    \u2191(\u2191f m)\n      (x,\n        (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                    (_ :\n                      \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                        \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n                (1, n))\n            x).snd)\n[PROOFSTEP]\nrefine' congr_arg (f m) (Prod.mk.eta.symm.trans _)\n[GOAL]\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx : CliffordAlgebra Q\n\u22a2 ((\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                  (_ :\n                    \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                      \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n              (1, n))\n          x).fst,\n      (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                  (_ :\n                    \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                      \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n              (1, n))\n          x).snd) =\n    (x,\n      (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                  (_ :\n                    \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                      \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n              (1, n))\n          x).snd)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_fst\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx : CliffordAlgebra Q\n\u22a2 (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        x).fst =\n    x\n[PROOFSTEP]\ninduction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx\n[GOAL]\ncase e_fst.hr\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nr : R\n\u22a2 (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        (\u2191(algebraMap R (CliffordAlgebra Q)) r)).fst =\n    \u2191(algebraMap R (CliffordAlgebra Q)) r\n[PROOFSTEP]\nsimp_rw [foldr_algebraMap, Prod.smul_mk, Algebra.algebraMap_eq_smul_one]\n[GOAL]\ncase e_fst.h_add\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm : M\nx y : CliffordAlgebra Q\nhx :\n  (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        x).fst =\n    x\nhy :\n  (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        y).fst =\n    y\n\u22a2 (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        (x + y)).fst =\n    x + y\n[PROOFSTEP]\nrw [map_add, Prod.fst_add, hx, hy]\n[GOAL]\ncase e_fst.h_mul_\u03b9\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : AddCommGroup N\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module R N\nQ : QuadraticForm R M\nf : M \u2192\u2097[R] CliffordAlgebra Q \u00d7 N \u2192\u2097[R] N\nhf : \u2200 (m : M) (x : CliffordAlgebra Q) (fx : N), \u2191(\u2191f m) (\u2191(\u03b9 Q) m * x, \u2191(\u2191f m) (x, fx)) = \u2191Q m \u2022 fx\nn : N\nm\u271d : M\nm : CliffordAlgebra Q\nx : M\nhx :\n  (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        m).fst =\n    m\n\u22a2 (\u2191(\u2191(foldr Q (foldr'Aux Q f)\n                (_ :\n                  \u2200 (v : M) (x_fx : CliffordAlgebra Q \u00d7 N),\n                    \u2191(\u2191(foldr'Aux Q f) v) (\u2191(\u2191(foldr'Aux Q f) v) x_fx) = \u2191Q v \u2022 x_fx))\n            (1, n))\n        (\u2191(\u03b9 Q) x * m)).fst =\n    \u2191(\u03b9 Q) x * m\n[PROOFSTEP]\nrw [foldr_mul, foldr_\u03b9, foldr'Aux_apply_apply, hx]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.CliffordAlgebra.Fold", "llama_tokens": 14319, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998714925402, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.514512571520136}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : \u2115), a \u2208 toFinset (factors n) \u2194 (fun p => if Prime p then padicValNat p n else 0) a \u2260 0\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn0)\n[GOAL]\ncase inl\n\u22a2 \u2200 (a : \u2115), a \u2208 toFinset (factors 0) \u2194 (fun p => if Prime p then padicValNat p 0 else 0) a \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 \u2200 (a : \u2115), a \u2208 toFinset (factors n) \u2194 (fun p => if Prime p then padicValNat p n else 0) a \u2260 0\n[PROOFSTEP]\nsimp only [mem_factors hn0, mem_toFinset, Ne.def, ite_eq_right_iff, not_forall, exists_prop, and_congr_right_iff]\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\n\u22a2 \u2200 (a : \u2115), Prime a \u2192 (a \u2223 n \u2194 \u00acpadicValNat a n = 0)\n[PROOFSTEP]\nrintro p hp\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\np : \u2115\nhp : Prime p\n\u22a2 p \u2223 n \u2194 \u00acpadicValNat p n = 0\n[PROOFSTEP]\nhaveI := fact_iff.mpr hp\n[GOAL]\ncase inr\nn : \u2115\nhn0 : n \u2260 0\np : \u2115\nhp : Prime p\nthis : Fact (Prime p)\n\u22a2 p \u2223 n \u2194 \u00acpadicValNat p n = 0\n[PROOFSTEP]\nexact dvd_iff_padicValNat_ne_zero hn0\n[GOAL]\nn p : \u2115\npp : Prime p\n\u22a2 \u2191(factorization n) p = padicValNat p n\n[PROOFSTEP]\nsimpa [factorization] using absurd pp\n[GOAL]\nn p : \u2115\n\u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn0)\n[GOAL]\ncase inl\np : \u2115\n\u22a2 count p (factors 0) = \u2191(factorization 0) p\n[PROOFSTEP]\nsimp [factorization, count]\n[GOAL]\ncase inr\nn p : \u2115\nhn0 : n > 0\n\u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\nn p : \u2115\nhn0 : n > 0\npp : Prime p\n\u22a2 count p (factors n) = \u2191(factorization n) p\ncase neg n p : \u2115 hn0 : n > 0 pp : \u00acPrime p \u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\ncase neg =>\n  rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]\n  simp [factorization, pp]\n[GOAL]\nn p : \u2115\nhn0 : n > 0\npp : \u00acPrime p\n\u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\ncase neg =>\n  rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]\n  simp [factorization, pp]\n[GOAL]\nn p : \u2115\nhn0 : n > 0\npp : \u00acPrime p\n\u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\nrw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]\n[GOAL]\nn p : \u2115\nhn0 : n > 0\npp : \u00acPrime p\n\u22a2 0 = \u2191(factorization n) p\n[PROOFSTEP]\nsimp [factorization, pp]\n[GOAL]\ncase pos\nn p : \u2115\nhn0 : n > 0\npp : Prime p\n\u22a2 count p (factors n) = \u2191(factorization n) p\n[PROOFSTEP]\nsimp only [factorization, coe_mk, pp, if_true]\n[GOAL]\ncase pos\nn p : \u2115\nhn0 : n > 0\npp : Prime p\n\u22a2 count p (factors n) = padicValNat p n\n[PROOFSTEP]\nrw [\u2190 PartENat.natCast_inj, padicValNat_def' pp.ne_one hn0,\n  UniqueFactorizationMonoid.multiplicity_eq_count_normalizedFactors pp hn0.ne']\n[GOAL]\ncase pos\nn p : \u2115\nhn0 : n > 0\npp : Prime p\n\u22a2 \u2191(count p (factors n)) = \u2191(Multiset.count (\u2191normalize p) (UniqueFactorizationMonoid.normalizedFactors n))\n[PROOFSTEP]\nsimp [factors_eq]\n[GOAL]\nn : \u2115\n\u22a2 factorization n = \u2191Multiset.toFinsupp \u2191(factors n)\n[PROOFSTEP]\next p\n[GOAL]\ncase h\nn p : \u2115\n\u22a2 \u2191(factorization n) p = \u2191(\u2191Multiset.toFinsupp \u2191(factors n)) p\n[PROOFSTEP]\nsimp\n[GOAL]\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 multiplicity p n = \u2191(\u2191(factorization n) p)\n[PROOFSTEP]\nsimp [factorization, pp, padicValNat_def' pp.ne_one hn.bot_lt]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\n\u22a2 (Finsupp.prod (factorization n) fun x x_1 => x ^ x_1) = n\n[PROOFSTEP]\nrw [factorization_eq_factors_multiset n]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\n\u22a2 (Finsupp.prod (\u2191Multiset.toFinsupp \u2191(factors n)) fun x x_1 => x ^ x_1) = n\n[PROOFSTEP]\nsimp only [\u2190 prod_toMultiset, factorization, Multiset.coe_prod, Multiset.toFinsupp_toMultiset]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\n\u22a2 List.prod (factors n) = n\n[PROOFSTEP]\nexact prod_factors hn\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : \u2115), \u2191(factorization a) p = \u2191(factorization b) p\n\u22a2 factors a ~ factors b\n[PROOFSTEP]\nsimpa only [List.perm_iff_count, factors_count_eq] using h\n[GOAL]\na : \u2115\nha : a \u2208 {x | x \u2260 0}\nb : \u2115\nhb : b \u2208 {x | x \u2260 0}\nh : factorization a = factorization b\np : \u2115\n\u22a2 \u2191(factorization a) p = \u2191(factorization b) p\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u22a2 factorization 0 = 0\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\n\u22a2 factorization 1 = 0\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\nn : \u2115\n\u22a2 (factorization n).support = toFinset (factors n)\n[PROOFSTEP]\nsimp [factorization]\n[GOAL]\nn p : \u2115\n\u22a2 p \u2208 (factorization n).support \u2194 p \u2208 factors n\n[PROOFSTEP]\nsimp only [support_factorization, List.mem_toFinset]\n[GOAL]\nn p : \u2115\n\u22a2 \u2191(factorization n) p = 0 \u2194 \u00acPrime p \u2228 \u00acp \u2223 n \u2228 n = 0\n[PROOFSTEP]\nrw [\u2190 not_mem_support_iff, support_factorization, mem_toFinset]\n[GOAL]\nn p : \u2115\n\u22a2 \u00acp \u2208 factors n \u2194 \u00acPrime p \u2228 \u00acp \u2223 n \u2228 n = 0\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\n\u22a2 \u00acp \u2208 factors 0 \u2194 \u00acPrime p \u2228 \u00acp \u2223 0 \u2228 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn p : \u2115\nhn : n \u2260 0\n\u22a2 \u00acp \u2208 factors n \u2194 \u00acPrime p \u2228 \u00acp \u2223 n \u2228 n = 0\n[PROOFSTEP]\nsimp [hn, Nat.mem_factors, not_and_or, -not_and]\n[GOAL]\nn p : \u2115\nhp : \u00acPrime p\n\u22a2 \u2191(factorization n) p = 0\n[PROOFSTEP]\nsimp [factorization_eq_zero_iff, hp]\n[GOAL]\nn p : \u2115\nh : \u00acp \u2223 n\n\u22a2 \u2191(factorization n) p = 0\n[PROOFSTEP]\nsimp [factorization_eq_zero_iff, h]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhn : n \u2260 0\nh : p \u2223 n\n\u22a2 0 < \u2191(factorization n) p\n[PROOFSTEP]\nrwa [\u2190 factors_count_eq, count_pos, mem_factors_iff_dvd hn hp]\n[GOAL]\np r i : \u2115\nhr : \u00acp \u2223 r\n\u22a2 \u2191(factorization (p * i + r)) p = 0\n[PROOFSTEP]\napply factorization_eq_zero_of_not_dvd\n[GOAL]\ncase h\np r i : \u2115\nhr : \u00acp \u2223 r\n\u22a2 \u00acp \u2223 p * i + r\n[PROOFSTEP]\nrwa [\u2190 Nat.dvd_add_iff_right (Dvd.intro i rfl)]\n[GOAL]\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\n\u22a2 \u00acp \u2223 r \u2194 \u2191(factorization (p * i + r)) p = 0\n[PROOFSTEP]\nrefine' \u27e8factorization_eq_zero_of_remainder i, fun h => _\u27e9\n[GOAL]\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\nh : \u2191(factorization (p * i + r)) p = 0\n\u22a2 \u00acp \u2223 r\n[PROOFSTEP]\nrw [factorization_eq_zero_iff] at h \n[GOAL]\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\nh : \u00acPrime p \u2228 \u00acp \u2223 p * i + r \u2228 p * i + r = 0\n\u22a2 \u00acp \u2223 r\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\nh : p \u2223 r\n\u22a2 Prime p \u2227 p \u2223 p * i + r \u2227 p * i + r \u2260 0\n[PROOFSTEP]\nrefine' \u27e8pp, _, _\u27e9\n[GOAL]\ncase refine'_1\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\nh : p \u2223 r\n\u22a2 p \u2223 p * i + r\n[PROOFSTEP]\nrwa [\u2190 Nat.dvd_add_iff_right (dvd_mul_right p i)]\n[GOAL]\ncase refine'_2\np r i : \u2115\npp : Prime p\nhr0 : r \u2260 0\nh : p \u2223 r\n\u22a2 p * i + r \u2260 0\n[PROOFSTEP]\ncontrapose! hr0\n[GOAL]\ncase refine'_2\np r i : \u2115\npp : Prime p\nh : p \u2223 r\nhr0 : p * i + r = 0\n\u22a2 r = 0\n[PROOFSTEP]\nexact (add_eq_zero_iff.mp hr0).2\n[GOAL]\nn : \u2115\n\u22a2 factorization n = 0 \u2194 n = 0 \u2228 n = 1\n[PROOFSTEP]\nrw [factorization_eq_factors_multiset n]\n[GOAL]\nn : \u2115\n\u22a2 \u2191Multiset.toFinsupp \u2191(factors n) = 0 \u2194 n = 0 \u2228 n = 1\n[PROOFSTEP]\nsimp [factorization, AddEquiv.map_eq_zero_iff, Multiset.coe_eq_zero]\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 factorization (a * b) = factorization a + factorization b\n[PROOFSTEP]\next p\n[GOAL]\ncase h\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\np : \u2115\n\u22a2 \u2191(factorization (a * b)) p = \u2191(factorization a + factorization b) p\n[PROOFSTEP]\nsimp only [add_apply, \u2190 factors_count_eq, perm_iff_count.mp (perm_factors_mul ha hb) p, count_append]\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 (factorization (a * b)).support = (factorization a).support \u222a (factorization b).support\n[PROOFSTEP]\next q\n[GOAL]\ncase a\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nq : \u2115\n\u22a2 q \u2208 (factorization (a * b)).support \u2194 q \u2208 (factorization a).support \u222a (factorization b).support\n[PROOFSTEP]\nsimp only [Finset.mem_union, factor_iff_mem_factorization]\n[GOAL]\ncase a\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nq : \u2115\n\u22a2 q \u2208 factors (a * b) \u2194 q \u2208 factors a \u2228 q \u2208 factors b\n[PROOFSTEP]\nexact mem_factors_mul ha hb\n[GOAL]\nn : \u2115\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\n\u22a2 (Finsupp.prod (factorization n) fun p x => f p) = \u220f p in toFinset (factors n), f p\n[PROOFSTEP]\napply prod_congr support_factorization\n[GOAL]\nn : \u2115\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\n\u22a2 \u2200 (x : \u2115), x \u2208 toFinset (factors n) \u2192 (fun p x => f p) x (\u2191(factorization n) x) = f x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\n\u22a2 factorization (Finset.prod S g) = \u2211 x in S, factorization (g x)\n[PROOFSTEP]\nclassical\next p\nrefine' Finset.induction_on' S ?_ ?_\n\u00b7 simp\n\u00b7 intro x T hxS hTS hxT IH\n  have hT : T.prod g \u2260 0 := prod_ne_zero_iff.mpr fun x hx => hS x (hTS hx)\n  simp [prod_insert hxT, sum_insert hxT, \u2190 IH, factorization_mul (hS x hxS) hT]\n[GOAL]\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\n\u22a2 factorization (Finset.prod S g) = \u2211 x in S, factorization (g x)\n[PROOFSTEP]\next p\n[GOAL]\ncase h\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\np : \u2115\n\u22a2 \u2191(factorization (Finset.prod S g)) p = \u2191(\u2211 x in S, factorization (g x)) p\n[PROOFSTEP]\nrefine' Finset.induction_on' S ?_ ?_\n[GOAL]\ncase h.refine'_1\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\np : \u2115\n\u22a2 \u2191(factorization (Finset.prod \u2205 g)) p = \u2191(\u2211 x in \u2205, factorization (g x)) p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\np : \u2115\n\u22a2 \u2200 {a : \u03b1} {s : Finset \u03b1},\n    a \u2208 S \u2192\n      s \u2286 S \u2192\n        \u00aca \u2208 s \u2192\n          \u2191(factorization (Finset.prod s g)) p = \u2191(\u2211 x in s, factorization (g x)) p \u2192\n            \u2191(factorization (Finset.prod (insert a s) g)) p = \u2191(\u2211 x in insert a s, factorization (g x)) p\n[PROOFSTEP]\nintro x T hxS hTS hxT IH\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\np : \u2115\nx : \u03b1\nT : Finset \u03b1\nhxS : x \u2208 S\nhTS : T \u2286 S\nhxT : \u00acx \u2208 T\nIH : \u2191(factorization (Finset.prod T g)) p = \u2191(\u2211 x in T, factorization (g x)) p\n\u22a2 \u2191(factorization (Finset.prod (insert x T) g)) p = \u2191(\u2211 x in insert x T, factorization (g x)) p\n[PROOFSTEP]\nhave hT : T.prod g \u2260 0 := prod_ne_zero_iff.mpr fun x hx => hS x (hTS hx)\n[GOAL]\ncase h.refine'_2\n\u03b1 : Type u_1\nS : Finset \u03b1\ng : \u03b1 \u2192 \u2115\nhS : \u2200 (x : \u03b1), x \u2208 S \u2192 g x \u2260 0\np : \u2115\nx : \u03b1\nT : Finset \u03b1\nhxS : x \u2208 S\nhTS : T \u2286 S\nhxT : \u00acx \u2208 T\nIH : \u2191(factorization (Finset.prod T g)) p = \u2191(\u2211 x in T, factorization (g x)) p\nhT : Finset.prod T g \u2260 0\n\u22a2 \u2191(factorization (Finset.prod (insert x T) g)) p = \u2191(\u2211 x in insert x T, factorization (g x)) p\n[PROOFSTEP]\nsimp [prod_insert hxT, sum_insert hxT, \u2190 IH, factorization_mul (hS x hxS) hT]\n[GOAL]\nn k : \u2115\n\u22a2 factorization (n ^ k) = k \u2022 factorization n\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 factorization (n ^ zero) = zero \u2022 factorization n\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase succ.inl\nk : \u2115\nih : factorization (0 ^ k) = k \u2022 factorization 0\n\u22a2 factorization (0 ^ succ k) = succ k \u2022 factorization 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.inr\nn k : \u2115\nih : factorization (n ^ k) = k \u2022 factorization n\nhn : n \u2260 0\n\u22a2 factorization (n ^ succ k) = succ k \u2022 factorization n\n[PROOFSTEP]\nrw [pow_succ, mul_comm, factorization_mul hn (pow_ne_zero _ hn), ih, succ_eq_one_add, add_smul, one_smul]\n[GOAL]\np : \u2115\nhp : Prime p\n\u22a2 Nat.factorization p = single p 1\n[PROOFSTEP]\next q\n[GOAL]\ncase h\np : \u2115\nhp : Prime p\nq : \u2115\n\u22a2 \u2191(Nat.factorization p) q = \u2191(single p 1) q\n[PROOFSTEP]\nrw [\u2190 factors_count_eq, factors_prime hp, single_apply, count_singleton', if_congr eq_comm]\n[GOAL]\ncase h.h_t\np : \u2115\nhp : Prime p\nq : \u2115\n\u22a2 1 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.h_e\np : \u2115\nhp : Prime p\nq : \u2115\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\np : \u2115\nhp : Prime p\n\u22a2 \u2191(Nat.factorization p) p = 1\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\np k : \u2115\nhp : Prime p\n\u22a2 Nat.factorization (p ^ k) = single p k\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\nn p k : \u2115\nhn : n \u2260 0\nh : factorization n = single p k\n\u22a2 n = p ^ k\n[PROOFSTEP]\nrw [\u2190 Nat.factorization_prod_pow_eq_self hn, h, Finsupp.prod_single_index]\n[GOAL]\nn p k : \u2115\nhn : n \u2260 0\nh : factorization n = single p k\n\u22a2 p ^ 0 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\np q : \u2115\nhp : Prime p\nh : \u2191(Nat.factorization p) q \u2260 0\n\u22a2 p = q\n[PROOFSTEP]\nsimpa [hp.factorization, single_apply] using h\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\n\u22a2 factorization (Finsupp.prod f fun x x_1 => x ^ x_1) = f\n[PROOFSTEP]\nhave h : \u2200 x : \u2115, x \u2208 f.support \u2192 x ^ f x \u2260 0 := fun p hp => pow_ne_zero _ (Prime.ne_zero (hf p hp))\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n\u22a2 factorization (Finsupp.prod f fun x x_1 => x ^ x_1) = f\n[PROOFSTEP]\nsimp only [Finsupp.prod, factorization_prod h]\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n\u22a2 \u2211 x in f.support, factorization (x ^ \u2191f x) = f\n[PROOFSTEP]\nconv =>\n  rhs\n  rw [(sum_single f).symm]\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n| \u2211 x in f.support, factorization (x ^ \u2191f x) = f\n[PROOFSTEP]\n  rhs\n  rw [(sum_single f).symm]\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n| \u2211 x in f.support, factorization (x ^ \u2191f x) = f\n[PROOFSTEP]\n  rhs\n  rw [(sum_single f).symm]\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n| \u2211 x in f.support, factorization (x ^ \u2191f x) = f\n[PROOFSTEP]\nrhs\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n| f\n[PROOFSTEP]\nrw [(sum_single f).symm]\n[GOAL]\nf : \u2115 \u2192\u2080 \u2115\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : \u2200 (x : \u2115), x \u2208 f.support \u2192 x ^ \u2191f x \u2260 0\n\u22a2 \u2211 x in f.support, factorization (x ^ \u2191f x) = Finsupp.sum f single\n[PROOFSTEP]\nexact sum_congr rfl fun p hp => Prime.factorization_pow (hf p hp)\n[GOAL]\nn : \u2115\nf : \u2115 \u2192\u2080 \u2115\nhn : n \u2260 0\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : f = factorization n\n\u22a2 (Finsupp.prod f fun x x_1 => x ^ x_1) = n\n[PROOFSTEP]\nrw [h, factorization_prod_pow_eq_self hn]\n[GOAL]\nn : \u2115\nf : \u2115 \u2192\u2080 \u2115\nhn : n \u2260 0\nhf : \u2200 (p : \u2115), p \u2208 f.support \u2192 Prime p\nh : (Finsupp.prod f fun x x_1 => x ^ x_1) = n\n\u22a2 f = factorization n\n[PROOFSTEP]\nrw [\u2190 h, prod_pow_factorization_eq_self hf]\n[GOAL]\nn : \u2115+\n\u22a2 \u2191(\u2191factorizationEquiv n) = factorization \u2191n\n[PROOFSTEP]\ncases n\n[GOAL]\ncase mk\nval\u271d : \u2115\nproperty\u271d : 0 < val\u271d\n\u22a2 \u2191(\u2191factorizationEquiv { val := val\u271d, property := property\u271d }) = factorization \u2191{ val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\nn p : \u2115\nhp : \u00acPrime p\n\u22a2 p ^ \u2191(factorization n) p = 1\n[PROOFSTEP]\nsimp [factorization_eq_zero_of_non_prime n hp]\n[GOAL]\nn p : \u2115\nhp : \u00acPrime p\n\u22a2 n / p ^ \u2191(factorization n) p = n\n[PROOFSTEP]\nsimp [factorization_eq_zero_of_non_prime n hp]\n[GOAL]\nn p : \u2115\n\u22a2 p ^ \u2191(factorization n) p \u2223 n\n[PROOFSTEP]\nby_cases hp : p.Prime\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\n\u22a2 p ^ \u2191(factorization n) p \u2223 n\ncase neg n p : \u2115 hp : \u00acPrime p \u22a2 p ^ \u2191(factorization n) p \u2223 n\n[PROOFSTEP]\ncase neg => simp [hp]\n[GOAL]\nn p : \u2115\nhp : \u00acPrime p\n\u22a2 p ^ \u2191(factorization n) p \u2223 n\n[PROOFSTEP]\ncase neg => simp [hp]\n[GOAL]\nn p : \u2115\nhp : \u00acPrime p\n\u22a2 p ^ \u2191(factorization n) p \u2223 n\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\n\u22a2 p ^ \u2191(factorization n) p \u2223 n\n[PROOFSTEP]\nrw [\u2190 factors_count_eq]\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\n\u22a2 p ^ count p (factors n) \u2223 n\n[PROOFSTEP]\napply dvd_of_factors_subperm (pow_ne_zero _ hp.ne_zero)\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\n\u22a2 factors (p ^ count p (factors n)) <+~ factors n\n[PROOFSTEP]\nrw [hp.factors_pow, List.subperm_ext_iff]\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\n\u22a2 \u2200 (x : \u2115), x \u2208 replicate (count p (factors n)) p \u2192 count x (replicate (count p (factors n)) p) \u2264 count x (factors n)\n[PROOFSTEP]\nintro q hq\n[GOAL]\ncase pos\nn p : \u2115\nhp : Prime p\nq : \u2115\nhq : q \u2208 replicate (count p (factors n)) p\n\u22a2 count q (replicate (count p (factors n)) p) \u2264 count q (factors n)\n[PROOFSTEP]\nsimp [List.eq_of_mem_replicate hq]\n[GOAL]\nn p : \u2115\n\u22a2 0 < p ^ \u2191(factorization n) p\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\nn p : \u2115\npp : Prime p\n\u22a2 0 < p ^ \u2191(factorization n) p\n[PROOFSTEP]\nsimp [pow_pos pp.pos]\n[GOAL]\ncase neg\nn p : \u2115\npp : \u00acPrime p\n\u22a2 0 < p ^ \u2191(factorization n) p\n[PROOFSTEP]\nsimp [pp]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\n\u22a2 0 < n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\ncases' em' p.Prime with pp pp\n[GOAL]\ncase inl\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 0 < n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nsimpa [Nat.factorization_eq_zero_of_non_prime n pp] using hn.bot_lt\n[GOAL]\ncase inr\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 0 < n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nexact Nat.div_pos (ord_proj_le p hn) (ord_proj_pos n p)\n[GOAL]\na b p : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 p ^ \u2191(factorization (a * b)) p = p ^ \u2191(factorization a) p * p ^ \u2191(factorization b) p\n[PROOFSTEP]\nsimp [factorization_mul ha hb, pow_add]\n[GOAL]\na b p : \u2115\n\u22a2 a * b / p ^ \u2191(factorization (a * b)) p = a / p ^ \u2191(factorization a) p * (b / p ^ \u2191(factorization b) p)\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\nb p : \u2115\n\u22a2 0 * b / p ^ \u2191(factorization (0 * b)) p = 0 / p ^ \u2191(factorization 0) p * (b / p ^ \u2191(factorization b) p)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\na b p : \u2115\nha : a \u2260 0\n\u22a2 a * b / p ^ \u2191(factorization (a * b)) p = a / p ^ \u2191(factorization a) p * (b / p ^ \u2191(factorization b) p)\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb)\n[GOAL]\ncase inr.inl\na p : \u2115\nha : a \u2260 0\n\u22a2 a * 0 / p ^ \u2191(factorization (a * 0)) p = a / p ^ \u2191(factorization a) p * (0 / p ^ \u2191(factorization 0) p)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\na b p : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a * b / p ^ \u2191(factorization (a * b)) p = a / p ^ \u2191(factorization a) p * (b / p ^ \u2191(factorization b) p)\n[PROOFSTEP]\nsimp only [ord_proj_mul p ha hb]\n[GOAL]\ncase inr.inr\na b p : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a * b / (p ^ \u2191(factorization a) p * p ^ \u2191(factorization b) p) =\n    a / p ^ \u2191(factorization a) p * (b / p ^ \u2191(factorization b) p)\n[PROOFSTEP]\nrw [mul_div_mul_comm_of_dvd_dvd (ord_proj_dvd a p) (ord_proj_dvd b p)]\n[GOAL]\nn p : \u2115\nh : p \u2208 (factorization n).support\n\u22a2 p \u2223 n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\nh : p \u2208 (factorization 0).support\n\u22a2 p \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn p : \u2115\nh : p \u2208 (factorization n).support\nhn : n \u2260 0\n\u22a2 p \u2223 n\n[PROOFSTEP]\nsimp [\u2190 mem_factors_iff_dvd hn (prime_of_mem_factorization h), factor_iff_mem_factorization.mp h]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\n\u22a2 \u2191(factorization n) p < n\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 \u2191(factorization n) p < n\ncase neg n p : \u2115 hn : n \u2260 0 pp : \u00acPrime p \u22a2 \u2191(factorization n) p < n\n[PROOFSTEP]\ncase neg =>\n  simp [factorization_eq_zero_of_non_prime n pp]\n  exact hn.bot_lt\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 \u2191(factorization n) p < n\n[PROOFSTEP]\ncase neg =>\n  simp [factorization_eq_zero_of_non_prime n pp]\n  exact hn.bot_lt\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 \u2191(factorization n) p < n\n[PROOFSTEP]\nsimp [factorization_eq_zero_of_non_prime n pp]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 0 < n\n[PROOFSTEP]\nexact hn.bot_lt\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 \u2191(factorization n) p < n\n[PROOFSTEP]\nrw [\u2190 pow_lt_iff_lt_right pp.two_le]\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 p ^ \u2191(factorization n) p < p ^ n\n[PROOFSTEP]\napply lt_of_le_of_lt (ord_proj_le p hn)\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 n < p ^ n\n[PROOFSTEP]\nexact lt_of_lt_of_le (lt_two_pow n) (pow_le_pow_of_le_left pp.two_le n)\n[GOAL]\nn p b : \u2115\nhb : n \u2264 p ^ b\n\u22a2 \u2191(factorization n) p \u2264 b\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np b : \u2115\nhb : 0 \u2264 p ^ b\n\u22a2 \u2191(factorization 0) p \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn p b : \u2115\nhb : n \u2264 p ^ b\nhn : n \u2260 0\n\u22a2 \u2191(factorization n) p \u2264 b\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\nn p b : \u2115\nhb : n \u2264 p ^ b\nhn : n \u2260 0\npp : Prime p\n\u22a2 \u2191(factorization n) p \u2264 b\n[PROOFSTEP]\nexact (pow_le_iff_le_right pp.two_le).1 (le_trans (ord_proj_le p hn) hb)\n[GOAL]\ncase neg\nn p b : \u2115\nhb : n \u2264 p ^ b\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 \u2191(factorization n) p \u2264 b\n[PROOFSTEP]\nsimp [factorization_eq_zero_of_non_prime n pp]\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 factorization d \u2264 factorization n \u2194 d \u2223 n\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 factorization d \u2264 factorization n \u2192 d \u2223 n\n[PROOFSTEP]\nintro hdn\n[GOAL]\ncase mp\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\nhdn : factorization d \u2264 factorization n\n\u22a2 d \u2223 n\n[PROOFSTEP]\nset K := n.factorization - d.factorization with hK\n[GOAL]\ncase mp\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\nhdn : factorization d \u2264 factorization n\nK : \u2115 \u2192\u2080 \u2115 := factorization n - factorization d\nhK : K = factorization n - factorization d\n\u22a2 d \u2223 n\n[PROOFSTEP]\nuse K.prod (\u00b7 ^ \u00b7)\n[GOAL]\ncase h\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\nhdn : factorization d \u2264 factorization n\nK : \u2115 \u2192\u2080 \u2115 := factorization n - factorization d\nhK : K = factorization n - factorization d\n\u22a2 n = d * Finsupp.prod K fun x x_1 => x ^ x_1\n[PROOFSTEP]\nrw [\u2190 factorization_prod_pow_eq_self hn, \u2190 factorization_prod_pow_eq_self hd, \u2190\n  Finsupp.prod_add_index' pow_zero pow_add, hK, add_tsub_cancel_of_le hdn]\n[GOAL]\ncase mpr\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 d \u2223 n \u2192 factorization d \u2264 factorization n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9\n[GOAL]\ncase mpr.intro\nd : \u2115\nhd : d \u2260 0\nc : \u2115\nhn : d * c \u2260 0\n\u22a2 factorization d \u2264 factorization (d * c)\n[PROOFSTEP]\nrw [factorization_mul hd (right_ne_zero_of_mul hn)]\n[GOAL]\ncase mpr.intro\nd : \u2115\nhd : d \u2260 0\nc : \u2115\nhn : d * c \u2260 0\n\u22a2 factorization d \u2264 factorization d + factorization c\n[PROOFSTEP]\nsimp\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 (\u2200 (p : \u2115), Prime p \u2192 \u2191(factorization d) p \u2264 \u2191(factorization n) p) \u2194 d \u2223 n\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd hd hn]\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 (\u2200 (p : \u2115), Prime p \u2192 \u2191(factorization d) p \u2264 \u2191(factorization n) p) \u2194 factorization d \u2264 factorization n\n[PROOFSTEP]\nrefine' \u27e8fun h p => (em p.Prime).elim (h p) fun hp => _, fun h p _ => h p\u27e9\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhn : n \u2260 0\nh : \u2200 (p : \u2115), Prime p \u2192 \u2191(factorization d) p \u2264 \u2191(factorization n) p\np : \u2115\nhp : \u00acPrime p\n\u22a2 \u2191(factorization d) p \u2264 \u2191(factorization n) p\n[PROOFSTEP]\nsimp_rw [factorization_eq_zero_of_non_prime _ hp, le_refl]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\nhp : Prime p\n\u22a2 \u00acp ^ (\u2191(factorization n) p + 1) \u2223 n\n[PROOFSTEP]\nintro h\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\nhp : Prime p\nh : p ^ (\u2191(factorization n) p + 1) \u2223 n\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd (pow_pos hp.pos _).ne' hn] at h \n[GOAL]\nn p : \u2115\nhn : n \u2260 0\nhp : Prime p\nh : factorization (p ^ (\u2191(factorization n) p + 1)) \u2264 factorization n\n\u22a2 False\n[PROOFSTEP]\nsimpa [hp.factorization] using h p\n[GOAL]\na b : \u2115\nhb : b \u2260 0\n\u22a2 factorization a \u2264 factorization (a * b)\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha)\n[GOAL]\ncase inl\nb : \u2115\nhb : b \u2260 0\n\u22a2 factorization 0 \u2264 factorization (0 * b)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\na b : \u2115\nhb : b \u2260 0\nha : a \u2260 0\n\u22a2 factorization a \u2264 factorization (a * b)\n[PROOFSTEP]\nrw [factorization_le_iff_dvd ha <| mul_ne_zero ha hb]\n[GOAL]\ncase inr\na b : \u2115\nhb : b \u2260 0\nha : a \u2260 0\n\u22a2 a \u2223 a * b\n[PROOFSTEP]\nexact Dvd.intro b rfl\n[GOAL]\na b : \u2115\nha : a \u2260 0\n\u22a2 factorization b \u2264 factorization (a * b)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\na b : \u2115\nha : a \u2260 0\n\u22a2 factorization b \u2264 factorization (b * a)\n[PROOFSTEP]\napply factorization_le_factorization_mul_left ha\n[GOAL]\np k n : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 p ^ k \u2223 n \u2194 k \u2264 \u2191(Nat.factorization n) p\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd (pow_pos pp.pos k).ne' hn, pp.factorization_pow, single_le_iff]\n[GOAL]\np k n : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 p ^ k \u2223 n \u2194 p ^ k \u2223 p ^ \u2191(Nat.factorization n) p\n[PROOFSTEP]\nrw [pow_dvd_pow_iff_le_right pp.one_lt, pp.pow_dvd_iff_le_factorization hn]\n[GOAL]\np n : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 p \u2223 n \u2194 p ^ 1 \u2223 n\n[PROOFSTEP]\nsimp\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhab : a < b\n\u22a2 \u2203 p, \u2191(factorization a) p < \u2191(factorization b) p\n[PROOFSTEP]\nhave hb : b \u2260 0 := (ha.bot_lt.trans hab).ne'\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhab : a < b\nhb : b \u2260 0\n\u22a2 \u2203 p, \u2191(factorization a) p < \u2191(factorization b) p\n[PROOFSTEP]\ncontrapose! hab\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nhab : \u2200 (p : \u2115), \u2191(factorization b) p \u2264 \u2191(factorization a) p\n\u22a2 b \u2264 a\n[PROOFSTEP]\nrw [\u2190 Finsupp.le_def, factorization_le_iff_dvd hb ha] at hab \n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nhab : b \u2223 a\n\u22a2 b \u2264 a\n[PROOFSTEP]\nexact le_of_dvd ha.bot_lt hab\n[GOAL]\nd n : \u2115\nh : d \u2223 n\n\u22a2 factorization (n / d) = factorization n - factorization d\n[PROOFSTEP]\nrcases eq_or_ne d 0 with (rfl | hd)\n[GOAL]\ncase inl\nn : \u2115\nh : 0 \u2223 n\n\u22a2 factorization (n / 0) = factorization n - factorization 0\n[PROOFSTEP]\nsimp [zero_dvd_iff.mp h]\n[GOAL]\ncase inr\nd n : \u2115\nh : d \u2223 n\nhd : d \u2260 0\n\u22a2 factorization (n / d) = factorization n - factorization d\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inr.inl\nd : \u2115\nhd : d \u2260 0\nh : d \u2223 0\n\u22a2 factorization (0 / d) = factorization 0 - factorization d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nd n : \u2115\nh : d \u2223 n\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 factorization (n / d) = factorization n - factorization d\n[PROOFSTEP]\napply add_left_injective d.factorization\n[GOAL]\ncase inr.inr.a\nd n : \u2115\nh : d \u2223 n\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 (fun x => x + factorization d) (factorization (n / d)) =\n    (fun x => x + factorization d) (factorization n - factorization d)\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase inr.inr.a\nd n : \u2115\nh : d \u2223 n\nhd : d \u2260 0\nhn : n \u2260 0\n\u22a2 factorization (n / d) + factorization d = factorization n - factorization d + factorization d\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le <| (Nat.factorization_le_iff_dvd hd hn).mpr h, \u2190\n  Nat.factorization_mul (Nat.div_pos (Nat.le_of_dvd hn.bot_lt h) hd.bot_lt).ne' hd, Nat.div_mul_cancel h]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhn : n \u2260 0\n\u22a2 \u00acp \u2223 n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nrw [Nat.Prime.dvd_iff_one_le_factorization hp (ord_compl_pos p hn).ne']\n[GOAL]\nn p : \u2115\nhp : Prime p\nhn : n \u2260 0\n\u22a2 \u00ac1 \u2264 \u2191(factorization (n / p ^ \u2191(factorization n) p)) p\n[PROOFSTEP]\nrw [Nat.factorization_div (Nat.ord_proj_dvd n p)]\n[GOAL]\nn p : \u2115\nhp : Prime p\nhn : n \u2260 0\n\u22a2 \u00ac1 \u2264 \u2191(factorization n - factorization (p ^ \u2191(factorization n) p)) p\n[PROOFSTEP]\nsimp [hp.factorization]\n[GOAL]\nn p : \u2115\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\n\u22a2 factorization (0 / p ^ \u2191(factorization 0) p) = Finsupp.erase p (factorization 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn p : \u2115\nhn : n \u2260 0\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\ncase neg\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\ncase neg =>\n  -- porting note: needed to solve side goal explicitly\n  rw [Finsupp.erase_of_not_mem_support]\n  \u00b7 simp [pp]\n  \u00b7 simp [mt prime_of_mem_factors pp]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\ncase neg =>\n  -- porting note: needed to solve side goal explicitly\n  rw [Finsupp.erase_of_not_mem_support]\n  \u00b7 simp [pp]\n  \u00b7 simp [mt prime_of_mem_factors pp]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\nrw [Finsupp.erase_of_not_mem_support]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = factorization n\n[PROOFSTEP]\nsimp [pp]\n[GOAL]\nn p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 \u00acp \u2208 (factorization n).support\n[PROOFSTEP]\nsimp [mt prime_of_mem_factors pp]\n[GOAL]\ncase pos\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 factorization (n / p ^ \u2191(factorization n) p) = Finsupp.erase p (factorization n)\n[PROOFSTEP]\next q\n[GOAL]\ncase pos.h\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\nq : \u2115\n\u22a2 \u2191(factorization (n / p ^ \u2191(factorization n) p)) q = \u2191(Finsupp.erase p (factorization n)) q\n[PROOFSTEP]\nrcases eq_or_ne q p with (rfl | hqp)\n[GOAL]\ncase pos.h.inl\nn : \u2115\nhn : n \u2260 0\nq : \u2115\npp : Prime q\n\u22a2 \u2191(factorization (n / q ^ \u2191(factorization n) q)) q = \u2191(Finsupp.erase q (factorization n)) q\n[PROOFSTEP]\nsimp only [Finsupp.erase_same, factorization_eq_zero_iff, not_dvd_ord_compl pp hn]\n[GOAL]\ncase pos.h.inl\nn : \u2115\nhn : n \u2260 0\nq : \u2115\npp : Prime q\n\u22a2 \u00acPrime q \u2228 True \u2228 n / q ^ \u2191(factorization n) q = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.h.inr\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\nq : \u2115\nhqp : q \u2260 p\n\u22a2 \u2191(factorization (n / p ^ \u2191(factorization n) p)) q = \u2191(Finsupp.erase p (factorization n)) q\n[PROOFSTEP]\nrw [Finsupp.erase_ne hqp, factorization_div (ord_proj_dvd n p)]\n[GOAL]\ncase pos.h.inr\nn p : \u2115\nhn : n \u2260 0\npp : Prime p\nq : \u2115\nhqp : q \u2260 p\n\u22a2 \u2191(factorization n - factorization (p ^ \u2191(factorization n) p)) q = \u2191(factorization n) q\n[PROOFSTEP]\nsimp [pp.factorization, hqp.symm]\n[GOAL]\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\n\u22a2 d \u2223 n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn0)\n[GOAL]\ncase inl\np d : \u2115\nhpd : \u00acp \u2223 d\nhdn : d \u2223 0\n\u22a2 d \u2223 0 / p ^ \u2191(factorization 0) p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\nhn0 : n \u2260 0\n\u22a2 d \u2223 n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nrcases eq_or_ne d 0 with (rfl | hd0)\n[GOAL]\ncase inr.inl\np n : \u2115\nhn0 : n \u2260 0\nhdn : 0 \u2223 n\nhpd : \u00acp \u2223 0\n\u22a2 0 \u2223 n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nsimp at hpd \n[GOAL]\ncase inr.inr\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\nhn0 : n \u2260 0\nhd0 : d \u2260 0\n\u22a2 d \u2223 n / p ^ \u2191(factorization n) p\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd hd0 (ord_compl_pos p hn0).ne', factorization_ord_compl]\n[GOAL]\ncase inr.inr\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\nhn0 : n \u2260 0\nhd0 : d \u2260 0\n\u22a2 factorization d \u2264 Finsupp.erase p (factorization n)\n[PROOFSTEP]\nintro q\n[GOAL]\ncase inr.inr\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\nhn0 : n \u2260 0\nhd0 : d \u2260 0\nq : \u2115\n\u22a2 \u2191(factorization d) q \u2264 \u2191(Finsupp.erase p (factorization n)) q\n[PROOFSTEP]\nrcases eq_or_ne q p with (rfl | hqp)\n[GOAL]\ncase inr.inr.inl\nd n : \u2115\nhdn : d \u2223 n\nhn0 : n \u2260 0\nhd0 : d \u2260 0\nq : \u2115\nhpd : \u00acq \u2223 d\n\u22a2 \u2191(factorization d) q \u2264 \u2191(Finsupp.erase q (factorization n)) q\n[PROOFSTEP]\nsimp [factorization_eq_zero_iff, hpd]\n[GOAL]\ncase inr.inr.inr\np d n : \u2115\nhdn : d \u2223 n\nhpd : \u00acp \u2223 d\nhn0 : n \u2260 0\nhd0 : d \u2260 0\nq : \u2115\nhqp : q \u2260 p\n\u22a2 \u2191(factorization d) q \u2264 \u2191(Finsupp.erase p (factorization n)) q\n[PROOFSTEP]\nsimp [hqp, (factorization_le_iff_dvd hd0 hn0).2 hdn q]\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\n\u22a2 d \u2223 n \u2194 factorization (n / d) = factorization n - factorization d\n[PROOFSTEP]\nrefine' \u27e8factorization_div, _\u27e9\n[GOAL]\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\n\u22a2 factorization (n / d) = factorization n - factorization d \u2192 d \u2223 n\n[PROOFSTEP]\nrcases eq_or_lt_of_le hdn with (rfl | hd_lt_n)\n[GOAL]\ncase inl\nd : \u2115\nhd : d \u2260 0\nhdn : d \u2264 d\n\u22a2 factorization (d / d) = factorization d - factorization d \u2192 d \u2223 d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\n\u22a2 factorization (n / d) = factorization n - factorization d \u2192 d \u2223 n\n[PROOFSTEP]\nhave h1 : n / d \u2260 0 := fun H => Nat.lt_asymm hd_lt_n ((Nat.div_eq_zero_iff hd.bot_lt).mp H)\n[GOAL]\ncase inr\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\nh1 : n / d \u2260 0\n\u22a2 factorization (n / d) = factorization n - factorization d \u2192 d \u2223 n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase inr\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\nh1 : n / d \u2260 0\nh : factorization (n / d) = factorization n - factorization d\n\u22a2 d \u2223 n\n[PROOFSTEP]\nrw [dvd_iff_le_div_mul n d]\n[GOAL]\ncase inr\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\nh1 : n / d \u2260 0\nh : factorization (n / d) = factorization n - factorization d\n\u22a2 n \u2264 n / d * d\n[PROOFSTEP]\nby_contra h2\n[GOAL]\ncase inr\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\nh1 : n / d \u2260 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : \u00acn \u2264 n / d * d\n\u22a2 False\n[PROOFSTEP]\ncases' exists_factorization_lt_of_lt (mul_ne_zero h1 hd) (not_le.mp h2) with p hp\n[GOAL]\ncase inr.intro\nd n : \u2115\nhd : d \u2260 0\nhdn : d \u2264 n\nhd_lt_n : d < n\nh1 : n / d \u2260 0\nh : factorization (n / d) = factorization n - factorization d\nh2 : \u00acn \u2264 n / d * d\np : \u2115\nhp : \u2191(factorization (n / d * d)) p < \u2191(factorization n) p\n\u22a2 False\n[PROOFSTEP]\nrwa [factorization_mul h1 hd, add_apply, \u2190 lt_tsub_iff_right, h, tsub_apply, lt_self_iff_false] at hp \n[GOAL]\na b : \u2115\nhb0 : b \u2260 0\nhab : a \u2223 b\np : \u2115\n\u22a2 p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrcases em' p.Prime with (pp | pp)\n[GOAL]\ncase inl\na b : \u2115\nhb0 : b \u2260 0\nhab : a \u2223 b\np : \u2115\npp : \u00acPrime p\n\u22a2 p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n[PROOFSTEP]\nsimp [pp]\n[GOAL]\ncase inr\na b : \u2115\nhb0 : b \u2260 0\nhab : a \u2223 b\np : \u2115\npp : Prime p\n\u22a2 p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha0)\n[GOAL]\ncase inr.inl\nb : \u2115\nhb0 : b \u2260 0\np : \u2115\npp : Prime p\nhab : 0 \u2223 b\n\u22a2 p ^ \u2191(factorization 0) p \u2223 p ^ \u2191(factorization b) p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\na b : \u2115\nhb0 : b \u2260 0\nhab : a \u2223 b\np : \u2115\npp : Prime p\nha0 : a \u2260 0\n\u22a2 p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrw [pow_dvd_pow_iff_le_right pp.one_lt]\n[GOAL]\ncase inr.inr\na b : \u2115\nhb0 : b \u2260 0\nhab : a \u2223 b\np : \u2115\npp : Prime p\nha0 : a \u2260 0\n\u22a2 \u2191(factorization a) p \u2264 \u2191(factorization b) p\n[PROOFSTEP]\nexact (factorization_le_iff_dvd ha0 hb0).2 hab p\n[GOAL]\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 (\u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p) \u2194 a \u2223 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun hab p => ord_proj_dvd_ord_proj_of_dvd hb0 hab p\u27e9\n[GOAL]\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd ha0 hb0]\n[GOAL]\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\n\u22a2 factorization a \u2264 factorization b\n[PROOFSTEP]\nintro q\n[GOAL]\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\nq : \u2115\n\u22a2 \u2191(factorization a) q \u2264 \u2191(factorization b) q\n[PROOFSTEP]\nrcases le_or_lt q 1 with (hq_le | hq1)\n[GOAL]\ncase inl\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\nq : \u2115\nhq_le : q \u2264 1\n\u22a2 \u2191(factorization a) q \u2264 \u2191(factorization b) q\n[PROOFSTEP]\ninterval_cases q\n[GOAL]\ncase inl.\u00ab0\u00bb\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\nq : \u2115\nhq_le : 0 \u2264 1\n\u22a2 \u2191(factorization a) 0 \u2264 \u2191(factorization b) 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.\u00ab1\u00bb\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\nq : \u2115\nhq_le : 1 \u2264 1\n\u22a2 \u2191(factorization a) 1 \u2264 \u2191(factorization b) 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\na b : \u2115\nha0 : a \u2260 0\nhb0 : b \u2260 0\nh : \u2200 (p : \u2115), p ^ \u2191(factorization a) p \u2223 p ^ \u2191(factorization b) p\nq : \u2115\nhq1 : 1 < q\n\u22a2 \u2191(factorization a) q \u2264 \u2191(factorization b) q\n[PROOFSTEP]\nexact (pow_dvd_pow_iff_le_right hq1).1 (h q)\n[GOAL]\na b : \u2115\nhab : a \u2223 b\np : \u2115\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrcases em' p.Prime with (pp | pp)\n[GOAL]\ncase inl\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : \u00acPrime p\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nsimp [pp, hab]\n[GOAL]\ncase inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb0)\n[GOAL]\ncase inr.inl\na p : \u2115\npp : Prime p\nhab : a \u2223 0\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 0 / p ^ \u2191(factorization 0) p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrcases eq_or_ne a 0 with (rfl | ha0)\n[GOAL]\ncase inr.inr.inl\nb p : \u2115\npp : Prime p\nhb0 : b \u2260 0\nhab : 0 \u2223 b\n\u22a2 0 / p ^ \u2191(factorization 0) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\ncases hb0 (zero_dvd_iff.1 hab)\n[GOAL]\ncase inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nhave ha := (Nat.div_pos (ord_proj_le p ha0) (ord_proj_pos a p)).ne'\n[GOAL]\ncase inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nhave hb := (Nat.div_pos (ord_proj_le p hb0) (ord_proj_pos b p)).ne'\n[GOAL]\ncase inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\nhb : b / p ^ \u2191(factorization b) p \u2260 0\n\u22a2 a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd ha hb, factorization_ord_compl a p, factorization_ord_compl b p]\n[GOAL]\ncase inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\nhb : b / p ^ \u2191(factorization b) p \u2260 0\n\u22a2 Finsupp.erase p (factorization a) \u2264 Finsupp.erase p (factorization b)\n[PROOFSTEP]\nintro q\n[GOAL]\ncase inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\nhb : b / p ^ \u2191(factorization b) p \u2260 0\nq : \u2115\n\u22a2 \u2191(Finsupp.erase p (factorization a)) q \u2264 \u2191(Finsupp.erase p (factorization b)) q\n[PROOFSTEP]\nrcases eq_or_ne q p with (rfl | hqp)\n[GOAL]\ncase inr.inr.inr.inl\na b : \u2115\nhab : a \u2223 b\nhb0 : b \u2260 0\nha0 : a \u2260 0\nq : \u2115\npp : Prime q\nha : a / q ^ \u2191(factorization a) q \u2260 0\nhb : b / q ^ \u2191(factorization b) q \u2260 0\n\u22a2 \u2191(Finsupp.erase q (factorization a)) q \u2264 \u2191(Finsupp.erase q (factorization b)) q\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\nhb : b / p ^ \u2191(factorization b) p \u2260 0\nq : \u2115\nhqp : q \u2260 p\n\u22a2 \u2191(Finsupp.erase p (factorization a)) q \u2264 \u2191(Finsupp.erase p (factorization b)) q\n[PROOFSTEP]\nsimp_rw [erase_ne hqp]\n[GOAL]\ncase inr.inr.inr.inr\na b : \u2115\nhab : a \u2223 b\np : \u2115\npp : Prime p\nhb0 : b \u2260 0\nha0 : a \u2260 0\nha : a / p ^ \u2191(factorization a) p \u2260 0\nhb : b / p ^ \u2191(factorization b) p \u2260 0\nq : \u2115\nhqp : q \u2260 p\n\u22a2 \u2191(factorization a) q \u2264 \u2191(factorization b) q\n[PROOFSTEP]\nexact (factorization_le_iff_dvd ha0 hb0).2 hab q\n[GOAL]\na b : \u2115\n\u22a2 (\u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p) \u2194 a \u2223 b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun hab p => ord_compl_dvd_ord_compl_of_dvd hab p\u27e9\n[GOAL]\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrcases eq_or_ne b 0 with (rfl | hb0)\n[GOAL]\ncase inl\na : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 0 / p ^ \u2191(factorization 0) p\n\u22a2 a \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\n\u22a2 a \u2223 b\n[PROOFSTEP]\nby_cases pa : a.Prime\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\n\u22a2 a \u2223 b\ncase neg\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : \u00acPrime a\n\u22a2 a \u2223 b\n[PROOFSTEP]\ncase neg => simpa [pa] using h a\n[GOAL]\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : \u00acPrime a\n\u22a2 a \u2223 b\n[PROOFSTEP]\ncase neg => simpa [pa] using h a\n[GOAL]\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : \u00acPrime a\n\u22a2 a \u2223 b\n[PROOFSTEP]\nsimpa [pa] using h a\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\n\u22a2 a \u2223 b\n[PROOFSTEP]\nby_cases pb : b.Prime\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\n\u22a2 a \u2223 b\ncase neg\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : \u00acPrime b\n\u22a2 a \u2223 b\n[PROOFSTEP]\ncase neg => simpa [pb] using h b\n[GOAL]\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : \u00acPrime b\n\u22a2 a \u2223 b\n[PROOFSTEP]\ncase neg => simpa [pb] using h b\n[GOAL]\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : \u00acPrime b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nsimpa [pb] using h b\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\n\u22a2 a \u2223 b\n[PROOFSTEP]\nrw [prime_dvd_prime_iff_eq pa pb]\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\n\u22a2 a = b\n[PROOFSTEP]\nby_contra hab\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\nhab : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\napply pa.ne_one\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\nhab : \u00aca = b\n\u22a2 a = 1\n[PROOFSTEP]\nrw [\u2190 Nat.dvd_one, \u2190 Nat.mul_dvd_mul_iff_left hb0.bot_lt, mul_one]\n[GOAL]\ncase pos\na b : \u2115\nh : \u2200 (p : \u2115), a / p ^ \u2191(factorization a) p \u2223 b / p ^ \u2191(factorization b) p\nhb0 : b \u2260 0\npa : Prime a\npb : Prime b\nhab : \u00aca = b\n\u22a2 b * a \u2223 b\n[PROOFSTEP]\nsimpa [Prime.factorization_self pb, Prime.factorization pa, hab] using h b\n[GOAL]\nn d : \u2115\n\u22a2 d \u2223 n \u2194 \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\nd : \u2115\n\u22a2 d \u2223 0 \u2194 \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn d : \u2115\nhn : n \u2260 0\n\u22a2 d \u2223 n \u2194 \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n\n[PROOFSTEP]\nrcases eq_or_ne d 0 with (rfl | hd)\n[GOAL]\ncase inr.inl\nn : \u2115\nhn : n \u2260 0\n\u22a2 0 \u2223 n \u2194 \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 0 \u2192 p ^ k \u2223 n\n[PROOFSTEP]\nsimp only [zero_dvd_iff, hn, false_iff_iff, not_forall]\n[GOAL]\ncase inr.inl\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2203 x x_1 h x_2, \u00acx ^ x_1 \u2223 n\n[PROOFSTEP]\nexact \u27e82, n, prime_two, dvd_zero _, mt (le_of_dvd hn.bot_lt) (lt_two_pow n).not_le\u27e9\n[GOAL]\ncase inr.inr\nn d : \u2115\nhn : n \u2260 0\nhd : d \u2260 0\n\u22a2 d \u2223 n \u2194 \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n\n[PROOFSTEP]\nrefine' \u27e8fun h p k _ hpkd => dvd_trans hpkd h, _\u27e9\n[GOAL]\ncase inr.inr\nn d : \u2115\nhn : n \u2260 0\nhd : d \u2260 0\n\u22a2 (\u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n) \u2192 d \u2223 n\n[PROOFSTEP]\nrw [\u2190 factorization_prime_le_iff_dvd hd hn]\n[GOAL]\ncase inr.inr\nn d : \u2115\nhn : n \u2260 0\nhd : d \u2260 0\n\u22a2 (\u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n) \u2192 \u2200 (p : \u2115), Prime p \u2192 \u2191(factorization d) p \u2264 \u2191(factorization n) p\n[PROOFSTEP]\nintro h p pp\n[GOAL]\ncase inr.inr\nn d : \u2115\nhn : n \u2260 0\nhd : d \u2260 0\nh : \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n\np : \u2115\npp : Prime p\n\u22a2 \u2191(factorization d) p \u2264 \u2191(factorization n) p\n[PROOFSTEP]\nsimp_rw [\u2190 pp.pow_dvd_iff_le_factorization hn]\n[GOAL]\ncase inr.inr\nn d : \u2115\nhn : n \u2260 0\nhd : d \u2260 0\nh : \u2200 (p k : \u2115), Prime p \u2192 p ^ k \u2223 d \u2192 p ^ k \u2223 n\np : \u2115\npp : Prime p\n\u22a2 p ^ \u2191(factorization d) p \u2223 n\n[PROOFSTEP]\nexact h p _ pp (ord_proj_dvd _ _)\n[GOAL]\nn : \u2115\n\u22a2 \u220f p in toFinset (factors n), p \u2223 n\n[PROOFSTEP]\nby_cases hn : n = 0\n[GOAL]\ncase pos\nn : \u2115\nhn : n = 0\n\u22a2 \u220f p in toFinset (factors n), p \u2223 n\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase pos\n\u22a2 \u220f p in toFinset (factors 0), p \u2223 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nn : \u2115\nhn : \u00acn = 0\n\u22a2 \u220f p in toFinset (factors n), p \u2223 n\n[PROOFSTEP]\nsimpa [prod_factors hn] using Multiset.toFinset_prod_dvd_prod (n.factors : Multiset \u2115)\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nlet dfac := a.factorization \u2293 b.factorization\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nlet d := dfac.prod Nat.pow\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nhave dfac_prime : \u2200 p : \u2115, p \u2208 dfac.support \u2192 Prime p :=\n  by\n  intro p hp\n  have : p \u2208 a.factors \u2227 p \u2208 b.factors := by simpa using hp\n  exact prime_of_mem_factors this.1\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\n\u22a2 \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\n[PROOFSTEP]\nintro p hp\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\np : \u2115\nhp : p \u2208 dfac.support\n\u22a2 Prime p\n[PROOFSTEP]\nhave : p \u2208 a.factors \u2227 p \u2208 b.factors := by simpa using hp\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\np : \u2115\nhp : p \u2208 dfac.support\n\u22a2 p \u2208 factors a \u2227 p \u2208 factors b\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\np : \u2115\nhp : p \u2208 dfac.support\nthis : p \u2208 factors a \u2227 p \u2208 factors b\n\u22a2 Prime p\n[PROOFSTEP]\nexact prime_of_mem_factors this.1\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nhave h1 : d.factorization = dfac := prod_pow_factorization_eq_self dfac_prime\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nhave hd_pos : d \u2260 0 := (factorizationEquiv.invFun \u27e8dfac, dfac_prime\u27e9).2.ne'\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nsuffices d = gcd a b by rwa [\u2190 this]\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\nthis : d = gcd a b\n\u22a2 factorization (gcd a b) = factorization a \u2293 factorization b\n[PROOFSTEP]\nrwa [\u2190 this]\n[GOAL]\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 d = gcd a b\n[PROOFSTEP]\napply gcd_greatest\n[GOAL]\ncase hda\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 d \u2223 a\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd hd_pos ha_pos, h1]\n[GOAL]\ncase hda\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 dfac \u2264 factorization a\n[PROOFSTEP]\nexact inf_le_left\n[GOAL]\ncase hdb\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 d \u2223 b\n[PROOFSTEP]\nrw [\u2190 factorization_le_iff_dvd hd_pos hb_pos, h1]\n[GOAL]\ncase hdb\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 dfac \u2264 factorization b\n[PROOFSTEP]\nexact inf_le_right\n[GOAL]\ncase hd\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\n\u22a2 \u2200 (e : \u2115), e \u2223 a \u2192 e \u2223 b \u2192 e \u2223 d\n[PROOFSTEP]\nintro e hea heb\n[GOAL]\ncase hd\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\ne : \u2115\nhea : e \u2223 a\nheb : e \u2223 b\n\u22a2 e \u2223 d\n[PROOFSTEP]\nrcases Decidable.eq_or_ne e 0 with (rfl | he_pos)\n[GOAL]\ncase hd.inl\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\nhea : 0 \u2223 a\nheb : 0 \u2223 b\n\u22a2 0 \u2223 d\n[PROOFSTEP]\nsimp only [zero_dvd_iff] at hea \n[GOAL]\ncase hd.inl\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\nheb : 0 \u2223 b\nhea : a = 0\n\u22a2 0 \u2223 d\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase hd.inr\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\ne : \u2115\nhea : e \u2223 a\nheb : e \u2223 b\nhe_pos : e \u2260 0\n\u22a2 e \u2223 d\n[PROOFSTEP]\nhave hea' := (factorization_le_iff_dvd he_pos ha_pos).mpr hea\n[GOAL]\ncase hd.inr\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\ne : \u2115\nhea : e \u2223 a\nheb : e \u2223 b\nhe_pos : e \u2260 0\nhea' : factorization e \u2264 factorization a\n\u22a2 e \u2223 d\n[PROOFSTEP]\nhave heb' := (factorization_le_iff_dvd he_pos hb_pos).mpr heb\n[GOAL]\ncase hd.inr\na b : \u2115\nha_pos : a \u2260 0\nhb_pos : b \u2260 0\ndfac : \u2115 \u2192\u2080 \u2115 := factorization a \u2293 factorization b\nd : \u2115 := Finsupp.prod dfac Nat.pow\ndfac_prime : \u2200 (p : \u2115), p \u2208 dfac.support \u2192 Prime p\nh1 : factorization d = dfac\nhd_pos : d \u2260 0\ne : \u2115\nhea : e \u2223 a\nheb : e \u2223 b\nhe_pos : e \u2260 0\nhea' : factorization e \u2264 factorization a\nheb' : factorization e \u2264 factorization b\n\u22a2 e \u2223 d\n[PROOFSTEP]\nsimp [\u2190 factorization_le_iff_dvd he_pos hd_pos, h1, hea', heb']\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 factorization (lcm a b) = factorization a \u2294 factorization b\n[PROOFSTEP]\nrw [\u2190 add_right_inj (a.gcd b).factorization, \u2190\n  factorization_mul (mt gcd_eq_zero_iff.1 fun h => ha h.1) (lcm_ne_zero ha hb), gcd_mul_lcm, factorization_gcd ha hb,\n  factorization_mul ha hb]\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 factorization a + factorization b = factorization a \u2293 factorization b + factorization a \u2294 factorization b\n[PROOFSTEP]\next1\n[GOAL]\ncase h\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\na\u271d : \u2115\n\u22a2 \u2191(factorization a + factorization b) a\u271d = \u2191(factorization a \u2293 factorization b + factorization a \u2294 factorization b) a\u271d\n[PROOFSTEP]\nexact (min_add_max _ _).symm\n[GOAL]\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\n\u22a2 Finset.prod (toFinset (factors (gcd m n))) f * Finset.prod (toFinset (factors (m * n))) f =\n    Finset.prod (toFinset (factors m)) f * Finset.prod (toFinset (factors n)) f\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hm0)\n[GOAL]\ncase inl\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm : \u2115\nf : \u2115 \u2192 \u03b2\n\u22a2 Finset.prod (toFinset (factors (gcd m 0))) f * Finset.prod (toFinset (factors (m * 0))) f =\n    Finset.prod (toFinset (factors m)) f * Finset.prod (toFinset (factors 0)) f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\n\u22a2 Finset.prod (toFinset (factors (gcd m n))) f * Finset.prod (toFinset (factors (m * n))) f =\n    Finset.prod (toFinset (factors m)) f * Finset.prod (toFinset (factors n)) f\n[PROOFSTEP]\nrcases eq_or_ne m 0 with (rfl | hn0)\n[GOAL]\ncase inr.inl\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nn : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\n\u22a2 Finset.prod (toFinset (factors (gcd 0 n))) f * Finset.prod (toFinset (factors (0 * n))) f =\n    Finset.prod (toFinset (factors 0)) f * Finset.prod (toFinset (factors n)) f\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 Finset.prod (toFinset (factors (gcd m n))) f * Finset.prod (toFinset (factors (m * n))) f =\n    Finset.prod (toFinset (factors m)) f * Finset.prod (toFinset (factors n)) f\n[PROOFSTEP]\nrw [\u2190 @Finset.prod_union_inter _ _ m.factors.toFinset n.factors.toFinset, mul_comm]\n[GOAL]\ncase inr.inr\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 Finset.prod (toFinset (factors (m * n))) f * Finset.prod (toFinset (factors (gcd m n))) f =\n    (\u220f x in toFinset (factors m) \u222a toFinset (factors n), f x) * \u220f x in toFinset (factors m) \u2229 toFinset (factors n), f x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr.inr.e_a.e_s\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 toFinset (factors (m * n)) = toFinset (factors m) \u222a toFinset (factors n)\n[PROOFSTEP]\napply factors_mul_toFinset\n[GOAL]\ncase inr.inr.e_a.e_s.ha\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 m \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.e_a.e_s.hb\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 n \u2260 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr.inr.e_a.e_s\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nm n : \u2115\nf : \u2115 \u2192 \u03b2\nhm0 : n \u2260 0\nhn0 : m \u2260 0\n\u22a2 toFinset (factors (gcd m n)) = toFinset (factors m) \u2229 toFinset (factors n)\n[PROOFSTEP]\nsimp only [\u2190 support_factorization, factorization_gcd hn0 hm0, Finsupp.support_inf]\n[GOAL]\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 {i | i \u2260 0 \u2227 p ^ i \u2223 n} = Set.Icc 1 (\u2191(factorization n) p)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\nx\u271d : \u2115\n\u22a2 x\u271d \u2208 {i | i \u2260 0 \u2227 p ^ i \u2223 n} \u2194 x\u271d \u2208 Set.Icc 1 (\u2191(factorization n) p)\n[PROOFSTEP]\nsimp [lt_succ_iff, one_le_iff_ne_zero, pp.pow_dvd_iff_le_factorization hn]\n[GOAL]\nn p : \u2115\npp : Prime p\n\u22a2 Icc 1 (\u2191(factorization n) p) = Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n)\n[PROOFSTEP]\nrcases eq_or_ne n 0 with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\npp : Prime p\n\u22a2 Icc 1 (\u2191(factorization 0) p) = Finset.filter (fun i => p ^ i \u2223 0) (Ico 1 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\n\u22a2 Icc 1 (\u2191(factorization n) p) = Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n)\n[PROOFSTEP]\next x\n[GOAL]\ncase inr.a\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\nx : (fun x => \u2115) p\n\u22a2 x \u2208 Icc 1 (\u2191(factorization n) p) \u2194 x \u2208 Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n)\n[PROOFSTEP]\nsimp only [mem_Icc, Finset.mem_filter, mem_Ico, and_assoc, and_congr_right_iff, pp.pow_dvd_iff_le_factorization hn,\n  iff_and_self]\n[GOAL]\ncase inr.a\nn p : \u2115\npp : Prime p\nhn : n \u2260 0\nx : (fun x => \u2115) p\n\u22a2 1 \u2264 x \u2192 x \u2264 \u2191(factorization n) p \u2192 x < n\n[PROOFSTEP]\nexact fun _ H => lt_of_le_of_lt H (factorization_lt p hn)\n[GOAL]\nn p : \u2115\npp : Prime p\n\u22a2 \u2191(factorization n) p = card (Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n))\n[PROOFSTEP]\nsimp [\u2190 Icc_factorization_eq_pow_dvd n pp]\n[GOAL]\nn p b : \u2115\npp : Prime p\nhn : n \u2260 0\nhb : n \u2264 p ^ b\n\u22a2 Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n) = Finset.filter (fun i => p ^ i \u2223 n) (Icc 1 b)\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nn p b : \u2115\npp : Prime p\nhn : n \u2260 0\nhb : n \u2264 p ^ b\nx : \u2115\n\u22a2 x \u2208 Finset.filter (fun i => p ^ i \u2223 n) (Ico 1 n) \u2194 x \u2208 Finset.filter (fun i => p ^ i \u2223 n) (Icc 1 b)\n[PROOFSTEP]\nsimp only [Finset.mem_filter, mem_Ico, mem_Icc, and_congr_left_iff, and_congr_right_iff]\n[GOAL]\ncase a\nn p b : \u2115\npp : Prime p\nhn : n \u2260 0\nhb : n \u2264 p ^ b\nx : \u2115\n\u22a2 p ^ x \u2223 n \u2192 1 \u2264 x \u2192 (x < n \u2194 x \u2264 b)\n[PROOFSTEP]\nrintro h1 -\n[GOAL]\ncase a\nn p b : \u2115\npp : Prime p\nhn : n \u2260 0\nhb : n \u2264 p ^ b\nx : \u2115\nh1 : p ^ x \u2223 n\n\u22a2 x < n \u2194 x \u2264 b\n[PROOFSTEP]\nsimp [lt_of_pow_dvd_right hn pp.two_le h1, (pow_le_iff_le_right pp.two_le).1 ((le_of_dvd hn.bot_lt h1).trans hb)]\n[GOAL]\np a b : \u2115\nhab : coprime a b\n\u22a2 \u2191(factorization (a * b)) p = \u2191(factorization a) p + \u2191(factorization b) p\n[PROOFSTEP]\nsimp only [\u2190 factors_count_eq, perm_iff_count.mp (perm_factors_mul_of_coprime hab), count_append]\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 factorization (a * b) = factorization a + factorization b\n[PROOFSTEP]\next q\n[GOAL]\ncase h\na b : \u2115\nhab : coprime a b\nq : \u2115\n\u22a2 \u2191(factorization (a * b)) q = \u2191(factorization a + factorization b) q\n[PROOFSTEP]\nrw [Finsupp.add_apply, factorization_mul_apply_of_coprime hab]\n[GOAL]\np a b : \u2115\nhab : coprime a b\nhpa : p \u2208 factors a\n\u22a2 \u2191(factorization (a * b)) p = \u2191(factorization a) p\n[PROOFSTEP]\nrw [factorization_mul_apply_of_coprime hab, \u2190 factors_count_eq, \u2190 factors_count_eq,\n  count_eq_zero_of_not_mem (coprime_factors_disjoint hab hpa), add_zero]\n[GOAL]\np a b : \u2115\nhab : coprime a b\nhpb : p \u2208 factors b\n\u22a2 \u2191(factorization (a * b)) p = \u2191(factorization b) p\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\np a b : \u2115\nhab : coprime a b\nhpb : p \u2208 factors b\n\u22a2 \u2191(factorization (b * a)) p = \u2191(factorization b) p\n[PROOFSTEP]\nexact factorization_eq_of_coprime_left (coprime_comm.mp hab) hpb\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 _root_.Disjoint (factorization a).support (factorization b).support\n[PROOFSTEP]\nsimpa only [support_factorization] using disjoint_toFinset_iff_disjoint.mpr (coprime_factors_disjoint hab)\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 (factorization (a * b)).support = (factorization a).support \u222a (factorization b).support\n[PROOFSTEP]\nrw [factorization_mul_of_coprime hab]\n[GOAL]\na b : \u2115\nhab : coprime a b\n\u22a2 (factorization a + factorization b).support = (factorization a).support \u222a (factorization b).support\n[PROOFSTEP]\nexact support_add_eq (factorization_disjoint_of_coprime hab)\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\n\u22a2 P (k + 2)\n[PROOFSTEP]\nlet p := (k + 2).minFac\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\n\u22a2 P (k + 2)\n[PROOFSTEP]\nhave hp : Prime p :=\n  minFac_prime\n    (succ_succ_ne_one k)\n      -- the awkward `let` stuff here is because `factorization` is noncomputable (Finsupp);\n            -- we get around this by using the computable `factors.count`, and rewriting when we want\n            -- to use the `factorization` API\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\n\u22a2 P (k + 2)\n[PROOFSTEP]\nlet t := (k + 2).factors.count p\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\n\u22a2 P (k + 2)\n[PROOFSTEP]\nhave ht : t = (k + 2).factorization p := factors_count_eq\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\n\u22a2 P (k + 2)\n[PROOFSTEP]\nhave hpt : p ^ t \u2223 k + 2 := by\n  rw [ht]\n  exact ord_proj_dvd _ _\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\n\u22a2 p ^ t \u2223 k + 2\n[PROOFSTEP]\nrw [ht]\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\n\u22a2 p ^ \u2191(factorization (k + 2)) p \u2223 k + 2\n[PROOFSTEP]\nexact ord_proj_dvd _ _\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\n\u22a2 P (k + 2)\n[PROOFSTEP]\nhave htp : 0 < t := by\n  rw [ht]\n  exact hp.factorization_pos_of_dvd (Nat.succ_ne_zero _) (minFac_dvd _)\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\n\u22a2 0 < t\n[PROOFSTEP]\nrw [ht]\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\n\u22a2 0 < \u2191(factorization (k + 2)) p\n[PROOFSTEP]\nexact hp.factorization_pos_of_dvd (Nat.succ_ne_zero _) (minFac_dvd _)\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 P (k + 2)\n[PROOFSTEP]\nconvert h ((k + 2) / p ^ t) p t hp _ _ _ using 1\n[GOAL]\ncase h.e'_1\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 k + 2 = p ^ t * ((k + 2) / p ^ t)\n[PROOFSTEP]\nrw [Nat.mul_div_cancel' hpt]\n[GOAL]\ncase convert_1\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 \u00acp \u2223 (k + 2) / p ^ t\n[PROOFSTEP]\nrw [Nat.dvd_div_iff hpt, \u2190 pow_succ, ht]\n[GOAL]\ncase convert_1\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 \u00acp ^ succ (\u2191(factorization (k + 2)) p) \u2223 k + 2\n[PROOFSTEP]\nexact pow_succ_factorization_not_dvd (k + 1).succ_ne_zero hp\n[GOAL]\ncase convert_2\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 0 < t\n[PROOFSTEP]\nexact htp\n[GOAL]\ncase convert_3\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 P ((k + 2) / p ^ t)\n[PROOFSTEP]\napply hk _ (Nat.div_lt_of_lt_mul _)\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 k + 2 < p ^ t * (k + 2)\n[PROOFSTEP]\nrw [lt_mul_iff_one_lt_left Nat.succ_pos', one_lt_pow_iff htp.ne]\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nh : (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\na n k : \u2115\nhk : (m : \u2115) \u2192 m < k + 2 \u2192 P m\np : \u2115 := minFac (k + 2)\nhp : Prime p\nt : \u2115 := count p (factors (k + 2))\nht : t = \u2191(factorization (k + 2)) p\nhpt : p ^ t \u2223 k + 2\nhtp : 0 < t\n\u22a2 1 < p\n[PROOFSTEP]\nexact hp.one_lt\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\n\u22a2 (a p n : \u2115) \u2192 Prime p \u2192 \u00acp \u2223 a \u2192 0 < n \u2192 P a \u2192 P (p ^ n * a)\n[PROOFSTEP]\nintro a p n hp' hpa hn hPa\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\n\u22a2 P (p ^ n * a)\n[PROOFSTEP]\nby_cases ha1 : a = 1\n[GOAL]\ncase pos\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\nha1 : a = 1\n\u22a2 P (p ^ n * a)\n[PROOFSTEP]\nrw [ha1, mul_one]\n[GOAL]\ncase pos\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\nha1 : a = 1\n\u22a2 P (p ^ n)\n[PROOFSTEP]\nexact hp p n hp' hn\n[GOAL]\ncase neg\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\nha1 : \u00aca = 1\n\u22a2 P (p ^ n * a)\n[PROOFSTEP]\nrefine' h (p ^ n) a (hp'.one_lt.trans_le (le_self_pow hn.ne' _)) _ _ (hp _ _ hp' hn) hPa\n[GOAL]\ncase neg.refine'_1\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\nha1 : \u00aca = 1\n\u22a2 1 < a\n[PROOFSTEP]\ncontrapose! hpa\n[GOAL]\ncase neg.refine'_1\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhn : 0 < n\nhPa : P a\nha1 : \u00aca = 1\nhpa : a \u2264 1\n\u22a2 p \u2223 a\n[PROOFSTEP]\nsimp [lt_one_iff.1 (lt_of_le_of_ne hpa ha1)]\n[GOAL]\ncase neg.refine'_2\nP : \u2115 \u2192 Sort u_1\nhp : (p n : \u2115) \u2192 Prime p \u2192 0 < n \u2192 P (p ^ n)\nh0 : P 0\nh1 : P 1\nh : (a b : \u2115) \u2192 1 < a \u2192 1 < b \u2192 coprime a b \u2192 P a \u2192 P b \u2192 P (a * b)\na p n : \u2115\nhp' : Prime p\nhpa : \u00acp \u2223 a\nhn : 0 < n\nhPa : P a\nha1 : \u00aca = 1\n\u22a2 coprime (p ^ n) a\n[PROOFSTEP]\nsimpa [hn, Prime.coprime_iff_not_dvd hp']\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nhp : (p : \u2115) \u2192 Prime p \u2192 P p\nh : (a b : \u2115) \u2192 P a \u2192 P b \u2192 P (a * b)\np n\u271d : \u2115\nhp' : Prime p\nn : \u2115\nhn : P (p ^ n)\n\u22a2 P (p ^ succ n)\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nP : \u2115 \u2192 Sort u_1\nh0 : P 0\nh1 : P 1\nhp : (p : \u2115) \u2192 Prime p \u2192 P p\nh : (a b : \u2115) \u2192 P a \u2192 P b \u2192 P (a * b)\np n\u271d : \u2115\nhp' : Prime p\nn : \u2115\nhn : P (p ^ n)\n\u22a2 P (p ^ n * p)\n[PROOFSTEP]\napply h _ _ hn (hp p hp')\n[GOAL]\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 \u2200 {n : \u2115}, n \u2260 0 \u2192 f n = Finsupp.prod (factorization n) fun p k => f (p ^ k)\n[PROOFSTEP]\napply Nat.recOnPosPrimePosCoprime\n[GOAL]\ncase hp\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 \u2200 (p n : \u2115), Prime p \u2192 0 < n \u2192 p ^ n \u2260 0 \u2192 f (p ^ n) = Finsupp.prod (factorization (p ^ n)) fun p k => f (p ^ k)\n[PROOFSTEP]\nrintro p k hp -\n  -\n      -- Porting note: replaced `simp` with `rw`\n[GOAL]\ncase hp\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\np k : \u2115\nhp : Prime p\n\u22a2 f (p ^ k) = Finsupp.prod (factorization (p ^ k)) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [Prime.factorization_pow hp, Finsupp.prod_single_index _]\n[GOAL]\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\np k : \u2115\nhp : Prime p\n\u22a2 f (p ^ 0) = 1\n[PROOFSTEP]\nrwa [pow_zero]\n[GOAL]\ncase h0\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 0 \u2260 0 \u2192 f 0 = Finsupp.prod (factorization 0) fun p k => f (p ^ k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h1\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 1 \u2260 0 \u2192 f 1 = Finsupp.prod (factorization 1) fun p k => f (p ^ k)\n[PROOFSTEP]\nrintro -\n[GOAL]\ncase h1\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 f 1 = Finsupp.prod (factorization 1) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [factorization_one, hf]\n[GOAL]\ncase h1\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 1 = Finsupp.prod 0 fun p k => f (p ^ k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\n\u22a2 \u2200 (a b : \u2115),\n    1 < a \u2192\n      1 < b \u2192\n        coprime a b \u2192\n          (a \u2260 0 \u2192 f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)) \u2192\n            (b \u2260 0 \u2192 f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)) \u2192\n              a * b \u2260 0 \u2192 f (a * b) = Finsupp.prod (factorization (a * b)) fun p k => f (p ^ k)\n[PROOFSTEP]\nintro a b _ _ hab ha hb hab_pos\n[GOAL]\ncase h\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\na b : \u2115\na\u271d\u00b9 : 1 < a\na\u271d : 1 < b\nhab : coprime a b\nha : a \u2260 0 \u2192 f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)\nhb : b \u2260 0 \u2192 f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)\nhab_pos : a * b \u2260 0\n\u22a2 f (a * b) = Finsupp.prod (factorization (a * b)) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [h_mult a b hab, ha (left_ne_zero_of_mul hab_pos), hb (right_ne_zero_of_mul hab_pos),\n  factorization_mul_of_coprime hab, \u2190 prod_add_index_of_disjoint]\n[GOAL]\ncase h.hd\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf : f 1 = 1\na b : \u2115\na\u271d\u00b9 : 1 < a\na\u271d : 1 < b\nhab : coprime a b\nha : a \u2260 0 \u2192 f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)\nhb : b \u2260 0 \u2192 f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)\nhab_pos : a * b \u2260 0\n\u22a2 _root_.Disjoint (factorization a).support (factorization b).support\n[PROOFSTEP]\nconvert factorization_disjoint_of_coprime hab\n[GOAL]\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 \u2200 {n : \u2115}, f n = Finsupp.prod (factorization n) fun p k => f (p ^ k)\n[PROOFSTEP]\napply Nat.recOnPosPrimePosCoprime\n[GOAL]\ncase hp\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 \u2200 (p n : \u2115), Prime p \u2192 0 < n \u2192 f (p ^ n) = Finsupp.prod (factorization (p ^ n)) fun p k => f (p ^ k)\n[PROOFSTEP]\nrintro p k hp -\n[GOAL]\ncase hp\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\np k : \u2115\nhp : Prime p\n\u22a2 f (p ^ k) = Finsupp.prod (factorization (p ^ k)) fun p k => f (p ^ k)\n[PROOFSTEP]\nsimp only [hp.factorization_pow]\n[GOAL]\ncase hp\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\np k : \u2115\nhp : Prime p\n\u22a2 f (p ^ k) = Finsupp.prod (single p k) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [prod_single_index _]\n[GOAL]\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\np k : \u2115\nhp : Prime p\n\u22a2 f (p ^ 0) = 1\n[PROOFSTEP]\nsimp [hf1]\n[GOAL]\ncase h0\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 f 0 = Finsupp.prod (factorization 0) fun p k => f (p ^ k)\n[PROOFSTEP]\nsimp [hf0]\n[GOAL]\ncase h1\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 f 1 = Finsupp.prod (factorization 1) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [factorization_one, hf1]\n[GOAL]\ncase h1\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 1 = Finsupp.prod 0 fun p k => f (p ^ k)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\n\u22a2 \u2200 (a b : \u2115),\n    1 < a \u2192\n      1 < b \u2192\n        coprime a b \u2192\n          (f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)) \u2192\n            (f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)) \u2192\n              f (a * b) = Finsupp.prod (factorization (a * b)) fun p k => f (p ^ k)\n[PROOFSTEP]\nintro a b _ _ hab ha hb\n[GOAL]\ncase h\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\na b : \u2115\na\u271d\u00b9 : 1 < a\na\u271d : 1 < b\nhab : coprime a b\nha : f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)\nhb : f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)\n\u22a2 f (a * b) = Finsupp.prod (factorization (a * b)) fun p k => f (p ^ k)\n[PROOFSTEP]\nrw [h_mult a b hab, ha, hb, factorization_mul_of_coprime hab, \u2190 prod_add_index_of_disjoint]\n[GOAL]\ncase h.hd\n\u03b2 : Type u_1\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\nh_mult : \u2200 (x y : \u2115), coprime x y \u2192 f (x * y) = f x * f y\nhf0 : f 0 = 1\nhf1 : f 1 = 1\na b : \u2115\na\u271d\u00b9 : 1 < a\na\u271d : 1 < b\nhab : coprime a b\nha : f a = Finsupp.prod (factorization a) fun p k => f (p ^ k)\nhb : f b = Finsupp.prod (factorization b) fun p k => f (p ^ k)\n\u22a2 _root_.Disjoint (factorization a).support (factorization b).support\n[PROOFSTEP]\nexact factorization_disjoint_of_coprime hab\n[GOAL]\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a = b \u2194 \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 a = b \u2192 \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\na : \u2115\nha hb : a \u2260 0\n\u22a2 \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\n\u22a2 (\u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b) \u2192 a = b\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\n\u22a2 a = b\n[PROOFSTEP]\nrefine' eq_of_factorization_eq ha hb fun p => _\n[GOAL]\ncase mpr\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\np : \u2115\n\u22a2 \u2191(factorization a) p = \u2191(factorization b) p\n[PROOFSTEP]\nby_cases pp : p.Prime\n[GOAL]\ncase pos\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\np : \u2115\npp : Prime p\n\u22a2 \u2191(factorization a) p = \u2191(factorization b) p\n[PROOFSTEP]\nsimp [factorization_def, pp, h p pp]\n[GOAL]\ncase neg\na b : \u2115\nha : a \u2260 0\nhb : b \u2260 0\nh : \u2200 (p : \u2115), Prime p \u2192 padicValNat p a = padicValNat p b\np : \u2115\npp : \u00acPrime p\n\u22a2 \u2191(factorization a) p = \u2191(factorization b) p\n[PROOFSTEP]\nsimp [factorization_eq_zero_of_non_prime, pp]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n = n\n[PROOFSTEP]\nconv =>\n  rhs\n  rw [\u2190 factorization_prod_pow_eq_self hn]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n| \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n = n\n[PROOFSTEP]\n  rhs\n  rw [\u2190 factorization_prod_pow_eq_self hn]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n| \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n = n\n[PROOFSTEP]\n  rhs\n  rw [\u2190 factorization_prod_pow_eq_self hn]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n| \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n = n\n[PROOFSTEP]\nrhs\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n| n\n[PROOFSTEP]\nrw [\u2190 factorization_prod_pow_eq_self hn]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n = Finsupp.prod (factorization n) fun x x_1 => x ^ x_1\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 (Finsupp.prod (factorization n) fun x x_1 => x ^ x_1) =\n    \u220f p in Finset.filter Prime (Finset.range m), p ^ padicValNat p n\n[PROOFSTEP]\napply Finset.prod_subset_one_on_sdiff\n[GOAL]\ncase h\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 (factorization n).support \u2286 Finset.filter Prime (Finset.range m)\n[PROOFSTEP]\nexact fun p hp =>\n  Finset.mem_filter.mpr\n    \u27e8Finset.mem_range.mpr (gt_of_gt_of_ge pr (le_of_mem_factorization hp)), prime_of_mem_factorization hp\u27e9\n[GOAL]\ncase hg\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 \u2200 (x : \u2115), x \u2208 Finset.filter Prime (Finset.range m) \\ (factorization n).support \u2192 x ^ padicValNat x n = 1\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase hg\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\np : \u2115\nhp : p \u2208 Finset.filter Prime (Finset.range m) \\ (factorization n).support\n\u22a2 p ^ padicValNat p n = 1\n[PROOFSTEP]\ncases' Finset.mem_sdiff.mp hp with hp1 hp2\n[GOAL]\ncase hg.intro\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\np : \u2115\nhp : p \u2208 Finset.filter Prime (Finset.range m) \\ (factorization n).support\nhp1 : p \u2208 Finset.filter Prime (Finset.range m)\nhp2 : \u00acp \u2208 (factorization n).support\n\u22a2 p ^ padicValNat p n = 1\n[PROOFSTEP]\nrw [\u2190 factorization_def n (Finset.mem_filter.mp hp1).2]\n[GOAL]\ncase hg.intro\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\np : \u2115\nhp : p \u2208 Finset.filter Prime (Finset.range m) \\ (factorization n).support\nhp1 : p \u2208 Finset.filter Prime (Finset.range m)\nhp2 : \u00acp \u2208 (factorization n).support\n\u22a2 p ^ \u2191(factorization n) p = 1\n[PROOFSTEP]\nsimp [Finsupp.not_mem_support_iff.mp hp2]\n[GOAL]\ncase hfg\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\n\u22a2 \u2200 (x : \u2115), x \u2208 (factorization n).support \u2192 (fun x x_1 => x ^ x_1) x (\u2191(factorization n) x) = x ^ padicValNat x n\n[PROOFSTEP]\nintro p hp\n[GOAL]\ncase hfg\nn : \u2115\nhn : n \u2260 0\nm : \u2115\npr : n < m\np : \u2115\nhp : p \u2208 (factorization n).support\n\u22a2 (fun x x_1 => x ^ x_1) p (\u2191(factorization n) p) = p ^ padicValNat p n\n[PROOFSTEP]\nsimp [factorization_def n (prime_of_mem_factorization hp)]\n[GOAL]\nn p : \u2115\n\u22a2 card (Finset.filter (fun e => p \u2223 e + 1) (Finset.range n)) = n / p\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase zero\np : \u2115\n\u22a2 card (Finset.filter (fun e => p \u2223 e + 1) (Finset.range zero)) = zero / p\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\np n : \u2115\nhn : card (Finset.filter (fun e => p \u2223 e + 1) (Finset.range n)) = n / p\n\u22a2 card (Finset.filter (fun e => p \u2223 e + 1) (Finset.range (succ n))) = succ n / p\n[PROOFSTEP]\nsimp [Nat.succ_div, add_ite, add_zero, Finset.range_succ, filter_insert, apply_ite card, card_insert_of_not_mem, hn]\n[GOAL]\nn p : \u2115\n\u22a2 card (Finset.filter (fun x => p \u2223 x) (Ioc 0 n)) = n / p\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\np : \u2115\n\u22a2 card (Finset.filter (fun x => p \u2223 x) (Ioc 0 zero)) = zero / p\n[PROOFSTEP]\nsimp\n  -- TODO: Golf away `h1` after Ya\u00ebl PRs a lemma asserting this\n[GOAL]\ncase succ\np n : \u2115\nIH : card (Finset.filter (fun x => p \u2223 x) (Ioc 0 n)) = n / p\n\u22a2 card (Finset.filter (fun x => p \u2223 x) (Ioc 0 (succ n))) = succ n / p\n[PROOFSTEP]\nhave h1 : Ioc 0 n.succ = insert n.succ (Ioc 0 n) :=\n  by\n  rcases n.eq_zero_or_pos with (rfl | hn)\n  \u00b7 simp\n  simp_rw [\u2190 Ico_succ_succ, Ico_insert_right (succ_le_succ hn.le), Ico_succ_right]\n[GOAL]\np n : \u2115\nIH : card (Finset.filter (fun x => p \u2223 x) (Ioc 0 n)) = n / p\n\u22a2 Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hn)\n[GOAL]\ncase inl\np : \u2115\nIH : card (Finset.filter (fun x => p \u2223 x) (Ioc 0 0)) = 0 / p\n\u22a2 Ioc 0 (succ 0) = insert (succ 0) (Ioc 0 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np n : \u2115\nIH : card (Finset.filter (fun x => p \u2223 x) (Ioc 0 n)) = n / p\nhn : n > 0\n\u22a2 Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)\n[PROOFSTEP]\nsimp_rw [\u2190 Ico_succ_succ, Ico_insert_right (succ_le_succ hn.le), Ico_succ_right]\n[GOAL]\ncase succ\np n : \u2115\nIH : card (Finset.filter (fun x => p \u2223 x) (Ioc 0 n)) = n / p\nh1 : Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)\n\u22a2 card (Finset.filter (fun x => p \u2223 x) (Ioc 0 (succ n))) = succ n / p\n[PROOFSTEP]\nsimp [Nat.succ_div, add_ite, add_zero, h1, filter_insert, apply_ite card, card_insert_eq_ite, IH, Finset.mem_filter,\n  mem_Ioc, not_le.2 (lt_add_one n), Nat.succ_eq_add_one]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Factorization.Basic", "llama_tokens": 40703, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.5145125646905157}}
{"text": "[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\n\u22a2 \u2191i \u2264 2 * \u2191i\n[PROOFSTEP]\nrw [Nat.succ_mul, Nat.one_mul]\n[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\n\u22a2 \u2191i \u2264 \u2191i + \u2191i\n[PROOFSTEP]\nexact Nat.le_add_left i i\n[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\nleft_le : \u2191i \u2264 left\nright_le : \u2191i \u2264 right\ni_le : i \u2264 i\nj\u271d j : { j // i \u2264 j }\nh : \u00ac\u2191i = \u2191\u2191j\na' : Array \u03b1 := Array.swap a i \u2191j\n\u22a2 \u2191\u2191j < Array.size a'\n[PROOFSTEP]\nrw [a.size_swap i j]\n[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\nleft_le : \u2191i \u2264 left\nright_le : \u2191i \u2264 right\ni_le : i \u2264 i\nj\u271d j : { j // i \u2264 j }\nh : \u00ac\u2191i = \u2191\u2191j\na' : Array \u03b1 := Array.swap a i \u2191j\n\u22a2 \u2191\u2191j < Array.size a\n[PROOFSTEP]\nexact j.1.2\n[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\nleft_le : \u2191i \u2264 left\nright_le : \u2191i \u2264 right\ni_le : i \u2264 i\nj\u271d j : { j // i \u2264 j }\nh : \u00ac\u2191i = \u2191\u2191j\na' : Array \u03b1 := Array.swap a i \u2191j\nj' : Fin (Array.size a') := { val := \u2191\u2191j, isLt := ?m.5009 }\n\u22a2 Array.size a' - \u2191\u2191j < Array.size a - \u2191i\n[PROOFSTEP]\nrw [a.size_swap i j]\n[GOAL]\n\u03b1 : Type ?u.337\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\nleft_le : \u2191i \u2264 left\nright_le : \u2191i \u2264 right\ni_le : i \u2264 i\nj\u271d j : { j // i \u2264 j }\nh : \u00ac\u2191i = \u2191\u2191j\na' : Array \u03b1 := Array.swap a i \u2191j\nj' : Fin (Array.size a') := { val := \u2191\u2191j, isLt := ?m.5009 }\n\u22a2 Array.size a - \u2191\u2191j < Array.size a - \u2191i\n[PROOFSTEP]\nexact Nat.sub_lt_sub_left i.2 <| Nat.lt_of_le_of_ne j.2 h\n[GOAL]\n\u03b1 : Type u_1\na : Array \u03b1\ni : Fin (Array.size a)\nleft : \u2115 := 2 * \u2191i + 1\nright : \u2115 := left + 1\nleft_le : \u2191i \u2264 left\nright_le : \u2191i \u2264 right\ni_le : i \u2264 i\nj : { j // i \u2264 j }\nh : \u00ac\u2191i = \u2191\u2191j\na' : Array \u03b1 := Array.swap a i \u2191j\nj' : Fin (Array.size a') := { val := \u2191\u2191j, isLt := (_ : \u2191\u2191j < Array.size a') }\nthis : Array.size a' - \u2191\u2191j < Array.size a - \u2191i\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => Array.size a - \u2191snd) instWellFoundedRelation).1\n    { fst := a', snd := j' } { fst := a, snd := i }\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type ?u.8277\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\ni0 : \u00ac\u2191i = 0\nthis : (\u2191i - 1) / 2 < \u2191i\nj : Fin (Array.size a) := { val := (\u2191i - 1) / 2, isLt := (_ : (\u2191i - 1) / 2 < Array.size a) }\na' : Array \u03b1 := Array.swap a i j\n\u22a2 \u2191j < Array.size a'\n[PROOFSTEP]\nrw [a.size_swap i j]\n[GOAL]\n\u03b1 : Type ?u.8277\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\ni0 : \u00ac\u2191i = 0\nthis : (\u2191i - 1) / 2 < \u2191i\nj : Fin (Array.size a) := { val := (\u2191i - 1) / 2, isLt := (_ : (\u2191i - 1) / 2 < Array.size a) }\na' : Array \u03b1 := Array.swap a i j\n\u22a2 \u2191j < Array.size a\n[PROOFSTEP]\nexact j.2\n[GOAL]\n\u03b1 : Type u_1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : Array \u03b1\ni : Fin (Array.size a)\ni0 : \u00ac\u2191i = 0\nthis : (\u2191i - 1) / 2 < \u2191i\nj : Fin (Array.size a) := { val := (\u2191i - 1) / 2, isLt := (_ : (\u2191i - 1) / 2 < Array.size a) }\nh\u271d : lt (Array.get a j) (Array.get a i) = true\na' : Array \u03b1 := Array.swap a i j\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => \u2191snd) instWellFoundedRelation).1\n    { fst := a', snd := { val := \u2191j, isLt := (_ : \u2191j < Array.size a') } } { fst := a, snd := i }\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type ?u.10435\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx : \u03b1\nn : \u2115 := size self\n\u22a2 n < Array.size (Array.push self.arr x)\n[PROOFSTEP]\nrw [Array.size_push]\n[GOAL]\n\u03b1 : Type ?u.10435\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx : \u03b1\nn : \u2115 := size self\n\u22a2 n < Array.size self.arr + 1\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\n\u03b1 : Type u_1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx : \u03b1\n\u22a2 size (insert self x) = size self + 1\n[PROOFSTEP]\nsimp [insert, size, size_heapifyUp]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ne : Array.size self.arr = 0\n\u22a2 size self = size self - 1\n[PROOFSTEP]\nsimp [size, e]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\n\u22a2 0 < Array.size self.arr\n[PROOFSTEP]\nrw [e]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\n\u22a2 0 < Nat.succ n\n[PROOFSTEP]\napply Nat.succ_pos\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\nh0 : 0 < Array.size self.arr\n\u22a2 n < Array.size self.arr\n[PROOFSTEP]\nrw [e]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\nh0 : 0 < Array.size self.arr\n\u22a2 n < Nat.succ n\n[PROOFSTEP]\napply Nat.lt_succ_self\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\nh0 : 0 < Array.size self.arr\nhn : n < Array.size self.arr\nhn0 : 0 < n\na : Array \u03b1 := Array.pop (Array.swap self.arr { val := 0, isLt := h0 } { val := n, isLt := hn })\n\u22a2 0 < Array.size a\n[PROOFSTEP]\nrwa [Array.size_pop, Array.size_swap, e]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\nh0 : 0 < Array.size self.arr\nhn : n < Array.size self.arr\nhn0 : 0 < n\na : Array \u03b1 := Array.pop (Array.swap self.arr { val := 0, isLt := h0 } { val := n, isLt := hn })\n\u22a2 size { arr := \u2191(heapifyDown lt a { val := 0, isLt := (_ : 0 < Array.size a) }) } = size self - 1\n[PROOFSTEP]\nsimp [size]\n[GOAL]\n\u03b1 : Type ?u.11234\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nn : \u2115\ne : Array.size self.arr = Nat.succ n\nh0 : 0 < Array.size self.arr\nhn : n < Array.size self.arr\nhn0 : \u00ac0 < n\n\u22a2 size { arr := Array.pop self.arr } = size self - 1\n[PROOFSTEP]\nsimp [size]\n[GOAL]\n\u03b1 : Type u_1\nx : \u03b1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ne : max self = some x\nh : \u00ac0 < Array.size self.arr\n\u22a2 False\n[PROOFSTEP]\nsimp [BinaryHeap.max, Array.get?, h] at e \n[GOAL]\n\u03b1 : Type ?u.16836\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx m : \u03b1\ne : max self = some m\na : Array \u03b1 := Array.set self.arr { val := 0, isLt := (_ : 0 < size self) } x\n\u22a2 0 < Array.size a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.16836\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx m : \u03b1\ne : max self = some m\na : Array \u03b1 := Array.set self.arr { val := 0, isLt := (_ : 0 < size self) } x\n\u22a2 0 < Array.size self.arr\n[PROOFSTEP]\nexact size_pos_of_max e\n[GOAL]\n\u03b1 : Type ?u.18030\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx m : \u03b1\ne : max self = some m\na : Array \u03b1 := Array.set self.arr { val := 0, isLt := (_ : 0 < size self) } x\n\u22a2 0 < Array.size a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.18030\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\nx m : \u03b1\ne : max self = some m\na : Array \u03b1 := Array.set self.arr { val := 0, isLt := (_ : 0 < size self) } x\n\u22a2 0 < Array.size self.arr\n[PROOFSTEP]\nexact size_pos_of_max e\n[GOAL]\n\u03b1 : Type ?u.19048\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ni : Fin (size self)\nx : \u03b1\n\u22a2 \u2191i < Array.size (Array.set self.arr i x)\n[PROOFSTEP]\nrw [self.1.size_set]\n[GOAL]\n\u03b1 : Type ?u.19048\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ni : Fin (size self)\nx : \u03b1\n\u22a2 \u2191i < Array.size self.arr\n[PROOFSTEP]\nexact i.2\n[GOAL]\n\u03b1 : Type ?u.19626\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ni : Fin (size self)\nx : \u03b1\n\u22a2 \u2191i < Array.size (Array.set self.arr i x)\n[PROOFSTEP]\nrw [self.1.size_set]\n[GOAL]\n\u03b1 : Type ?u.19626\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\nself : BinaryHeap \u03b1 lt\ni : Fin (size self)\nx : \u03b1\n\u22a2 \u2191i < Array.size self.arr\n[PROOFSTEP]\nexact i.2\n[GOAL]\n\u03b1 : Type ?u.20451\na\u271d : Array \u03b1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\ngt : \u03b1 \u2192 \u03b1 \u2192 Bool := fun y x => lt x y\na : BinaryHeap \u03b1 gt\nout : Array \u03b1\nx : \u03b1\ne : BinaryHeap.max a = some x\n\u22a2 BinaryHeap.size (BinaryHeap.popMax a) < BinaryHeap.size a\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.20451\na\u271d : Array \u03b1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\ngt : \u03b1 \u2192 \u03b1 \u2192 Bool := fun y x => lt x y\na : BinaryHeap \u03b1 gt\nout : Array \u03b1\nx : \u03b1\ne : BinaryHeap.max a = some x\n\u22a2 BinaryHeap.size a - 1 < BinaryHeap.size a\n[PROOFSTEP]\nexact Nat.sub_lt (BinaryHeap.size_pos_of_max e) Nat.zero_lt_one\n[GOAL]\n\u03b1 : Type u_1\nlt : \u03b1 \u2192 \u03b1 \u2192 Bool\na : BinaryHeap \u03b1 fun y x => lt x y\nout : Array \u03b1\ngt : \u03b1 \u2192 \u03b1 \u2192 Bool := fun y x => lt x y\nx : \u03b1\ne : BinaryHeap.max a = some x\nthis : BinaryHeap.size (BinaryHeap.popMax a) < BinaryHeap.size a\n\u22a2 (invImage (fun a => PSigma.casesOn a fun a snd => BinaryHeap.size a) instWellFoundedRelation).1\n    { fst := BinaryHeap.popMax a, snd := push out x } { fst := a, snd := out }\n[PROOFSTEP]\nassumption\n", "meta": {"mathlib_filename": "Mathlib.Data.BinaryHeap", "llama_tokens": 4068, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.6688802603710085, "lm_q1q2_score": 0.5144225821376093}}
{"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Mul \u03b1\na : \u03b1\nx\u271d : IsSquare (op a)\nc : \u03b1\u1d50\u1d52\u1d56\nhc : op a = c * c\n\u22a2 a = unop c * unop c\n[PROOFSTEP]\nrw [\u2190 unop_mul, \u2190 hc, unop_op]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Mul \u03b1\na : \u03b1\nx\u271d : IsSquare a\nc : \u03b1\nhc : a = c * c\n\u22a2 IsSquare (op a)\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nm : \u03b1\nf : F\n\u22a2 IsSquare m \u2192 IsSquare (\u2191f m)\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\nm : \u03b1\n\u22a2 IsSquare (\u2191f (m * m))\n[PROOFSTEP]\nexact \u27e8f m, by simp\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\nm : \u03b1\n\u22a2 \u2191f (m * m) = \u2191f m * \u2191f m\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn : \u2115\na m : \u03b1\n\u22a2 IsSquare m \u2194 \u2203 c, m = c ^ 2\n[PROOFSTEP]\nsimp [IsSquare, pow_two]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn\u271d : \u2115\na : \u03b1\nn : \u2115\n\u22a2 IsSquare a \u2192 IsSquare (a ^ n)\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn\u271d n : \u2115\na : \u03b1\n\u22a2 IsSquare ((a * a) ^ n)\n[PROOFSTEP]\nexact \u27e8a ^ n, (Commute.refl _).mul_pow _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\nn : \u2115\na : \u03b1\n\u22a2 Even n \u2192 \u2200 (a : \u03b1), IsSquare (a ^ n)\n[PROOFSTEP]\nrintro \u27e8n, rfl\u27e9 a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Monoid \u03b1\na\u271d : \u03b1\nn : \u2115\na : \u03b1\n\u22a2 IsSquare (a ^ (n + n))\n[PROOFSTEP]\nexact \u27e8a ^ n, pow_add _ _ _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\na : \u03b1\ninst\u271d : HasDistribNeg \u03b1\n\u22a2 Even n \u2192 \u2200 (a : \u03b1), (-a) ^ n = a ^ n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9 a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\na\u271d : \u03b1\ninst\u271d : HasDistribNeg \u03b1\nc : \u2115\na : \u03b1\n\u22a2 (-a) ^ (c + c) = a ^ (c + c)\n[PROOFSTEP]\nsimp_rw [\u2190 two_mul, pow_mul, neg_sq]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\nn : \u2115\na : \u03b1\ninst\u271d : HasDistribNeg \u03b1\nh : Even n\n\u22a2 (-1) ^ n = 1\n[PROOFSTEP]\nrw [h.neg_pow, one_pow]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : CommSemigroup \u03b1\na b : \u03b1\n\u22a2 IsSquare a \u2192 IsSquare b \u2192 IsSquare (a * b)\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9 \u27e8b, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : CommSemigroup \u03b1\na b : \u03b1\n\u22a2 IsSquare (a * a * (b * b))\n[PROOFSTEP]\nexact \u27e8a * b, mul_mul_mul_comm _ _ _ _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\n\u22a2 IsSquare a\u207b\u00b9 \u2194 IsSquare a\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nh : IsSquare a\u207b\u00b9\n\u22a2 IsSquare a\n[PROOFSTEP]\nrw [\u2190 isSquare_op_iff, \u2190 inv_inv a]\n[GOAL]\ncase refine'_1\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nh : IsSquare a\u207b\u00b9\n\u22a2 IsSquare (op a\u207b\u00b9\u207b\u00b9)\n[PROOFSTEP]\nexact h.map (MulEquiv.inv' \u03b1)\n[GOAL]\ncase refine'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nh : IsSquare a\n\u22a2 IsSquare a\u207b\u00b9\n[PROOFSTEP]\nexact ((isSquare_op_iff a).mpr h).map (MulEquiv.inv' \u03b1).symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\na : \u03b1\nn : \u2124\n\u22a2 IsSquare a \u2192 IsSquare (a ^ n)\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionMonoid \u03b1\nn : \u2124\na : \u03b1\n\u22a2 IsSquare ((a * a) ^ n)\n[PROOFSTEP]\nexact \u27e8a ^ n, (Commute.refl _).mul_zpow _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : DivisionMonoid \u03b1\na : \u03b1\ninst\u271d : HasDistribNeg \u03b1\nn : \u2124\n\u22a2 Even n \u2192 \u2200 (a : \u03b1), (-a) ^ n = a ^ n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9 a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : DivisionMonoid \u03b1\na\u271d : \u03b1\ninst\u271d : HasDistribNeg \u03b1\nc : \u2124\na : \u03b1\n\u22a2 (-a) ^ (c + c) = a ^ (c + c)\n[PROOFSTEP]\nexact zpow_bit0_neg _ _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : DivisionMonoid \u03b1\na : \u03b1\ninst\u271d : HasDistribNeg \u03b1\nn : \u2124\nh : Even n\n\u22a2 (-1) ^ n = 1\n[PROOFSTEP]\nrw [h.neg_zpow, one_zpow]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 Even |a| \u2194 Even a\n[PROOFSTEP]\ncases abs_choice a\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d : |a| = a\n\u22a2 Even |a| \u2194 Even a\n[PROOFSTEP]\nhave h : abs a = a := by assumption\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d : |a| = a\n\u22a2 |a| = a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d h : |a| = a\n\u22a2 Even |a| \u2194 Even a\n[PROOFSTEP]\nsimp only [h, even_neg]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d : |a| = -a\n\u22a2 Even |a| \u2194 Even a\n[PROOFSTEP]\nhave h : abs a = -a := by assumption\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d : |a| = -a\n\u22a2 |a| = -a\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : SubtractionMonoid \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\nh\u271d h : |a| = -a\n\u22a2 Even |a| \u2194 Even a\n[PROOFSTEP]\nsimp only [h, even_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionCommMonoid \u03b1\na b : \u03b1\nha : IsSquare a\nhb : IsSquare b\n\u22a2 IsSquare (a / b)\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : DivisionCommMonoid \u03b1\na b : \u03b1\nha : IsSquare a\nhb : IsSquare b\n\u22a2 IsSquare (a * b\u207b\u00b9)\n[PROOFSTEP]\nexact ha.mul hb.inv\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Group \u03b1\nn : \u2124\n\u22a2 Even n \u2192 \u2200 (a : \u03b1), IsSquare (a ^ n)\n[PROOFSTEP]\nrintro \u27e8n, rfl\u27e9 a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Group \u03b1\nn : \u2124\na : \u03b1\n\u22a2 IsSquare (a ^ (n + n))\n[PROOFSTEP]\nexact \u27e8a ^ n, zpow_add _ _ _\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nm n : \u03b1\nhm : Even m\nhn : Even n\n\u22a2 Even (m - n)\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hm\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nn : \u03b1\nhn : Even n\na : \u03b1\n\u22a2 Even (a + a - n)\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 := hn\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 Even (a + a - (b + b))\n[PROOFSTEP]\nrefine' \u27e8a - b, _\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a + a - (b + b) = a - b + (a - b)\n[PROOFSTEP]\nobtain h | h := le_total a b\n[GOAL]\ncase intro.intro.inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : a \u2264 b\n\u22a2 a + a - (b + b) = a - b + (a - b)\n[PROOFSTEP]\nrw [tsub_eq_zero_of_le h, tsub_eq_zero_of_le (add_le_add h h), add_zero]\n[GOAL]\ncase intro.intro.inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b3 : CanonicallyLinearOrderedAddMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : b \u2264 a\n\u22a2 a + a - (b + b) = a - b + (a - b)\n[PROOFSTEP]\nexact (tsub_add_tsub_comm h h).symm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm\u271d n m : \u03b1\n\u22a2 Even m \u2194 \u2203 c, m = 2 * c\n[PROOFSTEP]\nsimp [even_iff_exists_two_nsmul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n a : \u03b1\n\u22a2 Even a \u2194 2 \u2223 a\n[PROOFSTEP]\nsimp [Even, Dvd.dvd, two_mul]\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\u271d\ninst\u271d\u00b9 : Semiring \u03b2\nm n : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : Semiring \u03b1\n\u22a2 (Set.range fun x => 2 * x) = {a | Even a}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\u271d\ninst\u271d\u00b9 : Semiring \u03b2\nm n : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : Semiring \u03b1\nx : \u03b1\n\u22a2 (x \u2208 Set.range fun x => 2 * x) \u2194 x \u2208 {a | Even a}\n[PROOFSTEP]\nsimp [eq_comm, two_mul, Even]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 2 = 1 + 1\n[PROOFSTEP]\nrw [one_add_one_eq_two]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Even m\na : \u2115\nx\u271d : a + 1 \u2260 0\n\u22a2 Even (m ^ (a + 1))\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Even m\na : \u2115\nx\u271d : a + 1 \u2260 0\n\u22a2 Even (m * m ^ a)\n[PROOFSTEP]\nexact hm.mul_right _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n a b : \u03b1\n\u22a2 a = 2 * b + 1 \u2194 a = bit1 b\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n a b : \u03b1\n\u22a2 a = b + b + 1 \u2194 a = bit1 b\n[PROOFSTEP]\nrfl\n[GOAL]\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\u271d\ninst\u271d\u00b9 : Semiring \u03b2\nm n : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : Semiring \u03b1\n\u22a2 (Set.range fun x => 2 * x + 1) = {a | Odd a}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nF : Type u_1\n\u03b1\u271d : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\u271d\ninst\u271d\u00b9 : Semiring \u03b2\nm n : \u03b1\u271d\n\u03b1 : Type u_5\ninst\u271d : Semiring \u03b1\nx : \u03b1\n\u22a2 (x \u2208 Set.range fun x => 2 * x + 1) \u2194 x \u2208 {a | Odd a}\n[PROOFSTEP]\nsimp [Odd, eq_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Even m \u2192 Odd n \u2192 Odd (m + n)\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9 \u27e8n, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd (m + m + (2 * n + 1))\n[PROOFSTEP]\nexact \u27e8m + n, by rw [mul_add, \u2190 two_mul, add_assoc]\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 m + m + (2 * n + 1) = 2 * (m + n) + 1\n[PROOFSTEP]\nrw [mul_add, \u2190 two_mul, add_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhe : Even m\nho : Odd n\n\u22a2 Odd (n + m)\n[PROOFSTEP]\nsimp only [he.add_odd ho, add_comm n m]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\nhn : Even n\n\u22a2 Odd (m + n)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\nhn : Even n\n\u22a2 Odd (n + m)\n[PROOFSTEP]\nexact hn.add_odd hm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd m \u2192 Odd n \u2192 Even (m + n)\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9 \u27e8n, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Even (2 * m + 1 + (2 * n + 1))\n[PROOFSTEP]\nrefine' \u27e8n + m + 1, _\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 2 * m + 1 + (2 * n + 1) = n + m + 1 + (n + m + 1)\n[PROOFSTEP]\nrw [two_mul, two_mul]\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 m + m + 1 + (n + n + 1) = n + m + 1 + (n + m + 1)\n[PROOFSTEP]\nac_rfl\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd (m + (m + 1))\n[PROOFSTEP]\nsimp [\u2190 add_assoc]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd (m + 1 + m)\n[PROOFSTEP]\nsimp [add_comm _ m]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd (m + (1 + m))\n[PROOFSTEP]\nsimp [add_comm 1 m]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd (1 + m + m)\n[PROOFSTEP]\nsimp [add_comm 1 m]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Semiring \u03b2\nm n : \u03b1\ninst\u271d : RingHomClass F \u03b1 \u03b2\nf : F\n\u22a2 Odd m \u2192 Odd (\u2191f m)\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Semiring \u03b2\nn : \u03b1\ninst\u271d : RingHomClass F \u03b1 \u03b2\nf : F\nm : \u03b1\n\u22a2 Odd (\u2191f (2 * m + 1))\n[PROOFSTEP]\nexact \u27e8f m, by simp [two_mul]\u27e9\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b2 : Semiring \u03b1\ninst\u271d\u00b9 : Semiring \u03b2\nn : \u03b1\ninst\u271d : RingHomClass F \u03b1 \u03b2\nf : F\nm : \u03b1\n\u22a2 \u2191f (2 * m + 1) = 2 * \u2191f m + 1\n[PROOFSTEP]\nsimp [two_mul]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd m \u2192 Odd n \u2192 Odd (m * n)\n[PROOFSTEP]\nrintro \u27e8m, rfl\u27e9 \u27e8n, rfl\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 Odd ((2 * m + 1) * (2 * n + 1))\n[PROOFSTEP]\nrefine' \u27e82 * m * n + n + m, _\u27e9\n[GOAL]\ncase intro.intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\n\u22a2 (2 * m + 1) * (2 * n + 1) = 2 * (2 * m * n + n + m) + 1\n[PROOFSTEP]\nrw [mul_add, add_mul, mul_one, \u2190 add_assoc, one_mul, mul_assoc, \u2190 mul_add, \u2190 mul_add, \u2190 mul_assoc, \u2190 Nat.cast_two, \u2190\n  Nat.cast_comm]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\n\u22a2 Odd (m ^ 0)\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\n\u22a2 Odd 1\n[PROOFSTEP]\nexact odd_one\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\na : \u2115\n\u22a2 Odd (m ^ (a + 1))\n[PROOFSTEP]\nrw [pow_succ]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\nm n : \u03b1\nhm : Odd m\na : \u2115\n\u22a2 Odd (m * m ^ a)\n[PROOFSTEP]\nexact hm.mul <| Odd.pow hm\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\nn : \u2115\n\u22a2 Odd n \u2192 \u2200 (a : \u03b1), (-a) ^ n = -a ^ n\n[PROOFSTEP]\nrintro \u27e8c, rfl\u27e9 a\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na\u271d : \u03b1\nc : \u2115\na : \u03b1\n\u22a2 (-a) ^ (2 * c + 1) = -a ^ (2 * c + 1)\n[PROOFSTEP]\nsimp_rw [pow_add, pow_mul, neg_sq, pow_one, mul_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\na : \u03b1\nn : \u2115\nh : Odd n\n\u22a2 (-1) ^ n = -1\n[PROOFSTEP]\nrw [h.neg_pow, one_pow]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d : Nontrivial \u03b1\nn : \u03b1\nhn : Odd n\n\u22a2 0 < n\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := hn\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d : Nontrivial \u03b1\nk : \u03b1\n\u22a2 0 < 2 * k + 1\n[PROOFSTEP]\nrw [pos_iff_ne_zero, Ne.def, add_eq_zero_iff, not_and']\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d : Nontrivial \u03b1\nk : \u03b1\n\u22a2 1 = 0 \u2192 \u00ac2 * k = 0\n[PROOFSTEP]\nexact fun h => (one_ne_zero h).elim\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Even (-2)\n[PROOFSTEP]\nsimp only [even_neg, even_two]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nhp : Odd a\n\u22a2 Odd (-a)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hp\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nk : \u03b1\nhk : a = 2 * k + 1\n\u22a2 Odd (-a)\n[PROOFSTEP]\nuse-(k + 1)\n[GOAL]\ncase h\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nk : \u03b1\nhk : a = 2 * k + 1\n\u22a2 -a = 2 * -(k + 1) + 1\n[PROOFSTEP]\nrw [mul_neg, mul_add, neg_add, add_assoc, two_mul (1 : \u03b1), neg_add, neg_add_cancel_right, \u2190 neg_add, hk]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 Odd (-1)\n[PROOFSTEP]\nsimp\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Odd a\nhb : Even b\n\u22a2 Odd (a - b)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Odd a\nhb : Even b\n\u22a2 Odd (a + -b)\n[PROOFSTEP]\nexact ha.add_even hb.neg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Even a\nhb : Odd b\n\u22a2 Odd (a - b)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Even a\nhb : Odd b\n\u22a2 Odd (a + -b)\n[PROOFSTEP]\nexact ha.add_odd hb.neg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Odd a\nhb : Odd b\n\u22a2 Even (a - b)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : Ring \u03b1\na b : \u03b1\nn : \u2115\nha : Odd a\nhb : Odd b\n\u22a2 Even (a + -b)\n[PROOFSTEP]\nexact ha.add_odd hb.neg\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Ring \u03b1\na b : \u03b1\nn : \u2115\ninst\u271d : LinearOrder \u03b1\n\u22a2 Odd |a| \u2194 Odd a\n[PROOFSTEP]\ncases' abs_choice a with h h\n[GOAL]\ncase inl\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Ring \u03b1\na b : \u03b1\nn : \u2115\ninst\u271d : LinearOrder \u03b1\nh : |a| = a\n\u22a2 Odd |a| \u2194 Odd a\n[PROOFSTEP]\nsimp only [h, odd_neg]\n[GOAL]\ncase inr\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Ring \u03b1\na b : \u03b1\nn : \u2115\ninst\u271d : LinearOrder \u03b1\nh : |a| = -a\n\u22a2 Odd |a| \u2194 Odd a\n[PROOFSTEP]\nsimp only [h, odd_neg]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn : \u2115\nhn : Even n\na : R\n\u22a2 0 \u2264 a ^ n\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn : \u2115\na : R\nk : \u2115\nhk : n = k + k\n\u22a2 0 \u2264 a ^ n\n[PROOFSTEP]\nsimpa only [hk, two_mul] using pow_bit0_nonneg a k\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Even n\nha : a \u2260 0\n\u22a2 0 < a ^ n\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nha : a \u2260 0\nk : \u2115\nhk : n = k + k\n\u22a2 0 < a ^ n\n[PROOFSTEP]\nsimpa only [hk, two_mul] using pow_bit0_pos ha k\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Odd n\nha : a \u2264 0\n\u22a2 a ^ n \u2264 0\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nha : a \u2264 0\nk : \u2115\nhk : n = 2 * k + 1\n\u22a2 a ^ n \u2264 0\n[PROOFSTEP]\nsimpa only [hk, two_mul] using pow_bit1_nonpos_iff.mpr ha\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Odd n\nha : a < 0\n\u22a2 a ^ n < 0\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nha : a < 0\nk : \u2115\nhk : n = 2 * k + 1\n\u22a2 a ^ n < 0\n[PROOFSTEP]\nsimpa only [hk, two_mul] using pow_bit1_neg_iff.mpr ha\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Even n\nh\u2080 : 0 < n\nh : 0 < a ^ n\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\nrw [ha, zero_pow h\u2080] at h \n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Even n\nh\u2080 : 0 < n\nh : 0 < 0\nha : a = 0\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl 0 h\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn p : \u2115\nhp : Even p\na : R\n\u22a2 |a| ^ p = a ^ p\n[PROOFSTEP]\nrw [\u2190 abs_pow, abs_eq_self]\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na\u271d : R\nn p : \u2115\nhp : Even p\na : R\n\u22a2 0 \u2264 a ^ p\n[PROOFSTEP]\nexact hp.pow_nonneg _\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn : \u2115\nhn : Odd n\n\u22a2 StrictMono fun a => a ^ n\n[PROOFSTEP]\ncases' hn with k hk\n[GOAL]\ncase intro\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d : LinearOrderedRing R\na : R\nn k : \u2115\nhk : n = 2 * k + 1\n\u22a2 StrictMono fun a => a ^ n\n[PROOFSTEP]\nsimpa only [hk, two_mul] using strictMono_pow_bit1 _\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Parity", "llama_tokens": 11493, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.6548947223065754, "lm_q1q2_score": 0.5142944515866823}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a = 0\nhb : b = 0\n\u22a2 a + b = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a = 0\nhb : b \u2264 0\n\u22a2 a + b \u2264 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a = 0\nhb : b < 0\n\u22a2 a + b < 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a \u2264 0\nhb : b = 0\n\u22a2 a + b \u2264 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a < 0\nhb : b = 0\n\u22a2 a + b < 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : StrictOrderedRing \u03b1\na b : \u03b1\nha : a < 0\nhb : 0 < b\nthis : -b * a > 0\n\u22a2 0 < -(b * a)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedRing \u03b1\na b : \u03b1\nha : a \u2264 0\nhb : 0 < b\nthis : -b * a \u2265 0\n\u22a2 b * a \u2264 0\n[PROOFSTEP]\nsimpa\n  -- used alongside `mul_neg` and `mul_nonpos`, so has the same argument pattern for uniformity\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedSemiring \u03b1\na b : \u03b1\nha : a = 0\nx\u271d : 0 < b\n\u22a2 b * a = 0\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Semiring \u03b1\na b : \u03b1\nx\u271d : R a 0\nh : b = 0\n\u22a2 a * b = 0\n[PROOFSTEP]\nsimp [h]\n  -- used in the `nlinarith` normalization steps. The `_` argument is for uniformity.\n[GOAL]\n\u03b1 : Type u_1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : Semiring \u03b1\na b : \u03b1\nh : a = 0\nx\u271d : R b 0\n\u22a2 a * b = 0\n[PROOFSTEP]\nsimp [h]\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Linarith.Lemmas", "llama_tokens": 725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.7122321842389469, "lm_q1q2_score": 0.514048740280959}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\ns : Set E\nx\u271d y\u271d x y : E\n\u22a2 [x-[\ud835\udd5c]y] = (fun p => p.fst \u2022 x + p.snd \u2022 y) '' {p | 0 \u2264 p.fst \u2227 0 \u2264 p.snd \u2227 p.fst + p.snd = 1}\n[PROOFSTEP]\nsimp only [segment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : SMul \ud835\udd5c E\ns : Set E\nx\u271d y\u271d x y : E\n\u22a2 openSegment \ud835\udd5c x y = (fun p => p.fst \u2022 x + p.snd \u2022 y) '' {p | 0 < p.fst \u2227 0 < p.snd \u2227 p.fst + p.snd = 1}\n[PROOFSTEP]\nsimp only [openSegment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : MulActionWithZero \ud835\udd5c E\nx y : E\n\u22a2 1 \u2022 x + 0 \u2022 y = x\n[PROOFSTEP]\nrw [zero_smul, one_smul, add_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d\u00b9 y z\u271d x z : E\nx\u271d : z \u2208 [x-[\ud835\udd5c]x]\na b : \ud835\udd5c\nw\u271d\u00b9 : 0 \u2264 a\nw\u271d : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 x = z\n\u22a2 z \u2208 {x}\n[PROOFSTEP]\nsimpa only [(add_smul _ _ _).symm, mem_singleton_iff, hab, one_smul, eq_comm] using hz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\n\u22a2 insert x (insert y (openSegment \ud835\udd5c x y)) = [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nsimp only [subset_antisymm_iff, insert_subset_iff, left_mem_segment, right_mem_segment, openSegment_subset_segment,\n  true_and_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\n\u22a2 [x-[\ud835\udd5c]y] \u2286 insert x (insert y (openSegment \ud835\udd5c x y))\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 insert x (insert y (openSegment \ud835\udd5c x y))\n[PROOFSTEP]\nrefine' hb.eq_or_gt.imp _ fun hb' => ha.eq_or_gt.imp _ fun ha' => _\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 b = 0 \u2192 a \u2022 x + b \u2022 y = x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\na : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 0\nhab : a + 0 = 1\n\u22a2 a \u2022 x + 0 \u2022 y = x\n[PROOFSTEP]\nrw [\u2190 add_zero a, hab, one_smul, zero_smul, add_zero]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhb' : 0 < b\n\u22a2 a = 0 \u2192 a \u2022 x + b \u2022 y = y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\nb : \ud835\udd5c\nhb : 0 \u2264 b\nhb' : 0 < b\nha : 0 \u2264 0\nhab : 0 + b = 1\n\u22a2 0 \u2022 x + b \u2022 y = y\n[PROOFSTEP]\nrw [\u2190 zero_add b, hab, one_smul, zero_smul, zero_add]\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx\u271d y\u271d z x y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhb' : 0 < b\nha' : 0 < a\n\u22a2 a \u2022 x + b \u2022 y \u2208 openSegment \ud835\udd5c x y\n[PROOFSTEP]\nexact \u27e8a, b, ha', hb', hab, rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx y z : E\nhx : x \u2260 z\nhy : y \u2260 z\nhz : z \u2208 [x-[\ud835\udd5c]y]\n\u22a2 z \u2208 openSegment \ud835\udd5c x y\n[PROOFSTEP]\nrw [\u2190 insert_endpoints_openSegment] at hz \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx y z : E\nhx : x \u2260 z\nhy : y \u2260 z\nhz : z \u2208 insert x (insert y (openSegment \ud835\udd5c x y))\n\u22a2 z \u2208 openSegment \ud835\udd5c x y\n[PROOFSTEP]\nexact (hz.resolve_left hx.symm).resolve_left hy.symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\ns : Set E\nx y z : E\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 openSegment \ud835\udd5c x y \u2286 s \u2194 [x-[\ud835\udd5c]y] \u2286 s\n[PROOFSTEP]\nsimp only [\u2190 insert_endpoints_openSegment, insert_subset_iff, *, true_and_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2077 : OrderedRing \ud835\udd5c\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : Nontrivial \ud835\udd5c\ninst\u271d : DenselyOrdered \ud835\udd5c\nx z : E\nx\u271d : z \u2208 openSegment \ud835\udd5c x x\na b : \ud835\udd5c\nw\u271d\u00b9 : 0 < a\nw\u271d : 0 < b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 x = z\n\u22a2 z \u2208 {x}\n[PROOFSTEP]\nsimpa only [\u2190 add_smul, mem_singleton_iff, hab, one_smul, eq_comm] using hz\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2077 : OrderedRing \ud835\udd5c\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : Nontrivial \ud835\udd5c\ninst\u271d : DenselyOrdered \ud835\udd5c\nx z : E\nh : z = x\n\u22a2 z \u2208 openSegment \ud835\udd5c x x\n[PROOFSTEP]\nobtain \u27e8a, ha\u2080, ha\u2081\u27e9 := DenselyOrdered.dense (0 : \ud835\udd5c) 1 zero_lt_one\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2077 : OrderedRing \ud835\udd5c\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : Nontrivial \ud835\udd5c\ninst\u271d : DenselyOrdered \ud835\udd5c\nx z : E\nh : z = x\na : \ud835\udd5c\nha\u2080 : 0 < a\nha\u2081 : a < 1\n\u22a2 z \u2208 openSegment \ud835\udd5c x x\n[PROOFSTEP]\nrefine' \u27e8a, 1 - a, ha\u2080, sub_pos_of_lt ha\u2081, add_sub_cancel'_right _ _, _\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2077 : OrderedRing \ud835\udd5c\ninst\u271d\u2076 : AddCommGroup E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : AddCommGroup G\ninst\u271d\u00b3 : Module \ud835\udd5c E\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : Nontrivial \ud835\udd5c\ninst\u271d : DenselyOrdered \ud835\udd5c\nx z : E\nh : z = x\na : \ud835\udd5c\nha\u2080 : 0 < a\nha\u2081 : a < 1\n\u22a2 a \u2022 x + (1 - a) \u2022 x = z\n[PROOFSTEP]\nrw [\u2190 add_smul, add_sub_cancel'_right, one_smul, h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E\nx\u271d : z \u2208 [x-[\ud835\udd5c]y]\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 (fun \u03b8 => (a + b - \u03b8) \u2022 x + \u03b8 \u2022 y) b = a \u2022 x + b \u2022 y\n[PROOFSTEP]\nsimp only [add_sub_cancel]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E\nx\u271d : z \u2208 openSegment \ud835\udd5c x y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 (fun \u03b8 => (a + b - \u03b8) \u2022 x + \u03b8 \u2022 y) b = a \u2022 x + b \u2022 y\n[PROOFSTEP]\nsimp only [add_sub_cancel]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\n\u22a2 [x-[\ud835\udd5c]y] = (fun \u03b8 => x + \u03b8 \u2022 (y - x)) '' Icc 0 1\n[PROOFSTEP]\nconvert segment_eq_image \ud835\udd5c x y using 2\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Icc 0 1\n\u22a2 x + a\u271d\u00b9 \u2022 (y - x) = (1 - a\u271d\u00b9) \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nsimp only [smul_sub, sub_smul, one_smul]\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Icc 0 1\n\u22a2 x + (a\u271d\u00b9 \u2022 y - a\u271d\u00b9 \u2022 x) = x - a\u271d\u00b9 \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Icc 0 1\n\u22a2 x + (a\u271d\u00b9 \u2022 y - a\u271d\u00b9 \u2022 x) = x - a\u271d\u00b9 \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\n\u22a2 openSegment \ud835\udd5c x y = (fun \u03b8 => x + \u03b8 \u2022 (y - x)) '' Ioo 0 1\n[PROOFSTEP]\nconvert openSegment_eq_image \ud835\udd5c x y using 2\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Ioo 0 1\n\u22a2 x + a\u271d\u00b9 \u2022 (y - x) = (1 - a\u271d\u00b9) \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nsimp only [smul_sub, sub_smul, one_smul]\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Ioo 0 1\n\u22a2 x + (a\u271d\u00b9 \u2022 y - a\u271d\u00b9 \u2022 x) = x - a\u271d\u00b9 \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Ioo 0 1\n\u22a2 x + (a\u271d\u00b9 \u2022 y - a\u271d\u00b9 \u2022 x) = x - a\u271d\u00b9 \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\n\u22a2 [x-[\ud835\udd5c]y] = \u2191(AffineMap.lineMap x y) '' Icc 0 1\n[PROOFSTEP]\nconvert segment_eq_image \ud835\udd5c x y using 2\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Icc 0 1\n\u22a2 \u2191(AffineMap.lineMap x y) a\u271d\u00b9 = (1 - a\u271d\u00b9) \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nexact AffineMap.lineMap_apply_module _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\n\u22a2 openSegment \ud835\udd5c x y = \u2191(AffineMap.lineMap x y) '' Ioo 0 1\n[PROOFSTEP]\nconvert openSegment_eq_image \ud835\udd5c x y using 2\n[GOAL]\ncase h.e'_3.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E\na\u271d\u00b9 : \ud835\udd5c\na\u271d : a\u271d\u00b9 \u2208 Ioo 0 1\n\u22a2 \u2191(AffineMap.lineMap x y) a\u271d\u00b9 = (1 - a\u271d\u00b9) \u2022 x + a\u271d\u00b9 \u2022 y\n[PROOFSTEP]\nexact AffineMap.lineMap_apply_module _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nf : E \u2192\u1d43[\ud835\udd5c] F\na b : E\nx : F\n\u22a2 x \u2208 \u2191f '' [a-[\ud835\udd5c]b] \u2194 x \u2208 [\u2191f a-[\ud835\udd5c]\u2191f b]\n[PROOFSTEP]\nsimp_rw [segment_eq_image_lineMap, mem_image, exists_exists_and_eq_and, AffineMap.apply_lineMap]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nf : E \u2192\u1d43[\ud835\udd5c] F\na b : E\nx : F\n\u22a2 x \u2208 \u2191f '' openSegment \ud835\udd5c a b \u2194 x \u2208 openSegment \ud835\udd5c (\u2191f a) (\u2191f b)\n[PROOFSTEP]\nsimp_rw [openSegment_eq_image_lineMap, mem_image, exists_exists_and_eq_and, AffineMap.apply_lineMap]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\na x b c : E\n\u22a2 a + x \u2208 [a + b-[\ud835\udd5c]a + c] \u2194 x \u2208 [b-[\ud835\udd5c]c]\n[PROOFSTEP]\nsimp_rw [\u2190 vadd_eq_add, \u2190 vadd_segment, vadd_mem_vadd_set_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2075 : OrderedRing \ud835\udd5c\ninst\u271d\u2074 : AddCommGroup E\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : AddCommGroup G\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\na x b c : E\n\u22a2 a + x \u2208 openSegment \ud835\udd5c (a + b) (a + c) \u2194 x \u2208 openSegment \ud835\udd5c b c\n[PROOFSTEP]\nsimp_rw [\u2190 vadd_eq_add, \u2190 vadd_openSegment, vadd_mem_vadd_set_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : StrictOrderedCommRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : x \u2208 [y-[\ud835\udd5c]z]\n\u22a2 SameRay \ud835\udd5c (x - y) (z - x)\n[PROOFSTEP]\nrw [segment_eq_image'] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : StrictOrderedCommRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : x \u2208 (fun \u03b8 => y + \u03b8 \u2022 (z - y)) '' Icc 0 1\n\u22a2 SameRay \ud835\udd5c (x - y) (z - x)\n[PROOFSTEP]\nrcases h with \u27e8\u03b8, \u27e8h\u03b8\u2080, h\u03b8\u2081\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : StrictOrderedCommRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\n\u03b8 : \ud835\udd5c\nh\u03b8\u2080 : 0 \u2264 \u03b8\nh\u03b8\u2081 : \u03b8 \u2264 1\n\u22a2 SameRay \ud835\udd5c ((fun \u03b8 => y + \u03b8 \u2022 (z - y)) \u03b8 - y) (z - (fun \u03b8 => y + \u03b8 \u2022 (z - y)) \u03b8)\n[PROOFSTEP]\nsimpa only [add_sub_cancel', \u2190 sub_sub, sub_smul, one_smul] using\n  (SameRay.sameRay_nonneg_smul_left (z - y) h\u03b8\u2080).nonneg_smul_right (sub_nonneg.2 h\u03b8\u2081)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 midpoint \ud835\udd5c x y \u2208 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nrw [segment_eq_image_lineMap]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 midpoint \ud835\udd5c x y \u2208 \u2191(AffineMap.lineMap x y) '' Icc 0 1\n[PROOFSTEP]\nexact \u27e8\u215f2, \u27e8invOf_nonneg.mpr zero_le_two, invOf_le_one one_le_two\u27e9, rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 x \u2208 [x - y-[\ud835\udd5c]x + y]\n[PROOFSTEP]\nconvert @midpoint_mem_segment \ud835\udd5c _ _ _ _ _ _ _\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 x = midpoint \ud835\udd5c (x - y) (x + y)\n[PROOFSTEP]\nrw [midpoint_sub_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 x \u2208 [x + y-[\ud835\udd5c]x - y]\n[PROOFSTEP]\nconvert @midpoint_mem_segment \ud835\udd5c _ _ _ _ _ _ _\n[GOAL]\ncase h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module \ud835\udd5c E\nx\u271d y\u271d : E\ninst\u271d : Invertible 2\nx y : E\n\u22a2 x = midpoint \ud835\udd5c (x + y) (x - y)\n[PROOFSTEP]\nrw [midpoint_add_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 x \u2208 openSegment \ud835\udd5c x y \u2194 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 x \u2208 openSegment \ud835\udd5c x y \u2192 x = y\n[PROOFSTEP]\nrintro \u27e8a, b, _, hb, hab, hx\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\na b : \ud835\udd5c\nw\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : a \u2022 x + b \u2022 y = x\n\u22a2 x = y\n[PROOFSTEP]\nrefine' smul_right_injective _ hb.ne' ((add_right_inj (a \u2022 x)).1 _)\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\na b : \ud835\udd5c\nw\u271d : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : a \u2022 x + b \u2022 y = x\n\u22a2 a \u2022 x + (fun x x_1 => x \u2022 x_1) b x = a \u2022 x + (fun x x_1 => x \u2022 x_1) b y\n[PROOFSTEP]\nrw [hx, \u2190 add_smul, hab, one_smul]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 x = y \u2192 x \u2208 openSegment \ud835\udd5c x y\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 x \u2208 openSegment \ud835\udd5c x x\n[PROOFSTEP]\nrw [openSegment_same]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 x \u2208 {x}\n[PROOFSTEP]\nexact mem_singleton _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\nx y : E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\n\u22a2 y \u2208 openSegment \ud835\udd5c x y \u2194 x = y\n[PROOFSTEP]\nrw [openSegment_symm, left_mem_openSegment_iff, eq_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 x \u2208 [y-[\ud835\udd5c]z] \u2194 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 0 < a + b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 x \u2208 [y-[\ud835\udd5c]z] \u2192 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 0 < a + b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2203 a_1 b_1, 0 \u2264 a_1 \u2227 0 \u2264 b_1 \u2227 0 < a_1 + b_1 \u2227 (a_1 / (a_1 + b_1)) \u2022 y + (b_1 / (a_1 + b_1)) \u2022 z = a \u2022 y + b \u2022 z\n[PROOFSTEP]\nuse a, b, ha, hb\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 0 < a + b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = a \u2022 y + b \u2022 z\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 (\u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 0 < a + b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x) \u2192 x \u2208 [y-[\ud835\udd5c]z]\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z \u2208 [y-[\ud835\udd5c]z]\n[PROOFSTEP]\nrefine' \u27e8a / (a + b), b / (a + b), by positivity, by positivity, _, rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 0 \u2264 a / (a + b)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 0 \u2264 b / (a + b)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mpr.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : 0 < a + b\n\u22a2 a / (a + b) + b / (a + b) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hab.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 x \u2208 openSegment \ud835\udd5c y z \u2194 \u2203 a b, 0 < a \u2227 0 < b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 x \u2208 openSegment \ud835\udd5c y z \u2192 \u2203 a b, 0 < a \u2227 0 < b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 < b_1 \u2227 (a_1 / (a_1 + b_1)) \u2022 y + (b_1 / (a_1 + b_1)) \u2022 z = a \u2022 y + b \u2022 z\n[PROOFSTEP]\nuse a, b, ha, hb\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = a \u2022 y + b \u2022 z\n[PROOFSTEP]\nrw [hab, div_one, div_one]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 (\u2203 a b, 0 < a \u2227 0 < b \u2227 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z = x) \u2192 x \u2208 openSegment \ud835\udd5c y z\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\n\u22a2 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z \u2208 openSegment \ud835\udd5c y z\n[PROOFSTEP]\nhave hab : 0 < a + b := by positivity\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\n\u22a2 0 < a + b\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 (a / (a + b)) \u2022 y + (b / (a + b)) \u2022 z \u2208 openSegment \ud835\udd5c y z\n[PROOFSTEP]\nrefine' \u27e8a / (a + b), b / (a + b), by positivity, by positivity, _, rfl\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 0 < a / (a + b)\n[PROOFSTEP]\npositivity\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 0 < b / (a + b)\n[PROOFSTEP]\npositivity\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedSemifield \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\ny z : E\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : 0 < a + b\n\u22a2 a / (a + b) + b / (a + b) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, div_self hab.ne']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\n\u22a2 x \u2208 [y-[\ud835\udd5c]z] \u2194 SameRay \ud835\udd5c (x - y) (z - x)\n[PROOFSTEP]\nrefine' \u27e8sameRay_of_mem_segment, fun h => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : SameRay \ud835\udd5c (x - y) (z - x)\n\u22a2 x \u2208 [y-[\ud835\udd5c]z]\n[PROOFSTEP]\nrcases h.exists_eq_smul_add with \u27e8a, b, ha, hb, hab, hxy, hzx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : SameRay \ud835\udd5c (x - y) (z - x)\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhxy : x - y = a \u2022 (x - y + (z - x))\nhzx : z - x = b \u2022 (x - y + (z - x))\n\u22a2 x \u2208 [y-[\ud835\udd5c]z]\n[PROOFSTEP]\nrw [add_comm, sub_add_sub_cancel] at hxy hzx \n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : SameRay \ud835\udd5c (x - y) (z - x)\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhxy : x - y = a \u2022 (z - y)\nhzx : z - x = b \u2022 (z - y)\n\u22a2 x \u2208 [y-[\ud835\udd5c]z]\n[PROOFSTEP]\nrw [\u2190 mem_segment_translate _ (-x), neg_add_self]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : SameRay \ud835\udd5c (x - y) (z - x)\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhxy : x - y = a \u2022 (z - y)\nhzx : z - x = b \u2022 (z - y)\n\u22a2 0 \u2208 [-x + y-[\ud835\udd5c]-x + z]\n[PROOFSTEP]\nrefine' \u27e8b, a, hb, ha, add_comm a b \u25b8 hab, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx y z : E\nh : SameRay \ud835\udd5c (x - y) (z - x)\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhxy : x - y = a \u2022 (z - y)\nhzx : z - x = b \u2022 (z - y)\n\u22a2 b \u2022 (-x + y) + a \u2022 (-x + z) = 0\n[PROOFSTEP]\nrw [\u2190 sub_eq_neg_add, \u2190 neg_sub, hxy, \u2190 sub_eq_neg_add, hzx, smul_neg, smul_comm, neg_add_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z\u271d x y z : E\nhz : z \u2208 range \u2191(lineMap x y)\n\u22a2 openSegment \ud835\udd5c x y \u2286 insert z (openSegment \ud835\udd5c x z \u222a openSegment \ud835\udd5c z y)\n[PROOFSTEP]\nrcases hz with \u27e8c, rfl\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc : \ud835\udd5c\n\u22a2 openSegment \ud835\udd5c x y \u2286\n    insert (\u2191(lineMap x y) c) (openSegment \ud835\udd5c x (\u2191(lineMap x y) c) \u222a openSegment \ud835\udd5c (\u2191(lineMap x y) c) y)\n[PROOFSTEP]\nsimp only [openSegment_eq_image_lineMap, \u2190 mapsTo']\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc : \ud835\udd5c\n\u22a2 MapsTo (fun a => \u2191(lineMap x y) a) (Ioo 0 1)\n    (insert (\u2191(lineMap x y) c)\n      ((fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a\n        (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1))\n[PROOFSTEP]\nrintro a \u27e8h\u2080, h\u2081\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    insert (\u2191(lineMap x y) c)\n      ((fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a\n        (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1)\n[PROOFSTEP]\nrcases lt_trichotomy a c with (hac | rfl | hca)\n[GOAL]\ncase intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhac : a < c\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    insert (\u2191(lineMap x y) c)\n      ((fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a\n        (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1)\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.inl.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhac : a < c\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    (fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.intro.inl.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhac : a < c\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208 (fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1\n[PROOFSTEP]\nhave hc : 0 < c := h\u2080.trans hac\n[GOAL]\ncase intro.intro.inl.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhac : a < c\nhc : 0 < c\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208 (fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1\n[PROOFSTEP]\nrefine' \u27e8a / c, \u27e8div_pos h\u2080 hc, (div_lt_one hc).2 hac\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.inl.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhac : a < c\nhc : 0 < c\n\u22a2 (fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) (a / c) = (fun a => \u2191(lineMap x y) a) a\n[PROOFSTEP]\nsimp only [\u2190 homothety_eq_lineMap, \u2190 homothety_mul_apply, div_mul_cancel _ hc.ne']\n[GOAL]\ncase intro.intro.inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\na : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    insert (\u2191(lineMap x y) a)\n      ((fun a_1 => \u2191(lineMap x (\u2191(lineMap x y) a)) a_1) '' Ioo 0 1 \u222a\n        (fun a_1 => \u2191(lineMap (\u2191(lineMap x y) a) y) a_1) '' Ioo 0 1)\n[PROOFSTEP]\nleft\n[GOAL]\ncase intro.intro.inr.inl.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\na : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\n\u22a2 (fun a => \u2191(lineMap x y) a) a = \u2191(lineMap x y) a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    insert (\u2191(lineMap x y) c)\n      ((fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a\n        (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1)\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.inr.inr.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208\n    (fun a => \u2191(lineMap x (\u2191(lineMap x y) c)) a) '' Ioo 0 1 \u222a (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.inr.inr.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208 (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1\n[PROOFSTEP]\nhave hc : 0 < 1 - c := sub_pos.2 (hca.trans h\u2081)\n[GOAL]\ncase intro.intro.inr.inr.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\nhc : 0 < 1 - c\n\u22a2 (fun a => \u2191(lineMap x y) a) a \u2208 (fun a => \u2191(lineMap (\u2191(lineMap x y) c) y) a) '' Ioo 0 1\n[PROOFSTEP]\nsimp only [\u2190 lineMap_apply_one_sub y]\n[GOAL]\ncase intro.intro.inr.inr.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\nhc : 0 < 1 - c\n\u22a2 \u2191(lineMap y x) (1 - a) \u2208 (fun a => \u2191(lineMap y (\u2191(lineMap y x) (1 - c))) (1 - a)) '' Ioo 0 1\n[PROOFSTEP]\nrefine' \u27e8(a - c) / (1 - c), \u27e8div_pos (sub_pos.2 hca) hc, (div_lt_one hc).2 <| sub_lt_sub_right h\u2081 _\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.inr.inr.h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : Module \ud835\udd5c E\nx\u271d y\u271d z x y : E\nc a : \ud835\udd5c\nh\u2080 : 0 < a\nh\u2081 : a < 1\nhca : c < a\nhc : 0 < 1 - c\n\u22a2 (fun a => \u2191(lineMap y (\u2191(lineMap y x) (1 - c))) (1 - a)) ((a - c) / (1 - c)) = \u2191(lineMap y x) (1 - a)\n[PROOFSTEP]\nsimp only [\u2190 homothety_eq_lineMap, \u2190 homothety_mul_apply, sub_mul, one_mul, div_mul_cancel _ hc.ne',\n  sub_sub_sub_cancel_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] \u2286 Icc x y\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 Icc x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 x \u2264 a \u2022 x + b \u2022 y\ncase intro.intro.intro.intro.intro.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2264 y\n[PROOFSTEP]\ncalc\n  x = a \u2022 x + b \u2022 x := (Convex.combo_self hab _).symm\n  _ \u2264 a \u2022 x + b \u2022 y := by gcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 x \u2264 a \u2022 x + b \u2022 y\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.intro.intro.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2264 y\n[PROOFSTEP]\ncalc\n  a \u2022 x + b \u2022 y \u2264 a \u2022 y + b \u2022 y := by gcongr\n  _ = y := Convex.combo_self hab _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x \u2264 y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2264 a \u2022 y + b \u2022 y\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\n\u22a2 openSegment \ud835\udd5c x y \u2286 Ioo x y\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 Ioo x y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase intro.intro.intro.intro.intro.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 x < a \u2022 x + b \u2022 y\ncase intro.intro.intro.intro.intro.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y < y\n[PROOFSTEP]\ncalc\n  x = a \u2022 x + b \u2022 x := (Convex.combo_self hab _).symm\n  _ < a \u2022 x + b \u2022 y := by gcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 x < a \u2022 x + b \u2022 y\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.intro.intro.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y < y\n[PROOFSTEP]\ncalc\n  a \u2022 x + b \u2022 y < a \u2022 y + b \u2022 y := by gcongr\n  _ = y := Convex.combo_self hab _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\nx y : E\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y < a \u2022 y + b \u2022 y\n[PROOFSTEP]\ngcongr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\na b : \ud835\udd5c\nx y : E\n\u22a2 [x-[\ud835\udd5c]y] \u2286 uIcc x y\n[PROOFSTEP]\ncases' le_total x y with h h\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\na b : \ud835\udd5c\nx y : E\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] \u2286 uIcc x y\n[PROOFSTEP]\nrw [uIcc_of_le h]\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\na b : \ud835\udd5c\nx y : E\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] \u2286 Icc x y\n[PROOFSTEP]\nexact segment_subset_Icc h\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\na b : \ud835\udd5c\nx y : E\nh : y \u2264 x\n\u22a2 [x-[\ud835\udd5c]y] \u2286 uIcc x y\n[PROOFSTEP]\nrw [uIcc_of_ge h, segment_symm]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : LinearOrderedAddCommMonoid E\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : OrderedSMul \ud835\udd5c E\na b : \ud835\udd5c\nx y : E\nh : y \u2264 x\n\u22a2 [y-[\ud835\udd5c]x] \u2286 Icc y x\n[PROOFSTEP]\nexact segment_subset_Icc h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\n\u22a2 Icc x y \u2286 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nrintro z \u27e8hxz, hyz\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz : x \u2264 z\nhyz : z \u2264 y\n\u22a2 z \u2208 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nobtain rfl | h := (hxz.trans hyz).eq_or_lt\n[GOAL]\ncase intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx z\u271d z : \ud835\udd5c\nhxz : x \u2264 z\nhyz : z \u2264 x\n\u22a2 z \u2208 [x-[\ud835\udd5c]x]\n[PROOFSTEP]\nrw [segment_same]\n[GOAL]\ncase intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx z\u271d z : \ud835\udd5c\nhxz : x \u2264 z\nhyz : z \u2264 x\n\u22a2 z \u2208 {x}\n[PROOFSTEP]\nexact hyz.antisymm hxz\n[GOAL]\ncase intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz : x \u2264 z\nhyz : z \u2264 y\nh : x < y\n\u22a2 z \u2208 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nrw [\u2190 sub_nonneg] at hxz hyz \n[GOAL]\ncase intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz\u271d : x \u2264 z\nhxz : 0 \u2264 z - x\nhyz\u271d : z \u2264 y\nhyz : 0 \u2264 y - z\nh : x < y\n\u22a2 z \u2208 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nrw [\u2190 sub_pos] at h \n[GOAL]\ncase intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz\u271d : x \u2264 z\nhxz : 0 \u2264 z - x\nhyz\u271d : z \u2264 y\nhyz : 0 \u2264 y - z\nh\u271d : x < y\nh : 0 < y - x\n\u22a2 z \u2208 [x-[\ud835\udd5c]y]\n[PROOFSTEP]\nrefine' \u27e8(y - z) / (y - x), (z - x) / (y - x), div_nonneg hyz h.le, div_nonneg hxz h.le, _, _\u27e9\n[GOAL]\ncase intro.inr.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz\u271d : x \u2264 z\nhxz : 0 \u2264 z - x\nhyz\u271d : z \u2264 y\nhyz : 0 \u2264 y - z\nh\u271d : x < y\nh : 0 < y - x\n\u22a2 (y - z) / (y - x) + (z - x) / (y - x) = 1\n[PROOFSTEP]\nrw [\u2190 add_div, sub_add_sub_cancel, div_self h.ne']\n[GOAL]\ncase intro.inr.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z\u271d z : \ud835\udd5c\nhxz\u271d : x \u2264 z\nhxz : 0 \u2264 z - x\nhyz\u271d : z \u2264 y\nhyz : 0 \u2264 y - z\nh\u271d : x < y\nh : 0 < y - x\n\u22a2 ((y - z) / (y - x)) \u2022 x + ((z - x) / (y - x)) \u2022 y = z\n[PROOFSTEP]\nrw [smul_eq_mul, smul_eq_mul, \u2190 mul_div_right_comm, \u2190 mul_div_right_comm, \u2190 add_div, div_eq_iff h.ne', add_comm,\n  sub_mul, sub_mul, mul_comm x, sub_add_sub_cancel, mul_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)\n[PROOFSTEP]\ncases' le_total x y with h h\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : x \u2264 y\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)\n[PROOFSTEP]\nrw [segment_eq_Icc h, max_eq_right h, min_eq_left h]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx\u271d y\u271d z x y : \ud835\udd5c\nh : y \u2264 x\n\u22a2 [x-[\ud835\udd5c]y] = Icc (min x y) (max x y)\n[PROOFSTEP]\nrw [segment_symm, segment_eq_Icc h, max_eq_left h, min_eq_right h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nhxy : x \u2260 y\n\u22a2 openSegment \ud835\udd5c x y = Ioo (min x y) (max x y)\n[PROOFSTEP]\ncases' hxy.lt_or_lt with h h\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nhxy : x \u2260 y\nh : x < y\n\u22a2 openSegment \ud835\udd5c x y = Ioo (min x y) (max x y)\n[PROOFSTEP]\nrw [openSegment_eq_Ioo h, max_eq_right h.le, min_eq_left h.le]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nhxy : x \u2260 y\nh : y < x\n\u22a2 openSegment \ud835\udd5c x y = Ioo (min x y) (max x y)\n[PROOFSTEP]\nrw [openSegment_symm, openSegment_eq_Ioo h, max_eq_left h.le, min_eq_right h.le]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x \u2264 y\n\u22a2 z \u2208 Icc x y \u2194 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nrw [\u2190 segment_eq_Icc h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x \u2264 y\n\u22a2 z \u2208 [x-[\ud835\udd5c]y] \u2194 \u2203 a b, 0 \u2264 a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nsimp_rw [\u2190 exists_prop]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x \u2264 y\n\u22a2 z \u2208 [x-[\ud835\udd5c]y] \u2194 \u2203 a b _h _h _h, a * x + b * y = z\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 Ioo x y \u2194 \u2203 a b, 0 < a \u2227 0 < b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nrw [\u2190 openSegment_eq_Ioo h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 openSegment \ud835\udd5c x y \u2194 \u2203 a b, 0 < a \u2227 0 < b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nsimp_rw [\u2190 exists_prop]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 openSegment \ud835\udd5c x y \u2194 \u2203 a b _h _h _h, a * x + b * y = z\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 Ioc x y \u2194 \u2203 a b, 0 \u2264 a \u2227 0 < b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nrefine' \u27e8fun hz => _, _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\nhz : z \u2208 Ioc x y\n\u22a2 \u2203 a b, 0 \u2264 a \u2227 0 < b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := (Convex.mem_Icc h.le).1 (Ioc_subset_Icc_self hz)\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ioc x y\n\u22a2 \u2203 a_1 b_1, 0 \u2264 a_1 \u2227 0 < b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y\n[PROOFSTEP]\nobtain rfl | hb' := hb.eq_or_lt\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 0\nhab : a + 0 = 1\nhz : a * x + 0 * y \u2208 Ioc x y\n\u22a2 \u2203 a_1 b, 0 \u2264 a_1 \u2227 0 < b \u2227 a_1 + b = 1 \u2227 a_1 * x + b * y = a * x + 0 * y\n[PROOFSTEP]\nrw [add_zero] at hab \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 0\nhab : a = 1\nhz : a * x + 0 * y \u2208 Ioc x y\n\u22a2 \u2203 a_1 b, 0 \u2264 a_1 \u2227 0 < b \u2227 a_1 + b = 1 \u2227 a_1 * x + b * y = a * x + 0 * y\n[PROOFSTEP]\nrw [hab, one_mul, zero_mul, add_zero] at hz \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 0\nhab : a = 1\nhz : x \u2208 Ioc x y\n\u22a2 \u2203 a_1 b, 0 \u2264 a_1 \u2227 0 < b \u2227 a_1 + b = 1 \u2227 a_1 * x + b * y = a * x + 0 * y\n[PROOFSTEP]\nexact (hz.1.ne rfl).elim\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ioc x y\nhb' : 0 < b\n\u22a2 \u2203 a_1 b_1, 0 \u2264 a_1 \u2227 0 < b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y\n[PROOFSTEP]\nexact \u27e8a, b, ha, hb', hab, rfl\u27e9\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 (\u2203 a b, 0 \u2264 a \u2227 0 < b \u2227 a + b = 1 \u2227 a * x + b * y = z) \u2192 z \u2208 Ioc x y\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a * x + b * y \u2208 Ioc x y\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 < b\nha : 0 \u2264 0\nhab : 0 + b = 1\n\u22a2 0 * x + b * y \u2208 Ioc x y\n[PROOFSTEP]\nrw [zero_add] at hab \n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 < b\nha : 0 \u2264 0\nhab : b = 1\n\u22a2 0 * x + b * y \u2208 Ioc x y\n[PROOFSTEP]\nrwa [hab, one_mul, zero_mul, zero_add, right_mem_Ioc]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 < b\nhab : a + b = 1\nha' : 0 < a\n\u22a2 a * x + b * y \u2208 Ioc x y\n[PROOFSTEP]\nexact Ioo_subset_Ioc_self ((Convex.mem_Ioo h).2 \u27e8a, b, ha', hb, hab, rfl\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 z \u2208 Ico x y \u2194 \u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nrefine' \u27e8fun hz => _, _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\nhz : z \u2208 Ico x y\n\u22a2 \u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z\n[PROOFSTEP]\nobtain \u27e8a, b, ha, hb, hab, rfl\u27e9 := (Convex.mem_Icc h.le).1 (Ico_subset_Icc_self hz)\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y\n[PROOFSTEP]\nobtain rfl | ha' := ha.eq_or_lt\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : 0 + b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y\n[PROOFSTEP]\nrw [zero_add] at hab \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : 0 * x + b * y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y\n[PROOFSTEP]\nrw [hab, one_mul, zero_mul, zero_add] at hz \n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\nb : \ud835\udd5c\nhb : 0 \u2264 b\nha : 0 \u2264 0\nhab : b = 1\nhz : y \u2208 Ico x y\n\u22a2 \u2203 a b_1, 0 < a \u2227 0 \u2264 b_1 \u2227 a + b_1 = 1 \u2227 a * x + b_1 * y = 0 * x + b * y\n[PROOFSTEP]\nexact (hz.2.ne rfl).elim\n[GOAL]\ncase refine'_1.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a * x + b * y \u2208 Ico x y\nha' : 0 < a\n\u22a2 \u2203 a_1 b_1, 0 < a_1 \u2227 0 \u2264 b_1 \u2227 a_1 + b_1 = 1 \u2227 a_1 * x + b_1 * y = a * x + b * y\n[PROOFSTEP]\nexact \u27e8a, b, ha', hb, hab, rfl\u27e9\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y z : \ud835\udd5c\nh : x < y\n\u22a2 (\u2203 a b, 0 < a \u2227 0 \u2264 b \u2227 a + b = 1 \u2227 a * x + b * y = z) \u2192 z \u2208 Ico x y\n[PROOFSTEP]\nrintro \u27e8a, b, ha, hb, hab, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a * x + b * y \u2208 Ico x y\n[PROOFSTEP]\nobtain rfl | hb' := hb.eq_or_lt\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a + 0 = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y\n[PROOFSTEP]\nrw [add_zero] at hab \n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 0\nhab : a = 1\n\u22a2 a * x + 0 * y \u2208 Ico x y\n[PROOFSTEP]\nrwa [hab, one_mul, zero_mul, add_zero, left_mem_Ico]\n[GOAL]\ncase refine'_2.intro.intro.intro.intro.intro.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d : LinearOrderedField \ud835\udd5c\nx y : \ud835\udd5c\nh : x < y\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 \u2264 b\nhab : a + b = 1\nhb' : 0 < b\n\u22a2 a * x + b * y \u2208 Ico x y\n[PROOFSTEP]\nexact Ioo_subset_Ico_self ((Convex.mem_Ioo h).2 \u27e8a, b, ha, hb', hab, rfl\u27e9)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E \u00d7 F\n\u22a2 [x-[\ud835\udd5c]y] \u2286 [x.fst-[\ud835\udd5c]y.fst] \u00d7\u02e2 [x.snd-[\ud835\udd5c]y.snd]\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, hz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E \u00d7 F\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 z \u2208 [x.fst-[\ud835\udd5c]y.fst] \u00d7\u02e2 [x.snd-[\ud835\udd5c]y.snd]\n[PROOFSTEP]\nexact \u27e8\u27e8a, b, ha, hb, hab, congr_arg Prod.fst hz\u27e9, a, b, ha, hb, hab, congr_arg Prod.snd hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y : E \u00d7 F\n\u22a2 openSegment \ud835\udd5c x y \u2286 openSegment \ud835\udd5c x.fst y.fst \u00d7\u02e2 openSegment \ud835\udd5c x.snd y.snd\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, hz\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx y z : E \u00d7 F\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\n\u22a2 z \u2208 openSegment \ud835\udd5c x.fst y.fst \u00d7\u02e2 openSegment \ud835\udd5c x.snd y.snd\n[PROOFSTEP]\nexact \u27e8\u27e8a, b, ha, hb, hab, congr_arg Prod.fst hz\u27e9, a, b, ha, hb, hab, congr_arg Prod.snd hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\n\u22a2 (fun x => (x, y)) '' [x\u2081-[\ud835\udd5c]x\u2082] = [(x\u2081, y)-[\ud835\udd5c](x\u2082, y)]\n[PROOFSTEP]\next \u27e8x', y'\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\n\u22a2 (x', y') \u2208 (fun x => (x, y)) '' [x\u2081-[\ud835\udd5c]x\u2082] \u2194 (x', y') \u2208 [(x\u2081, y)-[\ud835\udd5c](x\u2082, y)]\n[PROOFSTEP]\nsimp_rw [Set.mem_image, segment, Set.mem_setOf, Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff, \u2190 exists_and_right,\n  @exists_comm E, exists_eq_left']\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\n\u22a2 (\u2203 b b_1 h h h, b \u2022 x\u2081 + b_1 \u2022 x\u2082 = x' \u2227 y = y') \u2194 \u2203 a b h h h, a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 a \u2022 y + b \u2022 y = y'\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 y = y' \u2194 a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 a \u2022 y + b \u2022 y = y'\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\n\u22a2 (fun y => (x, y)) '' [y\u2081-[\ud835\udd5c]y\u2082] = [(x, y\u2081)-[\ud835\udd5c](x, y\u2082)]\n[PROOFSTEP]\next \u27e8x', y'\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\n\u22a2 (x', y') \u2208 (fun y => (x, y)) '' [y\u2081-[\ud835\udd5c]y\u2082] \u2194 (x', y') \u2208 [(x, y\u2081)-[\ud835\udd5c](x, y\u2082)]\n[PROOFSTEP]\nsimp_rw [Set.mem_image, segment, Set.mem_setOf, Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff, \u2190 exists_and_right,\n  @exists_comm F, exists_eq_left']\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\n\u22a2 (\u2203 b b_1 h h h, x = x' \u2227 b \u2022 y\u2081 + b_1 \u2022 y\u2082 = y') \u2194 \u2203 a b h h h, a \u2022 x + b \u2022 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y'\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y' \u2194 a \u2022 x + b \u2022 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y'\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\n\u22a2 (fun x => (x, y)) '' openSegment \ud835\udd5c x\u2081 x\u2082 = openSegment \ud835\udd5c (x\u2081, y) (x\u2082, y)\n[PROOFSTEP]\next \u27e8x', y'\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\n\u22a2 (x', y') \u2208 (fun x => (x, y)) '' openSegment \ud835\udd5c x\u2081 x\u2082 \u2194 (x', y') \u2208 openSegment \ud835\udd5c (x\u2081, y) (x\u2082, y)\n[PROOFSTEP]\nsimp_rw [Set.mem_image, openSegment, Set.mem_setOf, Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff, \u2190 exists_and_right,\n  @exists_comm E, exists_eq_left']\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\n\u22a2 (\u2203 b b_1 h h h, b \u2022 x\u2081 + b_1 \u2022 x\u2082 = x' \u2227 y = y') \u2194 \u2203 a b h h h, a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 a \u2022 y + b \u2022 y = y'\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx\u2081 x\u2082 : E\ny : F\nx' : E\ny' : F\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 y = y' \u2194 a \u2022 x\u2081 + b \u2022 x\u2082 = x' \u2227 a \u2022 y + b \u2022 y = y'\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\n\u22a2 (fun y => (x, y)) '' openSegment \ud835\udd5c y\u2081 y\u2082 = openSegment \ud835\udd5c (x, y\u2081) (x, y\u2082)\n[PROOFSTEP]\next \u27e8x', y'\u27e9\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\n\u22a2 (x', y') \u2208 (fun y => (x, y)) '' openSegment \ud835\udd5c y\u2081 y\u2082 \u2194 (x', y') \u2208 openSegment \ud835\udd5c (x, y\u2081) (x, y\u2082)\n[PROOFSTEP]\nsimp_rw [Set.mem_image, openSegment, Set.mem_setOf, Prod.smul_mk, Prod.mk_add_mk, Prod.mk.inj_iff, \u2190 exists_and_right,\n  @exists_comm F, exists_eq_left']\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\n\u22a2 (\u2203 b b_1 h h h, x = x' \u2227 b \u2022 y\u2081 + b_1 \u2022 y\u2082 = y') \u2194 \u2203 a b h h h, a \u2022 x + b \u2022 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y'\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\nx : E\ny\u2081 y\u2082 : F\nx' : E\ny' : F\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y' \u2194 a \u2022 x + b \u2022 x = x' \u2227 a \u2022 y\u2081 + b \u2022 y\u2082 = y'\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\nx y : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 [x-[\ud835\udd5c]y] \u2286 pi s fun i => [x i-[\ud835\udd5c]y i]\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, hz\u27e9 i -\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\nx y z : (i : \u03b9) \u2192 \u03c0 i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\ni : \u03b9\n\u22a2 z i \u2208 (fun i => [x i-[\ud835\udd5c]y i]) i\n[PROOFSTEP]\nexact \u27e8a, b, ha, hb, hab, congr_fun hz i\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\nx y : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 openSegment \ud835\udd5c x y \u2286 pi s fun i => openSegment \ud835\udd5c (x i) (y i)\n[PROOFSTEP]\nrintro z \u27e8a, b, ha, hb, hab, hz\u27e9 i -\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\nx y z : (i : \u03b9) \u2192 \u03c0 i\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhz : a \u2022 x + b \u2022 y = z\ni : \u03b9\n\u22a2 z i \u2208 (fun i => openSegment \ud835\udd5c (x i) (y i)) i\n[PROOFSTEP]\nexact \u27e8a, b, ha, hb, hab, congr_fun hz i\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 update y i '' [x\u2081-[\ud835\udd5c]x\u2082] = [update y i x\u2081-[\ud835\udd5c]update y i x\u2082]\n[PROOFSTEP]\next z\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\n\u22a2 z \u2208 update y i '' [x\u2081-[\ud835\udd5c]x\u2082] \u2194 z \u2208 [update y i x\u2081-[\ud835\udd5c]update y i x\u2082]\n[PROOFSTEP]\nsimp_rw [Set.mem_image, segment, Set.mem_setOf, \u2190 update_smul, \u2190 update_add, update_eq_iff, \u2190 exists_and_right,\n  @exists_comm (\u03c0 i), exists_eq_left']\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\n\u22a2 (\u2203 b b_1 h h h, b \u2022 x\u2081 + b_1 \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 y x = z x) \u2194\n    \u2203 a b h h h, a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 (a \u2022 y + b \u2022 y) x = z x\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 (a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 y x = z x) \u2194\n    a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 (a \u2022 y + b \u2022 y) x = z x\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 update y i '' openSegment \ud835\udd5c x\u2081 x\u2082 = openSegment \ud835\udd5c (update y i x\u2081) (update y i x\u2082)\n[PROOFSTEP]\next z\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\n\u22a2 z \u2208 update y i '' openSegment \ud835\udd5c x\u2081 x\u2082 \u2194 z \u2208 openSegment \ud835\udd5c (update y i x\u2081) (update y i x\u2082)\n[PROOFSTEP]\nsimp_rw [Set.mem_image, openSegment, Set.mem_setOf, \u2190 update_smul, \u2190 update_add, update_eq_iff, \u2190 exists_and_right,\n  @exists_comm (\u03c0 i), exists_eq_left']\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\n\u22a2 (\u2203 b b_1 h h h, b \u2022 x\u2081 + b_1 \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 y x = z x) \u2194\n    \u2203 a b h h h, a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 (a \u2022 y + b \u2022 y) x = z x\n[PROOFSTEP]\nrefine' exists\u2085_congr fun a b ha hb hab => _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c0 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03c0 i)\ns : Set \u03b9\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nx\u2081 x\u2082 : \u03c0 i\ny z : (a : \u03b9) \u2192 \u03c0 a\na b : \ud835\udd5c\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 (a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 y x = z x) \u2194\n    a \u2022 x\u2081 + b \u2022 x\u2082 = z i \u2227 \u2200 (x : \u03b9), x \u2260 i \u2192 (a \u2022 y + b \u2022 y) x = z x\n[PROOFSTEP]\nrw [Convex.combo_self hab]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Convex.Segment", "llama_tokens": 37442, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8354835371034368, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5138957771559309}}
{"text": "[GOAL]\n\u03b9 \u03b1 : Type v\nU : \u03b9 \u2192 \u03b1\ninst\u271d : CompleteLattice \u03b1\ns : Cocone (diagram U)\n\u22a2 (cocone U).pt \u2264 s.pt\n[PROOFSTEP]\napply CompleteSemilatticeSup.sSup_le\n[GOAL]\ncase a\n\u03b9 \u03b1 : Type v\nU : \u03b9 \u2192 \u03b1\ninst\u271d : CompleteLattice \u03b1\ns : Cocone (diagram U)\n\u22a2 \u2200 (b : \u03b1), b \u2208 Set.range U \u2192 b \u2264 s.pt\n[PROOFSTEP]\nrintro _ \u27e8j, rfl\u27e9\n[GOAL]\ncase a.intro\n\u03b9 \u03b1 : Type v\nU : \u03b9 \u2192 \u03b1\ninst\u271d : CompleteLattice \u03b1\ns : Cocone (diagram U)\nj : \u03b9\n\u22a2 U j \u2264 s.pt\n[PROOFSTEP]\nexact (s.\u03b9.app (single j)).le\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Category.Pairwise", "llama_tokens": 236, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7799929002541067, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.5135609758411671}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 EquicontinuousAt F x\u2080 \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n[PROOFSTEP]\nrw [equicontinuousAt_iff_pair]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 (\u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U) \u2194\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 (\u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U) \u2192\n    \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\n\u22a2 (\u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5) \u2192\n    \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n[PROOFSTEP]\nintro \u03b5 h\u03b5\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n[PROOFSTEP]\nexact H _ (dist_mem_uniformity h\u03b5)\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (U : Set (\u03b1 \u00d7 \u03b1)), U \u2208 \ud835\udce4 \u03b1 \u2192 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n[PROOFSTEP]\nintro U hU\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n[PROOFSTEP]\nrcases mem_uniformity_dist.mp hU with \u27e8\u03b5, h\u03b5, h\u03b5U\u27e9\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n[PROOFSTEP]\nrefine' Exists.imp (fun V => And.imp_right fun h => _) (H _ h\u03b5)\n[GOAL]\ncase mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nx\u2080 : \u03b2\nH : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 U, U \u2208 \ud835\udcdd x\u2080 \u2227 \u2200 (x : \u03b2), x \u2208 U \u2192 \u2200 (x' : \u03b2), x' \u2208 U \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\nU : Set (\u03b1 \u00d7 \u03b1)\nhU : U \u2208 \ud835\udce4 \u03b1\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5U : \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 U\nV : Set \u03b2\nh : \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (x' : \u03b2), x' \u2208 V \u2192 \u2200 (i : \u03b9), dist (F i x) (F i x') < \u03b5\n\u22a2 \u2200 (x : \u03b2), x \u2208 V \u2192 \u2200 (y : \u03b2), y \u2208 V \u2192 \u2200 (i : \u03b9), (F i x, F i y) \u2208 U\n[PROOFSTEP]\nexact fun x hx x' hx' i => h\u03b5U (h _ hx _ hx' i)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nx\u2080 : \u03b2\nb : \u03b2 \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200\u1da0 (x : \u03b2) in \ud835\udcdd x\u2080, \u2200 (i : \u03b9), dist (F i x\u2080) (F i x) \u2264 b x\n\u22a2 EquicontinuousAt F x\u2080\n[PROOFSTEP]\nrw [Metric.equicontinuousAt_iff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nx\u2080 : \u03b2\nb : \u03b2 \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200\u1da0 (x : \u03b2) in \ud835\udcdd x\u2080, \u2200 (i : \u03b9), dist (F i x\u2080) (F i x) \u2264 b x\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b2) in \ud835\udcdd x\u2080, \u2200 (i : \u03b9), dist (F i x\u2080) (F i x) < \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : TopologicalSpace \u03b2\nx\u2080 : \u03b2\nb : \u03b2 \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd x\u2080) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200\u1da0 (x : \u03b2) in \ud835\udcdd x\u2080, \u2200 (i : \u03b9), dist (F i x\u2080) (F i x) \u2264 b x\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2200\u1da0 (x : \u03b2) in \ud835\udcdd x\u2080, \u2200 (i : \u03b9), dist (F i x\u2080) (F i x) < \u03b5\n[PROOFSTEP]\nfilter_upwards [Filter.mem_map.mp <| b_lim (Iio_mem_nhds \u03b50), H] using fun x hx\u2081 hx\u2082 i => (hx\u2082 i).trans_lt hx\u2081\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u22a2 UniformEquicontinuous F\n[PROOFSTEP]\nrw [Metric.uniformEquicontinuous_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u22a2 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x y : \u03b2), dist x y < \u03b4 \u2192 \u2200 (i : \u03b9), dist (F i x) (F i y) < \u03b5\n[PROOFSTEP]\nintro \u03b5 \u03b50\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x y : \u03b2), dist x y < \u03b4 \u2192 \u2200 (i : \u03b9), dist (F i x) (F i y) < \u03b5\n[PROOFSTEP]\nrcases tendsto_nhds_nhds.1 b_lim \u03b5 \u03b50 with \u27e8\u03b4, \u03b40, h\u03b4\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 {x : \u211d}, dist x 0 < \u03b4 \u2192 dist (b x) 0 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x y : \u03b2), dist x y < \u03b4 \u2192 \u2200 (i : \u03b9), dist (F i x) (F i y) < \u03b5\n[PROOFSTEP]\nrefine' \u27e8\u03b4, \u03b40, fun x y hxy i => _\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 {x : \u211d}, dist x 0 < \u03b4 \u2192 dist (b x) 0 < \u03b5\nx y : \u03b2\nhxy : dist x y < \u03b4\ni : \u03b9\n\u22a2 dist (F i x) (F i y) < \u03b5\n[PROOFSTEP]\ncalc\n  dist (F i x) (F i y) \u2264 b (dist x y) := H x y i\n  _ \u2264 |b (dist x y)| := (le_abs_self _)\n  _ = dist (b (dist x y)) 0 := by simp [Real.dist_eq]\n  _ < \u03b5 := h\u03b4 (by simpa only [Real.dist_eq, tsub_zero, abs_dist] using hxy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 {x : \u211d}, dist x 0 < \u03b4 \u2192 dist (b x) 0 < \u03b5\nx y : \u03b2\nhxy : dist x y < \u03b4\ni : \u03b9\n\u22a2 |b (dist x y)| = dist (b (dist x y)) 0\n[PROOFSTEP]\nsimp [Real.dist_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\n\u03b9 : Type u_4\ninst\u271d : PseudoMetricSpace \u03b2\nb : \u211d \u2192 \u211d\nb_lim : Tendsto b (\ud835\udcdd 0) (\ud835\udcdd 0)\nF : \u03b9 \u2192 \u03b2 \u2192 \u03b1\nH : \u2200 (x y : \u03b2) (i : \u03b9), dist (F i x) (F i y) \u2264 b (dist x y)\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 {x : \u211d}, dist x 0 < \u03b4 \u2192 dist (b x) 0 < \u03b5\nx y : \u03b2\nhxy : dist x y < \u03b4\ni : \u03b9\n\u22a2 dist (dist x y) 0 < \u03b4\n[PROOFSTEP]\nsimpa only [Real.dist_eq, tsub_zero, abs_dist] using hxy\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.Equicontinuity", "llama_tokens": 5108, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789086703225, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.513365659246197}}
{"text": "[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g h : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u03b8 : g \u27f6 h\n\u22a2 \u03b7 \u2297\u226b \u03b8 = \u03b7 \u226b \u03b8\n[PROOFSTEP]\ndsimp [bicategoricalComp]\n[GOAL]\nB : Type u\ninst\u271d : Bicategory B\na b c d e : B\nf g h : a \u27f6 b\n\u03b7 : f \u27f6 g\n\u03b8 : g \u27f6 h\n\u22a2 \u03b7 \u226b \ud835\udfd9 g \u226b \u03b8 = \u03b7 \u226b \u03b8\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Tactic.CategoryTheory.BicategoryCoherence", "llama_tokens": 167, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8198933183101078, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5133550293418997}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nx : \u03b1\n\u22a2 reverse (x ::\u1d65 xs) = snoc (reverse xs) x\n[PROOFSTEP]\ncases xs\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 reverse (x ::\u1d65 { val := val\u271d, property := property\u271d }) = snoc (reverse { val := val\u271d, property := property\u271d }) x\n[PROOFSTEP]\nsimp only [reverse, cons, toList_mk, List.reverse_cons, snoc]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 { val := List.reverse val\u271d ++ [x],\n      property := (_ : (fun l => List.length l = Nat.succ n) (List.reverse val\u271d ++ [x])) } =\n    append { val := List.reverse val\u271d, property := (_ : (fun l => List.length l = n) (List.reverse val\u271d)) }\n      { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nx : \u03b1\n\u22a2 reverse (snoc xs x) = x ::\u1d65 reverse xs\n[PROOFSTEP]\ncases xs\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 reverse (snoc { val := val\u271d, property := property\u271d } x) = x ::\u1d65 reverse { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nsimp only [reverse, snoc, cons, toList_mk]\n[GOAL]\ncase mk\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 {\n      val :=\n        List.reverse\n          (toList\n            (append { val := val\u271d, property := property\u271d }\n              { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })),\n      property :=\n        (_ :\n          (fun l => List.length l = n + 1)\n            (List.reverse\n              (toList\n                (append { val := val\u271d, property := property\u271d }\n                  { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })))) } =\n    { val := x :: List.reverse val\u271d, property := (_ : Nat.succ (List.length (List.reverse val\u271d)) = Nat.succ n) }\n[PROOFSTEP]\ncongr\n[GOAL]\ncase mk.e_val\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 List.reverse\n      (toList\n        (append { val := val\u271d, property := property\u271d }\n          { val := [x], property := (_ : Nat.succ (List.length []) = Nat.succ 0) })) =\n    x :: List.reverse val\u271d\n[PROOFSTEP]\nsimp [toList, (\u00b7 ++ \u00b7), Vector.append, Append.append]\n[GOAL]\ncase mk.e_val\n\u03b1 : Type u_1\nn : \u2115\nx : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 [] ++ [x] ++ List.reverse val\u271d = x :: List.reverse val\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nxs : Vector \u03b1 n\nval : \u03b1\n\u22a2 replicate (n + 1) val = snoc (replicate n val) val\n[PROOFSTEP]\nclear xs\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (n + 1) val = snoc (replicate n val) val\n[PROOFSTEP]\ninduction n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val\ncase succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val\n[PROOFSTEP]\ncase zero => rfl\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\n\u22a2 replicate (Nat.zero + 1) val = snoc (replicate Nat.zero val) val\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b1 : Type u_1\nn : \u2115\nval : \u03b1\nn\u271d : \u2115\nn_ih\u271d : replicate (n\u271d + 1) val = snoc (replicate n\u271d val) val\n\u22a2 replicate (Nat.succ n\u271d + 1) val = snoc (replicate (Nat.succ n\u271d) val) val\n[PROOFSTEP]\ncase succ n ih =>\n  rw [replicate_succ]\n  conv => {rhs; rw [replicate_succ]}\n  rw [snoc_cons, ih]\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 replicate (Nat.succ n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\ncase succ n ih =>\n  rw [replicate_succ]\n  conv => {rhs; rw [replicate_succ]}\n  rw [snoc_cons, ih]\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 replicate (Nat.succ n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\nrw [replicate_succ]\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\nconv => {rhs; rw [replicate_succ]}\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n| val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\n{rhs; rw [replicate_succ]}\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n| val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\n{rhs; rw [replicate_succ]}\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n| val ::\u1d65 replicate (n + 1) val = snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\nrhs\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n| snoc (replicate (Nat.succ n) val) val\n[PROOFSTEP]\nrw [replicate_succ]\n[GOAL]\n\u03b1 : Type u_1\nn\u271d : \u2115\nval : \u03b1\nn : \u2115\nih : replicate (n + 1) val = snoc (replicate n val) val\n\u22a2 val ::\u1d65 replicate (n + 1) val = snoc (val ::\u1d65 replicate n val) val\n[PROOFSTEP]\nrw [snoc_cons, ih]\n[GOAL]\n\u03b1 : Type ?u.2475\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_1\nn : \u2115\nv : Vector \u03b1 n\nnil : C Vector.nil\nsnoc : {n : \u2115} \u2192 (xs : Vector \u03b1 n) \u2192 (x : \u03b1) \u2192 C xs \u2192 C (Vector.snoc xs x)\n\u22a2 (fun {n} v => C (reverse v)) (reverse v) = C v\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.2475\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Sort u_1\nn\u271d : \u2115\nv : Vector \u03b1 n\u271d\nnil : C Vector.nil\nsnoc : {n : \u2115} \u2192 (xs : Vector \u03b1 n) \u2192 (x : \u03b1) \u2192 C xs \u2192 C (Vector.snoc xs x)\nn : \u2115\nx : \u03b1\nxs : Vector \u03b1 n\nr : C (reverse xs)\n\u22a2 C (Vector.snoc (reverse xs) x) = (fun {n} v => C (reverse v)) (x ::\u1d65 xs)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.2985\n\u03b2 : Type ?u.2988\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_1\nn : \u2115\nv : Vector \u03b1 n\nw : Vector \u03b2 n\nnil : C Vector.nil Vector.nil\nsnoc :\n  {n : \u2115} \u2192\n    (xs : Vector \u03b1 n) \u2192 (ys : Vector \u03b2 n) \u2192 (x : \u03b1) \u2192 (y : \u03b2) \u2192 C xs ys \u2192 C (Vector.snoc xs x) (Vector.snoc ys y)\n\u22a2 (fun {n} v w => C (reverse v) (reverse w)) (reverse v) (reverse w) = C v w\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.2985\n\u03b2 : Type ?u.2988\nC : {n : \u2115} \u2192 Vector \u03b1 n \u2192 Vector \u03b2 n \u2192 Sort u_1\nn\u271d : \u2115\nv : Vector \u03b1 n\u271d\nw : Vector \u03b2 n\u271d\nnil : C Vector.nil Vector.nil\nsnoc :\n  {n : \u2115} \u2192\n    (xs : Vector \u03b1 n) \u2192 (ys : Vector \u03b2 n) \u2192 (x : \u03b1) \u2192 (y : \u03b2) \u2192 C xs ys \u2192 C (Vector.snoc xs x) (Vector.snoc ys y)\nn : \u2115\nx : \u03b1\ny : \u03b2\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\nr : C (reverse xs) (reverse ys)\n\u22a2 C (Vector.snoc (reverse xs) x) (Vector.snoc (reverse ys) y) =\n    (fun {n} v w => C (reverse v) (reverse w)) (x ::\u1d65 xs) (y ::\u1d65 ys)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nx : \u03b1\n\u22a2 map f (snoc xs x) = snoc (map f xs) (f x)\n[PROOFSTEP]\ninduction xs using Vector.inductionOn\n[GOAL]\ncase h_nil\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nx : \u03b1\n\u22a2 map f (snoc nil x) = snoc (map f nil) (f x)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase h_cons\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d\nx : \u03b1\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d : map f (snoc w\u271d x) = snoc (map f w\u271d) (f x)\n\u22a2 map f (snoc (x\u271d ::\u1d65 w\u271d) x) = snoc (map f (x\u271d ::\u1d65 w\u271d)) (f x)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\n\u22a2 mapAccumr f (snoc xs x) s =\n    let q := f x s;\n    let r := mapAccumr f xs q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\ninduction xs using Vector.inductionOn\n[GOAL]\ncase h_nil\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\n\u22a2 mapAccumr f (snoc nil x) s =\n    let q := f x s;\n    let r := mapAccumr f nil q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_cons\n\u03b1 : Type u_2\nn : \u2115\nxs : Vector \u03b1 n\n\u03b1\u271d : Type\n\u03b2\u271d : Type u_1\nf : \u03b1 \u2192 \u03b1\u271d \u2192 \u03b1\u271d \u00d7 \u03b2\u271d\nx : \u03b1\ns : \u03b1\u271d\nn\u271d : \u2115\nx\u271d : \u03b1\nw\u271d : Vector \u03b1 n\u271d\na\u271d :\n  mapAccumr f (snoc w\u271d x) s =\n    let q := f x s;\n    let r := mapAccumr f w\u271d q.fst;\n    (r.fst, snoc r.snd q.snd)\n\u22a2 mapAccumr f (snoc (x\u271d ::\u1d65 w\u271d) x) s =\n    let q := f x s;\n    let r := mapAccumr f (x\u271d ::\u1d65 w\u271d) q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\nx : \u03b1\ny : \u03b2\n\u22a2 map\u2082 f (snoc xs x) (snoc ys y) = snoc (map\u2082 f xs ys) (f x y)\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\nx : \u03b1\ny : \u03b2\n\u22a2 map\u2082 f (snoc nil x) (snoc nil y) = snoc (map\u2082 f nil nil) (f x y)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase cons\n\u03b1 : Type u_2\nn : \u2115\n\u03b2 : Type u_3\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03b1\u271d : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\u271d\nx : \u03b1\ny : \u03b2\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d : map\u2082 f (snoc x\u271d x) (snoc y\u271d y) = snoc (map\u2082 f x\u271d y\u271d) (f x y)\n\u22a2 map\u2082 f (snoc (a\u271d\u00b9 ::\u1d65 x\u271d) x) (snoc (b\u271d ::\u1d65 y\u271d) y) = snoc (map\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d)) (f x y)\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03c6 : Type\nc : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6\nx : \u03b1\ny : \u03b2\n\u22a2 mapAccumr\u2082 f (snoc xs x) (snoc ys y) c =\n    let q := f x y c;\n    let r := mapAccumr\u2082 f xs ys q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\ninduction xs, ys using Vector.inductionOn\u2082\n[GOAL]\ncase nil\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03c6 : Type\nc : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6\nx : \u03b1\ny : \u03b2\n\u22a2 mapAccumr\u2082 f (snoc nil x) (snoc nil y) c =\n    let q := f x y c;\n    let r := mapAccumr\u2082 f nil nil q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\nsimp_all\n[GOAL]\ncase cons\n\u03b1 : Type\nn : \u2115\n\u03b2 : Type\nxs : Vector \u03b1 n\nys : Vector \u03b2 n\n\u03c3 \u03c6 : Type\nc : \u03c3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6\nx : \u03b1\ny : \u03b2\nn\u271d : \u2115\na\u271d\u00b9 : \u03b1\nb\u271d : \u03b2\nx\u271d : Vector \u03b1 n\u271d\ny\u271d : Vector \u03b2 n\u271d\na\u271d :\n  mapAccumr\u2082 f (snoc x\u271d x) (snoc y\u271d y) c =\n    let q := f x y c;\n    let r := mapAccumr\u2082 f x\u271d y\u271d q.fst;\n    (r.fst, snoc r.snd q.snd)\n\u22a2 mapAccumr\u2082 f (snoc (a\u271d\u00b9 ::\u1d65 x\u271d) x) (snoc (b\u271d ::\u1d65 y\u271d) y) c =\n    let q := f x y c;\n    let r := mapAccumr\u2082 f (a\u271d\u00b9 ::\u1d65 x\u271d) (b\u271d ::\u1d65 y\u271d) q.fst;\n    (r.fst, snoc r.snd q.snd)\n[PROOFSTEP]\nsimp_all\n", "meta": {"mathlib_filename": "Mathlib.Data.Vector.Snoc", "llama_tokens": 5181, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.6825737344123242, "lm_q1q2_score": 0.5133040307962482}}
{"text": "[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) (cos z)\n[PROOFSTEP]\nrw [Complex.cos, Complex.exp_eq_exp_\u2102]\n[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n[PROOFSTEP]\nhave := ((expSeries_div_hasSum_exp \u2102 (z * Complex.I)).add (expSeries_div_hasSum_exp \u2102 (-z * Complex.I))).div_const 2\n[GOAL]\nz : \u2102\nthis :\n  HasSum (fun i => ((z * I) ^ i / \u2191i ! + (-z * I) ^ i / \u2191i !) / 2) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n[PROOFSTEP]\nreplace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this\n[GOAL]\nz : \u2102\nthis :\n  HasSum ((fun i => ((z * I) ^ i / \u2191i ! + (-z * I) ^ i / \u2191i !) / 2) \u2218 \u2191(Nat.divModEquiv 2).symm)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n[PROOFSTEP]\ndsimp [Function.comp] at this \n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (x.fst * 2 + \u2191x.snd) / \u2191(x.fst * 2 + \u2191x.snd)! +\n          (-z * I) ^ (x.fst * 2 + \u2191x.snd) / \u2191(x.fst * 2 + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n[PROOFSTEP]\nsimp_rw [\u2190 mul_comm 2 _] at this \n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! +\n          (-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!) ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\n[PROOFSTEP]\nrefine' this.prod_fiberwise fun k => _\n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! +\n          (-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\nk : \u2115\n\u22a2 HasSum\n    (fun c =>\n      ((z * I) ^ (2 * (k, c).fst + \u2191(k, c).snd) / \u2191(2 * (k, c).fst + \u2191(k, c).snd)! +\n          (-z * I) ^ (2 * (k, c).fst + \u2191(k, c).snd) / \u2191(2 * (k, c).fst + \u2191(k, c).snd)!) /\n        2)\n    ((z * I) ^ (2 * k) / \u2191(2 * k)!)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! +\n          (-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\nk : \u2115\n\u22a2 HasSum (fun c => ((z * I) ^ (2 * k + \u2191c) / \u2191(2 * k + \u2191c)! + (-z * I) ^ (2 * k + \u2191c) / \u2191(2 * k + \u2191c)!) / 2)\n    ((z * I) ^ (2 * k) / \u2191(2 * k)!)\n[PROOFSTEP]\nconvert hasSum_fintype (_ : Fin 2 \u2192 \u2102) using 1\n[GOAL]\ncase h.e'_6\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! +\n          (-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\nk : \u2115\n\u22a2 (z * I) ^ (2 * k) / \u2191(2 * k)! =\n    Finset.sum Finset.univ fun b =>\n      ((z * I) ^ (2 * k + \u2191b) / \u2191(2 * k + \u2191b)! + (-z * I) ^ (2 * k + \u2191b) / \u2191(2 * k + \u2191b)!) / 2\n[PROOFSTEP]\nrw [Fin.sum_univ_two]\n[GOAL]\ncase h.e'_6\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! +\n          (-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) /\n        2)\n    ((_root_.exp \u2102 (z * I) + _root_.exp \u2102 (-z * I)) / 2)\nk : \u2115\n\u22a2 (z * I) ^ (2 * k) / \u2191(2 * k)! =\n    ((z * I) ^ (2 * k + \u21910) / \u2191(2 * k + \u21910)! + (-z * I) ^ (2 * k + \u21910) / \u2191(2 * k + \u21910)!) / 2 +\n      ((z * I) ^ (2 * k + \u21911) / \u2191(2 * k + \u21911)! + (-z * I) ^ (2 * k + \u21911) / \u2191(2 * k + \u21911)!) / 2\n[PROOFSTEP]\nsimp_rw [Fin.val_zero, Fin.val_one, add_zero, pow_succ', pow_mul, mul_pow, neg_sq, \u2190 two_mul, neg_mul, mul_neg, neg_div,\n  add_right_neg, zero_div, add_zero, mul_div_cancel_left _ (two_ne_zero : (2 : \u2102) \u2260 0)]\n[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) (sin z)\n[PROOFSTEP]\nrw [Complex.sin, Complex.exp_eq_exp_\u2102]\n[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n[PROOFSTEP]\nhave :=\n  (((expSeries_div_hasSum_exp \u2102 (-z * Complex.I)).sub (expSeries_div_hasSum_exp \u2102 (z * Complex.I))).mul_right\n        Complex.I).div_const\n    2\n[GOAL]\nz : \u2102\nthis :\n  HasSum (fun i => ((-z * I) ^ i / \u2191i ! - (z * I) ^ i / \u2191i !) * I / 2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n[PROOFSTEP]\nreplace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this\n[GOAL]\nz : \u2102\nthis :\n  HasSum ((fun i => ((-z * I) ^ i / \u2191i ! - (z * I) ^ i / \u2191i !) * I / 2) \u2218 \u2191(Nat.divModEquiv 2).symm)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n[PROOFSTEP]\ndsimp [Function.comp] at this \n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (x.fst * 2 + \u2191x.snd) / \u2191(x.fst * 2 + \u2191x.snd)! -\n            (z * I) ^ (x.fst * 2 + \u2191x.snd) / \u2191(x.fst * 2 + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n[PROOFSTEP]\nsimp_rw [\u2190 mul_comm 2 _] at this \n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! -\n            (z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n\u22a2 HasSum (fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I) ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\n[PROOFSTEP]\nrefine' this.prod_fiberwise fun k => _\n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! -\n            (z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\nk : \u2115\n\u22a2 HasSum\n    (fun c =>\n      ((-z * I) ^ (2 * (k, c).fst + \u2191(k, c).snd) / \u2191(2 * (k, c).fst + \u2191(k, c).snd)! -\n            (z * I) ^ (2 * (k, c).fst + \u2191(k, c).snd) / \u2191(2 * (k, c).fst + \u2191(k, c).snd)!) *\n          I /\n        2)\n    ((z * I) ^ (2 * k + 1) / \u2191(2 * k + 1)! / I)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! -\n            (z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\nk : \u2115\n\u22a2 HasSum (fun c => ((-z * I) ^ (2 * k + \u2191c) / \u2191(2 * k + \u2191c)! - (z * I) ^ (2 * k + \u2191c) / \u2191(2 * k + \u2191c)!) * I / 2)\n    ((z * I) ^ (2 * k + 1) / \u2191(2 * k + 1)! / I)\n[PROOFSTEP]\nconvert hasSum_fintype (_ : Fin 2 \u2192 \u2102) using 1\n[GOAL]\ncase h.e'_6\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! -\n            (z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\nk : \u2115\n\u22a2 (z * I) ^ (2 * k + 1) / \u2191(2 * k + 1)! / I =\n    Finset.sum Finset.univ fun b =>\n      ((-z * I) ^ (2 * k + \u2191b) / \u2191(2 * k + \u2191b)! - (z * I) ^ (2 * k + \u2191b) / \u2191(2 * k + \u2191b)!) * I / 2\n[PROOFSTEP]\nrw [Fin.sum_univ_two]\n[GOAL]\ncase h.e'_6\nz : \u2102\nthis :\n  HasSum\n    (fun x =>\n      ((-z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)! -\n            (z * I) ^ (2 * x.fst + \u2191x.snd) / \u2191(2 * x.fst + \u2191x.snd)!) *\n          I /\n        2)\n    ((_root_.exp \u2102 (-z * I) - _root_.exp \u2102 (z * I)) * I / 2)\nk : \u2115\n\u22a2 (z * I) ^ (2 * k + 1) / \u2191(2 * k + 1)! / I =\n    ((-z * I) ^ (2 * k + \u21910) / \u2191(2 * k + \u21910)! - (z * I) ^ (2 * k + \u21910) / \u2191(2 * k + \u21910)!) * I / 2 +\n      ((-z * I) ^ (2 * k + \u21911) / \u2191(2 * k + \u21911)! - (z * I) ^ (2 * k + \u21911) / \u2191(2 * k + \u21911)!) * I / 2\n[PROOFSTEP]\nsimp_rw [Fin.val_zero, Fin.val_one, add_zero, pow_succ', pow_mul, mul_pow, neg_sq, sub_self, zero_mul, zero_div,\n  zero_add, neg_mul, mul_neg, neg_div, \u2190 neg_add', \u2190 two_mul, neg_mul, neg_div, mul_assoc,\n  mul_div_cancel_left _ (two_ne_zero : (2 : \u2102) \u2260 0), Complex.div_I]\n[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (-1) ^ n * z ^ (2 * n) / \u2191(2 * n)!) (cos z)\n[PROOFSTEP]\nconvert Complex.hasSum_cos' z using 1\n[GOAL]\ncase h.e'_5\nz : \u2102\n\u22a2 (fun n => (-1) ^ n * z ^ (2 * n) / \u2191(2 * n)!) = fun n => (z * I) ^ (2 * n) / \u2191(2 * n)!\n[PROOFSTEP]\nsimp_rw [mul_pow, pow_mul, Complex.I_sq, mul_comm]\n[GOAL]\nz : \u2102\n\u22a2 HasSum (fun n => (-1) ^ n * z ^ (2 * n + 1) / \u2191(2 * n + 1)!) (sin z)\n[PROOFSTEP]\nconvert Complex.hasSum_sin' z using 1\n[GOAL]\ncase h.e'_5\nz : \u2102\n\u22a2 (fun n => (-1) ^ n * z ^ (2 * n + 1) / \u2191(2 * n + 1)!) = fun n => (z * I) ^ (2 * n + 1) / \u2191(2 * n + 1)! / I\n[PROOFSTEP]\nsimp_rw [mul_pow, pow_succ', pow_mul, Complex.I_sq, \u2190 mul_assoc, mul_div_assoc, div_right_comm,\n  div_self Complex.I_ne_zero, mul_comm _ ((-1 : \u2102) ^ _), mul_one_div, mul_div_assoc, mul_assoc]\n[GOAL]\nr : \u211d\n\u22a2 HasSum (fun n => (-1) ^ n * r ^ (2 * n) / \u2191(2 * n)!) (cos r)\n[PROOFSTEP]\nexact_mod_cast Complex.hasSum_cos r\n[GOAL]\nr : \u211d\n\u22a2 HasSum (fun n => (-1) ^ n * r ^ (2 * n + 1) / \u2191(2 * n + 1)!) (sin r)\n[PROOFSTEP]\nexact_mod_cast Complex.hasSum_sin r\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Trigonometric.Series", "llama_tokens": 5172, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.5129520162135262}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : HasContDiffBump E\ninst\u271d : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\nx : E\n\u22a2 ContDiffBump.normed f \u03bc (c - x) = ContDiffBump.normed f \u03bc (c + x)\n[PROOFSTEP]\nsimp_rw [f.normed_def, f.sub]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : HasContDiffBump E\ninst\u271d : MeasurableSpace E\nc : E\nf\u271d : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\nf : ContDiffBump 0\nx : E\n\u22a2 ContDiffBump.normed f \u03bc (-x) = ContDiffBump.normed f \u03bc x\n[PROOFSTEP]\nsimp_rw [f.normed_def, f.neg]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 0 < \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\nrefine' (integral_pos_iff_support_of_nonneg f.nonneg' f.integrable).mpr _\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 0 < \u2191\u2191\u03bc (support fun i => \u2191f i)\n[PROOFSTEP]\nrw [f.support_eq]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 0 < \u2191\u2191\u03bc (ball c f.rOut)\n[PROOFSTEP]\nexact measure_ball_pos \u03bc c f.rOut_pos\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 \u222b (x : E), ContDiffBump.normed f \u03bc x \u2202\u03bc = 1\n[PROOFSTEP]\nsimp_rw [ContDiffBump.normed, div_eq_mul_inv, mul_comm (f _), \u2190 smul_eq_mul, integral_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 (\u222b (x : E), \u2191f x \u2202\u03bc)\u207b\u00b9 \u2022 \u222b (x : E), \u2191f x \u2202\u03bc = 1\n[PROOFSTEP]\nexact inv_mul_cancel f.integral_pos.ne'\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 support (ContDiffBump.normed f \u03bc) = ball c f.rOut\n[PROOFSTEP]\nunfold ContDiffBump.normed\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 (support fun x => \u2191f x / \u222b (x : E), \u2191f x \u2202\u03bc) = ball c f.rOut\n[PROOFSTEP]\nrw [support_div, f.support_eq, support_const f.integral_pos.ne', inter_univ]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 tsupport (ContDiffBump.normed f \u03bc) = closedBall c f.rOut\n[PROOFSTEP]\nrw [tsupport, f.support_normed_eq, closure_ball _ f.rOut_pos.ne']\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 HasCompactSupport (ContDiffBump.normed f \u03bc)\n[PROOFSTEP]\nsimp only [HasCompactSupport, f.tsupport_normed_eq (\u03bc := \u03bc), isCompact_closedBall]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u03b9 : Type u_2\n\u03c6 : \u03b9 \u2192 ContDiffBump c\nl : Filter \u03b9\nh\u03c6 : Tendsto (fun i => (\u03c6 i).rOut) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i => support fun x => ContDiffBump.normed (\u03c6 i) \u03bc x) l (smallSets (\ud835\udcdd c))\n[PROOFSTEP]\nsimp_rw [NormedAddCommGroup.tendsto_nhds_zero, Real.norm_eq_abs, abs_eq_self.mpr (\u03c6 _).rOut_pos.le] at h\u03c6 \n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u03b9 : Type u_2\n\u03c6 : \u03b9 \u2192 ContDiffBump c\nl : Filter \u03b9\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, (\u03c6 x).rOut < \u03b5\n\u22a2 Tendsto (fun i => support fun x => ContDiffBump.normed (\u03c6 i) \u03bc x) l (smallSets (\ud835\udcdd c))\n[PROOFSTEP]\nrw [nhds_basis_ball.smallSets.tendsto_right_iff]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u03b9 : Type u_2\n\u03c6 : \u03b9 \u2192 ContDiffBump c\nl : Filter \u03b9\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, (\u03c6 x).rOut < \u03b5\n\u22a2 \u2200 (i : \u211d), 0 < i \u2192 \u2200\u1da0 (x : \u03b9) in l, (support fun x_1 => ContDiffBump.normed (\u03c6 x) \u03bc x_1) \u2208 \ud835\udcab ball c i\n[PROOFSTEP]\nrefine fun \u03b5 h\u03b5 \u21a6 (h\u03c6 \u03b5 h\u03b5).mono fun i hi \u21a6 ?_\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u03b9 : Type u_2\n\u03c6 : \u03b9 \u2192 ContDiffBump c\nl : Filter \u03b9\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, (\u03c6 x).rOut < \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\nhi : (\u03c6 i).rOut < \u03b5\n\u22a2 (support fun x => ContDiffBump.normed (\u03c6 i) \u03bc x) \u2208 \ud835\udcab ball c \u03b5\n[PROOFSTEP]\nrw [(\u03c6 i).support_normed_eq]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u03b9 : Type u_2\n\u03c6 : \u03b9 \u2192 ContDiffBump c\nl : Filter \u03b9\nh\u03c6 : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2200\u1da0 (x : \u03b9) in l, (\u03c6 x).rOut < \u03b5\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\nhi : (\u03c6 i).rOut < \u03b5\n\u22a2 ball c (\u03c6 i).rOut \u2208 \ud835\udcab ball c \u03b5\n[PROOFSTEP]\nexact ball_subset_ball hi.le\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : HasContDiffBump E\ninst\u271d\u2077 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2076 : BorelSpace E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b3 : IsOpenPosMeasure \u03bc\nX : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup X\ninst\u271d\u00b9 : NormedSpace \u211d X\ninst\u271d : CompleteSpace X\nz : X\n\u22a2 \u222b (x : E), ContDiffBump.normed f \u03bc x \u2022 z \u2202\u03bc = z\n[PROOFSTEP]\nsimp_rw [integral_smul_const, f.integral_normed (\u03bc := \u03bc), one_smul]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn)) \u2264 \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\ncalc\n  (\u03bc (closedBall c f.rIn)).toReal = \u222b x in closedBall c f.rIn, 1 \u2202\u03bc := by simp\n  _ = \u222b x in closedBall c f.rIn, f x \u2202\u03bc :=\n    (set_integral_congr (measurableSet_closedBall) (fun x hx \u21a6 (one_of_mem_closedBall f hx).symm))\n  _ \u2264 \u222b x, f x \u2202\u03bc := set_integral_le_integral f.integrable (eventually_of_forall (fun x \u21a6 f.nonneg))\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn)) = \u222b (x : E) in closedBall c f.rIn, 1 \u2202\u03bc\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\n\u22a2 ContDiffBump.normed f \u03bc x \u2264 1 / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn))\n[PROOFSTEP]\nrw [normed_def]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\n\u22a2 \u2191f x / \u222b (x : E), \u2191f x \u2202\u03bc \u2264 1 / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn))\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase hd\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn))\n[PROOFSTEP]\nexact ENNReal.toReal_pos (measure_closedBall_pos _ _ f.rIn_pos).ne' measure_closedBall_lt_top.ne\n[GOAL]\ncase hac\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\n\u22a2 \u2191f x \u2264 1\n[PROOFSTEP]\nexact f.le_one\n[GOAL]\ncase hbd\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn)) \u2264 \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\nexact f.measure_closedBall_le_integral \u03bc\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 \u222b (x : E), \u2191f x \u2202\u03bc \u2264 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\ncalc\n  \u222b x, f x \u2202\u03bc = \u222b x in closedBall c f.rOut, f x \u2202\u03bc :=\n    by\n    apply (set_integral_eq_integral_of_forall_compl_eq_zero (fun x hx \u21a6 ?_)).symm\n    apply f.zero_of_le_dist (le_of_lt _)\n    simpa using hx\n  _ \u2264 \u222b x in closedBall c f.rOut, 1 \u2202\u03bc :=\n    by\n    apply set_integral_mono f.integrable.integrableOn _ (fun x \u21a6 f.le_one)\n    simp [measure_closedBall_lt_top]\n  _ = (\u03bc (closedBall c f.rOut)).toReal := by simp\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 \u222b (x : E), \u2191f x \u2202\u03bc = \u222b (x : E) in closedBall c f.rOut, \u2191f x \u2202\u03bc\n[PROOFSTEP]\napply (set_integral_eq_integral_of_forall_compl_eq_zero (fun x hx \u21a6 ?_)).symm\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\nhx : \u00acx \u2208 closedBall c f.rOut\n\u22a2 \u2191f x = 0\n[PROOFSTEP]\napply f.zero_of_le_dist (le_of_lt _)\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\nx : E\nhx : \u00acx \u2208 closedBall c f.rOut\n\u22a2 f.rOut < dist x c\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 \u222b (x : E) in closedBall c f.rOut, \u2191f x \u2202\u03bc \u2264 \u222b (x : E) in closedBall c f.rOut, 1 \u2202\u03bc\n[PROOFSTEP]\napply set_integral_mono f.integrable.integrableOn _ (fun x \u21a6 f.le_one)\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 IntegrableOn (fun x => 1) (closedBall c f.rOut)\n[PROOFSTEP]\nsimp [measure_closedBall_lt_top]\n[GOAL]\nE : Type u_1\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : HasContDiffBump E\ninst\u271d\u2074 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u00b3 : BorelSpace E\ninst\u271d\u00b2 : FiniteDimensional \u211d E\ninst\u271d\u00b9 : IsLocallyFiniteMeasure \u03bc\ninst\u271d : IsOpenPosMeasure \u03bc\n\u22a2 \u222b (x : E) in closedBall c f.rOut, 1 \u2202\u03bc = ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut)) / K ^ finrank \u211d E \u2264 \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\nhave K_pos : 0 < K := by simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\n\u22a2 0 < K\n[PROOFSTEP]\nsimpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nK_pos : 0 < K\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut)) / K ^ finrank \u211d E \u2264 \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\napply le_trans _ (f.measure_closedBall_le_integral \u03bc)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nK_pos : 0 < K\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut)) / K ^ finrank \u211d E \u2264 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rIn))\n[PROOFSTEP]\nrw [div_le_iff (pow_pos K_pos _), addHaar_closedBall' _ _ f.rIn_pos.le, addHaar_closedBall' _ _ f.rOut_pos.le,\n  ENNReal.toReal_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (pow_nonneg f.rOut_pos.le _),\n  ENNReal.toReal_ofReal (pow_nonneg f.rIn_pos.le _), mul_assoc, mul_comm _ (K ^ _), \u2190 mul_assoc, \u2190 mul_pow,\n  mul_comm _ K]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nK_pos : 0 < K\n\u22a2 f.rOut ^ finrank \u211d E * ENNReal.toReal (\u2191\u2191\u03bc (closedBall 0 1)) \u2264\n    (K * f.rIn) ^ finrank \u211d E * ENNReal.toReal (\u2191\u2191\u03bc (closedBall 0 1))\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h.ha\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nK_pos : 0 < K\n\u22a2 0 \u2264 f.rOut\n[PROOFSTEP]\nexact f.rOut_pos.le\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\n\u22a2 ContDiffBump.normed f \u03bc x \u2264 K ^ finrank \u211d E / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\nhave K_pos : 0 < K := by simpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\n\u22a2 0 < K\n[PROOFSTEP]\nsimpa [f.rIn_pos, not_lt.2 f.rIn_pos.le] using mul_pos_iff.1 (f.rOut_pos.trans_le h)\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\n\u22a2 ContDiffBump.normed f \u03bc x \u2264 K ^ finrank \u211d E / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\nhave : f x / \u222b y, f y \u2202\u03bc \u2264 1 / \u222b y, f y \u2202\u03bc := by\n  gcongr\n  \u00b7 exact f.integral_pos.le\n  \u00b7 exact f.le_one\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\n\u22a2 \u2191f x / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 1 / \u222b (y : E), \u2191f y \u2202\u03bc\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase hc\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\n\u22a2 0 \u2264 \u222b (y : E), \u2191f y \u2202\u03bc\n[PROOFSTEP]\nexact f.integral_pos.le\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\n\u22a2 \u2191f x \u2264 1\n[PROOFSTEP]\nexact f.le_one\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\nthis : \u2191f x / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 1 / \u222b (y : E), \u2191f y \u2202\u03bc\n\u22a2 ContDiffBump.normed f \u03bc x \u2264 K ^ finrank \u211d E / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\napply this.trans\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\nthis : \u2191f x / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 1 / \u222b (y : E), \u2191f y \u2202\u03bc\n\u22a2 1 / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 K ^ finrank \u211d E / ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\nrw [div_le_div_iff f.integral_pos, one_mul, \u2190 div_le_iff' (pow_pos K_pos _)]\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\nthis : \u2191f x / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 1 / \u222b (y : E), \u2191f y \u2202\u03bc\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut)) / K ^ finrank \u211d E \u2264 \u222b (x : E), \u2191f x \u2202\u03bc\n[PROOFSTEP]\nexact f.measure_closedBall_div_le_integral \u03bc K h\n[GOAL]\nE : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : HasContDiffBump E\ninst\u271d\u2075 : MeasurableSpace E\nc : E\nf : ContDiffBump c\nx\u271d : E\nn : \u2115\u221e\n\u03bc : Measure E\ninst\u271d\u2074 : BorelSpace E\ninst\u271d\u00b3 : FiniteDimensional \u211d E\ninst\u271d\u00b2 : IsLocallyFiniteMeasure \u03bc\ninst\u271d\u00b9 : IsOpenPosMeasure \u03bc\ninst\u271d : IsAddHaarMeasure \u03bc\nK : \u211d\nh : f.rOut \u2264 K * f.rIn\nx : E\nK_pos : 0 < K\nthis : \u2191f x / \u222b (y : E), \u2191f y \u2202\u03bc \u2264 1 / \u222b (y : E), \u2191f y \u2202\u03bc\n\u22a2 0 < ENNReal.toReal (\u2191\u2191\u03bc (closedBall c f.rOut))\n[PROOFSTEP]\nexact ENNReal.toReal_pos (measure_closedBall_pos _ _ f.rOut_pos).ne' measure_closedBall_lt_top.ne\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.BumpFunction.Normed", "llama_tokens": 11540, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920068519376, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.5129519972857493}}
{"text": "[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX : (C \u00d7 D) \u00d7 E\n\u22a2 (\ud835\udfed ((C \u00d7 D) \u00d7 E)).obj X = (associator C D E \u22d9 inverseAssociator C D E).obj X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nX : C \u00d7 D \u00d7 E\n\u22a2 (inverseAssociator C D E \u22d9 associator C D E).obj X = (\ud835\udfed (C \u00d7 D \u00d7 E)).obj X\n[PROOFSTEP]\nsimp\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 IsEquivalence (associativity C D E).functor\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\n\u22a2 IsEquivalence (associativity C D E).inverse\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Products.Associator", "llama_tokens": 456, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006919925839875, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.512951988145189}}
{"text": "[GOAL]\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 ofNat (toNat (mk Nat\u03b1.zero f)) = mk Nat\u03b1.zero f\n[PROOFSTEP]\nrw [toNat, ofNat]\n[GOAL]\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 mk Nat\u03b1.zero Empty.elim = mk Nat\u03b1.zero f\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_f\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\n\u22a2 Empty.elim = f\n[PROOFSTEP]\next x\n[GOAL]\ncase e_f.h\nf : Nat\u03b2 Nat\u03b1.zero \u2192 WType Nat\u03b2\nx : Empty\n\u22a2 Empty.elim x = f x\n[PROOFSTEP]\ncases x\n[GOAL]\nf : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2\n\u22a2 ofNat (toNat (mk Nat\u03b1.succ f)) = mk Nat\u03b1.succ f\n[PROOFSTEP]\nsimp only [toNat, ofNat, leftInverse_nat (f ()), mk.injEq, heq_eq_eq, true_and]\n[GOAL]\nf : Nat\u03b2 Nat\u03b1.succ \u2192 WType Nat\u03b2\n\u22a2 (fun x => f ()) = f\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 toNat (ofNat (Nat.succ n)) = Nat.succ n\n[PROOFSTEP]\nrw [ofNat, toNat, rightInverse_nat n]\n[GOAL]\n\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\n\u22a2 ofList \u03b3 (toList \u03b3 (mk List\u03b1.nil f)) = mk List\u03b1.nil f\n[PROOFSTEP]\nsimp only [toList, ofList, mk.injEq, heq_eq_eq, true_and]\n[GOAL]\n\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\n\u22a2 PEmpty.elim = f\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b3 : Type u\nf : List\u03b2 \u03b3 List\u03b1.nil \u2192 WType (List\u03b2 \u03b3)\nx : PEmpty\n\u22a2 PEmpty.elim x = f x\n[PROOFSTEP]\ncases x\n[GOAL]\n\u03b3 : Type u\nx : \u03b3\nf : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3)\n\u22a2 ofList \u03b3 (toList \u03b3 (mk (List\u03b1.cons x) f)) = mk (List\u03b1.cons x) f\n[PROOFSTEP]\nsimp only [ofList, leftInverse_list (f PUnit.unit), mk.injEq, heq_eq_eq, true_and]\n[GOAL]\n\u03b3 : Type u\nx : \u03b3\nf : List\u03b2 \u03b3 (List\u03b1.cons x) \u2192 WType (List\u03b2 \u03b3)\n\u22a2 (fun x => f PUnit.unit) = f\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b3 : Type u\nhd : \u03b3\ntl : List \u03b3\n\u22a2 toList \u03b3 (ofList \u03b3 (hd :: tl)) = hd :: tl\n[PROOFSTEP]\nsimp only [toList, rightInverse_list tl]\n", "meta": {"mathlib_filename": "Mathlib.Data.W.Constructions", "llama_tokens": 833, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149978955811, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.5127221862587154}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\n\u22a2 NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure K) (C + \u03b5)\n[PROOFSTEP]\nrintro (h : H)\n  (h_in : h \u2208 K.topologicalClosure)\n    -- We first get rid of the easy case where `h = 0`.\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nby_cases hyp_h : h = 0\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : h = 0\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nrw [hyp_h]\n[GOAL]\ncase pos\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : h = 0\n\u22a2 \u2203 g, \u2191f g = 0 \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u20160\u2016\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : h = 0\n\u22a2 \u2191f 0 = 0 \u2227 \u20160\u2016 \u2264 (C + \u03b5) * \u20160\u2016\n[PROOFSTEP]\nsimp\n  /- The desired preimage will be constructed as the sum of a series. Convergence of\n      the series will be guaranteed by completeness of `G`. We first write `h` as the sum\n      of a sequence `v` of elements of `K` which starts close to `h` and then quickly goes to zero.\n      The sequence `b` below quantifies this. -/\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nset b : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nhave b_pos : \u2200 i, 0 < b i := by\n  intro i\n  field_simp [hC]\n  exact div_pos (mul_pos h\u03b5 (norm_pos_iff.mpr hyp_h)) (mul_pos (by norm_num : (0 : \u211d) < 2 ^ i * 2) hC)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\n\u22a2 \u2200 (i : \u2115), 0 < b i\n[PROOFSTEP]\nintro i\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\ni : \u2115\n\u22a2 0 < b i\n[PROOFSTEP]\nfield_simp [hC]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\ni : \u2115\n\u22a2 0 < \u03b5 * \u2016h\u2016 / (2 ^ i * 2 * C)\n[PROOFSTEP]\nexact div_pos (mul_pos h\u03b5 (norm_pos_iff.mpr hyp_h)) (mul_pos (by norm_num : (0 : \u211d) < 2 ^ i * 2) hC)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\ni : \u2115\n\u22a2 0 < 2 ^ i * 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nobtain\n  \u27e8v : \u2115 \u2192 H, lim_v : Tendsto (fun n : \u2115 => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h), v_in : \u2200 n, v n \u2208 K, hv\u2080 :\n    \u2016v 0 - h\u2016 < b 0, hv : \u2200 n > 0, \u2016v n\u2016 < b n\u27e9 :=\n  controlled_sum_of_mem_closure h_in b_pos\n[GOAL]\ncase neg.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nhave : \u2200 n, \u2203 m' : G, f m' = v n \u2227 \u2016m'\u2016 \u2264 C * \u2016v n\u2016 := fun n : \u2115 => hyp (v n) (v_in n)\n[GOAL]\ncase neg.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nthis : \u2200 (n : \u2115), \u2203 m', \u2191f m' = v n \u2227 \u2016m'\u2016 \u2264 C * \u2016v n\u2016\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nchoose u hu hnorm_u using this\n[GOAL]\ncase neg.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nset s : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n[GOAL]\ncase neg.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nhave : CauchySeq s :=\n  by\n  apply NormedAddCommGroup.cauchy_series_of_le_geometric'' (by norm_num) one_half_lt_one\n  rintro n (hn : n \u2265 1)\n  calc\n    \u2016u n\u2016 \u2264 C * \u2016v n\u2016 := hnorm_u n\n    _ \u2264 C * b n := by gcongr; exact (hv _ <| Nat.succ_le_iff.mp hn).le\n    _ = (1 / 2) ^ n * (\u03b5 * \u2016h\u2016 / 2) := by simp [mul_div_cancel' _ hC.ne.symm]\n    _ = \u03b5 * \u2016h\u2016 / 2 * (1 / 2) ^ n :=\n      mul_comm _\n        _\n          -- We now show that the limit `g` of `s` is the desired preimage.\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 CauchySeq s\n[PROOFSTEP]\napply NormedAddCommGroup.cauchy_series_of_le_geometric'' (by norm_num) one_half_lt_one\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 \u2200 (n : \u2115), n \u2265 ?m.14929 \u2192 \u2016u n\u2016 \u2264 ?m.14927 * (1 / 2) ^ n\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 \u211d\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 \u2115\n[PROOFSTEP]\nrintro n (hn : n \u2265 1)\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nn : \u2115\nhn : n \u2265 1\n\u22a2 \u2016u n\u2016 \u2264 ?m.14927 * (1 / 2) ^ n\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\n\u22a2 \u211d\n[PROOFSTEP]\ncalc\n  \u2016u n\u2016 \u2264 C * \u2016v n\u2016 := hnorm_u n\n  _ \u2264 C * b n := by gcongr; exact (hv _ <| Nat.succ_le_iff.mp hn).le\n  _ = (1 / 2) ^ n * (\u03b5 * \u2016h\u2016 / 2) := by simp [mul_div_cancel' _ hC.ne.symm]\n  _ = \u03b5 * \u2016h\u2016 / 2 * (1 / 2) ^ n :=\n    mul_comm _\n      _\n        -- We now show that the limit `g` of `s` is the desired preimage.\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nn : \u2115\nhn : n \u2265 1\n\u22a2 C * \u2016v n\u2016 \u2264 C * b n\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nn : \u2115\nhn : n \u2265 1\n\u22a2 \u2016v n\u2016 \u2264 b n\n[PROOFSTEP]\nexact (hv _ <| Nat.succ_le_iff.mp hn).le\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nn : \u2115\nhn : n \u2265 1\n\u22a2 C * b n = (1 / 2) ^ n * (\u03b5 * \u2016h\u2016 / 2)\n[PROOFSTEP]\nsimp [mul_div_cancel' _ hC.ne.symm]\n[GOAL]\ncase neg.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nobtain \u27e8g : G, hg\u27e9 := cauchySeq_tendsto_of_complete this\n[GOAL]\ncase neg.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nrefine' \u27e8g, _, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2191f g = h\n[PROOFSTEP]\nhave : f \u2218 s = fun n => \u2211 k in range (n + 1), v k := by\n  ext n\n  simp [map_sum, hu]\n    /- In the above equality, the left-hand-side converges to `f g` by continuity of `f` and\n          definition of `g` while the right-hand-side converges to `h` by construction of `v` so\n          `g` is indeed a preimage of `h`. -/\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2191f \u2218 s = fun n => \u2211 k in range (n + 1), v k\n[PROOFSTEP]\next n\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\n\u22a2 (\u2191f \u2218 s) n = \u2211 k in range (n + 1), v k\n[PROOFSTEP]\nsimp [map_sum, hu]\n  /- In the above equality, the left-hand-side converges to `f g` by continuity of `f` and\n        definition of `g` while the right-hand-side converges to `h` by construction of `v` so\n        `g` is indeed a preimage of `h`. -/\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nthis : \u2191f \u2218 s = fun n => \u2211 k in range (n + 1), v k\n\u22a2 \u2191f g = h\n[PROOFSTEP]\nrw [\u2190 this] at lim_v \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.refine'_1\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nlim_v : Tendsto (\u2191f \u2218 s) atTop (\ud835\udcdd h)\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nthis : \u2191f \u2218 s = fun n => \u2211 k in range (n + 1), v k\n\u22a2 \u2191f g = h\n[PROOFSTEP]\nexact tendsto_nhds_unique ((f.continuous.tendsto g).comp hg) lim_v\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nsuffices : \u2200 n, \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.refine'_2\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nthis : \u2200 (n : \u2115), \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n\u22a2 \u2016g\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\ncase this\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2200 (n : \u2115), \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nexact le_of_tendsto' (continuous_norm.continuousAt.tendsto.comp hg) this\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\n\u22a2 \u2200 (n : \u2115), \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nintro n\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\n\u22a2 \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nhave hnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016 :=\n  by\n  have :=\n    calc\n      \u2016v 0\u2016 \u2264 \u2016h\u2016 + \u2016v 0 - h\u2016 := norm_le_insert' _ _\n      _ \u2264 \u2016h\u2016 + b 0 := by gcongr\n  calc\n    \u2016u 0\u2016 \u2264 C * \u2016v 0\u2016 := hnorm_u 0\n    _ \u2264 C * (\u2016h\u2016 + b 0) := by gcongr\n    _ = C * b 0 + C * \u2016h\u2016 := by rw [add_comm, mul_add]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\n\u22a2 \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n[PROOFSTEP]\nhave :=\n  calc\n    \u2016v 0\u2016 \u2264 \u2016h\u2016 + \u2016v 0 - h\u2016 := norm_le_insert' _ _\n    _ \u2264 \u2016h\u2016 + b 0 := by gcongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\n\u22a2 \u2016h\u2016 + \u2016v 0 - h\u2016 \u2264 \u2016h\u2016 + b 0\n[PROOFSTEP]\ngcongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nthis : \u2016v 0\u2016 \u2264 \u2016h\u2016 + b 0\n\u22a2 \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n[PROOFSTEP]\ncalc\n  \u2016u 0\u2016 \u2264 C * \u2016v 0\u2016 := hnorm_u 0\n  _ \u2264 C * (\u2016h\u2016 + b 0) := by gcongr\n  _ = C * b 0 + C * \u2016h\u2016 := by rw [add_comm, mul_add]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nthis : \u2016v 0\u2016 \u2264 \u2016h\u2016 + b 0\n\u22a2 C * \u2016v 0\u2016 \u2264 C * (\u2016h\u2016 + b 0)\n[PROOFSTEP]\ngcongr\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nthis : \u2016v 0\u2016 \u2264 \u2016h\u2016 + b 0\n\u22a2 C * (\u2016h\u2016 + b 0) = C * b 0 + C * \u2016h\u2016\n[PROOFSTEP]\nrw [add_comm, mul_add]\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n\u22a2 \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nhave : (\u2211 k in range (n + 1), C * b k) \u2264 \u03b5 * \u2016h\u2016 :=\n  calc\n    (\u2211 k in range (n + 1), C * b k) = (\u2211 k in range (n + 1), (1 / 2 : \u211d) ^ k) * (\u03b5 * \u2016h\u2016 / 2) := by\n      simp only [mul_div_cancel' _ hC.ne.symm, \u2190 sum_mul]\n    _ \u2264 2 * (\u03b5 * \u2016h\u2016 / 2) := by gcongr; apply sum_geometric_two_le\n    _ = \u03b5 * \u2016h\u2016 := mul_div_cancel' _ two_ne_zero\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n\u22a2 \u2211 k in range (n + 1), C * b k = (\u2211 k in range (n + 1), (1 / 2) ^ k) * (\u03b5 * \u2016h\u2016 / 2)\n[PROOFSTEP]\nsimp only [mul_div_cancel' _ hC.ne.symm, \u2190 sum_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n\u22a2 (\u2211 k in range (n + 1), (1 / 2) ^ k) * (\u03b5 * \u2016h\u2016 / 2) \u2264 2 * (\u03b5 * \u2016h\u2016 / 2)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\n\u22a2 \u2211 k in range (n + 1), (1 / 2) ^ k \u2264 2\n[PROOFSTEP]\napply sum_geometric_two_le\n[GOAL]\ncase this\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 \u2016s n\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\ncalc\n  \u2016s n\u2016 \u2264 \u2211 k in range (n + 1), \u2016u k\u2016 := norm_sum_le _ _\n  _ = (\u2211 k in range n, \u2016u (k + 1)\u2016) + \u2016u 0\u2016 := (sum_range_succ' _ _)\n  _ \u2264 (\u2211 k in range n, C * \u2016v (k + 1)\u2016) + \u2016u 0\u2016 := by gcongr; apply hnorm_u\n  _ \u2264 (\u2211 k in range n, C * b (k + 1)) + (C * b 0 + C * \u2016h\u2016) := by gcongr with k; exact (hv _ k.succ_pos).le\n  _ = (\u2211 k in range (n + 1), C * b k) + C * \u2016h\u2016 := by rw [\u2190 add_assoc, sum_range_succ']\n  _ \u2264 (C + \u03b5) * \u2016h\u2016 := by\n    rw [add_comm, add_mul]\n    apply add_le_add_left this\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 \u2211 k in range n, \u2016u (k + 1)\u2016 + \u2016u 0\u2016 \u2264 \u2211 k in range n, C * \u2016v (k + 1)\u2016 + \u2016u 0\u2016\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase bc.h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\ni\u271d : \u2115\na\u271d : i\u271d \u2208 range n\n\u22a2 \u2016u (i\u271d + 1)\u2016 \u2264 C * \u2016v (i\u271d + 1)\u2016\n[PROOFSTEP]\napply hnorm_u\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 \u2211 k in range n, C * \u2016v (k + 1)\u2016 + \u2016u 0\u2016 \u2264 \u2211 k in range n, C * b (k + 1) + (C * b 0 + C * \u2016h\u2016)\n[PROOFSTEP]\ngcongr with k\n[GOAL]\ncase h\u2081.h.h\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\nk : \u2115\na\u271d : k \u2208 range n\n\u22a2 \u2016v (k + 1)\u2016 \u2264 b (k + 1)\n[PROOFSTEP]\nexact (hv _ k.succ_pos).le\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 \u2211 k in range n, C * b (k + 1) + (C * b 0 + C * \u2016h\u2016) = \u2211 k in range (n + 1), C * b k + C * \u2016h\u2016\n[PROOFSTEP]\nrw [\u2190 add_assoc, sum_range_succ']\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 \u2211 k in range (n + 1), C * b k + C * \u2016h\u2016 \u2264 (C + \u03b5) * \u2016h\u2016\n[PROOFSTEP]\nrw [add_comm, add_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : CompleteSpace G\nH : Type u_2\ninst\u271d : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : NormedAddGroupHom.SurjectiveOnWith f K C\nh : H\nh_in : h \u2208 AddSubgroup.topologicalClosure K\nhyp_h : \u00ach = 0\nb : \u2115 \u2192 \u211d := fun i => (1 / 2) ^ i * (\u03b5 * \u2016h\u2016 / 2) / C\nb_pos : \u2200 (i : \u2115), 0 < b i\nv : \u2115 \u2192 H\nlim_v : Tendsto (fun n => \u2211 k in range (n + 1), v k) atTop (\ud835\udcdd h)\nv_in : \u2200 (n : \u2115), v n \u2208 K\nhv\u2080 : \u2016v 0 - h\u2016 < b 0\nhv : \u2200 (n : \u2115), n > 0 \u2192 \u2016v n\u2016 < b n\nu : \u2115 \u2192 G\nhu : \u2200 (n : \u2115), \u2191f (u n) = v n\nhnorm_u : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 C * \u2016v n\u2016\ns : \u2115 \u2192 G := fun n => \u2211 k in range (n + 1), u k\nthis\u271d : CauchySeq s\ng : G\nhg : Tendsto s atTop (\ud835\udcdd g)\nn : \u2115\nhnorm\u2080 : \u2016u 0\u2016 \u2264 C * b 0 + C * \u2016h\u2016\nthis : \u2211 k in range (n + 1), C * b k \u2264 \u03b5 * \u2016h\u2016\n\u22a2 C * \u2016h\u2016 + \u2211 k in range (n + 1), C * b k \u2264 C * \u2016h\u2016 + \u03b5 * \u2016h\u2016\n[PROOFSTEP]\napply add_le_add_left this\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (k : K), \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\n\u22a2 NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure (NormedAddGroupHom.range j)) (C + \u03b5)\n[PROOFSTEP]\nreplace hyp : \u2200 h \u2208 j.range, \u2203 g, f g = h \u2227 \u2016g\u2016 \u2264 C * \u2016h\u2016\n[GOAL]\ncase hyp\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (k : K), \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\n\u22a2 \u2200 (h : H), h \u2208 NormedAddGroupHom.range j \u2192 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 C * \u2016h\u2016\n[PROOFSTEP]\nintro h h_in\n[GOAL]\ncase hyp\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (k : K), \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\nh : H\nh_in : h \u2208 NormedAddGroupHom.range j\n\u22a2 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 C * \u2016h\u2016\n[PROOFSTEP]\nrcases(j.mem_range _).mp h_in with \u27e8k, rfl\u27e9\n[GOAL]\ncase hyp.intro\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (k : K), \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\nk : K\nh_in : \u2191j k \u2208 NormedAddGroupHom.range j\n\u22a2 \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016\u2191j k\u2016\n[PROOFSTEP]\nrw [hj]\n[GOAL]\ncase hyp.intro\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (k : K), \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\nk : K\nh_in : \u2191j k \u2208 NormedAddGroupHom.range j\n\u22a2 \u2203 g, \u2191f g = \u2191j k \u2227 \u2016g\u2016 \u2264 C * \u2016k\u2016\n[PROOFSTEP]\nexact hyp k\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : CompleteSpace G\nH : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst\u271d : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : \u2200 (x : K), \u2016\u2191j x\u2016 = \u2016x\u2016\nC \u03b5 : \u211d\nhC : 0 < C\nh\u03b5 : 0 < \u03b5\nhyp : \u2200 (h : H), h \u2208 NormedAddGroupHom.range j \u2192 \u2203 g, \u2191f g = h \u2227 \u2016g\u2016 \u2264 C * \u2016h\u2016\n\u22a2 NormedAddGroupHom.SurjectiveOnWith f (AddSubgroup.topologicalClosure (NormedAddGroupHom.range j)) (C + \u03b5)\n[PROOFSTEP]\nexact controlled_closure_of_complete hC h\u03b5 hyp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Normed.Group.ControlledClosure", "llama_tokens": 25292, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.679178686187839, "lm_q1q2_score": 0.5127221614743148}}
{"text": "[GOAL]\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b1 : Sort u_3\nh : Finite \u03b1\nf : \u03b1 \u2243 \u03b2\n\u22a2 Finite \u03b2\n[PROOFSTEP]\ncases' h with n e\n[GOAL]\ncase intro\n\u03b1\u271d : Sort u_1\n\u03b2 : Sort u_2\n\u03b1 : Sort u_3\nf : \u03b1 \u2243 \u03b2\nn : \u2115\ne : \u03b1 \u2243 Fin n\n\u22a2 Finite \u03b2\n[PROOFSTEP]\nexact Finite.intro (f.symm.trans e)\n", "meta": {"mathlib_filename": "Mathlib.Data.Finite.Defs", "llama_tokens": 145, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.512582924239928}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i < r i\n[PROOFSTEP]\nrcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with \u27e8v, hsv, hvc, hcv\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i < r i\n[PROOFSTEP]\nhave := fun i => exists_lt_subset_ball (hvc i) (hcv i)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\nthis : \u2200 (i : \u03b9), \u2203 r', r' < r i \u2227 v i \u2286 ball (c i) r'\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i < r i\n[PROOFSTEP]\nchoose r' hlt hsub using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\nr' : \u03b9 \u2192 \u211d\nhlt : \u2200 (i : \u03b9), r' i < r i\nhsub : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r' i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i < r i\n[PROOFSTEP]\nexact \u27e8r', hsv.trans <| iUnion_mono <| hsub, hlt\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhr : \u2200 (i : \u03b9), 0 < r i\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i \u2208 Ioo 0 (r i)\n[PROOFSTEP]\nrcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with \u27e8v, hsv, hvc, hcv\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhr : \u2200 (i : \u03b9), 0 < r i\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i \u2208 Ioo 0 (r i)\n[PROOFSTEP]\nhave := fun i => exists_pos_lt_subset_ball (hr i) (hvc i) (hcv i)\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhr : \u2200 (i : \u03b9), 0 < r i\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\nthis : \u2200 (i : \u03b9), \u2203 r', r' \u2208 Ioo 0 (r i) \u2227 v i \u2286 ball (c i) r'\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i \u2208 Ioo 0 (r i)\n[PROOFSTEP]\nchoose r' hlt hsub using this\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nr : \u03b9 \u2192 \u211d\nhr : \u2200 (i : \u03b9), 0 < r i\nhs : IsClosed s\nuf : \u2200 (x : \u03b1), x \u2208 s \u2192 Set.Finite {i | x \u2208 ball (c i) (r i)}\nus : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nv : \u03b9 \u2192 Set \u03b1\nhsv : s \u2286 iUnion v\nhvc : \u2200 (i : \u03b9), IsClosed (v i)\nhcv : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r i)\nr' : \u03b9 \u2192 \u211d\nhlt : \u2200 (i : \u03b9), r' i \u2208 Ioo 0 (r i)\nhsub : \u2200 (i : \u03b9), v i \u2286 ball (c i) (r' i)\n\u22a2 \u2203 r', s \u2286 \u22c3 (i : \u03b9), ball (c i) (r' i) \u2227 \u2200 (i : \u03b9), r' i \u2208 Ioo 0 (r i)\n[PROOFSTEP]\nexact \u27e8r', hsv.trans <| iUnion_mono hsub, hlt\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\n\u22a2 \u2203 \u03b9 c r r',\n    (\u2200 (i : \u03b9), c i \u2208 s \u2227 0 < r i \u2227 r i < r' i \u2227 r' i < R (c i)) \u2227\n      (LocallyFinite fun i => ball (c i) (r' i)) \u2227 s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n[PROOFSTEP]\nhave : \u2200 x \u2208 s, (\ud835\udcdd x).HasBasis (fun r : \u211d => 0 < r \u2227 r < R x) fun r => ball x r := fun x hx =>\n  nhds_basis_uniformity (uniformity_basis_dist_lt (hR x hx))\n[GOAL]\n\u03b1 : Type u\n\u03b9 : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc : \u03b9 \u2192 \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r \u2227 r < R x) fun r => ball x r\n\u22a2 \u2203 \u03b9 c r r',\n    (\u2200 (i : \u03b9), c i \u2208 s \u2227 0 < r i \u2227 r i < r' i \u2227 r' i < R (c i)) \u2227\n      (LocallyFinite fun i => ball (c i) (r' i)) \u2227 s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n[PROOFSTEP]\nrcases refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set hs this with \u27e8\u03b9, c, r', hr', hsub', hfin\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b9\u271d : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc\u271d : \u03b9\u271d \u2192 \u03b1\nx : \u03b1\nr : \u211d\ns : Set \u03b1\nhs : IsClosed s\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r \u2227 r < R x) fun r => ball x r\n\u03b9 : Type u\nc : \u03b9 \u2192 \u03b1\nr' : \u03b9 \u2192 \u211d\nhr' : \u2200 (a : \u03b9), c a \u2208 s \u2227 0 < r' a \u2227 r' a < R (c a)\nhsub' : s \u2286 \u22c3 (a : \u03b9), ball (c a) (r' a)\nhfin : LocallyFinite fun a => ball (c a) (r' a)\n\u22a2 \u2203 \u03b9 c r r',\n    (\u2200 (i : \u03b9), c i \u2208 s \u2227 0 < r i \u2227 r i < r' i \u2227 r' i < R (c i)) \u2227\n      (LocallyFinite fun i => ball (c i) (r' i)) \u2227 s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n[PROOFSTEP]\nrcases exists_subset_iUnion_ball_radius_pos_lt (fun i => (hr' i).2.1) hs (fun x _ => hfin.point_finite x) hsub' with\n  \u27e8r, hsub, hlt\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b9\u271d : Type v\ninst\u271d\u00b9 : MetricSpace \u03b1\ninst\u271d : ProperSpace \u03b1\nc\u271d : \u03b9\u271d \u2192 \u03b1\nx : \u03b1\nr\u271d : \u211d\ns : Set \u03b1\nhs : IsClosed s\nR : \u03b1 \u2192 \u211d\nhR : \u2200 (x : \u03b1), x \u2208 s \u2192 0 < R x\nthis : \u2200 (x : \u03b1), x \u2208 s \u2192 Filter.HasBasis (\ud835\udcdd x) (fun r => 0 < r \u2227 r < R x) fun r => ball x r\n\u03b9 : Type u\nc : \u03b9 \u2192 \u03b1\nr' : \u03b9 \u2192 \u211d\nhr' : \u2200 (a : \u03b9), c a \u2208 s \u2227 0 < r' a \u2227 r' a < R (c a)\nhsub' : s \u2286 \u22c3 (a : \u03b9), ball (c a) (r' a)\nhfin : LocallyFinite fun a => ball (c a) (r' a)\nr : \u03b9 \u2192 \u211d\nhsub : s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\nhlt : \u2200 (i : \u03b9), r i \u2208 Ioo 0 (r' i)\n\u22a2 \u2203 \u03b9 c r r',\n    (\u2200 (i : \u03b9), c i \u2208 s \u2227 0 < r i \u2227 r i < r' i \u2227 r' i < R (c i)) \u2227\n      (LocallyFinite fun i => ball (c i) (r' i)) \u2227 s \u2286 \u22c3 (i : \u03b9), ball (c i) (r i)\n[PROOFSTEP]\nexact \u27e8\u03b9, c, r, r', fun i => \u27e8(hr' i).1, (hlt i).1, (hlt i).2, (hr' i).2.2\u27e9, hfin, hsub\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.ShrinkingLemma", "llama_tokens": 3879, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333246035907932, "lm_q2_score": 0.615087848460224, "lm_q1q2_score": 0.51256783749163}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 \u2203 a b, (\u2200 {d : A}, d \u2223 a \u2192 d \u2223 \u2191b \u2192 IsUnit d) \u2227 mk' K a b = x\n[PROOFSTEP]\nobtain \u27e8\u27e8b, b_nonzero\u27e9, a, hab\u27e9 := exists_integer_multiple (nonZeroDivisors A) x\n[GOAL]\ncase intro.mk.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nb : A\nb_nonzero : b \u2208 nonZeroDivisors A\na : A\nhab : \u2191(algebraMap A K) a = \u2191{ val := b, property := b_nonzero } \u2022 x\n\u22a2 \u2203 a b, (\u2200 {d : A}, d \u2223 a \u2192 d \u2223 \u2191b \u2192 IsUnit d) \u2227 mk' K a b = x\n[PROOFSTEP]\nobtain \u27e8a', b', c', no_factor, rfl, rfl\u27e9 :=\n  UniqueFactorizationMonoid.exists_reduced_factors' a b (mem_nonZeroDivisors_iff_ne_zero.mp b_nonzero)\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : \u2200 {d : A}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nb_nonzero : c' * b' \u2208 nonZeroDivisors A\nhab : \u2191(algebraMap A K) (c' * a') = \u2191{ val := c' * b', property := b_nonzero } \u2022 x\n\u22a2 \u2203 a b, (\u2200 {d : A}, d \u2223 a \u2192 d \u2223 \u2191b \u2192 IsUnit d) \u2227 mk' K a b = x\n[PROOFSTEP]\nobtain \u27e8_, b'_nonzero\u27e9 := mul_mem_nonZeroDivisors.mp b_nonzero\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : \u2200 {d : A}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nb_nonzero : c' * b' \u2208 nonZeroDivisors A\nhab : \u2191(algebraMap A K) (c' * a') = \u2191{ val := c' * b', property := b_nonzero } \u2022 x\nleft\u271d : c' \u2208 nonZeroDivisors A\nb'_nonzero : b' \u2208 nonZeroDivisors A\n\u22a2 \u2203 a b, (\u2200 {d : A}, d \u2223 a \u2192 d \u2223 \u2191b \u2192 IsUnit d) \u2227 mk' K a b = x\n[PROOFSTEP]\nrefine' \u27e8a', \u27e8b', b'_nonzero\u27e9, no_factor, _\u27e9\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : \u2200 {d : A}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nb_nonzero : c' * b' \u2208 nonZeroDivisors A\nhab : \u2191(algebraMap A K) (c' * a') = \u2191{ val := c' * b', property := b_nonzero } \u2022 x\nleft\u271d : c' \u2208 nonZeroDivisors A\nb'_nonzero : b' \u2208 nonZeroDivisors A\n\u22a2 mk' K a' { val := b', property := b'_nonzero } = x\n[PROOFSTEP]\nrefine' mul_left_cancel\u2080 (IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors b_nonzero) _\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : \u2200 {d : A}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nb_nonzero : c' * b' \u2208 nonZeroDivisors A\nhab : \u2191(algebraMap A K) (c' * a') = \u2191{ val := c' * b', property := b_nonzero } \u2022 x\nleft\u271d : c' \u2208 nonZeroDivisors A\nb'_nonzero : b' \u2208 nonZeroDivisors A\n\u22a2 \u2191(algebraMap A K) (c' * b') * mk' K a' { val := b', property := b'_nonzero } = \u2191(algebraMap A K) (c' * b') * x\n[PROOFSTEP]\nsimp only [Subtype.coe_mk, RingHom.map_mul, Algebra.smul_def] at *\n[GOAL]\ncase intro.mk.intro.intro.intro.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\na' b' c' : A\nno_factor : \u2200 {d : A}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d\nb_nonzero : c' * b' \u2208 nonZeroDivisors A\nleft\u271d : c' \u2208 nonZeroDivisors A\nb'_nonzero : b' \u2208 nonZeroDivisors A\nhab : \u2191(algebraMap A K) c' * \u2191(algebraMap A K) a' = \u2191(algebraMap A K) c' * \u2191(algebraMap A K) b' * x\n\u22a2 \u2191(algebraMap A K) c' * \u2191(algebraMap A K) b' * mk' K a' { val := b', property := b'_nonzero } =\n    \u2191(algebraMap A K) c' * \u2191(algebraMap A K) b' * x\n[PROOFSTEP]\nerw [\u2190 hab, mul_assoc, mk'_spec' _ a' \u27e8b', b'_nonzero\u27e9]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 \u2191(algebraMap A K) (num A x) / \u2191(algebraMap A K) \u2191(den A x) = x\n[PROOFSTEP]\nrw [\u2190 mk'_eq_div]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\n\u22a2 mk' K (num A x) (den A x) = x\n[PROOFSTEP]\napply mk'_num_den\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : x * \u2191(algebraMap A K) \u2191(den A y) = \u2191(algebraMap A K) (num A y)\n\u22a2 x = y\n[PROOFSTEP]\nsimpa only [mk'_num_den] using eq_mk'_iff_mul_eq.mpr h\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : x = y\n\u22a2 x = mk' K (num A y) (den A y)\n[PROOFSTEP]\nrw [h, mk'_num_den]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : y * \u2191(algebraMap A K) \u2191(den A x) = \u2191(algebraMap A K) (num A x)\n\u22a2 x = y\n[PROOFSTEP]\nsimpa only [eq_comm, mk'_num_den] using eq_mk'_iff_mul_eq.mpr h\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : x = y\n\u22a2 y = mk' K (num A x) (den A x)\n[PROOFSTEP]\nrw [h, mk'_num_den]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : num A y * \u2191(den A x) = num A x * \u2191(den A y)\n\u22a2 x = y\n[PROOFSTEP]\nsimpa only [mk'_num_den] using mk'_eq_of_eq' (S := K) h\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx y : K\nh : x = y\n\u22a2 num A y * \u2191(den A x) = num A x * \u2191(den A y)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nh : num A x = 0\n\u22a2 0 * \u2191(algebraMap (?m.160854 h) K) \u2191(den (?m.160854 h) x) = \u2191(algebraMap (?m.160854 h) K) (num (?m.160854 h) x)\n[PROOFSTEP]\nrw [zero_mul, h, RingHom.map_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nh : IsUnit \u2191(den A x)\n\u22a2 IsInteger A x\n[PROOFSTEP]\ncases' h with d hd\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\n\u22a2 IsInteger A x\n[PROOFSTEP]\nhave d_ne_zero : algebraMap A K (den A x) \u2260 0 := IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A x).2\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\nd_ne_zero : \u2191(algebraMap A K) \u2191(den A x) \u2260 0\n\u22a2 IsInteger A x\n[PROOFSTEP]\nuse\u2191d\u207b\u00b9 * num A x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\nd_ne_zero : \u2191(algebraMap A K) \u2191(den A x) \u2260 0\n\u22a2 \u2191(algebraMap A K) (\u2191d\u207b\u00b9 * num A x) = x\n[PROOFSTEP]\nrefine' _root_.trans _ (mk'_num_den A x)\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\nd_ne_zero : \u2191(algebraMap A K) \u2191(den A x) \u2260 0\n\u22a2 \u2191(algebraMap A K) (\u2191d\u207b\u00b9 * num A x) = mk' K (num A x) (den A x)\n[PROOFSTEP]\nrw [map_mul, map_units_inv, hd]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\nd_ne_zero : \u2191(algebraMap A K) \u2191(den A x) \u2260 0\n\u22a2 (\u2191(algebraMap A K) \u2191(den A x))\u207b\u00b9 * \u2191(algebraMap A K) (num A x) = mk' K (num A x) (den A x)\n[PROOFSTEP]\napply mul_left_cancel\u2080 d_ne_zero\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2079 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : Algebra R S\nP : Type u_3\ninst\u271d\u2076 : CommRing P\nA : Type u_4\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\nK : Type u_5\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nx : K\nd : A\u02e3\nhd : \u2191d = \u2191(den A x)\nd_ne_zero : \u2191(algebraMap A K) \u2191(den A x) \u2260 0\n\u22a2 \u2191(algebraMap A K) \u2191(den A x) * ((\u2191(algebraMap A K) \u2191(den A x))\u207b\u00b9 * \u2191(algebraMap A K) (num A x)) =\n    \u2191(algebraMap A K) \u2191(den A x) * mk' K (num A x) (den A x)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_inv_cancel d_ne_zero, one_mul, mk'_spec']\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.NumDen", "llama_tokens": 6359, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.5125615373569877}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedAddCommGroup E\nc : InnerProductSpace \ud835\udd5c E\nx : E\n\u22a2 0 \u2264 \u2191re (inner x x)\n[PROOFSTEP]\nrw [\u2190 InnerProductSpace.norm_sq_eq_inner]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedAddCommGroup E\nc : InnerProductSpace \ud835\udd5c E\nx : E\n\u22a2 0 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\napply sq_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b9 : IsROrC \ud835\udd5c\ninst\u271d : NormedAddCommGroup E\nc : InnerProductSpace \ud835\udd5c E\nx : E\nhx : inner x x = 0\n\u22a2 \u2016x\u2016 ^ 2 = 0\n[PROOFSTEP]\nrw [InnerProductSpace.norm_sq_eq_inner (\ud835\udd5c := \ud835\udd5c) x, hx, map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191im (inner x x) = 0\n[PROOFSTEP]\nrw [\u2190 @ofReal_inj \ud835\udd5c, im_eq_conj_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 I * (\u2191(starRingEnd \ud835\udd5c) (inner x x) - inner x x) / 2 = \u21910\n[PROOFSTEP]\nsimp [inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 inner x (y + z) = inner x y + inner x z\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_add_left, RingHom.map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) + \u2191(starRingEnd \ud835\udd5c) (inner z x) = inner x y + inner x z\n[PROOFSTEP]\nsimp only [inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191(normSq x) = inner x x\n[PROOFSTEP]\nrw [ext_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191re \u2191(normSq x) = \u2191re (inner x x) \u2227 \u2191im \u2191(normSq x) = \u2191im (inner x x)\n[PROOFSTEP]\nexact \u27e8by simp only [ofReal_re]; rfl, by simp only [inner_self_im, ofReal_im]\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191re \u2191(normSq x) = \u2191re (inner x x)\n[PROOFSTEP]\nsimp only [ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 normSq x = \u2191re (inner x x)\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191im \u2191(normSq x) = \u2191im (inner x x)\n[PROOFSTEP]\nsimp only [inner_self_im, ofReal_im]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191re (inner x y) = \u2191re (inner y x)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, conj_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191im (inner x y) = -\u2191im (inner y x)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, conj_im]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nr : \ud835\udd5c\n\u22a2 inner x (r \u2022 y) = r * inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nr : \ud835\udd5c\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (\u2191(starRingEnd \ud835\udd5c) r * inner y x) = r * inner x y\n[PROOFSTEP]\nsimp only [conj_conj, inner_conj_symm, RingHom.map_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 inner 0 x = 0\n[PROOFSTEP]\nrw [\u2190 zero_smul \ud835\udd5c (0 : F), inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) 0 * inner 0 x = 0\n[PROOFSTEP]\nsimp only [zero_mul, RingHom.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 inner x 0 = 0\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_zero_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) 0 = 0\n[PROOFSTEP]\nsimp only [RingHom.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 x = 0 \u2192 inner x x = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\n\u22a2 inner 0 0 = 0\n[PROOFSTEP]\nexact inner_zero_left _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 normSq x = 0 \u2194 inner x x = 0\n[PROOFSTEP]\nsimp only [normSq, ext_iff, map_zero, inner_self_im, eq_self_iff_true, and_true_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191(\u2191re (inner x x)) = inner x x\n[PROOFSTEP]\nrw [ext_iff, inner_self_im]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191re \u2191(\u2191re (inner x x)) = \u2191re (inner x x) \u2227 \u2191im \u2191(\u2191re (inner x x)) = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2016inner x y\u2016 = \u2016inner y x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, norm_conj]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner (-x) y = -inner x y\n[PROOFSTEP]\nrw [\u2190 neg_one_smul \ud835\udd5c x, inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (-1) * inner x y = -inner x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner x (-y) = -inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_neg_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (-inner y x) = -inner x y\n[PROOFSTEP]\nsimp only [RingHom.map_neg, inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 inner (x - y) z = inner x z - inner y z\n[PROOFSTEP]\nsimp [sub_eq_add_neg, inner_add_left, inner_neg_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y z : F\n\u22a2 inner x (y - z) = inner x y - inner x z\n[PROOFSTEP]\nsimp [sub_eq_add_neg, inner_add_right, inner_neg_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191re (inner x y * inner y x) = \u2016inner x y * inner y x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191re (inner y x * \u2191(starRingEnd \ud835\udd5c) (inner y x)) = \u2016inner y x * \u2191(starRingEnd \ud835\udd5c) (inner y x)\u2016\n[PROOFSTEP]\nexact re_eq_norm_of_mul_conj (inner y x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner (x + y) (x + y) = inner x x + inner x y + inner y x + inner y y\n[PROOFSTEP]\nsimp only [inner_add_left, inner_add_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner x x + inner y x + (inner x y + inner y y) = inner x x + inner x y + inner y x + inner y y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner (x - y) (x - y) = inner x x - inner x y - inner y x + inner y y\n[PROOFSTEP]\nsimp only [inner_sub_left, inner_sub_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner x x - inner y x - (inner x y - inner y y) = inner x x - inner x y - inner y x + inner y y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 normSq (inner x y \u2022 x - inner x x \u2022 y) = normSq x * (normSq x * normSq y - \u2016inner x y\u2016 ^ 2)\n[PROOFSTEP]\nrw [\u2190 @ofReal_inj \ud835\udd5c, ofReal_normSq_eq_inner_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner (inner x y \u2022 x - inner x x \u2022 y) (inner x y \u2022 x - inner x x \u2022 y) =\n    \u2191(normSq x * (normSq x * normSq y - \u2016inner x y\u2016 ^ 2))\n[PROOFSTEP]\nsimp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, \u2190 ofReal_normSq_eq_inner_self x,\n  \u2190 ofReal_normSq_eq_inner_self y]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 inner x y * (\u2191(starRingEnd \ud835\udd5c) (inner x y) * \u2191(normSq x)) - \u2191(normSq x) * (\u2191(starRingEnd \ud835\udd5c) (inner x y) * inner x y) -\n        inner x y * (\u2191(normSq x) * inner y x) +\n      \u2191(normSq x) * (\u2191(normSq x) * \u2191(normSq y)) =\n    \u2191(normSq x * (normSq x * normSq y) - normSq x * \u2016inner x y\u2016 ^ 2)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_conj, IsROrC.conj_mul, normSq_eq_def', mul_left_comm, \u2190 inner_conj_symm y, mul_conj,\n  normSq_eq_def']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191(\u2016inner x y\u2016 ^ 2) * \u2191(normSq x) - \u2191(normSq x) * \u2191(\u2016inner x y\u2016 ^ 2) - \u2191(normSq x) * \u2191(\u2016inner x y\u2016 ^ 2) +\n      \u2191(normSq x) * (\u2191(normSq x) * \u2191(normSq y)) =\n    \u2191(normSq x * (normSq x * normSq y) - normSq x * \u2016inner x y\u2016 ^ 2)\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191\u2016inner x y\u2016 ^ 2 * \u2191(normSq x) - \u2191(normSq x) * \u2191\u2016inner x y\u2016 ^ 2 - \u2191(normSq x) * \u2191\u2016inner x y\u2016 ^ 2 +\n      \u2191(normSq x) * (\u2191(normSq x) * \u2191(normSq y)) =\n    \u2191(normSq x) * (\u2191(normSq x) * \u2191(normSq y)) - \u2191(normSq x) * \u2191\u2016inner x y\u2016 ^ 2\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2016inner x y\u2016 * \u2016inner y x\u2016 \u2264 \u2191re (inner x x) * \u2191re (inner y y)\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | hx)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\ny : F\n\u22a2 \u2016inner 0 y\u2016 * \u2016inner y 0\u2016 \u2264 \u2191re (inner 0 0) * \u2191re (inner y y)\n[PROOFSTEP]\nsimpa only [inner_zero_left, map_zero, zero_mul, norm_zero] using le_rfl\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nhx : x \u2260 0\n\u22a2 \u2016inner x y\u2016 * \u2016inner y x\u2016 \u2264 \u2191re (inner x x) * \u2191re (inner y y)\n[PROOFSTEP]\nhave hx' : 0 < normSqF x := inner_self_nonneg.lt_of_ne' (mt normSq_eq_zero.1 hx)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nhx : x \u2260 0\nhx' : 0 < normSq x\n\u22a2 \u2016inner x y\u2016 * \u2016inner y x\u2016 \u2264 \u2191re (inner x x) * \u2191re (inner y y)\n[PROOFSTEP]\nrw [\u2190 sub_nonneg, \u2190 mul_nonneg_iff_right_nonneg_of_pos hx', \u2190 normSq, \u2190 normSq, norm_inner_symm y, \u2190 sq, \u2190\n  cauchy_schwarz_aux]\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nhx : x \u2260 0\nhx' : 0 < normSq x\n\u22a2 0 \u2264 normSq (inner x y \u2022 x - inner x x \u2022 y)\n[PROOFSTEP]\nexact inner_self_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 \u2191re (inner x x) = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_inner, \u2190 sqrt_mul inner_self_nonneg (re \u27eax, x\u27eb), sqrt_mul_self inner_self_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2016inner x y\u2016 * \u2016inner x y\u2016 = \u2016inner x y\u2016 * \u2016inner y x\u2016\n[PROOFSTEP]\nrw [norm_inner_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2191re (inner x x) * \u2191re (inner y y) = \u2016x\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016y\u2016)\n[PROOFSTEP]\nsimp only [inner_self_eq_norm_mul_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 \u2016x\u2016 * \u2016x\u2016 * (\u2016y\u2016 * \u2016y\u2016) = \u2016x\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016y\u2016)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) 0 = 0\n[PROOFSTEP]\nsimp only [sqrt_zero, inner_zero_right, map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nhave h\u2081 : \u2016\u27eax, y\u27eb\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016 := norm_inner_le_norm _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nhave h\u2082 : re \u27eax, y\u27eb \u2264 \u2016\u27eax, y\u27eb\u2016 := re_le_norm _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nhave h\u2083 : re \u27eax, y\u27eb \u2264 \u2016x\u2016 * \u2016y\u2016 := h\u2082.trans h\u2081\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nhave h\u2084 : re \u27eay, x\u27eb \u2264 \u2016x\u2016 * \u2016y\u2016 := by rwa [\u2190 inner_conj_symm, conj_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 \u2191re (inner y x) \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrwa [\u2190 inner_conj_symm, conj_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2084 : \u2191re (inner y x) \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nhave : \u2016x + y\u2016 * \u2016x + y\u2016 \u2264 (\u2016x\u2016 + \u2016y\u2016) * (\u2016x\u2016 + \u2016y\u2016) :=\n  by\n  simp only [\u2190 inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]\n  linarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2084 : \u2191re (inner y x) \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 \u2264 (\u2016x\u2016 + \u2016y\u2016) * (\u2016x\u2016 + \u2016y\u2016)\n[PROOFSTEP]\nsimp only [\u2190 inner_self_eq_norm_mul_norm, inner_add_add_self, mul_add, mul_comm, map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2084 : \u2191re (inner y x) \u2264 \u2016x\u2016 * \u2016y\u2016\n\u22a2 \u2191re (inner x x) + \u2191re (inner x y) + \u2191re (inner y x) + \u2191re (inner y y) \u2264\n    \u2191re (inner x x) + \u2016x\u2016 * \u2016y\u2016 + (\u2016x\u2016 * \u2016y\u2016 + \u2191re (inner y y))\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx y : F\nh\u2081 : \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2082 : \u2191re (inner x y) \u2264 \u2016inner x y\u2016\nh\u2083 : \u2191re (inner x y) \u2264 \u2016x\u2016 * \u2016y\u2016\nh\u2084 : \u2191re (inner y x) \u2264 \u2016x\u2016 * \u2016y\u2016\nthis : \u2016x + y\u2016 * \u2016x + y\u2016 \u2264 (\u2016x\u2016 + \u2016y\u2016) * (\u2016x\u2016 + \u2016y\u2016)\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (x + y) \u2264 (fun x => sqrt (\u2191re (inner x x))) x + (fun x => sqrt (\u2191re (inner x x))) y\n[PROOFSTEP]\nexact nonneg_le_nonneg_of_sq_le_sq (add_nonneg (sqrt_nonneg _) (sqrt_nonneg _)) this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nx : F\n\u22a2 (fun x => sqrt (\u2191re (inner x x))) (-x) = (fun x => sqrt (\u2191re (inner x x))) x\n[PROOFSTEP]\nsimp only [inner_neg_left, neg_neg, inner_neg_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nr : \ud835\udd5c\nx : F\n\u22a2 \u2016r \u2022 x\u2016 \u2264 \u2016r\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_inner, inner_smul_left, inner_smul_right, \u2190 mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nr : \ud835\udd5c\nx : F\n\u22a2 sqrt (\u2191re (\u2191(starRingEnd \ud835\udd5c) r * r * inner x x)) \u2264 \u2016r\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [IsROrC.conj_mul, ofReal_mul_re, sqrt_mul, \u2190 ofReal_normSq_eq_inner_self, ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nr : \ud835\udd5c\nx : F\n\u22a2 sqrt (\u2191IsROrC.normSq r) * sqrt (normSq x) \u2264 \u2016r\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimp [sqrt_normSq_eq_norm, IsROrC.sqrt_normSq_eq_norm]\n[GOAL]\ncase hx\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nr : \ud835\udd5c\nx : F\n\u22a2 0 \u2264 \u2191IsROrC.normSq r\n[PROOFSTEP]\nexact normSq_nonneg r\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nthis : NormedSpace \ud835\udd5c F := Core.toNormedSpace\nx : F\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner x x)\n[PROOFSTEP]\nhave h\u2081 : \u2016x\u2016 ^ 2 = sqrt (re (c.inner x x)) ^ 2 := rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nthis : NormedSpace \ud835\udd5c F := Core.toNormedSpace\nx : F\nh\u2081 : \u2016x\u2016 ^ 2 = sqrt (\u2191re (inner x x)) ^ 2\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner x x)\n[PROOFSTEP]\nhave h\u2082 : 0 \u2264 re (c.inner x x) := InnerProductSpace.Core.inner_self_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup F\ninst\u271d : Module \ud835\udd5c F\nc : Core \ud835\udd5c F\nthis : NormedSpace \ud835\udd5c F := Core.toNormedSpace\nx : F\nh\u2081 : \u2016x\u2016 ^ 2 = sqrt (\u2191re (inner x x)) ^ 2\nh\u2082 : 0 \u2264 \u2191re (inner x x)\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner x x)\n[PROOFSTEP]\nsimp [h\u2081, sq_sqrt, h\u2082]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x y = 0 \u2194 inner y x = 0\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) = 0 \u2194 inner y x = 0\n[PROOFSTEP]\nexact star_eq_zero\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191im (inner x x) = 0\n[PROOFSTEP]\nrw [\u2190 @ofReal_inj \ud835\udd5c, im_eq_conj_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 I * (\u2191(starRingEnd \ud835\udd5c) (inner x x) - inner x x) / 2 = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : E\n\u22a2 inner x (y + z) = inner x y + inner x z\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_add_left, RingHom.map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : E\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) + \u2191(starRingEnd \ud835\udd5c) (inner z x) = inner x y + inner x z\n[PROOFSTEP]\nsimp only [inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = \u2191re (inner y x)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, conj_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191im (inner x y) = -\u2191im (inner y x)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, conj_im]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nr : \u211d\n\u22a2 inner (\u2191r \u2022 x) y = r \u2022 inner x y\n[PROOFSTEP]\nrw [inner_smul_left, conj_ofReal, Algebra.smul_def]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nr : \u211d\n\u22a2 \u2191r * inner x y = \u2191(algebraMap \u211d \ud835\udd5c) r * inner x y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nr : \ud835\udd5c\n\u22a2 inner x (r \u2022 y) = r * inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_smul_left, RingHom.map_mul, conj_conj, inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nr : \u211d\n\u22a2 inner x (\u2191r \u2022 y) = r \u2022 inner x y\n[PROOFSTEP]\nrw [inner_smul_right, Algebra.smul_def]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nr : \u211d\n\u22a2 \u2191r * inner x y = \u2191(algebraMap \u211d \ud835\udd5c) r * inner x y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nv : \u03b9 \u2192 E\nx : E\n\u22a2 inner (sum l fun i a => a \u2022 v i) x = sum l fun i a => \u2191(starRingEnd \ud835\udd5c) a \u2022 inner (v i) x\n[PROOFSTEP]\nconvert _root_.sum_inner (\ud835\udd5c := \ud835\udd5c) l.support (fun a => l a \u2022 v a) x\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nv : \u03b9 \u2192 E\nx : E\n\u22a2 (sum l fun i a => \u2191(starRingEnd \ud835\udd5c) a \u2022 inner (v i) x) = \u2211 i in l.support, inner (\u2191l i \u2022 v i) x\n[PROOFSTEP]\nsimp only [inner_smul_left, Finsupp.sum, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nv : \u03b9 \u2192 E\nx : E\n\u22a2 inner x (sum l fun i a => a \u2022 v i) = sum l fun i a => a \u2022 inner x (v i)\n[PROOFSTEP]\nconvert _root_.inner_sum (\ud835\udd5c := \ud835\udd5c) l.support (fun a => l a \u2022 v a) x\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nv : \u03b9 \u2192 E\nx : E\n\u22a2 (sum l fun i a => a \u2022 inner x (v i)) = \u2211 i in l.support, inner x (\u2191l i \u2022 v i)\n[PROOFSTEP]\nsimp only [inner_smul_right, Finsupp.sum, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b1 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 E\nl : \u03a0\u2080 (i : \u03b9), \u03b1 i\nx : E\n\u22a2 inner (sum l f) x = sum l fun i a => inner (f i a) x\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [DFinsupp.sum, _root_.sum_inner, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec : DecidableEq \u03b9\n\u03b1 : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 AddZeroClass (\u03b1 i)\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b1 i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 E\nl : \u03a0\u2080 (i : \u03b9), \u03b1 i\nx : E\n\u22a2 inner x (sum l f) = sum l fun i a => inner x (f i a)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [DFinsupp.sum, _root_.inner_sum, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 inner 0 x = 0\n[PROOFSTEP]\nrw [\u2190 zero_smul \ud835\udd5c (0 : E), inner_smul_left, RingHom.map_zero, zero_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner 0 x) = 0\n[PROOFSTEP]\nsimp only [inner_zero_left, AddMonoidHom.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 inner x 0 = 0\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_zero_left, RingHom.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x 0) = 0\n[PROOFSTEP]\nsimp only [inner_zero_right, AddMonoidHom.map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 inner x x = \u2191\u2016x\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 inner_self_ofReal_re, \u2190 norm_sq_eq_inner, ofReal_pow]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x x) = \u2016inner x x\u2016\n[PROOFSTEP]\nconv_rhs => rw [\u2190 inner_self_ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n| \u2016inner x x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_self_ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n| \u2016inner x x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_self_ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n| \u2016inner x x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_self_ofReal_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x x) = \u2016\u2191(\u2191re (inner x x))\u2016\n[PROOFSTEP]\nsymm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2016\u2191(\u2191re (inner x x))\u2016 = \u2191re (inner x x)\n[PROOFSTEP]\nexact norm_of_nonneg inner_self_nonneg\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191\u2016inner x x\u2016 = inner x x\n[PROOFSTEP]\nrw [\u2190 inner_self_re_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191(\u2191re (inner x x)) = inner x x\n[PROOFSTEP]\nexact inner_self_ofReal_re _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 inner x x = 0 \u2194 x = 0\n[PROOFSTEP]\nrw [inner_self_eq_norm_sq_to_K, sq_eq_zero_iff, ofReal_eq_zero, norm_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x x) \u2264 0 \u2194 x = 0\n[PROOFSTEP]\nrw [\u2190 norm_sq_eq_inner, (sq_nonneg _).le_iff_eq, sq_eq_zero_iff, norm_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y\u2016 = \u2016inner y x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, norm_conj]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (-x) y = -inner x y\n[PROOFSTEP]\nrw [\u2190 neg_one_smul \ud835\udd5c x, inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (-1) * inner x y = -inner x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x (-y) = -inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, inner_neg_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (-inner y x) = -inner x y\n[PROOFSTEP]\nsimp only [RingHom.map_neg, inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (-x) (-y) = inner x y\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : E\n\u22a2 inner (x - y) z = inner x z - inner y z\n[PROOFSTEP]\nsimp [sub_eq_add_neg, inner_add_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : E\n\u22a2 inner x (y - z) = inner x y - inner x z\n[PROOFSTEP]\nsimp [sub_eq_add_neg, inner_add_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y * inner y x) = \u2016inner x y * inner y x\u2016\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, mul_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner y x * \u2191(starRingEnd \ud835\udd5c) (inner y x)) = \u2016inner y x * \u2191(starRingEnd \ud835\udd5c) (inner y x)\u2016\n[PROOFSTEP]\nexact re_eq_norm_of_mul_conj (inner y x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (x + y) (x + y) = inner x x + inner x y + inner y x + inner y y\n[PROOFSTEP]\nsimp only [inner_add_left, inner_add_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x x + inner y x + (inner x y + inner y y) = inner x x + inner x y + inner y x + inner y y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y\n[PROOFSTEP]\nhave : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190 inner_conj_symm]; rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner y x = inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2191(starRingEnd \u211d) (inner x y) = inner x y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner (x + y) (x + y) = inner x x + 2 * inner x y + inner y y\n[PROOFSTEP]\nsimp only [inner_add_add_self, this, add_left_inj]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner x x + inner x y + inner x y = inner x x + 2 * inner x y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (x - y) (x - y) = inner x x - inner x y - inner y x + inner y y\n[PROOFSTEP]\nsimp only [inner_sub_left, inner_sub_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x x - inner y x - (inner x y - inner y y) = inner x x - inner x y - inner y x + inner y y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y\n[PROOFSTEP]\nhave : \u27eay, x\u27eb_\u211d = \u27eax, y\u27eb_\u211d := by rw [\u2190 inner_conj_symm]; rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner y x = inner x y\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2191(starRingEnd \u211d) (inner x y) = inner x y\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner (x - y) (x - y) = inner x x - 2 * inner x y + inner y y\n[PROOFSTEP]\nsimp only [inner_sub_sub_self, this, add_left_inj]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nthis : inner y x = inner x y\n\u22a2 inner x x - inner x y - inner x y = inner x x - 2 * inner x y\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2200 (v : E), inner v x = inner v y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 @inner_self_eq_zero \ud835\udd5c, inner_sub_right, sub_eq_zero, h (x - y)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2200 (v : E), inner x v = inner y v\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 @inner_self_eq_zero \ud835\udd5c, inner_sub_left, sub_eq_zero, h (x - y)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner (x + y) (x + y) + inner (x - y) (x - y) = 2 * (inner x x + inner y y)\n[PROOFSTEP]\nsimp only [inner_add_add_self, inner_sub_sub_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x x + inner x y + inner y x + inner y y + (inner x x - inner x y - inner y x + inner y y) =\n    2 * (inner x x + inner y y)\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x y * inner x y \u2264 \u2016inner x y\u2016 * \u2016inner y x\u2016\n[PROOFSTEP]\nrw [real_inner_comm y, \u2190 norm_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner y x * inner y x \u2264 \u2016inner y x * inner y x\u2016\n[PROOFSTEP]\nexact le_abs_self _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\n\u22a2 LinearIndependent \ud835\udd5c v\n[PROOFSTEP]\nrw [linearIndependent_iff']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\n\u22a2 \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 \ud835\udd5c), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s g hg i hi\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nhave h' : g i * inner (v i) (v i) = inner (v i) (\u2211 j in s, g j \u2022 v j) :=\n  by\n  rw [inner_sum]\n  symm\n  convert Finset.sum_eq_single (\u03b2 := \ud835\udd5c) i ?_ ?_\n  \u00b7 rw [inner_smul_right]\n  \u00b7 intro j _hj hji\n    rw [inner_smul_right, ho i j hji.symm, mul_zero]\n  \u00b7 exact fun h => False.elim (h hi)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i * inner (v i) (v i) = inner (v i) (\u2211 j in s, g j \u2022 v j)\n[PROOFSTEP]\nrw [inner_sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i * inner (v i) (v i) = \u2211 i_1 in s, inner (v i) (g i_1 \u2022 v i_1)\n[PROOFSTEP]\nsymm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 i_1 in s, inner (v i) (g i_1 \u2022 v i_1) = g i * inner (v i) (v i)\n[PROOFSTEP]\nconvert Finset.sum_eq_single (\u03b2 := \ud835\udd5c) i ?_ ?_\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i * inner (v i) (v i) = inner (v i) (g i \u2022 v i)\n[PROOFSTEP]\nrw [inner_smul_right]\n[GOAL]\ncase convert_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2200 (b : \u03b9), b \u2208 s \u2192 b \u2260 i \u2192 inner (v i) (g b \u2022 v b) = 0\n[PROOFSTEP]\nintro j _hj hji\n[GOAL]\ncase convert_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\n_hj : j \u2208 s\nhji : j \u2260 i\n\u22a2 inner (v i) (g j \u2022 v j) = 0\n[PROOFSTEP]\nrw [inner_smul_right, ho i j hji.symm, mul_zero]\n[GOAL]\ncase convert_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u00aci \u2208 s \u2192 inner (v i) (g i \u2022 v i) = 0\n[PROOFSTEP]\nexact fun h => False.elim (h hi)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhz : \u2200 (i : \u03b9), v i \u2260 0\nho : \u2200 (i j : \u03b9), i \u2260 j \u2192 inner (v i) (v j) = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 \ud835\udd5c\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\nh' : g i * inner (v i) (v i) = inner (v i) (\u2211 j in s, g j \u2022 v j)\n\u22a2 g i = 0\n[PROOFSTEP]\nsimpa [hg, hz] using h'\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\n\u22a2 Orthonormal \ud835\udd5c v \u2194 \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\n\u22a2 Orthonormal \ud835\udd5c v \u2192 \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n[PROOFSTEP]\nintro hv i j\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni j : \u03b9\n\u22a2 inner (v i) (v j) = if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni j : \u03b9\nh : i = j\n\u22a2 inner (v i) (v j) = 1\n[PROOFSTEP]\nsimp [h, inner_self_eq_norm_sq_to_K, hv.1]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni j : \u03b9\nh : \u00aci = j\n\u22a2 inner (v i) (v j) = 0\n[PROOFSTEP]\nexact hv.2 h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\n\u22a2 (\u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0) \u2192 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mpr.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 \u2200 (i : \u03b9), \u2016v i\u2016 = 1\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mpr.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni : \u03b9\n\u22a2 \u2016v i\u2016 = 1\n[PROOFSTEP]\nhave h' : \u2016v i\u2016 ^ 2 = 1 ^ 2 := by simp [@norm_sq_eq_inner \ud835\udd5c, h i i]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni : \u03b9\n\u22a2 \u2016v i\u2016 ^ 2 = 1 ^ 2\n[PROOFSTEP]\nsimp [@norm_sq_eq_inner \ud835\udd5c, h i i]\n[GOAL]\ncase mpr.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni : \u03b9\nh' : \u2016v i\u2016 ^ 2 = 1 ^ 2\n\u22a2 \u2016v i\u2016 = 1\n[PROOFSTEP]\nhave h\u2081 : 0 \u2264 \u2016v i\u2016 := norm_nonneg _\n[GOAL]\ncase mpr.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni : \u03b9\nh' : \u2016v i\u2016 ^ 2 = 1 ^ 2\nh\u2081 : 0 \u2264 \u2016v i\u2016\n\u22a2 \u2016v i\u2016 = 1\n[PROOFSTEP]\nhave h\u2082 : (0 : \u211d) \u2264 1 := zero_le_one\n[GOAL]\ncase mpr.left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni : \u03b9\nh' : \u2016v i\u2016 ^ 2 = 1 ^ 2\nh\u2081 : 0 \u2264 \u2016v i\u2016\nh\u2082 : 0 \u2264 1\n\u22a2 \u2016v i\u2016 = 1\n[PROOFSTEP]\nrwa [sq_eq_sq h\u2081 h\u2082] at h' \n[GOAL]\ncase mpr.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 \u2200 {i j : \u03b9}, i \u2260 j \u2192 inner (v i) (v j) = 0\n[PROOFSTEP]\nintro i j hij\n[GOAL]\ncase mpr.right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nh : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhij : i \u2260 j\n\u22a2 inner (v i) (v j) = 0\n[PROOFSTEP]\nsimpa [hij] using h i j\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\n\u22a2 Orthonormal \ud835\udd5c Subtype.val \u2194 \u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0\n[PROOFSTEP]\nrw [orthonormal_iff_ite]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\n\u22a2 (\u2200 (i j : { x // x \u2208 s }), inner \u2191i \u2191j = if i = j then 1 else 0) \u2194\n    \u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\n\u22a2 (\u2200 (i j : { x // x \u2208 s }), inner \u2191i \u2191j = if i = j then 1 else 0) \u2192\n    \u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0\n[PROOFSTEP]\nintro h v hv w hw\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nh : \u2200 (i j : { x // x \u2208 s }), inner \u2191i \u2191j = if i = j then 1 else 0\nv : E\nhv : v \u2208 s\nw : E\nhw : w \u2208 s\n\u22a2 inner v w = if v = w then 1 else 0\n[PROOFSTEP]\nconvert h \u27e8v, hv\u27e9 \u27e8w, hw\u27e9 using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nh : \u2200 (i j : { x // x \u2208 s }), inner \u2191i \u2191j = if i = j then 1 else 0\nv : E\nhv : v \u2208 s\nw : E\nhw : w \u2208 s\n\u22a2 (if v = w then 1 else 0) = if { val := v, property := hv } = { val := w, property := hw } then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\n\u22a2 (\u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0) \u2192\n    \u2200 (i j : { x // x \u2208 s }), inner \u2191i \u2191j = if i = j then 1 else 0\n[PROOFSTEP]\nrintro h \u27e8v, hv\u27e9 \u27e8w, hw\u27e9\n[GOAL]\ncase mpr.mk.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nh : \u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0\nv : E\nhv : v \u2208 s\nw : E\nhw : w \u2208 s\n\u22a2 inner \u2191{ val := v, property := hv } \u2191{ val := w, property := hw } =\n    if { val := v, property := hv } = { val := w, property := hw } then 1 else 0\n[PROOFSTEP]\nconvert h v hv w hw using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nh : \u2200 (v : E), v \u2208 s \u2192 \u2200 (w : E), w \u2208 s \u2192 inner v w = if v = w then 1 else 0\nv : E\nhv : v \u2208 s\nw : E\nhw : w \u2208 s\n\u22a2 (if { val := v, property := hv } = { val := w, property := hw } then 1 else 0) = if v = w then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\ni : \u03b9\n\u22a2 inner (v i) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) = \u2191l i\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\ni : \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (v i) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) = \u2191l i\n[PROOFSTEP]\nsimpa [Finsupp.total_apply, Finsupp.inner_sum, orthonormal_iff_ite.mp hv] using Eq.symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 inner (v i) (\u2211 i in s, l i \u2022 v i) = l i\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (v i) (\u2211 i in s, l i \u2022 v i) = l i\n[PROOFSTEP]\nsimp [inner_sum, inner_smul_right, orthonormal_iff_ite.mp hv, hi]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\ni : \u03b9\n\u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) (v i) = \u2191(starRingEnd ((fun x => \ud835\udd5c) i)) (\u2191l i)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, hv.inner_right_finsupp]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 inner (\u2211 i in s, l i \u2022 v i) (v i) = \u2191(starRingEnd \ud835\udd5c) (l i)\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (\u2211 i in s, l i \u2022 v i) (v i) = \u2191(starRingEnd \ud835\udd5c) (l i)\n[PROOFSTEP]\nsimp only [sum_inner, inner_smul_left, orthonormal_iff_ite.mp hv, hi, mul_boole, Finset.sum_ite_eq', if_true]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192\u2080 \ud835\udd5c\n\u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2081) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2082) =\n    Finsupp.sum l\u2081 fun i y => \u2191(starRingEnd \ud835\udd5c) y * \u2191l\u2082 i\n[PROOFSTEP]\nsimp only [l\u2081.total_apply _, Finsupp.sum_inner, hv.inner_right_finsupp, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192\u2080 \ud835\udd5c\n\u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2081) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2082) =\n    Finsupp.sum l\u2082 fun i y => \u2191(starRingEnd ((fun x => \ud835\udd5c) i)) (\u2191l\u2081 i) * y\n[PROOFSTEP]\nsimp only [l\u2082.total_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 inner (\u2211 i in s, l\u2081 i \u2022 v i) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2211 i in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i\n[PROOFSTEP]\nsimp_rw [sum_inner, inner_smul_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * inner (v x) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2211 x in s, \u2191(starRingEnd \ud835\udd5c) (l\u2081 x) * l\u2082 x\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * inner (v i) (\u2211 i in s, l\u2082 i \u2022 v i) = \u2191(starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i\n[PROOFSTEP]\nrw [hv.inner_right_sum l\u2082 hi]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\na : \u03b9 \u2192 \u03b9 \u2192 \ud835\udd5c\n\u22a2 \u2211 i in s, \u2211 j in s, a i j \u2022 inner (v j) (v i) = \u2211 k in s, a k k\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\na : \u03b9 \u2192 \u03b9 \u2192 \ud835\udd5c\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 \u2211 i in s, \u2211 j in s, a i j \u2022 inner (v j) (v i) = \u2211 k in s, a k k\n[PROOFSTEP]\nsimp [orthonormal_iff_ite.mp hv, Finset.sum_ite_of_true]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 LinearIndependent \ud835\udd5c v\n[PROOFSTEP]\nrw [linearIndependent_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2200 (l : \u03b9 \u2192\u2080 \ud835\udd5c), \u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l = 0 \u2192 l = 0\n[PROOFSTEP]\nintro l hl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l = 0\n\u22a2 l = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l = 0\ni : \u03b9\n\u22a2 \u2191l i = \u21910 i\n[PROOFSTEP]\nhave key : \u27eav i, Finsupp.total \u03b9 E \ud835\udd5c v l\u27eb = \u27eav i, 0\u27eb := by rw [hl]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l = 0\ni : \u03b9\n\u22a2 inner (v i) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) = inner (v i) 0\n[PROOFSTEP]\nrw [hl]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l = 0\ni : \u03b9\nkey : inner (v i) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) = inner (v i) 0\n\u22a2 \u2191l i = \u21910 i\n[PROOFSTEP]\nsimpa only [hv.inner_right_finsupp, inner_zero_right] using key\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b9' : Type u_5\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nf : \u03b9' \u2192 \u03b9\nhf : Function.Injective f\n\u22a2 Orthonormal \ud835\udd5c (v \u2218 f)\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b9' : Type u_5\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nf : \u03b9' \u2192 \u03b9\nhf : Function.Injective f\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Orthonormal \ud835\udd5c (v \u2218 f)\n[PROOFSTEP]\nrw [orthonormal_iff_ite] at hv \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b9' : Type u_5\nv : \u03b9 \u2192 E\nf : \u03b9' \u2192 \u03b9\nhf : Function.Injective f\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 \u2200 (i j : \u03b9'), inner ((v \u2218 f) i) ((v \u2218 f) j) = if i = j then 1 else 0\n[PROOFSTEP]\nintro i j\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b9' : Type u_5\nv : \u03b9 \u2192 E\nf : \u03b9' \u2192 \u03b9\nhf : Function.Injective f\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9'\n\u22a2 inner ((v \u2218 f) i) ((v \u2218 f) j) = if i = j then 1 else 0\n[PROOFSTEP]\nconvert hv (f i) (f j) using 1\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b9' : Type u_5\nv : \u03b9 \u2192 E\nf : \u03b9' \u2192 \u03b9\nhf : Function.Injective f\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9'\n\u22a2 (if i = j then 1 else 0) = if f i = f j then 1 else 0\n[PROOFSTEP]\nsimp [hf.eq_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Function.Injective v\n\u22a2 Orthonormal \ud835\udd5c Subtype.val \u2194 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nlet f : \u03b9 \u2243 Set.range v := Equiv.ofInjective v hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Function.Injective v\nf : \u03b9 \u2243 \u2191(Set.range v) := Equiv.ofInjective v hv\n\u22a2 Orthonormal \ud835\udd5c Subtype.val \u2194 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nrefine' \u27e8fun h => h.comp f f.injective, fun h => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Function.Injective v\nf : \u03b9 \u2243 \u2191(Set.range v) := Equiv.ofInjective v hv\nh : Orthonormal \ud835\udd5c v\n\u22a2 Orthonormal \ud835\udd5c Subtype.val\n[PROOFSTEP]\nrw [\u2190 Equiv.self_comp_ofInjective_symm hv]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Function.Injective v\nf : \u03b9 \u2243 \u2191(Set.range v) := Equiv.ofInjective v hv\nh : Orthonormal \ud835\udd5c v\n\u22a2 Orthonormal \ud835\udd5c (v \u2218 \u2191(Equiv.ofInjective v hv).symm)\n[PROOFSTEP]\nexact h.comp f.symm f.symm.injective\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 s\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : l \u2208 Finsupp.supported \ud835\udd5c \ud835\udd5c s\n\u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) (v i) = 0\n[PROOFSTEP]\nrw [Finsupp.mem_supported'] at hl \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 s\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2200 (x : \u03b9), \u00acx \u2208 s \u2192 \u2191l x = 0\n\u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l) (v i) = 0\n[PROOFSTEP]\nsimp only [hv.inner_left_finsupp, hl i hi, map_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\n\u22a2 Orthonormal \ud835\udd5c w\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Orthonormal \ud835\udd5c w\n[PROOFSTEP]\nrw [orthonormal_iff_ite] at *\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 \u2200 (i j : \u03b9), inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nintro i j\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\ncases' hw i with hi hi\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = v i\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\ncases' hw j with hj hj\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = -v i\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\ncases' hw j with hj hj\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = v i\nhj : w j = v j\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nreplace hv := hv i j\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = v i\nhj : w j = -v j\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nreplace hv := hv i j\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = -v i\nhj : w j = v j\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nreplace hv := hv i j\n[GOAL]\ncase inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\nhv : \u2200 (i j : \u03b9), inner (v i) (v j) = if i = j then 1 else 0\ni j : \u03b9\nhi : w i = -v i\nhj : w j = -v j\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nreplace hv := hv i j\n[GOAL]\ncase inl.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = v j\nhv : inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs at hv \u22a2 with h\n[GOAL]\ncase inl.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = -v j\nhv : inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs at hv \u22a2 with h\n[GOAL]\ncase inr.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = v j\nhv : inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs at hv \u22a2 with h\n[GOAL]\ncase inr.inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = -v j\nhv : inner (v i) (v j) = if i = j then 1 else 0\n\u22a2 inner (w i) (w j) = if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs at hv \u22a2 with h\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = v j\nh : i = j\nhv : inner (v i) (v j) = 1\n\u22a2 inner (w i) (w j) = 1\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = v j\nh : \u00aci = j\nhv : inner (v i) (v j) = 0\n\u22a2 inner (w i) (w j) = 0\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = -v j\nh : i = j\nhv : inner (v i) (v j) = 1\n\u22a2 inner (w i) (w j) = 1\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = v i\nhj : w j = -v j\nh : \u00aci = j\nhv : inner (v i) (v j) = 0\n\u22a2 inner (w i) (w j) = 0\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = v j\nh : i = j\nhv : inner (v i) (v j) = 1\n\u22a2 inner (w i) (w j) = 1\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = v j\nh : \u00aci = j\nhv : inner (v i) (v j) = 0\n\u22a2 inner (w i) (w j) = 0\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = -v j\nh : i = j\nhv : inner (v i) (v j) = 1\n\u22a2 inner (w i) (w j) = 1\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv w : \u03b9 \u2192 E\nhw : \u2200 (i : \u03b9), w i = v i \u2228 w i = -v i\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\nhi : w i = -v i\nhj : w j = -v j\nh : \u00aci = j\nhv : inner (v i) (v j) = 0\n\u22a2 inner (w i) (w j) = 0\n[PROOFSTEP]\nsimpa only [hi, hj, h, inner_neg_right, inner_neg_left, neg_neg, eq_self_iff_true, neg_eq_zero] using hv\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nsimp [orthonormal_subtype_iff_ite]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nrw [orthonormal_subtype_iff_ite]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 \u2200 (v : E), v \u2208 \u22c3 (i : \u03b7), s i \u2192 \u2200 (w : E), w \u2208 \u22c3 (i : \u03b7), s i \u2192 inner v w = if v = w then 1 else 0\n[PROOFSTEP]\nrintro x \u27e8_, \u27e8i, rfl\u27e9, hxi\u27e9 y \u27e8_, \u27e8j, rfl\u27e9, hyj\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\nx : E\ni : \u03b7\nhxi : x \u2208 (fun i => s i) i\ny : E\nj : \u03b7\nhyj : y \u2208 (fun i => s i) j\n\u22a2 inner x y = if x = y then 1 else 0\n[PROOFSTEP]\nobtain \u27e8k, hik, hjk\u27e9 := hs i j\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\nx : E\ni : \u03b7\nhxi : x \u2208 (fun i => s i) i\ny : E\nj : \u03b7\nhyj : y \u2208 (fun i => s i) j\nk : \u03b7\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\n\u22a2 inner x y = if x = y then 1 else 0\n[PROOFSTEP]\nhave h_orth : Orthonormal \ud835\udd5c (fun x => x : s k \u2192 E) := h k\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\nx : E\ni : \u03b7\nhxi : x \u2208 (fun i => s i) i\ny : E\nj : \u03b7\nhyj : y \u2208 (fun i => s i) j\nk : \u03b7\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\nh_orth : Orthonormal \ud835\udd5c fun x => \u2191x\n\u22a2 inner x y = if x = y then 1 else 0\n[PROOFSTEP]\nrw [orthonormal_subtype_iff_ite] at h_orth \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\n\u03b7 : Type u_5\ns : \u03b7 \u2192 Set E\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), Orthonormal \ud835\udd5c fun x => \u2191x\nem\u271d : (a : Prop) \u2192 Decidable a\nx : E\ni : \u03b7\nhxi : x \u2208 (fun i => s i) i\ny : E\nj : \u03b7\nhyj : y \u2208 (fun i => s i) j\nk : \u03b7\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\nh_orth : \u2200 (v : E), v \u2208 s k \u2192 \u2200 (w : E), w \u2208 s k \u2192 inner v w = if v = w then 1 else 0\n\u22a2 inner x y = if x = y then 1 else 0\n[PROOFSTEP]\nexact h_orth x (hik hxi) y (hjk hyj)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set (Set E)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set E), a \u2208 s \u2192 Orthonormal \ud835\udd5c fun x => \u2191x\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nrw [Set.sUnion_eq_iUnion]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set (Set E)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set E), a \u2208 s \u2192 Orthonormal \ud835\udd5c fun x => \u2191x\n\u22a2 Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nexact orthonormal_iUnion_of_directed hs.directed_val (by simpa using h)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set (Set E)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set E), a \u2208 s \u2192 Orthonormal \ud835\udd5c fun x => \u2191x\n\u22a2 \u2200 (i : \u2191s), Orthonormal \ud835\udd5c fun x => \u2191x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2203 w _hw, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 (u : Set E), u \u2287 w \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = w\n[PROOFSTEP]\nhave := zorn_subset_nonempty {b | Orthonormal \ud835\udd5c (Subtype.val : b \u2192 E)} ?_ _ hs\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nthis :\n  \u2203 m, m \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2227 s \u2286 m \u2227 \u2200 (a : Set E), a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2192 m \u2286 a \u2192 a = m\n\u22a2 \u2203 w _hw, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 (u : Set E), u \u2287 w \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = w\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2200 (c : Set (Set E)),\n    c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val} \u2192\n      IsChain (fun x x_1 => x \u2286 x_1) c \u2192\n        Set.Nonempty c \u2192 \u2203 ub, ub \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2227 \u2200 (s : Set E), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nobtain \u27e8b, bi, sb, h\u27e9 := this\n[GOAL]\ncase refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 (a : Set E), a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2192 b \u2286 a \u2192 a = b\n\u22a2 \u2203 w _hw, Orthonormal \ud835\udd5c Subtype.val \u2227 \u2200 (u : Set E), u \u2287 w \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = w\n[PROOFSTEP]\nrefine' \u27e8b, sb, bi, _\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nb : Set E\nbi : b \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\nsb : s \u2286 b\nh : \u2200 (a : Set E), a \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2192 b \u2286 a \u2192 a = b\n\u22a2 \u2200 (u : Set E), u \u2287 b \u2192 Orthonormal \ud835\udd5c Subtype.val \u2192 u = b\n[PROOFSTEP]\nexact fun u hus hu => h u hu hus\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\n\u22a2 \u2200 (c : Set (Set E)),\n    c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val} \u2192\n      IsChain (fun x x_1 => x \u2286 x_1) c \u2192\n        Set.Nonempty c \u2192 \u2203 ub, ub \u2208 {b | Orthonormal \ud835\udd5c Subtype.val} \u2227 \u2200 (s : Set E), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nrefine' fun c hc cc _c0 => \u27e8\u22c3\u2080 c, _, _\u27e9\n[GOAL]\ncase refine_1.refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : Set.Nonempty c\n\u22a2 \u22c3\u2080 c \u2208 {b | Orthonormal \ud835\udd5c Subtype.val}\n[PROOFSTEP]\nexact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc\n[GOAL]\ncase refine_1.refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\ns : Set E\nhs : Orthonormal \ud835\udd5c Subtype.val\nc : Set (Set E)\nhc : c \u2286 {b | Orthonormal \ud835\udd5c Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : Set.Nonempty c\n\u22a2 \u2200 (s : Set E), s \u2208 c \u2192 s \u2286 \u22c3\u2080 c\n[PROOFSTEP]\nexact fun _ => Set.subset_sUnion_of_mem\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni : \u03b9\n\u22a2 v i \u2260 0\n[PROOFSTEP]\nhave : \u2016v i\u2016 \u2260 0 := by\n  rw [hv.1 i]\n  norm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni : \u03b9\n\u22a2 \u2016v i\u2016 \u2260 0\n[PROOFSTEP]\nrw [hv.1 i]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni : \u03b9\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni : \u03b9\nthis : \u2016v i\u2016 \u2260 0\n\u22a2 v i \u2260 0\n[PROOFSTEP]\nsimpa using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x x) = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nrw [@norm_eq_sqrt_inner \ud835\udd5c, \u2190 sqrt_mul inner_self_nonneg (re \u27eax, x\u27eb), sqrt_mul_self inner_self_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191re (inner x x) = \u2016x\u2016 ^ 2\n[PROOFSTEP]\nrw [pow_two, inner_self_eq_norm_mul_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\n\u22a2 inner x x = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nhave h := @inner_self_eq_norm_mul_norm \u211d F _ _ _ x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nh : \u2191re (inner x x) = \u2016x\u2016 * \u2016x\u2016\n\u22a2 inner x x = \u2016x\u2016 * \u2016x\u2016\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\n\u22a2 inner x x = \u2016x\u2016 ^ 2\n[PROOFSTEP]\nrw [pow_two, real_inner_self_eq_norm_mul_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrepeat' rw [sq (M := \u211d), \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [sq (M := \u211d), \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner (x + y) (x + y)) = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [sq (M := \u211d), \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner (x + y) (x + y)) = \u2191re (inner x x) + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [sq (M := \u211d), \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner (x + y) (x + y)) = \u2191re (inner x x) + 2 * \u2191re (inner x y) + \u2191re (inner y y)\n[PROOFSTEP]\nrw [sq (M := \u211d), \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner (x + y) (x + y)) = \u2191re (inner x x) + 2 * \u2191re (inner x y) + \u2191re (inner y y)\n[PROOFSTEP]\nrw [inner_add_add_self, two_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x x + inner x y + inner y x + inner y y) =\n    \u2191re (inner x x) + (\u2191re (inner x y) + \u2191re (inner x y)) + \u2191re (inner y y)\n[PROOFSTEP]\nsimp only [add_assoc, add_left_inj, add_right_inj, AddMonoidHom.map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner y x) = \u2191re (inner x y)\n[PROOFSTEP]\nrw [\u2190 inner_conj_symm, conj_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * inner x y + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nhave h := @norm_add_sq \u211d _ _ _ _ x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * inner x y + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrepeat' rw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 ^ 2 = \u2016x\u2016 ^ 2 + 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nexact norm_add_sq _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + 2 * inner x y + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nhave h := @norm_add_mul_self \u211d _ _ _ _ x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + 2 * inner x y + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 ^ 2 = \u2016x\u2016 ^ 2 - 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [sub_eq_add_neg, @norm_add_sq \ud835\udd5c _ _ _ _ x (-y), norm_neg, inner_neg_right, map_neg, mul_neg, sub_eq_add_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrepeat' rw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 ^ 2 = \u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 ^ 2 = \u2016x\u2016 ^ 2 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 ^ 2 = \u2016x\u2016 ^ 2 - 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 sq (M := \u211d)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x - y\u2016 ^ 2 = \u2016x\u2016 ^ 2 - 2 * \u2191re (inner x y) + \u2016y\u2016 ^ 2\n[PROOFSTEP]\nexact norm_sub_sq _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 - 2 * inner x y + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nhave h := @norm_sub_mul_self \u211d _ _ _ _ x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 - 2 * inner x y + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [norm_eq_sqrt_inner (\ud835\udd5c := \ud835\udd5c) x, norm_eq_sqrt_inner (\ud835\udd5c := \ud835\udd5c) y]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y\u2016 \u2264 sqrt (\u2191re (inner x x)) * sqrt (\u2191re (inner y y))\n[PROOFSTEP]\nletI : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nthis : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\n\u22a2 \u2016inner x y\u2016 \u2264 sqrt (\u2191re (inner x x)) * sqrt (\u2191re (inner y y))\n[PROOFSTEP]\nexact InnerProductSpace.Core.norm_inner_le_norm x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 + \u2016x - y\u2016 * \u2016x - y\u2016 = 2 * (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016)\n[PROOFSTEP]\nsimp only [\u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner (x + y) (x + y)) + \u2191re (inner (x - y) (x - y)) = 2 * (\u2191re (inner x x) + \u2191re (inner y y))\n[PROOFSTEP]\nrw [\u2190 re.map_add, parallelogram_law, two_mul, two_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x x + inner y y + (inner x x + inner y y)) =\n    \u2191re (inner x x) + \u2191re (inner y y) + (\u2191re (inner x x) + \u2191re (inner y y))\n[PROOFSTEP]\nsimp only [re.map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x + y\u2016 * \u2016x + y\u2016 - \u2016x\u2016 * \u2016x\u2016 - \u2016y\u2016 * \u2016y\u2016) / 2\n[PROOFSTEP]\nrw [@norm_add_mul_self \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016 - \u2016x\u2016 * \u2016x\u2016 - \u2016y\u2016 * \u2016y\u2016) / 2\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 2\n[PROOFSTEP]\nrw [@norm_sub_mul_self \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 - (\u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016)) / 2\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x + y\u2016 * \u2016x + y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 4\n[PROOFSTEP]\nrw [@norm_add_mul_self \ud835\udd5c, @norm_sub_mul_self \ud835\udd5c]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191re (inner x y) = (\u2016x\u2016 * \u2016x\u2016 + 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016 - (\u2016x\u2016 * \u2016x\u2016 - 2 * \u2191re (inner x y) + \u2016y\u2016 * \u2016y\u2016)) / 4\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191im (inner x y) = (\u2016x - I \u2022 y\u2016 * \u2016x - I \u2022 y\u2016 - \u2016x + I \u2022 y\u2016 * \u2016x + I \u2022 y\u2016) / 4\n[PROOFSTEP]\nsimp only [@norm_add_mul_self \ud835\udd5c, @norm_sub_mul_self \ud835\udd5c, inner_smul_right, I_mul_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191im (inner x y) =\n    (\u2016x\u2016 * \u2016x\u2016 - 2 * -\u2191im (inner x y) + \u2016I \u2022 y\u2016 * \u2016I \u2022 y\u2016 - (\u2016x\u2016 * \u2016x\u2016 + 2 * -\u2191im (inner x y) + \u2016I \u2022 y\u2016 * \u2016I \u2022 y\u2016)) / 4\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x y = (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4\n[PROOFSTEP]\nrw [\u2190 re_add_im \u27eax, y\u27eb, re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four,\n  im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2191((\u2016x + y\u2016 * \u2016x + y\u2016 - \u2016x - y\u2016 * \u2016x - y\u2016) / 4) + \u2191((\u2016x - I \u2022 y\u2016 * \u2016x - I \u2022 y\u2016 - \u2016x + I \u2022 y\u2016 * \u2016x + I \u2022 y\u2016) / 4) * I =\n    (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4\n[PROOFSTEP]\npush_cast\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 (\u2191\u2016x + y\u2016 * \u2191\u2016x + y\u2016 - \u2191\u2016x - y\u2016 * \u2191\u2016x - y\u2016) / 4 +\n      (\u2191\u2016x - I \u2022 y\u2016 * \u2191\u2016x - I \u2022 y\u2016 - \u2191\u2016x + I \u2022 y\u2016 * \u2191\u2016x + I \u2022 y\u2016) / 4 * I =\n    (\u2191\u2016x + y\u2016 ^ 2 - \u2191\u2016x - y\u2016 ^ 2 + (\u2191\u2016x - I \u2022 y\u2016 ^ 2 - \u2191\u2016x + I \u2022 y\u2016 ^ 2) * I) / 4\n[PROOFSTEP]\nsimp only [sq, \u2190 mul_div_right_comm, \u2190 add_div]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : x \u2260 0\nhy : y \u2260 0\nR : \u211d\nhx' : \u2016x\u2016 \u2260 0\nhy' : \u2016y\u2016 \u2260 0\n\u22a2 dist ((R / \u2016x\u2016) ^ 2 \u2022 x) ((R / \u2016y\u2016) ^ 2 \u2022 y) = sqrt (\u2016(R / \u2016x\u2016) ^ 2 \u2022 x - (R / \u2016y\u2016) ^ 2 \u2022 y\u2016 ^ 2)\n[PROOFSTEP]\nrw [dist_eq_norm, sqrt_sq (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : x \u2260 0\nhy : y \u2260 0\nR : \u211d\nhx' : \u2016x\u2016 \u2260 0\nhy' : \u2016y\u2016 \u2260 0\n\u22a2 \u2016(R / \u2016x\u2016) ^ 2 \u2022 x - (R / \u2016y\u2016) ^ 2 \u2022 y\u2016 ^ 2 = (R ^ 2 / (\u2016x\u2016 * \u2016y\u2016)) ^ 2 * \u2016x - y\u2016 ^ 2\n[PROOFSTEP]\nfield_simp [sq, norm_sub_mul_self_real, norm_smul, real_inner_smul_left, inner_smul_right,\n  Real.norm_of_nonneg (mul_self_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : x \u2260 0\nhy : y \u2260 0\nR : \u211d\nhx' : \u2016x\u2016 \u2260 0\nhy' : \u2016y\u2016 \u2260 0\n\u22a2 ((R * R * \u2016x\u2016 * (R * R * \u2016x\u2016) * (\u2016y\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016x\u2016)) -\n            \u2016x\u2016 * \u2016x\u2016 * (\u2016x\u2016 * \u2016x\u2016) * (2 * (R * R * (R * R * inner x y)))) *\n          (\u2016y\u2016 * \u2016y\u2016 * (\u2016y\u2016 * \u2016y\u2016)) +\n        R * R * \u2016y\u2016 * (R * R * \u2016y\u2016) * (\u2016x\u2016 * \u2016x\u2016 * (\u2016x\u2016 * \u2016x\u2016) * (\u2016y\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016x\u2016)))) *\n      (\u2016x\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016y\u2016)) =\n    R * R * (R * R) * (\u2016x\u2016 * \u2016x\u2016 - 2 * inner x y + \u2016y\u2016 * \u2016y\u2016) *\n      (\u2016x\u2016 * \u2016x\u2016 * (\u2016x\u2016 * \u2016x\u2016) * (\u2016y\u2016 * \u2016y\u2016 * (\u2016x\u2016 * \u2016x\u2016)) * (\u2016y\u2016 * \u2016y\u2016 * (\u2016y\u2016 * \u2016y\u2016)))\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : x \u2260 0\nhy : y \u2260 0\nR : \u211d\nhx' : \u2016x\u2016 \u2260 0\nhy' : \u2016y\u2016 \u2260 0\n\u22a2 sqrt ((R ^ 2 / (\u2016x\u2016 * \u2016y\u2016)) ^ 2 * \u2016x - y\u2016 ^ 2) = R ^ 2 / (\u2016x\u2016 * \u2016y\u2016) * dist x y\n[PROOFSTEP]\nrw [sqrt_mul (sq_nonneg _), sqrt_sq (norm_nonneg _),\n  sqrt_sq (div_nonneg (sq_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _))), dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4, 0 < \u03b4 \u2227 \u2200 \u2983x : F\u2984, \u2016x\u2016 = 1 \u2192 \u2200 \u2983y : F\u2984, \u2016y\u2016 = 1 \u2192 \u03b5 \u2264 \u2016x - y\u2016 \u2192 \u2016x + y\u2016 \u2264 2 - \u03b4\n[PROOFSTEP]\nrefine' \u27e82 - sqrt (4 - \u03b5 ^ 2), sub_pos_of_lt <| (sqrt_lt' zero_lt_two).2 _, fun x hx y hy hxy => _\u27e9\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 4 - \u03b5 ^ 2 < 2 ^ 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 0 < \u03b5 ^ 2\n[PROOFSTEP]\nexact pow_pos h\u03b5 _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nx : F\nhx : \u2016x\u2016 = 1\ny : F\nhy : \u2016y\u2016 = 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 2 - (2 - sqrt (4 - \u03b5 ^ 2))\n[PROOFSTEP]\nrw [sub_sub_cancel]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nx : F\nhx : \u2016x\u2016 = 1\ny : F\nhy : \u2016y\u2016 = 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 \u2264 sqrt (4 - \u03b5 ^ 2)\n[PROOFSTEP]\nrefine' le_sqrt_of_sq_le _\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nx : F\nhx : \u2016x\u2016 = 1\ny : F\nhy : \u2016y\u2016 = 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 \u2016x + y\u2016 ^ 2 \u2264 4 - \u03b5 ^ 2\n[PROOFSTEP]\nrw [sq, eq_sub_iff_add_eq.2 (parallelogram_law_with_norm \u211d x y), \u2190 sq \u2016x - y\u2016, hx, hy]\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nx : F\nhx : \u2016x\u2016 = 1\ny : F\nhy : \u2016y\u2016 = 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 2 * (1 * 1 + 1 * 1) - \u2016x - y\u2016 ^ 2 \u2264 4 - \u03b5 ^ 2\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nx : F\nhx : \u2016x\u2016 = 1\ny : F\nhy : \u2016y\u2016 = 1\nhxy : \u03b5 \u2264 \u2016x - y\u2016\n\u22a2 4 - \u2016x - y\u2016 ^ 2 \u2264 4 - \u03b5 ^ 2\n[PROOFSTEP]\nexact sub_le_sub_left (pow_le_pow_of_le_left h\u03b5.le hxy _) 4\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nx y : V\n\u22a2 inner (\u2191T y) x =\n    (inner (\u2191T (x + y)) (x + y) - inner (\u2191T (x - y)) (x - y) +\n          Complex.I * inner (\u2191T (x + Complex.I \u2022 y)) (x + Complex.I \u2022 y) -\n        Complex.I * inner (\u2191T (x - Complex.I \u2022 y)) (x - Complex.I \u2022 y)) /\n      4\n[PROOFSTEP]\nsimp only [map_add, map_sub, inner_add_left, inner_add_right, LinearMap.map_smul, inner_smul_left, inner_smul_right,\n  Complex.conj_I, \u2190 pow_two, Complex.I_sq, inner_sub_left, inner_sub_right, mul_add, \u2190 mul_assoc, mul_neg, neg_neg,\n  sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nx y : V\n\u22a2 inner (\u2191T y) x =\n    (inner (\u2191T x) x + inner (\u2191T y) x + (inner (\u2191T x) y + inner (\u2191T y) y) -\n            (inner (\u2191T x) x - (inner (\u2191T y) x + (inner (\u2191T x) y - inner (\u2191T y) y))) +\n          (Complex.I * inner (\u2191T x) x + inner (\u2191T y) x + (-inner (\u2191T x) y + Complex.I * inner (\u2191T y) y)) -\n        (Complex.I * inner (\u2191T x) x - (inner (\u2191T y) x + (-inner (\u2191T x) y - Complex.I * inner (\u2191T y) y)))) /\n      4\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nx y : V\n\u22a2 inner (\u2191T x) y =\n    (inner (\u2191T (x + y)) (x + y) - inner (\u2191T (x - y)) (x - y) -\n          Complex.I * inner (\u2191T (x + Complex.I \u2022 y)) (x + Complex.I \u2022 y) +\n        Complex.I * inner (\u2191T (x - Complex.I \u2022 y)) (x - Complex.I \u2022 y)) /\n      4\n[PROOFSTEP]\nsimp only [map_add, map_sub, inner_add_left, inner_add_right, LinearMap.map_smul, inner_smul_left, inner_smul_right,\n  Complex.conj_I, \u2190 pow_two, Complex.I_sq, inner_sub_left, inner_sub_right, mul_add, \u2190 mul_assoc, mul_neg, neg_neg,\n  sub_neg_eq_add, one_mul, neg_one_mul, mul_sub, sub_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nx y : V\n\u22a2 inner (\u2191T x) y =\n    (inner (\u2191T x) x + inner (\u2191T y) x + (inner (\u2191T x) y + inner (\u2191T y) y) -\n          (inner (\u2191T x) x - (inner (\u2191T y) x + (inner (\u2191T x) y - inner (\u2191T y) y)) +\n            (Complex.I * inner (\u2191T x) x + inner (\u2191T y) x + (-inner (\u2191T x) y + Complex.I * inner (\u2191T y) y))) +\n        (Complex.I * inner (\u2191T x) x - (inner (\u2191T y) x + (-inner (\u2191T x) y - Complex.I * inner (\u2191T y) y)))) /\n      4\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\n\u22a2 (\u2200 (x : V), inner (\u2191T x) x = 0) \u2194 T = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\n\u22a2 (\u2200 (x : V), inner (\u2191T x) x = 0) \u2192 T = 0\n[PROOFSTEP]\nintro hT\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nhT : \u2200 (x : V), inner (\u2191T x) x = 0\n\u22a2 T = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase mp.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nhT : \u2200 (x : V), inner (\u2191T x) x = 0\nx : V\n\u22a2 \u2191T x = \u21910 x\n[PROOFSTEP]\nsimp only [LinearMap.zero_apply, \u2190 @inner_self_eq_zero \u2102 V]\n[GOAL]\ncase mp.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nhT : \u2200 (x : V), inner (\u2191T x) x = 0\nx : V\n\u22a2 inner (\u2191T x) (\u2191T x) = 0\n[PROOFSTEP]\nsimp (config := { singlePass := true }) only [inner_map_polarization]\n[GOAL]\ncase mp.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nhT : \u2200 (x : V), inner (\u2191T x) x = 0\nx : V\n\u22a2 (inner (\u2191T (\u2191T x + x)) (\u2191T x + x) - inner (\u2191T (\u2191T x - x)) (\u2191T x - x) +\n          Complex.I * inner (\u2191T (\u2191T x + Complex.I \u2022 x)) (\u2191T x + Complex.I \u2022 x) -\n        Complex.I * inner (\u2191T (\u2191T x - Complex.I \u2022 x)) (\u2191T x - Complex.I \u2022 x)) /\n      4 =\n    0\n[PROOFSTEP]\nsimp only [hT]\n[GOAL]\ncase mp.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\nhT : \u2200 (x : V), inner (\u2191T x) x = 0\nx : V\n\u22a2 (0 - 0 + Complex.I * 0 - Complex.I * 0) / 4 = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nT : V \u2192\u2097[\u2102] V\n\u22a2 T = 0 \u2192 \u2200 (x : V), inner (\u2191T x) x = 0\n[PROOFSTEP]\nrintro rfl x\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nx : V\n\u22a2 inner (\u21910 x) x = 0\n[PROOFSTEP]\nsimp only [LinearMap.zero_apply, inner_zero_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nS T : V \u2192\u2097[\u2102] V\n\u22a2 (\u2200 (x : V), inner (\u2191S x) x = inner (\u2191T x) x) \u2194 S = T\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 inner_map_self_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nS T : V \u2192\u2097[\u2102] V\n\u22a2 (\u2200 (x : V), inner (\u2191S x) x = inner (\u2191T x) x) \u2194 \u2200 (x : V), inner (\u2191(S - T) x) x = 0\n[PROOFSTEP]\nrefine' forall_congr' fun x => _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nV : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u2102 V\nS T : V \u2192\u2097[\u2102] V\nx : V\n\u22a2 inner (\u2191S x) x = inner (\u2191T x) x \u2194 inner (\u2191(S - T) x) x = 0\n[PROOFSTEP]\nrw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nsimp [inner_eq_sum_norm_sq_div_four, \u2190 f.norm_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2192\u2097[\ud835\udd5c] E'\nh : \u2200 (x y : E), inner (\u2191f x) (\u2191f y) = inner x y\nx : E\n\u22a2 \u2016\u2191f x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nsimp only [@norm_eq_sqrt_inner \ud835\udd5c, h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : \u03b9 \u2192 E\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\n\u22a2 Orthonormal \ud835\udd5c (\u2191f \u2218 v) \u2194 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nclassical simp_rw [orthonormal_iff_ite, Function.comp_apply, LinearIsometry.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : \u03b9 \u2192 E\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\n\u22a2 Orthonormal \ud835\udd5c (\u2191f \u2218 v) \u2194 Orthonormal \ud835\udd5c v\n[PROOFSTEP]\nsimp_rw [orthonormal_iff_ite, Function.comp_apply, LinearIsometry.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nf : E \u2192\u2097\u1d62[\ud835\udd5c] E'\n\u22a2 Orthonormal \ud835\udd5c (\u2191f \u2218 v)\n[PROOFSTEP]\nrwa [f.orthonormal_comp_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2192\u2097[\ud835\udd5c] E'\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nhf : Orthonormal \ud835\udd5c (\u2191f \u2218 \u2191v)\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nclassical rw [\u2190 v.total_repr x, \u2190 v.total_repr y, Finsupp.apply_total, Finsupp.apply_total,\n  hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2192\u2097[\ud835\udd5c] E'\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nhf : Orthonormal \ud835\udd5c (\u2191f \u2218 \u2191v)\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nrw [\u2190 v.total_repr x, \u2190 v.total_repr y, Finsupp.apply_total, Finsupp.apply_total, hv.inner_finsupp_eq_sum_left,\n  hf.inner_finsupp_eq_sum_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2243\u2097[\ud835\udd5c] E'\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nhf : Orthonormal \ud835\udd5c (\u2191f \u2218 \u2191v)\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nrw [\u2190 LinearEquiv.coe_coe] at hf \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2243\u2097[\ud835\udd5c] E'\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nhf : Orthonormal \ud835\udd5c (\u2191\u2191f \u2218 \u2191v)\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nclassical rw [\u2190 v.total_repr x, \u2190 v.total_repr y, \u2190 LinearEquiv.coe_coe f, Finsupp.apply_total, Finsupp.apply_total,\n  hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nf : E \u2243\u2097[\ud835\udd5c] E'\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nhf : Orthonormal \ud835\udd5c (\u2191\u2191f \u2218 \u2191v)\nx y : E\n\u22a2 inner (\u2191f x) (\u2191f y) = inner x y\n[PROOFSTEP]\nrw [\u2190 v.total_repr x, \u2190 v.total_repr y, \u2190 LinearEquiv.coe_coe f, Finsupp.apply_total, Finsupp.apply_total,\n  hv.inner_finsupp_eq_sum_left, hf.inner_finsupp_eq_sum_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\n\u22a2 Orthonormal \ud835\udd5c (\u2191(Basis.equiv v v' e) \u2218 \u2191v)\n[PROOFSTEP]\nhave h : v.equiv v' e \u2218 v = v' \u2218 e := by\n  ext i\n  simp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\n\u22a2 \u2191(Basis.equiv v v' e) \u2218 \u2191v = \u2191v' \u2218 \u2191e\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9\n\u22a2 (\u2191(Basis.equiv v v' e) \u2218 \u2191v) i = (\u2191v' \u2218 \u2191e) i\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\nh : \u2191(Basis.equiv v v' e) \u2218 \u2191v = \u2191v' \u2218 \u2191e\n\u22a2 Orthonormal \ud835\udd5c (\u2191(Basis.equiv v v' e) \u2218 \u2191v)\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\nh : \u2191(Basis.equiv v v' e) \u2218 \u2191v = \u2191v' \u2218 \u2191e\n\u22a2 Orthonormal \ud835\udd5c (\u2191v' \u2218 \u2191e)\n[PROOFSTEP]\nclassical exact hv'.comp _ e.injective\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\nh : \u2191(Basis.equiv v v' e) \u2218 \u2191v = \u2191v' \u2218 \u2191e\n\u22a2 Orthonormal \ud835\udd5c (\u2191v' \u2218 \u2191e)\n[PROOFSTEP]\nexact hv'.comp _ e.injective\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\ni : \u03b9\n\u22a2 \u2191(equiv hv hv (Equiv.refl \u03b9)) (\u2191v i) = \u2191(LinearIsometryEquiv.refl \ud835\udd5c E) (\u2191v i)\n[PROOFSTEP]\nsimp only [Orthonormal.equiv_apply, Equiv.coe_refl, id.def, LinearIsometryEquiv.coe_refl]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\ni : \u03b9'\n\u22a2 \u2191(equiv hv hv' e) (\u2191(LinearIsometryEquiv.symm (equiv hv hv' e)) (\u2191v' i)) =\n    \u2191(equiv hv hv' e) (\u2191(equiv hv' hv e.symm) (\u2191v' i))\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.apply_symm_apply, Orthonormal.equiv_apply, e.apply_symm_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\n\u03b9' : Type u_5\n\u03b9'' : Type u_6\nE' : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E'\nE'' : Type u_8\ninst\u271d\u00b9 : NormedAddCommGroup E''\ninst\u271d : InnerProductSpace \ud835\udd5c E''\nv : Basis \u03b9 \ud835\udd5c E\nhv : Orthonormal \ud835\udd5c \u2191v\nv' : Basis \u03b9' \ud835\udd5c E'\nhv' : Orthonormal \ud835\udd5c \u2191v'\ne : \u03b9 \u2243 \u03b9'\nv'' : Basis \u03b9'' \ud835\udd5c E''\nhv'' : Orthonormal \ud835\udd5c \u2191v''\ne' : \u03b9' \u2243 \u03b9''\ni : \u03b9\n\u22a2 \u2191(LinearIsometryEquiv.trans (equiv hv hv' e) (equiv hv' hv'' e')) (\u2191v i) = \u2191(equiv hv hv'' (e.trans e')) (\u2191v i)\n[PROOFSTEP]\nsimp only [LinearIsometryEquiv.trans_apply, Orthonormal.equiv_apply, e.coe_trans, Function.comp_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 \u2194 inner x y = 0\n[PROOFSTEP]\nrw [@norm_add_mul_self \u211d, add_right_cancel_iff, add_right_eq_self, mul_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 2 = 0 \u2228 \u2191re (inner x y) = 0 \u2194 inner x y = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x + y\u2016 = sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016) \u2194 inner x y = 0\n[PROOFSTEP]\nrw [\u2190 norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero, eq_comm,\n  sqrt_eq_iff_mul_self_eq (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : inner x y = 0\n\u22a2 \u2016x + y\u2016 * \u2016x + y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [@norm_add_mul_self \ud835\udd5c, add_right_cancel_iff, add_right_eq_self, mul_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : inner x y = 0\n\u22a2 2 = 0 \u2228 \u2191re (inner x y) = 0\n[PROOFSTEP]\napply Or.inr\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : inner x y = 0\n\u22a2 \u2191re (inner x y) = 0\n[PROOFSTEP]\nsimp only [h, zero_re']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x - y\u2016 * \u2016x - y\u2016 = \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016 \u2194 inner x y = 0\n[PROOFSTEP]\nrw [@norm_sub_mul_self \u211d, add_right_cancel_iff, sub_eq_add_neg, add_right_eq_self, neg_eq_zero, mul_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 2 = 0 \u2228 \u2191re (inner x y) = 0 \u2194 inner x y = 0\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 \u2016x - y\u2016 = sqrt (\u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016) \u2194 inner x y = 0\n[PROOFSTEP]\nrw [\u2190 norm_sub_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero, eq_comm,\n  sqrt_eq_iff_mul_self_eq (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _)) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner (x + y) (x - y) = 0 \u2194 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nconv_rhs => rw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n| \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n| \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n| \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner (x + y) (x - y) = 0 \u2194 \u2016x\u2016 * \u2016x\u2016 = \u2016y\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp only [\u2190 @inner_self_eq_norm_mul_norm \u211d, inner_add_left, inner_sub_right, real_inner_comm y x, sub_eq_zero,\n  re_to_real]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x x + inner y x = inner y x + inner y y \u2194 inner x x = inner y y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x x + inner y x = inner y x + inner y y \u2192 inner x x = inner y y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x x + inner y x = inner y x + inner y y\n\u22a2 inner x x = inner y y\n[PROOFSTEP]\nrw [add_comm] at h \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner y x + inner x x = inner y x + inner y y\n\u22a2 inner x x = inner y y\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x x = inner y y \u2192 inner x x + inner y x = inner y x + inner y y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x x = inner y y\n\u22a2 inner x x + inner y x = inner y x + inner y y\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nv w : E\nh : inner v w = 0\n\u22a2 \u2016w - v\u2016 = \u2016w + v\u2016\n[PROOFSTEP]\nrw [\u2190 mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nv w : E\nh : inner v w = 0\n\u22a2 \u2016w - v\u2016 * \u2016w - v\u2016 = \u2016w + v\u2016 * \u2016w + v\u2016\n[PROOFSTEP]\nsimp only [h, \u2190 @inner_self_eq_norm_mul_norm \ud835\udd5c, sub_neg_eq_add, sub_zero, map_sub, zero_re', zero_sub, add_zero,\n  map_add, inner_add_right, inner_sub_left, inner_sub_right, inner_re_symm, zero_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 |inner x y / (\u2016x\u2016 * \u2016y\u2016)| \u2264 1\n[PROOFSTEP]\nrw [abs_div, abs_mul, abs_norm, abs_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 |inner x y| / (\u2016x\u2016 * \u2016y\u2016) \u2264 1\n[PROOFSTEP]\nexact\n  div_le_one_of_le (abs_real_inner_le_norm x y)\n    (mul_nonneg (norm_nonneg _) (norm_nonneg _))\n      -- porting note: was `(by positivity)`\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\n\u22a2 inner (r \u2022 x) x = r * (\u2016x\u2016 * \u2016x\u2016)\n[PROOFSTEP]\nrw [real_inner_smul_left, \u2190 real_inner_self_eq_norm_mul_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\n\u22a2 inner x (r \u2022 x) = r * (\u2016x\u2016 * \u2016x\u2016)\n[PROOFSTEP]\nrw [inner_smul_right, \u2190 real_inner_self_eq_norm_mul_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nhave hx' : \u2016x\u2016 \u2260 0 := by simp [hx]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\n\u22a2 \u2016x\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\nhx' : \u2016x\u2016 \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nhave hr' : \u2016r\u2016 \u2260 0 := by simp [hr]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\nhx' : \u2016x\u2016 \u2260 0\n\u22a2 \u2016r\u2016 \u2260 0\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\nhx' : \u2016x\u2016 \u2260 0\nhr' : \u2016r\u2016 \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nrw [inner_smul_right, norm_mul, \u2190 inner_self_re_eq_norm, inner_self_eq_norm_mul_norm, norm_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\nhx : x \u2260 0\nhr : r \u2260 0\nhx' : \u2016x\u2016 \u2260 0\nhr' : \u2016r\u2016 \u2260 0\n\u22a2 \u2016r\u2016 * (\u2016x\u2016 * \u2016x\u2016) / (\u2016x\u2016 * (\u2016r\u2016 * \u2016x\u2016)) = 1\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 div_div, mul_div_cancel _ hx', \u2190 div_div, mul_comm, mul_div_cancel _ hr', div_self hx']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\nhx : x \u2260 0\nhr : 0 < r\n\u22a2 inner x (r \u2022 x) / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nrw [real_inner_smul_self_right, norm_smul, Real.norm_eq_abs, \u2190 mul_assoc \u2016x\u2016, mul_comm _ |r|, mul_assoc,\n  abs_of_nonneg hr.le, div_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\nhx : x \u2260 0\nhr : 0 < r\n\u22a2 r * (\u2016x\u2016 * \u2016x\u2016) \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero hr.ne' (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\nhx : x \u2260 0\nhr : r < 0\n\u22a2 inner x (r \u2022 x) / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = -1\n[PROOFSTEP]\nrw [real_inner_smul_self_right, norm_smul, Real.norm_eq_abs, \u2190 mul_assoc \u2016x\u2016, mul_comm _ |r|, mul_assoc, abs_of_neg hr,\n  neg_mul, div_neg_eq_neg_div, div_self]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nr : \u211d\nhx : x \u2260 0\nhr : r < 0\n\u22a2 r * (\u2016x\u2016 * \u2016x\u2016) \u2260 0\n[PROOFSTEP]\nexact mul_ne_zero hr.ne (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nrefine' fun h => or_iff_not_imp_left.2 fun hx\u2080 => _\n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhx\u2080 : \u00acx = 0\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nhave : \u2016x\u2016 ^ 2 \u2260 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx\u2080)\n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\nhx\u2080 : \u00acx = 0\nthis : \u2016x\u2016 ^ 2 \u2260 0\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nrw [\u2190 sq_eq_sq (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)), mul_pow, \u2190 mul_right_inj' this, eq_comm, \u2190\n  sub_eq_zero, \u2190 mul_sub] at h \n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u271d : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2) = \u2016x\u2016 ^ 2 * \u2016inner x y\u2016 ^ 2\nh : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2 - \u2016inner x y\u2016 ^ 2) = 0\nhx\u2080 : \u00acx = 0\nthis : \u2016x\u2016 ^ 2 \u2260 0\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nsimp only [@norm_sq_eq_inner \ud835\udd5c] at h \n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u271d : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2) = \u2016x\u2016 ^ 2 * \u2016inner x y\u2016 ^ 2\nhx\u2080 : \u00acx = 0\nthis : \u2016x\u2016 ^ 2 \u2260 0\nh : \u2191re (inner x x) * (\u2191re (inner x x) * \u2191re (inner y y) - \u2016inner x y\u2016 ^ 2) = 0\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nletI : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u271d : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2) = \u2016x\u2016 ^ 2 * \u2016inner x y\u2016 ^ 2\nhx\u2080 : \u00acx = 0\nthis\u271d : \u2016x\u2016 ^ 2 \u2260 0\nh : \u2191re (inner x x) * (\u2191re (inner x x) * \u2191re (inner y y) - \u2016inner x y\u2016 ^ 2) = 0\nthis : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nerw [\u2190 InnerProductSpace.Core.cauchy_schwarz_aux, InnerProductSpace.Core.normSq_eq_zero, sub_eq_zero] at h \n[GOAL]\ncase tfae_1_to_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u271d : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2) = \u2016x\u2016 ^ 2 * \u2016inner x y\u2016 ^ 2\nhx\u2080 : \u00acx = 0\nthis\u271d : \u2016x\u2016 ^ 2 \u2260 0\nthis : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\nh : inner x y \u2022 x = inner x x \u2022 y\n\u22a2 y = (inner x y / inner x x) \u2022 x\n[PROOFSTEP]\nrw [div_eq_inv_mul, mul_smul, h, inv_smul_smul\u2080]\n[GOAL]\ncase tfae_1_to_2.hc\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u271d : \u2016x\u2016 ^ 2 * (\u2016x\u2016 ^ 2 * \u2016y\u2016 ^ 2) = \u2016x\u2016 ^ 2 * \u2016inner x y\u2016 ^ 2\nhx\u2080 : \u00acx = 0\nthis\u271d : \u2016x\u2016 ^ 2 \u2260 0\nthis : InnerProductSpace.Core \ud835\udd5c E := InnerProductSpace.toCore\nh : inner x y \u2022 x = inner x x \u2022 y\n\u22a2 inner x x \u2260 0\n[PROOFSTEP]\nrwa [inner_self_ne_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\ntfae_have 2 \u2192 3\n[GOAL]\ncase tfae_2_to_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\n\u22a2 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\nexact fun h => h.imp_right fun h' => \u27e8_, h'\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\ntfae_have 3 \u2192 1\n[GOAL]\ncase tfae_3_to_1\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\n\u22a2 (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2192 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrintro (rfl | \u27e8r, rfl\u27e9)\n[GOAL]\ncase tfae_3_to_1.inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ny : E\ntfae_1_to_2 : \u2016inner 0 y\u2016 = \u20160\u2016 * \u2016y\u2016 \u2192 0 = 0 \u2228 y = (inner 0 y / inner 0 0) \u2022 0\ntfae_2_to_3 : 0 = 0 \u2228 y = (inner 0 y / inner 0 0) \u2022 0 \u2192 0 = 0 \u2228 \u2203 r, y = r \u2022 0\n\u22a2 \u2016inner 0 y\u2016 = \u20160\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K, inner_self_eq_norm_mul_norm, sq, mul_left_comm]\n[GOAL]\ncase tfae_3_to_1.inr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nr : \ud835\udd5c\ntfae_1_to_2 : \u2016inner x (r \u2022 x)\u2016 = \u2016x\u2016 * \u2016r \u2022 x\u2016 \u2192 x = 0 \u2228 r \u2022 x = (inner x (r \u2022 x) / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 r \u2022 x = (inner x (r \u2022 x) / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r_1, r \u2022 x = r_1 \u2022 x\n\u22a2 \u2016inner x (r \u2022 x)\u2016 = \u2016x\u2016 * \u2016r \u2022 x\u2016\n[PROOFSTEP]\nsimp [inner_smul_right, norm_smul, inner_self_eq_norm_sq_to_K, inner_self_eq_norm_mul_norm, sq, mul_left_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\ntfae_3_to_1 : (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2192 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\ntfae_have 3 \u2194 4\n[GOAL]\ncase tfae_3_iff_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\ntfae_3_to_1 : (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2192 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\n\u22a2 (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2194 x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}\n[PROOFSTEP]\nsimp only [Submodule.mem_span_singleton, eq_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\ntfae_1_to_2 : \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016 \u2192 x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\ntfae_2_to_3 : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x \u2192 x = 0 \u2228 \u2203 r, y = r \u2022 x\ntfae_3_to_1 : (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2192 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\ntfae_3_iff_4 : (x = 0 \u2228 \u2203 r, y = r \u2022 x) \u2194 x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}\n\u22a2 List.TFAE\n    [\u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016, x = 0 \u2228 y = (inner x y / inner x x) \u2022 x, x = 0 \u2228 \u2203 r, y = r \u2022 x,\n      x = 0 \u2228 y \u2208 Submodule.span \ud835\udd5c {x}]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u2194 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1 \u2192 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nhave hx\u2080 : x \u2260 0 := fun h\u2080 => by simp [h\u2080] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nh\u2080 : x = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h\u2080] at h \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nhave hy\u2080 : y \u2260 0 := fun h\u2080 => by simp [h\u2080] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nh\u2080 : y = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h\u2080] at h \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nrefine' \u27e8hx\u2080, (norm_inner_eq_norm_iff hx\u2080 hy\u2080).1 <| eq_of_div_eq_one _\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh : \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 \u2016inner x y\u2016 / (\u2016x\u2016 * \u2016y\u2016) = 1\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 (x \u2260 0 \u2227 \u2203 r, r \u2260 0 \u2227 y = r \u2022 x) \u2192 \u2016inner x y / (\u2191\u2016x\u2016 * \u2191\u2016y\u2016)\u2016 = 1\n[PROOFSTEP]\nrintro \u27e8hx, \u27e8r, \u27e8hr, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x) / (\u2191\u2016x\u2016 * \u2191\u2016r \u2022 x\u2016)\u2016 = 1\n[PROOFSTEP]\nsimp only [norm_div, norm_mul, norm_ofReal, abs_norm]\n[GOAL]\ncase mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nhx : x \u2260 0\nr : \ud835\udd5c\nhr : r \u2260 0\n\u22a2 \u2016inner x (r \u2022 x)\u2016 / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nexact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2194 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nhave h\u2080' := h\u2080\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 h\u2080' : x \u2260 0\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2194 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nrw [\u2190 norm_ne_zero_iff, Ne.def, \u2190 @ofReal_eq_zero \ud835\udd5c] at h\u2080' \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2194 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2192 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\n\u22a2 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y \u2192 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n\u22a2 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nhave : x = 0 \u2228 y = (\u27eax, y\u27eb / \u27eax, x\u27eb : \ud835\udd5c) \u2022 x := ((@norm_inner_eq_norm_tfae \ud835\udd5c _ _ _ _ x y).out 0 1).1 (by simp [h])\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n\u22a2 \u2016inner x y\u2016 = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\nthis : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\n\u22a2 (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n[PROOFSTEP]\nrw [this.resolve_left h\u2080, h]\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\nthis : x = 0 \u2228 y = (inner x y / inner x x) \u2022 x\n\u22a2 (\u2191\u2016(\u2191\u2016x\u2016 * \u2191\u2016y\u2016 / inner x x) \u2022 x\u2016 / \u2191\u2016x\u2016) \u2022 x = (\u2191\u2016x\u2016 * \u2191\u2016y\u2016 / inner x x) \u2022 x\n[PROOFSTEP]\nsimp [norm_smul, inner_self_ofReal_norm, mul_div_cancel _ h\u2080']\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n[PROOFSTEP]\nconv_lhs => rw [\u2190 h, inner_smul_right, inner_self_eq_norm_sq_to_K]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n| inner x y\n[PROOFSTEP]\nrw [\u2190 h, inner_smul_right, inner_self_eq_norm_sq_to_K]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n| inner x y\n[PROOFSTEP]\nrw [\u2190 h, inner_smul_right, inner_self_eq_norm_sq_to_K]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n| inner x y\n[PROOFSTEP]\nrw [\u2190 h, inner_smul_right, inner_self_eq_norm_sq_to_K]\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\nh\u2080' : \u00ac\u2191\u2016x\u2016 = 0\nh : (\u2191\u2016y\u2016 / \u2191\u2016x\u2016) \u2022 x = y\n\u22a2 \u2191\u2016y\u2016 / \u2191\u2016x\u2016 * \u2191\u2016x\u2016 ^ 2 = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n[PROOFSTEP]\nfield_simp [sq, mul_left_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2194 \u2191\u2016y\u2016 \u2022 x = \u2191\u2016x\u2016 \u2022 y\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h\u2080)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ny : E\n\u22a2 inner 0 y = \u2191\u20160\u2016 * \u2191\u2016y\u2016 \u2194 \u2191\u2016y\u2016 \u2022 0 = \u2191\u20160\u2016 \u2022 y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\n\u22a2 inner x y = \u2191\u2016x\u2016 * \u2191\u2016y\u2016 \u2194 \u2191\u2016y\u2016 \u2022 x = \u2191\u2016x\u2016 \u2022 y\n[PROOFSTEP]\nrw [inner_eq_norm_mul_iff_div h\u2080, div_eq_inv_mul, mul_smul, inv_smul_eq_iff\u2080]\n[GOAL]\ncase inr.ha\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nh\u2080 : x \u2260 0\n\u22a2 \u2191\u2016x\u2016 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, ofReal_eq_zero, norm_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1 \u2194 x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1 \u2192 x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\n\u22a2 x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x\n[PROOFSTEP]\nhave hx\u2080 : x \u2260 0 := fun h\u2080 => by simp [h\u2080] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\nh\u2080 : x = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h\u2080] at h \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\nhx\u2080 : x \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x\n[PROOFSTEP]\nhave hy\u2080 : y \u2260 0 := fun h\u2080 => by simp [h\u2080] at h \n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\nhx\u2080 : x \u2260 0\nh\u2080 : y = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [h\u2080] at h \n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x\n[PROOFSTEP]\nrefine' \u27e8hx\u2080, \u2016y\u2016 / \u2016x\u2016, div_pos (norm_pos_iff.2 hy\u2080) (norm_pos_iff.2 hx\u2080), _\u27e9\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nh : inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\nhx\u2080 : x \u2260 0\nhy\u2080 : y \u2260 0\n\u22a2 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 x\n[PROOFSTEP]\nexact ((inner_eq_norm_mul_iff_div hx\u2080).1 (eq_of_div_eq_one h)).symm\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 (x \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 x) \u2192 inner x y / (\u2016x\u2016 * \u2016y\u2016) = 1\n[PROOFSTEP]\nrintro \u27e8hx, \u27e8r, \u27e8hr, rfl\u27e9\u27e9\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 < r\n\u22a2 inner x (r \u2022 x) / (\u2016x\u2016 * \u2016r \u2022 x\u2016) = 1\n[PROOFSTEP]\nexact real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul hx hr\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 inner x y / (\u2016x\u2016 * \u2016y\u2016) = -1 \u2194 x \u2260 0 \u2227 \u2203 r, r < 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nrw [\u2190 neg_eq_iff_eq_neg, \u2190 neg_div, \u2190 inner_neg_right, \u2190 norm_neg y, real_inner_div_norm_mul_norm_eq_one_iff,\n  (@neg_surjective \u211d _).exists]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\n\u22a2 (x \u2260 0 \u2227 \u2203 x_1, 0 < -x_1 \u2227 -y = -x_1 \u2022 x) \u2194 x \u2260 0 \u2227 \u2203 r, r < 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nrefine' Iff.rfl.and (exists_congr fun r => _)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nr : \u211d\n\u22a2 0 < -r \u2227 -y = -r \u2022 x \u2194 r < 0 \u2227 y = r \u2022 x\n[PROOFSTEP]\nrw [neg_pos, neg_smul, neg_inj]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 inner x y = 1 \u2194 x = y\n[PROOFSTEP]\nconvert inner_eq_norm_mul_iff (\ud835\udd5c := \ud835\udd5c) (E := E) using 2\n[GOAL]\ncase h.e'_1.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 1 = \u2191\u2016x\u2016 * \u2191\u2016y\u2016\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase h.e'_2.h.e'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 x = \u2191\u2016y\u2016 \u2022 x\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 y = \u2191\u2016x\u2016 \u2022 y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 inner x y < 1 \u2194 x \u2260 y\n[PROOFSTEP]\nconvert inner_lt_norm_mul_iff_real (F := F)\n[GOAL]\ncase h.e'_1.h.e'_4\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 1 = \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase h.e'_2.h.e'_2\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 x = \u2016y\u2016 \u2022 x\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase h.e'_2.h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : F\nhx : \u2016x\u2016 = 1\nhy : \u2016y\u2016 = 1\n\u22a2 y = \u2016x\u2016 \u2022 y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9\u2081 : Type u_4\ns\u2081 : Finset \u03b9\u2081\nw\u2081 : \u03b9\u2081 \u2192 \u211d\nv\u2081 : \u03b9\u2081 \u2192 F\nh\u2081 : \u2211 i in s\u2081, w\u2081 i = 0\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\nw\u2082 : \u03b9\u2082 \u2192 \u211d\nv\u2082 : \u03b9\u2082 \u2192 F\nh\u2082 : \u2211 i in s\u2082, w\u2082 i = 0\n\u22a2 inner (\u2211 i\u2081 in s\u2081, w\u2081 i\u2081 \u2022 v\u2081 i\u2081) (\u2211 i\u2082 in s\u2082, w\u2082 i\u2082 \u2022 v\u2082 i\u2082) =\n    (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (\u2016v\u2081 i\u2081 - v\u2082 i\u2082\u2016 * \u2016v\u2081 i\u2081 - v\u2082 i\u2082\u2016)) / 2\n[PROOFSTEP]\nsimp_rw [sum_inner, inner_sum, real_inner_smul_left, real_inner_smul_right,\n  real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two, \u2190 div_sub_div_same, \u2190 div_add_div_same,\n  mul_sub_left_distrib, left_distrib, Finset.sum_sub_distrib, Finset.sum_add_distrib, \u2190 Finset.mul_sum, \u2190\n  Finset.sum_mul, h\u2081, h\u2082, zero_mul, mul_zero, Finset.sum_const_zero, zero_add, zero_sub, Finset.mul_sum, neg_div,\n  Finset.sum_div, mul_div_assoc, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 \u2016\u2191(\u2191(inner\u209b\u2097 \ud835\udd5c) x) y\u2016 \u2264 1 * \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nsimp only [norm_inner_le_norm, one_mul, inner\u209b\u2097_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2016\u2191(innerSL \ud835\udd5c) x\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrefine' le_antisymm ((innerSL \ud835\udd5c x).op_norm_le_bound (norm_nonneg _) fun y => norm_inner_le_norm _ _) _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(innerSL \ud835\udd5c) x\u2016\n[PROOFSTEP]\nrcases eq_or_ne x 0 with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u22a2 \u20160\u2016 \u2264 \u2016\u2191(innerSL \ud835\udd5c) 0\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nh : x \u2260 0\n\u22a2 \u2016x\u2016 \u2264 \u2016\u2191(innerSL \ud835\udd5c) x\u2016\n[PROOFSTEP]\nrefine' (mul_le_mul_right (norm_pos_iff.2 h)).mp _\n[GOAL]\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nh : x \u2260 0\n\u22a2 \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016\u2191(innerSL \ud835\udd5c) x\u2016 * \u2016x\u2016\n[PROOFSTEP]\ncalc\n  \u2016x\u2016 * \u2016x\u2016 = \u2016(\u27eax, x\u27eb : \ud835\udd5c)\u2016 := by rw [\u2190 sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]\n  _ \u2264 \u2016innerSL \ud835\udd5c x\u2016 * \u2016x\u2016 := (innerSL \ud835\udd5c x).le_op_norm _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\nh : x \u2260 0\n\u22a2 \u2016x\u2016 * \u2016x\u2016 = \u2016inner x x\u2016\n[PROOFSTEP]\nrw [\u2190 sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\n\u22a2 \u2016\u2191(\u2191toSesqForm f) v\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016\n[PROOFSTEP]\nrefine' op_norm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\n\u22a2 \u2200 (x : E), \u2016\u2191(\u2191(\u2191toSesqForm f) v) x\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016\n[PROOFSTEP]\nintro x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\nx : E\n\u22a2 \u2016\u2191(\u2191(\u2191toSesqForm f) v) x\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016\n[PROOFSTEP]\nhave h\u2081 : \u2016f x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016 := le_op_norm _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\nx : E\nh\u2081 : \u2016\u2191f x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\n\u22a2 \u2016\u2191(\u2191(\u2191toSesqForm f) v) x\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016\n[PROOFSTEP]\nhave h\u2082 := @norm_inner_le_norm \ud835\udd5c E' _ _ _ v (f x)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\nx : E\nh\u2081 : \u2016\u2191f x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\nh\u2082 : \u2016inner v (\u2191f x)\u2016 \u2264 \u2016v\u2016 * \u2016\u2191f x\u2016\n\u22a2 \u2016\u2191(\u2191(\u2191toSesqForm f) v) x\u2016 \u2264 \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016\n[PROOFSTEP]\ncalc\n  \u2016\u27eav, f x\u27eb\u2016 \u2264 \u2016v\u2016 * \u2016f x\u2016 := h\u2082\n  _ \u2264 \u2016v\u2016 * (\u2016f\u2016 * \u2016x\u2016) := (mul_le_mul_of_nonneg_left h\u2081 (norm_nonneg v))\n  _ = \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016 := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nE' : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup E'\ninst\u271d : InnerProductSpace \ud835\udd5c E'\nf : E \u2192L[\ud835\udd5c] E'\nv : E'\nx : E\nh\u2081 : \u2016\u2191f x\u2016 \u2264 \u2016f\u2016 * \u2016x\u2016\nh\u2082 : \u2016inner v (\u2191f x)\u2016 \u2264 \u2016v\u2016 * \u2016\u2191f x\u2016\n\u22a2 \u2016v\u2016 * (\u2016f\u2016 * \u2016x\u2016) = \u2016f\u2016 * \u2016v\u2016 * \u2016x\u2016\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninst\u271d : NormedSpace \u211d E\nr : \u211d\nx y : E\n\u22a2 inner (r \u2022 x, y).fst (r \u2022 x, y).snd = r \u2022 inner (x, y).fst (x, y).snd\n[PROOFSTEP]\nsimp only [\u2190 algebraMap_smul \ud835\udd5c r x, algebraMap_eq_ofReal, inner_smul_real_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninst\u271d : NormedSpace \u211d E\nr : \u211d\nx y : E\n\u22a2 inner (x, r \u2022 y).fst (x, r \u2022 y).snd = r \u2022 inner (x, y).fst (x, y).snd\n[PROOFSTEP]\nsimp only [\u2190 algebraMap_smul \ud835\udd5c r y, algebraMap_eq_ofReal, inner_smul_real_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninst\u271d : NormedSpace \u211d E\nx y : E\n\u22a2 \u2016inner (x, y).fst (x, y).snd\u2016 \u2264 1 * \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nrw [one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninst\u271d : NormedSpace \u211d E\nx y : E\n\u22a2 \u2016inner (x, y).fst (x, y).snd\u2016 \u2264 \u2016x\u2016 * \u2016y\u2016\n[PROOFSTEP]\nexact norm_inner_le_norm x y\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nhave h\u2082 : (\u2211 i in s, \u2211 j in s, \u27eav i, x\u27eb * \u27eax, v j\u27eb * \u27eav j, v i\u27eb) = (\u2211 k in s, \u27eav k, x\u27eb * \u27eax, v k\u27eb : \ud835\udd5c) := by\n  classical exact hv.inner_left_right_finset\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\n[PROOFSTEP]\nclassical exact hv.inner_left_right_finset\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\n[PROOFSTEP]\nexact hv.inner_left_right_finset\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\n\u22a2 \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nhave h\u2083 : \u2200 z : \ud835\udd5c, re (z * conj z) = \u2016z\u2016 ^ 2 := by\n  intro z\n  simp only [mul_conj, normSq_eq_def']\n  norm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\n\u22a2 \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n[PROOFSTEP]\nintro z\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nz : \ud835\udd5c\n\u22a2 \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n[PROOFSTEP]\nsimp only [mul_conj, normSq_eq_def']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nz : \ud835\udd5c\n\u22a2 \u2191re \u2191(\u2016z\u2016 ^ 2) = \u2016z\u2016 ^ 2\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nsuffices hbf : \u2016x - \u2211 i in s, \u27eav i, x\u27eb \u2022 v i\u2016 ^ 2 = \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016\u27eav i, x\u27eb\u2016 ^ 2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\nhbf : \u2016x - \u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2 = \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n\u22a2 \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 sub_nonneg, \u2190 hbf]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\nhbf : \u2016x - \u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2 = \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n\u22a2 0 \u2264 \u2016x - \u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2\n[PROOFSTEP]\nsimp only [norm_nonneg, pow_nonneg]\n[GOAL]\ncase hbf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2016x - \u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2 = \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nrw [@norm_sub_sq \ud835\udd5c, sub_add]\n[GOAL]\ncase hbf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2016x\u2016 ^ 2 - (2 * \u2191re (inner x (\u2211 i in s, inner (v i) x \u2022 v i)) - \u2016\u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2) =\n    \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nclassical\nsimp only [@InnerProductSpace.norm_sq_eq_inner \ud835\udd5c, _root_.inner_sum, _root_.sum_inner]\nsimp only [inner_smul_right, two_mul, inner_smul_left, inner_conj_symm, \u2190 mul_assoc, h\u2082, add_sub_cancel, sub_right_inj]\nsimp only [map_sum, \u2190 inner_conj_symm x, \u2190 h\u2083]\n[GOAL]\ncase hbf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2016x\u2016 ^ 2 - (2 * \u2191re (inner x (\u2211 i in s, inner (v i) x \u2022 v i)) - \u2016\u2211 i in s, inner (v i) x \u2022 v i\u2016 ^ 2) =\n    \u2016x\u2016 ^ 2 - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [@InnerProductSpace.norm_sq_eq_inner \ud835\udd5c, _root_.inner_sum, _root_.sum_inner]\n[GOAL]\ncase hbf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2191re (inner x x) -\n      (2 * \u2191re (\u2211 i in s, inner x (inner (v i) x \u2022 v i)) -\n        \u2191re (\u2211 x_1 in s, \u2211 i in s, inner (inner (v i) x \u2022 v i) (inner (v x_1) x \u2022 v x_1))) =\n    \u2191re (inner x x) - \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [inner_smul_right, two_mul, inner_smul_left, inner_conj_symm, \u2190 mul_assoc, h\u2082, add_sub_cancel, sub_right_inj]\n[GOAL]\ncase hbf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\ns : Finset \u03b9\nhv : Orthonormal \ud835\udd5c v\nh\u2082 : \u2211 i in s, \u2211 j in s, inner (v i) x * inner x (v j) * inner (v j) (v i) = \u2211 k in s, inner (v k) x * inner x (v k)\nh\u2083 : \u2200 (z : \ud835\udd5c), \u2191re (z * \u2191(starRingEnd \ud835\udd5c) z) = \u2016z\u2016 ^ 2\n\u22a2 \u2191re (\u2211 x_1 in s, inner (v x_1) x * inner x (v x_1)) = \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [map_sum, \u2190 inner_conj_symm x, \u2190 h\u2083]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2211' (i : \u03b9), \u2016inner (v i) x\u2016 ^ 2 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nrefine' tsum_le_of_sum_le' _ fun s => hv.sum_inner_products_le x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 0 \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [norm_nonneg, pow_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 Summable fun i => \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nuse\u2a06 s : Finset \u03b9, \u2211 i in s, \u2016\u27eav i, x\u27eb\u2016 ^ 2\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 HasSum (fun i => \u2016inner (v i) x\u2016 ^ 2) (\u2a06 (s : Finset \u03b9), \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2)\n[PROOFSTEP]\napply hasSum_of_isLUB_of_nonneg\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2200 (i : \u03b9), 0 \u2264 \u2016inner (v i) x\u2016 ^ 2\n[PROOFSTEP]\nintro b\n[GOAL]\ncase h.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nb : \u03b9\n\u22a2 0 \u2264 \u2016inner (v b) x\u2016 ^ 2\n[PROOFSTEP]\nsimp only [norm_nonneg, pow_nonneg]\n[GOAL]\ncase h.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 IsLUB (Set.range fun s => \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2) (\u2a06 (s : Finset \u03b9), \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2)\n[PROOFSTEP]\nrefine' isLUB_ciSup _\n[GOAL]\ncase h.hf\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 BddAbove (Set.range fun s => \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2)\n[PROOFSTEP]\nuse\u2016x\u2016 ^ 2\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\n\u22a2 \u2016x\u2016 ^ 2 \u2208 upperBounds (Set.range fun s => \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2)\n[PROOFSTEP]\nrintro y \u27e8s, rfl\u27e9\n[GOAL]\ncase h.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nx : E\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 i in s, \u2016inner (v i) x\u2016 ^ 2) s \u2264 \u2016x\u2016 ^ 2\n[PROOFSTEP]\nexact hv.sum_inner_products_le x\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : \ud835\udd5c\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner x x)\n[PROOFSTEP]\nsimp only [inner]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : \ud835\udd5c\n\u22a2 \u2016x\u2016 ^ 2 = \u2191re (\u2191(starRingEnd \ud835\udd5c) x * x)\n[PROOFSTEP]\nrw [mul_comm, mul_conj, ofReal_re, normSq_eq_def']\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : \ud835\udd5c\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner y x) = inner x y\n[PROOFSTEP]\nsimp only [mul_comm, map_mul, starRingEnd_self_apply]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : \ud835\udd5c\n\u22a2 inner (x + y) z = inner x z + inner y z\n[PROOFSTEP]\nsimp only [add_mul, map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : \ud835\udd5c\n\u22a2 inner (z \u2022 x) y = \u2191(starRingEnd \ud835\udd5c) z * inner x y\n[PROOFSTEP]\nsimp only [mul_assoc, smul_eq_mul, map_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ni j : \u03b9\nhij : i \u2260 j\na : (fun _i => \ud835\udd5c) i\nb : (fun _i => \ud835\udd5c) j\n\u22a2 inner (\u2191((fun i => LinearIsometry.toSpanSingleton \ud835\udd5c E (_ : \u2016v i\u2016 = 1)) i) a)\n      (\u2191((fun i => LinearIsometry.toSpanSingleton \ud835\udd5c E (_ : \u2016v i\u2016 = 1)) j) b) =\n    0\n[PROOFSTEP]\nsimp [inner_smul_left, inner_smul_right, hv.2 hij]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ni j : \u03b9\nv : G i\nw : G j\n\u22a2 inner (\u2191(V i) v) (\u2191(V j) w) = if i = j then inner (\u2191(V i) v) (\u2191(V j) w) else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ni j : \u03b9\nv : G i\nw : G j\nh : i = j\n\u22a2 inner (\u2191(V i) v) (\u2191(V j) w) = inner (\u2191(V i) v) (\u2191(V j) w)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ni j : \u03b9\nv : G i\nw : G j\nh : \u00aci = j\n\u22a2 inner (\u2191(V i) v) (\u2191(V j) w) = 0\n[PROOFSTEP]\nexact hV h v w\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\n\u22a2 (DFinsupp.sum l fun j w => if i = j then inner (\u2191(V i) v) (\u2191(V j) w) else 0) = inner v (\u2191l i)\n[PROOFSTEP]\nsimp only [DFinsupp.sum, Submodule.coe_inner, Finset.sum_ite_eq, ite_eq_left_iff, DFinsupp.mem_support_toFun]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\n\u22a2 (if \u2191l i \u2260 0 then inner (\u2191(V i) v) (\u2191(V i) (\u2191l i)) else 0) = inner v (\u2191l i)\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : \u2191l i \u2260 0\n\u22a2 inner (\u2191(V i) v) (\u2191(V i) (\u2191l i)) = inner v (\u2191l i)\n[PROOFSTEP]\nsimp only [LinearIsometry.inner_map_map]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : \u00ac\u2191l i \u2260 0\n\u22a2 0 = inner v (\u2191l i)\n[PROOFSTEP]\nsimp only [of_not_not h, inner_zero_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : Fintype \u03b9\nl : (i : \u03b9) \u2192 G i\ni : \u03b9\nv : G i\n\u22a2 inner (\u2191(V i) v) (\u2211 j : \u03b9, \u2191(V j) (l j)) = inner v (l i)\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : Fintype \u03b9\nl : (i : \u03b9) \u2192 G i\ni : \u03b9\nv : G i\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (\u2191(V i) v) (\u2211 j : \u03b9, \u2191(V j) (l j)) = inner v (l i)\n[PROOFSTEP]\ncalc\n  \u27eaV i v, \u2211 j : \u03b9, V j (l j)\u27eb = \u2211 j : \u03b9, \u27eaV i v, V j (l j)\u27eb := by rw [inner_sum]\n  _ = \u2211 j, ite (i = j) \u27eaV i v, V j (l j)\u27eb 0 := (congr_arg (Finset.sum Finset.univ) <| funext fun j => hV.eq_ite v (l j))\n  _ = \u27eav, l i\u27eb := by simp only [Finset.sum_ite_eq, Finset.mem_univ, (V i).inner_map_map, if_true]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : Fintype \u03b9\nl : (i : \u03b9) \u2192 G i\ni : \u03b9\nv : G i\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (\u2191(V i) v) (\u2211 j : \u03b9, \u2191(V j) (l j)) = \u2211 j : \u03b9, inner (\u2191(V i) v) (\u2191(V j) (l j))\n[PROOFSTEP]\nrw [inner_sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : Fintype \u03b9\nl : (i : \u03b9) \u2192 G i\ni : \u03b9\nv : G i\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 (\u2211 j : \u03b9, if i = j then inner (\u2191(V i) v) (\u2191(V j) (l j)) else 0) = inner v (l i)\n[PROOFSTEP]\nsimp only [Finset.sum_ite_eq, Finset.mem_univ, (V i).inner_map_map, if_true]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\n\u22a2 inner (\u2211 i in s, \u2191(V i) (l\u2081 i)) (\u2211 j in s, \u2191(V j) (l\u2082 j)) = \u2211 i in s, inner (l\u2081 i) (l\u2082 i)\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (\u2211 i in s, \u2191(V i) (l\u2081 i)) (\u2211 j in s, \u2191(V j) (l\u2082 j)) = \u2211 i in s, inner (l\u2081 i) (l\u2082 i)\n[PROOFSTEP]\ncalc\n  \u27ea\u2211 i in s, V i (l\u2081 i), \u2211 j in s, V j (l\u2082 j)\u27eb = \u2211 j in s, \u2211 i in s, \u27eaV i (l\u2081 i), V j (l\u2082 j)\u27eb := by\n    simp only [_root_.sum_inner, _root_.inner_sum]\n  _ = \u2211 j in s, \u2211 i in s, ite (i = j) \u27eaV i (l\u2081 i), V j (l\u2082 j)\u27eb 0 :=\n    by\n    congr with i\n    congr with j\n    apply hV.eq_ite\n  _ = \u2211 i in s, \u27eal\u2081 i, l\u2082 i\u27eb := by\n    simp only [Finset.sum_ite_of_true, Finset.sum_ite_eq', LinearIsometry.inner_map_map, imp_self, imp_true_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 inner (\u2211 i in s, \u2191(V i) (l\u2081 i)) (\u2211 j in s, \u2191(V j) (l\u2082 j)) = \u2211 j in s, \u2211 i in s, inner (\u2191(V i) (l\u2081 i)) (\u2191(V j) (l\u2082 j))\n[PROOFSTEP]\nsimp only [_root_.sum_inner, _root_.inner_sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 \u2211 j in s, \u2211 i in s, inner (\u2191(V i) (l\u2081 i)) (\u2191(V j) (l\u2082 j)) =\n    \u2211 j in s, \u2211 i in s, if i = j then inner (\u2191(V i) (l\u2081 i)) (\u2191(V j) (l\u2082 j)) else 0\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\ni : \u03b9\n\u22a2 \u2211 i_1 in s, inner (\u2191(V i_1) (l\u2081 i_1)) (\u2191(V i) (l\u2082 i)) =\n    \u2211 i_1 in s, if i_1 = i then inner (\u2191(V i_1) (l\u2081 i_1)) (\u2191(V i) (l\u2082 i)) else 0\n[PROOFSTEP]\ncongr with j\n[GOAL]\ncase e_f.h.e_f.h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\ni j : \u03b9\n\u22a2 inner (\u2191(V j) (l\u2081 j)) (\u2191(V i) (l\u2082 i)) = if j = i then inner (\u2191(V j) (l\u2081 j)) (\u2191(V i) (l\u2082 i)) else 0\n[PROOFSTEP]\napply hV.eq_ite\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl\u2081 l\u2082 : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 (\u2211 j in s, \u2211 i in s, if i = j then inner (\u2191(V i) (l\u2081 i)) (\u2191(V j) (l\u2082 j)) else 0) = \u2211 i in s, inner (l\u2081 i) (l\u2082 i)\n[PROOFSTEP]\nsimp only [Finset.sum_ite_of_true, Finset.sum_ite_eq', LinearIsometry.inner_map_map, imp_self, imp_true_iff]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\n\u22a2 \u2016\u2211 i in s, \u2191(V i) (l i)\u2016 ^ 2 = \u2211 i in s, \u2016l i\u2016 ^ 2\n[PROOFSTEP]\nhave : ((\u2016\u2211 i in s, V i (l i)\u2016 : \u211d) : \ud835\udd5c) ^ 2 = \u2211 i in s, ((\u2016l i\u2016 : \u211d) : \ud835\udd5c) ^ 2 := by\n  simp only [\u2190 inner_self_eq_norm_sq_to_K, hV.inner_sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\n\u22a2 \u2191\u2016\u2211 i in s, \u2191(V i) (l i)\u2016 ^ 2 = \u2211 i in s, \u2191\u2016l i\u2016 ^ 2\n[PROOFSTEP]\nsimp only [\u2190 inner_self_eq_norm_sq_to_K, hV.inner_sum]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nl : (i : \u03b9) \u2192 G i\ns : Finset \u03b9\nthis : \u2191\u2016\u2211 i in s, \u2191(V i) (l i)\u2016 ^ 2 = \u2211 i in s, \u2191\u2016l i\u2016 ^ 2\n\u22a2 \u2016\u2211 i in s, \u2191(V i) (l i)\u2016 ^ 2 = \u2211 i in s, \u2016l i\u2016 ^ 2\n[PROOFSTEP]\nexact_mod_cast this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\n\u22a2 Orthonormal \ud835\udd5c fun a => \u2191(V a.fst) (v_family a.fst a.snd)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\n\u22a2 \u2200 (i : (i : \u03b9) \u00d7 \u03b1 i), \u2016(fun a => \u2191(V a.fst) (v_family a.fst a.snd)) i\u2016 = 1\n[PROOFSTEP]\nrintro \u27e8i, v\u27e9\n[GOAL]\ncase left.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv : \u03b1 i\n\u22a2 \u2016(fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v }\u2016 = 1\n[PROOFSTEP]\nsimpa only [LinearIsometry.norm_map] using (hv_family i).left v\n[GOAL]\ncase right\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\n\u22a2 \u2200 {i j : (i : \u03b9) \u00d7 \u03b1 i},\n    i \u2260 j \u2192 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) i) ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) j) = 0\n[PROOFSTEP]\nrintro \u27e8i, v\u27e9 \u27e8j, w\u27e9 hvw\n[GOAL]\ncase right.mk.mk\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nw : \u03b1 j\nhvw : { fst := i, snd := v } \u2260 { fst := j, snd := w }\n\u22a2 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v })\n      ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := j, snd := w }) =\n    0\n[PROOFSTEP]\nby_cases hij : i = j\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nw : \u03b1 j\nhvw : { fst := i, snd := v } \u2260 { fst := j, snd := w }\nhij : i = j\n\u22a2 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v })\n      ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := j, snd := w }) =\n    0\n[PROOFSTEP]\nsubst hij\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv w : \u03b1 i\nhvw : { fst := i, snd := v } \u2260 { fst := i, snd := w }\n\u22a2 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v })\n      ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := w }) =\n    0\n[PROOFSTEP]\nhave : v \u2260 w := fun h => by\n  subst h\n  exact hvw rfl\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv w : \u03b1 i\nhvw : { fst := i, snd := v } \u2260 { fst := i, snd := w }\nh : v = w\n\u22a2 False\n[PROOFSTEP]\nsubst h\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv : \u03b1 i\nhvw : { fst := i, snd := v } \u2260 { fst := i, snd := v }\n\u22a2 False\n[PROOFSTEP]\nexact hvw rfl\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv w : \u03b1 i\nhvw : { fst := i, snd := v } \u2260 { fst := i, snd := w }\nthis : v \u2260 w\n\u22a2 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v })\n      ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := w }) =\n    0\n[PROOFSTEP]\nsimpa only [LinearIsometry.inner_map_map] using (hv_family i).2 this\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 \u03b1 i \u2192 G i\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c (v_family i)\ni : \u03b9\nv : \u03b1 i\nj : \u03b9\nw : \u03b1 j\nhvw : { fst := i, snd := v } \u2260 { fst := j, snd := w }\nhij : \u00aci = j\n\u22a2 inner ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := i, snd := v })\n      ((fun a => \u2191(V a.fst) (v_family a.fst a.snd)) { fst := j, snd := w }) =\n    0\n[PROOFSTEP]\nexact hV hij (v_family i v) (v_family j w)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\n\u22a2 \u2016\u2211 i in s\u2081, \u2191(V i) (f i) - \u2211 i in s\u2082, \u2191(V i) (f i)\u2016 ^ 2 = \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nrw [\u2190 Finset.sum_sdiff_sub_sum_sdiff, sub_eq_add_neg, \u2190 Finset.sum_neg_distrib]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nlet F : \u2200 i, G i := fun i => if i \u2208 s\u2081 then f i else -f i\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nhave hF\u2081 : \u2200 i \u2208 s\u2081 \\ s\u2082, F i = f i := fun i hi => if_pos (Finset.sdiff_subset _ _ hi)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nhave hF\u2082 : \u2200 i \u2208 s\u2082 \\ s\u2081, F i = -f i := fun i hi => if_neg (Finset.mem_sdiff.mp hi).2\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nhave hF : \u2200 i, \u2016F i\u2016 = \u2016f i\u2016 := by\n  intro i\n  dsimp only\n  split_ifs <;> simp only [eq_self_iff_true, norm_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\n\u22a2 \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\n[PROOFSTEP]\nintro i\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\ni : \u03b9\n\u22a2 \u2016F i\u2016 = \u2016f i\u2016\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\ni : \u03b9\n\u22a2 \u2016if i \u2208 s\u2081 then f i else -f i\u2016 = \u2016f i\u2016\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\ni : \u03b9\nh\u271d : i \u2208 s\u2081\n\u22a2 \u2016f i\u2016 = \u2016f i\u2016\n[PROOFSTEP]\nsimp only [eq_self_iff_true, norm_neg]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\ni : \u03b9\nh\u271d : \u00aci \u2208 s\u2081\n\u22a2 \u2016-f i\u2016 = \u2016f i\u2016\n[PROOFSTEP]\nsimp only [eq_self_iff_true, norm_neg]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nhave :\n  \u2016(\u2211 i in s\u2081 \\ s\u2082, V i (F i)) + \u2211 i in s\u2082 \\ s\u2081, V i (F i)\u2016 ^ 2 =\n    (\u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2) + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2 :=\n  by\n  have hs : Disjoint (s\u2081 \\ s\u2082) (s\u2082 \\ s\u2081) := disjoint_sdiff_sdiff\n  simpa only [Finset.sum_union hs] using hV.norm_sum F (s\u2081 \\ s\u2082 \u222a s\u2082 \\ s\u2081)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\n\u22a2 \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\n[PROOFSTEP]\nhave hs : Disjoint (s\u2081 \\ s\u2082) (s\u2082 \\ s\u2081) := disjoint_sdiff_sdiff\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nhs : Disjoint (s\u2081 \\ s\u2082) (s\u2082 \\ s\u2081)\n\u22a2 \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\n[PROOFSTEP]\nsimpa only [Finset.sum_union hs] using hV.norm_sum F (s\u2081 \\ s\u2082 \u222a s\u2082 \\ s\u2081)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\n\u22a2 \u2016\u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) + \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nconvert this using 4\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2191(V x) (f x) = \u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\ni : \u03b9\nhi : i \u2208 s\u2081 \\ s\u2082\n\u22a2 \u2191(V i) (f i) = \u2191(V i) (F i)\n[PROOFSTEP]\nsimp only [hF\u2081 i hi]\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_6\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\n\u22a2 \u2211 x in s\u2082 \\ s\u2081, -\u2191(V x) (f x) = \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i hi => _\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_3.h.e'_6\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\ni : \u03b9\nhi : i \u2208 s\u2082 \\ s\u2081\n\u22a2 -\u2191(V i) (f i) = \u2191(V i) (F i)\n[PROOFSTEP]\nsimp only [hF\u2082 i hi, LinearIsometry.map_neg]\n[GOAL]\ncase h.e'_3.h.e'_5.a.h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 s\u2081 \\ s\u2082\n\u22a2 \u2016f x\u271d\u2016 = \u2016F x\u271d\u2016\n[PROOFSTEP]\nsimp only [hF]\n[GOAL]\ncase h.e'_3.h.e'_6.a.h.e'_5\n\ud835\udd5c : Type u_1\nE : Type u_2\nF\u271d : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\u271d\ninst\u271d\u00b2 : InnerProductSpace \u211d F\u271d\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nf : (i : \u03b9) \u2192 G i\ns\u2081 s\u2082 : Finset \u03b9\nF : (i : \u03b9) \u2192 G i := fun i => if i \u2208 s\u2081 then f i else -f i\nhF\u2081 : \u2200 (i : \u03b9), i \u2208 s\u2081 \\ s\u2082 \u2192 F i = f i\nhF\u2082 : \u2200 (i : \u03b9), i \u2208 s\u2082 \\ s\u2081 \u2192 F i = -f i\nhF : \u2200 (i : \u03b9), \u2016F i\u2016 = \u2016f i\u2016\nthis :\n  \u2016\u2211 i in s\u2081 \\ s\u2082, \u2191(V i) (F i) + \u2211 i in s\u2082 \\ s\u2081, \u2191(V i) (F i)\u2016 ^ 2 =\n    \u2211 i in s\u2081 \\ s\u2082, \u2016F i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016F i\u2016 ^ 2\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 s\u2082 \\ s\u2081\n\u22a2 \u2016f x\u271d\u2016 = \u2016F x\u271d\u2016\n[PROOFSTEP]\nsimp only [hF]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (Summable fun i => \u2191(V i) (f i)) \u2194 Summable fun i => \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nclassical\nclear dec_\u03b9\nsimp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]\nconstructor\n\u00b7 intro hf \u03b5 h\u03b5\n  obtain \u27e8a, H\u27e9 := hf _ (sqrt_pos.mpr h\u03b5)\n  use a\n  intro s\u2081 hs\u2081 s\u2082 hs\u2082\n  rw [\u2190 Finset.sum_sdiff_sub_sum_sdiff]\n  refine' (abs_sub _ _).trans_lt _\n  have : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ 2 := fun i : \u03b9 => sq_nonneg _\n  simp only [Finset.abs_sum_of_nonneg' this]\n  have : ((\u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2) + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2) < sqrt \u03b5 ^ 2 :=\n    by\n    rw [\u2190 hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _), abs_of_nonneg (norm_nonneg _)]\n    exact H s\u2081 hs\u2081 s\u2082 hs\u2082\n  have h\u03b7 := sq_sqrt (le_of_lt h\u03b5)\n  linarith\n\u00b7 intro hf \u03b5 h\u03b5\n  have h\u03b5' : 0 < \u03b5 ^ 2 / 2 := half_pos (sq_pos_of_pos h\u03b5)\n  obtain \u27e8a, H\u27e9 := hf _ h\u03b5'\n  use a\n  intro s\u2081 hs\u2081 s\u2082 hs\u2082\n  refine' (abs_lt_of_sq_lt_sq' _ (le_of_lt h\u03b5)).2\n  have has : a \u2264 s\u2081 \u2293 s\u2082 := le_inf hs\u2081 hs\u2082\n  rw [hV.norm_sq_diff_sum]\n  have Hs\u2081 : \u2211 x : \u03b9 in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2 :=\n    by\n    convert H _ hs\u2081 _ has\n    have : s\u2081 \u2293 s\u2082 \u2286 s\u2081 := Finset.inter_subset_left _ _\n    rw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n    \u00b7 simp\n    \u00b7 exact fun i => sq_nonneg _\n  have Hs\u2082 : \u2211 x : \u03b9 in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2 :=\n    by\n    convert H _ hs\u2082 _ has\n    have : s\u2081 \u2293 s\u2082 \u2286 s\u2082 := Finset.inter_subset_right _ _\n    rw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n    \u00b7 simp\n    \u00b7 exact fun i => sq_nonneg _\n  linarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (Summable fun i => \u2191(V i) (f i)) \u2194 Summable fun i => \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nclear dec_\u03b9\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (Summable fun i => \u2191(V i) (f i)) \u2194 Summable fun i => \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nsimp only [summable_iff_cauchySeq_finset, NormedAddCommGroup.cauchySeq_iff, Real.norm_eq_abs]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (\u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192\n        \u2203 N,\n          \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5) \u2194\n    \u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (\u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192\n        \u2203 N,\n          \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5) \u2192\n    \u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nintro hf \u03b5 h\u03b5\n[GOAL]\ncase mp\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nobtain \u27e8a, H\u27e9 := hf _ (sqrt_pos.mpr h\u03b5)\n[GOAL]\ncase mp.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\n\u22a2 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\n\u22a2 \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nintro s\u2081 hs\u2081 s\u2082 hs\u2082\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\n\u22a2 |\u2211 i in s\u2081, \u2016f i\u2016 ^ 2 - \u2211 i in s\u2082, \u2016f i\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nrw [\u2190 Finset.sum_sdiff_sub_sum_sdiff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\n\u22a2 |\u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 - \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nrefine' (abs_sub _ _).trans_lt _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\n\u22a2 |\u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2| + |\u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nhave : \u2200 i, 0 \u2264 \u2016f i\u2016 ^ 2 := fun i : \u03b9 => sq_nonneg _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n\u22a2 |\u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2| + |\u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2| < \u03b5\n[PROOFSTEP]\nsimp only [Finset.abs_sum_of_nonneg' this]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 + \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5\n[PROOFSTEP]\nhave : ((\u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2) + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2) < sqrt \u03b5 ^ 2 :=\n  by\n  rw [\u2190 hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _), abs_of_nonneg (norm_nonneg _)]\n  exact H s\u2081 hs\u2081 s\u2082 hs\u2082\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n\u22a2 \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < sqrt \u03b5 ^ 2\n[PROOFSTEP]\nrw [\u2190 hV.norm_sq_diff_sum, sq_lt_sq, abs_of_nonneg (sqrt_nonneg _), abs_of_nonneg (norm_nonneg _)]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n\u22a2 \u2016\u2211 i in s\u2081, \u2191(V i) (f i) - \u2211 i in s\u2082, \u2191(V i) (f i)\u2016 < sqrt \u03b5\n[PROOFSTEP]\nexact H s\u2081 hs\u2081 s\u2082 hs\u2082\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis\u271d : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\nthis : \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < sqrt \u03b5 ^ 2\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 + \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5\n[PROOFSTEP]\nhave h\u03b7 := sq_sqrt (le_of_lt h\u03b5)\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192\n      \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < sqrt \u03b5\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nthis\u271d : \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\nthis : \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < sqrt \u03b5 ^ 2\nh\u03b7 : sqrt \u03b5 ^ 2 = \u03b5\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 + \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\n\u22a2 (\u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192\n        \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5) \u2192\n    \u2200 (\u03b5 : \u211d),\n      \u03b5 > 0 \u2192\n        \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nintro hf \u03b5 h\u03b5\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nhave h\u03b5' : 0 < \u03b5 ^ 2 / 2 := half_pos (sq_pos_of_pos h\u03b5)\n[GOAL]\ncase mpr\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\n\u22a2 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nobtain \u27e8a, H\u27e9 := hf _ h\u03b5'\n[GOAL]\ncase mpr.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\n\u22a2 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\n\u22a2 \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 \u2016\u2211 b in m, \u2191(V b) (f b) - \u2211 b in n, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nintro s\u2081 hs\u2081 s\u2082 hs\u2082\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\n\u22a2 \u2016\u2211 b in s\u2081, \u2191(V b) (f b) - \u2211 b in s\u2082, \u2191(V b) (f b)\u2016 < \u03b5\n[PROOFSTEP]\nrefine' (abs_lt_of_sq_lt_sq' _ (le_of_lt h\u03b5)).2\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\n\u22a2 \u2016\u2211 b in s\u2081, \u2191(V b) (f b) - \u2211 b in s\u2082, \u2191(V b) (f b)\u2016 ^ 2 < \u03b5 ^ 2\n[PROOFSTEP]\nhave has : a \u2264 s\u2081 \u2293 s\u2082 := le_inf hs\u2081 hs\u2082\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\n\u22a2 \u2016\u2211 b in s\u2081, \u2191(V b) (f b) - \u2211 b in s\u2082, \u2191(V b) (f b)\u2016 ^ 2 < \u03b5 ^ 2\n[PROOFSTEP]\nrw [hV.norm_sq_diff_sum]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\n\u22a2 \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < \u03b5 ^ 2\n[PROOFSTEP]\nhave Hs\u2081 : \u2211 x : \u03b9 in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2 :=\n  by\n  convert H _ hs\u2081 _ has\n  have : s\u2081 \u2293 s\u2082 \u2286 s\u2081 := Finset.inter_subset_left _ _\n  rw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n  \u00b7 simp\n  \u00b7 exact fun i => sq_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n[PROOFSTEP]\nconvert H _ hs\u2081 _ has\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 = |\u2211 i in s\u2081, \u2016f i\u2016 ^ 2 - \u2211 i in s\u2081 \u2293 s\u2082, \u2016f i\u2016 ^ 2|\n[PROOFSTEP]\nhave : s\u2081 \u2293 s\u2082 \u2286 s\u2081 := Finset.inter_subset_left _ _\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2081\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 = |\u2211 i in s\u2081, \u2016f i\u2016 ^ 2 - \u2211 i in s\u2081 \u2293 s\u2082, \u2016f i\u2016 ^ 2|\n[PROOFSTEP]\nrw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2081\n\u22a2 \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 = \u2211 i in s\u2081 \\ (s\u2081 \u2293 s\u2082), \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2081\n\u22a2 \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nexact fun i => sq_nonneg _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n\u22a2 \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < \u03b5 ^ 2\n[PROOFSTEP]\nhave Hs\u2082 : \u2211 x : \u03b9 in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2 :=\n  by\n  convert H _ hs\u2082 _ has\n  have : s\u2081 \u2293 s\u2082 \u2286 s\u2082 := Finset.inter_subset_right _ _\n  rw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n  \u00b7 simp\n  \u00b7 exact fun i => sq_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n\u22a2 \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n[PROOFSTEP]\nconvert H _ hs\u2082 _ has\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n\u22a2 \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 = |\u2211 i in s\u2082, \u2016f i\u2016 ^ 2 - \u2211 i in s\u2081 \u2293 s\u2082, \u2016f i\u2016 ^ 2|\n[PROOFSTEP]\nhave : s\u2081 \u2293 s\u2082 \u2286 s\u2082 := Finset.inter_subset_right _ _\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2082\n\u22a2 \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 = |\u2211 i in s\u2082, \u2016f i\u2016 ^ 2 - \u2211 i in s\u2081 \u2293 s\u2082, \u2016f i\u2016 ^ 2|\n[PROOFSTEP]\nrw [\u2190 Finset.sum_sdiff this, add_tsub_cancel_right, Finset.abs_sum_of_nonneg']\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2082\n\u22a2 \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 = \u2211 i in s\u2082 \\ (s\u2081 \u2293 s\u2082), \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\nthis : s\u2081 \u2293 s\u2082 \u2286 s\u2082\n\u22a2 \u2200 (i : \u03b9), 0 \u2264 \u2016f i\u2016 ^ 2\n[PROOFSTEP]\nexact fun i => sq_nonneg _\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : IsROrC \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : CompleteSpace E\nf : (i : \u03b9) \u2192 G i\nhf :\n  \u2200 (\u03b5 : \u211d),\n    \u03b5 > 0 \u2192 \u2203 N, \u2200 (m : Finset \u03b9), N \u2264 m \u2192 \u2200 (n : Finset \u03b9), N \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nh\u03b5' : 0 < \u03b5 ^ 2 / 2\na : Finset \u03b9\nH : \u2200 (m : Finset \u03b9), a \u2264 m \u2192 \u2200 (n : Finset \u03b9), a \u2264 n \u2192 |\u2211 i in m, \u2016f i\u2016 ^ 2 - \u2211 i in n, \u2016f i\u2016 ^ 2| < \u03b5 ^ 2 / 2\ns\u2081 : Finset \u03b9\nhs\u2081 : a \u2264 s\u2081\ns\u2082 : Finset \u03b9\nhs\u2082 : a \u2264 s\u2082\nhas : a \u2264 s\u2081 \u2293 s\u2082\nHs\u2081 : \u2211 x in s\u2081 \\ s\u2082, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\nHs\u2082 : \u2211 x in s\u2082 \\ s\u2081, \u2016f x\u2016 ^ 2 < \u03b5 ^ 2 / 2\n\u22a2 \u2211 i in s\u2081 \\ s\u2082, \u2016f i\u2016 ^ 2 + \u2211 i in s\u2082 \\ s\u2081, \u2016f i\u2016 ^ 2 < \u03b5 ^ 2\n[PROOFSTEP]\nlinarith\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\n\u22a2 CompleteLattice.Independent V\n[PROOFSTEP]\nclassical!\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 CompleteLattice.Independent V\n[PROOFSTEP]\napply CompleteLattice.independent_of_dfinsupp_lsum_injective\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Function.Injective \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i))\n[PROOFSTEP]\nrefine LinearMap.ker_eq_bot.mp ?_\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 LinearMap.ker (\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) = \u22a5\n[PROOFSTEP]\nrw [Submodule.eq_bot_iff]\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 \u2200 (x : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }), x \u2208 LinearMap.ker (\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) \u2192 x = 0\n[PROOFSTEP]\nintro v hv\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : v \u2208 LinearMap.ker (\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i))\n\u22a2 v = 0\n[PROOFSTEP]\nrw [LinearMap.mem_ker] at hv \n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\n\u22a2 v = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase h.h.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\ni : \u03b9\n\u22a2 \u2191(\u2191v i) = \u2191(\u21910 i)\n[PROOFSTEP]\nsuffices \u27ea(v i : E), v i\u27eb = 0 by simpa only [inner_self_eq_zero] using this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\ni : \u03b9\nthis : inner \u2191(\u2191v i) \u2191(\u2191v i) = 0\n\u22a2 \u2191(\u2191v i) = \u2191(\u21910 i)\n[PROOFSTEP]\nsimpa only [inner_self_eq_zero] using this\n[GOAL]\ncase h.h.a\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\ni : \u03b9\n\u22a2 inner \u2191(\u2191v i) \u2191(\u2191v i) = 0\n[PROOFSTEP]\ncalc\n  \u27ea(v i : E), v i\u27eb = \u27ea(v i : E), DFinsupp.lsum \u2115 (fun i => (V i).subtype) v\u27eb := by\n    simpa only [DFinsupp.sumAddHom_apply, DFinsupp.lsum_apply_apply] using (hV.inner_right_dfinsupp v i (v i)).symm\n  _ = 0 := by simp only [hv, inner_zero_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\ni : \u03b9\n\u22a2 inner \u2191(\u2191v i) \u2191(\u2191v i) = inner (\u2191(\u2191v i)) (\u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v)\n[PROOFSTEP]\nsimpa only [DFinsupp.sumAddHom_apply, DFinsupp.lsum_apply_apply] using (hV.inner_right_dfinsupp v i (v i)).symm\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nem\u271d : (a : Prop) \u2192 Decidable a\nv : \u03a0\u2080 (i : \u03b9), { x // x \u2208 V i }\nhv : \u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v = 0\ni : \u03b9\n\u22a2 inner (\u2191(\u2191v i)) (\u2191(\u2191(DFinsupp.lsum \u2115) fun i => Submodule.subtype (V i)) v) = 0\n[PROOFSTEP]\nsimp only [hv, inner_zero_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u03b9 : Type u_4\ndec_\u03b9 : DecidableEq \u03b9\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV\u271d : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV\u271d : OrthogonalFamily \ud835\udd5c G V\u271d\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\nV : \u03b9 \u2192 Submodule \ud835\udd5c E\nhV : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 V i }) fun i => Submodule.subtype\u2097\u1d62 (V i)\nhV_sum : IsInternal fun i => V i\n\u03b1 : \u03b9 \u2192 Type u_6\nv_family : (i : \u03b9) \u2192 Basis (\u03b1 i) \ud835\udd5c { x // x \u2208 V i }\nhv_family : \u2200 (i : \u03b9), Orthonormal \ud835\udd5c \u2191(v_family i)\n\u22a2 Orthonormal \ud835\udd5c \u2191(collectedBasis hV_sum v_family)\n[PROOFSTEP]\nsimpa only [hV_sum.collectedBasis_coe] using hV.orthonormal_sigma_orthonormal hv_family\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\nsrc\u271d\u00b9 : Inner \u211d E := Inner.isROrCToReal \ud835\udd5c E\nsrc\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nx y z : E\n\u22a2 inner (x + y) z = inner x z + inner y z\n[PROOFSTEP]\nchange re \u27eax + y, z\u27eb = re \u27eax, z\u27eb + re \u27eay, z\u27eb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\nsrc\u271d\u00b9 : Inner \u211d E := Inner.isROrCToReal \ud835\udd5c E\nsrc\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nx y z : E\n\u22a2 \u2191re (inner (x + y) z) = \u2191re (inner x z) + \u2191re (inner y z)\n[PROOFSTEP]\nsimp only [inner_add_left, map_add]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\nsrc\u271d\u00b9 : Inner \u211d E := Inner.isROrCToReal \ud835\udd5c E\nsrc\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nx y : E\nr : \u211d\n\u22a2 inner (r \u2022 x) y = \u2191(starRingEnd \u211d) r * inner x y\n[PROOFSTEP]\nchange re \u27ea(r : \ud835\udd5c) \u2022 x, y\u27eb = r * re \u27eax, y\u27eb\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\nsrc\u271d\u00b9 : Inner \u211d E := Inner.isROrCToReal \ud835\udd5c E\nsrc\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nx y : E\nr : \u211d\n\u22a2 \u2191re (inner (\u2191r \u2022 x) y) = r * \u2191re (inner x y)\n[PROOFSTEP]\nsimp only [inner_smul_left, conj_ofReal, ofReal_mul_re]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\nx : E\n\u22a2 inner x (I \u2022 x) = 0\n[PROOFSTEP]\nsimp [real_inner_eq_re_inner \ud835\udd5c, inner_smul_right]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2076 : IsROrC \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : InnerProductSpace \u211d G\nf : G \u2243\u2097\u1d62[\u211d] \u2102\nx y : G\n\u22a2 inner x y = (\u2191(starRingEnd \u2102) (\u2191f x) * \u2191f y).re\n[PROOFSTEP]\nrw [\u2190 Complex.inner, f.inner_map_map]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\n\u22a2 reApplyInnerSelf T (c \u2022 x) = \u2016c\u2016 ^ 2 * reApplyInnerSelf T x\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.reApplyInnerSelf_apply, inner_smul_left, inner_smul_right,\n  \u2190 mul_assoc, mul_conj, normSq_eq_def', \u2190 smul_re, Algebra.smul_def (\u2016c\u2016 ^ 2) \u27eaT x, x\u27eb, algebraMap_eq_ofReal]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u22a2 Continuous (uncurry inner)\n[PROOFSTEP]\nlet inner' : E \u2192+ E \u2192+ \ud835\udd5c :=\n  { toFun := fun x => (inner\u209b\u2097 \ud835\udd5c x).toAddMonoidHom\n    map_zero' := by ext x; exact inner_zero_left _\n    map_add' := fun x y => by ext z; exact inner_add_left _ _ _ }\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\n\u22a2 (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : E\n\u22a2 \u2191((fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0) x = \u21910 x\n[PROOFSTEP]\nexact inner_zero_left _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : E\n\u22a2 ZeroHom.toFun\n      { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n      (x + y) =\n    ZeroHom.toFun\n        { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n          map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n        x +\n      ZeroHom.toFun\n        { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n          map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n        y\n[PROOFSTEP]\next z\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : E\n\u22a2 \u2191(ZeroHom.toFun\n          { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n            map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n          (x + y))\n      z =\n    \u2191(ZeroHom.toFun\n            { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n              map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n            x +\n          ZeroHom.toFun\n            { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n              map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n            y)\n      z\n[PROOFSTEP]\nexact inner_add_left _ _ _\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninner' : E \u2192+ E \u2192+ \ud835\udd5c :=\n  {\n    toZeroHom :=\n      { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) },\n    map_add' :=\n      (_ :\n        \u2200 (x y : E),\n          ZeroHom.toFun\n              { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n              (x + y) =\n            ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                x +\n              ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                y) }\n\u22a2 Continuous (uncurry inner)\n[PROOFSTEP]\nhave : Continuous fun p : E \u00d7 E => inner' p.1 p.2 := continuous_inner\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninner' : E \u2192+ E \u2192+ \ud835\udd5c :=\n  {\n    toZeroHom :=\n      { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) },\n    map_add' :=\n      (_ :\n        \u2200 (x y : E),\n          ZeroHom.toFun\n              { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n              (x + y) =\n            ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                x +\n              ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                y) }\nthis : Continuous fun p => \u2191(\u2191inner' p.fst) p.snd\n\u22a2 Continuous (uncurry inner)\n[PROOFSTEP]\nrw [Completion.toInner, inner, uncurry_curry _]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninner' : E \u2192+ E \u2192+ \ud835\udd5c :=\n  {\n    toZeroHom :=\n      { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) },\n    map_add' :=\n      (_ :\n        \u2200 (x y : E),\n          ZeroHom.toFun\n              { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n              (x + y) =\n            ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                x +\n              ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                y) }\nthis : Continuous fun p => \u2191(\u2191inner' p.fst) p.snd\n\u22a2 Continuous (DenseInducing.extend (_ : DenseInducing fun p => (\u2191E p.fst, \u2191E p.snd)) (uncurry inner))\n[PROOFSTEP]\nchange Continuous (((denseInducing_toCompl E).prod (denseInducing_toCompl E)).extend fun p : E \u00d7 E => inner' p.1 p.2)\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\ninner' : E \u2192+ E \u2192+ \ud835\udd5c :=\n  {\n    toZeroHom :=\n      { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n        map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) },\n    map_add' :=\n      (_ :\n        \u2200 (x y : E),\n          ZeroHom.toFun\n              { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n              (x + y) =\n            ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                x +\n              ZeroHom.toFun\n                { toFun := fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x),\n                  map_zero' := (_ : (fun x => LinearMap.toAddMonoidHom (\u2191(inner\u209b\u2097 \ud835\udd5c) x)) 0 = 0) }\n                y) }\nthis : Continuous fun p => \u2191(\u2191inner' p.fst) p.snd\n\u22a2 Continuous\n    (DenseInducing.extend (_ : DenseInducing fun p => (\u2191toCompl p.fst, \u2191toCompl p.snd)) fun p => \u2191(\u2191inner' p.fst) p.snd)\n[PROOFSTEP]\nexact (denseInducing_toCompl E).extend_Z_bilin (denseInducing_toCompl E) this\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx : Completion E\na : E\n\u22a2 \u2016\u2191E a\u2016 ^ 2 = \u2191re (inner (\u2191E a) (\u2191E a))\n[PROOFSTEP]\nsimp only [norm_coe, inner_coe, inner_self_eq_norm_sq]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : Completion E\na b : E\n\u22a2 \u2191(starRingEnd \ud835\udd5c) (inner (\u2191E b) (\u2191E a)) = inner (\u2191E a) (\u2191E b)\n[PROOFSTEP]\nsimp only [inner_coe, inner_conj_symm]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y z : Completion E\na b c : E\n\u22a2 inner (\u2191E a + \u2191E b) (\u2191E c) = inner (\u2191E a) (\u2191E c) + inner (\u2191E b) (\u2191E c)\n[PROOFSTEP]\nsimp only [\u2190 coe_add, inner_coe, inner_add_left]\n[GOAL]\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : IsROrC \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\ndec_E : DecidableEq E\nx y : Completion E\nc : \ud835\udd5c\na b : E\n\u22a2 inner (c \u2022 \u2191E a) (\u2191E b) = \u2191(starRingEnd \ud835\udd5c) c * inner (\u2191E a) (\u2191E b)\n[PROOFSTEP]\nsimp only [\u2190 coe_smul c a, inner_coe, inner_smul_left]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.InnerProductSpace.Basic", "llama_tokens": 154274, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5125586864499858}}
{"text": "[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nhf : LinearMap.ker f = \u22a5\n\u22a2 Function.Injective \u2191f\n[PROOFSTEP]\nconvert LinearMap.ker_eq_bot.1 hf\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 Mono f \u2194 Function.Injective \u2191f\n[PROOFSTEP]\nrw [mono_iff_ker_eq_bot, LinearMap.ker_eq_bot]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u22a2 Epi f \u2194 Function.Surjective \u2191f\n[PROOFSTEP]\nrw [epi_iff_range_eq_top, LinearMap.range_eq_top]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf\u271d : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nX\u271d Y\u271d : ModuleCat R\nf : X\u271d \u27f6 Y\u271d\nhf : Epi f\n\u22a2 Epi ((forget (ModuleCat R)).map f)\n[PROOFSTEP]\nerw [CategoryTheory.epi_iff_surjective, \u2190 epi_iff_surjective]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf\u271d : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nX\u271d Y\u271d : ModuleCat R\nf : X\u271d \u27f6 Y\u271d\nhf : Epi f\n\u22a2 Epi f\n[PROOFSTEP]\nexact hf\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf\u271d : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nX\u271d Y\u271d : ModuleCat R\nf : X\u271d \u27f6 Y\u271d\nhf : Mono f\n\u22a2 Mono ((forget (ModuleCat R)).map f)\n[PROOFSTEP]\nerw [CategoryTheory.mono_iff_injective, \u2190 mono_iff_injective]\n[GOAL]\nR : Type u\ninst\u271d\u00b2 : Ring R\nX Y : ModuleCat R\nf\u271d : X \u27f6 Y\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nX\u271d Y\u271d : ModuleCat R\nf : X\u271d \u27f6 Y\u271d\nhf : Mono f\n\u22a2 Mono f\n[PROOFSTEP]\nexact hf\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.ModuleCat.EpiMono", "llama_tokens": 810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.512269882551838}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\nhd : Summable d\n\u22a2 CauchySeq f\n[PROOFSTEP]\nrefine\n  EMetric.cauchySeq_iff_NNReal.2 fun \u03b5 \u03b5pos =>\n    ?_\n      -- Actually we need partial sums of `d` to be a Cauchy sequence\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\nhd : Summable d\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2203 N, \u2200 (n : \u2115), N \u2264 n \u2192 edist (f n) (f N) < \u2191\u03b5\n[PROOFSTEP]\nreplace hd : CauchySeq fun n : \u2115 => \u2211 x in range n, d x :=\n  let \u27e8_, H\u27e9 := hd\n  H.tendsto_sum_nat.cauchySeq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\n\u22a2 \u2203 N, \u2200 (n : \u2115), N \u2264 n \u2192 edist (f n) (f N) < \u2191\u03b5\n[PROOFSTEP]\nrefine (Metric.cauchySeq_iff'.1 hd \u03b5 (NNReal.coe_pos.2 \u03b5pos)).imp fun N hN n hn => ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u2191\u03b5\nn : \u2115\nhn : N \u2264 n\n\u22a2 edist (f n) (f N) < \u2191\u03b5\n[PROOFSTEP]\nspecialize hN n hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN n : \u2115\nhn : N \u2264 n\nhN : dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u2191\u03b5\n\u22a2 edist (f n) (f N) < \u2191\u03b5\n[PROOFSTEP]\nrw [dist_nndist, NNReal.nndist_eq, \u2190 sum_range_add_sum_Ico _ hn, add_tsub_cancel_left, NNReal.coe_lt_coe, max_lt_iff] at\n  hN \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN n : \u2115\nhn : N \u2264 n\nhN : \u2211 k in Ico N n, d k < \u03b5 \u2227 \u2211 x in range N, d x - (\u2211 k in range N, d k + \u2211 k in Ico N n, d k) < \u03b5\n\u22a2 edist (f n) (f N) < \u2191\u03b5\n[PROOFSTEP]\nrw [edist_comm]\n  -- Then use `hf` to simplify the goal to the same form\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN n : \u2115\nhn : N \u2264 n\nhN : \u2211 k in Ico N n, d k < \u03b5 \u2227 \u2211 x in range N, d x - (\u2211 k in range N, d k + \u2211 k in Ico N n, d k) < \u03b5\n\u22a2 edist (f N) (f n) < \u2191\u03b5\n[PROOFSTEP]\nrefine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ => hf _) ?_\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoEMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\nd : \u2115 \u2192 \u211d\u22650\nhf : \u2200 (n : \u2115), edist (f n) (f (Nat.succ n)) \u2264 \u2191(d n)\n\u03b5 : \u211d\u22650\n\u03b5pos : 0 < \u03b5\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN n : \u2115\nhn : N \u2264 n\nhN : \u2211 k in Ico N n, d k < \u03b5 \u2227 \u2211 x in range N, d x - (\u2211 k in range N, d k + \u2211 k in Ico N n, d k) < \u03b5\n\u22a2 \u2211 i in Ico N n, \u2191(d i) < \u2191\u03b5\n[PROOFSTEP]\nexact_mod_cast hN.1\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\n\u22a2 CauchySeq f\n[PROOFSTEP]\nrefine' Metric.cauchySeq_iff'.2 fun \u03b5 \u03b5pos => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f n) (f N) < \u03b5\n[PROOFSTEP]\nreplace hd : CauchySeq fun n : \u2115 => \u2211 x in range n, d x :=\n  let \u27e8_, H\u27e9 := hd\n  H.tendsto_sum_nat.cauchySeq\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhd : CauchySeq fun n => \u2211 x in range n, d x\n\u22a2 \u2203 N, \u2200 (n : \u2115), n \u2265 N \u2192 dist (f n) (f N) < \u03b5\n[PROOFSTEP]\nrefine' (Metric.cauchySeq_iff'.1 hd \u03b5 \u03b5pos).imp fun N hN n hn => _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u03b5\nn : \u2115\nhn : n \u2265 N\n\u22a2 dist (f n) (f N) < \u03b5\n[PROOFSTEP]\nhave hsum := hN n hn\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u03b5\nn : \u2115\nhn : n \u2265 N\nhsum : dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u03b5\n\u22a2 dist (f n) (f N) < \u03b5\n[PROOFSTEP]\nrw [Real.dist_eq, \u2190 sum_Ico_eq_sub _ hn] at hsum \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nhd : CauchySeq fun n => \u2211 x in range n, d x\nN : \u2115\nhN : \u2200 (n : \u2115), n \u2265 N \u2192 dist (\u2211 x in range n, d x) (\u2211 x in range N, d x) < \u03b5\nn : \u2115\nhn : n \u2265 N\nhsum : |\u2211 k in Ico N n, d k| < \u03b5\n\u22a2 dist (f n) (f N) < \u03b5\n[PROOFSTEP]\ncalc\n  dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _\n  _ \u2264 \u2211 x in Ico N n, d x := (dist_le_Ico_sum_of_dist_le hn fun _ _ => hf _)\n  _ \u2264 |\u2211 x in Ico N n, d x| := (le_abs_self _)\n  _ < \u03b5 := hsum\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na\u271d : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn : \u2115\n\u22a2 dist (f n) a \u2264 \u2211' (m : \u2115), d (n + m)\n[PROOFSTEP]\nrefine' le_of_tendsto (tendsto_const_nhds.dist ha) (eventually_atTop.2 \u27e8n, fun m hnm => _\u27e9)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na\u271d : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn m : \u2115\nhnm : m \u2265 n\n\u22a2 dist (f n) (f m) \u2264 \u2211' (m : \u2115), d (n + m)\n[PROOFSTEP]\nrefine' le_trans (dist_le_Ico_sum_of_dist_le hnm fun _ _ => hf _) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na\u271d : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn m : \u2115\nhnm : m \u2265 n\n\u22a2 \u2211 i in Ico n m, d i \u2264 \u2211' (m : \u2115), d (n + m)\n[PROOFSTEP]\nrw [sum_Ico_eq_sum_range]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na\u271d : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn m : \u2115\nhnm : m \u2265 n\n\u22a2 \u2211 k in range (m - n), d (n + k) \u2264 \u2211' (m : \u2115), d (n + m)\n[PROOFSTEP]\nrefine' sum_le_tsum (range _) (fun _ _ => le_trans dist_nonneg (hf _)) _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na\u271d : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\na : \u03b1\nha : Tendsto f atTop (\ud835\udcdd a)\nn m : \u2115\nhnm : m \u2265 n\n\u22a2 Summable fun k => d (n + k)\n[PROOFSTEP]\nexact hd.comp_injective (add_right_injective n)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nd : \u2115 \u2192 \u211d\nhf : \u2200 (n : \u2115), dist (f n) (f (Nat.succ n)) \u2264 d n\nhd : Summable d\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 dist (f 0) a \u2264 tsum d\n[PROOFSTEP]\nsimpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : PseudoMetricSpace \u03b1\nf : \u2115 \u2192 \u03b1\na : \u03b1\nh : Summable fun n => dist (f n) (f (Nat.succ n))\nha : Tendsto f atTop (\ud835\udcdd a)\n\u22a2 dist (f 0) a \u2264 \u2211' (n : \u2115), dist (f n) (f (Nat.succ n))\n[PROOFSTEP]\nsimpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.InfiniteSum.Real", "llama_tokens": 4078, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.512104137046595}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\n[PROOFSTEP]\nclassical\ninduction' I using Finset.induction_on with i I hiI ihI hI\n\u00b7 simp\nsimp only [Finset.mem_insert] at hI \nrw [Finset.set_biUnion_insert, hm, ihI, Finset.sum_insert hiI]\nexacts [fun i hi j hj hij => hI i (Or.inr hi) j (Or.inr hj) hij,\n  IsMetricSeparated.finset_iUnion_right fun j hj => hI i (Or.inl rfl) j (Or.inr hj) (ne_of_mem_of_not_mem hj hiI).symm]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\n[PROOFSTEP]\ninduction' I using Finset.induction_on with i I hiI ihI hI\n[GOAL]\ncase empty\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\nhI : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2205 \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\n\u22a2 \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 \u2205), s i) = \u2211 i in \u2205, \u2191\u03bc (s i)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase insert\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192\n    \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i I \u2192 \u2200 (j : \u03b9), j \u2208 insert i I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i I), s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)\n[PROOFSTEP]\nsimp only [Finset.mem_insert] at hI \n[GOAL]\ncase insert\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192\n    \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2191\u03bc (\u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 insert i I), s i_1) = \u2211 i in insert i I, \u2191\u03bc (s i)\n[PROOFSTEP]\nrw [Finset.set_biUnion_insert, hm, ihI, Finset.sum_insert hiI]\n[GOAL]\ncase insert\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192\n    \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 \u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ncase insert.a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nI\u271d : Finset \u03b9\ns : \u03b9 \u2192 Set X\nhI\u271d : \u2200 (i : \u03b9), i \u2208 I\u271d \u2192 \u2200 (j : \u03b9), j \u2208 I\u271d \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)\ni : \u03b9\nI : Finset \u03b9\nhiI : \u00aci \u2208 I\nihI :\n  (\u2200 (i : \u03b9), i \u2208 I \u2192 \u2200 (j : \u03b9), j \u2208 I \u2192 i \u2260 j \u2192 IsMetricSeparated (s i) (s j)) \u2192\n    \u2191\u03bc (\u22c3 (i : \u03b9) (_ : i \u2208 I), s i) = \u2211 i in I, \u2191\u03bc (s i)\nhI : \u2200 (i_1 : \u03b9), i_1 = i \u2228 i_1 \u2208 I \u2192 \u2200 (j : \u03b9), j = i \u2228 j \u2208 I \u2192 i_1 \u2260 j \u2192 IsMetricSeparated (s i_1) (s j)\n\u22a2 IsMetricSeparated (s i) (\u22c3 (x : \u03b9) (_ : x \u2208 I), s x)\n[PROOFSTEP]\nexacts [fun i hi j hj hij => hI i (Or.inr hi) j (Or.inr hj) hij,\n  IsMetricSeparated.finset_iUnion_right fun j hj => hI i (Or.inl rfl) j (Or.inr hj) (ne_of_mem_of_not_mem hj hiI).symm]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\n\u22a2 borel X \u2264 OuterMeasure.caratheodory \u03bc\n[PROOFSTEP]\nrw [borel_eq_generateFrom_isClosed]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\n\u22a2 MeasurableSpace.generateFrom {s | IsClosed s} \u2264 OuterMeasure.caratheodory \u03bc\n[PROOFSTEP]\nrefine' MeasurableSpace.generateFrom_le fun t ht => \u03bc.isCaratheodory_iff_le.2 fun s => _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nset S : \u2115 \u2192 Set X := fun n => {x \u2208 s | (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave n0 : \u2200 {n : \u2115}, (n\u207b\u00b9 : \u211d\u22650\u221e) \u2260 0 := fun {n} => ENNReal.inv_ne_zero.2 (ENNReal.nat_ne_top _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave Ssep : \u2200 n, IsMetricSeparated (S n) t := fun n =>\n  \u27e8n\u207b\u00b9, n0, fun x hx y hy => hx.2.trans <| infEdist_le_edist_of_mem hy\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave Ssep' : \u2200 n, IsMetricSeparated (S n) (s \u2229 t) := fun n => (Ssep n).mono Subset.rfl (inter_subset_right _ _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave S_sub : \u2200 n, S n \u2286 s \\ t := fun n => subset_inter (inter_subset_left _ _) (Ssep n).subset_compl_right\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave hSs : \u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s := fun n =>\n  calc\n    \u03bc (s \u2229 t) + \u03bc (S n) = \u03bc (s \u2229 t \u222a S n) := Eq.symm <| hm _ _ <| (Ssep' n).symm\n    _ \u2264 \u03bc (s \u2229 t \u222a s \\ t) := (\u03bc.mono <| union_subset_union_right _ <| S_sub n)\n    _ = \u03bc s := by rw [inter_union_diff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nn : \u2115\n\u22a2 \u2191\u03bc (s \u2229 t \u222a s \\ t) = \u2191\u03bc s\n[PROOFSTEP]\nrw [inter_union_diff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nhave iUnion_S : \u22c3 n, S n = s \\ t := by\n  refine' Subset.antisymm (iUnion_subset S_sub) _\n  rintro x \u27e8hxs, hxt\u27e9\n  rw [mem_iff_infEdist_zero_of_closed ht] at hxt \n  rcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9\n  exact\n    mem_iUnion.2\n      \u27e8n, hxs, hn.le\u27e9\n        /- Now we have `\u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s` and we need to prove\n            `\u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) \u2264 \u03bc s`. We can't pass to the limit because\n            `\u03bc` is only an outer measure. -/\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 \u22c3 (n : \u2115), S n = s \\ t\n[PROOFSTEP]\nrefine' Subset.antisymm (iUnion_subset S_sub) _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\n\u22a2 s \\ t \u2286 \u22c3 (n : \u2115), S n\n[PROOFSTEP]\nrintro x \u27e8hxs, hxt\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acx \u2208 t\n\u22a2 x \u2208 \u22c3 (n : \u2115), S n\n[PROOFSTEP]\nrw [mem_iff_infEdist_zero_of_closed ht] at hxt \n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\n\u22a2 x \u2208 \u22c3 (n : \u2115), S n\n[PROOFSTEP]\nrcases ENNReal.exists_inv_nat_lt hxt with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\nx : X\nhxs : x \u2208 s\nhxt : \u00acinfEdist x t = 0\nn : \u2115\nhn : (\u2191n)\u207b\u00b9 < infEdist x t\n\u22a2 x \u2208 \u22c3 (n : \u2115), S n\n[PROOFSTEP]\nexact\n  mem_iUnion.2\n    \u27e8n, hxs, hn.le\u27e9\n      /- Now we have `\u2200 n, \u03bc (s \u2229 t) + \u03bc (S n) \u2264 \u03bc s` and we need to prove\n          `\u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) \u2264 \u03bc s`. We can't pass to the limit because\n          `\u03bc` is only an outer measure. -/\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nby_cases htop : \u03bc (s \\ t) = \u221e\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nrw [htop, add_top, \u2190 htop]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nexact \u03bc.mono (diff_subset _ _)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\n[PROOFSTEP]\nsuffices : \u03bc (\u22c3 n, S n) \u2264 \u2a06 n, \u03bc (S n)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) \u2264 \u2191\u03bc s\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n[PROOFSTEP]\ncalc\n  \u03bc (s \u2229 t) + \u03bc (s \\ t) = \u03bc (s \u2229 t) + \u03bc (\u22c3 n, S n) := by rw [iUnion_S]\n  _ \u2264 \u03bc (s \u2229 t) + \u2a06 n, \u03bc (S n) := (add_le_add le_rfl this)\n  _ = \u2a06 n, \u03bc (s \u2229 t) + \u03bc (S n) := ENNReal.add_iSup\n  _ \u2264 \u03bc s := iSup_le hSs\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n\u22a2 \u2191\u03bc (s \u2229 t) + \u2191\u03bc (s \\ t) = \u2191\u03bc (s \u2229 t) + \u2191\u03bc (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nrw [iUnion_S]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n[PROOFSTEP]\nhave : \u2200 n, S n \u2286 S (n + 1) := fun n x hx => \u27e8hx.1, le_trans (ENNReal.inv_le_inv.2 <| Nat.cast_le.2 n.le_succ) hx.2\u27e9\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n[PROOFSTEP]\nclassical\n  -- Porting note: Added this to get the next tactic to work\nrefine' (\u03bc.iUnion_nat_of_monotone_of_tsum_ne_top this _).le;\nclear this\nrw [\u2190 tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top]\nsuffices : \u2200 a, (\u2211' k : \u2115, \u03bc (S (2 * k + 1 + a) \\ S (2 * k + a))) \u2260 \u221e\nexact \u27e8by simpa using this 0, by simpa using this 1\u27e9\nrefine' fun r => ne_top_of_le_ne_top htop _\nrw [\u2190 iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]\nintro n\nrw [\u2190 hm.finset_iUnion_of_pairwise_separated]\n\u00b7 exact \u03bc.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 \u27e8_, hx.1\u27e9)\nsuffices : \u2200 i j, i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\nexact fun i _ j _ hij =>\n  hij.lt_or_lt.elim (fun h => (this i j h).mono (inter_subset_left _ _) fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) fun h =>\n    (this j i h).symm.mono (fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) (inter_subset_left _ _)\nintro i j hj\nhave A : ((\u2191(2 * j + r))\u207b\u00b9 : \u211d\u22650\u221e) < (\u2191(2 * i + 1 + r))\u207b\u00b9 := by rw [ENNReal.inv_lt_inv, Nat.cast_lt]; linarith\nrefine' \u27e8(\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9, by simpa using A, fun x hx y hy => _\u27e9\nhave : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 := not_le.1 fun hle => hy.2 \u27e8hy.1, hle\u27e9\nrcases infEdist_lt_iff.mp this with \u27e8z, hzt, hyz\u27e9\nhave hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z := le_infEdist.1 hx.2 _ hzt\napply ENNReal.le_of_add_le_add_right hyz.ne_top\nrefine' le_trans _ (edist_triangle _ _ _)\nrefine' (add_le_add le_rfl hyz.le).trans (Eq.trans_le _ hxz)\nrw [tsub_add_cancel_of_le A.le]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2191\u03bc (\u22c3 (n : \u2115), S n) \u2264 \u2a06 (n : \u2115), \u2191\u03bc (S n)\n[PROOFSTEP]\nrefine' (\u03bc.iUnion_nat_of_monotone_of_tsum_ne_top this _).le\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (n : \u2115), S n \u2286 S (n + 1)\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4\n[PROOFSTEP]\nclear this\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (k + 1) \\ S k) \u2260 \u22a4\n[PROOFSTEP]\nrw [\u2190 tsum_even_add_odd ENNReal.summable ENNReal.summable, ENNReal.add_ne_top]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4\n[PROOFSTEP]\nsuffices : \u2200 a, (\u2211' k : \u2115, \u03bc (S (2 * k + 1 + a) \\ S (2 * k + a))) \u2260 \u221e\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4 \u2227 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n[PROOFSTEP]\nexact \u27e8by simpa using this 0, by simpa using this 1\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1) \\ S (2 * k)) \u2260 \u22a4\n[PROOFSTEP]\nsimpa using this 0\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nthis : \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + 1) \\ S (2 * k + 1)) \u2260 \u22a4\n[PROOFSTEP]\nsimpa using this 1\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\n\u22a2 \u2200 (a : \u2115), \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + a) \\ S (2 * k + a)) \u2260 \u22a4\n[PROOFSTEP]\nrefine' fun r => ne_top_of_le_ne_top htop _\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2211' (k : \u2115), \u2191\u03bc (S (2 * k + 1 + r) \\ S (2 * k + r)) \u2264 \u2191\u03bc (s \\ t)\n[PROOFSTEP]\nrw [\u2190 iUnion_S, ENNReal.tsum_eq_iSup_nat, iSup_le_iff]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr : \u2115\n\u22a2 \u2200 (i : \u2115), \u2211 a in Finset.range i, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nintro n\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2211 a in Finset.range n, \u2191\u03bc (S (2 * a + 1 + r) \\ S (2 * a + r)) \u2264 \u2191\u03bc (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nrw [\u2190 hm.finset_iUnion_of_pairwise_separated]\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2191\u03bc (\u22c3 (i : \u2115) (_ : i \u2208 Finset.range n), S (2 * i + 1 + r) \\ S (2 * i + r)) \u2264 \u2191\u03bc (\u22c3 (n : \u2115), S n)\n[PROOFSTEP]\nexact \u03bc.mono (iUnion_subset fun i => iUnion_subset fun _ x hx => mem_iUnion.2 \u27e8_, hx.1\u27e9)\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))\n[PROOFSTEP]\nsuffices : \u2200 i j, i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 Finset.range n \u2192\n      \u2200 (j : \u2115),\n        j \u2208 Finset.range n \u2192\n          i \u2260 j \u2192 IsMetricSeparated (S (2 * i + 1 + r) \\ S (2 * i + r)) (S (2 * j + 1 + r) \\ S (2 * j + r))\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n[PROOFSTEP]\nexact fun i _ j _ hij =>\n  hij.lt_or_lt.elim (fun h => (this i j h).mono (inter_subset_left _ _) fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) fun h =>\n    (this j i h).symm.mono (fun x hx => by exact \u27e8hx.1.1, hx.2\u27e9) (inter_subset_left _ _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\ni : \u2115\nx\u271d\u00b9 : i \u2208 Finset.range n\nj : \u2115\nx\u271d : j \u2208 Finset.range n\nhij : i \u2260 j\nh : i < j\nx : X\nhx : x \u2208 S (2 * j + 1 + r) \\ S (2 * j + r)\n\u22a2 x \u2208 s \\ S (2 * j + r)\n[PROOFSTEP]\nexact \u27e8hx.1.1, hx.2\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\nthis : \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\ni : \u2115\nx\u271d\u00b9 : i \u2208 Finset.range n\nj : \u2115\nx\u271d : j \u2208 Finset.range n\nhij : i \u2260 j\nh : j < i\nx : X\nhx : x \u2208 S (2 * i + 1 + r) \\ S (2 * i + r)\n\u22a2 x \u2208 s \\ S (2 * i + r)\n[PROOFSTEP]\nexact \u27e8hx.1.1, hx.2\u27e9\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n : \u2115\n\u22a2 \u2200 (i j : \u2115), i < j \u2192 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n[PROOFSTEP]\nintro i j hj\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n[PROOFSTEP]\nhave A : ((\u2191(2 * j + r))\u207b\u00b9 : \u211d\u22650\u221e) < (\u2191(2 * i + 1 + r))\u207b\u00b9 := by rw [ENNReal.inv_lt_inv, Nat.cast_lt]; linarith\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.inv_lt_inv, Nat.cast_lt]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\n\u22a2 2 * i + 1 + r < 2 * j + r\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n\u22a2 IsMetricSeparated (S (2 * i + 1 + r)) (s \\ S (2 * j + r))\n[PROOFSTEP]\nrefine' \u27e8(\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9, by simpa using A, fun x hx y hy => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2260 0\n[PROOFSTEP]\nsimpa using A\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y\n[PROOFSTEP]\nhave : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9 := not_le.1 fun hle => hy.2 \u27e8hy.1, hle\u27e9\n[GOAL]\ncase this\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y\n[PROOFSTEP]\nrcases infEdist_lt_iff.mp this with \u27e8z, hzt, hyz\u27e9\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y\n[PROOFSTEP]\nhave hxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z := le_infEdist.1 hx.2 _ hzt\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 \u2264 edist x y\n[PROOFSTEP]\napply ENNReal.le_of_add_le_add_right hyz.ne_top\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x y + edist y z\n[PROOFSTEP]\nrefine' le_trans _ (edist_triangle _ _ _)\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + edist y z \u2264 edist x z\n[PROOFSTEP]\nrefine' (add_le_add le_rfl hyz.le).trans (Eq.trans_le _ hxz)\n[GOAL]\ncase this.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u03bc : OuterMeasure X\nhm : IsMetric \u03bc\nt : Set X\nht : t \u2208 {s | IsClosed s}\ns : Set X\nS : \u2115 \u2192 Set X := fun n => {x | x \u2208 s \u2227 (\u2191n)\u207b\u00b9 \u2264 infEdist x t}\nn0 : \u2200 {n : \u2115}, (\u2191n)\u207b\u00b9 \u2260 0\nSsep : \u2200 (n : \u2115), IsMetricSeparated (S n) t\nSsep' : \u2200 (n : \u2115), IsMetricSeparated (S n) (s \u2229 t)\nS_sub : \u2200 (n : \u2115), S n \u2286 s \\ t\nhSs : \u2200 (n : \u2115), \u2191\u03bc (s \u2229 t) + \u2191\u03bc (S n) \u2264 \u2191\u03bc s\niUnion_S : \u22c3 (n : \u2115), S n = s \\ t\nhtop : \u00ac\u2191\u03bc (s \\ t) = \u22a4\nr n i j : \u2115\nhj : i < j\nA : (\u2191(2 * j + r))\u207b\u00b9 < (\u2191(2 * i + 1 + r))\u207b\u00b9\nx : X\nhx : x \u2208 S (2 * i + 1 + r)\ny : X\nhy : y \u2208 s \\ S (2 * j + r)\nthis : infEdist y t < (\u2191(2 * j + r))\u207b\u00b9\nz : X\nhzt : z \u2208 t\nhyz : edist y z < (\u2191(2 * j + r))\u207b\u00b9\nhxz : (\u2191(2 * i + 1 + r))\u207b\u00b9 \u2264 edist x z\n\u22a2 (\u2191(2 * i + 1 + r))\u207b\u00b9 - (\u2191(2 * j + r))\u207b\u00b9 + (\u2191(2 * j + r))\u207b\u00b9 = (\u2191(2 * i + 1 + r))\u207b\u00b9\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le A.le]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\n\u03bc : OuterMeasure X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nhm : IsMetric \u03bc\n\u22a2 inst\u271d\u00b9 \u2264 OuterMeasure.caratheodory \u03bc\n[PROOFSTEP]\nrw [BorelSpace.measurable_eq (\u03b1 := X)]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\n\u03bc : OuterMeasure X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nhm : IsMetric \u03bc\n\u22a2 borel X \u2264 OuterMeasure.caratheodory \u03bc\n[PROOFSTEP]\nexact hm.borel_le_caratheodory\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns : Set X\n\u22a2 \u03bc \u2264 pre m r \u2194 \u2200 (s : Set X), diam s \u2264 r \u2192 \u2191\u03bc s \u2264 m s\n[PROOFSTEP]\nsimp only [pre, le_boundedBy, extend, le_iInf_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\nk l : \u2115\nh : k \u2264 l\ns : Set X\nhs : diam s \u2264 (\u2191l)\u207b\u00b9\n\u22a2 (\u2191l)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto (fun r => \u2191(pre m r) s) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u2191(mkMetric' m) s))\n[PROOFSTEP]\nrw [\u2190 map_coe_Ioi_atBot, tendsto_map'_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto ((fun r => \u2191(pre m r) s) \u2218 Subtype.val) atBot (\ud835\udcdd (\u2191(mkMetric' m) s))\n[PROOFSTEP]\nsimp only [mkMetric', OuterMeasure.iSup_apply, iSup_subtype']\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto ((fun r => \u2191(pre m r) s) \u2218 Subtype.val) atBot (\ud835\udcdd (\u2a06 (i : { i // i > 0 }), \u2191(pre m \u2191i) s))\n[PROOFSTEP]\nexact tendsto_atBot_iSup fun r r' hr => mono_pre _ hr _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto (fun n => \u2191(pre m (\u2191n)\u207b\u00b9) s) atTop (\ud835\udcdd (\u2191(mkMetric' m) s))\n[PROOFSTEP]\nrefine' (tendsto_pre m s).comp (tendsto_inf.2 \u27e8ENNReal.tendsto_inv_nat_nhds_zero, _\u27e9)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 Tendsto (fun n => (\u2191n)\u207b\u00b9) atTop (\ud835\udcdf (Ioi 0))\n[PROOFSTEP]\nrefine' tendsto_principal.2 (eventually_of_forall fun n => _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\nn : \u2115\n\u22a2 (\u2191n)\u207b\u00b9 \u2208 Ioi 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns : Set X\nm : Set X \u2192 \u211d\u22650\u221e\n\u22a2 mkMetric' m = \u2a06 (n : \u2115), pre m (\u2191n)\u207b\u00b9\n[PROOFSTEP]\next1 s\n[GOAL]\ncase h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2191(\u2a06 (n : \u2115), pre m (\u2191n)\u207b\u00b9) s\n[PROOFSTEP]\nrw [iSup_apply]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\nm : Set X \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191(mkMetric' m) s = \u2a06 (i : \u2115), \u2191(pre m (\u2191i)\u207b\u00b9) s\n[PROOFSTEP]\nrefine'\n  tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s) (tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr\u271d : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns : Set X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nm : Set X \u2192 \u211d\u22650\u221e\nhcl : \u2200 (s : Set X), m (closure s) = m s\nr : \u211d\u22650\u221e\n\u22a2 trim (pre m r) = pre m r\n[PROOFSTEP]\nrefine' le_antisymm (le_pre.2 fun s hs => _) (le_trim _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr\u271d : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nm : Set X \u2192 \u211d\u22650\u221e\nhcl : \u2200 (s : Set X), m (closure s) = m s\nr : \u211d\u22650\u221e\ns : Set X\nhs : diam s \u2264 r\n\u22a2 \u2191(trim (pre m r)) s \u2264 m s\n[PROOFSTEP]\nrw [trim_eq_iInf]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr\u271d : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nm : Set X \u2192 \u211d\u22650\u221e\nhcl : \u2200 (s : Set X), m (closure s) = m s\nr : \u211d\u22650\u221e\ns : Set X\nhs : diam s \u2264 r\n\u22a2 \u2a05 (t : Set X) (_ : s \u2286 t) (_ : MeasurableSet t), \u2191(pre m r) t \u2264 m s\n[PROOFSTEP]\nrefine'\n  iInf_le_of_le (closure s) <|\n    iInf_le_of_le subset_closure <| iInf_le_of_le measurableSet_closure ((pre_le _).trans_eq (hcl _))\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\nm\u271d : Set X \u2192 \u211d\u22650\u221e\nr\u271d : \u211d\u22650\u221e\n\u03bc : OuterMeasure X\ns\u271d : Set X\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : OpensMeasurableSpace X\nm : Set X \u2192 \u211d\u22650\u221e\nhcl : \u2200 (s : Set X), m (closure s) = m s\nr : \u211d\u22650\u221e\ns : Set X\nhs : diam s \u2264 r\n\u22a2 diam (closure s) \u2264 r\n[PROOFSTEP]\nrwa [diam_closure]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\n\u22a2 IsMetric (mkMetric' m)\n[PROOFSTEP]\nrintro s t \u27e8r, r0, hr\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2191(mkMetric' m) (s \u222a t) = \u2191(mkMetric' m) s + \u2191(mkMetric' m) t\n[PROOFSTEP]\nrefine'\n  tendsto_nhds_unique_of_eventuallyEq (mkMetric'.tendsto_pre _ _)\n    ((mkMetric'.tendsto_pre _ _).add (mkMetric'.tendsto_pre _ _)) _\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : r \u2260 0\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t\n[PROOFSTEP]\nrw [\u2190 pos_iff_ne_zero] at r0 \n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 (fun r => \u2191(mkMetric'.pre m r) (s \u222a t)) =\u1da0[\ud835\udcdd[Ioi 0] 0] fun x => \u2191(mkMetric'.pre m x) s + \u2191(mkMetric'.pre m x) t\n[PROOFSTEP]\nfilter_upwards [Ioo_mem_nhdsWithin_Ioi \u27e8le_rfl, r0\u27e9]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u22a2 \u2200 (a : \u211d\u22650\u221e), a \u2208 Ioo 0 r \u2192 \u2191(mkMetric'.pre m a) (s \u222a t) = \u2191(mkMetric'.pre m a) s + \u2191(mkMetric'.pre m a) t\n[PROOFSTEP]\nrintro \u03b5 \u27e8_, \u03b5r\u27e9\n[GOAL]\ncase h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2191(mkMetric'.pre m \u03b5) (s \u222a t) = \u2191(mkMetric'.pre m \u03b5) s + \u2191(mkMetric'.pre m \u03b5) t\n[PROOFSTEP]\nrefine' boundedBy_union_of_top_of_nonempty_inter _\n[GOAL]\ncase h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\n\u22a2 \u2200 (u : Set X), Set.Nonempty (s \u2229 u) \u2192 Set.Nonempty (t \u2229 u) \u2192 extend (fun s x => m s) u = \u22a4\n[PROOFSTEP]\nrintro u \u27e8x, hxs, hxu\u27e9 \u27e8y, hyt, hyu\u27e9\n[GOAL]\ncase h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\n\u22a2 extend (fun s x => m s) u = \u22a4\n[PROOFSTEP]\nhave : \u03b5 < diam u := \u03b5r.trans_le ((hr x hxs y hyt).trans <| edist_le_diam_of_mem hxu hyu)\n[GOAL]\ncase h.intro.intro.intro.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : Set X \u2192 \u211d\u22650\u221e\ns t : Set X\nr : \u211d\u22650\u221e\nr0 : 0 < r\nhr : \u2200 (x : X), x \u2208 s \u2192 \u2200 (y : X), y \u2208 t \u2192 r \u2264 edist x y\n\u03b5 : \u211d\u22650\u221e\nleft\u271d : 0 < \u03b5\n\u03b5r : \u03b5 < r\nu : Set X\nx : X\nhxs : x \u2208 s\nhxu : x \u2208 u\ny : X\nhyt : y \u2208 t\nhyu : y \u2208 u\nthis : \u03b5 < diam u\n\u22a2 extend (fun s x => m s) u = \u22a4\n[PROOFSTEP]\nexact iInf_eq_top.2 fun h => (this.not_le h).elim\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\n\u22a2 mkMetric m\u2081 \u2264 c \u2022 mkMetric m\u2082\n[PROOFSTEP]\nclassical\nrcases(mem_nhdsWithin_Ici_iff_exists_Ico_subset' zero_lt_one).1 hle with \u27e8r, hr0, hr\u27e9\nrefine' fun s =>\n  le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)\n    (ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))\n    (mem_of_superset (Ioo_mem_nhdsWithin_Ioi \u27e8le_rfl, hr0\u27e9) fun r' hr' => _)\nsimp only [mem_setOf_eq, mkMetric'.pre, RingHom.id_apply]\nrw [\u2190 smul_eq_mul, \u2190 smul_apply, smul_boundedBy hc]\nrefine' le_boundedBy.2 (fun t => (boundedBy_le _).trans _) _\nsimp only [smul_eq_mul, Pi.smul_apply, extend, iInf_eq_if]\nsplit_ifs with ht\n\u00b7 apply hr\n  exact \u27e8zero_le _, ht.trans_lt hr'.2\u27e9\n\u00b7 simp [h0]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\n\u22a2 mkMetric m\u2081 \u2264 c \u2022 mkMetric m\u2082\n[PROOFSTEP]\nrcases(mem_nhdsWithin_Ici_iff_exists_Ico_subset' zero_lt_one).1 hle with \u27e8r, hr0, hr\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\n\u22a2 mkMetric m\u2081 \u2264 c \u2022 mkMetric m\u2082\n[PROOFSTEP]\nrefine' fun s =>\n  le_of_tendsto_of_tendsto (mkMetric'.tendsto_pre _ s)\n    (ENNReal.Tendsto.const_mul (mkMetric'.tendsto_pre _ s) (Or.inr hc))\n    (mem_of_superset (Ioo_mem_nhdsWithin_Ioi \u27e8le_rfl, hr0\u27e9) fun r' hr' => _)\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\n\u22a2 r' \u2208\n    {x |\n      (fun x =>\n          (fun r => \u2191(mkMetric'.pre (fun s => m\u2081 (diam s)) r) s) x \u2264\n            (fun b => \u2191(RingHom.id \u211d\u22650\u221e) c * \u2191(mkMetric'.pre (fun s => m\u2082 (diam s)) b) s) x)\n        x}\n[PROOFSTEP]\nsimp only [mem_setOf_eq, mkMetric'.pre, RingHom.id_apply]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\n\u22a2 \u2191(boundedBy (extend fun s x => m\u2081 (diam s))) s \u2264 c * \u2191(boundedBy (extend fun s x => m\u2082 (diam s))) s\n[PROOFSTEP]\nrw [\u2190 smul_eq_mul, \u2190 smul_apply, smul_boundedBy hc]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\n\u22a2 \u2191(boundedBy (extend fun s x => m\u2081 (diam s))) s \u2264 \u2191(boundedBy (c \u2022 extend fun s x => m\u2082 (diam s))) s\n[PROOFSTEP]\nrefine' le_boundedBy.2 (fun t => (boundedBy_le _).trans _) _\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\nt : Set X\n\u22a2 extend (fun s x => m\u2081 (diam s)) t \u2264 (c \u2022 extend fun s x => m\u2082 (diam s)) t\n[PROOFSTEP]\nsimp only [smul_eq_mul, Pi.smul_apply, extend, iInf_eq_if]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\nt : Set X\n\u22a2 (if diam t \u2264 r' then m\u2081 (diam t) else \u22a4) \u2264 c * if diam t \u2264 r' then m\u2082 (diam t) else \u22a4\n[PROOFSTEP]\nsplit_ifs with ht\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\nt : Set X\nht : diam t \u2264 r'\n\u22a2 m\u2081 (diam t) \u2264 c * m\u2082 (diam t)\n[PROOFSTEP]\napply hr\n[GOAL]\ncase pos.a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\nt : Set X\nht : diam t \u2264 r'\n\u22a2 diam t \u2208 Ico 0 r\n[PROOFSTEP]\nexact \u27e8zero_le _, ht.trans_lt hr'.2\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\nr : \u211d\u22650\u221e\nhr0 : r \u2208 Ioi 0\nhr : Ico 0 r \u2286 {x | (fun x => m\u2081 x \u2264 (c \u2022 m\u2082) x) x}\ns : Set X\nr' : \u211d\u22650\u221e\nhr' : r' \u2208 Ioo 0 r\nt : Set X\nht : \u00acdiam t \u2264 r'\n\u22a2 \u22a4 \u2264 c * \u22a4\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 (mkMetric fun x => \u22a4) = \u22a4\n[PROOFSTEP]\nsimp_rw [mkMetric, mkMetric', mkMetric'.pre, extend_top, boundedBy_top, eq_top_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u22a4 \u2264 \u2a06 (r : \u211d\u22650\u221e) (_ : r > 0), \u22a4\n[PROOFSTEP]\nrw [le_iSup_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\n\u22a2 \u2200 (b : OuterMeasure X), (\u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b) \u2192 \u22a4 \u2264 b\n[PROOFSTEP]\nintro b hb\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nb : OuterMeasure X\nhb : \u2200 (i : \u211d\u22650\u221e), \u2a06 (_ : i > 0), \u22a4 \u2264 b\n\u22a2 \u22a4 \u2264 b\n[PROOFSTEP]\nsimpa using hb \u22a4\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 mkMetric m\u2081 \u2264 mkMetric m\u2082\n[PROOFSTEP]\nconvert @mkMetric_mono_smul X _ _ m\u2082 _ ENNReal.one_ne_top one_ne_zero _\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 mkMetric m\u2082 = 1 \u2022 mkMetric m\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase convert_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] 1 \u2022 m\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2191(comap f) (mkMetric m) = mkMetric m\n[PROOFSTEP]\nsimp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, comap_iSup]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2a06 (i : \u211d\u22650\u221e) (_ : i > 0), \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) =\n    \u2a06 (r : \u211d\u22650\u221e) (_ : r > 0), boundedBy (extend fun s x => m (diam s))\n[PROOFSTEP]\nrefine' surjective_id.iSup_congr id fun \u03b5 => surjective_id.iSup_congr id fun h\u03b5 => _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s))\n[PROOFSTEP]\nrw [comap_boundedBy _ (H.imp _ id)]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 (boundedBy fun s => extend (fun s x => m (diam s)) (f '' s)) = boundedBy (extend fun s x => m (diam s))\n[PROOFSTEP]\ncongr with s : 1\n[GOAL]\ncase e_m.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 extend (fun s x => m (diam s)) (f '' s) = extend (fun s x => m (diam s)) s\n[PROOFSTEP]\napply extend_congr\n[GOAL]\ncase e_m.h.hP\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2194 diam s \u2264 id \u03b5\n[PROOFSTEP]\nsimp [hf.ediam_image]\n[GOAL]\ncase e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\n\u22a2 diam (f '' s) \u2264 \u03b5 \u2192 diam s \u2264 id \u03b5 \u2192 m (diam (f '' s)) = m (diam s)\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_m.h.hm\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\ns : Set X\nha\u271d : diam (f '' s) \u2264 \u03b5\nhb\u271d : diam s \u2264 id \u03b5\n\u22a2 m (diam (f '' s)) = m (diam s)\n[PROOFSTEP]\nsimp [hf.injective.subsingleton_image_iff, hf.ediam_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\n\u22a2 Monotone m \u2192 Monotone fun s => extend (fun s x => m (diam s)) \u2191s\n[PROOFSTEP]\nintro h_mono s t hst\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 (fun s => extend (fun s x => m (diam s)) \u2191s) s \u2264 (fun s => extend (fun s x => m (diam s)) \u2191s) t\n[PROOFSTEP]\nsimp only [extend, le_iInf_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\n\u22a2 diam \u2191t \u2264 \u03b5 \u2192 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)\n[PROOFSTEP]\nintro ht\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t)\n[PROOFSTEP]\napply le_trans _ (h_mono (diam_mono hst))\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : id \u03b5 > 0\nh_mono : Monotone m\ns t : { s // Set.Nonempty s }\nhst : s \u2264 t\nht : diam \u2191t \u2264 \u03b5\n\u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam ((fun a => \u2191a) s))\n[PROOFSTEP]\nsimp only [(diam_mono hst).trans ht, le_refl, ciInf_pos]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nhc' : c \u2260 0\n\u22a2 mkMetric (c \u2022 m) = c \u2022 mkMetric m\n[PROOFSTEP]\nsimp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, ENNReal.smul_iSup]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nhc' : c \u2260 0\n\u22a2 \u2a06 (r : \u211d\u22650\u221e) (_ : r > 0), boundedBy (extend fun s x => (c \u2022 m) (diam s)) =\n    c \u2022 \u2a06 (r : \u211d\u22650\u221e) (_ : r > 0), boundedBy (extend fun s x => m (diam s))\n[PROOFSTEP]\nsimp_rw [smul_iSup, smul_boundedBy hc, smul_extend _ hc', Pi.smul_apply]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 mkMetric (c \u2022 m) = c \u2022 mkMetric m\n[PROOFSTEP]\nrw [ENNReal.smul_def, ENNReal.smul_def, mkMetric_smul m ENNReal.coe_ne_top (ENNReal.coe_ne_zero.mpr hc)]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2192 Y\nhf : Isometry f\nH : Monotone m \u2228 Surjective f\n\u22a2 \u2191(map f) (mkMetric m) = \u2191(restrict (range f)) (mkMetric m)\n[PROOFSTEP]\nrw [\u2190 isometry_comap_mkMetric _ hf H, map_comap]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nf : X \u2243\u1d62 Y\n\u22a2 \u2191(map \u2191f) (mkMetric m) = mkMetric m\n[PROOFSTEP]\nrw [\u2190 isometryEquiv_comap_mkMetric _ f, map_comap_of_surjective f.surjective]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 trim (mkMetric m) = mkMetric m\n[PROOFSTEP]\nsimp only [mkMetric, mkMetric'.eq_iSup_nat, trim_iSup]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2a06 (i : \u2115), trim (mkMetric'.pre (fun s => m (diam s)) (\u2191i)\u207b\u00b9) = \u2a06 (n : \u2115), mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9\n[PROOFSTEP]\ncongr 1 with n : 1\n[GOAL]\ncase e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\n\u22a2 trim (mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9) = mkMetric'.pre (fun s => m (diam s)) (\u2191n)\u207b\u00b9\n[PROOFSTEP]\nrefine' mkMetric'.trim_pre _ (fun s => _) _\n[GOAL]\ncase e_s.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nn : \u2115\ns : Set X\n\u22a2 m (diam (closure s)) = m (diam s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2191(mkMetric m) = \u2191\u2191(Measure.mkMetric m)\n[PROOFSTEP]\nrw [\u2190 Measure.mkMetric_toOuterMeasure]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\n\u22a2 mkMetric m\u2081 \u2264 c \u2022 mkMetric m\u2082\n[PROOFSTEP]\nintro s _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\ns : Set X\na\u271d : MeasurableSet s\n\u22a2 \u2191\u2191(mkMetric m\u2081) s \u2264 \u2191\u2191(c \u2022 mkMetric m\u2082) s\n[PROOFSTEP]\nrw [\u2190 OuterMeasure.coe_mkMetric, coe_smul, \u2190 OuterMeasure.coe_mkMetric]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\nh0 : c \u2260 0\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] c \u2022 m\u2082\ns : Set X\na\u271d : MeasurableSet s\n\u22a2 \u2191(OuterMeasure.mkMetric m\u2081) s \u2264 (c \u2022 \u2191(OuterMeasure.mkMetric m\u2082)) s\n[PROOFSTEP]\nexact OuterMeasure.mkMetric_mono_smul hc h0 hle s\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u22a2 (mkMetric fun x => \u22a4) = \u22a4\n[PROOFSTEP]\napply toOuterMeasure_injective\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u22a2 \u2191(mkMetric fun x => \u22a4) = \u2191\u22a4\n[PROOFSTEP]\nrw [mkMetric_toOuterMeasure, OuterMeasure.mkMetric_top, toOuterMeasure_top]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 mkMetric m\u2081 \u2264 mkMetric m\u2082\n[PROOFSTEP]\nconvert @mkMetric_mono_smul X _ _ _ _ m\u2082 _ ENNReal.one_ne_top one_ne_zero _\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 mkMetric m\u2082 = 1 \u2022 mkMetric m\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase convert_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm\u2081 m\u2082 : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nhle : m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] m\u2082\n\u22a2 m\u2081 \u2264\u1da0[\ud835\udcdd[Ici 0] 0] 1 \u2022 m\u2082\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191\u2191(mkMetric m) s =\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nclassical\n  -- We mostly unfold the definitions but we need to switch the order of `\u2211'` and `\u2a05`\nsimp only [\u2190 OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric', OuterMeasure.iSup_apply,\n  OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend]\nrefine'\n  surjective_id.iSup_congr (fun r => r) fun r =>\n    iSup_congr_Prop Iff.rfl fun _ => surjective_id.iInf_congr _ fun t => iInf_congr_Prop Iff.rfl fun ht => _\ndsimp\nby_cases htr : \u2200 n, diam (t n) \u2264 r\n\u00b7 rw [iInf_eq_if, if_pos htr]\n  congr 1 with n : 1\n  simp only [iInf_eq_if, htr n, id, if_true, iSup_and']\n\u00b7 rw [iInf_eq_if, if_neg htr]\n  push_neg at htr ; rcases htr with \u27e8n, hn\u27e9\n  refine' ENNReal.tsum_eq_top_of_eq_top \u27e8n, _\u27e9\n  rw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]\n  exact hn.not_le\n  rcases diam_pos_iff.1 ((zero_le r).trans_lt hn) with \u27e8x, hx, -\u27e9\n  exact \u27e8x, hx\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2191\u2191(mkMetric m) s =\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nsimp only [\u2190 OuterMeasure.coe_mkMetric, OuterMeasure.mkMetric, OuterMeasure.mkMetric', OuterMeasure.iSup_apply,\n  OuterMeasure.mkMetric'.pre, OuterMeasure.boundedBy_apply, extend]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\n\u22a2 \u2a06 (i : \u211d\u22650\u221e) (_ : i > 0),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 i), m (diam (t n)) =\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nrefine'\n  surjective_id.iSup_congr (fun r => r) fun r =>\n    iSup_congr_Prop Iff.rfl fun _ => surjective_id.iInf_congr _ fun t => iInf_congr_Prop Iff.rfl fun ht => _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (id t n)), \u2a05 (_ : diam (id t n) \u2264 r), m (diam (id t n)) =\n    \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 (fun r => r) r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) =\n    \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nby_cases htr : \u2200 n, diam (t n) \u2264 r\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u2200 (n : \u2115), diam (t n) \u2264 r\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) =\n    \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nrw [iInf_eq_if, if_pos htr]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u2200 (n : \u2115), diam (t n) \u2264 r\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) =\n    \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\ncongr 1 with n : 1\n[GOAL]\ncase pos.e_f.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u2200 (n : \u2115), diam (t n) \u2264 r\nn : \u2115\n\u22a2 \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) = \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nsimp only [iInf_eq_if, htr n, id, if_true, iSup_and']\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u00ac\u2200 (n : \u2115), diam (t n) \u2264 r\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) =\n    \u2a05 (_ : \u2200 (n : \u2115), diam (t n) \u2264 r), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n))\n[PROOFSTEP]\nrw [iInf_eq_if, if_neg htr]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u00ac\u2200 (n : \u2115), diam (t n) \u2264 r\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) = \u22a4\n[PROOFSTEP]\npush_neg at htr \n[GOAL]\ncase neg\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nhtr : \u2203 n, r < diam (t n)\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) = \u22a4\n[PROOFSTEP]\nrcases htr with \u27e8n, hn\u27e9\n[GOAL]\ncase neg.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) = \u22a4\n[PROOFSTEP]\nrefine' ENNReal.tsum_eq_top_of_eq_top \u27e8n, _\u27e9\n[GOAL]\ncase neg.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\n\u22a2 \u2a06 (_ : Set.Nonempty (t n)), \u2a05 (_ : diam (t n) \u2264 r), m (diam (t n)) = \u22a4\n[PROOFSTEP]\nrw [iSup_eq_if, if_pos, iInf_eq_if, if_neg]\n[GOAL]\ncase neg.intro.hnc\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\n\u22a2 \u00acdiam (t n) \u2264 r\ncase neg.intro.hc\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\n\u22a2 Set.Nonempty (t n)\n[PROOFSTEP]\nexact hn.not_le\n[GOAL]\ncase neg.intro.hc\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\n\u22a2 Set.Nonempty (t n)\n[PROOFSTEP]\nrcases diam_pos_iff.1 ((zero_le r).trans_lt hn) with \u27e8x, hx, -\u27e9\n[GOAL]\ncase neg.intro.hc.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\ns : Set X\nr : \u211d\u22650\u221e\nx\u271d : r > 0\nt : \u2115 \u2192 Set X\nht : s \u2286 iUnion t\nn : \u2115\nhn : r < diam (t n)\nx : X\nhx : x \u2208 t n\n\u22a2 Set.Nonempty (t n)\n[PROOFSTEP]\nexact \u27e8x, hx\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u03bc : Measure X\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nh : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s)\n\u22a2 \u03bc \u2264 mkMetric m\n[PROOFSTEP]\nrw [\u2190 toOuterMeasure_le, mkMetric_toOuterMeasure]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u03bc : Measure X\n\u03b5 : \u211d\u22650\u221e\nh\u2080 : 0 < \u03b5\nh : \u2200 (s : Set X), diam s \u2264 \u03b5 \u2192 \u2191\u2191\u03bc s \u2264 m (diam s)\n\u22a2 \u2191\u03bc \u2264 OuterMeasure.mkMetric m\n[PROOFSTEP]\nexact OuterMeasure.le_mkMetric m \u03bc.toOuterMeasure \u03b5 h\u2080 h\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l\n[PROOFSTEP]\nhaveI : \u2200 n, Encodable (\u03b9 n) := fun n => Encodable.ofCountable _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l\n[PROOFSTEP]\nsimp only [mkMetric_apply]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u22a2 \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l\n[PROOFSTEP]\nrefine' iSup\u2082_le fun \u03b5 h\u03b5 => _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l\n[PROOFSTEP]\nrefine' le_of_forall_le_of_dense fun c hc => _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c\n[PROOFSTEP]\nrcases((frequently_lt_of_liminf_lt (by isBoundedDefault) hc).and_eventually\n      ((hr.eventually (gt_mem_nhds h\u03b5)).and (ht.and hst))).exists with\n  \u27e8n, hn, hrn, htn, hstn\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) l fun n => \u2211' (i : \u03b9 n), m (diam (t n i))\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c\n[PROOFSTEP]\nset u : \u2115 \u2192 Set X := fun j => \u22c3 b \u2208 decode\u2082 (\u03b9 n) j, t n b\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : s \u2286 iUnion t) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), m (diam (t n)) \u2264\n    c\n[PROOFSTEP]\nrefine' iInf\u2082_le_of_le u (by rwa [iUnion_decode\u2082]) _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\n\u22a2 s \u2286 iUnion u\n[PROOFSTEP]\nrwa [iUnion_decode\u2082]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\n\u22a2 \u2a05 (_ : \u2200 (n : \u2115), diam (u n) \u2264 \u03b5), \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c\n[PROOFSTEP]\nrefine' iInf_le_of_le (fun j => _) _\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\nj : \u2115\n\u22a2 diam (u j) \u2264 \u03b5\n[PROOFSTEP]\nrw [EMetric.diam_iUnion_mem_option]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\nj : \u2115\n\u22a2 \u2a06 (i : \u03b9 n) (_ : i \u2208 decode\u2082 (\u03b9 n) j), diam (t n i) \u2264 \u03b5\n[PROOFSTEP]\nexact iSup\u2082_le fun _ _ => (htn _).trans hrn.le\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (u n)), m (diam (u n)) \u2264 c\n[PROOFSTEP]\ncalc\n  (\u2211' j : \u2115, \u2a06 _ : (u j).Nonempty, m (diam (u j))) = _ :=\n    tsum_iUnion_decode\u2082 (fun t : Set X => \u2a06 _ : t.Nonempty, m (diam t)) (by simp) _\n  _ \u2264 \u2211' i : \u03b9 n, m (diam (t n i)) := (ENNReal.tsum_le_tsum fun b => iSup_le fun _ => le_rfl)\n  _ \u2264 c := hn.le\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2074 : EMetricSpace X\ninst\u271d\u00b3 : EMetricSpace Y\ninst\u271d\u00b2 : MeasurableSpace X\ninst\u271d\u00b9 : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\ninst\u271d : \u2200 (n : \u03b2), Countable (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\nthis : (n : \u03b2) \u2192 Encodable (\u03b9 n)\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : 0 < \u03b5\nc : \u211d\u22650\u221e\nhc : liminf (fun n => \u2211' (i : \u03b9 n), m (diam (t n i))) l < c\nn : \u03b2\nhn : \u2211' (i : \u03b9 n), m (diam (t n i)) < c\nhrn : r n < \u03b5\nhtn : \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhstn : s \u2286 \u22c3 (i : \u03b9 n), t n i\nu : \u2115 \u2192 Set X := fun j => \u22c3 (b : \u03b9 n) (_ : b \u2208 decode\u2082 (\u03b9 n) j), t n b\n\u22a2 (fun t => \u2a06 (_ : Set.Nonempty t), m (diam t)) \u2205 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\n\u03b2 : Type u_4\n\u03b9 : \u03b2 \u2192 Type u_5\nh\u03b9 : (n : \u03b2) \u2192 Fintype (\u03b9 n)\ns : Set X\nl : Filter \u03b2\nr : \u03b2 \u2192 \u211d\u22650\u221e\nhr : Tendsto r l (\ud835\udcdd 0)\nt : (n : \u03b2) \u2192 \u03b9 n \u2192 Set X\nht : \u2200\u1da0 (n : \u03b2) in l, \u2200 (i : \u03b9 n), diam (t n i) \u2264 r n\nhst : \u2200\u1da0 (n : \u03b2) in l, s \u2286 \u22c3 (i : \u03b9 n), t n i\nm : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e\n\u22a2 \u2191\u2191(mkMetric m) s \u2264 liminf (fun n => \u2211 i : \u03b9 n, m (diam (t n i))) l\n[PROOFSTEP]\nsimpa only [tsum_fintype] using mkMetric_le_liminf_tsum s r hr t ht hst m\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s = 0 \u2228 \u2191\u2191\u03bcH[d\u2081] s = \u22a4\n[PROOFSTEP]\nby_contra' H\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\n\u22a2 False\n[PROOFSTEP]\nsuffices \u2200 c : \u211d\u22650, c \u2260 0 \u2192 \u03bcH[d\u2082] s \u2264 c * \u03bcH[d\u2081] s\n  by\n  rcases ENNReal.exists_nnreal_pos_mul_lt H.2 H.1 with \u27e8c, hc0, hc\u27e9\n  exact hc.not_le (this c (pos_iff_ne_zero.1 hc0))\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nthis : \u2200 (c : \u211d\u22650), c \u2260 0 \u2192 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191c * \u2191\u2191\u03bcH[d\u2081] s\n\u22a2 False\n[PROOFSTEP]\nrcases ENNReal.exists_nnreal_pos_mul_lt H.2 H.1 with \u27e8c, hc0, hc\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nthis : \u2200 (c : \u211d\u22650), c \u2260 0 \u2192 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191c * \u2191\u2191\u03bcH[d\u2081] s\nc : \u211d\u22650\nhc0 : c > 0\nhc : \u2191c * \u2191\u2191\u03bcH[d\u2081] s < \u2191\u2191\u03bcH[d\u2082] s\n\u22a2 False\n[PROOFSTEP]\nexact hc.not_le (this c (pos_iff_ne_zero.1 hc0))\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\n\u22a2 \u2200 (c : \u211d\u22650), c \u2260 0 \u2192 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191c * \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nintro c hc\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191c * \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nrefine' le_iff'.1 (mkMetric_mono_smul ENNReal.coe_ne_top (by exact_mod_cast hc) _) s\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 \u2191c \u2260 0\n[PROOFSTEP]\nexact_mod_cast hc\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 (fun r => r ^ d\u2082) \u2264\u1da0[\ud835\udcdd[Ici 0] 0] \u2191c \u2022 fun r => r ^ d\u2081\n[PROOFSTEP]\nhave : 0 < ((c : \u211d\u22650\u221e) ^ (d\u2082 - d\u2081)\u207b\u00b9) :=\n  by\n  rw [ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne.def, ENNReal.coe_eq_zero, NNReal.rpow_eq_zero_iff]\n  exact mt And.left hc\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n[PROOFSTEP]\nrw [ENNReal.coe_rpow_of_ne_zero hc, pos_iff_ne_zero, Ne.def, ENNReal.coe_eq_zero, NNReal.rpow_eq_zero_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\n\u22a2 \u00ac(c = 0 \u2227 (d\u2082 - d\u2081)\u207b\u00b9 \u2260 0)\n[PROOFSTEP]\nexact mt And.left hc\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 (fun r => r ^ d\u2082) \u2264\u1da0[\ud835\udcdd[Ici 0] 0] \u2191c \u2022 fun r => r ^ d\u2081\n[PROOFSTEP]\nfilter_upwards [Ico_mem_nhdsWithin_Ici \u27e8le_rfl, this\u27e9]\n[GOAL]\ncase h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 \u2200 (a : \u211d\u22650\u221e), a \u2208 Ico 0 (\u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9) \u2192 a ^ d\u2082 \u2264 (\u2191c \u2022 fun r => r ^ d\u2081) a\n[PROOFSTEP]\nrintro r \u27e8hr\u2080, hrc\u27e9\n[GOAL]\ncase h.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\u221e\nhr\u2080 : 0 \u2264 r\nhrc : r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 r ^ d\u2082 \u2264 (\u2191c \u2022 fun r => r ^ d\u2081) r\n[PROOFSTEP]\nlift r to \u211d\u22650 using ne_top_of_lt hrc\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080 : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 \u2191r ^ d\u2082 \u2264 (\u2191c \u2022 fun r => r ^ d\u2081) \u2191r\n[PROOFSTEP]\nrw [Pi.smul_apply, smul_eq_mul, \u2190\n  ENNReal.div_le_iff_le_mul (Or.inr ENNReal.coe_ne_top) (Or.inr <| mt ENNReal.coe_eq_zero.1 hc)]\n[GOAL]\ncase h.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080 : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 \u2191r ^ d\u2082 / \u2191r ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nrcases eq_or_ne r 0 with (rfl | hr\u2080)\n[GOAL]\ncase h.intro.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : 0 \u2264 \u21910\nhrc : \u21910 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\n\u22a2 \u21910 ^ d\u2082 / \u21910 ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nrcases lt_or_le 0 d\u2082 with (h\u2082 | h\u2082)\n[GOAL]\ncase h.intro.intro.inl.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : 0 \u2264 \u21910\nhrc : \u21910 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nh\u2082 : 0 < d\u2082\n\u22a2 \u21910 ^ d\u2082 / \u21910 ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nsimp only [h\u2082, ENNReal.zero_rpow_of_pos, zero_le, ENNReal.zero_div, ENNReal.coe_zero]\n[GOAL]\ncase h.intro.intro.inl.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : 0 \u2264 \u21910\nhrc : \u21910 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nh\u2082 : d\u2082 \u2264 0\n\u22a2 \u21910 ^ d\u2082 / \u21910 ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nsimp only [h.trans_le h\u2082, ENNReal.div_top, zero_le, ENNReal.zero_rpow_of_neg, ENNReal.coe_zero]\n[GOAL]\ncase h.intro.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080\u271d : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : r \u2260 0\n\u22a2 \u2191r ^ d\u2082 / \u2191r ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nhave : (r : \u211d\u22650\u221e) \u2260 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using hr\u2080\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080\u271d : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : r \u2260 0\n\u22a2 \u2191r \u2260 0\n[PROOFSTEP]\nsimpa only [ENNReal.coe_eq_zero, Ne.def] using hr\u2080\n[GOAL]\ncase h.intro.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis\u271d : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080\u271d : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : r \u2260 0\nthis : \u2191r \u2260 0\n\u22a2 \u2191r ^ d\u2082 / \u2191r ^ d\u2081 \u2264 \u2191c\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_sub _ _ this ENNReal.coe_ne_top]\n[GOAL]\ncase h.intro.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis\u271d : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080\u271d : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : r \u2260 0\nthis : \u2191r \u2260 0\n\u22a2 \u2191r ^ (d\u2082 - d\u2081) \u2264 \u2191c\n[PROOFSTEP]\nrefine' (ENNReal.rpow_lt_rpow hrc (sub_pos.2 h)).le.trans _\n[GOAL]\ncase h.intro.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 < d\u2082\ns : Set X\nH : \u2191\u2191\u03bcH[d\u2082] s \u2260 0 \u2227 \u2191\u2191\u03bcH[d\u2081] s \u2260 \u22a4\nc : \u211d\u22650\nhc : c \u2260 0\nthis\u271d : 0 < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nr : \u211d\u22650\nhr\u2080\u271d : 0 \u2264 \u2191r\nhrc : \u2191r < \u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9\nhr\u2080 : r \u2260 0\nthis : \u2191r \u2260 0\n\u22a2 (\u2191c ^ (d\u2082 - d\u2081)\u207b\u00b9) ^ (d\u2082 - d\u2081) \u2264 \u2191c\n[PROOFSTEP]\nrw [\u2190 ENNReal.rpow_mul, inv_mul_cancel (sub_pos.2 h).ne', ENNReal.rpow_one]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh : d\u2081 \u2264 d\u2082\ns : Set X\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nrcases h.eq_or_lt with (rfl | h)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 : \u211d\ns : Set X\nh : d\u2081 \u2264 d\u2081\n\u22a2 \u2191\u2191\u03bcH[d\u2081] s \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nexact le_rfl\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh\u271d : d\u2081 \u2264 d\u2082\ns : Set X\nh : d\u2081 < d\u2082\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\ncases' hausdorffMeasure_zero_or_top h s with hs hs\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh\u271d : d\u2081 \u2264 d\u2082\ns : Set X\nh : d\u2081 < d\u2082\nhs : \u2191\u2191\u03bcH[d\u2082] s = 0\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh\u271d : d\u2081 \u2264 d\u2082\ns : Set X\nh : d\u2081 < d\u2082\nhs : \u2191\u2191\u03bcH[d\u2082] s = 0\n\u22a2 0 \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nexact zero_le _\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh\u271d : d\u2081 \u2264 d\u2082\ns : Set X\nh : d\u2081 < d\u2082\nhs : \u2191\u2191\u03bcH[d\u2081] s = \u22a4\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u2191\u2191\u03bcH[d\u2081] s\n[PROOFSTEP]\nrw [hs]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd\u2081 d\u2082 : \u211d\nh\u271d : d\u2081 \u2264 d\u2082\ns : Set X\nh : d\u2081 < d\u2082\nhs : \u2191\u2191\u03bcH[d\u2081] s = \u22a4\n\u22a2 \u2191\u2191\u03bcH[d\u2082] s \u2264 \u22a4\n[PROOFSTEP]\nexact le_top\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\n\u22a2 NoAtoms \u03bcH[d]\n[PROOFSTEP]\nrefine' \u27e8fun x => _\u27e9\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\nx : X\n\u22a2 \u2191\u2191\u03bcH[d] {x} = 0\n[PROOFSTEP]\nrw [\u2190 nonpos_iff_eq_zero, hausdorffMeasure_apply]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\nx : X\n\u22a2 \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    0\n[PROOFSTEP]\nrefine' iSup\u2082_le fun \u03b5 _ => iInf\u2082_le_of_le (fun _ => { x }) _ <| iInf_le_of_le (fun _ => _) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\nx : X\n\u03b5 : \u211d\u22650\u221e\nx\u271d : 0 < \u03b5\n\u22a2 {x} \u2286 \u22c3 (n : \u2115), (fun x_1 => {x}) n\n[PROOFSTEP]\nexact subset_iUnion (fun _ => { x } : \u2115 \u2192 Set X) 0\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\nx : X\n\u03b5 : \u211d\u22650\u221e\nx\u271d\u00b9 : 0 < \u03b5\nx\u271d : \u2115\n\u22a2 diam ((fun x_1 => {x}) x\u271d) \u2264 \u03b5\n[PROOFSTEP]\nsimp only [EMetric.diam_singleton, zero_le]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 < d\nx : X\n\u03b5 : \u211d\u22650\u221e\nx\u271d : 0 < \u03b5\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty ((fun x_1 => {x}) n)), diam ((fun x_2 => {x}) n) ^ d \u2264 0\n[PROOFSTEP]\nsimp [hd]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 \u2191\u2191\u03bcH[0] {x} = 1\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 1\n[PROOFSTEP]\nlet r : \u2115 \u2192 \u211d\u22650\u221e := fun _ => 0\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nr : \u2115 \u2192 \u211d\u22650\u221e := fun x => 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 1\n[PROOFSTEP]\nlet t : \u2115 \u2192 Unit \u2192 Set X := fun _ _ => { x }\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nr : \u2115 \u2192 \u211d\u22650\u221e := fun x => 0\nt : \u2115 \u2192 Unit \u2192 Set X := fun x_1 x_2 => {x}\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 1\n[PROOFSTEP]\nhave ht : \u2200\u1da0 n in atTop, \u2200 i, diam (t n i) \u2264 r n := by\n  simp only [imp_true_iff, eq_self_iff_true, diam_singleton, eventually_atTop, nonpos_iff_eq_zero, exists_const]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nr : \u2115 \u2192 \u211d\u22650\u221e := fun x => 0\nt : \u2115 \u2192 Unit \u2192 Set X := fun x_1 x_2 => {x}\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : Unit), diam (t n i) \u2264 r n\n[PROOFSTEP]\nsimp only [imp_true_iff, eq_self_iff_true, diam_singleton, eventually_atTop, nonpos_iff_eq_zero, exists_const]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nr : \u2115 \u2192 \u211d\u22650\u221e := fun x => 0\nt : \u2115 \u2192 Unit \u2192 Set X := fun x_1 x_2 => {x}\nht : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : Unit), diam (t n i) \u2264 r n\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 1\n[PROOFSTEP]\nsimpa [liminf_const] using hausdorffMeasure_le_liminf_sum 0 { x } r tendsto_const_nhds t ht\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] {x}\n[PROOFSTEP]\nrw [hausdorffMeasure_apply]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 1 \u2264\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n[PROOFSTEP]\nsuffices\n  (1 : \u211d\u22650\u221e) \u2264\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : { x } \u2286 \u22c3 n, t n) (_ : \u2200 n, diam (t n) \u2264 1), \u2211' n, \u2a06 _ : (t n).Nonempty, diam (t n) ^ (0 : \u211d)\n  by\n  apply le_trans this _\n  convert le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) (1 : \u211d\u22650\u221e) zero_lt_one\n  rfl\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nthis :\n  1 \u2264\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n\u22a2 1 \u2264\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n[PROOFSTEP]\napply le_trans this _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nthis :\n  1 \u2264\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0 \u2264\n    \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n[PROOFSTEP]\nconvert le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) (1 : \u211d\u22650\u221e) zero_lt_one\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nthis :\n  1 \u2264\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0 =\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 1 \u2264\n    \u2a05 (t : \u2115 \u2192 Set X) (_ : {x} \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 1),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ 0\n[PROOFSTEP]\nsimp only [ENNReal.rpow_zero, le_iInf_iff]\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\n\u22a2 \u2200 (i : \u2115 \u2192 Set X), {x} \u2286 \u22c3 (n : \u2115), i n \u2192 (\u2200 (n : \u2115), diam (i n) \u2264 1) \u2192 1 \u2264 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (i n)), 1\n[PROOFSTEP]\nintro t hst _\n[GOAL]\ncase a\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nt : \u2115 \u2192 Set X\nhst : {x} \u2286 \u22c3 (n : \u2115), t n\ni\u271d : \u2200 (n : \u2115), diam (t n) \u2264 1\n\u22a2 1 \u2264 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), 1\n[PROOFSTEP]\nrcases mem_iUnion.1 (hst (mem_singleton x)) with \u27e8m, hm\u27e9\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nt : \u2115 \u2192 Set X\nhst : {x} \u2286 \u22c3 (n : \u2115), t n\ni\u271d : \u2200 (n : \u2115), diam (t n) \u2264 1\nm : \u2115\nhm : x \u2208 t m\n\u22a2 1 \u2264 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), 1\n[PROOFSTEP]\nhave A : (t m).Nonempty := \u27e8x, hm\u27e9\n[GOAL]\ncase a.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nt : \u2115 \u2192 Set X\nhst : {x} \u2286 \u22c3 (n : \u2115), t n\ni\u271d : \u2200 (n : \u2115), diam (t n) \u2264 1\nm : \u2115\nhm : x \u2208 t m\nA : Set.Nonempty (t m)\n\u22a2 1 \u2264 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), 1\n[PROOFSTEP]\ncalc\n  (1 : \u211d\u22650\u221e) = \u2a06 h : (t m).Nonempty, 1 := by simp only [A, ciSup_pos]\n  _ \u2264 \u2211' n, \u2a06 h : (t n).Nonempty, 1 := ENNReal.le_tsum _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nx : X\nt : \u2115 \u2192 Set X\nhst : {x} \u2286 \u22c3 (n : \u2115), t n\ni\u271d : \u2200 (n : \u2115), diam (t n) \u2264 1\nm : \u2115\nhm : x \u2208 t m\nA : Set.Nonempty (t m)\n\u22a2 1 = \u2a06 (_ : Set.Nonempty (t m)), 1\n[PROOFSTEP]\nsimp only [A, ciSup_pos]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nh : Set.Nonempty s\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nrcases h with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nx : X\nhx : x \u2208 s\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\ncalc\n  (1 : \u211d\u22650\u221e) = \u03bcH[0] ({ x } : Set X) := (hausdorffMeasure_zero_singleton x).symm\n  _ \u2264 \u03bcH[0] s := measure_mono (singleton_subset_iff.2 hx)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nd : \u211d\nhd : 0 \u2264 d\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 1\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\nd : \u211d\nhd : 0 \u2264 d\nhs : Set.Subsingleton \u2205\n\u22a2 \u2191\u2191\u03bcH[d] \u2205 \u2264 1\n[PROOFSTEP]\nsimp only [measure_empty, zero_le]\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nd : \u211d\nhd : 0 \u2264 d\nx : X\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 1\n[PROOFSTEP]\nrw [(subsingleton_iff_singleton hx).1 hs]\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nd : \u211d\nhd : 0 \u2264 d\nx : X\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 1\n[PROOFSTEP]\nrcases eq_or_lt_of_le hd with (rfl | dpos)\n[GOAL]\ncase inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nx : X\nhx : x \u2208 s\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 1\n[PROOFSTEP]\nsimp only [le_refl, hausdorffMeasure_zero_singleton]\n[GOAL]\ncase inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nd : \u211d\nhd : 0 \u2264 d\nx : X\nhx : x \u2208 s\ndpos : 0 < d\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 1\n[PROOFSTEP]\nhaveI := noAtoms_hausdorff X dpos\n[GOAL]\ncase inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b3 : EMetricSpace X\ninst\u271d\u00b2 : EMetricSpace Y\ninst\u271d\u00b9 : MeasurableSpace X\ninst\u271d : BorelSpace X\ns : Set X\nhs : Set.Subsingleton s\nd : \u211d\nhd : 0 \u2264 d\nx : X\nhx : x \u2208 s\ndpos : 0 < d\nthis : NoAtoms \u03bcH[d]\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 1\n[PROOFSTEP]\nsimp only [zero_le, measure_singleton]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u2191C ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nrcases(zero_le C).eq_or_lt with (rfl | hC0)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty s with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\nt : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f \u2205\n\u22a2 \u2191\u2191\u03bcH[d] (f '' \u2205) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] \u2205\n[PROOFSTEP]\nsimp only [measure_empty, nonpos_iff_eq_zero, mul_zero, image_empty]\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nhave : f '' s = {f x} :=\n  haveI : (f '' s).Subsingleton := by simpa [diam_eq_zero_iff] using h.ediam_image_le\n  (subsingleton_iff_singleton (mem_image_of_mem f hx)).1 this\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\n\u22a2 Set.Subsingleton (f '' s)\n[PROOFSTEP]\nsimpa [diam_eq_zero_iff] using h.ediam_image_le\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\n\u22a2 \u2191\u2191\u03bcH[d] {f x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nrcases eq_or_lt_of_le hd with (rfl | h'd)\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {f x} \u2264 \u21910 ^ 0 * \u2191\u2191\u03bcH[\u2191r * 0] s\n[PROOFSTEP]\nsimp only [ENNReal.rpow_zero, one_mul, mul_zero]\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {f x} \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nrw [hausdorffMeasure_zero_singleton]\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\nhd : 0 \u2264 0\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nexact one_le_hausdorffMeasure_zero_of_nonempty \u27e8x, hx\u27e9\n[GOAL]\ncase inl.inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis : f '' s = {f x}\nh'd : 0 < d\n\u22a2 \u2191\u2191\u03bcH[d] {f x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nhaveI := noAtoms_hausdorff Y h'd\n[GOAL]\ncase inl.inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nr : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nh : HolderOnWith 0 r f s\nx : X\nhx : x \u2208 s\nthis\u271d : f '' s = {f x}\nh'd : 0 < d\nthis : NoAtoms \u03bcH[d]\n\u22a2 \u2191\u2191\u03bcH[d] {f x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nsimp only [zero_le, measure_singleton]\n  -- Now assume `C \u2260 0`\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u2191C ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nhave hCd0 : (C : \u211d\u22650\u221e) ^ d \u2260 0 := by simp [hC0.ne']\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\n\u22a2 \u2191C ^ d \u2260 0\n[PROOFSTEP]\nsimp [hC0.ne']\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u2191C ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nhave hCd : (C : \u211d\u22650\u221e) ^ d \u2260 \u221e := by simp [hd]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\n\u22a2 \u2191C ^ d \u2260 \u22a4\n[PROOFSTEP]\nsimp [hd]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u2191C ^ d * \u2191\u2191\u03bcH[\u2191r * d] s\n[PROOFSTEP]\nsimp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hCd0 hCd, \u2190 ENNReal.tsum_mul_left]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\n\u22a2 \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set Y) (_ : f '' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set X) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191C ^ d * diam (i_1 i) ^ (\u2191r * d)\n[PROOFSTEP]\nrefine' iSup_le fun R => iSup_le fun hR => _\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\n\u22a2 \u2a05 (t : \u2115 \u2192 Set Y) (_ : f '' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 R),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set X) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191C ^ d * diam (i_1 i) ^ (\u2191r * d)\n[PROOFSTEP]\nhave : Tendsto (fun d : \u211d\u22650\u221e => (C : \u211d\u22650\u221e) * d ^ (r : \u211d)) (\ud835\udcdd 0) (\ud835\udcdd 0) :=\n  ENNReal.tendsto_const_mul_rpow_nhds_zero_of_pos ENNReal.coe_ne_top hr\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 \u2a05 (t : \u2115 \u2192 Set Y) (_ : f '' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 R),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set X) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191C ^ d * diam (i_1 i) ^ (\u2191r * d)\n[PROOFSTEP]\nrcases ENNReal.nhds_zero_basis_Iic.eventually_iff.1 (this.eventually (gt_mem_nhds hR)) with \u27e8\u03b4, \u03b40, H\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\n\u22a2 \u2a05 (t : \u2115 \u2192 Set Y) (_ : f '' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 R),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set X) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191C ^ d * diam (i_1 i) ^ (\u2191r * d)\n[PROOFSTEP]\nrefine'\n  le_iSup\u2082_of_le \u03b4 \u03b40\n    (iInf\u2082_mono' fun t hst =>\n      \u27e8fun n => f '' (t n \u2229 s), _,\n        iInf_mono' fun ht\u03b4 => \u27e8fun n => (h.ediam_image_inter_le (t n)).trans (H (ht\u03b4 n)).le, _\u27e9\u27e9)\n[GOAL]\ncase inr.intro.intro.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\n\u22a2 f '' s \u2286 \u22c3 (n : \u2115), (fun n => f '' (t n \u2229 s)) n\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 iUnion_inter]\n[GOAL]\ncase inr.intro.intro.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\n\u22a2 f '' s \u2286 f '' ((\u22c3 (i : \u2115), t i) \u2229 s)\n[PROOFSTEP]\nexact image_subset _ (subset_inter hst Subset.rfl)\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty ((fun n => f '' (t n \u2229 s)) n)), diam ((fun n => f '' (t n \u2229 s)) n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191C ^ d * diam (t i) ^ (\u2191r * d)\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => _\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\nn : \u2115\n\u22a2 \u2a06 (_ : Set.Nonempty ((fun n => f '' (t n \u2229 s)) n)), diam ((fun n => f '' (t n \u2229 s)) n) ^ d \u2264\n    \u2a06 (_ : Set.Nonempty (t n)), \u2191C ^ d * diam (t n) ^ (\u2191r * d)\n[PROOFSTEP]\nsimp only [iSup_le_iff, nonempty_image_iff]\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\nn : \u2115\n\u22a2 Set.Nonempty (t n \u2229 s) \u2192 diam (f '' (t n \u2229 s)) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191C ^ d * diam (t n) ^ (\u2191r * d)\n[PROOFSTEP]\nintro hft\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\nn : \u2115\nhft : Set.Nonempty (t n \u2229 s)\n\u22a2 diam (f '' (t n \u2229 s)) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191C ^ d * diam (t n) ^ (\u2191r * d)\n[PROOFSTEP]\nsimp only [Nonempty.mono ((t n).inter_subset_left s) hft, ciSup_pos]\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\nn : \u2115\nhft : Set.Nonempty (t n \u2229 s)\n\u22a2 diam (f '' (t n \u2229 s)) ^ d \u2264 \u2191C ^ d * diam (t n) ^ (\u2191r * d)\n[PROOFSTEP]\nrw [ENNReal.rpow_mul, \u2190 ENNReal.mul_rpow_of_nonneg _ _ hd]\n[GOAL]\ncase inr.intro.intro.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nC r : \u211d\u22650\nf : X \u2192 Y\ns t\u271d : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : \u211d\nhd : 0 \u2264 d\nhC0 : 0 < C\nhCd0 : \u2191C ^ d \u2260 0\nhCd : \u2191C ^ d \u2260 \u22a4\nR : \u211d\u22650\u221e\nhR : 0 < R\nthis : Tendsto (fun d => \u2191C * d ^ \u2191r) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u03b4 : \u211d\u22650\u221e\n\u03b40 : 0 < \u03b4\nH : \u2200 \u2983x : \u211d\u22650\u221e\u2984, x \u2208 Iic \u03b4 \u2192 \u2191C * x ^ \u2191r < R\nt : \u2115 \u2192 Set X\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b4 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b4\nn : \u2115\nhft : Set.Nonempty (t n \u2229 s)\n\u22a2 diam (f '' (t n \u2229 s)) ^ d \u2264 (\u2191C * diam (t n) ^ \u2191r) ^ d\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (h.ediam_image_inter_le _) hd\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nK : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : LipschitzOnWith K f s\nd : \u211d\nhd : 0 \u2264 d\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimpa only [NNReal.coe_one, one_mul] using h.holderOnWith.hausdorffMeasure_image_le zero_lt_one hd\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) = NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsuffices \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 s : Set E, \u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u03bcH[d] s\n  by\n  refine' le_antisymm (this hr s) _\n  rw [\u2190 ENNReal.le_inv_smul_iff]\n  dsimp\n  rw [\u2190 NNReal.inv_rpow, \u2190 nnnorm_inv]\n  \u00b7 refine' Eq.trans_le _ (this (inv_ne_zero hr) (r \u2022 s))\n    rw [inv_smul_smul\u2080 hr]\n  \u00b7 simp [hr]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) = NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrefine' le_antisymm (this hr s) _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s \u2264 \u2191\u2191\u03bcH[d] (r \u2022 s)\n[PROOFSTEP]\nrw [\u2190 ENNReal.le_inv_smul_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 (NNReal.rpow \u2016r\u2016\u208a d)\u207b\u00b9 \u2022 \u2191\u2191\u03bcH[d] (r \u2022 s)\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 \u2191(\u2016r\u2016\u208a ^ d)\u207b\u00b9 * \u2191\u2191\u03bcH[d] (r \u2022 s)\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0\n[PROOFSTEP]\nrw [\u2190 NNReal.inv_rpow, \u2190 nnnorm_inv]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s \u2264 \u2191(\u2016r\u207b\u00b9\u2016\u208a ^ d) * \u2191\u2191\u03bcH[d] (r \u2022 s)\n[PROOFSTEP]\nrefine' Eq.trans_le _ (this (inv_ne_zero hr) (r \u2022 s))\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] s = \u2191\u2191\u03bcH[d] (r\u207b\u00b9 \u2022 r \u2022 s)\n[PROOFSTEP]\nrw [inv_smul_smul\u2080 hr]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\nthis : \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 NNReal.rpow \u2016r\u2016\u208a d \u2260 0\n[PROOFSTEP]\nsimp [hr]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr : \ud835\udd5c\nhr : r \u2260 0\ns : Set E\n\u22a2 \u2200 {r : \ud835\udd5c}, r \u2260 0 \u2192 \u2200 (s : Set E), \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nintro r _ s\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 NNReal.rpow \u2016r\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimp only [NNReal.rpow_eq_pow, ENNReal.smul_def, \u2190 ENNReal.coe_rpow_of_nonneg _ hd, smul_eq_mul]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u2070 : EMetricSpace X\ninst\u271d\u2079 : EMetricSpace Y\ninst\u271d\u2078 : MeasurableSpace X\ninst\u271d\u2077 : BorelSpace X\ninst\u271d\u2076 : MeasurableSpace Y\ninst\u271d\u2075 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nd : \u211d\nhd : 0 \u2264 d\nr\u271d : \ud835\udd5c\nhr : r\u271d \u2260 0\ns\u271d : Set E\nr : \ud835\udd5c\na\u271d : r \u2260 0\ns : Set E\n\u22a2 \u2191\u2191\u03bcH[d] (r \u2022 s) \u2264 \u2191\u2016r\u2016\u208a ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nexact (lipschitzWith_smul (\u03b2 := E) r).hausdorffMeasure_image_le hd s\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrcases eq_or_ne K 0 with (rfl | h0)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrcases eq_empty_or_nonempty (f \u207b\u00b9' s) with (hs | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nhs : f \u207b\u00b9' s = \u2205\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimp only [hs, measure_empty, zero_le]\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nhave : f \u207b\u00b9' s = { x } := by\n  haveI : Subsingleton X := hf.subsingleton\n  have : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _)\n  exact (subsingleton_iff_singleton hx).1 this\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 f \u207b\u00b9' s = {x}\n[PROOFSTEP]\nhaveI : Subsingleton X := hf.subsingleton\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : Subsingleton X\n\u22a2 f \u207b\u00b9' s = {x}\n[PROOFSTEP]\nhave : (f \u207b\u00b9' s).Subsingleton := subsingleton_univ.anti (subset_univ _)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : Subsingleton X\nthis : Set.Subsingleton (f \u207b\u00b9' s)\n\u22a2 f \u207b\u00b9' s = {x}\n[PROOFSTEP]\nexact (subsingleton_iff_singleton hx).1 this\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase inl.inr.intro\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrcases eq_or_lt_of_le hd with (rfl | h'd)\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u21910 ^ 0 * \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nsimp only [ENNReal.rpow_zero, one_mul, mul_zero]\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 \u2191\u2191\u03bcH[0] {x} \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nrw [hausdorffMeasure_zero_singleton]\n[GOAL]\ncase inl.inr.intro.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nhd : 0 \u2264 0\n\u22a2 1 \u2264 \u2191\u2191\u03bcH[0] s\n[PROOFSTEP]\nexact one_le_hausdorffMeasure_zero_of_nonempty \u27e8f x, hx\u27e9\n[GOAL]\ncase inl.inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis : f \u207b\u00b9' s = {x}\nh'd : 0 < d\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nhaveI := noAtoms_hausdorff X h'd\n[GOAL]\ncase inl.inr.intro.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhd : 0 \u2264 d\ns : Set Y\nhf : AntilipschitzWith 0 f\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nthis\u271d : f \u207b\u00b9' s = {x}\nh'd : 0 < d\nthis : NoAtoms \u03bcH[d]\n\u22a2 \u2191\u2191\u03bcH[d] {x} \u2264 \u21910 ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimp only [zero_le, measure_singleton]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nhave hKd0 : (K : \u211d\u22650\u221e) ^ d \u2260 0 := by simp [h0]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\n\u22a2 \u2191K ^ d \u2260 0\n[PROOFSTEP]\nsimp [h0]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nhave hKd : (K : \u211d\u22650\u221e) ^ d \u2260 \u221e := by simp [hd]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\n\u22a2 \u2191K ^ d \u2260 \u22a4\n[PROOFSTEP]\nsimp [hd]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) \u2264 \u2191K ^ d * \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hKd0 hKd, \u2190 ENNReal.tsum_mul_left]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u22a2 \u2a06 (r : \u211d\u22650\u221e) (_ : 0 < r),\n      \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 r),\n        \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set Y) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d\n[PROOFSTEP]\nrefine' iSup\u2082_le fun \u03b5 \u03b50 => _\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a06 (i : \u211d\u22650\u221e) (_ : 0 < i),\n      \u2a05 (i_1 : \u2115 \u2192 Set Y) (_ : s \u2286 \u22c3 (n : \u2115), i_1 n) (_ : \u2200 (n : \u2115), diam (i_1 n) \u2264 i),\n        \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (i_1 i)), \u2191K ^ d * diam (i_1 i) ^ d\n[PROOFSTEP]\nrefine' le_iSup\u2082_of_le (\u03b5 / K) (by simp [\u03b50.ne']) _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 0 < \u03b5 / \u2191K\n[PROOFSTEP]\nsimp [\u03b50.ne']\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2a05 (i : \u2115 \u2192 Set Y) (_ : s \u2286 \u22c3 (n : \u2115), i n) (_ : \u2200 (n : \u2115), diam (i n) \u2264 \u03b5 / \u2191K),\n      \u2211' (i_1 : \u2115), \u2a06 (_ : Set.Nonempty (i i_1)), \u2191K ^ d * diam (i i_1) ^ d\n[PROOFSTEP]\nrefine' le_iInf\u2082 fun t hst => le_iInf fun ht\u03b5 => _\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nhst : s \u2286 \u22c3 (n : \u2115), t n\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d\n[PROOFSTEP]\nreplace hst : f \u207b\u00b9' s \u2286 _ := preimage_mono hst\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 f \u207b\u00b9' \u22c3 (n : \u2115), t n\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d\n[PROOFSTEP]\nrw [preimage_iUnion] at hst \n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\n\u22a2 \u2a05 (t : \u2115 \u2192 Set X) (_ : f \u207b\u00b9' s \u2286 \u22c3 (n : \u2115), t n) (_ : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5),\n      \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (t n)), diam (t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d\n[PROOFSTEP]\nrefine' iInf\u2082_le_of_le _ hst (iInf_le_of_le (fun n => _) _)\n[GOAL]\ncase inr.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\n\u22a2 diam (f \u207b\u00b9' t n) \u2264 \u03b5\n[PROOFSTEP]\nexact (hf.ediam_preimage_le _).trans (ENNReal.mul_le_of_le_div' <| ht\u03b5 n)\n[GOAL]\ncase inr.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\n\u22a2 \u2211' (n : \u2115), \u2a06 (_ : Set.Nonempty (f \u207b\u00b9' t n)), diam (f \u207b\u00b9' t n) ^ d \u2264\n    \u2211' (i : \u2115), \u2a06 (_ : Set.Nonempty (t i)), \u2191K ^ d * diam (t i) ^ d\n[PROOFSTEP]\nrefine' ENNReal.tsum_le_tsum fun n => iSup_le_iff.2 fun hft => _\n[GOAL]\ncase inr.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2a06 (_ : Set.Nonempty (t n)), \u2191K ^ d * diam (t n) ^ d\n[PROOFSTEP]\nsimp only [nonempty_of_nonempty_preimage hft, ciSup_pos]\n[GOAL]\ncase inr.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 \u2191K ^ d * diam (t n) ^ d\n[PROOFSTEP]\nrw [\u2190 ENNReal.mul_rpow_of_nonneg _ _ hd]\n[GOAL]\ncase inr.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nK : \u211d\u22650\nd : \u211d\nhf : AntilipschitzWith K f\nhd : 0 \u2264 d\ns : Set Y\nh0 : K \u2260 0\nhKd0 : \u2191K ^ d \u2260 0\nhKd : \u2191K ^ d \u2260 \u22a4\n\u03b5 : \u211d\u22650\u221e\n\u03b50 : 0 < \u03b5\nt : \u2115 \u2192 Set Y\nht\u03b5 : \u2200 (n : \u2115), diam (t n) \u2264 \u03b5 / \u2191K\nhst : f \u207b\u00b9' s \u2286 \u22c3 (i : \u2115), f \u207b\u00b9' t i\nn : \u2115\nhft : Set.Nonempty (f \u207b\u00b9' t n)\n\u22a2 diam (f \u207b\u00b9' t n) ^ d \u2264 (\u2191K * diam (t n)) ^ d\n[PROOFSTEP]\nexact ENNReal.rpow_le_rpow (hf.ediam_preimage_le _) hd\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191\u2191\u03bcH[d] (f '' s) = \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimp only [hausdorffMeasure, \u2190 OuterMeasure.coe_mkMetric, \u2190 OuterMeasure.comap_apply]\n  -- porting note: this proof was slightly nicer before the port\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) \u2191(mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s\n[PROOFSTEP]\nsimp only [mkMetric_toOuterMeasure]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s\n[PROOFSTEP]\nhave : 0 \u2264 d \u2192 Monotone fun r \u21a6 @HPow.hPow \u211d\u22650\u221e \u211d \u211d\u22650\u221e instHPow r d := by\n  exact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\n\u22a2 0 \u2264 d \u2192 Monotone fun r => r ^ d\n[PROOFSTEP]\nexact fun hd x y hxy => ENNReal.rpow_le_rpow hxy hd\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis : 0 \u2264 d \u2192 Monotone fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s\n[PROOFSTEP]\nhave := OuterMeasure.isometry_comap_mkMetric (fun (r : \u211d\u22650\u221e) => r ^ d) hf (hd.imp_left this)\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set X\nthis\u271d : 0 \u2264 d \u2192 Monotone fun r => r ^ d\nthis : \u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d) = OuterMeasure.mkMetric fun r => r ^ d\n\u22a2 \u2191(\u2191(OuterMeasure.comap f) (OuterMeasure.mkMetric fun r => r ^ d)) s = \u2191(OuterMeasure.mkMetric fun r => r ^ d) s\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set Y\n\u22a2 \u2191\u2191\u03bcH[d] (f \u207b\u00b9' s) = \u2191\u2191\u03bcH[d] (s \u2229 range f)\n[PROOFSTEP]\nrw [\u2190 hf.hausdorffMeasure_image hd, image_preimage_eq_inter_range]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\n\u22a2 Measure.map f \u03bcH[d] = Measure.restrict \u03bcH[d] (range f)\n[PROOFSTEP]\next1 s hs\n[GOAL]\ncase h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\nf : X \u2192 Y\nd : \u211d\nhf : Isometry f\nhd : 0 \u2264 d \u2228 Surjective f\ns : Set Y\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.map f \u03bcH[d]) s = \u2191\u2191(Measure.restrict \u03bcH[d] (range f)) s\n[PROOFSTEP]\nrw [map_apply hf.continuous.measurable hs, Measure.restrict_apply hs, hf.hausdorffMeasure_preimage hd]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\ne : X \u2243\u1d62 Y\nd : \u211d\ns : Set Y\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191e \u207b\u00b9' s) = \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrw [\u2190 e.image_symm, e.symm.hausdorffMeasure_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\ne : X \u2243\u1d62 Y\nd : \u211d\n\u22a2 Measure.map \u2191e \u03bcH[d] = \u03bcH[d]\n[PROOFSTEP]\nrw [e.isometry.map_hausdorffMeasure (Or.inr e.surjective), e.surjective.range_eq, restrict_univ]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u03bcH[\u2191(Fintype.card \u03b9)] = volume\n[PROOFSTEP]\nclassical\n  -- it suffices to check that the two measures coincide on products of rational intervals\nrefine'\n  (pi_eq_generateFrom (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm) (fun _ => Real.isPiSystem_Ioo_rat)\n      (fun _ => Real.finiteSpanningSetsInIooRat _) _).symm\nsimp only [mem_iUnion, mem_singleton_iff]\n  -- fix such a product `s` of rational intervals, of the form `\u03a0 (a i, b i)`.\nintro s hs\nchoose a b H using hs\nobtain rfl : s = fun i => Ioo (\u03b1 := \u211d) (a i) (b i)\nexact funext fun i => (H i).2\nreplace H := fun i => (H i).1\napply\n  le_antisymm\n    _\n      -- first check that `volume s \u2264 \u03bcH s`\n\u00b7 have Hle : volume \u2264 (\u03bcH[Fintype.card \u03b9] : Measure (\u03b9 \u2192 \u211d)) :=\n    by\n    refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _\n    rw [ENNReal.rpow_nat_cast]\n    exact Real.volume_pi_le_diam_pow s\n  rw [\u2190 volume_pi_pi fun i => Ioo (a i : \u211d) (b i)]\n  exact\n    Measure.le_iff'.1 Hle\n      _\n        /- For the other inequality `\u03bcH s \u2264 volume s`, we use a covering of `s` by sets of small diameter\n            `1/n`, namely cubes with left-most point of the form `a i + f i / n` with `f i` ranging between\n            `0` and `\u2308(b i - a i) * n\u2309`. Their number is asymptotic to `n^d * \u03a0 (b i - a i)`. -/\nhave I : \u2200 i, 0 \u2264 (b i : \u211d) - a i := fun i => by simpa only [sub_nonneg, Rat.cast_le] using (H i).le\nlet \u03b3 := fun n : \u2115 => \u2200 i : \u03b9, Fin \u2308((b i : \u211d) - a i) * n\u2309\u208a\nlet t : \u2200 n : \u2115, \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) := fun n f => Set.pi univ fun i => Icc (a i + f i / n) (a i + (f i + 1) / n)\nhave A : Tendsto (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) atTop (\ud835\udcdd 0) := by simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]\nhave B : \u2200\u1da0 n in atTop, \u2200 i : \u03b3 n, diam (t n i) \u2264 1 / n :=\n  by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  intro f\n  refine' diam_pi_le_of_le fun b => _\n  simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl, add_sub_add_left_eq_sub,\n    add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]\nhave C : \u2200\u1da0 n in atTop, (Set.pi univ fun i : \u03b9 => Ioo (a i : \u211d) (b i)) \u2286 \u22c3 i : \u03b3 n, t n i :=\n  by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  have npos : (0 : \u211d) < n := Nat.cast_pos.2 hn\n  intro x hx\n  simp only [mem_Ioo, mem_univ_pi] at hx \n  simp only [mem_iUnion, mem_Ioo, mem_univ_pi]\n  let f : \u03b3 n := fun i =>\n    \u27e8\u230a(x i - a i) * n\u230b\u208a, by\n      apply Nat.floor_lt_ceil_of_lt_of_pos\n      \u00b7 refine' (mul_lt_mul_right npos).2 _\n        simp only [(hx i).right, sub_lt_sub_iff_right]\n      \u00b7 refine' mul_pos _ npos\n        simpa only [Rat.cast_lt, sub_pos] using H i\u27e9\n  refine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9\n  \u00b7\n    calc\n      (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n :=\n        by\n        refine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n        exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n      _ = x i := by field_simp [npos.ne']\n  \u00b7\n    calc\n      x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n      _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n        add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le)\ncalc\n  \u03bcH[Fintype.card \u03b9] (Set.pi univ fun i : \u03b9 => Ioo (a i : \u211d) (b i)) \u2264\n      liminf (fun n : \u2115 => \u2211 i : \u03b3 n, diam (t n i) ^ ((Fintype.card \u03b9) : \u211d)) atTop :=\n    hausdorffMeasure_le_liminf_sum _ (Set.pi univ fun i => Ioo (a i : \u211d) (b i)) (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) A t B C\n  _ \u2264 liminf (fun n : \u2115 => \u2211 i : \u03b3 n, (1 / (n : \u211d\u22650\u221e)) ^ Fintype.card \u03b9) atTop :=\n    by\n    refine' liminf_le_liminf _ _\n    \u00b7 filter_upwards [B] with _ hn\n      apply Finset.sum_le_sum fun i _ => _\n      simp only [ENNReal.rpow_nat_cast]\n      intros i _\n      exact pow_le_pow_of_le_left' (hn i) _\n    \u00b7 isBoundedDefault\n  _ = liminf (fun n : \u2115 => \u220f i : \u03b9, (\u2308((b i : \u211d) - a i) * n\u2309\u208a : \u211d\u22650\u221e) / n) atTop := by\n    simp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const, nsmul_eq_mul,\n      Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]\n  _ = \u220f i : \u03b9, volume (Ioo (a i : \u211d) (b i)) := by\n    simp only [Real.volume_Ioo]\n    apply Tendsto.liminf_eq\n    refine' ENNReal.tendsto_finset_prod_of_ne_top _ (fun i _ => _) fun i _ => _\n    \u00b7 apply\n        Tendsto.congr' _\n          ((ENNReal.continuous_ofReal.tendsto _).comp\n            ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_nat_cast_atTop_atTop))\n      apply eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n      intros n hn\n      simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply, ENNReal.ofReal_coe_nat]\n    \u00b7 simp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u03bcH[\u2191(Fintype.card \u03b9)] = volume\n[PROOFSTEP]\nrefine'\n  (pi_eq_generateFrom (fun _ => Real.borel_eq_generateFrom_Ioo_rat.symm) (fun _ => Real.isPiSystem_Ioo_rat)\n      (fun _ => Real.finiteSpanningSetsInIooRat _) _).symm\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), s i \u2208 \u22c3 (a : \u211a) (b : \u211a) (_ : a < b), {Ioo \u2191a \u2191b}) \u2192\n      \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_singleton_iff]\n  -- fix such a product `s` of rational intervals, of the form `\u03a0 (a i, b i)`.\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\n\u22a2 \u2200 (s : \u03b9 \u2192 Set \u211d),\n    (\u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2) \u2192 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\nhs : \u2200 (i : \u03b9), \u2203 i_1 i_2 h, s i = Ioo \u2191i_1 \u2191i_2\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)\n[PROOFSTEP]\nchoose a b H using hs\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ s) = \u220f i : \u03b9, \u2191\u2191volume (s i)\n[PROOFSTEP]\nobtain rfl : s = fun i => Ioo (\u03b1 := \u211d) (a i) (b i)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\ns : \u03b9 \u2192 Set \u211d\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, s i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 s = fun i => Ioo \u2191(a i) \u2191(b i)\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nexact funext fun i => (H i).2\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), \u2203 h, (fun i => Ioo \u2191(a i) \u2191(b i)) i = Ioo \u2191(a i) \u2191(b i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nreplace H := fun i => (H i).1\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) = \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\napply\n  le_antisymm\n    _\n      -- first check that `volume s \u2264 \u03bcH s`\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))\n[PROOFSTEP]\nhave Hle : volume \u2264 (\u03bcH[Fintype.card \u03b9] : Measure (\u03b9 \u2192 \u211d)) :=\n  by\n  refine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _\n  rw [ENNReal.rpow_nat_cast]\n  exact Real.volume_pi_le_diam_pow s\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n[PROOFSTEP]\nrefine' le_hausdorffMeasure _ _ \u221e ENNReal.coe_lt_top fun s _ => _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ \u2191(Fintype.card \u03b9)\n[PROOFSTEP]\nrw [ENNReal.rpow_nat_cast]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ns : Set (\u03b9 \u2192 \u211d)\nx\u271d : diam s \u2264 \u22a4\n\u22a2 \u2191\u2191volume s \u2264 diam s ^ Fintype.card \u03b9\n[PROOFSTEP]\nexact Real.volume_pi_le_diam_pow s\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))\n[PROOFSTEP]\nrw [\u2190 volume_pi_pi fun i => Ioo (a i : \u211d) (b i)]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nHle : volume \u2264 \u03bcH[\u2191(Fintype.card \u03b9)]\n\u22a2 \u2191\u2191volume (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i))\n[PROOFSTEP]\nexact\n  Measure.le_iff'.1 Hle\n    _\n      /- For the other inequality `\u03bcH s \u2264 volume s`, we use a covering of `s` by sets of small diameter\n          `1/n`, namely cubes with left-most point of the form `a i + f i / n` with `f i` ranging between\n          `0` and `\u2308(b i - a i) * n\u2309`. Their number is asymptotic to `n^d * \u03a0 (b i - a i)`. -/\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nhave I : \u2200 i, 0 \u2264 (b i : \u211d) - a i := fun i => by simpa only [sub_nonneg, Rat.cast_le] using (H i).le\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\ni : \u03b9\n\u22a2 0 \u2264 \u2191(b i) - \u2191(a i)\n[PROOFSTEP]\nsimpa only [sub_nonneg, Rat.cast_le] using (H i).le\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nlet \u03b3 := fun n : \u2115 => \u2200 i : \u03b9, Fin \u2308((b i : \u211d) - a i) * n\u2309\u208a\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nlet t : \u2200 n : \u2115, \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) := fun n f => Set.pi univ fun i => Icc (a i + f i / n) (a i + (f i + 1) / n)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nhave A : Tendsto (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) atTop (\ud835\udcdd 0) := by simp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\n\u22a2 Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [one_div, ENNReal.tendsto_inv_nat_nhds_zero]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nhave B : \u2200\u1da0 n in atTop, \u2200 i : \u03b3 n, diam (t n i) \u2264 1 / n :=\n  by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  intro f\n  refine' diam_pi_le_of_le fun b => _\n  simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl, add_sub_add_left_eq_sub,\n    add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\n\u22a2 \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n[PROOFSTEP]\nintro f\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\n\u22a2 diam (t n f) \u2264 1 / \u2191n\n[PROOFSTEP]\nrefine' diam_pi_le_of_le fun b => _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b\u271d : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b\u271d i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b\u271d i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b\u271d i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nn : \u2115\nhn : n \u2265 1\nf : \u03b3 n\nb : \u03b9\n\u22a2 diam (Icc (\u2191(a b) + \u2191\u2191(f b) / \u2191n) (\u2191(a b) + (\u2191\u2191(f b) + 1) / \u2191n)) \u2264 1 / \u2191n\n[PROOFSTEP]\nsimp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl, add_sub_add_left_eq_sub,\n  add_sub_cancel', ENNReal.ofReal_one, ENNReal.ofReal_coe_nat]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\nhave C : \u2200\u1da0 n in atTop, (Set.pi univ fun i : \u03b9 => Ioo (a i : \u211d) (b i)) \u2286 \u22c3 i : \u03b3 n, t n i :=\n  by\n  refine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n  have npos : (0 : \u211d) < n := Nat.cast_pos.2 hn\n  intro x hx\n  simp only [mem_Ioo, mem_univ_pi] at hx \n  simp only [mem_iUnion, mem_Ioo, mem_univ_pi]\n  let f : \u03b3 n := fun i =>\n    \u27e8\u230a(x i - a i) * n\u230b\u208a, by\n      apply Nat.floor_lt_ceil_of_lt_of_pos\n      \u00b7 refine' (mul_lt_mul_right npos).2 _\n        simp only [(hx i).right, sub_lt_sub_iff_right]\n      \u00b7 refine' mul_pos _ npos\n        simpa only [Rat.cast_lt, sub_pos] using H i\u27e9\n  refine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9\n  \u00b7\n    calc\n      (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n :=\n        by\n        refine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n        exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n      _ = x i := by field_simp [npos.ne']\n  \u00b7\n    calc\n      x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n      _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n        add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n[PROOFSTEP]\nhave npos : (0 : \u211d) < n := Nat.cast_pos.2 hn\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\n\u22a2 (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : x \u2208 Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)\n\u22a2 x \u2208 \u22c3 (i : \u03b3 n), t n i\n[PROOFSTEP]\nsimp only [mem_Ioo, mem_univ_pi] at hx \n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\n\u22a2 x \u2208 \u22c3 (i : \u03b3 n), t n i\n[PROOFSTEP]\nsimp only [mem_iUnion, mem_Ioo, mem_univ_pi]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\n\u22a2 \u2203 i, \u2200 (i_1 : \u03b9), x i_1 \u2208 Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191n) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191n)\n[PROOFSTEP]\nlet f : \u03b3 n := fun i =>\n  \u27e8\u230a(x i - a i) * n\u230b\u208a, by\n    apply Nat.floor_lt_ceil_of_lt_of_pos\n    \u00b7 refine' (mul_lt_mul_right npos).2 _\n      simp only [(hx i).right, sub_lt_sub_iff_right]\n    \u00b7 refine' mul_pos _ npos\n      simpa only [Rat.cast_lt, sub_pos] using H i\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\n[PROOFSTEP]\napply Nat.floor_lt_ceil_of_lt_of_pos\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 (x i - \u2191(a i)) * \u2191n < (\u2191(b i) - \u2191(a i)) * \u2191n\n[PROOFSTEP]\nrefine' (mul_lt_mul_right npos).2 _\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 x i - \u2191(a i) < \u2191(b i) - \u2191(a i)\n[PROOFSTEP]\nsimp only [(hx i).right, sub_lt_sub_iff_right]\n[GOAL]\ncase h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < (\u2191(b i) - \u2191(a i)) * \u2191n\n[PROOFSTEP]\nrefine' mul_pos _ npos\n[GOAL]\ncase h'\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\ni : \u03b9\n\u22a2 0 < \u2191(b i) - \u2191(a i)\n[PROOFSTEP]\nsimpa only [Rat.cast_lt, sub_pos] using H i\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\n\u22a2 \u2203 i, \u2200 (i_1 : \u03b9), x i_1 \u2208 Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191n) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191n)\n[PROOFSTEP]\nrefine' \u27e8f, fun i => \u27e8_, _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + \u2191\u2191(f i) / \u2191n \u2264 x i\n[PROOFSTEP]\ncalc\n  (a i : \u211d) + \u230a(x i - a i) * n\u230b\u208a / n \u2264 (a i : \u211d) + (x i - a i) * n / n :=\n    by\n    refine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n    exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n  _ = x i := by field_simp [npos.ne']\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a / \u2191n \u2264 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n\n[PROOFSTEP]\nrefine' add_le_add le_rfl ((div_le_div_right npos).2 _)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191\u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a \u2264 (x i - \u2191(a i)) * \u2191n\n[PROOFSTEP]\nexact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n = x i\n[PROOFSTEP]\nfield_simp [npos.ne']\n[GOAL]\ncase refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 x i \u2264 \u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n\n[PROOFSTEP]\ncalc\n  x i = (a i : \u211d) + (x i - a i) * n / n := by field_simp [npos.ne']\n  _ \u2264 (a i : \u211d) + (\u230a(x i - a i) * n\u230b\u208a + 1) / n :=\n    add_le_add le_rfl ((div_le_div_right npos).2 (Nat.lt_floor_add_one _).le)\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nn : \u2115\nhn : n \u2265 1\nnpos : 0 < \u2191n\nx : \u03b9 \u2192 \u211d\nhx : \u2200 (i : \u03b9), \u2191(a i) < x i \u2227 x i < \u2191(b i)\nf : \u03b3 n := fun i => { val := \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a, isLt := (_ : \u230a(x i - \u2191(a i)) * \u2191n\u230b\u208a < \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a) }\ni : \u03b9\n\u22a2 x i = \u2191(a i) + (x i - \u2191(a i)) * \u2191n / \u2191n\n[PROOFSTEP]\nfield_simp [npos.ne']\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 \u2191\u2191\u03bcH[\u2191(Fintype.card \u03b9)] (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2264 \u220f i : \u03b9, \u2191\u2191volume ((fun i => Ioo \u2191(a i) \u2191(b i)) i)\n[PROOFSTEP]\ncalc\n  \u03bcH[Fintype.card \u03b9] (Set.pi univ fun i : \u03b9 => Ioo (a i : \u211d) (b i)) \u2264\n      liminf (fun n : \u2115 => \u2211 i : \u03b3 n, diam (t n i) ^ ((Fintype.card \u03b9) : \u211d)) atTop :=\n    hausdorffMeasure_le_liminf_sum _ (Set.pi univ fun i => Ioo (a i : \u211d) (b i)) (fun n : \u2115 => 1 / (n : \u211d\u22650\u221e)) A t B C\n  _ \u2264 liminf (fun n : \u2115 => \u2211 i : \u03b3 n, (1 / (n : \u211d\u22650\u221e)) ^ Fintype.card \u03b9) atTop :=\n    by\n    refine' liminf_le_liminf _ _\n    \u00b7 filter_upwards [B] with _ hn\n      apply Finset.sum_le_sum fun i _ => _\n      simp only [ENNReal.rpow_nat_cast]\n      intros i _\n      exact pow_le_pow_of_le_left' (hn i) _\n    \u00b7 isBoundedDefault\n  _ = liminf (fun n : \u2115 => \u220f i : \u03b9, (\u2308((b i : \u211d) - a i) * n\u2309\u208a : \u211d\u22650\u221e) / n) atTop := by\n    simp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const, nsmul_eq_mul,\n      Fintype.card_pi, div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]\n  _ = \u220f i : \u03b9, volume (Ioo (a i : \u211d) (b i)) := by\n    simp only [Real.volume_Ioo]\n    apply Tendsto.liminf_eq\n    refine' ENNReal.tendsto_finset_prod_of_ne_top _ (fun i _ => _) fun i _ => _\n    \u00b7 apply\n        Tendsto.congr' _\n          ((ENNReal.continuous_ofReal.tendsto _).comp\n            ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_nat_cast_atTop_atTop))\n      apply eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n      intros n hn\n      simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply, ENNReal.ofReal_coe_nat]\n    \u00b7 simp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 liminf (fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)) atTop \u2264\n    liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop\n[PROOFSTEP]\nrefine' liminf_le_liminf _ _\n[GOAL]\ncase refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 \u2200\u1da0 (a_1 : \u2115) in atTop, \u2211 i : \u03b3 a_1, diam (t a_1 i) ^ \u2191(Fintype.card \u03b9) \u2264 \u2211 i : \u03b3 a_1, (1 / \u2191a_1) ^ Fintype.card \u03b9\n[PROOFSTEP]\nfilter_upwards [B] with _ hn\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\na\u271d : \u2115\nhn :\n  \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) \u2264 1 / \u2191a\u271d\n\u22a2 \u2211 i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a,\n      diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^\n        \u2191(Fintype.card \u03b9) \u2264\n    \u2211 i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a, (1 / \u2191a\u271d) ^ Fintype.card \u03b9\n[PROOFSTEP]\napply Finset.sum_le_sum fun i _ => _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\na\u271d : \u2115\nhn :\n  \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    i \u2208 Finset.univ \u2192\n      diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^\n          \u2191(Fintype.card \u03b9) \u2264\n        (1 / \u2191a\u271d) ^ Fintype.card \u03b9\n[PROOFSTEP]\nsimp only [ENNReal.rpow_nat_cast]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\na\u271d : \u2115\nhn :\n  \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) \u2264 1 / \u2191a\u271d\n\u22a2 \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    i \u2208 Finset.univ \u2192\n      diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^\n          Fintype.card \u03b9 \u2264\n        (1 / \u2191a\u271d) ^ Fintype.card \u03b9\n[PROOFSTEP]\nintros i _\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\na\u271d : \u2115\nhn :\n  \u2200 (i : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a),\n    diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) \u2264 1 / \u2191a\u271d\ni : (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191a\u271d\u2309\u208a\nx\u271d : i \u2208 Finset.univ\n\u22a2 diam (Set.pi univ fun i_1 => Icc (\u2191(a i_1) + \u2191\u2191(i i_1) / \u2191a\u271d) (\u2191(a i_1) + (\u2191\u2191(i i_1) + 1) / \u2191a\u271d)) ^ Fintype.card \u03b9 \u2264\n    (1 / \u2191a\u271d) ^ Fintype.card \u03b9\n[PROOFSTEP]\nexact pow_le_pow_of_le_left' (hn i) _\n[GOAL]\ncase refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2265 x_1) atTop fun n => \u2211 i : \u03b3 n, diam (t n i) ^ \u2191(Fintype.card \u03b9)\n[PROOFSTEP]\nisBoundedDefault\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 liminf (fun n => \u2211 i : \u03b3 n, (1 / \u2191n) ^ Fintype.card \u03b9) atTop =\n    liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop\n[PROOFSTEP]\nsimp only [Finset.card_univ, Nat.cast_prod, one_mul, Fintype.card_fin, Finset.sum_const, nsmul_eq_mul, Fintype.card_pi,\n  div_eq_mul_inv, Finset.prod_mul_distrib, Finset.prod_const]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f i : \u03b9, \u2191\u2191volume (Ioo \u2191(a i) \u2191(b i))\n[PROOFSTEP]\nsimp only [Real.volume_Ioo]\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 liminf (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop = \u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))\n[PROOFSTEP]\napply Tendsto.liminf_eq\n[GOAL]\ncase h\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\n\u22a2 Tendsto (fun n => \u220f i : \u03b9, \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (\u220f x : \u03b9, ENNReal.ofReal (\u2191(b x) - \u2191(a x))))\n[PROOFSTEP]\nrefine' ENNReal.tendsto_finset_prod_of_ne_top _ (fun i _ => _) fun i _ => _\n[GOAL]\ncase h.refine'_1\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Tendsto (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) atTop (\ud835\udcdd (ENNReal.ofReal (\u2191(b i) - \u2191(a i))))\n[PROOFSTEP]\napply\n  Tendsto.congr' _\n    ((ENNReal.continuous_ofReal.tendsto _).comp\n      ((tendsto_nat_ceil_mul_div_atTop (I i)).comp tendsto_nat_cast_atTop_atTop))\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast =\u1da0[atTop] fun n =>\n    \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n\n[PROOFSTEP]\napply eventually_atTop.2 \u27e81, fun n hn => _\u27e9\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 \u2200 (n : \u2115),\n    n \u2265 1 \u2192\n      (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n        (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n\n[PROOFSTEP]\nintros n hn\n[GOAL]\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\nn : \u2115\nhn : n \u2265 1\n\u22a2 (ENNReal.ofReal \u2218 (fun x => \u2191\u2308(\u2191(b i) - \u2191(a i)) * x\u2309\u208a / x) \u2218 Nat.cast) n =\n    (fun n => \u2191\u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a / \u2191n) n\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply, ENNReal.ofReal_coe_nat]\n[GOAL]\ncase h.refine'_2\n\u03b9\u271d : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2076 : EMetricSpace X\ninst\u271d\u2075 : EMetricSpace Y\ninst\u271d\u2074 : MeasurableSpace X\ninst\u271d\u00b3 : BorelSpace X\ninst\u271d\u00b2 : MeasurableSpace Y\ninst\u271d\u00b9 : BorelSpace Y\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\na b : \u03b9 \u2192 \u211a\nH : \u2200 (i : \u03b9), a i < b i\nI : \u2200 (i : \u03b9), 0 \u2264 \u2191(b i) - \u2191(a i)\n\u03b3 : \u2115 \u2192 Type u_4 := fun n => (i : \u03b9) \u2192 Fin \u2308(\u2191(b i) - \u2191(a i)) * \u2191n\u2309\u208a\nt : (n : \u2115) \u2192 \u03b3 n \u2192 Set (\u03b9 \u2192 \u211d) :=\n  fun n f => Set.pi univ fun i => Icc (\u2191(a i) + \u2191\u2191(f i) / \u2191n) (\u2191(a i) + (\u2191\u2191(f i) + 1) / \u2191n)\nA : Tendsto (fun n => 1 / \u2191n) atTop (\ud835\udcdd 0)\nB : \u2200\u1da0 (n : \u2115) in atTop, \u2200 (i : \u03b3 n), diam (t n i) \u2264 1 / \u2191n\nC : \u2200\u1da0 (n : \u2115) in atTop, (Set.pi univ fun i => Ioo \u2191(a i) \u2191(b i)) \u2286 \u22c3 (i : \u03b3 n), t n i\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 ENNReal.ofReal (\u2191(b i) - \u2191(a i)) \u2260 \u22a4\n[PROOFSTEP]\nsimp only [ENNReal.ofReal_ne_top, Ne.def, not_false_iff]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\n\u22a2 \u03bcH[1] = volume\n[PROOFSTEP]\nrw [\u2190 (volume_preserving_funUnique Unit \u211d).map_eq, \u2190 (hausdorffMeasure_measurePreserving_funUnique Unit \u211d 1).map_eq, \u2190\n  hausdorffMeasure_pi_real, Fintype.card_unit, Nat.cast_one]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2075 : EMetricSpace X\ninst\u271d\u2074 : EMetricSpace Y\ninst\u271d\u00b3 : MeasurableSpace X\ninst\u271d\u00b2 : BorelSpace X\ninst\u271d\u00b9 : MeasurableSpace Y\ninst\u271d : BorelSpace Y\n\u22a2 \u03bcH[2] = volume\n[PROOFSTEP]\nrw [\u2190 (volume_preserving_piFinTwo fun _ => \u211d).map_eq, \u2190\n  (hausdorffMeasure_measurePreserving_piFinTwo (fun _ => \u211d) _).map_eq, \u2190 hausdorffMeasure_pi_real, Fintype.card_fin,\n  Nat.cast_two]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 v) '' s) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nobtain rfl | hv := eq_or_ne v 0\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\ns : Set \u211d\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 0) '' s) = \u20160\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nhaveI := noAtoms_hausdorff E one_pos\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\ns : Set \u211d\nthis : NoAtoms \u03bcH[1]\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 0) '' s) = \u20160\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nobtain rfl | hs := s.eq_empty_or_nonempty\n[GOAL]\ncase inl.inl\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nthis : NoAtoms \u03bcH[1]\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 0) '' \u2205) = \u20160\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inl.inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\ns : Set \u211d\nthis : NoAtoms \u03bcH[1]\nhs : Set.Nonempty s\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 0) '' s) = \u20160\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 v) '' s) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nhave hn : \u2016v\u2016 \u2260 0 := norm_ne_zero_iff.mpr hv\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 v) '' s) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nsuffices \u03bcH[1] ((\u00b7 \u2022 \u00b7) \u2016v\u2016 '' (LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v) '' s)) = \u2016v\u2016\u208a \u2022 \u03bcH[1] s by\n  -- porting note: proof was shorter, could need some golf\n  simp only [hausdorffMeasure_real, nnreal_smul_coe_apply]\n  convert this\n  \u00b7 simp only [image_smul, LinearMap.toSpanSingleton_apply, Set.image_image]\n    ext e\n    simp\n    refine' \u27e8fun \u27e8x, h\u27e9 => \u27e8x, _\u27e9, fun \u27e8x, h\u27e9 => \u27e8x, _\u27e9\u27e9\n    \u00b7 rw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul\u2080 hn]\n      exact h\n    \u00b7 rw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul\u2080 hn] at h \n      exact h\n  \u00b7 exact hausdorffMeasure_real.symm\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 v) '' s) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nsimp only [hausdorffMeasure_real, nnreal_smul_coe_apply]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 \u2191\u2191\u03bcH[1] ((fun r => r \u2022 v) '' s) = \u2191\u2016v\u2016\u208a * \u2191\u2191volume s\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 (fun r => r \u2022 v) '' s = (fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)\n[PROOFSTEP]\nsimp only [image_smul, LinearMap.toSpanSingleton_apply, Set.image_image]\n[GOAL]\ncase h.e'_2.h.e'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 (fun r => r \u2022 v) '' s = (fun a => \u2016v\u2016 \u2022 a \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v) '' s\n[PROOFSTEP]\next e\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\n\u22a2 e \u2208 (fun r => r \u2022 v) '' s \u2194 e \u2208 (fun a => \u2016v\u2016 \u2022 a \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v) '' s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_3.h\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\n\u22a2 (\u2203 x, x \u2208 s \u2227 x \u2022 v = e) \u2194 \u2203 x, x \u2208 s \u2227 \u2016v\u2016 \u2022 x \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v = e\n[PROOFSTEP]\nrefine' \u27e8fun \u27e8x, h\u27e9 => \u27e8x, _\u27e9, fun \u27e8x, h\u27e9 => \u27e8x, _\u27e9\u27e9\n[GOAL]\ncase h.e'_2.h.e'_3.h.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\nx\u271d : \u2203 x, x \u2208 s \u2227 x \u2022 v = e\nx : \u211d\nh : x \u2208 s \u2227 x \u2022 v = e\n\u22a2 x \u2208 s \u2227 \u2016v\u2016 \u2022 x \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v = e\n[PROOFSTEP]\nrw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul\u2080 hn]\n[GOAL]\ncase h.e'_2.h.e'_3.h.refine'_1\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\nx\u271d : \u2203 x, x \u2208 s \u2227 x \u2022 v = e\nx : \u211d\nh : x \u2208 s \u2227 x \u2022 v = e\n\u22a2 x \u2208 s \u2227 x \u2022 v = e\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.e'_2.h.e'_3.h.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\nx\u271d : \u2203 x, x \u2208 s \u2227 \u2016v\u2016 \u2022 x \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v = e\nx : \u211d\nh : x \u2208 s \u2227 \u2016v\u2016 \u2022 x \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v = e\n\u22a2 x \u2208 s \u2227 x \u2022 v = e\n[PROOFSTEP]\nrw [smul_comm (norm _), smul_comm (norm _), inv_smul_smul\u2080 hn] at h \n[GOAL]\ncase h.e'_2.h.e'_3.h.refine'_2\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\ne : E\nx\u271d : \u2203 x, x \u2208 s \u2227 \u2016v\u2016 \u2022 x \u2022 \u2016v\u2016\u207b\u00b9 \u2022 v = e\nx : \u211d\nh : x \u2208 s \u2227 x \u2022 v = e\n\u22a2 x \u2208 s \u2227 x \u2022 v = e\n[PROOFSTEP]\nexact h\n[GOAL]\ncase h.e'_3.h.e'_1.h.e'_2.h.e'_3\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nthis : \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 volume = \u03bcH[1]\n[PROOFSTEP]\nexact hausdorffMeasure_real.symm\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\n\u22a2 \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nhave iso_smul : Isometry (LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) :=\n  by\n  refine' AddMonoidHomClass.isometry_of_norm _ fun x => (norm_smul _ _).trans _\n  rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hn, mul_one, LinearMap.id_apply]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\n\u22a2 Isometry \u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v))\n[PROOFSTEP]\nrefine' AddMonoidHomClass.isometry_of_norm _ fun x => (norm_smul _ _).trans _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\nx : \u211d\n\u22a2 \u2016\u2191LinearMap.id x\u2016 * \u2016\u2016v\u2016\u207b\u00b9 \u2022 v\u2016 = \u2016x\u2016\n[PROOFSTEP]\nrw [norm_smul, norm_inv, norm_norm, inv_mul_cancel hn, mul_one, LinearMap.id_apply]\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nv : E\ns : Set \u211d\nhv : v \u2260 0\nhn : \u2016v\u2016 \u2260 0\niso_smul : Isometry \u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v))\n\u22a2 \u2191\u2191\u03bcH[1] ((fun x x_1 => x \u2022 x_1) \u2016v\u2016 '' (\u2191(LinearMap.toSpanSingleton \u211d E (\u2016v\u2016\u207b\u00b9 \u2022 v)) '' s)) = \u2016v\u2016\u208a \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nrw [Set.image_smul, Measure.hausdorffMeasure_smul\u2080 zero_le_one hn, nnnorm_norm, NNReal.rpow_eq_pow, NNReal.rpow_one,\n  iso_smul.hausdorffMeasure_image (Or.inl <| zero_le_one' \u211d)]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(AffineMap.homothety x c) '' s) = NNReal.rpow \u2016c\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsuffices\n  \u03bcH[d] (IsometryEquiv.vaddConst x '' ((\u00b7 \u2022 \u00b7) c '' ((IsometryEquiv.vaddConst x).symm '' s))) =\n    NNReal.rpow \u2016c\u2016\u208a d \u2022 \u03bcH[d] s\n  by simpa only [Set.image_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\nthis :\n  \u2191\u2191\u03bcH[d]\n      (\u2191(IsometryEquiv.vaddConst x) ''\n        ((fun x x_1 => x \u2022 x_1) c '' (\u2191(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) =\n    NNReal.rpow \u2016c\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(AffineMap.homothety x c) '' s) = NNReal.rpow \u2016c\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nsimpa only [Set.image_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d]\n      (\u2191(IsometryEquiv.vaddConst x) ''\n        ((fun x x_1 => x \u2022 x_1) c '' (\u2191(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) =\n    NNReal.rpow \u2016c\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2191\u2191\u03bcH[d]\n      (\u2191(IsometryEquiv.vaddConst x) ''\n        ((fun x x_1 => x \u2022 x_1) c '' (\u2191(IsometryEquiv.symm (IsometryEquiv.vaddConst x)) '' s))) =\n    NNReal.rpow \u2016c\u2016\u208a d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrw [IsometryEquiv.hausdorffMeasure_image, Set.image_smul, Measure.hausdorffMeasure_smul\u2080 hd hc,\n  IsometryEquiv.hausdorffMeasure_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(AffineMap.homothety x c) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nchange \u03bcH[d] (AffineEquiv.homothetyUnitsMulHom x (Units.mk0 c hc) \u207b\u00b9' s) = _\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b2 : EMetricSpace X\ninst\u271d\u00b9\u00b9 : EMetricSpace Y\ninst\u271d\u00b9\u2070 : MeasurableSpace X\ninst\u271d\u2079 : BorelSpace X\ninst\u271d\u2078 : MeasurableSpace Y\ninst\u271d\u2077 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2076 : NormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nd : \u211d\nhd : 0 \u2264 d\nx : P\nc : \ud835\udd5c\nhc : c \u2260 0\ns : Set P\n\u22a2 \u2191\u2191\u03bcH[d] (\u2191(\u2191(AffineEquiv.homothetyUnitsMulHom x) (Units.mk0 c hc)) \u207b\u00b9' s) = NNReal.rpow \u2016c\u2016\u208a\u207b\u00b9 d \u2022 \u2191\u2191\u03bcH[d] s\n[PROOFSTEP]\nrw [\u2190 AffineEquiv.image_symm, AffineEquiv.coe_homothetyUnitsMulHom_apply_symm,\n  hausdorffMeasure_homothety_image hd x (_ : \ud835\udd5c\u02e3).isUnit.ne_zero, Units.val_inv_eq_inv_val, Units.val_mk0, nnnorm_inv]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(AffineMap.lineMap x y) '' s) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nsuffices \u03bcH[1] (IsometryEquiv.vaddConst x '' ((\u00b7 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u03bcH[1] s by\n  simpa only [Set.image_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\nthis : \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(AffineMap.lineMap x y) '' s) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nsimpa only [Set.image_image]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nborelize E\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\ns : Set \u211d\nthis\u271d\u00b9 : MeasurableSpace E := borel E\nthis\u271d : BorelSpace E\n\u22a2 \u2191\u2191\u03bcH[1] (\u2191(IsometryEquiv.vaddConst x) '' ((fun x_1 => x_1 \u2022 (y -\u1d65 x)) '' s)) = nndist x y \u2022 \u2191\u2191\u03bcH[1] s\n[PROOFSTEP]\nrw [IsometryEquiv.hausdorffMeasure_image, hausdorffMeasure_smul_right_image, nndist_eq_nnnorm_vsub' E]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\n\u22a2 \u2191\u2191\u03bcH[1] (affineSegment \u211d x y) = edist x y\n[PROOFSTEP]\nrw [affineSegment, hausdorffMeasure_lineMap_image, hausdorffMeasure_real, Real.volume_Icc, sub_zero, ENNReal.ofReal_one,\n  \u2190 Algebra.algebraMap_eq_smul_one]\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9\u00b9 : EMetricSpace X\ninst\u271d\u00b9\u2070 : EMetricSpace Y\ninst\u271d\u2079 : MeasurableSpace X\ninst\u271d\u2078 : BorelSpace X\ninst\u271d\u2077 : MeasurableSpace Y\ninst\u271d\u2076 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE : Type u_5\nP : Type u_6\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : MeasurableSpace P\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor E P\ninst\u271d : BorelSpace P\nx y : P\n\u22a2 \u2191(algebraMap \u211d\u22650 \u211d\u22650\u221e) (nndist x y) = edist x y\n[PROOFSTEP]\nexact (edist_nndist _ _).symm\n[GOAL]\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u2079 : EMetricSpace X\ninst\u271d\u2078 : EMetricSpace Y\ninst\u271d\u2077 : MeasurableSpace X\ninst\u271d\u2076 : BorelSpace X\ninst\u271d\u2075 : MeasurableSpace Y\ninst\u271d\u2074 : BorelSpace Y\n\ud835\udd5c : Type u_4\nE\u271d : Type u_5\nP : Type u_6\nE : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : MeasurableSpace E\ninst\u271d : BorelSpace E\nx y : E\n\u22a2 \u2191\u2191\u03bcH[1] (segment \u211d x y) = edist x y\n[PROOFSTEP]\nrw [\u2190 affineSegment_eq_segment, hausdorffMeasure_affineSegment]\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Hausdorff", "llama_tokens": 130899, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696748, "lm_q2_score": 0.6477982247516796, "lm_q1q2_score": 0.5121041370465949}}
{"text": "[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\n\u22a2 Function.Injective \u2191(autToPow K h\u03bc)\n[PROOFSTEP]\nintro f g hfg\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191(autToPow K h\u03bc) f = \u2191(autToPow K h\u03bc) g\n\u22a2 f = g\n[PROOFSTEP]\napply_fun Units.val at hfg \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg : \u2191(\u2191(autToPow K h\u03bc) f) = \u2191(\u2191(autToPow K h\u03bc) g)\n\u22a2 f = g\n[PROOFSTEP]\nsimp only [IsPrimitiveRoot.coe_autToPow_apply] at hfg \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhfg :\n  \u2191(Exists.choose (_ : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) =\n    \u2191(Exists.choose (_ : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m))\n\u22a2 f = g\n[PROOFSTEP]\nrevert hfg\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\n\u22a2 \u2191(Exists.choose (_ : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) =\n      \u2191(Exists.choose (_ : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m)) \u2192\n    f = g\n[PROOFSTEP]\ngeneralize_proofs hf' hg'\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\n\u22a2 \u2191(Exists.choose hf') = \u2191(Exists.choose hg') \u2192 f = g\n[PROOFSTEP]\nintro hfg\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\n\u22a2 f = g\n[PROOFSTEP]\nhave hf := hf'.choose_spec\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\n\u22a2 f = g\n[PROOFSTEP]\nhave hg := hg'.choose_spec\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 f = g\n[PROOFSTEP]\nrevert hf hg\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\n\u22a2 \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf' \u2192\n    \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg' \u2192 f = g\n[PROOFSTEP]\ngeneralize_proofs h\u03b6\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\n\u22a2 \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf' \u2192\n    \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg' \u2192 f = g\n[PROOFSTEP]\nintro hf hg\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 f = g\n[PROOFSTEP]\nsuffices f (h\u03bc.toRootsOfUnity : L\u02e3) = g (h\u03bc.toRootsOfUnity : L\u02e3)\n  by\n  apply AlgEquiv.coe_algHom_injective\n  apply (h\u03bc.powerBasis K).algHom_ext\n  exact this\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191g \u2191\u2191(toRootsOfUnity h\u03bc)\n\u22a2 f = g\n[PROOFSTEP]\napply AlgEquiv.coe_algHom_injective\n[GOAL]\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191g \u2191\u2191(toRootsOfUnity h\u03bc)\n\u22a2 \u2191f = \u2191g\n[PROOFSTEP]\napply (h\u03bc.powerBasis K).algHom_ext\n[GOAL]\ncase a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\nthis : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191g \u2191\u2191(toRootsOfUnity h\u03bc)\n\u22a2 \u2191\u2191f (IsPrimitiveRoot.powerBasis K h\u03bc).gen = \u2191\u2191g (IsPrimitiveRoot.powerBasis K h\u03bc).gen\n[PROOFSTEP]\nexact this\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : \u2191(Exists.choose hf') = \u2191(Exists.choose hg')\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191g \u2191\u2191(toRootsOfUnity h\u03bc)\n[PROOFSTEP]\nrw [ZMod.eq_iff_modEq_nat] at hfg \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191g \u2191\u2191(toRootsOfUnity h\u03bc)\n[PROOFSTEP]\nrefine' (hf.trans _).trans hg.symm\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf' = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n[PROOFSTEP]\nrw [\u2190 rootsOfUnity.coe_pow _ hf'.choose, \u2190 rootsOfUnity.coe_pow _ hg'.choose]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 \u2191\u2191(toRootsOfUnity h\u03bc ^ Exists.choose hf') = \u2191\u2191(toRootsOfUnity h\u03bc ^ Exists.choose hg')\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 toRootsOfUnity h\u03bc ^ Exists.choose hf' = toRootsOfUnity h\u03bc ^ Exists.choose hg'\n[PROOFSTEP]\nrw [pow_eq_pow_iff_modEq]\n[GOAL]\ncase e_self.e_self\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 Exists.choose hf' \u2261 Exists.choose hg' [MOD orderOf (toRootsOfUnity h\u03bc)]\n[PROOFSTEP]\nconvert hfg\n[GOAL]\ncase h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 orderOf (toRootsOfUnity h\u03bc) = \u2191n\n[PROOFSTEP]\nrw [h\u03bc.eq_orderOf]\n  -- Porting note: was `{occs := occurrences.pos [2]}`\n[GOAL]\ncase h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 orderOf (toRootsOfUnity h\u03bc) = orderOf \u03bc\n[PROOFSTEP]\nconv_rhs => rw [\u2190 h\u03bc.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n| orderOf \u03bc\n[PROOFSTEP]\nrw [\u2190 h\u03bc.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n| orderOf \u03bc\n[PROOFSTEP]\nrw [\u2190 h\u03bc.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n| orderOf \u03bc\n[PROOFSTEP]\nrw [\u2190 h\u03bc.toRootsOfUnity_coe_val]\n[GOAL]\ncase h.e'_1\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nf g : L \u2243\u2090[K] L\nhf' : \u2203 m, \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhg' : \u2203 m, \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ m\nhfg : Exists.choose hf' \u2261 Exists.choose hg' [MOD \u2191n]\nhf : \u2191f \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hf'\nhg : \u2191g \u2191\u2191(toRootsOfUnity h\u03bc) = \u2191\u2191(toRootsOfUnity h\u03bc) ^ Exists.choose hg'\n\u22a2 orderOf (toRootsOfUnity h\u03bc) = orderOf \u2191\u2191(toRootsOfUnity h\u03bc)\n[PROOFSTEP]\nrw [orderOf_units, orderOf_subgroup]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nt : (ZMod \u2191n)\u02e3\n\u22a2 minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n)).gen\n[PROOFSTEP]\nhaveI := IsCyclotomicExtension.neZero' n K L\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nt : (ZMod \u2191n)\u02e3\nthis : NeZero \u2191\u2191n\n\u22a2 minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n)).gen\n[PROOFSTEP]\nsimp only [IsPrimitiveRoot.powerBasis_gen]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nt : (ZMod \u2191n)\u02e3\nthis : NeZero \u2191\u2191n\n\u22a2 minpoly K (zeta n K L) = minpoly K (zeta n K L ^ ZMod.val \u2191t)\n[PROOFSTEP]\nhave hr :=\n  IsPrimitiveRoot.minpoly_eq_cyclotomic_of_irreducible\n    ((zeta_spec n K L).pow_of_coprime _ (ZMod.val_coe_unit_coprime t)) h\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nt : (ZMod \u2191n)\u02e3\nthis : NeZero \u2191\u2191n\nhr : cyclotomic (\u2191n) K = minpoly K (zeta n K L ^ ZMod.val \u2191t)\n\u22a2 minpoly K (zeta n K L) = minpoly K (zeta n K L ^ ZMod.val \u2191t)\n[PROOFSTEP]\nexact ((zeta_spec n K L).minpoly_eq_cyclotomic_of_irreducible h).symm.trans hr\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 (fun t =>\n        PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n))\n          (_ :\n            minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n              minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n)).gen))\n      (OneHom.toFun (\u2191src\u271d) f) =\n    f\n[PROOFSTEP]\nsimp only [MonoidHom.toFun_eq_coe]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n      (IsPrimitiveRoot.powerBasis K\n        (_ :\n          IsPrimitiveRoot\n            (zeta n K L ^ ZMod.val \u2191(\u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)) f)) \u2191n))\n      (_ :\n        minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n          minpoly K\n            (IsPrimitiveRoot.powerBasis K\n                (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191(OneHom.toFun (\u2191src\u271d) f)) \u2191n)).gen) =\n    f\n[PROOFSTEP]\napply AlgEquiv.coe_algHom_injective\n[GOAL]\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K\n          (_ :\n            IsPrimitiveRoot\n              (zeta n K L ^ ZMod.val \u2191(\u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)) f)) \u2191n))\n        (_ :\n          minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n            minpoly K\n              (IsPrimitiveRoot.powerBasis K\n                  (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191(OneHom.toFun (\u2191src\u271d) f)) \u2191n)).gen)) =\n    \u2191f\n[PROOFSTEP]\napply (h\u03b6.powerBasis K).algHom_ext\n[GOAL]\ncase a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 \u2191\u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n            (IsPrimitiveRoot.powerBasis K\n              (_ :\n                IsPrimitiveRoot\n                  (zeta n K L ^ ZMod.val \u2191(\u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)) f)) \u2191n))\n            (_ :\n              minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n                minpoly K\n                  (IsPrimitiveRoot.powerBasis K\n                      (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191(OneHom.toFun (\u2191src\u271d) f)) \u2191n)).gen))\n      (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    \u2191\u2191f (IsPrimitiveRoot.powerBasis K h\u03b6).gen\n[PROOFSTEP]\nsimp only [AlgHom.coe_coe]\n[GOAL]\ncase a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K\n            (_ :\n              IsPrimitiveRoot\n                (zeta n K L ^ ZMod.val \u2191(\u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)) f)) \u2191n))\n          (_ :\n            minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n              minpoly K\n                (IsPrimitiveRoot.powerBasis K\n                    (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191(OneHom.toFun (\u2191src\u271d) f)) \u2191n)).gen))\n      (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    \u2191f (IsPrimitiveRoot.powerBasis K h\u03b6).gen\n[PROOFSTEP]\nrw [PowerBasis.equivOfMinpoly_gen]\n[GOAL]\ncase a.h\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nf : L \u2243\u2090[K] L\n\u22a2 (IsPrimitiveRoot.powerBasis K\n        (_ :\n          IsPrimitiveRoot\n            (zeta n K L ^ ZMod.val \u2191(\u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)) f)) \u2191n)).gen =\n    \u2191f (IsPrimitiveRoot.powerBasis K h\u03b6).gen\n[PROOFSTEP]\nsimp only [IsPrimitiveRoot.powerBasis_gen, IsPrimitiveRoot.autToPow_spec]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\n\u22a2 OneHom.toFun (\u2191src\u271d)\n      ((fun t =>\n          PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n            (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n))\n            (_ :\n              minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n                minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n)).gen))\n        x) =\n    x\n[PROOFSTEP]\nsimp only [MonoidHom.toFun_eq_coe]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n))\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n))\n        (_ :\n          minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n            minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen)) =\n    x\n[PROOFSTEP]\ngeneralize_proofs _ h\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nhave key := h\u03b6.autToPow_spec K ((h\u03b6.powerBasis K).equivOfMinpoly ((h\u03bc x).powerBasis K) h)\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L)\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nhave := (h\u03b6.powerBasis K).equivOfMinpoly_gen ((h\u03bc x).powerBasis K) h\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L)\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nrw [h\u03b6.powerBasis_gen K] at this \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L)\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nrw [this, IsPrimitiveRoot.powerBasis_gen] at key \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nconv at key =>\n  congr; congr\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n  rfl; rfl\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\n  congr; congr\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n  rfl; rfl\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\n  congr; congr\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n  rfl; rfl\n  rw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^\n    ZMod.val\n      \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n          (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n            (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L\ncase a.a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| ZMod.val\n    \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n        (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\nrw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n[GOAL]\ncase a.a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| \u2191\u2191(IsPrimitiveRoot.toRootsOfUnity h\u03b6)\ncase a.a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| ZMod.val\n    \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n        (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| ZMod.val\n    \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n        (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  zeta n K L ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    zeta n K L ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n| zeta n K L ^ ZMod.val \u2191x\n[PROOFSTEP]\nrw [\u2190 h\u03b6.toRootsOfUnity_coe_val]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  \u2191\u2191(IsPrimitiveRoot.toRootsOfUnity h\u03b6) ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    \u2191\u2191(IsPrimitiveRoot.toRootsOfUnity h\u03b6) ^ ZMod.val \u2191x\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nsimp only [\u2190 rootsOfUnity.coe_pow] at key \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  \u2191\u2191(IsPrimitiveRoot.toRootsOfUnity h\u03b6 ^\n          ZMod.val\n            \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n                (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n                  (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))) =\n    \u2191\u2191(IsPrimitiveRoot.toRootsOfUnity h\u03b6 ^ ZMod.val \u2191x)\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nreplace key := rootsOfUnity.coe_injective key\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  IsPrimitiveRoot.toRootsOfUnity h\u03b6 ^\n      ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    IsPrimitiveRoot.toRootsOfUnity h\u03b6 ^ ZMod.val \u2191x\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nrw [pow_eq_pow_iff_modEq, \u2190 orderOf_subgroup, \u2190 orderOf_units, h\u03b6.toRootsOfUnity_coe_val, \u2190\n  (zeta_spec n K L).eq_orderOf, \u2190 ZMod.eq_iff_modEq_nat] at key \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  \u2191(ZMod.val\n        \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n            (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n              (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h))) =\n    \u2191(ZMod.val \u2191x)\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nsimp only [ZMod.nat_cast_val, ZMod.cast_id', id.def] at key \n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc\u271d : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh\u271d\u00b9 : Irreducible (cyclotomic (\u2191n) K)\nh\u03b6 : IsPrimitiveRoot (zeta n K L) \u2191n := zeta_spec n K L\nh\u03bc : \u2200 (t : (ZMod \u2191n)\u02e3), IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191t) \u2191n :=\n  fun t => IsPrimitiveRoot.pow_of_coprime h\u03b6 (ZMod.val \u2191t) (ZMod.val_coe_unit_coprime t)\nsrc\u271d : (L \u2243\u2090[K] L) \u2192* (ZMod \u2191n)\u02e3 := IsPrimitiveRoot.autToPow K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)\nx : (ZMod \u2191n)\u02e3\nh\u271d : IsPrimitiveRoot (zeta n K L) \u2191n\nh :\n  minpoly K (IsPrimitiveRoot.powerBasis K h\u03b6).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nthis :\n  \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)\n      (zeta n K L) =\n    (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)).gen\nkey :\n  \u2191(\u2191(IsPrimitiveRoot.autToPow K h\u03b6)\n        (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n          (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h)) =\n    \u2191x\n\u22a2 \u2191(IsPrimitiveRoot.autToPow K h\u271d)\n      (PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K h\u03b6)\n        (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L ^ ZMod.val \u2191x) \u2191n)) h) =\n    x\n[PROOFSTEP]\nexact Units.ext key\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 \u2191(fromZetaAut h\u03bc h) (zeta n K L) = \u03bc\n[PROOFSTEP]\nsimp_rw [fromZetaAut, autEquivPow_symm_apply]\n  -- Porting note: `generalize_proofs` did not generalize the same proofs, making the proof different.\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n))\n          (IsPrimitiveRoot.powerBasis K\n            (_ :\n              IsPrimitiveRoot\n                (zeta n K L ^\n                  ZMod.val\n                    \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n                        (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n                \u2191n))\n          (_ :\n            minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n              minpoly K\n                (IsPrimitiveRoot.powerBasis K\n                    (_ :\n                      IsPrimitiveRoot\n                        (zeta n K L ^\n                          ZMod.val\n                            \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n                                (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n                        \u2191n)).gen))\n      (zeta n K L) =\n    \u03bc\n[PROOFSTEP]\ngeneralize_proofs h1 h2\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh1 :\n  IsPrimitiveRoot\n    (zeta n K L ^\n      ZMod.val\n        \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n            (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n    \u2191n\nh2 :\n  minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K h1).gen\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n))\n          (IsPrimitiveRoot.powerBasis K h1) h2)\n      (zeta n K L) =\n    \u03bc\n[PROOFSTEP]\nnth_rewrite 4 [\u2190 (zeta_spec n K L).powerBasis_gen K]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh1 :\n  IsPrimitiveRoot\n    (zeta n K L ^\n      ZMod.val\n        \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n            (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n    \u2191n\nh2 :\n  minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K h1).gen\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n))\n          (IsPrimitiveRoot.powerBasis K h1) h2)\n      (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    \u03bc\n[PROOFSTEP]\nhave := Exists.choose_spec ((zeta_spec n K L).eq_pow_of_pow_eq_one h\u03bc.pow_eq_one n.pos)\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh1 :\n  IsPrimitiveRoot\n    (zeta n K L ^\n      ZMod.val\n        \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n            (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n    \u2191n\nh2 :\n  minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K h1).gen\nthis :\n  Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc) < \u2191n \u2227\n    zeta n K L ^ Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc) = \u03bc\n\u22a2 \u2191(PowerBasis.equivOfMinpoly (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n))\n          (IsPrimitiveRoot.powerBasis K h1) h2)\n      (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    \u03bc\n[PROOFSTEP]\nrw [PowerBasis.equivOfMinpoly_gen, h1.powerBasis_gen K, ZMod.coe_unitOfCoprime, ZMod.val_cast_of_lt this.1]\n[GOAL]\nn : \u2115+\nK : Type u_1\ninst\u271d\u2074 : Field K\nL : Type u_2\n\u03bc : L\ninst\u271d\u00b3 : CommRing L\ninst\u271d\u00b2 : IsDomain L\nh\u03bc : IsPrimitiveRoot \u03bc \u2191n\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : IsCyclotomicExtension {n} K L\nh : Irreducible (cyclotomic (\u2191n) K)\nh1 :\n  IsPrimitiveRoot\n    (zeta n K L ^\n      ZMod.val\n        \u2191(ZMod.unitOfCoprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc))\n            (_ : Nat.coprime (Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc)) \u2191n)))\n    \u2191n\nh2 :\n  minpoly K (IsPrimitiveRoot.powerBasis K (_ : IsPrimitiveRoot (zeta n K L) \u2191n)).gen =\n    minpoly K (IsPrimitiveRoot.powerBasis K h1).gen\nthis :\n  Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc) < \u2191n \u2227\n    zeta n K L ^ Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc) = \u03bc\n\u22a2 zeta n K L ^ Exists.choose (_ : \u2203 i, i < \u2191n \u2227 zeta n K L ^ i = \u03bc) = \u03bc\n[PROOFSTEP]\nexact this.2\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Cyclotomic.Gal", "llama_tokens": 33165, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673178375735, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.5119231959630364}}
{"text": "[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\n\u22a2 ContinuousAt (fun y => \u2221 y.fst y.snd.fst y.snd.snd) x\n[PROOFSTEP]\nlet f : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.1 -\u1d65 y.2.1, y.2.2 -\u1d65 y.2.1)\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\nf : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.fst -\u1d65 y.snd.fst, y.snd.snd -\u1d65 y.snd.fst)\n\u22a2 ContinuousAt (fun y => \u2221 y.fst y.snd.fst y.snd.snd) x\n[PROOFSTEP]\nhave hf1 : (f x).1 \u2260 0 := by simp [hx12]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\nf : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.fst -\u1d65 y.snd.fst, y.snd.snd -\u1d65 y.snd.fst)\n\u22a2 (f x).fst \u2260 0\n[PROOFSTEP]\nsimp [hx12]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\nf : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.fst -\u1d65 y.snd.fst, y.snd.snd -\u1d65 y.snd.fst)\nhf1 : (f x).fst \u2260 0\n\u22a2 ContinuousAt (fun y => \u2221 y.fst y.snd.fst y.snd.snd) x\n[PROOFSTEP]\nhave hf2 : (f x).2 \u2260 0 := by simp [hx32]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\nf : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.fst -\u1d65 y.snd.fst, y.snd.snd -\u1d65 y.snd.fst)\nhf1 : (f x).fst \u2260 0\n\u22a2 (f x).snd \u2260 0\n[PROOFSTEP]\nsimp [hx32]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nx : P \u00d7 P \u00d7 P\nhx12 : x.fst \u2260 x.snd.fst\nhx32 : x.snd.snd \u2260 x.snd.fst\nf : P \u00d7 P \u00d7 P \u2192 V \u00d7 V := fun y => (y.fst -\u1d65 y.snd.fst, y.snd.snd -\u1d65 y.snd.fst)\nhf1 : (f x).fst \u2260 0\nhf2 : (f x).snd \u2260 0\n\u22a2 ContinuousAt (fun y => \u2221 y.fst y.snd.fst y.snd.snd) x\n[PROOFSTEP]\nexact\n  (o.continuousAt_oangle hf1 hf2).comp\n    ((continuous_fst.vsub continuous_snd.fst).prod_mk (continuous_snd.snd.vsub continuous_snd.fst)).continuousAt\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 : P\n\u22a2 \u2221 p\u2081 p\u2081 p\u2082 = 0\n[PROOFSTEP]\nsimp [oangle]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 : P\n\u22a2 \u2221 p\u2081 p\u2082 p\u2082 = 0\n[PROOFSTEP]\nsimp [oangle]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2081 \u2260 p\u2082\n[PROOFSTEP]\nrw [\u2190 @vsub_ne_zero V]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2081 -\u1d65 p\u2082 \u2260 0\n[PROOFSTEP]\nexact o.left_ne_zero_of_oangle_ne_zero h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2083 \u2260 p\u2082\n[PROOFSTEP]\nrw [\u2190 @vsub_ne_zero V]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2083 -\u1d65 p\u2082 \u2260 0\n[PROOFSTEP]\nexact o.right_ne_zero_of_oangle_ne_zero h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2081 \u2260 p\u2083\n[PROOFSTEP]\nrw [\u2190 (vsub_left_injective p\u2082).ne_iff]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 \u2260 0\n\u22a2 p\u2081 -\u1d65 p\u2082 \u2260 p\u2083 -\u1d65 p\u2082\n[PROOFSTEP]\nexact o.ne_of_oangle_ne_zero h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = 1\n\u22a2 1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = -1\n\u22a2 -1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = -1\n\u22a2 -1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = -1\n\u22a2 -1 \u2260 0\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 \u2260 0 \u2227 \u2221 p\u2081 p\u2082 p\u2083 \u2260 \u2191\u03c0 \u2194 AffineIndependent \u211d ![p\u2081, p\u2082, p\u2083]\n[PROOFSTEP]\nrw [oangle, o.oangle_ne_zero_and_ne_pi_iff_linearIndependent,\n  affineIndependent_iff_linearIndependent_vsub \u211d _ (1 : Fin 3), \u2190\n  linearIndependent_equiv (finSuccAboveEquiv (1 : Fin 3)).toEquiv]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 LinearIndependent \u211d ![p\u2081 -\u1d65 p\u2082, p\u2083 -\u1d65 p\u2082] \u2194\n    LinearIndependent \u211d\n      ((fun i => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 1) \u2218 \u2191(finSuccAboveEquiv 1).toEquiv)\n[PROOFSTEP]\nconvert Iff.rfl\n[GOAL]\ncase h.e'_2.h.e'_4\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 (fun i => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 1) \u2218 \u2191(finSuccAboveEquiv 1).toEquiv =\n    ![p\u2081 -\u1d65 p\u2082, p\u2083 -\u1d65 p\u2082]\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_2.h.e'_4.h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\ni : Fin 2\n\u22a2 ((fun i => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 1) \u2218 \u2191(finSuccAboveEquiv 1).toEquiv) i =\n    Matrix.vecCons (p\u2081 -\u1d65 p\u2082) ![p\u2083 -\u1d65 p\u2082] i\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.e'_2.h.e'_4.h.head\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 ((fun i => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 1) \u2218 \u2191(finSuccAboveEquiv 1).toEquiv)\n      { val := 0, isLt := (_ : 0 < 2) } =\n    Matrix.vecCons (p\u2081 -\u1d65 p\u2082) ![p\u2083 -\u1d65 p\u2082] { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.h.e'_4.h.tail.head\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 ((fun i => Matrix.vecCons p\u2081 ![p\u2082, p\u2083] \u2191i -\u1d65 Matrix.vecCons p\u2081 ![p\u2082, p\u2083] 1) \u2218 \u2191(finSuccAboveEquiv 1).toEquiv)\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    Matrix.vecCons (p\u2081 -\u1d65 p\u2082) ![p\u2083 -\u1d65 p\u2082] { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2228 \u2221 p\u2081 p\u2082 p\u2083 = \u2191\u03c0 \u2194 Collinear \u211d {p\u2081, p\u2082, p\u2083}\n[PROOFSTEP]\nrw [\u2190 not_iff_not, not_or, oangle_ne_zero_and_ne_pi_iff_affineIndependent, affineIndependent_iff_not_collinear_set]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n\u22a2 AffineIndependent \u211d ![p\u2081, p\u2082, p\u2083] \u2194 AffineIndependent \u211d ![p\u2084, p\u2085, p\u2086]\n[PROOFSTEP]\nsimp_rw [\u2190 oangle_ne_zero_and_ne_pi_iff_affineIndependent, \u2190 Real.Angle.two_zsmul_ne_zero_iff, h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083} \u2194 Collinear \u211d {p\u2084, p\u2085, p\u2086}\n[PROOFSTEP]\nsimp_rw [\u2190 oangle_eq_zero_or_eq_pi_iff_collinear, \u2190 Real.Angle.two_zsmul_eq_zero_iff, h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : vectorSpan \u211d {p\u2081, p\u2082} = vectorSpan \u211d {p\u2084, p\u2085}\nh\u2083\u2082\u2086\u2085 : vectorSpan \u211d {p\u2083, p\u2082} = vectorSpan \u211d {p\u2086, p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n[PROOFSTEP]\nsimp_rw [vectorSpan_pair] at h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085 \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : Submodule.span \u211d {p\u2081 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2084 -\u1d65 p\u2085}\nh\u2083\u2082\u2086\u2085 : Submodule.span \u211d {p\u2083 -\u1d65 p\u2082} = Submodule.span \u211d {p\u2086 -\u1d65 p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n[PROOFSTEP]\nexact o.two_zsmul_oangle_of_span_eq_of_span_eq h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : affineSpan \u211d {p\u2081, p\u2082} \u2225 affineSpan \u211d {p\u2084, p\u2085}\nh\u2083\u2082\u2086\u2085 : affineSpan \u211d {p\u2083, p\u2082} \u2225 affineSpan \u211d {p\u2086, p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n[PROOFSTEP]\nrw [AffineSubspace.affineSpan_pair_parallel_iff_vectorSpan_eq] at h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085 \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 p\u2086 : P\nh\u2081\u2082\u2084\u2085 : vectorSpan \u211d {p\u2081, p\u2082} = vectorSpan \u211d {p\u2084, p\u2085}\nh\u2083\u2082\u2086\u2085 : vectorSpan \u211d {p\u2083, p\u2082} = vectorSpan \u211d {p\u2086, p\u2085}\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2084 p\u2085 p\u2086\n[PROOFSTEP]\nexact two_zsmul_oangle_of_vectorSpan_eq h\u2081\u2082\u2084\u2085 h\u2083\u2082\u2086\u2085\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : dist p\u2081 p\u2082 = dist p\u2081 p\u2083\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp_rw [dist_eq_norm_vsub V] at h \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nrw [oangle, oangle, \u2190 vsub_sub_vsub_cancel_left p\u2083 p\u2082 p\u2081, \u2190 vsub_sub_vsub_cancel_left p\u2082 p\u2083 p\u2081,\n  o.oangle_sub_eq_oangle_sub_rev_of_norm_eq h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : dist p\u2081 p\u2082 = dist p\u2081 p\u2083\n\u22a2 \u2221 p\u2083 p\u2081 p\u2082 = \u2191\u03c0 - 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nsimp_rw [dist_eq_norm_vsub V] at h \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 \u2221 p\u2083 p\u2081 p\u2082 = \u2191\u03c0 - 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [oangle, oangle]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 Orientation.oangle o (p\u2083 -\u1d65 p\u2081) (p\u2082 -\u1d65 p\u2081) = \u2191\u03c0 - 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082)\n[PROOFSTEP]\nconvert o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq _ h using 1\n[GOAL]\ncase h.e'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 Orientation.oangle o (p\u2083 -\u1d65 p\u2081) (p\u2082 -\u1d65 p\u2081) = Orientation.oangle o (p\u2081 -\u1d65 p\u2083) (p\u2081 -\u1d65 p\u2082)\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev p\u2081 p\u2083, \u2190 neg_vsub_eq_vsub_rev p\u2081 p\u2082, o.oangle_neg_neg]\n[GOAL]\ncase h.e'_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 \u2191\u03c0 - 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082) = \u2191\u03c0 - 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2083 - (p\u2081 -\u1d65 p\u2082)) (p\u2081 -\u1d65 p\u2083)\n[PROOFSTEP]\nrw [\u2190 o.oangle_sub_eq_oangle_sub_rev_of_norm_eq h]\n[GOAL]\ncase h.e'_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 \u2191\u03c0 - 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082) = \u2191\u03c0 - 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 p\u2082 - (p\u2081 -\u1d65 p\u2083))\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhn : p\u2082 \u2260 p\u2083\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 p\u2081 -\u1d65 p\u2082 \u2260 p\u2081 -\u1d65 p\u2083\n[PROOFSTEP]\nsimpa using hn\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : dist p\u2081 p\u2082 = dist p\u2081 p\u2083\n\u22a2 |Real.Angle.toReal (\u2221 p\u2081 p\u2082 p\u2083)| < \u03c0 / 2\n[PROOFSTEP]\nsimp_rw [dist_eq_norm_vsub V] at h \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 |Real.Angle.toReal (\u2221 p\u2081 p\u2082 p\u2083)| < \u03c0 / 2\n[PROOFSTEP]\nrw [oangle, \u2190 vsub_sub_vsub_cancel_left p\u2083 p\u2082 p\u2081]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016\n\u22a2 |Real.Angle.toReal (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 p\u2082 - (p\u2081 -\u1d65 p\u2083)))| < \u03c0 / 2\n[PROOFSTEP]\nexact o.abs_oangle_sub_right_toReal_lt_pi_div_two h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np p\u2081 p\u2082 : P\nh : Real.Angle.sign (\u2221 p\u2081 p p\u2082) = 0\n\u22a2 p\u2081 = p \u2228 p\u2082 = p \u2228 \u2220 p\u2081 p p\u2082 = 0 \u2228 \u2220 p\u2081 p p\u2082 = \u03c0\n[PROOFSTEP]\nconvert o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero h\n[GOAL]\ncase h.e'_1.a\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np p\u2081 p\u2082 : P\nh : Real.Angle.sign (\u2221 p\u2081 p p\u2082) = 0\n\u22a2 p\u2081 = p \u2194 p\u2081 -\u1d65 p = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_2.h.e'_1.a\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np p\u2081 p\u2082 : P\nh : Real.Angle.sign (\u2221 p\u2081 p p\u2082) = 0\n\u22a2 p\u2082 = p \u2194 p\u2082 -\u1d65 p = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2\n[PROOFSTEP]\nrw [angle, \u2190 InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)\n\u22a2 inner (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082) = 0\n[PROOFSTEP]\nexact o.inner_eq_zero_of_oangle_eq_pi_div_two h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)\n\u22a2 \u2220 p\u2083 p\u2082 p\u2081 = \u03c0 / 2\n[PROOFSTEP]\nrw [angle_comm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2)\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2\n[PROOFSTEP]\nexact angle_eq_pi_div_two_of_oangle_eq_pi_div_two h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(-\u03c0 / 2)\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2\n[PROOFSTEP]\nrw [angle, \u2190 InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(-\u03c0 / 2)\n\u22a2 inner (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082) = 0\n[PROOFSTEP]\nexact o.inner_eq_zero_of_oangle_eq_neg_pi_div_two h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(-\u03c0 / 2)\n\u22a2 \u2220 p\u2083 p\u2082 p\u2081 = \u03c0 / 2\n[PROOFSTEP]\nrw [angle_comm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(-\u03c0 / 2)\n\u22a2 \u2220 p\u2081 p\u2082 p\u2083 = \u03c0 / 2\n[PROOFSTEP]\nexact angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two h\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 -Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = Real.Angle.sign (\u2221 p\u2082 p\u2081 p\u2083)\n[PROOFSTEP]\nrw [eq_comm, oangle, oangle, \u2190 o.oangle_neg_neg, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev, \u2190\n  vsub_sub_vsub_cancel_left p\u2081 p\u2083 p\u2082, \u2190 neg_vsub_eq_vsub_rev p\u2083 p\u2082, sub_eq_add_neg, neg_vsub_eq_vsub_rev p\u2082 p\u2081,\n  add_comm, \u2190 @neg_one_smul \u211d]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 p\u2082 + -1 \u2022 (p\u2083 -\u1d65 p\u2082))) =\n    -Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082))\n[PROOFSTEP]\nnth_rw 2 [\u2190 one_smul \u211d (p\u2081 -\u1d65 p\u2082)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (1 \u2022 (p\u2081 -\u1d65 p\u2082) + -1 \u2022 (p\u2083 -\u1d65 p\u2082))) =\n    -Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082))\n[PROOFSTEP]\nrw [o.oangle_sign_smul_add_smul_right]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2191SignType.sign (-1) * Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082)) =\n    -Real.Angle.sign (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082))\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 -Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = Real.Angle.sign (\u2221 p\u2083 p\u2082 p\u2081)\n[PROOFSTEP]\nrw [oangle_rev, Real.Angle.sign_neg, neg_neg]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 -Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nrw [oangle_swap\u2081\u2083_sign, \u2190 oangle_swap\u2081\u2082_sign, oangle_swap\u2081\u2083_sign]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 Real.Angle.sign (\u2221 p\u2082 p\u2083 p\u2081) = Real.Angle.sign (\u2221 p\u2081 p\u2082 p\u2083)\n[PROOFSTEP]\nrw [\u2190 oangle_swap\u2081\u2082_sign, oangle_swap\u2081\u2083_sign]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2191\u03c0 \u2194 Sbtw \u211d p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [oangle_eq_pi_iff_angle_eq_pi, angle_eq_pi_iff_sbtw]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2221 p\u2083 p\u2082 p\u2081 = \u2191\u03c0\n[PROOFSTEP]\nrw [oangle_eq_pi_iff_oangle_rev_eq_pi, \u2190 h.oangle\u2081\u2082\u2083_eq_pi]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2221 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nby_cases hp\u2082p\u2081 : p\u2082 = p\u2081\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : p\u2082 = p\u2081\n\u22a2 \u2221 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nsimp [hp\u2082p\u2081]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : \u00acp\u2082 = p\u2081\n\u22a2 \u2221 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nby_cases hp\u2083p\u2081 : p\u2083 = p\u2081\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083p\u2081 : p\u2083 = p\u2081\n\u22a2 \u2221 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nsimp [hp\u2083p\u2081]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083p\u2081 : \u00acp\u2083 = p\u2081\n\u22a2 \u2221 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nrw [oangle_eq_zero_iff_angle_eq_zero hp\u2082p\u2081 hp\u2083p\u2081]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2082p\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083p\u2081 : \u00acp\u2083 = p\u2081\n\u22a2 \u2220 p\u2082 p\u2081 p\u2083 = 0\n[PROOFSTEP]\nexact h.angle\u2082\u2081\u2083_eq_zero_of_ne hp\u2082p\u2081\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 \u2221 p\u2083 p\u2081 p\u2082 = 0\n[PROOFSTEP]\nrw [oangle_eq_zero_iff_oangle_rev_eq_zero, h.oangle\u2082\u2081\u2083_eq_zero]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nby_cases hp\u2081p\u2082 : p\u2081 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : p\u2081 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp [hp\u2081p\u2082]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nby_cases hp\u2083p\u2082 : p\u2083 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2083p\u2082 : p\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp [hp\u2083p\u2082]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = 0 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nrw [oangle_eq_zero_iff_angle_eq_zero hp\u2081p\u2082 hp\u2083p\u2082, angle_eq_zero_iff_ne_and_wbtw]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nhp\u2081p\u2082 : \u00acp\u2081 = p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 p\u2081 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 p\u2083 \u2260 p\u2082 \u2227 Wbtw \u211d p\u2082 p\u2083 p\u2081 \u2194 Wbtw \u211d p\u2082 p\u2081 p\u2083 \u2228 Wbtw \u211d p\u2082 p\u2083 p\u2081\n[PROOFSTEP]\nsimp [hp\u2081p\u2082, hp\u2083p\u2082]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2081'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nby_cases hp\u2083p\u2082 : p\u2083 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2081'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2082 : p\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nsimp [hp\u2083p\u2082]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2081'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nby_cases hp\u2081'p\u2082 : p\u2081' = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2081'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhp\u2081'p\u2082 : p\u2081' = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrw [hp\u2081'p\u2082, wbtw_self_iff] at h \n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : p\u2081 = p\u2082\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhp\u2081'p\u2082 : p\u2081' = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nexact False.elim (hp\u2081p\u2082 h)\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2082 p\u2081 p\u2081'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhp\u2081'p\u2082 : \u00acp\u2081' = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrw [\u2190 oangle_add hp\u2081'p\u2082 hp\u2081p\u2082 hp\u2083p\u2082, h.oangle\u2083\u2081\u2082_eq_zero, zero_add]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2083' : P\nh : Wbtw \u211d p\u2082 p\u2083 p\u2083'\nhp\u2083p\u2082 : p\u2083 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083'\n[PROOFSTEP]\nrw [oangle_rev, h.oangle_eq_left hp\u2083p\u2082, \u2190 oangle_rev]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 (midpoint \u211d p\u2081 p\u2082) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nby_cases h : p\u2081 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : p\u2081 = p\u2082\n\u22a2 \u2221 (midpoint \u211d p\u2081 p\u2082) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u00acp\u2081 = p\u2082\n\u22a2 \u2221 (midpoint \u211d p\u2081 p\u2082) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nexact (sbtw_midpoint_of_ne \u211d h).symm.oangle_eq_left\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 (midpoint \u211d p\u2082 p\u2081) p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [midpoint_comm, oangle_midpoint_left]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 (midpoint \u211d p\u2083 p\u2082) = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nby_cases h : p\u2083 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : p\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 (midpoint \u211d p\u2083 p\u2082) = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\nh : \u00acp\u2083 = p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 (midpoint \u211d p\u2083 p\u2082) = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nexact (sbtw_midpoint_of_ne \u211d h).symm.oangle_eq_right\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 \u2221 p\u2081 p\u2082 (midpoint \u211d p\u2082 p\u2083) = \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [midpoint_comm, oangle_midpoint_right]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2081'\nhp\u2083p\u2082 : p\u2083 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083 + \u2191\u03c0\n[PROOFSTEP]\nrw [\u2190 h.oangle\u2081\u2082\u2083_eq_pi, oangle_add_swap h.left_ne h.right_ne hp\u2083p\u2082]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2083' : P\nh : Sbtw \u211d p\u2083 p\u2082 p\u2083'\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081 p\u2082 p\u2083' + \u2191\u03c0\n[PROOFSTEP]\nrw [\u2190 h.oangle\u2083\u2082\u2081_eq_pi, oangle_add hp\u2081p\u2082 h.right_ne h.left_ne]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 p\u2083' : P\nh\u2081 : Sbtw \u211d p\u2081 p\u2082 p\u2081'\nh\u2083 : Sbtw \u211d p\u2083 p\u2082 p\u2083'\n\u22a2 \u2221 p\u2081 p\u2082 p\u2083 = \u2221 p\u2081' p\u2082 p\u2083'\n[PROOFSTEP]\nrw [h\u2081.oangle_eq_add_pi_left h\u2083.left_ne, h\u2083.oangle_eq_add_pi_right h\u2081.right_ne, add_assoc, Real.Angle.coe_pi_add_coe_pi,\n  add_zero]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nby_cases hp\u2083p\u2082 : p\u2083 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : p\u2083 = p\u2082\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nsimp [hp\u2083p\u2082]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrcases h.wbtw_or_wbtw_or_wbtw with (hw | hw | hw)\n[GOAL]\ncase neg.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhw : Wbtw \u211d p\u2081 p\u2082 p\u2081'\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nhave hw' : Sbtw \u211d p\u2081 p\u2082 p\u2081' := \u27e8hw, hp\u2081p\u2082.symm, hp\u2081'p\u2082.symm\u27e9\n[GOAL]\ncase neg.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhw : Wbtw \u211d p\u2081 p\u2082 p\u2081'\nhw' : Sbtw \u211d p\u2081 p\u2082 p\u2081'\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrw [hw'.oangle_eq_add_pi_left hp\u2083p\u2082, smul_add, Real.Angle.two_zsmul_coe_pi, add_zero]\n[GOAL]\ncase neg.inr.inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhw : Wbtw \u211d p\u2082 p\u2081' p\u2081\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrw [hw.oangle_eq_left hp\u2081'p\u2082]\n[GOAL]\ncase neg.inr.inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2081' p\u2082 p\u2083 : P\nh : Collinear \u211d {p\u2081, p\u2082, p\u2081'}\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081'p\u2082 : p\u2081' \u2260 p\u2082\nhp\u2083p\u2082 : \u00acp\u2083 = p\u2082\nhw : Wbtw \u211d p\u2081' p\u2081 p\u2082\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081' p\u2082 p\u2083\n[PROOFSTEP]\nrw [hw.symm.oangle_eq_left hp\u2081p\u2082]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2083' : P\nh : Collinear \u211d {p\u2083, p\u2082, p\u2083'}\nhp\u2083p\u2082 : p\u2083 \u2260 p\u2082\nhp\u2083'p\u2082 : p\u2083' \u2260 p\u2082\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083'\n[PROOFSTEP]\nrw [oangle_rev, smul_neg, h.two_zsmul_oangle_eq_left hp\u2083p\u2082 hp\u2083'p\u2082, \u2190 smul_neg, \u2190 oangle_rev]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\n\u22a2 dist p\u2081 p = dist p\u2082 p \u2194 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nrefine' \u27e8fun hd => _, fun hr => _\u27e9\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\n\u22a2 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nhave hi : \u27eap\u2082 -\u1d65 p\u2081, p -\u1d65 midpoint \u211d p\u2081 p\u2082\u27eb = 0 :=\n  by\n  rw [@dist_eq_norm_vsub' V, @dist_eq_norm_vsub' V, \u2190 mul_self_inj (norm_nonneg _) (norm_nonneg _), \u2190\n    real_inner_self_eq_norm_mul_norm, \u2190 real_inner_self_eq_norm_mul_norm] at hd \n  simp_rw [vsub_midpoint, \u2190 vsub_sub_vsub_cancel_left p\u2082 p\u2081 p, inner_sub_left, inner_add_right, inner_smul_right, hd,\n    real_inner_comm (p -\u1d65 p\u2081)]\n  abel\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\n\u22a2 inner (p\u2082 -\u1d65 p\u2081) (p -\u1d65 midpoint \u211d p\u2081 p\u2082) = 0\n[PROOFSTEP]\nrw [@dist_eq_norm_vsub' V, @dist_eq_norm_vsub' V, \u2190 mul_self_inj (norm_nonneg _) (norm_nonneg _), \u2190\n  real_inner_self_eq_norm_mul_norm, \u2190 real_inner_self_eq_norm_mul_norm] at hd \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd\u271d\u00b9 : \u2016p -\u1d65 p\u2081\u2016 * \u2016p -\u1d65 p\u2081\u2016 = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd\u271d : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082)\n\u22a2 inner (p\u2082 -\u1d65 p\u2081) (p -\u1d65 midpoint \u211d p\u2081 p\u2082) = 0\n[PROOFSTEP]\nsimp_rw [vsub_midpoint, \u2190 vsub_sub_vsub_cancel_left p\u2082 p\u2081 p, inner_sub_left, inner_add_right, inner_smul_right, hd,\n  real_inner_comm (p -\u1d65 p\u2081)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd\u271d\u00b9 : \u2016p -\u1d65 p\u2081\u2016 * \u2016p -\u1d65 p\u2081\u2016 = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd\u271d : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082)\n\u22a2 \u215f2 * inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082) + \u215f2 * inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2082) -\n      (\u215f2 * inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2082) + \u215f2 * inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082)) =\n    0\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd\u271d\u00b9 : \u2016p -\u1d65 p\u2081\u2016 * \u2016p -\u1d65 p\u2081\u2016 = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd\u271d : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = \u2016p -\u1d65 p\u2082\u2016 * \u2016p -\u1d65 p\u2082\u2016\nhd : inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2081) = inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082)\n\u22a2 \u215f2 * inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082) + \u215f2 * inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2082) -\n      (\u215f2 * inner (p -\u1d65 p\u2081) (p -\u1d65 p\u2082) + \u215f2 * inner (p -\u1d65 p\u2082) (p -\u1d65 p\u2082)) =\n    0\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\nhi : inner (p\u2082 -\u1d65 p\u2081) (p -\u1d65 midpoint \u211d p\u2081 p\u2082) = 0\n\u22a2 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nrw [@Orientation.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two V _ _ _ o,\n  or_iff_right (vsub_ne_zero.2 h.symm)] at hi \n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\nhi : \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) = p -\u1d65 midpoint \u211d p\u2081 p\u2082\n\u22a2 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nrcases hi with \u27e8r, hr\u27e9\n[GOAL]\ncase refine'_1.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) = p -\u1d65 midpoint \u211d p\u2081 p\u2082\n\u22a2 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nrw [eq_comm, \u2190 eq_vadd_iff_vsub_eq] at hr \n[GOAL]\ncase refine'_1.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhd : dist p\u2081 p = dist p\u2082 p\nr : \u211d\nhr : p = r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082\n\u22a2 \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n[PROOFSTEP]\nexact \u27e8r, hr.symm\u27e9\n[GOAL]\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p : P\nh : p\u2081 \u2260 p\u2082\nhr : \u2203 r, r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = p\n\u22a2 dist p\u2081 p = dist p\u2082 p\n[PROOFSTEP]\nrcases hr with \u27e8r, rfl\u27e9\n[GOAL]\ncase refine'_2.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\nr : \u211d\n\u22a2 dist p\u2081 (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082) =\n    dist p\u2082 (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082)\n[PROOFSTEP]\nsimp_rw [@dist_eq_norm_vsub V, vsub_vadd_eq_vsub_sub, left_vsub_midpoint, right_vsub_midpoint, invOf_eq_inv, \u2190\n  neg_vsub_eq_vsub_rev p\u2082 p\u2081, \u2190 mul_self_inj (norm_nonneg _) (norm_nonneg _), \u2190 real_inner_self_eq_norm_mul_norm,\n  inner_sub_sub_self]\n[GOAL]\ncase refine'_2.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 : P\nh : p\u2081 \u2260 p\u2082\nr : \u211d\n\u22a2 inner (2\u207b\u00b9 \u2022 -(p\u2082 -\u1d65 p\u2081)) (2\u207b\u00b9 \u2022 -(p\u2082 -\u1d65 p\u2081)) -\n          inner (2\u207b\u00b9 \u2022 -(p\u2082 -\u1d65 p\u2081)) (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) -\n        inner (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) (2\u207b\u00b9 \u2022 -(p\u2082 -\u1d65 p\u2081)) +\n      inner (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) =\n    inner (2\u207b\u00b9 \u2022 (p\u2082 -\u1d65 p\u2081)) (2\u207b\u00b9 \u2022 (p\u2082 -\u1d65 p\u2081)) -\n          inner (2\u207b\u00b9 \u2022 (p\u2082 -\u1d65 p\u2081)) (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) -\n        inner (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) (2\u207b\u00b9 \u2022 (p\u2082 -\u1d65 p\u2081)) +\n      inner (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081)) (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081))\n[PROOFSTEP]\nsimp [-neg_vsub_eq_vsub_rev]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nby_cases hc\u2085\u2081\u2082 : Collinear \u211d ({ p\u2085, p\u2081, p\u2082 } : Set P)\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d ({ p\u2085, p\u2081, p\u2082, p\u2083, p\u2084 } : Set P) :=\n  (hc.collinear_insert_iff_of_ne (Set.mem_insert _ _) (Set.mem_insert_of_mem _ (Set.mem_insert _ _)) hp\u2081p\u2082).2 hc\u2085\u2081\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hc\u2085\u2083\u2084 : Collinear \u211d ({ p\u2085, p\u2083, p\u2084 } : Set P) :=\n  (hc.collinear_insert_iff_of_ne (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _)))\n        (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_singleton _)))) hp\u2083p\u2084).1\n    hc\u2085\u2081\u2082\u2083\u2084\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2085, p\u2081, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2085, p\u2083, p\u2084}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nrw [Set.insert_comm] at hc\u2085\u2081\u2082 hc\u2085\u2083\u2084 \n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hs\u2081\u2085\u2082 := oangle_eq_zero_or_eq_pi_iff_collinear.2 hc\u2085\u2081\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hs\u2083\u2085\u2084 := oangle_eq_zero_or_eq_pi_iff_collinear.2 hc\u2085\u2083\u2084\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : \u2221 p\u2081 p\u2085 p\u2082 = 0 \u2228 \u2221 p\u2081 p\u2085 p\u2082 = \u2191\u03c0\nhs\u2083\u2085\u2084 : \u2221 p\u2083 p\u2085 p\u2084 = 0 \u2228 \u2221 p\u2083 p\u2085 p\u2084 = \u2191\u03c0\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nrw [\u2190 Real.Angle.sign_eq_zero_iff] at hs\u2081\u2085\u2082 hs\u2083\u2085\u2084 \n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : Collinear \u211d {p\u2081, p\u2085, p\u2082}\nhc\u2085\u2081\u2082\u2083\u2084 : Collinear \u211d {p\u2085, p\u2081, p\u2082, p\u2083, p\u2084}\nhc\u2085\u2083\u2084 : Collinear \u211d {p\u2083, p\u2085, p\u2084}\nhs\u2081\u2085\u2082 : Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = 0\nhs\u2083\u2085\u2084 : Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084) = 0\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nrw [hs\u2081\u2085\u2082, hs\u2083\u2085\u2084]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nlet s : Set (P \u00d7 P \u00d7 P) :=\n  (fun x : line[\u211d, p\u2081, p\u2082] \u00d7 V => (x.1, p\u2085, x.2 +\u1d65 (x.1 : P))) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hco : IsConnected s :=\n  haveI : ConnectedSpace line[\u211d, p\u2081, p\u2082] := AddTorsor.connectedSpace _ _\n  (isConnected_univ.prod (isConnected_setOf_sameRay_and_ne_zero (vsub_ne_zero.2 hp\u2081p\u2082.symm))).image _\n    (continuous_fst.subtype_val.prod_mk\n        (continuous_const.prod_mk (continuous_snd.vadd continuous_fst.subtype_val))).continuousOn\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hf : ContinuousOn (fun p : P \u00d7 P \u00d7 P => \u2221 p.1 p.2.1 p.2.2) s :=\n  by\n  refine' ContinuousAt.continuousOn fun p hp => continuousAt_oangle _ _\n  all_goals\n    simp_rw [Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n    obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n    dsimp only at hp \u22a2\n    obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n    dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n    obtain \u27e8hvr, -\u27e9 := hv\n    rintro rfl\n    refine' hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan _).2 (collinear_pair _ _ _))\n  \u00b7 exact hq\n  \u00b7 refine' vadd_mem_of_mem_direction _ hq\n    rw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr \n    obtain \u27e8r, -, rfl\u27e9 := hvr\n    rw [direction_affineSpan]\n    exact smul_vsub_rev_mem_vectorSpan_pair _ _ _\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\n[PROOFSTEP]\nrefine' ContinuousAt.continuousOn fun p hp => continuousAt_oangle _ _\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.fst \u2260 p.snd.fst\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nall_goals\n  simp_rw [Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n  obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n  dsimp only at hp \u22a2\n  obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n  dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n  obtain \u27e8hvr, -\u27e9 := hv\n  rintro rfl\n  refine' hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan _).2 (collinear_pair _ _ _))\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.fst \u2260 p.snd.fst\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.fst = p.fst \u2227 p\u2085 = p.snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = p.snd.snd\n\u22a2 p.fst \u2260 p.snd.fst\n[PROOFSTEP]\nobtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n[GOAL]\ncase refine'_1.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227\n      \u2191x.fst = (q\u2081, q\u2085, q\u2082).fst \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = (q\u2081, q\u2085, q\u2082).snd.snd\n\u22a2 (q\u2081, q\u2085, q\u2082).fst \u2260 (q\u2081, q\u2085, q\u2082).snd.fst\n[PROOFSTEP]\ndsimp only at hp \u22a2\n[GOAL]\ncase refine'_1.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.fst = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.snd +\u1d65 \u2191x.fst = q\u2082\n\u22a2 q\u2081 \u2260 q\u2085\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n[GOAL]\ncase refine'_1.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : ({ val := q, property := hq }, v).snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 \u2191({ val := q, property := hq }, v).fst \u2260 p\u2085\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n[GOAL]\ncase refine'_1.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : v \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 q \u2260 p\u2085\n[PROOFSTEP]\nobtain \u27e8hvr, -\u27e9 := hv\n[GOAL]\ncase refine'_1.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\n\u22a2 q \u2260 p\u2085\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 False\n[PROOFSTEP]\nrefine' hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan _).2 (collinear_pair _ _ _))\n[GOAL]\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_prod, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n[GOAL]\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.fst = p.fst \u2227 p\u2085 = p.snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = p.snd.snd\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nobtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n[GOAL]\ncase refine'_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227\n      \u2191x.fst = (q\u2081, q\u2085, q\u2082).fst \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = (q\u2081, q\u2085, q\u2082).snd.snd\n\u22a2 (q\u2081, q\u2085, q\u2082).snd.snd \u2260 (q\u2081, q\u2085, q\u2082).snd.fst\n[PROOFSTEP]\ndsimp only at hp \u22a2\n[GOAL]\ncase refine'_2.mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, x.snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0} \u2227 \u2191x.fst = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.snd +\u1d65 \u2191x.fst = q\u2082\n\u22a2 q\u2082 \u2260 q\u2085\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : ({ val := q, property := hq }, v).snd \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 ({ val := q, property := hq }, v).snd +\u1d65 \u2191({ val := q, property := hq }, v).fst \u2260 p\u2085\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : v \u2208 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\n\u22a2 v +\u1d65 q \u2260 p\u2085\n[PROOFSTEP]\nobtain \u27e8hvr, -\u27e9 := hv\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\n\u22a2 v +\u1d65 q \u2260 p\u2085\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, v +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 False\n[PROOFSTEP]\nrefine' hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan _).2 (collinear_pair _ _ _))\n[GOAL]\ncase refine'_1.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n[PROOFSTEP]\nexact hq\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, v +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n[PROOFSTEP]\nrefine' vadd_mem_of_mem_direction _ hq\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, v +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 direction (affineSpan \u211d {p\u2081, p\u2082})\n[PROOFSTEP]\nrw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr \n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {v +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, v +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 v \u2208 direction (affineSpan \u211d {p\u2081, p\u2082})\n[PROOFSTEP]\nobtain \u27e8r, -, rfl\u27e9 := hvr\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.fst, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 direction (affineSpan \u211d {p\u2081, p\u2082})\n[PROOFSTEP]\nrw [direction_affineSpan]\n[GOAL]\ncase refine'_2.mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) :=\n  (fun x => (\u2191x.fst, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}\n[PROOFSTEP]\nexact smul_vsub_rev_mem_vectorSpan_pair _ _ _\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hsp : \u2200 p : P \u00d7 P \u00d7 P, p \u2208 s \u2192 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u03c0 :=\n  by\n  intro p hp\n  simp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n  obtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n  dsimp only at hp \u22a2\n  obtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n  dsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n  obtain \u27e8hvr, hv0\u27e9 := hv\n  rw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr \n  obtain \u27e8r, -, rfl\u27e9 := hvr\n  change q \u2208 line[\u211d, p\u2081, p\u2082] at hq \n  rw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]\n  refine'\n    affineIndependent_of_ne_of_mem_of_not_mem_of_mem _ hq\n      (fun h => hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan h).2 (collinear_pair _ _ _))) _\n  \u00b7 rwa [\u2190 @vsub_ne_zero V, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, neg_ne_zero]\n  \u00b7 refine' vadd_mem_of_mem_direction _ hq\n    rw [direction_affineSpan]\n    exact smul_vsub_rev_mem_vectorSpan_pair _ _ _\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\n\u22a2 \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nintro p hp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\np : P \u00d7 P \u00d7 P\nhp : p \u2208 s\n\u22a2 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff] at hp \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.snd \u2227 x.snd \u2260 0) \u2227 \u2191x.fst = p.fst \u2227 p\u2085 = p.snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = p.snd.snd\n\u22a2 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nobtain \u27e8q\u2081, q\u2085, q\u2082\u27e9 := p\n[GOAL]\ncase mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq\u2081 q\u2085 q\u2082 : P\nhp :\n  \u2203 x,\n    (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.snd \u2227 x.snd \u2260 0) \u2227\n      \u2191x.fst = (q\u2081, q\u2085, q\u2082).fst \u2227 p\u2085 = (q\u2081, q\u2085, q\u2082).snd.fst \u2227 x.snd +\u1d65 \u2191x.fst = (q\u2081, q\u2085, q\u2082).snd.snd\n\u22a2 \u2221 (q\u2081, q\u2085, q\u2082).fst (q\u2081, q\u2085, q\u2082).snd.fst (q\u2081, q\u2085, q\u2082).snd.snd \u2260 0 \u2227\n    \u2221 (q\u2081, q\u2085, q\u2082).fst (q\u2081, q\u2085, q\u2082).snd.fst (q\u2081, q\u2085, q\u2082).snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\ndsimp only at hp \u22a2\n[GOAL]\ncase mk.mk\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq\u2081 q\u2085 q\u2082 : P\nhp : \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.snd \u2227 x.snd \u2260 0) \u2227 \u2191x.fst = q\u2081 \u2227 p\u2085 = q\u2085 \u2227 x.snd +\u1d65 \u2191x.fst = q\u2082\n\u22a2 \u2221 q\u2081 q\u2085 q\u2082 \u2260 0 \u2227 \u2221 q\u2081 q\u2085 q\u2082 \u2260 \u2191\u03c0\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8q, hq\u27e9, v\u27e9, hv, rfl, rfl, rfl\u27e9 := hp\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) ({ val := q, property := hq }, v).snd \u2227 ({ val := q, property := hq }, v).snd \u2260 0\n\u22a2 \u2221 (\u2191({ val := q, property := hq }, v).fst) p\u2085\n        (({ val := q, property := hq }, v).snd +\u1d65 \u2191({ val := q, property := hq }, v).fst) \u2260\n      0 \u2227\n    \u2221 (\u2191({ val := q, property := hq }, v).fst) p\u2085\n        (({ val := q, property := hq }, v).snd +\u1d65 \u2191({ val := q, property := hq }, v).fst) \u2260\n      \u2191\u03c0\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk, Set.mem_setOf] at hv \u22a2\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhv : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0\n[PROOFSTEP]\nobtain \u27e8hvr, hv0\u27e9 := hv\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0\n[PROOFSTEP]\nrw [\u2190 exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp\u2081p\u2082.symm)] at hvr \n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nv : V\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nhvr : \u2203 r, 0 \u2264 r \u2227 r \u2022 (p\u2082 -\u1d65 p\u2081) = v\nhv0 : v \u2260 0\n\u22a2 \u2221 q p\u2085 (v +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (v +\u1d65 q) \u2260 \u2191\u03c0\n[PROOFSTEP]\nobtain \u27e8r, -, rfl\u27e9 := hvr\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0\n[PROOFSTEP]\nchange q \u2208 line[\u211d, p\u2081, p\u2082] at hq \n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 0 \u2227 \u2221 q p\u2085 (r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q) \u2260 \u2191\u03c0\n[PROOFSTEP]\nrw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 AffineIndependent \u211d ![q, p\u2085, r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q]\n[PROOFSTEP]\nrefine'\n  affineIndependent_of_ne_of_mem_of_not_mem_of_mem _ hq\n    (fun h => hc\u2085\u2081\u2082 ((collinear_insert_iff_of_mem_affineSpan h).2 (collinear_pair _ _ _))) _\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 q \u2260 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q\n[PROOFSTEP]\nrwa [\u2190 @vsub_ne_zero V, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, neg_ne_zero]\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) +\u1d65 q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n[PROOFSTEP]\nrefine' vadd_mem_of_mem_direction _ hq\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 direction (affineSpan \u211d {p\u2081, p\u2082})\n[PROOFSTEP]\nrw [direction_affineSpan]\n[GOAL]\ncase mk.mk.intro.mk.mk.intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nq : P\nr : \u211d\nhv0 : r \u2022 (p\u2082 -\u1d65 p\u2081) \u2260 0\nhq : q \u2208 affineSpan \u211d {p\u2081, p\u2082}\n\u22a2 r \u2022 (p\u2082 -\u1d65 p\u2081) \u2208 vectorSpan \u211d {p\u2081, p\u2082}\n[PROOFSTEP]\nexact smul_vsub_rev_mem_vectorSpan_pair _ _ _\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s :=\n  by\n  simp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff]\n  refine' \u27e8\u27e8\u27e8p\u2081, left_mem_affineSpan_pair _ _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9, \u27e8SameRay.rfl, vsub_ne_zero.2 hp\u2081p\u2082.symm\u27e9, _\u27e9\n  simp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n\u22a2 (p\u2081, p\u2085, p\u2082) \u2208 s\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.snd \u2227 x.snd \u2260 0) \u2227 \u2191x.fst = p\u2081 \u2227 True \u2227 x.snd +\u1d65 \u2191x.fst = p\u2082\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8p\u2081, left_mem_affineSpan_pair _ _ _\u27e9, p\u2082 -\u1d65 p\u2081\u27e9, \u27e8SameRay.rfl, vsub_ne_zero.2 hp\u2081p\u2082.symm\u27e9, _\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n\u22a2 \u2191({ val := p\u2081, property := (_ : p\u2081 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2082 -\u1d65 p\u2081).fst = p\u2081 \u2227\n    True \u2227\n      ({ val := p\u2081, property := (_ : p\u2081 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2082 -\u1d65 p\u2081).snd +\u1d65\n          \u2191({ val := p\u2081, property := (_ : p\u2081 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2082 -\u1d65 p\u2081).fst =\n        p\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nhave hp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s :=\n  by\n  simp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff]\n  refine'\n    \u27e8\u27e8\u27e8p\u2083,\n          hc.mem_affineSpan_of_mem_of_ne (Set.mem_insert _ _) (Set.mem_insert_of_mem _ (Set.mem_insert _ _))\n            (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _))) hp\u2081p\u2082\u27e9,\n        p\u2084 -\u1d65 p\u2083\u27e9,\n      \u27e8hr, vsub_ne_zero.2 hp\u2083p\u2084.symm\u27e9, _\u27e9\n  simp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 (p\u2083, p\u2085, p\u2084) \u2208 s\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_prod, Set.mem_setOf, Set.mem_univ, true_and_iff, Prod.ext_iff]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2203 x, (SameRay \u211d (p\u2082 -\u1d65 p\u2081) x.snd \u2227 x.snd \u2260 0) \u2227 \u2191x.fst = p\u2083 \u2227 True \u2227 x.snd +\u1d65 \u2191x.fst = p\u2084\n[PROOFSTEP]\nrefine'\n  \u27e8\u27e8\u27e8p\u2083,\n        hc.mem_affineSpan_of_mem_of_ne (Set.mem_insert _ _) (Set.mem_insert_of_mem _ (Set.mem_insert _ _))\n          (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_insert _ _))) hp\u2081p\u2082\u27e9,\n      p\u2084 -\u1d65 p\u2083\u27e9,\n    \u27e8hr, vsub_ne_zero.2 hp\u2083p\u2084.symm\u27e9, _\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\n\u22a2 \u2191({ val := p\u2083, property := (_ : p\u2083 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2084 -\u1d65 p\u2083).fst = p\u2083 \u2227\n    True \u2227\n      ({ val := p\u2083, property := (_ : p\u2083 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2084 -\u1d65 p\u2083).snd +\u1d65\n          \u2191({ val := p\u2083, property := (_ : p\u2083 \u2208 affineSpan \u211d {p\u2081, p\u2082}) }, p\u2084 -\u1d65 p\u2083).fst =\n        p\u2084\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\nhr : SameRay \u211d (p\u2082 -\u1d65 p\u2081) (p\u2084 -\u1d65 p\u2083)\nhc\u2085\u2081\u2082 : \u00acCollinear \u211d {p\u2085, p\u2081, p\u2082}\ns : Set (P \u00d7 P \u00d7 P) := (fun x => (\u2191x.fst, p\u2085, x.snd +\u1d65 \u2191x.fst)) '' Set.univ \u00d7\u02e2 {v | SameRay \u211d (p\u2082 -\u1d65 p\u2081) v \u2227 v \u2260 0}\nhco : IsConnected s\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) s\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 s \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2081p\u2082s : (p\u2081, p\u2085, p\u2082) \u2208 s\nhp\u2083p\u2084s : (p\u2083, p\u2085, p\u2084) \u2208 s\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2085 p\u2082) = Real.Angle.sign (\u2221 p\u2083 p\u2085 p\u2084)\n[PROOFSTEP]\nconvert Real.Angle.sign_eq_of_continuousOn hco hf hsp hp\u2083p\u2084s hp\u2081p\u2082s\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2082, p\u2083}\n[PROOFSTEP]\nsimpa using h.wbtw.collinear\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhne : p\u2081 \u2260 p\u2082\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2081, p\u2083}\n[PROOFSTEP]\nsimpa [Set.insert_comm p\u2082] using h.collinear\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhne : p\u2082 \u2260 p\u2083\n\u22a2 Real.Angle.sign (\u2221 p\u2082 p\u2084 p\u2083) = Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2083)\n[PROOFSTEP]\nsimp_rw [oangle_rev p\u2083, Real.Angle.sign_neg, h.symm.oangle_sign_eq_of_ne_left _ hne.symm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nby_cases h : p\u2081 = p\u2082\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : p\u2081 = p\u2082\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nlet sp : Set (P \u00d7 P \u00d7 P) := (fun p : P => (p\u2081, p, p\u2082)) '' {p | s.SSameSide p\u2083 p}\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hc : IsConnected sp :=\n  (isConnected_setOf_sSameSide hp\u2083p\u2084.2.1 hp\u2083p\u2084.nonempty).image _\n    (continuous_const.prod_mk (Continuous.Prod.mk_left _)).continuousOn\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hf : ContinuousOn (fun p : P \u00d7 P \u00d7 P => \u2221 p.1 p.2.1 p.2.2) sp :=\n  by\n  refine' ContinuousAt.continuousOn fun p hp => continuousAt_oangle _ _\n  all_goals\n    simp_rw [Set.mem_image, Set.mem_setOf] at hp \n    obtain \u27e8p', hp', rfl\u27e9 := hp\n    dsimp only\n    rintro rfl\n  \u00b7 exact hp'.2.2 hp\u2081\n  \u00b7 exact hp'.2.2 hp\u2082\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\n\u22a2 ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\n[PROOFSTEP]\nrefine' ContinuousAt.continuousOn fun p hp => continuousAt_oangle _ _\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : p \u2208 sp\n\u22a2 p.fst \u2260 p.snd.fst\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : p \u2208 sp\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nall_goals\n  simp_rw [Set.mem_image, Set.mem_setOf] at hp \n  obtain \u27e8p', hp', rfl\u27e9 := hp\n  dsimp only\n  rintro rfl\n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : p \u2208 sp\n\u22a2 p.fst \u2260 p.snd.fst\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_setOf] at hp \n[GOAL]\ncase refine'_1\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, SSameSide s p\u2083 x \u2227 (p\u2081, x, p\u2082) = p\n\u22a2 p.fst \u2260 p.snd.fst\n[PROOFSTEP]\nobtain \u27e8p', hp', rfl\u27e9 := hp\n[GOAL]\ncase refine'_1.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 (p\u2081, p', p\u2082).fst \u2260 (p\u2081, p', p\u2082).snd.fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_1.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 p\u2081 \u2260 p'\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : p \u2208 sp\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_setOf] at hp \n[GOAL]\ncase refine'_2\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, SSameSide s p\u2083 x \u2227 (p\u2081, x, p\u2082) = p\n\u22a2 p.snd.snd \u2260 p.snd.fst\n[PROOFSTEP]\nobtain \u27e8p', hp', rfl\u27e9 := hp\n[GOAL]\ncase refine'_2.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 (p\u2081, p', p\u2082).snd.snd \u2260 (p\u2081, p', p\u2082).snd.fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_2.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 p\u2082 \u2260 p'\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhp' : SSameSide s p\u2083 p\u2081\n\u22a2 False\n[PROOFSTEP]\nexact hp'.2.2 hp\u2081\n[GOAL]\ncase refine'_2.intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhp' : SSameSide s p\u2083 p\u2082\n\u22a2 False\n[PROOFSTEP]\nexact hp'.2.2 hp\u2082\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hsp : \u2200 p : P \u00d7 P \u00d7 P, p \u2208 sp \u2192 \u2221 p.1 p.2.1 p.2.2 \u2260 0 \u2227 \u2221 p.1 p.2.1 p.2.2 \u2260 \u03c0 :=\n  by\n  intro p hp\n  simp_rw [Set.mem_image, Set.mem_setOf] at hp \n  obtain \u27e8p', hp', rfl\u27e9 := hp\n  dsimp only\n  rw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]\n  exact affineIndependent_of_ne_of_mem_of_not_mem_of_mem h hp\u2081 hp'.2.2 hp\u2082\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\n\u22a2 \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 sp \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nintro p hp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\np : P \u00d7 P \u00d7 P\nhp : p \u2208 sp\n\u22a2 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nsimp_rw [Set.mem_image, Set.mem_setOf] at hp \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\np : P \u00d7 P \u00d7 P\nhp : \u2203 x, SSameSide s p\u2083 x \u2227 (p\u2081, x, p\u2082) = p\n\u22a2 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\nobtain \u27e8p', hp', rfl\u27e9 := hp\n[GOAL]\ncase intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 \u2221 (p\u2081, p', p\u2082).fst (p\u2081, p', p\u2082).snd.fst (p\u2081, p', p\u2082).snd.snd \u2260 0 \u2227\n    \u2221 (p\u2081, p', p\u2082).fst (p\u2081, p', p\u2082).snd.fst (p\u2081, p', p\u2082).snd.snd \u2260 \u2191\u03c0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 \u2221 p\u2081 p' p\u2082 \u2260 0 \u2227 \u2221 p\u2081 p' p\u2082 \u2260 \u2191\u03c0\n[PROOFSTEP]\nrw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]\n[GOAL]\ncase intro.intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\np' : P\nhp' : SSameSide s p\u2083 p'\n\u22a2 AffineIndependent \u211d ![p\u2081, p', p\u2082]\n[PROOFSTEP]\nexact affineIndependent_of_ne_of_mem_of_not_mem_of_mem h hp\u2081 hp'.2.2 hp\u2082\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 sp \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hp\u2083 : (p\u2081, p\u2083, p\u2082) \u2208 sp := Set.mem_image_of_mem _ (sSameSide_self_iff.2 \u27e8hp\u2083p\u2084.nonempty, hp\u2083p\u2084.2.1\u27e9)\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 sp \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2083 : (p\u2081, p\u2083, p\u2082) \u2208 sp\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hp\u2084 : (p\u2081, p\u2084, p\u2082) \u2208 sp := Set.mem_image_of_mem _ hp\u2083p\u2084\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SSameSide s p\u2083 p\u2084\nh : \u00acp\u2081 = p\u2082\nsp : Set (P \u00d7 P \u00d7 P) := (fun p => (p\u2081, p, p\u2082)) '' {p | SSameSide s p\u2083 p}\nhc : IsConnected sp\nhf : ContinuousOn (fun p => \u2221 p.fst p.snd.fst p.snd.snd) sp\nhsp : \u2200 (p : P \u00d7 P \u00d7 P), p \u2208 sp \u2192 \u2221 p.fst p.snd.fst p.snd.snd \u2260 0 \u2227 \u2221 p.fst p.snd.fst p.snd.snd \u2260 \u2191\u03c0\nhp\u2083 : (p\u2081, p\u2083, p\u2082) \u2208 sp\nhp\u2084 : (p\u2081, p\u2084, p\u2082) \u2208 sp\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nconvert Real.Angle.sign_eq_of_continuousOn hc hf hsp hp\u2083 hp\u2084\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SOppSide s p\u2083 p\u2084\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = -Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nhave hp\u2081p\u2083 : p\u2081 \u2260 p\u2083 := by rintro rfl; exact hp\u2083p\u2084.left_not_mem hp\u2081\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SOppSide s p\u2083 p\u2084\n\u22a2 p\u2081 \u2260 p\u2083\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SOppSide s p\u2081 p\u2084\n\u22a2 False\n[PROOFSTEP]\nexact hp\u2083p\u2084.left_not_mem hp\u2081\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : AffineSubspace \u211d P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083p\u2084 : SOppSide s p\u2083 p\u2084\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\n\u22a2 Real.Angle.sign (\u2221 p\u2081 p\u2084 p\u2082) = -Real.Angle.sign (\u2221 p\u2081 p\u2083 p\u2082)\n[PROOFSTEP]\nrw [\u2190 (hp\u2083p\u2084.symm.trans (sOppSide_pointReflection hp\u2081 hp\u2083p\u2084.left_not_mem)).oangle_sign_eq hp\u2081 hp\u2082, \u2190\n  oangle_rotate_sign p\u2081, \u2190 oangle_rotate_sign p\u2081, oangle_swap\u2081\u2083_sign,\n  (sbtw_pointReflection_of_ne \u211d hp\u2081p\u2083).symm.oangle_sign_eq _]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Angle.Oriented.Affine", "llama_tokens": 64793, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673178375734, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.5119231959630363}}
{"text": "[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\ni j : \u2115\n\u22a2 I ^ i \u2022 N F j \u2264 N F (i + j)\n[PROOFSTEP]\ninduction' i with _ ih\n[GOAL]\ncase zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nj : \u2115\n\u22a2 I ^ Nat.zero \u2022 N F j \u2264 N F (Nat.zero + j)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nj n\u271d : \u2115\nih : I ^ n\u271d \u2022 N F j \u2264 N F (n\u271d + j)\n\u22a2 I ^ Nat.succ n\u271d \u2022 N F j \u2264 N F (Nat.succ n\u271d + j)\n[PROOFSTEP]\nrw [pow_succ, mul_smul, Nat.succ_eq_add_one, add_assoc, add_comm 1, \u2190 add_assoc]\n[GOAL]\ncase succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nj n\u271d : \u2115\nih : I ^ n\u271d \u2022 N F j \u2264 N F (n\u271d + j)\n\u22a2 I \u2022 I ^ n\u271d \u2022 N F j \u2264 N F (n\u271d + j + 1)\n[PROOFSTEP]\nexact (Submodule.smul_mono_right ih).trans (F.smul_le _)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\ni j k : \u2115\n\u22a2 I ^ (i + k) \u2022 N F j \u2264 I ^ k \u2022 N F (i + j)\n[PROOFSTEP]\nrw [add_comm, pow_add, mul_smul]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\ni j k : \u2115\n\u22a2 I ^ k \u2022 I ^ i \u2022 N F j \u2264 I ^ k \u2022 N F (i + j)\n[PROOFSTEP]\nexact Submodule.smul_mono_right (F.pow_smul_le i j)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 sSup (N '' S) (i + 1) \u2264 sSup (N '' S) i\n[PROOFSTEP]\napply sSup_le_sSup_of_forall_exists_le _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2200 (x : Submodule R M), (x \u2208 Set.range fun f => \u2191f (i + 1)) \u2192 \u2203 y, (y \u2208 Set.range fun f => \u2191f i) \u2227 x \u2264 y\n[PROOFSTEP]\nrintro _ \u27e8\u27e8_, F, hF, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF\u271d F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\nF : Filtration I M\nhF : F \u2208 S\n\u22a2 \u2203 y,\n    (y \u2208 Set.range fun f => \u2191f i) \u2227 (fun f => \u2191f (i + 1)) { val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) } \u2264 y\n[PROOFSTEP]\nexact \u27e8_, \u27e8\u27e8_, F, hF, rfl\u27e9, rfl\u27e9, F.mono i\u27e9\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 I \u2022 sSup (N '' S) i \u2264 sSup (N '' S) (i + 1)\n[PROOFSTEP]\nrw [sSup_eq_iSup', iSup_apply, Submodule.smul_iSup, iSup_apply]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2a06 (i_1 : \u2191(N '' S)), I \u2022 \u2191i_1 i \u2264 \u2a06 (i_1 : \u2191(N '' S)), \u2191i_1 (i + 1)\n[PROOFSTEP]\napply iSup_mono _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2191(N '' S)), I \u2022 \u2191i_1 i \u2264 \u2191i_1 (i + 1)\n[PROOFSTEP]\nrintro \u27e8_, F, hF, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF\u271d F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\nF : Filtration I M\nhF : F \u2208 S\n\u22a2 I \u2022 \u2191{ val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) } i \u2264\n    \u2191{ val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) } (i + 1)\n[PROOFSTEP]\nexact F.smul_le i\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 sInf (N '' S) (i + 1) \u2264 sInf (N '' S) i\n[PROOFSTEP]\napply sInf_le_sInf_of_forall_exists_le _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2200 (x : Submodule R M), (x \u2208 Set.range fun f => \u2191f i) \u2192 \u2203 y, (y \u2208 Set.range fun f => \u2191f (i + 1)) \u2227 y \u2264 x\n[PROOFSTEP]\nrintro _ \u27e8\u27e8_, F, hF, rfl\u27e9, rfl\u27e9\n[GOAL]\ncase intro.mk.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF\u271d F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\nF : Filtration I M\nhF : F \u2208 S\n\u22a2 \u2203 y,\n    (y \u2208 Set.range fun f => \u2191f (i + 1)) \u2227 y \u2264 (fun f => \u2191f i) { val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) }\n[PROOFSTEP]\nexact \u27e8_, \u27e8\u27e8_, F, hF, rfl\u27e9, rfl\u27e9, F.mono i\u27e9\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 I \u2022 sInf (N '' S) i \u2264 sInf (N '' S) (i + 1)\n[PROOFSTEP]\nrw [sInf_eq_iInf', iInf_apply, iInf_apply]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 I \u2022 \u2a05 (i_1 : \u2191(N '' S)), \u2191i_1 i \u2264 \u2a05 (i_1 : \u2191(N '' S)), \u2191i_1 (i + 1)\n[PROOFSTEP]\nrefine' Submodule.smul_iInf_le.trans _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2a05 (i_1 : \u2191(N '' S)), I \u2022 \u2191i_1 i \u2264 \u2a05 (i_1 : \u2191(N '' S)), \u2191i_1 (i + 1)\n[PROOFSTEP]\napply iInf_mono _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\n\u22a2 \u2200 (i_1 : \u2191(N '' S)), I \u2022 \u2191i_1 i \u2264 \u2191i_1 (i + 1)\n[PROOFSTEP]\nrintro \u27e8_, F, hF, rfl\u27e9\n[GOAL]\ncase mk.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF\u271d F' : Filtration I M\nS : Set (Filtration I M)\ni : \u2115\nF : Filtration I M\nhF : F \u2208 S\n\u22a2 I \u2022 \u2191{ val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) } i \u2264\n    \u2191{ val := F.N, property := (_ : \u2203 a, a \u2208 S \u2227 a.N = F.N) } (i + 1)\n[PROOFSTEP]\nexact F.smul_le i\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\ni : \u2115\n\u22a2 (fun i => I ^ i \u2022 N) (i + 1) \u2264 (fun i => I ^ i \u2022 N) i\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\ni : \u2115\n\u22a2 I ^ (i + 1) \u2022 N \u2264 I ^ i \u2022 N\n[PROOFSTEP]\nrw [add_comm, pow_add, mul_smul]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\ni : \u2115\n\u22a2 I ^ 1 \u2022 I ^ i \u2022 N \u2264 I ^ i \u2022 N\n[PROOFSTEP]\nexact Submodule.smul_le_right\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\ni : \u2115\n\u22a2 I \u2022 (fun i => I ^ i \u2022 N) i \u2264 (fun i => I ^ i \u2022 N) (i + 1)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\ni : \u2115\n\u22a2 I \u2022 I ^ i \u2022 N \u2264 I ^ (i + 1) \u2022 N\n[PROOFSTEP]\nrw [add_comm, pow_add, mul_smul, pow_one]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\n\u22a2 Stable (stableFiltration I N)\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\n\u22a2 \u2200 (n : \u2115), n \u2265 0 \u2192 I \u2022 Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (n + 1)\n[PROOFSTEP]\nintro n _\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\nn : \u2115\na\u271d : n \u2265 0\n\u22a2 I \u2022 Ideal.Filtration.N (stableFiltration I N) n = Ideal.Filtration.N (stableFiltration I N) (n + 1)\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI\u271d : Ideal R\nF F' : Filtration I\u271d M\nI : Ideal R\nN : Submodule R M\nn : \u2115\na\u271d : n \u2265 0\n\u22a2 I \u2022 I ^ n \u2022 N = I ^ (n + 1) \u2022 N\n[PROOFSTEP]\nrw [add_comm, pow_add, mul_smul, pow_one]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 \u2203 n\u2080, \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n[PROOFSTEP]\nobtain \u27e8n\u2080, hn\u27e9 := h\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2203 n\u2080, \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n[PROOFSTEP]\nuse n\u2080\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n[PROOFSTEP]\nintro k\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nk : \u2115\n\u22a2 N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n[PROOFSTEP]\ninduction' k with _ ih\n[GOAL]\ncase h.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 N F (n\u2080 + Nat.zero) = I ^ Nat.zero \u2022 N F n\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn\u271d : \u2115\nih : N F (n\u2080 + n\u271d) = I ^ n\u271d \u2022 N F n\u2080\n\u22a2 N F (n\u2080 + Nat.succ n\u271d) = I ^ Nat.succ n\u271d \u2022 N F n\u2080\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, \u2190 add_assoc, \u2190 hn, ih, add_comm, pow_add, mul_smul, pow_one]\n[GOAL]\ncase h.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nn\u2080 : \u2115\nhn : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn\u271d : \u2115\nih : N F (n\u2080 + n\u271d) = I ^ n\u271d \u2022 N F n\u2080\n\u22a2 n\u2080 + n\u271d \u2265 n\u2080\n[PROOFSTEP]\nlinarith\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\n[PROOFSTEP]\nobtain \u27e8n\u2080, hn\u2080\u27e9 := h.exists_pow_smul_eq\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\n[PROOFSTEP]\nuse n\u2080\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\n\u22a2 \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\n[PROOFSTEP]\nintro n hn\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\n[PROOFSTEP]\nconvert hn\u2080 (n - n\u2080)\n[GOAL]\ncase h.e'_2.h.e'_8\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), N F (n\u2080 + k) = I ^ k \u2022 N F n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 n = n\u2080 + (n - n\u2080)\n[PROOFSTEP]\nrw [add_comm, tsub_add_cancel_of_le hn]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 Stable F \u2194 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\n[PROOFSTEP]\nrefine' \u27e8Stable.exists_pow_smul_eq_of_ge, fun h => \u27e8h.choose, fun n hn => _\u27e9\u27e9\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d : Stable F\nh : \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\nn : \u2115\nhn : n \u2265 Exists.choose h\n\u22a2 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nrw [h.choose_spec n hn, h.choose_spec (n + 1) (by linarith), smul_smul, \u2190 pow_succ, tsub_add_eq_add_tsub hn]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d : Stable F\nh : \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 N F n = I ^ (n - n\u2080) \u2022 N F n\u2080\nn : \u2115\nhn : n \u2265 Exists.choose h\n\u22a2 n + 1 \u2265 Exists.choose h\n[PROOFSTEP]\nlinarith\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\ne : N F 0 \u2264 N F' 0\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n\n[PROOFSTEP]\nobtain \u27e8n\u2080, hF\u27e9 := h\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n\n[PROOFSTEP]\nuse n\u2080\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\n\u22a2 N F (n + n\u2080) \u2264 N F' n\n[PROOFSTEP]\ninduction' n with n hn\n[GOAL]\ncase h.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 N F (Nat.zero + n\u2080) \u2264 N F' Nat.zero\n[PROOFSTEP]\nrefine' (F.antitone _).trans e\n[GOAL]\ncase h.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n\u22a2 0 \u2264 Nat.zero + n\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 N F (Nat.succ n + n\u2080) \u2264 N F' (Nat.succ n)\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, add_assoc, add_comm, add_comm 1 n, \u2190 hF]\n[GOAL]\ncase h.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 I \u2022 N F (n + n\u2080) \u2264 N F' (n + 1)\ncase h.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080\n[PROOFSTEP]\nexact (Submodule.smul_mono_right hn).trans (F'.smul_le _)\n[GOAL]\ncase h.succ.a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ne : N F 0 \u2264 N F' 0\nn\u2080 : \u2115\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nn : \u2115\nhn : N F (n + n\u2080) \u2264 N F' n\n\u22a2 n + n\u2080 \u2265 n\u2080\n[PROOFSTEP]\nsimp\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n \u2227 N F' (n + n\u2080) \u2264 N F n\n[PROOFSTEP]\nobtain \u27e8n\u2081, h\u2081\u27e9 := h.exists_forall_le (le_of_eq e)\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n \u2227 N F' (n + n\u2080) \u2264 N F n\n[PROOFSTEP]\nobtain \u27e8n\u2082, h\u2082\u27e9 := h'.exists_forall_le (le_of_eq e.symm)\n[GOAL]\ncase intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\nn\u2082 : \u2115\nh\u2082 : \u2200 (n : \u2115), N F' (n + n\u2082) \u2264 N F n\n\u22a2 \u2203 n\u2080, \u2200 (n : \u2115), N F (n + n\u2080) \u2264 N F' n \u2227 N F' (n + n\u2080) \u2264 N F n\n[PROOFSTEP]\nuse max n\u2081 n\u2082\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\nn\u2082 : \u2115\nh\u2082 : \u2200 (n : \u2115), N F' (n + n\u2082) \u2264 N F n\n\u22a2 \u2200 (n : \u2115), N F (n + max n\u2081 n\u2082) \u2264 N F' n \u2227 N F' (n + max n\u2081 n\u2082) \u2264 N F n\n[PROOFSTEP]\nintro n\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\nn\u2082 : \u2115\nh\u2082 : \u2200 (n : \u2115), N F' (n + n\u2082) \u2264 N F n\nn : \u2115\n\u22a2 N F (n + max n\u2081 n\u2082) \u2264 N F' n \u2227 N F' (n + max n\u2081 n\u2082) \u2264 N F n\n[PROOFSTEP]\nrefine' \u27e8(F.antitone _).trans (h\u2081 n), (F'.antitone _).trans (h\u2082 n)\u27e9\n[GOAL]\ncase h.refine'_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\nn\u2082 : \u2115\nh\u2082 : \u2200 (n : \u2115), N F' (n + n\u2082) \u2264 N F n\nn : \u2115\n\u22a2 n + n\u2081 \u2264 n + max n\u2081 n\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.refine'_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d h : Stable F\nh' : Stable F'\ne : N F 0 = N F' 0\nn\u2081 : \u2115\nh\u2081 : \u2200 (n : \u2115), N F (n + n\u2081) \u2264 N F' n\nn\u2082 : \u2115\nh\u2082 : \u2200 (n : \u2115), N F' (n + n\u2082) \u2264 N F n\nn : \u2115\n\u22a2 n + n\u2082 \u2264 n + max n\u2081 n\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\ni : \u2115\n\u22a2 \u2191(r \u2022 f) i \u2208 N F i\n[PROOFSTEP]\nrw [Subalgebra.smul_def, PolynomialModule.smul_apply]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\ni : \u2115\n\u22a2 \u2211 x in Finset.Nat.antidiagonal i, coeff (\u2191r) x.fst \u2022 \u2191f x.snd \u2208 N F i\n[PROOFSTEP]\napply Submodule.sum_mem\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\ni : \u2115\n\u22a2 \u2200 (c : \u2115 \u00d7 \u2115), c \u2208 Finset.Nat.antidiagonal i \u2192 coeff (\u2191r) c.fst \u2022 \u2191f c.snd \u2208 N F i\n[PROOFSTEP]\nrintro \u27e8j, k\u27e9 e\n[GOAL]\ncase a.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\ni j k : \u2115\ne : (j, k) \u2208 Finset.Nat.antidiagonal i\n\u22a2 coeff \u2191r (j, k).fst \u2022 \u2191f (j, k).snd \u2208 N F i\n[PROOFSTEP]\nrw [Finset.Nat.mem_antidiagonal] at e \n[GOAL]\ncase a.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\ni j k : \u2115\ne : (j, k).fst + (j, k).snd = i\n\u22a2 coeff \u2191r (j, k).fst \u2022 \u2191f (j, k).snd \u2208 N F i\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase a.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nr : { x // x \u2208 reesAlgebra I }\nf : PolynomialModule R M\nhf :\n  f \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i},\n              add_mem' :=\n                (_ :\n                  \u2200 {a b : PolynomialModule R M},\n                    a \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192\n                      b \u2208 {f | \u2200 (i : \u2115), \u2191f i \u2208 N F i} \u2192 \u2200 (i : \u2115), \u2191a i + \u2191b i \u2208 N F i) },\n          zero_mem' := (_ : \u2200 (i : \u2115), 0 \u2208 N F i) }.toAddSubsemigroup.carrier\nj k : \u2115\n\u22a2 coeff \u2191r (j, k).fst \u2022 \u2191f (j, k).snd \u2208 N F ((j, k).fst + (j, k).snd)\n[PROOFSTEP]\nexact F.pow_smul_le j k (Submodule.smul_mem_smul (r.2 j) (hf k))\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 Filtration.submodule (F \u2293 F') = Filtration.submodule F \u2293 Filtration.submodule F'\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nx\u271d : PolynomialModule R M\n\u22a2 x\u271d \u2208 Filtration.submodule (F \u2293 F') \u2194 x\u271d \u2208 Filtration.submodule F \u2293 Filtration.submodule F'\n[PROOFSTEP]\nexact forall_and\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i)) = (Filtration.submodule F).toAddSubmonoid\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i)) \u2264 (Filtration.submodule F).toAddSubmonoid\n[PROOFSTEP]\nrw [AddSubmonoid.closure_le, Set.iUnion_subset_iff]\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 \u2200 (i : \u2115), \u2191(single R i) '' \u2191(N F i) \u2286 \u2191(Filtration.submodule F).toAddSubmonoid\n[PROOFSTEP]\nrintro i _ \u27e8m, hm, rfl\u27e9 j\n[GOAL]\ncase a.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ni : \u2115\nm : M\nhm : m \u2208 \u2191(N F i)\nj : \u2115\n\u22a2 \u2191(\u2191(single R i) m) j \u2208 N F j\n[PROOFSTEP]\nrw [single_apply]\n[GOAL]\ncase a.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\ni : \u2115\nm : M\nhm : m \u2208 \u2191(N F i)\nj : \u2115\n\u22a2 (if i = j then m else 0) \u2208 N F j\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d : Stable F\ni : \u2115\nm : M\nhm : m \u2208 \u2191(N F i)\nj : \u2115\nh : i = j\n\u22a2 m \u2208 N F j\n[PROOFSTEP]\nrwa [\u2190 h]\n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh\u271d : Stable F\ni : \u2115\nm : M\nhm : m \u2208 \u2191(N F i)\nj : \u2115\nh : \u00aci = j\n\u22a2 0 \u2208 N F j\n[PROOFSTEP]\nexact (F.N j).zero_mem\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 (Filtration.submodule F).toAddSubmonoid \u2264 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nintro f hf\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nf : PolynomialModule R M\nhf : f \u2208 (Filtration.submodule F).toAddSubmonoid\n\u22a2 f \u2208 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nrw [\u2190 f.sum_single]\n[GOAL]\ncase a\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nf : PolynomialModule R M\nhf : f \u2208 (Filtration.submodule F).toAddSubmonoid\n\u22a2 Finsupp.sum f Finsupp.single \u2208 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\napply AddSubmonoid.sum_mem _ _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nf : PolynomialModule R M\nhf : f \u2208 (Filtration.submodule F).toAddSubmonoid\n\u22a2 \u2200 (c : \u2115), c \u2208 f.support \u2192 Finsupp.single c (\u2191f c) \u2208 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nrintro c -\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nf : PolynomialModule R M\nhf : f \u2208 (Filtration.submodule F).toAddSubmonoid\nc : \u2115\n\u22a2 Finsupp.single c (\u2191f c) \u2208 AddSubmonoid.closure (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nexact AddSubmonoid.subset_closure (Set.subset_iUnion _ c <| Set.mem_image_of_mem _ (hf c))\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i)) = Filtration.submodule F\n[PROOFSTEP]\nrw [\u2190 Submodule.span_closure, submodule_closure_single, Submodule.coe_toAddSubmonoid]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Filtration.submodule F) = Filtration.submodule F\n[PROOFSTEP]\nexact Submodule.span_eq (Filtration.submodule F)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 Filtration.submodule F =\n      Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2194\n    \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nrw [\u2190 submodule_span_single, \u2190 LE.le.le_iff_eq, Submodule.span_le, Set.iUnion_subset_iff]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 (\u2200 (i : \u2115),\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))) \u2194\n    \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2264\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nswap\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2264\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nexact Submodule.span_mono (Set.iUnion\u2082_subset_iUnion _ _)\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 (\u2200 (i : \u2115),\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))) \u2194\n    \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 (\u2200 (i : \u2115),\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))) \u2192\n    \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nintro H n hn\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nrefine' (F.smul_le n).antisymm _\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 N F (n + 1) \u2264 I \u2022 N F n\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\n\u22a2 x \u2208 I \u2022 N F n\n[PROOFSTEP]\nobtain \u27e8l, hl\u27e9 := (Finsupp.mem_span_iff_total _ _ _).mp (H _ \u27e8x, hx, rfl\u27e9)\n[GOAL]\ncase mp.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n          { x // x \u2208 reesAlgebra I } Subtype.val)\n      l =\n    \u2191(single R (n + 1)) x\n\u22a2 x \u2208 I \u2022 N F n\n[PROOFSTEP]\nreplace hl := congr_arg (fun f : \u2115 \u2192\u2080 M => f (n + 1)) hl\n[GOAL]\ncase mp.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  (fun f => \u2191f (n + 1))\n      (\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n            { x // x \u2208 reesAlgebra I } Subtype.val)\n        l) =\n    (fun f => \u2191f (n + 1)) (\u2191(single R (n + 1)) x)\n\u22a2 x \u2208 I \u2022 N F n\n[PROOFSTEP]\ndsimp only at hl \n[GOAL]\ncase mp.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    \u2191(\u2191(single R (n + 1)) x) (n + 1)\n\u22a2 x \u2208 I \u2022 N F n\n[PROOFSTEP]\nerw [Finsupp.single_eq_same] at hl \n[GOAL]\ncase mp.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\n\u22a2 x \u2208 I \u2022 N F n\n[PROOFSTEP]\nrw [\u2190 hl, Finsupp.total_apply, Finsupp.sum_apply]\n[GOAL]\ncase mp.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\n\u22a2 (Finsupp.sum l fun a\u2081 b => \u2191(b \u2022 \u2191a\u2081) (n + 1)) \u2208 I \u2022 N F n\n[PROOFSTEP]\napply Submodule.sum_mem _ _\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\n\u22a2 \u2200 (c : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))),\n    c \u2208 l.support \u2192 (fun a\u2081 b => \u2191(b \u2022 \u2191a\u2081) (n + 1)) c (\u2191l c) \u2208 I \u2022 N F n\n[PROOFSTEP]\nrintro \u27e8_, _, \u27e8n', rfl\u27e9, _, \u27e8hn', rfl\u27e9, m, hm, rfl\u27e9 -\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\n\u22a2 (fun a\u2081 b => \u2191(b \u2022 \u2191a\u2081) (n + 1))\n      { val := \u2191(single R n') m,\n        property :=\n          (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }\n      (\u2191l\n        { val := \u2191(single R n') m,\n          property :=\n            (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }) \u2208\n    I \u2022 N F n\n[PROOFSTEP]\ndsimp only [Subtype.coe_mk]\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\n\u22a2 \u2191(\u2191l\n            { val := \u2191(single R n') m,\n              property :=\n                (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) } \u2022\n          \u2191(single R n') m)\n      (n + 1) \u2208\n    I \u2022 N F n\n[PROOFSTEP]\nrw [Subalgebra.smul_def, smul_single_apply, if_pos (show n' \u2264 n + 1 by linarith)]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\n\u22a2 n' \u2264 n + 1\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\n\u22a2 coeff\n        (\u2191(\u2191l\n            { val := \u2191(single R n') m,\n              property :=\n                (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }))\n        (n + 1 - n') \u2022\n      m \u2208\n    I \u2022 N F n\n[PROOFSTEP]\nhave e : n' \u2264 n := by linarith\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\n\u22a2 n' \u2264 n\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\ne : n' \u2264 n\n\u22a2 coeff\n        (\u2191(\u2191l\n            { val := \u2191(single R n') m,\n              property :=\n                (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }))\n        (n + 1 - n') \u2022\n      m \u2208\n    I \u2022 N F n\n[PROOFSTEP]\nhave := F.pow_smul_le_pow_smul (n - n') n' 1\n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\ne : n' \u2264 n\nthis : I ^ (n - n' + 1) \u2022 N F n' \u2264 I ^ 1 \u2022 N F (n - n' + n')\n\u22a2 coeff\n        (\u2191(\u2191l\n            { val := \u2191(single R n') m,\n              property :=\n                (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }))\n        (n + 1 - n') \u2022\n      m \u2208\n    I \u2022 N F n\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le e, pow_one, add_comm _ 1, \u2190 add_tsub_assoc_of_le e, add_comm] at this \n[GOAL]\ncase mk.intro.intro.intro.intro.intro.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nH :\n  \u2200 (i : \u2115),\n    \u2191(single R i) '' \u2191(N F i) \u2286\n      \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nn : \u2115\nhn : n \u2265 n\u2080\nx : M\nhx : x \u2208 N F (n + 1)\nl : \u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2192\u2080 { x // x \u2208 reesAlgebra I }\nhl :\n  \u2191(\u2191(Finsupp.total (\u2191(\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) (PolynomialModule R M)\n              { x // x \u2208 reesAlgebra I } Subtype.val)\n          l)\n      (n + 1) =\n    x\nn' : \u2115\nhn' : n' \u2264 n\u2080\nm : M\nhm : m \u2208 \u2191(N F n')\ne : n' \u2264 n\nthis : I ^ (n + 1 - n') \u2022 N F n' \u2264 I \u2022 N F n\n\u22a2 coeff\n        (\u2191(\u2191l\n            { val := \u2191(single R n') m,\n              property :=\n                (_ : \u2203 t, (t \u2208 Set.range fun i => \u22c3 (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) \u2227 \u2191(single R n') m \u2208 t) }))\n        (n + 1 - n') \u2022\n      m \u2208\n    I \u2022 N F n\n[PROOFSTEP]\nexact this (Submodule.smul_mem_smul ((l _).2 <| n + 1 - n') hm)\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nn\u2080 : \u2115\n\u22a2 (\u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)) \u2192\n    \u2200 (i : \u2115),\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nlet F' := Submodule.span (reesAlgebra I) (\u22c3 i \u2264 n\u2080, single R i '' (F.N i : Set M))\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\n\u22a2 (\u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)) \u2192\n    \u2200 (i : \u2115),\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nintro hF i\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni : \u2115\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nhave : \u2200 i \u2264 n\u2080, single R i '' (F.N i : Set M) \u2286 F' := by\n  -- Porting note: Original proof was\n        -- `fun i hi => Set.Subset.trans (Set.subset_iUnion\u2082 i hi) Submodule.subset_span`\n  intro i hi\n  refine Set.Subset.trans ?_ Submodule.subset_span\n  refine @Set.subset_iUnion\u2082 _ _ _ (fun i => fun _ => \u2191((single R i) '' ((N F i) : Set M))) i ?_\n  exact hi\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni : \u2115\n\u22a2 \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\n[PROOFSTEP]\nintro i hi\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni\u271d i : \u2115\nhi : i \u2264 n\u2080\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\n[PROOFSTEP]\nrefine Set.Subset.trans ?_ Submodule.subset_span\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni\u271d i : \u2115\nhi : i \u2264 n\u2080\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286 \u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)\n[PROOFSTEP]\nrefine @Set.subset_iUnion\u2082 _ _ _ (fun i => fun _ => \u2191((single R i) '' ((N F i) : Set M))) i ?_\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni\u271d i : \u2115\nhi : i \u2264 n\u2080\n\u22a2 i \u2264 n\u2080\n[PROOFSTEP]\nexact hi\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\ni : \u2115\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\ninduction' i with j hj\n[GOAL]\ncase mpr.zero\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\n\u22a2 \u2191(single R Nat.zero) '' \u2191(N F Nat.zero) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nexact this _ (zero_le _)\n[GOAL]\ncase mpr.succ\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n\u22a2 \u2191(single R (Nat.succ j)) '' \u2191(N F (Nat.succ j)) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nby_cases hj' : j.succ \u2264 n\u2080\n[GOAL]\ncase pos\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : Nat.succ j \u2264 n\u2080\n\u22a2 \u2191(single R (Nat.succ j)) '' \u2191(N F (Nat.succ j)) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nexact this _ hj'\n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : \u00acNat.succ j \u2264 n\u2080\n\u22a2 \u2191(single R (Nat.succ j)) '' \u2191(N F (Nat.succ j)) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nsimp only [not_le, Nat.lt_succ_iff] at hj' \n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\n\u22a2 \u2191(single R (Nat.succ j)) '' \u2191(N F (Nat.succ j)) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nrw [Nat.succ_eq_add_one, \u2190 hF _ hj']\n[GOAL]\ncase neg\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\n\u22a2 \u2191(single R (j + 1)) '' \u2191(I \u2022 N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nrintro _ \u27e8m, hm, rfl\u27e9\n[GOAL]\ncase neg.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\n\u22a2 \u2191(single R (j + 1)) m \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nrefine' Submodule.smul_induction_on hm (fun r hr m' hm' => _) (fun x y hx hy => _)\n[GOAL]\ncase neg.intro.intro.refine'_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\nr : R\nhr : r \u2208 I\nm' : M\nhm' : m' \u2208 N F j\n\u22a2 \u2191(single R (j + 1)) (r \u2022 m') \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nrw [add_comm, \u2190 monomial_smul_single]\n[GOAL]\ncase neg.intro.intro.refine'_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\nr : R\nhr : r \u2208 I\nm' : M\nhm' : m' \u2208 N F j\n\u22a2 \u2191(monomial 1) r \u2022 \u2191(single R j) m' \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nexact F'.smul_mem \u27e8_, reesAlgebra.monomial_mem.mpr (by rwa [pow_one])\u27e9 (hj <| Set.mem_image_of_mem _ hm')\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\nr : R\nhr : r \u2208 I\nm' : M\nhm' : m' \u2208 N F j\n\u22a2 r \u2208 I ^ 1\n[PROOFSTEP]\nrwa [pow_one]\n[GOAL]\ncase neg.intro.intro.refine'_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\nx y : M\nhx :\n  \u2191(single R (j + 1)) x \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhy :\n  \u2191(single R (j + 1)) y \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n\u22a2 \u2191(single R (j + 1)) (x + y) \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nrw [map_add]\n[GOAL]\ncase neg.intro.intro.refine'_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\nn\u2080 : \u2115\nF' : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M) :=\n  Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\nhF : \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\nthis : \u2200 (i : \u2115), i \u2264 n\u2080 \u2192 \u2191(single R i) '' \u2191(N F i) \u2286 \u2191F'\nj : \u2115\nhj :\n  \u2191(single R j) '' \u2191(N F j) \u2286\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhj' : n\u2080 \u2264 j\nm : M\nhm : m \u2208 \u2191(I \u2022 N F j)\nx y : M\nhx :\n  \u2191(single R (j + 1)) x \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\nhy :\n  \u2191(single R (j + 1)) y \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n\u22a2 \u2191(single R (j + 1)) x + \u2191(single R (j + 1)) y \u2208\n    \u2191(Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n[PROOFSTEP]\nexact F'.add_mem hx hy\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 Submodule.FG (Filtration.submodule F) \u2194 Stable F\n[PROOFSTEP]\nclassical\ndelta Ideal.Filtration.Stable\nsimp_rw [\u2190 F.submodule_eq_span_le_iff_stable_ge]\nconstructor\n\u00b7 rintro H\n  refine H.stablizes_of_iSup_eq \u27e8fun n\u2080 => Submodule.span _ (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), single R i '' \u2191(F.N i)), ?_\u27e9 ?_\n  \u00b7 intro n m e\n    rw [Submodule.span_le, Set.iUnion\u2082_subset_iff]\n    intro i hi\n    refine Set.Subset.trans ?_ Submodule.subset_span\n    refine @Set.subset_iUnion\u2082 _ _ _ (fun i => fun _ => \u2191((single R i) '' ((N F i) : Set M))) i ?_\n    exact hi.trans e\n  \u00b7 dsimp\n    rw [\u2190 Submodule.span_iUnion, \u2190 submodule_span_single]\n    congr 1\n    ext\n    simp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]\n    constructor\n    \u00b7 rintro \u27e8-, i, -, e\u27e9; exact \u27e8i, e\u27e9\n    \u00b7 rintro \u27e8i, e\u27e9; exact \u27e8i, i, le_refl i, e\u27e9\n\u00b7 rintro \u27e8n, hn\u27e9\n  rw [hn]\n  simp_rw [Submodule.span_iUnion\u2082, \u2190 Finset.mem_range_succ_iff, iSup_subtype']\n  apply Submodule.fg_iSup\n  rintro \u27e8i, hi\u27e9\n  obtain \u27e8s, hs\u27e9 := hF' i\n  have :\n    Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =\n      Submodule.span _ (single R i '' (F.N i : Set M)) :=\n    by rw [Finset.coe_image, \u2190 Submodule.span_span_of_tower R, \u2190 Submodule.map_span, hs]; rfl\n  rw [Subtype.coe_mk, \u2190 this]\n  exact \u27e8_, rfl\u27e9\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 Submodule.FG (Filtration.submodule F) \u2194 Stable F\n[PROOFSTEP]\ndelta Ideal.Filtration.Stable\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 Submodule.FG (Filtration.submodule F) \u2194 \u2203 n\u2080, \u2200 (n : \u2115), n \u2265 n\u2080 \u2192 I \u2022 N F n = N F (n + 1)\n[PROOFSTEP]\nsimp_rw [\u2190 F.submodule_eq_span_le_iff_stable_ge]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 Submodule.FG (Filtration.submodule F) \u2194\n    \u2203 n\u2080,\n      Filtration.submodule F =\n        Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 Submodule.FG (Filtration.submodule F) \u2192\n    \u2203 n\u2080,\n      Filtration.submodule F =\n        Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n[PROOFSTEP]\nrintro H\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 \u2203 n\u2080,\n    Filtration.submodule F =\n      Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n[PROOFSTEP]\nrefine H.stablizes_of_iSup_eq \u27e8fun n\u2080 => Submodule.span _ (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), single R i '' \u2191(F.N i)), ?_\u27e9 ?_\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 Monotone fun n\u2080 => Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nintro n m e\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nn m : \u2115\ne : n \u2264 m\n\u22a2 (fun n\u2080 => Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) n \u2264\n    (fun n\u2080 => Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) m\n[PROOFSTEP]\nrw [Submodule.span_le, Set.iUnion\u2082_subset_iff]\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nn m : \u2115\ne : n \u2264 m\n\u22a2 \u2200 (i : \u2115),\n    i \u2264 n \u2192\n      \u2191(single R i) '' \u2191(N F i) \u2286\n        \u2191((fun n\u2080 => Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) m)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nn m : \u2115\ne : n \u2264 m\ni : \u2115\nhi : i \u2264 n\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286\n    \u2191((fun n\u2080 => Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i))) m)\n[PROOFSTEP]\nrefine Set.Subset.trans ?_ Submodule.subset_span\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nn m : \u2115\ne : n \u2264 m\ni : \u2115\nhi : i \u2264 n\n\u22a2 \u2191(single R i) '' \u2191(N F i) \u2286 \u22c3 (i : \u2115) (_ : i \u2264 m), \u2191(single R i) '' \u2191(N F i)\n[PROOFSTEP]\nrefine @Set.subset_iUnion\u2082 _ _ _ (fun i => fun _ => \u2191((single R i) '' ((N F i) : Set M))) i ?_\n[GOAL]\ncase mp.refine_1\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nn m : \u2115\ne : n \u2264 m\ni : \u2115\nhi : i \u2264 n\n\u22a2 i \u2264 m\n[PROOFSTEP]\nexact hi.trans e\n[GOAL]\ncase mp.refine_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 iSup\n      \u2191{\n          toFun := fun n\u2080 =>\n            Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)),\n          monotone' :=\n            (_ :\n              \u2200 \u2983n m : \u2115\u2984,\n                n \u2264 m \u2192\n                  (fun n\u2080 =>\n                        Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n                      n \u2264\n                    (fun n\u2080 =>\n                        Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)))\n                      m) } =\n    Filtration.submodule F\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase mp.refine_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 \u2a06 (n\u2080 : \u2115), Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), \u2191(single R i) '' \u2191(N F i)) =\n    Filtration.submodule F\n[PROOFSTEP]\nrw [\u2190 Submodule.span_iUnion, \u2190 submodule_span_single]\n[GOAL]\ncase mp.refine_2\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (i_1 : \u2115) (_ : i_1 \u2264 i), \u2191(single R i_1) '' \u2191(N F i_1)) =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase mp.refine_2.e_s\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\n\u22a2 \u22c3 (i : \u2115) (i_1 : \u2115) (_ : i_1 \u2264 i), \u2191(single R i_1) '' \u2191(N F i_1) = \u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i)\n[PROOFSTEP]\next\n[GOAL]\ncase mp.refine_2.e_s.h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\n\u22a2 x\u271d \u2208 \u22c3 (i : \u2115) (i_1 : \u2115) (_ : i_1 \u2264 i), \u2191(single R i_1) '' \u2191(N F i_1) \u2194 x\u271d \u2208 \u22c3 (i : \u2115), \u2191(single R i) '' \u2191(N F i)\n[PROOFSTEP]\nsimp only [Set.mem_iUnion, Set.mem_image, SetLike.mem_coe, exists_prop]\n[GOAL]\ncase mp.refine_2.e_s.h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\n\u22a2 (\u2203 i i_1, i_1 \u2264 i \u2227 \u2203 x, x \u2208 N F i_1 \u2227 \u2191(single R i_1) x = x\u271d) \u2194 \u2203 i x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp.refine_2.e_s.h.mp\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\n\u22a2 (\u2203 i i_1, i_1 \u2264 i \u2227 \u2203 x, x \u2208 N F i_1 \u2227 \u2191(single R i_1) x = x\u271d) \u2192 \u2203 i x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d\n[PROOFSTEP]\nrintro \u27e8-, i, -, e\u27e9\n[GOAL]\ncase mp.refine_2.e_s.h.mp.intro.intro.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\ni : \u2115\ne : \u2203 x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d\n\u22a2 \u2203 i x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d\n[PROOFSTEP]\nexact \u27e8i, e\u27e9\n[GOAL]\ncase mp.refine_2.e_s.h.mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\n\u22a2 (\u2203 i x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d) \u2192 \u2203 i i_1, i_1 \u2264 i \u2227 \u2203 x, x \u2208 N F i_1 \u2227 \u2191(single R i_1) x = x\u271d\n[PROOFSTEP]\nrintro \u27e8i, e\u27e9\n[GOAL]\ncase mp.refine_2.e_s.h.mpr.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nH : Submodule.FG (Filtration.submodule F)\nx\u271d : PolynomialModule R M\ni : \u2115\ne : \u2203 x, x \u2208 N F i \u2227 \u2191(single R i) x = x\u271d\n\u22a2 \u2203 i i_1, i_1 \u2264 i \u2227 \u2203 x, x \u2208 N F i_1 \u2227 \u2191(single R i_1) x = x\u271d\n[PROOFSTEP]\nexact \u27e8i, i, le_refl i, e\u27e9\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\n\u22a2 (\u2203 n\u2080,\n      Filtration.submodule F =\n        Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n\u2080), (fun a => \u2191(single R i) a) '' \u2191(N F i))) \u2192\n    Submodule.FG (Filtration.submodule F)\n[PROOFSTEP]\nrintro \u27e8n, hn\u27e9\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n\u22a2 Submodule.FG (Filtration.submodule F)\n[PROOFSTEP]\nrw [hn]\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n\u22a2 Submodule.FG\n    (Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i)))\n[PROOFSTEP]\nsimp_rw [Submodule.span_iUnion\u2082, \u2190 Finset.mem_range_succ_iff, iSup_subtype']\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n\u22a2 Submodule.FG\n    (\u2a06 (x : { i // i \u2208 Finset.range (Nat.succ n) }),\n      Submodule.span { x // x \u2208 reesAlgebra I } ((fun a => \u2191(single R \u2191x) a) '' \u2191(N F \u2191x)))\n[PROOFSTEP]\napply Submodule.fg_iSup\n[GOAL]\ncase mpr.intro.h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\n\u22a2 \u2200 (i : { i // i \u2208 Finset.range (Nat.succ n) }),\n    Submodule.FG (Submodule.span { x // x \u2208 reesAlgebra I } ((fun a => \u2191(single R \u2191i) a) '' \u2191(N F \u2191i)))\n[PROOFSTEP]\nrintro \u27e8i, hi\u27e9\n[GOAL]\ncase mpr.intro.h.mk\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\n\u22a2 Submodule.FG\n    (Submodule.span { x // x \u2208 reesAlgebra I }\n      ((fun a => \u2191(single R \u2191{ val := i, property := hi }) a) '' \u2191(N F \u2191{ val := i, property := hi })))\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := hF' i\n[GOAL]\ncase mpr.intro.h.mk.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\ns : Finset M\nhs : Submodule.span R \u2191s = N F i\n\u22a2 Submodule.FG\n    (Submodule.span { x // x \u2208 reesAlgebra I }\n      ((fun a => \u2191(single R \u2191{ val := i, property := hi }) a) '' \u2191(N F \u2191{ val := i, property := hi })))\n[PROOFSTEP]\nhave :\n  Submodule.span (reesAlgebra I) (s.image (lsingle R i) : Set (PolynomialModule R M)) =\n    Submodule.span _ (single R i '' (F.N i : Set M)) :=\n  by rw [Finset.coe_image, \u2190 Submodule.span_span_of_tower R, \u2190 Submodule.map_span, hs]; rfl\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\ns : Finset M\nhs : Submodule.span R \u2191s = N F i\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Finset.image (\u2191(lsingle R i)) s) =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nrw [Finset.coe_image, \u2190 Submodule.span_span_of_tower R, \u2190 Submodule.map_span, hs]\n[GOAL]\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\ns : Finset M\nhs : Submodule.span R \u2191s = N F i\n\u22a2 Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Submodule.map (lsingle R i) (N F i)) =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u2191(single R i) '' \u2191(N F i))\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr.intro.h.mk.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\ns : Finset M\nhs : Submodule.span R \u2191s = N F i\nthis :\n  Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Finset.image (\u2191(lsingle R i)) s) =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u2191(single R i) '' \u2191(N F i))\n\u22a2 Submodule.FG\n    (Submodule.span { x // x \u2208 reesAlgebra I }\n      ((fun a => \u2191(single R \u2191{ val := i, property := hi }) a) '' \u2191(N F \u2191{ val := i, property := hi })))\n[PROOFSTEP]\nrw [Subtype.coe_mk, \u2190 this]\n[GOAL]\ncase mpr.intro.h.mk.intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : Filtration I M\nh : Stable F\nhF' : \u2200 (i : \u2115), Submodule.FG (N F i)\nn : \u2115\nhn :\n  Filtration.submodule F =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u22c3 (i : \u2115) (_ : i \u2264 n), (fun a => \u2191(single R i) a) '' \u2191(N F i))\ni : \u2115\nhi : i \u2208 Finset.range (Nat.succ n)\ns : Finset M\nhs : Submodule.span R \u2191s = N F i\nthis :\n  Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Finset.image (\u2191(lsingle R i)) s) =\n    Submodule.span { x // x \u2208 reesAlgebra I } (\u2191(single R i) '' \u2191(N F i))\n\u22a2 Submodule.FG (Submodule.span { x // x \u2208 reesAlgebra I } \u2191(Finset.image (\u2191(lsingle R i)) s))\n[PROOFSTEP]\nexact \u27e8_, rfl\u27e9\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\n\u22a2 Stable F'\n[PROOFSTEP]\nhave := isNoetherian_of_isNoetherianRing_of_finite R M\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Stable F'\n[PROOFSTEP]\nrw [\u2190 submodule_fg_iff_stable] at hF \u22a2\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F' i)\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F i)\n[PROOFSTEP]\nany_goals intro i; exact IsNoetherian.noetherian _\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F' i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\ni : \u2115\n\u22a2 Submodule.FG (N F' i)\n[PROOFSTEP]\nexact IsNoetherian.noetherian _\n[GOAL]\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 \u2200 (i : \u2115), Submodule.FG (N F i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase hF'\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Stable F\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\ni : \u2115\n\u22a2 Submodule.FG (N F i)\n[PROOFSTEP]\nexact IsNoetherian.noetherian _\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis : IsNoetherian R M\n\u22a2 Submodule.FG (Filtration.submodule F')\n[PROOFSTEP]\nhave := isNoetherian_of_fg_of_noetherian _ hF\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : IsNoetherian { x // x \u2208 reesAlgebra I } { x // x \u2208 Filtration.submodule F }\n\u22a2 Submodule.FG (Filtration.submodule F')\n[PROOFSTEP]\nrw [isNoetherian_submodule] at this \n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F'\u271d : Filtration I M\nh : Stable F\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nhF : Submodule.FG (Filtration.submodule F)\nF' : Filtration I M\nhf : F' \u2264 F\nthis\u271d : IsNoetherian R M\nthis : \u2200 (s : Submodule { x // x \u2208 reesAlgebra I } (PolynomialModule R M)), s \u2264 Filtration.submodule F \u2192 Submodule.FG s\n\u22a2 Submodule.FG (Filtration.submodule F')\n[PROOFSTEP]\nexact this _ (OrderHomClass.mono (submoduleInfHom M I) hf)\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2194 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nlet N := (\u2a05 i : \u2115, I ^ i \u2022 \u22a4 : Submodule R M)\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2194 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nhave hN : \u2200 k, (I.stableFiltration \u22a4 \u2293 I.trivialFiltration N).N k = N := fun k =>\n  inf_eq_right.mpr ((iInf_le _ k).trans <| le_of_eq <| by simp)\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nk : \u2115\n\u22a2 I ^ k \u2022 \u22a4 = Filtration.N (stableFiltration I \u22a4) k\n[PROOFSTEP]\nsimp\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2194 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2192 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nhave := isNoetherian_of_isNoetherianRing_of_finite R M\n[GOAL]\ncase mp\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis : IsNoetherian R M\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2192 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nobtain \u27e8r, hr\u2081, hr\u2082\u27e9 :=\n  Submodule.exists_mem_and_smul_eq_self_of_fg_of_le_smul I N (IsNoetherian.noetherian N)\n    (by\n      obtain \u27e8k, hk\u27e9 := (I.stableFiltration_stable \u22a4).inter_right (I.trivialFiltration N)\n      have := hk k (le_refl _)\n      rw [hN, hN] at this \n      exact le_of_eq this.symm)\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis : IsNoetherian R M\n\u22a2 N \u2264 I \u2022 N\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := (I.stableFiltration_stable \u22a4).inter_right (I.trivialFiltration N)\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis : IsNoetherian R M\nk : \u2115\nhk :\n  \u2200 (n : \u2115),\n    n \u2265 k \u2192\n      I \u2022 Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) n =\n        Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) (n + 1)\n\u22a2 N \u2264 I \u2022 N\n[PROOFSTEP]\nhave := hk k (le_refl _)\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis\u271d : IsNoetherian R M\nk : \u2115\nhk :\n  \u2200 (n : \u2115),\n    n \u2265 k \u2192\n      I \u2022 Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) n =\n        Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) (n + 1)\nthis :\n  I \u2022 Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k =\n    Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) (k + 1)\n\u22a2 N \u2264 I \u2022 N\n[PROOFSTEP]\nrw [hN, hN] at this \n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis\u271d : IsNoetherian R M\nk : \u2115\nhk :\n  \u2200 (n : \u2115),\n    n \u2265 k \u2192\n      I \u2022 Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) n =\n        Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) (n + 1)\nthis : I \u2022 N = N\n\u22a2 N \u2264 I \u2022 N\n[PROOFSTEP]\nexact le_of_eq this.symm\n[GOAL]\ncase mp.intro.intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis : IsNoetherian R M\nr : R\nhr\u2081 : r \u2208 I\nhr\u2082 : \u2200 (n : M), n \u2208 N \u2192 r \u2022 n = n\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2192 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp.intro.intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nthis : IsNoetherian R M\nr : R\nhr\u2081 : r \u2208 I\nhr\u2082 : \u2200 (n : M), n \u2208 N \u2192 r \u2022 n = n\nH : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\n\u22a2 \u2203 r, \u2191r \u2022 x = x\n[PROOFSTEP]\nexact \u27e8\u27e8r, hr\u2081\u27e9, hr\u2082 _ H\u27e9\n[GOAL]\ncase mpr\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\n\u22a2 (\u2203 r, \u2191r \u2022 x = x) \u2192 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\n[PROOFSTEP]\nrintro \u27e8r, eq\u27e9\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\n\u22a2 x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\n[PROOFSTEP]\nrw [Submodule.mem_iInf]\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\n\u22a2 \u2200 (i : \u2115), x \u2208 I ^ i \u2022 \u22a4\n[PROOFSTEP]\nintro i\n[GOAL]\ncase mpr.intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\ni : \u2115\n\u22a2 x \u2208 I ^ i \u2022 \u22a4\n[PROOFSTEP]\ninduction' i with i hi\n[GOAL]\ncase mpr.intro.zero\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\n\u22a2 x \u2208 I ^ Nat.zero \u2022 \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr.intro.succ\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\ni : \u2115\nhi : x \u2208 I ^ i \u2022 \u22a4\n\u22a2 x \u2208 I ^ Nat.succ i \u2022 \u22a4\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, pow_add, \u2190 smul_smul, pow_one, \u2190 eq]\n[GOAL]\ncase mpr.intro.succ\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : Module.Finite R M\nx : M\nN : Submodule R M := \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nhN : \u2200 (k : \u2115), Filtration.N (stableFiltration I \u22a4 \u2293 trivialFiltration I N) k = N\nr : { x // x \u2208 I }\neq : \u2191r \u2022 x = x\ni : \u2115\nhi : x \u2208 I ^ i \u2022 \u22a4\n\u22a2 \u2191r \u2022 x \u2208 I \u2022 I ^ i \u2022 \u22a4\n[PROOFSTEP]\nexact Submodule.smul_mem_smul r.prop hi\n[GOAL]\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\n\u22a2 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\n\u22a2 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2264 \u22a5\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\nx : M\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := (I.mem_iInf_smul_pow_eq_bot_iff x).mp hx\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\nx : M\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nr : { x // x \u2208 I }\nhr : \u2191r \u2022 x = x\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nhave := LocalRing.isUnit_one_sub_self_of_mem_nonunits _ (LocalRing.le_maximalIdeal h r.prop)\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\nx : M\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nr : { x // x \u2208 I }\nhr : \u2191r \u2022 x = x\nthis : IsUnit (1 - \u2191r)\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\napply this.smul_left_cancel.mp\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : Module.Finite R M\nh : I \u2260 \u22a4\nx : M\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4\nr : { x // x \u2208 I }\nhr : \u2191r \u2022 x = x\nthis : IsUnit (1 - \u2191r)\n\u22a2 (1 - \u2191r) \u2022 x = (1 - \u2191r) \u2022 0\n[PROOFSTEP]\nsimp [sub_smul, hr]\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : LocalRing R\nh : I \u2260 \u22a4\n\u22a2 \u2a05 (i : \u2115), I ^ i = \u22a5\n[PROOFSTEP]\nconvert I.iInf_pow_smul_eq_bot_of_localRing (M := R) h\n[GOAL]\ncase h.e'_2.h.e'_4.h\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : LocalRing R\nh : I \u2260 \u22a4\nx\u271d : \u2115\n\u22a2 I ^ x\u271d = I ^ x\u271d \u2022 \u22a4\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_2.h.e'_4.h.h\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : LocalRing R\nh : I \u2260 \u22a4\nx\u271d : \u2115\ni : R\n\u22a2 i \u2208 I ^ x\u271d \u2194 i \u2208 I ^ x\u271d \u2022 \u22a4\n[PROOFSTEP]\nrw [smul_eq_mul, \u2190 Ideal.one_eq_top, mul_one]\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\n\u22a2 \u2a05 (i : \u2115), I ^ i = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\n\u22a2 \u2a05 (i : \u2115), I ^ i \u2264 \u22a5\n[PROOFSTEP]\nintro x hx\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nby_contra hx'\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\nhx' : \u00acx \u2208 \u22a5\n\u22a2 False\n[PROOFSTEP]\nhave := Ideal.mem_iInf_smul_pow_eq_bot_iff I x\n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\nhx' : \u00acx \u2208 \u22a5\nthis : x \u2208 \u2a05 (i : \u2115), I ^ i \u2022 \u22a4 \u2194 \u2203 r, \u2191r \u2022 x = x\n\u22a2 False\n[PROOFSTEP]\nsimp_rw [smul_eq_mul, \u2190 Ideal.one_eq_top, mul_one] at this \n[GOAL]\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\nhx' : \u00acx \u2208 \u22a5\nthis : x \u2208 \u2a05 (i : \u2115), I ^ i \u2194 \u2203 r, \u2191r * x = x\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 := this.mp hx\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\nhx' : \u00acx \u2208 \u22a5\nthis : x \u2208 \u2a05 (i : \u2115), I ^ i \u2194 \u2203 r, \u2191r * x = x\nr : { x // x \u2208 I }\nhr : \u2191r * x = x\n\u22a2 False\n[PROOFSTEP]\nhave := mul_right_cancel\u2080 hx' (hr.trans (one_mul x).symm)\n[GOAL]\ncase intro\nR M : Type u\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nI : Ideal R\nF F' : Filtration I M\ninst\u271d\u00b9 : IsNoetherianRing R\ninst\u271d : IsDomain R\nh : I \u2260 \u22a4\nx : R\nhx : x \u2208 \u2a05 (i : \u2115), I ^ i\nhx' : \u00acx \u2208 \u22a5\nthis\u271d : x \u2208 \u2a05 (i : \u2115), I ^ i \u2194 \u2203 r, \u2191r * x = x\nr : { x // x \u2208 I }\nhr : \u2191r * x = x\nthis : \u2191r = 1\n\u22a2 False\n[PROOFSTEP]\nexact I.eq_top_iff_one.not.mp h (this \u25b8 r.prop)\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Filtration", "llama_tokens": 47949, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.5118136260492318}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na\u271d b : \u03b1\ns : Finset \u03b2\nhs : Finset.Nonempty s\nf : \u03b2 \u2192 \u03b1\na : \u03b1\n\u22a2 sup' s hs f \u2264 a \u2194 sSup (f '' \u2191s) \u2264 a\n[PROOFSTEP]\nsimp [csSup_le_iff (s.finite_toSet.image f).bddAbove (hs.to_set.image f)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Finset \u03b1\nhs : Finset.Nonempty s\n\u22a2 sup' s hs id = sSup \u2191s\n[PROOFSTEP]\nrw [hs.sup'_eq_cSup_image, Set.image_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 sSup \u2191s \u2208 s\n[PROOFSTEP]\nrw [h.cSup_eq_max']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns\u271d t : Set \u03b1\na b : \u03b1\ns : Finset \u03b1\nh : Finset.Nonempty s\n\u22a2 max' s h \u2208 s\n[PROOFSTEP]\nexact s.max'_mem _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\ns t : Set \u03b1\na b : \u03b1\nh : Set.Nonempty s\nhs : Set.Finite s\n\u22a2 sSup s \u2208 s\n[PROOFSTEP]\nlift s to Finset \u03b1 using hs\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLinearOrder \u03b1\nt : Set \u03b1\na b : \u03b1\ns : Finset \u03b1\nh : Set.Nonempty \u2191s\n\u22a2 sSup \u2191s \u2208 \u2191s\n[PROOFSTEP]\nexact Finset.Nonempty.cSup_mem h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\n\u22a2 sup' s H f = sSup (f '' \u2191s)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\n\u22a2 sup' s H f \u2264 sSup (f '' \u2191s)\n[PROOFSTEP]\nrefine' Finset.sup'_le _ _ fun a ha => _\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nha : a \u2208 s\n\u22a2 f a \u2264 sSup (f '' \u2191s)\n[PROOFSTEP]\nrefine' le_csSup \u27e8s.sup' H f, _\u27e9 \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nha : a \u2208 s\n\u22a2 sup' s H f \u2208 upperBounds (f '' \u2191s)\n[PROOFSTEP]\nrintro i \u27e8j, hj, rfl\u27e9\n[GOAL]\ncase a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nha : a \u2208 s\nj : \u03b1\nhj : j \u2208 \u2191s\n\u22a2 f j \u2264 sup' s H f\n[PROOFSTEP]\nexact Finset.le_sup' _ hj\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\n\u22a2 sSup (f '' \u2191s) \u2264 sup' s H f\n[PROOFSTEP]\napply csSup_le ((coe_nonempty.mpr H).image _)\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (b : \u03b2), b \u2208 f '' \u2191s \u2192 b \u2264 sup' s H f\n[PROOFSTEP]\nrintro _ \u27e8a, ha, rfl\u27e9\n[GOAL]\ncase a.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b2\ns : Finset \u03b1\nH : Finset.Nonempty s\nf : \u03b1 \u2192 \u03b2\na : \u03b1\nha : a \u2208 \u2191s\n\u22a2 f a \u2264 sup' s H f\n[PROOFSTEP]\nexact Finset.le_sup' _ ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : ConditionallyCompleteLattice \u03b1\ns : Finset \u03b1\nH : Finset.Nonempty s\n\u22a2 sup' s H id = sSup \u2191s\n[PROOFSTEP]\nrw [sup'_eq_csSup_image s H, Set.image_id]\n", "meta": {"mathlib_filename": "Mathlib.Order.ConditionallyCompleteLattice.Finset", "llama_tokens": 1637, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599563, "lm_q2_score": 0.7248702702332475, "lm_q1q2_score": 0.5116025748830353}}
{"text": "[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.65, u_1} C\ninst\u271d\u00b9 : Category.{?u.69, u_2} D\nL : C \u2964 D\nW : MorphismProperty C\nE : Type u_3\ninst\u271d : Category.{?u.118, u_3} E\nh : StrictUniversalPropertyFixedTarget L W E\u1d52\u1d56\nF : C\u1d52\u1d56 \u2964 E\nhF : MorphismProperty.IsInvertedBy (MorphismProperty.op W) F\n\u22a2 L.op \u22d9 (fun F hF => (lift h F.rightOp (_ : MorphismProperty.IsInvertedBy W F.rightOp)).leftOp) F hF = F\n[PROOFSTEP]\nconvert congr_arg Functor.leftOp (h.fac F.rightOp hF.rightOp)\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.65, u_1} C\ninst\u271d\u00b9 : Category.{?u.69, u_2} D\nL : C \u2964 D\nW : MorphismProperty C\nE : Type u_3\ninst\u271d : Category.{?u.118, u_3} E\nh : StrictUniversalPropertyFixedTarget L W E\u1d52\u1d56\nF\u2081 F\u2082 : D\u1d52\u1d56 \u2964 E\neq : L.op \u22d9 F\u2081 = L.op \u22d9 F\u2082\n\u22a2 F\u2081 = F\u2082\n[PROOFSTEP]\nsuffices F\u2081.rightOp = F\u2082.rightOp by rw [\u2190 F\u2081.rightOp_leftOp_eq, \u2190 F\u2082.rightOp_leftOp_eq, this]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.65, u_1} C\ninst\u271d\u00b9 : Category.{?u.69, u_2} D\nL : C \u2964 D\nW : MorphismProperty C\nE : Type u_3\ninst\u271d : Category.{?u.118, u_3} E\nh : StrictUniversalPropertyFixedTarget L W E\u1d52\u1d56\nF\u2081 F\u2082 : D\u1d52\u1d56 \u2964 E\neq : L.op \u22d9 F\u2081 = L.op \u22d9 F\u2082\nthis : F\u2081.rightOp = F\u2082.rightOp\n\u22a2 F\u2081 = F\u2082\n[PROOFSTEP]\nrw [\u2190 F\u2081.rightOp_leftOp_eq, \u2190 F\u2082.rightOp_leftOp_eq, this]\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.65, u_1} C\ninst\u271d\u00b9 : Category.{?u.69, u_2} D\nL : C \u2964 D\nW : MorphismProperty C\nE : Type u_3\ninst\u271d : Category.{?u.118, u_3} E\nh : StrictUniversalPropertyFixedTarget L W E\u1d52\u1d56\nF\u2081 F\u2082 : D\u1d52\u1d56 \u2964 E\neq : L.op \u22d9 F\u2081 = L.op \u22d9 F\u2082\n\u22a2 F\u2081.rightOp = F\u2082.rightOp\n[PROOFSTEP]\nhave eq' := congr_arg Functor.rightOp eq\n[GOAL]\nC : Type u_1\nD : Type u_2\ninst\u271d\u00b2 : Category.{?u.65, u_1} C\ninst\u271d\u00b9 : Category.{?u.69, u_2} D\nL : C \u2964 D\nW : MorphismProperty C\nE : Type u_3\ninst\u271d : Category.{?u.118, u_3} E\nh : StrictUniversalPropertyFixedTarget L W E\u1d52\u1d56\nF\u2081 F\u2082 : D\u1d52\u1d56 \u2964 E\neq : L.op \u22d9 F\u2081 = L.op \u22d9 F\u2082\neq' : (L.op \u22d9 F\u2081).rightOp = (L.op \u22d9 F\u2082).rightOp\n\u22a2 F\u2081.rightOp = F\u2082.rightOp\n[PROOFSTEP]\nexact h.uniq _ _ eq'\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Localization.Opposite", "llama_tokens": 1099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085909370422, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.5115145816006043}}
{"text": "[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx y : R\nhx : x \u2208 {x | \u2191v x \u2264 1}\nhy : y \u2208 {x | \u2191v x \u2264 1}\n\u22a2 x * y \u2208 {x | \u2191v x \u2264 1}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, _root_.map_mul, mul_le_one' hx hy]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\n\u22a2 0 \u2208\n    {\n          toSubsemigroup :=\n            { carrier := {x | \u2191v x \u2264 1},\n              mul_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n          one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, _root_.map_zero, zero_le']\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx : R\nhx :\n  x \u2208\n    {\n            toSubmonoid :=\n              {\n                toSubsemigroup :=\n                  { carrier := {x | \u2191v x \u2264 1},\n                    mul_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                one_mem' := (_ : \u2191v 1 \u2264 1) },\n            add_mem' :=\n              (_ :\n                \u2200 {x y : R},\n                  x \u2208\n                      {\n                            toSubsemigroup :=\n                              { carrier := {x | \u2191v x \u2264 1},\n                                mul_mem' :=\n                                  (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                            one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                    y \u2208\n                        {\n                              toSubsemigroup :=\n                                { carrier := {x | \u2191v x \u2264 1},\n                                  mul_mem' :=\n                                    (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                              one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                      \u2191v (x + y) \u2264 1),\n            zero_mem' := (_ : \u2191v 0 \u2264 1) }.toSubmonoid.toSubsemigroup.carrier\n\u22a2 -x \u2208\n    {\n            toSubmonoid :=\n              {\n                toSubsemigroup :=\n                  { carrier := {x | \u2191v x \u2264 1},\n                    mul_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                one_mem' := (_ : \u2191v 1 \u2264 1) },\n            add_mem' :=\n              (_ :\n                \u2200 {x y : R},\n                  x \u2208\n                      {\n                            toSubsemigroup :=\n                              { carrier := {x | \u2191v x \u2264 1},\n                                mul_mem' :=\n                                  (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                            one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                    y \u2208\n                        {\n                              toSubsemigroup :=\n                                { carrier := {x | \u2191v x \u2264 1},\n                                  mul_mem' :=\n                                    (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                              one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                      \u2191v (x + y) \u2264 1),\n            zero_mem' := (_ : \u2191v 0 \u2264 1) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq] at hx \n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nx : R\nhx : \u2191v x \u2264 1\n\u22a2 -x \u2208\n    {\n            toSubmonoid :=\n              {\n                toSubsemigroup :=\n                  { carrier := {x | \u2191v x \u2264 1},\n                    mul_mem' := (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                one_mem' := (_ : \u2191v 1 \u2264 1) },\n            add_mem' :=\n              (_ :\n                \u2200 {x y : R},\n                  x \u2208\n                      {\n                            toSubsemigroup :=\n                              { carrier := {x | \u2191v x \u2264 1},\n                                mul_mem' :=\n                                  (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                            one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                    y \u2208\n                        {\n                              toSubsemigroup :=\n                                { carrier := {x | \u2191v x \u2264 1},\n                                  mul_mem' :=\n                                    (_ : \u2200 {x y : R}, x \u2208 {x | \u2191v x \u2264 1} \u2192 y \u2208 {x | \u2191v x \u2264 1} \u2192 \u2191v (x * y) \u2264 1) },\n                              one_mem' := (_ : \u2191v 1 \u2264 1) }.toSubsemigroup.carrier \u2192\n                      \u2191v (x + y) \u2264 1),\n            zero_mem' := (_ : \u2191v 0 \u2264 1) }.toSubmonoid.toSubsemigroup.carrier\n[PROOFSTEP]\nsimpa only [Set.mem_setOf_eq, map_neg]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx : O\nhx : IsUnit x\nu : O\u02e3\nhu : \u2191u = x\n\u22a2 1 \u2264 \u2191v (\u2191(algebraMap O R) x)\n[PROOFSTEP]\nrw [\u2190 v.map_one, \u2190 (algebraMap O R).map_one, \u2190 u.mul_inv, \u2190 mul_one (v (algebraMap O R x)), hu,\n  (algebraMap O R).map_mul, v.map_mul]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx : O\nhx : IsUnit x\nu : O\u02e3\nhu : \u2191u = x\n\u22a2 \u2191v (\u2191(algebraMap O R) x) * \u2191v (\u2191(algebraMap O R) \u2191u\u207b\u00b9) \u2264 \u2191v (\u2191(algebraMap O R) x) * 1\n[PROOFSTEP]\nexact mul_le_mul_left' (hv.2 (u\u207b\u00b9 : Units O)) _\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx : O\nhx : IsUnit (\u2191(algebraMap O R) x)\nhvx : \u2191v (\u2191(algebraMap O R) x) = 1\nu : ((fun x => R) x)\u02e3\nhu : \u2191u = \u2191(algebraMap O R) x\nh1 : \u2191v \u2191u \u2264 1\n\u22a2 \u2191v \u2191u\u207b\u00b9 \u2264 1\n[PROOFSTEP]\nrw [\u2190 one_mul (v _), \u2190 hvx, \u2190 v.map_mul, \u2190 hu, u.mul_inv, hu, hvx, v.map_one]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx : O\nhx : IsUnit (\u2191(algebraMap O R) x)\nhvx : \u2191v (\u2191(algebraMap O R) x) = 1\nu : ((fun x => R) x)\u02e3\nhu : \u2191u = \u2191(algebraMap O R) x\nh1 : \u2191v \u2191u \u2264 1\nh2 : \u2191v \u2191u\u207b\u00b9 \u2264 1\nr1 : O\nhr1 : \u2191(algebraMap O R) r1 = \u2191u\nr2 : O\nhr2 : \u2191(algebraMap O R) r2 = \u2191u\u207b\u00b9\n\u22a2 \u2191(algebraMap O R) (r1 * r2) = \u2191(algebraMap O R) 1\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_one, hr1, hr2, Units.mul_inv]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx : O\nhx : IsUnit (\u2191(algebraMap O R) x)\nhvx : \u2191v (\u2191(algebraMap O R) x) = 1\nu : ((fun x => R) x)\u02e3\nhu : \u2191u = \u2191(algebraMap O R) x\nh1 : \u2191v \u2191u \u2264 1\nh2 : \u2191v \u2191u\u207b\u00b9 \u2264 1\nr1 : O\nhr1 : \u2191(algebraMap O R) r1 = \u2191u\nr2 : O\nhr2 : \u2191(algebraMap O R) r2 = \u2191u\u207b\u00b9\n\u22a2 \u2191(algebraMap O R) (r2 * r1) = \u2191(algebraMap O R) 1\n[PROOFSTEP]\nrw [RingHom.map_mul, RingHom.map_one, hr1, hr2, Units.inv_mul]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx y : O\nh : x \u2223 y\n\u22a2 \u2191v (\u2191(algebraMap O R) y) \u2264 \u2191v (\u2191(algebraMap O R) x)\n[PROOFSTEP]\nlet \u27e8z, hz\u27e9 := h\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx y : O\nh : x \u2223 y\nz : O\nhz : y = x * z\n\u22a2 \u2191v (\u2191(algebraMap O R) y) \u2264 \u2191v (\u2191(algebraMap O R) x)\n[PROOFSTEP]\nrw [\u2190 mul_one (v (algebraMap O R x)), hz, RingHom.map_mul, v.map_mul]\n[GOAL]\nR : Type u\n\u0393\u2080 : Type v\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\nO : Type w\ninst\u271d\u00b3 : CommRing O\ninst\u271d\u00b2 : Algebra O R\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O R\nhv : Integers v O\nx y : O\nh : x \u2223 y\nz : O\nhz : y = x * z\n\u22a2 \u2191v (\u2191(algebraMap O R) x) * \u2191v (\u2191(algebraMap O R) z) \u2264 \u2191v (\u2191(algebraMap O R) x) * 1\n[PROOFSTEP]\nexact mul_le_mul_left' (hv.2 z) _\n[GOAL]\nF : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation F \u0393\u2080\nO : Type w\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O F\nhv : Integers v O\nx y : O\nh : \u2191v (\u2191(algebraMap O F) x) \u2264 \u2191v (\u2191(algebraMap O F) y)\nhy : \u2191(algebraMap O F) y \u2260 0\n\u22a2 \u2191v ((\u2191(algebraMap O F) y)\u207b\u00b9 * \u2191(algebraMap O F) x) \u2264 1\n[PROOFSTEP]\nrw [\u2190 v.map_one, \u2190 inv_mul_cancel hy, v.map_mul, v.map_mul]\n[GOAL]\nF : Type u\n\u0393\u2080 : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation F \u0393\u2080\nO : Type w\ninst\u271d\u00b9 : CommRing O\ninst\u271d : Algebra O F\nhv : Integers v O\nx y : O\nh : \u2191v (\u2191(algebraMap O F) x) \u2264 \u2191v (\u2191(algebraMap O F) y)\nhy : \u2191(algebraMap O F) y \u2260 0\n\u22a2 \u2191v (\u2191(algebraMap O F) y)\u207b\u00b9 * \u2191v (\u2191(algebraMap O F) x) \u2264 \u2191v (\u2191(algebraMap O F) y)\u207b\u00b9 * \u2191v (\u2191(algebraMap O F) y)\n[PROOFSTEP]\nexact mul_le_mul_left' h _\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Valuation.Integers", "llama_tokens": 4331, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8519528170040853, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.5113321634047916}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\n\u22a2 trace R S = 0\n[PROOFSTEP]\next s\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\ns : S\n\u22a2 \u2191(trace R S) s = \u21910 s\n[PROOFSTEP]\nsimp [trace_apply, LinearMap.trace, h]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : CommRing T\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9 : Fintype \u03b9\nb\u271d : Basis \u03b9 R S\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\ns : S\n\u22a2 \u2191(trace R S) s = Matrix.trace (\u2191(leftMulMatrix b) s)\n[PROOFSTEP]\nrw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, \u2190 toMatrix_lmul_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : CommRing T\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9 : Fintype \u03b9\nb\u271d : Basis \u03b9 R S\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\ns : S\n\u22a2 Matrix.trace (\u2191(toMatrix b b) (\u2191(lmul R S) s)) = Matrix.trace (\u2191(toMatrix b b) (mulLeft R s))\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\n\u22a2 \u2191(trace R S) (\u2191(algebraMap R S) x) = Fintype.card \u03b9 \u2022 x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 \u2191(trace R S) (\u2191(algebraMap R S) x) = Fintype.card \u03b9 \u2022 x\n[PROOFSTEP]\nrw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 \u2211 i : \u03b9, Matrix.diag (\u2191(toMatrix b b) (\u2191(lmul R S) (\u2191(algebraMap R S) x))) i = Fintype.card \u03b9 \u2022 x\n[PROOFSTEP]\nconvert Finset.sum_const x\n[GOAL]\ncase h.e'_2.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\nx\u271d : \u03b9\na\u271d : x\u271d \u2208 Finset.univ\n\u22a2 Matrix.diag (\u2191(toMatrix b b) (\u2191(lmul R S) (\u2191(algebraMap R S) x))) x\u271d = x\n[PROOFSTEP]\nsimp only [AlgHom.commutes, toMatrix_algebraMap, diag_apply, Matrix.scalar_apply_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : K\n\u22a2 \u2191(trace K L) (\u2191(algebraMap K L) x) = finrank K L \u2022 x\n[PROOFSTEP]\nby_cases H : \u2203 s : Finset L, Nonempty (Basis s K L)\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : K\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\n\u22a2 \u2191(trace K L) (\u2191(algebraMap K L) x) = finrank K L \u2022 x\n[PROOFSTEP]\nrw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : K\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\n\u22a2 \u2191(trace K L) (\u2191(algebraMap K L) x) = finrank K L \u2022 x\n[PROOFSTEP]\nsimp [trace_eq_zero_of_not_exists_basis K H, finrank_eq_zero_of_not_exists_basis_finset H]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\n\u22a2 \u2191(trace R S) (\u2191(trace S T) x) = \u2191(trace R T) x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis : DecidableEq \u03b9\n\u22a2 \u2191(trace R S) (\u2191(trace S T) x) = \u2191(trace R T) x\n[PROOFSTEP]\nhaveI := Classical.decEq \u03ba\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\n\u22a2 \u2191(trace R S) (\u2191(trace S T) x) = \u2191(trace R T) x\n[PROOFSTEP]\ncases nonempty_fintype \u03b9\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d : Fintype \u03b9\n\u22a2 \u2191(trace R S) (\u2191(trace S T) x) = \u2191(trace R T) x\n[PROOFSTEP]\ncases nonempty_fintype \u03ba\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d\u00b9 : Fintype \u03b9\nval\u271d : Fintype \u03ba\n\u22a2 \u2191(trace R S) (\u2191(trace S T) x) = \u2191(trace R T) x\n[PROOFSTEP]\nrw [trace_eq_matrix_trace (b.smul c), trace_eq_matrix_trace b, trace_eq_matrix_trace c, Matrix.trace, Matrix.trace,\n  Matrix.trace, \u2190 Finset.univ_product_univ, Finset.sum_product]\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d\u00b9 : Fintype \u03b9\nval\u271d : Fintype \u03ba\n\u22a2 \u2211 i : \u03b9, Matrix.diag (\u2191(leftMulMatrix b) (\u2211 i : \u03ba, Matrix.diag (\u2191(leftMulMatrix c) x) i)) i =\n    \u2211 x_1 : \u03b9, \u2211 y : \u03ba, Matrix.diag (\u2191(leftMulMatrix (Basis.smul b c)) x) (x_1, y)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i _ => _\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d\u00b9 : Fintype \u03b9\nval\u271d : Fintype \u03ba\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Matrix.diag (\u2191(leftMulMatrix b) (\u2211 i : \u03ba, Matrix.diag (\u2191(leftMulMatrix c) x) i)) i =\n    \u2211 y : \u03ba, Matrix.diag (\u2191(leftMulMatrix (Basis.smul b c)) x) (i, y)\n[PROOFSTEP]\nsimp only [AlgHom.map_sum, smul_leftMulMatrix, Finset.sum_apply, Matrix.diag]\n  -- Porting note: the `rw` was inside `simp only`, but it doesn't work anymore.\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d\u00b9 : Fintype \u03b9\nval\u271d : Fintype \u03ba\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Finset.sum Finset.univ (fun x_1 => \u2191(leftMulMatrix b) (\u2191(leftMulMatrix c) x x_1 x_1)) i i =\n    \u2211 x_1 : \u03ba, \u2191(leftMulMatrix b) (\u2191(leftMulMatrix c) x x_1 x_1) i i\n[PROOFSTEP]\nrw [Finset.sum_apply i (Finset.univ : Finset \u03ba) fun y => leftMulMatrix b (leftMulMatrix c x y y)]\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Finite \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx : T\nthis\u271d : DecidableEq \u03b9\nthis : DecidableEq \u03ba\nval\u271d\u00b9 : Fintype \u03b9\nval\u271d : Fintype \u03ba\ni : \u03b9\nx\u271d : i \u2208 Finset.univ\n\u22a2 Finset.sum Finset.univ (fun c_1 => \u2191(leftMulMatrix b) (\u2191(leftMulMatrix c) x c_1 c_1) i) i =\n    \u2211 x_1 : \u03ba, \u2191(leftMulMatrix b) (\u2191(leftMulMatrix c) x x_1 x_1) i i\n[PROOFSTEP]\napply Finset.sum_apply\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\n\u22a2 comp (trace R S) (\u2191R (trace S T)) = trace R T\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9\u271d \u03ba\u271d : Type w\ninst\u271d\u2074 : Fintype \u03b9\u271d\nb\u271d : Basis \u03b9\u271d R S\ninst\u271d\u00b3 : Algebra S T\ninst\u271d\u00b2 : IsScalarTower R S T\n\u03b9 : Type u_6\n\u03ba : Type u_7\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : Fintype \u03ba\nb : Basis \u03b9 R S\nc : Basis \u03ba S T\nx\u271d : T\n\u22a2 \u2191(comp (trace R S) (\u2191R (trace S T))) x\u271d = \u2191(trace R T) x\u271d\n[PROOFSTEP]\nrw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : CommRing T\ninst\u271d\u00b9\u2070 : Algebra R S\ninst\u271d\u2079 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2075 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u2074 : Algebra K T\ninst\u271d\u00b3 : Algebra L T\ninst\u271d\u00b2 : IsScalarTower K L T\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : FiniteDimensional L T\n\u22a2 comp (trace K L) (\u2191K (trace L T)) = trace K T\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : CommRing S\ninst\u271d\u00b9\u00b9 : CommRing T\ninst\u271d\u00b9\u2070 : Algebra R S\ninst\u271d\u2079 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2075 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u2074 : Algebra K T\ninst\u271d\u00b3 : Algebra L T\ninst\u271d\u00b2 : IsScalarTower K L T\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : FiniteDimensional L T\nx\u271d : T\n\u22a2 \u2191(comp (trace K L) (\u2191K (trace L T))) x\u271d = \u2191(trace K T) x\u271d\n[PROOFSTEP]\nrw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\nx : S \u00d7 T\n\u22a2 \u2191(trace R (S \u00d7 T)) x = \u2191(trace R S) x.fst + \u2191(trace R T) x.snd\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\nx : S \u00d7 T\n\u271d : Nontrivial R\n\u22a2 \u2191(trace R (S \u00d7 T)) x = \u2191(trace R S) x.fst + \u2191(trace R T) x.snd\n[PROOFSTEP]\nlet f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\nx : S \u00d7 T\n\u271d : Nontrivial R\nf : S \u00d7 T \u2192\u2097[R] Module.End R S \u00d7 Module.End R T :=\n  prodMap (AlgHom.toLinearMap (lmul R S)) (AlgHom.toLinearMap (lmul R T))\n\u22a2 \u2191(trace R (S \u00d7 T)) x = \u2191(trace R S) x.fst + \u2191(trace R T) x.snd\n[PROOFSTEP]\nhave : (lmul R (S \u00d7 T)).toLinearMap = (prodMapLinear R S T S T R).comp f := LinearMap.ext\u2082 Prod.mul_def\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\nx : S \u00d7 T\n\u271d : Nontrivial R\nf : S \u00d7 T \u2192\u2097[R] Module.End R S \u00d7 Module.End R T :=\n  prodMap (AlgHom.toLinearMap (lmul R S)) (AlgHom.toLinearMap (lmul R T))\nthis : AlgHom.toLinearMap (lmul R (S \u00d7 T)) = comp (prodMapLinear R S T S T R) f\n\u22a2 \u2191(trace R (S \u00d7 T)) x = \u2191(trace R S) x.fst + \u2191(trace R T) x.snd\n[PROOFSTEP]\nsimp_rw [trace, this]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\nx : S \u00d7 T\n\u271d : Nontrivial R\nf : S \u00d7 T \u2192\u2097[R] Module.End R S \u00d7 Module.End R T :=\n  prodMap (AlgHom.toLinearMap (lmul R S)) (AlgHom.toLinearMap (lmul R T))\nthis : AlgHom.toLinearMap (lmul R (S \u00d7 T)) = comp (prodMapLinear R S T S T R) f\n\u22a2 \u2191(comp (LinearMap.trace R (S \u00d7 T))\n          (comp (prodMapLinear R S T S T R) (prodMap (AlgHom.toLinearMap (lmul R S)) (AlgHom.toLinearMap (lmul R T)))))\n      x =\n    \u2191(comp (LinearMap.trace R S) (AlgHom.toLinearMap (lmul R S))) x.fst +\n      \u2191(comp (LinearMap.trace R T) (AlgHom.toLinearMap (lmul R T))) x.snd\n[PROOFSTEP]\nexact trace_prodMap' _ _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d\u00b3 : Module.Free R S\ninst\u271d\u00b2 : Module.Free R T\ninst\u271d\u00b9 : Module.Finite R S\ninst\u271d : Module.Finite R T\np : S \u00d7 T\n\u22a2 \u2191(trace R (S \u00d7 T)) p = \u2191(coprod (trace R S) (trace R T)) p\n[PROOFSTEP]\nrw [coprod_apply, trace_prod_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : CommRing S\ninst\u271d\u2077 : CommRing T\ninst\u271d\u2076 : Algebra R S\ninst\u271d\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9 : Fintype \u03b9\nb : Basis \u03b9 R S\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 \u2191(BilinForm.toMatrix b) (traceForm R S) i j = \u2191(trace R S) (\u2191b i * \u2191b j)\n[PROOFSTEP]\nrw [BilinForm.toMatrix_apply, traceForm_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nh : PowerBasis R S\n\u22a2 \u2191(BilinForm.toMatrix h.basis) (traceForm R S) = \u2191of fun i j => \u2191(trace R S) (h.gen ^ (\u2191i + \u2191j))\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing S\ninst\u271d\u2076 : CommRing T\ninst\u271d\u2075 : Algebra R S\ninst\u271d\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Field L\ninst\u271d\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nh : PowerBasis R S\ni\u271d x\u271d : Fin h.dim\n\u22a2 \u2191(BilinForm.toMatrix h.basis) (traceForm R S) i\u271d x\u271d = \u2191of (fun i j => \u2191(trace R S) (h.gen ^ (\u2191i + \u2191j))) i\u271d x\u271d\n[PROOFSTEP]\nrw [traceForm_toMatrix, of_apply, pow_add, h.basis_eq_pow, h.basis_eq_pow]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\n\u22a2 \u2191(Algebra.trace K S) pb.gen = -nextCoeff (minpoly K pb.gen)\n[PROOFSTEP]\nhave d_pos : 0 < pb.dim := PowerBasis.dim_pos pb\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\n\u22a2 \u2191(Algebra.trace K S) pb.gen = -nextCoeff (minpoly K pb.gen)\n[PROOFSTEP]\nhave d_pos' : 0 < (minpoly K pb.gen).natDegree := by simpa\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\n\u22a2 0 < natDegree (minpoly K pb.gen)\n[PROOFSTEP]\nsimpa\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\nd_pos' : 0 < natDegree (minpoly K pb.gen)\n\u22a2 \u2191(Algebra.trace K S) pb.gen = -nextCoeff (minpoly K pb.gen)\n[PROOFSTEP]\nhaveI : Nonempty (Fin pb.dim) := \u27e8\u27e80, d_pos\u27e9\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\nd_pos' : 0 < natDegree (minpoly K pb.gen)\nthis : Nonempty (Fin pb.dim)\n\u22a2 \u2191(Algebra.trace K S) pb.gen = -nextCoeff (minpoly K pb.gen)\n[PROOFSTEP]\nrw [trace_eq_matrix_trace pb.basis, trace_eq_neg_charpoly_coeff, charpoly_leftMulMatrix, \u2190 pb.natDegree_minpoly,\n  Fintype.card_fin, \u2190 nextCoeff_of_pos_natDegree _ d_pos']\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : Nontrivial S\npb : PowerBasis K S\nhf : Splits (algebraMap K F) (minpoly K pb.gen)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K S) pb.gen) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K pb.gen)))\n[PROOFSTEP]\nrw [PowerBasis.trace_gen_eq_nextCoeff_minpoly, RingHom.map_neg, \u2190 nextCoeff_map (algebraMap K F).injective,\n  sum_roots_eq_nextCoeff_of_monic_of_split ((minpoly.monic (PowerBasis.isIntegral_gen _)).map _)\n    ((splits_id_iff_splits _).2 hf),\n  neg_neg]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x) = 0\n[PROOFSTEP]\nrw [trace_eq_zero_of_not_exists_basis, LinearMap.zero_apply]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u00ac\u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef })\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhx : \u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef })\n\u22a2 IsIntegral K x\n[PROOFSTEP]\nobtain \u27e8s, \u27e8b\u27e9\u27e9 := hx\n[GOAL]\ncase h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 IsIntegral K x\n[PROOFSTEP]\nrefine' isIntegral_of_mem_of_FG K\u27eex\u27ef.toSubalgebra _ x _\n[GOAL]\ncase h.intro.intro.refine'_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 Submodule.FG (\u2191Subalgebra.toSubmodule K\u27eex\u27ef.toSubalgebra)\n[PROOFSTEP]\nexact (Submodule.fg_iff_finiteDimensional _).mpr (FiniteDimensional.of_fintype_basis b)\n[GOAL]\ncase h.intro.intro.refine'_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 x \u2208 K\u27eex\u27ef.toSubalgebra\n[PROOFSTEP]\nexact subset_adjoin K _ (Set.mem_singleton x)\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nhave injKxL := (algebraMap K\u27eex\u27ef L).injective\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nby_cases hx : IsIntegral K x\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nsimp [minpoly.eq_zero hx, trace_gen_eq_zero hx]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nhave hx' : IsIntegral K (AdjoinSimple.gen K x) := by\n  rwa [\u2190 isIntegral_algebraMap_iff injKxL, AdjoinSimple.algebraMap_gen]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 IsIntegral K (gen K x)\n[PROOFSTEP]\nrwa [\u2190 isIntegral_algebraMap_iff injKxL, AdjoinSimple.algebraMap_gen]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (gen K x)) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nrw [\u2190 adjoin.powerBasis_gen hx, (adjoin.powerBasis hx).trace_gen_eq_sum_roots]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K (adjoin.powerBasis hx).gen))) =\n    Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nrw [adjoin.powerBasis_gen hx, minpoly.eq_of_algebraMap_eq injKxL hx']\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K (adjoin.powerBasis hx).gen)\n[PROOFSTEP]\nrw [adjoin.powerBasis_gen hx, minpoly.eq_of_algebraMap_eq injKxL hx']\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 x = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (gen K x)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 x = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (gen K x)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.547142)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.547142)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 ?m.547142 = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (gen K x)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 ?m.547142 = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (gen K x)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 L\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 L\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.547142)\n[PROOFSTEP]\nexact hf\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : CommRing S\ninst\u271d\u2079 : CommRing T\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b3 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K S\ninst\u271d : Algebra K F\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (gen K x)\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n\u22a2 \u2191(Algebra.trace K L) x =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x)\n[PROOFSTEP]\nconv =>\n  lhs\n  rw [\u2190 IntermediateField.AdjoinSimple.algebraMap_gen K x]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n| \u2191(Algebra.trace K L) x =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x)\n[PROOFSTEP]\n  lhs\n  rw [\u2190 IntermediateField.AdjoinSimple.algebraMap_gen K x]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n| \u2191(Algebra.trace K L) x =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x)\n[PROOFSTEP]\n  lhs\n  rw [\u2190 IntermediateField.AdjoinSimple.algebraMap_gen K x]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n| \u2191(Algebra.trace K L) x =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x)\n[PROOFSTEP]\nlhs\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n| \u2191(Algebra.trace K L) x\n[PROOFSTEP]\nrw [\u2190 IntermediateField.AdjoinSimple.algebraMap_gen K x]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\n\u22a2 \u2191(Algebra.trace K L) (\u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)) =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x)\n[PROOFSTEP]\nrw [\u2190 @trace_trace _ _ K K\u27eex\u27ef _ _ _ _ _ _ _ _ _, trace_algebraMap, LinearMap.map_smul_of_tower]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\nhF : Splits (algebraMap K F) (minpoly K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(Algebra.trace K L) x) =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022\n      Multiset.sum (roots (Polynomial.map (algebraMap K ((fun x => F) (\u2191(Algebra.trace K L) x))) (minpoly K x)))\n[PROOFSTEP]\nrw [trace_eq_trace_adjoin K x, Algebra.smul_def, RingHom.map_mul, \u2190 Algebra.smul_def,\n  IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots _ hF]\n  -- Porting note: last `simp` was `IsScalarTower.algebraMap_smul` inside the `rw`.\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : CommRing T\ninst\u271d\u2079 : Algebra R S\ninst\u271d\u2078 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2074 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : Algebra K F\ninst\u271d : FiniteDimensional K L\nx : L\nhF : Splits (algebraMap K F) (minpoly K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(algebraMap \u2115 ((fun x => K) (AdjoinSimple.gen K x))) (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L)) *\n      Multiset.sum (roots (Polynomial.map (algebraMap K F) (minpoly K x))) =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022\n      Multiset.sum\n        (roots\n          (Polynomial.map\n            (algebraMap K\n              ((fun x => F)\n                (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 \u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (AdjoinSimple.gen K x))))\n            (minpoly K x)))\n[PROOFSTEP]\nsimp only [eq_natCast, map_natCast, nsmul_eq_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\n\u22a2 IsIntegral R (\u2191(trace L F) x)\n[PROOFSTEP]\nhave hx' : IsIntegral L x := isIntegral_of_isScalarTower hx\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n\u22a2 IsIntegral R (\u2191(trace L F) x)\n[PROOFSTEP]\nrw [\u2190 isIntegral_algebraMap_iff (algebraMap L (AlgebraicClosure F)).injective, trace_eq_sum_roots]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n\u22a2 IsIntegral R\n    (finrank { x_1 // x_1 \u2208 L\u27eex\u27ef } F \u2022\n      Multiset.sum\n        (roots (Polynomial.map (algebraMap L ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) (minpoly L x))))\n[PROOFSTEP]\nrefine' (IsIntegral.multiset_sum _).nsmul _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n\u22a2 \u2200 (x_1 : (fun x => AlgebraicClosure F) (\u2191(trace L F) x)),\n    x_1 \u2208 roots (Polynomial.map (algebraMap L ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) (minpoly L x)) \u2192\n      IsIntegral R x_1\n[PROOFSTEP]\nintro y hy\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\ny : (fun x => AlgebraicClosure F) (\u2191(trace L F) x)\nhy : y \u2208 roots (Polynomial.map (algebraMap L ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) (minpoly L x))\n\u22a2 IsIntegral R y\n[PROOFSTEP]\nrw [mem_roots_map (minpoly.ne_zero hx')] at hy \n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\ny : (fun x => AlgebraicClosure F) (\u2191(trace L F) x)\nhy : eval\u2082 (algebraMap L ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) y (minpoly L x) = 0\n\u22a2 IsIntegral R y\n[PROOFSTEP]\nuse minpoly R x, minpoly.monic hx\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\ny : (fun x => AlgebraicClosure F) (\u2191(trace L F) x)\nhy : eval\u2082 (algebraMap L ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) y (minpoly L x) = 0\n\u22a2 eval\u2082 (algebraMap R ((fun x => AlgebraicClosure F) (\u2191(trace L F) x))) y (minpoly R x) = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def] at hy \u22a2\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\ny : (fun x => AlgebraicClosure F) (\u2191(trace L F) x)\nhy : \u2191(aeval y) (minpoly L x) = 0\n\u22a2 \u2191(aeval y) (minpoly R x) = 0\n[PROOFSTEP]\nexact minpoly.aeval_of_isScalarTower R x y hy\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\ninst\u271d : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n\u22a2 Splits (algebraMap L (AlgebraicClosure F)) (minpoly L x)\n[PROOFSTEP]\napply IsAlgClosed.splits_codomain\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) pb.gen) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI := Classical.decEq E\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis : DecidableEq E := Classical.decEq E\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) pb.gen) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI : Fintype (L \u2192\u2090[K] E) := (PowerBasis.AlgHom.fintype pb)\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) pb.gen) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nrw [pb.trace_gen_eq_sum_roots hE, Fintype.sum_equiv pb.liftEquiv', Finset.sum_mem_multiset, Finset.sum_eq_multiset_sum,\n  Multiset.toFinset_val, Multiset.dedup_eq_self.mpr _, Multiset.map_id]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 Multiset.Nodup (roots (Polynomial.map (algebraMap K E) (minpoly K pb.gen)))\n[PROOFSTEP]\nexact\n  nodup_roots\n    ((separable_map _).mpr hfx)\n      -- Porting note: the following goal does not exist in mathlib3.\n[GOAL]\ncase f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 { x // x \u2208 roots (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) } \u2192 (fun x => E) pb.gen\n[PROOFSTEP]\nexact (fun x => x.1)\n[GOAL]\ncase hfg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 \u2200 (x : { x // x \u2208 roots (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) }), \u2191x = _root_.id \u2191x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hfg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nx : { x // x \u2208 roots (Polynomial.map (algebraMap K E) (minpoly K pb.gen)) }\n\u22a2 \u2191x = _root_.id \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 \u2200 (x : L \u2192\u2090[K] E), \u2191x pb.gen = \u2191(\u2191(PowerBasis.liftEquiv' pb) x)\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2077 : CommRing R\ninst\u271d\u00b9\u2076 : CommRing S\ninst\u271d\u00b9\u2075 : CommRing T\ninst\u271d\u00b9\u2074 : Algebra R S\ninst\u271d\u00b9\u00b3 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b2 : Field K\ninst\u271d\u00b9\u00b9 : Field L\ninst\u271d\u00b9\u2070 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2079 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2078 : Field F\ninst\u271d\u2077 : Algebra R L\ninst\u271d\u2076 : Algebra L F\ninst\u271d\u2075 : Algebra R F\ninst\u271d\u2074 : IsScalarTower R L F\ninst\u271d\u00b3 : Algebra K F\ninst\u271d\u00b2 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u00b9 : Field E\ninst\u271d : Algebra K E\npb : PowerBasis K L\nhE : Splits (algebraMap K E) (minpoly K pb.gen)\nhfx : Separable (minpoly K pb.gen)\nthis\u271d : DecidableEq E := Classical.decEq E\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u03c3 : L \u2192\u2090[K] E\n\u22a2 \u2191\u03c3 pb.gen = \u2191(\u2191(PowerBasis.liftEquiv' pb) \u03c3)\n[PROOFSTEP]\nrw [PowerBasis.liftEquiv'_apply_coe]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\n\u22a2 \u2211 \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = finrank L F \u2022 \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nhaveI : FiniteDimensional L F := FiniteDimensional.right K L F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis : FiniteDimensional L F\n\u22a2 \u2211 \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = finrank L F \u2022 \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nhaveI : IsSeparable L F := isSeparable_tower_top_of_isSeparable K L F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d : FiniteDimensional L F\nthis : IsSeparable L F\n\u22a2 \u2211 \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = finrank L F \u2022 \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 \u2211 \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = finrank L F \u2022 \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI : \u2200 f : L \u2192\u2090[K] E, Fintype (@AlgHom L F E _ _ _ _ f.toRingHom.toAlgebra) := ?_\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u2211 \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = finrank L F \u2022 \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E)\n[PROOFSTEP]\nrw [Fintype.sum_equiv algHomEquivSigma (fun \u03c3 : F \u2192\u2090[K] E => _) fun \u03c3 => \u03c3.1 pb.gen, \u2190 Finset.univ_sigma_univ,\n  Finset.sum_sigma, \u2190 Finset.sum_nsmul]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u2211 a : L \u2192\u2090[K] E, \u2211 s : F \u2192\u2090[L] E, \u2191{ fst := a, snd := s }.fst pb.gen = \u2211 x : L \u2192\u2090[K] E, finrank L F \u2022 \u2191x pb.gen\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u2200 (x : F \u2192\u2090[K] E), \u2191x (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma x).fst pb.gen\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun \u03c3 _ => _\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\n\u22a2 \u2211 s : F \u2192\u2090[L] E, \u2191{ fst := \u03c3, snd := s }.fst pb.gen = finrank L F \u2022 \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI : Algebra L E := \u03c3.toRingHom.toAlgebra\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 \u2211 s : F \u2192\u2090[L] E, \u2191{ fst := \u03c3, snd := s }.fst pb.gen = finrank L F \u2022 \u2191\u03c3 pb.gen\n[PROOFSTEP]\nsimp only [Finset.sum_const]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 Finset.card Finset.univ \u2022 \u2191\u03c3 pb.gen = finrank L F \u2022 \u2191\u03c3 pb.gen\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine_2.e_a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 Finset.card Finset.univ = finrank L F\n[PROOFSTEP]\nrw [\u2190 AlgHom.card L F E]\n[GOAL]\ncase refine_2.e_a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 Finset.card Finset.univ = Fintype.card (F \u2192\u2090[L] E)\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E)\n[PROOFSTEP]\nexact Finset.card_univ (\u03b1 := F \u2192\u2090[L] E)\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := fun \u03c3 => minpoly.AlgHom.fintype L F E\n\u22a2 \u2200 (x : F \u2192\u2090[K] E), \u2191x (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma x).fst pb.gen\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K F\ninst\u271d : IsSeparable K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := fun \u03c3 => minpoly.AlgHom.fintype L F E\n\u03c3 : F \u2192\u2090[K] E\n\u22a2 \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma \u03c3).fst pb.gen\n[PROOFSTEP]\nsimp only [algHomEquivSigma, Equiv.coe_fn_mk, AlgHom.restrictDomain, AlgHom.comp_apply, IsScalarTower.coe_toAlgHom']\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) x) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nhave hx := IsSeparable.isIntegral K x\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) x) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nlet pb := (adjoin.powerBasis hx)\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\n\u22a2 \u2191(algebraMap K E) (\u2191(Algebra.trace K L) x) = \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nrw [trace_eq_trace_adjoin K x, Algebra.smul_def, RingHom.map_mul, \u2190 adjoin.powerBasis_gen hx,\n  trace_eq_sum_embeddings_gen E pb (IsAlgClosed.splits_codomain _)]\n  -- Porting note: the following `convert` was `exact`, with `\u2190 algebra.smul_def, algebra_map_smul`\n  -- in the previous `rw`.\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\n\u22a2 \u2191(algebraMap K E) (\u2191(algebraMap \u2115 ((fun x => K) (adjoin.powerBasis hx).gen)) (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L)) *\n      \u2211 \u03c3 : { x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E, \u2191\u03c3 pb.gen =\n    \u2211 \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nconvert (sum_embeddings_eq_finrank_mul L E pb).symm\n[GOAL]\ncase h.e'_2.h.e\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\n\u22a2 HMul.hMul\n      (\u2191(algebraMap K E)\n        (\u2191(algebraMap \u2115 ((fun x => K) (adjoin.powerBasis hx).gen)) (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L))) =\n    HSMul.hSMul (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\n[PROOFSTEP]\next\n[GOAL]\ncase h.e'_2.h.e.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\nx\u271d : (fun x => E) (\u2191(Algebra.trace K { x_1 // x_1 \u2208 K\u27eex\u27ef }) (adjoin.powerBasis hx).gen)\n\u22a2 \u2191(algebraMap K E) (\u2191(algebraMap \u2115 ((fun x => K) (adjoin.powerBasis hx).gen)) (finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L)) * x\u271d =\n    finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L \u2022 x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\n\u22a2 Separable (minpoly K pb.gen)\n[PROOFSTEP]\nhaveI := isSeparable_tower_bot_of_isSeparable K K\u27eex\u27ef L\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis hx\nthis : IsSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 Separable (minpoly K pb.gen)\n[PROOFSTEP]\nexact IsSeparable.separable K _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2191(algebraMap K L) (\u2191(Algebra.trace K L) x) = \u2211 \u03c3 : L \u2243\u2090[K] L, \u2191\u03c3 x\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective L (AlgebraicClosure L)\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(Algebra.trace K L) x)) =\n    \u2191(algebraMap L (AlgebraicClosure L)) (\u2211 \u03c3 : L \u2243\u2090[K] L, \u2191\u03c3 x)\n[PROOFSTEP]\nrw [_root_.map_sum (algebraMap L (AlgebraicClosure L))]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(Algebra.trace K L) x)) =\n    \u2211 x_1 : L \u2243\u2090[K] L, \u2191(algebraMap L (AlgebraicClosure L)) (\u2191x_1 x)\n[PROOFSTEP]\nrw [\u2190 Fintype.sum_equiv (Normal.algHomEquivAut K (AlgebraicClosure L) L)]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(Algebra.trace K L) x)) =\n    \u2211 x : L \u2192\u2090[K] AlgebraicClosure L, ?a.f\u271d x\n[PROOFSTEP]\nrw [\u2190 trace_eq_sum_embeddings (AlgebraicClosure L)]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(Algebra.trace K L) x)) =\n    \u2191(algebraMap K (AlgebraicClosure L)) (\u2191(Algebra.trace K L) ?m.943761)\n[PROOFSTEP]\nsimp only [algebraMap_eq_smul_one]\n  -- Porting note: `smul_one_smul` was in the `simp only`.\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 (\u2191(Algebra.trace K L) x \u2022 1) \u2022 1 = \u2191(Algebra.trace K L) ?m.943761 \u2022 1\n[PROOFSTEP]\napply smul_one_smul\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u22a2 \u2200 (x_1 : L \u2192\u2090[K] AlgebraicClosure L),\n    \u2191x_1 x = \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(\u2191(Normal.algHomEquivAut K (AlgebraicClosure L) L) x_1) x)\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\ninst\u271d\u2076 : Algebra K F\ninst\u271d\u2075 : IsScalarTower K L F\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : IsAlgClosed E\nx : L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\n\u03c3 : L \u2192\u2090[K] AlgebraicClosure L\n\u22a2 \u2191\u03c3 x = \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(\u2191(Normal.algHomEquivAut K (AlgebraicClosure L) L) \u03c3) x)\n[PROOFSTEP]\nsimp only [Normal.algHomEquivAut, AlgHom.restrictNormal', Equiv.coe_fn_mk, AlgEquiv.coe_ofBijective,\n  AlgHom.restrictNormal_commutes, id.map_eq_id, RingHom.id_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\n\u03ba' : Type u_7\nb : Basis \u03ba A B\nf : \u03ba \u2243 \u03ba'\n\u22a2 traceMatrix A \u2191(Basis.reindex b f) = \u2191(reindex f f) (traceMatrix A \u2191b)\n[PROOFSTEP]\next (x y)\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\n\u03ba' : Type u_7\nb : Basis \u03ba A B\nf : \u03ba \u2243 \u03ba'\nx y : \u03ba'\n\u22a2 traceMatrix A (\u2191(Basis.reindex b f)) x y = \u2191(reindex f f) (traceMatrix A \u2191b) x y\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u22a2 traceMatrix A (vecMul b (Matrix.map P \u2191(algebraMap A B))) = P\u1d40 * traceMatrix A b * P\n[PROOFSTEP]\next (\u03b1 \u03b2)\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 : \u03ba\n\u22a2 traceMatrix A (vecMul b (Matrix.map P \u2191(algebraMap A B))) \u03b1 \u03b2 = (P\u1d40 * traceMatrix A b * P) \u03b1 \u03b2\n[PROOFSTEP]\nrw [traceMatrix_apply, vecMul, dotProduct, vecMul, dotProduct, Matrix.mul_apply, BilinForm.sum_left,\n  Fintype.sum_congr _ _ fun i : \u03ba =>\n    @BilinForm.sum_right _ _ _ _ _ _ _ _ (b i * P.map (algebraMap A B) i \u03b1) fun y : \u03ba =>\n      b y * P.map (algebraMap A B) y \u03b2,\n  sum_comm]\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 : \u03ba\n\u22a2 \u2211 y : \u03ba,\n      \u2211 x : \u03ba,\n        BilinForm.bilin (traceForm A B) (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b1)\n          (b y * Matrix.map P (\u2191(algebraMap A B)) y \u03b2) =\n    \u2211 j : \u03ba, (P\u1d40 * traceMatrix A b) \u03b1 j * P j \u03b2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.h.e_f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 : \u03ba\n\u22a2 (fun y =>\n      \u2211 x : \u03ba,\n        BilinForm.bilin (traceForm A B) (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b1)\n          (b y * Matrix.map P (\u2191(algebraMap A B)) y \u03b2)) =\n    fun j => (P\u1d40 * traceMatrix A b) \u03b1 j * P j \u03b2\n[PROOFSTEP]\next x\n[GOAL]\ncase a.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x : \u03ba\n\u22a2 \u2211 x_1 : \u03ba,\n      BilinForm.bilin (traceForm A B) (b x_1 * Matrix.map P (\u2191(algebraMap A B)) x_1 \u03b1)\n        (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b2) =\n    (P\u1d40 * traceMatrix A b) \u03b1 x * P x \u03b2\n[PROOFSTEP]\nrw [Matrix.mul_apply, sum_mul]\n[GOAL]\ncase a.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x : \u03ba\n\u22a2 \u2211 x_1 : \u03ba,\n      BilinForm.bilin (traceForm A B) (b x_1 * Matrix.map P (\u2191(algebraMap A B)) x_1 \u03b1)\n        (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b2) =\n    \u2211 x_1 : \u03ba, P\u1d40 \u03b1 x_1 * traceMatrix A b x_1 x * P x \u03b2\n[PROOFSTEP]\ncongr\n[GOAL]\ncase a.h.e_f.h.e_f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x : \u03ba\n\u22a2 (fun x_1 =>\n      BilinForm.bilin (traceForm A B) (b x_1 * Matrix.map P (\u2191(algebraMap A B)) x_1 \u03b1)\n        (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b2)) =\n    fun x_1 => P\u1d40 \u03b1 x_1 * traceMatrix A b x_1 x * P x \u03b2\n[PROOFSTEP]\next y\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 BilinForm.bilin (traceForm A B) (b y * Matrix.map P (\u2191(algebraMap A B)) y \u03b1)\n      (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b2) =\n    P\u1d40 \u03b1 y * traceMatrix A b y x * P x \u03b2\n[PROOFSTEP]\nrw [map_apply, traceForm_apply, mul_comm (b y), \u2190 smul_def]\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 \u2191(trace A B) (P y \u03b1 \u2022 b y * (b x * Matrix.map P (\u2191(algebraMap A B)) x \u03b2)) = P\u1d40 \u03b1 y * traceMatrix A b y x * P x \u03b2\n[PROOFSTEP]\nsimp only [id.smul_eq_mul, RingHom.id_apply, map_apply, transpose_apply, LinearMap.map_smul\u209b\u2097, traceForm_apply,\n  Algebra.smul_mul_assoc]\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 P y \u03b1 * \u2191(trace A B) (b y * (b x * \u2191(algebraMap A B) (P x \u03b2))) = P y \u03b1 * traceMatrix A b y x * P x \u03b2\n[PROOFSTEP]\nrw [mul_comm (b x), \u2190 smul_def]\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 P y \u03b1 * \u2191(trace A B) (b y * P x \u03b2 \u2022 b x) = P y \u03b1 * traceMatrix A b y x * P x \u03b2\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 P y \u03b1 * \u2191(trace A B) (b y * P x \u03b2 \u2022 b x) = P y \u03b1 * traceMatrix A b y x * P x \u03b2\n[PROOFSTEP]\nrw [mul_assoc]\n[GOAL]\ncase a.h.e_f.h.e_f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u03b1 \u03b2 x y : \u03ba\n\u22a2 P y \u03b1 * \u2191(trace A B) (b y * P x \u03b2 \u2022 b x) = P y \u03b1 * (traceMatrix A b y x * P x \u03b2)\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u22a2 traceMatrix A (mulVec (Matrix.map P \u2191(algebraMap A B)) b) = P * traceMatrix A b * P\u1d40\n[PROOFSTEP]\nrefine' AddEquiv.injective (transposeAddEquiv \u03ba \u03ba A) _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2079 : CommRing R\ninst\u271d\u00b9\u2078 : CommRing S\ninst\u271d\u00b9\u2077 : CommRing T\ninst\u271d\u00b9\u2076 : Algebra R S\ninst\u271d\u00b9\u2075 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2074 : Field K\ninst\u271d\u00b9\u00b3 : Field L\ninst\u271d\u00b9\u00b2 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b9 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2070 : Field F\ninst\u271d\u2079 : Algebra R L\ninst\u271d\u2078 : Algebra L F\ninst\u271d\u2077 : Algebra R F\ninst\u271d\u2076 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : Algebra A B\ninst\u271d\u00b2 : CommRing C\ninst\u271d\u00b9 : Algebra A C\ninst\u271d : Fintype \u03ba\nb : \u03ba \u2192 B\nP : Matrix \u03ba \u03ba A\n\u22a2 \u2191(transposeAddEquiv \u03ba \u03ba A) (traceMatrix A (mulVec (Matrix.map P \u2191(algebraMap A B)) b)) =\n    \u2191(transposeAddEquiv \u03ba \u03ba A) (P * traceMatrix A b * P\u1d40)\n[PROOFSTEP]\nrw [transposeAddEquiv_apply, transposeAddEquiv_apply, \u2190 vecMul_transpose, \u2190 transpose_map, traceMatrix_of_matrix_vecMul,\n  transpose_transpose, transpose_mul, transpose_transpose, transpose_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq \u03ba\nb : Basis \u03ba A B\n\u22a2 traceMatrix A \u2191b = \u2191(BilinForm.toMatrix b) (traceForm A B)\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2070 : CommRing R\ninst\u271d\u00b9\u2079 : CommRing S\ninst\u271d\u00b9\u2078 : CommRing T\ninst\u271d\u00b9\u2077 : Algebra R S\ninst\u271d\u00b9\u2076 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2075 : Field K\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u00b2 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field F\ninst\u271d\u00b9\u2070 : Algebra R L\ninst\u271d\u2079 : Algebra L F\ninst\u271d\u2078 : Algebra R F\ninst\u271d\u2077 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2076 : CommRing A\ninst\u271d\u2075 : CommRing B\ninst\u271d\u2074 : Algebra A B\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Fintype \u03ba\ninst\u271d : DecidableEq \u03ba\nb : Basis \u03ba A B\ni j : \u03ba\n\u22a2 traceMatrix A (\u2191b) i j = \u2191(BilinForm.toMatrix b) (traceForm A B) i j\n[PROOFSTEP]\nrw [traceMatrix_apply, traceForm_apply, traceForm_toMatrix]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\n\u22a2 mulVec (traceMatrix A \u2191b) (\u2191(Basis.equivFun b) z) = fun i => \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\next i\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 mulVec (traceMatrix A \u2191b) (\u2191(Basis.equivFun b) z) i = \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\nrw [\u2190 col_apply ((traceMatrix A b).mulVec (b.equivFun z)) i Unit.unit, col_mulVec, Matrix.mul_apply, traceMatrix]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 j : \u03b9, \u2191of (fun i j => BilinForm.bilin (traceForm A B) (\u2191b i) (\u2191b j)) i j * col (\u2191(Basis.equivFun b) z) j () =\n    \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\nsimp only [col_apply, traceForm_apply]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 x : \u03b9, \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x = \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  rfl\n  ext\n  rw [mul_comm _ (b.equivFun z _), \u2190 smul_eq_mul, of_apply, \u2190 LinearMap.map_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 x : \u03b9, \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [mul_comm _ (b.equivFun z _), \u2190 smul_eq_mul, of_apply, \u2190 LinearMap.map_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 x : \u03b9, \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [mul_comm _ (b.equivFun z _), \u2190 smul_eq_mul, of_apply, \u2190 LinearMap.map_smul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 x : \u03b9, \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| univ\ncase f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| fun x => \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| fun x => \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x * \u2191(Basis.equivFun b) z x\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni x\u271d : \u03b9\n| \u2191of (fun i j => \u2191(trace A B) (\u2191b i * \u2191b j)) i x\u271d * \u2191(Basis.equivFun b) z x\u271d\n[PROOFSTEP]\nrw [mul_comm _ (b.equivFun z _), \u2190 smul_eq_mul, of_apply, \u2190 LinearMap.map_smul]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 x : \u03b9, \u2191(trace A B) (\u2191(Basis.equivFun b) z x \u2022 (\u2191b i * \u2191b x)) = \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\nrw [\u2190 LinearMap.map_sum]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2191(trace A B) (\u2211 i_1 : \u03b9, \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)) = \u2191(trace A B) (z * \u2191b i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.h.e_6.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 i_1 : \u03b9, \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1) = z * \u2191b i\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  rfl\n  ext\n  rw [\u2190 mul_smul_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 i_1 : \u03b9, \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [\u2190 mul_smul_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 i_1 : \u03b9, \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)\n[PROOFSTEP]\n  congr\n  rfl\n  ext\n  rw [\u2190 mul_smul_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| \u2211 i_1 : \u03b9, \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase s\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| univ\ncase f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| fun i_1 => \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase f\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n| fun i_1 => \u2191(Basis.equivFun b) z i_1 \u2022 (\u2191b i * \u2191b i_1)\n[PROOFSTEP]\next\n[GOAL]\ncase f.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni x\u271d : \u03b9\n| \u2191(Basis.equivFun b) z x\u271d \u2022 (\u2191b i * \u2191b x\u271d)\n[PROOFSTEP]\nrw [\u2190 mul_smul_comm]\n[GOAL]\ncase h.h.e_6.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 x : \u03b9, \u2191b i * \u2191(Basis.equivFun b) z x \u2022 \u2191b x = z * \u2191b i\n[PROOFSTEP]\nrw [\u2190 Finset.mul_sum, mul_comm z]\n[GOAL]\ncase h.h.e_6.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2191b i * \u2211 x : \u03b9, \u2191(Basis.equivFun b) z x \u2022 \u2191b x = \u2191b i * z\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.h.e_6.h.e_a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\nb : Basis \u03b9 A B\nz : B\ni : \u03b9\n\u22a2 \u2211 x : \u03b9, \u2191(Basis.equivFun b) z x \u2022 \u2191b x = z\n[PROOFSTEP]\nrw [b.sum_equivFun]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\npb : PowerBasis A B\ne : Fin pb.dim \u2243 (B \u2192\u2090[A] C)\n\u22a2 embeddingsMatrixReindex A C (\u2191pb.basis) e = (vandermonde fun i => \u2191(\u2191e i) pb.gen)\u1d40\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2078 : CommRing R\ninst\u271d\u00b9\u2077 : CommRing S\ninst\u271d\u00b9\u2076 : CommRing T\ninst\u271d\u00b9\u2075 : Algebra R S\ninst\u271d\u00b9\u2074 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u00b3 : Field K\ninst\u271d\u00b9\u00b2 : Field L\ninst\u271d\u00b9\u00b9 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2070 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra R L\ninst\u271d\u2077 : Algebra L F\ninst\u271d\u2076 : Algebra R F\ninst\u271d\u2075 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra A B\ninst\u271d\u00b9 : CommRing C\ninst\u271d : Algebra A C\npb : PowerBasis A B\ne : Fin pb.dim \u2243 (B \u2192\u2090[A] C)\ni j : Fin pb.dim\n\u22a2 embeddingsMatrixReindex A C (\u2191pb.basis) e i j = (vandermonde fun i => \u2191(\u2191e i) pb.gen)\u1d40 i j\n[PROOFSTEP]\nsimp [embeddingsMatrixReindex, embeddingsMatrix]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u00b3 : CommRing R\ninst\u271d\u00b2\u00b2 : CommRing S\ninst\u271d\u00b2\u00b9 : CommRing T\ninst\u271d\u00b2\u2070 : Algebra R S\ninst\u271d\u00b9\u2079 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2078 : Field K\ninst\u271d\u00b9\u2077 : Field L\ninst\u271d\u00b9\u2076 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2075 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2074 : Field F\ninst\u271d\u00b9\u00b3 : Algebra R L\ninst\u271d\u00b9\u00b2 : Algebra L F\ninst\u271d\u00b9\u00b9 : Algebra R F\ninst\u271d\u00b9\u2070 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : CommRing C\ninst\u271d\u2075 : Algebra A C\nE : Type z\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Module.Finite K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nb : \u03ba \u2192 L\npb : PowerBasis K L\n\u22a2 Matrix.map (traceMatrix K b) \u2191(algebraMap K E) = embeddingsMatrix K E b * (embeddingsMatrix K E b)\u1d40\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u00b3 : CommRing R\ninst\u271d\u00b2\u00b2 : CommRing S\ninst\u271d\u00b2\u00b9 : CommRing T\ninst\u271d\u00b2\u2070 : Algebra R S\ninst\u271d\u00b9\u2079 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2078 : Field K\ninst\u271d\u00b9\u2077 : Field L\ninst\u271d\u00b9\u2076 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2075 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2074 : Field F\ninst\u271d\u00b9\u00b3 : Algebra R L\ninst\u271d\u00b9\u00b2 : Algebra L F\ninst\u271d\u00b9\u00b9 : Algebra R F\ninst\u271d\u00b9\u2070 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u2079 : CommRing A\ninst\u271d\u2078 : CommRing B\ninst\u271d\u2077 : Algebra A B\ninst\u271d\u2076 : CommRing C\ninst\u271d\u2075 : Algebra A C\nE : Type z\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Module.Finite K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nb : \u03ba \u2192 L\npb : PowerBasis K L\ni j : \u03ba\n\u22a2 Matrix.map (traceMatrix K b) (\u2191(algebraMap K E)) i j = (embeddingsMatrix K E b * (embeddingsMatrix K E b)\u1d40) i j\n[PROOFSTEP]\nsimp [trace_eq_sum_embeddings, embeddingsMatrix, Matrix.mul_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b2\u2074 : CommRing R\ninst\u271d\u00b2\u00b3 : CommRing S\ninst\u271d\u00b2\u00b2 : CommRing T\ninst\u271d\u00b2\u00b9 : Algebra R S\ninst\u271d\u00b2\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2079 : Field K\ninst\u271d\u00b9\u2078 : Field L\ninst\u271d\u00b9\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u00b9\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u00b9\u2075 : Field F\ninst\u271d\u00b9\u2074 : Algebra R L\ninst\u271d\u00b9\u00b3 : Algebra L F\ninst\u271d\u00b9\u00b2 : Algebra R F\ninst\u271d\u00b9\u00b9 : IsScalarTower R L F\nA : Type u\nB : Type v\nC : Type z\ninst\u271d\u00b9\u2070 : CommRing A\ninst\u271d\u2079 : CommRing B\ninst\u271d\u2078 : Algebra A B\ninst\u271d\u2077 : CommRing C\ninst\u271d\u2076 : Algebra A C\nE : Type z\ninst\u271d\u2075 : Field E\ninst\u271d\u2074 : Algebra K E\ninst\u271d\u00b3 : Module.Finite K L\ninst\u271d\u00b2 : IsSeparable K L\ninst\u271d\u00b9 : IsAlgClosed E\nb : \u03ba \u2192 L\npb : PowerBasis K L\ninst\u271d : Fintype \u03ba\ne : \u03ba \u2243 (L \u2192\u2090[K] E)\n\u22a2 Matrix.map (traceMatrix K b) \u2191(algebraMap K E) = embeddingsMatrixReindex K E b e * (embeddingsMatrixReindex K E b e)\u1d40\n[PROOFSTEP]\nrw [traceMatrix_eq_embeddingsMatrix_mul_trans, embeddingsMatrixReindex, reindex_apply, transpose_submatrix, \u2190\n  submatrix_mul_transpose_submatrix, \u2190 Equiv.coe_refl, Equiv.refl_symm]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\n\u22a2 det (traceMatrix K \u2191pb.basis) \u2260 0\n[PROOFSTEP]\nsuffices algebraMap K (AlgebraicClosure L) (det (traceMatrix K pb.basis)) \u2260 0\n  by\n  refine' mt (fun ht => _) this\n  rw [ht, RingHom.map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) \u2260 0\n\u22a2 det (traceMatrix K \u2191pb.basis) \u2260 0\n[PROOFSTEP]\nrefine' mt (fun ht => _) this\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) \u2260 0\nht : det (traceMatrix K \u2191pb.basis) = 0\n\u22a2 \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) = 0\n[PROOFSTEP]\nrw [ht, RingHom.map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\n\u22a2 \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) \u2260 0\n[PROOFSTEP]\nhaveI : FiniteDimensional K L := pb.finiteDimensional\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\n\u22a2 \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) \u2260 0\n[PROOFSTEP]\nlet e : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?_).symm\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\n\u22a2 \u2191(algebraMap K (AlgebraicClosure L)) (det (traceMatrix K \u2191pb.basis)) \u2260 0\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\n\u22a2 Fintype.card (L \u2192\u2090[K] AlgebraicClosure L) = pb.dim\n[PROOFSTEP]\nrw [RingHom.map_det, RingHom.mapMatrix_apply, traceMatrix_eq_embeddingsMatrixReindex_mul_trans K _ _ e,\n  embeddingsMatrixReindex_eq_vandermonde, det_mul, det_transpose]\n  -- Porting note: the following is necessary.\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\n\u22a2 det (vandermonde fun i => \u2191(\u2191e i) pb.gen) * det (vandermonde fun i => \u2191(\u2191e i) pb.gen)\u1d40\u1d40 \u2260 0\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\n\u22a2 Fintype.card (L \u2192\u2090[K] AlgebraicClosure L) = pb.dim\n[PROOFSTEP]\nhaveI := IsDomain.to_noZeroDivisors (AlgebraicClosure L)\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis\u271d : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\nthis : NoZeroDivisors (AlgebraicClosure L)\n\u22a2 det (vandermonde fun i => \u2191(\u2191e i) pb.gen) * det (vandermonde fun i => \u2191(\u2191e i) pb.gen)\u1d40\u1d40 \u2260 0\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\n\u22a2 Fintype.card (L \u2192\u2090[K] AlgebraicClosure L) = pb.dim\n[PROOFSTEP]\nrefine' mt mul_self_eq_zero.mp _\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis\u271d : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\nthis : NoZeroDivisors (AlgebraicClosure L)\n\u22a2 \u00acdet (vandermonde fun i => \u2191(\u2191e i) pb.gen) = 0\n[PROOFSTEP]\nsimp only [det_vandermonde, Finset.prod_eq_zero_iff, not_exists, sub_eq_zero]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis\u271d : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\nthis : NoZeroDivisors (AlgebraicClosure L)\n\u22a2 \u2200 (x : Fin pb.dim),\n    \u00ac(x \u2208 Finset.univ \u2227\n        \u2203 a,\n          a \u2208 Finset.Ioi x \u2227\n            \u2191(\u2191(Fintype.equivFinOfCardEq ?refine_1).symm a) pb.gen =\n              \u2191(\u2191(Fintype.equivFinOfCardEq ?refine_1).symm x) pb.gen)\n[PROOFSTEP]\nrintro i \u27e8_, j, hij, h\u27e9\n[GOAL]\ncase refine_2.intro.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis\u271d : FiniteDimensional K L\ne : Fin pb.dim \u2243 (L \u2192\u2090[K] AlgebraicClosure L) := (Fintype.equivFinOfCardEq ?refine_1).symm\nthis : NoZeroDivisors (AlgebraicClosure L)\ni : Fin pb.dim\nleft\u271d : i \u2208 Finset.univ\nj : Fin pb.dim\nhij : j \u2208 Finset.Ioi i\nh : \u2191(\u2191(Fintype.equivFinOfCardEq ?refine_1).symm j) pb.gen = \u2191(\u2191(Fintype.equivFinOfCardEq ?refine_1).symm i) pb.gen\n\u22a2 False\n[PROOFSTEP]\nexact (Finset.mem_Ioi.mp hij).ne' (e.injective <| pb.algHom_ext h)\n[GOAL]\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : CommRing S\ninst\u271d\u00b9\u00b2 : CommRing T\ninst\u271d\u00b9\u00b9 : Algebra R S\ninst\u271d\u00b9\u2070 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2076 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra L F\ninst\u271d\u00b2 : Algebra R F\ninst\u271d\u00b9 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d : IsSeparable K L\nthis : FiniteDimensional K L\n\u22a2 Fintype.card (L \u2192\u2090[K] AlgebraicClosure L) = pb.dim\n[PROOFSTEP]\nrw [AlgHom.card, pb.finrank]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\n\u22a2 det (\u2191(BilinForm.toMatrix b) (traceForm K L)) \u2260 0\n[PROOFSTEP]\nhaveI : FiniteDimensional K L := FiniteDimensional.of_fintype_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\n\u22a2 det (\u2191(BilinForm.toMatrix b) (traceForm K L)) \u2260 0\n[PROOFSTEP]\nlet pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable _ _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (\u2191(BilinForm.toMatrix b) (traceForm K L)) \u2260 0\n[PROOFSTEP]\nrw [\u2190 BilinForm.toMatrix_mul_basis_toMatrix pb.basis b, \u2190 det_comm' (pb.basis.toMatrix_mul_toMatrix_flip b) _, \u2190\n  Matrix.mul_assoc, det_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix pb.basis \u2191b)\u1d40) *\n      det (\u2191(BilinForm.toMatrix pb.basis) (traceForm K L)) \u2260\n    0\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 Basis.toMatrix b \u2191pb.basis * Basis.toMatrix pb.basis \u2191b = 1\n[PROOFSTEP]\nswap\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 Basis.toMatrix b \u2191pb.basis * Basis.toMatrix pb.basis \u2191b = 1\n[PROOFSTEP]\napply Basis.toMatrix_mul_toMatrix_flip\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix pb.basis \u2191b)\u1d40) *\n      det (\u2191(BilinForm.toMatrix pb.basis) (traceForm K L)) \u2260\n    0\n[PROOFSTEP]\nrefine' mul_ne_zero (isUnit_of_mul_eq_one _ ((b.toMatrix pb.basis)\u1d40 * b.toMatrix pb.basis).det _).ne_zero _\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix pb.basis \u2191b)\u1d40) *\n      det ((Basis.toMatrix b \u2191pb.basis)\u1d40 * Basis.toMatrix b \u2191pb.basis) =\n    1\n[PROOFSTEP]\ncalc\n  (pb.basis.toMatrix b * (pb.basis.toMatrix b)\u1d40).det * ((b.toMatrix pb.basis)\u1d40 * b.toMatrix pb.basis).det =\n      (pb.basis.toMatrix b * (b.toMatrix pb.basis * pb.basis.toMatrix b)\u1d40 * b.toMatrix pb.basis).det :=\n    by simp only [\u2190 det_mul, Matrix.mul_assoc, Matrix.transpose_mul]\n  _ = 1 := by simp only [Basis.toMatrix_mul_toMatrix_flip, Matrix.transpose_one, Matrix.mul_one, Matrix.det_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix pb.basis \u2191b)\u1d40) *\n      det ((Basis.toMatrix b \u2191pb.basis)\u1d40 * Basis.toMatrix b \u2191pb.basis) =\n    det\n      (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix b \u2191pb.basis * Basis.toMatrix pb.basis \u2191b)\u1d40 *\n        Basis.toMatrix b \u2191pb.basis)\n[PROOFSTEP]\nsimp only [\u2190 det_mul, Matrix.mul_assoc, Matrix.transpose_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det\n      (Basis.toMatrix pb.basis \u2191b * (Basis.toMatrix b \u2191pb.basis * Basis.toMatrix pb.basis \u2191b)\u1d40 *\n        Basis.toMatrix b \u2191pb.basis) =\n    1\n[PROOFSTEP]\nsimp only [Basis.toMatrix_mul_toMatrix_flip, Matrix.transpose_one, Matrix.mul_one, Matrix.det_one]\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2075 : CommRing R\ninst\u271d\u00b9\u2074 : CommRing S\ninst\u271d\u00b9\u00b3 : CommRing T\ninst\u271d\u00b9\u00b2 : Algebra R S\ninst\u271d\u00b9\u00b9 : Algebra R T\nK : Type u_4\nL : Type u_5\ninst\u271d\u00b9\u2070 : Field K\ninst\u271d\u2079 : Field L\ninst\u271d\u2078 : Algebra K L\n\u03b9 \u03ba : Type w\ninst\u271d\u2077 : Fintype \u03b9\nF : Type u_6\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra R L\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : Algebra R F\ninst\u271d\u00b2 : IsScalarTower R L F\npb\u271d : PowerBasis K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 K L\nthis : FiniteDimensional K L\npb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L\n\u22a2 det (\u2191(BilinForm.toMatrix pb.basis) (traceForm K L)) \u2260 0\n[PROOFSTEP]\nsimpa only [traceMatrix_of_basis] using det_traceMatrix_ne_zero' pb\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Trace", "llama_tokens": 72803, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799928900257127, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.5108132271144924}}
{"text": "[GOAL]\n\u03b1 : Sort u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : Quot r \u2192 Sort v\nh : \u2200 (a : \u03b1), Subsingleton (motive (mk r a))\nq : Quot r\nf : (a : \u03b1) \u2192 motive (mk r a)\n\u22a2 motive q\n[PROOFSTEP]\ninduction q using Quot.rec\n[GOAL]\ncase f\n\u03b1 : Sort u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : Quot r \u2192 Sort v\nh : \u2200 (a : \u03b1), Subsingleton (motive (mk r a))\nf : (a : \u03b1) \u2192 motive (mk r a)\na\u271d : \u03b1\n\u22a2 motive (mk r a\u271d)\ncase h\n\u03b1 : Sort u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : Quot r \u2192 Sort v\nh : \u2200 (a : \u03b1), Subsingleton (motive (mk r a))\nq : Quot r\nf : (a : \u03b1) \u2192 motive (mk r a)\na\u271d b\u271d : \u03b1\np\u271d : r a\u271d b\u271d\n\u22a2 (_ : mk r a\u271d = mk r b\u271d) \u25b8 ?m.408 a\u271d = ?m.408 b\u271d\n[PROOFSTEP]\napply f\n[GOAL]\ncase h\n\u03b1 : Sort u\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nmotive : Quot r \u2192 Sort v\nh : \u2200 (a : \u03b1), Subsingleton (motive (mk r a))\nq : Quot r\nf : (a : \u03b1) \u2192 motive (mk r a)\na\u271d b\u271d : \u03b1\np\u271d : r a\u271d b\u271d\n\u22a2 (_ : mk r a\u271d = mk r b\u271d) \u25b8 f a\u271d = f b\u271d\n[PROOFSTEP]\napply Subsingleton.elim\n", "meta": {"mathlib_filename": "Mathlib.Init.Quot", "llama_tokens": 471, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5106792235946853}}
{"text": "[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (l : L), \u00acf l \u2208 l\n[PROOFSTEP]\nclassical\nlet t : L \u2192 Finset P := fun l => Set.toFinset {p | p \u2209 l}\nsuffices \u2200 s : Finset L, s.card \u2264 (s.biUnion t).card by\n  -- Hall's marriage theorem\n  obtain \u27e8f, hf1, hf2\u27e9 := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this\n  exact \u27e8f, hf1, fun l => Set.mem_toFinset.mp (hf2 l)\u27e9\nintro s\nby_cases hs\u2080 :\n  s.card =\n    0\n      -- If `s = \u2205`, then `s.card = 0 \u2264 (s.bUnion t).card`\n\u00b7 simp_rw [hs\u2080, zero_le]\nby_cases hs\u2081 :\n  s.card =\n    1\n      -- If `s = {l}`, then pick a point `p \u2209 l`\n\u00b7 obtain \u27e8l, rfl\u27e9 := Finset.card_eq_one.mp hs\u2081\n  obtain \u27e8p, hl\u27e9 := exists_point l\n  rw [Finset.card_singleton, Finset.singleton_biUnion, Nat.one_le_iff_ne_zero]\n  exact Finset.card_ne_zero_of_mem (Set.mem_toFinset.mpr hl)\nsuffices (s.biUnion t)\u1d9c.card \u2264 s\u1d9c.card by\n  -- Rephrase in terms of complements (uses `h`)\n  rw [Finset.card_compl, Finset.card_compl, tsub_le_iff_left] at this \n  replace := h.trans this\n  rwa [\u2190 add_tsub_assoc_of_le s.card_le_univ, le_tsub_iff_left (le_add_left s.card_le_univ), add_le_add_iff_right] at\n    this \nhave hs\u2082 : (s.biUnion t)\u1d9c.card \u2264 1 := by\n  -- At most one line through two points of `s`\n  refine' Finset.card_le_one_iff.mpr @fun p\u2081 p\u2082 hp\u2081 hp\u2082 => _\n  simp_rw [Finset.mem_compl, Finset.mem_biUnion, not_exists, not_and, Set.mem_toFinset, Set.mem_setOf_eq,\n    Classical.not_not] at hp\u2081 hp\u2082 \n  obtain \u27e8l\u2081, l\u2082, hl\u2081, hl\u2082, hl\u2083\u27e9 := Finset.one_lt_card_iff.mp (Nat.one_lt_iff_ne_zero_and_ne_one.mpr \u27e8hs\u2080, hs\u2081\u27e9)\n  exact (eq_or_eq (hp\u2081 l\u2081 hl\u2081) (hp\u2082 l\u2081 hl\u2081) (hp\u2081 l\u2082 hl\u2082) (hp\u2082 l\u2082 hl\u2082)).resolve_right hl\u2083\nby_cases hs\u2083 : s\u1d9c.card = 0\n\u00b7 rw [hs\u2083, le_zero_iff]\n  rw [Finset.card_compl, tsub_eq_zero_iff_le, LE.le.le_iff_eq (Finset.card_le_univ _), eq_comm,\n    Finset.card_eq_iff_eq_univ] at hs\u2083 \u22a2\n  rw [hs\u2083]\n  rw [Finset.eq_univ_iff_forall] at hs\u2083 \u22a2\n  exact fun p =>\n    Exists.elim\n      (exists_line p)\n        -- If `s = univ`, then show `s.bUnion t = univ`\n      fun l hl => Finset.mem_biUnion.mpr \u27e8l, Finset.mem_univ l, Set.mem_toFinset.mpr hl\u27e9\n\u00b7 exact hs\u2082.trans (Nat.one_le_iff_ne_zero.mpr hs\u2083)\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (l : L), \u00acf l \u2208 l\n[PROOFSTEP]\nlet t : L \u2192 Finset P := fun l => Set.toFinset {p | p \u2209 l}\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (l : L), \u00acf l \u2208 l\n[PROOFSTEP]\nsuffices \u2200 s : Finset L, s.card \u2264 (s.biUnion t).card by\n  -- Hall's marriage theorem\n  obtain \u27e8f, hf1, hf2\u27e9 := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this\n  exact \u27e8f, hf1, fun l => Set.mem_toFinset.mp (hf2 l)\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\nthis : \u2200 (s : Finset L), Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (l : L), \u00acf l \u2208 l\n[PROOFSTEP]\nobtain \u27e8f, hf1, hf2\u27e9 := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\nthis : \u2200 (s : Finset L), Finset.card s \u2264 Finset.card (Finset.biUnion s t)\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (x : L), f x \u2208 t x\n\u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (l : L), \u00acf l \u2208 l\n[PROOFSTEP]\nexact \u27e8f, hf1, fun l => Set.mem_toFinset.mp (hf2 l)\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\n\u22a2 \u2200 (s : Finset L), Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nintro s\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nby_cases hs\u2080 :\n  s.card =\n    0\n      -- If `s = \u2205`, then `s.card = 0 \u2264 (s.bUnion t).card`\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : Finset.card s = 0\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nsimp_rw [hs\u2080, zero_le]\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nby_cases hs\u2081 :\n  s.card =\n    1\n      -- If `s = {l}`, then pick a point `p \u2209 l`\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : Finset.card s = 1\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nobtain \u27e8l, rfl\u27e9 := Finset.card_eq_one.mp hs\u2081\n[GOAL]\ncase pos.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\nl : L\nhs\u2080 : \u00acFinset.card {l} = 0\nhs\u2081 : Finset.card {l} = 1\n\u22a2 Finset.card {l} \u2264 Finset.card (Finset.biUnion {l} t)\n[PROOFSTEP]\nobtain \u27e8p, hl\u27e9 := exists_point l\n[GOAL]\ncase pos.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\nl : L\nhs\u2080 : \u00acFinset.card {l} = 0\nhs\u2081 : Finset.card {l} = 1\np : P\nhl : \u00acp \u2208 l\n\u22a2 Finset.card {l} \u2264 Finset.card (Finset.biUnion {l} t)\n[PROOFSTEP]\nrw [Finset.card_singleton, Finset.singleton_biUnion, Nat.one_le_iff_ne_zero]\n[GOAL]\ncase pos.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\nl : L\nhs\u2080 : \u00acFinset.card {l} = 0\nhs\u2081 : Finset.card {l} = 1\np : P\nhl : \u00acp \u2208 l\n\u22a2 Finset.card (t l) \u2260 0\n[PROOFSTEP]\nexact Finset.card_ne_zero_of_mem (Set.mem_toFinset.mpr hl)\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nsuffices (s.biUnion t)\u1d9c.card \u2264 s\u1d9c.card by\n  -- Rephrase in terms of complements (uses `h`)\n  rw [Finset.card_compl, Finset.card_compl, tsub_le_iff_left] at this \n  replace := h.trans this\n  rwa [\u2190 add_tsub_assoc_of_le s.card_le_univ, le_tsub_iff_left (le_add_left s.card_le_univ), add_le_add_iff_right] at\n    this \n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nthis : Finset.card (Finset.biUnion s t)\u1d9c \u2264 Finset.card s\u1d9c\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nrw [Finset.card_compl, Finset.card_compl, tsub_le_iff_left] at this \n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nthis\u271d : Fintype.card P - Finset.card (Finset.biUnion s t) \u2264 Fintype.card L - Finset.card s\nthis : Fintype.card P \u2264 Finset.card (Finset.biUnion s t) + (Fintype.card L - Finset.card s)\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nreplace := h.trans this\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nthis\u271d : Fintype.card P - Finset.card (Finset.biUnion s t) \u2264 Fintype.card L - Finset.card s\nthis : Fintype.card L \u2264 Finset.card (Finset.biUnion s t) + (Fintype.card L - Finset.card s)\n\u22a2 Finset.card s \u2264 Finset.card (Finset.biUnion s t)\n[PROOFSTEP]\nrwa [\u2190 add_tsub_assoc_of_le s.card_le_univ, le_tsub_iff_left (le_add_left s.card_le_univ), add_le_add_iff_right] at this \n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c \u2264 Finset.card s\u1d9c\n[PROOFSTEP]\nhave hs\u2082 : (s.biUnion t)\u1d9c.card \u2264 1 := by\n  -- At most one line through two points of `s`\n  refine' Finset.card_le_one_iff.mpr @fun p\u2081 p\u2082 hp\u2081 hp\u2082 => _\n  simp_rw [Finset.mem_compl, Finset.mem_biUnion, not_exists, not_and, Set.mem_toFinset, Set.mem_setOf_eq,\n    Classical.not_not] at hp\u2081 hp\u2082 \n  obtain \u27e8l\u2081, l\u2082, hl\u2081, hl\u2082, hl\u2083\u27e9 := Finset.one_lt_card_iff.mp (Nat.one_lt_iff_ne_zero_and_ne_one.mpr \u27e8hs\u2080, hs\u2081\u27e9)\n  exact (eq_or_eq (hp\u2081 l\u2081 hl\u2081) (hp\u2082 l\u2081 hl\u2081) (hp\u2081 l\u2082 hl\u2082) (hp\u2082 l\u2082 hl\u2082)).resolve_right hl\u2083\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\n[PROOFSTEP]\nrefine' Finset.card_le_one_iff.mpr @fun p\u2081 p\u2082 hp\u2081 hp\u2082 => _\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 (Finset.biUnion s t)\u1d9c\nhp\u2082 : p\u2082 \u2208 (Finset.biUnion s t)\u1d9c\n\u22a2 p\u2081 = p\u2082\n[PROOFSTEP]\nsimp_rw [Finset.mem_compl, Finset.mem_biUnion, not_exists, not_and, Set.mem_toFinset, Set.mem_setOf_eq,\n  Classical.not_not] at hp\u2081 hp\u2082 \n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\np\u2081 p\u2082 : P\nhp\u2081 : \u2200 (x : L), x \u2208 s \u2192 p\u2081 \u2208 x\nhp\u2082 : \u2200 (x : L), x \u2208 s \u2192 p\u2082 \u2208 x\n\u22a2 p\u2081 = p\u2082\n[PROOFSTEP]\nobtain \u27e8l\u2081, l\u2082, hl\u2081, hl\u2082, hl\u2083\u27e9 := Finset.one_lt_card_iff.mp (Nat.one_lt_iff_ne_zero_and_ne_one.mpr \u27e8hs\u2080, hs\u2081\u27e9)\n[GOAL]\ncase intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\np\u2081 p\u2082 : P\nhp\u2081 : \u2200 (x : L), x \u2208 s \u2192 p\u2081 \u2208 x\nhp\u2082 : \u2200 (x : L), x \u2208 s \u2192 p\u2082 \u2208 x\nl\u2081 l\u2082 : L\nhl\u2081 : l\u2081 \u2208 s\nhl\u2082 : l\u2082 \u2208 s\nhl\u2083 : l\u2081 \u2260 l\u2082\n\u22a2 p\u2081 = p\u2082\n[PROOFSTEP]\nexact (eq_or_eq (hp\u2081 l\u2081 hl\u2081) (hp\u2082 l\u2081 hl\u2081) (hp\u2081 l\u2082 hl\u2082) (hp\u2082 l\u2082 hl\u2082)).resolve_right hl\u2083\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c \u2264 Finset.card s\u1d9c\n[PROOFSTEP]\nby_cases hs\u2083 : s\u1d9c.card = 0\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : Finset.card s\u1d9c = 0\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c \u2264 Finset.card s\u1d9c\n[PROOFSTEP]\nrw [hs\u2083, le_zero_iff]\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : Finset.card s\u1d9c = 0\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c = 0\n[PROOFSTEP]\nrw [Finset.card_compl, tsub_eq_zero_iff_le, LE.le.le_iff_eq (Finset.card_le_univ _), eq_comm,\n  Finset.card_eq_iff_eq_univ] at hs\u2083 \u22a2\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : s = Finset.univ\n\u22a2 Finset.biUnion s t = Finset.univ\n[PROOFSTEP]\nrw [hs\u2083]\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : s = Finset.univ\n\u22a2 Finset.biUnion Finset.univ t = Finset.univ\n[PROOFSTEP]\nrw [Finset.eq_univ_iff_forall] at hs\u2083 \u22a2\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : \u2200 (x : L), x \u2208 s\n\u22a2 \u2200 (x : P), x \u2208 Finset.biUnion Finset.univ t\n[PROOFSTEP]\nexact fun p =>\n  Exists.elim\n    (exists_line p)\n      -- If `s = univ`, then show `s.bUnion t = univ`\n    fun l hl => Finset.mem_biUnion.mpr \u27e8l, Finset.mem_univ l, Set.mem_toFinset.mpr hl\u27e9\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : Nondegenerate P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card L \u2264 Fintype.card P\nt : L \u2192 Finset P := fun l => Set.toFinset {p | \u00acp \u2208 l}\ns : Finset L\nhs\u2080 : \u00acFinset.card s = 0\nhs\u2081 : \u00acFinset.card s = 1\nhs\u2082 : Finset.card (Finset.biUnion s t)\u1d9c \u2264 1\nhs\u2083 : \u00acFinset.card s\u1d9c = 0\n\u22a2 Finset.card (Finset.biUnion s t)\u1d9c \u2264 Finset.card s\u1d9c\n[PROOFSTEP]\nexact hs\u2082.trans (Nat.one_le_iff_ne_zero.mpr hs\u2083)\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 \u2211 p : P, lineCount L p = \u2211 l : L, pointCount P l\n[PROOFSTEP]\nclassical\nsimp only [lineCount, pointCount, Nat.card_eq_fintype_card, \u2190 Fintype.card_sigma]\napply Fintype.card_congr\ncalc\n  (\u03a3 p, { l : L // p \u2208 l }) \u2243 { x : P \u00d7 L // x.1 \u2208 x.2 } := (Equiv.subtypeProdEquivSigmaSubtype (\u00b7 \u2208 \u00b7)).symm\n  _ \u2243 { x : L \u00d7 P // x.2 \u2208 x.1 } := ((Equiv.prodComm P L).subtypeEquiv fun x => Iff.rfl)\n  _ \u2243 \u03a3 l, { p // p \u2208 l } := Equiv.subtypeProdEquivSigmaSubtype fun (l : L) (p : P) => p \u2208 l\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 \u2211 p : P, lineCount L p = \u2211 l : L, pointCount P l\n[PROOFSTEP]\nsimp only [lineCount, pointCount, Nat.card_eq_fintype_card, \u2190 Fintype.card_sigma]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 Fintype.card ((a : P) \u00d7 { l // a \u2208 l }) = Fintype.card ((a : L) \u00d7 { p // p \u2208 a })\n[PROOFSTEP]\napply Fintype.card_congr\n[GOAL]\ncase f\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 (a : P) \u00d7 { l // a \u2208 l } \u2243 (a : L) \u00d7 { p // p \u2208 a }\n[PROOFSTEP]\ncalc\n  (\u03a3 p, { l : L // p \u2208 l }) \u2243 { x : P \u00d7 L // x.1 \u2208 x.2 } := (Equiv.subtypeProdEquivSigmaSubtype (\u00b7 \u2208 \u00b7)).symm\n  _ \u2243 { x : L \u00d7 P // x.2 \u2208 x.1 } := ((Equiv.prodComm P L).subtypeEquiv fun x => Iff.rfl)\n  _ \u2243 \u03a3 l, { p // p \u2208 l } := Equiv.subtypeProdEquivSigmaSubtype fun (l : L) (p : P) => p \u2208 l\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\n\u22a2 pointCount P l \u2264 lineCount L p\n[PROOFSTEP]\nby_cases hf : Infinite { p : P // p \u2208 l }\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : Infinite { p // p \u2208 l }\n\u22a2 pointCount P l \u2264 lineCount L p\n[PROOFSTEP]\nexact (le_of_eq Nat.card_eq_zero_of_infinite).trans (zero_le (lineCount L p))\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : \u00acInfinite { p // p \u2208 l }\n\u22a2 pointCount P l \u2264 lineCount L p\n[PROOFSTEP]\nhaveI := fintypeOfNotInfinite hf\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : \u00acInfinite { p // p \u2208 l }\nthis : Fintype { p // p \u2208 l }\n\u22a2 pointCount P l \u2264 lineCount L p\n[PROOFSTEP]\ncases nonempty_fintype { l : L // p \u2208 l }\n[GOAL]\ncase neg.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : \u00acInfinite { p // p \u2208 l }\nthis : Fintype { p // p \u2208 l }\nval\u271d : Fintype { l // p \u2208 l }\n\u22a2 pointCount P l \u2264 lineCount L p\n[PROOFSTEP]\nrw [lineCount, pointCount, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card]\n[GOAL]\ncase neg.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : \u00acInfinite { p // p \u2208 l }\nthis : Fintype { p // p \u2208 l }\nval\u271d : Fintype { l // p \u2208 l }\n\u22a2 Fintype.card { p // p \u2208 l } \u2264 Fintype.card { l // p \u2208 l }\n[PROOFSTEP]\nhave : \u2200 p' : { p // p \u2208 l }, p \u2260 p' := fun p' hp' => h ((congr_arg (\u00b7 \u2208 l) hp').mpr p'.2)\n[GOAL]\ncase neg.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b2 : Membership P L\ninst\u271d\u00b9 : HasLines P L\np : P\nl : L\nh : \u00acp \u2208 l\ninst\u271d : Finite { l // p \u2208 l }\nhf : \u00acInfinite { p // p \u2208 l }\nthis\u271d : Fintype { p // p \u2208 l }\nval\u271d : Fintype { l // p \u2208 l }\nthis : \u2200 (p' : { p // p \u2208 l }), p \u2260 \u2191p'\n\u22a2 Fintype.card { p // p \u2208 l } \u2264 Fintype.card { l // p \u2208 l }\n[PROOFSTEP]\nexact\n  Fintype.card_le_of_injective (fun p' => \u27e8mkLine (this p'), (mkLine_ax (this p')).1\u27e9) fun p\u2081 p\u2082 hp =>\n    Subtype.ext\n      ((eq_or_eq p\u2081.2 p\u2082.2 (mkLine_ax (this p\u2081)).2\n            ((congr_arg _ (Subtype.ext_iff.mp hp)).mpr (mkLine_ax (this p\u2082)).2)).resolve_right\n        fun h' => (congr_arg (\u00acp \u2208 \u00b7) h').mp h (mkLine_ax (this p\u2081)).1)\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 Fintype.card P \u2264 Fintype.card L\n[PROOFSTEP]\nclassical\nby_contra hc\u2082\nobtain \u27e8f, hf\u2081, hf\u2082\u27e9 := Nondegenerate.exists_injective_of_card_le (le_of_not_le hc\u2082)\nhave :=\n  calc\n    \u2211 p, lineCount L p = \u2211 l, pointCount P l := sum_lineCount_eq_sum_pointCount P L\n    _ \u2264 \u2211 l, lineCount L (f l) := (Finset.sum_le_sum fun l _ => HasLines.pointCount_le_lineCount (hf\u2082 l))\n    _ = \u2211 p in Finset.univ.image f, lineCount L p :=\n      (Finset.sum_bij (fun l _ => f l) (fun l hl => Finset.mem_image_of_mem f hl) (fun l _ => rfl)\n        (fun l\u2081 l\u2082 hl\u2081 hl\u2082 hl\u2083 => hf\u2081 hl\u2083) fun p => by\n        rw [Finset.mem_image]\n        exact fun \u27e8a, \u27e8h, h'\u27e9\u27e9 => \u27e8a, \u27e8h, h'.symm\u27e9\u27e9)\n    _ < \u2211 p, lineCount L p :=\n      by\n      obtain \u27e8p, hp\u27e9 := not_forall.mp (mt (Fintype.card_le_of_surjective f) hc\u2082)\n      refine'\n        Finset.sum_lt_sum_of_subset (Finset.univ.image f).subset_univ (Finset.mem_univ p) _ _ fun p _ _ =>\n          zero_le (lineCount L p)\n      \u00b7 simpa only [Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff]\n      \u00b7 rw [lineCount, Nat.card_eq_fintype_card, Fintype.card_pos_iff]\n        obtain \u27e8l, _\u27e9 := @exists_line P L _ _ p\n        exact\n          let this := not_exists.mp hp l\n          \u27e8\u27e8mkLine this, (mkLine_ax this).2\u27e9\u27e9\nexact lt_irrefl _ this\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\n\u22a2 Fintype.card P \u2264 Fintype.card L\n[PROOFSTEP]\nby_contra hc\u2082\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8f, hf\u2081, hf\u2082\u27e9 := Nondegenerate.exists_injective_of_card_le (le_of_not_le hc\u2082)\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\n\u22a2 False\n[PROOFSTEP]\nhave :=\n  calc\n    \u2211 p, lineCount L p = \u2211 l, pointCount P l := sum_lineCount_eq_sum_pointCount P L\n    _ \u2264 \u2211 l, lineCount L (f l) := (Finset.sum_le_sum fun l _ => HasLines.pointCount_le_lineCount (hf\u2082 l))\n    _ = \u2211 p in Finset.univ.image f, lineCount L p :=\n      (Finset.sum_bij (fun l _ => f l) (fun l hl => Finset.mem_image_of_mem f hl) (fun l _ => rfl)\n        (fun l\u2081 l\u2082 hl\u2081 hl\u2082 hl\u2083 => hf\u2081 hl\u2083) fun p => by\n        rw [Finset.mem_image]\n        exact fun \u27e8a, \u27e8h, h'\u27e9\u27e9 => \u27e8a, \u27e8h, h'.symm\u27e9\u27e9)\n    _ < \u2211 p, lineCount L p :=\n      by\n      obtain \u27e8p, hp\u27e9 := not_forall.mp (mt (Fintype.card_le_of_surjective f) hc\u2082)\n      refine'\n        Finset.sum_lt_sum_of_subset (Finset.univ.image f).subset_univ (Finset.mem_univ p) _ _ fun p _ _ =>\n          zero_le (lineCount L p)\n      \u00b7 simpa only [Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff]\n      \u00b7 rw [lineCount, Nat.card_eq_fintype_card, Fintype.card_pos_iff]\n        obtain \u27e8l, _\u27e9 := @exists_line P L _ _ p\n        exact\n          let this := not_exists.mp hp l\n          \u27e8\u27e8mkLine this, (mkLine_ax this).2\u27e9\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\n\u22a2 p \u2208 Finset.image f Finset.univ \u2192 \u2203 a ha, p = (fun l x => f l) a ha\n[PROOFSTEP]\nrw [Finset.mem_image]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\n\u22a2 (\u2203 a, a \u2208 Finset.univ \u2227 f a = p) \u2192 \u2203 a ha, p = (fun l x => f l) a ha\n[PROOFSTEP]\nexact fun \u27e8a, \u27e8h, h'\u27e9\u27e9 => \u27e8a, \u27e8h, h'.symm\u27e9\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\n\u22a2 \u2211 p in Finset.image f Finset.univ, lineCount L p < \u2211 p : P, lineCount L p\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := not_forall.mp (mt (Fintype.card_le_of_surjective f) hc\u2082)\n[GOAL]\ncase intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\nhp : \u00ac\u2203 a, f a = p\n\u22a2 \u2211 p in Finset.image f Finset.univ, lineCount L p < \u2211 p : P, lineCount L p\n[PROOFSTEP]\nrefine'\n  Finset.sum_lt_sum_of_subset (Finset.univ.image f).subset_univ (Finset.mem_univ p) _ _ fun p _ _ =>\n    zero_le (lineCount L p)\n[GOAL]\ncase intro.refine'_1\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\nhp : \u00ac\u2203 a, f a = p\n\u22a2 \u00acp \u2208 Finset.image f Finset.univ\n[PROOFSTEP]\nsimpa only [Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff]\n[GOAL]\ncase intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\nhp : \u00ac\u2203 a, f a = p\n\u22a2 0 < lineCount L p\n[PROOFSTEP]\nrw [lineCount, Nat.card_eq_fintype_card, Fintype.card_pos_iff]\n[GOAL]\ncase intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\nhp : \u00ac\u2203 a, f a = p\n\u22a2 Nonempty { l // p \u2208 l }\n[PROOFSTEP]\nobtain \u27e8l, _\u27e9 := @exists_line P L _ _ p\n[GOAL]\ncase intro.refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\np : P\nhp : \u00ac\u2203 a, f a = p\nl : L\nh\u271d : \u00acp \u2208 l\n\u22a2 Nonempty { l // p \u2208 l }\n[PROOFSTEP]\nexact\n  let this := not_exists.mp hp l\n  \u27e8\u27e8mkLine this, (mkLine_ax this).2\u27e9\u27e9\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhc\u2082 : \u00acFintype.card P \u2264 Fintype.card L\nf : L \u2192 P\nhf\u2081 : Function.Injective f\nhf\u2082 : \u2200 (l : L), \u00acf l \u2208 l\nthis : \u2211 p : P, lineCount L p < \u2211 p : P, lineCount L p\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ this\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)\n[PROOFSTEP]\nclassical\nobtain \u27e8f, hf1, hf2\u27e9 := Nondegenerate.exists_injective_of_card_le (ge_of_eq h)\nhave hf3 := (Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf1, h.symm\u27e9\nrefine'\n  \u27e8f, hf3, fun l =>\n    (Finset.sum_eq_sum_iff_of_le fun l _ => HasLines.pointCount_le_lineCount (hf2 l)).mp\n      ((sum_lineCount_eq_sum_pointCount P L).symm.trans\n        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l _ => refl (lineCount L (f l)))\n            (fun l\u2081 l\u2082 hl\u2081 hl\u2082 hl => hf1 hl) fun p hp => _).symm)\n      l (Finset.mem_univ l)\u27e9\nobtain \u27e8l, rfl\u27e9 := hf3.2 p\nexact \u27e8l, Finset.mem_univ l, rfl\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)\n[PROOFSTEP]\nobtain \u27e8f, hf1, hf2\u27e9 := Nondegenerate.exists_injective_of_card_le (ge_of_eq h)\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), \u00acf l \u2208 l\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)\n[PROOFSTEP]\nhave hf3 := (Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf1, h.symm\u27e9\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), \u00acf l \u2208 l\nhf3 : Function.Bijective f\n\u22a2 \u2203 f, Function.Bijective f \u2227 \u2200 (l : L), pointCount P l = lineCount L (f l)\n[PROOFSTEP]\nrefine'\n  \u27e8f, hf3, fun l =>\n    (Finset.sum_eq_sum_iff_of_le fun l _ => HasLines.pointCount_le_lineCount (hf2 l)).mp\n      ((sum_lineCount_eq_sum_pointCount P L).symm.trans\n        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l _ => refl (lineCount L (f l)))\n            (fun l\u2081 l\u2082 hl\u2081 hl\u2082 hl => hf1 hl) fun p hp => _).symm)\n      l (Finset.mem_univ l)\u27e9\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), \u00acf l \u2208 l\nhf3 : Function.Bijective f\nl : L\np : P\nhp : p \u2208 Finset.univ\n\u22a2 \u2203 a ha, p = (fun l x => f l) a ha\n[PROOFSTEP]\nobtain \u27e8l, rfl\u27e9 := hf3.2 p\n[GOAL]\ncase intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nf : L \u2192 P\nhf1 : Function.Injective f\nhf2 : \u2200 (l : L), \u00acf l \u2208 l\nhf3 : Function.Bijective f\nl\u271d l : L\nhp : f l \u2208 Finset.univ\n\u22a2 \u2203 a ha, f l = (fun l x => f l) a ha\n[PROOFSTEP]\nexact \u27e8l, Finset.mem_univ l, rfl\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nclassical\nobtain \u27e8f, hf1, hf2\u27e9 := HasLines.exists_bijective_of_card_eq hPL\nlet s : Finset (P \u00d7 L) := Set.toFinset {i | i.1 \u2208 i.2}\nhave step1 : \u2211 i : P \u00d7 L, lineCount L i.1 = \u2211 i : P \u00d7 L, pointCount P i.2 :=\n  by\n  rw [\u2190 Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]\n  simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]\nhave step2 : \u2211 i in s, lineCount L i.1 = \u2211 i in s, pointCount P i.2 :=\n  by\n  rw [s.sum_finset_product Finset.univ fun p => Set.toFinset {l | p \u2208 l}]\n  rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset {p | p \u2208 l}]\n  refine'\n    (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _) (fun _ _ _ _ h => hf1.1 h)\n        fun p _ => _).symm\n  \u00b7 simp_rw [Finset.sum_const, Set.toFinset_card, \u2190 Nat.card_eq_fintype_card]\n    change pointCount P l \u2022 pointCount P l = lineCount L (f l) \u2022 lineCount L (f l)\n    rw [hf2]\n  \u00b7 obtain \u27e8l, hl\u27e9 := hf1.2 p\n    exact \u27e8l, Finset.mem_univ l, hl.symm\u27e9\n  all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl\nhave step3 : \u2211 i in s\u1d9c, lineCount L i.1 = \u2211 i in s\u1d9c, pointCount P i.2 := by\n  rwa [\u2190 s.sum_add_sum_compl, \u2190 s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1 \nrw [\u2190 Set.toFinset_compl] at step3 \nexact\n  ((Finset.sum_eq_sum_iff_of_le fun i hi => HasLines.pointCount_le_lineCount (by exact Set.mem_toFinset.mp hi)).mp\n      step3.symm (p, l) (Set.mem_toFinset.mpr hpl)).symm\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nobtain \u27e8f, hf1, hf2\u27e9 := HasLines.exists_bijective_of_card_eq hPL\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nlet s : Finset (P \u00d7 L) := Set.toFinset {i | i.1 \u2208 i.2}\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nhave step1 : \u2211 i : P \u00d7 L, lineCount L i.1 = \u2211 i : P \u00d7 L, pointCount P i.2 :=\n  by\n  rw [\u2190 Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]\n  simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\n\u22a2 \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n[PROOFSTEP]\nrw [\u2190 Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\n\u22a2 \u2211 y : L, \u2211 x : P, lineCount L (x, y).fst = \u2211 x : P, \u2211 y : L, pointCount P (x, y).snd\n[PROOFSTEP]\nsimp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nhave step2 : \u2211 i in s, lineCount L i.1 = \u2211 i in s, pointCount P i.2 :=\n  by\n  rw [s.sum_finset_product Finset.univ fun p => Set.toFinset {l | p \u2208 l}]\n  rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset {p | p \u2208 l}]\n  refine'\n    (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _) (fun _ _ _ _ h => hf1.1 h)\n        fun p _ => _).symm\n  \u00b7 simp_rw [Finset.sum_const, Set.toFinset_card, \u2190 Nat.card_eq_fintype_card]\n    change pointCount P l \u2022 pointCount P l = lineCount L (f l) \u2022 lineCount L (f l)\n    rw [hf2]\n  \u00b7 obtain \u27e8l, hl\u27e9 := hf1.2 p\n    exact \u27e8l, Finset.mem_univ l, hl.symm\u27e9\n  all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\n[PROOFSTEP]\nrw [s.sum_finset_product Finset.univ fun p => Set.toFinset {l | p \u2208 l}]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2211 c : P, \u2211 a in Set.toFinset {l | c \u2208 l}, lineCount L (c, a).fst = \u2211 i in s, pointCount P i.snd\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.fst \u2208 Finset.univ \u2227 p.snd \u2208 Set.toFinset {l | p.fst \u2208 l}\n[PROOFSTEP]\nrw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset {p | p \u2208 l}]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2211 c : P, \u2211 a in Set.toFinset {l | c \u2208 l}, lineCount L (c, a).fst =\n    \u2211 c : L, \u2211 a in Set.toFinset {p | p \u2208 c}, pointCount P (a, c).snd\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.snd \u2208 Finset.univ \u2227 p.fst \u2208 Set.toFinset {p_1 | p_1 \u2208 p.snd}\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.fst \u2208 Finset.univ \u2227 p.snd \u2208 Set.toFinset {l | p.fst \u2208 l}\n[PROOFSTEP]\nrefine'\n  (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _) (fun _ _ _ _ h => hf1.1 h)\n      fun p _ => _).symm\n[GOAL]\ncase refine'_1\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl\u271d : L\nhpl : \u00acp \u2208 l\u271d\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nl : L\nhl : l \u2208 Finset.univ\n\u22a2 \u2211 a in Set.toFinset {p | p \u2208 l}, pointCount P (a, l).snd =\n    \u2211 a in Set.toFinset {l_1 | (fun l x => f l) l hl \u2208 l_1}, lineCount L ((fun l x => f l) l hl, a).fst\n[PROOFSTEP]\nsimp_rw [Finset.sum_const, Set.toFinset_card, \u2190 Nat.card_eq_fintype_card]\n[GOAL]\ncase refine'_1\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl\u271d : L\nhpl : \u00acp \u2208 l\u271d\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nl : L\nhl : l \u2208 Finset.univ\n\u22a2 Nat.card \u2191{p | p \u2208 l} \u2022 pointCount P l = Nat.card \u2191{l_1 | f l \u2208 l_1} \u2022 lineCount L (f l)\n[PROOFSTEP]\nchange pointCount P l \u2022 pointCount P l = lineCount L (f l) \u2022 lineCount L (f l)\n[GOAL]\ncase refine'_1\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl\u271d : L\nhpl : \u00acp \u2208 l\u271d\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nl : L\nhl : l \u2208 Finset.univ\n\u22a2 pointCount P l \u2022 pointCount P l = lineCount L (f l) \u2022 lineCount L (f l)\n[PROOFSTEP]\nrw [hf2]\n[GOAL]\ncase refine'_2\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np\u271d : P\nl : L\nhpl : \u00acp\u271d \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\np : P\nx\u271d : p \u2208 Finset.univ\n\u22a2 \u2203 a ha, p = (fun l x => f l) a ha\n[PROOFSTEP]\nobtain \u27e8l, hl\u27e9 := hf1.2 p\n[GOAL]\ncase refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np\u271d : P\nl\u271d : L\nhpl : \u00acp\u271d \u2208 l\u271d\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\np : P\nx\u271d : p \u2208 Finset.univ\nl : L\nhl : f l = p\n\u22a2 \u2203 a ha, p = (fun l x => f l) a ha\n[PROOFSTEP]\nexact \u27e8l, Finset.mem_univ l, hl.symm\u27e9\n[GOAL]\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.snd \u2208 Finset.univ \u2227 p.fst \u2208 Set.toFinset {p_1 | p_1 \u2208 p.snd}\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.fst \u2208 Finset.univ \u2227 p.snd \u2208 Set.toFinset {l | p.fst \u2208 l}\n[PROOFSTEP]\nall_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl\n[GOAL]\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.snd \u2208 Finset.univ \u2227 p.fst \u2208 Set.toFinset {p_1 | p_1 \u2208 p.snd}\n[PROOFSTEP]\nsimp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]\n[GOAL]\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 {i | i.fst \u2208 i.snd} \u2194 p.fst \u2208 {p_1 | p_1 \u2208 p.snd}\n[PROOFSTEP]\nexact fun p => Iff.rfl\n[GOAL]\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 s \u2194 p.fst \u2208 Finset.univ \u2227 p.snd \u2208 Set.toFinset {l | p.fst \u2208 l}\n[PROOFSTEP]\nsimp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]\n[GOAL]\ncase h\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\n\u22a2 \u2200 (p : P \u00d7 L), p \u2208 {i | i.fst \u2208 i.snd} \u2194 p.snd \u2208 {l | p.fst \u2208 l}\n[PROOFSTEP]\nexact fun p => Iff.rfl\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nstep2 : \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nhave step3 : \u2211 i in s\u1d9c, lineCount L i.1 = \u2211 i in s\u1d9c, pointCount P i.2 := by\n  rwa [\u2190 s.sum_add_sum_compl, \u2190 s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1 \n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nstep2 : \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\n\u22a2 \u2211 i in s\u1d9c, lineCount L i.fst = \u2211 i in s\u1d9c, pointCount P i.snd\n[PROOFSTEP]\nrwa [\u2190 s.sum_add_sum_compl, \u2190 s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1 \n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nstep2 : \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\nstep3 : \u2211 i in s\u1d9c, lineCount L i.fst = \u2211 i in s\u1d9c, pointCount P i.snd\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_compl] at step3 \n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nstep2 : \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\nstep3 :\n  \u2211 i in Set.toFinset {i | i.fst \u2208 i.snd}\u1d9c, lineCount L i.fst =\n    \u2211 i in Set.toFinset {i | i.fst \u2208 i.snd}\u1d9c, pointCount P i.snd\n\u22a2 lineCount L p = pointCount P l\n[PROOFSTEP]\nexact\n  ((Finset.sum_eq_sum_iff_of_le fun i hi => HasLines.pointCount_le_lineCount (by exact Set.mem_toFinset.mp hi)).mp\n      step3.symm (p, l) (Set.mem_toFinset.mpr hpl)).symm\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nhPL : Fintype.card P = Fintype.card L\np : P\nl : L\nhpl : \u00acp \u2208 l\nf : L \u2192 P\nhf1 : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\ns : Finset (P \u00d7 L) := Set.toFinset {i | i.fst \u2208 i.snd}\nstep1 : \u2211 i : P \u00d7 L, lineCount L i.fst = \u2211 i : P \u00d7 L, pointCount P i.snd\nstep2 : \u2211 i in s, lineCount L i.fst = \u2211 i in s, pointCount P i.snd\nstep3 :\n  \u2211 i in Set.toFinset {i | i.fst \u2208 i.snd}\u1d9c, lineCount L i.fst =\n    \u2211 i in Set.toFinset {i | i.fst \u2208 i.snd}\u1d9c, pointCount P i.snd\ni : P \u00d7 L\nhi : i \u2208 Set.toFinset {i | i.fst \u2208 i.snd}\u1d9c\n\u22a2 \u00aci.fst \u2208 i.snd\n[PROOFSTEP]\nexact Set.mem_toFinset.mp hi\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nclassical\nobtain \u27e8f, _, hf2\u27e9 := HasLines.exists_bijective_of_card_eq h\nhaveI : Nontrivial L := \u27e8\u27e8l\u2081, l\u2082, hl\u27e9\u27e9\nhaveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)\nhave h\u2081 : \u2200 p : P, 0 < lineCount L p := fun p =>\n  Exists.elim (exists_ne p) fun q hq =>\n    (congr_arg _ Nat.card_eq_fintype_card).mpr (Fintype.card_pos_iff.mpr \u27e8\u27e8mkLine hq, (mkLine_ax hq).2\u27e9\u27e9)\nhave h\u2082 : \u2200 l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h\u2081 (f l))\nobtain \u27e8p, hl\u2081\u27e9 := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h\u2082 l\u2081))\nby_cases hl\u2082 : p \u2208 l\u2082\nexact \u27e8p, hl\u2081, hl\u2082\u27e9\nhave key' : Fintype.card { q : P // q \u2208 l\u2082 } = Fintype.card { l : L // p \u2208 l } :=\n  ((HasLines.lineCount_eq_pointCount h hl\u2082).trans Nat.card_eq_fintype_card).symm.trans Nat.card_eq_fintype_card\nhave : \u2200 q : { q // q \u2208 l\u2082 }, p \u2260 q := fun q hq => hl\u2082 ((congr_arg (\u00b7 \u2208 l\u2082) hq).mpr q.2)\nlet f : { q : P // q \u2208 l\u2082 } \u2192 { l : L // p \u2208 l } := fun q => \u27e8mkLine (this q), (mkLine_ax (this q)).1\u27e9\nhave hf : Function.Injective f := fun q\u2081 q\u2082 hq =>\n  Subtype.ext\n    ((eq_or_eq q\u2081.2 q\u2082.2 (mkLine_ax (this q\u2081)).2\n          ((congr_arg _ (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q\u2082)).2)).resolve_right\n      fun h => (congr_arg (\u00acp \u2208 \u00b7) h).mp hl\u2082 (mkLine_ax (this q\u2081)).1)\nhave key' := ((Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf, key'\u27e9).2\nobtain \u27e8q, hq\u27e9 := key' \u27e8l\u2081, hl\u2081\u27e9\nexact \u27e8q, (congr_arg _ (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nobtain \u27e8f, _, hf2\u27e9 := HasLines.exists_bijective_of_card_eq h\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhaveI : Nontrivial L := \u27e8\u27e8l\u2081, l\u2082, hl\u27e9\u27e9\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis : Nontrivial L\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhaveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave h\u2081 : \u2200 p : P, 0 < lineCount L p := fun p =>\n  Exists.elim (exists_ne p) fun q hq =>\n    (congr_arg _ Nat.card_eq_fintype_card).mpr (Fintype.card_pos_iff.mpr \u27e8\u27e8mkLine hq, (mkLine_ax hq).2\u27e9\u27e9)\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave h\u2082 : \u2200 l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h\u2081 (f l))\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nobtain \u27e8p, hl\u2081\u27e9 := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h\u2082 l\u2081))\n[GOAL]\ncase intro.intro.intro.mk\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nby_cases hl\u2082 : p \u2208 l\u2082\n[GOAL]\ncase pos\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : p \u2208 l\u2082\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nexact \u27e8p, hl\u2081, hl\u2082\u27e9\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave key' : Fintype.card { q : P // q \u2208 l\u2082 } = Fintype.card { l : L // p \u2208 l } :=\n  ((HasLines.lineCount_eq_pointCount h hl\u2082).trans Nat.card_eq_fintype_card).symm.trans Nat.card_eq_fintype_card\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d : Nontrivial L\nthis : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey' : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave : \u2200 q : { q // q \u2208 l\u2082 }, p \u2260 q := fun q hq => hl\u2082 ((congr_arg (\u00b7 \u2208 l\u2082) hq).mpr q.2)\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf : L \u2192 P\nleft\u271d : Function.Bijective f\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f l)\nthis\u271d\u00b9 : Nontrivial L\nthis\u271d : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey' : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\nthis : \u2200 (q : { q // q \u2208 l\u2082 }), p \u2260 \u2191q\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nlet f : { q : P // q \u2208 l\u2082 } \u2192 { l : L // p \u2208 l } := fun q => \u27e8mkLine (this q), (mkLine_ax (this q)).1\u27e9\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf\u271d : L \u2192 P\nleft\u271d : Function.Bijective f\u271d\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f\u271d l)\nthis\u271d\u00b9 : Nontrivial L\nthis\u271d : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey' : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\nthis : \u2200 (q : { q // q \u2208 l\u2082 }), p \u2260 \u2191q\nf : { q // q \u2208 l\u2082 } \u2192 { l // p \u2208 l } :=\n  fun q => { val := mkLine (_ : p \u2260 \u2191q), property := (_ : p \u2208 mkLine (_ : p \u2260 \u2191q)) }\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave hf : Function.Injective f := fun q\u2081 q\u2082 hq =>\n  Subtype.ext\n    ((eq_or_eq q\u2081.2 q\u2082.2 (mkLine_ax (this q\u2081)).2\n          ((congr_arg _ (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q\u2082)).2)).resolve_right\n      fun h => (congr_arg (\u00acp \u2208 \u00b7) h).mp hl\u2082 (mkLine_ax (this q\u2081)).1)\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf\u271d : L \u2192 P\nleft\u271d : Function.Bijective f\u271d\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f\u271d l)\nthis\u271d\u00b9 : Nontrivial L\nthis\u271d : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey' : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\nthis : \u2200 (q : { q // q \u2208 l\u2082 }), p \u2260 \u2191q\nf : { q // q \u2208 l\u2082 } \u2192 { l // p \u2208 l } :=\n  fun q => { val := mkLine (_ : p \u2260 \u2191q), property := (_ : p \u2208 mkLine (_ : p \u2260 \u2191q)) }\nhf : Function.Injective f\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nhave key' := ((Fintype.bijective_iff_injective_and_card f).mpr \u27e8hf, key'\u27e9).2\n[GOAL]\ncase neg\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf\u271d : L \u2192 P\nleft\u271d : Function.Bijective f\u271d\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f\u271d l)\nthis\u271d\u00b9 : Nontrivial L\nthis\u271d : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey'\u271d : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\nthis : \u2200 (q : { q // q \u2208 l\u2082 }), p \u2260 \u2191q\nf : { q // q \u2208 l\u2082 } \u2192 { l // p \u2208 l } :=\n  fun q => { val := mkLine (_ : p \u2260 \u2191q), property := (_ : p \u2208 mkLine (_ : p \u2260 \u2191q)) }\nhf : Function.Injective f\nkey' : Function.Surjective f\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nobtain \u27e8q, hq\u27e9 := key' \u27e8l\u2081, hl\u2081\u27e9\n[GOAL]\ncase neg.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : HasLines P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Fintype L\nh : Fintype.card P = Fintype.card L\nl\u2081 l\u2082 : L\nhl : l\u2081 \u2260 l\u2082\nf\u271d : L \u2192 P\nleft\u271d : Function.Bijective f\u271d\nhf2 : \u2200 (l : L), pointCount P l = lineCount L (f\u271d l)\nthis\u271d\u00b9 : Nontrivial L\nthis\u271d : Nontrivial P\nh\u2081 : \u2200 (p : P), 0 < lineCount L p\nh\u2082 : \u2200 (l : L), 0 < pointCount P l\np : P\nhl\u2081 : p \u2208 l\u2081\nhl\u2082 : \u00acp \u2208 l\u2082\nkey'\u271d : Fintype.card { q // q \u2208 l\u2082 } = Fintype.card { l // p \u2208 l }\nthis : \u2200 (q : { q // q \u2208 l\u2082 }), p \u2260 \u2191q\nf : { q // q \u2208 l\u2082 } \u2192 { l // p \u2208 l } :=\n  fun q => { val := mkLine (_ : p \u2260 \u2191q), property := (_ : p \u2208 mkLine (_ : p \u2260 \u2191q)) }\nhf : Function.Injective f\nkey' : Function.Surjective f\nq : { q // q \u2208 l\u2082 }\nhq : f q = { val := l\u2081, property := hl\u2081 }\n\u22a2 \u2203 p, p \u2208 l\u2081 \u2227 p \u2208 l\u2082\n[PROOFSTEP]\nexact \u27e8q, (congr_arg _ (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\ncases nonempty_fintype P\n[GOAL]\ncase intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d : Fintype P\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\ncases nonempty_fintype L\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nobtain \u27e8p\u2081, p\u2082, p\u2083, l\u2081, l\u2082, l\u2083, h\u2081\u2082, h\u2081\u2083, h\u2082\u2081, h\u2082\u2082, h\u2082\u2083, h\u2083\u2081, h\u2083\u2082, h\u2083\u2083\u27e9 := @exists_config P L _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave h := card_points_eq_card_lines P L\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nlet n := lineCount L p\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hp\u2082 : lineCount L p\u2082 = n := rfl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hl\u2081 : pointCount P l\u2081 = n := (HasLines.lineCount_eq_pointCount h h\u2082\u2081).symm.trans hp\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hp\u2083 : lineCount L p\u2083 = n := (HasLines.lineCount_eq_pointCount h h\u2083\u2081).trans hl\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hl\u2083 : pointCount P l\u2083 = n := (HasLines.lineCount_eq_pointCount h h\u2083\u2083).symm.trans hp\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hp\u2081 : lineCount L p\u2081 = n := (HasLines.lineCount_eq_pointCount h h\u2081\u2083).trans hl\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nhave hl\u2082 : pointCount P l\u2082 = n := (HasLines.lineCount_eq_pointCount h h\u2081\u2082).symm.trans hp\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\nhl\u2082 : pointCount P l\u2082 = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nsuffices \u2200 p : P, lineCount L p = n by exact (this p).trans (this q).symm\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\nhl\u2082 : pointCount P l\u2082 = n\nthis : \u2200 (p : P), lineCount L p = n\n\u22a2 lineCount L p = lineCount L q\n[PROOFSTEP]\nexact (this p).trans (this q).symm\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\nhl\u2082 : pointCount P l\u2082 = n\n\u22a2 \u2200 (p : P), lineCount L p = n\n[PROOFSTEP]\nrefine' fun p => or_not.elim (fun h\u2082 => _) fun h\u2082 => (HasLines.lineCount_eq_pointCount h h\u2082).trans hl\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u271d q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\nhl\u2082 : pointCount P l\u2082 = n\np : P\nh\u2082 : p \u2208 l\u2082\n\u22a2 lineCount L p = n\n[PROOFSTEP]\nrefine' or_not.elim (fun h\u2083 => _) fun h\u2083 => (HasLines.lineCount_eq_pointCount h h\u2083).trans hl\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u271d q : P\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2081\u2082 : \u00acp\u2081 \u2208 l\u2082\nh\u2081\u2083 : \u00acp\u2081 \u2208 l\u2083\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nh : Fintype.card P = Fintype.card L\nn : \u2115 := lineCount L p\u2082\nhp\u2082 : lineCount L p\u2082 = n\nhl\u2081 : pointCount P l\u2081 = n\nhp\u2083 : lineCount L p\u2083 = n\nhl\u2083 : pointCount P l\u2083 = n\nhp\u2081 : lineCount L p\u2081 = n\nhl\u2082 : pointCount P l\u2082 = n\np : P\nh\u2082 : p \u2208 l\u2082\nh\u2083 : p \u2208 l\u2083\n\u22a2 lineCount L p = n\n[PROOFSTEP]\nrw [(eq_or_eq h\u2082 h\u2082\u2082 h\u2083 h\u2082\u2083).resolve_right fun h => h\u2083\u2083 ((congr_arg (Membership.mem p\u2083) h).mp h\u2083\u2082)]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\nl m : L\n\u22a2 pointCount P l = pointCount P m\n[PROOFSTEP]\napply lineCount_eq_lineCount (Dual P)\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\nl : L\nq : P\nhq : \u00acq \u2208 l\n\u22a2 lineCount L q = pointCount P l\n[PROOFSTEP]\ncases nonempty_fintype P\n[GOAL]\ncase intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\nl : L\nq : P\nhq : \u00acq \u2208 l\nval\u271d : Fintype P\n\u22a2 lineCount L q = pointCount P l\n[PROOFSTEP]\ncases nonempty_fintype L\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\nl : L\nq : P\nhq : \u00acq \u2208 l\nval\u271d\u00b9 : Fintype P\nval\u271d : Fintype L\n\u22a2 lineCount L q = pointCount P l\n[PROOFSTEP]\nexact HasLines.lineCount_eq_pointCount (card_points_eq_card_lines P L) hq\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\n\u22a2 lineCount L p = order P L + 1\n[PROOFSTEP]\nclassical\nobtain \u27e8q, -, -, l, -, -, -, -, h, -\u27e9 := Classical.choose_spec (@exists_config P L _ _)\ncases nonempty_fintype { l : L // q \u2208 l }\nrw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q, lineCount,\n  Nat.card_eq_fintype_card, Nat.sub_add_cancel]\nexact Fintype.card_pos_iff.mpr \u27e8\u27e8l, h\u27e9\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\n\u22a2 lineCount L p = order P L + 1\n[PROOFSTEP]\nobtain \u27e8q, -, -, l, -, -, -, -, h, -\u27e9 := Classical.choose_spec (@exists_config P L _ _)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\n\u22a2 lineCount L p = order P L + 1\n[PROOFSTEP]\ncases nonempty_fintype { l : L // q \u2208 l }\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 lineCount L p = order P L + 1\n[PROOFSTEP]\nrw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q, lineCount,\n  Nat.card_eq_fintype_card, Nat.sub_add_cancel]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np q : P\nl : L\nh : q \u2208 l\nval\u271d : Fintype { l // q \u2208 l }\n\u22a2 1 \u2264 Fintype.card { l // q \u2208 l }\n[PROOFSTEP]\nexact Fintype.card_pos_iff.mpr \u27e8\u27e8l, h\u27e9\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\n\u22a2 1 < order P L\n[PROOFSTEP]\nobtain \u27e8p\u2081, p\u2082, p\u2083, l\u2081, l\u2082, l\u2083, -, -, h\u2082\u2081, h\u2082\u2082, h\u2082\u2083, h\u2083\u2081, h\u2083\u2082, h\u2083\u2083\u27e9 := @exists_config P L _ _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\n\u22a2 1 < order P L\n[PROOFSTEP]\ncases nonempty_fintype { p : P // p \u2208 l\u2082 }\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nval\u271d : Fintype { p // p \u2208 l\u2082 }\n\u22a2 1 < order P L\n[PROOFSTEP]\nrw [\u2190 add_lt_add_iff_right 1, \u2190 pointCount_eq _ l\u2082, pointCount, Nat.card_eq_fintype_card, Fintype.two_lt_card_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nval\u271d : Fintype { p // p \u2208 l\u2082 }\n\u22a2 \u2203 a b c, a \u2260 b \u2227 a \u2260 c \u2227 b \u2260 c\n[PROOFSTEP]\nsimp_rw [Ne, Subtype.ext_iff]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nval\u271d : Fintype { p // p \u2208 l\u2082 }\n\u22a2 \u2203 a b c, \u00ac\u2191a = \u2191b \u2227 \u00ac\u2191a = \u2191c \u2227 \u00ac\u2191b = \u2191c\n[PROOFSTEP]\nhave h := mkPoint_ax fun h => h\u2082\u2081 ((congr_arg _ h).mpr h\u2082\u2082)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np\u2081 p\u2082 p\u2083 : P\nl\u2081 l\u2082 l\u2083 : L\nh\u2082\u2081 : \u00acp\u2082 \u2208 l\u2081\nh\u2082\u2082 : p\u2082 \u2208 l\u2082\nh\u2082\u2083 : p\u2082 \u2208 l\u2083\nh\u2083\u2081 : \u00acp\u2083 \u2208 l\u2081\nh\u2083\u2082 : p\u2083 \u2208 l\u2082\nh\u2083\u2083 : \u00acp\u2083 \u2208 l\u2083\nval\u271d : Fintype { p // p \u2208 l\u2082 }\nh : mkPoint (_ : l\u2081 = l\u2082 \u2192 False) \u2208 l\u2081 \u2227 mkPoint (_ : l\u2081 = l\u2082 \u2192 False) \u2208 l\u2082\n\u22a2 \u2203 a b c, \u00ac\u2191a = \u2191b \u2227 \u00ac\u2191a = \u2191c \u2227 \u00ac\u2191b = \u2191c\n[PROOFSTEP]\nexact\n  \u27e8\u27e8mkPoint _, h.2\u27e9, \u27e8p\u2082, h\u2082\u2082\u27e9, \u27e8p\u2083, h\u2083\u2082\u27e9, ne_of_mem_of_not_mem h.1 h\u2082\u2081, ne_of_mem_of_not_mem h.1 h\u2083\u2081,\n    ne_of_mem_of_not_mem h\u2082\u2083 h\u2083\u2083\u27e9\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\np : P\n\u22a2 2 < lineCount L p\n[PROOFSTEP]\nsimpa only [lineCount_eq L p, Nat.succ_lt_succ_iff] using one_lt_order P L\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Finite P\ninst\u271d : Finite L\nl : L\n\u22a2 2 < pointCount P l\n[PROOFSTEP]\nsimpa only [pointCount_eq P l, Nat.succ_lt_succ_iff] using one_lt_order P L\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\ncases nonempty_fintype L\n[GOAL]\ncase intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nobtain \u27e8p, -\u27e9 := @exists_config P L _ _\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nlet \u03d5 : { q // q \u2260 p } \u2243 \u03a3 l : { l : L // p \u2208 l }, { q // q \u2208 l.1 \u2227 q \u2260 p } :=\n  { toFun := fun q => \u27e8\u27e8mkLine q.2, (mkLine_ax q.2).2\u27e9, q, (mkLine_ax q.2).1, q.2\u27e9\n    invFun := fun lq => \u27e8lq.2, lq.2.2.2\u27e9\n    left_inv := fun q => Subtype.ext rfl\n    right_inv := fun lq =>\n      Sigma.subtype_ext\n        (Subtype.ext ((eq_or_eq (mkLine_ax lq.2.2.2).1 (mkLine_ax lq.2.2.2).2 lq.2.2.1 lq.1.2).resolve_left lq.2.2.2))\n        rfl }\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nclassical\nhave h1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P :=\n  by\n  apply (eq_tsub_iff_add_eq_of_le (Nat.succ_le_of_lt (Fintype.card_pos_iff.mpr \u27e8p\u27e9))).mp\n  convert (Fintype.card_subtype_compl _).trans (congr_arg _ (Fintype.card_subtype_eq p))\nhave h2 : \u2200 l : { l : L // p \u2208 l }, Fintype.card { q // q \u2208 l.1 \u2227 q \u2260 p } = order P L :=\n  by\n  intro l\n  rw [\u2190 Fintype.card_congr (Equiv.subtypeSubtypeEquivSubtypeInter (\u00b7 \u2208 l.val) (\u00b7 \u2260 p)),\n    Fintype.card_subtype_compl fun x : Subtype (\u00b7 \u2208 l.val) => x.val = p, \u2190 Nat.card_eq_fintype_card]\n  refine' tsub_eq_of_eq_add ((pointCount_eq P l.1).trans _)\n  rw [\u2190 Fintype.card_subtype_eq (\u27e8p, l.2\u27e9 : { q : P // q \u2208 l.1 })]\n  simp_rw [Subtype.ext_iff_val]\nsimp_rw [\u2190 h1, Fintype.card_congr \u03d5, Fintype.card_sigma, h2, Finset.sum_const, Finset.card_univ]\nrw [\u2190 Nat.card_eq_fintype_card, \u2190 lineCount, lineCount_eq, smul_eq_mul, Nat.succ_mul, sq]\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nhave h1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P :=\n  by\n  apply (eq_tsub_iff_add_eq_of_le (Nat.succ_le_of_lt (Fintype.card_pos_iff.mpr \u27e8p\u27e9))).mp\n  convert (Fintype.card_subtype_compl _).trans (congr_arg _ (Fintype.card_subtype_eq p))\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\n\u22a2 Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\n[PROOFSTEP]\napply (eq_tsub_iff_add_eq_of_le (Nat.succ_le_of_lt (Fintype.card_pos_iff.mpr \u27e8p\u27e9))).mp\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\n\u22a2 Fintype.card { q // q \u2260 p } = Fintype.card P - Nat.succ 0\n[PROOFSTEP]\nconvert (Fintype.card_subtype_compl _).trans (congr_arg _ (Fintype.card_subtype_eq p))\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nhave h2 : \u2200 l : { l : L // p \u2208 l }, Fintype.card { q // q \u2208 l.1 \u2227 q \u2260 p } = order P L :=\n  by\n  intro l\n  rw [\u2190 Fintype.card_congr (Equiv.subtypeSubtypeEquivSubtypeInter (\u00b7 \u2208 l.val) (\u00b7 \u2260 p)),\n    Fintype.card_subtype_compl fun x : Subtype (\u00b7 \u2208 l.val) => x.val = p, \u2190 Nat.card_eq_fintype_card]\n  refine' tsub_eq_of_eq_add ((pointCount_eq P l.1).trans _)\n  rw [\u2190 Fintype.card_subtype_eq (\u27e8p, l.2\u27e9 : { q : P // q \u2208 l.1 })]\n  simp_rw [Subtype.ext_iff_val]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\n\u22a2 \u2200 (l : { l // p \u2208 l }), Fintype.card { q // q \u2208 \u2191l \u2227 q \u2260 p } = order P L\n[PROOFSTEP]\nintro l\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nl : { l // p \u2208 l }\n\u22a2 Fintype.card { q // q \u2208 \u2191l \u2227 q \u2260 p } = order P L\n[PROOFSTEP]\nrw [\u2190 Fintype.card_congr (Equiv.subtypeSubtypeEquivSubtypeInter (\u00b7 \u2208 l.val) (\u00b7 \u2260 p)),\n  Fintype.card_subtype_compl fun x : Subtype (\u00b7 \u2208 l.val) => x.val = p, \u2190 Nat.card_eq_fintype_card]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nl : { l // p \u2208 l }\n\u22a2 Nat.card { x // x \u2208 \u2191l } - Fintype.card { x // \u2191x = p } = order P L\n[PROOFSTEP]\nrefine' tsub_eq_of_eq_add ((pointCount_eq P l.1).trans _)\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nl : { l // p \u2208 l }\n\u22a2 order P L + 1 = order P L + Fintype.card { x // \u2191x = p }\n[PROOFSTEP]\nrw [\u2190 Fintype.card_subtype_eq (\u27e8p, l.2\u27e9 : { q : P // q \u2208 l.1 })]\n[GOAL]\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nl : { l // p \u2208 l }\n\u22a2 order P L + Fintype.card { x // x = { val := p, property := (_ : p \u2208 \u2191l) } } =\n    order P L + Fintype.card { x // \u2191x = p }\n[PROOFSTEP]\nsimp_rw [Subtype.ext_iff_val]\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nh2 : \u2200 (l : { l // p \u2208 l }), Fintype.card { q // q \u2208 \u2191l \u2227 q \u2260 p } = order P L\n\u22a2 Fintype.card P = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nsimp_rw [\u2190 h1, Fintype.card_congr \u03d5, Fintype.card_sigma, h2, Finset.sum_const, Finset.card_univ]\n[GOAL]\ncase intro.intro\nP : Type u_1\nL : Type u_2\ninst\u271d\u00b3 : Membership P L\ninst\u271d\u00b2 : ProjectivePlane P L\ninst\u271d\u00b9 : Fintype P\ninst\u271d : Finite L\nval\u271d : Fintype L\np : P\n\u03d5 : { q // q \u2260 p } \u2243 (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p } :=\n  {\n    toFun := fun q =>\n      { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n        snd :=\n          { val := \u2191q,\n            property :=\n              (_ : \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } },\n    invFun := fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) },\n    left_inv :=\n      (_ :\n        \u2200 (q : { q // q \u2260 p }),\n          (fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) })\n              ((fun q =>\n                  { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                    snd :=\n                      { val := \u2191q,\n                        property :=\n                          (_ :\n                            \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227\n                              \u2191q \u2260 p) } })\n                q) =\n            q),\n    right_inv :=\n      (_ :\n        \u2200 (lq : (l : { l // p \u2208 l }) \u00d7 { q // q \u2208 \u2191l \u2227 q \u2260 p }),\n          (fun q =>\n                { fst := { val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) },\n                  snd :=\n                    { val := \u2191q,\n                      property :=\n                        (_ :\n                          \u2191q \u2208 \u2191{ val := mkLine (_ : \u2191q \u2260 p), property := (_ : p \u2208 mkLine (_ : \u2191q \u2260 p)) } \u2227 \u2191q \u2260 p) } })\n              ((fun lq => { val := \u2191lq.snd, property := (_ : \u2191lq.snd \u2260 p) }) lq) =\n            lq) }\nh1 : Fintype.card { q // q \u2260 p } + 1 = Fintype.card P\nh2 : \u2200 (l : { l // p \u2208 l }), Fintype.card { q // q \u2208 \u2191l \u2227 q \u2260 p } = order P L\n\u22a2 Fintype.card { l // p \u2208 l } \u2022 order P L + 1 = order P L ^ 2 + order P L + 1\n[PROOFSTEP]\nrw [\u2190 Nat.card_eq_fintype_card, \u2190 lineCount, lineCount_eq, smul_eq_mul, Nat.succ_mul, sq]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Configuration", "llama_tokens": 47980, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.6688802603710085, "lm_q1q2_score": 0.5106792185552326}}
{"text": "[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\n\u22a2 Periodic (circleTransformDeriv R z w f) (2 * \u03c0)\n[PROOFSTEP]\nhave := periodic_circleMap\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nthis : \u2200 (c : \u2102) (R : \u211d), Periodic (circleMap c R) (2 * \u03c0)\n\u22a2 Periodic (circleTransformDeriv R z w f) (2 * \u03c0)\n[PROOFSTEP]\nsimp_rw [Periodic] at *\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nthis : \u2200 (c : \u2102) (R x : \u211d), circleMap c R (x + 2 * \u03c0) = circleMap c R x\n\u22a2 \u2200 (x : \u211d), circleTransformDeriv R z w f (x + 2 * \u03c0) = circleTransformDeriv R z w f x\n[PROOFSTEP]\nintro x\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nthis : \u2200 (c : \u2102) (R x : \u211d), circleMap c R (x + 2 * \u03c0) = circleMap c R x\nx : \u211d\n\u22a2 circleTransformDeriv R z w f (x + 2 * \u03c0) = circleTransformDeriv R z w f x\n[PROOFSTEP]\nsimp_rw [circleTransformDeriv, this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nthis : \u2200 (c : \u2102) (R x : \u211d), circleMap c R (x + 2 * \u03c0) = circleMap c R x\nx : \u211d\n\u22a2 (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 deriv (circleMap z R) (x + 2 * \u03c0) \u2022 ((circleMap z R x - w) ^ 2)\u207b\u00b9 \u2022 f (circleMap z R x) =\n    (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 deriv (circleMap z R) x \u2022 ((circleMap z R x - w) ^ 2)\u207b\u00b9 \u2022 f (circleMap z R x)\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nthis : \u2200 (c : \u2102) (R x : \u211d), circleMap c R (x + 2 * \u03c0) = circleMap c R x\nx : \u211d\n\u22a2 deriv (circleMap z R) (x + 2 * \u03c0) = deriv (circleMap z R) x\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\n\u22a2 circleTransformDeriv R z w f = fun \u03b8 => (circleMap z R \u03b8 - w)\u207b\u00b9 \u2022 circleTransform R z w f \u03b8\n[PROOFSTEP]\next\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nx\u271d : \u211d\n\u22a2 circleTransformDeriv R z w f x\u271d = (circleMap z R x\u271d - w)\u207b\u00b9 \u2022 circleTransform R z w f x\u271d\n[PROOFSTEP]\nsimp_rw [circleTransformDeriv, circleTransform, \u2190 mul_smul, \u2190 mul_assoc]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nx\u271d : \u211d\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 * deriv (circleMap z R) x\u271d * ((circleMap z R x\u271d - w) ^ 2)\u207b\u00b9) \u2022 f (circleMap z R x\u271d) =\n    ((circleMap z R x\u271d - w)\u207b\u00b9 * (2 * \u2191\u03c0 * I)\u207b\u00b9 * deriv (circleMap z R) x\u271d * (circleMap z R x\u271d - w)\u207b\u00b9) \u2022\n      f (circleMap z R x\u271d)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nx\u271d : \u211d\n\u22a2 ((\u2191\u03c0)\u207b\u00b9 * I\u207b\u00b9 * deriv (circleMap z R) x\u271d * (-(circleMap z R x\u271d * w * 2) + circleMap z R x\u271d ^ 2 + w ^ 2)\u207b\u00b9 *\n        (\u2191(Int.ofNat 1) / \u21912)) \u2022\n      f (circleMap z R x\u271d) =\n    ((\u2191\u03c0)\u207b\u00b9 * I\u207b\u00b9 * deriv (circleMap z R) x\u271d * (circleMap z R x\u271d - w)\u207b\u00b9 ^ 2 * (\u2191(Int.ofNat 1) / \u21912)) \u2022\n      f (circleMap z R x\u271d)\n[PROOFSTEP]\nrw [inv_pow]\n[GOAL]\ncase h\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nx\u271d : \u211d\n\u22a2 ((\u2191\u03c0)\u207b\u00b9 * I\u207b\u00b9 * deriv (circleMap z R) x\u271d * (-(circleMap z R x\u271d * w * 2) + circleMap z R x\u271d ^ 2 + w ^ 2)\u207b\u00b9 *\n        (\u2191(Int.ofNat 1) / \u21912)) \u2022\n      f (circleMap z R x\u271d) =\n    ((\u2191\u03c0)\u207b\u00b9 * I\u207b\u00b9 * deriv (circleMap z R) x\u271d * ((circleMap z R x\u271d - w) ^ 2)\u207b\u00b9 * (\u2191(Int.ofNat 1) / \u21912)) \u2022\n      f (circleMap z R x\u271d)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase h.e_a.e_a.e_a.e_a\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\nx\u271d : \u211d\n\u22a2 -(circleMap z R x\u271d * w * 2) + circleMap z R x\u271d ^ 2 + w ^ 2 = (circleMap z R x\u271d - w) ^ 2\n[PROOFSTEP]\nring\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, circleTransform R z w f \u03b8 = (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(z, R), (z - w)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nsimp_rw [circleTransform, circleIntegral, deriv_circleMap, circleMap]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR : \u211d\nz w : \u2102\nf : \u2102 \u2192 E\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0,\n      (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 ((0 + \u2191R * exp (\u2191\u03b8 * I)) * I) \u2022 (z + \u2191R * exp (\u2191\u03b8 * I) - w)\u207b\u00b9 \u2022 f (z + \u2191R * exp (\u2191\u03b8 * I)) =\n    (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022\n      \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, ((0 + \u2191R * exp (\u2191\u03b8 * I)) * I) \u2022 (z + \u2191R * exp (\u2191\u03b8 * I) - w)\u207b\u00b9 \u2022 f (z + \u2191R * exp (\u2191\u03b8 * I))\n[PROOFSTEP]\nsimp\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous (circleTransform R z w f)\n[PROOFSTEP]\napply_rules [Continuous.smul, continuous_const]\n[GOAL]\ncase hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => deriv (circleMap z R) x\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)\n[PROOFSTEP]\nsimp_rw [deriv_circleMap]\n[GOAL]\ncase hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => circleMap 0 R x * I\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)\n[PROOFSTEP]\napply_rules [Continuous.mul, continuous_circleMap 0 R, continuous_const]\n[GOAL]\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => (circleMap z R x - w)\u207b\u00b9\n[PROOFSTEP]\napply continuous_circleMap_inv hw\n[GOAL]\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun x => f (circleMap z R x)\n[PROOFSTEP]\napply ContinuousOn.comp_continuous hf (continuous_circleMap z R)\n[GOAL]\ncase hg.hg.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 \u2200 (x : \u211d), circleMap z R x \u2208 sphere z R\n[PROOFSTEP]\nexact fun _ => (circleMap_mem_sphere _ hR.le) _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous (circleTransformDeriv R z w f)\n[PROOFSTEP]\nrw [circleTransformDeriv_eq]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w\u271d : \u2102\nR : \u211d\nhR : 0 < R\nf : \u2102 \u2192 E\nz w : \u2102\nhf : ContinuousOn f (sphere z R)\nhw : w \u2208 ball z R\n\u22a2 Continuous fun \u03b8 => (circleMap z R \u03b8 - w)\u207b\u00b9 \u2022 circleTransform R z w f \u03b8\n[PROOFSTEP]\nexact (continuous_circleMap_inv hw).smul (continuous_circleTransform hR hf hw)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun w => (circleMap z R w.snd - w.fst)\u207b\u00b9 ^ 2) (closedBall z r \u00d7\u02e2 univ)\n[PROOFSTEP]\nsimp_rw [\u2190 one_div]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun w => (1 / (circleMap z R w.snd - w.fst)) ^ 2) (closedBall z r \u00d7\u02e2 univ)\n[PROOFSTEP]\napply_rules [ContinuousOn.pow, ContinuousOn.div, continuousOn_const]\n[GOAL]\ncase hf.hg\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => circleMap z R x.snd - x.fst) (closedBall z r \u00d7\u02e2 univ)\ncase hf.h\u2080\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 \u2200 (x : \u2102 \u00d7 \u211d), x \u2208 closedBall z r \u00d7\u02e2 univ \u2192 circleMap z R x.snd - x.fst \u2260 0\n[PROOFSTEP]\nrefine'\n  ((continuous_circleMap z R).continuousOn.comp continuousOn_snd fun _ => And.right).sub\n    (continuousOn_id.comp continuousOn_fst fun _ => And.left)\n[GOAL]\ncase hf.h\u2080\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 \u2200 (x : \u2102 \u00d7 \u211d), x \u2208 closedBall z r \u00d7\u02e2 univ \u2192 circleMap z R x.snd - x.fst \u2260 0\n[PROOFSTEP]\nsimp only [mem_prod, Ne.def, and_imp, Prod.forall]\n[GOAL]\ncase hf.h\u2080\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 \u2200 (a : \u2102) (b : \u211d), a \u2208 closedBall z r \u2192 b \u2208 univ \u2192 \u00accircleMap z R b - a = 0\n[PROOFSTEP]\nintro a b ha _\n[GOAL]\ncase hf.h\u2080\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz a : \u2102\nb : \u211d\nha : a \u2208 closedBall z r\na\u271d : b \u2208 univ\n\u22a2 \u00accircleMap z R b - a = 0\n[PROOFSTEP]\nhave ha2 : a \u2208 ball z R := by simp at *; linarith\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz a : \u2102\nb : \u211d\nha : a \u2208 closedBall z r\na\u271d : b \u2208 univ\n\u22a2 a \u2208 ball z R\n[PROOFSTEP]\nsimp at *\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz a : \u2102\nb : \u211d\nha : dist a z \u2264 r\na\u271d : True\n\u22a2 dist a z < R\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase hf.h\u2080\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz a : \u2102\nb : \u211d\nha : a \u2208 closedBall z r\na\u271d : b \u2208 univ\nha2 : a \u2208 ball z R\n\u22a2 \u00accircleMap z R b - a = 0\n[PROOFSTEP]\nexact sub_ne_zero.2 (circleMap_ne_mem_ball ha2 b)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 (\u22a4 : Set \u211d)) :=\n  by\n  apply_rules [ContinuousOn.smul, continuousOn_const]\n  simp only [deriv_circleMap]\n  have c := (continuous_circleMap 0 R).continuousOn (s := \u22a4)\n  apply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const]\n  simp_rw [\u2190 inv_pow]\n  apply continuousOn_prod_circle_transform_function hr\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\napply_rules [ContinuousOn.smul, continuousOn_const]\n[GOAL]\ncase hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => deriv (circleMap z R) x.snd) (closedBall z r \u00d7\u02e2 \u22a4)\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.snd - x.fst) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\nsimp only [deriv_circleMap]\n[GOAL]\ncase hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.snd * I) (closedBall z r \u00d7\u02e2 \u22a4)\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.snd - x.fst) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\nhave c := (continuous_circleMap 0 R).continuousOn (s := \u22a4)\n[GOAL]\ncase hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nc : ContinuousOn (circleMap 0 R) \u22a4\n\u22a2 ContinuousOn (fun x => circleMap 0 R x.snd * I) (closedBall z r \u00d7\u02e2 \u22a4)\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.snd - x.fst) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\napply_rules [ContinuousOn.mul, c.comp continuousOn_snd fun _ => And.right, continuousOn_const]\n[GOAL]\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => ((circleMap z R x.snd - x.fst) ^ 2)\u207b\u00b9) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\nsimp_rw [\u2190 inv_pow]\n[GOAL]\ncase hg.hg.hf\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\n\u22a2 ContinuousOn (fun x => (circleMap z R x.snd - x.fst)\u207b\u00b9 ^ 2) (closedBall z r \u00d7\u02e2 \u22a4)\n[PROOFSTEP]\napply continuousOn_prod_circle_transform_function hr\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' continuous_abs.continuousOn (s := \u22a4).comp this _\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 MapsTo (fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a4\n[PROOFSTEP]\nshow MapsTo _ _ (\u22a4 : Set \u2102)\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nz : \u2102\nthis : ContinuousOn (circleTransformBoundingFunction R z) (closedBall z r \u00d7\u02e2 \u22a4)\n\u22a2 MapsTo (fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ) \u22a4\n[PROOFSTEP]\nsimp [MapsTo]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nhave cts := continuousOn_abs_circleTransformBoundingFunction hr z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nhave comp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) := by\n  apply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\n\u22a2 IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n[PROOFSTEP]\napply_rules [IsCompact.prod, ProperSpace.isCompact_closedBall z r, isCompact_uIcc]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nhave none : (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]).Nonempty := (nonempty_closedBall.2 hr').prod nonempty_uIcc\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nhave :=\n  IsCompact.exists_isMaxOn comp none\n    (cts.mono (by intro z; simp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]; tauto))\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\n\u22a2 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2286 closedBall z r \u00d7\u02e2 univ\n[PROOFSTEP]\nintro z\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 z \u2208 closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2192 z \u2208 closedBall z\u271d r \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp only [mem_prod, mem_closedBall, mem_univ, and_true_iff, and_imp]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d\u00b9 w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz\u271d : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z\u271d t) (closedBall z\u271d r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z\u271d r \u00d7\u02e2 [[0, 2 * \u03c0]])\nz : \u2102 \u00d7 \u211d\n\u22a2 dist z.fst z\u271d \u2264 r \u2192 z.snd \u2208 [[0, 2 * \u03c0]] \u2192 dist z.fst z\u271d \u2264 r\n[PROOFSTEP]\ntauto\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      IsMaxOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nsimp only [IsMaxOn, IsMaxFilter] at this \n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR r : \u211d\nhr : r < R\nhr' : 0 \u2264 r\nz : \u2102\ncts : ContinuousOn (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) (closedBall z r \u00d7\u02e2 univ)\ncomp : IsCompact (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nnone : Set.Nonempty (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])\nthis :\n  \u2203 x,\n    x \u2208 closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]] \u2227\n      \u2200\u1da0 (x_1 : \u2102 \u00d7 \u211d) in \ud835\udcdf (closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]]),\n        (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x_1 \u2264\n          (\u2191abs \u2218 fun t => circleTransformBoundingFunction R z t) x\n\u22a2 \u2203 x,\n    \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n      \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264 \u2191abs (circleTransformBoundingFunction R z \u2191x)\n[PROOFSTEP]\nsimpa [SetCoe.forall, Subtype.coe_mk, SetCoe.exists]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nobtain \u27e8r, hr, hrx\u27e9 := exists_lt_mem_ball_of_mem_ball hx\n[GOAL]\ncase intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nobtain \u27e8\u03b5', h\u03b5', H\u27e9 := exists_ball_subset_ball hrx\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e8a, b\u27e9, \u27e8ha, hb\u27e9\u27e9, hab\u27e9 := abs_circleTransformBoundingFunction_le hr (pos_of_mem_ball hrx).le z\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nlet V : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun _ => 1) \u03b8\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nhave funccomp : ContinuousOn (fun r => abs (f r)) (sphere z R) :=\n  by\n  have cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn\n  apply cabs.comp hf; rw [MapsTo]; tauto\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\n[PROOFSTEP]\nhave cabs : ContinuousOn abs \u22a4 := by apply continuous_abs.continuousOn\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\n\u22a2 ContinuousOn \u2191abs \u22a4\n[PROOFSTEP]\napply continuous_abs.continuousOn\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\n[PROOFSTEP]\napply cabs.comp hf\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 MapsTo f (sphere z R) \u22a4\n[PROOFSTEP]\nrw [MapsTo]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\ncabs : ContinuousOn \u2191abs \u22a4\n\u22a2 \u2200 \u2983x : \u2102\u2984, x \u2208 sphere z R \u2192 f x \u2208 \u22a4\n[PROOFSTEP]\ntauto\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nhave sbou := IsCompact.exists_isMaxOn (isCompact_sphere z R) (NormedSpace.sphere_nonempty.2 hR.le) funccomp\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nsbou : \u2203 x, x \u2208 sphere z R \u2227 IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) x\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nobtain \u27e8X, HX, HX2\u27e9 := sbou\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2203 B \u03b5, 0 < \u03b5 \u2227 ball x \u03b5 \u2286 ball z R \u2227 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5 \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 B\n[PROOFSTEP]\nrefine' \u27e8abs (V b a) * abs (f X), \u03b5', h\u03b5', Subset.trans H (ball_subset_ball hr.le), _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\n\u22a2 \u2200 (t : \u211d) (y : \u2102), y \u2208 ball x \u03b5' \u2192 \u2016circleTransformDeriv R z y f t\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nintro y v hv\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nobtain \u27e8y1, hy1, hfun\u27e9 := Periodic.exists_mem_Ico\u2080 (circleTransformDeriv_periodic R z v f) Real.two_pi_pos y\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nhave hy2 : y1 \u2208 [[0, 2 * \u03c0]] := by\n  convert Ico_subset_Icc_self hy1 using 1\n  simp [uIcc_of_le Real.two_pi_pos.le]\n[GOAL]\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 y1 \u2208 [[0, 2 * \u03c0]]\n[PROOFSTEP]\nconvert Ico_subset_Icc_self hy1 using 1\n[GOAL]\ncase h.e'_5\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\n\u22a2 [[0, 2 * \u03c0]] = Icc 0 (2 * \u03c0)\n[PROOFSTEP]\nsimp [uIcc_of_le Real.two_pi_pos.le]\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\nHX2 : IsMaxOn (fun r => \u2191abs (f r)) (sphere z R) X\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nsimp only [IsMaxOn, IsMaxFilter, eventually_principal, mem_sphere_iff_norm, norm_eq_abs] at HX2 \n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nhave :=\n  mul_le_mul (hab \u27e8\u27e8v, y1\u27e9, \u27e8ball_subset_closedBall (H hv), hy2\u27e9\u27e9)\n    (HX2 (circleMap z R y1) (circleMap_mem_sphere z hR.le y1)) (Complex.abs.nonneg _) (Complex.abs.nonneg _)\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).fst \u2208 closedBall z r \u2227 (v, y1).snd \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y\u2016 \u2264 \u2191abs (V b a) * \u2191abs (f X)\n[PROOFSTEP]\nsimp_rw [hfun]\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhx : x \u2208 ball z R\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\nhrx : x \u2208 ball z r\n\u03b5' : \u211d\nh\u03b5' : \u03b5' > 0\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nhab :\n  \u2200 (y : \u2191(closedBall z r \u00d7\u02e2 [[0, 2 * \u03c0]])),\n    \u2191abs (circleTransformBoundingFunction R z \u2191y) \u2264\n      \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) })\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\nHX : X \u2208 sphere z R\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhfun : circleTransformDeriv R z v f y = circleTransformDeriv R z v f y1\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nthis :\n  \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (v, y1), property := (_ : (v, y1).fst \u2208 closedBall z r \u2227 (v, y1).snd \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs\n        (circleTransformBoundingFunction R z\n          \u2191{ val := (a, b), property := (_ : (a, b).fst \u2208 closedBall z r \u2227 (a, b).snd \u2208 [[0, 2 * \u03c0]]) }) *\n      \u2191abs (f X)\n\u22a2 \u2016circleTransformDeriv R z v f y1\u2016 \u2264 \u2191abs (circleTransformDeriv R z a (fun x => 1) b) * \u2191abs (f X)\n[PROOFSTEP]\nsimp only [circleTransformBoundingFunction, circleTransformDeriv, norm_eq_abs, Algebra.id.smul_eq_mul, deriv_circleMap,\n  map_mul, abs_circleMap_zero, abs_I, mul_one, \u2190 mul_assoc, mul_inv_rev, inv_I, abs_neg, abs_inv, abs_ofReal, one_mul,\n  abs_two, abs_pow, mem_ball, gt_iff_lt, Subtype.coe_mk, SetCoe.forall, mem_prod, mem_closedBall, and_imp, Prod.forall,\n  NormedSpace.sphere_nonempty, mem_sphere_iff_norm] at *\n[GOAL]\ncase intro.intro.intro.intro.intro.mk.mk.intro.intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR\u271d : \u211d\nz\u271d w : \u2102\nR : \u211d\nhR : 0 < R\nz x : \u2102\nf : \u2102 \u2192 \u2102\nhf : ContinuousOn f (sphere z R)\nr : \u211d\nhr : r < R\n\u03b5' : \u211d\nH : ball x \u03b5' \u2286 ball z r\na : \u2102\nb : \u211d\nha : (a, b).fst \u2208 closedBall z r\nhb : (a, b).snd \u2208 [[0, 2 * \u03c0]]\nV : \u211d \u2192 \u2102 \u2192 \u2102 := fun \u03b8 w => circleTransformDeriv R z w (fun x => 1) \u03b8\nfunccomp : ContinuousOn (fun r => \u2191abs (f r)) (sphere z R)\nX : \u2102\ny : \u211d\nv : \u2102\nhv : v \u2208 ball x \u03b5'\ny1 : \u211d\nhy1 : y1 \u2208 Ico 0 (2 * \u03c0)\nhy2 : y1 \u2208 [[0, 2 * \u03c0]]\nHX2 : \u2200 (x : \u2102), \u2191abs (x - z) = R \u2192 \u2191abs (f x) \u2264 \u2191abs (f X)\nhx : dist x z < R\nhrx : dist x z < r\nh\u03b5' : 0 < \u03b5'\nhab :\n  \u2200 (a_1 : \u2102) (b_1 : \u211d),\n    dist a_1 z \u2264 r \u2192\n      b_1 \u2208 [[0, 2 * \u03c0]] \u2192\n        \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b_1 - a_1) ^ 2)\u207b\u00b9 \u2264\n          \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9\nHX : \u2191abs (X - z) = R\nhfun :\n  -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y * I * ((circleMap z R y - v) ^ 2)\u207b\u00b9 * f (circleMap z R y) =\n    -I * (\u2191\u03c0)\u207b\u00b9 * 2\u207b\u00b9 * circleMap 0 R y1 * I * ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * f (circleMap z R y1)\nthis :\n  \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)\n\u22a2 \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R y1 - v) ^ 2)\u207b\u00b9 * \u2191abs (f (circleMap z R y1)) \u2264\n    \u2191abs (-I) * \u2191abs (\u2191\u03c0)\u207b\u00b9 * \u2191abs 2\u207b\u00b9 * |R| * \u2191abs ((circleMap z R b - a) ^ 2)\u207b\u00b9 * \u2191abs (f X)\n[PROOFSTEP]\nexact this\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.CircleTransform", "llama_tokens": 21786, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324803738429, "lm_q2_score": 0.6261241772283035, "lm_q1q2_score": 0.5104367660238615}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\nx : S\n\u22a2 \u2191(norm R) x = 1\n[PROOFSTEP]\nrw [norm_apply, LinearMap.det]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\nx : S\n\u22a2 \u2191(if H : \u2203 s, Nonempty (Basis { x // x \u2208 s } R S) then\n          detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose H } R S))))\n        else 1)\n      (\u2191(lmul R S) x) =\n    1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\nx : S\nh\u271d : \u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\n\u22a2 \u2191(detAux (Trunc.mk (Nonempty.some (_ : Nonempty (Basis { x // x \u2208 Exists.choose h\u271d } R S))))) (\u2191(lmul R S) x) = 1\n[PROOFSTEP]\ntrivial\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\nx : S\nh\u271d : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)\n\u22a2 \u21911 (\u2191(lmul R S) x) = 1\n[PROOFSTEP]\ntrivial\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00acModule.Finite R S\nx : S\n\u22a2 \u2191(norm R) x = 1\n[PROOFSTEP]\nrefine norm_eq_one_of_not_exists_basis _ (mt ?_ h) _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00acModule.Finite R S\nx : S\n\u22a2 (\u2203 s, Nonempty (Basis { x // x \u2208 s } R S)) \u2192 Module.Finite R S\n[PROOFSTEP]\nrintro \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nh : \u00acModule.Finite R S\nx : S\ns : Finset S\nb : Basis { x // x \u2208 s } R S\n\u22a2 Module.Finite R S\n[PROOFSTEP]\nexact Module.Finite.of_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\ns : S\n\u22a2 \u2191(norm R) s = det (\u2191(leftMulMatrix b) s)\n[PROOFSTEP]\nrw [norm_apply, \u2190 LinearMap.det_toMatrix b, \u2190 toMatrix_lmul_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\nb : Basis \u03b9 R S\ns : S\n\u22a2 det (\u2191(toMatrix b b) (\u2191(lmul R S) s)) = det (\u2191(toMatrix b b) (mulLeft R s))\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\n\u22a2 \u2191(norm R) (\u2191(algebraMap R S) x) = x ^ Fintype.card \u03b9\n[PROOFSTEP]\nhaveI := Classical.decEq \u03b9\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 \u2191(norm R) (\u2191(algebraMap R S) x) = x ^ Fintype.card \u03b9\n[PROOFSTEP]\nrw [norm_apply, \u2190 det_toMatrix b, lmul_algebraMap]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 det (\u2191(toMatrix b b) (\u2191(lsmul R R ((fun x => S) x)) x)) = x ^ Fintype.card \u03b9\n[PROOFSTEP]\nconvert @det_diagonal _ _ _ _ _ fun _ : \u03b9 => x\n[GOAL]\ncase h.e'_2.h.e'_6\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 \u2191(toMatrix b b) (\u2191(lsmul R R ((fun x => S) x)) x) = diagonal fun x_1 => x\n[PROOFSTEP]\next (i j)\n[GOAL]\ncase h.e'_2.h.e'_6.a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 \u2191(toMatrix b b) (\u2191(lsmul R R ((fun x => S) x)) x) i j = diagonal (fun x_1 => x) i j\n[PROOFSTEP]\nrw [toMatrix_lsmul, Matrix.diagonal]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Fintype \u03b9\nb : Basis \u03b9 R S\nx : R\nthis : DecidableEq \u03b9\n\u22a2 x ^ Fintype.card \u03b9 = \u220f i : \u03b9, x\n[PROOFSTEP]\nrw [Finset.prod_const, Finset.card_univ]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL\u271d : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\u271d\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\u271d\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\nL : Type u_7\ninst\u271d\u00b9 : Ring L\ninst\u271d : Algebra K L\nx : K\n\u22a2 \u2191(norm K) (\u2191(algebraMap K L) x) = x ^ finrank K L\n[PROOFSTEP]\nby_cases H : \u2203 s : Finset L, Nonempty (Basis s K L)\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL\u271d : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\u271d\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\u271d\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\nL : Type u_7\ninst\u271d\u00b9 : Ring L\ninst\u271d : Algebra K L\nx : K\nH : \u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\n\u22a2 \u2191(norm K) (\u2191(algebraMap K L) x) = x ^ finrank K L\n[PROOFSTEP]\nrw [norm_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL\u271d : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\u271d\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\u271d\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\nL : Type u_7\ninst\u271d\u00b9 : Ring L\ninst\u271d : Algebra K L\nx : K\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\n\u22a2 \u2191(norm K) (\u2191(algebraMap K L) x) = x ^ finrank K L\n[PROOFSTEP]\nrw [norm_eq_one_of_not_exists_basis K H, finrank_eq_zero_of_not_exists_basis, pow_zero]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL\u271d : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\u271d\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\u271d\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\nL : Type u_7\ninst\u271d\u00b9 : Ring L\ninst\u271d : Algebra K L\nx : K\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\n\u22a2 \u00ac\u2203 s, Nonempty (Basis (\u2191\u2191s) K L)\n[PROOFSTEP]\nrintro \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL\u271d : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\u271d\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\u271d\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\nL : Type u_7\ninst\u271d\u00b9 : Ring L\ninst\u271d : Algebra K L\nx : K\nH : \u00ac\u2203 s, Nonempty (Basis { x // x \u2208 s } K L)\ns : Finset L\nb : Basis (\u2191\u2191s) K L\n\u22a2 False\n[PROOFSTEP]\nexact H \u27e8s, \u27e8b\u27e9\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\npb : PowerBasis R S\n\u22a2 \u2191(norm R) pb.gen = (-1) ^ pb.dim * coeff (minpoly R pb.gen) 0\n[PROOFSTEP]\nrw [norm_eq_matrix_det pb.basis, det_eq_sign_charpoly_coeff, charpoly_leftMulMatrix, Fintype.card_fin]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhf : Splits (algebraMap R F) (minpoly R pb.gen)\n\u22a2 \u2191(algebraMap R F) (\u2191(norm R) pb.gen) = Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)))\n[PROOFSTEP]\nhaveI := Module.nontrivial R F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhf : Splits (algebraMap R F) (minpoly R pb.gen)\nthis : Nontrivial R\n\u22a2 \u2191(algebraMap R F) (\u2191(norm R) pb.gen) = Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)))\n[PROOFSTEP]\nhave := minpoly.monic pb.isIntegral_gen\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhf : Splits (algebraMap R F) (minpoly R pb.gen)\nthis\u271d : Nontrivial R\nthis : Monic (minpoly R pb.gen)\n\u22a2 \u2191(algebraMap R F) (\u2191(norm R) pb.gen) = Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)))\n[PROOFSTEP]\nrw [PowerBasis.norm_gen_eq_coeff_zero_minpoly, \u2190 pb.natDegree_minpoly, RingHom.map_mul, \u2190 coeff_map,\n  prod_roots_eq_coeff_zero_of_monic_of_split (this.map _) ((splits_id_iff_splits _).2 hf), this.natDegree_map, map_pow,\n  \u2190 mul_assoc, \u2190 mul_pow]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\n\u03b9 : Type w\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhf : Splits (algebraMap R F) (minpoly R pb.gen)\nthis\u271d : Nontrivial R\nthis : Monic (minpoly R pb.gen)\n\u22a2 (\u2191(algebraMap R F) (-1) * -1) ^ natDegree (minpoly R pb.gen) *\n      Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen))) =\n    Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)))\n[PROOFSTEP]\nsimp only [map_neg, _root_.map_one, neg_mul, neg_neg, one_pow, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : Nontrivial S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\n\u22a2 \u2191(norm R) 0 = 0\n[PROOFSTEP]\nnontriviality\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b3 : Finite \u03b9\ninst\u271d\u00b2 : Nontrivial S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\n\u271d : Nontrivial ((fun x => R) 0)\n\u22a2 \u2191(norm R) 0 = 0\n[PROOFSTEP]\nrw [norm_apply, coe_lmul_eq_mul, map_zero, LinearMap.det_zero' (Module.Free.chooseBasis R S)]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\n\u22a2 \u2191(norm R) x = 0 \u2194 x = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\n\u22a2 \u2191(norm R) x = 0 \u2192 x = 0\ncase mpr\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\n\u22a2 x = 0 \u2192 \u2191(norm R) x = 0\n[PROOFSTEP]\nlet b := Module.Free.chooseBasis R S\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\n\u22a2 \u2191(norm R) x = 0 \u2192 x = 0\ncase mpr\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\n\u22a2 x = 0 \u2192 \u2191(norm R) x = 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\n\u22a2 x = 0 \u2192 \u2191(norm R) x = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\n\u22a2 \u2191(norm R) 0 = 0\n[PROOFSTEP]\nexact norm_zero\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\n\u22a2 \u2191(norm R) x = 0 \u2192 x = 0\n[PROOFSTEP]\nletI := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n\u22a2 \u2191(norm R) x = 0 \u2192 x = 0\n[PROOFSTEP]\nrw [norm_eq_matrix_det b, \u2190 Matrix.exists_mulVec_eq_zero_iff]\n[GOAL]\ncase mp\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\n\u22a2 (\u2203 v x_1, mulVec (\u2191(leftMulMatrix b) x) v = 0) \u2192 x = 0\n[PROOFSTEP]\nrintro \u27e8v, v_ne, hv\u27e9\n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nv_ne : v \u2260 0\nhv : mulVec (\u2191(leftMulMatrix b) x) v = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrw [\u2190 b.equivFun.apply_symm_apply v, b.equivFun_symm_apply, b.equivFun_apply, leftMulMatrix_mulVec_repr] at hv \n[GOAL]\ncase mp.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nv_ne : v \u2260 0\nhv : \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) = 0\n\u22a2 x = 0\n[PROOFSTEP]\nrefine (mul_eq_zero.mp (b.ext_elem fun i => ?_)).resolve_right (show \u2211 i, v i \u2022 b i \u2260 0 from ?_)\n[GOAL]\ncase mp.intro.intro.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nv_ne : v \u2260 0\nhv : \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) = 0\ni : Module.Free.ChooseBasisIndex R S\n\u22a2 \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) i = \u2191(\u2191b.repr 0) i\n[PROOFSTEP]\nsimpa only [LinearEquiv.map_zero, Pi.zero_apply] using congr_fun hv i\n[GOAL]\ncase mp.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nv_ne : v \u2260 0\nhv : \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) = 0\n\u22a2 \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i \u2260 0\n[PROOFSTEP]\ncontrapose! v_ne with sum_eq\n[GOAL]\ncase mp.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nhv : \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) = 0\nsum_eq : \u2211 x : Module.Free.ChooseBasisIndex R S, v x \u2022 \u2191(Module.Free.chooseBasis R S) x = 0\n\u22a2 v = 0\n[PROOFSTEP]\napply b.equivFun.symm.injective\n[GOAL]\ncase mp.intro.intro.refine_2.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : IsDomain R\ninst\u271d\u00b2 : IsDomain S\ninst\u271d\u00b9 : Module.Free R S\ninst\u271d : Module.Finite R S\nx : S\nb : Basis (Module.Free.ChooseBasisIndex R S) R S := Module.Free.chooseBasis R S\nthis : DecidableEq (Module.Free.ChooseBasisIndex R S) := Classical.decEq (Module.Free.ChooseBasisIndex R S)\nv : Module.Free.ChooseBasisIndex R S \u2192 (fun x => R) x\nhv : \u2191(\u2191b.repr (x * \u2211 i : Module.Free.ChooseBasisIndex R S, v i \u2022 \u2191b i)) = 0\nsum_eq : \u2211 x : Module.Free.ChooseBasisIndex R S, v x \u2022 \u2191(Module.Free.chooseBasis R S) x = 0\n\u22a2 \u2191(LinearEquiv.symm (Basis.equivFun b)) v = \u2191(LinearEquiv.symm (Basis.equivFun b)) 0\n[PROOFSTEP]\nrw [b.equivFun_symm_apply, sum_eq, LinearEquiv.map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDomain S\nb : Basis \u03b9 R S\nx : S\n\u22a2 \u2191(norm R) x = 0 \u2194 x = 0\n[PROOFSTEP]\nhaveI : Module.Free R S := Module.Free.of_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDomain S\nb : Basis \u03b9 R S\nx : S\nthis : Module.Free R S\n\u22a2 \u2191(norm R) x = 0 \u2194 x = 0\n[PROOFSTEP]\nhaveI : Module.Finite R S := Module.Finite.of_basis b\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b2 : Finite \u03b9\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDomain S\nb : Basis \u03b9 R S\nx : S\nthis\u271d : Module.Free R S\nthis : Module.Finite R S\n\u22a2 \u2191(norm R) x = 0 \u2194 x = 0\n[PROOFSTEP]\nexact norm_eq_zero_iff\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\n\u22a2 \u2191(norm K) x = \u2191(norm K) (AdjoinSimple.gen K x) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n[PROOFSTEP]\nletI := isSeparable_tower_top_of_isSeparable K K\u27eex\u27ef L\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n\u22a2 \u2191(norm K) x = \u2191(norm K) (AdjoinSimple.gen K x) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n[PROOFSTEP]\nlet pbL := Field.powerBasisOfFiniteOfSeparable K\u27eex\u27ef L\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbL : PowerBasis { x_1 // x_1 \u2208 K\u27eex\u27ef } L := Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n\u22a2 \u2191(norm K) x = \u2191(norm K) (AdjoinSimple.gen K x) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n[PROOFSTEP]\nlet pbx := IntermediateField.adjoin.powerBasis (IsSeparable.isIntegral K x)\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbL : PowerBasis { x_1 // x_1 \u2208 K\u27eex\u27ef } L := Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbx : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis (_ : IsIntegral K x)\n\u22a2 \u2191(norm K) x = \u2191(norm K) (AdjoinSimple.gen K x) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n[PROOFSTEP]\nrw [\u2190 AdjoinSimple.algebraMap_gen K x, norm_eq_matrix_det (pbx.basis.smul pbL.basis) _, smul_leftMulMatrix_algebraMap,\n  det_blockDiagonal, norm_eq_matrix_det pbx.basis]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbL : PowerBasis { x_1 // x_1 \u2208 K\u27eex\u27ef } L := Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbx : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis (_ : IsIntegral K x)\n\u22a2 \u220f k : Fin pbL.dim, det (\u2191(leftMulMatrix pbx.basis) (AdjoinSimple.gen K x)) =\n    det\n        (\u2191(leftMulMatrix pbx.basis)\n          (AdjoinSimple.gen K (\u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)))) ^\n      finrank { x_1 // x_1 \u2208 K\u27ee\u2191(algebraMap { x_2 // x_2 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\u27ef } L\n[PROOFSTEP]\nsimp only [Finset.card_fin, Finset.prod_const]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbL : PowerBasis { x_1 // x_1 \u2208 K\u27eex\u27ef } L := Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbx : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis (_ : IsIntegral K x)\n\u22a2 det (\u2191(leftMulMatrix (adjoin.powerBasis (_ : IsIntegral K x)).basis) (AdjoinSimple.gen K x)) ^\n      (Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L).dim =\n    det\n        (\u2191(leftMulMatrix (adjoin.powerBasis (_ : IsIntegral K x)).basis)\n          (AdjoinSimple.gen K (\u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)))) ^\n      finrank { x_1 // x_1 \u2208 K\u27ee\u2191(algebraMap { x_2 // x_2 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\u27ef } L\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Ring S\ninst\u271d\u2077 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Field L\ninst\u271d\u2074 : Field F\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : Algebra K F\n\u03b9 : Type w\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nx : L\nthis : IsSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L := isSeparable_tower_top_of_isSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbL : PowerBasis { x_1 // x_1 \u2208 K\u27eex\u27ef } L := Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L\npbx : PowerBasis K { x_1 // x_1 \u2208 K\u27eex\u27ef } := adjoin.powerBasis (_ : IsIntegral K x)\n\u22a2 (Field.powerBasisOfFiniteOfSeparable { x_1 // x_1 \u2208 K\u27eex\u27ef } L).dim =\n    finrank { x_1 // x_1 \u2208 K\u27ee\u2191(algebraMap { x_2 // x_2 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\u27ef } L\n[PROOFSTEP]\nrw [\u2190 PowerBasis.finrank, AdjoinSimple.algebraMap_gen K x]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(norm K) (AdjoinSimple.gen K x) = 1\n[PROOFSTEP]\nrw [norm_eq_one_of_not_exists_basis]\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u00acIsIntegral K x\n\u22a2 \u00ac\u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef })\n[PROOFSTEP]\ncontrapose! hx\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhx : \u2203 s, Nonempty (Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef })\n\u22a2 IsIntegral K x\n[PROOFSTEP]\nobtain \u27e8s, \u27e8b\u27e9\u27e9 := hx\n[GOAL]\ncase h.intro.intro\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 IsIntegral K x\n[PROOFSTEP]\nrefine isIntegral_of_mem_of_FG K\u27eex\u27ef.toSubalgebra ?_ x ?_\n[GOAL]\ncase h.intro.intro.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 Submodule.FG (\u2191Subalgebra.toSubmodule K\u27eex\u27ef.toSubalgebra)\n[PROOFSTEP]\nexact (Submodule.fg_iff_finiteDimensional _).mpr (of_fintype_basis b)\n[GOAL]\ncase h.intro.intro.refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\ns : Finset { x_1 // x_1 \u2208 K\u27eex\u27ef }\nb : Basis { x_1 // x_1 \u2208 s } K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 x \u2208 K\u27eex\u27ef.toSubalgebra\n[PROOFSTEP]\nexact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nhave injKxL := (algebraMap K\u27eex\u27ef L).injective\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nby_cases hx : IsIntegral K x\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : \u00acIsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nsimp [minpoly.eq_zero hx, IntermediateField.AdjoinSimple.norm_gen_eq_one hx]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nhave hx' : IsIntegral K (AdjoinSimple.gen K x) := by\n  rwa [\u2190 isIntegral_algebraMap_iff injKxL, AdjoinSimple.algebraMap_gen]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\n\u22a2 IsIntegral K (AdjoinSimple.gen K x)\n[PROOFSTEP]\nrwa [\u2190 isIntegral_algebraMap_iff injKxL, AdjoinSimple.algebraMap_gen]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) (AdjoinSimple.gen K x)) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nrw [\u2190 adjoin.powerBasis_gen hx, PowerBasis.norm_gen_eq_prod_roots]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K (adjoin.powerBasis hx).gen))) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x)))\n[PROOFSTEP]\nrw [adjoin.powerBasis_gen hx, minpoly.eq_of_algebraMap_eq injKxL hx']\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K (adjoin.powerBasis hx).gen)\n[PROOFSTEP]\nrw [adjoin.powerBasis_gen hx, minpoly.eq_of_algebraMap_eq injKxL hx']\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 x = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 x = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.762940)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.762940)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 ?m.762940 = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 ?m.762940 = \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L) (AdjoinSimple.gen K x)\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 L\n[PROOFSTEP]\ntry simp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 L\n[PROOFSTEP]\nsimp only [AdjoinSimple.algebraMap_gen _ _]\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.762940)\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 ?m.762940 = x\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 L\n[PROOFSTEP]\ntry exact hf\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 Splits (algebraMap K F) (minpoly K ?m.762940)\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 ?m.762940 = x\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 L\n[PROOFSTEP]\nexact hf\n[GOAL]\ncase pos.hf\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : Ring S\ninst\u271d\u2075 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Field L\ninst\u271d\u00b2 : Field F\ninst\u271d\u00b9 : Algebra K L\ninst\u271d : Algebra K F\n\u03b9 : Type w\nx : L\nhf : Splits (algebraMap K F) (minpoly K x)\ninjKxL : Function.Injective \u2191(algebraMap { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\nhx : IsIntegral K x\nhx' : IsIntegral K (AdjoinSimple.gen K x)\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\n\u22a2 \u2191(algebraMap R F) (\u2191(norm R) pb.gen) = \u220f \u03c3 : S \u2192\u2090[R] F, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nletI := Classical.decEq F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 \u2191(algebraMap R F) (\u2191(norm R) pb.gen) = \u220f \u03c3 : S \u2192\u2090[R] F, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nrw [PowerBasis.norm_gen_eq_prod_roots pb hE]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen))) = \u220f \u03c3 : S \u2192\u2090[R] F, \u2191\u03c3 pb.gen\n[PROOFSTEP]\nrw [@Fintype.prod_equiv (S \u2192\u2090[R] F) _ _ (PowerBasis.AlgHom.fintype pb) _ _ pb.liftEquiv' (fun \u03c3 => \u03c3 pb.gen)\n    (fun x => x) ?_]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 Multiset.prod (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen))) =\n    \u220f x : { y // y \u2208 roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)) }, \u2191x\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 \u2200 (x : S \u2192\u2090[R] F), (fun \u03c3 => \u2191\u03c3 pb.gen) x = (fun x => \u2191x) (\u2191(PowerBasis.liftEquiv' pb) x)\n[PROOFSTEP]\nrw [Finset.prod_mem_multiset, Finset.prod_eq_multiset_prod, Multiset.toFinset_val, Multiset.dedup_eq_self.mpr,\n  Multiset.map_id]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 Multiset.Nodup (roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)))\n[PROOFSTEP]\nexact nodup_roots hfx.map\n[GOAL]\ncase hfg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 \u2200 (x : { x // x \u2208 roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)) }), \u2191x = _root_.id \u2191x\n[PROOFSTEP]\nintro x\n[GOAL]\ncase hfg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\nx : { x // x \u2208 roots (Polynomial.map (algebraMap R F) (minpoly R pb.gen)) }\n\u22a2 \u2191x = _root_.id \u2191x\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u22a2 \u2200 (x : S \u2192\u2090[R] F), (fun \u03c3 => \u2191\u03c3 pb.gen) x = (fun x => \u2191x) (\u2191(PowerBasis.liftEquiv' pb) x)\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Ring S\ninst\u271d\u2078 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : Field F\ninst\u271d\u2074 : Algebra K L\ninst\u271d\u00b3 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b2 : Field E\ninst\u271d\u00b9 : Algebra K E\ninst\u271d : Algebra R F\npb : PowerBasis R S\nhE : Splits (algebraMap R F) (minpoly R pb.gen)\nhfx : Separable (minpoly R pb.gen)\nthis : DecidableEq F := Classical.decEq F\n\u03c3 : S \u2192\u2090[R] F\n\u22a2 (fun \u03c3 => \u2191\u03c3 pb.gen) \u03c3 = (fun x => \u2191x) (\u2191(PowerBasis.liftEquiv' pb) \u03c3)\n[PROOFSTEP]\nsimp only [PowerBasis.liftEquiv'_apply_coe]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhF : Splits (algebraMap K F) (minpoly K x)\n\u22a2 \u2191(algebraMap K F) (\u2191(norm K) x) =\n    Multiset.prod (roots (Polynomial.map (algebraMap K F) (minpoly K x))) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L\n[PROOFSTEP]\nrw [norm_eq_norm_adjoin K x, map_pow, IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots _ hF]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\n\u22a2 \u220f \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = (\u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen) ^ finrank L F\n[PROOFSTEP]\nhaveI : FiniteDimensional L F := FiniteDimensional.right K L F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis : FiniteDimensional L F\n\u22a2 \u220f \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = (\u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen) ^ finrank L F\n[PROOFSTEP]\nhaveI : IsSeparable L F := isSeparable_tower_top_of_isSeparable K L F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d : FiniteDimensional L F\nthis : IsSeparable L F\n\u22a2 \u220f \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = (\u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen) ^ finrank L F\n[PROOFSTEP]\nletI : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 \u220f \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = (\u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen) ^ finrank L F\n[PROOFSTEP]\nletI : \u2200 f : L \u2192\u2090[K] E, Fintype (@AlgHom L F E _ _ _ _ f.toRingHom.toAlgebra) := ?_\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u220f \u03c3 : F \u2192\u2090[K] E, \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = (\u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 pb.gen) ^ finrank L F\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E)\n[PROOFSTEP]\nrw [Fintype.prod_equiv algHomEquivSigma (fun \u03c3 : F \u2192\u2090[K] E => _) fun \u03c3 => \u03c3.1 pb.gen, \u2190 Finset.univ_sigma_univ,\n  Finset.prod_sigma, \u2190 Finset.prod_pow]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u220f a : L \u2192\u2090[K] E, \u220f s : F \u2192\u2090[L] E, \u2191{ fst := a, snd := s }.fst pb.gen = \u220f x : L \u2192\u2090[K] E, \u2191x pb.gen ^ finrank L F\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u22a2 \u2200 (x : F \u2192\u2090[K] E), \u2191x (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma x).fst pb.gen\ncase refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : IsSeparable L F\nthis : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\n\u22a2 (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E)\n[PROOFSTEP]\nrefine Finset.prod_congr rfl fun \u03c3 _ => ?_\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\n\u22a2 \u220f s : F \u2192\u2090[L] E, \u2191{ fst := \u03c3, snd := s }.fst pb.gen = \u2191\u03c3 pb.gen ^ finrank L F\n[PROOFSTEP]\nletI : Algebra L E := \u03c3.toRingHom.toAlgebra\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 \u220f s : F \u2192\u2090[L] E, \u2191{ fst := \u03c3, snd := s }.fst pb.gen = \u2191\u03c3 pb.gen ^ finrank L F\n[PROOFSTEP]\nsimp_rw [Finset.prod_const]\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 \u2191\u03c3 pb.gen ^ Finset.card Finset.univ = \u2191\u03c3 pb.gen ^ finrank L F\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine_2.e_a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : IsSeparable L F\nthis\u271d\u00b9 : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis\u271d : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := ?refine_1\n\u03c3 : L \u2192\u2090[K] E\nx\u271d : \u03c3 \u2208 Finset.univ\nthis : Algebra L E := RingHom.toAlgebra \u2191\u03c3\n\u22a2 Finset.card Finset.univ = finrank L F\n[PROOFSTEP]\nexact AlgHom.card L F E\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := fun \u03c3 => minpoly.AlgHom.fintype L F E\n\u22a2 \u2200 (x : F \u2192\u2090[K] E), \u2191x (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma x).fst pb.gen\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra L F\ninst\u271d\u00b3 : IsScalarTower K L F\ninst\u271d\u00b2 : IsAlgClosed E\ninst\u271d\u00b9 : IsSeparable K F\ninst\u271d : FiniteDimensional K F\npb : PowerBasis K L\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : Fintype (L \u2192\u2090[K] E) := PowerBasis.AlgHom.fintype pb\nthis : (f : L \u2192\u2090[K] E) \u2192 Fintype (F \u2192\u2090[L] E) := fun \u03c3 => minpoly.AlgHom.fintype L F E\n\u03c3 : F \u2192\u2090[K] E\n\u22a2 \u2191\u03c3 (\u2191(algebraMap L F) pb.gen) = \u2191(\u2191algHomEquivSigma \u03c3).fst pb.gen\n[PROOFSTEP]\nsimp only [algHomEquivSigma, Equiv.coe_fn_mk, AlgHom.restrictDomain, AlgHom.comp_apply, IsScalarTower.coe_toAlgHom']\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nx : L\n\u22a2 \u2191(algebraMap K E) (\u2191(norm K) x) = \u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nhave hx := IsSeparable.isIntegral K x\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nx : L\nhx : IsIntegral K x\n\u22a2 \u2191(algebraMap K E) (\u2191(norm K) x) = \u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nrw [norm_eq_norm_adjoin K x, RingHom.map_pow, \u2190 adjoin.powerBasis_gen hx,\n  norm_eq_prod_embeddings_gen E (adjoin.powerBasis hx) (IsAlgClosed.splits_codomain _)]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nx : L\nhx : IsIntegral K x\n\u22a2 (\u220f \u03c3 : { x_1 // x_1 \u2208 K\u27eex\u27ef } \u2192\u2090[K] E, \u2191\u03c3 (adjoin.powerBasis hx).gen) ^ finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L =\n    \u220f \u03c3 : L \u2192\u2090[K] E, \u2191\u03c3 x\n[PROOFSTEP]\nexact (prod_embeddings_eq_finrank_pow L (L := K\u27eex\u27ef) E (adjoin.powerBasis hx)).symm\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nx : L\nhx : IsIntegral K x\n\u22a2 Separable (minpoly K (adjoin.powerBasis hx).gen)\n[PROOFSTEP]\nhaveI := isSeparable_tower_bot_of_isSeparable K K\u27eex\u27ef L\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : IsAlgClosed E\nx : L\nhx : IsIntegral K x\nthis : IsSeparable K { x_1 // x_1 \u2208 K\u27eex\u27ef }\n\u22a2 Separable (minpoly K (adjoin.powerBasis hx).gen)\n[PROOFSTEP]\nexact IsSeparable.separable K _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(algebraMap K L) (\u2191(norm K) x) = \u220f \u03c3 : L \u2243\u2090[K] L, \u2191\u03c3 x\n[PROOFSTEP]\napply NoZeroSMulDivisors.algebraMap_injective L (AlgebraicClosure L)\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(norm K) x)) =\n    \u2191(algebraMap L (AlgebraicClosure L)) (\u220f \u03c3 : L \u2243\u2090[K] L, \u2191\u03c3 x)\n[PROOFSTEP]\nrw [map_prod (algebraMap L (AlgebraicClosure L))]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(norm K) x)) =\n    \u220f x_1 : L \u2243\u2090[K] L, \u2191(algebraMap L (AlgebraicClosure L)) (\u2191x_1 x)\n[PROOFSTEP]\nrw [\u2190 Fintype.prod_equiv (Normal.algHomEquivAut K (AlgebraicClosure L) L)]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(norm K) x)) = \u220f x : L \u2192\u2090[K] AlgebraicClosure L, ?a.f\u271d x\n[PROOFSTEP]\nrw [\u2190 norm_eq_prod_embeddings]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(algebraMap K L) (\u2191(norm K) x)) =\n    \u2191(algebraMap K (AlgebraicClosure L)) (\u2191(norm K) ?a.x\u271d)\ncase a.x\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 L\n[PROOFSTEP]\nsimp only [algebraMap_eq_smul_one, smul_one_smul]\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2191(norm K) x \u2022 1 = \u2191(norm K) ?a.x\u271d \u2022 1\ncase a.x\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 L\n[PROOFSTEP]\nrfl\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u22a2 \u2200 (x_1 : L \u2192\u2090[K] AlgebraicClosure L),\n    \u2191x_1 x = \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(\u2191(Normal.algHomEquivAut K (AlgebraicClosure L) L) x_1) x)\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase a.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b9 : CommRing R\ninst\u271d\u00b9\u2070 : Ring S\ninst\u271d\u2079 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field F\ninst\u271d\u2075 : Algebra K L\ninst\u271d\u2074 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u00b3 : Field E\ninst\u271d\u00b2 : Algebra K E\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsGalois K L\nx : L\n\u03c3 : L \u2192\u2090[K] AlgebraicClosure L\n\u22a2 \u2191\u03c3 x = \u2191(algebraMap L (AlgebraicClosure L)) (\u2191(\u2191(Normal.algHomEquivAut K (AlgebraicClosure L) L) \u03c3) x)\n[PROOFSTEP]\nsimp only [Normal.algHomEquivAut, AlgHom.restrictNormal', Equiv.coe_fn_mk, AlgEquiv.coe_ofBijective,\n  AlgHom.restrictNormal_commutes, id.map_eq_id, RingHom.id_apply]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\n\u22a2 IsIntegral R (\u2191(norm K) x)\n[PROOFSTEP]\nhave hx' : IsIntegral K x := isIntegral_of_isScalarTower hx\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\n\u22a2 IsIntegral R (\u2191(norm K) x)\n[PROOFSTEP]\nrw [\u2190 isIntegral_algebraMap_iff (algebraMap K (AlgebraicClosure L)).injective, norm_eq_prod_roots]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\n\u22a2 IsIntegral R\n    (Multiset.prod (roots (Polynomial.map (algebraMap K (AlgebraicClosure L)) (minpoly K x))) ^\n      finrank { x_1 // x_1 \u2208 K\u27eex\u27ef } L)\n[PROOFSTEP]\nrefine' (IsIntegral.multiset_prod fun y hy => _).pow _\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\ny : (fun x => AlgebraicClosure L) (\u2191(norm K) x)\nhy : y \u2208 roots (Polynomial.map (algebraMap K (AlgebraicClosure L)) (minpoly K x))\n\u22a2 IsIntegral R y\n[PROOFSTEP]\nrw [mem_roots_map (minpoly.ne_zero hx')] at hy \n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\ny : (fun x => AlgebraicClosure L) (\u2191(norm K) x)\nhy : eval\u2082 (algebraMap K (AlgebraicClosure L)) y (minpoly K x) = 0\n\u22a2 IsIntegral R y\n[PROOFSTEP]\nuse minpoly R x, minpoly.monic hx\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\ny : (fun x => AlgebraicClosure L) (\u2191(norm K) x)\nhy : eval\u2082 (algebraMap K (AlgebraicClosure L)) y (minpoly K x) = 0\n\u22a2 eval\u2082 (algebraMap R ((fun x => AlgebraicClosure L) (\u2191(norm K) x))) y (minpoly R x) = 0\n[PROOFSTEP]\nrw [\u2190 aeval_def] at hy \u22a2\n[GOAL]\ncase right\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\ny : (fun x => AlgebraicClosure L) (\u2191(norm K) x)\nhy : \u2191(aeval y) (minpoly K x) = 0\n\u22a2 \u2191(aeval y) (minpoly R x) = 0\n[PROOFSTEP]\nexact minpoly.aeval_of_isScalarTower R x y hy\n[GOAL]\ncase hF\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : Ring S\ninst\u271d\u00b9\u00b2 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : Field F\ninst\u271d\u2078 : Algebra K L\ninst\u271d\u2077 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra K E\ninst\u271d\u2074 : Algebra R L\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : IsScalarTower R K L\ninst\u271d\u00b9 : IsSeparable K L\ninst\u271d : FiniteDimensional K L\nx : L\nhx : IsIntegral R x\nhx' : IsIntegral K x\n\u22a2 Splits (algebraMap K (AlgebraicClosure L)) (minpoly K x)\n[PROOFSTEP]\napply IsAlgClosed.splits_codomain\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = \u2191(norm K) x\n[PROOFSTEP]\nby_cases hKF : FiniteDimensional K F\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = \u2191(norm K) x\n[PROOFSTEP]\nlet A := AlgebraicClosure K\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = \u2191(norm K) x\n[PROOFSTEP]\napply (algebraMap K A).injective\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nhaveI : FiniteDimensional L F := FiniteDimensional.right K L F\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis : FiniteDimensional L F\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nhaveI : FiniteDimensional K L := FiniteDimensional.left K L F\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d : FiniteDimensional L F\nthis : FiniteDimensional K L\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nhaveI : IsSeparable K L := isSeparable_tower_bot_of_isSeparable K L F\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b9 : FiniteDimensional L F\nthis\u271d : FiniteDimensional K L\nthis : IsSeparable K L\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nhaveI : IsSeparable L F := isSeparable_tower_top_of_isSeparable K L F\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b2 : FiniteDimensional L F\nthis\u271d\u00b9 : FiniteDimensional K L\nthis\u271d : IsSeparable K L\nthis : IsSeparable L F\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nletI :\n  \u2200 \u03c3 : L \u2192\u2090[K] A,\n    haveI := \u03c3.toRingHom.toAlgebra\n    Fintype (F \u2192\u2090[L] A) :=\n  fun _ => inferInstance\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : FiniteDimensional K L\nthis\u271d\u00b9 : IsSeparable K L\nthis\u271d : IsSeparable L F\nthis : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u2191(algebraMap K A) (\u2191(norm K) x)\n[PROOFSTEP]\nrw [norm_eq_prod_embeddings K A (_ : F),\n  Fintype.prod_equiv algHomEquivSigma (fun \u03c3 : F \u2192\u2090[K] A => \u03c3 x) (fun \u03c0 : \u03a3 f : L \u2192\u2090[K] A, _ => (\u03c0.2 : F \u2192 A) x)\n    fun _ => rfl]\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : FiniteDimensional K L\nthis\u271d\u00b9 : IsSeparable K L\nthis\u271d : IsSeparable L F\nthis : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u220f x_1 : (f : L \u2192\u2090[K] A) \u00d7 (F \u2192\u2090[L] A), \u2191x_1.snd x\n[PROOFSTEP]\nsuffices\n  \u2200 \u03c3 : L \u2192\u2090[K] A,\n    haveI := \u03c3.toRingHom.toAlgebra\n    \u220f \u03c0 : F \u2192\u2090[L] A, \u03c0 x = \u03c3 (norm L x)\n  by simp_rw [\u2190 Finset.univ_sigma_univ, Finset.prod_sigma, this, norm_eq_prod_embeddings]\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u2074 : FiniteDimensional L F\nthis\u271d\u00b3 : FiniteDimensional K L\nthis\u271d\u00b2 : IsSeparable K L\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\nthis : \u2200 (\u03c3 : L \u2192\u2090[K] A), \u220f \u03c0 : F \u2192\u2090[L] A, \u2191\u03c0 x = \u2191\u03c3 (\u2191(norm L) x)\n\u22a2 \u2191(algebraMap K A) (\u2191(norm K) (\u2191(norm L) x)) = \u220f x_1 : (f : L \u2192\u2090[K] A) \u00d7 (F \u2192\u2090[L] A), \u2191x_1.snd x\n[PROOFSTEP]\nsimp_rw [\u2190 Finset.univ_sigma_univ, Finset.prod_sigma, this, norm_eq_prod_embeddings]\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : FiniteDimensional K L\nthis\u271d\u00b9 : IsSeparable K L\nthis\u271d : IsSeparable L F\nthis : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u22a2 \u2200 (\u03c3 : L \u2192\u2090[K] A), \u220f \u03c0 : F \u2192\u2090[L] A, \u2191\u03c0 x = \u2191\u03c3 (\u2191(norm L) x)\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u00b3 : FiniteDimensional L F\nthis\u271d\u00b2 : FiniteDimensional K L\nthis\u271d\u00b9 : IsSeparable K L\nthis\u271d : IsSeparable L F\nthis : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u03c3 : L \u2192\u2090[K] A\n\u22a2 \u220f \u03c0 : F \u2192\u2090[L] A, \u2191\u03c0 x = \u2191\u03c3 (\u2191(norm L) x)\n[PROOFSTEP]\nletI : Algebra L A := \u03c3.toRingHom.toAlgebra\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u2074 : FiniteDimensional L F\nthis\u271d\u00b3 : FiniteDimensional K L\nthis\u271d\u00b2 : IsSeparable K L\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u03c3 : L \u2192\u2090[K] A\nthis : Algebra L A := RingHom.toAlgebra \u2191\u03c3\n\u22a2 \u220f \u03c0 : F \u2192\u2090[L] A, \u2191\u03c0 x = \u2191\u03c3 (\u2191(norm L) x)\n[PROOFSTEP]\nrw [\u2190 norm_eq_prod_embeddings L A (_ : F)]\n[GOAL]\ncase pos.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : FiniteDimensional K F\nA : Type u_4 := AlgebraicClosure K\nthis\u271d\u2074 : FiniteDimensional L F\nthis\u271d\u00b3 : FiniteDimensional K L\nthis\u271d\u00b2 : IsSeparable K L\nthis\u271d\u00b9 : IsSeparable L F\nthis\u271d : (\u03c3 : L \u2192\u2090[K] A) \u2192 Fintype (F \u2192\u2090[L] A) := fun x => inferInstance\n\u03c3 : L \u2192\u2090[K] A\nthis : Algebra L A := RingHom.toAlgebra \u2191\u03c3\n\u22a2 \u2191(algebraMap L A) (\u2191(norm L) x) = \u2191\u03c3 (\u2191(norm L) x)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = \u2191(norm K) x\n[PROOFSTEP]\nrw [norm_eq_one_of_not_module_finite hKF]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = 1\n[PROOFSTEP]\nby_cases hKL : FiniteDimensional K L\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\nhKL : FiniteDimensional K L\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = 1\n[PROOFSTEP]\nhave hLF : \u00acFiniteDimensional L F := by\n  refine' (mt _) hKF\n  intro hKF\n  exact FiniteDimensional.trans K L F\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\nhKL : FiniteDimensional K L\n\u22a2 \u00acFiniteDimensional L F\n[PROOFSTEP]\nrefine' (mt _) hKF\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\nhKL : FiniteDimensional K L\n\u22a2 FiniteDimensional L F \u2192 FiniteDimensional K F\n[PROOFSTEP]\nintro hKF\n[GOAL]\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF\u271d : \u00acFiniteDimensional K F\nhKL : FiniteDimensional K L\nhKF : FiniteDimensional L F\n\u22a2 FiniteDimensional K F\n[PROOFSTEP]\nexact FiniteDimensional.trans K L F\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\nhKL : FiniteDimensional K L\nhLF : \u00acFiniteDimensional L F\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = 1\n[PROOFSTEP]\nrw [norm_eq_one_of_not_module_finite hLF, _root_.map_one]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Ring S\ninst\u271d\u00b9\u2070 : Algebra R S\nK : Type u_4\nL : Type u_5\nF : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Algebra K L\ninst\u271d\u2075 : Algebra K F\n\u03b9 : Type w\nE : Type u_7\ninst\u271d\u2074 : Field E\ninst\u271d\u00b3 : Algebra K E\ninst\u271d\u00b2 : Algebra L F\ninst\u271d\u00b9 : IsScalarTower K L F\ninst\u271d : IsSeparable K F\nx : F\nhKF : \u00acFiniteDimensional K F\nhKL : \u00acFiniteDimensional K L\n\u22a2 \u2191(norm K) (\u2191(norm L) x) = 1\n[PROOFSTEP]\nrw [norm_eq_one_of_not_module_finite hKL]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Norm", "llama_tokens": 46861, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104789086703225, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.5104190365174097}}
{"text": "[GOAL]\n\u03b1\u271d \u03b2\u271d : Type u\ns\u271d : Set \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Set \u03b2\u271d\ng : Set (\u03b1\u271d \u2192 \u03b2\u271d)\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\n\u22a2 image2 f s t = Seq.seq (f <$> s) fun x => t\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1\u271d \u03b2\u271d : Type u\ns\u271d : Set \u03b1\u271d\nf\u271d : \u03b1\u271d \u2192 Set \u03b2\u271d\ng : Set (\u03b1\u271d \u2192 \u03b2\u271d)\n\u03b1 \u03b2 \u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\nx\u271d : \u03b3\n\u22a2 x\u271d \u2208 image2 f s t \u2194 x\u271d \u2208 Seq.seq (f <$> s) fun x => t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 \u03b2 : Type u\ns : Set \u03b1\nf : \u03b1 \u2192 Set \u03b2\ng : Set (\u03b1 \u2192 \u03b2)\n\u03b1\u271d \u03b2\u271d \u03b3\u271d : Type ?u.7831\nx\u271d\u00b2 : Set \u03b1\u271d\nx\u271d\u00b9 : \u03b1\u271d \u2192 Set \u03b2\u271d\nx\u271d : \u03b2\u271d \u2192 Set \u03b3\u271d\n\u22a2 x\u271d\u00b2 >>= x\u271d\u00b9 >>= x\u271d = x\u271d\u00b2 >>= fun x => x\u271d\u00b9 x >>= x\u271d\n[PROOFSTEP]\nsimp only [bind_def, biUnion_iUnion]\n[GOAL]\n\u03b1 \u03b2\u271d : Type u\ns : Set \u03b1\nf : \u03b1 \u2192 Set \u03b2\u271d\ng : Set (\u03b1 \u2192 \u03b2\u271d)\n\u03b2 : Set \u03b1\n\u03b3 : Set \u2191\u03b2\n\u22a2 Lean.Internal.coeM \u03b3 \u2286 \u03b2\n[PROOFSTEP]\nintro _ \u27e8_, \u27e8\u27e8\u27e8_, ha\u27e9, rfl\u27e9, _, \u27e8_, rfl\u27e9, _\u27e9\u27e9\n[GOAL]\n\u03b1 \u03b2\u271d : Type u\ns : Set \u03b1\nf : \u03b1 \u2192 Set \u03b2\u271d\ng : Set (\u03b1 \u2192 \u03b2\u271d)\n\u03b2 : Set \u03b1\n\u03b3 : Set \u2191\u03b2\na\u271d val\u271d : \u03b1\nha : val\u271d \u2208 \u03b2\nh\u271d : { val := val\u271d, property := ha } \u2208 \u03b3\nright\u271d : a\u271d \u2208 (fun h => (fun a => pure (CoeT.coe a)) { val := val\u271d, property := ha }) h\u271d\n\u22a2 a\u271d \u2208 \u03b2\n[PROOFSTEP]\nconvert ha\n[GOAL]\n\u03b1 \u03b2\u271d : Type u\ns : Set \u03b1\nf : \u03b1 \u2192 Set \u03b2\u271d\ng : Set (\u03b1 \u2192 \u03b2\u271d)\n\u03b2 : Set \u03b1\n\u03b3 : Set \u2191\u03b2\na : \u03b1\nha : a \u2208 Lean.Internal.coeM \u03b3\n\u22a2 { val := a, property := (_ : a \u2208 \u03b2) } \u2208 \u03b3\n[PROOFSTEP]\nrcases ha with \u27e8_, \u27e8_, rfl\u27e9, _, \u27e8ha, rfl\u27e9, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 \u03b2\u271d : Type u\ns : Set \u03b1\nf : \u03b1 \u2192 Set \u03b2\u271d\ng : Set (\u03b1 \u2192 \u03b2\u271d)\n\u03b2 : Set \u03b1\n\u03b3 : Set \u2191\u03b2\na : \u03b1\nw\u271d : \u2191\u03b2\nha : w\u271d \u2208 \u03b3\nright\u271d : a \u2208 (fun h => (fun a => pure (CoeT.coe a)) w\u271d) ha\n\u22a2 { val := a, property := (_ : a \u2208 \u03b2) } \u2208 \u03b3\n[PROOFSTEP]\nconvert ha\n", "meta": {"mathlib_filename": "Mathlib.Data.Set.Functor", "llama_tokens": 1008, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8056321796478255, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5102956913380823}}
{"text": "[GOAL]\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R\n[PROOFSTEP]\nrcases Nat.eq_zero_or_pos n with (rfl | hnpos)\n[GOAL]\ncase inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d : CommRing R\nhdiv : \u00acp \u2223 0\n\u22a2 \u2191(expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R * cyclotomic 0 R\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R\n[PROOFSTEP]\nhaveI := NeZero.of_pos hnpos\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R\n[PROOFSTEP]\nsuffices expand \u2124 p (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124 by\n  rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, Polynomial.map_mul, map_cyclotomic_int, map_cyclotomic]\n[GOAL]\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis\u271d : NeZero n\nthis : \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, Polynomial.map_mul, map_cyclotomic_int, map_cyclotomic]\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 * cyclotomic n \u2124\n[PROOFSTEP]\nrefine'\n  eq_of_monic_of_dvd_of_natDegree_le ((cyclotomic.monic _ \u2124).mul (cyclotomic.monic _ _))\n    ((cyclotomic.monic n \u2124).expand hp.pos) _ _\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u2124 * cyclotomic n \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124)\n[PROOFSTEP]\nrefine'\n  (IsPrimitive.Int.dvd_iff_map_cast_dvd_map_cast _ _\n        (IsPrimitive.mul (cyclotomic.isPrimitive (n * p) \u2124) (cyclotomic.isPrimitive n \u2124))\n        ((cyclotomic.monic n \u2124).expand hp.pos).isPrimitive).2\n    _\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 map (Int.castRingHom \u211a) (cyclotomic (n * p) \u2124 * cyclotomic n \u2124) \u2223\n    map (Int.castRingHom \u211a) (\u2191(expand \u2124 p) (cyclotomic n \u2124))\n[PROOFSTEP]\nrw [Polynomial.map_mul, map_cyclotomic_int, map_cyclotomic_int, map_expand, map_cyclotomic_int]\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a * cyclotomic n \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nrefine' IsCoprime.mul_dvd (cyclotomic.isCoprime_rat fun h => _) _ _\n[GOAL]\ncase inr.refine'_1.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n\n\u22a2 False\n[PROOFSTEP]\nreplace h : n * p = n * 1 := by simp [h]\n[GOAL]\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n\n\u22a2 n * p = n * 1\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase inr.refine'_1.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nh : n * p = n * 1\n\u22a2 False\n[PROOFSTEP]\nexact Nat.Prime.ne_one hp (mul_left_cancel\u2080 hnpos.ne' h)\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nhave hpos : 0 < n * p := mul_pos hnpos hp.pos\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\n\u22a2 cyclotomic (n * p) \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nhave hprim := Complex.isPrimitiveRoot_exp _ hpos.ne'\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 cyclotomic (n * p) \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nrw [cyclotomic_eq_minpoly_rat hprim hpos]\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nrefine' @minpoly.dvd \u211a \u2102 _ _ algebraRat _ _ _\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 \u2191(aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)))) (\u2191(expand \u211a p) (cyclotomic n \u211a)) = 0\n[PROOFSTEP]\nrw [aeval_def, \u2190 eval_map, map_expand, map_cyclotomic, expand_eval, \u2190 IsRoot.def, @isRoot_cyclotomic_iff]\n[GOAL]\ncase inr.refine'_1.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)) ^ p) n\n[PROOFSTEP]\nconvert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n)\n[GOAL]\ncase h.e'_4\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 n = n * p / p\n[PROOFSTEP]\nrw [Nat.mul_div_cancel _ (Nat.Prime.pos hp)]\n[GOAL]\ncase inr.refine'_1.refine'_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 cyclotomic n \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nhave hprim := Complex.isPrimitiveRoot_exp _ hnpos.ne.symm\n[GOAL]\ncase inr.refine'_1.refine'_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 cyclotomic n \u211a \u2223 \u2191(expand \u211a p) (cyclotomic n \u211a)\n[PROOFSTEP]\nrw [cyclotomic_eq_minpoly_rat hprim hnpos]\n[GOAL]\ncase inr.refine'_1.refine'_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) \u2223\n    \u2191(expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))\n[PROOFSTEP]\nrefine' @minpoly.dvd \u211a \u2102 _ _ algebraRat _ _ _\n[GOAL]\ncase inr.refine'_1.refine'_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 \u2191(aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))\n      (\u2191(expand \u211a p) (minpoly \u211a (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)))) =\n    0\n[PROOFSTEP]\nrw [aeval_def, \u2190 eval_map, map_expand, expand_eval, \u2190 IsRoot.def, \u2190 cyclotomic_eq_minpoly_rat hprim hnpos,\n  map_cyclotomic, @isRoot_cyclotomic_iff]\n[GOAL]\ncase inr.refine'_1.refine'_3\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n)) n\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191n) ^ p) n\n[PROOFSTEP]\nexact IsPrimitiveRoot.pow_of_prime hprim hp hdiv\n[GOAL]\ncase inr.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : \u00acp \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhnpos : n > 0\nthis : NeZero n\n\u22a2 natDegree (\u2191(expand \u2124 p) (cyclotomic n \u2124)) \u2264 natDegree (cyclotomic (n * p) \u2124 * cyclotomic n \u2124)\n[PROOFSTEP]\nrw [natDegree_expand, natDegree_cyclotomic, natDegree_mul (cyclotomic_ne_zero _ \u2124) (cyclotomic_ne_zero _ \u2124),\n  natDegree_cyclotomic, natDegree_cyclotomic, mul_comm n, Nat.totient_mul ((Nat.Prime.coprime_iff_not_dvd hp).2 hdiv),\n  Nat.totient_prime hp, mul_comm (p - 1), \u2190 Nat.mul_succ, Nat.sub_one, Nat.succ_pred_eq_of_pos hp.pos]\n[GOAL]\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R\n[PROOFSTEP]\nrcases n.eq_zero_or_pos with (rfl | hzero)\n[GOAL]\ncase inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d : CommRing R\nhdiv : p \u2223 0\n\u22a2 \u2191(expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R\n[PROOFSTEP]\nhaveI := NeZero.of_pos hzero\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R\n[PROOFSTEP]\nsuffices expand \u2124 p (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 by\n  rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, map_cyclotomic_int, map_cyclotomic]\n[GOAL]\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis\u271d : NeZero n\nthis : \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124\n\u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, map_cyclotomic_int, map_cyclotomic]\n[GOAL]\ncase inr\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\n\u22a2 \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124\n[PROOFSTEP]\nrefine' eq_of_monic_of_dvd_of_natDegree_le (cyclotomic.monic _ _) ((cyclotomic.monic n \u2124).expand hp.pos) _ _\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\n\u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124)\n[PROOFSTEP]\nhave hpos := Nat.mul_pos hzero hp.pos\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\n\u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124)\n[PROOFSTEP]\nhave hprim := Complex.isPrimitiveRoot_exp _ hpos.ne.symm\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124)\n[PROOFSTEP]\nrw [cyclotomic_eq_minpoly hprim hpos]\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 minpoly \u2124 (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124)\n[PROOFSTEP]\nrefine' minpoly.isIntegrallyClosed_dvd (hprim.isIntegral hpos) _\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 \u2191(aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)))) (\u2191(expand \u2124 p) (cyclotomic n \u2124)) = 0\n[PROOFSTEP]\nrw [aeval_def, \u2190 eval_map, map_expand, map_cyclotomic, expand_eval, \u2190 IsRoot.def, @isRoot_cyclotomic_iff]\n[GOAL]\ncase inr.refine'_1\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)) ^ p) n\n[PROOFSTEP]\nconvert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n)\n[GOAL]\ncase h.e'_4\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : n * p > 0\nhprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p)\n\u22a2 n = n * p / p\n[PROOFSTEP]\nrw [Nat.mul_div_cancel _ hp.pos]\n[GOAL]\ncase inr.refine'_2\np n : \u2115\nhp : Nat.Prime p\nhdiv : p \u2223 n\nR : Type u_1\ninst\u271d : CommRing R\nhzero : n > 0\nthis : NeZero n\n\u22a2 natDegree (\u2191(expand \u2124 p) (cyclotomic n \u2124)) \u2264 natDegree (cyclotomic (n * p) \u2124)\n[PROOFSTEP]\nrw [natDegree_expand, natDegree_cyclotomic, natDegree_cyclotomic, mul_comm n, Nat.totient_mul_of_prime_of_dvd hp hdiv,\n  mul_comm]\n[GOAL]\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn m : \u2115\nhmn : m \u2264 n\nh : Irreducible (cyclotomic (p ^ n) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nrcases m.eq_zero_or_pos with (rfl | hm)\n[GOAL]\ncase inl\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn : \u2115\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 \u2264 n\n\u22a2 Irreducible (cyclotomic (p ^ 0) R)\n[PROOFSTEP]\nsimpa using irreducible_X_sub_C (1 : R)\n[GOAL]\ncase inr\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nn m : \u2115\nhmn : m \u2264 n\nh : Irreducible (cyclotomic (p ^ n) R)\nhm : m > 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn\n[GOAL]\ncase inr.intro\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhmn : m \u2264 m + k\nh : Irreducible (cyclotomic (p ^ (m + k)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\ninduction' k with k hk\n[GOAL]\ncase inr.intro.zero\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk : \u2115\nhmn\u271d : m \u2264 m + k\nh\u271d : Irreducible (cyclotomic (p ^ (m + k)) R)\nhmn : m \u2264 m + Nat.zero\nh : Irreducible (cyclotomic (p ^ (m + Nat.zero)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk\u271d : \u2115\nhmn\u271d : m \u2264 m + k\u271d\nh\u271d : Irreducible (cyclotomic (p ^ (m + k\u271d)) R)\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + Nat.succ k\nh : Irreducible (cyclotomic (p ^ (m + Nat.succ k)) R)\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nhave : m + k \u2260 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne'\n[GOAL]\ncase inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk\u271d : \u2115\nhmn\u271d : m \u2264 m + k\u271d\nh\u271d : Irreducible (cyclotomic (p ^ (m + k\u271d)) R)\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + Nat.succ k\nh : Irreducible (cyclotomic (p ^ (m + Nat.succ k)) R)\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nrw [Nat.add_succ, pow_succ', \u2190 cyclotomic_expand_eq_cyclotomic hp <| dvd_pow_self p this] at h \n[GOAL]\ncase inr.intro.succ\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk\u271d : \u2115\nhmn\u271d : m \u2264 m + k\u271d\nh\u271d : Irreducible (cyclotomic (p ^ (m + k\u271d)) R)\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + Nat.succ k\nh : Irreducible (\u2191(expand R p) (cyclotomic (p ^ (m + k)) R))\nthis : m + k \u2260 0\n\u22a2 Irreducible (cyclotomic (p ^ m) R)\n[PROOFSTEP]\nexact hk (by linarith) (of_irreducible_expand hp.ne_zero h)\n[GOAL]\np : \u2115\nhp : Nat.Prime p\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\nm : \u2115\nhm : m > 0\nk\u271d : \u2115\nhmn\u271d : m \u2264 m + k\u271d\nh\u271d : Irreducible (cyclotomic (p ^ (m + k\u271d)) R)\nk : \u2115\nhk : m \u2264 m + k \u2192 Irreducible (cyclotomic (p ^ (m + k)) R) \u2192 Irreducible (cyclotomic (p ^ m) R)\nhmn : m \u2264 m + Nat.succ k\nh : Irreducible (\u2191(expand R p) (cyclotomic (p ^ (m + k)) R))\nthis : m + k \u2260 0\n\u22a2 m \u2264 m + k\n[PROOFSTEP]\nlinarith\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ (p - 1)\n[PROOFSTEP]\nletI : Algebra (ZMod p) R := ZMod.algebra _ _\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ (p - 1)\n[PROOFSTEP]\nsuffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1) by\n  rw [\u2190 map_cyclotomic _ (algebraMap (ZMod p) R), \u2190 map_cyclotomic _ (algebraMap (ZMod p) R), this, Polynomial.map_pow]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis\u271d : Algebra (ZMod p) R := ZMod.algebra R p\nthis : cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1)\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ (p - 1)\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic _ (algebraMap (ZMod p) R), \u2190 map_cyclotomic _ (algebraMap (ZMod p) R), this, Polynomial.map_pow]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1)\n[PROOFSTEP]\napply mul_right_injective\u2080 (cyclotomic_ne_zero n <| ZMod p)\n[GOAL]\ncase a\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 (fun x x_1 => x * x_1) (cyclotomic n (ZMod p)) (cyclotomic (n * p) (ZMod p)) =\n    (fun x x_1 => x * x_1) (cyclotomic n (ZMod p)) (cyclotomic n (ZMod p) ^ (p - 1))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic n (ZMod p) * cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) * cyclotomic n (ZMod p) ^ (p - 1)\n[PROOFSTEP]\nrw [\u2190 pow_succ, tsub_add_cancel_of_le hp.out.one_lt.le, mul_comm, \u2190 ZMod.expand_card]\n[GOAL]\ncase a\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) (ZMod p) * cyclotomic n (ZMod p) = \u2191(expand (ZMod p) p) (cyclotomic n (ZMod p))\n[PROOFSTEP]\nconv_rhs => rw [\u2190 map_cyclotomic_int]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(expand (ZMod p) p) (cyclotomic n (ZMod p))\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic_int]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(expand (ZMod p) p) (cyclotomic n (ZMod p))\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic_int]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n| \u2191(expand (ZMod p) p) (cyclotomic n (ZMod p))\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic_int]\n[GOAL]\ncase a\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : \u00acp \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) (ZMod p) * cyclotomic n (ZMod p) =\n    \u2191(expand (ZMod p) p) (map (Int.castRingHom (ZMod p)) (cyclotomic n \u2124))\n[PROOFSTEP]\nrw [\u2190 map_expand, cyclotomic_expand_eq_cyclotomic_mul hp.out hn, Polynomial.map_mul, map_cyclotomic, map_cyclotomic]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : p \u2223 n\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ p\n[PROOFSTEP]\nletI : Algebra (ZMod p) R := ZMod.algebra _ _\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : p \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ p\n[PROOFSTEP]\nsuffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p by\n  rw [\u2190 map_cyclotomic _ (algebraMap (ZMod p) R), \u2190 map_cyclotomic _ (algebraMap (ZMod p) R), this, Polynomial.map_pow]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : p \u2223 n\nthis\u271d : Algebra (ZMod p) R := ZMod.algebra R p\nthis : cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p\n\u22a2 cyclotomic (n * p) R = cyclotomic n R ^ p\n[PROOFSTEP]\nrw [\u2190 map_cyclotomic _ (algebraMap (ZMod p) R), \u2190 map_cyclotomic _ (algebraMap (ZMod p) R), this, Polynomial.map_pow]\n[GOAL]\nR : Type u_1\np n : \u2115\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhn : p \u2223 n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n\u22a2 cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p\n[PROOFSTEP]\nrw [\u2190 ZMod.expand_card, \u2190 map_cyclotomic_int n, \u2190 map_expand, cyclotomic_expand_eq_cyclotomic hp.out hn, map_cyclotomic,\n  mul_comm]\n[GOAL]\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\nx\u271d : 0 < 1\n\u22a2 cyclotomic (p ^ 1 * m) R = cyclotomic m R ^ (p ^ 1 - p ^ (1 - 1))\n[PROOFSTEP]\nrw [pow_one, Nat.sub_self, pow_zero, mul_comm, cyclotomic_mul_prime_eq_pow_of_not_dvd R hm]\n[GOAL]\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\n\u22a2 cyclotomic (p ^ (a + 2) * m) R = cyclotomic m R ^ (p ^ (a + 2) - p ^ (a + 2 - 1))\n[PROOFSTEP]\nhave hdiv : p \u2223 p ^ a.succ * m := \u27e8p ^ a * m, by rw [\u2190 mul_assoc, pow_succ]\u27e9\n[GOAL]\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\n\u22a2 p ^ Nat.succ a * m = p * (p ^ a * m)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, pow_succ]\n[GOAL]\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 cyclotomic (p ^ (a + 2) * m) R = cyclotomic m R ^ (p ^ (a + 2) - p ^ (a + 2 - 1))\n[PROOFSTEP]\nrw [pow_succ, mul_assoc, mul_comm, cyclotomic_mul_prime_dvd_eq_pow R hdiv, cyclotomic_mul_prime_pow_eq _ _ a.succ_pos, \u2190\n  pow_mul]\n[GOAL]\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 cyclotomic m R ^ ((p ^ Nat.succ a - p ^ (Nat.succ a - 1)) * p) = cyclotomic m R ^ (p * p ^ (a + 1) - p ^ (a + 2 - 1))\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 \u00acp \u2223 m\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 (p ^ Nat.succ a - p ^ (Nat.succ a - 1)) * p = p * p ^ (a + 1) - p ^ (a + 2 - 1)\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 \u00acp \u2223 m\n[PROOFSTEP]\nsimp only [tsub_zero, Nat.succ_sub_succ_eq_sub]\n[GOAL]\ncase e_a\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 (p ^ Nat.succ a - p ^ a) * p = p * p ^ (a + 1) - p ^ (a + 1)\nR : Type u_1\np m : \u2115\ninst\u271d\u00b2 : Fact (Nat.Prime p)\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R p\nhm : \u00acp \u2223 m\na : \u2115\nx\u271d : 0 < a + 2\nhdiv : p \u2223 p ^ Nat.succ a * m\n\u22a2 \u00acp \u2223 m\n[PROOFSTEP]\nrwa [Nat.mul_sub_right_distrib, mul_comm, pow_succ']\n[GOAL]\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\n\u22a2 IsRoot (cyclotomic (p ^ k * m) R) \u03bc \u2194 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nrcases k.eq_zero_or_pos with (rfl | hk)\n[GOAL]\ncase inl\nm p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\n\u22a2 IsRoot (cyclotomic (p ^ 0 * m) R) \u03bc \u2194 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nrw [pow_zero, one_mul, isRoot_cyclotomic_iff]\n[GOAL]\ncase inr\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\n\u22a2 IsRoot (cyclotomic (p ^ k * m) R) \u03bc \u2194 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase inr.refine'_1\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : IsRoot (cyclotomic (p ^ k * m) R) \u03bc\n\u22a2 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nrw [IsRoot.def, cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, eval_pow] at h \n[GOAL]\ncase inr.refine'_1\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : eval \u03bc (cyclotomic m R) ^ (p ^ k - p ^ (k - 1)) = 0\n\u22a2 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nreplace h := pow_eq_zero h\n[GOAL]\ncase inr.refine'_1\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : eval \u03bc (cyclotomic m R) = 0\n\u22a2 IsPrimitiveRoot \u03bc m\n[PROOFSTEP]\nrwa [\u2190 IsRoot.def, isRoot_cyclotomic_iff] at h \n[GOAL]\ncase inr.refine'_2\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : IsPrimitiveRoot \u03bc m\n\u22a2 IsRoot (cyclotomic (p ^ k * m) R) \u03bc\n[PROOFSTEP]\nrw [\u2190 isRoot_cyclotomic_iff, IsRoot.def] at h \n[GOAL]\ncase inr.refine'_2\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : eval \u03bc (cyclotomic m R) = 0\n\u22a2 IsRoot (cyclotomic (p ^ k * m) R) \u03bc\n[PROOFSTEP]\nrw [cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, IsRoot.def, eval_pow, h, zero_pow]\n[GOAL]\ncase inr.refine'_2\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : eval \u03bc (cyclotomic m R) = 0\n\u22a2 0 < p ^ k - p ^ (k - 1)\n[PROOFSTEP]\nsimp only [tsub_pos_iff_lt]\n[GOAL]\ncase inr.refine'_2\nm k p : \u2115\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\n\u03bc : R\ninst\u271d : NeZero \u2191m\nhk : k > 0\nh : eval \u03bc (cyclotomic m R) = 0\n\u22a2 p ^ (k - 1) < p ^ k\n[PROOFSTEP]\napply pow_strictMono_right hp.out.one_lt (Nat.pred_lt hk.ne')\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Cyclotomic.Expand", "llama_tokens": 13294, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149758396752, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.510144817451602}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : DecidableRel r\ninst\u271d\u00b2 : IsTrans \u03b1 r\ninst\u271d\u00b9 : IsAntisymm \u03b1 r\ninst\u271d : IsTotal \u03b1 r\ns : Multiset \u03b1\na : \u03b1\n\u22a2 a \u2208 sort r s \u2194 a \u2208 s\n[PROOFSTEP]\nrw [\u2190 mem_coe, sort_eq]\n", "meta": {"mathlib_filename": "Mathlib.Data.Multiset.Sort", "llama_tokens": 112, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7745833737577159, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.5099992385426556}}
{"text": "[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nne : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\np : \u03b9 \u2192 Prop\nf : (i : Subtype p) \u2192 \u03b1 i.val\n\u22a2 \u2203 g, (fun i => g i.val) = f\n[PROOFSTEP]\nhaveI : DecidablePred p := fun i \u21a6 Classical.propDecidable (p i)\n[GOAL]\n\u03b9 : Sort u_1\n\u03b1 : \u03b9 \u2192 Sort u_2\nne : \u2200 (i : \u03b9), Nonempty (\u03b1 i)\np : \u03b9 \u2192 Prop\nf : (i : Subtype p) \u2192 \u03b1 i.val\nthis : DecidablePred p\n\u22a2 \u2203 g, (fun i => g i.val) = f\n[PROOFSTEP]\nexact \u27e8fun i => if hi : p i then f \u27e8i, hi\u27e9 else Classical.choice (ne i), funext fun i \u21a6 dif_pos i.2\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Tactic.Lift", "llama_tokens": 250, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8289387914176259, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5098701835671475}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b \u2264 c \u2194 a \u2264 b + c\n[PROOFSTEP]\nrw [tsub_le_iff_right, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b \u2264 c \u2194 a - c \u2264 b\n[PROOFSTEP]\nrw [tsub_le_iff_left, tsub_le_iff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b - c \u2264 a + (b - c)\n[PROOFSTEP]\nrw [tsub_le_iff_left, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b \u2264 a + (c + (b - c))\n[PROOFSTEP]\nexact add_le_add_left le_add_tsub a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b - c \u2264 a - c + b\n[PROOFSTEP]\nrw [add_comm, add_comm _ b]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 b + a - c \u2264 b + (a - c)\n[PROOFSTEP]\nexact add_tsub_le_assoc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b \u2264 a + c + (b - c)\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b \u2264 a + (c + (b - c))\n[PROOFSTEP]\nexact add_le_add_left le_add_tsub a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b \u2264 a - c + (b + c)\n[PROOFSTEP]\nrw [add_comm a, add_comm (a - c)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 b + a \u2264 b + c + (a - c)\n[PROOFSTEP]\nexact add_le_add_add_tsub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a - c \u2264 a - b + (b - c)\n[PROOFSTEP]\nrw [tsub_le_iff_left, \u2190 add_assoc, add_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a \u2264 c + (b - c) + (a - b)\n[PROOFSTEP]\nexact le_add_tsub.trans (add_le_add_right le_add_tsub _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 c - a - (c - b) \u2264 b - a\n[PROOFSTEP]\nrw [tsub_le_iff_left, tsub_le_iff_left, add_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 c \u2264 c - b + (a + (b - a))\n[PROOFSTEP]\nexact le_tsub_add.trans (add_le_add_left le_add_tsub _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b - (c + d) \u2264 a - c + (b - d)\n[PROOFSTEP]\nrw [add_comm c, tsub_le_iff_left, add_assoc, \u2190 tsub_le_iff_left, \u2190 tsub_le_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b - d - c \u2264 a - c + (b - d)\n[PROOFSTEP]\nrefine' (tsub_le_tsub_right add_tsub_le_assoc c).trans _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + (b - d) - c \u2264 a - c + (b - d)\n[PROOFSTEP]\nrw [add_comm a, add_comm (a - c)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 b - d + a - c \u2264 b - d + (a - c)\n[PROOFSTEP]\nexact add_tsub_le_assoc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b - (a + c) \u2264 b - c\n[PROOFSTEP]\nrw [tsub_le_iff_left, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + b \u2264 a + (c + (b - c))\n[PROOFSTEP]\nexact add_le_add_left le_add_tsub _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + c - (b + c) \u2264 a - b\n[PROOFSTEP]\nrw [tsub_le_iff_left, add_right_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : Preorder \u03b1\ninst\u271d\u00b3 : AddCommSemigroup \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na b c d : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a + c \u2264 b + (a - b) + c\n[PROOFSTEP]\nexact add_le_add_right le_add_tsub c\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\n\u22a2 a \u2264 a + b - b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\n\u22a2 a \u2264 b + a - b\n[PROOFSTEP]\nexact hb.le_add_tsub_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a + b \u2264 c\n\u22a2 b + a \u2264 c\n[PROOFSTEP]\nrwa [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : AddCommMonoid \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b \u2264 0 \u2194 a \u2264 b\n[PROOFSTEP]\nrw [tsub_le_iff_left, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c\u271d d b a c : \u03b1\n\u22a2 b - a - c = b - (a + c)\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c\u271d d b a c : \u03b1\n\u22a2 b - a - c \u2264 b - (a + c)\n[PROOFSTEP]\nrw [tsub_le_iff_left, tsub_le_iff_left, \u2190 add_assoc, \u2190 tsub_le_iff_left]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c\u271d d b a c : \u03b1\n\u22a2 b - (a + c) \u2264 b - a - c\n[PROOFSTEP]\nrw [tsub_le_iff_left, add_assoc, \u2190 tsub_le_iff_left, \u2190 tsub_le_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a - (b + c) = a - c - b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a - (c + b) = a - c - b\n[PROOFSTEP]\napply tsub_add_eq_tsub_tsub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\n\u22a2 a - b - c = a - c - b\n[PROOFSTEP]\nrw [\u2190 tsub_add_eq_tsub_tsub, tsub_add_eq_tsub_tsub_swap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a = c + b\n\u22a2 c \u2264 a - b\n[PROOFSTEP]\nrw [h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a = c + b\n\u22a2 c \u2264 c + b - b\n[PROOFSTEP]\nexact hb.le_add_tsub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a = b + c\n\u22a2 a = c + b\n[PROOFSTEP]\nrw [add_comm, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\n\u22a2 a + b = a + b\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a - b < c\n\u22a2 a < b + c\n[PROOFSTEP]\nrw [lt_iff_le_and_ne, \u2190 tsub_le_iff_left]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a - b < c\n\u22a2 a - b \u2264 c \u2227 a \u2260 b + c\n[PROOFSTEP]\nrefine' \u27e8h.le, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhb : AddLECancellable b\nh : a - b < c\n\u22a2 a \u2260 b + c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhb : AddLECancellable b\nh : b + c - b < c\n\u22a2 False\n[PROOFSTEP]\nsimp [hb] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a < b + c\n[PROOFSTEP]\nrw [lt_iff_le_and_ne, \u2190 tsub_le_iff_right]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a - c \u2264 b \u2227 a \u2260 b + c\n[PROOFSTEP]\nrefine' \u27e8h.le, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a - c < b\n\u22a2 a \u2260 b + c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhc : AddLECancellable c\nh : b + c - c < b\n\u22a2 False\n[PROOFSTEP]\nsimp [hc] at h \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nhc : AddLECancellable c\nh : a + c < b\n\u22a2 a \u2260 b - c\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\nb c d : \u03b1\nhc : AddLECancellable c\nh : b - c + c < b\n\u22a2 False\n[PROOFSTEP]\nexact h.not_le le_tsub_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : AddCommSemigroup \u03b1\ninst\u271d\u00b9 : Sub \u03b1\ninst\u271d : OrderedSub \u03b1\na b c d : \u03b1\nha : AddLECancellable a\nh : a + c < b\n\u22a2 c + a < b\n[PROOFSTEP]\nrwa [add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na c b : \u03b1\n\u22a2 a + c - (b + c) = a - b\n[PROOFSTEP]\nrefine' add_tsub_add_le_tsub_right.antisymm (tsub_le_iff_right.2 <| le_of_add_le_add_right _)\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na c b : \u03b1\n\u22a2 \u03b1\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na c b : \u03b1\n\u22a2 a + ?refine'_1 \u2264 a + c - (b + c) + b + ?refine'_1\n[PROOFSTEP]\nexact c\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na c b : \u03b1\n\u22a2 a + c \u2264 a + c - (b + c) + b + c\n[PROOFSTEP]\nrw [add_assoc]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na c b : \u03b1\n\u22a2 a + c \u2264 a + c - (b + c) + (b + c)\n[PROOFSTEP]\nexact le_tsub_add\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : PartialOrder \u03b1\ninst\u271d\u2074 : AddCommSemigroup \u03b1\ninst\u271d\u00b3 : Sub \u03b1\ninst\u271d\u00b2 : OrderedSub \u03b1\na\u271d b\u271d c\u271d d : \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 a + b - (a + c) = b - c\n[PROOFSTEP]\nrw [add_comm a b, add_comm a c, add_tsub_add_eq_tsub_right]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Sub.Defs", "llama_tokens": 7281, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.7154240018510026, "lm_q1q2_score": 0.5095479907553885}}
{"text": "[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => u n / \u2191n) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave lnonneg : 0 \u2264 l := by\n  rcases hlim 2 one_lt_two with \u27e8c, _, ctop, clim\u27e9\n  have : Tendsto (fun n => u 0 / c n) atTop (\ud835\udcdd 0) := tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)\n  apply le_of_tendsto_of_tendsto' this clim fun n => _\n  simp_rw [div_eq_inv_mul]\n  exact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 0 \u2264 l\n[PROOFSTEP]\nrcases hlim 2 one_lt_two with \u27e8c, _, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nc : \u2115 \u2192 \u2115\nleft\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 2 * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 0 \u2264 l\n[PROOFSTEP]\nhave : Tendsto (fun n => u 0 / c n) atTop (\ud835\udcdd 0) := tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nc : \u2115 \u2192 \u2115\nleft\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 2 * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nthis : Tendsto (fun n => u 0 / \u2191(c n)) atTop (\ud835\udcdd 0)\n\u22a2 0 \u2264 l\n[PROOFSTEP]\napply le_of_tendsto_of_tendsto' this clim fun n => _\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nc : \u2115 \u2192 \u2115\nleft\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 2 * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nthis : Tendsto (fun n => u 0 / \u2191(c n)) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), u 0 / \u2191(c n) \u2264 u (c n) / \u2191(c n)\n[PROOFSTEP]\nsimp_rw [div_eq_inv_mul]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nc : \u2115 \u2192 \u2115\nleft\u271d : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 2 * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nthis : Tendsto (fun n => u 0 / \u2191(c n)) atTop (\ud835\udcdd 0)\n\u22a2 \u2200 (n : \u2115), (\u2191(c n))\u207b\u00b9 * u 0 \u2264 (\u2191(c n))\u207b\u00b9 * u (c n)\n[PROOFSTEP]\nexact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u22a2 Tendsto (fun n => u n / \u2191n) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave A : \u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192 \u2200\u1da0 n in atTop, u n - n * l \u2264 \u03b5 * (1 + \u03b5 + l) * n :=\n  by\n  intro \u03b5 \u03b5pos\n  rcases hlim (1 + \u03b5) ((lt_add_iff_pos_right _).2 \u03b5pos) with \u27e8c, cgrowth, ctop, clim\u27e9\n  have L : \u2200\u1da0 n in atTop, u (c n) - c n * l \u2264 \u03b5 * c n :=\n    by\n    rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n    filter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n    have cnpos : 0 < c n := cnpos'\n    calc\n      u (c n) - c n * l = (u (c n) / c n - l) * c n := by\n        simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]\n      _ \u2264 \u03b5 * c n := by\n        refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n        simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n        exact le_trans (le_abs_self _) hn\n  obtain \u27e8a, ha\u27e9 : \u2203 a : \u2115, \u2200 b : \u2115, a \u2264 b \u2192 (c (b + 1) : \u211d) \u2264 (1 + \u03b5) * c b \u2227 u (c b) - c b * l \u2264 \u03b5 * c b :=\n    eventually_atTop.1 (cgrowth.and L)\n  let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)\n  filter_upwards [Ici_mem_atTop M] with n hn\n  have exN : \u2203 N, n < c N := by\n    rcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n    exact \u27e8N, by linarith only [hN]\u27e9\n  let N := Nat.find exN\n  have ncN : n < c N := Nat.find_spec exN\n  have aN : a + 1 \u2264 N := by\n    by_contra' h\n    have cNM : c N \u2264 M := by\n      apply le_max'\n      apply mem_image_of_mem\n      exact mem_range.2 h\n    exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n  have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN\n  have cNn : c (N - 1) \u2264 n := by\n    have : N - 1 < N := Nat.pred_lt Npos.ne'\n    simpa only [not_lt] using Nat.find_min exN this\n  have IcN : (c N : \u211d) \u2264 (1 + \u03b5) * c (N - 1) :=\n    by\n    have A : a \u2264 N - 1 := by\n      apply @Nat.le_of_add_le_add_right a 1 (N - 1)\n      rw [Nat.sub_add_cancel Npos]\n      exact aN\n    have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n    have := (ha _ A).1\n    rwa [B] at this \n  calc\n    u n - n * l \u2264 u (c N) - c (N - 1) * l := by\n      apply sub_le_sub (hmono ncN.le)\n      apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg\n    _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring\n    _ \u2264 \u03b5 * c N + \u03b5 * c (N - 1) * l := by\n      apply add_le_add\n      \u00b7 apply (ha _ _).2\n        exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN\n      \u00b7 apply mul_le_mul_of_nonneg_right _ lnonneg\n        linarith only [IcN]\n    _ \u2264 \u03b5 * ((1 + \u03b5) * c (N - 1)) + \u03b5 * c (N - 1) * l := (add_le_add (mul_le_mul_of_nonneg_left IcN \u03b5pos.le) le_rfl)\n    _ = \u03b5 * (1 + \u03b5 + l) * c (N - 1) := by ring\n    _ \u2264 \u03b5 * (1 + \u03b5 + l) * n := by\n      refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n      apply mul_nonneg \u03b5pos.le\n      linarith only [\u03b5pos, lnonneg]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nrcases hlim (1 + \u03b5) ((lt_add_iff_pos_right _).2 \u03b5pos) with \u27e8c, cgrowth, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave L : \u2200\u1da0 n in atTop, u (c n) - c n * l \u2264 \u03b5 * c n :=\n  by\n  rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n  filter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n  have cnpos : 0 < c n := cnpos'\n  calc\n    u (c n) - c n * l = (u (c n) / c n - l) * c n := by\n      simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]\n    _ \u2264 \u03b5 * c n := by\n      refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n      simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n      exact le_trans (le_abs_self _) hn\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nrw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nfilter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\n\u22a2 u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nhave cnpos : 0 < c n := cnpos'\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\ncalc\n  u (c n) - c n * l = (u (c n) / c n - l) * c n := by\n    simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]\n  _ \u2264 \u03b5 * c n := by\n    refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n    simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n    exact le_trans (le_abs_self _) hn\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 u (c n) - \u2191(c n) * l = (u (c n) / \u2191(c n) - l) * \u2191(c n)\n[PROOFSTEP]\nsimp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 (u (c n) / \u2191(c n) - l) * \u2191(c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 u (c n) / \u2191(c n) - l \u2264 \u03b5\n[PROOFSTEP]\nsimp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\nhn : |u (c n) / \u2191(c n) - l| \u2264 \u03b5\n\u22a2 u (c n) / \u2191(c n) - l \u2264 \u03b5\n[PROOFSTEP]\nexact le_trans (le_abs_self _) hn\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a : \u2115, \u2200 b : \u2115, a \u2264 b \u2192 (c (b + 1) : \u211d) \u2264 (1 + \u03b5) * c b \u2227 u (c b) - c b * l \u2264 \u03b5 * c b :=\n  eventually_atTop.1 (cgrowth.and L)\n[GOAL]\ncase intro.intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nlet M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\n\u22a2 Finset.Nonempty (image (fun i => c i) (range (a + 1)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop M] with n hn\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave exN : \u2203 N, n < c N := by\n  rcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n  exact \u27e8N, by linarith only [hN]\u27e9\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\n\u22a2 \u2203 N, n < c N\n[PROOFSTEP]\nrcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nN : \u2115\nhN : n + 1 \u2264 c N\n\u22a2 \u2203 N, n < c N\n[PROOFSTEP]\nexact \u27e8N, by linarith only [hN]\u27e9\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nN : \u2115\nhN : n + 1 \u2264 c N\n\u22a2 n < c N\n[PROOFSTEP]\nlinarith only [hN]\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nlet N := Nat.find exN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave ncN : n < c N := Nat.find_spec exN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave aN : a + 1 \u2264 N := by\n  by_contra' h\n  have cNM : c N \u2264 M := by\n    apply le_max'\n    apply mem_image_of_mem\n    exact mem_range.2 h\n  exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\n\u22a2 a + 1 \u2264 N\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 False\n[PROOFSTEP]\nhave cNM : c N \u2264 M := by\n  apply le_max'\n  apply mem_image_of_mem\n  exact mem_range.2 h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 c N \u2264 M\n[PROOFSTEP]\napply le_max'\n[GOAL]\ncase H2\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 c N \u2208 image (fun i => c i) (range (a + 1))\n[PROOFSTEP]\napply mem_image_of_mem\n[GOAL]\ncase H2.h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 N \u2208 range (a + 1)\n[PROOFSTEP]\nexact mem_range.2 h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\ncNM : c N \u2264 M\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave cNn : c (N - 1) \u2264 n := by\n  have : N - 1 < N := Nat.pred_lt Npos.ne'\n  simpa only [not_lt] using Nat.find_min exN this\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 c (N - 1) \u2264 n\n[PROOFSTEP]\nhave : N - 1 < N := Nat.pred_lt Npos.ne'\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\nthis : N - 1 < N\n\u22a2 c (N - 1) \u2264 n\n[PROOFSTEP]\nsimpa only [not_lt] using Nat.find_min exN this\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nhave IcN : (c N : \u211d) \u2264 (1 + \u03b5) * c (N - 1) :=\n  by\n  have A : a \u2264 N - 1 := by\n    apply @Nat.le_of_add_le_add_right a 1 (N - 1)\n    rw [Nat.sub_add_cancel Npos]\n    exact aN\n  have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n  have := (ha _ A).1\n  rwa [B] at this \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nhave A : a \u2264 N - 1 := by\n  apply @Nat.le_of_add_le_add_right a 1 (N - 1)\n  rw [Nat.sub_add_cancel Npos]\n  exact aN\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\n\u22a2 a \u2264 N - 1\n[PROOFSTEP]\napply @Nat.le_of_add_le_add_right a 1 (N - 1)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\n\u22a2 a + 1 \u2264 N - 1 + 1\n[PROOFSTEP]\nrw [Nat.sub_add_cancel Npos]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\n\u22a2 a + 1 \u2264 N\n[PROOFSTEP]\nexact aN\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nA : a \u2264 N - 1\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nhave B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nA : a \u2264 N - 1\nB : N - 1 + 1 = N\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nhave := (ha _ A).1\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nA : a \u2264 N - 1\nB : N - 1 + 1 = N\nthis : \u2191(c (N - 1 + 1)) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nrwa [B] at this \n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\ncalc\n  u n - n * l \u2264 u (c N) - c (N - 1) * l := by\n    apply sub_le_sub (hmono ncN.le)\n    apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg\n  _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring\n  _ \u2264 \u03b5 * c N + \u03b5 * c (N - 1) * l := by\n    apply add_le_add\n    \u00b7 apply (ha _ _).2\n      exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN\n    \u00b7 apply mul_le_mul_of_nonneg_right _ lnonneg\n      linarith only [IcN]\n  _ \u2264 \u03b5 * ((1 + \u03b5) * c (N - 1)) + \u03b5 * c (N - 1) * l := (add_le_add (mul_le_mul_of_nonneg_left IcN \u03b5pos.le) le_rfl)\n  _ = \u03b5 * (1 + \u03b5 + l) * c (N - 1) := by ring\n  _ \u2264 \u03b5 * (1 + \u03b5 + l) * n := by\n    refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n    apply mul_nonneg \u03b5pos.le\n    linarith only [\u03b5pos, lnonneg]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 u n - \u2191n * l \u2264 u (c N) - \u2191(c (N - 1)) * l\n[PROOFSTEP]\napply sub_le_sub (hmono ncN.le)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u2191(c (N - 1)) * l \u2264 \u2191n * l\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 u (c N) - \u2191(c (N - 1)) * l = u (c N) - \u2191(c N) * l + (\u2191(c N) - \u2191(c (N - 1))) * l\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 u (c N) - \u2191(c N) * l + (\u2191(c N) - \u2191(c (N - 1))) * l \u2264 \u03b5 * \u2191(c N) + \u03b5 * \u2191(c (N - 1)) * l\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 u (c N) - \u2191(c N) * l \u2264 \u03b5 * \u2191(c N)\n[PROOFSTEP]\napply (ha _ _).2\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 a \u2264 N\n[PROOFSTEP]\nexact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 a \u2264 a + 1\n[PROOFSTEP]\nsimp only [le_add_iff_nonneg_right, zero_le']\n[GOAL]\ncase h\u2082\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 (\u2191(c N) - \u2191(c (N - 1))) * l \u2264 \u03b5 * \u2191(c (N - 1)) * l\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_right _ lnonneg\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u2191(c N) - \u2191(c (N - 1)) \u2264 \u03b5 * \u2191(c (N - 1))\n[PROOFSTEP]\nlinarith only [IcN]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u03b5 * ((1 + \u03b5) * \u2191(c (N - 1))) + \u03b5 * \u2191(c (N - 1)) * l = \u03b5 * (1 + \u03b5 + l) * \u2191(c (N - 1))\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u03b5 * (1 + \u03b5 + l) * \u2191(c (N - 1)) \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 0 \u2264 \u03b5 * (1 + \u03b5 + l)\n[PROOFSTEP]\napply mul_nonneg \u03b5pos.le\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, u (c n) - \u2191(c n) * l \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 u (c b) - \u2191(c b) * l \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\ncNn : c (N - 1) \u2264 n\nIcN : \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 0 \u2264 1 + \u03b5 + l\n[PROOFSTEP]\nlinarith only [\u03b5pos, lnonneg]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u22a2 Tendsto (fun n => u n / \u2191n) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nhave B : \u2200 \u03b5 : \u211d, 0 < \u03b5 \u2192 \u2200\u1da0 n : \u2115 in atTop, (n : \u211d) * l - u n \u2264 \u03b5 * (1 + l) * n :=\n  by\n  intro \u03b5 \u03b5pos\n  rcases hlim (1 + \u03b5) ((lt_add_iff_pos_right _).2 \u03b5pos) with \u27e8c, cgrowth, ctop, clim\u27e9\n  have L : \u2200\u1da0 n : \u2115 in atTop, (c n : \u211d) * l - u (c n) \u2264 \u03b5 * c n :=\n    by\n    rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n    filter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n    have cnpos : 0 < c n := cnpos'\n    calc\n      (c n : \u211d) * l - u (c n) = -(u (c n) / c n - l) * c n := by\n        simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]\n      _ \u2264 \u03b5 * c n := by\n        refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n        simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n        exact le_trans (neg_le_abs_self _) hn\n  obtain \u27e8a, ha\u27e9 : \u2203 a : \u2115, \u2200 b : \u2115, a \u2264 b \u2192 (c (b + 1) : \u211d) \u2264 (1 + \u03b5) * c b \u2227 (c b : \u211d) * l - u (c b) \u2264 \u03b5 * c b :=\n    eventually_atTop.1 (cgrowth.and L)\n  let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)\n  filter_upwards [Ici_mem_atTop M] with n hn\n  have exN : \u2203 N, n < c N := by\n    rcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n    exact \u27e8N, by linarith only [hN]\u27e9\n  let N := Nat.find exN\n  have ncN : n < c N := Nat.find_spec exN\n  have aN : a + 1 \u2264 N := by\n    by_contra' h\n    have cNM : c N \u2264 M := by\n      apply le_max'\n      apply mem_image_of_mem\n      exact mem_range.2 h\n    exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n  have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN\n  have aN' : a \u2264 N - 1 := by\n    apply @Nat.le_of_add_le_add_right a 1 (N - 1)\n    rw [Nat.sub_add_cancel Npos]\n    exact aN\n  have cNn : c (N - 1) \u2264 n := by\n    have : N - 1 < N := Nat.pred_lt Npos.ne'\n    simpa only [not_lt] using Nat.find_min exN this\n  calc\n    (n : \u211d) * l - u n \u2264 c N * l - u (c (N - 1)) :=\n      by\n      refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _\n      exact neg_le_neg (hmono cNn)\n    _ \u2264 (1 + \u03b5) * c (N - 1) * l - u (c (N - 1)) :=\n      by\n      refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl\n      have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n      have := (ha _ aN').1\n      rwa [B] at this \n    _ = c (N - 1) * l - u (c (N - 1)) + \u03b5 * c (N - 1) * l := by ring\n    _ \u2264 \u03b5 * c (N - 1) + \u03b5 * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)\n    _ = \u03b5 * (1 + l) * c (N - 1) := by ring\n    _ \u2264 \u03b5 * (1 + l) * n := by\n      refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n      exact mul_nonneg \u03b5pos.le (add_nonneg zero_le_one lnonneg)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u22a2 \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nintro \u03b5 \u03b5pos\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nrcases hlim (1 + \u03b5) ((lt_add_iff_pos_right _).2 \u03b5pos) with \u27e8c, cgrowth, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave L : \u2200\u1da0 n : \u2115 in atTop, (c n : \u211d) * l - u (c n) \u2264 \u03b5 * c n :=\n  by\n  rw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n  filter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n  have cnpos : 0 < c n := cnpos'\n  calc\n    (c n : \u211d) * l - u (c n) = -(u (c n) / c n - l) * c n := by\n      simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]\n    _ \u2264 \u03b5 * c n := by\n      refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n      simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n      exact le_trans (neg_le_abs_self _) hn\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nrw [\u2190 tendsto_sub_nhds_zero_iff, \u2190 Asymptotics.isLittleO_one_iff \u211d, Asymptotics.isLittleO_iff] at clim \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nfilter_upwards [clim \u03b5pos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\n\u22a2 \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nhave cnpos : 0 < c n := cnpos'\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\ncalc\n  (c n : \u211d) * l - u (c n) = -(u (c n) / c n - l) * c n := by\n    simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]\n  _ \u2264 \u03b5 * c n := by\n    refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n    simp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n    exact le_trans (neg_le_abs_self _) hn\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 \u2191(c n) * l - u (c n) = -(u (c n) / \u2191(c n) - l) * \u2191(c n)\n[PROOFSTEP]\nsimp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 -(u (c n) / \u2191(c n) - l) * \u2191(c n) \u2264 \u03b5 * \u2191(c n)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\nhn : \u2016u (c n) / \u2191(c n) - l\u2016 \u2264 \u03b5 * \u20161\u2016\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\n\u22a2 -(u (c n) / \u2191(c n) - l) \u2264 \u03b5\n[PROOFSTEP]\nsimp only [mul_one, Real.norm_eq_abs, abs_one] at hn \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim\u271d : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nclim : \u2200 \u2983c_1 : \u211d\u2984, 0 < c_1 \u2192 \u2200\u1da0 (x : \u2115) in atTop, \u2016u (c x) / \u2191(c x) - l\u2016 \u2264 c_1 * \u20161\u2016\nn : \u2115\ncnpos' : n \u2208 c \u207b\u00b9' Set.Ioi 0\ncnpos : 0 < c n\nhn : |u (c n) / \u2191(c n) - l| \u2264 \u03b5\n\u22a2 -(u (c n) / \u2191(c n) - l) \u2264 \u03b5\n[PROOFSTEP]\nexact le_trans (neg_le_abs_self _) hn\n[GOAL]\ncase intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 : \u2203 a : \u2115, \u2200 b : \u2115, a \u2264 b \u2192 (c (b + 1) : \u211d) \u2264 (1 + \u03b5) * c b \u2227 (c b : \u211d) * l - u (c b) \u2264 \u03b5 * c b :=\n  eventually_atTop.1 (cgrowth.and L)\n[GOAL]\ncase intro.intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nlet M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\n\u22a2 Finset.Nonempty (image (fun i => c i) (range (a + 1)))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nfilter_upwards [Ici_mem_atTop M] with n hn\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave exN : \u2203 N, n < c N := by\n  rcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n  exact \u27e8N, by linarith only [hN]\u27e9\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\n\u22a2 \u2203 N, n < c N\n[PROOFSTEP]\nrcases(tendsto_atTop.1 ctop (n + 1)).exists with \u27e8N, hN\u27e9\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nN : \u2115\nhN : n + 1 \u2264 c N\n\u22a2 \u2203 N, n < c N\n[PROOFSTEP]\nexact \u27e8N, by linarith only [hN]\u27e9\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nN : \u2115\nhN : n + 1 \u2264 c N\n\u22a2 n < c N\n[PROOFSTEP]\nlinarith only [hN]\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nlet N := Nat.find exN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave ncN : n < c N := Nat.find_spec exN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave aN : a + 1 \u2264 N := by\n  by_contra' h\n  have cNM : c N \u2264 M := by\n    apply le_max'\n    apply mem_image_of_mem\n    exact mem_range.2 h\n  exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\n\u22a2 a + 1 \u2264 N\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 False\n[PROOFSTEP]\nhave cNM : c N \u2264 M := by\n  apply le_max'\n  apply mem_image_of_mem\n  exact mem_range.2 h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 c N \u2264 M\n[PROOFSTEP]\napply le_max'\n[GOAL]\ncase H2\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 c N \u2208 image (fun i => c i) (range (a + 1))\n[PROOFSTEP]\napply mem_image_of_mem\n[GOAL]\ncase H2.h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\n\u22a2 N \u2208 range (a + 1)\n[PROOFSTEP]\nexact mem_range.2 h\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\nh : Nat.find exN < a + 1\ncNM : c N \u2264 M\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave aN' : a \u2264 N - 1 := by\n  apply @Nat.le_of_add_le_add_right a 1 (N - 1)\n  rw [Nat.sub_add_cancel Npos]\n  exact aN\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 a \u2264 N - 1\n[PROOFSTEP]\napply @Nat.le_of_add_le_add_right a 1 (N - 1)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 a + 1 \u2264 N - 1 + 1\n[PROOFSTEP]\nrw [Nat.sub_add_cancel Npos]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\n\u22a2 a + 1 \u2264 N\n[PROOFSTEP]\nexact aN\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nhave cNn : c (N - 1) \u2264 n := by\n  have : N - 1 < N := Nat.pred_lt Npos.ne'\n  simpa only [not_lt] using Nat.find_min exN this\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\n\u22a2 c (N - 1) \u2264 n\n[PROOFSTEP]\nhave : N - 1 < N := Nat.pred_lt Npos.ne'\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\nthis : N - 1 < N\n\u22a2 c (N - 1) \u2264 n\n[PROOFSTEP]\nsimpa only [not_lt] using Nat.find_min exN this\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\ncalc\n  (n : \u211d) * l - u n \u2264 c N * l - u (c (N - 1)) :=\n    by\n    refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _\n    exact neg_le_neg (hmono cNn)\n  _ \u2264 (1 + \u03b5) * c (N - 1) * l - u (c (N - 1)) :=\n    by\n    refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl\n    have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n    have := (ha _ aN').1\n    rwa [B] at this \n  _ = c (N - 1) * l - u (c (N - 1)) + \u03b5 * c (N - 1) * l := by ring\n  _ \u2264 \u03b5 * c (N - 1) + \u03b5 * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)\n  _ = \u03b5 * (1 + l) * c (N - 1) := by ring\n  _ \u2264 \u03b5 * (1 + l) * n := by\n    refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n    exact mul_nonneg \u03b5pos.le (add_nonneg zero_le_one lnonneg)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u2191n * l - u n \u2264 \u2191(c N) * l - u (c (N - 1))\n[PROOFSTEP]\nrefine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 -u n \u2264 -u (c (N - 1))\n[PROOFSTEP]\nexact neg_le_neg (hmono cNn)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u2191(c N) * l - u (c (N - 1)) \u2264 (1 + \u03b5) * \u2191(c (N - 1)) * l - u (c (N - 1))\n[PROOFSTEP]\nrefine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nhave B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\nB : N - 1 + 1 = N\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nhave := (ha _ aN').1\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\nB : N - 1 + 1 = N\nthis : \u2191(c (N - 1 + 1)) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n\u22a2 \u2191(c N) \u2264 (1 + \u03b5) * \u2191(c (N - 1))\n[PROOFSTEP]\nrwa [B] at this \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 (1 + \u03b5) * \u2191(c (N - 1)) * l - u (c (N - 1)) = \u2191(c (N - 1)) * l - u (c (N - 1)) + \u03b5 * \u2191(c (N - 1)) * l\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u03b5 * \u2191(c (N - 1)) + \u03b5 * \u2191(c (N - 1)) * l = \u03b5 * (1 + l) * \u2191(c (N - 1))\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 \u03b5 * (1 + l) * \u2191(c (N - 1)) \u2264 \u03b5 * (1 + l) * \u2191n\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\n\u03b5 : \u211d\n\u03b5pos : 0 < \u03b5\nc : \u2115 \u2192 \u2115\ncgrowth : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 (1 + \u03b5) * \u2191(c n)\nctop : Tendsto c atTop atTop\nclim : Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nL : \u2200\u1da0 (n : \u2115) in atTop, \u2191(c n) * l - u (c n) \u2264 \u03b5 * \u2191(c n)\na : \u2115\nha : \u2200 (b : \u2115), a \u2264 b \u2192 \u2191(c (b + 1)) \u2264 (1 + \u03b5) * \u2191(c b) \u2227 \u2191(c b) * l - u (c b) \u2264 \u03b5 * \u2191(c b)\nM : \u2115 := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))\nn : \u2115\nhn :\n  n \u2208 Set.Ici (max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1)))))\nexN : \u2203 N, n < c N\nN : \u2115 := Nat.find exN\nncN : n < c N\naN : a + 1 \u2264 N\nNpos : 0 < N\naN' : a \u2264 N - 1\ncNn : c (N - 1) \u2264 n\n\u22a2 0 \u2264 \u03b5 * (1 + l)\n[PROOFSTEP]\nexact mul_nonneg \u03b5pos.le (add_nonneg zero_le_one lnonneg)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\n\u22a2 Tendsto (fun n => u n / \u2191n) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nrefine' tendsto_order.2 \u27e8fun d hd => _, fun d hd => _\u27e9\n[GOAL]\ncase refine'_1\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, d < u b / \u2191b\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 : \u2203 \u03b5 : \u211d, d + \u03b5 * (1 + l) < l \u2227 0 < \u03b5 :=\n  by\n  have L : Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d + 0 * (1 + l))) :=\n    by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)\n  simp only [zero_mul, add_zero] at L \n  exact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u22a2 \u2203 \u03b5, d + \u03b5 * (1 + l) < l \u2227 0 < \u03b5\n[PROOFSTEP]\nhave L : Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d + 0 * (1 + l))) :=\n  by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u22a2 Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd (d + 0 * (1 + l)))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u22a2 Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd 0) (\ud835\udcdd (d + 0 * (1 + l)))\n[PROOFSTEP]\nexact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\nL : Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd (d + 0 * (1 + l)))\n\u22a2 \u2203 \u03b5, d + \u03b5 * (1 + l) < l \u2227 0 < \u03b5\n[PROOFSTEP]\nsimp only [zero_mul, add_zero] at L \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\nL : Tendsto (fun \u03b5 => d + \u03b5 * (1 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd d)\n\u22a2 \u2203 \u03b5, d + \u03b5 * (1 + l) < l \u2227 0 < \u03b5\n[PROOFSTEP]\nexact (((tendsto_order.1 L).2 l hd).and self_mem_nhdsWithin).exists\n[GOAL]\ncase refine'_1.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, d < u b / \u2191b\n[PROOFSTEP]\nfilter_upwards [B \u03b5 \u03b5pos, Ioi_mem_atTop 0] with n hn npos\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 d < u n / \u2191n\n[PROOFSTEP]\nsimp_rw [div_eq_inv_mul]\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 d < (\u2191n)\u207b\u00b9 * u n\n[PROOFSTEP]\ncalc\n  d < (n : \u211d)\u207b\u00b9 * n * (l - \u03b5 * (1 + l)) := by\n    rw [inv_mul_cancel, one_mul]\n    \u00b7 linarith only [h\u03b5]\n    \u00b7 exact Nat.cast_ne_zero.2 (ne_of_gt npos)\n  _ = (n : \u211d)\u207b\u00b9 * (n * l - \u03b5 * (1 + l) * n) := by ring\n  _ \u2264 (n : \u211d)\u207b\u00b9 * u n := by\n    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))\n    linarith only [hn]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 d < (\u2191n)\u207b\u00b9 * \u2191n * (l - \u03b5 * (1 + l))\n[PROOFSTEP]\nrw [inv_mul_cancel, one_mul]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 d < l - \u03b5 * (1 + l)\n[PROOFSTEP]\nlinarith only [h\u03b5]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact Nat.cast_ne_zero.2 (ne_of_gt npos)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * \u2191n * (l - \u03b5 * (1 + l)) = (\u2191n)\u207b\u00b9 * (\u2191n * l - \u03b5 * (1 + l) * \u2191n)\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * (\u2191n * l - \u03b5 * (1 + l) * \u2191n) \u2264 (\u2191n)\u207b\u00b9 * u n\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d < l\n\u03b5 : \u211d\nh\u03b5 : d + \u03b5 * (1 + l) < l\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 \u2191n * l - \u03b5 * (1 + l) * \u2191n \u2264 u n\n[PROOFSTEP]\nlinarith only [hn]\n[GOAL]\ncase refine'_2\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, u b / \u2191b < d\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 : \u2203 \u03b5 : \u211d, l + \u03b5 * (1 + \u03b5 + l) < d \u2227 0 < \u03b5 :=\n  by\n  have L : Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd[>] 0) (\ud835\udcdd (l + 0 * (1 + 0 + l))) :=\n    by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))\n  simp only [zero_mul, add_zero] at L \n  exact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u22a2 \u2203 \u03b5, l + \u03b5 * (1 + \u03b5 + l) < d \u2227 0 < \u03b5\n[PROOFSTEP]\nhave L : Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd[>] 0) (\ud835\udcdd (l + 0 * (1 + 0 + l))) :=\n  by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u22a2 Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd (l + 0 * (1 + 0 + l)))\n[PROOFSTEP]\napply Tendsto.mono_left _ nhdsWithin_le_nhds\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u22a2 Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd 0) (\ud835\udcdd (l + 0 * (1 + 0 + l)))\n[PROOFSTEP]\nexact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\nL : Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd (l + 0 * (1 + 0 + l)))\n\u22a2 \u2203 \u03b5, l + \u03b5 * (1 + \u03b5 + l) < d \u2227 0 < \u03b5\n[PROOFSTEP]\nsimp only [zero_mul, add_zero] at L \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\nL : Tendsto (fun \u03b5 => l + \u03b5 * (1 + \u03b5 + l)) (\ud835\udcdd[Set.Ioi 0] 0) (\ud835\udcdd l)\n\u22a2 \u2203 \u03b5, l + \u03b5 * (1 + \u03b5 + l) < d \u2227 0 < \u03b5\n[PROOFSTEP]\nexact (((tendsto_order.1 L).2 d hd).and self_mem_nhdsWithin).exists\n[GOAL]\ncase refine'_2.intro.intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\n\u22a2 \u2200\u1da0 (b : \u2115) in atTop, u b / \u2191b < d\n[PROOFSTEP]\nfilter_upwards [A \u03b5 \u03b5pos, Ioi_mem_atTop 0] with n hn npos\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 u n / \u2191n < d\n[PROOFSTEP]\nsimp_rw [div_eq_inv_mul]\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * u n < d\n[PROOFSTEP]\ncalc\n  (n : \u211d)\u207b\u00b9 * u n \u2264 (n : \u211d)\u207b\u00b9 * (n * l + \u03b5 * (1 + \u03b5 + l) * n) :=\n    by\n    refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))\n    linarith only [hn]\n  _ = (n : \u211d)\u207b\u00b9 * n * (l + \u03b5 * (1 + \u03b5 + l)) := by ring\n  _ < d := by\n    rwa [inv_mul_cancel, one_mul]\n    exact Nat.cast_ne_zero.2 (ne_of_gt npos)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * u n \u2264 (\u2191n)\u207b\u00b9 * (\u2191n * l + \u03b5 * (1 + \u03b5 + l) * \u2191n)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 u n \u2264 \u2191n * l + \u03b5 * (1 + \u03b5 + l) * \u2191n\n[PROOFSTEP]\nlinarith only [hn]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * (\u2191n * l + \u03b5 * (1 + \u03b5 + l) * \u2191n) = (\u2191n)\u207b\u00b9 * \u2191n * (l + \u03b5 * (1 + \u03b5 + l))\n[PROOFSTEP]\nring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 (\u2191n)\u207b\u00b9 * \u2191n * (l + \u03b5 * (1 + \u03b5 + l)) < d\n[PROOFSTEP]\nrwa [inv_mul_cancel, one_mul]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nhlim :\n  \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\nlnonneg : 0 \u2264 l\nA : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nB : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2191n * l - u n \u2264 \u03b5 * (1 + l) * \u2191n\nd : \u211d\nhd : d > l\n\u03b5 : \u211d\nh\u03b5 : l + \u03b5 * (1 + \u03b5 + l) < d\n\u03b5pos : 0 < \u03b5\nn : \u2115\nhn : u n - \u2191n * l \u2264 \u03b5 * (1 + \u03b5 + l) * \u2191n\nnpos : n \u2208 Set.Ioi 0\n\u22a2 \u2191n \u2260 0\n[PROOFSTEP]\nexact Nat.cast_ne_zero.2 (ne_of_gt npos)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\n\u22a2 Tendsto (fun n => u n / \u2191n) atTop (\ud835\udcdd l)\n[PROOFSTEP]\napply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\n\u22a2 \u2200 (a : \u211d),\n    1 < a \u2192\n      \u2203 c,\n        (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n          Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nintro a ha\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\n\u22a2 \u2203 c,\n    (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n      Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 : \u2203 k, c k < a := ((tendsto_order.1 clim).2 a ha).exists\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\n\u22a2 \u2203 c,\n    (\u2200\u1da0 (n : \u2115) in atTop, \u2191(c (n + 1)) \u2264 a * \u2191(c n)) \u2227\n      Tendsto c atTop atTop \u2227 Tendsto (fun n => u (c n) / \u2191(c n)) atTop (\ud835\udcdd l)\n[PROOFSTEP]\nrefine'\n  \u27e8fun n => \u230ac k ^ n\u230b\u208a, _, (tendsto_nat_floor_atTop (\u03b1 := \u211d)).comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k\u27e9\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191((fun n => \u230ac k ^ n\u230b\u208a) (n + 1)) \u2264 a * \u2191((fun n => \u230ac k ^ n\u230b\u208a) n)\n[PROOFSTEP]\nhave H : \u2200 n : \u2115, (0 : \u211d) < \u230ac k ^ n\u230b\u208a := by\n  intro n\n  refine' zero_lt_one.trans_le _\n  simp only [Real.rpow_nat_cast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le (cone k).le n]\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\n\u22a2 \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n[PROOFSTEP]\nintro n\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nn : \u2115\n\u22a2 0 < \u2191\u230ac k ^ n\u230b\u208a\n[PROOFSTEP]\nrefine' zero_lt_one.trans_le _\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nn : \u2115\n\u22a2 1 \u2264 \u2191\u230ac k ^ n\u230b\u208a\n[PROOFSTEP]\nsimp only [Real.rpow_nat_cast, Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le (cone k).le n]\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191((fun n => \u230ac k ^ n\u230b\u208a) (n + 1)) \u2264 a * \u2191((fun n => \u230ac k ^ n\u230b\u208a) n)\n[PROOFSTEP]\nhave A :\n  Tendsto (fun n : \u2115 => (\u230ac k ^ (n + 1)\u230b\u208a : \u211d) / c k ^ (n + 1) * c k / (\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop\n    (\ud835\udcdd (1 * c k / 1)) :=\n  by\n  refine' Tendsto.div (Tendsto.mul _ tendsto_const_nhds) _ one_ne_zero\n  \u00b7 refine' tendsto_nat_floor_div_atTop.comp _\n    exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)\n  \u00b7 refine' tendsto_nat_floor_div_atTop.comp _\n    exact tendsto_pow_atTop_atTop_of_one_lt (cone k)\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (1 * c k / 1))\n[PROOFSTEP]\nrefine' Tendsto.div (Tendsto.mul _ tendsto_const_nhds) _ one_ne_zero\n[GOAL]\ncase refine'_1\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1)) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine' tendsto_nat_floor_div_atTop.comp _\n[GOAL]\ncase refine'_1\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 Tendsto (fun n => c k ^ (n + 1)) atTop atTop\n[PROOFSTEP]\nexact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_atTop_nat 1)\n[GOAL]\ncase refine'_2\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 Tendsto (fun n => \u2191\u230ac k ^ n\u230b\u208a / c k ^ n) atTop (\ud835\udcdd 1)\n[PROOFSTEP]\nrefine' tendsto_nat_floor_div_atTop.comp _\n[GOAL]\ncase refine'_2\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\n\u22a2 Tendsto (fun n => c k ^ n) atTop atTop\n[PROOFSTEP]\nexact tendsto_pow_atTop_atTop_of_one_lt (cone k)\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (1 * c k / 1))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191((fun n => \u230ac k ^ n\u230b\u208a) (n + 1)) \u2264 a * \u2191((fun n => \u230ac k ^ n\u230b\u208a) n)\n[PROOFSTEP]\nhave B : Tendsto (fun n : \u2115 => (\u230ac k ^ (n + 1)\u230b\u208a : \u211d) / \u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (c k)) :=\n  by\n  simp only [one_mul, div_one] at A \n  convert A using 1\n  ext1 n\n  field_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']\n  ring\n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (1 * c k / 1))\n\u22a2 Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (c k))\n[PROOFSTEP]\nsimp only [one_mul, div_one] at A \n[GOAL]\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (c k))\n\u22a2 Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (c k))\n[PROOFSTEP]\nconvert A using 1\n[GOAL]\ncase h.e'_3\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (c k))\n\u22a2 (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) = fun n =>\n    \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h.e'_3.h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (c k))\nn : \u2115\n\u22a2 \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a = \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)\n[PROOFSTEP]\nfield_simp [(zero_lt_one.trans (cone k)).ne', (H n).ne']\n[GOAL]\ncase h.e'_3.h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (c k))\nn : \u2115\n\u22a2 \u2191\u230ac k ^ (n + 1)\u230b\u208a * (c k ^ (n + 1) * \u2191\u230ac k ^ n\u230b\u208a) = \u2191\u230ac k ^ (n + 1)\u230b\u208a * c k * c k ^ n * \u2191\u230ac k ^ n\u230b\u208a\n[PROOFSTEP]\nring\n[GOAL]\ncase intro\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (1 * c k / 1))\nB : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (c k))\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191((fun n => \u230ac k ^ n\u230b\u208a) (n + 1)) \u2264 a * \u2191((fun n => \u230ac k ^ n\u230b\u208a) n)\n[PROOFSTEP]\nfilter_upwards [(tendsto_order.1 B).2 a hk] with n hn\n[GOAL]\ncase h\nu : \u2115 \u2192 \u211d\nl : \u211d\nhmono : Monotone u\nc : \u2115 \u2192 \u211d\ncone : \u2200 (k : \u2115), 1 < c k\nclim : Tendsto c atTop (\ud835\udcdd 1)\nhc : \u2200 (k : \u2115), Tendsto (fun n => u \u230ac k ^ n\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd l)\na : \u211d\nha : 1 < a\nk : \u2115\nhk : c k < a\nH : \u2200 (n : \u2115), 0 < \u2191\u230ac k ^ n\u230b\u208a\nA : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / c k ^ (n + 1) * c k / (\u2191\u230ac k ^ n\u230b\u208a / c k ^ n)) atTop (\ud835\udcdd (1 * c k / 1))\nB : Tendsto (fun n => \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a) atTop (\ud835\udcdd (c k))\nn : \u2115\nhn : \u2191\u230ac k ^ (n + 1)\u230b\u208a / \u2191\u230ac k ^ n\u230b\u208a < a\n\u22a2 \u2191\u230ac k ^ (n + 1)\u230b\u208a \u2264 a * \u2191\u230ac k ^ n\u230b\u208a\n[PROOFSTEP]\nexact (div_le_iff (H n)).1 hn.le\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nhave cpos : 0 < c := zero_lt_one.trans hc\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nhave A : (0 : \u211d) < c\u207b\u00b9 ^ 2 := sq_pos_of_pos (inv_pos.2 cpos)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nhave B : c ^ 2 * ((1 : \u211d) - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 :=\n  by\n  rw [\u2190 div_eq_mul_inv, \u2190 div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]\n  swap\n  \u00b7 exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)\n  have : c ^ 3 = c ^ 2 * c := by ring\n  simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]\n  rw [mul_assoc, mul_comm c, \u2190 mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]\n  simpa using pow_le_pow hc.le one_le_two\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 div_eq_mul_inv, \u2190 div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 c ^ 2 * (c - 1) \u2264 c ^ 3 * (1 - c\u207b\u00b9 ^ 2)\nN : \u2115 j : \u211d hj : 0 < j c : \u211d hc : 1 < c cpos : 0 < c A : 0 < c\u207b\u00b9 ^ 2 \u22a2 0 < 1 - c\u207b\u00b9 ^ 2\n[PROOFSTEP]\nswap\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 0 < 1 - c\u207b\u00b9 ^ 2\n[PROOFSTEP]\nexact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 c ^ 2 * (c - 1) \u2264 c ^ 3 * (1 - c\u207b\u00b9 ^ 2)\n[PROOFSTEP]\nhave : c ^ 3 = c ^ 2 * c := by ring\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\n\u22a2 c ^ 3 = c ^ 2 * c\n[PROOFSTEP]\nring\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nthis : c ^ 3 = c ^ 2 * c\n\u22a2 c ^ 2 * (c - 1) \u2264 c ^ 3 * (1 - c\u207b\u00b9 ^ 2)\n[PROOFSTEP]\nsimp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nthis : c ^ 3 = c ^ 2 * c\n\u22a2 c ^ 2 * c * (c ^ 2)\u207b\u00b9 \u2264 c ^ 2\n[PROOFSTEP]\nrw [mul_assoc, mul_comm c, \u2190 mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nthis : c ^ 3 = c ^ 2 * c\n\u22a2 c \u2264 c ^ 2\n[PROOFSTEP]\nsimpa using pow_le_pow hc.le one_le_two\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / (c ^ i) ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\ncalc\n  (\u2211 i in (range N).filter fun i => j < c ^ i, (1 : \u211d) / (c ^ i) ^ 2) \u2264\n      \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, (1 : \u211d) / (c ^ i) ^ 2 :=\n    by\n    refine' sum_le_sum_of_subset_of_nonneg _ fun i _hi _hident => div_nonneg zero_le_one (sq_nonneg _)\n    intro i hi\n    simp only [mem_filter, mem_range] at hi \n    simp only [hi.1, mem_Ico, and_true_iff]\n    apply Nat.floor_le_of_le\n    apply le_of_lt\n    rw [div_lt_iff (Real.log_pos hc), \u2190 Real.log_pow]\n    exact Real.log_lt_log hj hi.2\n  _ = \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, (c\u207b\u00b9 ^ 2) ^ i :=\n    by\n    congr 1 with i\n    simp [\u2190 pow_mul, mul_comm]\n  _ \u2264 (c\u207b\u00b9 ^ 2) ^ \u230aReal.log j / Real.log c\u230b\u208a / ((1 : \u211d) - c\u207b\u00b9 ^ 2) :=\n    by\n    apply geom_sum_Ico_le_of_lt_one (sq_nonneg _)\n    rw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]\n    exact inv_lt_one hc\n  _ \u2264 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1) / ((1 : \u211d) - c\u207b\u00b9 ^ 2) :=\n    by\n    apply div_le_div _ _ _ le_rfl\n    \u00b7 apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)\n    \u00b7 rw [\u2190 Real.rpow_nat_cast]\n      apply Real.rpow_le_rpow_of_exponent_ge A\n      \u00b7 exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)\n      \u00b7 exact (Nat.sub_one_lt_floor _).le\n    \u00b7 simpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)\n  _ = c ^ 2 * ((1 : \u211d) - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2 :=\n    by\n    have I : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = (1 : \u211d) / j ^ 2 :=\n      by\n      apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)\n      \u00b7 rw [one_div]\n        exact inv_pos.2 (sq_pos_of_pos hj)\n      rw [Real.log_rpow A]\n      simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]\n      field_simp [(Real.log_pos hc).ne']\n      ring\n    rw [Real.rpow_sub A, I]\n    have : c ^ 2 - 1 \u2260 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'\n    field_simp [hj.ne', (zero_lt_one.trans hc).ne']\n    ring\n  _ \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2 := by\n    apply div_le_div _ B (sq_pos_of_pos hj) le_rfl\n    exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 \u2211 i in filter (fun i => j < c ^ i) (range N), 1 / (c ^ i) ^ 2 \u2264\n    \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, 1 / (c ^ i) ^ 2\n[PROOFSTEP]\nrefine' sum_le_sum_of_subset_of_nonneg _ fun i _hi _hident => div_nonneg zero_le_one (sq_nonneg _)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 filter (fun i => j < c ^ i) (range N) \u2286 Ico \u230aReal.log j / Real.log c\u230b\u208a N\n[PROOFSTEP]\nintro i hi\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i \u2208 filter (fun i => j < c ^ i) (range N)\n\u22a2 i \u2208 Ico \u230aReal.log j / Real.log c\u230b\u208a N\n[PROOFSTEP]\nsimp only [mem_filter, mem_range] at hi \n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < c ^ i\n\u22a2 i \u2208 Ico \u230aReal.log j / Real.log c\u230b\u208a N\n[PROOFSTEP]\nsimp only [hi.1, mem_Ico, and_true_iff]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < c ^ i\n\u22a2 \u230aReal.log j / Real.log c\u230b\u208a \u2264 i\n[PROOFSTEP]\napply Nat.floor_le_of_le\n[GOAL]\ncase h\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < c ^ i\n\u22a2 Real.log j / Real.log c \u2264 \u2191i\n[PROOFSTEP]\napply le_of_lt\n[GOAL]\ncase h.a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < c ^ i\n\u22a2 Real.log j / Real.log c < \u2191i\n[PROOFSTEP]\nrw [div_lt_iff (Real.log_pos hc), \u2190 Real.log_pow]\n[GOAL]\ncase h.a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < c ^ i\n\u22a2 Real.log j < Real.log (c ^ i)\n[PROOFSTEP]\nexact Real.log_lt_log hj hi.2\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, 1 / (c ^ i) ^ 2 = \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, (c\u207b\u00b9 ^ 2) ^ i\n[PROOFSTEP]\ncongr 1 with i\n[GOAL]\ncase e_f.h\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\ni : \u2115\n\u22a2 1 / (c ^ i) ^ 2 = (c\u207b\u00b9 ^ 2) ^ i\n[PROOFSTEP]\nsimp [\u2190 pow_mul, mul_comm]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 \u2211 i in Ico \u230aReal.log j / Real.log c\u230b\u208a N, (c\u207b\u00b9 ^ 2) ^ i \u2264 (c\u207b\u00b9 ^ 2) ^ \u230aReal.log j / Real.log c\u230b\u208a / (1 - c\u207b\u00b9 ^ 2)\n[PROOFSTEP]\napply geom_sum_Ico_le_of_lt_one (sq_nonneg _)\n[GOAL]\ncase h'x\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 c\u207b\u00b9 ^ 2 < 1\n[PROOFSTEP]\nrw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]\n[GOAL]\ncase h'x\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 c\u207b\u00b9 < 1\n[PROOFSTEP]\nexact inv_lt_one hc\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (c\u207b\u00b9 ^ 2) ^ \u230aReal.log j / Real.log c\u230b\u208a / (1 - c\u207b\u00b9 ^ 2) \u2264 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c\u207b\u00b9 ^ 2)\n[PROOFSTEP]\napply div_le_div _ _ _ le_rfl\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 0 \u2264 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1)\n[PROOFSTEP]\napply Real.rpow_nonneg_of_nonneg (sq_nonneg _)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (c\u207b\u00b9 ^ 2) ^ \u230aReal.log j / Real.log c\u230b\u208a \u2264 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1)\n[PROOFSTEP]\nrw [\u2190 Real.rpow_nat_cast]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (c\u207b\u00b9 ^ 2) ^ \u2191\u230aReal.log j / Real.log c\u230b\u208a \u2264 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1)\n[PROOFSTEP]\napply Real.rpow_le_rpow_of_exponent_ge A\n[GOAL]\ncase hx1\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 c\u207b\u00b9 ^ 2 \u2264 1\n[PROOFSTEP]\nexact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)\n[GOAL]\ncase hyz\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 Real.log j / Real.log c - 1 \u2264 \u2191\u230aReal.log j / Real.log c\u230b\u208a\n[PROOFSTEP]\nexact (Nat.sub_one_lt_floor _).le\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 0 < 1 - c\u207b\u00b9 ^ 2\n[PROOFSTEP]\nsimpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c\u207b\u00b9 ^ 2) = c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nhave I : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = (1 : \u211d) / j ^ 2 :=\n  by\n  apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)\n  \u00b7 rw [one_div]\n    exact inv_pos.2 (sq_pos_of_pos hj)\n  rw [Real.log_rpow A]\n  simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]\n  field_simp [(Real.log_pos hc).ne']\n  ring\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2\n[PROOFSTEP]\napply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 1 / j ^ 2 \u2208 Set.Ioi 0\n[PROOFSTEP]\nrw [one_div]\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 (j ^ 2)\u207b\u00b9 \u2208 Set.Ioi 0\n[PROOFSTEP]\nexact inv_pos.2 (sq_pos_of_pos hj)\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 Real.log ((c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c)) = Real.log (1 / j ^ 2)\n[PROOFSTEP]\nrw [Real.log_rpow A]\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 Real.log j / Real.log c * Real.log (c\u207b\u00b9 ^ 2) = Real.log (1 / j ^ 2)\n[PROOFSTEP]\nsimp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_one, mul_neg, neg_inj]\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 Real.log j / Real.log c * (\u21912 * Real.log c) = \u21912 * Real.log j\n[PROOFSTEP]\nfield_simp [(Real.log_pos hc).ne']\n[GOAL]\ncase a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 Real.log j * (2 * Real.log c) = 2 * Real.log j * Real.log c\n[PROOFSTEP]\nring\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\nI : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2\n\u22a2 (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c\u207b\u00b9 ^ 2) = c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nrw [Real.rpow_sub A, I]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\nI : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2\n\u22a2 1 / j ^ 2 / (c\u207b\u00b9 ^ 2) ^ 1 / (1 - c\u207b\u00b9 ^ 2) = c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nhave : c ^ 2 - 1 \u2260 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\nI : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2\nthis : c ^ 2 - 1 \u2260 0\n\u22a2 1 / j ^ 2 / (c\u207b\u00b9 ^ 2) ^ 1 / (1 - c\u207b\u00b9 ^ 2) = c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nfield_simp [hj.ne', (zero_lt_one.trans hc).ne']\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\nI : (c\u207b\u00b9 ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2\nthis : c ^ 2 - 1 \u2260 0\n\u22a2 (c ^ 2 - 1) * j ^ 2 = j ^ 2 * (c ^ 2 - 1)\n[PROOFSTEP]\nring\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 / j ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\napply div_le_div _ B (sq_pos_of_pos hj) le_rfl\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c\u207b\u00b9 ^ 2\nB : c ^ 2 * (1 - c\u207b\u00b9 ^ 2)\u207b\u00b9 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n\u22a2 0 \u2264 c ^ 3 * (c - 1)\u207b\u00b9\n[PROOFSTEP]\nexact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)\n[GOAL]\nc : \u211d\nhc : 1 < c\ni : \u2115\n\u22a2 (1 - c\u207b\u00b9) * c ^ i \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nhave cpos : 0 < c := zero_lt_one.trans hc\n[GOAL]\nc : \u211d\nhc : 1 < c\ni : \u2115\ncpos : 0 < c\n\u22a2 (1 - c\u207b\u00b9) * c ^ i \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nrcases Nat.eq_zero_or_pos i with (rfl | hi)\n[GOAL]\ncase inl\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\n\u22a2 (1 - c\u207b\u00b9) * c ^ 0 \u2264 \u2191\u230ac ^ 0\u230b\u208a\n[PROOFSTEP]\nsimp only [pow_zero, Nat.floor_one, Nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg, cpos.le]\n[GOAL]\ncase inr\nc : \u211d\nhc : 1 < c\ni : \u2115\ncpos : 0 < c\nhi : i > 0\n\u22a2 (1 - c\u207b\u00b9) * c ^ i \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nhave hident : 1 \u2264 i := hi\n[GOAL]\ncase inr\nc : \u211d\nhc : 1 < c\ni : \u2115\ncpos : 0 < c\nhi : i > 0\nhident : 1 \u2264 i\n\u22a2 (1 - c\u207b\u00b9) * c ^ i \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\ncalc\n  (1 - c\u207b\u00b9) * c ^ i = c ^ i - c ^ i * c\u207b\u00b9 := by ring\n  _ \u2264 c ^ i - 1 := by\n    simpa only [\u2190 div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using pow_le_pow hc.le hident\n  _ \u2264 \u230ac ^ i\u230b\u208a := (Nat.sub_one_lt_floor _).le\n[GOAL]\nc : \u211d\nhc : 1 < c\ni : \u2115\ncpos : 0 < c\nhi : i > 0\nhident : 1 \u2264 i\n\u22a2 (1 - c\u207b\u00b9) * c ^ i = c ^ i - c ^ i * c\u207b\u00b9\n[PROOFSTEP]\nring\n[GOAL]\nc : \u211d\nhc : 1 < c\ni : \u2115\ncpos : 0 < c\nhi : i > 0\nhident : 1 \u2264 i\n\u22a2 c ^ i - c ^ i * c\u207b\u00b9 \u2264 c ^ i - 1\n[PROOFSTEP]\nsimpa only [\u2190 div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using pow_le_pow hc.le hident\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\n\u22a2 \u2211 i in filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / j ^ 2\n[PROOFSTEP]\nhave cpos : 0 < c := zero_lt_one.trans hc\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\n\u22a2 \u2211 i in filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / j ^ 2\n[PROOFSTEP]\nhave A : 0 < 1 - c\u207b\u00b9 := sub_pos.2 (inv_lt_one hc)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2211 i in filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264 c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / j ^ 2\n[PROOFSTEP]\ncalc\n  (\u2211 i in (range N).filter (j < \u230ac ^ \u00b7\u230b\u208a), (1 : \u211d) / (\u230ac ^ i\u230b\u208a : \u211d) ^ 2) \u2264\n      \u2211 i in (range N).filter (j < c ^ \u00b7), (1 : \u211d) / (\u230ac ^ i\u230b\u208a : \u211d) ^ 2 :=\n    by\n    apply sum_le_sum_of_subset_of_nonneg\n    \u00b7 intro i hi\n      simp only [mem_filter, mem_range] at hi \n      simpa only [hi.1, mem_filter, mem_range, true_and_iff] using hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))\n    \u00b7 intro i _hi _hident\n      exact div_nonneg zero_le_one (sq_nonneg _)\n  _ \u2264 \u2211 i in (range N).filter (j < c ^ \u00b7), (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * ((1 : \u211d) / (c ^ i) ^ 2) :=\n    by\n    refine' sum_le_sum fun i _hi => _\n    rw [mul_div_assoc', mul_one, div_le_div_iff]; rotate_left\n    \u00b7 apply sq_pos_of_pos\n      refine' zero_lt_one.trans_le _\n      simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]\n    \u00b7 exact sq_pos_of_pos (pow_pos cpos _)\n    rw [one_mul, \u2190 mul_pow]\n    apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)\n    rw [\u2190 div_eq_inv_mul, le_div_iff A, mul_comm]\n    exact mul_pow_le_nat_floor_pow hc i\n  _ \u2264 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (c ^ 3 * (c - 1)\u207b\u00b9) / j ^ 2 :=\n    by\n    rw [\u2190 mul_sum, \u2190 mul_div_assoc']\n    refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)\n    exact sum_div_pow_sq_le_div_sq N hj hc\n  _ = c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / j ^ 2 := by\n    congr 1\n    field_simp [cpos.ne', (sub_pos.2 hc).ne']\n    ring!\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2211 i in filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264\n    \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2\n[PROOFSTEP]\napply sum_le_sum_of_subset_of_nonneg\n[GOAL]\ncase h\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N) \u2286 filter (fun x => j < c ^ x) (range N)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\nhi : i \u2208 filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\u22a2 i \u2208 filter (fun x => j < c ^ x) (range N)\n[PROOFSTEP]\nsimp only [mem_filter, mem_range] at hi \n[GOAL]\ncase h\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\nhi : i < N \u2227 j < \u2191\u230ac ^ i\u230b\u208a\n\u22a2 i \u2208 filter (fun x => j < c ^ x) (range N)\n[PROOFSTEP]\nsimpa only [hi.1, mem_filter, mem_range, true_and_iff] using hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))\n[GOAL]\ncase hf\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2200 (i : \u2115),\n    i \u2208 filter (fun x => j < c ^ x) (range N) \u2192 \u00aci \u2208 filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N) \u2192 0 \u2264 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2\n[PROOFSTEP]\nintro i _hi _hident\n[GOAL]\ncase hf\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n_hident : \u00aci \u2208 filter (fun x => j < \u2191\u230ac ^ x\u230b\u208a) (range N)\n\u22a2 0 \u2264 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2\n[PROOFSTEP]\nexact div_nonneg zero_le_one (sq_nonneg _)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264\n    \u2211 i in filter (fun x => j < c ^ x) (range N), (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (1 / (c ^ i) ^ 2)\n[PROOFSTEP]\nrefine' sum_le_sum fun i _hi => _\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 1 / \u2191\u230ac ^ i\u230b\u208a ^ 2 \u2264 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (1 / (c ^ i) ^ 2)\n[PROOFSTEP]\nrw [mul_div_assoc', mul_one, div_le_div_iff]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 1 * (c ^ i) ^ 2 \u2264 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * \u2191\u230ac ^ i\u230b\u208a ^ 2\ncase b0\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 0 < \u2191\u230ac ^ i\u230b\u208a ^ 2\ncase d0\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 0 < (c ^ i) ^ 2\n[PROOFSTEP]\nrotate_left\n[GOAL]\ncase b0\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 0 < \u2191\u230ac ^ i\u230b\u208a ^ 2\n[PROOFSTEP]\napply sq_pos_of_pos\n[GOAL]\ncase b0.ha\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 0 < \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nrefine' zero_lt_one.trans_le _\n[GOAL]\ncase b0.ha\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 1 \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nsimp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]\n[GOAL]\ncase d0\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 0 < (c ^ i) ^ 2\n[PROOFSTEP]\nexact sq_pos_of_pos (pow_pos cpos _)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 1 * (c ^ i) ^ 2 \u2264 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * \u2191\u230ac ^ i\u230b\u208a ^ 2\n[PROOFSTEP]\nrw [one_mul, \u2190 mul_pow]\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 (c ^ i) ^ 2 \u2264 ((1 - c\u207b\u00b9)\u207b\u00b9 * \u2191\u230ac ^ i\u230b\u208a) ^ 2\n[PROOFSTEP]\napply pow_le_pow_of_le_left (pow_nonneg cpos.le _)\n[GOAL]\ncase hab\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 c ^ i \u2264 (1 - c\u207b\u00b9)\u207b\u00b9 * \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nrw [\u2190 div_eq_inv_mul, le_div_iff A, mul_comm]\n[GOAL]\ncase hab\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\ni : \u2115\n_hi : i \u2208 filter (fun x => j < c ^ x) (range N)\n\u22a2 (1 - c\u207b\u00b9) * c ^ i \u2264 \u2191\u230ac ^ i\u230b\u208a\n[PROOFSTEP]\nexact mul_pow_le_nat_floor_pow hc i\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2211 i in filter (fun x => j < c ^ x) (range N), (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (1 / (c ^ i) ^ 2) \u2264\n    (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (c ^ 3 * (c - 1)\u207b\u00b9) / j ^ 2\n[PROOFSTEP]\nrw [\u2190 mul_sum, \u2190 mul_div_assoc']\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * \u2211 x in filter (fun x => j < c ^ x) (range N), 1 / (c ^ x) ^ 2 \u2264\n    (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 \u2211 x in filter (fun x => j < c ^ x) (range N), 1 / (c ^ x) ^ 2 \u2264 c ^ 3 * (c - 1)\u207b\u00b9 / j ^ 2\n[PROOFSTEP]\nexact sum_div_pow_sq_le_div_sq N hj hc\n[GOAL]\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (c ^ 3 * (c - 1)\u207b\u00b9) / j ^ 2 = c ^ 5 * (c - 1)\u207b\u00b9 ^ 3 / j ^ 2\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 (1 - c\u207b\u00b9)\u207b\u00b9 ^ 2 * (c ^ 3 * (c - 1)\u207b\u00b9) = c ^ 5 * (c - 1)\u207b\u00b9 ^ 3\n[PROOFSTEP]\nfield_simp [cpos.ne', (sub_pos.2 hc).ne']\n[GOAL]\ncase e_a\nN : \u2115\nj : \u211d\nhj : 0 < j\nc : \u211d\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c\u207b\u00b9\n\u22a2 c ^ 2 * c ^ 3 * (c - 1) ^ 3 = c ^ 5 * ((c - 1) ^ 2 * (c - 1))\n[PROOFSTEP]\nring!\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecificLimits.FloorPow", "llama_tokens": 94642, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303285397349, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5092794850842155}}
{"text": "[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na : k\n\u22a2 slope f a a = 0\n[PROOFSTEP]\nrw [slope, sub_self, inv_zero, zero_smul]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\n\u22a2 (b - a) \u2022 slope f a b = f b -\u1d65 f a\n[PROOFSTEP]\nrcases eq_or_ne a b with (rfl | hne)\n[GOAL]\ncase inl\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na : k\n\u22a2 (a - a) \u2022 slope f a a = f a -\u1d65 f a\n[PROOFSTEP]\nrw [sub_self, zero_smul, vsub_self]\n[GOAL]\ncase inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\nhne : a \u2260 b\n\u22a2 (b - a) \u2022 slope f a b = f b -\u1d65 f a\n[PROOFSTEP]\nrw [slope, smul_inv_smul\u2080 (sub_ne_zero.2 hne.symm)]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\n\u22a2 (b - a) \u2022 slope f a b +\u1d65 f a = f b\n[PROOFSTEP]\nrw [sub_smul_slope, vsub_vadd]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\nc : PE\n\u22a2 (slope fun x => f x +\u1d65 c) = slope f\n[PROOFSTEP]\next a b\n[GOAL]\ncase h.h\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\nc : PE\na b : k\n\u22a2 slope (fun x => f x +\u1d65 c) a b = slope f a b\n[PROOFSTEP]\nsimp only [slope, vadd_vsub_vadd_cancel_right, vsub_eq_sub]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 E\na b : k\nh : a \u2260 b\n\u22a2 slope (fun x => (x - a) \u2022 f x) a b = f b\n[PROOFSTEP]\nsimp [slope, inv_smul_smul\u2080 (sub_ne_zero.2 h.symm)]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\nh : slope f a b = 0\n\u22a2 f a = f b\n[PROOFSTEP]\nrw [\u2190 sub_smul_slope_vadd f a b, h, smul_zero, zero_vadd]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u2076 : Field k\ninst\u271d\u2075 : AddCommGroup E\ninst\u271d\u2074 : Module k E\ninst\u271d\u00b3 : AddTorsor E PE\nF : Type u_4\nPF : Type u_5\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module k F\ninst\u271d : AddTorsor F PF\nf : PE \u2192\u1d43[k] PF\ng : k \u2192 PE\na b : k\n\u22a2 slope (\u2191f \u2218 g) a b = \u2191f.linear (slope g a b)\n[PROOFSTEP]\nsimp only [slope, (\u00b7 \u2218 \u00b7), f.linear.map_smul, f.linearMap_vsub]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\n\u22a2 slope f a b = slope f b a\n[PROOFSTEP]\nrw [slope, slope, \u2190 neg_vsub_eq_vsub_rev, smul_neg, \u2190 neg_smul, neg_inv, neg_sub]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n[PROOFSTEP]\nby_cases hab : a = b\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : a = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n[PROOFSTEP]\nsubst hab\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((a - a) / (c - a)) \u2022 slope f a a + ((c - a) / (c - a)) \u2022 slope f a c = slope f a c\n[PROOFSTEP]\nrw [sub_self, zero_div, zero_smul, zero_add]\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c\n[PROOFSTEP]\nby_cases hac : a = c\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : a = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c\n[PROOFSTEP]\nsimp [hac]\n[GOAL]\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na c : k\nhac : \u00aca = c\n\u22a2 ((c - a) / (c - a)) \u2022 slope f a c = slope f a c\n[PROOFSTEP]\nrw [div_self (sub_ne_zero.2 <| Ne.symm hac), one_smul]\n[GOAL]\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n[PROOFSTEP]\nby_cases hbc : b = c\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : b = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n[PROOFSTEP]\nsubst hbc\n[GOAL]\ncase pos\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b : k\nhab : \u00aca = b\n\u22a2 ((b - a) / (b - a)) \u2022 slope f a b + ((b - b) / (b - a)) \u2022 slope f b b = slope f a b\n[PROOFSTEP]\nsimp [sub_ne_zero.2 (Ne.symm hab)]\n[GOAL]\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((b - a) / (c - a)) \u2022 slope f a b + ((c - b) / (c - a)) \u2022 slope f b c = slope f a c\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase neg\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nhab : \u00aca = b\nhbc : \u00acb = c\n\u22a2 ((c - b) / (c - a)) \u2022 slope f b c + ((b - a) / (c - a)) \u2022 slope f a b = slope f a c\n[PROOFSTEP]\nsimp_rw [slope, div_eq_inv_mul, mul_smul, \u2190 smul_add, smul_inv_smul\u2080 (sub_ne_zero.2 <| Ne.symm hab),\n  smul_inv_smul\u2080 (sub_ne_zero.2 <| Ne.symm hbc), vsub_add_vsub_cancel]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b c : k\nh : a \u2260 c\n\u22a2 \u2191(lineMap (slope f a b) (slope f b c)) ((c - b) / (c - a)) = slope f a c\n[PROOFSTEP]\nfield_simp [sub_ne_zero.2 h.symm, \u2190 sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c, lineMap_apply_module]\n[GOAL]\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b r : k\n\u22a2 \u2191(lineMap (slope f (\u2191(lineMap a b) r) b) (slope f a (\u2191(lineMap a b) r))) r = slope f a b\n[PROOFSTEP]\nobtain rfl | hab : a = b \u2228 a \u2260 b := Classical.em _\n[GOAL]\ncase inl\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na r : k\n\u22a2 \u2191(lineMap (slope f (\u2191(lineMap a a) r) a) (slope f a (\u2191(lineMap a a) r))) r = slope f a a\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b r : k\nhab : a \u2260 b\n\u22a2 \u2191(lineMap (slope f (\u2191(lineMap a b) r) b) (slope f a (\u2191(lineMap a b) r))) r = slope f a b\n[PROOFSTEP]\nrw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b]\n[GOAL]\ncase inr\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b r : k\nhab : a \u2260 b\n\u22a2 \u2191(lineMap (slope f b (\u2191(lineMap a b) r)) (slope f (\u2191(lineMap a b) r) a)) r = slope f b a\n[PROOFSTEP]\nconvert lineMap_slope_slope_sub_div_sub f b (lineMap a b r) a hab.symm using 2\n[GOAL]\ncase h.e'_2.h.e'_6\nk : Type u_1\nE : Type u_2\nPE : Type u_3\ninst\u271d\u00b3 : Field k\ninst\u271d\u00b2 : AddCommGroup E\ninst\u271d\u00b9 : Module k E\ninst\u271d : AddTorsor E PE\nf : k \u2192 PE\na b r : k\nhab : a \u2260 b\n\u22a2 r = (a - \u2191(lineMap a b) r) / (a - b)\n[PROOFSTEP]\nrw [lineMap_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, \u2190 sub_sub, sub_sub_cancel]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Slope", "llama_tokens": 4243, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303285397348, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.5092794850842155}}
{"text": "[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\n\u22a2 x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\n[PROOFSTEP]\nexact (pure_le_nhds _).trans\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 2 \u2192 3\n[GOAL]\ncase tfae_2_to_3\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\n\u22a2 pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\n[PROOFSTEP]\nexact fun h s hso hy => h (hso.mem_nhds hy)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 3 \u2192 4\n[GOAL]\ncase tfae_3_to_4\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\n\u22a2 (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\n[PROOFSTEP]\nexact fun h s hsc hx => of_not_not fun hy => h s\u1d9c hsc.isOpen_compl hy hx\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 4 \u2192 5\n[GOAL]\ncase tfae_4_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\n\u22a2 (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\n[PROOFSTEP]\nexact fun h => h _ isClosed_closure (subset_closure <| mem_singleton _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 6 \u2194 5\n[GOAL]\ncase tfae_6_iff_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\n\u22a2 closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\n[PROOFSTEP]\nexact isClosed_closure.closure_subset_iff.trans singleton_subset_iff\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 5 \u2194 7\n[GOAL]\ncase tfae_5_iff_7\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\n\u22a2 y \u2208 closure {x} \u2194 ClusterPt y (pure x)\n[PROOFSTEP]\nrw [mem_closure_iff_clusterPt, principal_singleton]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\ntfae_5_iff_7 : y \u2208 closure {x} \u2194 ClusterPt y (pure x)\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_have 5 \u2192 1\n[GOAL]\ncase tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\ntfae_5_iff_7 : y \u2208 closure {x} \u2194 ClusterPt y (pure x)\n\u22a2 y \u2208 closure {x} \u2192 x \u2933 y\n[PROOFSTEP]\nrefine' fun h => (nhds_basis_opens _).ge_iff.2 _\n[GOAL]\ncase tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\ntfae_5_iff_7 : y \u2208 closure {x} \u2194 ClusterPt y (pure x)\nh : y \u2208 closure {x}\n\u22a2 \u2200 (i' : Set X), y \u2208 i' \u2227 IsOpen i' \u2192 i' \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrintro s \u27e8hy, ho\u27e9\n[GOAL]\ncase tfae_5_to_1.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns\u271d : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\ntfae_5_iff_7 : y \u2208 closure {x} \u2194 ClusterPt y (pure x)\nh : y \u2208 closure {x}\ns : Set X\nhy : y \u2208 s\nho : IsOpen s\n\u22a2 s \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrcases mem_closure_iff.1 h s ho hy with \u27e8z, hxs, rfl : z = x\u27e9\n[GOAL]\ncase tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y\u271d z\u271d : X\ns\u271d : Set X\nf : X \u2192 Y\ny : X\ns : Set X\nhy : y \u2208 s\nho : IsOpen s\nz : X\nhxs : z \u2208 s\ntfae_1_to_2 : z \u2933 y \u2192 pure z \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure z \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 z \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 z \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 z \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 z \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {z}\ntfae_6_iff_5 : closure {y} \u2286 closure {z} \u2194 y \u2208 closure {z}\ntfae_5_iff_7 : y \u2208 closure {z} \u2194 ClusterPt y (pure z)\nh : y \u2208 closure {z}\n\u22a2 s \u2208 \ud835\udcdd z\n[PROOFSTEP]\nexact ho.mem_nhds hxs\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nx y : X\ntfae_1_to_2 : x \u2933 y \u2192 pure x \u2264 \ud835\udcdd y\ntfae_2_to_3 : pure x \u2264 \ud835\udcdd y \u2192 \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s\ntfae_3_to_4 : (\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2192 \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s\ntfae_4_to_5 : (\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2192 y \u2208 closure {x}\ntfae_6_iff_5 : closure {y} \u2286 closure {x} \u2194 y \u2208 closure {x}\ntfae_5_iff_7 : y \u2208 closure {x} \u2194 ClusterPt y (pure x)\ntfae_5_to_1 : y \u2208 closure {x} \u2192 x \u2933 y\n\u22a2 TFAE\n    [x \u2933 y, pure x \u2264 \ud835\udcdd y, \u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s, \u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s,\n      y \u2208 closure {x}, closure {y} \u2286 closure {x}, ClusterPt y (pure x)]\n[PROOFSTEP]\ntfae_finish\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nhf : Inducing f\n\u22a2 f x \u2933 f y \u2194 x \u2933 y\n[PROOFSTEP]\nsimp only [specializes_iff_mem_closure, hf.closure_eq_preimage_closure_image, image_singleton, mem_preimage]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nx\u2081 x\u2082 : X\ny\u2081 y\u2082 : Y\n\u22a2 (x\u2081, y\u2081) \u2933 (x\u2082, y\u2082) \u2194 x\u2081 \u2933 x\u2082 \u2227 y\u2081 \u2933 y\u2082\n[PROOFSTEP]\nsimp only [Specializes, nhds_prod_eq, prod_le_prod]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf\u271d : X \u2192 Y\nf g : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 f \u2933 g \u2194 \u2200 (i : \u03b9), f i \u2933 g i\n[PROOFSTEP]\nsimp only [Specializes, nhds_pi, pi_le_pi]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 \u00acx \u2933 y \u2194 \u2203 S, IsOpen S \u2227 y \u2208 S \u2227 \u00acx \u2208 S\n[PROOFSTEP]\nrw [specializes_iff_forall_open]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (\u00ac\u2200 (s : Set X), IsOpen s \u2192 y \u2208 s \u2192 x \u2208 s) \u2194 \u2203 S, IsOpen S \u2227 y \u2208 S \u2227 \u00acx \u2208 S\n[PROOFSTEP]\npush_neg\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (\u2203 s, IsOpen s \u2227 y \u2208 s \u2227 \u00acx \u2208 s) \u2194 \u2203 S, IsOpen S \u2227 y \u2208 S \u2227 \u00acx \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 \u00acx \u2933 y \u2194 \u2203 S, IsClosed S \u2227 x \u2208 S \u2227 \u00acy \u2208 S\n[PROOFSTEP]\nrw [specializes_iff_forall_closed]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (\u00ac\u2200 (s : Set X), IsClosed s \u2192 x \u2208 s \u2192 y \u2208 s) \u2194 \u2203 S, IsClosed S \u2227 x \u2208 S \u2227 \u00acy \u2208 S\n[PROOFSTEP]\npush_neg\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (\u2203 s, IsClosed s \u2227 x \u2208 s \u2227 \u00acy \u2208 s) \u2194 \u2203 S, IsClosed S \u2227 x \u2208 S \u2227 \u00acy \u2208 S\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (x ~\u1d62 y) \u2194 \u2200 (s : Set X), IsOpen s \u2192 (x \u2208 s \u2194 y \u2208 s)\n[PROOFSTEP]\nsimp only [inseparable_iff_specializes_and, specializes_iff_forall_open, \u2190 forall_and, \u2190 iff_def, Iff.comm]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 \u00ac(x ~\u1d62 y) \u2194 \u2203 s, IsOpen s \u2227 Xor' (x \u2208 s) (y \u2208 s)\n[PROOFSTEP]\nsimp [inseparable_iff_forall_open, \u2190 xor_iff_not_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (x ~\u1d62 y) \u2194 \u2200 (s : Set X), IsClosed s \u2192 (x \u2208 s \u2194 y \u2208 s)\n[PROOFSTEP]\nsimp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, \u2190 forall_and, \u2190 iff_def]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 x \u2933 y \u2227 y \u2933 x \u2194 x \u2208 closure {y} \u2227 y \u2208 closure {x}\n[PROOFSTEP]\nsimp only [specializes_iff_mem_closure, and_comm]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\n\u22a2 (x ~\u1d62 y) \u2194 closure {x} = closure {y}\n[PROOFSTEP]\nsimp only [inseparable_iff_specializes_and, specializes_iff_closure_subset, \u2190 subset_antisymm_iff, eq_comm]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nhf : Inducing f\n\u22a2 (f x ~\u1d62 f y) \u2194 (x ~\u1d62 y)\n[PROOFSTEP]\nsimp only [inseparable_iff_specializes_and, hf.specializes_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nx\u2081 x\u2082 : X\ny\u2081 y\u2082 : Y\n\u22a2 ((x\u2081, y\u2081) ~\u1d62 (x\u2082, y\u2082)) \u2194 (x\u2081 ~\u1d62 x\u2082) \u2227 (y\u2081 ~\u1d62 y\u2082)\n[PROOFSTEP]\nsimp only [Inseparable, nhds_prod_eq, prod_inj]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf\u271d : X \u2192 Y\nf g : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 (f ~\u1d62 g) \u2194 \u2200 (i : \u03b9), f i ~\u1d62 g i\n[PROOFSTEP]\nsimp only [Inseparable, nhds_pi, funext_iff, pi_inj]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\n\u22a2 mk \u207b\u00b9' (mk '' s) = s\n[PROOFSTEP]\nrefine' Subset.antisymm _ (subset_preimage_image _ _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\n\u22a2 mk \u207b\u00b9' (mk '' s) \u2286 s\n[PROOFSTEP]\nrintro x \u27e8y, hys, hxy\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsOpen s\nx y : X\nhys : y \u2208 s\nhxy : mk y = mk x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact ((mk_eq_mk.1 hxy).mem_open_iff hs).1 hys\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns\u271d : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\ns : Set X\nhs : IsOpen s\n\u22a2 IsOpen (mk \u207b\u00b9' (mk '' s))\n[PROOFSTEP]\nrwa [preimage_image_mk_open hs]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsClosed s\n\u22a2 mk \u207b\u00b9' (mk '' s) = s\n[PROOFSTEP]\nrefine' Subset.antisymm _ (subset_preimage_image _ _)\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsClosed s\n\u22a2 mk \u207b\u00b9' (mk '' s) \u2286 s\n[PROOFSTEP]\nrintro x \u27e8y, hys, hxy\u27e9\n[GOAL]\ncase intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nhs : IsClosed s\nx y : X\nhys : y \u2208 s\nhxy : mk y = mk x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact ((mk_eq_mk.1 hxy).mem_closed_iff hs).1 hys\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n\u22a2 IsClosed (Set.range mk)\n[PROOFSTEP]\nrw [range_mk]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n\u22a2 IsClosed univ\n[PROOFSTEP]\nexact isClosed_univ\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n\u22a2 Filter.map mk (\ud835\udcdd x) = \ud835\udcdd (mk x)\n[PROOFSTEP]\nrw [\u2190 comap_mk_nhds_mk, map_comap_of_surjective surjective_mk]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n\u22a2 Filter.map mk (\ud835\udcdd\u02e2 s) = \ud835\udcdd\u02e2 (mk '' s)\n[PROOFSTEP]\nrw [\u2190 comap_mk_nhdsSet_image, map_comap_of_surjective surjective_mk]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n\u22a2 comap mk (\ud835\udcdd\u02e2 t) = \ud835\udcdd\u02e2 (mk \u207b\u00b9' t)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n| comap mk (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nrw [\u2190 image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n| comap mk (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nrw [\u2190 image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\n| comap mk (\ud835\udcdd\u02e2 t)\n[PROOFSTEP]\nrw [\u2190 image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf : X \u2192 Y\nt : Set (SeparationQuotient X)\nx : X\ny : Y\n\u22a2 Filter.map (Prod.map mk mk) (\ud835\udcdd (x, y)) = \ud835\udcdd (mk x, mk y)\n[PROOFSTEP]\nrw [nhds_prod_eq, \u2190 prod_map_map_eq', map_mk_nhds, map_mk_nhds, nhds_prod_eq]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y z : X\ns\u271d : Set X\nf : X \u2192 Y\nt s : Set (SeparationQuotient X)\nx : X\n\u22a2 Filter.map mk (\ud835\udcdd[mk \u207b\u00b9' s] x) = \ud835\udcdd[s] mk x\n[PROOFSTEP]\nrw [nhdsWithin, \u2190 comap_principal, Filter.push_pull, nhdsWithin, map_mk_nhds]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y z : X\ns : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 \u03b1\nhf : \u2200 (x y : X), (x ~\u1d62 y) \u2192 f x = f y\nx : X\nl : Filter \u03b1\n\u22a2 Tendsto (lift f hf) (\ud835\udcdd (mk x)) l \u2194 Tendsto f (\ud835\udcdd x) l\n[PROOFSTEP]\nsimp only [\u2190 map_mk_nhds, tendsto_map'_iff, lift_comp_mk]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 \u03b1\nhf : \u2200 (x y : X), (x ~\u1d62 y) \u2192 f x = f y\nx : X\ns : Set (SeparationQuotient X)\nl : Filter \u03b1\n\u22a2 Tendsto (lift f hf) (\ud835\udcdd[s] mk x) l \u2194 Tendsto f (\ud835\udcdd[mk \u207b\u00b9' s] x) l\n[PROOFSTEP]\nsimp only [\u2190 map_mk_nhdsWithin_preimage, tendsto_map'_iff, lift_comp_mk]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y\nhf : \u2200 (x y : X), (x ~\u1d62 y) \u2192 f x = f y\ns : Set (SeparationQuotient X)\n\u22a2 ContinuousOn (lift f hf) s \u2194 ContinuousOn f (mk \u207b\u00b9' s)\n[PROOFSTEP]\nsimp only [ContinuousOn, surjective_mk.forall, continuousWithinAt_lift, mem_preimage]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y\nhf : \u2200 (x y : X), (x ~\u1d62 y) \u2192 f x = f y\n\u22a2 Continuous (lift f hf) \u2194 Continuous f\n[PROOFSTEP]\nsimp only [continuous_iff_continuousOn_univ, continuousOn_lift, preimage_univ]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 \u03b1\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\nx : X\ny : Y\nl : Filter \u03b1\n\u22a2 Tendsto (uncurry (lift\u2082 f hf)) (\ud835\udcdd (mk x, mk y)) l \u2194 Tendsto (uncurry f) (\ud835\udcdd (x, y)) l\n[PROOFSTEP]\nrw [\u2190 map_prod_map_mk_nhds, tendsto_map'_iff]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 \u03b1\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\nx : X\ny : Y\nl : Filter \u03b1\n\u22a2 Tendsto (uncurry (lift\u2082 f hf) \u2218 Prod.map mk mk) (\ud835\udcdd (x, y)) l \u2194 Tendsto (uncurry f) (\ud835\udcdd (x, y)) l\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 \u03b1\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\nx : X\ny : Y\ns : Set (SeparationQuotient X \u00d7 SeparationQuotient Y)\nl : Filter \u03b1\n\u22a2 Tendsto (uncurry (lift\u2082 f hf)) (\ud835\udcdd[s] (mk x, mk y)) l \u2194 Tendsto (uncurry f) (\ud835\udcdd[Prod.map mk mk \u207b\u00b9' s] (x, y)) l\n[PROOFSTEP]\nrw [nhdsWithin, \u2190 map_prod_map_mk_nhds, \u2190 Filter.push_pull, comap_principal]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx\u271d y\u271d z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 \u03b1\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\nx : X\ny : Y\ns : Set (SeparationQuotient X \u00d7 SeparationQuotient Y)\nl : Filter \u03b1\n\u22a2 Tendsto (uncurry (lift\u2082 f hf)) (Filter.map (Prod.map mk mk) (\ud835\udcdd (x, y) \u2293 \ud835\udcdf (Prod.map mk mk \u207b\u00b9' s))) l \u2194\n    Tendsto (uncurry f) (\ud835\udcdd[Prod.map mk mk \u207b\u00b9' s] (x, y)) l\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 Z\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\ns : Set (SeparationQuotient X \u00d7 SeparationQuotient Y)\n\u22a2 ContinuousOn (uncurry (lift\u2082 f hf)) s \u2194 ContinuousOn (uncurry f) (Prod.map mk mk \u207b\u00b9' s)\n[PROOFSTEP]\nsimp_rw [ContinuousOn, (surjective_mk.Prod_map surjective_mk).forall, Prod.forall, Prod.map, continuousWithinAt_lift\u2082]\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns\u271d : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 Z\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\ns : Set (SeparationQuotient X \u00d7 SeparationQuotient Y)\n\u22a2 (\u2200 (a : X) (b : Y), (mk a, mk b) \u2208 s \u2192 ContinuousWithinAt (uncurry f) (Prod.map mk mk \u207b\u00b9' s) (a, b)) \u2194\n    \u2200 (a : X) (b : Y),\n      (a, b) \u2208 (fun x => (mk x.fst, mk x.snd)) \u207b\u00b9' s \u2192\n        ContinuousWithinAt (uncurry f) ((fun x => (mk x.fst, mk x.snd)) \u207b\u00b9' s) (a, b)\n[PROOFSTEP]\nrfl\n[GOAL]\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nx y z : X\ns : Set X\nf\u271d : X \u2192 Y\nt : Set (SeparationQuotient X)\nf : X \u2192 Y \u2192 Z\nhf : \u2200 (a : X) (b : Y) (c : X) (d : Y), (a ~\u1d62 c) \u2192 (b ~\u1d62 d) \u2192 f a b = f c d\n\u22a2 Continuous (uncurry (lift\u2082 f hf)) \u2194 Continuous (uncurry f)\n[PROOFSTEP]\nsimp only [continuous_iff_continuousOn_univ, continuousOn_lift\u2082, preimage_univ]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Inseparable", "llama_tokens": 15960, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673223709251, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.5089558947572167}}
{"text": "[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 : Type ?u.90} (I\u03b1 : Mul \u03b1) (I\u03b2 : Mul \u03b2), Function.Injective MulHom.toFun\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1\u271d \u03b2\u271d : Type ?u.90\nI\u03b1\u271d : Mul \u03b1\u271d\nI\u03b2\u271d : Mul \u03b2\u271d\n\u22a2 Function.Injective MulHom.toFun\n[PROOFSTEP]\napply @FunLike.coe_injective\n[GOAL]\n\u22a2 \u2200 {\u03b1 : Type ?u.90} (I : Mul \u03b1), (MulHom.id \u03b1).toFun = id\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u22a2 \u2200 {\u03b1 \u03b2 \u03b3 : Type ?u.90} (I\u03b1 : Mul \u03b1) (I\u03b2 : Mul \u03b2) (I\u03b3 : Mul \u03b3) (f : \u03b1 \u2192\u2099* \u03b2) (g : \u03b2 \u2192\u2099* \u03b3),\n    (MulHom.comp g f).toFun = g.toFun \u2218 f.toFun\n[PROOFSTEP]\naesop_cat\n[GOAL]\nX Y : Type u\ninst\u271d\u00b9 : Mul X\ninst\u271d : Mul Y\ne : X \u2243* Y\n\u22a2 toMulHom e \u226b toMulHom (symm e) = \ud835\udfd9 (MagmaCat.of X)\n[PROOFSTEP]\next\n[GOAL]\ncase w\nX Y : Type u\ninst\u271d\u00b9 : Mul X\ninst\u271d : Mul Y\ne : X \u2243* Y\nx\u271d : (forget MagmaCat).obj (MagmaCat.of X)\n\u22a2 \u2191(toMulHom e \u226b toMulHom (symm e)) x\u271d = \u2191(\ud835\udfd9 (MagmaCat.of X)) x\u271d\n[PROOFSTEP]\nsimp_rw [comp_apply, toMulHom_eq_coe, MagmaCat.MulEquiv_coe_eq, symm_apply_apply, id_apply]\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MagmaCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MagmaCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nskip\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MagmaCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MagmaCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget MagmaCat).map f)\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MagmaCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MagmaCat).map f)\ni : (forget MagmaCat).obj X \u2245 (forget MagmaCat).obj Y := asIso ((forget MagmaCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet e : X \u2243* Y := { f, i.toEquiv with }\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : MagmaCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget MagmaCat).map f)\ni : (forget MagmaCat).obj X \u2245 (forget MagmaCat).obj Y := asIso ((forget MagmaCat).map f)\ne : \u2191X \u2243* \u2191Y :=\n  let src := i.toEquiv;\n  {\n    toEquiv :=\n      { toFun := f.toFun, invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n        right_inv := (_ : Function.RightInverse src.invFun src.toFun) },\n    map_mul' := (_ : \u2200 (x y : \u2191X), MulHom.toFun f (x * y) = MulHom.toFun f x * MulHom.toFun f y) }\n\u22a2 IsIso f\n[PROOFSTEP]\nexact \u27e8(IsIso.of_iso e.toMagmaCatIso).1\u27e9\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : SemigroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget SemigroupCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nskip\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : SemigroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget SemigroupCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet i := asIso ((forget SemigroupCat).map f)\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : SemigroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget SemigroupCat).map f)\ni : (forget SemigroupCat).obj X \u2245 (forget SemigroupCat).obj Y := asIso ((forget SemigroupCat).map f)\n\u22a2 IsIso f\n[PROOFSTEP]\nlet e : X \u2243* Y := { f, i.toEquiv with }\n[GOAL]\nX\u271d Y\u271d : Type u\nX Y : SemigroupCat\nf : X \u27f6 Y\nx\u271d : IsIso ((forget SemigroupCat).map f)\ni : (forget SemigroupCat).obj X \u2245 (forget SemigroupCat).obj Y := asIso ((forget SemigroupCat).map f)\ne : \u2191X \u2243* \u2191Y :=\n  let src := i.toEquiv;\n  {\n    toEquiv :=\n      { toFun := f.toFun, invFun := src.invFun, left_inv := (_ : Function.LeftInverse src.invFun src.toFun),\n        right_inv := (_ : Function.RightInverse src.invFun src.toFun) },\n    map_mul' := (_ : \u2200 (x y : \u2191X), MulHom.toFun f (x * y) = MulHom.toFun f x * MulHom.toFun f y) }\n\u22a2 IsIso f\n[PROOFSTEP]\nexact \u27e8(IsIso.of_iso e.toSemigroupCatIso).1\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Category.SemigroupCat.Basic", "llama_tokens": 1609, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673087708699, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5089558748999461}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na : \u03b1\n\u22a2 |(-a)| = |a|\n[PROOFSTEP]\nrw [abs_eq_max_neg, max_comm, neg_neg, abs_eq_max_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : |a| = b\n\u22a2 a = b \u2228 a = -b\n[PROOFSTEP]\nsimpa only [\u2190 h, eq_comm (a := |a|), neg_eq_iff_eq_neg] using abs_choice a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 |a| = |b| \u2194 a = b \u2228 a = -b\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : |a| = |b|\n\u22a2 a = b \u2228 a = -b\n[PROOFSTEP]\nobtain rfl | rfl := eq_or_eq_neg_of_abs_eq h\n[GOAL]\ncase refine'_1.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\nb : \u03b1\nh : |(|b|)| = |b|\n\u22a2 |b| = b \u2228 |b| = -b\n[PROOFSTEP]\nsimpa only [neg_eq_iff_eq_neg (a := |b|), neg_inj, or_comm] using abs_choice b\n[GOAL]\ncase refine'_1.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\nb : \u03b1\nh : |(-|b|)| = |b|\n\u22a2 -|b| = b \u2228 -|b| = -b\n[PROOFSTEP]\nsimpa only [neg_eq_iff_eq_neg (a := |b|), neg_inj, or_comm] using abs_choice b\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a = b \u2228 a = -b\n\u22a2 |a| = |b|\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase refine'_2.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a = b\n\u22a2 |a| = |b|\n[PROOFSTEP]\nsimp [h, abs_neg]\n[GOAL]\ncase refine'_2.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : AddGroup \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nh : a = -b\n\u22a2 |a| = |b|\n[PROOFSTEP]\nsimp [h, abs_neg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nha : 0 \u2264 a\nhab : a \u2264 b\n\u22a2 |a| \u2264 |b|\n[PROOFSTEP]\nrwa [abs_of_nonneg ha, abs_of_nonneg (ha.trans hab)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 0 < |a| \u2194 a \u2260 0\n[PROOFSTEP]\nrcases lt_trichotomy a 0 with (ha | rfl | ha)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nha : a < 0\n\u22a2 0 < |a| \u2194 a \u2260 0\n[PROOFSTEP]\nsimp [abs_of_neg ha, neg_pos, ha.ne, ha]\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nb c : \u03b1\n\u22a2 0 < |0| \u2194 0 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\nha : 0 < a\n\u22a2 0 < |a| \u2194 a \u2260 0\n[PROOFSTEP]\nsimp [abs_of_pos ha, ha, ha.ne.symm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\n\u22a2 -|a| \u2264 a\n[PROOFSTEP]\ncases' le_total 0 a with h h\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\nh : 0 \u2264 a\n\u22a2 -|a| \u2264 a\n[PROOFSTEP]\ncalc\n  -|a| = -a := congr_arg Neg.neg (abs_of_nonneg h)\n  _ \u2264 0 := (neg_nonpos.mpr h)\n  _ \u2264 a := h\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\nh : a \u2264 0\n\u22a2 -|a| \u2264 a\n[PROOFSTEP]\ncalc\n  -|a| = - -a := congr_arg Neg.neg (abs_of_nonpos h)\n  _ \u2264 a := (neg_neg a).le\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\n\u22a2 0 \u2264 a + |a|\n[PROOFSTEP]\nrw [\u2190 add_right_neg a]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\n\u22a2 a + -a \u2264 a + |a|\n[PROOFSTEP]\napply add_le_add_left\n[GOAL]\ncase bc\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\n\u22a2 -a \u2264 |a|\n[PROOFSTEP]\nexact neg_le_abs_self a\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b2 : AddGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b c a : \u03b1\n\u22a2 -|a| \u2264 -a\n[PROOFSTEP]\nsimpa using neg_abs_le_self (-a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nha : a \u2264 0\nhab : b \u2264 a\n\u22a2 |a| \u2264 |b|\n[PROOFSTEP]\nrw [abs_of_nonpos ha, abs_of_nonpos (hab.trans ha)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\nha : a \u2264 0\nhab : b \u2264 a\n\u22a2 -a \u2264 -b\n[PROOFSTEP]\nexact neg_le_neg_iff.mpr hab\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 -a < b \u2227 a < b \u2194 -b < a \u2227 a < b\n[PROOFSTEP]\nrw [neg_lt]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 max a b - min a b = |a - b|\n[PROOFSTEP]\ncases' le_total a b with ab ba\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nab : a \u2264 b\n\u22a2 max a b - min a b = |a - b|\n[PROOFSTEP]\nrw [max_eq_right ab, min_eq_left ab, abs_of_nonpos, neg_sub]\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nab : a \u2264 b\n\u22a2 a - b \u2264 0\n[PROOFSTEP]\nrwa [sub_nonpos]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nba : b \u2264 a\n\u22a2 max a b - min a b = |a - b|\n[PROOFSTEP]\nrw [max_eq_left ba, min_eq_right ba, abs_of_nonneg]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nba : b \u2264 a\n\u22a2 0 \u2264 a - b\n[PROOFSTEP]\nrwa [sub_nonneg]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 max a b - min a b = |b - a|\n[PROOFSTEP]\nrw [abs_sub_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b3 : AddGroup \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na\u271d b\u271d c : \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 max a b - min a b = |a - b|\n[PROOFSTEP]\nexact max_sub_min_eq_abs' _ _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a| \u2264 b \u2194 -b \u2264 a \u2227 a \u2264 b\n[PROOFSTEP]\nrw [abs_le', and_comm, @neg_le \u03b1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 a \u2264 |b| \u2194 b \u2264 -a \u2228 a \u2264 b\n[PROOFSTEP]\nrw [le_abs, or_comm, @le_neg \u03b1]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |a| \u2264 |b| + |b + a|\n[PROOFSTEP]\nsimpa using abs_add (-b) (b + a)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |a - b| \u2264 |a| + |b|\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 abs_neg b]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |a + -b| \u2264 |a| + |(-b)|\n[PROOFSTEP]\nexact abs_add a _\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a - b| \u2264 c \u2194 a - b \u2264 c \u2227 b - a \u2264 c\n[PROOFSTEP]\nrw [abs_le, neg_le_sub_iff_le_add, sub_le_iff_le_add', and_comm, sub_le_iff_le_add']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\n\u22a2 |a - b| < c \u2194 a - b < c \u2227 b - a < c\n[PROOFSTEP]\nrw [@abs_lt \u03b1, neg_lt_sub_iff_lt_add', sub_lt_iff_lt_add', and_comm, sub_lt_iff_lt_add']\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |a| = |a - b + b|\n[PROOFSTEP]\nrw [sub_add_cancel]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |b| - |a| \u2264 |a - b|\n[PROOFSTEP]\nrw [abs_sub_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c d a b : \u03b1\n\u22a2 |b| - |a| \u2264 |b - a|\n[PROOFSTEP]\napply abs_sub_abs_le_abs_sub\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nhb : 0 \u2264 b\n\u22a2 |a| = b \u2194 a = b \u2228 a = -b\n[PROOFSTEP]\nrefine' \u27e8eq_or_eq_neg_of_abs_eq, _\u27e9\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nhb : 0 \u2264 b\n\u22a2 a = b \u2228 a = -b \u2192 |a| = b\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na c d : \u03b1\nhb : 0 \u2264 a\n\u22a2 |a| = a\n[PROOFSTEP]\nsimp only [abs_neg, abs_of_nonneg hb]\n[GOAL]\ncase inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nb c d : \u03b1\nhb : 0 \u2264 b\n\u22a2 |(-b)| = b\n[PROOFSTEP]\nsimp only [abs_neg, abs_of_nonneg hb]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 b \u2264 max |a| |c|\n[PROOFSTEP]\nsimp [hbc.trans (le_abs_self c)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na b c d : \u03b1\nhab : a \u2264 b\nhbc : b \u2264 c\n\u22a2 -b \u2264 max |a| |c|\n[PROOFSTEP]\nsimp [((@neg_le_neg_iff \u03b1 ..).mpr hab).trans (neg_le_abs_self a)]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 |a - c| = |a - b + (b - c)|\n[PROOFSTEP]\nrw [sub_add_sub_cancel]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Order.Group.Abs", "llama_tokens": 5312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772883, "lm_q2_score": 0.668880247169804, "lm_q1q2_score": 0.5087841965726573}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2194 LinearIndependent K (Projectivization.rep \u2218 f)\n[PROOFSTEP]\nrefine' \u27e8_, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2192 LinearIndependent K (Projectivization.rep \u2218 f)\n[PROOFSTEP]\nrintro \u27e8ff, hff, hh\u27e9\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\n[PROOFSTEP]\nchoose a ha using fun i : \u03b9 => exists_smul_eq_mk_rep K (ff i) (hff i)\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\n[PROOFSTEP]\nconvert hh.units_smul a\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0)) = a \u2022 ff\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_4.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0)) i = (a \u2022 ff) i\n[PROOFSTEP]\nexact (ha i).symm\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 Independent f\n[PROOFSTEP]\nconvert Independent.mk _ _ h\n[GOAL]\ncase h.e'_7.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\nx\u271d : \u03b9\n\u22a2 f x\u271d = mk K ((Projectivization.rep \u2218 f) x\u271d) (_ : ?m.12147 x\u271d \u2260 0)\n[PROOFSTEP]\nsimp only [mk_rep, Function.comp_apply]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 \u2200 (i : \u03b9), (Projectivization.rep \u2218 f) i \u2260 0\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 f) i \u2260 0\n[PROOFSTEP]\napply rep_nonzero\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2194 CompleteLattice.Independent fun i => Projectivization.submodule (f i)\n[PROOFSTEP]\nrefine' \u27e8_, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2192 CompleteLattice.Independent fun i => Projectivization.submodule (f i)\n[PROOFSTEP]\nrintro \u27e8f, hf, hi\u27e9\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 V\nhf : \u2200 (i : \u03b9), f i \u2260 0\nhi : LinearIndependent K f\n\u22a2 CompleteLattice.Independent fun i => Projectivization.submodule ((fun i => mk K (f i) (_ : f i \u2260 0)) i)\n[PROOFSTEP]\nsimp only [submodule_mk]\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 V\nhf : \u2200 (i : \u03b9), f i \u2260 0\nhi : LinearIndependent K f\n\u22a2 CompleteLattice.Independent fun i => Submodule.span K {f i}\n[PROOFSTEP]\nexact (CompleteLattice.independent_iff_linearIndependent_of_ne_zero (R := K) hf).mpr hi\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : CompleteLattice.Independent fun i => Projectivization.submodule (f i)\n\u22a2 Independent f\n[PROOFSTEP]\nrw [independent_iff]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : CompleteLattice.Independent fun i => Projectivization.submodule (f i)\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 f)\n[PROOFSTEP]\nrefine' h.linearIndependent (Projectivization.submodule \u2218 f) (fun i => _) fun i => _\n[GOAL]\ncase refine'_2.refine'_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : CompleteLattice.Independent fun i => Projectivization.submodule (f i)\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 f) i \u2208 (Projectivization.submodule \u2218 f) i\n[PROOFSTEP]\nsimpa only [Function.comp_apply, submodule_eq] using Submodule.mem_span_singleton_self _\n[GOAL]\ncase refine'_2.refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : CompleteLattice.Independent fun i => Projectivization.submodule (f i)\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 f) i \u2260 0\n[PROOFSTEP]\nexact rep_nonzero (f i)\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Dependent f \u2194 \u00acLinearIndependent K (Projectivization.rep \u2218 f)\n[PROOFSTEP]\nrefine' \u27e8_, fun h => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Dependent f \u2192 \u00acLinearIndependent K (Projectivization.rep \u2218 f)\n[PROOFSTEP]\nrintro \u27e8ff, hff, hh1\u27e9\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh1 : \u00acLinearIndependent K ff\n\u22a2 \u00acLinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\n[PROOFSTEP]\ncontrapose! hh1\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh1 : LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\n\u22a2 LinearIndependent K ff\n[PROOFSTEP]\nchoose a ha using fun i : \u03b9 => exists_smul_eq_mk_rep K (ff i) (hff i)\n[GOAL]\ncase refine'_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh1 : LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\n\u22a2 LinearIndependent K ff\n[PROOFSTEP]\nconvert hh1.units_smul a\u207b\u00b9\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh1 : LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\n\u22a2 ff = a\u207b\u00b9 \u2022 Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0)\n[PROOFSTEP]\next i\n[GOAL]\ncase h.e'_4.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh1 : LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0))\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = Projectivization.rep (mk K (ff i) (_ : ff i \u2260 0))\ni : \u03b9\n\u22a2 ff i = (a\u207b\u00b9 \u2022 Projectivization.rep \u2218 fun i => mk K (ff i) (_ : ff i \u2260 0)) i\n[PROOFSTEP]\nsimp only [\u2190 ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : \u00acLinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 Dependent f\n[PROOFSTEP]\nconvert Dependent.mk _ _ h\n[GOAL]\ncase h.e'_7.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : \u00acLinearIndependent K (Projectivization.rep \u2218 f)\nx\u271d : \u03b9\n\u22a2 f x\u271d = mk K ((Projectivization.rep \u2218 f) x\u271d) (_ : ?m.30442 x\u271d \u2260 0)\n[PROOFSTEP]\nsimp only [mk_rep, Function.comp_apply]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : \u00acLinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 \u2200 (i : \u03b9), (Projectivization.rep \u2218 f) i \u2260 0\n[PROOFSTEP]\nexact fun i => rep_nonzero (f i)\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Dependent f \u2194 \u00acIndependent f\n[PROOFSTEP]\nrw [dependent_iff, independent_iff]\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2194 \u00acDependent f\n[PROOFSTEP]\nrw [dependent_iff_not_independent, Classical.not_not]\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 Dependent ![u, v] \u2194 u = v\n[PROOFSTEP]\nrw [dependent_iff_not_independent, independent_iff, linearIndependent_fin2, Function.comp_apply, Matrix.cons_val_one,\n  Matrix.head_cons, Ne.def]\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00ac(\u00acProjectivization.rep v = 0 \u2227 \u2200 (a : K), a \u2022 Projectivization.rep v \u2260 (Projectivization.rep \u2218 ![u, v]) 0) \u2194 u = v\n[PROOFSTEP]\nsimp only [Matrix.cons_val_zero, not_and, not_forall, Classical.not_not, Function.comp_apply, \u2190\n  mk_eq_mk_iff' K _ _ (rep_nonzero u) (rep_nonzero v), mk_rep, imp_iff_right_iff]\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 \u00acProjectivization.rep v = 0 \u2228 u = v\n[PROOFSTEP]\nexact Or.inl (rep_nonzero v)\n[GOAL]\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nu v : \u2119 K V\n\u22a2 Independent ![u, v] \u2194 u \u2260 v\n[PROOFSTEP]\nrw [independent_iff_not_dependent, dependent_pair_iff_eq u v]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.ProjectiveSpace.Independence", "llama_tokens": 5048, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.63341027059799, "lm_q1q2_score": 0.5087385316876399}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nX Y Z : C\nf : X \u27f6 Y\ng : Y \u27f6 Z\n\u22a2 f \u2297\u226b g = f \u226b g\n[PROOFSTEP]\nsimp [monoidalComp]\n[GOAL]\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : MonoidalCategory C\nU V W X Y : C\nf : U \u27f6 V \u2297 W \u2297 X\ng : (V \u2297 W) \u2297 X \u27f6 Y\n\u22a2 f \u2297\u226b g = f \u226b (\u03b1_ V W X).inv \u226b g\n[PROOFSTEP]\nsimp [monoidalComp]\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\ninst\u271d\u00b9 : MonoidalCategory C\u271d\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf g : X \u27f6 Y\nw : f \u226b \ud835\udfd9 Y = g\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using w\n[GOAL]\nC\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\ninst\u271d\u00b9 : MonoidalCategory C\u271d\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf g : X \u27f6 Y\nw : f = g \u226b \ud835\udfd9 Y\n\u22a2 f = g\n[PROOFSTEP]\nsimpa using w\n", "meta": {"mathlib_filename": "Mathlib.Tactic.CategoryTheory.Coherence", "llama_tokens": 422, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.508738526129557}}
{"text": "[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nc d : C\nf : c \u27f6 d\n\u03b3 : c \u27f6 c\n\u22a2 (fun \u03b4 => f \u226b \u03b4 \u226b inv f) ((fun \u03b3 => inv f \u226b \u03b3 \u226b f) \u03b3) = \u03b3\n[PROOFSTEP]\nsimp_rw [Category.assoc, comp_inv, Category.comp_id, \u2190 Category.assoc, comp_inv, Category.id_comp]\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nc d : C\nf : c \u27f6 d\n\u03b4 : d \u27f6 d\n\u22a2 (fun \u03b3 => inv f \u226b \u03b3 \u226b f) ((fun \u03b4 => f \u226b \u03b4 \u226b inv f) \u03b4) = \u03b4\n[PROOFSTEP]\nsimp_rw [Category.assoc, inv_comp, \u2190 Category.assoc, inv_comp, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d : Groupoid C\nc d : C\nf : c \u27f6 d\n\u03b3\u2081 \u03b3\u2082 : c \u27f6 c\n\u22a2 Equiv.toFun\n      { toFun := fun \u03b3 => inv f \u226b \u03b3 \u226b f, invFun := fun \u03b4 => f \u226b \u03b4 \u226b inv f,\n        left_inv := (_ : \u2200 (\u03b3 : c \u27f6 c), (fun \u03b4 => f \u226b \u03b4 \u226b inv f) ((fun \u03b3 => inv f \u226b \u03b3 \u226b f) \u03b3) = \u03b3),\n        right_inv := (_ : \u2200 (\u03b4 : d \u27f6 d), (fun \u03b3 => inv f \u226b \u03b3 \u226b f) ((fun \u03b4 => f \u226b \u03b4 \u226b inv f) \u03b4) = \u03b4) }\n      (\u03b3\u2081 * \u03b3\u2082) =\n    Equiv.toFun\n        { toFun := fun \u03b3 => inv f \u226b \u03b3 \u226b f, invFun := fun \u03b4 => f \u226b \u03b4 \u226b inv f,\n          left_inv := (_ : \u2200 (\u03b3 : c \u27f6 c), (fun \u03b4 => f \u226b \u03b4 \u226b inv f) ((fun \u03b3 => inv f \u226b \u03b3 \u226b f) \u03b3) = \u03b3),\n          right_inv := (_ : \u2200 (\u03b4 : d \u27f6 d), (fun \u03b3 => inv f \u226b \u03b3 \u226b f) ((fun \u03b4 => f \u226b \u03b4 \u226b inv f) \u03b4) = \u03b4) }\n        \u03b3\u2081 *\n      Equiv.toFun\n        { toFun := fun \u03b3 => inv f \u226b \u03b3 \u226b f, invFun := fun \u03b4 => f \u226b \u03b4 \u226b inv f,\n          left_inv := (_ : \u2200 (\u03b3 : c \u27f6 c), (fun \u03b4 => f \u226b \u03b4 \u226b inv f) ((fun \u03b3 => inv f \u226b \u03b3 \u226b f) \u03b3) = \u03b3),\n          right_inv := (_ : \u2200 (\u03b4 : d \u27f6 d), (fun \u03b3 => inv f \u226b \u03b3 \u226b f) ((fun \u03b4 => f \u226b \u03b4 \u226b inv f) \u03b4) = \u03b4) }\n        \u03b3\u2082\n[PROOFSTEP]\nsimp only [vertexGroup_mul, inv_eq_inv, Category.assoc, IsIso.hom_inv_id_assoc]\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Groupoid.VertexGroup", "llama_tokens": 808, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.5087215099211105}}
{"text": "[GOAL]\n\u03b1 : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\n\u22a2 IsClosed \u2191(direction s) \u2194 IsClosed \u2191s\n[PROOFSTEP]\nrcases s.eq_bot_or_nonempty with (rfl | \u27e8x, hx\u27e9)\n[GOAL]\ncase inl\n\u03b1 : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\n\u22a2 IsClosed \u2191(direction \u22a5) \u2194 IsClosed \u2191\u22a5\n[PROOFSTEP]\nsimp [isClosed_singleton]\n[GOAL]\ncase inr.intro\n\u03b1 : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed \u2191(direction s) \u2194 IsClosed \u2191s\n[PROOFSTEP]\nrw [\u2190 (IsometryEquiv.vaddConst x).toHomeomorph.symm.isClosed_image, AffineSubspace.coe_direction_eq_vsub_set_right hx]\n[GOAL]\ncase inr.intro\n\u03b1 : Type ?u.597\nV : Type ?u.600\nP : Type ?u.603\nW : Type u_2\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\ns : AffineSubspace \ud835\udd5c Q\nx : Q\nhx : x \u2208 \u2191s\n\u22a2 IsClosed ((fun x_1 => x_1 -\u1d65 x) '' \u2191s) \u2194\n    IsClosed (\u2191(Homeomorph.symm (IsometryEquiv.toHomeomorph (IsometryEquiv.vaddConst x))) '' \u2191s)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type ?u.5475\nV : Type u_2\nP : Type u_1\nW : Type ?u.5484\nQ : Type ?u.5487\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist p\u2081 (\u2191(homothety p\u2081 c) p\u2082) = \u2016c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [homothety_def, dist_eq_norm_vsub V]\n[GOAL]\n\u03b1 : Type ?u.5475\nV : Type u_2\nP : Type u_1\nW : Type ?u.5484\nQ : Type ?u.5487\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 \u2016p\u2081 -\u1d65 \u2191(c \u2022 (AffineMap.id \ud835\udd5c P -\u1d65 const \ud835\udd5c P p\u2081) +\u1d65 const \ud835\udd5c P p\u2081) p\u2082\u2016 = \u2016c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nsimp [norm_smul, \u2190 dist_eq_norm_vsub V, dist_comm]\n[GOAL]\n\u03b1 : Type ?u.14492\nV : Type u_2\nP : Type u_1\nW : Type ?u.14501\nQ : Type ?u.14504\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist (\u2191(homothety p\u2081 c) p\u2082) p\u2081 = \u2016c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [dist_comm, dist_center_homothety]\n[GOAL]\n\u03b1 : Type ?u.16780\nV : Type u_3\nP : Type u_1\nW : Type ?u.16789\nQ : Type ?u.16792\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc\u2081 c\u2082 : \ud835\udd5c\n\u22a2 dist (\u2191(lineMap p\u2081 p\u2082) c\u2081) (\u2191(lineMap p\u2081 p\u2082) c\u2082) = dist c\u2081 c\u2082 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [dist_comm p\u2081 p\u2082]\n  -- Porting note: was `simp only [lineMap_apply, dist_eq_norm_vsub, vadd_vsub_vadd_cancel_right,`\n    -- `\u2190 sub_smul, norm_smul, vsub_eq_sub]`\n[GOAL]\n\u03b1 : Type ?u.16780\nV : Type u_3\nP : Type u_1\nW : Type ?u.16789\nQ : Type ?u.16792\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc\u2081 c\u2082 : \ud835\udd5c\n\u22a2 dist (\u2191(lineMap p\u2081 p\u2082) c\u2081) (\u2191(lineMap p\u2081 p\u2082) c\u2082) = dist c\u2081 c\u2082 * dist p\u2082 p\u2081\n[PROOFSTEP]\nrw [lineMap_apply, lineMap_apply, dist_eq_norm_vsub V, vadd_vsub_vadd_cancel_right, \u2190 sub_smul, norm_smul, \u2190\n  vsub_eq_sub, \u2190 dist_eq_norm_vsub V, \u2190 dist_eq_norm_vsub \ud835\udd5c]\n[GOAL]\n\u03b1 : Type ?u.28725\nV : Type u_3\nP : Type u_1\nW : Type ?u.28734\nQ : Type ?u.28737\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist (\u2191(lineMap p\u2081 p\u2082) c) p\u2081 = \u2016c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [\u2190 dist_zero_right, \u2190 dist_lineMap_lineMap, lineMap_apply_zero]\n[GOAL]\n\u03b1 : Type ?u.37626\nV : Type u_3\nP : Type u_1\nW : Type ?u.37635\nQ : Type ?u.37638\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist (\u2191(lineMap p\u2081 p\u2082) c) p\u2082 = \u20161 - c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm', \u2190 dist_lineMap_lineMap, lineMap_apply_one]\n[GOAL]\n\u03b1 : Type ?u.47659\nV : Type u_2\nP : Type u_1\nW : Type ?u.47668\nQ : Type ?u.47671\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist (\u2191(homothety p\u2081 c) p\u2082) p\u2082 = \u20161 - c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [homothety_eq_lineMap, dist_lineMap_right]\n[GOAL]\n\u03b1 : Type ?u.51220\nV : Type u_2\nP : Type u_1\nW : Type ?u.51229\nQ : Type ?u.51232\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : P\nc : \ud835\udd5c\n\u22a2 dist p\u2082 (\u2191(homothety p\u2081 c) p\u2082) = \u20161 - c\u2016 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [dist_comm, dist_homothety_self]\n[GOAL]\n\u03b1 : Type ?u.54860\nV : Type u_2\nP : Type u_1\nW : Type ?u.54869\nQ : Type ?u.54872\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 : P\n\u22a2 dist p\u2081 (midpoint \ud835\udd5c p\u2081 p\u2082) = \u20162\u2016\u207b\u00b9 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [midpoint, dist_comm, dist_lineMap_left, invOf_eq_inv, \u2190 norm_inv]\n[GOAL]\n\u03b1 : Type ?u.59357\nV : Type u_2\nP : Type u_1\nW : Type ?u.59366\nQ : Type ?u.59369\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 : P\n\u22a2 dist (midpoint \ud835\udd5c p\u2081 p\u2082) p\u2081 = \u20162\u2016\u207b\u00b9 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [dist_comm, dist_left_midpoint]\n[GOAL]\n\u03b1 : Type ?u.63410\nV : Type u_2\nP : Type u_1\nW : Type ?u.63419\nQ : Type ?u.63422\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 : P\n\u22a2 dist (midpoint \ud835\udd5c p\u2081 p\u2082) p\u2082 = \u20162\u2016\u207b\u00b9 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [midpoint_comm, dist_midpoint_left, dist_comm]\n[GOAL]\n\u03b1 : Type ?u.68130\nV : Type u_2\nP : Type u_1\nW : Type ?u.68139\nQ : Type ?u.68142\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 : P\n\u22a2 dist p\u2082 (midpoint \ud835\udd5c p\u2081 p\u2082) = \u20162\u2016\u207b\u00b9 * dist p\u2081 p\u2082\n[PROOFSTEP]\nrw [dist_comm, dist_midpoint_right]\n[GOAL]\n\u03b1 : Type ?u.72183\nV : Type u_2\nP : Type u_1\nW : Type ?u.72192\nQ : Type ?u.72195\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 dist (midpoint \ud835\udd5c p\u2081 p\u2082) (midpoint \ud835\udd5c p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / \u20162\u2016\n[PROOFSTEP]\nrw [dist_eq_norm_vsub V, dist_eq_norm_vsub V, dist_eq_norm_vsub V, midpoint_vsub_midpoint]\n[GOAL]\n\u03b1 : Type ?u.72183\nV : Type u_2\nP : Type u_1\nW : Type ?u.72192\nQ : Type ?u.72195\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2016midpoint \ud835\udd5c (p\u2081 -\u1d65 p\u2083) (p\u2082 -\u1d65 p\u2084)\u2016 \u2264 (\u2016p\u2081 -\u1d65 p\u2083\u2016 + \u2016p\u2082 -\u1d65 p\u2084\u2016) / \u20162\u2016\n[PROOFSTEP]\ntry infer_instance\n[GOAL]\n\u03b1 : Type ?u.72183\nV : Type u_2\nP : Type u_1\nW : Type ?u.72192\nQ : Type ?u.72195\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2016midpoint \ud835\udd5c (p\u2081 -\u1d65 p\u2083) (p\u2082 -\u1d65 p\u2084)\u2016 \u2264 (\u2016p\u2081 -\u1d65 p\u2083\u2016 + \u2016p\u2082 -\u1d65 p\u2084\u2016) / \u20162\u2016\n[PROOFSTEP]\ninfer_instance\n[GOAL]\n\u03b1 : Type ?u.72183\nV : Type u_2\nP : Type u_1\nW : Type ?u.72192\nQ : Type ?u.72195\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2016midpoint \ud835\udd5c (p\u2081 -\u1d65 p\u2083) (p\u2082 -\u1d65 p\u2084)\u2016 \u2264 (\u2016p\u2081 -\u1d65 p\u2083\u2016 + \u2016p\u2082 -\u1d65 p\u2084\u2016) / \u20162\u2016\n[PROOFSTEP]\nrw [midpoint_eq_smul_add, norm_smul, invOf_eq_inv, norm_inv, \u2190 div_eq_inv_mul]\n[GOAL]\n\u03b1 : Type ?u.72183\nV : Type u_2\nP : Type u_1\nW : Type ?u.72192\nQ : Type ?u.72195\ninst\u271d\u2079 : SeminormedAddCommGroup V\ninst\u271d\u2078 : PseudoMetricSpace P\ninst\u271d\u2077 : NormedAddTorsor V P\ninst\u271d\u2076 : NormedAddCommGroup W\ninst\u271d\u2075 : MetricSpace Q\ninst\u271d\u2074 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b3 : NormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c V\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c W\ninst\u271d : Invertible 2\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 \u2016p\u2081 -\u1d65 p\u2083 + (p\u2082 -\u1d65 p\u2084)\u2016 / \u20162\u2016 \u2264 (\u2016p\u2081 -\u1d65 p\u2083\u2016 + \u2016p\u2082 -\u1d65 p\u2084\u2016) / \u20162\u2016\n[PROOFSTEP]\nexact div_le_div_of_le_of_nonneg (norm_add_le _ _) (norm_nonneg _)\n[GOAL]\n\u03b1 : Type ?u.79698\nV : Type u_2\nP : Type u_1\nW : Type ?u.79707\nQ : Type ?u.79710\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type ?u.79957\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np q : P\n\u22a2 dist (\u2191(Equiv.pointReflection p) q) p = dist p q\n[PROOFSTEP]\nsimp [dist_eq_norm_vsub V, Equiv.pointReflection_vsub_left (G := V)]\n[GOAL]\n\u03b1 : Type ?u.82730\nV : Type u_2\nP : Type u_1\nW : Type ?u.82739\nQ : Type ?u.82742\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_3\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np q : P\n\u22a2 dist (\u2191(Equiv.pointReflection p) q) q = \u20162\u2016 * dist p q\n[PROOFSTEP]\nsimp [dist_eq_norm_vsub V, Equiv.pointReflection_vsub_right (G := V), nsmul_eq_smul_cast \ud835\udd5c, norm_smul]\n[GOAL]\n\u03b1 : Type ?u.88928\nV : Type ?u.88931\nP : Type ?u.88934\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\np\u2081 p\u2082 : Q\nh : p\u2081 \u2260 p\u2082\nc\u2081 c\u2082 : \ud835\udd5c\n\u22a2 dist c\u2081 c\u2082 \u2264 \u2191(nndist p\u2081 p\u2082)\u207b\u00b9 * dist (\u2191(lineMap p\u2081 p\u2082) c\u2081) (\u2191(lineMap p\u2081 p\u2082) c\u2082)\n[PROOFSTEP]\nrw [dist_lineMap_lineMap, NNReal.coe_inv, \u2190 dist_nndist, mul_left_comm, inv_mul_cancel (dist_ne_zero.2 h), mul_one]\n[GOAL]\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) y \u2208 s\n[PROOFSTEP]\nrw [(NormedAddCommGroup.nhds_basis_norm_lt (1 : \ud835\udd5c)).eventually_iff]\n[GOAL]\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\ncases' eq_or_ne y x with h h\n[GOAL]\ncase inl\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y = x\n\u22a2 0 < 1 \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < 1} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nsimp [h.symm, interior_subset hy]\n[GOAL]\ncase inr\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nhave hxy : 0 < \u2016y -\u1d65 x\u2016 := by rwa [norm_pos_iff, vsub_ne_zero]\n[GOAL]\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\n\u22a2 0 < \u2016y -\u1d65 x\u2016\n[PROOFSTEP]\nrwa [norm_pos_iff, vsub_ne_zero]\n[GOAL]\ncase inr\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nobtain \u27e8u, hu\u2081, hu\u2082, hu\u2083\u27e9 := mem_interior.mp hy\n[GOAL]\ncase inr.intro.intro.intro\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nobtain \u27e8\u03b5, h\u03b5, hy\u03b5\u27e9 := Metric.isOpen_iff.mp hu\u2082 y hu\u2083\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u22a2 \u2203 i, 0 < i \u2227 \u2200 \u2983x_1 : \ud835\udd5c\u2984, x_1 \u2208 {y | \u2016y - 1\u2016 < i} \u2192 \u2191(homothety x x_1) y \u2208 s\n[PROOFSTEP]\nrefine' \u27e8\u03b5 / \u2016y -\u1d65 x\u2016, div_pos h\u03b5 hxy, fun \u03b4 (h\u03b4 : \u2016\u03b4 - 1\u2016 < \u03b5 / \u2016y -\u1d65 x\u2016) => hu\u2081 (hy\u03b5 _)\u27e9\n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4 : \u2016\u03b4 - 1\u2016 < \u03b5 / \u2016y -\u1d65 x\u2016\n\u22a2 \u2191(homothety x \u03b4) y \u2208 Metric.ball y \u03b5\n[PROOFSTEP]\nrw [lt_div_iff hxy, \u2190 norm_smul, sub_smul, one_smul] at h\u03b4 \n[GOAL]\ncase inr.intro.intro.intro.intro.intro\n\u03b1 : Type ?u.92785\nV : Type ?u.92788\nP : Type ?u.92791\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns : Set Q\ny : Q\nhy : y \u2208 interior s\nh : y \u2260 x\nhxy : 0 < \u2016y -\u1d65 x\u2016\nu : Set Q\nhu\u2081 : u \u2286 s\nhu\u2082 : IsOpen u\nhu\u2083 : y \u2208 u\n\u03b5 : \u211d\nh\u03b5 : \u03b5 > 0\nhy\u03b5 : Metric.ball y \u03b5 \u2286 u\n\u03b4 : \ud835\udd5c\nh\u03b4\u271d : \u2016\u03b4 - 1\u2016 * \u2016y -\u1d65 x\u2016 < \u03b5\nh\u03b4 : \u2016\u03b4 \u2022 (y -\u1d65 x) - (y -\u1d65 x)\u2016 < \u03b5\n\u22a2 \u2191(homothety x \u03b4) y \u2208 Metric.ball y \u03b5\n[PROOFSTEP]\nrwa [homothety_apply, Metric.mem_ball, dist_eq_norm_vsub W, vadd_vsub_eq_sub_vsub]\n[GOAL]\n\u03b1 : Type ?u.102650\nV : Type ?u.102653\nP : Type ?u.102656\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : Set.Finite t\nh : t \u2286 interior s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) '' t \u2286 s\n[PROOFSTEP]\nsuffices \u2200 y \u2208 t, \u2200\u1da0 \u03b4 in \ud835\udcdd (1 : \ud835\udd5c), homothety x \u03b4 y \u2208 s\n  by\n  simp_rw [Set.image_subset_iff]\n  exact (Filter.eventually_all_finite ht).mpr this\n[GOAL]\n\u03b1 : Type ?u.102650\nV : Type ?u.102653\nP : Type ?u.102656\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : Set.Finite t\nh : t \u2286 interior s\nthis : \u2200 (y : Q), y \u2208 t \u2192 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) y \u2208 s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) '' t \u2286 s\n[PROOFSTEP]\nsimp_rw [Set.image_subset_iff]\n[GOAL]\n\u03b1 : Type ?u.102650\nV : Type ?u.102653\nP : Type ?u.102656\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : Set.Finite t\nh : t \u2286 interior s\nthis : \u2200 (y : Q), y \u2208 t \u2192 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) y \u2208 s\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, t \u2286 (fun a => \u2191(homothety x \u03b4) a) \u207b\u00b9' s\n[PROOFSTEP]\nexact (Filter.eventually_all_finite ht).mpr this\n[GOAL]\n\u03b1 : Type ?u.102650\nV : Type ?u.102653\nP : Type ?u.102656\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : Set.Finite t\nh : t \u2286 interior s\n\u22a2 \u2200 (y : Q), y \u2208 t \u2192 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) y \u2208 s\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type ?u.102650\nV : Type ?u.102653\nP : Type ?u.102656\nW : Type u_3\nQ : Type u_1\ninst\u271d\u2078 : SeminormedAddCommGroup V\ninst\u271d\u2077 : PseudoMetricSpace P\ninst\u271d\u2076 : NormedAddTorsor V P\ninst\u271d\u2075 : NormedAddCommGroup W\ninst\u271d\u2074 : MetricSpace Q\ninst\u271d\u00b3 : NormedAddTorsor W Q\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : NormedField \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c V\ninst\u271d : NormedSpace \ud835\udd5c W\nx : Q\ns t : Set Q\nht : Set.Finite t\nh : t \u2286 interior s\ny : Q\nhy : y \u2208 t\n\u22a2 \u2200\u1da0 (\u03b4 : \ud835\udd5c) in \ud835\udcdd 1, \u2191(homothety x \u03b4) y \u2208 s\n[PROOFSTEP]\nexact eventually_homothety_mem_of_mem_interior \ud835\udd5c x (h hy)\n[GOAL]\n\u03b1 : Type ?u.107060\nV : Type u_1\nP : Type ?u.107066\nW : Type ?u.107069\nQ : Type ?u.107072\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\np\u2081 p\u2082 p\u2083 p\u2084 : V\n\u22a2 dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / 2\n[PROOFSTEP]\nhave := dist_midpoint_midpoint_le' (\ud835\udd5c := \u211d) p\u2081 p\u2082 p\u2083 p\u2084\n[GOAL]\n\u03b1 : Type ?u.107060\nV : Type u_1\nP : Type ?u.107066\nW : Type ?u.107069\nQ : Type ?u.107072\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\np\u2081 p\u2082 p\u2083 p\u2084 : V\nthis : dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / \u20162\u2016\n\u22a2 dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / 2\n[PROOFSTEP]\nrw [Real.norm_eq_abs, abs_two] at this \n[GOAL]\n\u03b1 : Type ?u.107060\nV : Type u_1\nP : Type ?u.107066\nW : Type ?u.107069\nQ : Type ?u.107072\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\np\u2081 p\u2082 p\u2083 p\u2084 : V\nthis : dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / 2\n\u22a2 dist (midpoint \u211d p\u2081 p\u2082) (midpoint \u211d p\u2083 p\u2084) \u2264 (dist p\u2081 p\u2083 + dist p\u2082 p\u2084) / 2\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\n\u22a2 (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) 0 = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\nx y : V\n\u22a2 (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) (midpoint \u211d x y) =\n    midpoint \u211d ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) x)\n      ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\nx y : V\n\u22a2 f (midpoint \u211d x y +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P) =\n    midpoint \u211d (f (x +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P))\n      (f (y +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P))\n[PROOFSTEP]\nconv_lhs => rw [(midpoint_self \u211d (Classical.arbitrary P)).symm, midpoint_vadd_midpoint, h, h, midpoint_vsub_midpoint]\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\nx y : V\n| f (midpoint \u211d x y +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P)\n[PROOFSTEP]\nrw [(midpoint_self \u211d (Classical.arbitrary P)).symm, midpoint_vadd_midpoint, h, h, midpoint_vsub_midpoint]\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\nx y : V\n| f (midpoint \u211d x y +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P)\n[PROOFSTEP]\nrw [(midpoint_self \u211d (Classical.arbitrary P)).symm, midpoint_vadd_midpoint, h, h, midpoint_vsub_midpoint]\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\nx y : V\n| f (midpoint \u211d x y +\u1d65 Classical.arbitrary P) -\u1d65 f (Classical.arbitrary P)\n[PROOFSTEP]\nrw [(midpoint_self \u211d (Classical.arbitrary P)).symm, midpoint_vadd_midpoint, h, h, midpoint_vsub_midpoint]\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\n\u22a2 Continuous\n    \u2191(AddMonoidHom.ofMapMidpoint \u211d \u211d\n        (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c))\n        (_ :\n          \u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f (Classical.arbitrary P)))) (f (0 +\u1d65 Classical.arbitrary P)) =\n            0)\n        (_ :\n          \u2200 (x y : V),\n            (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) (midpoint \u211d x y) =\n              midpoint \u211d ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) x)\n                ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) y)))\n[PROOFSTEP]\napply_rules [Continuous.vadd, Continuous.vsub, continuous_const, hfc.comp, continuous_id]\n[GOAL]\n\u03b1 : Type ?u.111434\nV : Type ?u.111437\nP : Type ?u.111440\nW : Type ?u.111443\nQ : Type ?u.111446\ninst\u271d\u2077 : SeminormedAddCommGroup V\ninst\u271d\u2076 : PseudoMetricSpace P\ninst\u271d\u2075 : NormedAddTorsor V P\ninst\u271d\u2074 : NormedAddCommGroup W\ninst\u271d\u00b3 : MetricSpace Q\ninst\u271d\u00b2 : NormedAddTorsor W Q\ninst\u271d\u00b9 : NormedSpace \u211d V\ninst\u271d : NormedSpace \u211d W\nf : P \u2192 Q\nh : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)\nhfc : Continuous f\nc : P := Classical.arbitrary P\np : P\n\u22a2 f p =\n    \u2191\u2191(AddMonoidHom.toRealLinearMap\n              (AddMonoidHom.ofMapMidpoint \u211d \u211d\n                (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c))\n                (_ :\n                  \u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f (Classical.arbitrary P))))\n                      (f (0 +\u1d65 Classical.arbitrary P)) =\n                    0)\n                (_ :\n                  \u2200 (x y : V),\n                    (\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c))\n                        (midpoint \u211d x y) =\n                      midpoint \u211d\n                        ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) x)\n                        ((\u2191(AffineEquiv.symm (AffineEquiv.vaddConst \u211d (f c))) \u2218 f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) y)))\n              (_ : Continuous ((fun i => (f \u2218 \u2191(AffineEquiv.vaddConst \u211d c)) i) -\u1d65 fun i => f c)))\n        (p -\u1d65 c) +\u1d65\n      f c\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.AddTorsor", "llama_tokens": 15993, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.5087214992415904}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nb : \u2102\nhb : b \u2260 0\n\u22a2 (fun x => 0 ^ x) =\u1da0[\ud835\udcdd b] 0\n[PROOFSTEP]\nsuffices : \u2200\u1da0 x : \u2102 in \ud835\udcdd b, x \u2260 0\n[GOAL]\n\u03b1 : Type u_1\nb : \u2102\nhb : b \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd b, x \u2260 0\n\u22a2 (fun x => 0 ^ x) =\u1da0[\ud835\udcdd b] 0\ncase this \u03b1 : Type u_1 b : \u2102 hb : b \u2260 0 \u22a2 \u2200\u1da0 (x : \u2102) in \ud835\udcdd b, x \u2260 0\n[PROOFSTEP]\nexact\n  this.mono fun x hx => by\n    dsimp only\n    rw [zero_cpow hx, Pi.zero_apply]\n[GOAL]\n\u03b1 : Type u_1\nb : \u2102\nhb : b \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd b, x \u2260 0\nx : \u2102\nhx : x \u2260 0\n\u22a2 (fun x => 0 ^ x) x = OfNat.ofNat 0 x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nb : \u2102\nhb : b \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd b, x \u2260 0\nx : \u2102\nhx : x \u2260 0\n\u22a2 0 ^ x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [zero_cpow hx, Pi.zero_apply]\n[GOAL]\ncase this\n\u03b1 : Type u_1\nb : \u2102\nhb : b \u2260 0\n\u22a2 \u2200\u1da0 (x : \u2102) in \ud835\udcdd b, x \u2260 0\n[PROOFSTEP]\nexact IsOpen.eventually_mem isOpen_ne hb\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\n\u22a2 (fun x => x ^ b) =\u1da0[\ud835\udcdd a] fun x => Complex.exp (Complex.log x * b)\n[PROOFSTEP]\nsuffices : \u2200\u1da0 x : \u2102 in \ud835\udcdd a, x \u2260 0\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd a, x \u2260 0\n\u22a2 (fun x => x ^ b) =\u1da0[\ud835\udcdd a] fun x => Complex.exp (Complex.log x * b)\ncase this \u03b1 : Type u_1 a b : \u2102 ha : a \u2260 0 \u22a2 \u2200\u1da0 (x : \u2102) in \ud835\udcdd a, x \u2260 0\n[PROOFSTEP]\nexact\n  this.mono fun x hx => by\n    dsimp only\n    rw [cpow_def_of_ne_zero hx]\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd a, x \u2260 0\nx : \u2102\nhx : x \u2260 0\n\u22a2 (fun x => x ^ b) x = (fun x => Complex.exp (Complex.log x * b)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\nthis : \u2200\u1da0 (x : \u2102) in \ud835\udcdd a, x \u2260 0\nx : \u2102\nhx : x \u2260 0\n\u22a2 x ^ b = Complex.exp (Complex.log x * b)\n[PROOFSTEP]\nrw [cpow_def_of_ne_zero hx]\n[GOAL]\ncase this\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\n\u22a2 \u2200\u1da0 (x : \u2102) in \ud835\udcdd a, x \u2260 0\n[PROOFSTEP]\nexact IsOpen.eventually_mem isOpen_ne ha\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => Complex.exp (Complex.log x.fst * x.snd)\n[PROOFSTEP]\nsuffices : \u2200\u1da0 x : \u2102 \u00d7 \u2102 in \ud835\udcdd p, x.1 \u2260 0\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\nthis : \u2200\u1da0 (x : \u2102 \u00d7 \u2102) in \ud835\udcdd p, x.fst \u2260 0\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => Complex.exp (Complex.log x.fst * x.snd)\ncase this \u03b1 : Type u_1 p : \u2102 \u00d7 \u2102 hp_fst : p.fst \u2260 0 \u22a2 \u2200\u1da0 (x : \u2102 \u00d7 \u2102) in \ud835\udcdd p, x.fst \u2260 0\n[PROOFSTEP]\nexact\n  this.mono fun x hx => by\n    dsimp only\n    rw [cpow_def_of_ne_zero hx]\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\nthis : \u2200\u1da0 (x : \u2102 \u00d7 \u2102) in \ud835\udcdd p, x.fst \u2260 0\nx : \u2102 \u00d7 \u2102\nhx : x.fst \u2260 0\n\u22a2 (fun x => x.fst ^ x.snd) x = (fun x => Complex.exp (Complex.log x.fst * x.snd)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\nthis : \u2200\u1da0 (x : \u2102 \u00d7 \u2102) in \ud835\udcdd p, x.fst \u2260 0\nx : \u2102 \u00d7 \u2102\nhx : x.fst \u2260 0\n\u22a2 x.fst ^ x.snd = Complex.exp (Complex.log x.fst * x.snd)\n[PROOFSTEP]\nrw [cpow_def_of_ne_zero hx]\n[GOAL]\ncase this\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\n\u22a2 \u2200\u1da0 (x : \u2102 \u00d7 \u2102) in \ud835\udcdd p, x.fst \u2260 0\n[PROOFSTEP]\nrefine' IsOpen.eventually_mem _ hp_fst\n[GOAL]\ncase this\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\n\u22a2 IsOpen fun x => x.fst = 0 \u2192 False\n[PROOFSTEP]\nchange IsOpen {x : \u2102 \u00d7 \u2102 | x.1 = 0}\u1d9c\n[GOAL]\ncase this\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\n\u22a2 IsOpen {x | x.fst = 0}\u1d9c\n[PROOFSTEP]\nrw [isOpen_compl_iff]\n[GOAL]\ncase this\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : p.fst \u2260 0\n\u22a2 IsClosed {x | x.fst = 0}\n[PROOFSTEP]\nexact isClosed_eq continuous_fst continuous_const\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\n\u22a2 ContinuousAt (fun x => a ^ x) b\n[PROOFSTEP]\nhave cpow_eq : (fun x : \u2102 => a ^ x) = fun x => exp (log a * x) :=\n  by\n  ext1 b\n  rw [cpow_def_of_ne_zero ha]\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\n\u22a2 (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n[PROOFSTEP]\next1 b\n[GOAL]\ncase h\n\u03b1 : Type u_1\na b\u271d : \u2102\nha : a \u2260 0\nb : \u2102\n\u22a2 a ^ b = Complex.exp (Complex.log a * b)\n[PROOFSTEP]\nrw [cpow_def_of_ne_zero ha]\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => a ^ x) b\n[PROOFSTEP]\nrw [cpow_eq]\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nha : a \u2260 0\ncpow_eq : (fun x => a ^ x) = fun x => Complex.exp (Complex.log a * x)\n\u22a2 ContinuousAt (fun x => Complex.exp (Complex.log a * x)) b\n[PROOFSTEP]\nexact continuous_exp.continuousAt.comp (ContinuousAt.mul continuousAt_const continuousAt_id)\n[GOAL]\n\u03b1 : Type u_1\na b : \u2102\nh : b \u2260 0\n\u22a2 ContinuousAt (fun x => a ^ x) b\n[PROOFSTEP]\nby_cases ha : a = 0\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na b : \u2102\nh : b \u2260 0\nha : a = 0\n\u22a2 ContinuousAt (fun x => a ^ x) b\n[PROOFSTEP]\nrw [ha, continuousAt_congr (zero_cpow_eq_nhds h)]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\na b : \u2102\nh : b \u2260 0\nha : a = 0\n\u22a2 ContinuousAt 0 b\n[PROOFSTEP]\nexact continuousAt_const\n[GOAL]\ncase neg\n\u03b1 : Type u_1\na b : \u2102\nh : b \u2260 0\nha : \u00aca = 0\n\u22a2 ContinuousAt (fun x => a ^ x) b\n[PROOFSTEP]\nexact continuousAt_const_cpow ha\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) p\n[PROOFSTEP]\nhave hp_fst_ne_zero : p.fst \u2260 0 := by\n  intro h\n  cases' hp_fst with hp_fst hp_fst <;>\n    \u00b7 rw [h] at hp_fst \n      simp at hp_fst \n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\n\u22a2 p.fst \u2260 0\n[PROOFSTEP]\nintro h\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\nh : p.fst = 0\n\u22a2 False\n[PROOFSTEP]\ncases' hp_fst with hp_fst hp_fst\n[GOAL]\ncase inl\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nh : p.fst = 0\nhp_fst : 0 < p.fst.re\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hp_fst \n[GOAL]\ncase inl\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nh : p.fst = 0\nhp_fst : 0 < 0.re\n\u22a2 False\n[PROOFSTEP]\nsimp at hp_fst \n[GOAL]\ncase inr\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nh : p.fst = 0\nhp_fst : p.fst.im \u2260 0\n\u22a2 False\n[PROOFSTEP]\nrw [h] at hp_fst \n[GOAL]\ncase inr\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nh : p.fst = 0\nhp_fst : 0.im \u2260 0\n\u22a2 False\n[PROOFSTEP]\nsimp at hp_fst \n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\nhp_fst_ne_zero : p.fst \u2260 0\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) p\n[PROOFSTEP]\nrw [continuousAt_congr (cpow_eq_nhds' hp_fst_ne_zero)]\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\nhp_fst_ne_zero : p.fst \u2260 0\n\u22a2 ContinuousAt (fun x => Complex.exp (Complex.log x.fst * x.snd)) p\n[PROOFSTEP]\nrefine' continuous_exp.continuousAt.comp _\n[GOAL]\n\u03b1 : Type u_1\np : \u2102 \u00d7 \u2102\nhp_fst : 0 < p.fst.re \u2228 p.fst.im \u2260 0\nhp_fst_ne_zero : p.fst \u2260 0\n\u22a2 ContinuousAt (fun x => Complex.log x.fst * x.snd) p\n[PROOFSTEP]\nexact\n  ContinuousAt.mul (ContinuousAt.comp (continuousAt_clog hp_fst) continuous_fst.continuousAt)\n    continuous_snd.continuousAt\n[GOAL]\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : a \u2260 0 \u2228 b \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\ncases h with\n| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf\n| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf\n[GOAL]\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : a \u2260 0 \u2228 b \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\ncases h with\n| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf\n| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : a \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\n\n| inl h => exact (continuousAt_const_cpow h).tendsto.comp hf\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : a \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\nexact (continuousAt_const_cpow h).tendsto.comp hf\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : b \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\n\n| inr h => exact (continuousAt_const_cpow' h).tendsto.comp hf\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u2102\na b : \u2102\nhf : Tendsto f l (\ud835\udcdd b)\nh : b \u2260 0\n\u22a2 Tendsto (fun x => a ^ f x) l (\ud835\udcdd (a ^ b))\n[PROOFSTEP]\nexact (continuousAt_const_cpow' h).tendsto.comp hf\n[GOAL]\na b : \u211d\nh : a \u2260 0\n\u22a2 ContinuousAt (rpow a) b\n[PROOFSTEP]\nhave : rpow a = fun x : \u211d => ((a : \u2102) ^ (x : \u2102)).re := by\n  ext1 x\n  rw [rpow_eq_pow, rpow_def]\n[GOAL]\na b : \u211d\nh : a \u2260 0\n\u22a2 rpow a = fun x => (\u2191a ^ \u2191x).re\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\na b : \u211d\nh : a \u2260 0\nx : \u211d\n\u22a2 rpow a x = (\u2191a ^ \u2191x).re\n[PROOFSTEP]\nrw [rpow_eq_pow, rpow_def]\n[GOAL]\na b : \u211d\nh : a \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (rpow a) b\n[PROOFSTEP]\nrw [this]\n[GOAL]\na b : \u211d\nh : a \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (fun x => (\u2191a ^ \u2191x).re) b\n[PROOFSTEP]\nrefine' Complex.continuous_re.continuousAt.comp _\n[GOAL]\na b : \u211d\nh : a \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (fun x => \u2191a ^ \u2191x) b\n[PROOFSTEP]\nrefine' (continuousAt_const_cpow _).comp Complex.continuous_ofReal.continuousAt\n[GOAL]\na b : \u211d\nh : a \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 \u2191a \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\na b : \u211d\nh : b \u2260 0\n\u22a2 ContinuousAt (rpow a) b\n[PROOFSTEP]\nhave : rpow a = fun x : \u211d => ((a : \u2102) ^ (x : \u2102)).re := by\n  ext1 x\n  rw [rpow_eq_pow, rpow_def]\n[GOAL]\na b : \u211d\nh : b \u2260 0\n\u22a2 rpow a = fun x => (\u2191a ^ \u2191x).re\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h\na b : \u211d\nh : b \u2260 0\nx : \u211d\n\u22a2 rpow a x = (\u2191a ^ \u2191x).re\n[PROOFSTEP]\nrw [rpow_eq_pow, rpow_def]\n[GOAL]\na b : \u211d\nh : b \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (rpow a) b\n[PROOFSTEP]\nrw [this]\n[GOAL]\na b : \u211d\nh : b \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (fun x => (\u2191a ^ \u2191x).re) b\n[PROOFSTEP]\nrefine' Complex.continuous_re.continuousAt.comp _\n[GOAL]\na b : \u211d\nh : b \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 ContinuousAt (fun x => \u2191a ^ \u2191x) b\n[PROOFSTEP]\nrefine' (continuousAt_const_cpow' _).comp Complex.continuous_ofReal.continuousAt\n[GOAL]\na b : \u211d\nh : b \u2260 0\nthis : rpow a = fun x => (\u2191a ^ \u2191x).re\n\u22a2 \u2191b \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : p.fst < 0\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => exp (log x.fst * x.snd) * cos (x.snd * \u03c0)\n[PROOFSTEP]\nsuffices : \u2200\u1da0 x : \u211d \u00d7 \u211d in \ud835\udcdd p, x.1 < 0\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : p.fst < 0\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, x.fst < 0\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => exp (log x.fst * x.snd) * cos (x.snd * \u03c0)\ncase this p : \u211d \u00d7 \u211d hp_fst : p.fst < 0 \u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, x.fst < 0\n[PROOFSTEP]\nexact\n  this.mono fun x hx => by\n    dsimp only\n    rw [rpow_def_of_neg hx]\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : p.fst < 0\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, x.fst < 0\nx : \u211d \u00d7 \u211d\nhx : x.fst < 0\n\u22a2 (fun x => x.fst ^ x.snd) x = (fun x => exp (log x.fst * x.snd) * cos (x.snd * \u03c0)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : p.fst < 0\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, x.fst < 0\nx : \u211d \u00d7 \u211d\nhx : x.fst < 0\n\u22a2 x.fst ^ x.snd = exp (log x.fst * x.snd) * cos (x.snd * \u03c0)\n[PROOFSTEP]\nrw [rpow_def_of_neg hx]\n[GOAL]\ncase this\np : \u211d \u00d7 \u211d\nhp_fst : p.fst < 0\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, x.fst < 0\n[PROOFSTEP]\nexact IsOpen.eventually_mem (isOpen_lt continuous_fst continuous_const) hp_fst\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.fst\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => exp (log x.fst * x.snd)\n[PROOFSTEP]\nsuffices : \u2200\u1da0 x : \u211d \u00d7 \u211d in \ud835\udcdd p, 0 < x.1\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.fst\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.fst\n\u22a2 (fun x => x.fst ^ x.snd) =\u1da0[\ud835\udcdd p] fun x => exp (log x.fst * x.snd)\ncase this p : \u211d \u00d7 \u211d hp_fst : 0 < p.fst \u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.fst\n[PROOFSTEP]\nexact\n  this.mono fun x hx => by\n    dsimp only\n    rw [rpow_def_of_pos hx]\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.fst\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.fst\nx : \u211d \u00d7 \u211d\nhx : 0 < x.fst\n\u22a2 (fun x => x.fst ^ x.snd) x = (fun x => exp (log x.fst * x.snd)) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.fst\nthis : \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.fst\nx : \u211d \u00d7 \u211d\nhx : 0 < x.fst\n\u22a2 x.fst ^ x.snd = exp (log x.fst * x.snd)\n[PROOFSTEP]\nrw [rpow_def_of_pos hx]\n[GOAL]\ncase this\np : \u211d \u00d7 \u211d\nhp_fst : 0 < p.fst\n\u22a2 \u2200\u1da0 (x : \u211d \u00d7 \u211d) in \ud835\udcdd p, 0 < x.fst\n[PROOFSTEP]\nexact IsOpen.eventually_mem (isOpen_lt continuous_const continuous_fst) hp_fst\n[GOAL]\np : \u211d \u00d7 \u211d\nhp : p.fst \u2260 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\nrw [ne_iff_lt_or_gt] at hp \n[GOAL]\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0 \u2228 p.fst > 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\ncases hp with\n| inl hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]\n  refine' ContinuousAt.mul _ (continuous_cos.continuousAt.comp _)\n  \u00b7 refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n    refine' (continuousAt_log _).comp continuous_fst.continuousAt\n    exact hp.ne\n  \u00b7 exact continuous_snd.continuousAt.mul continuousAt_const\n| inr hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_pos hp)]\n  refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n  refine' (continuousAt_log _).comp continuous_fst.continuousAt\n  exact hp.lt.ne.symm\n[GOAL]\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0 \u2228 p.fst > 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\ncases hp with\n| inl hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]\n  refine' ContinuousAt.mul _ (continuous_cos.continuousAt.comp _)\n  \u00b7 refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n    refine' (continuousAt_log _).comp continuous_fst.continuousAt\n    exact hp.ne\n  \u00b7 exact continuous_snd.continuousAt.mul continuousAt_const\n| inr hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_pos hp)]\n  refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n  refine' (continuousAt_log _).comp continuous_fst.continuousAt\n  exact hp.lt.ne.symm\n[GOAL]\ncase inl\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\n\n| inl hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]\n  refine' ContinuousAt.mul _ (continuous_cos.continuousAt.comp _)\n  \u00b7 refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n    refine' (continuousAt_log _).comp continuous_fst.continuousAt\n    exact hp.ne\n  \u00b7 exact continuous_snd.continuousAt.mul continuousAt_const\n[GOAL]\ncase inl\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\nrw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]\n[GOAL]\ncase inl\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun x => exp (log x.fst * x.snd) * cos (x.snd * \u03c0)) p\n[PROOFSTEP]\nrefine' ContinuousAt.mul _ (continuous_cos.continuousAt.comp _)\n[GOAL]\ncase inl.refine'_1\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun x => exp (log x.fst * x.snd)) p\n[PROOFSTEP]\nrefine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n[GOAL]\ncase inl.refine'_1\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun x => log x.fst) p\n[PROOFSTEP]\nrefine' (continuousAt_log _).comp continuous_fst.continuousAt\n[GOAL]\ncase inl.refine'_1\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 p.fst \u2260 0\n[PROOFSTEP]\nexact hp.ne\n[GOAL]\ncase inl.refine'_2\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst < 0\n\u22a2 ContinuousAt (fun x => x.snd * \u03c0) p\n[PROOFSTEP]\nexact continuous_snd.continuousAt.mul continuousAt_const\n[GOAL]\ncase inr\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst > 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\n\n| inr hp =>\n  rw [continuousAt_congr (rpow_eq_nhds_of_pos hp)]\n  refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n  refine' (continuousAt_log _).comp continuous_fst.continuousAt\n  exact hp.lt.ne.symm\n[GOAL]\ncase inr\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst > 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\nrw [continuousAt_congr (rpow_eq_nhds_of_pos hp)]\n[GOAL]\ncase inr\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst > 0\n\u22a2 ContinuousAt (fun x => exp (log x.fst * x.snd)) p\n[PROOFSTEP]\nrefine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)\n[GOAL]\ncase inr\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst > 0\n\u22a2 ContinuousAt (fun x => log x.fst) p\n[PROOFSTEP]\nrefine' (continuousAt_log _).comp continuous_fst.continuousAt\n[GOAL]\ncase inr\np : \u211d \u00d7 \u211d\nhp\u271d : p.fst \u2260 0\nhp : p.fst > 0\n\u22a2 p.fst \u2260 0\n[PROOFSTEP]\nexact hp.lt.ne.symm\n[GOAL]\np : \u211d \u00d7 \u211d\nhp : 0 < p.snd\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) p\n[PROOFSTEP]\ncases' p with x y\n[GOAL]\ncase mk\nx y : \u211d\nhp : 0 < (x, y).snd\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\ndsimp only at hp \n[GOAL]\ncase mk\nx y : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nobtain hx | rfl := ne_or_eq x 0\n[GOAL]\ncase mk.inl\nx y : \u211d\nhp : 0 < y\nhx : x \u2260 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nexact continuousAt_rpow_of_ne (x, y) hx\n[GOAL]\ncase mk.inr\ny : \u211d\nhp : 0 < y\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nhave A : Tendsto (fun p : \u211d \u00d7 \u211d => exp (log p.1 * p.2)) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n  tendsto_exp_atBot.comp ((tendsto_log_nhdsWithin_zero.comp tendsto_fst).atBot_mul hp tendsto_snd)\n[GOAL]\ncase mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => exp (log p.fst * p.snd)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nhave B : Tendsto (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[\u2260] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0) :=\n  squeeze_zero_norm (fun p => abs_rpow_le_exp_log_mul p.1 p.2) A\n[GOAL]\ncase mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => exp (log p.fst * p.snd)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nhave C : Tendsto (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0) :=\n  by\n  rw [nhdsWithin_singleton, tendsto_pure, pure_prod, eventually_map]\n  exact (lt_mem_nhds hp).mono fun y hy => zero_rpow hy.ne'\n[GOAL]\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => exp (log p.fst * p.snd)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0)\n[PROOFSTEP]\nrw [nhdsWithin_singleton, tendsto_pure, pure_prod, eventually_map]\n[GOAL]\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => exp (log p.fst * p.snd)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (a : \u211d) in \ud835\udcdd y, (0, a).fst ^ (0, a).snd = 0\n[PROOFSTEP]\nexact (lt_mem_nhds hp).mono fun y hy => zero_rpow hy.ne'\n[GOAL]\ncase mk.inr\ny : \u211d\nhp : 0 < y\nA : Tendsto (fun p => exp (log p.fst * p.snd)) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nB : Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}\u1d9c] 0 \u00d7\u02e2 \ud835\udcdd y) (\ud835\udcdd 0)\nC : Tendsto (fun p => p.fst ^ p.snd) (\ud835\udcdd[{0}] 0 \u00d7\u02e2 \ud835\udcdd y) (pure 0)\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nsimpa only [\u2190 sup_prod, \u2190 nhdsWithin_union, compl_union_self, nhdsWithin_univ, nhds_prod_eq, ContinuousAt,\n  zero_rpow hp.ne'] using B.sup (C.mono_right (pure_le_nhds _))\n[GOAL]\nx q : \u211d\nh : x \u2260 0 \u2228 0 < q\n\u22a2 ContinuousAt (fun x => x ^ q) x\n[PROOFSTEP]\nchange ContinuousAt ((fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) \u2218 fun y : \u211d => (y, q)) x\n[GOAL]\nx q : \u211d\nh : x \u2260 0 \u2228 0 < q\n\u22a2 ContinuousAt ((fun p => p.fst ^ p.snd) \u2218 fun y => (y, q)) x\n[PROOFSTEP]\napply ContinuousAt.comp\n[GOAL]\ncase hg\nx q : \u211d\nh : x \u2260 0 \u2228 0 < q\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, q)\n[PROOFSTEP]\nexact continuousAt_rpow (x, q) h\n[GOAL]\ncase hf\nx q : \u211d\nh : x \u2260 0 \u2228 0 < q\n\u22a2 ContinuousAt (fun y => (y, q)) x\n[PROOFSTEP]\nexact (continuous_id'.prod_mk continuous_const).continuousAt\n[GOAL]\n\u03b1 : Type u_1\nl : Filter \u03b1\nf : \u03b1 \u2192 \u211d\nx p : \u211d\nhf : Tendsto f l (\ud835\udcdd x)\nh : x \u2260 0 \u2228 0 \u2264 p\nh0 : 0 = p\n\u22a2 Tendsto (fun a => f a ^ 0) l (\ud835\udcdd (x ^ 0))\n[PROOFSTEP]\nsimp [tendsto_const_nhds]\n[GOAL]\nz : \u2102\nhz : 0 < z.re\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (0, z)\n[PROOFSTEP]\nhave hz\u2080 : z \u2260 0 := ne_of_apply_ne re hz.ne'\n[GOAL]\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (0, z)\n[PROOFSTEP]\nrw [ContinuousAt, zero_cpow hz\u2080, tendsto_zero_iff_norm_tendsto_zero]\n[GOAL]\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 Tendsto (fun e => \u2016e.fst ^ e.snd\u2016) (\ud835\udcdd (0, z)) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero (fun _ => norm_nonneg _) (fun _ => abs_cpow_le _ _) _\n[GOAL]\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 Tendsto (fun x => \u2191abs x.fst ^ x.snd.re / Real.exp (arg x.fst * x.snd.im)) (\ud835\udcdd (0, z)) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, \u2190 Real.exp_neg]\n[GOAL]\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 Tendsto (fun x => \u2191abs x.fst ^ x.snd.re * Real.exp (-(arg x.fst * x.snd.im))) (\ud835\udcdd (0, z)) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' Tendsto.zero_mul_isBoundedUnder_le _ _\n[GOAL]\ncase refine'_1\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 Tendsto (fun x => \u2191abs x.fst ^ x.snd.re) (\ud835\udcdd (0, z)) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert (continuous_fst.norm.tendsto ((0 : \u2102), z)).rpow ((continuous_re.comp continuous_snd).tendsto _) _\n[GOAL]\ncase h.e'_5.h.e'_3\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 0 = \u2016(0, z).fst\u2016 ^ (re \u2218 Prod.snd) (0, z)\n[PROOFSTEP]\nsimp [hz, Real.zero_rpow hz.ne']\n[GOAL]\ncase refine'_1\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 \u2016(0, z).fst\u2016 \u2260 0 \u2228 0 < (re \u2218 Prod.snd) (0, z)\n[PROOFSTEP]\nsimp [hz, Real.zero_rpow hz.ne']\n[GOAL]\ncase refine'_2\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd (0, z)) ((fun x => \u2016x\u2016) \u2218 fun x => Real.exp (-(arg x.fst * x.snd.im)))\n[PROOFSTEP]\nsimp only [Function.comp, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]\n[GOAL]\ncase refine'_2\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd (0, z)) fun x => Real.exp (-(arg x.fst * x.snd.im))\n[PROOFSTEP]\nrcases exists_gt |im z| with \u27e8C, hC\u27e9\n[GOAL]\ncase refine'_2.intro\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\nC : \u211d\nhC : |z.im| < C\n\u22a2 IsBoundedUnder (fun x x_1 => x \u2264 x_1) (\ud835\udcdd (0, z)) fun x => Real.exp (-(arg x.fst * x.snd.im))\n[PROOFSTEP]\nrefine' \u27e8Real.exp (\u03c0 * C), eventually_map.2 _\u27e9\n[GOAL]\ncase refine'_2.intro\nz : \u2102\nhz : 0 < z.re\nhz\u2080 : z \u2260 0\nC : \u211d\nhC : |z.im| < C\n\u22a2 \u2200\u1da0 (a : \u2102 \u00d7 \u2102) in \ud835\udcdd (0, z), (fun x x_1 => x \u2264 x_1) (Real.exp (-(arg a.fst * a.snd.im))) (Real.exp (\u03c0 * C))\n[PROOFSTEP]\nrefine'\n  (((continuous_im.comp continuous_snd).abs.tendsto (_, z)).eventually (gt_mem_nhds hC)).mono fun z hz =>\n    Real.exp_le_exp.2 <| (neg_le_abs_self _).trans _\n[GOAL]\ncase refine'_2.intro\nz\u271d : \u2102\nhz\u271d : 0 < z\u271d.re\nhz\u2080 : z\u271d \u2260 0\nC : \u211d\nhC : |z\u271d.im| < C\nz : \u2102 \u00d7 \u2102\nhz : |(im \u2218 Prod.snd) z| < C\n\u22a2 |arg z.fst * z.snd.im| \u2264 \u03c0 * C\n[PROOFSTEP]\nrw [_root_.abs_mul]\n[GOAL]\ncase refine'_2.intro\nz\u271d : \u2102\nhz\u271d : 0 < z\u271d.re\nhz\u2080 : z\u271d \u2260 0\nC : \u211d\nhC : |z\u271d.im| < C\nz : \u2102 \u00d7 \u2102\nhz : |(im \u2218 Prod.snd) z| < C\n\u22a2 |arg z.fst| * |z.snd.im| \u2264 \u03c0 * C\n[PROOFSTEP]\nexact mul_le_mul (abs_le.2 \u27e8(neg_pi_lt_arg _).le, arg_le_pi _\u27e9) hz.le (_root_.abs_nonneg _) Real.pi_pos.le\n[GOAL]\np : \u2102 \u00d7 \u2102\nh\u2081 : 0 \u2264 p.fst.re \u2228 p.fst.im \u2260 0\nh\u2082 : 0 < p.snd.re\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) p\n[PROOFSTEP]\ncases' p with z w\n[GOAL]\ncase mk\nz w : \u2102\nh\u2081 : 0 \u2264 (z, w).fst.re \u2228 (z, w).fst.im \u2260 0\nh\u2082 : 0 < (z, w).snd.re\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (z, w)\n[PROOFSTEP]\nrw [\u2190 not_lt_zero_iff, lt_iff_le_and_ne, not_and_or, Ne.def, Classical.not_not, not_le_zero_iff] at h\u2081 \n[GOAL]\ncase mk\nz w : \u2102\nh\u2081 : (0 < (z, w).fst.re \u2228 (z, w).fst.im \u2260 0) \u2228 (z, w).fst = 0\nh\u2082 : 0 < (z, w).snd.re\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (z, w)\n[PROOFSTEP]\nrcases h\u2081 with (h\u2081 | (rfl : z = 0))\n[GOAL]\ncase mk.inl\nz w : \u2102\nh\u2082 : 0 < (z, w).snd.re\nh\u2081 : 0 < (z, w).fst.re \u2228 (z, w).fst.im \u2260 0\n\u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (z, w)\ncase mk.inr w : \u2102 h\u2082 : 0 < (0, w).snd.re \u22a2 ContinuousAt (fun x => x.fst ^ x.snd) (0, w)\n[PROOFSTEP]\nexacts [continuousAt_cpow h\u2081, continuousAt_cpow_zero_of_re_pos h\u2082]\n[GOAL]\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nrcases lt_trichotomy (0 : \u211d) x with (hx | rfl | hx)\n[GOAL]\ncase inl\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : 0 < x\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nhave : ContinuousAt (fun p => \u27e8\u2191p.1, p.2\u27e9 : \u211d \u00d7 \u2102 \u2192 \u2102 \u00d7 \u2102) (x, y) :=\n  continuous_ofReal.continuousAt.prod_map continuousAt_id\n[GOAL]\ncase inl\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : 0 < x\nthis : ContinuousAt (fun p => (\u2191p.fst, p.snd)) (x, y)\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nrefine' (continuousAt_cpow (Or.inl _)).comp this\n[GOAL]\ncase inl\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : 0 < x\nthis : ContinuousAt (fun p => (\u2191p.fst, p.snd)) (x, y)\n\u22a2 0 < (\u2191(x, y).fst, (x, y).snd).fst.re\n[PROOFSTEP]\nrwa [ofReal_re]\n[GOAL]\ncase inr.inl\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nhave A : ContinuousAt (fun p => p.1 ^ p.2 : \u2102 \u00d7 \u2102 \u2192 \u2102) \u27e8\u2191(0 : \u211d), y\u27e9 :=\n  by\n  rw [ofReal_zero]\n  apply continuousAt_cpow_zero_of_re_pos\n  tauto\n[GOAL]\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (\u21910, y)\n[PROOFSTEP]\nrw [ofReal_zero]\n[GOAL]\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\napply continuousAt_cpow_zero_of_re_pos\n[GOAL]\ncase hz\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\n\u22a2 0 < y.re\n[PROOFSTEP]\ntauto\n[GOAL]\ncase inr.inl\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\nA : ContinuousAt (fun p => p.fst ^ p.snd) (\u21910, y)\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nhave B : ContinuousAt (fun p => \u27e8\u2191p.1, p.2\u27e9 : \u211d \u00d7 \u2102 \u2192 \u2102 \u00d7 \u2102) \u27e80, y\u27e9 :=\n  continuous_ofReal.continuousAt.prod_map continuousAt_id\n[GOAL]\ncase inr.inl\ny : \u2102\nh : 0 < y.re \u2228 0 \u2260 0\nA : ContinuousAt (fun p => p.fst ^ p.snd) (\u21910, y)\nB : ContinuousAt (fun p => (\u2191p.fst, p.snd)) (0, y)\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (0, y)\n[PROOFSTEP]\nexact @ContinuousAt.comp (\u211d \u00d7 \u2102) (\u2102 \u00d7 \u2102) \u2102 _ _ _ _ (fun p => \u27e8\u2191p.1, p.2\u27e9) \u27e80, y\u27e9 A B\n[GOAL]\ncase inr.inr\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nsuffices ContinuousAt (fun p => (-(p.1 : \u2102)) ^ p.2 * exp (\u03c0 * I * p.2) : \u211d \u00d7 \u2102 \u2192 \u2102) (x, y)\n  by\n  refine' this.congr (eventually_of_mem (prod_mem_nhds (Iio_mem_nhds hx) univ_mem) _)\n  exact fun p hp => (ofReal_cpow_of_nonpos (le_of_lt hp.1) p.2).symm\n[GOAL]\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nthis : ContinuousAt (fun p => (-\u2191p.fst) ^ p.snd * exp (\u2191\u03c0 * I * p.snd)) (x, y)\n\u22a2 ContinuousAt (fun p => \u2191p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nrefine' this.congr (eventually_of_mem (prod_mem_nhds (Iio_mem_nhds hx) univ_mem) _)\n[GOAL]\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nthis : ContinuousAt (fun p => (-\u2191p.fst) ^ p.snd * exp (\u2191\u03c0 * I * p.snd)) (x, y)\n\u22a2 \u2200 (x : \u211d \u00d7 \u2102),\n    x \u2208 Set.Iio 0 \u00d7\u02e2 Set.univ \u2192 (fun p => (-\u2191p.fst) ^ p.snd * exp (\u2191\u03c0 * I * p.snd)) x = (fun p => \u2191p.fst ^ p.snd) x\n[PROOFSTEP]\nexact fun p hp => (ofReal_cpow_of_nonpos (le_of_lt hp.1) p.2).symm\n[GOAL]\ncase inr.inr\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\n\u22a2 ContinuousAt (fun p => (-\u2191p.fst) ^ p.snd * exp (\u2191\u03c0 * I * p.snd)) (x, y)\n[PROOFSTEP]\nhave A : ContinuousAt (fun p => \u27e8-\u2191p.1, p.2\u27e9 : \u211d \u00d7 \u2102 \u2192 \u2102 \u00d7 \u2102) (x, y) :=\n  ContinuousAt.prod_map continuous_ofReal.continuousAt.neg continuousAt_id\n[GOAL]\ncase inr.inr\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nA : ContinuousAt (fun p => (-\u2191p.fst, p.snd)) (x, y)\n\u22a2 ContinuousAt (fun p => (-\u2191p.fst) ^ p.snd * exp (\u2191\u03c0 * I * p.snd)) (x, y)\n[PROOFSTEP]\napply ContinuousAt.mul\n[GOAL]\ncase inr.inr.hf\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nA : ContinuousAt (fun p => (-\u2191p.fst, p.snd)) (x, y)\n\u22a2 ContinuousAt (fun x => (-\u2191x.fst) ^ x.snd) (x, y)\n[PROOFSTEP]\nrefine' (continuousAt_cpow (Or.inl _)).comp A\n[GOAL]\ncase inr.inr.hf\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nA : ContinuousAt (fun p => (-\u2191p.fst, p.snd)) (x, y)\n\u22a2 0 < (-\u2191(x, y).fst, (x, y).snd).fst.re\n[PROOFSTEP]\nrwa [neg_re, ofReal_re, neg_pos]\n[GOAL]\ncase inr.inr.hg\nx : \u211d\ny : \u2102\nh : 0 < y.re \u2228 x \u2260 0\nhx : x < 0\nA : ContinuousAt (fun p => (-\u2191p.fst, p.snd)) (x, y)\n\u22a2 ContinuousAt (fun x => exp (\u2191\u03c0 * I * x.snd)) (x, y)\n[PROOFSTEP]\nexact (continuous_exp.comp (continuous_const.mul continuous_snd)).continuousAt\n[GOAL]\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nhave :\n  (fun p : \u211d\u22650 \u00d7 \u211d => p.1 ^ p.2) = Real.toNNReal \u2218 (fun p : \u211d \u00d7 \u211d => p.1 ^ p.2) \u2218 fun p : \u211d\u22650 \u00d7 \u211d => (p.1.1, p.2) :=\n  by\n  ext p\n  erw [coe_rpow, Real.coe_toNNReal _ (Real.rpow_nonneg_of_nonneg p.1.2 _)]\n  rfl\n[GOAL]\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\n\u22a2 (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n[PROOFSTEP]\next p\n[GOAL]\ncase h.a\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\np : \u211d\u22650 \u00d7 \u211d\n\u22a2 \u2191(p.fst ^ p.snd) = \u2191((toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)) p)\n[PROOFSTEP]\nerw [coe_rpow, Real.coe_toNNReal _ (Real.rpow_nonneg_of_nonneg p.1.2 _)]\n[GOAL]\ncase h.a\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\np : \u211d\u22650 \u00d7 \u211d\n\u22a2 \u2191p.fst ^ p.snd = \u2191p.fst ^ ((fun p => (\u2191p.fst, p.snd)) p).snd\n[PROOFSTEP]\nrfl\n[GOAL]\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (x, y)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 ContinuousAt (toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)) (x, y)\n[PROOFSTEP]\nrefine' continuous_real_toNNReal.continuousAt.comp (ContinuousAt.comp _ _)\n[GOAL]\ncase refine'_1\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 ContinuousAt (fun p => p.fst ^ p.snd) (\u2191(x, y).fst, (x, y).snd)\n[PROOFSTEP]\napply Real.continuousAt_rpow\n[GOAL]\ncase refine'_1.h\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 (\u2191(x, y).fst, (x, y).snd).fst \u2260 0 \u2228 0 < (\u2191(x, y).fst, (x, y).snd).snd\n[PROOFSTEP]\nsimp only [Ne.def] at h \n[GOAL]\ncase refine'_1.h\nx : \u211d\u22650\ny : \u211d\nh : \u00acx = 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 (\u2191(x, y).fst, (x, y).snd).fst \u2260 0 \u2228 0 < (\u2191(x, y).fst, (x, y).snd).snd\n[PROOFSTEP]\nrw [\u2190 NNReal.coe_eq_zero x] at h \n[GOAL]\ncase refine'_1.h\nx : \u211d\u22650\ny : \u211d\nh : \u00ac\u2191x = 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 (\u2191(x, y).fst, (x, y).snd).fst \u2260 0 \u2228 0 < (\u2191(x, y).fst, (x, y).snd).snd\n[PROOFSTEP]\nexact h\n[GOAL]\ncase refine'_2\nx : \u211d\u22650\ny : \u211d\nh : x \u2260 0 \u2228 0 < y\nthis : (fun p => p.fst ^ p.snd) = toNNReal \u2218 (fun p => p.fst ^ p.snd) \u2218 fun p => (\u2191p.fst, p.snd)\n\u22a2 ContinuousAt (fun p => (\u2191p.fst, p.snd)) (x, y)\n[PROOFSTEP]\nexact ((continuous_subtype_val.comp continuous_fst).prod_mk continuous_snd).continuousAt\n[GOAL]\nx y : \u211d\u22650\nhy : 1 < y\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nobtain \u27e8m, hm\u27e9 := add_one_pow_unbounded_of_pos x (tsub_pos_of_lt hy)\n[GOAL]\ncase intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < (y - 1 + 1) ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le hy.le] at hm \n[GOAL]\ncase intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nrefine' eventually_atTop.2 \u27e8m + 1, fun n hn => _\u27e9\n[GOAL]\ncase intro\nx y : \u211d\u22650\nhy : 1 < y\nm : \u2115\nhm : x < y ^ m\nn : \u2115\nhn : n \u2265 m + 1\n\u22a2 x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nsimpa only [NNReal.rpow_one_div_le_iff (Nat.cast_pos.2 <| m.succ_pos.trans_le hn), NNReal.rpow_nat_cast] using\n  hm.le.trans (pow_le_pow hy.le (m.le_succ.trans hn))\n[GOAL]\nx : \u211d\u22650\ny : \u211d\nh\u271d\u00b9 : x \u2260 0 \u2228 0 \u2264 y\nh\u271d : 0 \u2264 y\nh : 0 = y\n\u22a2 ContinuousAt (fun z => z ^ 0) x\n[PROOFSTEP]\nsimp only [rpow_zero, continuousAt_const]\n[GOAL]\nx : \u211d\u22650\u221e\nhx : x \u2260 \u22a4\ny : \u211d\u22650\u221e\nhy : 1 < y\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nlift x to \u211d\u22650 using hx\n[GOAL]\ncase intro\ny : \u211d\u22650\u221e\nhy : 1 < y\nx : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nby_cases y = \u221e\n[GOAL]\ncase intro\ny : \u211d\u22650\u221e\nhy : 1 < y\nx : \u211d\u22650\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nby_cases y = \u221e\n[GOAL]\ncase pos\ny : \u211d\u22650\u221e\nhy : 1 < y\nx : \u211d\u22650\nh : y = \u22a4\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nexact eventually_of_forall fun n => h.symm \u25b8 le_top\n[GOAL]\ncase neg\ny : \u211d\u22650\u221e\nhy : 1 < y\nx : \u211d\u22650\nh : \u00acy = \u22a4\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 y\n[PROOFSTEP]\nlift y to \u211d\u22650 using h\n[GOAL]\ncase neg.intro\nx y : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 \u2191y\n[PROOFSTEP]\nhave := NNReal.eventually_pow_one_div_le x (by exact_mod_cast hy : 1 < y)\n[GOAL]\nx y : \u211d\u22650\nhy : 1 < \u2191y\n\u22a2 1 < y\n[PROOFSTEP]\nexact_mod_cast hy\n[GOAL]\ncase neg.intro\nx y : \u211d\u22650\nhy : 1 < \u2191y\nthis : \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, \u2191x ^ (1 / \u2191n) \u2264 \u2191y\n[PROOFSTEP]\nrefine' this.congr (eventually_of_forall fun n => _)\n[GOAL]\ncase neg.intro\nx y : \u211d\u22650\nhy : 1 < \u2191y\nthis : \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\nn : \u2115\n\u22a2 x ^ (1 / \u2191n) \u2264 y \u2194 \u2191x ^ (1 / \u2191n) \u2264 \u2191y\n[PROOFSTEP]\nrw [coe_rpow_of_nonneg x (by positivity : 0 \u2264 (1 / n : \u211d)), coe_le_coe]\n[GOAL]\nx y : \u211d\u22650\nhy : 1 < \u2191y\nthis : \u2200\u1da0 (n : \u2115) in atTop, x ^ (1 / \u2191n) \u2264 y\nn : \u2115\n\u22a2 0 \u2264 1 / \u2191n\n[PROOFSTEP]\npositivity\n[GOAL]\nx : \u211d\u22650\u221e\ny : \u211d\nh : 0 < y\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nby_cases hx : x = \u22a4\n[GOAL]\ncase pos\nx : \u211d\u22650\u221e\ny : \u211d\nh : 0 < y\nhx : x = \u22a4\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nrw [hx, ContinuousAt]\n[GOAL]\ncase pos\nx : \u211d\u22650\u221e\ny : \u211d\nh : 0 < y\nhx : x = \u22a4\n\u22a2 Tendsto (fun a => a ^ y) (\ud835\udcdd \u22a4) (\ud835\udcdd (\u22a4 ^ y))\n[PROOFSTEP]\nconvert ENNReal.tendsto_rpow_at_top h\n[GOAL]\ncase h.e'_5.h.e'_3\nx : \u211d\u22650\u221e\ny : \u211d\nh : 0 < y\nhx : x = \u22a4\n\u22a2 \u22a4 ^ y = \u22a4\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nx : \u211d\u22650\u221e\ny : \u211d\nh : 0 < y\nhx : \u00acx = \u22a4\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nlift x to \u211d\u22650 using hx\n[GOAL]\ncase neg.intro\ny : \u211d\nh : 0 < y\nx : \u211d\u22650\n\u22a2 ContinuousAt (fun a => a ^ y) \u2191x\n[PROOFSTEP]\nrw [continuousAt_coe_iff]\n[GOAL]\ncase neg.intro\ny : \u211d\nh : 0 < y\nx : \u211d\u22650\n\u22a2 ContinuousAt ((fun a => a ^ y) \u2218 some) x\n[PROOFSTEP]\nconvert continuous_coe.continuousAt.comp (NNReal.continuousAt_rpow_const (Or.inr h.le)) using 1\n[GOAL]\ncase h.e'_5\ny : \u211d\nh : 0 < y\nx : \u211d\u22650\n\u22a2 (fun a => a ^ y) \u2218 some = some \u2218 fun z => z ^ y\n[PROOFSTEP]\next1 x\n[GOAL]\ncase h.e'_5.h\ny : \u211d\nh : 0 < y\nx\u271d x : \u211d\u22650\n\u22a2 ((fun a => a ^ y) \u2218 some) x = (some \u2218 fun z => z ^ y) x\n[PROOFSTEP]\nsimp [coe_rpow_of_nonneg _ h.le]\n[GOAL]\ny : \u211d\n\u22a2 Continuous fun a => a ^ y\n[PROOFSTEP]\nrefine continuous_iff_continuousAt.2 fun x => ?_\n[GOAL]\ny : \u211d\nx : \u211d\u22650\u221e\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nrcases lt_trichotomy (0 : \u211d) y with (hy | rfl | hy)\n[GOAL]\ncase inl\ny : \u211d\nx : \u211d\u22650\u221e\nhy : 0 < y\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nexact continuousAt_rpow_const_of_pos hy\n[GOAL]\ncase inr.inl\nx : \u211d\u22650\u221e\n\u22a2 ContinuousAt (fun a => a ^ 0) x\n[PROOFSTEP]\nsimp only [rpow_zero]\n[GOAL]\ncase inr.inl\nx : \u211d\u22650\u221e\n\u22a2 ContinuousAt (fun a => 1) x\n[PROOFSTEP]\nexact continuousAt_const\n[GOAL]\ncase inr.inr\ny : \u211d\nx : \u211d\u22650\u221e\nhy : y < 0\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nobtain \u27e8z, hz\u27e9 : \u2203 z, y = -z := \u27e8-y, (neg_neg _).symm\u27e9\n[GOAL]\ncase inr.inr.intro\ny : \u211d\nx : \u211d\u22650\u221e\nhy : y < 0\nz : \u211d\nhz : y = -z\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nhave z_pos : 0 < z := by simpa [hz] using hy\n[GOAL]\ny : \u211d\nx : \u211d\u22650\u221e\nhy : y < 0\nz : \u211d\nhz : y = -z\n\u22a2 0 < z\n[PROOFSTEP]\nsimpa [hz] using hy\n[GOAL]\ncase inr.inr.intro\ny : \u211d\nx : \u211d\u22650\u221e\nhy : y < 0\nz : \u211d\nhz : y = -z\nz_pos : 0 < z\n\u22a2 ContinuousAt (fun a => a ^ y) x\n[PROOFSTEP]\nsimp_rw [hz, rpow_neg]\n[GOAL]\ncase inr.inr.intro\ny : \u211d\nx : \u211d\u22650\u221e\nhy : y < 0\nz : \u211d\nhz : y = -z\nz_pos : 0 < z\n\u22a2 ContinuousAt (fun a => (a ^ z)\u207b\u00b9) x\n[PROOFSTEP]\nexact continuous_inv.continuousAt.comp (continuousAt_rpow_const_of_pos z_pos)\n[GOAL]\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ny : \u211d\nhy : 0 < y\n\u22a2 Tendsto (fun x => c * x ^ y) (\ud835\udcdd 0) (\ud835\udcdd 0)\n[PROOFSTEP]\nconvert ENNReal.Tendsto.const_mul (ENNReal.continuous_rpow_const.tendsto 0) _\n[GOAL]\ncase h.e'_5.h.e'_3\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ny : \u211d\nhy : 0 < y\n\u22a2 0 = c * 0 ^ y\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase convert_3\nc : \u211d\u22650\u221e\nhc : c \u2260 \u22a4\ny : \u211d\nhy : 0 < y\n\u22a2 0 ^ y \u2260 0 \u2228 c \u2260 \u22a4\n[PROOFSTEP]\nexact Or.inr hc\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Pow.Continuity", "llama_tokens": 19808, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117940706734, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5085003845447704}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 IsIntegrallyClosed R \u2194 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nlet e : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\n\u22a2 IsIntegrallyClosed R \u2194 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\n\u22a2 IsIntegrallyClosed R \u2192 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nrintro \u27e8cl\u27e9\n[GOAL]\ncase mp.mk\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : FractionRing R}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = x\n\u22a2 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nrefine' fun hx => _\n[GOAL]\ncase mp.mk\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : FractionRing R}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = x\nx\u271d : K\nhx : IsIntegral R x\u271d\n\u22a2 \u2203 y, \u2191(algebraMap R K) y = x\u271d\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := cl ((isIntegral_algEquiv e).mpr hx)\n[GOAL]\ncase mp.mk.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : FractionRing R}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = x\nx\u271d : K\nhx : IsIntegral R x\u271d\ny : R\nhy : \u2191(algebraMap R (FractionRing R)) y = \u2191e x\u271d\n\u22a2 \u2203 y, \u2191(algebraMap R K) y = x\u271d\n[PROOFSTEP]\nexact \u27e8y, e.algebraMap_eq_apply.mp hy\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\n\u22a2 (\u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x) \u2192 IsIntegrallyClosed R\n[PROOFSTEP]\nrintro cl\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n\u22a2 IsIntegrallyClosed R\n[PROOFSTEP]\nrefine' \u27e8fun hx => _\u27e9\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\nx\u271d : FractionRing R\nhx : IsIntegral R x\u271d\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = x\u271d\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := cl ((isIntegral_algEquiv e.symm).mpr hx)\n[GOAL]\ncase mpr.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ne : K \u2243\u2090[R] FractionRing R := IsLocalization.algEquiv R\u2070 K (FractionRing R)\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\nx\u271d : FractionRing R\nhx : IsIntegral R x\u271d\ny : R\nhy : \u2191(algebraMap R K) y = \u2191(AlgEquiv.symm e) x\u271d\n\u22a2 \u2203 y, \u2191(algebraMap R (FractionRing R)) y = x\u271d\n[PROOFSTEP]\nexact \u27e8y, e.symm.algebraMap_eq_apply.mp hy\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 (\u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x) \u2194 IsIntegralClosure R R K\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 (\u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x) \u2192 IsIntegralClosure R R K\n[PROOFSTEP]\nintro cl\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n\u22a2 IsIntegralClosure R R K\n[PROOFSTEP]\nrefine' \u27e8IsFractionRing.injective R K, \u27e8cl, _\u27e9\u27e9\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\nx\u271d : K\n\u22a2 (\u2203 y, \u2191(algebraMap R K) y = x\u271d) \u2192 IsIntegral R x\u271d\n[PROOFSTEP]\nrintro \u27e8y, y_eq\u27e9\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\nx\u271d : K\ny : R\ny_eq : \u2191(algebraMap R K) y = x\u271d\n\u22a2 IsIntegral R x\u271d\n[PROOFSTEP]\nrw [\u2190 y_eq]\n[GOAL]\ncase mp.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ncl : \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\nx\u271d : K\ny : R\ny_eq : \u2191(algebraMap R K) y = x\u271d\n\u22a2 IsIntegral R (\u2191(algebraMap R K) y)\n[PROOFSTEP]\nexact isIntegral_algebraMap\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\n\u22a2 IsIntegralClosure R R K \u2192 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nrintro \u27e8-, cl\u27e9 x hx\n[GOAL]\ncase mpr.mk\nR : Type u_1\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : IsDomain R\nK : Type u_2\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra R K\ninst\u271d : IsFractionRing R K\ncl : \u2200 {x : K}, IsIntegral R x \u2194 \u2203 y, \u2191(algebraMap R K) y = x\nx : K\nhx : IsIntegral R x\n\u22a2 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nexact cl.mp hx\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\n\u22a2 integralClosure R K = \u22a5 \u2194 IsIntegrallyClosed R\n[PROOFSTEP]\nrefine' eq_bot_iff.trans _\n[GOAL]\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\n\u22a2 integralClosure R K \u2264 \u22a5 \u2194 IsIntegrallyClosed R\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\n\u22a2 integralClosure R K \u2264 \u22a5 \u2192 IsIntegrallyClosed R\n[PROOFSTEP]\nrw [isIntegrallyClosed_iff K]\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\n\u22a2 integralClosure R K \u2264 \u22a5 \u2192 \u2200 {x : K}, IsIntegral R x \u2192 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\nh : integralClosure R K \u2264 \u22a5\nx : K\nhx : IsIntegral R x\n\u22a2 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nexact Set.mem_range.mp (Algebra.mem_bot.mp (h hx))\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\n\u22a2 IsIntegrallyClosed R \u2192 integralClosure R K \u2264 \u22a5\n[PROOFSTEP]\nintro h x hx\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\nh : IsIntegrallyClosed R\nx : K\nhx : x \u2208 integralClosure R K\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nrw [Algebra.mem_bot, Set.mem_range]\n[GOAL]\ncase mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\nid : IsDomain R\niic : IsIntegrallyClosed R\nK : Type u_2\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra R K\nifr : IsFractionRing R K\nh : IsIntegrallyClosed R\nx : K\nhx : x \u2208 integralClosure R K\n\u22a2 \u2203 y, \u2191(algebraMap R K) y = x\n[PROOFSTEP]\nexact isIntegral_iff.mp hx\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.IntegrallyClosed", "llama_tokens": 4150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.5081952718453935}}
{"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) (\u03c3 X) (\u03c3 Y) (PushQuiver.arrow f)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 \u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)\n[PROOFSTEP]\nrw [\u2190 h X, \u2190 h Y]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 \u03c6.obj X \u27f6 \u03c6.obj Y\n[PROOFSTEP]\nexact \u03c6.map f\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u22a2 of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h = \u03c6\n[PROOFSTEP]\nfapply Prefunctor.ext\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u22a2 \u2200 (X : V), (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X = \u03c6.obj X\n[PROOFSTEP]\nrintro X\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX : V\n\u22a2 (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X = \u03c6.obj X\n[PROOFSTEP]\nsimp only [Prefunctor.comp_obj]\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX : V\n\u22a2 (lift \u03c3 \u03c6 \u03c4 h).obj ((of \u03c3).obj X) = \u03c6.obj X\n[PROOFSTEP]\napply Eq.symm\n[GOAL]\ncase h_obj.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX : V\n\u22a2 \u03c6.obj X = (lift \u03c3 \u03c6 \u03c4 h).obj ((of \u03c3).obj X)\n[PROOFSTEP]\nexact h X\n[GOAL]\ncase h_map\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u22a2 \u2200 (X Y : V) (f : X \u27f6 Y),\n    (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).map f =\n      Eq.recOn (_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y)\n        (Eq.recOn (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) (\u03c6.map f))\n[PROOFSTEP]\nrintro X Y f\n[GOAL]\ncase h_map\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).map f =\n    Eq.recOn (_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y)\n      (Eq.recOn (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) (\u03c6.map f))\n[PROOFSTEP]\nsimp only [Prefunctor.comp_map]\n[GOAL]\ncase h_map\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 (lift \u03c3 \u03c6 \u03c4 h).map ((of \u03c3).map f) =\n    (_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f\n[PROOFSTEP]\napply eq_of_heq\n[GOAL]\ncase h_map.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq ((lift \u03c3 \u03c6 \u03c4 h).map ((of \u03c3).map f))\n    ((_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f)\n[PROOFSTEP]\niterate 2 apply (cast_heq _ _).trans\n[GOAL]\ncase h_map.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq ((lift \u03c3 \u03c6 \u03c4 h).map ((of \u03c3).map f))\n    ((_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f)\n[PROOFSTEP]\napply (cast_heq _ _).trans\n[GOAL]\ncase h_map.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))\n    ((_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f)\n[PROOFSTEP]\napply (cast_heq _ _).trans\n[GOAL]\ncase h_map.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq (\u03c6.map f) ((_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f)\n[PROOFSTEP]\napply HEq.symm\n[GOAL]\ncase h_map.h.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq ((_ : \u03c6.obj Y = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj Y) \u25b8 (_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f) (\u03c6.map f)\n[PROOFSTEP]\napply (eqRec_heq _ _).trans\n[GOAL]\ncase h_map.h.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 HEq ((_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f) (\u03c6.map f)\n[PROOFSTEP]\nhave : \u2200 {\u03b1 \u03b3} {\u03b2 : \u03b1 \u2192 \u03b3 \u2192 Sort _} {a a'} (p : a = a') g (b : \u03b2 a g), HEq (p \u25b8 b) b :=\n  by\n  intros\n  subst_vars\n  rfl\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u22a2 \u2200 {\u03b1 : Sort ?u.2529} {\u03b3 : Sort ?u.2531} {\u03b2 : \u03b1 \u2192 \u03b3 \u2192 Sort ?u.2533} {a a' : \u03b1} (p : a = a') (g : \u03b3) (b : \u03b2 a g),\n    HEq (p \u25b8 b) b\n[PROOFSTEP]\nintros\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u03b1\u271d : Sort ?u.2529\n\u03b3\u271d : Sort ?u.2531\n\u03b2\u271d : \u03b1\u271d \u2192 \u03b3\u271d \u2192 Sort ?u.2533\na\u271d a'\u271d : \u03b1\u271d\np\u271d : a\u271d = a'\u271d\ng\u271d : \u03b3\u271d\nb\u271d : \u03b2\u271d a\u271d g\u271d\n\u22a2 HEq (p\u271d \u25b8 b\u271d) b\u271d\n[PROOFSTEP]\nsubst_vars\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\n\u03b1\u271d : Sort ?u.2529\n\u03b3\u271d : Sort ?u.2531\n\u03b2\u271d : \u03b1\u271d \u2192 \u03b3\u271d \u2192 Sort ?u.2533\na'\u271d : \u03b1\u271d\ng\u271d : \u03b3\u271d\nb\u271d : \u03b2\u271d a'\u271d g\u271d\n\u22a2 HEq ((_ : a'\u271d = a'\u271d) \u25b8 b\u271d) b\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h_map.h.h\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\nX Y : V\nf : X \u27f6 Y\nthis :\n  \u2200 {\u03b1 : Sort ?u.2529} {\u03b3 : Sort ?u.2531} {\u03b2 : \u03b1 \u2192 \u03b3 \u2192 Sort ?u.2533} {a a' : \u03b1} (p : a = a') (g : \u03b3) (b : \u03b2 a g),\n    HEq (p \u25b8 b) b\n\u22a2 HEq ((_ : \u03c6.obj X = (of \u03c3 \u22d9q lift \u03c3 \u03c6 \u03c4 h).obj X) \u25b8 \u03c6.map f) (\u03c6.map f)\n[PROOFSTEP]\napply this\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\n\u22a2 \u03a6 = lift \u03c3 \u03c6 \u03c4 h\n[PROOFSTEP]\ndsimp only [of, lift]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\n\u22a2 \u03a6 =\n    { obj := \u03c4,\n      map :=\n        @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n          id\n            (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n              (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }\n[PROOFSTEP]\nfapply Prefunctor.ext\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\n\u22a2 \u2200 (X : Push \u03c3),\n    \u03a6.obj X =\n      { obj := \u03c4,\n            map :=\n              @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                id\n                  (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                    (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n        X\n[PROOFSTEP]\nintro X\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\nX : Push \u03c3\n\u22a2 \u03a6.obj X =\n    { obj := \u03c4,\n          map :=\n            @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n              id\n                (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                  (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n      X\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h_obj\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\nX : Push \u03c3\n\u22a2 \u03a6.obj X = \u03c4 X\n[PROOFSTEP]\nrw [\u03a6\u2080]\n[GOAL]\ncase h_map\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\n\u22a2 \u2200 (X Y : Push \u03c3) (f : X \u27f6 Y),\n    \u03a6.map f =\n      Eq.recOn\n        (_ :\n          { obj := \u03c4,\n                  map :=\n                    @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                      id\n                        (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                          (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n              Y =\n            \u03a6.obj Y)\n        (Eq.recOn\n          (_ :\n            { obj := \u03c4,\n                    map :=\n                      @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                        id\n                          (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                            (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n                X =\n              \u03a6.obj X)\n          ({ obj := \u03c4,\n                map :=\n                  @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                    id\n                      (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                        (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.map\n            f))\n[PROOFSTEP]\nrintro _ _ \u27e8\u27e9\n[GOAL]\ncase h_map.arrow\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03c6 : V \u2964q W'\n\u03c4 : W \u2192 W'\nh : \u2200 (x : V), \u03c6.obj x = \u03c4 (\u03c3 x)\n\u03a6 : Push \u03c3 \u2964q W'\n\u03a6\u2080 : \u03a6.obj = \u03c4\n\u03a6comp : of \u03c3 \u22d9q \u03a6 = \u03c6\nX\u271d Y\u271d : V\nf\u271d : X\u271d \u27f6 Y\u271d\n\u22a2 \u03a6.map (PushQuiver.arrow f\u271d) =\n    Eq.recOn\n      (_ :\n        { obj := \u03c4,\n                map :=\n                  @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                    id\n                      (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                        (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n            (\u03c3 Y\u271d) =\n          \u03a6.obj (\u03c3 Y\u271d))\n      (Eq.recOn\n        (_ :\n          { obj := \u03c4,\n                  map :=\n                    @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                      id\n                        (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                          (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.obj\n              (\u03c3 X\u271d) =\n            \u03a6.obj (\u03c3 X\u271d))\n        ({ obj := \u03c4,\n              map :=\n                @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03c4 X \u27f6 \u03c4 Y) fun X Y f =>\n                  id\n                    (Eq.mpr (_ : (\u03c4 (\u03c3 X) \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)))\n                      (Eq.mpr (_ : (\u03c6.obj X \u27f6 \u03c4 (\u03c3 Y)) = (\u03c6.obj X \u27f6 \u03c6.obj Y)) (\u03c6.map f))) }.map\n          (PushQuiver.arrow f\u271d)))\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase h_map.arrow\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03a6 : Push \u03c3 \u2964q W'\nX\u271d Y\u271d : V\nf\u271d : X\u271d \u27f6 Y\u271d\nh : \u2200 (x : V), (of \u03c3 \u22d9q \u03a6).obj x = \u03a6.obj (\u03c3 x)\n\u22a2 \u03a6.map (PushQuiver.arrow f\u271d) =\n    Eq.recOn\n      (_ :\n        { obj := \u03a6.obj,\n                map :=\n                  @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03a6.obj X \u27f6 \u03a6.obj Y) fun X Y f =>\n                    id\n                      (Eq.mpr (_ : (\u03a6.obj (\u03c3 X) \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)))\n                        (Eq.mpr (_ : ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 (of \u03c3 \u22d9q \u03a6).obj Y))\n                          ((of \u03c3 \u22d9q \u03a6).map f))) }.obj\n            (\u03c3 Y\u271d) =\n          \u03a6.obj (\u03c3 Y\u271d))\n      (Eq.recOn\n        (_ :\n          { obj := \u03a6.obj,\n                  map :=\n                    @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03a6.obj X \u27f6 \u03a6.obj Y) fun X Y f =>\n                      id\n                        (Eq.mpr (_ : (\u03a6.obj (\u03c3 X) \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)))\n                          (Eq.mpr (_ : ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 (of \u03c3 \u22d9q \u03a6).obj Y))\n                            ((of \u03c3 \u22d9q \u03a6).map f))) }.obj\n              (\u03c3 X\u271d) =\n            \u03a6.obj (\u03c3 X\u271d))\n        ({ obj := \u03a6.obj,\n              map :=\n                @PushQuiver.rec V inst\u271d\u00b9 W \u03c3 (fun X Y x => \u03a6.obj X \u27f6 \u03a6.obj Y) fun X Y f =>\n                  id\n                    (Eq.mpr (_ : (\u03a6.obj (\u03c3 X) \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)))\n                      (Eq.mpr (_ : ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 \u03a6.obj (\u03c3 Y)) = ((of \u03c3 \u22d9q \u03a6).obj X \u27f6 (of \u03c3 \u22d9q \u03a6).obj Y))\n                        ((of \u03c3 \u22d9q \u03a6).map f))) }.map\n          (PushQuiver.arrow f\u271d)))\n[PROOFSTEP]\nsimp only [Prefunctor.comp_map, cast_eq]\n[GOAL]\ncase h_map.arrow\nV : Type u_1\ninst\u271d\u00b9 : Quiver V\nW : Type u_2\n\u03c3 : V \u2192 W\nW' : Type u_3\ninst\u271d : Quiver W'\n\u03a6 : Push \u03c3 \u2964q W'\nX\u271d Y\u271d : V\nf\u271d : X\u271d \u27f6 Y\u271d\nh : \u2200 (x : V), (of \u03c3 \u22d9q \u03a6).obj x = \u03a6.obj (\u03c3 x)\n\u22a2 \u03a6.map (PushQuiver.arrow f\u271d) =\n    id\n      (Eq.mpr (_ : (\u03a6.obj (\u03c3 X\u271d) \u27f6 \u03a6.obj (\u03c3 Y\u271d)) = ((of \u03c3 \u22d9q \u03a6).obj X\u271d \u27f6 \u03a6.obj (\u03c3 Y\u271d)))\n        (Eq.mpr (_ : ((of \u03c3 \u22d9q \u03a6).obj X\u271d \u27f6 \u03a6.obj (\u03c3 Y\u271d)) = ((of \u03c3 \u22d9q \u03a6).obj X\u271d \u27f6 (of \u03c3 \u22d9q \u03a6).obj Y\u271d))\n          (\u03a6.map ((of \u03c3).map f\u271d))))\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.Quiver.Push", "llama_tokens": 7797, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8311430394931455, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.5081458715750179}}
{"text": "[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\ninst\u271d : Nontrivial R\ni j : n\n\u22a2 natDegree (charmatrix M i j) = if i = j then 1 else 0\n[PROOFSTEP]\nby_cases i = j\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\ninst\u271d : Nontrivial R\ni j : n\n\u22a2 natDegree (charmatrix M i j) = if i = j then 1 else 0\n[PROOFSTEP]\nby_cases i = j\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\ninst\u271d : Nontrivial R\ni j : n\nh : i = j\n\u22a2 natDegree (charmatrix M i j) = if i = j then 1 else 0\n[PROOFSTEP]\nsimp [h, \u2190 degree_eq_iff_natDegree_eq_of_pos (Nat.succ_pos 0)]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\ninst\u271d : Nontrivial R\ni j : n\nh : \u00aci = j\n\u22a2 natDegree (charmatrix M i j) = if i = j then 1 else 0\n[PROOFSTEP]\nsimp [h, \u2190 degree_eq_iff_natDegree_eq_of_pos (Nat.succ_pos 0)]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\ni j : n\n\u22a2 natDegree (charmatrix M i j) \u2264 if i = j then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\ni j : n\nh : i = j\n\u22a2 natDegree (charmatrix M i j) \u2264 1\n[PROOFSTEP]\nsimp [h, natDegree_X_sub_C_le]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\ni j : n\nh : \u00aci = j\n\u22a2 natDegree (charmatrix M i j) \u2264 0\n[PROOFSTEP]\nsimp [h, natDegree_X_sub_C_le]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < \u2191(Fintype.card n - 1)\n[PROOFSTEP]\nrw [charpoly, det_apply', \u2190 insert_erase (mem_univ (Equiv.refl n)), sum_insert (not_mem_erase (Equiv.refl n) univ),\n  add_comm]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\n\u22a2 degree\n      (\u2211 x in Finset.erase univ (Equiv.refl n), \u2191\u2191(\u2191Equiv.Perm.sign x) * \u220f i : n, charmatrix M (\u2191x i) i +\n          \u2191\u2191(\u2191Equiv.Perm.sign (Equiv.refl n)) * \u220f i : n, charmatrix M (\u2191(Equiv.refl n) i) i -\n        \u220f i : n, (X - \u2191C (M i i))) <\n    \u2191(Fintype.card n - 1)\n[PROOFSTEP]\nsimp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id.def, Int.cast_one, Units.val_one, add_sub_cancel,\n  Equiv.coe_refl]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\n\u22a2 degree (\u2211 x in Finset.erase univ (Equiv.refl n), \u2191\u2191(\u2191Equiv.Perm.sign x) * \u220f x_1 : n, charmatrix M (\u2191x x_1) x_1) <\n    \u2191(Fintype.card n - 1)\n[PROOFSTEP]\nrw [\u2190 mem_degreeLT]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\n\u22a2 \u2211 x in Finset.erase univ (Equiv.refl n), \u2191\u2191(\u2191Equiv.Perm.sign x) * \u220f x_1 : n, charmatrix M (\u2191x x_1) x_1 \u2208\n    degreeLT R (Fintype.card n - 1)\n[PROOFSTEP]\napply Submodule.sum_mem (degreeLT R (Fintype.card n - 1))\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\n\u22a2 \u2200 (c : n \u2243 n),\n    c \u2208 Finset.erase univ (Equiv.refl n) \u2192\n      \u2191\u2191(\u2191Equiv.Perm.sign c) * \u220f x : n, charmatrix M (\u2191c x) x \u2208 degreeLT R (Fintype.card n - 1)\n[PROOFSTEP]\nintro c hc\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2191\u2191(\u2191Equiv.Perm.sign c) * \u220f x : n, charmatrix M (\u2191c x) x \u2208 degreeLT R (Fintype.card n - 1)\n[PROOFSTEP]\nrw [\u2190 C_eq_int_cast, C_mul']\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2191\u2191(\u2191Equiv.Perm.sign c) \u2022 \u220f x : n, charmatrix M (\u2191c x) x \u2208 degreeLT R (Fintype.card n - 1)\n[PROOFSTEP]\napply Submodule.smul_mem (degreeLT R (Fintype.card n - 1)) \u2191\u2191(Equiv.Perm.sign c)\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u220f x : n, charmatrix M (\u2191c x) x \u2208 degreeLT R (Fintype.card n - 1)\n[PROOFSTEP]\nrw [mem_degreeLT]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 degree (\u220f x : n, charmatrix M (\u2191c x) x) < \u2191(Fintype.card n - 1)\n[PROOFSTEP]\napply lt_of_le_of_lt degree_le_natDegree _\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2191(natDegree (\u220f x : n, charmatrix M (\u2191c x) x)) < \u2191(Fintype.card n - 1)\n[PROOFSTEP]\nrw [Nat.cast_withBot, Nat.cast_withBot]\n  -- porting note: added\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2191(natDegree (\u220f x : n, charmatrix M (\u2191c x) x)) < \u2191(Fintype.card n - 1)\n[PROOFSTEP]\nrw [WithBot.coe_lt_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 natDegree (\u220f x : n, charmatrix M (\u2191c x) x) < Fintype.card n - 1\n[PROOFSTEP]\napply lt_of_le_of_lt _ (Equiv.Perm.fixed_point_card_lt_of_ne_one (ne_of_mem_erase hc))\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 natDegree (\u220f x : n, charmatrix M (\u2191c x) x) \u2264 card (filter (fun x => \u2191c x = x) univ)\n[PROOFSTEP]\napply le_trans (Polynomial.natDegree_prod_le univ fun i : n => charmatrix M (c i) i) _\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2211 i : n, natDegree (charmatrix M (\u2191c i) i) \u2264 card (filter (fun x => \u2191c x = x) univ)\n[PROOFSTEP]\nrw [card_eq_sum_ones]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2211 i : n, natDegree (charmatrix M (\u2191c i) i) \u2264 \u2211 x in filter (fun x => \u2191c x = x) univ, 1\n[PROOFSTEP]\nrw [sum_filter]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2211 i : n, natDegree (charmatrix M (\u2191c i) i) \u2264 \u2211 a : n, if \u2191c a = a then 1 else 0\n[PROOFSTEP]\napply sum_le_sum\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\n\u22a2 \u2200 (i : n), i \u2208 univ \u2192 natDegree (charmatrix M (\u2191c i) i) \u2264 if \u2191c i = i then 1 else 0\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nc : n \u2243 n\nhc : c \u2208 Finset.erase univ (Equiv.refl n)\ni\u271d : n\na\u271d : i\u271d \u2208 univ\n\u22a2 natDegree (charmatrix M (\u2191c i\u271d) i\u271d) \u2264 if \u2191c i\u271d = i\u271d then 1 else 0\n[PROOFSTEP]\napply charmatrix_apply_natDegree_le\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 coeff (charpoly M) k = coeff (\u220f i : n, (X - \u2191C (M i i))) k\n[PROOFSTEP]\napply eq_of_sub_eq_zero\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 coeff (charpoly M) k - coeff (\u220f i : n, (X - \u2191C (M i i))) k = 0\n[PROOFSTEP]\nrw [\u2190 coeff_sub]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 coeff (charpoly M - \u220f i : n, (X - \u2191C (M i i))) k = 0\n[PROOFSTEP]\napply Polynomial.coeff_eq_zero_of_degree_lt\n[GOAL]\ncase h.h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < \u2191k\n[PROOFSTEP]\napply lt_of_lt_of_le (charpoly_sub_diagonal_degree_lt M) ?_\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 \u2191(Fintype.card n - 1) \u2264 \u2191k\n[PROOFSTEP]\nrw [Nat.cast_withBot, Nat.cast_withBot]\n  -- porting note: added\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 \u2191(Fintype.card n - 1) \u2264 \u2191k\n[PROOFSTEP]\nrw [WithBot.coe_le_coe]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM : Matrix n n R\nk : \u2115\nh : Fintype.card n - 1 \u2264 k\n\u22a2 Fintype.card n - 1 \u2264 k\n[PROOFSTEP]\napply h\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nh : Fintype.card n = 0\nM : Matrix n n R\n\u22a2 det M = 1\n[PROOFSTEP]\nrw [Fintype.card_eq_zero_iff] at h \n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nh : IsEmpty n\nM : Matrix n n R\n\u22a2 det M = 1\n[PROOFSTEP]\nsuffices M = 1 by simp [this]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nh : IsEmpty n\nM : Matrix n n R\nthis : M = 1\n\u22a2 det M = 1\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nh : IsEmpty n\nM : Matrix n n R\n\u22a2 M = 1\n[PROOFSTEP]\next i\n[GOAL]\ncase a.h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nh : IsEmpty n\nM : Matrix n n R\ni x\u271d : n\n\u22a2 M i x\u271d = OfNat.ofNat 1 i x\u271d\n[PROOFSTEP]\nexact h.elim i\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\n\u22a2 degree (charpoly M) = \u2191(Fintype.card n)\n[PROOFSTEP]\nby_cases h : Fintype.card n = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n\u22a2 degree (charpoly M) = \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n\u22a2 degree (charpoly M) = \u21910\n[PROOFSTEP]\nunfold charpoly\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n\u22a2 degree (det (charmatrix M)) = \u21910\n[PROOFSTEP]\nrw [det_of_card_zero]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n\u22a2 degree 1 = \u21910\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : Fintype.card n = 0\n\u22a2 Fintype.card n = 0\n[PROOFSTEP]\nassumption\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 degree (charpoly M) = \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel M.charpoly (\u220f i : n, (X - C (M i i)))]\n  -- porting note: added `\u2191` in front of `Fintype.card n`\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i)) + \u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n[PROOFSTEP]\nhave h1 : (\u220f i : n, (X - C (M i i))).degree = \u2191(Fintype.card n) :=\n  by\n  rw [degree_eq_iff_natDegree_eq_of_pos (Nat.pos_of_ne_zero h), natDegree_prod']\n  simp_rw [natDegree_X_sub_C]\n  rw [\u2190 Finset.card_univ, sum_const, smul_eq_mul, mul_one]\n  simp_rw [(monic_X_sub_C _).leadingCoeff]\n  simp\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [degree_eq_iff_natDegree_eq_of_pos (Nat.pos_of_ne_zero h), natDegree_prod']\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u2211 i : n, natDegree (X - \u2191C (M i i)) = Fintype.card n\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u220f i : n, leadingCoeff (X - \u2191C (M i i)) \u2260 0\n[PROOFSTEP]\nsimp_rw [natDegree_X_sub_C]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u2211 x : n, 1 = Fintype.card n\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u220f i : n, leadingCoeff (X - \u2191C (M i i)) \u2260 0\n[PROOFSTEP]\nrw [\u2190 Finset.card_univ, sum_const, smul_eq_mul, mul_one]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u220f i : n, leadingCoeff (X - \u2191C (M i i)) \u2260 0\n[PROOFSTEP]\nsimp_rw [(monic_X_sub_C _).leadingCoeff]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\n\u22a2 \u220f x : n, 1 \u2260 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i)) + \u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [degree_add_eq_right_of_degree_lt]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < degree (\u220f i : n, (X - \u2191C (M i i)))\n[PROOFSTEP]\nexact h1\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < degree (\u220f i : n, (X - \u2191C (M i i)))\n[PROOFSTEP]\nrw [h1]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < \u2191(Fintype.card n)\n[PROOFSTEP]\napply lt_trans (charpoly_sub_diagonal_degree_lt M)\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 \u2191(Fintype.card n - 1) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [Nat.cast_withBot, Nat.cast_withBot]\n  -- porting note: added\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 \u2191(Fintype.card n - 1) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [WithBot.coe_lt_coe]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 Fintype.card n - 1 < Fintype.card n\n[PROOFSTEP]\nrw [\u2190 Nat.pred_eq_sub_one]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 Nat.pred (Fintype.card n) < Fintype.card n\n[PROOFSTEP]\napply Nat.pred_lt\n[GOAL]\ncase neg.a\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nontrivial R\nM : Matrix n n R\nh : \u00acFintype.card n = 0\nh1 : degree (\u220f i : n, (X - \u2191C (M i i))) = \u2191(Fintype.card n)\n\u22a2 Fintype.card n \u2260 0\n[PROOFSTEP]\napply h\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nnontriviality R\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nby_cases h : Fintype.card n = 0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : Fintype.card n = 0\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nrw [charpoly, det_of_card_zero h]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : Fintype.card n = 0\n\u22a2 Monic 1\n[PROOFSTEP]\napply monic_one\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nhave mon : (\u220f i : n, (X - C (M i i))).Monic :=\n  by\n  apply monic_prod_of_monic univ fun i : n => X - C (M i i)\n  simp [monic_X_sub_C]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\n\u22a2 Monic (\u220f i : n, (X - \u2191C (M i i)))\n[PROOFSTEP]\napply monic_prod_of_monic univ fun i : n => X - C (M i i)\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\n\u22a2 \u2200 (i : n), i \u2208 univ \u2192 Monic (X - \u2191C (M i i))\n[PROOFSTEP]\nsimp [monic_X_sub_C]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : Monic (\u220f i : n, (X - \u2191C (M i i)))\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nrw [\u2190 sub_add_cancel (\u220f i : n, (X - C (M i i))) M.charpoly] at mon \n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : Monic (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M)\n\u22a2 Monic (charpoly M)\n[PROOFSTEP]\nrw [Monic] at *\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 leadingCoeff (charpoly M) = 1\n[PROOFSTEP]\nrwa [leadingCoeff_add_of_degree_lt] at mon \n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 degree (\u220f i : n, (X - \u2191C (M i i)) - charpoly M) < degree (charpoly M)\n[PROOFSTEP]\nrw [charpoly_degree_eq_dim]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 degree (\u220f i : n, (X - \u2191C (M i i)) - charpoly M) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [\u2190 neg_sub]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 degree (-(charpoly M - \u220f i : n, (X - \u2191C (M i i)))) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [degree_neg]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 degree (charpoly M - \u220f i : n, (X - \u2191C (M i i))) < \u2191(Fintype.card n)\n[PROOFSTEP]\napply lt_trans (charpoly_sub_diagonal_degree_lt M)\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 \u2191(Fintype.card n - 1) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [Nat.cast_withBot, Nat.cast_withBot]\n  -- porting note: added\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 \u2191(Fintype.card n - 1) < \u2191(Fintype.card n)\n[PROOFSTEP]\nrw [WithBot.coe_lt_coe]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 Fintype.card n - 1 < Fintype.card n\n[PROOFSTEP]\nrw [\u2190 Nat.pred_eq_sub_one]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 Nat.pred (Fintype.card n) < Fintype.card n\n[PROOFSTEP]\napply Nat.pred_lt\n[GOAL]\ncase neg.a\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u271d : Nontrivial R\nh : \u00acFintype.card n = 0\nmon : leadingCoeff (\u220f i : n, (X - \u2191C (M i i)) - charpoly M + charpoly M) = 1\n\u22a2 Fintype.card n \u2260 0\n[PROOFSTEP]\napply h\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nonempty n\nM : Matrix n n R\n\u22a2 trace M = -coeff (charpoly M) (Fintype.card n - 1)\n[PROOFSTEP]\nrw [charpoly_coeff_eq_prod_coeff_of_le _ le_rfl, Fintype.card,\n  prod_X_sub_C_coeff_card_pred univ (fun i : n => M i i) Fintype.card_pos, neg_neg, trace]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\ninst\u271d : Nonempty n\nM : Matrix n n R\n\u22a2 \u2211 i : n, diag M i = \u2211 i : n, M i i\n[PROOFSTEP]\nsimp_rw [diag_apply]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 eval (\u2191(scalar n) r) (\u2191matPolyEquiv M) i j = eval r (M i j)\n[PROOFSTEP]\nunfold Polynomial.eval\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 eval\u2082 (RingHom.id ((fun x => Matrix n n R) r)) (\u2191(scalar n) r) (\u2191matPolyEquiv M) i j = eval\u2082 (RingHom.id R) r (M i j)\n[PROOFSTEP]\nrw [Polynomial.eval\u2082_def, Polynomial.eval\u2082_def]\n  -- porting note: was `unfold eval\u2082`\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 sum (\u2191matPolyEquiv M) (fun e a => \u2191(RingHom.id ((fun x => Matrix n n R) r)) a * \u2191(scalar n) r ^ e) i j =\n    sum (M i j) fun e a => \u2191(RingHom.id R) a * r ^ e\n[PROOFSTEP]\ntrans Polynomial.sum (matPolyEquiv M) fun (e : \u2115) (a : Matrix n n R) => (a * (scalar n) r ^ e) i j\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 sum (\u2191matPolyEquiv M) (fun e a => \u2191(RingHom.id ((fun x => Matrix n n R) r)) a * \u2191(scalar n) r ^ e) i j =\n    sum (\u2191matPolyEquiv M) fun e a => (a * \u2191(scalar n) r ^ e) i j\n[PROOFSTEP]\nunfold Polynomial.sum\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 Finset.sum (support (\u2191matPolyEquiv M))\n      (fun n_1 =>\n        (fun e a => \u2191(RingHom.id ((fun x => Matrix n n R) r)) a * \u2191(scalar n) r ^ e) n_1 (coeff (\u2191matPolyEquiv M) n_1))\n      i j =\n    \u2211 n_1 in support (\u2191matPolyEquiv M), (fun e a => (a * \u2191(scalar n) r ^ e) i j) n_1 (coeff (\u2191matPolyEquiv M) n_1)\n[PROOFSTEP]\nsimp only [sum_apply]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 \u2211 c in support (\u2191matPolyEquiv M), (\u2191(RingHom.id (Matrix n n R)) (coeff (\u2191matPolyEquiv M) c) * \u2191(scalar n) r ^ c) i j =\n    \u2211 x in support (\u2191matPolyEquiv M), (coeff (\u2191matPolyEquiv M) x * \u2191(scalar n) r ^ x) i j\n[PROOFSTEP]\ndsimp\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 (sum (\u2191matPolyEquiv M) fun e a => (a * \u2191(scalar n) r ^ e) i j) = sum (M i j) fun e a => \u2191(RingHom.id R) a * r ^ e\n[PROOFSTEP]\nsimp_rw [\u2190 RingHom.map_pow, \u2190 (scalar.commute _ _).eq]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 (sum (\u2191matPolyEquiv M) fun e a => (\u2191(scalar n) (r ^ e) * a) i j) = sum (M i j) fun e a => \u2191(RingHom.id R) a * r ^ e\n[PROOFSTEP]\nsimp only [coe_scalar, Matrix.one_mul, RingHom.id_apply, Pi.smul_apply, smul_eq_mul, Algebra.smul_mul_assoc]\n  -- porting note: the `have` was present and unused also in the original\n      --have h : \u2200 x : \u2115, (fun (e : \u2115) (a : R) => r ^ e * a) x 0 = 0 := by simp\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 (sum (\u2191matPolyEquiv M) fun e a => (r ^ e \u2022 a) i j) = sum (M i j) fun e a => a * r ^ e\n[PROOFSTEP]\nsimp only [Polynomial.sum, matPolyEquiv_coeff_apply, mul_comm]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 \u2211 x in support (\u2191matPolyEquiv M), (r ^ x \u2022 coeff (\u2191matPolyEquiv M) x) i j =\n    \u2211 x in support (M i j), r ^ x * coeff (M i j) x\n[PROOFSTEP]\nsimp only [smul_apply, matPolyEquiv_coeff_apply, smul_eq_mul]\n  -- porting note: added\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 \u2211 x in support (\u2191matPolyEquiv M), r ^ x * coeff (M i j) x = \u2211 x in support (M i j), r ^ x * coeff (M i j) x\n[PROOFSTEP]\napply (Finset.sum_subset (support_subset_support_matPolyEquiv _ _ _) _).symm\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni j : n\n\u22a2 \u2200 (x : \u2115), x \u2208 support (\u2191matPolyEquiv M) \u2192 \u00acx \u2208 support (M i j) \u2192 r ^ x * coeff (M i j) x = 0\n[PROOFSTEP]\nintro n _hn h'n\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn\u271d G : Type v\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : Fintype n\u271d\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n\u271d n\u271d R\nM : Matrix n\u271d n\u271d R[X]\nr : R\ni j : n\u271d\nn : \u2115\n_hn : n \u2208 support (\u2191matPolyEquiv M)\nh'n : \u00acn \u2208 support (M i j)\n\u22a2 r ^ n * coeff (M i j) n = 0\n[PROOFSTEP]\nrw [not_mem_support_iff] at h'n \n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn\u271d G : Type v\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : Fintype n\u271d\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n\u271d n\u271d R\nM : Matrix n\u271d n\u271d R[X]\nr : R\ni j : n\u271d\nn : \u2115\n_hn : n \u2208 support (\u2191matPolyEquiv M)\nh'n : coeff (M i j) n = 0\n\u22a2 r ^ n * coeff (M i j) n = 0\n[PROOFSTEP]\nsimp only [h'n, zero_mul]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn\u271d G : Type v\ninst\u271d\u00b2 : DecidableEq n\u271d\ninst\u271d\u00b9 : Fintype n\u271d\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n\u271d n\u271d R\nM : Matrix n\u271d n\u271d R[X]\nr : R\ni j : n\u271d\nn : \u2115\n_hn : n \u2208 support (\u2191matPolyEquiv M)\nh'n : coeff (M i j) n = 0\n\u22a2 r ^ n * 0 = 0\n[PROOFSTEP]\nsimp only [mul_zero]\n  -- porting note: added\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\n\u22a2 eval r (det M) = det (eval (\u2191(scalar n) r) (\u2191matPolyEquiv M))\n[PROOFSTEP]\nrw [Polynomial.eval, \u2190 coe_eval\u2082RingHom, RingHom.map_det]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\n\u22a2 det (\u2191(RingHom.mapMatrix (eval\u2082RingHom (RingHom.id R) r)) M) = det (eval (\u2191(scalar n) r) (\u2191matPolyEquiv M))\n[PROOFSTEP]\napply congr_arg det\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\n\u22a2 \u2191(RingHom.mapMatrix (eval\u2082RingHom (RingHom.id R) r)) M = eval (\u2191(scalar n) r) (\u2191matPolyEquiv M)\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni\u271d x\u271d : n\n\u22a2 \u2191(RingHom.mapMatrix (eval\u2082RingHom (RingHom.id R) r)) M i\u271d x\u271d = eval (\u2191(scalar n) r) (\u2191matPolyEquiv M) i\u271d x\u271d\n[PROOFSTEP]\nsymm\n  -- porting note: `exact` was `convert`\n[GOAL]\ncase a.h\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d : Matrix n n R\nM : Matrix n n R[X]\nr : R\ni\u271d x\u271d : n\n\u22a2 eval (\u2191(scalar n) r) (\u2191matPolyEquiv M) i\u271d x\u271d = \u2191(RingHom.mapMatrix (eval\u2082RingHom (RingHom.id R) r)) M i\u271d x\u271d\n[PROOFSTEP]\nexact matPolyEquiv_eval _ _ _ _\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u22a2 det M = (-1) ^ Fintype.card n * coeff (charpoly M) 0\n[PROOFSTEP]\nrw [coeff_zero_eq_eval_zero, charpoly, eval_det, matPolyEquiv_charmatrix, \u2190 det_smul]\n[GOAL]\nR : Type u\ninst\u271d\u00b3 : CommRing R\nn G : Type v\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d : DecidableEq \u03b1\nM\u271d M : Matrix n n R\n\u22a2 det M = det (-1 \u2022 eval (\u2191(scalar n) 0) (X - \u2191C M))\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\n\u22a2 \u2191matPolyEquiv (\u2191(RingHom.mapMatrix \u2191(expand K k)) (charmatrix (M ^ k))) = X ^ k - \u2191C (M ^ k)\n[PROOFSTEP]\next m i j\n[GOAL]\ncase a.a.h\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\n\u22a2 coeff (\u2191matPolyEquiv (\u2191(RingHom.mapMatrix \u2191(expand K k)) (charmatrix (M ^ k)))) m i j =\n    coeff (X ^ k - \u2191C (M ^ k)) m i j\n[PROOFSTEP]\nrw [coeff_sub, coeff_C, matPolyEquiv_coeff_apply, RingHom.mapMatrix_apply, Matrix.map_apply, AlgHom.coe_toRingHom,\n  DMatrix.sub_apply, coeff_X_pow]\n[GOAL]\ncase a.a.h\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\n\u22a2 coeff (\u2191(expand K k) (charmatrix (M ^ k) i j)) m = ite (m = k) 1 0 i j - ite (m = 0) (M ^ k) 0 i j\n[PROOFSTEP]\nby_cases hij : i = j\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\n\u22a2 coeff (\u2191(expand K k) (charmatrix (M ^ k) i j)) m = ite (m = k) 1 0 i j - ite (m = 0) (M ^ k) 0 i j\n[PROOFSTEP]\nrw [hij, charmatrix_apply_eq, AlgHom.map_sub, expand_C, expand_X, coeff_sub, coeff_X_pow, coeff_C]\n  -- porting note: the second `Matrix.` was `DMatrix.`\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\n\u22a2 ((if m = k then 1 else 0) - if m = 0 then (M ^ k) j j else 0) = ite (m = k) 1 0 j j - ite (m = 0) (M ^ k) 0 j j\n[PROOFSTEP]\nsplit_ifs with mp m0\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\nmp : m = k\nm0 : m = 0\n\u22a2 1 - (M ^ k) j j = OfNat.ofNat 1 j j - (M ^ k) j j\n[PROOFSTEP]\nsimp only [Matrix.one_apply_eq, Matrix.zero_apply]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\nmp : m = k\nm0 : \u00acm = 0\n\u22a2 1 - 0 = OfNat.ofNat 1 j j - OfNat.ofNat 0 j j\n[PROOFSTEP]\nsimp only [Matrix.one_apply_eq, Matrix.zero_apply]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\nmp : \u00acm = k\nh\u271d : m = 0\n\u22a2 0 - (M ^ k) j j = OfNat.ofNat 0 j j - (M ^ k) j j\n[PROOFSTEP]\nsimp only [Matrix.one_apply_eq, Matrix.zero_apply]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : i = j\nmp : \u00acm = k\nh\u271d : \u00acm = 0\n\u22a2 0 - 0 = OfNat.ofNat 0 j j - OfNat.ofNat 0 j j\n[PROOFSTEP]\nsimp only [Matrix.one_apply_eq, Matrix.zero_apply]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\n\u22a2 coeff (\u2191(expand K k) (charmatrix (M ^ k) i j)) m = ite (m = k) 1 0 i j - ite (m = 0) (M ^ k) 0 i j\n[PROOFSTEP]\nrw [charmatrix_apply_ne _ _ _ hij, AlgHom.map_neg, expand_C, coeff_neg, coeff_C]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\n\u22a2 (-if m = 0 then (M ^ k) i j else 0) = ite (m = k) 1 0 i j - ite (m = 0) (M ^ k) 0 i j\n[PROOFSTEP]\nsplit_ifs with m0 mp\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\nm0 : m = 0\nmp : m = k\n\u22a2 -(M ^ k) i j = OfNat.ofNat 1 i j - (M ^ k) i j\n[PROOFSTEP]\nsimp only [hij, zero_sub, Matrix.zero_apply, sub_zero, neg_zero, Matrix.one_apply_ne, Ne.def, not_false_iff]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\nm0 : m = 0\nmp : \u00acm = k\n\u22a2 -(M ^ k) i j = OfNat.ofNat 0 i j - (M ^ k) i j\n[PROOFSTEP]\nsimp only [hij, zero_sub, Matrix.zero_apply, sub_zero, neg_zero, Matrix.one_apply_ne, Ne.def, not_false_iff]\n[GOAL]\ncase pos\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\nm0 : \u00acm = 0\nh\u271d : m = k\n\u22a2 -0 = OfNat.ofNat 1 i j - OfNat.ofNat 0 i j\n[PROOFSTEP]\nsimp only [hij, zero_sub, Matrix.zero_apply, sub_zero, neg_zero, Matrix.one_apply_ne, Ne.def, not_false_iff]\n[GOAL]\ncase neg\nR : Type u\ninst\u271d\u2075 : CommRing R\nn G : Type v\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b2 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\nK : Type u_1\nk : \u2115\ninst\u271d : Field K\nM : Matrix n n K\nm : \u2115\ni j : n\nhij : \u00aci = j\nm0 : \u00acm = 0\nh\u271d : \u00acm = k\n\u22a2 -0 = OfNat.ofNat 0 i j - OfNat.ofNat 0 i j\n[PROOFSTEP]\nsimp only [hij, zero_sub, Matrix.zero_apply, sub_zero, neg_zero, Matrix.one_apply_ne, Ne.def, not_false_iff]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM\u271d : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nM : Matrix n n R\nk : \u2115\n\u22a2 M ^ k = \u2191(aeval M) (X ^ k %\u2098 charpoly M)\n[PROOFSTEP]\nrw [\u2190 aeval_eq_aeval_mod_charpoly, map_pow, aeval_X]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\n\u22a2 coeff (charpoly M) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\ndelta charpoly\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\n\u22a2 coeff (det (charmatrix M)) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\nrw [Matrix.det_apply, finset_sum_coeff]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\n\u22a2 \u2211 b : Equiv.Perm n, coeff (\u2191Equiv.Perm.sign b \u2022 \u220f i : n, charmatrix M (\u2191b i) i) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\napply sum_mem\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\n\u22a2 \u2200 (c : Equiv.Perm n),\n    c \u2208 univ \u2192 coeff (\u2191Equiv.Perm.sign c \u2022 \u220f i : n, charmatrix M (\u2191c i) i) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\nrintro c -\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\n\u22a2 coeff (\u2191Equiv.Perm.sign c \u2022 \u220f i : n, charmatrix M (\u2191c i) i) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\nrw [coeff_smul, Submodule.smul_mem_iff']\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\n\u22a2 coeff (\u220f i : n, charmatrix M (\u2191c i) i) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\nhave : \u2211 x : n, 1 = Fintype.card n := by rw [Finset.sum_const, card_univ, smul_eq_mul, mul_one]\n[GOAL]\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\n\u22a2 \u2211 x : n, 1 = Fintype.card n\n[PROOFSTEP]\nrw [Finset.sum_const, card_univ, smul_eq_mul, mul_one]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\n\u22a2 coeff (\u220f i : n, charmatrix M (\u2191c i) i) k \u2208 I ^ (Fintype.card n - k)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\n\u22a2 coeff (\u220f i : n, charmatrix M (\u2191c i) i) k \u2208 I ^ (\u2211 x : n, 1 - k)\n[PROOFSTEP]\napply coeff_prod_mem_ideal_pow_tsub\n[GOAL]\ncase h.h\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\n\u22a2 \u2200 (i : n), i \u2208 univ \u2192 \u2200 (k : \u2115), coeff (charmatrix M (\u2191c i) i) k \u2208 I ^ (1 - k)\n[PROOFSTEP]\nrintro i - (_ | k)\n[GOAL]\ncase h.h.zero\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\n\u22a2 coeff (charmatrix M (\u2191c i) i) Nat.zero \u2208 I ^ (1 - Nat.zero)\n[PROOFSTEP]\nrw [Nat.zero_eq]\n  -- porting note: `rw [Nat.zero_eq]` was not present\n[GOAL]\ncase h.h.zero\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\n\u22a2 coeff (charmatrix M (\u2191c i) i) 0 \u2208 I ^ (1 - 0)\n[PROOFSTEP]\nrw [tsub_zero, pow_one, charmatrix_apply, coeff_sub, coeff_X_mul_zero, coeff_C_zero, zero_sub]\n[GOAL]\ncase h.h.zero\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\n\u22a2 -M (\u2191c i) i \u2208 I\n[PROOFSTEP]\napply neg_mem\n[GOAL]\ncase h.h.zero.a\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\n\u22a2 M (\u2191c i) i \u2208 I\n[PROOFSTEP]\nexact h (c i) i\n[GOAL]\ncase h.h.succ\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk\u271d : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\nk : \u2115\n\u22a2 coeff (charmatrix M (\u2191c i) i) (Nat.succ k) \u2208 I ^ (1 - Nat.succ k)\n[PROOFSTEP]\nrw [Nat.succ_eq_one_add, tsub_self_add, pow_zero, Ideal.one_eq_top]\n[GOAL]\ncase h.h.succ\nR : Type u\ninst\u271d\u2074 : CommRing R\nn G : Type v\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 \u03b2 : Type v\ninst\u271d\u00b9 : DecidableEq \u03b1\nM : Matrix n n R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nI : Ideal R\nh : \u2200 (i j : n), M i j \u2208 I\nk\u271d : \u2115\nc : Equiv.Perm n\nthis : \u2211 x : n, 1 = Fintype.card n\ni : n\nk : \u2115\n\u22a2 coeff (charmatrix M (\u2191c i) i) (1 + k) \u2208 \u22a4\n[PROOFSTEP]\nexact Submodule.mem_top\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff", "llama_tokens": 24926, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424373085145, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.5077248034767761}}
{"text": "[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    \u222b (x : \u211d) in z.re..w.re, \u222b (y : \u211d) in z.im..w.im, I \u2022 \u2191(f' (\u2191x + \u2191y * I)) 1 - \u2191(f' (\u2191x + \u2191y * I)) I\n[PROOFSTEP]\nset e : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := equivRealProdClm.symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    \u222b (x : \u211d) in z.re..w.re, \u222b (y : \u211d) in z.im..w.im, I \u2022 \u2191(f' (\u2191x + \u2191y * I)) 1 - \u2191(f' (\u2191x + \u2191y * I)) I\n[PROOFSTEP]\nhave he : \u2200 x y : \u211d, \u2191x + \u2191y * I = e (x, y) := fun x y => (mk_eq_add_mul_I x y).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe : \u2200 (x y : \u211d), \u2191x + \u2191y * I = \u2191e (x, y)\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    \u222b (x : \u211d) in z.re..w.re, \u222b (y : \u211d) in z.im..w.im, I \u2022 \u2191(f' (\u2191x + \u2191y * I)) 1 - \u2191(f' (\u2191x + \u2191y * I)) I\n[PROOFSTEP]\nhave he\u2081 : e (1, 0) = 1 := rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe : \u2200 (x y : \u211d), \u2191x + \u2191y * I = \u2191e (x, y)\nhe\u2081 : \u2191e (1, 0) = 1\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    \u222b (x : \u211d) in z.re..w.re, \u222b (y : \u211d) in z.im..w.im, I \u2022 \u2191(f' (\u2191x + \u2191y * I)) 1 - \u2191(f' (\u2191x + \u2191y * I)) I\n[PROOFSTEP]\nhave he\u2082 : e (0, 1) = I := rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe : \u2200 (x y : \u211d), \u2191x + \u2191y * I = \u2191e (x, y)\nhe\u2081 : \u2191e (1, 0) = 1\nhe\u2082 : \u2191e (0, 1) = I\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    \u222b (x : \u211d) in z.re..w.re, \u222b (y : \u211d) in z.im..w.im, I \u2022 \u2191(f' (\u2191x + \u2191y * I)) 1 - \u2191(f' (\u2191x + \u2191y * I)) I\n[PROOFSTEP]\nsimp only [he] at *\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nset F : \u211d \u00d7 \u211d \u2192 E := f \u2218 e\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nset F' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => (f' (e p)).comp (e : \u211d \u00d7 \u211d \u2192L[\u211d] \u2102)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nhave hF' : \u2200 p : \u211d \u00d7 \u211d, (-(I \u2022 F' p)) (1, 0) + F' p (0, 1) = -(I \u2022 f' (e p) 1 - f' (e p) I) :=\n  by\n  rintro \u27e8x, y\u27e9\n  simp only [ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply,\n    ContinuousLinearEquiv.coe_coe, he\u2081, he\u2082, neg_add_eq_sub, neg_sub]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\n\u22a2 \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase mk\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nx y : \u211d\n\u22a2 \u2191(-(I \u2022 F' (x, y))) (1, 0) + \u2191(F' (x, y)) (0, 1) = -(I \u2022 \u2191(f' (\u2191e (x, y))) 1 - \u2191(f' (\u2191e (x, y))) I)\n[PROOFSTEP]\nsimp only [ContinuousLinearMap.neg_apply, ContinuousLinearMap.smul_apply, ContinuousLinearMap.comp_apply,\n  ContinuousLinearEquiv.coe_coe, he\u2081, he\u2082, neg_add_eq_sub, neg_sub]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nset R : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nset t : Set (\u211d \u00d7 \u211d) := e \u207b\u00b9' s\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nrw [uIcc_comm z.im] at Hc Hi \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nrw [min_comm z.im, max_comm z.im] at Hd \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nhave hR : e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R := rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nhave htc : ContinuousOn F R := Hc.comp e.continuousOn hR.ge\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nhave htd :\n  \u2200 p \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u02e2 Ioo (min w.im z.im) (max w.im z.im) \\ t, HasFDerivAt F (F' p) p :=\n  fun p hp => (Hd (e p) hp).comp p e.hasFDerivAt\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u02e2 Ioo (min w.im z.im) (max w.im z.im) \\ t \u2192 HasFDerivAt F (F' p) p\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, y))) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, y)) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) 1 -\n          \u2191(f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, y))) I\n[PROOFSTEP]\nsimp_rw [\u2190 intervalIntegral.integral_smul, intervalIntegral.integral_symm w.im z.im, \u2190 intervalIntegral.integral_neg, \u2190\n  hF']\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u02e2 Ioo (min w.im z.im) (max w.im z.im) \\ t \u2192 HasFDerivAt F (F' p) p\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, z.im))) -\n          \u222b (x : \u211d) in z.re..w.re, f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, w.im))) +\n        \u222b (x : \u211d) in w.im..z.im, -(I \u2022 f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (w.re, x)))) -\n      \u222b (x : \u211d) in w.im..z.im, -(I \u2022 f (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (z.re, x))) =\n    \u222b (x : \u211d) in z.re..w.re,\n      \u222b (x_1 : \u211d) in w.im..z.im,\n        \u2191(-(I \u2022\n                  ContinuousLinearMap.comp (f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n                    \u2191(ContinuousLinearEquiv.symm equivRealProdClm)))\n            (1, 0) +\n          \u2191(ContinuousLinearMap.comp (f' (\u2191(ContinuousLinearEquiv.symm equivRealProdClm) (x, x_1)))\n                \u2191(ContinuousLinearEquiv.symm equivRealProdClm))\n            (0, 1)\n[PROOFSTEP]\nrefine'\n  (integral2_divergence_prod_of_hasFDerivWithinAt_off_countable (fun p => -(I \u2022 F p)) F (fun p => -(I \u2022 F' p)) F' z.re\n      w.im w.re z.im t (hs.preimage e.injective) (htc.const_smul _).neg htc (fun p hp => ((htd p hp).const_smul I).neg)\n      htd _).symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u02e2 Ioo (min w.im z.im) (max w.im z.im) \\ t \u2192 HasFDerivAt F (F' p) p\n\u22a2 IntegrableOn (fun x => \u2191((fun p => -(I \u2022 F' p)) x) (1, 0) + \u2191(F' x) (0, 1)) ([[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]])\n[PROOFSTEP]\nrw [\u2190 (volume_preserving_equiv_real_prod.symm _).integrableOn_comp_preimage (MeasurableEquiv.measurableEmbedding _)] at\n  Hi \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nf' : \u2102 \u2192 \u2102 \u2192L[\u211d] E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min w.im z.im) (max w.im z.im) \\ s \u2192 HasFDerivAt f (f' x) x\nHi :\n  IntegrableOn ((fun z => I \u2022 \u2191(f' z) 1 - \u2191(f' z) I) \u2218 \u2191(MeasurableEquiv.symm measurableEquivRealProd))\n    (\u2191(MeasurableEquiv.symm measurableEquivRealProd) \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]))\ne : (\u211d \u00d7 \u211d) \u2243L[\u211d] \u2102 := ContinuousLinearEquiv.symm equivRealProdClm\nhe\u2081 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (1, 0) = 1\nhe\u2082 : \u2191(ContinuousLinearEquiv.symm equivRealProdClm) (0, 1) = I\nhe : \u211d \u2192 \u211d \u2192 True\nF : \u211d \u00d7 \u211d \u2192 E := f \u2218 \u2191e\nF' : \u211d \u00d7 \u211d \u2192 \u211d \u00d7 \u211d \u2192L[\u211d] E := fun p => ContinuousLinearMap.comp (f' (\u2191e p)) \u2191e\nhF' : \u2200 (p : \u211d \u00d7 \u211d), \u2191(-(I \u2022 F' p)) (1, 0) + \u2191(F' p) (0, 1) = -(I \u2022 \u2191(f' (\u2191e p)) 1 - \u2191(f' (\u2191e p)) I)\nR : Set (\u211d \u00d7 \u211d) := [[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]]\nt : Set (\u211d \u00d7 \u211d) := \u2191e \u207b\u00b9' s\nhR : \u2191e \u207b\u00b9' ([[z.re, w.re]] \u00d7\u2102 [[w.im, z.im]]) = R\nhtc : ContinuousOn F R\nhtd :\n  \u2200 (p : \u211d \u00d7 \u211d),\n    p \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u02e2 Ioo (min w.im z.im) (max w.im z.im) \\ t \u2192 HasFDerivAt F (F' p) p\n\u22a2 IntegrableOn (fun x => \u2191((fun p => -(I \u2022 F' p)) x) (1, 0) + \u2191(F' x) (0, 1)) ([[z.re, w.re]] \u00d7\u02e2 [[w.im, z.im]])\n[PROOFSTEP]\nsimpa only [hF'] using Hi.neg\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz w : \u2102\nHd : DifferentiableOn \u211d f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHi : IntegrableOn (fun z => I \u2022 \u2191(fderiv \u211d f z) 1 - \u2191(fderiv \u211d f z) I) ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nx : \u2102\nhx : x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ \u2205\n\u22a2 [[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]] \u2208 \ud835\udcdd x\n[PROOFSTEP]\nsimpa only [\u2190 mem_interior_iff_mem_nhds, interior_reProdIm, uIcc, interior_Icc] using hx.1\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    0\n[PROOFSTEP]\nhave : \u2200 z, I \u2022 (fderiv \u2102 f z).restrictScalars \u211d 1 = (fderiv \u2102 f z).restrictScalars \u211d I := fun z \u21a6 by\n  rw [(fderiv \u2102 f _).coe_restrictScalars', \u2190 (fderiv \u2102 f _).map_smul, smul_eq_mul, mul_one]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz\u271d w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z\u271d.re, w.re]] \u00d7\u2102 [[z\u271d.im, w.im]])\nHd :\n  \u2200 (x : \u2102),\n    x \u2208 Ioo (min z\u271d.re w.re) (max z\u271d.re w.re) \u00d7\u2102 Ioo (min z\u271d.im w.im) (max z\u271d.im w.im) \\ s \u2192 DifferentiableAt \u2102 f x\nz : \u2102\n\u22a2 I \u2022 \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) 1 =\n    \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) I\n[PROOFSTEP]\nrw [(fderiv \u2102 f _).coe_restrictScalars', \u2190 (fderiv \u2102 f _).map_smul, smul_eq_mul, mul_one]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 DifferentiableAt \u2102 f x\nthis :\n  \u2200 (z : \u2102),\n    I \u2022 \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) 1 =\n      \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) I\n\u22a2 (((\u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191z.im * I)) - \u222b (x : \u211d) in z.re..w.re, f (\u2191x + \u2191w.im * I)) +\n        I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191w.re + \u2191y * I)) -\n      I \u2022 \u222b (y : \u211d) in z.im..w.im, f (\u2191z.re + \u2191y * I) =\n    0\n[PROOFSTEP]\nrefine\n  (integral_boundary_rect_of_hasFDerivAt_real_off_countable f (fun z => (fderiv \u2102 f z).restrictScalars \u211d) z w s hs Hc\n        (fun x hx => (Hd x hx).hasFDerivAt.restrictScalars \u211d) ?_).trans\n    ?_\n[GOAL]\ncase refine_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 DifferentiableAt \u2102 f x\nthis :\n  \u2200 (z : \u2102),\n    I \u2022 \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) 1 =\n      \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) I\n\u22a2 IntegrableOn\n    (fun z =>\n      I \u2022 \u2191((fun z => ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) z) 1 -\n        \u2191((fun z => ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) z) I)\n    ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase refine_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nz w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nHc : ContinuousOn f ([[z.re, w.re]] \u00d7\u2102 [[z.im, w.im]])\nHd :\n  \u2200 (x : \u2102), x \u2208 Ioo (min z.re w.re) (max z.re w.re) \u00d7\u2102 Ioo (min z.im w.im) (max z.im w.im) \\ s \u2192 DifferentiableAt \u2102 f x\nthis :\n  \u2200 (z : \u2102),\n    I \u2022 \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) 1 =\n      \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f z)) I\n\u22a2 \u222b (x : \u211d) in z.re..w.re,\n      \u222b (y : \u211d) in z.im..w.im,\n        I \u2022 \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f (\u2191x + \u2191y * I))) 1 -\n          \u2191(ContinuousLinearMap.restrictScalars \u211d (fderiv \u2102 f (\u2191x + \u2191y * I))) I =\n    0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nr R : \u211d\nh0 : 0 < r\nhle : r \u2264 R\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ ball c r)\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c r) \\ s \u2192 DifferentiableAt \u2102 f z\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nset A := closedBall c R \\ ball c r\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nr R : \u211d\nh0 : 0 < r\nhle : r \u2264 R\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c r) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c R \\ ball c r\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 : \u2203 a, Real.exp a = r\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nr R : \u211d\nh0 : 0 < r\nhle : r \u2264 R\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c r) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c R \\ ball c r\nhc : ContinuousOn f A\n\u22a2 \u2203 a, Real.exp a = r\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nhle : Real.exp a \u2264 R\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c R \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nexact \u27e8Real.log r, Real.exp_log h0\u27e9\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nhle : Real.exp a \u2264 R\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c R \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nobtain \u27e8b, rfl\u27e9 : \u2203 b, Real.exp b = R\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nhle : Real.exp a \u2264 R\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c R \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 \u2203 b, Real.exp b = R\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : Real.exp a \u2264 Real.exp b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, Real.exp b), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nexact \u27e8Real.log R, Real.exp_log (h0.trans_le hle)\u27e9\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : Real.exp a \u2264 Real.exp b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, Real.exp b), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nrw [Real.exp_le_exp] at hle \n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 (\u222e (z : \u2102) in C(c, Real.exp b), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nsuffices (\u222b \u03b8 in (0)..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8)) = \u222b \u03b8 in (0)..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n  by\n  simpa only [circleIntegral, add_sub_cancel', ofReal_exp, \u2190 exp_add, smul_smul, \u2190 div_eq_mul_inv,\n    mul_div_cancel_left _ (circleMap_ne_center (Real.exp_pos _).ne'), circleMap_sub_center, deriv_circleMap]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nthis :\n  \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n\u22a2 (\u222e (z : \u2102) in C(c, Real.exp b), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, Real.exp a), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nsimpa only [circleIntegral, add_sub_cancel', ofReal_exp, \u2190 exp_add, smul_smul, \u2190 div_eq_mul_inv,\n  mul_div_cancel_left _ (circleMap_ne_center (Real.exp_pos _).ne'), circleMap_sub_center, deriv_circleMap]\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nset R := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nset g : \u2102 \u2192 \u2102 := (\u00b7 + \u00b7) c \u2218 exp\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nhave hdg : Differentiable \u2102 g := differentiable_exp.const_add _\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nreplace hs : (g \u207b\u00b9' s).Countable := (hs.preimage (add_right_injective c)).preimage_cexp\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nhave h_maps : MapsTo g R A := by rintro z \u27e8h, -\u27e9; simpa [dist_eq, abs_exp, hle] using h.symm\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\n\u22a2 MapsTo g R A\n[PROOFSTEP]\nrintro z \u27e8h, -\u27e9\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nz : \u2102\nh : z \u2208 re \u207b\u00b9' [[a, b]]\n\u22a2 g z \u2208 A\n[PROOFSTEP]\nsimpa [dist_eq, abs_exp, hle] using h.symm\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nreplace hc : ContinuousOn (f \u2218 g) R\n[GOAL]\ncase hc\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nhc : ContinuousOn f A\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\n\u22a2 ContinuousOn (f \u2218 g) R\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\nhc : ContinuousOn (f \u2218 g) R\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nexact hc.comp hdg.continuous.continuousOn h_maps\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\nhc : ContinuousOn (f \u2218 g) R\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nreplace hd :\n  \u2200 z \u2208 Ioo (min a b) (max a b) \u00d7\u2102 Ioo (min 0 (2 * \u03c0)) (max 0 (2 * \u03c0)) \\ g \u207b\u00b9' s, DifferentiableAt \u2102 (f \u2218 g) z\n[GOAL]\ncase hd\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\nhc : ContinuousOn (f \u2218 g) R\n\u22a2 \u2200 (z : \u2102), z \u2208 Ioo (min a b) (max a b) \u00d7\u2102 Ioo (min 0 (2 * \u03c0)) (max 0 (2 * \u03c0)) \\ g \u207b\u00b9' s \u2192 DifferentiableAt \u2102 (f \u2218 g) z\n[PROOFSTEP]\nrefine' fun z hz => (hd (g z) \u27e8_, hz.2\u27e9).comp z (hdg _)\n[GOAL]\ncase hd\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nhd : \u2200 (z : \u2102), z \u2208 (ball c (Real.exp b) \\ closedBall c (Real.exp a)) \\ s \u2192 DifferentiableAt \u2102 f z\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\nhc : ContinuousOn (f \u2218 g) R\nz : \u2102\nhz : z \u2208 Ioo (min a b) (max a b) \u00d7\u2102 Ioo (min 0 (2 * \u03c0)) (max 0 (2 * \u03c0)) \\ g \u207b\u00b9' s\n\u22a2 g z \u2208 ball c (Real.exp b) \\ closedBall c (Real.exp a)\n[PROOFSTEP]\nsimpa [dist_eq, abs_exp, hle, and_comm] using hz.1.1\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\na : \u211d\nh0 : 0 < Real.exp a\nb : \u211d\nhle : a \u2264 b\nA : Set \u2102 := closedBall c (Real.exp b) \\ ball c (Real.exp a)\nR : Set \u2102 := [[a, b]] \u00d7\u2102 [[0, 2 * \u03c0]]\ng : \u2102 \u2192 \u2102 := (fun x x_1 => x + x_1) c \u2218 exp\nhdg : Differentiable \u2102 g\nhs : Set.Countable (g \u207b\u00b9' s)\nh_maps : MapsTo g R A\nhc : ContinuousOn (f \u2218 g) R\nhd :\n  \u2200 (z : \u2102), z \u2208 Ioo (min a b) (max a b) \u00d7\u2102 Ioo (min 0 (2 * \u03c0)) (max 0 (2 * \u03c0)) \\ g \u207b\u00b9' s \u2192 DifferentiableAt \u2102 (f \u2218 g) z\n\u22a2 \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp b) \u03b8) =\n    \u222b (\u03b8 : \u211d) in 0 ..2 * \u03c0, I \u2022 f (circleMap c (Real.exp a) \u03b8)\n[PROOFSTEP]\nsimpa [circleMap, exp_periodic _, sub_eq_zero, \u2190 exp_add] using\n  integral_boundary_rect_eq_zero_of_differentiable_on_off_countable _ \u27e8a, 0\u27e9 \u27e8b, 2 * \u03c0\u27e9 _ hs hc hd\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 y\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 norm_le_zero_iff]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 0\n[PROOFSTEP]\nrefine' le_of_forall_le_of_dense fun \u03b5 \u03b50 => _\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8\u03b4, \u03b40, h\u03b4\u27e9 : \u2203 \u03b4 > (0 : \u211d), \u2200 z \u2208 closedBall c \u03b4 \\ { c }, dist (f z) y < \u03b5 / (2 * \u03c0)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nexact ((nhdsWithin_hasBasis nhds_basis_closedBall _).tendsto_iff nhds_basis_ball).1 hy _ (div_pos \u03b50 Real.two_pi_pos)\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nobtain \u27e8r, hr0, hr\u03b4, hrR\u27e9 : \u2203 r, 0 < r \u2227 r \u2264 \u03b4 \u2227 r \u2264 R := \u27e8min \u03b4 R, lt_min \u03b40 h0, min_le_left _ _, min_le_right _ _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nhave hsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ { c } :=\n  diff_subset_diff_right (singleton_subset_iff.2 <| mem_ball_self hr0)\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nhave hsub' : ball c R \\ closedBall c r \u2286 ball c R \\ { c } :=\n  diff_subset_diff_right (singleton_subset_iff.2 <| mem_closedBall_self hr0.le)\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\nhave hzne : \u2200 z \u2208 sphere c r, z \u2260 c := fun z hz => ne_of_mem_of_not_mem hz fun h => hr0.ne' <| dist_self c \u25b8 Eq.symm h\n[GOAL]\ncase intro.intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 \u2264 \u03b5\n[PROOFSTEP]\ncalc\n  \u2016(\u222e z in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 =\n      \u2016(\u222e z in C(c, r), (z - c)\u207b\u00b9 \u2022 f z) - \u222e z in C(c, r), (z - c)\u207b\u00b9 \u2022 y\u2016 :=\n    by\n    congr 2\n    \u00b7\n      exact\n        circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0 hrR hs (hc.mono hsub)\n          fun z hz => hd z \u27e8hsub' hz.1, hz.2\u27e9\n    \u00b7 simp [hr0.ne']\n  _ = \u2016\u222e z in C(c, r), (z - c)\u207b\u00b9 \u2022 (f z - y)\u2016 := by\n    simp only [smul_sub]\n    have hc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r) :=\n      (continuousOn_id.sub continuousOn_const).inv\u2080 fun z hz => sub_ne_zero.2 <| hzne _ hz\n    rw [circleIntegral.integral_sub] <;> refine' (hc'.smul _).circleIntegrable hr0.le\n    \u00b7 exact hc.mono <| subset_inter (sphere_subset_closedBall.trans <| closedBall_subset_closedBall hrR) hzne\n    \u00b7 exact continuousOn_const\n  _ \u2264 2 * \u03c0 * r * (r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))) :=\n    by\n    refine' circleIntegral.norm_integral_le_of_norm_le_const hr0.le fun z hz => _\n    specialize hzne z hz\n    rw [mem_sphere, dist_eq_norm] at hz \n    rw [norm_smul, norm_inv, hz, \u2190 dist_eq_norm]\n    refine' mul_le_mul_of_nonneg_left (h\u03b4 _ \u27e8_, hzne\u27e9).le (inv_nonneg.2 hr0.le)\n    rwa [mem_closedBall_iff_norm, hz]\n  _ = \u03b5 := by field_simp [hr0.ne', Real.two_pi_pos.ne']; ac_rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) - (2 * \u2191\u03c0 * I) \u2022 y\u2016 =\n    \u2016(\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z) - \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 y\u2016\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - c)\u207b\u00b9 \u2022 f z) = \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z\n[PROOFSTEP]\nexact\n  circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0 hrR hs (hc.mono hsub) fun z hz =>\n    hd z \u27e8hsub' hz.1, hz.2\u27e9\n[GOAL]\ncase e_a.e_a\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 (2 * \u2191\u03c0 * I) \u2022 y = \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 y\n[PROOFSTEP]\nsimp [hr0.ne']\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z) - \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 y\u2016 =\n    \u2016\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 (f z - y)\u2016\n[PROOFSTEP]\nsimp only [smul_sub]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z) - \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 y\u2016 =\n    \u2016\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z - (z - c)\u207b\u00b9 \u2022 y\u2016\n[PROOFSTEP]\nhave hc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r) :=\n  (continuousOn_id.sub continuousOn_const).inv\u2080 fun z hz => sub_ne_zero.2 <| hzne _ hz\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nhc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r)\n\u22a2 \u2016(\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z) - \u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 y\u2016 =\n    \u2016\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 f z - (z - c)\u207b\u00b9 \u2022 y\u2016\n[PROOFSTEP]\nrw [circleIntegral.integral_sub]\n[GOAL]\ncase hf\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nhc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r)\n\u22a2 CircleIntegrable (fun z => (z - c)\u207b\u00b9 \u2022 f z) c r\n[PROOFSTEP]\nrefine' (hc'.smul _).circleIntegrable hr0.le\n[GOAL]\ncase hg\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nhc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r)\n\u22a2 CircleIntegrable (fun z => (z - c)\u207b\u00b9 \u2022 y) c r\n[PROOFSTEP]\nrefine' (hc'.smul _).circleIntegrable hr0.le\n[GOAL]\ncase hf\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nhc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r)\n\u22a2 ContinuousOn (fun x => f x) (sphere c r)\n[PROOFSTEP]\nexact hc.mono <| subset_inter (sphere_subset_closedBall.trans <| closedBall_subset_closedBall hrR) hzne\n[GOAL]\ncase hg\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nhc' : ContinuousOn (fun z => (z - c)\u207b\u00b9) (sphere c r)\n\u22a2 ContinuousOn (fun x => y) (sphere c r)\n[PROOFSTEP]\nexact continuousOn_const\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 \u2016\u222e (z : \u2102) in C(c, r), (z - c)\u207b\u00b9 \u2022 (f z - y)\u2016 \u2264 2 * \u03c0 * r * (r\u207b\u00b9 * (\u03b5 / (2 * \u03c0)))\n[PROOFSTEP]\nrefine' circleIntegral.norm_integral_le_of_norm_le_const hr0.le fun z hz => _\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\nz : \u2102\nhz : z \u2208 sphere c r\n\u22a2 \u2016(z - c)\u207b\u00b9 \u2022 (f z - y)\u2016 \u2264 r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))\n[PROOFSTEP]\nspecialize hzne z hz\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nz : \u2102\nhz : z \u2208 sphere c r\nhzne : z \u2260 c\n\u22a2 \u2016(z - c)\u207b\u00b9 \u2022 (f z - y)\u2016 \u2264 r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))\n[PROOFSTEP]\nrw [mem_sphere, dist_eq_norm] at hz \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nz : \u2102\nhz : \u2016z - c\u2016 = r\nhzne : z \u2260 c\n\u22a2 \u2016(z - c)\u207b\u00b9 \u2022 (f z - y)\u2016 \u2264 r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))\n[PROOFSTEP]\nrw [norm_smul, norm_inv, hz, \u2190 dist_eq_norm]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nz : \u2102\nhz : \u2016z - c\u2016 = r\nhzne : z \u2260 c\n\u22a2 r\u207b\u00b9 * dist (f z) y \u2264 r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left (h\u03b4 _ \u27e8_, hzne\u27e9).le (inv_nonneg.2 hr0.le)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nz : \u2102\nhz : \u2016z - c\u2016 = r\nhzne : z \u2260 c\n\u22a2 z \u2208 closedBall c \u03b4\n[PROOFSTEP]\nrwa [mem_closedBall_iff_norm, hz]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 2 * \u03c0 * r * (r\u207b\u00b9 * (\u03b5 / (2 * \u03c0))) = \u03b5\n[PROOFSTEP]\nfield_simp [hr0.ne', Real.two_pi_pos.ne']\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc : \u2102\nR : \u211d\nh0 : 0 < R\nf : \u2102 \u2192 E\ny : E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : \u2200 (z : \u2102), z \u2208 (ball c R \\ {c}) \\ s \u2192 DifferentiableAt \u2102 f z\nhy : Tendsto f (\ud835\udcdd[{c}\u1d9c] c) (\ud835\udcdd y)\n\u03b5 : \u211d\n\u03b50 : 0 < \u03b5\n\u03b4 : \u211d\n\u03b40 : \u03b4 > 0\nh\u03b4 : \u2200 (z : \u2102), z \u2208 closedBall c \u03b4 \\ {c} \u2192 dist (f z) y < \u03b5 / (2 * \u03c0)\nr : \u211d\nhr0 : 0 < r\nhr\u03b4 : r \u2264 \u03b4\nhrR : r \u2264 R\nhsub : closedBall c R \\ ball c r \u2286 closedBall c R \\ {c}\nhsub' : ball c R \\ closedBall c r \u2286 ball c R \\ {c}\nhzne : \u2200 (z : \u2102), z \u2208 sphere c r \u2192 z \u2260 c\n\u22a2 2 * \u03c0 * r * \u03b5 = \u03b5 * (r * (2 * \u03c0))\n[PROOFSTEP]\nac_rfl\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nh0 : 0 \u2264 R\nf : \u2102 \u2192 E\nc : \u2102\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (z : \u2102), z \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f z\n\u22a2 (\u222e (z : \u2102) in C(c, R), f z) = 0\n[PROOFSTEP]\nrcases h0.eq_or_lt with (rfl | h0)\n[GOAL]\ncase inl\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf : \u2102 \u2192 E\nc : \u2102\ns : Set \u2102\nhs : Set.Countable s\nh0 : 0 \u2264 0\nhc : ContinuousOn f (closedBall c 0)\nhd : \u2200 (z : \u2102), z \u2208 ball c 0 \\ s \u2192 DifferentiableAt \u2102 f z\n\u22a2 (\u222e (z : \u2102) in C(c, 0), f z) = 0\n[PROOFSTEP]\napply circleIntegral.integral_radius_zero\n[GOAL]\ncase inr\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nh0\u271d : 0 \u2264 R\nf : \u2102 \u2192 E\nc : \u2102\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (z : \u2102), z \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f z\nh0 : 0 < R\n\u22a2 (\u222e (z : \u2102) in C(c, R), f z) = 0\n[PROOFSTEP]\ncalc\n  (\u222e z in C(c, R), f z) = \u222e z in C(c, R), (z - c)\u207b\u00b9 \u2022 (z - c) \u2022 f z :=\n    (circleIntegral.integral_sub_inv_smul_sub_smul _ _ _ _).symm\n  _ = (2 * \u2191\u03c0 * I : \u2102) \u2022 (c - c) \u2022 f c :=\n    (circleIntegral_sub_center_inv_smul_of_differentiable_on_off_countable h0 hs\n      ((continuousOn_id.sub continuousOn_const).smul hc) fun z hz => (differentiableAt_id.sub_const _).smul (hd z hz))\n  _ = 0 := by rw [sub_self, zero_smul, smul_zero]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nh0\u271d : 0 \u2264 R\nf : \u2102 \u2192 E\nc : \u2102\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (z : \u2102), z \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f z\nh0 : 0 < R\n\u22a2 (2 * \u2191\u03c0 * I) \u2022 (c - c) \u2022 f c = 0\n[PROOFSTEP]\nrw [sub_self, zero_smul, smul_zero]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hR : 0 < R := dist_nonneg.trans_lt hw.1\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nset F : \u2102 \u2192 E := dslope f w\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hws : (insert w s).Countable := hs.insert w\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hcF : ContinuousOn F (closedBall c R) := (continuousOn_dslope <| closedBall_mem_nhds_of_mem hw.1).2 \u27e8hc, hd _ hw\u27e9\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hdF : \u2200 z \u2208 ball (c : \u2102) R \\ insert w s, DifferentiableAt \u2102 F z := fun z hz =>\n  (differentiableAt_dslope_of_ne (ne_of_mem_of_not_mem (mem_insert _ _) hz.2).symm).2\n    (hd _ (diff_subset_diff_right (subset_insert _ _) hz))\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave HI := circleIntegral_eq_zero_of_differentiable_on_off_countable hR.le hws hcF hdF\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hne : \u2200 z \u2208 sphere c R, z \u2260 w := fun z hz => ne_of_mem_of_not_mem hz (ne_of_lt hw.1)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\nhne : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 z \u2260 w\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hFeq : EqOn F (fun z => (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w) (sphere c R) := fun z hz \u21a6\n  calc\n    F z = (z - w)\u207b\u00b9 \u2022 (f z - f w) := update_noteq (hne z hz) _ _\n    _ = (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w := smul_sub _ _ _\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\nhne : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 z \u2260 w\nhFeq : EqOn F (fun z => (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w) (sphere c R)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nhave hc' : ContinuousOn (fun z => (z - w)\u207b\u00b9) (sphere c R) :=\n  (continuousOn_id.sub continuousOn_const).inv\u2080 fun z hz => sub_ne_zero.2 <| hne z hz\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\nhne : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 z \u2260 w\nhFeq : EqOn F (fun z => (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w) (sphere c R)\nhc' : ContinuousOn (fun z => (z - w)\u207b\u00b9) (sphere c R)\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nrw [\u2190 circleIntegral.integral_sub_inv_of_mem_ball hw.1, \u2190 circleIntegral.integral_smul_const, \u2190 sub_eq_zero, \u2190\n  circleIntegral.integral_sub, \u2190 circleIntegral.integral_congr hR.le hFeq, HI]\n[GOAL]\ncase hf\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\nhne : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 z \u2260 w\nhFeq : EqOn F (fun z => (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w) (sphere c R)\nhc' : ContinuousOn (fun z => (z - w)\u207b\u00b9) (sphere c R)\n\u22a2 CircleIntegrable (fun z => (z - w)\u207b\u00b9 \u2022 f z) c R\ncase hg\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R \\ s\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nF : \u2102 \u2192 E := dslope f w\nhws : Set.Countable (insert w s)\nhcF : ContinuousOn F (closedBall c R)\nhdF : \u2200 (z : \u2102), z \u2208 ball c R \\ insert w s \u2192 DifferentiableAt \u2102 F z\nHI : (\u222e (z : \u2102) in C(c, R), F z) = 0\nhne : \u2200 (z : \u2102), z \u2208 sphere c R \u2192 z \u2260 w\nhFeq : EqOn F (fun z => (z - w)\u207b\u00b9 \u2022 f z - (z - w)\u207b\u00b9 \u2022 f w) (sphere c R)\nhc' : ContinuousOn (fun z => (z - w)\u207b\u00b9) (sphere c R)\n\u22a2 CircleIntegrable (fun z => (z - w)\u207b\u00b9 \u2022 f w) c R\n[PROOFSTEP]\nexacts [(hc'.smul (hc.mono sphere_subset_closedBall)).circleIntegrable hR.le,\n  (hc'.smul continuousOn_const).circleIntegrable hR.le]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nhave hR : 0 < R := dist_nonneg.trans_lt hw\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nsuffices w \u2208 closure (ball c R \\ s) by\n  lift R to \u211d\u22650 using hR.le\n  have A : ContinuousAt (fun w => (2 * \u03c0 * I : \u2102)\u207b\u00b9 \u2022 \u222e z in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) w :=\n    by\n    have := hasFPowerSeriesOn_cauchy_integral ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR\n    refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)\n    rwa [Metric.emetric_ball_nnreal]\n  have B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)\n  refine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)\n  intro z hz\n  rw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd, inv_smul_smul\u2080]\n  simp [Real.pi_ne_zero, I_ne_zero]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nthis : w \u2208 closure (ball c R \\ s)\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nlift R to \u211d\u22650 using hR.le\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nhave A : ContinuousAt (fun w => (2 * \u03c0 * I : \u2102)\u207b\u00b9 \u2022 \u222e z in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) w :=\n  by\n  have := hasFPowerSeriesOn_cauchy_integral ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR\n  refine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)\n  rwa [Metric.emetric_ball_nnreal]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\n\u22a2 ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\n[PROOFSTEP]\nhave := hasFPowerSeriesOn_cauchy_integral ((hc.mono sphere_subset_closedBall).circleIntegrable R.coe_nonneg) hR\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis\u271d : w \u2208 closure (ball c \u2191R \\ s)\nthis :\n  HasFPowerSeriesOnBall (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) (cauchyPowerSeries f c \u2191R) c\n    \u2191R\n\u22a2 ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\n[PROOFSTEP]\nrefine' this.continuousOn.continuousAt (EMetric.isOpen_ball.mem_nhds _)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis\u271d : w \u2208 closure (ball c \u2191R \\ s)\nthis :\n  HasFPowerSeriesOnBall (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) (cauchyPowerSeries f c \u2191R) c\n    \u2191R\n\u22a2 w \u2208 EMetric.ball c \u2191R\n[PROOFSTEP]\nrwa [Metric.emetric_ball_nnreal]\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\nA : ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nhave B : ContinuousAt f w := hc.continuousAt (closedBall_mem_nhds_of_mem hw)\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\nA : ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\nB : ContinuousAt f w\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nrefine' tendsto_nhds_unique_of_frequently_eq A B ((mem_closure_iff_frequently.1 this).mono _)\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\nA : ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\nB : ContinuousAt f w\n\u22a2 \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - x)\u207b\u00b9 \u2022 f z) = f x\n[PROOFSTEP]\nintro z hz\n[GOAL]\ncase intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\nA : ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\nB : ContinuousAt f w\nz : \u2102\nhz : z \u2208 ball c \u2191R \\ s\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z_1 : \u2102) in C(c, \u2191R), (z_1 - z)\u207b\u00b9 \u2022 f z_1) = f z\n[PROOFSTEP]\nrw [circleIntegral_sub_inv_smul_of_differentiable_on_off_countable_aux hs hz hc hd, inv_smul_smul\u2080]\n[GOAL]\ncase intro.hc\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nR : \u211d\u22650\nhw : w \u2208 ball c \u2191R\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (x : \u2102), x \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < \u2191R\nthis : w \u2208 closure (ball c \u2191R \\ s)\nA : ContinuousAt (fun w => (2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - w)\u207b\u00b9 \u2022 f z) w\nB : ContinuousAt f w\nz : \u2102\nhz : z \u2208 ball c \u2191R \\ s\n\u22a2 2 * \u2191\u03c0 * I \u2260 0\n[PROOFSTEP]\nsimp [Real.pi_ne_zero, I_ne_zero]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\n\u22a2 w \u2208 closure (ball c R \\ s)\n[PROOFSTEP]\nrefine'\n  mem_closure_iff_nhds.2 fun t ht =>\n    _\n      -- TODO: generalize to any vector space over `\u211d`\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\n\u22a2 Set.Nonempty (t \u2229 (ball c R \\ s))\n[PROOFSTEP]\nset g : \u211d \u2192 \u2102 := fun x => w + ofReal x\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\n\u22a2 Set.Nonempty (t \u2229 (ball c R \\ s))\n[PROOFSTEP]\nhave : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w) := (continuous_const.add continuous_ofReal).tendsto' 0 w (add_zero _)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\n\u22a2 Set.Nonempty (t \u2229 (ball c R \\ s))\n[PROOFSTEP]\nrcases mem_nhds_iff_exists_Ioo_subset.1 (this <| inter_mem ht <| isOpen_ball.mem_nhds hw) with \u27e8l, u, hlu\u2080, hlu_sub\u27e9\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\n\u22a2 Set.Nonempty (t \u2229 (ball c R \\ s))\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : (Ioo l u \\ g \u207b\u00b9' s).Nonempty :=\n  by\n  refine' nonempty_diff.2 fun hsub => _\n  have : (Ioo l u).Countable := (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub\n  rw [\u2190 Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu\u2080.1.trans hlu\u2080.2)] at this \n  exact this.not_lt Cardinal.aleph0_lt_continuum\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\n\u22a2 Set.Nonempty (Ioo l u \\ g \u207b\u00b9' s)\n[PROOFSTEP]\nrefine' nonempty_diff.2 fun hsub => _\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\nhsub : Ioo l u \u2286 g \u207b\u00b9' s\n\u22a2 False\n[PROOFSTEP]\nhave : (Ioo l u).Countable := (hs.preimage ((add_right_injective w).comp ofReal_injective)).mono hsub\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis\u271d : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\nhsub : Ioo l u \u2286 g \u207b\u00b9' s\nthis : Set.Countable (Ioo l u)\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Cardinal.le_aleph0_iff_set_countable, Cardinal.mk_Ioo_real (hlu\u2080.1.trans hlu\u2080.2)] at this \n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis\u271d : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\nhsub : Ioo l u \u2286 g \u207b\u00b9' s\nthis : Cardinal.continuum \u2264 Cardinal.aleph0\n\u22a2 False\n[PROOFSTEP]\nexact this.not_lt Cardinal.aleph0_lt_continuum\n[GOAL]\ncase intro.intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\nhR : 0 < R\nt : Set \u2102\nht : t \u2208 \ud835\udcdd w\ng : \u211d \u2192 \u2102 := fun x => w + \u2191ofReal x\nthis : Tendsto g (\ud835\udcdd 0) (\ud835\udcdd w)\nl u : \u211d\nhlu\u2080 : 0 \u2208 Ioo l u\nhlu_sub : Ioo l u \u2286 g \u207b\u00b9' (t \u2229 ball c R)\nx : \u211d\nhx : x \u2208 Ioo l u \\ g \u207b\u00b9' s\n\u22a2 Set.Nonempty (t \u2229 (ball c R \\ s))\n[PROOFSTEP]\nexact \u27e8g x, (hlu_sub hx.1).1, (hlu_sub hx.1).2, hx.2\u27e9\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 (\u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = (2 * \u2191\u03c0 * I) \u2022 f w\n[PROOFSTEP]\nrw [\u2190 two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd, smul_inv_smul\u2080]\n[GOAL]\ncase hc\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (x : \u2102), x \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f x\n\u22a2 2 * \u2191\u03c0 * I \u2260 0\n[PROOFSTEP]\nsimp [Real.pi_ne_zero, I_ne_zero]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\nhf : DiffContOnCl \u2102 f (ball c R)\nhw : w \u2208 ball c R\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nhave hR : 0 < R := not_le.mp (ball_eq_empty.not.mp (Set.nonempty_of_mem hw).ne_empty)\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\nhf : DiffContOnCl \u2102 f (ball c R)\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, R), (z - w)\u207b\u00b9 \u2022 f z) = f w\n[PROOFSTEP]\nrefine' two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable countable_empty hw _ _\n[GOAL]\ncase refine'_1\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\nhf : DiffContOnCl \u2102 f (ball c R)\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 ContinuousOn (fun z => f z) (closedBall c R)\n[PROOFSTEP]\nsimpa only [closure_ball c hR.ne.symm] using hf.continuousOn\n[GOAL]\ncase refine'_2\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\nf : \u2102 \u2192 E\nhf : DiffContOnCl \u2102 f (ball c R)\nhw : w \u2208 ball c R\nhR : 0 < R\n\u22a2 \u2200 (x : \u2102), x \u2208 ball c R \\ \u2205 \u2192 DifferentiableAt \u2102 (fun z => f z) x\n[PROOFSTEP]\nsimpa only [diff_empty] using fun z hz => hf.differentiableAt isOpen_ball hz\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\nc w : \u2102\ns : Set \u2102\nhs : Set.Countable s\nhw : w \u2208 ball c R\nf : \u2102 \u2192 \u2102\nhc : ContinuousOn f (closedBall c R)\nhd : \u2200 (z : \u2102), z \u2208 ball c R \\ s \u2192 DifferentiableAt \u2102 f z\n\u22a2 (\u222e (z : \u2102) in C(c, R), f z / (z - w)) = 2 * \u2191\u03c0 * I * f w\n[PROOFSTEP]\nsimpa only [smul_eq_mul, div_eq_inv_mul] using\n  circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\u22650\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (z : \u2102), z \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f z\nhR : 0 < R\nw : \u2102\nhw : w \u2208 EMetric.ball 0 \u2191R\n\u22a2 HasSum (fun n => \u2191(cauchyPowerSeries f c (\u2191R) n) fun x => w) (f (c + w))\n[PROOFSTEP]\nhave hw' : c + w \u2208 ball c R := by\n  simpa only [add_mem_ball_iff_norm, \u2190 coe_nnnorm, mem_emetric_ball_zero_iff, NNReal.coe_lt_coe,\n    ENNReal.coe_lt_coe] using hw\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\u22650\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (z : \u2102), z \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f z\nhR : 0 < R\nw : \u2102\nhw : w \u2208 EMetric.ball 0 \u2191R\n\u22a2 c + w \u2208 ball c \u2191R\n[PROOFSTEP]\nsimpa only [add_mem_ball_iff_norm, \u2190 coe_nnnorm, mem_emetric_ball_zero_iff, NNReal.coe_lt_coe, ENNReal.coe_lt_coe] using\n  hw\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\u22650\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (z : \u2102), z \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f z\nhR : 0 < R\nw : \u2102\nhw : w \u2208 EMetric.ball 0 \u2191R\nhw' : c + w \u2208 ball c \u2191R\n\u22a2 HasSum (fun n => \u2191(cauchyPowerSeries f c (\u2191R) n) fun x => w) (f (c + w))\n[PROOFSTEP]\nrw [\u2190 two_pi_I_inv_smul_circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw' hc hd]\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nR : \u211d\u22650\nc : \u2102\nf : \u2102 \u2192 E\ns : Set \u2102\nhs : Set.Countable s\nhc : ContinuousOn f (closedBall c \u2191R)\nhd : \u2200 (z : \u2102), z \u2208 ball c \u2191R \\ s \u2192 DifferentiableAt \u2102 f z\nhR : 0 < R\nw : \u2102\nhw : w \u2208 EMetric.ball 0 \u2191R\nhw' : c + w \u2208 ball c \u2191R\n\u22a2 HasSum (fun n => \u2191(cauchyPowerSeries f c (\u2191R) n) fun x => w)\n    ((2 * \u2191\u03c0 * I)\u207b\u00b9 \u2022 \u222e (z : \u2102) in C(c, \u2191R), (z - (c + w))\u207b\u00b9 \u2022 f z)\n[PROOFSTEP]\nexact (hasFPowerSeriesOn_cauchy_integral ((hc.mono sphere_subset_closedBall).circleIntegrable R.2) hR).hasSum hw\n[GOAL]\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\n\u22a2 AnalyticAt \u2102 f z\n[PROOFSTEP]\nrcases nhds_basis_closedBall.mem_iff.1 hz with \u27e8R, hR0, hRs\u27e9\n[GOAL]\ncase intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\nhR0 : 0 < R\nhRs : closedBall z R \u2286 s\n\u22a2 AnalyticAt \u2102 f z\n[PROOFSTEP]\nlift R to \u211d\u22650 using hR0.le\n[GOAL]\ncase intro.intro.intro\nE : Type u\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\ns : Set \u2102\nf : \u2102 \u2192 E\nz : \u2102\nhd : DifferentiableOn \u2102 f s\nhz : s \u2208 \ud835\udcdd z\nR : \u211d\u22650\nhR0 : 0 < \u2191R\nhRs : closedBall z \u2191R \u2286 s\n\u22a2 AnalyticAt \u2102 f z\n[PROOFSTEP]\nexact ((hd.mono hRs).hasFPowerSeriesOnBall hR0).analyticAt\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.CauchyIntegral", "llama_tokens": 51729, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8459424334245618, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.5077248011456729}}
{"text": "[GOAL]\nC : FloatCfg\ne : \u2124\nm : \u2115\n\u22a2 Decidable (ValidFinite e m)\n[PROOFSTEP]\nunfold ValidFinite\n[GOAL]\nC : FloatCfg\ne : \u2124\nm : \u2115\n\u22a2 Decidable (emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m) - \u2191prec) emin)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : FloatCfg\n\u22a2 emin \u2264 emin + \u2191prec - 1\n[PROOFSTEP]\nrw [add_sub_assoc]\n[GOAL]\nC : FloatCfg\n\u22a2 emin \u2264 emin + (\u2191prec - 1)\n[PROOFSTEP]\napply le_add_of_nonneg_right\n[GOAL]\ncase h\nC : FloatCfg\n\u22a2 0 \u2264 \u2191prec - 1\n[PROOFSTEP]\napply sub_nonneg_of_le\n[GOAL]\ncase h.a\nC : FloatCfg\n\u22a2 1 \u2264 \u2191prec\n[PROOFSTEP]\napply Int.ofNat_le_ofNat_of_le\n[GOAL]\ncase h.a.a\nC : FloatCfg\n\u22a2 1 \u2264 prec\n[PROOFSTEP]\nexact C.precPos\n[GOAL]\nC : FloatCfg\n\u22a2 0 < 2\n[PROOFSTEP]\ndecide\n[GOAL]\nC : FloatCfg\nthis : prec \u2264 2 * emax\n\u22a2 emin + \u2191prec - 1 \u2264 \u2191emax\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_le] at this \n[GOAL]\nC : FloatCfg\nthis : \u2191prec \u2264 \u2191(2 * emax)\n\u22a2 emin + \u2191prec - 1 \u2264 \u2191emax\n[PROOFSTEP]\nrw [\u2190 sub_nonneg] at *\n[GOAL]\nC : FloatCfg\nthis\u271d : \u2191prec \u2264 \u2191(2 * emax)\nthis : 0 \u2264 \u2191(2 * emax) - \u2191prec\n\u22a2 0 \u2264 \u2191emax - (emin + \u2191prec - 1)\n[PROOFSTEP]\nsimp only [emin, emax] at *\n[GOAL]\nC : FloatCfg\nthis\u271d : \u2191prec \u2264 \u2191(2 * FloatCfg.emax)\nthis : 0 \u2264 \u2191(2 * FloatCfg.emax) - \u2191prec\n\u22a2 0 \u2264 \u2191FloatCfg.emax - (1 - \u2191FloatCfg.emax + \u2191prec - 1)\n[PROOFSTEP]\nring_nf\n[GOAL]\nC : FloatCfg\nthis\u271d : \u2191prec \u2264 \u2191(2 * FloatCfg.emax)\nthis : 0 \u2264 \u2191(2 * FloatCfg.emax) - \u2191prec\n\u22a2 0 \u2264 \u2191FloatCfg.emax * 2 - \u2191prec\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\nC : FloatCfg\nthis\u271d : \u2191prec \u2264 \u2191(2 * FloatCfg.emax)\nthis : 0 \u2264 \u2191(2 * FloatCfg.emax) - \u2191prec\n\u22a2 0 \u2264 2 * \u2191FloatCfg.emax - \u2191prec\n[PROOFSTEP]\nassumption\n[GOAL]\nC : FloatCfg\n\u22a2 emin = max (emin + \u2191(Nat.size 0) - \u2191prec) emin\n[PROOFSTEP]\nrw [max_eq_right]\n[GOAL]\nC : FloatCfg\n\u22a2 emin + \u2191(Nat.size 0) - \u2191prec \u2264 emin\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\nC : FloatCfg\nn d : \u2115+\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nlet e\u2081 : \u2124 := n.1.size - d.1.size - prec\n[GOAL]\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\ncases' h\u2081 : Int.shift2 d.1 n.1 (e\u2081 + prec) with d\u2081 n\u2081\n[GOAL]\ncase mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nlet e\u2082 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\n[GOAL]\ncase mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nlet e\u2083 := max e\u2082 emin\n[GOAL]\ncase mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\ne\u2083 : \u2124 := max e\u2082 emin\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\ncases' h\u2082 : Int.shift2 d.1 n.1 (e\u2083 + prec) with d\u2082 n\u2082\n[GOAL]\ncase mk.mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\ne\u2083 : \u2124 := max e\u2082 emin\nd\u2082 n\u2082 : \u2115\nh\u2082 : Int.shift2 (\u2191d) (\u2191n) (e\u2083 + \u2191prec) = (d\u2082, n\u2082)\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nlet r := mkRat n\u2082 d\u2082\n[GOAL]\ncase mk.mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\ne\u2083 : \u2124 := max e\u2082 emin\nd\u2082 n\u2082 : \u2115\nh\u2082 : Int.shift2 (\u2191d) (\u2191n) (e\u2083 + \u2191prec) = (d\u2082, n\u2082)\nr : \u211a := mkRat (\u2191n\u2082) d\u2082\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nlet m := r.floor\n[GOAL]\ncase mk.mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\ne\u2083 : \u2124 := max e\u2082 emin\nd\u2082 n\u2082 : \u2115\nh\u2082 : Int.shift2 (\u2191d) (\u2191n) (e\u2083 + \u2191prec) = (d\u2082, n\u2082)\nr : \u211a := mkRat (\u2191n\u2082) d\u2082\nm : \u2124 := Rat.floor r\n\u22a2 Float \u00d7 Bool\n[PROOFSTEP]\nrefine' (Float.finite Bool.false e\u2083 (Int.toNat m) _, r.den = 1)\n[GOAL]\ncase mk.mk\nC : FloatCfg\nn d : \u2115+\ne\u2081 : \u2124 := \u2191(Nat.size \u2191n) - \u2191(Nat.size \u2191d) - \u2191prec\nd\u2081 n\u2081 : \u2115\nh\u2081 : Int.shift2 (\u2191d) (\u2191n) (e\u2081 + \u2191prec) = (d\u2081, n\u2081)\ne\u2082 : \u2124 := if n\u2081 < d\u2081 then e\u2081 - 1 else e\u2081\ne\u2083 : \u2124 := max e\u2082 emin\nd\u2082 n\u2082 : \u2115\nh\u2082 : Int.shift2 (\u2191d) (\u2191n) (e\u2083 + \u2191prec) = (d\u2082, n\u2082)\nr : \u211a := mkRat (\u2191n\u2082) d\u2082\nm : \u2124 := Rat.floor r\n\u22a2 ValidFinite e\u2083 (Int.toNat m)\n[PROOFSTEP]\nexact lcProof\n[GOAL]\nC : FloatCfg\ne : \u2124\nm : \u2115\nv : ValidFinite e m\nm' : \u2115 := Nat.succ m\nss : Nat.size m' = Nat.size m\n\u22a2 ValidFinite e m'\n[PROOFSTEP]\nunfold ValidFinite at *\n[GOAL]\nC : FloatCfg\ne : \u2124\nm : \u2115\nv : emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m) - \u2191prec) emin\nm' : \u2115 := Nat.succ m\nss : Nat.size m' = Nat.size m\n\u22a2 emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m') - \u2191prec) emin\n[PROOFSTEP]\nrw [ss]\n[GOAL]\nC : FloatCfg\ne : \u2124\nm : \u2115\nv : emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m) - \u2191prec) emin\nm' : \u2115 := Nat.succ m\nss : Nat.size m' = Nat.size m\n\u22a2 emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m) - \u2191prec) emin\n[PROOFSTEP]\nexact v\n[GOAL]\nC : FloatCfg\ne : \u2124\nm m' : \u2115\nv : ValidFinite e (Nat.succ m')\nss : Nat.size m' = Nat.size (Nat.succ m')\n\u22a2 ValidFinite e m'\n[PROOFSTEP]\nunfold ValidFinite at *\n[GOAL]\nC : FloatCfg\ne : \u2124\nm m' : \u2115\nv : emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size (Nat.succ m')) - \u2191prec) emin\nss : Nat.size m' = Nat.size (Nat.succ m')\n\u22a2 emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size m') - \u2191prec) emin\n[PROOFSTEP]\nrw [ss]\n[GOAL]\nC : FloatCfg\ne : \u2124\nm m' : \u2115\nv : emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size (Nat.succ m')) - \u2191prec) emin\nss : Nat.size m' = Nat.size (Nat.succ m')\n\u22a2 emin \u2264 e + \u2191prec - 1 \u2227 e + \u2191prec - 1 \u2264 \u2191emax \u2227 e = max (e + \u2191(Nat.size (Nat.succ m')) - \u2191prec) emin\n[PROOFSTEP]\nexact v\n", "meta": {"mathlib_filename": "Mathlib.Data.FP.Basic", "llama_tokens": 3265, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467548438126, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.5077031762404488}}
{"text": "[GOAL]\nM : Type u_1\ninst\u271d : MulOneClass M\na b : M\nh : 1 * a = b\n\u22a2 a = b\n[PROOFSTEP]\nrwa [one_mul] at h \n", "meta": {"mathlib_filename": "Mathlib.Tactic.Linarith.Preprocessing", "llama_tokens": 59, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7879311956428946, "lm_q2_score": 0.6442251133170356, "lm_q1q2_score": 0.5076050637990711}}
{"text": "[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\n\u22a2 oangle o y z = 2 \u2022 oangle o (y - x) (z - x)\n[PROOFSTEP]\nhave hy : y \u2260 0 := by\n  rintro rfl\n  rw [norm_zero, norm_eq_zero] at hxy \n  exact hxyne hxy\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\n\u22a2 y \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx z : V\nhxzne : x \u2260 z\nhxz : \u2016x\u2016 = \u2016z\u2016\nhxyne : x \u2260 0\nhxy : \u2016x\u2016 = \u20160\u2016\n\u22a2 False\n[PROOFSTEP]\nrw [norm_zero, norm_eq_zero] at hxy \n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx z : V\nhxzne : x \u2260 z\nhxz : \u2016x\u2016 = \u2016z\u2016\nhxyne : x \u2260 0\nhxy : x = 0\n\u22a2 False\n[PROOFSTEP]\nexact hxyne hxy\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\n\u22a2 oangle o y z = 2 \u2022 oangle o (y - x) (z - x)\n[PROOFSTEP]\nhave hx : x \u2260 0 := norm_ne_zero_iff.1 (hxy.symm \u25b8 norm_ne_zero_iff.2 hy)\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\n\u22a2 oangle o y z = 2 \u2022 oangle o (y - x) (z - x)\n[PROOFSTEP]\nhave hz : z \u2260 0 := norm_ne_zero_iff.1 (hxz \u25b8 norm_ne_zero_iff.2 hx)\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 oangle o y z = 2 \u2022 oangle o (y - x) (z - x)\n[PROOFSTEP]\ncalc\n  o.oangle y z = o.oangle x z - o.oangle x y := (o.oangle_sub_left hx hy hz).symm\n  _ = \u03c0 - (2 : \u2124) \u2022 o.oangle (x - z) x - (\u03c0 - (2 : \u2124) \u2022 o.oangle (x - y) x) := by\n    rw [o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxzne.symm hxz.symm,\n      o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxyne.symm hxy.symm]\n  _ = (2 : \u2124) \u2022 (o.oangle (x - y) x - o.oangle (x - z) x) := by abel\n  _ = (2 : \u2124) \u2022 o.oangle (x - y) (x - z) := by\n    rw [o.oangle_sub_right (sub_ne_zero_of_ne hxyne) (sub_ne_zero_of_ne hxzne) hx]\n  _ = (2 : \u2124) \u2022 o.oangle (y - x) (z - x) := by rw [\u2190 oangle_neg_neg, neg_sub, neg_sub]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 oangle o x z - oangle o x y = \u2191\u03c0 - 2 \u2022 oangle o (x - z) x - (\u2191\u03c0 - 2 \u2022 oangle o (x - y) x)\n[PROOFSTEP]\nrw [o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxzne.symm hxz.symm,\n  o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hxyne.symm hxy.symm]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 \u2191\u03c0 - 2 \u2022 oangle o (x - z) x - (\u2191\u03c0 - 2 \u2022 oangle o (x - y) x) = 2 \u2022 (oangle o (x - y) x - oangle o (x - z) x)\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 \u2191\u03c0 - 2 \u2022 oangle o (x - z) x - (\u2191\u03c0 - 2 \u2022 oangle o (x - y) x) = 2 \u2022 (oangle o (x - y) x - oangle o (x - z) x)\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 2 \u2022 (oangle o (x - y) x - oangle o (x - z) x) = 2 \u2022 oangle o (x - y) (x - z)\n[PROOFSTEP]\nrw [o.oangle_sub_right (sub_ne_zero_of_ne hxyne) (sub_ne_zero_of_ne hxzne) hx]\n[GOAL]\nV : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup V\ninst\u271d\u00b9 : InnerProductSpace \u211d V\ninst\u271d : Fact (finrank \u211d V = 2)\no : Orientation \u211d V (Fin 2)\nx y z : V\nhxyne : x \u2260 y\nhxzne : x \u2260 z\nhxy : \u2016x\u2016 = \u2016y\u2016\nhxz : \u2016x\u2016 = \u2016z\u2016\nhy : y \u2260 0\nhx : x \u2260 0\nhz : z \u2260 0\n\u22a2 2 \u2022 oangle o (x - y) (x - z) = 2 \u2022 oangle o (y - x) (z - x)\n[PROOFSTEP]\nrw [\u2190 oangle_neg_neg, neg_sub, neg_sub]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 \u2221 p\u2081 s.center p\u2083 = 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [mem_sphere, @dist_eq_norm_vsub V] at hp\u2081 hp\u2082 hp\u2083 \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 \u2221 p\u2081 s.center p\u2083 = 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083\n[PROOFSTEP]\nrw [oangle, oangle, o.oangle_eq_two_zsmul_oangle_sub_of_norm_eq_real _ _ hp\u2082 hp\u2081 hp\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 s.center - (p\u2082 -\u1d65 s.center)) (p\u2083 -\u1d65 s.center - (p\u2082 -\u1d65 s.center)) =\n    2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2082)\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 p\u2082 -\u1d65 s.center \u2260 p\u2081 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 p\u2082 -\u1d65 s.center \u2260 p\u2083 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2084 : p\u2084 \u2208 s\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n[PROOFSTEP]\nrw [mem_sphere, @dist_eq_norm_vsub V] at hp\u2081 hp\u2082 hp\u2083 hp\u2084 \n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n[PROOFSTEP]\nrw [oangle, oangle, \u2190 vsub_sub_vsub_cancel_right p\u2081 p\u2082 s.center, \u2190 vsub_sub_vsub_cancel_right p\u2084 p\u2082 s.center,\n  o.two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq _ _ _ _ hp\u2082 hp\u2083 hp\u2081 hp\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 s.center - (p\u2083 -\u1d65 s.center)) (p\u2084 -\u1d65 s.center - (p\u2083 -\u1d65 s.center)) =\n    2 \u2022 Orientation.oangle o (p\u2081 -\u1d65 p\u2083) (p\u2084 -\u1d65 p\u2083)\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 p\u2082 -\u1d65 s.center \u2260 p\u2081 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 p\u2082 -\u1d65 s.center \u2260 p\u2084 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 p\u2083 -\u1d65 s.center \u2260 p\u2081 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 p\u2083 -\u1d65 s.center \u2260 p\u2084 -\u1d65 s.center\n[PROOFSTEP]\nsimp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n[PROOFSTEP]\nobtain \u27e8s, hs\u27e9 := cospherical_iff_exists_sphere.1 h\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\ns : Sphere P\nhs : {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere s.center s.radius\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n[PROOFSTEP]\nsimp_rw [Set.insert_subset_iff, Set.singleton_subset_iff, Sphere.mem_coe] at hs \n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\ns : Sphere P\nhs : p\u2081 \u2208 s \u2227 p\u2082 \u2208 s \u2227 p\u2083 \u2208 s \u2227 p\u2084 \u2208 s\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n[PROOFSTEP]\nexact Sphere.two_zsmul_oangle_eq hs.1 hs.2.1 hs.2.2.1 hs.2.2.2 hp\u2082p\u2081 hp\u2082p\u2084 hp\u2083p\u2081 hp\u2083p\u2084\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 s.center p\u2082 = \u2191\u03c0 - 2 \u2022 \u2221 s.center p\u2082 p\u2081\n[PROOFSTEP]\nrw [oangle_eq_pi_sub_two_zsmul_oangle_of_dist_eq h.symm (dist_center_eq_dist_center_of_mem_sphere' hp\u2082 hp\u2081)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 \u2221 p\u2081 s.center p\u2082 = \u2191\u03c0 - 2 \u2022 \u2221 p\u2082 p\u2081 s.center\n[PROOFSTEP]\nrw [oangle_eq_pi_sub_two_zsmul_oangle_center_left hp\u2081 hp\u2082 h,\n  oangle_eq_oangle_of_dist_eq (dist_center_eq_dist_center_of_mem_sphere' hp\u2082 hp\u2081)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\n\u22a2 2 \u2022 \u2221 p\u2083 p\u2081 s.center + 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 = \u2191\u03c0\n[PROOFSTEP]\nrw [\u2190 oangle_center_eq_two_zsmul_oangle hp\u2081 hp\u2082 hp\u2083 hp\u2082p\u2081 hp\u2082p\u2083,\n  oangle_eq_pi_sub_two_zsmul_oangle_center_right hp\u2081 hp\u2083 hp\u2081p\u2083, add_sub_cancel'_right]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 (Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) / 2) \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n[PROOFSTEP]\nobtain \u27e8r, hr\u27e9 :=\n  (dist_eq_iff_eq_smul_rotation_pi_div_two_vadd_midpoint h).1 (dist_center_eq_dist_center_of_mem_sphere hp\u2081 hp\u2082)\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n\u22a2 (Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) / 2) \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n[PROOFSTEP]\nrw [\u2190 hr, \u2190 oangle_midpoint_rev_left, oangle, vadd_vsub_assoc]\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n\u22a2 (Real.Angle.tan\n            (Orientation.oangle o (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\n              (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) + (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081))) /\n          2) \u2022\n        \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65\n      midpoint \u211d p\u2081 p\u2082 =\n    r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082\n[PROOFSTEP]\nnth_rw 1 [show p\u2082 -\u1d65 p\u2081 = (2 : \u211d) \u2022 (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081) by simp]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n\u22a2 p\u2082 -\u1d65 p\u2081 = 2 \u2022 (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n\u22a2 (Real.Angle.tan\n            (Orientation.oangle o (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\n              (r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (2 \u2022 (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)) + (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081))) /\n          2) \u2022\n        \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65\n      midpoint \u211d p\u2081 p\u2082 =\n    r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082\n[PROOFSTEP]\nrw [map_smul, smul_smul, add_comm, o.tan_oangle_add_right_smul_rotation_pi_div_two, mul_div_cancel _ (two_ne_zero' \u211d)]\n[GOAL]\ncase intro.h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\nr : \u211d\nhr : r \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2082 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2082 = s.center\n\u22a2 midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 \u2260 0\n[PROOFSTEP]\nsimpa using h.symm\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 ((Real.Angle.tan (\u2221 p\u2081 p\u2082 p\u2083))\u207b\u00b9 / 2) \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (p\u2083 -\u1d65 p\u2081) +\u1d65 midpoint \u211d p\u2081 p\u2083 = s.center\n[PROOFSTEP]\nconvert tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center hp\u2081 hp\u2083 hp\u2081p\u2083\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_5.h.e'_5\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 (Real.Angle.tan (\u2221 p\u2081 p\u2082 p\u2083))\u207b\u00b9 = Real.Angle.tan (\u2221 p\u2083 p\u2081 s.center)\n[PROOFSTEP]\nconvert (Real.Angle.tan_eq_inv_of_two_zsmul_add_two_zsmul_eq_pi _).symm\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_5.h.e'_5.convert_3\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2083 + 2 \u2022 \u2221 p\u2083 p\u2081 s.center = \u2191\u03c0\n[PROOFSTEP]\nrw [add_comm, two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi hp\u2081 hp\u2082 hp\u2083 hp\u2081p\u2082.symm hp\u2082p\u2083 hp\u2081p\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 dist p\u2081 p\u2082 / Real.Angle.cos (\u2221 p\u2082 p\u2081 s.center) / 2 = s.radius\n[PROOFSTEP]\nrw [div_right_comm, div_eq_mul_inv _ (2 : \u211d), mul_comm,\n  show (2 : \u211d)\u207b\u00b9 * dist p\u2081 p\u2082 = dist p\u2081 (midpoint \u211d p\u2081 p\u2082) by simp, \u2190 mem_sphere.1 hp\u2081, \u2190\n  tan_div_two_smul_rotation_pi_div_two_vadd_midpoint_eq_center hp\u2081 hp\u2082 h, \u2190 oangle_midpoint_rev_left, oangle,\n  vadd_vsub_assoc, show p\u2082 -\u1d65 p\u2081 = (2 : \u211d) \u2022 (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081) by simp, map_smul, smul_smul,\n  div_mul_cancel _ (two_ne_zero' \u211d), @dist_eq_norm_vsub' V, @dist_eq_norm_vsub' V, vadd_vsub_assoc, add_comm,\n  o.oangle_add_right_smul_rotation_pi_div_two, Real.Angle.cos_coe, Real.cos_arctan]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 2\u207b\u00b9 * dist p\u2081 p\u2082 = dist p\u2081 (midpoint \u211d p\u2081 p\u2082)\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 p\u2082 -\u1d65 p\u2081 = 2 \u2022 (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 / (\u21911 / Real.sqrt (\u21911 + Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) ^ 2)) =\n    \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 +\n        Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\u2016\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 \u2260 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 / (1 / Real.sqrt (1 + Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) ^ 2)) =\n    \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 +\n        Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) \u2022 \u2191(Orientation.rotation o \u2191(\u03c0 / 2)) (midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081)\u2016\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 \u2260 0\n[PROOFSTEP]\nrw [one_div, div_inv_eq_mul, \u2190 mul_self_inj (mul_nonneg (norm_nonneg _) (Real.sqrt_nonneg _)) (norm_nonneg _),\n  norm_add_sq_eq_norm_sq_add_norm_sq_real (o.inner_smul_rotation_pi_div_two_right _ _), \u2190 mul_assoc, mul_comm,\n  mul_comm _ (Real.sqrt _), \u2190 mul_assoc, \u2190 mul_assoc, Real.mul_self_sqrt (add_nonneg zero_le_one (sq_nonneg _)),\n  norm_smul, LinearIsometryEquiv.norm_map]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 (1 + Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) ^ 2) * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 =\n    \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n      \u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n        (\u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016)\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 \u2260 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase h\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081 \u2260 0\n[PROOFSTEP]\nsimpa using h.symm\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 (1 + Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) ^ 2) * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 =\n    \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n      \u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n        (\u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016)\n[PROOFSTEP]\nconv_rhs => rw [\u2190 mul_assoc, mul_comm _ \u2016Real.Angle.tan _\u2016, \u2190 mul_assoc, Real.norm_eq_abs, abs_mul_abs_self]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n| \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n    \u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n      (\u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm _ \u2016Real.Angle.tan _\u2016, \u2190 mul_assoc, Real.norm_eq_abs, abs_mul_abs_self]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n| \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n    \u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n      (\u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm _ \u2016Real.Angle.tan _\u2016, \u2190 mul_assoc, Real.norm_eq_abs, abs_mul_abs_self]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n| \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n    \u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n      (\u2016Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center)\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, mul_comm _ \u2016Real.Angle.tan _\u2016, \u2190 mul_assoc, Real.norm_eq_abs, abs_mul_abs_self]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 (1 + Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) ^ 2) * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 =\n    \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 +\n      Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) * Real.Angle.tan (\u2221 p\u2082 p\u2081 s.center) * \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016 *\n        \u2016midpoint \u211d p\u2081 p\u2082 -\u1d65 p\u2081\u2016\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nh : p\u2081 \u2260 p\u2082\n\u22a2 dist p\u2081 p\u2082 / Real.Angle.cos (\u2221 p\u2082 p\u2081 s.center) = 2 * s.radius\n[PROOFSTEP]\nrw [\u2190 dist_div_cos_oangle_center_div_two_eq_radius hp\u2081 hp\u2082 h, mul_div_cancel' _ (two_ne_zero' \u211d)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 dist p\u2081 p\u2083 / |Real.Angle.sin (\u2221 p\u2081 p\u2082 p\u2083)| / 2 = s.radius\n[PROOFSTEP]\nconvert dist_div_cos_oangle_center_div_two_eq_radius hp\u2081 hp\u2083 hp\u2081p\u2083\n[GOAL]\ncase h.e'_2.h.e'_5.h.e'_6\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 |Real.Angle.sin (\u2221 p\u2081 p\u2082 p\u2083)| = Real.Angle.cos (\u2221 p\u2083 p\u2081 s.center)\n[PROOFSTEP]\nrw [\u2190\n  Real.Angle.abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi\n    (two_zsmul_oangle_center_add_two_zsmul_oangle_eq_pi hp\u2081 hp\u2082 hp\u2083 hp\u2081p\u2082.symm hp\u2082p\u2083 hp\u2081p\u2083),\n  _root_.abs_of_nonneg (Real.Angle.cos_nonneg_iff_abs_toReal_le_pi_div_two.2 _)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 |Real.Angle.toReal (\u2221 p\u2083 p\u2081 s.center)| \u2264 \u03c0 / 2\n[PROOFSTEP]\nexact (abs_oangle_center_right_toReal_lt_pi_div_two hp\u2081 hp\u2083).le\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2081p\u2082 : p\u2081 \u2260 p\u2082\nhp\u2081p\u2083 : p\u2081 \u2260 p\u2083\nhp\u2082p\u2083 : p\u2082 \u2260 p\u2083\n\u22a2 dist p\u2081 p\u2083 / |Real.Angle.sin (\u2221 p\u2081 p\u2082 p\u2083)| = 2 * s.radius\n[PROOFSTEP]\nrw [\u2190 dist_div_sin_oangle_div_two_eq_radius hp\u2081 hp\u2082 hp\u2083 hp\u2081p\u2082 hp\u2081p\u2083 hp\u2082p\u2083, mul_div_cancel' _ (two_ne_zero' \u211d)]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt\u2081 t\u2082 : Triangle \u211d P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh\u2081 : Simplex.points t\u2081 i\u2081 = Simplex.points t\u2082 i\u2081\nh\u2083 : Simplex.points t\u2081 i\u2083 = Simplex.points t\u2082 i\u2083\nh\u2082 :\n  2 \u2022 \u2221 (Simplex.points t\u2081 i\u2081) (Simplex.points t\u2081 i\u2082) (Simplex.points t\u2081 i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t\u2082 i\u2081) (Simplex.points t\u2082 i\u2082) (Simplex.points t\u2082 i\u2083)\n\u22a2 Simplex.circumsphere t\u2081 = Simplex.circumsphere t\u2082\n[PROOFSTEP]\nrw [t\u2081.circumsphere_eq_of_dist_of_oangle h\u2081\u2082 h\u2081\u2083 h\u2082\u2083, t\u2082.circumsphere_eq_of_dist_of_oangle h\u2081\u2082 h\u2081\u2083 h\u2082\u2083,\n  -- Porting note: was `congrm \u27e8((_ : \u211d)\u207b\u00b9 / 2) \u2022 _ +\u1d65 _, _ / _ / 2\u27e9` and five more linesReal.Angle.tan_eq_of_two_zsmul_eq\n    h\u2082,\n  Real.Angle.abs_sin_eq_of_two_zsmul_eq h\u2082, h\u2081, h\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nlet t'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2081 : t'p i\u2081 = t.points i\u2081 := by simp [h\u2081\u2082]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\n\u22a2 t'p i\u2081 = Simplex.points t i\u2081\n[PROOFSTEP]\nsimp [h\u2081\u2082]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2082 : t'p i\u2082 = p := by simp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\n\u22a2 t'p i\u2082 = p\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2083 : t'p i\u2083 = t.points i\u2083 := by simp [h\u2082\u2083.symm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\n\u22a2 t'p i\u2083 = Simplex.points t i\u2083\n[PROOFSTEP]\nsimp [h\u2082\u2083.symm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave ha : AffineIndependent \u211d t'p :=\n  by\n  rw [affineIndependent_iff_not_collinear_of_ne h\u2081\u2082 h\u2081\u2083 h\u2082\u2083, h\u2081, h\u2082, h\u2083, collinear_iff_of_two_zsmul_oangle_eq h, \u2190\n    affineIndependent_iff_not_collinear_of_ne h\u2081\u2082 h\u2081\u2083 h\u2082\u2083]\n  exact t.Independent\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\n\u22a2 AffineIndependent \u211d t'p\n[PROOFSTEP]\nrw [affineIndependent_iff_not_collinear_of_ne h\u2081\u2082 h\u2081\u2083 h\u2082\u2083, h\u2081, h\u2082, h\u2083, collinear_iff_of_two_zsmul_oangle_eq h, \u2190\n  affineIndependent_iff_not_collinear_of_ne h\u2081\u2082 h\u2081\u2083 h\u2082\u2083]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\n\u22a2 AffineIndependent \u211d t.points\n[PROOFSTEP]\nexact t.Independent\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nlet t' : Triangle \u211d P := \u27e8t'p, ha\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2081' : t'.points i\u2081 = t.points i\u2081 := h\u2081\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2082' : t'.points i\u2082 = p := h\u2082\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\nh\u2082' : Simplex.points t' i\u2082 = p\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h\u2083' : t'.points i\u2083 = t.points i\u2083 := h\u2083\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\nh\u2082' : Simplex.points t' i\u2082 = p\nh\u2083' : Simplex.points t' i\u2083 = Simplex.points t i\u2083\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nhave h' :\n  (2 : \u2124) \u2022 \u2221 (t'.points i\u2081) (t'.points i\u2082) (t'.points i\u2083) = (2 : \u2124) \u2022 \u2221 (t.points i\u2081) (t.points i\u2082) (t.points i\u2083) := by\n  rwa [h\u2081', h\u2082', h\u2083']\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\nh\u2082' : Simplex.points t' i\u2082 = p\nh\u2083' : Simplex.points t' i\u2083 = Simplex.points t i\u2083\n\u22a2 2 \u2022 \u2221 (Simplex.points t' i\u2081) (Simplex.points t' i\u2082) (Simplex.points t' i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\n[PROOFSTEP]\nrwa [h\u2081', h\u2082', h\u2083']\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\nh\u2082' : Simplex.points t' i\u2082 = p\nh\u2083' : Simplex.points t' i\u2083 = Simplex.points t i\u2083\nh' :\n  2 \u2022 \u2221 (Simplex.points t' i\u2081) (Simplex.points t' i\u2082) (Simplex.points t' i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\n\u22a2 p \u2208 Simplex.circumsphere t\n[PROOFSTEP]\nrw [\u2190 circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq h\u2081\u2082 h\u2081\u2083 h\u2082\u2083 h\u2081' h\u2083' h', \u2190 h\u2082']\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\nt : Triangle \u211d P\np : P\ni\u2081 i\u2082 i\u2083 : Fin 3\nh\u2081\u2082 : i\u2081 \u2260 i\u2082\nh\u2081\u2083 : i\u2081 \u2260 i\u2083\nh\u2082\u2083 : i\u2082 \u2260 i\u2083\nh :\n  2 \u2022 \u2221 (Simplex.points t i\u2081) p (Simplex.points t i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\nt'p : Fin 3 \u2192 P := Function.update t.points i\u2082 p\nh\u2081 : t'p i\u2081 = Simplex.points t i\u2081\nh\u2082 : t'p i\u2082 = p\nh\u2083 : t'p i\u2083 = Simplex.points t i\u2083\nha : AffineIndependent \u211d t'p\nt' : Triangle \u211d P := { points := t'p, Independent := ha }\nh\u2081' : Simplex.points t' i\u2081 = Simplex.points t i\u2081\nh\u2082' : Simplex.points t' i\u2082 = p\nh\u2083' : Simplex.points t' i\u2083 = Simplex.points t i\u2083\nh' :\n  2 \u2022 \u2221 (Simplex.points t' i\u2081) (Simplex.points t' i\u2082) (Simplex.points t' i\u2083) =\n    2 \u2022 \u2221 (Simplex.points t i\u2081) (Simplex.points t i\u2082) (Simplex.points t i\u2083)\n\u22a2 Simplex.points t' i\u2082 \u2208 Simplex.circumsphere t'\n[PROOFSTEP]\nexact Simplex.mem_circumsphere _ _\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nhave hn' : \u00acCollinear \u211d ({ p\u2081, p\u2083, p\u2084 } : Set P) := by rwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\n[PROOFSTEP]\nrwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nlet t\u2081 : Affine.Triangle \u211d P := \u27e8![p\u2081, p\u2082, p\u2084], affineIndependent_iff_not_collinear_set.2 hn\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nlet t\u2082 : Affine.Triangle \u211d P := \u27e8![p\u2081, p\u2083, p\u2084], affineIndependent_iff_not_collinear_set.2 hn'\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrw [cospherical_iff_exists_sphere]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 \u2203 s, {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere s.center s.radius\n[PROOFSTEP]\nrefine' \u27e8t\u2082.circumsphere, _\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius\n[PROOFSTEP]\nsimp_rw [Set.insert_subset_iff, Set.singleton_subset_iff]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 p\u2081 \u2208\n      Metric.sphere\n        (Affine.Simplex.circumsphere\n            { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).center\n        (Affine.Simplex.circumsphere\n            { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).radius \u2227\n    p\u2082 \u2208\n        Metric.sphere\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).center\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).radius \u2227\n      p\u2083 \u2208\n          Metric.sphere\n            (Affine.Simplex.circumsphere\n                { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).center\n            (Affine.Simplex.circumsphere\n                { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).radius \u2227\n        p\u2084 \u2208\n          Metric.sphere\n            (Affine.Simplex.circumsphere\n                { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).center\n            (Affine.Simplex.circumsphere\n                { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).radius\n[PROOFSTEP]\nrefine' \u27e8t\u2082.mem_circumsphere 0, _, t\u2082.mem_circumsphere 1, t\u2082.mem_circumsphere 2\u27e9\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 p\u2082 \u2208\n    Metric.sphere\n      (Affine.Simplex.circumsphere\n          { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).center\n      (Affine.Simplex.circumsphere\n          { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }).radius\n[PROOFSTEP]\nrw [Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq (by decide : (0 : Fin 3) \u2260 1)\n    (by decide : (0 : Fin 3) \u2260 2) (by decide) (show t\u2082.points 0 = t\u2081.points 0 from rfl) rfl h.symm]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 0 \u2260 1\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 0 \u2260 2\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 1 \u2260 2\n[PROOFSTEP]\ndecide\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2082, p\u2084]) }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2081, p\u2083, p\u2084]) }\n\u22a2 p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2081).center (Affine.Simplex.circumsphere t\u2081).radius\n[PROOFSTEP]\nexact t\u2081.mem_circumsphere 1\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nby_cases hc : Collinear \u211d ({ p\u2081, p\u2082, p\u2084 } : Set P)\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nby_cases he : p\u2081 = p\u2084\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrw [he, Set.insert_eq_self.2 (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_singleton _)))]\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\n\u22a2 Cospherical {p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nby_cases hl : Collinear \u211d ({ p\u2082, p\u2083, p\u2084 } : Set P)\n[GOAL]\ncase pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : Collinear \u211d {p\u2082, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nexact Or.inr hl\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrw [or_iff_left hl]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nlet t : Affine.Triangle \u211d P := \u27e8![p\u2082, p\u2083, p\u2084], affineIndependent_iff_not_collinear_set.2 hl\u27e9\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\nt : Affine.Triangle \u211d P := { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }\n\u22a2 Cospherical {p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrw [cospherical_iff_exists_sphere]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\nt : Affine.Triangle \u211d P := { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }\n\u22a2 \u2203 s, {p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere s.center s.radius\n[PROOFSTEP]\nrefine' \u27e8t.circumsphere, _\u27e9\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\nt : Affine.Triangle \u211d P := { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }\n\u22a2 {p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere (Affine.Simplex.circumsphere t).center (Affine.Simplex.circumsphere t).radius\n[PROOFSTEP]\nsimp_rw [Set.insert_subset_iff, Set.singleton_subset_iff]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : p\u2081 = p\u2084\nhl : \u00acCollinear \u211d {p\u2082, p\u2083, p\u2084}\nt : Affine.Triangle \u211d P := { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }\n\u22a2 p\u2082 \u2208\n      Metric.sphere\n        (Affine.Simplex.circumsphere\n            { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).center\n        (Affine.Simplex.circumsphere\n            { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).radius \u2227\n    p\u2083 \u2208\n        Metric.sphere\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).center\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).radius \u2227\n      p\u2084 \u2208\n        Metric.sphere\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).center\n          (Affine.Simplex.circumsphere\n              { points := ![p\u2082, p\u2083, p\u2084], Independent := (_ : AffineIndependent \u211d ![p\u2082, p\u2083, p\u2084]) }).radius\n[PROOFSTEP]\nexact \u27e8t.mem_circumsphere 0, t.mem_circumsphere 1, t.mem_circumsphere 2\u27e9\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : \u00acp\u2081 = p\u2084\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nhave hc' : Collinear \u211d ({ p\u2081, p\u2083, p\u2084 } : Set P) := by rwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : \u00acp\u2081 = p\u2084\n\u22a2 Collinear \u211d {p\u2081, p\u2083, p\u2084}\n[PROOFSTEP]\nrwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : \u00acp\u2081 = p\u2084\nhc' : Collinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrefine' Or.inr _\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2084}\nhe : \u00acp\u2081 = p\u2084\nhc' : Collinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrw [Set.insert_comm p\u2081 p\u2082] at hc \n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2082, p\u2081, p\u2084}\nhe : \u00acp\u2081 = p\u2084\nhc' : Collinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrwa [Set.insert_comm p\u2081 p\u2082,\n  hc'.collinear_insert_iff_of_ne (Set.mem_insert _ _)\n    (Set.mem_insert_of_mem _ (Set.mem_insert_of_mem _ (Set.mem_singleton _))) he]\n[GOAL]\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nexact Or.inl (cospherical_of_two_zsmul_oangle_eq_of_not_collinear h hc)\n[GOAL]\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\n\u22a2 Concyclic {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nrcases cospherical_or_collinear_of_two_zsmul_oangle_eq h with (hc | hc)\n[GOAL]\ncase inl\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}\n\u22a2 Concyclic {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nexact Or.inl \u27e8hc, coplanar_of_fact_finrank_eq_two _\u27e9\n[GOAL]\ncase inr\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhc : Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n\u22a2 Concyclic {p\u2081, p\u2082, p\u2083, p\u2084} \u2228 Collinear \u211d {p\u2081, p\u2082, p\u2083, p\u2084}\n[PROOFSTEP]\nexact Or.inr hc\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Angle.Sphere", "llama_tokens": 33438, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.787931185683219, "lm_q2_score": 0.6442250928250375, "lm_q1q2_score": 0.5076050412365136}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nr : R\n\u22a2 \u2191(aeval (IsLocalization.Away.invSelf r)) (\u2191C r * X - 1) = 0\n[PROOFSTEP]\nsimp only [map_sub, map_mul, aeval_C, aeval_X, IsLocalization.Away.mul_invSelf, aeval_one, sub_self]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Localization.Away.AdjoinRoot", "llama_tokens": 108, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.8104789086703225, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.5074604398520989}}
{"text": "[GOAL]\na : { x // x \u2208 circle }\nsrc\u271d : \u2102 \u2243\u2097[\u211d] \u2102 := DistribMulAction.toLinearEquiv \u211d \u2102 a\nx : \u2102\n\u22a2 \u2191Complex.abs (\u2191a * x) = \u2191Complex.abs x\n[PROOFSTEP]\nrw [map_mul, abs_coe_circle, one_mul]\n[GOAL]\na b : { x // x \u2208 circle }\n\u22a2 LinearIsometryEquiv.trans (\u2191rotation a) (\u2191rotation b) = \u2191rotation (b * a)\n[PROOFSTEP]\next1\n[GOAL]\ncase h\na b : { x // x \u2208 circle }\nx\u271d : \u2102\n\u22a2 \u2191(LinearIsometryEquiv.trans (\u2191rotation a) (\u2191rotation b)) x\u271d = \u2191(\u2191rotation (b * a)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191rotation a \u2260 conjLie\n[PROOFSTEP]\nintro h\n[GOAL]\na : { x // x \u2208 circle }\nh : \u2191rotation a = conjLie\n\u22a2 False\n[PROOFSTEP]\nhave h1 : rotation a 1 = conj 1 := LinearIsometryEquiv.congr_fun h 1\n[GOAL]\na : { x // x \u2208 circle }\nh : \u2191rotation a = conjLie\nh1 : \u2191(\u2191rotation a) 1 = \u2191(starRingEnd \u2102) 1\n\u22a2 False\n[PROOFSTEP]\nhave hI : rotation a I = conj I := LinearIsometryEquiv.congr_fun h I\n[GOAL]\na : { x // x \u2208 circle }\nh : \u2191rotation a = conjLie\nh1 : \u2191(\u2191rotation a) 1 = \u2191(starRingEnd \u2102) 1\nhI : \u2191(\u2191rotation a) I = \u2191(starRingEnd \u2102) I\n\u22a2 False\n[PROOFSTEP]\nrw [rotation_apply, RingHom.map_one, mul_one] at h1 \n[GOAL]\na : { x // x \u2208 circle }\nh : \u2191rotation a = conjLie\nh1 : \u2191a = 1\nhI : \u2191(\u2191rotation a) I = \u2191(starRingEnd \u2102) I\n\u22a2 False\n[PROOFSTEP]\nrw [rotation_apply, conj_I, \u2190 neg_one_mul, mul_left_inj' I_ne_zero, h1, eq_neg_self_iff] at hI \n[GOAL]\na : { x // x \u2208 circle }\nh : \u2191rotation a = conjLie\nh1 : \u2191a = 1\nhI : 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero hI\n[GOAL]\ne : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\n\u22a2 \u2191e 1 / \u2191(\u2191Complex.abs (\u2191e 1)) \u2208 circle\n[PROOFSTEP]\nsimp\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191(rotationOf (\u2191rotation a)) = \u2191a\n[PROOFSTEP]\nsimp\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh\u2083 : \u2200 (z : \u2102), z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\nz : \u2102\n\u22a2 (\u2191f z).re = z.re\n[PROOFSTEP]\nsimpa [ext_iff, add_re, add_im, conj_re, conj_im, \u2190 two_mul, show (2 : \u211d) \u2260 0 by simp [two_ne_zero]] using (h\u2083 z).symm\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh\u2083 : \u2200 (z : \u2102), z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\nz : \u2102\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nsimp [two_ne_zero]\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh\u2082 : \u2200 (z : \u2102), (\u2191f z).re = z.re\nz : \u2102\n\u22a2 (\u2191f z).im = z.im \u2228 (\u2191f z).im = -z.im\n[PROOFSTEP]\nhave h\u2081 := f.norm_map z\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh\u2082 : \u2200 (z : \u2102), (\u2191f z).re = z.re\nz : \u2102\nh\u2081 : \u2016\u2191f z\u2016 = \u2016z\u2016\n\u22a2 (\u2191f z).im = z.im \u2228 (\u2191f z).im = -z.im\n[PROOFSTEP]\nsimp only [Complex.abs_def, norm_eq_abs] at h\u2081 \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh\u2082 : \u2200 (z : \u2102), (\u2191f z).re = z.re\nz : \u2102\nh\u2081 : Real.sqrt (\u2191normSq (\u2191f z)) = Real.sqrt (\u2191normSq z)\n\u22a2 (\u2191f z).im = z.im \u2228 (\u2191f z).im = -z.im\n[PROOFSTEP]\nrwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z, h\u2082, add_left_cancel_iff,\n  mul_self_eq_mul_self_iff] at h\u2081 \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nhave : \u2016f z - 1\u2016 = \u2016z - 1\u2016 := by rw [\u2190 f.norm_map (z - 1), f.map_sub, h]\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\n\u22a2 \u2016\u2191f z - 1\u2016 = \u2016z - 1\u2016\n[PROOFSTEP]\nrw [\u2190 f.norm_map (z - 1), f.map_sub, h]\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : \u2016\u2191f z - 1\u2016 = \u2016z - 1\u2016\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\napply_fun fun x => x ^ 2 at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : \u2016\u2191f z - 1\u2016 ^ 2 = \u2016z - 1\u2016 ^ 2\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nsimp only [norm_eq_abs, \u2190 normSq_eq_abs] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : \u2191normSq (\u2191f z - 1) = \u2191normSq (z - 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nrw [\u2190 ofReal_inj, \u2190 mul_conj, \u2190 mul_conj] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : (\u2191f z - 1) * \u2191(starRingEnd \u2102) (\u2191f z - 1) = (z - 1) * \u2191(starRingEnd \u2102) (z - 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nrw [RingHom.map_sub, RingHom.map_sub] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : (\u2191f z - 1) * (\u2191(starRingEnd \u2102) (\u2191f z) - \u2191(starRingEnd \u2102) 1) = (z - 1) * (\u2191(starRingEnd \u2102) z - \u2191(starRingEnd \u2102) 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nsimp only [sub_mul, mul_sub, one_mul, mul_one] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis :\n  \u2191f z * \u2191(starRingEnd \u2102) (\u2191f z) - \u2191(starRingEnd \u2102) (\u2191f z) - (\u2191f z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1) =\n    z * \u2191(starRingEnd \u2102) z - \u2191(starRingEnd \u2102) z - (z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nrw [mul_conj, normSq_eq_abs, \u2190 norm_eq_abs, LinearIsometry.norm_map] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - \u2191(starRingEnd \u2102) (\u2191f z) - (\u2191f z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1) =\n    z * \u2191(starRingEnd \u2102) z - \u2191(starRingEnd \u2102) z - (z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nrw [mul_conj, normSq_eq_abs, \u2190 norm_eq_abs] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - \u2191(starRingEnd \u2102) (\u2191f z) - (\u2191f z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1) =\n    \u2191(\u2016z\u2016 ^ 2) - \u2191(starRingEnd \u2102) z - (z * \u2191(starRingEnd \u2102) 1 - \u2191(starRingEnd \u2102) 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nsimp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, RingHom.map_one, norm_eq_abs] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : \u2191(starRingEnd \u2102) (\u2191f z) + (\u2191f z - 1) = \u2191(starRingEnd \u2102) z + (z - 1)\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nsimp only [add_sub, sub_left_inj] at this \n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\nthis : \u2191(starRingEnd \u2102) (\u2191f z) + \u2191f z = \u2191(starRingEnd \u2102) z + z\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nrw [add_comm, \u2190 this, add_comm]\n[GOAL]\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\n\u22a2 (\u2191f z).re = z.re\n[PROOFSTEP]\napply LinearIsometry.re_apply_eq_re_of_add_conj_eq\n[GOAL]\ncase h\u2083\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\n\u22a2 \u2200 (z : \u2102), z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\nintro z\n[GOAL]\ncase h\u2083\nf : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz\u271d z : \u2102\n\u22a2 z + \u2191(starRingEnd \u2102) z = \u2191f z + \u2191(starRingEnd ((fun x => \u2102) z)) (\u2191f z)\n[PROOFSTEP]\napply LinearIsometry.im_apply_eq_im h\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 f = LinearIsometryEquiv.refl \u211d \u2102 \u2228 f = conjLie\n[PROOFSTEP]\nhave h0 : f I = I \u2228 f I = -I :=\n  by\n  simp only [ext_iff, \u2190 and_or_left, neg_re, I_re, neg_im, neg_zero]\n  constructor\n  \u00b7 rw [\u2190 I_re]\n    exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I\n  \u00b7 apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry\n    intro z\n    rw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 \u2191f I = I \u2228 \u2191f I = -I\n[PROOFSTEP]\nsimp only [ext_iff, \u2190 and_or_left, neg_re, I_re, neg_im, neg_zero]\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 (\u2191f I).re = 0 \u2227 ((\u2191f I).im = I.im \u2228 (\u2191f I).im = -I.im)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 (\u2191f I).re = 0\n[PROOFSTEP]\nrw [\u2190 I_re]\n[GOAL]\ncase left\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 (\u2191f I).re = I.re\n[PROOFSTEP]\nexact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I\n[GOAL]\ncase right\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 (\u2191f I).im = I.im \u2228 (\u2191f I).im = -I.im\n[PROOFSTEP]\napply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry\n[GOAL]\ncase right.h\u2082\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\n\u22a2 \u2200 (z : \u2102), (\u2191(LinearIsometryEquiv.toLinearIsometry f) z).re = z.re\n[PROOFSTEP]\nintro z\n[GOAL]\ncase right.h\u2082\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nz : \u2102\n\u22a2 (\u2191(LinearIsometryEquiv.toLinearIsometry f) z).re = z.re\n[PROOFSTEP]\nrw [@LinearIsometry.re_apply_eq_re f.toLinearIsometry h]\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\n\u22a2 f = LinearIsometryEquiv.refl \u211d \u2102 \u2228 f = conjLie\n[PROOFSTEP]\nrefine' h0.imp (fun h' : f I = I => _) fun h' : f I = -I => _\n[GOAL]\ncase refine'_1\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\n\u22a2 f = LinearIsometryEquiv.refl \u211d \u2102\n[PROOFSTEP]\napply LinearIsometryEquiv.toLinearEquiv_injective\n[GOAL]\ncase refine'_1.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\n\u22a2 f.toLinearEquiv = (LinearIsometryEquiv.refl \u211d \u2102).toLinearEquiv\n[PROOFSTEP]\napply Complex.basisOneI.ext'\n[GOAL]\ncase refine'_1.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\n\u22a2 \u2200 (i : Fin 2), \u2191f.toLinearEquiv (\u2191basisOneI i) = \u2191(LinearIsometryEquiv.refl \u211d \u2102).toLinearEquiv (\u2191basisOneI i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine'_1.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\ni : Fin 2\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI i) = \u2191(LinearIsometryEquiv.refl \u211d \u2102).toLinearEquiv (\u2191basisOneI i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase refine'_1.a.head\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) =\n    \u2191(LinearIsometryEquiv.refl \u211d \u2102).toLinearEquiv (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase refine'_1.a.tail.head\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = I\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    \u2191(LinearIsometryEquiv.refl \u211d \u2102).toLinearEquiv (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase refine'_2\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\n\u22a2 f = conjLie\n[PROOFSTEP]\napply LinearIsometryEquiv.toLinearEquiv_injective\n[GOAL]\ncase refine'_2.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\n\u22a2 f.toLinearEquiv = conjLie.toLinearEquiv\n[PROOFSTEP]\napply Complex.basisOneI.ext'\n[GOAL]\ncase refine'_2.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\n\u22a2 \u2200 (i : Fin 2), \u2191f.toLinearEquiv (\u2191basisOneI i) = \u2191conjLie.toLinearEquiv (\u2191basisOneI i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase refine'_2.a\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\ni : Fin 2\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI i) = \u2191conjLie.toLinearEquiv (\u2191basisOneI i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase refine'_2.a.head\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) }) =\n    \u2191conjLie.toLinearEquiv (\u2191basisOneI { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\ncase refine'_2.a.tail.head\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\nh : \u2191f 1 = 1\nh0 : \u2191f I = I \u2228 \u2191f I = -I\nh' : \u2191f I = -I\n\u22a2 \u2191f.toLinearEquiv (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    \u2191conjLie.toLinearEquiv (\u2191basisOneI { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nsimp [h, h']\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\n\u22a2 \u2203 a, f = \u2191rotation a \u2228 f = LinearIsometryEquiv.trans conjLie (\u2191rotation a)\n[PROOFSTEP]\nlet a : circle := \u27e8f 1, by rw [mem_circle_iff_abs, \u2190 Complex.norm_eq_abs, f.norm_map, norm_one]\u27e9\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\n\u22a2 \u2191f 1 \u2208 circle\n[PROOFSTEP]\nrw [mem_circle_iff_abs, \u2190 Complex.norm_eq_abs, f.norm_map, norm_one]\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\n\u22a2 \u2203 a, f = \u2191rotation a \u2228 f = LinearIsometryEquiv.trans conjLie (\u2191rotation a)\n[PROOFSTEP]\nuse a\n[GOAL]\ncase h\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\n\u22a2 f = \u2191rotation a \u2228 f = LinearIsometryEquiv.trans conjLie (\u2191rotation a)\n[PROOFSTEP]\nhave : (f.trans (rotation a).symm) 1 = 1 := by simpa using rotation_apply a\u207b\u00b9 (f 1)\n[GOAL]\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\n\u22a2 \u2191(LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a))) 1 = 1\n[PROOFSTEP]\nsimpa using rotation_apply a\u207b\u00b9 (f 1)\n[GOAL]\ncase h\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\nthis : \u2191(LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a))) 1 = 1\n\u22a2 f = \u2191rotation a \u2228 f = LinearIsometryEquiv.trans conjLie (\u2191rotation a)\n[PROOFSTEP]\nrefine' (linear_isometry_complex_aux this).imp (fun h\u2081 => _) fun h\u2082 => _\n[GOAL]\ncase h.refine'_1\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\nthis : \u2191(LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a))) 1 = 1\nh\u2081 : LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a)) = LinearIsometryEquiv.refl \u211d \u2102\n\u22a2 f = \u2191rotation a\n[PROOFSTEP]\nsimpa using eq_mul_of_inv_mul_eq h\u2081\n[GOAL]\ncase h.refine'_2\nf : \u2102 \u2243\u2097\u1d62[\u211d] \u2102\na : { x // x \u2208 circle } := { val := \u2191f 1, property := (_ : \u2191f 1 \u2208 circle) }\nthis : \u2191(LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a))) 1 = 1\nh\u2082 : LinearIsometryEquiv.trans f (LinearIsometryEquiv.symm (\u2191rotation a)) = conjLie\n\u22a2 f = LinearIsometryEquiv.trans conjLie (\u2191rotation a)\n[PROOFSTEP]\nexact eq_mul_of_inv_mul_eq h\u2082\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0\n[PROOFSTEP]\nsimp [pow_two, \u2190 normSq_apply]\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) \u2191(\u2191rotation a).toLinearEquiv =\n    \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0))\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\na : { x // x \u2208 circle }\ni j : Fin 2\n\u22a2 \u2191(LinearMap.toMatrix basisOneI basisOneI) (\u2191(\u2191rotation a).toLinearEquiv) i j =\n    \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0)) i j\n[PROOFSTEP]\nsimp [LinearMap.toMatrix_apply]\n[GOAL]\ncase a.h\na : { x // x \u2208 circle }\ni j : Fin 2\n\u22a2 Matrix.vecCons ((\u2191a).re * (Matrix.vecCons 1 ![I] j).re - (\u2191a).im * (Matrix.vecCons 1 ![I] j).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] j).im + (\u2191a).im * (Matrix.vecCons 1 ![I] j).re] i =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] j) (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] j) i\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase a.h.head\na : { x // x \u2208 circle }\nj : Fin 2\n\u22a2 Matrix.vecCons ((\u2191a).re * (Matrix.vecCons 1 ![I] j).re - (\u2191a).im * (Matrix.vecCons 1 ![I] j).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] j).im + (\u2191a).im * (Matrix.vecCons 1 ![I] j).re]\n      { val := 0, isLt := (_ : 0 < 2) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] j) (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] j)\n      { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase a.h.tail.head\na : { x // x \u2208 circle }\nj : Fin 2\n\u22a2 Matrix.vecCons ((\u2191a).re * (Matrix.vecCons 1 ![I] j).re - (\u2191a).im * (Matrix.vecCons 1 ![I] j).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] j).im + (\u2191a).im * (Matrix.vecCons 1 ![I] j).re]\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] j) (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] j)\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nfin_cases j\n[GOAL]\ncase a.h.head.head\na : { x // x \u2208 circle }\n\u22a2 Matrix.vecCons\n      ((\u2191a).re * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).re -\n        (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).im +\n          (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).re]\n      { val := 0, isLt := (_ : 0 < 2) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] { val := 0, isLt := (_ : 0 < 2) })\n      (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] { val := 0, isLt := (_ : 0 < 2) }) { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.head.tail.head\na : { x // x \u2208 circle }\n\u22a2 Matrix.vecCons\n      ((\u2191a).re * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).re -\n        (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).im +\n          (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).re]\n      { val := 0, isLt := (_ : 0 < 2) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      { val := 0, isLt := (_ : 0 < 2) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.tail.head.head\na : { x // x \u2208 circle }\n\u22a2 Matrix.vecCons\n      ((\u2191a).re * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).re -\n        (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).im +\n          (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 0, isLt := (_ : 0 < 2) }).re]\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] { val := 0, isLt := (_ : 0 < 2) })\n      (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] { val := 0, isLt := (_ : 0 < 2) })\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.h.tail.head.tail.head\na : { x // x \u2208 circle }\n\u22a2 Matrix.vecCons\n      ((\u2191a).re * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).re -\n        (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).im)\n      ![(\u2191a).re * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).im +\n          (\u2191a).im * (Matrix.vecCons 1 ![I] { val := 1, isLt := (_ : (fun a => a < 2) 1) }).re]\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n    Matrix.vecCons (Matrix.vecCons (\u2191a).re ![-(\u2191a).im] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      (fun i => Matrix.vecCons (\u2191a).im ![(\u2191a).re] { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n      { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n[PROOFSTEP]\nsimp\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191LinearMap.det \u2191(\u2191rotation a).toLinearEquiv = 1\n[PROOFSTEP]\nrw [\u2190 LinearMap.det_toMatrix basisOneI, toMatrix_rotation, Matrix.det_fin_two]\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0)) 0 0 *\n        \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0)) 1 1 -\n      \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0)) 0 1 *\n        \u2191(Matrix.planeConformalMatrix (\u2191a).re (\u2191a).im (_ : (\u2191a).re ^ 2 + (\u2191a).im ^ 2 \u2260 0)) 1 0 =\n    1\n[PROOFSTEP]\nsimp [\u2190 normSq_apply]\n[GOAL]\na : { x // x \u2208 circle }\n\u22a2 \u2191LinearEquiv.det (\u2191rotation a).toLinearEquiv = 1\n[PROOFSTEP]\nrw [\u2190 Units.eq_iff, LinearEquiv.coe_det, det_rotation, Units.val_one]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Complex.Isometry", "llama_tokens": 9999, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.815232489352, "lm_q2_score": 0.6224593382055109, "lm_q1q2_score": 0.5074490758056771}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nr : \ud835\udd5c\nhr\u2080 : r \u2260 0\nhr : \u2016r\u2016 < 1\nhs : \u2200\u1da0 (n : \u2115) in atTop, x + r ^ n \u2022 y \u2208 s\n\u22a2 y \u2208 tangentConeAt \ud835\udd5c s x\n[PROOFSTEP]\nrefine \u27e8fun n \u21a6 (r ^ n)\u207b\u00b9, fun n \u21a6 r ^ n \u2022 y, hs, ?_, ?_\u27e9\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nr : \ud835\udd5c\nhr\u2080 : r \u2260 0\nhr : \u2016r\u2016 < 1\nhs : \u2200\u1da0 (n : \u2115) in atTop, x + r ^ n \u2022 y \u2208 s\n\u22a2 Tendsto (fun n => \u2016(fun n => (r ^ n)\u207b\u00b9) n\u2016) atTop atTop\n[PROOFSTEP]\nsimp only [norm_inv, norm_pow, \u2190 inv_pow]\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nr : \ud835\udd5c\nhr\u2080 : r \u2260 0\nhr : \u2016r\u2016 < 1\nhs : \u2200\u1da0 (n : \u2115) in atTop, x + r ^ n \u2022 y \u2208 s\n\u22a2 Tendsto (fun n => \u2016r\u2016\u207b\u00b9 ^ n) atTop atTop\n[PROOFSTEP]\nexact tendsto_pow_atTop_atTop_of_one_lt <| one_lt_inv (norm_pos_iff.2 hr\u2080) hr\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nr : \ud835\udd5c\nhr\u2080 : r \u2260 0\nhr : \u2016r\u2016 < 1\nhs : \u2200\u1da0 (n : \u2115) in atTop, x + r ^ n \u2022 y \u2208 s\n\u22a2 Tendsto (fun n => (fun n => (r ^ n)\u207b\u00b9) n \u2022 (fun n => r ^ n \u2022 y) n) atTop (\ud835\udcdd y)\n[PROOFSTEP]\nsimp only [inv_smul_smul\u2080 (pow_ne_zero _ hr\u2080), tendsto_const_nhds]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nh : s \u2286 t\n\u22a2 tangentConeAt \ud835\udd5c s x \u2286 tangentConeAt \ud835\udd5c t x\n[PROOFSTEP]\nrintro y \u27e8c, d, ds, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : s \u2286 t\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 tangentConeAt \ud835\udd5c t x\n[PROOFSTEP]\nexact \u27e8c, d, mem_of_superset ds fun n hn => h hn, ctop, clim\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave A : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0) := tendsto_inv_atTop_zero.comp hc\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave B : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016) := (continuous_norm.tendsto _).comp hd\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave C : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd (0 * \u2016y\u2016)) := A.mul B\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd (0 * \u2016y\u2016))\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [zero_mul] at C \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : \u2200\u1da0 n in l, \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016 :=\n  by\n  refine (eventually_ne_of_tendsto_norm_atTop hc 0).mono fun n hn => ?_\n  rw [norm_smul, \u2190 mul_assoc, inv_mul_cancel, one_mul]\n  rwa [Ne.def, norm_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\n\u22a2 \u2200\u1da0 (n : \u03b1) in l, \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016\n[PROOFSTEP]\nrefine (eventually_ne_of_tendsto_norm_atTop hc 0).mono fun n hn => ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\nn : \u03b1\nhn : c n \u2260 0\n\u22a2 \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016\n[PROOFSTEP]\nrw [norm_smul, \u2190 mul_assoc, inv_mul_cancel, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\nn : \u03b1\nhn : c n \u2260 0\n\u22a2 \u2016c n\u2016 \u2260 0\n[PROOFSTEP]\nrwa [Ne.def, norm_eq_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\nthis : \u2200\u1da0 (n : \u03b1) in l, \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nhave D : Tendsto (fun n => \u2016d n\u2016) l (\ud835\udcdd 0) := Tendsto.congr' this C\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\nthis : \u2200\u1da0 (n : \u03b1) in l, \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016\nD : Tendsto (fun n => \u2016d n\u2016) l (\ud835\udcdd 0)\n\u22a2 Tendsto d l (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_zero_iff_norm_tendsto_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u03b1 : Type u_5\nl : Filter \u03b1\nc : \u03b1 \u2192 \ud835\udd5c\nd : \u03b1 \u2192 E\nhc : Tendsto (fun n => \u2016c n\u2016) l atTop\nhd : Tendsto (fun n => c n \u2022 d n) l (\ud835\udcdd y)\nA : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9) l (\ud835\udcdd 0)\nB : Tendsto (fun n => \u2016c n \u2022 d n\u2016) l (\ud835\udcdd \u2016y\u2016)\nC : Tendsto (fun n => \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016) l (\ud835\udcdd 0)\nthis : \u2200\u1da0 (n : \u03b1) in l, \u2016c n\u2016\u207b\u00b9 * \u2016c n \u2022 d n\u2016 = \u2016d n\u2016\nD : Tendsto (fun n => \u2016d n\u2016) l (\ud835\udcdd 0)\n\u22a2 Tendsto (fun e => \u2016d e\u2016) l (\ud835\udcdd 0)\n[PROOFSTEP]\nexact D\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\n\u22a2 tangentConeAt \ud835\udd5c s x \u2286 tangentConeAt \ud835\udd5c t x\n[PROOFSTEP]\nrintro y \u27e8c, d, ds, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 y \u2208 tangentConeAt \ud835\udd5c t x\n[PROOFSTEP]\nrefine' \u27e8c, d, _, ctop, clim\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 t\n[PROOFSTEP]\nsuffices : Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x)\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\nthis : Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x)\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 t\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x)\n[PROOFSTEP]\nexact tendsto_principal.1 (tendsto_inf.1 this).2\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x)\n[PROOFSTEP]\nrefine' (tendsto_inf.2 \u27e8_, tendsto_principal.2 ds\u27e9).mono_right h\n[GOAL]\ncase this\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t : Set E\nh : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\ny : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nclim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y)\n\u22a2 Tendsto (fun a => x + d a) atTop (\ud835\udcdd x)\n[PROOFSTEP]\nsimpa only [add_zero] using tendsto_const_nhds.add (tangentConeAt.lim_zero atTop ctop clim)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\n\u22a2 \u2191(LinearMap.inl \ud835\udd5c E F) '' tangentConeAt \ud835\udd5c s x \u2286 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrintro _ \u27e8v, \u27e8c, d, hd, hc, hy\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\n\u22a2 \u2191(LinearMap.inl \ud835\udd5c E F) v \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nhave : \u2200 n, \u2203 d', y + d' \u2208 t \u2227 \u2016c n \u2022 d'\u2016 < ((1 : \u211d) / 2) ^ n :=\n  by\n  intro n\n  rcases mem_closure_iff_nhds.1 ht _ (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with\n    \u27e8z, hz, hzt\u27e9\n  exact \u27e8z - y, by simpa using hzt, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\n\u22a2 \u2200 (n : \u2115), \u2203 d', y + d' \u2208 t \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nn : \u2115\n\u22a2 \u2203 d', y + d' \u2208 t \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nrcases mem_closure_iff_nhds.1 ht _ (eventually_nhds_norm_smul_sub_lt (c n) y (pow_pos one_half_pos n)) with \u27e8z, hz, hzt\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nn : \u2115\nz : F\nhz : z \u2208 {x | (fun y_1 => \u2016c n \u2022 (y_1 - y)\u2016 < (1 / 2) ^ n) x}\nhzt : z \u2208 t\n\u22a2 \u2203 d', y + d' \u2208 t \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nexact \u27e8z - y, by simpa using hzt, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nn : \u2115\nz : F\nhz : z \u2208 {x | (fun y_1 => \u2016c n \u2022 (y_1 - y)\u2016 < (1 / 2) ^ n) x}\nhzt : z \u2208 t\n\u22a2 y + (z - y) \u2208 t\n[PROOFSTEP]\nsimpa using hzt\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nn : \u2115\nz : F\nhz : z \u2208 {x | (fun y_1 => \u2016c n \u2022 (y_1 - y)\u2016 < (1 / 2) ^ n) x}\nhzt : z \u2208 t\n\u22a2 \u2016c n \u2022 (z - y)\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nsimpa using hz\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nthis : \u2200 (n : \u2115), \u2203 d', y + d' \u2208 t \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.inl \ud835\udd5c E F) v \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nchoose d' hd' using this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.inl \ud835\udd5c E F) v \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrefine' \u27e8c, fun n => (d n, d' n), _, hc, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (x, y) + (fun n => (d n, d' n)) n \u2208 s \u00d7\u02e2 t\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => c n \u2022 (fun n => (d n, d' n)) n) atTop (\ud835\udcdd (\u2191(LinearMap.inl \ud835\udd5c E F) v))\n[PROOFSTEP]\nshow \u2200\u1da0 n in atTop, (x, y) + (d n, d' n) \u2208 s \u00d7\u02e2 t\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (x, y) + (d n, d' n) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nfilter_upwards [hd] with n hn\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\nn : \u2115\nhn : x + d n \u2208 s\n\u22a2 (x, y) + (d n, d' n) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [hn, (hd' n).1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => c n \u2022 (fun n => (d n, d' n)) n) atTop (\ud835\udcdd (\u2191(LinearMap.inl \ud835\udd5c E F) v))\n[PROOFSTEP]\napply Tendsto.prod_mk_nhds hy _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun c_1 => c c_1 \u2022 ((fun n => (d n, d' n)) c_1).snd) atTop (\ud835\udcdd (\u21910 v))\n[PROOFSTEP]\nrefine' squeeze_zero_norm (fun n => (hd' n).2.le) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nht : y \u2208 closure t\nv : E\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E\nhd : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd v)\nd' : \u2115 \u2192 F\nhd' : \u2200 (n : \u2115), y + d' n \u2208 t \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_pow_atTop_nhds_0_of_lt_1 one_half_pos.le one_half_lt_one\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\n\u22a2 \u2191(LinearMap.inr \ud835\udd5c E F) '' tangentConeAt \ud835\udd5c t y \u2286 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrintro _ \u27e8w, \u27e8c, d, hd, hc, hy\u27e9, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\n\u22a2 \u2191(LinearMap.inr \ud835\udd5c E F) w \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nhave : \u2200 n, \u2203 d', x + d' \u2208 s \u2227 \u2016c n \u2022 d'\u2016 < ((1 : \u211d) / 2) ^ n :=\n  by\n  intro n\n  rcases mem_closure_iff_nhds.1 hs _ (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with\n    \u27e8z, hz, hzs\u27e9\n  exact \u27e8z - x, by simpa using hzs, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\n\u22a2 \u2200 (n : \u2115), \u2203 d', x + d' \u2208 s \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\n\u22a2 \u2203 d', x + d' \u2208 s \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nrcases mem_closure_iff_nhds.1 hs _ (eventually_nhds_norm_smul_sub_lt (c n) x (pow_pos one_half_pos n)) with \u27e8z, hz, hzs\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nz : E\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s\n\u22a2 \u2203 d', x + d' \u2208 s \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nexact \u27e8z - x, by simpa using hzs, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nz : E\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s\n\u22a2 x + (z - x) \u2208 s\n[PROOFSTEP]\nsimpa using hzs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nz : E\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s\n\u22a2 \u2016c n \u2022 (z - x)\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nsimpa using hz\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nthis : \u2200 (n : \u2115), \u2203 d', x + d' \u2208 s \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.inr \ud835\udd5c E F) w \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nchoose d' hd' using this\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.inr \ud835\udd5c E F) w \u2208 tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrefine' \u27e8c, fun n => (d' n, d n), _, hc, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (x, y) + (fun n => (d' n, d n)) n \u2208 s \u00d7\u02e2 t\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => c n \u2022 (fun n => (d' n, d n)) n) atTop (\ud835\udcdd (\u2191(LinearMap.inr \ud835\udd5c E F) w))\n[PROOFSTEP]\nshow \u2200\u1da0 n in atTop, (x, y) + (d' n, d n) \u2208 s \u00d7\u02e2 t\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, (x, y) + (d' n, d n) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nfilter_upwards [hd] with n hn\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\nn : \u2115\nhn : y + d n \u2208 t\n\u22a2 (x, y) + (d' n, d n) \u2208 s \u00d7\u02e2 t\n[PROOFSTEP]\nsimp [hn, (hd' n).1]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => c n \u2022 (fun n => (d' n, d n)) n) atTop (\ud835\udcdd (\u2191(LinearMap.inr \ud835\udd5c E F) w))\n[PROOFSTEP]\napply Tendsto.prod_mk_nhds _ hy\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun c_1 => c c_1 \u2022 ((fun n => (d' n, d n)) c_1).fst) atTop (\ud835\udcdd (\u21910 w))\n[PROOFSTEP]\nrefine' squeeze_zero_norm (fun n => (hd' n).2.le) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : x \u2208 closure s\nw : F\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 F\nhd : \u2200\u1da0 (n : \u2115) in atTop, y + d n \u2208 t\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 E\nhd' : \u2200 (n : \u2115), x + d' n \u2208 s \u2227 \u2016c n \u2022 d' n\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun n => (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_pow_atTop_nhds_0_of_lt_1 one_half_pos.le one_half_lt_one\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\n\u22a2 MapsTo (\u2191(LinearMap.single i)) (tangentConeAt \ud835\udd5c (s i) (x i)) (tangentConeAt \ud835\udd5c (Set.pi univ s) x)\n[PROOFSTEP]\nrintro w \u27e8c, d, hd, hc, hy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\n\u22a2 \u2191(LinearMap.single i) w \u2208 tangentConeAt \ud835\udd5c (Set.pi univ s) x\n[PROOFSTEP]\nhave : \u2200 (n) (j) (_ : j \u2260 i), \u2203 d', x j + d' \u2208 s j \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2 : \u211d) ^ n :=\n  by\n  intro n j hj\n  rcases mem_closure_iff_nhds.1 (hi j hj) _ (eventually_nhds_norm_smul_sub_lt (c n) (x j) (pow_pos one_half_pos n)) with\n    \u27e8z, hz, hzs\u27e9\n  exact \u27e8z - x j, by simpa using hzs, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\n\u22a2 \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2203 d', x j + d' \u2208 s j \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nintro n j hj\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nj : \u03b9\nhj : j \u2260 i\n\u22a2 \u2203 d', x j + d' \u2208 s j \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nrcases mem_closure_iff_nhds.1 (hi j hj) _ (eventually_nhds_norm_smul_sub_lt (c n) (x j) (pow_pos one_half_pos n)) with\n  \u27e8z, hz, hzs\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nj : \u03b9\nhj : j \u2260 i\nz : E j\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x j)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s j\n\u22a2 \u2203 d', x j + d' \u2208 s j \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nexact \u27e8z - x j, by simpa using hzs, by simpa using hz\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nj : \u03b9\nhj : j \u2260 i\nz : E j\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x j)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s j\n\u22a2 x j + (z - x j) \u2208 s j\n[PROOFSTEP]\nsimpa using hzs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nn : \u2115\nj : \u03b9\nhj : j \u2260 i\nz : E j\nhz : z \u2208 {x_1 | (fun y => \u2016c n \u2022 (y - x j)\u2016 < (1 / 2) ^ n) x_1}\nhzs : z \u2208 s j\n\u22a2 \u2016c n \u2022 (z - x j)\u2016 < (1 / 2) ^ n\n[PROOFSTEP]\nsimpa using hz\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nthis : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2203 d', x j + d' \u2208 s j \u2227 \u2016c n \u2022 d'\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.single i) w \u2208 tangentConeAt \ud835\udd5c (Set.pi univ s) x\n[PROOFSTEP]\nchoose! d' hd's hcd' using this\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\n\u22a2 \u2191(LinearMap.single i) w \u2208 tangentConeAt \ud835\udd5c (Set.pi univ s) x\n[PROOFSTEP]\nrefine' \u27e8c, fun n => Function.update (d' n) i (d n), hd.mono fun n hn j _ => _, hc, tendsto_pi_nhds.2 fun j => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d\u00b9 y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nn : \u2115\nhn : x i + d n \u2208 s i\nj : \u03b9\nx\u271d : j \u2208 univ\n\u22a2 (x + (fun n => Function.update (d' n) i (d n)) n) j \u2208 s j\n[PROOFSTEP]\nrcases em (j = i) with (rfl | hj)\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d\u00b9 y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nc : \u2115 \u2192 \ud835\udd5c\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nn : \u2115\nj : \u03b9\nx\u271d : j \u2208 univ\nhi : \u2200 (j_1 : \u03b9), j_1 \u2260 j \u2192 x j_1 \u2208 closure (s j_1)\nw : E j\nd : \u2115 \u2192 E j\nhd : \u2200\u1da0 (n : \u2115) in atTop, x j + d n \u2208 s j\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nhd's : \u2200 (n : \u2115) (j_1 : \u03b9), j_1 \u2260 j \u2192 x j_1 + d' n j_1 \u2208 s j_1\nhcd' : \u2200 (n : \u2115) (j_1 : \u03b9), j_1 \u2260 j \u2192 \u2016c n \u2022 d' n j_1\u2016 < (1 / 2) ^ n\nhn : x j + d n \u2208 s j\n\u22a2 (x + (fun n => Function.update (d' n) j (d n)) n) j \u2208 s j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.intro.intro.refine'_1.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d\u00b9 y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nn : \u2115\nhn : x i + d n \u2208 s i\nj : \u03b9\nx\u271d : j \u2208 univ\nhj : \u00acj = i\n\u22a2 (x + (fun n => Function.update (d' n) i (d n)) n) j \u2208 s j\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nj : \u03b9\n\u22a2 Tendsto (fun i_1 => (c i_1 \u2022 (fun n => Function.update (d' n) i (d n)) i_1) j) atTop (\ud835\udcdd (\u2191(LinearMap.single i) w j))\n[PROOFSTEP]\nrcases em (j = i) with (rfl | hj)\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.inl\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nc : \u2115 \u2192 \ud835\udd5c\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nj : \u03b9\nhi : \u2200 (j_1 : \u03b9), j_1 \u2260 j \u2192 x j_1 \u2208 closure (s j_1)\nw : E j\nd : \u2115 \u2192 E j\nhd : \u2200\u1da0 (n : \u2115) in atTop, x j + d n \u2208 s j\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nhd's : \u2200 (n : \u2115) (j_1 : \u03b9), j_1 \u2260 j \u2192 x j_1 + d' n j_1 \u2208 s j_1\nhcd' : \u2200 (n : \u2115) (j_1 : \u03b9), j_1 \u2260 j \u2192 \u2016c n \u2022 d' n j_1\u2016 < (1 / 2) ^ n\n\u22a2 Tendsto (fun i => (c i \u2022 (fun n => Function.update (d' n) j (d n)) i) j) atTop (\ud835\udcdd (\u2191(LinearMap.single j) w j))\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nj : \u03b9\nhj : \u00acj = i\n\u22a2 Tendsto (fun i_1 => (c i_1 \u2022 (fun n => Function.update (d' n) i (d n)) i_1) j) atTop (\ud835\udcdd (\u2191(LinearMap.single i) w j))\n[PROOFSTEP]\nsuffices Tendsto (fun n => c n \u2022 d' n j) atTop (\ud835\udcdd 0) by simpa [hj]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nj : \u03b9\nhj : \u00acj = i\nthis : Tendsto (fun n => c n \u2022 d' n j) atTop (\ud835\udcdd 0)\n\u22a2 Tendsto (fun i_1 => (c i_1 \u2022 (fun n => Function.update (d' n) i (d n)) i_1) j) atTop (\ud835\udcdd (\u2191(LinearMap.single i) w j))\n[PROOFSTEP]\nsimpa [hj]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nj : \u03b9\nhj : \u00acj = i\n\u22a2 Tendsto (fun n => c n \u2022 d' n j) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero_norm (fun n => (hcd' n j hj).le) _\n[GOAL]\ncase intro.intro.intro.intro.refine'_2.inr\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : DecidableEq \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\ni : \u03b9\nhi : \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2208 closure (s j)\nw : E i\nc : \u2115 \u2192 \ud835\udd5c\nd : \u2115 \u2192 E i\nhd : \u2200\u1da0 (n : \u2115) in atTop, x i + d n \u2208 s i\nhc : Tendsto (fun n => \u2016c n\u2016) atTop atTop\nhy : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd w)\nd' : \u2115 \u2192 (j : \u03b9) \u2192 E j\nhd's : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 x j + d' n j \u2208 s j\nhcd' : \u2200 (n : \u2115) (j : \u03b9), j \u2260 i \u2192 \u2016c n \u2022 d' n j\u2016 < (1 / 2) ^ n\nj : \u03b9\nhj : \u00acj = i\n\u22a2 Tendsto (fun n => (1 / 2) ^ n) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nexact tendsto_pow_atTop_nhds_0_of_lt_1 one_half_pos.le one_half_lt_one\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\n\u22a2 y - x \u2208 tangentConeAt \u211d s x\n[PROOFSTEP]\nrefine mem_tangentConeAt_of_pow_smul one_half_pos.ne' (by norm_num) ?_\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\n\u22a2 \u20161 / 2\u2016 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\n\u22a2 \u2200\u1da0 (n : \u2115) in atTop, x + (1 / 2) ^ n \u2022 (y - x) \u2208 s\n[PROOFSTEP]\nrefine (eventually_ne_atTop 0).mono fun n hn \u21a6 (h ?_)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 x + (1 / 2) ^ n \u2022 (y - x) \u2208 openSegment \u211d x y\n[PROOFSTEP]\nrw [openSegment_eq_image]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 x + (1 / 2) ^ n \u2022 (y - x) \u2208 (fun \u03b8 => (1 - \u03b8) \u2022 x + \u03b8 \u2022 y) '' Ioo 0 1\n[PROOFSTEP]\nrefine \u27e8(1 / 2) ^ n, \u27e8?_, ?_\u27e9, ?_\u27e9\n[GOAL]\ncase refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 0 < (1 / 2) ^ n\n[PROOFSTEP]\nexact pow_pos one_half_pos _\n[GOAL]\ncase refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 (1 / 2) ^ n < 1\n[PROOFSTEP]\nexact pow_lt_one one_half_pos.le one_half_lt_one hn\n[GOAL]\ncase refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 (fun \u03b8 => (1 - \u03b8) \u2022 x + \u03b8 \u2022 y) ((1 / 2) ^ n) = x + (1 / 2) ^ n \u2022 (y - x)\n[PROOFSTEP]\nsimp only [sub_smul, one_smul, smul_sub]\n[GOAL]\ncase refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 x - (1 / 2) ^ n \u2022 x + (1 / 2) ^ n \u2022 y = x + ((1 / 2) ^ n \u2022 y - (1 / 2) ^ n \u2022 x)\n[PROOFSTEP]\nabel\n[GOAL]\ncase refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nx y : G\nh : openSegment \u211d x y \u2286 s\nn : \u2115\nhn : n \u2260 0\n\u22a2 x - (1 / 2) ^ n \u2022 x + (1 / 2) ^ n \u2022 y = x + ((1 / 2) ^ n \u2022 y - (1 / 2) ^ n \u2022 x)\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c univ x\n[PROOFSTEP]\nrw [uniqueDiffWithinAt_iff, tangentCone_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c univ) \u2227 x \u2208 closure univ\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nh : UniqueDiffWithinAt \ud835\udd5c s x\nst : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\n\u22a2 UniqueDiffWithinAt \ud835\udd5c t x\n[PROOFSTEP]\nsimp only [uniqueDiffWithinAt_iff] at *\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nst : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\nh : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t x)) \u2227 x \u2208 closure t\n[PROOFSTEP]\nrw [mem_closure_iff_nhdsWithin_neBot] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nst : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x\nh : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 NeBot (\ud835\udcdd[s] x)\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t x)) \u2227 NeBot (\ud835\udcdd[t] x)\n[PROOFSTEP]\nexact \u27e8h.1.mono <| Submodule.span_mono <| tangentCone_mono_nhds st, h.2.mono st\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\nh : s \u2208 \ud835\udcdd x\n\u22a2 UniqueDiffWithinAt \ud835\udd5c s x\n[PROOFSTEP]\nsimpa only [univ_inter] using uniqueDiffWithinAt_univ.inter h\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : UniqueDiffWithinAt \ud835\udd5c s x\nht : UniqueDiffWithinAt \ud835\udd5c t y\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)\n[PROOFSTEP]\nrw [uniqueDiffWithinAt_iff] at hs ht \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\nht : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2227 y \u2208 closure t\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y))) \u2227 (x, y) \u2208 closure (s \u00d7\u02e2 t)\n[PROOFSTEP]\nrw [closure_prod_eq]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\nht : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2227 y \u2208 closure t\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y))) \u2227 (x, y) \u2208 closure s \u00d7\u02e2 closure t\n[PROOFSTEP]\nrefine' \u27e8_, hs.2, ht.2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\nht : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2227 y \u2208 closure t\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)))\n[PROOFSTEP]\nhave : _ \u2264 Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)) :=\n  Submodule.span_mono (union_subset (subset_tangentCone_prod_left ht.2) (subset_tangentCone_prod_right hs.2))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\nht : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2227 y \u2208 closure t\nthis :\n  Submodule.span \ud835\udd5c (\u2191(LinearMap.inl \ud835\udd5c E F) '' tangentConeAt \ud835\udd5c s x \u222a \u2191(LinearMap.inr \ud835\udd5c E F) '' tangentConeAt \ud835\udd5c t y) \u2264\n    Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y))\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)))\n[PROOFSTEP]\nrw [LinearMap.span_inl_union_inr, SetLike.le_def] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y\u271d : E\ns t\u271d : Set E\nt : Set F\ny : F\nhs : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) \u2227 x \u2208 closure s\nht : Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2227 y \u2208 closure t\nthis :\n  \u2200 \u2983x_1 : E \u00d7 F\u2984,\n    x_1 \u2208 Submodule.prod (Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c s x)) (Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c t y)) \u2192\n      x_1 \u2208 Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y))\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s \u00d7\u02e2 t) (x, y)))\n[PROOFSTEP]\nexact (hs.1.prod ht.1).mono this\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (Set.pi univ s) x\n[PROOFSTEP]\nclassical\nsimp only [uniqueDiffWithinAt_iff, closure_pi_set] at h \u22a2\nrefine' \u27e8(dense_pi univ fun i _ => (h i).1).mono _, fun i _ => (h i).2\u27e9\nnorm_cast\nsimp only [\u2190 Submodule.iSup_map_single, iSup_le_iff, LinearMap.map_span, Submodule.span_le, \u2190 mapsTo']\nexact fun i => (mapsTo_tangentCone_pi fun j _ => (h j).2).mono Subset.rfl Submodule.subset_span\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (Set.pi univ s) x\n[PROOFSTEP]\nsimp only [uniqueDiffWithinAt_iff, closure_pi_set] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i))) \u2227 x i \u2208 closure (s i)\n\u22a2 Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (Set.pi univ s) x)) \u2227 x \u2208 Set.pi univ fun i => closure (s i)\n[PROOFSTEP]\nrefine' \u27e8(dense_pi univ fun i _ => (h i).1).mono _, fun i _ => (h i).2\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i))) \u2227 x i \u2208 closure (s i)\n\u22a2 (Set.pi univ fun i => \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i)))) \u2286\n    \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (Set.pi univ s) x))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i))) \u2227 x i \u2208 closure (s i)\n\u22a2 (Submodule.pi univ fun i => Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i))) \u2264\n    Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (Set.pi univ s) x)\n[PROOFSTEP]\nsimp only [\u2190 Submodule.iSup_map_single, iSup_le_iff, LinearMap.map_span, Submodule.span_le, \u2190 mapsTo']\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nh : \u2200 (i : \u03b9), Dense \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (s i) (x i))) \u2227 x i \u2208 closure (s i)\n\u22a2 \u2200 (i : \u03b9),\n    MapsTo (fun a => \u2191(LinearMap.single i) a) (tangentConeAt \ud835\udd5c (s i) (x i))\n      \u2191(Submodule.span \ud835\udd5c (tangentConeAt \ud835\udd5c (Set.pi univ s) x))\n[PROOFSTEP]\nexact fun i => (mapsTo_tangentCone_pi fun j _ => (h j).2).mono Subset.rfl Submodule.subset_span\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (Set.pi I s) x\n[PROOFSTEP]\nclassical\nrw [\u2190 Set.univ_pi_piecewise_univ]\nrefine' UniqueDiffWithinAt.univ_pi \u03b9 E _ _ fun i => _\nby_cases hi : i \u2208 I <;> simp [*, uniqueDiffWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (Set.pi I s) x\n[PROOFSTEP]\nrw [\u2190 Set.univ_pi_piecewise_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (Set.pi univ (piecewise I s fun x => univ)) x\n[PROOFSTEP]\nrefine' UniqueDiffWithinAt.univ_pi \u03b9 E _ _ fun i => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\ni : \u03b9\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (piecewise I s (fun x => univ) i) (x i)\n[PROOFSTEP]\nby_cases hi : i \u2208 I\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\ni : \u03b9\nhi : i \u2208 I\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (piecewise I s (fun x => univ) i) (x i)\n[PROOFSTEP]\nsimp [*, uniqueDiffWithinAt_univ]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE\u271d : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\u271d\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\u271d\nF : Type u_3\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u2074 : NormedAddCommGroup G\ninst\u271d\u00b3 : NormedSpace \u211d G\nx\u271d y : E\u271d\ns\u271d t : Set E\u271d\n\u03b9 : Type u_5\ninst\u271d\u00b2 : Finite \u03b9\nE : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\ns : (i : \u03b9) \u2192 Set (E i)\nx : (i : \u03b9) \u2192 E i\nI : Set \u03b9\nh : \u2200 (i : \u03b9), i \u2208 I \u2192 UniqueDiffWithinAt \ud835\udd5c (s i) (x i)\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (piecewise I s (fun x => univ) i) (x i)\n[PROOFSTEP]\nsimp [*, uniqueDiffWithinAt_univ]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nhs : Set.Nonempty (interior s)\nx : G\nhx : x \u2208 closure s\n\u22a2 UniqueDiffWithinAt \u211d s x\n[PROOFSTEP]\nrcases hs with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : y \u2208 interior s\n\u22a2 UniqueDiffWithinAt \u211d s x\n[PROOFSTEP]\nsuffices y - x \u2208 interior (tangentConeAt \u211d s x)\n  by\n  refine' \u27e8Dense.of_closure _, hx\u27e9\n  simp [(Submodule.span \u211d (tangentConeAt \u211d s x)).eq_top_of_nonempty_interior'\n      \u27e8y - x, interior_mono Submodule.subset_span this\u27e9]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : y \u2208 interior s\nthis : y - x \u2208 interior (tangentConeAt \u211d s x)\n\u22a2 UniqueDiffWithinAt \u211d s x\n[PROOFSTEP]\nrefine' \u27e8Dense.of_closure _, hx\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : y \u2208 interior s\nthis : y - x \u2208 interior (tangentConeAt \u211d s x)\n\u22a2 Dense (closure \u2191(Submodule.span \u211d (tangentConeAt \u211d s x)))\n[PROOFSTEP]\nsimp [(Submodule.span \u211d (tangentConeAt \u211d s x)).eq_top_of_nonempty_interior'\n    \u27e8y - x, interior_mono Submodule.subset_span this\u27e9]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : y \u2208 interior s\n\u22a2 y - x \u2208 interior (tangentConeAt \u211d s x)\n[PROOFSTEP]\nrw [mem_interior_iff_mem_nhds]\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : y \u2208 interior s\n\u22a2 tangentConeAt \u211d s x \u2208 \ud835\udcdd (y - x)\n[PROOFSTEP]\nreplace hy : interior s \u2208 \ud835\udcdd y := IsOpen.mem_nhds isOpen_interior hy\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : interior s \u2208 \ud835\udcdd y\n\u22a2 tangentConeAt \u211d s x \u2208 \ud835\udcdd (y - x)\n[PROOFSTEP]\napply mem_of_superset ((isOpenMap_sub_right x).image_mem_nhds hy)\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : interior s \u2208 \ud835\udcdd y\n\u22a2 (fun x_1 => x_1 - x) '' interior s \u2286 tangentConeAt \u211d s x\n[PROOFSTEP]\nrintro _ \u27e8z, zs, rfl\u27e9\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : interior s \u2208 \ud835\udcdd y\nz : G\nzs : z \u2208 interior s\n\u22a2 (fun x_1 => x_1 - x) z \u2208 tangentConeAt \u211d s x\n[PROOFSTEP]\nrefine' mem_tangentCone_of_openSegment_subset (Subset.trans _ interior_subset)\n[GOAL]\ncase intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx\u271d y\u271d : E\ns\u271d t : Set E\ns : Set G\nconv : Convex \u211d s\nx : G\nhx : x \u2208 closure s\ny : G\nhy : interior s \u2208 \ud835\udcdd y\nz : G\nzs : z \u2208 interior s\n\u22a2 openSegment \u211d x z \u2286 interior s\n[PROOFSTEP]\nexact conv.openSegment_closure_interior_subset_interior hx zs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 Set.Nonempty (interior (Ici a))\n[PROOFSTEP]\nsimp only [interior_Ici, nonempty_Ioi]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 Set.Nonempty (interior (Iic a))\n[PROOFSTEP]\nsimp only [interior_Iic, nonempty_Iio]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na b : \u211d\nhab : a < b\n\u22a2 Set.Nonempty (interior (Icc a b))\n[PROOFSTEP]\nsimp only [interior_Icc, nonempty_Ioo, hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na b : \u211d\nhab : a < b\n\u22a2 Set.Nonempty (interior (Ico a b))\n[PROOFSTEP]\nsimp only [interior_Ico, nonempty_Ioo, hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na b : \u211d\nhab : \u00aca < b\n\u22a2 UniqueDiffOn \u211d (Ico a b)\n[PROOFSTEP]\nsimp only [Ico_eq_empty hab, uniqueDiffOn_empty]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na b : \u211d\nhab : a < b\n\u22a2 Set.Nonempty (interior (Ioc a b))\n[PROOFSTEP]\nsimp only [interior_Ioc, nonempty_Ioo, hab]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na b : \u211d\nhab : \u00aca < b\n\u22a2 UniqueDiffOn \u211d (Ioc a b)\n[PROOFSTEP]\nsimp only [Ioc_eq_empty hab, uniqueDiffOn_empty]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 Set.Nonempty (interior (Ioi a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 a \u2208 closure (Ioi a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 Set.Nonempty (interior (Iio a))\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \u211d G\nx y : E\ns t : Set E\na : \u211d\n\u22a2 a \u2208 closure (Iio a)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Analysis.Calculus.TangentCone", "llama_tokens": 43127, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324803738429, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.5074490645075312}}
{"text": "[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\ncases' isEmpty_or_nonempty \u03b2 with h\u03b2 h\u03b2\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave : IsEmpty \u03b1 := Function.isEmpty f\n[GOAL]\ncase inl\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : IsEmpty \u03b2\nthis : IsEmpty \u03b1\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nexact \u27e8_, ((Equiv.equivEmpty \u03b1).trans (Equiv.equivEmpty \u03b2).symm).bijective\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nset F : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c\n    monotone' := fun s t hst =>\n      compl_subset_compl.mpr <| image_subset _ <| compl_subset_compl.mpr <| image_subset _ hst }\n    -- Porting note: dot notation `F.lfp` doesn't work here\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nset s : Set \u03b1 := OrderHom.lfp F\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave hs : (g '' (f '' s)\u1d9c)\u1d9c = s := F.map_lfp\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave hns : g '' (f '' s)\u1d9c = s\u1d9c := compl_injective (by simp [hs])\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\n\u22a2 (g '' (f '' s)\u1d9c)\u1d9c = s\u1d9c\u1d9c\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nset g' := invFun g\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave g'g : LeftInverse g' g := leftInverse_invFun hg\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave hg'ns : g' '' s\u1d9c = (f '' s)\u1d9c := by rw [\u2190 hns, g'g.image_image]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\n\u22a2 g' '' s\u1d9c = (f '' s)\u1d9c\n[PROOFSTEP]\nrw [\u2190 hns, g'g.image_image]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nset h : \u03b1 \u2192 \u03b2 := s.piecewise f g'\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave : Surjective h := by rw [\u2190 range_iff_surjective, range_piecewise, hg'ns, union_compl_self]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\n\u22a2 Surjective h\n[PROOFSTEP]\nrw [\u2190 range_iff_surjective, range_piecewise, hg'ns, union_compl_self]\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nhave : Injective h := by\n  refine' (injective_piecewise_iff _).2 \u27e8hf.injOn _, _, _\u27e9\n  \u00b7 intro x hx y hy hxy\n    obtain \u27e8x', _, rfl\u27e9 : x \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n    obtain \u27e8y', _, rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n    rw [g'g _, g'g _] at hxy \n    rw [hxy]\n  \u00b7 intro x hx y hy hxy\n    obtain \u27e8y', hy', rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n    rw [g'g _] at hxy \n    exact hy' \u27e8x, hx, hxy\u27e9\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\n\u22a2 Injective h\n[PROOFSTEP]\nrefine' (injective_piecewise_iff _).2 \u27e8hf.injOn _, _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\n\u22a2 InjOn g' s\u1d9c\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\u1d9c\ny : \u03b1\nhy : y \u2208 s\u1d9c\nhxy : g' x = g' y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8x', _, rfl\u27e9 : x \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\u1d9c\ny : \u03b1\nhy : y \u2208 s\u1d9c\nhxy : g' x = g' y\n\u22a2 x \u2208 g '' (f '' s)\u1d9c\n[PROOFSTEP]\nrwa [hns]\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\ny : \u03b1\nhy : y \u2208 s\u1d9c\nx' : \u03b2\nleft\u271d : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\nhxy : g' (g x') = g' y\n\u22a2 g x' = y\n[PROOFSTEP]\nobtain \u27e8y', _, rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\ny : \u03b1\nhy : y \u2208 s\u1d9c\nx' : \u03b2\nleft\u271d : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\nhxy : g' (g x') = g' y\n\u22a2 y \u2208 g '' (f '' s)\u1d9c\n[PROOFSTEP]\nrwa [hns]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : g' (g x') = g' (g y')\n\u22a2 g x' = g y'\n[PROOFSTEP]\nrw [g'g _, g'g _] at hxy \n[GOAL]\ncase refine'_1.intro.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx' : \u03b2\nleft\u271d\u00b9 : x' \u2208 (f '' s)\u1d9c\nhx : g x' \u2208 s\u1d9c\ny' : \u03b2\nleft\u271d : y' \u2208 (f '' s)\u1d9c\nhy : g y' \u2208 s\u1d9c\nhxy : x' = y'\n\u22a2 g x' = g y'\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), \u00acy \u2208 s \u2192 f x \u2260 g' y\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : \u00acy \u2208 s\nhxy : f x = g' y\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8y', hy', rfl\u27e9 : y \u2208 g '' (f '' s)\u1d9c := by rwa [hns]\n[GOAL]\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : \u00acy \u2208 s\nhxy : f x = g' y\n\u22a2 y \u2208 g '' (f '' s)\u1d9c\n[PROOFSTEP]\nrwa [hns]\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : \u00acg y' \u2208 s\nhxy : f x = g' (g y')\n\u22a2 False\n[PROOFSTEP]\nrw [g'g _] at hxy \n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis : Surjective h\nx : \u03b1\nhx : x \u2208 s\ny' : \u03b2\nhy' : y' \u2208 (f '' s)\u1d9c\nhy : \u00acg y' \u2208 s\nhxy : f x = y'\n\u22a2 False\n[PROOFSTEP]\nexact hy' \u27e8x, hx, hxy\u27e9\n[GOAL]\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b1\nhf : Injective f\nhg : Injective g\nh\u03b2 : Nonempty \u03b2\nF : Set \u03b1 \u2192o Set \u03b1 :=\n  { toFun := fun s => (g '' (f '' s)\u1d9c)\u1d9c,\n    monotone' := (_ : \u2200 (s t : Set \u03b1), s \u2264 t \u2192 (g '' (f '' s)\u1d9c)\u1d9c \u2286 (g '' (f '' t)\u1d9c)\u1d9c) }\ns : Set \u03b1 := \u2191OrderHom.lfp F\nhs : (g '' (f '' s)\u1d9c)\u1d9c = s\nhns : g '' (f '' s)\u1d9c = s\u1d9c\ng' : \u03b1 \u2192 \u03b2 := invFun g\ng'g : LeftInverse g' g\nhg'ns : g' '' s\u1d9c = (f '' s)\u1d9c\nh : \u03b1 \u2192 \u03b2 := piecewise s f g'\nthis\u271d : Surjective h\nthis : Injective h\n\u22a2 \u2203 h, Bijective h\n[PROOFSTEP]\nexact \u27e8h, \u2039Injective h\u203a, \u2039Surjective h\u203a\u27e9\n[GOAL]\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\n\u22a2 \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\n[PROOFSTEP]\nsimpa only [ne_eq, not_exists, not_forall, not_and] using h\n[GOAL]\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 insert f s\ny : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 insert f s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\ncases' hx with hx hx\n[GOAL]\ncase inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 insert f s\nhx : x = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\ncases' hy with hy hy\n[GOAL]\ncase inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 insert f s\nhx : x \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\ncases' hy with hy hy\n[GOAL]\ncase inl.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x = f\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\nsubst x\n[GOAL]\ncase inl.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\ny : (i : \u03b9) \u2192 \u03b2 i\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), f i = y i \u2192 f = y\n[PROOFSTEP]\nexact fun i e => (hf i y hy e.symm).elim\n[GOAL]\ncase inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y = f\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\nsubst y\n[GOAL]\ncase inr.inl\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = f i \u2192 x = f\n[PROOFSTEP]\nexact fun i e => (hf i x hx e).elim\n[GOAL]\ncase inr.inr\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\nh\u271d : \u00ac\u2203 i, \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nh : \u2200 (i : \u03b9), \u2203 y, \u2200 (x : (i : \u03b9) \u2192 \u03b2 i), x \u2208 s \u2192 x i \u2260 y\nf : (x : \u03b9) \u2192 \u03b2 x\nhf : \u2200 (x : \u03b9) (x_1 : (i : \u03b9) \u2192 \u03b2 i), x_1 \u2208 s \u2192 x_1 x \u2260 f x\nx y : (i : \u03b9) \u2192 \u03b2 i\nhx : x \u2208 s\nhy : y \u2208 s\n\u22a2 \u2200 (i : \u03b9), x i = y i \u2192 x = y\n[PROOFSTEP]\nexact hs x hx y hy\n[GOAL]\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\n\u22a2 a = b\n[PROOFSTEP]\nlet \u27e8sa, ea\u27e9 := hf a\n[GOAL]\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\n\u22a2 a = b\n[PROOFSTEP]\nlet \u27e8sb, eb\u27e9 := hf b\n[GOAL]\n\u03b9 : Type u\n\u03b2 : \u03b9 \u2192 Type v\nI : Nonempty \u03b9\ns : Set ((i : \u03b9) \u2192 \u03b2 i)\nhs : s \u2208 Function.Embedding.sets \u03b2\nms : \u2200 (a : Set ((i : \u03b9) \u2192 \u03b2 i)), a \u2208 Function.Embedding.sets \u03b2 \u2192 s \u2286 a \u2192 a = s\ni : \u03b9\ne : \u2200 (y : \u03b2 i), \u2203 x, x \u2208 s \u2227 x i = y\nf : \u03b2 i \u2192 (i : \u03b9) \u2192 \u03b2 i\nhf : \u2200 (x : \u03b2 i), f x \u2208 s \u2227 f x i = x\nj : \u03b9\na b : \u03b2 i\ne' : (fun a => f a j) a = (fun a => f a j) b\nsa : f a \u2208 s\nea : f a i = a\nsb : f b \u2208 s\neb : f b i = b\n\u22a2 a = b\n[PROOFSTEP]\nrw [\u2190 ea, \u2190 eb, hs _ sa _ sb _ e']\n", "meta": {"mathlib_filename": "Mathlib.SetTheory.Cardinal.SchroederBernstein", "llama_tokens": 12034, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.819893353516963, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5073339649129261}}
{"text": "[GOAL]\na b m n k : \u2115\nS : Set \u2115\n\u22a2 S = Set.univ \u2192 0 \u2208 S \u2227 \u2200 (k : \u2115), k \u2208 S \u2192 k + 1 \u2208 S\n[PROOFSTEP]\nrintro rfl\n[GOAL]\na b m n k : \u2115\n\u22a2 0 \u2208 Set.univ \u2227 \u2200 (k : \u2115), k \u2208 Set.univ \u2192 k + 1 \u2208 Set.univ\n[PROOFSTEP]\nsimp\n[GOAL]\na\u271d b m n\u271d k n d : \u2115\nhnd : d \u2223 n\na : \u2115\n\u22a2 a < n / d \u2194 d * a < n\n[PROOFSTEP]\nrcases d.eq_zero_or_pos with (rfl | hd0)\n[GOAL]\ncase inl\na\u271d b m n\u271d k n a : \u2115\nhnd : 0 \u2223 n\n\u22a2 a < n / 0 \u2194 0 * a < n\n[PROOFSTEP]\nsimp [zero_dvd_iff.mp hnd]\n[GOAL]\ncase inr\na\u271d b m n\u271d k n d : \u2115\nhnd : d \u2223 n\na : \u2115\nhd0 : d > 0\n\u22a2 a < n / d \u2194 d * a < n\n[PROOFSTEP]\nrw [\u2190 mul_lt_mul_left hd0, \u2190 Nat.eq_mul_of_div_eq_right hnd rfl]\n[GOAL]\na b m n\u271d k n d : \u2115\n\u22a2 d * (n / d) = n \u2194 d * (n / d) = d * (n / d) + n % d\n[PROOFSTEP]\nrw [div_add_mod]\n[GOAL]\na b m n\u271d k n d : \u2115\n\u22a2 d * (n / d) = d * (n / d) + n % d \u2194 d \u2223 n\n[PROOFSTEP]\nrw [self_eq_add_right, dvd_iff_mod_eq_zero]\n  -- porting note: new lemma\n[GOAL]\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\n\u22a2 n / x = n / y \u2194 x = y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\n\u22a2 n / x = n / y \u2192 x = y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\nh : n / x = n / y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' hn]\n[GOAL]\ncase mp\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\nh : n / x = n / y\n\u22a2 n * x = n * y\n[PROOFSTEP]\napply Nat.eq_mul_of_div_eq_left (dvd_mul_of_dvd_left hy x)\n[GOAL]\ncase mp\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\nh : n / x = n / y\n\u22a2 n * x / y = n\n[PROOFSTEP]\nrw [eq_comm, mul_comm, Nat.mul_div_assoc _ hy]\n[GOAL]\ncase mp\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\nh : n / x = n / y\n\u22a2 n = x * (n / y)\n[PROOFSTEP]\nexact Nat.eq_mul_of_div_eq_right hx h\n[GOAL]\ncase mpr\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\n\u22a2 x = y \u2192 n / x = n / y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\na b m n\u271d k n x y : \u2115\nhn : n \u2260 0\nhx : x \u2223 n\nhy : y \u2223 n\nh : x = y\n\u22a2 n / x = n / y\n[PROOFSTEP]\nrw [h]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhb : 0 < b\nh : a / b = 0\n\u22a2 a < b\n[PROOFSTEP]\nrw [\u2190 mod_add_div a b, h, mul_zero, add_zero]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhb : 0 < b\nh : a / b = 0\n\u22a2 a % b < b\n[PROOFSTEP]\nexact mod_lt _ hb\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhb : 0 < b\nh : a < b\n\u22a2 a / b = 0\n[PROOFSTEP]\nrw [\u2190 mul_right_inj' hb.ne', \u2190 @add_left_cancel_iff _ _ _ (a % b), mod_add_div, mod_eq_of_lt h, mul_zero, add_zero]\n[GOAL]\na b m n\u271d k n : \u2115\n\u22a2 \u00ac2 \u2223 bit1 n\n[PROOFSTEP]\nrw [bit1, Nat.dvd_add_right two_dvd_bit0, Nat.dvd_one]\n  -- Porting note: was `cc`\n[GOAL]\na b m n\u271d k n : \u2115\n\u22a2 \u00ac2 = 1\n[PROOFSTEP]\ndecide\n[GOAL]\na b m\u271d n\u271d k\u271d k m n : \u2115\nh\u2081 : k \u2223 m\nh\u2082 : k \u2223 n\n\u22a2 k \u2223 m - n\n[PROOFSTEP]\ncases' le_total n m with H H\n[GOAL]\ncase inl\na b m\u271d n\u271d k\u271d k m n : \u2115\nh\u2081 : k \u2223 m\nh\u2082 : k \u2223 n\nH : n \u2264 m\n\u22a2 k \u2223 m - n\n[PROOFSTEP]\nexact dvd_sub H h\u2081 h\u2082\n[GOAL]\ncase inr\na b m\u271d n\u271d k\u271d k m n : \u2115\nh\u2081 : k \u2223 m\nh\u2082 : k \u2223 n\nH : m \u2264 n\n\u22a2 k \u2223 m - n\n[PROOFSTEP]\nrw [tsub_eq_zero_iff_le.mpr H]\n[GOAL]\ncase inr\na b m\u271d n\u271d k\u271d k m n : \u2115\nh\u2081 : k \u2223 m\nh\u2082 : k \u2223 n\nH : m \u2264 n\n\u22a2 k \u2223 0\n[PROOFSTEP]\nexact dvd_zero k\n[GOAL]\na\u271d b m n k a : \u2115\n\u22a2 (a + 1) / 0 = a / 0 + if 0 \u2223 a + 1 then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\na b m n k : \u2115\n\u22a2 (0 + 1) / 1 = 0 / 1 + if 1 \u2223 0 + 1 then 1 else 0\n[PROOFSTEP]\nsimp\n[GOAL]\na b\u271d m n k b : \u2115\n\u22a2 (0 + 1) / (b + 2) = 0 / (b + 2) + if b + 2 \u2223 0 + 1 then 1 else 0\n[PROOFSTEP]\nhave hb2 : b + 2 > 1 := by simp\n[GOAL]\na b\u271d m n k b : \u2115\n\u22a2 b + 2 > 1\n[PROOFSTEP]\nsimp\n[GOAL]\na b\u271d m n k b : \u2115\nhb2 : b + 2 > 1\n\u22a2 (0 + 1) / (b + 2) = 0 / (b + 2) + if b + 2 \u2223 0 + 1 then 1 else 0\n[PROOFSTEP]\nsimp [ne_of_gt hb2, div_eq_of_lt hb2]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n\u22a2 (a + 1 + 1) / (b + 1) = (a + 1) / (b + 1) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nrw [Nat.div_eq]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (a + 1) / (b + 1) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nconv_rhs => rw [Nat.div_eq]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n| (a + 1) / (b + 1) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nrw [Nat.div_eq]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n| (a + 1) / (b + 1) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nrw [Nat.div_eq]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n| (a + 1) / (b + 1) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nrw [Nat.div_eq]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nby_cases hb_eq_a : b = a + 1\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : b = a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nsimp [hb_eq_a, le_refl]\n[GOAL]\ncase neg\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nby_cases hb_le_a1 : b \u2264 a + 1\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave hb_le_a : b \u2264 a := le_of_lt_succ (lt_of_le_of_ne hb_le_a1 hb_eq_a)\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave h\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 := \u27e8succ_pos _, (add_le_add_iff_right _).2 hb_le_a1\u27e9\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave h\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 := \u27e8succ_pos _, (add_le_add_iff_right _).2 hb_le_a\u27e9\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\nh\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave dvd_iff : b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1 := by\n  rw [Nat.dvd_add_iff_left (dvd_refl (b + 1)), \u2190 add_tsub_add_eq_tsub_right a 1 b, add_comm (_ - _), add_assoc,\n    tsub_add_cancel_of_le (succ_le_succ hb_le_a), add_comm 1]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\nh\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1\n\u22a2 b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1\n[PROOFSTEP]\nrw [Nat.dvd_add_iff_left (dvd_refl (b + 1)), \u2190 add_tsub_add_eq_tsub_right a 1 b, add_comm (_ - _), add_assoc,\n  tsub_add_cancel_of_le (succ_le_succ hb_le_a), add_comm 1]\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\nh\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1\ndvd_iff : b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave wf : a - b < a + 1 := lt_succ_of_le tsub_le_self\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\nh\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1\ndvd_iff : b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1\nwf : a - b < a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nrw [if_pos h\u2081, if_pos h\u2082, @add_tsub_add_eq_tsub_right, \u2190 tsub_add_eq_add_tsub hb_le_a,\n  have := wf\n  succ_div (a - b),\n  @add_tsub_add_eq_tsub_right]\n[GOAL]\ncase pos\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : b \u2264 a + 1\nhb_le_a : b \u2264 a\nh\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1\nh\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1\ndvd_iff : b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1\nwf : a - b < a + 1\n\u22a2 ((a - b) / (b + 1) + if b + 1 \u2223 a - b + 1 then 1 else 0) + 1 =\n    (a - b) / (b + 1) + 1 + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nsimp [dvd_iff, succ_eq_add_one, add_comm 1, add_assoc]\n[GOAL]\ncase neg\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : \u00acb \u2264 a + 1\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave hba : \u00acb \u2264 a := not_le_of_gt (lt_trans (lt_succ_self a) (lt_of_not_ge hb_le_a1))\n[GOAL]\ncase neg\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : \u00acb \u2264 a + 1\nhba : \u00acb \u2264 a\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nhave hb_dvd_a : \u00acb + 1 \u2223 a + 2 := fun h => hb_le_a1 (le_of_succ_le_succ (le_of_dvd (succ_pos _) h))\n[GOAL]\ncase neg\na\u271d b\u271d m n k a b : \u2115\nhb_eq_a : \u00acb = a + 1\nhb_le_a1 : \u00acb \u2264 a + 1\nhba : \u00acb \u2264 a\nhb_dvd_a : \u00acb + 1 \u2223 a + 2\n\u22a2 (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 then (a + 1 + 1 - (b + 1)) / (b + 1) + 1 else 0) =\n    (if 0 < b + 1 \u2227 b + 1 \u2264 a + 1 then (a + 1 - (b + 1)) / (b + 1) + 1 else 0) + if b + 1 \u2223 a + 1 + 1 then 1 else 0\n[PROOFSTEP]\nsimp [hba, hb_le_a1, hb_dvd_a]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhba : b \u2223 a + 1\n\u22a2 (a + 1) / b = a / b + 1\n[PROOFSTEP]\nrw [succ_div, if_pos hba]\n[GOAL]\na\u271d b\u271d m n k a b : \u2115\nhba : \u00acb \u2223 a + 1\n\u22a2 (a + 1) / b = a / b\n[PROOFSTEP]\nrw [succ_div, if_neg hba, add_zero]\n[GOAL]\na\u271d b\u271d m n k a b c : \u2115\nh : a * b \u2223 c\nha : a = 0\n\u22a2 b \u2223 c / a\n[PROOFSTEP]\nsimp [ha]\n[GOAL]\na\u271d b\u271d m n k a b c : \u2115\nh : a * b \u2223 c\nha\u271d : \u00aca = 0\nha : 0 < a\nh1 : \u2203 d, c = a * b * d\nd : \u2115\nhd : c = a * b * d\n\u22a2 c = a * (b * d)\n[PROOFSTEP]\nsimpa [mul_assoc] using hd\n[GOAL]\na\u271d b\u271d m n k a b c x\u271d\u00b9 x\u271d : \u2115\ndvd : x\u271d\u00b9 \u2223 0\ndvd2 : 0 \u2223 x\u271d\n\u22a2 x\u271d / (0 / x\u271d\u00b9) / x\u271d\u00b9 = x\u271d / 0\n[PROOFSTEP]\nsimp\n[GOAL]\na\u271d\u00b9 b\u271d m n k a\u271d b c a x\u271d : \u2115\ndvd : 0 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\n\u22a2 x\u271d / ((a + 1) / 0) / 0 = x\u271d / (a + 1)\n[PROOFSTEP]\nsimp at dvd \n[GOAL]\na\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)\n[PROOFSTEP]\nhave a_split : a + 1 \u2260 0 := succ_ne_zero a\n[GOAL]\na\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)\n[PROOFSTEP]\nhave c_split : c + 1 \u2260 0 := succ_ne_zero c\n[GOAL]\na\u271d\u00b9 b\u271d m n k a\u271d b c\u271d a c x\u271d : \u2115\ndvd : c + 1 \u2223 a + 1\ndvd2 : a + 1 \u2223 x\u271d\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\n\u22a2 x\u271d / ((a + 1) / (c + 1)) / (c + 1) = x\u271d / (a + 1)\n[PROOFSTEP]\nrcases dvd2 with \u27e8k, rfl\u27e9\n[GOAL]\ncase intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\ndvd : c + 1 \u2223 a + 1\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk : \u2115\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)\n[PROOFSTEP]\nrcases dvd with \u27e8k2, pr\u27e9\n[GOAL]\ncase intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)\n[PROOFSTEP]\nhave k2_nonzero : k2 \u2260 0 := fun k2_zero => by simp [k2_zero] at pr \n[GOAL]\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\nk2_zero : k2 = 0\n\u22a2 False\n[PROOFSTEP]\nsimp [k2_zero] at pr \n[GOAL]\ncase intro.intro\na\u271d\u00b9 b\u271d m n k\u271d a\u271d b c\u271d a c : \u2115\na_split : a + 1 \u2260 0\nc_split : c + 1 \u2260 0\nk k2 : \u2115\npr : a + 1 = (c + 1) * k2\nk2_nonzero : k2 \u2260 0\n\u22a2 (a + 1) * k / ((a + 1) / (c + 1)) / (c + 1) = (a + 1) * k / (a + 1)\n[PROOFSTEP]\nrw [Nat.mul_div_cancel_left k (Nat.pos_of_ne_zero a_split), pr, Nat.mul_div_cancel_left k2 (Nat.pos_of_ne_zero c_split),\n  Nat.mul_comm ((c + 1) * k2) k, \u2190 Nat.mul_assoc k (c + 1) k2, Nat.mul_div_cancel _ (Nat.pos_of_ne_zero k2_nonzero),\n  Nat.mul_div_cancel _ (Nat.pos_of_ne_zero c_split)]\n[GOAL]\na b m n k : \u2115\nh : a < b + n\n\u22a2 n \u2223 a \u2192 n \u2223 b \u2192 a \u2264 b\n[PROOFSTEP]\nrintro \u27e8a, rfl\u27e9\n  \u27e8b, rfl\u27e9\n      -- porting note: Needed to give an explicit argument to `mul_add_one`\n[GOAL]\ncase intro.intro\nm n k a b : \u2115\nh : n * a < n * b + n\n\u22a2 n * a \u2264 n * b\n[PROOFSTEP]\nrw [\u2190 mul_add_one n] at h \n[GOAL]\ncase intro.intro\nm n k a b : \u2115\nh : n * a < n * (b + 1)\n\u22a2 n * a \u2264 n * b\n[PROOFSTEP]\nexact mul_le_mul_left' (lt_succ_iff.1 <| lt_of_mul_lt_mul_left h bot_le) _\n[GOAL]\na b m\u271d n\u271d k m n : \u2115\n\u22a2 m % n / n = 0\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\na b m\u271d n k m : \u2115\n\u22a2 m % zero / zero = 0\n[PROOFSTEP]\nexact (m % 0).div_zero\n[GOAL]\ncase succ\na b m\u271d n k m n\u271d : \u2115\n\u22a2 m % succ n\u271d / succ n\u271d = 0\n[PROOFSTEP]\ncase succ n => exact Nat.div_eq_zero (m.mod_lt n.succ_pos)\n[GOAL]\na b m\u271d n\u271d k m n : \u2115\n\u22a2 m % succ n / succ n = 0\n[PROOFSTEP]\ncase succ n => exact Nat.div_eq_zero (m.mod_lt n.succ_pos)\n[GOAL]\na b m\u271d n\u271d k m n : \u2115\n\u22a2 m % succ n / succ n = 0\n[PROOFSTEP]\nexact Nat.div_eq_zero (m.mod_lt n.succ_pos)\n[GOAL]\na\u271d b m n\u271d k n a : \u2115\nha : 0 < a\n\u22a2 (\u2203 k, a * k < n \u2227 n < a * (k + 1)) \u2194 \u00aca \u2223 n\n[PROOFSTEP]\nrefine'\n  \u27e8fun \u27e8k, hk1, hk2\u27e9 => not_dvd_of_between_consec_multiples hk1 hk2, fun han =>\n    \u27e8n / a, \u27e8lt_of_le_of_ne (mul_div_le n a) _, lt_mul_div_succ _ ha\u27e9\u27e9\u27e9\n[GOAL]\na\u271d b m n\u271d k n a : \u2115\nha : 0 < a\nhan : \u00aca \u2223 n\n\u22a2 a * (n / a) \u2260 n\n[PROOFSTEP]\nexact mt (Dvd.intro (n / a)) han\n[GOAL]\na b m\u271d n\u271d\u00b9 k m n\u271d : \u2115\nh : m = n\u271d\nn : \u2115\n\u22a2 m \u2223 n \u2194 n\u271d \u2223 n\n[PROOFSTEP]\nrw [h]\n[GOAL]\na b m\u271d n\u271d\u00b9 k m n\u271d : \u2115\nh : m = n\u271d\nn : \u2115\n\u22a2 n \u2223 m \u2194 n \u2223 n\u271d\n[PROOFSTEP]\nrw [h]\n[GOAL]\na\u271d b\u271d m n k a b d : \u2115\nhdb : d \u2223 b\nh : a < b\n\u22a2 a / d < b / d\n[PROOFSTEP]\nrw [Nat.lt_div_iff_mul_lt hdb]\n[GOAL]\na\u271d b\u271d m n k a b d : \u2115\nhdb : d \u2223 b\nh : a < b\n\u22a2 d * (a / d) < b\n[PROOFSTEP]\nexact lt_of_le_of_lt (mul_div_le a d) h\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Order.Lemmas", "llama_tokens": 9055, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8006920020959544, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5071665321723362}}
{"text": "[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u22a2 vectorSpan k \u2205 = \u22a5\n[PROOFSTEP]\nrw [vectorSpan_def, vsub_empty, Submodule.span_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\n\u22a2 vectorSpan k {p} = \u22a5\n[PROOFSTEP]\nsimp [vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 Set.Nonempty (spanPoints k s) \u2194 Set.Nonempty s\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 Set.Nonempty (spanPoints k s) \u2192 Set.Nonempty s\n[PROOFSTEP]\ncontrapose\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 \u00acSet.Nonempty s \u2192 \u00acSet.Nonempty (spanPoints k s)\n[PROOFSTEP]\nrw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty]\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 s = \u2205 \u2192 spanPoints k s = \u2205\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\nh : s = \u2205\n\u22a2 spanPoints k s = \u2205\n[PROOFSTEP]\nsimp [h, spanPoints]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 Set.Nonempty s \u2192 Set.Nonempty (spanPoints k s)\n[PROOFSTEP]\nexact fun h => h.mono (subset_spanPoints _ _)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhp : p \u2208 spanPoints k s\nhv : v \u2208 vectorSpan k s\n\u22a2 v +\u1d65 p \u2208 spanPoints k s\n[PROOFSTEP]\nrcases hp with \u27e8p2, \u27e8hp2, \u27e8v2, \u27e8hv2, hv2p\u27e9\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v +\u1d65 p \u2208 spanPoints k s\n[PROOFSTEP]\nrw [hv2p, vadd_vadd]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np : P\nv : V\nhv : v \u2208 vectorSpan k s\np2 : P\nhp2 : p2 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p = v2 +\u1d65 p2\n\u22a2 v + v2 +\u1d65 p2 \u2208 spanPoints k s\n[PROOFSTEP]\nexact \u27e8p2, hp2, v + v2, (vectorSpan k s).add_mem hv hv2, rfl\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 : P\nhp1 : p1 \u2208 spanPoints k s\nhp2 : p2 \u2208 spanPoints k s\n\u22a2 p1 -\u1d65 p2 \u2208 vectorSpan k s\n[PROOFSTEP]\nrcases hp1 with \u27e8p1a, \u27e8hp1a, \u27e8v1, \u27e8hv1, hv1p\u27e9\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 : P\nhp2 : p2 \u2208 spanPoints k s\np1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\n\u22a2 p1 -\u1d65 p2 \u2208 vectorSpan k s\n[PROOFSTEP]\nrcases hp2 with \u27e8p2a, \u27e8hp2a, \u27e8v2, \u27e8hv2, hv2p\u27e9\u27e9\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 p1 -\u1d65 p2 \u2208 vectorSpan k s\n[PROOFSTEP]\nrw [hv1p, hv2p, vsub_vadd_eq_vsub_sub (v1 +\u1d65 p1a), vadd_vsub_assoc, add_comm, add_sub_assoc]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 p1a -\u1d65 p2a + (v1 - v2) \u2208 vectorSpan k s\n[PROOFSTEP]\nhave hv1v2 : v1 - v2 \u2208 vectorSpan k s := by\n  rw [sub_eq_add_neg]\n  apply (vectorSpan k s).add_mem hv1\n  rw [\u2190 neg_one_smul k v2]\n  exact (vectorSpan k s).smul_mem (-1 : k) hv2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 v1 - v2 \u2208 vectorSpan k s\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 v1 + -v2 \u2208 vectorSpan k s\n[PROOFSTEP]\napply (vectorSpan k s).add_mem hv1\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 -v2 \u2208 vectorSpan k s\n[PROOFSTEP]\nrw [\u2190 neg_one_smul k v2]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\n\u22a2 -1 \u2022 v2 \u2208 vectorSpan k s\n[PROOFSTEP]\nexact (vectorSpan k s).smul_mem (-1 : k) hv2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\nhv1v2 : v1 - v2 \u2208 vectorSpan k s\n\u22a2 p1a -\u1d65 p2a + (v1 - v2) \u2208 vectorSpan k s\n[PROOFSTEP]\nrefine' (vectorSpan k s).add_mem _ hv1v2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p2 p1a : P\nhp1a : p1a \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhv1p : p1 = v1 +\u1d65 p1a\np2a : P\nhp2a : p2a \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhv2p : p2 = v2 +\u1d65 p2a\nhv1v2 : v1 - v2 \u2208 vectorSpan k s\n\u22a2 p1a -\u1d65 p2a \u2208 vectorSpan k s\n[PROOFSTEP]\nexact vsub_mem_vectorSpan k hp1a hp2a\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np q : AffineSubspace k P\nx\u271d : p.carrier = q.carrier\n\u22a2 p = q\n[PROOFSTEP]\ncases p\n[GOAL]\ncase mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nq : AffineSubspace k P\ncarrier\u271d : Set P\nsmul_vsub_vadd_mem\u271d :\n  \u2200 (c : k) {p1 p2 p3 : P}, p1 \u2208 carrier\u271d \u2192 p2 \u2208 carrier\u271d \u2192 p3 \u2208 carrier\u271d \u2192 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 carrier\u271d\nx\u271d : { carrier := carrier\u271d, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d }.carrier = q.carrier\n\u22a2 { carrier := carrier\u271d, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d } = q\n[PROOFSTEP]\ncases q\n[GOAL]\ncase mk.mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ncarrier\u271d\u00b9 : Set P\nsmul_vsub_vadd_mem\u271d\u00b9 :\n  \u2200 (c : k) {p1 p2 p3 : P}, p1 \u2208 carrier\u271d\u00b9 \u2192 p2 \u2208 carrier\u271d\u00b9 \u2192 p3 \u2208 carrier\u271d\u00b9 \u2192 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 carrier\u271d\u00b9\ncarrier\u271d : Set P\nsmul_vsub_vadd_mem\u271d :\n  \u2200 (c : k) {p1 p2 p3 : P}, p1 \u2208 carrier\u271d \u2192 p2 \u2208 carrier\u271d \u2192 p3 \u2208 carrier\u271d \u2192 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 carrier\u271d\nx\u271d :\n  { carrier := carrier\u271d\u00b9, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d\u00b9 }.carrier =\n    { carrier := carrier\u271d, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d }.carrier\n\u22a2 { carrier := carrier\u271d\u00b9, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d\u00b9 } =\n    { carrier := carrier\u271d, smul_vsub_vadd_mem := smul_vsub_vadd_mem\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nintro a b ha hb\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\na b : V\nha : a \u2208 \u2191s -\u1d65 \u2191s\nhb : b \u2208 \u2191s -\u1d65 \u2191s\n\u22a2 a + b \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nrcases ha with \u27e8p1, p2, hp1, hp2, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nb : V\nhb : b \u2208 \u2191s -\u1d65 \u2191s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 + b \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nrcases hb with \u27e8p3, p4, hp3, hp4, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\np3 p4 : P\nhp3 : p3 \u2208 \u2191s\nhp4 : p4 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 + (fun x x_1 => x -\u1d65 x_1) p3 p4 \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nrw [\u2190 vadd_vsub_assoc]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\np3 p4 : P\nhp3 : p3 \u2208 \u2191s\nhp4 : p4 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 +\u1d65 p3 -\u1d65 p4 \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nrefine' vsub_mem_vsub _ hp4\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\np3 p4 : P\nhp3 : p3 \u2208 \u2191s\nhp4 : p4 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 +\u1d65 p3 \u2208 \u2191s\n[PROOFSTEP]\nconvert s.smul_vsub_vadd_mem 1 hp1 hp2 hp3\n[GOAL]\ncase h.e'_4.h.e'_5\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\np3 p4 : P\nhp3 : p3 \u2208 \u2191s\nhp4 : p4 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 = 1 \u2022 (p1 -\u1d65 p2)\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 0 \u2208 { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier\n[PROOFSTEP]\ncases' h with p hp\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s\n\u22a2 0 \u2208 { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier\n[PROOFSTEP]\nexact vsub_self p \u25b8 vsub_mem_vsub hp hp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 \u2200 (c : k) {x : V},\n    x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n              zero_mem' :=\n                (_ :\n                  0 \u2208\n                    { carrier := \u2191s -\u1d65 \u2191s,\n                        add_mem' :=\n                          (_ :\n                            \u2200 {a b : V},\n                              a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier \u2192\n      c \u2022 x \u2208\n        {\n              toAddSubsemigroup :=\n                { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n              zero_mem' :=\n                (_ :\n                  0 \u2208\n                    { carrier := \u2191s -\u1d65 \u2191s,\n                        add_mem' :=\n                          (_ :\n                            \u2200 {a b : V},\n                              a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nintro c v hv\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nc : k\nv : V\nhv :\n  v \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n          zero_mem' :=\n            (_ :\n              0 \u2208\n                { carrier := \u2191s -\u1d65 \u2191s,\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : V},\n                          a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier\n\u22a2 c \u2022 v \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n          zero_mem' :=\n            (_ :\n              0 \u2208\n                { carrier := \u2191s -\u1d65 \u2191s,\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : V},\n                          a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrcases hv with \u27e8p1, p2, hp1, hp2, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nc : k\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\n\u22a2 c \u2022 (fun x x_1 => x -\u1d65 x_1) p1 p2 \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n          zero_mem' :=\n            (_ :\n              0 \u2208\n                { carrier := \u2191s -\u1d65 \u2191s,\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : V},\n                          a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrw [\u2190 vadd_vsub (c \u2022 (p1 -\u1d65 p2)) p2]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nc : k\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p2 -\u1d65 p2 \u2208\n    {\n          toAddSubsemigroup :=\n            { carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) },\n          zero_mem' :=\n            (_ :\n              0 \u2208\n                { carrier := \u2191s -\u1d65 \u2191s,\n                    add_mem' :=\n                      (_ :\n                        \u2200 {a b : V},\n                          a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) }.carrier) }.toAddSubsemigroup.carrier\n[PROOFSTEP]\nrefine' vsub_mem_vsub _ hp2\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nc : k\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p2 \u2208 \u2191s\n[PROOFSTEP]\nexact s.smul_vsub_vadd_mem c hp1 hp2 hp2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 directionOfNonempty h = direction s\n[PROOFSTEP]\nrefine le_antisymm ?_ (Submodule.span_le.2 Set.Subset.rfl)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 directionOfNonempty h \u2264 direction s\n[PROOFSTEP]\nrw [\u2190 SetLike.coe_subset_coe, directionOfNonempty, direction, Submodule.coe_set_mk, AddSubmonoid.coe_set_mk]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 \u2191{ carrier := \u2191s -\u1d65 \u2191s, add_mem' := (_ : \u2200 {a b : V}, a \u2208 \u2191s -\u1d65 \u2191s \u2192 b \u2208 \u2191s -\u1d65 \u2191s \u2192 a + b \u2208 \u2191s -\u1d65 \u2191s) } \u2286\n    \u2191(vectorSpan k \u2191s)\n[PROOFSTEP]\nexact (vsub_set_subset_vectorSpan k _)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nv : V\n\u22a2 v \u2208 direction s \u2194 \u2203 p1, p1 \u2208 s \u2227 \u2203 p2, p2 \u2208 s \u2227 v = p1 -\u1d65 p2\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_direction_eq_vsub_set h]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nv : V\n\u22a2 v \u2208 \u2191s -\u1d65 \u2191s \u2194 \u2203 p1, p1 \u2208 s \u2227 \u2203 p2, p2 \u2208 s \u2227 v = p1 -\u1d65 p2\n[PROOFSTEP]\nexact\n  \u27e8fun \u27e8p1, p2, hp1, hp2, hv\u27e9 => \u27e8p1, hp1, p2, hp2, hv.symm\u27e9, fun \u27e8p1, hp1, p2, hp2, hv\u27e9 => \u27e8p1, p2, hp1, hp2, hv.symm\u27e9\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nhv : v \u2208 direction s\np : P\nhp : p \u2208 s\n\u22a2 v +\u1d65 p \u2208 s\n[PROOFSTEP]\nrw [mem_direction_iff_eq_vsub \u27e8p, hp\u27e9] at hv \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nhv : \u2203 p1, p1 \u2208 s \u2227 \u2203 p2, p2 \u2208 s \u2227 v = p1 -\u1d65 p2\np : P\nhp : p \u2208 s\n\u22a2 v +\u1d65 p \u2208 s\n[PROOFSTEP]\nrcases hv with \u27e8p1, hp1, p2, hp2, hv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p \u2208 s\np1 : P\nhp1 : p1 \u2208 s\np2 : P\nhp2 : p2 \u2208 s\nhv : v = p1 -\u1d65 p2\n\u22a2 v +\u1d65 p \u2208 s\n[PROOFSTEP]\nrw [hv]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p \u2208 s\np1 : P\nhp1 : p1 \u2208 s\np2 : P\nhp2 : p2 \u2208 s\nhv : v = p1 -\u1d65 p2\n\u22a2 p1 -\u1d65 p2 +\u1d65 p \u2208 s\n[PROOFSTEP]\nconvert s.smul_vsub_vadd_mem 1 hp1 hp2 hp\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p \u2208 s\np1 : P\nhp1 : p1 \u2208 s\np2 : P\nhp2 : p2 \u2208 s\nhv : v = p1 -\u1d65 p2\n\u22a2 p1 -\u1d65 p2 +\u1d65 p \u2208 s \u2194 1 \u2022 (p1 -\u1d65 p2) +\u1d65 p \u2208 s.carrier\n[PROOFSTEP]\nrw [one_smul]\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p \u2208 s\np1 : P\nhp1 : p1 \u2208 s\np2 : P\nhp2 : p2 \u2208 s\nhv : v = p1 -\u1d65 p2\n\u22a2 p1 -\u1d65 p2 +\u1d65 p \u2208 s \u2194 p1 -\u1d65 p2 +\u1d65 p \u2208 s.carrier\n[PROOFSTEP]\nexact s.mem_coe k P _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\np : P\nhp : p \u2208 s\nh : v +\u1d65 p \u2208 s\n\u22a2 v \u2208 direction s\n[PROOFSTEP]\nsimpa using vsub_mem_direction h hp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nhv : v \u2208 direction s\np : P\n\u22a2 v +\u1d65 p \u2208 s \u2194 p \u2208 s\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => vadd_mem_of_mem_direction hv h\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nhv : v \u2208 direction s\np : P\nh : v +\u1d65 p \u2208 s\n\u22a2 p \u2208 s\n[PROOFSTEP]\nconvert vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) h\n[GOAL]\ncase h.e'_4\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nhv : v \u2208 direction s\np : P\nh : v +\u1d65 p \u2208 s\n\u22a2 p = -v +\u1d65 (v +\u1d65 p)\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\n\u22a2 \u2191(direction s) = (fun x => x -\u1d65 p) '' \u2191s\n[PROOFSTEP]\nrw [coe_direction_eq_vsub_set \u27e8p, hp\u27e9]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\n\u22a2 \u2191s -\u1d65 \u2191s = (fun x => x -\u1d65 p) '' \u2191s\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\n\u22a2 \u2191s -\u1d65 \u2191s \u2264 (fun x => x -\u1d65 p) '' \u2191s\n[PROOFSTEP]\nrintro v \u27e8p1, p2, hp1, hp2, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np1 p2 : P\nhp1 : p1 \u2208 \u2191s\nhp2 : p2 \u2208 \u2191s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p2 \u2208 (fun x => x -\u1d65 p) '' \u2191s\n[PROOFSTEP]\nexact \u27e8p1 -\u1d65 p2 +\u1d65 p, vadd_mem_of_mem_direction (vsub_mem_direction hp1 hp2) hp, vadd_vsub _ _\u27e9\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\n\u22a2 (fun x => x -\u1d65 p) '' \u2191s \u2264 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nrintro v \u27e8p2, hp2, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np2 : P\nhp2 : p2 \u2208 \u2191s\n\u22a2 (fun x => x -\u1d65 p) p2 \u2208 \u2191s -\u1d65 \u2191s\n[PROOFSTEP]\nexact \u27e8p2, p, hp2, hp, rfl\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\n\u22a2 \u2191(direction s) = (fun x x_1 => x -\u1d65 x_1) p '' \u2191s\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 \u2191(direction s) \u2194 v \u2208 (fun x x_1 => x -\u1d65 x_1) p '' \u2191s\n[PROOFSTEP]\nrw [SetLike.mem_coe, \u2190 Submodule.neg_mem_iff, \u2190 SetLike.mem_coe, coe_direction_eq_vsub_set_right hp,\n  Set.mem_image_iff_bex, Set.mem_image_iff_bex]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 (\u2203 x x_1, x -\u1d65 p = -v) \u2194 \u2203 x x_1, (fun x x_2 => x -\u1d65 x_2) p x = v\n[PROOFSTEP]\nconv_lhs =>\n  congr\n  ext\n  rw [\u2190 neg_vsub_eq_vsub_rev, neg_inj]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n| \u2203 x x_1, x -\u1d65 p = -v\n[PROOFSTEP]\n  congr\n  ext\n  rw [\u2190 neg_vsub_eq_vsub_rev, neg_inj]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n| \u2203 x x_1, x -\u1d65 p = -v\n[PROOFSTEP]\n  congr\n  ext\n  rw [\u2190 neg_vsub_eq_vsub_rev, neg_inj]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n| \u2203 x x_1, x -\u1d65 p = -v\n[PROOFSTEP]\ncongr\n[GOAL]\ncase p\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n| fun x => \u2203 x_1, x -\u1d65 p = -v\n[PROOFSTEP]\next\n[GOAL]\ncase p.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\nx\u271d : P\n| \u2203 x, x\u271d -\u1d65 p = -v\n[PROOFSTEP]\nrw [\u2190 neg_vsub_eq_vsub_rev, neg_inj]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 direction s \u2194 \u2203 p2, p2 \u2208 s \u2227 v = p2 -\u1d65 p\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_direction_eq_vsub_set_right hp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 (fun x => x -\u1d65 p) '' \u2191s \u2194 \u2203 p2, p2 \u2208 s \u2227 v = p2 -\u1d65 p\n[PROOFSTEP]\nexact \u27e8fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9, fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 direction s \u2194 \u2203 p2, p2 \u2208 s \u2227 v = p -\u1d65 p2\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe, coe_direction_eq_vsub_set_left hp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 (fun x x_1 => x -\u1d65 x_1) p '' \u2191s \u2194 \u2203 p2, p2 \u2208 s \u2227 v = p -\u1d65 p2\n[PROOFSTEP]\nexact \u27e8fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9, fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np2 : P\n\u22a2 p2 -\u1d65 p \u2208 direction s \u2194 p2 \u2208 s\n[PROOFSTEP]\nrw [mem_direction_iff_eq_vsub_right hp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np2 : P\n\u22a2 (\u2203 p2_1, p2_1 \u2208 s \u2227 p2 -\u1d65 p = p2_1 -\u1d65 p) \u2194 p2 \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np2 : P\n\u22a2 p -\u1d65 p2 \u2208 direction s \u2194 p2 \u2208 s\n[PROOFSTEP]\nrw [mem_direction_iff_eq_vsub_left hp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\np2 : P\n\u22a2 (\u2203 p2_1, p2_1 \u2208 s \u2227 p -\u1d65 p2 = p -\u1d65 p2_1) \u2194 p2 \u2208 s\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\n\u22a2 s1 = s2\n[PROOFSTEP]\next p\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\n\u22a2 p \u2208 s1 \u2194 p \u2208 s2\n[PROOFSTEP]\nhave hq1 := Set.mem_of_mem_inter_left hn.some_mem\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\n\u22a2 p \u2208 s1 \u2194 p \u2208 s2\n[PROOFSTEP]\nhave hq2 := Set.mem_of_mem_inter_right hn.some_mem\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\n\u22a2 p \u2208 s1 \u2194 p \u2208 s2\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\n\u22a2 p \u2208 s1 \u2192 p \u2208 s2\n[PROOFSTEP]\nintro hp\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s1\n\u22a2 p \u2208 s2\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p hn.some]\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s1\n\u22a2 p -\u1d65 Set.Nonempty.some hn +\u1d65 Set.Nonempty.some hn \u2208 s2\n[PROOFSTEP]\nrefine' vadd_mem_of_mem_direction _ hq2\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s1\n\u22a2 p -\u1d65 Set.Nonempty.some hn \u2208 direction s2\n[PROOFSTEP]\nrw [\u2190 hd]\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s1\n\u22a2 p -\u1d65 Set.Nonempty.some hn \u2208 direction s1\n[PROOFSTEP]\nexact vsub_mem_direction hp hq1\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\n\u22a2 p \u2208 s2 \u2192 p \u2208 s1\n[PROOFSTEP]\nintro hp\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s2\n\u22a2 p \u2208 s1\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p hn.some]\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s2\n\u22a2 p -\u1d65 Set.Nonempty.some hn +\u1d65 Set.Nonempty.some hn \u2208 s1\n[PROOFSTEP]\nrefine' vadd_mem_of_mem_direction _ hq1\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s2\n\u22a2 p -\u1d65 Set.Nonempty.some hn \u2208 direction s1\n[PROOFSTEP]\nrw [hd]\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\nhd : direction s1 = direction s2\nhn : Set.Nonempty (\u2191s1 \u2229 \u2191s2)\np : P\nhq1 : Set.Nonempty.some hn \u2208 \u2191s1\nhq2 : Set.Nonempty.some hn \u2208 \u2191s2\nhp : p \u2208 s2\n\u22a2 p -\u1d65 Set.Nonempty.some hn \u2208 direction s2\n[PROOFSTEP]\nexact vsub_mem_direction hp hq2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na : { x // x \u2208 s }\n\u22a2 0 +\u1d65 a = a\n[PROOFSTEP]\next\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na : { x // x \u2208 s }\n\u22a2 \u2191(0 +\u1d65 a) = \u2191a\n[PROOFSTEP]\nexact zero_vadd _ _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na b : { x // x \u2208 direction s }\nc : { x // x \u2208 s }\n\u22a2 a + b +\u1d65 c = a +\u1d65 (b +\u1d65 c)\n[PROOFSTEP]\next\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na b : { x // x \u2208 direction s }\nc : { x // x \u2208 s }\n\u22a2 \u2191(a + b +\u1d65 c) = \u2191(a +\u1d65 (b +\u1d65 c))\n[PROOFSTEP]\napply add_vadd\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\n\u22a2 Nonempty { x // x \u2208 s }\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na b : { x // x \u2208 s }\n\u22a2 a -\u1d65 b +\u1d65 b = a\n[PROOFSTEP]\next\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na b : { x // x \u2208 s }\n\u22a2 \u2191(a -\u1d65 b +\u1d65 b) = \u2191a\n[PROOFSTEP]\napply AddTorsor.vsub_vadd'\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na : { x // x \u2208 direction s }\nb : { x // x \u2208 s }\n\u22a2 a +\u1d65 b -\u1d65 b = a\n[PROOFSTEP]\next\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\ns : AffineSubspace k P\ninst\u271d : Nonempty { x // x \u2208 s }\na : { x // x \u2208 direction s }\nb : { x // x \u2208 s }\n\u22a2 \u2191(a +\u1d65 b -\u1d65 b) = \u2191a\n[PROOFSTEP]\napply AddTorsor.vadd_vsub'\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np1 p2 p3 : P\nhp1 : p1 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\nhp2 : p2 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\nhp3 : p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\n[PROOFSTEP]\nrcases hp1 with \u27e8v1, hv1, hp1\u27e9\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np1 p2 p3 : P\nhp2 : p2 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\nhp3 : p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\n[PROOFSTEP]\nrcases hp2 with \u27e8v2, hv2, hp2\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np1 p2 p3 : P\nhp3 : p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\nv2 : V\nhv2 : v2 \u2208 direction\nhp2 : p2 = v2 +\u1d65 p\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\n[PROOFSTEP]\nrcases hp3 with \u27e8v3, hv3, hp3\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np1 p2 p3 : P\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\nv2 : V\nhv2 : v2 \u2208 direction\nhp2 : p2 = v2 +\u1d65 p\nv3 : V\nhv3 : v3 \u2208 direction\nhp3 : p3 = v3 +\u1d65 p\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 {q | \u2203 v, v \u2208 direction \u2227 q = v +\u1d65 p}\n[PROOFSTEP]\nuse c \u2022 (v1 - v2) + v3, direction.add_mem (direction.smul_mem c (direction.sub_mem hv1 hv2)) hv3\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nc : k\np1 p2 p3 : P\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\nv2 : V\nhv2 : v2 \u2208 direction\nhp2 : p2 = v2 +\u1d65 p\nv3 : V\nhv3 : v3 \u2208 direction\nhp3 : p3 = v3 +\u1d65 p\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 = c \u2022 (v1 - v2) + v3 +\u1d65 p\n[PROOFSTEP]\nsimp [hp1, hp2, hp3, vadd_vadd]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\n\u22a2 AffineSubspace.direction (mk' p direction) = direction\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\n\u22a2 v \u2208 AffineSubspace.direction (mk' p direction) \u2194 v \u2208 direction\n[PROOFSTEP]\nrw [mem_direction_iff_eq_vsub (mk'_nonempty _ _)]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\n\u22a2 (\u2203 p1, p1 \u2208 mk' p direction \u2227 \u2203 p2, p2 \u2208 mk' p direction \u2227 v = p1 -\u1d65 p2) \u2194 v \u2208 direction\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\n\u22a2 (\u2203 p1, p1 \u2208 mk' p direction \u2227 \u2203 p2, p2 \u2208 mk' p direction \u2227 v = p1 -\u1d65 p2) \u2192 v \u2208 direction\n[PROOFSTEP]\nrintro \u27e8p1, \u27e8v1, hv1, hp1\u27e9, p2, \u27e8v2, hv2, hp2\u27e9, hv\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\np1 : P\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\np2 : P\nhv : v = p1 -\u1d65 p2\nv2 : V\nhv2 : v2 \u2208 direction\nhp2 : p2 = v2 +\u1d65 p\n\u22a2 v \u2208 direction\n[PROOFSTEP]\nrw [hv, hp1, hp2, vadd_vsub_vadd_cancel_right]\n[GOAL]\ncase h.mp.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\np1 : P\nv1 : V\nhv1 : v1 \u2208 direction\nhp1 : p1 = v1 +\u1d65 p\np2 : P\nhv : v = p1 -\u1d65 p2\nv2 : V\nhv2 : v2 \u2208 direction\nhp2 : p2 = v2 +\u1d65 p\n\u22a2 v1 - v2 \u2208 direction\n[PROOFSTEP]\nexact direction.sub_mem hv1 hv2\n[GOAL]\ncase h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np : P\ndirection : Submodule k V\nv : V\n\u22a2 v \u2208 direction \u2192 \u2203 p1, p1 \u2208 mk' p direction \u2227 \u2203 p2, p2 \u2208 mk' p direction \u2227 v = p1 -\u1d65 p2\n[PROOFSTEP]\nexact fun hv => \u27e8v +\u1d65 p, vadd_mem_mk' _ hv, p, self_mem_mk' _ _, (vadd_vsub _ _).symm\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\ndirection : Submodule k V\n\u22a2 p\u2082 \u2208 mk' p\u2081 direction \u2194 p\u2082 -\u1d65 p\u2081 \u2208 direction\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\ndirection : Submodule k V\nh : p\u2082 \u2208 mk' p\u2081 direction\n\u22a2 p\u2082 -\u1d65 p\u2081 \u2208 direction\n[PROOFSTEP]\nrw [\u2190 direction_mk' p\u2081 direction]\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\ndirection : Submodule k V\nh : p\u2082 \u2208 mk' p\u2081 direction\n\u22a2 p\u2082 -\u1d65 p\u2081 \u2208 AffineSubspace.direction (mk' p\u2081 direction)\n[PROOFSTEP]\nexact vsub_mem_direction h (self_mem_mk' _ _)\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\ndirection : Submodule k V\nh : p\u2082 -\u1d65 p\u2081 \u2208 direction\n\u22a2 p\u2082 \u2208 mk' p\u2081 direction\n[PROOFSTEP]\nrw [\u2190 vsub_vadd p\u2082 p\u2081]\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 : P\ndirection : Submodule k V\nh : p\u2082 -\u1d65 p\u2081 \u2208 direction\n\u22a2 p\u2082 -\u1d65 p\u2081 +\u1d65 p\u2081 \u2208 mk' p\u2081 direction\n[PROOFSTEP]\nexact vadd_mem_mk' p\u2081 h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\n\u22a2 spanPoints k s \u2286 \u2191s1\n[PROOFSTEP]\nrintro p \u27e8p1, hp1, v, hv, hp\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\n\u22a2 p \u2208 \u2191s1\n[PROOFSTEP]\nrw [hp]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\n\u22a2 v +\u1d65 p1 \u2208 \u2191s1\n[PROOFSTEP]\nhave hp1s1 : p1 \u2208 (s1 : Set P) := Set.mem_of_mem_of_subset hp1 h\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\nhp1s1 : p1 \u2208 \u2191s1\n\u22a2 v +\u1d65 p1 \u2208 \u2191s1\n[PROOFSTEP]\nrefine' vadd_mem_of_mem_direction _ hp1s1\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\nhp1s1 : p1 \u2208 \u2191s1\n\u22a2 v \u2208 direction s1\n[PROOFSTEP]\nhave hs : vectorSpan k s \u2264 s1.direction := vectorSpan_mono k h\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\nhp1s1 : p1 \u2208 \u2191s1\nhs : vectorSpan k s \u2264 direction s1\n\u22a2 v \u2208 direction s1\n[PROOFSTEP]\nrw [SetLike.le_def] at hs \n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\nhp1s1 : p1 \u2208 \u2191s1\nhs : \u2200 \u2983x : V\u2984, x \u2208 vectorSpan k s \u2192 x \u2208 direction s1\n\u22a2 v \u2208 direction s1\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\ns1 : AffineSubspace k P\nh : s \u2286 \u2191s1\np p1 : P\nhp1 : p1 \u2208 s\nv : V\nhv : v \u2208 vectorSpan k s\nhp : p = v +\u1d65 p1\nhp1s1 : p1 \u2208 \u2191s1\nhs : \u2200 \u2983x : V\u2984, x \u2208 vectorSpan k s \u2192 x \u2208 direction s1\n\u22a2 v \u2208 \u2191(direction s1)\n[PROOFSTEP]\nexact Set.mem_of_mem_of_subset hv hs\n[GOAL]\nk : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\ns : Submodule k V\n\u22a2 AffineSubspace.direction (toAffineSubspace s) = s\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\ns : Submodule k V\nx : V\n\u22a2 x \u2208 AffineSubspace.direction (toAffineSubspace s) \u2194 x \u2208 s\n[PROOFSTEP]\nsimp [\u2190 s.toAffineSubspace.vadd_mem_iff_mem_direction _ s.zero_mem]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nQ : AffineSubspace k P\np\u2080 p\u2081 : P\nc : k\nh\u2080 : p\u2080 \u2208 Q\nh\u2081 : p\u2081 \u2208 Q\n\u22a2 \u2191(lineMap p\u2080 p\u2081) c \u2208 Q\n[PROOFSTEP]\nrw [AffineMap.lineMap_apply]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\nQ : AffineSubspace k P\np\u2080 p\u2081 : P\nc : k\nh\u2080 : p\u2080 \u2208 Q\nh\u2081 : p\u2081 \u2208 Q\n\u22a2 c \u2022 (p\u2081 -\u1d65 p\u2080) +\u1d65 p\u2080 \u2208 Q\n[PROOFSTEP]\nexact Q.smul_vsub_vadd_mem c h\u2081 h\u2080 h\u2080\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 AffineSubspace.direction (affineSpan k s) = vectorSpan k s\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 AffineSubspace.direction (affineSpan k s) \u2264 vectorSpan k s\n[PROOFSTEP]\nrefine' Submodule.span_le.2 _\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 \u2191(affineSpan k s) -\u1d65 \u2191(affineSpan k s) \u2286 \u2191(vectorSpan k s)\n[PROOFSTEP]\nrintro v \u27e8p1, p3, \u27e8p2, hp2, v1, hv1, hp1\u27e9, \u27e8p4, hp4, v2, hv2, hp3\u27e9, rfl\u27e9\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p1 p3 \u2208 \u2191(vectorSpan k s)\n[PROOFSTEP]\nsimp only [SetLike.mem_coe]\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 p1 -\u1d65 p3 \u2208 vectorSpan k s\n[PROOFSTEP]\nrw [hp1, hp3, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc]\n[GOAL]\ncase a.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\np1 p3 p2 : P\nhp2 : p2 \u2208 s\nv1 : V\nhv1 : v1 \u2208 vectorSpan k s\nhp1 : p1 = v1 +\u1d65 p2\np4 : P\nhp4 : p4 \u2208 s\nv2 : V\nhv2 : v2 \u2208 vectorSpan k s\nhp3 : p3 = v2 +\u1d65 p4\n\u22a2 v1 + (p2 -\u1d65 p4) - v2 \u2208 vectorSpan k s\n[PROOFSTEP]\nexact (vectorSpan k s).sub_mem ((vectorSpan k s).add_mem hv1 (vsub_mem_vectorSpan k hp2 hp4)) hv2\n[GOAL]\ncase a\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : Set P\n\u22a2 vectorSpan k s \u2264 AffineSubspace.direction (affineSpan k s)\n[PROOFSTEP]\nexact vectorSpan_mono k (subset_spanPoints k s)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\nsrc\u271d : PartialOrder (AffineSubspace k P) := PartialOrder.lift SetLike.coe (_ : Function.Injective SetLike.coe)\ns : Set (AffineSubspace k P)\nc : k\np1 p2 p3 : P\nhp1 : p1 \u2208 \u22c2 (s' : AffineSubspace k P) (_ : s' \u2208 s), \u2191s'\nhp2 : p2 \u2208 \u22c2 (s' : AffineSubspace k P) (_ : s' \u2208 s), \u2191s'\nhp3 : p3 \u2208 \u22c2 (s' : AffineSubspace k P) (_ : s' \u2208 s), \u2191s'\ns2 : AffineSubspace k P\nhs2 : s2 \u2208 s\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 \u2191s2\n[PROOFSTEP]\nrw [Set.mem_iInter\u2082] at *\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\nsrc\u271d : PartialOrder (AffineSubspace k P) := PartialOrder.lift SetLike.coe (_ : Function.Injective SetLike.coe)\ns : Set (AffineSubspace k P)\nc : k\np1 p2 p3 : P\nhp1 : \u2200 (i : AffineSubspace k P), i \u2208 s \u2192 p1 \u2208 \u2191i\nhp2 : \u2200 (i : AffineSubspace k P), i \u2208 s \u2192 p2 \u2208 \u2191i\nhp3 : \u2200 (i : AffineSubspace k P), i \u2208 s \u2192 p3 \u2208 \u2191i\ns2 : AffineSubspace k P\nhs2 : s2 \u2208 s\n\u22a2 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 \u2191s2\n[PROOFSTEP]\nexact s2.smul_vsub_vadd_mem c (hp1 s2 hs2) (hp2 s2 hs2) (hp3 s2 hs2)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS\u271d : AffineSpace V P\nsrc\u271d : PartialOrder (AffineSubspace k P) := PartialOrder.lift SetLike.coe (_ : Function.Injective SetLike.coe)\nS : Set (AffineSubspace k P)\ns1 : AffineSubspace k P\nhs1 : \u2200 (b : AffineSubspace k P), b \u2208 S \u2192 s1 \u2264 b\n\u22a2 s1 \u2264 sInf S\n[PROOFSTEP]\nrefine' Set.subset_sInter (t := (s1 : Set P)) _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS\u271d : AffineSpace V P\nsrc\u271d : PartialOrder (AffineSubspace k P) := PartialOrder.lift SetLike.coe (_ : Function.Injective SetLike.coe)\nS : Set (AffineSubspace k P)\ns1 : AffineSubspace k P\nhs1 : \u2200 (b : AffineSubspace k P), b \u2208 S \u2192 s1 \u2264 b\n\u22a2 \u2200 (t' : Set P), (t' \u2208 range fun s' => \u22c2 (_ : s' \u2208 S), \u2191s') \u2192 \u2191s1 \u2286 t'\n[PROOFSTEP]\nrintro t \u27e8s, _hs, rfl\u27e9\n[GOAL]\ncase intro.refl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS\u271d : AffineSpace V P\nsrc\u271d : PartialOrder (AffineSubspace k P) := PartialOrder.lift SetLike.coe (_ : Function.Injective SetLike.coe)\nS : Set (AffineSubspace k P)\ns1 : AffineSubspace k P\nhs1 : \u2200 (b : AffineSubspace k P), b \u2208 S \u2192 s1 \u2264 b\ns : AffineSubspace k P\n\u22a2 \u2191s1 \u2286 (fun s' => \u22c2 (_ : s' \u2208 S), \u2191s') s\n[PROOFSTEP]\nexact Set.subset_iInter (hs1 s)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\ns1 s2 : AffineSubspace k P\n\u22a2 s1 < s2 \u2194 s1 \u2264 s2 \u2227 \u2203 p, p \u2208 s2 \u2227 \u00acp \u2208 s1\n[PROOFSTEP]\nrw [lt_iff_le_not_le, not_le_iff_exists]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p : P\n\u22a2 \u2191(affineSpan k {p}) = {p}\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p x : P\n\u22a2 x \u2208 \u2191(affineSpan k {p}) \u2194 x \u2208 {p}\n[PROOFSTEP]\nrw [mem_coe, \u2190 vsub_right_mem_direction_iff_mem (mem_affineSpan k (Set.mem_singleton p)) _, direction_affineSpan]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p x : P\n\u22a2 x -\u1d65 p \u2208 vectorSpan k {p} \u2194 x \u2208 {p}\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u22a2 p\u2081 \u2208 affineSpan k {p\u2082} \u2194 p\u2081 = p\u2082\n[PROOFSTEP]\nsimp [\u2190 mem_coe]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u22a2 direction \u22a4 = \u22a4\n[PROOFSTEP]\ncases' S.Nonempty with p\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p : P\n\u22a2 direction \u22a4 = \u22a4\n[PROOFSTEP]\next v\n[GOAL]\ncase intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p : P\nv : V\n\u22a2 v \u2208 direction \u22a4 \u2194 v \u2208 \u22a4\n[PROOFSTEP]\nrefine' \u27e8imp_intro Submodule.mem_top, fun _hv => _\u27e9\n[GOAL]\ncase intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p : P\nv : V\n_hv : v \u2208 \u22a4\n\u22a2 v \u2208 direction \u22a4\n[PROOFSTEP]\nhave hpv : (v +\u1d65 p -\u1d65 p : V) \u2208 (\u22a4 : AffineSubspace k P).direction := vsub_mem_direction (mem_top k V _) (mem_top k V _)\n[GOAL]\ncase intro.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 p : P\nv : V\n_hv : v \u2208 \u22a4\nhpv : v +\u1d65 p -\u1d65 p \u2208 direction \u22a4\n\u22a2 v \u2208 direction \u22a4\n[PROOFSTEP]\nrwa [vadd_vsub] at hpv \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u22a2 \u22a5 \u2260 \u22a4\n[PROOFSTEP]\nintro contra\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ncontra : \u22a5 = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 ext_iff, bot_coe, top_coe] at contra \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ncontra : \u2205 = univ\n\u22a2 False\n[PROOFSTEP]\nexact Set.empty_ne_univ contra\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh : affineSpan k s = \u22a4\n\u22a2 Set.Nonempty s\n[PROOFSTEP]\nrw [Set.nonempty_iff_ne_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh : affineSpan k s = \u22a4\n\u22a2 s \u2260 \u2205\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nh : affineSpan k \u2205 = \u22a4\n\u22a2 False\n[PROOFSTEP]\nrw [AffineSubspace.span_empty] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nh : \u22a5 = \u22a4\n\u22a2 False\n[PROOFSTEP]\nexact bot_ne_top k V P h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh : affineSpan k s = \u22a4\n\u22a2 vectorSpan k s = \u22a4\n[PROOFSTEP]\nrw [\u2190 direction_affineSpan, h, direction_top]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\n\u22a2 affineSpan k s = \u22a4 \u2194 vectorSpan k s = \u22a4\n[PROOFSTEP]\nrefine' \u27e8vectorSpan_eq_top_of_affineSpan_eq_top k V P, _\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\n\u22a2 vectorSpan k s = \u22a4 \u2192 affineSpan k s = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = \u22a4\n\u22a2 affineSpan k s = \u22a4\n[PROOFSTEP]\nsuffices Nonempty (affineSpan k s) by\n  obtain \u27e8p, hp : p \u2208 affineSpan k s\u27e9 := this\n  rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = \u22a4\nthis : Nonempty { x // x \u2208 affineSpan k s }\n\u22a2 affineSpan k s = \u22a4\n[PROOFSTEP]\nobtain \u27e8p, hp : p \u2208 affineSpan k s\u27e9 := this\n[GOAL]\ncase intro.mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = \u22a4\np : P\nhp : p \u2208 affineSpan k s\n\u22a2 affineSpan k s = \u22a4\n[PROOFSTEP]\nrw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nhs : Set.Nonempty s\nh : vectorSpan k s = \u22a4\n\u22a2 Nonempty { x // x \u2208 affineSpan k s }\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := hs\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh : vectorSpan k s = \u22a4\nx : P\nhx : x \u2208 s\n\u22a2 Nonempty { x // x \u2208 affineSpan k s }\n[PROOFSTEP]\nexact \u27e8\u27e8x, mem_affineSpan k hx\u27e9\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\ninst\u271d : Nontrivial P\n\u22a2 affineSpan k s = \u22a4 \u2194 vectorSpan k s = \u22a4\n[PROOFSTEP]\ncases' s.eq_empty_or_nonempty with hs hs\n[GOAL]\ncase inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\ninst\u271d : Nontrivial P\nhs : s = \u2205\n\u22a2 affineSpan k s = \u22a4 \u2194 vectorSpan k s = \u22a4\n[PROOFSTEP]\nsimp [hs, subsingleton_iff_bot_eq_top, AddTorsor.subsingleton_iff V P, not_subsingleton]\n[GOAL]\ncase inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\ninst\u271d : Nontrivial P\nhs : Set.Nonempty s\n\u22a2 affineSpan k s = \u22a4 \u2194 vectorSpan k s = \u22a4\n[PROOFSTEP]\nrw [affineSpan_eq_top_iff_vectorSpan_eq_top_of_nonempty k V P hs]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh : affineSpan k (range p) = \u22a4\n\u22a2 0 < Fintype.card \u03b9\n[PROOFSTEP]\nobtain \u27e8-, \u27e8i, -\u27e9\u27e9 := nonempty_of_affineSpan_eq_top k V P h\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u03b9 : Type u_4\ninst\u271d : Fintype \u03b9\np : \u03b9 \u2192 P\nh : affineSpan k (range p) = \u22a4\ni : \u03b9\n\u22a2 0 < Fintype.card \u03b9\n[PROOFSTEP]\nexact Fintype.card_pos_iff.mpr \u27e8i\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\n\u22a2 direction \u22a5 = \u22a5\n[PROOFSTEP]\nrw [direction_eq_vectorSpan, bot_coe, vectorSpan_def, vsub_empty, Submodule.span_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nQ : AffineSubspace k P\n\u22a2 Set.Nonempty \u2191Q \u2194 Q \u2260 \u22a5\n[PROOFSTEP]\nrw [nonempty_iff_ne_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nQ : AffineSubspace k P\n\u22a2 \u2191Q \u2260 \u2205 \u2194 Q \u2260 \u22a5\n[PROOFSTEP]\nexact not_congr Q.coe_eq_bot_iff\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nQ : AffineSubspace k P\n\u22a2 Q = \u22a5 \u2228 Set.Nonempty \u2191Q\n[PROOFSTEP]\nrw [nonempty_iff_ne_bot]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nQ : AffineSubspace k P\n\u22a2 Q = \u22a5 \u2228 Q \u2260 \u22a5\n[PROOFSTEP]\napply eq_or_ne\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\n\u22a2 Subsingleton P\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := AffineSubspace.nonempty_of_affineSpan_eq_top k V P h\u2082\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\n\u22a2 Subsingleton P\n[PROOFSTEP]\nhave : s = { p } := Subset.antisymm (fun q hq => h\u2081 hq hp) (by simp [hp])\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\n\u22a2 {p} \u2286 s\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\nthis : s = {p}\n\u22a2 Subsingleton P\n[PROOFSTEP]\nrw [this, \u2190 AffineSubspace.ext_iff, AffineSubspace.coe_affineSpan_singleton, AffineSubspace.top_coe, eq_comm, \u2190\n  subsingleton_iff_singleton (mem_univ _)] at h\u2082 \n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\np : P\nh\u2082 : Set.Subsingleton univ\nhp : p \u2208 s\nthis : s = {p}\n\u22a2 Subsingleton P\n[PROOFSTEP]\nexact subsingleton_of_univ_subsingleton h\u2082\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\n\u22a2 s = univ\n[PROOFSTEP]\nobtain \u27e8p, hp\u27e9 := AffineSubspace.nonempty_of_affineSpan_eq_top k V P h\u2082\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\n\u22a2 s = univ\n[PROOFSTEP]\nhave : s = { p } := Subset.antisymm (fun q hq => h\u2081 hq hp) (by simp [hp])\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\n\u22a2 {p} \u2286 s\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\nthis : s = {p}\n\u22a2 s = univ\n[PROOFSTEP]\nrw [this, eq_comm, \u2190 subsingleton_iff_singleton (mem_univ p), subsingleton_univ_iff]\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : Set P\nh\u2081 : Set.Subsingleton s\nh\u2082 : affineSpan k s = \u22a4\np : P\nhp : p \u2208 s\nthis : s = {p}\n\u22a2 Subsingleton P\n[PROOFSTEP]\nexact subsingleton_of_subsingleton_span_eq_top h\u2081 h\u2082\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 direction s = \u22a4 \u2194 s = \u22a4\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 direction s = \u22a4 \u2192 s = \u22a4\n[PROOFSTEP]\nintro hd\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nhd : direction s = \u22a4\n\u22a2 s = \u22a4\n[PROOFSTEP]\nrw [\u2190 direction_top k V P] at hd \n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nhd : direction s = direction \u22a4\n\u22a2 s = \u22a4\n[PROOFSTEP]\nrefine' ext_of_direction_eq hd _\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\nhd : direction s = direction \u22a4\n\u22a2 Set.Nonempty (\u2191s \u2229 \u2191\u22a4)\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nh : Set.Nonempty \u2191s\n\u22a2 s = \u22a4 \u2192 direction s = \u22a4\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\nh : Set.Nonempty \u2191\u22a4\n\u22a2 direction \u22a4 = \u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\n\u22a2 direction (s1 \u2293 s2) \u2264 direction s1 \u2293 direction s2\n[PROOFSTEP]\nsimp only [direction_eq_vectorSpan, vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\n\u22a2 Submodule.span k (\u2191(s1 \u2293 s2) -\u1d65 \u2191(s1 \u2293 s2)) \u2264 Submodule.span k (\u2191s1 -\u1d65 \u2191s1) \u2293 Submodule.span k (\u2191s2 -\u1d65 \u2191s2)\n[PROOFSTEP]\nexact\n  le_inf (sInf_le_sInf fun p hp => trans (vsub_self_mono (inter_subset_left _ _)) hp)\n    (sInf_le_sInf fun p hp => trans (vsub_self_mono (inter_subset_right _ _)) hp)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns\u2081 s\u2082 : AffineSubspace k P\np : P\nh\u2081 : p \u2208 s\u2081\nh\u2082 : p \u2208 s\u2082\n\u22a2 direction (s\u2081 \u2293 s\u2082) = direction s\u2081 \u2293 direction s\u2082\n[PROOFSTEP]\next v\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns\u2081 s\u2082 : AffineSubspace k P\np : P\nh\u2081 : p \u2208 s\u2081\nh\u2082 : p \u2208 s\u2082\nv : V\n\u22a2 v \u2208 direction (s\u2081 \u2293 s\u2082) \u2194 v \u2208 direction s\u2081 \u2293 direction s\u2082\n[PROOFSTEP]\nrw [Submodule.mem_inf, \u2190 vadd_mem_iff_mem_direction v h\u2081, \u2190 vadd_mem_iff_mem_direction v h\u2082, \u2190\n  vadd_mem_iff_mem_direction v ((mem_inf_iff p s\u2081 s\u2082).2 \u27e8h\u2081, h\u2082\u27e9), mem_inf_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh : s1 \u2264 s2\n\u22a2 direction s1 \u2264 direction s2\n[PROOFSTEP]\nsimp only [direction_eq_vectorSpan, vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh : s1 \u2264 s2\n\u22a2 Submodule.span k (\u2191s1 -\u1d65 \u2191s1) \u2264 Submodule.span k (\u2191s2 -\u1d65 \u2191s2)\n[PROOFSTEP]\nexact vectorSpan_mono k h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh : s1 < s2\nhn : Set.Nonempty \u2191s1\n\u22a2 direction s1 < direction s2\n[PROOFSTEP]\ncases' hn with p hp\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh : s1 < s2\np : P\nhp : p \u2208 \u2191s1\n\u22a2 direction s1 < direction s2\n[PROOFSTEP]\nrw [lt_iff_le_and_exists] at h \n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh : s1 \u2264 s2 \u2227 \u2203 p, p \u2208 s2 \u2227 \u00acp \u2208 s1\np : P\nhp : p \u2208 \u2191s1\n\u22a2 direction s1 < direction s2\n[PROOFSTEP]\nrcases h with \u27e8hle, p2, hp2, hp2s1\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s1\nhle : s1 \u2264 s2\np2 : P\nhp2 : p2 \u2208 s2\nhp2s1 : \u00acp2 \u2208 s1\n\u22a2 direction s1 < direction s2\n[PROOFSTEP]\nrw [SetLike.lt_iff_le_and_exists]\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s1\nhle : s1 \u2264 s2\np2 : P\nhp2 : p2 \u2208 s2\nhp2s1 : \u00acp2 \u2208 s1\n\u22a2 direction s1 \u2264 direction s2 \u2227 \u2203 x, x \u2208 direction s2 \u2227 \u00acx \u2208 direction s1\n[PROOFSTEP]\nuse direction_le hle, p2 -\u1d65 p, vsub_mem_direction hp2 (hle hp)\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s1\nhle : s1 \u2264 s2\np2 : P\nhp2 : p2 \u2208 s2\nhp2s1 : \u00acp2 \u2208 s1\n\u22a2 \u00acp2 -\u1d65 p \u2208 direction s1\n[PROOFSTEP]\nintro hm\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s1\nhle : s1 \u2264 s2\np2 : P\nhp2 : p2 \u2208 s2\nhp2s1 : \u00acp2 \u2208 s1\nhm : p2 -\u1d65 p \u2208 direction s1\n\u22a2 False\n[PROOFSTEP]\nrw [vsub_right_mem_direction_iff_mem hp p2] at hm \n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\np : P\nhp : p \u2208 \u2191s1\nhle : s1 \u2264 s2\np2 : P\nhp2 : p2 \u2208 s2\nhp2s1 : \u00acp2 \u2208 s1\nhm : p2 \u2208 s1\n\u22a2 False\n[PROOFSTEP]\nexact hp2s1 hm\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\n\u22a2 direction s1 \u2294 direction s2 \u2264 direction (s1 \u2294 s2)\n[PROOFSTEP]\nsimp only [direction_eq_vectorSpan, vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\n\u22a2 Submodule.span k (\u2191s1 -\u1d65 \u2191s1) \u2294 Submodule.span k (\u2191s2 -\u1d65 \u2191s2) \u2264 Submodule.span k (\u2191(s1 \u2294 s2) -\u1d65 \u2191(s1 \u2294 s2))\n[PROOFSTEP]\nexact\n  sup_le (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_left : s1 \u2264 s1 \u2294 s2)) hp)\n    (sInf_le_sInf fun p hp => Set.Subset.trans (vsub_self_mono (le_sup_right : s2 \u2264 s1 \u2294 s2)) hp)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\n\u22a2 direction s1 \u2294 direction s2 < direction (s1 \u2294 s2)\n[PROOFSTEP]\ncases' h1 with p1 hp1\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh2 : Set.Nonempty \u2191s2\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\n\u22a2 direction s1 \u2294 direction s2 < direction (s1 \u2294 s2)\n[PROOFSTEP]\ncases' h2 with p2 hp2\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\n\u22a2 direction s1 \u2294 direction s2 < direction (s1 \u2294 s2)\n[PROOFSTEP]\nrw [SetLike.lt_iff_le_and_exists]\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\n\u22a2 direction s1 \u2294 direction s2 \u2264 direction (s1 \u2294 s2) \u2227 \u2203 x, x \u2208 direction (s1 \u2294 s2) \u2227 \u00acx \u2208 direction s1 \u2294 direction s2\n[PROOFSTEP]\nuse sup_direction_le s1 s2, p2 -\u1d65 p1,\n  vsub_mem_direction ((le_sup_right : s2 \u2264 s1 \u2294 s2) hp2) ((le_sup_left : s1 \u2264 s1 \u2294 s2) hp1)\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\n\u22a2 \u00acp2 -\u1d65 p1 \u2208 direction s1 \u2294 direction s2\n[PROOFSTEP]\nintro h\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nh : p2 -\u1d65 p1 \u2208 direction s1 \u2294 direction s2\n\u22a2 False\n[PROOFSTEP]\nrw [Submodule.mem_sup] at h \n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nh : \u2203 y, y \u2208 direction s1 \u2227 \u2203 z, z \u2208 direction s2 \u2227 y + z = p2 -\u1d65 p1\n\u22a2 False\n[PROOFSTEP]\nrcases h with \u27e8v1, hv1, v2, hv2, hv1v2\u27e9\n[GOAL]\ncase right.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nv1 : V\nhv1 : v1 \u2208 direction s1\nv2 : V\nhv2 : v2 \u2208 direction s2\nhv1v2 : v1 + v2 = p2 -\u1d65 p1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, sub_eq_add_neg, neg_vsub_eq_vsub_rev, add_comm v1, add_assoc, \u2190 vadd_vsub_assoc, \u2190 neg_neg v2,\n  add_comm, \u2190 sub_eq_add_neg, \u2190 vsub_vadd_eq_vsub_sub, vsub_eq_zero_iff_eq] at hv1v2 \n[GOAL]\ncase right.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nv1 : V\nhv1 : v1 \u2208 direction s1\nv2 : V\nhv2 : v2 \u2208 direction s2\nhv1v2\u271d : v1 + v2 = p2 -\u1d65 p1\nhv1v2 : v1 +\u1d65 p1 = -v2 +\u1d65 p2\n\u22a2 False\n[PROOFSTEP]\nrefine' Set.Nonempty.ne_empty _ he\n[GOAL]\ncase right.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nv1 : V\nhv1 : v1 \u2208 direction s1\nv2 : V\nhv2 : v2 \u2208 direction s2\nhv1v2\u271d : v1 + v2 = p2 -\u1d65 p1\nhv1v2 : v1 +\u1d65 p1 = -v2 +\u1d65 p2\n\u22a2 Set.Nonempty (\u2191s1 \u2229 \u2191s2)\n[PROOFSTEP]\nuse v1 +\u1d65 p1, vadd_mem_of_mem_direction hv1 hp1\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nv1 : V\nhv1 : v1 \u2208 direction s1\nv2 : V\nhv2 : v2 \u2208 direction s2\nhv1v2\u271d : v1 + v2 = p2 -\u1d65 p1\nhv1v2 : v1 +\u1d65 p1 = -v2 +\u1d65 p2\n\u22a2 v1 +\u1d65 p1 \u2208 \u2191s2\n[PROOFSTEP]\nrw [hv1v2]\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nhe : \u2191s1 \u2229 \u2191s2 = \u2205\np1 : P\nhp1 : p1 \u2208 \u2191s1\np2 : P\nhp2 : p2 \u2208 \u2191s2\nv1 : V\nhv1 : v1 \u2208 direction s1\nv2 : V\nhv2 : v2 \u2208 direction s2\nhv1v2\u271d : v1 + v2 = p2 -\u1d65 p1\nhv1v2 : v1 +\u1d65 p1 = -v2 +\u1d65 p2\n\u22a2 -v2 +\u1d65 p2 \u2208 \u2191s2\n[PROOFSTEP]\nexact vadd_mem_of_mem_direction (Submodule.neg_mem _ hv2) hp2\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : direction s1 \u2294 direction s2 = \u22a4\n\u22a2 Set.Nonempty (\u2191s1 \u2229 \u2191s2)\n[PROOFSTEP]\nby_contra h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : direction s1 \u2294 direction s2 = \u22a4\nh : \u00acSet.Nonempty (\u2191s1 \u2229 \u2191s2)\n\u22a2 False\n[PROOFSTEP]\nrw [Set.not_nonempty_iff_eq_empty] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : direction s1 \u2294 direction s2 = \u22a4\nh : \u2191s1 \u2229 \u2191s2 = \u2205\n\u22a2 False\n[PROOFSTEP]\nhave hlt := sup_direction_lt_of_nonempty_of_inter_empty h1 h2 h\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : direction s1 \u2294 direction s2 = \u22a4\nh : \u2191s1 \u2229 \u2191s2 = \u2205\nhlt : direction s1 \u2294 direction s2 < direction (s1 \u2294 s2)\n\u22a2 False\n[PROOFSTEP]\nrw [hd] at hlt \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : direction s1 \u2294 direction s2 = \u22a4\nh : \u2191s1 \u2229 \u2191s2 = \u2205\nhlt : \u22a4 < direction (s1 \u2294 s2)\n\u22a2 False\n[PROOFSTEP]\nexact not_top_lt hlt\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\n\u22a2 \u2203 p, \u2191s1 \u2229 \u2191s2 = {p}\n[PROOFSTEP]\ncases' inter_nonempty_of_nonempty_of_sup_direction_eq_top h1 h2 hd.sup_eq_top with p hp\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\n\u22a2 \u2203 p, \u2191s1 \u2229 \u2191s2 = {p}\n[PROOFSTEP]\nuse p\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\n\u22a2 \u2191s1 \u2229 \u2191s2 = {p}\n[PROOFSTEP]\next q\n[GOAL]\ncase h.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\n\u22a2 q \u2208 \u2191s1 \u2229 \u2191s2 \u2194 q \u2208 {p}\n[PROOFSTEP]\nrw [Set.mem_singleton_iff]\n[GOAL]\ncase h.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\n\u22a2 q \u2208 \u2191s1 \u2229 \u2191s2 \u2194 q = p\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\n\u22a2 q \u2208 \u2191s1 \u2229 \u2191s2 \u2192 q = p\n[PROOFSTEP]\nrintro \u27e8hq1, hq2\u27e9\n[GOAL]\ncase h.h.mp.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\nhq1 : q \u2208 \u2191s1\nhq2 : q \u2208 \u2191s2\n\u22a2 q = p\n[PROOFSTEP]\nhave hqp : q -\u1d65 p \u2208 s1.direction \u2293 s2.direction := \u27e8vsub_mem_direction hq1 hp.1, vsub_mem_direction hq2 hp.2\u27e9\n[GOAL]\ncase h.h.mp.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\nhq1 : q \u2208 \u2191s1\nhq2 : q \u2208 \u2191s2\nhqp : q -\u1d65 p \u2208 direction s1 \u2293 direction s2\n\u22a2 q = p\n[PROOFSTEP]\nrwa [hd.inf_eq_bot, Submodule.mem_bot, vsub_eq_zero_iff_eq] at hqp \n[GOAL]\ncase h.h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns1 s2 : AffineSubspace k P\nh1 : Set.Nonempty \u2191s1\nh2 : Set.Nonempty \u2191s2\nhd : IsCompl (direction s1) (direction s2)\np : P\nhp : p \u2208 \u2191s1 \u2229 \u2191s2\nq : P\n\u22a2 q = p \u2192 q \u2208 \u2191s1 \u2229 \u2191s2\n[PROOFSTEP]\nexact fun h => h.symm \u25b8 hp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\n\u22a2 affineSpan k \u2191s = s\n[PROOFSTEP]\nrefine' le_antisymm _ (subset_spanPoints _ _)\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\n\u22a2 affineSpan k \u2191s \u2264 s\n[PROOFSTEP]\nrintro p \u27e8p1, hp1, v, hv, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b2 : Ring k\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module k V\nS : AffineSpace V P\np\u2081 p\u2082 : P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\nv : V\nhv : v \u2208 vectorSpan k \u2191s\n\u22a2 v +\u1d65 p1 \u2208 \u2191s\n[PROOFSTEP]\nexact vadd_mem_of_mem_direction hv hp1\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 vectorSpan k s = Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s)\n[PROOFSTEP]\nrw [vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (s -\u1d65 s) = Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s)\n[PROOFSTEP]\nrefine' le_antisymm _ (Submodule.span_mono _)\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (s -\u1d65 s) \u2264 Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s)\n[PROOFSTEP]\nrw [Submodule.span_le]\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 s -\u1d65 s \u2286 \u2191(Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s))\n[PROOFSTEP]\nrintro v \u27e8p1, p2, hp1, hp2, hv\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : (fun x x_1 => x -\u1d65 x_1) p1 p2 = v\n\u22a2 v \u2208 \u2191(Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s))\n[PROOFSTEP]\nsimp_rw [\u2190 vsub_sub_vsub_cancel_left p1 p2 p] at hv \n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : p -\u1d65 p2 - (p -\u1d65 p1) = v\n\u22a2 v \u2208 \u2191(Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' s))\n[PROOFSTEP]\nrw [\u2190 hv, SetLike.mem_coe, Submodule.mem_span]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : p -\u1d65 p2 - (p -\u1d65 p1) = v\n\u22a2 \u2200 (p_1 : Submodule k V), (fun x x_1 => x -\u1d65 x_1) p '' s \u2286 \u2191p_1 \u2192 p -\u1d65 p2 - (p -\u1d65 p1) \u2208 p_1\n[PROOFSTEP]\nexact fun m hm => Submodule.sub_mem _ (hm \u27e8p2, hp2, rfl\u27e9) (hm \u27e8p1, hp1, rfl\u27e9)\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 (fun x x_1 => x -\u1d65 x_1) p '' s \u2286 s -\u1d65 s\n[PROOFSTEP]\nrintro v \u27e8p2, hp2, hv\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np2 : P\nhp2 : p2 \u2208 s\nhv : (fun x x_1 => x -\u1d65 x_1) p p2 = v\n\u22a2 v \u2208 s -\u1d65 s\n[PROOFSTEP]\nexact \u27e8p, p2, hp, hp2, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 vectorSpan k s = Submodule.span k ((fun x => x -\u1d65 p) '' s)\n[PROOFSTEP]\nrw [vectorSpan_def]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (s -\u1d65 s) = Submodule.span k ((fun x => x -\u1d65 p) '' s)\n[PROOFSTEP]\nrefine' le_antisymm _ (Submodule.span_mono _)\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (s -\u1d65 s) \u2264 Submodule.span k ((fun x => x -\u1d65 p) '' s)\n[PROOFSTEP]\nrw [Submodule.span_le]\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 s -\u1d65 s \u2286 \u2191(Submodule.span k ((fun x => x -\u1d65 p) '' s))\n[PROOFSTEP]\nrintro v \u27e8p1, p2, hp1, hp2, hv\u27e9\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : (fun x x_1 => x -\u1d65 x_1) p1 p2 = v\n\u22a2 v \u2208 \u2191(Submodule.span k ((fun x => x -\u1d65 p) '' s))\n[PROOFSTEP]\nsimp_rw [\u2190 vsub_sub_vsub_cancel_right p1 p2 p] at hv \n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : p1 -\u1d65 p - (p2 -\u1d65 p) = v\n\u22a2 v \u2208 \u2191(Submodule.span k ((fun x => x -\u1d65 p) '' s))\n[PROOFSTEP]\nrw [\u2190 hv, SetLike.mem_coe, Submodule.mem_span]\n[GOAL]\ncase refine'_1.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np1 p2 : P\nhp1 : p1 \u2208 s\nhp2 : p2 \u2208 s\nhv : p1 -\u1d65 p - (p2 -\u1d65 p) = v\n\u22a2 \u2200 (p_1 : Submodule k V), (fun x => x -\u1d65 p) '' s \u2286 \u2191p_1 \u2192 p1 -\u1d65 p - (p2 -\u1d65 p) \u2208 p_1\n[PROOFSTEP]\nexact fun m hm => Submodule.sub_mem _ (hm \u27e8p1, hp1, rfl\u27e9) (hm \u27e8p2, hp2, rfl\u27e9)\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 (fun x => x -\u1d65 p) '' s \u2286 s -\u1d65 s\n[PROOFSTEP]\nrintro v \u27e8p2, hp2, hv\u27e9\n[GOAL]\ncase refine'_2.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\nv : V\np2 : P\nhp2 : p2 \u2208 s\nhv : (fun x => x -\u1d65 p) p2 = v\n\u22a2 v \u2208 s -\u1d65 s\n[PROOFSTEP]\nexact \u27e8p2, p, hp2, hp, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 vectorSpan k s = Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' (s \\ {p}))\n[PROOFSTEP]\nconv_lhs =>\n  rw [vectorSpan_eq_span_vsub_set_left k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton,\n    Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (insert ((fun x x_1 => x -\u1d65 x_1) p p) ((fun x x_1 => x -\u1d65 x_1) p '' (s \\ {p}))) =\n    Submodule.span k ((fun x x_1 => x -\u1d65 x_1) p '' (s \\ {p}))\n[PROOFSTEP]\nsimp [Submodule.span_insert_eq_span]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 vectorSpan k s = Submodule.span k ((fun x => x -\u1d65 p) '' (s \\ {p}))\n[PROOFSTEP]\nconv_lhs =>\n  rw [vectorSpan_eq_span_vsub_set_right k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton,\n    Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n| vectorSpan k s\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k hp, \u2190 Set.insert_eq_of_mem hp, \u2190 Set.insert_diff_singleton, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : P\nhp : p \u2208 s\n\u22a2 Submodule.span k (insert (p -\u1d65 p) ((fun x => x -\u1d65 p) '' (s \\ {p}))) =\n    Submodule.span k ((fun x => x -\u1d65 p) '' (s \\ {p}))\n[PROOFSTEP]\nsimp [Submodule.span_insert_eq_span]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2075 : Ring k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq P\ninst\u271d : DecidableEq V\ns : Finset P\np : P\nhp : p \u2208 s\n\u22a2 vectorSpan k \u2191s = Submodule.span k \u2191(Finset.image (fun x => x -\u1d65 p) (Finset.erase s p))\n[PROOFSTEP]\nsimp [vectorSpan_eq_span_vsub_set_right_ne _ (Finset.mem_coe.mpr hp)]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 vectorSpan k (p '' s) = Submodule.span k ((fun x x_1 => x -\u1d65 x_1) (p i) '' (p '' (s \\ {i})))\n[PROOFSTEP]\nconv_lhs =>\n  rw [vectorSpan_eq_span_vsub_set_left k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n  Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Submodule.span k (insert ((fun x x_1 => x -\u1d65 x_1) (p i) (p i)) ((fun x x_1 => x -\u1d65 x_1) (p i) '' (p '' (s \\ {i})))) =\n    Submodule.span k ((fun x x_1 => x -\u1d65 x_1) (p i) '' (p '' (s \\ {i})))\n[PROOFSTEP]\nsimp [Submodule.span_insert_eq_span]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 vectorSpan k (p '' s) = Submodule.span k ((fun x => x -\u1d65 p i) '' (p '' (s \\ {i})))\n[PROOFSTEP]\nconv_lhs =>\n  rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n    Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n| vectorSpan k (p '' s)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), \u2190 Set.insert_eq_of_mem hi, \u2190\n  Set.insert_diff_singleton, Set.image_insert_eq, Set.image_insert_eq]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ns : Set \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 Submodule.span k (insert (p i -\u1d65 p i) ((fun x => x -\u1d65 p i) '' (p '' (s \\ {i})))) =\n    Submodule.span k ((fun x => x -\u1d65 p i) '' (p '' (s \\ {i})))\n[PROOFSTEP]\nsimp [Submodule.span_insert_eq_span]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 vectorSpan k (range p) = Submodule.span k (range fun i => p i0 -\u1d65 p i)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_left k (Set.mem_range_self i0), \u2190 Set.range_comp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 Submodule.span k (range ((fun x x_1 => x -\u1d65 x_1) (p i0) \u2218 p)) = Submodule.span k (range fun i => p i0 -\u1d65 p i)\n[PROOFSTEP]\ncongr\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 vectorSpan k (range p) = Submodule.span k (range fun i => p i -\u1d65 p i0)\n[PROOFSTEP]\nrw [vectorSpan_eq_span_vsub_set_right k (Set.mem_range_self i0), \u2190 Set.range_comp]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni0 : \u03b9\n\u22a2 Submodule.span k (range ((fun x => x -\u1d65 p i0) \u2218 p)) = Submodule.span k (range fun i => p i -\u1d65 p i0)\n[PROOFSTEP]\ncongr\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\n\u22a2 vectorSpan k (range p) = Submodule.span k (range fun i => p i\u2080 -\u1d65 p \u2191i)\n[PROOFSTEP]\nrw [\u2190 Set.image_univ, vectorSpan_image_eq_span_vsub_set_left_ne k _ (Set.mem_univ i\u2080)]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\n\u22a2 Submodule.span k ((fun x x_1 => x -\u1d65 x_1) (p i\u2080) '' (p '' (univ \\ {i\u2080}))) =\n    Submodule.span k (range fun i => p i\u2080 -\u1d65 p \u2191i)\n[PROOFSTEP]\ncongr with v\n[GOAL]\ncase e_s.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 v \u2208 (fun x x_1 => x -\u1d65 x_1) (p i\u2080) '' (p '' (univ \\ {i\u2080})) \u2194 v \u2208 range fun i => p i\u2080 -\u1d65 p \u2191i\n[PROOFSTEP]\nsimp only [Set.mem_range, Set.mem_image, Set.mem_diff, Set.mem_singleton_iff, Subtype.exists, Subtype.coe_mk]\n[GOAL]\ncase e_s.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 p i\u2080 -\u1d65 x = v) \u2194 \u2203 a h, p i\u2080 -\u1d65 p a = v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase e_s.h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 p i\u2080 -\u1d65 x = v) \u2192 \u2203 a h, p i\u2080 -\u1d65 p a = v\n[PROOFSTEP]\nrintro \u27e8x, \u27e8i\u2081, \u27e8\u27e8_, hi\u2081\u27e9, rfl\u27e9\u27e9, hv\u27e9\n[GOAL]\ncase e_s.h.mp.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\ni\u2081 : \u03b9\nleft\u271d : i\u2081 \u2208 univ\nhi\u2081 : \u00aci\u2081 = i\u2080\nhv : p i\u2080 -\u1d65 p i\u2081 = v\n\u22a2 \u2203 a h, p i\u2080 -\u1d65 p a = v\n[PROOFSTEP]\nexact \u27e8i\u2081, hi\u2081, hv\u27e9\n[GOAL]\ncase e_s.h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 a h, p i\u2080 -\u1d65 p a = v) \u2192 \u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 p i\u2080 -\u1d65 x = v\n[PROOFSTEP]\nexact fun \u27e8i\u2081, hi\u2081, hv\u27e9 => \u27e8p i\u2081, \u27e8i\u2081, \u27e8Set.mem_univ _, hi\u2081\u27e9, rfl\u27e9, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\n\u22a2 vectorSpan k (range p) = Submodule.span k (range fun i => p \u2191i -\u1d65 p i\u2080)\n[PROOFSTEP]\nrw [\u2190 Set.image_univ, vectorSpan_image_eq_span_vsub_set_right_ne k _ (Set.mem_univ i\u2080)]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\n\u22a2 Submodule.span k ((fun x => x -\u1d65 p i\u2080) '' (p '' (univ \\ {i\u2080}))) = Submodule.span k (range fun i => p \u2191i -\u1d65 p i\u2080)\n[PROOFSTEP]\ncongr with v\n[GOAL]\ncase e_s.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 v \u2208 (fun x => x -\u1d65 p i\u2080) '' (p '' (univ \\ {i\u2080})) \u2194 v \u2208 range fun i => p \u2191i -\u1d65 p i\u2080\n[PROOFSTEP]\nsimp only [Set.mem_range, Set.mem_image, Set.mem_diff, Set.mem_singleton_iff, Subtype.exists, Subtype.coe_mk]\n[GOAL]\ncase e_s.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 x -\u1d65 p i\u2080 = v) \u2194 \u2203 a h, p a -\u1d65 p i\u2080 = v\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase e_s.h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 x -\u1d65 p i\u2080 = v) \u2192 \u2203 a h, p a -\u1d65 p i\u2080 = v\n[PROOFSTEP]\nrintro \u27e8x, \u27e8i\u2081, \u27e8\u27e8_, hi\u2081\u27e9, rfl\u27e9\u27e9, hv\u27e9\n[GOAL]\ncase e_s.h.mp.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\ni\u2081 : \u03b9\nleft\u271d : i\u2081 \u2208 univ\nhi\u2081 : \u00aci\u2081 = i\u2080\nhv : p i\u2081 -\u1d65 p i\u2080 = v\n\u22a2 \u2203 a h, p a -\u1d65 p i\u2080 = v\n[PROOFSTEP]\nexact \u27e8i\u2081, hi\u2081, hv\u27e9\n[GOAL]\ncase e_s.h.mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : \u03b9 \u2192 P\ni\u2080 : \u03b9\nv : V\n\u22a2 (\u2203 a h, p a -\u1d65 p i\u2080 = v) \u2192 \u2203 x, (\u2203 x_1, (x_1 \u2208 univ \u2227 \u00acx_1 = i\u2080) \u2227 p x_1 = x) \u2227 x -\u1d65 p i\u2080 = v\n[PROOFSTEP]\nexact fun \u27e8i\u2081, hi\u2081, hv\u27e9 => \u27e8p i\u2081, \u27e8i\u2081, \u27e8Set.mem_univ _, hi\u2081\u27e9, rfl\u27e9, hv\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 affineSpan k s = \u22a5 \u2194 s = \u2205\n[PROOFSTEP]\nrw [\u2190 not_iff_not, \u2190 Ne.def, \u2190 Ne.def, \u2190 nonempty_iff_ne_bot, affineSpan_nonempty, nonempty_iff_ne_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 \u22a5 < affineSpan k s \u2194 Set.Nonempty s\n[PROOFSTEP]\nrw [bot_lt_iff_ne_bot, nonempty_iff_ne_empty]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 affineSpan k s \u2260 \u22a5 \u2194 s \u2260 \u2205\n[PROOFSTEP]\nexact (affineSpan_eq_bot _).not\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : (x : P) \u2192 x \u2208 affineSpan k s \u2192 Prop\nHs : \u2200 (y : P) (hys : y \u2208 s), p y (_ : y \u2208 \u2191(affineSpan k s))\nHc :\n  \u2200 (c : k) (u : P) (hu : u \u2208 affineSpan k s) (v : P) (hv : v \u2208 affineSpan k s) (w : P) (hw : w \u2208 affineSpan k s),\n    p u hu \u2192 p v hv \u2192 p w hw \u2192 p (c \u2022 (u -\u1d65 v) +\u1d65 w) (_ : c \u2022 (u -\u1d65 v) +\u1d65 w \u2208 (affineSpan k s).carrier)\nx : P\nh : x \u2208 affineSpan k s\n\u22a2 p x h\n[PROOFSTEP]\nrefine' Exists.elim _ fun (hx : x \u2208 affineSpan k s) (hc : p x hx) => hc\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : (x : P) \u2192 x \u2208 affineSpan k s \u2192 Prop\nHs : \u2200 (y : P) (hys : y \u2208 s), p y (_ : y \u2208 \u2191(affineSpan k s))\nHc :\n  \u2200 (c : k) (u : P) (hu : u \u2208 affineSpan k s) (v : P) (hv : v \u2208 affineSpan k s) (w : P) (hw : w \u2208 affineSpan k s),\n    p u hu \u2192 p v hv \u2192 p w hw \u2192 p (c \u2022 (u -\u1d65 v) +\u1d65 w) (_ : c \u2022 (u -\u1d65 v) +\u1d65 w \u2208 (affineSpan k s).carrier)\nx : P\nh : x \u2208 affineSpan k s\n\u22a2 \u2203 x_1, p x x_1\n[PROOFSTEP]\nrefine' @affineSpan_induction k V P _ _ _ _ _ _ (fun y => \u2203 z, p y z) h _ _\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : (x : P) \u2192 x \u2208 affineSpan k s \u2192 Prop\nHs : \u2200 (y : P) (hys : y \u2208 s), p y (_ : y \u2208 \u2191(affineSpan k s))\nHc :\n  \u2200 (c : k) (u : P) (hu : u \u2208 affineSpan k s) (v : P) (hv : v \u2208 affineSpan k s) (w : P) (hw : w \u2208 affineSpan k s),\n    p u hu \u2192 p v hv \u2192 p w hw \u2192 p (c \u2022 (u -\u1d65 v) +\u1d65 w) (_ : c \u2022 (u -\u1d65 v) +\u1d65 w \u2208 (affineSpan k s).carrier)\nx : P\nh : x \u2208 affineSpan k s\n\u22a2 \u2200 (x : P), x \u2208 s \u2192 (fun y => \u2203 z, p y z) x\n[PROOFSTEP]\nexact fun y hy => \u27e8subset_affineSpan _ _ hy, Hs y hy\u27e9\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\np : (x : P) \u2192 x \u2208 affineSpan k s \u2192 Prop\nHs : \u2200 (y : P) (hys : y \u2208 s), p y (_ : y \u2208 \u2191(affineSpan k s))\nHc :\n  \u2200 (c : k) (u : P) (hu : u \u2208 affineSpan k s) (v : P) (hv : v \u2208 affineSpan k s) (w : P) (hw : w \u2208 affineSpan k s),\n    p u hu \u2192 p v hv \u2192 p w hw \u2192 p (c \u2022 (u -\u1d65 v) +\u1d65 w) (_ : c \u2022 (u -\u1d65 v) +\u1d65 w \u2208 (affineSpan k s).carrier)\nx : P\nh : x \u2208 affineSpan k s\n\u22a2 \u2200 (c : k) (u v w : P),\n    (fun y => \u2203 z, p y z) u \u2192\n      (fun y => \u2203 z, p y z) v \u2192 (fun y => \u2203 z, p y z) w \u2192 (fun y => \u2203 z, p y z) (c \u2022 (u -\u1d65 v) +\u1d65 w)\n[PROOFSTEP]\nexact fun c u v w hu hv hw =>\n  Exists.elim hu fun hu' hu =>\n    Exists.elim hv fun hv' hv =>\n      Exists.elim hw fun hw' hw => \u27e8AffineSubspace.smul_vsub_vadd_mem _ _ hu' hv' hw', Hc _ _ _ _ _ _ _ hu hv hw\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\nA : Set P\ninst\u271d : Nonempty \u2191A\n\u22a2 affineSpan k (Subtype.val \u207b\u00b9' A) = \u22a4\n[PROOFSTEP]\nrw [eq_top_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\nA : Set P\ninst\u271d : Nonempty \u2191A\n\u22a2 \u22a4 \u2264 affineSpan k (Subtype.val \u207b\u00b9' A)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9 -\n[GOAL]\ncase mk\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\nA : Set P\ninst\u271d : Nonempty \u2191A\nx : P\nhx : x \u2208 \u2191(affineSpan k A)\n\u22a2 { val := x, property := hx } \u2208 \u2191(affineSpan k (Subtype.val \u207b\u00b9' A))\n[PROOFSTEP]\nrefine'\n  affineSpan_induction' (fun y hy => _) (fun c u hu v hv w hw => _) hx (p := fun y hy =>\n    \u27e8y, hy\u27e9 \u2208 (affineSpan k (((\u2191) : { z // z \u2208 affineSpan k A } \u2192 P) \u207b\u00b9' A)))\n    -- porting note: Lean couldn't infer the motive\n[GOAL]\ncase mk.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\nA : Set P\ninst\u271d : Nonempty \u2191A\nx : P\nhx : x \u2208 \u2191(affineSpan k A)\ny : P\nhy : y \u2208 A\n\u22a2 (fun y hy => { val := y, property := hy } \u2208 affineSpan k (Subtype.val \u207b\u00b9' A)) y (_ : y \u2208 \u2191(affineSpan k A))\n[PROOFSTEP]\nexact subset_affineSpan _ _ hy\n[GOAL]\ncase mk.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2074 : Ring k\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : Module k V\ninst\u271d\u00b9 : AffineSpace V P\n\u03b9 : Type u_4\nA : Set P\ninst\u271d : Nonempty \u2191A\nx : P\nhx : x \u2208 \u2191(affineSpan k A)\nc : k\nu : P\nhu : u \u2208 affineSpan k A\nv : P\nhv : v \u2208 affineSpan k A\nw : P\nhw : w \u2208 affineSpan k A\n\u22a2 (fun y hy => { val := y, property := hy } \u2208 affineSpan k (Subtype.val \u207b\u00b9' A)) u hu \u2192\n    (fun y hy => { val := y, property := hy } \u2208 affineSpan k (Subtype.val \u207b\u00b9' A)) v hv \u2192\n      (fun y hy => { val := y, property := hy } \u2208 affineSpan k (Subtype.val \u207b\u00b9' A)) w hw \u2192\n        (fun y hy => { val := y, property := hy } \u2208 affineSpan k (Subtype.val \u207b\u00b9' A)) (c \u2022 (u -\u1d65 v) +\u1d65 w)\n          (_ : c \u2022 (u -\u1d65 v) +\u1d65 w \u2208 (affineSpan k A).carrier)\n[PROOFSTEP]\nexact AffineSubspace.smul_vsub_vadd_mem _ _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 affineSpan k ({p} \u222a (fun v => v +\u1d65 p) '' s) = \u22a4\n[PROOFSTEP]\nconvert ext_of_direction_eq _ \u27e8p, mem_affineSpan k (Set.mem_union_left _ (Set.mem_singleton _)), mem_top k V p\u27e9\n[GOAL]\ncase convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 direction (affineSpan k ({p} \u222a (fun v => v +\u1d65 p) '' s)) = direction \u22a4\n[PROOFSTEP]\nrw [direction_affineSpan, direction_top,\n  vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ (Set.mem_singleton _) : p \u2208 _), eq_top_iff, \u2190 h]\n[GOAL]\ncase convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 Submodule.span k (range Subtype.val) \u2264 Submodule.span k ((fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s))\n[PROOFSTEP]\napply Submodule.span_mono\n[GOAL]\ncase convert_1.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\n\u22a2 range Subtype.val \u2286 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)\n[PROOFSTEP]\nrintro v \u27e8v', rfl\u27e9\n[GOAL]\ncase convert_1.h.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' \u2208 (fun x => x -\u1d65 p) '' ({p} \u222a (fun v => v +\u1d65 p) '' s)\n[PROOFSTEP]\nuse(v' : V) +\u1d65 p\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = \u22a4\nv' : { x // x \u2208 s }\n\u22a2 \u2191v' +\u1d65 p \u2208 {p} \u222a (fun v => v +\u1d65 p) '' s \u2227 (fun x => x -\u1d65 p) (\u2191v' +\u1d65 p) = \u2191v'\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\n\u22a2 vectorSpan k {p\u2081, p\u2082} = Submodule.span k {p\u2081 -\u1d65 p\u2082}\n[PROOFSTEP]\nsimp_rw [vectorSpan_eq_span_vsub_set_left k (mem_insert p\u2081 _), image_pair, vsub_self, Submodule.span_insert_zero]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\n\u22a2 vectorSpan k {p\u2081, p\u2082} = Submodule.span k {p\u2082 -\u1d65 p\u2081}\n[PROOFSTEP]\nrw [pair_comm, vectorSpan_pair]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nv : V\n\u22a2 v \u2208 vectorSpan k {p\u2081, p\u2082} \u2194 \u2203 r, r \u2022 (p\u2081 -\u1d65 p\u2082) = v\n[PROOFSTEP]\nrw [vectorSpan_pair, Submodule.mem_span_singleton]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nv : V\n\u22a2 v \u2208 vectorSpan k {p\u2081, p\u2082} \u2194 \u2203 r, r \u2022 (p\u2082 -\u1d65 p\u2081) = v\n[PROOFSTEP]\nrw [vectorSpan_pair_rev, Submodule.mem_span_singleton]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nv : V\n\u22a2 v +\u1d65 p\u2081 \u2208 affineSpan k {p\u2081, p\u2082} \u2194 \u2203 r, r \u2022 (p\u2082 -\u1d65 p\u2081) = v\n[PROOFSTEP]\nrw [vadd_mem_iff_mem_direction _ (left_mem_affineSpan_pair _ _ _), direction_affineSpan, mem_vectorSpan_pair_rev]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\nv : V\n\u22a2 v +\u1d65 p\u2082 \u2208 affineSpan k {p\u2081, p\u2082} \u2194 \u2203 r, r \u2022 (p\u2081 -\u1d65 p\u2082) = v\n[PROOFSTEP]\nrw [vadd_mem_iff_mem_direction _ (right_mem_affineSpan_pair _ _ _), direction_affineSpan, mem_vectorSpan_pair]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\n\u22a2 affineSpan k {p\u2081, p\u2082} \u2264 s\n[PROOFSTEP]\nrw [affineSpan_le, Set.insert_subset_iff, Set.singleton_subset_iff]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np\u2081 p\u2082 : P\ns : AffineSubspace k P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\n\u22a2 p\u2081 \u2208 \u2191s \u2227 p\u2082 \u2208 \u2191s\n[PROOFSTEP]\nexact \u27e8hp\u2081, hp\u2082\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : P\nps : Set P\n\u22a2 affineSpan k (insert p \u2191(affineSpan k ps)) = affineSpan k (insert p ps)\n[PROOFSTEP]\nrw [Set.insert_eq, Set.insert_eq, span_union, span_union, affineSpan_coe]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : P\nps : Set P\nh : p \u2208 affineSpan k ps\n\u22a2 affineSpan k (insert p ps) = affineSpan k ps\n[PROOFSTEP]\nrw [\u2190 mem_coe] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : P\nps : Set P\nh : p \u2208 \u2191(affineSpan k ps)\n\u22a2 affineSpan k (insert p ps) = affineSpan k ps\n[PROOFSTEP]\nrw [\u2190 affineSpan_insert_affineSpan, Set.insert_eq_of_mem h, affineSpan_coe]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\np : P\nps : Set P\nh : p \u2208 affineSpan k ps\n\u22a2 vectorSpan k (insert p ps) = vectorSpan k ps\n[PROOFSTEP]\nsimp_rw [\u2190 direction_affineSpan, affineSpan_insert_eq_affineSpan _ h]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 direction (s1 \u2294 s2) = direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 direction (s1 \u2294 s2) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nchange (affineSpan k ((s1 : Set P) \u222a s2)).direction \u2264 _\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 direction (affineSpan k (\u2191s1 \u222a \u2191s2)) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nrw [\u2190 mem_coe] at hp1 \n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\n\u22a2 direction (affineSpan k (\u2191s1 \u222a \u2191s2)) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nrw [direction_affineSpan, vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ hp1), Submodule.span_le]\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\n\u22a2 (fun x => x -\u1d65 p1) '' (\u2191s1 \u222a \u2191s2) \u2286 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1})\n[PROOFSTEP]\nrintro v \u27e8p3, hp3, rfl\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s1 \u222a \u2191s2\n\u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1})\n[PROOFSTEP]\ncases' hp3 with hp3 hp3\n[GOAL]\ncase refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s1\n\u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1})\n[PROOFSTEP]\nrw [sup_assoc, sup_comm, SetLike.mem_coe, Submodule.mem_sup]\n[GOAL]\ncase refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s1\n\u22a2 \u2203 y, y \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s1 \u2227 y + z = (fun x => x -\u1d65 p1) p3\n[PROOFSTEP]\nuse 0, Submodule.zero_mem _, p3 -\u1d65 p1, vsub_mem_direction hp3 hp1\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s1\n\u22a2 0 + (p3 -\u1d65 p1) = (fun x => x -\u1d65 p1) p3\n[PROOFSTEP]\nrw [zero_add]\n[GOAL]\ncase refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1})\n[PROOFSTEP]\nrw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup]\n[GOAL]\ncase refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 \u2203 y, y \u2208 direction s1 \u2227 \u2203 z, z \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 y + z = (fun x => x -\u1d65 p1) p3\n[PROOFSTEP]\nuse 0, Submodule.zero_mem _, p3 -\u1d65 p1\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 0 + (p3 -\u1d65 p1) = (fun x => x -\u1d65 p1) p3\n[PROOFSTEP]\nrw [and_comm, zero_add]\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 p3 -\u1d65 p1 = (fun x => x -\u1d65 p1) p3 \u2227 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nuse rfl\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nrw [\u2190 vsub_add_vsub_cancel p3 p2 p1, Submodule.mem_sup]\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 \u2191s1\nhp2 : p2 \u2208 s2\np3 : P\nhp3 : p3 \u2208 \u2191s2\n\u22a2 \u2203 y, y \u2208 direction s2 \u2227 \u2203 z, z \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 y + z = p3 -\u1d65 p2 + (p2 -\u1d65 p1)\n[PROOFSTEP]\nuse p3 -\u1d65 p2, vsub_mem_direction hp3 hp2, p2 -\u1d65 p1, Submodule.mem_span_singleton_self _\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2264 direction (s1 \u2294 s2)\n[PROOFSTEP]\nrefine' sup_le (sup_direction_le _ _) _\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 Submodule.span k {p2 -\u1d65 p1} \u2264 direction (s1 \u2294 s2)\n[PROOFSTEP]\nrw [direction_eq_vectorSpan, vectorSpan_def]\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s1\nhp2 : p2 \u2208 s2\n\u22a2 Submodule.span k {p2 -\u1d65 p1} \u2264 Submodule.span k (\u2191(s1 \u2294 s2) -\u1d65 \u2191(s1 \u2294 s2))\n[PROOFSTEP]\nexact\n  sInf_le_sInf fun p hp =>\n    Set.Subset.trans\n      (Set.singleton_subset_iff.2\n        (vsub_mem_vsub (mem_spanPoints k p2 _ (Set.mem_union_right _ hp2))\n          (mem_spanPoints k p1 _ (Set.mem_union_left _ hp1))))\n      hp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 direction (affineSpan k (insert p2 \u2191s)) = Submodule.span k {p2 -\u1d65 p1} \u2294 direction s\n[PROOFSTEP]\nrw [sup_comm, \u2190 Set.union_singleton, \u2190 coe_affineSpan_singleton k V p2]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 direction (affineSpan k (\u2191s \u222a \u2191(affineSpan k {p2}))) = direction s \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nchange (s \u2294 affineSpan k { p2 }).direction = _\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 direction (s \u2294 affineSpan k {p2}) = direction s \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nrw [direction_sup hp1 (mem_affineSpan k (Set.mem_singleton _)), direction_affineSpan]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 direction s \u2294 vectorSpan k {p2} \u2294 Submodule.span k {p2 -\u1d65 p1} = direction s \u2294 Submodule.span k {p2 -\u1d65 p1}\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 s\np2 p : P\n\u22a2 p \u2208 affineSpan k (insert p2 \u2191s) \u2194 \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nrw [\u2190 mem_coe] at hp1 \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\n\u22a2 p \u2208 affineSpan k (insert p2 \u2191s) \u2194 \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nrw [\u2190 vsub_right_mem_direction_iff_mem (mem_affineSpan k (Set.mem_insert_of_mem _ hp1)),\n  direction_affineSpan_insert hp1, Submodule.mem_sup]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\n\u22a2 (\u2203 y, y \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s \u2227 y + z = p -\u1d65 p1) \u2194\n    \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\n\u22a2 (\u2203 y, y \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s \u2227 y + z = p -\u1d65 p1) \u2192\n    \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nrintro \u27e8v1, hv1, v2, hv2, hp\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv1 : V\nhv1 : v1 \u2208 Submodule.span k {p2 -\u1d65 p1}\nv2 : V\nhv2 : v2 \u2208 direction s\nhp : v1 + v2 = p -\u1d65 p1\n\u22a2 \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nrw [Submodule.mem_span_singleton] at hv1 \n[GOAL]\ncase mp.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv1 : V\nhv1 : \u2203 a, a \u2022 (p2 -\u1d65 p1) = v1\nv2 : V\nhv2 : v2 \u2208 direction s\nhp : v1 + v2 = p -\u1d65 p1\n\u22a2 \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nrcases hv1 with \u27e8r, rfl\u27e9\n[GOAL]\ncase mp.intro.intro.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv2 : V\nhv2 : v2 \u2208 direction s\nr : k\nhp : r \u2022 (p2 -\u1d65 p1) + v2 = p -\u1d65 p1\n\u22a2 \u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0\n[PROOFSTEP]\nuse r, v2 +\u1d65 p1, vadd_mem_of_mem_direction hv2 hp1\n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv2 : V\nhv2 : v2 \u2208 direction s\nr : k\nhp : r \u2022 (p2 -\u1d65 p1) + v2 = p -\u1d65 p1\n\u22a2 p = r \u2022 (p2 -\u1d65 p1) +\u1d65 (v2 +\u1d65 p1)\n[PROOFSTEP]\nsymm at hp \n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv2 : V\nhv2 : v2 \u2208 direction s\nr : k\nhp : p -\u1d65 p1 = r \u2022 (p2 -\u1d65 p1) + v2\n\u22a2 p = r \u2022 (p2 -\u1d65 p1) +\u1d65 (v2 +\u1d65 p1)\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 vsub_vadd_eq_vsub_sub, vsub_eq_zero_iff_eq] at hp \n[GOAL]\ncase h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\nv2 : V\nhv2 : v2 \u2208 direction s\nr : k\nhp\u271d : p -\u1d65 p1 = r \u2022 (p2 -\u1d65 p1) + v2\nhp : p = r \u2022 (p2 -\u1d65 p1) + v2 +\u1d65 p1\n\u22a2 p = r \u2022 (p2 -\u1d65 p1) +\u1d65 (v2 +\u1d65 p1)\n[PROOFSTEP]\nrw [hp, vadd_vadd]\n[GOAL]\ncase mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 p : P\n\u22a2 (\u2203 r p0 _hp0, p = r \u2022 (p2 -\u1d65 p1) +\u1d65 p0) \u2192\n    \u2203 y, y \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s \u2227 y + z = p -\u1d65 p1\n[PROOFSTEP]\nrintro \u27e8r, p3, hp3, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 : P\nr : k\np3 : P\nhp3 : p3 \u2208 s\n\u22a2 \u2203 y, y \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s \u2227 y + z = r \u2022 (p2 -\u1d65 p1) +\u1d65 p3 -\u1d65 p1\n[PROOFSTEP]\nuse r \u2022 (p2 -\u1d65 p1), Submodule.mem_span_singleton.2 \u27e8r, rfl\u27e9, p3 -\u1d65 p1, vsub_mem_direction hp3 hp1\n[GOAL]\ncase right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 : P\nhp1 : p1 \u2208 \u2191s\np2 : P\nr : k\np3 : P\nhp3 : p3 \u2208 s\n\u22a2 r \u2022 (p2 -\u1d65 p1) + (p3 -\u1d65 p1) = r \u2022 (p2 -\u1d65 p1) +\u1d65 p3 -\u1d65 p1\n[PROOFSTEP]\nrw [vadd_vsub_assoc, add_comm]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\n\u22a2 Submodule.map f.linear (vectorSpan k s) = vectorSpan k (\u2191f '' s)\n[PROOFSTEP]\nrw [vectorSpan_def, vectorSpan_def, f.image_vsub_image, Submodule.span_image]\n  -- porting note: Lean unfolds things too far with `simp` here.\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\n\u22a2 \u2200 (c : k) {p1 p2 p3 : P\u2082}, p1 \u2208 \u2191f '' \u2191s \u2192 p2 \u2208 \u2191f '' \u2191s \u2192 p3 \u2208 \u2191f '' \u2191s \u2192 c \u2022 (p1 -\u1d65 p2) +\u1d65 p3 \u2208 \u2191f '' \u2191s\n[PROOFSTEP]\nrintro t - - - \u27e8p\u2081, h\u2081, rfl\u27e9 \u27e8p\u2082, h\u2082, rfl\u27e9 \u27e8p\u2083, h\u2083, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\n\u22a2 t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083 \u2208 \u2191f '' \u2191s\n[PROOFSTEP]\nuse t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083\n[GOAL]\ncase h\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\n\u22a2 t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 \u2191s \u2227 \u2191f (t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083) = t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083\n[PROOFSTEP]\nsuffices t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 s by {\n  simp only [SetLike.mem_coe, true_and, this]\n  rw [AffineMap.map_vadd, map_smul, AffineMap.linearMap_vsub]\n}\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\nthis : t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 s\n\u22a2 t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 \u2191s \u2227 \u2191f (t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083) = t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083\n[PROOFSTEP]\n{ simp only [SetLike.mem_coe, true_and, this]\n  rw [AffineMap.map_vadd, map_smul, AffineMap.linearMap_vsub]\n}\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\nthis : t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 s\n\u22a2 t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 \u2191s \u2227 \u2191f (t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083) = t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083\n[PROOFSTEP]\nsimp only [SetLike.mem_coe, true_and, this]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\nthis : t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 s\n\u22a2 \u2191f (t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083) = t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083\n[PROOFSTEP]\nrw [AffineMap.map_vadd, map_smul, AffineMap.linearMap_vsub]\n[GOAL]\ncase h\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nt : k\np\u2081 : P\u2081\nh\u2081 : p\u2081 \u2208 \u2191s\np\u2082 : P\u2081\nh\u2082 : p\u2082 \u2208 \u2191s\np\u2083 : P\u2081\nh\u2083 : p\u2083 \u2208 \u2191s\n\u22a2 t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083 \u2208 s\n[PROOFSTEP]\nexact s.smul_vsub_vadd_mem t h\u2081 h\u2082 h\u2083\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf\u271d f : P\u2081 \u2192\u1d43[k] P\u2082\nx : P\u2082\ns : AffineSubspace k P\u2081\n\u22a2 x \u2208 map f s \u2194 \u2203 y, y \u2208 s \u2227 \u2191f y = x\n[PROOFSTEP]\nsimpa only [bex_def] using mem_image_iff_bex\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\n\u22a2 map f s = \u22a5 \u2194 s = \u22a5\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nh : map f s = \u22a5\n\u22a2 s = \u22a5\n[PROOFSTEP]\nrwa [\u2190 coe_eq_bot_iff, coe_map, image_eq_empty, coe_eq_bot_iff] at h \n[GOAL]\ncase refine'_2\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\nh : s = \u22a5\n\u22a2 map f s = \u22a5\n[PROOFSTEP]\nrw [h, map_bot]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2081\n\u22a2 direction (map f s) = Submodule.map f.linear (direction s)\n[PROOFSTEP]\nrw [direction_eq_vectorSpan, direction_eq_vectorSpan, coe_map, AffineMap.vectorSpan_image_eq_submodule_map]\n  -- porting note: again, Lean unfolds too aggressively with `simp`\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\n\u22a2 map f (affineSpan k s) = affineSpan k (\u2191f '' s)\n[PROOFSTEP]\nrcases s.eq_empty_or_nonempty with (rfl | \u27e8p, hp\u27e9)\n[GOAL]\ncase inl\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\n\u22a2 map f (affineSpan k \u2205) = affineSpan k (\u2191f '' \u2205)\n[PROOFSTEP]\nrw [image_empty, span_empty, span_empty, map_bot]\n  -- porting note: I don't know exactly why this `simp` was broken.\n[GOAL]\ncase inr.intro\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\np : P\u2081\nhp : p \u2208 s\n\u22a2 map f (affineSpan k s) = affineSpan k (\u2191f '' s)\n[PROOFSTEP]\napply ext_of_direction_eq\n[GOAL]\ncase inr.intro.hd\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\np : P\u2081\nhp : p \u2208 s\n\u22a2 direction (map f (affineSpan k s)) = direction (affineSpan k (\u2191f '' s))\n[PROOFSTEP]\nsimp [direction_affineSpan]\n[GOAL]\ncase inr.intro.hn\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\np : P\u2081\nhp : p \u2208 s\n\u22a2 Set.Nonempty (\u2191(map f (affineSpan k s)) \u2229 \u2191(affineSpan k (\u2191f '' s)))\n[PROOFSTEP]\nexact \u27e8f p, mem_image_of_mem f (subset_affineSpan k _ hp), subset_affineSpan k _ (mem_image_of_mem f hp)\u27e9\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\nhf : Function.Surjective \u2191f\n\u22a2 AffineSubspace.map f \u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 AffineSubspace.ext_iff]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\nhf : Function.Surjective \u2191f\n\u22a2 \u2191(AffineSubspace.map f \u22a4) = \u2191\u22a4\n[PROOFSTEP]\nexact image_univ_of_surjective hf\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\nhf : Function.Surjective \u2191f\nh : affineSpan k s = \u22a4\n\u22a2 affineSpan k (\u2191f '' s) = \u22a4\n[PROOFSTEP]\nrw [\u2190 AffineSubspace.map_span, h, map_top_of_surjective f hf]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\n\u22a2 affineSpan k s = \u22a4 \u2194 affineSpan k (\u2191e '' s) = \u22a4\n[PROOFSTEP]\nrefine' \u27e8(e : P\u2081 \u2192\u1d43[k] P\u2082).span_eq_top_of_surjective e.surjective, _\u27e9\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\n\u22a2 affineSpan k (\u2191e '' s) = \u22a4 \u2192 affineSpan k s = \u22a4\n[PROOFSTEP]\nintro h\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 affineSpan k s = \u22a4\n[PROOFSTEP]\nhave : s = e.symm '' (e '' s) := by rw [\u2190 image_comp]; simp\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 s = \u2191(symm e) '' (\u2191e '' s)\n[PROOFSTEP]\nrw [\u2190 image_comp]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\n\u22a2 s = \u2191(symm e) \u2218 \u2191e '' s\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k s = \u22a4\n[PROOFSTEP]\nrw [this]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : Set P\u2081\ne : P\u2081 \u2243\u1d43[k] P\u2082\nh : affineSpan k (\u2191e '' s) = \u22a4\nthis : s = \u2191(symm e) '' (\u2191e '' s)\n\u22a2 affineSpan k (\u2191(symm e) '' (\u2191e '' s)) = \u22a4\n[PROOFSTEP]\nexact (e.symm : P\u2082 \u2192\u1d43[k] P\u2081).span_eq_top_of_surjective e.symm.surjective h\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2082\nt : k\np\u2081 p\u2082 p\u2083 : P\u2081\nhp\u2081 : \u2191f p\u2081 \u2208 s\nhp\u2082 : \u2191f p\u2082 \u2208 s\nhp\u2083 : \u2191f p\u2083 \u2208 s\n\u22a2 \u2191f (t \u2022 (p\u2081 -\u1d65 p\u2082) +\u1d65 p\u2083) \u2208 s\n[PROOFSTEP]\nrw [AffineMap.map_vadd, LinearMap.map_smul, AffineMap.linearMap_vsub]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\ns : AffineSubspace k P\u2082\nt : k\np\u2081 p\u2082 p\u2083 : P\u2081\nhp\u2081 : \u2191f p\u2081 \u2208 s\nhp\u2082 : \u2191f p\u2082 \u2208 s\nhp\u2083 : \u2191f p\u2083 \u2208 s\n\u22a2 t \u2022 (\u2191f p\u2081 -\u1d65 \u2191f p\u2082) +\u1d65 \u2191f p\u2083 \u2208 s\n[PROOFSTEP]\napply s.smul_vsub_vadd_mem _ hp\u2081 hp\u2082 hp\u2083\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\n\u22a2 comap f \u22a4 = \u22a4\n[PROOFSTEP]\nrw [\u2190 ext_iff]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2192\u1d43[k] P\u2082\n\u22a2 \u2191(comap f \u22a4) = \u2191\u22a4\n[PROOFSTEP]\nexact preimage_univ (f := f)\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\nV\u2083 : Type u_6\nP\u2083 : Type u_7\ninst\u271d\u2079 : Ring k\ninst\u271d\u2078 : AddCommGroup V\u2081\ninst\u271d\u2077 : Module k V\u2081\ninst\u271d\u2076 : AffineSpace V\u2081 P\u2081\ninst\u271d\u2075 : AddCommGroup V\u2082\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AffineSpace V\u2082 P\u2082\ninst\u271d\u00b2 : AddCommGroup V\u2083\ninst\u271d\u00b9 : Module k V\u2083\ninst\u271d : AffineSpace V\u2083 P\u2083\nf : P\u2081 \u2243\u1d43[k] P\u2082\ns : Set P\u2082\n\u22a2 comap (\u2191f) (affineSpan k s) = affineSpan k (\u2191f \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 map_symm, map_span, AffineEquiv.coe_coe, f.image_symm]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : s\u2081 \u2225 s\u2082\n\u22a2 s\u2082 \u2225 s\u2081\n[PROOFSTEP]\nrcases h with \u27e8v, rfl\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 map (\u2191(constVAdd k P v)) s\u2081 \u2225 s\u2081\n[PROOFSTEP]\nrefine' \u27e8-v, _\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 s\u2081 = map (\u2191(constVAdd k P (-v))) (map (\u2191(constVAdd k P v)) s\u2081)\n[PROOFSTEP]\nrw [map_map, \u2190 coe_trans_to_affineMap, \u2190 constVAdd_add, neg_add_self, constVAdd_zero, coe_refl_to_affineMap, map_id]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\n\u22a2 s = map (\u2191(constVAdd k P 0)) s\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 s\u2083 : AffineSubspace k P\nh\u2081\u2082 : s\u2081 \u2225 s\u2082\nh\u2082\u2083 : s\u2082 \u2225 s\u2083\n\u22a2 s\u2081 \u2225 s\u2083\n[PROOFSTEP]\nrcases h\u2081\u2082 with \u27e8v\u2081\u2082, rfl\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2083 : AffineSubspace k P\nv\u2081\u2082 : V\nh\u2082\u2083 : map (\u2191(constVAdd k P v\u2081\u2082)) s\u2081 \u2225 s\u2083\n\u22a2 s\u2081 \u2225 s\u2083\n[PROOFSTEP]\nrcases h\u2082\u2083 with \u27e8v\u2082\u2083, rfl\u27e9\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv\u2081\u2082 v\u2082\u2083 : V\n\u22a2 s\u2081 \u2225 map (\u2191(constVAdd k P v\u2082\u2083)) (map (\u2191(constVAdd k P v\u2081\u2082)) s\u2081)\n[PROOFSTEP]\nrefine' \u27e8v\u2082\u2083 + v\u2081\u2082, _\u27e9\n[GOAL]\ncase intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv\u2081\u2082 v\u2082\u2083 : V\n\u22a2 map (\u2191(constVAdd k P v\u2082\u2083)) (map (\u2191(constVAdd k P v\u2081\u2082)) s\u2081) = map (\u2191(constVAdd k P (v\u2082\u2083 + v\u2081\u2082))) s\u2081\n[PROOFSTEP]\nrw [map_map, \u2190 coe_trans_to_affineMap, \u2190 constVAdd_add]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : s\u2081 \u2225 s\u2082\n\u22a2 direction s\u2081 = direction s\u2082\n[PROOFSTEP]\nrcases h with \u27e8v, rfl\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nv : V\n\u22a2 direction s\u2081 = direction (map (\u2191(constVAdd k P v)) s\u2081)\n[PROOFSTEP]\nsimp\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\n\u22a2 s \u2225 \u22a5 \u2194 s = \u22a5\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => h \u25b8 Parallel.refl _\u27e9\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nh : s \u2225 \u22a5\n\u22a2 s = \u22a5\n[PROOFSTEP]\nrcases h with \u27e8v, h\u27e9\n[GOAL]\ncase intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\nv : V\nh : \u22a5 = map (\u2191(constVAdd k P v)) s\n\u22a2 s = \u22a5\n[PROOFSTEP]\nrwa [eq_comm, map_eq_bot_iff] at h \n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\n\u22a2 \u22a5 \u2225 s \u2194 s = \u22a5\n[PROOFSTEP]\nrw [parallel_comm, parallel_bot_iff_eq_bot]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\n\u22a2 s\u2081 \u2225 s\u2082 \u2194 direction s\u2081 = direction s\u2082 \u2227 (s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5)\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.direction_eq, _, _\u27e9, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : s\u2081 \u2225 s\u2082\n\u22a2 s\u2081 = \u22a5 \u2192 s\u2082 = \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2082 : AffineSubspace k P\nh : \u22a5 \u2225 s\u2082\n\u22a2 s\u2082 = \u22a5\n[PROOFSTEP]\nexact bot_parallel_iff_eq_bot.1 h\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : s\u2081 \u2225 s\u2082\n\u22a2 s\u2082 = \u22a5 \u2192 s\u2081 = \u22a5\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 : AffineSubspace k P\nh : s\u2081 \u2225 \u22a5\n\u22a2 s\u2081 = \u22a5\n[PROOFSTEP]\nexact parallel_bot_iff_eq_bot.1 h\n[GOAL]\ncase refine'_3\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nh : direction s\u2081 = direction s\u2082 \u2227 (s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5)\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nrcases h with \u27e8hd, hb\u27e9\n[GOAL]\ncase refine'_3.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nby_cases hs\u2081 : s\u2081 = \u22a5\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : s\u2081 = \u22a5\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nrw [hs\u2081, bot_parallel_iff_eq_bot]\n[GOAL]\ncase pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : s\u2081 = \u22a5\n\u22a2 s\u2082 = \u22a5\n[PROOFSTEP]\nexact hb.1 hs\u2081\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nhave hs\u2082 : s\u2082 \u2260 \u22a5 := hb.not.1 hs\u2081\n[GOAL]\ncase neg\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nrcases(nonempty_iff_ne_bot s\u2081).2 hs\u2081 with \u27e8p\u2081, hp\u2081\u27e9\n[GOAL]\ncase neg.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nrcases(nonempty_iff_ne_bot s\u2082).2 hs\u2082 with \u27e8p\u2082, hp\u2082\u27e9\n[GOAL]\ncase neg.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 \u2191s\u2082\n\u22a2 s\u2081 \u2225 s\u2082\n[PROOFSTEP]\nrefine' \u27e8p\u2082 -\u1d65 p\u2081, (eq_iff_direction_eq_of_mem hp\u2082 _).2 _\u27e9\n[GOAL]\ncase neg.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 \u2191s\u2082\n\u22a2 p\u2082 \u2208 map (\u2191(constVAdd k P (p\u2082 -\u1d65 p\u2081))) s\u2081\n[PROOFSTEP]\nrw [mem_map]\n[GOAL]\ncase neg.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 \u2191s\u2082\n\u22a2 \u2203 y, y \u2208 s\u2081 \u2227 \u2191\u2191(constVAdd k P (p\u2082 -\u1d65 p\u2081)) y = p\u2082\n[PROOFSTEP]\nrefine' \u27e8p\u2081, hp\u2081, _\u27e9\n[GOAL]\ncase neg.intro.intro.refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 \u2191s\u2082\n\u22a2 \u2191\u2191(constVAdd k P (p\u2082 -\u1d65 p\u2081)) p\u2081 = p\u2082\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.intro.intro.refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : AffineSubspace k P\nhd : direction s\u2081 = direction s\u2082\nhb : s\u2081 = \u22a5 \u2194 s\u2082 = \u22a5\nhs\u2081 : \u00acs\u2081 = \u22a5\nhs\u2082 : s\u2082 \u2260 \u22a5\np\u2081 : P\nhp\u2081 : p\u2081 \u2208 \u2191s\u2081\np\u2082 : P\nhp\u2082 : p\u2082 \u2208 \u2191s\u2082\n\u22a2 direction s\u2082 = direction (map (\u2191(constVAdd k P (p\u2082 -\u1d65 p\u2081))) s\u2081)\n[PROOFSTEP]\nsimpa using hd.symm\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\nh : affineSpan k s\u2081 \u2225 affineSpan k s\u2082\n\u22a2 vectorSpan k s\u2081 = vectorSpan k s\u2082\n[PROOFSTEP]\nsimp_rw [\u2190 direction_affineSpan]\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\nh : affineSpan k s\u2081 \u2225 affineSpan k s\u2082\n\u22a2 direction (affineSpan k s\u2081) = direction (affineSpan k s\u2082)\n[PROOFSTEP]\nexact h.direction_eq\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\n\u22a2 affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194 vectorSpan k s\u2081 = vectorSpan k s\u2082 \u2227 (s\u2081 = \u2205 \u2194 s\u2082 = \u2205)\n[PROOFSTEP]\nrepeat\n  rw [\u2190 direction_affineSpan, \u2190 affineSpan_eq_bot k]\n    -- porting note: more issues with `simp`\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\n\u22a2 affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194 vectorSpan k s\u2081 = vectorSpan k s\u2082 \u2227 (s\u2081 = \u2205 \u2194 s\u2082 = \u2205)\n[PROOFSTEP]\nrw [\u2190 direction_affineSpan, \u2190 affineSpan_eq_bot k]\n  -- porting note: more issues with `simp`\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\n\u22a2 affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194 direction (affineSpan k s\u2081) = vectorSpan k s\u2082 \u2227 (affineSpan k s\u2081 = \u22a5 \u2194 s\u2082 = \u2205)\n[PROOFSTEP]\nrw [\u2190 direction_affineSpan, \u2190 affineSpan_eq_bot k]\n  -- porting note: more issues with `simp`\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\n\u22a2 affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194\n    direction (affineSpan k s\u2081) = direction (affineSpan k s\u2082) \u2227 (affineSpan k s\u2081 = \u22a5 \u2194 affineSpan k s\u2082 = \u22a5)\n[PROOFSTEP]\nrw [\u2190 direction_affineSpan, \u2190 affineSpan_eq_bot k]\n  -- porting note: more issues with `simp`\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns\u2081 s\u2082 : Set P\n\u22a2 affineSpan k s\u2081 \u2225 affineSpan k s\u2082 \u2194\n    direction (affineSpan k s\u2081) = direction (affineSpan k s\u2082) \u2227 (affineSpan k s\u2081 = \u22a5 \u2194 affineSpan k s\u2082 = \u22a5)\n[PROOFSTEP]\nexact parallel_iff_direction_eq_and_eq_bot_iff_eq_bot\n[GOAL]\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\np\u2081 p\u2082 p\u2083 p\u2084 : P\n\u22a2 affineSpan k {p\u2081, p\u2082} \u2225 affineSpan k {p\u2083, p\u2084} \u2194 vectorSpan k {p\u2081, p\u2082} = vectorSpan k {p\u2083, p\u2084}\n[PROOFSTEP]\nsimp [affineSpan_parallel_iff_vectorSpan_eq_and_eq_empty_iff_eq_empty, \u2190 not_nonempty_iff_eq_empty]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.AffineSubspace", "llama_tokens": 80150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624789529375, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.5070073538019262}}
{"text": "[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 A \u2208 unitaryGroup n \u03b1 \u2194 A * star A = 1\n[PROOFSTEP]\nrefine' \u27e8And.right, fun hA => \u27e8_, hA\u27e9\u27e9\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\nhA : A * star A = 1\n\u22a2 star A * A = 1\n[PROOFSTEP]\nsimpa only [mul_eq_one_comm] using hA\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 A \u2208 unitaryGroup n \u03b1 \u2194 star A * A = 1\n[PROOFSTEP]\nrefine' \u27e8And.left, fun hA => \u27e8hA, _\u27e9\u27e9\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\nhA : star A * A = 1\n\u22a2 A * star A = 1\n[PROOFSTEP]\nrwa [mul_eq_one_comm] at hA \n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d A : Matrix n n \u03b1\nhA : A \u2208 unitaryGroup n \u03b1\n\u22a2 det A \u2208 unitary \u03b1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d A : Matrix n n \u03b1\nhA : A \u2208 unitaryGroup n \u03b1\n\u22a2 star (det A) * det A = 1\n[PROOFSTEP]\nsimpa [star, det_transpose] using congr_arg det hA.1\n[GOAL]\ncase right\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d A : Matrix n n \u03b1\nhA : A \u2208 unitaryGroup n \u03b1\n\u22a2 det A * star (det A) = 1\n[PROOFSTEP]\nsimpa [star, det_transpose] using congr_arg det hA.2\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA : { x // x \u2208 unitaryGroup n \u03b1 }\nsrc\u271d : (fun x => (n \u2192 \u03b1) \u2192\u2097[\u03b1] n \u2192 \u03b1) \u2191A := \u2191Matrix.toLin' \u2191A\nx : n \u2192 \u03b1\n\u22a2 \u2191(comp (toLin' A\u207b\u00b9) (toLin' A)) x = \u2191(toLin' (A\u207b\u00b9 * A)) x\n[PROOFSTEP]\nrw [\u2190 toLin'_mul]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA : { x // x \u2208 unitaryGroup n \u03b1 }\nsrc\u271d : (fun x => (n \u2192 \u03b1) \u2192\u2097[\u03b1] n \u2192 \u03b1) \u2191A := \u2191Matrix.toLin' \u2191A\nx : n \u2192 \u03b1\n\u22a2 \u2191(toLin' (A\u207b\u00b9 * A)) x = x\n[PROOFSTEP]\nrw [mul_left_inv, toLin'_one, id_apply]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA : { x // x \u2208 unitaryGroup n \u03b1 }\nsrc\u271d : (fun x => (n \u2192 \u03b1) \u2192\u2097[\u03b1] n \u2192 \u03b1) \u2191A := \u2191Matrix.toLin' \u2191A\nx : n \u2192 \u03b1\n\u22a2 \u2191(comp (toLin' A) (toLin' A\u207b\u00b9)) x = \u2191(toLin' (A * A\u207b\u00b9)) x\n[PROOFSTEP]\nrw [\u2190 toLin'_mul]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA : { x // x \u2208 unitaryGroup n \u03b1 }\nsrc\u271d : (fun x => (n \u2192 \u03b1) \u2192\u2097[\u03b1] n \u2192 \u03b1) \u2191A := \u2191Matrix.toLin' \u2191A\nx : n \u2192 \u03b1\n\u22a2 \u2191(toLin' (A * A\u207b\u00b9)) x = x\n[PROOFSTEP]\nrw [mul_right_inv, toLin'_one, id_apply]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 \u2191(toGL 1) = \u21911\n[PROOFSTEP]\nsimp only [coe_toGL, toLin'_one]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA : Matrix n n \u03b1\n\u22a2 LinearMap.id = \u21911\n[PROOFSTEP]\nrfl\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA B : { x // x \u2208 unitaryGroup n \u03b1 }\n\u22a2 \u2191(toGL (A * B)) = \u2191(toGL A * toGL B)\n[PROOFSTEP]\nsimp only [coe_toGL, toLin'_mul]\n[GOAL]\nn : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b9 : CommRing \u03b1\ninst\u271d : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\nA B : { x // x \u2208 unitaryGroup n \u03b1 }\n\u22a2 comp (toLin' A) (toLin' B) = \u2191(toGL A * toGL B)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\n\u22a2 A \u2208 orthogonalGroup n \u03b2 \u2194 A * star A = 1\n[PROOFSTEP]\nrefine' \u27e8And.right, fun hA => \u27e8_, hA\u27e9\u27e9\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\nhA : A * star A = 1\n\u22a2 star A * A = 1\n[PROOFSTEP]\nsimpa only [mul_eq_one_comm] using hA\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\n\u22a2 A \u2208 orthogonalGroup n \u03b2 \u2194 star A * A = 1\n[PROOFSTEP]\nrefine' \u27e8And.left, fun hA => \u27e8hA, _\u27e9\u27e9\n[GOAL]\nn : Type u\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\n\u03b1 : Type v\ninst\u271d\u00b2 : CommRing \u03b1\ninst\u271d\u00b9 : StarRing \u03b1\nA\u271d : Matrix n n \u03b1\n\u03b2 : Type v\ninst\u271d : CommRing \u03b2\nA : Matrix n n \u03b2\nhA : star A * A = 1\n\u22a2 A * star A = 1\n[PROOFSTEP]\nrwa [mul_eq_one_comm] at hA \n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.UnitaryGroup", "llama_tokens": 2497, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.5068736940502424}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : OrderedCancelCommMonoid \u03b1\nhs : IsPwo s\nht : IsPwo t\n\u22a2 IsPwo (s * t)\n[PROOFSTEP]\nrw [\u2190 image_mul_prod]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : OrderedCancelCommMonoid \u03b1\nhs : IsPwo s\nht : IsPwo t\n\u22a2 IsPwo ((fun x => x.fst * x.snd) '' s \u00d7\u02e2 t)\n[PROOFSTEP]\nexact (hs.prod ht).image_of_monotone (monotone_fst.mul' monotone_snd)\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\nhs : IsWf s\nht : IsWf t\nhsn : Set.Nonempty s\nhtn : Set.Nonempty t\n\u22a2 min (_ : IsWf (s * t)) (_ : Set.Nonempty (s * t)) = min hs hsn * min ht htn\n[PROOFSTEP]\nrefine' le_antisymm (IsWf.min_le _ _ (mem_mul.2 \u27e8_, _, hs.min_mem _, ht.min_mem _, rfl\u27e9)) _\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\nhs : IsWf s\nht : IsWf t\nhsn : Set.Nonempty s\nhtn : Set.Nonempty t\n\u22a2 min hs hsn * min ht htn \u2264 min (_ : IsWf (s * t)) (_ : Set.Nonempty (s * t))\n[PROOFSTEP]\nrw [IsWf.le_min_iff]\n[GOAL]\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\nhs : IsWf s\nht : IsWf t\nhsn : Set.Nonempty s\nhtn : Set.Nonempty t\n\u22a2 \u2200 (b : \u03b1), b \u2208 s * t \u2192 min hs hsn * min ht htn \u2264 b\n[PROOFSTEP]\nrintro _ \u27e8x, y, hx, hy, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\ns t : Set \u03b1\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\nhs : IsWf s\nht : IsWf t\nhsn : Set.Nonempty s\nhtn : Set.Nonempty t\nx y : \u03b1\nhx : x \u2208 s\nhy : y \u2208 t\n\u22a2 min hs hsn * min ht htn \u2264 (fun x x_1 => x * x_1) x y\n[PROOFSTEP]\nexact mul_le_mul' (hs.min_le _ hx) (ht.min_le _ hy)\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsPwo s\nht : Set.IsPwo t\na : \u03b1\nu : Set \u03b1\nhu : Set.IsPwo u\nx : \u03b1 \u00d7 \u03b1\n\u22a2 x \u2208 mulAntidiagonal hs ht a \u2194 x.fst \u2208 s \u2227 x.snd \u2208 t \u2227 x.fst * x.snd = a\n[PROOFSTEP]\nsimp only [mulAntidiagonal, Set.Finite.mem_toFinset, Set.mem_mulAntidiagonal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsPwo s\nht : Set.IsPwo t\na : \u03b1\nu : Set \u03b1\nhu : Set.IsPwo u\nx : \u03b1 \u00d7 \u03b1\n\u22a2 Prod.swap x \u2208 mulAntidiagonal hs ht a \u2194 x \u2208 mulAntidiagonal ht hs a\n[PROOFSTEP]\nsimp only [mem_mulAntidiagonal, Prod.fst_swap, Prod.snd_swap, Set.swap_mem_mulAntidiagonal_aux, Set.mem_mulAntidiagonal]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsPwo s\nht : Set.IsPwo t\na\u271d : \u03b1\nu : Set \u03b1\nhu : Set.IsPwo u\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nx\u271d : a \u2208 {a | Finset.Nonempty (mulAntidiagonal hs ht a)}\nb : \u03b1 \u00d7 \u03b1\nhb : b \u2208 mulAntidiagonal hs ht a\n\u22a2 a \u2208 s * t\n[PROOFSTEP]\nrw [mem_mulAntidiagonal] at hb \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : OrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsPwo s\nht : Set.IsPwo t\na\u271d : \u03b1\nu : Set \u03b1\nhu : Set.IsPwo u\nx : \u03b1 \u00d7 \u03b1\na : \u03b1\nx\u271d : a \u2208 {a | Finset.Nonempty (mulAntidiagonal hs ht a)}\nb : \u03b1 \u00d7 \u03b1\nhb : b.fst \u2208 s \u2227 b.snd \u2208 t \u2227 b.fst * b.snd = a\n\u22a2 a \u2208 s * t\n[PROOFSTEP]\nexact \u27e8b.1, b.2, hb\u27e9\n[GOAL]\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\n\u22a2 mulAntidiagonal (_ : Set.IsPwo s) (_ : Set.IsPwo t) (Set.IsWf.min hs hns * Set.IsWf.min ht hnt) =\n    {(Set.IsWf.min hs hns, Set.IsWf.min ht hnt)}\n[PROOFSTEP]\next \u27e8a, b\u27e9\n[GOAL]\ncase a.mk\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na\u271d : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : \u03b1\n\u22a2 (a, b) \u2208 mulAntidiagonal (_ : Set.IsPwo s) (_ : Set.IsPwo t) (Set.IsWf.min hs hns * Set.IsWf.min ht hnt) \u2194\n    (a, b) \u2208 {(Set.IsWf.min hs hns, Set.IsWf.min ht hnt)}\n[PROOFSTEP]\nsimp only [mem_mulAntidiagonal, mem_singleton, Prod.ext_iff]\n[GOAL]\ncase a.mk\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na\u271d : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : \u03b1\n\u22a2 a \u2208 s \u2227 b \u2208 t \u2227 a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt \u2194 a = Set.IsWf.min hs hns \u2227 b = Set.IsWf.min ht hnt\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase a.mk.mp\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na\u271d : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : \u03b1\n\u22a2 a \u2208 s \u2227 b \u2208 t \u2227 a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt \u2192 a = Set.IsWf.min hs hns \u2227 b = Set.IsWf.min ht hnt\n[PROOFSTEP]\nrintro \u27e8has, hat, hst\u27e9\n[GOAL]\ncase a.mk.mp.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na\u271d : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : \u03b1\nhas : a \u2208 s\nhat : b \u2208 t\nhst : a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n\u22a2 a = Set.IsWf.min hs hns \u2227 b = Set.IsWf.min ht hnt\n[PROOFSTEP]\nobtain rfl := (hs.min_le hns has).eq_of_not_lt fun hlt => (mul_lt_mul_of_lt_of_le hlt <| ht.min_le hnt hat).ne' hst\n[GOAL]\ncase a.mk.mp.intro.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\nb : \u03b1\nhat : b \u2208 t\nhas : Set.IsWf.min hs hns \u2208 s\nhst : Set.IsWf.min hs hns * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n\u22a2 Set.IsWf.min hs hns = Set.IsWf.min hs hns \u2227 b = Set.IsWf.min ht hnt\n[PROOFSTEP]\nexact \u27e8rfl, mul_left_cancel hst\u27e9\n[GOAL]\ncase a.mk.mpr\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na\u271d : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\na b : \u03b1\n\u22a2 a = Set.IsWf.min hs hns \u2227 b = Set.IsWf.min ht hnt \u2192 a \u2208 s \u2227 b \u2208 t \u2227 a * b = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n[PROOFSTEP]\nrintro \u27e8rfl, rfl\u27e9\n[GOAL]\ncase a.mk.mpr.intro\n\u03b1\u271d : Type u_1\ninst\u271d\u00b9 : OrderedCancelCommMonoid \u03b1\u271d\ns\u271d t\u271d : Set \u03b1\u271d\nhs\u271d : Set.IsPwo s\u271d\nht\u271d : Set.IsPwo t\u271d\na : \u03b1\u271d\nu : Set \u03b1\u271d\nhu : Set.IsPwo u\nx : \u03b1\u271d \u00d7 \u03b1\u271d\n\u03b1 : Type u_2\ninst\u271d : LinearOrderedCancelCommMonoid \u03b1\ns t : Set \u03b1\nhs : Set.IsWf s\nht : Set.IsWf t\nhns : Set.Nonempty s\nhnt : Set.Nonempty t\n\u22a2 Set.IsWf.min hs hns \u2208 s \u2227\n    Set.IsWf.min ht hnt \u2208 t \u2227 Set.IsWf.min hs hns * Set.IsWf.min ht hnt = Set.IsWf.min hs hns * Set.IsWf.min ht hnt\n[PROOFSTEP]\nexact \u27e8hs.min_mem _, ht.min_mem _, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Data.Finset.MulAntidiagonal", "llama_tokens": 3792, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.5068687362246832}}
{"text": "[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\ninst\u271d\u2075 : Semiring R\u2081\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9 : TopologicalSpace M\u2082\ninst\u271d : T2Space M\u2082\nf : M\u2081 \u2192 M\u2082\n\u22a2 IsCompactOperator f \u2194 \u2203 V, V \u2208 \ud835\udcdd 0 \u2227 IsCompact (closure (f '' V))\n[PROOFSTEP]\nrw [isCompactOperator_iff_exists_mem_nhds_image_subset_compact]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\ninst\u271d\u2075 : Semiring R\u2081\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace M\u2081\ninst\u271d\u00b2 : AddCommMonoid M\u2081\ninst\u271d\u00b9 : TopologicalSpace M\u2082\ninst\u271d : T2Space M\u2082\nf : M\u2081 \u2192 M\u2082\n\u22a2 (\u2203 V, V \u2208 \ud835\udcdd 0 \u2227 \u2203 K, IsCompact K \u2227 f '' V \u2286 K) \u2194 \u2203 V, V \u2208 \ud835\udcdd 0 \u2227 IsCompact (closure (f '' V))\n[PROOFSTEP]\nexact\n  \u27e8fun \u27e8V, hV, K, hK, hKV\u27e9 => \u27e8V, hV, isCompact_closure_of_subset_compact hK hKV\u27e9, fun \u27e8V, hV, hVc\u27e9 =>\n    \u27e8V, hV, closure (f '' V), hVc, subset_closure\u27e9\u27e9\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2077 : SeminormedRing \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nS : Set M\u2081\nhS : IsVonNBounded \ud835\udd5c\u2081 S\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrS : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 \u2191f \u207b\u00b9' K\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nthis : c \u2260 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))\n\u22a2 \u2191f '' S \u2286 \u2191\u03c3\u2081\u2082 c \u2022 K\n[PROOFSTEP]\nrw [image_subset_iff, preimage_smul_set\u209b\u2097 _ _ _ f this.isUnit]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2077 : SeminormedRing \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : AddCommMonoid M\u2081\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nS : Set M\u2081\nhS : IsVonNBounded \ud835\udd5c\u2081 S\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrS : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 S \u2286 a \u2022 \u2191f \u207b\u00b9' K\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nthis : c \u2260 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))\n\u22a2 S \u2286 c \u2022 \u2191f \u207b\u00b9' K\n[PROOFSTEP]\nexact hrS c hc.le\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2076 : SeminormedRing \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2075 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2074 : TopologicalSpace M\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nS : Set M\u2081\nhS : Metric.Bounded S\n\u22a2 IsVonNBounded \ud835\udd5c\u2081 S\n[PROOFSTEP]\nrwa [NormedSpace.isVonNBounded_iff, \u2190 Metric.bounded_iff_isBounded]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2077 : SeminormedRing \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\nM\u2083 : Type u_5\ninst\u271d\u2076 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b2 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d : T2Space M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nS : Set M\u2081\nhS : Metric.Bounded S\n\u22a2 IsVonNBounded \ud835\udd5c\u2081 S\n[PROOFSTEP]\nrwa [NormedSpace.isVonNBounded_iff, \u2190 Metric.bounded_iff_isBounded]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nR\u2084 : Type u_4\ninst\u271d\u00b9\u00b2 : Semiring R\u2081\ninst\u271d\u00b9\u00b9 : Semiring R\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\u2083\ninst\u271d\u2079 : CommSemiring R\u2084\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\nM\u2084 : Type u_8\ninst\u271d\u2078 : TopologicalSpace M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : TopologicalSpace M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : TopologicalSpace M\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : TopologicalAddGroup M\u2084\nf g : M\u2081 \u2192 M\u2084\nhf : IsCompactOperator f\nhg : IsCompactOperator g\n\u22a2 IsCompactOperator (f - g)\n[PROOFSTEP]\nrw [sub_eq_add_neg]\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\nR\u2084 : Type u_4\ninst\u271d\u00b9\u00b2 : Semiring R\u2081\ninst\u271d\u00b9\u00b9 : Semiring R\u2082\ninst\u271d\u00b9\u2070 : CommSemiring R\u2083\ninst\u271d\u2079 : CommSemiring R\u2084\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084\n\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\nM\u2084 : Type u_8\ninst\u271d\u2078 : TopologicalSpace M\u2081\ninst\u271d\u2077 : AddCommMonoid M\u2081\ninst\u271d\u2076 : TopologicalSpace M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommGroup M\u2083\ninst\u271d\u00b2 : TopologicalSpace M\u2084\ninst\u271d\u00b9 : AddCommGroup M\u2084\ninst\u271d : TopologicalAddGroup M\u2084\nf g : M\u2081 \u2192 M\u2084\nhf : IsCompactOperator f\nhg : IsCompactOperator g\n\u22a2 IsCompactOperator (f + -g)\n[PROOFSTEP]\nexact hf.add hg.neg\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\n\u22a2 IsCompactOperator (f \u2218 \u2191g)\n[PROOFSTEP]\nhave := g.continuous.tendsto 0\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd (\u2191g 0))\n\u22a2 IsCompactOperator (f \u2218 \u2191g)\n[PROOFSTEP]\nrw [map_zero] at this \n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\nhf : IsCompactOperator f\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsCompactOperator (f \u2218 \u2191g)\n[PROOFSTEP]\nrcases hf with \u27e8K, hK, hKf\u27e9\n[GOAL]\ncase intro.intro\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2079 : Semiring R\u2081\ninst\u271d\u2078 : Semiring R\u2082\ninst\u271d\u2077 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2076 : TopologicalSpace M\u2081\ninst\u271d\u2075 : TopologicalSpace M\u2082\ninst\u271d\u2074 : TopologicalSpace M\u2083\ninst\u271d\u00b3 : AddCommMonoid M\u2081\ninst\u271d\u00b2 : Module R\u2081 M\u2081\ninst\u271d\u00b9 : AddCommMonoid M\u2082\ninst\u271d : Module R\u2082 M\u2082\nf : M\u2082 \u2192 M\u2083\ng : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nthis : Tendsto (\u2191g) (\ud835\udcdd 0) (\ud835\udcdd 0)\nK : Set M\u2083\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 IsCompactOperator (f \u2218 \u2191g)\n[PROOFSTEP]\nexact \u27e8K, hK, this hKf\u27e9\n[GOAL]\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2077 : Semiring R\u2081\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b3 : TopologicalSpace M\u2082\ninst\u271d\u00b2 : TopologicalSpace M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2081\ninst\u271d : Module R\u2081 M\u2081\nf : M\u2081 \u2192 M\u2082\nhf : IsCompactOperator f\ng : M\u2082 \u2192 M\u2083\nhg : Continuous g\n\u22a2 IsCompactOperator (g \u2218 f)\n[PROOFSTEP]\nrcases hf with \u27e8K, hK, hKf\u27e9\n[GOAL]\ncase intro.intro\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2077 : Semiring R\u2081\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b3 : TopologicalSpace M\u2082\ninst\u271d\u00b2 : TopologicalSpace M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2081\ninst\u271d : Module R\u2081 M\u2081\nf : M\u2081 \u2192 M\u2082\ng : M\u2082 \u2192 M\u2083\nhg : Continuous g\nK : Set M\u2082\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 IsCompactOperator (g \u2218 f)\n[PROOFSTEP]\nrefine' \u27e8g '' K, hK.image hg, mem_of_superset hKf _\u27e9\n[GOAL]\ncase intro.intro\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2077 : Semiring R\u2081\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b3 : TopologicalSpace M\u2082\ninst\u271d\u00b2 : TopologicalSpace M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2081\ninst\u271d : Module R\u2081 M\u2081\nf : M\u2081 \u2192 M\u2082\ng : M\u2082 \u2192 M\u2083\nhg : Continuous g\nK : Set M\u2082\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 f \u207b\u00b9' K \u2286 g \u2218 f \u207b\u00b9' (g '' K)\n[PROOFSTEP]\nrw [preimage_comp]\n[GOAL]\ncase intro.intro\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u2077 : Semiring R\u2081\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\nM\u2081 : Type u_4\nM\u2082 : Type u_5\nM\u2083 : Type u_6\ninst\u271d\u2074 : TopologicalSpace M\u2081\ninst\u271d\u00b3 : TopologicalSpace M\u2082\ninst\u271d\u00b2 : TopologicalSpace M\u2083\ninst\u271d\u00b9 : AddCommMonoid M\u2081\ninst\u271d : Module R\u2081 M\u2081\nf : M\u2081 \u2192 M\u2082\ng : M\u2082 \u2192 M\u2083\nhg : Continuous g\nK : Set M\u2082\nhK : IsCompact K\nhKf : f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 f \u207b\u00b9' K \u2286 f \u207b\u00b9' (g \u207b\u00b9' (g '' K))\n[PROOFSTEP]\nexact preimage_mono (subset_preimage_image _ _)\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nletI : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace _\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nthis : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nhaveI : UniformAddGroup M\u2082 := comm_topologicalAddGroup_is_uniform\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\n\u22a2 Continuous \u2191f\n[PROOFSTEP]\nrefine' continuous_of_continuousAt_zero f fun U hU => _\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd (\u2191f 0)\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [map_zero] at hU \n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nhf : IsCompactOperator \u2191f\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrcases hf with\n  \u27e8K, hK, hKf\u27e9\n    -- But any compact set is totally bounded, hence Von-Neumann bounded. Thus, `K` absorbs `U`.\n      -- This gives `r > 0` such that `\u2200 a : \ud835\udd5c\u2082, r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U`.\n[GOAL]\ncase intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrcases hK.totallyBounded.isVonNBounded \ud835\udd5c\u2082 hU with\n  \u27e8r, hr, hrU\u27e9\n    -- Choose `c : \ud835\udd5c\u2082` with `r < \u2016c\u2016`.\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrcases NormedField.exists_lt_norm \ud835\udd5c\u2081 r with \u27e8c, hc\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave hcnz : c \u2260 0 := ne_zero_of_norm_ne_zero (hr.trans hc).ne.symm\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nsuffices (\u03c3\u2081\u2082 <| c\u207b\u00b9) \u2022 K \u2286 U by\n  refine' mem_of_superset _ this\n  have : IsUnit c\u207b\u00b9 := hcnz.isUnit.inv\n  rwa [mem_map, preimage_smul_set\u209b\u2097 _ _ _ f this, set_smul_mem_nhds_zero_iff (inv_ne_zero hcnz)]\n    -- Since `\u03c3\u2081\u2082 c\u207b\u00b9` = `(\u03c3\u2081\u2082 c)\u207b\u00b9`, we have to prove that `K \u2286 \u03c3\u2081\u2082 c \u2022 U`.\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d\u00b9 : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis\u271d : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\nthis : \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2286 U\n\u22a2 U \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' mem_of_superset _ this\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d\u00b9 : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis\u271d : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\nthis : \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2286 U\n\u22a2 \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave : IsUnit c\u207b\u00b9 := hcnz.isUnit.inv\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d\u00b2 : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis\u271d\u00b9 : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\nthis\u271d : \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2286 U\nthis : IsUnit c\u207b\u00b9\n\u22a2 \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2208 map (\u2191f) (\ud835\udcdd 0)\n[PROOFSTEP]\nrwa [mem_map, preimage_smul_set\u209b\u2097 _ _ _ f this, set_smul_mem_nhds_zero_iff (inv_ne_zero hcnz)]\n  -- Since `\u03c3\u2081\u2082 c\u207b\u00b9` = `(\u03c3\u2081\u2082 c)\u207b\u00b9`, we have to prove that `K \u2286 \u03c3\u2081\u2082 c \u2022 U`.\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\n\u22a2 \u2191\u03c3\u2081\u2082 c\u207b\u00b9 \u2022 K \u2286 U\n[PROOFSTEP]\nrw [map_inv\u2080, \u2190 subset_set_smul_iff\u2080 ((map_ne_zero \u03c3\u2081\u2082).mpr hcnz)]\n  -- But `\u03c3\u2081\u2082` is isometric, so `\u2016\u03c3\u2081\u2082 c\u2016 = \u2016c\u2016 > r`, which concludes the argument since\n    -- `\u2200 a : \ud835\udd5c\u2082, r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U`.\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\n\u22a2 K \u2286 \u2191\u03c3\u2081\u2082 c \u2022 U\n[PROOFSTEP]\nrefine' hrU (\u03c3\u2081\u2082 c) _\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\n\u22a2 r \u2264 \u2016\u2191\u03c3\u2081\u2082 c\u2016\n[PROOFSTEP]\nrw [RingHomIsometric.is_iso]\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9\u2070 : RingHomIsometric \u03c3\u2081\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2079 : TopologicalSpace M\u2081\ninst\u271d\u2078 : AddCommGroup M\u2081\ninst\u271d\u2077 : TopologicalSpace M\u2082\ninst\u271d\u2076 : AddCommGroup M\u2082\ninst\u271d\u2075 : Module \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b3 : TopologicalAddGroup M\u2081\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2081 M\u2081\ninst\u271d\u00b9 : TopologicalAddGroup M\u2082\ninst\u271d : ContinuousSMul \ud835\udd5c\u2082 M\u2082\nf : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082\nthis\u271d : UniformSpace M\u2082 := TopologicalAddGroup.toUniformSpace M\u2082\nthis : UniformAddGroup M\u2082\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nK : Set M\u2082\nhK : IsCompact K\nhKf : \u2191f \u207b\u00b9' K \u2208 \ud835\udcdd 0\nr : \u211d\nhr : 0 < r\nhrU : \u2200 (a : \ud835\udd5c\u2082), r \u2264 \u2016a\u2016 \u2192 K \u2286 a \u2022 U\nc : \ud835\udd5c\u2081\nhc : r < \u2016c\u2016\nhcnz : c \u2260 0\n\u22a2 r \u2264 \u2016c\u2016\n[PROOFSTEP]\nexact hc.le\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\n\u22a2 IsClosed {f | IsCompactOperator \u2191f}\n[PROOFSTEP]\nrefine' isClosed_of_closure_subset _\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\n\u22a2 closure {f | IsCompactOperator \u2191f} \u2286 {f | IsCompactOperator \u2191f}\n[PROOFSTEP]\nrintro u hu\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu : u \u2208 closure {f | IsCompactOperator \u2191f}\n\u22a2 u \u2208 {f | IsCompactOperator \u2191f}\n[PROOFSTEP]\nrw [mem_closure_iff_nhds_zero] at hu \n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\n\u22a2 u \u2208 {f | IsCompactOperator \u2191f}\n[PROOFSTEP]\nsuffices TotallyBounded (u '' Metric.closedBall 0 1)\n  by\n  change IsCompactOperator (u : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n  rw [isCompactOperator_iff_isCompact_closure_image_closedBall (u : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) zero_lt_one]\n  exact isCompact_of_totallyBounded_isClosed this.closure isClosed_closure\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nthis : TotallyBounded (\u2191u '' closedBall 0 1)\n\u22a2 u \u2208 {f | IsCompactOperator \u2191f}\n[PROOFSTEP]\nchange IsCompactOperator (u : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nthis : TotallyBounded (\u2191u '' closedBall 0 1)\n\u22a2 IsCompactOperator \u2191\u2191u\n[PROOFSTEP]\nrw [isCompactOperator_iff_isCompact_closure_image_closedBall (u : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) zero_lt_one]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nthis : TotallyBounded (\u2191u '' closedBall 0 1)\n\u22a2 IsCompact (closure (\u2191\u2191u '' closedBall 0 1))\n[PROOFSTEP]\nexact isCompact_of_totallyBounded_isClosed this.closure isClosed_closure\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\n\u22a2 TotallyBounded (\u2191u '' closedBall 0 1)\n[PROOFSTEP]\nrw [totallyBounded_iff_subset_finite_iUnion_nhds_zero]\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\n\u22a2 \u2200 (U : Set M\u2082), U \u2208 \ud835\udcdd 0 \u2192 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nintro U hU\n[GOAL]\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nrcases exists_nhds_zero_half hU with \u27e8V, hV, hVU\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nlet SV : Set M\u2081 \u00d7 Set M\u2082 := \u27e8closedBall 0 1, -V\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nrcases hu {f | \u2200 x \u2208 SV.1, f x \u2208 SV.2}\n    (ContinuousLinearMap.hasBasis_nhds_zero.mem_of_mem\n      \u27e8NormedSpace.isVonNBounded_closedBall _ _ _, neg_mem_nhds_zero M\u2082 hV\u27e9) with\n  \u27e8v, hv, huv\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nrcases totallyBounded_iff_subset_finite_iUnion_nhds_zero.mp (hv.isCompact_closure_image_closedBall 1).totallyBounded V\n    hV with\n  \u27e8T, hT, hTv\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nhave hTv : v '' closedBall 0 1 \u2286 _ := subset_closure.trans hTv\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nhTv : \u2191v '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\n\u22a2 \u2203 t, Set.Finite t \u2227 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 t), y +\u1d65 U\n[PROOFSTEP]\nrefine' \u27e8T, hT, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nhTv : \u2191v '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\n\u22a2 \u2191u '' closedBall 0 1 \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 U\n[PROOFSTEP]\nrw [image_subset_iff, preimage_iUnion\u2082] at hTv \u22a2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nhTv : closedBall 0 1 \u2286 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191v \u207b\u00b9' (i +\u1d65 V)\n\u22a2 closedBall 0 1 \u2286 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191u \u207b\u00b9' (i +\u1d65 U)\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nhTv : closedBall 0 1 \u2286 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191v \u207b\u00b9' (i +\u1d65 V)\nx : M\u2081\nhx : x \u2208 closedBall 0 1\n\u22a2 x \u2208 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191u \u207b\u00b9' (i +\u1d65 U)\n[PROOFSTEP]\nspecialize hTv hx\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nhTv : x \u2208 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191v \u207b\u00b9' (i +\u1d65 V)\n\u22a2 x \u2208 \u22c3 (i : M\u2082) (_ : i \u2208 T), \u2191u \u207b\u00b9' (i +\u1d65 U)\n[PROOFSTEP]\nrw [mem_iUnion\u2082] at hTv \u22a2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv\u271d : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nhTv : \u2203 i j, x \u2208 \u2191v \u207b\u00b9' (i +\u1d65 V)\n\u22a2 \u2203 i j, x \u2208 \u2191u \u207b\u00b9' (i +\u1d65 U)\n[PROOFSTEP]\nrcases hTv with \u27e8t, ht, htx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : x \u2208 \u2191v \u207b\u00b9' (t +\u1d65 V)\n\u22a2 \u2203 i j, x \u2208 \u2191u \u207b\u00b9' (i +\u1d65 U)\n[PROOFSTEP]\nrefine' \u27e8t, ht, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : x \u2208 \u2191v \u207b\u00b9' (t +\u1d65 V)\n\u22a2 x \u2208 \u2191u \u207b\u00b9' (t +\u1d65 U)\n[PROOFSTEP]\nrw [mem_preimage, mem_vadd_set_iff_neg_vadd_mem, vadd_eq_add, neg_add_eq_sub] at htx \u22a2\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : \u2191v x - t \u2208 V\n\u22a2 \u2191u x - t \u2208 U\n[PROOFSTEP]\nconvert hVU _ htx _ (huv x hx) using 1\n[GOAL]\ncase h.e'_4\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : \u2191v x - t \u2208 V\n\u22a2 \u2191u x - t = \u2191v x - t + -\u2191(v - u) x\n[PROOFSTEP]\nrw [ContinuousLinearMap.sub_apply]\n[GOAL]\ncase h.e'_4\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : \u2191v x - t \u2208 V\n\u22a2 \u2191u x - t = \u2191v x - t + -(\u2191v x - \u2191u x)\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_4\n\ud835\udd5c\u2081 : Type u_1\n\ud835\udd5c\u2082 : Type u_2\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2081\ninst\u271d\u2079 : NormedField \ud835\udd5c\u2082\n\u03c3\u2081\u2082 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\nM\u2081 : Type u_3\nM\u2082 : Type u_4\ninst\u271d\u2078 : SeminormedAddCommGroup M\u2081\ninst\u271d\u2077 : AddCommGroup M\u2082\ninst\u271d\u2076 : NormedSpace \ud835\udd5c\u2081 M\u2081\ninst\u271d\u2075 : Module \ud835\udd5c\u2082 M\u2082\ninst\u271d\u2074 : UniformSpace M\u2082\ninst\u271d\u00b3 : UniformAddGroup M\u2082\ninst\u271d\u00b2 : ContinuousConstSMul \ud835\udd5c\u2082 M\u2082\ninst\u271d\u00b9 : T2Space M\u2082\ninst\u271d : CompleteSpace M\u2082\nu : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhu\u271d : u \u2208 closure {f | IsCompactOperator \u2191f}\nhu : \u2200 (U : Set (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)), U \u2208 \ud835\udcdd 0 \u2192 \u2203 y, y \u2208 {f | IsCompactOperator \u2191f} \u2227 y - u \u2208 U\nU : Set M\u2082\nhU : U \u2208 \ud835\udcdd 0\nV : Set M\u2082\nhV : V \u2208 \ud835\udcdd 0\nhVU : \u2200 (v : M\u2082), v \u2208 V \u2192 \u2200 (w : M\u2082), w \u2208 V \u2192 v + w \u2208 U\nSV : Set M\u2081 \u00d7 Set M\u2082 := (closedBall 0 1, -V)\nv : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\nhv : v \u2208 {f | IsCompactOperator \u2191f}\nhuv : v - u \u2208 {f | \u2200 (x : M\u2081), x \u2208 SV.fst \u2192 \u2191f x \u2208 SV.snd}\nT : Set M\u2082\nhT : Set.Finite T\nhTv : closure (\u2191\u2191v '' closedBall 0 1) \u2286 \u22c3 (y : M\u2082) (_ : y \u2208 T), y +\u1d65 V\nx : M\u2081\nhx : x \u2208 closedBall 0 1\nt : M\u2082\nht : t \u2208 T\nhtx : \u2191v x - t \u2208 V\n\u22a2 \u2191u x - t = \u2191v x - t + -(\u2191v x - \u2191u x)\n[PROOFSTEP]\nabel\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.CompactOperator", "llama_tokens": 24864, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580903722561, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5068587164102937}}
{"text": "[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 V', V' \u2208 \ud835\udce4 \u03b2 \u2227 V' \u2286 V \u2227 \u2200 (g : C(\u03b1, \u03b2)), g \u2208 compactConvNhd K V' f \u2192 compactConvNhd K V' g \u2286 compactConvNhd K V f\n[PROOFSTEP]\nobtain \u27e8V', h\u2081, h\u2082\u27e9 := comp_mem_uniformity_sets hV\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nh\u2081 : V' \u2208 \ud835\udce4 \u03b2\nh\u2082 : V' \u25cb V' \u2286 V\n\u22a2 \u2203 V', V' \u2208 \ud835\udce4 \u03b2 \u2227 V' \u2286 V \u2227 \u2200 (g : C(\u03b1, \u03b2)), g \u2208 compactConvNhd K V' f \u2192 compactConvNhd K V' g \u2286 compactConvNhd K V f\n[PROOFSTEP]\nexact\n  \u27e8V', h\u2081, Subset.trans (subset_comp_self_of_mem_uniformity h\u2081) h\u2082, fun g hg g' hg' =>\n    compactConvNhd_mono f h\u2082 (compactConvNhd_mem_comp f hg hg')\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 {i j : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)},\n    IsCompact i.fst \u2227 i.snd \u2208 \ud835\udce4 \u03b2 \u2192\n      IsCompact j.fst \u2227 j.snd \u2208 \ud835\udce4 \u03b2 \u2192\n        \u2203 k,\n          (IsCompact k.fst \u2227 k.snd \u2208 \ud835\udce4 \u03b2) \u2227\n            compactConvNhd k.fst k.snd f \u2286 compactConvNhd i.fst i.snd f \u2229 compactConvNhd j.fst j.snd f\n[PROOFSTEP]\nrintro \u27e8K\u2081, V\u2081\u27e9 \u27e8K\u2082, V\u2082\u27e9 \u27e8hK\u2081, hV\u2081\u27e9 \u27e8hK\u2082, hV\u2082\u27e9\n[GOAL]\ncase mk.mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK\u2081 : Set \u03b1\nV\u2081 : Set (\u03b2 \u00d7 \u03b2)\nK\u2082 : Set \u03b1\nV\u2082 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2081 : IsCompact (K\u2081, V\u2081).fst\nhV\u2081 : (K\u2081, V\u2081).snd \u2208 \ud835\udce4 \u03b2\nhK\u2082 : IsCompact (K\u2082, V\u2082).fst\nhV\u2082 : (K\u2082, V\u2082).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 k,\n    (IsCompact k.fst \u2227 k.snd \u2208 \ud835\udce4 \u03b2) \u2227\n      compactConvNhd k.fst k.snd f \u2286\n        compactConvNhd (K\u2081, V\u2081).fst (K\u2081, V\u2081).snd f \u2229 compactConvNhd (K\u2082, V\u2082).fst (K\u2082, V\u2082).snd f\n[PROOFSTEP]\nexact \u27e8\u27e8K\u2081 \u222a K\u2082, V\u2081 \u2229 V\u2082\u27e9, \u27e8hK\u2081.union hK\u2082, Filter.inter_mem hV\u2081 hV\u2082\u27e9, compactConvNhd_subset_inter f K\u2081 K\u2082 V\u2081 V\u2082\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\n\u22a2 Y \u2208 FilterBasis.filter (compactConvergenceFilterBasis f) \u2194 \u2203 K V _hK _hV, compactConvNhd K V f \u2286 Y\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\n\u22a2 Y \u2208 FilterBasis.filter (compactConvergenceFilterBasis f) \u2192 \u2203 K V _hK _hV, compactConvNhd K V f \u2286 Y\n[PROOFSTEP]\nrintro \u27e8X, \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9, hY\u27e9\n[GOAL]\ncase mp.intro.intro.intro.mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\nhY : (fun KV => compactConvNhd KV.fst KV.snd f) (K, V) \u2286 Y\n\u22a2 \u2203 K V _hK _hV, compactConvNhd K V f \u2286 Y\n[PROOFSTEP]\nexact \u27e8K, V, hK, hV, hY\u27e9\n[GOAL]\ncase mpr\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\n\u22a2 (\u2203 K V _hK _hV, compactConvNhd K V f \u2286 Y) \u2192 Y \u2208 FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nrintro \u27e8K, V, hK, hV, hY\u27e9\n[GOAL]\ncase mpr.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhY : compactConvNhd K V f \u2286 Y\n\u22a2 Y \u2208 FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nexact \u27e8compactConvNhd K V f, \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9, hY\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \ud835\udcdd f = FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nrw [TopologicalSpace.nhds_mkOfNhds_filterBasis]\n[GOAL]\ncase h\u2080\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (x : C(\u03b1, \u03b2)) (n : Set C(\u03b1, \u03b2)), n \u2208 compactConvergenceFilterBasis x \u2192 x \u2208 n\n[PROOFSTEP]\nrintro g - \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9\n[GOAL]\ncase h\u2081\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (x : C(\u03b1, \u03b2)) (n : Set C(\u03b1, \u03b2)),\n    n \u2208 compactConvergenceFilterBasis x \u2192\n      \u2203 n\u2081,\n        n\u2081 \u2208 compactConvergenceFilterBasis x \u2227\n          n\u2081 \u2286 n \u2227 \u2200 (x' : C(\u03b1, \u03b2)), x' \u2208 n\u2081 \u2192 \u2203 n\u2082, n\u2082 \u2208 compactConvergenceFilterBasis x' \u2227 n\u2082 \u2286 n\n[PROOFSTEP]\nrintro g - \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9\n[GOAL]\ncase h\u2080.intro.mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf g : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 g \u2208 (fun KV => compactConvNhd KV.fst KV.snd g) (K, V)\n[PROOFSTEP]\nexact self_mem_compactConvNhd g hV\n[GOAL]\ncase h\u2081.intro.mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf g : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 n\u2081,\n    n\u2081 \u2208 compactConvergenceFilterBasis g \u2227\n      n\u2081 \u2286 (fun KV => compactConvNhd KV.fst KV.snd g) (K, V) \u2227\n        \u2200 (x' : C(\u03b1, \u03b2)),\n          x' \u2208 n\u2081 \u2192 \u2203 n\u2082, n\u2082 \u2208 compactConvergenceFilterBasis x' \u2227 n\u2082 \u2286 (fun KV => compactConvNhd KV.fst KV.snd g) (K, V)\n[PROOFSTEP]\nobtain \u27e8V', hV', h\u2081, h\u2082\u27e9 := compactConvNhd_nhd_basis g hV\n[GOAL]\ncase h\u2081.intro.mk.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf g : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nh\u2081 : V' \u2286 (K, V).snd\nh\u2082 :\n  \u2200 (g_1 : C(\u03b1, \u03b2)),\n    g_1 \u2208 compactConvNhd ?m.4688 V' g \u2192 compactConvNhd ?m.4688 V' g_1 \u2286 compactConvNhd ?m.4688 (K, V).snd g\n\u22a2 \u2203 n\u2081,\n    n\u2081 \u2208 compactConvergenceFilterBasis g \u2227\n      n\u2081 \u2286 (fun KV => compactConvNhd KV.fst KV.snd g) (K, V) \u2227\n        \u2200 (x' : C(\u03b1, \u03b2)),\n          x' \u2208 n\u2081 \u2192 \u2203 n\u2082, n\u2082 \u2208 compactConvergenceFilterBasis x' \u2227 n\u2082 \u2286 (fun KV => compactConvNhd KV.fst KV.snd g) (K, V)\n[PROOFSTEP]\nexact\n  \u27e8compactConvNhd K V' g, \u27e8\u27e8K, V'\u27e9, \u27e8hK, hV'\u27e9, rfl\u27e9, compactConvNhd_mono g h\u2081, fun g' hg' =>\n    \u27e8compactConvNhd K V' g', \u27e8\u27e8K, V'\u27e9, \u27e8hK, hV'\u27e9, rfl\u27e9, h\u2082 g' hg'\u27e9\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 Tendsto F p (\ud835\udcdd f) \u2194 \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n[PROOFSTEP]\nsimp only [(hasBasis_nhds_compactConvergence f).tendsto_right_iff, TendstoUniformlyOn, and_imp, Prod.forall]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 (\u2200 (a : Set \u03b1) (b : Set (\u03b2 \u00d7 \u03b2)), IsCompact a \u2192 b \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (x : \u03b9) in p, F x \u2208 compactConvNhd a b f) \u2194\n    \u2200 (K : Set \u03b1), IsCompact K \u2192 \u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191f x, \u2191(F n) x) \u2208 u\n[PROOFSTEP]\nrefine' forall_congr' fun K => _\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nK : Set \u03b1\n\u22a2 (\u2200 (b : Set (\u03b2 \u00d7 \u03b2)), IsCompact K \u2192 b \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (x : \u03b9) in p, F x \u2208 compactConvNhd K b f) \u2194\n    IsCompact K \u2192 \u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191f x, \u2191(F n) x) \u2208 u\n[PROOFSTEP]\nrw [forall_swap]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nK : Set \u03b1\n\u22a2 (IsCompact K \u2192 \u2200 (x : Set (\u03b2 \u00d7 \u03b2)), x \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (x_1 : \u03b9) in p, F x_1 \u2208 compactConvNhd K x f) \u2194\n    IsCompact K \u2192 \u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191f x, \u2191(F n) x) \u2208 u\n[PROOFSTEP]\nexact forall\u2083_congr fun _hK V _hV => Iff.rfl\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nhf : f \u2208 CompactOpen.gen K U\n\u22a2 \u2203 V, V \u2208 \ud835\udce4 \u03b2 \u2227 IsOpen V \u2227 compactConvNhd K V f \u2286 CompactOpen.gen K U\n[PROOFSTEP]\nobtain \u27e8V, hV\u2081, hV\u2082, hV\u2083\u27e9 := lebesgue_number_of_compact_open (hK.image f.continuous) hU hf\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nhf : f \u2208 CompactOpen.gen K U\nV : Set (\u03b2 \u00d7 \u03b2)\nhV\u2081 : V \u2208 \ud835\udce4 \u03b2\nhV\u2082 : IsOpen V\nhV\u2083 : \u2200 (x : \u03b2), x \u2208 \u2191f '' K \u2192 ball x V \u2286 U\n\u22a2 \u2203 V, V \u2208 \ud835\udce4 \u03b2 \u2227 IsOpen V \u2227 compactConvNhd K V f \u2286 CompactOpen.gen K U\n[PROOFSTEP]\nrefine' \u27e8V, hV\u2081, hV\u2082, _\u27e9\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nhf : f \u2208 CompactOpen.gen K U\nV : Set (\u03b2 \u00d7 \u03b2)\nhV\u2081 : V \u2208 \ud835\udce4 \u03b2\nhV\u2082 : IsOpen V\nhV\u2083 : \u2200 (x : \u03b2), x \u2208 \u2191f '' K \u2192 ball x V \u2286 U\n\u22a2 compactConvNhd K V f \u2286 CompactOpen.gen K U\n[PROOFSTEP]\nrintro g hg _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nhf : f \u2208 CompactOpen.gen K U\nV : Set (\u03b2 \u00d7 \u03b2)\nhV\u2081 : V \u2208 \ud835\udce4 \u03b2\nhV\u2082 : IsOpen V\nhV\u2083 : \u2200 (x : \u03b2), x \u2208 \u2191f '' K \u2192 ball x V \u2286 U\ng : C(\u03b1, \u03b2)\nhg : g \u2208 compactConvNhd K V f\nx : \u03b1\nhx : x \u2208 K\n\u22a2 \u2191g x \u2208 U\n[PROOFSTEP]\nexact hV\u2083 (f x) \u27e8x, hx, rfl\u27e9 (hg x hx)\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nobtain \u27e8W, hW\u2081, hW\u2084, hW\u2082, hW\u2083\u27e9 := comp_open_symm_mem_uniformity_sets hV\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nobtain \u27e8Z, hZ\u2081, hZ\u2084, hZ\u2082, hZ\u2083\u27e9 := comp_open_symm_mem_uniformity_sets hW\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nlet U : \u03b1 \u2192 Set \u03b1 := fun x => f \u207b\u00b9' ball (f x) Z\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nhave hU : \u2200 x, IsOpen (U x) := fun x => f.continuous.isOpen_preimage _ (isOpen_ball _ hZ\u2084)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nhave hUK : K \u2286 \u22c3 x : K, U (x : K) := by\n  intro x hx\n  simp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]\n  exact \u27e8(\u27e8x, hx\u27e9 : K), by simp [hx, mem_ball_self (f x) hZ\u2081]\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\n\u22a2 K \u2286 \u22c3 (x : \u2191K), U \u2191x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nx : \u03b1\nhx : x \u2208 K\n\u22a2 x \u2208 \u22c3 (x : \u2191K), U \u2191x\n[PROOFSTEP]\nsimp only [exists_prop, mem_iUnion, iUnion_coe_set, mem_preimage]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nx : \u03b1\nhx : x \u2208 K\n\u22a2 \u2203 i, i \u2208 K \u2227 \u2191f x \u2208 ball (\u2191f i) Z\n[PROOFSTEP]\nexact \u27e8(\u27e8x, hx\u27e9 : K), by simp [hx, mem_ball_self (f x) hZ\u2081]\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nx : \u03b1\nhx : x \u2208 K\n\u22a2 \u2191{ val := x, property := hx } \u2208 K \u2227 \u2191f x \u2208 ball (\u2191f \u2191{ val := x, property := hx }) Z\n[PROOFSTEP]\nsimp [hx, mem_ball_self (f x) hZ\u2081]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nobtain \u27e8t, ht\u27e9 := hK.elim_finite_subcover _ (fun x : K => hU x.val) hUK\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nlet C : t \u2192 Set \u03b1 := fun i => K \u2229 closure (U ((i : K) : \u03b1))\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nhave hC : K \u2286 \u22c3 i, C i := by\n  rw [\u2190 K.inter_iUnion, subset_inter_iff]\n  refine' \u27e8Subset.rfl, ht.trans _\u27e9\n  simp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n  exact fun x hx\u2081 hx\u2082 => subset_iUnion_of_subset (\u27e8_, hx\u2082\u27e9 : t) (by simp [subset_closure])\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\n\u22a2 K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\n[PROOFSTEP]\nrw [\u2190 K.inter_iUnion, subset_inter_iff]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\n\u22a2 K \u2286 K \u2227 K \u2286 \u22c3 (i : { x // x \u2208 t }), closure (U \u2191\u2191i)\n[PROOFSTEP]\nrefine' \u27e8Subset.rfl, ht.trans _\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\n\u22a2 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i \u2286 \u22c3 (i : { x // x \u2208 t }), closure (U \u2191\u2191i)\n[PROOFSTEP]\nsimp only [SetCoe.forall, Subtype.coe_mk, iUnion_subset_iff]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\n\u22a2 \u2200 (x : \u03b1) (h : x \u2208 K),\n    { val := x, property := h } \u2208 t \u2192 \u2191f \u207b\u00b9' ball (\u2191f x) Z \u2286 \u22c3 (i : { x // x \u2208 t }), closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) Z)\n[PROOFSTEP]\nexact fun x hx\u2081 hx\u2082 => subset_iUnion_of_subset (\u27e8_, hx\u2082\u27e9 : t) (by simp [subset_closure])\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\n\u22a2 \u2191f \u207b\u00b9' ball (\u2191f x) Z \u2286 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191{ val := { val := x, property := hx\u2081 }, property := hx\u2082 }) Z)\n[PROOFSTEP]\nsimp [subset_closure]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nhave hfC : \u2200 i : t, C i \u2286 f \u207b\u00b9' ball (f ((i : K) : \u03b1)) W :=\n  by\n  simp only [\u2190 image_subset_iff, \u2190 mem_preimage]\n  rintro \u27e8\u27e8x, hx\u2081\u27e9, hx\u2082\u27e9\n  have hZW : closure (ball (f x) Z) \u2286 ball (f x) W := by\n    intro y hy\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := UniformSpace.mem_closure_iff_ball.mp hy hZ\u2081\n    exact ball_mono hZ\u2083 _ (mem_ball_comp hz\u2082 ((mem_ball_symmetry hZ\u2082).mp hz\u2081))\n  calc\n    f '' (K \u2229 closure (U x)) \u2286 f '' closure (U x) := image_subset _ (inter_subset_right _ _)\n    _ \u2286 closure (f '' U x) := f.continuous.continuousOn.image_closure\n    _ \u2286 closure (ball (f x) Z) := by\n      apply closure_mono\n      simp only [image_subset_iff]\n      rfl\n    _ \u2286 ball (f x) W := hZW\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\n\u22a2 \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\n[PROOFSTEP]\nsimp only [\u2190 image_subset_iff, \u2190 mem_preimage]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\n\u22a2 \u2200 (i : { x // x \u2208 t }), \u2191f '' (K \u2229 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) Z)) \u2286 ball (\u2191f \u2191\u2191i) W\n[PROOFSTEP]\nrintro \u27e8\u27e8x, hx\u2081\u27e9, hx\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\n\u22a2 \u2191f '' (K \u2229 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191{ val := { val := x, property := hx\u2081 }, property := hx\u2082 }) Z)) \u2286\n    ball (\u2191f \u2191\u2191{ val := { val := x, property := hx\u2081 }, property := hx\u2082 }) W\n[PROOFSTEP]\nhave hZW : closure (ball (f x) Z) \u2286 ball (f x) W := by\n  intro y hy\n  obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := UniformSpace.mem_closure_iff_ball.mp hy hZ\u2081\n  exact ball_mono hZ\u2083 _ (mem_ball_comp hz\u2082 ((mem_ball_symmetry hZ\u2082).mp hz\u2081))\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\n\u22a2 closure (ball (\u2191f x) Z) \u2286 ball (\u2191f x) W\n[PROOFSTEP]\nintro y hy\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\ny : \u03b2\nhy : y \u2208 closure (ball (\u2191f x) Z)\n\u22a2 y \u2208 ball (\u2191f x) W\n[PROOFSTEP]\nobtain \u27e8z, hz\u2081, hz\u2082\u27e9 := UniformSpace.mem_closure_iff_ball.mp hy hZ\u2081\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\ny : \u03b2\nhy : y \u2208 closure (ball (\u2191f x) Z)\nz : \u03b2\nhz\u2081 : z \u2208 ball y Z\nhz\u2082 : z \u2208 ball (\u2191f x) Z\n\u22a2 y \u2208 ball (\u2191f x) W\n[PROOFSTEP]\nexact ball_mono hZ\u2083 _ (mem_ball_comp hz\u2082 ((mem_ball_symmetry hZ\u2082).mp hz\u2081))\n[GOAL]\ncase mk.mk\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\nhZW : closure (ball (\u2191f x) Z) \u2286 ball (\u2191f x) W\n\u22a2 \u2191f '' (K \u2229 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191{ val := { val := x, property := hx\u2081 }, property := hx\u2082 }) Z)) \u2286\n    ball (\u2191f \u2191\u2191{ val := { val := x, property := hx\u2081 }, property := hx\u2082 }) W\n[PROOFSTEP]\ncalc\n  f '' (K \u2229 closure (U x)) \u2286 f '' closure (U x) := image_subset _ (inter_subset_right _ _)\n  _ \u2286 closure (f '' U x) := f.continuous.continuousOn.image_closure\n  _ \u2286 closure (ball (f x) Z) := by\n    apply closure_mono\n    simp only [image_subset_iff]\n    rfl\n  _ \u2286 ball (f x) W := hZW\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\nhZW : closure (ball (\u2191f x) Z) \u2286 ball (\u2191f x) W\n\u22a2 closure (\u2191f '' U x) \u2286 closure (ball (\u2191f x) Z)\n[PROOFSTEP]\napply closure_mono\n[GOAL]\ncase h\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\nhZW : closure (ball (\u2191f x) Z) \u2286 ball (\u2191f x) W\n\u22a2 \u2191f '' U x \u2286 ball (\u2191f x) Z\n[PROOFSTEP]\nsimp only [image_subset_iff]\n[GOAL]\ncase h\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nx : \u03b1\nhx\u2081 : x \u2208 K\nhx\u2082 : { val := x, property := hx\u2081 } \u2208 t\nhZW : closure (ball (\u2191f x) Z) \u2286 ball (\u2191f x) W\n\u22a2 \u2191f \u207b\u00b9' ball (\u2191f x) Z \u2286 (fun a => \u2191f a) \u207b\u00b9' ball (\u2191f x) Z\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nhfC : \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\n\u22a2 \u2203 \u03b9 x C _hC U _hU,\n    f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2227 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd K V f\n[PROOFSTEP]\nrefine'\n  \u27e8t, t.fintypeCoeSort, C, fun i => hK.inter_right isClosed_closure, fun i => ball (f ((i : K) : \u03b1)) W, fun i =>\n    isOpen_ball _ hW\u2084, by simp [CompactOpen.gen, hfC], fun g hg x hx => hW\u2083 (mem_compRel.mpr _)\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nhfC : \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\n\u22a2 f \u2208 \u22c2 (i : { x // x \u2208 t }), CompactOpen.gen (C i) ((fun i => ball (\u2191f \u2191\u2191i) W) i)\n[PROOFSTEP]\nsimp [CompactOpen.gen, hfC]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nhfC : \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\ng : C(\u03b1, \u03b2)\nhg : g \u2208 \u22c2 (i : { x // x \u2208 t }), CompactOpen.gen (C i) ((fun i => ball (\u2191f \u2191\u2191i) W) i)\nx : \u03b1\nhx : x \u2208 K\n\u22a2 \u2203 z, (\u2191f x, z) \u2208 W \u2227 (z, \u2191g x) \u2208 W\n[PROOFSTEP]\nsimp only [mem_iInter, CompactOpen.gen, mem_setOf_eq, image_subset_iff] at hg \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nhfC : \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\ng : C(\u03b1, \u03b2)\nx : \u03b1\nhx : x \u2208 K\nhg : \u2200 (i : { x // x \u2208 t }), K \u2229 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) Z) \u2286 (fun a => \u2191g a) \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\n\u22a2 \u2203 z, (\u2191f x, z) \u2208 W \u2227 (z, \u2191g x) \u2208 W\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := mem_iUnion.mp (hC hx)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nW : Set (\u03b2 \u00d7 \u03b2)\nhW\u2081 : W \u2208 \ud835\udce4 \u03b2\nhW\u2084 : IsOpen W\nhW\u2082 : SymmetricRel W\nhW\u2083 : W \u25cb W \u2286 V\nZ : Set (\u03b2 \u00d7 \u03b2)\nhZ\u2081 : Z \u2208 \ud835\udce4 \u03b2\nhZ\u2084 : IsOpen Z\nhZ\u2082 : SymmetricRel Z\nhZ\u2083 : Z \u25cb Z \u2286 W\nU : \u03b1 \u2192 Set \u03b1 := fun x => \u2191f \u207b\u00b9' ball (\u2191f x) Z\nhU : \u2200 (x : \u03b1), IsOpen (U x)\nhUK : K \u2286 \u22c3 (x : \u2191K), U \u2191x\nt : Finset \u2191K\nht : K \u2286 \u22c3 (i : \u2191K) (_ : i \u2208 t), U \u2191i\nC : { x // x \u2208 t } \u2192 Set \u03b1 := fun i => K \u2229 closure (U \u2191\u2191i)\nhC : K \u2286 \u22c3 (i : { x // x \u2208 t }), C i\nhfC : \u2200 (i : { x // x \u2208 t }), C i \u2286 \u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\ng : C(\u03b1, \u03b2)\nx : \u03b1\nhx : x \u2208 K\nhg : \u2200 (i : { x // x \u2208 t }), K \u2229 closure (\u2191f \u207b\u00b9' ball (\u2191f \u2191\u2191i) Z) \u2286 (fun a => \u2191g a) \u207b\u00b9' ball (\u2191f \u2191\u2191i) W\ny : { x // x \u2208 t }\nhy : x \u2208 C y\n\u22a2 \u2203 z, (\u2191f x, z) \u2208 W \u2227 (z, \u2191g x) \u2208 W\n[PROOFSTEP]\nexact \u27e8f y, (mem_ball_symmetry hW\u2082).mp (hfC y hy), mem_preimage.mp (hg y hy)\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 compactOpen = compactConvergenceTopology\n[PROOFSTEP]\nrw [compactConvergenceTopology, ContinuousMap.compactOpen]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 TopologicalSpace.generateFrom {m | \u2203 s x u x, m = CompactOpen.gen s u} =\n    TopologicalSpace.mkOfNhds fun f => FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nrefine' le_antisymm _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 TopologicalSpace.generateFrom {m | \u2203 s x u x, m = CompactOpen.gen s u} \u2264\n    TopologicalSpace.mkOfNhds fun f => FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nrefine' fun X hX => isOpen_iff_forall_mem_open.mpr fun f hf => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\n\u22a2 \u2203 t, t \u2286 X \u2227 IsOpen t \u2227 f \u2208 t\n[PROOFSTEP]\nhave hXf : X \u2208 (compactConvergenceFilterBasis f).filter :=\n  by\n  rw [\u2190 nhds_compactConvergence]\n  exact @IsOpen.mem_nhds C(\u03b1, \u03b2) compactConvergenceTopology _ _ hX hf\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\n\u22a2 X \u2208 FilterBasis.filter (compactConvergenceFilterBasis f)\n[PROOFSTEP]\nrw [\u2190 nhds_compactConvergence]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\n\u22a2 X \u2208 \ud835\udcdd f\n[PROOFSTEP]\nexact @IsOpen.mem_nhds C(\u03b1, \u03b2) compactConvergenceTopology _ _ hX hf\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\nhXf : X \u2208 FilterBasis.filter (compactConvergenceFilterBasis f)\n\u22a2 \u2203 t, t \u2286 X \u2227 IsOpen t \u2227 f \u2208 t\n[PROOFSTEP]\nobtain \u27e8-, \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9, hXf\u27e9 := hXf\n[GOAL]\ncase refine'_1.intro.intro.intro.mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\nhXf : (fun KV => compactConvNhd KV.fst KV.snd f) (K, V) \u2286 X\n\u22a2 \u2203 t, t \u2286 X \u2227 IsOpen t \u2227 f \u2208 t\n[PROOFSTEP]\nobtain \u27e8\u03b9, h\u03b9, C, hC, U, hU, h\u2081, h\u2082\u27e9 := iInter_compactOpen_gen_subset_compactConvNhd f hK hV\n[GOAL]\ncase refine'_1.intro.intro.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\nhXf : (fun KV => compactConvNhd KV.fst KV.snd f) (K, V) \u2286 X\n\u03b9 : Type u\u2081\nh\u03b9 : Fintype \u03b9\nC : \u03b9 \u2192 Set \u03b1\nhC : \u2200 (i : \u03b9), IsCompact (C i)\nU : \u03b9 \u2192 Set \u03b2\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nh\u2081 : f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i)\nh\u2082 : \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd (K, V).fst (K, V).snd f\n\u22a2 \u2203 t, t \u2286 X \u2227 IsOpen t \u2227 f \u2208 t\n[PROOFSTEP]\nhaveI := h\u03b9\n[GOAL]\ncase refine'_1.intro.intro.intro.mk.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set C(\u03b1, \u03b2)\nhX : IsOpen X\nf : C(\u03b1, \u03b2)\nhf : f \u2208 X\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact (K, V).fst\nhV : (K, V).snd \u2208 \ud835\udce4 \u03b2\nhXf : (fun KV => compactConvNhd KV.fst KV.snd f) (K, V) \u2286 X\n\u03b9 : Type u\u2081\nh\u03b9 : Fintype \u03b9\nC : \u03b9 \u2192 Set \u03b1\nhC : \u2200 (i : \u03b9), IsCompact (C i)\nU : \u03b9 \u2192 Set \u03b2\nhU : \u2200 (i : \u03b9), IsOpen (U i)\nh\u2081 : f \u2208 \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i)\nh\u2082 : \u22c2 (i : \u03b9), CompactOpen.gen (C i) (U i) \u2286 compactConvNhd (K, V).fst (K, V).snd f\nthis : Fintype \u03b9\n\u22a2 \u2203 t, t \u2286 X \u2227 IsOpen t \u2227 f \u2208 t\n[PROOFSTEP]\nexact\n  \u27e8\u22c2 i, CompactOpen.gen (C i) (U i), h\u2082.trans hXf, isOpen_iInter fun i => ContinuousMap.isOpen_gen (hC i) (hU i), h\u2081\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 (TopologicalSpace.mkOfNhds fun f => FilterBasis.filter (compactConvergenceFilterBasis f)) \u2264\n    TopologicalSpace.generateFrom {m | \u2203 s x u x, m = CompactOpen.gen s u}\n[PROOFSTEP]\nsimp only [TopologicalSpace.le_generateFrom_iff_subset_isOpen, and_imp, exists_prop, forall_exists_index,\n  setOf_subset_setOf]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : Set C(\u03b1, \u03b2)) (x : Set \u03b1), IsCompact x \u2192 \u2200 (x_1 : Set \u03b2), IsOpen x_1 \u2192 a = CompactOpen.gen x x_1 \u2192 IsOpen a\n[PROOFSTEP]\nrintro - K hK U hU rfl f hf\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nf : C(\u03b1, \u03b2)\nhf : f \u2208 CompactOpen.gen K U\n\u22a2 CompactOpen.gen K U \u2208 (fun f => FilterBasis.filter (compactConvergenceFilterBasis f)) f\n[PROOFSTEP]\nobtain \u27e8V, hV, _hV', hVf\u27e9 := compactConvNhd_subset_compactOpen f hK hU hf\n[GOAL]\ncase refine'_2.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nK : Set \u03b1\nhK : IsCompact K\nU : Set \u03b2\nhU : IsOpen U\nf : C(\u03b1, \u03b2)\nhf : f \u2208 CompactOpen.gen K U\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\n_hV' : IsOpen V\nhVf : compactConvNhd K V f \u2286 CompactOpen.gen K U\n\u22a2 CompactOpen.gen K U \u2208 (fun f => FilterBasis.filter (compactConvergenceFilterBasis f)) f\n[PROOFSTEP]\nexact Filter.mem_of_superset (FilterBasis.mem_filter_of_mem _ \u27e8\u27e8K, V\u27e9, \u27e8hK, hV\u27e9, rfl\u27e9) hVf\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 HasBasis compactConvergenceUniformity (fun p => IsCompact p.fst \u2227 p.snd \u2208 \ud835\udce4 \u03b2) fun p =>\n    {fg | \u2200 (x : \u03b1), x \u2208 p.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 p.snd}\n[PROOFSTEP]\nrefine' Filter.hasBasis_biInf_principal _ compactConvNhd_compact_entourage_nonempty\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 DirectedOn ((fun KV => {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1)\n    {KV | IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2}\n[PROOFSTEP]\nrintro \u27e8K\u2081, V\u2081\u27e9 \u27e8hK\u2081, hV\u2081\u27e9 \u27e8K\u2082, V\u2082\u27e9 \u27e8hK\u2082, hV\u2082\u27e9\n[GOAL]\ncase mk.intro.mk.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK\u2081 : Set \u03b1\nV\u2081 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2081 : IsCompact (K\u2081, V\u2081).fst\nhV\u2081 : (K\u2081, V\u2081).snd \u2208 \ud835\udce4 \u03b2\nK\u2082 : Set \u03b1\nV\u2082 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2082 : IsCompact (K\u2082, V\u2082).fst\nhV\u2082 : (K\u2082, V\u2082).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 z,\n    z \u2208 {KV | IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2} \u2227\n      ((fun KV => {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) (K\u2081, V\u2081)\n          z \u2227\n        ((fun KV => {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) (K\u2082, V\u2082)\n          z\n[PROOFSTEP]\nrefine' \u27e8\u27e8K\u2081 \u222a K\u2082, V\u2081 \u2229 V\u2082\u27e9, \u27e8hK\u2081.union hK\u2082, Filter.inter_mem hV\u2081 hV\u2082\u27e9, _\u27e9\n[GOAL]\ncase mk.intro.mk.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK\u2081 : Set \u03b1\nV\u2081 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2081 : IsCompact (K\u2081, V\u2081).fst\nhV\u2081 : (K\u2081, V\u2081).snd \u2208 \ud835\udce4 \u03b2\nK\u2082 : Set \u03b1\nV\u2082 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2082 : IsCompact (K\u2082, V\u2082).fst\nhV\u2082 : (K\u2082, V\u2082).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 ((fun KV => {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) (K\u2081, V\u2081)\n      (K\u2081 \u222a K\u2082, V\u2081 \u2229 V\u2082) \u2227\n    ((fun KV => {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) (K\u2082, V\u2082)\n      (K\u2081 \u222a K\u2082, V\u2081 \u2229 V\u2082)\n[PROOFSTEP]\nsimp only [le_eq_subset, Prod.forall, setOf_subset_setOf, ge_iff_le, Order.Preimage, \u2190 forall_and, mem_inter_iff,\n  mem_union]\n[GOAL]\ncase mk.intro.mk.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK\u2081 : Set \u03b1\nV\u2081 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2081 : IsCompact (K\u2081, V\u2081).fst\nhV\u2081 : (K\u2081, V\u2081).snd \u2208 \ud835\udce4 \u03b2\nK\u2082 : Set \u03b1\nV\u2082 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2082 : IsCompact (K\u2082, V\u2082).fst\nhV\u2082 : (K\u2082, V\u2082).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2200 (x x_1 : C(\u03b1, \u03b2)),\n    (\u2200 (x_2 : \u03b1), x_2 \u2208 K\u2081 \u2228 x_2 \u2208 K\u2082 \u2192 (\u2191x x_2, \u2191x_1 x_2) \u2208 V\u2081 \u2227 (\u2191x x_2, \u2191x_1 x_2) \u2208 V\u2082) \u2192\n      \u2200 (x_3 : \u03b1), (x_3 \u2208 K\u2081 \u2192 (\u2191x x_3, \u2191x_1 x_3) \u2208 V\u2081) \u2227 (x_3 \u2208 K\u2082 \u2192 (\u2191x x_3, \u2191x_1 x_3) \u2208 V\u2082)\n[PROOFSTEP]\nexact fun f g => forall_imp fun x => by tauto\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nK\u2081 : Set \u03b1\nV\u2081 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2081 : IsCompact (K\u2081, V\u2081).fst\nhV\u2081 : (K\u2081, V\u2081).snd \u2208 \ud835\udce4 \u03b2\nK\u2082 : Set \u03b1\nV\u2082 : Set (\u03b2 \u00d7 \u03b2)\nhK\u2082 : IsCompact (K\u2082, V\u2082).fst\nhV\u2082 : (K\u2082, V\u2082).snd \u2208 \ud835\udce4 \u03b2\nf g : C(\u03b1, \u03b2)\nx : \u03b1\n\u22a2 (x \u2208 K\u2081 \u2228 x \u2208 K\u2082 \u2192 (\u2191f x, \u2191g x) \u2208 V\u2081 \u2227 (\u2191f x, \u2191g x) \u2208 V\u2082) \u2192\n    (x \u2208 K\u2081 \u2192 (\u2191f x, \u2191g x) \u2208 V\u2081) \u2227 (x \u2208 K\u2082 \u2192 (\u2191f x, \u2191g x) \u2208 V\u2082)\n[PROOFSTEP]\ntauto\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\n\u22a2 X \u2208 compactConvergenceUniformity \u2194 \u2203 K V _hK _hV, {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\n[PROOFSTEP]\nsimp only [hasBasis_compactConvergenceUniformity_aux.mem_iff, exists_prop, Prod.exists, and_assoc]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \ud835\udcdf idRel \u2264 compactConvergenceUniformity\n[PROOFSTEP]\nsimp only [compactConvergenceUniformity, and_imp, Filter.le_principal_iff, Prod.forall, Filter.mem_principal,\n  mem_setOf_eq, le_iInf_iff, idRel_subset]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : Set \u03b1) (b : Set (\u03b2 \u00d7 \u03b2)), IsCompact a \u2192 b \u2208 \ud835\udce4 \u03b2 \u2192 \u2200 (a_3 : C(\u03b1, \u03b2)) (x : \u03b1), x \u2208 a \u2192 (\u2191a_3 x, \u2191a_3 x) \u2208 b\n[PROOFSTEP]\nexact fun K V _hK hV f x _hx => refl_mem_uniformity hV\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 Tendsto Prod.swap compactConvergenceUniformity compactConvergenceUniformity\n[PROOFSTEP]\nsimp only [compactConvergenceUniformity, and_imp, Prod.forall, mem_setOf_eq, Prod.fst_swap, Filter.tendsto_principal,\n  Prod.snd_swap, Filter.tendsto_iInf]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (a : Set \u03b1) (b : Set (\u03b2 \u00d7 \u03b2)),\n    IsCompact a \u2192\n      b \u2208 \ud835\udce4 \u03b2 \u2192\n        \u2200\u1da0 (a_3 : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) in\n          \u2a05 (KV : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)) (_ : IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2),\n            \ud835\udcdf {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd},\n          \u2200 (x : \u03b1), x \u2208 a \u2192 (\u2191a_3.snd x, \u2191a_3.fst x) \u2208 b\n[PROOFSTEP]\nintro K V hK hV\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2200\u1da0 (a : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) in\n    \u2a05 (KV : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)) (_ : IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2),\n      \ud835\udcdf {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd},\n    \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a.snd x, \u2191a.fst x) \u2208 V\n[PROOFSTEP]\nobtain \u27e8V', hV', hsymm, hsub\u27e9 := symm_of_uniformity hV\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhsymm : \u2200 (a b : \u03b2), (a, b) \u2208 V' \u2192 (b, a) \u2208 V'\nhsub : V' \u2286 V\n\u22a2 \u2200\u1da0 (a : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) in\n    \u2a05 (KV : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)) (_ : IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2),\n      \ud835\udcdf {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd},\n    \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a.snd x, \u2191a.fst x) \u2208 V\n[PROOFSTEP]\nlet X := {fg : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2) | \u2200 x : \u03b1, x \u2208 K \u2192 (fg.1 x, fg.2 x) \u2208 V'}\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhsymm : \u2200 (a b : \u03b2), (a, b) \u2208 V' \u2192 (b, a) \u2208 V'\nhsub : V' \u2286 V\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\n\u22a2 \u2200\u1da0 (a : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) in\n    \u2a05 (KV : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)) (_ : IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2),\n      \ud835\udcdf {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd},\n    \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a.snd x, \u2191a.fst x) \u2208 V\n[PROOFSTEP]\nhave hX : X \u2208 compactConvergenceUniformity := (mem_compactConvergenceUniformity X).mpr \u27e8K, V', hK, hV', by simp\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhsymm : \u2200 (a b : \u03b2), (a, b) \u2208 V' \u2192 (b, a) \u2208 V'\nhsub : V' \u2286 V\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\n\u22a2 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2286 X\n[PROOFSTEP]\nsimp\n[GOAL]\ncase intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhsymm : \u2200 (a b : \u03b2), (a, b) \u2208 V' \u2192 (b, a) \u2208 V'\nhsub : V' \u2286 V\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\nhX : X \u2208 compactConvergenceUniformity\n\u22a2 \u2200\u1da0 (a : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) in\n    \u2a05 (KV : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)) (_ : IsCompact KV.fst \u2227 KV.snd \u2208 \ud835\udce4 \u03b2),\n      \ud835\udcdf {fg | \u2200 (x : \u03b1), x \u2208 KV.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 KV.snd},\n    \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a.snd x, \u2191a.fst x) \u2208 V\n[PROOFSTEP]\nexact Filter.eventually_of_mem hX fun fg hfg x hx => hsub (hsymm _ _ (hfg x hx))\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX : X \u2208 compactConvergenceUniformity\n\u22a2 X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s\n[PROOFSTEP]\nobtain \u27e8K, V, hK, hV, hX\u27e9 := (mem_compactConvergenceUniformity X).mp hX\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\n\u22a2 X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s\n[PROOFSTEP]\nobtain \u27e8V', hV', hcomp\u27e9 := comp_mem_uniformity_sets hV\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\n\u22a2 X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s\n[PROOFSTEP]\nlet h := fun s : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) => s \u25cb s\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\n\u22a2 X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s\n[PROOFSTEP]\nsuffices h {fg : C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2) | \u2200 x \u2208 K, (fg.1 x, fg.2 x) \u2208 V'} \u2208 compactConvergenceUniformity.lift' h\n  by\n  apply Filter.mem_of_superset this\n  rintro \u27e8f, g\u27e9 \u27e8z, hz\u2081, hz\u2082\u27e9\n  refine' hX fun x hx => hcomp _\n  exact \u27e8z x, hz\u2081 x hx, hz\u2082 x hx\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\nthis : h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 Filter.lift' compactConvergenceUniformity h\n\u22a2 X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s\n[PROOFSTEP]\napply Filter.mem_of_superset this\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\nthis : h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 Filter.lift' compactConvergenceUniformity h\n\u22a2 h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2286 X\n[PROOFSTEP]\nrintro \u27e8f, g\u27e9 \u27e8z, hz\u2081, hz\u2082\u27e9\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\nthis : h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 Filter.lift' compactConvergenceUniformity h\nf g z : C(\u03b1, \u03b2)\nhz\u2081 : ((f, g).fst, z) \u2208 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\nhz\u2082 : (z, (f, g).snd) \u2208 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\n\u22a2 (f, g) \u2208 X\n[PROOFSTEP]\nrefine' hX fun x hx => hcomp _\n[GOAL]\ncase mk.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\nthis : h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 Filter.lift' compactConvergenceUniformity h\nf g z : C(\u03b1, \u03b2)\nhz\u2081 : ((f, g).fst, z) \u2208 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\nhz\u2082 : (z, (f, g).snd) \u2208 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'}\nx : \u03b1\nhx : x \u2208 K\n\u22a2 (\u2191(f, g).fst x, \u2191(f, g).snd x) \u2208 V' \u25cb V'\n[PROOFSTEP]\nexact \u27e8z x, hz\u2081 x hx, hz\u2082 x hx\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\n\u22a2 h {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 Filter.lift' compactConvergenceUniformity h\n[PROOFSTEP]\napply Filter.mem_lift'\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.ht\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nX : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))\nhX\u271d : X \u2208 compactConvergenceUniformity\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nhK : IsCompact K\nhV : V \u2208 \ud835\udce4 \u03b2\nhX : {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V} \u2286 X\nV' : Set (\u03b2 \u00d7 \u03b2)\nhV' : V' \u2208 \ud835\udce4 \u03b2\nhcomp : V' \u25cb V' \u2286 V\nh : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u2192 Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) := fun s => s \u25cb s\n\u22a2 {fg | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 V'} \u2208 compactConvergenceUniformity\n[PROOFSTEP]\nexact (mem_compactConvergenceUniformity _).mpr \u27e8K, V', hK, hV', Subset.refl _\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (s : Set C(\u03b1, \u03b2)),\n    IsOpen s \u2194\n      \u2200 (x : C(\u03b1, \u03b2)),\n        x \u2208 s \u2192\n          {p | p.fst = x \u2192 p.snd \u2208 s} \u2208\n            { uniformity := compactConvergenceUniformity, refl := (_ : \ud835\udcdf idRel \u2264 compactConvergenceUniformity),\n                symm := (_ : Tendsto Prod.swap compactConvergenceUniformity compactConvergenceUniformity),\n                comp :=\n                  (_ :\n                    \u2200 (X : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))),\n                      X \u2208 compactConvergenceUniformity \u2192\n                        X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s) }.uniformity\n[PROOFSTEP]\nrw [compactOpen_eq_compactConvergence]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u22a2 \u2200 (s : Set C(\u03b1, \u03b2)),\n    IsOpen s \u2194\n      \u2200 (x : C(\u03b1, \u03b2)),\n        x \u2208 s \u2192\n          {p | p.fst = x \u2192 p.snd \u2208 s} \u2208\n            { uniformity := compactConvergenceUniformity, refl := (_ : \ud835\udcdf idRel \u2264 compactConvergenceUniformity),\n                symm := (_ : Tendsto Prod.swap compactConvergenceUniformity compactConvergenceUniformity),\n                comp :=\n                  (_ :\n                    \u2200 (X : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))),\n                      X \u2208 compactConvergenceUniformity \u2192\n                        X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s) }.uniformity\n[PROOFSTEP]\nrefine' fun Y => forall\u2082_congr fun f hf => _\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\nhf : f \u2208 Y\n\u22a2 Y \u2208 (fun f => FilterBasis.filter (compactConvergenceFilterBasis f)) f \u2194\n    {p | p.fst = f \u2192 p.snd \u2208 Y} \u2208\n      { uniformity := compactConvergenceUniformity, refl := (_ : \ud835\udcdf idRel \u2264 compactConvergenceUniformity),\n          symm := (_ : Tendsto Prod.swap compactConvergenceUniformity compactConvergenceUniformity),\n          comp :=\n            (_ :\n              \u2200 (X : Set (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2))),\n                X \u2208 compactConvergenceUniformity \u2192\n                  X \u2208 Filter.lift' compactConvergenceUniformity fun s => s \u25cb s) }.uniformity\n[PROOFSTEP]\nsimp only [mem_compactConvergence_nhd_filter, mem_compactConvergenceUniformity, Prod.forall, setOf_subset_setOf,\n  compactConvNhd]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\nhf : f \u2208 Y\n\u22a2 (\u2203 K V h h, {g | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191f x, \u2191g x) \u2208 V} \u2286 Y) \u2194\n    \u2203 K V h h, \u2200 (a b : C(\u03b1, \u03b2)), (\u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a x, \u2191b x) \u2208 V) \u2192 a = f \u2192 b \u2208 Y\n[PROOFSTEP]\nrefine' exists\u2084_congr fun K V _hK _hV => \u27e8_, fun hY g hg => hY f g hg rfl\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf\u271d : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nf : C(\u03b1, \u03b2)\nhf : f \u2208 Y\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\n_hK : IsCompact K\n_hV : V \u2208 \ud835\udce4 \u03b2\n\u22a2 {g | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191f x, \u2191g x) \u2208 V} \u2286 Y \u2192\n    \u2200 (a b : C(\u03b1, \u03b2)), (\u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191a x, \u2191b x) \u2208 V) \u2192 a = f \u2192 b \u2208 Y\n[PROOFSTEP]\nrintro hY g\u2081 g\u2082 hg\u2081 rfl\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\nY : Set C(\u03b1, \u03b2)\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\n_hK : IsCompact K\n_hV : V \u2208 \ud835\udce4 \u03b2\ng\u2081 g\u2082 : C(\u03b1, \u03b2)\nhg\u2081 : \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191g\u2081 x, \u2191g\u2082 x) \u2208 V\nhf : g\u2081 \u2208 Y\nhY : {g | \u2200 (x : \u03b1), x \u2208 K \u2192 (\u2191g\u2081 x, \u2191g x) \u2208 V} \u2286 Y\n\u22a2 g\u2082 \u2208 Y\n[PROOFSTEP]\nexact hY hg\u2081\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\n\u22a2 HasBasis (\ud835\udce4 C(\u03b1, \u03b2)) (fun p => IsCompact p.fst \u2227 pi p.snd) fun p =>\n    {fg | \u2200 (x : \u03b1), x \u2208 p.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s p.snd}\n[PROOFSTEP]\nrefine' hasBasis_compactConvergenceUniformity.to_hasBasis _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\n\u22a2 \u2200 (i : Set \u03b1 \u00d7 Set (\u03b2 \u00d7 \u03b2)),\n    IsCompact i.fst \u2227 i.snd \u2208 \ud835\udce4 \u03b2 \u2192\n      \u2203 i',\n        (IsCompact i'.fst \u2227 pi i'.snd) \u2227\n          {fg | \u2200 (x : \u03b1), x \u2208 i'.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s i'.snd} \u2286\n            {fg | \u2200 (x : \u03b1), x \u2208 i.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 i.snd}\n[PROOFSTEP]\nrintro \u27e8t\u2081, t\u2082\u27e9 \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase refine'_1.mk.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\nt\u2081 : Set \u03b1\nt\u2082 : Set (\u03b2 \u00d7 \u03b2)\nh\u2081 : IsCompact (t\u2081, t\u2082).fst\nh\u2082 : (t\u2081, t\u2082).snd \u2208 \ud835\udce4 \u03b2\n\u22a2 \u2203 i',\n    (IsCompact i'.fst \u2227 pi i'.snd) \u2227\n      {fg | \u2200 (x : \u03b1), x \u2208 i'.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s i'.snd} \u2286\n        {fg | \u2200 (x : \u03b1), x \u2208 (t\u2081, t\u2082).fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 (t\u2081, t\u2082).snd}\n[PROOFSTEP]\nrcases h.mem_iff.1 h\u2082 with \u27e8i, hpi, hi\u27e9\n[GOAL]\ncase refine'_1.mk.intro.intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\nt\u2081 : Set \u03b1\nt\u2082 : Set (\u03b2 \u00d7 \u03b2)\nh\u2081 : IsCompact (t\u2081, t\u2082).fst\nh\u2082 : (t\u2081, t\u2082).snd \u2208 \ud835\udce4 \u03b2\ni : \u03b9\nhpi : pi i\nhi : s i \u2286 (t\u2081, t\u2082).snd\n\u22a2 \u2203 i',\n    (IsCompact i'.fst \u2227 pi i'.snd) \u2227\n      {fg | \u2200 (x : \u03b1), x \u2208 i'.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s i'.snd} \u2286\n        {fg | \u2200 (x : \u03b1), x \u2208 (t\u2081, t\u2082).fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 (t\u2081, t\u2082).snd}\n[PROOFSTEP]\nexact \u27e8(t\u2081, i), \u27e8h\u2081, hpi\u27e9, fun fg hfg x hx => hi (hfg _ hx)\u27e9\n[GOAL]\ncase refine'_2\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\n\u22a2 \u2200 (i' : Set \u03b1 \u00d7 \u03b9),\n    IsCompact i'.fst \u2227 pi i'.snd \u2192\n      \u2203 i,\n        (IsCompact i.fst \u2227 i.snd \u2208 \ud835\udce4 \u03b2) \u2227\n          {fg | \u2200 (x : \u03b1), x \u2208 i.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 i.snd} \u2286\n            {fg | \u2200 (x : \u03b1), x \u2208 i'.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s i'.snd}\n[PROOFSTEP]\nrintro \u27e8t, i\u27e9 \u27e8ht, hi\u27e9\n[GOAL]\ncase refine'_2.mk.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u_1\npi : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set (\u03b2 \u00d7 \u03b2)\nh : HasBasis (\ud835\udce4 \u03b2) pi s\nt : Set \u03b1\ni : \u03b9\nht : IsCompact (t, i).fst\nhi : pi (t, i).snd\n\u22a2 \u2203 i_1,\n    (IsCompact i_1.fst \u2227 i_1.snd \u2208 \ud835\udce4 \u03b2) \u2227\n      {fg | \u2200 (x : \u03b1), x \u2208 i_1.fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 i_1.snd} \u2286\n        {fg | \u2200 (x : \u03b1), x \u2208 (t, i).fst \u2192 (\u2191fg.fst x, \u2191fg.snd x) \u2208 s (t, i).snd}\n[PROOFSTEP]\nexact \u27e8(t, s i), \u27e8ht, h.mem_of_mem hi\u27e9, Subset.rfl\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\n\u22a2 Tendsto F p (\ud835\udcdd f) \u2194 \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n[PROOFSTEP]\nrw [compactOpen_eq_compactConvergence, tendsto_iff_forall_compact_tendstoUniformlyOn']\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh : TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\n\u22a2 Tendsto F p (\ud835\udcdd f)\n[PROOFSTEP]\nrw [tendsto_iff_forall_compact_tendstoUniformlyOn]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh : TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\n\u22a2 \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n[PROOFSTEP]\nintro K hK\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh : TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n[PROOFSTEP]\nrw [\u2190 tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK\u271d : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh : TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\nK : Set \u03b1\nhK : IsCompact K\n\u22a2 TendstoLocallyUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n[PROOFSTEP]\nexact h.tendstoLocallyUniformlyOn\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh\u03b1 : \u2200 (x : \u03b1), \u2203 n, IsCompact n \u2227 n \u2208 \ud835\udcdd x\nh : Tendsto F p (\ud835\udcdd f)\n\u22a2 TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\n[PROOFSTEP]\nrw [tendsto_iff_forall_compact_tendstoUniformlyOn] at h \n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh\u03b1 : \u2200 (x : \u03b1), \u2203 n, IsCompact n \u2227 n \u2208 \ud835\udcdd x\nh : \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\n\u22a2 TendstoLocallyUniformly (fun i a => \u2191(F i) a) (\u2191f) p\n[PROOFSTEP]\nintro V hV x\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh\u03b1 : \u2200 (x : \u03b1), \u2203 n, IsCompact n \u2227 n \u2208 \ud835\udcdd x\nh : \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nx : \u03b1\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 t \u2192 (\u2191f y, (fun i a => \u2191(F i) a) n y) \u2208 V\n[PROOFSTEP]\nobtain \u27e8n, hn\u2081, hn\u2082\u27e9 := h\u03b1 x\n[GOAL]\ncase intro.intro\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : UniformSpace \u03b2\nK : Set \u03b1\nV\u271d : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\nh\u03b1 : \u2200 (x : \u03b1), \u2203 n, IsCompact n \u2227 n \u2208 \ud835\udcdd x\nh : \u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K\nV : Set (\u03b2 \u00d7 \u03b2)\nhV : V \u2208 \ud835\udce4 \u03b2\nx : \u03b1\nn : Set \u03b1\nhn\u2081 : IsCompact n\nhn\u2082 : n \u2208 \ud835\udcdd x\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x \u2227 \u2200\u1da0 (n : \u03b9) in p, \u2200 (y : \u03b1), y \u2208 t \u2192 (\u2191f y, (fun i a => \u2191(F i) a) n y) \u2208 V\n[PROOFSTEP]\nexact \u27e8n, hn\u2082, h n hn\u2081 V hV\u27e9\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\ninst\u271d : CompactSpace \u03b1\n\u22a2 Tendsto F p (\ud835\udcdd f) \u2194 TendstoUniformly (fun i a => \u2191(F i) a) (\u2191f) p\n[PROOFSTEP]\nrw [tendsto_iff_forall_compact_tendstoUniformlyOn, \u2190 tendstoUniformlyOn_univ]\n[GOAL]\n\u03b1 : Type u\u2081\n\u03b2 : Type u\u2082\ninst\u271d\u00b2 : TopologicalSpace \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b2\nK : Set \u03b1\nV : Set (\u03b2 \u00d7 \u03b2)\nf : C(\u03b1, \u03b2)\n\u03b9 : Type u\u2083\np : Filter \u03b9\nF : \u03b9 \u2192 C(\u03b1, \u03b2)\ninst\u271d : CompactSpace \u03b1\n\u22a2 (\u2200 (K : Set \u03b1), IsCompact K \u2192 TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p K) \u2194\n    TendstoUniformlyOn (fun i a => \u2191(F i) a) (\u2191f) p univ\n[PROOFSTEP]\nexact \u27e8fun h => h univ isCompact_univ, fun h K _hK => h.mono (subset_univ K)\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.UniformSpace.CompactConvergence", "llama_tokens": 36908, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.5062980564016972}}
{"text": "[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 predictablePart f \u2131 \u03bc 0 = 0\n[PROOFSTEP]\nsimp_rw [predictablePart, Finset.range_zero, Finset.sum_empty]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 martingalePart f \u2131 \u03bc = fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])\n[PROOFSTEP]\nunfold martingalePart predictablePart\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\n\u22a2 (fun n => f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i]) = fun n =>\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])\n[PROOFSTEP]\next1 n\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d n : \u2115\n\u22a2 f n - \u2211 i in Finset.range n, \u03bc[f (i + 1) - f i|\u2191\u2131 i] =\n    f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])\n[PROOFSTEP]\nrw [Finset.eq_sum_range_sub f n, \u2190 add_sub, \u2190 Finset.sum_sub_distrib]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\nhf_int : \u2200 (n : \u2115), Integrable (f n)\nn : \u2115\n\u22a2 Integrable (martingalePart f \u2131 \u03bc n)\n[PROOFSTEP]\nrw [martingalePart_eq_sum]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\nhf_int : \u2200 (n : \u2115), Integrable (f n)\nn : \u2115\n\u22a2 Integrable ((fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) n)\n[PROOFSTEP]\nexact (hf_int 0).add (integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\n\u22a2 Martingale (martingalePart f \u2131 \u03bc) \u2131 \u03bc\n[PROOFSTEP]\nrefine'\n  \u27e8adapted_martingalePart hf, fun i j hij => _\u27e9\n    -- \u22a2 \u03bc[martingalePart f \u2131 \u03bc j | \u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[PROOFSTEP]\nhave h_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i]) :=\n  by\n  rw [martingalePart_eq_sum]\n  refine' (condexp_add (hf_int 0) _).trans _\n  \u00b7 exact integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp\n  refine' (EventuallyEq.add EventuallyEq.rfl (condexp_finset_sum fun i _ => _)).trans _\n  \u00b7 exact ((hf_int _).sub (hf_int _)).sub integrable_condexp\n  refine' EventuallyEq.add _ _\n  \u00b7 rw [condexp_of_stronglyMeasurable (\u2131.le _) _ (hf_int 0)]\n    \u00b7 exact (hf 0).mono (\u2131.mono (zero_le i))\n  \u00b7 exact eventuallyEq_sum fun k _ => condexp_sub ((hf_int _).sub (hf_int _)) integrable_condexp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n[PROOFSTEP]\nrw [martingalePart_eq_sum]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[(fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n[PROOFSTEP]\nrefine' (condexp_add (hf_int 0) _).trans _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 Integrable (\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]))\n[PROOFSTEP]\nexact integrable_finset_sum' _ fun i _ => ((hf_int _).sub (hf_int _)).sub integrable_condexp\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 \u03bc[f 0|\u2191\u2131 i] + \u03bc[\u2211 i in Finset.range j, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n[PROOFSTEP]\nrefine' (EventuallyEq.add EventuallyEq.rfl (condexp_finset_sum fun i _ => _)).trans _\n[GOAL]\ncase refine'_2.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni\u271d j : \u2115\nhij : i\u271d \u2264 j\ni : \u2115\nx\u271d : i \u2208 Finset.range j\n\u22a2 Integrable (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])\n[PROOFSTEP]\nexact ((hf_int _).sub (hf_int _)).sub integrable_condexp\n[GOAL]\ncase refine'_2.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      (\u03bc[f 0|\u2191\u2131 i]) x +\n        Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n[PROOFSTEP]\nrefine' EventuallyEq.add _ _\n[GOAL]\ncase refine'_2.refine'_2.refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x => (\u03bc[f 0|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => f 0 x\n[PROOFSTEP]\nrw [condexp_of_stronglyMeasurable (\u2131.le _) _ (hf_int 0)]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 StronglyMeasurable (f 0)\n[PROOFSTEP]\nexact (hf 0).mono (\u2131.mono (zero_le i))\n[GOAL]\ncase refine'_2.refine'_2.refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\n\u22a2 (fun x =>\n      Finset.sum (Finset.range j) (fun i_1 => \u03bc[f (i_1 + 1) - f i_1 - \u03bc[f (i_1 + 1) - f i_1|\u2191\u2131 i_1]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x\n[PROOFSTEP]\nexact eventuallyEq_sum fun k _ => condexp_sub ((hf_int _).sub (hf_int _)) integrable_condexp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[PROOFSTEP]\nrefine' h_eq_sum.trans _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[PROOFSTEP]\nhave h_ge : \u2200 k, i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] 0 :=\n  by\n  intro k hk\n  have : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] := condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k)\n  filter_upwards [this] with x hx\n  rw [Pi.sub_apply, Pi.zero_apply, hx, sub_self]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\n\u22a2 \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n[PROOFSTEP]\nintro k hk\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n[PROOFSTEP]\nhave : \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2131 i] := condexp_condexp_of_le (\u2131.mono hk) (\u2131.le k)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\n\u22a2 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n[PROOFSTEP]\nfilter_upwards [this] with x hx\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nk : \u2115\nhk : i \u2264 k\nthis : \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] \u03bc[f (k + 1) - f k|\u2191\u2131 i]\nx : \u03a9\nhx : (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x\n\u22a2 (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [Pi.sub_apply, Pi.zero_apply, hx, sub_self]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[PROOFSTEP]\nhave h_lt :\n  \u2200 k, k < i \u2192 \u03bc[f (k + 1) - f k|\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2131 k]|\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2131 k] :=\n  by\n  refine' fun k hk => EventuallyEq.sub _ _\n  \u00b7 rw [condexp_of_stronglyMeasurable]\n    \u00b7 exact ((hf (k + 1)).mono (\u2131.mono (Nat.succ_le_of_lt hk))).sub ((hf k).mono (\u2131.mono hk.le))\n    \u00b7 exact (hf_int _).sub (hf_int _)\n  \u00b7 rw [condexp_of_stronglyMeasurable]\n    \u00b7 exact stronglyMeasurable_condexp.mono (\u2131.mono hk.le)\n    \u00b7 exact integrable_condexp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\n\u22a2 \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n[PROOFSTEP]\nrefine' fun k hk => EventuallyEq.sub _ _\n[GOAL]\ncase refine'_1\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (f (k + 1) - f k) x\n[PROOFSTEP]\nrw [condexp_of_stronglyMeasurable]\n[GOAL]\ncase refine'_1.hf\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (f (k + 1) - f k)\n[PROOFSTEP]\nexact ((hf (k + 1)).mono (\u2131.mono (Nat.succ_le_of_lt hk))).sub ((hf k).mono (\u2131.mono hk.le))\n[GOAL]\ncase refine'_1.hfi\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (f (k + 1) - f k)\n[PROOFSTEP]\nexact (hf_int _).sub (hf_int _)\n[GOAL]\ncase refine'_2\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 (fun x => (\u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc] fun x => (\u03bc[f (k + 1) - f k|\u2191\u2131 k]) x\n[PROOFSTEP]\nrw [condexp_of_stronglyMeasurable]\n[GOAL]\ncase refine'_2.hf\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 StronglyMeasurable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])\n[PROOFSTEP]\nexact stronglyMeasurable_condexp.mono (\u2131.mono hk.le)\n[GOAL]\ncase refine'_2.hfi\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nk : \u2115\nhk : k < i\n\u22a2 Integrable (\u03bc[f (k + 1) - f k|\u2191\u2131 k])\n[PROOFSTEP]\nexact integrable_condexp\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc] martingalePart f \u2131 \u03bc i\n[PROOFSTEP]\nrw [martingalePart_eq_sum]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) =\u1d50[\u03bc]\n    (fun n => f 0 + \u2211 i in Finset.range n, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i])) i\n[PROOFSTEP]\nrefine' EventuallyEq.add EventuallyEq.rfl _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.range j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => Finset.sum (Finset.range i) (fun i => f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) x\n[PROOFSTEP]\nrw [\u2190 Finset.sum_range_add_sum_Ico _ hij, \u2190\n  add_zero (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2131 i]))]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x =>\n      (\u2211 k in Finset.range i, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) +\n          \u2211 k in Finset.Ico i j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]))\n        x) =\u1d50[\u03bc]\n    fun x => (\u2211 i in Finset.range i, (f (i + 1) - f i - \u03bc[f (i + 1) - f i|\u2191\u2131 i]) + 0) x\n[PROOFSTEP]\nrefine' (eventuallyEq_sum fun k hk => h_lt k (Finset.mem_range.mp hk)).add _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 (fun x => Finset.sum (Finset.Ico i j) (fun k => \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i]) x) =\u1d50[\u03bc]\n    fun x => OfNat.ofNat 0 x\n[PROOFSTEP]\nrefine' (eventuallyEq_sum fun k hk => h_ge k (Finset.mem_Ico.mp hk).1).trans _\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 \u2211 i in Finset.Ico i j, 0 =\u1d50[\u03bc] fun x => OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp only [Finset.sum_const_zero, Pi.zero_apply]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn : \u2115\nhf : Adapted \u2131 f\nhf_int : \u2200 (n : \u2115), Integrable (f n)\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\ni j : \u2115\nhij : i \u2264 j\nh_eq_sum :\n  \u03bc[martingalePart f \u2131 \u03bc j|\u2191\u2131 i] =\u1d50[\u03bc]\n    f 0 + \u2211 k in Finset.range j, (\u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i])\nh_ge : \u2200 (k : \u2115), i \u2264 k \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] 0\nh_lt :\n  \u2200 (k : \u2115),\n    k < i \u2192 \u03bc[f (k + 1) - f k|\u2191\u2131 i] - \u03bc[\u03bc[f (k + 1) - f k|\u2191\u2131 k]|\u2191\u2131 i] =\u1d50[\u03bc] f (k + 1) - f k - \u03bc[f (k + 1) - f k|\u2191\u2131 k]\n\u22a2 0 =\u1d50[\u03bc] fun x => 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n\n[PROOFSTEP]\nset h := f - martingalePart (f + g) \u2131 \u03bc with hhdef\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n\n[PROOFSTEP]\nhave hh : h = predictablePart (f + g) \u2131 \u03bc - g := by\n  rw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc), martingalePart_add_predictablePart]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\n\u22a2 h = predictablePart (f + g) \u2131 \u03bc - g\n[PROOFSTEP]\nrw [hhdef, sub_eq_sub_iff_add_eq_add, add_comm (predictablePart (f + g) \u2131 \u03bc), martingalePart_add_predictablePart]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n\n[PROOFSTEP]\nhave hhpred : Adapted \u2131 fun n => h (n + 1) := by\n  rw [hh]\n  exact adapted_predictablePart.sub hg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 Adapted \u2131 fun n => h (n + 1)\n[PROOFSTEP]\nrw [hh]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\n\u22a2 Adapted \u2131 fun n => (predictablePart (f + g) \u2131 \u03bc - g) (n + 1)\n[PROOFSTEP]\nexact adapted_predictablePart.sub hg\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n\n[PROOFSTEP]\nhave hhmgle : Martingale h \u2131 \u03bc :=\n  hf.sub\n    (martingale_martingalePart (hf.adapted.add <| Predictable.adapted hg <| hg0.symm \u25b8 stronglyMeasurable_zero) fun n =>\n      (hf.integrable n).add <| hgint n)\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] f n\n[PROOFSTEP]\nrefine' (eventuallyEq_iff_sub.2 _).symm\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u22a2 f n - martingalePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] 0\n[PROOFSTEP]\nfilter_upwards [hhmgle.eq_zero_of_predictable hhpred n] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f - martingalePart (f + g) \u2131 \u03bc) n \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9\n[PROOFSTEP]\nrw [Pi.sub_apply] at h\u03c9 \n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = OfNat.ofNat 0 \u03c9\n[PROOFSTEP]\nrw [h\u03c9, Pi.sub_apply, martingalePart]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\nh : \u2115 \u2192 \u03a9 \u2192 E := f - martingalePart (f + g) \u2131 \u03bc\nhhdef : h = f - martingalePart (f + g) \u2131 \u03bc\nhh : h = predictablePart (f + g) \u2131 \u03bc - g\nhhpred : Adapted \u2131 fun n => h (n + 1)\nhhmgle : Martingale h \u2131 \u03bc\n\u03c9 : \u03a9\nh\u03c9 : (f n - martingalePart (f + g) \u2131 \u03bc n) \u03c9 = (f - martingalePart (f + g) \u2131 \u03bc) 0 \u03c9\n\u22a2 (f 0 - ((f + g) 0 - predictablePart (f + g) \u2131 \u03bc 0)) \u03c9 = OfNat.ofNat 0 \u03c9\n[PROOFSTEP]\nsimp [hg0]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u22a2 predictablePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] g n\n[PROOFSTEP]\nfilter_upwards [martingalePart_add_ae_eq hf hg hg0 hgint n] with \u03c9 h\u03c9\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 predictablePart (f + g) \u2131 \u03bc n \u03c9 = g n \u03c9\n[PROOFSTEP]\nrw [\u2190 add_right_inj (f n \u03c9)]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 + g n \u03c9\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n| f n \u03c9 + g n \u03c9\n[PROOFSTEP]\nrw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n| f n \u03c9 + g n \u03c9\n[PROOFSTEP]\nrw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n| f n \u03c9 + g n \u03c9\n[PROOFSTEP]\nrw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u2115 m0\nn\u271d : \u2115\ninst\u271d : SigmaFiniteFiltration \u03bc \u2131\nf g : \u2115 \u2192 \u03a9 \u2192 E\nhf : Martingale f \u2131 \u03bc\nhg : Adapted \u2131 fun n => g (n + 1)\nhg0 : g 0 = 0\nhgint : \u2200 (n : \u2115), Integrable (g n)\nn : \u2115\n\u03c9 : \u03a9\nh\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9\n\u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = (martingalePart (f + g) \u2131 \u03bc + predictablePart (f + g) \u2131 \u03bc) n \u03c9\n[PROOFSTEP]\nrw [Pi.add_apply, Pi.add_apply, h\u03c9]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nsimp_rw [predictablePart, Finset.sum_apply, Finset.sum_range_succ_sub_sum]\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |(\u03bc[f (i + 1) - f i|\u2191\u2131 i]) \u03c9| \u2264 \u2191R\n[PROOFSTEP]\nexact ae_all_iff.2 fun i => ae_bdd_condexp_of_ae_bdd <| ae_all_iff.1 hbdd i\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u22a2 \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R)\n[PROOFSTEP]\nfilter_upwards [hbdd, predictablePart_bdd_difference \u2131 hbdd] with \u03c9 h\u03c9\u2081 h\u03c9\u2082 i\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |martingalePart f \u2131 \u03bc (i + 1) \u03c9 - martingalePart f \u2131 \u03bc i \u03c9| \u2264 \u2191(2 * R)\n[PROOFSTEP]\nsimp only [two_mul, martingalePart, Pi.sub_apply]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)\n[PROOFSTEP]\nhave :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| :=\n  by\n  ring_nf\n    -- `ring` suggests `ring_nf` despite proving the goal\n[GOAL]\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n[PROOFSTEP]\nring_nf\n  -- `ring` suggests `ring_nf` despite proving the goal\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase h\n\u03a9 : Type u_1\nE : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : CompleteSpace E\nf\u271d : \u2115 \u2192 \u03a9 \u2192 E\n\u2131\u271d : Filtration \u2115 m0\nn : \u2115\nR : \u211d\u22650\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u2131 : Filtration \u2115 m0\nhbdd : \u2200\u1d50 (\u03c9 : \u03a9) \u2202\u03bc, \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\n\u03c9 : \u03a9\nh\u03c9\u2081 : \u2200 (i : \u2115), |f (i + 1) \u03c9 - f i \u03c9| \u2264 \u2191R\nh\u03c9\u2082 : \u2200 (i : \u2115), |predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9| \u2264 \u2191R\ni : \u2115\nthis :\n  |f (i + 1) \u03c9 - predictablePart f \u2131 \u03bc (i + 1) \u03c9 - (f i \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| =\n    |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)|\n\u22a2 |f (i + 1) \u03c9 - f i \u03c9 - (predictablePart f \u2131 \u03bc (i + 1) \u03c9 - predictablePart f \u2131 \u03bc i \u03c9)| \u2264 \u2191(R + R)\n[PROOFSTEP]\nexact (abs_sub _ _).trans (add_le_add (h\u03c9\u2081 i) (h\u03c9\u2082 i))\n", "meta": {"mathlib_filename": "Mathlib.Probability.Martingale.Centering", "llama_tokens": 24799, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.808067208930584, "lm_q2_score": 0.6261241632752915, "lm_q1q2_score": 0.5059504050618621}}
{"text": "[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\n\u22a2 2 = 0\n[PROOFSTEP]\nrw [\u2190 Nat.cast_two, CharP.cast_eq_zero]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\nx : R\n\u22a2 x + x = 0\n[PROOFSTEP]\nrw [\u2190 two_smul R x, two_eq_zero, zero_smul]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\n\u22a2 bit0 = 0\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\nx\u271d : R\n\u22a2 bit0 x\u271d = OfNat.ofNat 0 x\u271d\n[PROOFSTEP]\nexact add_self_eq_zero _\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\nx : R\n\u22a2 bit0 x = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\n\u22a2 bit1 = 1\n[PROOFSTEP]\nfunext\n[GOAL]\ncase h\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\nx\u271d : R\n\u22a2 bit1 x\u271d = OfNat.ofNat 1 x\u271d\n[PROOFSTEP]\nsimp [bit1]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Semiring R\ninst\u271d : CharP R 2\nx : R\n\u22a2 bit1 x = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R 2\nx : R\n\u22a2 -x = x\n[PROOFSTEP]\nrw [neg_eq_iff_add_eq_zero, \u2190 two_smul R x, two_eq_zero, zero_smul]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : CharP R 2\nx y : R\n\u22a2 x - y = x + y\n[PROOFSTEP]\nrw [sub_eq_add_neg, neg_eq]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CharP R 2\nx y : R\n\u22a2 (x + y) * (x + y) = x * x + y * y\n[PROOFSTEP]\nrw [\u2190 pow_two, \u2190 pow_two, \u2190 pow_two, add_sq]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CharP R 2\nl : List R\n\u22a2 List.sum l * List.sum l = List.sum (List.map (fun x => x * x) l)\n[PROOFSTEP]\nsimp_rw [\u2190 pow_two, list_sum_sq]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CharP R 2\nl : Multiset R\n\u22a2 Multiset.sum l * Multiset.sum l = Multiset.sum (Multiset.map (fun x => x * x) l)\n[PROOFSTEP]\nsimp_rw [\u2190 pow_two, multiset_sum_sq]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CharP R 2\ns : Finset \u03b9\nf : \u03b9 \u2192 R\n\u22a2 (\u2211 i in s, f i) * \u2211 i in s, f i = \u2211 i in s, f i * f i\n[PROOFSTEP]\nsimp_rw [\u2190 pow_two, sum_sq]\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\n\u22a2 -1 = 1 \u2194 ringChar R = 2\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => @CharTwo.neg_eq _ _ (ringChar.of_eq h) 1\u27e9\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : -1 = 1\n\u22a2 ringChar R = 2\n[PROOFSTEP]\nrw [eq_comm, \u2190 sub_eq_zero, sub_neg_eq_add, \u2190 Nat.cast_one, \u2190 Nat.cast_add] at h \n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh\u271d : 1 = -1\nh : \u2191(1 + 1) = 0\n\u22a2 ringChar R = 2\n[PROOFSTEP]\nexact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one\n[GOAL]\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\n\u22a2 orderOf (-1) = if ringChar R = 2 then 1 else 2\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : ringChar R = 2\n\u22a2 orderOf (-1) = 1\n[PROOFSTEP]\nrw [neg_one_eq_one_iff.2 h, orderOf_one]\n[GOAL]\ncase neg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 orderOf (-1) = 2\n[PROOFSTEP]\napply orderOf_eq_prime\n[GOAL]\ncase neg.hg\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 (-1) ^ 2 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.hg1\nR : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : Nontrivial R\nh : \u00acringChar R = 2\n\u22a2 -1 \u2260 1\n[PROOFSTEP]\nsimpa [neg_one_eq_one_iff] using h\n", "meta": {"mathlib_filename": "Mathlib.Algebra.CharP.Two", "llama_tokens": 1857, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.6442251133170356, "lm_q1q2_score": 0.5059155095112564}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nh' : Submodule.IsPrincipal (maximalIdeal R)\nI : Ideal R\nhI : I \u2260 \u22a5\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nclassical\nobtain \u27e8x, hx : _ = Ideal.span _\u27e9 := h'\nby_cases hI' : I = \u22a4\n\u00b7 use 0; rw [pow_zero, hI', Ideal.one_eq_top]\nhave H : \u2200 r : R, \u00acIsUnit r \u2194 x \u2223 r := fun r => (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton\nhave : x \u2260 0 := by\n  rintro rfl\n  apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n  simp [hx]\nhave hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx\nhave H' : \u2200 r : R, r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n : \u2115, Associated (x ^ n) r :=\n  by\n  intro r hr\u2081 hr\u2082\n  obtain \u27e8f, hf\u2081, rfl, hf\u2082\u27e9 := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr\u2081).mp hr\u2082\n  have : \u2200 b \u2208 f, Associated x b := by\n    intro b hb\n    exact Irreducible.associated_of_dvd hx' (hf\u2081 b hb) ((H b).mp (hf\u2081 b hb).1)\n  clear hr\u2081 hr\u2082 hf\u2081\n  induction' f using Multiset.induction with fa fs fh\n  \u00b7 exact (hf\u2082 rfl).elim\n  rcases eq_or_ne fs \u2205 with (rfl | hf')\n  \u00b7 use 1\n    rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]\n    exact this _ (Multiset.mem_cons_self _ _)\n  \u00b7 obtain \u27e8n, hn\u27e9 := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)\n    use n + 1\n    rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]\n    exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn\nhave : \u2203 n : \u2115, x ^ n \u2208 I :=\n  by\n  obtain \u27e8r, hr\u2081, hr\u2082\u27e9 : \u2203 r : R, r \u2208 I \u2227 r \u2260 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h\n  obtain \u27e8n, u, rfl\u27e9 := H' r hr\u2082 (le_maximalIdeal hI' hr\u2081)\n  use n\n  rwa [\u2190 I.unit_mul_mem_iff_mem u.isUnit, mul_comm]\nuse Nat.find this\napply le_antisymm\n\u00b7 change \u2200 s \u2208 I, s \u2208 _\n  by_contra hI''\n  push_neg at hI'' \n  obtain \u27e8s, hs\u2081, hs\u2082\u27e9 := hI''\n  apply hs\u2082\n  by_cases hs\u2083 : s = 0; \u00b7 rw [hs\u2083]; exact zero_mem _\n  obtain \u27e8n, u, rfl\u27e9 := H' s hs\u2083 (le_maximalIdeal hI' hs\u2081)\n  rw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs\u2081 \u22a2\n  apply Ideal.pow_le_pow (Nat.find_min' this hs\u2081)\n  apply Ideal.pow_mem_pow\n  exact (H _).mpr (dvd_refl _)\n\u00b7 rw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]\n  exact Nat.find_spec this\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nh' : Submodule.IsPrincipal (maximalIdeal R)\nI : Ideal R\nhI : I \u2260 \u22a5\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nobtain \u27e8x, hx : _ = Ideal.span _\u27e9 := h'\n[GOAL]\ncase mk.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nby_cases hI' : I = \u22a4\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : I = \u22a4\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : I = \u22a4\n\u22a2 I = maximalIdeal R ^ 0\n[PROOFSTEP]\nrw [pow_zero, hI', Ideal.one_eq_top]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nhave H : \u2200 r : R, \u00acIsUnit r \u2194 x \u2223 r := fun r => (SetLike.ext_iff.mp hx r).trans Ideal.mem_span_singleton\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nhave : x \u2260 0 := by\n  rintro rfl\n  apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n  simp [hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\n\u22a2 x \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nhI' : \u00acI = \u22a4\nhx : maximalIdeal R = Ideal.span {0}\nH : \u2200 (r : R), \u00acIsUnit r \u2194 0 \u2223 r\n\u22a2 False\n[PROOFSTEP]\napply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nhI' : \u00acI = \u22a4\nhx : maximalIdeal R = Ideal.span {0}\nH : \u2200 (r : R), \u00acIsUnit r \u2194 0 \u2223 r\n\u22a2 maximalIdeal R = \u22a5\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nhave hx' := DiscreteValuationRing.irreducible_of_span_eq_maximalIdeal x this hx\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nhave H' : \u2200 r : R, r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n : \u2115, Associated (x ^ n) r :=\n  by\n  intro r hr\u2081 hr\u2082\n  obtain \u27e8f, hf\u2081, rfl, hf\u2082\u27e9 := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr\u2081).mp hr\u2082\n  have : \u2200 b \u2208 f, Associated x b := by\n    intro b hb\n    exact Irreducible.associated_of_dvd hx' (hf\u2081 b hb) ((H b).mp (hf\u2081 b hb).1)\n  clear hr\u2081 hr\u2082 hf\u2081\n  induction' f using Multiset.induction with fa fs fh\n  \u00b7 exact (hf\u2082 rfl).elim\n  rcases eq_or_ne fs \u2205 with (rfl | hf')\n  \u00b7 use 1\n    rw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]\n    exact this _ (Multiset.mem_cons_self _ _)\n  \u00b7 obtain \u27e8n, hn\u27e9 := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)\n    use n + 1\n    rw [pow_add, Multiset.prod_cons, mul_comm, pow_one]\n    exact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\n\u22a2 \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\n[PROOFSTEP]\nintro r hr\u2081 hr\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nr : R\nhr\u2081 : r \u2260 0\nhr\u2082 : r \u2208 nonunits R\n\u22a2 \u2203 n, Associated (x ^ n) r\n[PROOFSTEP]\nobtain \u27e8f, hf\u2081, rfl, hf\u2082\u27e9 := (WfDvdMonoid.not_unit_iff_exists_factors_eq r hr\u2081).mp hr\u2082\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2081 : \u2200 (b : R), b \u2208 f \u2192 Irreducible b\nhf\u2082 : f \u2260 \u2205\nhr\u2081 : Multiset.prod f \u2260 0\nhr\u2082 : Multiset.prod f \u2208 nonunits R\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod f)\n[PROOFSTEP]\nhave : \u2200 b \u2208 f, Associated x b := by\n  intro b hb\n  exact Irreducible.associated_of_dvd hx' (hf\u2081 b hb) ((H b).mp (hf\u2081 b hb).1)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2081 : \u2200 (b : R), b \u2208 f \u2192 Irreducible b\nhf\u2082 : f \u2260 \u2205\nhr\u2081 : Multiset.prod f \u2260 0\nhr\u2082 : Multiset.prod f \u2208 nonunits R\n\u22a2 \u2200 (b : R), b \u2208 f \u2192 Associated x b\n[PROOFSTEP]\nintro b hb\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2081 : \u2200 (b : R), b \u2208 f \u2192 Irreducible b\nhf\u2082 : f \u2260 \u2205\nhr\u2081 : Multiset.prod f \u2260 0\nhr\u2082 : Multiset.prod f \u2208 nonunits R\nb : R\nhb : b \u2208 f\n\u22a2 Associated x b\n[PROOFSTEP]\nexact Irreducible.associated_of_dvd hx' (hf\u2081 b hb) ((H b).mp (hf\u2081 b hb).1)\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2081 : \u2200 (b : R), b \u2208 f \u2192 Irreducible b\nhf\u2082 : f \u2260 \u2205\nhr\u2081 : Multiset.prod f \u2260 0\nhr\u2082 : Multiset.prod f \u2208 nonunits R\nthis : \u2200 (b : R), b \u2208 f \u2192 Associated x b\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod f)\n[PROOFSTEP]\nclear hr\u2081 hr\u2082 hf\u2081\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082 : f \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 f \u2192 Associated x b\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod f)\n[PROOFSTEP]\ninduction' f using Multiset.induction with fa fs fh\n[GOAL]\ncase intro.intro.intro.empty\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nhf\u2082 : 0 \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 0 \u2192 Associated x b\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod 0)\n[PROOFSTEP]\nexact (hf\u2082 rfl).elim\n[GOAL]\ncase intro.intro.intro.cons\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfs : Multiset R\nfh : fs \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 fs \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod fs)\nhf\u2082 : fa ::\u2098 fs \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 fs \u2192 Associated x b\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod (fa ::\u2098 fs))\n[PROOFSTEP]\nrcases eq_or_ne fs \u2205 with (rfl | hf')\n[GOAL]\ncase intro.intro.intro.cons.inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfh : \u2205 \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 \u2205 \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod \u2205)\nhf\u2082 : fa ::\u2098 \u2205 \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 \u2205 \u2192 Associated x b\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod (fa ::\u2098 \u2205))\n[PROOFSTEP]\nuse 1\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfh : \u2205 \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 \u2205 \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod \u2205)\nhf\u2082 : fa ::\u2098 \u2205 \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 \u2205 \u2192 Associated x b\n\u22a2 Associated (x ^ 1) (Multiset.prod (fa ::\u2098 \u2205))\n[PROOFSTEP]\nrw [pow_one, Multiset.prod_cons, Multiset.empty_eq_zero, Multiset.prod_zero, mul_one]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfh : \u2205 \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 \u2205 \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod \u2205)\nhf\u2082 : fa ::\u2098 \u2205 \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 \u2205 \u2192 Associated x b\n\u22a2 Associated x fa\n[PROOFSTEP]\nexact this _ (Multiset.mem_cons_self _ _)\n[GOAL]\ncase intro.intro.intro.cons.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfs : Multiset R\nfh : fs \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 fs \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod fs)\nhf\u2082 : fa ::\u2098 fs \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 fs \u2192 Associated x b\nhf' : fs \u2260 \u2205\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod (fa ::\u2098 fs))\n[PROOFSTEP]\nobtain \u27e8n, hn\u27e9 := fh hf' fun b hb => this _ (Multiset.mem_cons_of_mem hb)\n[GOAL]\ncase intro.intro.intro.cons.inr.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfs : Multiset R\nfh : fs \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 fs \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod fs)\nhf\u2082 : fa ::\u2098 fs \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 fs \u2192 Associated x b\nhf' : fs \u2260 \u2205\nn : \u2115\nhn : Associated (x ^ n) (Multiset.prod fs)\n\u22a2 \u2203 n, Associated (x ^ n) (Multiset.prod (fa ::\u2098 fs))\n[PROOFSTEP]\nuse n + 1\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfs : Multiset R\nfh : fs \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 fs \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod fs)\nhf\u2082 : fa ::\u2098 fs \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 fs \u2192 Associated x b\nhf' : fs \u2260 \u2205\nn : \u2115\nhn : Associated (x ^ n) (Multiset.prod fs)\n\u22a2 Associated (x ^ (n + 1)) (Multiset.prod (fa ::\u2098 fs))\n[PROOFSTEP]\nrw [pow_add, Multiset.prod_cons, mul_comm, pow_one]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d\u00b9 : x \u2260 0\nhx' : Irreducible x\nf : Multiset R\nhf\u2082\u271d : f \u2260 \u2205\nthis\u271d : \u2200 (b : R), b \u2208 f \u2192 Associated x b\nfa : R\nfs : Multiset R\nfh : fs \u2260 \u2205 \u2192 (\u2200 (b : R), b \u2208 fs \u2192 Associated x b) \u2192 \u2203 n, Associated (x ^ n) (Multiset.prod fs)\nhf\u2082 : fa ::\u2098 fs \u2260 \u2205\nthis : \u2200 (b : R), b \u2208 fa ::\u2098 fs \u2192 Associated x b\nhf' : fs \u2260 \u2205\nn : \u2115\nhn : Associated (x ^ n) (Multiset.prod fs)\n\u22a2 Associated (x * x ^ n) (fa * Multiset.prod fs)\n[PROOFSTEP]\nexact Associated.mul_mul (this _ (Multiset.mem_cons_self _ _)) hn\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nhave : \u2203 n : \u2115, x ^ n \u2208 I :=\n  by\n  obtain \u27e8r, hr\u2081, hr\u2082\u27e9 : \u2203 r : R, r \u2208 I \u2227 r \u2260 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h\n  obtain \u27e8n, u, rfl\u27e9 := H' r hr\u2082 (le_maximalIdeal hI' hr\u2081)\n  use n\n  rwa [\u2190 I.unit_mul_mem_iff_mem u.isUnit, mul_comm]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\n\u22a2 \u2203 n, x ^ n \u2208 I\n[PROOFSTEP]\nobtain \u27e8r, hr\u2081, hr\u2082\u27e9 : \u2203 r : R, r \u2208 I \u2227 r \u2260 0 := by by_contra' h; apply hI; rw [eq_bot_iff]; exact h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\n\u22a2 \u2203 r, r \u2208 I \u2227 r \u2260 0\n[PROOFSTEP]\nby_contra' h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh\u271d : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nh : \u2200 (r : R), r \u2208 I \u2192 r = 0\n\u22a2 False\n[PROOFSTEP]\napply hI\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh\u271d : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nh : \u2200 (r : R), r \u2208 I \u2192 r = 0\n\u22a2 I = \u22a5\n[PROOFSTEP]\nrw [eq_bot_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh\u271d : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nh : \u2200 (r : R), r \u2208 I \u2192 r = 0\n\u22a2 I \u2264 \u22a5\n[PROOFSTEP]\nexact h\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nr : R\nhr\u2081 : r \u2208 I\nhr\u2082 : r \u2260 0\n\u22a2 \u2203 n, x ^ n \u2208 I\n[PROOFSTEP]\nobtain \u27e8n, u, rfl\u27e9 := H' r hr\u2082 (le_maximalIdeal hI' hr\u2081)\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nn : \u2115\nu : R\u02e3\nhr\u2081 : x ^ n * \u2191u \u2208 I\nhr\u2082 : x ^ n * \u2191u \u2260 0\n\u22a2 \u2203 n, x ^ n \u2208 I\n[PROOFSTEP]\nuse n\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nn : \u2115\nu : R\u02e3\nhr\u2081 : x ^ n * \u2191u \u2208 I\nhr\u2082 : x ^ n * \u2191u \u2260 0\n\u22a2 x ^ n \u2208 I\n[PROOFSTEP]\nrwa [\u2190 I.unit_mul_mem_iff_mem u.isUnit, mul_comm]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nuse Nat.find this\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 I = maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 I \u2264 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nchange \u2200 s \u2208 I, s \u2208 _\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 \u2200 (s : R), s \u2208 I \u2192 s \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nby_contra hI''\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nhI'' : \u00ac\u2200 (s : R), s \u2208 I \u2192 s \u2208 maximalIdeal R ^ Nat.find this\n\u22a2 False\n[PROOFSTEP]\npush_neg at hI'' \n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nhI'' : \u2203 s, s \u2208 I \u2227 \u00acs \u2208 maximalIdeal R ^ Nat.find this\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8s, hs\u2081, hs\u2082\u27e9 := hI''\n[GOAL]\ncase h.a.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\ns : R\nhs\u2081 : s \u2208 I\nhs\u2082 : \u00acs \u2208 maximalIdeal R ^ Nat.find this\n\u22a2 False\n[PROOFSTEP]\napply hs\u2082\n[GOAL]\ncase h.a.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\ns : R\nhs\u2081 : s \u2208 I\nhs\u2082 : \u00acs \u2208 maximalIdeal R ^ Nat.find this\n\u22a2 s \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nby_cases hs\u2083 : s = 0\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\ns : R\nhs\u2081 : s \u2208 I\nhs\u2082 : \u00acs \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : s = 0\n\u22a2 s \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nrw [hs\u2083]\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\ns : R\nhs\u2081 : s \u2208 I\nhs\u2082 : \u00acs \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : s = 0\n\u22a2 0 \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nexact zero_mem _\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\ns : R\nhs\u2081 : s \u2208 I\nhs\u2082 : \u00acs \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : \u00acs = 0\n\u22a2 s \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nobtain \u27e8n, u, rfl\u27e9 := H' s hs\u2083 (le_maximalIdeal hI' hs\u2081)\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhs\u2081 : x ^ n * \u2191u \u2208 I\nhs\u2082 : \u00acx ^ n * \u2191u \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : \u00acx ^ n * \u2191u = 0\n\u22a2 x ^ n * \u2191u \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nrw [mul_comm, Ideal.unit_mul_mem_iff_mem _ u.isUnit] at hs\u2081 \u22a2\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhs\u2081 : x ^ n \u2208 I\nhs\u2082 : \u00acx ^ n * \u2191u \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : \u00acx ^ n * \u2191u = 0\n\u22a2 x ^ n \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\napply Ideal.pow_le_pow (Nat.find_min' this hs\u2081)\n[GOAL]\ncase neg.intro.intro.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhs\u2081 : x ^ n \u2208 I\nhs\u2082 : \u00acx ^ n * \u2191u \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : \u00acx ^ n * \u2191u = 0\n\u22a2 x ^ n \u2208 maximalIdeal R ^ n\n[PROOFSTEP]\napply Ideal.pow_mem_pow\n[GOAL]\ncase neg.intro.intro.a.hx\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\nn : \u2115\nu : R\u02e3\nhs\u2081 : x ^ n \u2208 I\nhs\u2082 : \u00acx ^ n * \u2191u \u2208 maximalIdeal R ^ Nat.find this\nhs\u2083 : \u00acx ^ n * \u2191u = 0\n\u22a2 x \u2208 maximalIdeal R\n[PROOFSTEP]\nexact (H _).mpr (dvd_refl _)\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 maximalIdeal R ^ Nat.find this \u2264 I\n[PROOFSTEP]\nrw [hx, Ideal.span_singleton_pow, Ideal.span_le, Set.singleton_subset_iff]\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nI : Ideal R\nhI : I \u2260 \u22a5\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : \u00acI = \u22a4\nH : \u2200 (r : R), \u00acIsUnit r \u2194 x \u2223 r\nthis\u271d : x \u2260 0\nhx' : Irreducible x\nH' : \u2200 (r : R), r \u2260 0 \u2192 r \u2208 nonunits R \u2192 \u2203 n, Associated (x ^ n) r\nthis : \u2203 n, x ^ n \u2208 I\n\u22a2 x ^ Nat.find this \u2208 \u2191I\n[PROOFSTEP]\nexact Nat.find_spec this\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nclassical\nby_cases ne_bot : maximalIdeal R = \u22a5\n\u00b7 rw [ne_bot]; infer_instance\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 : \u2203 a \u2208 maximalIdeal R, a \u2260 (0 : R) := by by_contra' h'; apply ne_bot; rwa [eq_bot_iff]\nhave hle : Ideal.span { a } \u2264 maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]\nhave : (Ideal.span { a }).radical = maximalIdeal R :=\n  by\n  rw [Ideal.radical_eq_sInf]\n  apply le_antisymm\n  \u00b7 exact sInf_le \u27e8hle, inferInstance\u27e9\n  \u00b7 refine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha\u2082 _)).ge\n    rw [\u2190 Ideal.span_singleton_eq_bot, eq_bot_iff, \u2190 e]; exact hI.1\nhave : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span { a } := by rw [\u2190 this]; apply Ideal.exists_radical_pow_le_of_fg;\n  exact IsNoetherian.noetherian _\ncases' hn : Nat.find this with n\n\u00b7 have := Nat.find_spec this\n  rw [hn, pow_zero, Ideal.one_eq_top] at this \n  exact (Ideal.IsMaximal.ne_top inferInstance (eq_top_iff.mpr <| this.trans hle)).elim\nobtain \u27e8b, hb\u2081, hb\u2082\u27e9 : \u2203 b \u2208 maximalIdeal R ^ n, \u00acb \u2208 Ideal.span { a } := by by_contra' h'; rw [Nat.find_eq_iff] at hn ;\n  exact hn.2 n n.lt_succ_self fun x hx => h' x hx\nhave hb\u2083 : \u2200 m \u2208 maximalIdeal R, \u2203 k : R, k * a = b * m :=\n  by\n  intro m hm; rw [\u2190 Ideal.mem_span_singleton']; apply Nat.find_spec this\n  rw [hn, pow_succ']; exact Ideal.mul_mem_mul hb\u2081 hm\nhave hb\u2084 : b \u2260 0 := by rintro rfl; apply hb\u2082; exact zero_mem _\nlet K := FractionRing R\nlet x : K := algebraMap R K b / algebraMap R K a\nlet M := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nhave ha\u2083 : algebraMap R K a \u2260 0 := IsFractionRing.to_map_eq_zero_iff.not.mpr ha\u2082\nby_cases hx : \u2200 y \u2208 M, x * y \u2208 M\n\u00b7 have := isIntegral_of_smul_mem_submodule M ?_ ?_ x hx\n  \u00b7 obtain \u27e8y, e\u27e9 := IsIntegrallyClosed.algebraMap_eq_of_integral this\n    refine' (hb\u2082 (Ideal.mem_span_singleton'.mpr \u27e8y, _\u27e9)).elim\n    apply IsFractionRing.injective R K\n    rw [map_mul, e, div_mul_cancel _ ha\u2083]\n  \u00b7 rw [Submodule.ne_bot_iff]; refine' \u27e8_, \u27e8a, ha\u2081, rfl\u27e9, _\u27e9\n    exact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha\u2082\n  \u00b7 apply Submodule.FG.map; exact IsNoetherian.noetherian _\n\u00b7 have : (M.map (DistribMulAction.toLinearMap R K x)).comap (Algebra.linearMap R K) = \u22a4 :=\n    by\n    by_contra h; apply hx\n    rintro m' \u27e8m, hm, rfl : algebraMap R K m = m'\u27e9\n    obtain \u27e8k, hk\u27e9 := hb\u2083 m hm\n    have hk' : x * algebraMap R K m = algebraMap R K k := by\n      rw [\u2190 mul_div_right_comm, \u2190 map_mul, \u2190 hk, map_mul, mul_div_cancel _ ha\u2083]\n    exact \u27e8k, le_maximalIdeal h \u27e8_, \u27e8_, hm, rfl\u27e9, hk'\u27e9, hk'.symm\u27e9\n  obtain \u27e8y, hy\u2081, hy\u2082\u27e9 : \u2203 y \u2208 maximalIdeal R, b * y = a :=\n    by\n    rw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this \n    obtain \u27e8_, \u27e8y, hy, rfl\u27e9, hy' : x * algebraMap R K y = algebraMap R K 1\u27e9 := this\n    rw [map_one, \u2190 mul_div_right_comm, div_eq_one_iff_eq ha\u2083, \u2190 map_mul] at hy' \n    exact \u27e8y, hy, IsFractionRing.injective R K hy'\u27e9\n  refine' \u27e8\u27e8y, _\u27e9\u27e9\n  apply le_antisymm\n  \u00b7 intro m hm; obtain \u27e8k, hk\u27e9 := hb\u2083 m hm; rw [\u2190 hy\u2082, mul_comm, mul_assoc] at hk \n    rw [\u2190 mul_left_cancel\u2080 hb\u2084 hk, mul_comm]; exact Ideal.mem_span_singleton'.mpr \u27e8_, rfl\u27e9\n  \u00b7 rwa [Submodule.span_le, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nby_cases ne_bot : maximalIdeal R = \u22a5\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : maximalIdeal R = \u22a5\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nrw [ne_bot]\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : maximalIdeal R = \u22a5\n\u22a2 Submodule.IsPrincipal \u22a5\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nobtain \u27e8a, ha\u2081, ha\u2082\u27e9 : \u2203 a \u2208 maximalIdeal R, a \u2260 (0 : R) := by by_contra' h'; apply ne_bot; rwa [eq_bot_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\n\u22a2 \u2203 a, a \u2208 maximalIdeal R \u2227 a \u2260 0\n[PROOFSTEP]\nby_contra' h'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\nh' : \u2200 (a : R), a \u2208 maximalIdeal R \u2192 a = 0\n\u22a2 False\n[PROOFSTEP]\napply ne_bot\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\nh' : \u2200 (a : R), a \u2208 maximalIdeal R \u2192 a = 0\n\u22a2 maximalIdeal R = \u22a5\n[PROOFSTEP]\nrwa [eq_bot_iff]\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave hle : Ideal.span { a } \u2264 maximalIdeal R := by rwa [Ideal.span_le, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\n\u22a2 Ideal.span {a} \u2264 maximalIdeal R\n[PROOFSTEP]\nrwa [Ideal.span_le, Set.singleton_subset_iff]\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave : (Ideal.span { a }).radical = maximalIdeal R :=\n  by\n  rw [Ideal.radical_eq_sInf]\n  apply le_antisymm\n  \u00b7 exact sInf_le \u27e8hle, inferInstance\u27e9\n  \u00b7 refine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha\u2082 _)).ge\n    rw [\u2190 Ideal.span_singleton_eq_bot, eq_bot_iff, \u2190 e]; exact hI.1\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\n\u22a2 Ideal.radical (Ideal.span {a}) = maximalIdeal R\n[PROOFSTEP]\nrw [Ideal.radical_eq_sInf]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\n\u22a2 sInf {J | Ideal.span {a} \u2264 J \u2227 Ideal.IsPrime J} = maximalIdeal R\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\n\u22a2 sInf {J | Ideal.span {a} \u2264 J \u2227 Ideal.IsPrime J} \u2264 maximalIdeal R\n[PROOFSTEP]\nexact sInf_le \u27e8hle, inferInstance\u27e9\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\n\u22a2 maximalIdeal R \u2264 sInf {J | Ideal.span {a} \u2264 J \u2227 Ideal.IsPrime J}\n[PROOFSTEP]\nrefine' le_sInf fun I hI => (eq_maximalIdeal <| hI.2.isMaximal (fun e => ha\u2082 _)).ge\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nI : Ideal R\nhI : I \u2208 {J | Ideal.span {a} \u2264 J \u2227 Ideal.IsPrime J}\ne : I = \u22a5\n\u22a2 a = 0\n[PROOFSTEP]\nrw [\u2190 Ideal.span_singleton_eq_bot, eq_bot_iff, \u2190 e]\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nI : Ideal R\nhI : I \u2208 {J | Ideal.span {a} \u2264 J \u2227 Ideal.IsPrime J}\ne : I = \u22a5\n\u22a2 Ideal.span {a} \u2264 I\n[PROOFSTEP]\nexact hI.1\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis : Ideal.radical (Ideal.span {a}) = maximalIdeal R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span { a } := by rw [\u2190 this]; apply Ideal.exists_radical_pow_le_of_fg;\n  exact IsNoetherian.noetherian _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis : Ideal.radical (Ideal.span {a}) = maximalIdeal R\n\u22a2 \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis : Ideal.radical (Ideal.span {a}) = maximalIdeal R\n\u22a2 \u2203 n, Ideal.radical (Ideal.span {a}) ^ n \u2264 Ideal.span {a}\n[PROOFSTEP]\napply Ideal.exists_radical_pow_le_of_fg\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis : Ideal.radical (Ideal.span {a}) = maximalIdeal R\n\u22a2 Ideal.FG (Ideal.radical (Ideal.span {a}))\n[PROOFSTEP]\nexact IsNoetherian.noetherian _\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\ncases' hn : Nat.find this with n\n[GOAL]\ncase neg.intro.intro.zero\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nhn : Nat.find this = Nat.zero\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave := Nat.find_spec this\n[GOAL]\ncase neg.intro.intro.zero\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nhn : Nat.find this\u271d = Nat.zero\nthis : maximalIdeal R ^ Nat.find this\u271d \u2264 Ideal.span {a}\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nrw [hn, pow_zero, Ideal.one_eq_top] at this \n[GOAL]\ncase neg.intro.intro.zero\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nhn : Nat.find this\u271d = Nat.zero\nthis : \u22a4 \u2264 Ideal.span {a}\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nexact (Ideal.IsMaximal.ne_top inferInstance (eq_top_iff.mpr <| this.trans hle)).elim\n[GOAL]\ncase neg.intro.intro.succ\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nobtain \u27e8b, hb\u2081, hb\u2082\u27e9 : \u2203 b \u2208 maximalIdeal R ^ n, \u00acb \u2208 Ideal.span { a } := by by_contra' h'; rw [Nat.find_eq_iff] at hn ;\n  exact hn.2 n n.lt_succ_self fun x hx => h' x hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\n\u22a2 \u2203 b, b \u2208 maximalIdeal R ^ n \u2227 \u00acb \u2208 Ideal.span {a}\n[PROOFSTEP]\nby_contra' h'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nh' : \u2200 (b : R), b \u2208 maximalIdeal R ^ n \u2192 b \u2208 Ideal.span {a}\n\u22a2 False\n[PROOFSTEP]\nrw [Nat.find_eq_iff] at hn \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn :\n  maximalIdeal R ^ Nat.succ n \u2264 Ideal.span {a} \u2227 \u2200 (n_1 : \u2115), n_1 < Nat.succ n \u2192 \u00acmaximalIdeal R ^ n_1 \u2264 Ideal.span {a}\nh' : \u2200 (b : R), b \u2208 maximalIdeal R ^ n \u2192 b \u2208 Ideal.span {a}\n\u22a2 False\n[PROOFSTEP]\nexact hn.2 n n.lt_succ_self fun x hx => h' x hx\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave hb\u2083 : \u2200 m \u2208 maximalIdeal R, \u2203 k : R, k * a = b * m :=\n  by\n  intro m hm; rw [\u2190 Ideal.mem_span_singleton']; apply Nat.find_spec this\n  rw [hn, pow_succ']; exact Ideal.mul_mem_mul hb\u2081 hm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\n\u22a2 \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\n[PROOFSTEP]\nintro m hm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 \u2203 k, k * a = b * m\n[PROOFSTEP]\nrw [\u2190 Ideal.mem_span_singleton']\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 b * m \u2208 Ideal.span {a}\n[PROOFSTEP]\napply Nat.find_spec this\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 b * m \u2208 maximalIdeal R ^ Nat.find this\n[PROOFSTEP]\nrw [hn, pow_succ']\n[GOAL]\ncase a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 b * m \u2208 maximalIdeal R ^ n * maximalIdeal R\n[PROOFSTEP]\nexact Ideal.mul_mem_mul hb\u2081 hm\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave hb\u2084 : b \u2260 0 := by rintro rfl; apply hb\u2082; exact zero_mem _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\n\u22a2 b \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nhb\u2081 : 0 \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00ac0 \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = 0 * m\n\u22a2 False\n[PROOFSTEP]\napply hb\u2082\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nhb\u2081 : 0 \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00ac0 \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = 0 * m\n\u22a2 0 \u2208 Ideal.span {a}\n[PROOFSTEP]\nexact zero_mem _\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nlet K := FractionRing R\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nlet x : K := algebraMap R K b / algebraMap R K a\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nlet M := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave ha\u2083 : algebraMap R K a \u2260 0 := IsFractionRing.to_map_eq_zero_iff.not.mpr ha\u2082\n[GOAL]\ncase neg.intro.intro.succ.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nby_cases hx : \u2200 y \u2208 M, x * y \u2208 M\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave := isIntegral_of_smul_mem_submodule M ?_ ?_ x hx\n[GOAL]\ncase pos.refine_3\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : IsIntegral R x\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nobtain \u27e8y, e\u27e9 := IsIntegrallyClosed.algebraMap_eq_of_integral this\n[GOAL]\ncase pos.refine_3.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : IsIntegral R x\ny : R\ne : \u2191(algebraMap R (FractionRing R)) y = x\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nrefine' (hb\u2082 (Ideal.mem_span_singleton'.mpr \u27e8y, _\u27e9)).elim\n[GOAL]\ncase pos.refine_3.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : IsIntegral R x\ny : R\ne : \u2191(algebraMap R (FractionRing R)) y = x\n\u22a2 y * a = b\n[PROOFSTEP]\napply IsFractionRing.injective R K\n[GOAL]\ncase pos.refine_3.intro.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : IsIntegral R x\ny : R\ne : \u2191(algebraMap R (FractionRing R)) y = x\n\u22a2 \u2191(algebraMap R K) (y * a) = \u2191(algebraMap R K) b\n[PROOFSTEP]\nrw [map_mul, e, div_mul_cancel _ ha\u2083]\n[GOAL]\ncase pos.refine_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 M \u2260 \u22a5\n[PROOFSTEP]\nrw [Submodule.ne_bot_iff]\n[GOAL]\ncase pos.refine_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 \u2203 x, x \u2208 M \u2227 x \u2260 0\n[PROOFSTEP]\nrefine' \u27e8_, \u27e8a, ha\u2081, rfl\u27e9, _\u27e9\n[GOAL]\ncase pos.refine_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 \u2191(Algebra.linearMap R K) a \u2260 0\n[PROOFSTEP]\nexact (IsFractionRing.to_map_eq_zero_iff (K := K)).not.mpr ha\u2082\n[GOAL]\ncase pos.refine_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 Submodule.FG M\n[PROOFSTEP]\napply Submodule.FG.map\n[GOAL]\ncase pos.refine_2.hs\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 Submodule.FG (maximalIdeal R)\n[PROOFSTEP]\nexact IsNoetherian.noetherian _\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nhave : (M.map (DistribMulAction.toLinearMap R K x)).comap (Algebra.linearMap R K) = \u22a4 :=\n  by\n  by_contra h; apply hx\n  rintro m' \u27e8m, hm, rfl : algebraMap R K m = m'\u27e9\n  obtain \u27e8k, hk\u27e9 := hb\u2083 m hm\n  have hk' : x * algebraMap R K m = algebraMap R K k := by\n    rw [\u2190 mul_div_right_comm, \u2190 map_mul, \u2190 hk, map_mul, mul_div_cancel _ ha\u2083]\n  exact \u27e8k, le_maximalIdeal h \u27e8_, \u27e8_, hm, rfl\u27e9, hk'\u27e9, hk'.symm\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n\u22a2 Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\n[PROOFSTEP]\nby_contra h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\n\u22a2 False\n[PROOFSTEP]\napply hx\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\n\u22a2 \u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\n[PROOFSTEP]\nrintro m' \u27e8m, hm, rfl : algebraMap R K m = m'\u27e9\n[GOAL]\ncase intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\nm : R\nhm : m \u2208 \u2191(maximalIdeal R)\n\u22a2 x * \u2191(algebraMap R K) m \u2208 M\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hb\u2083 m hm\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\nm : R\nhm : m \u2208 \u2191(maximalIdeal R)\nk : R\nhk : k * a = b * m\n\u22a2 x * \u2191(algebraMap R K) m \u2208 M\n[PROOFSTEP]\nhave hk' : x * algebraMap R K m = algebraMap R K k := by\n  rw [\u2190 mul_div_right_comm, \u2190 map_mul, \u2190 hk, map_mul, mul_div_cancel _ ha\u2083]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\nm : R\nhm : m \u2208 \u2191(maximalIdeal R)\nk : R\nhk : k * a = b * m\n\u22a2 x * \u2191(algebraMap R K) m = \u2191(algebraMap R K) k\n[PROOFSTEP]\nrw [\u2190 mul_div_right_comm, \u2190 map_mul, \u2190 hk, map_mul, mul_div_cancel _ ha\u2083]\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nh : \u00acSubmodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\nm : R\nhm : m \u2208 \u2191(maximalIdeal R)\nk : R\nhk : k * a = b * m\nhk' : x * \u2191(algebraMap R K) m = \u2191(algebraMap R K) k\n\u22a2 x * \u2191(algebraMap R K) m \u2208 M\n[PROOFSTEP]\nexact \u27e8k, le_maximalIdeal h \u27e8_, \u27e8_, hm, rfl\u27e9, hk'\u27e9, hk'.symm\u27e9\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nobtain \u27e8y, hy\u2081, hy\u2082\u27e9 : \u2203 y \u2208 maximalIdeal R, b * y = a :=\n  by\n  rw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this \n  obtain \u27e8_, \u27e8y, hy, rfl\u27e9, hy' : x * algebraMap R K y = algebraMap R K 1\u27e9 := this\n  rw [map_one, \u2190 mul_div_right_comm, div_eq_one_iff_eq ha\u2083, \u2190 map_mul] at hy' \n  exact \u27e8y, hy, IsFractionRing.injective R K hy'\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\n\u22a2 \u2203 y, y \u2208 maximalIdeal R \u2227 b * y = a\n[PROOFSTEP]\nrw [Ideal.eq_top_iff_one, Submodule.mem_comap] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : \u2191(Algebra.linearMap R K) 1 \u2208 Submodule.map (DistribMulAction.toLinearMap R K x) M\n\u22a2 \u2203 y, y \u2208 maximalIdeal R \u2227 b * y = a\n[PROOFSTEP]\nobtain \u27e8_, \u27e8y, hy, rfl\u27e9, hy' : x * algebraMap R K y = algebraMap R K 1\u27e9 := this\n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\ny : R\nhy : y \u2208 \u2191(maximalIdeal R)\nhy' : x * \u2191(algebraMap R K) y = \u2191(algebraMap R K) 1\n\u22a2 \u2203 y, y \u2208 maximalIdeal R \u2227 b * y = a\n[PROOFSTEP]\nrw [map_one, \u2190 mul_div_right_comm, div_eq_one_iff_eq ha\u2083, \u2190 map_mul] at hy' \n[GOAL]\ncase intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\ny : R\nhy : y \u2208 \u2191(maximalIdeal R)\nhy' : \u2191(algebraMap R K) (b * y) = \u2191(algebraMap R K) a\n\u22a2 \u2203 y, y \u2208 maximalIdeal R \u2227 b * y = a\n[PROOFSTEP]\nexact \u27e8y, hy, IsFractionRing.injective R K hy'\u27e9\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nrefine' \u27e8\u27e8y, _\u27e9\u27e9\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\n\u22a2 maximalIdeal R = Submodule.span R {y}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase neg.intro.intro.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\n\u22a2 maximalIdeal R \u2264 Submodule.span R {y}\n[PROOFSTEP]\nintro m hm\n[GOAL]\ncase neg.intro.intro.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 m \u2208 Submodule.span R {y}\n[PROOFSTEP]\nobtain \u27e8k, hk\u27e9 := hb\u2083 m hm\n[GOAL]\ncase neg.intro.intro.a.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\nm : R\nhm : m \u2208 maximalIdeal R\nk : R\nhk : k * a = b * m\n\u22a2 m \u2208 Submodule.span R {y}\n[PROOFSTEP]\nrw [\u2190 hy\u2082, mul_comm, mul_assoc] at hk \n[GOAL]\ncase neg.intro.intro.a.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\nm : R\nhm : m \u2208 maximalIdeal R\nk : R\nhk : b * (y * k) = b * m\n\u22a2 m \u2208 Submodule.span R {y}\n[PROOFSTEP]\nrw [\u2190 mul_left_cancel\u2080 hb\u2084 hk, mul_comm]\n[GOAL]\ncase neg.intro.intro.a.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\nm : R\nhm : m \u2208 maximalIdeal R\nk : R\nhk : b * (y * k) = b * m\n\u22a2 k * y \u2208 Submodule.span R {y}\n[PROOFSTEP]\nexact Ideal.mem_span_singleton'.mpr \u27e8_, rfl\u27e9\n[GOAL]\ncase neg.intro.intro.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK\u271d : Type u_2\ninst\u271d\u2075 : Field K\u271d\ninst\u271d\u2074 : Algebra R K\u271d\ninst\u271d\u00b3 : IsFractionRing R K\u271d\ninst\u271d\u00b2 : LocalRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : IsDedekindDomain R\nne_bot : \u00acmaximalIdeal R = \u22a5\na : R\nha\u2081 : a \u2208 maximalIdeal R\nha\u2082 : a \u2260 0\nhle : Ideal.span {a} \u2264 maximalIdeal R\nthis\u271d\u00b9 : Ideal.radical (Ideal.span {a}) = maximalIdeal R\nthis\u271d : \u2203 n, maximalIdeal R ^ n \u2264 Ideal.span {a}\nn : \u2115\nhn : Nat.find this\u271d = Nat.succ n\nb : R\nhb\u2081 : b \u2208 maximalIdeal R ^ n\nhb\u2082 : \u00acb \u2208 Ideal.span {a}\nhb\u2083 : \u2200 (m : R), m \u2208 maximalIdeal R \u2192 \u2203 k, k * a = b * m\nhb\u2084 : b \u2260 0\nK : Type u_1 := FractionRing R\nx : K := \u2191(algebraMap R K) b / \u2191(algebraMap R K) a\nM : Submodule R K := Submodule.map (Algebra.linearMap R K) (maximalIdeal R)\nha\u2083 : \u2191(algebraMap R K) a \u2260 0\nhx : \u00ac\u2200 (y : K), y \u2208 M \u2192 x * y \u2208 M\nthis : Submodule.comap (Algebra.linearMap R K) (Submodule.map (DistribMulAction.toLinearMap R K x) M) = \u22a4\ny : R\nhy\u2081 : y \u2208 maximalIdeal R\nhy\u2082 : b * y = a\n\u22a2 Submodule.span R {y} \u2264 maximalIdeal R\n[PROOFSTEP]\nrwa [Submodule.span_le, Set.singleton_subset_iff]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\nhave ne_bot := Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\nclassical\nrw [finrank_eq_one_iff']\ntfae_have 1 \u2192 2\n\u00b7 intro; infer_instance\ntfae_have 2 \u2192 1\n\u00b7 intro\n  haveI := IsBezout.toGCDDomain R\n  haveI : UniqueFactorizationMonoid R := ufm_of_gcd_of_wfDvdMonoid\n  apply DiscreteValuationRing.of_ufd_of_unique_irreducible\n  \u00b7 obtain \u27e8x, hx\u2081, hx\u2082\u27e9 := Ring.exists_not_isUnit_of_not_isField h\n    obtain \u27e8p, hp\u2081, -\u27e9 := WfDvdMonoid.exists_irreducible_factor hx\u2082 hx\u2081\n    exact \u27e8p, hp\u2081\u27e9\n  \u00b7 exact ValuationRing.unique_irreducible\ntfae_have 1 \u2192 4\n\u00b7 intro H\n  exact \u27e8inferInstance, ((DiscreteValuationRing.iff_pid_with_one_nonzero_prime R).mp H).2\u27e9\ntfae_have 4 \u2192 3\n\u00b7 rintro \u27e8h\u2081, h\u2082\u27e9;\n  exact\n    { h\u2081 with\n      maximalOfPrime := fun hI hI' =>\n        ExistsUnique.unique h\u2082 \u27e8ne_bot, inferInstance\u27e9 \u27e8hI, hI'\u27e9 \u25b8 maximalIdeal.isMaximal R, }\ntfae_have 3 \u2192 5\n\u00b7 intro h; exact maximalIdeal_isPrincipal_of_isDedekindDomain R\ntfae_have 5 \u2192 6\n\u00b7 rintro \u27e8x, hx\u27e9\n  have : x \u2208 maximalIdeal R := by rw [hx]; exact Submodule.subset_span (Set.mem_singleton x)\n  let x' : maximalIdeal R := \u27e8x, this\u27e9\n  use Submodule.Quotient.mk x'\n  constructor\n  swap\n  \u00b7 intro e\n    rw [Submodule.Quotient.mk_eq_zero] at e \n    apply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n    apply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (maximalIdeal R)\n    \u00b7 exact \u27e8{ x }, (Finset.coe_singleton x).symm \u25b8 hx.symm\u27e9\n    \u00b7 conv_lhs => rw [hx]\n      rw [Submodule.mem_smul_top_iff] at e \n      rwa [Submodule.span_le, Set.singleton_subset_iff]\n    \u00b7 rw [LocalRing.jacobson_eq_maximalIdeal (\u22a5 : Ideal R) bot_ne_top]\n  \u00b7 refine' fun w => Quotient.inductionOn' w fun y => _\n    obtain \u27e8y, hy\u27e9 := y\n    rw [hx, Submodule.mem_span_singleton] at hy \n    obtain \u27e8a, rfl\u27e9 := hy\n    exact \u27e8Ideal.Quotient.mk _ a, rfl\u27e9\ntfae_have 6 \u2192 5\n\u00b7 rintro \u27e8x, hx, hx'\u27e9\n  induction x using Quotient.inductionOn' with\n  | h x => ?_\n  use x\n  apply le_antisymm\n  swap; \u00b7 rw [Submodule.span_le, Set.singleton_subset_iff]; exact x.prop\n  have h\u2081 :\n    (Ideal.span {(x : R)} : Ideal R) \u2294 maximalIdeal R \u2264 Ideal.span {(x : R)} \u2294 maximalIdeal R \u2022 maximalIdeal R :=\n    by\n    refine' sup_le le_sup_left _\n    rintro m hm\n    obtain \u27e8c, hc\u27e9 := hx' (Submodule.Quotient.mk \u27e8m, hm\u27e9)\n    induction c using Quotient.inductionOn' with\n    | h c => ?_\n    rw [\u2190 sub_sub_cancel (c * x) m]\n    apply sub_mem _ _\n    \u00b7 refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr \u27e8c, rfl\u27e9)\n    \u00b7 have := (Submodule.Quotient.eq _).mp hc\n      rw [Submodule.mem_smul_top_iff] at this \n      exact Ideal.mem_sup_right this\n  have h\u2082 : maximalIdeal R \u2264 (\u22a5 : Ideal R).jacobson :=\n    by\n    rw [LocalRing.jacobson_eq_maximalIdeal]\n    exact bot_ne_top\n  have := Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson (IsNoetherian.noetherian _) h\u2082 h\u2081\n  rw [Submodule.bot_smul, sup_bot_eq] at this \n  rw [\u2190 sup_eq_left, eq_comm]\n  exact le_sup_left.antisymm (h\u2081.trans <| le_of_eq this)\ntfae_have 5 \u2192 7\n\u00b7 exact exists_maximalIdeal_pow_eq_of_principal R h\ntfae_have 7 \u2192 2\n\u00b7 rw [ValuationRing.iff_ideal_total]\n  intro H\n  constructor\n  intro I J\n  let _ := Classical.decEq (Ideal R)\n  by_cases hI : I = \u22a5; \u00b7 subst hI; left; exact bot_le\n  by_cases hJ : J = \u22a5; \u00b7 subst hJ; right; exact bot_le\n  obtain \u27e8n, rfl\u27e9 := H I hI\n  obtain \u27e8m, rfl\u27e9 := H J hJ\n  cases' le_total m n with h' h'\n  \u00b7 left; exact Ideal.pow_le_pow h'\n  \u00b7 right; exact Ideal.pow_le_pow h'\ntfae_finish\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), FiniteDimensional.finrank (ResidueField R) (CotangentSpace R) = 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\nrw [finrank_eq_one_iff']\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 1 \u2192 2\n[GOAL]\ncase tfae_1_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\n\u22a2 DiscreteValuationRing R \u2192 ValuationRing R\n[PROOFSTEP]\nintro\n[GOAL]\ncase tfae_1_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\n\u271d : DiscreteValuationRing R\n\u22a2 ValuationRing R\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 2 \u2192 1\n[GOAL]\ncase tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u22a2 ValuationRing R \u2192 DiscreteValuationRing R\n[PROOFSTEP]\nintro\n[GOAL]\ncase tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\n\u22a2 DiscreteValuationRing R\n[PROOFSTEP]\nhaveI := IsBezout.toGCDDomain R\n[GOAL]\ncase tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis : GCDMonoid R\n\u22a2 DiscreteValuationRing R\n[PROOFSTEP]\nhaveI : UniqueFactorizationMonoid R := ufm_of_gcd_of_wfDvdMonoid\n[GOAL]\ncase tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis\u271d : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n\u22a2 DiscreteValuationRing R\n[PROOFSTEP]\napply DiscreteValuationRing.of_ufd_of_unique_irreducible\n[GOAL]\ncase tfae_2_to_1.h\u2081\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis\u271d : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n\u22a2 \u2203 p, Irreducible p\n[PROOFSTEP]\nobtain \u27e8x, hx\u2081, hx\u2082\u27e9 := Ring.exists_not_isUnit_of_not_isField h\n[GOAL]\ncase tfae_2_to_1.h\u2081.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis\u271d : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx\u2081 : x \u2260 0\nhx\u2082 : \u00acIsUnit x\n\u22a2 \u2203 p, Irreducible p\n[PROOFSTEP]\nobtain \u27e8p, hp\u2081, -\u27e9 := WfDvdMonoid.exists_irreducible_factor hx\u2082 hx\u2081\n[GOAL]\ncase tfae_2_to_1.h\u2081.intro.intro.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis\u271d : GCDMonoid R\nthis : UniqueFactorizationMonoid R\nx : R\nhx\u2081 : x \u2260 0\nhx\u2082 : \u00acIsUnit x\np : R\nhp\u2081 : Irreducible p\n\u22a2 \u2203 p, Irreducible p\n[PROOFSTEP]\nexact \u27e8p, hp\u2081\u27e9\n[GOAL]\ncase tfae_2_to_1.h\u2082\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\n\u271d : ValuationRing R\nthis\u271d : GCDMonoid R\nthis : UniqueFactorizationMonoid R\n\u22a2 \u2200 \u2983p q : R\u2984, Irreducible p \u2192 Irreducible q \u2192 Associated p q\n[PROOFSTEP]\nexact ValuationRing.unique_irreducible\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 1 \u2192 4\n[GOAL]\ncase tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\n\u22a2 DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n[PROOFSTEP]\nintro H\n[GOAL]\ncase tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\nH : DiscreteValuationRing R\n\u22a2 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n[PROOFSTEP]\nexact \u27e8inferInstance, ((DiscreteValuationRing.iff_pid_with_one_nonzero_prime R).mp H).2\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 4 \u2192 3\n[GOAL]\ncase tfae_4_to_3\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n\u22a2 (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\n[PROOFSTEP]\nrintro \u27e8h\u2081, h\u2082\u27e9\n[GOAL]\ncase tfae_4_to_3.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\nh\u2081 : IsIntegrallyClosed R\nh\u2082 : \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\n\u22a2 IsDedekindDomain R\n[PROOFSTEP]\nexact\n  { h\u2081 with\n    maximalOfPrime := fun hI hI' =>\n      ExistsUnique.unique h\u2082 \u27e8ne_bot, inferInstance\u27e9 \u27e8hI, hI'\u27e9 \u25b8 maximalIdeal.isMaximal R, }\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 3 \u2192 5\n[GOAL]\ncase tfae_3_to_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\n\u22a2 IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase tfae_3_to_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh\u271d : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\nh : IsDedekindDomain R\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nexact maximalIdeal_isPrincipal_of_isDedekindDomain R\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 5 \u2192 6\n[GOAL]\ncase tfae_5_to_6\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase tfae_5_to_6.mk.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n\u22a2 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nhave : x \u2208 maximalIdeal R := by rw [hx]; exact Submodule.subset_span (Set.mem_singleton x)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n\u22a2 x \u2208 maximalIdeal R\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\n\u22a2 x \u2208 Submodule.span R {x}\n[PROOFSTEP]\nexact Submodule.subset_span (Set.mem_singleton x)\n[GOAL]\ncase tfae_5_to_6.mk.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\n\u22a2 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nlet x' : maximalIdeal R := \u27e8x, this\u27e9\n[GOAL]\ncase tfae_5_to_6.mk.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n[PROOFSTEP]\nuse Submodule.Quotient.mk x'\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 \u2203 _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Submodule.Quotient.mk x' = w\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Submodule.Quotient.mk x' = w\ncase h.w\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 Submodule.Quotient.mk x' \u2260 0\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.w\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 Submodule.Quotient.mk x' \u2260 0\n[PROOFSTEP]\nintro e\n[GOAL]\ncase h.w\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : Submodule.Quotient.mk x' = 0\n\u22a2 False\n[PROOFSTEP]\nrw [Submodule.Quotient.mk_eq_zero] at e \n[GOAL]\ncase h.w\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 False\n[PROOFSTEP]\napply Ring.ne_bot_of_isMaximal_of_not_isField (maximalIdeal.isMaximal R) h\n[GOAL]\ncase h.w\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 maximalIdeal R = \u22a5\n[PROOFSTEP]\napply Submodule.eq_bot_of_le_smul_of_le_jacobson_bot (maximalIdeal R)\n[GOAL]\ncase h.w.hN\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 Submodule.FG (maximalIdeal R)\n[PROOFSTEP]\nexact \u27e8{ x }, (Finset.coe_singleton x).symm \u25b8 hx.symm\u27e9\n[GOAL]\ncase h.w.hIN\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 maximalIdeal R \u2264 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nconv_lhs => rw [hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n| maximalIdeal R\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n| maximalIdeal R\n[PROOFSTEP]\nrw [hx]\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n| maximalIdeal R\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h.w.hIN\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 Submodule.span R {x} \u2264 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrw [Submodule.mem_smul_top_iff] at e \n[GOAL]\ncase h.w.hIN\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : \u2191x' \u2208 maximalIdeal R \u2022 maximalIdeal R\n\u22a2 Submodule.span R {x} \u2264 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrwa [Submodule.span_le, Set.singleton_subset_iff]\n[GOAL]\ncase h.w.hIjac\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\ne : x' \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 maximalIdeal R \u2264 Ideal.jacobson \u22a5\n[PROOFSTEP]\nrw [LocalRing.jacobson_eq_maximalIdeal (\u22a5 : Ideal R) bot_ne_top]\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\n\u22a2 \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Submodule.Quotient.mk x' = w\n[PROOFSTEP]\nrefine' fun w => Quotient.inductionOn' w fun y => _\n[GOAL]\ncase h.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : { x // x \u2208 maximalIdeal R }\n\u22a2 \u2203 c, c \u2022 Submodule.Quotient.mk x' = Quotient.mk'' y\n[PROOFSTEP]\nobtain \u27e8y, hy\u27e9 := y\n[GOAL]\ncase h.h.mk\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy : y \u2208 maximalIdeal R\n\u22a2 \u2203 c, c \u2022 Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy }\n[PROOFSTEP]\nrw [hx, Submodule.mem_span_singleton] at hy \n[GOAL]\ncase h.h.mk\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\ny : R\nhy\u271d : y \u2208 maximalIdeal R\nhy : \u2203 a, a \u2022 x = y\n\u22a2 \u2203 c, c \u2022 Submodule.Quotient.mk x' = Quotient.mk'' { val := y, property := hy\u271d }\n[PROOFSTEP]\nobtain \u27e8a, rfl\u27e9 := hy\n[GOAL]\ncase h.h.mk.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\nx : R\nhx : maximalIdeal R = Submodule.span R {x}\nthis : x \u2208 maximalIdeal R\nx' : { x // x \u2208 maximalIdeal R } := { val := x, property := this }\nw : CotangentSpace R\na : R\nhy : a \u2022 x \u2208 maximalIdeal R\n\u22a2 \u2203 c, c \u2022 Submodule.Quotient.mk x' = Quotient.mk'' { val := a \u2022 x, property := hy }\n[PROOFSTEP]\nexact \u27e8Ideal.Quotient.mk _ a, rfl\u27e9\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 6 \u2192 5\n[GOAL]\ncase tfae_6_to_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\n\u22a2 (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nrintro \u27e8x, hx, hx'\u27e9\n[GOAL]\ncase tfae_6_to_5.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : CotangentSpace R\nhx : x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 x = w\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\ninduction x using Quotient.inductionOn' with\n| h x => ?_\n[GOAL]\ncase tfae_6_to_5.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : CotangentSpace R\nhx : x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 x = w\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\ninduction x using Quotient.inductionOn' with\n| h x => ?_\n[GOAL]\ncase tfae_6_to_5.intro.intro.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 Submodule.IsPrincipal (maximalIdeal R)\n[PROOFSTEP]\nuse x\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 maximalIdeal R = Submodule.span R {\u2191x}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 Submodule.span R {\u2191x} \u2264 maximalIdeal R\n[PROOFSTEP]\nswap\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 Submodule.span R {\u2191x} \u2264 maximalIdeal R\n[PROOFSTEP]\nrw [Submodule.span_le, Set.singleton_subset_iff]\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 \u2191x \u2208 \u2191(maximalIdeal R)\n[PROOFSTEP]\nexact x.prop\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\n[PROOFSTEP]\nhave h\u2081 : (Ideal.span {(x : R)} : Ideal R) \u2294 maximalIdeal R \u2264 Ideal.span {(x : R)} \u2294 maximalIdeal R \u2022 maximalIdeal R :=\n  by\n  refine' sup_le le_sup_left _\n  rintro m hm\n  obtain \u27e8c, hc\u27e9 := hx' (Submodule.Quotient.mk \u27e8m, hm\u27e9)\n  induction c using Quotient.inductionOn' with\n  | h c => ?_\n  rw [\u2190 sub_sub_cancel (c * x) m]\n  apply sub_mem _ _\n  \u00b7 refine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr \u27e8c, rfl\u27e9)\n  \u00b7 have := (Submodule.Quotient.eq _).mp hc\n    rw [Submodule.mem_smul_top_iff] at this \n    exact Ideal.mem_sup_right this\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrefine' sup_le le_sup_left _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\n\u22a2 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrintro m hm\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\n\u22a2 m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nobtain \u27e8c, hc\u27e9 := hx' (Submodule.Quotient.mk \u27e8m, hm\u27e9)\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : ResidueField R\nhc : c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\ninduction c using Quotient.inductionOn' with\n| h c => ?_\n[GOAL]\ncase intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : ResidueField R\nhc : c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\ninduction c using Quotient.inductionOn' with\n| h c => ?_\n[GOAL]\ncase intro.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrw [\u2190 sub_sub_cancel (c * x) m]\n[GOAL]\ncase intro.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 c * \u2191x - (c * \u2191x - m) \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\napply sub_mem _ _\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 c * \u2191x \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrefine' Ideal.mem_sup_left (Ideal.mem_span_singleton'.mpr \u27e8c, rfl\u27e9)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\n\u22a2 c * \u2191x - m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nhave := (Submodule.Quotient.eq _).mp hc\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : (fun x x_1 => x \u2022 x_1) c x - { val := m, property := hm } \u2208 maximalIdeal R \u2022 \u22a4\n\u22a2 c * \u2191x - m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nrw [Submodule.mem_smul_top_iff] at this \n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nm : R\nhm : m \u2208 maximalIdeal R\nc : R\nhc : Quotient.mk'' c \u2022 Quotient.mk'' x = Submodule.Quotient.mk { val := m, property := hm }\nthis : \u2191((fun x x_1 => x \u2022 x_1) c x - { val := m, property := hm }) \u2208 maximalIdeal R \u2022 maximalIdeal R\n\u22a2 c * \u2191x - m \u2208 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n[PROOFSTEP]\nexact Ideal.mem_sup_right this\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\n[PROOFSTEP]\nhave h\u2082 : maximalIdeal R \u2264 (\u22a5 : Ideal R).jacobson :=\n  by\n  rw [LocalRing.jacobson_eq_maximalIdeal]\n  exact bot_ne_top\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n\u22a2 maximalIdeal R \u2264 Ideal.jacobson \u22a5\n[PROOFSTEP]\nrw [LocalRing.jacobson_eq_maximalIdeal]\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\n\u22a2 \u22a5 \u2260 \u22a4\n[PROOFSTEP]\nexact bot_ne_top\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\nh\u2082 : maximalIdeal R \u2264 Ideal.jacobson \u22a5\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\n[PROOFSTEP]\nhave := Submodule.smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson (IsNoetherian.noetherian _) h\u2082 h\u2081\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\nh\u2082 : maximalIdeal R \u2264 Ideal.jacobson \u22a5\nthis : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R = Ideal.span {\u2191x} \u2294 \u22a5 \u2022 maximalIdeal R\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\n[PROOFSTEP]\nrw [Submodule.bot_smul, sup_bot_eq] at this \n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\nh\u2082 : maximalIdeal R \u2264 Ideal.jacobson \u22a5\nthis : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R = Ideal.span {\u2191x}\n\u22a2 maximalIdeal R \u2264 Submodule.span R {\u2191x}\n[PROOFSTEP]\nrw [\u2190 sup_eq_left, eq_comm]\n[GOAL]\ncase h.a\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\nx : { x // x \u2208 maximalIdeal R }\nhx : Quotient.mk'' x \u2260 0\nhx' : \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 Quotient.mk'' x = w\nh\u2081 : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2264 Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R\nh\u2082 : maximalIdeal R \u2264 Ideal.jacobson \u22a5\nthis : Ideal.span {\u2191x} \u2294 maximalIdeal R \u2022 maximalIdeal R = Ideal.span {\u2191x}\n\u22a2 Submodule.span R {\u2191x} = Submodule.span R {\u2191x} \u2294 maximalIdeal R\n[PROOFSTEP]\nexact le_sup_left.antisymm (h\u2081.trans <| le_of_eq this)\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 5 \u2192 7\n[GOAL]\ncase tfae_5_to_7\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n[PROOFSTEP]\nexact exists_maximalIdeal_pow_eq_of_principal R h\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_have 7 \u2192 2\n[GOAL]\ncase tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R\n[PROOFSTEP]\nrw [ValuationRing.iff_ideal_total]\n[GOAL]\ncase tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nintro H\n[GOAL]\ncase tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 \u2200 (a b : Ideal R), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintro I J\n[GOAL]\ncase tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nlet _ := Classical.decEq (Ideal R)\n[GOAL]\ncase tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nby_cases hI : I = \u22a5\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : I = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nsubst hI\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J \u2228 J \u2264 \u22a5\n[PROOFSTEP]\nleft\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nby_cases hJ : J = \u22a5\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : J = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nsubst hJ\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 \u22a5 \u2228 \u22a5 \u2264 I\n[PROOFSTEP]\nright\n[GOAL]\ncase pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 \u22a5 \u2264 I\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : \u00acJ = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := H I hI\n[GOAL]\ncase neg.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhJ : \u00acJ = \u22a5\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 J \u2228 J \u2264 maximalIdeal R ^ n\n[PROOFSTEP]\nobtain \u27e8m, rfl\u27e9 := H J hJ\n[GOAL]\ncase neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m \u2228 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n\n[PROOFSTEP]\ncases' le_total m n with h' h'\n[GOAL]\ncase neg.intro.intro.inl\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\nh' : m \u2264 n\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m \u2228 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n\n[PROOFSTEP]\nleft\n[GOAL]\ncase neg.intro.intro.inl.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\nh' : m \u2264 n\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m\n[PROOFSTEP]\nexact Ideal.pow_le_pow h'\n[GOAL]\ncase neg.intro.intro.inr\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\nh' : n \u2264 m\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m \u2228 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n\n[PROOFSTEP]\nright\n[GOAL]\ncase neg.intro.intro.inr.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\nh' : n \u2264 m\n\u22a2 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n\n[PROOFSTEP]\nexact Ideal.pow_le_pow h'\n[GOAL]\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\nh : \u00acIsField R\nne_bot : maximalIdeal R \u2260 \u22a5\ntfae_1_to_2 : DiscreteValuationRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 DiscreteValuationRing R\ntfae_1_to_4 : DiscreteValuationRing R \u2192 IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_6 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w\ntfae_6_to_5 : (\u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w) \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\ntfae_7_to_2 : (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R\n\u22a2 List.TFAE\n    [DiscreteValuationRing R, ValuationRing R, IsDedekindDomain R, IsIntegrallyClosed R \u2227 \u2203! P, P \u2260 \u22a5 \u2227 Ideal.IsPrime P,\n      Submodule.IsPrincipal (maximalIdeal R), \u2203 v _n, \u2200 (w : CotangentSpace R), \u2203 c, c \u2022 v = w,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n]\n[PROOFSTEP]\ntfae_finish\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.DiscreteValuationRing.TFAE", "llama_tokens": 94129, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.5059154987828951}}
{"text": "[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.24757, u_1} J\ninst\u271d\u00b9 : Category.{?u.24761, u_2} C\ninst\u271d : HasZeroMorphisms C\nF : J \u2964 ShortComplex C\n\u22a2 whiskerLeft F \u03c0\u2081To\u03c0\u2082 \u226b whiskerLeft F \u03c0\u2082To\u03c0\u2083 = 0\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.24757, u_1} J\ninst\u271d\u00b9 : Category.{?u.24761, u_2} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : J \u2964 ShortComplex C\n\u03c6 : X\u271d \u27f6 Y\u271d\n\u22a2 whiskerRight \u03c6 \u03c0\u2081 \u226b\n      ((fun F => mk (whiskerLeft F \u03c0\u2081To\u03c0\u2082) (whiskerLeft F \u03c0\u2082To\u03c0\u2083) (_ : whiskerLeft F \u03c0\u2081To\u03c0\u2082 \u226b whiskerLeft F \u03c0\u2082To\u03c0\u2083 = 0))\n          Y\u271d).f =\n    ((fun F => mk (whiskerLeft F \u03c0\u2081To\u03c0\u2082) (whiskerLeft F \u03c0\u2082To\u03c0\u2083) (_ : whiskerLeft F \u03c0\u2081To\u03c0\u2082 \u226b whiskerLeft F \u03c0\u2082To\u03c0\u2083 = 0))\n          X\u271d).f \u226b\n      whiskerRight \u03c6 \u03c0\u2082\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.24757, u_1} J\ninst\u271d\u00b9 : Category.{?u.24761, u_2} C\ninst\u271d : HasZeroMorphisms C\nX\u271d Y\u271d : J \u2964 ShortComplex C\n\u03c6 : X\u271d \u27f6 Y\u271d\n\u22a2 whiskerRight \u03c6 \u03c0\u2082 \u226b\n      ((fun F => mk (whiskerLeft F \u03c0\u2081To\u03c0\u2082) (whiskerLeft F \u03c0\u2082To\u03c0\u2083) (_ : whiskerLeft F \u03c0\u2081To\u03c0\u2082 \u226b whiskerLeft F \u03c0\u2082To\u03c0\u2083 = 0))\n          Y\u271d).g =\n    ((fun F => mk (whiskerLeft F \u03c0\u2081To\u03c0\u2082) (whiskerLeft F \u03c0\u2082To\u03c0\u2083) (_ : whiskerLeft F \u03c0\u2081To\u03c0\u2082 \u226b whiskerLeft F \u03c0\u2082To\u03c0\u2083 = 0))\n          X\u271d).g \u226b\n      whiskerRight \u03c6 \u03c0\u2083\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : ShortComplex (J \u2964 C)\n\u22a2 \u2200 {X Y : J} (f : X \u27f6 Y),\n    ((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2081.map f \u226b\n        ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2081.obj x)) Y).hom =\n      ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2081.obj x)) X).hom \u226b\n        ((functor J C \u22d9 inverse J C).obj x\u271d).X\u2081.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : ShortComplex (J \u2964 C)\n\u22a2 \u2200 {X Y : J} (f : X \u27f6 Y),\n    ((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2082.map f \u226b\n        ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2082.obj x)) Y).hom =\n      ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2082.obj x)) X).hom \u226b\n        ((functor J C \u22d9 inverse J C).obj x\u271d).X\u2082.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : ShortComplex (J \u2964 C)\n\u22a2 \u2200 {X Y : J} (f : X \u27f6 Y),\n    ((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2083.map f \u226b\n        ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2083.obj x)) Y).hom =\n      ((fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2083.obj x)) X).hom \u226b\n        ((functor J C \u22d9 inverse J C).obj x\u271d).X\u2083.map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : ShortComplex (J \u2964 C)\n\u22a2 (NatIso.ofComponents fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2081.obj x)).hom \u226b\n      ((functor J C \u22d9 inverse J C).obj x\u271d).f =\n    ((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).f \u226b\n      (NatIso.ofComponents fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2082.obj x)).hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : ShortComplex (J \u2964 C)\n\u22a2 (NatIso.ofComponents fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2082.obj x)).hom \u226b\n      ((functor J C \u22d9 inverse J C).obj x\u271d).g =\n    ((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).g \u226b\n      (NatIso.ofComponents fun x => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x\u271d).X\u2083.obj x)).hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.53514, u_1} J\ninst\u271d\u00b9 : Category.{?u.53518, u_2} C\ninst\u271d : HasZeroMorphisms C\n\u22a2 \u2200 {X Y : ShortComplex (J \u2964 C)} (f : X \u27f6 Y),\n    (\ud835\udfed (ShortComplex (J \u2964 C))).map f \u226b\n        ((fun x =>\n              isoMk (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2081.obj x_1))\n                (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2082.obj x_1))\n                (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2083.obj x_1)))\n            Y).hom =\n      ((fun x =>\n              isoMk (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2081.obj x_1))\n                (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2082.obj x_1))\n                (NatIso.ofComponents fun x_1 => Iso.refl (((\ud835\udfed (ShortComplex (J \u2964 C))).obj x).X\u2083.obj x_1)))\n            X).hom \u226b\n        (functor J C \u22d9 inverse J C).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.111338, u_1} J\ninst\u271d\u00b9 : Category.{?u.111342, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d\u00b9 : J \u2964 ShortComplex C\nx\u271d : J\n\u22a2 (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).X\u2081).hom \u226b (((\ud835\udfed (J \u2964 ShortComplex C)).obj x\u271d\u00b9).obj x\u271d).f =\n    (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).f \u226b (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).X\u2082).hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.111338, u_1} J\ninst\u271d\u00b9 : Category.{?u.111342, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d\u00b9 : J \u2964 ShortComplex C\nx\u271d : J\n\u22a2 (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).X\u2082).hom \u226b (((\ud835\udfed (J \u2964 ShortComplex C)).obj x\u271d\u00b9).obj x\u271d).g =\n    (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).g \u226b (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d\u00b9).obj x\u271d).X\u2083).hom\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.111338, u_1} J\ninst\u271d\u00b9 : Category.{?u.111342, u_2} C\ninst\u271d : HasZeroMorphisms C\nx\u271d : J \u2964 ShortComplex C\n\u22a2 \u2200 {X Y : J} (f : X \u27f6 Y),\n    ((inverse J C \u22d9 functor J C).obj x\u271d).map f \u226b\n        ((fun x =>\n              isoMk (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2081)\n                (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2082)\n                (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2083))\n            Y).hom =\n      ((fun x =>\n              isoMk (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2081)\n                (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2082)\n                (Iso.refl (((inverse J C \u22d9 functor J C).obj x\u271d).obj x).X\u2083))\n            X).hom \u226b\n        ((\ud835\udfed (J \u2964 ShortComplex C)).obj x\u271d).map f\n[PROOFSTEP]\naesop_cat\n[GOAL]\nJ : Type u_1\nC : Type u_2\ninst\u271d\u00b2 : Category.{?u.111338, u_1} J\ninst\u271d\u00b9 : Category.{?u.111342, u_2} C\ninst\u271d : HasZeroMorphisms C\n\u22a2 \u2200 {X Y : J \u2964 ShortComplex C} (f : X \u27f6 Y),\n    (inverse J C \u22d9 functor J C).map f \u226b\n        ((fun x =>\n              NatIso.ofComponents fun x_1 =>\n                isoMk (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2081)\n                  (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2082)\n                  (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2083))\n            Y).hom =\n      ((fun x =>\n              NatIso.ofComponents fun x_1 =>\n                isoMk (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2081)\n                  (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2082)\n                  (Iso.refl (((inverse J C \u22d9 functor J C).obj x).obj x_1).X\u2083))\n            X).hom \u226b\n        (\ud835\udfed (J \u2964 ShortComplex C)).map f\n[PROOFSTEP]\naesop_cat\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Homology.ShortComplex.FunctorEquivalence", "llama_tokens": 3857, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744850834648, "lm_q2_score": 0.6187804407739559, "lm_q1q2_score": 0.5058991002454863}}
{"text": "[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nhave A : \u22c3 n : \u2115, closure (f '' ball 0 n) = Set.univ :=\n  by\n  refine' Subset.antisymm (subset_univ _) fun y _ => _\n  rcases surj y with \u27e8x, hx\u27e9\n  rcases exists_nat_gt \u2016x\u2016 with \u27e8n, hn\u27e9\n  refine' mem_iUnion.2 \u27e8n, subset_closure _\u27e9\n  refine' (mem_image _ _ _).2 \u27e8x, \u27e8_, hx\u27e9\u27e9\n  rwa [mem_ball, dist_eq_norm, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\n\u22a2 \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\n[PROOFSTEP]\nrefine' Subset.antisymm (subset_univ _) fun y _ => _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\ny : F\nx\u271d : y \u2208 Set.univ\n\u22a2 y \u2208 \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n)\n[PROOFSTEP]\nrcases surj y with \u27e8x, hx\u27e9\n[GOAL]\ncase intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\ny : F\nx\u271d : y \u2208 Set.univ\nx : E\nhx : \u2191f x = y\n\u22a2 y \u2208 \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n)\n[PROOFSTEP]\nrcases exists_nat_gt \u2016x\u2016 with \u27e8n, hn\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\ny : F\nx\u271d : y \u2208 Set.univ\nx : E\nhx : \u2191f x = y\nn : \u2115\nhn : \u2016x\u2016 < \u2191n\n\u22a2 y \u2208 \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n)\n[PROOFSTEP]\nrefine' mem_iUnion.2 \u27e8n, subset_closure _\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\ny : F\nx\u271d : y \u2208 Set.univ\nx : E\nhx : \u2191f x = y\nn : \u2115\nhn : \u2016x\u2016 < \u2191n\n\u22a2 y \u2208 \u2191f '' ball 0 \u2191n\n[PROOFSTEP]\nrefine' (mem_image _ _ _).2 \u27e8x, \u27e8_, hx\u27e9\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\ny : F\nx\u271d : y \u2208 Set.univ\nx : E\nhx : \u2191f x = y\nn : \u2115\nhn : \u2016x\u2016 < \u2191n\n\u22a2 x \u2208 ball 0 \u2191n\n[PROOFSTEP]\nrwa [mem_ball, dist_eq_norm, sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nhave : \u2203 (n : \u2115) (x : _), x \u2208 interior (closure (f '' ball 0 n)) :=\n  nonempty_interior_of_iUnion_of_closed (fun n => isClosed_closure) A\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nthis : \u2203 n x, x \u2208 interior (closure (\u2191f '' ball 0 \u2191n))\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nsimp only [mem_interior_iff_mem_nhds, Metric.mem_nhds_iff] at this \n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nthis : \u2203 n x \u03b5, \u03b5 > 0 \u2227 ball x \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nrcases this with \u27e8n, a, \u03b5, \u27e8\u03b5pos, H\u27e9\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nrcases NormedField.exists_one_lt_norm \ud835\udd5c with \u27e8c, hc\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 \u2203 C, C \u2265 0 \u2227 \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nrefine' \u27e8(\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * n, _, fun y => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n \u2265 0\n[PROOFSTEP]\nrefine' mul_nonneg (mul_nonneg (mul_nonneg _ (norm_nonneg _)) (by norm_num)) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_1.refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 0 \u2264 (\u03b5 / 2)\u207b\u00b9\ncase intro.intro.intro.intro.intro.refine'_1.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 0 \u2264 \u2191n\n[PROOFSTEP]\nexacts [inv_nonneg.2 (div_nonneg (le_of_lt \u03b5pos) (by norm_num)), n.cast_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\n\u22a2 0 \u2264 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro.intro.intro.intro.refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : y = 0\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nuse 0\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : y = 0\n\u22a2 dist (\u2191f 0) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u20160\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrcases rescale_to_shell hc (half_pos \u03b5pos) hy with \u27e8d, hd, ydlt, -, dinv\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nlet \u03b4 := \u2016d\u2016 * \u2016y\u2016 / 4\n[GOAL]\ncase neg.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave \u03b4pos : 0 < \u03b4 := div_pos (mul_pos (norm_pos_iff.2 hd) (norm_pos_iff.2 hy)) (by norm_num)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u22a2 0 < 4\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase neg.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave : a + d \u2022 y \u2208 ball a \u03b5 := by simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self \u03b5pos)]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\n\u22a2 a + d \u2022 y \u2208 ball a \u03b5\n[PROOFSTEP]\nsimp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self \u03b5pos)]\n[GOAL]\ncase neg.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrcases Metric.mem_closure_iff.1 (H this) _ \u03b4pos with \u27e8z\u2081, z\u2081im, h\u2081\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nh\u2081 : dist (a + d \u2022 y) z\u2081 < \u03b4\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 z\u2081im with \u27e8x\u2081, hx\u2081, xz\u2081\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nh\u2081 : dist (a + d \u2022 y) z\u2081 < \u03b4\nx\u2081 : E\nhx\u2081 : x\u2081 \u2208 ball 0 \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 xz\u2081] at h\u2081 \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : x\u2081 \u2208 ball 0 \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrw [mem_ball, dist_eq_norm, sub_zero] at hx\u2081 \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave : a \u2208 ball a \u03b5 := by\n  simp\n  exact \u03b5pos\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\n\u22a2 a \u2208 ball a \u03b5\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\n\u22a2 0 < \u03b5\n[PROOFSTEP]\nexact \u03b5pos\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrcases Metric.mem_closure_iff.1 (H this) _ \u03b4pos with \u27e8z\u2082, z\u2082im, h\u2082\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nh\u2082 : dist a z\u2082 < \u03b4\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrcases(mem_image _ _ _).1 z\u2082im with \u27e8x\u2082, hx\u2082, xz\u2082\u27e9\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nh\u2082 : dist a z\u2082 < \u03b4\nx\u2082 : E\nhx\u2082 : x\u2082 \u2208 ball 0 \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 xz\u2082] at h\u2082 \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : x\u2082 \u2208 ball 0 \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrw [mem_ball, dist_eq_norm, sub_zero] at hx\u2082 \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nlet x := x\u2081 - x\u2082\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave I : \u2016f x - d \u2022 y\u2016 \u2264 2 * \u03b4 :=\n  calc\n    \u2016f x - d \u2022 y\u2016 = \u2016f x\u2081 - (a + d \u2022 y) - (f x\u2082 - a)\u2016 := by\n      congr 1\n      simp only [f.map_sub]\n      abel\n    _ \u2264 \u2016f x\u2081 - (a + d \u2022 y)\u2016 + \u2016f x\u2082 - a\u2016 := (norm_sub_le _ _)\n    _ \u2264 \u03b4 + \u03b4 := by\n      apply add_le_add\n      \u00b7 rw [\u2190 dist_eq_norm, dist_comm]\n        exact le_of_lt h\u2081\n      \u00b7 rw [\u2190 dist_eq_norm, dist_comm]\n        exact le_of_lt h\u2082\n    _ = 2 * \u03b4 := (two_mul _).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2016\u2191f x - d \u2022 y\u2016 = \u2016\u2191f x\u2081 - (a + d \u2022 y) - (\u2191f x\u2082 - a)\u2016\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2191f x - d \u2022 y = \u2191f x\u2081 - (a + d \u2022 y) - (\u2191f x\u2082 - a)\n[PROOFSTEP]\nsimp only [f.map_sub]\n[GOAL]\ncase e_a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2191f x\u2081 - \u2191f x\u2082 - d \u2022 y = \u2191f x\u2081 - (a + d \u2022 y) - (\u2191f x\u2082 - a)\n[PROOFSTEP]\nabel\n[GOAL]\ncase e_a\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2191f x\u2081 - \u2191f x\u2082 - d \u2022 y = \u2191f x\u2081 - (a + d \u2022 y) - (\u2191f x\u2082 - a)\n[PROOFSTEP]\nabel\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2016\u2191f x\u2081 - (a + d \u2022 y)\u2016 + \u2016\u2191f x\u2082 - a\u2016 \u2264 \u03b4 + \u03b4\n[PROOFSTEP]\napply add_le_add\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2016\u2191f x\u2081 - (a + d \u2022 y)\u2016 \u2264 \u03b4\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm, dist_comm]\n[GOAL]\ncase h\u2081\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 dist (a + d \u2022 y) (\u2191f x\u2081) \u2264 \u03b4\n[PROOFSTEP]\nexact le_of_lt h\u2081\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 \u2016\u2191f x\u2082 - a\u2016 \u2264 \u03b4\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm, dist_comm]\n[GOAL]\ncase h\u2082\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\n\u22a2 dist a (\u2191f x\u2082) \u2264 \u03b4\n[PROOFSTEP]\nexact le_of_lt h\u2082\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave J : \u2016f (d\u207b\u00b9 \u2022 x) - y\u2016 \u2264 1 / 2 * \u2016y\u2016 :=\n  calc\n    \u2016f (d\u207b\u00b9 \u2022 x) - y\u2016 = \u2016d\u207b\u00b9 \u2022 f x - (d\u207b\u00b9 * d) \u2022 y\u2016 := by rwa [f.map_smul _, inv_mul_cancel, one_smul]\n    _ = \u2016d\u207b\u00b9 \u2022 (f x - d \u2022 y)\u2016 := by rw [mul_smul, smul_sub]\n    _ = \u2016d\u2016\u207b\u00b9 * \u2016f x - d \u2022 y\u2016 := by rw [norm_smul, norm_inv]\n    _ \u2264 \u2016d\u2016\u207b\u00b9 * (2 * \u03b4) := by\n      apply mul_le_mul_of_nonneg_left I\n      rw [inv_nonneg]\n      exact norm_nonneg _\n    _ = \u2016d\u2016\u207b\u00b9 * \u2016d\u2016 * \u2016y\u2016 / 2 := by\n      simp only\n      ring\n    _ = \u2016y\u2016 / 2 := by\n      rw [inv_mul_cancel, one_mul]\n      simp [norm_eq_zero, hd]\n    _ = 1 / 2 * \u2016y\u2016 := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016\u2191f (d\u207b\u00b9 \u2022 x) - y\u2016 = \u2016d\u207b\u00b9 \u2022 \u2191f x - (d\u207b\u00b9 * d) \u2022 y\u2016\n[PROOFSTEP]\nrwa [f.map_smul _, inv_mul_cancel, one_smul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u207b\u00b9 \u2022 \u2191f x - (d\u207b\u00b9 * d) \u2022 y\u2016 = \u2016d\u207b\u00b9 \u2022 (\u2191f x - d \u2022 y)\u2016\n[PROOFSTEP]\nrw [mul_smul, smul_sub]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u207b\u00b9 \u2022 (\u2191f x - d \u2022 y)\u2016 = \u2016d\u2016\u207b\u00b9 * \u2016\u2191f x - d \u2022 y\u2016\n[PROOFSTEP]\nrw [norm_smul, norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u2016\u207b\u00b9 * \u2016\u2191f x - d \u2022 y\u2016 \u2264 \u2016d\u2016\u207b\u00b9 * (2 * \u03b4)\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left I\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 0 \u2264 \u2016d\u2016\u207b\u00b9\n[PROOFSTEP]\nrw [inv_nonneg]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 0 \u2264 \u2016d\u2016\n[PROOFSTEP]\nexact norm_nonneg _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u2016\u207b\u00b9 * (2 * \u03b4) = \u2016d\u2016\u207b\u00b9 * \u2016d\u2016 * \u2016y\u2016 / 2\n[PROOFSTEP]\nsimp only\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u2016\u207b\u00b9 * (2 * (\u2016d\u2016 * \u2016y\u2016 / 4)) = \u2016d\u2016\u207b\u00b9 * \u2016d\u2016 * \u2016y\u2016 / 2\n[PROOFSTEP]\nring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u2016\u207b\u00b9 * \u2016d\u2016 * \u2016y\u2016 / 2 = \u2016y\u2016 / 2\n[PROOFSTEP]\nrw [inv_mul_cancel, one_mul]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016d\u2016 \u2260 0\n[PROOFSTEP]\nsimp [norm_eq_zero, hd]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\n\u22a2 \u2016y\u2016 / 2 = 1 / 2 * \u2016y\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : \u2016\u2191f (d\u207b\u00b9 \u2022 x) - y\u2016 \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm] at J \n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nhave K : \u2016d\u207b\u00b9 \u2022 x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016 :=\n  calc\n    \u2016d\u207b\u00b9 \u2022 x\u2016 = \u2016d\u2016\u207b\u00b9 * \u2016x\u2081 - x\u2082\u2016 := by rw [norm_smul, norm_inv]\n    _ \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016 * (n + n) :=\n      by\n      refine' mul_le_mul dinv _ (norm_nonneg _) _\n      \u00b7 exact le_trans (norm_sub_le _ _) (add_le_add (le_of_lt hx\u2081) (le_of_lt hx\u2082))\n      \u00b7 apply mul_nonneg (mul_nonneg _ (norm_nonneg _)) (norm_nonneg _)\n        exact inv_nonneg.2 (le_of_lt (half_pos \u03b5pos))\n    _ = (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016 := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2016d\u207b\u00b9 \u2022 x\u2016 = \u2016d\u2016\u207b\u00b9 * \u2016x\u2081 - x\u2082\u2016\n[PROOFSTEP]\nrw [norm_smul, norm_inv]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2016d\u2016\u207b\u00b9 * \u2016x\u2081 - x\u2082\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016 * (\u2191n + \u2191n)\n[PROOFSTEP]\nrefine' mul_le_mul dinv _ (norm_nonneg _) _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2016x\u2081 - x\u2082\u2016 \u2264 \u2191n + \u2191n\n[PROOFSTEP]\nexact le_trans (norm_sub_le _ _) (add_le_add (le_of_lt hx\u2081) (le_of_lt hx\u2082))\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 0 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n[PROOFSTEP]\napply mul_nonneg (mul_nonneg _ (norm_nonneg _)) (norm_nonneg _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 0 \u2264 (\u03b5 / 2)\u207b\u00b9\n[PROOFSTEP]\nexact inv_nonneg.2 (le_of_lt (half_pos \u03b5pos))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\n\u22a2 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016 * (\u2191n + \u2191n) = (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase neg.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d : CompleteSpace F\nsurj : Surjective \u2191f\nA : \u22c3 (n : \u2115), closure (\u2191f '' ball 0 \u2191n) = Set.univ\nn : \u2115\na : F\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nH : ball a \u03b5 \u2286 closure ((fun a => \u2191f a) '' ball 0 \u2191n)\nc : \ud835\udd5c\nhc : 1 < \u2016c\u2016\ny : F\nhy : \u00acy = 0\nd : \ud835\udd5c\nhd : d \u2260 0\nydlt : \u2016d \u2022 y\u2016 < \u03b5 / 2\ndinv : \u2016d\u2016\u207b\u00b9 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * \u2016y\u2016\n\u03b4 : \u211d := \u2016d\u2016 * \u2016y\u2016 / 4\n\u03b4pos : 0 < \u03b4\nthis\u271d : a + d \u2022 y \u2208 ball a \u03b5\nz\u2081 : F\nz\u2081im : z\u2081 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2081 : E\nh\u2081 : dist (a + d \u2022 y) (\u2191f x\u2081) < \u03b4\nhx\u2081 : \u2016x\u2081\u2016 < \u2191n\nxz\u2081 : \u2191f x\u2081 = z\u2081\nthis : a \u2208 ball a \u03b5\nz\u2082 : F\nz\u2082im : z\u2082 \u2208 (fun a => \u2191f a) '' ball 0 \u2191n\nx\u2082 : E\nh\u2082 : dist a (\u2191f x\u2082) < \u03b4\nhx\u2082 : \u2016x\u2082\u2016 < \u2191n\nxz\u2082 : \u2191f x\u2082 = z\u2082\nx : E := x\u2081 - x\u2082\nI : \u2016\u2191f x - d \u2022 y\u2016 \u2264 2 * \u03b4\nJ : dist (\u2191f (d\u207b\u00b9 \u2022 x)) y \u2264 1 / 2 * \u2016y\u2016\nK : \u2016d\u207b\u00b9 \u2022 x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n\u22a2 \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 (\u03b5 / 2)\u207b\u00b9 * \u2016c\u2016 * 2 * \u2191n * \u2016y\u2016\n[PROOFSTEP]\nexact \u27e8d\u207b\u00b9 \u2022 x, J, K\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nobtain \u27e8C, C0, hC\u27e9 := exists_approx_preimage_norm_le f surj\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\nhC : \u2200 (y : F), \u2203 x, dist (\u2191f x) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nchoose g hg using hC\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nlet h y := y - f (g y)\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nhave hle : \u2200 y, \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016 := by\n  intro y\n  rw [\u2190 dist_eq_norm, dist_comm]\n  exact (hg y).1\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\n\u22a2 \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\n[PROOFSTEP]\nintro y\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\ny : F\n\u22a2 \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 dist_eq_norm, dist_comm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\ny : F\n\u22a2 dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016\n[PROOFSTEP]\nexact (hg y).1\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\n\u22a2 \u2203 C, C > 0 \u2227 \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n[PROOFSTEP]\nrefine' \u27e82 * C + 1, by linarith, fun y => _\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\n\u22a2 2 * C + 1 > 0\n[PROOFSTEP]\nlinarith\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave hnle : \u2200 n : \u2115, \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016 := by\n  intro n\n  induction' n with n IH\n  \u00b7 simp only [one_div, Nat.zero_eq, one_mul, iterate_zero_apply, pow_zero, le_rfl]\n  \u00b7 rw [iterate_succ']\n    apply le_trans (hle _) _\n    rw [pow_succ, mul_assoc]\n    apply mul_le_mul_of_nonneg_left IH\n    norm_num\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\n\u22a2 \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\n\u22a2 \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\n\u22a2 \u2016h^[Nat.zero] y\u2016 \u2264 (1 / 2) ^ Nat.zero * \u2016y\u2016\n[PROOFSTEP]\nsimp only [one_div, Nat.zero_eq, one_mul, iterate_zero_apply, pow_zero, le_rfl]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\nIH : \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 \u2016h^[Nat.succ n] y\u2016 \u2264 (1 / 2) ^ Nat.succ n * \u2016y\u2016\n[PROOFSTEP]\nrw [iterate_succ']\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\nIH : \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 \u2016(h \u2218 h^[n]) y\u2016 \u2264 (1 / 2) ^ Nat.succ n * \u2016y\u2016\n[PROOFSTEP]\napply le_trans (hle _) _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\nIH : \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 1 / 2 * \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ Nat.succ n * \u2016y\u2016\n[PROOFSTEP]\nrw [pow_succ, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\nIH : \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 1 / 2 * \u2016h^[n] y\u2016 \u2264 1 / 2 * ((1 / 2) ^ n * \u2016y\u2016)\n[PROOFSTEP]\napply mul_le_mul_of_nonneg_left IH\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nn : \u2115\nIH : \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nlet u n := g (h^[n] y)\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave ule : \u2200 n, \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016) := by\n  intro n\n  apply le_trans (hg _).2 _\n  calc\n    C * \u2016h^[n] y\u2016 \u2264 C * ((1 / 2) ^ n * \u2016y\u2016) := mul_le_mul_of_nonneg_left (hnle n) C0\n    _ = (1 / 2) ^ n * (C * \u2016y\u2016) := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\n\u22a2 \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nn : \u2115\n\u22a2 \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n[PROOFSTEP]\napply le_trans (hg _).2 _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nn : \u2115\n\u22a2 C * \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n[PROOFSTEP]\ncalc\n  C * \u2016h^[n] y\u2016 \u2264 C * ((1 / 2) ^ n * \u2016y\u2016) := mul_le_mul_of_nonneg_left (hnle n) C0\n  _ = (1 / 2) ^ n * (C * \u2016y\u2016) := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nn : \u2115\n\u22a2 C * ((1 / 2) ^ n * \u2016y\u2016) = (1 / 2) ^ n * (C * \u2016y\u2016)\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave sNu : Summable fun n => \u2016u n\u2016 :=\n  by\n  refine' summable_of_nonneg_of_le (fun n => norm_nonneg _) ule _\n  exact Summable.mul_right _ (summable_geometric_of_lt_1 (by norm_num) (by norm_num))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n\u22a2 Summable fun n => \u2016u n\u2016\n[PROOFSTEP]\nrefine' summable_of_nonneg_of_le (fun n => norm_nonneg _) ule _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n\u22a2 Summable fun b => (1 / 2) ^ b * (C * \u2016y\u2016)\n[PROOFSTEP]\nexact Summable.mul_right _ (summable_geometric_of_lt_1 (by norm_num) (by norm_num))\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave su : Summable u := summable_of_summable_norm sNu\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nlet x := tsum u\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave x_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016 :=\n  calc\n    \u2016x\u2016 \u2264 \u2211' n, \u2016u n\u2016 := norm_tsum_le_tsum_norm sNu\n    _ \u2264 \u2211' n, (1 / 2) ^ n * (C * \u2016y\u2016) := (tsum_le_tsum ule sNu (Summable.mul_right _ summable_geometric_two))\n    _ = (\u2211' n, (1 / 2) ^ n) * (C * \u2016y\u2016) := tsum_mul_right\n    _ = 2 * C * \u2016y\u2016 := by rw [tsum_geometric_two, mul_assoc]\n    _ \u2264 2 * C * \u2016y\u2016 + \u2016y\u2016 := (le_add_of_nonneg_right (norm_nonneg y))\n    _ = (2 * C + 1) * \u2016y\u2016 := by ring\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\n\u22a2 (\u2211' (n : \u2115), (1 / 2) ^ n) * (C * \u2016y\u2016) = 2 * C * \u2016y\u2016\n[PROOFSTEP]\nrw [tsum_geometric_two, mul_assoc]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\n\u22a2 2 * C * \u2016y\u2016 + \u2016y\u2016 = (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nring\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave fsumeq : \u2200 n : \u2115, f (\u2211 i in Finset.range n, u i) = y - h^[n] y :=\n  by\n  intro n\n  induction' n with n IH\n  \u00b7 simp [f.map_zero]\n  \u00b7 rw [sum_range_succ, f.map_add, IH, iterate_succ_apply', sub_add]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n\u22a2 \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\n[PROOFSTEP]\nintro n\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nn : \u2115\n\u22a2 \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\n[PROOFSTEP]\ninduction' n with n IH\n[GOAL]\ncase zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n\u22a2 \u2191f (\u2211 i in Finset.range Nat.zero, u i) = y - h^[Nat.zero] y\n[PROOFSTEP]\nsimp [f.map_zero]\n[GOAL]\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nn : \u2115\nIH : \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\n\u22a2 \u2191f (\u2211 i in Finset.range (Nat.succ n), u i) = y - h^[Nat.succ n] y\n[PROOFSTEP]\nrw [sum_range_succ, f.map_add, IH, iterate_succ_apply', sub_add]\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x) := su.hasSum.tendsto_sum_nat\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave L\u2081 : Tendsto (fun n => f (\u2211 i in Finset.range n, u i)) atTop (\ud835\udcdd (f x)) := (f.continuous.tendsto _).comp this\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => \u2191f (\u2211 i in Finset.range n, u i)) atTop (\ud835\udcdd (\u2191f x))\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nsimp only [fsumeq] at L\u2081 \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave L\u2082 : Tendsto (fun n => y - h^[n] y) atTop (\ud835\udcdd (y - 0)) :=\n  by\n  refine' tendsto_const_nhds.sub _\n  rw [tendsto_iff_norm_tendsto_zero]\n  simp only [sub_zero]\n  refine' squeeze_zero (fun _ => norm_nonneg _) hnle _\n  rw [\u2190 zero_mul \u2016y\u2016]\n  refine' (_root_.tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun n => y - h^[n] y) atTop (\ud835\udcdd (y - 0))\n[PROOFSTEP]\nrefine' tendsto_const_nhds.sub _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun n => h^[n] y) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [tendsto_iff_norm_tendsto_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun e => \u2016h^[e] y - 0\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nsimp only [sub_zero]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun e => \u2016(fun y => y - \u2191f (g y))^[e] y\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrefine' squeeze_zero (fun _ => norm_nonneg _) hnle _\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun t => (1 / 2) ^ t * \u2016y\u2016) atTop (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 zero_mul \u2016y\u2016]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 Tendsto (fun t => (1 / 2) ^ t * \u2016y\u2016) atTop (\ud835\udcdd (0 * \u2016y\u2016))\n[PROOFSTEP]\nrefine' (_root_.tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 0 \u2264 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\n\u22a2 1 / 2 < 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\nL\u2082 : Tendsto (fun n => y - h^[n] y) atTop (\ud835\udcdd (y - 0))\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nhave feq : f x = y - 0 := tendsto_nhds_unique L\u2081 L\u2082\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\nL\u2082 : Tendsto (fun n => y - h^[n] y) atTop (\ud835\udcdd (y - 0))\nfeq : \u2191f x = y - 0\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nrw [sub_zero] at feq \n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\nC : \u211d\nC0 : C \u2265 0\ng : F \u2192 E\nhg : \u2200 (y : F), dist (\u2191f (g y)) y \u2264 1 / 2 * \u2016y\u2016 \u2227 \u2016g y\u2016 \u2264 C * \u2016y\u2016\nh : F \u2192 F := fun y => y - \u2191f (g y)\nhle : \u2200 (y : F), \u2016h y\u2016 \u2264 1 / 2 * \u2016y\u2016\ny : F\nhnle : \u2200 (n : \u2115), \u2016h^[n] y\u2016 \u2264 (1 / 2) ^ n * \u2016y\u2016\nu : \u2115 \u2192 E := fun n => g (h^[n] y)\nule : \u2200 (n : \u2115), \u2016u n\u2016 \u2264 (1 / 2) ^ n * (C * \u2016y\u2016)\nsNu : Summable fun n => \u2016u n\u2016\nsu : Summable u\nx : E := tsum u\nx_ineq : \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\nfsumeq : \u2200 (n : \u2115), \u2191f (\u2211 i in Finset.range n, u i) = y - h^[n] y\nthis : Tendsto (fun n => \u2211 i in Finset.range n, u i) atTop (\ud835\udcdd x)\nL\u2081 : Tendsto (fun n => y - (fun y => y - \u2191f (g y))^[n] y) atTop (\ud835\udcdd (\u2191f (\u2211' (n : \u2115), g ((fun y => y - \u2191f (g y))^[n] y))))\nL\u2082 : Tendsto (fun n => y - h^[n] y) atTop (\ud835\udcdd (y - 0))\nfeq : \u2191f x = y\n\u22a2 \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 (2 * C + 1) * \u2016y\u2016\n[PROOFSTEP]\nexact \u27e8x, feq, x_ineq\u27e9\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\n\u22a2 IsOpenMap \u2191f\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen (\u2191f '' s)\n[PROOFSTEP]\nrcases exists_preimage_norm_le f surj with \u27e8C, Cpos, hC\u27e9\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\n\u22a2 IsOpen (\u2191f '' s)\n[PROOFSTEP]\nrefine' isOpen_iff.2 fun y yfs => _\n[GOAL]\ncase intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 ball y \u03b5 \u2286 \u2191f '' s\n[PROOFSTEP]\nrcases mem_image_iff_bex.1 yfs with \u27e8x, xs, fxy\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 ball y \u03b5 \u2286 \u2191f '' s\n[PROOFSTEP]\nrcases isOpen_iff.1 hs x xs with \u27e8\u03b5, \u03b5pos, h\u03b5\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\n\u22a2 \u2203 \u03b5, \u03b5 > 0 \u2227 ball y \u03b5 \u2286 \u2191f '' s\n[PROOFSTEP]\nrefine' \u27e8\u03b5 / C, div_pos \u03b5pos Cpos, fun z hz => _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\n\u22a2 z \u2208 \u2191f '' s\n[PROOFSTEP]\nrcases hC (z - y) with \u27e8w, wim, wnorm\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\n\u22a2 z \u2208 \u2191f '' s\n[PROOFSTEP]\nhave : f (x + w) = z := by rw [f.map_add, wim, fxy, add_sub_cancel'_right]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\n\u22a2 \u2191f (x + w) = z\n[PROOFSTEP]\nrw [f.map_add, wim, fxy, add_sub_cancel'_right]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 z \u2208 \u2191f '' s\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 \u2191f (x + w) \u2208 \u2191f '' s\n[PROOFSTEP]\nhave : x + w \u2208 ball x \u03b5 :=\n  calc\n    dist (x + w) x = \u2016w\u2016 := by\n      rw [dist_eq_norm]\n      simp\n    _ \u2264 C * \u2016z - y\u2016 := wnorm\n    _ < C * (\u03b5 / C) := by\n      apply mul_lt_mul_of_pos_left _ Cpos\n      rwa [mem_ball, dist_eq_norm] at hz \n    _ = \u03b5 := mul_div_cancel' _ (ne_of_gt Cpos)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 dist (x + w) x = \u2016w\u2016\n[PROOFSTEP]\nrw [dist_eq_norm]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 \u2016x + w - x\u2016 = \u2016w\u2016\n[PROOFSTEP]\nsimp\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 C * \u2016z - y\u2016 < C * (\u03b5 / C)\n[PROOFSTEP]\napply mul_lt_mul_of_pos_left _ Cpos\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis : \u2191f (x + w) = z\n\u22a2 \u2016z - y\u2016 < \u03b5 / C\n[PROOFSTEP]\nrwa [mem_ball, dist_eq_norm] at hz \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nsurj : Surjective \u2191f\ns : Set E\nhs : IsOpen s\nC : \u211d\nCpos : C > 0\nhC : \u2200 (y : F), \u2203 x, \u2191f x = y \u2227 \u2016x\u2016 \u2264 C * \u2016y\u2016\ny : F\nyfs : y \u2208 \u2191f '' s\nx : E\nxs : x \u2208 s\nfxy : \u2191f x = y\n\u03b5 : \u211d\n\u03b5pos : \u03b5 > 0\nh\u03b5 : ball x \u03b5 \u2286 s\nz : F\nhz : z \u2208 ball y (\u03b5 / C)\nw : E\nwim : \u2191f w = z - y\nwnorm : \u2016w\u2016 \u2264 C * \u2016z - y\u2016\nthis\u271d : \u2191f (x + w) = z\nthis : x + w \u2208 ball x \u03b5\n\u22a2 \u2191f (x + w) \u2208 \u2191f '' s\n[PROOFSTEP]\nexact Set.mem_image_of_mem _ (h\u03b5 this)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhsurj : LinearMap.range f = \u22a4\n\u22a2 \u2203 fsymm, 0 < fsymm.nnnorm\n[PROOFSTEP]\nchoose C hC fsymm h using exists_preimage_norm_le _ (LinearMap.range_eq_top.mp hsurj)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhsurj : LinearMap.range f = \u22a4\nC : \u211d\nhC : C > 0\nfsymm : F \u2192 E\nh : \u2200 (y : F), \u2191f (fsymm y) = y \u2227 \u2016fsymm y\u2016 \u2264 C * \u2016y\u2016\n\u22a2 \u2203 fsymm, 0 < fsymm.nnnorm\n[PROOFSTEP]\nuse{\n    toFun := fsymm\n    nnnorm := \u27e8C, hC.lt.le\u27e9\n    bound' := fun y => (h y).2\n    right_inv' := fun y => (h y).1 }\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhsurj : LinearMap.range f = \u22a4\nC : \u211d\nhC : C > 0\nfsymm : F \u2192 E\nh : \u2200 (y : F), \u2191f (fsymm y) = y \u2227 \u2016fsymm y\u2016 \u2264 C * \u2016y\u2016\n\u22a2 0 <\n    { toFun := fsymm, nnnorm := { val := C, property := (_ : 0 \u2264 C) }, bound' := (_ : \u2200 (y : F), \u2016fsymm y\u2016 \u2264 C * \u2016y\u2016),\n        right_inv' := (_ : \u2200 (y : F), \u2191f (fsymm y) = y) }.nnnorm\n[PROOFSTEP]\nexact hC\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhsurj : LinearMap.range f = \u22a4\n\u22a2 0 < (nonlinearRightInverseOfSurjective f hsurj).nnnorm\n[PROOFSTEP]\nrw [nonlinearRightInverseOfSurjective]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhsurj : LinearMap.range f = \u22a4\n\u22a2 0 < (Classical.choose (_ : \u2203 fsymm, 0 < fsymm.nnnorm)).nnnorm\n[PROOFSTEP]\nexact Classical.choose_spec (exists_nonlinearRightInverse_of_surjective f hsurj)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ne : E \u2243\u2097[\ud835\udd5c] F\nh : Continuous \u2191e\n\u22a2 Continuous \u2191(symm e)\n[PROOFSTEP]\nrw [continuous_def]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ne : E \u2243\u2097[\ud835\udd5c] F\nh : Continuous \u2191e\n\u22a2 \u2200 (s : Set E), IsOpen s \u2192 IsOpen (\u2191(symm e) \u207b\u00b9' s)\n[PROOFSTEP]\nintro s hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ne : E \u2243\u2097[\ud835\udd5c] F\nh : Continuous \u2191e\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen (\u2191(symm e) \u207b\u00b9' s)\n[PROOFSTEP]\nrw [\u2190 e.image_eq_preimage]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ne : E \u2243\u2097[\ud835\udd5c] F\nh : Continuous \u2191e\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen (\u2191e '' s)\n[PROOFSTEP]\nrw [\u2190 e.coe_coe] at h \u22a2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ne : E \u2243\u2097[\ud835\udd5c] F\nh : Continuous \u2191\u2191e\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen (\u2191\u2191e '' s)\n[PROOFSTEP]\nexact ContinuousLinearMap.isOpenMap (\ud835\udd5c := \ud835\udd5c) \u27e8\u2191e, h\u27e9 e.surjective s hs\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhinj : ker f = \u22a5\nhsurj : LinearMap.range f = \u22a4\n\u22a2 Continuous \u2191(LinearEquiv.ofBijective \u2191f (_ : Injective \u2191\u2191f \u2227 Surjective \u2191\u2191f))\n[PROOFSTEP]\nconvert f.continuous\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhinj : ker f = \u22a5\nhsurj : LinearMap.range f = \u22a4\n\u22a2 \u2191(ofBijective f hinj hsurj) = f\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nhinj : ker f = \u22a5\nhsurj : LinearMap.range f = \u22a4\nx\u271d : E\n\u22a2 \u2191\u2191(ofBijective f hinj hsurj) x\u271d = \u2191f x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 ker (coprod f (Submodule.subtypeL G)) = \u22a5\n[PROOFSTEP]\nrw [ker_coprod_of_disjoint_range]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 Submodule.prod (ker f) (ker (Submodule.subtypeL G)) = \u22a5\n[PROOFSTEP]\nrw [hker, Submodule.ker_subtypeL, Submodule.prod_bot]\n[GOAL]\ncase hd\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 Disjoint (LinearMap.range f) (LinearMap.range (Submodule.subtypeL G))\n[PROOFSTEP]\nrw [Submodule.range_subtypeL]\n[GOAL]\ncase hd\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 Disjoint (LinearMap.range f) G\n[PROOFSTEP]\nexact h.disjoint\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 LinearMap.range (coprod f (Submodule.subtypeL G)) = \u22a4\n[PROOFSTEP]\nrw [range_coprod, Submodule.range_subtypeL, h.sup_eq_top]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 LinearMap.range f = Submodule.map (\u2191\u2191(coprodSubtypeLEquivOfIsCompl f h hker)) (Submodule.prod \u22a4 \u22a5)\n[PROOFSTEP]\nrw [coprodSubtypeLEquivOfIsCompl, ContinuousLinearEquiv.coe_ofBijective, coe_coprod, LinearMap.coprod_map_prod,\n  Submodule.map_bot, sup_bot_eq, Submodule.map_top]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b2 : CompleteSpace F\ninst\u271d\u00b9 : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\ninst\u271d : CompleteSpace { x // x \u2208 G }\nhker : ker f = \u22a5\n\u22a2 LinearMap.range f = LinearMap.range \u2191f\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed \u2191G\nhker : ker f = \u22a5\n\u22a2 IsClosed \u2191(LinearMap.range f)\n[PROOFSTEP]\nhaveI : CompleteSpace G := hG.completeSpace_coe\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed \u2191G\nhker : ker f = \u22a5\nthis : CompleteSpace { x // x \u2208 G }\n\u22a2 IsClosed \u2191(LinearMap.range f)\n[PROOFSTEP]\nlet g := coprodSubtypeLEquivOfIsCompl f h hker\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed \u2191G\nhker : ker f = \u22a5\nthis : CompleteSpace { x // x \u2208 G }\ng : (E \u00d7 { x // x \u2208 G }) \u2243L[\ud835\udd5c] F := coprodSubtypeLEquivOfIsCompl f h hker\n\u22a2 IsClosed \u2191(LinearMap.range f)\n[PROOFSTEP]\nrw [range_eq_map_coprodSubtypeLEquivOfIsCompl f h hker]\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed \u2191G\nhker : ker f = \u22a5\nthis : CompleteSpace { x // x \u2208 G }\ng : (E \u00d7 { x // x \u2208 G }) \u2243L[\ud835\udd5c] F := coprodSubtypeLEquivOfIsCompl f h hker\n\u22a2 IsClosed \u2191(Submodule.map (\u2191\u2191(coprodSubtypeLEquivOfIsCompl f h hker)) (Submodule.prod \u22a4 \u22a5))\n[PROOFSTEP]\napply g.toHomeomorph.isClosed_image.2\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf\u271d : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\nf : E \u2192L[\ud835\udd5c] F\nG : Submodule \ud835\udd5c F\nh : IsCompl (LinearMap.range f) G\nhG : IsClosed \u2191G\nhker : ker f = \u22a5\nthis : CompleteSpace { x // x \u2208 G }\ng : (E \u00d7 { x // x \u2208 G }) \u2243L[\ud835\udd5c] F := coprodSubtypeLEquivOfIsCompl f h hker\n\u22a2 IsClosed \u2191(Submodule.prod \u22a4 \u22a5)\n[PROOFSTEP]\nexact isClosed_univ.prod isClosed_singleton\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nletI : CompleteSpace g.graph := completeSpace_coe_iff_isComplete.mpr hg.isComplete\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\nthis : CompleteSpace { x // x \u2208 graph g } := Iff.mpr completeSpace_coe_iff_isComplete (IsClosed.isComplete hg)\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nlet \u03c6\u2080 : E \u2192\u2097[\ud835\udd5c] E \u00d7 F := LinearMap.id.prod g\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\nthis : CompleteSpace { x // x \u2208 graph g } := Iff.mpr completeSpace_coe_iff_isComplete (IsClosed.isComplete hg)\n\u03c6\u2080 : E \u2192\u2097[\ud835\udd5c] E \u00d7 F := prod id g\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nhave : Function.LeftInverse Prod.fst \u03c6\u2080 := fun x => rfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\nthis\u271d : CompleteSpace { x // x \u2208 graph g } := Iff.mpr completeSpace_coe_iff_isComplete (IsClosed.isComplete hg)\n\u03c6\u2080 : E \u2192\u2097[\ud835\udd5c] E \u00d7 F := prod id g\nthis : LeftInverse Prod.fst \u2191\u03c6\u2080\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nlet \u03c6 : E \u2243\u2097[\ud835\udd5c] g.graph := (LinearEquiv.ofLeftInverse this).trans (LinearEquiv.ofEq _ _ g.graph_eq_range_prod.symm)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\nthis\u271d : CompleteSpace { x // x \u2208 graph g } := Iff.mpr completeSpace_coe_iff_isComplete (IsClosed.isComplete hg)\n\u03c6\u2080 : E \u2192\u2097[\ud835\udd5c] E \u00d7 F := prod id g\nthis : LeftInverse Prod.fst \u2191\u03c6\u2080\n\u03c6 : E \u2243\u2097[\ud835\udd5c] { x // x \u2208 graph g } :=\n  LinearEquiv.trans (LinearEquiv.ofLeftInverse this)\n    (LinearEquiv.ofEq (range \u03c6\u2080) (graph g) (_ : range (prod id g) = graph g))\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nlet \u03c8 : g.graph \u2243L[\ud835\udd5c] E := \u03c6.symm.toContinuousLinearEquivOfContinuous continuous_subtype_val.fst\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(graph g)\nthis\u271d : CompleteSpace { x // x \u2208 graph g } := Iff.mpr completeSpace_coe_iff_isComplete (IsClosed.isComplete hg)\n\u03c6\u2080 : E \u2192\u2097[\ud835\udd5c] E \u00d7 F := prod id g\nthis : LeftInverse Prod.fst \u2191\u03c6\u2080\n\u03c6 : E \u2243\u2097[\ud835\udd5c] { x // x \u2208 graph g } :=\n  LinearEquiv.trans (LinearEquiv.ofLeftInverse this)\n    (LinearEquiv.ofEq (range \u03c6\u2080) (graph g) (_ : range (prod id g) = graph g))\n\u03c8 : { x // x \u2208 graph g } \u2243L[\ud835\udd5c] E :=\n  LinearEquiv.toContinuousLinearEquivOfContinuous (LinearEquiv.symm \u03c6) (_ : Continuous fun a => (\u2191a).fst)\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nexact (continuous_subtype_val.comp \u03c8.symm.continuous).snd\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u22a2 Continuous \u2191g\n[PROOFSTEP]\nrefine' g.continuous_of_isClosed_graph (IsSeqClosed.isClosed _)\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u22a2 IsSeqClosed \u2191(graph g)\n[PROOFSTEP]\nrintro \u03c6 \u27e8x, y\u27e9 h\u03c6g h\u03c6\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\n\u22a2 (x, y) \u2208 \u2191(graph g)\n[PROOFSTEP]\nrefine' hg (Prod.fst \u2218 \u03c6) x y ((continuous_fst.tendsto _).comp h\u03c6) _\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\n\u22a2 Tendsto (\u2191g \u2218 Prod.fst \u2218 \u03c6) atTop (\ud835\udcdd y)\n[PROOFSTEP]\nhave : g \u2218 Prod.fst \u2218 \u03c6 = Prod.snd \u2218 \u03c6 := by\n  ext n\n  exact (h\u03c6g n).symm\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\n\u22a2 \u2191g \u2218 Prod.fst \u2218 \u03c6 = Prod.snd \u2218 \u03c6\n[PROOFSTEP]\next n\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\nn : \u2115\n\u22a2 (\u2191g \u2218 Prod.fst \u2218 \u03c6) n = (Prod.snd \u2218 \u03c6) n\n[PROOFSTEP]\nexact (h\u03c6g n).symm\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\nthis : \u2191g \u2218 Prod.fst \u2218 \u03c6 = Prod.snd \u2218 \u03c6\n\u22a2 Tendsto (\u2191g \u2218 Prod.fst \u2218 \u03c6) atTop (\ud835\udcdd y)\n[PROOFSTEP]\nrw [this]\n[GOAL]\ncase mk\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u03c6 : \u2115 \u2192 E \u00d7 F\nx : E\ny : F\nh\u03c6g : \u2200 (n : \u2115), \u03c6 n \u2208 \u2191(graph g)\nh\u03c6 : Tendsto \u03c6 atTop (\ud835\udcdd (x, y))\nthis : \u2191g \u2218 Prod.fst \u2218 \u03c6 = Prod.snd \u2218 \u03c6\n\u22a2 Tendsto (Prod.snd \u2218 \u03c6) atTop (\ud835\udcdd y)\n[PROOFSTEP]\nexact (continuous_snd.tendsto _).comp h\u03c6\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(LinearMap.graph g)\n\u22a2 \u2191(ofIsClosedGraph hg) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : IsClosed \u2191(LinearMap.graph g)\nx\u271d : E\n\u22a2 \u2191\u2191(ofIsClosedGraph hg) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\n\u22a2 \u2191(ofSeqClosedGraph hg) = g\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\ud835\udd5c : Type u_1\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\nf : E \u2192L[\ud835\udd5c] F\ninst\u271d\u00b9 : CompleteSpace F\ninst\u271d : CompleteSpace E\ng : E \u2192\u2097[\ud835\udd5c] F\nhg : \u2200 (u : \u2115 \u2192 E) (x : E) (y : F), Tendsto u atTop (\ud835\udcdd x) \u2192 Tendsto (\u2191g \u2218 u) atTop (\ud835\udcdd y) \u2192 y = \u2191g x\nx\u271d : E\n\u22a2 \u2191\u2191(ofSeqClosedGraph hg) x\u271d = \u2191g x\u271d\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.Analysis.NormedSpace.Banach", "llama_tokens": 72290, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8376199471193039, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.5058653506292013}}
{"text": "[GOAL]\nn : \u2115\n\u22a2 \u2200 (a b c : QuaternionGroup n), a * b * c = a * (b * c)\n[PROOFSTEP]\nrintro (i | i) (j | j) (k | k)\n[GOAL]\ncase a.a.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a i * a j * a k = a i * (a j * a k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase a.a.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a i * a j * xa k = a i * (a j * xa k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase a.xa.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a i * xa j * a k = a i * (xa j * a k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase a.xa.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a i * xa j * xa k = a i * (xa j * xa k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase xa.a.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa i * a j * a k = xa i * (a j * a k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase xa.a.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa i * a j * xa k = xa i * (a j * xa k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase xa.xa.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa i * xa j * a k = xa i * (xa j * a k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase xa.xa.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa i * xa j * xa k = xa i * (xa j * xa k)\n[PROOFSTEP]\nsimp only [(\u00b7 * \u00b7), mul]\n[GOAL]\ncase a.a.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a (i + j + k) = a (i + (j + k))\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase a.a.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa (k - (i + j)) = xa (k - j - i)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase a.xa.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa (j - i + k) = xa (j + k - i)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase a.xa.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a (\u2191n + k - (j - i)) = a (i + (\u2191n + k - j))\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase xa.a.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa (i + j + k) = xa (i + (j + k))\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase xa.a.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a (\u2191n + k - (i + j)) = a (\u2191n + (k - j) - i)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase xa.xa.a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 a (\u2191n + j - i + k) = a (\u2191n + (j + k) - i)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase xa.xa.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa (k - (\u2191n + j - i)) = xa (i + (\u2191n + k - j))\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase xa.xa.xa\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 xa (k + (-\u2191n - j) + i) = xa (k + (\u2191n - j) + i)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase xa.xa.xa.e_a.e_a.e_a.e_a\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 -\u2191n = \u2191n\n[PROOFSTEP]\ncalc\n  -(n : ZMod (2 * n)) = 0 - n := by rw [zero_sub]\n  _ = 2 * n - n := by norm_cast; simp\n  _ = n := by ring\n[GOAL]\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 -\u2191n = 0 - \u2191n\n[PROOFSTEP]\nrw [zero_sub]\n[GOAL]\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 0 - \u2191n = 2 * \u2191n - \u2191n\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 \u2191(Int.subNatNat 0 n) = \u2191(Int.subNatNat (2 * n) n)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\ni j k : ZMod (2 * n)\n\u22a2 2 * \u2191n - \u2191n = \u2191n\n[PROOFSTEP]\nring\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : QuaternionGroup n), 1 * a = a\n[PROOFSTEP]\nrintro (i | i)\n[GOAL]\ncase a\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 1 * a i = a i\n[PROOFSTEP]\nexact congr_arg a (zero_add i)\n[GOAL]\ncase xa\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 1 * xa i = xa i\n[PROOFSTEP]\nexact congr_arg xa (sub_zero i)\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : QuaternionGroup n), a * 1 = a\n[PROOFSTEP]\nrintro (i | i)\n[GOAL]\ncase a\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 a i * 1 = a i\n[PROOFSTEP]\nexact congr_arg a (add_zero i)\n[GOAL]\ncase xa\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 xa i * 1 = xa i\n[PROOFSTEP]\nexact congr_arg xa (add_zero i)\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (a : QuaternionGroup n), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nrintro (i | i)\n[GOAL]\ncase a\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 (a i)\u207b\u00b9 * a i = 1\n[PROOFSTEP]\nexact congr_arg a (neg_add_self i)\n[GOAL]\ncase xa\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 (xa i)\u207b\u00b9 * xa i = 1\n[PROOFSTEP]\nexact congr_arg a (sub_self (n + i))\n[GOAL]\nn : \u2115\n\u22a2 Function.LeftInverse\n    (fun i =>\n      match i with\n      | a j => Sum.inl j\n      | xa j => Sum.inr j)\n    fun i =>\n    match i with\n    | Sum.inl j => a j\n    | Sum.inr j => xa j\n[PROOFSTEP]\nrintro (x | x)\n[GOAL]\ncase inl\nn : \u2115\nx : ZMod (2 * n)\n\u22a2 (fun i =>\n        match i with\n        | a j => Sum.inl j\n        | xa j => Sum.inr j)\n      ((fun i =>\n          match i with\n          | Sum.inl j => a j\n          | Sum.inr j => xa j)\n        (Sum.inl x)) =\n    Sum.inl x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr\nn : \u2115\nx : ZMod (2 * n)\n\u22a2 (fun i =>\n        match i with\n        | a j => Sum.inl j\n        | xa j => Sum.inr j)\n      ((fun i =>\n          match i with\n          | Sum.inl j => a j\n          | Sum.inr j => xa j)\n        (Sum.inr x)) =\n    Sum.inr x\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 Function.RightInverse\n    (fun i =>\n      match i with\n      | a j => Sum.inl j\n      | xa j => Sum.inr j)\n    fun i =>\n    match i with\n    | Sum.inl j => a j\n    | Sum.inr j => xa j\n[PROOFSTEP]\nrintro (x | x)\n[GOAL]\ncase a\nn : \u2115\nx : ZMod (2 * n)\n\u22a2 (fun i =>\n        match i with\n        | Sum.inl j => a j\n        | Sum.inr j => xa j)\n      ((fun i =>\n          match i with\n          | a j => Sum.inl j\n          | xa j => Sum.inr j)\n        (a x)) =\n    a x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase xa\nn : \u2115\nx : ZMod (2 * n)\n\u22a2 (fun i =>\n        match i with\n        | Sum.inl j => a j\n        | Sum.inr j => xa j)\n      ((fun i =>\n          match i with\n          | a j => Sum.inl j\n          | xa j => Sum.inr j)\n        (xa x)) =\n    xa x\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 Function.LeftInverse\n    (fun i =>\n      match i with\n      | DihedralGroup.r j => a j\n      | DihedralGroup.sr j => xa j)\n    fun i =>\n    match i with\n    | a j => DihedralGroup.r j\n    | xa j => DihedralGroup.sr j\n[PROOFSTEP]\nrintro (k | k)\n[GOAL]\ncase a\nn : \u2115\nk : ZMod (2 * 0)\n\u22a2 (fun i =>\n        match i with\n        | DihedralGroup.r j => a j\n        | DihedralGroup.sr j => xa j)\n      ((fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j)\n        (a k)) =\n    a k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase xa\nn : \u2115\nk : ZMod (2 * 0)\n\u22a2 (fun i =>\n        match i with\n        | DihedralGroup.r j => a j\n        | DihedralGroup.sr j => xa j)\n      ((fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j)\n        (xa k)) =\n    xa k\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 Function.RightInverse\n    (fun i =>\n      match i with\n      | DihedralGroup.r j => a j\n      | DihedralGroup.sr j => xa j)\n    fun i =>\n    match i with\n    | a j => DihedralGroup.r j\n    | xa j => DihedralGroup.sr j\n[PROOFSTEP]\nrintro (k | k)\n[GOAL]\ncase r\nn : \u2115\nk : ZMod 0\n\u22a2 (fun i =>\n        match i with\n        | a j => DihedralGroup.r j\n        | xa j => DihedralGroup.sr j)\n      ((fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j)\n        (DihedralGroup.r k)) =\n    DihedralGroup.r k\n[PROOFSTEP]\nrfl\n[GOAL]\ncase sr\nn : \u2115\nk : ZMod 0\n\u22a2 (fun i =>\n        match i with\n        | a j => DihedralGroup.r j\n        | xa j => DihedralGroup.sr j)\n      ((fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j)\n        (DihedralGroup.sr k)) =\n    DihedralGroup.sr k\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\n\u22a2 \u2200 (x y : QuaternionGroup 0),\n    Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (x * y) =\n      Equiv.toFun\n          {\n            toFun := fun i =>\n              match i with\n              | a j => DihedralGroup.r j\n              | xa j => DihedralGroup.sr j,\n            invFun := fun i =>\n              match i with\n              | DihedralGroup.r j => a j\n              | DihedralGroup.sr j => xa j,\n            left_inv :=\n              (_ :\n                \u2200 (x : QuaternionGroup 0),\n                  (fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      ((fun i =>\n                          match i with\n                          | a j => DihedralGroup.r j\n                          | xa j => DihedralGroup.sr j)\n                        x) =\n                    x),\n            right_inv :=\n              (_ :\n                \u2200 (x : DihedralGroup 0),\n                  (fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      ((fun i =>\n                          match i with\n                          | DihedralGroup.r j => a j\n                          | DihedralGroup.sr j => xa j)\n                        x) =\n                    x) }\n          x *\n        Equiv.toFun\n          {\n            toFun := fun i =>\n              match i with\n              | a j => DihedralGroup.r j\n              | xa j => DihedralGroup.sr j,\n            invFun := fun i =>\n              match i with\n              | DihedralGroup.r j => a j\n              | DihedralGroup.sr j => xa j,\n            left_inv :=\n              (_ :\n                \u2200 (x : QuaternionGroup 0),\n                  (fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      ((fun i =>\n                          match i with\n                          | a j => DihedralGroup.r j\n                          | xa j => DihedralGroup.sr j)\n                        x) =\n                    x),\n            right_inv :=\n              (_ :\n                \u2200 (x : DihedralGroup 0),\n                  (fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      ((fun i =>\n                          match i with\n                          | DihedralGroup.r j => a j\n                          | DihedralGroup.sr j => xa j)\n                        x) =\n                    x) }\n          y\n[PROOFSTEP]\nrintro (k | k) (l | l)\n[GOAL]\ncase a.a\nn : \u2115\nk l : ZMod (2 * 0)\n\u22a2 Equiv.toFun\n      {\n        toFun := fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j,\n        invFun := fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j,\n        left_inv :=\n          (_ :\n            \u2200 (x : QuaternionGroup 0),\n              (fun i =>\n                    match i with\n                    | DihedralGroup.r j => a j\n                    | DihedralGroup.sr j => xa j)\n                  ((fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    x) =\n                x),\n        right_inv :=\n          (_ :\n            \u2200 (x : DihedralGroup 0),\n              (fun i =>\n                    match i with\n                    | a j => DihedralGroup.r j\n                    | xa j => DihedralGroup.sr j)\n                  ((fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    x) =\n                x) }\n      (a k * a l) =\n    Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (a k) *\n      Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (a l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase a.xa\nn : \u2115\nk l : ZMod (2 * 0)\n\u22a2 Equiv.toFun\n      {\n        toFun := fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j,\n        invFun := fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j,\n        left_inv :=\n          (_ :\n            \u2200 (x : QuaternionGroup 0),\n              (fun i =>\n                    match i with\n                    | DihedralGroup.r j => a j\n                    | DihedralGroup.sr j => xa j)\n                  ((fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    x) =\n                x),\n        right_inv :=\n          (_ :\n            \u2200 (x : DihedralGroup 0),\n              (fun i =>\n                    match i with\n                    | a j => DihedralGroup.r j\n                    | xa j => DihedralGroup.sr j)\n                  ((fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    x) =\n                x) }\n      (a k * xa l) =\n    Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (a k) *\n      Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (xa l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase xa.a\nn : \u2115\nk l : ZMod (2 * 0)\n\u22a2 Equiv.toFun\n      {\n        toFun := fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j,\n        invFun := fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j,\n        left_inv :=\n          (_ :\n            \u2200 (x : QuaternionGroup 0),\n              (fun i =>\n                    match i with\n                    | DihedralGroup.r j => a j\n                    | DihedralGroup.sr j => xa j)\n                  ((fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    x) =\n                x),\n        right_inv :=\n          (_ :\n            \u2200 (x : DihedralGroup 0),\n              (fun i =>\n                    match i with\n                    | a j => DihedralGroup.r j\n                    | xa j => DihedralGroup.sr j)\n                  ((fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    x) =\n                x) }\n      (xa k * a l) =\n    Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (xa k) *\n      Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (a l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase xa.xa\nn : \u2115\nk l : ZMod (2 * 0)\n\u22a2 Equiv.toFun\n      {\n        toFun := fun i =>\n          match i with\n          | a j => DihedralGroup.r j\n          | xa j => DihedralGroup.sr j,\n        invFun := fun i =>\n          match i with\n          | DihedralGroup.r j => a j\n          | DihedralGroup.sr j => xa j,\n        left_inv :=\n          (_ :\n            \u2200 (x : QuaternionGroup 0),\n              (fun i =>\n                    match i with\n                    | DihedralGroup.r j => a j\n                    | DihedralGroup.sr j => xa j)\n                  ((fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    x) =\n                x),\n        right_inv :=\n          (_ :\n            \u2200 (x : DihedralGroup 0),\n              (fun i =>\n                    match i with\n                    | a j => DihedralGroup.r j\n                    | xa j => DihedralGroup.sr j)\n                  ((fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    x) =\n                x) }\n      (xa k * xa l) =\n    Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (xa k) *\n      Equiv.toFun\n        {\n          toFun := fun i =>\n            match i with\n            | a j => DihedralGroup.r j\n            | xa j => DihedralGroup.sr j,\n          invFun := fun i =>\n            match i with\n            | DihedralGroup.r j => a j\n            | DihedralGroup.sr j => xa j,\n          left_inv :=\n            (_ :\n              \u2200 (x : QuaternionGroup 0),\n                (fun i =>\n                      match i with\n                      | DihedralGroup.r j => a j\n                      | DihedralGroup.sr j => xa j)\n                    ((fun i =>\n                        match i with\n                        | a j => DihedralGroup.r j\n                        | xa j => DihedralGroup.sr j)\n                      x) =\n                  x),\n          right_inv :=\n            (_ :\n              \u2200 (x : DihedralGroup 0),\n                (fun i =>\n                      match i with\n                      | a j => DihedralGroup.r j\n                      | xa j => DihedralGroup.sr j)\n                    ((fun i =>\n                        match i with\n                        | DihedralGroup.r j => a j\n                        | DihedralGroup.sr j => xa j)\n                      x) =\n                  x) }\n        (xa l)\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\n\u22a2 a 0 \u2260 xa 0\n[PROOFSTEP]\nrevert n\n[GOAL]\n\u22a2 \u2200 {n : \u2115}, a 0 \u2260 xa 0\n[PROOFSTEP]\nsimp\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\n\u22a2 Fintype.card (QuaternionGroup n) = 4 * n\n[PROOFSTEP]\nrw [\u2190 Fintype.card_eq.mpr \u27e8fintypeHelper\u27e9, Fintype.card_sum, ZMod.card, two_mul]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\n\u22a2 n + n + (n + n) = 4 * n\n[PROOFSTEP]\nring\n[GOAL]\nn k : \u2115\n\u22a2 a 1 ^ k = a \u2191k\n[PROOFSTEP]\ninduction' k with k IH\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 a 1 ^ Nat.zero = a \u2191Nat.zero\n[PROOFSTEP]\nrw [Nat.cast_zero]\n[GOAL]\ncase zero\nn : \u2115\n\u22a2 a 1 ^ Nat.zero = a 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\nn k : \u2115\nIH : a 1 ^ k = a \u2191k\n\u22a2 a 1 ^ Nat.succ k = a \u2191(Nat.succ k)\n[PROOFSTEP]\nrw [pow_succ, IH, a_mul_a]\n[GOAL]\ncase succ\nn k : \u2115\nIH : a 1 ^ k = a \u2191k\n\u22a2 a (1 + \u2191k) = a \u2191(Nat.succ k)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase succ.e_a\nn k : \u2115\nIH : a 1 ^ k = a \u2191k\n\u22a2 1 + \u2191k = \u2191(Nat.succ k)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase succ.e_a\nn k : \u2115\nIH : a 1 ^ k = a \u2191k\n\u22a2 \u2191(1 + k) = \u2191(Nat.succ k)\n[PROOFSTEP]\nrw [Nat.one_add]\n[GOAL]\nn : \u2115\n\u22a2 a 1 ^ (2 * n) = 1\n[PROOFSTEP]\nrw [a_one_pow, one_def]\n[GOAL]\nn : \u2115\n\u22a2 a \u2191(2 * n) = a 0\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nn : \u2115\n\u22a2 \u2191(2 * n) = 0\n[PROOFSTEP]\nexact ZMod.nat_cast_self _\n[GOAL]\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 xa i ^ 2 = a \u2191n\n[PROOFSTEP]\nsimp [sq]\n[GOAL]\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 xa i ^ 4 = 1\n[PROOFSTEP]\nrw [pow_succ, pow_succ, sq, xa_mul_xa, xa_mul_a, xa_mul_xa, add_sub_cancel, add_sub_assoc, add_sub_cancel']\n[GOAL]\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 a (\u2191n + \u2191n) = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 a \u2191(n + n) = 1\n[PROOFSTEP]\nrw [\u2190 two_mul]\n[GOAL]\nn : \u2115\ni : ZMod (2 * n)\n\u22a2 a \u2191(2 * n) = 1\n[PROOFSTEP]\nsimp [one_def]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n\u22a2 orderOf (xa i) = 4\n[PROOFSTEP]\nchange _ = 2 ^ 2\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n\u22a2 orderOf (xa i) = 2 ^ 2\n[PROOFSTEP]\nhaveI : Fact (Nat.Prime 2) := Fact.mk Nat.prime_two\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n\u22a2 orderOf (xa i) = 2 ^ 2\n[PROOFSTEP]\napply orderOf_eq_prime_pow\n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n\u22a2 \u00acxa i ^ 2 ^ 1 = 1\n[PROOFSTEP]\nintro h\n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : xa i ^ 2 ^ 1 = 1\n\u22a2 False\n[PROOFSTEP]\nsimp only [pow_one, xa_sq] at h \n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : a \u2191n = 1\n\u22a2 False\n[PROOFSTEP]\ninjection h with h'\n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : \u2191n = 0\n\u22a2 False\n[PROOFSTEP]\napply_fun ZMod.val at h' \n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ZMod.val \u2191n = ZMod.val 0\n\u22a2 False\n[PROOFSTEP]\napply_fun (\u00b7 / n) at h' \n[GOAL]\ncase hnot\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ZMod.val \u2191n / n = ZMod.val 0 / n\n\u22a2 False\n[PROOFSTEP]\nsimp only [ZMod.val_nat_cast, ZMod.val_zero, Nat.zero_div, Nat.mod_mul_left_div_self, Nat.div_self (NeZero.pos n)] at h' \n[GOAL]\ncase hfin\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n\u22a2 xa i ^ 2 ^ (1 + 1) = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\n\u22a2 IsCyclic (QuaternionGroup 1)\n[PROOFSTEP]\napply isCyclic_of_orderOf_eq_card\n[GOAL]\ncase hx\nn : \u2115\n\u22a2 orderOf ?x = Fintype.card (QuaternionGroup 1)\ncase x n : \u2115 \u22a2 QuaternionGroup 1\n[PROOFSTEP]\nrw [card, mul_one]\n[GOAL]\ncase hx\nn : \u2115\n\u22a2 orderOf ?x = 4\ncase x\nn : \u2115\n\u22a2 QuaternionGroup 1\ncase x\nn : \u2115\n\u22a2 QuaternionGroup 1\ncase x n : \u2115 \u22a2 QuaternionGroup 1\n[PROOFSTEP]\nexact orderOf_xa 0\n[GOAL]\nn : \u2115\n\u22a2 orderOf (a 1) = 2 * n\n[PROOFSTEP]\ncases' eq_zero_or_neZero n with hn hn\n[GOAL]\ncase inl\nn : \u2115\nhn : n = 0\n\u22a2 orderOf (a 1) = 2 * n\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase inl\n\u22a2 orderOf (a 1) = 2 * 0\n[PROOFSTEP]\nsimp_rw [mul_zero, orderOf_eq_zero_iff']\n[GOAL]\ncase inl\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 QuaternionGroup.a 1 ^ n \u2260 1\n[PROOFSTEP]\nintro n h\n[GOAL]\ncase inl\nn : \u2115\nh : 0 < n\n\u22a2 a 1 ^ n \u2260 1\n[PROOFSTEP]\nrw [one_def, a_one_pow]\n[GOAL]\ncase inl\nn : \u2115\nh : 0 < n\n\u22a2 a \u2191n \u2260 a 0\n[PROOFSTEP]\napply mt a.inj\n[GOAL]\ncase inl\nn : \u2115\nh : 0 < n\n\u22a2 \u00ac\u2191n = 0\n[PROOFSTEP]\nhaveI : CharZero (ZMod (2 * 0)) := ZMod.charZero\n[GOAL]\ncase inl\nn : \u2115\nh : 0 < n\nthis : CharZero (ZMod (2 * 0))\n\u22a2 \u00ac\u2191n = 0\n[PROOFSTEP]\nsimpa using h.ne'\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\n\u22a2 orderOf (a 1) = 2 * n\n[PROOFSTEP]\napply (Nat.le_of_dvd (NeZero.pos _) (orderOf_dvd_of_pow_eq_one (@a_one_pow_n n))).lt_or_eq.resolve_left\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\n\u22a2 \u00acorderOf (a 1) < 2 * n\n[PROOFSTEP]\nintro h\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\nh : orderOf (a 1) < 2 * n\n\u22a2 False\n[PROOFSTEP]\nhave h1 : (a 1 : QuaternionGroup n) ^ orderOf (a 1) = 1 := pow_orderOf_eq_one _\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\nh : orderOf (a 1) < 2 * n\nh1 : a 1 ^ orderOf (a 1) = 1\n\u22a2 False\n[PROOFSTEP]\nrw [a_one_pow] at h1 \n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\nh : orderOf (a 1) < 2 * n\nh1 : a \u2191(orderOf (a 1)) = 1\n\u22a2 False\n[PROOFSTEP]\ninjection h1 with h2\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\nh : orderOf (a 1) < 2 * n\nh2 : \u2191(orderOf (a 1)) = 0\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 ZMod.val_eq_zero, ZMod.val_nat_cast, Nat.mod_eq_of_lt h] at h2 \n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\nh : orderOf (a 1) < 2 * n\nh2 : orderOf (a 1) = 0\n\u22a2 False\n[PROOFSTEP]\nexact absurd h2.symm (orderOf_pos _).ne\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n\u22a2 orderOf (a i) = 2 * n / Nat.gcd (2 * n) (ZMod.val i)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 ZMod.nat_cast_zmod_val i]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n| orderOf (a i)\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val i]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n| orderOf (a i)\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val i]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n| orderOf (a i)\n[PROOFSTEP]\nrw [\u2190 ZMod.nat_cast_zmod_val i]\n[GOAL]\nn : \u2115\ninst\u271d : NeZero n\ni : ZMod (2 * n)\n\u22a2 orderOf (a \u2191(ZMod.val i)) = 2 * n / Nat.gcd (2 * n) (ZMod.val i)\n[PROOFSTEP]\nrw [\u2190 a_one_pow, orderOf_pow, orderOf_a_one]\n[GOAL]\nn : \u2115\n\u22a2 Monoid.exponent (QuaternionGroup n) = 2 * lcm n 2\n[PROOFSTEP]\nrw [\u2190 normalize_eq 2, \u2190 lcm_mul_left, normalize_eq]\n[GOAL]\nn : \u2115\n\u22a2 Monoid.exponent (QuaternionGroup n) = lcm (2 * n) (2 * 2)\n[PROOFSTEP]\nnorm_num\n[GOAL]\nn : \u2115\n\u22a2 Monoid.exponent (QuaternionGroup n) = lcm (2 * n) 4\n[PROOFSTEP]\ncases' eq_zero_or_neZero n with hn hn\n[GOAL]\ncase inl\nn : \u2115\nhn : n = 0\n\u22a2 Monoid.exponent (QuaternionGroup n) = lcm (2 * n) 4\n[PROOFSTEP]\nsubst hn\n[GOAL]\ncase inl\n\u22a2 Monoid.exponent (QuaternionGroup 0) = lcm (2 * 0) 4\n[PROOFSTEP]\nsimp only [lcm_zero_left, mul_zero]\n[GOAL]\ncase inl\n\u22a2 Monoid.exponent (QuaternionGroup 0) = 0\n[PROOFSTEP]\nexact Monoid.exponent_eq_zero_of_order_zero orderOf_a_one\n[GOAL]\ncase inr\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (QuaternionGroup n) = lcm (2 * n) 4\n[PROOFSTEP]\napply Nat.dvd_antisymm\n[GOAL]\ncase inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 Monoid.exponent (QuaternionGroup n) \u2223 lcm (2 * n) 4\n[PROOFSTEP]\napply Monoid.exponent_dvd_of_forall_pow_eq_one\n[GOAL]\ncase inr.a.hG\nn : \u2115\nhn : NeZero n\n\u22a2 \u2200 (g : QuaternionGroup n), g ^ lcm (2 * n) 4 = 1\n[PROOFSTEP]\nrintro (m | m)\n[GOAL]\ncase inr.a.hG.a\nn : \u2115\nhn : NeZero n\nm : ZMod (2 * n)\n\u22a2 a m ^ lcm (2 * n) 4 = 1\n[PROOFSTEP]\nrw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_a]\n[GOAL]\ncase inr.a.hG.a\nn : \u2115\nhn : NeZero n\nm : ZMod (2 * n)\n\u22a2 2 * n / Nat.gcd (2 * n) (ZMod.val m) \u2223 lcm (2 * n) 4\n[PROOFSTEP]\nrefine' Nat.dvd_trans \u27e8gcd (2 * n) m.val, _\u27e9 (dvd_lcm_left (2 * n) 4)\n[GOAL]\ncase inr.a.hG.a\nn : \u2115\nhn : NeZero n\nm : ZMod (2 * n)\n\u22a2 2 * n = 2 * n / Nat.gcd (2 * n) (ZMod.val m) * gcd (2 * n) (ZMod.val m)\n[PROOFSTEP]\nexact (Nat.div_mul_cancel (Nat.gcd_dvd_left (2 * n) m.val)).symm\n[GOAL]\ncase inr.a.hG.xa\nn : \u2115\nhn : NeZero n\nm : ZMod (2 * n)\n\u22a2 xa m ^ lcm (2 * n) 4 = 1\n[PROOFSTEP]\nrw [\u2190 orderOf_dvd_iff_pow_eq_one, orderOf_xa]\n[GOAL]\ncase inr.a.hG.xa\nn : \u2115\nhn : NeZero n\nm : ZMod (2 * n)\n\u22a2 4 \u2223 lcm (2 * n) 4\n[PROOFSTEP]\nexact dvd_lcm_right (2 * n) 4\n[GOAL]\ncase inr.a\nn : \u2115\nhn : NeZero n\n\u22a2 lcm (2 * n) 4 \u2223 Monoid.exponent (QuaternionGroup n)\n[PROOFSTEP]\napply lcm_dvd\n[GOAL]\ncase inr.a.hab\nn : \u2115\nhn : NeZero n\n\u22a2 2 * n \u2223 Monoid.exponent (QuaternionGroup n)\n[PROOFSTEP]\nconvert Monoid.order_dvd_exponent (a 1)\n[GOAL]\ncase h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 2 * n = orderOf (a 1)\n[PROOFSTEP]\nexact orderOf_a_one.symm\n[GOAL]\ncase inr.a.hcb\nn : \u2115\nhn : NeZero n\n\u22a2 4 \u2223 Monoid.exponent (QuaternionGroup n)\n[PROOFSTEP]\nconvert Monoid.order_dvd_exponent (xa (0 : ZMod (2 * n)))\n[GOAL]\ncase h.e'_3\nn : \u2115\nhn : NeZero n\n\u22a2 4 = orderOf (xa 0)\n[PROOFSTEP]\nexact (orderOf_xa 0).symm\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.SpecificGroups.Quaternion", "llama_tokens": 13339, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8479677583778257, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.5057560780602484}}
{"text": "[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : Fin n \u2192 \u2102\nR \u03b8 : Fin n \u2192 \u211d\n\u22a2 torusMap c R \u03b8 - c = torusMap 0 R \u03b8\n[PROOFSTEP]\next1 i\n[GOAL]\ncase h\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : Fin n \u2192 \u2102\nR \u03b8 : Fin n \u2192 \u211d\ni : Fin n\n\u22a2 (torusMap c R \u03b8 - c) i = torusMap 0 R \u03b8 i\n[PROOFSTEP]\nsimp [torusMap]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nc : Fin n \u2192 \u2102\nR \u03b8 : Fin n \u2192 \u211d\n\u22a2 torusMap c R \u03b8 = c \u2194 R = 0\n[PROOFSTEP]\nsimp [funext_iff, torusMap, exp_ne_zero]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\na : E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\n\u22a2 TorusIntegrable (fun x => a) c R\n[PROOFSTEP]\nsimp [TorusIntegrable, measure_Icc_lt_top]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 TorusIntegrable f c 0\n[PROOFSTEP]\nrw [TorusIntegrable, torusMap_zero_radius]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d : NormedAddCommGroup E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 IntegrableOn (fun \u03b8 => f (const (Fin n \u2192 \u211d) c \u03b8)) (Icc 0 fun x => 2 * \u03c0)\n[PROOFSTEP]\napply torusIntegrable_const (f c) c 0\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\ninst\u271d : NormedSpace \u2102 E\nhf : TorusIntegrable f c R\n\u22a2 IntegrableOn (fun \u03b8 => (\u220f i : Fin n, \u2191(R i) * exp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8)) (Icc 0 fun x => 2 * \u03c0)\n[PROOFSTEP]\nrefine' (hf.norm.const_mul (\u220f i, |R i|)).mono' _ _\n[GOAL]\ncase refine'_1\nn : \u2115\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\ninst\u271d : NormedSpace \u2102 E\nhf : TorusIntegrable f c R\n\u22a2 AEStronglyMeasurable (fun \u03b8 => (\u220f i : Fin n, \u2191(R i) * exp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8))\n    (Measure.restrict volume (Icc 0 fun x => 2 * \u03c0))\n[PROOFSTEP]\nrefine (Continuous.aestronglyMeasurable ?_).smul hf.1\n[GOAL]\ncase refine'_1\nn : \u2115\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\ninst\u271d : NormedSpace \u2102 E\nhf : TorusIntegrable f c R\n\u22a2 Continuous fun \u03b8 => \u220f i : Fin n, \u2191(R i) * exp (\u2191(\u03b8 i) * I) * I\n[PROOFSTEP]\ncontinuity\n[GOAL]\ncase refine'_2\nn : \u2115\nE : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\ninst\u271d : NormedSpace \u2102 E\nhf : TorusIntegrable f c R\n\u22a2 \u2200\u1d50 (a : Fin n \u2192 \u211d) \u2202Measure.restrict volume (Icc 0 fun x => 2 * \u03c0),\n    \u2016(\u220f i : Fin n, \u2191(R i) * exp (\u2191(a i) * I) * I) \u2022 f (torusMap c R a)\u2016 \u2264 (\u220f i : Fin n, |R i|) * \u2016f (torusMap c R a)\u2016\n[PROOFSTEP]\nsimp [norm_smul, map_prod]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nhn : n \u2260 0\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, 0), f x) = 0\n[PROOFSTEP]\nsimp only [torusIntegral, Pi.zero_apply, ofReal_zero, mul_zero, zero_mul, Fin.prod_const, zero_pow' n hn, zero_smul,\n  integral_zero]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), -f x) = -\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x\n[PROOFSTEP]\nsimp [torusIntegral, integral_neg]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x + g x) = (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x) + \u222f (x : Fin n \u2192 \u2102) in T(c, R), g x\n[PROOFSTEP]\nsimpa only [torusIntegral, smul_add, Pi.add_apply] using integral_add hf.function_integrable hg.function_integrable\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x - g x) = (\u222f (x : Fin n \u2192 \u2102) in T(c, R), f x) - \u222f (x : Fin n \u2192 \u2102) in T(c, R), g x\n[PROOFSTEP]\nsimpa only [sub_eq_add_neg, \u2190 torusIntegral_neg] using torusIntegral_add hf hg.neg\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u2102 E\ninst\u271d\u00b3 : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\n\ud835\udd5c : Type u_2\ninst\u271d\u00b2 : IsROrC \ud835\udd5c\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : SMulCommClass \ud835\udd5c \u2102 E\na : \ud835\udd5c\nf : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\n\u22a2 (\u222f (x : Fin n \u2192 \u2102) in T(c, R), a \u2022 f x) = a \u2022 \u222f (x : Fin n \u2192 \u2102) in T(c, R), f x\n[PROOFSTEP]\nsimp only [torusIntegral, integral_smul, \u2190 smul_comm a (_ : \u2102) (_ : E)]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nC : \u211d\nhf : \u2200 (\u03b8 : Fin n \u2192 \u211d), \u2016f (torusMap c R \u03b8)\u2016 \u2264 C\n\u03b8 : Fin n \u2192 \u211d\nx\u271d : \u03b8 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 \u2016(\u220f i : Fin n, \u2191(R i) * exp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8)\u2016 = (\u220f i : Fin n, |R i|) * \u2016f (torusMap c R \u03b8)\u2016\n[PROOFSTEP]\nsimp [norm_smul]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf g : (Fin n \u2192 \u2102) \u2192 E\nc : Fin n \u2192 \u2102\nR : Fin n \u2192 \u211d\nC : \u211d\nhf : \u2200 (\u03b8 : Fin n \u2192 \u211d), \u2016f (torusMap c R \u03b8)\u2016 \u2264 C\n\u22a2 (\u220f i : Fin n, |R i|) * C * ENNReal.toReal (\u2191\u2191volume (Icc 0 fun x => 2 * \u03c0)) = ((2 * \u03c0) ^ n * \u220f i : Fin n, |R i|) * C\n[PROOFSTEP]\nsimp only [Pi.zero_def, Real.volume_Icc_pi_toReal fun _ => Real.two_pi_pos.le, sub_zero, Fin.prod_const, mul_assoc,\n  mul_comm ((2 * \u03c0) ^ (n : \u2115))]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 0 \u2192 \u2102) \u2192 E\nc : Fin 0 \u2192 \u2102\nR : Fin 0 \u2192 \u211d\n\u22a2 (\u222f (x : Fin 0 \u2192 \u2102) in T(c, R), f x) = f c\n[PROOFSTEP]\nsimp only [torusIntegral, Fin.prod_univ_zero, one_smul, Subsingleton.elim (fun _ : Fin 0 => 2 * \u03c0) 0, Icc_self,\n  Measure.restrict_singleton, volume_pi, integral_smul_measure, integral_dirac,\n  Measure.pi_of_empty (fun _ : Fin 0 \u21a6 volume) 0, Measure.dirac_apply_of_mem (mem_singleton _),\n  Subsingleton.elim (torusMap c R 0) c]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\n\u22a2 (\u222f (x : Fin 1 \u2192 \u2102) in T(c, R), f x) = \u222e (z : \u2102) in C(c 0, R 0), f fun x => z\n[PROOFSTEP]\nhave H\u2081 : (((MeasurableEquiv.funUnique _ _).symm) \u207b\u00b9' Icc 0 fun _ => 2 * \u03c0) = Icc 0 (2 * \u03c0) :=\n  (OrderIso.funUnique (Fin 1) \u211d).symm.preimage_Icc _ _\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\n\u22a2 (\u222f (x : Fin 1 \u2192 \u2102) in T(c, R), f x) = \u222e (z : \u2102) in C(c 0, R 0), f fun x => z\n[PROOFSTEP]\nhave H\u2082 : torusMap c R = fun \u03b8 _ \u21a6 circleMap (c 0) (R 0) (\u03b8 0) :=\n  by\n  ext \u03b8 i : 2\n  rw [Subsingleton.elim i 0]; rfl\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\n\u22a2 torusMap c R = fun \u03b8 x => circleMap (c 0) (R 0) (\u03b8 0)\n[PROOFSTEP]\next \u03b8 i : 2\n[GOAL]\ncase h.h\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\n\u03b8 : Fin 1 \u2192 \u211d\ni : Fin 1\n\u22a2 torusMap c R \u03b8 i = circleMap (c 0) (R 0) (\u03b8 0)\n[PROOFSTEP]\nrw [Subsingleton.elim i 0]\n[GOAL]\ncase h.h\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\n\u03b8 : Fin 1 \u2192 \u211d\ni : Fin 1\n\u22a2 torusMap c R \u03b8 0 = circleMap (c 0) (R 0) (\u03b8 0)\n[PROOFSTEP]\nrfl\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\nH\u2082 : torusMap c R = fun \u03b8 x => circleMap (c 0) (R 0) (\u03b8 0)\n\u22a2 (\u222f (x : Fin 1 \u2192 \u2102) in T(c, R), f x) = \u222e (z : \u2102) in C(c 0, R 0), f fun x => z\n[PROOFSTEP]\nrw [torusIntegral, circleIntegral, intervalIntegral.integral_of_le Real.two_pi_pos.le,\n  Measure.restrict_congr_set Ioc_ae_eq_Icc, \u2190\n  ((volume_preserving_funUnique (Fin 1) \u211d).symm _).set_integral_preimage_emb (MeasurableEquiv.measurableEmbedding _),\n  H\u2081, H\u2082]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin 1 \u2192 \u2102) \u2192 E\nc : Fin 1 \u2192 \u2102\nR : Fin 1 \u2192 \u211d\nH\u2081 : (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0)\nH\u2082 : torusMap c R = fun \u03b8 x => circleMap (c 0) (R 0) (\u03b8 0)\n\u22a2 \u222b (x : \u211d) in Icc 0 (2 * \u03c0),\n      (\u220f i : Fin 1, \u2191(R i) * exp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) x i) * I) * I) \u2022\n        f ((fun \u03b8 x => circleMap (c 0) (R 0) (\u03b8 0)) (\u2191(MeasurableEquiv.symm (MeasurableEquiv.funUnique (Fin 1) \u211d)) x)) =\n    \u222b (x : \u211d) in Icc 0 (2 * \u03c0), deriv (circleMap (c 0) (R 0)) x \u2022 f fun x_1 => circleMap (c 0) (R 0) x\n[PROOFSTEP]\nsimp [circleMap_zero]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)\n[PROOFSTEP]\nset e : \u211d \u00d7 \u211d\u207f \u2243\u1d50 \u211d\u207f\u207a\u00b9 := (MeasurableEquiv.piFinSuccAboveEquiv (fun _ => \u211d) i).symm\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)\n[PROOFSTEP]\nhave hem : MeasurePreserving e := (volume_preserving_piFinSuccAboveEquiv (fun _ : Fin (n + 1) => \u211d) i).symm _\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)\n[PROOFSTEP]\nhave he\u03c0 : (e \u207b\u00b9' Icc 0 fun _ => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc (0 : \u211d\u207f) fun _ => 2 * \u03c0 :=\n  ((OrderIso.piFinSuccAboveIso (fun _ => \u211d) i).symm.preimage_Icc _ _).trans (Icc_prod_eq _ _)\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c i, R i), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i x y)\n[PROOFSTEP]\nrw [torusIntegral, \u2190 hem.map_eq, set_integral_map_equiv, he\u03c0, Measure.volume_eq_prod, set_integral_prod,\n  circleIntegral_def_Icc]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 \u222b (x : \u211d) in Icc 0 (2 * \u03c0),\n      \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n        (\u220f i : Fin (n + 1), \u2191(R i) * exp (\u2191(\u2191e (x, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (x, y))) =\n    \u222b (\u03b8 : \u211d) in Icc 0 (2 * \u03c0),\n      deriv (circleMap (c i) (R i)) \u03b8 \u2022\n        \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i),\n          f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Icc fun \u03b8 _ => _\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 \u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u220f i : Fin (n + 1), \u2191(R i) * exp (\u2191(\u2191e (\u03b8, y) i) * I) * I) \u2022 f (torusMap c R (\u2191e (\u03b8, y))) =\n    deriv (circleMap (c i) (R i)) \u03b8 \u2022\n      \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succAbove i, R \u2218 Fin.succAbove i), f (Fin.insertNth i (circleMap (c i) (R i) \u03b8) y)\n[PROOFSTEP]\nsimp only [torusIntegral, \u2190 integral_smul, deriv_circleMap, i.prod_univ_succAbove _, smul_smul, torusMap,\n  circleMap_zero]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u22a2 (\u222b (y : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * exp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i) * I) * I *\n          \u220f i_1 : Fin n,\n            \u2191(R (Fin.succAbove i i_1)) *\n                exp\n                  (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y)\n                        (Fin.succAbove i i_1)) *\n                    I) *\n              I) \u2022\n        f fun i_1 =>\n          c i_1 +\n            \u2191(R i_1) *\n              exp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, y) i_1) * I)) =\n    \u222b (a : Fin n \u2192 \u211d) in Icc 0 fun x => 2 * \u03c0,\n      (\u2191(R i) * exp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * exp (\u2191(a i_1) * I) * I) \u2022\n        f\n          (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n            (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * exp (\u2191(a i_1) * I))\n[PROOFSTEP]\nrefine' set_integral_congr measurableSet_Icc fun \u0398 _ => _\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 ((\u2191(R i) * exp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398) i) * I) * I *\n        \u220f i_1 : Fin n,\n          \u2191(R (Fin.succAbove i i_1)) *\n              exp\n                (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398)\n                      (Fin.succAbove i i_1)) *\n                  I) *\n            I) \u2022\n      f fun i_1 =>\n        c i_1 +\n          \u2191(R i_1) *\n            exp (\u2191(\u2191(MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)) (\u03b8, \u0398) i_1) * I)) =\n    (\u2191(R i) * exp (\u2191\u03b8 * I) * I * \u220f i_1 : Fin n, \u2191((R \u2218 Fin.succAbove i) i_1) * exp (\u2191(\u0398 i_1) * I) * I) \u2022\n      f\n        (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n          (c \u2218 Fin.succAbove i) i_1 + \u2191((R \u2218 Fin.succAbove i) i_1) * exp (\u2191(\u0398 i_1) * I))\n[PROOFSTEP]\nsimp only [MeasurableEquiv.piFinSuccAboveEquiv_symm_apply, i.insertNth_apply_same, i.insertNth_apply_succAbove, (\u00b7 \u2218 \u00b7)]\n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 ((\u2191(R i) * exp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * exp (\u2191(\u0398 x) * I) * I) \u2022\n      f fun i_1 => c i_1 + \u2191(R i_1) * exp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =\n    (\u2191(R i) * exp (\u2191\u03b8 * I) * I * \u220f x : Fin n, \u2191(R (Fin.succAbove i x)) * exp (\u2191(\u0398 x) * I) * I) \u2022\n      f\n        (Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n          c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * exp (\u2191(\u0398 i_1) * I))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u03b8 : \u211d\nx\u271d\u00b9 : \u03b8 \u2208 Icc 0 (2 * \u03c0)\n\u0398 : Fin n \u2192 \u211d\nx\u271d : \u0398 \u2208 Icc 0 fun x => 2 * \u03c0\n\u22a2 (fun i_1 => c i_1 + \u2191(R i_1) * exp (\u2191(Fin.insertNth i \u03b8 \u0398 i_1) * I)) =\n    Fin.insertNth i (circleMap (c i) (R i) \u03b8) fun i_1 =>\n      c (Fin.succAbove i i_1) + \u2191(R (Fin.succAbove i i_1)) * exp (\u2191(\u0398 i_1) * I)\n[PROOFSTEP]\nsimp only [funext_iff, i.forall_iff_succAbove, circleMap, Fin.insertNth_apply_same, eq_self_iff_true,\n  Fin.insertNth_apply_succAbove, imp_true_iff, and_self_iff]\n[GOAL]\ncase hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * exp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)\n[PROOFSTEP]\nhave := hf.function_integrable\n[GOAL]\ncase hf\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\ni : Fin (n + 1)\ne : \u211d \u00d7 (Fin n \u2192 \u211d) \u2243\u1d50 (Fin (n + 1) \u2192 \u211d) := MeasurableEquiv.symm (MeasurableEquiv.piFinSuccAboveEquiv (fun x => \u211d) i)\nhem : MeasurePreserving \u2191e\nhe\u03c0 : (\u2191e \u207b\u00b9' Icc 0 fun x => 2 * \u03c0) = Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0\nthis :\n  IntegrableOn (fun \u03b8 => (\u220f i : Fin (n + 1), \u2191(R i) * exp (\u2191(\u03b8 i) * I) * I) \u2022 f (torusMap c R \u03b8)) (Icc 0 fun x => 2 * \u03c0)\n\u22a2 IntegrableOn (fun x => (\u220f i : Fin (n + 1), \u2191(R i) * exp (\u2191(\u2191e x i) * I) * I) \u2022 f (torusMap c R (\u2191e x)))\n    (Icc 0 (2 * \u03c0) \u00d7\u02e2 Icc 0 fun x => 2 * \u03c0)\n[PROOFSTEP]\nrwa [\u2190 hem.integrableOn_comp_preimage e.measurableEmbedding, he\u03c0] at this \n[GOAL]\nn : \u2115\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u2102 E\ninst\u271d : CompleteSpace E\nf\u271d g : (Fin n \u2192 \u2102) \u2192 E\nc\u271d : Fin n \u2192 \u2102\nR\u271d : Fin n \u2192 \u211d\nf : (Fin (n + 1) \u2192 \u2102) \u2192 E\nc : Fin (n + 1) \u2192 \u2102\nR : Fin (n + 1) \u2192 \u211d\nhf : TorusIntegrable f c R\n\u22a2 (\u222f (x : Fin (n + 1) \u2192 \u2102) in T(c, R), f x) =\n    \u222e (x : \u2102) in C(c 0, R 0), \u222f (y : Fin n \u2192 \u2102) in T(c \u2218 Fin.succ, R \u2218 Fin.succ), f (Fin.cons x y)\n[PROOFSTEP]\nsimpa using torusIntegral_succAbove hf 0\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Integral.TorusIntegral", "llama_tokens": 11801, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117983401363, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.5054366906147488}}
{"text": "[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\n\u22a2 Subgraph.Connected H \u2194 Subgraph.Preconnected H \u2227 Set.Nonempty H.verts\n[PROOFSTEP]\nrw [H.connected_iff', connected_iff, H.preconnected_iff, Set.nonempty_coe_sort]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Connected H\n\u22a2 Subgraph.Preconnected H\n[PROOFSTEP]\nrw [H.connected_iff] at h \n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Preconnected H \u2227 Set.Nonempty H.verts\n\u22a2 Subgraph.Preconnected H\n[PROOFSTEP]\nexact h.1\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Connected H\n\u22a2 Set.Nonempty H.verts\n[PROOFSTEP]\nrw [H.connected_iff] at h \n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Preconnected H \u2227 Set.Nonempty H.verts\n\u22a2 Set.Nonempty H.verts\n[PROOFSTEP]\nexact h.2\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv : V\n\u22a2 Subgraph.Connected (SimpleGraph.singletonSubgraph G v)\n[PROOFSTEP]\nrefine \u27e8\u27e8?_\u27e9\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv : V\n\u22a2 Preconnected (Subgraph.coe (SimpleGraph.singletonSubgraph G v))\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv a : V\nha : a \u2208 (SimpleGraph.singletonSubgraph G v).verts\nb : V\nhb : b \u2208 (SimpleGraph.singletonSubgraph G v).verts\n\u22a2 Reachable (Subgraph.coe (SimpleGraph.singletonSubgraph G v)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nsimp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb \n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv a : V\nha\u271d : a \u2208 (SimpleGraph.singletonSubgraph G v).verts\nb : V\nhb\u271d : b \u2208 (SimpleGraph.singletonSubgraph G v).verts\nha : a = v\nhb : b = v\n\u22a2 Reachable (Subgraph.coe (SimpleGraph.singletonSubgraph G v)) { val := a, property := ha\u271d }\n    { val := b, property := hb\u271d }\n[PROOFSTEP]\nsubst_vars\n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nb : V\nha hb : b = b\n\u22a2 Reachable (Subgraph.coe (SimpleGraph.singletonSubgraph G b)) { val := b, property := (_ : b = b) }\n    { val := b, property := (_ : b = b) }\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : SimpleGraph.Adj G v w\n\u22a2 Subgraph.Connected (subgraphOfAdj G hvw)\n[PROOFSTEP]\nrefine \u27e8\u27e8?_\u27e9\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : SimpleGraph.Adj G v w\n\u22a2 Preconnected (Subgraph.coe (subgraphOfAdj G hvw))\n[PROOFSTEP]\nrintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9\n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : SimpleGraph.Adj G v w\na : V\nha : a \u2208 (subgraphOfAdj G hvw).verts\nb : V\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nsimp only [subgraphOfAdj_verts, Set.mem_insert_iff, Set.mem_singleton_iff] at ha hb \n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\nhvw : SimpleGraph.Adj G v w\na : V\nha\u271d : a \u2208 (subgraphOfAdj G hvw).verts\nb : V\nhb\u271d : b \u2208 (subgraphOfAdj G hvw).verts\nha : a = v \u2228 a = w\nhb : b = v \u2228 b = w\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha\u271d } { val := b, property := hb\u271d }\n[PROOFSTEP]\nobtain rfl | rfl := ha\n[GOAL]\ncase mk.mk.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nw a b : V\nhvw : SimpleGraph.Adj G a w\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb\u271d : b \u2208 (subgraphOfAdj G hvw).verts\nhb : b = a \u2228 b = w\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb\u271d }\n[PROOFSTEP]\nobtain rfl | rfl := hb\n[GOAL]\ncase mk.mk.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv a b : V\nhvw : SimpleGraph.Adj G v a\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb\u271d : b \u2208 (subgraphOfAdj G hvw).verts\nhb : b = v \u2228 b = a\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb\u271d }\n[PROOFSTEP]\nobtain rfl | rfl := hb\n[GOAL]\ncase mk.mk.inl.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nw b : V\nhvw : SimpleGraph.Adj G b w\nha hb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := b, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nfirst\n| rfl\n| (apply Adj.reachable; simp)\n[GOAL]\ncase mk.mk.inl.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nw b : V\nhvw : SimpleGraph.Adj G b w\nha hb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := b, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.inl.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G a b\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nfirst\n| rfl\n| (apply Adj.reachable; simp)\n[GOAL]\ncase mk.mk.inl.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G a b\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.inl.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G a b\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\napply Adj.reachable\n[GOAL]\ncase mk.mk.inl.inr.h\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G a b\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 SimpleGraph.Adj (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.inr.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G b a\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nfirst\n| rfl\n| (apply Adj.reachable; simp)\n[GOAL]\ncase mk.mk.inr.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G b a\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mk.mk.inr.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G b a\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\napply Adj.reachable\n[GOAL]\ncase mk.mk.inr.inl.h\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\na b : V\nhvw : SimpleGraph.Adj G b a\nha : a \u2208 (subgraphOfAdj G hvw).verts\nhb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 SimpleGraph.Adj (Subgraph.coe (subgraphOfAdj G hvw)) { val := a, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.inr.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv b : V\nhvw : SimpleGraph.Adj G v b\nha hb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := b, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nfirst\n| rfl\n| (apply Adj.reachable; simp)\n[GOAL]\ncase mk.mk.inr.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv b : V\nhvw : SimpleGraph.Adj G v b\nha hb : b \u2208 (subgraphOfAdj G hvw).verts\n\u22a2 Reachable (Subgraph.coe (subgraphOfAdj G hvw)) { val := b, property := ha } { val := b, property := hb }\n[PROOFSTEP]\nrfl\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nhuv : SimpleGraph.Adj G u v\n\u22a2 Subgraph.Connected (induce \u22a4 {u, v})\n[PROOFSTEP]\nrw [\u2190 subgraphOfAdj_eq_induce huv]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nhuv : SimpleGraph.Adj G u v\n\u22a2 Subgraph.Connected (subgraphOfAdj G huv)\n[PROOFSTEP]\nexact subgraphOfAdj_connected huv\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH H' : Subgraph G\nhle : H \u2264 H'\nhv : H.verts = H'.verts\nh : Subgraph.Connected H\n\u22a2 Subgraph.Connected H'\n[PROOFSTEP]\nrw [\u2190 Subgraph.copy_eq H' H.verts hv H'.Adj rfl]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH H' : Subgraph G\nhle : H \u2264 H'\nhv : H.verts = H'.verts\nh : Subgraph.Connected H\n\u22a2 Subgraph.Connected (copy H' H.verts hv H'.Adj (_ : H'.Adj = H'.Adj))\n[PROOFSTEP]\nrefine \u27e8h.coe.mono ?_\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH H' : Subgraph G\nhle : H \u2264 H'\nhv : H.verts = H'.verts\nh : Subgraph.Connected H\n\u22a2 Subgraph.coe H \u2264 Subgraph.coe (copy H' H.verts hv H'.Adj (_ : H'.Adj = H'.Adj))\n[PROOFSTEP]\nrintro \u27e8v, hv\u27e9 \u27e8w, hw\u27e9 hvw\n[GOAL]\ncase mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH H' : Subgraph G\nhle : H \u2264 H'\nhv\u271d : H.verts = H'.verts\nh : Subgraph.Connected H\nv : V\nhv : v \u2208 H.verts\nw : V\nhw : w \u2208 H.verts\nhvw : SimpleGraph.Adj (Subgraph.coe H) { val := v, property := hv } { val := w, property := hw }\n\u22a2 SimpleGraph.Adj (Subgraph.coe (copy H' H.verts hv\u271d H'.Adj (_ : H'.Adj = H'.Adj))) { val := v, property := hv }\n    { val := w, property := hw }\n[PROOFSTEP]\nexact hle.2 hvw\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH H' : Subgraph G\nhle : \u2200 (v w : V), Adj H v w \u2192 Adj H' v w\nhv : H.verts = H'.verts\nh : Subgraph.Connected H\n\u22a2 Subgraph.Connected H'\n[PROOFSTEP]\nexact h.mono \u27e8hv.le, hle\u27e9 hv\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nhn : Set.Nonempty (H \u2293 K).verts\n\u22a2 Subgraph.Connected (H \u2294 K)\n[PROOFSTEP]\nrw [Subgraph.connected_iff', connected_iff_exists_forall_reachable]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nhn : Set.Nonempty (H \u2293 K).verts\n\u22a2 \u2203 v, \u2200 (w : \u2191(H \u2294 K).verts), Reachable (Subgraph.coe (H \u2294 K)) v w\n[PROOFSTEP]\nobtain \u27e8u, hu, hu'\u27e9 := hn\n[GOAL]\ncase intro.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nu : V\nhu : u \u2208 H.verts\nhu' : u \u2208 K.verts\n\u22a2 \u2203 v, \u2200 (w : \u2191(H \u2294 K).verts), Reachable (Subgraph.coe (H \u2294 K)) v w\n[PROOFSTEP]\nexists \u27e8u, Or.inl hu\u27e9\n[GOAL]\ncase intro.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nu : V\nhu : u \u2208 H.verts\nhu' : u \u2208 K.verts\n\u22a2 \u2200 (w : \u2191(H \u2294 K).verts), Reachable (Subgraph.coe (H \u2294 K)) { val := u, property := (_ : u \u2208 H.verts \u2228 u \u2208 K.verts) } w\n[PROOFSTEP]\nrintro \u27e8v, (hv | hv)\u27e9\n[GOAL]\ncase intro.intro.mk.inl\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nu : V\nhu : u \u2208 H.verts\nhu' : u \u2208 K.verts\nv : V\nhv : v \u2208 H.verts\n\u22a2 Reachable (Subgraph.coe (H \u2294 K)) { val := u, property := (_ : u \u2208 H.verts \u2228 u \u2208 K.verts) }\n    { val := v, property := (_ : v \u2208 H.verts \u2228 v \u2208 K.verts) }\n[PROOFSTEP]\nexact Reachable.map (Subgraph.inclusion (le_sup_left : H \u2264 H \u2294 K)) (hH \u27e8u, hu\u27e9 \u27e8v, hv\u27e9)\n[GOAL]\ncase intro.intro.mk.inr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nhH : Subgraph.Connected H\nhK : Subgraph.Connected K\nu : V\nhu : u \u2208 H.verts\nhu' : u \u2208 K.verts\nv : V\nhv : v \u2208 K.verts\n\u22a2 Reachable (Subgraph.coe (H \u2294 K)) { val := u, property := (_ : u \u2208 H.verts \u2228 u \u2208 K.verts) }\n    { val := v, property := (_ : v \u2208 H.verts \u2228 v \u2208 K.verts) }\n[PROOFSTEP]\nexact Reachable.map (Subgraph.inclusion (le_sup_right : K \u2264 H \u2294 K)) (hK \u27e8u, hu'\u27e9 \u27e8v, hv\u27e9)\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\np : Walk G u v\n\u22a2 Subgraph.Connected (Walk.toSubgraph p)\n[PROOFSTEP]\ninduction p with\n| nil => apply singletonSubgraph_connected\n| cons h p ih =>\n  apply (subgraphOfAdj_connected h).sup ih\n  rename_i w _\n  exists w\n  simp\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\np : Walk G u v\n\u22a2 Subgraph.Connected (Walk.toSubgraph p)\n[PROOFSTEP]\ninduction p with\n| nil => apply singletonSubgraph_connected\n| cons h p ih =>\n  apply (subgraphOfAdj_connected h).sup ih\n  rename_i w _\n  exists w\n  simp\n[GOAL]\ncase nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d : V\n\u22a2 Subgraph.Connected (Walk.toSubgraph Walk.nil)\n[PROOFSTEP]\n\n| nil => apply singletonSubgraph_connected\n[GOAL]\ncase nil\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d : V\n\u22a2 Subgraph.Connected (Walk.toSubgraph Walk.nil)\n[PROOFSTEP]\napply singletonSubgraph_connected\n[GOAL]\ncase cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d v\u271d w\u271d : V\nh : SimpleGraph.Adj G u\u271d v\u271d\np : Walk G v\u271d w\u271d\nih : Subgraph.Connected (Walk.toSubgraph p)\n\u22a2 Subgraph.Connected (Walk.toSubgraph (Walk.cons h p))\n[PROOFSTEP]\n\n| cons h p ih =>\n  apply (subgraphOfAdj_connected h).sup ih\n  rename_i w _\n  exists w\n  simp\n[GOAL]\ncase cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d v\u271d w\u271d : V\nh : SimpleGraph.Adj G u\u271d v\u271d\np : Walk G v\u271d w\u271d\nih : Subgraph.Connected (Walk.toSubgraph p)\n\u22a2 Subgraph.Connected (Walk.toSubgraph (Walk.cons h p))\n[PROOFSTEP]\napply (subgraphOfAdj_connected h).sup ih\n[GOAL]\ncase cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d v\u271d w\u271d : V\nh : SimpleGraph.Adj G u\u271d v\u271d\np : Walk G v\u271d w\u271d\nih : Subgraph.Connected (Walk.toSubgraph p)\n\u22a2 Set.Nonempty (subgraphOfAdj G h \u2293 Walk.toSubgraph p).verts\n[PROOFSTEP]\nrename_i w _\n[GOAL]\ncase cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d w w\u271d : V\nh : SimpleGraph.Adj G u\u271d w\np : Walk G w w\u271d\nih : Subgraph.Connected (Walk.toSubgraph p)\n\u22a2 Set.Nonempty (subgraphOfAdj G h \u2293 Walk.toSubgraph p).verts\n[PROOFSTEP]\nexists w\n[GOAL]\ncase cons\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v u\u271d w w\u271d : V\nh : SimpleGraph.Adj G u\u271d w\np : Walk G w w\u271d\nih : Subgraph.Connected (Walk.toSubgraph p)\n\u22a2 w \u2208 (subgraphOfAdj G h \u2293 Walk.toSubgraph p).verts\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\ns t : Set V\nsconn : Subgraph.Connected (induce H s)\ntconn : Subgraph.Connected (induce H t)\nsintert : Set.Nonempty (s \u2293 t)\n\u22a2 Subgraph.Connected (induce H (s \u222a t))\n[PROOFSTEP]\nrefine (sconn.sup tconn sintert).mono ?_ ?_\n[GOAL]\ncase refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\ns t : Set V\nsconn : Subgraph.Connected (induce H s)\ntconn : Subgraph.Connected (induce H t)\nsintert : Set.Nonempty (s \u2293 t)\n\u22a2 induce H s \u2294 induce H t \u2264 induce H (s \u222a t)\n[PROOFSTEP]\napply le_induce_union\n[GOAL]\ncase refine_2\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\ns t : Set V\nsconn : Subgraph.Connected (induce H s)\ntconn : Subgraph.Connected (induce H t)\nsintert : Set.Nonempty (s \u2293 t)\n\u22a2 (induce H s \u2294 induce H t).verts = (induce H (s \u222a t)).verts\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nHconn : Subgraph.Connected H\nKconn : Subgraph.Connected K\nu v : V\nuH : u \u2208 H.verts\nvK : v \u2208 K.verts\nhuv : SimpleGraph.Adj G u v\n\u22a2 Subgraph.Connected (induce \u22a4 {u, v} \u2294 H \u2294 K)\n[PROOFSTEP]\nrefine ((top_induce_pair_connected_of_adj huv).sup Hconn ?_).sup Kconn ?_\n[GOAL]\ncase refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nHconn : Subgraph.Connected H\nKconn : Subgraph.Connected K\nu v : V\nuH : u \u2208 H.verts\nvK : v \u2208 K.verts\nhuv : SimpleGraph.Adj G u v\n\u22a2 Set.Nonempty (induce \u22a4 {u, v} \u2293 H).verts\n[PROOFSTEP]\nexact \u27e8u, by simp [uH]\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nHconn : Subgraph.Connected H\nKconn : Subgraph.Connected K\nu v : V\nuH : u \u2208 H.verts\nvK : v \u2208 K.verts\nhuv : SimpleGraph.Adj G u v\n\u22a2 u \u2208 (induce \u22a4 {u, v} \u2293 H).verts\n[PROOFSTEP]\nsimp [uH]\n[GOAL]\ncase refine_2\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nHconn : Subgraph.Connected H\nKconn : Subgraph.Connected K\nu v : V\nuH : u \u2208 H.verts\nvK : v \u2208 K.verts\nhuv : SimpleGraph.Adj G u v\n\u22a2 Set.Nonempty ((induce \u22a4 {u, v} \u2294 H) \u2293 K).verts\n[PROOFSTEP]\nexact \u27e8v, by simp [vK]\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH K : Subgraph G\nHconn : Subgraph.Connected H\nKconn : Subgraph.Connected K\nu v : V\nuH : u \u2208 H.verts\nvK : v \u2208 K.verts\nhuv : SimpleGraph.Adj G u v\n\u22a2 v \u2208 ((induce \u22a4 {u, v} \u2294 H) \u2293 K).verts\n[PROOFSTEP]\nsimp [vK]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\n\u22a2 Subgraph.Preconnected H \u2194 \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\n\u22a2 Subgraph.Preconnected H \u2192 \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\n[PROOFSTEP]\nintro hc u v hu hv\n[GOAL]\ncase mp\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhc : Subgraph.Preconnected H\nu v : V\nhu : u \u2208 H.verts\nhv : v \u2208 H.verts\n\u22a2 \u2203 p, Walk.toSubgraph p \u2264 H\n[PROOFSTEP]\nrefine (hc \u27e8_, hu\u27e9 \u27e8_, hv\u27e9).elim fun p => ?_\n[GOAL]\ncase mp\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhc : Subgraph.Preconnected H\nu v : V\nhu : u \u2208 H.verts\nhv : v \u2208 H.verts\np : Walk (Subgraph.coe H) { val := u, property := hu } { val := v, property := hv }\n\u22a2 \u2203 p, Walk.toSubgraph p \u2264 H\n[PROOFSTEP]\nexists p.map (Subgraph.hom _)\n[GOAL]\ncase mp\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhc : Subgraph.Preconnected H\nu v : V\nhu : u \u2208 H.verts\nhv : v \u2208 H.verts\np : Walk (Subgraph.coe H) { val := u, property := hu } { val := v, property := hv }\n\u22a2 Walk.toSubgraph (Walk.map (Subgraph.hom H) p) \u2264 H\n[PROOFSTEP]\nsimp [coeSubgraph_le]\n[GOAL]\ncase mpr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\n\u22a2 (\u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H) \u2192 Subgraph.Preconnected H\n[PROOFSTEP]\nintro hw\n[GOAL]\ncase mpr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhw : \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\n\u22a2 Subgraph.Preconnected H\n[PROOFSTEP]\nrw [Subgraph.preconnected_iff]\n[GOAL]\ncase mpr\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhw : \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\n\u22a2 Preconnected (Subgraph.coe H)\n[PROOFSTEP]\nrintro \u27e8u, hu\u27e9 \u27e8v, hv\u27e9\n[GOAL]\ncase mpr.mk.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhw : \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\nu : V\nhu : u \u2208 H.verts\nv : V\nhv : v \u2208 H.verts\n\u22a2 Reachable (Subgraph.coe H) { val := u, property := hu } { val := v, property := hv }\n[PROOFSTEP]\nobtain \u27e8p, h\u27e9 := hw hu hv\n[GOAL]\ncase mpr.mk.mk.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nhw : \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\nu : V\nhu : u \u2208 H.verts\nv : V\nhv : v \u2208 H.verts\np : Walk G u v\nh : Walk.toSubgraph p \u2264 H\n\u22a2 Reachable (Subgraph.coe H) { val := u, property := hu } { val := v, property := hv }\n[PROOFSTEP]\nexact\n  Reachable.map (Subgraph.inclusion h)\n    (p.toSubgraph_connected \u27e8_, p.start_mem_verts_toSubgraph\u27e9 \u27e8_, p.end_mem_verts_toSubgraph\u27e9)\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\n\u22a2 Subgraph.Connected H \u2194 Set.Nonempty H.verts \u2227 \u2200 {u v : V}, u \u2208 H.verts \u2192 v \u2208 H.verts \u2192 \u2203 p, Walk.toSubgraph p \u2264 H\n[PROOFSTEP]\nrw [H.connected_iff, preconnected_iff_forall_exists_walk_subgraph, and_comm]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\n\u22a2 Connected (induce s G) \u2194 Subgraph.Connected (Subgraph.induce \u22a4 s)\n[PROOFSTEP]\nrw [induce_eq_coe_induce_top, \u2190 Subgraph.connected_iff']\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns t : Set V\nsconn : Connected (induce s G)\ntconn : Connected (induce t G)\nsintert : Set.Nonempty (s \u2229 t)\n\u22a2 Connected (induce (s \u222a t) G)\n[PROOFSTEP]\nrw [connected_induce_iff] at sconn tconn \u22a2\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nsintert : Set.Nonempty (s \u2229 t)\n\u22a2 Subgraph.Connected (Subgraph.induce \u22a4 (s \u222a t))\n[PROOFSTEP]\nexact Subgraph.induce_union_connected sconn tconn sintert\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nhuv : Adj G u v\n\u22a2 Connected (induce {u, v} G)\n[PROOFSTEP]\nrw [connected_induce_iff]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\nhuv : Adj G u v\n\u22a2 Subgraph.Connected (Subgraph.induce \u22a4 {u, v})\n[PROOFSTEP]\nexact Subgraph.top_induce_pair_connected_of_adj huv\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Connected H\n\u22a2 Connected (SimpleGraph.induce H.verts G)\n[PROOFSTEP]\nrw [connected_induce_iff]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Connected H\n\u22a2 Subgraph.Connected (induce \u22a4 H.verts)\n[PROOFSTEP]\nexact h.mono le_induce_top_verts (by exact rfl)\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nH : Subgraph G\nh : Subgraph.Connected H\n\u22a2 H.verts = (induce \u22a4 H.verts).verts\n[PROOFSTEP]\nexact rfl\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\np : Walk G u v\n\u22a2 Connected (induce {v_1 | v_1 \u2208 support p} G)\n[PROOFSTEP]\nrw [\u2190 p.verts_toSubgraph]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v : V\np : Walk G u v\n\u22a2 Connected (induce (Walk.toSubgraph p).verts G)\n[PROOFSTEP]\nexact p.toSubgraph_connected.induce_verts\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Connected (induce s G)\ntconn : Connected (induce t G)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 Connected (induce (s \u222a t) G)\n[PROOFSTEP]\nrw [connected_induce_iff] at sconn tconn \u22a2\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 Subgraph.Connected (Subgraph.induce \u22a4 (s \u222a t))\n[PROOFSTEP]\napply (sconn.adj_union tconn hv hw ha).mono\n[GOAL]\ncase hle\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 Subgraph.induce \u22a4 {v, w} \u2294 Subgraph.induce \u22a4 s \u2294 Subgraph.induce \u22a4 t \u2264 Subgraph.induce \u22a4 (s \u222a t)\n[PROOFSTEP]\nsimp only [Set.mem_singleton_iff, sup_le_iff, Subgraph.le_induce_union_left, Subgraph.le_induce_union_right, and_true, \u2190\n  Subgraph.subgraphOfAdj_eq_induce ha]\n[GOAL]\ncase hle\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 subgraphOfAdj G ha \u2264 Subgraph.induce \u22a4 (s \u222a t)\n[PROOFSTEP]\napply subgraphOfAdj_le_of_adj\n[GOAL]\ncase hle.h\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 Subgraph.Adj (Subgraph.induce \u22a4 (s \u222a t)) v w\n[PROOFSTEP]\nsimp [hv, hw, ha]\n[GOAL]\ncase hv\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 (Subgraph.induce \u22a4 {v, w} \u2294 Subgraph.induce \u22a4 s \u2294 Subgraph.induce \u22a4 t).verts = (Subgraph.induce \u22a4 (s \u222a t)).verts\n[PROOFSTEP]\nsimp only [Set.mem_singleton_iff, sup_le_iff, Subgraph.verts_sup, Subgraph.induce_verts]\n[GOAL]\ncase hv\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 {v, w} \u222a s \u222a t = s \u222a t\n[PROOFSTEP]\nrw [Set.union_assoc]\n[GOAL]\ncase hv\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nv w : V\ns t : Set V\nsconn : Subgraph.Connected (Subgraph.induce \u22a4 s)\ntconn : Subgraph.Connected (Subgraph.induce \u22a4 t)\nhv : v \u2208 s\nhw : w \u2208 t\nha : Adj G v w\n\u22a2 {v, w} \u222a (s \u222a t) = s \u222a t\n[PROOFSTEP]\nsimp [Set.insert_subset_iff, Set.singleton_subset_iff, hv, hw]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\nu : V\nhu : u \u2208 s\npatches :\n  \u2200 {v : V}, v \u2208 s \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n\u22a2 Connected (induce s G)\n[PROOFSTEP]\nrw [connected_iff_exists_forall_reachable]\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\nu : V\nhu : u \u2208 s\npatches :\n  \u2200 {v : V}, v \u2208 s \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n\u22a2 \u2203 v, \u2200 (w : \u2191s), Reachable (induce s G) v w\n[PROOFSTEP]\nrefine \u27e8\u27e8u, hu\u27e9, ?_\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\nu : V\nhu : u \u2208 s\npatches :\n  \u2200 {v : V}, v \u2208 s \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n\u22a2 \u2200 (w : \u2191s), Reachable (induce s G) { val := u, property := hu } w\n[PROOFSTEP]\nrintro \u27e8v, hv\u27e9\n[GOAL]\ncase mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\nu : V\nhu : u \u2208 s\npatches :\n  \u2200 {v : V}, v \u2208 s \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\nv : V\nhv : v \u2208 s\n\u22a2 Reachable (induce s G) { val := u, property := hu } { val := v, property := hv }\n[PROOFSTEP]\nobtain \u27e8sv, svs, hu', hv', uv\u27e9 := patches hv\n[GOAL]\ncase mk.intro.intro.intro.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\ns : Set V\nu : V\nhu : u \u2208 s\npatches :\n  \u2200 {v : V}, v \u2208 s \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\nv : V\nhv : v \u2208 s\nsv : Set V\nsvs : sv \u2286 s\nhu' : u \u2208 sv\nhv' : v \u2208 sv\nuv : Reachable (induce sv G) { val := u, property := hu' } { val := v, property := hv' }\n\u22a2 Reachable (induce s G) { val := u, property := hu } { val := v, property := hv }\n[PROOFSTEP]\nexact uv.map (induceHomOfLE _ svs).toHom\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSn : Set.Nonempty S\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\n\u22a2 Connected (induce (\u22c3\u2080 S) G)\n[PROOFSTEP]\nobtain \u27e8s, sS\u27e9 := Sn\n[GOAL]\ncase intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\n\u22a2 Connected (induce (\u22c3\u2080 S) G)\n[PROOFSTEP]\nobtain \u27e8v, vs\u27e9 := (Sc sS).nonempty\n[GOAL]\ncase intro.intro.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\nv : V\nvs : v \u2208 s\n\u22a2 Connected (induce (\u22c3\u2080 S) G)\n[PROOFSTEP]\napply G.induce_connected_of_patches _ (Set.subset_sUnion_of_mem sS vs)\n[GOAL]\ncase intro.intro.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\nv : V\nvs : v \u2208 s\n\u22a2 \u2200 {v_1 : V},\n    v_1 \u2208 \u22c3\u2080 S \u2192 \u2203 s' x hu' hv', Reachable (induce s' G) { val := v, property := hu' } { val := v_1, property := hv' }\n[PROOFSTEP]\nrintro w hw\n[GOAL]\ncase intro.intro.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\nv : V\nvs : v \u2208 s\nw : V\nhw : w \u2208 \u22c3\u2080 S\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := v, property := hu' } { val := w, property := hv' }\n[PROOFSTEP]\nsimp only [Set.mem_sUnion, exists_prop] at hw \n[GOAL]\ncase intro.intro.mk\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\nv : V\nvs : v \u2208 s\nw : V\nhw : \u2203 t, t \u2208 S \u2227 w \u2208 t\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := v, property := hu' } { val := w, property := hv' }\n[PROOFSTEP]\nobtain \u27e8t, tS, wt\u27e9 := hw\n[GOAL]\ncase intro.intro.mk.intro.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nS : Set (Set V)\nSnd : \u2200 {s t : Set V}, s \u2208 S \u2192 t \u2208 S \u2192 Set.Nonempty (s \u2229 t)\nSc : \u2200 {s : Set V}, s \u2208 S \u2192 Connected (induce s G)\ns : Set V\nsS : s \u2208 S\nv : V\nvs : v \u2208 s\nw : V\nt : Set V\ntS : t \u2208 S\nwt : w \u2208 t\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := v, property := hu' } { val := w, property := hv' }\n[PROOFSTEP]\nrefine\n  \u27e8s \u222a t, Set.union_subset (Set.subset_sUnion_of_mem sS) (Set.subset_sUnion_of_mem tS), Or.inl vs, Or.inr wt,\n    induce_union_connected (Sc sS) (Sc tS) (Snd sS tS) _ _\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\ntn : Finset.Nonempty t\n\u22a2 \u2203 t', t \u2286 t' \u2227 Connected (induce (\u2191t') G)\n[PROOFSTEP]\nclassical\nobtain \u27e8u, ut\u27e9 := tn\nrefine \u27e8t.biUnion (fun v => (Gpc u v).some.support.toFinset), fun v vt => ?_, ?_\u27e9\n\u00b7 simp only [Finset.mem_biUnion, List.mem_toFinset, exists_prop]\n  refine \u27e8v, vt, Walk.end_mem_support _\u27e9\n\u00b7 apply G.induce_connected_of_patches u\n  \u00b7 simp only [Finset.coe_biUnion, Finset.mem_coe, List.coe_toFinset, Set.mem_iUnion, Set.mem_setOf_eq,\n      Walk.start_mem_support, exists_prop, and_true]\n    exact \u27e8u, ut\u27e9\n  intros v hv\n  simp only [Finset.mem_coe, Finset.mem_biUnion, List.mem_toFinset, exists_prop] at hv \n  obtain \u27e8w, wt, hw\u27e9 := hv\n  refine \u27e8{x | x \u2208 (Gpc u w).some.support}, ?_, ?_\u27e9\n  \u00b7 simp only [Finset.coe_biUnion, Finset.mem_coe, List.coe_toFinset]\n    exact fun x xw => Set.mem_iUnion\u2082.mpr \u27e8w, wt, xw\u27e9\n  \u00b7 simp only [Set.mem_setOf_eq, Walk.start_mem_support, exists_true_left]\n    refine \u27e8hw, Walk.connected_induce_support _ _ _\u27e9\n[GOAL]\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\ntn : Finset.Nonempty t\n\u22a2 \u2203 t', t \u2286 t' \u2227 Connected (induce (\u2191t') G)\n[PROOFSTEP]\nobtain \u27e8u, ut\u27e9 := tn\n[GOAL]\ncase intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\n\u22a2 \u2203 t', t \u2286 t' \u2227 Connected (induce (\u2191t') G)\n[PROOFSTEP]\nrefine \u27e8t.biUnion (fun v => (Gpc u v).some.support.toFinset), fun v vt => ?_, ?_\u27e9\n[GOAL]\ncase intro.refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv : V\nvt : v \u2208 t\n\u22a2 v \u2208 Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v)))\n[PROOFSTEP]\nsimp only [Finset.mem_biUnion, List.mem_toFinset, exists_prop]\n[GOAL]\ncase intro.refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv : V\nvt : v \u2208 t\n\u22a2 \u2203 a, a \u2208 t \u2227 v \u2208 Walk.support (Nonempty.some (_ : Reachable G u a))\n[PROOFSTEP]\nrefine \u27e8v, vt, Walk.end_mem_support _\u27e9\n[GOAL]\ncase intro.refine_2\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\n\u22a2 Connected (induce (\u2191(Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v))))) G)\n[PROOFSTEP]\napply G.induce_connected_of_patches u\n[GOAL]\ncase intro.refine_2.hu\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\n\u22a2 u \u2208 \u2191(Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v))))\n[PROOFSTEP]\nsimp only [Finset.coe_biUnion, Finset.mem_coe, List.coe_toFinset, Set.mem_iUnion, Set.mem_setOf_eq,\n  Walk.start_mem_support, exists_prop, and_true]\n[GOAL]\ncase intro.refine_2.hu\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\n\u22a2 \u2203 i, i \u2208 t\n[PROOFSTEP]\nexact \u27e8u, ut\u27e9\n[GOAL]\ncase intro.refine_2.patches\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\n\u22a2 \u2200 {v : V},\n    v \u2208 \u2191(Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v)))) \u2192\n      \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n[PROOFSTEP]\nintros v hv\n[GOAL]\ncase intro.refine_2.patches\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv : V\nhv : v \u2208 \u2191(Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v))))\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n[PROOFSTEP]\nsimp only [Finset.mem_coe, Finset.mem_biUnion, List.mem_toFinset, exists_prop] at hv \n[GOAL]\ncase intro.refine_2.patches\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv : V\nhv : \u2203 a, a \u2208 t \u2227 v \u2208 Walk.support (Nonempty.some (_ : Reachable G u a))\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n[PROOFSTEP]\nobtain \u27e8w, wt, hw\u27e9 := hv\n[GOAL]\ncase intro.refine_2.patches.intro.intro\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv w : V\nwt : w \u2208 t\nhw : v \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))\n\u22a2 \u2203 s' x hu' hv', Reachable (induce s' G) { val := u, property := hu' } { val := v, property := hv' }\n[PROOFSTEP]\nrefine \u27e8{x | x \u2208 (Gpc u w).some.support}, ?_, ?_\u27e9\n[GOAL]\ncase intro.refine_2.patches.intro.intro.refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv w : V\nwt : w \u2208 t\nhw : v \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))\n\u22a2 {x | x \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))} \u2286\n    \u2191(Finset.biUnion t fun v => List.toFinset (Walk.support (Nonempty.some (_ : Reachable G u v))))\n[PROOFSTEP]\nsimp only [Finset.coe_biUnion, Finset.mem_coe, List.coe_toFinset]\n[GOAL]\ncase intro.refine_2.patches.intro.intro.refine_1\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv w : V\nwt : w \u2208 t\nhw : v \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))\n\u22a2 {x | x \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))} \u2286\n    \u22c3 (x : V) (_ : x \u2208 t), {a | a \u2208 Walk.support (Nonempty.some (_ : Reachable G u x))}\n[PROOFSTEP]\nexact fun x xw => Set.mem_iUnion\u2082.mpr \u27e8w, wt, xw\u27e9\n[GOAL]\ncase intro.refine_2.patches.intro.intro.refine_2\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv w : V\nwt : w \u2208 t\nhw : v \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))\n\u22a2 \u2203 hu' hv',\n    Reachable (induce {x | x \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))} G) { val := u, property := hu' }\n      { val := v, property := hv' }\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, Walk.start_mem_support, exists_true_left]\n[GOAL]\ncase intro.refine_2.patches.intro.intro.refine_2\nV : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nGpc : Preconnected G\nt : Finset V\nu : V\nut : u \u2208 t\nv w : V\nwt : w \u2208 t\nhw : v \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))\n\u22a2 \u2203 hv',\n    Reachable (induce {x | x \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))} G)\n      { val := u, property := (_ : u \u2208 {x | x \u2208 Walk.support (Nonempty.some (_ : Reachable G u w))}) }\n      { val := v, property := hv' }\n[PROOFSTEP]\nrefine \u27e8hw, Walk.connected_induce_support _ _ _\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph", "llama_tokens": 16711, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929104825006, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.505278012122228}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nN : Type u_2\ninst\u271d : Zero N\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : N \u2192 N \u2192 Prop\nhbot : \u2200 \u2983n : N\u2984, \u00acs n 0\nhs : WellFounded s\nx : \u03b1 \u2192\u2080 N\nh : \u2200 (a : \u03b1), a \u2208 x.support \u2192 Acc (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) a\n\u22a2 Acc (Finsupp.Lex r s) x\n[PROOFSTEP]\nrw [lex_eq_invImage_dfinsupp_lex]\n[GOAL]\n\u03b1 : Type u_1\nN : Type u_2\ninst\u271d : Zero N\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : N \u2192 N \u2192 Prop\nhbot : \u2200 \u2983n : N\u2984, \u00acs n 0\nhs : WellFounded s\nx : \u03b1 \u2192\u2080 N\nh : \u2200 (a : \u03b1), a \u2208 x.support \u2192 Acc (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) a\n\u22a2 Acc (InvImage (DFinsupp.Lex r fun x => s) toDFinsupp) x\n[PROOFSTEP]\nclassical\nrefine' InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ _)\nsimpa only [toDFinsupp_support] using h\n[GOAL]\n\u03b1 : Type u_1\nN : Type u_2\ninst\u271d : Zero N\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : N \u2192 N \u2192 Prop\nhbot : \u2200 \u2983n : N\u2984, \u00acs n 0\nhs : WellFounded s\nx : \u03b1 \u2192\u2080 N\nh : \u2200 (a : \u03b1), a \u2208 x.support \u2192 Acc (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) a\n\u22a2 Acc (InvImage (DFinsupp.Lex r fun x => s) toDFinsupp) x\n[PROOFSTEP]\nrefine' InvImage.accessible toDFinsupp (DFinsupp.Lex.acc (fun _ => hbot) (fun _ => hs) _ _)\n[GOAL]\n\u03b1 : Type u_1\nN : Type u_2\ninst\u271d : Zero N\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : N \u2192 N \u2192 Prop\nhbot : \u2200 \u2983n : N\u2984, \u00acs n 0\nhs : WellFounded s\nx : \u03b1 \u2192\u2080 N\nh : \u2200 (a : \u03b1), a \u2208 x.support \u2192 Acc (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) a\n\u22a2 \u2200 (i : \u03b1), i \u2208 DFinsupp.support (toDFinsupp x) \u2192 Acc (r\u1d9c \u2293 fun x x_1 => x \u2260 x_1) i\n[PROOFSTEP]\nsimpa only [toDFinsupp_support] using h\n", "meta": {"mathlib_filename": "Mathlib.Data.Finsupp.WellFounded", "llama_tokens": 749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.5052780107999085}}
{"text": "[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 bernsteinPolynomial \u2124 3 2 = 3 * X ^ 2 - 3 * X ^ 3\n[PROOFSTEP]\nsimp [bernsteinPolynomial, choose]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 (1 + 1 + 1) * X ^ 2 * (1 - X) = 3 * X ^ 2 - 3 * X ^ 3\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 3 * X ^ 2 * (1 - X) = 3 * X ^ 2 - 3 * X ^ 3\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : n < \u03bd\n\u22a2 bernsteinPolynomial R n \u03bd = 0\n[PROOFSTEP]\nsimp [bernsteinPolynomial, Nat.choose_eq_zero_of_lt h]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\nS : Type u_2\ninst\u271d : CommRing S\nf : R \u2192+* S\nn \u03bd : \u2115\n\u22a2 Polynomial.map f (bernsteinPolynomial R n \u03bd) = bernsteinPolynomial S n \u03bd\n[PROOFSTEP]\nsimp [bernsteinPolynomial]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 Polynomial.comp (bernsteinPolynomial R n \u03bd) (1 - X) = bernsteinPolynomial R n (n - \u03bd)\n[PROOFSTEP]\nsimp [bernsteinPolynomial, h, tsub_tsub_assoc, mul_right_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 bernsteinPolynomial R n \u03bd = Polynomial.comp (bernsteinPolynomial R n (n - \u03bd)) (1 - X)\n[PROOFSTEP]\nsimp [\u2190 flip _ _ _ h, Polynomial.comp_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 Polynomial.eval 0 (bernsteinPolynomial R n \u03bd) = if \u03bd = 0 then 1 else 0\n[PROOFSTEP]\nrw [bernsteinPolynomial]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 Polynomial.eval 0 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = if \u03bd = 0 then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd = 0\n\u22a2 Polynomial.eval 0 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 1\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 Polynomial.eval 0 (\u2191(choose n 0) * X ^ 0 * (1 - X) ^ (n - 0)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u00ac\u03bd = 0\n\u22a2 Polynomial.eval 0 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 0\n[PROOFSTEP]\nsimp [zero_pow (Nat.pos_of_ne_zero h)]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 Polynomial.eval 1 (bernsteinPolynomial R n \u03bd) = if \u03bd = n then 1 else 0\n[PROOFSTEP]\nrw [bernsteinPolynomial]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 Polynomial.eval 1 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = if \u03bd = n then 1 else 0\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd = n\n\u22a2 Polynomial.eval 1 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 1\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\n\u03bd : \u2115\n\u22a2 Polynomial.eval 1 (\u2191(choose \u03bd \u03bd) * X ^ \u03bd * (1 - X) ^ (\u03bd - \u03bd)) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u00ac\u03bd = n\n\u22a2 Polynomial.eval 1 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 0\n[PROOFSTEP]\nobtain w | w := (n - \u03bd).eq_zero_or_pos\n[GOAL]\ncase neg.inl\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u00ac\u03bd = n\nw : n - \u03bd = 0\n\u22a2 Polynomial.eval 1 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 0\n[PROOFSTEP]\nsimp [Nat.choose_eq_zero_of_lt ((tsub_eq_zero_iff_le.mp w).lt_of_ne (Ne.symm h))]\n[GOAL]\ncase neg.inr\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u00ac\u03bd = n\nw : n - \u03bd > 0\n\u22a2 Polynomial.eval 1 (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) = 0\n[PROOFSTEP]\nsimp [zero_pow w]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R (n + 1) (\u03bd + 1)) =\n    (\u2191n + 1) * (bernsteinPolynomial R n \u03bd - bernsteinPolynomial R n (\u03bd + 1))\n[PROOFSTEP]\nrw [bernsteinPolynomial]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191Polynomial.derivative (\u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n + 1 - (\u03bd + 1))) =\n    (\u2191n + 1) * (bernsteinPolynomial R n \u03bd - bernsteinPolynomial R n (\u03bd + 1))\n[PROOFSTEP]\nsuffices\n  ((n + 1).choose (\u03bd + 1) : R[X]) * ((\u2191(\u03bd + 1 : \u2115) : R[X]) * X ^ \u03bd) * (1 - X) ^ (n - \u03bd) -\n      ((n + 1).choose (\u03bd + 1) : R[X]) * X ^ (\u03bd + 1) * ((\u2191(n - \u03bd) : R[X]) * (1 - X) ^ (n - \u03bd - 1)) =\n    (\u2191(n + 1) : R[X]) *\n      ((n.choose \u03bd : R[X]) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) -\n        (n.choose (\u03bd + 1) : R[X]) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n  by\n  simpa [Polynomial.derivative_pow, \u2190 sub_eq_add_neg, Nat.succ_sub_succ_eq_sub, Polynomial.derivative_mul,\n    Polynomial.derivative_nat_cast, zero_mul, Nat.cast_add, algebraMap.coe_one, Polynomial.derivative_X, mul_one,\n    zero_add, Polynomial.derivative_sub, Polynomial.derivative_one, zero_sub, mul_neg, Nat.sub_zero,\n    bernsteinPolynomial, map_add, map_natCast, Nat.cast_one]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nthis :\n  \u2191(choose (n + 1) (\u03bd + 1)) * (\u2191(\u03bd + 1) * X ^ \u03bd) * (1 - X) ^ (n - \u03bd) -\n      \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (\u2191(n - \u03bd) * (1 - X) ^ (n - \u03bd - 1)) =\n    \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) - \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n\u22a2 \u2191Polynomial.derivative (\u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n + 1 - (\u03bd + 1))) =\n    (\u2191n + 1) * (bernsteinPolynomial R n \u03bd - bernsteinPolynomial R n (\u03bd + 1))\n[PROOFSTEP]\nsimpa [Polynomial.derivative_pow, \u2190 sub_eq_add_neg, Nat.succ_sub_succ_eq_sub, Polynomial.derivative_mul,\n  Polynomial.derivative_nat_cast, zero_mul, Nat.cast_add, algebraMap.coe_one, Polynomial.derivative_X, mul_one,\n  zero_add, Polynomial.derivative_sub, Polynomial.derivative_one, zero_sub, mul_neg, Nat.sub_zero, bernsteinPolynomial,\n  map_add, map_natCast, Nat.cast_one]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * (\u2191(\u03bd + 1) * X ^ \u03bd) * (1 - X) ^ (n - \u03bd) -\n      \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (\u2191(n - \u03bd) * (1 - X) ^ (n - \u03bd - 1)) =\n    \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) - \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nconv_rhs =>\n  rw [mul_sub]\n    -- We'll prove the two terms match up separately.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n| \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) - \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nrw [mul_sub]\n    -- We'll prove the two terms match up separately.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n| \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) - \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nrw [mul_sub]\n    -- We'll prove the two terms match up separately.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n| \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) - \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nrw [mul_sub]\n  -- We'll prove the two terms match up separately.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * (\u2191(\u03bd + 1) * X ^ \u03bd) * (1 - X) ^ (n - \u03bd) -\n      \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (\u2191(n - \u03bd) * (1 - X) ^ (n - \u03bd - 1)) =\n    \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)) -\n      \u2191(n + 1) * (\u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nrefine' congr (congr_arg Sub.sub _) _\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * (\u2191(\u03bd + 1) * X ^ \u03bd) * (1 - X) ^ (n - \u03bd) =\n    \u2191(n + 1) * (\u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd))\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc]\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * \u2191(\u03bd + 1) * X ^ \u03bd * (1 - X) ^ (n - \u03bd) =\n    \u2191(n + 1) * \u2191(choose n \u03bd) * X ^ \u03bd * (1 - X) ^ (n - \u03bd)\n[PROOFSTEP]\nrefine' congr (congr_arg (\u00b7 * \u00b7) (congr (congr_arg (\u00b7 * \u00b7) _) rfl)) rfl\n[GOAL]\ncase refine'_1\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * \u2191(\u03bd + 1) = \u2191(n + 1) * \u2191(choose n \u03bd)\n[PROOFSTEP]\nexact_mod_cast congr_arg (fun m : \u2115 => (m : R[X])) (Nat.succ_mul_choose_eq n \u03bd).symm\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * (\u2191(n - \u03bd) * (1 - X) ^ (n - \u03bd - 1)) =\n    \u2191(n + 1) * (\u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1) * (1 - X) ^ (n - (\u03bd + 1)))\n[PROOFSTEP]\nrw [\u2190 tsub_add_eq_tsub_tsub, \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\ncase refine'_2\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * \u2191(n - \u03bd) * (1 - X) ^ (n - (\u03bd + 1)) =\n    \u2191(n + 1) * (\u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1)) * (1 - X) ^ (n - (\u03bd + 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine'_2.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) * \u2191(n - \u03bd) = \u2191(n + 1) * (\u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1))\n[PROOFSTEP]\nrw [mul_comm, \u2190 mul_assoc, \u2190 mul_assoc]\n[GOAL]\ncase refine'_2.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(n - \u03bd) * \u2191(choose (n + 1) (\u03bd + 1)) * X ^ (\u03bd + 1) = \u2191(n + 1) * \u2191(choose n (\u03bd + 1)) * X ^ (\u03bd + 1)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine'_2.e_a.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191(n - \u03bd) * \u2191(choose (n + 1) (\u03bd + 1)) = \u2191(n + 1) * \u2191(choose n (\u03bd + 1))\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase refine'_2.e_a.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191((n - \u03bd) * choose (n + 1) (\u03bd + 1)) = \u2191((n + 1) * choose n (\u03bd + 1))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase refine'_2.e_a.e_a.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 (n - \u03bd) * choose (n + 1) (\u03bd + 1) = (n + 1) * choose n (\u03bd + 1)\n[PROOFSTEP]\nconvert (Nat.choose_mul_succ_eq n (\u03bd + 1)).symm using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 (n - \u03bd) * choose (n + 1) (\u03bd + 1) = choose (n + 1) (\u03bd + 1) * (n + 1 - (\u03bd + 1))\n[PROOFSTEP]\nrw [mul_comm, Nat.succ_sub_succ_eq_sub]\n[GOAL]\ncase h.e'_3\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 (n + 1) * choose n (\u03bd + 1) = choose n (\u03bd + 1) * (n + 1)\n[PROOFSTEP]\napply mul_comm\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R n (\u03bd + 1)) =\n    \u2191n * (bernsteinPolynomial R (n - 1) \u03bd - bernsteinPolynomial R (n - 1) (\u03bd + 1))\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\n\u03bd : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R Nat.zero (\u03bd + 1)) =\n    \u2191Nat.zero * (bernsteinPolynomial R (Nat.zero - 1) \u03bd - bernsteinPolynomial R (Nat.zero - 1) (\u03bd + 1))\n[PROOFSTEP]\nsimp [bernsteinPolynomial]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\n\u03bd n\u271d : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R (Nat.succ n\u271d) (\u03bd + 1)) =\n    \u2191(Nat.succ n\u271d) * (bernsteinPolynomial R (Nat.succ n\u271d - 1) \u03bd - bernsteinPolynomial R (Nat.succ n\u271d - 1) (\u03bd + 1))\n[PROOFSTEP]\nrw [Nat.cast_succ]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\n\u03bd n\u271d : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R (Nat.succ n\u271d) (\u03bd + 1)) =\n    (\u2191n\u271d + 1) * (bernsteinPolynomial R (Nat.succ n\u271d - 1) \u03bd - bernsteinPolynomial R (Nat.succ n\u271d - 1) (\u03bd + 1))\n[PROOFSTEP]\napply derivative_succ_aux\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2191Polynomial.derivative (bernsteinPolynomial R n 0) = -\u2191n * bernsteinPolynomial R (n - 1) 0\n[PROOFSTEP]\nsimp [bernsteinPolynomial, Polynomial.derivative_pow]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\n\u22a2 k < \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0\n[PROOFSTEP]\ncases' \u03bd with \u03bd\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\n\u22a2 k < Nat.zero \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n Nat.zero)) = 0\n[PROOFSTEP]\nrintro \u27e8\u27e9\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k < Nat.succ \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\n[PROOFSTEP]\nrw [Nat.lt_succ_iff]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn k \u03bd : \u2115\n\u22a2 k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\n[PROOFSTEP]\ninduction' k with k ih generalizing n \u03bd\n[GOAL]\ncase succ.zero\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d n \u03bd : \u2115\n\u22a2 Nat.zero \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[Nat.zero] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\n[PROOFSTEP]\nsimp [eval_at_0]\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[Nat.succ k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\n[PROOFSTEP]\nsimp only [derivative_succ, Int.coe_nat_eq_zero, mul_eq_zero, Function.comp_apply, Function.iterate_succ,\n  Polynomial.iterate_derivative_sub, Polynomial.iterate_derivative_nat_cast_mul, Polynomial.eval_mul,\n  Polynomial.eval_nat_cast, Polynomial.eval_sub]\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\n\u22a2 Nat.succ k \u2264 \u03bd \u2192\n    \u2191n *\n        (Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) \u03bd)) -\n          Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n      0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 \u2191n *\n      (Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) \u03bd)) -\n        Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) (\u03bd + 1)))) =\n    0\n[PROOFSTEP]\napply mul_eq_zero_of_right\n[GOAL]\ncase succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) \u03bd)) -\n      Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) (\u03bd + 1))) =\n    0\n[PROOFSTEP]\nrw [ih _ _ (Nat.le_of_succ_le h), sub_zero]\n[GOAL]\ncase succ.succ.h\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\n\u22a2 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R (n - 1) \u03bd)) = 0\n[PROOFSTEP]\nconvert ih _ _ (Nat.pred_le_pred h)\n[GOAL]\ncase h.e'_2.h.e'_4.h.h.e'_4.h.e'_4\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d k : \u2115\nih : \u2200 (n \u03bd : \u2115), k \u2264 \u03bd \u2192 Polynomial.eval 0 ((\u2191Polynomial.derivative)^[k] (bernsteinPolynomial R n (Nat.succ \u03bd))) = 0\nn \u03bd : \u2115\nh : Nat.succ k \u2264 \u03bd\ne_2\u271d : Ring.toSemiring = CommSemiring.toSemiring\n\u22a2 \u03bd = Nat.succ (Nat.pred \u03bd)\n[PROOFSTEP]\nexact (Nat.succ_pred_eq_of_pos (k.succ_pos.trans_le h)).symm\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\n[PROOFSTEP]\nby_cases h : \u03bd \u2264 n\n[GOAL]\ncase pos\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\n[PROOFSTEP]\ninduction' \u03bd with \u03bd ih generalizing n\n[GOAL]\ncase pos.zero\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd : \u2115\nh\u271d : \u03bd \u2264 n\u271d\nn : \u2115\nh : Nat.zero \u2264 n\n\u22a2 eval 0 ((\u2191derivative)^[Nat.zero] (bernsteinPolynomial R n Nat.zero)) =\n    eval (\u2191(n - (Nat.zero - 1))) (pochhammer R Nat.zero)\n[PROOFSTEP]\nsimp [eval_at_0]\n[GOAL]\ncase pos.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\n\u22a2 eval 0 ((\u2191derivative)^[Nat.succ \u03bd] (bernsteinPolynomial R n (Nat.succ \u03bd))) =\n    eval (\u2191(n - (Nat.succ \u03bd - 1))) (pochhammer R (Nat.succ \u03bd))\n[PROOFSTEP]\nhave h' : \u03bd \u2264 n - 1 := le_tsub_of_add_le_right h\n[GOAL]\ncase pos.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\n\u22a2 eval 0 ((\u2191derivative)^[Nat.succ \u03bd] (bernsteinPolynomial R n (Nat.succ \u03bd))) =\n    eval (\u2191(n - (Nat.succ \u03bd - 1))) (pochhammer R (Nat.succ \u03bd))\n[PROOFSTEP]\nsimp only [derivative_succ, ih (n - 1) h', iterate_derivative_succ_at_0_eq_zero, Nat.succ_sub_succ_eq_sub, tsub_zero,\n  sub_zero, iterate_derivative_sub, iterate_derivative_nat_cast_mul, eval_one, eval_mul, eval_add, eval_sub, eval_X,\n  eval_comp, eval_nat_cast, Function.comp_apply, Function.iterate_succ, pochhammer_succ_left]\n[GOAL]\ncase pos.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\n\u22a2 \u2191n * eval (\u2191(n - 1 - (\u03bd - 1))) (pochhammer R \u03bd) = \u2191(n - \u03bd) * eval (\u2191(n - \u03bd) + 1) (pochhammer R \u03bd)\n[PROOFSTEP]\nobtain rfl | h'' := \u03bd.eq_zero_or_pos\n[GOAL]\ncase pos.succ.inl\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd : \u2115\nh\u271d : \u03bd \u2264 n\u271d\nn : \u2115\nih : \u2200 (n : \u2115), 0 \u2264 n \u2192 eval 0 ((\u2191derivative)^[0] (bernsteinPolynomial R n 0)) = eval (\u2191(n - (0 - 1))) (pochhammer R 0)\nh : Nat.succ 0 \u2264 n\nh' : 0 \u2264 n - 1\n\u22a2 \u2191n * eval (\u2191(n - 1 - (0 - 1))) (pochhammer R 0) = \u2191(n - 0) * eval (\u2191(n - 0) + 1) (pochhammer R 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase pos.succ.inr\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\nh'' : \u03bd > 0\n\u22a2 \u2191n * eval (\u2191(n - 1 - (\u03bd - 1))) (pochhammer R \u03bd) = \u2191(n - \u03bd) * eval (\u2191(n - \u03bd) + 1) (pochhammer R \u03bd)\n[PROOFSTEP]\nhave : n - 1 - (\u03bd - 1) = n - \u03bd := by\n  rw [gt_iff_lt, \u2190 Nat.succ_le_iff] at h'' \n  rw [\u2190 tsub_add_eq_tsub_tsub, add_comm, tsub_add_cancel_of_le h'']\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\nh'' : \u03bd > 0\n\u22a2 n - 1 - (\u03bd - 1) = n - \u03bd\n[PROOFSTEP]\nrw [gt_iff_lt, \u2190 Nat.succ_le_iff] at h'' \n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\nh'' : Nat.succ 0 \u2264 \u03bd\n\u22a2 n - 1 - (\u03bd - 1) = n - \u03bd\n[PROOFSTEP]\nrw [\u2190 tsub_add_eq_tsub_tsub, add_comm, tsub_add_cancel_of_le h'']\n[GOAL]\ncase pos.succ.inr\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\nh'' : \u03bd > 0\nthis : n - 1 - (\u03bd - 1) = n - \u03bd\n\u22a2 \u2191n * eval (\u2191(n - 1 - (\u03bd - 1))) (pochhammer R \u03bd) = \u2191(n - \u03bd) * eval (\u2191(n - \u03bd) + 1) (pochhammer R \u03bd)\n[PROOFSTEP]\nrw [this, pochhammer_eval_succ]\n[GOAL]\ncase pos.succ.inr\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d \u03bd\u271d : \u2115\nh\u271d : \u03bd\u271d \u2264 n\u271d\n\u03bd : \u2115\nih : \u2200 (n : \u2115), \u03bd \u2264 n \u2192 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\nn : \u2115\nh : Nat.succ \u03bd \u2264 n\nh' : \u03bd \u2264 n - 1\nh'' : \u03bd > 0\nthis : n - 1 - (\u03bd - 1) = n - \u03bd\n\u22a2 \u2191n * eval (\u2191(n - \u03bd)) (pochhammer R \u03bd) = (\u2191(n - \u03bd) + \u2191\u03bd) * eval (\u2191(n - \u03bd)) (pochhammer R \u03bd)\n[PROOFSTEP]\nrw_mod_cast [tsub_add_cancel_of_le (h'.trans n.pred_le)]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u00ac\u03bd \u2264 n\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\n[PROOFSTEP]\nsimp only [not_le] at h \n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : n < \u03bd\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) = eval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd)\n[PROOFSTEP]\nrw [tsub_eq_zero_iff_le.mpr (Nat.le_pred_of_lt h), eq_zero_of_lt R h]\n[GOAL]\ncase neg\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : n < \u03bd\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] 0) = eval (\u21910) (pochhammer R \u03bd)\n[PROOFSTEP]\nsimp [pos_iff_ne_zero.mp (pos_of_gt h)]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 eval 0 ((\u2191derivative)^[\u03bd] (bernsteinPolynomial R n \u03bd)) \u2260 0\n[PROOFSTEP]\nsimp only [Int.coe_nat_eq_zero, bernsteinPolynomial.iterate_derivative_at_0, Ne.def, Nat.cast_eq_zero]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 \u00aceval (\u2191(n - (\u03bd - 1))) (pochhammer R \u03bd) = 0\n[PROOFSTEP]\nsimp only [\u2190 pochhammer_eval_cast]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 \u00ac\u2191(eval (n - (\u03bd - 1)) (pochhammer \u2115 \u03bd)) = 0\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 \u00aceval (n - (\u03bd - 1)) (pochhammer \u2115 \u03bd) = 0\n[PROOFSTEP]\napply ne_of_gt\n[GOAL]\ncase h\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 0 < eval (n - (\u03bd - 1)) (pochhammer \u2115 \u03bd)\n[PROOFSTEP]\nobtain rfl | h' := Nat.eq_zero_or_pos \u03bd\n[GOAL]\ncase h.inl\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn : \u2115\nh : 0 \u2264 n\n\u22a2 0 < eval (n - (0 - 1)) (pochhammer \u2115 0)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\nh' : \u03bd > 0\n\u22a2 0 < eval (n - (\u03bd - 1)) (pochhammer \u2115 \u03bd)\n[PROOFSTEP]\nrw [\u2190 Nat.succ_pred_eq_of_pos h'] at h \n[GOAL]\ncase h.inr\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : Nat.succ (Nat.pred \u03bd) \u2264 n\nh' : \u03bd > 0\n\u22a2 0 < eval (n - (\u03bd - 1)) (pochhammer \u2115 \u03bd)\n[PROOFSTEP]\nexact pochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h))\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\n\u22a2 k < n - \u03bd \u2192 eval 1 ((\u2191derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0\n[PROOFSTEP]\nintro w\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (bernsteinPolynomial R n \u03bd)) = 0\n[PROOFSTEP]\nrw [flip' _ _ _ (tsub_pos_iff_lt.mp (pos_of_gt w)).le]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd k : \u2115\nw : k < n - \u03bd\n\u22a2 eval 1 ((\u2191derivative)^[k] (comp (bernsteinPolynomial R n (n - \u03bd)) (1 - X))) = 0\n[PROOFSTEP]\nsimp [Polynomial.eval_comp, iterate_derivative_at_0_eq_zero_of_lt R n w]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 eval 1 ((\u2191derivative)^[n - \u03bd] (bernsteinPolynomial R n \u03bd)) = (-1) ^ (n - \u03bd) * eval (\u2191\u03bd + 1) (pochhammer R (n - \u03bd))\n[PROOFSTEP]\nrw [flip' _ _ _ h]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 eval 1 ((\u2191derivative)^[n - \u03bd] (comp (bernsteinPolynomial R n (n - \u03bd)) (1 - X))) =\n    (-1) ^ (n - \u03bd) * eval (\u2191\u03bd + 1) (pochhammer R (n - \u03bd))\n[PROOFSTEP]\nsimp [Polynomial.eval_comp, h]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 (-1) ^ (n - \u03bd) * eval (\u2191(n - (n - \u03bd - 1))) (pochhammer R (n - \u03bd)) =\n    (-1) ^ (n - \u03bd) * eval (\u2191\u03bd + 1) (pochhammer R (n - \u03bd))\n[PROOFSTEP]\nobtain rfl | h' := h.eq_or_lt\n[GOAL]\ncase inl\nR : Type u_1\ninst\u271d : CommRing R\n\u03bd : \u2115\nh : \u03bd \u2264 \u03bd\n\u22a2 (-1) ^ (\u03bd - \u03bd) * eval (\u2191(\u03bd - (\u03bd - \u03bd - 1))) (pochhammer R (\u03bd - \u03bd)) =\n    (-1) ^ (\u03bd - \u03bd) * eval (\u2191\u03bd + 1) (pochhammer R (\u03bd - \u03bd))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\nh' : \u03bd < n\n\u22a2 (-1) ^ (n - \u03bd) * eval (\u2191(n - (n - \u03bd - 1))) (pochhammer R (n - \u03bd)) =\n    (-1) ^ (n - \u03bd) * eval (\u2191\u03bd + 1) (pochhammer R (n - \u03bd))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase inr.e_a.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\nh' : \u03bd < n\n\u22a2 \u2191(n - (n - \u03bd - 1)) = \u2191\u03bd + 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase inr.e_a.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\nh' : \u03bd < n\n\u22a2 \u2191(n - (n - \u03bd - 1)) = \u2191(\u03bd + 1)\n[PROOFSTEP]\nrw [\u2190 tsub_add_eq_tsub_tsub, tsub_tsub_cancel_of_le (Nat.succ_le_iff.mpr h')]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 eval 1 ((\u2191derivative)^[n - \u03bd] (bernsteinPolynomial R n \u03bd)) \u2260 0\n[PROOFSTEP]\nrw [bernsteinPolynomial.iterate_derivative_at_1 _ _ _ h, Ne.def, neg_one_pow_mul_eq_zero_iff, \u2190 Nat.cast_succ, \u2190\n  pochhammer_eval_cast, \u2190 Nat.cast_zero, Nat.cast_inj]\n[GOAL]\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CharZero R\nn \u03bd : \u2115\nh : \u03bd \u2264 n\n\u22a2 \u00aceval (Nat.succ \u03bd) (pochhammer \u2115 (n - \u03bd)) = 0\n[PROOFSTEP]\nexact (pochhammer_pos _ _ (Nat.succ_pos \u03bd)).ne'\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\nh : k \u2264 n + 1\n\u22a2 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\n[PROOFSTEP]\ninduction' k with k ih\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\nh\u271d : k \u2264 n + 1\nh : Nat.zero \u2264 n + 1\n\u22a2 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\n[PROOFSTEP]\nsimp [Nat.zero_eq]\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\nn k : \u2115\nh\u271d : k \u2264 n + 1\nh : Nat.zero \u2264 n + 1\n\u22a2 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\n[PROOFSTEP]\napply linearIndependent_empty_type\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nih : k \u2264 n + 1 \u2192 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\nh : Nat.succ k \u2264 n + 1\n\u22a2 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\n[PROOFSTEP]\napply linearIndependent_fin_succ'.mpr\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nih : k \u2264 n + 1 \u2192 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\nh : Nat.succ k \u2264 n + 1\n\u22a2 LinearIndependent \u211a (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd) \u2227\n    \u00acbernsteinPolynomial \u211a n \u2191(Fin.last k) \u2208 span \u211a (Set.range (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd))\n[PROOFSTEP]\nfconstructor\n[GOAL]\ncase succ.left\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nih : k \u2264 n + 1 \u2192 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\nh : Nat.succ k \u2264 n + 1\n\u22a2 LinearIndependent \u211a (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd)\n[PROOFSTEP]\nexact ih (le_of_lt h)\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nih : k \u2264 n + 1 \u2192 LinearIndependent \u211a fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd\nh : Nat.succ k \u2264 n + 1\n\u22a2 \u00acbernsteinPolynomial \u211a n \u2191(Fin.last k) \u2208 span \u211a (Set.range (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd))\n[PROOFSTEP]\nclear ih\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : Nat.succ k \u2264 n + 1\n\u22a2 \u00acbernsteinPolynomial \u211a n \u2191(Fin.last k) \u2208 span \u211a (Set.range (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd))\n[PROOFSTEP]\nsimp only [Nat.succ_eq_add_one, add_le_add_iff_right] at h \n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\n\u22a2 \u00acbernsteinPolynomial \u211a n \u2191(Fin.last k) \u2208 span \u211a (Set.range (Fin.init fun \u03bd => bernsteinPolynomial \u211a n \u2191\u03bd))\n[PROOFSTEP]\nsimp only [Fin.val_last, Fin.init_def]\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\n\u22a2 \u00acbernsteinPolynomial \u211a n k \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191(Fin.castSucc k_1))\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\n\u22a2 \u00acbernsteinPolynomial \u211a n k \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n[PROOFSTEP]\napply not_mem_span_of_apply_not_mem_span_image (@Polynomial.derivative \u211a _ ^ (n - k))\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\n\u22a2 \u00ac\u2191(derivative ^ (n - k)) (bernsteinPolynomial \u211a n k) \u2208\n      span \u211a (\u2191(derivative ^ (n - k)) '' Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n[PROOFSTEP]\nsimp only [not_exists, not_and, Submodule.mem_map, Submodule.span_image]\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\n\u22a2 \u2200 (x : \u211a[X]),\n    x \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1) \u2192\n      \u00ac\u2191(derivative ^ (n - k)) x = \u2191(derivative ^ (n - k)) (bernsteinPolynomial \u211a n k)\n[PROOFSTEP]\nintro p m\n[GOAL]\ncase succ.right\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 \u00ac\u2191(derivative ^ (n - k)) p = \u2191(derivative ^ (n - k)) (bernsteinPolynomial \u211a n k)\n[PROOFSTEP]\napply_fun Polynomial.eval (1 : \u211a)\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 eval 1 (\u2191(derivative ^ (n - k)) p) \u2260 eval 1 (\u2191(derivative ^ (n - k)) (bernsteinPolynomial \u211a n k))\n[PROOFSTEP]\nsimp only [LinearMap.pow_apply]\n  -- The right hand side is nonzero,\n        -- so it will suffice to show the left hand side is always zero.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 eval 1 ((\u2191derivative)^[n - k] p) \u2260 eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n k))\n[PROOFSTEP]\nsuffices (Polynomial.derivative^[n - k] p).eval 1 = 0 by\n  rw [this]\n  exact (iterate_derivative_at_1_ne_zero \u211a n k h).symm\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\nthis : eval 1 ((\u2191derivative)^[n - k] p) = 0\n\u22a2 eval 1 ((\u2191derivative)^[n - k] p) \u2260 eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n k))\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\nthis : eval 1 ((\u2191derivative)^[n - k] p) = 0\n\u22a2 0 \u2260 eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n k))\n[PROOFSTEP]\nexact (iterate_derivative_at_1_ne_zero \u211a n k h).symm\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 eval 1 ((\u2191derivative)^[n - k] p) = 0\n[PROOFSTEP]\nrefine span_induction m ?_ ?_ ?_ ?_\n[GOAL]\ncase refine_1\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 \u2200 (x : \u211a[X]), (x \u2208 Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1) \u2192 eval 1 ((\u2191derivative)^[n - k] x) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_1\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 \u2200 (a : Fin k), eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n \u2191a)) = 0\n[PROOFSTEP]\nrintro \u27e8a, w\u27e9\n[GOAL]\ncase refine_1.mk\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\na : \u2115\nw : a < k\n\u22a2 eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n \u2191{ val := a, isLt := w })) = 0\n[PROOFSTEP]\nsimp only [Fin.val_mk]\n[GOAL]\ncase refine_1.mk\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\na : \u2115\nw : a < k\n\u22a2 eval 1 ((\u2191derivative)^[n - k] (bernsteinPolynomial \u211a n a)) = 0\n[PROOFSTEP]\nrw [iterate_derivative_at_1_eq_zero_of_lt \u211a n ((tsub_lt_tsub_iff_left_of_le h).mpr w)]\n[GOAL]\ncase refine_2\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 eval 1 ((\u2191derivative)^[n - k] 0) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine_3\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 \u2200 (x y : \u211a[X]),\n    eval 1 ((\u2191derivative)^[n - k] x) = 0 \u2192\n      eval 1 ((\u2191derivative)^[n - k] y) = 0 \u2192 eval 1 ((\u2191derivative)^[n - k] (x + y)) = 0\n[PROOFSTEP]\nintro x y hx hy\n[GOAL]\ncase refine_3\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\nx y : \u211a[X]\nhx : eval 1 ((\u2191derivative)^[n - k] x) = 0\nhy : eval 1 ((\u2191derivative)^[n - k] y) = 0\n\u22a2 eval 1 ((\u2191derivative)^[n - k] (x + y)) = 0\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase refine_4\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d : k\u271d \u2264 n + 1\nk : \u2115\nh : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\n\u22a2 \u2200 (a : \u211a) (x : \u211a[X]), eval 1 ((\u2191derivative)^[n - k] x) = 0 \u2192 eval 1 ((\u2191derivative)^[n - k] (a \u2022 x)) = 0\n[PROOFSTEP]\nintro a x h\n[GOAL]\ncase refine_4\nR : Type u_1\ninst\u271d : CommRing R\nn k\u271d : \u2115\nh\u271d\u00b9 : k\u271d \u2264 n + 1\nk : \u2115\nh\u271d : k \u2264 n\np : \u211a[X]\nm : p \u2208 span \u211a (Set.range fun k_1 => bernsteinPolynomial \u211a n \u2191k_1)\na : \u211a\nx : \u211a[X]\nh : eval 1 ((\u2191derivative)^[n - k] x) = 0\n\u22a2 eval 1 ((\u2191derivative)^[n - k] (a \u2022 x)) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd = (X + (1 - X)) ^ n\n[PROOFSTEP]\nrw [add_pow]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 m in Finset.range (n + 1), X ^ m * (1 - X) ^ (n - m) * \u2191(choose n m)\n[PROOFSTEP]\nsimp only [bernsteinPolynomial, mul_comm, mul_assoc, mul_left_comm]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 (X + (1 - X)) ^ n = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\nlet x : MvPolynomial Bool R := MvPolynomial.X true\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\nlet y : MvPolynomial Bool R := MvPolynomial.X false\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\nhave pderiv_true_x : pderiv true x = 1 := by rw [pderiv_X]; rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 \u2191(pderiv true) x = 1\n[PROOFSTEP]\nrw [pderiv_X]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 Pi.single true 1 true = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\nhave pderiv_true_y : pderiv true y = 0 := by rw [pderiv_X]; rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 \u2191(pderiv true) y = 0\n[PROOFSTEP]\nrw [pderiv_X]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 Pi.single true 1 false = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\nlet e : Bool \u2192 R[X] := fun i =>\n  cond i X\n    (1 - X)\n      -- Start with `(x+y)^n = (x+y)^n`,\n        -- take the `x`-derivative, evaluate at `x=X, y=1-X`, and multiply by `X`:\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd = n \u2022 X\n[PROOFSTEP]\ntrans MvPolynomial.aeval e (pderiv true ((x + y) ^ n)) * X\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) ((x + y) ^ n)) * X\n[PROOFSTEP]\nhave w :\n  \u2200 k : \u2115,\n    k \u2022 bernsteinPolynomial R n k =\n      (k : R[X]) * Polynomial.X ^ (k - 1) * (1 - Polynomial.X) ^ (n - k) * (n.choose k : R[X]) * Polynomial.X :=\n  by\n  rintro (_ | k)\n  \u00b7 simp\n  \u00b7 rw [bernsteinPolynomial]\n    simp only [\u2190 nat_cast_mul, Nat.succ_eq_add_one, Nat.add_succ_sub_one, add_zero, pow_succ]\n    push_cast\n    ring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2200 (k : \u2115), k \u2022 bernsteinPolynomial R n k = \u2191k * X ^ (k - 1) * (1 - X) ^ (n - k) * \u2191(choose n k) * X\n[PROOFSTEP]\nrintro (_ | k)\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 Nat.zero \u2022 bernsteinPolynomial R n Nat.zero =\n    \u2191Nat.zero * X ^ (Nat.zero - 1) * (1 - X) ^ (n - Nat.zero) * \u2191(choose n Nat.zero) * X\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 Nat.succ k \u2022 bernsteinPolynomial R n (Nat.succ k) =\n    \u2191(Nat.succ k) * X ^ (Nat.succ k - 1) * (1 - X) ^ (n - Nat.succ k) * \u2191(choose n (Nat.succ k)) * X\n[PROOFSTEP]\nrw [bernsteinPolynomial]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 Nat.succ k \u2022 (\u2191(choose n (Nat.succ k)) * X ^ Nat.succ k * (1 - X) ^ (n - Nat.succ k)) =\n    \u2191(Nat.succ k) * X ^ (Nat.succ k - 1) * (1 - X) ^ (n - Nat.succ k) * \u2191(choose n (Nat.succ k)) * X\n[PROOFSTEP]\nsimp only [\u2190 nat_cast_mul, Nat.succ_eq_add_one, Nat.add_succ_sub_one, add_zero, pow_succ]\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 \u2191(k + 1) * (\u2191(choose n (k + 1)) * (X * X ^ k) * (1 - X) ^ (n - (k + 1))) =\n    \u2191(k + 1) * X ^ k * (1 - X) ^ (n - (k + 1)) * \u2191(choose n (k + 1)) * X\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 (\u2191k + 1) * (\u2191(choose n (k + 1)) * (X * X ^ k) * (1 - X) ^ (n - (k + 1))) =\n    (\u2191k + 1) * X ^ k * (1 - X) ^ (n - (k + 1)) * \u2191(choose n (k + 1)) * X\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw : \u2200 (k : \u2115), k \u2022 bernsteinPolynomial R n k = \u2191k * X ^ (k - 1) * (1 - X) ^ (n - k) * \u2191(choose n k) * X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) ((x + y) ^ n)) * X\n[PROOFSTEP]\nrw [add_pow, (pderiv true).map_sum, (MvPolynomial.aeval e).map_sum, Finset.sum_mul]\n  -- Step inside the sum:\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw : \u2200 (k : \u2115), k \u2022 bernsteinPolynomial R n k = \u2191k * X ^ (k - 1) * (1 - X) ^ (n - k) * \u2191(choose n k) * X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd =\n    \u2211 x_1 in Finset.range (n + 1),\n      \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (x ^ x_1 * y ^ (n - x_1) * \u2191(choose n x_1))) * X\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun k _ => (w k).trans _\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw : \u2200 (k : \u2115), k \u2022 bernsteinPolynomial R n k = \u2191k * X ^ (k - 1) * (1 - X) ^ (n - k) * \u2191(choose n k) * X\nk : \u2115\nx\u271d : k \u2208 Finset.range (n + 1)\n\u22a2 \u2191k * X ^ (k - 1) * (1 - X) ^ (n - k) * \u2191(choose n k) * X =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (x ^ k * y ^ (n - k) * \u2191(choose n k))) * X\n[PROOFSTEP]\nsimp only [pderiv_true_x, pderiv_true_y, Algebra.id.smul_eq_mul, nsmul_eq_mul, Bool.cond_true, Bool.cond_false,\n  add_zero, mul_one, mul_zero, smul_zero, MvPolynomial.aeval_X, MvPolynomial.pderiv_mul, Derivation.leibniz_pow,\n  Derivation.map_coe_nat, map_natCast, map_pow, map_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) ((x + y) ^ n)) * X = n \u2022 X\n[PROOFSTEP]\nrw [(pderiv true).leibniz_pow, (pderiv true).map_add, pderiv_true_x, pderiv_true_y]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2191(MvPolynomial.aeval e) (n \u2022 (x + y) ^ (n - 1) \u2022 (1 + 0)) * X = n \u2022 X\n[PROOFSTEP]\nsimp only [Algebra.id.smul_eq_mul, nsmul_eq_mul, map_natCast, map_pow, map_add, map_mul, Bool.cond_true,\n  Bool.cond_false, MvPolynomial.aeval_X, add_sub_cancel'_right, one_pow, add_zero, mul_one]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nlet x : MvPolynomial Bool R := MvPolynomial.X true\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nlet y : MvPolynomial Bool R := MvPolynomial.X false\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nhave pderiv_true_x : pderiv true x = 1 := by rw [pderiv_X]; rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 \u2191(pderiv true) x = 1\n[PROOFSTEP]\nrw [pderiv_X]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\n\u22a2 Pi.single true 1 true = 1\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nhave pderiv_true_y : pderiv true y = 0 := by rw [pderiv_X]; rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 \u2191(pderiv true) y = 0\n[PROOFSTEP]\nrw [pderiv_X]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\n\u22a2 Pi.single true 1 false = 0\n[PROOFSTEP]\nrfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nlet e : Bool \u2192 R[X] := fun i =>\n  cond i X\n    (1 - X)\n      -- Start with `(x+y)^n = (x+y)^n`,\n        -- take the second `x`-derivative, evaluate at `x=X, y=1-X`, and multiply by `X`:\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\ntrans\n  MvPolynomial.aeval e (pderiv true (pderiv true ((x + y) ^ n))) *\n    X ^\n      2\n        -- On the left hand side we'll use the binomial theorem, then simplify.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (\u2191(pderiv true) ((x + y) ^ n))) * X ^ 2\n[PROOFSTEP]\nhave w :\n  \u2200 k : \u2115,\n    (k * (k - 1)) \u2022 bernsteinPolynomial R n k =\n      (n.choose k : R[X]) *\n          ((1 - Polynomial.X) ^ (n - k) * ((k : R[X]) * ((\u2191(k - 1) : R[X]) * Polynomial.X ^ (k - 1 - 1)))) *\n        Polynomial.X ^ 2 :=\n  by\n  rintro (_ | _ | k)\n  \u00b7 simp\n  \u00b7 simp\n  \u00b7 rw [bernsteinPolynomial]\n    simp only [\u2190 nat_cast_mul, Nat.succ_eq_add_one, Nat.add_succ_sub_one, add_zero, pow_succ]\n    push_cast\n    ring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2200 (k : \u2115),\n    (k * (k - 1)) \u2022 bernsteinPolynomial R n k =\n      \u2191(choose n k) * ((1 - X) ^ (n - k) * (\u2191k * (\u2191(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2\n[PROOFSTEP]\nrintro (_ | _ | k)\n[GOAL]\ncase zero\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 (Nat.zero * (Nat.zero - 1)) \u2022 bernsteinPolynomial R n Nat.zero =\n    \u2191(choose n Nat.zero) * ((1 - X) ^ (n - Nat.zero) * (\u2191Nat.zero * (\u2191(Nat.zero - 1) * X ^ (Nat.zero - 1 - 1)))) * X ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.zero\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 (Nat.succ Nat.zero * (Nat.succ Nat.zero - 1)) \u2022 bernsteinPolynomial R n (Nat.succ Nat.zero) =\n    \u2191(choose n (Nat.succ Nat.zero)) *\n        ((1 - X) ^ (n - Nat.succ Nat.zero) *\n          (\u2191(Nat.succ Nat.zero) * (\u2191(Nat.succ Nat.zero - 1) * X ^ (Nat.succ Nat.zero - 1 - 1)))) *\n      X ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 (Nat.succ (Nat.succ k) * (Nat.succ (Nat.succ k) - 1)) \u2022 bernsteinPolynomial R n (Nat.succ (Nat.succ k)) =\n    \u2191(choose n (Nat.succ (Nat.succ k))) *\n        ((1 - X) ^ (n - Nat.succ (Nat.succ k)) *\n          (\u2191(Nat.succ (Nat.succ k)) * (\u2191(Nat.succ (Nat.succ k) - 1) * X ^ (Nat.succ (Nat.succ k) - 1 - 1)))) *\n      X ^ 2\n[PROOFSTEP]\nrw [bernsteinPolynomial]\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 (Nat.succ (Nat.succ k) * (Nat.succ (Nat.succ k) - 1)) \u2022\n      (\u2191(choose n (Nat.succ (Nat.succ k))) * X ^ Nat.succ (Nat.succ k) * (1 - X) ^ (n - Nat.succ (Nat.succ k))) =\n    \u2191(choose n (Nat.succ (Nat.succ k))) *\n        ((1 - X) ^ (n - Nat.succ (Nat.succ k)) *\n          (\u2191(Nat.succ (Nat.succ k)) * (\u2191(Nat.succ (Nat.succ k) - 1) * X ^ (Nat.succ (Nat.succ k) - 1 - 1)))) *\n      X ^ 2\n[PROOFSTEP]\nsimp only [\u2190 nat_cast_mul, Nat.succ_eq_add_one, Nat.add_succ_sub_one, add_zero, pow_succ]\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 \u2191((k + 1 + 1) * (k + 1)) * (\u2191(choose n (k + 1 + 1)) * (X * (X * X ^ k)) * (1 - X) ^ (n - (k + 1 + 1))) =\n    \u2191(choose n (k + 1 + 1)) * ((1 - X) ^ (n - (k + 1 + 1)) * (\u2191(k + 1 + 1) * (\u2191(k + 1) * X ^ k))) * (X * (X * X ^ 0))\n[PROOFSTEP]\npush_cast\n[GOAL]\ncase succ.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nk : \u2115\n\u22a2 (\u2191k + 1 + 1) * (\u2191k + 1) * (\u2191(choose n (k + 1 + 1)) * (X * (X * X ^ k)) * (1 - X) ^ (n - (k + 1 + 1))) =\n    \u2191(choose n (k + 1 + 1)) * ((1 - X) ^ (n - (k + 1 + 1)) * ((\u2191k + 1 + 1) * ((\u2191k + 1) * X ^ k))) * (X * (X * X ^ 0))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw :\n  \u2200 (k : \u2115),\n    (k * (k - 1)) \u2022 bernsteinPolynomial R n k =\n      \u2191(choose n k) * ((1 - X) ^ (n - k) * (\u2191k * (\u2191(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (\u2191(pderiv true) ((x + y) ^ n))) * X ^ 2\n[PROOFSTEP]\nrw [add_pow, (pderiv true).map_sum, (pderiv true).map_sum, (MvPolynomial.aeval e).map_sum, Finset.sum_mul]\n  -- Step inside the sum:\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw :\n  \u2200 (k : \u2115),\n    (k * (k - 1)) \u2022 bernsteinPolynomial R n k =\n      \u2191(choose n k) * ((1 - X) ^ (n - k) * (\u2191k * (\u2191(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd =\n    \u2211 x_1 in Finset.range (n + 1),\n      \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (\u2191(pderiv true) (x ^ x_1 * y ^ (n - x_1) * \u2191(choose n x_1)))) * X ^ 2\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun k _ => (w k).trans _\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\nw :\n  \u2200 (k : \u2115),\n    (k * (k - 1)) \u2022 bernsteinPolynomial R n k =\n      \u2191(choose n k) * ((1 - X) ^ (n - k) * (\u2191k * (\u2191(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2\nk : \u2115\nx\u271d : k \u2208 Finset.range (n + 1)\n\u22a2 \u2191(choose n k) * ((1 - X) ^ (n - k) * (\u2191k * (\u2191(k - 1) * X ^ (k - 1 - 1)))) * X ^ 2 =\n    \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (\u2191(pderiv true) (x ^ k * y ^ (n - k) * \u2191(choose n k)))) * X ^ 2\n[PROOFSTEP]\nsimp only [pderiv_true_x, pderiv_true_y, Algebra.id.smul_eq_mul, nsmul_eq_mul, Bool.cond_true, Bool.cond_false,\n  add_zero, zero_add, mul_zero, smul_zero, mul_one, MvPolynomial.aeval_X, MvPolynomial.pderiv_X_self,\n  MvPolynomial.pderiv_X_of_ne, Derivation.leibniz_pow, Derivation.leibniz, Derivation.map_coe_nat, map_natCast, map_pow,\n  map_mul, map_add]\n  -- On the right hand side, we'll just simplify.\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\nx : MvPolynomial Bool R := MvPolynomial.X true\ny : MvPolynomial Bool R := MvPolynomial.X false\npderiv_true_x : \u2191(pderiv true) x = 1\npderiv_true_y : \u2191(pderiv true) y = 0\ne : Bool \u2192 R[X] := fun i => bif i then X else 1 - X\n\u22a2 \u2191(MvPolynomial.aeval e) (\u2191(pderiv true) (\u2191(pderiv true) ((x + y) ^ n))) * X ^ 2 = (n * (n - 1)) \u2022 X ^ 2\n[PROOFSTEP]\nsimp only [pderiv_one, pderiv_mul, (pderiv _).leibniz_pow, (pderiv _).map_coe_nat, (pderiv true).map_add, pderiv_true_x,\n  pderiv_true_y, Algebra.id.smul_eq_mul, add_zero, mul_one, Derivation.map_smul_of_tower, map_nsmul, map_pow, map_add,\n  Bool.cond_true, Bool.cond_false, MvPolynomial.aeval_X, add_sub_cancel'_right, one_pow, smul_smul, smul_one_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (n \u2022 X - \u2191\u03bd) ^ 2 * bernsteinPolynomial R n \u03bd = n \u2022 X * (1 - X)\n[PROOFSTEP]\nhave p :\n  ((((Finset.range (n + 1)).sum fun \u03bd => (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd) +\n        (1 - (2 * n) \u2022 Polynomial.X) * (Finset.range (n + 1)).sum fun \u03bd => \u03bd \u2022 bernsteinPolynomial R n \u03bd) +\n      n ^ 2 \u2022 X ^ 2 * (Finset.range (n + 1)).sum fun \u03bd => bernsteinPolynomial R n \u03bd) =\n    _ :=\n  rfl\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (n \u2022 X - \u2191\u03bd) ^ 2 * bernsteinPolynomial R n \u03bd = n \u2022 X * (1 - X)\n[PROOFSTEP]\nconv at p =>\n  lhs\n  rw [Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_add_distrib, \u2190 Finset.sum_add_distrib]\n  simp only [\u2190 nat_cast_mul]\n  simp only [\u2190 mul_assoc]\n  simp only [\u2190 add_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\n  lhs\n  rw [Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_add_distrib, \u2190 Finset.sum_add_distrib]\n  simp only [\u2190 nat_cast_mul]\n  simp only [\u2190 mul_assoc]\n  simp only [\u2190 add_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\n  lhs\n  rw [Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_add_distrib, \u2190 Finset.sum_add_distrib]\n  simp only [\u2190 nat_cast_mul]\n  simp only [\u2190 mul_assoc]\n  simp only [\u2190 add_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\nlhs\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n      (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n    n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\nrw [Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_add_distrib, \u2190 Finset.sum_add_distrib]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n    ((x * (x - 1)) \u2022 bernsteinPolynomial R n x + (\u21911 - (2 * n) \u2022 X) * x \u2022 bernsteinPolynomial R n x +\n      n ^ 2 \u2022 X ^ 2 * bernsteinPolynomial R n x)\n[PROOFSTEP]\nsimp only [\u2190 nat_cast_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n    (\u2191(x * (x - 1)) * bernsteinPolynomial R n x + (\u21911 - \u2191(2 * n) * X) * (\u2191x * bernsteinPolynomial R n x) +\n      \u2191(n ^ 2) * X ^ 2 * bernsteinPolynomial R n x)\n[PROOFSTEP]\nsimp only [\u2190 mul_assoc]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n    (\u2191(x * (x - 1)) * bernsteinPolynomial R n x + (\u21911 - \u2191(2 * n) * X) * \u2191x * bernsteinPolynomial R n x +\n      \u2191(n ^ 2) * X ^ 2 * bernsteinPolynomial R n x)\n[PROOFSTEP]\nsimp only [\u2190 add_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (n \u2022 X - \u2191\u03bd) ^ 2 * bernsteinPolynomial R n \u03bd = n \u2022 X * (1 - X)\n[PROOFSTEP]\nconv at p =>\n  rhs\n  rw [sum, sum_smul, sum_mul_smul, \u2190 nat_cast_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\n  rhs\n  rw [sum, sum_smul, sum_mul_smul, \u2190 nat_cast_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\n  rhs\n  rw [sum, sum_smul, sum_mul_smul, \u2190 nat_cast_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\nrhs\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n        (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n      n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n| \u2211 \u03bd in Finset.range (n + 1), (\u03bd * (\u03bd - 1)) \u2022 bernsteinPolynomial R n \u03bd +\n      (\u21911 - (2 * n) \u2022 X) * \u2211 \u03bd in Finset.range (n + 1), \u03bd \u2022 bernsteinPolynomial R n \u03bd +\n    n ^ 2 \u2022 X ^ 2 * \u2211 \u03bd in Finset.range (n + 1), bernsteinPolynomial R n \u03bd\n[PROOFSTEP]\nrw [sum, sum_smul, sum_mul_smul, \u2190 nat_cast_mul]\n[GOAL]\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\n\u22a2 \u2211 \u03bd in Finset.range (n + 1), (n \u2022 X - \u2191\u03bd) ^ 2 * bernsteinPolynomial R n \u03bd = n \u2022 X * (1 - X)\n[PROOFSTEP]\ncalc\n  _ = _ := Finset.sum_congr rfl fun k m => ?_\n  _ = _ := p\n  _ = _ := ?_\n[GOAL]\ncase calc_1\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nk : \u2115\nm : k \u2208 Finset.range (n + 1)\n\u22a2 (n \u2022 X - \u2191k) ^ 2 * bernsteinPolynomial R n k =\n    (\u2191(k * (k - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191k + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n k\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase calc_1.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nk : \u2115\nm : k \u2208 Finset.range (n + 1)\n\u22a2 (n \u2022 X - \u2191k) ^ 2 = \u2191(k * (k - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191k + \u2191(n ^ 2) * X ^ 2\n[PROOFSTEP]\nsimp only [\u2190 nat_cast_mul, push_cast]\n[GOAL]\ncase calc_1.e_a\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nk : \u2115\nm : k \u2208 Finset.range (n + 1)\n\u22a2 (\u2191n * X - \u2191k) ^ 2 = \u2191k * \u2191(k - 1) + (1 - 2 * \u2191n * X) * \u2191k + \u2191n ^ 2 * X ^ 2\n[PROOFSTEP]\ncases k\n[GOAL]\ncase calc_1.e_a.zero\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nm : Nat.zero \u2208 Finset.range (n + 1)\n\u22a2 (\u2191n * X - \u2191Nat.zero) ^ 2 = \u2191Nat.zero * \u2191(Nat.zero - 1) + (1 - 2 * \u2191n * X) * \u2191Nat.zero + \u2191n ^ 2 * X ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase calc_1.e_a.zero\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nm : Nat.zero \u2208 Finset.range (n + 1)\n\u22a2 (\u2191n * X) ^ 2 = \u2191n ^ 2 * X ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase calc_1.e_a.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nn\u271d : \u2115\nm : Nat.succ n\u271d \u2208 Finset.range (n + 1)\n\u22a2 (\u2191n * X - \u2191(Nat.succ n\u271d)) ^ 2 =\n    \u2191(Nat.succ n\u271d) * \u2191(Nat.succ n\u271d - 1) + (1 - 2 * \u2191n * X) * \u2191(Nat.succ n\u271d) + \u2191n ^ 2 * X ^ 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase calc_1.e_a.succ\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\nn\u271d : \u2115\nm : Nat.succ n\u271d \u2208 Finset.range (n + 1)\n\u22a2 (\u2191n * X - (\u2191n\u271d + 1)) ^ 2 = (\u2191n\u271d + 1) * \u2191n\u271d + (1 - 2 * \u2191n * X) * (\u2191n\u271d + 1) + \u2191n ^ 2 * X ^ 2\n[PROOFSTEP]\nring\n[GOAL]\ncase calc_2\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\n\u22a2 \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1 = n \u2022 X * (1 - X)\n[PROOFSTEP]\nsimp only [\u2190 nat_cast_mul, push_cast]\n[GOAL]\ncase calc_2\nR : Type u_1\ninst\u271d : CommRing R\nn : \u2115\np :\n  \u2211 x in Finset.range (n + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * n) * X) * \u2191x + \u2191(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n    \u2191(n * (n - 1)) * X ^ 2 + (\u21911 - (2 * n) \u2022 X) * n \u2022 X + n ^ 2 \u2022 X ^ 2 * 1\n\u22a2 \u2191n * \u2191(n - 1) * X ^ 2 + (1 - 2 * \u2191n * X) * (\u2191n * X) + \u2191n ^ 2 * X ^ 2 * 1 = \u2191n * X * (1 - X)\n[PROOFSTEP]\ncases n\n[GOAL]\ncase calc_2.zero\nR : Type u_1\ninst\u271d : CommRing R\np :\n  \u2211 x in Finset.range (Nat.zero + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * Nat.zero) * X) * \u2191x + \u2191(Nat.zero ^ 2) * X ^ 2) * bernsteinPolynomial R Nat.zero x =\n    \u2191(Nat.zero * (Nat.zero - 1)) * X ^ 2 + (\u21911 - (2 * Nat.zero) \u2022 X) * Nat.zero \u2022 X + Nat.zero ^ 2 \u2022 X ^ 2 * 1\n\u22a2 \u2191Nat.zero * \u2191(Nat.zero - 1) * X ^ 2 + (1 - 2 * \u2191Nat.zero * X) * (\u2191Nat.zero * X) + \u2191Nat.zero ^ 2 * X ^ 2 * 1 =\n    \u2191Nat.zero * X * (1 - X)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase calc_2.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d : \u2115\np :\n  \u2211 x in Finset.range (Nat.succ n\u271d + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * Nat.succ n\u271d) * X) * \u2191x + \u2191(Nat.succ n\u271d ^ 2) * X ^ 2) *\n        bernsteinPolynomial R (Nat.succ n\u271d) x =\n    \u2191(Nat.succ n\u271d * (Nat.succ n\u271d - 1)) * X ^ 2 + (\u21911 - (2 * Nat.succ n\u271d) \u2022 X) * Nat.succ n\u271d \u2022 X +\n      Nat.succ n\u271d ^ 2 \u2022 X ^ 2 * 1\n\u22a2 \u2191(Nat.succ n\u271d) * \u2191(Nat.succ n\u271d - 1) * X ^ 2 + (1 - 2 * \u2191(Nat.succ n\u271d) * X) * (\u2191(Nat.succ n\u271d) * X) +\n      \u2191(Nat.succ n\u271d) ^ 2 * X ^ 2 * 1 =\n    \u2191(Nat.succ n\u271d) * X * (1 - X)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase calc_2.succ\nR : Type u_1\ninst\u271d : CommRing R\nn\u271d : \u2115\np :\n  \u2211 x in Finset.range (Nat.succ n\u271d + 1),\n      (\u2191(x * (x - 1)) + (\u21911 - \u2191(2 * Nat.succ n\u271d) * X) * \u2191x + \u2191(Nat.succ n\u271d ^ 2) * X ^ 2) *\n        bernsteinPolynomial R (Nat.succ n\u271d) x =\n    \u2191(Nat.succ n\u271d * (Nat.succ n\u271d - 1)) * X ^ 2 + (\u21911 - (2 * Nat.succ n\u271d) \u2022 X) * Nat.succ n\u271d \u2022 X +\n      Nat.succ n\u271d ^ 2 \u2022 X ^ 2 * 1\n\u22a2 (\u2191n\u271d + 1) * \u2191n\u271d * X ^ 2 + (1 - 2 * (\u2191n\u271d + 1) * X) * ((\u2191n\u271d + 1) * X) + (\u2191n\u271d + 1) ^ 2 * X ^ 2 = (\u2191n\u271d + 1) * X * (1 - X)\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Bernstein", "llama_tokens": 36664, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929002541068, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.5052780054962928}}
{"text": "[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : IsUnit (r \u2022 \u2191\u2191\u2090 s - a)\n\u22a2 (\u2191\u2191\u2090 s - r\u207b\u00b9 \u2022 a) * r \u2022 \u2191(IsUnit.unit h)\u207b\u00b9 = 1\n[PROOFSTEP]\nrw [mul_smul_comm, \u2190 smul_mul_assoc, smul_sub, smul_inv_smul, h.mul_val_inv]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : IsUnit (r \u2022 \u2191\u2191\u2090 s - a)\n\u22a2 r \u2022 \u2191(IsUnit.unit h)\u207b\u00b9 * (\u2191\u2191\u2090 s - r\u207b\u00b9 \u2022 a) = 1\n[PROOFSTEP]\nrw [smul_mul_assoc, \u2190 mul_smul_comm, smul_sub, smul_inv_smul, h.val_inv_mul]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\na : A\n\u22a2 \u00acr \u2208 \u03c3 a \u2194 IsUnit (\u2191\u2191\u2090 r - a)\n[PROOFSTEP]\napply not_iff_not.mp\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\na : A\n\u22a2 \u00ac\u00acr \u2208 \u03c3 a \u2194 \u00acIsUnit (\u2191\u2191\u2090 r - a)\n[PROOFSTEP]\nsimp [Set.not_not_mem, mem_iff]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\n\u22a2 0 \u2208 \u03c3 a \u2194 \u00acIsUnit a\n[PROOFSTEP]\nrw [mem_iff, map_zero, zero_sub, IsUnit.neg_iff]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\n\u22a2 \u00ac0 \u2208 \u03c3 a \u2194 IsUnit a\n[PROOFSTEP]\nrw [zero_mem_iff, Classical.not_not]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\na b c : A\nh\u2081 : (\u2191\u2191\u2090 r - a) * b = 1\nh\u2082 : c * (\u2191\u2191\u2090 r - a) = 1\n\u22a2 b * (\u2191\u2191\u2090 r - a) = 1\n[PROOFSTEP]\nrwa [\u2190 left_inv_eq_right_inv h\u2082 h\u2081]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Subsingleton A\na : A\n\u22a2 resolventSet R a = univ\n[PROOFSTEP]\nsimp_rw [resolventSet, Subsingleton.elim (algebraMap R A _ - a) 1, isUnit_one, Set.setOf_true]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra R A\ninst\u271d : Subsingleton A\na : A\n\u22a2 \u03c3 a = \u2205\n[PROOFSTEP]\nrw [spectrum, resolventSet_of_subsingleton, Set.compl_univ]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\n\u22a2 r \u2022 resolvent a s = resolvent (r\u207b\u00b9 \u2022 a) (r\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nby_cases h : s \u2208 spectrum R a\n[GOAL]\ncase pos\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : s \u2208 \u03c3 a\n\u22a2 r \u2022 resolvent a s = resolvent (r\u207b\u00b9 \u2022 a) (r\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nrw [mem_iff] at h \n[GOAL]\ncase pos\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acIsUnit (\u2191\u2191\u2090 s - a)\n\u22a2 r \u2022 resolvent a s = resolvent (r\u207b\u00b9 \u2022 a) (r\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nsimp only [resolvent, Algebra.algebraMap_eq_smul_one] at *\n[GOAL]\ncase pos\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acIsUnit (s \u2022 1 - a)\n\u22a2 r \u2022 Ring.inverse (s \u2022 1 - a) = Ring.inverse ((r\u207b\u00b9 \u2022 s) \u2022 1 - r\u207b\u00b9 \u2022 a)\n[PROOFSTEP]\nrw [smul_assoc, \u2190 smul_sub]\n[GOAL]\ncase pos\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acIsUnit (s \u2022 1 - a)\n\u22a2 r \u2022 Ring.inverse (s \u2022 1 - a) = Ring.inverse (r\u207b\u00b9 \u2022 (s \u2022 1 - a))\n[PROOFSTEP]\nhave h' : \u00acIsUnit (r\u207b\u00b9 \u2022 (s \u2022 (1 : A) - a)) := fun hu => h (by simpa only [smul_inv_smul] using IsUnit.smul r hu)\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acIsUnit (s \u2022 1 - a)\nhu : IsUnit (r\u207b\u00b9 \u2022 (s \u2022 1 - a))\n\u22a2 IsUnit (s \u2022 1 - a)\n[PROOFSTEP]\nsimpa only [smul_inv_smul] using IsUnit.smul r hu\n[GOAL]\ncase pos\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acIsUnit (s \u2022 1 - a)\nh' : \u00acIsUnit (r\u207b\u00b9 \u2022 (s \u2022 1 - a))\n\u22a2 r \u2022 Ring.inverse (s \u2022 1 - a) = Ring.inverse (r\u207b\u00b9 \u2022 (s \u2022 1 - a))\n[PROOFSTEP]\nsimp only [Ring.inverse_non_unit _ h, Ring.inverse_non_unit _ h', smul_zero]\n[GOAL]\ncase neg\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acs \u2208 \u03c3 a\n\u22a2 r \u2022 resolvent a s = resolvent (r\u207b\u00b9 \u2022 a) (r\u207b\u00b9 \u2022 s)\n[PROOFSTEP]\nsimp only [resolvent]\n[GOAL]\ncase neg\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acs \u2208 \u03c3 a\n\u22a2 r \u2022 Ring.inverse (\u2191\u2191\u2090 s - a) = Ring.inverse (\u2191\u2191\u2090 (r\u207b\u00b9 \u2022 s) - r\u207b\u00b9 \u2022 a)\n[PROOFSTEP]\nhave h' : IsUnit (r \u2022 algebraMap R A (r\u207b\u00b9 \u2022 s) - a) := by\n  simpa [Algebra.algebraMap_eq_smul_one, smul_assoc] using not_mem_iff.mp h\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acs \u2208 \u03c3 a\n\u22a2 IsUnit (r \u2022 \u2191\u2191\u2090 (r\u207b\u00b9 \u2022 s) - a)\n[PROOFSTEP]\nsimpa [Algebra.algebraMap_eq_smul_one, smul_assoc] using not_mem_iff.mp h\n[GOAL]\ncase neg\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acs \u2208 \u03c3 a\nh' : IsUnit (r \u2022 \u2191\u2191\u2090 (r\u207b\u00b9 \u2022 s) - a)\n\u22a2 r \u2022 Ring.inverse (\u2191\u2191\u2090 s - a) = Ring.inverse (\u2191\u2191\u2090 (r\u207b\u00b9 \u2022 s) - r\u207b\u00b9 \u2022 a)\n[PROOFSTEP]\nrw [\u2190 h'.subInvSMul_val, \u2190 (not_mem_iff.mp h).unit_spec, Ring.inverse_unit, Ring.inverse_unit, \u2190\n  h'.subInvSMul.inv_eq_val_inv, h'.subInvSMul_inv]\n[GOAL]\ncase neg\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\ns : R\na : A\nh : \u00acs \u2208 \u03c3 a\nh' : IsUnit (r \u2022 \u2191\u2191\u2090 (r\u207b\u00b9 \u2022 s) - a)\n\u22a2 r \u2022 \u2191(IsUnit.unit (_ : IsUnit (\u2191\u2191\u2090 s - a)))\u207b\u00b9 = r \u2022 \u2191(IsUnit.unit h')\u207b\u00b9\n[PROOFSTEP]\nsimp only [Algebra.algebraMap_eq_smul_one, smul_assoc, smul_inv_smul]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\n\u22a2 r \u2022 resolvent a \u2191r = resolvent (r\u207b\u00b9 \u2022 a) 1\n[PROOFSTEP]\nsimpa only [Units.smul_def, Algebra.id.smul_eq_mul, Units.inv_mul] using @units_smul_resolvent _ _ _ _ _ r r a\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : \u2191r \u2208 resolventSet R \u2191a\n\u22a2 \u2191r\u207b\u00b9 \u2208 resolventSet R \u2191a\u207b\u00b9\n[PROOFSTEP]\nrw [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one, \u2190 Units.smul_def] at h \u22a2\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\n\u22a2 IsUnit (r\u207b\u00b9 \u2022 1 - \u2191a\u207b\u00b9)\n[PROOFSTEP]\nrw [IsUnit.smul_sub_iff_sub_inv_smul, inv_inv, IsUnit.sub_iff]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\n\u22a2 IsUnit (r \u2022 \u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nhave h\u2081 : (a : A) * (r \u2022 (\u2191a\u207b\u00b9 : A) - 1) = r \u2022 (1 : A) - a := by rw [mul_sub, mul_smul_comm, a.mul_inv, mul_one]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\n\u22a2 \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\n[PROOFSTEP]\nrw [mul_sub, mul_smul_comm, a.mul_inv, mul_one]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\nh\u2081 : \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\n\u22a2 IsUnit (r \u2022 \u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nhave h\u2082 : (r \u2022 (\u2191a\u207b\u00b9 : A) - 1) * a = r \u2022 (1 : A) - a := by rw [sub_mul, smul_mul_assoc, a.inv_mul, one_mul]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\nh\u2081 : \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\n\u22a2 (r \u2022 \u2191a\u207b\u00b9 - 1) * \u2191a = r \u2022 1 - \u2191a\n[PROOFSTEP]\nrw [sub_mul, smul_mul_assoc, a.inv_mul, one_mul]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\nh\u2081 : \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\nh\u2082 : (r \u2022 \u2191a\u207b\u00b9 - 1) * \u2191a = r \u2022 1 - \u2191a\n\u22a2 IsUnit (r \u2022 \u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nhave hcomm : Commute (a : A) (r \u2022 (\u2191a\u207b\u00b9 : A) - 1) := by rwa [\u2190 h\u2082] at h\u2081 \n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\nh\u2081 : \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\nh\u2082 : (r \u2022 \u2191a\u207b\u00b9 - 1) * \u2191a = r \u2022 1 - \u2191a\n\u22a2 Commute (\u2191a) (r \u2022 \u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nrwa [\u2190 h\u2082] at h\u2081 \n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\nr : R\u02e3\na : A\u02e3\nh : IsUnit (r \u2022 1 - \u2191a)\nh\u2081 : \u2191a * (r \u2022 \u2191a\u207b\u00b9 - 1) = r \u2022 1 - \u2191a\nh\u2082 : (r \u2022 \u2191a\u207b\u00b9 - 1) * \u2191a = r \u2022 1 - \u2191a\nhcomm : Commute (\u2191a) (r \u2022 \u2191a\u207b\u00b9 - 1)\n\u22a2 IsUnit (r \u2022 \u2191a\u207b\u00b9 - 1)\n[PROOFSTEP]\nexact (hcomm.isUnit_mul_iff.mp (h\u2081.symm \u25b8 h)).2\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\u02e3\n\u22a2 0 \u2208 resolventSet R \u2191a\n[PROOFSTEP]\nsimpa only [mem_resolventSet_iff, \u2190 not_mem_iff, zero_not_mem_iff] using a.isUnit\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr s : R\n\u22a2 r + s \u2208 \u03c3 a \u2194 r \u2208 \u03c3 (-\u2191\u2191\u2090 s + a)\n[PROOFSTEP]\nsimp only [mem_iff, sub_neg_eq_add, \u2190 sub_sub, map_add]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr s : R\n\u22a2 r + s \u2208 \u03c3 (\u2191\u2191\u2090 s + a) \u2194 r \u2208 \u03c3 a\n[PROOFSTEP]\nrw [add_mem_iff, neg_add_cancel_left]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\ns : R\nr : R\u02e3\n\u22a2 r \u2022 s \u2208 \u03c3 (r \u2022 a) \u2194 s \u2208 \u03c3 a\n[PROOFSTEP]\nsimp only [mem_iff, not_iff_not, Algebra.algebraMap_eq_smul_one, smul_assoc, \u2190 smul_sub, isUnit_smul_iff]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\n\u22a2 \u03c3 (r \u2022 a) = r \u2022 \u03c3 a\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\n\u22a2 x \u2208 \u03c3 (r \u2022 a) \u2194 x \u2208 r \u2022 \u03c3 a\n[PROOFSTEP]\nhave x_eq : x = r \u2022 r\u207b\u00b9 \u2022 x := by simp\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\n\u22a2 x = r \u2022 r\u207b\u00b9 \u2022 x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\n\u22a2 x \u2208 \u03c3 (r \u2022 a) \u2194 x \u2208 r \u2022 \u03c3 a\n[PROOFSTEP]\nnth_rw 1 [x_eq]\n[GOAL]\ncase h\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\n\u22a2 r \u2022 r\u207b\u00b9 \u2022 x \u2208 \u03c3 (r \u2022 a) \u2194 x \u2208 r \u2022 \u03c3 a\n[PROOFSTEP]\nrw [smul_mem_smul_iff]\n[GOAL]\ncase h\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\n\u22a2 r\u207b\u00b9 \u2022 x \u2208 \u03c3 a \u2194 x \u2208 r \u2022 \u03c3 a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\n\u22a2 r\u207b\u00b9 \u2022 x \u2208 \u03c3 a \u2192 x \u2208 r \u2022 \u03c3 a\n[PROOFSTEP]\nexact fun h => \u27e8r\u207b\u00b9 \u2022 x, \u27e8h, show r \u2022 r\u207b\u00b9 \u2022 x = x by simp\u27e9\u27e9\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\nh : r\u207b\u00b9 \u2022 x \u2208 \u03c3 a\n\u22a2 r \u2022 r\u207b\u00b9 \u2022 x = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mpr\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\n\u22a2 x \u2208 r \u2022 \u03c3 a \u2192 r\u207b\u00b9 \u2022 x \u2208 \u03c3 a\n[PROOFSTEP]\nrintro \u27e8w, _, (x'_eq : r \u2022 w = x)\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\u02e3\nx : R\nx_eq : x = r \u2022 r\u207b\u00b9 \u2022 x\nw : R\nleft\u271d : w \u2208 \u03c3 a\nx'_eq : r \u2022 w = x\n\u22a2 r\u207b\u00b9 \u2022 x \u2208 \u03c3 a\n[PROOFSTEP]\nsimpa [\u2190 x'_eq]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\n\u22a2 \u2191r \u2208 \u03c3 (a * b) \u2194 \u2191r \u2208 \u03c3 (b * a)\n[PROOFSTEP]\nhave h\u2081 : \u2200 x y : A, IsUnit (1 - x * y) \u2192 IsUnit (1 - y * x) :=\n  by\n  refine' fun x y h => \u27e8\u27e81 - y * x, 1 + y * h.unit.inv * x, _, _\u27e9, rfl\u27e9\n  calc\n    (1 - y * x) * (1 + y * (IsUnit.unit h).inv * x) = 1 - y * x + y * ((1 - x * y) * h.unit.inv) * x := by noncomm_ring\n    _ = 1 := by simp only [Units.inv_eq_val_inv, IsUnit.mul_val_inv, mul_one, sub_add_cancel]\n  calc\n    (1 + y * (IsUnit.unit h).inv * x) * (1 - y * x) = 1 - y * x + y * (h.unit.inv * (1 - x * y)) * x := by noncomm_ring\n    _ = 1 := by simp only [Units.inv_eq_val_inv, IsUnit.val_inv_mul, mul_one, sub_add_cancel]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\n\u22a2 \u2200 (x y : A), IsUnit (1 - x * y) \u2192 IsUnit (1 - y * x)\n[PROOFSTEP]\nrefine' fun x y h => \u27e8\u27e81 - y * x, 1 + y * h.unit.inv * x, _, _\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 (1 - y * x) * (1 + y * (IsUnit.unit h).inv * x) = 1\ncase refine'_2\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 (1 + y * (IsUnit.unit h).inv * x) * (1 - y * x) = 1\n[PROOFSTEP]\ncalc\n  (1 - y * x) * (1 + y * (IsUnit.unit h).inv * x) = 1 - y * x + y * ((1 - x * y) * h.unit.inv) * x := by noncomm_ring\n  _ = 1 := by simp only [Units.inv_eq_val_inv, IsUnit.mul_val_inv, mul_one, sub_add_cancel]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 (1 - y * x) * (1 + y * (IsUnit.unit h).inv * x) = 1 - y * x + y * ((1 - x * y) * (IsUnit.unit h).inv) * x\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 1 - y * x + y * ((1 - x * y) * (IsUnit.unit h).inv) * x = 1\n[PROOFSTEP]\nsimp only [Units.inv_eq_val_inv, IsUnit.mul_val_inv, mul_one, sub_add_cancel]\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 (1 + y * (IsUnit.unit h).inv * x) * (1 - y * x) = 1\n[PROOFSTEP]\ncalc\n  (1 + y * (IsUnit.unit h).inv * x) * (1 - y * x) = 1 - y * x + y * (h.unit.inv * (1 - x * y)) * x := by noncomm_ring\n  _ = 1 := by simp only [Units.inv_eq_val_inv, IsUnit.val_inv_mul, mul_one, sub_add_cancel]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 (1 + y * (IsUnit.unit h).inv * x) * (1 - y * x) = 1 - y * x + y * ((IsUnit.unit h).inv * (1 - x * y)) * x\n[PROOFSTEP]\nnoncomm_ring\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nx y : A\nh : IsUnit (1 - x * y)\n\u22a2 1 - y * x + y * ((IsUnit.unit h).inv * (1 - x * y)) * x = 1\n[PROOFSTEP]\nsimp only [Units.inv_eq_val_inv, IsUnit.val_inv_mul, mul_one, sub_add_cancel]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nh\u2081 : \u2200 (x y : A), IsUnit (1 - x * y) \u2192 IsUnit (1 - y * x)\n\u22a2 \u2191r \u2208 \u03c3 (a * b) \u2194 \u2191r \u2208 \u03c3 (b * a)\n[PROOFSTEP]\nhave := Iff.intro (h\u2081 (r\u207b\u00b9 \u2022 a) b) (h\u2081 b (r\u207b\u00b9 \u2022 a))\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nh\u2081 : \u2200 (x y : A), IsUnit (1 - x * y) \u2192 IsUnit (1 - y * x)\nthis : IsUnit (1 - r\u207b\u00b9 \u2022 a * b) \u2194 IsUnit (1 - b * r\u207b\u00b9 \u2022 a)\n\u22a2 \u2191r \u2208 \u03c3 (a * b) \u2194 \u2191r \u2208 \u03c3 (b * a)\n[PROOFSTEP]\nrw [mul_smul_comm r\u207b\u00b9 b a] at this \n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na b : A\nr : R\u02e3\nh\u2081 : \u2200 (x y : A), IsUnit (1 - x * y) \u2192 IsUnit (1 - y * x)\nthis : IsUnit (1 - r\u207b\u00b9 \u2022 a * b) \u2194 IsUnit (1 - r\u207b\u00b9 \u2022 (b * a))\n\u22a2 \u2191r \u2208 \u03c3 (a * b) \u2194 \u2191r \u2208 \u03c3 (b * a)\n[PROOFSTEP]\nsimpa only [mem_iff, not_iff_not, Algebra.algebraMap_eq_smul_one, \u2190 Units.smul_def, IsUnit.smul_sub_iff_sub_inv_smul,\n  smul_mul_assoc]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : InvolutiveStar R\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nr : R\na : A\n\u22a2 star r \u2208 resolventSet R a \u2194 r \u2208 resolventSet R (star a)\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : InvolutiveStar R\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nr : R\na : A\nh : star r \u2208 resolventSet R a\n\u22a2 r \u2208 resolventSet R (star a)\n[PROOFSTEP]\nsimpa only [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one, star_sub, star_smul, star_star, star_one] using\n  IsUnit.star h\n[GOAL]\ncase refine'_2\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : InvolutiveStar R\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\nr : R\na : A\nh : r \u2208 resolventSet R (star a)\n\u22a2 star r \u2208 resolventSet R a\n[PROOFSTEP]\nsimpa only [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one, star_sub, star_smul, star_star, star_one] using\n  IsUnit.star h\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : InvolutiveStar R\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\na : A\n\u22a2 \u03c3 (star a) = star (\u03c3 a)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nR : Type u\nA : Type v\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : InvolutiveStar R\ninst\u271d\u00b9 : StarRing A\ninst\u271d : StarModule R A\na : A\nx\u271d : R\n\u22a2 x\u271d \u2208 \u03c3 (star a) \u2194 x\u271d \u2208 star (\u03c3 a)\n[PROOFSTEP]\nsimpa only [Set.mem_star, mem_iff, not_iff_not] using star_mem_resolventSet_iff.symm\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr x : R\n\u22a2 x \u2208 {r} + \u03c3 a \u2194 x \u2208 \u03c3 (\u2191\u2191\u2090 r + a)\n[PROOFSTEP]\nrw [singleton_add, image_add_left, mem_preimage]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr x : R\n\u22a2 (fun x x_1 => x + x_1) (-r) x \u2208 \u03c3 a \u2194 x \u2208 \u03c3 (\u2191\u2191\u2090 r + a)\n[PROOFSTEP]\nsimp only\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr x : R\n\u22a2 -r + x \u2208 \u03c3 a \u2194 x \u2208 \u03c3 (\u2191\u2191\u2090 r + a)\n[PROOFSTEP]\nrw [add_comm, add_mem_iff, map_neg, neg_neg]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nx : R\n\u22a2 x \u2208 -\u03c3 a \u2194 x \u2208 \u03c3 (-a)\n[PROOFSTEP]\nsimp only [mem_neg, mem_iff, map_neg, \u2190 neg_add', IsUnit.neg_iff, sub_neg_eq_add]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\n\u22a2 {r} - \u03c3 a = \u03c3 (\u2191\u2191\u2090 r - a)\n[PROOFSTEP]\nrw [sub_eq_add_neg, neg_eq, singleton_add_eq, sub_eq_add_neg]\n[GOAL]\nR : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr : R\n\u22a2 \u03c3 a - {r} = \u03c3 (a - \u2191\u2191\u2090 r)\n[PROOFSTEP]\nsimpa only [neg_sub, neg_eq] using congr_arg Neg.neg (singleton_sub_eq a r)\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\n\u22a2 \u03c3 0 = {0}\n[PROOFSTEP]\nrefine' Set.Subset.antisymm _ (by simp [Algebra.algebraMap_eq_smul_one, mem_iff])\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\n\u22a2 {0} \u2286 \u03c3 0\n[PROOFSTEP]\nsimp [Algebra.algebraMap_eq_smul_one, mem_iff]\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\n\u22a2 \u03c3 0 \u2286 {0}\n[PROOFSTEP]\nrw [spectrum, Set.compl_subset_comm]\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\n\u22a2 {0}\u1d9c \u2286 resolventSet \ud835\udd5c 0\n[PROOFSTEP]\nintro k hk\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\nhk : k \u2208 {0}\u1d9c\n\u22a2 k \u2208 resolventSet \ud835\udd5c 0\n[PROOFSTEP]\nrw [Set.mem_compl_singleton_iff] at hk \n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\nhk : k \u2260 0\n\u22a2 k \u2208 resolventSet \ud835\udd5c 0\n[PROOFSTEP]\nhave : IsUnit (Units.mk0 k hk \u2022 (1 : A)) := IsUnit.smul (Units.mk0 k hk) isUnit_one\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\nhk : k \u2260 0\nthis : IsUnit (Units.mk0 k hk \u2022 1)\n\u22a2 k \u2208 resolventSet \ud835\udd5c 0\n[PROOFSTEP]\nsimpa [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one]\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\n\u22a2 \u03c3 (\u2191\u2191\u2090 k) = {k}\n[PROOFSTEP]\nrw [\u2190 add_zero (\u2191\u2090 k), \u2190 singleton_add_eq, zero_eq, Set.singleton_add_singleton, add_zero]\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\n\u22a2 \u03c3 1 = \u03c3 (\u2191\u2191\u2090 1)\n[PROOFSTEP]\nrw [Algebra.algebraMap_eq_smul_one, one_smul]\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\na : A\nha : Set.Nonempty (\u03c3 a)\n\u22a2 \u03c3 (k \u2022 a) = k \u2022 \u03c3 a\n[PROOFSTEP]\nrcases eq_or_ne k 0 with (rfl | h)\n[GOAL]\ncase inl\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\na : A\nha : Set.Nonempty (\u03c3 a)\n\u22a2 \u03c3 (0 \u2022 a) = 0 \u2022 \u03c3 a\n[PROOFSTEP]\nsimpa [ha, zero_smul_set] using (show {(0 : \ud835\udd5c)} = (0 : Set \ud835\udd5c) from rfl)\n[GOAL]\ncase inr\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b3 : Field \ud835\udd5c\ninst\u271d\u00b2 : Ring A\ninst\u271d\u00b9 : Algebra \ud835\udd5c A\ninst\u271d : Nontrivial A\nk : \ud835\udd5c\na : A\nha : Set.Nonempty (\u03c3 a)\nh : k \u2260 0\n\u22a2 \u03c3 (k \u2022 a) = k \u2022 \u03c3 a\n[PROOFSTEP]\nexact unit_smul_eq_smul a (Units.mk0 k h)\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na b : A\n\u22a2 \u03c3 (a * b) \\ {0} = \u03c3 (b * a) \\ {0}\n[PROOFSTEP]\nsuffices h : \u2200 x y : A, \u03c3 (x * y) \\ {0} \u2286 \u03c3 (y * x) \\ {0}\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na b : A\nh : \u2200 (x y : A), \u03c3 (x * y) \\ {0} \u2286 \u03c3 (y * x) \\ {0}\n\u22a2 \u03c3 (a * b) \\ {0} = \u03c3 (b * a) \\ {0}\n[PROOFSTEP]\nexact Set.eq_of_subset_of_subset (h a b) (h b a)\n[GOAL]\ncase h\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na b : A\n\u22a2 \u2200 (x y : A), \u03c3 (x * y) \\ {0} \u2286 \u03c3 (y * x) \\ {0}\n[PROOFSTEP]\nrintro _ _ k \u27e8k_mem, k_neq\u27e9\n[GOAL]\ncase h.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na b x\u271d y\u271d : A\nk : \ud835\udd5c\nk_mem : k \u2208 \u03c3 (x\u271d * y\u271d)\nk_neq : \u00ack \u2208 {0}\n\u22a2 k \u2208 \u03c3 (y\u271d * x\u271d) \\ {0}\n[PROOFSTEP]\nchange ((Units.mk0 k k_neq) : \ud835\udd5c) \u2208 _ at k_mem \n[GOAL]\ncase h.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na b x\u271d y\u271d : A\nk : \ud835\udd5c\nk_neq : \u00ack \u2208 {0}\nk_mem : \u2191(Units.mk0 k k_neq) \u2208 \u03c3 (x\u271d * y\u271d)\n\u22a2 k \u2208 \u03c3 (y\u271d * x\u271d) \\ {0}\n[PROOFSTEP]\nexact \u27e8unit_mem_mul_iff_mem_swap_mul.mp k_mem, k_neq\u27e9\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\n\u22a2 (\u03c3 \u2191a)\u207b\u00b9 = \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nrefine' Set.eq_of_subset_of_subset (fun k hk => _) fun k hk => _\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 (\u03c3 \u2191a)\u207b\u00b9\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nrw [Set.mem_inv] at hk \n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nhave : k \u2260 0 := by simpa only [inv_inv] using inv_ne_zero (ne_zero_of_mem_of_unit hk)\n[GOAL]\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\n\u22a2 k \u2260 0\n[PROOFSTEP]\nsimpa only [inv_inv] using inv_ne_zero (ne_zero_of_mem_of_unit hk)\n[GOAL]\ncase refine'_1\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : k \u2260 0\n\u22a2 k \u2208 \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nlift k to \ud835\udd5c\u02e3 using isUnit_iff_ne_zero.mpr this\n[GOAL]\ncase refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : (\u2191k)\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 Units.val_inv_eq_inv_val k] at hk \n[GOAL]\ncase refine'_1.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k\u207b\u00b9 \u2208 \u03c3 \u2191a\nthis : \u2191k \u2260 0\n\u22a2 \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9\n[PROOFSTEP]\nexact inv_mem_iff.mp hk\n[GOAL]\ncase refine'_2\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\nhk : k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 k \u2208 (\u03c3 \u2191a)\u207b\u00b9\n[PROOFSTEP]\nlift k to \ud835\udd5c\u02e3 using isUnit_iff_ne_zero.mpr (ne_zero_of_mem_of_unit hk)\n[GOAL]\ncase refine'_2.intro\n\ud835\udd5c : Type u\nA : Type v\ninst\u271d\u00b2 : Field \ud835\udd5c\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra \ud835\udd5c A\na : A\u02e3\nk : \ud835\udd5c\u02e3\nhk : \u2191k \u2208 \u03c3 \u2191a\u207b\u00b9\n\u22a2 \u2191k \u2208 (\u03c3 \u2191a)\u207b\u00b9\n[PROOFSTEP]\nsimpa only [Units.val_inv_eq_inv_val] using inv_mem_iff.mp hk\n[GOAL]\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Algebra R A\ninst\u271d\u00b2 : Ring B\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : AlgHomClass F R A B\n\u03c6 : F\na : A\nr : R\nh : r \u2208 resolventSet R a\n\u22a2 r \u2208 resolventSet R (\u2191\u03c6 a)\n[PROOFSTEP]\nsimpa only [map_sub, AlgHomClass.commutes] using h.map \u03c6\n[GOAL]\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : AlgHomClass F R A R\ninst\u271d : Nontrivial R\n\u03c6 : F\na : A\n\u22a2 \u2191\u03c6 a \u2208 \u03c3 a\n[PROOFSTEP]\nhave h : \u2191\u2090 (\u03c6 a) - a \u2208 RingHom.ker (\u03c6 : A \u2192+* R) := by\n  simp only [RingHom.mem_ker, map_sub, RingHom.coe_coe, AlgHomClass.commutes, Algebra.id.map_eq_id, RingHom.id_apply,\n    sub_self]\n[GOAL]\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : AlgHomClass F R A R\ninst\u271d : Nontrivial R\n\u03c6 : F\na : A\n\u22a2 \u2191\u2191\u2090 (\u2191\u03c6 a) - a \u2208 RingHom.ker \u2191\u03c6\n[PROOFSTEP]\nsimp only [RingHom.mem_ker, map_sub, RingHom.coe_coe, AlgHomClass.commutes, Algebra.id.map_eq_id, RingHom.id_apply,\n  sub_self]\n[GOAL]\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : AlgHomClass F R A R\ninst\u271d : Nontrivial R\n\u03c6 : F\na : A\nh : \u2191\u2191\u2090 (\u2191\u03c6 a) - a \u2208 RingHom.ker \u2191\u03c6\n\u22a2 \u2191\u03c6 a \u2208 \u03c3 a\n[PROOFSTEP]\nsimp only [spectrum.mem_iff, \u2190 mem_nonunits_iff, coe_subset_nonunits (RingHom.ker_ne_top (\u03c6 : A \u2192+* R)) h]\n[GOAL]\nF : Type u_1\nR : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u2075 : CommSemiring R\ninst\u271d\u2074 : Ring A\ninst\u271d\u00b3 : Ring B\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R B\ninst\u271d : AlgEquivClass F R A B\nf : F\na : A\n\u22a2 spectrum R a \u2286 spectrum R (\u2191f a)\n[PROOFSTEP]\nsimpa only [AlgEquiv.coe_algHom, AlgEquiv.coe_coe_symm_apply_coe_apply] using\n  AlgHom.spectrum_apply_subset (f : A \u2243\u2090[R] B).symm (f a)\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Algebra.Spectrum", "llama_tokens": 13705, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059511841119, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.5052329032593972}}
{"text": "[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 \u2200 (x y : V),\n    inner (\u2191(Real.Angle.cos \u03b8 \u2022 LinearMap.id + Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv) x)\n        (\u2191(Real.Angle.cos \u03b8 \u2022 LinearMap.id + Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv) y) =\n      inner x y\n[PROOFSTEP]\nintro x y\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx y : V\n\u22a2 inner (\u2191(Real.Angle.cos \u03b8 \u2022 LinearMap.id + Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv) x)\n      (\u2191(Real.Angle.cos \u03b8 \u2022 LinearMap.id + Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv) y) =\n    inner x y\n[PROOFSTEP]\nsimp only [IsROrC.conj_to_real, id.def, LinearMap.smul_apply, LinearMap.add_apply, LinearMap.id_coe,\n  LinearEquiv.coe_coe, LinearIsometryEquiv.coe_toLinearEquiv, Orientation.areaForm_rightAngleRotation_left,\n  Orientation.inner_rightAngleRotation_left, Orientation.inner_rightAngleRotation_right, inner_add_left,\n  inner_smul_left, inner_add_right, inner_smul_right]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx y : V\n\u22a2 Real.Angle.cos \u03b8 * (Real.Angle.cos \u03b8 * inner x y + Real.Angle.sin \u03b8 * \u2191(\u2191(areaForm o) x) y) +\n      Real.Angle.sin \u03b8 * (Real.Angle.cos \u03b8 * -\u2191(\u2191(areaForm o) x) y + Real.Angle.sin \u03b8 * - -inner x y) =\n    inner x y\n[PROOFSTEP]\nlinear_combination inner (\ud835\udd5c := \u211d) x y * \u03b8.cos_sq_add_sin_sq\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 LinearMap.comp (rotationAux o \u03b8).toLinearMap\n      (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv) =\n    LinearMap.id\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191(LinearMap.comp (rotationAux o \u03b8).toLinearMap\n          (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv))\n      x =\n    \u2191LinearMap.id x\n[PROOFSTEP]\nconvert congr_arg (fun t : \u211d => t \u2022 x) \u03b8.cos_sq_add_sin_sq using 1\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191(LinearMap.comp (rotationAux o \u03b8).toLinearMap\n          (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv))\n      x =\n    (Real.Angle.cos \u03b8 ^ 2 + Real.Angle.sin \u03b8 ^ 2) \u2022 x\n[PROOFSTEP]\nsimp only [o.rightAngleRotation_rightAngleRotation, o.rotationAux_apply, Function.comp_apply, id.def,\n  LinearEquiv.coe_coe, LinearIsometry.coe_toLinearMap, LinearIsometryEquiv.coe_toLinearEquiv, map_smul, map_sub,\n  LinearMap.coe_comp, LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply, \u2190 mul_smul, add_smul, smul_add,\n  smul_neg, smul_sub, mul_comm, sq]\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8 * Real.Angle.cos \u03b8) \u2022 x + (Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x -\n      ((Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x +\n        -((Real.Angle.sin \u03b8 * Real.Angle.sin \u03b8) \u2022 x)) =\n    (Real.Angle.cos \u03b8 * Real.Angle.cos \u03b8) \u2022 x + (Real.Angle.sin \u03b8 * Real.Angle.sin \u03b8) \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8 * Real.Angle.cos \u03b8) \u2022 x + (Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x -\n      ((Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x +\n        -((Real.Angle.sin \u03b8 * Real.Angle.sin \u03b8) \u2022 x)) =\n    (Real.Angle.cos \u03b8 * Real.Angle.cos \u03b8) \u2022 x + (Real.Angle.sin \u03b8 * Real.Angle.sin \u03b8) \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191LinearMap.id x = 1 \u2022 x\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 LinearMap.comp (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv)\n      (rotationAux o \u03b8).toLinearMap =\n    LinearMap.id\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191(LinearMap.comp (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv)\n          (rotationAux o \u03b8).toLinearMap)\n      x =\n    \u2191LinearMap.id x\n[PROOFSTEP]\nconvert congr_arg (fun t : \u211d => t \u2022 x) \u03b8.cos_sq_add_sin_sq using 1\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191(LinearMap.comp (Real.Angle.cos \u03b8 \u2022 LinearMap.id - Real.Angle.sin \u03b8 \u2022 \u2191(rightAngleRotation o).toLinearEquiv)\n          (rotationAux o \u03b8).toLinearMap)\n      x =\n    (Real.Angle.cos \u03b8 ^ 2 + Real.Angle.sin \u03b8 ^ 2) \u2022 x\n[PROOFSTEP]\nsimp only [o.rightAngleRotation_rightAngleRotation, o.rotationAux_apply, Function.comp_apply, id.def,\n  LinearEquiv.coe_coe, LinearIsometry.coe_toLinearMap, LinearIsometryEquiv.coe_toLinearEquiv, map_add, map_smul,\n  LinearMap.coe_comp, LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply, add_smul, smul_neg, smul_sub,\n  smul_smul]\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8 * Real.Angle.cos \u03b8) \u2022 x - (Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.sin \u03b8 * Real.Angle.cos \u03b8) \u2022 \u2191(rightAngleRotation o) x -\n        -((Real.Angle.sin \u03b8 * Real.Angle.sin \u03b8) \u2022 x)) =\n    Real.Angle.cos \u03b8 ^ 2 \u2022 x + Real.Angle.sin \u03b8 ^ 2 \u2022 x\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 Real.Angle.cos \u03b8 ^ 2 \u2022 x - (Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x - -(Real.Angle.sin \u03b8 ^ 2 \u2022 x)) =\n    Real.Angle.cos \u03b8 ^ 2 \u2022 x + Real.Angle.sin \u03b8 ^ 2 \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 Real.Angle.cos \u03b8 ^ 2 \u2022 x - (Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.cos \u03b8 * Real.Angle.sin \u03b8) \u2022 \u2191(rightAngleRotation o) x - -(Real.Angle.sin \u03b8 ^ 2 \u2022 x)) =\n    Real.Angle.cos \u03b8 ^ 2 \u2022 x + Real.Angle.sin \u03b8 ^ 2 \u2022 x\n[PROOFSTEP]\nabel\n[GOAL]\ncase h.e'_3\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 \u2191LinearMap.id x = 1 \u2022 x\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2191(rotation o \u03b8).toLinearEquiv =\n    \u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n      (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]])\n[PROOFSTEP]\napply (o.basisRightAngleRotation x hx).ext\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2200 (i : Fin 2),\n    \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i) =\n      \u2191(\u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n            (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]]))\n        (\u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nintro i\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\ni : Fin 2\n\u22a2 \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i) =\n    \u2191(\u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n          (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]]))\n      (\u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 0, isLt := (_ : 0 < 2) }) =\n    \u2191(\u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n          (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]]))\n      (\u2191(basisRightAngleRotation o x hx) { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nrw [Matrix.toLin_self]\n[GOAL]\ncase head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 0, isLt := (_ : 0 < 2) }) =\n    Finset.sum Finset.univ fun j =>\n      \u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]] j\n          { val := 0, isLt := (_ : 0 < 2) } \u2022\n        \u2191(basisRightAngleRotation o x hx) j\n[PROOFSTEP]\nsimp [rotation_apply, Fin.sum_univ_succ]\n[GOAL]\ncase tail.head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    \u2191(\u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n          (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]]))\n      (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nrw [Matrix.toLin_self]\n[GOAL]\ncase tail.head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o \u03b8).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    Finset.sum Finset.univ fun j =>\n      \u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]] j\n          { val := 1, isLt := (_ : (fun a => a < 2) 1) } \u2022\n        \u2191(basisRightAngleRotation o x hx) j\n[PROOFSTEP]\nsimp [rotation_apply, Fin.sum_univ_succ, add_comm]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 \u2191LinearMap.det \u2191(rotation o \u03b8).toLinearEquiv = 1\n[PROOFSTEP]\nhaveI : Nontrivial V := FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank \u211d V = 2) _)\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nthis : Nontrivial V\n\u22a2 \u2191LinearMap.det \u2191(rotation o \u03b8).toLinearEquiv = 1\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x, x \u2260 (0 : V) := exists_ne (0 : V)\n[GOAL]\ncase intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2191LinearMap.det \u2191(rotation o \u03b8).toLinearEquiv = 1\n[PROOFSTEP]\nrw [o.rotation_eq_matrix_toLin \u03b8 hx]\n[GOAL]\ncase intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2191LinearMap.det\n      (\u2191(Matrix.toLin (basisRightAngleRotation o x hx) (basisRightAngleRotation o x hx))\n        (\u2191Matrix.of ![![Real.Angle.cos \u03b8, -Real.Angle.sin \u03b8], ![Real.Angle.sin \u03b8, Real.Angle.cos \u03b8]])) =\n    1\n[PROOFSTEP]\nsimpa [sq] using \u03b8.cos_sq_add_sin_sq\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191LinearEquiv.det (rotation o \u03b8).toLinearEquiv) = \u21911\n[PROOFSTEP]\nsimpa only [LinearEquiv.coe_det, Units.val_one] using o.det_rotation \u03b8\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 LinearIsometryEquiv.symm (rotation o \u03b8) = rotation o (-\u03b8)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx\u271d : V\n\u22a2 \u2191(LinearIsometryEquiv.symm (rotation o \u03b8)) x\u271d = \u2191(rotation o (-\u03b8)) x\u271d\n[PROOFSTEP]\nsimp [o.rotation_apply, o.rotation_symm_apply, sub_eq_add_neg]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u22a2 rotation o 0 = LinearIsometryEquiv.refl \u211d V\n[PROOFSTEP]\next\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx\u271d : V\n\u22a2 \u2191(rotation o 0) x\u271d = \u2191(LinearIsometryEquiv.refl \u211d V) x\u271d\n[PROOFSTEP]\nsimp [rotation]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u22a2 rotation o \u2191\u03c0 = LinearIsometryEquiv.neg \u211d\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 \u2191(rotation o \u2191\u03c0) x = \u2191(LinearIsometryEquiv.neg \u211d) x\n[PROOFSTEP]\nsimp [rotation]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 \u2191(rotation o \u2191\u03c0) x = -x\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u22a2 rotation o \u2191(\u03c0 / 2) = rightAngleRotation o\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 \u2191(rotation o \u2191(\u03c0 / 2)) x = \u2191(rightAngleRotation o) x\n[PROOFSTEP]\nsimp [rotation]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8\u2081 \u03b8\u2082 : Real.Angle\nx : V\n\u22a2 \u2191(rotation o \u03b8\u2081) (\u2191(rotation o \u03b8\u2082) x) = \u2191(rotation o (\u03b8\u2081 + \u03b8\u2082)) x\n[PROOFSTEP]\nsimp only [o.rotation_apply, \u2190 mul_smul, Real.Angle.cos_add, Real.Angle.sin_add, add_smul, sub_smul,\n  LinearIsometryEquiv.trans_apply, smul_add, LinearIsometryEquiv.map_add, LinearIsometryEquiv.map_smul,\n  rightAngleRotation_rightAngleRotation, smul_neg]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8\u2081 \u03b8\u2082 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 x + (Real.Angle.cos \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.sin \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n        -((Real.Angle.sin \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 x)) =\n    (Real.Angle.cos \u03b8\u2081 * Real.Angle.cos \u03b8\u2082) \u2022 x - (Real.Angle.sin \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 x +\n      ((Real.Angle.sin \u03b8\u2081 * Real.Angle.cos \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x +\n        (Real.Angle.cos \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nring_nf\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8\u2081 \u03b8\u2082 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 x + (Real.Angle.cos \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.cos \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x +\n        -((Real.Angle.sin \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 x)) =\n    (Real.Angle.cos \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 x - (Real.Angle.sin \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 x +\n      ((Real.Angle.cos \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n        (Real.Angle.cos \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8\u2081 \u03b8\u2082 : Real.Angle\nx : V\n\u22a2 (Real.Angle.cos \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 x + (Real.Angle.cos \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n      ((Real.Angle.cos \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x +\n        -((Real.Angle.sin \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 x)) =\n    (Real.Angle.cos \u03b8\u2082 * Real.Angle.cos \u03b8\u2081) \u2022 x - (Real.Angle.sin \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 x +\n      ((Real.Angle.cos \u03b8\u2082 * Real.Angle.sin \u03b8\u2081) \u2022 \u2191(rightAngleRotation o) x +\n        (Real.Angle.cos \u03b8\u2081 * Real.Angle.sin \u03b8\u2082) \u2022 \u2191(rightAngleRotation o) x)\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8\u2081 \u03b8\u2082 : Real.Angle\nx\u271d : V\n\u22a2 \u2191(LinearIsometryEquiv.trans (rotation o \u03b8\u2081) (rotation o \u03b8\u2082)) x\u271d = \u2191(rotation o (\u03b8\u2082 + \u03b8\u2081)) x\u271d\n[PROOFSTEP]\nrw [\u2190 rotation_rotation, LinearIsometryEquiv.trans_apply]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191(kahler o) (\u2191(rotation o \u03b8) x)) y = \u2191(starRingEnd \u2102) \u2191(Real.Angle.expMapCircle \u03b8) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [o.rotation_apply, map_add, map_mul, LinearMap.map_smul\u209b\u2097, RingHom.id_apply, LinearMap.add_apply,\n  LinearMap.smul_apply, real_smul, kahler_rightAngleRotation_left, Real.Angle.coe_expMapCircle, Complex.conj_ofReal,\n  conj_I]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(Real.Angle.cos \u03b8) * \u2191(\u2191(kahler o) x) y + \u2191(Real.Angle.sin \u03b8) * (-I * \u2191(\u2191(kahler o) x) y) =\n    (\u2191(Real.Angle.cos \u03b8) + \u2191(Real.Angle.sin \u03b8) * -I) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nx : V\n\u22a2 -\u2191(rotation o \u03b8) x = \u2191(rotation o (\u2191\u03c0 + \u03b8)) x\n[PROOFSTEP]\nrw [\u2190 o.rotation_pi_apply, rotation_rotation]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 -\u2191(rotation o \u2191(-\u03c0 / 2)) x = \u2191(rotation o \u2191(\u03c0 / 2)) x\n[PROOFSTEP]\nrw [neg_rotation, \u2190 Real.Angle.coe_add, neg_div, \u2190 sub_eq_add_neg, sub_half]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191(kahler o) (\u2191(rotation o \u03b8) x)) y = \u2191(Real.Angle.expMapCircle (-\u03b8)) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [Real.Angle.expMapCircle_neg, coe_inv_circle_eq_conj, kahler_rotation_left]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191(kahler o) x) (\u2191(rotation o \u03b8) y) = \u2191(Real.Angle.expMapCircle \u03b8) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nsimp only [o.rotation_apply, map_add, LinearMap.map_smul\u209b\u2097, RingHom.id_apply, real_smul,\n  kahler_rightAngleRotation_right, Real.Angle.coe_expMapCircle]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(Real.Angle.cos \u03b8) * \u2191(\u2191(kahler o) x) y + \u2191(Real.Angle.sin \u03b8) * (I * \u2191(\u2191(kahler o) x) y) =\n    (\u2191(Real.Angle.cos \u03b8) + \u2191(Real.Angle.sin \u03b8) * I) * \u2191(\u2191(kahler o) x) y\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) y = oangle o x y - \u03b8\n[PROOFSTEP]\nsimp only [oangle, o.kahler_rotation_left']\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(arg (\u2191(Real.Angle.expMapCircle (-\u03b8)) * \u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) - \u03b8\n[PROOFSTEP]\nrw [Complex.arg_mul_coe_angle, Real.Angle.arg_expMapCircle]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 -\u03b8 + \u2191(arg (\u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) - \u03b8\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 -\u03b8 + \u2191(arg (\u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) - \u03b8\n[PROOFSTEP]\nabel\n[GOAL]\ncase hx\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(Real.Angle.expMapCircle (-\u03b8)) \u2260 0\n[PROOFSTEP]\nexact ne_zero_of_mem_circle _\n[GOAL]\ncase hy\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191(kahler o) x) y \u2260 0\n[PROOFSTEP]\nexact o.kahler_ne_zero hx hy\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o x (\u2191(rotation o \u03b8) y) = oangle o x y + \u03b8\n[PROOFSTEP]\nsimp only [oangle, o.kahler_rotation_right]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(arg (\u2191(Real.Angle.expMapCircle \u03b8) * \u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) + \u03b8\n[PROOFSTEP]\nrw [Complex.arg_mul_coe_angle, Real.Angle.arg_expMapCircle]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u03b8 + \u2191(arg (\u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) + \u03b8\n[PROOFSTEP]\nabel\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u03b8 + \u2191(arg (\u2191(\u2191(kahler o) x) y)) = \u2191(arg (\u2191(\u2191(kahler o) x) y)) + \u03b8\n[PROOFSTEP]\nabel\n[GOAL]\ncase hx\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(Real.Angle.expMapCircle \u03b8) \u2260 0\n[PROOFSTEP]\nexact ne_zero_of_mem_circle _\n[GOAL]\ncase hy\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(\u2191(kahler o) x) y \u2260 0\n[PROOFSTEP]\nexact o.kahler_ne_zero hx hy\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) x = -\u03b8\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o x (\u2191(rotation o \u03b8) x) = \u03b8\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x = 0\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nsimp [hx]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : \u00acx = 0\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nsimp [hy]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 oangle o y (\u2191(rotation o (oangle o x y)) x) = 0\n[PROOFSTEP]\nrw [oangle_rev]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 -oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nsimp\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : x = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : x = 0\nhy : y = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : x = 0\nhy : \u00acy = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 oangle o (\u2191(rotation o \u03b8) x) (\u2191(rotation o \u03b8) y) = oangle o x y\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(rotation o \u03b8) x = x \u2194 \u03b8 = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u2191(rotation o \u03b8) x = x \u2192 \u03b8 = 0\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\nh : \u2191(rotation o \u03b8) x = x\n\u22a2 \u03b8 = 0\n[PROOFSTEP]\nrw [eq_comm]\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\nh : \u2191(rotation o \u03b8) x = x\n\u22a2 0 = \u03b8\n[PROOFSTEP]\nsimpa [hx, h] using o.oangle_rotation_right hx hx \u03b8\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 \u03b8 = 0 \u2192 \u2191(rotation o \u03b8) x = x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\nh : \u03b8 = 0\n\u22a2 \u2191(rotation o \u03b8) x = x\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\n\u22a2 x = \u2191(rotation o \u03b8) x \u2194 \u03b8 = 0\n[PROOFSTEP]\nrw [\u2190 o.rotation_eq_self_iff_angle_eq_zero hx, eq_comm]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u03b8 : Real.Angle\n\u22a2 \u2191(rotation o \u03b8) x = x \u2194 x = 0 \u2228 \u03b8 = 0\n[PROOFSTEP]\nby_cases h : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u03b8 : Real.Angle\nh : x = 0\n\u22a2 \u2191(rotation o \u03b8) x = x \u2194 x = 0 \u2228 \u03b8 = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u03b8 : Real.Angle\nh : \u00acx = 0\n\u22a2 \u2191(rotation o \u03b8) x = x \u2194 x = 0 \u2228 \u03b8 = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u03b8 : Real.Angle\n\u22a2 x = \u2191(rotation o \u03b8) x \u2194 x = 0 \u2228 \u03b8 = 0\n[PROOFSTEP]\nrw [\u2190 rotation_eq_self_iff, eq_comm]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 \u2191(rotation o (oangle o x y)) x = y \u2194 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 \u2191(rotation o (oangle o x y)) x = y \u2192 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : \u2191(rotation o (oangle o x y)) x = y\n\u22a2 \u2016x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nrw [\u2190 h, LinearIsometryEquiv.norm_map]\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 \u2016x\u2016 = \u2016y\u2016 \u2192 \u2191(rotation o (oangle o x y)) x = y\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 \u2191(rotation o (oangle o x y)) x = y\n[PROOFSTEP]\nrw [o.eq_iff_oangle_eq_zero_of_norm_eq]\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 oangle o (\u2191(rotation o (oangle o x y)) x) y = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : \u2016x\u2016 = \u2016y\u2016\n\u22a2 \u2016\u2191(rotation o (oangle o x y)) x\u2016 = \u2016y\u2016\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o x y = \u03b8 \u2194 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nhave hp := div_pos (norm_pos_iff.2 hy) (norm_pos_iff.2 hx)\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nhp : 0 < \u2016y\u2016 / \u2016x\u2016\n\u22a2 oangle o x y = \u03b8 \u2194 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nhp : 0 < \u2016y\u2016 / \u2016x\u2016\n\u22a2 oangle o x y = \u03b8 \u2192 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\nhp : 0 < \u2016y\u2016 / \u2016x\u2016\n\u22a2 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o (oangle o x y)) x\n[PROOFSTEP]\nrw [\u2190 LinearIsometryEquiv.map_smul, \u2190 o.oangle_smul_left_of_pos x y hp, eq_comm, rotation_oangle_eq_iff_norm_eq,\n  norm_smul, Real.norm_of_nonneg hp.le, div_mul_cancel _ (norm_ne_zero_iff.2 hx)]\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nhp : 0 < \u2016y\u2016 / \u2016x\u2016\n\u22a2 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2192 oangle o x y = \u03b8\n[PROOFSTEP]\nintro hye\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nhp : 0 < \u2016y\u2016 / \u2016x\u2016\nhye : y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x\n\u22a2 oangle o x y = \u03b8\n[PROOFSTEP]\nrw [hye, o.oangle_smul_right_of_pos _ _ hp, o.oangle_rotation_self_right hx]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o x y = \u03b8 \u2194 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 oangle o x y = \u03b8 \u2192 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nh : oangle o x y = \u03b8\n\u22a2 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nrw [o.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero hx hy] at h \n[GOAL]\ncase mp\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\nh : y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x\n\u22a2 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x\n[PROOFSTEP]\nexact \u27e8\u2016y\u2016 / \u2016x\u2016, div_pos (norm_pos_iff.2 hy) (norm_pos_iff.2 hx), h\u27e9\n[GOAL]\ncase mpr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nhx : x \u2260 0\nhy : y \u2260 0\n\u03b8 : Real.Angle\n\u22a2 (\u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2192 oangle o x y = \u03b8\n[PROOFSTEP]\nrintro \u27e8r, hr, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : x \u2260 0\n\u03b8 : Real.Angle\nr : \u211d\nhr : 0 < r\nhy : r \u2022 \u2191(rotation o \u03b8) x \u2260 0\n\u22a2 oangle o x (r \u2022 \u2191(rotation o \u03b8) x) = \u03b8\n[PROOFSTEP]\nrw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right hx]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 oangle o x y = \u03b8 \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : x = 0\n\u22a2 oangle o x y = \u03b8 \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hx, eq_comm]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\n\u22a2 oangle o x y = \u03b8 \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 oangle o x y = \u03b8 \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hy, eq_comm]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 oangle o x y = \u03b8 \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nrw [o.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero hx hy]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2194 x \u2260 0 \u2227 y \u2260 0 \u2227 y = (\u2016y\u2016 / \u2016x\u2016) \u2022 \u2191(rotation o \u03b8) x \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\n\u22a2 oangle o x y = \u03b8 \u2194 (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : x = 0\n\u22a2 oangle o x y = \u03b8 \u2194 (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hx, eq_comm]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\n\u22a2 oangle o x y = \u03b8 \u2194 (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : y = 0\n\u22a2 oangle o x y = \u03b8 \u2194 (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hy, eq_comm]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 oangle o x y = \u03b8 \u2194 (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nrw [o.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero hx hy]\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u03b8 : Real.Angle\nhx : \u00acx = 0\nhy : \u00acy = 0\n\u22a2 (\u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2194\n    (x \u2260 0 \u2227 y \u2260 0 \u2227 \u2203 r, 0 < r \u2227 y = r \u2022 \u2191(rotation o \u03b8) x) \u2228 \u03b8 = 0 \u2227 (x = 0 \u2228 y = 0)\n[PROOFSTEP]\nsimp [hx, hy]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\n\u22a2 \u2203 \u03b8, f = rotation o \u03b8\n[PROOFSTEP]\nhaveI : Nontrivial V := FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank \u211d V = 2) _)\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\n\u22a2 \u2203 \u03b8, f = rotation o \u03b8\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 : \u2203 x, x \u2260 (0 : V) := exists_ne (0 : V)\n[GOAL]\ncase intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2203 \u03b8, f = rotation o \u03b8\n[PROOFSTEP]\nuse o.oangle x (f x)\n[GOAL]\ncase h\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 f = rotation o (oangle o x (\u2191f x))\n[PROOFSTEP]\napply LinearIsometryEquiv.toLinearEquiv_injective\n[GOAL]\ncase h.a\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 f.toLinearEquiv = (rotation o (oangle o x (\u2191f x))).toLinearEquiv\n[PROOFSTEP]\napply LinearEquiv.toLinearMap_injective\n[GOAL]\ncase h.a.a\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2191f.toLinearEquiv = \u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv\n[PROOFSTEP]\napply (o.basisRightAngleRotation x hx).ext\n[GOAL]\ncase h.a.a\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2200 (i : Fin 2),\n    \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i) =\n      \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.a.a\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\ni : Fin 2\n\u22a2 \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i) =\n    \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nsymm\n[GOAL]\ncase h.a.a\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\ni : Fin 2\n\u22a2 \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i) =\n    \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) i)\n[PROOFSTEP]\nfin_cases i\n[GOAL]\ncase h.a.a.head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv\n      (\u2191(basisRightAngleRotation o x hx) { val := 0, isLt := (_ : 0 < 2) }) =\n    \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 0, isLt := (_ : 0 < 2) })\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.a.a.tail.head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv\n      (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nhave : o.oangle (J x) (f (J x)) = o.oangle x (f x) := by\n  simp only [oangle, o.linearIsometryEquiv_comp_rightAngleRotation f hd, o.kahler_comp_rightAngleRotation]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis : Nontrivial V\nx : V\nhx : x \u2260 0\n\u22a2 oangle o (\u2191(rightAngleRotation o) x) (\u2191f (\u2191(rightAngleRotation o) x)) = oangle o x (\u2191f x)\n[PROOFSTEP]\nsimp only [oangle, o.linearIsometryEquiv_comp_rightAngleRotation f hd, o.kahler_comp_rightAngleRotation]\n[GOAL]\ncase h.a.a.tail.head\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nf : V \u2243\u2097\u1d62[\u211d] V\nhd : 0 < \u2191LinearMap.det \u2191f.toLinearEquiv\nthis\u271d : Nontrivial V\nx : V\nhx : x \u2260 0\nthis : oangle o (\u2191(rightAngleRotation o) x) (\u2191f (\u2191(rightAngleRotation o) x)) = oangle o x (\u2191f x)\n\u22a2 \u2191\u2191(rotation o (oangle o x (\u2191f x))).toLinearEquiv\n      (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) }) =\n    \u2191\u2191f.toLinearEquiv (\u2191(basisRightAngleRotation o x hx) { val := 1, isLt := (_ : (fun a => a < 2) 1) })\n[PROOFSTEP]\nsimp [\u2190 this]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nf : V \u2243\u2097\u1d62[\u211d] V'\nx : V'\n\u22a2 \u2191(rotation (\u2191(map (Fin 2) f.toLinearEquiv) o) \u03b8) x = \u2191f (\u2191(rotation o \u03b8) (\u2191(LinearIsometryEquiv.symm f) x))\n[PROOFSTEP]\nsimp [rotation_apply, o.rightAngleRotation_map]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nz : \u2102\n\u22a2 \u2191(rotation Complex.orientation \u03b8) z = \u2191(Real.Angle.expMapCircle \u03b8) * z\n[PROOFSTEP]\nsimp only [rotation_apply, Complex.rightAngleRotation, Real.Angle.coe_expMapCircle, real_smul]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nz : \u2102\n\u22a2 \u2191(Real.Angle.cos \u03b8) * z + \u2191(Real.Angle.sin \u03b8) * (I * z) = (\u2191(Real.Angle.cos \u03b8) + \u2191(Real.Angle.sin \u03b8) * I) * z\n[PROOFSTEP]\nring\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\nf : V \u2243\u2097\u1d62[\u211d] \u2102\nhf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx : V\n\u22a2 \u2191f (\u2191(rotation o \u03b8) x) = \u2191(Real.Angle.expMapCircle \u03b8) * \u2191f x\n[PROOFSTEP]\nrw [\u2190 Complex.rotation, \u2190 hf, o.rotation_map, LinearIsometryEquiv.symm_apply_apply]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\n\u03b8 : Real.Angle\n\u22a2 \u2200 (x : V), \u2191(rotation (-o) \u03b8) x = \u2191(rotation o (-\u03b8)) x\n[PROOFSTEP]\nsimp [rotation_apply]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 inner (\u2191(rotation o \u2191(\u03c0 / 2)) x) x = 0\n[PROOFSTEP]\nrw [rotation_pi_div_two, inner_rightAngleRotation_self]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 inner x (\u2191(rotation o \u2191(\u03c0 / 2)) x) = 0\n[PROOFSTEP]\nrw [real_inner_comm, inner_rotation_pi_div_two_left]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\n\u22a2 inner (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) x = 0\n[PROOFSTEP]\nrw [inner_smul_left, inner_rotation_pi_div_two_left, mul_zero]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\n\u22a2 inner x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = 0\n[PROOFSTEP]\nrw [real_inner_comm, inner_smul_rotation_pi_div_two_left]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\n\u22a2 inner (\u2191(rotation o \u2191(\u03c0 / 2)) x) (r \u2022 x) = 0\n[PROOFSTEP]\nrw [inner_smul_right, inner_rotation_pi_div_two_left, mul_zero]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\n\u22a2 inner (r \u2022 x) (\u2191(rotation o \u2191(\u03c0 / 2)) x) = 0\n[PROOFSTEP]\nrw [real_inner_comm, inner_rotation_pi_div_two_left_smul]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr\u2081 r\u2082 : \u211d\n\u22a2 inner (r\u2081 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) (r\u2082 \u2022 x) = 0\n[PROOFSTEP]\nrw [inner_smul_right, inner_smul_rotation_pi_div_two_left, mul_zero]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr\u2081 r\u2082 : \u211d\n\u22a2 inner (r\u2082 \u2022 x) (r\u2081 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = 0\n[PROOFSTEP]\nrw [real_inner_comm, inner_smul_rotation_pi_div_two_smul_left]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 inner x y = 0 \u2194 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n[PROOFSTEP]\nrw [\u2190 o.eq_zero_or_oangle_eq_iff_inner_eq_zero]\n[GOAL]\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\n\u22a2 x = 0 \u2228 y = 0 \u2228 oangle o x y = \u2191(\u03c0 / 2) \u2228 oangle o x y = \u2191(-\u03c0 / 2) \u2194 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n[PROOFSTEP]\nrefine' \u27e8fun h => _, fun h => _\u27e9\n[GOAL]\ncase refine'_1\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : x = 0 \u2228 y = 0 \u2228 oangle o x y = \u2191(\u03c0 / 2) \u2228 oangle o x y = \u2191(-\u03c0 / 2)\n\u22a2 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n[PROOFSTEP]\nrcases h with (rfl | rfl | h | h)\n[GOAL]\ncase refine'_1.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\ny : V\n\u22a2 0 = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) 0 = y\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\ncase refine'_1.inr.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\n\u22a2 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0\n[PROOFSTEP]\nexact Or.inr \u27e80, zero_smul _ _\u27e9\n[GOAL]\ncase refine'_1.inr.inr.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(\u03c0 / 2)\n\u22a2 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n[PROOFSTEP]\nobtain \u27e8r, _, rfl\u27e9 :=\n  (o.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero (o.left_ne_zero_of_oangle_eq_pi_div_two h)\n        (o.right_ne_zero_of_oangle_eq_pi_div_two h) _).1\n    h\n[GOAL]\ncase refine'_1.inr.inr.inl.intro.intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nleft\u271d : 0 < r\nh : oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2)\n\u22a2 x = 0 \u2228 \u2203 r_1, r_1 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x\n[PROOFSTEP]\nexact Or.inr \u27e8r, rfl\u27e9\n[GOAL]\ncase refine'_1.inr.inr.inr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : oangle o x y = \u2191(-\u03c0 / 2)\n\u22a2 x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n[PROOFSTEP]\nobtain \u27e8r, _, rfl\u27e9 :=\n  (o.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero (o.left_ne_zero_of_oangle_eq_neg_pi_div_two h)\n        (o.right_ne_zero_of_oangle_eq_neg_pi_div_two h) _).1\n    h\n[GOAL]\ncase refine'_1.inr.inr.inr.intro.intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nleft\u271d : 0 < r\nh : oangle o x (r \u2022 \u2191(rotation o \u2191(-\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n\u22a2 x = 0 \u2228 \u2203 r_1, r_1 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = r \u2022 \u2191(rotation o \u2191(-\u03c0 / 2)) x\n[PROOFSTEP]\nrefine' Or.inr \u27e8-r, _\u27e9\n[GOAL]\ncase refine'_1.inr.inr.inr.intro.intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nleft\u271d : 0 < r\nh : oangle o x (r \u2022 \u2191(rotation o \u2191(-\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n\u22a2 -r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = r \u2022 \u2191(rotation o \u2191(-\u03c0 / 2)) x\n[PROOFSTEP]\nrw [neg_smul, \u2190 smul_neg, o.neg_rotation_pi_div_two]\n[GOAL]\ncase refine'_2\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx y : V\nh : x = 0 \u2228 \u2203 r, r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = y\n\u22a2 x = 0 \u2228 y = 0 \u2228 oangle o x y = \u2191(\u03c0 / 2) \u2228 oangle o x y = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nrcases h with (rfl | \u27e8r, rfl\u27e9)\n[GOAL]\ncase refine'_2.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\ny : V\n\u22a2 0 = 0 \u2228 y = 0 \u2228 oangle o 0 y = \u2191(\u03c0 / 2) \u2228 oangle o 0 y = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nexact Or.inl rfl\n[GOAL]\ncase refine'_2.inr.intro\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\n\u22a2 x = 0 \u2228\n    r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : x = 0\n\u22a2 x = 0 \u2228\n    r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nexact Or.inl hx\n[GOAL]\ncase neg\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : \u00acx = 0\n\u22a2 x = 0 \u2228\n    r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nrcases lt_trichotomy r 0 with (hr | rfl | hr)\n[GOAL]\ncase neg.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : \u00acx = 0\nhr : r < 0\n\u22a2 x = 0 \u2228\n    r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nrefine' Or.inr (Or.inr (Or.inr _))\n[GOAL]\ncase neg.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : \u00acx = 0\nhr : r < 0\n\u22a2 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nrw [o.oangle_smul_right_of_neg _ _ hr, o.neg_rotation_pi_div_two, o.oangle_rotation_self_right hx]\n[GOAL]\ncase neg.inr.inl\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nhx : \u00acx = 0\n\u22a2 x = 0 \u2228\n    0 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (0 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (0 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nexact Or.inr (Or.inl (zero_smul _ _))\n[GOAL]\ncase neg.inr.inr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : \u00acx = 0\nhr : 0 < r\n\u22a2 x = 0 \u2228\n    r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x = 0 \u2228\n      oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2) \u2228 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(-\u03c0 / 2)\n[PROOFSTEP]\nrefine' Or.inr (Or.inr (Or.inl _))\n[GOAL]\ncase neg.inr.inr\nV : Type u_1\nV' : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup V\ninst\u271d\u2074 : NormedAddCommGroup V'\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : InnerProductSpace \u211d V'\ninst\u271d\u00b9 : Fact (finrank \u211d V = 2)\ninst\u271d : Fact (finrank \u211d V' = 2)\no : Orientation \u211d V (Fin 2)\nx : V\nr : \u211d\nhx : \u00acx = 0\nhr : 0 < r\n\u22a2 oangle o x (r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) = \u2191(\u03c0 / 2)\n[PROOFSTEP]\nrw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right hx]\n", "meta": {"mathlib_filename": "Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation", "llama_tokens": 37645, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936484231889, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.5045411589634982}}
{"text": "[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : Preorder \u03b4\nf g : \u03b1 \u2192Co \u03b2\nh : (fun f => f.toFun) f = (fun f => f.toFun) g\n\u22a2 f = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f\n[GOAL]\ncase mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : Preorder \u03b4\ng : \u03b1 \u2192Co \u03b2\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmonotone'\u271d : Monotone toFun\u271d\ncontinuous_toFun\u271d : Continuous { toFun := toFun\u271d, monotone' := monotone'\u271d }.toFun\nh :\n  (fun f => f.toFun)\n      { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, continuous_toFun := continuous_toFun\u271d } =\n    (fun f => f.toFun) g\n\u22a2 { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, continuous_toFun := continuous_toFun\u271d } = g\n[PROOFSTEP]\nobtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g\n[GOAL]\ncase mk.mk.mk.mk\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : Preorder \u03b4\ntoFun\u271d\u00b9 : \u03b1 \u2192 \u03b2\nmonotone'\u271d\u00b9 : Monotone toFun\u271d\u00b9\ncontinuous_toFun\u271d\u00b9 : Continuous { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }.toFun\ntoFun\u271d : \u03b1 \u2192 \u03b2\nmonotone'\u271d : Monotone toFun\u271d\ncontinuous_toFun\u271d : Continuous { toFun := toFun\u271d, monotone' := monotone'\u271d }.toFun\nh :\n  (fun f => f.toFun)\n      { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, continuous_toFun := continuous_toFun\u271d\u00b9 } =\n    (fun f => f.toFun)\n      { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, continuous_toFun := continuous_toFun\u271d }\n\u22a2 { toOrderHom := { toFun := toFun\u271d\u00b9, monotone' := monotone'\u271d\u00b9 }, continuous_toFun := continuous_toFun\u271d\u00b9 } =\n    { toOrderHom := { toFun := toFun\u271d, monotone' := monotone'\u271d }, continuous_toFun := continuous_toFun\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : Preorder \u03b4\ng : \u03b2 \u2192Co \u03b3\nf\u2081 f\u2082 : \u03b1 \u2192Co \u03b2\nhg : Injective \u2191g\nh : comp g f\u2081 = comp g f\u2082\na : \u03b1\n\u22a2 \u2191g (\u2191f\u2081 a) = \u2191g (\u2191f\u2082 a)\n[PROOFSTEP]\nrw [\u2190 comp_apply, h, comp_apply]\n", "meta": {"mathlib_filename": "Mathlib.Topology.Order.Hom.Basic", "llama_tokens": 1224, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833841649232, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5045286606075899}}
{"text": "[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : e.toLocalEquiv = e'.toLocalEquiv\nh\u2082 : e.baseSet = e'.baseSet\n\u22a2 e = e'\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F proj\ntoLocalEquiv\u271d : LocalEquiv Z (B \u00d7 F)\nopen_target\u271d : IsOpen toLocalEquiv\u271d.target\nbaseSet\u271d : Set B\nopen_baseSet\u271d : IsOpen baseSet\u271d\nsource_eq\u271d : toLocalEquiv\u271d.source = proj \u207b\u00b9' baseSet\u271d\ntarget_eq\u271d : toLocalEquiv\u271d.target = baseSet\u271d \u00d7\u02e2 univ\nproj_toFun\u271d : \u2200 (p : Z), p \u2208 toLocalEquiv\u271d.source \u2192 (\u2191toLocalEquiv\u271d p).fst = proj p\nh\u2081 :\n  { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.toLocalEquiv =\n    e'.toLocalEquiv\nh\u2082 :\n  { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.baseSet =\n    e'.baseSet\n\u22a2 { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n      source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d } =\n    e'\n[PROOFSTEP]\ncases e'\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\ntoLocalEquiv\u271d\u00b9 : LocalEquiv Z (B \u00d7 F)\nopen_target\u271d\u00b9 : IsOpen toLocalEquiv\u271d\u00b9.target\nbaseSet\u271d\u00b9 : Set B\nopen_baseSet\u271d\u00b9 : IsOpen baseSet\u271d\u00b9\nsource_eq\u271d\u00b9 : toLocalEquiv\u271d\u00b9.source = proj \u207b\u00b9' baseSet\u271d\u00b9\ntarget_eq\u271d\u00b9 : toLocalEquiv\u271d\u00b9.target = baseSet\u271d\u00b9 \u00d7\u02e2 univ\nproj_toFun\u271d\u00b9 : \u2200 (p : Z), p \u2208 toLocalEquiv\u271d\u00b9.source \u2192 (\u2191toLocalEquiv\u271d\u00b9 p).fst = proj p\ntoLocalEquiv\u271d : LocalEquiv Z (B \u00d7 F)\nopen_target\u271d : IsOpen toLocalEquiv\u271d.target\nbaseSet\u271d : Set B\nopen_baseSet\u271d : IsOpen baseSet\u271d\nsource_eq\u271d : toLocalEquiv\u271d.source = proj \u207b\u00b9' baseSet\u271d\ntarget_eq\u271d : toLocalEquiv\u271d.target = baseSet\u271d \u00d7\u02e2 univ\nproj_toFun\u271d : \u2200 (p : Z), p \u2208 toLocalEquiv\u271d.source \u2192 (\u2191toLocalEquiv\u271d p).fst = proj p\nh\u2081 :\n  { toLocalEquiv := toLocalEquiv\u271d\u00b9, open_target := open_target\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n        source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 }.toLocalEquiv =\n    { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.toLocalEquiv\nh\u2082 :\n  { toLocalEquiv := toLocalEquiv\u271d\u00b9, open_target := open_target\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n        source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 }.baseSet =\n    { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.baseSet\n\u22a2 { toLocalEquiv := toLocalEquiv\u271d\u00b9, open_target := open_target\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n      source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 } =\n    { toLocalEquiv := toLocalEquiv\u271d, open_target := open_target\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n      source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\n\u22a2 e = e'\n[PROOFSTEP]\next1 <;> [ext1; exact h\u2083]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\n\u22a2 e = e'\n[PROOFSTEP]\next1\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\n\u22a2 e.toLocalEquiv = e'.toLocalEquiv\n[PROOFSTEP]\next1\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\n\u22a2 e.baseSet = e'.baseSet\n[PROOFSTEP]\nexact h\u2083\n[GOAL]\ncase h\u2081.h\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\nx\u271d : Z\n\u22a2 \u2191e.toLocalEquiv x\u271d = \u2191e'.toLocalEquiv x\u271d\n[PROOFSTEP]\napply h\u2081\n[GOAL]\ncase h\u2081.hsymm\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\nx\u271d : B \u00d7 F\n\u22a2 \u2191(LocalEquiv.symm e.toLocalEquiv) x\u271d = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\u271d\n[PROOFSTEP]\napply h\u2082\n[GOAL]\ncase h\u2081.hs\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\nh\u2081 : \u2200 (x : Z), \u2191e x = \u2191e' x\nh\u2082 : \u2200 (x : B \u00d7 F), \u2191(LocalEquiv.symm e.toLocalEquiv) x = \u2191(LocalEquiv.symm e'.toLocalEquiv) x\nh\u2083 : e.baseSet = e'.baseSet\n\u22a2 e.source = e'.source\n[PROOFSTEP]\nrw [e.source_eq, e'.source_eq, h\u2083]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ninst\u271d : Nonempty F\ne e' : Pretrivialization F proj\nh : e.toLocalEquiv = e'.toLocalEquiv\n\u22a2 e = e'\n[PROOFSTEP]\nrefine ext' _ _ h ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ninst\u271d : Nonempty F\ne e' : Pretrivialization F proj\nh : e.toLocalEquiv = e'.toLocalEquiv\n\u22a2 e.baseSet = e'.baseSet\n[PROOFSTEP]\nsimpa only [fst_image_prod, univ_nonempty, target_eq] using congr_arg (Prod.fst '' LocalEquiv.target \u00b7) h\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\n\u22a2 x \u2208 e.source \u2194 proj x \u2208 e.baseSet\n[PROOFSTEP]\nrw [e.source_eq, mem_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\nx : B \u00d7 F\n\u22a2 x \u2208 e.target \u2194 x.fst \u2208 e.baseSet\n[PROOFSTEP]\nrw [e.target_eq, prod_univ, mem_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\nx : B \u00d7 F\nhx : x \u2208 e.target\n\u22a2 proj (\u2191(LocalEquiv.symm e.toLocalEquiv) x) = x.fst\n[PROOFSTEP]\nhave := (e.coe_fst (e.map_target hx)).symm\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\nx : B \u00d7 F\nhx : x \u2208 e.target\nthis : proj (\u2191(LocalEquiv.symm e.toLocalEquiv) x) = (\u2191e (\u2191(LocalEquiv.symm e.toLocalEquiv) x)).fst\n\u22a2 proj (\u2191(LocalEquiv.symm e.toLocalEquiv) x) = x.fst\n[PROOFSTEP]\nrwa [\u2190 e.coe_coe, e.right_inv hx] at this \n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d x : Z\nex : x \u2208 e.source\n\u22a2 \u2191(LocalEquiv.symm e.toLocalEquiv) (proj x, (\u2191e x).snd) = x\n[PROOFSTEP]\nrw [\u2190 e.coe_fst ex, Prod.mk.eta, \u2190 e.coe_coe, e.left_inv ex]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\n\u22a2 \u2191(LocalEquiv.symm e.toLocalEquiv) \u207b\u00b9' (proj \u207b\u00b9' e.baseSet) \u2229 e.target = e.target\n[PROOFSTEP]\nrefine' inter_eq_right_iff_subset.mpr fun x hx => _\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\nx : B \u00d7 F\nhx : x \u2208 e.target\n\u22a2 x \u2208 \u2191(LocalEquiv.symm e.toLocalEquiv) \u207b\u00b9' (proj \u207b\u00b9' e.baseSet)\n[PROOFSTEP]\nsimp only [mem_preimage, LocalEquiv.invFun_as_coe, e.proj_symm_apply hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\nx : B \u00d7 F\nhx : x \u2208 e.target\n\u22a2 x.fst \u2208 e.baseSet\n[PROOFSTEP]\nexact e.mem_target.mp hx\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\ns : Set B\n\u22a2 \u2191(LocalEquiv.symm e.toLocalEquiv) \u207b\u00b9' (proj \u207b\u00b9' s) \u2229 e.baseSet \u00d7\u02e2 univ = (s \u2229 e.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\next \u27e8x, y\u27e9\n[GOAL]\ncase h.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\ns : Set B\nx : B\ny : F\n\u22a2 (x, y) \u2208 \u2191(LocalEquiv.symm e.toLocalEquiv) \u207b\u00b9' (proj \u207b\u00b9' s) \u2229 e.baseSet \u00d7\u02e2 univ \u2194 (x, y) \u2208 (s \u2229 e.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nsuffices x \u2208 e.baseSet \u2192 (proj (e.toLocalEquiv.symm (x, y)) \u2208 s \u2194 x \u2208 s) by\n  simpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\ns : Set B\nx : B\ny : F\nthis : x \u2208 e.baseSet \u2192 (proj (\u2191(LocalEquiv.symm e.toLocalEquiv) (x, y)) \u2208 s \u2194 x \u2208 s)\n\u22a2 (x, y) \u2208 \u2191(LocalEquiv.symm e.toLocalEquiv) \u207b\u00b9' (proj \u207b\u00b9' s) \u2229 e.baseSet \u00d7\u02e2 univ \u2194 (x, y) \u2208 (s \u2229 e.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nsimpa only [prod_mk_mem_set_prod_eq, mem_inter_iff, and_true_iff, mem_univ, and_congr_left_iff]\n[GOAL]\ncase h.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\ns : Set B\nx : B\ny : F\n\u22a2 x \u2208 e.baseSet \u2192 (proj (\u2191(LocalEquiv.symm e.toLocalEquiv) (x, y)) \u2208 s \u2194 x \u2208 s)\n[PROOFSTEP]\nintro h\n[GOAL]\ncase h.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx\u271d : Z\ns : Set B\nx : B\ny : F\nh : x \u2208 e.baseSet\n\u22a2 proj (\u2191(LocalEquiv.symm e.toLocalEquiv) (x, y)) \u2208 s \u2194 x \u2208 s\n[PROOFSTEP]\nrw [e.proj_symm_apply' h]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne f : Pretrivialization F proj\n\u22a2 f.target \u2229 \u2191(LocalEquiv.symm f.toLocalEquiv) \u207b\u00b9' e.source = (e.baseSet \u2229 f.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne f : Pretrivialization F proj\n\u22a2 (LocalEquiv.trans (LocalEquiv.symm f.toLocalEquiv) e.toLocalEquiv).source = (e.baseSet \u2229 f.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrw [LocalEquiv.trans_source, LocalEquiv.symm_source, e.target_inter_preimage_symm_source_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\n\u22a2 LocalEquiv.symm (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv) =\n    LocalEquiv.trans (LocalEquiv.symm e'.toLocalEquiv) e.toLocalEquiv\n[PROOFSTEP]\nrw [LocalEquiv.trans_symm_eq_symm_trans_symm, LocalEquiv.symm_symm]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\n\u22a2 (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).source = (e.baseSet \u2229 e'.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrw [LocalEquiv.trans_source, e'.source_eq, LocalEquiv.symm_source, e.target_eq, inter_comm, e.preimage_symm_proj_inter,\n  inter_comm]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne e' : Pretrivialization F proj\n\u22a2 (LocalEquiv.trans (LocalEquiv.symm e.toLocalEquiv) e'.toLocalEquiv).target = (e.baseSet \u2229 e'.baseSet) \u00d7\u02e2 univ\n[PROOFSTEP]\nrw [\u2190 LocalEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Pretrivialization F TotalSpace.proj\nb : B\nhb : b \u2208 e.baseSet\ny : F\n\u22a2 { proj := b, snd := Pretrivialization.symm e b y } = \u2191(LocalEquiv.symm e.toLocalEquiv) (b, y)\n[PROOFSTEP]\nsimp only [e.symm_apply hb, TotalSpace.mk_cast (e.proj_symm_apply' hb), TotalSpace.eta]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Pretrivialization F TotalSpace.proj\nz : TotalSpace F E\nhz : z.proj \u2208 e.baseSet\n\u22a2 Pretrivialization.symm e z.proj (\u2191e z).snd = z.snd\n[PROOFSTEP]\nrw [e.symm_apply hz, cast_eq_iff_heq, e.mk_proj_snd' hz, e.symm_apply_apply (e.mem_source.mpr hz)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b2 : TopologicalSpace B\ninst\u271d\u00b9 : TopologicalSpace F\nproj : Z \u2192 B\ne\u271d : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny\u271d : E b\u271d\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Pretrivialization F TotalSpace.proj\nb : B\nhb : b \u2208 e.baseSet\ny : F\n\u22a2 \u2191e { proj := b, snd := Pretrivialization.symm e b y } = (b, y)\n[PROOFSTEP]\nrw [e.mk_symm hb, e.apply_symm_apply (e.mk_mem_target.mpr hb)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne e' : Trivialization F proj\nh\u2081 : e.toLocalHomeomorph = e'.toLocalHomeomorph\nh\u2082 : e.baseSet = e'.baseSet\n\u22a2 e = e'\n[PROOFSTEP]\ncases e\n[GOAL]\ncase mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F proj\ntoLocalHomeomorph\u271d : LocalHomeomorph Z (B \u00d7 F)\nbaseSet\u271d : Set B\nopen_baseSet\u271d : IsOpen baseSet\u271d\nsource_eq\u271d : toLocalHomeomorph\u271d.source = proj \u207b\u00b9' baseSet\u271d\ntarget_eq\u271d : toLocalHomeomorph\u271d.target = baseSet\u271d \u00d7\u02e2 univ\nproj_toFun\u271d : \u2200 (p : Z), p \u2208 toLocalHomeomorph\u271d.source \u2192 (\u2191toLocalHomeomorph\u271d p).fst = proj p\nh\u2081 :\n  { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.toLocalHomeomorph =\n    e'.toLocalHomeomorph\nh\u2082 :\n  { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.baseSet =\n    e'.baseSet\n\u22a2 { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n      source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d } =\n    e'\n[PROOFSTEP]\ncases e'\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ntoLocalHomeomorph\u271d\u00b9 : LocalHomeomorph Z (B \u00d7 F)\nbaseSet\u271d\u00b9 : Set B\nopen_baseSet\u271d\u00b9 : IsOpen baseSet\u271d\u00b9\nsource_eq\u271d\u00b9 : toLocalHomeomorph\u271d\u00b9.source = proj \u207b\u00b9' baseSet\u271d\u00b9\ntarget_eq\u271d\u00b9 : toLocalHomeomorph\u271d\u00b9.target = baseSet\u271d\u00b9 \u00d7\u02e2 univ\nproj_toFun\u271d\u00b9 : \u2200 (p : Z), p \u2208 toLocalHomeomorph\u271d\u00b9.source \u2192 (\u2191toLocalHomeomorph\u271d\u00b9 p).fst = proj p\ntoLocalHomeomorph\u271d : LocalHomeomorph Z (B \u00d7 F)\nbaseSet\u271d : Set B\nopen_baseSet\u271d : IsOpen baseSet\u271d\nsource_eq\u271d : toLocalHomeomorph\u271d.source = proj \u207b\u00b9' baseSet\u271d\ntarget_eq\u271d : toLocalHomeomorph\u271d.target = baseSet\u271d \u00d7\u02e2 univ\nproj_toFun\u271d : \u2200 (p : Z), p \u2208 toLocalHomeomorph\u271d.source \u2192 (\u2191toLocalHomeomorph\u271d p).fst = proj p\nh\u2081 :\n  { toLocalHomeomorph := toLocalHomeomorph\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n        source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 }.toLocalHomeomorph =\n    { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.toLocalHomeomorph\nh\u2082 :\n  { toLocalHomeomorph := toLocalHomeomorph\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n        source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 }.baseSet =\n    { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n        source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }.baseSet\n\u22a2 { toLocalHomeomorph := toLocalHomeomorph\u271d\u00b9, baseSet := baseSet\u271d\u00b9, open_baseSet := open_baseSet\u271d\u00b9,\n      source_eq := source_eq\u271d\u00b9, target_eq := target_eq\u271d\u00b9, proj_toFun := proj_toFun\u271d\u00b9 } =\n    { toLocalHomeomorph := toLocalHomeomorph\u271d, baseSet := baseSet\u271d, open_baseSet := open_baseSet\u271d,\n      source_eq := source_eq\u271d, target_eq := target_eq\u271d, proj_toFun := proj_toFun\u271d }\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne e' : Trivialization F proj\nh : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e'\n\u22a2 e = e'\n[PROOFSTEP]\next1\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne e' : Trivialization F proj\nh : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e'\n\u22a2 e.toLocalHomeomorph = e'.toLocalHomeomorph\ncase h\u2082\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne e' : Trivialization F proj\nh : (fun e => toPretrivialization e) e = (fun e => toPretrivialization e) e'\n\u22a2 e.baseSet = e'.baseSet\n[PROOFSTEP]\nexacts [LocalHomeomorph.toLocalEquiv_injective (congr_arg Pretrivialization.toLocalEquiv h),\n  congr_arg Pretrivialization.baseSet h]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u22a2 x \u2208 e.source \u2194 proj x \u2208 e.baseSet\n[PROOFSTEP]\nrw [e.source_eq, mem_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nex : x \u2208 e.source\n\u22a2 map proj (\ud835\udcdd x) = \ud835\udcdd (proj x)\n[PROOFSTEP]\nrw [\u2190 e.coe_fst ex, \u2190 map_congr (e.coe_fst_eventuallyEq_proj ex), \u2190 map_map, \u2190 e.coe_coe, e.map_nhds_eq ex,\n  map_fst_nhds]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\n\u22a2 Tendsto f l (\ud835\udcdd z) \u2194 Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nrw [e.nhds_eq_comap_inf_principal hz, tendsto_inf, tendsto_comap_iff, Prod.tendsto_iff, coe_coe, tendsto_principal,\n  coe_fst _ hz]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\n\u22a2 ((Tendsto (fun n => ((\u2191e \u2218 f) n).fst) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun n => ((\u2191e \u2218 f) n).snd) l (\ud835\udcdd (\u2191e z).snd)) \u2227\n      \u2200\u1da0 (a : \u03b1) in l, f a \u2208 e.source) \u2194\n    Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nby_cases hl : \u2200\u1da0 x in l, f x \u2208 e.source\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\nhl : \u2200\u1da0 (x : \u03b1) in l, f x \u2208 e.source\n\u22a2 ((Tendsto (fun n => ((\u2191e \u2218 f) n).fst) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun n => ((\u2191e \u2218 f) n).snd) l (\ud835\udcdd (\u2191e z).snd)) \u2227\n      \u2200\u1da0 (a : \u03b1) in l, f a \u2208 e.source) \u2194\n    Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nsimp only [hl, and_true]\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\nhl : \u2200\u1da0 (x : \u03b1) in l, f x \u2208 e.source\n\u22a2 Tendsto (fun n => ((\u2191e \u2218 f) n).fst) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun n => ((\u2191e \u2218 f) n).snd) l (\ud835\udcdd (\u2191e z).snd) \u2194\n    Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nrefine (tendsto_congr' ?_).and Iff.rfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\nhl : \u2200\u1da0 (x : \u03b1) in l, f x \u2208 e.source\n\u22a2 (fun n => ((\u2191e \u2218 f) n).fst) =\u1da0[l] proj \u2218 f\n[PROOFSTEP]\nexact hl.mono fun x \u21a6 e.coe_fst\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\nhl : \u00ac\u2200\u1da0 (x : \u03b1) in l, f x \u2208 e.source\n\u22a2 ((Tendsto (fun n => ((\u2191e \u2218 f) n).fst) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun n => ((\u2191e \u2218 f) n).snd) l (\ud835\udcdd (\u2191e z).snd)) \u2227\n      \u2200\u1da0 (a : \u03b1) in l, f a \u2208 e.source) \u2194\n    Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2227 Tendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nsimp only [hl, and_false, false_iff, not_and]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 e.source\nhl : \u00ac\u2200\u1da0 (x : \u03b1) in l, f x \u2208 e.source\n\u22a2 Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2192 \u00acTendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nrw [e.source_eq] at hl hz \n[GOAL]\ncase neg\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\n\u03b1 : Type u_6\nl : Filter \u03b1\nf : \u03b1 \u2192 Z\nz : Z\nhz : z \u2208 proj \u207b\u00b9' e.baseSet\nhl : \u00ac\u2200\u1da0 (x : \u03b1) in l, f x \u2208 proj \u207b\u00b9' e.baseSet\n\u22a2 Tendsto (proj \u2218 f) l (\ud835\udcdd (proj z)) \u2192 \u00acTendsto (fun x => (\u2191e (f x)).snd) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nexact fun h _ \u21a6 hl <| h <| e.open_baseSet.mem_nhds hz\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx z : Z\nhz : z \u2208 e.source\n\u22a2 \ud835\udcdd z = comap proj (\ud835\udcdd (proj z)) \u2293 comap (Prod.snd \u2218 \u2191e) (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nrefine eq_of_forall_le_iff fun l \u21a6 ?_\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx z : Z\nhz : z \u2208 e.source\nl : Filter Z\n\u22a2 l \u2264 \ud835\udcdd z \u2194 l \u2264 comap proj (\ud835\udcdd (proj z)) \u2293 comap (Prod.snd \u2218 \u2191e) (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nrw [le_inf_iff, \u2190 tendsto_iff_comap, \u2190 tendsto_iff_comap]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx z : Z\nhz : z \u2208 e.source\nl : Filter Z\n\u22a2 l \u2264 \ud835\udcdd z \u2194 Tendsto proj l (\ud835\udcdd (proj z)) \u2227 Tendsto (Prod.snd \u2218 \u2191e) l (\ud835\udcdd (\u2191e z).snd)\n[PROOFSTEP]\nexact e.tendsto_nhds_iff hz\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nb : B\nhb : b \u2208 e.baseSet\np : F\n\u22a2 \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) (b, p) \u2208 proj \u207b\u00b9' {b}\n[PROOFSTEP]\nrw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_6\ninst\u271d : TopologicalSpace Z'\nh : Z' \u2243\u209c Z\n\u22a2 (LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph).toLocalEquiv.source =\n    proj \u2218 \u2191h \u207b\u00b9' e.baseSet\n[PROOFSTEP]\nsimp [source_eq, preimage_preimage, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_6\ninst\u271d : TopologicalSpace Z'\nh : Z' \u2243\u209c Z\n\u22a2 (LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph).toLocalEquiv.target = e.baseSet \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp [target_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_6\ninst\u271d : TopologicalSpace Z'\nh : Z' \u2243\u209c Z\np : Z'\nhp : p \u2208 (LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph).toLocalEquiv.source\n\u22a2 (\u2191(LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph) p).fst = (proj \u2218 \u2191h) p\n[PROOFSTEP]\nhave hp : h p \u2208 e.source := by simpa using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_6\ninst\u271d : TopologicalSpace Z'\nh : Z' \u2243\u209c Z\np : Z'\nhp : p \u2208 (LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph).toLocalEquiv.source\n\u22a2 \u2191h p \u2208 e.source\n[PROOFSTEP]\nsimpa using hp\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nZ' : Type u_6\ninst\u271d : TopologicalSpace Z'\nh : Z' \u2243\u209c Z\np : Z'\nhp\u271d : p \u2208 (LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph).toLocalEquiv.source\nhp : \u2191h p \u2208 e.source\n\u22a2 (\u2191(LocalHomeomorph.trans (Homeomorph.toLocalHomeomorph h) e.toLocalHomeomorph) p).fst = (proj \u2218 \u2191h) p\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : Z \u2192 X\nz : Z\ne : Trivialization F proj\nhe : proj z \u2208 e.baseSet\nhf : ContinuousAt (f \u2218 \u2191(LocalEquiv.symm e.toLocalEquiv)) (\u2191e z)\n\u22a2 ContinuousAt f z\n[PROOFSTEP]\nhave hez : z \u2208 e.toLocalEquiv.symm.target :=\n  by\n  rw [LocalEquiv.symm_target, e.mem_source]\n  exact he\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : Z \u2192 X\nz : Z\ne : Trivialization F proj\nhe : proj z \u2208 e.baseSet\nhf : ContinuousAt (f \u2218 \u2191(LocalEquiv.symm e.toLocalEquiv)) (\u2191e z)\n\u22a2 z \u2208 (LocalEquiv.symm e.toLocalEquiv).target\n[PROOFSTEP]\nrw [LocalEquiv.symm_target, e.mem_source]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : Z \u2192 X\nz : Z\ne : Trivialization F proj\nhe : proj z \u2208 e.baseSet\nhf : ContinuousAt (f \u2218 \u2191(LocalEquiv.symm e.toLocalEquiv)) (\u2191e z)\n\u22a2 proj z \u2208 e.baseSet\n[PROOFSTEP]\nexact he\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : Z \u2192 X\nz : Z\ne : Trivialization F proj\nhe : proj z \u2208 e.baseSet\nhf : ContinuousAt (f \u2218 \u2191(LocalEquiv.symm e.toLocalEquiv)) (\u2191e z)\nhez : z \u2208 (LocalEquiv.symm e.toLocalEquiv).target\n\u22a2 ContinuousAt f z\n[PROOFSTEP]\nrwa [e.toLocalHomeomorph.symm.continuousAt_iff_continuousAt_comp_right hez, LocalHomeomorph.symm_symm]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : X \u2192 Z\nx : X\ne : Trivialization F proj\nhf_proj : ContinuousAt (proj \u2218 f) x\nhe : proj (f x) \u2208 e.baseSet\nhf : ContinuousAt (\u2191e \u2218 f) x\n\u22a2 ContinuousAt f x\n[PROOFSTEP]\nrw [e.continuousAt_iff_continuousAt_comp_left]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : X \u2192 Z\nx : X\ne : Trivialization F proj\nhf_proj : ContinuousAt (proj \u2218 f) x\nhe : proj (f x) \u2208 e.baseSet\nhf : ContinuousAt (\u2191e \u2218 f) x\n\u22a2 ContinuousAt (\u2191e.toLocalHomeomorph \u2218 f) x\n[PROOFSTEP]\nexact hf\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : X \u2192 Z\nx : X\ne : Trivialization F proj\nhf_proj : ContinuousAt (proj \u2218 f) x\nhe : proj (f x) \u2208 e.baseSet\nhf : ContinuousAt (\u2191e \u2218 f) x\n\u22a2 f \u207b\u00b9' e.source \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrw [e.source_eq, \u2190 preimage_comp]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\nX : Type u_6\ninst\u271d : TopologicalSpace X\nf : X \u2192 Z\nx : X\ne : Trivialization F proj\nhf_proj : ContinuousAt (proj \u2218 f) x\nhe : proj (f x) \u2208 e.baseSet\nhf : ContinuousAt (\u2191e \u2218 f) x\n\u22a2 proj \u2218 f \u207b\u00b9' e.baseSet \u2208 \ud835\udcdd x\n[PROOFSTEP]\nexact hf_proj.preimage_mem_nhds (e.open_baseSet.mem_nhds he)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\n\u22a2 ContinuousOn (fun z => TotalSpace.mk' F z.fst (Trivialization.symm e z.fst z.snd)) (e.baseSet \u00d7\u02e2 univ)\n[PROOFSTEP]\nhave : \u2200 z \u2208 e.baseSet \u00d7\u02e2 (univ : Set F), TotalSpace.mk z.1 (e.symm z.1 z.2) = e.toLocalHomeomorph.symm z :=\n  by\n  rintro x \u27e8hx : x.1 \u2208 e.baseSet, _\u27e9\n  rw [e.mk_symm hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\n\u22a2 \u2200 (z : B \u00d7 F),\n    z \u2208 e.baseSet \u00d7\u02e2 univ \u2192\n      { proj := z.fst, snd := Trivialization.symm e z.fst z.snd } = \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) z\n[PROOFSTEP]\nrintro x \u27e8hx : x.1 \u2208 e.baseSet, _\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\nx : B \u00d7 F\nhx : x.fst \u2208 e.baseSet\nright\u271d : x.snd \u2208 univ\n\u22a2 { proj := x.fst, snd := Trivialization.symm e x.fst x.snd } = \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x\n[PROOFSTEP]\nrw [e.mk_symm hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\nthis :\n  \u2200 (z : B \u00d7 F),\n    z \u2208 e.baseSet \u00d7\u02e2 univ \u2192\n      { proj := z.fst, snd := Trivialization.symm e z.fst z.snd } = \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) z\n\u22a2 ContinuousOn (fun z => TotalSpace.mk' F z.fst (Trivialization.symm e z.fst z.snd)) (e.baseSet \u00d7\u02e2 univ)\n[PROOFSTEP]\nrefine' ContinuousOn.congr _ this\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\nthis :\n  \u2200 (z : B \u00d7 F),\n    z \u2208 e.baseSet \u00d7\u02e2 univ \u2192\n      { proj := z.fst, snd := Trivialization.symm e z.fst z.snd } = \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) z\n\u22a2 ContinuousOn (fun x => \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x) (e.baseSet \u00d7\u02e2 univ)\n[PROOFSTEP]\nrw [\u2190 e.target_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ninst\u271d : (x : B) \u2192 Zero (E x)\ne : Trivialization F TotalSpace.proj\nthis :\n  \u2200 (z : B \u00d7 F),\n    z \u2208 e.baseSet \u00d7\u02e2 univ \u2192\n      { proj := z.fst, snd := Trivialization.symm e z.fst z.snd } = \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) z\n\u22a2 ContinuousOn (fun x => \u2191(LocalHomeomorph.symm e.toLocalHomeomorph) x) e.target\n[PROOFSTEP]\nexact e.toLocalHomeomorph.continuousOn_symm\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nF' : Type u_6\ninst\u271d : TopologicalSpace F'\ne : Trivialization F proj\nh : F \u2243\u209c F'\n\u22a2 (LocalHomeomorph.transHomeomorph e.toLocalHomeomorph\n          (Homeomorph.prodCongr (Homeomorph.refl B) h)).toLocalEquiv.target =\n    e.baseSet \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp [target_eq, prod_univ, preimage_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 (b, coordChange e\u2081 e\u2082 b x) = \u2191e\u2082 (\u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x))\n[PROOFSTEP]\nrefine' Prod.ext _ rfl\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 (b, coordChange e\u2081 e\u2082 b x).fst = (\u2191e\u2082 (\u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x))).fst\n[PROOFSTEP]\nrw [e\u2082.coe_fst', \u2190 e\u2081.coe_fst', e\u2081.apply_symm_apply' h\u2081]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 proj (\u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x)) \u2208 e\u2081.baseSet\n[PROOFSTEP]\nrwa [e\u2081.proj_symm_apply' h\u2081]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 proj (\u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x)) \u2208 e\u2082.baseSet\n[PROOFSTEP]\nrwa [e\u2081.proj_symm_apply' h\u2081]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\ne\u2081 e\u2082 : Trivialization F proj\np : Z\nh : proj p \u2208 e\u2081.baseSet\n\u22a2 coordChange e\u2081 e\u2082 (proj p) (\u2191e\u2081 p).snd = (\u2191e\u2082 p).snd\n[PROOFSTEP]\nrw [coordChange, e\u2081.symm_apply_mk_proj (e\u2081.mem_source.2 h)]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne : Trivialization F proj\nb : B\nh : b \u2208 e.baseSet\nx : F\n\u22a2 coordChange e e b x = x\n[PROOFSTEP]\nrw [coordChange, e.apply_symm_apply' h]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 e\u2083 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 coordChange e\u2082 e\u2083 b (coordChange e\u2081 e\u2082 b x) = coordChange e\u2081 e\u2083 b x\n[PROOFSTEP]\nrw [coordChange, e\u2081.mk_coordChange _ h\u2081 h\u2082, \u2190 e\u2082.coe_coe, e\u2082.left_inv, coordChange]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 e\u2083 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 \u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x) \u2208 e\u2082.source\n[PROOFSTEP]\nrwa [e\u2082.mem_source, e\u2081.proj_symm_apply' h\u2081]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\n\u22a2 Continuous (coordChange e\u2081 e\u2082 b)\n[PROOFSTEP]\nrefine'\n  continuous_snd.comp\n    (e\u2082.toLocalHomeomorph.continuousOn.comp_continuous (e\u2081.toLocalHomeomorph.continuousOn_symm.comp_continuous _ _) _)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\n\u22a2 Continuous fun x => (b, x)\n[PROOFSTEP]\nexact continuous_const.prod_mk continuous_id\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\n\u22a2 \u2200 (x : F), (b, x) \u2208 e\u2081.target\n[PROOFSTEP]\nexact fun x => e\u2081.mem_target.2 h\u2081\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\n\u22a2 \u2200 (x : F), \u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x) \u2208 e\u2082.source\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_3\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 \u2191(LocalHomeomorph.symm e\u2081.toLocalHomeomorph) (b, x) \u2208 e\u2082.source\n[PROOFSTEP]\nrwa [e\u2082.mem_source, e\u2081.proj_symm_apply' h\u2081]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 coordChange e\u2082 e\u2081 b (coordChange e\u2081 e\u2082 b x) = x\n[PROOFSTEP]\nsimp only [*, coordChange_coordChange, coordChange_same_apply]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b3 : TopologicalSpace B\ninst\u271d\u00b2 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b9 : TopologicalSpace Z\ninst\u271d : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb\u271d : B\ny : E b\u271d\ne\u2081 e\u2082 : Trivialization F proj\nb : B\nh\u2081 : b \u2208 e\u2081.baseSet\nh\u2082 : b \u2208 e\u2082.baseSet\nx : F\n\u22a2 coordChange e\u2081 e\u2082 b (coordChange e\u2082 e\u2081 b x) = x\n[PROOFSTEP]\nsimp only [*, coordChange_coordChange, coordChange_same_apply]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne : Trivialization F proj\ns : Set B\nx : Z\nhx : x \u2208 e.source\n\u22a2 \u2191e.toLocalHomeomorph x \u2208 s \u00d7\u02e2 univ \u2194 x \u2208 proj \u207b\u00b9' s\n[PROOFSTEP]\nsimp [e.coe_fst', hx]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne : Trivialization F proj\ns : Set B\nhs : IsOpen s\n\u22a2 (LocalHomeomorph.symm\n          (LocalHomeomorph.IsImage.restr\n            (_ : LocalHomeomorph.IsImage (LocalHomeomorph.symm e.toLocalHomeomorph) (s \u00d7\u02e2 univ) (proj \u207b\u00b9' s))\n            (_ :\n              IsOpen\n                ((LocalHomeomorph.symm e.toLocalHomeomorph).toLocalEquiv.source \u2229 s \u00d7\u02e2 univ)))).toLocalEquiv.source =\n    proj \u207b\u00b9' (e.baseSet \u2229 s)\n[PROOFSTEP]\nsimp [source_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne : Trivialization F proj\ns : Set B\nhs : IsOpen s\n\u22a2 (LocalHomeomorph.symm\n          (LocalHomeomorph.IsImage.restr\n            (_ : LocalHomeomorph.IsImage (LocalHomeomorph.symm e.toLocalHomeomorph) (s \u00d7\u02e2 univ) (proj \u207b\u00b9' s))\n            (_ :\n              IsOpen\n                ((LocalHomeomorph.symm e.toLocalHomeomorph).toLocalEquiv.source \u2229 s \u00d7\u02e2 univ)))).toLocalEquiv.target =\n    (e.baseSet \u2229 s) \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp [target_eq, prod_univ]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne : Trivialization F proj\ns : Set B\n\u22a2 e.source \u2229 frontier (proj \u207b\u00b9' s) = proj \u207b\u00b9' (e.baseSet \u2229 frontier s)\n[PROOFSTEP]\nrw [\u2190 (e.isImage_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq, (e.isImage_preimage_prod _).preimage_eq,\n  e.source_eq, preimage_inter]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\n\u22a2 e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s)\n[PROOFSTEP]\nrw [e.frontier_preimage, e'.frontier_preimage, Hs]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\n\u22a2 EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph) (e.source \u2229 frontier (proj \u207b\u00b9' s))\n[PROOFSTEP]\nrwa [e.frontier_preimage]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\n\u22a2 (LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n          (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n          (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n          (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n          (_ :\n            EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph)\n              (e.source \u2229 frontier (proj \u207b\u00b9' s)))).toLocalEquiv.source =\n    proj \u207b\u00b9' Set.ite s e.baseSet e'.baseSet\n[PROOFSTEP]\nsimp [source_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\n\u22a2 (LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n          (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n          (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n          (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n          (_ :\n            EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph)\n              (e.source \u2229 frontier (proj \u207b\u00b9' s)))).toLocalEquiv.target =\n    Set.ite s e.baseSet e'.baseSet \u00d7\u02e2 univ\n[PROOFSTEP]\nsimp [target_eq, prod_univ]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\np : Z\n\u22a2 p \u2208\n      (LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n            (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n            (_ :\n              EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph)\n                (e.source \u2229 frontier (proj \u207b\u00b9' s)))).toLocalEquiv.source \u2192\n    (\u2191(LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n              (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n              (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n              (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n              (_ : EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph) (e.source \u2229 frontier (proj \u207b\u00b9' s))))\n          p).fst =\n      proj p\n[PROOFSTEP]\nrintro\n  (\u27e8he, hs\u27e9 | \u27e8he, hs\u27e9)\n      -- porting note: was `<;> simp [*]`\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\np : Z\nhe : p \u2208 e.source\nhs : p \u2208 proj \u207b\u00b9' s\n\u22a2 (\u2191(LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n            (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n            (_ : EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph) (e.source \u2229 frontier (proj \u207b\u00b9' s))))\n        p).fst =\n    proj p\n[PROOFSTEP]\nsimp [piecewise_eq_of_mem _ _ _ hs, *]\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\ns : Set B\nHs : e.baseSet \u2229 frontier s = e'.baseSet \u2229 frontier s\nHeq : EqOn (\u2191e) (\u2191e') (proj \u207b\u00b9' (e.baseSet \u2229 frontier s))\np : Z\nhe : p \u2208 e'.source\nhs : \u00acp \u2208 proj \u207b\u00b9' s\n\u22a2 (\u2191(LocalHomeomorph.piecewise e.toLocalHomeomorph e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ)\n            (_ : LocalHomeomorph.IsImage e.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : LocalHomeomorph.IsImage e'.toLocalHomeomorph (proj \u207b\u00b9' s) (s \u00d7\u02e2 univ))\n            (_ : e.source \u2229 frontier (proj \u207b\u00b9' s) = e'.source \u2229 frontier (proj \u207b\u00b9' s))\n            (_ : EqOn (\u2191e.toLocalHomeomorph) (\u2191e'.toLocalHomeomorph) (e.source \u2229 frontier (proj \u207b\u00b9' s))))\n        p).fst =\n    proj p\n[PROOFSTEP]\nsimp [piecewise_eq_of_not_mem _ _ _ hs, *]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\na : B\nHe : a \u2208 e.baseSet\nHe' : a \u2208 e'.baseSet\nHeq : \u2200 (p : Z), proj p = a \u2192 \u2191e p = \u2191e' p\nx : B\nhx : x \u2208 frontier (Iic a)\n\u22a2 x \u2208 e.baseSet \u2194 x \u2208 e'.baseSet\n[PROOFSTEP]\nobtain rfl : x = a := mem_singleton_iff.1 (frontier_Iic_subset _ hx)\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\nx : B\nHe : x \u2208 e.baseSet\nHe' : x \u2208 e'.baseSet\nHeq : \u2200 (p : Z), proj p = x \u2192 \u2191e p = \u2191e' p\nhx : x \u2208 frontier (Iic x)\n\u22a2 x \u2208 e.baseSet \u2194 x \u2208 e'.baseSet\n[PROOFSTEP]\nsimp [He, He']\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\na : B\nHe : a \u2208 e.baseSet\nHe' : a \u2208 e'.baseSet\n\u22a2 \u2200 (p : Z), proj p = a \u2192 \u2191e p = \u2191(transFiberHomeomorph e' (Trivialization.coordChangeHomeomorph e' e He' He)) p\n[PROOFSTEP]\nrintro p rfl\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\np : Z\nHe : proj p \u2208 e.baseSet\nHe' : proj p \u2208 e'.baseSet\n\u22a2 \u2191e p = \u2191(transFiberHomeomorph e' (Trivialization.coordChangeHomeomorph e' e He' He)) p\n[PROOFSTEP]\next1\n[GOAL]\ncase h\u2081\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\np : Z\nHe : proj p \u2208 e.baseSet\nHe' : proj p \u2208 e'.baseSet\n\u22a2 (\u2191e p).fst = (\u2191(transFiberHomeomorph e' (Trivialization.coordChangeHomeomorph e' e He' He)) p).fst\n[PROOFSTEP]\nsimp [e.coe_fst', e'.coe_fst', *]\n[GOAL]\ncase h\u2082\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2076 : TopologicalSpace B\ninst\u271d\u2075 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u2074 : TopologicalSpace Z\ninst\u271d\u00b3 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d\u00b2 : TopologicalSpace B'\ninst\u271d\u00b9 : LinearOrder B\ninst\u271d : OrderTopology B\ne e' : Trivialization F proj\np : Z\nHe : proj p \u2208 e.baseSet\nHe' : proj p \u2208 e'.baseSet\n\u22a2 (\u2191e p).snd = (\u2191(transFiberHomeomorph e' (Trivialization.coordChangeHomeomorph e' e He' He)) p).snd\n[PROOFSTEP]\nsimp [coordChange_apply_snd, *]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\n\u22a2 Disjoint e.source e'.source\n[PROOFSTEP]\nrw [e.source_eq, e'.source_eq]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\n\u22a2 Disjoint (proj \u207b\u00b9' e.baseSet) (proj \u207b\u00b9' e'.baseSet)\n[PROOFSTEP]\nexact H.preimage _\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\n\u22a2 Disjoint e.target e'.target\n[PROOFSTEP]\nrw [e.target_eq, e'.target_eq, disjoint_iff_inf_le]\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\n\u22a2 e.baseSet \u00d7\u02e2 univ \u2293 e'.baseSet \u00d7\u02e2 univ \u2264 \u22a5\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx\u271d : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\nx : B \u00d7 F\nhx : x \u2208 e.baseSet \u00d7\u02e2 univ \u2293 e'.baseSet \u00d7\u02e2 univ\n\u22a2 x \u2208 \u22a5\n[PROOFSTEP]\nexact H.le_bot \u27e8hx.1.1, hx.2.1\u27e9\n[GOAL]\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\n\u22a2 \u2200 (p : Z),\n    p \u2208\n        (LocalHomeomorph.disjointUnion e.toLocalHomeomorph e'.toLocalHomeomorph (_ : Disjoint e.source e'.source)\n              (_ : Disjoint e.target e'.target)).toLocalEquiv.source \u2192\n      (\u2191(LocalHomeomorph.disjointUnion e.toLocalHomeomorph e'.toLocalHomeomorph (_ : Disjoint e.source e'.source)\n                (_ : Disjoint e.target e'.target))\n            p).fst =\n        proj p\n[PROOFSTEP]\nrintro p (hp | hp')\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp : p \u2208 e.source\n\u22a2 (\u2191(LocalHomeomorph.disjointUnion e.toLocalHomeomorph e'.toLocalHomeomorph (_ : Disjoint e.source e'.source)\n            (_ : Disjoint e.target e'.target))\n        p).fst =\n    proj p\n[PROOFSTEP]\nshow (e.source.piecewise e e' p).1 = proj p\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp : p \u2208 e.source\n\u22a2 (Set.piecewise e.source (\u2191e) (\u2191e') p).fst = proj p\n[PROOFSTEP]\nrw [piecewise_eq_of_mem, e.coe_fst]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp : p \u2208 e.source\n\u22a2 p \u2208 e.source\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase inl.hi\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp : p \u2208 e.source\n\u22a2 p \u2208 e.source\n[PROOFSTEP]\nexact hp\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp' : p \u2208 e'.source\n\u22a2 (\u2191(LocalHomeomorph.disjointUnion e.toLocalHomeomorph e'.toLocalHomeomorph (_ : Disjoint e.source e'.source)\n            (_ : Disjoint e.target e'.target))\n        p).fst =\n    proj p\n[PROOFSTEP]\nshow (e.source.piecewise e e' p).1 = proj p\n[GOAL]\ncase inr\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp' : p \u2208 e'.source\n\u22a2 (Set.piecewise e.source (\u2191e) (\u2191e') p).fst = proj p\n[PROOFSTEP]\nrw [piecewise_eq_of_not_mem, e'.coe_fst hp']\n[GOAL]\ncase inr.hi\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp' : p \u2208 e'.source\n\u22a2 \u00acp \u2208 e.source\n[PROOFSTEP]\nsimp only [source_eq] at hp' \u22a2\n[GOAL]\ncase inr.hi\n\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u2074 : TopologicalSpace B\ninst\u271d\u00b3 : TopologicalSpace F\nproj : Z \u2192 B\ninst\u271d\u00b2 : TopologicalSpace Z\ninst\u271d\u00b9 : TopologicalSpace (TotalSpace F E)\ne\u271d : Trivialization F proj\nx : Z\ne'\u271d : Trivialization F TotalSpace.proj\nx' : TotalSpace F E\nb : B\ny : E b\nB' : Type u_6\ninst\u271d : TopologicalSpace B'\ne e' : Trivialization F proj\nH : Disjoint e.baseSet e'.baseSet\np : Z\nhp' : p \u2208 proj \u207b\u00b9' e'.baseSet\n\u22a2 \u00acp \u2208 proj \u207b\u00b9' e.baseSet\n[PROOFSTEP]\nexact fun h => H.le_bot \u27e8h, hp'\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.FiberBundle.Trivialization", "llama_tokens": 31632, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8152324938410784, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.504449916141063}}
{"text": "[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u22a2 \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => g \u2022 x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nlet \u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\n\u22a2 \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => g \u2022 x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nhave meas_\u03c0 : Measurable \u03c0 := continuous_quotient_mk'.measurable\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\u22a2 \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => g \u2022 x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nhave \ud835\udcd5meas : NullMeasurableSet \ud835\udcd5 \u03bc := h\ud835\udcd5.nullMeasurableSet\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u22a2 \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => g \u2022 x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nhave meas_\u03c0A : MeasurableSet (\u03c0 \u207b\u00b9' A) := measurableSet_preimage meas_\u03c0 hA\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\nmeas_\u03c0A : MeasurableSet (\u03c0 \u207b\u00b9' A)\n\u22a2 \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => g \u2022 x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nrw [Measure.map_apply meas_\u03c0 hA, Measure.map_apply meas_\u03c0 (measurableSet_preimage (measurable_const_smul g) hA),\n  Measure.restrict_apply\u2080' \ud835\udcd5meas, Measure.restrict_apply\u2080' \ud835\udcd5meas]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\nmeas_\u03c0A : MeasurableSet (\u03c0 \u207b\u00b9' A)\n\u22a2 \u2191\u2191\u03bc (\u03c0 \u207b\u00b9' ((fun x x_1 => x \u2022 x_1) g \u207b\u00b9' A) \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0 \u207b\u00b9' A \u2229 \ud835\udcd5)\n[PROOFSTEP]\nset \u03c0_preA := \u03c0 \u207b\u00b9' A\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\n\u22a2 \u2191\u2191\u03bc (\u03c0 \u207b\u00b9' ((fun x x_1 => x \u2022 x_1) g \u207b\u00b9' A) \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nhave : \u03c0 \u207b\u00b9' ((fun x : G \u29f8 \u0393 => g \u2022 x) \u207b\u00b9' A) = (g * \u00b7) \u207b\u00b9' \u03c0_preA := by ext1; simp\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\n\u22a2 \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nx\u271d : G\n\u22a2 x\u271d \u2208 \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) \u2194 x\u271d \u2208 (fun x => g * x) \u207b\u00b9' \u03c0_preA\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc (\u03c0 \u207b\u00b9' ((fun x x_1 => x \u2022 x_1) g \u207b\u00b9' A) \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nhave : \u03bc ((g * \u00b7) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u03bc (\u03c0_preA \u2229 (g\u207b\u00b9 * \u00b7) \u207b\u00b9' \ud835\udcd5) :=\n  by\n  trans \u03bc ((g * \u00b7) \u207b\u00b9' (\u03c0_preA \u2229 (g\u207b\u00b9 * \u00b7) \u207b\u00b9' \ud835\udcd5))\n  \u00b7 rw [preimage_inter]\n    congr 2\n    simp [Set.preimage]\n  rw [measure_preimage_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\n[PROOFSTEP]\ntrans \u03bc ((g * \u00b7) \u207b\u00b9' (\u03c0_preA \u2229 (g\u207b\u00b9 * \u00b7) \u207b\u00b9' \ud835\udcd5))\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5))\n[PROOFSTEP]\nrw [preimage_inter]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) =\n    \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 (fun x => g * x) \u207b\u00b9' ((fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5))\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase e_a.e_a\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \ud835\udcd5 = (fun x => g * x) \u207b\u00b9' ((fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\n[PROOFSTEP]\nsimp [Set.preimage]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\n[PROOFSTEP]\nrw [measure_preimage_mul]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\n\u22a2 \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nrw [this]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\n\u22a2 \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nhave h\ud835\udcd5_translate_fundom : IsFundamentalDomain (Subgroup.opposite \u0393) (g \u2022 \ud835\udcd5) \u03bc := h\ud835\udcd5.smul_of_comm g\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u22a2 \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 \ud835\udcd5)\n[PROOFSTEP]\nrw [h\ud835\udcd5.measure_set_eq h\ud835\udcd5_translate_fundom meas_\u03c0A, \u2190 preimage_smul_inv]\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u22a2 \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 \u2022 x) \u207b\u00b9' \ud835\udcd5)\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u22a2 \u2200 (g : { x // x \u2208 \u2191Subgroup.opposite \u0393 }), (fun x => g \u2022 x) \u207b\u00b9' \u03c0_preA = \u03c0_preA\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u22a2 \u2200 (g : { x // x \u2208 \u2191Subgroup.opposite \u0393 }), (fun x => g \u2022 x) \u207b\u00b9' \u03c0_preA = \u03c0_preA\n[PROOFSTEP]\nrintro \u27e8\u03b3, \u03b3_in_\u0393\u27e9\n[GOAL]\ncase mk\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u03b3 : G\u1d50\u1d52\u1d56\n\u03b3_in_\u0393 : \u03b3 \u2208 \u2191Subgroup.opposite \u0393\n\u22a2 (fun x => { val := \u03b3, property := \u03b3_in_\u0393 } \u2022 x) \u207b\u00b9' \u03c0_preA = \u03c0_preA\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.h\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u03b3 : G\u1d50\u1d52\u1d56\n\u03b3_in_\u0393 : \u03b3 \u2208 \u2191Subgroup.opposite \u0393\nx : G\n\u22a2 x \u2208 (fun x => { val := \u03b3, property := \u03b3_in_\u0393 } \u2022 x) \u207b\u00b9' \u03c0_preA \u2194 x \u2208 \u03c0_preA\n[PROOFSTEP]\nhave : \u03c0 (x * MulOpposite.unop \u03b3) = \u03c0 x := by simpa [QuotientGroup.eq'] using \u03b3_in_\u0393\n[GOAL]\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u03b3 : G\u1d50\u1d52\u1d56\n\u03b3_in_\u0393 : \u03b3 \u2208 \u2191Subgroup.opposite \u0393\nx : G\n\u22a2 \u03c0 (x * MulOpposite.unop \u03b3) = \u03c0 x\n[PROOFSTEP]\nsimpa [QuotientGroup.eq'] using \u03b3_in_\u0393\n[GOAL]\ncase mk.h\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d\u00b9 : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis\u271d : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u03b3 : G\u1d50\u1d52\u1d56\n\u03b3_in_\u0393 : \u03b3 \u2208 \u2191Subgroup.opposite \u0393\nx : G\nthis : \u03c0 (x * MulOpposite.unop \u03b3) = \u03c0 x\n\u22a2 x \u2208 (fun x => { val := \u03b3, property := \u03b3_in_\u0393 } \u2022 x) \u207b\u00b9' \u03c0_preA \u2194 x \u2208 \u03c0_preA\n[PROOFSTEP]\nsimp only [(\u00b7 \u2022 \u00b7), \u2190 this, mem_preimage]\n[GOAL]\ncase mk.h\nG : Type u_1\ninst\u271d\u2079 : Group G\ninst\u271d\u2078 : MeasurableSpace G\ninst\u271d\u2077 : TopologicalSpace G\ninst\u271d\u2076 : TopologicalGroup G\ninst\u271d\u2075 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2074 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u00b3 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\ng : G\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u03c0 : G \u2192 G \u29f8 \u0393 := QuotientGroup.mk\nmeas_\u03c0 : Measurable \u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u03c0_preA : Set G := \u03c0 \u207b\u00b9' A\nmeas_\u03c0A : MeasurableSet \u03c0_preA\nthis\u271d\u00b9 : \u03c0 \u207b\u00b9' ((fun x => g \u2022 x) \u207b\u00b9' A) = (fun x => g * x) \u207b\u00b9' \u03c0_preA\nthis\u271d : \u2191\u2191\u03bc ((fun x => g * x) \u207b\u00b9' \u03c0_preA \u2229 \ud835\udcd5) = \u2191\u2191\u03bc (\u03c0_preA \u2229 (fun x => g\u207b\u00b9 * x) \u207b\u00b9' \ud835\udcd5)\nh\ud835\udcd5_translate_fundom : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } (g \u2022 \ud835\udcd5)\n\u03b3 : G\u1d50\u1d52\u1d56\n\u03b3_in_\u0393 : \u03b3 \u2208 \u2191Subgroup.opposite \u0393\nx : G\nthis : \u03c0 (x * MulOpposite.unop \u03b3) = \u03c0 x\n\u22a2 \u2191(SMul.smul { val := \u03b3, property := \u03b3_in_\u0393 } x) \u2208 A \u2194 \u2191(x * MulOpposite.unop \u03b3) \u2208 A\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\n\u22a2 map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) =\n    map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)\n[PROOFSTEP]\napply Measure.ext\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\n\u22a2 \u2200 (s : Set (G \u29f8 \u0393)),\n    MeasurableSet s \u2192\n      \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) s =\n        \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) s\n[PROOFSTEP]\nintro A hA\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nobtain \u27e8x\u2081, h\u27e9 := @Quotient.exists_rep _ (QuotientGroup.leftRel \u0393) x\n[GOAL]\ncase h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nhaveI := h\ud835\udcd5.smulInvariantMeasure_map\n[GOAL]\ncase h.intro\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5)) A\n[PROOFSTEP]\nconvert measure_preimage_smul x\u2081 ((Measure.map QuotientGroup.mk) (\u03bc.restrict \ud835\udcd5)) A using 1\n[GOAL]\ncase h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (fun x_1 => x * x_1) (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))) A =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)\n[PROOFSTEP]\nrw [\u2190 h, Measure.map_apply]\n[GOAL]\ncase h.e'_2\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 \u2191\u2191(map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))\n      ((fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x) \u207b\u00b9' A) =\n    \u2191\u2191(map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5)) ((fun x => x\u2081 \u2022 x) \u207b\u00b9' A)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.e'_2.hf\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 Measurable fun x => Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 * x\n[PROOFSTEP]\nexact measurable_const_mul _\n[GOAL]\ncase h.e'_2.hs\nG : Type u_1\ninst\u271d\u00b9\u2070 : Group G\ninst\u271d\u2079 : MeasurableSpace G\ninst\u271d\u2078 : TopologicalSpace G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2075 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2074 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u00b3 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsMulLeftInvariant \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nx : G \u29f8 \u0393\nA : Set (G \u29f8 \u0393)\nhA : MeasurableSet A\nx\u2081 : G\nh : Quotient.mk (QuotientGroup.leftRel \u0393) x\u2081 = x\nthis : SMulInvariantMeasure G (G \u29f8 \u0393) (map QuotientGroup.mk (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 MeasurableSet A\n[PROOFSTEP]\nexact hA\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nlet \u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nhave meas_\u03c0 : Measurable \u03c0 := continuous_quotient_mk'.measurable\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\nmeas_\u03c0 : Measurable \u2191\u03c0\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nhave \ud835\udcd5meas : NullMeasurableSet \ud835\udcd5 \u03bc := h\ud835\udcd5.nullMeasurableSet\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\nmeas_\u03c0 : Measurable \u2191\u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nhaveI := Fact.mk h\ud835\udcd5_finite\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\nmeas_\u03c0 : Measurable \u2191\u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\nthis : Fact (\u2191\u2191\u03bc \ud835\udcd5 < \u22a4)\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nhaveI : (Measure.map (QuotientGroup.mk' \u0393) (\u03bc.restrict \ud835\udcd5)).IsMulLeftInvariant := h\ud835\udcd5.isMulLeftInvariant_map\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\nmeas_\u03c0 : Measurable \u2191\u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\nthis\u271d : Fact (\u2191\u2191\u03bc \ud835\udcd5 < \u22a4)\nthis : IsMulLeftInvariant (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) \u2022 haarMeasure K\n[PROOFSTEP]\nrw [Measure.haarMeasure_unique (Measure.map (QuotientGroup.mk' \u0393) (\u03bc.restrict \ud835\udcd5)) K, Measure.map_apply meas_\u03c0,\n  Measure.restrict_apply\u2080' \ud835\udcd5meas, inter_comm]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\n\u03c0 : G \u2192* G \u29f8 \u0393 := QuotientGroup.mk' \u0393\nmeas_\u03c0 : Measurable \u2191\u03c0\n\ud835\udcd5meas : NullMeasurableSet \ud835\udcd5\nthis\u271d : Fact (\u2191\u2191\u03bc \ud835\udcd5 < \u22a4)\nthis : IsMulLeftInvariant (map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5))\n\u22a2 MeasurableSet \u2191K\n[PROOFSTEP]\nexact K.isCompact.measurableSet\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\nc : \u211d\u22650\nh : \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) = \u2191c\n\u22a2 map (\u2191(QuotientGroup.mk' \u0393)) (Measure.restrict \u03bc \ud835\udcd5) = c \u2022 haarMeasure K\n[PROOFSTEP]\nrw [h\ud835\udcd5.map_restrict_quotient K h\ud835\udcd5_finite, h]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b9\u00b2 : Group G\ninst\u271d\u00b9\u00b9 : MeasurableSpace G\ninst\u271d\u00b9\u2070 : TopologicalSpace G\ninst\u271d\u2079 : TopologicalGroup G\ninst\u271d\u2078 : BorelSpace G\n\u03bc : Measure G\n\u0393 : Subgroup G\n\ud835\udcd5 : Set G\nh\ud835\udcd5 : IsFundamentalDomain { x // x \u2208 \u2191Subgroup.opposite \u0393 } \ud835\udcd5\ninst\u271d\u2077 : Countable { x // x \u2208 \u0393 }\ninst\u271d\u2076 : MeasurableSpace (G \u29f8 \u0393)\ninst\u271d\u2075 : BorelSpace (G \u29f8 \u0393)\ninst\u271d\u2074 : T2Space (G \u29f8 \u0393)\ninst\u271d\u00b3 : SecondCountableTopology (G \u29f8 \u0393)\nK : PositiveCompacts (G \u29f8 \u0393)\ninst\u271d\u00b2 : Subgroup.Normal \u0393\ninst\u271d\u00b9 : IsHaarMeasure \u03bc\ninst\u271d : IsMulRightInvariant \u03bc\nh\ud835\udcd5_finite : \u2191\u2191\u03bc \ud835\udcd5 < \u22a4\nc : \u211d\u22650\nh : \u2191\u2191\u03bc (\ud835\udcd5 \u2229 \u2191(QuotientGroup.mk' \u0393) \u207b\u00b9' \u2191K) = \u2191c\n\u22a2 \u2191c \u2022 haarMeasure K = c \u2022 haarMeasure K\n[PROOFSTEP]\nrfl\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Measure.Haar.Quotient", "llama_tokens": 20172, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8791467706759584, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.5043475193184005}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue \u2124 S\nx : \u2124\u02e3\n\u22a2 \u2191abv \u2191x = 1\n[PROOFSTEP]\nrcases Int.units_eq_one_or x with (rfl | rfl)\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue \u2124 S\n\u22a2 \u2191abv \u21911 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue \u2124 S\n\u22a2 \u2191abv \u2191(-1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommRing S\ninst\u271d : Nontrivial R\nabv : AbsoluteValue R S\nx : \u2124\u02e3\n\u22a2 \u2191abv \u2191\u2191x = 1\n[PROOFSTEP]\nrcases Int.units_eq_one_or x with (rfl | rfl)\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommRing S\ninst\u271d : Nontrivial R\nabv : AbsoluteValue R S\n\u22a2 \u2191abv \u2191\u21911 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : LinearOrderedCommRing S\ninst\u271d : Nontrivial R\nabv : AbsoluteValue R S\n\u22a2 \u2191abv \u2191\u2191(-1) = 1\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue R S\nx : \u2124\u02e3\ny : R\n\u22a2 \u2191abv (x \u2022 y) = \u2191abv y\n[PROOFSTEP]\nrcases Int.units_eq_one_or x with (rfl | rfl)\n[GOAL]\ncase inl\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue R S\ny : R\n\u22a2 \u2191abv (1 \u2022 y) = \u2191abv y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : LinearOrderedCommRing S\nabv : AbsoluteValue R S\ny : R\n\u22a2 \u2191abv (-1 \u2022 y) = \u2191abv y\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Int.AbsoluteValue", "llama_tokens": 814, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.5042443636973905}}
{"text": "[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u22a2 det M = \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M (\u2191\u03c3 i) i\n[PROOFSTEP]\nsimp [det_apply, Units.smul_def]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u22a2 det (diagonal d) = \u220f i : n, d i\n[PROOFSTEP]\nrw [det_apply']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, diagonal d (\u2191\u03c3 i) i = \u220f i : n, d i\n[PROOFSTEP]\nrefine' (Finset.sum_eq_single 1 _ _).trans _\n[GOAL]\ncase refine'_1\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u22a2 \u2200 (b : Perm n), b \u2208 univ \u2192 b \u2260 1 \u2192 \u2191\u2191(\u2191sign b) * \u220f i : n, diagonal d (\u2191b i) i = 0\n[PROOFSTEP]\nrintro \u03c3 - h2\n[GOAL]\ncase refine'_1\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u03c3 : Perm n\nh2 : \u03c3 \u2260 1\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, diagonal d (\u2191\u03c3 i) i = 0\n[PROOFSTEP]\ncases' not_forall.1 (mt Equiv.ext h2) with x h3\n[GOAL]\ncase refine'_1.intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u03c3 : Perm n\nh2 : \u03c3 \u2260 1\nx : n\nh3 : \u00ac\u2191\u03c3 x = \u21911 x\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, diagonal d (\u2191\u03c3 i) i = 0\n[PROOFSTEP]\nconvert mul_zero (\u03b5 \u03c3)\n[GOAL]\ncase h.e'_2.h.e'_6\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u03c3 : Perm n\nh2 : \u03c3 \u2260 1\nx : n\nh3 : \u00ac\u2191\u03c3 x = \u21911 x\n\u22a2 \u220f i : n, diagonal d (\u2191\u03c3 i) i = 0\n[PROOFSTEP]\napply Finset.prod_eq_zero (mem_univ x)\n[GOAL]\ncase h.e'_2.h.e'_6\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u03c3 : Perm n\nh2 : \u03c3 \u2260 1\nx : n\nh3 : \u00ac\u2191\u03c3 x = \u21911 x\n\u22a2 diagonal d (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nexact if_neg h3\n[GOAL]\ncase refine'_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u22a2 \u00ac1 \u2208 univ \u2192 \u2191\u2191(\u2191sign 1) * \u220f i : n, diagonal d (\u21911 i) i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nd : n \u2192 R\n\u22a2 \u2191\u2191(\u2191sign 1) * \u220f i : n, diagonal d (\u21911 i) i = \u220f i : n, d i\n[PROOFSTEP]\nsimp\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\n\u22a2 det 1 = 1\n[PROOFSTEP]\nrw [\u2190 diagonal_one]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\n\u22a2 det (diagonal fun x => 1) = 1\n[PROOFSTEP]\nsimp [-diagonal_one]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsEmpty n\nA : Matrix n n R\n\u22a2 det A = 1\n[PROOFSTEP]\nsimp [det_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsEmpty n\n\u22a2 det = const (Matrix n n R) 1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsEmpty n\nx\u271d : Matrix n n R\n\u22a2 det x\u271d = const (Matrix n n R) 1 x\u271d\n[PROOFSTEP]\nexact det_isEmpty\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2077 : DecidableEq n\u271d\ninst\u271d\u2076 : Fintype n\u271d\ninst\u271d\u2075 : DecidableEq m\ninst\u271d\u2074 : Fintype m\nR : Type v\ninst\u271d\u00b3 : CommRing R\nn : Type u_3\ninst\u271d\u00b2 : Unique n\ninst\u271d\u00b9 : DecidableEq n\ninst\u271d : Fintype n\nA : Matrix n n R\n\u22a2 det A = A default default\n[PROOFSTEP]\nsimp [det_apply, univ_unique]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Subsingleton n\nA : Matrix n n R\nk : n\n\u22a2 det A = A k k\n[PROOFSTEP]\nhave := uniqueOfSubsingleton k\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Subsingleton n\nA : Matrix n n R\nk : n\nthis : Unique n\n\u22a2 det A = A k k\n[PROOFSTEP]\nconvert det_unique A\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f x : n, M (\u2191\u03c3 x) (p x) * N (p x) x = 0\n[PROOFSTEP]\nobtain \u27e8i, j, hpij, hij\u27e9 : \u2203 i j, p i = p j \u2227 i \u2260 j :=\n  by\n  rw [\u2190 Finite.injective_iff_bijective, Injective] at H \n  push_neg at H \n  exact H\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\n\u22a2 \u2203 i j, p i = p j \u2227 i \u2260 j\n[PROOFSTEP]\nrw [\u2190 Finite.injective_iff_bijective, Injective] at H \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00ac\u2200 \u2983a\u2081 a\u2082 : n\u2984, p a\u2081 = p a\u2082 \u2192 a\u2081 = a\u2082\n\u22a2 \u2203 i j, p i = p j \u2227 i \u2260 j\n[PROOFSTEP]\npush_neg at H \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : Exists fun \u2983a\u2081\u2984 => Exists fun \u2983a\u2082\u2984 => p a\u2081 = p a\u2082 \u2227 a\u2081 \u2260 a\u2082\n\u22a2 \u2203 i j, p i = p j \u2227 i \u2260 j\n[PROOFSTEP]\nexact H\n[GOAL]\ncase intro.intro.intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\ni j : n\nhpij : p i = p j\nhij : i \u2260 j\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f x : n, M (\u2191\u03c3 x) (p x) * N (p x) x = 0\n[PROOFSTEP]\nexact\n  sum_involution (fun \u03c3 _ => \u03c3 * Equiv.swap i j)\n    (fun \u03c3 _ =>\n      by\n      have : (\u220f x, M (\u03c3 x) (p x)) = \u220f x, M ((\u03c3 * Equiv.swap i j) x) (p x) :=\n        Fintype.prod_equiv (swap i j) _ _ (by simp [apply_swap_eq_self hpij])\n      simp [this, sign_swap hij, -sign_swap', prod_mul_distrib])\n    (fun \u03c3 _ _ => (not_congr mul_swap_eq_iff).mpr hij) (fun _ _ => mem_univ _) fun \u03c3 _ => mul_swap_involutive i j \u03c3\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\ni j : n\nhpij : p i = p j\nhij : i \u2260 j\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : n, M (\u2191\u03c3 x) (p x) * N (p x) x +\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => \u03c3 * Equiv.swap i j) \u03c3 x\u271d)) *\n        \u220f x : n, M (\u2191((fun \u03c3 x => \u03c3 * Equiv.swap i j) \u03c3 x\u271d) x) (p x) * N (p x) x =\n    0\n[PROOFSTEP]\nhave : (\u220f x, M (\u03c3 x) (p x)) = \u220f x, M ((\u03c3 * Equiv.swap i j) x) (p x) :=\n  Fintype.prod_equiv (swap i j) _ _ (by simp [apply_swap_eq_self hpij])\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\ni j : n\nhpij : p i = p j\nhij : i \u2260 j\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u22a2 \u2200 (x : n), M (\u2191\u03c3 x) (p x) = M (\u2191(\u03c3 * Equiv.swap i j) (\u2191(Equiv.swap i j) x)) (p (\u2191(Equiv.swap i j) x))\n[PROOFSTEP]\nsimp [apply_swap_eq_self hpij]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\np : n \u2192 n\nH : \u00acBijective p\ni j : n\nhpij : p i = p j\nhij : i \u2260 j\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\nthis : \u220f x : n, M (\u2191\u03c3 x) (p x) = \u220f x : n, M (\u2191(\u03c3 * Equiv.swap i j) x) (p x)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : n, M (\u2191\u03c3 x) (p x) * N (p x) x +\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => \u03c3 * Equiv.swap i j) \u03c3 x\u271d)) *\n        \u220f x : n, M (\u2191((fun \u03c3 x => \u03c3 * Equiv.swap i j) \u03c3 x\u271d) x) (p x) * N (p x) x =\n    0\n[PROOFSTEP]\nsimp [this, sign_swap hij, -sign_swap', prod_mul_distrib]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u22a2 det (M * N) = \u2211 p : n \u2192 n, \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M (\u2191\u03c3 i) (p i) * N (p i) i\n[PROOFSTEP]\nsimp only [det_apply', mul_apply, prod_univ_sum, mul_sum, Fintype.piFinset_univ]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u22a2 \u2211 x : Perm n, \u2211 x_1 : n \u2192 n, \u2191\u2191(\u2191sign x) * \u220f x_2 : n, M (\u2191x x_2) (x_1 x_2) * N (x_1 x_2) x_2 =\n    \u2211 x : n \u2192 n, \u2211 x_1 : Perm n, \u2191\u2191(\u2191sign x_1) * \u220f x_2 : n, M (\u2191x_1 x_2) (x x_2) * N (x x_2) x_2\n[PROOFSTEP]\nrw [Finset.sum_comm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\nf : n \u2192 n\nx\u271d : f \u2208 univ\nhbij : \u00acf \u2208 filter Bijective univ\n\u22a2 \u00acBijective fun i => f i\n[PROOFSTEP]\nsimpa only [true_and_iff, mem_filter, mem_univ] using hbij\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\nx\u271d\u00b3 x\u271d\u00b2 : n \u2192 n\nx\u271d\u00b9 : x\u271d\u00b3 \u2208 filter Bijective univ\nx\u271d : x\u271d\u00b2 \u2208 filter Bijective univ\nh : (fun p h => ofBijective p (_ : Bijective p)) x\u271d\u00b3 x\u271d\u00b9 = (fun p h => ofBijective p (_ : Bijective p)) x\u271d\u00b2 x\u271d\n\u22a2 x\u271d\u00b3 = x\u271d\u00b2\n[PROOFSTEP]\ninjection h\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u22a2 \u2211 \u03c4 : Perm n, \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M (\u2191\u03c3 i) (\u2191\u03c4 i) * N (\u2191\u03c4 i) i =\n    \u2211 \u03c3 : Perm n, \u2211 \u03c4 : Perm n, (\u220f i : n, N (\u2191\u03c3 i) i) * \u2191\u2191(\u2191sign \u03c4) * \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j)\n[PROOFSTEP]\nsimp only [mul_comm, mul_left_comm, prod_mul_distrib, mul_assoc]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\n\u22a2 (\u220f i : n, N (\u2191\u03c3 i) i) * \u2191\u2191(\u2191sign \u03c4) * \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) =\n    (\u220f i : n, N (\u2191\u03c3 i) i) * (\u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4))) *\n      \u220f i : n, M (\u2191(\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4) i) i\n[PROOFSTEP]\nhave : (\u220f j, M (\u03c4 j) (\u03c3 j)) = \u220f j, M ((\u03c4 * \u03c3\u207b\u00b9) j) j :=\n  by\n  rw [\u2190 (\u03c3\u207b\u00b9 : _ \u2243 _).prod_comp]\n  simp only [Equiv.Perm.coe_mul, apply_inv_self, Function.comp_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\n\u22a2 \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n[PROOFSTEP]\nrw [\u2190 (\u03c3\u207b\u00b9 : _ \u2243 _).prod_comp]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\n\u22a2 \u220f i : n, M (\u2191\u03c4 (\u2191\u03c3\u207b\u00b9 i)) (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 i)) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n[PROOFSTEP]\nsimp only [Equiv.Perm.coe_mul, apply_inv_self, Function.comp_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n\u22a2 (\u220f i : n, N (\u2191\u03c3 i) i) * \u2191\u2191(\u2191sign \u03c4) * \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) =\n    (\u220f i : n, N (\u2191\u03c3 i) i) * (\u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4))) *\n      \u220f i : n, M (\u2191(\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4) i) i\n[PROOFSTEP]\nhave h : \u03b5 \u03c3 * \u03b5 (\u03c4 * \u03c3\u207b\u00b9) = \u03b5 \u03c4 :=\n  calc\n    \u03b5 \u03c3 * \u03b5 (\u03c4 * \u03c3\u207b\u00b9) = \u03b5 (\u03c4 * \u03c3\u207b\u00b9 * \u03c3) := by\n      rw [mul_comm, sign_mul (\u03c4 * \u03c3\u207b\u00b9)]\n      simp only [Int.cast_mul, Units.val_mul]\n    _ = \u03b5 \u03c4 := by simp only [inv_mul_cancel_right]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9)) = \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9 * \u03c3))\n[PROOFSTEP]\nrw [mul_comm, sign_mul (\u03c4 * \u03c3\u207b\u00b9)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n\u22a2 \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9)) * \u2191\u2191(\u2191sign \u03c3) = \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9) * \u2191sign \u03c3)\n[PROOFSTEP]\nsimp only [Int.cast_mul, Units.val_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\n\u22a2 \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9 * \u03c3)) = \u2191\u2191(\u2191sign \u03c4)\n[PROOFSTEP]\nsimp only [inv_mul_cancel_right]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\nh : \u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9)) = \u2191\u2191(\u2191sign \u03c4)\n\u22a2 (\u220f i : n, N (\u2191\u03c3 i) i) * \u2191\u2191(\u2191sign \u03c4) * \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) =\n    (\u220f i : n, N (\u2191\u03c3 i) i) * (\u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4))) *\n      \u220f i : n, M (\u2191(\u2191(Equiv.mulRight \u03c3\u207b\u00b9) \u03c4) i) i\n[PROOFSTEP]\nsimp_rw [Equiv.coe_mulRight, h]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u03c3 : Perm n\nx\u271d : \u03c3 \u2208 univ\n\u03c4 : Perm n\nthis : \u220f j : n, M (\u2191\u03c4 j) (\u2191\u03c3 j) = \u220f j : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) j) j\nh : \u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign (\u03c4 * \u03c3\u207b\u00b9)) = \u2191\u2191(\u2191sign \u03c4)\n\u22a2 (\u220f x : n, N (\u2191\u03c3 x) x) * \u2191\u2191(\u2191sign \u03c4) * \u220f x : n, M (\u2191\u03c4 x) (\u2191\u03c3 x) =\n    (\u220f x : n, N (\u2191\u03c3 x) x) * \u2191\u2191(\u2191sign \u03c4) * \u220f x : n, M (\u2191(\u03c4 * \u03c3\u207b\u00b9) x) x\n[PROOFSTEP]\nsimp only [this]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix n n R\n\u22a2 \u2211 \u03c3 : Perm n, \u2211 \u03c4 : Perm n, (\u220f i : n, N (\u2191\u03c3 i) i) * (\u2191\u2191(\u2191sign \u03c3) * \u2191\u2191(\u2191sign \u03c4)) * \u220f i : n, M (\u2191\u03c4 i) i = det M * det N\n[PROOFSTEP]\nsimp only [det_apply', Finset.mul_sum, mul_comm, mul_left_comm, mul_assoc]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N : Matrix m m R\n\u22a2 det (M * N) = det (N * M)\n[PROOFSTEP]\nrw [det_mul, det_mul, mul_comm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N P : Matrix m m R\n\u22a2 det (M * (N * P)) = det (N * (M * P))\n[PROOFSTEP]\nrw [\u2190 Matrix.mul_assoc, \u2190 Matrix.mul_assoc, det_mul, det_mul_comm M N, \u2190 det_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM N P : Matrix m m R\n\u22a2 det (M * N * P) = det (M * P * N)\n[PROOFSTEP]\nrw [Matrix.mul_assoc, Matrix.mul_assoc, det_mul, det_mul_comm N P, \u2190 det_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : (Matrix m m R)\u02e3\nN : Matrix m m R\n\u22a2 det (\u2191M * N * \u2191M\u207b\u00b9) = det N\n[PROOFSTEP]\nrw [det_mul_right_comm, Units.mul_inv, one_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u22a2 det M\u1d40 = det M\n[PROOFSTEP]\nrw [det_apply', det_apply']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M\u1d40 (\u2191\u03c3 i) i = \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M (\u2191\u03c3 i) i\n[PROOFSTEP]\nrefine' Fintype.sum_bijective _ inv_involutive.bijective _ _ _\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u22a2 \u2200 (x : Perm n), \u2191\u2191(\u2191sign x) * \u220f i : n, M\u1d40 (\u2191x i) i = \u2191\u2191(\u2191sign x\u207b\u00b9) * \u220f i : n, M (\u2191x\u207b\u00b9 i) i\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M\u1d40 (\u2191\u03c3 i) i = \u2191\u2191(\u2191sign \u03c3\u207b\u00b9) * \u220f i : n, M (\u2191\u03c3\u207b\u00b9 i) i\n[PROOFSTEP]\nrw [sign_inv]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M\u1d40 (\u2191\u03c3 i) i = \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, M (\u2191\u03c3\u207b\u00b9 i) i\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u03c3 : Perm n\n\u22a2 \u220f i : n, M\u1d40 (\u2191\u03c3 i) i = \u220f i : n, M (\u2191\u03c3\u207b\u00b9 i) i\n[PROOFSTEP]\napply Fintype.prod_equiv \u03c3\n[GOAL]\ncase e_a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u03c3 : Perm n\n\u22a2 \u2200 (x : n), M\u1d40 (\u2191\u03c3 x) x = M (\u2191\u03c3\u207b\u00b9 (\u2191\u03c3 x)) (\u2191\u03c3 x)\n[PROOFSTEP]\nintros\n[GOAL]\ncase e_a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\n\u03c3 : Perm n\nx\u271d : n\n\u22a2 M\u1d40 (\u2191\u03c3 x\u271d) x\u271d = M (\u2191\u03c3\u207b\u00b9 (\u2191\u03c3 x\u271d)) (\u2191\u03c3 x\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\n\u03c3 : Perm n\nM : Matrix n n R\n\u22a2 \u2191sign \u03c3 \u2022 \u2191detRowAlternating M = \u2191\u2191(\u2191sign \u03c3) * det M\n[PROOFSTEP]\nsimp [Units.smul_def]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 det (submatrix A \u2191e \u2191e) = det A\n[PROOFSTEP]\nrw [det_apply', det_apply']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2211 \u03c3 : Perm n, \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2211 \u03c3 : Perm m, \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191\u03c3 i) i\n[PROOFSTEP]\napply Fintype.sum_equiv (Equiv.permCongr e)\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u22a2 \u2200 (x : Perm n),\n    \u2191\u2191(\u2191sign x) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191x i) i =\n      \u2191\u2191(\u2191sign (\u2191(permCongr e) x)) * \u220f i : m, A (\u2191(\u2191(permCongr e) x) i) i\n[PROOFSTEP]\nintro \u03c3\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i =\n    \u2191\u2191(\u2191sign (\u2191(permCongr e) \u03c3)) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i\n[PROOFSTEP]\nrw [Equiv.Perm.sign_permCongr e \u03c3]\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u2191\u2191(\u2191sign \u03c3) * \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e_a\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u220f i : n, submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = \u220f i : m, A (\u2191(\u2191(permCongr e) \u03c3) i) i\n[PROOFSTEP]\napply Fintype.prod_equiv e\n[GOAL]\ncase h.e_a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\n\u22a2 \u2200 (x : n), submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 x) x = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e x)) (\u2191e x)\n[PROOFSTEP]\nintro i\n[GOAL]\ncase h.e_a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\ne : n \u2243 m\nA : Matrix m m R\n\u03c3 : Perm n\ni : n\n\u22a2 submatrix A (\u2191e) (\u2191e) (\u2191\u03c3 i) i = A (\u2191(\u2191(permCongr e) \u03c3) (\u2191e i)) (\u2191e i)\n[PROOFSTEP]\nrw [Equiv.permCongr_apply, Equiv.symm_apply_apply, submatrix_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\n\u03c3 : Perm n\n\u22a2 det (PEquiv.toMatrix (toPEquiv \u03c3)) = \u2191\u2191(\u2191sign \u03c3)\n[PROOFSTEP]\nrw [\u2190 Matrix.mul_one (\u03c3.toPEquiv.toMatrix : Matrix n n R), PEquiv.toPEquiv_mul_matrix, det_permute, det_one, mul_one]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\nc : R\n\u22a2 det (c \u2022 A) = det ((diagonal fun x => c) * A)\n[PROOFSTEP]\nrw [smul_eq_diagonal_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\nc : R\n\u22a2 det (diagonal fun x => c) * det A = c ^ Fintype.card n * det A\n[PROOFSTEP]\nsimp [card_univ]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2078 : DecidableEq n\ninst\u271d\u2077 : Fintype n\ninst\u271d\u2076 : DecidableEq m\ninst\u271d\u2075 : Fintype m\nR : Type v\ninst\u271d\u2074 : CommRing R\n\u03b1 : Type u_3\ninst\u271d\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : DistribMulAction \u03b1 R\ninst\u271d\u00b9 : IsScalarTower \u03b1 R R\ninst\u271d : SMulCommClass \u03b1 R R\nc : \u03b1\nA : Matrix n n R\n\u22a2 det (c \u2022 A) = c ^ Fintype.card n \u2022 det A\n[PROOFSTEP]\nrw [\u2190 smul_one_smul R c A, det_smul, smul_pow, one_pow, smul_mul_assoc, one_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\n\u22a2 det (-A) = (-1) ^ Fintype.card n * det A\n[PROOFSTEP]\nrw [\u2190 det_smul, neg_one_smul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\n\u22a2 det (-A) = (-1) ^ Fintype.card n \u2022 det A\n[PROOFSTEP]\nrw [\u2190 det_smul_of_tower, Units.neg_smul, one_smul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nv : n \u2192 R\nA : Matrix n n R\n\u22a2 (\u2191of fun i j => v j * A i j) = A * diagonal v\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nv : n \u2192 R\nA : Matrix n n R\ni\u271d x\u271d : n\n\u22a2 \u2191of (fun i j => v j * A i j) i\u271d x\u271d = (A * diagonal v) i\u271d x\u271d\n[PROOFSTEP]\nsimp [mul_comm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nv : n \u2192 R\nA : Matrix n n R\n\u22a2 det (A * diagonal v) = (\u220f i : n, v i) * det A\n[PROOFSTEP]\nrw [det_mul, det_diagonal, mul_comm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2075 : DecidableEq n\ninst\u271d\u2074 : Fintype n\ninst\u271d\u00b3 : DecidableEq m\ninst\u271d\u00b2 : Fintype m\nR : Type v\ninst\u271d\u00b9 : CommRing R\nS : Type w\ninst\u271d : CommRing S\nf : R \u2192+* S\nM : Matrix n n R\n\u22a2 \u2191f (det M) = det (\u2191(RingHom.mapMatrix f) M)\n[PROOFSTEP]\nsimp [Matrix.det_apply', f.map_sum, f.map_prod]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\nj : n\nh : \u2200 (i : n), A i j = 0\n\u22a2 det A = 0\n[PROOFSTEP]\nrw [\u2190 det_transpose]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\nj : n\nh : \u2200 (i : n), A i j = 0\n\u22a2 det A\u1d40 = 0\n[PROOFSTEP]\nexact det_eq_zero_of_row_eq_zero j h\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\ni j : n\ni_ne_j : i \u2260 j\nhij : \u2200 (k : n), M k i = M k j\n\u22a2 det M = 0\n[PROOFSTEP]\nrw [\u2190 det_transpose, det_zero_of_row_eq i_ne_j]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\ni j : n\ni_ne_j : i \u2260 j\nhij : \u2200 (k : n), M k i = M k j\n\u22a2 M\u1d40 i = M\u1d40 j\n[PROOFSTEP]\nexact funext hij\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\nu v : n \u2192 R\n\u22a2 det (updateColumn M j (u + v)) = det (updateColumn M j u) + det (updateColumn M j v)\n[PROOFSTEP]\nrw [\u2190 det_transpose, \u2190 updateRow_transpose, det_updateRow_add]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\nu v : n \u2192 R\n\u22a2 det (updateRow M\u1d40 j u) + det (updateRow M\u1d40 j v) = det (updateColumn M j u) + det (updateColumn M j v)\n[PROOFSTEP]\nsimp [updateRow_transpose, det_transpose]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n \u2192 R\n\u22a2 det (updateColumn M j (s \u2022 u)) = s * det (updateColumn M j u)\n[PROOFSTEP]\nrw [\u2190 det_transpose, \u2190 updateRow_transpose, det_updateRow_smul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n \u2192 R\n\u22a2 s * det (updateRow M\u1d40 j u) = s * det (updateColumn M j u)\n[PROOFSTEP]\nsimp [updateRow_transpose, det_transpose]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n \u2192 R\n\u22a2 det (updateColumn (s \u2022 M) j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u)\n[PROOFSTEP]\nrw [\u2190 det_transpose, \u2190 updateRow_transpose, transpose_smul, det_updateRow_smul']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n \u2192 R\n\u22a2 s ^ (Fintype.card n - 1) * det (updateRow M\u1d40 j u) = s ^ (Fintype.card n - 1) * det (updateColumn M j u)\n[PROOFSTEP]\nsimp [updateRow_transpose, det_transpose]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B C : Matrix n n R\nhC : det C = 1\nhA : A = B * C\n\u22a2 det B * det C = det B\n[PROOFSTEP]\nrw [hC, mul_one]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B C : Matrix n n R\nhC : det C = 1\nhA : A = C * B\n\u22a2 det C * det B = det B\n[PROOFSTEP]\nrw [hC, one_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 det (updateRow A i (A i + A j)) = det A\n[PROOFSTEP]\nsimp [det_updateRow_add, det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 det (updateColumn A i fun k => A k i + A k j) = det A\n[PROOFSTEP]\nrw [\u2190 det_transpose, \u2190 updateRow_transpose, \u2190 det_transpose A]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\n\u22a2 det (updateRow A\u1d40 i fun k => A k i + A k j) = det A\u1d40\n[PROOFSTEP]\nexact det_updateRow_add_self A\u1d40 hij\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\nc : R\n\u22a2 det (updateRow A i (A i + c \u2022 A j)) = det A\n[PROOFSTEP]\nsimp [det_updateRow_add, det_updateRow_smul, det_zero_of_row_eq hij (updateRow_self.trans (updateRow_ne hij.symm).symm)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\nc : R\n\u22a2 det (updateColumn A i fun k => A k i + c \u2022 A k j) = det A\n[PROOFSTEP]\nrw [\u2190 det_transpose, \u2190 updateRow_transpose, \u2190 det_transpose A]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni j : n\nhij : i \u2260 j\nc : R\n\u22a2 det (updateRow A\u1d40 i fun k => A k i + c \u2022 A k j) = det A\u1d40\n[PROOFSTEP]\nexact det_updateRow_add_smul_self A\u1d40 hij c\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\ns : Finset n\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\ninduction s using Finset.induction_on generalizing B with\n| empty =>\n  rintro c hs k - A_eq\n  have : \u2200 i, c i = 0 := by\n    intro i\n    specialize hs i\n    contrapose! hs\n    simp [hs]\n  congr\n  ext i j\n  rw [A_eq, this, zero_mul, add_zero]\n| @insert i s _hi ih =>\n  intro c hs k hk A_eq\n  have hAi : A i = B i + c i \u2022 B k := funext (A_eq i)\n  rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]\n  \u00b7 exact mt (fun h => show k \u2208 insert i s from h \u25b8 Finset.mem_insert_self _ _) hk\n  \u00b7 intro i' hi'\n    rw [Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 rfl\n    \u00b7 exact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i)\n  \u00b7 exact k\n  \u00b7 exact fun h => hk (Finset.mem_insert_of_mem h)\n  \u00b7 intro i' j'\n    rw [updateRow_apply, Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 simp [hi'i]\n    rw [A_eq, updateRow_ne fun h : k = i => hk <| h \u25b8 Finset.mem_insert_self k s]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\ns : Finset n\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\ninduction s using Finset.induction_on generalizing B with\n| empty =>\n  rintro c hs k - A_eq\n  have : \u2200 i, c i = 0 := by\n    intro i\n    specialize hs i\n    contrapose! hs\n    simp [hs]\n  congr\n  ext i j\n  rw [A_eq, this, zero_mul, add_zero]\n| @insert i s _hi ih =>\n  intro c hs k hk A_eq\n  have hAi : A i = B i + c i \u2022 B k := funext (A_eq i)\n  rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]\n  \u00b7 exact mt (fun h => show k \u2208 insert i s from h \u25b8 Finset.mem_insert_self _ _) hk\n  \u00b7 intro i' hi'\n    rw [Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 rfl\n    \u00b7 exact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i)\n  \u00b7 exact k\n  \u00b7 exact fun h => hk (Finset.mem_insert_of_mem h)\n  \u00b7 intro i' j'\n    rw [updateRow_apply, Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 simp [hi'i]\n    rw [A_eq, updateRow_ne fun h : k = i => hk <| h \u25b8 Finset.mem_insert_self k s]\n[GOAL]\ncase empty\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 \u2205 \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\n\n| empty =>\n  rintro c hs k - A_eq\n  have : \u2200 i, c i = 0 := by\n    intro i\n    specialize hs i\n    contrapose! hs\n    simp [hs]\n  congr\n  ext i j\n  rw [A_eq, this, zero_mul, add_zero]\n[GOAL]\ncase empty\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 \u2205 \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\nrintro c hs k - A_eq\n[GOAL]\ncase empty\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\n\u22a2 det A = det B\n[PROOFSTEP]\nhave : \u2200 i, c i = 0 := by\n  intro i\n  specialize hs i\n  contrapose! hs\n  simp [hs]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\n\u22a2 \u2200 (i : n), c i = 0\n[PROOFSTEP]\nintro i\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\ni : n\n\u22a2 c i = 0\n[PROOFSTEP]\nspecialize hs i\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : \u00aci \u2208 \u2205 \u2192 c i = 0\n\u22a2 c i = 0\n[PROOFSTEP]\ncontrapose! hs\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : c i \u2260 0\n\u22a2 \u00aci \u2208 \u2205 \u2227 c i \u2260 0\n[PROOFSTEP]\nsimp [hs]\n[GOAL]\ncase empty\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nthis : \u2200 (i : n), c i = 0\n\u22a2 det A = det B\n[PROOFSTEP]\ncongr\n[GOAL]\ncase empty.e_M\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nthis : \u2200 (i : n), c i = 0\n\u22a2 A = B\n[PROOFSTEP]\next i j\n[GOAL]\ncase empty.e_M.a.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA B : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i : n), \u00aci \u2208 \u2205 \u2192 c i = 0\nk : n\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nthis : \u2200 (i : n), c i = 0\ni j : n\n\u22a2 A i j = B i j\n[PROOFSTEP]\nrw [A_eq, this, zero_mul, add_zero]\n[GOAL]\ncase insert\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0) \u2192\n      \u2200 (k : n), \u00ack \u2208 insert i s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\n\n| @insert i s _hi ih =>\n  intro c hs k hk A_eq\n  have hAi : A i = B i + c i \u2022 B k := funext (A_eq i)\n  rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]\n  \u00b7 exact mt (fun h => show k \u2208 insert i s from h \u25b8 Finset.mem_insert_self _ _) hk\n  \u00b7 intro i' hi'\n    rw [Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 rfl\n    \u00b7 exact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i)\n  \u00b7 exact k\n  \u00b7 exact fun h => hk (Finset.mem_insert_of_mem h)\n  \u00b7 intro i' j'\n    rw [updateRow_apply, Function.update_apply]\n    split_ifs with hi'i\n    \u00b7 simp [hi'i]\n    rw [A_eq, updateRow_ne fun h : k = i => hk <| h \u25b8 Finset.mem_insert_self k s]\n[GOAL]\ncase insert\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\n\u22a2 \u2200 (c : n \u2192 R),\n    (\u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0) \u2192\n      \u2200 (k : n), \u00ack \u2208 insert i s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\n[PROOFSTEP]\nintro c hs k hk A_eq\n[GOAL]\ncase insert\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\n\u22a2 det A = det B\n[PROOFSTEP]\nhave hAi : A i = B i + c i \u2022 B k := funext (A_eq i)\n[GOAL]\ncase insert\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 det A = det B\n[PROOFSTEP]\nrw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]\n[GOAL]\ncase insert.hij\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 i \u2260 k\n[PROOFSTEP]\nexact mt (fun h => show k \u2208 insert i s from h \u25b8 Finset.mem_insert_self _ _) hk\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 \u2200 (i_1 : n), \u00aci_1 \u2208 s \u2192 update c i 0 i_1 = 0\n[PROOFSTEP]\nintro i' hi'\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' : n\nhi' : \u00aci' \u2208 s\n\u22a2 update c i 0 i' = 0\n[PROOFSTEP]\nrw [Function.update_apply]\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' : n\nhi' : \u00aci' \u2208 s\n\u22a2 (if i' = i then 0 else c i') = 0\n[PROOFSTEP]\nsplit_ifs with hi'i\n[GOAL]\ncase pos\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' : n\nhi' : \u00aci' \u2208 s\nhi'i : i' = i\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' : n\nhi' : \u00aci' \u2208 s\nhi'i : \u00aci' = i\n\u22a2 c i' = 0\n[PROOFSTEP]\nexact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i)\n[GOAL]\ncase insert.k\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 n\n[PROOFSTEP]\nexact k\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 \u00ack \u2208 s\n[PROOFSTEP]\nexact fun h => hk (Finset.mem_insert_of_mem h)\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\n\u22a2 \u2200 (i_1 j : n), A i_1 j = updateRow B i (A i) i_1 j + update c i 0 i_1 * updateRow B i (A i) k j\n[PROOFSTEP]\nintro i' j'\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' j' : n\n\u22a2 A i' j' = updateRow B i (A i) i' j' + update c i 0 i' * updateRow B i (A i) k j'\n[PROOFSTEP]\nrw [updateRow_apply, Function.update_apply]\n[GOAL]\ncase insert.x\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' j' : n\n\u22a2 A i' j' = (if i' = i then A i j' else B i' j') + (if i' = i then 0 else c i') * updateRow B i (A i) k j'\n[PROOFSTEP]\nsplit_ifs with hi'i\n[GOAL]\ncase pos\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' j' : n\nhi'i : i' = i\n\u22a2 A i' j' = A i j' + 0 * updateRow B i (A i) k j'\n[PROOFSTEP]\nsimp [hi'i]\n[GOAL]\ncase neg\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : \u00aci \u2208 s\nih :\n  \u2200 {B : Matrix n n R} (c : n \u2192 R),\n    (\u2200 (i : n), \u00aci \u2208 s \u2192 c i = 0) \u2192 \u2200 (k : n), \u00ack \u2208 s \u2192 (\u2200 (i j : n), A i j = B i j + c i * B k j) \u2192 det A = det B\nB : Matrix n n R\nc : n \u2192 R\nhs : \u2200 (i_1 : n), \u00aci_1 \u2208 insert i s \u2192 c i_1 = 0\nk : n\nhk : \u00ack \u2208 insert i s\nA_eq : \u2200 (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i \u2022 B k\ni' j' : n\nhi'i : \u00aci' = i\n\u22a2 A i' j' = B i' j' + c i' * updateRow B i (A i) k j'\n[PROOFSTEP]\nrw [A_eq, updateRow_ne fun h : k = i => hk <| h \u25b8 Finset.mem_insert_self k s]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\n\u22a2 \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\n[PROOFSTEP]\nrefine' Fin.induction _ (fun k ih => _) k\n[GOAL]\ncase refine'_1\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\n\u22a2 \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), 0 < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\n[PROOFSTEP]\nintro c hc M N h0 hsucc\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\n\u22a2 \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\n[PROOFSTEP]\nintro c hc M N h0 hsucc\n[GOAL]\ncase refine'_1\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), 0 < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\n\u22a2 det M = det N\n[PROOFSTEP]\ncongr\n[GOAL]\ncase refine'_1.e_M\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), 0 < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\n\u22a2 M = N\n[PROOFSTEP]\next i j\n[GOAL]\ncase refine'_1.e_M.a.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), 0 < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\ni j : Fin (Nat.succ n)\n\u22a2 M i j = N i j\n[PROOFSTEP]\nrefine' Fin.cases (h0 j) (fun i => _) i\n[GOAL]\ncase refine'_1.e_M.a.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk : Fin (n + 1)\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), 0 < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\ni\u271d j : Fin (Nat.succ n)\ni : Fin n\n\u22a2 M (Fin.succ i) j = N (Fin.succ i) j\n[PROOFSTEP]\nrw [hsucc, hc i (Fin.succ_pos _), zero_mul, add_zero]\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\n\u22a2 det M = det N\n[PROOFSTEP]\nset M' := updateRow M k.succ (N k.succ) with hM'\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\n\u22a2 det M = det N\n[PROOFSTEP]\nhave hM : M = updateRow M' k.succ (M' k.succ + c k \u2022 M (Fin.castSucc k)) :=\n  by\n  ext i j\n  by_cases hi : i = k.succ\n  \u00b7 simp [hi, hM', hsucc, updateRow_self]\n  rw [updateRow_ne hi, hM', updateRow_ne hi]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\n\u22a2 M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\n[PROOFSTEP]\next i j\n[GOAL]\ncase a.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\ni j : Fin (Nat.succ n)\n\u22a2 M i j = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k)) i j\n[PROOFSTEP]\nby_cases hi : i = k.succ\n[GOAL]\ncase pos\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\ni j : Fin (Nat.succ n)\nhi : i = Fin.succ k\n\u22a2 M i j = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k)) i j\n[PROOFSTEP]\nsimp [hi, hM', hsucc, updateRow_self]\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\ni j : Fin (Nat.succ n)\nhi : \u00aci = Fin.succ k\n\u22a2 M i j = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k)) i j\n[PROOFSTEP]\nrw [updateRow_ne hi, hM', updateRow_ne hi]\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\n\u22a2 det M = det N\n[PROOFSTEP]\nhave k_ne_succ : (Fin.castSucc k) \u2260 k.succ := (Fin.castSucc_lt_succ k).ne\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\n\u22a2 det M = det N\n[PROOFSTEP]\nhave M_k : M (Fin.castSucc k) = M' (Fin.castSucc k) := (updateRow_ne k_ne_succ).symm\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\n\u22a2 det M = det N\n[PROOFSTEP]\nrw [hM, M_k, det_updateRow_add_smul_self M' k_ne_succ.symm, ih (Function.update c k 0)]\n[GOAL]\ncase refine'_2._hc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\n\u22a2 \u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 update c k 0 i = 0\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase refine'_2._hc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nhi : Fin.castSucc k < Fin.succ i\n\u22a2 update c k 0 i = 0\n[PROOFSTEP]\nrw [Fin.lt_iff_val_lt_val, Fin.coe_castSucc, Fin.val_succ, Nat.lt_succ_iff] at hi \n[GOAL]\ncase refine'_2._hc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nhi : \u2191k \u2264 \u2191i\n\u22a2 update c k 0 i = 0\n[PROOFSTEP]\nrw [Function.update_apply]\n[GOAL]\ncase refine'_2._hc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nhi : \u2191k \u2264 \u2191i\n\u22a2 (if i = k then 0 else c i) = 0\n[PROOFSTEP]\nsplit_ifs with hik\n[GOAL]\ncase pos\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nhi : \u2191k \u2264 \u2191i\nhik : i = k\n\u22a2 0 = 0\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nhi : \u2191k \u2264 \u2191i\nhik : \u00aci = k\n\u22a2 c i = 0\n[PROOFSTEP]\nexact hc _ (Fin.succ_lt_succ_iff.mpr (lt_of_le_of_ne hi (Ne.symm hik)))\n[GOAL]\ncase refine'_2._h0\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\n\u22a2 \u2200 (j : Fin (Nat.succ n)), M' 0 j = N 0 j\n[PROOFSTEP]\nrwa [hM', updateRow_ne (Fin.succ_ne_zero _).symm]\n[GOAL]\ncase refine'_2._hsucc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\n\u22a2 \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M' (Fin.succ i) j = N (Fin.succ i) j + update c k 0 i * M' (Fin.castSucc i) j\n[PROOFSTEP]\nintro i j\n[GOAL]\ncase refine'_2._hsucc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\n\u22a2 M' (Fin.succ i) j = N (Fin.succ i) j + update c k 0 i * M' (Fin.castSucc i) j\n[PROOFSTEP]\nrw [Function.update_apply]\n[GOAL]\ncase refine'_2._hsucc\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\n\u22a2 M' (Fin.succ i) j = N (Fin.succ i) j + (if i = k then 0 else c i) * M' (Fin.castSucc i) j\n[PROOFSTEP]\nsplit_ifs with hik\n[GOAL]\ncase pos\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : i = k\n\u22a2 M' (Fin.succ i) j = N (Fin.succ i) j + 0 * M' (Fin.castSucc i) j\n[PROOFSTEP]\nrw [zero_mul, add_zero, hM', hik, updateRow_self]\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\n\u22a2 M' (Fin.succ i) j = N (Fin.succ i) j + c i * M' (Fin.castSucc i) j\n[PROOFSTEP]\nrw [hM', updateRow_ne ((Fin.succ_injective _).ne hik), hsucc]\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\n\u22a2 N (Fin.succ i) j + c i * M (Fin.castSucc i) j =\n    N (Fin.succ i) j + c i * updateRow M (Fin.succ k) (N (Fin.succ k)) (Fin.castSucc i) j\n[PROOFSTEP]\nby_cases hik2 : k < i\n[GOAL]\ncase pos\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\nhik2 : k < i\n\u22a2 N (Fin.succ i) j + c i * M (Fin.castSucc i) j =\n    N (Fin.succ i) j + c i * updateRow M (Fin.succ k) (N (Fin.succ k)) (Fin.castSucc i) j\n[PROOFSTEP]\nsimp [hc i (Fin.succ_lt_succ_iff.mpr hik2)]\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\nhik2 : \u00ack < i\n\u22a2 N (Fin.succ i) j + c i * M (Fin.castSucc i) j =\n    N (Fin.succ i) j + c i * updateRow M (Fin.succ k) (N (Fin.succ k)) (Fin.castSucc i) j\n[PROOFSTEP]\nrw [updateRow_ne]\n[GOAL]\ncase neg\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\nhik2 : \u00ack < i\n\u22a2 Fin.castSucc i \u2260 Fin.succ k\n[PROOFSTEP]\napply ne_of_lt\n[GOAL]\ncase neg.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nk\u271d : Fin (n + 1)\nk : Fin n\nih :\n  \u2200 (c : Fin n \u2192 R),\n    (\u2200 (i : Fin n), Fin.castSucc k < Fin.succ i \u2192 c i = 0) \u2192\n      \u2200 {M N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R},\n        (\u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j) \u2192\n          (\u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j) \u2192\n            det M = det N\nc : Fin n \u2192 R\nhc : \u2200 (i : Fin n), Fin.succ k < Fin.succ i \u2192 c i = 0\nM N : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nh0 : \u2200 (j : Fin (Nat.succ n)), M 0 j = N 0 j\nhsucc : \u2200 (i : Fin n) (j : Fin (Nat.succ n)), M (Fin.succ i) j = N (Fin.succ i) j + c i * M (Fin.castSucc i) j\nM' : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R := updateRow M (Fin.succ k) (N (Fin.succ k))\nhM' : M' = updateRow M (Fin.succ k) (N (Fin.succ k))\nhM : M = updateRow M' (Fin.succ k) (M' (Fin.succ k) + c k \u2022 M (Fin.castSucc k))\nk_ne_succ : Fin.castSucc k \u2260 Fin.succ k\nM_k : M (Fin.castSucc k) = M' (Fin.castSucc k)\ni : Fin n\nj : Fin (Nat.succ n)\nhik : \u00aci = k\nhik2 : \u00ack < i\n\u22a2 Fin.castSucc i < Fin.succ k\n[PROOFSTEP]\nrwa [Fin.lt_iff_val_lt_val, Fin.coe_castSucc, Fin.val_succ, Nat.lt_succ_iff, \u2190 not_lt]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA B : Matrix (Fin (n + 1)) (Fin (n + 1)) R\nc : Fin n \u2192 R\nA_zero : \u2200 (i : Fin (n + 1)), A i 0 = B i 0\nA_succ : \u2200 (i : Fin (n + 1)) (j : Fin n), A i (Fin.succ j) = B i (Fin.succ j) + c j * A i (Fin.castSucc j)\n\u22a2 det A = det B\n[PROOFSTEP]\nrw [\u2190 det_transpose A, \u2190 det_transpose B]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA B : Matrix (Fin (n + 1)) (Fin (n + 1)) R\nc : Fin n \u2192 R\nA_zero : \u2200 (i : Fin (n + 1)), A i 0 = B i 0\nA_succ : \u2200 (i : Fin (n + 1)) (j : Fin n), A i (Fin.succ j) = B i (Fin.succ j) + c j * A i (Fin.castSucc j)\n\u22a2 det A\u1d40 = det B\u1d40\n[PROOFSTEP]\nexact det_eq_of_forall_row_eq_smul_add_pred c A_zero fun i j => A_succ j i\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\n\u22a2 det (blockDiagonal M) = \u220f k : o, det (M k)\n[PROOFSTEP]\nsimp_rw [det_apply']\n  -- The right hand side is a product of sums, rewrite it as a sum of products.\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\n\u22a2 \u2211 x : Perm (n \u00d7 o), \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u220f x : o, \u2211 x_1 : Perm n, \u2191\u2191(\u2191sign x_1) * \u220f x_2 : n, M x (\u2191x_1 x_2) x_2\n[PROOFSTEP]\nrw [Finset.prod_sum]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\n\u22a2 \u2211 x : Perm (n \u00d7 o), \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u2211 p in pi univ fun x => univ,\n      \u220f x in attach univ, \u2191\u2191(\u2191sign (p \u2191x (_ : \u2191x \u2208 univ))) * \u220f x_1 : n, M (\u2191x) (\u2191(p \u2191x (_ : \u2191x \u2208 univ)) x_1) x_1\n[PROOFSTEP]\nsimp_rw [Finset.prod_attach_univ, Finset.univ_pi_univ]\n  -- We claim that the only permutations contributing to the sum are those that\n    -- preserve their second component.\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\n\u22a2 \u2211 x : Perm (n \u00d7 o), \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u2211 x : (a : o) \u2192 a \u2208 univ \u2192 Perm n,\n      \u220f x_1 : o,\n        \u2191\u2191(\u2191sign (x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ))) *\n          \u220f x_2 : n, M x_1 (\u2191(x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ)) x_2) x_2\n[PROOFSTEP]\nlet preserving_snd : Finset (Equiv.Perm (n \u00d7 o)) := Finset.univ.filter fun \u03c3 => \u2200 x, (\u03c3 x).snd = x.snd\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\n\u22a2 \u2211 x : Perm (n \u00d7 o), \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u2211 x : (a : o) \u2192 a \u2208 univ \u2192 Perm n,\n      \u220f x_1 : o,\n        \u2191\u2191(\u2191sign (x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ))) *\n          \u220f x_2 : n, M x_1 (\u2191(x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ)) x_2) x_2\n[PROOFSTEP]\nhave mem_preserving_snd : \u2200 {\u03c3 : Equiv.Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 x, (\u03c3 x).snd = x.snd := fun {\u03c3} =>\n  Finset.mem_filter.trans \u27e8fun h => h.2, fun h => \u27e8Finset.mem_univ _, h\u27e9\u27e9\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2211 x : Perm (n \u00d7 o), \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u2211 x : (a : o) \u2192 a \u2208 univ \u2192 Perm n,\n      \u220f x_1 : o,\n        \u2191\u2191(\u2191sign (x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ))) *\n          \u220f x_2 : n, M x_1 (\u2191(x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ)) x_2) x_2\n[PROOFSTEP]\nrw [\u2190 Finset.sum_subset (Finset.subset_univ preserving_snd) _]\n  -- And that these are in bijection with `o \u2192 Equiv.Perm m`.\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2211 x in preserving_snd, \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 =\n    \u2211 x : (a : o) \u2192 a \u2208 univ \u2192 Perm n,\n      \u220f x_1 : o,\n        \u2191\u2191(\u2191sign (x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ))) *\n          \u220f x_2 : n, M x_1 (\u2191(x x_1 (_ : \u2191{ val := x_1, property := (_ : x_1 \u2208 univ) } \u2208 univ)) x_2) x_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (x : Perm (n \u00d7 o)), x \u2208 univ \u2192 \u00acx \u2208 preserving_snd \u2192 \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nrw [(Finset.sum_bij\n      (fun (\u03c3 : \u2200 k : o, k \u2208 Finset.univ \u2192 Equiv.Perm n) _ => prodCongrLeft fun k => \u03c3 k (Finset.mem_univ k)) _ _ _\n      _).symm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (a : (k : o) \u2192 k \u2208 univ \u2192 Perm n) (ha : a \u2208 univ),\n    (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha \u2208 preserving_snd\n[PROOFSTEP]\nintro \u03c3 _\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u271d : \u03c3 \u2208 univ\n\u22a2 (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d \u2208 preserving_snd\n[PROOFSTEP]\nrw [mem_preserving_snd]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u271d : \u03c3 \u2208 univ\n\u22a2 \u2200 (x : n \u00d7 o), (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d) x).snd = x.snd\n[PROOFSTEP]\nrintro \u27e8-, x\u27e9\n[GOAL]\ncase mk\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u271d : \u03c3 \u2208 univ\nfst\u271d : n\nx : o\n\u22a2 (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d) (fst\u271d, x)).snd = (fst\u271d, x).snd\n[PROOFSTEP]\nsimp only [prodCongrLeft_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (a : (k : o) \u2192 k \u2208 univ \u2192 Perm n) (ha : a \u2208 univ),\n    \u220f x : o,\n        \u2191\u2191(\u2191sign (a x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ))) *\n          \u220f x_1 : n, M x (\u2191(a x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ)) x_1) x_1 =\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha)) *\n        \u220f x : n \u00d7 o, blockDiagonal M (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha) x) x\n[PROOFSTEP]\nintro \u03c3 _\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u271d : \u03c3 \u2208 univ\n\u22a2 \u220f x : o,\n      \u2191\u2191(\u2191sign (\u03c3 x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ))) *\n        \u220f x_1 : n, M x (\u2191(\u03c3 x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ)) x_1) x_1 =\n    \u2191\u2191(\u2191sign ((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d)) *\n      \u220f x : n \u00d7 o, blockDiagonal M (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d) x) x\n[PROOFSTEP]\nrw [Finset.prod_mul_distrib, \u2190 Finset.univ_product_univ, Finset.prod_product_right]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u271d : \u03c3 \u2208 univ\n\u22a2 (\u220f x : o, \u2191\u2191(\u2191sign (\u03c3 x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ)))) *\n      \u220f x : o, \u220f x_1 : n, M x (\u2191(\u03c3 x (_ : \u2191{ val := x, property := (_ : x \u2208 univ) } \u2208 univ)) x_1) x_1 =\n    \u2191\u2191(\u2191sign ((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d)) *\n      \u220f y : o, \u220f x : n, blockDiagonal M (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u271d) (x, y)) (x, y)\n[PROOFSTEP]\nsimp only [sign_prodCongrLeft, Units.coe_prod, Int.cast_prod, blockDiagonal_apply_eq, prodCongrLeft_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (a\u2081 a\u2082 : (k : o) \u2192 k \u2208 univ \u2192 Perm n) (ha\u2081 : a\u2081 \u2208 univ) (ha\u2082 : a\u2082 \u2208 univ),\n    (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a\u2081 ha\u2081 =\n        (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a\u2082 ha\u2082 \u2192\n      a\u2081 = a\u2082\n[PROOFSTEP]\nintro \u03c3 \u03c3' _ _ eq\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq :\n  (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u2081\u271d =\n    (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3' ha\u2082\u271d\n\u22a2 \u03c3 = \u03c3'\n[PROOFSTEP]\next x hx k\n[GOAL]\ncase h.h.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq :\n  (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3 ha\u2081\u271d =\n    (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) \u03c3' ha\u2082\u271d\nx : o\nhx : x \u2208 univ\nk : n\n\u22a2 \u2191(\u03c3 x hx) k = \u2191(\u03c3' x hx) k\n[PROOFSTEP]\nsimp only at eq \n[GOAL]\ncase h.h.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq : (prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) = prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)\nx : o\nhx : x \u2208 univ\nk : n\n\u22a2 \u2191(\u03c3 x hx) k = \u2191(\u03c3' x hx) k\n[PROOFSTEP]\nhave :\n  \u2200 k x,\n    prodCongrLeft (fun k => \u03c3 k (Finset.mem_univ _)) (k, x) =\n      prodCongrLeft (fun k => \u03c3' k (Finset.mem_univ _)) (k, x) :=\n  fun k x => by rw [eq]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq : (prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) = prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)\nx\u271d : o\nhx : x\u271d \u2208 univ\nk\u271d k : n\nx : o\n\u22a2 \u2191(prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) (k, x) = \u2191(prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)) (k, x)\n[PROOFSTEP]\nrw [eq]\n[GOAL]\ncase h.h.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq : (prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) = prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)\nx : o\nhx : x \u2208 univ\nk : n\nthis :\n  \u2200 (k : n) (x : o),\n    \u2191(prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) (k, x) = \u2191(prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)) (k, x)\n\u22a2 \u2191(\u03c3 x hx) k = \u2191(\u03c3' x hx) k\n[PROOFSTEP]\nsimp only [prodCongrLeft_apply, Prod.mk.inj_iff] at this \n[GOAL]\ncase h.h.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 \u03c3' : (k : o) \u2192 k \u2208 univ \u2192 Perm n\nha\u2081\u271d : \u03c3 \u2208 univ\nha\u2082\u271d : \u03c3' \u2208 univ\neq : (prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) = prodCongrLeft fun k => \u03c3' k (_ : k \u2208 univ)\nx : o\nhx : x \u2208 univ\nk : n\nthis : \u2200 (k : n) (x : o), \u2191(\u03c3 x (_ : x \u2208 univ)) k = \u2191(\u03c3' x (_ : x \u2208 univ)) k \u2227 True\n\u22a2 \u2191(\u03c3 x hx) k = \u2191(\u03c3' x hx) k\n[PROOFSTEP]\nexact (this k x).1\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (b : n \u00d7 o \u2243 n \u00d7 o), b \u2208 preserving_snd \u2192 \u2203 a ha, b = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nintro \u03c3 h\u03c3\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u03c3 \u2208 preserving_snd\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nrw [mem_preserving_snd] at h\u03c3 \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nhave h\u03c3' : \u2200 x, (\u03c3\u207b\u00b9 x).snd = x.snd := by\n  intro x\n  conv_rhs => rw [\u2190 Perm.apply_inv_self \u03c3 x, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\n[PROOFSTEP]\nintro x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nx : n \u00d7 o\n\u22a2 (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Perm.apply_inv_self \u03c3 x, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nx : n \u00d7 o\n| x.snd\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 x, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nx : n \u00d7 o\n| x.snd\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 x, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nx : n \u00d7 o\n| x.snd\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 x, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nhave mk_apply_eq : \u2200 k x, ((\u03c3 (x, k)).fst, k) = \u03c3 (x, k) :=\n  by\n  intro k x\n  ext\n  \u00b7 simp only\n  \u00b7 simp only [h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\n\u22a2 \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\n[PROOFSTEP]\nintro k x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nk : o\nx : n\n\u22a2 ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nk : o\nx : n\n\u22a2 ((\u2191\u03c3 (x, k)).fst, k).fst = (\u2191\u03c3 (x, k)).fst\n[PROOFSTEP]\nsimp only\n[GOAL]\ncase h\u2082\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nk : o\nx : n\n\u22a2 ((\u2191\u03c3 (x, k)).fst, k).snd = (\u2191\u03c3 (x, k)).snd\n[PROOFSTEP]\nsimp only [h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nhave mk_inv_apply_eq : \u2200 k x, ((\u03c3\u207b\u00b9 (x, k)).fst, k) = \u03c3\u207b\u00b9 (x, k) :=\n  by\n  intro k x\n  conv_lhs => rw [\u2190 Perm.apply_inv_self \u03c3 (x, k)]\n  ext\n  \u00b7 simp only [apply_inv_self]\n  \u00b7 simp only [h\u03c3']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\n\u22a2 \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n[PROOFSTEP]\nintro k x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n\u22a2 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n[PROOFSTEP]\nconv_lhs => rw [\u2190 Perm.apply_inv_self \u03c3 (x, k)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n| ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 (x, k)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n| ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 (x, k)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n| ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)\n[PROOFSTEP]\nrw [\u2190 Perm.apply_inv_self \u03c3 (x, k)]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n\u22a2 ((\u2191\u03c3\u207b\u00b9 (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 (x, k)))).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n\u22a2 ((\u2191\u03c3\u207b\u00b9 (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 (x, k)))).fst, k).fst = (\u2191\u03c3\u207b\u00b9 (x, k)).fst\n[PROOFSTEP]\nsimp only [apply_inv_self]\n[GOAL]\ncase h\u2082\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nk : o\nx : n\n\u22a2 ((\u2191\u03c3\u207b\u00b9 (\u2191\u03c3 (\u2191\u03c3\u207b\u00b9 (x, k)))).fst, k).snd = (\u2191\u03c3\u207b\u00b9 (x, k)).snd\n[PROOFSTEP]\nsimp only [h\u03c3']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ)) a ha\n[PROOFSTEP]\nrefine' \u27e8fun k _ => \u27e8fun x => (\u03c3 (x, k)).fst, fun x => (\u03c3\u207b\u00b9 (x, k)).fst, _, _\u27e9, _, _\u27e9\n[GOAL]\ncase refine'_1\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : o\nx\u271d : k \u2208 univ\n\u22a2 LeftInverse (fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst) fun x => (\u2191\u03c3 (x, k)).fst\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_1\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : o\nx\u271d : k \u2208 univ\nx : n\n\u22a2 (fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst) ((fun x => (\u2191\u03c3 (x, k)).fst) x) = x\n[PROOFSTEP]\nsimp only [mk_apply_eq, inv_apply_self]\n[GOAL]\ncase refine'_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : o\nx\u271d : k \u2208 univ\n\u22a2 Function.RightInverse (fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst) fun x => (\u2191\u03c3 (x, k)).fst\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : o\nx\u271d : k \u2208 univ\nx : n\n\u22a2 (fun x => (\u2191\u03c3 (x, k)).fst) ((fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst) x) = x\n[PROOFSTEP]\nsimp only [mk_inv_apply_eq, apply_inv_self]\n[GOAL]\ncase refine'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n\u22a2 (fun k x =>\n      { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n        left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n        right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) }) \u2208\n    univ\n[PROOFSTEP]\napply Finset.mem_univ\n[GOAL]\ncase refine'_4\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\n\u22a2 \u03c3 =\n    (fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ))\n      (fun k x =>\n        { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n          left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n          right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) })\n      (_ :\n        (fun k x =>\n            { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n              left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n              right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) }) \u2208\n          univ)\n[PROOFSTEP]\next \u27e8k, x\u27e9\n[GOAL]\ncase refine'_4.H.mk.h\u2081\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : n\nx : o\n\u22a2 (\u2191\u03c3 (k, x)).fst =\n    (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ))\n            (fun k x =>\n              { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n                left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n                right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) })\n            (_ :\n              (fun k x =>\n                  { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n                    left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n                    right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) }) \u2208\n                univ))\n        (k, x)).fst\n[PROOFSTEP]\nsimp only [coe_fn_mk, prodCongrLeft_apply]\n[GOAL]\ncase refine'_4.H.mk.h\u2082\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : n \u00d7 o \u2243 n \u00d7 o\nh\u03c3 : \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nh\u03c3' : \u2200 (x : n \u00d7 o), (\u2191\u03c3\u207b\u00b9 x).snd = x.snd\nmk_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3 (x, k)).fst, k) = \u2191\u03c3 (x, k)\nmk_inv_apply_eq : \u2200 (k : o) (x : n), ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k) = \u2191\u03c3\u207b\u00b9 (x, k)\nk : n\nx : o\n\u22a2 (\u2191\u03c3 (k, x)).snd =\n    (\u2191((fun \u03c3 x => prodCongrLeft fun k => \u03c3 k (_ : k \u2208 univ))\n            (fun k x =>\n              { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n                left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n                right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) })\n            (_ :\n              (fun k x =>\n                  { toFun := fun x => (\u2191\u03c3 (x, k)).fst, invFun := fun x => (\u2191\u03c3\u207b\u00b9 (x, k)).fst,\n                    left_inv := (_ : \u2200 (x : n), (\u2191\u03c3\u207b\u00b9 ((\u2191\u03c3 (x, k)).fst, k)).fst = x),\n                    right_inv := (_ : \u2200 (x : n), (\u2191\u03c3 ((\u2191\u03c3\u207b\u00b9 (x, k)).fst, k)).fst = x) }) \u2208\n                univ))\n        (k, x)).snd\n[PROOFSTEP]\nsimp only [prodCongrLeft_apply, h\u03c3]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2200 (x : Perm (n \u00d7 o)), x \u2208 univ \u2192 \u00acx \u2208 preserving_snd \u2192 \u2191\u2191(\u2191sign x) * \u220f x_1 : n \u00d7 o, blockDiagonal M (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nintro \u03c3 _ h\u03c3\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : Perm (n \u00d7 o)\na\u271d : \u03c3 \u2208 univ\nh\u03c3 : \u00ac\u03c3 \u2208 preserving_snd\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : n \u00d7 o, blockDiagonal M (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nrw [mem_preserving_snd] at h\u03c3 \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : Perm (n \u00d7 o)\na\u271d : \u03c3 \u2208 univ\nh\u03c3 : \u00ac\u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : n \u00d7 o, blockDiagonal M (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nobtain \u27e8\u27e8k, x\u27e9, hkx\u27e9 := not_forall.mp h\u03c3\n[GOAL]\ncase intro.mk\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : Perm (n \u00d7 o)\na\u271d : \u03c3 \u2208 univ\nh\u03c3 : \u00ac\u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nk : n\nx : o\nhkx : \u00ac(\u2191\u03c3 (k, x)).snd = (k, x).snd\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : n \u00d7 o, blockDiagonal M (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nrw [Finset.prod_eq_zero (Finset.mem_univ (k, x)), mul_zero]\n[GOAL]\ncase intro.mk\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : Perm (n \u00d7 o)\na\u271d : \u03c3 \u2208 univ\nh\u03c3 : \u00ac\u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nk : n\nx : o\nhkx : \u00ac(\u2191\u03c3 (k, x)).snd = (k, x).snd\n\u22a2 blockDiagonal M (\u2191\u03c3 (k, x)) (k, x) = 0\n[PROOFSTEP]\nrw [\u2190 @Prod.mk.eta _ _ (\u03c3 (k, x)), blockDiagonal_apply_ne]\n[GOAL]\ncase intro.mk.h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2076 : DecidableEq n\ninst\u271d\u2075 : Fintype n\ninst\u271d\u2074 : DecidableEq m\ninst\u271d\u00b3 : Fintype m\nR : Type v\ninst\u271d\u00b2 : CommRing R\no : Type u_3\ninst\u271d\u00b9 : Fintype o\ninst\u271d : DecidableEq o\nM : o \u2192 Matrix n n R\npreserving_snd : Finset (Perm (n \u00d7 o)) := filter (fun \u03c3 => \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd) univ\nmem_preserving_snd : \u2200 {\u03c3 : Perm (n \u00d7 o)}, \u03c3 \u2208 preserving_snd \u2194 \u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\n\u03c3 : Perm (n \u00d7 o)\na\u271d : \u03c3 \u2208 univ\nh\u03c3 : \u00ac\u2200 (x : n \u00d7 o), (\u2191\u03c3 x).snd = x.snd\nk : n\nx : o\nhkx : \u00ac(\u2191\u03c3 (k, x)).snd = (k, x).snd\n\u22a2 (\u2191\u03c3 (k, x)).snd \u2260 x\n[PROOFSTEP]\nexact hkx\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 det (fromBlocks A B 0 D) = det A * det D\n[PROOFSTEP]\nclassical\nsimp_rw [det_apply']\nconvert Eq.symm <| sum_subset (\u03b2 := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_\nrw [sum_mul_sum]\nsimp_rw [univ_product_univ]\nrw [(sum_bij (fun (\u03c3 : Perm m \u00d7 Perm n) _ => Equiv.sumCongr \u03c3.fst \u03c3.snd) _ _ _ _).symm]\n\u00b7 intro \u03c3\u2081\u2082 h\n  simp only\n  erw [Set.mem_toFinset, MonoidHom.mem_range]\n  use \u03c3\u2081\u2082\n  simp only [sumCongrHom_apply]\n\u00b7 simp only [forall_prop_of_true, Prod.forall, mem_univ]\n  intro \u03c3\u2081 \u03c3\u2082\n  rw [Fintype.prod_sum_type]\n  simp_rw [Equiv.sumCongr_apply, Sum.map_inr, Sum.map_inl, fromBlocks_apply\u2081\u2081, fromBlocks_apply\u2082\u2082]\n  rw [mul_mul_mul_comm]\n  congr\n  rw [sign_sumCongr, Units.val_mul, Int.cast_mul]\n\u00b7 intro \u03c3\u2081 \u03c3\u2082 h\u2081 h\u2082\n  dsimp only\n  intro h\n  have h2 : \u2200 x, Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd x = Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd x :=\n    by\n    intro x\n    exact congr_fun (congr_arg toFun h) x\n  simp only [Sum.map_inr, Sum.map_inl, Perm.sumCongr_apply, Sum.forall, Sum.inl.injEq, Sum.inr.injEq] at h2 \n  ext x\n  \u00b7 exact h2.left x\n  \u00b7 exact h2.right x\n\u00b7 intro \u03c3 h\u03c3\n  erw [Set.mem_toFinset, MonoidHom.mem_range] at h\u03c3 \n  obtain \u27e8\u03c3\u2081\u2082, h\u03c3\u2081\u2082\u27e9 := h\u03c3\n  use \u03c3\u2081\u2082\n  rw [\u2190 h\u03c3\u2081\u2082]\n  simp\n\u00b7 rintro \u03c3 - h\u03c3n\n  have h1 : \u00ac\u2200 x, \u2203 y, Sum.inl y = \u03c3 (Sum.inl x) :=\n    by\n    rw [Set.mem_toFinset] at h\u03c3n \n    simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] using\n      mt mem_sumCongrHom_range_of_perm_mapsTo_inl h\u03c3n\n  obtain \u27e8a, ha\u27e9 := not_forall.mp h1\n  cases' hx : \u03c3 (Sum.inl a) with a2 b\n  \u00b7 have hn := (not_exists.mp ha) a2\n    exact absurd hx.symm hn\n  \u00b7 rw [Finset.prod_eq_zero (Finset.mem_univ (Sum.inl a)), mul_zero]\n    rw [hx, fromBlocks_apply\u2082\u2081, zero_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 det (fromBlocks A B 0 D) = det A * det D\n[PROOFSTEP]\nsimp_rw [det_apply']\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 x : Perm (m \u2295 n), \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 =\n    (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1\n[PROOFSTEP]\nconvert Eq.symm <| sum_subset (\u03b2 := R) (subset_univ ((sumCongrHom m n).range : Set (Perm (Sum m n))).toFinset) ?_\n[GOAL]\ncase h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 (\u2211 x : Perm m, \u2191\u2191(\u2191sign x) * \u220f x_1 : m, A (\u2191x x_1) x_1) * \u2211 x : Perm n, \u2191\u2191(\u2191sign x) * \u220f x_1 : n, D (\u2191x x_1) x_1 =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nrw [sum_mul_sum]\n[GOAL]\ncase h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p in univ \u00d7\u02e2 univ, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nsimp_rw [univ_product_univ]\n[GOAL]\ncase h.e'_3\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2211 p : Perm m \u00d7 Perm n, (\u2191\u2191(\u2191sign p.fst) * \u220f x : m, A (\u2191p.fst x) x) * (\u2191\u2191(\u2191sign p.snd) * \u220f x : n, D (\u2191p.snd x) x) =\n    \u2211 x in Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)),\n      \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nrw [(sum_bij (fun (\u03c3 : Perm m \u00d7 Perm n) _ => Equiv.sumCongr \u03c3.fst \u03c3.snd) _ _ _ _).symm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n[PROOFSTEP]\nintro \u03c3\u2081\u2082 h\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 h \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n[PROOFSTEP]\nsimp only\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n[PROOFSTEP]\nerw [Set.mem_toFinset, MonoidHom.mem_range]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2203 x, \u2191(sumCongrHom m n) x = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd\n[PROOFSTEP]\nuse \u03c3\u2081\u2082\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh : \u03c3\u2081\u2082 \u2208 univ\n\u22a2 \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = Equiv.sumCongr \u03c3\u2081\u2082.fst \u03c3\u2081\u2082.snd\n[PROOFSTEP]\nsimp only [sumCongrHom_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m \u00d7 Perm n) (ha : a \u2208 univ),\n    (\u2191\u2191(\u2191sign a.fst) * \u220f x : m, A (\u2191a.fst x) x) * (\u2191\u2191(\u2191sign a.snd) * \u220f x : n, D (\u2191a.snd x) x) =\n      \u2191\u2191(\u2191sign ((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha)) *\n        \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191((fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha) x) x\n[PROOFSTEP]\nsimp only [forall_prop_of_true, Prod.forall, mem_univ]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a : Perm m) (b : Perm n),\n    (\u2191\u2191(\u2191sign a) * \u220f x : m, A (\u2191a x) x) * (\u2191\u2191(\u2191sign b) * \u220f x : n, D (\u2191b x) x) =\n      \u2191\u2191(\u2191sign (Equiv.sumCongr a b)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr a b) x) x\n[PROOFSTEP]\nintro \u03c3\u2081 \u03c3\u2082\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) x) x\n[PROOFSTEP]\nrw [Fintype.prod_sum_type]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) *\n      ((\u220f a\u2081 : m, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inl a\u2081)) (Sum.inl a\u2081)) *\n        \u220f a\u2082 : n, fromBlocks A B 0 D (\u2191(Equiv.sumCongr \u03c3\u2081 \u03c3\u2082) (Sum.inr a\u2082)) (Sum.inr a\u2082))\n[PROOFSTEP]\nsimp_rw [Equiv.sumCongr_apply, Sum.map_inr, Sum.map_inl, fromBlocks_apply\u2081\u2081, fromBlocks_apply\u2082\u2082]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 (\u2191\u2191(\u2191sign \u03c3\u2081) * \u220f x : m, A (\u2191\u03c3\u2081 x) x) * (\u2191\u2191(\u2191sign \u03c3\u2082) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)\n[PROOFSTEP]\nrw [mul_mul_mul_comm]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x) =\n    \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082)) * ((\u220f x : m, A (\u2191\u03c3\u2081 x) x) * \u220f x : n, D (\u2191\u03c3\u2082 x) x)\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 : Perm m\n\u03c3\u2082 : Perm n\n\u22a2 \u2191\u2191(\u2191sign \u03c3\u2081) * \u2191\u2191(\u2191sign \u03c3\u2082) = \u2191\u2191(\u2191sign (Equiv.sumCongr \u03c3\u2081 \u03c3\u2082))\n[PROOFSTEP]\nrw [sign_sumCongr, Units.val_mul, Int.cast_mul]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (a\u2081 a\u2082 : Perm m \u00d7 Perm n) (ha\u2081 : a\u2081 \u2208 univ) (ha\u2082 : a\u2082 \u2208 univ),\n    (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2081 ha\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a\u2082 ha\u2082 \u2192 a\u2081 = a\u2082\n[PROOFSTEP]\nintro \u03c3\u2081 \u03c3\u2082 h\u2081 h\u2082\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081 h\u2081 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2082 h\u2082 \u2192 \u03c3\u2081 = \u03c3\u2082\n[PROOFSTEP]\ndsimp only\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\n\u22a2 Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd \u2192 \u03c3\u2081 = \u03c3\u2082\n[PROOFSTEP]\nintro h\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\n\u22a2 \u03c3\u2081 = \u03c3\u2082\n[PROOFSTEP]\nhave h2 : \u2200 x, Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd x = Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd x :=\n  by\n  intro x\n  exact congr_fun (congr_arg toFun h) x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\n\u22a2 \u2200 (x : m \u2295 n), \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x\n[PROOFSTEP]\nintro x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nx : m \u2295 n\n\u22a2 \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x\n[PROOFSTEP]\nexact congr_fun (congr_arg toFun h) x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : \u2200 (x : m \u2295 n), \u2191(Perm.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd) x = \u2191(Perm.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd) x\n\u22a2 \u03c3\u2081 = \u03c3\u2082\n[PROOFSTEP]\nsimp only [Sum.map_inr, Sum.map_inl, Perm.sumCongr_apply, Sum.forall, Sum.inl.injEq, Sum.inr.injEq] at h2 \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\n\u22a2 \u03c3\u2081 = \u03c3\u2082\n[PROOFSTEP]\next x\n[GOAL]\ncase h\u2081.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : m\n\u22a2 \u2191\u03c3\u2081.fst x = \u2191\u03c3\u2082.fst x\n[PROOFSTEP]\nexact h2.left x\n[GOAL]\ncase h\u2082.H\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3\u2081 \u03c3\u2082 : Perm m \u00d7 Perm n\nh\u2081 : \u03c3\u2081 \u2208 univ\nh\u2082 : \u03c3\u2082 \u2208 univ\nh : Equiv.sumCongr \u03c3\u2081.fst \u03c3\u2081.snd = Equiv.sumCongr \u03c3\u2082.fst \u03c3\u2082.snd\nh2 : (\u2200 (a : m), \u2191\u03c3\u2081.fst a = \u2191\u03c3\u2082.fst a) \u2227 \u2200 (b : n), \u2191\u03c3\u2081.snd b = \u2191\u03c3\u2082.snd b\nx : n\n\u22a2 \u2191\u03c3\u2081.snd x = \u2191\u03c3\u2082.snd x\n[PROOFSTEP]\nexact h2.right x\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (b : m \u2295 n \u2243 m \u2295 n),\n    b \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192 \u2203 a ha, b = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha\n[PROOFSTEP]\nintro \u03c3 h\u03c3\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha\n[PROOFSTEP]\nerw [Set.mem_toFinset, MonoidHom.mem_range] at h\u03c3 \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\nh\u03c3 : \u2203 x, \u2191(sumCongrHom m n) x = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha\n[PROOFSTEP]\nobtain \u27e8\u03c3\u2081\u2082, h\u03c3\u2081\u2082\u27e9 := h\u03c3\n[GOAL]\ncase intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 a ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) a ha\n[PROOFSTEP]\nuse \u03c3\u2081\u2082\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u03c3 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha\n[PROOFSTEP]\nrw [\u2190 h\u03c3\u2081\u2082]\n[GOAL]\ncase h\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : m \u2295 n \u2243 m \u2295 n\n\u03c3\u2081\u2082 : Perm m \u00d7 Perm n\nh\u03c3\u2081\u2082 : \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = \u03c3\n\u22a2 \u2203 ha, \u2191(sumCongrHom m n) \u03c3\u2081\u2082 = (fun \u03c3 x => Equiv.sumCongr \u03c3.fst \u03c3.snd) \u03c3\u2081\u2082 ha\n[PROOFSTEP]\nsimp\n[GOAL]\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u22a2 \u2200 (x : Perm (m \u2295 n)),\n    x \u2208 univ \u2192\n      \u00acx \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n)) \u2192\n        \u2191\u2191(\u2191sign x) * \u220f x_1 : m \u2295 n, fromBlocks A B 0 D (\u2191x x_1) x_1 = 0\n[PROOFSTEP]\nrintro \u03c3 - h\u03c3n\n[GOAL]\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nhave h1 : \u00ac\u2200 x, \u2203 y, Sum.inl y = \u03c3 (Sum.inl x) :=\n  by\n  rw [Set.mem_toFinset] at h\u03c3n \n  simpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] using\n    mt mem_sumCongrHom_range_of_perm_mapsTo_inl h\u03c3n\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\n[PROOFSTEP]\nrw [Set.mem_toFinset] at h\u03c3n \n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 \u2191(MonoidHom.range (sumCongrHom m n))\n\u22a2 \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\n[PROOFSTEP]\nsimpa only [Set.MapsTo, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] using\n  mt mem_sumCongrHom_range_of_perm_mapsTo_inl h\u03c3n\n[GOAL]\ncase convert_2\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := not_forall.mp h1\n[GOAL]\ncase convert_2.intro\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\ncases' hx : \u03c3 (Sum.inl a) with a2 b\n[GOAL]\ncase convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nhave hn := (not_exists.mp ha) a2\n[GOAL]\ncase convert_2.intro.inl\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\na2 : m\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inl a2\nhn : \u00acSum.inl a2 = \u2191\u03c3 (Sum.inl a)\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nexact absurd hx.symm hn\n[GOAL]\ncase convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 \u2191\u2191(\u2191sign \u03c3) * \u220f x : m \u2295 n, fromBlocks A B 0 D (\u2191\u03c3 x) x = 0\n[PROOFSTEP]\nrw [Finset.prod_eq_zero (Finset.mem_univ (Sum.inl a)), mul_zero]\n[GOAL]\ncase convert_2.intro.inr\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nB : Matrix m n R\nD : Matrix n n R\n\u03c3 : Perm (m \u2295 n)\nh\u03c3n : \u00ac\u03c3 \u2208 Set.toFinset \u2191(MonoidHom.range (sumCongrHom m n))\nh1 : \u00ac\u2200 (x : m), \u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl x)\na : m\nha : \u00ac\u2203 y, Sum.inl y = \u2191\u03c3 (Sum.inl a)\nb : n\nhx : \u2191\u03c3 (Sum.inl a) = Sum.inr b\n\u22a2 fromBlocks A B 0 D (\u2191\u03c3 (Sum.inl a)) (Sum.inl a) = 0\n[PROOFSTEP]\nrw [hx, fromBlocks_apply\u2082\u2081, zero_apply]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix m m R\nC : Matrix n m R\nD : Matrix n n R\n\u22a2 det (fromBlocks A 0 C D) = det A * det D\n[PROOFSTEP]\nrw [\u2190 det_transpose, fromBlocks_transpose, transpose_zero, det_fromBlocks_zero\u2082\u2081, det_transpose, det_transpose]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\n\u22a2 det A = \u2211 i : Fin (Nat.succ n), (-1) ^ \u2191i * A i 0 * det (submatrix A (Fin.succAbove i) Fin.succ)\n[PROOFSTEP]\nrw [Matrix.det_apply, Finset.univ_perm_fin_succ, \u2190 Finset.univ_product_univ]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\n\u22a2 \u2211 \u03c3 in Finset.map (Equiv.toEmbedding decomposeFin.symm) (univ \u00d7\u02e2 univ), \u2191sign \u03c3 \u2022 \u220f i : Fin (Nat.succ n), A (\u2191\u03c3 i) i =\n    \u2211 i : Fin (Nat.succ n), (-1) ^ \u2191i * A i 0 * det (submatrix A (Fin.succAbove i) Fin.succ)\n[PROOFSTEP]\nsimp only [Finset.sum_map, Equiv.toEmbedding_apply, Finset.sum_product, Matrix.submatrix]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\n\u22a2 \u2211 x : Fin (Nat.succ n),\n      \u2211 y : Perm (Fin n),\n        \u2191sign (\u2191decomposeFin.symm (x, y)) \u2022 \u220f x_1 : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (x, y)) x_1) x_1 =\n    \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * A x 0 * det (\u2191of fun i j => A (Fin.succAbove x i) (Fin.succ j))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i _ => Fin.cases _ (fun i => _) i\n[GOAL]\ncase refine'_1\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nx\u271d : i \u2208 univ\n\u22a2 \u2211 y : Perm (Fin n), \u2191sign (\u2191decomposeFin.symm (0, y)) \u2022 \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (0, y)) x) x =\n    (-1) ^ \u21910 * A 0 0 * det (\u2191of fun i j => A (Fin.succAbove 0 i) (Fin.succ j))\n[PROOFSTEP]\nsimp only [Fin.prod_univ_succ, Matrix.det_apply, Finset.mul_sum, Equiv.Perm.decomposeFin_symm_apply_zero, Fin.val_zero,\n  one_mul, Equiv.Perm.decomposeFin.symm_sign, Equiv.swap_self, if_true, id.def, eq_self_iff_true,\n  Equiv.Perm.decomposeFin_symm_apply_succ, Fin.succAbove_zero, Equiv.coe_refl, pow_zero, mul_smul_comm, of_apply]\n  -- `univ_perm_fin_succ` gives a different embedding of `Perm (Fin n)` into\n    -- `Perm (Fin n.succ)` than the determinant of the submatrix we want,\n    -- permute `A` so that we get the correct one.\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d : i\u271d \u2208 univ\ni : Fin n\n\u22a2 \u2211 y : Perm (Fin n),\n      \u2191sign (\u2191decomposeFin.symm (Fin.succ i, y)) \u2022\n        \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, y)) x) x =\n    (-1) ^ \u2191(Fin.succ i) * A (Fin.succ i) 0 * det (\u2191of fun i_1 j => A (Fin.succAbove (Fin.succ i) i_1) (Fin.succ j))\n[PROOFSTEP]\nhave : (-1 : R) ^ (i : \u2115) = (Perm.sign i.cycleRange) := by simp [Fin.sign_cycleRange]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d : i\u271d \u2208 univ\ni : Fin n\n\u22a2 (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n[PROOFSTEP]\nsimp [Fin.sign_cycleRange]\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u22a2 \u2211 y : Perm (Fin n),\n      \u2191sign (\u2191decomposeFin.symm (Fin.succ i, y)) \u2022\n        \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, y)) x) x =\n    (-1) ^ \u2191(Fin.succ i) * A (Fin.succ i) 0 * det (\u2191of fun i_1 j => A (Fin.succAbove (Fin.succ i) i_1) (Fin.succ j))\n[PROOFSTEP]\nrw [Fin.val_succ, pow_succ, this, mul_assoc, mul_assoc, mul_left_comm (\u03b5 _), \u2190 det_permute, Matrix.det_apply,\n  Finset.mul_sum, Finset.mul_sum]\n  -- now we just need to move the corresponding parts to the same place\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u22a2 \u2211 y : Perm (Fin n),\n      \u2191sign (\u2191decomposeFin.symm (Fin.succ i, y)) \u2022\n        \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, y)) x) x =\n    \u2211 x : Perm (Fin n),\n      -1 *\n        (A (Fin.succ i) 0 *\n          \u2191sign x \u2022\n            \u220f i_1 : Fin n,\n              \u2191of (fun i_2 j => A (Fin.succAbove (Fin.succ i) i_2) (Fin.succ j)) (\u2191(Fin.cycleRange i) (\u2191x i_1)) i_1)\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun \u03c3 _ => _\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d\u00b9 : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u03c3 : Perm (Fin n)\nx\u271d : \u03c3 \u2208 univ\n\u22a2 \u2191sign (\u2191decomposeFin.symm (Fin.succ i, \u03c3)) \u2022 \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, \u03c3)) x) x =\n    -1 *\n      (A (Fin.succ i) 0 *\n        \u2191sign \u03c3 \u2022\n          \u220f i_1 : Fin n,\n            \u2191of (fun i_2 j => A (Fin.succAbove (Fin.succ i) i_2) (Fin.succ j)) (\u2191(Fin.cycleRange i) (\u2191\u03c3 i_1)) i_1)\n[PROOFSTEP]\nrw [Equiv.Perm.decomposeFin.symm_sign, if_neg (Fin.succ_ne_zero i)]\n[GOAL]\ncase refine'_2\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d\u00b9 : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u03c3 : Perm (Fin n)\nx\u271d : \u03c3 \u2208 univ\n\u22a2 (-1 * \u2191sign \u03c3) \u2022 \u220f x : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, \u03c3)) x) x =\n    -1 *\n      (A (Fin.succ i) 0 *\n        \u2191sign \u03c3 \u2022\n          \u220f i_1 : Fin n,\n            \u2191of (fun i_2 j => A (Fin.succAbove (Fin.succ i) i_2) (Fin.succ j)) (\u2191(Fin.cycleRange i) (\u2191\u03c3 i_1)) i_1)\n[PROOFSTEP]\ncalc\n  ((-1 * Perm.sign \u03c3 : \u2124) \u2022 \u220f i', A (Perm.decomposeFin.symm (Fin.succ i, \u03c3) i') i') =\n      (-1 * Perm.sign \u03c3 : \u2124) \u2022\n        (A (Fin.succ i) 0 * \u220f i', A ((Fin.succ i).succAbove (Fin.cycleRange i (\u03c3 i'))) i'.succ) :=\n    by\n    simp only [Fin.prod_univ_succ, Fin.succAbove_cycleRange, Equiv.Perm.decomposeFin_symm_apply_zero,\n      Equiv.Perm.decomposeFin_symm_apply_succ]\n  _ =\n      -1 *\n        (A (Fin.succ i) 0 * (Perm.sign \u03c3 : \u2124) \u2022 \u220f i', A ((Fin.succ i).succAbove (Fin.cycleRange i (\u03c3 i'))) i'.succ) :=\n    by\n    simp [mul_assoc, mul_comm, _root_.neg_mul, one_mul, zsmul_eq_mul, neg_inj, neg_smul, Fin.succAbove_cycleRange,\n      mul_left_comm]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d\u00b9 : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u03c3 : Perm (Fin n)\nx\u271d : \u03c3 \u2208 univ\n\u22a2 (-1 * \u2191(\u2191sign \u03c3)) \u2022 \u220f i' : Fin (Nat.succ n), A (\u2191(\u2191decomposeFin.symm (Fin.succ i, \u03c3)) i') i' =\n    (-1 * \u2191(\u2191sign \u03c3)) \u2022\n      (A (Fin.succ i) 0 * \u220f i' : Fin n, A (Fin.succAbove (Fin.succ i) (\u2191(Fin.cycleRange i) (\u2191\u03c3 i'))) (Fin.succ i'))\n[PROOFSTEP]\nsimp only [Fin.prod_univ_succ, Fin.succAbove_cycleRange, Equiv.Perm.decomposeFin_symm_apply_zero,\n  Equiv.Perm.decomposeFin_symm_apply_succ]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni\u271d : Fin (Nat.succ n)\nx\u271d\u00b9 : i\u271d \u2208 univ\ni : Fin n\nthis : (-1) ^ \u2191i = \u2191\u2191(\u2191sign (Fin.cycleRange i))\n\u03c3 : Perm (Fin n)\nx\u271d : \u03c3 \u2208 univ\n\u22a2 (-1 * \u2191(\u2191sign \u03c3)) \u2022\n      (A (Fin.succ i) 0 * \u220f i' : Fin n, A (Fin.succAbove (Fin.succ i) (\u2191(Fin.cycleRange i) (\u2191\u03c3 i'))) (Fin.succ i')) =\n    -1 *\n      (A (Fin.succ i) 0 *\n        \u2191(\u2191sign \u03c3) \u2022 \u220f i' : Fin n, A (Fin.succAbove (Fin.succ i) (\u2191(Fin.cycleRange i) (\u2191\u03c3 i'))) (Fin.succ i'))\n[PROOFSTEP]\nsimp [mul_assoc, mul_comm, _root_.neg_mul, one_mul, zsmul_eq_mul, neg_inj, neg_smul, Fin.succAbove_cycleRange,\n  mul_left_comm]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\n\u22a2 det A = \u2211 j : Fin (Nat.succ n), (-1) ^ \u2191j * A 0 j * det (submatrix A Fin.succ (Fin.succAbove j))\n[PROOFSTEP]\nrw [\u2190 det_transpose A, det_succ_column_zero]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\n\u22a2 \u2211 i : Fin (Nat.succ n), (-1) ^ \u2191i * A\u1d40 i 0 * det (submatrix A\u1d40 (Fin.succAbove i) Fin.succ) =\n    \u2211 j : Fin (Nat.succ n), (-1) ^ \u2191j * A 0 j * det (submatrix A Fin.succ (Fin.succAbove j))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i _ => _\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * det (submatrix A\u1d40 (Fin.succAbove i) Fin.succ) =\n    (-1) ^ \u2191i * A 0 i * det (submatrix A Fin.succ (Fin.succAbove i))\n[PROOFSTEP]\nrw [\u2190 det_transpose]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ \u2191i * A\u1d40 i 0 * det (submatrix A\u1d40 (Fin.succAbove i) Fin.succ)\u1d40 =\n    (-1) ^ \u2191i * A 0 i * det (submatrix A Fin.succ (Fin.succAbove i))\n[PROOFSTEP]\nsimp only [transpose_apply, transpose_submatrix, transpose_transpose]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\n\u22a2 det A = \u2211 j : Fin (Nat.succ n), (-1) ^ (\u2191i + \u2191j) * A i j * det (submatrix A (Fin.succAbove i) (Fin.succAbove j))\n[PROOFSTEP]\nsimp_rw [pow_add, mul_assoc, \u2190 mul_sum]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\n\u22a2 det A =\n    (-1) ^ \u2191i * \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * (A i x * det (submatrix A (Fin.succAbove i) (Fin.succAbove x)))\n[PROOFSTEP]\nhave : det A = (-1 : R) ^ (i : \u2115) * (Perm.sign i.cycleRange\u207b\u00b9) * det A := by\n  calc\n    det A = \u2191((-1 : \u2124\u02e3) ^ (i : \u2115) * (-1 : \u2124\u02e3) ^ (i : \u2115) : \u2124\u02e3) * det A := by simp\n    _ = (-1 : R) ^ (i : \u2115) * (Perm.sign i.cycleRange\u207b\u00b9) * det A := by simp [-Int.units_mul_self]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\n\u22a2 det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n[PROOFSTEP]\ncalc\n  det A = \u2191((-1 : \u2124\u02e3) ^ (i : \u2115) * (-1 : \u2124\u02e3) ^ (i : \u2115) : \u2124\u02e3) * det A := by simp\n  _ = (-1 : R) ^ (i : \u2115) * (Perm.sign i.cycleRange\u207b\u00b9) * det A := by simp [-Int.units_mul_self]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\n\u22a2 det A = \u2191\u2191((-1) ^ \u2191i * (-1) ^ \u2191i) * det A\n[PROOFSTEP]\nsimp\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\n\u22a2 \u2191\u2191((-1) ^ \u2191i * (-1) ^ \u2191i) * det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n[PROOFSTEP]\nsimp [-Int.units_mul_self]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n\u22a2 det A =\n    (-1) ^ \u2191i * \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * (A i x * det (submatrix A (Fin.succAbove i) (Fin.succAbove x)))\n[PROOFSTEP]\nrw [this, mul_assoc]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n\u22a2 (-1) ^ \u2191i * (\u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A) =\n    (-1) ^ \u2191i * \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * (A i x * det (submatrix A (Fin.succAbove i) (Fin.succAbove x)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n\u22a2 \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A =\n    \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * (A i x * det (submatrix A (Fin.succAbove i) (Fin.succAbove x)))\n[PROOFSTEP]\nrw [\u2190 det_permute, det_succ_row_zero]\n[GOAL]\ncase e_a\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\n\u22a2 \u2211 j : Fin (Nat.succ n),\n      (-1) ^ \u2191j * A (\u2191(Fin.cycleRange i)\u207b\u00b9 0) j *\n        det (submatrix (fun i_1 => A (\u2191(Fin.cycleRange i)\u207b\u00b9 i_1)) Fin.succ (Fin.succAbove j)) =\n    \u2211 x : Fin (Nat.succ n), (-1) ^ \u2191x * (A i x * det (submatrix A (Fin.succAbove i) (Fin.succAbove x)))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun j _ => _\n[GOAL]\ncase e_a\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\nj : Fin (Nat.succ n)\nx\u271d : j \u2208 univ\n\u22a2 (-1) ^ \u2191j * A (\u2191(Fin.cycleRange i)\u207b\u00b9 0) j *\n      det (submatrix (fun i_1 => A (\u2191(Fin.cycleRange i)\u207b\u00b9 i_1)) Fin.succ (Fin.succAbove j)) =\n    (-1) ^ \u2191j * (A i j * det (submatrix A (Fin.succAbove i) (Fin.succAbove j)))\n[PROOFSTEP]\nrw [mul_assoc, Matrix.submatrix, Matrix.submatrix]\n[GOAL]\ncase e_a\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\nj : Fin (Nat.succ n)\nx\u271d : j \u2208 univ\n\u22a2 (-1) ^ \u2191j *\n      (A (\u2191(Fin.cycleRange i)\u207b\u00b9 0) j *\n        det (\u2191of fun i_1 j_1 => A (\u2191(Fin.cycleRange i)\u207b\u00b9 (Fin.succ i_1)) (Fin.succAbove j j_1))) =\n    (-1) ^ \u2191j * (A i j * det (\u2191of fun i_1 j_1 => A (Fin.succAbove i i_1) (Fin.succAbove j j_1)))\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_a.e_a.e_a\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\nj : Fin (Nat.succ n)\nx\u271d : j \u2208 univ\n\u22a2 \u2191(Fin.cycleRange i)\u207b\u00b9 0 = i\n[PROOFSTEP]\nrw [Equiv.Perm.inv_def, Fin.cycleRange_symm_zero]\n[GOAL]\ncase e_a.e_a.e_a.e_M.h.e_6.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\nj : Fin (Nat.succ n)\nx\u271d : j \u2208 univ\n\u22a2 (fun i_1 j_1 => A (\u2191(Fin.cycleRange i)\u207b\u00b9 (Fin.succ i_1)) (Fin.succAbove j j_1)) = fun i_1 j_1 =>\n    A (Fin.succAbove i i_1) (Fin.succAbove j j_1)\n[PROOFSTEP]\next i' j'\n[GOAL]\ncase e_a.e_a.e_a.e_M.h.e_6.h.h.h\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\ni : Fin (Nat.succ n)\nthis : det A = (-1) ^ \u2191i * \u2191\u2191(\u2191sign (Fin.cycleRange i)\u207b\u00b9) * det A\nj : Fin (Nat.succ n)\nx\u271d : j \u2208 univ\ni' j' : Fin n\n\u22a2 A (\u2191(Fin.cycleRange i)\u207b\u00b9 (Fin.succ i')) (Fin.succAbove j j') = A (Fin.succAbove i i') (Fin.succAbove j j')\n[PROOFSTEP]\nrw [Equiv.Perm.inv_def, Fin.cycleRange_symm_succ]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj : Fin (Nat.succ n)\n\u22a2 det A = \u2211 i : Fin (Nat.succ n), (-1) ^ (\u2191i + \u2191j) * A i j * det (submatrix A (Fin.succAbove i) (Fin.succAbove j))\n[PROOFSTEP]\nrw [\u2190 det_transpose, det_succ_row _ j]\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj : Fin (Nat.succ n)\n\u22a2 \u2211 j_1 : Fin (Nat.succ n), (-1) ^ (\u2191j + \u2191j_1) * A\u1d40 j j_1 * det (submatrix A\u1d40 (Fin.succAbove j) (Fin.succAbove j_1)) =\n    \u2211 i : Fin (Nat.succ n), (-1) ^ (\u2191i + \u2191j) * A i j * det (submatrix A (Fin.succAbove i) (Fin.succAbove j))\n[PROOFSTEP]\nrefine' Finset.sum_congr rfl fun i _ => _\n[GOAL]\nm : Type u_1\nn\u271d : Type u_2\ninst\u271d\u2074 : DecidableEq n\u271d\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nn : \u2115\nA : Matrix (Fin (Nat.succ n)) (Fin (Nat.succ n)) R\nj i : Fin (Nat.succ n)\nx\u271d : i \u2208 univ\n\u22a2 (-1) ^ (\u2191j + \u2191i) * A\u1d40 j i * det (submatrix A\u1d40 (Fin.succAbove j) (Fin.succAbove i)) =\n    (-1) ^ (\u2191i + \u2191j) * A i j * det (submatrix A (Fin.succAbove i) (Fin.succAbove j))\n[PROOFSTEP]\nrw [add_comm, \u2190 det_transpose, transpose_apply, transpose_submatrix, transpose_transpose]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix (Fin 2) (Fin 2) R\n\u22a2 det A = A 0 0 * A 1 1 - A 0 1 * A 1 0\n[PROOFSTEP]\nsimp [Matrix.det_succ_row_zero, Fin.sum_univ_succ]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix (Fin 2) (Fin 2) R\n\u22a2 A 0 0 * A 1 1 + -(A 0 1 * A 1 0) = A 0 0 * A 1 1 - A 0 1 * A 1 0\n[PROOFSTEP]\nring\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix (Fin 3) (Fin 3) R\n\u22a2 det A =\n    A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1 - A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0 +\n        A 0 2 * A 1 0 * A 2 1 -\n      A 0 2 * A 1 1 * A 2 0\n[PROOFSTEP]\nsimp [Matrix.det_succ_row_zero, Fin.sum_univ_succ]\n[GOAL]\nm : Type u_1\nn : Type u_2\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : Fintype n\ninst\u271d\u00b2 : DecidableEq m\ninst\u271d\u00b9 : Fintype m\nR : Type v\ninst\u271d : CommRing R\nA : Matrix (Fin 3) (Fin 3) R\n\u22a2 A 0 0 * (A 1 1 * A 2 2 + -(A 1 2 * A 2 1)) +\n      (-(A 0 1 * (A 1 0 * A 2 2 + -(A 1 2 * A 2 0))) + A 0 2 * (A 1 0 * A 2 1 + -(A 1 1 * A 2 0))) =\n    A 0 0 * A 1 1 * A 2 2 - A 0 0 * A 1 2 * A 2 1 - A 0 1 * A 1 0 * A 2 2 + A 0 1 * A 1 2 * A 2 0 +\n        A 0 2 * A 1 0 * A 2 1 -\n      A 0 2 * A 1 1 * A 2 0\n[PROOFSTEP]\nring\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.Matrix.Determinant", "llama_tokens": 90256, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956580952177051, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.5039780038892546}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na\u271d b a : \u03b1\ns : Sym \u03b1 n\n\u22a2 \u2191Multiset.card (a ::\u2098 \u2191s) = Nat.succ n\n[PROOFSTEP]\nrw [Multiset.card_cons, s.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nv : Vector \u03b1 n\n\u22a2 ofVector (a ::\u1d65 v) = a ::\u209b ofVector v\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 ofVector (a ::\u1d65 { val := val\u271d, property := property\u271d }) = a ::\u209b ofVector { val := val\u271d, property := property\u271d }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nv : Vector \u03b1 n\n\u22a2 \u2191(a ::\u209b ofVector v) = \u2191(ofVector (a ::\u1d65 v))\n[PROOFSTEP]\ncases v\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na\u271d b a : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 \u2191(a ::\u209b ofVector { val := val\u271d, property := property\u271d }) = \u2191(ofVector (a ::\u1d65 { val := val\u271d, property := property\u271d }))\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\nm : Multiset \u03b1\nhc : \u2191Multiset.card m = n + 1\na : \u03b1\nh : a \u2208 m\n\u22a2 \u2191Multiset.card (Multiset.erase m a) = n\n[PROOFSTEP]\nrw [Multiset.card_erase_of_mem h, hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\nm : Multiset \u03b1\nhc : \u2191Multiset.card m = n + 1\na : \u03b1\nh : a \u2208 m\n\u22a2 Nat.pred (n + 1) = n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\nx\u271d : List \u03b1\n\u22a2 List.length x\u271d = n \u2194 \u2191Multiset.card (Quotient.mk (List.isSetoid \u03b1) x\u271d) = n\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\nx\u271d\u00b9 x\u271d : { l // List.length l = n }\n\u22a2 Setoid.r x\u271d\u00b9 x\u271d \u2194 \u2191x\u271d\u00b9 \u2248 \u2191x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\na : \u03b1\ns : Sym \u03b1 n\n\u22a2 (a :: \u2191symEquivSym' s) = \u2191symEquivSym' (a ::\u209b s)\n[PROOFSTEP]\nrcases s with \u27e8\u27e8l\u27e9, _\u27e9\n[GOAL]\ncase mk.mk\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns : Sym \u03b1\u271d n\u271d\na\u271d b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\na : \u03b1\nval\u271d : Multiset \u03b1\nl : List \u03b1\nproperty\u271d : \u2191Multiset.card (Quot.mk Setoid.r l) = n\n\u22a2 (a :: \u2191symEquivSym' { val := Quot.mk Setoid.r l, property := property\u271d }) =\n    \u2191symEquivSym' (a ::\u209b { val := Quot.mk Setoid.r l, property := property\u271d })\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\n\u22a2 s = replicate n a \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\n[PROOFSTEP]\nerw [Subtype.ext_iff, Multiset.eq_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\n\u22a2 (\u2191Multiset.card \u2191s = n \u2227 \u2200 (b : \u03b1), b \u2208 \u2191s \u2192 b = a) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a\n[PROOFSTEP]\nexact and_iff_right s.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 (Nat.succ n)\n\u22a2 \u2203 a s', s = a ::\u209b s'\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := exists_mem s\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na\u271d b : \u03b1\ns : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\n\u22a2 \u2203 a s', s = a ::\u209b s'\n[PROOFSTEP]\nclassical exact \u27e8a, s.erase a ha, (cons_erase ha).symm\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na\u271d b : \u03b1\ns : Sym \u03b1 (Nat.succ n)\na : \u03b1\nha : a \u2208 s\n\u22a2 \u2203 a s', s = a ::\u209b s'\n[PROOFSTEP]\nexact \u27e8a, s.erase a ha, (cons_erase ha).symm\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns : Sym \u03b1 n\u271d\na b : \u03b1\ninst\u271d : Subsingleton \u03b1\nn : \u2115\n\u22a2 \u2200 (a b : Sym \u03b1 n), a = b\n[PROOFSTEP]\ncases n\n[GOAL]\ncase zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\ninst\u271d : Subsingleton \u03b1\n\u22a2 \u2200 (a b : Sym \u03b1 Nat.zero), a = b\n[PROOFSTEP]\nsimp\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\ninst\u271d : Subsingleton \u03b1\nn\u271d : \u2115\n\u22a2 \u2200 (a b : Sym \u03b1 (Nat.succ n\u271d)), a = b\n[PROOFSTEP]\nintro s s'\n[GOAL]\ncase succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ninst\u271d : Subsingleton \u03b1\nn\u271d : \u2115\ns s' : Sym \u03b1 (Nat.succ n\u271d)\n\u22a2 s = s'\n[PROOFSTEP]\nobtain \u27e8b, -\u27e9 := exists_mem s\n[GOAL]\ncase succ.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b\u271d : \u03b1\ninst\u271d : Subsingleton \u03b1\nn\u271d : \u2115\ns s' : Sym \u03b1 (Nat.succ n\u271d)\nb : \u03b1\n\u22a2 s = s'\n[PROOFSTEP]\nrw [eq_replicate_of_subsingleton b s', eq_replicate_of_subsingleton b s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1 n\u271d\na b : \u03b1\nn : \u2115\ninst\u271d : IsEmpty \u03b1\ns : Sym \u03b1 (Nat.succ n)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8a, -\u27e9 := exists_mem s\n[GOAL]\ncase intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1 n\u271d\na\u271d b : \u03b1\nn : \u2115\ninst\u271d : IsEmpty \u03b1\ns : Sym \u03b1 (Nat.succ n)\na : \u03b1\n\u22a2 False\n[PROOFSTEP]\nexact isEmptyElim a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns : Sym \u03b1 n\u271d\na b : \u03b1\nn : \u2115\nf : \u03b1 \u2192 \u03b2\nx : Sym \u03b1 n\n\u22a2 \u2191Multiset.card (Multiset.map f \u2191x) = n\n[PROOFSTEP]\nsimpa [Multiset.card_map] using x.property\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 map (fun x => x) s = s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 \u2191(map (fun x => x) s) = \u2191s\n[PROOFSTEP]\nsimp [Sym.map]\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 { val := \u2191s, property := (_ : (fun s => \u2191Multiset.card s = n) \u2191s) } = s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 map id s = s\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 \u2191(map id s) = \u2191s\n[PROOFSTEP]\nsimp [Sym.map]\n[GOAL]\ncase h\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\nn : \u2115\ns : Sym \u03b1 n\n\u22a2 { val := \u2191s, property := (_ : (fun s => \u2191Multiset.card s = n) \u2191s) } = s\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nn : \u2115\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Sym \u03b1 n\n\u22a2 \u2191(map g (map f s)) = \u2191(map (g \u2218 f) s)\n[PROOFSTEP]\ndsimp only [Sym.map]\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\nn\u271d n' m : \u2115\ns\u271d : Sym \u03b1\u271d n\u271d\na b : \u03b1\u271d\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\nn : \u2115\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\ns : Sym \u03b1 n\n\u22a2 Multiset.map g (Multiset.map f \u2191s) = Multiset.map (g \u2218 f) \u2191s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na b : \u03b1\nf : \u03b1 \u2192 \u03b2\nm : Multiset \u03b1\nhc : \u2191Multiset.card m = n\n\u22a2 \u2191Multiset.card (Multiset.map f m) = n\n[PROOFSTEP]\nsimp [hc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\ne : \u03b1 \u2243 \u03b2\nx : Sym \u03b1 n\n\u22a2 map (\u2191e.symm) (map (\u2191e) x) = x\n[PROOFSTEP]\nrw [map_map, Equiv.symm_comp_self, map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\ne : \u03b1 \u2243 \u03b2\nx : Sym \u03b2 n\n\u22a2 map (\u2191e) (map (\u2191e.symm) x) = x\n[PROOFSTEP]\nrw [map_map, Equiv.self_comp_symm, map_id]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 n\n\u22a2 \u2191Multiset.card (Multiset.attach \u2191s) = n\n[PROOFSTEP]\nconv_rhs => rw [\u2190 s.2, \u2190 Multiset.card_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 n\n| n\n[PROOFSTEP]\nrw [\u2190 s.2, \u2190 Multiset.card_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 n\n| n\n[PROOFSTEP]\nrw [\u2190 s.2, \u2190 Multiset.card_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 n\n| n\n[PROOFSTEP]\nrw [\u2190 s.2, \u2190 Multiset.card_attach]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns : Sym \u03b1 n\ns' : Sym \u03b1 n'\n\u22a2 \u2191Multiset.card (\u2191s + \u2191s') = n + n'\n[PROOFSTEP]\nrw [map_add, s.2, s'.2]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns s' : Sym \u03b1 n'\n\u22a2 append s s' = \u2191(Sym.cast (_ : n' + n' = n' + n')) (append s' s)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na b : \u03b1\ns s' : Sym \u03b1 n'\n\u22a2 \u2191(append s s') = \u2191(\u2191(Sym.cast (_ : n' + n' = n' + n')) (append s' s))\n[PROOFSTEP]\nsimp [append, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns\u271d : Sym \u03b1 n\na\u271d b\u271d a b : \u03b1\ni : Fin (n + 1)\ns : Sym \u03b1 (n - \u2191i)\n\u22a2 a \u2208 fill b i s \u2194 \u2191i \u2260 0 \u2227 a = b \u2228 a \u2208 s\n[PROOFSTEP]\nrw [fill, mem_cast, mem_append_iff, or_comm, mem_replicate]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 \u2191card \u2191m < n + 1\n[PROOFSTEP]\nrw [m.2, Nat.lt_succ_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 \u2191card (filter ((fun x x_1 => x \u2260 x_1) a) \u2191m) + \u2191{ val := count a \u2191m, isLt := (_ : count a \u2191m < n + 1) } = \u2191card \u2191m\n[PROOFSTEP]\nrw [\u2190 countp_eq_card_filter, add_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 \u2191{ val := count a \u2191m, isLt := (_ : count a \u2191m < n + 1) } + countp ((fun x x_1 => x \u2260 x_1) a) \u2191m = \u2191card \u2191m\n[PROOFSTEP]\nsimp only [eq_comm, Ne.def, count]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 countp (fun x => a = x) \u2191m + countp (fun x => \u00aca = x) \u2191m = \u2191card \u2191m\n[PROOFSTEP]\nrw [\u2190 card_eq_countp_add_countp _ _]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m : \u2115\ns : Sym \u03b1 n\na b : \u03b1\nm\u2081 m\u2082 : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u2191m\u2081.snd = \u2191m\u2082.snd\n\u22a2 \u2191m\u2081.fst = \u2191m\u2082.fst\n[PROOFSTEP]\nrw [\u2190 Nat.sub_sub_self (Nat.le_of_lt_succ m\u2081.1.is_lt), \u2190 m\u2081.2.2, val_eq_coe, h, \u2190 val_eq_coe, m\u2082.2.2,\n  Nat.sub_sub_self (Nat.le_of_lt_succ m\u2082.1.is_lt)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 \u2191(fill a (filterNe a m).fst (filterNe a m).snd) = \u2191m\n[PROOFSTEP]\nrw [coe_fill, filterNe, \u2190 val_eq_coe, Subtype.coe_mk, Fin.val_mk]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 filter ((fun x x_1 => x \u2260 x_1) a) \u2191m + \u2191(replicate (count a \u2191m) a) = \u2191m\n[PROOFSTEP]\next b\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\nb : \u03b1\n\u22a2 count b (filter ((fun x x_1 => x \u2260 x_1) a) \u2191m + \u2191(replicate (count a \u2191m) a)) = count b \u2191m\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\nb : \u03b1\n\u22a2 count b (filter (fun x => \u00aca = x) \u2191m + \u2191(replicate (count a \u2191m) a)) = count b \u2191m\n[PROOFSTEP]\nrw [count_add, count_filter, Sym.coe_replicate, count_replicate]\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\nb : \u03b1\n\u22a2 ((if \u00aca = b then count b \u2191m else 0) + if b = a then count a \u2191m else 0) = count b \u2191m\n[PROOFSTEP]\nobtain rfl | h := eq_or_ne a b\n[GOAL]\ncase a.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\n\u22a2 ((if \u00aca = a then count a \u2191m else 0) + if a = a then count a \u2191m else 0) = count a \u2191m\n[PROOFSTEP]\nrw [if_pos rfl, if_neg (not_not.2 rfl), zero_add]\n[GOAL]\ncase a.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : Sym \u03b1 n\nb : \u03b1\nh : a \u2260 b\n\u22a2 ((if \u00aca = b then count b \u2191m else 0) + if b = a then count a \u2191m else 0) = count b \u2191m\n[PROOFSTEP]\nrw [if_pos h, if_neg h.symm, add_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\n\u22a2 \u2191(filterNe a (fill a m.fst m.snd)).snd = \u2191m.snd\n[PROOFSTEP]\nrw [filterNe, \u2190 val_eq_coe, Subtype.coe_mk, val_eq_coe, coe_fill]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\n\u22a2 filter ((fun x x_1 => x \u2260 x_1) a) (\u2191m.snd + \u2191(replicate (\u2191m.fst) a)) = \u2191m.snd\n[PROOFSTEP]\nrw [filter_add, filter_eq_self.2, add_right_eq_self, eq_zero_iff_forall_not_mem]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\n\u22a2 \u2200 (a_1 : \u03b1), \u00aca_1 \u2208 filter ((fun x x_1 => x \u2260 x_1) a) \u2191(replicate (\u2191m.fst) a)\n[PROOFSTEP]\nintro b hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\nb : \u03b1\nhb : b \u2208 filter ((fun x x_1 => x \u2260 x_1) a) \u2191(replicate (\u2191m.fst) a)\n\u22a2 False\n[PROOFSTEP]\nrw [mem_filter, Sym.mem_coe, mem_replicate] at hb \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b\u271d : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\nb : \u03b1\nhb : (\u2191m.fst \u2260 0 \u2227 b = a) \u2227 (fun x x_1 => x \u2260 x_1) a b\n\u22a2 False\n[PROOFSTEP]\nexact hb.2 hb.1.2.symm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn n' m\u271d : \u2115\ns : Sym \u03b1 n\na\u271d b : \u03b1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nm : (i : Fin (n + 1)) \u00d7 Sym \u03b1 (n - \u2191i)\nh : \u00aca \u2208 m.snd\n\u22a2 \u2200 (a_1 : \u03b1), a_1 \u2208 \u2191m.snd \u2192 (fun x x_1 => x \u2260 x_1) a a_1\n[PROOFSTEP]\nexact fun a ha ha' => h <| ha'.symm \u25b8 ha\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) (Nat.succ n)\n\u22a2 decode (encode s) = s\n[PROOFSTEP]\nby_cases h : none \u2208 s\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) (Nat.succ n)\nh : none \u2208 s\n\u22a2 decode (encode s) = s\n[PROOFSTEP]\nsimp [h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) (Nat.succ n)\nh : \u00acnone \u2208 s\n\u22a2 decode (encode s) = s\n[PROOFSTEP]\nsimp only [decode, h, not_false_iff, encode_of_not_none_mem, Embedding.some_apply, map_map, comp_apply, Option.some_get]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) (Nat.succ n)\nh : \u00acnone \u2208 s\n\u22a2 map (fun x => \u2191x) (attach s) = s\n[PROOFSTEP]\nconvert s.attach_map_coe\n[GOAL]\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) n \u2295 Sym \u03b1 (Nat.succ n)\n\u22a2 encode (decode s) = s\n[PROOFSTEP]\nobtain s | s := s\n[GOAL]\ncase inl\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym (Option \u03b1) n\n\u22a2 encode (decode (Sum.inl s)) = Sum.inl s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\n\u22a2 encode (decode (Sum.inr s)) = Sum.inr s\n[PROOFSTEP]\nunfold SymOptionSuccEquiv.encode\n[GOAL]\ncase inr\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\n\u22a2 (if h : none \u2208 decode (Sum.inr s) then Sum.inl (erase (decode (Sum.inr s)) none h)\n    else Sum.inr (map (fun o => Option.get \u2191o (_ : Option.isSome \u2191o = true)) (attach (decode (Sum.inr s))))) =\n    Sum.inr s\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : none \u2208 decode (Sum.inr s)\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8a, _, ha\u27e9 := Multiset.mem_map.mp h\n[GOAL]\ncase pos.intro.intro\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : none \u2208 decode (Sum.inr s)\na : \u03b1\nleft\u271d : a \u2208 \u2191s\nha : \u2191Embedding.some a = none\n\u22a2 False\n[PROOFSTEP]\nexact Option.some_ne_none _ ha\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 Sum.inr (map (fun o => Option.get \u2191o (_ : Option.isSome \u2191o = true)) (attach (decode (Sum.inr s)))) = Sum.inr s\n[PROOFSTEP]\nrefine' congr_arg Sum.inr _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 map (fun o => Option.get \u2191o (_ : Option.isSome \u2191o = true)) (attach (decode (Sum.inr s))) = s\n[PROOFSTEP]\nrefine' map_injective (Option.some_injective _) _ _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 map some (map (fun o => Option.get \u2191o (_ : Option.isSome \u2191o = true)) (attach (decode (Sum.inr s)))) = map some s\n[PROOFSTEP]\nrefine' Eq.trans _ (Eq.trans (SymOptionSuccEquiv.decode (Sum.inr s)).attach_map_coe _)\n[GOAL]\ncase neg.refine'_1\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 map some (map (fun o => Option.get \u2191o (_ : Option.isSome \u2191o = true)) (attach (decode (Sum.inr s)))) =\n    map Subtype.val (attach (decode (Sum.inr s)))\ncase neg.refine'_2\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 decode (Sum.inr s) = map some s\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.refine'_2\n\u03b1 : Type u_1\nn : \u2115\ninst\u271d : DecidableEq \u03b1\ns : Sym \u03b1 (Nat.succ n)\nh : \u00acnone \u2208 decode (Sum.inr s)\n\u22a2 decode (Sum.inr s) = map some s\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.Sym.Basic", "llama_tokens": 8971, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.5036350814409133}}
{"text": "[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 LinearIndependent R v \u2194 \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\n[PROOFSTEP]\nsimp [LinearIndependent, LinearMap.ker_eq_bot']\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhf : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhis : i \u2208 s\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (\u2211 i in s, Finsupp.single i (g i)) = 0\n[PROOFSTEP]\nsimpa only [LinearMap.map_sum, Finsupp.total_single] using hg\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhf : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhis : i \u2208 s\nh : \u2211 i in s, Finsupp.single i (g i) = 0\n\u22a2 g i = \u2191(Finsupp.lapply i) (Finsupp.single i (g i))\n[PROOFSTEP]\n{rw [Finsupp.lapply_apply, Finsupp.single_eq_same]\n}\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhf : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhis : i \u2208 s\nh : \u2211 i in s, Finsupp.single i (g i) = 0\n\u22a2 g i = \u2191(Finsupp.lapply i) (Finsupp.single i (g i))\n[PROOFSTEP]\nrw [Finsupp.lapply_apply, Finsupp.single_eq_same]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhf : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhis : i \u2208 s\nh : \u2211 i in s, Finsupp.single i (g i) = 0\nj : \u03b9\n_hjs : j \u2208 s\nhji : j \u2260 i\n\u22a2 \u2191(Finsupp.lapply i) (Finsupp.single j (g j)) = 0\n[PROOFSTEP]\nrw [Finsupp.lapply_apply, Finsupp.single_eq_of_ne hji]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 LinearIndependent R v \u2194\n    \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nclassical exact\n  linearIndependent_iff'.trans\n    \u27e8fun H s g hg hv i => if his : i \u2208 s then H s g hv i his else hg i his, fun H s g hg i hi =>\n      by\n      convert\n        H s (fun j => if j \u2208 s then g j else 0) (fun j hj => if_neg hj)\n          (by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i\n      exact (if_pos hi).symm\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 LinearIndependent R v \u2194\n    \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nexact\n  linearIndependent_iff'.trans\n    \u27e8fun H s g hg hv i => if his : i \u2208 s then H s g hv i his else hg i his, fun H s g hg i hi =>\n      by\n      convert\n        H s (fun j => if j \u2208 s then g j else 0) (fun j hj => if_neg hj)\n          (by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i\n      exact (if_pos hi).symm\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nH : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nconvert\n  H s (fun j => if j \u2208 s then g j else 0) (fun j hj => if_neg hj)\n    (by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nH : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 i in s, (fun j => if j \u2208 s then g j else 0) i \u2022 v i = 0\n[PROOFSTEP]\nsimp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nH : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i = if i \u2208 s then g i else 0\n[PROOFSTEP]\nexact (if_pos hi).symm\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 \u00acLinearIndependent R v \u2194 \u2203 s g, \u2211 i in s, g i \u2022 v i = 0 \u2227 \u2203 i, i \u2208 s \u2227 g i \u2260 0\n[PROOFSTEP]\nrw [linearIndependent_iff']\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 (\u00ac\u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0) \u2194\n    \u2203 s g, \u2211 i in s, g i \u2022 v i = 0 \u2227 \u2203 i, i \u2208 s \u2227 g i \u2260 0\n[PROOFSTEP]\nsimp only [exists_prop, not_forall]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\n\u22a2 LinearIndependent R v \u2194 \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nrefine'\n  \u27e8fun H g => by simpa using linearIndependent_iff'.1 H Finset.univ g, fun H =>\n    linearIndependent_iff''.2 fun s g hg hs i => H _ _ _\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : LinearIndependent R v\ng : \u03b9 \u2192 R\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nsimpa using linearIndependent_iff'.1 H Finset.univ g\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = 0\n[PROOFSTEP]\nrw [\u2190 hs]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\n\u22a2 \u2211 i : \u03b9, g i \u2022 v i = \u2211 i in s, g i \u2022 v i\n[PROOFSTEP]\nrefine' (Finset.sum_subset (Finset.subset_univ _) fun i _ hi => _).symm\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\nH : \u2200 (g : \u03b9 \u2192 R), \u2211 i : \u03b9, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhs : \u2211 i in s, g i \u2022 v i = 0\ni\u271d i : \u03b9\nx\u271d : i \u2208 Finset.univ\nhi : \u00aci \u2208 s\n\u22a2 g i \u2022 v i = 0\n[PROOFSTEP]\nrw [hg i hi, zero_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M'\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : DecidableEq \u03b9\n\u22a2 LinearIndependent R v \u2194\n    LinearMap.ker (\u2191(LinearMap.lsum R (fun x => R) \u2115) fun i => LinearMap.smulRight LinearMap.id (v i)) = \u22a5\n[PROOFSTEP]\nsimp [Fintype.linearIndependent_iff, LinearMap.ker_eq_bot', funext_iff]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Fintype \u03b9\n\u22a2 \u00acLinearIndependent R v \u2194 \u2203 g, \u2211 i : \u03b9, g i \u2022 v i = 0 \u2227 \u2203 i, g i \u2260 0\n[PROOFSTEP]\nsimpa using not_iff_not.2 Fintype.linearIndependent_iff\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ni : \u03b9\nhv : LinearIndependent R v\nh : v i = 0\n\u22a2 1 = 0\n[PROOFSTEP]\nsuffices (Finsupp.single i 1 : \u03b9 \u2192\u2080 R) i = 0 by simpa\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ni : \u03b9\nhv : LinearIndependent R v\nh : v i = 0\nthis : \u2191(Finsupp.single i 1) i = 0\n\u22a2 1 = 0\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ni : \u03b9\nhv : LinearIndependent R v\nh : v i = 0\n\u22a2 \u2191(Finsupp.single i 1) i = 0\n[PROOFSTEP]\nrw [linearIndependent_iff.1 hv (Finsupp.single i 1)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ni : \u03b9\nhv : LinearIndependent R v\nh : v i = 0\n\u22a2 \u21910 i = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : AddCommMonoid M'\ninst\u271d\u2074 : AddCommMonoid M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ni : \u03b9\nhv : LinearIndependent R v\nh : v i = 0\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t : R\nh' : s \u2022 x + t \u2022 y = 0\n\u22a2 s = 0 \u2227 t = 0\n[PROOFSTEP]\nhave := linearIndependent_iff'.1 h Finset.univ ![s, t]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t : R\nh' : s \u2022 x + t \u2022 y = 0\nthis :\n  \u2211 i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons s ![t] i \u2022 Matrix.vecCons x ![y] i = 0 \u2192\n    \u2200 (i : Fin (Nat.succ (Nat.succ 0))), i \u2208 Finset.univ \u2192 Matrix.vecCons s ![t] i = 0\n\u22a2 s = 0 \u2227 t = 0\n[PROOFSTEP]\nsimp only [Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.head_cons, h', Finset.mem_univ,\n  forall_true_left] at this \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t : R\nh' : s \u2022 x + t \u2022 y = 0\nthis : \u2200 (i : Fin (Nat.succ (Nat.succ 0))), Matrix.vecCons s ![t] i = 0\n\u22a2 s = 0 \u2227 t = 0\n[PROOFSTEP]\nexact \u27e8this 0, this 1\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\n\u22a2 LinearIndependent R (v \u2218 f)\n[PROOFSTEP]\nrw [linearIndependent_iff, Finsupp.total_comp]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\n\u22a2 \u2200 (l : \u03b9' \u2192\u2080 R), \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 \u2192 l = 0\n[PROOFSTEP]\nintro l hl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\n\u22a2 l = 0\n[PROOFSTEP]\nhave h_map_domain : \u2200 x, (Finsupp.mapDomain f l) (f x) = 0 := by\n  rw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl]; simp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\n\u22a2 \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0\n[PROOFSTEP]\nrw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\n\u22a2 \u2200 (x : \u03b9'), \u21910 (f x) = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\nh_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0\n\u22a2 l = 0\n[PROOFSTEP]\next x\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\nh_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0\nx : \u03b9'\n\u22a2 \u2191l x = \u21910 x\n[PROOFSTEP]\nconvert h_map_domain x\n[GOAL]\ncase h.e'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nh : LinearIndependent R v\nf : \u03b9' \u2192 \u03b9\nhf : Injective f\nl : \u03b9' \u2192\u2080 R\nhl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0\nh_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0\nx : \u03b9'\n\u22a2 \u2191l x = \u2191(Finsupp.mapDomain f l) (f x)\n[PROOFSTEP]\nrw [Finsupp.mapDomain_apply hf]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ni : LinearIndependent R v\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nsimpa using i.comp _ (rangeSplitting_injective v)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nf : M \u2192\u2097[R] M'\nhf_inj : Disjoint (span R (range v)) (LinearMap.ker f)\n\u22a2 LinearIndependent R (\u2191f \u2218 v)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, \u2190 Set.image_univ, Finsupp.span_image_eq_map_total, map_inf_eq_map_inf_comap,\n  map_le_iff_le_comap, comap_bot, Finsupp.supported_univ, top_inf_eq] at hf_inj \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nf : M \u2192\u2097[R] M'\nhf_inj : comap (Finsupp.total \u03b9 M R v) (LinearMap.ker f) \u2264 LinearMap.ker (Finsupp.total \u03b9 M R v)\n\u22a2 LinearIndependent R (\u2191f \u2218 v)\n[PROOFSTEP]\nunfold LinearIndependent at hv \u22a2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearMap.ker (Finsupp.total \u03b9 M R v) = \u22a5\nf : M \u2192\u2097[R] M'\nhf_inj : comap (Finsupp.total \u03b9 M R v) (LinearMap.ker f) \u2264 LinearMap.ker (Finsupp.total \u03b9 M R v)\n\u22a2 LinearMap.ker (Finsupp.total \u03b9 M' R (\u2191f \u2218 v)) = \u22a5\n[PROOFSTEP]\nrw [hv, le_bot_iff] at hf_inj \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearMap.ker (Finsupp.total \u03b9 M R v) = \u22a5\nf : M \u2192\u2097[R] M'\nhf_inj : comap (Finsupp.total \u03b9 M R v) (LinearMap.ker f) = \u22a5\n\u22a2 LinearMap.ker (Finsupp.total \u03b9 M' R (\u2191f \u2218 v)) = \u22a5\n[PROOFSTEP]\nhaveI : Inhabited M := \u27e80\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearMap.ker (Finsupp.total \u03b9 M R v) = \u22a5\nf : M \u2192\u2097[R] M'\nhf_inj : comap (Finsupp.total \u03b9 M R v) (LinearMap.ker f) = \u22a5\nthis : Inhabited M\n\u22a2 LinearMap.ker (Finsupp.total \u03b9 M' R (\u2191f \u2218 v)) = \u22a5\n[PROOFSTEP]\nrw [Finsupp.total_comp, @Finsupp.lmapDomain_total _ _ R _ _ _ _ _ _ _ _ _ _ f, LinearMap.ker_comp, hf_inj]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearMap.ker (Finsupp.total \u03b9 M R v) = \u22a5\nf : M \u2192\u2097[R] M'\nhf_inj : comap (Finsupp.total \u03b9 M R v) (LinearMap.ker f) = \u22a5\nthis : Inhabited M\n\u22a2 \u2200 (i : \u03b9), \u2191f (v i) = \u2191f (v i)\n[PROOFSTEP]\nexact fun _ => rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nf : M \u2192\u2097[R] M'\nhv : LinearIndependent R (\u2191f \u2218 v)\n\u22a2 Disjoint (span R (range v)) (LinearMap.ker f)\n[PROOFSTEP]\nrw [LinearIndependent, Finsupp.total_comp, Finsupp.lmapDomain_total R _ f (fun _ \u21a6 rfl), LinearMap.ker_comp] at hv \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nf : M \u2192\u2097[R] M'\nhv : comap (Finsupp.total \u03b9 M R fun x => v x) (LinearMap.ker f) = \u22a5\n\u22a2 Disjoint (span R (range v)) (LinearMap.ker f)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, \u2190 Set.image_univ, Finsupp.span_image_eq_map_total, map_inf_eq_map_inf_comap, hv, inf_bot_eq,\n  map_bot]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nf : M \u2192\u2097[R] M'\nhf_inj : LinearMap.ker f = \u22a5\n\u22a2 Disjoint (span R (range v)) (LinearMap.ker f)\n[PROOFSTEP]\nsimp [hf_inj]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nf : M \u2192\u2097[R] M'\nhfv : LinearIndependent R (\u2191f \u2218 v)\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhis : i \u2208 s\n\u22a2 \u2211 i in s, g i \u2022 \u2191f (v i) = 0\n[PROOFSTEP]\nsimp_rw [\u2190 f.map_smul, \u2190 f.map_sum, hg, f.map_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nf : M \u2192\u2097[R] M'\nhf_inj : ker f = \u22a5\nh : LinearIndependent R v\n\u22a2 Disjoint (span R (Set.range v)) (ker f)\n[PROOFSTEP]\nsimp only [hf_inj, disjoint_bot_right]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli : LinearIndependent R v\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\n\u22a2 LinearIndependent R (Fin.cons x v)\n[PROOFSTEP]\nrw [Fintype.linearIndependent_iff] at hli \u22a2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\n\u22a2 \u2200 (g : Fin (m + 1) \u2192 R), \u2211 i : Fin (m + 1), g i \u2022 Fin.cons x v i = 0 \u2192 \u2200 (i : Fin (m + 1)), g i = 0\n[PROOFSTEP]\nrintro g total_eq j\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\ntotal_eq : \u2211 i : Fin (m + 1), g i \u2022 Fin.cons x v i = 0\nj : Fin (m + 1)\n\u22a2 g j = 0\n[PROOFSTEP]\nsimp_rw [Fin.sum_univ_succ, Fin.cons_zero, Fin.cons_succ] at total_eq \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\nj : Fin (m + 1)\ntotal_eq : g 0 \u2022 x + \u2211 x : Fin m, g (Fin.succ x) \u2022 v x = 0\n\u22a2 g j = 0\n[PROOFSTEP]\nhave : g 0 = 0 := by\n  refine' x_ortho (g 0) \u27e8\u2211 i : Fin m, g i.succ \u2022 v i, _\u27e9 total_eq\n  exact sum_mem fun i _ => smul_mem _ _ (subset_span \u27e8i, rfl\u27e9)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\nj : Fin (m + 1)\ntotal_eq : g 0 \u2022 x + \u2211 x : Fin m, g (Fin.succ x) \u2022 v x = 0\n\u22a2 g 0 = 0\n[PROOFSTEP]\nrefine' x_ortho (g 0) \u27e8\u2211 i : Fin m, g i.succ \u2022 v i, _\u27e9 total_eq\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\nj : Fin (m + 1)\ntotal_eq : g 0 \u2022 x + \u2211 x : Fin m, g (Fin.succ x) \u2022 v x = 0\n\u22a2 \u2211 i : Fin m, g (Fin.succ i) \u2022 v i \u2208 span R (range v)\n[PROOFSTEP]\nexact sum_mem fun i _ => smul_mem _ _ (subset_span \u27e8i, rfl\u27e9)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\nj : Fin (m + 1)\ntotal_eq : g 0 \u2022 x + \u2211 x : Fin m, g (Fin.succ x) \u2022 v x = 0\nthis : g 0 = 0\n\u22a2 g j = 0\n[PROOFSTEP]\nrw [this, zero_smul, zero_add] at total_eq \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nm : \u2115\nx : M\nv : Fin m \u2192 M\nhli\u271d : LinearIndependent R v\nhli : \u2200 (g : Fin m \u2192 R), \u2211 i : Fin m, g i \u2022 v i = 0 \u2192 \u2200 (i : Fin m), g i = 0\nx_ortho : \u2200 (c : R) (y : { x // x \u2208 span R (range v) }), c \u2022 x + \u2191y = 0 \u2192 c = 0\ng : Fin (m + 1) \u2192 R\nj : Fin (m + 1)\ntotal_eq : \u2211 x : Fin m, g (Fin.succ x) \u2022 v x = 0\nthis : g 0 = 0\n\u22a2 g j = 0\n[PROOFSTEP]\nexact Fin.cases this (hli _ total_eq) j\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\n\u22a2 LinearIndependent R v\n[PROOFSTEP]\nrefine' linearIndependent_iff'.mpr fun s g hg i hi => hinj _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (fun r => r \u2022 1) (g i) = (fun r => r \u2022 1) 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i \u2022 1 = 0 \u2022 1\n[PROOFSTEP]\nrw [zero_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i \u2022 1 = 0\n[PROOFSTEP]\nrefine' (linearIndependent_iff'.mp li : _) _ (g \u00b7 \u2022 (1 : K)) _ i hi\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 i in s, (fun x => g x \u2022 1) i \u2022 v i = 0\n[PROOFSTEP]\nsimp_rw [smul_assoc, one_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Semiring R\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : AddCommMonoid M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\ninst\u271d\u00b3 : Semiring K\ninst\u271d\u00b2 : SMulWithZero R K\ninst\u271d\u00b9 : Module K M\ninst\u271d : IsScalarTower R K M\nhinj : Injective fun r => r \u2022 1\nli : LinearIndependent K v\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 \u2211 x in s, g x \u2022 v x = 0\n[PROOFSTEP]\nexact hg\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nlet f : t.map (Embedding.subtype s) \u2192 s := fun x =>\n  \u27e8x.1, by\n    obtain \u27e8x, h\u27e9 := x\n    rw [Finset.mem_map] at h \n    obtain \u27e8a, _ha, rfl\u27e9 := h\n    simp only [Subtype.coe_prop, Embedding.coe_subtype]\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nx : { x // x \u2208 Finset.map (Embedding.subtype s) t }\n\u22a2 \u2191x \u2208 s\n[PROOFSTEP]\nobtain \u27e8x, h\u27e9 := x\n[GOAL]\ncase mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nx : M\nh : x \u2208 Finset.map (Embedding.subtype s) t\n\u22a2 \u2191{ val := x, property := h } \u2208 s\n[PROOFSTEP]\nrw [Finset.mem_map] at h \n[GOAL]\ncase mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nx : M\nh\u271d : x \u2208 Finset.map (Embedding.subtype s) t\nh : \u2203 a, a \u2208 t \u2227 \u2191(Embedding.subtype s) a = x\n\u22a2 \u2191{ val := x, property := h\u271d } \u2208 s\n[PROOFSTEP]\nobtain \u27e8a, _ha, rfl\u27e9 := h\n[GOAL]\ncase mk.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\na : Subtype s\n_ha : a \u2208 t\nh : \u2191(Embedding.subtype s) a \u2208 Finset.map (Embedding.subtype s) t\n\u22a2 \u2191{ val := \u2191(Embedding.subtype s) a, property := h } \u2208 s\n[PROOFSTEP]\nsimp only [Subtype.coe_prop, Embedding.coe_subtype]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nconvert LinearIndependent.comp li f ?_\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\n\u22a2 Injective f\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9 \u27e8y, hy\u27e9\n[GOAL]\ncase mk.mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\nx : M\nhx : x \u2208 Finset.map (Embedding.subtype s) t\ny : M\nhy : y \u2208 Finset.map (Embedding.subtype s) t\n\u22a2 f { val := x, property := hx } = f { val := y, property := hy } \u2192\n    { val := x, property := hx } = { val := y, property := hy }\n[PROOFSTEP]\nrw [Finset.mem_map] at hx hy \n[GOAL]\ncase mk.mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\nx : M\nhx\u271d : x \u2208 Finset.map (Embedding.subtype s) t\nhx : \u2203 a, a \u2208 t \u2227 \u2191(Embedding.subtype s) a = x\ny : M\nhy\u271d : y \u2208 Finset.map (Embedding.subtype s) t\nhy : \u2203 a, a \u2208 t \u2227 \u2191(Embedding.subtype s) a = y\n\u22a2 f { val := x, property := hx\u271d } = f { val := y, property := hy\u271d } \u2192\n    { val := x, property := hx\u271d } = { val := y, property := hy\u271d }\n[PROOFSTEP]\nobtain \u27e8a, _ha, rfl\u27e9 := hx\n[GOAL]\ncase mk.mk.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y\u271d : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\ny : M\nhy\u271d : y \u2208 Finset.map (Embedding.subtype s) t\nhy : \u2203 a, a \u2208 t \u2227 \u2191(Embedding.subtype s) a = y\na : Subtype s\n_ha : a \u2208 t\nhx : \u2191(Embedding.subtype s) a \u2208 Finset.map (Embedding.subtype s) t\n\u22a2 f { val := \u2191(Embedding.subtype s) a, property := hx } = f { val := y, property := hy\u271d } \u2192\n    { val := \u2191(Embedding.subtype s) a, property := hx } = { val := y, property := hy\u271d }\n[PROOFSTEP]\nobtain \u27e8b, _hb, rfl\u27e9 := hy\n[GOAL]\ncase mk.mk.intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b\u271d : R\nx y : M\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\nf : { x // x \u2208 Finset.map (Embedding.subtype s) t } \u2192 \u2191s := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }\na : Subtype s\n_ha : a \u2208 t\nhx : \u2191(Embedding.subtype s) a \u2208 Finset.map (Embedding.subtype s) t\nb : Subtype s\n_hb : b \u2208 t\nhy : \u2191(Embedding.subtype s) b \u2208 Finset.map (Embedding.subtype s) t\n\u22a2 f { val := \u2191(Embedding.subtype s) a, property := hx } = f { val := \u2191(Embedding.subtype s) b, property := hy } \u2192\n    { val := \u2191(Embedding.subtype s) a, property := hx } = { val := \u2191(Embedding.subtype s) b, property := hy }\n[PROOFSTEP]\nsimp only [imp_self, Subtype.mk_eq_mk]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\n\u22a2 \u2200 (s : Set M), LinearIndependent R Subtype.val \u2192 #\u2191s \u2264 \u2191n\n[PROOFSTEP]\nintro s li\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\ns : Set M\nli : LinearIndependent R Subtype.val\n\u22a2 #\u2191s \u2264 \u2191n\n[PROOFSTEP]\napply Cardinal.card_le_of\n[GOAL]\ncase H\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\ns : Set M\nli : LinearIndependent R Subtype.val\n\u22a2 \u2200 (s_1 : Finset \u2191s), Finset.card s_1 \u2264 n\n[PROOFSTEP]\nintro t\n[GOAL]\ncase H\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\n\u22a2 Finset.card t \u2264 n\n[PROOFSTEP]\nrw [\u2190 Finset.card_map (Embedding.subtype s)]\n[GOAL]\ncase H\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\n\u22a2 Finset.card (Finset.map (Embedding.subtype s) t) \u2264 n\n[PROOFSTEP]\napply H\n[GOAL]\ncase H.a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nn : \u2115\nH : \u2200 (s : Finset M), (LinearIndependent R fun i => \u2191i) \u2192 Finset.card s \u2264 n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset \u2191s\n\u22a2 LinearIndependent R fun i => \u2191i\n[PROOFSTEP]\napply linearIndependent_finset_map_embedding_subtype _ li\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 LinearIndependent R (v \u2218 Subtype.val) \u2194\n    \u2200 (l : \u03b9 \u2192\u2080 R), l \u2208 Finsupp.supported R R s \u2192 \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\n[PROOFSTEP]\nsimp only [linearIndependent_iff, (\u00b7 \u2218 \u00b7), Finsupp.mem_supported, Finsupp.total_apply, Set.subset_def, Finset.mem_coe]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 (\u2200 (l : \u2191s \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0 \u2192 l = 0) \u2194\n    \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 (\u2200 (l : \u2191s \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0 \u2192 l = 0) \u2192\n    \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\n[PROOFSTEP]\nintro h l hl\u2081 hl\u2082\n[GOAL]\ncase mp\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u2191s \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0 \u2192 l = 0\nl : \u03b9 \u2192\u2080 R\nhl\u2081 : \u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s\nhl\u2082 : (Finsupp.sum l fun i a => a \u2022 v i) = 0\n\u22a2 l = 0\n[PROOFSTEP]\nhave := h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl\u2081).trans hl\u2082)\n[GOAL]\ncase mp\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u2191s \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0 \u2192 l = 0\nl : \u03b9 \u2192\u2080 R\nhl\u2081 : \u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s\nhl\u2082 : (Finsupp.sum l fun i a => a \u2022 v i) = 0\nthis : Finsupp.subtypeDomain s l = 0\n\u22a2 l = 0\n[PROOFSTEP]\nexact (Finsupp.subtypeDomain_eq_zero_iff hl\u2081).1 this\n[GOAL]\ncase mpr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 (\u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0) \u2192\n    \u2200 (l : \u2191s \u2192\u2080 R), (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0 \u2192 l = 0\n[PROOFSTEP]\nintro h l hl\n[GOAL]\ncase mpr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 l = 0\n[PROOFSTEP]\nrefine' Finsupp.embDomain_eq_zero.1 (h (l.embDomain <| Function.Embedding.subtype s) _ _)\n[GOAL]\ncase mpr.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 (Finsupp.embDomain (Embedding.subtype s) l).support \u2192 x \u2208 s\n[PROOFSTEP]\nsuffices \u2200 i hi, \u00acl \u27e8i, hi\u27e9 = 0 \u2192 i \u2208 s by simpa\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\nthis : \u2200 (i : \u03b9) (hi : i \u2208 s), \u00ac\u2191l { val := i, property := hi } = 0 \u2192 i \u2208 s\n\u22a2 \u2200 (x : \u03b9), x \u2208 (Finsupp.embDomain (Embedding.subtype s) l).support \u2192 x \u2208 s\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase mpr.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 \u2200 (i : \u03b9) (hi : i \u2208 s), \u00ac\u2191l { val := i, property := hi } = 0 \u2192 i \u2208 s\n[PROOFSTEP]\nintros\n[GOAL]\ncase mpr.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\ni\u271d : \u03b9\nhi\u271d : i\u271d \u2208 s\na\u271d : \u00ac\u2191l { val := i\u271d, property := hi\u271d } = 0\n\u22a2 i\u271d \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase mpr.refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 (Finsupp.sum (Finsupp.embDomain (Embedding.subtype s) l) fun i a => a \u2022 v i) = 0\n[PROOFSTEP]\nrwa [Finsupp.embDomain_eq_mapDomain, Finsupp.sum_mapDomain_index]\n[GOAL]\ncase mpr.refine'_2.h_zero\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 \u2200 (b : \u03b9), 0 \u2022 v b = 0\ncase mpr.refine'_2.h_add\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), (\u2200 (x : \u03b9), x \u2208 l.support \u2192 x \u2208 s) \u2192 (Finsupp.sum l fun i a => a \u2022 v i) = 0 \u2192 l = 0\nl : \u2191s \u2192\u2080 R\nhl : (Finsupp.sum l fun i a => a \u2022 v \u2191i) = 0\n\u22a2 \u2200 (b : \u03b9) (m\u2081 m\u2082 : R), (m\u2081 + m\u2082) \u2022 v b = m\u2081 \u2022 v b + m\u2082 \u2022 v b\n[PROOFSTEP]\nexacts [fun _ => zero_smul _ _, fun _ _ _ => add_smul _ _ _]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 \u00acLinearIndependent R (v \u2218 Subtype.val) \u2194 \u2203 f, f \u2208 Finsupp.supported R R s \u2227 \u2191(Finsupp.total \u03b9 M R v) f = 0 \u2227 f \u2260 0\n[PROOFSTEP]\nsimp [linearIndependent_comp_subtype, and_left_comm]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\n\u22a2 (LinearIndependent R fun x => \u2191x) \u2194\n    \u2200 (l : M \u2192\u2080 R), l \u2208 Finsupp.supported R R s \u2192 \u2191(Finsupp.total M M R id) l = 0 \u2192 l = 0\n[PROOFSTEP]\napply @linearIndependent_comp_subtype _ _ _ id\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set \u03b9\n\u22a2 LinearIndependent R (v \u2218 Subtype.val) \u2194 Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total \u03b9 M R v))\n[PROOFSTEP]\nrw [linearIndependent_comp_subtype, LinearMap.disjoint_ker]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\n\u22a2 (LinearIndependent R fun x => \u2191x) \u2194 Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total M M R id))\n[PROOFSTEP]\napply @linearIndependent_comp_subtype_disjoint _ _ _ id\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\n\u22a2 (LinearIndependent R fun x => \u2191x) \u2194 LinearMap.ker (Finsupp.totalOn M M R id s) = \u22a5\n[PROOFSTEP]\nrw [Finsupp.totalOn, LinearMap.ker, LinearMap.comap_codRestrict, Submodule.map_bot, comap_bot, LinearMap.ker_comp,\n  linearIndependent_subtype_disjoint, disjoint_iff_inf_le, \u2190 map_comap_subtype, map_le_iff_le_comap, comap_bot,\n  ker_subtype, le_bot_iff]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nsimp [linearIndependent_subtype_disjoint]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nt s : Set M\nh : t \u2286 s\n\u22a2 (LinearIndependent R fun x => \u2191x) \u2192 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nsimp only [linearIndependent_subtype_disjoint]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nt s : Set M\nh : t \u2286 s\n\u22a2 Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total M M R id)) \u2192\n    Disjoint (Finsupp.supported R R t) (LinearMap.ker (Finsupp.total M M R id))\n[PROOFSTEP]\nexact Disjoint.mono_left (Finsupp.supported_mono h)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nby_cases h\u03b7 : Nonempty \u03b7\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : Nonempty \u03b7\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nskip\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : Nonempty \u03b7\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' linearIndependent_of_finite (\u22c3 i, s i) fun t ht ft => _\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : Nonempty \u03b7\nt : Set M\nht : t \u2286 \u22c3 (i : \u03b7), s i\nft : Set.Finite t\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrcases finite_subset_iUnion ft ht with \u27e8I, fi, hI\u27e9\n[GOAL]\ncase pos.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : Nonempty \u03b7\nt : Set M\nht : t \u2286 \u22c3 (i : \u03b7), s i\nft : Set.Finite t\nI : Set \u03b7\nfi : Set.Finite I\nhI : t \u2286 \u22c3 (i : \u03b7) (_ : i \u2208 I), s i\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrcases hs.finset_le fi.toFinset with \u27e8i, hi\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : Nonempty \u03b7\nt : Set M\nht : t \u2286 \u22c3 (i : \u03b7), s i\nft : Set.Finite t\nI : Set \u03b7\nfi : Set.Finite I\nhI : t \u2286 \u22c3 (i : \u03b7) (_ : i \u2208 I), s i\ni : \u03b7\nhi : \u2200 (i_1 : \u03b7), i_1 \u2208 Finite.toFinset fi \u2192 s i_1 \u2286 s i\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nexact (h i).mono (Subset.trans hI <| iUnion\u2082_subset fun j hj => hi j (fi.mem_toFinset.2 hj))\n[GOAL]\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : \u00acNonempty \u03b7\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine (linearIndependent_empty R M).mono (t := iUnion (s \u00b7)) ?_\n[GOAL]\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : \u00acNonempty \u03b7\n\u22a2 \u22c3 (x : \u03b7), s x \u2286 \u2205\n[PROOFSTEP]\nrintro _ \u27e8_, \u27e8i, _\u27e9, _\u27e9\n[GOAL]\ncase neg.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : \u03b7 \u2192 Set M\nhs : Directed (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (i : \u03b7), LinearIndependent R fun x => \u2191x\nh\u03b7 : \u00acNonempty \u03b7\na\u271d : M\nw\u271d : Set M\nright\u271d : a\u271d \u2208 w\u271d\ni : \u03b7\nh\u271d : (fun x => s x) i = w\u271d\n\u22a2 a\u271d \u2208 \u2205\n[PROOFSTEP]\nexact h\u03b7 \u27e8i\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set M), a \u2208 s \u2192 LinearIndependent R Subtype.val\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrw [sUnion_eq_iUnion]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set M), a \u2208 s \u2192 LinearIndependent R Subtype.val\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nexact linearIndependent_iUnion_of_directed hs.directed_val (by simpa using h)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set (Set M)\nhs : DirectedOn (fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : Set M), a \u2208 s \u2192 LinearIndependent R Subtype.val\n\u22a2 \u2200 (i : \u2191s), LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : Set \u03b7\nt : \u03b7 \u2192 Set M\nhs : DirectedOn (t \u207b\u00b9'o fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : \u03b7), a \u2208 s \u2192 LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrw [biUnion_eq_iUnion]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : Set \u03b7\nt : \u03b7 \u2192 Set M\nhs : DirectedOn (t \u207b\u00b9'o fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : \u03b7), a \u2208 s \u2192 LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nexact linearIndependent_iUnion_of_directed (directed_comp.2 <| hs.directed_val) (by simpa using h)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M'\ninst\u271d\u00b3 : AddCommMonoid M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\ns : Set \u03b7\nt : \u03b7 \u2192 Set M\nhs : DirectedOn (t \u207b\u00b9'o fun x x_1 => x \u2286 x_1) s\nh : \u2200 (a : \u03b7), a \u2208 s \u2192 LinearIndependent R fun x => \u2191x\n\u22a2 \u2200 (i : \u2191s), LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\n\u22a2 Injective v\n[PROOFSTEP]\nintro i j hij\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\n\u22a2 i = j\n[PROOFSTEP]\nlet l : \u03b9 \u2192\u2080 R := Finsupp.single i (1 : R) - Finsupp.single j 1\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\n\u22a2 i = j\n[PROOFSTEP]\nhave h_total : Finsupp.total \u03b9 M R v l = 0 :=\n  by\n  simp_rw [LinearMap.map_sub, Finsupp.total_apply]\n  simp [hij]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = 0\n[PROOFSTEP]\nsimp_rw [LinearMap.map_sub, Finsupp.total_apply]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\n\u22a2 ((Finsupp.sum (Finsupp.single i 1) fun i a => a \u2022 v i) - Finsupp.sum (Finsupp.single j 1) fun i a => a \u2022 v i) = 0\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 i = j\n[PROOFSTEP]\nhave h_single_eq : Finsupp.single i (1 : R) = Finsupp.single j 1 :=\n  by\n  rw [linearIndependent_iff] at hv \n  simp [eq_add_of_sub_eq' (hv l h_total)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i 1 = Finsupp.single j 1\n[PROOFSTEP]\nrw [linearIndependent_iff] at hv \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i 1 = Finsupp.single j 1\n[PROOFSTEP]\nsimp [eq_add_of_sub_eq' (hv l h_total)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ni j : \u03b9\nhij : v i = v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i 1 - Finsupp.single j 1\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i 1 = Finsupp.single j 1\n\u22a2 i = j\n[PROOFSTEP]\nsimpa [Finsupp.single_eq_single_iff] using h_single_eq\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 M\nhf : LinearIndependent R f\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nnontriviality R\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 M\nhf : LinearIndependent R f\n\u271d : Nontrivial R\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nexact (linearIndependent_subtype_range hf.injective).2 hf\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9'\u271d : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b9'\ng : \u03b9' \u2192 M\nhs : LinearIndependent R fun x => g (f \u2191x)\n\u22a2 LinearIndependent R fun x => g \u2191x\n[PROOFSTEP]\nnontriviality R\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9'\u271d : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b9'\ng : \u03b9' \u2192 M\nhs : LinearIndependent R fun x => g (f \u2191x)\n\u271d : Nontrivial R\n\u22a2 LinearIndependent R fun x => g \u2191x\n[PROOFSTEP]\nhave : InjOn f s := injOn_iff_injective.2 hs.injective.of_comp\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9'\u271d : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ns : Set \u03b9\nf : \u03b9 \u2192 \u03b9'\ng : \u03b9' \u2192 M\nhs : LinearIndependent R fun x => g (f \u2191x)\n\u271d : Nontrivial R\nthis : InjOn f s\n\u22a2 LinearIndependent R fun x => g \u2191x\n[PROOFSTEP]\nexact (linearIndependent_equiv' (Equiv.Set.imageOfInjOn f s this) rfl).1 hs\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\ns : Set \u03b9\nf : \u03b9 \u2192 M\nhs : LinearIndependent R fun x => f \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nconvert LinearIndependent.image_of_comp s f id hs\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : LinearIndependent R v\nw : \u03b9 \u2192 G\n\u22a2 LinearIndependent R (w \u2022 v)\n[PROOFSTEP]\nrw [linearIndependent_iff''] at hv \u22a2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\n\u22a2 \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 (w \u2022 v) i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nintro s g hgs hsum i\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 g i = 0\n[PROOFSTEP]\nrefine' (smul_eq_zero_iff_eq (w i)).1 _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 w i \u2022 g i = 0\n[PROOFSTEP]\nrefine' hv s (fun i => w i \u2022 g i) (fun i hi => _) _ i\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni\u271d i : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 (fun i => w i \u2022 g i) i = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni\u271d i : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 w i \u2022 g i = 0\n[PROOFSTEP]\nexact (hgs i hi).symm \u25b8 smul_zero _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 \u2211 i in s, (fun i => w i \u2022 g i) i \u2022 v i = 0\n[PROOFSTEP]\nrw [\u2190 hsum, Finset.sum_congr rfl _]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 (fun i => w i \u2022 g i) x \u2022 v x = g x \u2022 (w \u2022 v) x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u00b9\u2070 : Ring R\ninst\u271d\u2079 : AddCommGroup M\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : Module R M'\ninst\u271d\u2074 : Module R M''\na b : R\nx y : M\nG : Type u_8\nhG : Group G\ninst\u271d\u00b3 : DistribMulAction G R\ninst\u271d\u00b2 : DistribMulAction G M\ninst\u271d\u00b9 : IsScalarTower G R M\ninst\u271d : SMulCommClass G R M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 G\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni x\u271d : \u03b9\na\u271d : x\u271d \u2208 s\n\u22a2 (fun i => w i \u2022 g i) x\u271d \u2022 v x\u271d = g x\u271d \u2022 (w \u2022 v) x\u271d\n[PROOFSTEP]\nerw [Pi.smul_apply, smul_assoc, smul_comm]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : LinearIndependent R v\nw : \u03b9 \u2192 R\u02e3\n\u22a2 LinearIndependent R (w \u2022 v)\n[PROOFSTEP]\nrw [linearIndependent_iff''] at hv \u22a2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\n\u22a2 \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 (w \u2022 v) i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\n[PROOFSTEP]\nintro s g hgs hsum i\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 g i = 0\n[PROOFSTEP]\nrw [\u2190 (w i).mul_left_eq_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 g i * \u2191(w i) = 0\n[PROOFSTEP]\nrefine' hv s (fun i => g i \u2022 (w i : R)) (fun i hi => _) _ i\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni\u271d i : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 (fun i => g i \u2022 \u2191(w i)) i = 0\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni\u271d i : \u03b9\nhi : \u00aci \u2208 s\n\u22a2 g i \u2022 \u2191(w i) = 0\n[PROOFSTEP]\nexact (hgs i hi).symm \u25b8 zero_smul _ _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 \u2211 i in s, (fun i => g i \u2022 \u2191(w i)) i \u2022 v i = 0\n[PROOFSTEP]\nrw [\u2190 hsum, Finset.sum_congr rfl _]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni : \u03b9\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 (fun i => g i \u2022 \u2191(w i)) x \u2022 v x = g x \u2022 (w \u2022 v) x\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni x\u271d : \u03b9\na\u271d : x\u271d \u2208 s\n\u22a2 (fun i => g i \u2022 \u2191(w i)) x\u271d \u2022 v x\u271d = g x\u271d \u2022 (w \u2022 v) x\u271d\n[PROOFSTEP]\nerw [Pi.smul_apply, smul_assoc]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nhv : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), (\u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0) \u2192 \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9), g i = 0\nw : \u03b9 \u2192 R\u02e3\ns : Finset \u03b9\ng : \u03b9 \u2192 R\nhgs : \u2200 (i : \u03b9), \u00aci \u2208 s \u2192 g i = 0\nhsum : \u2211 i in s, g i \u2022 (w \u2022 v) i = 0\ni x\u271d : \u03b9\na\u271d : x\u271d \u2208 s\n\u22a2 g x\u271d \u2022 \u2191(w x\u271d) \u2022 v x\u271d = g x\u271d \u2022 w x\u271d \u2022 v x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\n\u22a2 s = s' \u2227 t = t'\n[PROOFSTEP]\nhave : (s - s') \u2022 x + (t - t') \u2022 y = 0 :=\n  by\n  rw [\u2190 sub_eq_zero_of_eq h', \u2190 sub_eq_zero]\n  simp only [sub_smul]\n  abel\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\n\u22a2 (s - s') \u2022 x + (t - t') \u2022 y = 0\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero_of_eq h', \u2190 sub_eq_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\n\u22a2 (s - s') \u2022 x + (t - t') \u2022 y - (s \u2022 x + t \u2022 y - (s' \u2022 x + t' \u2022 y)) = 0\n[PROOFSTEP]\nsimp only [sub_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\n\u22a2 s \u2022 x - s' \u2022 x + (t \u2022 y - t' \u2022 y) - (s \u2022 x + t \u2022 y - (s' \u2022 x + t' \u2022 y)) = 0\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\n\u22a2 s \u2022 x - s' \u2022 x + (t \u2022 y - t' \u2022 y) - (s \u2022 x + t \u2022 y - (s' \u2022 x + t' \u2022 y)) = 0\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y\u271d x y : M\nh : LinearIndependent R ![x, y]\ns t s' t' : R\nh' : s \u2022 x + t \u2022 y = s' \u2022 x + t' \u2022 y\nthis : (s - s') \u2022 x + (t - t') \u2022 y = 0\n\u22a2 s = s' \u2227 t = t'\n[PROOFSTEP]\nsimpa [sub_eq_zero] using h.eq_zero_of_pair this\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\n\u22a2 Maximal i \u2194 \u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\n\u22a2 Maximal i \u2192 \u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j\n[PROOFSTEP]\nrintro p \u03ba w i' j rfl\n[GOAL]\ncase mp\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03ba : Type v\nw : \u03ba \u2192 M\ni' : LinearIndependent R w\nj : \u03b9 \u2192 \u03ba\ni : LinearIndependent R (w \u2218 j)\np : Maximal i\n\u22a2 Surjective j\n[PROOFSTEP]\nspecialize p (range w) i'.coe_range (range_comp_subset_range _ _)\n[GOAL]\ncase mp\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03ba : Type v\nw : \u03ba \u2192 M\ni' : LinearIndependent R w\nj : \u03b9 \u2192 \u03ba\ni : LinearIndependent R (w \u2218 j)\np : range (w \u2218 j) = range w\n\u22a2 Surjective j\n[PROOFSTEP]\nrw [range_comp, \u2190 @image_univ _ _ w] at p \n[GOAL]\ncase mp\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\n\u03ba : Type v\nw : \u03ba \u2192 M\ni' : LinearIndependent R w\nj : \u03b9 \u2192 \u03ba\ni : LinearIndependent R (w \u2218 j)\np : w '' range j = w '' univ\n\u22a2 Surjective j\n[PROOFSTEP]\nexact range_iff_surjective.mp (image_injective.mpr i'.injective p)\n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\n\u22a2 (\u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j) \u2192 Maximal i\n[PROOFSTEP]\nintro p w i' h\n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\np : \u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\n\u22a2 range v = w\n[PROOFSTEP]\nspecialize\n  p w ((\u2191) : w \u2192 M) i' (fun i => \u27e8v i, range_subset_iff.mp h i\u27e9)\n    (by\n      ext\n      simp)\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\np : \u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\n\u22a2 (Subtype.val \u2218 fun i => { val := v i, property := (_ : v i \u2208 w) }) = v\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\np : \u2200 (\u03ba : Type v) (w : \u03ba \u2192 M), LinearIndependent R w \u2192 \u2200 (j : \u03b9 \u2192 \u03ba), w \u2218 j = v \u2192 Surjective j\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\nx\u271d : \u03b9\n\u22a2 (Subtype.val \u2218 fun i => { val := v i, property := (_ : v i \u2208 w) }) x\u271d = v x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\np : Surjective fun i => { val := v i, property := (_ : v i \u2208 w) }\n\u22a2 range v = w\n[PROOFSTEP]\nhave q := congr_arg (fun s => ((\u2191) : w \u2192 M) '' s) p.range_eq\n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\np : Surjective fun i => { val := v i, property := (_ : v i \u2208 w) }\nq :\n  (fun s => Subtype.val '' s) (range fun i => { val := v i, property := (_ : v i \u2208 w) }) =\n    (fun s => Subtype.val '' s) univ\n\u22a2 range v = w\n[PROOFSTEP]\ndsimp at q \n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\np : Surjective fun i => { val := v i, property := (_ : v i \u2208 w) }\nq : (Subtype.val '' range fun i => { val := v i, property := (_ : v i \u2208 w) }) = Subtype.val '' univ\n\u22a2 range v = w\n[PROOFSTEP]\nrw [\u2190 image_univ, image_image] at q \n[GOAL]\ncase mpr\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR\u271d : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9\u271d \u2192 M\u271d\ninst\u271d\u00b9\u2070 : Ring R\u271d\ninst\u271d\u2079 : AddCommGroup M\u271d\ninst\u271d\u2078 : AddCommGroup M'\ninst\u271d\u2077 : AddCommGroup M''\ninst\u271d\u2076 : Module R\u271d M\u271d\ninst\u271d\u2075 : Module R\u271d M'\ninst\u271d\u2074 : Module R\u271d M''\na b : R\u271d\nx y : M\u271d\n\u03b9 : Type w\nR : Type u\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : Nontrivial R\nM : Type v\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v \u2264 w\np : Surjective fun i => { val := v i, property := (_ : v i \u2208 w) }\nq : (fun x => \u2191{ val := v x, property := (_ : v x \u2208 w) }) '' univ = Subtype.val '' univ\n\u22a2 range v = w\n[PROOFSTEP]\nsimpa using q\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\n\u22a2 i = j\n[PROOFSTEP]\nlet l : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 i = j\n[PROOFSTEP]\nhave h_total : Finsupp.total \u03b9 M R v l = 0 :=\n  by\n  simp_rw [LinearMap.map_sub, Finsupp.total_apply]\n  simp [h]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = 0\n[PROOFSTEP]\nsimp_rw [LinearMap.map_sub, Finsupp.total_apply]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\n\u22a2 ((Finsupp.sum (Finsupp.single i c) fun i a => a \u2022 v i) - Finsupp.sum (Finsupp.single j d) fun i a => a \u2022 v i) = 0\n[PROOFSTEP]\nsimp [h]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 i = j\n[PROOFSTEP]\nhave h_single_eq : Finsupp.single i c = Finsupp.single j d :=\n  by\n  rw [linearIndependent_iff] at li \n  simp [eq_add_of_sub_eq' (li l h_total)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i c = Finsupp.single j d\n[PROOFSTEP]\nrw [linearIndependent_iff] at li \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l = 0 \u2192 l = 0\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 Finsupp.single i c = Finsupp.single j d\n[PROOFSTEP]\nsimp [eq_add_of_sub_eq' (li l h_total)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\n\u22a2 i = j\n[PROOFSTEP]\nrcases(Finsupp.single_eq_single_iff _ _ _ _).mp h_single_eq with (\u27e8H, _\u27e9 | \u27e8hc, _\u27e9)\n[GOAL]\ncase inl.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nH : i = j\nright\u271d : c = d\n\u22a2 i = j\n[PROOFSTEP]\nexact H\n[GOAL]\ncase inr.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM\u271d : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\u271d\ninst\u271d\u2078 : Ring R\ninst\u271d\u2077 : AddCommGroup M\u271d\ninst\u271d\u2076 : AddCommGroup M'\ninst\u271d\u2075 : AddCommGroup M''\ninst\u271d\u2074 : Module R M\u271d\ninst\u271d\u00b3 : Module R M'\ninst\u271d\u00b2 : Module R M''\na b : R\nx y : M\u271d\nM : Type u_8\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nv : \u03b9 \u2192 M\nli : LinearIndependent R v\nc d : R\ni j : \u03b9\nhc\u271d : c \u2260 0\nh : c \u2022 v i = d \u2022 v j\nl : \u03b9 \u2192\u2080 R := Finsupp.single i c - Finsupp.single j d\nh_total : \u2191(Finsupp.total \u03b9 M R v) l = 0\nh_single_eq : Finsupp.single i c = Finsupp.single j d\nhc : c = 0\nright\u271d : d = 0\n\u22a2 i = j\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\n\u22a2 Disjoint (span R (v '' s)) (span R (v '' t))\n[PROOFSTEP]\nsimp only [disjoint_def, Finsupp.mem_span_image_iff_total]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\n\u22a2 \u2200 (x : M),\n    (\u2203 l, l \u2208 Finsupp.supported R R s \u2227 \u2191(Finsupp.total \u03b9 M R v) l = x) \u2192\n      (\u2203 l, l \u2208 Finsupp.supported R R t \u2227 \u2191(Finsupp.total \u03b9 M R v) l = x) \u2192 x = 0\n[PROOFSTEP]\nrintro _ \u27e8l\u2081, hl\u2081, rfl\u27e9 \u27e8l\u2082, hl\u2082, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\nl\u2081 : \u03b9 \u2192\u2080 R\nhl\u2081 : l\u2081 \u2208 Finsupp.supported R R s\nl\u2082 : \u03b9 \u2192\u2080 R\nhl\u2082 : l\u2082 \u2208 Finsupp.supported R R t\nH : \u2191(Finsupp.total \u03b9 M R v) l\u2082 = \u2191(Finsupp.total \u03b9 M R v) l\u2081\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l\u2081 = 0\n[PROOFSTEP]\nrw [hv.injective_total.eq_iff] at H \n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\nl\u2081 : \u03b9 \u2192\u2080 R\nhl\u2081 : l\u2081 \u2208 Finsupp.supported R R s\nl\u2082 : \u03b9 \u2192\u2080 R\nhl\u2082 : l\u2082 \u2208 Finsupp.supported R R t\nH : l\u2082 = l\u2081\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l\u2081 = 0\n[PROOFSTEP]\nsubst l\u2082\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\nl\u2081 : \u03b9 \u2192\u2080 R\nhl\u2081 : l\u2081 \u2208 Finsupp.supported R R s\nhl\u2082 : l\u2081 \u2208 Finsupp.supported R R t\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l\u2081 = 0\n[PROOFSTEP]\nhave : l\u2081 = 0 := Submodule.disjoint_def.mp (Finsupp.disjoint_supported_supported hs) _ hl\u2081 hl\u2082\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set \u03b9\nhs : Disjoint s t\nl\u2081 : \u03b9 \u2192\u2080 R\nhl\u2081 : l\u2081 \u2208 Finsupp.supported R R s\nhl\u2082 : l\u2081 \u2208 Finsupp.supported R R t\nthis : l\u2081 = 0\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l\u2081 = 0\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\n\u22a2 \u00acv x \u2208 span R (v '' s)\n[PROOFSTEP]\nhave h' : v x \u2208 Submodule.span R (v '' { x }) :=\n  by\n  rw [Set.image_singleton]\n  exact mem_span_singleton_self (v x)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\n\u22a2 v x \u2208 span R (v '' {x})\n[PROOFSTEP]\nrw [Set.image_singleton]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\n\u22a2 v x \u2208 span R {v x}\n[PROOFSTEP]\nexact mem_span_singleton_self (v x)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\nh' : v x \u2208 span R (v '' {x})\n\u22a2 \u00acv x \u2208 span R (v '' s)\n[PROOFSTEP]\nintro w\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\nh' : v x \u2208 span R (v '' {x})\nw : v x \u2208 span R (v '' s)\n\u22a2 False\n[PROOFSTEP]\napply LinearIndependent.ne_zero x hv\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\nh' : v x \u2208 span R (v '' {x})\nw : v x \u2208 span R (v '' s)\n\u22a2 v x = 0\n[PROOFSTEP]\nrefine' disjoint_def.1 (hv.disjoint_span_image _) (v x) h' w\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\ns : Set \u03b9\nx : \u03b9\nh : \u00acx \u2208 s\nh' : v x \u2208 span R (v '' {x})\nw : v x \u2208 span R (v '' s)\n\u22a2 Disjoint {x} s\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 f.support\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) f \u2260 v x\n[PROOFSTEP]\nreplace h : x \u2209 (f.support : Set \u03b9) := h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) f \u2260 v x\n[PROOFSTEP]\nhave p := hv.not_mem_span_image h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\np : \u00acv x \u2208 span R (v '' \u2191f.support)\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) f \u2260 v x\n[PROOFSTEP]\nintro w\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\np : \u00acv x \u2208 span R (v '' \u2191f.support)\nw : \u2191(Finsupp.total \u03b9 M R v) f = v x\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 w] at p \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\np : \u00ac\u2191(Finsupp.total \u03b9 M R v) f \u2208 span R (v '' \u2191f.support)\nw : \u2191(Finsupp.total \u03b9 M R v) f = v x\n\u22a2 False\n[PROOFSTEP]\nrw [Finsupp.span_image_eq_map_total] at p \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\np : \u00ac\u2191(Finsupp.total \u03b9 M R v) f \u2208 Submodule.map (Finsupp.total \u03b9 M R v) (Finsupp.supported R R \u2191f.support)\nw : \u2191(Finsupp.total \u03b9 M R v) f = v x\n\u22a2 False\n[PROOFSTEP]\nsimp only [not_exists, not_and, mem_map] at p \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nx : \u03b9\nf : \u03b9 \u2192\u2080 R\nh : \u00acx \u2208 \u2191f.support\nw : \u2191(Finsupp.total \u03b9 M R v) f = v x\np : \u2200 (x : \u03b9 \u2192\u2080 R), x \u2208 Finsupp.supported R R \u2191f.support \u2192 \u00ac\u2191(Finsupp.total \u03b9 M R v) x = \u2191(Finsupp.total \u03b9 M R v) f\n\u22a2 False\n[PROOFSTEP]\nexact p f (f.mem_supported_support R) rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\n\u22a2 LinearIndependent R v \u2194\n    LinearIndependent R (v \u2218 Sum.inl) \u2227\n      LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (range (v \u2218 Sum.inl))) (span R (range (v \u2218 Sum.inr)))\n[PROOFSTEP]\nclassical\nrw [range_comp v, range_comp v]\nrefine' \u27e8_, _\u27e9\n\u00b7 intro h\n  refine' \u27e8h.comp _ Sum.inl_injective, h.comp _ Sum.inr_injective, _\u27e9\n  refine'\n    h.disjoint_span_image\n      _\n        -- Porting note: `isCompl_range_inl_range_inr.1` timeouts.\n  exact IsCompl.disjoint isCompl_range_inl_range_inr\nrintro \u27e8hl, hr, hlr\u27e9\nrw [linearIndependent_iff'] at *\nintro s g hg i hi\nhave :\n  ((\u2211 i in s.preimage Sum.inl (Sum.inl_injective.injOn _), (fun x => g x \u2022 v x) (Sum.inl i)) +\n      \u2211 i in s.preimage Sum.inr (Sum.inr_injective.injOn _), (fun x => g x \u2022 v x) (Sum.inr i)) =\n    0 :=\n  by\n  -- Porting note: `g` must be specified.\n  rw [Finset.sum_preimage' (g := fun x => g x \u2022 v x), Finset.sum_preimage' (g := fun x => g x \u2022 v x), \u2190\n    Finset.sum_union, \u2190 Finset.filter_or]\n  \u00b7 simpa only [\u2190 mem_union, range_inl_union_range_inr, mem_univ, Finset.filter_True]\n  \u00b7\n    -- Porting note: Here was one `exact`, but timeouted.\n    refine Finset.disjoint_filter.2 fun x _ hx => disjoint_left.1 ?_ hx\n    exact IsCompl.disjoint isCompl_range_inl_range_inr\n\u00b7 rw [\u2190 eq_neg_iff_add_eq_zero] at this \n  rw [disjoint_def'] at hlr \n  have A := by\n    refine hlr _ (sum_mem fun i _ => ?_) _ (neg_mem <| sum_mem fun i _ => ?_) this\n    \u00b7 exact smul_mem _ _ (subset_span \u27e8Sum.inl i, mem_range_self _, rfl\u27e9)\n    \u00b7 exact smul_mem _ _ (subset_span \u27e8Sum.inr i, mem_range_self _, rfl\u27e9)\n  cases' i with i i\n  \u00b7 exact hl _ _ A i (Finset.mem_preimage.2 hi)\n  \u00b7 rw [this, neg_eq_zero] at A \n    exact hr _ _ A i (Finset.mem_preimage.2 hi)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\n\u22a2 LinearIndependent R v \u2194\n    LinearIndependent R (v \u2218 Sum.inl) \u2227\n      LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (range (v \u2218 Sum.inl))) (span R (range (v \u2218 Sum.inr)))\n[PROOFSTEP]\nrw [range_comp v, range_comp v]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\n\u22a2 LinearIndependent R v \u2194\n    LinearIndependent R (v \u2218 Sum.inl) \u2227\n      LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n[PROOFSTEP]\nrefine' \u27e8_, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\n\u22a2 LinearIndependent R v \u2192\n    LinearIndependent R (v \u2218 Sum.inl) \u2227\n      LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n[PROOFSTEP]\nintro h\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nh : LinearIndependent R v\n\u22a2 LinearIndependent R (v \u2218 Sum.inl) \u2227\n    LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n[PROOFSTEP]\nrefine' \u27e8h.comp _ Sum.inl_injective, h.comp _ Sum.inr_injective, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nh : LinearIndependent R v\n\u22a2 Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n[PROOFSTEP]\nrefine'\n  h.disjoint_span_image\n    _\n      -- Porting note: `isCompl_range_inl_range_inr.1` timeouts.\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nh : LinearIndependent R v\n\u22a2 Disjoint (range Sum.inl) (range Sum.inr)\n[PROOFSTEP]\nexact IsCompl.disjoint isCompl_range_inl_range_inr\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\n\u22a2 LinearIndependent R (v \u2218 Sum.inl) \u2227\n      LinearIndependent R (v \u2218 Sum.inr) \u2227 Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr)) \u2192\n    LinearIndependent R v\n[PROOFSTEP]\nrintro \u27e8hl, hr, hlr\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : LinearIndependent R (v \u2218 Sum.inl)\nhr : LinearIndependent R (v \u2218 Sum.inr)\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n\u22a2 LinearIndependent R v\n[PROOFSTEP]\nrw [linearIndependent_iff'] at *\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\n\u22a2 \u2200 (s : Finset (\u03b9 \u2295 \u03b9')) (g : \u03b9 \u2295 \u03b9' \u2192 R), \u2211 i in s, g i \u2022 v i = 0 \u2192 \u2200 (i : \u03b9 \u2295 \u03b9'), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nintro s g hg i hi\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\n\u22a2 g i = 0\n[PROOFSTEP]\nhave :\n  ((\u2211 i in s.preimage Sum.inl (Sum.inl_injective.injOn _), (fun x => g x \u2022 v x) (Sum.inl i)) +\n      \u2211 i in s.preimage Sum.inr (Sum.inr_injective.injOn _), (fun x => g x \u2022 v x) (Sum.inr i)) =\n    0 :=\n  by\n  -- Porting note: `g` must be specified.\n  rw [Finset.sum_preimage' (g := fun x => g x \u2022 v x), Finset.sum_preimage' (g := fun x => g x \u2022 v x), \u2190\n    Finset.sum_union, \u2190 Finset.filter_or]\n  \u00b7 simpa only [\u2190 mem_union, range_inl_union_range_inr, mem_univ, Finset.filter_True]\n  \u00b7\n    -- Porting note: Here was one `exact`, but timeouted.\n    refine Finset.disjoint_filter.2 fun x _ hx => disjoint_left.1 ?_ hx\n    exact IsCompl.disjoint isCompl_range_inl_range_inr\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\n\u22a2 \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) +\n      \u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i) =\n    0\n[PROOFSTEP]\nrw [Finset.sum_preimage' (g := fun x => g x \u2022 v x), Finset.sum_preimage' (g := fun x => g x \u2022 v x), \u2190 Finset.sum_union,\n  \u2190 Finset.filter_or]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\n\u22a2 \u2211 x in Finset.filter (fun a => a \u2208 range Sum.inl \u2228 a \u2208 range Sum.inr) s, g x \u2022 v x = 0\n[PROOFSTEP]\nsimpa only [\u2190 mem_union, range_inl_union_range_inr, mem_univ, Finset.filter_True]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\n\u22a2 Disjoint (Finset.filter (fun x => x \u2208 range Sum.inl) s) (Finset.filter (fun x => x \u2208 range Sum.inr) s)\n[PROOFSTEP]\nrefine Finset.disjoint_filter.2 fun x _ hx => disjoint_left.1 ?_ hx\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d\u00b9 y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nx : \u03b9 \u2295 \u03b9'\nx\u271d : x \u2208 s\nhx : x \u2208 range Sum.inl\n\u22a2 Disjoint (range Sum.inl) (range Sum.inr)\n[PROOFSTEP]\nexact IsCompl.disjoint isCompl_range_inl_range_inr\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) +\n      \u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i) =\n    0\n\u22a2 g i = 0\n[PROOFSTEP]\nrw [\u2190 eq_neg_iff_add_eq_zero] at this \n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : Disjoint (span R (v '' range Sum.inl)) (span R (v '' range Sum.inr))\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\n\u22a2 g i = 0\n[PROOFSTEP]\nrw [disjoint_def'] at hlr \n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\n\u22a2 g i = 0\n[PROOFSTEP]\nhave A := by\n  refine hlr _ (sum_mem fun i _ => ?_) _ (neg_mem <| sum_mem fun i _ => ?_) this\n  \u00b7 exact smul_mem _ _ (subset_span \u27e8Sum.inl i, mem_range_self _, rfl\u27e9)\n  \u00b7 exact smul_mem _ _ (subset_span \u27e8Sum.inr i, mem_range_self _, rfl\u27e9)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\n\u22a2 ?m.395655\n[PROOFSTEP]\nrefine hlr _ (sum_mem fun i _ => ?_) _ (neg_mem <| sum_mem fun i _ => ?_) this\n[GOAL]\ncase refine_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni\u271d : \u03b9 \u2295 \u03b9'\nhi : i\u271d \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\ni : \u03b9\nx\u271d : i \u2208 Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s))\n\u22a2 (fun x => g x \u2022 v x) (Sum.inl i) \u2208 span R (v '' range Sum.inl)\n[PROOFSTEP]\nexact smul_mem _ _ (subset_span \u27e8Sum.inl i, mem_range_self _, rfl\u27e9)\n[GOAL]\ncase refine_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni\u271d : \u03b9 \u2295 \u03b9'\nhi : i\u271d \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\ni : \u03b9'\nx\u271d : i \u2208 Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s))\n\u22a2 (fun x => g x \u2022 v x) (Sum.inr i) \u2208 span R (v '' range Sum.inr)\n[PROOFSTEP]\nexact smul_mem _ _ (subset_span \u27e8Sum.inr i, mem_range_self _, rfl\u27e9)\n[GOAL]\ncase refine'_2.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\ni : \u03b9 \u2295 \u03b9'\nhi : i \u2208 s\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\nA : \u2211 c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl c) = 0\n\u22a2 g i = 0\n[PROOFSTEP]\ncases' i with i i\n[GOAL]\ncase refine'_2.intro.intro.inl\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\nA : \u2211 c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl c) = 0\ni : \u03b9\nhi : Sum.inl i \u2208 s\n\u22a2 g (Sum.inl i) = 0\n[PROOFSTEP]\nexact hl _ _ A i (Finset.mem_preimage.2 hi)\n[GOAL]\ncase refine'_2.intro.intro.inr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\nA : \u2211 c in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl c) = 0\ni : \u03b9'\nhi : Sum.inr i \u2208 s\n\u22a2 g (Sum.inr i) = 0\n[PROOFSTEP]\nrw [this, neg_eq_zero] at A \n[GOAL]\ncase refine'_2.intro.intro.inr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2295 \u03b9' \u2192 M\nhl : \u2200 (s : Finset \u03b9) (g : \u03b9 \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inl) i = 0 \u2192 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nhr : \u2200 (s : Finset \u03b9') (g : \u03b9' \u2192 R), \u2211 i in s, g i \u2022 (v \u2218 Sum.inr) i = 0 \u2192 \u2200 (i : \u03b9'), i \u2208 s \u2192 g i = 0\nhlr : \u2200 (x : M), x \u2208 span R (v '' range Sum.inl) \u2192 \u2200 (y : M), y \u2208 span R (v '' range Sum.inr) \u2192 x = y \u2192 x = 0\ns : Finset (\u03b9 \u2295 \u03b9')\ng : \u03b9 \u2295 \u03b9' \u2192 R\nhg : \u2211 i in s, g i \u2022 v i = 0\nthis :\n  \u2211 i in Finset.preimage s Sum.inl (_ : InjOn Sum.inl (Sum.inl \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inl i) =\n    -\u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i)\nA : \u2211 i in Finset.preimage s Sum.inr (_ : InjOn Sum.inr (Sum.inr \u207b\u00b9' \u2191s)), (fun x => g x \u2022 v x) (Sum.inr i) = 0\ni : \u03b9'\nhi : Sum.inr i \u2208 s\n\u22a2 g (Sum.inr i) = 0\n[PROOFSTEP]\nexact hr _ _ A i (Finset.mem_preimage.2 hi)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\nhst : Disjoint (span R s) (span R t)\n\u22a2 Disjoint (span R (range fun x => \u2191x)) (span R (range fun x => \u2191x))\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\nhst : Disjoint (span R s) (span R t)\n\u22a2 range (Sum.elim (fun x => \u2191x) fun x => \u2191x) = s \u222a t\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nclassical\nrw [iUnion_eq_iUnion_finset f]\napply linearIndependent_iUnion_of_directed\n\u00b7 apply directed_of_sup\n  exact fun t\u2081 t\u2082 ht => iUnion_mono fun i => iUnion_subset_iUnion_const fun h => ht h\nintro t\ninduction' t using Finset.induction_on with i s his ih\n\u00b7 refine' (linearIndependent_empty R M).mono _\n  simp\n\u00b7 rw [Finset.set_biUnion_insert]\n  refine' (hl _).union ih _\n  rw [span_iUnion\u2082]\n  exact hd i s s.finite_toSet his\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrw [iUnion_eq_iUnion_finset f]\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\napply linearIndependent_iUnion_of_directed\n[GOAL]\ncase hs\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 Directed (fun x x_1 => x \u2286 x_1) fun i => \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 i), f i_1\n[PROOFSTEP]\napply directed_of_sup\n[GOAL]\ncase hs.H\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 \u2200 \u2983i j : Finset \u03b9\u2984, i \u2264 j \u2192 \u22c3 (i_1 : \u03b9) (_ : i_1 \u2208 i), f i_1 \u2286 \u22c3 (i : \u03b9) (_ : i \u2208 j), f i\n[PROOFSTEP]\nexact fun t\u2081 t\u2082 ht => iUnion_mono fun i => iUnion_subset_iUnion_const fun h => ht h\n[GOAL]\ncase h\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 \u2200 (i : Finset \u03b9), LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nintro t\n[GOAL]\ncase h\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\nt : Finset \u03b9\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\ninduction' t using Finset.induction_on with i s his ih\n[GOAL]\ncase h.empty\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' (linearIndependent_empty R M).mono _\n[GOAL]\ncase h.empty\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\n\u22a2 \u22c3 (i : \u03b9) (_ : i \u2208 \u2205), f i \u2286 \u2205\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.insert\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrw [Finset.set_biUnion_insert]\n[GOAL]\ncase h.insert\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' (hl _).union ih _\n[GOAL]\ncase h.insert\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R (f i)) (span R (\u22c3 (x : \u03b9) (_ : x \u2208 s), f x))\n[PROOFSTEP]\nrw [span_iUnion\u2082]\n[GOAL]\ncase h.insert\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9\u271d \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b9 : Type u_8\nf : \u03b9 \u2192 Set M\nhl : \u2200 (i : \u03b9), LinearIndependent R fun x => \u2191x\nhd : \u2200 (i : \u03b9) (t : Set \u03b9), Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 t), span R (f i))\ni : \u03b9\ns : Finset \u03b9\nhis : \u00aci \u2208 s\nih : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R (f i)) (\u2a06 (i : \u03b9) (_ : i \u2208 s), span R (f i))\n[PROOFSTEP]\nexact hd i s s.finite_toSet his\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u22a2 LinearIndependent R fun ji => f ji.fst ji.snd\n[PROOFSTEP]\nnontriviality R\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\n\u22a2 LinearIndependent R fun ji => f ji.fst ji.snd\n[PROOFSTEP]\napply LinearIndependent.of_subtype_range\n[GOAL]\ncase hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\n\u22a2 Injective fun ji => f ji.fst ji.snd\n[PROOFSTEP]\nrintro \u27e8x\u2081, x\u2082\u27e9 \u27e8y\u2081, y\u2082\u27e9 hxy\n[GOAL]\ncase hf.mk.mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := y\u2081, snd := y\u2082 }\n[PROOFSTEP]\nby_cases h_cases : x\u2081 = y\u2081\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : x\u2081 = y\u2081\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := y\u2081, snd := y\u2082 }\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := y\u2081, snd := y\u2082 }\n[PROOFSTEP]\nsubst h_cases\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 y\u2082 : \u03b9s x\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := y\u2082 }\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := x\u2081, snd := y\u2082 }\n[PROOFSTEP]\napply Sigma.eq\n[GOAL]\ncase pos.a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 y\u2082 : \u03b9s x\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := y\u2082 }\n\u22a2 Eq.recOn ?pos.h\u2081\u271d { fst := x\u2081, snd := x\u2082 }.snd = { fst := x\u2081, snd := y\u2082 }.snd\ncase pos.h\u2081\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 y\u2082 : \u03b9s x\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := y\u2082 }\n\u22a2 { fst := x\u2081, snd := x\u2082 }.fst = { fst := x\u2081, snd := y\u2082 }.fst\n[PROOFSTEP]\nrw [LinearIndependent.injective (hindep _) hxy]\n[GOAL]\ncase pos.h\u2081\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 y\u2082 : \u03b9s x\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := y\u2082 }\n\u22a2 { fst := x\u2081, snd := x\u2082 }.fst = { fst := x\u2081, snd := y\u2082 }.fst\ncase pos.h\u2081\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 y\u2082 : \u03b9s x\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := y\u2082 }\n\u22a2 { fst := x\u2081, snd := x\u2082 }.fst = { fst := x\u2081, snd := y\u2082 }.fst\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := y\u2081, snd := y\u2082 }\n[PROOFSTEP]\nhave h0 : f x\u2081 x\u2082 = 0 :=\n  by\n  apply\n    disjoint_def.1 (hd x\u2081 { y\u2081 } (finite_singleton y\u2081) fun h => h_cases (eq_of_mem_singleton h)) (f x\u2081 x\u2082)\n      (subset_span (mem_range_self _))\n  rw [iSup_singleton]\n  simp only at hxy \n  rw [hxy]\n  exact subset_span (mem_range_self y\u2082)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 f x\u2081 x\u2082 = 0\n[PROOFSTEP]\napply\n  disjoint_def.1 (hd x\u2081 { y\u2081 } (finite_singleton y\u2081) fun h => h_cases (eq_of_mem_singleton h)) (f x\u2081 x\u2082)\n    (subset_span (mem_range_self _))\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 f x\u2081 x\u2082 \u2208 \u2a06 (i : \u03b7) (_ : i \u2208 {y\u2081}), span R (range (f i))\n[PROOFSTEP]\nrw [iSup_singleton]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 f x\u2081 x\u2082 \u2208 span R (range (f y\u2081))\n[PROOFSTEP]\nsimp only at hxy \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : f x\u2081 x\u2082 = f y\u2081 y\u2082\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 f x\u2081 x\u2082 \u2208 span R (range (f y\u2081))\n[PROOFSTEP]\nrw [hxy]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : f x\u2081 x\u2082 = f y\u2081 y\u2082\nh_cases : \u00acx\u2081 = y\u2081\n\u22a2 f y\u2081 y\u2082 \u2208 span R (range (f y\u2081))\n[PROOFSTEP]\nexact subset_span (mem_range_self y\u2082)\n[GOAL]\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\nx\u2081 : \u03b7\nx\u2082 : \u03b9s x\u2081\ny\u2081 : \u03b7\ny\u2082 : \u03b9s y\u2081\nhxy : (fun ji => f ji.fst ji.snd) { fst := x\u2081, snd := x\u2082 } = (fun ji => f ji.fst ji.snd) { fst := y\u2081, snd := y\u2082 }\nh_cases : \u00acx\u2081 = y\u2081\nh0 : f x\u2081 x\u2082 = 0\n\u22a2 { fst := x\u2081, snd := x\u2082 } = { fst := y\u2081, snd := y\u2082 }\n[PROOFSTEP]\nexact False.elim ((hindep x\u2081).ne_zero _ h0)\n[GOAL]\ncase a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\nrw [range_sigma_eq_iUnion_range]\n[GOAL]\ncase a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\n\u03b7 : Type u_8\n\u03b9s : \u03b7 \u2192 Type u_9\nf : (j : \u03b7) \u2192 \u03b9s j \u2192 M\nhindep : \u2200 (j : \u03b7), LinearIndependent R (f j)\nhd :\n  \u2200 (i : \u03b7) (t : Set \u03b7),\n    Set.Finite t \u2192 \u00aci \u2208 t \u2192 Disjoint (span R (range (f i))) (\u2a06 (i : \u03b7) (_ : i \u2208 t), span R (range (f i)))\n\u271d : Nontrivial R\n\u22a2 LinearIndependent R Subtype.val\n[PROOFSTEP]\napply linearIndependent_iUnion_finite_subtype (fun j => (hindep j).to_subtype_range) hd\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 (\u03b9 \u2192\u2080 R) \u2243\u2097[R] { x // x \u2208 span R (range v) }\n[PROOFSTEP]\napply LinearEquiv.ofBijective (LinearMap.codRestrict (span R (range v)) (Finsupp.total \u03b9 M R v) _)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 Bijective \u2191(LinearMap.codRestrict (span R (range v)) (Finsupp.total \u03b9 M R v) ?m.425197)\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 \u2200 (c : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) c \u2208 span R (range v)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 Injective \u2191(LinearMap.codRestrict (span R (range v)) (Finsupp.total \u03b9 M R v) ?m.425197)\n[PROOFSTEP]\nrw [\u2190 LinearMap.ker_eq_bot, LinearMap.ker_codRestrict]\n[GOAL]\ncase left\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 LinearMap.ker (Finsupp.total \u03b9 M R v) = \u22a5\ncase left.hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 \u2200 (c : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) c \u2208 span R (range v)\n[PROOFSTEP]\napply hv\n[GOAL]\ncase left.hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 \u2200 (c : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) c \u2208 span R (range v)\n[PROOFSTEP]\nintro l\n[GOAL]\ncase left.hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l \u2208 span R (range v)\n[PROOFSTEP]\nrw [\u2190 Finsupp.range_total]\n[GOAL]\ncase left.hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l \u2208 LinearMap.range (Finsupp.total \u03b9 M R v)\n[PROOFSTEP]\nrw [LinearMap.mem_range]\n[GOAL]\ncase left.hf\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\n\u22a2 \u2203 y, \u2191(Finsupp.total \u03b9 M R v) y = \u2191(Finsupp.total \u03b9 M R v) l\n[PROOFSTEP]\napply mem_range_self l\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 Surjective\n    \u2191(LinearMap.codRestrict (span R (range v)) (Finsupp.total \u03b9 M R v)\n        (_ : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191(Finsupp.total \u03b9 M R v) l \u2208 span R (range v)))\n[PROOFSTEP]\nrw [\u2190 LinearMap.range_eq_top, LinearMap.range_eq_map, LinearMap.map_codRestrict, \u2190 LinearMap.range_le_iff_comap,\n  range_subtype, Submodule.map_top]\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 span R (range v) \u2264 LinearMap.range (Finsupp.total \u03b9 M R v)\n[PROOFSTEP]\nrw [Finsupp.range_total]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\n\u22a2 LinearMap.ker (repr hv) = \u22a5\n[PROOFSTEP]\nrw [LinearIndependent.repr, LinearEquiv.ker]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\n\u22a2 LinearMap.range (repr hv) = \u22a4\n[PROOFSTEP]\nrw [LinearIndependent.repr, LinearEquiv.range]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\n\u22a2 \u2191(repr hv) x = l\n[PROOFSTEP]\nhave : \u2191((LinearIndependent.totalEquiv hv : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] span R (range v)) l) = Finsupp.total \u03b9 M R v l := rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191(Finsupp.total \u03b9 M R v) l\n\u22a2 \u2191(repr hv) x = l\n[PROOFSTEP]\nhave : (LinearIndependent.totalEquiv hv : (\u03b9 \u2192\u2080 R) \u2192\u2097[R] span R (range v)) l = x :=\n  by\n  rw [eq] at this \n  exact Subtype.ext_iff.2 this\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191(Finsupp.total \u03b9 M R v) l\n\u22a2 \u2191\u2191(totalEquiv hv) l = x\n[PROOFSTEP]\nrw [eq] at this \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191x\n\u22a2 \u2191\u2191(totalEquiv hv) l = x\n[PROOFSTEP]\nexact Subtype.ext_iff.2 this\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis\u271d : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191(Finsupp.total \u03b9 M R v) l\nthis : \u2191\u2191(totalEquiv hv) l = x\n\u22a2 \u2191(repr hv) x = l\n[PROOFSTEP]\nrw [\u2190 LinearEquiv.symm_apply_apply hv.totalEquiv l]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis\u271d : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191(Finsupp.total \u03b9 M R v) l\nthis : \u2191\u2191(totalEquiv hv) l = x\n\u22a2 \u2191(repr hv) x = \u2191(LinearEquiv.symm (totalEquiv hv)) (\u2191(totalEquiv hv) l)\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\nl : \u03b9 \u2192\u2080 R\nx : { x // x \u2208 span R (range v) }\neq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191x\nthis\u271d : \u2191(\u2191\u2191(totalEquiv hv) l) = \u2191(Finsupp.total \u03b9 M R v) l\nthis : \u2191\u2191(totalEquiv hv) l = x\n\u22a2 \u2191(repr hv) (\u2191\u2191(totalEquiv hv) l) = \u2191(LinearEquiv.symm (totalEquiv hv)) (\u2191(totalEquiv hv) l)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ni : \u03b9\nx : { x // x \u2208 span R (range v) }\nhx : \u2191x = v i\n\u22a2 \u2191(repr hv) x = Finsupp.single i 1\n[PROOFSTEP]\napply hv.repr_eq\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ni : \u03b9\nx : { x // x \u2208 span R (range v) }\nhx : \u2191x = v i\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1) = \u2191x\n[PROOFSTEP]\nsimp [Finsupp.total_single, hx]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\n\u22a2 Span.repr R (range v) x = Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)) (\u2191(repr hv) x)\n[PROOFSTEP]\nhave p : (Span.repr R (Set.range v) x).equivMapDomain (Equiv.ofInjective _ hv.injective).symm = hv.repr x :=\n  by\n  apply (LinearIndependent.totalEquiv hv).injective\n  ext\n  simp only [LinearIndependent.totalEquiv_apply_coe, Equiv.self_comp_ofInjective_symm, LinearIndependent.total_repr,\n    Finsupp.total_equivMapDomain, Span.finsupp_total_repr]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\n\u22a2 Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)).symm (Span.repr R (range v) x) = \u2191(repr hv) x\n[PROOFSTEP]\napply (LinearIndependent.totalEquiv hv).injective\n[GOAL]\ncase a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\n\u22a2 \u2191(totalEquiv hv) (Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)).symm (Span.repr R (range v) x)) =\n    \u2191(totalEquiv hv) (\u2191(repr hv) x)\n[PROOFSTEP]\next\n[GOAL]\ncase a.a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\n\u22a2 \u2191(\u2191(totalEquiv hv) (Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)).symm (Span.repr R (range v) x))) =\n    \u2191(\u2191(totalEquiv hv) (\u2191(repr hv) x))\n[PROOFSTEP]\nsimp only [LinearIndependent.totalEquiv_apply_coe, Equiv.self_comp_ofInjective_symm, LinearIndependent.total_repr,\n  Finsupp.total_equivMapDomain, Span.finsupp_total_repr]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\np : Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)).symm (Span.repr R (range v) x) = \u2191(repr hv) x\n\u22a2 Span.repr R (range v) x = Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)) (\u2191(repr hv) x)\n[PROOFSTEP]\next \u27e8_, \u27e8i, rfl\u27e9\u27e9\n[GOAL]\ncase h.mk.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ninst\u271d : Nontrivial R\nx : { x // x \u2208 span R (range v) }\np : Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)).symm (Span.repr R (range v) x) = \u2191(repr hv) x\ni : \u03b9\n\u22a2 \u2191(Span.repr R (range v) x) { val := v i, property := (_ : \u2203 y, v y = v i) } =\n    \u2191(Finsupp.equivMapDomain (Equiv.ofInjective v (_ : Injective v)) (\u2191(repr hv) x))\n      { val := v i, property := (_ : \u2203 y, v y = v i) }\n[PROOFSTEP]\nsimp [\u2190 p]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nha : a \u2022 v i \u2208 span R (v '' (univ \\ {i}))\n\u22a2 a = 0\n[PROOFSTEP]\nrw [Finsupp.span_image_eq_map_total, mem_map] at ha \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nha : \u2203 y, y \u2208 Finsupp.supported R R (univ \\ {i}) \u2227 \u2191(Finsupp.total \u03b9 M R v) y = a \u2022 v i\n\u22a2 a = 0\n[PROOFSTEP]\nrcases ha with \u27e8l, hl, e\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nl : \u03b9 \u2192\u2080 R\nhl : l \u2208 Finsupp.supported R R (univ \\ {i})\ne : \u2191(Finsupp.total \u03b9 M R v) l = a \u2022 v i\n\u22a2 a = 0\n[PROOFSTEP]\nrw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nl : \u03b9 \u2192\u2080 R\nhl : l \u2208 Finsupp.supported R R (univ \\ {i})\ne : \u2191(Finsupp.total \u03b9 M R v) l = a \u2022 v i\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (l - Finsupp.single i a) = 0\n[PROOFSTEP]\nsimp [e]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nl : \u03b9 \u2192\u2080 R\nhl : Finsupp.single i a \u2208 Finsupp.supported R R (univ \\ {i})\ne : \u2191(Finsupp.total \u03b9 M R v) l = a \u2022 v i\n\u22a2 a = 0\n[PROOFSTEP]\nby_contra hn\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nl : \u03b9 \u2192\u2080 R\nhl : Finsupp.single i a \u2208 Finsupp.supported R R (univ \\ {i})\ne : \u2191(Finsupp.total \u03b9 M R v) l = a \u2022 v i\nhn : \u00aca = 0\n\u22a2 False\n[PROOFSTEP]\nexact (not_mem_of_mem_diff (hl <| by simp [hn])) (mem_singleton _)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na\u271d b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\na : R\nl : \u03b9 \u2192\u2080 R\nhl : Finsupp.single i a \u2208 Finsupp.supported R R (univ \\ {i})\ne : \u2191(Finsupp.total \u03b9 M R v) l = a \u2022 v i\nhn : \u00aca = 0\n\u22a2 i \u2208 \u2191(Finsupp.single i a).support\n[PROOFSTEP]\nsimp [hn]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\n\u22a2 l = 0\n[PROOFSTEP]\next i\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\n\u22a2 \u2191l i = \u21910 i\n[PROOFSTEP]\nsimp only [Finsupp.zero_apply]\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\n\u22a2 \u2191l i = 0\n[PROOFSTEP]\nby_contra hn\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' hn (H i _ _)\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\n\u22a2 \u2191l i \u2022 v i \u2208 span R (v '' (univ \\ {i}))\n[PROOFSTEP]\nrefine' (Finsupp.mem_span_image_iff_total R).2 \u27e8Finsupp.single i (l i) - l, _, _\u27e9\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\n\u22a2 Finsupp.single i (\u2191l i) - l \u2208 Finsupp.supported R R (univ \\ {i})\n[PROOFSTEP]\nrw [Finsupp.mem_supported']\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\n\u22a2 \u2200 (x : \u03b9), \u00acx \u2208 univ \\ {i} \u2192 \u2191(Finsupp.single i (\u2191l i) - l) x = 0\n[PROOFSTEP]\nintro j hj\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\nj : \u03b9\nhj : \u00acj \u2208 univ \\ {i}\n\u22a2 \u2191(Finsupp.single i (\u2191l i) - l) j = 0\n[PROOFSTEP]\nhave hij : j = i :=\n  Classical.not_not.1 fun hij : j \u2260 i => hj ((mem_diff _).2 \u27e8mem_univ _, fun h => hij (eq_of_mem_singleton h)\u27e9)\n[GOAL]\ncase h.refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\nj : \u03b9\nhj : \u00acj \u2208 univ \\ {i}\nhij : j = i\n\u22a2 \u2191(Finsupp.single i (\u2191l i) - l) j = 0\n[PROOFSTEP]\nsimp [hij]\n[GOAL]\ncase h.refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : \u2200 (i : \u03b9) (a : R), a \u2022 v i \u2208 span R (v '' (univ \\ {i})) \u2192 a = 0\nl : \u03b9 \u2192\u2080 R\nhl : \u2191(Finsupp.total \u03b9 M R v) l = 0\ni : \u03b9\nhn : \u00ac\u2191l i = 0\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i (\u2191l i) - l) = \u2191l i \u2022 v i\n[PROOFSTEP]\nsimp [hl]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\n\u22a2 CompleteLattice.Independent fun i => span R {v i}\n[PROOFSTEP]\nrefine' CompleteLattice.independent_def.mp fun i => _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\n\u22a2 Disjoint ((fun i => span R {v i}) i) (\u2a06 (j : \u03b9) (_ : j \u2260 i), (fun i => span R {v i}) j)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\n\u22a2 (fun i => span R {v i}) i \u2293 \u2a06 (j : \u03b9) (_ : j \u2260 i), (fun i => span R {v i}) j \u2264 \u22a5\n[PROOFSTEP]\nintro m hm\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nm : M\nhm : m \u2208 (fun i => span R {v i}) i \u2293 \u2a06 (j : \u03b9) (_ : j \u2260 i), (fun i => span R {v i}) j\n\u22a2 m \u2208 \u22a5\n[PROOFSTEP]\nsimp only [mem_inf, mem_span_singleton, iSup_subtype', \u2190 span_range_eq_iSup] at hm \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nm : M\nhm : (\u2203 a, a \u2022 v i = m) \u2227 m \u2208 span R (range fun i_1 => v \u2191i_1)\n\u22a2 m \u2208 \u22a5\n[PROOFSTEP]\nobtain \u27e8\u27e8r, rfl\u27e9, hm\u27e9 := hm\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\n\u22a2 r \u2022 v i \u2208 \u22a5\n[PROOFSTEP]\nsuffices r = 0 by simp [this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\nthis : r = 0\n\u22a2 r \u2022 v i \u2208 \u22a5\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\n\u22a2 r = 0\n[PROOFSTEP]\napply linearIndependent_iff_not_smul_mem_span.mp hv i\n[GOAL]\ncase intro.intro.a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\n\u22a2 r \u2022 v i \u2208 span R (v '' (univ \\ {i}))\n[PROOFSTEP]\nsuffices v '' (univ \\ { i }) = range fun j : { j // j \u2260 i } => v j by rwa [this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\nthis : v '' (univ \\ {i}) = range fun j => v \u2191j\n\u22a2 r \u2022 v i \u2208 span R (v '' (univ \\ {i}))\n[PROOFSTEP]\nrwa [this]\n[GOAL]\ncase intro.intro.a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\n\u22a2 v '' (univ \\ {i}) = range fun j => v \u2191j\n[PROOFSTEP]\next\n[GOAL]\ncase intro.intro.a.h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv\u271d hv : LinearIndependent R v\ni : \u03b9\nr : R\nhm : r \u2022 v i \u2208 span R (range fun i_1 => v \u2191i_1)\nx\u271d : M\n\u22a2 x\u271d \u2208 v '' (univ \\ {i}) \u2194 x\u271d \u2208 range fun j => v \u2191j\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nlet indep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 (\u2191) : I \u2192 M)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nlet X := { I : Set \u03b9 // indep I }\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nlet r : X \u2192 X \u2192 Prop := fun I J => I.1 \u2286 J.1\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nhave key : \u2200 c : Set X, IsChain r c \u2192 indep (\u22c3 (I : X) (_ : I \u2208 c), I) :=\n  by\n  intro c hc\n  dsimp\n  rw [linearIndependent_comp_subtype]\n  intro f hsupport hsum\n  rcases eq_empty_or_nonempty c with (rfl | hn)\n  \u00b7 simpa using hsupport\n  haveI : IsRefl X r := \u27e8fun _ => Set.Subset.refl _\u27e9\n  obtain \u27e8I, _I_mem, hI\u27e9 : \u2203 I \u2208 c, (f.support : Set \u03b9) \u2286 I :=\n    hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport\n  exact linearIndependent_comp_subtype.mp I.2 f hI hsum\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\n\u22a2 \u2200 (c : Set X), IsChain r c \u2192 indep (\u22c3 (I : X) (_ : I \u2208 c), \u2191I)\n[PROOFSTEP]\nintro c hc\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\n\u22a2 indep (\u22c3 (I : X) (_ : I \u2208 c), \u2191I)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\n\u22a2 LinearIndependent R (s \u2218 Subtype.val)\n[PROOFSTEP]\nrw [linearIndependent_comp_subtype]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\n\u22a2 \u2200 (l : \u03b9 \u2192\u2080 R),\n    l \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 c), \u2191I) \u2192\n      \u2191(Finsupp.total \u03b9 M R s) l = 0 \u2192 l = 0\n[PROOFSTEP]\nintro f hsupport hsum\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\nf : \u03b9 \u2192\u2080 R\nhsupport : f \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 c), \u2191I)\nhsum : \u2191(Finsupp.total \u03b9 M R s) f = 0\n\u22a2 f = 0\n[PROOFSTEP]\nrcases eq_empty_or_nonempty c with (rfl | hn)\n[GOAL]\ncase inl\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nf : \u03b9 \u2192\u2080 R\nhsum : \u2191(Finsupp.total \u03b9 M R s) f = 0\nhc : IsChain r \u2205\nhsupport : f \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 \u2205), \u2191I)\n\u22a2 f = 0\n[PROOFSTEP]\nsimpa using hsupport\n[GOAL]\ncase inr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\nf : \u03b9 \u2192\u2080 R\nhsupport : f \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 c), \u2191I)\nhsum : \u2191(Finsupp.total \u03b9 M R s) f = 0\nhn : Set.Nonempty c\n\u22a2 f = 0\n[PROOFSTEP]\nhaveI : IsRefl X r := \u27e8fun _ => Set.Subset.refl _\u27e9\n[GOAL]\ncase inr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\nf : \u03b9 \u2192\u2080 R\nhsupport : f \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 c), \u2191I)\nhsum : \u2191(Finsupp.total \u03b9 M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\n\u22a2 f = 0\n[PROOFSTEP]\nobtain \u27e8I, _I_mem, hI\u27e9 : \u2203 I \u2208 c, (f.support : Set \u03b9) \u2286 I :=\n  hc.directedOn.exists_mem_subset_of_finset_subset_biUnion hn hsupport\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nc : Set X\nhc : IsChain r c\nf : \u03b9 \u2192\u2080 R\nhsupport : f \u2208 Finsupp.supported R R (\u22c3 (I : { I // LinearIndependent R (s \u2218 Subtype.val) }) (_ : I \u2208 c), \u2191I)\nhsum : \u2191(Finsupp.total \u03b9 M R s) f = 0\nhn : Set.Nonempty c\nthis : IsRefl X r\nI : X\n_I_mem : I \u2208 c\nhI : \u2191f.support \u2286 \u2191I\n\u22a2 f = 0\n[PROOFSTEP]\nexact linearIndependent_comp_subtype.mp I.2 f hI hsum\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nkey : \u2200 (c : Set X), IsChain r c \u2192 indep (\u22c3 (I : X) (_ : I \u2208 c), \u2191I)\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nhave trans : Transitive r := fun I J K => Set.Subset.trans\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nkey : \u2200 (c : Set X), IsChain r c \u2192 indep (\u22c3 (I : X) (_ : I \u2208 c), \u2191I)\ntrans : Transitive r\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nobtain \u27e8\u27e8I, hli : indep I\u27e9, hmax : \u2200 a, r \u27e8I, hli\u27e9 a \u2192 r a \u27e8I, hli\u27e9\u27e9 :=\n  @exists_maximal_of_chains_bounded _ r\n    (fun c hc => \u27e8\u27e8\u22c3 I \u2208 c, (I : Set \u03b9), key c hc\u27e9, fun I => Set.subset_biUnion_of_mem\u27e9) @trans\n[GOAL]\ncase intro.mk\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nindep : Set \u03b9 \u2192 Prop := fun I => LinearIndependent R (s \u2218 Subtype.val)\nX : Type u' := { I // indep I }\nr : X \u2192 X \u2192 Prop := fun I J => \u2191I \u2286 \u2191J\nkey : \u2200 (c : Set X), IsChain r c \u2192 indep (\u22c3 (I : X) (_ : I \u2208 c), \u2191I)\ntrans : Transitive r\nI : Set \u03b9\nhli : indep I\nhmax : \u2200 (a : X), r { val := I, property := hli } a \u2192 r a { val := I, property := hli }\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n[PROOFSTEP]\nexact \u27e8I, hli, fun J hsub hli => Set.Subset.antisymm hsub (hmax \u27e8J, hli\u27e9 hsub)\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nclassical\nrcases exists_maximal_independent' R s with \u27e8I, hIlinind, hImaximal\u27e9\nuse I, hIlinind\nintro i hi\nspecialize hImaximal (I \u222a { i }) (by simp)\nset J := I \u222a { i } with hJ\nhave memJ : \u2200 {x}, x \u2208 J \u2194 x = i \u2228 x \u2208 I := by simp [hJ]\nhave hiJ : i \u2208 J := by simp\nhave h := by\n  refine mt hImaximal ?_\n  \u00b7 intro h2\n    rw [h2] at hi \n    exact absurd hiJ hi\nobtain \u27e8f, supp_f, sum_f, f_ne\u27e9 := linearDependent_comp_subtype.mp h\nhave hfi : f i \u2260 0 := by\n  contrapose hIlinind\n  refine' linearDependent_comp_subtype.mpr \u27e8f, _, sum_f, f_ne\u27e9\n  simp only [Finsupp.mem_supported, hJ] at supp_f \u22a2\n  rintro x hx\n  refine' (memJ.mp (supp_f hx)).resolve_left _\n  rintro rfl\n  exact hIlinind (Finsupp.mem_support_iff.mp hx)\nuse f i, hfi\nhave hfi' : i \u2208 f.support := Finsupp.mem_support_iff.mpr hfi\nrw [\u2190 Finset.insert_erase hfi', Finset.sum_insert (Finset.not_mem_erase _ _), add_eq_zero_iff_eq_neg] at sum_f \nrw [sum_f]\nrefine' neg_mem (sum_mem fun c hc => smul_mem _ _ (subset_span \u27e8c, _, rfl\u27e9))\nexact (memJ.mp (supp_f (Finset.erase_subset _ _ hc))).resolve_left (Finset.ne_of_mem_erase hc)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nrcases exists_maximal_independent' R s with \u27e8I, hIlinind, hImaximal\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\nhImaximal : \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n\u22a2 \u2203 I, (LinearIndependent R fun x => s \u2191x) \u2227 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nuse I, hIlinind\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\nhImaximal : \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\n\u22a2 \u2200 (i : \u03b9), \u00aci \u2208 I \u2192 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\nhImaximal : \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nspecialize hImaximal (I \u222a { i }) (by simp)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\nhImaximal : \u2200 (J : Set \u03b9), I \u2286 J \u2192 (LinearIndependent R fun x => s \u2191x) \u2192 I = J\ni : \u03b9\nhi : \u00aci \u2208 I\n\u22a2 I \u2286 I \u222a {i}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = I \u222a {i}\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nset J := I \u222a { i } with hJ\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nhave memJ : \u2200 {x}, x \u2208 J \u2194 x = i \u2228 x \u2208 I := by simp [hJ]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\n\u22a2 \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\n[PROOFSTEP]\nsimp [hJ]\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nhave hiJ : i \u2208 J := by simp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\n\u22a2 i \u2208 J\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nhave h := by\n  refine mt hImaximal ?_\n  \u00b7 intro h2\n    rw [h2] at hi \n    exact absurd hiJ hi\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\n\u22a2 ?m.535465\n[PROOFSTEP]\nrefine mt hImaximal ?_\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\n\u22a2 \u00acI = J\n[PROOFSTEP]\nintro h2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh2 : I = J\n\u22a2 False\n[PROOFSTEP]\nrw [h2] at hi \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nJ : Set \u03b9 := I \u222a {i}\nhi : \u00aci \u2208 J\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh2 : I = J\n\u22a2 False\n[PROOFSTEP]\nexact absurd hiJ hi\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nobtain \u27e8f, supp_f, sum_f, f_ne\u27e9 := linearDependent_comp_subtype.mp h\n[GOAL]\ncase right.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nhave hfi : f i \u2260 0 := by\n  contrapose hIlinind\n  refine' linearDependent_comp_subtype.mpr \u27e8f, _, sum_f, f_ne\u27e9\n  simp only [Finsupp.mem_supported, hJ] at supp_f \u22a2\n  rintro x hx\n  refine' (memJ.mp (supp_f hx)).resolve_left _\n  rintro rfl\n  exact hIlinind (Finsupp.mem_support_iff.mp hx)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\n\u22a2 \u2191f i \u2260 0\n[PROOFSTEP]\ncontrapose hIlinind\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhIlinind : \u00ac\u2191f i \u2260 0\n\u22a2 \u00acLinearIndependent R fun x => s \u2191x\n[PROOFSTEP]\nrefine' linearDependent_comp_subtype.mpr \u27e8f, _, sum_f, f_ne\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhIlinind : \u00ac\u2191f i \u2260 0\n\u22a2 f \u2208 Finsupp.supported R R I\n[PROOFSTEP]\nsimp only [Finsupp.mem_supported, hJ] at supp_f \u22a2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhIlinind : \u00ac\u2191f i \u2260 0\nsupp_f : \u2191f.support \u2286 I \u222a {i}\n\u22a2 \u2191f.support \u2286 I\n[PROOFSTEP]\nrintro x hx\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhIlinind : \u00ac\u2191f i \u2260 0\nsupp_f : \u2191f.support \u2286 I \u222a {i}\nx : \u03b9\nhx : x \u2208 \u2191f.support\n\u22a2 x \u2208 I\n[PROOFSTEP]\nrefine' (memJ.mp (supp_f hx)).resolve_left _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhIlinind : \u00ac\u2191f i \u2260 0\nsupp_f : \u2191f.support \u2286 I \u222a {i}\nx : \u03b9\nhx : x \u2208 \u2191f.support\n\u22a2 \u00acx = i\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx\u271d y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nf : \u03b9 \u2192\u2080 R\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nx : \u03b9\nhx : x \u2208 \u2191f.support\nhi : \u00acx \u2208 I\nJ : Set \u03b9 := I \u222a {x}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {x}\nmemJ : \u2200 {x_1 : \u03b9}, x_1 \u2208 J \u2194 x_1 = x \u2228 x_1 \u2208 I\nhiJ : x \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nhIlinind : \u00ac\u2191f x \u2260 0\nsupp_f : \u2191f.support \u2286 I \u222a {x}\n\u22a2 False\n[PROOFSTEP]\nexact hIlinind (Finsupp.mem_support_iff.mp hx)\n[GOAL]\ncase right.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\n\u22a2 \u2203 a, a \u2260 0 \u2227 a \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nuse f i, hfi\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\n\u22a2 \u2191f i \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nhave hfi' : i \u2208 f.support := Finsupp.mem_support_iff.mpr hfi\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2211 i in f.support, \u2191f i \u2022 s i = 0\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\nhfi' : i \u2208 f.support\n\u22a2 \u2191f i \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nrw [\u2190 Finset.insert_erase hfi', Finset.sum_insert (Finset.not_mem_erase _ _), add_eq_zero_iff_eq_neg] at sum_f \n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2191f i \u2022 s i = -\u2211 x in Finset.erase f.support i, \u2191f x \u2022 s x\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\nhfi' : i \u2208 f.support\n\u22a2 \u2191f i \u2022 s i \u2208 span R (s '' I)\n[PROOFSTEP]\nrw [sum_f]\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2191f i \u2022 s i = -\u2211 x in Finset.erase f.support i, \u2191f x \u2022 s x\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\nhfi' : i \u2208 f.support\n\u22a2 -\u2211 x in Finset.erase f.support i, \u2191f x \u2022 s x \u2208 span R (s '' I)\n[PROOFSTEP]\nrefine' neg_mem (sum_mem fun c hc => smul_mem _ _ (subset_span \u27e8c, _, rfl\u27e9))\n[GOAL]\ncase right\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns : \u03b9 \u2192 M\nI : Set \u03b9\nhIlinind : LinearIndependent R fun x => s \u2191x\ni : \u03b9\nhi : \u00aci \u2208 I\nJ : Set \u03b9 := I \u222a {i}\nhImaximal : (LinearIndependent R fun x => s \u2191x) \u2192 I = J\nhJ : J = I \u222a {i}\nmemJ : \u2200 {x : \u03b9}, x \u2208 J \u2194 x = i \u2228 x \u2208 I\nhiJ : i \u2208 J\nh : \u00acLinearIndependent R fun x => s \u2191x\nf : \u03b9 \u2192\u2080 R\nsupp_f : f \u2208 Finsupp.supported R R J\nsum_f : \u2191f i \u2022 s i = -\u2211 x in Finset.erase f.support i, \u2191f x \u2022 s x\nf_ne : f \u2260 0\nhfi : \u2191f i \u2260 0\nhfi' : i \u2208 f.support\nc : \u03b9\nhc : c \u2208 Finset.erase f.support i\n\u22a2 c \u2208 I\n[PROOFSTEP]\nexact (memJ.mp (supp_f (Finset.erase_subset _ _ hc))).resolve_left (Finset.ne_of_mem_erase hc)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\nintro i\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nlet repr : (span R (range (v \u2218 f)) : Type _) \u2192 \u03b9' \u2192\u2080 R := (hv.comp f f.injective).repr\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nlet l := (repr \u27e8v i, hss (mem_range_self i)\u27e9).mapDomain f\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nhave h_total_l : Finsupp.total \u03b9 M R v l = v i := by\n  dsimp only []\n  rw [Finsupp.total_mapDomain]\n  rw [(hv.comp f f.injective).total_repr]\n    -- Porting note: `rfl` isn't necessary.\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = v i\n[PROOFSTEP]\ndsimp only []\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9 M R v)\n      (Finsupp.mapDomain (\u2191f)\n        (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n          { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })) =\n    v i\n[PROOFSTEP]\nrw [Finsupp.total_mapDomain]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\n\u22a2 \u2191(Finsupp.total \u03b9' M R (v \u2218 \u2191f))\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    v i\n[PROOFSTEP]\nrw [(hv.comp f f.injective).total_repr]\n  -- Porting note: `rfl` isn't necessary.\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nhave h_total_eq : (Finsupp.total \u03b9 M R v) l = (Finsupp.total \u03b9 M R v) (Finsupp.single i 1) := by\n  rw [h_total_l, Finsupp.total_single, one_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\n\u22a2 \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\n[PROOFSTEP]\nrw [h_total_l, Finsupp.total_single, one_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nhave l_eq : l = _ := LinearMap.ker_eq_bot.1 hv h_total_eq\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq : l = Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\ndsimp only [] at l_eq \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.mapDomain (\u2191f)\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nrw [\u2190 Finsupp.embDomain_eq_mapDomain] at l_eq \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nrcases Finsupp.single_of_embDomain_single (repr \u27e8v i, _\u27e9) f i (1 : R) zero_ne_one.symm l_eq with \u27e8i', hi'\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2203 a, \u2191f a = i\n[PROOFSTEP]\nuse i'\n[GOAL]\ncase h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\nhv : LinearIndependent R v\nf : \u03b9' \u21aa \u03b9\nhss : range v \u2286 \u2191(span R (range (v \u2218 \u2191f)))\ni : \u03b9\nrepr : { x // x \u2208 span R (range (v \u2218 \u2191f)) } \u2192 \u03b9' \u2192\u2080 R := \u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\nl : \u03b9 \u2192\u2080 R := Finsupp.mapDomain (\u2191f) (repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) })\nh_total_l : \u2191(Finsupp.total \u03b9 M R v) l = v i\nh_total_eq : \u2191(Finsupp.total \u03b9 M R v) l = \u2191(Finsupp.total \u03b9 M R v) (Finsupp.single i 1)\nl_eq :\n  Finsupp.embDomain f\n      (\u2191(LinearIndependent.repr (_ : LinearIndependent R (v \u2218 \u2191f)))\n        { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) }) =\n    Finsupp.single i 1\ni' : \u03b9'\nhi' : repr { val := v i, property := (_ : v i \u2208 \u2191(span R (range (v \u2218 \u2191f)))) } = Finsupp.single i' 1 \u2227 \u2191f i' = i\n\u22a2 \u2191f i' = i\n[PROOFSTEP]\nexact hi'.2\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\n\u22a2 s = t\n[PROOFSTEP]\nlet f : t \u21aa s := \u27e8fun x => \u27e8x.1, h x.2\u27e9, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\n\u22a2 s = t\n[PROOFSTEP]\nhave h_surj : Surjective f := by\n  apply surjective_of_linearIndependent_of_span hs f _\n  convert hst <;> simp [comp]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\n\u22a2 Surjective \u2191f\n[PROOFSTEP]\napply surjective_of_linearIndependent_of_span hs f _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\n\u22a2 (range fun x => \u2191x) \u2286 \u2191(span R (range ((fun x => \u2191x) \u2218 \u2191f)))\n[PROOFSTEP]\nconvert hst\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\n\u22a2 (range fun x => \u2191x) = s\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\ncase h.e'_4.h.e'_4.h.e'_6\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\n\u22a2 range ((fun x => \u2191x) \u2218 \u2191f) = t\n[PROOFSTEP]\nsimp [comp]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s = t\n[PROOFSTEP]\nshow s = t\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s = t\n[PROOFSTEP]\napply Subset.antisymm _ h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s \u2286 t\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\n\u22a2 x \u2208 t\n[PROOFSTEP]\nrcases h_surj \u27e8x, hx\u27e9 with \u27e8y, hy\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x \u2208 t\n[PROOFSTEP]\nconvert y.mem\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n  { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n    inj' :=\n      (_ :\n        \u2200 (a b : \u2191t),\n          (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n            a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x = \u2191y\n[PROOFSTEP]\nrw [\u2190 Subtype.mk.inj hy]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun x => \u2191x\nhf_inj : Disjoint (span R s) (ker f)\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrw [\u2190 @Subtype.range_coe _ s] at hf_inj \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun x => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' (hs.map hf_inj).to_subtype_range' _\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nf : M \u2192\u2097[R] M'\nhs : LinearIndependent R fun x => \u2191x\nhf_inj : Disjoint (span R (Set.range Subtype.val)) (ker f)\n\u22a2 Set.range (\u2191f \u2218 fun x => \u2191x) = \u2191f '' s\n[PROOFSTEP]\nsimp [Set.range_comp f]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' (hs.image_subtype _).union (ht.image_subtype _) _ <;> [simp; simp; skip]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 LinearIndependent R fun x => \u2191x\n[PROOFSTEP]\nrefine' (hs.image_subtype _).union (ht.image_subtype _) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R s) (ker (inl R M M'))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R t) (ker (inr R M M'))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase refine'_3\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R (\u2191(inl R M M') '' s)) (span R (\u2191(inr R M M') '' t))\n[PROOFSTEP]\nskip\n[GOAL]\ncase refine'_3\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (span R (\u2191(inl R M M') '' s)) (span R (\u2191(inr R M M') '' t))\n[PROOFSTEP]\nsimp only [span_image]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\ns : Set M\nt : Set M'\nhs : LinearIndependent R fun x => \u2191x\nht : LinearIndependent R fun x => \u2191x\n\u22a2 Disjoint (Submodule.map (inl R M M') (span R s)) (Submodule.map (inr R M M') (span R t))\n[PROOFSTEP]\nsimp [disjoint_iff, prod_inf_prod]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nv' : \u03b9' \u2192 M'\nhv : LinearIndependent R v\nhv' : LinearIndependent R v'\n\u22a2 Disjoint (span R (Set.range (\u2191(inl R M M') \u2218 v))) (span R (Set.range (\u2191(inr R M M') \u2218 v')))\n[PROOFSTEP]\nrefine' isCompl_range_inl_inr.disjoint.mono _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nv' : \u03b9' \u2192 M'\nhv : LinearIndependent R v\nhv' : LinearIndependent R v'\n\u22a2 span R (Set.range (\u2191(inl R M M') \u2218 v)) \u2264 LinearMap.range (inl R M M')\n[PROOFSTEP]\nsimp only [span_le, range_coe, range_comp_subset_range]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv\u271d : \u03b9 \u2192 M\ninst\u271d\u2076 : Ring R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : AddCommGroup M''\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module R M'\ninst\u271d : Module R M''\na b : R\nx y : M\nv : \u03b9 \u2192 M\nv' : \u03b9' \u2192 M'\nhv : LinearIndependent R v\nhv' : LinearIndependent R v'\n\u22a2 span R (Set.range (\u2191(inr R M M') \u2218 v')) \u2264 LinearMap.range (inr R M M')\n[PROOFSTEP]\nsimp only [span_le, range_coe, range_comp_subset_range]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\n\u22a2 LinearIndependent L fun f => \u2191f\n[PROOFSTEP]\nletI := Classical.decEq (G \u2192* L)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\n\u22a2 LinearIndependent L fun f => \u2191f\n[PROOFSTEP]\nletI : MulAction L L := DistribMulAction.toMulAction\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\n\u22a2 LinearIndependent L fun f => \u2191f\n[PROOFSTEP]\nexact\n  linearIndependent_iff'.2\n      -- To do this, we use `Finset` induction,\n          -- Porting note: `False.elim` \u2192 `fun h => False.elim <| Finset.not_mem_empty _ h`\n    fun s =>\n    Finset.induction_on s (fun g _hg i h => False.elim <| Finset.not_mem_empty _ h)\n      fun a s has ih g hg =>\n        -- Here\n                -- * `a` is a new character we will insert into the `Finset` of characters `s`,\n                -- * `ih` is the fact that only the trivial linear combination of characters in `s` is zero\n                -- * `hg` is the fact that `g` are the coefficients of a linear combination summing to zero\n                -- and it remains to prove that `g` vanishes on `insert a s`.\n                -- We now make the key calculation:\n                -- For any character `i` in the original `Finset`, we have `g i \u2022 i = g i \u2022 a` as functions\n                -- on the monoid `G`.\n      have h1 : \u2200 i \u2208 s, (g i \u2022 (i : G \u2192 L)) = g i \u2022 (a : G \u2192 L) := fun i his =>\n        funext\n          fun x : G =>\n            -- We prove these expressions are equal by showing\n                        -- the differences of their values on each monoid element `x` is zero\n          eq_of_sub_eq_zero <|\n            ih (fun j => g j * j x - g j * a x)\n              (funext fun y : G =>\n                calc\n                  -- After that, it's just a chase scene.\n                  (\u2211 i in s, ((g i * i x - g i * a x) \u2022 (i : G \u2192 L))) y = \u2211 i in s, (g i * i x - g i * a x) * i y :=\n                    Finset.sum_apply _ _ _\n                  _ = \u2211 i in s, (g i * i x * i y - g i * a x * i y) := (Finset.sum_congr rfl fun _ _ => sub_mul _ _ _)\n                  _ = (\u2211 i in s, g i * i x * i y) - \u2211 i in s, g i * a x * i y := Finset.sum_sub_distrib\n                  _ = (g a * a x * a y + \u2211 i in s, g i * i x * i y) - (g a * a x * a y + \u2211 i in s, g i * a x * i y) :=\n                    by rw [add_sub_add_left_eq_sub]\n                  _ = (\u2211 i in insert a s, g i * i x * i y) - \u2211 i in insert a s, g i * a x * i y := by\n                    rw [Finset.sum_insert has, Finset.sum_insert has]\n                  _ = (\u2211 i in insert a s, g i * i (x * y)) - \u2211 i in insert a s, a x * (g i * i y) :=\n                    (congr (congr_arg Sub.sub (Finset.sum_congr rfl fun i _ => by rw [i.map_mul, mul_assoc]))\n                      (Finset.sum_congr rfl fun _ _ => by rw [mul_assoc, mul_left_comm]))\n                  _ =\n                      (\u2211 i in insert a s, (g i \u2022 (i : G \u2192 L))) (x * y) -\n                        a x * (\u2211 i in insert a s, (g i \u2022 (i : G \u2192 L))) y :=\n                    by rw [Finset.sum_apply, Finset.sum_apply, Finset.mul_sum]; rfl\n                  _ = 0 - a x * 0 := by rw [hg]; rfl\n                  _ = 0 := by rw [mul_zero, sub_zero])\n              i his\n      have h2 : \u2200 i : G \u2192* L, i \u2208 s \u2192 \u2203 y, i y \u2260 a y := fun i his =>\n        Classical.by_contradiction fun h =>\n          have hia : i = a := MonoidHom.ext fun y => Classical.by_contradiction fun hy => h \u27e8y, hy\u27e9\n          has <| hia \u25b8 his\n      have h3 : \u2200 i \u2208 s, g i = 0 := fun i his =>\n        let \u27e8y, hy\u27e9 := h2 i his\n        have h : g i \u2022 i y = g i \u2022 a y := congr_fun (h1 i his) y\n        Or.resolve_right (mul_eq_zero.1 <| by rw [mul_sub, sub_eq_zero]; exact h)\n          (sub_ne_zero_of_ne hy)\n            -- And so, using the fact that the linear combination over `s` and over `insert a s` both\n                    -- vanish, we deduce that `g a = 0`.\n      have h4 : g a = 0 :=\n        calc\n          g a = g a * 1 := (mul_one _).symm\n          _ = (g a \u2022 (a : G \u2192 L)) 1 := by rw [\u2190 a.map_one]; rfl\n          _ = (\u2211 i in insert a s, (g i \u2022 (i : G \u2192 L))) 1 :=\n            by\n            rw [Finset.sum_eq_single a]\n            \u00b7 intro i his hia\n              rw [Finset.mem_insert] at his \n              rw [h3 i (his.resolve_left hia), zero_smul]\n            \u00b7 intro haas\n              exfalso\n              apply haas\n              exact Finset.mem_insert_self a s\n          _ = 0 := by rw [hg];\n            rfl\n              -- Now we're done; the last two facts together imply that `g` vanishes on every element\n                      -- of `insert a s`.\n      (Finset.forall_mem_insert _ _ _).2 \u27e8h4, h3\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 \u2211 i in s, g i * \u2191i x * \u2191i y - \u2211 i in s, g i * \u2191a x * \u2191i y =\n    g a * \u2191a x * \u2191a y + \u2211 i in s, g i * \u2191i x * \u2191i y - (g a * \u2191a x * \u2191a y + \u2211 i in s, g i * \u2191a x * \u2191i y)\n[PROOFSTEP]\nrw [add_sub_add_left_eq_sub]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 g a * \u2191a x * \u2191a y + \u2211 i in s, g i * \u2191i x * \u2191i y - (g a * \u2191a x * \u2191a y + \u2211 i in s, g i * \u2191a x * \u2191i y) =\n    \u2211 i in insert a s, g i * \u2191i x * \u2191i y - \u2211 i in insert a s, g i * \u2191a x * \u2191i y\n[PROOFSTEP]\nrw [Finset.sum_insert has, Finset.sum_insert has]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d\u00b9 y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni\u271d : G \u2192* L\nhis : i\u271d \u2208 s\nx y : G\ni : G \u2192* L\nx\u271d : i \u2208 insert a s\n\u22a2 g i * \u2191i x * \u2191i y = g i * \u2191i (x * y)\n[PROOFSTEP]\nrw [i.map_mul, mul_assoc]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d\u00b2 y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\nx\u271d\u00b9 : G \u2192* L\nx\u271d : x\u271d\u00b9 \u2208 insert a s\n\u22a2 g x\u271d\u00b9 * \u2191a x * \u2191x\u271d\u00b9 y = \u2191a x * (g x\u271d\u00b9 * \u2191x\u271d\u00b9 y)\n[PROOFSTEP]\nrw [mul_assoc, mul_left_comm]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 \u2211 i in insert a s, g i * \u2191i (x * y) - \u2211 i in insert a s, \u2191a x * (g i * \u2191i y) =\n    Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) (x * y) - \u2191a x * Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) y\n[PROOFSTEP]\nrw [Finset.sum_apply, Finset.sum_apply, Finset.mul_sum]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 \u2211 i in insert a s, g i * \u2191i (x * y) - \u2211 i in insert a s, \u2191a x * (g i * \u2191i y) =\n    \u2211 c in insert a s, (g c \u2022 \u2191c) (x * y) - \u2211 x_1 in insert a s, \u2191a x * (g x_1 \u2022 \u2191x_1) y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) (x * y) - \u2191a x * Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) y =\n    0 - \u2191a x * 0\n[PROOFSTEP]\nrw [hg]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 OfNat.ofNat 0 (x * y) - \u2191a x * OfNat.ofNat 0 y = 0 - \u2191a x * 0\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx\u271d y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\ni : G \u2192* L\nhis : i \u2208 s\nx y : G\n\u22a2 0 - \u2191a x * 0 = 0\n[PROOFSTEP]\nrw [mul_zero, sub_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\ni : G \u2192* L\nhis : i \u2208 s\ny : G\nhy : \u2191i y \u2260 \u2191a y\nh : g i \u2022 \u2191i y = g i \u2022 \u2191a y\n\u22a2 g i * (\u2191i y - \u2191a y) = 0\n[PROOFSTEP]\nrw [mul_sub, sub_eq_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y\u271d : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\ni : G \u2192* L\nhis : i \u2208 s\ny : G\nhy : \u2191i y \u2260 \u2191a y\nh : g i \u2022 \u2191i y = g i \u2022 \u2191a y\n\u22a2 g i * \u2191i y = g i * \u2191a y\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 g a * 1 = (g a \u2022 \u2191a) 1\n[PROOFSTEP]\nrw [\u2190 a.map_one]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 g a * \u2191a 1 = (g a \u2022 \u2191a) 1\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 (g a \u2022 \u2191a) 1 = Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) 1\n[PROOFSTEP]\nrw [Finset.sum_eq_single a]\n[GOAL]\ncase h\u2080\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 \u2200 (b : G \u2192* L), b \u2208 insert a s \u2192 b \u2260 a \u2192 g b \u2022 \u2191b = 0\n[PROOFSTEP]\nintro i his hia\n[GOAL]\ncase h\u2080\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ni : G \u2192* L\nhis : i \u2208 insert a s\nhia : i \u2260 a\n\u22a2 g i \u2022 \u2191i = 0\n[PROOFSTEP]\nrw [Finset.mem_insert] at his \n[GOAL]\ncase h\u2080\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ni : G \u2192* L\nhis : i = a \u2228 i \u2208 s\nhia : i \u2260 a\n\u22a2 g i \u2022 \u2191i = 0\n[PROOFSTEP]\nrw [h3 i (his.resolve_left hia), zero_smul]\n[GOAL]\ncase h\u2081\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 \u00aca \u2208 insert a s \u2192 g a \u2022 \u2191a = 0\n[PROOFSTEP]\nintro haas\n[GOAL]\ncase h\u2081\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\nhaas : \u00aca \u2208 insert a s\n\u22a2 g a \u2022 \u2191a = 0\n[PROOFSTEP]\nexfalso\n[GOAL]\ncase h\u2081.h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\nhaas : \u00aca \u2208 insert a s\n\u22a2 False\n[PROOFSTEP]\napply haas\n[GOAL]\ncase h\u2081.h\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\nhaas : \u00aca \u2208 insert a s\n\u22a2 a \u2208 insert a s\n[PROOFSTEP]\nexact Finset.mem_insert_self a s\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 Finset.sum (insert a s) (fun i => g i \u2022 \u2191i) 1 = 0\n[PROOFSTEP]\nrw [hg]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2079 : Ring R\ninst\u271d\u2078 : AddCommGroup M\ninst\u271d\u2077 : AddCommGroup M'\ninst\u271d\u2076 : AddCommGroup M''\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R M'\ninst\u271d\u00b3 : Module R M''\na\u271d b : R\nx y : M\nG : Type u_8\ninst\u271d\u00b2 : Monoid G\nL : Type u_9\ninst\u271d\u00b9 : CommRing L\ninst\u271d : NoZeroDivisors L\nthis\u271d : DecidableEq (G \u2192* L) := Classical.decEq (G \u2192* L)\nthis : MulAction L L := DistribMulAction.toMulAction\ns\u271d : Finset (G \u2192* L)\na : G \u2192* L\ns : Finset (G \u2192* L)\nhas : \u00aca \u2208 s\nih : \u2200 (g : (G \u2192* L) \u2192 L), \u2211 i in s, g i \u2022 \u2191i = 0 \u2192 \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\ng : (G \u2192* L) \u2192 L\nhg : \u2211 i in insert a s, g i \u2022 \u2191i = 0\nh1 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i \u2022 \u2191i = g i \u2022 \u2191a\nh2 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 \u2203 y, \u2191i y \u2260 \u2191a y\nh3 : \u2200 (i : G \u2192* L), i \u2208 s \u2192 g i = 0\n\u22a2 OfNat.ofNat 0 1 = 0\n[PROOFSTEP]\nrfl\n  -- Now we're done; the last two facts together imply that `g` vanishes on every element\n          -- of `insert a s`.\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t u : Set M\nhl : LinearIndependent R Subtype.val\nhsu : s \u2286 u\nhtu : t \u2286 u\nhst : span R s \u2264 span R t\n\u22a2 s \u2286 t\n[PROOFSTEP]\nhave :=\n  eq_of_linearIndependent_of_span_subtype (hl.mono (Set.union_subset hsu htu)) (Set.subset_union_right _ _)\n    (Set.union_subset (Set.Subset.trans subset_span hst) subset_span)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t u : Set M\nhl : LinearIndependent R Subtype.val\nhsu : s \u2286 u\nhtu : t \u2286 u\nhst : span R s \u2264 span R t\nthis : s \u222a t = t\n\u22a2 s \u2286 t\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t u : Set M\nhl : LinearIndependent R Subtype.val\nhsu : s \u2286 u\nhtu : t \u2286 u\nhst : span R s \u2264 span R t\nthis : s \u222a t = t\n\u22a2 s \u2286 s \u222a t\n[PROOFSTEP]\napply Set.subset_union_left\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Nontrivial R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : NoZeroSMulDivisors R M\ninst\u271d\u00b9 : Module R M'\nv\u271d : \u03b9 \u2192 M\ns t : Set M\nx y z : M\nv : \u03b9 \u2192 M\ninst\u271d : Unique \u03b9\n\u22a2 LinearIndependent R v \u2194 v default \u2260 0\n[PROOFSTEP]\nsimp only [linearIndependent_iff, Finsupp.total_unique, smul_eq_zero]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Nontrivial R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : NoZeroSMulDivisors R M\ninst\u271d\u00b9 : Module R M'\nv\u271d : \u03b9 \u2192 M\ns t : Set M\nx y z : M\nv : \u03b9 \u2192 M\ninst\u271d : Unique \u03b9\n\u22a2 (\u2200 (l : \u03b9 \u2192\u2080 R), \u2191l default = 0 \u2228 v default = 0 \u2192 l = 0) \u2194 v default \u2260 0\n[PROOFSTEP]\nrefine' \u27e8fun h hv => _, fun hv l hl => Finsupp.unique_ext <| hl.resolve_right hv\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Nontrivial R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : NoZeroSMulDivisors R M\ninst\u271d\u00b9 : Module R M'\nv\u271d : \u03b9 \u2192 M\ns t : Set M\nx y z : M\nv : \u03b9 \u2192 M\ninst\u271d : Unique \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191l default = 0 \u2228 v default = 0 \u2192 l = 0\nhv : v default = 0\n\u22a2 False\n[PROOFSTEP]\nhave := h (Finsupp.single default 1) (Or.inr hv)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : Nontrivial R\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : AddCommGroup M'\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : NoZeroSMulDivisors R M\ninst\u271d\u00b9 : Module R M'\nv\u271d : \u03b9 \u2192 M\ns t : Set M\nx y z : M\nv : \u03b9 \u2192 M\ninst\u271d : Unique \u03b9\nh : \u2200 (l : \u03b9 \u2192\u2080 R), \u2191l default = 0 \u2228 v default = 0 \u2192 l = 0\nhv : v default = 0\nthis : Finsupp.single default 1 = 0\n\u22a2 False\n[PROOFSTEP]\nexact one_ne_zero (Finsupp.single_eq_zero.1 this)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 x \u2208 span K (insert y s) \u2192 \u00acx \u2208 span K s \u2192 y \u2208 span K (insert x s)\n[PROOFSTEP]\nsimp [mem_span_insert]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 \u2200 (x_1 : K) (x_2 : V), x_2 \u2208 span K s \u2192 x = x_1 \u2022 y + x_2 \u2192 \u00acx \u2208 span K s \u2192 \u2203 a z, z \u2208 span K s \u2227 y = a \u2022 x + z\n[PROOFSTEP]\nrintro a z hz rfl h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z\u271d : V\na : K\nz : V\nhz : z \u2208 span K s\nh : \u00aca \u2022 y + z \u2208 span K s\n\u22a2 \u2203 a_1 z_1, z_1 \u2208 span K s \u2227 y = a_1 \u2022 (a \u2022 y + z) + z_1\n[PROOFSTEP]\nrefine' \u27e8a\u207b\u00b9, -a\u207b\u00b9 \u2022 z, smul_mem _ _ hz, _\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z\u271d : V\na : K\nz : V\nhz : z \u2208 span K s\nh : \u00aca \u2022 y + z \u2208 span K s\n\u22a2 y = a\u207b\u00b9 \u2022 (a \u2022 y + z) + -a\u207b\u00b9 \u2022 z\n[PROOFSTEP]\nhave a0 : a \u2260 0 := by\n  rintro rfl\n  simp_all\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z\u271d : V\na : K\nz : V\nhz : z \u2208 span K s\nh : \u00aca \u2022 y + z \u2208 span K s\n\u22a2 a \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z\u271d z : V\nhz : z \u2208 span K s\nh : \u00ac0 \u2022 y + z \u2208 span K s\n\u22a2 False\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z\u271d : V\na : K\nz : V\nhz : z \u2208 span K s\nh : \u00aca \u2022 y + z \u2208 span K s\na0 : a \u2260 0\n\u22a2 y = a\u207b\u00b9 \u2022 (a \u2022 y + z) + -a\u207b\u00b9 \u2022 z\n[PROOFSTEP]\nsimp [a0, smul_add, smul_smul]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 LinearIndependent K v \u2194 \u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))\n[PROOFSTEP]\napply linearIndependent_iff_not_smul_mem_span.trans\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 (\u2200 (i : \u03b9) (a : K), a \u2022 v i \u2208 span K (v '' (univ \\ {i})) \u2192 a = 0) \u2194 \u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 (\u2200 (i : \u03b9) (a : K), a \u2022 v i \u2208 span K (v '' (univ \\ {i})) \u2192 a = 0) \u2192 \u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))\n[PROOFSTEP]\nintro h i h_in_span\n[GOAL]\ncase mp\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nh : \u2200 (i : \u03b9) (a : K), a \u2022 v i \u2208 span K (v '' (univ \\ {i})) \u2192 a = 0\ni : \u03b9\nh_in_span : v i \u2208 span K (v '' (univ \\ {i}))\n\u22a2 False\n[PROOFSTEP]\napply one_ne_zero (h i 1 (by simp [h_in_span]))\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nh : \u2200 (i : \u03b9) (a : K), a \u2022 v i \u2208 span K (v '' (univ \\ {i})) \u2192 a = 0\ni : \u03b9\nh_in_span : v i \u2208 span K (v '' (univ \\ {i}))\n\u22a2 1 \u2022 v i \u2208 span K (v '' (univ \\ {i}))\n[PROOFSTEP]\nsimp [h_in_span]\n[GOAL]\ncase mpr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 (\u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))) \u2192 \u2200 (i : \u03b9) (a : K), a \u2022 v i \u2208 span K (v '' (univ \\ {i})) \u2192 a = 0\n[PROOFSTEP]\nintro h i a ha\n[GOAL]\ncase mpr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nh : \u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))\ni : \u03b9\na : K\nha : a \u2022 v i \u2208 span K (v '' (univ \\ {i}))\n\u22a2 a = 0\n[PROOFSTEP]\nby_contra ha'\n[GOAL]\ncase mpr\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nh : \u2200 (i : \u03b9), \u00acv i \u2208 span K (v '' (univ \\ {i}))\ni : \u03b9\na : K\nha : a \u2022 v i \u2208 span K (v '' (univ \\ {i}))\nha' : \u00aca = 0\n\u22a2 False\n[PROOFSTEP]\nexact False.elim (h _ ((smul_mem_iff _ ha').1 ha))\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00acx \u2208 span K s\n\u22a2 LinearIndependent K fun b => \u2191b\n[PROOFSTEP]\nrw [\u2190 union_singleton]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00acx \u2208 span K s\n\u22a2 LinearIndependent K fun b => \u2191b\n[PROOFSTEP]\nhave x0 : x \u2260 0 := mt (by rintro rfl; apply zero_mem (span K s)) hx\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00acx \u2208 span K s\n\u22a2 x = 0 \u2192 x \u2208 span K s\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00ac0 \u2208 span K s\n\u22a2 0 \u2208 span K s\n[PROOFSTEP]\napply zero_mem (span K s)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00acx \u2208 span K s\nx0 : x \u2260 0\n\u22a2 LinearIndependent K fun b => \u2191b\n[PROOFSTEP]\napply hs.union (linearIndependent_singleton x0)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K fun b => \u2191b\nhx : \u00acx \u2208 span K s\nx0 : x \u2260 0\n\u22a2 Disjoint (span K s) (span K {x})\n[PROOFSTEP]\nrwa [disjoint_span_singleton' x0]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 (LinearIndependent K fun o => Option.casesOn' o x v) \u2194 LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\nrw [\u2190 linearIndependent_equiv (Equiv.optionEquivSumPUnit.{_, u'} \u03b9).symm, linearIndependent_sum, @range_unique _ PUnit,\n  @linearIndependent_unique_iff PUnit, disjoint_span_singleton]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 LinearIndependent K (((fun o => Option.casesOn' o x v) \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inl) \u2227\n      (((fun o => Option.casesOn' o x v) \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inr) default \u2260 0 \u2227\n        ((((fun o => Option.casesOn' o x v) \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inr) default \u2208\n            span K (range (((fun o => Option.casesOn' o x v) \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inl)) \u2192\n          (((fun o => Option.casesOn' o x v) \u2218 \u2191(Equiv.optionEquivSumPUnit \u03b9).symm) \u2218 Sum.inr) default = 0) \u2194\n    LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\ndsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 (LinearIndependent K fun x => v x) \u2227 \u00acx = 0 \u2227 (x \u2208 span K (range fun x => v x) \u2192 x = 0) \u2194\n    LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8h.1, fun hx => h.2.1 <| h.2.2 hx\u27e9, fun h => \u27e8h.1, _, fun hx => (h.2 hx).elim\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nh : LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n\u22a2 \u00acx = 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\ny z : V\nh : LinearIndependent K v \u2227 \u00ac0 \u2208 span K (range v)\n\u22a2 False\n[PROOFSTEP]\nexact h.2 (zero_mem _)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nv : Option \u03b9 \u2192 V\n\u22a2 LinearIndependent K v \u2194 LinearIndependent K (v \u2218 some) \u2227 \u00acv none \u2208 span K (range (v \u2218 some))\n[PROOFSTEP]\nsimp only [\u2190 linearIndependent_option', Option.casesOn'_none_coe]\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9\u271d \u2192 V\ns\u271d t : Set V\nx y z : V\n\u03b9 : Type u_8\ns : Set \u03b9\na : \u03b9\nf : \u03b9 \u2192 V\nhas : \u00aca \u2208 s\n\u22a2 (LinearIndependent K fun x => f \u2191x) \u2194 (LinearIndependent K fun x => f \u2191x) \u2227 \u00acf a \u2208 span K (f '' s)\n[PROOFSTEP]\nclassical\nrw [\u2190 linearIndependent_equiv ((Equiv.optionEquivSumPUnit _).trans (Equiv.Set.insert has).symm),\n  linearIndependent_option]\n  -- Porting note: `simp [(\u00b7 \u2218 \u00b7), range_comp f]` \u2192 `simp [(\u00b7 \u2218 \u00b7)]; erw [range_comp f ..]; simp`\n    -- https://github.com/leanprover-community/mathlib4/issues/5164\nsimp only [(\u00b7 \u2218 \u00b7)]\nerw [range_comp f ((\u2191) : s \u2192 \u03b9)]\nsimp\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9\u271d \u2192 V\ns\u271d t : Set V\nx y z : V\n\u03b9 : Type u_8\ns : Set \u03b9\na : \u03b9\nf : \u03b9 \u2192 V\nhas : \u00aca \u2208 s\n\u22a2 (LinearIndependent K fun x => f \u2191x) \u2194 (LinearIndependent K fun x => f \u2191x) \u2227 \u00acf a \u2208 span K (f '' s)\n[PROOFSTEP]\nrw [\u2190 linearIndependent_equiv ((Equiv.optionEquivSumPUnit _).trans (Equiv.Set.insert has).symm),\n  linearIndependent_option]\n  -- Porting note: `simp [(\u00b7 \u2218 \u00b7), range_comp f]` \u2192 `simp [(\u00b7 \u2218 \u00b7)]; erw [range_comp f ..]; simp`\n    -- https://github.com/leanprover-community/mathlib4/issues/5164\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9\u271d \u2192 V\ns\u271d t : Set V\nx y z : V\n\u03b9 : Type u_8\ns : Set \u03b9\na : \u03b9\nf : \u03b9 \u2192 V\nhas : \u00aca \u2208 s\n\u22a2 LinearIndependent K (((fun x => f \u2191x) \u2218 \u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm)) \u2218 some) \u2227\n      \u00ac((fun x => f \u2191x) \u2218 \u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm)) none \u2208\n          span K\n            (range (((fun x => f \u2191x) \u2218 \u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm)) \u2218 some)) \u2194\n    (LinearIndependent K fun x => f \u2191x) \u2227 \u00acf a \u2208 span K (f '' s)\n[PROOFSTEP]\nsimp only [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9\u271d \u2192 V\ns\u271d t : Set V\nx y z : V\n\u03b9 : Type u_8\ns : Set \u03b9\na : \u03b9\nf : \u03b9 \u2192 V\nhas : \u00aca \u2208 s\n\u22a2 (LinearIndependent K fun x => f \u2191(\u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm) (some x))) \u2227\n      \u00acf \u2191(\u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm) none) \u2208\n          span K (range fun x => f \u2191(\u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm) (some x))) \u2194\n    (LinearIndependent K fun x => f \u2191x) \u2227 \u00acf a \u2208 span K (f '' s)\n[PROOFSTEP]\nerw [range_comp f ((\u2191) : s \u2192 \u03b9)]\n[GOAL]\n\u03b9\u271d : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9\u271d \u2192 V\ns\u271d t : Set V\nx y z : V\n\u03b9 : Type u_8\ns : Set \u03b9\na : \u03b9\nf : \u03b9 \u2192 V\nhas : \u00aca \u2208 s\n\u22a2 (LinearIndependent K fun x => f \u2191(\u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm) (some x))) \u2227\n      \u00acf \u2191(\u2191((Equiv.optionEquivSumPUnit \u2191s).trans (Equiv.Set.insert has).symm) none) \u2208 span K (f '' range Subtype.val) \u2194\n    (LinearIndependent K fun x => f \u2191x) \u2227 \u00acf a \u2208 span K (f '' s)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhxs : \u00acx \u2208 s\n\u22a2 (LinearIndependent K fun x => id \u2191x) \u2227 \u00acid x \u2208 span K (id '' s) \u2194 (LinearIndependent K fun b => \u2191b) \u2227 \u00acx \u2208 span K s\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nn : \u2115\nv : Fin n \u2192 V\n\u22a2 LinearIndependent K (Fin.cons x v) \u2194 LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\nrw [\u2190 linearIndependent_equiv (finSuccEquiv n).symm, linearIndependent_option]\n  -- Porting note: `convert Iff.rfl; ...` \u2192 `exact Iff.rfl`\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nn : \u2115\nv : Fin n \u2192 V\n\u22a2 LinearIndependent K ((Fin.cons x v \u2218 \u2191(finSuccEquiv n).symm) \u2218 some) \u2227\n      \u00ac(Fin.cons x v \u2218 \u2191(finSuccEquiv n).symm) none \u2208 span K (range ((Fin.cons x v \u2218 \u2191(finSuccEquiv n).symm) \u2218 some)) \u2194\n    LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\nexact Iff.rfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nn : \u2115\nv : Fin n \u2192 V\n\u22a2 LinearIndependent K (Fin.snoc v x) \u2194 LinearIndependent K v \u2227 \u00acx \u2208 span K (range v)\n[PROOFSTEP]\nerw [Fin.snoc_eq_cons_rotate, linearIndependent_equiv, linearIndependent_fin_cons]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nn : \u2115\nv : Fin (n + 1) \u2192 V\n\u22a2 LinearIndependent K v \u2194 LinearIndependent K (Fin.tail v) \u2227 \u00acv 0 \u2208 span K (range (Fin.tail v))\n[PROOFSTEP]\nrw [\u2190 linearIndependent_fin_cons, Fin.cons_self_tail]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv\u271d : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nn : \u2115\nv : Fin (n + 1) \u2192 V\n\u22a2 LinearIndependent K v \u2194 LinearIndependent K (Fin.init v) \u2227 \u00acv (Fin.last n) \u2208 span K (range (Fin.init v))\n[PROOFSTEP]\nrw [\u2190 linearIndependent_fin_snoc, Fin.snoc_init_self]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : Fin 2 \u2192 V\n\u22a2 LinearIndependent K f \u2194 f 1 \u2260 0 \u2227 \u2200 (a : K), a \u2022 f 1 \u2260 f 0\n[PROOFSTEP]\nrw [linearIndependent_fin_succ, linearIndependent_unique_iff, range_unique, mem_span_singleton, not_exists,\n  show Fin.tail f default = f 1 by rw [\u2190 Fin.succ_zero_eq_one]; rfl]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : Fin 2 \u2192 V\n\u22a2 Fin.tail f default = f 1\n[PROOFSTEP]\nrw [\u2190 Fin.succ_zero_eq_one]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nf : Fin 2 \u2192 V\n\u22a2 Fin.tail f default = f (Fin.succ 0)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\n\u22a2 \u2203 b x, s \u2286 b \u2227 t \u2286 \u2191(span K b) \u2227 LinearIndependent K Subtype.val\n[PROOFSTEP]\nhave := by\n  refine zorn_subset_nonempty {b | b \u2286 t \u2227 LinearIndependent K ((\u2191) : b \u2192 V)} ?_ _ \u27e8hst, hs\u27e9\n  \u00b7 refine' fun c hc cc _c0 => \u27e8\u22c3\u2080 c, \u27e8_, _\u27e9, fun x => _\u27e9\n    \u00b7 exact sUnion_subset fun x xc => (hc xc).1\n    \u00b7 exact linearIndependent_sUnion_of_directed cc.directedOn fun x xc => (hc xc).2\n    \u00b7 exact subset_sUnion_of_mem\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\n\u22a2 ?m.1310755\n[PROOFSTEP]\nrefine zorn_subset_nonempty {b | b \u2286 t \u2227 LinearIndependent K ((\u2191) : b \u2192 V)} ?_ _ \u27e8hst, hs\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\n\u22a2 \u2200 (c : Set (Set V)),\n    c \u2286 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192\n      IsChain (fun x x_1 => x \u2286 x_1) c \u2192\n        Set.Nonempty c \u2192 \u2203 ub, ub \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2227 \u2200 (s : Set V), s \u2208 c \u2192 s \u2286 ub\n[PROOFSTEP]\nrefine' fun c hc cc _c0 => \u27e8\u22c3\u2080 c, \u27e8_, _\u27e9, fun x => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nc : Set (Set V)\nhc : c \u2286 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : Set.Nonempty c\n\u22a2 \u22c3\u2080 c \u2286 t\n[PROOFSTEP]\nexact sUnion_subset fun x xc => (hc xc).1\n[GOAL]\ncase refine'_2\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nc : Set (Set V)\nhc : c \u2286 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : Set.Nonempty c\n\u22a2 LinearIndependent K Subtype.val\n[PROOFSTEP]\nexact linearIndependent_sUnion_of_directed cc.directedOn fun x xc => (hc xc).2\n[GOAL]\ncase refine'_3\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nc : Set (Set V)\nhc : c \u2286 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val}\ncc : IsChain (fun x x_1 => x \u2286 x_1) c\n_c0 : Set.Nonempty c\nx : Set V\n\u22a2 x \u2208 c \u2192 x \u2286 \u22c3\u2080 c\n[PROOFSTEP]\nexact subset_sUnion_of_mem\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nthis :\n  \u2203 m,\n    m \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2227\n      s \u2286 m \u2227 \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 m \u2286 a \u2192 a = m\n\u22a2 \u2203 b x, s \u2286 b \u2227 t \u2286 \u2191(span K b) \u2227 LinearIndependent K Subtype.val\n[PROOFSTEP]\nrcases this with \u27e8b, \u27e8bt, bi\u27e9, sb, h\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nb : Set V\nbt : b \u2286 t\nbi : LinearIndependent K Subtype.val\nsb : s \u2286 b\nh : \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 b \u2286 a \u2192 a = b\n\u22a2 \u2203 b x, s \u2286 b \u2227 t \u2286 \u2191(span K b) \u2227 LinearIndependent K Subtype.val\n[PROOFSTEP]\nrefine' \u27e8b, bt, sb, fun x xt => _, bi\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nb : Set V\nbt : b \u2286 t\nbi : LinearIndependent K Subtype.val\nsb : s \u2286 b\nh : \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 b \u2286 a \u2192 a = b\nx : V\nxt : x \u2208 t\n\u22a2 x \u2208 \u2191(span K b)\n[PROOFSTEP]\nby_contra hn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nb : Set V\nbt : b \u2286 t\nbi : LinearIndependent K Subtype.val\nsb : s \u2286 b\nh : \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 b \u2286 a \u2192 a = b\nx : V\nxt : x \u2208 t\nhn : \u00acx \u2208 \u2191(span K b)\n\u22a2 False\n[PROOFSTEP]\napply hn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nb : Set V\nbt : b \u2286 t\nbi : LinearIndependent K Subtype.val\nsb : s \u2286 b\nh : \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 b \u2286 a \u2192 a = b\nx : V\nxt : x \u2208 t\nhn : \u00acx \u2208 \u2191(span K b)\n\u22a2 x \u2208 \u2191(span K b)\n[PROOFSTEP]\nrw [\u2190 h _ \u27e8insert_subset_iff.2 \u27e8xt, bt\u27e9, bi.insert hn\u27e9 (subset_insert _ _)]\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx\u271d y z : V\nhs : LinearIndependent K Subtype.val\nhst : s \u2286 t\nb : Set V\nbt : b \u2286 t\nbi : LinearIndependent K Subtype.val\nsb : s \u2286 b\nh : \u2200 (a : Set V), a \u2208 {b | b \u2286 t \u2227 LinearIndependent K Subtype.val} \u2192 b \u2286 a \u2192 a = b\nx : V\nxt : x \u2208 t\nhn : \u00acx \u2208 \u2191(span K b)\n\u22a2 x \u2208 \u2191(span K (insert x b))\n[PROOFSTEP]\nexact subset_span (mem_insert _ _)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\n\u22a2 \u2203 b x, span K b = span K t \u2227 LinearIndependent K Subtype.val\n[PROOFSTEP]\nobtain \u27e8b, hb\u2081, -, hb\u2082, hb\u2083\u27e9 := exists_linearIndependent_extension (linearIndependent_empty K V) (Set.empty_subset t)\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nb : Set V\nhb\u2081 : b \u2286 t\nhb\u2082 : t \u2286 \u2191(span K b)\nhb\u2083 : LinearIndependent K Subtype.val\n\u22a2 \u2203 b x, span K b = span K t \u2227 LinearIndependent K Subtype.val\n[PROOFSTEP]\nexact \u27e8b, hb\u2081, (span_eq_of_le _ hb\u2082 (Submodule.span_mono hb\u2081)).symm, hb\u2083\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\n\u22a2 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\nclassical\nhave :\n  \u2200 t : Finset V,\n    \u2200 s' : Finset V,\n      \u2191s' \u2286 s \u2192\n        s \u2229 \u2191t = \u2205 \u2192\n          s \u2286 (span K \u2191(s' \u222a t) : Submodule K V) \u2192 \u2203 t' : Finset V, \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 t'.card = (s' \u222a t).card :=\n  fun t =>\n  Finset.induction_on t\n    (fun s' hs' _ hss' =>\n      have : s = \u2191s' := eq_of_linearIndependent_of_span_subtype hs hs' <| by simpa using hss'\n      \u27e8s', by simp [this]\u27e9)\n    fun b\u2081 t hb\u2081t ih s' hs' hst hss' =>\n    have hb\u2081s : b\u2081 \u2209 s := fun h =>\n      by\n      have : b\u2081 \u2208 s \u2229 \u2191(insert b\u2081 t) := \u27e8h, Finset.mem_insert_self _ _\u27e9\n      rwa [hst] at this \n    have hb\u2081s' : b\u2081 \u2209 s' := fun h => hb\u2081s <| hs' h\n    have hst : s \u2229 \u2191t = \u2205 :=\n      eq_empty_of_subset_empty <|\n        -- Porting note: `-inter_subset_left, -subset_inter_iff` required.Subset.trans\n          (by simp [inter_subset_inter, Subset.refl, -inter_subset_left, -subset_inter_iff]) (le_of_eq hst)\n    Classical.by_cases (p := s \u2286 (span K \u2191(s' \u222a t) : Submodule K V))\n      (fun this =>\n        let \u27e8u, hust, hsu, Eq\u27e9 := ih _ hs' hst this\n        have hb\u2081u : b\u2081 \u2209 u := fun h => (hust h).elim hb\u2081s hb\u2081t\n        \u27e8insert b\u2081 u, by simp [insert_subset_insert hust], Subset.trans hsu (by simp), by simp [Eq, hb\u2081t, hb\u2081s', hb\u2081u]\u27e9)\n      fun this =>\n      let \u27e8b\u2082, hb\u2082s, hb\u2082t\u27e9 := not_subset.mp this\n      have hb\u2082t' : b\u2082 \u2209 s' \u222a t := fun h => hb\u2082t <| subset_span h\n      have : s \u2286 (span K \u2191(insert b\u2082 s' \u222a t) : Submodule K V) := fun b\u2083 hb\u2083 =>\n        by\n        have : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V) := by\n          -- Porting note: Too many theorems to be excluded, so\n                      --               `simp only` is shorter.simp only [insert_eq, union_subset_union, Subset.refl,\n            subset_union_right, Finset.union_insert, Finset.coe_insert]\n        have hb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V)) := span_mono this (hss' hb\u2083)\n        have : s \u2286 (span K (insert b\u2081 \u2191(s' \u222a t)) : Submodule K V) := by\n          simpa [insert_eq, -singleton_union, -union_singleton] using hss'\n        have hb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t)) := by exact mem_span_insert_exchange (this hb\u2082s) hb\u2082t\n        rw [span_insert_eq_span hb\u2081] at hb\u2083 ; simpa using hb\u2083\n      let \u27e8u, hust, hsu, eq\u27e9 := ih _ (by simp [insert_subset_iff, hb\u2082s, hs']) hst this\n      \u27e8u, Subset.trans hust <| union_subset_union (Subset.refl _) (by simp [subset_insert]), hsu, by\n        simp [eq, Finset.card_insert_of_not_mem hb\u2082t', hb\u2081t, hb\u2081s']\u27e9\nhave eq : ((t.filter fun x => x \u2208 s) \u222a t.filter fun x => x \u2209 s) = t :=\n  by\n  ext1 x\n  by_cases x \u2208 s <;> simp [*]\napply\n  Exists.elim\n    (this (t.filter fun x => x \u2209 s) (t.filter fun x => x \u2208 s) (by simp [Set.subset_def])\n      (by simp (config := { contextual := true }) [Set.ext_iff]) (by rwa [eq]))\nintro u h\nexact\n  \u27e8u, Subset.trans h.1 (by simp (config := { contextual := true }) [subset_def, and_imp, or_imp]), h.2.1, by\n    simp only [h.2.2, eq]\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\n\u22a2 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\nhave :\n  \u2200 t : Finset V,\n    \u2200 s' : Finset V,\n      \u2191s' \u2286 s \u2192\n        s \u2229 \u2191t = \u2205 \u2192\n          s \u2286 (span K \u2191(s' \u222a t) : Submodule K V) \u2192 \u2203 t' : Finset V, \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 t'.card = (s' \u222a t).card :=\n  fun t =>\n  Finset.induction_on t\n    (fun s' hs' _ hss' =>\n      have : s = \u2191s' := eq_of_linearIndependent_of_span_subtype hs hs' <| by simpa using hss'\n      \u27e8s', by simp [this]\u27e9)\n    fun b\u2081 t hb\u2081t ih s' hs' hst hss' =>\n    have hb\u2081s : b\u2081 \u2209 s := fun h =>\n      by\n      have : b\u2081 \u2208 s \u2229 \u2191(insert b\u2081 t) := \u27e8h, Finset.mem_insert_self _ _\u27e9\n      rwa [hst] at this \n    have hb\u2081s' : b\u2081 \u2209 s' := fun h => hb\u2081s <| hs' h\n    have hst : s \u2229 \u2191t = \u2205 :=\n      eq_empty_of_subset_empty <|\n        -- Porting note: `-inter_subset_left, -subset_inter_iff` required.Subset.trans\n          (by simp [inter_subset_inter, Subset.refl, -inter_subset_left, -subset_inter_iff]) (le_of_eq hst)\n    Classical.by_cases (p := s \u2286 (span K \u2191(s' \u222a t) : Submodule K V))\n      (fun this =>\n        let \u27e8u, hust, hsu, Eq\u27e9 := ih _ hs' hst this\n        have hb\u2081u : b\u2081 \u2209 u := fun h => (hust h).elim hb\u2081s hb\u2081t\n        \u27e8insert b\u2081 u, by simp [insert_subset_insert hust], Subset.trans hsu (by simp), by simp [Eq, hb\u2081t, hb\u2081s', hb\u2081u]\u27e9)\n      fun this =>\n      let \u27e8b\u2082, hb\u2082s, hb\u2082t\u27e9 := not_subset.mp this\n      have hb\u2082t' : b\u2082 \u2209 s' \u222a t := fun h => hb\u2082t <| subset_span h\n      have : s \u2286 (span K \u2191(insert b\u2082 s' \u222a t) : Submodule K V) := fun b\u2083 hb\u2083 =>\n        by\n        have : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V) := by\n          -- Porting note: Too many theorems to be excluded, so\n                      --               `simp only` is shorter.simp only [insert_eq, union_subset_union, Subset.refl,\n            subset_union_right, Finset.union_insert, Finset.coe_insert]\n        have hb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V)) := span_mono this (hss' hb\u2083)\n        have : s \u2286 (span K (insert b\u2081 \u2191(s' \u222a t)) : Submodule K V) := by\n          simpa [insert_eq, -singleton_union, -union_singleton] using hss'\n        have hb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t)) := by exact mem_span_insert_exchange (this hb\u2082s) hb\u2082t\n        rw [span_insert_eq_span hb\u2081] at hb\u2083 ; simpa using hb\u2083\n      let \u27e8u, hust, hsu, eq\u27e9 := ih _ (by simp [insert_subset_iff, hb\u2082s, hs']) hst this\n      \u27e8u, Subset.trans hust <| union_subset_union (Subset.refl _) (by simp [subset_insert]), hsu, by\n        simp [eq, Finset.card_insert_of_not_mem hb\u2082t', hb\u2081t, hb\u2081s']\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b9 : Set V\nx y z : V\nt\u271d : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t\u271d)\nt s' : Finset V\nhs' : \u2191s' \u2286 s\nx\u271d : s \u2229 \u2191\u2205 = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a \u2205))\n\u22a2 s \u2286 \u2191(span K \u2191s')\n[PROOFSTEP]\nsimpa using hss'\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b9 : Set V\nx y z : V\nt\u271d : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t\u271d)\nt s' : Finset V\nhs' : \u2191s' \u2286 s\nx\u271d : s \u2229 \u2191\u2205 = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a \u2205))\nthis : s = \u2191s'\n\u22a2 \u2191s' \u2286 s \u222a \u2191\u2205 \u2227 s \u2286 \u2191s' \u2227 Finset.card s' = Finset.card (s' \u222a \u2205)\n[PROOFSTEP]\nsimp [this]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nh : b\u2081 \u2208 s\n\u22a2 False\n[PROOFSTEP]\nhave : b\u2081 \u2208 s \u2229 \u2191(insert b\u2081 t) := \u27e8h, Finset.mem_insert_self _ _\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nh : b\u2081 \u2208 s\nthis : b\u2081 \u2208 s \u2229 \u2191(insert b\u2081 t)\n\u22a2 False\n[PROOFSTEP]\nrwa [hst] at this \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\n\u22a2 s \u2229 \u2191t \u2286 s \u2229 \u2191(insert b\u2081 t)\n[PROOFSTEP]\nsimp [inter_subset_inter, Subset.refl, -inter_subset_left, -subset_inter_iff]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis : s \u2286 \u2191(span K \u2191(s' \u222a t))\nu : Finset V\nhust : \u2191u \u2286 s \u222a \u2191t\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (s' \u222a t)\nhb\u2081u : \u00acb\u2081 \u2208 u\n\u22a2 \u2191(insert b\u2081 u) \u2286 s \u222a \u2191(insert b\u2081 t)\n[PROOFSTEP]\nsimp [insert_subset_insert hust]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis : s \u2286 \u2191(span K \u2191(s' \u222a t))\nu : Finset V\nhust : \u2191u \u2286 s \u222a \u2191t\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (s' \u222a t)\nhb\u2081u : \u00acb\u2081 \u2208 u\n\u22a2 \u2191u \u2286 \u2191(insert b\u2081 u)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis : s \u2286 \u2191(span K \u2191(s' \u222a t))\nu : Finset V\nhust : \u2191u \u2286 s \u222a \u2191t\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (s' \u222a t)\nhb\u2081u : \u00acb\u2081 \u2208 u\n\u22a2 Finset.card (insert b\u2081 u) = Finset.card (s' \u222a insert b\u2081 t)\n[PROOFSTEP]\nsimp [Eq, hb\u2081t, hb\u2081s', hb\u2081u]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083 : b\u2083 \u2208 s\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nhave : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V) := by\n  -- Porting note: Too many theorems to be excluded, so\n              --               `simp only` is shorter.simp only [insert_eq, union_subset_union, Subset.refl,\n    subset_union_right, Finset.union_insert, Finset.coe_insert]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083 : b\u2083 \u2208 s\n\u22a2 \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\n[PROOFSTEP]\nsimp only [insert_eq, union_subset_union, Subset.refl, subset_union_right, Finset.union_insert, Finset.coe_insert]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083 : b\u2083 \u2208 s\nthis : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nhave hb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t) : Set V)) := span_mono this (hss' hb\u2083)\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t)))\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nhave : s \u2286 (span K (insert b\u2081 \u2191(s' \u222a t)) : Submodule K V) := by\n  simpa [insert_eq, -singleton_union, -union_singleton] using hss'\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t)))\n\u22a2 s \u2286 \u2191(span K (insert b\u2081 \u2191(s' \u222a t)))\n[PROOFSTEP]\nsimpa [insert_eq, -singleton_union, -union_singleton] using hss'\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d\u00b9 : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis\u271d : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t)))\nthis : s \u2286 \u2191(span K (insert b\u2081 \u2191(s' \u222a t)))\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nhave hb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t)) := by exact mem_span_insert_exchange (this hb\u2082s) hb\u2082t\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d\u00b9 : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis\u271d : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t)))\nthis : s \u2286 \u2191(span K (insert b\u2081 \u2191(s' \u222a t)))\n\u22a2 b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t))\n[PROOFSTEP]\nexact mem_span_insert_exchange (this hb\u2082s) hb\u2082t\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d\u00b9 : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis\u271d : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2081 (insert b\u2082 \u2191(s' \u222a t)))\nthis : s \u2286 \u2191(span K (insert b\u2081 \u2191(s' \u222a t)))\nhb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t))\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nrw [span_insert_eq_span hb\u2081] at hb\u2083 \n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d\u00b9 : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nb\u2083 : V\nhb\u2083\u271d : b\u2083 \u2208 s\nthis\u271d : \u2191(s' \u222a insert b\u2081 t) \u2286 insert b\u2081 (insert b\u2082 \u2191(s' \u222a t))\nhb\u2083 : b\u2083 \u2208 span K (insert b\u2082 \u2191(s' \u222a t))\nthis : s \u2286 \u2191(span K (insert b\u2081 \u2191(s' \u222a t)))\nhb\u2081 : b\u2081 \u2208 span K (insert b\u2082 \u2191(s' \u222a t))\n\u22a2 b\u2083 \u2208 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n[PROOFSTEP]\nsimpa using hb\u2083\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nthis : s \u2286 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\n\u22a2 \u2191(insert b\u2082 s') \u2286 s\n[PROOFSTEP]\nsimp [insert_subset_iff, hb\u2082s, hs']\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nthis : s \u2286 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\nu : Finset V\nhust : \u2191u \u2286 s \u222a \u2191t\nhsu : s \u2286 \u2191u\neq : Finset.card u = Finset.card (insert b\u2082 s' \u222a t)\n\u22a2 \u2191t \u2286 \u2191(insert b\u2081 t)\n[PROOFSTEP]\nsimp [subset_insert]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d\u00b2 : Set V\nx y z : V\nt\u271d\u00b9 : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst\u271d\u00b9 : s \u2286 \u2191(span K \u2191t\u271d\u00b9)\nt\u271d : Finset V\nb\u2081 : V\nt : Finset V\nhb\u2081t : \u00acb\u2081 \u2208 t\nih :\n  \u2200 (s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\ns' : Finset V\nhs' : \u2191s' \u2286 s\nhst\u271d : s \u2229 \u2191(insert b\u2081 t) = \u2205\nhss' : s \u2286 \u2191(span K \u2191(s' \u222a insert b\u2081 t))\nhb\u2081s : \u00acb\u2081 \u2208 s\nhb\u2081s' : \u00acb\u2081 \u2208 s'\nhst : s \u2229 \u2191t = \u2205\nthis\u271d : \u00acs \u2286 \u2191(span K \u2191(s' \u222a t))\nb\u2082 : V\nhb\u2082s : b\u2082 \u2208 s\nhb\u2082t : \u00acb\u2082 \u2208 \u2191(span K \u2191(s' \u222a t))\nhb\u2082t' : \u00acb\u2082 \u2208 s' \u222a t\nthis : s \u2286 \u2191(span K \u2191(insert b\u2082 s' \u222a t))\nu : Finset V\nhust : \u2191u \u2286 s \u222a \u2191t\nhsu : s \u2286 \u2191u\neq : Finset.card u = Finset.card (insert b\u2082 s' \u222a t)\n\u22a2 Finset.card u = Finset.card (s' \u222a insert b\u2081 t)\n[PROOFSTEP]\nsimp [eq, Finset.card_insert_of_not_mem hb\u2082t', hb\u2081t, hb\u2081s']\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\n\u22a2 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\nhave eq : ((t.filter fun x => x \u2208 s) \u222a t.filter fun x => x \u2209 s) = t :=\n  by\n  ext1 x\n  by_cases x \u2208 s <;> simp [*]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\n\u22a2 Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n[PROOFSTEP]\next1 x\n[GOAL]\ncase a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx\u271d y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\nx : V\n\u22a2 x \u2208 Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t \u2194 x \u2208 t\n[PROOFSTEP]\nby_cases x \u2208 s\n[GOAL]\ncase a\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx\u271d y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\nx : V\n\u22a2 x \u2208 Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t \u2194 x \u2208 t\n[PROOFSTEP]\nby_cases x \u2208 s\n[GOAL]\ncase pos\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx\u271d y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\nx : V\nh : x \u2208 s\n\u22a2 x \u2208 Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t \u2194 x \u2208 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx\u271d y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\nx : V\nh : \u00acx \u2208 s\n\u22a2 x \u2208 Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t \u2194 x \u2208 t\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n\u22a2 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\napply\n  Exists.elim\n    (this (t.filter fun x => x \u2209 s) (t.filter fun x => x \u2208 s) (by simp [Set.subset_def])\n      (by simp (config := { contextual := true }) [Set.ext_iff]) (by rwa [eq]))\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n\u22a2 \u2191(Finset.filter (fun x => x \u2208 s) t) \u2286 s\n[PROOFSTEP]\nsimp [Set.subset_def]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n\u22a2 s \u2229 \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) = \u2205\n[PROOFSTEP]\nsimp (config := { contextual := true }) [Set.ext_iff]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n\u22a2 s \u2286 \u2191(span K \u2191(Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t))\n[PROOFSTEP]\nrwa [eq]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\n\u22a2 \u2200 (a : Finset V),\n    \u2191a \u2286 s \u222a \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) \u2227\n        s \u2286 \u2191a \u2227 Finset.card a = Finset.card (Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t) \u2192\n      \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\nintro u h\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\nu : Finset V\nh :\n  \u2191u \u2286 s \u222a \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) \u2227\n    s \u2286 \u2191u \u2227 Finset.card u = Finset.card (Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t)\n\u22a2 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card t\n[PROOFSTEP]\nexact\n  \u27e8u, Subset.trans h.1 (by simp (config := { contextual := true }) [subset_def, and_imp, or_imp]), h.2.1, by\n    simp only [h.2.2, eq]\u27e9\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\nu : Finset V\nh :\n  \u2191u \u2286 s \u222a \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) \u2227\n    s \u2286 \u2191u \u2227 Finset.card u = Finset.card (Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t)\n\u22a2 s \u222a \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) \u2286 s \u222a \u2191t\n[PROOFSTEP]\nsimp (config := { contextual := true }) [subset_def, and_imp, or_imp]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t\u271d : Set V\nx y z : V\nt : Finset V\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K \u2191t)\nthis :\n  \u2200 (t s' : Finset V),\n    \u2191s' \u2286 s \u2192\n      s \u2229 \u2191t = \u2205 \u2192 s \u2286 \u2191(span K \u2191(s' \u222a t)) \u2192 \u2203 t', \u2191t' \u2286 s \u222a \u2191t \u2227 s \u2286 \u2191t' \u2227 Finset.card t' = Finset.card (s' \u222a t)\neq : Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t = t\nu : Finset V\nh :\n  \u2191u \u2286 s \u222a \u2191(Finset.filter (fun x => \u00acx \u2208 s) t) \u2227\n    s \u2286 \u2191u \u2227 Finset.card u = Finset.card (Finset.filter (fun x => x \u2208 s) t \u222a Finset.filter (fun x => \u00acx \u2208 s) t)\n\u22a2 Finset.card u = Finset.card t\n[PROOFSTEP]\nsimp only [h.2.2, eq]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K t)\n\u22a2 s \u2286 \u2191(span K \u2191(Finite.toFinset ht))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K t)\n\u22a2 s \u2286 \u2191(span K t)\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K t)\nthis\u271d : s \u2286 \u2191(span K \u2191(Finite.toFinset ht))\nu : Finset V\n_hust : \u2191u \u2286 s \u222a \u2191(Finite.toFinset ht)\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n\u22a2 Finset.card (Finite.toFinset this) \u2264 Finset.card (Finite.toFinset ht)\n[PROOFSTEP]\nrw [\u2190 Eq]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K t)\nthis\u271d : s \u2286 \u2191(span K \u2191(Finite.toFinset ht))\nu : Finset V\n_hust : \u2191u \u2286 s \u222a \u2191(Finite.toFinset ht)\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n\u22a2 Finset.card (Finite.toFinset this) \u2264 Finset.card u\n[PROOFSTEP]\nexact Finset.card_le_of_subset <| Finset.coe_subset.mp <| by simp [hsu]\n[GOAL]\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\ninst\u271d\u2074 : DivisionRing K\ninst\u271d\u00b3 : AddCommGroup V\ninst\u271d\u00b2 : AddCommGroup V'\ninst\u271d\u00b9 : Module K V\ninst\u271d : Module K V'\nv : \u03b9 \u2192 V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => \u2191x\nhst : s \u2286 \u2191(span K t)\nthis\u271d : s \u2286 \u2191(span K \u2191(Finite.toFinset ht))\nu : Finset V\n_hust : \u2191u \u2286 s \u222a \u2191(Finite.toFinset ht)\nhsu : s \u2286 \u2191u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n\u22a2 \u2191(Finite.toFinset this) \u2286 \u2191u\n[PROOFSTEP]\nsimp [hsu]\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.LinearIndependent", "llama_tokens": 186254, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.5035921678580136}}
{"text": "[GOAL]\nx : \u211d\nh : Liouville x\n\u22a2 Irrational x\n[PROOFSTEP]\nrintro\n  \u27e8\u27e8a, b, bN0, cop\u27e9, rfl\u27e9\n      -- clear up the mess of constructions of rationals\n[GOAL]\ncase intro.mk'\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville \u2191(Rat.mk' a b)\n\u22a2 False\n[PROOFSTEP]\nrw [Rat.cast_mk', \u2190 div_eq_mul_inv] at h \n[GOAL]\ncase intro.mk'\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\n\u22a2 False\n[PROOFSTEP]\nrcases h (b + 1) with\n  \u27e8p, q, q1, a0, a1\u27e9\n    -- A few useful inequalities\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a / \u2191b - \u2191p / \u2191q| < 1 / \u2191q ^ (b + 1)\n\u22a2 False\n[PROOFSTEP]\nhave qR0 : (0 : \u211d) < q := Int.cast_pos.mpr (zero_lt_one.trans q1)\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a / \u2191b - \u2191p / \u2191q| < 1 / \u2191q ^ (b + 1)\nqR0 : 0 < \u2191q\n\u22a2 False\n[PROOFSTEP]\nhave b0 : (b : \u211d) \u2260 0 := Nat.cast_ne_zero.mpr bN0\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a / \u2191b - \u2191p / \u2191q| < 1 / \u2191q ^ (b + 1)\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave bq0 : (0 : \u211d) < b * q := mul_pos (Nat.cast_pos.mpr bN0.bot_lt) qR0\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a / \u2191b - \u2191p / \u2191q| < 1 / \u2191q ^ (b + 1)\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\n\u22a2 False\n[PROOFSTEP]\nreplace a1 : |a * q - b * p| * q ^ (b + 1) < b * q\n[GOAL]\ncase a1\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a / \u2191b - \u2191p / \u2191q| < 1 / \u2191q ^ (b + 1)\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\n\u22a2 |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\n[PROOFSTEP]\nrw [div_sub_div _ _ b0 qR0.ne', abs_div, div_lt_div_iff (abs_pos.mpr bq0.ne') (pow_pos qR0 _), abs_of_pos bq0,\n  one_mul] at a1 \n[GOAL]\ncase a1\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\na1 : |\u2191a * \u2191q - \u2191b * \u2191p| * \u2191q ^ (b + 1) < \u2191b * \u2191q\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\n\u22a2 |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\n[PROOFSTEP]\nexact_mod_cast a1\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\na1 : |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\n\u22a2 False\n[PROOFSTEP]\nreplace a0 : a * q - \u2191b * p \u2260 0\n[GOAL]\ncase a0\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0 : \u2191a / \u2191b \u2260 \u2191p / \u2191q\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\na1 : |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\n\u22a2 a * q - \u2191b * p \u2260 0\n[PROOFSTEP]\nrw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : \u211d), \u2190 sub_eq_zero] at a0 \n[GOAL]\ncase a0\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\na0\u271d : \u00ac\u2191a * \u2191q = \u2191b * \u2191p\na0 : \u00ac\u2191a * \u2191q - \u2191b * \u2191p = 0\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\na1 : |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\n\u22a2 a * q - \u2191b * p \u2260 0\n[PROOFSTEP]\nexact_mod_cast a0\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np q : \u2124\nq1 : 1 < q\nqR0 : 0 < \u2191q\nb0 : \u2191b \u2260 0\nbq0 : 0 < \u2191b * \u2191q\na1 : |a * q - \u2191b * p| * q ^ (b + 1) < \u2191b * q\na0 : a * q - \u2191b * p \u2260 0\n\u22a2 False\n[PROOFSTEP]\nlift q to \u2115 using (zero_lt_one.trans q1).le\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np : \u2124\nb0 : \u2191b \u2260 0\nq : \u2115\nq1 : 1 < \u2191q\nqR0 : 0 < \u2191\u2191q\nbq0 : 0 < \u2191b * \u2191\u2191q\na1 : |a * \u2191q - \u2191b * p| * \u2191q ^ (b + 1) < \u2191b * \u2191q\na0 : a * \u2191q - \u2191b * p \u2260 0\n\u22a2 False\n[PROOFSTEP]\nhave ap : 0 < |a * \u2191q - \u2191b * p| := abs_pos.mpr a0\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np : \u2124\nb0 : \u2191b \u2260 0\nq : \u2115\nq1 : 1 < \u2191q\nqR0 : 0 < \u2191\u2191q\nbq0 : 0 < \u2191b * \u2191\u2191q\na1 : |a * \u2191q - \u2191b * p| * \u2191q ^ (b + 1) < \u2191b * \u2191q\na0 : a * \u2191q - \u2191b * p \u2260 0\nap : 0 < |a * \u2191q - \u2191b * p|\n\u22a2 False\n[PROOFSTEP]\nlift |a * \u2191q - \u2191b * p| to \u2115 using abs_nonneg (a * \u2191q - \u2191b * p) with e he\n[GOAL]\ncase intro.mk'.intro.intro.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np : \u2124\nb0 : \u2191b \u2260 0\nq : \u2115\nq1 : 1 < \u2191q\nqR0 : 0 < \u2191\u2191q\nbq0 : 0 < \u2191b * \u2191\u2191q\na0 : a * \u2191q - \u2191b * p \u2260 0\ne : \u2115\nhe : \u2191e = |a * \u2191q - \u2191b * p|\na1\u271d a1 : \u2191e * \u2191q ^ (b + 1) < \u2191b * \u2191q\nap\u271d ap : 0 < \u2191e\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_mul, \u2190 Int.coe_nat_pow, \u2190 Int.ofNat_mul, Int.ofNat_lt] at a1 \n[GOAL]\ncase intro.mk'.intro.intro.intro.intro.intro.intro\na : \u2124\nb : \u2115\nbN0 : b \u2260 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (\u2191a / \u2191b)\np : \u2124\nb0 : \u2191b \u2260 0\nq : \u2115\nq1 : 1 < \u2191q\nqR0 : 0 < \u2191\u2191q\nbq0 : 0 < \u2191b * \u2191\u2191q\na0 : a * \u2191q - \u2191b * p \u2260 0\ne : \u2115\nhe : \u2191e = |a * \u2191q - \u2191b * p|\na1\u271d : \u2191e * \u2191q ^ (b + 1) < \u2191b * \u2191q\na1 : e * q ^ (b + 1) < b * q\nap\u271d ap : 0 < \u2191e\n\u22a2 False\n[PROOFSTEP]\nexact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (Int.coe_nat_pos.mp ap) (Int.ofNat_lt.mp q1)).le\n[GOAL]\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (z : Z) (a : N), 1 \u2264 d a * (dist \u03b1 (j z a) * A)\n[PROOFSTEP]\nhave me0 : 0 < max (1 / \u03b5) M :=\n  lt_max_iff.mpr\n    (Or.inl (one_div_pos.mpr e0))\n      -- The maximum between `1 / \u03b5` and `M` works\n[GOAL]\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (z : Z) (a : N), 1 \u2264 d a * (dist \u03b1 (j z a) * A)\n[PROOFSTEP]\nrefine'\n  \u27e8max (1 / \u03b5) M, me0, fun z a => _\u27e9\n    -- First, let's deal with the easy case in which we are far away from `\u03b1`\n[GOAL]\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\n\u22a2 1 \u2264 d a * (dist \u03b1 (j z a) * max (1 / \u03b5) M)\n[PROOFSTEP]\nby_cases dm1 : 1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n[GOAL]\ncase pos\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : 1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n\u22a2 1 \u2264 d a * (dist \u03b1 (j z a) * max (1 / \u03b5) M)\n[PROOFSTEP]\nexact one_le_mul_of_one_le_of_one_le (d0 a) dm1\n[GOAL]\ncase neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n\u22a2 1 \u2264 d a * (dist \u03b1 (j z a) * max (1 / \u03b5) M)\n[PROOFSTEP]\nhave : j z a \u2208 closedBall \u03b1 \u03b5 :=\n  by\n  refine' mem_closedBall'.mp (le_trans _ ((one_div_le me0 e0).mpr (le_max_left _ _)))\n  exact (le_div_iff me0).mpr (not_le.mp dm1).le\n[GOAL]\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n\u22a2 j z a \u2208 closedBall \u03b1 \u03b5\n[PROOFSTEP]\nrefine' mem_closedBall'.mp (le_trans _ ((one_div_le me0 e0).mpr (le_max_left _ _)))\n[GOAL]\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n\u22a2 dist \u03b1 (j z a) \u2264 1 / max (1 / \u03b5) M\n[PROOFSTEP]\nexact (le_div_iff me0).mpr (not_le.mp dm1).le\n[GOAL]\ncase neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\nthis : j z a \u2208 closedBall \u03b1 \u03b5\n\u22a2 1 \u2264 d a * (dist \u03b1 (j z a) * max (1 / \u03b5) M)\n[PROOFSTEP]\nrefine'\n  (L this).trans\n    _\n      -- remove a common factor and use the Lipschitz assumption `B`\n[GOAL]\ncase neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\nthis : j z a \u2208 closedBall \u03b1 \u03b5\n\u22a2 d a * dist (f \u03b1) (f (j z a)) \u2264 d a * (dist \u03b1 (j z a) * max (1 / \u03b5) M)\n[PROOFSTEP]\nrefine' mul_le_mul_of_nonneg_left ((B this).trans _) (zero_le_one.trans (d0 a))\n[GOAL]\ncase neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst\u271d : PseudoMetricSpace R\nd : N \u2192 \u211d\nj : Z \u2192 N \u2192 R\nf : R \u2192 R\n\u03b1 : R\n\u03b5 M : \u211d\nd0 : \u2200 (a : N), 1 \u2264 d a\ne0 : 0 < \u03b5\nB : \u2200 \u2983y : R\u2984, y \u2208 closedBall \u03b1 \u03b5 \u2192 dist (f \u03b1) (f y) \u2264 dist \u03b1 y * M\nL : \u2200 \u2983z : Z\u2984 \u2983a : N\u2984, j z a \u2208 closedBall \u03b1 \u03b5 \u2192 1 \u2264 d a * dist (f \u03b1) (f (j z a))\nme0 : 0 < max (1 / \u03b5) M\nz : Z\na : N\ndm1 : \u00ac1 \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\nthis : j z a \u2208 closedBall \u03b1 \u03b5\n\u22a2 dist \u03b1 (j z a) * M \u2264 dist \u03b1 (j z a) * max (1 / \u03b5) M\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_left (le_max_right _ M) dist_nonneg\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfa : eval \u03b1 (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nset fR : \u211d[X] := map (algebraMap \u2124 \u211d) f\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nobtain fR0 : fR \u2260 0 := fun fR0 =>\n  (map_injective (algebraMap \u2124 \u211d) fun _ _ A => Int.cast_inj.mp A).ne f0\n    (fR0.trans (Polynomial.map_zero _).symm)\n      -- reformulating assumption `fa`: `\u03b1` is a root of `fR`.\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nhave ar : \u03b1 \u2208 (fR.roots.toFinset : Set \u211d) :=\n  Finset.mem_coe.mpr\n    (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.def.mpr fa)))\n      -- Since the polynomial `fR` has finitely many roots, there is a closed interval centered at `\u03b1`\n        -- such that `\u03b1` is the only root of `fR` in the interval.\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nobtain \u27e8\u03b6, z0, U\u27e9 : \u2203 \u03b6 > 0, closedBall \u03b1 \u03b6 \u2229 fR.roots.toFinset = { \u03b1 } :=\n  @exists_closedBall_inter_eq_singleton_of_discrete _ _ _ discrete_of_t1_of_finite _ ar\n[GOAL]\ncase intro.intro\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nobtain \u27e8xm, -, hM\u27e9 : \u2203 xm : \u211d, xm \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6) \u2227 IsMaxOn (|fR.derivative.eval \u00b7|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm :=\n  IsCompact.exists_isMaxOn isCompact_Icc \u27e8\u03b1, (sub_lt_self \u03b1 z0).le, (lt_add_of_pos_right \u03b1 z0).le\u27e9\n    (continuous_abs.comp fR.derivative.continuous_aeval).continuousOn\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\n\u22a2 \u2203 A, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|\u03b1 - \u2191a / (\u2191b + 1)| * A)\n[PROOFSTEP]\nrefine'\n  @exists_one_le_pow_mul_dist \u2124 \u2115 \u211d _ _ _ (fun y => fR.eval y) \u03b1 \u03b6 |fR.derivative.eval xm| _ z0 (fun y hy => _)\n    fun z a hq =>\n    _\n      -- 1: the denominators are positive -- essentially by definition;\n[GOAL]\ncase intro.intro.intro.intro.refine'_1\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\n\u22a2 \u2200 (a : \u2115), 1 \u2264 (\u2191a + 1) ^ natDegree f\n[PROOFSTEP]\nexact fun a =>\n  one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg)\n    _\n      -- 2: the polynomial `fR` is Lipschitz at `\u03b1` -- as its derivative continuous;\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny : \u211d\nhy : y \u2208 closedBall \u03b1 \u03b6\n\u22a2 dist ((fun y => eval y fR) \u03b1) ((fun y => eval y fR) y) \u2264 dist \u03b1 y * |eval xm (\u2191derivative fR)|\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny : \u211d\nhy : y \u2208 closedBall \u03b1 \u03b6\n\u22a2 dist ((fun y => eval y fR) \u03b1) ((fun y => eval y fR) y) \u2264 |eval xm (\u2191derivative fR)| * dist \u03b1 y\n[PROOFSTEP]\nrw [Real.closedBall_eq_Icc] at hy \n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny : \u211d\nhy : y \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\n\u22a2 dist ((fun y => eval y fR) \u03b1) ((fun y => eval y fR) y) \u2264 |eval xm (\u2191derivative fR)| * dist \u03b1 y\n[PROOFSTEP]\nrefine'\n  Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiableAt) (fun y h => by rw [fR.deriv]; exact hM h)\n    (convex_Icc _ _) hy (mem_Icc_iff_abs_le.mp _)\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny\u271d : \u211d\nhy : y\u271d \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\ny : \u211d\nh : y \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\n\u22a2 \u2016deriv (fun x => eval x fR) y\u2016 \u2264 |eval xm (\u2191derivative fR)|\n[PROOFSTEP]\nrw [fR.deriv]\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny\u271d : \u211d\nhy : y\u271d \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\ny : \u211d\nh : y \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\n\u22a2 \u2016eval y (\u2191derivative fR)\u2016 \u2264 |eval xm (\u2191derivative fR)|\n[PROOFSTEP]\nexact hM h\n[GOAL]\ncase intro.intro.intro.intro.refine'_2\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\ny : \u211d\nhy : y \u2208 Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)\n\u22a2 |\u03b1 - \u03b1| \u2264 \u03b6\n[PROOFSTEP]\nexact\n  @mem_closedBall_self \u211d _ \u03b1 \u03b6\n    (le_of_lt z0)\n      -- 3: the weird inequality of Liouville type with powers of the denominators.\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / (\u2191a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 1 \u2264 (\u2191a + 1) ^ natDegree f * dist ((fun y => eval y fR) \u03b1) ((fun y => eval y fR) (\u2191z / (\u2191a + 1)))\n[PROOFSTEP]\nshow 1 \u2264 (a + 1 : \u211d) ^ f.natDegree * |eval \u03b1 fR - eval ((z : \u211d) / (a + 1)) fR|\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / (\u2191a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 1 \u2264 (\u2191a + 1) ^ natDegree f * |eval \u03b1 fR - eval (\u2191z / (\u2191a + 1)) fR|\n[PROOFSTEP]\nrw [fa, zero_sub, abs_neg]\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / (\u2191a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 1 \u2264 (\u2191a + 1) ^ natDegree f * |eval (\u2191z / (\u2191a + 1)) fR|\n[PROOFSTEP]\nrw [show (a + 1 : \u211d) = ((a + 1 : \u2115) : \u2124) by norm_cast] at hq \u22a2\n  -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,\n      -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / (\u2191a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 \u2191a + 1 = \u2191\u2191(a + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 \u2191a + 1 = \u2191\u2191(a + 1)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\n\u22a2 1 \u2264 \u2191\u2191(a + 1) ^ natDegree f * |eval (\u2191z / \u2191\u2191(a + 1)) fR|\n[PROOFSTEP]\nrefine'\n  one_le_pow_mul_abs_eval_div (Int.coe_nat_succ_pos a) fun hy =>\n    _\n      -- As the evaluation of the polynomial vanishes, we found a root of `fR` that is rational.\n          -- We know that `\u03b1` is the only root of `fR` in our interval, and `\u03b1` is irrational:\n          -- follow your nose.\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\nhy : eval (\u2191z / \u2191\u2191(a + 1)) (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 False\n[PROOFSTEP]\nrefine' (irrational_iff_ne_rational \u03b1).mp ha z (a + 1) (mem_singleton_iff.mp _).symm\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\nhy : eval (\u2191z / \u2191\u2191(a + 1)) (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 \u2191z / \u2191(\u2191a + 1) \u2208 {\u03b1}\n[PROOFSTEP]\nrefine' U.subset _\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\nhy : eval (\u2191z / \u2191\u2191(a + 1)) (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 \u2191z / \u2191(\u2191a + 1) \u2208 closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR))\n[PROOFSTEP]\nrefine' \u27e8hq, Finset.mem_coe.mp (Multiset.mem_toFinset.mpr _)\u27e9\n[GOAL]\ncase intro.intro.intro.intro.refine'_3\n\u03b1 : \u211d\nha : Irrational \u03b1\nf : \u2124[X]\nf0 : f \u2260 0\nfR : \u211d[X] := map (algebraMap \u2124 \u211d) f\nfa : eval \u03b1 fR = 0\nfR0 : fR \u2260 0\nar : \u03b1 \u2208 \u2191(Multiset.toFinset (roots fR))\n\u03b6 : \u211d\nz0 : \u03b6 > 0\nU : closedBall \u03b1 \u03b6 \u2229 \u2191(Multiset.toFinset (roots fR)) = {\u03b1}\nxm : \u211d\nhM : IsMaxOn (fun x => |eval x (\u2191derivative fR)|) (Icc (\u03b1 - \u03b6) (\u03b1 + \u03b6)) xm\nz : \u2124\na : \u2115\nhq : \u2191z / \u2191\u2191(a + 1) \u2208 closedBall \u03b1 \u03b6\nhy : eval (\u2191z / \u2191\u2191(a + 1)) (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 \u2191z / \u2191(\u2191a + 1) \u2208 roots fR\n[PROOFSTEP]\nexact (mem_roots fR0).mpr (IsRoot.def.mpr hy)\n[GOAL]\nx : \u211d\nlx : Liouville x\n\u22a2 Transcendental \u2124 x\n[PROOFSTEP]\nrintro\n  \u27e8f : \u2124[X], f0, ef0\u27e9\n      -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.\n[GOAL]\ncase intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : \u2191(aeval x) f = 0\n\u22a2 False\n[PROOFSTEP]\nreplace ef0 : (f.map (algebraMap \u2124 \u211d)).eval x = 0\n[GOAL]\ncase ef0\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : \u2191(aeval x) f = 0\n\u22a2 eval x (map (algebraMap \u2124 \u211d) f) = 0\n[PROOFSTEP]\nrwa [aeval_def, \u2190 eval_map] at ef0 \n[GOAL]\ncase intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8A, hA, h\u27e9 :\n  \u2203 A : \u211d, 0 < A \u2227 \u2200 (a : \u2124) (b : \u2115), (1 : \u211d) \u2264 ((b : \u211d) + 1) ^ f.natDegree * (|x - a / (b + 1)| * A) :=\n  exists_pos_real_of_irrational_root lx.irrational f0 ef0\n[GOAL]\ncase intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\n\u22a2 False\n[PROOFSTEP]\nrcases pow_unbounded_of_one_lt A (lt_add_one 1) with\n  \u27e8r, hn\u27e9\n    -- Use the Liouville property, with exponent `r + deg f`.\n[GOAL]\ncase intro.intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\n\u22a2 False\n[PROOFSTEP]\nobtain \u27e8a, b, b1, -, a1\u27e9 : \u2203 a b : \u2124, 1 < b \u2227 x \u2260 a / b \u2227 |x - a / b| < 1 / (b : \u211d) ^ (r + f.natDegree) :=\n  lx (r + f.natDegree)\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : |x - \u2191a / \u2191b| < 1 / \u2191b ^ (r + natDegree f)\n\u22a2 False\n[PROOFSTEP]\nhave b0 : (0 : \u211d) < b :=\n  zero_lt_one.trans\n    (by rw [\u2190 Int.cast_one]; exact Int.cast_lt.mpr b1)\n      -- Prove that `b ^ f.nat_degree * abs (x - a / b)` is strictly smaller than itself\n        -- recall, this is a proof by contradiction!\n[GOAL]\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : |x - \u2191a / \u2191b| < 1 / \u2191b ^ (r + natDegree f)\n\u22a2 1 < \u2191b\n[PROOFSTEP]\nrw [\u2190 Int.cast_one]\n[GOAL]\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : |x - \u2191a / \u2191b| < 1 / \u2191b ^ (r + natDegree f)\n\u22a2 \u21911 < \u2191b\n[PROOFSTEP]\nexact Int.cast_lt.mpr b1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : |x - \u2191a / \u2191b| < 1 / \u2191b ^ (r + natDegree f)\nb0 : 0 < \u2191b\n\u22a2 False\n[PROOFSTEP]\nrefine'\n  lt_irrefl ((b : \u211d) ^ f.natDegree * |x - \u2191a / \u2191b|)\n    _\n      -- clear denominators at `a1`\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : |x - \u2191a / \u2191b| < 1 / \u2191b ^ (r + natDegree f)\nb0 : 0 < \u2191b\n\u22a2 \u2191b ^ natDegree f * |x - \u2191a / \u2191b| < \u2191b ^ natDegree f * |x - \u2191a / \u2191b|\n[PROOFSTEP]\nrw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1 \n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 \u2191b ^ natDegree f * |x - \u2191a / \u2191b| < \u2191b ^ natDegree f * |x - \u2191a / \u2191b|\n[PROOFSTEP]\nrefine'\n  (_ : (b : \u211d) ^ f.natDegree * |x - a / b| < 1 / A).trans_le\n    _\n      -- This branch of the proof uses the Liouville condition and the Archimedean property\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 \u2191b ^ natDegree f * |x - \u2191a / \u2191b| < 1 / A\n[PROOFSTEP]\nrefine' (lt_div_iff' hA).mpr _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 A * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\n[PROOFSTEP]\nrefine' lt_of_le_of_lt _ a1\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 A * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) \u2264 \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|)\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 A \u2264 \u2191b ^ r\n[PROOFSTEP]\nrefine' hn.le.trans _\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 (1 + 1) ^ r \u2264 \u2191b ^ r\n[PROOFSTEP]\nrw [one_add_one_eq_two]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 2 ^ r \u2264 \u2191b ^ r\n[PROOFSTEP]\ngcongr\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.h.hab\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 2 \u2264 \u2191b\n[PROOFSTEP]\nexact\n  Int.cast_two.symm.le.trans\n    (Int.cast_le.mpr (Int.add_one_le_iff.mpr b1))\n      -- this branch of the proof exploits the \"integrality\" of evaluations of polynomials\n        -- at ratios of integers.\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na b : \u2124\nb1 : 1 < b\na1 : \u2191b ^ r * (\u2191b ^ natDegree f * |x - \u2191a / \u2191b|) < 1\nb0 : 0 < \u2191b\n\u22a2 1 / A \u2264 \u2191b ^ natDegree f * |x - \u2191a / \u2191b|\n[PROOFSTEP]\nlift b to \u2115 using zero_le_one.trans b1.le\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nh : \u2200 (a : \u2124) (b : \u2115), 1 \u2264 (\u2191b + 1) ^ natDegree f * (|x - \u2191a / (\u2191b + 1)| * A)\nr : \u2115\nhn : A < (1 + 1) ^ r\na : \u2124\nb : \u2115\nb1 : 1 < \u2191b\na1 : \u2191\u2191b ^ r * (\u2191\u2191b ^ natDegree f * |x - \u2191a / \u2191\u2191b|) < 1\nb0 : 0 < \u2191\u2191b\n\u22a2 1 / A \u2264 \u2191\u2191b ^ natDegree f * |x - \u2191a / \u2191\u2191b|\n[PROOFSTEP]\nspecialize h a b.pred\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2.intro\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nr : \u2115\nhn : A < (1 + 1) ^ r\na : \u2124\nb : \u2115\nb1 : 1 < \u2191b\na1 : \u2191\u2191b ^ r * (\u2191\u2191b ^ natDegree f * |x - \u2191a / \u2191\u2191b|) < 1\nb0 : 0 < \u2191\u2191b\nh : 1 \u2264 (\u2191(Nat.pred b) + 1) ^ natDegree f * (|x - \u2191a / (\u2191(Nat.pred b) + 1)| * A)\n\u22a2 1 / A \u2264 \u2191\u2191b ^ natDegree f * |x - \u2191a / \u2191\u2191b|\n[PROOFSTEP]\nrwa [\u2190 Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), \u2190 mul_assoc, \u2190 div_le_iff hA] at h \n[GOAL]\nx : \u211d\nlx : Liouville x\nf : \u2124[X]\nf0 : f \u2260 0\nef0 : eval x (map (algebraMap \u2124 \u211d) f) = 0\nA : \u211d\nhA : 0 < A\nr : \u2115\nhn : A < (1 + 1) ^ r\na : \u2124\nb : \u2115\nb1 : 1 < \u2191b\na1 : \u2191\u2191b ^ r * (\u2191\u2191b ^ natDegree f * |x - \u2191a / \u2191\u2191b|) < 1\nb0 : 0 < \u2191\u2191b\nh : 1 \u2264 \u2191(Nat.succ (Nat.pred b)) ^ natDegree f * (|x - \u2191a / \u2191(Nat.succ (Nat.pred b))| * A)\n\u22a2 1 < b\n[PROOFSTEP]\nexact Int.ofNat_lt.mp b1\n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.Liouville.Basic", "llama_tokens": 19024, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8333246035907933, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.5032712559355451}}
{"text": "[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u2191(GlueData.\u03b9 D.toGlueData i) \u207b\u00b9' U)\n[PROOFSTEP]\ndelta CategoryTheory.GlueData.\u03b9\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen U \u2194 \u2200 (i : D.J), IsOpen (\u2191(Multicoequalizer.\u03c0 (GlueData.diagram D.toGlueData) i) \u207b\u00b9' U)\n[PROOFSTEP]\nsimp_rw [\u2190 Multicoequalizer.\u03b9_sigma\u03c0 \ud835\udda3.diagram]\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen U \u2194\n    \u2200 (i : D.J),\n      IsOpen\n        (\u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b\n              Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9'\n          U)\n[PROOFSTEP]\nrw [\u2190 (homeoOfIso (Multicoequalizer.isoCoequalizer \ud835\udda3.diagram).symm).isOpen_preimage]\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U) \u2194\n    \u2200 (i : D.J),\n      IsOpen\n        (\u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b\n              Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9'\n          U)\n[PROOFSTEP]\nrw [coequalizer_isOpen_iff]\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen\n      (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            WalkingParallelPair.one) \u207b\u00b9'\n        (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J),\n      IsOpen\n        (\u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b\n              Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9'\n          U)\n[PROOFSTEP]\ndsimp only [GlueData.diagram_l, GlueData.diagram_left, GlueData.diagram_r, GlueData.diagram_right, parallelPair_obj_one]\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 IsOpen\n      (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            WalkingParallelPair.one) \u207b\u00b9'\n        (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)) \u2194\n    \u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\n[PROOFSTEP]\nrw [colimit_isOpen_iff.{_, u}]\n  -- porting note: changed `.{u}` to `.{_,u}`.  fun fact: the proof\n                                   -- breaks down if this `rw` is merged with the `rw` above.\n[GOAL]\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u2191(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u2191(colimit.\u03b9\n                (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n                  (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n                WalkingParallelPair.one) \u207b\u00b9'\n            (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))) \u2194\n    \u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 (\u2200 (j : Discrete D.J),\n      IsOpen\n        (\u2191(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u2191(colimit.\u03b9\n                (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n                  (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n                WalkingParallelPair.one) \u207b\u00b9'\n            (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))) \u2192\n    \u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\n[PROOFSTEP]\nintro h j\n[GOAL]\ncase mp\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\nh :\n  \u2200 (j : Discrete D.J),\n    IsOpen\n      (\u2191(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n        (\u2191(colimit.\u03b9\n              (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n                (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n              WalkingParallelPair.one) \u207b\u00b9'\n          (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))\nj : D.J\n\u22a2 IsOpen (\u2191(Sigma.\u03b9 D.U j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\n[PROOFSTEP]\nexact h \u27e8j\u27e9\n[GOAL]\ncase mpr\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\n\u22a2 (\u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)) \u2192\n    \u2200 (j : Discrete D.J),\n      IsOpen\n        (\u2191(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n          (\u2191(colimit.\u03b9\n                (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n                  (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n                WalkingParallelPair.one) \u207b\u00b9'\n            (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))\n[PROOFSTEP]\nintro h j\n[GOAL]\ncase mpr\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\nh : \u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\nj : Discrete D.J\n\u22a2 IsOpen\n    (\u2191(colimit.\u03b9 (Discrete.functor D.U) j) \u207b\u00b9'\n      (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            WalkingParallelPair.one) \u207b\u00b9'\n        (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))\n[PROOFSTEP]\ncases j\n[GOAL]\ncase mpr.mk\nD : GlueData\nU : Set \u2191(GlueData.glued D.toGlueData)\nh : \u2200 (i : D.J), IsOpen (\u2191(Sigma.\u03b9 D.U i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) \u207b\u00b9' U)\nas\u271d : D.J\n\u22a2 IsOpen\n    (\u2191(colimit.\u03b9 (Discrete.functor D.U) { as := as\u271d }) \u207b\u00b9'\n      (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            WalkingParallelPair.one) \u207b\u00b9'\n        (\u2191(homeoOfIso (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).symm) \u207b\u00b9' U)))\n[PROOFSTEP]\napply h\n[GOAL]\nD : GlueData\n\u22a2 \u2200 {x y : (i : D.J) \u00d7 \u2191(GlueData.U D.toGlueData i)}, Rel D x y \u2192 Rel D y x\n[PROOFSTEP]\nrintro a b (\u27e8\u27e8\u27e9\u27e9 | \u27e8x, e\u2081, e\u2082\u27e9)\n[GOAL]\ncase inl.refl\nD : GlueData\na : (i : D.J) \u00d7 \u2191(GlueData.U D.toGlueData i)\n\u22a2 Rel D a a\ncase inr.intro.intro\nD : GlueData\na b : (i : D.J) \u00d7 \u2191(GlueData.U D.toGlueData i)\nx : \u2191(GlueData.V D.toGlueData (a.fst, b.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData a.fst b.fst) x = a.snd\ne\u2082 : \u2191(GlueData.f D.toGlueData b.fst a.fst) (\u2191(GlueData.t D.toGlueData a.fst b.fst) x) = b.snd\n\u22a2 Rel D b a\n[PROOFSTEP]\nexacts [Or.inl rfl, Or.inr \u27e8D.t _ _ x, by simp [e\u2081, e\u2082]\u27e9]\n[GOAL]\nD : GlueData\na b : (i : D.J) \u00d7 \u2191(GlueData.U D.toGlueData i)\nx : \u2191(GlueData.V D.toGlueData (a.fst, b.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData a.fst b.fst) x = a.snd\ne\u2082 : \u2191(GlueData.f D.toGlueData b.fst a.fst) (\u2191(GlueData.t D.toGlueData a.fst b.fst) x) = b.snd\n\u22a2 \u2191(GlueData.f D.toGlueData b.fst a.fst) (\u2191(GlueData.t D.toGlueData a.fst b.fst) x) = b.snd \u2227\n    \u2191(GlueData.f D.toGlueData a.fst b.fst)\n        (\u2191(GlueData.t D.toGlueData b.fst a.fst) (\u2191(GlueData.t D.toGlueData a.fst b.fst) x)) =\n      a.snd\n[PROOFSTEP]\nsimp [e\u2081, e\u2082]\n[GOAL]\nD : GlueData\n\u22a2 \u2200 {x y z : (i : D.J) \u00d7 \u2191(GlueData.U D.toGlueData i)}, Rel D x y \u2192 Rel D y z \u2192 Rel D x z\n[PROOFSTEP]\nrintro \u27e8i, a\u27e9 \u27e8j, b\u27e9 \u27e8k, c\u27e9 (\u27e8\u27e8\u27e9\u27e9 | \u27e8x, e\u2081, e\u2082\u27e9)\n[GOAL]\ncase mk.mk.mk.inl.refl\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c } \u2192 Rel D { fst := i, snd := a } { fst := k, snd := c }\ncase mk.mk.mk.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\n\u22a2 Rel D { fst := j, snd := b } { fst := k, snd := c } \u2192 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nexact id\n[GOAL]\ncase mk.mk.mk.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\n\u22a2 Rel D { fst := j, snd := b } { fst := k, snd := c } \u2192 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nrintro (\u27e8\u27e8\u27e9\u27e9 | \u27e8y, e\u2083, e\u2084\u27e9)\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inl.refl\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\n\u22a2 Rel D { fst := i, snd := a } { fst := j, snd := b }\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nexact Or.inr \u27e8x, e\u2081, e\u2082\u27e9\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nlet z := (pullbackIsoProdSubtype (D.f j i) (D.f j k)).inv \u27e8\u27e8_, _\u27e9, e\u2082.trans e\u2083.symm\u27e9\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (fun x => (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)))\n  { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n    property :=\n      (_ :\n        \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n          \u2191(GlueData.f D.toGlueData j k)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) } :=\n  \u2191(pullbackIsoProdSubtype (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)).inv\n    { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n      property :=\n        (_ :\n          \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n            \u2191(GlueData.f D.toGlueData j k)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) }\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nhave eq\u2081 : (D.t j i) ((pullback.fst : _  /-(D.f j k)-/ \u27f6 D.V (j, i)) z) = x := by simp\n[GOAL]\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (fun x => (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)))\n  { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n    property :=\n      (_ :\n        \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n          \u2191(GlueData.f D.toGlueData j k)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) } :=\n  \u2191(pullbackIsoProdSubtype (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)).inv\n    { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n      property :=\n        (_ :\n          \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n            \u2191(GlueData.f D.toGlueData j k)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) }\n\u22a2 \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (fun x => (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)))\n  { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n    property :=\n      (_ :\n        \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n          \u2191(GlueData.f D.toGlueData j k)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) } :=\n  \u2191(pullbackIsoProdSubtype (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)).inv\n    { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n      property :=\n        (_ :\n          \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n            \u2191(GlueData.f D.toGlueData j k)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) }\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nhave eq\u2082 : (pullback.snd : _ \u27f6 D.V _) z = y := pullbackIsoProdSubtype_inv_snd_apply _ _ _\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (fun x => (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)))\n  { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n    property :=\n      (_ :\n        \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n          \u2191(GlueData.f D.toGlueData j k)\n            (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) } :=\n  \u2191(pullbackIsoProdSubtype (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k)).inv\n    { val := (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y),\n      property :=\n        (_ :\n          \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n            \u2191(GlueData.f D.toGlueData j k)\n              (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x, y).snd) }\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\neq\u2082 : \u2191pullback.snd z = y\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nclear_value z\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\neq\u2082 : \u2191pullback.snd z = y\n\u22a2 Rel D { fst := i, snd := a } { fst := k, snd := c }\n[PROOFSTEP]\nright\n[GOAL]\ncase mk.mk.mk.inr.intro.intro.inr.intro.intro.h\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\neq\u2082 : \u2191pullback.snd z = y\n\u22a2 \u2203 x,\n    \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := k, snd := c }.fst) x = { fst := i, snd := a }.snd \u2227\n      \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := i, snd := a }.fst)\n          (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := k, snd := c }.fst) x) =\n        { fst := k, snd := c }.snd\n[PROOFSTEP]\nuse(pullback.fst : _ \u27f6 D.V (i, k)) (D.t' _ _ _ z)\n[GOAL]\ncase h\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x = { fst := i, snd := a }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := i, snd := a }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := j, snd := b }.fst) x) =\n    { fst := j, snd := b }.snd\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y = { fst := j, snd := b }.snd\ne\u2084 :\n  \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := j, snd := b }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := j, snd := b }.fst { fst := k, snd := c }.fst) y) =\n    { fst := k, snd := c }.snd\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\neq\u2082 : \u2191pullback.snd z = y\n\u22a2 \u2191(GlueData.f D.toGlueData { fst := i, snd := a }.fst { fst := k, snd := c }.fst)\n        (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z)) =\n      { fst := i, snd := a }.snd \u2227\n    \u2191(GlueData.f D.toGlueData { fst := k, snd := c }.fst { fst := i, snd := a }.fst)\n        (\u2191(GlueData.t D.toGlueData { fst := i, snd := a }.fst { fst := k, snd := c }.fst)\n          (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z))) =\n      { fst := k, snd := c }.snd\n[PROOFSTEP]\ndsimp only at *\n[GOAL]\ncase h\nD : GlueData\ni : D.J\na : \u2191(GlueData.U D.toGlueData i)\nj : D.J\nb : \u2191(GlueData.U D.toGlueData j)\nk : D.J\nc : \u2191(GlueData.U D.toGlueData k)\nx : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := a }.fst, { fst := j, snd := b }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) x = a\ne\u2082 : \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) x) = b\ny : \u2191(GlueData.V D.toGlueData ({ fst := j, snd := b }.fst, { fst := k, snd := c }.fst))\ne\u2083 : \u2191(GlueData.f D.toGlueData j k) y = b\ne\u2084 : \u2191(GlueData.f D.toGlueData k j) (\u2191(GlueData.t D.toGlueData j k) y) = c\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\neq\u2081 : \u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z) = x\neq\u2082 : \u2191pullback.snd z = y\n\u22a2 \u2191(GlueData.f D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z)) = a \u2227\n    \u2191(GlueData.f D.toGlueData k i)\n        (\u2191(GlueData.t D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z))) =\n      c\n[PROOFSTEP]\nsubsts eq\u2081 eq\u2082 e\u2081 e\u2083 e\u2084\n[GOAL]\ncase h\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\n\u22a2 \u2191(GlueData.f D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z)) =\n      \u2191(GlueData.f D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z)) \u2227\n    \u2191(GlueData.f D.toGlueData k i)\n        (\u2191(GlueData.t D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z))) =\n      \u2191(GlueData.f D.toGlueData k j) (\u2191(GlueData.t D.toGlueData j k) (\u2191pullback.snd z))\n[PROOFSTEP]\nhave h\u2081 : D.t' j i k \u226b pullback.fst \u226b D.f i k = pullback.fst \u226b D.t j i \u226b D.f i j := by rw [\u2190 \ud835\udda3.t_fac_assoc]; congr 1;\n  exact pullback.condition\n[GOAL]\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\n\u22a2 GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n[PROOFSTEP]\nrw [\u2190 \ud835\udda3.t_fac_assoc]\n[GOAL]\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\n\u22a2 GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    GlueData.t' D.toGlueData j i k \u226b pullback.snd \u226b GlueData.f D.toGlueData i j\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\n\u22a2 pullback.fst \u226b GlueData.f D.toGlueData i k = pullback.snd \u226b GlueData.f D.toGlueData i j\n[PROOFSTEP]\nexact pullback.condition\n[GOAL]\ncase h\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n\u22a2 \u2191(GlueData.f D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z)) =\n      \u2191(GlueData.f D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z)) \u2227\n    \u2191(GlueData.f D.toGlueData k i)\n        (\u2191(GlueData.t D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z))) =\n      \u2191(GlueData.f D.toGlueData k j) (\u2191(GlueData.t D.toGlueData j k) (\u2191pullback.snd z))\n[PROOFSTEP]\nhave h\u2082 : D.t' j i k \u226b pullback.fst \u226b D.t i k \u226b D.f k i = pullback.snd \u226b D.t j k \u226b D.f k j :=\n  by\n  rw [\u2190 \ud835\udda3.t_fac_assoc]\n  apply @Epi.left_cancellation _ _ _ _ (D.t' k j i)\n  rw [\ud835\udda3.cocycle_assoc, \ud835\udda3.t_fac_assoc, \ud835\udda3.t_inv_assoc]\n  exact pullback.condition.symm\n[GOAL]\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n\u22a2 GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.t D.toGlueData i k \u226b GlueData.f D.toGlueData k i =\n    pullback.snd \u226b GlueData.t D.toGlueData j k \u226b GlueData.f D.toGlueData k j\n[PROOFSTEP]\nrw [\u2190 \ud835\udda3.t_fac_assoc]\n[GOAL]\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n\u22a2 GlueData.t' D.toGlueData j i k \u226b GlueData.t' D.toGlueData i k j \u226b pullback.snd \u226b GlueData.f D.toGlueData k i =\n    pullback.snd \u226b GlueData.t D.toGlueData j k \u226b GlueData.f D.toGlueData k j\n[PROOFSTEP]\napply @Epi.left_cancellation _ _ _ _ (D.t' k j i)\n[GOAL]\ncase a\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n\u22a2 GlueData.t' D.toGlueData k j i \u226b\n      GlueData.t' D.toGlueData j i k \u226b GlueData.t' D.toGlueData i k j \u226b pullback.snd \u226b GlueData.f D.toGlueData k i =\n    GlueData.t' D.toGlueData k j i \u226b pullback.snd \u226b GlueData.t D.toGlueData j k \u226b GlueData.f D.toGlueData k j\n[PROOFSTEP]\nrw [\ud835\udda3.cocycle_assoc, \ud835\udda3.t_fac_assoc, \ud835\udda3.t_inv_assoc]\n[GOAL]\ncase a\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\n\u22a2 pullback.snd \u226b GlueData.f D.toGlueData k i = pullback.fst \u226b GlueData.f D.toGlueData k j\n[PROOFSTEP]\nexact pullback.condition.symm\n[GOAL]\ncase h\nD : GlueData\ni j k : D.J\nz : (forget TopCat).obj (pullback (GlueData.f D.toGlueData j i) (GlueData.f D.toGlueData j k))\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z))) =\n    \u2191(GlueData.f D.toGlueData j k) (\u2191pullback.snd z)\nh\u2081 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.f D.toGlueData i k =\n    pullback.fst \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j\nh\u2082 :\n  GlueData.t' D.toGlueData j i k \u226b pullback.fst \u226b GlueData.t D.toGlueData i k \u226b GlueData.f D.toGlueData k i =\n    pullback.snd \u226b GlueData.t D.toGlueData j k \u226b GlueData.f D.toGlueData k j\n\u22a2 \u2191(GlueData.f D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z)) =\n      \u2191(GlueData.f D.toGlueData i j) (\u2191(GlueData.t D.toGlueData j i) (\u2191pullback.fst z)) \u2227\n    \u2191(GlueData.f D.toGlueData k i)\n        (\u2191(GlueData.t D.toGlueData i k) (\u2191pullback.fst (\u2191(GlueData.t' D.toGlueData j i k) z))) =\n      \u2191(GlueData.f D.toGlueData k j) (\u2191(GlueData.t D.toGlueData j k) (\u2191pullback.snd z))\n[PROOFSTEP]\nexact \u27e8ContinuousMap.congr_fun h\u2081 z, ContinuousMap.congr_fun h\u2082 z\u27e9\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh : \u2191(GlueData.\u03c0 D.toGlueData) x = \u2191(GlueData.\u03c0 D.toGlueData) y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\ndelta GlueData.\u03c0 Multicoequalizer.sigma\u03c0 at h \n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  \u2191(coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u226b\n          (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).inv)\n      x =\n    \u2191(coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u226b\n          (Multicoequalizer.isoCoequalizer (GlueData.diagram D.toGlueData)).inv)\n      y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nreplace h := (TopCat.mono_iff_injective (Multicoequalizer.isoCoequalizer \ud835\udda3.diagram).inv).mp inferInstance h\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nlet diagram := parallelPair \ud835\udda3.diagram.fstSigmaMap \ud835\udda3.diagram.sndSigmaMap \u22d9 forget _\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nhave : colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y :=\n  by\n  dsimp only [coequalizer.\u03c0, ContinuousMap.toFun_eq_coe] at h \n  rw [\u2190 \u03b9_preservesColimitsIso_hom, forget_map_eq_coe, types_comp_apply, h]\n  simp\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\n\u22a2 colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y\n[PROOFSTEP]\ndsimp only [coequalizer.\u03c0, ContinuousMap.toFun_eq_coe] at h \n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  \u2191(colimit.\u03b9\n          (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n          one)\n      x =\n    \u2191(colimit.\u03b9\n          (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n          one)\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\n\u22a2 colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y\n[PROOFSTEP]\nrw [\u2190 \u03b9_preservesColimitsIso_hom, forget_map_eq_coe, types_comp_apply, h]\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  \u2191(colimit.\u03b9\n          (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n          one)\n      x =\n    \u2191(colimit.\u03b9\n          (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n          one)\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\n\u22a2 (preservesColimitIso (forget TopCat)\n          (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n            (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))).hom\n      (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            one)\n        y) =\n    (\u2191(colimit.\u03b9\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n            one) \u226b\n        (preservesColimitIso (forget TopCat)\n            (parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n              (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))).hom)\n      y\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\nthis : colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nhave :\n  (colimit.\u03b9 diagram _ \u226b colim.map _ \u226b (colimit.isoColimitCocone _).hom) _ =\n    (colimit.\u03b9 diagram _ \u226b colim.map _ \u226b (colimit.isoColimitCocone _).hom) _ :=\n  (congr_arg\n      (colim.map (diagramIsoParallelPair diagram).hom \u226b (colimit.isoColimitCocone (Types.coequalizerColimit _ _)).hom)\n      this :\n    _)\n    -- Porting note: was\n      -- simp only [eqToHom_refl, types_comp_apply, colimit.\u03b9_map_assoc,\n      --   diagramIsoParallelPair_hom_app, colimit.isoColimitCocone_\u03b9_hom, types_id_apply] at this\n      -- See https://github.com/leanprover-community/mathlib4/issues/5026\n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\nthis\u271d : colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y\nthis :\n  (colimit.\u03b9 diagram one \u226b\n        colim.map (diagramIsoParallelPair diagram).hom \u226b\n          (colimit.isoColimitCocone\n              (Types.coequalizerColimit (diagram.map WalkingParallelPairHom.left)\n                (diagram.map WalkingParallelPairHom.right))).hom)\n      x =\n    (colimit.\u03b9 diagram one \u226b\n        colim.map (diagramIsoParallelPair diagram).hom \u226b\n          (colimit.isoColimitCocone\n              (Types.coequalizerColimit (diagram.map WalkingParallelPairHom.left)\n                (diagram.map WalkingParallelPairHom.right))).hom)\n      y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nrw [colimit.\u03b9_map_assoc, diagramIsoParallelPair_hom_app, eqToHom_refl, colimit.isoColimitCocone_\u03b9_hom, types_comp_apply,\n  types_id_apply, types_comp_apply, types_id_apply] at this \n[GOAL]\nD : GlueData\nx y : \u2191(\u2210 D.U)\nh :\n  (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      x =\n    (coequalizer.\u03c0 (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n          (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))).1\n      y\ndiagram : WalkingParallelPair \u2964 Type u :=\n  parallelPair (MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) \u22d9\n    forget TopCat\nthis\u271d : colimit.\u03b9 diagram one x = colimit.\u03b9 diagram one y\nthis :\n  NatTrans.app\n      (Types.coequalizerColimit (diagram.map WalkingParallelPairHom.left)\n            (diagram.map WalkingParallelPairHom.right)).cocone.\u03b9\n      one x =\n    NatTrans.app\n      (Types.coequalizerColimit (diagram.map WalkingParallelPairHom.left)\n            (diagram.map WalkingParallelPairHom.right)).cocone.\u03b9\n      one y\n\u22a2 EqvGen\n    (Types.CoequalizerRel \u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      \u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)))\n    x y\n[PROOFSTEP]\nexact Quot.eq.1 this\n[GOAL]\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y \u2194 Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y \u2192 Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\ndelta GlueData.\u03b9\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\n\u22a2 \u2191(Multicoequalizer.\u03c0 (GlueData.diagram D.toGlueData) i) x =\n      \u2191(Multicoequalizer.\u03c0 (GlueData.diagram D.toGlueData) j) y \u2192\n    Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nsimp_rw [\u2190 Multicoequalizer.\u03b9_sigma\u03c0]\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\n\u22a2 \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n      \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y \u2192\n    Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 Rel D { fst := i, snd := x } { fst := j, snd := y }\n[PROOFSTEP]\nrw [\u2190 show _ = Sigma.mk i x from ConcreteCategory.congr_hom (sigmaIsoSigma.{_, u} D.U).inv_hom_id _]\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 Rel D (\u2191((sigmaIsoSigma D.U).inv \u226b (sigmaIsoSigma D.U).hom) { fst := i, snd := x }) { fst := j, snd := y }\n[PROOFSTEP]\nrw [\u2190 show _ = Sigma.mk j y from ConcreteCategory.congr_hom (sigmaIsoSigma.{_, u} D.U).inv_hom_id _]\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 Rel D (\u2191((sigmaIsoSigma D.U).inv \u226b (sigmaIsoSigma D.U).hom) { fst := i, snd := x })\n    (\u2191((sigmaIsoSigma D.U).inv \u226b (sigmaIsoSigma D.U).hom) { fst := j, snd := y })\n[PROOFSTEP]\nchange InvImage D.Rel (sigmaIsoSigma.{_, u} D.U).hom _ _\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom) (\u2191(sigmaIsoSigma D.U).inv { fst := i, snd := x })\n    (\u2191(sigmaIsoSigma D.U).inv { fst := j, snd := y })\n[PROOFSTEP]\nsimp only [TopCat.sigmaIsoSigma_inv_apply]\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom) (\u2191(sigmaIsoSigma D.U).inv { fst := i, snd := x })\n    (\u2191(sigmaIsoSigma D.U).inv { fst := j, snd := y })\n[PROOFSTEP]\nrw [\u2190 (InvImage.equivalence _ _ D.rel_equiv).eqvGen_iff]\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 EqvGen (InvImage (Rel D) \u2191(sigmaIsoSigma D.U).hom) (\u2191(sigmaIsoSigma D.U).inv { fst := i, snd := x })\n    (\u2191(sigmaIsoSigma D.U).inv { fst := j, snd := y })\n[PROOFSTEP]\nrefine' EqvGen.mono _ (D.eqvGen_of_\u03c0_eq h : _)\n[GOAL]\ncase mp\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\n\u22a2 \u2200 (a b : (forget TopCat).obj (\u2210 D.U)),\n    Types.CoequalizerRel (\u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData)))\n        (\u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))) a b \u2192\n      InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom) a b\n[PROOFSTEP]\nrintro _ _ \u27e8x\u27e9\n[GOAL]\ncase mp.Rel\nD : GlueData\ni j : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom) (\u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData)) x)\n    (\u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData)) x)\n[PROOFSTEP]\nrw [\u2190 show (sigmaIsoSigma.{u, u} _).inv _ = x from ConcreteCategory.congr_hom (sigmaIsoSigma.{u, u} _).hom_inv_id x]\n[GOAL]\ncase mp.Rel\nD : GlueData\ni j : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom)\n    (\u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv\n        (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).hom x)))\n    (\u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv\n        (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).hom x)))\n[PROOFSTEP]\ngeneralize (sigmaIsoSigma.{u, u} D.V).hom x = x'\n[GOAL]\ncase mp.Rel\nD : GlueData\ni j : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\nx' : (forget TopCat).obj (of ((i : D.J \u00d7 D.J) \u00d7 \u2191(GlueData.V D.toGlueData i)))\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom)\n    (\u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv x'))\n    (\u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv x'))\n[PROOFSTEP]\nobtain \u27e8\u27e8i, j\u27e9, y\u27e9 := x'\n[GOAL]\ncase mp.Rel.mk.mk\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 InvImage (Rel D) (\u2191(sigmaIsoSigma D.U).hom)\n    (\u2191(MultispanIndex.fstSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv { fst := (i, j), snd := y }))\n    (\u2191(MultispanIndex.sndSigmaMap (GlueData.diagram D.toGlueData))\n      (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv { fst := (i, j), snd := y }))\n[PROOFSTEP]\nunfold InvImage MultispanIndex.fstSigmaMap MultispanIndex.sndSigmaMap\n[GOAL]\ncase mp.Rel.mk.mk\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 Rel D\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(Sigma.desc fun b =>\n            MultispanIndex.fst (GlueData.diagram D.toGlueData) b \u226b\n              Sigma.\u03b9 (GlueData.diagram D.toGlueData).right (MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) b))\n        (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv { fst := (i, j), snd := y })))\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(Sigma.desc fun b =>\n            MultispanIndex.snd (GlueData.diagram D.toGlueData) b \u226b\n              Sigma.\u03b9 (GlueData.diagram D.toGlueData).right (MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) b))\n        (\u2191(sigmaIsoSigma (GlueData.diagram D.toGlueData).left).inv { fst := (i, j), snd := y })))\n[PROOFSTEP]\nsimp only [Opens.inclusion_apply, TopCat.comp_app, sigmaIsoSigma_inv_apply, Cofan.mk_\u03b9_app]\n[GOAL]\ncase mp.Rel.mk.mk\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 Rel D\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(Sigma.desc fun b =>\n            MultispanIndex.fst (GlueData.diagram D.toGlueData) b \u226b\n              Sigma.\u03b9 (GlueData.diagram D.toGlueData).right (MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) b))\n        (\u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).left (i, j)) y)))\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(Sigma.desc fun b =>\n            MultispanIndex.snd (GlueData.diagram D.toGlueData) b \u226b\n              Sigma.\u03b9 (GlueData.diagram D.toGlueData).right (MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) b))\n        (\u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).left (i, j)) y)))\n[PROOFSTEP]\nrw [\u2190 comp_apply, colimit.\u03b9_desc, \u2190 comp_apply, colimit.\u03b9_desc]\n[GOAL]\ncase mp.Rel.mk.mk\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 Rel D\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(NatTrans.app\n            (Cofan.mk (\u2210 (GlueData.diagram D.toGlueData).right) fun b =>\n                MultispanIndex.fst (GlueData.diagram D.toGlueData) b \u226b\n                  Sigma.\u03b9 (GlueData.diagram D.toGlueData).right\n                    (MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) b)).\u03b9\n            { as := (i, j) })\n        y))\n    (\u2191(sigmaIsoSigma D.U).hom\n      (\u2191(NatTrans.app\n            (Cofan.mk (\u2210 (GlueData.diagram D.toGlueData).right) fun b =>\n                MultispanIndex.snd (GlueData.diagram D.toGlueData) b \u226b\n                  Sigma.\u03b9 (GlueData.diagram D.toGlueData).right\n                    (MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) b)).\u03b9\n            { as := (i, j) })\n        y))\n[PROOFSTEP]\nerw [sigmaIsoSigma_hom_\u03b9_apply, sigmaIsoSigma_hom_\u03b9_apply]\n[GOAL]\ncase mp.Rel.mk.mk\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 Rel D\n    { fst := MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n      snd := \u2191(MultispanIndex.fst (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }\n    { fst := MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n      snd := \u2191(MultispanIndex.snd (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }\n[PROOFSTEP]\nexact Or.inr \u27e8y, by dsimp [GlueData.diagram]; simp only [true_and]; rfl\u27e9\n[GOAL]\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 \u2191(GlueData.f D.toGlueData\n            { fst := MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                snd := \u2191(MultispanIndex.fst (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst\n            { fst := MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                snd := \u2191(MultispanIndex.snd (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst)\n        y =\n      { fst := MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n          snd := \u2191(MultispanIndex.fst (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.snd \u2227\n    \u2191(GlueData.f D.toGlueData\n            { fst := MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                snd := \u2191(MultispanIndex.snd (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst\n            { fst := MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                snd := \u2191(MultispanIndex.fst (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst)\n        (\u2191(GlueData.t D.toGlueData\n              { fst := MultispanIndex.fstFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                  snd := \u2191(MultispanIndex.fst (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst\n              { fst := MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n                  snd := \u2191(MultispanIndex.snd (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.fst)\n          y) =\n      { fst := MultispanIndex.sndFrom (GlueData.diagram D.toGlueData) { as := (i, j) }.as,\n          snd := \u2191(MultispanIndex.snd (GlueData.diagram D.toGlueData) { as := (i, j) }.as) y }.snd\n[PROOFSTEP]\ndsimp [GlueData.diagram]\n[GOAL]\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 \u2191(GlueData.f D.toGlueData i j) y = \u2191(GlueData.f D.toGlueData i j) y \u2227\n    \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) y) =\n      \u2191(GlueData.t D.toGlueData i j \u226b GlueData.f D.toGlueData j i) y\n[PROOFSTEP]\nsimp only [true_and]\n[GOAL]\nD : GlueData\ni\u271d j\u271d : D.J\nx\u271d : \u2191(GlueData.U D.toGlueData i\u271d)\ny\u271d : \u2191(GlueData.U D.toGlueData j\u271d)\nh :\n  \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right i\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) x\u271d =\n    \u2191(Sigma.\u03b9 (GlueData.diagram D.toGlueData).right j\u271d \u226b Multicoequalizer.sigma\u03c0 (GlueData.diagram D.toGlueData)) y\u271d\nx : \u2191(\u2210 (GlueData.diagram D.toGlueData).left)\ni j : D.J\ny : \u2191(GlueData.V D.toGlueData (i, j))\n\u22a2 \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) y) =\n    \u2191(GlueData.t D.toGlueData i j \u226b GlueData.f D.toGlueData j i) y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\n\u22a2 Rel D { fst := i, snd := x } { fst := j, snd := y } \u2192 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y\n[PROOFSTEP]\nrintro (\u27e8\u27e8\u27e9\u27e9 | \u27e8z, e\u2081, e\u2082\u27e9)\n[GOAL]\ncase mpr.inl.refl\nD : GlueData\ni : D.J\nx : \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) x\ncase mpr.inr.intro.intro\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nz : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := j, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst) z = { fst := i, snd := x }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst) z) =\n    { fst := j, snd := y }.snd\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y\n[PROOFSTEP]\nrfl\n[GOAL]\ncase mpr.inr.intro.intro\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nz : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := j, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst) z = { fst := i, snd := x }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := j, snd := y }.fst { fst := i, snd := x }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := x }.fst { fst := j, snd := y }.fst) z) =\n    { fst := j, snd := y }.snd\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y\n[PROOFSTEP]\ndsimp only at *\n  -- porting note: there were `subst e\u2081` and `subst e\u2082`, instead of the `rw`\n[GOAL]\ncase mpr.inr.intro.intro\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nz : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := j, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) z = x\ne\u2082 : \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) z) = y\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData j) y\n[PROOFSTEP]\nrw [\u2190 e\u2081, \u2190 e\u2082] at *\n[GOAL]\ncase mpr.inr.intro.intro\nD : GlueData\ni j : D.J\nx : \u2191(GlueData.U D.toGlueData i)\ny : \u2191(GlueData.U D.toGlueData j)\nz : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := j, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) z = \u2191(GlueData.f D.toGlueData i j) z\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) z) =\n    \u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) z)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) (\u2191(GlueData.f D.toGlueData i j) z) =\n    \u2191(GlueData.\u03b9 D.toGlueData j) (\u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) z))\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\ni : D.J\n\u22a2 Function.Injective \u2191(GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nintro x y h\n[GOAL]\nD : GlueData\ni : D.J\nx y : (forget TopCat).obj (GlueData.U D.toGlueData i)\nh : \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) y\n\u22a2 x = y\n[PROOFSTEP]\nrcases(D.\u03b9_eq_iff_rel _ _ _ _).mp h with (\u27e8\u27e8\u27e9\u27e9 | \u27e8_, e\u2081, e\u2082\u27e9)\n[GOAL]\ncase inl.refl\nD : GlueData\ni : D.J\nx : (forget TopCat).obj (GlueData.U D.toGlueData i)\nh : \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) x\n\u22a2 x = x\n[PROOFSTEP]\nrfl\n[GOAL]\ncase inr.intro.intro\nD : GlueData\ni : D.J\nx y : (forget TopCat).obj (GlueData.U D.toGlueData i)\nh : \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) y\nw\u271d : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := i, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := x }.fst { fst := i, snd := y }.fst) w\u271d = { fst := i, snd := x }.snd\ne\u2082 :\n  \u2191(GlueData.f D.toGlueData { fst := i, snd := y }.fst { fst := i, snd := x }.fst)\n      (\u2191(GlueData.t D.toGlueData { fst := i, snd := x }.fst { fst := i, snd := y }.fst) w\u271d) =\n    { fst := i, snd := y }.snd\n\u22a2 x = y\n[PROOFSTEP]\ndsimp only at *\n  -- porting note: there were `cases e\u2081` and `cases e\u2082`, instead of the `rw`\n[GOAL]\ncase inr.intro.intro\nD : GlueData\ni : D.J\nx y : (forget TopCat).obj (GlueData.U D.toGlueData i)\nh : \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) y\nw\u271d : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := i, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i i) w\u271d = x\ne\u2082 : \u2191(GlueData.f D.toGlueData i i) (\u2191(GlueData.t D.toGlueData i i) w\u271d) = y\n\u22a2 x = y\n[PROOFSTEP]\nrw [\u2190 e\u2081, \u2190 e\u2082]\n[GOAL]\ncase inr.intro.intro\nD : GlueData\ni : D.J\nx y : (forget TopCat).obj (GlueData.U D.toGlueData i)\nh : \u2191(GlueData.\u03b9 D.toGlueData i) x = \u2191(GlueData.\u03b9 D.toGlueData i) y\nw\u271d : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x }.fst, { fst := i, snd := y }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i i) w\u271d = x\ne\u2082 : \u2191(GlueData.f D.toGlueData i i) (\u2191(GlueData.t D.toGlueData i i) w\u271d) = y\n\u22a2 \u2191(GlueData.f D.toGlueData i i) w\u271d = \u2191(GlueData.f D.toGlueData i i) (\u2191(GlueData.t D.toGlueData i i) w\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 Set.range \u2191(GlueData.\u03b9 D.toGlueData i) \u2229 Set.range \u2191(GlueData.\u03b9 D.toGlueData j) =\n    Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\n\u22a2 x \u2208 Set.range \u2191(GlueData.\u03b9 D.toGlueData i) \u2229 Set.range \u2191(GlueData.\u03b9 D.toGlueData j) \u2194\n    x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\n\u22a2 x \u2208 Set.range \u2191(GlueData.\u03b9 D.toGlueData i) \u2229 Set.range \u2191(GlueData.\u03b9 D.toGlueData j) \u2192\n    x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nrintro \u27e8\u27e8x\u2081, eq\u2081\u27e9, \u27e8x\u2082, eq\u2082\u27e9\u27e9\n[GOAL]\ncase h.mp.intro.intro.intro\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\nx\u2082 : (forget TopCat).obj (GlueData.U D.toGlueData j)\neq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData j) x\u2082 = x\n\u22a2 x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nobtain \u27e8\u27e8\u27e9\u27e9 | \u27e8y, e\u2081, -\u27e9 := (D.\u03b9_eq_iff_rel _ _ _ _).mp (eq\u2081.trans eq\u2082.symm)\n[GOAL]\ncase h.mp.intro.intro.intro.inl.refl\nD : GlueData\ni : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 eq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\n\u22a2 x \u2208 Set.range \u2191(GlueData.f D.toGlueData i i \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nexact\n  \u27e8inv (D.f i i) x\u2081, by\n    -- Porting note: was `simp [eq\u2081]`\n            -- See https://github.com/leanprover-community/mathlib4/issues/5026\n    rw [TopCat.comp_app]\n    erw [CategoryTheory.IsIso.inv_hom_id_apply]\n    rw [eq\u2081]\u27e9\n[GOAL]\nD : GlueData\ni : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 eq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\n\u22a2 \u2191(GlueData.f D.toGlueData i i \u226b GlueData.\u03b9 D.toGlueData i) (\u2191(inv (GlueData.f D.toGlueData i i)) x\u2081) = x\n[PROOFSTEP]\nrw [TopCat.comp_app]\n[GOAL]\nD : GlueData\ni : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 eq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) (\u2191(GlueData.f D.toGlueData i i) (\u2191(inv (GlueData.f D.toGlueData i i)) x\u2081)) = x\n[PROOFSTEP]\nerw [CategoryTheory.IsIso.inv_hom_id_apply]\n[GOAL]\nD : GlueData\ni : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 eq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\n[PROOFSTEP]\nrw [eq\u2081]\n[GOAL]\ncase h.mp.intro.intro.intro.inr.intro.intro\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\nx\u2082 : (forget TopCat).obj (GlueData.U D.toGlueData j)\neq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData j) x\u2082 = x\ny : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x\u2081 }.fst, { fst := j, snd := x\u2082 }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData { fst := i, snd := x\u2081 }.fst { fst := j, snd := x\u2082 }.fst) y = { fst := i, snd := x\u2081 }.snd\n\u22a2 x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\ndsimp only at *\n[GOAL]\ncase h.mp.intro.intro.intro.inr.intro.intro\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\neq\u2081 : \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 = x\nx\u2082 : (forget TopCat).obj (GlueData.U D.toGlueData j)\neq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData j) x\u2082 = x\ny : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x\u2081 }.fst, { fst := j, snd := x\u2082 }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) y = x\u2081\n\u22a2 x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nsubsts eq\u2081\n[GOAL]\ncase h.mp.intro.intro.intro.inr.intro.intro\nD : GlueData\ni j : D.J\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\nx\u2082 : (forget TopCat).obj (GlueData.U D.toGlueData j)\ny : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x\u2081 }.fst, { fst := j, snd := x\u2082 }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) y = x\u2081\neq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData j) x\u2082 = \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081 \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nexact \u27e8y, by simp [e\u2081]\u27e9\n[GOAL]\nD : GlueData\ni j : D.J\nx\u2081 : (forget TopCat).obj (GlueData.U D.toGlueData i)\nx\u2082 : (forget TopCat).obj (GlueData.U D.toGlueData j)\ny : \u2191(GlueData.V D.toGlueData ({ fst := i, snd := x\u2081 }.fst, { fst := j, snd := x\u2082 }.fst))\ne\u2081 : \u2191(GlueData.f D.toGlueData i j) y = x\u2081\neq\u2082 : \u2191(GlueData.\u03b9 D.toGlueData j) x\u2082 = \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081\n\u22a2 \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i) y = \u2191(GlueData.\u03b9 D.toGlueData i) x\u2081\n[PROOFSTEP]\nsimp [e\u2081]\n[GOAL]\ncase h.mpr\nD : GlueData\ni j : D.J\nx : (forget TopCat).obj (GlueData.glued D.toGlueData)\n\u22a2 x \u2208 Set.range \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i) \u2192\n    x \u2208 Set.range \u2191(GlueData.\u03b9 D.toGlueData i) \u2229 Set.range \u2191(GlueData.\u03b9 D.toGlueData j)\n[PROOFSTEP]\nrintro \u27e8x, hx\u27e9\n[GOAL]\ncase h.mpr.intro\nD : GlueData\ni j : D.J\nx\u271d : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (i, j))\nhx : \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i) x = x\u271d\n\u22a2 x\u271d \u2208 Set.range \u2191(GlueData.\u03b9 D.toGlueData i) \u2229 Set.range \u2191(GlueData.\u03b9 D.toGlueData j)\n[PROOFSTEP]\nexact \u27e8\u27e8D.f i j x, hx\u27e9, \u27e8D.f j i (D.t _ _ x), by simp [\u2190 hx]\u27e9\u27e9\n[GOAL]\nD : GlueData\ni j : D.J\nx\u271d : (forget TopCat).obj (GlueData.glued D.toGlueData)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (i, j))\nhx : \u2191(GlueData.f D.toGlueData i j \u226b GlueData.\u03b9 D.toGlueData i) x = x\u271d\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) (\u2191(GlueData.f D.toGlueData j i) (\u2191(GlueData.t D.toGlueData i j) x)) = x\u271d\n[PROOFSTEP]\nsimp [\u2190 hx]\n[GOAL]\nD : GlueData\ni j : D.J\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' Set.range \u2191(GlueData.\u03b9 D.toGlueData i) = Set.range \u2191(GlueData.f D.toGlueData j i)\n[PROOFSTEP]\nrw [\u2190 Set.preimage_image_eq (Set.range (D.f j i)) (D.\u03b9_injective j), \u2190 Set.image_univ, \u2190 Set.image_univ, \u2190\n  Set.image_comp, \u2190 coe_comp, Set.image_univ, Set.image_univ, \u2190 image_inter, Set.preimage_range_inter]\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U)\n[PROOFSTEP]\nhave : D.f _ _ \u207b\u00b9' (\ud835\udda3.\u03b9 j \u207b\u00b9' (\ud835\udda3.\u03b9 i '' U)) = (D.t j i \u226b D.f _ _) \u207b\u00b9' U :=\n  by\n  ext x\n  conv_rhs => rw [\u2190 Set.preimage_image_eq U (D.\u03b9_injective _)]\n  generalize \ud835\udda3.\u03b9 i '' U = U'\n  simp\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\n\u22a2 x \u2208 \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) \u2194\n    x \u2208 \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n[PROOFSTEP]\nconv_rhs => rw [\u2190 Set.preimage_image_eq U (D.\u03b9_injective _)]\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\n| x \u2208 \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 Set.preimage_image_eq U (D.\u03b9_injective _)]\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\n| x \u2208 \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 Set.preimage_image_eq U (D.\u03b9_injective _)]\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\n| x \u2208 \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 Set.preimage_image_eq U (D.\u03b9_injective _)]\n[GOAL]\ncase h\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\n\u22a2 x \u2208 \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) \u2194\n    x \u2208\n      \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9'\n        (\u2191(GlueData.\u03b9 D.toGlueData i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U))\n[PROOFSTEP]\ngeneralize \ud835\udda3.\u03b9 i '' U = U'\n[GOAL]\ncase h\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nx : (forget TopCat).obj (GlueData.V D.toGlueData (j, i))\nU' : Set ((forget TopCat).obj (GlueData.glued D.toGlueData))\n\u22a2 x \u2208 \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' U') \u2194\n    x \u2208 \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) \u207b\u00b9' U')\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nthis :\n  \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U)\n[PROOFSTEP]\nrw [\u2190 this, Set.image_preimage_eq_inter_range]\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nthis :\n  \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) =\n    \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) \u2229 Set.range \u2191(GlueData.f D.toGlueData j i)\n[PROOFSTEP]\nsymm\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nthis :\n  \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) \u2229 Set.range \u2191(GlueData.f D.toGlueData j i) =\n    \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)\n[PROOFSTEP]\napply Set.inter_eq_self_of_subset_left\n[GOAL]\ncase a\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nthis :\n  \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) \u2286 Set.range \u2191(GlueData.f D.toGlueData j i)\n[PROOFSTEP]\nrw [\u2190 D.preimage_range i j]\n[GOAL]\ncase a\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\nthis :\n  \u2191(GlueData.f D.toGlueData j i) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U)) =\n    \u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) \u2286\n    \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' Set.range \u2191(GlueData.\u03b9 D.toGlueData i)\n[PROOFSTEP]\nexact Set.preimage_mono (Set.image_subset_range _ _)\n[GOAL]\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' U) =\n    \u2191(GlueData.t D.toGlueData i j \u226b GlueData.f D.toGlueData j i) '' (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U)\n[PROOFSTEP]\nconvert D.preimage_image_eq_image i j U using 1\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.t D.toGlueData i j \u226b GlueData.f D.toGlueData j i) '' (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U)\n[PROOFSTEP]\nrw [coe_comp, coe_comp]\n  -- porting note: `show` was not needed, since `rw [\u2190 Set.image_image]` worked.\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.f D.toGlueData j i) \u2218 \u2191(GlueData.t D.toGlueData i j) '' (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.f D.toGlueData i j) \u2218 \u2191(GlueData.t D.toGlueData j i) \u207b\u00b9' U)\n[PROOFSTEP]\nshow (fun x => ((forget TopCat).map _ ((forget TopCat).map _ x))) '' _ = _\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 (fun x => (forget TopCat).map (GlueData.f D.toGlueData j i) ((forget TopCat).map (GlueData.t D.toGlueData i j) x)) ''\n      (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.f D.toGlueData i j) \u2218 \u2191(GlueData.t D.toGlueData j i) \u207b\u00b9' U)\n[PROOFSTEP]\nrw [\u2190 Set.image_image]\n  -- porting note: `congr 1` was here, instead of `congr_arg`, however, it did nothing.\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 (forget TopCat).map (GlueData.f D.toGlueData j i) ''\n      ((fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x) '' (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U)) =\n    \u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.f D.toGlueData i j) \u2218 \u2191(GlueData.t D.toGlueData j i) \u207b\u00b9' U)\n[PROOFSTEP]\nrefine congr_arg ?_ ?_\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 (fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x) '' (\u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U) =\n    \u2191(GlueData.f D.toGlueData i j) \u2218 \u2191(GlueData.t D.toGlueData j i) \u207b\u00b9' U\n[PROOFSTEP]\nrw [\u2190 Set.eq_preimage_iff_image_eq, Set.preimage_preimage]\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U =\n    (fun x =>\n        (\u2191(GlueData.f D.toGlueData i j) \u2218 \u2191(GlueData.t D.toGlueData j i))\n          ((forget TopCat).map (GlueData.t D.toGlueData i j) x)) \u207b\u00b9'\n      U\ncase h.e'_3.hf\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 Function.Bijective fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x\n[PROOFSTEP]\nchange _ = (D.t i j \u226b D.t j i \u226b _) \u207b\u00b9' _\n[GOAL]\ncase h.e'_3\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2191(GlueData.f D.toGlueData i j) \u207b\u00b9' U =\n    \u2191(GlueData.t D.toGlueData i j \u226b GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' U\ncase h.e'_3.hf\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 Function.Bijective fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x\n[PROOFSTEP]\nrw [\ud835\udda3.t_inv_assoc]\n[GOAL]\ncase h.e'_3.hf\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 Function.Bijective fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x\n[PROOFSTEP]\nrw [\u2190 isIso_iff_bijective]\n[GOAL]\ncase h.e'_3.hf\nD : GlueData\ni j : D.J\nU : Set \u2191(GlueData.U D.toGlueData i)\n\u22a2 IsIso fun x => (forget TopCat).map (GlueData.t D.toGlueData i j) x\n[PROOFSTEP]\napply (forget TopCat).map_isIso\n[GOAL]\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\n\u22a2 IsOpen (\u2191(GlueData.\u03b9 D.toGlueData i) '' \u2191U)\n[PROOFSTEP]\nrw [isOpen_iff]\n[GOAL]\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\n\u22a2 \u2200 (i_1 : D.J), IsOpen (\u2191(GlueData.\u03b9 D.toGlueData i_1) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' \u2191U))\n[PROOFSTEP]\nintro j\n[GOAL]\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\nj : D.J\n\u22a2 IsOpen (\u2191(GlueData.\u03b9 D.toGlueData j) \u207b\u00b9' (\u2191(GlueData.\u03b9 D.toGlueData i) '' \u2191U))\n[PROOFSTEP]\nrw [preimage_image_eq_image]\n[GOAL]\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\nj : D.J\n\u22a2 IsOpen (\u2191(GlueData.f D.toGlueData j i) '' (\u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' \u2191U))\n[PROOFSTEP]\napply (D.f_open _ _).isOpenMap\n[GOAL]\ncase a\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\nj : D.J\n\u22a2 IsOpen (\u2191(GlueData.t D.toGlueData j i \u226b GlueData.f D.toGlueData i j) \u207b\u00b9' \u2191U)\n[PROOFSTEP]\napply (D.t j i \u226b D.f i j).continuous_toFun.isOpen_preimage\n[GOAL]\ncase a.a\nD : GlueData\ni : D.J\nU : Opens \u2191(GlueData.U D.toGlueData i)\nj : D.J\n\u22a2 IsOpen \u2191U\n[PROOFSTEP]\nexact U.isOpen\n[GOAL]\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\n\u22a2 \u2191(t h i j) (\u2191(t h j i) x) = x\n[PROOFSTEP]\nhave := h.cocycle j i j x ?_\n[GOAL]\ncase refine_2\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\nthis :\n  \u2191(\u2191(t h i j) { val := \u2191(\u2191(t h j i) x), property := (_ : \u2191(\u2191(t h j i) x) \u2208 V h i j) }) =\n    \u2191(\u2191(t h j j) { val := \u2191x, property := ?refine_1 })\n\u22a2 \u2191(t h i j) (\u2191(t h j i) x) = x\ncase refine_1 D : GlueData h : MkCore i j : h.J x : { x // x \u2208 V h j i } \u22a2 \u2191x \u2208 V h j j\n[PROOFSTEP]\nrw [h.t_id] at this \n[GOAL]\ncase refine_2\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\nthis :\n  \u2191(\u2191(t h i j) { val := \u2191(\u2191(t h j i) x), property := (_ : \u2191(\u2191(t h j i) x) \u2208 V h i j) }) =\n    \u2191(id { val := \u2191x, property := ?refine_1 })\n\u22a2 \u2191(t h i j) (\u2191(t h j i) x) = x\ncase refine_1\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\n\u22a2 \u2191x \u2208 V h j j\ncase refine_1 D : GlueData h : MkCore i j : h.J x : { x // x \u2208 V h j i } \u22a2 \u2191x \u2208 V h j j\n[PROOFSTEP]\nconvert Subtype.eq this\n[GOAL]\ncase refine_1\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\n\u22a2 \u2191x \u2208 V h j j\ncase refine_1\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\n\u22a2 \u2191x \u2208 V h j j\ncase refine_1 D : GlueData h : MkCore i j : h.J x : { x // x \u2208 V h j i } \u22a2 \u2191x \u2208 V h j j\n[PROOFSTEP]\nrw [h.V_id]\n[GOAL]\ncase refine_1\nD : GlueData\nh : MkCore\ni j : h.J\nx : { x // x \u2208 V h j i }\n\u22a2 \u2191x \u2208 \u22a4\n[PROOFSTEP]\ntrivial\n[GOAL]\nD : GlueData\nh : MkCore\ni j : h.J\n\u22a2 IsIso (MkCore.t h i j)\n[PROOFSTEP]\nuse h.t j i\n[GOAL]\ncase h\nD : GlueData\nh : MkCore\ni j : h.J\n\u22a2 MkCore.t h i j \u226b MkCore.t h j i = \ud835\udfd9 ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i j)) \u2227\n    MkCore.t h j i \u226b MkCore.t h i j = \ud835\udfd9 ((Opens.toTopCat (MkCore.U h j)).obj (MkCore.V h j i))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.left\nD : GlueData\nh : MkCore\ni j : h.J\n\u22a2 MkCore.t h i j \u226b MkCore.t h j i = \ud835\udfd9 ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i j))\n[PROOFSTEP]\next1\n[GOAL]\ncase h.right\nD : GlueData\nh : MkCore\ni j : h.J\n\u22a2 MkCore.t h j i \u226b MkCore.t h i j = \ud835\udfd9 ((Opens.toTopCat (MkCore.U h j)).obj (MkCore.V h j i))\n[PROOFSTEP]\next1\n[GOAL]\ncase h.left.w\nD : GlueData\nh : MkCore\ni j : h.J\nx\u271d : (forget TopCat).obj ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i j))\n\u22a2 \u2191(MkCore.t h i j \u226b MkCore.t h j i) x\u271d = \u2191(\ud835\udfd9 ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i j))) x\u271d\ncase h.right.w\nD : GlueData\nh : MkCore\ni j : h.J\nx\u271d : (forget TopCat).obj ((Opens.toTopCat (MkCore.U h j)).obj (MkCore.V h j i))\n\u22a2 \u2191(MkCore.t h j i \u226b MkCore.t h i j) x\u271d = \u2191(\ud835\udfd9 ((Opens.toTopCat (MkCore.U h j)).obj (MkCore.V h j i))) x\u271d\n[PROOFSTEP]\nexacts [h.t_inv _ _ _, h.t_inv _ _ _]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 pullback (Opens.inclusion (V h i j)) (Opens.inclusion (V h i k)) \u27f6\n    pullback (Opens.inclusion (V h j k)) (Opens.inclusion (V h j i))\n[PROOFSTEP]\nrefine' (pullbackIsoProdSubtype _ _).hom \u226b \u27e8_, _\u27e9 \u226b (pullbackIsoProdSubtype _ _).inv\n[GOAL]\ncase refine'_1\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 \u2191(of { p // \u2191(Opens.inclusion (V h i j)) p.fst = \u2191(Opens.inclusion (V h i k)) p.snd }) \u2192\n    \u2191(of { p // \u2191(Opens.inclusion (V h j k)) p.fst = \u2191(Opens.inclusion (V h j i)) p.snd })\n[PROOFSTEP]\nintro x\n[GOAL]\ncase refine'_1\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(of { p // \u2191(Opens.inclusion (V h i j)) p.fst = \u2191(Opens.inclusion (V h i k)) p.snd })\n\u22a2 \u2191(of { p // \u2191(Opens.inclusion (V h j k)) p.fst = \u2191(Opens.inclusion (V h j i)) p.snd })\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u27e8(h.t i j x.1.1).1, _\u27e9, h.t i j x.1.1\u27e9, rfl\u27e9\n[GOAL]\ncase refine'_1\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(of { p // \u2191(Opens.inclusion (V h i j)) p.fst = \u2191(Opens.inclusion (V h i k)) p.snd })\n\u22a2 \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k\n[PROOFSTEP]\nrcases x with \u27e8\u27e8\u27e8x, hx\u27e9, \u27e8x', hx'\u27e9\u27e9, rfl : x = x'\u27e9\n[GOAL]\ncase refine'_1.mk.mk.mk.mk\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(U h i)\nhx : x \u2208 V h i j\nhx' : x \u2208 V h i k\n\u22a2 \u2191(\u2191(t h i j)\n        (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }), property := (_ : x = x) }).fst) \u2208\n    V h j k\n[PROOFSTEP]\nexact h.t_inter _ \u27e8x, hx\u27e9 hx'\n[GOAL]\ncase refine'_2\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 Continuous fun x =>\n    {\n      val :=\n        ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) }, \u2191(t h i j) (\u2191x).fst),\n      property :=\n        (_ :\n          \u2191(Opens.inclusion (V h j k))\n              ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) },\n                  \u2191(t h i j) (\u2191x).fst).fst =\n            \u2191(Opens.inclusion (V h j k))\n              ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) },\n                  \u2191(t h i j) (\u2191x).fst).fst) }\n[PROOFSTEP]\nhave : Continuous (h.t i j) := map_continuous (self := ContinuousMap.toContinuousMapClass) _\n[GOAL]\ncase refine'_2\nD : GlueData\nh : MkCore\ni j k : h.J\nthis : Continuous \u2191(t h i j)\n\u22a2 Continuous fun x =>\n    {\n      val :=\n        ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) }, \u2191(t h i j) (\u2191x).fst),\n      property :=\n        (_ :\n          \u2191(Opens.inclusion (V h j k))\n              ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) },\n                  \u2191(t h i j) (\u2191x).fst).fst =\n            \u2191(Opens.inclusion (V h j k))\n              ({ val := \u2191(\u2191(t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(t h i j) (\u2191x).fst) \u2208 V h j k) },\n                  \u2191(t h i j) (\u2191x).fst).fst) }\n[PROOFSTEP]\nexact ((Continuous.subtype_mk (by continuity) _).prod_mk (by continuity)).subtype_mk _\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\nthis : Continuous \u2191(t h i j)\n\u22a2 Continuous fun x => \u2191(\u2191(t h i j) (\u2191x).fst)\n[PROOFSTEP]\ncontinuity\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\nthis : Continuous \u2191(t h i j)\n\u22a2 Continuous fun x => \u2191(t h i j) (\u2191x).fst\n[PROOFSTEP]\ncontinuity\n[GOAL]\nD : GlueData\nh : MkCore\ni : h.J\n\u22a2 IsIso ((fun i j => Opens.inclusion (MkCore.V h i j)) i i)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nD : GlueData\nh : MkCore\ni : h.J\n\u22a2 IsIso (Opens.inclusion (MkCore.V h i i))\n[PROOFSTEP]\nexact (h.V_id i).symm \u25b8 IsIso.of_iso (Opens.inclusionTopIso (h.U i))\n[GOAL]\nD : GlueData\nh : MkCore\ni : h.J\n\u22a2 MkCore.t h i i = \ud835\udfd9 ((fun i => (Opens.toTopCat (MkCore.U h i.fst)).obj (MkCore.V h i.fst i.snd)) (i, i))\n[PROOFSTEP]\next\n[GOAL]\ncase w\nD : GlueData\nh : MkCore\ni : h.J\nx\u271d : (forget TopCat).obj ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i i))\n\u22a2 \u2191(MkCore.t h i i) x\u271d = \u2191(\ud835\udfd9 ((fun i => (Opens.toTopCat (MkCore.U h i.fst)).obj (MkCore.V h i.fst i.snd)) (i, i))) x\u271d\n[PROOFSTEP]\nrw [h.t_id]\n[GOAL]\ncase w\nD : GlueData\nh : MkCore\ni : h.J\nx\u271d : (forget TopCat).obj ((Opens.toTopCat (MkCore.U h i)).obj (MkCore.V h i i))\n\u22a2 id x\u271d = \u2191(\ud835\udfd9 ((fun i => (Opens.toTopCat (MkCore.U h i.fst)).obj (MkCore.V h i.fst i.snd)) (i, i))) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 MkCore.t' h i j k \u226b pullback.snd = pullback.fst \u226b MkCore.t h i j\n[PROOFSTEP]\ndelta MkCore.t'\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom \u226b\n        (ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                  \u2191(MkCore.t h i j) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n          (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).inv) \u226b\n      pullback.snd =\n    pullback.fst \u226b MkCore.t h i j\n[PROOFSTEP]\nrw [Category.assoc, Category.assoc, pullbackIsoProdSubtype_inv_snd, \u2190 Iso.eq_inv_comp,\n  pullbackIsoProdSubtype_inv_fst_assoc]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 (ContinuousMap.mk fun x =>\n        {\n          val :=\n            ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n              \u2191(MkCore.t h i j) (\u2191x).fst),\n          property :=\n            (_ :\n              \u2191(Opens.inclusion (MkCore.V h j k))\n                  ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                      \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                \u2191(Opens.inclusion (MkCore.V h j k))\n                  ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                      \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n      pullbackSnd (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i)) =\n    pullbackFst (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k)) \u226b MkCore.t h i j\n[PROOFSTEP]\next \u27e8\u27e8\u27e8x, hx\u27e9, \u27e8x', hx'\u27e9\u27e9, rfl : x = x'\u27e9\n[GOAL]\ncase w.mk.mk.mk.mk\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(MkCore.U h i)\nhx : x \u2208 MkCore.V h i j\nhx' : x \u2208 MkCore.V h i k\n\u22a2 \u2191((ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                  \u2191(MkCore.t h i j) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n          pullbackSnd (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i)))\n      { val := ({ val := x, property := hx }, { val := x, property := hx' }), property := (_ : x = x) } =\n    \u2191(pullbackFst (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k)) \u226b MkCore.t h i j)\n      { val := ({ val := x, property := hx }, { val := x, property := hx' }), property := (_ : x = x) }\n[PROOFSTEP]\nrfl\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 MkCore.t' h i j k \u226b MkCore.t' h j k i \u226b MkCore.t' h k i j =\n    \ud835\udfd9 (pullback ((fun i j => Opens.inclusion (MkCore.V h i j)) i j) ((fun i j => Opens.inclusion (MkCore.V h i j)) i k))\n[PROOFSTEP]\ndelta MkCore.t'\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom \u226b\n        (ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                  \u2191(MkCore.t h i j) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n          (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).inv) \u226b\n      ((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).hom \u226b\n          (ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                    \u2191(MkCore.t h j k) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n            (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).inv) \u226b\n        (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).hom \u226b\n          (ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                    \u2191(MkCore.t h k i) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h i j))\n                        ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                            \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h i j))\n                        ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                            \u2191(MkCore.t h k i) (\u2191x).fst).fst) }) \u226b\n            (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).inv =\n    \ud835\udfd9 (pullback ((fun i j => Opens.inclusion (MkCore.V h i j)) i j) ((fun i j => Opens.inclusion (MkCore.V h i j)) i k))\n[PROOFSTEP]\nsimp_rw [\u2190 Category.assoc]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((((((((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom \u226b\n                    ContinuousMap.mk fun x =>\n                      {\n                        val :=\n                          ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                            \u2191(MkCore.t h i j) (\u2191x).fst),\n                        property :=\n                          (_ :\n                            \u2191(Opens.inclusion (MkCore.V h j k))\n                                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                                      property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                                    \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                              \u2191(Opens.inclusion (MkCore.V h j k))\n                                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                                      property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                                    \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n                  (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).inv) \u226b\n                (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).hom) \u226b\n              ContinuousMap.mk fun x =>\n                {\n                  val :=\n                    ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                      \u2191(MkCore.t h j k) (\u2191x).fst),\n                  property :=\n                    (_ :\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                          ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                              \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                        \u2191(Opens.inclusion (MkCore.V h k i))\n                          ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                              \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n            (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).inv) \u226b\n          (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).hom) \u226b\n        ContinuousMap.mk fun x =>\n          {\n            val :=\n              ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                  property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                \u2191(MkCore.t h k i) (\u2191x).fst),\n            property :=\n              (_ :\n                \u2191(Opens.inclusion (MkCore.V h i j))\n                    ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                        \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                  \u2191(Opens.inclusion (MkCore.V h i j))\n                    ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                        \u2191(MkCore.t h k i) (\u2191x).fst).fst) }) \u226b\n      (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).inv =\n    \ud835\udfd9 (pullback (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k)))\n[PROOFSTEP]\nrw [Iso.comp_inv_eq]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((((((((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom \u226b\n                  ContinuousMap.mk fun x =>\n                    {\n                      val :=\n                        ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst),\n                      property :=\n                        (_ :\n                          \u2191(Opens.inclusion (MkCore.V h j k))\n                              ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                                  \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                            \u2191(Opens.inclusion (MkCore.V h j k))\n                              ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                                  \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n                (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).inv) \u226b\n              (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h j k)) (Opens.inclusion (MkCore.V h j i))).hom) \u226b\n            ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                    \u2191(MkCore.t h j k) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n          (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).inv) \u226b\n        (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h k i)) (Opens.inclusion (MkCore.V h k j))).hom) \u226b\n      ContinuousMap.mk fun x =>\n        {\n          val :=\n            ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst), property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n              \u2191(MkCore.t h k i) (\u2191x).fst),\n          property :=\n            (_ :\n              \u2191(Opens.inclusion (MkCore.V h i j))\n                  ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                      \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                \u2191(Opens.inclusion (MkCore.V h i j))\n                  ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                      \u2191(MkCore.t h k i) (\u2191x).fst).fst) }) =\n    \ud835\udfd9 (pullback (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))) \u226b\n      (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom\n[PROOFSTEP]\nsimp only [Iso.inv_hom_id_assoc, Category.assoc, Category.id_comp]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom \u226b\n      (ContinuousMap.mk fun x =>\n          {\n            val :=\n              ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                  property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                \u2191(MkCore.t h i j) (\u2191x).fst),\n            property :=\n              (_ :\n                \u2191(Opens.inclusion (MkCore.V h j k))\n                    ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                        \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                    ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                        \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n        (ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                  \u2191(MkCore.t h j k) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h k i))\n                      ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                          \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h k i))\n                      ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                          \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n          ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                  \u2191(MkCore.t h k i) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h i j))\n                      ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                          \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h i j))\n                      ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                          \u2191(MkCore.t h k i) (\u2191x).fst).fst) }) =\n    (pullbackIsoProdSubtype (Opens.inclusion (MkCore.V h i j)) (Opens.inclusion (MkCore.V h i k))).hom\n[PROOFSTEP]\nrw [\u2190 Iso.eq_inv_comp, Iso.inv_hom_id]\n[GOAL]\nD : GlueData\nh : MkCore\ni j k : h.J\n\u22a2 ((ContinuousMap.mk fun x =>\n        {\n          val :=\n            ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst), property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n              \u2191(MkCore.t h i j) (\u2191x).fst),\n          property :=\n            (_ :\n              \u2191(Opens.inclusion (MkCore.V h j k))\n                  ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                      \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                \u2191(Opens.inclusion (MkCore.V h j k))\n                  ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                      \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n      (ContinuousMap.mk fun x =>\n          {\n            val :=\n              ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                  property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                \u2191(MkCore.t h j k) (\u2191x).fst),\n            property :=\n              (_ :\n                \u2191(Opens.inclusion (MkCore.V h k i))\n                    ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                        \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                  \u2191(Opens.inclusion (MkCore.V h k i))\n                    ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                        \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n        ContinuousMap.mk fun x =>\n          {\n            val :=\n              ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                  property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                \u2191(MkCore.t h k i) (\u2191x).fst),\n            property :=\n              (_ :\n                \u2191(Opens.inclusion (MkCore.V h i j))\n                    ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                        \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                  \u2191(Opens.inclusion (MkCore.V h i j))\n                    ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                        \u2191(MkCore.t h k i) (\u2191x).fst).fst) }) =\n    \ud835\udfd9 (of { p // \u2191(Opens.inclusion (MkCore.V h i j)) p.fst = \u2191(Opens.inclusion (MkCore.V h i k)) p.snd })\n[PROOFSTEP]\next1 \u27e8\u27e8\u27e8x, hx\u27e9, \u27e8x', hx'\u27e9\u27e9, rfl : x = x'\u27e9\n[GOAL]\ncase w.mk.mk.mk.mk\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(MkCore.U h i)\nhx : x \u2208 MkCore.V h i j\nhx' : x \u2208 MkCore.V h i k\n\u22a2 \u2191((ContinuousMap.mk fun x =>\n            {\n              val :=\n                ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                    property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                  \u2191(MkCore.t h i j) (\u2191x).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({ val := \u2191(\u2191(MkCore.t h i j) (\u2191x).fst),\n                            property := (_ : \u2191(\u2191(MkCore.t h i j) (\u2191x).fst) \u2208 MkCore.V h j k) },\n                          \u2191(MkCore.t h i j) (\u2191x).fst).fst) }) \u226b\n          (ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                    \u2191(MkCore.t h j k) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst) }) \u226b\n            ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                    \u2191(MkCore.t h k i) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h i j))\n                        ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                            \u2191(MkCore.t h k i) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h i j))\n                        ({ val := \u2191(\u2191(MkCore.t h k i) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h k i) (\u2191x).fst) \u2208 MkCore.V h i j) },\n                            \u2191(MkCore.t h k i) (\u2191x).fst).fst) })\n      { val := ({ val := x, property := hx }, { val := x, property := hx' }), property := (_ : x = x) } =\n    \u2191(\ud835\udfd9 (of { p // \u2191(Opens.inclusion (MkCore.V h i j)) p.fst = \u2191(Opens.inclusion (MkCore.V h i k)) p.snd }))\n      { val := ({ val := x, property := hx }, { val := x, property := hx' }), property := (_ : x = x) }\n[PROOFSTEP]\nrw [comp_app, ContinuousMap.coe_mk, comp_app, id_app, ContinuousMap.coe_mk, Subtype.mk_eq_mk, Prod.mk.inj_iff,\n  Subtype.mk_eq_mk, Subtype.ext_iff, and_self_iff]\n[GOAL]\ncase w.mk.mk.mk.mk\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(MkCore.U h i)\nhx : x \u2208 MkCore.V h i j\nhx' : x \u2208 MkCore.V h i k\n\u22a2 \u2191(\u2191(MkCore.t h k i)\n        (\u2191(\u2191(ContinuousMap.mk fun x =>\n                  {\n                    val :=\n                      ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                          property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                        \u2191(MkCore.t h j k) (\u2191x).fst),\n                    property :=\n                      (_ :\n                        \u2191(Opens.inclusion (MkCore.V h k i))\n                            ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                  property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                                \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                          \u2191(Opens.inclusion (MkCore.V h k i))\n                            ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                  property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                                \u2191(MkCore.t h j k) (\u2191x).fst).fst) })\n              {\n                val :=\n                  ({\n                      val :=\n                        \u2191(\u2191(MkCore.t h i j)\n                            (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                  property := (_ : x = x) }).fst),\n                      property :=\n                        (_ :\n                          \u2191(\u2191(MkCore.t h i j)\n                                (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                      property := (_ : x = x) }).fst) \u2208\n                            MkCore.V h j k) },\n                    \u2191(MkCore.t h i j)\n                      (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                            property := (_ : x = x) }).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                        ({\n                              val :=\n                                \u2191(\u2191(MkCore.t h i j)\n                                    (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                          property := (_ : x = x) }).fst),\n                              property :=\n                                (_ :\n                                  \u2191(\u2191(MkCore.t h i j)\n                                        (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                              property := (_ : x = x) }).fst) \u2208\n                                    MkCore.V h j k) },\n                            \u2191(MkCore.t h i j)\n                              (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                    property := (_ : x = x) }).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h j k))\n                        ({\n                              val :=\n                                \u2191(\u2191(MkCore.t h i j)\n                                    (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                          property := (_ : x = x) }).fst),\n                              property :=\n                                (_ :\n                                  \u2191(\u2191(MkCore.t h i j)\n                                        (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                              property := (_ : x = x) }).fst) \u2208\n                                    MkCore.V h j k) },\n                            \u2191(MkCore.t h i j)\n                              (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                    property := (_ : x = x) }).fst).fst) })).fst) =\n    x\n[PROOFSTEP]\nconvert congr_arg Subtype.val (h.t_inv k i \u27e8x, hx'\u27e9) using 3\n[GOAL]\ncase h.e'_2.h.e'_3.h.e'_6\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(MkCore.U h i)\nhx : x \u2208 MkCore.V h i j\nhx' : x \u2208 MkCore.V h i k\n\u22a2 (\u2191(\u2191(ContinuousMap.mk fun x =>\n              {\n                val :=\n                  ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                      property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                    \u2191(MkCore.t h j k) (\u2191x).fst),\n                property :=\n                  (_ :\n                    \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                        ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                              property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                            \u2191(MkCore.t h j k) (\u2191x).fst).fst) })\n          {\n            val :=\n              ({\n                  val :=\n                    \u2191(\u2191(MkCore.t h i j)\n                        (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                              property := (_ : x = x) }).fst),\n                  property :=\n                    (_ :\n                      \u2191(\u2191(MkCore.t h i j)\n                            (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                  property := (_ : x = x) }).fst) \u2208\n                        MkCore.V h j k) },\n                \u2191(MkCore.t h i j)\n                  (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                        property := (_ : x = x) }).fst),\n            property :=\n              (_ :\n                \u2191(Opens.inclusion (MkCore.V h j k))\n                    ({\n                          val :=\n                            \u2191(\u2191(MkCore.t h i j)\n                                (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                      property := (_ : x = x) }).fst),\n                          property :=\n                            (_ :\n                              \u2191(\u2191(MkCore.t h i j)\n                                    (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                          property := (_ : x = x) }).fst) \u2208\n                                MkCore.V h j k) },\n                        \u2191(MkCore.t h i j)\n                          (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                property := (_ : x = x) }).fst).fst =\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                    ({\n                          val :=\n                            \u2191(\u2191(MkCore.t h i j)\n                                (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                      property := (_ : x = x) }).fst),\n                          property :=\n                            (_ :\n                              \u2191(\u2191(MkCore.t h i j)\n                                    (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                          property := (_ : x = x) }).fst) \u2208\n                                MkCore.V h j k) },\n                        \u2191(MkCore.t h i j)\n                          (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                property := (_ : x = x) }).fst).fst) })).fst =\n    \u2191(MkCore.t h i k) { val := x, property := hx' }\n[PROOFSTEP]\nrefine Subtype.ext ?_\n[GOAL]\ncase h.e'_2.h.e'_3.h.e'_6\nD : GlueData\nh : MkCore\ni j k : h.J\nx : \u2191(MkCore.U h i)\nhx : x \u2208 MkCore.V h i j\nhx' : x \u2208 MkCore.V h i k\n\u22a2 \u2191(\u2191(\u2191(ContinuousMap.mk fun x =>\n                {\n                  val :=\n                    ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                        property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                      \u2191(MkCore.t h j k) (\u2191x).fst),\n                  property :=\n                    (_ :\n                      \u2191(Opens.inclusion (MkCore.V h k i))\n                          ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                              \u2191(MkCore.t h j k) (\u2191x).fst).fst =\n                        \u2191(Opens.inclusion (MkCore.V h k i))\n                          ({ val := \u2191(\u2191(MkCore.t h j k) (\u2191x).fst),\n                                property := (_ : \u2191(\u2191(MkCore.t h j k) (\u2191x).fst) \u2208 MkCore.V h k i) },\n                              \u2191(MkCore.t h j k) (\u2191x).fst).fst) })\n            {\n              val :=\n                ({\n                    val :=\n                      \u2191(\u2191(MkCore.t h i j)\n                          (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                property := (_ : x = x) }).fst),\n                    property :=\n                      (_ :\n                        \u2191(\u2191(MkCore.t h i j)\n                              (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                    property := (_ : x = x) }).fst) \u2208\n                          MkCore.V h j k) },\n                  \u2191(MkCore.t h i j)\n                    (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                          property := (_ : x = x) }).fst),\n              property :=\n                (_ :\n                  \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({\n                            val :=\n                              \u2191(\u2191(MkCore.t h i j)\n                                  (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                        property := (_ : x = x) }).fst),\n                            property :=\n                              (_ :\n                                \u2191(\u2191(MkCore.t h i j)\n                                      (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                            property := (_ : x = x) }).fst) \u2208\n                                  MkCore.V h j k) },\n                          \u2191(MkCore.t h i j)\n                            (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                  property := (_ : x = x) }).fst).fst =\n                    \u2191(Opens.inclusion (MkCore.V h j k))\n                      ({\n                            val :=\n                              \u2191(\u2191(MkCore.t h i j)\n                                  (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                        property := (_ : x = x) }).fst),\n                            property :=\n                              (_ :\n                                \u2191(\u2191(MkCore.t h i j)\n                                      (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                            property := (_ : x = x) }).fst) \u2208\n                                  MkCore.V h j k) },\n                          \u2191(MkCore.t h i j)\n                            (\u2191{ val := ({ val := x, property := hx }, { val := x, property := hx' }),\n                                  property := (_ : x = x) }).fst).fst) })).fst =\n    \u2191(\u2191(MkCore.t h i k) { val := x, property := hx' })\n[PROOFSTEP]\nexact h.cocycle i j k \u27e8x, hx\u27e9 hx'\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni j : J\n\u22a2 Continuous fun x =>\n    { val := { val := \u2191\u2191x, property := (_ : \u2191x \u2208 (fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i j) },\n      property := (_ : \u2191\u2191x \u2208 U i) }\n[PROOFSTEP]\nrefine Continuous.subtype_mk ?_ ?_\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni j : J\n\u22a2 Continuous fun x =>\n    { val := \u2191\u2191x, property := (_ : \u2191x \u2208 (fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i j) }\n[PROOFSTEP]\nrefine Continuous.subtype_mk ?_ ?_\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni j : J\n\u22a2 Continuous fun x => \u2191\u2191x\n[PROOFSTEP]\ncontinuity\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : J\n\u22a2 (fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i i = \u22a4\n[PROOFSTEP]\next\n  -- porting note: no longer needed `cases U i`!\n[GOAL]\ncase h.h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : J\nx\u271d : \u2191((Opens.toTopCat (of \u03b1)).obj (U i))\n\u22a2 x\u271d \u2208 \u2191((fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i i) \u2194 x\u271d \u2208 \u2191\u22a4\n[PROOFSTEP]\nsimp\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : J\n\u22a2 \u2191((fun i j =>\n          ContinuousMap.mk fun x =>\n            {\n              val :=\n                { val := \u2191\u2191x, property := (_ : \u2191x \u2208 (fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i j) },\n              property := (_ : \u2191\u2191x \u2208 U i) })\n        i i) =\n    id\n[PROOFSTEP]\next\n[GOAL]\ncase h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : J\nx\u271d :\n  (forget TopCat).obj\n    ((Opens.toTopCat ((fun i => (Opens.toTopCat (of \u03b1)).obj (U i)) i)).obj\n      ((fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i i))\n\u22a2 \u2191((fun i j =>\n            ContinuousMap.mk fun x =>\n              {\n                val :=\n                  { val := \u2191\u2191x, property := (_ : \u2191x \u2208 (fun i j => (Opens.map (Opens.inclusion (U i))).obj (U j)) i j) },\n                property := (_ : \u2191\u2191x \u2208 U i) })\n          i i)\n      x\u271d =\n    id x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\n\u22a2 \u2200 (a : (GlueData.diagram (ofOpenSubsets U).toGlueData).L),\n    MultispanIndex.fst (GlueData.diagram (ofOpenSubsets U).toGlueData) a \u226b\n        (fun x => Opens.inclusion (U x)) (MultispanIndex.fstFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) a) =\n      MultispanIndex.snd (GlueData.diagram (ofOpenSubsets U).toGlueData) a \u226b\n        (fun x => Opens.inclusion (U x)) (MultispanIndex.sndFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) a)\n[PROOFSTEP]\nrintro \u27e8i, j\u27e9\n[GOAL]\ncase mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni j : (ofOpenSubsets U).toGlueData.J\n\u22a2 MultispanIndex.fst (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j) \u226b\n      (fun x => Opens.inclusion (U x)) (MultispanIndex.fstFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j)) =\n    MultispanIndex.snd (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j) \u226b\n      (fun x => Opens.inclusion (U x)) (MultispanIndex.sndFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j))\n[PROOFSTEP]\next x\n[GOAL]\ncase mk.w\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni j : (ofOpenSubsets U).toGlueData.J\nx : (forget TopCat).obj (MultispanIndex.left (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j))\n\u22a2 \u2191(MultispanIndex.fst (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j) \u226b\n          (fun x => Opens.inclusion (U x))\n            (MultispanIndex.fstFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j)))\n      x =\n    \u2191(MultispanIndex.snd (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j) \u226b\n          (fun x => Opens.inclusion (U x))\n            (MultispanIndex.sndFrom (GlueData.diagram (ofOpenSubsets U).toGlueData) (i, j)))\n      x\n[PROOFSTEP]\nrfl\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\n\u22a2 Function.Injective \u2191(fromOpenSubsetsGlue U)\n[PROOFSTEP]\nintro x y e\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx y : (forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData)\ne : \u2191(fromOpenSubsetsGlue U) x = \u2191(fromOpenSubsetsGlue U) y\n\u22a2 x = y\n[PROOFSTEP]\nobtain \u27e8i, \u27e8x, hx\u27e9, rfl\u27e9 := (ofOpenSubsets U).\u03b9_jointly_surjective x\n[GOAL]\ncase intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ny : (forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\ne :\n  \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx }) =\n    \u2191(fromOpenSubsetsGlue U) y\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx } = y\n[PROOFSTEP]\nobtain \u27e8j, \u27e8y, hy\u27e9, rfl\u27e9 := (ofOpenSubsets U).\u03b9_jointly_surjective y\n[GOAL]\ncase intro.intro.mk.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\ny : \u2191(of \u03b1)\nhy : y \u2208 U j\ne :\n  \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx }) =\n    \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData j) { val := y, property := hy })\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx } =\n    \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData j) { val := y, property := hy }\n[PROOFSTEP]\nerw [\u03b9_fromOpenSubsetsGlue_apply, \u03b9_fromOpenSubsetsGlue_apply] at e \n[GOAL]\ncase intro.intro.mk.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\ny : \u2191(of \u03b1)\nhy : y \u2208 U j\ne : \u2191(Opens.inclusion (U i)) { val := x, property := hx } = \u2191(Opens.inclusion (U j)) { val := y, property := hy }\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx } =\n    \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData j) { val := y, property := hy }\n[PROOFSTEP]\nchange x = y at e \n[GOAL]\ncase intro.intro.mk.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\ny : \u2191(of \u03b1)\nhy : y \u2208 U j\ne : x = y\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx } =\n    \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData j) { val := y, property := hy }\n[PROOFSTEP]\nsubst e\n[GOAL]\ncase intro.intro.mk.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\nhy : x \u2208 U j\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx } =\n    \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData j) { val := x, property := hy }\n[PROOFSTEP]\nrw [(ofOpenSubsets U).\u03b9_eq_iff_rel]\n[GOAL]\ncase intro.intro.mk.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\nhy : x \u2208 U j\n\u22a2 Rel (ofOpenSubsets U) { fst := i, snd := { val := x, property := hx } }\n    { fst := j, snd := { val := x, property := hy } }\n[PROOFSTEP]\nright\n[GOAL]\ncase intro.intro.mk.intro.intro.mk.h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx : x \u2208 U i\nj : (ofOpenSubsets U).toGlueData.J\nhy : x \u2208 U j\n\u22a2 \u2203 x_1,\n    \u2191(GlueData.f (ofOpenSubsets U).toGlueData { fst := i, snd := { val := x, property := hx } }.fst\n              { fst := j, snd := { val := x, property := hy } }.fst)\n          x_1 =\n        { fst := i, snd := { val := x, property := hx } }.snd \u2227\n      \u2191(GlueData.f (ofOpenSubsets U).toGlueData { fst := j, snd := { val := x, property := hy } }.fst\n              { fst := i, snd := { val := x, property := hx } }.fst)\n          (\u2191(GlueData.t (ofOpenSubsets U).toGlueData { fst := i, snd := { val := x, property := hx } }.fst\n                { fst := j, snd := { val := x, property := hy } }.fst)\n            x_1) =\n        { fst := j, snd := { val := x, property := hy } }.snd\n[PROOFSTEP]\nexact \u27e8\u27e8\u27e8x, hx\u27e9, hy\u27e9, rfl, rfl\u27e9\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\n\u22a2 IsOpenMap \u2191(fromOpenSubsetsGlue U)\n[PROOFSTEP]\nintro s hs\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : IsOpen s\n\u22a2 IsOpen (\u2191(fromOpenSubsetsGlue U) '' s)\n[PROOFSTEP]\nrw [(ofOpenSubsets U).isOpen_iff] at hs \n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\n\u22a2 IsOpen (\u2191(fromOpenSubsetsGlue U) '' s)\n[PROOFSTEP]\nrw [isOpen_iff_forall_mem_open]\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\n\u22a2 \u2200 (x : (forget TopCat).obj (of \u03b1)),\n    x \u2208 \u2191(fromOpenSubsetsGlue U) '' s \u2192 \u2203 t, t \u2286 \u2191(fromOpenSubsetsGlue U) '' s \u2227 IsOpen t \u2227 x \u2208 t\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\nx : (forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData)\nhx : x \u2208 s\n\u22a2 \u2203 t, t \u2286 \u2191(fromOpenSubsetsGlue U) '' s \u2227 IsOpen t \u2227 \u2191(fromOpenSubsetsGlue U) x \u2208 t\n[PROOFSTEP]\nobtain \u27e8i, \u27e8x, hx'\u27e9, rfl\u27e9 := (ofOpenSubsets U).\u03b9_jointly_surjective x\n[GOAL]\ncase intro.intro.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 \u2203 t,\n    t \u2286 \u2191(fromOpenSubsetsGlue U) '' s \u2227\n      IsOpen t \u2227\n        \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208 t\n[PROOFSTEP]\nuse fromOpenSubsetsGlue U '' s \u2229 Set.range (@Opens.inclusion (TopCat.of \u03b1) (U i))\n[GOAL]\ncase h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 \u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i)) \u2286 \u2191(fromOpenSubsetsGlue U) '' s \u2227\n    IsOpen (\u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i))) \u2227\n      \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208\n        \u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i))\n[PROOFSTEP]\nuse Set.inter_subset_left _ _\n[GOAL]\ncase right\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 IsOpen (\u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i))) \u2227\n    \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208\n      \u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase right.left\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 IsOpen (\u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i)))\n[PROOFSTEP]\nerw [\u2190 Set.image_preimage_eq_inter_range]\n[GOAL]\ncase right.left\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 IsOpen (\u2191(Opens.inclusion (U i)) '' (\u2191(Opens.inclusion (U i)) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) '' s)))\n[PROOFSTEP]\napply (Opens.openEmbedding (X := TopCat.of \u03b1) (U i)).isOpenMap\n[GOAL]\ncase right.left.a\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 IsOpen (\u2191(Opens.inclusion (U i)) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) '' s))\n[PROOFSTEP]\nconvert hs i using 1\n[GOAL]\ncase h.e'_3.h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\ne_1\u271d :\n  (forget TopCat).obj ((Opens.toTopCat (of \u03b1)).obj (U i)) =\n    (forget TopCat).obj (GlueData.U (ofOpenSubsets U).toGlueData i)\n\u22a2 \u2191(Opens.inclusion (U i)) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) '' s) = \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s\n[PROOFSTEP]\nerw [\u2190 \u03b9_fromOpenSubsetsGlue, coe_comp, Set.preimage_comp]\n  --  porting note: `congr 1` did nothing, so I replaced it with `apply congr_arg`\n[GOAL]\ncase h.e'_3.h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\ne_1\u271d :\n  (forget TopCat).obj ((Opens.toTopCat (of \u03b1)).obj (U i)) =\n    (forget TopCat).obj (GlueData.U (ofOpenSubsets U).toGlueData i)\n\u22a2 \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) '' s)) =\n    \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s\n[PROOFSTEP]\napply congr_arg\n[GOAL]\ncase h.e'_3.h.h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\ne_1\u271d :\n  (forget TopCat).obj ((Opens.toTopCat (of \u03b1)).obj (U i)) =\n    (forget TopCat).obj (GlueData.U (ofOpenSubsets U).toGlueData i)\n\u22a2 \u2191(fromOpenSubsetsGlue U) \u207b\u00b9' (\u2191(fromOpenSubsetsGlue U) '' s) = s\n[PROOFSTEP]\nrefine' Set.preimage_image_eq _ (fromOpenSubsetsGlue_injective U)\n[GOAL]\ncase right.right\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208\n    \u2191(fromOpenSubsetsGlue U) '' s \u2229 Set.range \u2191(Opens.inclusion (U i))\n[PROOFSTEP]\nrefine'\n  \u27e8Set.mem_image_of_mem _ hx, _\u27e9\n    -- porting note: another `rw \u21a6 erw`\n        -- See above.\n[GOAL]\ncase right.right\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208\n    Set.range \u2191(Opens.inclusion (U i))\n[PROOFSTEP]\nerw [\u03b9_fromOpenSubsetsGlue_apply]\n[GOAL]\ncase right.right\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ns : Set ((forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData))\nhs : \u2200 (i : (ofOpenSubsets U).toGlueData.J), IsOpen (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) \u207b\u00b9' s)\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\nhx : \u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' } \u2208 s\n\u22a2 \u2191(Opens.inclusion (U i)) { val := x, property := hx' } \u2208 Set.range \u2191(Opens.inclusion (U i))\n[PROOFSTEP]\nexact Set.mem_range_self _\n[GOAL]\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\n\u22a2 Set.range \u2191(fromOpenSubsetsGlue U) = \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\next\n[GOAL]\ncase h\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx\u271d : (forget TopCat).obj (of \u03b1)\n\u22a2 x\u271d \u2208 Set.range \u2191(fromOpenSubsetsGlue U) \u2194 x\u271d \u2208 \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx\u271d : (forget TopCat).obj (of \u03b1)\n\u22a2 x\u271d \u2208 Set.range \u2191(fromOpenSubsetsGlue U) \u2192 x\u271d \u2208 \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\nrintro \u27e8x, rfl\u27e9\n[GOAL]\ncase h.mp.intro\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx : (forget TopCat).obj (GlueData.glued (ofOpenSubsets U).toGlueData)\n\u22a2 \u2191(fromOpenSubsetsGlue U) x \u2208 \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\nobtain \u27e8i, \u27e8x, hx'\u27e9, rfl\u27e9 := (ofOpenSubsets U).\u03b9_jointly_surjective x\n[GOAL]\ncase h.mp.intro.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\n\u22a2 \u2191(fromOpenSubsetsGlue U) (\u2191(GlueData.\u03b9 (ofOpenSubsets U).toGlueData i) { val := x, property := hx' }) \u2208\n    \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\nerw [\u03b9_fromOpenSubsetsGlue_apply]\n[GOAL]\ncase h.mp.intro.intro.intro.mk\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\ni : (ofOpenSubsets U).toGlueData.J\nx : \u2191(of \u03b1)\nhx' : x \u2208 U i\n\u22a2 \u2191(Opens.inclusion (U i)) { val := x, property := hx' } \u2208 \u22c3 (i : J), \u2191(U i)\n[PROOFSTEP]\nexact Set.subset_iUnion _ i hx'\n[GOAL]\ncase h.mpr\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx\u271d : (forget TopCat).obj (of \u03b1)\n\u22a2 x\u271d \u2208 \u22c3 (i : J), \u2191(U i) \u2192 x\u271d \u2208 Set.range \u2191(fromOpenSubsetsGlue U)\n[PROOFSTEP]\nrintro \u27e8_, \u27e8i, rfl\u27e9, hx\u27e9\n[GOAL]\ncase h.mpr.intro.intro.intro\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx\u271d : (forget TopCat).obj (of \u03b1)\ni : J\nhx : x\u271d \u2208 (fun i => \u2191(U i)) i\n\u22a2 x\u271d \u2208 Set.range \u2191(fromOpenSubsetsGlue U)\n[PROOFSTEP]\nrename_i x\n[GOAL]\ncase h.mpr.intro.intro.intro\nD : GlueData\n\u03b1 : Type u\ninst\u271d : TopologicalSpace \u03b1\nJ : Type u\nU : J \u2192 Opens \u03b1\nx : (forget TopCat).obj (of \u03b1)\ni : J\nhx : x \u2208 (fun i => \u2191(U i)) i\n\u22a2 x \u2208 Set.range \u2191(fromOpenSubsetsGlue U)\n[PROOFSTEP]\nrefine' \u27e8(ofOpenSubsets U).toGlueData.\u03b9 i \u27e8x, hx\u27e9, \u03b9_fromOpenSubsetsGlue_apply _ _ _\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Gluing", "llama_tokens": 68674, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7853085758631158, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.5030968200349505}}
{"text": "[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\n_h : v \u2260 w\n\u22a2 \u00acAdj \u22a5 v w \u2192 Fintype.card \u2191(commonNeighbors \u22a5 v w) = 0\n[PROOFSTEP]\nsimp only [card_eq_zero, Fintype.card_ofFinset, forall_true_left, not_false_iff, bot_adj]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\n_h : v \u2260 w\n\u22a2 filter (fun x => x \u2208 commonNeighbors \u22a5 v w) univ = \u2205\n[PROOFSTEP]\next\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\n_h : v \u2260 w\na\u271d : V\n\u22a2 a\u271d \u2208 filter (fun x => x \u2208 commonNeighbors \u22a5 v w) univ \u2194 a\u271d \u2208 \u2205\n[PROOFSTEP]\nsimp [mem_commonNeighbors]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : Adj \u22a4 v w\n\u22a2 Fintype.card \u2191(commonNeighbors \u22a4 v w) = Fintype.card V - 2\n[PROOFSTEP]\nrw [card_commonNeighbors_top]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : Adj \u22a4 v w\n\u22a2 v \u2260 w\n[PROOFSTEP]\nexact h\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) = 2 * k - Fintype.card \u2191(commonNeighbors G v w)\n[PROOFSTEP]\napply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) + Fintype.card \u2191(commonNeighbors G v w) =\n    2 * k - Fintype.card \u2191(commonNeighbors G v w) + Fintype.card \u2191(commonNeighbors G v w)\n[PROOFSTEP]\nrw [Nat.sub_add_cancel, \u2190 Set.toFinset_card]\n  -- porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the\n    -- instance arguments:\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) + Finset.card (Set.toFinset (commonNeighbors G v w)) = 2 * k\n[PROOFSTEP]\nsimp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_), \u2190 neighborFinset_def, Finset.card_union_add_card_inter,\n  card_neighborFinset_eq_degree, h.regular.degree_eq, two_mul]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 Fintype.card \u2191(commonNeighbors G v w) \u2264 2 * k\n[PROOFSTEP]\napply le_trans (card_commonNeighbors_le_degree_left _ _ _)\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 degree G v \u2264 2 * k\n[PROOFSTEP]\nsimp [h.regular.degree_eq, two_mul]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\nhne : v \u2260 w\nha : \u00acAdj G v w\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) = 2 * k - \u03bc\n[PROOFSTEP]\nrw [\u2190 h.of_not_adj v w hne ha]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\nhne : v \u2260 w\nha : \u00acAdj G v w\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) = 2 * k - Fintype.card \u2191(commonNeighbors G v w)\n[PROOFSTEP]\napply h.card_neighborFinset_union_eq\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\nha : Adj G v w\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) = 2 * k - \u2113\n[PROOFSTEP]\nrw [\u2190 h.of_adj v w ha]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : IsSRGWith G n k \u2113 \u03bc\nha : Adj G v w\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w) = 2 * k - Fintype.card \u2191(commonNeighbors G v w)\n[PROOFSTEP]\napply h.card_neighborFinset_union_eq\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\n\u22a2 (neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w}) =\n    ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})\n[PROOFSTEP]\next\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w a\u271d : V\n\u22a2 a\u271d \u2208 (neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w}) \u2194\n    a\u271d \u2208 ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})\n[PROOFSTEP]\nrw [\u2190 not_iff_not]\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w a\u271d : V\n\u22a2 \u00aca\u271d \u2208 (neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w}) \u2194\n    \u00aca\u271d \u2208 ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})\n[PROOFSTEP]\nsimp [imp_iff_not_or, or_assoc, or_comm, or_left_comm]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : Adj G v w\n\u22a2 ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v}) = (neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c\n[PROOFSTEP]\next\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : Adj G v w\na\u271d : V\n\u22a2 a\u271d \u2208 ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v}) \u2194\n    a\u271d \u2208 (neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c\n[PROOFSTEP]\nsimp only [and_imp, mem_union, mem_sdiff, mem_compl, and_iff_left_iff_imp, mem_neighborFinset, mem_inter, mem_singleton]\n[GOAL]\ncase a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv w : V\nh : Adj G v w\na\u271d : V\n\u22a2 \u00acAdj G v a\u271d \u2192 \u00acAdj G w a\u271d \u2192 \u00ac(a\u271d = w \u2228 a\u271d = v)\n[PROOFSTEP]\nrintro hnv hnw (rfl | rfl)\n[GOAL]\ncase a.inl\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nv a\u271d : V\nhnv : \u00acAdj G v a\u271d\nh : Adj G v a\u271d\nhnw : \u00acAdj G a\u271d a\u271d\n\u22a2 False\n[PROOFSTEP]\nexact hnv h\n[GOAL]\ncase a.inr\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nw a\u271d : V\nhnw : \u00acAdj G w a\u271d\nh : Adj G a\u271d w\nhnv : \u00acAdj G a\u271d a\u271d\n\u22a2 False\n[PROOFSTEP]\napply hnw\n[GOAL]\ncase a.inr\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nw a\u271d : V\nhnw : \u00acAdj G w a\u271d\nh : Adj G a\u271d w\nhnv : \u00acAdj G a\u271d a\u271d\n\u22a2 Adj G w a\u271d\n[PROOFSTEP]\nrwa [adj_comm]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 IsRegularOfDegree G\u1d9c (n - k - 1)\n[PROOFSTEP]\nrw [\u2190 h.card, Nat.sub_sub, add_comm, \u2190 Nat.sub_sub]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 IsRegularOfDegree G\u1d9c (Fintype.card V - 1 - k)\n[PROOFSTEP]\nexact h.regular.compl\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : Adj G\u1d9c v w\n\u22a2 Fintype.card \u2191(commonNeighbors G\u1d9c v w) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nsimp only [\u2190 Set.toFinset_card, commonNeighbors, Set.toFinset_inter, neighborSet_compl, Set.toFinset_diff,\n  Set.toFinset_singleton, Set.toFinset_compl, \u2190 neighborFinset_def]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : Adj G\u1d9c v w\n\u22a2 Finset.card ((neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w})) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nsimp_rw [compl_neighborFinset_sdiff_inter_eq]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : Adj G\u1d9c v w\n\u22a2 Finset.card (((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nhave hne : v \u2260 w := ne_of_adj _ ha\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : Adj G\u1d9c v w\nhne : v \u2260 w\n\u22a2 Finset.card (((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nrw [compl_adj] at ha \n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\n\u22a2 Finset.card (((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) \\ ({w} \u222a {v})) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nrw [card_sdiff, \u2190 insert_eq, card_insert_of_not_mem, card_singleton, \u2190 Finset.compl_union]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\n\u22a2 Finset.card (neighborFinset G v \u222a neighborFinset G w)\u1d9c - (1 + 1) = n - (2 * k - \u03bc) - 2\n[PROOFSTEP]\nrw [card_compl, h.card_neighborFinset_union_of_not_adj hne ha.2, \u2190 h.card]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\n\u22a2 \u00acw \u2208 {v}\n[PROOFSTEP]\nsimp only [hne.symm, not_false_iff, mem_singleton]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\n\u22a2 {w} \u222a {v} \u2286 (neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c\n[PROOFSTEP]\nintro u\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\nu : V\n\u22a2 u \u2208 {w} \u222a {v} \u2192 u \u2208 (neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c\n[PROOFSTEP]\nsimp only [mem_union, mem_compl, mem_neighborFinset, mem_inter, mem_singleton]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nha : v \u2260 w \u2227 \u00acAdj G v w\nhne : v \u2260 w\nu : V\n\u22a2 u = w \u2228 u = v \u2192 \u00acAdj G v u \u2227 \u00acAdj G w u\n[PROOFSTEP]\nrintro (rfl | rfl)\n[GOAL]\ncase inl\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv u : V\nha : v \u2260 u \u2227 \u00acAdj G v u\nhne : v \u2260 u\n\u22a2 \u00acAdj G v u \u2227 \u00acAdj G u u\n[PROOFSTEP]\nsimpa [adj_comm] using ha.2\n[GOAL]\ncase inr\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nw u : V\nha : u \u2260 w \u2227 \u00acAdj G u w\nhne : u \u2260 w\n\u22a2 \u00acAdj G u u \u2227 \u00acAdj G w u\n[PROOFSTEP]\nsimpa [adj_comm] using ha.2\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nhna : \u00acAdj G\u1d9c v w\n\u22a2 Fintype.card \u2191(commonNeighbors G\u1d9c v w) = n - (2 * k - \u2113)\n[PROOFSTEP]\nsimp only [\u2190 Set.toFinset_card, commonNeighbors, Set.toFinset_inter, neighborSet_compl, Set.toFinset_diff,\n  Set.toFinset_singleton, Set.toFinset_compl, \u2190 neighborFinset_def]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nhna : \u00acAdj G\u1d9c v w\n\u22a2 Finset.card ((neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w})) = n - (2 * k - \u2113)\n[PROOFSTEP]\nsimp only [not_and, Classical.not_not, compl_adj] at hna \n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nhna : v \u2260 w \u2192 Adj G v w\n\u22a2 Finset.card ((neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w})) = n - (2 * k - \u2113)\n[PROOFSTEP]\nhave h2' := hna hn\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nhna : v \u2260 w \u2192 Adj G v w\nh2' : Adj G v w\n\u22a2 Finset.card ((neighborFinset G v)\u1d9c \\ {v} \u2229 ((neighborFinset G w)\u1d9c \\ {w})) = n - (2 * k - \u2113)\n[PROOFSTEP]\nsimp_rw [compl_neighborFinset_sdiff_inter_eq, sdiff_compl_neighborFinset_inter_eq h2']\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nhna : v \u2260 w \u2192 Adj G v w\nh2' : Adj G v w\n\u22a2 Finset.card ((neighborFinset G v)\u1d9c \u2229 (neighborFinset G w)\u1d9c) = n - (2 * k - \u2113)\n[PROOFSTEP]\nrwa [\u2190 Finset.compl_union, card_compl, h.card_neighborFinset_union_of_adj, \u2190 h.card]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nhn : 0 < n\n\u22a2 k * (k - \u2113 - 1) = (n - k - 1) * \u03bc\n[PROOFSTEP]\nrw [\u2190 h.card, Fintype.card_pos_iff] at hn \n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nhn : Nonempty V\n\u22a2 k * (k - \u2113 - 1) = (n - k - 1) * \u03bc\n[PROOFSTEP]\nobtain \u27e8v\u27e9 := hn\n[GOAL]\ncase intro\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 k * (k - \u2113 - 1) = (n - k - 1) * \u03bc\n[PROOFSTEP]\nconvert card_mul_eq_card_mul G.Adj (s := G.neighborFinset v) (t := G\u1d9c.neighborFinset v) _ _\n[GOAL]\ncase h.e'_2.h.e'_5\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 k = Finset.card (neighborFinset G v)\n[PROOFSTEP]\nsimp [h.regular v]\n[GOAL]\ncase h.e'_3.h.e'_5\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 n - k - 1 = Finset.card (neighborFinset G\u1d9c v)\n[PROOFSTEP]\nsimp [h.compl.regular v]\n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 \u2200 (a : V), a \u2208 neighborFinset G v \u2192 Finset.card (bipartiteAbove G.Adj (neighborFinset G\u1d9c v) a) = k - \u2113 - 1\n[PROOFSTEP]\nintro w hw\n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : w \u2208 neighborFinset G v\n\u22a2 Finset.card (bipartiteAbove G.Adj (neighborFinset G\u1d9c v) w) = k - \u2113 - 1\n[PROOFSTEP]\nrw [mem_neighborFinset] at hw \n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\n\u22a2 Finset.card (bipartiteAbove G.Adj (neighborFinset G\u1d9c v) w) = k - \u2113 - 1\n[PROOFSTEP]\nsimp_rw [bipartiteAbove, show G.Adj w = fun a => G.Adj w a by rfl, \u2190 mem_neighborFinset, filter_mem_eq_inter]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\n\u22a2 Adj G w = fun a => Adj G w a\n[PROOFSTEP]\nrfl\n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\n\u22a2 Finset.card (neighborFinset G\u1d9c v \u2229 neighborFinset G w) = k - \u2113 - 1\n[PROOFSTEP]\nhave s : { v } \u2286 G.neighborFinset w \\ G.neighborFinset v :=\n  by\n  rw [singleton_subset_iff, mem_sdiff, mem_neighborFinset]\n  exact \u27e8hw.symm, G.not_mem_neighborFinset_self v\u27e9\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\n\u22a2 {v} \u2286 neighborFinset G w \\ neighborFinset G v\n[PROOFSTEP]\nrw [singleton_subset_iff, mem_sdiff, mem_neighborFinset]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\n\u22a2 Adj G w v \u2227 \u00acv \u2208 neighborFinset G v\n[PROOFSTEP]\nexact \u27e8hw.symm, G.not_mem_neighborFinset_self v\u27e9\n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 Finset.card (neighborFinset G\u1d9c v \u2229 neighborFinset G w) = k - \u2113 - 1\n[PROOFSTEP]\nrw [inter_comm, neighborFinset_compl, inter_sdiff, \u2190 sdiff_eq_inter_compl, card_sdiff s, card_singleton, \u2190\n  sdiff_inter_self_left, card_sdiff (by apply inter_subset_left)]\n[GOAL]\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 neighborFinset G w \u2229 neighborFinset G v \u2286 neighborFinset G w\n[PROOFSTEP]\napply inter_subset_left\n[GOAL]\ncase intro.convert_3\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 Finset.card (neighborFinset G w) - Finset.card (neighborFinset G w \u2229 neighborFinset G v) - 1 = k - \u2113 - 1\n[PROOFSTEP]\ncongr\n[GOAL]\ncase intro.convert_3.e_a.e_a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 Finset.card (neighborFinset G w) = k\n[PROOFSTEP]\nsimp [h.regular w]\n[GOAL]\ncase intro.convert_3.e_a.e_a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 Finset.card (neighborFinset G w \u2229 neighborFinset G v) = \u2113\n[PROOFSTEP]\nsimp_rw [inter_comm, neighborFinset_def, \u2190 Set.toFinset_inter, \u2190 h.of_adj v w hw, \u2190 Set.toFinset_card]\n[GOAL]\ncase intro.convert_3.e_a.e_a\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : Adj G v w\ns : {v} \u2286 neighborFinset G w \\ neighborFinset G v\n\u22a2 Finset.card (Set.toFinset (neighborSet G v \u2229 neighborSet G w)) = Finset.card (Set.toFinset (commonNeighbors G v w))\n[PROOFSTEP]\ncongr!\n[GOAL]\ncase intro.convert_4\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 \u2200 (b : V), b \u2208 neighborFinset G\u1d9c v \u2192 Finset.card (bipartiteBelow G.Adj (neighborFinset G v) b) = \u03bc\n[PROOFSTEP]\nintro w hw\n[GOAL]\ncase intro.convert_4\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : w \u2208 neighborFinset G\u1d9c v\n\u22a2 Finset.card (bipartiteBelow G.Adj (neighborFinset G v) w) = \u03bc\n[PROOFSTEP]\nsimp_rw [neighborFinset_compl, mem_sdiff, mem_compl, mem_singleton, mem_neighborFinset, \u2190 Ne.def] at hw \n[GOAL]\ncase intro.convert_4\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : \u00acAdj G v w \u2227 w \u2260 v\n\u22a2 Finset.card (bipartiteBelow G.Adj (neighborFinset G v) w) = \u03bc\n[PROOFSTEP]\nsimp_rw [bipartiteBelow, adj_comm, \u2190 mem_neighborFinset, filter_mem_eq_inter, neighborFinset_def, \u2190 Set.toFinset_inter,\n  \u2190 h.of_not_adj v w hw.2.symm hw.1, \u2190 Set.toFinset_card]\n[GOAL]\ncase intro.convert_4\nV : Type u\ninst\u271d\u00b2 : Fintype V\ninst\u271d\u00b9 : DecidableEq V\nG : SimpleGraph V\ninst\u271d : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhw : \u00acAdj G v w \u2227 w \u2260 v\n\u22a2 Finset.card (Set.toFinset (neighborSet G v \u2229 neighborSet G w)) = Finset.card (Set.toFinset (commonNeighbors G v w))\n[PROOFSTEP]\ncongr!\n[GOAL]\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\n\u22a2 adjMatrix \u03b1 G ^ 2 = k \u2022 1 + \u2113 \u2022 adjMatrix \u03b1 G + \u03bc \u2022 adjMatrix \u03b1 G\u1d9c\n[PROOFSTEP]\next v w\n[GOAL]\ncase a.h\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\n\u22a2 (adjMatrix \u03b1 G ^ 2) v w = (k \u2022 1 + \u2113 \u2022 adjMatrix \u03b1 G + \u03bc \u2022 adjMatrix \u03b1 G\u1d9c) v w\n[PROOFSTEP]\nsimp only [adjMatrix_pow_apply_eq_card_walk, Set.coe_setOf, Matrix.add_apply, Matrix.smul_apply, adjMatrix_apply,\n  compl_adj]\n[GOAL]\ncase a.h\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\n\u22a2 \u2191(Fintype.card { x // Walk.length x = 2 }) =\n    (k \u2022 OfNat.ofNat 1 v w + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if v \u2260 w \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nrw [Fintype.card_congr (G.walkLengthTwoEquivCommonNeighbors v w)]\n[GOAL]\ncase a.h\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) =\n    (k \u2022 OfNat.ofNat 1 v w + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if v \u2260 w \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nobtain rfl | hn := eq_or_ne v w\n[GOAL]\ncase a.h.inl\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v v)) =\n    (k \u2022 OfNat.ofNat 1 v v + \u2113 \u2022 if Adj G v v then 1 else 0) + \u03bc \u2022 if v \u2260 v \u2227 \u00acAdj G v v then 1 else 0\n[PROOFSTEP]\nrw [\u2190 Set.toFinset_card]\n[GOAL]\ncase a.h.inl\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv : V\n\u22a2 \u2191(Finset.card (Set.toFinset (commonNeighbors G v v))) =\n    (k \u2022 OfNat.ofNat 1 v v + \u2113 \u2022 if Adj G v v then 1 else 0) + \u03bc \u2022 if v \u2260 v \u2227 \u00acAdj G v v then 1 else 0\n[PROOFSTEP]\nsimp [commonNeighbors, \u2190 neighborFinset_def, h.regular v]\n[GOAL]\ncase a.h.inr\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) =\n    (k \u2022 OfNat.ofNat 1 v w + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if v \u2260 w \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nsimp only [Matrix.one_apply_ne' hn.symm, ne_eq, hn]\n[GOAL]\ncase a.h.inr\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) =\n    (k \u2022 0 + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if True \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nby_cases ha : G.Adj v w\n[GOAL]\ncase pos\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nha : Adj G v w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) =\n    (k \u2022 0 + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if True \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nsimp only [ha, ite_true, ite_false, add_zero, zero_add, nsmul_eq_mul, smul_zero, mul_one]\n[GOAL]\ncase neg\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nha : \u00acAdj G v w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) =\n    (k \u2022 0 + \u2113 \u2022 if Adj G v w then 1 else 0) + \u03bc \u2022 if True \u2227 \u00acAdj G v w then 1 else 0\n[PROOFSTEP]\nsimp only [ha, ite_true, ite_false, add_zero, zero_add, nsmul_eq_mul, smul_zero, mul_one]\n[GOAL]\ncase pos\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nha : Adj G v w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) = \u2191\u2113\n[PROOFSTEP]\nrw [h.of_adj v w ha]\n[GOAL]\ncase neg\nV : Type u\ninst\u271d\u00b3 : Fintype V\ninst\u271d\u00b2 : DecidableEq V\nG : SimpleGraph V\ninst\u271d\u00b9 : DecidableRel G.Adj\nn k \u2113 \u03bc : \u2115\n\u03b1 : Type u_1\ninst\u271d : Semiring \u03b1\nh : IsSRGWith G n k \u2113 \u03bc\nv w : V\nhn : v \u2260 w\nha : \u00acAdj G v w\n\u22a2 \u2191(Fintype.card \u2191(commonNeighbors G v w)) = \u2191\u03bc\n[PROOFSTEP]\nrw [h.of_not_adj v w hn ha]\n", "meta": {"mathlib_filename": "Mathlib.Combinatorics.SimpleGraph.StronglyRegular", "llama_tokens": 12881, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.5030063611749895}}
{"text": "[GOAL]\n\u03b1 : Type u\ninst\u271d : Group \u03b1\nx y : SingleObj \u03b1\nf : x \u27f6 y\n\u22a2 inv f = f\u207b\u00b9\n[PROOFSTEP]\napply IsIso.inv_eq_of_hom_inv_id\n[GOAL]\ncase hom_inv_id\n\u03b1 : Type u\ninst\u271d : Group \u03b1\nx y : SingleObj \u03b1\nf : x \u27f6 y\n\u22a2 f \u226b f\u207b\u00b9 = \ud835\udfd9 x\n[PROOFSTEP]\nrw [comp_as_mul, inv_mul_self, id_as_one]\n[GOAL]\n\u03b1\u271d \u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\n\u22a2 Function.LeftInverse\n    (fun f =>\n      {\n        toOneHom :=\n          { toFun := fun x => f.map (\u2191(toEnd \u03b1) x), map_one' := (_ : f.map (\ud835\udfd9 (star \u03b1)) = \ud835\udfd9 (f.obj (star \u03b1))) },\n        map_mul' := (_ : \u2200 (x y : \u03b1), f.map (y \u226b x) = f.map y \u226b f.map x) })\n    fun f => Functor.mk { obj := id, map := fun {X Y} => \u2191f }\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b1\u271d \u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\n\u22a2 Function.RightInverse\n    (fun f =>\n      {\n        toOneHom :=\n          { toFun := fun x => f.map (\u2191(toEnd \u03b1) x), map_one' := (_ : f.map (\ud835\udfd9 (star \u03b1)) = \ud835\udfd9 (f.obj (star \u03b1))) },\n        map_mul' := (_ : \u2200 (x y : \u03b1), f.map (y \u226b x) = f.map y \u226b f.map x) })\n    fun f => Functor.mk { obj := id, map := fun {X Y} => \u2191f }\n[PROOFSTEP]\naesop_cat\n[GOAL]\n\u03b1 : Type u\nC : Type ?u.20048\nG : Type ?u.20052\ninst\u271d\u00b9 : Category.{?u.20049, ?u.20048} C\ninst\u271d : Group G\nf : C \u2192 G\n\u22a2 \u2200 (X : C),\n    { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map (\ud835\udfd9 X) =\n      \ud835\udfd9 ({ obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.obj X)\n[PROOFSTEP]\nintro\n[GOAL]\n\u03b1 : Type u\nC : Type ?u.20048\nG : Type ?u.20052\ninst\u271d\u00b9 : Category.{?u.20049, ?u.20048} C\ninst\u271d : Group G\nf : C \u2192 G\nX\u271d : C\n\u22a2 { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map (\ud835\udfd9 X\u271d) =\n    \ud835\udfd9 ({ obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.obj X\u271d)\n[PROOFSTEP]\nsimp only [SingleObj.id_as_one, mul_right_inv]\n[GOAL]\n\u03b1 : Type u\nC : Type ?u.20048\nG : Type ?u.20052\ninst\u271d\u00b9 : Category.{?u.20049, ?u.20048} C\ninst\u271d : Group G\nf : C \u2192 G\n\u22a2 \u2200 {X Y Z : C} (f_1 : X \u27f6 Y) (g : Y \u27f6 Z),\n    { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map (f_1 \u226b g) =\n      { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map f_1 \u226b\n        { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map g\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u\nC : Type ?u.20048\nG : Type ?u.20052\ninst\u271d\u00b9 : Category.{?u.20049, ?u.20048} C\ninst\u271d : Group G\nf : C \u2192 G\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map (f\u271d \u226b g\u271d) =\n    { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map f\u271d \u226b\n      { obj := fun x => (), map := fun {x y} x_1 => f y * (f x)\u207b\u00b9 }.map g\u271d\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b1 : Type u\nC : Type ?u.20048\nG : Type ?u.20052\ninst\u271d\u00b9 : Category.{?u.20049, ?u.20048} C\ninst\u271d : Group G\nf : C \u2192 G\nX\u271d Y\u271d Z\u271d : C\nf\u271d : X\u271d \u27f6 Y\u271d\ng\u271d : Y\u271d \u27f6 Z\u271d\n\u22a2 f Z\u271d * (f X\u271d)\u207b\u00b9 = (f Y\u271d * (f X\u271d)\u207b\u00b9) \u226b (f Z\u271d * (f Y\u271d)\u207b\u00b9)\n[PROOFSTEP]\nrw [SingleObj.comp_as_mul, \u2190 mul_assoc, mul_left_inj, mul_assoc, inv_mul_self, mul_one]\n[GOAL]\n\u03b1 : Type u\ninst\u271d : Monoid \u03b1\n\u22a2 \u03b1 \u2243* End (SingleObj.star \u03b1)\n[PROOFSTEP]\nexact SingleObj.toEnd \u03b1\n[GOAL]\nX\u271d Y\u271d : MonCat\na\u2081\u271d a\u2082\u271d : X\u271d \u27f6 Y\u271d\nh : toCat.map a\u2081\u271d = toCat.map a\u2082\u271d\n\u22a2 a\u2081\u271d = a\u2082\u271d\n[PROOFSTEP]\nsimpa [toCat] using h\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.SingleObj", "llama_tokens": 1752, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.808067204308405, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.5029889715599621}}
{"text": "[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2264 snorm (\u2191\u2191f) 2 \u03bc\n[PROOFSTEP]\nrw [lpMeas_coe, \u2190 ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), \u2190 Lp.norm_def, \u2190 Lp.norm_def,\n  Submodule.norm_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 \u2016\u2191(condexpL2 E \ud835\udd5c hm) f\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nexact norm_condexpL2_le hm f\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 \u2016\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)\u2016 \u2264 \u2016f\u2016\n[PROOFSTEP]\nrw [Lp.norm_def, Lp.norm_def, \u2190 lpMeas_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 ENNReal.toReal (snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc) \u2264 ENNReal.toReal (snorm (\u2191\u2191f) 2 \u03bc)\n[PROOFSTEP]\nrefine' (ENNReal.toReal_le_toReal _ (Lp.snorm_ne_top _)).mpr (snorm_condexpL2_le hm f)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\n\u22a2 snorm (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) 2 \u03bc \u2260 \u22a4\n[PROOFSTEP]\nexact Lp.snorm_ne_top _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\n\u22a2 \u2191(\u2191(condexpL2 E \ud835\udd5c hm) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nrw [condexpL2]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nhaveI : Fact (m \u2264 m0) := \u27e8hm\u27e9\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nhave h_mem : indicatorConstLp 2 (hm s hs) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc := mem_lpMeas_indicatorConstLp hm hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nlet ind := (\u27e8indicatorConstLp 2 (hm s hs) h\u03bcs c, h_mem\u27e9 : lpMeas E \ud835\udd5c m 2 \u03bc)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nhave h_coe_ind : (ind : \u03b1 \u2192\u2082[\u03bc] E) = indicatorConstLp 2 (hm s hs) h\u03bcs c := by rfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\n\u22a2 \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nhave h_orth_mem := orthogonalProjection_mem_subspace_eq_self ind\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E\nthis : Fact (m \u2264 m0)\nh_mem : indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c \u2208 lpMeas E \ud835\udd5c m 2 \u03bc\nind : { x // x \u2208 lpMeas E \ud835\udd5c m 2 \u03bc } := { val := indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c, property := h_mem }\nh_coe_ind : \u2191ind = indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\nh_orth_mem : \u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) \u2191ind = ind\n\u22a2 \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c)) =\n    indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs c\n[PROOFSTEP]\nrw [\u2190 h_coe_ind, h_orth_mem]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 Lp E 2 }\nhg : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 inner (\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) g = inner f g\n[PROOFSTEP]\nsymm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 Lp E 2 }\nhg : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 inner f g = inner (\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f)) g\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, \u2190 inner_sub_left, condexpL2]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf g : { x // x \u2208 Lp E 2 }\nhg : AEStronglyMeasurable' m (\u2191\u2191g) \u03bc\n\u22a2 inner (f - \u2191(\u2191(orthogonalProjection (lpMeas E \ud835\udd5c m 2 \u03bc)) f)) g = 0\n[PROOFSTEP]\nsimp only [mem_lpMeas_iff_aeStronglyMeasurable'.mpr hg, orthogonalProjection_inner_eq_zero f g]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 L2.inner_indicatorConstLp_one (\ud835\udd5c := \ud835\udd5c) (hm s hs) h\u03bcs f]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f) x \u2202\u03bc = inner (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs 1) f\n[PROOFSTEP]\nhave h_eq_inner :\n  \u222b x in s, (condexpL2 \ud835\udd5c \ud835\udd5c hm f : \u03b1 \u2192 \ud835\udd5c) x \u2202\u03bc = inner (indicatorConstLp 2 (hm s hs) h\u03bcs (1 : \ud835\udd5c)) (condexpL2 \ud835\udd5c \ud835\udd5c hm f) :=\n  by rw [L2.inner_indicatorConstLp_one (hm s hs) h\u03bcs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f) x \u2202\u03bc =\n    inner (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs 1) \u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f)\n[PROOFSTEP]\nrw [L2.inner_indicatorConstLp_one (hm s hs) h\u03bcs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp \ud835\udd5c 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh_eq_inner :\n  \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f) x \u2202\u03bc =\n    inner (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs 1) \u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f)\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) f) x \u2202\u03bc = inner (indicatorConstLp 2 (_ : MeasurableSet s) h\u03bcs 1) f\n[PROOFSTEP]\nrw [h_eq_inner, \u2190 inner_condexpL2_left_eq_right, condexpL2_indicator_of_measurable hm hs h\u03bcs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nlet h_meas := lpMeas.aeStronglyMeasurable' (condexpL2 \u211d \u211d hm f)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nlet g := h_meas.choose\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nhave hg_meas : StronglyMeasurable[m] g := h_meas.choose_spec.1\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nhave hg_eq : g =\u1d50[\u03bc] condexpL2 \u211d \u211d hm f := h_meas.choose_spec.2.symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nhave hg_eq_restrict : g =\u1d50[\u03bc.restrict s] condexpL2 \u211d \u211d hm f := ae_restrict_of_ae hg_eq\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nhave hg_nnnorm_eq : (fun x => (\u2016g x\u2016\u208a : \u211d\u22650\u221e)) =\u1d50[\u03bc.restrict s] fun x => (\u2016(condexpL2 \u211d \u211d hm f : \u03b1 \u2192 \u211d) x\u2016\u208a : \u211d\u22650\u221e) :=\n  by\n  refine' hg_eq_restrict.mono fun x hx => _\n  dsimp only\n  simp_rw [hx]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\nrefine' hg_eq_restrict.mono fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) x = (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n\u22a2 \u2191\u2016Exists.choose (_ : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc) x\u2016\u208a = \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\nsimp_rw [hx]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nrw [lintegral_congr_ae hg_nnnorm_eq.symm]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1) in s, \u2191\u2016g a\u2016\u208a \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_nnnorm_le_of_forall_fin_meas_integral_eq hm (Lp.stronglyMeasurable f) _ _ _ _ hs h\u03bcs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun x => \u2191\u2191f x) s\n[PROOFSTEP]\nexact integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 StronglyMeasurable fun a => g a\n[PROOFSTEP]\nexact hg_meas\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (fun a => g a) s\n[PROOFSTEP]\nrw [IntegrableOn, integrable_congr hg_eq_restrict]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\n[PROOFSTEP]\nexact integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs f\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \u2191\u2191\u03bc t < \u22a4 \u2192 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191f x \u2202\u03bc\n[PROOFSTEP]\nintro t ht h\u03bct\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 integral_condexpL2_eq_of_fin_meas_real f ht h\u03bct.ne]\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d ?m.662736) f) x \u2202\u03bc\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' (\u2191(condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1d50[Measure.restrict \u03bc s] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t < \u22a4\n\u22a2 m \u2264 m0\n[PROOFSTEP]\nexact set_integral_congr_ae (hm t ht) (hg_eq.mono fun x hx _ => hx)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) =\u1d50[Measure.restrict \u03bc s] 0\n[PROOFSTEP]\nsuffices h_nnnorm_eq_zero : \u222b\u207b x in s, \u2016(condexpL2 \u211d \u211d hm f : \u03b1 \u2192 \u211d) x\u2016\u208a \u2202\u03bc = 0\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) =\u1d50[Measure.restrict \u03bc s] 0\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff] at h_nnnorm_eq_zero \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) =\u1d50[Measure.restrict \u03bc s] 0\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\nrefine' h_nnnorm_eq_zero.mono fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) x = OfNat.ofNat 0 x\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x = OfNat.ofNat 0 x\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\ndsimp only at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a = OfNat.ofNat 0 x\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x = OfNat.ofNat 0 x\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\nrw [Pi.zero_apply] at hx \u22a2\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : (fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a = 0\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x = 0\n[PROOFSTEP]\nrwa [ENNReal.coe_eq_zero, nnnorm_eq_zero] at hx \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n[PROOFSTEP]\nrefine' Measurable.coe_nnreal_ennreal (Measurable.nnnorm _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n[PROOFSTEP]\nrw [lpMeas_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nh_nnnorm_eq_zero : \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n\u22a2 Measurable fun x => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).measurable\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc = 0\n[PROOFSTEP]\nrefine' le_antisymm _ (zero_le _)\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) f) x\u2016\u208a \u2202\u03bc \u2264 0\n[PROOFSTEP]\nrefine' (lintegral_nnnorm_condexpL2_le hs h\u03bcs f).trans (le_of_eq _)\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\u2191\u2191f x\u2016\u208a \u2202\u03bc = 0\n[PROOFSTEP]\nrw [lintegral_eq_zero_iff]\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 (fun x => \u2191\u2016\u2191\u2191f x\u2016\u208a) =\u1d50[Measure.restrict \u03bc s] 0\n[PROOFSTEP]\nrefine' hf.mono fun x hx => _\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : \u2191\u2191f x = OfNat.ofNat 0 x\n\u22a2 (fun x => \u2191\u2016\u2191\u2191f x\u2016\u208a) x = OfNat.ofNat 0 x\n[PROOFSTEP]\ndsimp only\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : \u2191\u2191f x = OfNat.ofNat 0 x\n\u22a2 \u2191\u2016\u2191\u2191f x\u2016\u208a = OfNat.ofNat 0 x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\nx : \u03b1\nhx : \u2191\u2191f x = OfNat.ofNat 0 x\n\u22a2 \u2191\u2016OfNat.ofNat 0 x\u2016\u208a = OfNat.ofNat 0 x\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h_nnnorm_eq_zero\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nf : { x // x \u2208 Lp \u211d 2 }\nhf : \u2191\u2191f =\u1d50[Measure.restrict \u03bc s] 0\n\u22a2 Measurable fun x => \u2191\u2016\u2191\u2191f x\u2016\u208a\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).ennnorm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrefine' (lintegral_nnnorm_condexpL2_le ht h\u03bct _).trans (le_of_eq _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in t, \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a \u2202\u03bc = \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nhave h_eq :\n  \u222b\u207b x in t, \u2016(indicatorConstLp 2 hs h\u03bcs (1 : \u211d)) x\u2016\u208a \u2202\u03bc = \u222b\u207b x in t, s.indicator (fun _ => (1 : \u211d\u22650\u221e)) x \u2202\u03bc :=\n  by\n  refine' lintegral_congr_ae (ae_restrict_of_ae _)\n  refine' (@indicatorConstLp_coeFn _ _ _ 2 _ _ _ hs h\u03bcs (1 : \u211d)).mono fun x hx => _\n  dsimp only\n  rw [hx]\n  classical\n  simp_rw [Set.indicator_apply]\n  split_ifs <;> simp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (x : \u03b1) in t, \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, Set.indicator s (fun x => 1) x \u2202\u03bc\n[PROOFSTEP]\nrefine' lintegral_congr_ae (ae_restrict_of_ae _)\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, (fun x => \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a) x = (fun x => Set.indicator s (fun x => 1) x) x\n[PROOFSTEP]\nrefine' (@indicatorConstLp_coeFn _ _ _ 2 _ _ _ hs h\u03bcs (1 : \u211d)).mono fun x hx => _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\n\u22a2 (fun x => \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a) x = (fun x => Set.indicator s (fun x => 1) x) x\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\n\u22a2 \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a = Set.indicator s (fun x => 1) x\n[PROOFSTEP]\nrw [hx]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\n\u22a2 \u2191\u2016Set.indicator s (fun x => 1) x\u2016\u208a = Set.indicator s (fun x => 1) x\n[PROOFSTEP]\nclassical\nsimp_rw [Set.indicator_apply]\nsplit_ifs <;> simp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\n\u22a2 \u2191\u2016Set.indicator s (fun x => 1) x\u2016\u208a = Set.indicator s (fun x => 1) x\n[PROOFSTEP]\nsimp_rw [Set.indicator_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\n\u22a2 \u2191\u2016if x \u2208 s then 1 else 0\u2016\u208a = if x \u2208 s then 1 else 0\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\nh\u271d : x \u2208 s\n\u22a2 \u2191\u20161\u2016\u208a = 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : \u03b1\nhx : \u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x = Set.indicator s (fun x => 1) x\nh\u271d : \u00acx \u2208 s\n\u22a2 \u2191\u20160\u2016\u208a = 0\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_eq : \u222b\u207b (x : \u03b1) in t, \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, Set.indicator s (fun x => 1) x \u2202\u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in t, \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a \u2202\u03bc = \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nrw [h_eq, lintegral_indicator _ hs, lintegral_const, Measure.restrict_restrict hs]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nh_eq : \u222b\u207b (x : \u03b1) in t, \u2191\u2016\u2191\u2191(indicatorConstLp 2 hs h\u03bcs 1) x\u2016\u208a \u2202\u03bc = \u222b\u207b (x : \u03b1) in t, Set.indicator s (fun x => 1) x \u2202\u03bc\n\u22a2 1 * \u2191\u2191(Measure.restrict \u03bc (s \u2229 t)) Set.univ = \u2191\u2191\u03bc (s \u2229 t)\n[PROOFSTEP]\nsimp only [one_mul, Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n[PROOFSTEP]\nrw [lpMeas_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n[PROOFSTEP]\nhave h_mem_Lp : Mem\u2112p (fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb) 2 \u03bc := by refine' Mem\u2112p.const_inner _ _;\n  rw [lpMeas_coe]; exact Lp.mem\u2112p _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\n[PROOFSTEP]\nrefine' Mem\u2112p.const_inner _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => \u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) 2\n[PROOFSTEP]\nrw [lpMeas_coe]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\n\u22a2 Mem\u2112p (fun a => \u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a) 2\n[PROOFSTEP]\nexact Lp.mem\u2112p _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n[PROOFSTEP]\nhave h_eq : h_mem_Lp.toLp _ =\u1d50[\u03bc] fun a => \u27eac, (condexpL2 E \ud835\udd5c hm f : \u03b1 \u2192 E) a\u27eb := h_mem_Lp.coeFn_toLp\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ h_eq\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) =\u1d50[\u03bc]\n    \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)\n[PROOFSTEP]\nrefine'\n  Lp.ae_eq_of_forall_set_integral_eq' \ud835\udd5c hm _ _ two_ne_zero ENNReal.coe_ne_top\n    (fun s _ h\u03bcs => integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _) _ _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s\n[PROOFSTEP]\nintro s _ h\u03bcs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 IntegrableOn (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) s\n[PROOFSTEP]\nrw [IntegrableOn, integrable_congr (ae_restrict_of_ae h_eq)]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\na\u271d : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 Integrable fun x => (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) x\n[PROOFSTEP]\nexact (integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _).const_inner _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s,\n            \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))\n              x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe, integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs.ne, integral_congr_ae (ae_restrict_of_ae h_eq),\n  lpMeas_coe, \u2190 L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c (\u2191(condexpL2 E \ud835\udd5c hm f)) (hm s hs) c h\u03bcs.ne, \u2190\n  inner_condexpL2_left_eq_right, condexpL2_indicator_of_measurable _ hs,\n  L2.inner_indicatorConstLp_eq_set_integral_inner \ud835\udd5c f (hm s hs) c h\u03bcs.ne,\n  set_integral_congr_ae (hm s hs) ((Mem\u2112p.coeFn_toLp ((Lp.mem\u2112p f).const_inner c)).mono fun x hx _ => hx)]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))) \u03bc\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe]\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)))) \u03bc\n[PROOFSTEP]\nexact lpMeas.aeStronglyMeasurable' _\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp)) \u03bc\n[PROOFSTEP]\nrefine' AEStronglyMeasurable'.congr _ h_eq.symm\n[GOAL]\ncase refine'_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E 2 }\nc : E\nh_mem_Lp : Mem\u2112p (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) 2\nh_eq :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) h_mem_Lp) =\u1d50[\u03bc] fun a =>\n    inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => inner c (\u2191\u2191\u2191(\u2191(condexpL2 E \ud835\udd5c hm) f) a)) \u03bc\n[PROOFSTEP]\nexact (lpMeas.aeStronglyMeasurable' _).const_inner _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x \u2202\u03bc = \u222b (x : \u03b1) in s, \u2191\u2191f x \u2202\u03bc\n[PROOFSTEP]\nrw [\u2190 sub_eq_zero, lpMeas_coe, \u2190\n  integral_sub' (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)\n    (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs)]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u222b (a : \u03b1) in s, (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a \u2202\u03bc = 0\n[PROOFSTEP]\nrefine' integral_eq_zero_of_forall_integral_inner_eq_zero \ud835\udd5c _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun a => (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) a\n[PROOFSTEP]\nrw [integrable_congr (ae_restrict_of_ae (Lp.coeFn_sub (\u2191(condexpL2 E' \ud835\udd5c hm f)) f).symm)]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 Integrable fun x => \u2191\u2191(\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - f) x\n[PROOFSTEP]\nexact integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\n\u22a2 \u2200 (c : E'), \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0\n[PROOFSTEP]\nintro c\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c ((\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) - \u2191\u2191f) x) \u2202\u03bc = 0\n[PROOFSTEP]\nsimp_rw [Pi.sub_apply, inner_sub_right]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (x : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) x) - inner c (\u2191\u2191f x) \u2202\u03bc = 0\n[PROOFSTEP]\nrw [integral_sub ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)\n    ((integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs).const_inner c)]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0\n[PROOFSTEP]\nhave h_ae_eq_f := Mem\u2112p.coeFn_toLp (E := \ud835\udd5c) ((Lp.mem\u2112p f).const_inner c)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (a : \u03b1) in s, inner c (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a) \u2202\u03bc - \u222b (a : \u03b1) in s, inner c (\u2191\u2191f a) \u2202\u03bc = 0\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe, sub_eq_zero, \u2190 set_integral_congr_ae (hm s hs) ((condexpL2_const_inner hm f c).mono fun x hx _ => hx),\n  \u2190 set_integral_congr_ae (hm s hs) (h_ae_eq_f.mono fun x hx _ => hx)]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b9 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2070 : CompleteSpace E\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u2077 : CompleteSpace E'\ninst\u271d\u2076 : NormedSpace \u211d E'\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9 : NormedSpace \u211d G'\ninst\u271d : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nhm : m \u2264 m0\nf : { x // x \u2208 Lp E' 2 }\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : E'\nh_ae_eq_f :\n  \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) =\u1d50[\u03bc] fun a => inner c (\u2191\u2191f a)\n\u22a2 \u222b (x : \u03b1) in s,\n      \u2191\u2191\u2191(\u2191(condexpL2 \ud835\udd5c \ud835\udd5c hm) (Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2))) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(Mem\u2112p.toLp (fun a => inner c (\u2191\u2191f a)) (_ : Mem\u2112p (fun a => inner c (\u2191\u2191f a)) 2)) x \u2202\u03bc\n[PROOFSTEP]\nexact integral_condexpL2_eq_of_fin_meas_real _ hs h\u03bcs\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) =\u1d50[\u03bc] \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))\n[PROOFSTEP]\nrefine'\n  Lp.ae_eq_of_forall_set_integral_eq' \ud835\udd5c' hm _ _ two_ne_zero ENNReal.coe_ne_top\n    (fun s _ h\u03bcs => integrableOn_condexpL2_of_measure_ne_top hm h\u03bcs.ne _)\n    (fun s _ h\u03bcs => integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs.ne) _ _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 \u2200 (s : Set \u03b1),\n    MeasurableSet s \u2192\n      \u2191\u2191\u03bc s < \u22a4 \u2192\n        \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n          \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc\n[PROOFSTEP]\nintro s hs h\u03bcs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns\u271d t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s < \u22a4\n\u22a2 \u222b (x : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f)) x \u2202\u03bc =\n    \u222b (x : \u03b1) in s, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) x \u2202\u03bc\n[PROOFSTEP]\nrw [T.set_integral_compLp _ (hm s hs),\n  T.integral_comp_comm (integrableOn_Lp_of_measure_ne_top _ fact_one_le_two_ennreal.elim h\u03bcs.ne), \u2190 lpMeas_coe, \u2190\n  lpMeas_coe, integral_condexpL2_eq hm f hs h\u03bcs.ne, integral_condexpL2_eq hm (T.compLp f) hs h\u03bcs.ne,\n  T.set_integral_compLp _ (hm s hs),\n  T.integral_comp_comm (integrableOn_Lp_of_measure_ne_top f fact_one_le_two_ennreal.elim h\u03bcs.ne)]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f))) \u03bc\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 E'' \ud835\udd5c' hm) (compLp T f))) \u03bc\n[PROOFSTEP]\nexact lpMeas.aeStronglyMeasurable' _\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc\n[PROOFSTEP]\nhave h_coe := T.coeFn_compLp (condexpL2 E' \ud835\udd5c hm f : \u03b1 \u2192\u2082[\u03bc] E')\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a = \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc\n[PROOFSTEP]\nrw [\u2190 EventuallyEq] at h_coe \n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f))) \u03bc\n[PROOFSTEP]\nrefine' AEStronglyMeasurable'.congr _ h_coe.symm\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nT : E' \u2192L[\u211d] E''\nf : { x // x \u2208 Lp E' 2 }\nh_coe : (fun a => \u2191\u2191(compLp T \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f)) a) =\u1d50[\u03bc] fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)\n\u22a2 AEStronglyMeasurable' m (fun a => \u2191T (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) f) a)) \u03bc\n[PROOFSTEP]\nexact (lpMeas.aeStronglyMeasurable' (condexpL2 E' \ud835\udd5c hm f)).continuous_comp T.continuous\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nrw [indicatorConstLp_eq_toSpanSingleton_compLp hs h\u03bcs x]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nhave h_comp := condexpL2_comp_continuousLinearMap \u211d \ud835\udd5c hm (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs (1 : \u211d))\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe] at h_comp \n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nrefine' h_comp.trans _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (compLp (toSpanSingleton \u211d x) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n\u22a2 \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) =\u1d50[\u03bc] fun a =>\n    \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nexact (toSpanSingleton \u211d x).coeFn_compLp _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\n    compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\next1\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nrw [\u2190 lpMeas_coe]\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nrefine' (condexpL2_indicator_ae_eq_smul \ud835\udd5c hm hs h\u03bcs x).trans _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nhave h_comp := (toSpanSingleton \u211d x).coeFn_compLp (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192\u2082[\u03bc] \u211d)\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  \u2200\u1d50 (a : \u03b1) \u2202\u03bc,\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) a =\n      \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\n\u22a2 (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nrw [\u2190 EventuallyEq] at h_comp \n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  (fun a => \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) a) =\u1d50[\u03bc] fun a =>\n    \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\n\u22a2 (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) =\u1d50[\u03bc]\n    \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ h_comp.symm\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  (fun a => \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) a) =\u1d50[\u03bc] fun a =>\n    \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\n\u22a2 (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) =\u1d50[\u03bc] fun a =>\n    \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\n[PROOFSTEP]\nrefine' eventually_of_forall fun y => _\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nh_comp :\n  (fun a => \u2191\u2191(compLp (toSpanSingleton \u211d x) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) a) =\u1d50[\u03bc] fun a =>\n    \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\ny : \u03b1\n\u22a2 (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) y =\n    (fun a => \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)) y\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\na : \u03b1\nha :\n  \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) a =\n    (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) a\nx\u271d : a \u2208 t\n\u22a2 \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) a\u2016\u208a =\n    \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\u2016\u208a\n[PROOFSTEP]\nrw [ha]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\u2016\u208a \u2202\u03bc =\n    (\u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a \u2202\u03bc) * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_smul, ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a * \u2191\u2016x\u2016\u208a \u2202\u03bc =\n    (\u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a \u2202\u03bc) * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrw [lintegral_mul_const, lpMeas_coe]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2075 : CompleteSpace E\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b3 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b2 : CompleteSpace E'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \ud835\udd5c F\ninst\u271d\u2078 : NormedAddCommGroup G\ninst\u271d\u2077 : NormedAddCommGroup G'\ninst\u271d\u2076 : NormedSpace \u211d G'\ninst\u271d\u2075 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2074 : IsROrC \ud835\udd5c'\ninst\u271d\u00b3 : NormedAddCommGroup E''\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b9 : CompleteSpace E''\ninst\u271d : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Measurable fun a => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).ennnorm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' lintegral_le_of_forall_fin_meas_le' hm (\u03bc s * \u2016x\u2016\u208a) _ fun t ht h\u03bct => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 AEMeasurable fun a => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) a\u2016\u208a\n[PROOFSTEP]\nrw [lpMeas_coe]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 AEMeasurable fun a => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) a\u2016\u208a\n[PROOFSTEP]\nexact (Lp.aestronglyMeasurable _).ennnorm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' (set_lintegral_nnnorm_condexpL2_indicator_le hm hs h\u03bcs x ht h\u03bct).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) * \u2191\u2016x\u2016\u208a \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nexact mul_le_mul_right' (measure_mono (Set.inter_subset_left _ _)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x))\n[PROOFSTEP]\nrefine' integrable_of_forall_fin_meas_le' hm (\u03bc s * \u2016x\u2016\u208a) (ENNReal.mul_lt_top h\u03bcs ENNReal.coe_ne_top) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 AEStronglyMeasurable (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x))) \u03bc\n[PROOFSTEP]\nrw [lpMeas_coe]\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 AEStronglyMeasurable (\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x))) \u03bc\n[PROOFSTEP]\nexact Lp.aestronglyMeasurable _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191\u03bc s_1 \u2260 \u22a4 \u2192\n        \u222b\u207b (x_1 : \u03b1) in s_1, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 E' \ud835\udd5c hm) (indicatorConstLp 2 hs h\u03bcs x)) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' fun t ht h\u03bct => (set_lintegral_nnnorm_condexpL2_indicator_le hm hs h\u03bcs x ht h\u03bct).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\nhm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E'\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) * \u2191\u2016x\u2016\u208a \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nexact mul_le_mul_right' (measure_mono (Set.inter_subset_left _ _)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) \u03bc\n[PROOFSTEP]\nhave h : AEStronglyMeasurable' m (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) \u03bc :=\n  aeStronglyMeasurable'_condexpL2 _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) \u03bc\n[PROOFSTEP]\nrw [condexpIndSMul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))) \u03bc\n[PROOFSTEP]\nsuffices AEStronglyMeasurable' m (toSpanSingleton \u211d x \u2218 condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1)) \u03bc\n  by\n  refine' AEStronglyMeasurable'.congr this _\n  refine' EventuallyEq.trans _ (coeFn_compLpL _ _).symm\n  rfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nthis : AEStronglyMeasurable' m (\u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 AEStronglyMeasurable' m\n    (\u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))) \u03bc\n[PROOFSTEP]\nrefine' AEStronglyMeasurable'.congr this _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nthis : AEStronglyMeasurable' m (\u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\u1d50[\u03bc]\n    \u2191\u2191(\u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)))\n[PROOFSTEP]\nrefine' EventuallyEq.trans _ (coeFn_compLpL _ _).symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nthis : AEStronglyMeasurable' m (\u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\u1d50[\u03bc] fun a =>\n    \u2191(toSpanSingleton \u211d x) (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191(toSpanSingleton \u211d x) \u2218 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n[PROOFSTEP]\nexact AEStronglyMeasurable'.continuous_comp (toSpanSingleton \u211d x).continuous h\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 condexpIndSMul hm hs h\u03bcs (x + y) = condexpIndSMul hm hs h\u03bcs x + condexpIndSMul hm hs h\u03bcs y\n[PROOFSTEP]\nsimp_rw [condexpIndSMul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx y : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (x + y))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) +\n      \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d y)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\nrw [toSpanSingleton_add, add_compLpL, add_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 condexpIndSMul hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndSMul hm hs h\u03bcs x\n[PROOFSTEP]\nsimp_rw [condexpIndSMul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \u211d\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d (c \u2022 x))) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) =\n    c \u2022 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\nrw [toSpanSingleton_smul, smul_compLpL, smul_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b9 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u2070 : NormedAddCommGroup E\ninst\u271d\u00b9\u2079 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2078 : CompleteSpace E\ninst\u271d\u00b9\u2077 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2076 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2075 : CompleteSpace E'\ninst\u271d\u00b9\u2074 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup G\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G'\ninst\u271d\u2079 : NormedSpace \u211d G'\ninst\u271d\u2078 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2077 : IsROrC \ud835\udd5c'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2074 : CompleteSpace E''\ninst\u271d\u00b3 : NormedSpace \u211d E''\ninst\u271d\u00b2 : NormedSpace \u211d G\nhm : m \u2264 m0\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : SMulCommClass \u211d \ud835\udd5c F\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nc : \ud835\udd5c\nx : F\n\u22a2 condexpIndSMul hm hs h\u03bcs (c \u2022 x) = c \u2022 condexpIndSMul hm hs h\u03bcs x\n[PROOFSTEP]\nrw [condexpIndSMul, condexpIndSMul, toSpanSingleton_smul', (toSpanSingleton \u211d x).smul_compLpL c, smul_apply]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\na : \u03b1\nha : \u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a = (fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x) a\nx\u271d : a \u2208 t\n\u22a2 \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a = \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\u2016\u208a\n[PROOFSTEP]\nrw [ha]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\u2016\u208a \u2202\u03bc =\n    (\u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a \u2202\u03bc) * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nsimp_rw [nnnorm_smul, ENNReal.coe_mul]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a * \u2191\u2016x\u2016\u208a \u2202\u03bc =\n    (\u222b\u207b (a : \u03b1) in t, \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a \u2202\u03bc) * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrw [lintegral_mul_const, lpMeas_coe]\n[GOAL]\ncase hf\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 Measurable fun a => \u2191\u2016\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\u2016\u208a\n[PROOFSTEP]\nexact (Lp.stronglyMeasurable _).ennnorm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' lintegral_le_of_forall_fin_meas_le' hm (\u03bc s * \u2016x\u2016\u208a) _ fun t ht h\u03bct => _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 AEMeasurable fun a => \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) a\u2016\u208a\n[PROOFSTEP]\nexact (Lp.aestronglyMeasurable _).ennnorm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u222b\u207b (x_1 : \u03b1) in t, \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' (set_lintegral_nnnorm_condexpIndSMul_le hm hs h\u03bcs x ht h\u03bct).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) * \u2191\u2016x\u2016\u208a \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nexact mul_le_mul_right' (measure_mono (Set.inter_subset_left _ _)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\n\u22a2 Integrable \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\n[PROOFSTEP]\nrefine' integrable_of_forall_fin_meas_le' hm (\u03bc s * \u2016x\u2016\u208a) (ENNReal.mul_lt_top h\u03bcs ENNReal.coe_ne_top) _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\n\u22a2 AEStronglyMeasurable (\u2191\u2191(condexpIndSMul hm hs h\u03bcs x)) \u03bc\n[PROOFSTEP]\nexact Lp.aestronglyMeasurable _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 \u2260 \u22a4 \u2192 \u222b\u207b (x_1 : \u03b1) in s_1, \u2191\u2016\u2191\u2191(condexpIndSMul hm hs h\u03bcs x) x_1\u2016\u208a \u2202\u03bc \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nrefine' fun t ht h\u03bct => (set_lintegral_nnnorm_condexpIndSMul_le hm hs h\u03bcs x ht h\u03bct).trans _\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : G\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 \u2191\u2191\u03bc (s \u2229 t) * \u2191\u2016x\u2016\u208a \u2264 \u2191\u2191\u03bc s * \u2191\u2016x\u2016\u208a\n[PROOFSTEP]\nexact mul_le_mul_right' (measure_mono (Set.inter_subset_left _ _)) _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nx : G\n\u22a2 condexpIndSMul hm (_ : MeasurableSet \u2205) (_ : \u2191\u2191\u03bc \u2205 \u2260 \u22a4) x = 0\n[PROOFSTEP]\nrw [condexpIndSMul, indicatorConstLp_empty]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nx : G\n\u22a2 \u2191(compLpL 2 \u03bc (toSpanSingleton \u211d x)) \u2191(\u2191(condexpL2 \u211d \u211d hm) 0) = 0\n[PROOFSTEP]\nsimp only [Submodule.coe_zero, ContinuousLinearMap.map_zero]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\n\u22a2 ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s)) \u2022 1 = ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s))\n[PROOFSTEP]\nrw [smul_eq_mul, mul_one]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2078 : NormedAddCommGroup E\ninst\u271d\u00b9\u2077 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2076 : CompleteSpace E\ninst\u271d\u00b9\u2075 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2074 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u00b3 : CompleteSpace E'\ninst\u271d\u00b9\u00b2 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedAddCommGroup G'\ninst\u271d\u2077 : NormedSpace \u211d G'\ninst\u271d\u2076 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2075 : IsROrC \ud835\udd5c'\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b2 : CompleteSpace E''\ninst\u271d\u00b9 : NormedSpace \u211d E''\ninst\u271d : NormedSpace \u211d G\nhm : m \u2264 m0\nhs : MeasurableSet s\nht : MeasurableSet t\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nh\u03bct : \u2191\u2191\u03bc t \u2260 \u22a4\nx : G'\n\u22a2 (\u222b (a : \u03b1) in s, \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 ht h\u03bct 1)) a \u2202\u03bc) \u2022 x = ENNReal.toReal (\u2191\u2191\u03bc (t \u2229 s)) \u2022 x\n[PROOFSTEP]\nrw [set_integral_condexpL2_indicator hs ht h\u03bcs h\u03bct]\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\nhave h : AEStronglyMeasurable' m (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) \u03bc :=\n  aeStronglyMeasurable'_condexpL2 _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\nrefine' EventuallyLE.trans_eq _ h.ae_eq_mk.symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[\u03bc] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h\n[PROOFSTEP]\nrefine' @ae_le_of_ae_le_trim _ _ _ _ _ _ hm (0 : \u03b1 \u2192 \u211d) _ _\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 0 \u2264\u1d50[Measure.trim \u03bc hm] AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h\n[PROOFSTEP]\nrefine' ae_nonneg_of_forall_set_integral_nonneg_of_sigmaFinite _ _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) s_1\n[PROOFSTEP]\nrintro t - -\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 IntegrableOn (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h) t\n[PROOFSTEP]\nrefine @Integrable.integrableOn _ _ m _ _ _ _ ?_\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)\n[PROOFSTEP]\nrefine' Integrable.trim hm _ _\n[GOAL]\ncase refine'_1.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)\n[PROOFSTEP]\nrw [integrable_congr h.ae_eq_mk.symm]\n[GOAL]\ncase refine'_1.refine'_1\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 Integrable \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))\n[PROOFSTEP]\nexact integrable_condexpL2_indicator hm hs h\u03bcs _\n[GOAL]\ncase refine'_1.refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\n\u22a2 StronglyMeasurable (AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h)\n[PROOFSTEP]\nexact h.stronglyMeasurable_mk\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\n\u22a2 \u2200 (s_1 : Set \u03b1),\n    MeasurableSet s_1 \u2192\n      \u2191\u2191(Measure.trim \u03bc hm) s_1 < \u22a4 \u2192\n        0 \u2264\n          \u222b (x : \u03b1) in s_1,\n            AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm\n[PROOFSTEP]\nintro t ht h\u03bct\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264\n    \u222b (x : \u03b1) in t,\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202Measure.trim \u03bc hm\n[PROOFSTEP]\nrw [\u2190 set_integral_trim hm h.stronglyMeasurable_mk ht]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc\n[PROOFSTEP]\nhave h_ae : \u2200\u1d50 x \u2202\u03bc, x \u2208 t \u2192 h.mk _ x = (condexpL2 \u211d \u211d hm (indicatorConstLp 2 hs h\u03bcs 1) : \u03b1 \u2192 \u211d) x :=\n  by\n  filter_upwards [h.ae_eq_mk] with x hx\n  exact fun _ => hx.symm\n[GOAL]\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n[PROOFSTEP]\nfilter_upwards [h.ae_eq_mk] with x hx\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nx : \u03b1\nhx :\n  \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x =\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x\n\u22a2 x \u2208 t \u2192\n    AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n      \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n[PROOFSTEP]\nexact fun _ => hx.symm\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 \u222b (x : \u03b1) in t, AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x \u2202\u03bc\n[PROOFSTEP]\nrw [set_integral_congr_ae (hm t ht) h_ae, set_integral_condexpL2_indicator ht hs ((le_trim hm).trans_lt h\u03bct).ne h\u03bcs]\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u2070 : IsROrC \ud835\udd5c\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9\u2077 : CompleteSpace E\ninst\u271d\u00b9\u2076 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2075 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2074 : CompleteSpace E'\ninst\u271d\u00b9\u00b3 : NormedSpace \u211d E'\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u2070 : NormedAddCommGroup G\ninst\u271d\u2079 : NormedAddCommGroup G'\ninst\u271d\u2078 : NormedSpace \u211d G'\ninst\u271d\u2077 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t\u271d : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2076 : IsROrC \ud835\udd5c'\ninst\u271d\u2075 : NormedAddCommGroup E''\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u00b3 : CompleteSpace E''\ninst\u271d\u00b2 : NormedSpace \u211d E''\ninst\u271d\u00b9 : NormedSpace \u211d G\nhm\u271d hm : m \u2264 m0\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nh : AEStronglyMeasurable' m (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) \u03bc\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u2191\u2191(Measure.trim \u03bc hm) t < \u22a4\nh_ae :\n  \u2200\u1d50 (x : \u03b1) \u2202\u03bc,\n    x \u2208 t \u2192\n      AEStronglyMeasurable'.mk (\u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1))) h x =\n        \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) x\n\u22a2 0 \u2264 ENNReal.toReal (\u2191\u2191\u03bc (s \u2229 t))\n[PROOFSTEP]\nexact ENNReal.toReal_nonneg\n[GOAL]\n\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264\u1d50[\u03bc] \u2191\u2191(condexpIndSMul hm hs h\u03bcs x)\n[PROOFSTEP]\nrefine' EventuallyLE.trans_eq _ (condexpIndSMul_ae_eq_smul hm hs h\u03bcs x).symm\n[GOAL]\n\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\n\u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nfilter_upwards [condexpL2_indicator_nonneg hm hs h\u03bcs] with a ha\n[GOAL]\ncase h\n\u03b1 : Type u_1\nE\u271d : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b2\u00b3 : IsROrC \ud835\udd5c\ninst\u271d\u00b2\u00b2 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2\u00b9 : InnerProductSpace \ud835\udd5c E\u271d\ninst\u271d\u00b2\u2070 : CompleteSpace E\u271d\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E'\ninst\u271d\u00b9\u2078 : InnerProductSpace \ud835\udd5c E'\ninst\u271d\u00b9\u2077 : CompleteSpace E'\ninst\u271d\u00b9\u2076 : NormedSpace \u211d E'\ninst\u271d\u00b9\u2075 : NormedAddCommGroup F\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b9\u00b2 : NormedAddCommGroup G'\ninst\u271d\u00b9\u00b9 : NormedSpace \u211d G'\ninst\u271d\u00b9\u2070 : CompleteSpace G'\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\nE'' : Type u_8\n\ud835\udd5c' : Type u_9\ninst\u271d\u2079 : IsROrC \ud835\udd5c'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : InnerProductSpace \ud835\udd5c' E''\ninst\u271d\u2076 : CompleteSpace E''\ninst\u271d\u2075 : NormedSpace \u211d E''\ninst\u271d\u2074 : NormedSpace \u211d G\nhm : m \u2264 m0\nE : Type u_10\ninst\u271d\u00b3 : NormedLatticeAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : OrderedSMul \u211d E\ninst\u271d : SigmaFinite (Measure.trim \u03bc hm)\nhs : MeasurableSet s\nh\u03bcs : \u2191\u2191\u03bc s \u2260 \u22a4\nx : E\nhx : 0 \u2264 x\na : \u03b1\nha : OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a\n\u22a2 OfNat.ofNat 0 a \u2264 \u2191\u2191\u2191(\u2191(condexpL2 \u211d \u211d hm) (indicatorConstLp 2 hs h\u03bcs 1)) a \u2022 x\n[PROOFSTEP]\nexact smul_nonneg ha hx\n", "meta": {"mathlib_filename": "Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2", "llama_tokens": 95484, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8128673178375734, "lm_q2_score": 0.6187804267137442, "lm_q1q2_score": 0.5029863857931904}}
{"text": "[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nby_cases A : \u2203 i \u2208 s, z i = 0 \u2227 w i \u2260 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2203 i, i \u2208 s \u2227 z i = 0 \u2227 w i \u2260 0\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nrcases A with \u27e8i, his, hzi, hwi\u27e9\n[GOAL]\ncase pos.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\ni : \u03b9\nhis : i \u2208 s\nhzi : z i = 0\nhwi : w i \u2260 0\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nrw [prod_eq_zero his]\n[GOAL]\ncase pos.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\ni : \u03b9\nhis : i \u2208 s\nhzi : z i = 0\nhwi : w i \u2260 0\n\u22a2 0 \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nexact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)\n[GOAL]\ncase pos.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\ni : \u03b9\nhis : i \u2208 s\nhzi : z i = 0\nhwi : w i \u2260 0\n\u22a2 z i ^ w i = 0\n[PROOFSTEP]\nrw [hzi]\n[GOAL]\ncase pos.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\ni : \u03b9\nhis : i \u2208 s\nhzi : z i = 0\nhwi : w i \u2260 0\n\u22a2 0 ^ w i = 0\n[PROOFSTEP]\nexact zero_rpow hwi\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u00ac\u2203 i, i \u2208 s \u2227 z i = 0 \u2227 w i \u2260 0\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nsimp only [not_exists, not_and, Ne.def, Classical.not_not] at A \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nhave := convexOn_exp.map_sum_le hw hw' fun i _ => Set.mem_univ <| log (z i)\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : exp (\u2211 i in s, w i \u2022 log (z i)) \u2264 \u2211 i in s, w i \u2022 exp (log (z i))\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nsimp only [exp_sum, (\u00b7 \u2218 \u00b7), smul_eq_mul, mul_comm (w _) (log _)] at this \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nconvert this using 1 <;> [apply prod_congr rfl; apply sum_congr rfl]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u220f i in s, z i ^ w i \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u220f i in s, z i ^ w i = \u220f x in s, exp (log (z x) * w x)\n[PROOFSTEP]\napply prod_congr rfl\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u2211 i in s, w i * z i = \u2211 x in s, w x * exp (log (z x))\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 z x ^ w x = exp (log (z x) * w x)\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 w x * z x = w x * exp (log (z x))\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\n\u22a2 z i ^ w i = exp (log (z i) * w i)\n[PROOFSTEP]\ncases' eq_or_lt_of_le (hz i hi) with hz hz\n[GOAL]\ncase h.e'_3.inl\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\nhz : 0 = z i\n\u22a2 z i ^ w i = exp (log (z i) * w i)\n[PROOFSTEP]\nsimp [A i hi hz.symm]\n[GOAL]\ncase h.e'_3.inr\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\nhz : 0 < z i\n\u22a2 z i ^ w i = exp (log (z i) * w i)\n[PROOFSTEP]\nexact rpow_def_of_pos hz _\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i * z i = w i * exp (log (z i))\n[PROOFSTEP]\ncases' eq_or_lt_of_le (hz i hi) with hz hz\n[GOAL]\ncase h.e'_4.inl\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\nhz : 0 = z i\n\u22a2 w i * z i = w i * exp (log (z i))\n[PROOFSTEP]\nsimp [A i hi hz.symm]\n[GOAL]\ncase h.e'_4.inr\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nA : \u2200 (x : \u03b9), x \u2208 s \u2192 z x = 0 \u2192 w x = 0\nthis : \u220f x in s, exp (log (z x) * w x) \u2264 \u2211 x in s, w x * exp (log (z x))\ni : \u03b9\nhi : i \u2208 s\nhz : 0 < z i\n\u22a2 w i * z i = w i * exp (log (z i))\n[PROOFSTEP]\nrw [exp_log hz]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u220f i in s, z i ^ w i = \u220f i in s, x ^ w i\n[PROOFSTEP]\nrefine' prod_congr rfl fun i hi => _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\n\u22a2 z i ^ w i = x ^ w i\n[PROOFSTEP]\ncases' eq_or_ne (w i) 0 with h\u2080 h\u2080\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\nh\u2080 : w i = 0\n\u22a2 z i ^ w i = x ^ w i\n[PROOFSTEP]\nrw [h\u2080, rpow_zero, rpow_zero]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\nh\u2080 : w i \u2260 0\n\u22a2 z i ^ w i = x ^ w i\n[PROOFSTEP]\nrw [hx i hi h\u2080]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u220f i in s, x ^ w i = x\n[PROOFSTEP]\nrw [\u2190 rpow_sum_of_nonneg _ hw, hw', rpow_one]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nhave : (\u2211 i in s, w i) \u2260 0 := by\n  rw [hw']\n  exact one_ne_zero\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2211 i in s, w i \u2260 0\n[PROOFSTEP]\nrw [hw']\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nexact one_ne_zero\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\nthis : \u2211 i in s, w i \u2260 0\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nobtain \u27e8i, his, hi\u27e9 := exists_ne_zero_of_sum_ne_zero this\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\nthis : \u2211 i in s, w i \u2260 0\ni : \u03b9\nhis : i \u2208 s\nhi : w i \u2260 0\n\u22a2 0 \u2264 x\n[PROOFSTEP]\nrw [\u2190 hx i his hi]\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\nthis : \u2211 i in s, w i \u2260 0\ni : \u03b9\nhis : i \u2208 s\nhi : w i \u2260 0\n\u22a2 0 \u2264 z i\n[PROOFSTEP]\nexact hz i his\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw' : \u2211 i in s, w i = 1\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2211 i in s, w i * z i = \u2211 i in s, w i * x\n[PROOFSTEP]\nrefine' sum_congr rfl fun i hi => _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw' : \u2211 i in s, w i = 1\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\n\u22a2 w i * z i = w i * x\n[PROOFSTEP]\ncases' eq_or_ne (w i) 0 with hwi hwi\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw' : \u2211 i in s, w i = 1\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\nhwi : w i = 0\n\u22a2 w i * z i = w i * x\n[PROOFSTEP]\nrw [hwi, zero_mul, zero_mul]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw' : \u2211 i in s, w i = 1\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\ni : \u03b9\nhi : i \u2208 s\nhwi : w i \u2260 0\n\u22a2 w i * z i = w i * x\n[PROOFSTEP]\nrw [hx i hi hwi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw' : \u2211 i in s, w i = 1\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2211 i in s, w i * x = x\n[PROOFSTEP]\nrw [\u2190 sum_mul, hw', one_mul]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u220f i in s, z i ^ w i = \u2211 i in s, w i * z i\n[PROOFSTEP]\nrw [geom_mean_weighted_of_constant, arith_mean_weighted_of_constant]\n[GOAL]\ncase x\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u211d\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hw'\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2211 i in s, w i = 1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hw\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hw'\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2211 i in s, w i = 1\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hz\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\n[PROOFSTEP]\nassumption\n[GOAL]\ncase hx\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\nx : \u211d\nhw : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 w i\nhw' : \u2211 i in s, w i = 1\nhz : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 z i\nhx : \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n\u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 w i \u2260 0 \u2192 z i = x\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\n\u22a2 \u220f i in s, z i ^ \u2191(w i) \u2264 \u2211 i in s, w i * z i\n[PROOFSTEP]\nexact_mod_cast\n  Real.geom_mean_le_arith_mean_weighted _ _ _ (fun i _ => (w i).coe_nonneg) (by assumption_mod_cast) fun i _ =>\n    (z i).coe_nonneg\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw z : \u03b9 \u2192 \u211d\u22650\nhw' : \u2211 i in s, w i = 1\n\u22a2 \u2211 i in s, \u2191(w i) = 1\n[PROOFSTEP]\nassumption_mod_cast\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 p\u2081 p\u2082 : \u211d\u22650\n\u22a2 w\u2081 + w\u2082 = 1 \u2192 p\u2081 ^ \u2191w\u2081 * p\u2082 ^ \u2191w\u2082 \u2264 w\u2081 * p\u2081 + w\u2082 * p\u2082\n[PROOFSTEP]\nsimpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty, Fintype.univ_of_isEmpty,\n  Fin.cons_succ, Fin.cons_zero, add_zero, mul_one] using geom_mean_le_arith_mean_weighted univ ![w\u2081, w\u2082] ![p\u2081, p\u2082]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 w\u2083 p\u2081 p\u2082 p\u2083 : \u211d\u22650\n\u22a2 w\u2081 + w\u2082 + w\u2083 = 1 \u2192 p\u2081 ^ \u2191w\u2081 * p\u2082 ^ \u2191w\u2082 * p\u2083 ^ \u2191w\u2083 \u2264 w\u2081 * p\u2081 + w\u2082 * p\u2082 + w\u2083 * p\u2083\n[PROOFSTEP]\nsimpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty, Fintype.univ_of_isEmpty,\n  Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, \u2190 add_assoc, mul_assoc] using\n  geom_mean_le_arith_mean_weighted univ ![w\u2081, w\u2082, w\u2083] ![p\u2081, p\u2082, p\u2083]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 w\u2083 w\u2084 p\u2081 p\u2082 p\u2083 p\u2084 : \u211d\u22650\n\u22a2 w\u2081 + w\u2082 + w\u2083 + w\u2084 = 1 \u2192 p\u2081 ^ \u2191w\u2081 * p\u2082 ^ \u2191w\u2082 * p\u2083 ^ \u2191w\u2083 * p\u2084 ^ \u2191w\u2084 \u2264 w\u2081 * p\u2081 + w\u2082 * p\u2082 + w\u2083 * p\u2083 + w\u2084 * p\u2084\n[PROOFSTEP]\nsimpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty, Fintype.univ_of_isEmpty,\n  Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, \u2190 add_assoc, mul_assoc] using\n  geom_mean_le_arith_mean_weighted univ ![w\u2081, w\u2082, w\u2083, w\u2084] ![p\u2081, p\u2082, p\u2083, p\u2084]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 p\u2081 p\u2082 : \u211d\nhw\u2081 : 0 \u2264 w\u2081\nhw\u2082 : 0 \u2264 w\u2082\nhp\u2081 : 0 \u2264 p\u2081\nhp\u2082 : 0 \u2264 p\u2082\nhw : w\u2081 + w\u2082 = 1\n\u22a2 \u2191({ val := w\u2081, property := hw\u2081 } + { val := w\u2082, property := hw\u2082 }) = \u21911\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nw\u2081 w\u2082 w\u2083 w\u2084 p\u2081 p\u2082 p\u2083 p\u2084 : \u211d\nhw\u2081 : 0 \u2264 w\u2081\nhw\u2082 : 0 \u2264 w\u2082\nhw\u2083 : 0 \u2264 w\u2083\nhw\u2084 : 0 \u2264 w\u2084\nhp\u2081 : 0 \u2264 p\u2081\nhp\u2082 : 0 \u2264 p\u2082\nhp\u2083 : 0 \u2264 p\u2083\nhp\u2084 : 0 \u2264 p\u2084\nhw : w\u2081 + w\u2082 + w\u2083 + w\u2084 = 1\n\u22a2 \u2191({ val := w\u2081, property := hw\u2081 } + { val := w\u2082, property := hw\u2082 } + { val := w\u2083, property := hw\u2083 } +\n        { val := w\u2084, property := hw\u2084 }) =\n    \u21911\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b p q : \u211d\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhpq : IsConjugateExponent p q\n\u22a2 a * b \u2264 a ^ p / p + b ^ q / q\n[PROOFSTEP]\nsimpa [\u2190 rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using\n  geom_mean_le_arith_mean2_weighted hpq.one_div_nonneg hpq.symm.one_div_nonneg (rpow_nonneg_of_nonneg ha p)\n    (rpow_nonneg_of_nonneg hb q) hpq.inv_add_inv_conj\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 a * b \u2264 a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q\n[PROOFSTEP]\nnth_rw 1 [\u2190 Real.coe_toNNReal p hpq.nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 a * b \u2264 a ^ \u2191(Real.toNNReal p) / Real.toNNReal p + b ^ q / Real.toNNReal q\n[PROOFSTEP]\nnth_rw 1 [\u2190 Real.coe_toNNReal q hpq.symm.nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 a * b \u2264 a ^ \u2191(Real.toNNReal p) / Real.toNNReal p + b ^ \u2191(Real.toNNReal q) / Real.toNNReal q\n[PROOFSTEP]\nexact young_inequality a b hpq.one_lt_nnreal hpq.inv_add_inv_conj_nnreal\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 a * b \u2264 a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nby_cases h : a = \u22a4 \u2228 b = \u22a4\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 a * b \u2264 a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nrefine' le_trans le_top (le_of_eq _)\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 \u22a4 = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nrepeat rw [div_eq_mul_inv]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 \u22a4 = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4 \u2228 b = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\ncases' h with h h\n[GOAL]\ncase pos.inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos.inr\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : b = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\nrw [h]\n[GOAL]\ncase pos.inl\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a = \u22a4\n\u22a2 \u22a4 = \u22a4 ^ p * (ENNReal.ofReal p)\u207b\u00b9 + b ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\nsimp [h, hpq.pos, hpq.symm.pos]\n[GOAL]\ncase pos.inr\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : b = \u22a4\n\u22a2 \u22a4 = a ^ p * (ENNReal.ofReal p)\u207b\u00b9 + \u22a4 ^ q * (ENNReal.ofReal q)\u207b\u00b9\n[PROOFSTEP]\nsimp [h, hpq.pos, hpq.symm.pos]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : \u00ac(a = \u22a4 \u2228 b = \u22a4)\n\u22a2 a * b \u2264 a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\npush_neg at h \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a \u2260 \u22a4 \u2227 b \u2260 \u22a4\n\u22a2 a * b \u2264 a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q\n[PROOFSTEP]\nrw [\u2190 coe_toNNReal h.left, \u2190 coe_toNNReal h.right, \u2190 coe_mul, coe_rpow_of_nonneg _ hpq.nonneg,\n  coe_rpow_of_nonneg _ hpq.symm.nonneg, ENNReal.ofReal, ENNReal.ofReal, \u2190\n  @coe_div (Real.toNNReal p) _ (by simp [hpq.pos]), \u2190 @coe_div (Real.toNNReal q) _ (by simp [hpq.symm.pos]), \u2190 coe_add,\n  coe_le_coe]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a \u2260 \u22a4 \u2227 b \u2260 \u22a4\n\u22a2 Real.toNNReal p \u2260 0\n[PROOFSTEP]\nsimp [hpq.pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a \u2260 \u22a4 \u2227 b \u2260 \u22a4\n\u22a2 Real.toNNReal q \u2260 0\n[PROOFSTEP]\nsimp [hpq.symm.pos]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nh : a \u2260 \u22a4 \u2227 b \u2260 \u22a4\n\u22a2 ENNReal.toNNReal a * ENNReal.toNNReal b \u2264\n    ENNReal.toNNReal a ^ p / Real.toNNReal p + ENNReal.toNNReal b ^ q / Real.toNNReal q\n[PROOFSTEP]\nexact NNReal.young_inequality_real a.toNNReal b.toNNReal hpq\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\n\u22a2 \u2211 i in s, f i * g i \u2264 1\n[PROOFSTEP]\nhave hp_ne_zero : Real.toNNReal p \u2260 0 := (zero_lt_one.trans hpq.one_lt_nnreal).ne.symm\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\n\u22a2 \u2211 i in s, f i * g i \u2264 1\n[PROOFSTEP]\nhave hq_ne_zero : Real.toNNReal q \u2260 0 := (zero_lt_one.trans hpq.symm.one_lt_nnreal).ne.symm\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\nhq_ne_zero : Real.toNNReal q \u2260 0\n\u22a2 \u2211 i in s, f i * g i \u2264 1\n[PROOFSTEP]\ncalc\n  \u2211 i in s, f i * g i \u2264 \u2211 i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) :=\n    Finset.sum_le_sum fun i _ => young_inequality_real (f i) (g i) hpq\n  _ = (\u2211 i in s, f i ^ p) / Real.toNNReal p + (\u2211 i in s, g i ^ q) / Real.toNNReal q := by\n    rw [sum_add_distrib, sum_div, sum_div]\n  _ \u2264 1 / Real.toNNReal p + 1 / Real.toNNReal q :=\n    by\n    refine' add_le_add _ _\n    \u00b7 rwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]\n    \u00b7 rwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]\n  _ = 1 := hpq.inv_add_inv_conj_nnreal\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\nhq_ne_zero : Real.toNNReal q \u2260 0\n\u22a2 \u2211 i in s, (f i ^ p / Real.toNNReal p + g i ^ q / Real.toNNReal q) =\n    (\u2211 i in s, f i ^ p) / Real.toNNReal p + (\u2211 i in s, g i ^ q) / Real.toNNReal q\n[PROOFSTEP]\nrw [sum_add_distrib, sum_div, sum_div]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\nhq_ne_zero : Real.toNNReal q \u2260 0\n\u22a2 (\u2211 i in s, f i ^ p) / Real.toNNReal p + (\u2211 i in s, g i ^ q) / Real.toNNReal q \u2264\n    1 / Real.toNNReal p + 1 / Real.toNNReal q\n[PROOFSTEP]\nrefine' add_le_add _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\nhq_ne_zero : Real.toNNReal q \u2260 0\n\u22a2 (\u2211 i in s, f i ^ p) / Real.toNNReal p \u2264 1 / Real.toNNReal p\n[PROOFSTEP]\nrwa [div_le_iff hp_ne_zero, div_mul_cancel _ hp_ne_zero]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p \u2264 1\nhg : \u2211 i in s, g i ^ q \u2264 1\nhp_ne_zero : Real.toNNReal p \u2260 0\nhq_ne_zero : Real.toNNReal q \u2260 0\n\u22a2 (\u2211 i in s, g i ^ q) / Real.toNNReal q \u2264 1 / Real.toNNReal q\n[PROOFSTEP]\nrwa [div_le_iff hq_ne_zero, div_mul_cancel _ hq_ne_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp only [hf, hpq.ne_zero, one_div, sum_eq_zero_iff, zero_rpow, zero_mul, inv_eq_zero, Ne.def, not_false_iff,\n  le_zero_iff, mul_eq_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p = 0\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 f x = 0 \u2228 g x = 0\n[PROOFSTEP]\nintro i his\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p = 0\ni : \u03b9\nhis : i \u2208 s\n\u22a2 f i = 0 \u2228 g i = 0\n[PROOFSTEP]\nleft\n[GOAL]\ncase h\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p = 0\ni : \u03b9\nhis : i \u2208 s\n\u22a2 f i = 0\n[PROOFSTEP]\nrw [sum_eq_zero_iff] at hf \n[GOAL]\ncase h\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2200 (x : \u03b9), x \u2208 s \u2192 f x ^ p = 0\ni : \u03b9\nhis : i \u2208 s\n\u22a2 f i = 0\n[PROOFSTEP]\nexact (rpow_eq_zero_iff.mp (hf i his)).left\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nby_cases hF_zero : \u2211 i in s, f i ^ p = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u2211 i in s, f i ^ p = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nexact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hF_zero\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nby_cases hG_zero : \u2211 i in s, g i ^ q = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u2211 i in s, g i ^ q = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\ncalc\n  \u2211 i in s, f i * g i = \u2211 i in s, g i * f i := by\n    congr with i\n    rw [mul_comm]\n  _ \u2264 (\u2211 i in s, g i ^ q) ^ (1 / q) * (\u2211 i in s, f i ^ p) ^ (1 / p) :=\n    (inner_le_Lp_mul_Lp_of_norm_eq_zero s g f hpq.symm hG_zero)\n  _ = (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q) := mul_comm _ _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u2211 i in s, g i ^ q = 0\n\u22a2 \u2211 i in s, f i * g i = \u2211 i in s, g i * f i\n[PROOFSTEP]\ncongr with i\n[GOAL]\ncase e_f.h.a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u2211 i in s, g i ^ q = 0\ni : \u03b9\n\u22a2 \u2191(f i * g i) = \u2191(g i * f i)\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nlet f' i := f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nlet g' i := g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsuffices (\u2211 i in s, f' i * g' i) \u2264 1\n  by\n  simp_rw [div_mul_div_comm, \u2190 sum_div] at this \n  rwa [div_le_iff, one_mul] at this \n  refine' mul_ne_zero _ _\n  \u00b7 rw [Ne.def, rpow_eq_zero_iff, not_and_or]\n    exact Or.inl hF_zero\n  \u00b7 rw [Ne.def, rpow_eq_zero_iff, not_and_or]\n    exact Or.inl hG_zero\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : \u2211 i in s, f' i * g' i \u2264 1\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp_rw [div_mul_div_comm, \u2190 sum_div] at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nrwa [div_le_iff, one_mul] at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\n[PROOFSTEP]\nrefine' mul_ne_zero _ _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, rpow_eq_zero_iff, not_and_or]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 \u00ac\u2211 i in s, f i ^ p = 0 \u2228 \u00ac1 / p \u2260 0\n[PROOFSTEP]\nexact Or.inl hF_zero\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\n[PROOFSTEP]\nrw [Ne.def, rpow_eq_zero_iff, not_and_or]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\nthis : (\u2211 i in s, f i * g i) / ((\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)) \u2264 1\n\u22a2 \u00ac\u2211 i in s, g i ^ q = 0 \u2228 \u00ac1 / q \u2260 0\n[PROOFSTEP]\nexact Or.inl hG_zero\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\n\u22a2 \u2211 i in s, f' i * g' i \u2264 1\n[PROOFSTEP]\nrefine' inner_le_Lp_mul_Lp_of_norm_le_one s f' g' hpq (le_of_eq _) (le_of_eq _)\n[GOAL]\ncase neg.refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\n\u22a2 \u2211 i in s, f' i ^ p = 1\n[PROOFSTEP]\nsimp_rw [div_rpow, \u2190 sum_div, \u2190 rpow_mul, one_div, inv_mul_cancel hpq.ne_zero, rpow_one, div_self hF_zero]\n[GOAL]\ncase neg.refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhF_zero : \u00ac\u2211 i in s, f i ^ p = 0\nhG_zero : \u00ac\u2211 i in s, g i ^ q = 0\nf' : \u03b9 \u2192 \u211d\u22650 := fun i => f i / (\u2211 i in s, f i ^ p) ^ (1 / p)\ng' : \u03b9 \u2192 \u211d\u22650 := fun i => g i / (\u2211 i in s, g i ^ q) ^ (1 / q)\n\u22a2 \u2211 i in s, g' i ^ q = 1\n[PROOFSTEP]\nsimp_rw [div_rpow, \u2190 sum_div, \u2190 rpow_mul, one_div, inv_mul_cancel hpq.symm.ne_zero, rpow_one, div_self hG_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\n\u22a2 (Summable fun i => f i * g i) \u2227\n    \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave H\u2081 : \u2200 s : Finset \u03b9, \u2211 i in s, f i * g i \u2264 (\u2211' i, f i ^ p) ^ (1 / p) * (\u2211' i, g i ^ q) ^ (1 / q) :=\n  by\n  intro s\n  refine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)\n  \u00b7 rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]\n    exact sum_le_tsum _ (fun _ _ => zero_le _) hf\n  \u00b7 rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]\n    exact sum_le_tsum _ (fun _ _ => zero_le _) hg\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\n\u22a2 \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset \u03b9\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nrefine' le_trans (inner_le_Lp_mul_Lq s f g hpq) (mul_le_mul _ _ bot_le bot_le)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.pos)]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset \u03b9\n\u22a2 \u2211 i in s, f i ^ p \u2264 \u2211' (i : \u03b9), f i ^ p\n[PROOFSTEP]\nexact sum_le_tsum _ (fun _ _ => zero_le _) hf\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2264 (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nrw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr hpq.symm.pos)]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\ns : Finset \u03b9\n\u22a2 \u2211 i in s, g i ^ q \u2264 \u2211' (i : \u03b9), g i ^ q\n[PROOFSTEP]\nexact sum_le_tsum _ (fun _ _ => zero_le _) hg\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 (Summable fun i => f i * g i) \u2227\n    \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave bdd : BddAbove (Set.range fun s => \u2211 i in s, f i * g i) :=\n  by\n  refine' \u27e8(\u2211' i, f i ^ p) ^ (1 / p) * (\u2211' i, g i ^ q) ^ (1 / q), _\u27e9\n  rintro a \u27e8s, rfl\u27e9\n  exact H\u2081 s\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 BddAbove (Set.range fun s => \u2211 i in s, f i * g i)\n[PROOFSTEP]\nrefine' \u27e8(\u2211' i, f i ^ p) ^ (1 / p) * (\u2211' i, g i ^ q) ^ (1 / q), _\u27e9\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q) \u2208\n    upperBounds (Set.range fun s => \u2211 i in s, f i * g i)\n[PROOFSTEP]\nrintro a \u27e8s, rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 i in s, f i * g i) s \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nexact H\u2081 s\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => \u2211 i in s, f i * g i)\n\u22a2 (Summable fun i => f i * g i) \u2227\n    \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave H\u2082 : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ q\nH\u2081 : \u2200 (s : Finset \u03b9), \u2211 i in s, f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nbdd : BddAbove (Set.range fun s => \u2211 i in s, f i * g i)\nH\u2082 : Summable fun i => f i * g i\n\u22a2 (Summable fun i => f i * g i) \u2227\n    \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nexact \u27e8H\u2082, tsum_le_of_sum_le H\u2082 H\u2081\u27e9\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\n\u22a2 \u2203 C, C \u2264 A * B \u2227 HasSum (fun i => f i * g i) C\n[PROOFSTEP]\nobtain \u27e8H\u2081, H\u2082\u27e9 := inner_le_Lp_mul_Lq_tsum hpq hf.summable hg.summable\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 \u2203 C, C \u2264 A * B \u2227 HasSum (fun i => f i * g i) C\n[PROOFSTEP]\nhave hA : A = (\u2211' i : \u03b9, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [hf.tsum_eq, rpow_inv_rpow_self hpq.ne_zero]\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n\u22a2 \u2203 C, C \u2264 A * B \u2227 HasSum (fun i => f i * g i) C\n[PROOFSTEP]\nhave hB : B = (\u2211' i : \u03b9, g i ^ q) ^ (1 / q) := by rw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n\u22a2 B = (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nrw [hg.tsum_eq, rpow_inv_rpow_self hpq.symm.ne_zero]\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 \u2203 C, C \u2264 A * B \u2227 HasSum (fun i => f i * g i) C\n[PROOFSTEP]\nrefine' \u27e8\u2211' i, f i * g i, _, _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 \u2211' (i : \u03b9), f i * g i \u2264 A * B\n[PROOFSTEP]\nsimpa [hA, hB] using H\u2082\n[GOAL]\ncase intro.refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ q) (B ^ q)\nH\u2081 : Summable fun i => f i * g i\nH\u2082 : \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n\u22a2 HasSum (fun i => f i * g i) (\u2211' (i : \u03b9), f i * g i)\n[PROOFSTEP]\nsimpa only [rpow_self_rpow_inv hpq.ne_zero] using H\u2081.hasSum\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\ncases' eq_or_lt_of_le hp with hp hp\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 = p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nsimp [\u2190 hp]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nlet q : \u211d := p / (p - 1)\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\n\u22a2 Real.IsConjugateExponent p q\n[PROOFSTEP]\nrw [Real.isConjugateExponent_iff hp]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hp\u2081 : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hq : 1 / q * p = p - 1 := by\n  rw [\u2190 hpq.div_conj_eq_sub_one]\n  ring\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 1 / q * p = p - 1\n[PROOFSTEP]\nrw [\u2190 hpq.div_conj_eq_sub_one]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 1 / q * p = p / q\n[PROOFSTEP]\nring\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nsimpa only [NNReal.mul_rpow, \u2190 NNReal.rpow_mul, hp\u2081, hq, one_mul, one_rpow, rpow_one, Pi.one_apply, sum_const,\n  Nat.smul_one_eq_coe] using NNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 IsGreatest ((fun g => \u2211 i in s, f i * g i) '' {g | \u2211 i in s, g i ^ q \u2264 1}) ((\u2211 i in s, f i ^ p) ^ (1 / p))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) \u2208 (fun g => \u2211 i in s, f i * g i) '' {g | \u2211 i in s, g i ^ q \u2264 1}\n[PROOFSTEP]\nuse fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)\n[GOAL]\ncase h\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 (fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) \u2208 {g | \u2211 i in s, g i ^ q \u2264 1} \u2227\n    ((fun g => \u2211 i in s, f i * g i) fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) =\n      (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nby_cases hf : \u2211 i in s, f i ^ p = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u2211 i in s, f i ^ p = 0\n\u22a2 (fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) \u2208 {g | \u2211 i in s, g i ^ q \u2264 1} \u2227\n    ((fun g => \u2211 i in s, f i * g i) fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) =\n      (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimp [hf, hpq.ne_zero, hpq.symm.ne_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\n\u22a2 (fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) \u2208 {g | \u2211 i in s, g i ^ q \u2264 1} \u2227\n    ((fun g => \u2211 i in s, f i * g i) fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) =\n      (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave A : p + q - q \u2260 0 := by simp [hpq.ne_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\n\u22a2 p + q - q \u2260 0\n[PROOFSTEP]\nsimp [hpq.ne_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\n\u22a2 (fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) \u2208 {g | \u2211 i in s, g i ^ q \u2264 1} \u2227\n    ((fun g => \u2211 i in s, f i * g i) fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) =\n      (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave B : \u2200 y : \u211d\u22650, y * y ^ p / y = y ^ p :=\n  by\n  refine' fun y => mul_div_cancel_left_of_imp fun h => _\n  simp [h, hpq.ne_zero]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\n\u22a2 \u2200 (y : \u211d\u22650), y * y ^ p / y = y ^ p\n[PROOFSTEP]\nrefine' fun y => mul_div_cancel_left_of_imp fun h => _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\ny : \u211d\u22650\nh : y = 0\n\u22a2 y ^ p = 0\n[PROOFSTEP]\nsimp [h, hpq.ne_zero]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\nB : \u2200 (y : \u211d\u22650), y * y ^ p / y = y ^ p\n\u22a2 (fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) \u2208 {g | \u2211 i in s, g i ^ q \u2264 1} \u2227\n    ((fun g => \u2211 i in s, f i * g i) fun i => f i ^ p / f i / (\u2211 i in s, f i ^ p) ^ (1 / q)) =\n      (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, div_rpow, \u2190 sum_div, \u2190 rpow_mul, div_mul_cancel _ hpq.symm.ne_zero, rpow_one,\n  div_le_iff hf, one_mul, hpq.mul_eq_add, \u2190 rpow_sub' _ A, _root_.add_sub_cancel, le_refl, true_and_iff, \u2190\n  mul_div_assoc, B]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\nB : \u2200 (y : \u211d\u22650), y * y ^ p / y = y ^ p\n\u22a2 (\u2211 i in s, f i ^ p) / (\u2211 i in s, f i ^ p) ^ (1 / q) = (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [div_eq_iff, \u2190 rpow_add hf, hpq.inv_add_inv_conj, rpow_one]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nhf : \u00ac\u2211 i in s, f i ^ p = 0\nA : p + q - q \u2260 0\nB : \u2200 (y : \u211d\u22650), y * y ^ p / y = y ^ p\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / q) \u2260 0\n[PROOFSTEP]\nsimpa [hpq.symm.ne_zero] using hf\n[GOAL]\ncase right\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) \u2208 upperBounds ((fun g => \u2211 i in s, f i * g i) '' {g | \u2211 i in s, g i ^ q \u2264 1})\n[PROOFSTEP]\nrintro _ \u27e8g, hg, rfl\u27e9\n[GOAL]\ncase right.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\ng : \u03b9 \u2192 \u211d\u22650\nhg : g \u2208 {g | \u2211 i in s, g i ^ q \u2264 1}\n\u22a2 (fun g => \u2211 i in s, f i * g i) g \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\napply le_trans (inner_le_Lp_mul_Lq s f g hpq)\n[GOAL]\ncase right.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nf : \u03b9 \u2192 \u211d\u22650\np q : \u211d\nhpq : Real.IsConjugateExponent p q\ng : \u03b9 \u2192 \u211d\u22650\nhg : g \u2208 {g | \u2211 i in s, g i ^ q \u2264 1}\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimpa only [mul_one] using mul_le_mul_left' (NNReal.rpow_le_one hg (le_of_lt hpq.symm.one_div_pos)) _\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrcases eq_or_lt_of_le hp with (rfl | hp)\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nhp : 1 \u2264 1\n\u22a2 (\u2211 i in s, (f i + g i) ^ 1) ^ (1 / 1) \u2264 (\u2211 i in s, f i ^ 1) ^ (1 / 1) + (\u2211 i in s, g i ^ 1) ^ (1 / 1)\n[PROOFSTEP]\nsimp [Finset.sum_add_distrib]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave hpq := Real.isConjugateExponent_conjugateExponent hp\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave := isGreatest_Lp s (f + g) hpq\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nthis :\n  IsGreatest ((fun g_1 => \u2211 i in s, (f + g) i * g_1 i) '' {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1})\n    ((\u2211 i in s, (f + g) i ^ p) ^ (1 / p))\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimp only [Pi.add_apply, add_mul, sum_add_distrib] at this \n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nthis :\n  IsGreatest\n    ((fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) '' {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1})\n    ((\u2211 x in s, (f x + g x) ^ p) ^ (1 / p))\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrcases this.1 with \u27e8\u03c6, h\u03c6, H\u27e9\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nthis :\n  IsGreatest\n    ((fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) '' {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1})\n    ((\u2211 x in s, (f x + g x) ^ p) ^ (1 / p))\n\u03c6 : \u03b9 \u2192 \u211d\u22650\nh\u03c6 : \u03c6 \u2208 {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1}\nH : (fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) \u03c6 = (\u2211 x in s, (f x + g x) ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [\u2190 H]\n[GOAL]\ncase inr.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nthis :\n  IsGreatest\n    ((fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) '' {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1})\n    ((\u2211 x in s, (f x + g x) ^ p) ^ (1 / p))\n\u03c6 : \u03b9 \u2192 \u211d\u22650\nh\u03c6 : \u03c6 \u2208 {g | \u2211 i in s, g i ^ Real.conjugateExponent p \u2264 1}\nH : (fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) \u03c6 = (\u2211 x in s, (f x + g x) ^ p) ^ (1 / p)\n\u22a2 (fun a => \u2211 x in s, f x * a x + \u2211 x in s, g x * a x) \u03c6 \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nexact add_le_add ((isGreatest_Lp s f hpq).2 \u27e8\u03c6, h\u03c6, rfl\u27e9) ((isGreatest_Lp s g hpq).2 \u27e8\u03c6, h\u03c6, rfl\u27e9)\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave pos : 0 < p := lt_of_lt_of_le zero_lt_one hp\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave H\u2081 : \u2200 s : Finset \u03b9, (\u2211 i in s, (f i + g i) ^ p) \u2264 ((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p :=\n  by\n  intro s\n  rw [\u2190 NNReal.rpow_one_div_le_iff pos]\n  refine' le_trans (Lp_add_le s f g hp) (add_le_add _ _) <;> rw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)] <;>\n    refine' sum_le_tsum _ (fun _ _ => zero_le _) _\n  exacts [hf, hg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\n\u22a2 \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n[PROOFSTEP]\nintro s\n[GOAL]\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n[PROOFSTEP]\nrw [\u2190 NNReal.rpow_one_div_le_iff pos]\n[GOAL]\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrefine' le_trans (Lp_add_le s f g hp) (add_le_add _ _)\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 (\u2211 i in s, g i ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [NNReal.rpow_le_rpow_iff (one_div_pos.mpr pos)]\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 \u2211 i in s, f i ^ p \u2264 \u2211' (i : \u03b9), f i ^ p\n[PROOFSTEP]\nrefine' sum_le_tsum _ (fun _ _ => zero_le _) _\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 \u2211 i in s, g i ^ p \u2264 \u2211' (i : \u03b9), g i ^ p\n[PROOFSTEP]\nrefine' sum_le_tsum _ (fun _ _ => zero_le _) _\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => f i ^ p\ncase refine'_2\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\ns : Finset \u03b9\n\u22a2 Summable fun i => g i ^ p\n[PROOFSTEP]\nexacts [hf, hg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave bdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p) :=\n  by\n  refine' \u27e8((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p, _\u27e9\n  rintro a \u27e8s, rfl\u27e9\n  exact H\u2081 s\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\n[PROOFSTEP]\nrefine' \u27e8((\u2211' i, f i ^ p) ^ (1 / p) + (\u2211' i, g i ^ p) ^ (1 / p)) ^ p, _\u27e9\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n\u22a2 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p \u2208\n    upperBounds (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\n[PROOFSTEP]\nrintro a \u27e8s, rfl\u27e9\n[GOAL]\ncase intro\n\u03b9 : Type u\ns\u271d : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\ns : Finset \u03b9\n\u22a2 (fun s => \u2211 i in s, (f i + g i) ^ p) s \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n[PROOFSTEP]\nexact H\u2081 s\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave H\u2082 : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrefine' \u27e8H\u2082, _\u27e9\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [NNReal.rpow_one_div_le_iff pos]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : Summable fun i => f i ^ p\nhg : Summable fun i => g i ^ p\npos : 0 < p\nH\u2081 :\n  \u2200 (s : Finset \u03b9), \u2211 i in s, (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\nbdd : BddAbove (Set.range fun s => \u2211 i in s, (f i + g i) ^ p)\nH\u2082 : Summable fun i => (f i + g i) ^ p\n\u22a2 \u2211' (i : \u03b9), (f i + g i) ^ p \u2264 ((\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)) ^ p\n[PROOFSTEP]\nrefine' tsum_le_of_sum_le H\u2082 H\u2081\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\n\u22a2 \u2203 C, C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nhave hp' : p \u2260 0 := (lt_of_lt_of_le zero_lt_one hp).ne'\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\n\u22a2 \u2203 C, C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nobtain \u27e8H\u2081, H\u2082\u27e9 := Lp_add_le_tsum hp hf.summable hg.summable\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n\u22a2 \u2203 C, C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nhave hA : A = (\u2211' i : \u03b9, f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp']\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n\u22a2 A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [hf.tsum_eq, rpow_inv_rpow_self hp']\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n\u22a2 \u2203 C, C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nhave hB : B = (\u2211' i : \u03b9, g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp']\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\n\u22a2 B = (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nrw [hg.tsum_eq, rpow_inv_rpow_self hp']\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n\u22a2 \u2203 C, C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nrefine' \u27e8(\u2211' i, (f i + g i) ^ p) ^ (1 / p), _, _\u27e9\n[GOAL]\ncase intro.refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n\u22a2 (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 A + B\n[PROOFSTEP]\nsimpa [hA, hB] using H\u2082\n[GOAL]\ncase intro.refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\np : \u211d\nhp : 1 \u2264 p\nhf : HasSum (fun i => f i ^ p) (A ^ p)\nhg : HasSum (fun i => g i ^ p) (B ^ p)\nhp' : p \u2260 0\nH\u2081 : Summable fun i => (f i + g i) ^ p\nH\u2082 : (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\nhA : A = (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p)\nhB : B = (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n\u22a2 HasSum (fun i => (f i + g i) ^ p) (((\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p)) ^ p)\n[PROOFSTEP]\nsimpa only [rpow_self_rpow_inv hp'] using H\u2081.hasSum\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) * (\u2211 i in s, |g i| ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave :=\n  NNReal.coe_le_coe.2\n    (NNReal.inner_le_Lp_mul_Lq s (fun i => \u27e8_, abs_nonneg (f i)\u27e9) (fun i => \u27e8_, abs_nonneg (g i)\u27e9) hpq)\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nthis :\n  \u2191(\u2211 i in s,\n        (fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i *\n          (fun i => { val := |g i|, property := (_ : 0 \u2264 |g i|) }) i) \u2264\n    \u2191((\u2211 i in s, (fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i ^ p) ^ (1 / p) *\n        (\u2211 i in s, (fun i => { val := |g i|, property := (_ : 0 \u2264 |g i|) }) i ^ q) ^ (1 / q))\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) * (\u2211 i in s, |g i| ^ q) ^ (1 / q)\n[PROOFSTEP]\npush_cast at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nthis : \u2211 x in s, |f x| * |g x| \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) * (\u2211 x in s, |g x| ^ q) ^ (1 / q)\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) * (\u2211 i in s, |g i| ^ q) ^ (1 / q)\n[PROOFSTEP]\nrefine' le_trans (sum_le_sum fun i _ => _) this\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nthis : \u2211 x in s, |f x| * |g x| \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) * (\u2211 x in s, |g x| ^ q) ^ (1 / q)\ni : \u03b9\nx\u271d : i \u2208 s\n\u22a2 f i * g i \u2264 |f i| * |g i|\n[PROOFSTEP]\nsimp only [\u2190 abs_mul, le_abs_self]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, |f i|) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\nhave := NNReal.coe_le_coe.2 (NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => \u27e8_, abs_nonneg (f i)\u27e9) hp)\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis :\n  \u2191((\u2211 i in s, (fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i) ^ p) \u2264\n    \u2191(\u2191(card s) ^ (p - 1) * \u2211 i in s, (fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i ^ p)\n\u22a2 (\u2211 i in s, |f i|) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\npush_cast at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis : (\u2211 x in s, |f x|) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 x in s, |f x| ^ p\n\u22a2 (\u2211 i in s, |f i|) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\nexact this\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, |f i + g i| ^ p) ^ (1 / p) \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) + (\u2211 i in s, |g i| ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave := NNReal.coe_le_coe.2 (NNReal.Lp_add_le s (fun i => \u27e8_, abs_nonneg (f i)\u27e9) (fun i => \u27e8_, abs_nonneg (g i)\u27e9) hp)\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis :\n  \u2191((\u2211 i in s,\n          ((fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i +\n              (fun i => { val := |g i|, property := (_ : 0 \u2264 |g i|) }) i) ^\n            p) ^\n        (1 / p)) \u2264\n    \u2191((\u2211 i in s, (fun i => { val := |f i|, property := (_ : 0 \u2264 |f i|) }) i ^ p) ^ (1 / p) +\n        (\u2211 i in s, (fun i => { val := |g i|, property := (_ : 0 \u2264 |g i|) }) i ^ p) ^ (1 / p))\n\u22a2 (\u2211 i in s, |f i + g i| ^ p) ^ (1 / p) \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) + (\u2211 i in s, |g i| ^ p) ^ (1 / p)\n[PROOFSTEP]\npush_cast at this \n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis : (\u2211 x in s, (|f x| + |g x|) ^ p) ^ (1 / p) \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) + (\u2211 x in s, |g x| ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, |f i + g i| ^ p) ^ (1 / p) \u2264 (\u2211 i in s, |f i| ^ p) ^ (1 / p) + (\u2211 i in s, |g i| ^ p) ^ (1 / p)\n[PROOFSTEP]\nrefine' le_trans (rpow_le_rpow _ (sum_le_sum fun i _ => _) _) this\n[GOAL]\ncase refine'_1\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis : (\u2211 x in s, (|f x| + |g x|) ^ p) ^ (1 / p) \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) + (\u2211 x in s, |g x| ^ p) ^ (1 / p)\n\u22a2 0 \u2264 \u2211 i in s, |f i + g i| ^ p\n[PROOFSTEP]\nsimp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add, rpow_le_rpow]\n[GOAL]\ncase refine'_2\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis : (\u2211 x in s, (|f x| + |g x|) ^ p) ^ (1 / p) \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) + (\u2211 x in s, |g x| ^ p) ^ (1 / p)\ni : \u03b9\nx\u271d : i \u2208 s\n\u22a2 |f i + g i| ^ p \u2264 (|f i| + |g i|) ^ p\n[PROOFSTEP]\nsimp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add, rpow_le_rpow]\n[GOAL]\ncase refine'_3\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nthis : (\u2211 x in s, (|f x| + |g x|) ^ p) ^ (1 / p) \u2264 (\u2211 x in s, |f x| ^ p) ^ (1 / p) + (\u2211 x in s, |g x| ^ p) ^ (1 / p)\n\u22a2 0 \u2264 1 / p\n[PROOFSTEP]\nsimp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add, rpow_le_rpow]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nconvert inner_le_Lp_mul_Lq s f g hpq using 3\n[GOAL]\ncase h.e'_4.h.e'_5.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, f i ^ p = \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_4.h.e'_6.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, g i ^ q = \u2211 i in s, |g i| ^ q\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_4.h.e'_5.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 f x ^ p = |f x| ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4.h.e'_6.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 g x ^ q = |g x| ^ q\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4.h.e'_5.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i ^ p = |f i| ^ p\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi, hg i hi]\n[GOAL]\ncase h.e'_4.h.e'_6.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i ^ q = |g i| ^ q\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi, hg i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhf_sum : Summable fun i => f i ^ p\nhg_sum : Summable fun i => g i ^ q\n\u22a2 (Summable fun i => f i * g i) \u2227\n    \u2211' (i : \u03b9), f i * g i \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\ng : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhg_sum : Summable fun i => g i ^ q\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : Summable fun i => (fun i => \u2191(f i)) i ^ p\n\u22a2 (Summable fun i => (fun i => \u2191(f i)) i * g i) \u2227\n    \u2211' (i : \u03b9), (fun i => \u2191(f i)) i * g i \u2264\n      (\u2211' (i : \u03b9), (fun i => \u2191(f i)) i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nlift g to \u03b9 \u2192 \u211d\u22650 using hg\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : Summable fun i => (fun i => \u2191(f i)) i ^ p\ng : \u03b9 \u2192 \u211d\u22650\nhg_sum : Summable fun i => (fun i => \u2191(g i)) i ^ q\n\u22a2 (Summable fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) \u2227\n    \u2211' (i : \u03b9), (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i \u2264\n      (\u2211' (i : \u03b9), (fun i => \u2191(f i)) i ^ p) ^ (1 / p) * (\u2211' (i : \u03b9), (fun i => \u2191(g i)) i ^ q) ^ (1 / q)\n[PROOFSTEP]\nnorm_cast at *\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf g : \u03b9 \u2192 \u211d\u22650\nhf_sum : Summable fun a => f a ^ p\nhg_sum : Summable fun a => g a ^ q\n\u22a2 (Summable fun a => f a * g a) \u2227\n    \u2211' (a : \u03b9), f a * g a \u2264 (\u2211' (a : \u03b9), f a ^ p) ^ (1 / p) * (\u2211' (a : \u03b9), g a ^ q) ^ (1 / q)\n[PROOFSTEP]\nexact NNReal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhf_sum : HasSum (fun i => f i ^ p) (A ^ p)\nhg_sum : HasSum (fun i => g i ^ q) (B ^ q)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A * B \u2227 HasSum (fun i => f i * g i) C\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\ng : \u03b9 \u2192 \u211d\np q : \u211d\nhpq : IsConjugateExponent p q\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhg_sum : HasSum (fun i => g i ^ q) (B ^ q)\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (A ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A * B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * g i) C\n[PROOFSTEP]\nlift g to \u03b9 \u2192 \u211d\u22650 using hg\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (A ^ p)\ng : \u03b9 \u2192 \u211d\u22650\nhg_sum : HasSum (fun i => (fun i => \u2191(g i)) i ^ q) (B ^ q)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A * B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) C\n[PROOFSTEP]\nlift A to \u211d\u22650 using hA\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nB : \u211d\nhB : 0 \u2264 B\nf g : \u03b9 \u2192 \u211d\u22650\nhg_sum : HasSum (fun i => (fun i => \u2191(g i)) i ^ q) (B ^ q)\nA : \u211d\u22650\nhf_sum : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (\u2191A ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A * B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) C\n[PROOFSTEP]\nlift B to \u211d\u22650 using hB\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf g : \u03b9 \u2192 \u211d\u22650\nA : \u211d\u22650\nhf_sum : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (\u2191A ^ p)\nB : \u211d\u22650\nhg_sum : HasSum (fun i => (fun i => \u2191(g i)) i ^ q) (\u2191B ^ q)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A * \u2191B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) C\n[PROOFSTEP]\nnorm_cast at hf_sum hg_sum \n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhf_sum : HasSum (fun a => f a ^ p) (A ^ p)\nhg_sum : HasSum (fun a => g a ^ q) (B ^ q)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A * \u2191B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) C\n[PROOFSTEP]\nobtain \u27e8C, hC, H\u27e9 := NNReal.inner_le_Lp_mul_Lq_hasSum hpq hf_sum hg_sum\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhf_sum : HasSum (fun a => f a ^ p) (A ^ p)\nhg_sum : HasSum (fun a => g a ^ q) (B ^ q)\nC : \u211d\u22650\nhC : C \u2264 A * B\nH : HasSum (fun i => f i * g i) C\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A * \u2191B \u2227 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) C\n[PROOFSTEP]\nrefine' \u27e8C, C.prop, hC, _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhpq : IsConjugateExponent p q\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhf_sum : HasSum (fun a => f a ^ p) (A ^ p)\nhg_sum : HasSum (fun a => g a ^ q) (B ^ q)\nC : \u211d\u22650\nhC : C \u2264 A * B\nH : HasSum (fun i => f i * g i) C\n\u22a2 HasSum (fun i => (fun i => \u2191(f i)) i * (fun i => \u2191(g i)) i) \u2191C\n[PROOFSTEP]\nnorm_cast\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nconvert rpow_sum_le_const_mul_sum_rpow s f hp using 2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 \u2211 i in s, f i = \u2211 i in s, |f i|\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 \u2211 i in s, f i ^ p = \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 f x = |f x|\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 f x ^ p = |f x| ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i = |f i|\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi]\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i ^ p = |f i| ^ p\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nconvert Lp_add_le s f g hp using 2 <;> [skip; congr 1; congr 1]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nconvert Lp_add_le s f g hp using 2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, (f i + g i) ^ p = \u2211 i in s, |f i + g i| ^ p\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) = (\u2211 i in s, |f i| ^ p) ^ (1 / p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 (\u2211 i in s, g i ^ p) ^ (1 / p) = (\u2211 i in s, |g i| ^ p) ^ (1 / p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, (f i + g i) ^ p = \u2211 i in s, |f i + g i| ^ p\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_4.h.e'_5.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, f i ^ p = \u2211 i in s, |f i| ^ p\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_4.h.e'_6.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2211 i in s, g i ^ p = \u2211 i in s, |g i| ^ p\n[PROOFSTEP]\napply sum_congr rfl\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 (f x + g x) ^ p = |f x + g x| ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4.h.e'_5.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 f x ^ p = |f x| ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_4.h.e'_6.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\n\u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 g x ^ p = |g x| ^ p\n[PROOFSTEP]\nintro i hi\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (f i + g i) ^ p = |f i + g i| ^ p\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]\n[GOAL]\ncase h.e'_4.h.e'_5.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i ^ p = |f i| ^ p\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]\n[GOAL]\ncase h.e'_4.h.e'_6.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 f i\nhg : \u2200 (i : \u03b9), i \u2208 s \u2192 0 \u2264 g i\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i ^ p = |g i| ^ p\n[PROOFSTEP]\nsimp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhf_sum : Summable fun i => f i ^ p\nhg_sum : Summable fun i => g i ^ p\n\u22a2 (Summable fun i => (f i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211' (i : \u03b9), f i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\ng : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nhg_sum : Summable fun i => g i ^ p\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : Summable fun i => (fun i => \u2191(f i)) i ^ p\n\u22a2 (Summable fun i => ((fun i => \u2191(f i)) i + g i) ^ p) \u2227\n    (\u2211' (i : \u03b9), ((fun i => \u2191(f i)) i + g i) ^ p) ^ (1 / p) \u2264\n      (\u2211' (i : \u03b9), (fun i => \u2191(f i)) i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nlift g to \u03b9 \u2192 \u211d\u22650 using hg\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf : \u03b9 \u2192 \u211d\u22650\nhf_sum : Summable fun i => (fun i => \u2191(f i)) i ^ p\ng : \u03b9 \u2192 \u211d\u22650\nhg_sum : Summable fun i => (fun i => \u2191(g i)) i ^ p\n\u22a2 (Summable fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) \u2227\n    (\u2211' (i : \u03b9), ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) ^ (1 / p) \u2264\n      (\u2211' (i : \u03b9), (fun i => \u2191(f i)) i ^ p) ^ (1 / p) + (\u2211' (i : \u03b9), (fun i => \u2191(g i)) i ^ p) ^ (1 / p)\n[PROOFSTEP]\nnorm_cast0 at *\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nf g : \u03b9 \u2192 \u211d\u22650\nhp : 1 \u2264 p\nhf_sum : Summable fun a => f a ^ p\nhg_sum : Summable fun a => g a ^ p\n\u22a2 (Summable fun a => (f a + g a) ^ p) \u2227\n    (\u2211' (a : \u03b9), (f a + g a) ^ p) ^ (1 / p) \u2264 (\u2211' (a : \u03b9), f a ^ p) ^ (1 / p) + (\u2211' (a : \u03b9), g a ^ p) ^ (1 / p)\n[PROOFSTEP]\nexact NNReal.Lp_add_le_tsum hp hf_sum hg_sum\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhf : \u2200 (i : \u03b9), 0 \u2264 f i\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nhfA : HasSum (fun i => f i ^ p) (A ^ p)\nhgB : HasSum (fun i => g i ^ p) (B ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A + B \u2227 HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nlift f to \u03b9 \u2192 \u211d\u22650 using hf\n[GOAL]\ncase intro\n\u03b9 : Type u\ns : Finset \u03b9\ng : \u03b9 \u2192 \u211d\np q : \u211d\nhp : 1 \u2264 p\nhg : \u2200 (i : \u03b9), 0 \u2264 g i\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nhgB : HasSum (fun i => g i ^ p) (B ^ p)\nf : \u03b9 \u2192 \u211d\u22650\nhfA : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (A ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A + B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + g i) ^ p) (C ^ p)\n[PROOFSTEP]\nlift g to \u03b9 \u2192 \u211d\u22650 using hg\n[GOAL]\ncase intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nA B : \u211d\nhA : 0 \u2264 A\nhB : 0 \u2264 B\nf : \u03b9 \u2192 \u211d\u22650\nhfA : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (A ^ p)\ng : \u03b9 \u2192 \u211d\u22650\nhgB : HasSum (fun i => (fun i => \u2191(g i)) i ^ p) (B ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 A + B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (C ^ p)\n[PROOFSTEP]\nlift A to \u211d\u22650 using hA\n[GOAL]\ncase intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nB : \u211d\nhB : 0 \u2264 B\nf g : \u03b9 \u2192 \u211d\u22650\nhgB : HasSum (fun i => (fun i => \u2191(g i)) i ^ p) (B ^ p)\nA : \u211d\u22650\nhfA : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (\u2191A ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A + B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (C ^ p)\n[PROOFSTEP]\nlift B to \u211d\u22650 using hB\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf g : \u03b9 \u2192 \u211d\u22650\nA : \u211d\u22650\nhfA : HasSum (fun i => (fun i => \u2191(f i)) i ^ p) (\u2191A ^ p)\nB : \u211d\u22650\nhgB : HasSum (fun i => (fun i => \u2191(g i)) i ^ p) (\u2191B ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A + \u2191B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (C ^ p)\n[PROOFSTEP]\nnorm_cast at hfA hgB \n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A + \u2191B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (C ^ p)\n[PROOFSTEP]\nobtain \u27e8C, hC\u2081, hC\u2082\u27e9 := NNReal.Lp_add_le_hasSum hp hfA hgB\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : \u211d\u22650\nhC\u2081 : C \u2264 A + B\nhC\u2082 : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n\u22a2 \u2203 C, 0 \u2264 C \u2227 C \u2264 \u2191A + \u2191B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (C ^ p)\n[PROOFSTEP]\nuse C\n[GOAL]\ncase h\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : \u211d\u22650\nhC\u2081 : C \u2264 A + B\nhC\u2082 : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n\u22a2 0 \u2264 \u2191C \u2227 \u2191C \u2264 \u2191A + \u2191B \u2227 HasSum (fun i => ((fun i => \u2191(f i)) i + (fun i => \u2191(g i)) i) ^ p) (\u2191C ^ p)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase h\n\u03b9 : Type u\ns : Finset \u03b9\np q : \u211d\nhp : 1 \u2264 p\nf g : \u03b9 \u2192 \u211d\u22650\nA B : \u211d\u22650\nhfA : HasSum (fun a => f a ^ p) (A ^ p)\nhgB : HasSum (fun a => g a ^ p) (B ^ p)\nC : \u211d\u22650\nhC\u2081 : C \u2264 A + B\nhC\u2082 : HasSum (fun i => (f i + g i) ^ p) (C ^ p)\n\u22a2 0 \u2264 C \u2227 C \u2264 A + B \u2227 HasSum (fun a => (f a + g a) ^ p) (C ^ p)\n[PROOFSTEP]\nexact \u27e8zero_le _, hC\u2081, hC\u2082\u27e9\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nby_cases H : (\u2211 i in s, f i ^ p) ^ (1 / p) = 0 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = 0\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) = 0 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nreplace H : (\u2200 i \u2208 s, f i = 0) \u2228 \u2200 i \u2208 s, g i = 0\n[GOAL]\ncase H\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) = 0 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = 0\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0) \u2228 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n[PROOFSTEP]\nsimpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,\n  sum_eq_zero_iff_of_nonneg] using H\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0) \u2228 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave : \u2200 i \u2208 s, f i * g i = 0 := fun i hi => by cases' H with H H <;> simp [H i hi]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0) \u2228 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i * g i = 0\n[PROOFSTEP]\ncases' H with H H\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\ni : \u03b9\nhi : i \u2208 s\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0\n\u22a2 f i * g i = 0\n[PROOFSTEP]\nsimp [H i hi]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\ni : \u03b9\nhi : i \u2208 s\nH : \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\n\u22a2 f i * g i = 0\n[PROOFSTEP]\nsimp [H i hi]\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0) \u2228 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nthis : \u2200 (i : \u03b9), i \u2208 s \u2192 f i * g i = 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave : \u2211 i in s, f i * g i = \u2211 i in s, 0 := sum_congr rfl this\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i = 0) \u2228 \u2200 (i : \u03b9), i \u2208 s \u2192 g i = 0\nthis\u271d : \u2200 (i : \u03b9), i \u2208 s \u2192 f i * g i = 0\nthis : \u2211 i in s, f i * g i = \u2211 i in s, 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp [this]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = 0 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = 0)\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\npush_neg at H \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nby_cases H' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = \u22a4\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = \u22a4\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\ncases' H' with H' H'\n[GOAL]\ncase pos.inl\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]\n[GOAL]\ncase pos.inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2211 i in s, g i ^ q) ^ (1 / q) = \u22a4\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp [H', -one_div, -sum_eq_zero_iff, -rpow_eq_zero_iff, H]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = \u22a4)\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nreplace H' : (\u2200 i \u2208 s, f i \u2260 \u22a4) \u2227 \u2200 i \u2208 s, g i \u2260 \u22a4\n[GOAL]\ncase H'\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ q) ^ (1 / q) = \u22a4)\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\n[PROOFSTEP]\nsimpa [ENNReal.rpow_eq_top_iff, asymm hpq.pos, asymm hpq.symm.pos, hpq.pos, hpq.symm.pos, ENNReal.sum_eq_top_iff,\n  not_or] using H'\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nhave :=\n  ENNReal.coe_le_coe.2\n    (@NNReal.inner_le_Lp_mul_Lq _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _ _ hpq)\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2191(\u2211 i in s, (fun i => ENNReal.toNNReal (f i)) i * (fun i => ENNReal.toNNReal (g i)) i) \u2264\n    \u2191((\u2211 i in s, (fun i => ENNReal.toNNReal (f i)) i ^ p) ^ (1 / p) *\n        (\u2211 i in s, (fun i => ENNReal.toNNReal (g i)) i ^ q) ^ (1 / q))\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nsimp [\u2190 ENNReal.coe_rpow_of_nonneg, le_of_lt hpq.pos, le_of_lt hpq.one_div_pos, le_of_lt hpq.symm.pos,\n  le_of_lt hpq.symm.one_div_pos] at this \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nconvert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nconvert this using 1 <;> [skip; congr 2]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q)\n[PROOFSTEP]\nconvert this using 1\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x))\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) * (\u2211 i in s, g i ^ q) ^ (1 / q) =\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n[PROOFSTEP]\ncongr 2\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x))\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i ^ p = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 1 / p = p\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, g i ^ q = \u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 1 / q = q\u207b\u00b9\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i * g i = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x))\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_3\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i * g i = \u2191(ENNReal.toNNReal (f i)) * \u2191(ENNReal.toNNReal (g i))\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, f i ^ p = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i ^ p = \u2191(ENNReal.toNNReal (f i)) ^ p\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\n\u22a2 \u2211 i in s, g i ^ q = \u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_4.e_a.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhpq : Real.IsConjugateExponent p q\nH : (\u2211 i in s, f i ^ p) ^ (1 / p) \u2260 0 \u2227 (\u2211 i in s, g i ^ q) ^ (1 / q) \u2260 0\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) * \u2191(ENNReal.toNNReal (g x)) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ p\u207b\u00b9 * (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ q) ^ q\u207b\u00b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i ^ q = \u2191(ENNReal.toNNReal (g i)) ^ q\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul, WithTop.coe_mul.symm]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\ncases' eq_or_lt_of_le hp with hp hp\n[GOAL]\ncase inl\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 = p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nsimp [\u2190 hp]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nlet q : \u211d := p / (p - 1)\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hpq : p.IsConjugateExponent q := by rw [Real.isConjugateExponent_iff hp]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\n\u22a2 Real.IsConjugateExponent p q\n[PROOFSTEP]\nrw [Real.isConjugateExponent_iff hp]\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hp\u2081 : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nhave hq : 1 / q * p = p - 1 := by\n  rw [\u2190 hpq.div_conj_eq_sub_one]\n  ring\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 1 / q * p = p - 1\n[PROOFSTEP]\nrw [\u2190 hpq.div_conj_eq_sub_one]\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\n\u22a2 1 / q * p = p / q\n[PROOFSTEP]\nring\n[GOAL]\ncase inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q\u271d : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nq : \u211d := p / (p - 1)\nhpq : Real.IsConjugateExponent p q\nhp\u2081 : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n\u22a2 (\u2211 i in s, f i) ^ p \u2264 \u2191(card s) ^ (p - 1) * \u2211 i in s, f i ^ p\n[PROOFSTEP]\nsimpa only [ENNReal.mul_rpow_of_nonneg _ _ hpq.nonneg, \u2190 ENNReal.rpow_mul, hp\u2081, hq, coe_one, one_mul, one_rpow,\n  rpow_one, Pi.one_apply, sum_const, Nat.smul_one_eq_coe] using\n  ENNReal.rpow_le_rpow (inner_le_Lp_mul_Lq s 1 f hpq.symm) hpq.nonneg\n[GOAL]\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nby_cases H' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4\n[GOAL]\ncase pos\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\ncases' H' with H' H'\n[GOAL]\ncase pos.inl\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : (\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimp [H', -one_div]\n[GOAL]\ncase pos.inr\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nsimp [H', -one_div]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4)\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave pos : 0 < p := lt_of_lt_of_le zero_lt_one hp\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4)\npos : 0 < p\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nreplace H' : (\u2200 i \u2208 s, f i \u2260 \u22a4) \u2227 \u2200 i \u2208 s, g i \u2260 \u22a4\n[GOAL]\ncase H'\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\nH' : \u00ac((\u2211 i in s, f i ^ p) ^ (1 / p) = \u22a4 \u2228 (\u2211 i in s, g i ^ p) ^ (1 / p) = \u22a4)\npos : 0 < p\n\u22a2 (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\n[PROOFSTEP]\nsimpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top_iff, not_or] using H'\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nhave :=\n  ENNReal.coe_le_coe.2 (@NNReal.Lp_add_le _ s (fun i => ENNReal.toNNReal (f i)) (fun i => ENNReal.toNNReal (g i)) _ hp)\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  \u2191((\u2211 i in s, ((fun i => ENNReal.toNNReal (f i)) i + (fun i => ENNReal.toNNReal (g i)) i) ^ p) ^ (1 / p)) \u2264\n    \u2191((\u2211 i in s, (fun i => ENNReal.toNNReal (f i)) i ^ p) ^ (1 / p) +\n        (\u2211 i in s, (fun i => ENNReal.toNNReal (g i)) i ^ p) ^ (1 / p))\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\npush_cast [\u2190 ENNReal.coe_rpow_of_nonneg, le_of_lt pos, le_of_lt (one_div_pos.2 pos)] at this \n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nconvert this using 2 <;> [skip; congr 1; congr 1]\n[GOAL]\ncase neg\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, (f i + g i) ^ p) ^ (1 / p) \u2264 (\u2211 i in s, f i ^ p) ^ (1 / p) + (\u2211 i in s, g i ^ p) ^ (1 / p)\n[PROOFSTEP]\nconvert this using 2\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 \u2211 i in s, (f i + g i) ^ p = \u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p\n[PROOFSTEP]\nskip\n[GOAL]\ncase h.e'_4.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, f i ^ p) ^ (1 / p) = (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_4.h.e'_6\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 (\u2211 i in s, g i ^ p) ^ (1 / p) = (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 \u2211 i in s, (f i + g i) ^ p = \u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_3.h.e'_5\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (f i + g i) ^ p = (\u2191(ENNReal.toNNReal (f i)) + \u2191(ENNReal.toNNReal (g i))) ^ p\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi]\n[GOAL]\ncase h.e'_4.h.e'_5.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 \u2211 i in s, f i ^ p = \u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_4.h.e'_5.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 f i ^ p = \u2191(ENNReal.toNNReal (f i)) ^ p\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi]\n[GOAL]\ncase h.e'_4.h.e'_6.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\n\u22a2 \u2211 i in s, g i ^ p = \u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p\n[PROOFSTEP]\nrefine Finset.sum_congr rfl fun i hi => ?_\n[GOAL]\ncase h.e'_4.h.e'_6.e_a\n\u03b9 : Type u\ns : Finset \u03b9\nf g : \u03b9 \u2192 \u211d\u22650\u221e\np q : \u211d\nhp : 1 \u2264 p\npos : 0 < p\nH' : (\u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2260 \u22a4) \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 g i \u2260 \u22a4\nthis :\n  (\u2211 x in s, (\u2191(ENNReal.toNNReal (f x)) + \u2191(ENNReal.toNNReal (g x))) ^ p) ^ (1 / p) \u2264\n    (\u2211 x in s, \u2191(ENNReal.toNNReal (f x)) ^ p) ^ (1 / p) + (\u2211 x in s, \u2191(ENNReal.toNNReal (g x)) ^ p) ^ (1 / p)\ni : \u03b9\nhi : i \u2208 s\n\u22a2 g i ^ p = \u2191(ENNReal.toNNReal (g i)) ^ p\n[PROOFSTEP]\nsimp [H'.1 i hi, H'.2 i hi]\n", "meta": {"mathlib_filename": "Mathlib.Analysis.MeanInequalities", "llama_tokens": 62944, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8175744806385543, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.5028801340206301}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\n\u22a2 dickson k a 2 = X ^ 2 - \u2191C a * (3 - \u2191k)\n[PROOFSTEP]\nsimp only [dickson, sq]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\n\u22a2 dickson k a (n + 2) = X * dickson k a (n + 1) - \u2191C a * dickson k a n\n[PROOFSTEP]\nrw [dickson]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\nh : 2 \u2264 n\n\u22a2 dickson k a n = X * dickson k a (n - 1) - \u2191C a * dickson k a (n - 2)\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := Nat.exists_eq_add_of_le h\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 dickson k a (2 + n) = X * dickson k a (2 + n - 1) - \u2191C a * dickson k a (2 + n - 2)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 dickson k a (n + 2) = X * dickson k a (n + 2 - 1) - \u2191C a * dickson k a (n + 2 - 2)\n[PROOFSTEP]\nexact dickson_add_two k a n\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nf : R \u2192+* S\n\u22a2 map f (dickson k a 0) = dickson k (\u2191f a) 0\n[PROOFSTEP]\nsimp_rw [dickson_zero, Polynomial.map_sub, Polynomial.map_nat_cast, Polynomial.map_ofNat]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nf : R \u2192+* S\n\u22a2 map f (dickson k a 1) = dickson k (\u2191f a) 1\n[PROOFSTEP]\nsimp only [dickson_one, map_X]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f (dickson k a (n + 2)) = dickson k (\u2191f a) (n + 2)\n[PROOFSTEP]\nsimp only [dickson_add_two, Polynomial.map_sub, Polynomial.map_mul, map_X, map_C]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nf : R \u2192+* S\nn : \u2115\n\u22a2 X * map f (dickson k a (n + 1)) - \u2191C (\u2191f a) * map f (dickson k a n) =\n    X * dickson k (\u2191f a) (n + 1) - \u2191C (\u2191f a) * dickson k (\u2191f a) n\n[PROOFSTEP]\nrw [map_dickson f n, map_dickson f (n + 1)]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\n\u22a2 dickson 2 0 0 = X ^ 0\n[PROOFSTEP]\nsimp only [dickson_zero, pow_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\n\u22a2 3 - \u21912 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\n\u22a2 dickson 2 0 1 = X ^ 1\n[PROOFSTEP]\nsimp only [dickson_one, pow_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\n\u22a2 dickson 2 0 (n + 2) = X ^ (n + 2)\n[PROOFSTEP]\nsimp only [dickson_add_two, C_0, zero_mul, sub_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nn : \u2115\n\u22a2 X * dickson 2 0 (n + 1) = X ^ (n + 2)\n[PROOFSTEP]\nrw [dickson_two_zero (n + 1), pow_add X (n + 1) 1, mul_comm, pow_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\n\u22a2 eval (x + y) (dickson 1 1 0) = x ^ 0 + y ^ 0\n[PROOFSTEP]\nsuffices eval (x + y) 2 = 2 by convert this <;> norm_num\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nthis : eval (x + y) 2 = 2\n\u22a2 eval (x + y) (dickson 1 1 0) = x ^ 0 + y ^ 0\n[PROOFSTEP]\nconvert this\n[GOAL]\ncase h.e'_2.h.e'_4\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nthis : eval (x + y) 2 = 2\n\u22a2 dickson 1 1 0 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nthis : eval (x + y) 2 = 2\n\u22a2 x ^ 0 + y ^ 0 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\n\u22a2 eval (x + y) 2 = 2\n[PROOFSTEP]\nexact eval_nat_cast\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\n\u22a2 eval (x + y) (dickson 1 1 1) = x ^ 1 + y ^ 1\n[PROOFSTEP]\nsimp only [eval_X, dickson_one, pow_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n\u22a2 eval (x + y) (dickson 1 1 (n + 2)) = x ^ (n + 2) + y ^ (n + 2)\n[PROOFSTEP]\nsimp only [eval_sub, eval_mul, dickson_one_one_eval_add_inv x y h _, eval_X, dickson_add_two, C_1, eval_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n\u22a2 (x + y) * (x ^ (n + 1) + y ^ (n + 1)) - 1 * (x ^ n + y ^ n) = x ^ (n + 2) + y ^ (n + 2)\n[PROOFSTEP]\nconv_lhs => simp only [pow_succ, add_mul, mul_add, h, \u2190 mul_assoc, mul_comm y x, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n| (x + y) * (x ^ (n + 1) + y ^ (n + 1)) - 1 * (x ^ n + y ^ n)\n[PROOFSTEP]\nsimp only [pow_succ, add_mul, mul_add, h, \u2190 mul_assoc, mul_comm y x, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n| (x + y) * (x ^ (n + 1) + y ^ (n + 1)) - 1 * (x ^ n + y ^ n)\n[PROOFSTEP]\nsimp only [pow_succ, add_mul, mul_add, h, \u2190 mul_assoc, mul_comm y x, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n| (x + y) * (x ^ (n + 1) + y ^ (n + 1)) - 1 * (x ^ n + y ^ n)\n[PROOFSTEP]\nsimp only [pow_succ, add_mul, mul_add, h, \u2190 mul_assoc, mul_comm y x, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na x y : R\nh : x * y = 1\nn : \u2115\n\u22a2 x * x * x ^ n + x ^ n + (y ^ n + y * y * y ^ n) - (x ^ n + y ^ n) = x ^ (n + 2) + y ^ (n + 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 2 * \u2191C \u215f2 = 1\n[PROOFSTEP]\nrw [two_mul, \u2190 C_add, invOf_two_add_invOf_two, C_1]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 \u2191C \u215f2 * 2 = 1\n[PROOFSTEP]\nrw [mul_comm, two_mul_C_half_eq_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 dickson 1 1 0 = 2 * comp (Chebyshev.T R 0) (\u2191C \u215f2 * X)\n[PROOFSTEP]\nsimp only [Chebyshev.T_zero, mul_one, one_comp, dickson_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 3 - \u21911 = 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 dickson 1 1 1 = 2 * comp (Chebyshev.T R 1) (\u2191C \u215f2 * X)\n[PROOFSTEP]\nrw [dickson_one, Chebyshev.T_one, X_comp, \u2190 mul_assoc, two_mul_C_half_eq_one, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 dickson 1 1 (n + 2) = 2 * comp (Chebyshev.T R (n + 2)) (\u2191C \u215f2 * X)\n[PROOFSTEP]\nrw [dickson_add_two, C_1, Chebyshev.T_add_two, dickson_one_one_eq_chebyshev_T (n + 1), dickson_one_one_eq_chebyshev_T n,\n  sub_comp, mul_comp, mul_comp, X_comp, ofNat_comp]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * (2 * comp (Chebyshev.T R (n + 1)) (\u2191C \u215f2 * X)) - 1 * (2 * comp (Chebyshev.T R n) (\u2191C \u215f2 * X)) =\n    2 * (\u21912 * (\u2191C \u215f2 * X) * comp (Chebyshev.T R (n + 1)) (\u2191C \u215f2 * X) - comp (Chebyshev.T R n) (\u2191C \u215f2 * X))\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc, Nat.cast_ofNat, two_mul_C_half_eq_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * 2 * comp (Chebyshev.T R (n + 1)) (\u2191C \u215f2 * X) - 1 * 2 * comp (Chebyshev.T R n) (\u2191C \u215f2 * X) =\n    2 * (1 * X * comp (Chebyshev.T R (n + 1)) (\u2191C \u215f2 * X) - comp (Chebyshev.T R n) (\u2191C \u215f2 * X))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 Chebyshev.T R n = \u2191C \u215f2 * comp (dickson 1 1 n) (2 * X)\n[PROOFSTEP]\nrw [dickson_one_one_eq_chebyshev_T, mul_comp, ofNat_comp, comp_assoc, mul_comp, C_comp, X_comp]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 Chebyshev.T R n = \u2191C \u215f2 * (\u21912 * comp (Chebyshev.T R n) (\u2191C \u215f2 * (2 * X)))\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc, Nat.cast_ofNat, C_half_mul_two_eq_one, one_mul, comp_X]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\n\u22a2 dickson 1 1 (m * n) = comp (dickson 1 1 m) (dickson 1 1 n)\n[PROOFSTEP]\nhave h : (1 : R) = Int.castRingHom R 1\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\n\u22a2 1 = \u2191(Int.castRingHom R) 1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 dickson 1 1 (m * n) = comp (dickson 1 1 m) (dickson 1 1 n)\n[PROOFSTEP]\nsimp only [eq_intCast, Int.cast_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 dickson 1 1 (m * n) = comp (dickson 1 1 m) (dickson 1 1 n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 dickson 1 (\u2191(Int.castRingHom R) 1) (m * n) =\n    comp (dickson 1 (\u2191(Int.castRingHom R) 1) m) (dickson 1 (\u2191(Int.castRingHom R) 1) n)\n[PROOFSTEP]\nsimp only [\u2190 map_dickson (Int.castRingHom R), \u2190 map_comp]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 map (Int.castRingHom R) (dickson 1 1 (m * n)) = map (Int.castRingHom R) (comp (dickson 1 1 m) (dickson 1 1 n))\n[PROOFSTEP]\ncongr 1\n[GOAL]\ncase e_a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 dickson 1 1 (m * n) = comp (dickson 1 1 m) (dickson 1 1 n)\n[PROOFSTEP]\napply map_injective (Int.castRingHom \u211a) Int.cast_injective\n[GOAL]\ncase e_a.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 map (Int.castRingHom \u211a) (dickson 1 1 (m * n)) = map (Int.castRingHom \u211a) (comp (dickson 1 1 m) (dickson 1 1 n))\n[PROOFSTEP]\nsimp only [map_dickson, map_comp, eq_intCast, Int.cast_one, dickson_one_one_eq_chebyshev_T, Chebyshev.T_mul, two_mul, \u2190\n  add_comp]\n[GOAL]\ncase e_a.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 comp (comp (Chebyshev.T \u211a m + Chebyshev.T \u211a m) (Chebyshev.T \u211a n)) (\u2191C \u215f2 * X) =\n    comp (comp (Chebyshev.T \u211a m + Chebyshev.T \u211a m) (\u2191C \u215f2 * X)) (comp (Chebyshev.T \u211a n + Chebyshev.T \u211a n) (\u2191C \u215f2 * X))\n[PROOFSTEP]\nsimp only [\u2190 two_mul, \u2190 comp_assoc]\n[GOAL]\ncase e_a.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 comp (comp (2 * Chebyshev.T \u211a m) (Chebyshev.T \u211a n)) (\u2191C \u215f2 * X) =\n    comp (comp (comp (2 * Chebyshev.T \u211a m) (\u2191C \u215f2 * X)) (2 * Chebyshev.T \u211a n)) (\u2191C \u215f2 * X)\n[PROOFSTEP]\napply eval\u2082_congr rfl rfl\n[GOAL]\ncase e_a.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 comp (2 * Chebyshev.T \u211a m) (Chebyshev.T \u211a n) = comp (comp (2 * Chebyshev.T \u211a m) (\u2191C \u215f2 * X)) (2 * Chebyshev.T \u211a n)\n[PROOFSTEP]\nrw [comp_assoc]\n[GOAL]\ncase e_a.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 comp (2 * Chebyshev.T \u211a m) (Chebyshev.T \u211a n) = comp (2 * Chebyshev.T \u211a m) (comp (\u2191C \u215f2 * X) (2 * Chebyshev.T \u211a n))\n[PROOFSTEP]\napply eval\u2082_congr rfl _ rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\nh : 1 = \u2191(Int.castRingHom R) 1\n\u22a2 Chebyshev.T \u211a n = comp (\u2191C \u215f2 * X) (2 * Chebyshev.T \u211a n)\n[PROOFSTEP]\nrw [mul_comp, C_comp, X_comp, \u2190 mul_assoc, C_half_mul_two_eq_one, one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nk : \u2115\na : R\nm n : \u2115\n\u22a2 comp (dickson 1 1 m) (dickson 1 1 n) = comp (dickson 1 1 n) (dickson 1 1 m)\n[PROOFSTEP]\nrw [\u2190 dickson_one_one_mul, mul_comm, dickson_one_one_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 dickson 1 1 p = X ^ p\n[PROOFSTEP]\nobtain \u27e8K, _, _, H\u27e9 : \u2203 (K : Type) (_ : Field K), \u2203 _ : CharP K p, Infinite K :=\n  by\n  let K := FractionRing (Polynomial (ZMod p))\n  let f : ZMod p \u2192+* K := (algebraMap _ (FractionRing _)).comp C\n  have : CharP K p := by\n    rw [\u2190 f.charP_iff_charP]\n    infer_instance\n  haveI : Infinite K :=\n    Infinite.of_injective (algebraMap (Polynomial (ZMod p)) (FractionRing (Polynomial (ZMod p))))\n      (IsFractionRing.injective _ _)\n  refine' \u27e8K, _, _, _\u27e9 <;> infer_instance\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\n\u22a2 \u2203 K x x, Infinite K\n[PROOFSTEP]\nlet K := FractionRing (Polynomial (ZMod p))\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\n\u22a2 \u2203 K x x, Infinite K\n[PROOFSTEP]\nlet f : ZMod p \u2192+* K := (algebraMap _ (FractionRing _)).comp C\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\n\u22a2 \u2203 K x x, Infinite K\n[PROOFSTEP]\nhave : CharP K p := by\n  rw [\u2190 f.charP_iff_charP]\n  infer_instance\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\n\u22a2 CharP K p\n[PROOFSTEP]\nrw [\u2190 f.charP_iff_charP]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\n\u22a2 CharP (ZMod p) p\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\nthis : CharP K p\n\u22a2 \u2203 K x x, Infinite K\n[PROOFSTEP]\nhaveI : Infinite K :=\n  Infinite.of_injective (algebraMap (Polynomial (ZMod p)) (FractionRing (Polynomial (ZMod p))))\n    (IsFractionRing.injective _ _)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\nthis\u271d : CharP K p\nthis : Infinite K\n\u22a2 \u2203 K x x, Infinite K\n[PROOFSTEP]\nrefine' \u27e8K, _, _, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\nthis\u271d : CharP K p\nthis : Infinite K\n\u22a2 Field K\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\nthis\u271d : CharP K p\nthis : Infinite K\n\u22a2 CharP K p\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase refine'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type := FractionRing (ZMod p)[X]\nf : ZMod p \u2192+* K := RingHom.comp (algebraMap (ZMod p)[X] (FractionRing (ZMod p)[X])) C\nthis\u271d : CharP K p\nthis : Infinite K\n\u22a2 Infinite K\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 dickson 1 1 p = X ^ p\n[PROOFSTEP]\napply map_injective (ZMod.castHom (dvd_refl p) K) (RingHom.injective _)\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 map (ZMod.castHom (_ : p \u2223 p) K) (dickson 1 1 p) = map (ZMod.castHom (_ : p \u2223 p) K) (X ^ p)\n[PROOFSTEP]\nrw [map_dickson, Polynomial.map_pow, map_X]\n[GOAL]\ncase intro.intro.intro.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 dickson 1 (\u2191(ZMod.castHom (_ : p \u2223 p) K) 1) p = X ^ p\n[PROOFSTEP]\napply eq_of_infinite_eval_eq\n[GOAL]\ncase intro.intro.intro.a.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 Set.Infinite {x | eval x (dickson 1 (\u2191(ZMod.castHom (_ : p \u2223 p) K) 1) p) = eval x (X ^ p)}\n[PROOFSTEP]\napply @Set.Infinite.mono _ {x : K | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n[GOAL]\ncase intro.intro.intro.a.h.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0} \u2286 {x | eval x (dickson 1 (\u2191(ZMod.castHom (_ : p \u2223 p) K) 1) p) = eval x (X ^ p)}\n[PROOFSTEP]\nrintro _ \u27e8x, rfl, hx\u27e9\n[GOAL]\ncase intro.intro.intro.a.h.h.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\nx : K\nhx : x \u2260 0\n\u22a2 x + x\u207b\u00b9 \u2208 {x | eval x (dickson 1 (\u2191(ZMod.castHom (_ : p \u2223 p) K) 1) p) = eval x (X ^ p)}\n[PROOFSTEP]\nsimp only [eval_X, eval_pow, Set.mem_setOf_eq, @add_pow_char K _ p,\n  dickson_one_one_eval_add_inv _ _ (mul_inv_cancel hx), inv_pow, ZMod.castHom_apply, ZMod.cast_one']\n  -- Now we need to show that the set of such `x` is infinite.\n    -- If the set is finite, then we will show that `K` is also finite.\n[GOAL]\ncase intro.intro.intro.a.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\n\u22a2 Set.Infinite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n[PROOFSTEP]\nintro h\n[GOAL]\ncase intro.intro.intro.a.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Infinite K\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Set.infinite_univ_iff] at H \n[GOAL]\ncase intro.intro.intro.a.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 False\n[PROOFSTEP]\napply H\n[GOAL]\ncase intro.intro.intro.a.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 Set.Finite Set.univ\n[PROOFSTEP]\nsuffices (Set.univ : Set K) = {x : K | \u2203 y : K, x = y + y\u207b\u00b9 \u2227 y \u2260 0} >>= fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n  by\n  rw [this]\n  clear this\n  refine'\n    h.biUnion fun x _ =>\n      _\n        -- The following quadratic polynomial has as solutions the `y` for which `x = y + y\u207b\u00b9`.\n  let \u03c6 : K[X] := X ^ 2 - C x * X + 1\n  have h\u03c6 : \u03c6 \u2260 0 := by\n    intro H\n    have : \u03c6.eval 0 = 0 := by rw [H, eval_zero]\n    simpa [eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul, mul_zero, sq, zero_add, one_ne_zero]\n  classical\n  convert (\u03c6.roots \u222a {0}).toFinset.finite_toSet using 1\n  ext1 y\n  simp only [Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union, mem_roots h\u03c6, IsRoot,\n    eval_add, eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one, Multiset.mem_singleton]\n  by_cases hy : y = 0\n  \u00b7 simp only [hy, eq_self_iff_true, or_true_iff]\n  apply or_congr _ Iff.rfl\n  rw [\u2190 mul_left_inj' hy, eq_comm, \u2190 sub_eq_zero, add_mul, inv_mul_cancel hy]\n  apply eq_iff_eq_cancel_right.mpr\n  ring\n    -- Finally, we prove the claim that our finite union of finite sets covers all of `K`.\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nthis :\n  Set.univ = do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n\u22a2 Set.Finite Set.univ\n[PROOFSTEP]\nrw [this]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nthis :\n  Set.univ = do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n\u22a2 Set.Finite do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n[PROOFSTEP]\nclear this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 Set.Finite do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n[PROOFSTEP]\nrefine'\n  h.biUnion fun x _ =>\n    _\n      -- The following quadratic polynomial has as solutions the `y` for which `x = y + y\u207b\u00b9`.\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 Set.Finite ((fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x)\n[PROOFSTEP]\nlet \u03c6 : K[X] := X ^ 2 - C x * X + 1\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\n\u22a2 Set.Finite ((fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x)\n[PROOFSTEP]\nhave h\u03c6 : \u03c6 \u2260 0 := by\n  intro H\n  have : \u03c6.eval 0 = 0 := by rw [H, eval_zero]\n  simpa [eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul, mul_zero, sq, zero_add, one_ne_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\n\u22a2 \u03c6 \u2260 0\n[PROOFSTEP]\nintro H\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH\u271d : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nH : \u03c6 = 0\n\u22a2 False\n[PROOFSTEP]\nhave : \u03c6.eval 0 = 0 := by rw [H, eval_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH\u271d : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nH : \u03c6 = 0\n\u22a2 eval 0 \u03c6 = 0\n[PROOFSTEP]\nrw [H, eval_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH\u271d : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nH : \u03c6 = 0\nthis : eval 0 \u03c6 = 0\n\u22a2 False\n[PROOFSTEP]\nsimpa [eval_X, eval_one, eval_pow, eval_sub, sub_zero, eval_add, eval_mul, mul_zero, sq, zero_add, one_ne_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\n\u22a2 Set.Finite ((fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x)\n[PROOFSTEP]\nclassical\nconvert (\u03c6.roots \u222a {0}).toFinset.finite_toSet using 1\next1 y\nsimp only [Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union, mem_roots h\u03c6, IsRoot, eval_add,\n  eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one, Multiset.mem_singleton]\nby_cases hy : y = 0\n\u00b7 simp only [hy, eq_self_iff_true, or_true_iff]\napply or_congr _ Iff.rfl\nrw [\u2190 mul_left_inj' hy, eq_comm, \u2190 sub_eq_zero, add_mul, inv_mul_cancel hy]\napply eq_iff_eq_cancel_right.mpr\nring\n  -- Finally, we prove the claim that our finite union of finite sets covers all of `K`.\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\n\u22a2 Set.Finite ((fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x)\n[PROOFSTEP]\nconvert (\u03c6.roots \u222a {0}).toFinset.finite_toSet using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\n\u22a2 (fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x = \u2191(Multiset.toFinset (roots \u03c6 \u222a {0}))\n[PROOFSTEP]\next1 y\n[GOAL]\ncase h.e'_2.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\n\u22a2 y \u2208 (fun x => {y | x = y + y\u207b\u00b9 \u2228 y = 0}) x \u2194 y \u2208 \u2191(Multiset.toFinset (roots \u03c6 \u222a {0}))\n[PROOFSTEP]\nsimp only [Multiset.mem_toFinset, Set.mem_setOf_eq, Finset.mem_coe, Multiset.mem_union, mem_roots h\u03c6, IsRoot, eval_add,\n  eval_sub, eval_pow, eval_mul, eval_X, eval_C, eval_one, Multiset.mem_singleton]\n[GOAL]\ncase h.e'_2.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\n\u22a2 x = y + y\u207b\u00b9 \u2228 y = 0 \u2194 y ^ 2 - x * y + 1 = 0 \u2228 y = 0\n[PROOFSTEP]\nby_cases hy : y = 0\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\nhy : y = 0\n\u22a2 x = y + y\u207b\u00b9 \u2228 y = 0 \u2194 y ^ 2 - x * y + 1 = 0 \u2228 y = 0\n[PROOFSTEP]\nsimp only [hy, eq_self_iff_true, or_true_iff]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\nhy : \u00acy = 0\n\u22a2 x = y + y\u207b\u00b9 \u2228 y = 0 \u2194 y ^ 2 - x * y + 1 = 0 \u2228 y = 0\n[PROOFSTEP]\napply or_congr _ Iff.rfl\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\nhy : \u00acy = 0\n\u22a2 x = y + y\u207b\u00b9 \u2194 y ^ 2 - x * y + 1 = 0\n[PROOFSTEP]\nrw [\u2190 mul_left_inj' hy, eq_comm, \u2190 sub_eq_zero, add_mul, inv_mul_cancel hy]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\nhy : \u00acy = 0\n\u22a2 y * y + 1 - x * y = 0 \u2194 y ^ 2 - x * y + 1 = 0\n[PROOFSTEP]\napply eq_iff_eq_cancel_right.mpr\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nx\u271d : x \u2208 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u03c6 : K[X] := X ^ 2 - \u2191C x * X + 1\nh\u03c6 : \u03c6 \u2260 0\ny : K\nhy : \u00acy = 0\n\u22a2 y * y + 1 - x * y = y ^ 2 - x * y + 1\n[PROOFSTEP]\nring\n  -- Finally, we prove the claim that our finite union of finite sets covers all of `K`.\n[GOAL]\ncase intro.intro.intro.a.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 Set.univ = do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n[PROOFSTEP]\napply (Set.eq_univ_of_forall _).symm\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n\u22a2 \u2200 (x : K),\n    x \u2208 do\n      let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n      {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n[PROOFSTEP]\nintro x\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\n\u22a2 x \u2208 do\n    let x \u2190 {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\n    {y | x = y + y\u207b\u00b9 \u2228 y = 0}\n[PROOFSTEP]\nsimp only [exists_prop, Set.mem_iUnion, Set.bind_def, Ne.def, Set.mem_setOf_eq]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\n\u22a2 \u2203 i, (\u2203 y, i = y + y\u207b\u00b9 \u2227 \u00acy = 0) \u2227 (i = x + x\u207b\u00b9 \u2228 x = 0)\n[PROOFSTEP]\nby_cases hx : x = 0\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nhx : x = 0\n\u22a2 \u2203 i, (\u2203 y, i = y + y\u207b\u00b9 \u2227 \u00acy = 0) \u2227 (i = x + x\u207b\u00b9 \u2228 x = 0)\n[PROOFSTEP]\nsimp only [hx, and_true_iff, eq_self_iff_true, inv_zero, or_true_iff]\n[GOAL]\ncase pos\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nhx : x = 0\n\u22a2 \u2203 i y, i = y + y\u207b\u00b9 \u2227 \u00acy = 0\n[PROOFSTEP]\nexact \u27e8_, 1, rfl, one_ne_zero\u27e9\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nhx : \u00acx = 0\n\u22a2 \u2203 i, (\u2203 y, i = y + y\u207b\u00b9 \u2227 \u00acy = 0) \u2227 (i = x + x\u207b\u00b9 \u2228 x = 0)\n[PROOFSTEP]\nsimp only [hx, or_false_iff, exists_eq_right]\n[GOAL]\ncase neg\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nK : Type\nw\u271d\u00b9 : Field K\nw\u271d : CharP K p\nH : Set.Infinite Set.univ\nh : Set.Finite {x | \u2203 y, x = y + y\u207b\u00b9 \u2227 y \u2260 0}\nx : K\nhx : \u00acx = 0\n\u22a2 \u2203 y, x + x\u207b\u00b9 = y + y\u207b\u00b9 \u2227 \u00acy = 0\n[PROOFSTEP]\nexact \u27e8_, rfl, hx\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\n\u22a2 dickson 1 1 p = X ^ p\n[PROOFSTEP]\nhave h : (1 : R) = ZMod.castHom (dvd_refl p) R 1\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\n\u22a2 1 = \u2191(ZMod.castHom (_ : p \u2223 p) R) 1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nh : 1 = \u2191(ZMod.castHom (_ : p \u2223 p) R) 1\n\u22a2 dickson 1 1 p = X ^ p\n[PROOFSTEP]\nsimp only [ZMod.castHom_apply, ZMod.cast_one']\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\nk : \u2115\na : R\np : \u2115\ninst\u271d\u00b9 : Fact (Nat.Prime p)\ninst\u271d : CharP R p\nh : 1 = \u2191(ZMod.castHom (_ : p \u2223 p) R) 1\n\u22a2 dickson 1 1 p = X ^ p\n[PROOFSTEP]\nrw [h, \u2190 map_dickson (ZMod.castHom (dvd_refl p) R), dickson_one_one_zmod_p, Polynomial.map_pow, map_X]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Dickson", "llama_tokens": 18483, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7718435083355187, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.5027440291389833}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u22a2 prod (a :: l) = foldl (fun x x_1 => x * x_1) (a * 1) l\n[PROOFSTEP]\nsimp only [List.prod, foldl_cons, one_mul, mul_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u22a2 prod (l\u2081 ++ l\u2082) = foldl (fun x x_1 => x * x_1) (foldl (fun x x_1 => x * x_1) 1 l\u2081 * 1) l\u2082\n[PROOFSTEP]\nsimp [List.prod]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u22a2 prod (concat l a) = prod l * a\n[PROOFSTEP]\nrw [concat_eq_append, prod_append, prod_singleton]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List (List M)\n\u22a2 prod (join l) = prod (map prod l)\n[PROOFSTEP]\ninduction l <;> [rfl; simp only [*, List.join, map, prod_append, prod_cons]]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List (List M)\n\u22a2 prod (join l) = prod (map prod l)\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u22a2 prod (join []) = prod (map prod [])\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nhead\u271d : List M\ntail\u271d : List (List M)\ntail_ih\u271d : prod (join tail\u271d) = prod (map prod tail\u271d)\n\u22a2 prod (join (head\u271d :: tail\u271d)) = prod (map prod (head\u271d :: tail\u271d))\n[PROOFSTEP]\nsimp only [*, List.join, map, prod_append, prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na\u271d a : M\nl : List M\n\u22a2 prod (a :: l) = foldr (fun x x_1 => x * x_1) 1 (a :: l)\n[PROOFSTEP]\nrw [prod_cons, foldr_cons, prod_eq_foldr]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d : M\nn : \u2115\na : M\n\u22a2 prod (replicate n a) = a ^ n\n[PROOFSTEP]\ninduction' n with n ih\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d a : M\n\u22a2 prod (replicate Nat.zero a) = a ^ Nat.zero\n[PROOFSTEP]\nrw [pow_zero]\n[GOAL]\ncase zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d a : M\n\u22a2 prod (replicate Nat.zero a) = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase succ\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d a : M\nn : \u2115\nih : prod (replicate n a) = a ^ n\n\u22a2 prod (replicate (Nat.succ n) a) = a ^ Nat.succ n\n[PROOFSTEP]\nrw [replicate_succ, prod_cons, ih, pow_succ]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\nm : M\nh : \u2200 (x : M), x \u2208 l \u2192 x = m\n\u22a2 prod l = m ^ length l\n[PROOFSTEP]\nrw [\u2190 prod_replicate, \u2190 List.eq_replicate.mpr \u27e8rfl, h\u27e9]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na\u271d : M\nl\u271d : List \u03b9\nr : M \u2192 N \u2192 Prop\nf : \u03b9 \u2192 M\ng : \u03b9 \u2192 N\nh\u2081 : r 1 1\nh\u2082 : \u2200 \u2983i : \u03b9\u2984 \u2983a : M\u2984 \u2983b : N\u2984, r a b \u2192 r (f i * a) (g i * b)\na : \u03b9\nl : List \u03b9\nhl : r (prod (map f l)) (prod (map g l))\n\u22a2 r (prod (map f (a :: l))) (prod (map g (a :: l)))\n[PROOFSTEP]\nsimp only [map_cons, prod_cons, h\u2082 hl]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\nF : Type u_9\ninst\u271d : MonoidHomClass F M N\nf : F\n\u22a2 prod (map (\u2191f) l) = \u2191f (prod l)\n[PROOFSTEP]\nsimp only [prod, foldl_map, \u2190 map_one f]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\nF : Type u_9\ninst\u271d : MonoidHomClass F M N\nf : F\n\u22a2 foldl (fun x y => x * \u2191f y) (\u2191f 1) l = \u2191f (foldl (fun x x_1 => x * x_1) 1 l)\n[PROOFSTEP]\nexact l.foldl_hom f (\u00b7 * \u00b7) (\u00b7 * f \u00b7) 1 (fun x y => (map_mul f x y).symm)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List \u03b9\nf : M \u2192 N \u2192 P\nhf : \u2200 (a b : M) (c d : N), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf\u2081 : \u03b9 \u2192 M\nf\u2082 : \u03b9 \u2192 N\n\u22a2 prod (map (fun i => f (f\u2081 i) (f\u2082 i)) l) = f (prod (map f\u2081 l)) (prod (map f\u2082 l))\n[PROOFSTEP]\nsimp only [prod, foldl_map]\n  -- Porting note: next 3 lines used to be\n    -- convert l.foldl_hom\u2082 (fun a b => f a b) _ _ _ _ _ fun a b i => _\n    -- \u00b7 exact hf'.symm\n    -- \u00b7 exact hf _ _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List \u03b9\nf : M \u2192 N \u2192 P\nhf : \u2200 (a b : M) (c d : N), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf\u2081 : \u03b9 \u2192 M\nf\u2082 : \u03b9 \u2192 N\n\u22a2 foldl (fun x y => x * f (f\u2081 y) (f\u2082 y)) 1 l = f (foldl (fun x y => x * f\u2081 y) 1 l) (foldl (fun x y => x * f\u2082 y) 1 l)\n[PROOFSTEP]\nrw [\u2190 l.foldl_hom\u2082 (fun a b => f a b), hf']\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List \u03b9\nf : M \u2192 N \u2192 P\nhf : \u2200 (a b : M) (c d : N), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf\u2081 : \u03b9 \u2192 M\nf\u2082 : \u03b9 \u2192 N\n\u22a2 \u2200 (a : M) (b : N) (i : \u03b9), f (a * f\u2081 i) (b * f\u2082 i) = f a b * f (f\u2081 i) (f\u2082 i)\n[PROOFSTEP]\nintros\n[GOAL]\ncase h\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List \u03b9\nf : M \u2192 N \u2192 P\nhf : \u2200 (a b : M) (c d : N), f (a * b) (c * d) = f a c * f b d\nhf' : f 1 1 = 1\nf\u2081 : \u03b9 \u2192 M\nf\u2082 : \u03b9 \u2192 N\na\u271d : M\nb\u271d : N\ni\u271d : \u03b9\n\u22a2 f (a\u271d * f\u2081 i\u271d) (b\u271d * f\u2082 i\u271d) = f a\u271d b\u271d * f (f\u2081 i\u271d) (f\u2082 i\u271d)\n[PROOFSTEP]\nexact hf _ _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : HasDistribNeg \u03b1\nl : List \u03b1\n\u22a2 prod (map Neg.neg l) = (-1) ^ length l * prod l\n[PROOFSTEP]\nsimpa only [id_eq, neg_mul, one_mul, map_const', prod_replicate, map_id] using @prod_map_mul \u03b1 \u03b1 _ l (fun _ => -1) id\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG\u271d : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nL : List \u03b9\nf : \u03b9 \u2192 M\nG : Type u_9\ninst\u271d : MonoidHomClass G M N\ng : G\n\u22a2 prod (map (\u2191g \u2218 f) L) = \u2191g (prod (map f L))\n[PROOFSTEP]\nrw [\u2190 prod_hom, map_map]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nx\u271d : \u2200 (m : M), m \u2208 [] \u2192 IsUnit m\n\u22a2 IsUnit (prod [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nu : \u2200 (m : M), m \u2208 h :: t \u2192 IsUnit m\n\u22a2 IsUnit (prod (h :: t))\n[PROOFSTEP]\nsimp only [List.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nu : \u2200 (m : M), m \u2208 h :: t \u2192 IsUnit m\n\u22a2 IsUnit (h * prod t)\n[PROOFSTEP]\nexact IsUnit.mul (u h (mem_cons_self h t)) (prod_isUnit fun m mt => u m (mem_cons_of_mem h mt))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d : CommMonoid \u03b1\nL : List \u03b1\n\u22a2 IsUnit (prod L) \u2194 \u2200 (m : \u03b1), m \u2208 L \u2192 IsUnit m\n[PROOFSTEP]\nrefine' \u27e8fun h => _, prod_isUnit\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d : CommMonoid \u03b1\nL : List \u03b1\nh : IsUnit (prod L)\n\u22a2 \u2200 (m : \u03b1), m \u2208 L \u2192 IsUnit m\n[PROOFSTEP]\ninduction' L with m L ih\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d : CommMonoid \u03b1\nL : List \u03b1\nh\u271d : IsUnit (prod L)\nh : IsUnit (prod [])\n\u22a2 \u2200 (m : \u03b1), m \u2208 [] \u2192 IsUnit m\n[PROOFSTEP]\nexact fun m' h' => False.elim (not_mem_nil m' h')\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d : CommMonoid \u03b1\nL\u271d : List \u03b1\nh\u271d : IsUnit (prod L\u271d)\nm : \u03b1\nL : List \u03b1\nih : IsUnit (prod L) \u2192 \u2200 (m : \u03b1), m \u2208 L \u2192 IsUnit m\nh : IsUnit (prod (m :: L))\n\u22a2 \u2200 (m_1 : \u03b1), m_1 \u2208 m :: L \u2192 IsUnit m_1\n[PROOFSTEP]\nrw [prod_cons, IsUnit.mul_iff] at h \n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u03b1 : Type u_9\ninst\u271d : CommMonoid \u03b1\nL\u271d : List \u03b1\nh\u271d : IsUnit (prod L\u271d)\nm : \u03b1\nL : List \u03b1\nih : IsUnit (prod L) \u2192 \u2200 (m : \u03b1), m \u2208 L \u2192 IsUnit m\nh : IsUnit m \u2227 IsUnit (prod L)\n\u22a2 \u2200 (m_1 : \u03b1), m_1 \u2208 m :: L \u2192 IsUnit m_1\n[PROOFSTEP]\nexact fun m' h' => Or.elim (eq_or_mem_of_mem_cons h') (fun H => H.substr h.1) fun H => ih h.2 _ H\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\ni : \u2115\n\u22a2 prod (take i []) * prod (drop i []) = prod []\n[PROOFSTEP]\nsimp [Nat.zero_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nL : List M\n\u22a2 prod (take 0 L) * prod (drop 0 L) = prod L\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nn : \u2115\n\u22a2 prod (take (n + 1) (h :: t)) * prod (drop (n + 1) (h :: t)) = prod (h :: t)\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nn : \u2115\n\u22a2 prod (h :: take n t) * prod (drop n t) = prod (h :: t)\n[PROOFSTEP]\nrw [prod_cons, prod_cons, mul_assoc, prod_take_mul_prod_drop t]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\ni : \u2115\np : i < length []\n\u22a2 prod (take (i + 1) []) = prod (take i []) * nthLe [] i p\n[PROOFSTEP]\ncases p\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nn : \u2115\np : n + 1 < length (h :: t)\n\u22a2 prod (take (n + 1 + 1) (h :: t)) = prod (take (n + 1) (h :: t)) * nthLe (h :: t) (n + 1) p\n[PROOFSTEP]\ndsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nn : \u2115\np : n + 1 < length (h :: t)\n\u22a2 prod (h :: take (n + 1) t) = prod (h :: take n t) * nthLe (h :: t) (n + 1) p\n[PROOFSTEP]\nrw [prod_cons, prod_cons, prod_take_succ t n (Nat.lt_of_succ_lt_succ p), mul_assoc, nthLe_cons,\n  dif_neg (Nat.add_one_ne_zero _)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na h : M\nt : List M\nn : \u2115\np : n + 1 < length (h :: t)\n\u22a2 h * (prod (take n t) * nthLe t n (_ : n < length t)) =\n    h * (prod (take n t) * nthLe t (n + 1 - 1) (_ : n + 1 - 1 < length t))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nL : List M\nh : prod L \u2260 1\n\u22a2 0 < length L\n[PROOFSTEP]\ncases L\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : prod [] \u2260 1\n\u22a2 0 < length []\n[PROOFSTEP]\ncontrapose h\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : \u00ac0 < length []\n\u22a2 \u00acprod [] \u2260 1\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na head\u271d : M\ntail\u271d : List M\nh : prod (head\u271d :: tail\u271d) \u2260 1\n\u22a2 0 < length (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d x : M\nxs : List M\na : M\n\u22a2 prod (set (x :: xs) 0 a) =\n    (prod (take 0 (x :: xs)) * if 0 < length (x :: xs) then a else 1) * prod (drop (0 + 1) (x :: xs))\n[PROOFSTEP]\nsimp [set]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d x : M\nxs : List M\ni : \u2115\na : M\n\u22a2 prod (set (x :: xs) (i + 1) a) =\n    (prod (take (i + 1) (x :: xs)) * if i + 1 < length (x :: xs) then a else 1) * prod (drop (i + 1 + 1) (x :: xs))\n[PROOFSTEP]\nsimp [set, prod_set xs i a, mul_assoc, Nat.succ_eq_add_one]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nx\u271d\u00b9 : \u2115\nx\u271d : M\n\u22a2 prod (set [] x\u271d\u00b9 x\u271d) = (prod (take x\u271d\u00b9 []) * if x\u271d\u00b9 < length [] then x\u271d else 1) * prod (drop (x\u271d\u00b9 + 1) [])\n[PROOFSTEP]\nsimp [set, (Nat.zero_le _).not_lt, Nat.zero_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\n\u22a2 Option.getD (get? l 0) 1 * prod (tail l) = prod l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\n\u22a2 Option.getD (get? [] 0) 1 * prod (tail []) = prod []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na head\u271d : M\ntail\u271d : List M\n\u22a2 Option.getD (get? (head\u271d :: tail\u271d) 0) 1 * prod (tail (head\u271d :: tail\u271d)) = prod (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d : Inhabited M\nl : List M\nh : l \u2260 []\n\u22a2 headI l * prod (tail l) = prod l\n[PROOFSTEP]\ncases l <;> [contradiction; simp]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d : Inhabited M\nl : List M\nh : l \u2260 []\n\u22a2 headI l * prod (tail l) = prod l\n[PROOFSTEP]\ncases l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\ninst\u271d : Inhabited M\nh : [] \u2260 []\n\u22a2 headI [] * prod (tail []) = prod []\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\ninst\u271d\u00b9 : Monoid P\nl l\u2081 l\u2082 : List M\na : M\ninst\u271d : Inhabited M\nhead\u271d : M\ntail\u271d : List M\nh : head\u271d :: tail\u271d \u2260 []\n\u22a2 headI (head\u271d :: tail\u271d) * prod (tail (head\u271d :: tail\u271d)) = prod (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\ny : M\nh : \u2200 (x : M), x \u2208 l \u2192 Commute y x\n\u22a2 Commute y (prod l)\n[PROOFSTEP]\ninduction' l with z l IH\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\ny : M\nh\u271d : \u2200 (x : M), x \u2208 l \u2192 Commute y x\nh : \u2200 (x : M), x \u2208 [] \u2192 Commute y x\n\u22a2 Commute y (prod [])\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\nl\u271d : List M\ny : M\nh\u271d : \u2200 (x : M), x \u2208 l\u271d \u2192 Commute y x\nz : M\nl : List M\nIH : (\u2200 (x : M), x \u2208 l \u2192 Commute y x) \u2192 Commute y (prod l)\nh : \u2200 (x : M), x \u2208 z :: l \u2192 Commute y x\n\u22a2 Commute y (prod (z :: l))\n[PROOFSTEP]\nrw [List.forall_mem_cons] at h \n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\nl\u271d : List M\ny : M\nh\u271d : \u2200 (x : M), x \u2208 l\u271d \u2192 Commute y x\nz : M\nl : List M\nIH : (\u2200 (x : M), x \u2208 l \u2192 Commute y x) \u2192 Commute y (prod l)\nh : Commute y z \u2227 \u2200 (x : M), x \u2208 l \u2192 Commute y x\n\u22a2 Commute y (prod (z :: l))\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\nl\u271d : List M\ny : M\nh\u271d : \u2200 (x : M), x \u2208 l\u271d \u2192 Commute y x\nz : M\nl : List M\nIH : (\u2200 (x : M), x \u2208 l \u2192 Commute y x) \u2192 Commute y (prod l)\nh : Commute y z \u2227 \u2200 (x : M), x \u2208 l \u2192 Commute y x\n\u22a2 Commute y (z * prod l)\n[PROOFSTEP]\nexact Commute.mul_right h.1 (IH h.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\nh : Forall\u2082 (fun x x_1 => x \u2264 x_1) l\u2081 l\u2082\n\u22a2 prod l\u2081 \u2264 prod l\u2082\n[PROOFSTEP]\ninduction' h with a b la lb hab ih ih'\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\n\u22a2 prod [] \u2264 prod []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na\u271d : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\na b : M\nla lb : List M\nhab : a \u2264 b\nih : Forall\u2082 (fun x x_1 => x \u2264 x_1) la lb\nih' : prod la \u2264 prod lb\n\u22a2 prod (a :: la) \u2264 prod (b :: lb)\n[PROOFSTEP]\nsimpa only [prod_cons] using mul_le_mul' hab ih'\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\nh : l\u2081 <+ l\u2082\nh\u2081 : \u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a\n\u22a2 prod l\u2081 \u2264 prod l\u2082\n[PROOFSTEP]\ninduction h\n[GOAL]\ncase slnil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\nh\u2081 : \u2200 (a : M), a \u2208 [] \u2192 1 \u2264 a\n\u22a2 prod [] \u2264 prod []\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List M\na\u271d\u00b9 : M\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : (\u2200 (a : M), a \u2208 l\u2082\u271d \u2192 1 \u2264 a) \u2192 prod l\u2081\u271d \u2264 prod l\u2082\u271d\nh\u2081 : \u2200 (a : M), a \u2208 a\u271d\u00b9 :: l\u2082\u271d \u2192 1 \u2264 a\n\u22a2 prod l\u2081\u271d \u2264 prod (a\u271d\u00b9 :: l\u2082\u271d)\ncase cons\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List M\na\u271d\u00b9 : M\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : (\u2200 (a : M), a \u2208 l\u2082\u271d \u2192 1 \u2264 a) \u2192 prod l\u2081\u271d \u2264 prod l\u2082\u271d\nh\u2081 : \u2200 (a : M), a \u2208 a\u271d\u00b9 :: l\u2082\u271d \u2192 1 \u2264 a\n\u22a2 prod (a\u271d\u00b9 :: l\u2081\u271d) \u2264 prod (a\u271d\u00b9 :: l\u2082\u271d)\n[PROOFSTEP]\ncase slnil => rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\nh\u2081 : \u2200 (a : M), a \u2208 [] \u2192 1 \u2264 a\n\u22a2 prod [] \u2264 prod []\n[PROOFSTEP]\ncase slnil => rfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d l\u2082\u271d : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 : List M\nh\u2081 : \u2200 (a : M), a \u2208 [] \u2192 1 \u2264 a\n\u22a2 prod [] \u2264 prod []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List M\na\u271d\u00b9 : M\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : (\u2200 (a : M), a \u2208 l\u2082\u271d \u2192 1 \u2264 a) \u2192 prod l\u2081\u271d \u2264 prod l\u2082\u271d\nh\u2081 : \u2200 (a : M), a \u2208 a\u271d\u00b9 :: l\u2082\u271d \u2192 1 \u2264 a\n\u22a2 prod l\u2081\u271d \u2264 prod (a\u271d\u00b9 :: l\u2082\u271d)\ncase cons\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List M\na\u271d\u00b9 : M\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : (\u2200 (a : M), a \u2208 l\u2082\u271d \u2192 1 \u2264 a) \u2192 prod l\u2081\u271d \u2264 prod l\u2082\u271d\nh\u2081 : \u2200 (a : M), a \u2208 a\u271d\u00b9 :: l\u2082\u271d \u2192 1 \u2264 a\n\u22a2 prod (a\u271d\u00b9 :: l\u2081\u271d) \u2264 prod (a\u271d\u00b9 :: l\u2082\u271d)\n[PROOFSTEP]\ncase cons l\u2081 l\u2082 a _ ih' =>\n  simp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n  exact (ih' h\u2081.2).trans (le_mul_of_one_le_left' h\u2081.1)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : \u2200 (a_1 : M), a_1 \u2208 a :: l\u2082 \u2192 1 \u2264 a_1\n\u22a2 prod l\u2081 \u2264 prod (a :: l\u2082)\n[PROOFSTEP]\ncase cons l\u2081 l\u2082 a _ ih' =>\n  simp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n  exact (ih' h\u2081.2).trans (le_mul_of_one_le_left' h\u2081.1)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : \u2200 (a_1 : M), a_1 \u2208 a :: l\u2082 \u2192 1 \u2264 a_1\n\u22a2 prod l\u2081 \u2264 prod (a :: l\u2082)\n[PROOFSTEP]\nsimp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : 1 \u2264 a \u2227 \u2200 (x : M), x \u2208 l\u2082 \u2192 1 \u2264 x\n\u22a2 prod l\u2081 \u2264 a * prod l\u2082\n[PROOFSTEP]\nexact (ih' h\u2081.2).trans (le_mul_of_one_le_left' h\u2081.1)\n[GOAL]\ncase cons\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081 l\u2082 l\u2081\u271d l\u2082\u271d : List M\na\u271d\u00b9 : M\na\u271d : l\u2081\u271d <+ l\u2082\u271d\na_ih\u271d : (\u2200 (a : M), a \u2208 l\u2082\u271d \u2192 1 \u2264 a) \u2192 prod l\u2081\u271d \u2264 prod l\u2082\u271d\nh\u2081 : \u2200 (a : M), a \u2208 a\u271d\u00b9 :: l\u2082\u271d \u2192 1 \u2264 a\n\u22a2 prod (a\u271d\u00b9 :: l\u2081\u271d) \u2264 prod (a\u271d\u00b9 :: l\u2082\u271d)\n[PROOFSTEP]\ncase cons\u2082 l\u2081 l\u2082 a _ ih' =>\n  simp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n  exact mul_le_mul_left' (ih' h\u2081.2) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : \u2200 (a_1 : M), a_1 \u2208 a :: l\u2082 \u2192 1 \u2264 a_1\n\u22a2 prod (a :: l\u2081) \u2264 prod (a :: l\u2082)\n[PROOFSTEP]\ncase cons\u2082 l\u2081 l\u2082 a _ ih' =>\n  simp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n  exact mul_le_mul_left' (ih' h\u2081.2) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : \u2200 (a_1 : M), a_1 \u2208 a :: l\u2082 \u2192 1 \u2264 a_1\n\u22a2 prod (a :: l\u2081) \u2264 prod (a :: l\u2082)\n[PROOFSTEP]\nsimp only [prod_cons, forall_mem_cons] at h\u2081 \u22a2\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl l\u2081\u271d\u00b9 l\u2082\u271d\u00b9 : List M\na\u271d\u00b9 : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u2081\u271d l\u2082\u271d l\u2081 l\u2082 : List M\na : M\na\u271d : l\u2081 <+ l\u2082\nih' : (\u2200 (a : M), a \u2208 l\u2082 \u2192 1 \u2264 a) \u2192 prod l\u2081 \u2264 prod l\u2082\nh\u2081 : 1 \u2264 a \u2227 \u2200 (x : M), x \u2208 l\u2082 \u2192 1 \u2264 x\n\u22a2 a * prod l\u2081 \u2264 a * prod l\u2082\n[PROOFSTEP]\nexact mul_le_mul_left' (ih' h\u2081.2) _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nf g : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i\n\u22a2 Forall\u2082 (fun x x_1 => x \u2264 x_1) (map f l) (map g l)\n[PROOFSTEP]\nsimpa\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081 : \u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i\nh\u2082 : \u2203 i, i \u2208 l \u2227 f i < g i\n\u22a2 prod (map f l) < prod (map g l)\n[PROOFSTEP]\ninduction' l with i l ihl\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081\u271d : \u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i\nh\u2082\u271d : \u2203 i, i \u2208 l \u2227 f i < g i\nh\u2081 : \u2200 (i : \u03b9), i \u2208 [] \u2192 f i \u2264 g i\nh\u2082 : \u2203 i, i \u2208 [] \u2227 f i < g i\n\u22a2 prod (map f []) < prod (map g [])\n[PROOFSTEP]\nrcases h\u2082 with \u27e8_, \u27e8\u27e9, _\u27e9\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u271d : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081\u271d : \u2200 (i : \u03b9), i \u2208 l\u271d \u2192 f i \u2264 g i\nh\u2082\u271d : \u2203 i, i \u2208 l\u271d \u2227 f i < g i\ni : \u03b9\nl : List \u03b9\nihl : (\u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i) \u2192 (\u2203 i, i \u2208 l \u2227 f i < g i) \u2192 prod (map f l) < prod (map g l)\nh\u2081 : \u2200 (i_1 : \u03b9), i_1 \u2208 i :: l \u2192 f i_1 \u2264 g i_1\nh\u2082 : \u2203 i_1, i_1 \u2208 i :: l \u2227 f i_1 < g i_1\n\u22a2 prod (map f (i :: l)) < prod (map g (i :: l))\n[PROOFSTEP]\nsimp only [forall_mem_cons, exists_mem_cons, map_cons, prod_cons] at h\u2081 h\u2082 \u22a2\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u271d : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081\u271d : \u2200 (i : \u03b9), i \u2208 l\u271d \u2192 f i \u2264 g i\nh\u2082\u271d : \u2203 i, i \u2208 l\u271d \u2227 f i < g i\ni : \u03b9\nl : List \u03b9\nihl : (\u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i) \u2192 (\u2203 i, i \u2208 l \u2227 f i < g i) \u2192 prod (map f l) < prod (map g l)\nh\u2081 : f i \u2264 g i \u2227 \u2200 (x : \u03b9), x \u2208 l \u2192 f x \u2264 g x\nh\u2082 : f i < g i \u2228 \u2203 x, x \u2208 l \u2227 f x < g x\n\u22a2 f i * prod (map f l) < g i * prod (map g l)\n[PROOFSTEP]\ncases h\u2082\n[GOAL]\ncase cons.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u271d : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081\u271d : \u2200 (i : \u03b9), i \u2208 l\u271d \u2192 f i \u2264 g i\nh\u2082 : \u2203 i, i \u2208 l\u271d \u2227 f i < g i\ni : \u03b9\nl : List \u03b9\nihl : (\u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i) \u2192 (\u2203 i, i \u2208 l \u2227 f i < g i) \u2192 prod (map f l) < prod (map g l)\nh\u2081 : f i \u2264 g i \u2227 \u2200 (x : \u03b9), x \u2208 l \u2192 f x \u2264 g x\nh\u271d : f i < g i\n\u22a2 f i * prod (map f l) < g i * prod (map g l)\n[PROOFSTEP]\nexact mul_lt_mul_of_lt_of_le \u2039_\u203a (prod_le_prod' h\u2081.2)\n[GOAL]\ncase cons.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d\u00b9 l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : Preorder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl\u271d : List \u03b9\nf g : \u03b9 \u2192 M\nh\u2081\u271d : \u2200 (i : \u03b9), i \u2208 l\u271d \u2192 f i \u2264 g i\nh\u2082 : \u2203 i, i \u2208 l\u271d \u2227 f i < g i\ni : \u03b9\nl : List \u03b9\nihl : (\u2200 (i : \u03b9), i \u2208 l \u2192 f i \u2264 g i) \u2192 (\u2203 i, i \u2208 l \u2227 f i < g i) \u2192 prod (map f l) < prod (map g l)\nh\u2081 : f i \u2264 g i \u2227 \u2200 (x : \u03b9), x \u2208 l \u2192 f x \u2264 g x\nh\u271d : \u2203 x, x \u2208 l \u2227 f x < g x\n\u22a2 f i * prod (map f l) < g i * prod (map g l)\n[PROOFSTEP]\nexact mul_lt_mul_of_le_of_lt h\u2081.1 <| ihl h\u2081.2 \u2039_\u203a\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b2 : Preorder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nn : M\nh : \u2200 (x : M), x \u2208 l \u2192 x \u2264 n\n\u22a2 prod l \u2264 n ^ length l\n[PROOFSTEP]\nsimpa only [map_id'', map_const', prod_replicate] using prod_le_prod' h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nf g : \u03b9 \u2192 M\nh : prod (map f l) < prod (map g l)\n\u22a2 \u2203 i, i \u2208 l \u2227 f i < g i\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2075 : Monoid M\ninst\u271d\u2074 : Monoid N\ninst\u271d\u00b3 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b2 : LinearOrder M\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nf g : \u03b9 \u2192 M\nh : \u2200 (i : \u03b9), i \u2208 l \u2192 g i \u2264 f i\n\u22a2 prod (map g l) \u2264 prod (map f l)\n[PROOFSTEP]\nexact prod_le_prod' h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : LinearOrder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nhl : l \u2260 []\nf g : \u03b9 \u2192 M\nh : prod (map f l) \u2264 prod (map g l)\n\u22a2 \u2203 x, x \u2208 l \u2227 f x \u2264 g x\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2077 : Monoid M\ninst\u271d\u2076 : Monoid N\ninst\u271d\u2075 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u2074 : LinearOrder M\ninst\u271d\u00b3 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d\u00b2 : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\ninst\u271d\u00b9 : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List \u03b9\nhl : l \u2260 []\nf g : \u03b9 \u2192 M\nh : \u2200 (x : \u03b9), x \u2208 l \u2192 g x < f x\n\u22a2 prod (map g l) < prod (map f l)\n[PROOFSTEP]\nexact prod_lt_prod_of_ne_nil hl _ _ h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081 : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod l\n[PROOFSTEP]\ninduction' l with hd tl ih\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhl\u2081 : \u2200 (x : M), x \u2208 [] \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 prod (hd :: tl)\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u2074 : Monoid M\ninst\u271d\u00b3 : Monoid N\ninst\u271d\u00b2 : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nl : List M\nhl\u2081\u271d : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 1 \u2264 prod tl\nhl\u2081 : \u2200 (x : M), x \u2208 hd :: tl \u2192 1 \u2264 x\n\u22a2 1 \u2264 hd * prod tl\n[PROOFSTEP]\nexact one_le_mul (hl\u2081 hd (mem_cons_self hd tl)) (ih fun x h => hl\u2081 x (mem_cons_of_mem hd h))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL : List M\u2080\nh : 0 \u2208 L\n\u22a2 prod L = 0\n[PROOFSTEP]\ninduction' L with a L ihL\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL : List M\u2080\nh\u271d : 0 \u2208 L\nh : 0 \u2208 []\n\u22a2 prod [] = 0\n[PROOFSTEP]\nexact absurd h (not_mem_nil _)\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL\u271d : List M\u2080\nh\u271d : 0 \u2208 L\u271d\na : M\u2080\nL : List M\u2080\nihL : 0 \u2208 L \u2192 prod L = 0\nh : 0 \u2208 a :: L\n\u22a2 prod (a :: L) = 0\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL\u271d : List M\u2080\nh\u271d : 0 \u2208 L\u271d\na : M\u2080\nL : List M\u2080\nihL : 0 \u2208 L \u2192 prod L = 0\nh : 0 \u2208 a :: L\n\u22a2 a * prod L = 0\n[PROOFSTEP]\ncases' mem_cons.1 h with ha hL\n[GOAL]\ncase cons.inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL\u271d : List M\u2080\nh\u271d : 0 \u2208 L\u271d\na : M\u2080\nL : List M\u2080\nihL : 0 \u2208 L \u2192 prod L = 0\nh : 0 \u2208 a :: L\nha : 0 = a\n\u22a2 a * prod L = 0\ncase cons.inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : MonoidWithZero M\u2080\nL\u271d : List M\u2080\nh\u271d : 0 \u2208 L\u271d\na : M\u2080\nL : List M\u2080\nihL : 0 \u2208 L \u2192 prod L = 0\nh : 0 \u2208 a :: L\nhL : 0 \u2208 L\n\u22a2 a * prod L = 0\n[PROOFSTEP]\nexacts [mul_eq_zero_of_left ha.symm _, mul_eq_zero_of_right _ (ihL hL)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\nL : List M\u2080\n\u22a2 prod L = 0 \u2194 0 \u2208 L\n[PROOFSTEP]\ninduction' L with a L ihL\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\n\u22a2 prod [] = 0 \u2194 0 \u2208 []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : MonoidWithZero M\u2080\ninst\u271d\u00b9 : Nontrivial M\u2080\ninst\u271d : NoZeroDivisors M\u2080\na : M\u2080\nL : List M\u2080\nihL : prod L = 0 \u2194 0 \u2208 L\n\u22a2 prod (a :: L) = 0 \u2194 0 \u2208 a :: L\n[PROOFSTEP]\nrw [prod_cons, mul_eq_zero, ihL, mem_cons, eq_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Group G\n\u22a2 (prod [])\u207b\u00b9 = prod (reverse (map (fun x => x\u207b\u00b9) []))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Group G\nx : G\nxs : List G\n\u22a2 (prod (x :: xs))\u207b\u00b9 = prod (reverse (map (fun x => x\u207b\u00b9) (x :: xs)))\n[PROOFSTEP]\nsimp [prod_inv_reverse xs]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Group G\n\u22a2 \u2200 (L : List G), prod (reverse L) = (prod (map (fun x => x\u207b\u00b9) L))\u207b\u00b9\n[PROOFSTEP]\nsimp [prod_inv_reverse]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Group G\nx : G\nxs : List G\nx\u271d : 0 < length (x :: xs)\n\u22a2 prod (drop (0 + 1) (x :: xs)) = (nthLe (x :: xs) 0 x\u271d)\u207b\u00b9 * prod (drop 0 (x :: xs))\n[PROOFSTEP]\nsimp [nthLe]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\n\u22a2 (prod [])\u207b\u00b9 = prod (map (fun x => x\u207b\u00b9) [])\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\nx : G\nxs : List G\n\u22a2 (prod (x :: xs))\u207b\u00b9 = prod (map (fun x => x\u207b\u00b9) (x :: xs))\n[PROOFSTEP]\nsimp [mul_comm, prod_inv xs]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\nL : List G\nn : \u2115\na : G\n\u22a2 prod (set L n a) = prod L * if hn : n < length L then (nthLe L n hn)\u207b\u00b9 * a else 1\n[PROOFSTEP]\nrefine (prod_set L n a).trans ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\nL : List G\nn : \u2115\na : G\n\u22a2 (prod (take n L) * if n < length L then a else 1) * prod (drop (n + 1) L) =\n    prod L * if hn : n < length L then (nthLe L n hn)\u207b\u00b9 * a else 1\n[PROOFSTEP]\nsplit_ifs with hn\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\nL : List G\nn : \u2115\na : G\nhn : n < length L\n\u22a2 prod (take n L) * a * prod (drop (n + 1) L) = prod L * ((nthLe L n hn)\u207b\u00b9 * a)\n[PROOFSTEP]\nrw [mul_comm _ a, mul_assoc a, prod_drop_succ L n hn, mul_comm _ (drop n L).prod, \u2190 mul_assoc (take n L).prod,\n  prod_take_mul_prod_drop, mul_comm a, mul_assoc]\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup G\nL : List G\nn : \u2115\na : G\nhn : \u00acn < length L\n\u22a2 prod (take n L) * 1 * prod (drop (n + 1) L) = prod L * 1\n[PROOFSTEP]\nsimp only [take_all_of_le (le_of_not_lt hn), prod_nil, mul_one, drop_eq_nil_of_le ((le_of_not_lt hn).trans n.le_succ)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\n\u22a2 L = L'\n[PROOFSTEP]\nrefine ext_get h fun i h\u2081 h\u2082 => ?_\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }\n[PROOFSTEP]\nhave : (L.take (i + 1)).prod = (L'.take (i + 1)).prod := h' _ (Nat.succ_le_of_lt h\u2081)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take (i + 1) L) = prod (take (i + 1) L')\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }\n[PROOFSTEP]\nrw [prod_take_succ L i h\u2081, prod_take_succ L' i h\u2082, h' i (le_of_lt h\u2081)] at this \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : LeftCancelMonoid M\nL L' : List M\nh : length L = length L'\nh' : \u2200 (i : \u2115), i \u2264 length L \u2192 prod (take i L) = prod (take i L')\ni : \u2115\nh\u2081 : i < length L\nh\u2082 : i < length L'\nthis : prod (take i L') * nthLe L i h\u2081 = prod (take i L') * nthLe L' i h\u2082\n\u22a2 get L { val := i, isLt := h\u2081 } = get L' { val := i, isLt := h\u2082 }\n[PROOFSTEP]\nconvert mul_left_cancel this\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nL : List M\n\u22a2 Monotone fun i => prod (take i L)\n[PROOFSTEP]\nrefine' monotone_nat_of_le_succ fun n => _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nL : List M\nn : \u2115\n\u22a2 prod (take n L) \u2264 prod (take (n + 1) L)\n[PROOFSTEP]\ncases' lt_or_le n L.length with h h\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nL : List M\nn : \u2115\nh : n < length L\n\u22a2 prod (take n L) \u2264 prod (take (n + 1) L)\n[PROOFSTEP]\nrw [prod_take_succ _ _ h]\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nL : List M\nn : \u2115\nh : n < length L\n\u22a2 prod (take n L) \u2264 prod (take n L) * nthLe L n h\n[PROOFSTEP]\nexact le_self_mul\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CanonicallyOrderedMonoid M\nL : List M\nn : \u2115\nh : length L \u2264 n\n\u22a2 prod (take n L) \u2264 prod (take (n + 1) L)\n[PROOFSTEP]\nsimp [take_all_of_le h, take_all_of_le (le_trans h (Nat.le_succ _))]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nb : M\nh : \u2200 (x : M), x \u2208 [b] \u2192 1 < x\nx\u271d : [b] \u2260 []\n\u22a2 1 < prod [b]\n[PROOFSTEP]\nsimpa using h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nhl\u2081 : \u2200 (x : M), x \u2208 a :: b :: l \u2192 1 < x\nx\u271d : a :: b :: l \u2260 []\n\u22a2 1 < prod (a :: b :: l)\n[PROOFSTEP]\nsimp only [forall_eq_or_imp, List.mem_cons] at hl\u2081 \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\n\u22a2 1 < prod (a :: b :: l)\n[PROOFSTEP]\nrw [List.prod_cons]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\n\u22a2 1 < a * prod (b :: l)\n[PROOFSTEP]\napply one_lt_mul_of_lt_of_le' hl\u2081.1\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\n\u22a2 1 \u2264 prod (b :: l)\n[PROOFSTEP]\napply le_of_lt ((b :: l).one_lt_prod_of_one_lt _ (l.cons_ne_nil b))\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\n\u22a2 \u2200 (x : M), x \u2208 b :: l \u2192 1 < x\n[PROOFSTEP]\nintro x hx\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\nx : M\nhx : x \u2208 b :: l\n\u22a2 1 < x\n[PROOFSTEP]\ncases hx\n[GOAL]\ncase head\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\n\u22a2 1 < b\n[PROOFSTEP]\nexact hl\u2081.2.1\n[GOAL]\ncase tail\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\na b : M\nl : List M\nx\u271d : a :: b :: l \u2260 []\nhl\u2081 : 1 < a \u2227 1 < b \u2227 \u2200 (a : M), a \u2208 l \u2192 1 < a\nx : M\na\u271d : Mem x l\n\u22a2 1 < x\n[PROOFSTEP]\nexact hl\u2081.2.2 _ \u2039_\u203a\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nl : List M\nhl\u2081 : \u2200 (x : M), x \u2208 l \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 l \u2192 x \u2264 prod l\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhl\u2081 : \u2200 (x : M), x \u2208 [] \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 [] \u2192 x \u2264 prod []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : \u2200 (x : M), x \u2208 head\u271d :: tail\u271d \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 head\u271d :: tail\u271d \u2192 x \u2264 prod (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp_rw [prod_cons, forall_mem_cons] at hl\u2081 \u22a2\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 head\u271d \u2264 head\u271d * prod tail\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 head\u271d * prod tail\u271d\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.left\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 head\u271d \u2264 head\u271d * prod tail\u271d\ncase cons.right\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 head\u271d * prod tail\u271d\n[PROOFSTEP]\ncase cons.left => exact le_mul_of_one_le_right' (one_le_prod_of_one_le hl\u2081.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 head\u271d \u2264 head\u271d * prod tail\u271d\n[PROOFSTEP]\ncase cons.left => exact le_mul_of_one_le_right' (one_le_prod_of_one_le hl\u2081.2)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 head\u271d \u2264 head\u271d * prod tail\u271d\n[PROOFSTEP]\nexact le_mul_of_one_le_right' (one_le_prod_of_one_le hl\u2081.2)\n[GOAL]\ncase cons.right\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhead\u271d : M\ntail\u271d : List M\ntail_ih\u271d : (\u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 prod tail\u271d\nhl\u2081 : 1 \u2264 head\u271d \u2227 \u2200 (x : M), x \u2208 tail\u271d \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 tail\u271d \u2192 x \u2264 head\u271d * prod tail\u271d\n[PROOFSTEP]\ncase cons.right hd tl ih => exact fun x H => le_mul_of_one_le_of_le hl\u2081.1 (ih hl\u2081.right x H)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tl \u2192 x \u2264 prod tl\nhl\u2081 : 1 \u2264 hd \u2227 \u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 tl \u2192 x \u2264 hd * prod tl\n[PROOFSTEP]\ncase cons.right hd tl ih => exact fun x H => le_mul_of_one_le_of_le hl\u2081.1 (ih hl\u2081.right x H)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : OrderedCommMonoid M\nhd : M\ntl : List M\nih : (\u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x) \u2192 \u2200 (x : M), x \u2208 tl \u2192 x \u2264 prod tl\nhl\u2081 : 1 \u2264 hd \u2227 \u2200 (x : M), x \u2208 tl \u2192 1 \u2264 x\n\u22a2 \u2200 (x : M), x \u2208 tl \u2192 x \u2264 hd * prod tl\n[PROOFSTEP]\nexact fun x H => le_mul_of_one_le_of_le hl\u2081.1 (ih hl\u2081.right x H)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl : List M\nhl : \u2200 (x : M), x \u2208 l \u2192 x = 1\n\u22a2 prod l = 1\n[PROOFSTEP]\ninduction' l with i l hil\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl : List M\nhl\u271d : \u2200 (x : M), x \u2208 l \u2192 x = 1\nhl : \u2200 (x : M), x \u2208 [] \u2192 x = 1\n\u22a2 prod [] = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl\u271d : List M\nhl\u271d : \u2200 (x : M), x \u2208 l\u271d \u2192 x = 1\ni : M\nl : List M\nhil : (\u2200 (x : M), x \u2208 l \u2192 x = 1) \u2192 prod l = 1\nhl : \u2200 (x : M), x \u2208 i :: l \u2192 x = 1\n\u22a2 prod (i :: l) = 1\n[PROOFSTEP]\nrw [List.prod_cons, hil fun x hx => hl _ (mem_cons_of_mem i hx), hl _ (mem_cons_self i l), one_mul]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl : List M\nh : prod l \u2260 1\n\u22a2 \u2203 x, x \u2208 l \u2227 x \u2260 1\n[PROOFSTEP]\nsimpa only [not_forall, exists_prop] using mt prod_eq_one h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : LinearOrder N\nf : M \u2192 N\nh0 : f 0 \u2264 0\nhadd : \u2200 (x y : M), f (x + y) \u2264 max (f x) (f y)\nl : List M\n\u22a2 f (sum l) \u2264 foldr max 0 (map f l)\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : LinearOrder N\nf : M \u2192 N\nh0 : f 0 \u2264 0\nhadd : \u2200 (x y : M), f (x + y) \u2264 max (f x) (f y)\n\u22a2 f (sum []) \u2264 foldr max 0 (map f [])\n[PROOFSTEP]\nsimpa using h0\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : LinearOrder N\nf : M \u2192 N\nh0 : f 0 \u2264 0\nhadd : \u2200 (x y : M), f (x + y) \u2264 max (f x) (f y)\nhd : M\ntl : List M\nIH : f (sum tl) \u2264 foldr max 0 (map f tl)\n\u22a2 f (sum (hd :: tl)) \u2264 foldr max 0 (map f (hd :: tl))\n[PROOFSTEP]\nsimp only [List.sum_cons, List.foldr_map, List.foldr] at IH \u22a2\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b2 : AddMonoid M\ninst\u271d\u00b9 : AddMonoid N\ninst\u271d : LinearOrder N\nf : M \u2192 N\nh0 : f 0 \u2264 0\nhadd : \u2200 (x y : M), f (x + y) \u2264 max (f x) (f y)\nhd : M\ntl : List M\nIH : f (sum tl) \u2264 foldr (fun x y => max (f x) y) 0 tl\n\u22a2 f (hd + sum tl) \u2264 max (f hd) (foldr (fun x y => max (f x) y) 0 tl)\n[PROOFSTEP]\nexact (hadd _ _).trans (max_le_max le_rfl IH)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : CommMonoid M\na b : M\nl : List M\nh : a \u2208 b :: l\n\u22a2 a * prod (List.erase (b :: l) a) = prod (b :: l)\n[PROOFSTEP]\nobtain rfl | \u27e8ne, h\u27e9 := Decidable.List.eq_or_ne_mem_of_mem h\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : CommMonoid M\na : M\nl : List M\nh : a \u2208 a :: l\n\u22a2 a * prod (List.erase (a :: l) a) = prod (a :: l)\n[PROOFSTEP]\nsimp only [List.erase, if_pos, prod_cons, beq_self_eq_true]\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq M\ninst\u271d : CommMonoid M\na b : M\nl : List M\nh\u271d : a \u2208 b :: l\nne : a \u2260 b\nh : a \u2208 l\n\u22a2 a * prod (List.erase (b :: l) a) = prod (b :: l)\n[PROOFSTEP]\nsimp only [List.erase, beq_false_of_ne ne.symm, prod_cons, prod_erase h, mul_left_comm a b]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 M\na b : \u03b9\nl : List \u03b9\nh : a \u2208 b :: l\n\u22a2 f a * prod (map f (List.erase (b :: l) a)) = prod (map f (b :: l))\n[PROOFSTEP]\nobtain rfl | \u27e8ne, h\u27e9 := Decidable.List.eq_or_ne_mem_of_mem h\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 M\na : \u03b9\nl : List \u03b9\nh : a \u2208 a :: l\n\u22a2 f a * prod (map f (List.erase (a :: l) a)) = prod (map f (a :: l))\n[PROOFSTEP]\nsimp only [map, erase_cons_head, prod_cons]\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : CommMonoid M\nf : \u03b9 \u2192 M\na b : \u03b9\nl : List \u03b9\nh\u271d : a \u2208 b :: l\nne : a \u2260 b\nh : a \u2208 l\n\u22a2 f a * prod (map f (List.erase (b :: l) a)) = prod (map f (b :: l))\n[PROOFSTEP]\nsimp only [map, erase_cons_tail _ ne.symm, prod_cons, prod_map_erase _ h, mul_left_comm (f a) (f b)]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : StrictOrderedSemiring R\nl : List R\nh : \u2200 (a : R), a \u2208 l \u2192 0 < a\n\u22a2 0 < prod l\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : StrictOrderedSemiring R\nl : List R\nh\u271d : \u2200 (a : R), a \u2208 l \u2192 0 < a\nh : \u2200 (a : R), a \u2208 [] \u2192 0 < a\n\u22a2 0 < prod []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : StrictOrderedSemiring R\nl\u271d : List R\nh\u271d : \u2200 (a : R), a \u2208 l\u271d \u2192 0 < a\na : R\nl : List R\nih : (\u2200 (a : R), a \u2208 l \u2192 0 < a) \u2192 0 < prod l\nh : \u2200 (a_1 : R), a_1 \u2208 a :: l \u2192 0 < a_1\n\u22a2 0 < prod (a :: l)\n[PROOFSTEP]\nrw [prod_cons]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : StrictOrderedSemiring R\nl\u271d : List R\nh\u271d : \u2200 (a : R), a \u2208 l\u271d \u2192 0 < a\na : R\nl : List R\nih : (\u2200 (a : R), a \u2208 l \u2192 0 < a) \u2192 0 < prod l\nh : \u2200 (a_1 : R), a_1 \u2208 a :: l \u2192 0 < a_1\n\u22a2 0 < a * prod l\n[PROOFSTEP]\nexact mul_pos (h _ <| mem_cons_self _ _) (ih fun a ha => h a <| mem_cons_of_mem _ ha)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b9 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d : Nontrivial \u03b1\n\u22a2 0 < prod [] \u2194 \u2200 (x : \u03b1), x \u2208 [] \u2192 0 < x\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\n\u03b1 : Type u_9\ninst\u271d\u00b9 : CanonicallyOrderedCommSemiring \u03b1\ninst\u271d : Nontrivial \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 0 < prod (x :: xs) \u2194 \u2200 (x_1 : \u03b1), x_1 \u2208 x :: xs \u2192 0 < x_1\n[PROOFSTEP]\nsimp_rw [prod_cons, forall_mem_cons, CanonicallyOrderedCommSemiring.mul_pos, list_prod_pos]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nL : List \u2115\n\u22a2 headI L + sum (tail L) = sum L\n[PROOFSTEP]\ncases L\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\n\u22a2 headI [] + sum (tail []) = sum []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nhead\u271d : \u2115\ntail\u271d : List \u2115\n\u22a2 headI (head\u271d :: tail\u271d) + sum (tail (head\u271d :: tail\u271d)) = sum (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nL : List \u2115\n\u22a2 sum (tail L) = sum L - headI L\n[PROOFSTEP]\nrw [\u2190 headI_add_tail_sum L, add_comm, @add_tsub_cancel_right]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : DivInvMonoid \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 alternatingProd (a :: b :: l) = a / b * alternatingProd l\n[PROOFSTEP]\nrw [div_eq_mul_inv, alternatingProd_cons_cons']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup \u03b1\na : \u03b1\n\u22a2 alternatingProd [a] = a * (alternatingProd [])\u207b\u00b9\n[PROOFSTEP]\nrw [alternatingProd_nil, inv_one, mul_one, alternatingProd_singleton]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup \u03b1\na b : \u03b1\nl : List \u03b1\n\u22a2 alternatingProd (a :: b :: l) = a * (alternatingProd (b :: l))\u207b\u00b9\n[PROOFSTEP]\nrw [alternatingProd_cons_cons', alternatingProd_cons' b l, mul_inv, inv_inv, mul_assoc]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : CommGroup \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 alternatingProd (a :: l) = a / alternatingProd l\n[PROOFSTEP]\nrw [div_eq_mul_inv, alternatingProd_cons']\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nl : List \u2115\nn : \u2115\n\u22a2 sum l % n = sum (map (fun x => x % n) l) % n\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn : \u2115\n\u22a2 sum [] % n = sum (map (fun x => x % n) []) % n\n[PROOFSTEP]\nsimp [Nat.add_mod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn head\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : sum tail\u271d % n = sum (map (fun x => x % n) tail\u271d) % n\n\u22a2 sum (head\u271d :: tail\u271d) % n = sum (map (fun x => x % n) (head\u271d :: tail\u271d)) % n\n[PROOFSTEP]\nsimp [Nat.add_mod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nl : List \u2115\nn : \u2115\n\u22a2 prod l % n = prod (map (fun x => x % n) l) % n\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn : \u2115\n\u22a2 prod [] % n = prod (map (fun x => x % n) []) % n\n[PROOFSTEP]\nsimp [Nat.mul_mod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn head\u271d : \u2115\ntail\u271d : List \u2115\ntail_ih\u271d : prod tail\u271d % n = prod (map (fun x => x % n) tail\u271d) % n\n\u22a2 prod (head\u271d :: tail\u271d) % n = prod (map (fun x => x % n) (head\u271d :: tail\u271d)) % n\n[PROOFSTEP]\nsimp [Nat.mul_mod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nl : List \u2124\nn : \u2124\n\u22a2 sum l % n = sum (map (fun x => x % n) l) % n\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn : \u2124\n\u22a2 sum [] % n = sum (map (fun x => x % n) []) % n\n[PROOFSTEP]\nsimp [Int.add_emod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn head\u271d : \u2124\ntail\u271d : List \u2124\ntail_ih\u271d : sum tail\u271d % n = sum (map (fun x => x % n) tail\u271d) % n\n\u22a2 sum (head\u271d :: tail\u271d) % n = sum (map (fun x => x % n) (head\u271d :: tail\u271d)) % n\n[PROOFSTEP]\nsimp [Int.add_emod, *]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nl : List \u2124\nn : \u2124\n\u22a2 prod l % n = prod (map (fun x => x % n) l) % n\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn : \u2124\n\u22a2 prod [] % n = prod (map (fun x => x % n) []) % n\n[PROOFSTEP]\nsimp [Int.mul_emod, *]\n[GOAL]\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\nn head\u271d : \u2124\ntail\u271d : List \u2124\ntail_ih\u271d : prod tail\u271d % n = prod (map (fun x => x % n) tail\u271d) % n\n\u22a2 prod (head\u271d :: tail\u271d) % n = prod (map (fun x => x % n) (head\u271d :: tail\u271d)) % n\n[PROOFSTEP]\nsimp [Int.mul_emod, *]\n", "meta": {"mathlib_filename": "Mathlib.Data.List.BigOperators.Basic", "llama_tokens": 36438, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.5027106525872485}}
{"text": "[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nt : Term L \u03b1\n\u22a2 relabel id t = t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n_a\u271d : \u03b1\n\u22a2 relabel id (var _a\u271d) = var _a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), relabel id (_ts\u271d a) = _ts\u271d a\n\u22a2 relabel id (func _f\u271d _ts\u271d) = func _f\u271d _ts\u271d\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nt : Term L \u03b1\n\u22a2 relabel g (relabel f t) = relabel (g \u2218 f) t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\n_a\u271d : \u03b1\n\u22a2 relabel g (relabel f (var _a\u271d)) = relabel (g \u2218 f) (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2\ng : \u03b2 \u2192 \u03b3\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), relabel g (relabel f (_ts\u271d a)) = relabel (g \u2218 f) (_ts\u271d a)\n\u22a2 relabel g (relabel f (func _f\u271d _ts\u271d)) = relabel (g \u2218 f) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp [ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\ng : \u03b1 \u2243 \u03b2\nt : Term L \u03b1\n\u22a2 relabel (\u2191g.symm) (relabel (\u2191g) t) = t\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\ng : \u03b1 \u2243 \u03b2\nt : Term L \u03b2\n\u22a2 relabel (\u2191g) (relabel (\u2191g.symm) t) = t\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n\u22a2 Function.LeftInverse varsToConstants constantsToVars\n[PROOFSTEP]\nintro t\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nt : Term (L[[\u03b3]]) \u03b1\n\u22a2 varsToConstants (constantsToVars t) = t\n[PROOFSTEP]\ninduction' t with _ n f _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n_a\u271d : \u03b1\n\u22a2 varsToConstants (constantsToVars (var _a\u271d)) = var _a\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nf : Functions (L[[\u03b3]]) n\n_ts\u271d : Fin n \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin n), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\n\u22a2 varsToConstants (constantsToVars (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\ncases n\n[GOAL]\ncase func.zero\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nf : Functions (L[[\u03b3]]) Nat.zero\n_ts\u271d : Fin Nat.zero \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin Nat.zero), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\n\u22a2 varsToConstants (constantsToVars (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\ncases f\n[GOAL]\ncase func.zero.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n_ts\u271d : Fin Nat.zero \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin Nat.zero), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\nval\u271d : Functions L Nat.zero\n\u22a2 varsToConstants (constantsToVars (func (Sum.inl val\u271d) _ts\u271d)) = func (Sum.inl val\u271d) _ts\u271d\n[PROOFSTEP]\nsimp [constantsToVars, varsToConstants, ih]\n[GOAL]\ncase func.zero.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n_ts\u271d : Fin Nat.zero \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin Nat.zero), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\nval\u271d : Functions (constantsOn \u03b3) Nat.zero\n\u22a2 varsToConstants (constantsToVars (func (Sum.inr val\u271d) _ts\u271d)) = func (Sum.inr val\u271d) _ts\u271d\n[PROOFSTEP]\nsimp [constantsToVars, varsToConstants, Constants.term]\n[GOAL]\ncase func.succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\nf : Functions (L[[\u03b3]]) (Nat.succ n\u271d)\n_ts\u271d : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin (Nat.succ n\u271d)), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\n\u22a2 varsToConstants (constantsToVars (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\ncases' f with f f\n[GOAL]\ncase func.succ.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\n_ts\u271d : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin (Nat.succ n\u271d)), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\nf : Functions L (Nat.succ n\u271d)\n\u22a2 varsToConstants (constantsToVars (func (Sum.inl f) _ts\u271d)) = func (Sum.inl f) _ts\u271d\n[PROOFSTEP]\nsimp [constantsToVars, varsToConstants, ih]\n[GOAL]\ncase func.succ.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\n_ts\u271d : Fin (Nat.succ n\u271d) \u2192 Term (L[[\u03b3]]) \u03b1\nih : \u2200 (a : Fin (Nat.succ n\u271d)), varsToConstants (constantsToVars (_ts\u271d a)) = _ts\u271d a\nf : Functions (constantsOn \u03b3) (Nat.succ n\u271d)\n\u22a2 varsToConstants (constantsToVars (func (Sum.inr f) _ts\u271d)) = func (Sum.inr f) _ts\u271d\n[PROOFSTEP]\nexact isEmptyElim f\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n\u22a2 Function.RightInverse varsToConstants constantsToVars\n[PROOFSTEP]\nintro t\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nt : Term L (\u03b3 \u2295 \u03b1)\n\u22a2 constantsToVars (varsToConstants t) = t\n[PROOFSTEP]\ninduction' t with x n f _ ih\n[GOAL]\ncase var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nx : \u03b3 \u2295 \u03b1\n\u22a2 constantsToVars (varsToConstants (var x)) = var x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase var.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nval\u271d : \u03b3\n\u22a2 constantsToVars (varsToConstants (var (Sum.inl val\u271d))) = var (Sum.inl val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase var.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nval\u271d : \u03b1\n\u22a2 constantsToVars (varsToConstants (var (Sum.inr val\u271d))) = var (Sum.inr val\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nf : Functions L n\n_ts\u271d : Fin n \u2192 Term L (\u03b3 \u2295 \u03b1)\nih : \u2200 (a : Fin n), constantsToVars (varsToConstants (_ts\u271d a)) = _ts\u271d a\n\u22a2 constantsToVars (varsToConstants (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\ncases n\n[GOAL]\ncase func.zero\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nf : Functions L Nat.zero\n_ts\u271d : Fin Nat.zero \u2192 Term L (\u03b3 \u2295 \u03b1)\nih : \u2200 (a : Fin Nat.zero), constantsToVars (varsToConstants (_ts\u271d a)) = _ts\u271d a\n\u22a2 constantsToVars (varsToConstants (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\nsimp [varsToConstants, constantsToVars, ih]\n[GOAL]\ncase func.succ\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\nf : Functions L (Nat.succ n\u271d)\n_ts\u271d : Fin (Nat.succ n\u271d) \u2192 Term L (\u03b3 \u2295 \u03b1)\nih : \u2200 (a : Fin (Nat.succ n\u271d)), constantsToVars (varsToConstants (_ts\u271d a)) = _ts\u271d a\n\u22a2 constantsToVars (varsToConstants (func f _ts\u271d)) = func f _ts\u271d\n[PROOFSTEP]\nsimp [varsToConstants, constantsToVars, ih]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n\u22a2 onTerm (LHom.id L) = id\n[PROOFSTEP]\next t\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nt : Term L \u03b1\n\u22a2 onTerm (LHom.id L) t = id t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase h.var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n_a\u271d : \u03b1\n\u22a2 onTerm (LHom.id L) (var _a\u271d) = id (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), onTerm (LHom.id L) (_ts\u271d a) = id (_ts\u271d a)\n\u22a2 onTerm (LHom.id L) (func _f\u271d _ts\u271d) = id (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp_rw [onTerm, ih]\n[GOAL]\ncase h.func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), onTerm (LHom.id L) (_ts\u271d a) = id (_ts\u271d a)\n\u22a2 (func (onFunction (LHom.id L) _f\u271d) fun i => id (_ts\u271d i)) = id (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\n\u22a2 onTerm (comp \u03c6 \u03c8) = onTerm \u03c6 \u2218 onTerm \u03c8\n[PROOFSTEP]\next t\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nt : Term L \u03b1\n\u22a2 onTerm (comp \u03c6 \u03c8) t = (onTerm \u03c6 \u2218 onTerm \u03c8) t\n[PROOFSTEP]\ninduction' t with _ _ _ _ ih\n[GOAL]\ncase h.var\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\n_a\u271d : \u03b1\n\u22a2 onTerm (comp \u03c6 \u03c8) (var _a\u271d) = (onTerm \u03c6 \u2218 onTerm \u03c8) (var _a\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), onTerm (comp \u03c6 \u03c8) (_ts\u271d a) = (onTerm \u03c6 \u2218 onTerm \u03c8) (_ts\u271d a)\n\u22a2 onTerm (comp \u03c6 \u03c8) (func _f\u271d _ts\u271d) = (onTerm \u03c6 \u2218 onTerm \u03c8) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nsimp_rw [onTerm, ih]\n[GOAL]\ncase h.func\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nl\u271d : \u2115\n_f\u271d : Functions L l\u271d\n_ts\u271d : Fin l\u271d \u2192 Term L \u03b1\nih : \u2200 (a : Fin l\u271d), onTerm (comp \u03c6 \u03c8) (_ts\u271d a) = (onTerm \u03c6 \u2218 onTerm \u03c8) (_ts\u271d a)\n\u22a2 (func (onFunction (comp \u03c6 \u03c8) _f\u271d) fun i => (onTerm \u03c6 \u2218 onTerm \u03c8) (_ts\u271d i)) = (onTerm \u03c6 \u2218 onTerm \u03c8) (func _f\u271d _ts\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n\u03c6 : L \u2243\u1d38 L'\n\u22a2 Function.LeftInverse (LHom.onTerm \u03c6.invLHom) (LHom.onTerm \u03c6.toLHom)\n[PROOFSTEP]\nrw [Function.leftInverse_iff_comp, \u2190 LHom.comp_onTerm, \u03c6.left_inv, LHom.id_onTerm]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\n\u03c6 : L \u2243\u1d38 L'\n\u22a2 Function.RightInverse (LHom.onTerm \u03c6.invLHom) (LHom.onTerm \u03c6.toLHom)\n[PROOFSTEP]\nrw [Function.rightInverse_iff_comp, \u2190 LHom.comp_onTerm, \u03c6.right_inv, LHom.id_onTerm]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d n : \u2115\nh : n \u2264 n\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 castLE h \u03c6 = \u03c6\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n : \u2115\nh\u271d : n \u2264 n\nn\u271d : \u2115\nh : n\u271d \u2264 n\u271d\n\u22a2 castLE h falsum = falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n : \u2115\nh\u271d : n \u2264 n\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nh : n\u271d \u2264 n\u271d\n\u22a2 castLE h (equal t\u2081\u271d t\u2082\u271d) = equal t\u2081\u271d t\u2082\u271d\n[PROOFSTEP]\nsimp [Fin.castLE_of_eq]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n : \u2115\nh\u271d : n \u2264 n\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nh : n\u271d \u2264 n\u271d\n\u22a2 castLE h (rel R\u271d ts\u271d) = rel R\u271d ts\u271d\n[PROOFSTEP]\nsimp [Fin.castLE_of_eq]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n : \u2115\nh\u271d : n \u2264 n\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 (h : n\u271d \u2264 n\u271d), castLE h f\u2081\u271d = f\u2081\u271d\nih2 : \u2200 (h : n\u271d \u2264 n\u271d), castLE h f\u2082\u271d = f\u2082\u271d\nh : n\u271d \u2264 n\u271d\n\u22a2 castLE h (imp f\u2081\u271d f\u2082\u271d) = imp f\u2081\u271d f\u2082\u271d\n[PROOFSTEP]\nsimp [Fin.castLE_of_eq, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n : \u2115\nh\u271d : n \u2264 n\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 (h : n\u271d + 1 \u2264 n\u271d + 1), castLE h f\u271d = f\u271d\nh : n\u271d \u2264 n\u271d\n\u22a2 castLE h (all f\u271d) = all f\u271d\n[PROOFSTEP]\nsimp [Fin.castLE_of_eq, ih3]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d k m n : \u2115\nkm : k \u2264 m\nmn : m \u2264 n\n\u03c6 : BoundedFormula L \u03b1 k\n\u22a2 castLE mn (castLE km \u03c6) = castLE (_ : k \u2264 n) \u03c6\n[PROOFSTEP]\nrevert m n\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\n\u22a2 \u2200 {m n : \u2115} (km : k \u2264 m) (mn : m \u2264 n), castLE mn (castLE km \u03c6) = castLE (_ : k \u2264 n) \u03c6\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k n\u271d : \u2115\n\u22a2 \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km falsum) = castLE (_ : n\u271d \u2264 n) falsum\n[PROOFSTEP]\nintro m n km mn\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km (equal t\u2081\u271d t\u2082\u271d)) = castLE (_ : n\u271d \u2264 n) (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nintro m n km mn\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km (rel R\u271d ts\u271d)) = castLE (_ : n\u271d \u2264 n) (rel R\u271d ts\u271d)\n[PROOFSTEP]\nintro m n km mn\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u2081\u271d) = castLE (_ : n\u271d \u2264 n) f\u2081\u271d\nih2 : \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u2082\u271d) = castLE (_ : n\u271d \u2264 n) f\u2082\u271d\n\u22a2 \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km (imp f\u2081\u271d f\u2082\u271d)) = castLE (_ : n\u271d \u2264 n) (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nintro m n km mn\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {m n : \u2115} (km : n\u271d + 1 \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u271d) = castLE (_ : n\u271d + 1 \u2264 n) f\u271d\n\u22a2 \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km (all f\u271d)) = castLE (_ : n\u271d \u2264 n) (all f\u271d)\n[PROOFSTEP]\nintro m n km mn\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d m n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 castLE mn (castLE km falsum) = castLE (_ : n\u271d \u2264 n) falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 castLE mn (castLE km (equal t\u2081\u271d t\u2082\u271d)) = castLE (_ : n\u271d \u2264 n) (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 castLE mn (castLE km (rel R\u271d ts\u271d)) = castLE (_ : n\u271d \u2264 n) (rel R\u271d ts\u271d)\n[PROOFSTEP]\nsimp only [castLE, eq_self_iff_true, heq_iff_eq, true_and_iff]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 rel R\u271d (Term.relabel (Sum.map id (Fin.castLE mn)) \u2218 Term.relabel (Sum.map id (Fin.castLE km)) \u2218 ts\u271d) =\n    rel R\u271d (Term.relabel (Sum.map id (Fin.castLE (_ : n\u271d \u2264 n))) \u2218 ts\u271d)\n[PROOFSTEP]\nrw [\u2190 Function.comp.assoc, Term.relabel_comp_relabel]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 rel R\u271d (Term.relabel (Sum.map id (Fin.castLE mn) \u2218 Sum.map id (Fin.castLE km)) \u2218 ts\u271d) =\n    rel R\u271d (Term.relabel (Sum.map id (Fin.castLE (_ : n\u271d \u2264 n))) \u2218 ts\u271d)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u2081\u271d) = castLE (_ : n\u271d \u2264 n) f\u2081\u271d\nih2 : \u2200 {m n : \u2115} (km : n\u271d \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u2082\u271d) = castLE (_ : n\u271d \u2264 n) f\u2082\u271d\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 castLE mn (castLE km (imp f\u2081\u271d f\u2082\u271d)) = castLE (_ : n\u271d \u2264 n) (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nsimp [ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 k n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : \u2200 {m n : \u2115} (km : n\u271d + 1 \u2264 m) (mn : m \u2264 n), castLE mn (castLE km f\u271d) = castLE (_ : n\u271d + 1 \u2264 n) f\u271d\nm n : \u2115\nkm : n\u271d \u2264 m\nmn : m \u2264 n\n\u22a2 castLE mn (castLE km (all f\u271d)) = castLE (_ : n\u271d \u2264 n) (all f\u271d)\n[PROOFSTEP]\nsimp only [castLE, ih3]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d n n' m : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\nx\u271d : \u2115\n\u22a2 x\u271d + 1 + n' \u2264 x\u271d + n' + 1\n[PROOFSTEP]\nrw [add_assoc, add_comm 1, add_assoc]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) \u03c6) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) \u03c6\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn n\u271d : \u2115\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) falsum) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (equal t\u2081\u271d t\u2082\u271d)) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nsimp [mapTermRel]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (rel R\u271d ts\u271d)) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) (rel R\u271d ts\u271d)\n[PROOFSTEP]\nsimp [mapTermRel]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 :\n  mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f\u2081\u271d) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) f\u2081\u271d\nih2 :\n  mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f\u2082\u271d) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) f\u2082\u271d\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (imp f\u2081\u271d f\u2082\u271d)) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nsimp [mapTermRel, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 : \u2115\nL'' : Language\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2192 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2192 Relations L' n\nft' : (n : \u2115) \u2192 Term L' (\u03b2 \u2295 Fin n) \u2192 Term L'' (\u03b3 \u2295 Fin n)\nfr' : (n : \u2115) \u2192 Relations L' n \u2192 Relations L'' n\nn n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 :\n  mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) f\u271d) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) f\u271d\n\u22a2 mapTermRel ft' fr' (fun x => id) (mapTermRel ft fr (fun x => id) (all f\u271d)) =\n    mapTermRel (fun x => ft' x \u2218 ft x) (fun x => fr' x \u2218 fr x) (fun x => id) (all f\u271d)\n[PROOFSTEP]\nsimp [mapTermRel, ih3]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d n : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) \u03c6 = \u03c6\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n n\u271d : \u2115\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) falsum = falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) (equal t\u2081\u271d t\u2082\u271d) = equal t\u2081\u271d t\u2082\u271d\n[PROOFSTEP]\nsimp [mapTermRel]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) (rel R\u271d ts\u271d) = rel R\u271d ts\u271d\n[PROOFSTEP]\nsimp [mapTermRel]\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f\u2081\u271d = f\u2081\u271d\nih2 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f\u2082\u271d = f\u2082\u271d\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) (imp f\u2081\u271d f\u2082\u271d) = imp f\u2081\u271d f\u2082\u271d\n[PROOFSTEP]\nsimp [mapTermRel, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d\u00b9 n n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f\u271d = f\u271d\n\u22a2 mapTermRel (fun x => id) (fun x => id) (fun x => id) (all f\u271d) = all f\u271d\n[PROOFSTEP]\nsimp [mapTermRel, ih3]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2243 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2243 Relations L' n\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 mapTermRel (fun n => \u2191(ft n).symm) (fun n => \u2191(fr n).symm) (fun x => id)\n      (mapTermRel (fun n => \u2191(ft n)) (fun n => \u2191(fr n)) (fun x => id) \u03c6) =\n    \u03c6\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn\u271d : \u2115\nft : (n : \u2115) \u2192 Term L (\u03b1 \u2295 Fin n) \u2243 Term L' (\u03b2 \u2295 Fin n)\nfr : (n : \u2115) \u2192 Relations L n \u2243 Relations L' n\nn : \u2115\n\u03c6 : BoundedFormula L' \u03b2 n\n\u22a2 mapTermRel (fun n => \u2191(ft n)) (fun n => \u2191(fr n)) (fun x => id)\n      (mapTermRel (fun n => \u2191(ft n).symm) (fun n => \u2191(fr n).symm) (fun x => id) \u03c6) =\n    \u03c6\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\n\u22a2 Sum.elim v xs \u2218 relabelAux g m = Sum.elim (Sum.elim v (xs \u2218 castAdd m) \u2218 g) (xs \u2218 natAdd n)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : \u03b1 \u2295 Fin m\n\u22a2 (Sum.elim v xs \u2218 relabelAux g m) x = Sum.elim (Sum.elim v (xs \u2218 castAdd m) \u2218 g) (xs \u2218 natAdd n) x\n[PROOFSTEP]\ncases' x with x x\n[GOAL]\ncase h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : \u03b1\n\u22a2 (Sum.elim v xs \u2218 relabelAux g m) (Sum.inl x) = Sum.elim (Sum.elim v (xs \u2218 castAdd m) \u2218 g) (xs \u2218 natAdd n) (Sum.inl x)\n[PROOFSTEP]\nsimp only [BoundedFormula.relabelAux, Function.comp_apply, Sum.map_inl, Sum.elim_inl]\n[GOAL]\ncase h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : \u03b1\n\u22a2 Sum.elim v xs (Sum.map id (\u2191finSumFinEquiv) (\u2191(Equiv.sumAssoc \u03b2 (Fin n) (Fin m)) (Sum.inl (g x)))) =\n    Sum.elim v (xs \u2218 castAdd m) (g x)\n[PROOFSTEP]\ncases' g x with l r\n[GOAL]\ncase h.inl.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : \u03b1\nl : \u03b2\n\u22a2 Sum.elim v xs (Sum.map id (\u2191finSumFinEquiv) (\u2191(Equiv.sumAssoc \u03b2 (Fin n) (Fin m)) (Sum.inl (Sum.inl l)))) =\n    Sum.elim v (xs \u2218 castAdd m) (Sum.inl l)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inl.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : \u03b1\nr : Fin n\n\u22a2 Sum.elim v xs (Sum.map id (\u2191finSumFinEquiv) (\u2191(Equiv.sumAssoc \u03b2 (Fin n) (Fin m)) (Sum.inl (Sum.inr r)))) =\n    Sum.elim v (xs \u2218 castAdd m) (Sum.inr r)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn m : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nv : \u03b2 \u2192 M\nxs : Fin (n + m) \u2192 M\nx : Fin m\n\u22a2 (Sum.elim v xs \u2218 relabelAux g m) (Sum.inr x) = Sum.elim (Sum.elim v (xs \u2218 castAdd m) \u2218 g) (xs \u2218 natAdd n) (Sum.inr x)\n[PROOFSTEP]\nsimp [BoundedFormula.relabelAux]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k : \u2115\n\u22a2 relabelAux Sum.inl k = Sum.map id (natAdd n)\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k : \u2115\nx : \u03b1 \u2295 Fin k\n\u22a2 relabelAux Sum.inl k x = Sum.map id (natAdd n) x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k : \u2115\nval\u271d : \u03b1\n\u22a2 relabelAux Sum.inl k (Sum.inl val\u271d) = Sum.map id (natAdd n) (Sum.inl val\u271d)\n[PROOFSTEP]\nsimp [relabelAux]\n[GOAL]\ncase h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn k : \u2115\nval\u271d : Fin k\n\u22a2 relabelAux Sum.inl k (Sum.inr val\u271d) = Sum.map id (natAdd n) (Sum.inr val\u271d)\n[PROOFSTEP]\nsimp [relabelAux]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 k\n\u22a2 relabel g (BoundedFormula.not \u03c6) = BoundedFormula.not (relabel g \u03c6)\n[PROOFSTEP]\nsimp [BoundedFormula.not]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 (k + 1)\n\u22a2 relabel g (all \u03c6) = all (relabel g \u03c6)\n[PROOFSTEP]\nrw [relabel, mapTermRel, relabel]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 (k + 1)\n\u22a2 all\n      (castLE (_ : n + k + 1 \u2265 n + (k + 1))\n        (mapTermRel (fun x t => Term.relabel (relabelAux g x) t) (fun x => id)\n          (fun x => castLE (_ : n + x + 1 \u2265 n + (x + 1))) \u03c6)) =\n    all\n      (mapTermRel (fun x t => Term.relabel (relabelAux g x) t) (fun x => id)\n        (fun x => castLE (_ : n + x + 1 \u2265 n + (x + 1))) \u03c6)\n[PROOFSTEP]\nsimp\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ng : \u03b1 \u2192 \u03b2 \u2295 Fin n\nk : \u2115\n\u03c6 : BoundedFormula L \u03b1 (k + 1)\n\u22a2 relabel g (BoundedFormula.ex \u03c6) = BoundedFormula.ex (relabel g \u03c6)\n[PROOFSTEP]\nsimp [BoundedFormula.ex]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 relabel Sum.inl \u03c6 = castLE (_ : 0 + n \u2265 n) \u03c6\n[PROOFSTEP]\nsimp only [relabel, relabelAux_sum_inl]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\n\u03c6 : BoundedFormula L \u03b1 n\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) \u03c6 =\n    castLE (_ : 0 + n \u2265 n) \u03c6\n[PROOFSTEP]\ninduction' \u03c6 with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) falsum =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) (equal t\u2081\u271d t\u2082\u271d) =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nsimp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) (rel R\u271d ts\u271d) =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) (rel R\u271d ts\u271d)\n[PROOFSTEP]\nsimp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]\n[GOAL]\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 (fun i => Term.relabel (Sum.map id (Fin.cast (_ : n\u271d = 0 + n\u271d))) (ts\u271d i)) =\n    Term.relabel (Sum.map id (Fin.cast (_ : n\u271d = 0 + n\u271d))) \u2218 ts\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 :\n  mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) f\u2081\u271d =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) f\u2081\u271d\nih2 :\n  mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) f\u2082\u271d =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) f\u2082\u271d\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) (imp f\u2081\u271d f\u2082\u271d) =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nsimp [mapTermRel, ih1, ih2]\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 :\n  mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) f\u271d =\n    castLE (_ : 0 + (n\u271d + 1) \u2265 n\u271d + 1) f\u271d\n\u22a2 mapTermRel (fun x t => Term.relabel (Sum.map id (natAdd 0)) t) (fun x => id)\n      (fun x => castLE (_ : 0 + x + 1 \u2265 0 + (x + 1))) (all f\u271d) =\n    castLE (_ : 0 + n\u271d \u2265 n\u271d) (all f\u271d)\n[PROOFSTEP]\nsimp [mapTermRel, ih3, castLE]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsAtomic (all \u03c6)\n\u22a2 False\n[PROOFSTEP]\ncases con\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsAtomic (BoundedFormula.ex \u03c6)\n\u22a2 False\n[PROOFSTEP]\ncases con\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsQF (all \u03c6)\n\u22a2 False\n[PROOFSTEP]\ncases' con with _ con\n[GOAL]\ncase of_isAtomic\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsAtomic (all \u03c6)\n\u22a2 False\n[PROOFSTEP]\nexact \u03c6.not_all_isAtomic con\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsQF (BoundedFormula.ex \u03c6)\n\u22a2 False\n[PROOFSTEP]\ncases' con with _ con _ _ con\n[GOAL]\ncase of_isAtomic\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsAtomic (BoundedFormula.ex \u03c6)\n\u22a2 False\n[PROOFSTEP]\nexact \u03c6.not_ex_isAtomic con\n[GOAL]\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 : BoundedFormula L \u03b1 (n + 1)\ncon : IsQF (all (BoundedFormula.not \u03c6))\nh\u2082\u271d : IsQF \u22a5\n\u22a2 False\n[PROOFSTEP]\nexact not_all_isQF _ con\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nm : \u2115\n\u03c6 : BoundedFormula L \u03b1 m\nh\u271d : IsPrenex \u03c6\nf : \u03b1 \u2192 \u03b2 \u2295 Fin n\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nx\u271d : IsPrenex \u03c6\u271d\nh : IsPrenex (BoundedFormula.relabel f \u03c6\u271d)\n\u22a2 IsPrenex (BoundedFormula.relabel f (BoundedFormula.all \u03c6\u271d))\n[PROOFSTEP]\nsimp [h.all]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\nm : \u2115\n\u03c6 : BoundedFormula L \u03b1 m\nh\u271d : IsPrenex \u03c6\nf : \u03b1 \u2192 \u03b2 \u2295 Fin n\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nx\u271d : IsPrenex \u03c6\u271d\nh : IsPrenex (BoundedFormula.relabel f \u03c6\u271d)\n\u22a2 IsPrenex (BoundedFormula.relabel f (BoundedFormula.ex \u03c6\u271d))\n[PROOFSTEP]\nsimp [h.ex]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6 : IsQF \u03c6\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImpRight \u03c6 \u03c8)\n[PROOFSTEP]\ninduction' h\u03c8 with _ _ h\u03c8 _ _ _ ih1 _ _ _ ih2\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsQF \u03c6\u271d\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\n\u22a2 IsPrenex (toPrenexImpRight \u03c6 \u03c6\u271d)\n[PROOFSTEP]\nrw [h\u03c8.toPrenexImpRight]\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsQF \u03c6\u271d\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\n\u22a2 IsPrenex (imp \u03c6 \u03c6\u271d)\n[PROOFSTEP]\nexact (h\u03c6.imp h\u03c8).isPrenex\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh\u271d : IsPrenex \u03c6\u271d\nih1 : \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)}, IsQF \u03c6 \u2192 IsPrenex (toPrenexImpRight \u03c6 \u03c6\u271d)\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\n\u22a2 IsPrenex (toPrenexImpRight \u03c6 (all \u03c6\u271d))\n[PROOFSTEP]\nexact (ih1 h\u03c6.liftAt).all\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b2 \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6\u271d\u00b9 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6\u271d : IsQF \u03c6\u271d\u00b9\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh\u271d : IsPrenex \u03c6\u271d\nih2 : \u2200 {\u03c6 : BoundedFormula L \u03b1 (n\u271d + 1)}, IsQF \u03c6 \u2192 IsPrenex (toPrenexImpRight \u03c6 \u03c6\u271d)\n\u03c6 : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\n\u22a2 IsPrenex (toPrenexImpRight \u03c6 (BoundedFormula.ex \u03c6\u271d))\n[PROOFSTEP]\nexact (ih2 h\u03c6.liftAt).ex\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8\u271d : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8 : BoundedFormula L \u03b1 n\nh\u03c6 : IsPrenex \u03c6\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImp \u03c6 \u03c8)\n[PROOFSTEP]\ninduction' h\u03c6 with _ _ h\u03c6 _ _ _ ih1 _ _ _ ih2\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d\u00b9 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8\u271d : BoundedFormula L \u03b1 n\nh\u03c8\u271d : IsPrenex \u03c8\u271d\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\u271d\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImp \u03c6\u271d \u03c8)\n[PROOFSTEP]\nrw [h\u03c6.toPrenexImp]\n[GOAL]\ncase of_isQF\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d\u00b9 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8\u271d : BoundedFormula L \u03b1 n\nh\u03c8\u271d : IsPrenex \u03c8\u271d\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 n\u271d\nh\u03c6 : IsQF \u03c6\u271d\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImpRight \u03c6\u271d \u03c8)\n[PROOFSTEP]\nexact isPrenex_toPrenexImpRight h\u03c6 h\u03c8\n[GOAL]\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d\u00b9 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8\u271d : BoundedFormula L \u03b1 n\nh\u03c8\u271d : IsPrenex \u03c8\u271d\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh\u271d : IsPrenex \u03c6\u271d\nih1 : \u2200 {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)}, IsPrenex \u03c8 \u2192 IsPrenex (toPrenexImp \u03c6\u271d \u03c8)\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImp (all \u03c6\u271d) \u03c8)\n[PROOFSTEP]\nexact (ih1 h\u03c8.liftAt).ex\n[GOAL]\ncase ex\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d\u00b9 \u03c8\u271d\u00b9 : BoundedFormula L \u03b1 l\n\u03b8 : BoundedFormula L \u03b1 (Nat.succ l)\nv : \u03b1 \u2192 M\nxs : Fin l \u2192 M\n\u03c6 \u03c8\u271d : BoundedFormula L \u03b1 n\nh\u03c8\u271d : IsPrenex \u03c8\u271d\nn\u271d : \u2115\n\u03c6\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nh\u271d : IsPrenex \u03c6\u271d\nih2 : \u2200 {\u03c8 : BoundedFormula L \u03b1 (n\u271d + 1)}, IsPrenex \u03c8 \u2192 IsPrenex (toPrenexImp \u03c6\u271d \u03c8)\n\u03c8 : BoundedFormula L \u03b1 n\u271d\nh\u03c8 : IsPrenex \u03c8\n\u22a2 IsPrenex (toPrenexImp (BoundedFormula.ex \u03c6\u271d) \u03c8)\n[PROOFSTEP]\nexact (ih2 h\u03c8.liftAt).all\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\n\u22a2 onBoundedFormula (LHom.id L) = id\n[PROOFSTEP]\next f\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nf : BoundedFormula L \u03b1 n\n\u22a2 onBoundedFormula (LHom.id L) f = id f\n[PROOFSTEP]\ninduction' f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase h.falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\n\u22a2 onBoundedFormula (LHom.id L) falsum = id falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 onBoundedFormula (LHom.id L) (equal t\u2081\u271d t\u2082\u271d) = id (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nrw [onBoundedFormula, LHom.id_onTerm, id.def, id.def, id.def, Term.bdEqual]\n[GOAL]\ncase h.rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 onBoundedFormula (LHom.id L) (rel R\u271d ts\u271d) = id (rel R\u271d ts\u271d)\n[PROOFSTEP]\nrw [onBoundedFormula, LHom.id_onTerm]\n[GOAL]\ncase h.rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 Relations.boundedFormula (onRelation (LHom.id L) R\u271d) (id \u2218 ts\u271d) = id (rel R\u271d ts\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : onBoundedFormula (LHom.id L) f\u2081\u271d = id f\u2081\u271d\nih2 : onBoundedFormula (LHom.id L) f\u2082\u271d = id f\u2082\u271d\n\u22a2 onBoundedFormula (LHom.id L) (imp f\u2081\u271d f\u2082\u271d) = id (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nrw [onBoundedFormula, ih1, ih2, id.def, id.def, id.def]\n[GOAL]\ncase h.all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn n\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : onBoundedFormula (LHom.id L) f\u271d = id f\u271d\n\u22a2 onBoundedFormula (LHom.id L) (all f\u271d) = id (all f\u271d)\n[PROOFSTEP]\nrw [onBoundedFormula, ih3, id.def, id.def]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) = onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8\n[PROOFSTEP]\next f\n[GOAL]\ncase h\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nf : BoundedFormula L \u03b1 n\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) f = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) f\n[PROOFSTEP]\ninduction' f with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3\n[GOAL]\ncase h.falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d : \u2115\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) falsum = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) falsum\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) (equal t\u2081\u271d t\u2082\u271d) = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) (equal t\u2081\u271d t\u2082\u271d)\n[PROOFSTEP]\nsimp only [onBoundedFormula, comp_onTerm, Function.comp_apply]\n[GOAL]\ncase h.rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) (rel R\u271d ts\u271d) = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) (rel R\u271d ts\u271d)\n[PROOFSTEP]\nsimp only [onBoundedFormula, comp_onRelation, comp_onTerm, Function.comp_apply]\n[GOAL]\ncase h.rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d l\u271d : \u2115\nR\u271d : Relations L l\u271d\nts\u271d : Fin l\u271d \u2192 Term L (\u03b1 \u2295 Fin n\u271d)\n\u22a2 Relations.boundedFormula (onRelation \u03c6 (onRelation \u03c8 R\u271d)) ((onTerm \u03c6 \u2218 onTerm \u03c8) \u2218 ts\u271d) =\n    Relations.boundedFormula (onRelation \u03c6 (onRelation \u03c8 R\u271d)) (onTerm \u03c6 \u2218 onTerm \u03c8 \u2218 ts\u271d)\n[PROOFSTEP]\nrfl\n[GOAL]\ncase h.imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d : \u2115\nf\u2081\u271d f\u2082\u271d : BoundedFormula L \u03b1 n\u271d\nih1 : onBoundedFormula (comp \u03c6 \u03c8) f\u2081\u271d = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) f\u2081\u271d\nih2 : onBoundedFormula (comp \u03c6 \u03c8) f\u2082\u271d = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) f\u2082\u271d\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) (imp f\u2081\u271d f\u2082\u271d) = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) (imp f\u2081\u271d f\u2082\u271d)\n[PROOFSTEP]\nsimp only [onBoundedFormula, Function.comp_apply, ih1, ih2, eq_self_iff_true, and_self_iff]\n[GOAL]\ncase h.all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\nL'' : Language\n\u03c6 : L' \u2192\u1d38 L''\n\u03c8 : L \u2192\u1d38 L'\nn\u271d : \u2115\nf\u271d : BoundedFormula L \u03b1 (n\u271d + 1)\nih3 : onBoundedFormula (comp \u03c6 \u03c8) f\u271d = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) f\u271d\n\u22a2 onBoundedFormula (comp \u03c6 \u03c8) (all f\u271d) = (onBoundedFormula \u03c6 \u2218 onBoundedFormula \u03c8) (all f\u271d)\n[PROOFSTEP]\nsimp only [ih3, onBoundedFormula, Function.comp_apply]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\n\u03c6 : L \u2243\u1d38 L'\n\u22a2 Function.LeftInverse (LHom.onBoundedFormula \u03c6.invLHom) (LHom.onBoundedFormula \u03c6.toLHom)\n[PROOFSTEP]\nrw [Function.leftInverse_iff_comp, \u2190 LHom.comp_onBoundedFormula, \u03c6.left_inv, LHom.id_onBoundedFormula]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\n\u03c6 : L \u2243\u1d38 L'\n\u22a2 Function.RightInverse (LHom.onBoundedFormula \u03c6.invLHom) (LHom.onBoundedFormula \u03c6.toLHom)\n[PROOFSTEP]\nrw [Function.rightInverse_iff_comp, \u2190 LHom.comp_onBoundedFormula, \u03c6.right_inv, LHom.id_onBoundedFormula]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\n\u22a2 distinctConstantsTheory L s =\n    \u22c3 (t : Finset \u2191s), distinctConstantsTheory L \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) t)\n[PROOFSTEP]\nclassical\nsimp only [distinctConstantsTheory]\nrw [\u2190 image_iUnion, \u2190 iUnion_inter]\nrefine' congr rfl (congr (congr rfl _) rfl)\next \u27e8i, j\u27e9\nsimp only [prod_mk_mem_set_prod_eq, Finset.coe_map, Function.Embedding.coe_subtype, mem_iUnion, mem_image,\n  Finset.mem_coe, Subtype.exists, Subtype.coe_mk, exists_and_right, exists_eq_right]\nrefine' \u27e8fun h => \u27e8{\u27e8i, h.1\u27e9, \u27e8j, h.2\u27e9}, \u27e8h.1, _\u27e9, \u27e8h.2, _\u27e9\u27e9, _\u27e9\n\u00b7 simp\n\u00b7 simp\n\u00b7 rintro \u27e8t, \u27e8is, _\u27e9, \u27e8js, _\u27e9\u27e9\n  exact \u27e8is, js\u27e9\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\n\u22a2 distinctConstantsTheory L s =\n    \u22c3 (t : Finset \u2191s), distinctConstantsTheory L \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) t)\n[PROOFSTEP]\nsimp only [distinctConstantsTheory]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\n\u22a2 (fun ab =>\n        Formula.not (Term.equal (Constants.term (Language.con L ab.fst)) (Constants.term (Language.con L ab.snd)))) ''\n      (s \u00d7\u02e2 s \u2229 (diagonal \u03b1)\u1d9c) =\n    \u22c3 (t : Finset \u2191s),\n      (fun ab =>\n          Formula.not (Term.equal (Constants.term (Language.con L ab.fst)) (Constants.term (Language.con L ab.snd)))) ''\n        (\u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) t) \u00d7\u02e2\n            \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) t) \u2229\n          (diagonal \u03b1)\u1d9c)\n[PROOFSTEP]\nrw [\u2190 image_iUnion, \u2190 iUnion_inter]\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\n\u22a2 (fun ab =>\n        Formula.not (Term.equal (Constants.term (Language.con L ab.fst)) (Constants.term (Language.con L ab.snd)))) ''\n      (s \u00d7\u02e2 s \u2229 (diagonal \u03b1)\u1d9c) =\n    (fun ab =>\n        Formula.not (Term.equal (Constants.term (Language.con L ab.fst)) (Constants.term (Language.con L ab.snd)))) ''\n      ((\u22c3 (i : Finset \u2191s),\n          \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i) \u00d7\u02e2\n            \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i)) \u2229\n        (diagonal \u03b1)\u1d9c)\n[PROOFSTEP]\nrefine' congr rfl (congr (congr rfl _) rfl)\n[GOAL]\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\n\u22a2 s \u00d7\u02e2 s =\n    \u22c3 (i : Finset \u2191s),\n      \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i) \u00d7\u02e2\n        \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i)\n[PROOFSTEP]\next \u27e8i, j\u27e9\n[GOAL]\ncase h.mk\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\n\u22a2 (i, j) \u2208 s \u00d7\u02e2 s \u2194\n    (i, j) \u2208\n      \u22c3 (i : Finset \u2191s),\n        \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i) \u00d7\u02e2\n          \u2191(Finset.map (Function.Embedding.subtype fun x => x \u2208 s) i)\n[PROOFSTEP]\nsimp only [prod_mk_mem_set_prod_eq, Finset.coe_map, Function.Embedding.coe_subtype, mem_iUnion, mem_image,\n  Finset.mem_coe, Subtype.exists, Subtype.coe_mk, exists_and_right, exists_eq_right]\n[GOAL]\ncase h.mk\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\n\u22a2 i \u2208 s \u2227 j \u2208 s \u2194\n    \u2203 i_1, (\u2203 x, { val := i, property := (_ : i \u2208 s) } \u2208 i_1) \u2227 \u2203 x, { val := j, property := (_ : j \u2208 s) } \u2208 i_1\n[PROOFSTEP]\nrefine' \u27e8fun h => \u27e8{\u27e8i, h.1\u27e9, \u27e8j, h.2\u27e9}, \u27e8h.1, _\u27e9, \u27e8h.2, _\u27e9\u27e9, _\u27e9\n[GOAL]\ncase h.mk.refine'_1\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\nh : i \u2208 s \u2227 j \u2208 s\n\u22a2 { val := i, property := (_ : i \u2208 s) } \u2208 {{ val := i, property := (_ : i \u2208 s) }, { val := j, property := (_ : j \u2208 s) }}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mk.refine'_2\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\nh : i \u2208 s \u2227 j \u2208 s\n\u22a2 { val := j, property := (_ : j \u2208 s) } \u2208 {{ val := i, property := (_ : i \u2208 s) }, { val := j, property := (_ : j \u2208 s) }}\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h.mk.refine'_3\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\n\u22a2 (\u2203 i_1, (\u2203 x, { val := i, property := (_ : i \u2208 s) } \u2208 i_1) \u2227 \u2203 x, { val := j, property := (_ : j \u2208 s) } \u2208 i_1) \u2192\n    i \u2208 s \u2227 j \u2208 s\n[PROOFSTEP]\nrintro \u27e8t, \u27e8is, _\u27e9, \u27e8js, _\u27e9\u27e9\n[GOAL]\ncase h.mk.refine'_3.intro.intro.intro.intro\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : Structure L M\ninst\u271d\u00b9 : Structure L N\ninst\u271d : Structure L P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn : \u2115\ns : Set \u03b1\ni j : \u03b1\nt : Finset \u2191s\nis : i \u2208 s\nh\u271d\u00b9 : { val := i, property := (_ : i \u2208 s) } \u2208 t\njs : j \u2208 s\nh\u271d : { val := j, property := (_ : j \u2208 s) } \u2208 t\n\u22a2 i \u2208 s \u2227 j \u2208 s\n[PROOFSTEP]\nexact \u27e8is, js\u27e9\n", "meta": {"mathlib_filename": "Mathlib.ModelTheory.Syntax", "llama_tokens": 31219, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879991, "lm_q2_score": 0.7090191337850932, "lm_q1q2_score": 0.5027081233544889}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n\u22a2 StrictMonoOn f (s \u222a t)\n[PROOFSTEP]\nhave A : \u2200 x, x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s := by\n  intro x hx hxc\n  cases hx\n  \u00b7 assumption\n  rcases eq_or_lt_of_le hxc with (rfl | h'x)\n  \u00b7 exact hs.1\n  exact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n[PROOFSTEP]\nintro x hx hxc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : x \u2264 c\n\u22a2 x \u2208 s\n[PROOFSTEP]\ncases hx\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxc with (rfl | h'x)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nx : \u03b1\nh\u271d : x \u2208 t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x \u2264 x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact hs.1\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 t\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n\u22a2 StrictMonoOn f (s \u222a t)\n[PROOFSTEP]\nhave B : \u2200 x, x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t := by\n  intro x hx hxc\n  match hx with\n  | Or.inr hx => exact hx\n  | Or.inl hx =>\n    rcases eq_or_lt_of_le hxc with (rfl | h'x)\n    \u00b7 exact ht.1\n    exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\n[PROOFSTEP]\nintro x hx hxc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : c \u2264 x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nmatch hx with\n| Or.inr hx => exact hx\n| Or.inl hx =>\n  rcases eq_or_lt_of_le hxc with (rfl | h'x)\n  \u00b7 exact ht.1\n  exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 t\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\n\u22a2 x \u2208 t\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxc with (rfl | h'x)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nhx\u271d : c \u2208 s \u222a t\nhxc : c \u2264 c\nhx : c \u2208 s\n\u22a2 c \u2208 t\n[PROOFSTEP]\nexact ht.1\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\nh'x : c < x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\n\u22a2 StrictMonoOn f (s \u222a t)\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\n\u22a2 f x < f y\n[PROOFSTEP]\nrcases lt_or_le x c with (hxc | hcx)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhxc : x < c\n\u22a2 f x < f y\n[PROOFSTEP]\nhave xs : x \u2208 s := A _ hx hxc.le\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhxc : x < c\nxs : x \u2208 s\n\u22a2 f x < f y\n[PROOFSTEP]\nrcases lt_or_le y c with (hyc | hcy)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhxc : x < c\nxs : x \u2208 s\nhyc : y < c\n\u22a2 f x < f y\n[PROOFSTEP]\nexact h\u2081 xs (A _ hy hyc.le) hxy\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhxc : x < c\nxs : x \u2208 s\nhcy : c \u2264 y\n\u22a2 f x < f y\n[PROOFSTEP]\nexact (h\u2081 xs hs.1 hxc).trans_le (h\u2082.monotoneOn ht.1 (B _ hy hcy) hcy)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhcx : c \u2264 x\n\u22a2 f x < f y\n[PROOFSTEP]\nhave xt : x \u2208 t := B _ hx hcx\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhcx : c \u2264 x\nxt : x \u2208 t\n\u22a2 f x < f y\n[PROOFSTEP]\nhave yt : y \u2208 t := B _ hy (hcx.trans hxy.le)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : StrictMonoOn f s\nh\u2082 : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x < y\nhcx : c \u2264 x\nxt : x \u2208 t\nyt : y \u2208 t\n\u22a2 f x < f y\n[PROOFSTEP]\nexact h\u2082 xt yt hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMonoOn f (Iic a)\nh\u2082 : StrictMonoOn f (Ici a)\n\u22a2 StrictMono f\n[PROOFSTEP]\nrw [\u2190 strictMonoOn_univ, \u2190 @Iic_union_Ici _ _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nh\u2081 : StrictMonoOn f (Iic a)\nh\u2082 : StrictMonoOn f (Ici a)\n\u22a2 StrictMonoOn f (Iic a \u222a Ici a)\n[PROOFSTEP]\nexact StrictMonoOn.union h\u2081 h\u2082 isGreatest_Iic isLeast_Ici\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n\u22a2 MonotoneOn f (s \u222a t)\n[PROOFSTEP]\nhave A : \u2200 x, x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s := by\n  intro x hx hxc\n  cases hx\n  \u00b7 assumption\n  rcases eq_or_lt_of_le hxc with (rfl | h'x)\n  \u00b7 exact hs.1\n  exact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n[PROOFSTEP]\nintro x hx hxc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : x \u2264 c\n\u22a2 x \u2208 s\n[PROOFSTEP]\ncases hx\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 s\n\u22a2 x \u2208 s\n[PROOFSTEP]\nassumption\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\n\u22a2 x \u2208 s\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxc with (rfl | h'x)\n[GOAL]\ncase inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nx : \u03b1\nh\u271d : x \u2208 t\nhs : IsGreatest s x\nht : IsLeast t x\nhxc : x \u2264 x\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact hs.1\n[GOAL]\ncase inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 s\n[PROOFSTEP]\nexact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : \u03b1\nhxc : x \u2264 c\nh\u271d : x \u2208 t\nh'x : x < c\n\u22a2 x \u2208 t\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n\u22a2 MonotoneOn f (s \u222a t)\n[PROOFSTEP]\nhave B : \u2200 x, x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t := by\n  intro x hx hxc\n  match hx with\n  | Or.inr hx => exact hx\n  | Or.inl hx =>\n    rcases eq_or_lt_of_le hxc with (rfl | h'x)\n    \u00b7 exact ht.1\n    exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\n\u22a2 \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\n[PROOFSTEP]\nintro x hx hxc\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx : x \u2208 s \u222a t\nhxc : c \u2264 x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nmatch hx with\n| Or.inr hx => exact hx\n| Or.inl hx =>\n  rcases eq_or_lt_of_le hxc with (rfl | h'x)\n  \u00b7 exact ht.1\n  exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 t\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact hx\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\n\u22a2 x \u2208 t\n[PROOFSTEP]\nrcases eq_or_lt_of_le hxc with (rfl | h'x)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nhx\u271d : c \u2208 s \u222a t\nhxc : c \u2264 c\nhx : c \u2208 s\n\u22a2 c \u2208 t\n[PROOFSTEP]\nexact ht.1\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nx : \u03b1\nhx\u271d : x \u2208 s \u222a t\nhxc : c \u2264 x\nhx : x \u2208 s\nh'x : c < x\n\u22a2 x \u2208 t\n[PROOFSTEP]\nexact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\n\u22a2 MonotoneOn f (s \u222a t)\n[PROOFSTEP]\nintro x hx y hy hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nrcases lt_or_le x c with (hxc | hcx)\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nhave xs : x \u2208 s := A _ hx hxc.le\n[GOAL]\ncase inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nrcases lt_or_le y c with (hyc | hcy)\n[GOAL]\ncase inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhyc : y < c\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nexact h\u2081 xs (A _ hy hyc.le) hxy\n[GOAL]\ncase inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhxc : x < c\nxs : x \u2208 s\nhcy : c \u2264 y\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nexact (h\u2081 xs hs.1 hxc.le).trans (h\u2082 ht.1 (B _ hy hcy) hcy)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nhave xt : x \u2208 t := B _ hx hcx\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nhave yt : y \u2208 t := B _ hy (hcx.trans hxy)\n[GOAL]\ncase inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nc : \u03b1\nh\u2081 : MonotoneOn f s\nh\u2082 : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nA : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 x \u2264 c \u2192 x \u2208 s\nB : \u2200 (x : \u03b1), x \u2208 s \u222a t \u2192 c \u2264 x \u2192 x \u2208 t\nx : \u03b1\nhx : x \u2208 s \u222a t\ny : \u03b1\nhy : y \u2208 s \u222a t\nhxy : x \u2264 y\nhcx : c \u2264 x\nxt : x \u2208 t\nyt : y \u2208 t\n\u22a2 f x \u2264 f y\n[PROOFSTEP]\nexact h\u2082 xt yt hxy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nh\u2081 : MonotoneOn f (Iic a)\nh\u2082 : MonotoneOn f (Ici a)\n\u22a2 Monotone f\n[PROOFSTEP]\nrw [\u2190 monotoneOn_univ, \u2190 @Iic_union_Ici _ _ a]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : Preorder \u03b2\na : \u03b1\nf : \u03b1 \u2192 \u03b2\nh\u2081 : MonotoneOn f (Iic a)\nh\u2082 : MonotoneOn f (Ici a)\n\u22a2 MonotoneOn f (Iic a \u222a Ici a)\n[PROOFSTEP]\nexact MonotoneOn.union_right h\u2081 h\u2082 isGreatest_Iic isLeast_Ici\n", "meta": {"mathlib_filename": "Mathlib.Order.Monotone.Union", "llama_tokens": 10126, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.6584175139669998, "lm_q1q2_score": 0.5026910815535859}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np : MvPolynomial \u03c3 R\n\u22a2 Polynomial.eval x (eval\u2082 f g p) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\n[PROOFSTEP]\napply induction_on p\n[GOAL]\ncase h_C\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (a : R),\n    Polynomial.eval x (eval\u2082 f g (\u2191C a)) =\n      eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (\u2191C a)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase h_add\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (p q : MvPolynomial \u03c3 R),\n    Polynomial.eval x (eval\u2082 f g p) =\n        eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p \u2192\n      Polynomial.eval x (eval\u2082 f g q) =\n          eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q \u2192\n        Polynomial.eval x (eval\u2082 f g (p + q)) =\n          eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)\n[PROOFSTEP]\nintro p q hp hq\n[GOAL]\ncase h_add\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np\u271d p q : MvPolynomial \u03c3 R\nhp :\n  Polynomial.eval x (eval\u2082 f g p) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\nhq :\n  Polynomial.eval x (eval\u2082 f g q) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q\n\u22a2 Polynomial.eval x (eval\u2082 f g (p + q)) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)\n[PROOFSTEP]\nsimp [hp, hq]\n[GOAL]\ncase h_X\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np : MvPolynomial \u03c3 R\n\u22a2 \u2200 (p : MvPolynomial \u03c3 R) (n : \u03c3),\n    Polynomial.eval x (eval\u2082 f g p) =\n        eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p \u2192\n      Polynomial.eval x (eval\u2082 f g (p * X n)) =\n        eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)\n[PROOFSTEP]\nintro p n hp\n[GOAL]\ncase h_X\nR : Type u_1\nS : Type u_2\n\u03c3 : Type u_3\nx : S\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\nf : R \u2192+* Polynomial S\ng : \u03c3 \u2192 Polynomial S\np\u271d p : MvPolynomial \u03c3 R\nn : \u03c3\nhp :\n  Polynomial.eval x (eval\u2082 f g p) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\n\u22a2 Polynomial.eval x (eval\u2082 f g (p * X n)) =\n    eval\u2082 (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)\n[PROOFSTEP]\nsimp [hp]\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 \u2191(eval x) (Polynomial.eval q (\u2191(finSuccEquiv R n) f)) = \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f\n[PROOFSTEP]\nsimp only [finSuccEquiv_apply, coe_eval\u2082Hom, polynomial_eval_eval\u2082, eval_eval\u2082]\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C)))\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f\n[PROOFSTEP]\nconv in RingHom.comp _ _ => {\n  refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n  simp}\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n| RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C))\n[PROOFSTEP]\n{ refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n  simp}\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n| RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C))\n[PROOFSTEP]\n{ refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n  simp}\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n| RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C))\n[PROOFSTEP]\nrefine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\nr : R\n\u22a2 \u2191(RingHom.comp (eval x) (RingHom.comp (Polynomial.evalRingHom q) (RingHom.comp Polynomial.C C))) r = \u2191(RingHom.id R) r\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 eval\u2082 (RingHom.id R)\n      (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f\n[PROOFSTEP]\nsimp only [eval\u2082_id]\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 \u2191(eval fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) f =\n    \u2191(eval fun i => Fin.cases (\u2191(eval x) q) x i) f\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_a.e_f\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\n\u22a2 (fun s => \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) s))) = fun i =>\n    Fin.cases (\u2191(eval x) q) x i\n[PROOFSTEP]\nfunext i\n[GOAL]\ncase e_a.e_f.h\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) i)) = Fin.cases (\u2191(eval x) q) x i\n[PROOFSTEP]\nrefine Fin.cases (by simp) (by simp) i\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) 0)) = Fin.cases (\u2191(eval x) q) x 0\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nn : \u2115\nx : Fin n \u2192 R\ninst\u271d : CommSemiring R\nf : MvPolynomial (Fin (n + 1)) R\nq : MvPolynomial (Fin n) R\ni : Fin (n + 1)\n\u22a2 \u2200 (i : Fin n),\n    \u2191(eval x) (Polynomial.eval q (Fin.cases Polynomial.X (fun k => \u2191Polynomial.C (X k)) (Fin.succ i))) =\n      Fin.cases (\u2191(eval x) q) x (Fin.succ i)\n[PROOFSTEP]\nsimp\n", "meta": {"mathlib_filename": "Mathlib.Data.MvPolynomial.Polynomial", "llama_tokens": 3104, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7981867777396211, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.5026777756790307}}
{"text": "[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\u2081\ninst\u271d\u2074 : AddCommGroup V\u2082\ninst\u271d\u00b3 : Module k V\u2081\ninst\u271d\u00b2 : Module k V\u2082\ninst\u271d\u00b9 : AddTorsor V\u2081 P\u2081\ninst\u271d : AddTorsor V\u2082 P\u2082\nE : AffineSubspace k P\u2081\nEne : Nonempty { x // x \u2208 E }\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\n\u22a2 Nonempty { x // x \u2208 map \u03c6 E }\n[PROOFSTEP]\nobtain \u27e8x, hx\u27e9 := id Ene\n[GOAL]\ncase intro.mk\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\u2081\ninst\u271d\u2074 : AddCommGroup V\u2082\ninst\u271d\u00b3 : Module k V\u2081\ninst\u271d\u00b2 : Module k V\u2082\ninst\u271d\u00b9 : AddTorsor V\u2081 P\u2081\ninst\u271d : AddTorsor V\u2082 P\u2082\nE : AffineSubspace k P\u2081\nEne : Nonempty { x // x \u2208 E }\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nx : P\u2081\nhx : x \u2208 E\n\u22a2 Nonempty { x // x \u2208 map \u03c6 E }\n[PROOFSTEP]\nrefine' \u27e8\u27e8\u03c6 x, AffineSubspace.mem_map.mpr \u27e8x, hx, rfl\u27e9\u27e9\u27e9\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 { x // x \u2208 E } \u2192\u1d43[k] { x // x \u2208 F }\n[PROOFSTEP]\nrefine' \u27e8_, _, _\u27e9\n[GOAL]\ncase refine'_1\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 { x // x \u2208 E } \u2192 { x // x \u2208 F }\n[PROOFSTEP]\nexact fun x => \u27e8\u03c6 x, hEF <| AffineSubspace.mem_map.mpr \u27e8x, x.property, rfl\u27e9\u27e9\n[GOAL]\ncase refine'_2\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 { x // x \u2208 AffineSubspace.direction E } \u2192\u2097[k] { x // x \u2208 AffineSubspace.direction F }\n[PROOFSTEP]\nrefine' \u03c6.linear.restrict (_ : E.direction \u2264 F.direction.comap \u03c6.linear)\n[GOAL]\ncase refine'_2\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 AffineSubspace.direction E \u2264 Submodule.comap \u03c6.linear (AffineSubspace.direction F)\n[PROOFSTEP]\nrw [\u2190 Submodule.map_le_iff_le_comap, \u2190 AffineSubspace.map_direction]\n[GOAL]\ncase refine'_2\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 AffineSubspace.direction (AffineSubspace.map \u03c6 E) \u2264 AffineSubspace.direction F\n[PROOFSTEP]\nexact AffineSubspace.direction_le hEF\n[GOAL]\ncase refine'_3\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 \u2200 (p : { x // x \u2208 E }) (v : { x // x \u2208 AffineSubspace.direction E }),\n    { val := \u2191\u03c6 \u2191(v +\u1d65 p), property := (_ : \u2191\u03c6 \u2191(v +\u1d65 p) \u2208 \u2191F) } =\n      \u2191(LinearMap.restrict \u03c6.linear\n              (_ : AffineSubspace.direction E \u2264 Submodule.comap \u03c6.linear (AffineSubspace.direction F)))\n          v +\u1d65\n        { val := \u2191\u03c6 \u2191p, property := (_ : \u2191\u03c6 \u2191p \u2208 \u2191F) }\n[PROOFSTEP]\nintro p v\n[GOAL]\ncase refine'_3\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\np : { x // x \u2208 E }\nv : { x // x \u2208 AffineSubspace.direction E }\n\u22a2 { val := \u2191\u03c6 \u2191(v +\u1d65 p), property := (_ : \u2191\u03c6 \u2191(v +\u1d65 p) \u2208 \u2191F) } =\n    \u2191(LinearMap.restrict \u03c6.linear\n            (_ : AffineSubspace.direction E \u2264 Submodule.comap \u03c6.linear (AffineSubspace.direction F)))\n        v +\u1d65\n      { val := \u2191\u03c6 \u2191p, property := (_ : \u2191\u03c6 \u2191p \u2208 \u2191F) }\n[PROOFSTEP]\nsimp only [Subtype.ext_iff, Subtype.coe_mk, AffineSubspace.coe_vadd]\n[GOAL]\ncase refine'_3\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\np : { x // x \u2208 E }\nv : { x // x \u2208 AffineSubspace.direction E }\n\u22a2 \u2191\u03c6 (\u2191v +\u1d65 \u2191p) =\n    \u2191(\u2191(LinearMap.restrict \u03c6.linear\n              (_ : AffineSubspace.direction E \u2264 Submodule.comap \u03c6.linear (AffineSubspace.direction F)))\n          v) +\u1d65\n      \u2191\u03c6 \u2191p\n[PROOFSTEP]\napply AffineMap.map_vadd\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\u2081\ninst\u271d\u2074 : AddCommGroup V\u2082\ninst\u271d\u00b3 : Module k V\u2081\ninst\u271d\u00b2 : Module k V\u2082\ninst\u271d\u00b9 : AddTorsor V\u2081 P\u2081\ninst\u271d : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 AffineSubspace.direction E \u2264 Submodule.comap \u03c6.linear (AffineSubspace.direction F)\n[PROOFSTEP]\nrw [\u2190 Submodule.map_le_iff_le_comap, \u2190 AffineSubspace.map_direction]\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2076 : Ring k\ninst\u271d\u2075 : AddCommGroup V\u2081\ninst\u271d\u2074 : AddCommGroup V\u2082\ninst\u271d\u00b3 : Module k V\u2081\ninst\u271d\u00b2 : Module k V\u2082\ninst\u271d\u00b9 : AddTorsor V\u2081 P\u2081\ninst\u271d : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 AffineSubspace.direction (AffineSubspace.map \u03c6 E) \u2264 AffineSubspace.direction F\n[PROOFSTEP]\nexact AffineSubspace.direction_le hEF\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nh\u03c6 : Function.Injective \u2191\u03c6\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\n\u22a2 Function.Injective \u2191(restrict \u03c6 hEF)\n[PROOFSTEP]\nintro x y h\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nh\u03c6 : Function.Injective \u2191\u03c6\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\nx y : { x // x \u2208 E }\nh : \u2191(restrict \u03c6 hEF) x = \u2191(restrict \u03c6 hEF) y\n\u22a2 x = y\n[PROOFSTEP]\nsimp only [Subtype.ext_iff, Subtype.coe_mk, AffineMap.restrict.coe_apply] at h \u22a2\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nh\u03c6 : Function.Injective \u2191\u03c6\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nhEF : AffineSubspace.map \u03c6 E \u2264 F\nx y : { x // x \u2208 E }\nh : \u2191\u03c6 \u2191x = \u2191\u03c6 \u2191y\n\u22a2 \u2191x = \u2191y\n[PROOFSTEP]\nexact h\u03c6 h\n[GOAL]\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nh : AffineSubspace.map \u03c6 E = F\n\u22a2 Function.Surjective \u2191(restrict \u03c6 (_ : AffineSubspace.map \u03c6 E \u2264 F))\n[PROOFSTEP]\nrintro \u27e8x, hx : x \u2208 F\u27e9\n[GOAL]\ncase mk\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nh : AffineSubspace.map \u03c6 E = F\nx : P\u2082\nhx : x \u2208 F\n\u22a2 \u2203 a, \u2191(restrict \u03c6 (_ : AffineSubspace.map \u03c6 E \u2264 F)) a = { val := x, property := hx }\n[PROOFSTEP]\nrw [\u2190 h, AffineSubspace.mem_map] at hx \n[GOAL]\ncase mk\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nh : AffineSubspace.map \u03c6 E = F\nx : P\u2082\nhx\u271d : x \u2208 F\nhx : \u2203 y, y \u2208 E \u2227 \u2191\u03c6 y = x\n\u22a2 \u2203 a, \u2191(restrict \u03c6 (_ : AffineSubspace.map \u03c6 E \u2264 F)) a = { val := x, property := hx\u271d }\n[PROOFSTEP]\nobtain \u27e8y, hy, rfl\u27e9 := hx\n[GOAL]\ncase mk.intro.intro\nk : Type u_1\nV\u2081 : Type u_2\nP\u2081 : Type u_3\nV\u2082 : Type u_4\nP\u2082 : Type u_5\ninst\u271d\u2078 : Ring k\ninst\u271d\u2077 : AddCommGroup V\u2081\ninst\u271d\u2076 : AddCommGroup V\u2082\ninst\u271d\u2075 : Module k V\u2081\ninst\u271d\u2074 : Module k V\u2082\ninst\u271d\u00b3 : AddTorsor V\u2081 P\u2081\ninst\u271d\u00b2 : AddTorsor V\u2082 P\u2082\n\u03c6 : P\u2081 \u2192\u1d43[k] P\u2082\nE : AffineSubspace k P\u2081\nF : AffineSubspace k P\u2082\ninst\u271d\u00b9 : Nonempty { x // x \u2208 E }\ninst\u271d : Nonempty { x // x \u2208 F }\nh : AffineSubspace.map \u03c6 E = F\ny : P\u2081\nhy : y \u2208 E\nhx : \u2191\u03c6 y \u2208 F\n\u22a2 \u2203 a, \u2191(restrict \u03c6 (_ : AffineSubspace.map \u03c6 E \u2264 F)) a = { val := \u2191\u03c6 y, property := hx }\n[PROOFSTEP]\nexact \u27e8\u27e8y, hy\u27e9, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.LinearAlgebra.AffineSpace.Restrict", "llama_tokens": 5418, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.5026716130395124}}
{"text": "[GOAL]\nb : Bool\nh : Decidable (b = true)\n\u22a2 decide (b = true) = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\nh : Decidable (false = true)\n\u22a2 decide (false = true) = false\n[PROOFSTEP]\nexact decide_eq_false $ \u03bb j => by cases j\n[GOAL]\nh : Decidable (false = true)\nj : false = true\n\u22a2 False\n[PROOFSTEP]\ncases j\n[GOAL]\ncase true\nh : Decidable (true = true)\n\u22a2 decide (true = true) = true\n[PROOFSTEP]\nexact decide_eq_true $ rfl\n[GOAL]\np : Prop\ninst\u271d : Decidable p\n\u22a2 (decide \u00acp) = !decide p\n[PROOFSTEP]\nby_cases p\n[GOAL]\np : Prop\ninst\u271d : Decidable p\n\u22a2 (decide \u00acp) = !decide p\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np : Prop\ninst\u271d : Decidable p\nh : p\n\u22a2 (decide \u00acp) = !decide p\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\np : Prop\ninst\u271d : Decidable p\nh : \u00acp\n\u22a2 (decide \u00acp) = !decide p\n[PROOFSTEP]\nsimp [*]\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : \u00acp\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : \u00acp\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : p\nh : q\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : p\nh : \u00acq\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : \u00acp\nh : q\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : \u00acp\nh : \u00acq\n\u22a2 decide (p \u2227 q) = (decide p && decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : p\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : \u00acp\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh : \u00acp\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nby_cases q\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : p\nh : q\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : p\nh : \u00acq\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase pos\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : \u00acp\nh : q\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\np q : Prop\ninst\u271d\u00b9 : Decidable p\ninst\u271d : Decidable q\nh\u271d : \u00acp\nh : \u00acq\n\u22a2 decide (p \u2228 q) = (decide p || decide q)\n[PROOFSTEP]\nsimp [*]\n[GOAL]\na b : Bool\n\u22a2 a = b \u2194 (a = true \u2194 b = true)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 false = b \u2194 (false = true \u2194 b = true)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 true = b \u2194 (true = true \u2194 b = true)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 false = false \u2194 (false = true \u2194 false = true)\n[PROOFSTEP]\nsimp\n  -- Porting note: new theorem\n[GOAL]\ncase false.true\n\u22a2 false = true \u2194 (false = true \u2194 true = true)\n[PROOFSTEP]\nsimp\n  -- Porting note: new theorem\n[GOAL]\ncase true.false\n\u22a2 true = false \u2194 (true = true \u2194 false = true)\n[PROOFSTEP]\nsimp\n  -- Porting note: new theorem\n[GOAL]\ncase true.true\n\u22a2 true = true \u2194 (true = true \u2194 true = true)\n[PROOFSTEP]\nsimp\n  -- Porting note: new theorem\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : BEq \u03b1\ninst\u271d : LawfulBEq \u03b1\na b : \u03b1\n\u22a2 (a == b) = true \u2194 (b == a) = true\n[PROOFSTEP]\nsimp [@eq_comm \u03b1]\n[GOAL]\nb : Bool\n\u22a2 b = false \u2228 b = true\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 false = false \u2228 false = true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u22a2 true = false \u2228 true = true\n[PROOFSTEP]\nsimp\n[GOAL]\np : Bool \u2192 Prop\nh : \u2200 (b : Bool), p b\n\u22a2 p false \u2227 p true\n[PROOFSTEP]\nsimp [h]\n[GOAL]\np : Bool \u2192 Prop\nx\u271d : p false \u2227 p true\nb : Bool\nh\u2081 : p false\nh\u2082 : p true\n\u22a2 p b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\np : Bool \u2192 Prop\nx\u271d : p false \u2227 p true\nh\u2081 : p false\nh\u2082 : p true\n\u22a2 p false\n[PROOFSTEP]\nassumption\n[GOAL]\ncase true\np : Bool \u2192 Prop\nx\u271d : p false \u2227 p true\nh\u2081 : p false\nh\u2082 : p true\n\u22a2 p true\n[PROOFSTEP]\nassumption\n[GOAL]\np : Bool \u2192 Prop\nx\u271d : \u2203 b, p b\nb : Bool\nh : p b\n\u22a2 p false \u2228 p true\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\np : Bool \u2192 Prop\nx\u271d : \u2203 b, p b\nh : p false\n\u22a2 p false \u2228 p true\ncase true p : Bool \u2192 Prop x\u271d : \u2203 b, p b h : p true \u22a2 p false \u2228 p true\n[PROOFSTEP]\nexact Or.inl h\n[GOAL]\ncase true\np : Bool \u2192 Prop\nx\u271d : \u2203 b, p b\nh : p true\n\u22a2 p false \u2228 p true\n[PROOFSTEP]\nexact Or.inr h\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nt e : \u03b1\n\u22a2 (bif b then t else e) = if b = true then t else e\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nt e : \u03b1\n\u22a2 (bif false then t else e) = if false = true then t else e\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\n\u03b1 : Type u_1\nt e : \u03b1\n\u22a2 (bif true then t else e) = if true = true then t else e\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\np : Prop\ninst\u271d : Decidable p\nt e : \u03b1\n\u22a2 (bif decide p then t else e) = if p then t else e\n[PROOFSTEP]\nby_cases p\n[GOAL]\n\u03b1 : Type u_1\np : Prop\ninst\u271d : Decidable p\nt e : \u03b1\n\u22a2 (bif decide p then t else e) = if p then t else e\n[PROOFSTEP]\nby_cases p\n[GOAL]\ncase pos\n\u03b1 : Type u_1\np : Prop\ninst\u271d : Decidable p\nt e : \u03b1\nh : p\n\u22a2 (bif decide p then t else e) = if p then t else e\n[PROOFSTEP]\nsimp [*]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\np : Prop\ninst\u271d : Decidable p\nt e : \u03b1\nh : \u00acp\n\u22a2 (bif decide p then t else e) = if p then t else e\n[PROOFSTEP]\nsimp [*]\n[GOAL]\n\u03b1 : Type u_1\nb : Bool\nt e : \u03b1\n\u22a2 (bif !b then t else e) = bif b then e else t\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\nt e : \u03b1\n\u22a2 (bif !false then t else e) = bif false then e else t\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u03b1 : Type u_1\nt e : \u03b1\n\u22a2 (bif !true then t else e) = bif true then e else t\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, (a = true \u2194 b = true) \u2194 a = b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a : Bool}, a \u2260 false \u2192 a = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a : Bool}, a \u2260 true \u2192 a = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), (a || b) = (b || a)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b c : Bool), (a || (b || c)) = (b || (a || c))\n[PROOFSTEP]\ndecide\n[GOAL]\na b : Bool\nH : a = true\n\u22a2 (a || b) = true\n[PROOFSTEP]\nsimp [H]\n[GOAL]\na b : Bool\nH : b = true\n\u22a2 (a || b) = true\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\nH : b = true\n\u22a2 (false || b) = true\n[PROOFSTEP]\nsimp [H]\n[GOAL]\ncase true\nb : Bool\nH : b = true\n\u22a2 (true || b) = true\n[PROOFSTEP]\nsimp [H]\n[GOAL]\n\u22a2 \u2200 (a b : Bool), (a && b) = (b && a)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b c : Bool), (a && (b && c)) = (b && (a && c))\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, (a && b) = true \u2192 a = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, a = true \u2192 b = true \u2192 (a && b) = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, (a && b) = true \u2192 b = true\n[PROOFSTEP]\ndecide\n[GOAL]\na b c : Bool\n\u22a2 (a && (b || c)) = (a && b || a && c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 (false && (b || c)) = (false && b || false && c)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\nb c : Bool\n\u22a2 (true && (b || c)) = (true && b || true && c)\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : Bool\n\u22a2 ((a || b) && c) = (a && c || b && c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 ((false || b) && c) = (false && c || b && c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb c : Bool\n\u22a2 ((true || b) && c) = (true && c || b && c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\nc : Bool\n\u22a2 ((false || false) && c) = (false && c || false && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.true\nc : Bool\n\u22a2 ((false || true) && c) = (false && c || true && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.false\nc : Bool\n\u22a2 ((true || false) && c) = (true && c || false && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.true\nc : Bool\n\u22a2 ((true || true) && c) = (true && c || true && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.false.false\n\u22a2 ((false || false) && false) = (false && false || false && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.false.true\n\u22a2 ((false || false) && true) = (false && true || false && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.false\n\u22a2 ((false || true) && false) = (false && false || true && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.true\n\u22a2 ((false || true) && true) = (false && true || true && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.false\n\u22a2 ((true || false) && false) = (true && false || false && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.true\n\u22a2 ((true || false) && true) = (true && true || false && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.false\n\u22a2 ((true || true) && false) = (true && false || true && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.true\n\u22a2 ((true || true) && true) = (true && true || true && true)\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : Bool\n\u22a2 (a || b && c) = ((a || b) && (a || c))\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 (false || b && c) = ((false || b) && (false || c))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\nb c : Bool\n\u22a2 (true || b && c) = ((true || b) && (true || c))\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : Bool\n\u22a2 (a && b || c) = ((a || c) && (b || c))\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 (false && b || c) = ((false || c) && (b || c))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb c : Bool\n\u22a2 (true && b || c) = ((true || c) && (b || c))\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\nc : Bool\n\u22a2 (false && false || c) = ((false || c) && (false || c))\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.true\nc : Bool\n\u22a2 (false && true || c) = ((false || c) && (true || c))\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.false\nc : Bool\n\u22a2 (true && false || c) = ((true || c) && (false || c))\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.true\nc : Bool\n\u22a2 (true && true || c) = ((true || c) && (true || c))\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.false.false\n\u22a2 (false && false || false) = ((false || false) && (false || false))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.false.true\n\u22a2 (false && false || true) = ((false || true) && (false || true))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.false\n\u22a2 (false && true || false) = ((false || false) && (true || false))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.true\n\u22a2 (false && true || true) = ((false || true) && (true || true))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.false\n\u22a2 (true && false || false) = ((true || false) && (false || false))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.true\n\u22a2 (true && false || true) = ((true || true) && (false || true))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.false\n\u22a2 (true && true || false) = ((true || false) && (true || false))\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.true\n\u22a2 (true && true || true) = ((true || true) && (true || true))\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, a = !b \u2194 a \u2260 b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, (!decide (a = b)) = true \u2194 a \u2260 b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, \u00aca = !b \u2194 a = b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, \u00ac(!a) = b \u2194 a = b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (b : Bool), (!b) \u2260 b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (b : Bool), b \u2260 !b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), a = b \u2228 a = !b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {b : Bool}, (!b) = true \u2194 \u00acb = true\n[PROOFSTEP]\nsimp\n[GOAL]\na : Bool\n\u22a2 (!decide (a = false)) = true \u2192 a = true\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u22a2 (!decide (false = false)) = true \u2192 false = true\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true\n\u22a2 (!decide (true = false)) = true \u2192 true = true\n[PROOFSTEP]\ndecide\n[GOAL]\na : Bool\n\u22a2 (!decide (a = true)) = true \u2192 a = false\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\n\u22a2 (!decide (false = true)) = true \u2192 false = false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true\n\u22a2 (!decide (true = true)) = true \u2192 true = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x : Bool), (x && !x) = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x : Bool), (!x && x) = false\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x : Bool), (x || !x) = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x : Bool), (!x || x) = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), xor a b = xor b a\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b c : Bool), xor (xor a b) c = xor a (xor b c)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b c : Bool), xor a (xor b c) = xor b (xor a c)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a : Bool), xor (!a) a = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a : Bool), (xor a !a) = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), (xor (!a) !b) = xor a b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a : Bool), xor false a = a\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a : Bool), xor a false = a\n[PROOFSTEP]\ndecide\n[GOAL]\na b c : Bool\n\u22a2 (a && xor b c) = xor (a && b) (a && c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 (false && xor b c) = xor (false && b) (false && c)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true\nb c : Bool\n\u22a2 (true && xor b c) = xor (true && b) (true && c)\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : Bool\n\u22a2 (xor a b && c) = xor (a && c) (b && c)\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb c : Bool\n\u22a2 (xor false b && c) = xor (false && c) (b && c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb c : Bool\n\u22a2 (xor true b && c) = xor (true && c) (b && c)\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\nc : Bool\n\u22a2 (xor false false && c) = xor (false && c) (false && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.true\nc : Bool\n\u22a2 (xor false true && c) = xor (false && c) (true && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.false\nc : Bool\n\u22a2 (xor true false && c) = xor (true && c) (false && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase true.true\nc : Bool\n\u22a2 (xor true true && c) = xor (true && c) (true && c)\n[PROOFSTEP]\ncases c\n[GOAL]\ncase false.false.false\n\u22a2 (xor false false && false) = xor (false && false) (false && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.false.true\n\u22a2 (xor false false && true) = xor (false && true) (false && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.false\n\u22a2 (xor false true && false) = xor (false && false) (true && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase false.true.true\n\u22a2 (xor false true && true) = xor (false && true) (true && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.false\n\u22a2 (xor true false && false) = xor (true && false) (false && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.false.true\n\u22a2 (xor true false && true) = xor (true && true) (false && true)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.false\n\u22a2 (xor true true && false) = xor (true && false) (true && false)\n[PROOFSTEP]\nsimp\n[GOAL]\ncase true.true.true\n\u22a2 (xor true true && true) = xor (true && true) (true && true)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u22a2 \u2200 {x y : Bool}, xor x y = true \u2194 x \u2260 y\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), (!(a && b)) = (!a || !b)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), (!(a || b)) = (!a && !b)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {a b : Bool}, (!a) = !b \u2192 a = b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a : Bool), a \u2264 a\n[PROOFSTEP]\nunfold LE.le\n[GOAL]\n\u22a2 \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b c : Bool), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nunfold LE.le\n[GOAL]\n\u22a2 \u2200 (a b c : Bool),\n    { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n      { le := fun a b => a = false \u2228 b = true }.1 b c \u2192 { le := fun a b => a = false \u2228 b = true }.1 a c\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nunfold LE.le Preorder.toLE\n[GOAL]\n\u22a2 \u2200 (a b : Bool),\n    (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n              (_ :\n                \u2200 (a b c : Bool),\n                  { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                    { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                      { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n        a b \u2192\n      (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                (_ :\n                  \u2200 (a b c : Bool),\n                    { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                      { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                        { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n          b a \u2192\n        a = b\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (a b : Bool), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nunfold LE.le Preorder.toLE PartialOrder.toPreorder\n[GOAL]\n\u22a2 \u2200 (a b : Bool),\n    (PartialOrder.mk\n                (_ :\n                  \u2200 (a b : Bool),\n                    (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                              (_ :\n                                \u2200 (a b c : Bool),\n                                  { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                    { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                      { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                        a b \u2192\n                      (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                                (_ :\n                                  \u2200 (a b c : Bool),\n                                    { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                      { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                        { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                          b a \u2192\n                        a = b)).1.1.1\n        a b \u2228\n      (PartialOrder.mk\n                (_ :\n                  \u2200 (a b : Bool),\n                    (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                              (_ :\n                                \u2200 (a b c : Bool),\n                                  { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                    { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                      { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                        a b \u2192\n                      (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                                (_ :\n                                  \u2200 (a b c : Bool),\n                                    { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                      { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                        { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                          b a \u2192\n                        a = b)).1.1.1\n        b a\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 DecidableRel fun x x_1 => x \u2264 x_1\n[PROOFSTEP]\nunfold LE.le Preorder.toLE PartialOrder.toPreorder\n[GOAL]\n\u22a2 DecidableRel fun x x_1 =>\n    (PartialOrder.mk\n              (_ :\n                \u2200 (a b : Bool),\n                  (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                            (_ :\n                              \u2200 (a b c : Bool),\n                                { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                  { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                    { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                      a b \u2192\n                    (Preorder.mk (_ : \u2200 (a : Bool), { le := fun a b => a = false \u2228 b = true }.1 a a)\n                              (_ :\n                                \u2200 (a b c : Bool),\n                                  { le := fun a b => a = false \u2228 b = true }.1 a b \u2192\n                                    { le := fun a b => a = false \u2228 b = true }.1 b c \u2192\n                                      { le := fun a b => a = false \u2228 b = true }.1 a c)).1.1\n                        b a \u2192\n                      a = b)).1.1.1\n      x x_1\n[PROOFSTEP]\nexact inferInstance\n[GOAL]\na b : Bool\n\u22a2 min a b = if a \u2264 b then a else b\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 min false b = if false \u2264 b then false else b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 min true b = if true \u2264 b then true else b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 min false false = if false \u2264 false then false else false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase false.true\n\u22a2 min false true = if false \u2264 true then false else true\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.false\n\u22a2 min true false = if true \u2264 false then true else false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.true\n\u22a2 min true true = if true \u2264 true then true else true\n[PROOFSTEP]\ndecide\n[GOAL]\na b : Bool\n\u22a2 max a b = if a \u2264 b then b else a\n[PROOFSTEP]\ncases a\n[GOAL]\ncase false\nb : Bool\n\u22a2 max false b = if false \u2264 b then b else false\n[PROOFSTEP]\ncases b\n[GOAL]\ncase true\nb : Bool\n\u22a2 max true b = if true \u2264 b then b else true\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false.false\n\u22a2 max false false = if false \u2264 false then false else false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase false.true\n\u22a2 max false true = if false \u2264 true then true else false\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.false\n\u22a2 max true false = if true \u2264 false then false else true\n[PROOFSTEP]\ndecide\n[GOAL]\ncase true.true\n\u22a2 max true true = if true \u2264 true then true else true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {x y : Bool}, x < y \u2194 x = false \u2227 y = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {x y : Bool}, x \u2264 y \u2194 x = true \u2192 y = true\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x y : Bool), (x && y) \u2264 x\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x y : Bool), (x && y) \u2264 y\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {x y z : Bool}, x \u2264 y \u2192 x \u2264 z \u2192 x \u2264 (y && z)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x y : Bool), x \u2264 (x || y)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 (x y : Bool), y \u2264 (x || y)\n[PROOFSTEP]\ndecide\n[GOAL]\n\u22a2 \u2200 {x y z : Bool}, x \u2264 z \u2192 y \u2264 z \u2192 (x || y) \u2264 z\n[PROOFSTEP]\ndecide\n[GOAL]\nn m : \u2115\nh : n \u2264 m\n\u22a2 ofNat n \u2264 ofNat m\n[PROOFSTEP]\nsimp only [ofNat, ne_eq, _root_.decide_not]\n[GOAL]\nn m : \u2115\nh : n \u2264 m\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\ncases Nat.decEq n 0 with\n| isTrue hn => rw [decide_eq_true hn]; exact false_le\n| isFalse hn =>\n  cases Nat.decEq m 0 with\n  | isFalse hm => rw [decide_eq_false hm]; exact le_true\n  | isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\nn m : \u2115\nh : n \u2264 m\nx\u271d : Decidable (n = 0)\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\ncases Nat.decEq n 0 with\n| isTrue hn => rw [decide_eq_true hn]; exact false_le\n| isFalse hn =>\n  cases Nat.decEq m 0 with\n  | isFalse hm => rw [decide_eq_false hm]; exact le_true\n  | isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\ncase isTrue\nn m : \u2115\nh : n \u2264 m\nhn : n = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\n\n| isTrue hn => rw [decide_eq_true hn]; exact false_le\n[GOAL]\ncase isTrue\nn m : \u2115\nh : n \u2264 m\nhn : n = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\nrw [decide_eq_true hn]\n[GOAL]\ncase isTrue\nn m : \u2115\nh : n \u2264 m\nhn : n = 0\n\u22a2 (!true) \u2264 !decide (m = 0)\n[PROOFSTEP]\nexact false_le\n[GOAL]\ncase isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\n\n| isFalse hn =>\n  cases Nat.decEq m 0 with\n  | isFalse hm => rw [decide_eq_false hm]; exact le_true\n  | isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\ncase isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\ncases Nat.decEq m 0 with\n| isFalse hm => rw [decide_eq_false hm]; exact le_true\n| isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\ncase isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nx\u271d : Decidable (m = 0)\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\ncases Nat.decEq m 0 with\n| isFalse hm => rw [decide_eq_false hm]; exact le_true\n| isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\ncase isFalse.isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\n\n| isFalse hm => rw [decide_eq_false hm]; exact le_true\n[GOAL]\ncase isFalse.isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\nrw [decide_eq_false hm]\n[GOAL]\ncase isFalse.isFalse\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nhm : \u00acm = 0\n\u22a2 (!decide (n = 0)) \u2264 !false\n[PROOFSTEP]\nexact le_true\n[GOAL]\ncase isFalse.isTrue\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\n\n| isTrue hm => subst hm; have h := le_antisymm h (Nat.zero_le n); contradiction\n[GOAL]\ncase isFalse.isTrue\nn m : \u2115\nh : n \u2264 m\nhn : \u00acn = 0\nhm : m = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (m = 0)\n[PROOFSTEP]\nsubst hm\n[GOAL]\ncase isFalse.isTrue\nn : \u2115\nhn : \u00acn = 0\nh : n \u2264 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (0 = 0)\n[PROOFSTEP]\nhave h := le_antisymm h (Nat.zero_le n)\n[GOAL]\ncase isFalse.isTrue\nn : \u2115\nhn : \u00acn = 0\nh\u271d : n \u2264 0\nh : n = 0\n\u22a2 (!decide (n = 0)) \u2264 !decide (0 = 0)\n[PROOFSTEP]\ncontradiction\n[GOAL]\nb\u2080 b\u2081 : Bool\nh : b\u2080 \u2264 b\u2081\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\ncases h with\n| inl h => subst h; exact Nat.zero_le _\n| inr h => subst h; cases b\u2080 <;> simp\n[GOAL]\nb\u2080 b\u2081 : Bool\nh : b\u2080 \u2264 b\u2081\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\ncases h with\n| inl h => subst h; exact Nat.zero_le _\n| inr h => subst h; cases b\u2080 <;> simp\n[GOAL]\ncase inl\nb\u2080 b\u2081 : Bool\nh : b\u2080 = false\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\n\n| inl h => subst h; exact Nat.zero_le _\n[GOAL]\ncase inl\nb\u2080 b\u2081 : Bool\nh : b\u2080 = false\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inl\nb\u2081 : Bool\n\u22a2 toNat false \u2264 toNat b\u2081\n[PROOFSTEP]\nexact Nat.zero_le _\n[GOAL]\ncase inr\nb\u2080 b\u2081 : Bool\nh : b\u2081 = true\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\n\n| inr h => subst h; cases b\u2080 <;> simp\n[GOAL]\ncase inr\nb\u2080 b\u2081 : Bool\nh : b\u2081 = true\n\u22a2 toNat b\u2080 \u2264 toNat b\u2081\n[PROOFSTEP]\nsubst h\n[GOAL]\ncase inr\nb\u2080 : Bool\n\u22a2 toNat b\u2080 \u2264 toNat true\n[PROOFSTEP]\ncases b\u2080\n[GOAL]\ncase inr.false\n\u22a2 toNat false \u2264 toNat true\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.true\n\u22a2 toNat true \u2264 toNat true\n[PROOFSTEP]\nsimp\n[GOAL]\nb : Bool\n\u22a2 ofNat (toNat b) = b\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u22a2 ofNat (toNat false) = false\n[PROOFSTEP]\nrfl\n[GOAL]\ncase true\n\u22a2 ofNat (toNat true) = true\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nx y : Bool\nhxy : f x = f y\n\u22a2 x = y\n[PROOFSTEP]\ncases x\n[GOAL]\ncase false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\ny : Bool\nhxy : f false = f y\n\u22a2 false = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\ny : Bool\nhxy : f true = f y\n\u22a2 true = y\n[PROOFSTEP]\ncases y\n[GOAL]\ncase false.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f false\n\u22a2 false = false\ncase false.true\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f false = f true\n\u22a2 false = true\ncase true.false\n\u03b1 : Sort u_1\nf : Bool \u2192 \u03b1\nH : f false \u2260 f true\nhxy : f true = f false\n\u22a2 true = false\ncase true.true \u03b1 : Sort u_1 f : Bool \u2192 \u03b1 H : f false \u2260 f true hxy : f true = f true \u22a2 true = true\n[PROOFSTEP]\nexacts [rfl, (H hxy).elim, (H hxy.symm).elim, rfl]\n[GOAL]\nf : Bool \u2192 Bool\nx : Bool\n\u22a2 f (f (f x)) = f x\n[PROOFSTEP]\ncases x\n[GOAL]\ncase false\nf : Bool \u2192 Bool\n\u22a2 f (f (f false)) = f false\n[PROOFSTEP]\ncases h\u2081 : f true\n[GOAL]\ncase true\nf : Bool \u2192 Bool\n\u22a2 f (f (f true)) = f true\n[PROOFSTEP]\ncases h\u2081 : f true\n[GOAL]\ncase false.false\nf : Bool \u2192 Bool\nh\u2081 : f true = false\n\u22a2 f (f (f false)) = f false\n[PROOFSTEP]\ncases h\u2082 : f false\n[GOAL]\ncase false.true\nf : Bool \u2192 Bool\nh\u2081 : f true = true\n\u22a2 f (f (f false)) = f false\n[PROOFSTEP]\ncases h\u2082 : f false\n[GOAL]\ncase true.false\nf : Bool \u2192 Bool\nh\u2081 : f true = false\n\u22a2 f (f false) = false\n[PROOFSTEP]\ncases h\u2082 : f false\n[GOAL]\ncase true.true\nf : Bool \u2192 Bool\nh\u2081 : f true = true\n\u22a2 f (f true) = true\n[PROOFSTEP]\ncases h\u2082 : f false\n[GOAL]\ncase false.false.false\nf : Bool \u2192 Bool\nh\u2081 : f true = false\nh\u2082 : f false = false\n\u22a2 f (f false) = false\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase false.false.true\nf : Bool \u2192 Bool\nh\u2081 : f true = false\nh\u2082 : f false = true\n\u22a2 f (f true) = true\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase false.true.false\nf : Bool \u2192 Bool\nh\u2081 : f true = true\nh\u2082 : f false = false\n\u22a2 f (f false) = false\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase false.true.true\nf : Bool \u2192 Bool\nh\u2081 : f true = true\nh\u2082 : f false = true\n\u22a2 f (f true) = true\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase true.false.false\nf : Bool \u2192 Bool\nh\u2081 : f true = false\nh\u2082 : f false = false\n\u22a2 f false = false\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase true.false.true\nf : Bool \u2192 Bool\nh\u2081 : f true = false\nh\u2082 : f false = true\n\u22a2 f true = false\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase true.true.false\nf : Bool \u2192 Bool\nh\u2081 : f true = true\nh\u2082 : f false = false\n\u22a2 f (f true) = true\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n[GOAL]\ncase true.true.true\nf : Bool \u2192 Bool\nh\u2081 : f true = true\nh\u2082 : f false = true\n\u22a2 f (f true) = true\n[PROOFSTEP]\nsimp only [h\u2081, h\u2082]\n", "meta": {"mathlib_filename": "Mathlib.Data.Bool.Basic", "llama_tokens": 12880, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7799929104825006, "lm_q2_score": 0.6442250928250375, "lm_q1q2_score": 0.5024910051584601}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a = \u2191u * (\u2191u\u207b\u00b9 * a)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\nx\u271d : a \u2223 b * \u2191u\nc : \u03b1\nEq : b * \u2191u = a * c\n\u22a2 b = a * (c * \u2191u\u207b\u00b9)\n[PROOFSTEP]\nrw [\u2190 mul_assoc, \u2190 Eq, Units.mul_inv_cancel_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\nh : a \u2223 b\n\u22a2 a * \u2191u * \u2191u\u207b\u00b9 = a\n[PROOFSTEP]\nrw [mul_assoc, u.mul_inv, mul_one]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a \u2223 \u2191u * b \u2194 a \u2223 b\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a \u2223 b * \u2191u \u2194 a \u2223 b\n[PROOFSTEP]\napply dvd_mul_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191u * a \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\nrw [mul_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a * \u2191u \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\napply mul_right_dvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b u : \u03b1\nhu : IsUnit u\n\u22a2 u \u2223 a\n[PROOFSTEP]\nrcases hu with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191u \u2223 a\n[PROOFSTEP]\napply Units.coe_dvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b u : \u03b1\nhu : IsUnit u\n\u22a2 a \u2223 b * u \u2194 a \u2223 b\n[PROOFSTEP]\nrcases hu with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a \u2223 b * \u2191u \u2194 a \u2223 b\n[PROOFSTEP]\napply Units.dvd_mul_right\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b u : \u03b1\nhu : IsUnit u\n\u22a2 a * u \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\nrcases hu with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : Monoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a * \u2191u \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\napply Units.mul_right_dvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b u : \u03b1\nhu : IsUnit u\n\u22a2 a \u2223 u * b \u2194 a \u2223 b\n[PROOFSTEP]\nrcases hu with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 a \u2223 \u2191u * b \u2194 a \u2223 b\n[PROOFSTEP]\napply Units.dvd_mul_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b u : \u03b1\nhu : IsUnit u\n\u22a2 u * a \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\nrcases hu with \u27e8u, rfl\u27e9\n[GOAL]\ncase intro\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\na b : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191u * a \u2223 b \u2194 a \u2223 b\n[PROOFSTEP]\napply Units.mul_left_dvd\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d : CommMonoid \u03b1\nx : \u03b1\nx\u271d : x \u2223 1\ny : \u03b1\nh : 1 = x * y\n\u22a2 y * x = 1\n[PROOFSTEP]\nrw [h, mul_comm]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Divisibility.Units", "llama_tokens": 1235, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.6992544273261176, "lm_q1q2_score": 0.5024803104568933}}
{"text": "[GOAL]\n\u03b1 \u03b2 : DistLatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n        map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) } \u226b\n      {\n        toSupHom :=\n          { toFun := \u2191(OrderIso.symm e),\n            map_sup' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n        map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) } =\n    \ud835\udfd9 \u03b1\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : DistLatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget DistLatCat).obj \u03b1\n\u22a2 \u2191({ toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n            map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) } \u226b\n          {\n            toSupHom :=\n              { toFun := \u2191(OrderIso.symm e),\n                map_sup' :=\n                  (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n            map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) })\n      x\u271d =\n    \u2191(\ud835\udfd9 \u03b1) x\u271d\n[PROOFSTEP]\nexact e.symm_apply_apply _\n[GOAL]\n\u03b1 \u03b2 : DistLatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\n\u22a2 {\n        toSupHom :=\n          { toFun := \u2191(OrderIso.symm e),\n            map_sup' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n        map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) } \u226b\n      { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n        map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) } =\n    \ud835\udfd9 \u03b2\n[PROOFSTEP]\next\n[GOAL]\ncase w\n\u03b1 \u03b2 : DistLatCat\ne : \u2191\u03b1 \u2243o \u2191\u03b2\nx\u271d : (forget DistLatCat).obj \u03b2\n\u22a2 \u2191({\n            toSupHom :=\n              { toFun := \u2191(OrderIso.symm e),\n                map_sup' :=\n                  (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2294 b) = \u2191(OrderIso.symm e) a \u2294 \u2191(OrderIso.symm e) b) },\n            map_inf' := (_ : \u2200 (a b : \u2191\u03b2), \u2191(OrderIso.symm e) (a \u2293 b) = \u2191(OrderIso.symm e) a \u2293 \u2191(OrderIso.symm e) b) } \u226b\n          { toSupHom := { toFun := \u2191e, map_sup' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2294 b) = \u2191e a \u2294 \u2191e b) },\n            map_inf' := (_ : \u2200 (a b : \u2191\u03b1), \u2191e (a \u2293 b) = \u2191e a \u2293 \u2191e b) })\n      x\u271d =\n    \u2191(\ud835\udfd9 \u03b2) x\u271d\n[PROOFSTEP]\nexact e.apply_symm_apply _\n", "meta": {"mathlib_filename": "Mathlib.Order.Category.DistLatCat", "llama_tokens": 1347, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8267117983401363, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.5023623119176175}}
{"text": "[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\n\u22a2 Continuous fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd))\n[PROOFSTEP]\nexact\n  ((map_continuous HSpace.hmul).comp\n        ((continuous_fst.comp continuous_fst).prod_mk (continuous_fst.comp continuous_snd))).prod_mk\n    ((map_continuous HSpace.hmul).comp\n      ((continuous_snd.comp continuous_fst).prod_mk (continuous_snd.comp continuous_snd)))\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\n\u22a2 \u2191(ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd))) ((e, e), e, e) = (e, e)\n[PROOFSTEP]\nsimp only [ContinuousMap.coe_mk, Prod.mk.inj_iff]\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\n\u22a2 \u2191hmul (e, e) = e \u2227 \u2191hmul (e, e) = e\n[PROOFSTEP]\nexact \u27e8HSpace.hmul_e_e, HSpace.hmul_e_e\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nlet G : I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (HSpace.eHmul (p.1, p.2.1), HSpace.eHmul (p.1, p.2.2))\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nhave hG : Continuous G :=\n  (Continuous.comp HSpace.eHmul.1.1.2 (continuous_fst.prod_mk (continuous_fst.comp continuous_snd))).prod_mk\n    (Continuous.comp HSpace.eHmul.1.1.2 (continuous_fst.prod_mk (continuous_snd.comp continuous_snd)))\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nuse!\u27e8G, hG\u27e9\n[GOAL]\ncase map_zero_left\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (x : X \u00d7 Y),\n    ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n        x\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase map_zero_left.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\nx : X\ny : Y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk G) (0, x, y) =\n    \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n          (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n      (x, y)\n[PROOFSTEP]\nexacts [Prod.mk.inj_iff.mpr \u27e8HSpace.eHmul.1.2 x, HSpace.eHmul.1.2 y\u27e9]\n[GOAL]\ncase map_one_left\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (x : X \u00d7 Y), ContinuousMap.toFun (ContinuousMap.mk G) (1, x) = \u2191(ContinuousMap.id (X \u00d7 Y)) x\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase map_one_left.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\nx : X\ny : Y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk G) (1, x, y) = \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nexact Prod.mk.inj_iff.mpr \u27e8HSpace.eHmul.1.3 x, HSpace.eHmul.1.3 y\u27e9\n[GOAL]\ncase prop'\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (t : \u2191I) (x : X \u00d7 Y),\n    x \u2208 {(e, e)} \u2192\n      \u2191(ContinuousMap.mk fun x =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk G,\n                      map_zero_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                              \u2191(comp\n                                    (ContinuousMap.mk fun p =>\n                                      (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                    (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                                x),\n                      map_one_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                              \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n                  (t, x))\n            x =\n          \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n            x \u2227\n        \u2191(ContinuousMap.mk fun x =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk G,\n                      map_zero_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                              \u2191(comp\n                                    (ContinuousMap.mk fun p =>\n                                      (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                    (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                                x),\n                      map_one_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                              \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n                  (t, x))\n            x =\n          \u2191(ContinuousMap.id (X \u00d7 Y)) x\n[PROOFSTEP]\nrintro t \u27e8x, y\u27e9 h\n[GOAL]\ncase prop'.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\nt : \u2191I\nx : X\ny : Y\nh : (x, y) \u2208 {(e, e)}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n        (x, y) \u2227\n    \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nreplace h := Prod.mk.inj_iff.mp (Set.mem_singleton_iff.mp h)\n[GOAL]\ncase prop'.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191eHmul (p.fst, p.snd.fst), \u2191eHmul (p.fst, p.snd.snd))\nhG : Continuous G\nt : \u2191I\nx : X\ny : Y\nh : x = e \u2227 y = e\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n        (x, y) \u2227\n    \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (const (X \u00d7 Y) (e, e)) (ContinuousMap.id (X \u00d7 Y))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nexact\n  \u27e8Prod.mk.inj_iff.mpr\n      \u27e8HomotopyRel.eq_fst HSpace.eHmul t (Set.mem_singleton_iff.mpr h.1),\n        HomotopyRel.eq_fst HSpace.eHmul t (Set.mem_singleton_iff.mpr h.2)\u27e9,\n    Prod.mk.inj_iff.mpr \u27e8(HSpace.eHmul.2 t x h.1).2, (HSpace.eHmul.2 t y h.2).2\u27e9\u27e9\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nlet G : I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (HSpace.hmulE (p.1, p.2.1), HSpace.hmulE (p.1, p.2.2))\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nhave hG : Continuous G :=\n  (Continuous.comp HSpace.hmulE.1.1.2 (continuous_fst.prod_mk (continuous_fst.comp continuous_snd))).prod_mk\n    (Continuous.comp HSpace.hmulE.1.1.2 (continuous_fst.prod_mk (continuous_snd.comp continuous_snd)))\n[GOAL]\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 HomotopyRel\n    (comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n      (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n    (ContinuousMap.id (X \u00d7 Y)) {(e, e)}\n[PROOFSTEP]\nuse!\u27e8G, hG\u27e9\n[GOAL]\ncase map_zero_left\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (x : X \u00d7 Y),\n    ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n        x\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase map_zero_left.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\nx : X\ny : Y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk G) (0, x, y) =\n    \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n          (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n      (x, y)\n[PROOFSTEP]\nexacts [Prod.mk.inj_iff.mpr \u27e8HSpace.hmulE.1.2 x, HSpace.hmulE.1.2 y\u27e9]\n[GOAL]\ncase map_one_left\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (x : X \u00d7 Y), ContinuousMap.toFun (ContinuousMap.mk G) (1, x) = \u2191(ContinuousMap.id (X \u00d7 Y)) x\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9\n[GOAL]\ncase map_one_left.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\nx : X\ny : Y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk G) (1, x, y) = \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nexact Prod.mk.inj_iff.mpr \u27e8HSpace.hmulE.1.3 x, HSpace.hmulE.1.3 y\u27e9\n[GOAL]\ncase prop'\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\n\u22a2 \u2200 (t : \u2191I) (x : X \u00d7 Y),\n    x \u2208 {(e, e)} \u2192\n      \u2191(ContinuousMap.mk fun x =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk G,\n                      map_zero_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                              \u2191(comp\n                                    (ContinuousMap.mk fun p =>\n                                      (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                    (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                                x),\n                      map_one_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                              \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n                  (t, x))\n            x =\n          \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n            x \u2227\n        \u2191(ContinuousMap.mk fun x =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk G,\n                      map_zero_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                              \u2191(comp\n                                    (ContinuousMap.mk fun p =>\n                                      (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                    (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                                x),\n                      map_one_left :=\n                        (_ :\n                          \u2200 (x : X \u00d7 Y),\n                            ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                              \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n                  (t, x))\n            x =\n          \u2191(ContinuousMap.id (X \u00d7 Y)) x\n[PROOFSTEP]\nrintro t \u27e8x, y\u27e9 h\n[GOAL]\ncase prop'.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\nt : \u2191I\nx : X\ny : Y\nh : (x, y) \u2208 {(e, e)}\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n        (x, y) \u2227\n    \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nreplace h := Prod.mk.inj_iff.mp (Set.mem_singleton_iff.mp h)\n[GOAL]\ncase prop'.mk\nX : Type u\nY : Type v\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace Y\ninst\u271d\u00b9 : HSpace X\ninst\u271d : HSpace Y\nG : \u2191I \u00d7 X \u00d7 Y \u2192 X \u00d7 Y := fun p => (\u2191hmulE (p.fst, p.snd.fst), \u2191hmulE (p.fst, p.snd.snd))\nhG : Continuous G\nt : \u2191I\nx : X\ny : Y\nh : x = e \u2227 y = e\n\u22a2 \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(comp (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n            (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n        (x, y) \u2227\n    \u2191(ContinuousMap.mk fun x =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk G,\n                  map_zero_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (0, x) =\n                          \u2191(comp\n                                (ContinuousMap.mk fun p => (\u2191hmul (p.fst.fst, p.snd.fst), \u2191hmul (p.fst.snd, p.snd.snd)))\n                                (prodMk (ContinuousMap.id (X \u00d7 Y)) (const (X \u00d7 Y) (e, e))))\n                            x),\n                  map_one_left :=\n                    (_ :\n                      \u2200 (x : X \u00d7 Y),\n                        ContinuousMap.toFun (ContinuousMap.mk G) (1, x) =\n                          \u2191(ContinuousMap.id (X \u00d7 Y)) x) }.toContinuousMap\n              (t, x))\n        (x, y) =\n      \u2191(ContinuousMap.id (X \u00d7 Y)) (x, y)\n[PROOFSTEP]\nexact\n  \u27e8Prod.mk.inj_iff.mpr\n      \u27e8HomotopyRel.eq_fst HSpace.hmulE t (Set.mem_singleton_iff.mpr h.1),\n        HomotopyRel.eq_fst HSpace.hmulE t (Set.mem_singleton_iff.mpr h.2)\u27e9,\n    Prod.mk.inj_iff.mpr \u27e8(HSpace.hmulE.2 t x h.1).2, (HSpace.hmulE.2 t y h.2).2\u27e9\u27e9\n[GOAL]\nM : Type u\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\n\u22a2 comp (ContinuousMap.mk (Function.uncurry Mul.mul)) (prodMk (const M 1) (ContinuousMap.id M)) = ContinuousMap.id M\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nM : Type u\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\na\u271d : M\n\u22a2 \u2191(comp (ContinuousMap.mk (Function.uncurry Mul.mul)) (prodMk (const M 1) (ContinuousMap.id M))) a\u271d =\n    \u2191(ContinuousMap.id M) a\u271d\n[PROOFSTEP]\napply one_mul\n[GOAL]\nM : Type u\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\n\u22a2 comp (ContinuousMap.mk (Function.uncurry Mul.mul)) (prodMk (ContinuousMap.id M) (const M 1)) = ContinuousMap.id M\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nM : Type u\ninst\u271d\u00b2 : MulOneClass M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\na\u271d : M\n\u22a2 \u2191(comp (ContinuousMap.mk (Function.uncurry Mul.mul)) (prodMk (ContinuousMap.id M) (const M 1))) a\u271d =\n    \u2191(ContinuousMap.id M) a\u271d\n[PROOFSTEP]\napply mul_one\n[GOAL]\nG G' : Type u\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : TopologicalSpace G'\ninst\u271d\u00b9 : Group G'\ninst\u271d : TopologicalGroup G'\n\u22a2 hSpace (G \u00d7 G') = HSpace.prod G G'\n[PROOFSTEP]\nsimp only [HSpace.prod]\n[GOAL]\nG G' : Type u\ninst\u271d\u2075 : TopologicalSpace G\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : TopologicalSpace G'\ninst\u271d\u00b9 : Group G'\ninst\u271d : TopologicalGroup G'\n\u22a2 hSpace (G \u00d7 G') =\n    { hmul := ContinuousMap.mk fun p => (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)),\n      e := (HSpace.e, HSpace.e),\n      hmul_e_e :=\n        (_ :\n          \u2191(ContinuousMap.mk fun p => (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n              ((HSpace.e, HSpace.e), HSpace.e, HSpace.e) =\n            (HSpace.e, HSpace.e)),\n      eHmul :=\n        {\n          toHomotopy :=\n            {\n              toContinuousMap :=\n                ContinuousMap.mk fun p => (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)),\n              map_zero_left :=\n                (_ :\n                  \u2200 (x : G \u00d7 G'),\n                    ContinuousMap.toFun\n                        (ContinuousMap.mk fun p => (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                        (0, x) =\n                      \u2191(comp\n                            (ContinuousMap.mk fun p =>\n                              (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                            (prodMk (const (G \u00d7 G') (HSpace.e, HSpace.e)) (ContinuousMap.id (G \u00d7 G'))))\n                        x),\n              map_one_left :=\n                (_ :\n                  \u2200 (x : G \u00d7 G'),\n                    ContinuousMap.toFun\n                        (ContinuousMap.mk fun p => (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                        (1, x) =\n                      \u2191(ContinuousMap.id (G \u00d7 G')) x) },\n          prop' :=\n            (_ :\n              \u2200 (t : \u2191I) (x : G \u00d7 G'),\n                x \u2208 {(HSpace.e, HSpace.e)} \u2192\n                  \u2191(ContinuousMap.mk fun x =>\n                            ContinuousMap.toFun\n                              {\n                                  toContinuousMap :=\n                                    ContinuousMap.mk fun p =>\n                                      (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)),\n                                  map_zero_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                                            (0, x) =\n                                          \u2191(comp\n                                                (ContinuousMap.mk fun p =>\n                                                  (\u2191HSpace.hmul (p.fst.fst, p.snd.fst),\n                                                    \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                                                (prodMk (const (G \u00d7 G') (HSpace.e, HSpace.e))\n                                                  (ContinuousMap.id (G \u00d7 G'))))\n                                            x),\n                                  map_one_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                                            (1, x) =\n                                          \u2191(ContinuousMap.id (G \u00d7 G')) x) }.toContinuousMap\n                              (t, x))\n                        x =\n                      \u2191(comp\n                            (ContinuousMap.mk fun p =>\n                              (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                            (prodMk (const (G \u00d7 G') (HSpace.e, HSpace.e)) (ContinuousMap.id (G \u00d7 G'))))\n                        x \u2227\n                    \u2191(ContinuousMap.mk fun x =>\n                            ContinuousMap.toFun\n                              {\n                                  toContinuousMap :=\n                                    ContinuousMap.mk fun p =>\n                                      (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)),\n                                  map_zero_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                                            (0, x) =\n                                          \u2191(comp\n                                                (ContinuousMap.mk fun p =>\n                                                  (\u2191HSpace.hmul (p.fst.fst, p.snd.fst),\n                                                    \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                                                (prodMk (const (G \u00d7 G') (HSpace.e, HSpace.e))\n                                                  (ContinuousMap.id (G \u00d7 G'))))\n                                            x),\n                                  map_one_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.eHmul (p.fst, p.snd.fst), \u2191HSpace.eHmul (p.fst, p.snd.snd)))\n                                            (1, x) =\n                                          \u2191(ContinuousMap.id (G \u00d7 G')) x) }.toContinuousMap\n                              (t, x))\n                        x =\n                      \u2191(ContinuousMap.id (G \u00d7 G')) x) },\n      hmulE :=\n        {\n          toHomotopy :=\n            {\n              toContinuousMap :=\n                ContinuousMap.mk fun p => (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)),\n              map_zero_left :=\n                (_ :\n                  \u2200 (x : G \u00d7 G'),\n                    ContinuousMap.toFun\n                        (ContinuousMap.mk fun p => (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                        (0, x) =\n                      \u2191(comp\n                            (ContinuousMap.mk fun p =>\n                              (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                            (prodMk (ContinuousMap.id (G \u00d7 G')) (const (G \u00d7 G') (HSpace.e, HSpace.e))))\n                        x),\n              map_one_left :=\n                (_ :\n                  \u2200 (x : G \u00d7 G'),\n                    ContinuousMap.toFun\n                        (ContinuousMap.mk fun p => (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                        (1, x) =\n                      \u2191(ContinuousMap.id (G \u00d7 G')) x) },\n          prop' :=\n            (_ :\n              \u2200 (t : \u2191I) (x : G \u00d7 G'),\n                x \u2208 {(HSpace.e, HSpace.e)} \u2192\n                  \u2191(ContinuousMap.mk fun x =>\n                            ContinuousMap.toFun\n                              {\n                                  toContinuousMap :=\n                                    ContinuousMap.mk fun p =>\n                                      (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)),\n                                  map_zero_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                                            (0, x) =\n                                          \u2191(comp\n                                                (ContinuousMap.mk fun p =>\n                                                  (\u2191HSpace.hmul (p.fst.fst, p.snd.fst),\n                                                    \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                                                (prodMk (ContinuousMap.id (G \u00d7 G'))\n                                                  (const (G \u00d7 G') (HSpace.e, HSpace.e))))\n                                            x),\n                                  map_one_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                                            (1, x) =\n                                          \u2191(ContinuousMap.id (G \u00d7 G')) x) }.toContinuousMap\n                              (t, x))\n                        x =\n                      \u2191(comp\n                            (ContinuousMap.mk fun p =>\n                              (\u2191HSpace.hmul (p.fst.fst, p.snd.fst), \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                            (prodMk (ContinuousMap.id (G \u00d7 G')) (const (G \u00d7 G') (HSpace.e, HSpace.e))))\n                        x \u2227\n                    \u2191(ContinuousMap.mk fun x =>\n                            ContinuousMap.toFun\n                              {\n                                  toContinuousMap :=\n                                    ContinuousMap.mk fun p =>\n                                      (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)),\n                                  map_zero_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                                            (0, x) =\n                                          \u2191(comp\n                                                (ContinuousMap.mk fun p =>\n                                                  (\u2191HSpace.hmul (p.fst.fst, p.snd.fst),\n                                                    \u2191HSpace.hmul (p.fst.snd, p.snd.snd)))\n                                                (prodMk (ContinuousMap.id (G \u00d7 G'))\n                                                  (const (G \u00d7 G') (HSpace.e, HSpace.e))))\n                                            x),\n                                  map_one_left :=\n                                    (_ :\n                                      \u2200 (x : G \u00d7 G'),\n                                        ContinuousMap.toFun\n                                            (ContinuousMap.mk fun p =>\n                                              (\u2191HSpace.hmulE (p.fst, p.snd.fst), \u2191HSpace.hmulE (p.fst, p.snd.snd)))\n                                            (1, x) =\n                                          \u2191(ContinuousMap.id (G \u00d7 G')) x) }.toContinuousMap\n                              (t, x))\n                        x =\n                      \u2191(ContinuousMap.id (G \u00d7 G')) x) } }\n[PROOFSTEP]\nrfl\n[GOAL]\n\u22a2 Continuous fun p => 2 * \u2191p.fst\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u22a2 Continuous fun p => 1 + \u2191p.snd\n[PROOFSTEP]\ncontinuity\n[GOAL]\n\u03b8 : \u2191I\n\u22a2 2 * \u2191(0, \u03b8).fst / (1 + \u2191(0, \u03b8).snd) \u2264 0\n[PROOFSTEP]\nsimp only [coe_zero, mul_zero, zero_div, le_refl]\n[GOAL]\n\u03b8 : \u2191I\n\u22a2 1 * (1 + \u2191(1, \u03b8).snd) \u2264 2 * \u2191(1, \u03b8).fst\n[PROOFSTEP]\ndsimp only\n[GOAL]\n\u03b8 : \u2191I\n\u22a2 1 * (1 + \u2191\u03b8) \u2264 2 * \u21911\n[PROOFSTEP]\nrw [coe_one, one_mul, mul_one, add_comm, \u2190 one_add_one_eq_two]\n[GOAL]\n\u03b8 : \u2191I\n\u22a2 \u2191\u03b8 + 1 \u2264 1 + 1\n[PROOFSTEP]\nsimp only [add_le_add_iff_right]\n[GOAL]\n\u03b8 : \u2191I\n\u22a2 \u2191\u03b8 \u2264 1\n[PROOFSTEP]\nexact le_one _\n[GOAL]\nt : \u2191I\n\u22a2 \u2191(qRight (t, 0)) = if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1\n[PROOFSTEP]\nsimp only [qRight, coe_zero, add_zero, div_one]\n[GOAL]\nt : \u2191I\n\u22a2 \u2191(Set.projIcc 0 1 qRight.proof_1 (2 * \u2191t)) = if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191(Set.projIcc 0 1 qRight.proof_1 (2 * \u2191t)) = 2 * \u2191t\n[PROOFSTEP]\nrw [Set.projIcc_of_mem _ ((mul_pos_mem_iff zero_lt_two).2 _)]\n[GOAL]\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191t \u2208 Set.Icc 0 (1 / 2)\n[PROOFSTEP]\nrefine' \u27e8t.2.1, _\u27e9\n[GOAL]\nt : \u2191I\nh\u271d : \u2191t \u2264 1 / 2\n\u22a2 \u2191t \u2264 1 / 2\n[PROOFSTEP]\ntauto\n[GOAL]\ncase neg\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191(Set.projIcc 0 1 qRight.proof_1 (2 * \u2191t)) = 1\n[PROOFSTEP]\nrw [(Set.projIcc_eq_right _).2]\n[GOAL]\ncase neg\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 1 \u2264 2 * \u2191t\n[PROOFSTEP]\nlinarith\n[GOAL]\nt : \u2191I\nh\u271d : \u00ac\u2191t \u2264 1 / 2\n\u22a2 0 < 1\n[PROOFSTEP]\nexact zero_lt_one\n[GOAL]\nt : \u2191I\n\u22a2 qRight (t, 1) = Set.projIcc 0 1 (_ : 0 \u2264 1) \u2191t\n[PROOFSTEP]\nrw [qRight]\n[GOAL]\nt : \u2191I\n\u22a2 Set.projIcc 0 1 qRight.proof_1 (2 * \u2191(t, 1).fst / (1 + \u2191(t, 1).snd)) = Set.projIcc 0 1 (_ : 0 \u2264 1) \u2191t\n[PROOFSTEP]\ncongr\n[GOAL]\ncase e_x\nt : \u2191I\n\u22a2 2 * \u2191(t, 1).fst / (1 + \u2191(t, 1).snd) = \u2191t\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase e_x\nt : \u2191I\n\u22a2 2 * \u2191t / 2 = \u2191t\n[PROOFSTEP]\napply mul_div_cancel_left\n[GOAL]\ncase e_x.ha\nt : \u2191I\n\u22a2 2 \u2260 0\n[PROOFSTEP]\nexact two_ne_zero\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b8 : \u2191I\n\u03b3 : Path x y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk fun t => \u2191\u03b3 (qRight (t, \u03b8))) 0 = x\n[PROOFSTEP]\ndsimp only\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b8 : \u2191I\n\u03b3 : Path x y\n\u22a2 \u2191\u03b3 (qRight (0, \u03b8)) = x\n[PROOFSTEP]\nrw [qRight_zero_left, \u03b3.source]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b8 : \u2191I\n\u03b3 : Path x y\n\u22a2 ContinuousMap.toFun (ContinuousMap.mk fun t => \u2191\u03b3 (qRight (t, \u03b8))) 1 = y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b8 : \u2191I\n\u03b3 : Path x y\n\u22a2 \u2191\u03b3 (qRight (1, \u03b8)) = y\n[PROOFSTEP]\nrw [qRight_one_left, \u03b3.target]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\n\u22a2 delayReflRight 0 \u03b3 = trans \u03b3 (refl y)\n[PROOFSTEP]\next t\n[GOAL]\ncase a.h\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\n\u22a2 \u2191(delayReflRight 0 \u03b3) t = \u2191(trans \u03b3 (refl y)) t\n[PROOFSTEP]\nsimp only [delayReflRight, trans_apply, refl_extend, Path.coe_mk_mk, Function.comp_apply, refl_apply]\n[GOAL]\ncase a.h\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = if h : \u2191t \u2264 1 / 2 then \u2191\u03b3 { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) } else y\n[PROOFSTEP]\nsplit_ifs with h\n[GOAL]\ncase pos\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\ncase neg X : Type u inst\u271d : TopologicalSpace X x y : X \u03b3 : Path x y t : \u2191I h : \u00ac\u2191t \u2264 1 / 2 \u22a2 \u2191\u03b3 (qRight (t, 0)) = y\n[PROOFSTEP]\nswap\n[GOAL]\ncase neg\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = y\ncase pos\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\nconv_rhs => rw [\u2190 \u03b3.target]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n| y\n[PROOFSTEP]\nrw [\u2190 \u03b3.target]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n| y\n[PROOFSTEP]\nrw [\u2190 \u03b3.target]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n| y\n[PROOFSTEP]\nrw [\u2190 \u03b3.target]\n[GOAL]\ncase neg\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 1\ncase pos\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\nall_goals apply congr_arg \u03b3; ext1; rw [qRight_zero_right]\n[GOAL]\ncase neg\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 1\n[PROOFSTEP]\napply congr_arg \u03b3\n[GOAL]\ncase neg\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 qRight (t, 0) = 1\n[PROOFSTEP]\next1\n[GOAL]\ncase neg.a\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 \u2191(qRight (t, 0)) = \u21911\n[PROOFSTEP]\nrw [qRight_zero_right]\n[GOAL]\ncase pos\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 \u2191\u03b3 (qRight (t, 0)) = \u2191\u03b3 { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\napply congr_arg \u03b3\n[GOAL]\ncase pos\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 qRight (t, 0) = { val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\next1\n[GOAL]\ncase pos.a\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 \u2191(qRight (t, 0)) = \u2191{ val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\nrw [qRight_zero_right]\n[GOAL]\ncase neg.a\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u00ac\u2191t \u2264 1 / 2\n\u22a2 (if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1) = \u21911\ncase pos.a\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\nh : \u2191t \u2264 1 / 2\n\u22a2 (if \u2191t \u2264 1 / 2 then 2 * \u2191t else 1) = \u2191{ val := 2 * \u2191t, property := (_ : 2 * \u2191t \u2208 I) }\n[PROOFSTEP]\nexacts [if_neg h, if_pos h]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\n\u22a2 delayReflRight 1 \u03b3 = \u03b3\n[PROOFSTEP]\next t\n[GOAL]\ncase a.h\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\nt : \u2191I\n\u22a2 \u2191(delayReflRight 1 \u03b3) t = \u2191\u03b3 t\n[PROOFSTEP]\nexact congr_arg \u03b3 (qRight_one_right t)\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\n\u22a2 delayReflLeft 0 \u03b3 = trans (refl x) \u03b3\n[PROOFSTEP]\nsimp only [delayReflLeft, delayReflRight_zero, trans_symm, refl_symm, Path.symm_symm]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx y : X\n\u03b3 : Path x y\n\u22a2 delayReflLeft 1 \u03b3 = \u03b3\n[PROOFSTEP]\nsimp only [delayReflLeft, delayReflRight_one, Path.symm_symm]\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx\u271d y x : X\n\u22a2 \u2200 (t : \u2191I) (x_1 : Path x x),\n    x_1 \u2208 {refl x} \u2192\n      \u2191(ContinuousMap.mk fun x_2 =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk fun p => delayReflLeft p.fst p.snd,\n                      map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 0 \u03b3 = trans (refl x) \u03b3),\n                      map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 1 \u03b3 = \u03b3) }.toContinuousMap\n                  (t, x_2))\n            x_1 =\n          \u2191(comp (ContinuousMap.mk fun \u03c1 => trans \u03c1.fst \u03c1.snd)\n                (prodMk (const (Path x x) (refl x)) (ContinuousMap.id (Path x x))))\n            x_1 \u2227\n        \u2191(ContinuousMap.mk fun x_2 =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk fun p => delayReflLeft p.fst p.snd,\n                      map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 0 \u03b3 = trans (refl x) \u03b3),\n                      map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 1 \u03b3 = \u03b3) }.toContinuousMap\n                  (t, x_2))\n            x_1 =\n          \u2191(ContinuousMap.id (Path x x)) x_1\n[PROOFSTEP]\nrintro t _ (rfl : _ = _)\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx\u271d y x : X\nt : \u2191I\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk fun p => delayReflLeft p.fst p.snd,\n                  map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 0 \u03b3 = trans (refl x) \u03b3),\n                  map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 1 \u03b3 = \u03b3) }.toContinuousMap\n              (t, x_1))\n        (refl x) =\n      \u2191(comp (ContinuousMap.mk fun \u03c1 => trans \u03c1.fst \u03c1.snd)\n            (prodMk (const (Path x x) (refl x)) (ContinuousMap.id (Path x x))))\n        (refl x) \u2227\n    \u2191(ContinuousMap.mk fun x_1 =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk fun p => delayReflLeft p.fst p.snd,\n                  map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 0 \u03b3 = trans (refl x) \u03b3),\n                  map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflLeft 1 \u03b3 = \u03b3) }.toContinuousMap\n              (t, x_1))\n        (refl x) =\n      \u2191(ContinuousMap.id (Path x x)) (refl x)\n[PROOFSTEP]\nexact \u27e8refl_trans_refl.symm, rfl\u27e9\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx\u271d y x : X\n\u22a2 \u2200 (t : \u2191I) (x_1 : Path x x),\n    x_1 \u2208 {refl x} \u2192\n      \u2191(ContinuousMap.mk fun x_2 =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk fun p => delayReflRight p.fst p.snd,\n                      map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 0 \u03b3 = trans \u03b3 (refl x)),\n                      map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 1 \u03b3 = \u03b3) }.toContinuousMap\n                  (t, x_2))\n            x_1 =\n          \u2191(comp (ContinuousMap.mk fun \u03c1 => trans \u03c1.fst \u03c1.snd)\n                (prodMk (ContinuousMap.id (Path x x)) (const (Path x x) (refl x))))\n            x_1 \u2227\n        \u2191(ContinuousMap.mk fun x_2 =>\n                ContinuousMap.toFun\n                  { toContinuousMap := ContinuousMap.mk fun p => delayReflRight p.fst p.snd,\n                      map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 0 \u03b3 = trans \u03b3 (refl x)),\n                      map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 1 \u03b3 = \u03b3) }.toContinuousMap\n                  (t, x_2))\n            x_1 =\n          \u2191(ContinuousMap.id (Path x x)) x_1\n[PROOFSTEP]\nrintro t _ (rfl : _ = _)\n[GOAL]\nX : Type u\ninst\u271d : TopologicalSpace X\nx\u271d y x : X\nt : \u2191I\n\u22a2 \u2191(ContinuousMap.mk fun x_1 =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk fun p => delayReflRight p.fst p.snd,\n                  map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 0 \u03b3 = trans \u03b3 (refl x)),\n                  map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 1 \u03b3 = \u03b3) }.toContinuousMap\n              (t, x_1))\n        (refl x) =\n      \u2191(comp (ContinuousMap.mk fun \u03c1 => trans \u03c1.fst \u03c1.snd)\n            (prodMk (ContinuousMap.id (Path x x)) (const (Path x x) (refl x))))\n        (refl x) \u2227\n    \u2191(ContinuousMap.mk fun x_1 =>\n            ContinuousMap.toFun\n              { toContinuousMap := ContinuousMap.mk fun p => delayReflRight p.fst p.snd,\n                  map_zero_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 0 \u03b3 = trans \u03b3 (refl x)),\n                  map_one_left := (_ : \u2200 (\u03b3 : Path x x), delayReflRight 1 \u03b3 = \u03b3) }.toContinuousMap\n              (t, x_1))\n        (refl x) =\n      \u2191(ContinuousMap.id (Path x x)) (refl x)\n[PROOFSTEP]\nexact \u27e8refl_trans_refl.symm, rfl\u27e9\n", "meta": {"mathlib_filename": "Mathlib.Topology.Homotopy.HSpaces", "llama_tokens": 18857, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7931059414036511, "lm_q2_score": 0.63341027751814, "lm_q1q2_score": 0.5023614544457724}}
{"text": "[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\n\u03b1 : Type u_3\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : Group \u03b1\nh\u2081 : UniformContinuous fun p => p.fst * p.snd\nh\u2082 : UniformContinuous fun p => p\u207b\u00b9\n\u22a2 UniformContinuous fun p => p.fst / p.snd\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv] using h\u2081.comp (uniformContinuous_fst.prod_mk (h\u2082.comp uniformContinuous_snd))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => (f x)\u207b\u00b9\n[PROOFSTEP]\nhave : UniformContinuous fun x => 1 / f x := uniformContinuous_const.div hf\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nthis : UniformContinuous fun x => 1 / f x\n\u22a2 UniformContinuous fun x => (f x)\u207b\u00b9\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nhg : UniformContinuous g\n\u22a2 UniformContinuous fun x => f x * g x\n[PROOFSTEP]\nhave : UniformContinuous fun x => f x / (g x)\u207b\u00b9 := hf.div hg.inv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf g : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nhg : UniformContinuous g\nthis : UniformContinuous fun x => f x / (g x)\u207b\u00b9\n\u22a2 UniformContinuous fun x => f x * g x\n[PROOFSTEP]\nsimp_all\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => f x ^ 0\n[PROOFSTEP]\nsimp_rw [pow_zero]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\n\u22a2 UniformContinuous fun x => 1\n[PROOFSTEP]\nexact uniformContinuous_const\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x ^ (n + 1)\n[PROOFSTEP]\nsimp_rw [pow_succ]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x * f x ^ n\n[PROOFSTEP]\nexact hf.mul (hf.pow_const n)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x ^ \u2191n\n[PROOFSTEP]\nsimp_rw [zpow_ofNat]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x ^ n\n[PROOFSTEP]\nexact hf.pow_const _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => f x ^ Int.negSucc n\n[PROOFSTEP]\nsimp_rw [zpow_negSucc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : UniformSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : UniformContinuous f\nn : \u2115\n\u22a2 UniformContinuous fun x => (f x ^ (n + 1))\u207b\u00b9\n[PROOFSTEP]\nexact (hf.pow_const _).inv\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na : \u03b1\n\u22a2 \ud835\udce4 \u03b1 = map (fun x => (x.fst * a, x.snd * a)) (map (fun x => (x.fst * a\u207b\u00b9, x.snd * a\u207b\u00b9)) (\ud835\udce4 \u03b1))\n[PROOFSTEP]\nsimp [Filter.map_map, (\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na : \u03b1\n\u22a2 comap (fun x => (x.fst * a, x.snd * a)) (\ud835\udce4 \u03b1) = \ud835\udce4 \u03b1\n[PROOFSTEP]\nnth_rewrite 1 [\u2190 uniformity_translate_mul a, comap_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na : \u03b1\n\u22a2 \ud835\udce4 \u03b1 = \ud835\udce4 \u03b1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na : \u03b1\n\u22a2 Function.Injective fun x => (x.fst * a, x.snd * a)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na : \u03b1\n\u22a2 Function.Injective fun x => (x.fst * a, x.snd * a)\n[PROOFSTEP]\nrintro \u27e8p\u2081, p\u2082\u27e9 \u27e8q\u2081, q\u2082\u27e9\n[GOAL]\ncase mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\na p\u2081 p\u2082 q\u2081 q\u2082 : \u03b1\n\u22a2 (fun x => (x.fst * a, x.snd * a)) (p\u2081, p\u2082) = (fun x => (x.fst * a, x.snd * a)) (q\u2081, q\u2082) \u2192 (p\u2081, p\u2082) = (q\u2081, q\u2082)\n[PROOFSTEP]\nsimp only [Prod.mk.injEq, mul_left_inj, imp_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\n\u03b9 : Sort u_3\nus' : \u03b9 \u2192 UniformSpace \u03b2\nh' : \u2200 (i : \u03b9), UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nrw [\u2190 sInf_range]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\n\u03b9 : Sort u_3\nus' : \u03b9 \u2192 UniformSpace \u03b2\nh' : \u2200 (i : \u03b9), UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nexact uniformGroup_sInf (Set.forall_range_iff.mpr h')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\nu\u2081 u\u2082 : UniformSpace \u03b2\nh\u2081 : UniformGroup \u03b2\nh\u2082 : UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nrw [inf_eq_iInf]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\nu\u2081 u\u2082 : UniformSpace \u03b2\nh\u2081 : UniformGroup \u03b2\nh\u2082 : UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nrefine' uniformGroup_iInf fun b => _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\nu\u2081 u\u2082 : UniformSpace \u03b2\nh\u2081 : UniformGroup \u03b2\nh\u2082 : UniformGroup \u03b2\nb : Bool\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\ncases b\n[GOAL]\ncase false\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\nu\u2081 u\u2082 : UniformSpace \u03b2\nh\u2081 : UniformGroup \u03b2\nh\u2082 : UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nassumption\n[GOAL]\ncase true\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : Group \u03b2\nu\u2081 u\u2082 : UniformSpace \u03b2\nh\u2081 : UniformGroup \u03b2\nh\u2082 : UniformGroup \u03b2\n\u22a2 UniformGroup \u03b2\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\ninst\u271d\u00b3 : Group \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Group \u03b3\nu : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformGroup \u03b3\nF : Type u_4\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\nf : F\n\u22a2 UniformContinuous fun p => p.fst / p.snd\n[PROOFSTEP]\nletI : UniformSpace \u03b2 := u.comap f\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\ninst\u271d\u00b3 : Group \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Group \u03b3\nu : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformGroup \u03b3\nF : Type u_4\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\nf : F\nthis : UniformSpace \u03b2 := UniformSpace.comap (\u2191f) u\n\u22a2 UniformContinuous fun p => p.fst / p.snd\n[PROOFSTEP]\nrefine' uniformContinuous_comap' _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\ninst\u271d\u00b3 : Group \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Group \u03b3\nu : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformGroup \u03b3\nF : Type u_4\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\nf : F\nthis : UniformSpace \u03b2 := UniformSpace.comap (\u2191f) u\n\u22a2 UniformContinuous (\u2191f \u2218 fun p => p.fst / p.snd)\n[PROOFSTEP]\nsimp_rw [Function.comp, map_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\ninst\u271d\u00b3 : Group \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Group \u03b3\nu : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformGroup \u03b3\nF : Type u_4\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\nf : F\nthis : UniformSpace \u03b2 := UniformSpace.comap (\u2191f) u\n\u22a2 UniformContinuous fun x => \u2191f x.fst / \u2191f x.snd\n[PROOFSTEP]\nchange UniformContinuous ((fun p : \u03b3 \u00d7 \u03b3 => p.1 / p.2) \u2218 Prod.map f f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\ninst\u271d\u00b3 : Group \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : Group \u03b3\nu : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformGroup \u03b3\nF : Type u_4\ninst\u271d : MonoidHomClass F \u03b2 \u03b3\nf : F\nthis : UniformSpace \u03b2 := UniformSpace.comap (\u2191f) u\n\u22a2 UniformContinuous ((fun p => p.fst / p.snd) \u2218 Prod.map \u2191f \u2191f)\n[PROOFSTEP]\nexact uniformContinuous_div.comp (uniformContinuous_comap.prod_map uniformContinuous_comap)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (fun x => x.snd / x.fst) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [nhds_eq_comap_uniformity, Filter.comap_comap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrefine' le_antisymm (Filter.map_le_iff_le_comap.1 _) _\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 map (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 s \u2208 map (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrcases mem_uniformity_of_uniformContinuous_invariant uniformContinuous_div hs with \u27e8t, ht, hts\u27e9\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a / c, b / c) \u2208 s\n\u22a2 s \u2208 map (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrefine' mem_map.2 (mem_of_superset ht _)\n[GOAL]\ncase refine'_1.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a / c, b / c) \u2208 s\n\u22a2 t \u2286 (Prod.mk 1 \u2218 fun x => x.snd / x.fst) \u207b\u00b9' s\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase refine'_1.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a / c, b / c) \u2208 s\na b : \u03b1\n\u22a2 (a, b) \u2208 t \u2192 (a, b) \u2208 (Prod.mk 1 \u2218 fun x => x.snd / x.fst) \u207b\u00b9' s\n[PROOFSTEP]\nsimpa [subset_def] using hts a b a\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 comap (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1) \u2264 \ud835\udce4 \u03b1\n[PROOFSTEP]\nintro s hs\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\n\u22a2 s \u2208 comap (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrcases mem_uniformity_of_uniformContinuous_invariant uniformContinuous_mul hs with \u27e8t, ht, hts\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a * c, b * c) \u2208 s\n\u22a2 s \u2208 comap (Prod.mk 1 \u2218 fun x => x.snd / x.fst) (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrefine' \u27e8_, ht, _\u27e9\n[GOAL]\ncase refine'_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a * c, b * c) \u2208 s\n\u22a2 (Prod.mk 1 \u2218 fun x => x.snd / x.fst) \u207b\u00b9' t \u2286 s\n[PROOFSTEP]\nrintro \u27e8a, b\u27e9\n[GOAL]\ncase refine'_2.intro.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set (\u03b1 \u00d7 \u03b1)\nhs : s \u2208 \ud835\udce4 \u03b1\nt : Set (\u03b1 \u00d7 \u03b1)\nht : t \u2208 \ud835\udce4 \u03b1\nhts : \u2200 (a b c : \u03b1), (a, b) \u2208 t \u2192 (a * c, b * c) \u2208 s\na b : \u03b1\n\u22a2 (a, b) \u2208 (Prod.mk 1 \u2218 fun x => x.snd / x.fst) \u207b\u00b9' t \u2192 (a, b) \u2208 s\n[PROOFSTEP]\nsimpa [subset_def] using hts 1 (b / a) a\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (fun x => x.fst / x.snd) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 comap_swap_uniformity, uniformity_eq_comap_nhds_one, comap_comap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 comap ((fun x => x.snd / x.fst) \u2218 Prod.swap) (\ud835\udcdd 1) = comap (fun x => x.fst / x.snd) (\ud835\udcdd 1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\nG : Type u_3\ninst\u271d : Group G\nu v : UniformSpace G\nhu : UniformGroup G\nhv : UniformGroup G\nh : \ud835\udcdd 1 = \ud835\udcdd 1\n\u22a2 \ud835\udce4 G = \ud835\udce4 G\n[PROOFSTEP]\nrw [@uniformity_eq_comap_nhds_one _ u _ hu, @uniformity_eq_comap_nhds_one _ v _ hv, h]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udcdd 1)\n\u22a2 IsCountablyGenerated (\ud835\udce4 \u03b1)\n[PROOFSTEP]\nrw [uniformity_eq_comap_nhds_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : UniformGroup \u03b1\ninst\u271d : IsCountablyGenerated (\ud835\udcdd 1)\n\u22a2 IsCountablyGenerated (comap (fun x => x.snd / x.fst) (\ud835\udcdd 1))\n[PROOFSTEP]\nexact Filter.comap.isCountablyGenerated _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (fun x => x.fst\u207b\u00b9 * x.snd) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 comap_uniformity_mulOpposite, uniformity_eq_comap_nhds_one, \u2190 op_one, \u2190 comap_unop_nhds, comap_comap, comap_comap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 comap (unop \u2218 (fun x => x.snd / x.fst) \u2218 fun p => (op p.fst, op p.snd)) (\ud835\udcdd 1) = comap (fun x => x.fst\u207b\u00b9 * x.snd) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp [(\u00b7 \u2218 \u00b7)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 \ud835\udce4 \u03b1 = comap (fun x => x.snd\u207b\u00b9 * x.fst) (\ud835\udcdd 1)\n[PROOFSTEP]\nrw [\u2190 comap_swap_uniformity, uniformity_eq_comap_inv_mul_nhds_one, comap_comap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u22a2 comap ((fun x => x.fst\u207b\u00b9 * x.snd) \u2218 Prod.swap) (\ud835\udcdd 1) = comap (fun x => x.snd\u207b\u00b9 * x.fst) (\ud835\udcdd 1)\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun i => {x | x.snd / x.fst \u2208 U i}\n[PROOFSTEP]\nrw [uniformity_eq_comap_nhds_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (Filter.comap (fun x => x.snd / x.fst) (\ud835\udcdd 1)) p fun i => {x | x.snd / x.fst \u2208 U i}\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun i => {x | x.fst\u207b\u00b9 * x.snd \u2208 U i}\n[PROOFSTEP]\nrw [uniformity_eq_comap_inv_mul_nhds_one]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (Filter.comap (fun x => x.fst\u207b\u00b9 * x.snd) (\ud835\udcdd 1)) p fun i => {x | x.fst\u207b\u00b9 * x.snd \u2208 U i}\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun i => {x | x.fst / x.snd \u2208 U i}\n[PROOFSTEP]\nrw [uniformity_eq_comap_nhds_one_swapped]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (Filter.comap (fun x => x.fst / x.snd) (\ud835\udcdd 1)) p fun i => {x | x.fst / x.snd \u2208 U i}\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (\ud835\udce4 \u03b1) p fun i => {x | x.snd\u207b\u00b9 * x.fst \u2208 U i}\n[PROOFSTEP]\nrw [uniformity_eq_comap_inv_mul_nhds_one_swapped]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Sort u_3\np : \u03b9 \u2192 Prop\nU : \u03b9 \u2192 Set \u03b1\nh : HasBasis (\ud835\udcdd 1) p U\n\u22a2 HasBasis (Filter.comap (fun x => x.snd\u207b\u00b9 * x.fst) (\ud835\udcdd 1)) p fun i => {x | x.snd\u207b\u00b9 * x.fst \u2208 U i}\n[PROOFSTEP]\nexact h.comap _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\nx y : \u03b1\nthis : Embedding fun a => a * (y / x)\n\u22a2 (x, y) \u2208 \u22c2\u2080 (\ud835\udce4 \u03b1).sets \u2194 x / y \u2208 closure {1}\n[PROOFSTEP]\nrw [this.closure_eq_preimage_closure_image, uniformity_eq_comap_nhds_one \u03b1, sInter_comap_sets]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\nx y : \u03b1\nthis : Embedding fun a => a * (y / x)\n\u22a2 (x, y) \u2208 \u22c2 (U : Set \u03b1) (_ : U \u2208 \ud835\udcdd 1), (fun x => x.snd / x.fst) \u207b\u00b9' U \u2194\n    x / y \u2208 (fun a => a * (y / x)) \u207b\u00b9' closure ((fun a => a * (y / x)) '' {1})\n[PROOFSTEP]\nsimp [mem_closure_iff_nhds, inter_singleton_nonempty, sub_eq_add_neg, add_assoc]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\n\u22a2 UniformContinuous \u2191f\n[PROOFSTEP]\nhave : ((fun x : \u03b2 \u00d7 \u03b2 => x.2 / x.1) \u2218 fun x : \u03b1 \u00d7 \u03b1 => (f x.1, f x.2)) = fun x : \u03b1 \u00d7 \u03b1 => f (x.2 / x.1) := by ext;\n  simp only [Function.comp_apply, map_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun x => (\u2191f x.fst, \u2191f x.snd)) = fun x => \u2191f (x.snd / x.fst)\n[PROOFSTEP]\next\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\nx\u271d : \u03b1 \u00d7 \u03b1\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun x => (\u2191f x.fst, \u2191f x.snd)) x\u271d = \u2191f (x\u271d.snd / x\u271d.fst)\n[PROOFSTEP]\nsimp only [Function.comp_apply, map_div]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\nthis : ((fun x => x.snd / x.fst) \u2218 fun x => (\u2191f x.fst, \u2191f x.snd)) = fun x => \u2191f (x.snd / x.fst)\n\u22a2 UniformContinuous \u2191f\n[PROOFSTEP]\nrw [UniformContinuous, uniformity_eq_comap_nhds_one \u03b1, uniformity_eq_comap_nhds_one \u03b2, tendsto_comap_iff, this]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\nthis : ((fun x => x.snd / x.fst) \u2218 fun x => (\u2191f x.fst, \u2191f x.snd)) = fun x => \u2191f (x.snd / x.fst)\n\u22a2 Tendsto (fun x => \u2191f (x.snd / x.fst)) (comap (fun x => x.snd / x.fst) (\ud835\udcdd 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nexact Tendsto.comp h tendsto_comap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nhf : ContinuousAt (\u2191f) 1\n\u22a2 Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimpa using hf.tendsto\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : UniformSpace \u03b1\ninst\u271d\u2076 : Group \u03b1\ninst\u271d\u2075 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : DiscreteTopology \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\n\u22a2 UniformContinuous \u2191f \u2194 IsOpen \u2191(MonoidHom.ker \u2191f)\n[PROOFSTEP]\nrefine' \u27e8fun hf => _, fun hf => _\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : UniformSpace \u03b1\ninst\u271d\u2076 : Group \u03b1\ninst\u271d\u2075 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : DiscreteTopology \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nhf : UniformContinuous \u2191f\n\u22a2 IsOpen \u2191(MonoidHom.ker \u2191f)\n[PROOFSTEP]\napply (isOpen_discrete ({1} : Set \u03b2)).preimage hf.continuous\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : UniformSpace \u03b1\ninst\u271d\u2076 : Group \u03b1\ninst\u271d\u2075 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : DiscreteTopology \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nhf : IsOpen \u2191(MonoidHom.ker \u2191f)\n\u22a2 UniformContinuous \u2191f\n[PROOFSTEP]\napply uniformContinuous_of_continuousAt_one\n[GOAL]\ncase refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : UniformSpace \u03b1\ninst\u271d\u2076 : Group \u03b1\ninst\u271d\u2075 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : DiscreteTopology \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nhf : IsOpen \u2191(MonoidHom.ker \u2191f)\n\u22a2 ContinuousAt (\u2191f) 1\n[PROOFSTEP]\nrw [ContinuousAt, nhds_discrete \u03b2, map_one, tendsto_pure]\n[GOAL]\ncase refine'_2.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2077 : UniformSpace \u03b1\ninst\u271d\u2076 : Group \u03b1\ninst\u271d\u2075 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u2074 : UniformSpace \u03b2\ninst\u271d\u00b3 : DiscreteTopology \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nhf : IsOpen \u2191(MonoidHom.ker \u2191f)\n\u22a2 \u2200\u1da0 (x : \u03b1) in \ud835\udcdd 1, \u2191f x = 1\n[PROOFSTEP]\nexact hf.mem_nhds (map_one f)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2076 : UniformSpace \u03b1\ninst\u271d\u2075 : Group \u03b1\ninst\u271d\u2074 : UniformGroup \u03b1\nhom : Type u_3\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : Group \u03b2\ninst\u271d\u00b9 : UniformGroup \u03b2\ninst\u271d : MonoidHomClass hom \u03b1 \u03b2\nf : hom\nh : Continuous \u2191f\nthis : Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd (\u2191f 1))\n\u22a2 Tendsto (\u2191f) (\ud835\udcdd 1) (\ud835\udcdd 1)\n[PROOFSTEP]\nrwa [map_one] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\ns : Set \u03b1\n\u22a2 (\u2200 (i : Set \u03b1),\n      i \u2208 \ud835\udcdd 1 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), {x | (x, y) \u2208 {x | x.snd\u207b\u00b9 * x.fst \u2208 id i}}) \u2194\n    \u2200 (U : Set \u03b1), U \u2208 \ud835\udcdd 1 \u2192 \u2203 t, Set.Finite t \u2227 s \u2286 \u22c3 (y : \u03b1) (_ : y \u2208 t), y \u2022 U\n[PROOFSTEP]\nsimp [\u2190 preimage_smul_inv, preimage]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Type u_3\nl : Filter \u03b9\nl' : Filter \u03b2\nf f' : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ng g' : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhf : UniformCauchySeqOn f l s\nhf' : UniformCauchySeqOn f' l s\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (m : \u03b9 \u00d7 \u03b9) in l \u00d7\u02e2 l, \u2200 (x : \u03b2), x \u2208 s \u2192 ((f * f') m.fst x, (f * f') m.snd x) \u2208 u\n[PROOFSTEP]\nsimpa using (uniformContinuous_mul.comp_uniformCauchySeqOn (hf.prod' hf')) u hu\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : UniformSpace \u03b1\ninst\u271d\u00b9 : Group \u03b1\ninst\u271d : UniformGroup \u03b1\n\u03b9 : Type u_3\nl : Filter \u03b9\nl' : Filter \u03b2\nf f' : \u03b9 \u2192 \u03b2 \u2192 \u03b1\ng g' : \u03b2 \u2192 \u03b1\ns : Set \u03b2\nhf : UniformCauchySeqOn f l s\nhf' : UniformCauchySeqOn f' l s\nu : Set (\u03b1 \u00d7 \u03b1)\nhu : u \u2208 \ud835\udce4 \u03b1\n\u22a2 \u2200\u1da0 (m : \u03b9 \u00d7 \u03b9) in l \u00d7\u02e2 l, \u2200 (x : \u03b2), x \u2208 s \u2192 ((f / f') m.fst x, (f / f') m.snd x) \u2208 u\n[PROOFSTEP]\nsimpa using (uniformContinuous_div.comp_uniformCauchySeqOn (hf.prod' hf')) u hu\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\n\u22a2 Tendsto (fun p => p.snd / p.fst) (\ud835\udcdf idRel) (pure 1)\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : Tendsto (fun p => (p.snd / p.fst)\u207b\u00b9) (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) (\ud835\udcdd 1\u207b\u00b9)\n\u22a2 Tendsto Prod.swap (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1))\n[PROOFSTEP]\nsimpa [tendsto_comap_iff]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nU : Set G\nH : U \u2208 \ud835\udcdd 1\n\u22a2 U \u2208 map (fun p => p.snd / p.fst) (Filter.lift' (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) fun s => s \u25cb s)\n[PROOFSTEP]\nrcases exists_nhds_one_split H with \u27e8V, V_nhds, V_mul\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nU : Set G\nH : U \u2208 \ud835\udcdd 1\nV : Set G\nV_nhds : V \u2208 \ud835\udcdd 1\nV_mul : \u2200 (v : G), v \u2208 V \u2192 \u2200 (w : G), w \u2208 V \u2192 v * w \u2208 U\n\u22a2 U \u2208 map (fun p => p.snd / p.fst) (Filter.lift' (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) fun s => s \u25cb s)\n[PROOFSTEP]\nrefine mem_map.2 (mem_of_superset (mem_lift' <| preimage_mem_comap V_nhds) ?_)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nU : Set G\nH : U \u2208 \ud835\udcdd 1\nV : Set G\nV_nhds : V \u2208 \ud835\udcdd 1\nV_mul : \u2200 (v : G), v \u2208 V \u2192 \u2200 (w : G), w \u2208 V \u2192 v * w \u2208 U\n\u22a2 (fun p => p.snd / p.fst) \u207b\u00b9' V \u25cb (fun p => p.snd / p.fst) \u207b\u00b9' V \u2286 (fun p => p.snd / p.fst) \u207b\u00b9' U\n[PROOFSTEP]\nrintro \u27e8x, y\u27e9 \u27e8z, hz\u2081, hz\u2082\u27e9\n[GOAL]\ncase intro.intro.mk.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nU : Set G\nH : U \u2208 \ud835\udcdd 1\nV : Set G\nV_nhds : V \u2208 \ud835\udcdd 1\nV_mul : \u2200 (v : G), v \u2208 V \u2192 \u2200 (w : G), w \u2208 V \u2192 v * w \u2208 U\nx y z : G\nhz\u2081 : ((x, y).fst, z) \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' V\nhz\u2082 : (z, (x, y).snd) \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' V\n\u22a2 (x, y) \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' U\n[PROOFSTEP]\nsimpa using V_mul _ hz\u2082 _ hz\u2081\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nS : Set G\n\u22a2 IsOpen S \u2194\n    \u2200 (x : G),\n      x \u2208 S \u2192\n        {p | p.fst = x \u2192 p.snd \u2208 S} \u2208\n          { uniformity := comap (fun p => p.snd / p.fst) (\ud835\udcdd 1),\n              refl := (_ : \ud835\udcdf idRel \u2264 comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)),\n              symm :=\n                (_ : Tendsto Prod.swap (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1))),\n              comp :=\n                (_ :\n                  (Filter.lift' (comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) fun s => s \u25cb s) \u2264\n                    comap (fun p => p.snd / p.fst) (\ud835\udcdd 1)) }.uniformity\n[PROOFSTEP]\nsimp only [isOpen_iff_mem_nhds, \u2190 mem_comap_prod_mk, comap_comap, (\u00b7 \u2218 \u00b7), nhds_translation_div]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : CompactSpace G\n\u22a2 UniformContinuous fun p => p.fst / p.snd\n[PROOFSTEP]\napply CompactSpace.uniformContinuous_of_continuous\n[GOAL]\ncase h\nG : Type u_1\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : TopologicalSpace G\ninst\u271d\u00b9 : TopologicalGroup G\ninst\u271d : CompactSpace G\n\u22a2 Continuous fun p => p.fst / p.snd\n[PROOFSTEP]\nexact continuous_div'\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\n\u22a2 IsClosed \u2191H\n[PROOFSTEP]\nobtain \u27e8V, V_in, VH\u27e9 : \u2203 (V : Set G), V \u2208 \ud835\udcdd (1 : G) \u2227 V \u2229 (H : Set G) = {1}\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\n\u22a2 \u2203 V, V \u2208 \ud835\udcdd 1 \u2227 V \u2229 \u2191H = {1}\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\n\u22a2 IsClosed \u2191H\n[PROOFSTEP]\nexact nhds_inter_eq_singleton_of_mem_discrete H.one_mem\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\n\u22a2 IsClosed \u2191H\n[PROOFSTEP]\nhaveI : SeparatedSpace G := separated_iff_t2.mpr \u2039_\u203a\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis : SeparatedSpace G\n\u22a2 IsClosed \u2191H\n[PROOFSTEP]\nhave : (fun p : G \u00d7 G => p.2 / p.1) \u207b\u00b9' V \u2208 \ud835\udce4 G := preimage_mem_comap V_in\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\n\u22a2 IsClosed \u2191H\n[PROOFSTEP]\napply isClosed_of_spaced_out this\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\n\u22a2 Set.Pairwise \u2191H fun x y => \u00ac(x, y) \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' V\n[PROOFSTEP]\nintro h h_in h' h'_in\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\nh : G\nh_in : h \u2208 \u2191H\nh' : G\nh'_in : h' \u2208 \u2191H\n\u22a2 h \u2260 h' \u2192 (fun x y => \u00ac(x, y) \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' V) h h'\n[PROOFSTEP]\ncontrapose!\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\nh : G\nh_in : h \u2208 \u2191H\nh' : G\nh'_in : h' \u2208 \u2191H\n\u22a2 \u00ac\u00ac(h, h') \u2208 (fun p => p.snd / p.fst) \u207b\u00b9' V \u2192 h = h'\n[PROOFSTEP]\nsimp only [Set.mem_preimage, not_not]\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\nh : G\nh_in : h \u2208 \u2191H\nh' : G\nh'_in : h' \u2208 \u2191H\n\u22a2 h' / h \u2208 V \u2192 h = h'\n[PROOFSTEP]\nrintro (hyp : h' / h \u2208 V)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d : SeparatedSpace G\nthis : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\nh : G\nh_in : h \u2208 \u2191H\nh' : G\nh'_in : h' \u2208 \u2191H\nhyp : h' / h \u2208 V\n\u22a2 h = h'\n[PROOFSTEP]\nhave : h' / h \u2208 ({1} : Set G) := VH \u25b8 Set.mem_inter hyp (H.div_mem h'_in h_in)\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Subgroup G\ninst\u271d : DiscreteTopology { x // x \u2208 H }\nV : Set G\nV_in : V \u2208 \ud835\udcdd 1\nVH : V \u2229 \u2191H = {1}\nthis\u271d\u00b9 : SeparatedSpace G\nthis\u271d : (fun p => p.snd / p.fst) \u207b\u00b9' V \u2208 \ud835\udce4 G\nh : G\nh_in : h \u2208 \u2191H\nh' : G\nh'_in : h' \u2208 \u2191H\nhyp : h' / h \u2208 V\nthis : h' / h \u2208 {1}\n\u22a2 h = h'\n[PROOFSTEP]\nexact (eq_of_div_eq_one this).symm\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Type u_2\ninst\u271d : Group H\nf : H \u2192* G\nhf : Function.Injective \u2191f\nhf' : DiscreteTopology { x // x \u2208 range f }\n\u22a2 Tendsto (\u2191f) cofinite (cocompact G)\n[PROOFSTEP]\nreplace hf : Function.Injective f.rangeRestrict := by simpa\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Type u_2\ninst\u271d : Group H\nf : H \u2192* G\nhf : Function.Injective \u2191f\nhf' : DiscreteTopology { x // x \u2208 range f }\n\u22a2 Function.Injective \u2191(rangeRestrict f)\n[PROOFSTEP]\nsimpa\n[GOAL]\nG : Type u_1\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : TopologicalGroup G\ninst\u271d\u00b9 : T2Space G\nH : Type u_2\ninst\u271d : Group H\nf : H \u2192* G\nhf' : DiscreteTopology { x // x \u2208 range f }\nhf : Function.Injective \u2191(rangeRestrict f)\n\u22a2 Tendsto (\u2191f) cofinite (cocompact G)\n[PROOFSTEP]\nexact f.range.tendsto_coe_cofinite_of_discrete.comp hf.tendsto_cofinite\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\n\u22a2 UniformGroup G\n[PROOFSTEP]\nhave :\n  Tendsto ((fun p : G \u00d7 G => p.1 / p.2) \u2218 fun p : (G \u00d7 G) \u00d7 G \u00d7 G => (p.1.2 / p.1.1, p.2.2 / p.2.1))\n    (comap (fun p : (G \u00d7 G) \u00d7 G \u00d7 G => (p.1.2 / p.1.1, p.2.2 / p.2.1)) ((\ud835\udcdd 1).prod (\ud835\udcdd 1))) (\ud835\udcdd (1 / 1)) :=\n  (tendsto_fst.div' tendsto_snd).comp tendsto_comap\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto ((fun p => p.fst / p.snd) \u2218 fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst))\n    (comap (fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd (1 / 1))\n\u22a2 UniformGroup G\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase uniformContinuous_div\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto ((fun p => p.fst / p.snd) \u2218 fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst))\n    (comap (fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd (1 / 1))\n\u22a2 UniformContinuous fun p => p.fst / p.snd\n[PROOFSTEP]\nrw [UniformContinuous, uniformity_prod_eq_prod, tendsto_map'_iff, uniformity_eq_comap_nhds_one' G, tendsto_comap_iff,\n  prod_comap_comap_eq]\n[GOAL]\ncase uniformContinuous_div\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto ((fun p => p.fst / p.snd) \u2218 fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst))\n    (comap (fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd (1 / 1))\n\u22a2 Tendsto\n    ((fun p => p.snd / p.fst) \u2218\n      (fun x => (x.fst.fst / x.fst.snd, x.snd.fst / x.snd.snd)) \u2218 fun p =>\n        ((p.fst.fst, p.snd.fst), p.fst.snd, p.snd.snd))\n    (comap (fun p => (p.fst.snd / p.fst.fst, p.snd.snd / p.snd.fst)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp only [Function.comp, div_eq_mul_inv, mul_inv_rev, inv_inv, mul_comm, mul_left_comm] at *\n[GOAL]\ncase uniformContinuous_div\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto (fun x => x.snd.fst * (x.fst.snd * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9))\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd (1 * 1\u207b\u00b9))\n\u22a2 Tendsto (fun x => x.fst.snd * (x.snd.fst * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9))\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp only [inv_one, mul_one, \u2190 mul_assoc] at this \n[GOAL]\ncase uniformContinuous_div\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto (fun x => x.snd.fst * x.fst.snd * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9)\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun x => x.fst.snd * (x.snd.fst * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9))\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimp_rw [\u2190 mul_assoc, mul_comm]\n[GOAL]\ncase uniformContinuous_div\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis :\n  Tendsto (fun x => x.snd.fst * x.fst.snd * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9)\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (Filter.prod (\ud835\udcdd 1) (\ud835\udcdd 1))) (\ud835\udcdd 1)\n\u22a2 Tendsto (fun x => x.snd.fst * x.fst.snd * x.fst.fst\u207b\u00b9 * x.snd.snd\u207b\u00b9)\n    (comap (fun p => (p.fst.snd * p.fst.fst\u207b\u00b9, p.snd.snd * p.snd.fst\u207b\u00b9)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) (\ud835\udcdd 1)\n[PROOFSTEP]\nassumption\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\n\u22a2 T2Space G \u2194 IsClosed {1}\n[PROOFSTEP]\nhaveI : UniformGroup G := comm_topologicalGroup_is_uniform\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\n\u22a2 T2Space G \u2194 IsClosed {1}\n[PROOFSTEP]\nrw [\u2190 separated_iff_t2, separatedSpace_iff, \u2190 closure_eq_iff_isClosed]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\n\u22a2 \ud835\udce2 G = idRel \u2194 closure {1} = {1}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\n\u22a2 \ud835\udce2 G = idRel \u2192 closure {1} = {1}\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\n\u22a2 closure {1} = {1} \u2192 \ud835\udce2 G = idRel\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : \ud835\udce2 G = idRel\n\u22a2 closure {1} = {1}\n[PROOFSTEP]\napply Subset.antisymm\n[GOAL]\ncase mp.h\u2081\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : \ud835\udce2 G = idRel\n\u22a2 closure {1} \u2286 {1}\n[PROOFSTEP]\nintro x x_in\n[GOAL]\ncase mp.h\u2081\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : \ud835\udce2 G = idRel\nx : G\nx_in : x \u2208 closure {1}\n\u22a2 x \u2208 {1}\n[PROOFSTEP]\nhave := group_separationRel x 1\n[GOAL]\ncase mp.h\u2081\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis\u271d : UniformGroup G\nh : \ud835\udce2 G = idRel\nx : G\nx_in : x \u2208 closure {1}\nthis : (x, 1) \u2208 \ud835\udce2 G \u2194 x / 1 \u2208 closure {1}\n\u22a2 x \u2208 {1}\n[PROOFSTEP]\nrw [div_one] at this \n[GOAL]\ncase mp.h\u2081\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis\u271d : UniformGroup G\nh : \ud835\udce2 G = idRel\nx : G\nx_in : x \u2208 closure {1}\nthis : (x, 1) \u2208 \ud835\udce2 G \u2194 x \u2208 closure {1}\n\u22a2 x \u2208 {1}\n[PROOFSTEP]\nrw [\u2190 this, h] at x_in \n[GOAL]\ncase mp.h\u2081\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis\u271d : UniformGroup G\nh : \ud835\udce2 G = idRel\nx : G\nx_in : (x, 1) \u2208 idRel\nthis : (x, 1) \u2208 \ud835\udce2 G \u2194 x \u2208 closure {1}\n\u22a2 x \u2208 {1}\n[PROOFSTEP]\nrwa [mem_singleton_iff]\n[GOAL]\ncase mp.h\u2082\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : \ud835\udce2 G = idRel\n\u22a2 {1} \u2286 closure {1}\n[PROOFSTEP]\nexact subset_closure\n[GOAL]\ncase mpr\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : closure {1} = {1}\n\u22a2 \ud835\udce2 G = idRel\n[PROOFSTEP]\next p\n[GOAL]\ncase mpr.h\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : closure {1} = {1}\np : G \u00d7 G\n\u22a2 p \u2208 \ud835\udce2 G \u2194 p \u2208 idRel\n[PROOFSTEP]\ncases' p with x y\n[GOAL]\ncase mpr.h.mk\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : closure {1} = {1}\nx y : G\n\u22a2 (x, y) \u2208 \ud835\udce2 G \u2194 (x, y) \u2208 idRel\n[PROOFSTEP]\nrw [group_separationRel x, h, mem_singleton_iff, div_eq_one]\n[GOAL]\ncase mpr.h.mk\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nthis : UniformGroup G\nh : closure {1} = {1}\nx y : G\n\u22a2 x = y \u2194 (x, y) \u2208 idRel\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\n\u22a2 T2Space G\n[PROOFSTEP]\nrw [TopologicalGroup.t2Space_iff_one_closed, \u2190 isOpen_compl_iff, isOpen_iff_mem_nhds]\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\n\u22a2 \u2200 (a : G), a \u2208 {1}\u1d9c \u2192 {1}\u1d9c \u2208 \ud835\udcdd a\n[PROOFSTEP]\nintro x x_not\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\n\u22a2 {1}\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nhave : x \u2260 1 := mem_compl_singleton_iff.mp x_not\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\n\u22a2 {1}\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrcases H x this with \u27e8U, U_in, xU\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nU_in : U \u2208 \ud835\udcdd 1\nxU : \u00acx \u2208 U\n\u22a2 {1}\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrw [\u2190 nhds_one_symm G] at U_in \n[GOAL]\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nU_in : U \u2208 comap Inv.inv (\ud835\udcdd 1)\nxU : \u00acx \u2208 U\n\u22a2 {1}\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrcases U_in with \u27e8W, W_in, UW\u27e9\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\n\u22a2 {1}\u1d9c \u2208 \ud835\udcdd x\n[PROOFSTEP]\nrw [\u2190 nhds_translation_mul_inv]\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\n\u22a2 {1}\u1d9c \u2208 comap (fun y => y * x\u207b\u00b9) (\ud835\udcdd 1)\n[PROOFSTEP]\nuse W, W_in\n[GOAL]\ncase right\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\n\u22a2 (fun y => y * x\u207b\u00b9) \u207b\u00b9' W \u2286 {1}\u1d9c\n[PROOFSTEP]\nrw [subset_compl_comm]\n[GOAL]\ncase right\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\n\u22a2 {1} \u2286 ((fun y => y * x\u207b\u00b9) \u207b\u00b9' W)\u1d9c\n[PROOFSTEP]\nsuffices x\u207b\u00b9 \u2209 W by simpa\n[GOAL]\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis\u271d : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\nthis : \u00acx\u207b\u00b9 \u2208 W\n\u22a2 {1} \u2286 ((fun y => y * x\u207b\u00b9) \u207b\u00b9' W)\u1d9c\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase right\nG : Type u_1\ninst\u271d\u00b2 : CommGroup G\ninst\u271d\u00b9 : TopologicalSpace G\ninst\u271d : TopologicalGroup G\nH : \u2200 (x : G), x \u2260 1 \u2192 \u2203 U, U \u2208 \ud835\udcdd 1 \u2227 \u00acx \u2208 U\nx : G\nx_not : x \u2208 {1}\u1d9c\nthis : x \u2260 1\nU : Set G\nxU : \u00acx \u2208 U\nW : Set G\nW_in : W \u2208 \ud835\udcdd 1\nUW : Inv.inv \u207b\u00b9' W \u2286 U\n\u22a2 \u00acx\u207b\u00b9 \u2208 W\n[PROOFSTEP]\nexact fun h => xU (UW h)\n[GOAL]\nG\u271d : Type u_1\ninst\u271d\u2074 : CommGroup G\u271d\ninst\u271d\u00b3 : TopologicalSpace G\u271d\ninst\u271d\u00b2 : TopologicalGroup G\u271d\nG : Type u_2\nu : UniformSpace G\ninst\u271d\u00b9 : Group G\ninst\u271d : UniformGroup G\n\u22a2 TopologicalGroup.toUniformSpace G = u\n[PROOFSTEP]\next : 1\n[GOAL]\ncase a\nG\u271d : Type u_1\ninst\u271d\u2074 : CommGroup G\u271d\ninst\u271d\u00b3 : TopologicalSpace G\u271d\ninst\u271d\u00b2 : TopologicalGroup G\u271d\nG : Type u_2\nu : UniformSpace G\ninst\u271d\u00b9 : Group G\ninst\u271d : UniformGroup G\n\u22a2 \ud835\udce4 G = \ud835\udce4 G\n[PROOFSTEP]\nrw [uniformity_eq_comap_nhds_one' G, uniformity_eq_comap_nhds_one G]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave comm : ((fun x : \u03b1 \u00d7 \u03b1 => x.2 / x.1) \u2218 fun t : \u03b2 \u00d7 \u03b2 => (e t.1, e t.2)) = e \u2218 fun t : \u03b2 \u00d7 \u03b2 => t.2 / t.1 :=\n  by\n  ext t\n  change e t.2 / e t.1 = e (t.2 / t.1)\n  rw [\u2190 map_div e t.2 t.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n[PROOFSTEP]\next t\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) t = (\u2191e \u2218 fun t => t.snd / t.fst) t\n[PROOFSTEP]\nchange e t.2 / e t.1 = e (t.2 / t.1)\n[GOAL]\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\nt : \u03b2 \u00d7 \u03b2\n\u22a2 \u2191e t.snd / \u2191e t.fst = \u2191e (t.snd / t.fst)\n[PROOFSTEP]\nrw [\u2190 map_div e t.2 t.1]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)\n[PROOFSTEP]\nhave lim : Tendsto (fun x : \u03b1 \u00d7 \u03b1 => x.2 / x.1) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (e 1)) := by\n  simpa using (continuous_div'.comp (@continuous_swap \u03b1 \u03b1 _ _)).tendsto (x\u2080, x\u2080)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\n\u22a2 Tendsto (fun x => x.snd / x.fst) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (\u2191e 1))\n[PROOFSTEP]\nsimpa using (continuous_div'.comp (@continuous_swap \u03b1 \u03b1 _ _)).tendsto (x\u2080, x\u2080)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nhom : Type u_3\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : Group \u03b1\ninst\u271d\u00b3 : TopologicalGroup \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : Group \u03b2\ninst\u271d : MonoidHomClass hom \u03b2 \u03b1\ne : hom\nde : DenseInducing \u2191e\nx\u2080 : \u03b1\ncomm : ((fun x => x.snd / x.fst) \u2218 fun t => (\u2191e t.fst, \u2191e t.snd)) = \u2191e \u2218 fun t => t.snd / t.fst\nlim : Tendsto (fun x => x.snd / x.fst) (\ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd (\u2191e 1))\n\u22a2 Tendsto (fun t => t.snd / t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 1)\n[PROOFSTEP]\nsimpa using de.tendsto_comap_nhds_nhds lim comm\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nlet Nx := \ud835\udcdd x\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nlet ee := fun u : \u03b2 \u00d7 \u03b2 => (e u.1, e u.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nhave lim1 : Tendsto (fun a : \u03b2 \u00d7 \u03b2 => (a.2 - a.1, y\u2081)) (comap e Nx \u00d7\u02e2 comap e Nx) (\ud835\udcdd (0, y\u2081)) :=\n  by\n  have :=\n    Tendsto.prod_mk (tendsto_sub_comap_self de x\u2080)\n      (tendsto_const_nhds : Tendsto (fun _ : \u03b2 \u00d7 \u03b2 => y\u2081) (comap ee <| \ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd y\u2081))\n  rw [nhds_prod_eq, prod_comap_comap_eq, \u2190 nhds_prod_eq]\n  exact (this : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\n\u22a2 Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\n[PROOFSTEP]\nhave :=\n  Tendsto.prod_mk (tendsto_sub_comap_self de x\u2080)\n    (tendsto_const_nhds : Tendsto (fun _ : \u03b2 \u00d7 \u03b2 => y\u2081) (comap ee <| \ud835\udcdd (x\u2080, x\u2080)) (\ud835\udcdd y\u2081))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nthis : Tendsto (fun x => (x.snd - x.fst, y\u2081)) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd y\u2081)\n\u22a2 Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\n[PROOFSTEP]\nrw [nhds_prod_eq, prod_comap_comap_eq, \u2190 nhds_prod_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nthis : Tendsto (fun x => (x.snd - x.fst, y\u2081)) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd y\u2081)\n\u22a2 Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd y\u2081)\n[PROOFSTEP]\nexact (this : _)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nhave lim2 : Tendsto (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0) := by simpa using h\u03c6.tendsto (0, y\u2081)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\n\u22a2 Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using h\u03c6.tendsto (0, y\u2081)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\nlim2 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0)\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nhave lim := lim2.comp lim1\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\nlim2 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0)\nlim : Tendsto ((fun p => \u2191(\u2191\u03c6 p.fst) p.snd) \u2218 fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd 0)\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nrw [tendsto_prod_self_iff] at lim \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\nlim2 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0)\nlim :\n  \u2200 (W : Set G),\n    W \u2208 \ud835\udcdd 0 \u2192\n      \u2203 U,\n        U \u2208 comap (\u2191e) Nx \u2227\n          \u2200 (x x' : \u03b2), x \u2208 U \u2192 x' \u2208 U \u2192 ((fun p => \u2191(\u2191\u03c6 p.fst) p.snd) \u2218 fun a => (a.snd - a.fst, y\u2081)) (x, x') \u2208 W\n\u22a2 \u2203 U\u2082,\n    U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W'\n[PROOFSTEP]\nsimp_rw [ball_mem_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2081 : \u03b4\nNx : Filter \u03b1 := \ud835\udcdd x\u2080\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nlim1 : Tendsto (fun a => (a.snd - a.fst, y\u2081)) (comap (\u2191e) Nx \u00d7\u02e2 comap (\u2191e) Nx) (\ud835\udcdd (0, y\u2081))\nlim2 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, y\u2081)) (\ud835\udcdd 0)\nlim :\n  \u2200 (W : Set G),\n    W \u2208 \ud835\udcdd 0 \u2192\n      \u2203 U,\n        U \u2208 comap (\u2191e) Nx \u2227\n          \u2200 (x x' : \u03b2), x \u2208 U \u2192 x' \u2208 U \u2192 ((fun p => \u2191(\u2191\u03c6 p.fst) p.snd) \u2218 fun a => (a.snd - a.fst, y\u2081)) (x, x') \u2208 W\n\u22a2 \u2203 U\u2082, U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 \u2200 (a b : \u03b2), a \u2208 U\u2082 \u2192 b \u2208 U\u2082 \u2192 \u2191(\u2191\u03c6 (b - a)) y\u2081 \u2208 W'\n[PROOFSTEP]\nexact lim W' W'_nhd\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nlet ee := fun u : \u03b2 \u00d7 \u03b2 => (e u.1, e u.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nlet ff := fun u : \u03b4 \u00d7 \u03b4 => (f u.1, f u.2)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nhave lim_\u03c6 : Filter.Tendsto (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0) := by simpa using h\u03c6.tendsto (0, 0)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\n\u22a2 Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n[PROOFSTEP]\nsimpa using h\u03c6.tendsto (0, 0)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nhave lim_\u03c6_sub_sub :\n  Tendsto (fun p : (\u03b2 \u00d7 \u03b2) \u00d7 \u03b4 \u00d7 \u03b4 => (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))\n    ((comap ee <| \ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 (comap ff <| \ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0) :=\n  by\n  have lim_sub_sub :\n    Tendsto (fun p : (\u03b2 \u00d7 \u03b2) \u00d7 \u03b4 \u00d7 \u03b4 => (p.1.2 - p.1.1, p.2.2 - p.2.1)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n      (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0) :=\n    by\n    have := Filter.prod_mono (tendsto_sub_comap_self de x\u2080) (tendsto_sub_comap_self df y\u2080)\n    rwa [prod_map_map_eq] at this \n  rw [\u2190 nhds_prod_eq] at lim_sub_sub \n  exact Tendsto.comp lim_\u03c6 lim_sub_sub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\n[PROOFSTEP]\nhave lim_sub_sub :\n  Tendsto (fun p : (\u03b2 \u00d7 \u03b2) \u00d7 \u03b4 \u00d7 \u03b4 => (p.1.2 - p.1.1, p.2.2 - p.2.1)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n    (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0) :=\n  by\n  have := Filter.prod_mono (tendsto_sub_comap_self de x\u2080) (tendsto_sub_comap_self df y\u2080)\n  rwa [prod_map_map_eq] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\n\u22a2 Tendsto (fun p => (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n    (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0)\n[PROOFSTEP]\nhave := Filter.prod_mono (tendsto_sub_comap_self de x\u2080) (tendsto_sub_comap_self df y\u2080)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nthis :\n  map (fun t => t.snd - t.fst) (comap (fun p => (\u2191e p.fst, \u2191e p.snd)) (\ud835\udcdd (x\u2080, x\u2080))) \u00d7\u02e2\n      map (fun t => t.snd - t.fst) (comap (fun p => (\u2191f p.fst, \u2191f p.snd)) (\ud835\udcdd (y\u2080, y\u2080))) \u2264\n    \ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0\n\u22a2 Tendsto (fun p => (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n    (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0)\n[PROOFSTEP]\nrwa [prod_map_map_eq] at this \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_sub_sub :\n  Tendsto (fun p => (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n    (\ud835\udcdd 0 \u00d7\u02e2 \ud835\udcdd 0)\n\u22a2 Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\n[PROOFSTEP]\nrw [\u2190 nhds_prod_eq] at lim_sub_sub \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_sub_sub :\n  Tendsto (fun p => (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst)) (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080)))\n    (\ud835\udcdd (0, 0))\n\u22a2 Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\n[PROOFSTEP]\nexact Tendsto.comp lim_\u03c6 lim_sub_sub\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrcases exists_nhds_zero_quarter W'_nhd with \u27e8W, W_nhd, W4\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nhave :\n  \u2203 U\u2081 \u2208 comap e (\ud835\udcdd x\u2080),\n    \u2203 V\u2081 \u2208 comap f (\ud835\udcdd y\u2080),\n      \u2200 (x) (_ : x \u2208 U\u2081) (x') (_ : x' \u2208 U\u2081),\n        \u2200 (y) (_ : y \u2208 V\u2081) (y') (_ : y' \u2208 V\u2081), (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) (x' - x, y' - y) \u2208 W :=\n  by\n  rcases tendsto_prod_iff.1 lim_\u03c6_sub_sub W W_nhd with \u27e8U, U_in, V, V_in, H\u27e9\n  rw [nhds_prod_eq, \u2190 prod_comap_comap_eq, mem_prod_same_iff] at U_in V_in \n  rcases U_in with \u27e8U\u2081, U\u2081_in, HU\u2081\u27e9\n  rcases V_in with \u27e8V\u2081, V\u2081_in, HV\u2081\u27e9\n  exists U\u2081, U\u2081_in, V\u2081, V\u2081_in\n  intro x x_in x' x'_in y y_in y' y'_in\n  exact H _ _ (HU\u2081 (mk_mem_prod x_in x'_in)) (HV\u2081 (mk_mem_prod y_in y'_in))\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\n\u22a2 \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nrcases tendsto_prod_iff.1 lim_\u03c6_sub_sub W W_nhd with \u27e8U, U_in, V, V_in, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nU_in : U \u2208 comap ee (\ud835\udcdd (x\u2080, x\u2080))\nV : Set (\u03b4 \u00d7 \u03b4)\nV_in : V \u2208 comap ff (\ud835\udcdd (y\u2080, y\u2080))\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\n\u22a2 \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nrw [nhds_prod_eq, \u2190 prod_comap_comap_eq, mem_prod_same_iff] at U_in V_in \n[GOAL]\ncase intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nU_in : \u2203 t, t \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227 t \u00d7\u02e2 t \u2286 U\nV : Set (\u03b4 \u00d7 \u03b4)\nV_in : \u2203 t, t \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227 t \u00d7\u02e2 t \u2286 V\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\n\u22a2 \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nrcases U_in with \u27e8U\u2081, U\u2081_in, HU\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nV : Set (\u03b4 \u00d7 \u03b4)\nV_in : \u2203 t, t \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227 t \u00d7\u02e2 t \u2286 V\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\nU\u2081 : Set \u03b2\nU\u2081_in : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU\u2081 : U\u2081 \u00d7\u02e2 U\u2081 \u2286 U\n\u22a2 \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nrcases V_in with \u27e8V\u2081, V\u2081_in, HV\u2081\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nV : Set (\u03b4 \u00d7 \u03b4)\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\nU\u2081 : Set \u03b2\nU\u2081_in : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU\u2081 : U\u2081 \u00d7\u02e2 U\u2081 \u2286 U\nV\u2081 : Set \u03b4\nV\u2081_in : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV\u2081 : V\u2081 \u00d7\u02e2 V\u2081 \u2286 V\n\u22a2 \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nexists U\u2081, U\u2081_in, V\u2081, V\u2081_in\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nV : Set (\u03b4 \u00d7 \u03b4)\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\nU\u2081 : Set \u03b2\nU\u2081_in : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU\u2081 : U\u2081 \u00d7\u02e2 U\u2081 \u2286 U\nV\u2081 : Set \u03b4\nV\u2081_in : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV\u2081 : V\u2081 \u00d7\u02e2 V\u2081 \u2286 V\n\u22a2 \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nintro x x_in x' x'_in y y_in y' y'_in\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU : Set (\u03b2 \u00d7 \u03b2)\nV : Set (\u03b4 \u00d7 \u03b4)\nH :\n  \u2200 (x : \u03b2 \u00d7 \u03b2) (y : \u03b4 \u00d7 \u03b4),\n    x \u2208 U \u2192 y \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) ((x, y).fst.snd - (x, y).fst.fst, (x, y).snd.snd - (x, y).snd.fst) \u2208 W\nU\u2081 : Set \u03b2\nU\u2081_in : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU\u2081 : U\u2081 \u00d7\u02e2 U\u2081 \u2286 U\nV\u2081 : Set \u03b4\nV\u2081_in : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV\u2081 : V\u2081 \u00d7\u02e2 V\u2081 \u2286 V\nx : \u03b2\nx_in : x \u2208 U\u2081\nx' : \u03b2\nx'_in : x' \u2208 U\u2081\ny : \u03b4\ny_in : y \u2208 V\u2081\ny' : \u03b4\ny'_in : y' \u2208 V\u2081\n\u22a2 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n[PROOFSTEP]\nexact H _ _ (HU\u2081 (mk_mem_prod x_in x'_in)) (HV\u2081 (mk_mem_prod y_in y'_in))\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nthis :\n  \u2203 U\u2081,\n    U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V\u2081,\n        V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U\u2081 \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrcases this with \u27e8U\u2081, U\u2081_nhd, V\u2081, V\u2081_nhd, H\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nobtain \u27e8x\u2081, x\u2081_in\u27e9 : U\u2081.Nonempty := (de.comap_nhds_neBot _).nonempty_of_mem U\u2081_nhd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nobtain \u27e8y\u2081, y\u2081_in\u27e9 : V\u2081.Nonempty := (df.comap_nhds_neBot _).nonempty_of_mem V\u2081_nhd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nhave cont_flip : Continuous fun p : \u03b4 \u00d7 \u03b2 => \u03c6.flip p.1 p.2 :=\n  by\n  show Continuous ((fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) \u2218 Prod.swap)\n  exact h\u03c6.comp continuous_swap\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\n\u22a2 Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\n[PROOFSTEP]\nshow Continuous ((fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) \u2218 Prod.swap)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\n\u22a2 Continuous ((fun p => \u2191(\u2191\u03c6 p.fst) p.snd) \u2218 Prod.swap)\n[PROOFSTEP]\nexact h\u03c6.comp continuous_swap\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrcases extend_Z_bilin_aux de h\u03c6 W_nhd x\u2080 y\u2081 with \u27e8U\u2082, U\u2082_nhd, HU\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrcases extend_Z_bilin_aux df cont_flip W_nhd y\u2080 x\u2081 with \u27e8V\u2082, V\u2082_nhd, HV\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\n\u22a2 \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nexists U\u2081 \u2229 U\u2082, inter_mem U\u2081_nhd U\u2082_nhd, V\u2081 \u2229 V\u2082, inter_mem V\u2081_nhd V\u2082_nhd\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\n\u22a2 \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2229 U\u2082 \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U\u2081 \u2229 U\u2082 \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V\u2081 \u2229 V\u2082 \u2192\n              \u2200 (y' : \u03b4),\n                y' \u2208 V\u2081 \u2229 V\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrintro x \u27e8xU\u2081, xU\u2082\u27e9 x' \u27e8x'U\u2081, x'U\u2082\u27e9 y \u27e8yV\u2081, yV\u2082\u27e9 y' \u27e8y'V\u2081, y'V\u2082\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\n\u22a2 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nhave key_formula : \u03c6 x' y' - \u03c6 x y = \u03c6 (x' - x) y\u2081 + \u03c6 (x' - x) (y' - y\u2081) + \u03c6 x\u2081 (y' - y) + \u03c6 (x - x\u2081) (y' - y) := by\n  simp; abel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\n\u22a2 \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\n\u22a2 \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y =\n    \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y' + (\u2191(\u2191\u03c6 x\u2081) y' - \u2191(\u2191\u03c6 x\u2081) y) + (\u2191(\u2191\u03c6 x) y' - \u2191(\u2191\u03c6 x\u2081) y' - (\u2191(\u2191\u03c6 x) y - \u2191(\u2191\u03c6 x\u2081) y))\n[PROOFSTEP]\nabel\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\n\u22a2 \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y =\n    \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y' + (\u2191(\u2191\u03c6 x\u2081) y' - \u2191(\u2191\u03c6 x\u2081) y) + (\u2191(\u2191\u03c6 x) y' - \u2191(\u2191\u03c6 x\u2081) y' - (\u2191(\u2191\u03c6 x) y - \u2191(\u2191\u03c6 x\u2081) y))\n[PROOFSTEP]\nabel\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\n\u22a2 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n[PROOFSTEP]\nrw [key_formula]\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\n\u22a2 \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y) \u2208 W'\n[PROOFSTEP]\nhave h\u2081 := HU x xU\u2082 x' x'U\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\nh\u2081 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\n\u22a2 \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y) \u2208 W'\n[PROOFSTEP]\nhave h\u2082 := H x xU\u2081 x' x'U\u2081 y\u2081 y\u2081_in y' y'V\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\nh\u2081 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nh\u2082 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y\u2081) \u2208 W\n\u22a2 \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y) \u2208 W'\n[PROOFSTEP]\nhave h\u2083 := HV y yV\u2082 y' y'V\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\nh\u2081 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nh\u2082 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y\u2081) \u2208 W\nh\u2083 : (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (y' - y, x\u2081) \u2208 W\n\u22a2 \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y) \u2208 W'\n[PROOFSTEP]\nhave h\u2084 := H x\u2081 x\u2081_in x xU\u2081 y yV\u2081 y' y'V\u2081\n[GOAL]\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nee : (u : \u03b2 \u00d7 \u03b2) \u2192 (fun x => \u03b1) u.fst \u00d7 (fun x => \u03b1) u.snd := fun u => (\u2191e u.fst, \u2191e u.snd)\nff : (u : \u03b4 \u00d7 \u03b4) \u2192 (fun x => \u03b3) u.fst \u00d7 (fun x => \u03b3) u.snd := fun u => (\u2191f u.fst, \u2191f u.snd)\nlim_\u03c6 : Tendsto (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (\ud835\udcdd (0, 0)) (\ud835\udcdd 0)\nlim_\u03c6_sub_sub :\n  Tendsto (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (p.fst.snd - p.fst.fst, p.snd.snd - p.snd.fst))\n    (comap ee (\ud835\udcdd (x\u2080, x\u2080)) \u00d7\u02e2 comap ff (\ud835\udcdd (y\u2080, y\u2080))) (\ud835\udcdd 0)\nW : Set G\nW_nhd : W \u2208 \ud835\udcdd 0\nW4 : \u2200 {v w s t : G}, v \u2208 W \u2192 w \u2208 W \u2192 s \u2208 W \u2192 t \u2208 W \u2192 v + w + s + t \u2208 W'\nU\u2081 : Set \u03b2\nU\u2081_nhd : U\u2081 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV\u2081 : Set \u03b4\nV\u2081_nhd : V\u2081 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nH :\n  \u2200 (x : \u03b2),\n    x \u2208 U\u2081 \u2192\n      \u2200 (x' : \u03b2), x' \u2208 U\u2081 \u2192 \u2200 (y : \u03b4), y \u2208 V\u2081 \u2192 \u2200 (y' : \u03b4), y' \u2208 V\u2081 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y) \u2208 W\nx\u2081 : \u03b2\nx\u2081_in : x\u2081 \u2208 U\u2081\ny\u2081 : \u03b4\ny\u2081_in : y\u2081 \u2208 V\u2081\ncont_flip : Continuous fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd\nU\u2082 : Set \u03b2\nU\u2082_nhd : U\u2082 \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nHU : \u2200 (x : \u03b2), x \u2208 U\u2082 \u2192 \u2200 (x' : \u03b2), x' \u2208 U\u2082 \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nV\u2082 : Set \u03b4\nV\u2082_nhd : V\u2082 \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nHV : \u2200 (x : \u03b4), x \u2208 V\u2082 \u2192 \u2200 (x' : \u03b4), x' \u2208 V\u2082 \u2192 (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (x' - x, x\u2081) \u2208 W\nx : \u03b2\nxU\u2081 : x \u2208 U\u2081\nxU\u2082 : x \u2208 U\u2082\nx' : \u03b2\nx'U\u2081 : x' \u2208 U\u2081\nx'U\u2082 : x' \u2208 U\u2082\ny : \u03b4\nyV\u2081 : y \u2208 V\u2081\nyV\u2082 : y \u2208 V\u2082\ny' : \u03b4\ny'V\u2081 : y' \u2208 V\u2081\ny'V\u2082 : y' \u2208 V\u2082\nkey_formula :\n  \u2191(\u2191\u03c6 x') y' - \u2191(\u2191\u03c6 x) y = \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y)\nh\u2081 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y\u2081) \u2208 W\nh\u2082 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x' - x, y' - y\u2081) \u2208 W\nh\u2083 : (fun p => \u2191(\u2191(AddMonoidHom.flip \u03c6) p.fst) p.snd) (y' - y, x\u2081) \u2208 W\nh\u2084 : (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x - x\u2081, y' - y) \u2208 W\n\u22a2 \u2191(\u2191\u03c6 (x' - x)) y\u2081 + \u2191(\u2191\u03c6 (x' - x)) (y' - y\u2081) + \u2191(\u2191\u03c6 x\u2081) (y' - y) + \u2191(\u2191\u03c6 (x - x\u2081)) (y' - y) \u2208 W'\n[PROOFSTEP]\nexact W4 h\u2081 h\u2082 h\u2083 h\u2084\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\n\u22a2 Continuous (extend (_ : DenseInducing fun p => (\u2191e p.fst, \u2191f p.snd)) fun p => \u2191(\u2191\u03c6 p.fst) p.snd)\n[PROOFSTEP]\nrefine' continuous_extend_of_cauchy _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\n\u22a2 \u2200 (b : \u03b1 \u00d7 \u03b3), Cauchy (map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd b)))\n[PROOFSTEP]\nrintro \u27e8x\u2080, y\u2080\u27e9\n[GOAL]\ncase mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 Cauchy (map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))))\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 NeBot (map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))))\n[PROOFSTEP]\napply NeBot.map\n[GOAL]\ncase mk.left.hf\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 NeBot (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\napply comap_neBot\n[GOAL]\ncase mk.left.hf.hm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 \u2200 (t : Set (\u03b1 \u00d7 \u03b3)), t \u2208 \ud835\udcdd (x\u2080, y\u2080) \u2192 \u2203 a, (\u2191e a.fst, \u2191f a.snd) \u2208 t\n[PROOFSTEP]\nintro U h\n[GOAL]\ncase mk.left.hf.hm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nU : Set (\u03b1 \u00d7 \u03b3)\nh : U \u2208 \ud835\udcdd (x\u2080, y\u2080)\n\u22a2 \u2203 a, (\u2191e a.fst, \u2191f a.snd) \u2208 U\n[PROOFSTEP]\nrcases mem_closure_iff_nhds.1 ((de.prod df).dense (x\u2080, y\u2080)) U h with \u27e8x, x_in, \u27e8z, z_x\u27e9\u27e9\n[GOAL]\ncase mk.left.hf.hm.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nU : Set (\u03b1 \u00d7 \u03b3)\nh : U \u2208 \ud835\udcdd (x\u2080, y\u2080)\nx : \u03b1 \u00d7 \u03b3\nx_in : x \u2208 U\nz : \u03b2 \u00d7 \u03b4\nz_x : (fun p => (\u2191e p.fst, \u2191f p.snd)) z = x\n\u22a2 \u2203 a, (\u2191e a.fst, \u2191f a.snd) \u2208 U\n[PROOFSTEP]\nexists z\n[GOAL]\ncase mk.left.hf.hm.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nU : Set (\u03b1 \u00d7 \u03b3)\nh : U \u2208 \ud835\udcdd (x\u2080, y\u2080)\nx : \u03b1 \u00d7 \u03b3\nx_in : x \u2208 U\nz : \u03b2 \u00d7 \u03b4\nz_x : (fun p => (\u2191e p.fst, \u2191f p.snd)) z = x\n\u22a2 (\u2191e z.fst, \u2191f z.snd) \u2208 U\n[PROOFSTEP]\naesop\n[GOAL]\ncase mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))) \u00d7\u02e2\n      map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))) \u2264\n    \ud835\udce4 G\n[PROOFSTEP]\nsuffices\n  map (fun p : (\u03b2 \u00d7 \u03b4) \u00d7 \u03b2 \u00d7 \u03b4 => (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) p.2 - (fun p : \u03b2 \u00d7 \u03b4 => \u03c6 p.1 p.2) p.1)\n      (comap (fun p : (\u03b2 \u00d7 \u03b4) \u00d7 \u03b2 \u00d7 \u03b4 => ((e p.1.1, f p.1.2), (e p.2.1, f p.2.2))) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080))) \u2264\n    \ud835\udcdd 0\n  by rwa [uniformity_eq_comap_nhds_zero G, prod_map_map_eq, \u2190 map_le_iff_le_comap, Filter.map_map, prod_comap_comap_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nthis :\n  map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080))) \u2264\n    \ud835\udcdd 0\n\u22a2 map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))) \u00d7\u02e2\n      map (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (comap (fun p => (\u2191e p.fst, \u2191f p.snd)) (\ud835\udcdd (x\u2080, y\u2080))) \u2264\n    \ud835\udce4 G\n[PROOFSTEP]\nrwa [uniformity_eq_comap_nhds_zero G, prod_map_map_eq, \u2190 map_le_iff_le_comap, Filter.map_map, prod_comap_comap_eq]\n[GOAL]\ncase mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\n\u22a2 map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080))) \u2264\n    \ud835\udcdd 0\n[PROOFSTEP]\nintro W' W'_nhd\n[GOAL]\ncase mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nhave key := extend_Z_bilin_key de df h\u03c6 W'_nhd x\u2080 y\u2080\n[GOAL]\ncase mk.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nkey :\n  \u2203 U,\n    U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080) \u2227\n      \u2203 V,\n        V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080) \u2227\n          \u2200 (x : \u03b2),\n            x \u2208 U \u2192\n              \u2200 (x' : \u03b2),\n                x' \u2208 U \u2192\n                  \u2200 (y : \u03b4),\n                    y \u2208 V \u2192\n                      \u2200 (y' : \u03b4),\n                        y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrcases key with \u27e8U, U_nhd, V, V_nhd, h\u27e9\n[GOAL]\ncase mk.right.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nU_nhd : U \u2208 comap (\u2191e) (\ud835\udcdd x\u2080)\nV : Set \u03b4\nV_nhd : V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrw [mem_comap] at U_nhd \n[GOAL]\ncase mk.right.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nU_nhd : \u2203 t, t \u2208 \ud835\udcdd x\u2080 \u2227 \u2191e \u207b\u00b9' t \u2286 U\nV : Set \u03b4\nV_nhd : V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrcases U_nhd with \u27e8U', U'_nhd, U'_sub\u27e9\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nV_nhd : V \u2208 comap (\u2191f) (\ud835\udcdd y\u2080)\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrw [mem_comap] at V_nhd \n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nV_nhd : \u2203 t, t \u2208 \ud835\udcdd y\u2080 \u2227 \u2191f \u207b\u00b9' t \u2286 V\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrcases V_nhd with \u27e8V', V'_nhd, V'_sub\u27e9\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 W' \u2208\n    map (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst)\n      (comap (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) (\ud835\udcdd (x\u2080, y\u2080) \u00d7\u02e2 \ud835\udcdd (x\u2080, y\u2080)))\n[PROOFSTEP]\nrw [mem_map, mem_comap, nhds_prod_eq]\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 \u2203 t,\n    t \u2208 (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) \u00d7\u02e2 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227\n      (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' t \u2286\n        (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst) \u207b\u00b9' W'\n[PROOFSTEP]\nexists (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V'\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V' \u2208 (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) \u00d7\u02e2 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227\n    (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V' \u2286\n      (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst) \u207b\u00b9' W'\n[PROOFSTEP]\nrw [mem_prod_same_iff]\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 (\u2203 t, t \u2208 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227 t \u00d7\u02e2 t \u2286 (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V') \u2227\n    (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V' \u2286\n      (fun p => (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.snd - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) p.fst) \u207b\u00b9' W'\n[PROOFSTEP]\nsimp only [exists_prop]\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 (\u2203 t, t \u2208 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227 t \u00d7\u02e2 t \u2286 (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V') \u2227\n    (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V' \u2286\n      (fun p => \u2191(\u2191\u03c6 p.snd.fst) p.snd.snd - \u2191(\u2191\u03c6 p.fst.fst) p.fst.snd) \u207b\u00b9' W'\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227 t \u00d7\u02e2 t \u2286 (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V'\n[PROOFSTEP]\nhave := prod_mem_prod U'_nhd V'_nhd\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nthis : U' \u00d7\u02e2 V' \u2208 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080\n\u22a2 \u2203 t, t \u2208 \ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080 \u2227 t \u00d7\u02e2 t \u2286 (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V'\n[PROOFSTEP]\ntauto\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\n\u22a2 (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V' \u2286\n    (fun p => \u2191(\u2191\u03c6 p.snd.fst) p.snd.snd - \u2191(\u2191\u03c6 p.fst.fst) p.fst.snd) \u207b\u00b9' W'\n[PROOFSTEP]\nintro p h'\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\np : (\u03b2 \u00d7 \u03b4) \u00d7 \u03b2 \u00d7 \u03b4\nh' : p \u2208 (fun p => ((\u2191e p.fst.fst, \u2191f p.fst.snd), \u2191e p.snd.fst, \u2191f p.snd.snd)) \u207b\u00b9' (U' \u00d7\u02e2 V') \u00d7\u02e2 U' \u00d7\u02e2 V'\n\u22a2 p \u2208 (fun p => \u2191(\u2191\u03c6 p.snd.fst) p.snd.snd - \u2191(\u2191\u03c6 p.fst.fst) p.fst.snd) \u207b\u00b9' W'\n[PROOFSTEP]\nsimp only [Set.mem_preimage, Set.prod_mk_mem_set_prod_eq] at h' \n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\np : (\u03b2 \u00d7 \u03b4) \u00d7 \u03b2 \u00d7 \u03b4\nh' : (\u2191e p.fst.fst \u2208 U' \u2227 \u2191f p.fst.snd \u2208 V') \u2227 \u2191e p.snd.fst \u2208 U' \u2227 \u2191f p.snd.snd \u2208 V'\n\u22a2 p \u2208 (fun p => \u2191(\u2191\u03c6 p.snd.fst) p.snd.snd - \u2191(\u2191\u03c6 p.fst.fst) p.fst.snd) \u207b\u00b9' W'\n[PROOFSTEP]\nrcases p with \u27e8\u27e8x, y\u27e9, \u27e8x', y'\u27e9\u27e9\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right.mk.mk.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nx : \u03b2\ny : \u03b4\nx' : \u03b2\ny' : \u03b4\nh' :\n  (\u2191e ((x, y), x', y').fst.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').fst.snd \u2208 V') \u2227\n    \u2191e ((x, y), x', y').snd.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').snd.snd \u2208 V'\n\u22a2 ((x, y), x', y') \u2208 (fun p => \u2191(\u2191\u03c6 p.snd.fst) p.snd.snd - \u2191(\u2191\u03c6 p.fst.fst) p.fst.snd) \u207b\u00b9' W'\n[PROOFSTEP]\napply h\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right.mk.mk.mk.x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nx : \u03b2\ny : \u03b4\nx' : \u03b2\ny' : \u03b4\nh' :\n  (\u2191e ((x, y), x', y').fst.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').fst.snd \u2208 V') \u2227\n    \u2191e ((x, y), x', y').snd.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').snd.snd \u2208 V'\n\u22a2 x \u2208 U\n[PROOFSTEP]\ntauto\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right.mk.mk.mk.x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nx : \u03b2\ny : \u03b4\nx' : \u03b2\ny' : \u03b4\nh' :\n  (\u2191e ((x, y), x', y').fst.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').fst.snd \u2208 V') \u2227\n    \u2191e ((x, y), x', y').snd.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').snd.snd \u2208 V'\n\u22a2 x' \u2208 U\n[PROOFSTEP]\ntauto\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right.mk.mk.mk.x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nx : \u03b2\ny : \u03b4\nx' : \u03b2\ny' : \u03b4\nh' :\n  (\u2191e ((x, y), x', y').fst.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').fst.snd \u2208 V') \u2227\n    \u2191e ((x, y), x', y').snd.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').snd.snd \u2208 V'\n\u22a2 y \u2208 V\n[PROOFSTEP]\ntauto\n[GOAL]\ncase mk.right.intro.intro.intro.intro.intro.intro.intro.intro.right.mk.mk.mk.x\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nG : Type u_5\ninst\u271d\u00b9\u2076 : TopologicalSpace \u03b1\ninst\u271d\u00b9\u2075 : AddCommGroup \u03b1\ninst\u271d\u00b9\u2074 : TopologicalAddGroup \u03b1\ninst\u271d\u00b9\u00b3 : TopologicalSpace \u03b2\ninst\u271d\u00b9\u00b2 : AddCommGroup \u03b2\ninst\u271d\u00b9\u00b9 : TopologicalAddGroup \u03b2\ninst\u271d\u00b9\u2070 : TopologicalSpace \u03b3\ninst\u271d\u2079 : AddCommGroup \u03b3\ninst\u271d\u2078 : TopologicalAddGroup \u03b3\ninst\u271d\u2077 : TopologicalSpace \u03b4\ninst\u271d\u2076 : AddCommGroup \u03b4\ninst\u271d\u2075 : TopologicalAddGroup \u03b4\ninst\u271d\u2074 : UniformSpace G\ninst\u271d\u00b3 : AddCommGroup G\ninst\u271d\u00b2 : UniformAddGroup G\ninst\u271d\u00b9 : SeparatedSpace G\ninst\u271d : CompleteSpace G\ne : \u03b2 \u2192+ \u03b1\nde : DenseInducing \u2191e\nf : \u03b4 \u2192+ \u03b3\ndf : DenseInducing \u2191f\n\u03c6 : \u03b2 \u2192+ \u03b4 \u2192+ G\nh\u03c6 : Continuous fun p => \u2191(\u2191\u03c6 p.fst) p.snd\nW'\u271d : Set G\nW'_nhd\u271d : W'\u271d \u2208 \ud835\udcdd 0\nx\u2080 : \u03b1\ny\u2080 : \u03b3\nW' : Set G\nW'_nhd : W' \u2208 \ud835\udcdd 0\nU : Set \u03b2\nV : Set \u03b4\nh :\n  \u2200 (x : \u03b2),\n    x \u2208 U \u2192\n      \u2200 (x' : \u03b2),\n        x' \u2208 U \u2192\n          \u2200 (y : \u03b4),\n            y \u2208 V \u2192\n              \u2200 (y' : \u03b4), y' \u2208 V \u2192 (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x', y') - (fun p => \u2191(\u2191\u03c6 p.fst) p.snd) (x, y) \u2208 W'\nU' : Set \u03b1\nU'_nhd : U' \u2208 \ud835\udcdd x\u2080\nU'_sub : \u2191e \u207b\u00b9' U' \u2286 U\nV' : Set \u03b3\nV'_nhd : V' \u2208 \ud835\udcdd y\u2080\nV'_sub : \u2191f \u207b\u00b9' V' \u2286 V\nx : \u03b2\ny : \u03b4\nx' : \u03b2\ny' : \u03b4\nh' :\n  (\u2191e ((x, y), x', y').fst.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').fst.snd \u2208 V') \u2227\n    \u2191e ((x, y), x', y').snd.fst \u2208 U' \u2227 \u2191f ((x, y), x', y').snd.snd \u2208 V'\n\u22a2 y' \u2208 V\n[PROOFSTEP]\ntauto\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nletI : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nletI : UniformSpace G := TopologicalGroup.toUniformSpace G\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis : UniformSpace G := TopologicalGroup.toUniformSpace G\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nhaveI : (\ud835\udce4 (G \u29f8 N)).IsCountablyGenerated := comap.isCountablyGenerated _ _\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nobtain \u27e8u, hu, u_mul\u27e9 := TopologicalGroup.exists_antitone_basis_nhds_one G\n[GOAL]\ncase intro.intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nobtain \u27e8hv, v_anti\u27e9 := @HasAntitoneBasis.map _ _ _ _ _ _ ((\u2191) : G \u2192 G \u29f8 N) hu\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (Filter.map mk (\ud835\udcdd 1)) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nrw [\u2190 QuotientGroup.nhds_eq N 1, QuotientGroup.mk_one] at hv \n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nrefine'\n  UniformSpace.complete_of_cauchySeq_tendsto fun x hx =>\n    _\n      /- Given `n : \u2115`, for sufficiently large `a b : \u2115`, given any lift of `x b`, we can find a lift\n          of `x a` such that the quotient of the lifts lies in `u n`. -/\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave key\u2080 :\n  \u2200 i j : \u2115, \u2203 M : \u2115, j < M \u2227 \u2200 a b : \u2115, M \u2264 a \u2192 M \u2264 b \u2192 \u2200 g : G, x b = g \u2192 \u2203 g' : G, g / g' \u2208 u i \u2227 x a = g' :=\n  by\n  have h\ud835\udce4GN : (\ud835\udce4 (G \u29f8 N)).HasBasis (fun _ => True) fun i => {x | x.snd / x.fst \u2208 (\u2191) '' u i} := by\n    simpa [uniformity_eq_comap_nhds_one'] using hv.comap _\n  rw [h\ud835\udce4GN.cauchySeq_iff] at hx \n  simp only [ge_iff_le, mem_setOf_eq, forall_true_left, mem_image] at hx \n  intro i j\n  rcases hx i with \u27e8M, hM\u27e9\n  refine' \u27e8max j M + 1, (le_max_left _ _).trans_lt (lt_add_one _), fun a b ha hb g hg => _\u27e9\n  obtain \u27e8y, y_mem, hy\u27e9 :=\n    hM a (((le_max_right j _).trans (lt_add_one _).le).trans ha) b\n      (((le_max_right j _).trans (lt_add_one _).le).trans hb)\n  refine' \u27e8y\u207b\u00b9 * g, by simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, _\u27e9\n  rw [QuotientGroup.mk_mul, QuotientGroup.mk_inv, hy, hg, inv_div, div_mul_cancel']\n    /- Inductively construct a subsequence `\u03c6 : \u2115 \u2192 \u2115` using `key\u2080` so that if `a b : \u2115` exceed\n        `\u03c6 (n + 1)`, then we may find lifts whose quotients lie within `u n`. -/\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\n\u22a2 \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nhave h\ud835\udce4GN : (\ud835\udce4 (G \u29f8 N)).HasBasis (fun _ => True) fun i => {x | x.snd / x.fst \u2208 (\u2191) '' u i} := by\n  simpa [uniformity_eq_comap_nhds_one'] using hv.comap _\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\n\u22a2 HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\n[PROOFSTEP]\nsimpa [uniformity_eq_comap_nhds_one'] using hv.comap _\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\n\u22a2 \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nrw [h\ud835\udce4GN.cauchySeq_iff] at hx \n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : \u2200 (i : \u2115), True \u2192 \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 (x m, x n) \u2208 {x | x.snd / x.fst \u2208 mk '' u i}\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\n\u22a2 \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nsimp only [ge_iff_le, mem_setOf_eq, forall_true_left, mem_image] at hx \n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\n\u22a2 \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nintro i j\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j : \u2115\n\u22a2 \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nrcases hx i with \u27e8M, hM\u27e9\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j M : \u2115\nhM : \u2200 (m : \u2115), M \u2264 m \u2192 \u2200 (n : \u2115), M \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\n\u22a2 \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nrefine' \u27e8max j M + 1, (le_max_left _ _).trans_lt (lt_add_one _), fun a b ha hb g hg => _\u27e9\n[GOAL]\ncase intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j M : \u2115\nhM : \u2200 (m : \u2115), M \u2264 m \u2192 \u2200 (n : \u2115), M \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\na b : \u2115\nha : max j M + 1 \u2264 a\nhb : max j M + 1 \u2264 b\ng : G\nhg : x b = \u2191g\n\u22a2 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nobtain \u27e8y, y_mem, hy\u27e9 :=\n  hM a (((le_max_right j _).trans (lt_add_one _).le).trans ha) b (((le_max_right j _).trans (lt_add_one _).le).trans hb)\n[GOAL]\ncase intro.intro.intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j M : \u2115\nhM : \u2200 (m : \u2115), M \u2264 m \u2192 \u2200 (n : \u2115), M \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\na b : \u2115\nha : max j M + 1 \u2264 a\nhb : max j M + 1 \u2264 b\ng : G\nhg : x b = \u2191g\ny : G\ny_mem : y \u2208 u i\nhy : \u2191y = x b / x a\n\u22a2 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n[PROOFSTEP]\nrefine' \u27e8y\u207b\u00b9 * g, by simpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem, _\u27e9\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j M : \u2115\nhM : \u2200 (m : \u2115), M \u2264 m \u2192 \u2200 (n : \u2115), M \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\na b : \u2115\nha : max j M + 1 \u2264 a\nhb : max j M + 1 \u2264 b\ng : G\nhg : x b = \u2191g\ny : G\ny_mem : y \u2208 u i\nhy : \u2191y = x b / x a\n\u22a2 g / (y\u207b\u00b9 * g) \u2208 u i\n[PROOFSTEP]\nsimpa only [div_eq_mul_inv, mul_inv_rev, inv_inv, mul_inv_cancel_left] using y_mem\n[GOAL]\ncase intro.intro.intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nh\ud835\udce4GN : HasBasis (\ud835\udce4 (G \u29f8 N)) (fun x => True) fun i => {x | x.snd / x.fst \u2208 mk '' u i}\nhx : \u2200 (i : \u2115), \u2203 N_1, \u2200 (m : \u2115), N_1 \u2264 m \u2192 \u2200 (n : \u2115), N_1 \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\ni j M : \u2115\nhM : \u2200 (m : \u2115), M \u2264 m \u2192 \u2200 (n : \u2115), M \u2264 n \u2192 \u2203 x_1, x_1 \u2208 u i \u2227 \u2191x_1 = x n / x m\na b : \u2115\nha : max j M + 1 \u2264 a\nhb : max j M + 1 \u2264 b\ng : G\nhg : x b = \u2191g\ny : G\ny_mem : y \u2208 u i\nhy : \u2191y = x b / x a\n\u22a2 x a = \u2191(y\u207b\u00b9 * g)\n[PROOFSTEP]\nrw [QuotientGroup.mk_mul, QuotientGroup.mk_inv, hy, hg, inv_div, div_mul_cancel']\n  /- Inductively construct a subsequence `\u03c6 : \u2115 \u2192 \u2115` using `key\u2080` so that if `a b : \u2115` exceed\n      `\u03c6 (n + 1)`, then we may find lifts whose quotients lie within `u n`. -/\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nset \u03c6 : \u2115 \u2192 \u2115 := fun n => Nat.recOn n (choose <| key\u2080 0 0) fun k yk => choose <| key\u2080 (k + 1) yk\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave h\u03c6 :\n  \u2200 n : \u2115,\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 a b : \u2115, \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 g : G, x b = g \u2192 \u2203 g' : G, g / g' \u2208 u (n + 1) \u2227 x a = g' :=\n  fun n =>\n  choose_spec\n    (key\u2080 (n + 1) (\u03c6 n))\n      /- Inductively construct a sequence `x' n : G` of lifts of `x (\u03c6 (n + 1))` such that quotients of\n          successive terms lie in `x' n / x' (n + 1) \u2208 u (n + 1)`. We actually need the proofs that each\n          term is a lift to construct the next term, so we use a \u03a3-type. -/\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nset x' : \u2200 n, PSigma fun g : G => x (\u03c6 (n + 1)) = g := fun n =>\n  Nat.recOn n\n    \u27e8choose (QuotientGroup.mk_surjective (x (\u03c6 1))), (choose_spec (QuotientGroup.mk_surjective (x (\u03c6 1)))).symm\u27e9\n    fun k hk =>\n    \u27e8choose <| (h\u03c6 k).2 _ _ (h\u03c6 (k + 1)).1.le le_rfl hk.fst hk.snd,\n      (choose_spec <| (h\u03c6 k).2 _ _ (h\u03c6 (k + 1)).1.le le_rfl hk.fst hk.snd).2\u27e9\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave hx' : \u2200 n : \u2115, (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1) := fun n =>\n  (choose_spec <| (h\u03c6 n).2 _ _ (h\u03c6 (n + 1)).1.le le_rfl (x' n).fst (x' n).snd).1\n    /- The sequence `x'` is Cauchy. This is where we exploit the condition on `u`. The key idea\n        is to show by decreasing induction that `x' m / x' n \u2208 u m` if `m \u2264 n`. -/\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nhave x'_cauchy : CauchySeq fun n => (x' n).fst :=\n  by\n  have h\ud835\udce4G : (\ud835\udce4 G).HasBasis (fun _ => True) fun i => {x | x.snd / x.fst \u2208 u i} := by\n    simpa [uniformity_eq_comap_nhds_one'] using hu.toHasBasis.comap _\n  rw [h\ud835\udce4G.cauchySeq_iff']\n  simp only [ge_iff_le, mem_setOf_eq, forall_true_left]\n  exact fun m =>\n    \u27e8m, fun n hmn =>\n      Nat.decreasingInduction' (fun k _ _ hk => u_mul k \u27e8_, _, hx' k, hk, div_mul_div_cancel' _ _ _\u27e9) hmn\n        (by simpa only [div_self'] using mem_of_mem_nhds (hu.mem _))\u27e9\n      /- Since `G` is complete, `x'` converges to some `x\u2080`, and so the image of this sequence under\n          the quotient map converges to `\u2191x\u2080`. The image of `x'` is a convergent subsequence of `x`, and\n          since `x` is Cauchy, this implies it converges. -/\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\n\u22a2 CauchySeq fun n => (x' n).fst\n[PROOFSTEP]\nhave h\ud835\udce4G : (\ud835\udce4 G).HasBasis (fun _ => True) fun i => {x | x.snd / x.fst \u2208 u i} := by\n  simpa [uniformity_eq_comap_nhds_one'] using hu.toHasBasis.comap _\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\n\u22a2 HasBasis (\ud835\udce4 G) (fun x => True) fun i => {x | x.snd / x.fst \u2208 u i}\n[PROOFSTEP]\nsimpa [uniformity_eq_comap_nhds_one'] using hu.toHasBasis.comap _\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nh\ud835\udce4G : HasBasis (\ud835\udce4 G) (fun x => True) fun i => {x | x.snd / x.fst \u2208 u i}\n\u22a2 CauchySeq fun n => (x' n).fst\n[PROOFSTEP]\nrw [h\ud835\udce4G.cauchySeq_iff']\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nh\ud835\udce4G : HasBasis (\ud835\udce4 G) (fun x => True) fun i => {x | x.snd / x.fst \u2208 u i}\n\u22a2 \u2200 (i : \u2115), True \u2192 \u2203 N_1, \u2200 (n : \u2115), n \u2265 N_1 \u2192 ((x' n).fst, (x' N_1).fst) \u2208 {x | x.snd / x.fst \u2208 u i}\n[PROOFSTEP]\nsimp only [ge_iff_le, mem_setOf_eq, forall_true_left]\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nh\ud835\udce4G : HasBasis (\ud835\udce4 G) (fun x => True) fun i => {x | x.snd / x.fst \u2208 u i}\n\u22a2 \u2200 (i : \u2115),\n    \u2203 N_1,\n      \u2200 (n : \u2115),\n        N_1 \u2264 n \u2192\n          (Nat.rec { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n                  (fun k hk =>\n                    { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n                      snd :=\n                        (_ :\n                          x (\u03c6 (k + 1 + 1)) =\n                            \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) })\n                  N_1).fst /\n              (Nat.rec\n                  { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n                  (fun k hk =>\n                    { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n                      snd :=\n                        (_ :\n                          x (\u03c6 (k + 1 + 1)) =\n                            \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) })\n                  n).fst \u2208\n            u i\n[PROOFSTEP]\nexact fun m =>\n  \u27e8m, fun n hmn =>\n    Nat.decreasingInduction' (fun k _ _ hk => u_mul k \u27e8_, _, hx' k, hk, div_mul_div_cancel' _ _ _\u27e9) hmn\n      (by simpa only [div_self'] using mem_of_mem_nhds (hu.mem _))\u27e9\n    /- Since `G` is complete, `x'` converges to some `x\u2080`, and so the image of this sequence under\n        the quotient map converges to `\u2191x\u2080`. The image of `x'` is a convergent subsequence of `x`, and\n        since `x` is Cauchy, this implies it converges. -/\n[GOAL]\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nh\ud835\udce4G : HasBasis (\ud835\udce4 G) (fun x => True) fun i => {x | x.snd / x.fst \u2208 u i}\nm n : \u2115\nhmn : m \u2264 n\n\u22a2 (Nat.rec { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n          (fun k hk =>\n            { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n              snd :=\n                (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) })\n          n).fst /\n      (Nat.rec { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n          (fun k hk =>\n            { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n              snd :=\n                (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) })\n          n).fst \u2208\n    u n\n[PROOFSTEP]\nsimpa only [div_self'] using mem_of_mem_nhds (hu.mem _)\n[GOAL]\ncase intro.intro.mk\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nx'_cauchy : CauchySeq fun n => (x' n).fst\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrcases cauchySeq_tendsto_of_complete x'_cauchy with \u27e8x\u2080, hx\u2080\u27e9\n[GOAL]\ncase intro.intro.mk.intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nx'_cauchy : CauchySeq fun n => (x' n).fst\nx\u2080 : G\nhx\u2080 : Tendsto (fun n => (x' n).fst) atTop (\ud835\udcdd x\u2080)\n\u22a2 \u2203 a, Tendsto x atTop (\ud835\udcdd a)\n[PROOFSTEP]\nrefine'\n  \u27e8\u2191x\u2080, tendsto_nhds_of_cauchySeq_of_subseq hx (strictMono_nat_of_lt_succ fun n => (h\u03c6 (n + 1)).1).tendsto_atTop _\u27e9\n[GOAL]\ncase intro.intro.mk.intro\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nx'_cauchy : CauchySeq fun n => (x' n).fst\nx\u2080 : G\nhx\u2080 : Tendsto (fun n => (x' n).fst) atTop (\ud835\udcdd x\u2080)\n\u22a2 Tendsto (x \u2218 fun n => \u03c6 (n + 1)) atTop (\ud835\udcdd \u2191x\u2080)\n[PROOFSTEP]\nconvert ((continuous_coinduced_rng : Continuous ((\u2191) : G \u2192 G \u29f8 N)).tendsto x\u2080).comp hx\u2080\n[GOAL]\ncase h.e'_3\nG : Type u\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : TopologicalSpace G\ninst\u271d\u00b3 : TopologicalGroup G\ninst\u271d\u00b2 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal N\ninst\u271d : CompleteSpace G\nthis\u271d\u00b9 : UniformSpace (G \u29f8 N) := TopologicalGroup.toUniformSpace (G \u29f8 N)\nthis\u271d : UniformSpace G := TopologicalGroup.toUniformSpace G\nthis : IsCountablyGenerated (\ud835\udce4 (G \u29f8 N))\nu : \u2115 \u2192 Set G\nhu : HasAntitoneBasis (\ud835\udcdd 1) u\nu_mul : \u2200 (n : \u2115), u (n + 1) * u (n + 1) \u2286 u n\nhv : HasBasis (\ud835\udcdd 1) (fun x => True) fun n => mk '' u n\nv_anti : Antitone fun n => mk '' u n\nx : \u2115 \u2192 G \u29f8 N\nhx : CauchySeq x\nkey\u2080 : \u2200 (i j : \u2115), \u2203 M, j < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u i \u2227 x a = \u2191g'\n\u03c6 : \u2115 \u2192 \u2115 :=\n  fun n =>\n    Nat.recOn n\n      (choose (_ : \u2203 M, 0 < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u 0 \u2227 x a = \u2191g'))\n      fun k yk =>\n      choose (_ : \u2203 M, yk < M \u2227 \u2200 (a b : \u2115), M \u2264 a \u2192 M \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (k + 1) \u2227 x a = \u2191g')\nh\u03c6 :\n  \u2200 (n : \u2115),\n    \u03c6 n < \u03c6 (n + 1) \u2227\n      \u2200 (a b : \u2115), \u03c6 (n + 1) \u2264 a \u2192 \u03c6 (n + 1) \u2264 b \u2192 \u2200 (g : G), x b = \u2191g \u2192 \u2203 g', g / g' \u2208 u (n + 1) \u2227 x a = \u2191g'\nx' : (n : \u2115) \u2192 (g : G) \u00d7' x (\u03c6 (n + 1)) = \u2191g :=\n  fun n =>\n    Nat.recOn n { fst := choose (_ : \u2203 a, \u2191a = x (\u03c6 1)), snd := (_ : x (\u03c6 1) = \u2191(choose (_ : \u2203 a, \u2191a = x (\u03c6 1)))) }\n      fun k hk =>\n      { fst := choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'),\n        snd := (_ : x (\u03c6 (k + 1 + 1)) = \u2191(choose (_ : \u2203 g', hk.fst / g' \u2208 u (k + 1) \u2227 x (\u03c6 (k + 1 + 1)) = \u2191g'))) }\nhx' : \u2200 (n : \u2115), (x' n).fst / (x' (n + 1)).fst \u2208 u (n + 1)\nx'_cauchy : CauchySeq fun n => (x' n).fst\nx\u2080 : G\nhx\u2080 : Tendsto (fun n => (x' n).fst) atTop (\ud835\udcdd x\u2080)\n\u22a2 (x \u2218 fun n => \u03c6 (n + 1)) = mk \u2218 fun n => (x' n).fst\n[PROOFSTEP]\nexact funext fun n => (x' n).snd\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nus : UniformSpace G\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d : Subgroup.Normal N\nhG : CompleteSpace G\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\nrw [\u2190 @UniformGroup.toUniformSpace_eq _ us _ _] at hG \n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nus : UniformSpace G\ninst\u271d\u00b2 : UniformGroup G\ninst\u271d\u00b9 : FirstCountableTopology G\nN : Subgroup G\ninst\u271d : Subgroup.Normal N\nhG : CompleteSpace G\n\u22a2 CompleteSpace (G \u29f8 N)\n[PROOFSTEP]\ninfer_instance\n", "meta": {"mathlib_filename": "Mathlib.Topology.Algebra.UniformGroup", "llama_tokens": 117750, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676284, "lm_q2_score": 0.6654105653819835, "lm_q1q2_score": 0.5023284082269213}}
{"text": "[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d : PartialOrder \u03b1\nx y : \u03b1\n\u22a2 x \u2264 y \u2192 y = x \u2194 \u00acx < y\n[PROOFSTEP]\nrw [lt_iff_le_and_ne, not_and, Classical.not_not, eq_comm]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 min a b = if b \u2264 a then b else a\n[PROOFSTEP]\nrw [min_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 (if a \u2264 b then a else b) = if b \u2264 a then b else a\n[PROOFSTEP]\nrcases lt_trichotomy a b with (lt | eq | gt)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nlt : a < b\n\u22a2 (if a \u2264 b then a else b) = if b \u2264 a then b else a\n[PROOFSTEP]\nrw [if_pos lt.le, if_neg (not_le.mpr lt)]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\neq : a = b\n\u22a2 (if a \u2264 b then a else b) = if b \u2264 a then b else a\n[PROOFSTEP]\nrw [if_pos eq.le, if_pos eq.ge, eq]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\ngt : b < a\n\u22a2 (if a \u2264 b then a else b) = if b \u2264 a then b else a\n[PROOFSTEP]\nrw [if_neg (not_le.mpr gt.gt), if_pos gt.le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 max a b = if b \u2264 a then a else b\n[PROOFSTEP]\nrw [max_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\n\u22a2 (if a \u2264 b then b else a) = if b \u2264 a then a else b\n[PROOFSTEP]\nrcases lt_trichotomy a b with (lt | eq | gt)\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nlt : a < b\n\u22a2 (if a \u2264 b then b else a) = if b \u2264 a then a else b\n[PROOFSTEP]\nrw [if_pos lt.le, if_neg (not_le.mpr lt)]\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\neq : a = b\n\u22a2 (if a \u2264 b then b else a) = if b \u2264 a then a else b\n[PROOFSTEP]\nrw [if_pos eq.le, if_pos eq.ge, eq]\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\ngt : b < a\n\u22a2 (if a \u2264 b then b else a) = if b \u2264 a then a else b\n[PROOFSTEP]\nrw [if_neg (not_le.mpr gt.gt), if_pos gt.le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : PartialOrder \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ncomm : Commutative f\nassoc : \u2200 (a b c : \u03b1), f (f a b) c \u2264 f a (f b c)\na b c : \u03b1\n\u22a2 f a (f b c) \u2264 f (f a b) c\n[PROOFSTEP]\nrw [comm, comm b, comm _ c, comm a]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : PartialOrder \u03b1\nf : \u03b1 \u2192 \u03b1 \u2192 \u03b1\ncomm : Commutative f\nassoc : \u2200 (a b c : \u03b1), f (f a b) c \u2264 f a (f b c)\na b c : \u03b1\n\u22a2 f (f c b) a \u2264 f c (f b a)\n[PROOFSTEP]\nexact assoc _ _ _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d\u00b9 A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle\u271d B_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nh : toLE = toLE\n\u22a2 mk le_refl\u271d\u00b9 le_trans\u271d\u00b9 = mk le_refl\u271d le_trans\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nB_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n\u22a2 mk le_refl\u271d\u00b9 le_trans\u271d\u00b9 = mk le_refl\u271d le_trans\u271d\n[PROOFSTEP]\nhave : A_lt = B_lt := by\n  funext a b\n  show (LT.mk A_lt).lt a b = (LT.mk B_lt).lt a b\n  rw [A_iff, B_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nB_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n\u22a2 A_lt = B_lt\n[PROOFSTEP]\nfunext a b\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nB_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\na b : \u03b1\n\u22a2 A_lt a b = B_lt a b\n[PROOFSTEP]\nshow (LT.mk A_lt).lt a b = (LT.mk B_lt).lt a b\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nB_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\na b : \u03b1\n\u22a2 (a < b) = (a < b)\n[PROOFSTEP]\nrw [A_iff, B_iff]\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nB_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nthis : A_lt = B_lt\n\u22a2 mk le_refl\u271d\u00b9 le_trans\u271d\u00b9 = mk le_refl\u271d le_trans\u271d\n[PROOFSTEP]\ncases this\n[GOAL]\ncase refl.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nle\u271d A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nA_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nB_iff : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n\u22a2 mk le_refl\u271d\u00b9 le_trans\u271d\u00b9 = mk le_refl\u271d le_trans\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : PartialOrder \u03b1\nh : toPreorder = toPreorder\n\u22a2 A = B\n[PROOFSTEP]\ncases A\n[GOAL]\ncase mk\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nB : PartialOrder \u03b1\ntoPreorder\u271d : Preorder \u03b1\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nh : toPreorder = toPreorder\n\u22a2 mk le_antisymm\u271d = B\n[PROOFSTEP]\ncases B\n[GOAL]\ncase mk.mk\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ntoPreorder\u271d\u00b9 : Preorder \u03b1\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\ntoPreorder\u271d : Preorder \u03b1\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nh : toPreorder = toPreorder\n\u22a2 mk le_antisymm\u271d\u00b9 = mk le_antisymm\u271d\n[PROOFSTEP]\ncases h\n[GOAL]\ncase mk.mk.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ntoPreorder\u271d : Preorder \u03b1\nle_antisymm\u271d\u00b9 le_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n\u22a2 mk le_antisymm\u271d\u00b9 = mk le_antisymm\u271d\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_le B_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nB_decidableEq : DecidableEq \u03b1\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nh : toPartialOrder = toPartialOrder\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d B_decidableLE B_decidableEq B_decidableLT\n[PROOFSTEP]\ncases h\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nB_decidableEq : DecidableEq \u03b1\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d B_decidableLE B_decidableEq B_decidableLT\n[PROOFSTEP]\nobtain rfl : A_decidableLE = B_decidableLE := Subsingleton.elim _ _\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nB_decidableEq : DecidableEq \u03b1\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE B_decidableEq B_decidableLT\n[PROOFSTEP]\nobtain rfl : A_decidableEq = B_decidableEq := Subsingleton.elim _ _\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_decidableLT : DecidableRel fun x x_1 => x < x_1\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq B_decidableLT\n[PROOFSTEP]\nobtain rfl : A_decidableLT = B_decidableLT := Subsingleton.elim _ _\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\nhave : A_min = B_min := by\n  funext a b\n  exact (A_min_def _ _).trans (B_min_def _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\n\u22a2 A_min = B_min\n[PROOFSTEP]\nfunext a b\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\na b : \u03b1\n\u22a2 A_min a b = B_min a b\n[PROOFSTEP]\nexact (A_min_def _ _).trans (B_min_def _ _).symm\n[GOAL]\ncase refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min B_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nthis : A_min = B_min\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\ncases this\n[GOAL]\ncase refl.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\nhave : A_max = B_max := by\n  funext a b\n  exact (A_max_def _ _).trans (B_max_def _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\n\u22a2 A_max = B_max\n[PROOFSTEP]\nfunext a b\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\na b : \u03b1\n\u22a2 A_max a b = B_max a b\n[PROOFSTEP]\nexact (A_max_def _ _).trans (B_max_def _ _).symm\n[GOAL]\ncase refl.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nthis : A_max = B_max\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\ncases this\n[GOAL]\ncase refl.refl.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\nhave : A_compare = B_compare := by\n  funext a b\n  exact (A_compare_canonical _ _).trans (B_compare_canonical _ _).symm\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n\u22a2 A_compare = B_compare\n[PROOFSTEP]\nfunext a b\n[GOAL]\ncase h.h\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\na b : \u03b1\n\u22a2 A_compare a b = B_compare a b\n[PROOFSTEP]\nexact (A_compare_canonical _ _).trans (B_compare_canonical _ _).symm\n[GOAL]\ncase refl.refl.refl\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nA_le A_lt : \u03b1 \u2192 \u03b1 \u2192 Prop\nle_refl\u271d\u00b9 : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d\u00b9 : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d\u00b9 : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nA_min A_max : \u03b1 \u2192 \u03b1 \u2192 \u03b1\nA_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_total\u271d\u00b9 : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nA_decidableLE : DecidableRel fun x x_1 => x \u2264 x_1\nA_decidableEq : DecidableEq \u03b1\nA_decidableLT : DecidableRel fun x x_1 => x < x_1\nA_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nA_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nA_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_compare : \u03b1 \u2192 \u03b1 \u2192 Ordering\nle_refl\u271d : \u2200 (a : \u03b1), a \u2264 a\nle_trans\u271d : \u2200 (a b c : \u03b1), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\nlt_iff_le_not_le\u271d : \u2200 (a b : \u03b1), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\nle_antisymm\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2192 b \u2264 a \u2192 a = b\nle_total\u271d : \u2200 (a b : \u03b1), a \u2264 b \u2228 b \u2264 a\nB_compare_canonical : \u2200 (a b : \u03b1), compare a b = compareOfLessAndEq a b\nB_min_def : \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\nB_max_def : \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\nthis : A_compare = B_compare\n\u22a2 mk le_total\u271d\u00b9 A_decidableLE A_decidableEq A_decidableLT = mk le_total\u271d A_decidableLE A_decidableEq A_decidableLT\n[PROOFSTEP]\ncongr\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\next x y\n[GOAL]\ncase a.le.h.h.a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : Preorder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y : \u03b1\n\u22a2 x \u2264 y \u2194 x \u2264 y\n[PROOFSTEP]\nexact H x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : PartialOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\next x y\n[GOAL]\ncase a.a.le.h.h.a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : PartialOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y : \u03b1\n\u22a2 x \u2264 y \u2194 x \u2264 y\n[PROOFSTEP]\nexact H x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\n\u22a2 A = B\n[PROOFSTEP]\next x y\n[GOAL]\ncase a.a.a.le.h.h.a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\nA B : LinearOrder \u03b1\nH : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y\nx y : \u03b1\n\u22a2 x \u2264 y \u2194 x \u2264 y\n[PROOFSTEP]\nexact H x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : PartialOrder \u03b1\u1d52\u1d48 := inferInstanceAs (PartialOrder \u03b1\u1d52\u1d48)\na b : \u03b1\u1d52\u1d48\n\u22a2 max a b = if a \u2264 b then a else b\n[PROOFSTEP]\nrw [max_comm, max_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : PartialOrder \u03b1\u1d52\u1d48 := inferInstanceAs (PartialOrder \u03b1\u1d52\u1d48)\na b : \u03b1\u1d52\u1d48\n\u22a2 (if b \u2264 a then a else b) = if a \u2264 b then a else b\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : PartialOrder \u03b1\u1d52\u1d48 := inferInstanceAs (PartialOrder \u03b1\u1d52\u1d48)\na b : \u03b1\u1d52\u1d48\n\u22a2 min a b = if a \u2264 b then b else a\n[PROOFSTEP]\nrw [min_comm, min_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type u_3\ninst\u271d : LinearOrder \u03b1\nsrc\u271d : PartialOrder \u03b1\u1d52\u1d48 := inferInstanceAs (PartialOrder \u03b1\u1d52\u1d48)\na b : \u03b1\u1d52\u1d48\n\u22a2 (if b \u2264 a then b else a) = if a \u2264 b then b else a\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9\u271d : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9\u271d \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b9 : Type u\n\u03b1 : \u03b9 \u2192 Type v\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03b1 i)\nx y : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 x < y \u2194 x \u2264 y \u2227 \u2203 i, x i < y i\n[PROOFSTEP]\nsimp (config := { contextual := true }) [lt_iff_le_not_le, Pi.le_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03c0 i)\na b c : (i : \u03b9) \u2192 \u03c0 i\ninst\u271d : Nonempty \u03b9\nh : a \u227a b\n\u22a2 a < b\n[PROOFSTEP]\ninhabit \u03b9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03c0 i)\na b c : (i : \u03b9) \u2192 \u03c0 i\ninst\u271d : Nonempty \u03b9\nh : a \u227a b\ninhabited_h : Inhabited \u03b9\n\u22a2 a < b\n[PROOFSTEP]\nexact Pi.lt_def.2 \u27e8le_of_strongLT h, default, h _\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 update x i a \u2264 update y i b \u2194 a \u2264 b \u2227 \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2264 y j\n[PROOFSTEP]\nsimp (config := { contextual := true }) [update_le_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 update x i a \u2264 update x i b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [update_le_update_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 x \u2264 update x i a \u2194 x i \u2264 a\n[PROOFSTEP]\nsimp [le_update_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 update x i a \u2264 x \u2194 a \u2264 x i\n[PROOFSTEP]\nsimp [update_le_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 x < update x i a \u2194 x i < a\n[PROOFSTEP]\nsimp [lt_iff_le_not_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 Preorder (\u03c0 i)\nx y : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\na b : \u03c0 i\n\u22a2 update x i a < x \u2194 a < x i\n[PROOFSTEP]\nsimp [lt_iff_le_not_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Nonempty \u03b2\na b : \u03b1\n\u22a2 const \u03b2 a \u2264 const \u03b2 b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp [Pi.le_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Nonempty \u03b2\na b : \u03b1\n\u22a2 const \u03b2 a < const \u03b2 b \u2194 a < b\n[PROOFSTEP]\nsimpa [Pi.lt_def] using le_of_lt\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\np : \u03b1 \u2192 Prop\nx\u271d y\u271d x y : \u03b1\n\u22a2 min x y = if x < y then x else y\n[PROOFSTEP]\nrw [min_comm, min_def, \u2190 ite_not]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\np : \u03b1 \u2192 Prop\nx\u271d y\u271d x y : \u03b1\n\u22a2 (if \u00acy \u2264 x then x else y) = if x < y then x else y\n[PROOFSTEP]\nsimp only [not_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\np : \u03b1 \u2192 Prop\nx\u271d y\u271d x y : \u03b1\n\u22a2 max x y = if x < y then y else x\n[PROOFSTEP]\nrw [max_comm, max_def, \u2190 ite_not]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\np : \u03b1 \u2192 Prop\nx\u271d y\u271d x y : \u03b1\n\u22a2 (if \u00acy \u2264 x then y else x) = if x < y then y else x\n[PROOFSTEP]\nsimp only [not_le]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\n\u22a2 compare (f a) (f b) = compareOfLessAndEq a b\n[PROOFSTEP]\nhave h := LinearOrder.compare_eq_compareOfLessAndEq (f a) (f b)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\n\u22a2 compare (f a) (f b) = compareOfLessAndEq a b\n[PROOFSTEP]\nsimp only [h, compareOfLessAndEq]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\n\u22a2 (if f a < f b then Ordering.lt else if f a = f b then Ordering.eq else Ordering.gt) =\n    if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b9 : f a < f b\nh\u271d : a < b\n\u22a2 Ordering.lt = Ordering.lt\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b9 : f a < f b\nh\u271d : a < b\n\u22a2 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b9 : f a < f b\nh\u271d : a < b\n\u22a2 Ordering.lt = Ordering.lt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : f a < f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : f a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : f a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : f a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : f a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 Ordering.eq = Ordering.eq\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 Ordering.eq = Ordering.eq\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 Ordering.eq = Ordering.eq\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : \u00acf a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : \u00acf a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : \u00acf a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : \u00acf a = f b\nh\u271d : a < b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\nrfl\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 Ordering.gt = Ordering.gt\n[PROOFSTEP]\ntry\n  (first\n    | rfl\n    | contradiction)\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 Ordering.gt = Ordering.gt\n[PROOFSTEP]\nfirst\n| rfl\n| contradiction\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 Ordering.gt = Ordering.gt\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 False\n[PROOFSTEP]\nhave : \u00acf a = f b := by rename_i h; exact inj.ne h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\n\u22a2 \u00acf a = f b\n[PROOFSTEP]\nrename_i h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh\u271d\u00b3 : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : f a = f b\nh\u271d : \u00aca < b\nh : \u00aca = b\n\u22a2 \u00acf a = f b\n[PROOFSTEP]\nexact inj.ne h\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : f a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : \u00aca = b\nthis : \u00acf a = f b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 False\n[PROOFSTEP]\nhave : f a = f b := by rename_i h; exact congrArg f h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\n\u22a2 f a = f b\n[PROOFSTEP]\nrename_i h\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh\u271d\u00b3 : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b2 : \u00acf a < f b\nh\u271d\u00b9 : \u00acf a = f b\nh\u271d : \u00aca < b\nh : a = b\n\u22a2 f a = f b\n[PROOFSTEP]\nexact congrArg f h\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\ninst\u271d : Decidable (a < b)\nh : compare (f a) (f b) = compareOfLessAndEq (f a) (f b)\nh\u271d\u00b3 : \u00acf a < f b\nh\u271d\u00b2 : \u00acf a = f b\nh\u271d\u00b9 : \u00aca < b\nh\u271d : a = b\nthis : f a = f b\n\u22a2 False\n[PROOFSTEP]\ncontradiction\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\n\u22a2 \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nintros x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 min x y = if x \u2264 y then x else y\n[PROOFSTEP]\napply inj\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (min x y) = f (if x \u2264 y then x else y)\n[PROOFSTEP]\nrw [apply_ite f]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (min x y) = if x \u2264 y then f x else f y\n[PROOFSTEP]\nexact (hinf _ _).trans (min_def _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\n\u22a2 \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nintros x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 max x y = if x \u2264 y then y else x\n[PROOFSTEP]\napply inj\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (max x y) = f (if x \u2264 y then y else x)\n[PROOFSTEP]\nrw [apply_ite f]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.96324\n\u03b2 : Type ?u.96321\ninst\u271d\u00b2 : LinearOrder \u03b2\ninst\u271d\u00b9 : Sup \u03b1\ninst\u271d : Inf \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ninstOrd\u03b1 : Ord \u03b1 := { compare := fun a b => compare (f a) (f b) }\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (max x y) = if x \u2264 y then f y else f x\n[PROOFSTEP]\nexact (hsup _ _).trans (max_def _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\n\u22a2 \u2200 (a b : \u03b1), min a b = if a \u2264 b then a else b\n[PROOFSTEP]\nintros x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 min x y = if x \u2264 y then x else y\n[PROOFSTEP]\napply inj\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (min x y) = f (if x \u2264 y then x else y)\n[PROOFSTEP]\nrw [apply_ite f]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (min x y) = if x \u2264 y then f x else f y\n[PROOFSTEP]\nexact (hinf _ _).trans (min_def _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\n\u22a2 \u2200 (a b : \u03b1), max a b = if a \u2264 b then b else a\n[PROOFSTEP]\nintros x y\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 max x y = if x \u2264 y then y else x\n[PROOFSTEP]\napply inj\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (max x y) = f (if x \u2264 y then y else x)\n[PROOFSTEP]\nrw [apply_ite f]\n[GOAL]\ncase a\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2\u271d : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : Type ?u.97808\n\u03b2 : Type ?u.97805\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : Sup \u03b1\ninst\u271d\u00b9 : Inf \u03b1\ninst\u271d : Ord \u03b1\nf : \u03b1 \u2192 \u03b2\ninj : Injective f\nhsup : \u2200 (x y : \u03b1), f (x \u2294 y) = max (f x) (f y)\nhinf : \u2200 (x y : \u03b1), f (x \u2293 y) = min (f x) (f y)\ncompare_f : \u2200 (a b : \u03b1), compare a b = compare (f a) (f b)\ndecidableLE : (x y : \u03b1) \u2192 Decidable (f x \u2264 f y) := fun x y => inferInstance\ndecidableLT : (x y : \u03b1) \u2192 Decidable (f x < f y) := fun x y => inferInstance\ndecidableEq : (x y : \u03b1) \u2192 Decidable (x = y) := fun x y => decidable_of_iff (f x = f y) (_ : f x = f y \u2194 x = y)\nsrc\u271d : PartialOrder \u03b1 := PartialOrder.lift f inj\nx y : \u03b1\n\u22a2 f (max x y) = if x \u2264 y then f y else f x\n[PROOFSTEP]\nexact (hsup _ _).trans (max_def _ _)\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 x < y \u2194 x.fst < y.fst \u2227 x.snd \u2264 y.snd \u2228 x.fst \u2264 y.fst \u2227 x.snd < y.snd\n[PROOFSTEP]\nrefine' \u27e8fun h \u21a6 _, _\u27e9\n[GOAL]\ncase refine'_1\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : x < y\n\u22a2 x.fst < y.fst \u2227 x.snd \u2264 y.snd \u2228 x.fst \u2264 y.fst \u2227 x.snd < y.snd\n[PROOFSTEP]\nby_cases h\u2081 : y.1 \u2264 x.1\n[GOAL]\ncase pos\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : x < y\nh\u2081 : y.fst \u2264 x.fst\n\u22a2 x.fst < y.fst \u2227 x.snd \u2264 y.snd \u2228 x.fst \u2264 y.fst \u2227 x.snd < y.snd\n[PROOFSTEP]\nexact Or.inr \u27e8h.1.1, LE.le.lt_of_not_le h.1.2 fun h\u2082 \u21a6 h.2 \u27e8h\u2081, h\u2082\u27e9\u27e9\n[GOAL]\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : x < y\nh\u2081 : \u00acy.fst \u2264 x.fst\n\u22a2 x.fst < y.fst \u2227 x.snd \u2264 y.snd \u2228 x.fst \u2264 y.fst \u2227 x.snd < y.snd\n[PROOFSTEP]\nexact Or.inl \u27e8LE.le.lt_of_not_le h.1.1 h\u2081, h.1.2\u27e9\n[GOAL]\ncase refine'_2\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 x.fst < y.fst \u2227 x.snd \u2264 y.snd \u2228 x.fst \u2264 y.fst \u2227 x.snd < y.snd \u2192 x < y\n[PROOFSTEP]\nrintro (\u27e8h\u2081, h\u2082\u27e9 | \u27e8h\u2081, h\u2082\u27e9)\n[GOAL]\ncase refine'_2.inl.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh\u2081 : x.fst < y.fst\nh\u2082 : x.snd \u2264 y.snd\n\u22a2 x < y\n[PROOFSTEP]\nexact \u27e8\u27e8h\u2081.le, h\u2082\u27e9, fun h \u21a6 h\u2081.not_le h.1\u27e9\n[GOAL]\ncase refine'_2.inr.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh\u2081 : x.fst \u2264 y.fst\nh\u2082 : x.snd < y.snd\n\u22a2 x < y\n[PROOFSTEP]\nexact \u27e8\u27e8h\u2081, h\u2082.le\u27e9, fun h \u21a6 h\u2082.not_le h.2\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LT \u03b1\n\u22a2 DenselyOrdered \u03b1\u1d52\u1d48 \u2192 DenselyOrdered \u03b1\n[PROOFSTEP]\nconvert @OrderDual.denselyOrdered \u03b1\u1d52\u1d48 _\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a < b \u2192 \u2203 a_2, a < a_2 \u2227 a_2 < b\n[PROOFSTEP]\nsimp_rw [Prod.lt_iff]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\n\u22a2 a.fst < b.fst \u2227 a.snd \u2264 b.snd \u2228 a.fst \u2264 b.fst \u2227 a.snd < b.snd \u2192\n    \u2203 a_2,\n      (a.fst < a_2.fst \u2227 a.snd \u2264 a_2.snd \u2228 a.fst \u2264 a_2.fst \u2227 a.snd < a_2.snd) \u2227\n        (a_2.fst < b.fst \u2227 a_2.snd \u2264 b.snd \u2228 a_2.fst \u2264 b.fst \u2227 a_2.snd < b.snd)\n[PROOFSTEP]\nrintro (\u27e8h\u2081, h\u2082\u27e9 | \u27e8h\u2081, h\u2082\u27e9)\n[GOAL]\ncase inl.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\nh\u2081 : a.fst < b.fst\nh\u2082 : a.snd \u2264 b.snd\n\u22a2 \u2203 a_1,\n    (a.fst < a_1.fst \u2227 a.snd \u2264 a_1.snd \u2228 a.fst \u2264 a_1.fst \u2227 a.snd < a_1.snd) \u2227\n      (a_1.fst < b.fst \u2227 a_1.snd \u2264 b.snd \u2228 a_1.fst \u2264 b.fst \u2227 a_1.snd < b.snd)\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between h\u2081\n[GOAL]\ncase inl.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\nh\u2081 : a.fst < b.fst\nh\u2082 : a.snd \u2264 b.snd\nc : \u03b1\nha : a.fst < c\nhb : c < b.fst\n\u22a2 \u2203 a_1,\n    (a.fst < a_1.fst \u2227 a.snd \u2264 a_1.snd \u2228 a.fst \u2264 a_1.fst \u2227 a.snd < a_1.snd) \u2227\n      (a_1.fst < b.fst \u2227 a_1.snd \u2264 b.snd \u2228 a_1.fst \u2264 b.fst \u2227 a_1.snd < b.snd)\n[PROOFSTEP]\nexact \u27e8(c, _), Or.inl \u27e8ha, h\u2082\u27e9, Or.inl \u27e8hb, le_rfl\u27e9\u27e9\n[GOAL]\ncase inr.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\nh\u2081 : a.fst \u2264 b.fst\nh\u2082 : a.snd < b.snd\n\u22a2 \u2203 a_1,\n    (a.fst < a_1.fst \u2227 a.snd \u2264 a_1.snd \u2228 a.fst \u2264 a_1.fst \u2227 a.snd < a_1.snd) \u2227\n      (a_1.fst < b.fst \u2227 a_1.snd \u2264 b.snd \u2228 a_1.fst \u2264 b.fst \u2227 a_1.snd < b.snd)\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between h\u2082\n[GOAL]\ncase inr.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d\u00b3 : Preorder \u03b1\ninst\u271d\u00b2 : Preorder \u03b2\ninst\u271d\u00b9 : DenselyOrdered \u03b1\ninst\u271d : DenselyOrdered \u03b2\na b : \u03b1 \u00d7 \u03b2\nh\u2081 : a.fst \u2264 b.fst\nh\u2082 : a.snd < b.snd\nc : \u03b2\nha : a.snd < c\nhb : c < b.snd\n\u22a2 \u2203 a_1,\n    (a.fst < a_1.fst \u2227 a.snd \u2264 a_1.snd \u2228 a.fst \u2264 a_1.fst \u2227 a.snd < a_1.snd) \u2227\n      (a_1.fst < b.fst \u2227 a_1.snd \u2264 b.snd \u2228 a_1.fst \u2264 b.fst \u2227 a_1.snd < b.snd)\n[PROOFSTEP]\nexact \u27e8(_, c), Or.inr \u27e8h\u2081, ha\u27e9, Or.inr \u27e8le_rfl, hb\u27e9\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 a < b \u2192 \u2203 a_2, a < a_2 \u2227 a_2 < b\n[PROOFSTEP]\nclassical\nsimp_rw [Pi.lt_def]\nrintro \u27e8hab, i, hi\u27e9\nobtain \u27e8c, ha, hb\u27e9 := exists_between hi\nexact\n  \u27e8Function.update a i c, \u27e8le_update_iff.2 \u27e8ha.le, fun _ _ \u21a6 le_rfl\u27e9, i, by rwa [update_same]\u27e9,\n    update_le_iff.2 \u27e8hb.le, fun _ _ \u21a6 hab _\u27e9, i, by rwa [update_same]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 a < b \u2192 \u2203 a_2, a < a_2 \u2227 a_2 < b\n[PROOFSTEP]\nsimp_rw [Pi.lt_def]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\n\u22a2 (a \u2264 b \u2227 \u2203 i, a i < b i) \u2192 \u2203 a_2, (a \u2264 a_2 \u2227 \u2203 i, a i < a_2 i) \u2227 a_2 \u2264 b \u2227 \u2203 i, a_2 i < b i\n[PROOFSTEP]\nrintro \u27e8hab, i, hi\u27e9\n[GOAL]\ncase intro.intro\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\nhab : a \u2264 b\ni : \u03b9\nhi : a i < b i\n\u22a2 \u2203 a_1, (a \u2264 a_1 \u2227 \u2203 i, a i < a_1 i) \u2227 a_1 \u2264 b \u2227 \u2203 i, a_1 i < b i\n[PROOFSTEP]\nobtain \u27e8c, ha, hb\u27e9 := exists_between hi\n[GOAL]\ncase intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\nhab : a \u2264 b\ni : \u03b9\nhi : a i < b i\nc : \u03b1 i\nha : a i < c\nhb : c < b i\n\u22a2 \u2203 a_1, (a \u2264 a_1 \u2227 \u2203 i, a i < a_1 i) \u2227 a_1 \u2264 b \u2227 \u2203 i, a_1 i < b i\n[PROOFSTEP]\nexact\n  \u27e8Function.update a i c, \u27e8le_update_iff.2 \u27e8ha.le, fun _ _ \u21a6 le_rfl\u27e9, i, by rwa [update_same]\u27e9,\n    update_le_iff.2 \u27e8hb.le, fun _ _ \u21a6 hab _\u27e9, i, by rwa [update_same]\u27e9\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\nhab : a \u2264 b\ni : \u03b9\nhi : a i < b i\nc : \u03b1 i\nha : a i < c\nhb : c < b i\n\u22a2 a i < update a i c i\n[PROOFSTEP]\nrwa [update_same]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1\u271d : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\n\u03b1 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Preorder (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), DenselyOrdered (\u03b1 i)\na b : (i : \u03b9) \u2192 \u03b1 i\nhab : a \u2264 b\ni : \u03b9\nhi : a i < b i\nc : \u03b1 i\nha : a i < c\nhb : c < b i\n\u22a2 update a i c i < b i\n[PROOFSTEP]\nrwa [update_same]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\n\u22a2 x = y \u2228 y = z \u2228 x = z\n[PROOFSTEP]\nby_contra hne\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : \u00ac(x = y \u2228 y = z \u2228 x = z)\n\u22a2 False\n[PROOFSTEP]\nsimp only [not_or, \u2190 Ne.def] at hne \n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\n\u22a2 False\n[PROOFSTEP]\ncases' hne.1.lt_or_lt with h\u2081 h\u2081\n[GOAL]\ncase inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.1.lt_or_lt with h\u2082 h\u2082\n[GOAL]\ncase inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.1.lt_or_lt with h\u2082 h\u2082\n[GOAL]\ncase inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : y < z\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.2.lt_or_lt with h\u2083 h\u2083\n[GOAL]\ncase inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : z < y\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.2.lt_or_lt with h\u2083 h\u2083\n[GOAL]\ncase inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : y < z\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.2.lt_or_lt with h\u2083 h\u2083\n[GOAL]\ncase inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : z < y\n\u22a2 False\n[PROOFSTEP]\ncases' hne.2.2.lt_or_lt with h\u2083 h\u2083\n[GOAL]\ncase inl.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : y < z\nh\u2083 : x < z\n\u22a2 False\ncase inl.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : y < z\nh\u2083 : z < x\n\u22a2 False\ncase inl.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : z < y\nh\u2083 : x < z\n\u22a2 False\ncase inl.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : x < y\nh\u2082 : z < y\nh\u2083 : z < x\n\u22a2 False\ncase inr.inl.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : y < z\nh\u2083 : x < z\n\u22a2 False\ncase inr.inl.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : y < z\nh\u2083 : z < x\n\u22a2 False\ncase inr.inr.inl\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : z < y\nh\u2083 : x < z\n\u22a2 False\ncase inr.inr.inr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ninst\u271d : LinearOrder \u03b1\nh : \u2200 \u2983x y z : \u03b1\u2984, x < y \u2192 y < z \u2192 False\nx y z : \u03b1\nhne : x \u2260 y \u2227 y \u2260 z \u2227 x \u2260 z\nh\u2081 : y < x\nh\u2082 : z < y\nh\u2083 : z < x\n\u22a2 False\n[PROOFSTEP]\nexacts [h h\u2081 h\u2082, h h\u2082 h\u2083, h h\u2083 h\u2082, h h\u2083 h\u2081, h h\u2081 h\u2083, h h\u2082 h\u2083, h h\u2081 h\u2083, h h\u2082 h\u2081]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\n\u22a2 \u2200 (a : PUnit), a \u2264 a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\na\u271d : PUnit\n\u22a2 a\u271d \u2264 a\u271d\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\n\u22a2 \u2200 (a b c : PUnit), a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\na\u271d\u00b2 b\u271d c\u271d : PUnit\na\u271d\u00b9 : a\u271d\u00b2 \u2264 b\u271d\na\u271d : b\u271d \u2264 c\u271d\n\u22a2 a\u271d\u00b2 \u2264 c\u271d\n[PROOFSTEP]\ntrivial\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\n\u22a2 \u2200 (a b : PUnit), a < b \u2194 a \u2264 b \u2227 \u00acb \u2264 a\n[PROOFSTEP]\nsimp only [not_true, and_false, forall_const]\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\n\u22a2 \u2200 (a b : PUnit), a \u2264 b \u2192 b \u2264 a \u2192 a = b\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\na\u271d\u00b2 b\u271d : PUnit\na\u271d\u00b9 : a\u271d\u00b2 \u2264 b\u271d\na\u271d : b\u271d \u2264 a\u271d\u00b2\n\u22a2 a\u271d\u00b2 = b\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\n\u22a2 \u2200 (a b : PUnit), a \u2264 b \u2228 b \u2264 a\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03c0 : \u03b9 \u2192 Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : PUnit\na\u271d b\u271d : PUnit\n\u22a2 a\u271d \u2264 b\u271d \u2228 b\u271d \u2264 a\u271d\n[PROOFSTEP]\nexact Or.inl trivial\n", "meta": {"mathlib_filename": "Mathlib.Order.Basic", "llama_tokens": 41941, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.5019364292116514}}
{"text": "[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nn : \u2115\nh : Pairwise R (a :: l)\n\u22a2 Pairwise R (List.drop (n + 1) (a :: l))\n[PROOFSTEP]\nrw [List.drop]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nn : \u2115\nh : Pairwise R (a :: l)\n\u22a2 Pairwise R (List.drop n l)\n[PROOFSTEP]\nexact Pairwise.drop (pairwise_cons.mp h).right\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : \u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b\np : Pairwise R l\n\u22a2 Pairwise S l\n[PROOFSTEP]\ninduction p with\n| nil => constructor\n| @cons a l r _ ih =>\n  constructor\n  \u00b7 exact BAll.imp_right (fun x h \u21a6 H (mem_cons_self _ _) (mem_cons_of_mem _ h)) r\n  \u00b7 exact ih fun {a b} m m' \u21a6 H (mem_cons_of_mem _ m) (mem_cons_of_mem _ m')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : \u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b\np : Pairwise R l\n\u22a2 Pairwise S l\n[PROOFSTEP]\ninduction p with\n| nil => constructor\n| @cons a l r _ ih =>\n  constructor\n  \u00b7 exact BAll.imp_right (fun x h \u21a6 H (mem_cons_self _ _) (mem_cons_of_mem _ h)) r\n  \u00b7 exact ih fun {a b} m m' \u21a6 H (mem_cons_of_mem _ m) (mem_cons_of_mem _ m')\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : \u2200 {a b : \u03b1}, a \u2208 [] \u2192 b \u2208 [] \u2192 R a b \u2192 S a b\n\u22a2 Pairwise S []\n[PROOFSTEP]\n\n| nil => constructor\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl : List \u03b1\nH : \u2200 {a b : \u03b1}, a \u2208 [] \u2192 b \u2208 [] \u2192 R a b \u2192 S a b\n\u22a2 Pairwise S []\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 : \u03b1\nl\u271d\u00b9 : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nr : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\na\u271d : Pairwise R l\nih : (\u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b) \u2192 Pairwise S l\nH : \u2200 {a_1 b : \u03b1}, a_1 \u2208 a :: l \u2192 b \u2208 a :: l \u2192 R a_1 b \u2192 S a_1 b\n\u22a2 Pairwise S (a :: l)\n[PROOFSTEP]\n\n| @cons a l r _ ih =>\n  constructor\n  \u00b7 exact BAll.imp_right (fun x h \u21a6 H (mem_cons_self _ _) (mem_cons_of_mem _ h)) r\n  \u00b7 exact ih fun {a b} m m' \u21a6 H (mem_cons_of_mem _ m) (mem_cons_of_mem _ m')\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 : \u03b1\nl\u271d\u00b9 : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nr : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\na\u271d : Pairwise R l\nih : (\u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b) \u2192 Pairwise S l\nH : \u2200 {a_1 b : \u03b1}, a_1 \u2208 a :: l \u2192 b \u2208 a :: l \u2192 R a_1 b \u2192 S a_1 b\n\u22a2 Pairwise S (a :: l)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 : \u03b1\nl\u271d\u00b9 : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nr : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\na\u271d : Pairwise R l\nih : (\u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b) \u2192 Pairwise S l\nH : \u2200 {a_1 b : \u03b1}, a_1 \u2208 a :: l \u2192 b \u2208 a :: l \u2192 R a_1 b \u2192 S a_1 b\n\u22a2 \u2200 (a' : \u03b1), a' \u2208 l \u2192 S a a'\n[PROOFSTEP]\nexact BAll.imp_right (fun x h \u21a6 H (mem_cons_self _ _) (mem_cons_of_mem _ h)) r\n[GOAL]\ncase cons.a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 : \u03b1\nl\u271d\u00b9 : List \u03b1\nS : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nr : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\na\u271d : Pairwise R l\nih : (\u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 S a b) \u2192 Pairwise S l\nH : \u2200 {a_1 b : \u03b1}, a_1 \u2208 a :: l \u2192 b \u2208 a :: l \u2192 R a_1 b \u2192 S a_1 b\n\u22a2 Pairwise S l\n[PROOFSTEP]\nexact ih fun {a b} m m' \u21a6 H (mem_cons_of_mem _ m) (mem_cons_of_mem _ m')\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nx\u271d : Pairwise R l \u2227 Pairwise S l\nhR : Pairwise R l\nhS : Pairwise S l\n\u22a2 Pairwise (fun a b => R a b \u2227 S a b) l\n[PROOFSTEP]\ninduction' hR with a l R1 R2 IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nx\u271d\u00b9 : Pairwise R l \u2227 Pairwise S l\nhS\u271d : Pairwise S l\nx\u271d : Pairwise R [] \u2227 Pairwise S []\nhS : Pairwise S []\n\u22a2 Pairwise (fun a b => R a b \u2227 S a b) []\n[PROOFSTEP]\nsimp only [Pairwise.nil, pairwise_cons] at *\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nx\u271d\u00b9 : Pairwise R l\u271d \u2227 Pairwise S l\u271d\nhS\u271d : Pairwise S l\u271d\na : \u03b1\nl : List \u03b1\nR1 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nR2 : Pairwise R l\nIH : Pairwise R l \u2227 Pairwise S l \u2192 Pairwise S l \u2192 Pairwise (fun a b => R a b \u2227 S a b) l\nx\u271d : Pairwise R (a :: l) \u2227 Pairwise S (a :: l)\nhS : Pairwise S (a :: l)\n\u22a2 Pairwise (fun a b => R a b \u2227 S a b) (a :: l)\n[PROOFSTEP]\nsimp only [Pairwise.nil, pairwise_cons] at *\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nx\u271d\u00b9 : Pairwise R l\u271d \u2227 Pairwise S l\u271d\nhS\u271d : Pairwise S l\u271d\na : \u03b1\nl : List \u03b1\nR1 : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a'\nR2 : Pairwise R l\nIH : Pairwise R l \u2227 Pairwise S l \u2192 Pairwise S l \u2192 Pairwise (fun a b => R a b \u2227 S a b) l\nx\u271d : ((\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 Pairwise R l) \u2227 (\u2200 (a' : \u03b1), a' \u2208 l \u2192 S a a') \u2227 Pairwise S l\nhS : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 S a a') \u2227 Pairwise S l\n\u22a2 (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a' \u2227 S a a') \u2227 Pairwise (fun a b => R a b \u2227 S a b) l\n[PROOFSTEP]\nexact \u27e8fun b bl => \u27e8R1 b bl, hS.1 b bl\u27e9, IH \u27e8R2, hS.2\u27e9 hS.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nH : \u2200 (x y : \u03b1), R x y\n\u22a2 Pairwise R l\n[PROOFSTEP]\ninduction l <;> [exact Pairwise.nil; simp only [*, pairwise_cons, forall\u2082_true_iff, and_true_iff]]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nH : \u2200 (x y : \u03b1), R x y\n\u22a2 Pairwise R l\n[PROOFSTEP]\ninduction l\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH : \u2200 (x y : \u03b1), R x y\n\u22a2 Pairwise R []\n[PROOFSTEP]\nexact Pairwise.nil\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH : \u2200 (x y : \u03b1), R x y\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\ntail_ih\u271d : Pairwise R tail\u271d\n\u22a2 Pairwise R (head\u271d :: tail\u271d)\n[PROOFSTEP]\nsimp only [*, pairwise_cons, forall\u2082_true_iff, and_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\n\u22a2 \u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 (R a b \u2194 a \u2208 l \u2227 b \u2208 l \u2227 R a b)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [true_and_iff, iff_self_iff, forall\u2082_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\n\u22a2 \u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 (R a b \u2194 a \u2208 l \u2192 b \u2208 l \u2192 R a b)\n[PROOFSTEP]\nsimp (config := { contextual := true }) only [forall_prop_of_true, iff_self_iff, forall\u2082_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nh\u2081 : \u2200 (x : \u03b1), x \u2208 l \u2192 R x x\nh\u2082 : Pairwise R l\nh\u2083 : Pairwise (flip R) l\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\n[PROOFSTEP]\ninduction' l with a l ih\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l \u2192 R x x\nh\u2082\u271d : Pairwise R l\nh\u2083\u271d : Pairwise (flip R) l\nh\u2081 : \u2200 (x : \u03b1), x \u2208 [] \u2192 R x x\nh\u2082 : Pairwise R []\nh\u2083 : Pairwise (flip R) []\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 [] \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 [] \u2192 R x y\n[PROOFSTEP]\nexact forall_mem_nil _\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\na : \u03b1\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nh\u2081 : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 R x x\nh\u2082 : Pairwise R (a :: l)\nh\u2083 : Pairwise (flip R) (a :: l)\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 a :: l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 a :: l \u2192 R x y\n[PROOFSTEP]\nrw [pairwise_cons] at h\u2082 h\u2083 \n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\na : \u03b1\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nh\u2081 : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 R x x\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R a a') \u2227 Pairwise (flip R) l\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x \u2208 a :: l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 a :: l \u2192 R x y\n[PROOFSTEP]\nsimp only [mem_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\na : \u03b1\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nh\u2081 : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 R x x\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R a a') \u2227 Pairwise (flip R) l\n\u22a2 \u2200 \u2983x : \u03b1\u2984, x = a \u2228 x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y = a \u2228 y \u2208 l \u2192 R x y\n[PROOFSTEP]\nrintro x (rfl | hx) y (rfl | hy)\n[GOAL]\ncase cons.inl.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\ny : \u03b1\nh\u2081 : \u2200 (x : \u03b1), x \u2208 y :: l \u2192 R x x\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R y a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R y a') \u2227 Pairwise (flip R) l\n\u22a2 R y y\n[PROOFSTEP]\nexact h\u2081 _ (l.mem_cons_self _)\n[GOAL]\ncase cons.inl.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nx : \u03b1\nh\u2081 : \u2200 (x_1 : \u03b1), x_1 \u2208 x :: l \u2192 R x_1 x_1\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R x a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R x a') \u2227 Pairwise (flip R) l\ny : \u03b1\nhy : y \u2208 l\n\u22a2 R x y\n[PROOFSTEP]\nexact h\u2082.1 _ hy\n[GOAL]\ncase cons.inr.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nx : \u03b1\nhx : x \u2208 l\ny : \u03b1\nh\u2081 : \u2200 (x : \u03b1), x \u2208 y :: l \u2192 R x x\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R y a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R y a') \u2227 Pairwise (flip R) l\n\u22a2 R x y\n[PROOFSTEP]\nexact h\u2083.1 _ hx\n[GOAL]\ncase cons.inr.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nh\u2081\u271d : \u2200 (x : \u03b1), x \u2208 l\u271d \u2192 R x x\nh\u2082\u271d : Pairwise R l\u271d\nh\u2083\u271d : Pairwise (flip R) l\u271d\na : \u03b1\nl : List \u03b1\nih : (\u2200 (x : \u03b1), x \u2208 l \u2192 R x x) \u2192 Pairwise R l \u2192 Pairwise (flip R) l \u2192 \u2200 \u2983x : \u03b1\u2984, x \u2208 l \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 l \u2192 R x y\nh\u2081 : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 R x x\nh\u2082 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 Pairwise R l\nh\u2083 : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 flip R a a') \u2227 Pairwise (flip R) l\nx : \u03b1\nhx : x \u2208 l\ny : \u03b1\nhy : y \u2208 l\n\u22a2 R x y\n[PROOFSTEP]\nexact ih (fun x hx => h\u2081 _ <| mem_cons_of_mem _ hx) h\u2082.2 h\u2083.2 hx hy\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nH : Symmetric R\nH\u2081 : \u2200 (x : \u03b1), x \u2208 l \u2192 R x x\nH\u2082 : Pairwise R l\n\u22a2 Pairwise (flip fun x => R x) l\n[PROOFSTEP]\nrwa [H.flip_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 l \u2192 \u2200 \u2983b : \u03b1\u2984, b \u2208 l \u2192 a \u2260 b \u2192 R a b\n[PROOFSTEP]\napply Pairwise.forall_of_forall\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Symmetric fun x y => x \u2260 y \u2192 R x y\n[PROOFSTEP]\nexact fun a b h hne => hR (h hne.symm)\n[GOAL]\ncase H\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2260 x \u2192 R x x\n[PROOFSTEP]\nexact fun _ _ hx => (hx rfl).elim\n[GOAL]\ncase H\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\n\u22a2 Pairwise (fun x y => x \u2260 y \u2192 R x y) l\n[PROOFSTEP]\nexact hl.imp (@fun a b h _ => by exact h)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\nhR : Symmetric R\nhl : Pairwise R l\na b : \u03b1\nh : R a b\nx\u271d : a \u2260 b\n\u22a2 R a b\n[PROOFSTEP]\nexact h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\nR : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\n\u22a2 Pairwise R [a]\n[PROOFSTEP]\nsimp [Pairwise.nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\na b : \u03b1\n\u22a2 Pairwise R [a, b] \u2194 R a b\n[PROOFSTEP]\nsimp only [pairwise_cons, mem_singleton, forall_eq, forall_prop_of_false (not_mem_nil _), forall_true_iff, Pairwise.nil,\n  and_true_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ns : Symmetric R\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Pairwise R (l\u2081 ++ l\u2082) \u2194 Pairwise R (l\u2082 ++ l\u2081)\n[PROOFSTEP]\nhave : \u2200 l\u2081 l\u2082 : List \u03b1, (\u2200 x : \u03b1, x \u2208 l\u2081 \u2192 \u2200 y : \u03b1, y \u2208 l\u2082 \u2192 R x y) \u2192 \u2200 x : \u03b1, x \u2208 l\u2082 \u2192 \u2200 y : \u03b1, y \u2208 l\u2081 \u2192 R x y :=\n  fun l\u2081 l\u2082 a x xm y ym \u21a6 s (a y ym x xm)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ns : Symmetric R\nl\u2081 l\u2082 : List \u03b1\nthis :\n  \u2200 (l\u2081 l\u2082 : List \u03b1), (\u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) \u2192 \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2081 \u2192 R x y\n\u22a2 Pairwise R (l\u2081 ++ l\u2082) \u2194 Pairwise R (l\u2082 ++ l\u2081)\n[PROOFSTEP]\nsimp only [pairwise_append, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ns : Symmetric R\nl\u2081 l\u2082 : List \u03b1\nthis :\n  \u2200 (l\u2081 l\u2082 : List \u03b1), (\u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) \u2192 \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2081 \u2192 R x y\n\u22a2 (Pairwise R l\u2081 \u2227 Pairwise R l\u2082 \u2227 \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a b) \u2194\n    Pairwise R l\u2081 \u2227 Pairwise R l\u2082 \u2227 \u2200 (a : \u03b1), a \u2208 l\u2082 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 R a b\n[PROOFSTEP]\nrw [Iff.intro (this l\u2081 l\u2082) (this l\u2082 l\u2081)]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ns : Symmetric R\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 Pairwise R (l\u2081 ++ ([a] ++ l\u2082)) \u2194 Pairwise R ([a] ++ l\u2081 ++ l\u2082)\n[PROOFSTEP]\nrw [\u2190 append_assoc, pairwise_append, @pairwise_append _ _ ([a] ++ l\u2081), pairwise_append_comm s]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ns : Symmetric R\na : \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (Pairwise R ([a] ++ l\u2081) \u2227 Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 l\u2081 ++ [a] \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b) \u2194\n    Pairwise R ([a] ++ l\u2081) \u2227 Pairwise R l\u2082 \u2227 \u2200 (a_1 : \u03b1), a_1 \u2208 [a] ++ l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 R a_1 b\n[PROOFSTEP]\nsimp only [mem_append, or_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nf : \u03b2 \u2192 \u03b1\n\u22a2 Pairwise R (map f []) \u2194 Pairwise (fun a b => R (f a) (f b)) []\n[PROOFSTEP]\nsimp only [map, Pairwise.nil]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\nl : List \u03b2\n\u22a2 Pairwise R (map f (b :: l)) \u2194 Pairwise (fun a b => R (f a) (f b)) (b :: l)\n[PROOFSTEP]\nsimp only [map, pairwise_cons, mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082, pairwise_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l\n[PROOFSTEP]\nlet _S (a a' : \u03b2) := \u2200 b \u2208 f a, \u2200 b' \u2208 f a', R b b'\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b') l\n[PROOFSTEP]\nsimp only [Option.mem_def]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\nl : List \u03b2\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n[PROOFSTEP]\ninduction' l with a l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\n\u22a2 Pairwise R (filterMap f []) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') []\n[PROOFSTEP]\nsimp only [filterMap, Pairwise.nil]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)\n[PROOFSTEP]\ncases' e : f a with b\n[GOAL]\ncase cons.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\ne : f a = none\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)\n[PROOFSTEP]\nrw [filterMap_cons_none _ _ e, propext IH, pairwise_cons]\n[GOAL]\ncase cons.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\ne : f a = none\n\u22a2 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l \u2194\n    (\u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') \u2227\n      Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n[PROOFSTEP]\nsimp only [e, forall_prop_of_false not_false, forall\u2083_true_iff, true_and_iff]\n[GOAL]\ncase cons.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise R (filterMap f (a :: l)) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)\n[PROOFSTEP]\nrw [filterMap_cons_some _ _ _ e]\n[GOAL]\ncase cons.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise R (b :: filterMap f l) \u2194\n    Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') (a :: l)\n[PROOFSTEP]\nsimp only [pairwise_cons, mem_filterMap, forall_exists_index, and_imp, IH, e, Option.some.injEq, forall_eq',\n  and_congr_left_iff]\n[GOAL]\ncase cons.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\n\u22a2 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l \u2192\n    ((\u2200 (a' : \u03b1) (x : \u03b2), x \u2208 l \u2192 f x = some a' \u2192 R b a') \u2194 \u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b')\n[PROOFSTEP]\nintro _\n[GOAL]\ncase cons.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 : \u03b1\nl\u271d : List \u03b1\nf : \u03b2 \u2192 Option \u03b1\n_S : \u03b2 \u2192 \u03b2 \u2192 Prop := fun a a' => \u2200 (b : \u03b1), b \u2208 f a \u2192 \u2200 (b' : \u03b1), b' \u2208 f a' \u2192 R b b'\na : \u03b2\nl : List \u03b2\nIH : Pairwise R (filterMap f l) \u2194 Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\nb : \u03b1\ne : f a = some b\na\u271d : Pairwise (fun a a' => \u2200 (b : \u03b1), f a = some b \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b') l\n\u22a2 (\u2200 (a' : \u03b1) (x : \u03b2), x \u2208 l \u2192 f x = some a' \u2192 R b a') \u2194 \u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (b' : \u03b1), f a' = some b' \u2192 R b b'\n[PROOFSTEP]\nexact \u27e8fun h a ha b hab => h _ _ ha hab, fun h a b ha hab => h _ ha _ hab\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\n\u22a2 Pairwise R (filter (fun b => decide (p b)) l) \u2194 Pairwise (fun x y => p x \u2192 p y \u2192 R x y) l\n[PROOFSTEP]\nrw [\u2190 filterMap_eq_filter, pairwise_filterMap]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\n\u22a2 Pairwise\n      (fun a a' =>\n        \u2200 (b : \u03b1),\n          b \u2208 Option.guard (fun x => decide (p x) = true) a \u2192\n            \u2200 (b' : \u03b1), b' \u2208 Option.guard (fun x => decide (p x) = true) a' \u2192 R b b')\n      l \u2194\n    Pairwise (fun x y => p x \u2192 p y \u2192 R x y) l\n[PROOFSTEP]\napply Pairwise.iff\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\n\u22a2 \u2200 (a b : \u03b1),\n    (\u2200 (b_1 : \u03b1),\n        b_1 \u2208 Option.guard (fun x => decide (p x) = true) a \u2192\n          \u2200 (b' : \u03b1), b' \u2208 Option.guard (fun x => decide (p x) = true) b \u2192 R b_1 b') \u2194\n      p a \u2192 p b \u2192 R a b\n[PROOFSTEP]\nintros\n[GOAL]\ncase H\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nl : List \u03b1\na\u271d b\u271d : \u03b1\n\u22a2 (\u2200 (b : \u03b1),\n      b \u2208 Option.guard (fun x => decide (p x) = true) a\u271d \u2192\n        \u2200 (b' : \u03b1), b' \u2208 Option.guard (fun x => decide (p x) = true) b\u271d \u2192 R b b') \u2194\n    p a\u271d \u2192 p b\u271d \u2192 R a\u271d b\u271d\n[PROOFSTEP]\nsimp only [decide_eq_true_eq, Option.mem_def, Option.guard_eq_some, and_imp, forall_eq']\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl : List \u03b2\nh : \u2200 (x : \u03b2), x \u2208 l \u2192 p x\n\u22a2 Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\n[PROOFSTEP]\ninduction' l with a l ihl\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l \u2192 p x\nh : \u2200 (x : \u03b2), x \u2208 [] \u2192 p x\n\u22a2 Pairwise R (pmap f [] h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\n\u22a2 Pairwise R (pmap f (a :: l) h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) (a :: l)\n[PROOFSTEP]\nobtain \u27e8_, hl\u27e9 : p a \u2227 \u2200 b, b \u2208 l \u2192 p b := by simpa using h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\n\u22a2 p a \u2227 \u2200 (b : \u03b2), b \u2208 l \u2192 p b\n[PROOFSTEP]\nsimpa using h\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\nleft\u271d : p a\nhl : \u2200 (b : \u03b2), b \u2208 l \u2192 p b\n\u22a2 Pairwise R (pmap f (a :: l) h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) (a :: l)\n[PROOFSTEP]\nsimp only [ihl hl, pairwise_cons, bex_imp, pmap, and_congr_left_iff, mem_pmap]\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\nleft\u271d : p a\nhl : \u2200 (b : \u03b2), b \u2208 l \u2192 p b\n\u22a2 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l \u2192\n    ((\u2200 (a' : \u03b1) (x : \u03b2) (h_1 : x \u2208 l), f x (_ : p x) = a' \u2192 R (f a (_ : p a)) a') \u2194\n      \u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : p a'), R (f a h\u2081) (f a' h\u2082))\n[PROOFSTEP]\nrefine' fun _ => \u27e8fun H b hb _ hpb => H _ _ hb rfl, _\u27e9\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\nleft\u271d : p a\nhl : \u2200 (b : \u03b2), b \u2208 l \u2192 p b\nx\u271d : Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\n\u22a2 (\u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : p a'), R (f a h\u2081) (f a' h\u2082)) \u2192\n    \u2200 (a' : \u03b1) (x : \u03b2) (h_1 : x \u2208 l), f x (_ : p x) = a' \u2192 R (f a (_ : p a)) a'\n[PROOFSTEP]\nrintro H _ b hb rfl\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d\u00b9 : List \u03b1\np : \u03b2 \u2192 Prop\nf : (b : \u03b2) \u2192 p b \u2192 \u03b1\nl\u271d : List \u03b2\nh\u271d : \u2200 (x : \u03b2), x \u2208 l\u271d \u2192 p x\na : \u03b2\nl : List \u03b2\nihl :\n  \u2200 (h : \u2200 (x : \u03b2), x \u2208 l \u2192 p x),\n    Pairwise R (pmap f l h) \u2194 Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nh : \u2200 (x : \u03b2), x \u2208 a :: l \u2192 p x\nleft\u271d : p a\nhl : \u2200 (b : \u03b2), b \u2208 l \u2192 p b\nx\u271d : Pairwise (fun b\u2081 b\u2082 => \u2200 (h\u2081 : p b\u2081) (h\u2082 : p b\u2082), R (f b\u2081 h\u2081) (f b\u2082 h\u2082)) l\nH : \u2200 (a' : \u03b2), a' \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : p a'), R (f a h\u2081) (f a' h\u2082)\nb : \u03b2\nhb : b \u2208 l\n\u22a2 R (f a (_ : p a)) (f b (_ : p b))\n[PROOFSTEP]\nexact H b hb _ _\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nhl : Pairwise R l\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 l \u2192 p x\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\nhS : \u2200 \u2983x : \u03b1\u2984 (hx : p x) \u2983y : \u03b1\u2984 (hy : p y), R x y \u2192 S (f x hx) (f y hy)\n\u22a2 Pairwise S (List.pmap f l h)\n[PROOFSTEP]\nrefine' (pairwise_pmap h).2 (Pairwise.imp_of_mem _ hl)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nhl : Pairwise R l\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 l \u2192 p x\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\nhS : \u2200 \u2983x : \u03b1\u2984 (hx : p x) \u2983y : \u03b1\u2984 (hy : p y), R x y \u2192 S (f x hx) (f y hy)\n\u22a2 \u2200 {a b : \u03b1}, a \u2208 l \u2192 b \u2208 l \u2192 R a b \u2192 \u2200 (h\u2081 : p a) (h\u2082 : p b), S (f a h\u2081) (f b h\u2082)\n[PROOFSTEP]\nintros\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nhl : Pairwise R l\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 l \u2192 p x\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\nhS : \u2200 \u2983x : \u03b1\u2984 (hx : p x) \u2983y : \u03b1\u2984 (hy : p y), R x y \u2192 S (f x hx) (f y hy)\na\u271d\u00b3 b\u271d : \u03b1\na\u271d\u00b2 : a\u271d\u00b3 \u2208 l\na\u271d\u00b9 : b\u271d \u2208 l\na\u271d : R a\u271d\u00b3 b\u271d\nh\u2081\u271d : p a\u271d\u00b3\nh\u2082\u271d : p b\u271d\n\u22a2 S (f a\u271d\u00b3 h\u2081\u271d) (f b\u271d h\u2082\u271d)\n[PROOFSTEP]\napply hS\n[GOAL]\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S\u271d T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nhl : Pairwise R l\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nh : \u2200 (x : \u03b1), x \u2208 l \u2192 p x\nS : \u03b2 \u2192 \u03b2 \u2192 Prop\nhS : \u2200 \u2983x : \u03b1\u2984 (hx : p x) \u2983y : \u03b1\u2984 (hy : p y), R x y \u2192 S (f x hx) (f y hy)\na\u271d\u00b3 b\u271d : \u03b1\na\u271d\u00b2 : a\u271d\u00b3 \u2208 l\na\u271d\u00b9 : b\u271d \u2208 l\na\u271d : R a\u271d\u00b3 b\u271d\nh\u2081\u271d : p a\u271d\u00b3\nh\u2082\u271d : p b\u271d\n\u22a2 R a\u271d\u00b3 b\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\nL : List (List \u03b1)\n\u22a2 Pairwise R (join L) \u2194\n    (\u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L\n[PROOFSTEP]\ninduction' L with l L IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\n\u22a2 Pairwise R (join []) \u2194\n    (\u2200 (l : List \u03b1), l \u2208 [] \u2192 Pairwise R l) \u2227 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) []\n[PROOFSTEP]\nsimp only [join, Pairwise.nil, forall_prop_of_false (not_mem_nil _), forall_const, and_self_iff]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nL : List (List \u03b1)\nIH :\n  Pairwise R (join L) \u2194\n    (\u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L\n\u22a2 Pairwise R (join (l :: L)) \u2194\n    (\u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 Pairwise R l_1) \u2227\n      Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) (l :: L)\n[PROOFSTEP]\nhave :\n  (\u2200 x : \u03b1, x \u2208 l \u2192 \u2200 (y : \u03b1) (x_1 : List \u03b1), x_1 \u2208 L \u2192 y \u2208 x_1 \u2192 R x y) \u2194\n    \u2200 a' : List \u03b1, a' \u2208 L \u2192 \u2200 x : \u03b1, x \u2208 l \u2192 \u2200 y : \u03b1, y \u2208 a' \u2192 R x y :=\n  \u27e8fun h a b c d e => h c d e a b, fun h c d e a b => h a b c d e\u27e9\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nL : List (List \u03b1)\nIH :\n  Pairwise R (join L) \u2194\n    (\u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L\nthis :\n  (\u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1) (x_1 : List \u03b1), x_1 \u2208 L \u2192 y \u2208 x_1 \u2192 R x y) \u2194\n    \u2200 (a' : List \u03b1), a' \u2208 L \u2192 \u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1), y \u2208 a' \u2192 R x y\n\u22a2 Pairwise R (join (l :: L)) \u2194\n    (\u2200 (l_1 : List \u03b1), l_1 \u2208 l :: L \u2192 Pairwise R l_1) \u2227\n      Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) (l :: L)\n[PROOFSTEP]\nsimp only [join, pairwise_append, IH, mem_join, exists_imp, and_imp, this, forall_mem_cons, pairwise_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nL : List (List \u03b1)\nIH :\n  Pairwise R (join L) \u2194\n    (\u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227 Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L\nthis :\n  (\u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1) (x_1 : List \u03b1), x_1 \u2208 L \u2192 y \u2208 x_1 \u2192 R x y) \u2194\n    \u2200 (a' : List \u03b1), a' \u2208 L \u2192 \u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1), y \u2208 a' \u2192 R x y\n\u22a2 (Pairwise R l \u2227\n      ((\u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227\n          Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L) \u2227\n        \u2200 (a' : List \u03b1), a' \u2208 L \u2192 \u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1), y \u2208 a' \u2192 R x y) \u2194\n    (Pairwise R l \u2227 \u2200 (l : List \u03b1), l \u2208 L \u2192 Pairwise R l) \u2227\n      (\u2200 (a' : List \u03b1), a' \u2208 L \u2192 \u2200 (x : \u03b1), x \u2208 l \u2192 \u2200 (y : \u03b1), y \u2208 a' \u2192 R x y) \u2227\n        Pairwise (fun l\u2081 l\u2082 => \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 \u2200 (y : \u03b1), y \u2208 l\u2082 \u2192 R x y) L\n[PROOFSTEP]\nsimp only [and_assoc, and_comm, and_left_comm]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR\u271d S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\nR : \u03b2 \u2192 \u03b2 \u2192 Prop\nl : List \u03b1\nf : \u03b1 \u2192 List \u03b2\n\u22a2 Pairwise R (List.bind l f) \u2194\n    (\u2200 (a : \u03b1), a \u2208 l \u2192 Pairwise R (f a)) \u2227 Pairwise (fun a\u2081 a\u2082 => \u2200 (x : \u03b2), x \u2208 f a\u2081 \u2192 \u2200 (y : \u03b2), y \u2208 f a\u2082 \u2192 R x y) l\n[PROOFSTEP]\nsimp [List.bind, List.pairwise_join, List.mem_map, List.pairwise_map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r l\n[PROOFSTEP]\ninduction' l with hd tl IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhr : \u2200 (a : \u03b1), 1 < count a [] \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 [] \u2192 \u2200 (b : \u03b1), b \u2208 [] \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r []\n[PROOFSTEP]\nsimp\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r (hd :: tl)\n[PROOFSTEP]\nrw [List.pairwise_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 (\u2200 (a' : \u03b1), a' \u2208 tl \u2192 r hd a') \u2227 Pairwise r tl\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase cons.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 \u2200 (a' : \u03b1), a' \u2208 tl \u2192 r hd a'\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase cons.left\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\n\u22a2 r hd x\n[PROOFSTEP]\nby_cases H : hd = x\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\nH : hd = x\n\u22a2 r hd x\n[PROOFSTEP]\nrw [H]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\nH : hd = x\n\u22a2 r x x\n[PROOFSTEP]\nrefine' hr _ _\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\nH : hd = x\n\u22a2 1 < count x (hd :: tl)\n[PROOFSTEP]\nsimpa [count_cons, H, Nat.succ_lt_succ_iff, count_pos] using hx\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\nH : \u00achd = x\n\u22a2 r hd x\n[PROOFSTEP]\nexact h hd (mem_cons_self _ _) x (mem_cons_of_mem _ hx) H\n[GOAL]\ncase cons.right\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r tl\n[PROOFSTEP]\nrefine' IH _ _\n[GOAL]\ncase cons.right.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 \u2200 (a : \u03b1), 1 < count a tl \u2192 r a a\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase cons.right.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : 1 < count x tl\n\u22a2 r x x\n[PROOFSTEP]\nrefine' hr _ _\n[GOAL]\ncase cons.right.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : 1 < count x tl\n\u22a2 1 < count x (hd :: tl)\n[PROOFSTEP]\nrw [count_cons]\n[GOAL]\ncase cons.right.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : 1 < count x tl\n\u22a2 1 < if x = hd then succ (count x tl) else count x tl\n[PROOFSTEP]\nsplit_ifs\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d\u00b9 : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : 1 < count x tl\nh\u271d : x = hd\n\u22a2 1 < succ (count x tl)\n[PROOFSTEP]\nexact hx.trans (Nat.lt_succ_self _)\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d\u00b9 : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : 1 < count x tl\nh\u271d : \u00acx = hd\n\u22a2 1 < count x tl\n[PROOFSTEP]\nexact hx\n[GOAL]\ncase cons.right.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\n\u22a2 \u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b\n[PROOFSTEP]\nintro x hx y hy\n[GOAL]\ncase cons.right.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableEq \u03b1\nl : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr\u271d : \u2200 (a : \u03b1), 1 < count a l \u2192 r a a\nh\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\nhd : \u03b1\ntl : List \u03b1\nIH : (\u2200 (a : \u03b1), 1 < count a tl \u2192 r a a) \u2192 (\u2200 (a : \u03b1), a \u2208 tl \u2192 \u2200 (b : \u03b1), b \u2208 tl \u2192 a \u2260 b \u2192 r a b) \u2192 Pairwise r tl\nhr : \u2200 (a : \u03b1), 1 < count a (hd :: tl) \u2192 r a a\nh : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 \u2200 (b : \u03b1), b \u2208 hd :: tl \u2192 a \u2260 b \u2192 r a b\nx : \u03b1\nhx : x \u2208 tl\ny : \u03b1\nhy : y \u2208 tl\n\u22a2 x \u2260 y \u2192 r x y\n[PROOFSTEP]\nexact h x (mem_cons_of_mem _ hx) y (mem_cons_of_mem _ hy)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 r a b\n\u22a2 Pairwise r l\n[PROOFSTEP]\nclassical\nrefine' pairwise_of_reflexive_on_dupl_of_forall_ne (fun a ha' => _) fun a ha b hb _ => h a ha b hb\nhave ha := List.one_le_count_iff_mem.1 ha'.le\nexact h a ha a ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 r a b\n\u22a2 Pairwise r l\n[PROOFSTEP]\nrefine' pairwise_of_reflexive_on_dupl_of_forall_ne (fun a ha' => _) fun a ha b hb _ => h a ha b hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 r a b\na : \u03b1\nha' : 1 < count a l\n\u22a2 r a a\n[PROOFSTEP]\nhave ha := List.one_le_count_iff_mem.1 ha'.le\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 r a b\na : \u03b1\nha' : 1 < count a l\nha : a \u2208 l\n\u22a2 r a a\n[PROOFSTEP]\nexact h a ha a ha\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : Reflexive r\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r l\n[PROOFSTEP]\nclassical exact pairwise_of_reflexive_on_dupl_of_forall_ne (fun _ _ => hr _) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d l : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nhr : Reflexive r\nh : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 a \u2260 b \u2192 r a b\n\u22a2 Pairwise r l\n[PROOFSTEP]\nexact pairwise_of_reflexive_on_dupl_of_forall_ne (fun _ _ => hr _) h\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\n\u22a2 Pairwise R [] \u2194 \u2200 (i j : Fin (length [])), i < j \u2192 R (get [] i) (get [] j)\n[PROOFSTEP]\nsimp only [Pairwise.nil, true_iff_iff]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\n\u22a2 \u2200 (i j : Fin (length [])), i < j \u2192 R (get [] i) (get [] j)\n[PROOFSTEP]\nexact fun i j _h => (Nat.not_lt_zero j).elim j.2\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 Pairwise R (a :: l) \u2194 \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\n[PROOFSTEP]\nrw [pairwise_cons, pairwise_iff_get]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 ((\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)) \u2194\n    \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\n[PROOFSTEP]\nrefine' \u27e8fun H i j hij => _, fun H => \u27e8fun a' m => _, fun i j hij => _\u27e9\u27e9\n[GOAL]\ncase refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\ni j : Fin (length (a :: l))\nhij : i < j\n\u22a2 R (get (a :: l) i) (get (a :: l) j)\n[PROOFSTEP]\ncases' j with j hj\n[GOAL]\ncase refine'_1.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : \u2115\nhj : j < length (a :: l)\nhij : i < { val := j, isLt := hj }\n\u22a2 R (get (a :: l) i) (get (a :: l) { val := j, isLt := hj })\n[PROOFSTEP]\ncases' j with j\n[GOAL]\ncase refine'_1.mk.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\ni : Fin (length (a :: l))\nhj : zero < length (a :: l)\nhij : i < { val := zero, isLt := hj }\n\u22a2 R (get (a :: l) i) (get (a :: l) { val := zero, isLt := hj })\n[PROOFSTEP]\nexact (Nat.not_lt_zero _).elim hij\n[GOAL]\ncase refine'_1.mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\ni : Fin (length (a :: l))\nj : \u2115\nhj : succ j < length (a :: l)\nhij : i < { val := succ j, isLt := hj }\n\u22a2 R (get (a :: l) i) (get (a :: l) { val := succ j, isLt := hj })\n[PROOFSTEP]\ncases' i with i hi\n[GOAL]\ncase refine'_1.mk.succ.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\nj : \u2115\nhj : succ j < length (a :: l)\ni : \u2115\nhi : i < length (a :: l)\nhij : { val := i, isLt := hi } < { val := succ j, isLt := hj }\n\u22a2 R (get (a :: l) { val := i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase refine'_1.mk.succ.mk.zero\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\nj : \u2115\nhj : succ j < length (a :: l)\nhi : zero < length (a :: l)\nhij : { val := zero, isLt := hi } < { val := succ j, isLt := hj }\n\u22a2 R (get (a :: l) { val := zero, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })\n[PROOFSTEP]\nexact H.1 _ (get_mem l _ _)\n[GOAL]\ncase refine'_1.mk.succ.mk.succ\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 R a a') \u2227 \u2200 (i j : Fin (length l)), i < j \u2192 R (get l i) (get l j)\nj : \u2115\nhj : succ j < length (a :: l)\ni : \u2115\nhi : succ i < length (a :: l)\nhij : { val := succ i, isLt := hi } < { val := succ j, isLt := hj }\n\u22a2 R (get (a :: l) { val := succ i, isLt := hi }) (get (a :: l) { val := succ j, isLt := hj })\n[PROOFSTEP]\nexact H.2 _ _ (Nat.lt_of_succ_lt_succ hij)\n[GOAL]\ncase refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\na' : \u03b1\nm : a' \u2208 l\n\u22a2 R a a'\n[PROOFSTEP]\nrcases get_of_mem m with \u27e8n, h, rfl\u27e9\n[GOAL]\ncase refine'_2.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n \u2208 l\n\u22a2 R a (get l n)\n[PROOFSTEP]\nhave := H \u27e80, show 0 < (a :: l).length from Nat.succ_pos _\u27e9 \u27e8n.succ, Nat.succ_lt_succ n.2\u27e9 (Nat.succ_pos n)\n[GOAL]\ncase refine'_2.intro.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\nn : Fin (length l)\nm : get l n \u2208 l\nthis :\n  R (get (a :: l) { val := 0, isLt := (_ : 0 < length (a :: l)) })\n    (get (a :: l) { val := \u2191(Fin.succ n), isLt := (_ : succ \u2191n < succ (length l)) })\n\u22a2 R a (get l n)\n[PROOFSTEP]\nsimpa\n[GOAL]\ncase refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\na : \u03b1\nl : List \u03b1\nH : \u2200 (i j : Fin (length (a :: l))), i < j \u2192 R (get (a :: l) i) (get (a :: l) j)\ni j : Fin (length l)\nhij : i < j\n\u22a2 R (get l i) (get l j)\n[PROOFSTEP]\nsimpa using H i.succ j.succ (show i.1.succ < j.1.succ from Nat.succ_lt_succ hij)\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\na : \u03b1\u271d\nl : List \u03b1\u271d\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx : \u03b1\nhx : r x x\n\u22a2 Pairwise r (replicate 0 x)\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop\na : \u03b1\u271d\nl : List \u03b1\u271d\n\u03b1 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nx : \u03b1\nhx : r x x\nn : \u2115\n\u22a2 Pairwise r (replicate (n + 1) x)\n[PROOFSTEP]\nsimp only [replicate, add_eq, add_zero, pairwise_cons, mem_replicate, ne_eq, and_imp, forall_eq_apply_imp_iff', hx,\n  implies_true, pairwise_replicate hx n, and_self]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\n\u22a2 pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))\n[PROOFSTEP]\nhave h' : \u2200 b : \u03b2, b \u2208 pwFilter (fun x y : \u03b2 => R (f x) (f y)) xs \u2192 R (f x) (f b) := fun b hb =>\n  h _ (by rw [pwFilter_map f xs]; apply mem_map_of_mem _ hb)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nb : \u03b2\nhb : b \u2208 pwFilter (fun x y => R (f x) (f y)) xs\n\u22a2 f b \u2208 pwFilter R (map f xs)\n[PROOFSTEP]\nrw [pwFilter_map f xs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nb : \u03b2\nhb : b \u2208 pwFilter (fun x y => R (f x) (f y)) xs\n\u22a2 f b \u2208 map f (pwFilter (fun x y => R (f x) (f y)) xs)\n[PROOFSTEP]\napply mem_map_of_mem _ hb\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nh' : \u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\n\u22a2 pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))\n[PROOFSTEP]\nrw [map, pwFilter_cons_of_pos h, pwFilter_cons_of_pos h', pwFilter_map f xs, map]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\n\u22a2 pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))\n[PROOFSTEP]\nhave h' : \u00ac\u2200 b : \u03b2, b \u2208 pwFilter (fun x y : \u03b2 => R (f x) (f y)) xs \u2192 R (f x) (f b) := fun hh =>\n  h fun a ha => by\n    rw [pwFilter_map f xs, mem_map] at ha \n    rcases ha with \u27e8b, hb\u2080, hb\u2081\u27e9\n    subst a\n    exact hh _ hb\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nhh : \u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\na : \u03b1\nha : a \u2208 pwFilter R (map f xs)\n\u22a2 R (f x) a\n[PROOFSTEP]\nrw [pwFilter_map f xs, mem_map] at ha \n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nhh : \u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\na : \u03b1\nha : \u2203 a_1, a_1 \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2227 f a_1 = a\n\u22a2 R (f x) a\n[PROOFSTEP]\nrcases ha with \u27e8b, hb\u2080, hb\u2081\u27e9\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nhh : \u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\na : \u03b1\nb : \u03b2\nhb\u2080 : b \u2208 pwFilter (fun x y => R (f x) (f y)) xs\nhb\u2081 : f b = a\n\u22a2 R (f x) a\n[PROOFSTEP]\nsubst a\n[GOAL]\ncase intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nhh : \u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\nb : \u03b2\nhb\u2080 : b \u2208 pwFilter (fun x y => R (f x) (f y)) xs\n\u22a2 R (f x) (f b)\n[PROOFSTEP]\nexact hh _ hb\u2080\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nf : \u03b2 \u2192 \u03b1\nx : \u03b2\nxs : List \u03b2\nh : \u00ac\u2200 (b : \u03b1), b \u2208 pwFilter R (map f xs) \u2192 R (f x) b\nh' : \u00ac\u2200 (b : \u03b2), b \u2208 pwFilter (fun x y => R (f x) (f y)) xs \u2192 R (f x) (f b)\n\u22a2 pwFilter R (map f (x :: xs)) = map f (pwFilter (fun x y => R (f x) (f y)) (x :: xs))\n[PROOFSTEP]\nrw [map, pwFilter_cons_of_neg h, pwFilter_cons_of_neg h', pwFilter_map f xs]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\n\u22a2 pwFilter R (x :: l) <+ x :: l\n[PROOFSTEP]\nby_cases h : \u2200 y \u2208 pwFilter R l, R x y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 pwFilter R (x :: l) <+ x :: l\n[PROOFSTEP]\nrw [pwFilter_cons_of_pos h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 x :: pwFilter R l <+ x :: l\n[PROOFSTEP]\nexact (pwFilter_sublist l).cons_cons _\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 pwFilter R (x :: l) <+ x :: l\n[PROOFSTEP]\nrw [pwFilter_cons_of_neg h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 pwFilter R l <+ x :: l\n[PROOFSTEP]\nexact sublist_cons_of_sublist _ (pwFilter_sublist l)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\n\u22a2 Pairwise R (pwFilter R (x :: l))\n[PROOFSTEP]\nby_cases h : \u2200 y \u2208 pwFilter R l, R x y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 Pairwise R (pwFilter R (x :: l))\n[PROOFSTEP]\nrw [pwFilter_cons_of_pos h]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 Pairwise R (x :: pwFilter R l)\n[PROOFSTEP]\nexact pairwise_cons.2 \u27e8h, pairwise_pwFilter l\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 Pairwise R (pwFilter R (x :: l))\n[PROOFSTEP]\nrw [pwFilter_cons_of_neg h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nx : \u03b1\nl : List \u03b1\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 Pairwise R (pwFilter R l)\n[PROOFSTEP]\nexact pairwise_pwFilter l\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nl : List \u03b1\np : Pairwise R l\n\u22a2 pwFilter R l = l\n[PROOFSTEP]\ninduction' l with x l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nl : List \u03b1\np\u271d : Pairwise R l\np : Pairwise R []\n\u22a2 pwFilter R [] = []\n[PROOFSTEP]\nrfl\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableRel R\nl\u271d : List \u03b1\np\u271d : Pairwise R l\u271d\nx : \u03b1\nl : List \u03b1\nIH : Pairwise R l \u2192 pwFilter R l = l\np : Pairwise R (x :: l)\n\u22a2 pwFilter R (x :: l) = x :: l\n[PROOFSTEP]\ncases' pairwise_cons.1 p with al p\n[GOAL]\ncase cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u271d\u00b9 : List \u03b1\ninst\u271d : DecidableRel R\nl\u271d : List \u03b1\np\u271d\u00b9 : Pairwise R l\u271d\nx : \u03b1\nl : List \u03b1\nIH : Pairwise R l \u2192 pwFilter R l = l\np\u271d : Pairwise R (x :: l)\nal : \u2200 (a' : \u03b1), a' \u2208 l \u2192 R x a'\np : Pairwise R l\n\u22a2 pwFilter R (x :: l) = x :: l\n[PROOFSTEP]\nrw [pwFilter_cons_of_pos (BAll.imp_left (pwFilter_subset l) al), IH p]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\nl : List \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n[PROOFSTEP]\ninduction' l with x l IH\n[GOAL]\ncase nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na : \u03b1\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R [] \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 [] \u2192 R a b\n[PROOFSTEP]\nexact fun _ _ h => (not_mem_nil _ h).elim\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 x :: l \u2192 R a b\n[PROOFSTEP]\nsimp only [forall_mem_cons]\n[GOAL]\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x\n[PROOFSTEP]\nby_cases h : \u2200 y \u2208 pwFilter R l, R x y\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x\n[PROOFSTEP]\nsimp only [pwFilter_cons_of_pos h, forall_mem_cons, and_imp]\n[GOAL]\ncase pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 R a x \u2192 (\u2200 (x : \u03b1), x \u2208 pwFilter R l \u2192 R a x) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x\n[PROOFSTEP]\nexact fun r H => \u27e8r, IH H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R (x :: l) \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x\n[PROOFSTEP]\nrw [pwFilter_cons_of_neg h]\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\n\u22a2 (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 R a x \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 R a x\n[PROOFSTEP]\nrefine' fun H => \u27e8_, IH H\u27e9\n[GOAL]\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\n\u22a2 R a x\n[PROOFSTEP]\ncases' e : find? (fun y => \u00acR x y) (pwFilter R l) with k\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = none\n\u22a2 R a x\n[PROOFSTEP]\nrefine' h.elim (BAll.imp_right _ (find?_eq_none.1 e))\n[GOAL]\ncase neg.none\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = none\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 pwFilter R l \u2192 \u00ac(decide \u00acR x x_1) = true \u2192 R x x_1\n[PROOFSTEP]\nexact fun y _ => by simp\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = none\ny : \u03b1\nx\u271d : y \u2208 pwFilter R l\n\u22a2 \u00ac(decide \u00acR x y) = true \u2192 R x y\n[PROOFSTEP]\nsimp\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\n\u22a2 R a x\n[PROOFSTEP]\nhave := find?_some e\n[GOAL]\ncase neg.some\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 R a x\n[PROOFSTEP]\nexact (neg_trans (H k (find?_mem e))).resolve_right (by simpa)\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nR S T : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d : \u03b1\nl\u271d : List \u03b1\ninst\u271d : DecidableRel R\nneg_trans : \u2200 {x y z : \u03b1}, R x z \u2192 R x y \u2228 R y z\na x : \u03b1\nl : List \u03b1\nIH : (\u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b) \u2192 \u2200 (b : \u03b1), b \u2208 l \u2192 R a b\nh : \u00ac\u2200 (y : \u03b1), y \u2208 pwFilter R l \u2192 R x y\nH : \u2200 (b : \u03b1), b \u2208 pwFilter R l \u2192 R a b\nk : \u03b1\ne : find? (fun y => decide \u00acR x y) (pwFilter R l) = some k\nthis : (decide \u00acR x k) = true\n\u22a2 \u00acR x k\n[PROOFSTEP]\nsimpa\n", "meta": {"mathlib_filename": "Mathlib.Data.List.Pairwise", "llama_tokens": 33397, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.6893056040203135, "lm_q1q2_score": 0.5017975565200936}}
{"text": "[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\na b c d : G\nhab : Setoid.r a b\nhcd : Setoid.r c d\n\u22a2 Setoid.r (a * c) (b * d)\n[PROOFSTEP]\nrw [leftRel_eq] at hab hcd \u22a2\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\na b c d : G\nhab : a\u207b\u00b9 * b \u2208 N\nhcd : c\u207b\u00b9 * d \u2208 N\n\u22a2 (fun x y => x\u207b\u00b9 * y \u2208 N) (a * c) (b * d)\n[PROOFSTEP]\ndsimp only\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\na b c d : G\nhab : a\u207b\u00b9 * b \u2208 N\nhcd : c\u207b\u00b9 * d \u2208 N\n\u22a2 (a * c)\u207b\u00b9 * (b * d) \u2208 N\n[PROOFSTEP]\ncalc\n  (a * c)\u207b\u00b9 * (b * d) = c\u207b\u00b9 * (a\u207b\u00b9 * b) * c\u207b\u00b9\u207b\u00b9 * (c\u207b\u00b9 * d) := by\n    simp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left]\n  _ \u2208 N := N.mul_mem (nN.conj_mem _ hab _) hcd\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\na b c d : G\nhab : a\u207b\u00b9 * b \u2208 N\nhcd : c\u207b\u00b9 * d \u2208 N\n\u22a2 (a * c)\u207b\u00b9 * (b * d) = c\u207b\u00b9 * (a\u207b\u00b9 * b) * c\u207b\u00b9\u207b\u00b9 * (c\u207b\u00b9 * d)\n[PROOFSTEP]\nsimp only [mul_inv_rev, mul_assoc, inv_mul_cancel_left]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\nx y : G\n\u22a2 x\u207b\u00b9 * y \u2208 N \u2194 \u2203 z, z \u2208 N \u2227 x * z = y\n[PROOFSTEP]\nsimp only [\u2190 _root_.eq_inv_mul_iff_mul_eq, exists_prop, exists_eq_right]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN\u271d : Subgroup G\nnN\u271d : Subgroup.Normal N\u271d\nH : Type v\ninst\u271d : Group H\nN : Subgroup G\nnN : Subgroup.Normal N\nx : G\n\u22a2 \u2191x = 1 \u2194 x \u2208 N\n[PROOFSTEP]\nrefine' QuotientGroup.eq.trans _\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN\u271d : Subgroup G\nnN\u271d : Subgroup.Normal N\u271d\nH : Type v\ninst\u271d : Group H\nN : Subgroup G\nnN : Subgroup.Normal N\nx : G\n\u22a2 x\u207b\u00b9 * 1 \u2208 N \u2194 x \u2208 N\n[PROOFSTEP]\nrw [mul_one, Subgroup.inv_mem_iff]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nI : Type w\ninst\u271d\u00b9 : Group I\nf : G \u2192* H\ninst\u271d : Subgroup.Normal (MonoidHom.range f)\ng : H \u2192* I\nh : MonoidHom.comp (mk' (MonoidHom.range f)) (Subgroup.subtype (MonoidHom.ker g)) = 1\nx : H\nhx : x \u2208 MonoidHom.ker g\n\u22a2 \u2191x = 1\n[PROOFSTEP]\nexact FunLike.congr_fun h \u27e8x, hx\u27e9\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN\u271d : Subgroup G\nnN\u271d : Subgroup.Normal N\u271d\nH : Type v\ninst\u271d : Group H\nN : Subgroup G\nnN : Subgroup.Normal N\nx y : G\n\u22a2 \u2191x = \u2191y \u2194 x / y \u2208 N\n[PROOFSTEP]\nrefine' eq_comm.trans (QuotientGroup.eq.trans _)\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN\u271d : Subgroup G\nnN\u271d : Subgroup.Normal N\u271d\nH : Type v\ninst\u271d : Group H\nN : Subgroup G\nnN : Subgroup.Normal N\nx y : G\n\u22a2 y\u207b\u00b9 * x \u2208 N \u2194 x / y \u2208 N\n[PROOFSTEP]\nrw [nN.mem_comm_iff, div_eq_mul_inv]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nHN : \u2200 (x : G), x \u2208 N \u2192 \u2191\u03c6 x = 1\nx y : G\nh : \u2191(QuotientGroup.con N) x y\n\u22a2 \u2191(Con.ker \u03c6) x y\n[PROOFSTEP]\nsimp only [QuotientGroup.con, leftRel_apply, Con.rel_mk] at h \n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nHN : \u2200 (x : G), x \u2208 N \u2192 \u2191\u03c6 x = 1\nx y : G\nh : x\u207b\u00b9 * y \u2208 N\n\u22a2 \u2191(Con.ker \u03c6) x y\n[PROOFSTEP]\nrw [Con.ker_rel]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nHN : \u2200 (x : G), x \u2208 N \u2192 \u2191\u03c6 x = 1\nx y : G\nh : x\u207b\u00b9 * y \u2208 N\n\u22a2 \u2191\u03c6 x = \u2191\u03c6 y\n[PROOFSTEP]\ncalc\n  \u03c6 x = \u03c6 (y * (x\u207b\u00b9 * y)\u207b\u00b9) := by rw [mul_inv_rev, inv_inv, mul_inv_cancel_left]\n  _ = \u03c6 y := by rw [\u03c6.map_mul, HN _ (N.inv_mem h), mul_one]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nHN : \u2200 (x : G), x \u2208 N \u2192 \u2191\u03c6 x = 1\nx y : G\nh : x\u207b\u00b9 * y \u2208 N\n\u22a2 \u2191\u03c6 x = \u2191\u03c6 (y * (x\u207b\u00b9 * y)\u207b\u00b9)\n[PROOFSTEP]\nrw [mul_inv_rev, inv_inv, mul_inv_cancel_left]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nHN : \u2200 (x : G), x \u2208 N \u2192 \u2191\u03c6 x = 1\nx y : G\nh : x\u207b\u00b9 * y \u2208 N\n\u22a2 \u2191\u03c6 (y * (x\u207b\u00b9 * y)\u207b\u00b9) = \u2191\u03c6 y\n[PROOFSTEP]\nrw [\u03c6.map_mul, HN _ (N.inv_mem h), mul_one]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b9 : Group H\nM : Subgroup H\ninst\u271d : Subgroup.Normal M\nf : G \u2192* H\nh : N \u2264 Subgroup.comap f M\n\u22a2 G \u29f8 N \u2192* H \u29f8 M\n[PROOFSTEP]\nrefine' QuotientGroup.lift N ((mk' M).comp f) _\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b9 : Group H\nM : Subgroup H\ninst\u271d : Subgroup.Normal M\nf : G \u2192* H\nh : N \u2264 Subgroup.comap f M\n\u22a2 \u2200 (x : G), x \u2208 N \u2192 \u2191(MonoidHom.comp (mk' M) f) x = 1\n[PROOFSTEP]\nintro x hx\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b9 : Group H\nM : Subgroup H\ninst\u271d : Subgroup.Normal M\nf : G \u2192* H\nh : N \u2264 Subgroup.comap f M\nx : G\nhx : x \u2208 N\n\u22a2 \u2191(MonoidHom.comp (mk' M) f) x = 1\n[PROOFSTEP]\nrefine' QuotientGroup.eq.2 _\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b9 : Group H\nM : Subgroup H\ninst\u271d : Subgroup.Normal M\nf : G \u2192* H\nh : N \u2264 Subgroup.comap f M\nx : G\nhx : x \u2208 N\n\u22a2 (\u2191f x)\u207b\u00b9 * 1 \u2208 M\n[PROOFSTEP]\nrw [mul_one, Subgroup.inv_mem_iff]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b9 : Group H\nM : Subgroup H\ninst\u271d : Subgroup.Normal M\nf : G \u2192* H\nh : N \u2264 Subgroup.comap f M\nx : G\nhx : x \u2208 N\n\u22a2 \u2191f x \u2208 M\n[PROOFSTEP]\nexact h hx\n[GOAL]\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\nI : Type u_1\ninst\u271d\u00b2 : Group I\nM : Subgroup H\nO : Subgroup I\ninst\u271d\u00b9 : Subgroup.Normal M\ninst\u271d : Subgroup.Normal O\nf : G \u2192* H\ng : H \u2192* I\nhf : N \u2264 Subgroup.comap f M\nhg : M \u2264 Subgroup.comap g O\nhgf : optParam (N \u2264 Subgroup.comap (MonoidHom.comp g f) O) (_ : N \u2264 Subgroup.comap (MonoidHom.comp g f) O)\nx : G \u29f8 N\n\u22a2 \u2191(map M O g hg) (\u2191(map N M f hf) x) = \u2191(map N O (MonoidHom.comp g f) hgf) x\n[PROOFSTEP]\nrefine' induction_on' x fun x => _\n[GOAL]\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\nI : Type u_1\ninst\u271d\u00b2 : Group I\nM : Subgroup H\nO : Subgroup I\ninst\u271d\u00b9 : Subgroup.Normal M\ninst\u271d : Subgroup.Normal O\nf : G \u2192* H\ng : H \u2192* I\nhf : N \u2264 Subgroup.comap f M\nhg : M \u2264 Subgroup.comap g O\nhgf : optParam (N \u2264 Subgroup.comap (MonoidHom.comp g f) O) (_ : N \u2264 Subgroup.comap (MonoidHom.comp g f) O)\nx\u271d : G \u29f8 N\nx : G\n\u22a2 \u2191(map M O g hg) (\u2191(map N M f hf) \u2191x) = \u2191(map N O (MonoidHom.comp g f) hgf) \u2191x\n[PROOFSTEP]\nsimp only [map_mk, MonoidHom.comp_apply]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\ne : G \u2243* H\nhe : Subgroup.map (\u2191e) G' = H'\nsrc\u271d : G \u29f8 G' \u2192* H \u29f8 H' := map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H')\nx : G \u29f8 G'\n\u22a2 \u2191(map H' G' \u2191(MulEquiv.symm e) (_ : H' \u2264 Subgroup.comap (\u2191(MulEquiv.symm e)) G'))\n      (\u2191(map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H')) x) =\n    x\n[PROOFSTEP]\nrw [map_map G' H' G' e e.symm (he \u25b8 G'.le_comap_map (e : G \u2192* H)) (he \u25b8 (G'.map_equiv_eq_comap_symm e).le)]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\ne : G \u2243* H\nhe : Subgroup.map (\u2191e) G' = H'\nsrc\u271d : G \u29f8 G' \u2192* H \u29f8 H' := map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H')\nx : G \u29f8 G'\n\u22a2 \u2191(map G' G' (MonoidHom.comp \u2191(MulEquiv.symm e) \u2191e)\n          (_ : G' \u2264 Subgroup.comap (MonoidHom.comp \u2191(MulEquiv.symm e) \u2191e) G'))\n      x =\n    x\n[PROOFSTEP]\nsimp only [map_map, \u2190 MulEquiv.coe_monoidHom_trans, MulEquiv.self_trans_symm, MulEquiv.coe_monoidHom_refl, map_id_apply]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\ne : G \u2243* H\nhe : Subgroup.map (\u2191e) G' = H'\nsrc\u271d : G \u29f8 G' \u2192* H \u29f8 H' := map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H')\nx : H \u29f8 H'\n\u22a2 \u2191(map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H'))\n      (\u2191(map H' G' \u2191(MulEquiv.symm e) (_ : H' \u2264 Subgroup.comap (\u2191(MulEquiv.symm e)) G')) x) =\n    x\n[PROOFSTEP]\nrw [map_map H' G' H' e.symm e (he \u25b8 (G'.map_equiv_eq_comap_symm e).le) (he \u25b8 G'.le_comap_map (e : G \u2192* H))]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\ne : G \u2243* H\nhe : Subgroup.map (\u2191e) G' = H'\nsrc\u271d : G \u29f8 G' \u2192* H \u29f8 H' := map G' H' \u2191e (_ : G' \u2264 Subgroup.comap (\u2191e) H')\nx : H \u29f8 H'\n\u22a2 \u2191(map H' H' (MonoidHom.comp \u2191e \u2191(MulEquiv.symm e))\n          (_ : H' \u2264 Subgroup.comap (MonoidHom.comp \u2191e \u2191(MulEquiv.symm e)) H'))\n      x =\n    x\n[PROOFSTEP]\nsimp only [\u2190 MulEquiv.coe_monoidHom_trans, MulEquiv.symm_trans_self, MulEquiv.coe_monoidHom_refl, map_id_apply]\n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\nhe : optParam (Subgroup.map (\u2191(MulEquiv.refl G)) G' = G') (_ : Subgroup.map (MonoidHom.id G) G' = G')\n\u22a2 congr G' G' (MulEquiv.refl G) he = MulEquiv.refl (G \u29f8 G')\n[PROOFSTEP]\next \u27e8x\u27e9\n[GOAL]\ncase h.mk\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\nG' : Subgroup G\nH' : Subgroup H\ninst\u271d\u00b9 : Subgroup.Normal G'\ninst\u271d : Subgroup.Normal H'\nhe : optParam (Subgroup.map (\u2191(MulEquiv.refl G)) G' = G') (_ : Subgroup.map (MonoidHom.id G) G' = G')\nx\u271d : G \u29f8 G'\nx : G\n\u22a2 \u2191(congr G' G' (MulEquiv.refl G) he) (Quot.mk Setoid.r x) = \u2191(MulEquiv.refl (G \u29f8 G')) (Quot.mk Setoid.r x)\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\na\u271d b\u271d : G \u29f8 ker \u03c6\na b : G\nh : \u2191\u03c6 a = \u2191\u03c6 b\n\u22a2 Setoid.r a b\n[PROOFSTEP]\nrw [leftRel_apply, mem_ker, \u03c6.map_mul, \u2190 h, \u03c6.map_inv, inv_mul_self]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\ng : G\nhg : g \u2208 ker \u03c6\n\u22a2 g \u2208 ker (rangeRestrict \u03c6)\n[PROOFSTEP]\nrwa [ker_rangeRestrict]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\na\u271d b\u271d : G \u29f8 ker \u03c6\na b : G\nh : \u2191(rangeRestrict \u03c6) a = \u2191(rangeRestrict \u03c6) b\n\u22a2 Setoid.r a b\n[PROOFSTEP]\nrw [leftRel_apply, \u2190 ker_rangeRestrict, mem_ker, \u03c6.rangeRestrict.map_mul, \u2190 h, \u03c6.rangeRestrict.map_inv, inv_mul_self]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\n\u22a2 Surjective \u2191(rangeKerLift \u03c6)\n[PROOFSTEP]\nrintro \u27e8_, g, rfl\u27e9\n[GOAL]\ncase mk.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\ng : G\n\u22a2 \u2203 a, \u2191(rangeKerLift \u03c6) a = { val := \u2191\u03c6 g, property := (_ : \u2203 y, \u2191\u03c6 y = \u2191\u03c6 g) }\n[PROOFSTEP]\nuse mk g\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\ng : G\n\u22a2 \u2191(rangeKerLift \u03c6) \u2191g = { val := \u2191\u03c6 g, property := (_ : \u2203 y, \u2191\u03c6 y = \u2191\u03c6 g) }\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\n\u03c8 : H \u2192 G\nh\u03c6 : Function.RightInverse \u03c8 \u2191\u03c6\nsrc\u271d : G \u29f8 ker \u03c6 \u2192* H := kerLift \u03c6\nx : G \u29f8 ker \u03c6\n\u22a2 \u2191(kerLift \u03c6) ((mk \u2218 \u03c8) (\u2191(kerLift \u03c6) x)) = \u2191(kerLift \u03c6) x\n[PROOFSTEP]\nrw [Function.comp_apply, kerLift_mk', h\u03c6]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nA' A B' B : Subgroup G\nhAN : Subgroup.Normal (Subgroup.subgroupOf A' A)\nhBN : Subgroup.Normal (Subgroup.subgroupOf B' B)\nh' : A' = B'\nh : A = B\n\u22a2 MonoidHom.comp (quotientMapSubgroupOfOfLe (_ : B' \u2264 A') (_ : B \u2264 A))\n      (quotientMapSubgroupOfOfLe (_ : A' \u2264 B') (_ : A \u2264 B)) =\n    MonoidHom.id ({ x // x \u2208 A } \u29f8 Subgroup.subgroupOf A' A)\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.h.mk\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nA' A B' B : Subgroup G\nhAN : Subgroup.Normal (Subgroup.subgroupOf A' A)\nhBN : Subgroup.Normal (Subgroup.subgroupOf B' B)\nh' : A' = B'\nh : A = B\nx : G\nhx : x \u2208 A\n\u22a2 \u2191(MonoidHom.comp\n          (MonoidHom.comp (quotientMapSubgroupOfOfLe (_ : B' \u2264 A') (_ : B \u2264 A))\n            (quotientMapSubgroupOfOfLe (_ : A' \u2264 B') (_ : A \u2264 B)))\n          (mk' (Subgroup.subgroupOf A' A)))\n      { val := x, property := hx } =\n    \u2191(MonoidHom.comp (MonoidHom.id ({ x // x \u2208 A } \u29f8 Subgroup.subgroupOf A' A)) (mk' (Subgroup.subgroupOf A' A)))\n      { val := x, property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nA' A B' B : Subgroup G\nhAN : Subgroup.Normal (Subgroup.subgroupOf A' A)\nhBN : Subgroup.Normal (Subgroup.subgroupOf B' B)\nh' : A' = B'\nh : A = B\n\u22a2 MonoidHom.comp (quotientMapSubgroupOfOfLe (_ : A' \u2264 B') (_ : A \u2264 B))\n      (quotientMapSubgroupOfOfLe (_ : B' \u2264 A') (_ : B \u2264 A)) =\n    MonoidHom.id ({ x // x \u2208 B } \u29f8 Subgroup.subgroupOf B' B)\n[PROOFSTEP]\next \u27e8x, hx\u27e9\n[GOAL]\ncase h.h.mk\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nA' A B' B : Subgroup G\nhAN : Subgroup.Normal (Subgroup.subgroupOf A' A)\nhBN : Subgroup.Normal (Subgroup.subgroupOf B' B)\nh' : A' = B'\nh : A = B\nx : G\nhx : x \u2208 B\n\u22a2 \u2191(MonoidHom.comp\n          (MonoidHom.comp (quotientMapSubgroupOfOfLe (_ : A' \u2264 B') (_ : A \u2264 B))\n            (quotientMapSubgroupOfOfLe (_ : B' \u2264 A') (_ : B \u2264 A)))\n          (mk' (Subgroup.subgroupOf B' B)))\n      { val := x, property := hx } =\n    \u2191(MonoidHom.comp (MonoidHom.id ({ x // x \u2208 B } \u29f8 Subgroup.subgroupOf B' B)) (mk' (Subgroup.subgroupOf B' B)))\n      { val := x, property := hx }\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng\u271d : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\ng : A\nx\u271d : g \u2208 range (zpowGroupHom n)\nh : A\nhg : h ^ n = g\n\u22a2 \u2191(zpowGroupHom n) (\u2191f h) = \u2191f g\n[PROOFSTEP]\nsimp only [\u2190 hg, map_zpow, zpowGroupHom_apply]\n[GOAL]\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\n\u22a2 MulEquiv.refl (A \u29f8 range (zpowGroupHom n)) = equivQuotientZPowOfEquiv (MulEquiv.refl A) n\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\nx : A \u29f8 range (zpowGroupHom n)\n\u22a2 \u2191(MulEquiv.refl (A \u29f8 range (zpowGroupHom n))) x = \u2191(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) x\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq' x]\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\nx : A \u29f8 range (zpowGroupHom n)\n\u22a2 \u2191(MulEquiv.refl (A \u29f8 range (zpowGroupHom n))) (Quotient.mk'' (Quotient.out' x)) =\n    \u2191(equivQuotientZPowOfEquiv (MulEquiv.refl A) n) (Quotient.mk'' (Quotient.out' x))\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\n\u22a2 MulEquiv.trans (equivQuotientZPowOfEquiv e n) (equivQuotientZPowOfEquiv d n) =\n    equivQuotientZPowOfEquiv (MulEquiv.trans e d) n\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\nx : A \u29f8 range (zpowGroupHom n)\n\u22a2 \u2191(MulEquiv.trans (equivQuotientZPowOfEquiv e n) (equivQuotientZPowOfEquiv d n)) x =\n    \u2191(equivQuotientZPowOfEquiv (MulEquiv.trans e d) n) x\n[PROOFSTEP]\nrw [\u2190 Quotient.out_eq' x]\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u2074 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b3 : Group H\n\u03c6 : G \u2192* H\nA B C : Type u\ninst\u271d\u00b2 : CommGroup A\ninst\u271d\u00b9 : CommGroup B\ninst\u271d : CommGroup C\nf : A \u2192* B\ng : B \u2192* A\ne : A \u2243* B\nd : B \u2243* C\nn : \u2124\nx : A \u29f8 range (zpowGroupHom n)\n\u22a2 \u2191(MulEquiv.trans (equivQuotientZPowOfEquiv e n) (equivQuotientZPowOfEquiv d n)) (Quotient.mk'' (Quotient.out' x)) =\n    \u2191(equivQuotientZPowOfEquiv (MulEquiv.trans e d) n) (Quotient.mk'' (Quotient.out' x))\n[PROOFSTEP]\nrfl\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\n\u22a2 \u2200 (a : { x // x \u2208 H \u2294 N }), \u2203 a_1, \u2191\u03c6 a_1 = Quotient.mk'' a\n[PROOFSTEP]\nrintro \u27e8y, hy : y \u2208 (H \u2294 N)\u27e9\n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\ny : G\nhy : y \u2208 H \u2294 N\n\u22a2 \u2203 a, \u2191\u03c6 a = Quotient.mk'' { val := y, property := hy }\n[PROOFSTEP]\nrw [\u2190 SetLike.mem_coe] at hy \n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\ny : G\nhy\u271d : y \u2208 H \u2294 N\nhy : y \u2208 \u2191(H \u2294 N)\n\u22a2 \u2203 a, \u2191\u03c6 a = Quotient.mk'' { val := y, property := hy\u271d }\n[PROOFSTEP]\nrw [mul_normal H N] at hy \n[GOAL]\ncase mk\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\ny : G\nhy\u271d : y \u2208 H \u2294 N\nhy : y \u2208 \u2191H * \u2191N\n\u22a2 \u2203 a, \u2191\u03c6 a = Quotient.mk'' { val := y, property := hy\u271d }\n[PROOFSTEP]\nrcases hy with \u27e8h, n, hh, hn, rfl\u27e9\n[GOAL]\ncase mk.intro.intro.intro.intro\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\n\u22a2 \u2203 a, \u2191\u03c6 a = Quotient.mk'' { val := (fun x x_1 => x * x_1) h n, property := hy }\n[PROOFSTEP]\nuse\u27e8h, hh\u27e9\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\n\u22a2 \u2191\u03c6 { val := h, property := hh } = Quotient.mk'' { val := (fun x x_1 => x * x_1) h n, property := hy }\n[PROOFSTEP]\nlet _ : Setoid \u2191(H \u2294 N) :=\n  (@leftRel \u2191(H \u2294 N) (H \u2294 N : Subgroup G).toGroup (N.subgroupOf (H \u2294 N)))\n    -- porting note: Lean couldn't find this automatically\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\nx\u271d : Setoid { x // x \u2208 H \u2294 N } := leftRel (subgroupOf N (H \u2294 N))\n\u22a2 \u2191\u03c6 { val := h, property := hh } = Quotient.mk'' { val := (fun x x_1 => x * x_1) h n, property := hy }\n[PROOFSTEP]\nrefine Quotient.eq.mpr ?_\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\nx\u271d : Setoid { x // x \u2208 H \u2294 N } := leftRel (subgroupOf N (H \u2294 N))\n\u22a2 \u2191(inclusion (_ : H \u2264 H \u2294 N)) { val := h, property := hh } \u2248 { val := (fun x x_1 => x * x_1) h n, property := hy }\n[PROOFSTEP]\nchange Setoid.r _ _\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\nx\u271d : Setoid { x // x \u2208 H \u2294 N } := leftRel (subgroupOf N (H \u2294 N))\n\u22a2 Setoid.r (\u2191(inclusion (_ : H \u2264 H \u2294 N)) { val := h, property := hh })\n    { val := (fun x x_1 => x * x_1) h n, property := hy }\n[PROOFSTEP]\nrw [leftRel_apply]\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\nx\u271d : Setoid { x // x \u2208 H \u2294 N } := leftRel (subgroupOf N (H \u2294 N))\n\u22a2 (\u2191(inclusion (_ : H \u2264 H \u2294 N)) { val := h, property := hh })\u207b\u00b9 *\n      { val := (fun x x_1 => x * x_1) h n, property := hy } \u2208\n    subgroupOf N (H \u2294 N)\n[PROOFSTEP]\nchange h\u207b\u00b9 * (h * n) \u2208 N\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\nx : { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N)\nh n : G\nhh : h \u2208 \u2191H\nhn : n \u2208 \u2191N\nhy : (fun x x_1 => x * x_1) h n \u2208 H \u2294 N\nx\u271d : Setoid { x // x \u2208 H \u2294 N } := leftRel (subgroupOf N (H \u2294 N))\n\u22a2 h\u207b\u00b9 * (h * n) \u2208 N\n[PROOFSTEP]\nrwa [\u2190 mul_assoc, inv_mul_self, one_mul]\n[GOAL]\nG : Type u\ninst\u271d\u00b2 : Group G\nN\u271d : Subgroup G\nnN : Normal N\u271d\nH\u271d : Type v\ninst\u271d\u00b9 : Group H\u271d\n\u03c6\u271d : G \u2192* H\u271d\nH N : Subgroup G\ninst\u271d : Normal N\n\u03c6 : { x // x \u2208 H } \u2192* { x // x \u2208 H \u2294 N } \u29f8 subgroupOf N (H \u2294 N) :=\n  MonoidHom.comp (mk' (subgroupOf N (H \u2294 N))) (inclusion (_ : H \u2264 H \u2294 N))\n\u03c6_surjective : Surjective \u2191\u03c6\n\u22a2 subgroupOf N H = ker \u03c6\n[PROOFSTEP]\nsimp [\u2190 comap_ker]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\n\u22a2 \u2200 (x : G \u29f8 N), x \u2208 Subgroup.map (mk' N) M \u2192 \u2191(map N M (MonoidHom.id G) h) x = 1\n[PROOFSTEP]\nrintro _ \u27e8x, hx, rfl\u27e9\n[GOAL]\ncase intro.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\nx : G\nhx : x \u2208 \u2191M\n\u22a2 \u2191(map N M (MonoidHom.id G) h) (\u2191(mk' N) x) = 1\n[PROOFSTEP]\nrw [map_mk' N M _ _ x]\n[GOAL]\ncase intro.intro\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\nx : G\nhx : x \u2208 \u2191M\n\u22a2 \u2191(\u2191(MonoidHom.id G) x) = 1\n[PROOFSTEP]\nexact (QuotientGroup.eq_one_iff _).mpr hx\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\n\u22a2 MonoidHom.comp (map M (Subgroup.map (mk' N) M) (mk' N) (_ : M \u2264 Subgroup.comap (mk' N) (Subgroup.map (mk' N) M)))\n      (quotientQuotientEquivQuotientAux N M h) =\n    MonoidHom.id ((G \u29f8 N) \u29f8 Subgroup.map (mk' N) M)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h.h\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp\n          (MonoidHom.comp\n            (MonoidHom.comp\n              (map M (Subgroup.map (mk' N) M) (mk' N) (_ : M \u2264 Subgroup.comap (mk' N) (Subgroup.map (mk' N) M)))\n              (quotientQuotientEquivQuotientAux N M h))\n            (mk' (Subgroup.map (mk' N) M)))\n          (mk' N))\n      x\u271d =\n    \u2191(MonoidHom.comp (MonoidHom.comp (MonoidHom.id ((G \u29f8 N) \u29f8 Subgroup.map (mk' N) M)) (mk' (Subgroup.map (mk' N) M)))\n          (mk' N))\n      x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\n\u22a2 MonoidHom.comp (quotientQuotientEquivQuotientAux N M h)\n      (map M (Subgroup.map (mk' N) M) (mk' N) (_ : M \u2264 Subgroup.comap (mk' N) (Subgroup.map (mk' N) M))) =\n    MonoidHom.id (G \u29f8 M)\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\nM : Subgroup G\nnM : Subgroup.Normal M\nh : N \u2264 M\nx\u271d : G\n\u22a2 \u2191(MonoidHom.comp\n          (MonoidHom.comp (quotientQuotientEquivQuotientAux N M h)\n            (map M (Subgroup.map (mk' N) M) (mk' N) (_ : M \u2264 Subgroup.comap (mk' N) (Subgroup.map (mk' N) M))))\n          (mk' M))\n      x\u271d =\n    \u2191(MonoidHom.comp (MonoidHom.id (G \u29f8 M)) (mk' M)) x\u271d\n[PROOFSTEP]\nsimp\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\n\u22a2 Subsingleton (G \u29f8 \u22a4)\n[PROOFSTEP]\ndsimp [HasQuotient.Quotient, QuotientGroup.instHasQuotientSubgroup, Quotient]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\n\u22a2 Subsingleton (Quot Setoid.r)\n[PROOFSTEP]\nrw [leftRel_eq]\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d : Group H\n\u03c6 : G \u2192* H\n\u22a2 Subsingleton (Quot fun x y => x\u207b\u00b9 * y \u2208 \u22a4)\n[PROOFSTEP]\nexact Trunc.instSubsingletonTrunc\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH\u271d : Type v\ninst\u271d : Group H\u271d\n\u03c6 : G \u2192* H\u271d\nH : Subgroup G\nh : Subsingleton (G \u29f8 H)\nx : G\nx\u271d : x \u2208 \u22a4\n\u22a2 x \u2208 H\n[PROOFSTEP]\nhave this : 1\u207b\u00b9 * x \u2208 H := QuotientGroup.eq.1 (Subsingleton.elim _ _)\n[GOAL]\nG : Type u\ninst\u271d\u00b9 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH\u271d : Type v\ninst\u271d : Group H\u271d\n\u03c6 : G \u2192* H\u271d\nH : Subgroup G\nh : Subsingleton (G \u29f8 H)\nx : G\nx\u271d : x \u2208 \u22a4\nthis : 1\u207b\u00b9 * x \u2208 H\n\u22a2 x \u2208 H\n[PROOFSTEP]\nrwa [inv_one, one_mul] at this \n[GOAL]\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* H\nH\u2081 : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal H\u2081\nH\u2082 : Subgroup (G \u29f8 H\u2081)\ninst\u271d : Subgroup.Normal H\u2082\n\u22a2 Subgroup.comap (mk' H\u2081) (Subgroup.comap (mk' H\u2082) (Subgroup.center ((G \u29f8 H\u2081) \u29f8 H\u2082))) =\n    Subgroup.comap (mk' (Subgroup.comap (mk' H\u2081) H\u2082)) (Subgroup.center (G \u29f8 Subgroup.comap (mk' H\u2081) H\u2082))\n[PROOFSTEP]\next x\n[GOAL]\ncase h\nG : Type u\ninst\u271d\u00b3 : Group G\nN : Subgroup G\nnN : Subgroup.Normal N\nH : Type v\ninst\u271d\u00b2 : Group H\n\u03c6 : G \u2192* H\nH\u2081 : Subgroup G\ninst\u271d\u00b9 : Subgroup.Normal H\u2081\nH\u2082 : Subgroup (G \u29f8 H\u2081)\ninst\u271d : Subgroup.Normal H\u2082\nx : G\n\u22a2 x \u2208 Subgroup.comap (mk' H\u2081) (Subgroup.comap (mk' H\u2082) (Subgroup.center ((G \u29f8 H\u2081) \u29f8 H\u2082))) \u2194\n    x \u2208 Subgroup.comap (mk' (Subgroup.comap (mk' H\u2081) H\u2082)) (Subgroup.center (G \u29f8 Subgroup.comap (mk' H\u2081) H\u2082))\n[PROOFSTEP]\nsimp only [mk'_apply, Subgroup.mem_comap, Subgroup.mem_center_iff, forall_mk, \u2190 mk_mul, eq_iff_div_mem, mk_div]\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.QuotientGroup", "llama_tokens": 14555, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461390043208003, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.501652899700165}}
{"text": "[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 Group (MulAut M)\n[PROOFSTEP]\nrefine'\n  { mul := fun g h => MulEquiv.trans h g\n    one := MulEquiv.refl M\n    inv := MulEquiv.symm\n    div := fun g h => MulEquiv.trans h.symm g\n    npow := @npowRec _ \u27e8MulEquiv.refl M\u27e9 \u27e8fun g h => MulEquiv.trans h g\u27e9\n    zpow := @zpowRec _ \u27e8MulEquiv.refl M\u27e9 \u27e8fun g h => MulEquiv.trans h g\u27e9 \u27e8MulEquiv.symm\u27e9 .. }\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a b c : MulAut M), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a : MulAut M), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a : MulAut M), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (x : MulAut M), npowRec 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (n : \u2115) (x : MulAut M), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a b : MulAut M), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a : MulAut M), zpowRec 0 a = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (n : \u2115) (a : MulAut M), zpowRec (Int.ofNat (Nat.succ n)) a = a * zpowRec (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (n : \u2115) (a : MulAut M), zpowRec (Int.negSucc n) a = (zpowRec (\u2191(Nat.succ n)) a)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_10\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (a : MulAut M), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d c\u271d : MulAut M\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_3\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_4\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nx\u271d : MulAut M\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_5\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\nx\u271d : MulAut M\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_6\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d : MulAut M\n\u22a2 a\u271d / b\u271d = a\u271d * b\u271d\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_7\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\n\u22a2 zpowRec 0 a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_8\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\n\u22a2 zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_9\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\n\u22a2 zpowRec (Int.negSucc n\u271d) a\u271d = (zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_10\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\n\u22a2 a\u271d\u207b\u00b9 * a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d c\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d * b\u271d * c\u271d) x\u271d = \u2191(a\u271d * (b\u271d * c\u271d)) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_1.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d c\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d * b\u271d * c\u271d) x\u271d = \u2191(a\u271d * (b\u271d * c\u271d)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(1 * a\u271d) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_2.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(1 * a\u271d) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d * 1) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_3.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d * 1) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nx\u271d\u00b9 : MulAut M\nx\u271d : M\n\u22a2 \u2191(npowRec 0 x\u271d\u00b9) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_4.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nx\u271d\u00b9 : MulAut M\nx\u271d : M\n\u22a2 \u2191(npowRec 0 x\u271d\u00b9) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\nx\u271d\u00b9 : MulAut M\nx\u271d : M\n\u22a2 \u2191(npowRec (n\u271d + 1) x\u271d\u00b9) x\u271d = \u2191(x\u271d\u00b9 * npowRec n\u271d x\u271d\u00b9) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_5.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\nx\u271d\u00b9 : MulAut M\nx\u271d : M\n\u22a2 \u2191(npowRec (n\u271d + 1) x\u271d\u00b9) x\u271d = \u2191(x\u271d\u00b9 * npowRec n\u271d x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d / b\u271d) x\u271d = \u2191(a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_6.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d b\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d / b\u271d) x\u271d = \u2191(a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec 0 a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_7.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec 0 a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d) x\u271d = \u2191(a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_8.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d) x\u271d = \u2191(a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec (Int.negSucc n\u271d) a\u271d) x\u271d = \u2191(zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_9.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nn\u271d : \u2115\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(zpowRec (Int.negSucc n\u271d) a\u271d) x\u271d = \u2191(zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\na\u271d : MulAut M\nx\u271d : M\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\napply Equiv.left_inv\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 MulAut M \u2192* Equiv.Perm M\n[PROOFSTEP]\nrefine' { toFun := MulEquiv.toEquiv, .. }\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 1.toEquiv = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 \u2200 (x y : MulAut M),\n    OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := ?refine'_1 } (x * y) =\n      OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := ?refine'_1 } x *\n        OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := ?refine'_1 } y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\n\u22a2 1.toEquiv = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Mul M\nx\u271d y\u271d : MulAut M\n\u22a2 OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } (x\u271d * y\u271d) =\n    OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } x\u271d *\n      OneHom.toFun { toFun := MulEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng x\u271d : G\n\u22a2 (fun h => g\u207b\u00b9 * h * g) ((fun h => g * h * g\u207b\u00b9) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp only [mul_assoc, inv_mul_cancel_left, mul_left_inv, mul_one]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng x\u271d : G\n\u22a2 (fun h => g * h * g\u207b\u00b9) ((fun h => g\u207b\u00b9 * h * g) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp only [mul_assoc, mul_inv_cancel_left, mul_right_inv, mul_one]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng : G\n\u22a2 \u2200 (x y : G),\n    Equiv.toFun\n        { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n          left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n          right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) }\n        (x * y) =\n      Equiv.toFun\n          { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n            left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n            right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) }\n          x *\n        Equiv.toFun\n          { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n            left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n            right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) }\n          y\n[PROOFSTEP]\nsimp only [mul_assoc, inv_mul_cancel_left, forall_const]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\n\u22a2 (fun g =>\n        {\n          toEquiv :=\n            { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n              left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n              right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n          map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n      1 =\n    1\n[PROOFSTEP]\next\n[GOAL]\ncase h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\nx\u271d : G\n\u22a2 \u2191((fun g =>\n            {\n              toEquiv :=\n                { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                  left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                  right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n              map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n          1)\n      x\u271d =\n    \u21911 x\u271d\n[PROOFSTEP]\nsimp only [one_mul, inv_one, mul_one, one_apply]\n[GOAL]\ncase h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\nx\u271d : G\n\u22a2 \u2191{\n          toEquiv :=\n            { toFun := fun h => h, invFun := fun h => h, left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n              right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) },\n          map_mul' :=\n            (_ :\n              \u2200 (x y : G),\n                Equiv.toFun\n                    { toFun := fun h => h, invFun := fun h => h,\n                      left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                      right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                    (x * y) =\n                  Equiv.toFun\n                      { toFun := fun h => h, invFun := fun h => h,\n                        left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                        right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                      x *\n                    Equiv.toFun\n                      { toFun := fun h => h, invFun := fun h => h,\n                        left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                        right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                      y) }\n      x\u271d =\n    x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng\u2081 g\u2082 : G\n\u22a2 OneHom.toFun\n      {\n        toFun := fun g =>\n          {\n            toEquiv :=\n              { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n            map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n        map_one' :=\n          (_ :\n            (fun g =>\n                  {\n                    toEquiv :=\n                      { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                        left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                        right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                    map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                1 =\n              1) }\n      (g\u2081 * g\u2082) =\n    OneHom.toFun\n        {\n          toFun := fun g =>\n            {\n              toEquiv :=\n                { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                  left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                  right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n              map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n          map_one' :=\n            (_ :\n              (fun g =>\n                    {\n                      toEquiv :=\n                        { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                          left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                          right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                      map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                  1 =\n                1) }\n        g\u2081 *\n      OneHom.toFun\n        {\n          toFun := fun g =>\n            {\n              toEquiv :=\n                { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                  left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                  right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n              map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n          map_one' :=\n            (_ :\n              (fun g =>\n                    {\n                      toEquiv :=\n                        { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                          left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                          right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                      map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                  1 =\n                1) }\n        g\u2082\n[PROOFSTEP]\next h\n[GOAL]\ncase h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng\u2081 g\u2082 h : G\n\u22a2 \u2191(OneHom.toFun\n          {\n            toFun := fun g =>\n              {\n                toEquiv :=\n                  { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                    left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                    right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n            map_one' :=\n              (_ :\n                (fun g =>\n                      {\n                        toEquiv :=\n                          { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                            left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                            right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                        map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                    1 =\n                  1) }\n          (g\u2081 * g\u2082))\n      h =\n    \u2191(OneHom.toFun\n            {\n              toFun := fun g =>\n                {\n                  toEquiv :=\n                    { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                      left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                      right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                  map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n              map_one' :=\n                (_ :\n                  (fun g =>\n                        {\n                          toEquiv :=\n                            { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                              left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                              right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                          map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                      1 =\n                    1) }\n            g\u2081 *\n          OneHom.toFun\n            {\n              toFun := fun g =>\n                {\n                  toEquiv :=\n                    { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                      left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                      right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                  map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) },\n              map_one' :=\n                (_ :\n                  (fun g =>\n                        {\n                          toEquiv :=\n                            { toFun := fun h => g * h * g\u207b\u00b9, invFun := fun h => g\u207b\u00b9 * h * g,\n                              left_inv := (_ : \u2200 (x : G), g\u207b\u00b9 * (g * x * g\u207b\u00b9) * g = x),\n                              right_inv := (_ : \u2200 (x : G), g * (g\u207b\u00b9 * x * g) * g\u207b\u00b9 = x) },\n                          map_mul' := (_ : \u2200 (a a_1 : G), g * (a * a_1) * g\u207b\u00b9 = g * a * g\u207b\u00b9 * (g * a_1 * g\u207b\u00b9)) })\n                      1 =\n                    1) }\n            g\u2082)\n      h\n[PROOFSTEP]\nshow g\u2081 * g\u2082 * h * (g\u2081 * g\u2082)\u207b\u00b9 = g\u2081 * (g\u2082 * h * g\u2082\u207b\u00b9) * g\u2081\u207b\u00b9\n[GOAL]\ncase h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Mul M\ninst\u271d : Group G\ng\u2081 g\u2082 h : G\n\u22a2 g\u2081 * g\u2082 * h * (g\u2081 * g\u2082)\u207b\u00b9 = g\u2081 * (g\u2082 * h * g\u2082\u207b\u00b9) * g\u2081\u207b\u00b9\n[PROOFSTEP]\nsimp only [mul_assoc, mul_inv_rev]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 Group (AddAut A)\n[PROOFSTEP]\nrefine'\n  { mul := fun g h => AddEquiv.trans h g\n    one := AddEquiv.refl A\n    inv := AddEquiv.symm\n    div := fun g h => AddEquiv.trans h.symm g\n    npow := @npowRec _ \u27e8AddEquiv.refl A\u27e9 \u27e8fun g h => AddEquiv.trans h g\u27e9\n    zpow := @zpowRec _ \u27e8AddEquiv.refl A\u27e9 \u27e8fun g h => AddEquiv.trans h g\u27e9 \u27e8AddEquiv.symm\u27e9 .. }\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a b c : AddAut A), a * b * c = a * (b * c)\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a : AddAut A), 1 * a = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_3\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a : AddAut A), a * 1 = a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_4\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (x : AddAut A), npowRec 0 x = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_5\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (n : \u2115) (x : AddAut A), npowRec (n + 1) x = x * npowRec n x\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_6\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a b : AddAut A), a / b = a * b\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_7\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a : AddAut A), zpowRec 0 a = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_8\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (n : \u2115) (a : AddAut A), zpowRec (Int.ofNat (Nat.succ n)) a = a * zpowRec (Int.ofNat n) a\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_9\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (n : \u2115) (a : AddAut A), zpowRec (Int.negSucc n) a = (zpowRec (\u2191(Nat.succ n)) a)\u207b\u00b9\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_10\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (a : AddAut A), a\u207b\u00b9 * a = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d c\u271d : AddAut A\n\u22a2 a\u271d * b\u271d * c\u271d = a\u271d * (b\u271d * c\u271d)\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\n\u22a2 1 * a\u271d = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_3\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\n\u22a2 a\u271d * 1 = a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_4\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nx\u271d : AddAut A\n\u22a2 npowRec 0 x\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_5\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\nx\u271d : AddAut A\n\u22a2 npowRec (n\u271d + 1) x\u271d = x\u271d * npowRec n\u271d x\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_6\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d : AddAut A\n\u22a2 a\u271d / b\u271d = a\u271d * b\u271d\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_7\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\n\u22a2 zpowRec 0 a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_8\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\n\u22a2 zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d = a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_9\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\n\u22a2 zpowRec (Int.negSucc n\u271d) a\u271d = (zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_10\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\n\u22a2 a\u271d\u207b\u00b9 * a\u271d = 1\n[PROOFSTEP]\next\n[GOAL]\ncase refine'_1.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d c\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d * b\u271d * c\u271d) x\u271d = \u2191(a\u271d * (b\u271d * c\u271d)) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_1.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d c\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d * b\u271d * c\u271d) x\u271d = \u2191(a\u271d * (b\u271d * c\u271d)) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(1 * a\u271d) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_2.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(1 * a\u271d) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_3.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d * 1) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_3.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d * 1) x\u271d = \u2191a\u271d x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_4.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nx\u271d\u00b9 : AddAut A\nx\u271d : A\n\u22a2 \u2191(npowRec 0 x\u271d\u00b9) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_4.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nx\u271d\u00b9 : AddAut A\nx\u271d : A\n\u22a2 \u2191(npowRec 0 x\u271d\u00b9) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_5.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\nx\u271d\u00b9 : AddAut A\nx\u271d : A\n\u22a2 \u2191(npowRec (n\u271d + 1) x\u271d\u00b9) x\u271d = \u2191(x\u271d\u00b9 * npowRec n\u271d x\u271d\u00b9) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_5.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\nx\u271d\u00b9 : AddAut A\nx\u271d : A\n\u22a2 \u2191(npowRec (n\u271d + 1) x\u271d\u00b9) x\u271d = \u2191(x\u271d\u00b9 * npowRec n\u271d x\u271d\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_6.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d / b\u271d) x\u271d = \u2191(a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_6.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d b\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d / b\u271d) x\u271d = \u2191(a\u271d * b\u271d\u207b\u00b9) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_7.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec 0 a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_7.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec 0 a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_8.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d) x\u271d = \u2191(a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d) x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_8.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec (Int.ofNat (Nat.succ n\u271d)) a\u271d) x\u271d = \u2191(a\u271d * zpowRec (Int.ofNat n\u271d) a\u271d) x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_9.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec (Int.negSucc n\u271d) a\u271d) x\u271d = \u2191(zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_9.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nn\u271d : \u2115\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(zpowRec (Int.negSucc n\u271d) a\u271d) x\u271d = \u2191(zpowRec (\u2191(Nat.succ n\u271d)) a\u271d)\u207b\u00b9 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\ntry rfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_10.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\na\u271d : AddAut A\nx\u271d : A\n\u22a2 \u2191(a\u271d\u207b\u00b9 * a\u271d) x\u271d = \u21911 x\u271d\n[PROOFSTEP]\napply Equiv.left_inv\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 AddAut A \u2192* Equiv.Perm A\n[PROOFSTEP]\nrefine' { toFun := AddEquiv.toEquiv, .. }\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 1.toEquiv = 1\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 \u2200 (x y : AddAut A),\n    OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := ?refine'_1 } (x * y) =\n      OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := ?refine'_1 } x *\n        OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := ?refine'_1 } y\n[PROOFSTEP]\nintros\n[GOAL]\ncase refine'_1\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\n\u22a2 1.toEquiv = 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase refine'_2\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d : Add A\nx\u271d y\u271d : AddAut A\n\u22a2 OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } (x\u271d * y\u271d) =\n    OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } x\u271d *\n      OneHom.toFun { toFun := AddEquiv.toEquiv, map_one' := (_ : 1.toEquiv = 1.toEquiv) } y\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng x\u271d : G\n\u22a2 (fun h => -g + h + g) ((fun h => g + h + -g) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp only [add_assoc, neg_add_cancel_left, add_left_neg, add_zero]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng x\u271d : G\n\u22a2 (fun h => g + h + -g) ((fun h => -g + h + g) x\u271d) = x\u271d\n[PROOFSTEP]\nsimp only [add_assoc, add_neg_cancel_left, add_right_neg, add_zero]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng : G\n\u22a2 \u2200 (x y : G),\n    Equiv.toFun\n        { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n          left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n          right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) }\n        (x + y) =\n      Equiv.toFun\n          { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n            left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n            right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) }\n          x +\n        Equiv.toFun\n          { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n            left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n            right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) }\n          y\n[PROOFSTEP]\nsimp only [add_assoc, neg_add_cancel_left, forall_const]\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\n\u22a2 (fun g =>\n        \u2191Additive.ofMul\n          {\n            toEquiv :=\n              { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n            map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n      0 =\n    0\n[PROOFSTEP]\napply Additive.toMul.injective\n[GOAL]\ncase a\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\n\u22a2 \u2191Additive.toMul\n      ((fun g =>\n          \u2191Additive.ofMul\n            {\n              toEquiv :=\n                { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                  left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                  right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n              map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n        0) =\n    \u2191Additive.toMul 0\n[PROOFSTEP]\next\n[GOAL]\ncase a.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 \u2191(\u2191Additive.toMul\n          ((fun g =>\n              \u2191Additive.ofMul\n                {\n                  toEquiv :=\n                    { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                      left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                      right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                  map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n            0))\n      x\u271d =\n    \u2191(\u2191Additive.toMul 0) x\u271d\n[PROOFSTEP]\nsimp only [zero_add, neg_zero, add_zero, toMul_ofMul, toMul_zero, one_apply]\n[GOAL]\ncase a.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\nx\u271d : G\n\u22a2 \u2191{\n          toEquiv :=\n            { toFun := fun h => h, invFun := fun h => h, left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n              right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) },\n          map_add' :=\n            (_ :\n              \u2200 (x y : G),\n                Equiv.toFun\n                    { toFun := fun h => h, invFun := fun h => h,\n                      left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                      right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                    (x + y) =\n                  Equiv.toFun\n                      { toFun := fun h => h, invFun := fun h => h,\n                        left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                        right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                      x +\n                    Equiv.toFun\n                      { toFun := fun h => h, invFun := fun h => h,\n                        left_inv := (_ : Function.LeftInverse (fun h => h) fun h => h),\n                        right_inv := (_ : Function.RightInverse (fun h => h) fun h => h) }\n                      y) }\n      x\u271d =\n    x\u271d\n[PROOFSTEP]\nrfl\n[GOAL]\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng\u2081 g\u2082 : G\n\u22a2 ZeroHom.toFun\n      {\n        toFun := fun g =>\n          \u2191Additive.ofMul\n            {\n              toEquiv :=\n                { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                  left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                  right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n              map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n        map_zero' :=\n          (_ :\n            (fun g =>\n                  \u2191Additive.ofMul\n                    {\n                      toEquiv :=\n                        { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                          left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                          right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                      map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                0 =\n              0) }\n      (g\u2081 + g\u2082) =\n    ZeroHom.toFun\n        {\n          toFun := fun g =>\n            \u2191Additive.ofMul\n              {\n                toEquiv :=\n                  { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                    left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                    right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n          map_zero' :=\n            (_ :\n              (fun g =>\n                    \u2191Additive.ofMul\n                      {\n                        toEquiv :=\n                          { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                            left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                            right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                        map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                  0 =\n                0) }\n        g\u2081 +\n      ZeroHom.toFun\n        {\n          toFun := fun g =>\n            \u2191Additive.ofMul\n              {\n                toEquiv :=\n                  { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                    left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                    right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n          map_zero' :=\n            (_ :\n              (fun g =>\n                    \u2191Additive.ofMul\n                      {\n                        toEquiv :=\n                          { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                            left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                            right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                        map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                  0 =\n                0) }\n        g\u2082\n[PROOFSTEP]\napply Additive.toMul.injective\n[GOAL]\ncase a\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng\u2081 g\u2082 : G\n\u22a2 \u2191Additive.toMul\n      (ZeroHom.toFun\n        {\n          toFun := fun g =>\n            \u2191Additive.ofMul\n              {\n                toEquiv :=\n                  { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                    left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                    right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n          map_zero' :=\n            (_ :\n              (fun g =>\n                    \u2191Additive.ofMul\n                      {\n                        toEquiv :=\n                          { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                            left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                            right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                        map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                  0 =\n                0) }\n        (g\u2081 + g\u2082)) =\n    \u2191Additive.toMul\n      (ZeroHom.toFun\n          {\n            toFun := fun g =>\n              \u2191Additive.ofMul\n                {\n                  toEquiv :=\n                    { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                      left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                      right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                  map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n            map_zero' :=\n              (_ :\n                (fun g =>\n                      \u2191Additive.ofMul\n                        {\n                          toEquiv :=\n                            { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                              left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                              right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                          map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                    0 =\n                  0) }\n          g\u2081 +\n        ZeroHom.toFun\n          {\n            toFun := fun g =>\n              \u2191Additive.ofMul\n                {\n                  toEquiv :=\n                    { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                      left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                      right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                  map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n            map_zero' :=\n              (_ :\n                (fun g =>\n                      \u2191Additive.ofMul\n                        {\n                          toEquiv :=\n                            { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                              left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                              right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                          map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                    0 =\n                  0) }\n          g\u2082)\n[PROOFSTEP]\next h\n[GOAL]\ncase a.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng\u2081 g\u2082 h : G\n\u22a2 \u2191(\u2191Additive.toMul\n          (ZeroHom.toFun\n            {\n              toFun := fun g =>\n                \u2191Additive.ofMul\n                  {\n                    toEquiv :=\n                      { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                        left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                        right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                    map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n              map_zero' :=\n                (_ :\n                  (fun g =>\n                        \u2191Additive.ofMul\n                          {\n                            toEquiv :=\n                              { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                                left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                                right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                            map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                      0 =\n                    0) }\n            (g\u2081 + g\u2082)))\n      h =\n    \u2191(\u2191Additive.toMul\n          (ZeroHom.toFun\n              {\n                toFun := fun g =>\n                  \u2191Additive.ofMul\n                    {\n                      toEquiv :=\n                        { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                          left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                          right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                      map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n                map_zero' :=\n                  (_ :\n                    (fun g =>\n                          \u2191Additive.ofMul\n                            {\n                              toEquiv :=\n                                { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                                  left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                                  right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                              map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                        0 =\n                      0) }\n              g\u2081 +\n            ZeroHom.toFun\n              {\n                toFun := fun g =>\n                  \u2191Additive.ofMul\n                    {\n                      toEquiv :=\n                        { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                          left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                          right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                      map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) },\n                map_zero' :=\n                  (_ :\n                    (fun g =>\n                          \u2191Additive.ofMul\n                            {\n                              toEquiv :=\n                                { toFun := fun h => g + h + -g, invFun := fun h => -g + h + g,\n                                  left_inv := (_ : \u2200 (x : G), -g + (g + x + -g) + g = x),\n                                  right_inv := (_ : \u2200 (x : G), g + (-g + x + g) + -g = x) },\n                              map_add' := (_ : \u2200 (a a_1 : G), g + (a + a_1) + -g = g + a + -g + (g + a_1 + -g)) })\n                        0 =\n                      0) }\n              g\u2082))\n      h\n[PROOFSTEP]\nshow g\u2081 + g\u2082 + h + -(g\u2081 + g\u2082) = g\u2081 + (g\u2082 + h + -g\u2082) + -g\u2081\n[GOAL]\ncase a.h\nA : Type u_1\nM : Type u_2\nG : Type u_3\ninst\u271d\u00b9 : Add A\ninst\u271d : AddGroup G\ng\u2081 g\u2082 h : G\n\u22a2 g\u2081 + g\u2082 + h + -(g\u2081 + g\u2082) = g\u2081 + (g\u2082 + h + -g\u2082) + -g\u2081\n[PROOFSTEP]\nsimp only [add_assoc, neg_add_rev]\n", "meta": {"mathlib_filename": "Mathlib.Algebra.Hom.Aut", "llama_tokens": 19363, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.8104788995148791, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.5015084849820551}}
{"text": "[GOAL]\nd x y : \u2124\n\u22a2 IsPell { re := x, im := y } \u2194 { re := x, im := y } * star { re := x, im := y } = 1\n[PROOFSTEP]\nsimp [Zsqrtd.ext, IsPell, mul_comm]\n[GOAL]\nd x y : \u2124\n\u22a2 x * x - y * (d * y) = 1 \u2194 x * x + -(y * (d * y)) = 1\n[PROOFSTEP]\nring_nf\n[GOAL]\nd x y : \u2124\n\u22a2 IsPell { re := x, im := y } \u2194 { re := x, im := y } \u2208 unitary (\u2124\u221ad)\n[PROOFSTEP]\nrw [unitary.mem_iff, isPell_norm, mul_comm (star _), and_self_iff]\n[GOAL]\nd : \u2124\nb c : \u2124\u221ad\nhb : IsPell b\nhc : IsPell c\n\u22a2 b * c * star (b * c) = 1\n[PROOFSTEP]\nsimp [mul_comm, mul_left_comm c, mul_assoc, star_mul, isPell_norm.1 hb, isPell_norm.1 hc]\n[GOAL]\nd x y : \u2124\n\u22a2 IsPell { re := x, im := y } \u2194 IsPell (star { re := x, im := y })\n[PROOFSTEP]\nsimp [IsPell, Zsqrtd.star_mk]\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 0 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 xn a1 1 = a\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 yn a1 1 = 1\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 a \u2264 a * a\n[PROOFSTEP]\nhave := @Nat.mul_le_mul_left 1 a a (le_of_lt a1)\n[GOAL]\na : \u2115\na1 : 1 < a\nthis : a * 1 \u2264 a * a\n\u22a2 a \u2264 a * a\n[PROOFSTEP]\nrwa [mul_one] at this \n[GOAL]\na : \u2115\na1 : 1 < a\nthis : 1 \u2264 a * a\n\u22a2 \u2191(Pell.d a1) = az a * az a - 1\n[PROOFSTEP]\nrw [Pell.d, Int.ofNat_sub this]\n[GOAL]\na : \u2115\na1 : 1 < a\nthis : 1 \u2264 a * a\n\u22a2 \u2191(a * a) - \u21911 = az a * az a - 1\n[PROOFSTEP]\nrfl\n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nh : IsPell { re := \u2191x, im := \u2191y }\n\u22a2 \u2191(x * x - Pell.d a1 * y * y) = \u21911\n[PROOFSTEP]\nrw [Int.ofNat_sub (Int.le_of_ofNat_le_ofNat <| Int.le.intro_sub _ h)]\n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nh : IsPell { re := \u2191x, im := \u2191y }\n\u22a2 \u2191(x * x) - \u2191(Pell.d a1 * y * y) = \u21911\n[PROOFSTEP]\nexact h\n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nh : x * x - Pell.d a1 * y * y = 1\n\u22a2 \u2191(x * x) - \u2191(Pell.d a1 * y * y) = 1\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_sub <| le_of_lt <| Nat.lt_of_sub_eq_succ h, h]\n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nh : x * x - Pell.d a1 * y * y = 1\n\u22a2 \u21911 = 1\n[PROOFSTEP]\nrfl\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 pellZd a1 (n + 1) = pellZd a1 n * { re := \u2191a, im := 1 }\n[PROOFSTEP]\nsimp [Zsqrtd.ext]\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 az a * az a - \u2191(Pell.d a1) * 1 * 1 = 1\n[PROOFSTEP]\nsimp [dz_val]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 IsPell (pellZd a1 (n + 1))\n[PROOFSTEP]\nlet o := isPell_one a1\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\no : IsPell { re := \u2191a, im := 1 } := isPell_one a1\n\u22a2 IsPell (pellZd a1 (n + 1))\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\no : IsPell { re := \u2191a, im := 1 } := isPell_one a1\n\u22a2 IsPell (pellZd a1 n * { re := \u2191a, im := 1 })\n[PROOFSTEP]\nexact Pell.isPell_mul (isPell_pellZd n) o\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\npn : xz a1 n * xz a1 n - \u2191(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n\u22a2 \u2191(xn a1 n * xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = 1\n[PROOFSTEP]\nrepeat' rw [Int.ofNat_mul]; exact pn\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\npn : xz a1 n * xz a1 n - \u2191(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n\u22a2 \u2191(xn a1 n * xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = 1\n[PROOFSTEP]\nrw [Int.ofNat_mul]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\npn : xz a1 n * xz a1 n - \u2191(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n\u22a2 \u2191(xn a1 n) * \u2191(xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = 1\n[PROOFSTEP]\nexact pn\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\npn : xz a1 n * xz a1 n - \u2191(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\nh : \u2191(xn a1 n * xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = 1\nhl : Pell.d a1 * yn a1 n * yn a1 n \u2264 xn a1 n * xn a1 n\n\u22a2 \u2191(xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n) = \u21911\n[PROOFSTEP]\nrw [Int.ofNat_sub hl]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\npn : xz a1 n * xz a1 n - \u2191(Pell.d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\nh : \u2191(xn a1 n * xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = 1\nhl : Pell.d a1 * yn a1 n * yn a1 n \u2264 xn a1 n * xn a1 n\n\u22a2 \u2191(xn a1 n * xn a1 n) - \u2191(Pell.d a1 * yn a1 n * yn a1 n) = \u21911\n[PROOFSTEP]\nexact h\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\n\u22a2 n * n + 1 = a * a\n[PROOFSTEP]\nrw [\u2190 h]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\n\u22a2 Pell.d a1 + 1 = a * a\n[PROOFSTEP]\nexact Nat.succ_pred_eq_of_pos (asq_pos a1)\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis : n * n + 1 = a * a\n\u22a2 n * n < a * a\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis : n * n + 1 = a * a\n\u22a2 n * n < n * n + 1\n[PROOFSTEP]\nexact Nat.lt_succ_self _\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis : n * n + 1 = a * a\nna : n < a\n\u22a2 (n + 1) * (n + 1) \u2264 n * n + 1\n[PROOFSTEP]\nrw [this]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis : n * n + 1 = a * a\nna : n < a\n\u22a2 (n + 1) * (n + 1) \u2264 a * a\n[PROOFSTEP]\nexact Nat.mul_self_le_mul_self na\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis\u271d : n * n + 1 = a * a\nna : n < a\nthis : (n + 1) * (n + 1) \u2264 n * n + 1\n\u22a2 n + n + (n * n + 1) \u2264 0 + (n * n + 1)\n[PROOFSTEP]\nring_nf at this \u22a2\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis\u271d : n * n + 1 = a * a\nna : n < a\nthis : 1 + n * 2 + n ^ 2 \u2264 1 + n ^ 2\n\u22a2 1 + n * 2 + n ^ 2 \u2264 1 + n ^ 2\n[PROOFSTEP]\nassumption\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : Pell.d a1 = n * n\nthis\u271d\u00b9 : n * n + 1 = a * a\nna : n < a\nthis\u271d : (n + 1) * (n + 1) \u2264 n * n + 1\nthis : n + n \u2264 0\n\u22a2 Pell.d a1 = 0\n[PROOFSTEP]\nrwa [Nat.eq_zero_of_le_zero ((Nat.le_add_left _ _).trans this)] at h \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 a ^ (n + 1) \u2264 xn a1 (n + 1)\n[PROOFSTEP]\nsimp [_root_.pow_succ']\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 a ^ n * a \u2264 xn a1 n * a + Pell.d a1 * yn a1 n\n[PROOFSTEP]\nexact le_trans (Nat.mul_le_mul_right _ (xn_ge_a_pow n)) (Nat.le_add_right _ _)\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 n + 1 < a ^ (n + 1)\n[PROOFSTEP]\nhave IH := n_lt_a_pow n\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\n\u22a2 n + 1 < a ^ (n + 1)\n[PROOFSTEP]\nhave : a ^ n + a ^ n \u2264 a ^ n * a := by\n  rw [\u2190 mul_two]\n  exact Nat.mul_le_mul_left _ a1\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\n\u22a2 a ^ n + a ^ n \u2264 a ^ n * a\n[PROOFSTEP]\nrw [\u2190 mul_two]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\n\u22a2 a ^ n * 2 \u2264 a ^ n * a\n[PROOFSTEP]\nexact Nat.mul_le_mul_left _ a1\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\nthis : a ^ n + a ^ n \u2264 a ^ n * a\n\u22a2 n + 1 < a ^ (n + 1)\n[PROOFSTEP]\nsimp [_root_.pow_succ']\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\nthis : a ^ n + a ^ n \u2264 a ^ n * a\n\u22a2 n + 1 < a ^ n * a\n[PROOFSTEP]\nrefine' lt_of_lt_of_le _ this\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nIH : n < a ^ n\nthis : a ^ n + a ^ n \u2264 a ^ n * a\n\u22a2 n + 1 < a ^ n + a ^ n\n[PROOFSTEP]\nexact add_lt_add_of_lt_of_le IH (lt_of_le_of_lt (Nat.zero_le _) IH)\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\n\u22a2 Pell.d a1 * (0 + 1) * (0 + 1) \u2264 1 * (a + 0) * (a + 0)\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\n\u22a2 Pell.d a1 \u2264 a * a\n[PROOFSTEP]\nexact Nat.pred_le _\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\n\u22a2 1 \u2264 b * { re := \u2191a, im := -1 }\n[PROOFSTEP]\nrw [\u2190 a1m]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\n\u22a2 { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } \u2264 b * { re := \u2191a, im := -1 }\n[PROOFSTEP]\nexact mul_le_mul_of_nonneg_right ha am1p\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\n\u22a2 b * { re := \u2191a, im := -1 } \u2264 pellZd a1 n\n[PROOFSTEP]\nhave t := mul_le_mul_of_nonneg_right h am1p\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\nt : b * { re := \u2191a, im := -1 } \u2264 pellZd a1 (n + 1) * { re := \u2191a, im := -1 }\n\u22a2 b * { re := \u2191a, im := -1 } \u2264 pellZd a1 n\n[PROOFSTEP]\nrwa [pellZd_succ, mul_assoc, a1m, mul_one] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\nm : \u2115\ne : b * { re := \u2191a, im := -1 } = pellZd a1 m\n\u22a2 b = pellZd a1 (m + 1)\n[PROOFSTEP]\nrw [show b = b * \u27e8a, -1\u27e9 * \u27e8a, 1\u27e9 by rw [mul_assoc, Eq.trans (mul_comm _ _) a1m]; simp, pellZd_succ, e]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\nm : \u2115\ne : b * { re := \u2191a, im := -1 } = pellZd a1 m\n\u22a2 b = b * { re := \u2191a, im := -1 } * { re := \u2191a, im := 1 }\n[PROOFSTEP]\nrw [mul_assoc, Eq.trans (mul_comm _ _) a1m]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : { re := \u2191a, im := 1 } \u2264 b\nm : \u2115\ne : b * { re := \u2191a, im := -1 } = pellZd a1 m\n\u22a2 b = b * 1\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nb : \u2124\u221a\u2191(Pell.d a1)\nh1 : 1 \u2264 b\nhp : IsPell b\nh : b \u2264 pellZd a1 (n + 1)\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 b\nh1l : 1 < b\n\u22a2 False\n[PROOFSTEP]\ncases' b with x y\n[GOAL]\ncase mk\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\n\u22a2 False\n[PROOFSTEP]\nexact by\n  have bm : (_ * \u27e8_, _\u27e9 : \u2124\u221ad a1) = 1 := Pell.isPell_norm.1 hp\n  have y0l : (0 : \u2124\u221ad a1) < \u27e8x - x, y - -y\u27e9 :=\n    sub_lt_sub h1l fun hn : (1 : \u2124\u221ad a1) \u2264 \u27e8x, -y\u27e9 =>\n      by\n      have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)\n      erw [bm, mul_one] at t \n      exact h1l t\n  have yl2 : (\u27e8_, _\u27e9 : \u2124\u221a_) < \u27e8_, _\u27e9 :=\n    show (\u27e8x, y\u27e9 - \u27e8x, -y\u27e9 : \u2124\u221ad a1) < \u27e8a, 1\u27e9 - \u27e8a, -1\u27e9 from\n      sub_lt_sub ha fun hn : (\u27e8x, -y\u27e9 : \u2124\u221ad a1) \u2264 \u27e8a, -1\u27e9 =>\n        by\n        have t := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p\n        erw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t \n        exact ha t\n  simp at y0l ; simp at yl2 \n  exact\n    match y, y0l, (yl2 : (\u27e8_, _\u27e9 : \u2124\u221a_) < \u27e8_, _\u27e9) with\n    | 0, y0l, _ => y0l (le_refl 0)\n    | (y + 1 : \u2115), _, yl2 =>\n      yl2\n        (Zsqrtd.le_of_le_le (by simp [sub_eq_add_neg])\n          (let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)\n          add_le_add t t))\n    | Int.negSucc _, y0l, _ => y0l trivial\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\n\u22a2 False\n[PROOFSTEP]\nhave bm : (_ * \u27e8_, _\u27e9 : \u2124\u221ad a1) = 1 := Pell.isPell_norm.1 hp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\n\u22a2 False\n[PROOFSTEP]\nhave y0l : (0 : \u2124\u221ad a1) < \u27e8x - x, y - -y\u27e9 :=\n  sub_lt_sub h1l fun hn : (1 : \u2124\u221ad a1) \u2264 \u27e8x, -y\u27e9 =>\n    by\n    have t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)\n    erw [bm, mul_one] at t \n    exact h1l t\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\nhn : 1 \u2264 { re := x, im := -y }\n\u22a2 False\n[PROOFSTEP]\nhave t := mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\nhn : 1 \u2264 { re := x, im := -y }\nt : { re := x, im := y } * 1 \u2264 { re := x, im := y } * { re := x, im := -y }\n\u22a2 False\n[PROOFSTEP]\nerw [bm, mul_one] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\nhn : 1 \u2264 { re := x, im := -y }\nt : { re := x, im := y } \u2264 1\n\u22a2 False\n[PROOFSTEP]\nexact h1l t\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := x - x, im := y - -y }\n\u22a2 False\n[PROOFSTEP]\nhave yl2 : (\u27e8_, _\u27e9 : \u2124\u221a_) < \u27e8_, _\u27e9 :=\n  show (\u27e8x, y\u27e9 - \u27e8x, -y\u27e9 : \u2124\u221ad a1) < \u27e8a, 1\u27e9 - \u27e8a, -1\u27e9 from\n    sub_lt_sub ha fun hn : (\u27e8x, -y\u27e9 : \u2124\u221ad a1) \u2264 \u27e8a, -1\u27e9 =>\n      by\n      have t := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p\n      erw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t \n      exact ha t\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := x - x, im := y - -y }\nhn : { re := x, im := -y } \u2264 { re := \u2191a, im := -1 }\n\u22a2 False\n[PROOFSTEP]\nhave t := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hn (le_trans zero_le_one h1)) a1p\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := x - x, im := y - -y }\nhn : { re := x, im := -y } \u2264 { re := \u2191a, im := -1 }\nt :\n  { re := x, im := y } * { re := x, im := -y } * { re := \u2191a, im := 1 } \u2264\n    { re := x, im := y } * { re := \u2191a, im := -1 } * { re := \u2191a, im := 1 }\n\u22a2 False\n[PROOFSTEP]\nerw [bm, one_mul, mul_assoc, Eq.trans (mul_comm _ _) a1m, mul_one] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := x - x, im := y - -y }\nhn : { re := x, im := -y } \u2264 { re := \u2191a, im := -1 }\nt : { re := \u2191a, im := 1 } \u2264 { re := x, im := y }\n\u22a2 False\n[PROOFSTEP]\nexact ha t\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := x - x, im := y - -y }\nyl2 :\n  { re := { re := x, im := y }.re + (-{ re := x, im := -y }).re,\n      im := { re := x, im := y }.im + (-{ re := x, im := -y }).im } <\n    { re := { re := \u2191a, im := 1 }.re + (-{ re := \u2191a, im := -1 }).re,\n      im := { re := \u2191a, im := 1 }.im + (-{ re := \u2191a, im := -1 }).im }\n\u22a2 False\n[PROOFSTEP]\nsimp at y0l \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\nyl2 :\n  { re := { re := x, im := y }.re + (-{ re := x, im := -y }).re,\n      im := { re := x, im := y }.im + (-{ re := x, im := -y }).im } <\n    { re := { re := \u2191a, im := 1 }.re + (-{ re := \u2191a, im := -1 }).re,\n      im := { re := \u2191a, im := 1 }.im + (-{ re := \u2191a, im := -1 }).im }\ny0l : 0 < { re := 0, im := y + y }\n\u22a2 False\n[PROOFSTEP]\nsimp at yl2 \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y : \u2124\nh1 : 1 \u2264 { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := y }\nh1l : 1 < { re := x, im := y }\nbm : { re := x, im := y } * { re := { re := x, im := y }.re, im := -{ re := x, im := y }.im } = 1\ny0l : 0 < { re := 0, im := y + y }\nyl2 : { re := 0, im := y + y } < { re := 0, im := 1 + 1 }\n\u22a2 False\n[PROOFSTEP]\nexact\n  match y, y0l, (yl2 : (\u27e8_, _\u27e9 : \u2124\u221a_) < \u27e8_, _\u27e9) with\n  | 0, y0l, _ => y0l (le_refl 0)\n  | (y + 1 : \u2115), _, yl2 =>\n    yl2\n      (Zsqrtd.le_of_le_le (by simp [sub_eq_add_neg])\n        (let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)\n        add_le_add t t))\n  | Int.negSucc _, y0l, _ => y0l trivial\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\na1p : 0 \u2264 { re := \u2191a, im := 1 }\nam1p : 0 \u2264 { re := \u2191a, im := -1 }\na1m : { re := \u2191a, im := 1 } * { re := \u2191a, im := -1 } = 1\nx y\u271d : \u2124\ny0l : 0 < { re := 0, im := y\u271d + y\u271d }\nyl2\u271d : { re := 0, im := y\u271d + y\u271d } < { re := 0, im := 1 + 1 }\ny : \u2115\nx\u271d : 0 < { re := 0, im := \u2191(y + 1) + \u2191(y + 1) }\nyl2 : { re := 0, im := \u2191(y + 1) + \u2191(y + 1) } < { re := 0, im := 1 + 1 }\nh1 : 1 \u2264 { re := x, im := \u2191(y + 1) }\nhp : IsPell { re := x, im := \u2191(y + 1) }\nh : { re := x, im := \u2191(y + 1) } \u2264 pellZd a1 (n + 1)\nha : \u00ac{ re := \u2191a, im := 1 } \u2264 { re := x, im := \u2191(y + 1) }\nh1l : 1 < { re := x, im := \u2191(y + 1) }\nbm : { re := x, im := \u2191(y + 1) } * { re := { re := x, im := \u2191(y + 1) }.re, im := -{ re := x, im := \u2191(y + 1) }.im } = 1\n\u22a2 0 \u2264 0\n[PROOFSTEP]\nsimp [sub_eq_add_neg]\n[GOAL]\na : \u2115\na1 : 1 < a\nb : \u2124\u221a\u2191(Pell.d a1)\nb1 : 1 \u2264 b\nhp : IsPell b\nn : \u2115\nh : b \u2264 \u2191n\n\u22a2 \u2191n \u2264 pellZd a1 n\n[PROOFSTEP]\nrw [Zsqrtd.coe_nat_val]\n[GOAL]\na : \u2115\na1 : 1 < a\nb : \u2124\u221a\u2191(Pell.d a1)\nb1 : 1 \u2264 b\nhp : IsPell b\nn : \u2115\nh : b \u2264 \u2191n\n\u22a2 { re := \u2191n, im := 0 } \u2264 pellZd a1 n\n[PROOFSTEP]\nexact Zsqrtd.le_of_le_le (Int.ofNat_le_ofNat_of_le <| le_of_lt <| n_lt_xn _ _) (Int.ofNat_zero_le _)\n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nhp\u271d : x * x - Pell.d a1 * y * y = 1\nhp : 0 - Pell.d a1 * y * y = 1\n\u22a2 1 \u2264 { re := \u21910, im := \u2191y }\n[PROOFSTEP]\nrw [zero_tsub] at hp \n[GOAL]\na : \u2115\na1 : 1 < a\nx y : \u2115\nhp\u271d : x * x - Pell.d a1 * y * y = 1\nhp : 0 = 1\n\u22a2 1 \u2264 { re := \u21910, im := \u2191y }\n[PROOFSTEP]\ncontradiction\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\n\u22a2 pellZd a1 (m + (n + 1)) = pellZd a1 m * pellZd a1 (n + 1)\n[PROOFSTEP]\nrw [\u2190 add_assoc, pellZd_succ, pellZd_succ, pellZd_add _ n, \u2190 mul_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\n\u22a2 xn a1 (m + n) = xn a1 m * xn a1 n + Pell.d a1 * yn a1 m * yn a1 n\n[PROOFSTEP]\ninjection pellZd_add a1 m n with h _\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nim_eq\u271d : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 xn a1 (m + n) = xn a1 m * xn a1 n + Pell.d a1 * yn a1 m * yn a1 n\n[PROOFSTEP]\nzify\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nim_eq\u271d : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 \u2191(xn a1 (m + n)) = \u2191(xn a1 m) * \u2191(xn a1 n) + \u2191(Pell.d a1) * \u2191(yn a1 m) * \u2191(yn a1 n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nim_eq\u271d : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im =\n    \u2191(xn a1 m) * \u2191(xn a1 n) + \u2191(Pell.d a1) * \u2191(yn a1 m) * \u2191(yn a1 n)\n[PROOFSTEP]\nsimp [pellZd]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\n\u22a2 yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n\n[PROOFSTEP]\ninjection pellZd_add a1 m n with _ h\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nre_eq\u271d : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nh : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 yn a1 (m + n) = xn a1 m * yn a1 n + yn a1 m * xn a1 n\n[PROOFSTEP]\nzify\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nre_eq\u271d : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nh : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 \u2191(yn a1 (m + n)) = \u2191(xn a1 m) * \u2191(yn a1 n) + \u2191(yn a1 m) * \u2191(xn a1 n)\n[PROOFSTEP]\nrw [h]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nre_eq\u271d : \u2191(xn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).re + \u2191(Pell.d a1) * (pellZd a1 m).im * (pellZd a1 n).im\nh : \u2191(yn a1 (m + n)) = (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re\n\u22a2 (pellZd a1 m).re * (pellZd a1 n).im + (pellZd a1 m).im * (pellZd a1 n).re =\n    \u2191(xn a1 m) * \u2191(yn a1 n) + \u2191(yn a1 m) * \u2191(xn a1 n)\n[PROOFSTEP]\nsimp [pellZd]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\n\u22a2 pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)\n[PROOFSTEP]\nlet t := pellZd_add a1 n (m - n)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\nt : pellZd a1 (n + (m - n)) = pellZd a1 n * pellZd a1 (m - n) := pellZd_add a1 n (m - n)\n\u22a2 pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)\n[PROOFSTEP]\nrw [add_tsub_cancel_of_le h] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\nt : pellZd a1 m = pellZd a1 n * pellZd a1 (m - n)\n\u22a2 pellZd a1 (m - n) = pellZd a1 m * star (pellZd a1 n)\n[PROOFSTEP]\nrw [t, mul_comm (pellZd _ n) _, mul_assoc, isPell_norm.1 (isPell_pellZd _ _), mul_one]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\n\u22a2 xz a1 (m - n) = xz a1 m * xz a1 n - \u2191(Pell.d a1) * yz a1 m * yz a1 n\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 mul_neg]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\n\u22a2 xz a1 (m - n) = xz a1 m * xz a1 n + \u2191(Pell.d a1) * yz a1 m * -yz a1 n\n[PROOFSTEP]\nexact congr_arg Zsqrtd.re (pellZd_sub a1 h)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\n\u22a2 yz a1 (m - n) = xz a1 n * yz a1 m - xz a1 m * yz a1 n\n[PROOFSTEP]\nrw [sub_eq_add_neg, \u2190 mul_neg, mul_comm, add_comm]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : n \u2264 m\n\u22a2 yz a1 (m - n) = xz a1 m * -yz a1 n + yz a1 m * xz a1 n\n[PROOFSTEP]\nexact congr_arg Zsqrtd.im (pellZd_sub a1 h)\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\n\u22a2 k \u2223 1\n[PROOFSTEP]\nlet p := pell_eq a1 n\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 1\n[PROOFSTEP]\nrw [\u2190 p]\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nx\u271d : Nat.Prime k\nkx : k \u2223 xn a1 n\nky : k \u2223 yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n\u22a2 k \u2223 xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n\n[PROOFSTEP]\nexact Nat.dvd_sub (le_of_lt <| Nat.lt_of_sub_eq_succ p) (kx.mul_left _) (ky.mul_left _)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\n\u22a2 yn a1 m < yn a1 (n + 1)\n[PROOFSTEP]\nhave : yn a1 m \u2264 yn a1 n :=\n  Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_y hl) fun e => by rw [e]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\ne : m = n\n\u22a2 yn a1 m \u2264 yn a1 n\n[PROOFSTEP]\nrw [e]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : yn a1 m \u2264 yn a1 n\n\u22a2 yn a1 m < yn a1 (n + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : yn a1 m \u2264 yn a1 n\n\u22a2 yn a1 m < xn a1 n + yn a1 n * a\n[PROOFSTEP]\nrefine' lt_of_le_of_lt _ (Nat.lt_add_of_pos_left <| x_pos a1 n)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : yn a1 m \u2264 yn a1 n\n\u22a2 yn a1 m \u2264 yn a1 n * a\n[PROOFSTEP]\nrw [\u2190 mul_one (yn a1 m)]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : yn a1 m \u2264 yn a1 n\n\u22a2 yn a1 m * 1 \u2264 yn a1 n * a\n[PROOFSTEP]\nexact mul_le_mul this (le_of_lt a1) (Nat.zero_le _) (Nat.zero_le _)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\n\u22a2 xn a1 m < xn a1 (n + 1)\n[PROOFSTEP]\nhave : xn a1 m \u2264 xn a1 n :=\n  Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl) fun e => by rw [e]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\ne : m = n\n\u22a2 xn a1 m \u2264 xn a1 n\n[PROOFSTEP]\nrw [e]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : xn a1 m \u2264 xn a1 n\n\u22a2 xn a1 m < xn a1 (n + 1)\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : xn a1 m \u2264 xn a1 n\n\u22a2 xn a1 m < xn a1 n * a + Pell.d a1 * yn a1 n\n[PROOFSTEP]\nrefine' lt_of_lt_of_le (lt_of_le_of_lt this _) (Nat.le_add_right _ _)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : xn a1 m \u2264 xn a1 n\n\u22a2 xn a1 n < xn a1 n * a\n[PROOFSTEP]\nhave t := Nat.mul_lt_mul_of_pos_left a1 (x_pos a1 n)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : m < n + 1\nthis : xn a1 m \u2264 xn a1 n\nt : xn a1 n * 1 < xn a1 n * a\n\u22a2 xn a1 n < xn a1 n * a\n[PROOFSTEP]\nrwa [mul_one] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\n\u22a2 yn a1 n \u2223 yn a1 (n * (k + 1))\n[PROOFSTEP]\nrw [Nat.mul_succ, yn_add]\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\n\u22a2 yn a1 n \u2223 xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n\n[PROOFSTEP]\nexact dvd_add (dvd_mul_left _ _) ((y_mul_dvd _ k).mul_right _)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\n\u22a2 False\n[PROOFSTEP]\nhave co : Nat.coprime (yn a1 m) (xn a1 (m * (n / m))) :=\n  Nat.coprime.symm <| (xy_coprime a1 _).coprime_dvd_right (y_mul_dvd a1 m (n / m))\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\n\u22a2 False\n[PROOFSTEP]\nhave m0 : 0 < m :=\n  m.eq_zero_or_pos.resolve_left fun e => by\n    rw [e, Nat.mod_zero] at hp ; rw [e] at h \n    exact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e, Nat.mod_zero] at hp \n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False\n[PROOFSTEP]\nrw [e] at h \n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 0 \u2223 yn a1 n\nhp : n > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\ne : m = 0\n\u22a2 False\n[PROOFSTEP]\nexact _root_.ne_of_lt (strictMono_y a1 hp) (eq_zero_of_zero_dvd h).symm\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 yn a1 n\nhp : n % m > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div n m, yn_add] at h \n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nh : yn a1 m \u2223 xn a1 (n % m) * yn a1 (m * (n / m)) + yn a1 (n % m) * xn a1 (m * (n / m))\nhp : n % m > 0\nco : coprime (yn a1 m) (xn a1 (m * (n / m)))\nm0 : 0 < m\n\u22a2 False\n[PROOFSTEP]\nexact\n  not_le_of_gt (strictMono_y _ <| Nat.mod_lt n m0)\n    (Nat.le_of_dvd (strictMono_y _ hp) <|\n      co.dvd_of_dvd_mul_right <| (Nat.dvd_add_iff_right <| (y_mul_dvd _ _ _).mul_left _).2 h)\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nx\u271d : m \u2223 n\nk : \u2115\ne : n = m * k\n\u22a2 yn a1 m \u2223 yn a1 n\n[PROOFSTEP]\nrw [e]\n[GOAL]\na : \u2115\na1 : 1 < a\nm n : \u2115\nx\u271d : m \u2223 n\nk : \u2115\ne : n = m * k\n\u22a2 yn a1 m \u2223 yn a1 (m * k)\n[PROOFSTEP]\napply y_mul_dvd\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 xn a1 (n * 0) \u2261 xn a1 n ^ 0 [MOD yn a1 n ^ 2] \u2227 yn a1 (n * 0) \u2261 0 * xn a1 n ^ (0 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 xn a1 (n * 0) \u2261 xn a1 n ^ 0 [MOD yn a1 n ^ 2]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase right\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yn a1 (n * 0) \u2261 0 * xn a1 n ^ (0 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nsimp\n[GOAL]\ncase left\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 1 \u2261 1 [MOD yn a1 n ^ 2]\n[PROOFSTEP]\nexact Nat.ModEq.refl _\n[GOAL]\ncase right\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 0 \u2261 0 [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nexact Nat.ModEq.refl _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\n\u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227\n    yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nlet \u27e8hx, hy\u27e9 := xy_modEq_yn n k\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227\n    yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nhave L : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] :=\n  (hx.mul_right _).add <|\n    modEq_zero_iff_dvd.2 <| by\n      rw [_root_.pow_succ']\n      exact\n        mul_dvd_mul_right\n          (dvd_mul_of_dvd_right\n            (modEq_zero_iff_dvd.1 <| (hy.of_dvd <| by simp [_root_.pow_succ']).trans <| modEq_zero_iff_dvd.2 <| by simp)\n            _)\n          _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 yn a1 n ^ 2 \u2223 Pell.d a1 * yn a1 (n * k) * yn a1 n\n[PROOFSTEP]\nrw [_root_.pow_succ']\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 yn a1 n ^ 1 * yn a1 n \u2223 Pell.d a1 * yn a1 (n * k) * yn a1 n\n[PROOFSTEP]\nexact\n  mul_dvd_mul_right\n    (dvd_mul_of_dvd_right\n      (modEq_zero_iff_dvd.1 <| (hy.of_dvd <| by simp [_root_.pow_succ']).trans <| modEq_zero_iff_dvd.2 <| by simp) _)\n    _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 yn a1 n ^ 1 \u2223 yn a1 n ^ 3\n[PROOFSTEP]\nsimp [_root_.pow_succ']\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 yn a1 n ^ 1 \u2223 k * xn a1 n ^ (k - 1) * yn a1 n\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227\n    yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nhave R :\n  xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD\n    yn a1 n ^ 3] :=\n  ModEq.add\n      (by\n        rw [_root_.pow_succ']\n        exact hx.mul_right' _) <|\n    by\n    have : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n := by\n      cases' k with k <;> simp [_root_.pow_succ']; ring_nf\n    rw [\u2190 this]\n    exact hy.mul_right _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 xn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nrw [_root_.pow_succ']\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 xn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 2 * yn a1 n]\n[PROOFSTEP]\nexact hx.mul_right' _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nhave : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n := by\n  cases' k with k <;> simp [_root_.pow_succ']; ring_nf\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n\n[PROOFSTEP]\ncases' k with k\n[GOAL]\ncase zero\na : \u2115\na1 : 1 < a\nn : \u2115\nhx : xn a1 (n * zero) \u2261 xn a1 n ^ zero [MOD yn a1 n ^ 2]\nhy : yn a1 (n * zero) \u2261 zero * xn a1 n ^ (zero - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * zero) * xn a1 n + Pell.d a1 * yn a1 (n * zero) * yn a1 n \u2261 xn a1 n ^ zero * xn a1 n + 0 [MOD yn a1 n ^ 2]\n\u22a2 zero * xn a1 n ^ (zero - 1) * yn a1 n * xn a1 n = zero * xn a1 n ^ zero * yn a1 n\n[PROOFSTEP]\nsimp [_root_.pow_succ']\n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * succ k) \u2261 xn a1 n ^ succ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * succ k) \u2261 succ k * xn a1 n ^ (succ k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL :\n  xn a1 (n * succ k) * xn a1 n + Pell.d a1 * yn a1 (n * succ k) * yn a1 n \u2261 xn a1 n ^ succ k * xn a1 n + 0 [MOD\n    yn a1 n ^ 2]\n\u22a2 succ k * xn a1 n ^ (succ k - 1) * yn a1 n * xn a1 n = succ k * xn a1 n ^ succ k * yn a1 n\n[PROOFSTEP]\nsimp [_root_.pow_succ']\n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * succ k) \u2261 xn a1 n ^ succ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * succ k) \u2261 succ k * xn a1 n ^ (succ k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL :\n  xn a1 (n * succ k) * xn a1 n + Pell.d a1 * yn a1 (n * succ k) * yn a1 n \u2261 xn a1 n ^ succ k * xn a1 n + 0 [MOD\n    yn a1 n ^ 2]\n\u22a2 succ k * xn a1 n ^ k * yn a1 n * xn a1 n = succ k * (xn a1 n ^ k * xn a1 n) * yn a1 n\n[PROOFSTEP]\nring_nf\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nthis : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n\n\u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nrw [\u2190 this]\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nthis : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n\n\u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nexact hy.mul_right _\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nR :\n  xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD\n    yn a1 n ^ 3]\n\u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227\n    yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3]\n[PROOFSTEP]\nrw [add_tsub_cancel_right, Nat.mul_succ, xn_add, yn_add, pow_succ' (xn _ n), Nat.succ_mul,\n  add_comm (k * xn _ n ^ k) (xn _ n ^ k), right_distrib]\n[GOAL]\na : \u2115\na1 : 1 < a\nn k : \u2115\nhx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2]\nhy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2]\nR :\n  xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD\n    yn a1 n ^ 3]\n\u22a2 xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n [MOD yn a1 n ^ 2] \u2227\n    xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD\n      yn a1 n ^ 3]\n[PROOFSTEP]\nexact \u27e8L, R\u27e9\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yn a1 n * yn a1 n \u2223 yn a1 n ^ 3\n[PROOFSTEP]\nsimp [_root_.pow_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yn a1 n * yn a1 n \u2223 yn a1 n * xn a1 n ^ (yn a1 n - 1) * yn a1 n\n[PROOFSTEP]\nsimp [mul_dvd_mul_left, mul_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0 : n = 0\n\u22a2 yn a1 n \u2223 t\n[PROOFSTEP]\nrwa [n0] at nt \u22a2\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\n\u22a2 yn a1 n \u2223 t\n[PROOFSTEP]\nlet \u27e8k, ke\u27e9 := nt\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\n\u22a2 yn a1 n \u2223 t\n[PROOFSTEP]\nhave : yn a1 n \u2223 k * xn a1 n ^ (k - 1) :=\n  Nat.dvd_of_mul_dvd_mul_right (strictMono_y a1 n0l) <|\n    modEq_zero_iff_dvd.1 <| by\n      have xm := (xy_modEq_yn a1 n k).right; rw [\u2190 ke] at xm \n      exact (xm.of_dvd <| by simp [_root_.pow_succ]).symm.trans h.modEq_zero_nat\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\n\u22a2 k * xn a1 n ^ (k - 1) * yn a1 n \u2261 0 [MOD yn a1 n * yn a1 n]\n[PROOFSTEP]\nhave xm := (xy_modEq_yn a1 n k).right\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\nxm : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 k * xn a1 n ^ (k - 1) * yn a1 n \u2261 0 [MOD yn a1 n * yn a1 n]\n[PROOFSTEP]\nrw [\u2190 ke] at xm \n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\nxm : yn a1 t \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 k * xn a1 n ^ (k - 1) * yn a1 n \u2261 0 [MOD yn a1 n * yn a1 n]\n[PROOFSTEP]\nexact (xm.of_dvd <| by simp [_root_.pow_succ]).symm.trans h.modEq_zero_nat\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\nxm : yn a1 t \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3]\n\u22a2 yn a1 n * yn a1 n \u2223 yn a1 n ^ 3\n[PROOFSTEP]\nsimp [_root_.pow_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\nthis : yn a1 n \u2223 k * xn a1 n ^ (k - 1)\n\u22a2 yn a1 n \u2223 t\n[PROOFSTEP]\nrw [ke]\n[GOAL]\na : \u2115\na1 : 1 < a\nn t : \u2115\nh : yn a1 n * yn a1 n \u2223 yn a1 t\nnt : n \u2223 t\nn0l : 0 < n\nk : \u2115\nke : t = n * k\nthis : yn a1 n \u2223 k * xn a1 n ^ (k - 1)\n\u22a2 yn a1 n \u2223 n * k\n[PROOFSTEP]\nexact dvd_mul_of_dvd_right (((xy_coprime _ _).pow_left _).symm.dvd_of_dvd_mul_right this) _\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 pellZd a1 (n + 2) + pellZd a1 n = \u2191(2 * a) * pellZd a1 (n + 1)\n[PROOFSTEP]\nhave : (1 : \u2124\u221a(d a1)) + \u27e8a, 1\u27e9 * \u27e8a, 1\u27e9 = \u27e8a, 1\u27e9 * (2 * a) :=\n  by\n  rw [Zsqrtd.coe_nat_val]\n  change (\u27e8_, _\u27e9 : \u2124\u221a(d a1)) = \u27e8_, _\u27e9\n  rw [dz_val]\n  dsimp [az]\n  rw [Zsqrtd.ext]\n  dsimp\n  constructor <;> ring_nf\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 1 + { re := \u2191a, im := 1 } * { re := \u2191a, im := 1 } = { re := \u2191a, im := 1 } * (2 * \u2191a)\n[PROOFSTEP]\nrw [Zsqrtd.coe_nat_val]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 1 + { re := \u2191a, im := 1 } * { re := \u2191a, im := 1 } = { re := \u2191a, im := 1 } * (2 * { re := \u2191a, im := 0 })\n[PROOFSTEP]\nchange (\u27e8_, _\u27e9 : \u2124\u221a(d a1)) = \u27e8_, _\u27e9\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 { re := 1.re + ({ re := \u2191a, im := 1 } * { re := \u2191a, im := 1 }).re,\n      im := 1.im + ({ re := \u2191a, im := 1 } * { re := \u2191a, im := 1 }).im } =\n    {\n      re :=\n        { re := \u2191a, im := 1 }.re * (2 * { re := \u2191a, im := 0 }).re +\n          \u2191(Pell.d a1) * { re := \u2191a, im := 1 }.im * (2 * { re := \u2191a, im := 0 }).im,\n      im :=\n        { re := \u2191a, im := 1 }.re * (2 * { re := \u2191a, im := 0 }).im +\n          { re := \u2191a, im := 1 }.im * (2 * { re := \u2191a, im := 0 }).re }\n[PROOFSTEP]\nrw [dz_val]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 { re := 1.re + ({ re := \u2191a, im := 1 } * { re := \u2191a, im := 1 }).re,\n      im := 1.im + ({ re := \u2191a, im := 1 } * { re := \u2191a, im := 1 }).im } =\n    {\n      re :=\n        { re := \u2191a, im := 1 }.re * (2 * { re := \u2191a, im := 0 }).re +\n          (az a * az a - 1) * { re := \u2191a, im := 1 }.im * (2 * { re := \u2191a, im := 0 }).im,\n      im :=\n        { re := \u2191a, im := 1 }.re * (2 * { re := \u2191a, im := 0 }).im +\n          { re := \u2191a, im := 1 }.im * (2 * { re := \u2191a, im := 0 }).re }\n[PROOFSTEP]\ndsimp [az]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 { re := 1 + (\u2191a * \u2191a + (\u2191a * \u2191a - 1) * 1 * 1), im := 0 + (\u2191a * 1 + 1 * \u2191a) } =\n    { re := \u2191a * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) + (\u2191a * \u2191a - 1) * 1 * (\u21912 * 0 + 0 * \u2191a),\n      im := \u2191a * (\u21912 * 0 + 0 * \u2191a) + 1 * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) }\n[PROOFSTEP]\nrw [Zsqrtd.ext]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 { re := 1 + (\u2191a * \u2191a + (\u2191a * \u2191a - 1) * 1 * 1), im := 0 + (\u2191a * 1 + 1 * \u2191a) }.re =\n      { re := \u2191a * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) + (\u2191a * \u2191a - 1) * 1 * (\u21912 * 0 + 0 * \u2191a),\n          im := \u2191a * (\u21912 * 0 + 0 * \u2191a) + 1 * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) }.re \u2227\n    { re := 1 + (\u2191a * \u2191a + (\u2191a * \u2191a - 1) * 1 * 1), im := 0 + (\u2191a * 1 + 1 * \u2191a) }.im =\n      { re := \u2191a * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) + (\u2191a * \u2191a - 1) * 1 * (\u21912 * 0 + 0 * \u2191a),\n          im := \u2191a * (\u21912 * 0 + 0 * \u2191a) + 1 * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) }.im\n[PROOFSTEP]\ndsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 1 + (\u2191a * \u2191a + (\u2191a * \u2191a - 1) * 1 * 1) =\n      \u2191a * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) + (\u2191a * \u2191a - 1) * 1 * (\u21912 * 0 + 0 * \u2191a) \u2227\n    0 + (\u2191a * 1 + 1 * \u2191a) = \u2191a * (\u21912 * 0 + 0 * \u2191a) + 1 * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 1 + (\u2191a * \u2191a + (\u2191a * \u2191a - 1) * 1 * 1) = \u2191a * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0) + (\u2191a * \u2191a - 1) * 1 * (\u21912 * 0 + 0 * \u2191a)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase right\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 0 + (\u2191a * 1 + 1 * \u2191a) = \u2191a * (\u21912 * 0 + 0 * \u2191a) + 1 * (\u21912 * \u2191a + (\u2191a * \u2191a - 1) * 0 * 0)\n[PROOFSTEP]\nring_nf\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nthis : 1 + { re := \u2191a, im := 1 } * { re := \u2191a, im := 1 } = { re := \u2191a, im := 1 } * (2 * \u2191a)\n\u22a2 pellZd a1 (n + 2) + pellZd a1 n = \u2191(2 * a) * pellZd a1 (n + 1)\n[PROOFSTEP]\nsimpa [mul_add, mul_comm, mul_left_comm, add_comm] using congr_arg (\u00b7 * pellZd a1 n) this\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\nhave := pellZd_succ_succ a1 n\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nthis : pellZd a1 (n + 2) + pellZd a1 n = \u2191(2 * a) * pellZd a1 (n + 1)\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\nunfold pellZd at this \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nthis :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) } + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) } =\n    \u2191(2 * a) * { re := \u2191(xn a1 (n + 1)), im := \u2191(yn a1 (n + 1)) }\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\nerw [Zsqrtd.smul_val (2 * a : \u2115)] at this \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nthis :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) } + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) } =\n    { re := \u2191(2 * a) * \u2191(xn a1 (n + 1)), im := \u2191(2 * a) * \u2191(yn a1 (n + 1)) }\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\ninjection this with h\u2081 h\u2082\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\nconstructor <;> apply Int.ofNat.inj <;> [simpa using h\u2081; simpa using h\u2082]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) \u2227 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1)\n[PROOFSTEP]\napply Int.ofNat.inj\n[GOAL]\ncase right\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)\n[PROOFSTEP]\napply Int.ofNat.inj\n[GOAL]\ncase left.x\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 Int.ofNat (xn a1 (n + 2) + xn a1 n) = Int.ofNat (2 * a * xn a1 (n + 1))\n[PROOFSTEP]\nsimpa using h\u2081\n[GOAL]\ncase right.x\na : \u2115\na1 : 1 < a\nn : \u2115\nh\u2081 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.re + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.re =\n    \u2191(2 * a) * \u2191(xn a1 (n + 1))\nh\u2082 :\n  { re := \u2191(xn a1 (n + 2)), im := \u2191(yn a1 (n + 2)) }.im + { re := \u2191(xn a1 n), im := \u2191(yn a1 n) }.im =\n    \u2191(2 * a) * \u2191(yn a1 (n + 1))\n\u22a2 Int.ofNat (yn a1 (n + 2) + yn a1 n) = Int.ofNat (2 * a * yn a1 (n + 1))\n[PROOFSTEP]\nsimpa using h\u2082\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 xz a1 (n + 2) + xz a1 n = \u2191(2 * a) * xz a1 (n + 1)\n[PROOFSTEP]\ndelta xz\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 \u2191(xn a1 (n + 2)) + \u2191(xn a1 n) = \u2191(2 * a) * \u2191(xn a1 (n + 1))\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_add, \u2190 Int.ofNat_mul, xn_succ_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yz a1 (n + 2) + yz a1 n = \u2191(2 * a) * yz a1 (n + 1)\n[PROOFSTEP]\ndelta yz\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 \u2191(yn a1 (n + 2)) + \u2191(yn a1 n) = \u2191(2 * a) * \u2191(yn a1 (n + 1))\n[PROOFSTEP]\nrw [\u2190 Int.ofNat_add, \u2190 Int.ofNat_mul, yn_succ_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 yn a1 0 \u2261 0 [MOD a - 1]\n[PROOFSTEP]\nsimp [Nat.ModEq.refl]\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 yn a1 1 \u2261 1 [MOD a - 1]\n[PROOFSTEP]\nsimp [Nat.ModEq.refl]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yn a1 (n + 2) + yn a1 n \u2261 n + 2 + n [MOD a - 1]\n[PROOFSTEP]\nrw [yn_succ_succ, (by ring : n + 2 + n = 2 * (n + 1))]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 n + 2 + n = 2 * (n + 1)\n[PROOFSTEP]\nring\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 2 * a * yn a1 (n + 1) \u2261 2 * (n + 1) [MOD a - 1]\n[PROOFSTEP]\nexact ((modEq_sub a1.le).mul_left 2).mul (yn_modEq_a_sub_one (n + 1))\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 yn a1 0 \u2261 0 [MOD 2]\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\n\u22a2 yn a1 1 \u2261 1 [MOD 2]\n[PROOFSTEP]\nsimp\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 yn a1 (n + 2) + yn a1 n \u2261 n + 2 + n [MOD 2]\n[PROOFSTEP]\nrw [yn_succ_succ, mul_assoc, (by ring : n + 2 + n = 2 * (n + 1))]\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 n + 2 + n = 2 * (n + 1)\n[PROOFSTEP]\nring\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\n\u22a2 2 * (a * yn a1 (n + 1)) \u2261 2 * (n + 1) [MOD 2]\n[PROOFSTEP]\nexact (dvd_mul_right 2 _).modEq_zero_nat.trans (dvd_mul_right 2 _).zero_modEq_nat\n[GOAL]\na : \u2115\na1 : 1 < a\ny2 y1 y0 yn1 yn0 xn1 xn0 ay a2 : \u2124\n\u22a2 (a2 * yn1 - yn0) * ay + y2 - (a2 * xn1 - xn0) = y2 - a2 * y1 + y0 + a2 * (yn1 * ay + y1 - xn1) - (yn0 * ay + y0 - xn0)\n[PROOFSTEP]\nring\n[GOAL]\na : \u2115\na1 : 1 < a\ny : \u2115\n\u22a2 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 yz a1 0 * (\u2191a - \u2191y) + \u2191(y ^ 0) - xz a1 0\n[PROOFSTEP]\nsimp [xz, yz, Int.ofNat_zero, Int.ofNat_one]\n[GOAL]\na : \u2115\na1 : 1 < a\ny : \u2115\n\u22a2 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 yz a1 1 * (\u2191a - \u2191y) + \u2191(y ^ 1) - xz a1 1\n[PROOFSTEP]\nsimp [xz, yz, Int.ofNat_zero, Int.ofNat_one]\n[GOAL]\na : \u2115\na1 : 1 < a\ny n : \u2115\n\u22a2 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 yz a1 (n + 2) * (\u2191a - \u2191y) + \u2191(y ^ (n + 2)) - xz a1 (n + 2)\n[PROOFSTEP]\nhave : (2 * a * y - y * y - 1 : \u2124) \u2223 \u2191(y ^ (n + 2)) - \u2191(2 * a) * \u2191(y ^ (n + 1)) + \u2191(y ^ n) :=\n  \u27e8-\u2191(y ^ n),\n    by\n    simp [_root_.pow_succ, mul_add, Int.ofNat_mul, show ((2 : \u2115) : \u2124) = 2 from rfl, mul_comm, mul_left_comm]\n    ring\u27e9\n[GOAL]\na : \u2115\na1 : 1 < a\ny n : \u2115\n\u22a2 \u2191(y ^ (n + 2)) - \u2191(2 * a) * \u2191(y ^ (n + 1)) + \u2191(y ^ n) = (2 * \u2191a * \u2191y - \u2191y * \u2191y - 1) * -\u2191(y ^ n)\n[PROOFSTEP]\nsimp [_root_.pow_succ, mul_add, Int.ofNat_mul, show ((2 : \u2115) : \u2124) = 2 from rfl, mul_comm, mul_left_comm]\n[GOAL]\na : \u2115\na1 : 1 < a\ny n : \u2115\n\u22a2 \u2191y * (\u2191y * \u2191y ^ n) - \u2191a * (\u2191y * \u2191y ^ n * 2) + \u2191y ^ n = -(\u2191y ^ n * (\u2191a * (\u2191y * 2) - \u2191y * \u2191y - 1))\n[PROOFSTEP]\nring\n[GOAL]\na : \u2115\na1 : 1 < a\ny n : \u2115\nthis : 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 \u2191(y ^ (n + 2)) - \u2191(2 * a) * \u2191(y ^ (n + 1)) + \u2191(y ^ n)\n\u22a2 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 yz a1 (n + 2) * (\u2191a - \u2191y) + \u2191(y ^ (n + 2)) - xz a1 (n + 2)\n[PROOFSTEP]\nrw [xz_succ_succ, yz_succ_succ, x_sub_y_dvd_pow_lem \u2191(y ^ (n + 2)) \u2191(y ^ (n + 1)) \u2191(y ^ n)]\n[GOAL]\na : \u2115\na1 : 1 < a\ny n : \u2115\nthis : 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223 \u2191(y ^ (n + 2)) - \u2191(2 * a) * \u2191(y ^ (n + 1)) + \u2191(y ^ n)\n\u22a2 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1 \u2223\n    \u2191(y ^ (n + 2)) - \u2191(2 * a) * \u2191(y ^ (n + 1)) + \u2191(y ^ n) +\n        \u2191(2 * a) * (yz a1 (n + 1) * (\u2191a - \u2191y) + \u2191(y ^ (n + 1)) - xz a1 (n + 1)) -\n      (yz a1 n * (\u2191a - \u2191y) + \u2191(y ^ n) - xz a1 n)\n[PROOFSTEP]\nexact _root_.dvd_sub (dvd_add this <| (x_sub_y_dvd_pow _ (n + 1)).mul_left _) (x_sub_y_dvd_pow _ n)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 n \u2223 Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j\n[PROOFSTEP]\nhave h1 : d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (d a1 * yn a1 n * yn a1 n + 1) * xn a1 j := by\n  simp [add_mul, mul_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (Pell.d a1 * yn a1 n * yn a1 n + 1) * xn a1 j\n[PROOFSTEP]\nsimp [add_mul, mul_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (Pell.d a1 * yn a1 n * yn a1 n + 1) * xn a1 j\n\u22a2 xn a1 n \u2223 Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j\n[PROOFSTEP]\nhave h2 : d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n :=\n  by\n  zify at *\n  apply add_eq_of_eq_sub' (Eq.symm (pell_eqz a1 n))\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (Pell.d a1 * yn a1 n * yn a1 n + 1) * xn a1 j\n\u22a2 Pell.d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n\n[PROOFSTEP]\nzify at *\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 :\n  \u2191(Pell.d a1) * \u2191(yn a1 n) * (\u2191(yn a1 n) * \u2191(xn a1 j)) + \u2191(xn a1 j) =\n    (\u2191(Pell.d a1) * \u2191(yn a1 n) * \u2191(yn a1 n) + 1) * \u2191(xn a1 j)\n\u22a2 \u2191(Pell.d a1) * \u2191(yn a1 n) * \u2191(yn a1 n) + 1 = \u2191(xn a1 n) * \u2191(xn a1 n)\n[PROOFSTEP]\napply add_eq_of_eq_sub' (Eq.symm (pell_eqz a1 n))\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = (Pell.d a1 * yn a1 n * yn a1 n + 1) * xn a1 j\nh2 : Pell.d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n\n\u22a2 xn a1 n \u2223 Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j\n[PROOFSTEP]\nrw [h2] at h1 \n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = xn a1 n * xn a1 n * xn a1 j\nh2 : Pell.d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n\n\u22a2 xn a1 n \u2223 Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j\n[PROOFSTEP]\nrw [h1, mul_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh1 : Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j = xn a1 n * xn a1 n * xn a1 j\nh2 : Pell.d a1 * yn a1 n * yn a1 n + 1 = xn a1 n * xn a1 n\n\u22a2 xn a1 n \u2223 xn a1 n * (xn a1 n * xn a1 j)\n[PROOFSTEP]\nexact dvd_mul_right _ _\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 (2 * n + j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrw [two_mul, add_assoc, xn_add, add_assoc, \u2190 zero_add 0]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 n * xn a1 (n + j) + (Pell.d a1 * yn a1 n * yn a1 (n + j) + xn a1 j) \u2261 0 + 0 [MOD xn a1 n]\n[PROOFSTEP]\nrefine' (dvd_mul_right (xn a1 n) (xn a1 (n + j))).modEq_zero_nat.add _\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 Pell.d a1 * yn a1 n * yn a1 (n + j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrw [yn_add, left_distrib, add_assoc, \u2190 zero_add 0]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 Pell.d a1 * yn a1 n * (xn a1 n * yn a1 j) + (Pell.d a1 * yn a1 n * (yn a1 n * xn a1 j) + xn a1 j) \u2261 0 + 0 [MOD\n    xn a1 n]\n[PROOFSTEP]\nexact ((dvd_mul_right _ _).mul_left _).modEq_zero_nat.add (xn_modEq_x2n_add_lem _ _ _).modEq_zero_nat\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nhave h1 : xz a1 n \u2223 d a1 * yz a1 n * yz a1 (n - j) + xz a1 j :=\n  by\n  rw [yz_sub _ h, mul_sub_left_distrib, sub_add_eq_add_sub]\n  exact\n    dvd_sub\n      (by\n        delta xz; delta yz\n        rw [mul_comm (xn _ _ : \u2124)]\n        exact_mod_cast (xn_modEq_x2n_add_lem _ n j))\n      ((dvd_mul_right _ _).mul_left _)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n[PROOFSTEP]\nrw [yz_sub _ h, mul_sub_left_distrib, sub_add_eq_add_sub]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * (xz a1 j * yz a1 n) + xz a1 j - \u2191(Pell.d a1) * yz a1 n * (xz a1 n * yz a1 j)\n[PROOFSTEP]\nexact\n  dvd_sub\n    (by\n      delta xz; delta yz\n      rw [mul_comm (xn _ _ : \u2124)]\n      exact_mod_cast (xn_modEq_x2n_add_lem _ n j))\n    ((dvd_mul_right _ _).mul_left _)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * (xz a1 j * yz a1 n) + xz a1 j\n[PROOFSTEP]\ndelta xz\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 \u2191(xn a1 n) \u2223 \u2191(Pell.d a1) * yz a1 n * (\u2191(xn a1 j) * yz a1 n) + \u2191(xn a1 j)\n[PROOFSTEP]\ndelta yz\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 \u2191(xn a1 n) \u2223 \u2191(Pell.d a1) * \u2191(yn a1 n) * (\u2191(xn a1 j) * \u2191(yn a1 n)) + \u2191(xn a1 j)\n[PROOFSTEP]\nrw [mul_comm (xn _ _ : \u2124)]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\n\u22a2 \u2191(xn a1 n) \u2223 \u2191(Pell.d a1) * \u2191(yn a1 n) * (\u2191(yn a1 n) * \u2191(xn a1 j)) + \u2191(xn a1 j)\n[PROOFSTEP]\nexact_mod_cast (xn_modEq_x2n_add_lem _ n j)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\nh1 : xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n\u22a2 xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrw [two_mul, add_tsub_assoc_of_le h, xn_add, add_assoc, \u2190 zero_add 0]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\nh1 : xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n\u22a2 xn a1 n * xn a1 (n - j) + (Pell.d a1 * yn a1 n * yn a1 (n - j) + xn a1 j) \u2261 0 + 0 [MOD xn a1 n]\n[PROOFSTEP]\nexact (dvd_mul_right _ _).modEq_zero_nat.add (Int.coe_nat_dvd.1 <| by simpa [xz, yz] using h1).modEq_zero_nat\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 n\nh1 : xz a1 n \u2223 \u2191(Pell.d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n\u22a2 \u2191(xn a1 n) \u2223 \u2191(Pell.d a1 * yn a1 n * yn a1 (n - j) + xn a1 j)\n[PROOFSTEP]\nsimpa [xz, yz] using h1\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\njn : n \u2264 j\n\u22a2 xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nhave : 2 * n - j + j \u2264 n + j := by rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\njn : n \u2264 j\n\u22a2 2 * n - j + j \u2264 n + j\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le h, two_mul]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\njn : n \u2264 j\n\u22a2 n + n \u2264 n + j\n[PROOFSTEP]\nexact Nat.add_le_add_left jn _\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\njn : n \u2264 j\nthis : 2 * n - j + j \u2264 n + j\n\u22a2 xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nlet t := xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\njn : n \u2264 j\nthis : 2 * n - j + j \u2264 n + j\nt : xn a1 (2 * n - (2 * n - j)) + xn a1 (2 * n - j) \u2261 0 [MOD xn a1 n] :=\n  xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)\n\u22a2 xn a1 (2 * n - j) + xn a1 j \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrwa [tsub_tsub_cancel_of_le h, add_comm] at t \n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 (4 * n + j) + xn a1 (2 * n + j) \u2261 xn a1 j + xn a1 (2 * n + j) [MOD xn a1 n]\n[PROOFSTEP]\nrefine' @ModEq.trans _ _ 0 _ _ (by rw [add_comm]; exact (xn_modEq_x2n_add _ _ _).symm)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 0 \u2261 xn a1 j + xn a1 (2 * n + j) [MOD xn a1 n]\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 0 \u2261 xn a1 (2 * n + j) + xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\nexact (xn_modEq_x2n_add _ _ _).symm\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 (4 * n + j) + xn a1 (2 * n + j) \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrw [show 4 * n = 2 * n + 2 * n from right_distrib 2 2 n, add_assoc]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\n\u22a2 xn a1 (2 * n + (2 * n + j)) + xn a1 (2 * n + j) \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\napply xn_modEq_x2n_add\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh : j \u2264 2 * n\n\u22a2 2 * n \u2264 2 * n\n[PROOFSTEP]\nrw [Nat.succ_mul]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh h' : j \u2264 2 * n\n\u22a2 xn a1 (4 * n - j) + xn a1 (2 * n - j) \u2261 xn a1 j + xn a1 (2 * n - j) [MOD xn a1 n]\n[PROOFSTEP]\nrefine' @ModEq.trans _ _ 0 _ _ (by rw [add_comm]; exact (xn_modEq_x2n_sub _ h).symm)\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh h' : j \u2264 2 * n\n\u22a2 0 \u2261 xn a1 j + xn a1 (2 * n - j) [MOD xn a1 n]\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh h' : j \u2264 2 * n\n\u22a2 0 \u2261 xn a1 (2 * n - j) + xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\nexact (xn_modEq_x2n_sub _ h).symm\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh h' : j \u2264 2 * n\n\u22a2 xn a1 (4 * n - j) + xn a1 (2 * n - j) \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\nrw [show 4 * n = 2 * n + 2 * n from right_distrib 2 2 n, add_tsub_assoc_of_le h']\n[GOAL]\na : \u2115\na1 : 1 < a\nn j : \u2115\nh h' : j \u2264 2 * n\n\u22a2 xn a1 (2 * n + (2 * n - j)) + xn a1 (2 * n - j) \u2261 0 [MOD xn a1 n]\n[PROOFSTEP]\napply xn_modEq_x2n_add\n[GOAL]\na : \u2115\na1 : 1 < a\ni n j : \u2115\nij : i < j + 1\njn : j + 1 < n\n\u22a2 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nsuffices xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n from\n  (lt_or_eq_of_le (Nat.le_of_succ_le_succ ij)).elim (fun h => lt_trans (eq_of_xn_modEq_lem1 h (le_of_lt jn)) this)\n    fun h => by rw [h]; exact this\n[GOAL]\na : \u2115\na1 : 1 < a\ni n j : \u2115\nij : i < j + 1\njn : j + 1 < n\nthis : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n\u22a2 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nrw [h]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n j : \u2115\nij : i < j + 1\njn : j + 1 < n\nthis : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n\u22a2 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nexact this\n[GOAL]\na : \u2115\na1 : 1 < a\ni n j : \u2115\nij : i < j + 1\njn : j + 1 < n\n\u22a2 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (strictMono_x _ (Nat.lt_of_succ_lt jn)), Nat.mod_eq_of_lt (strictMono_x _ jn)]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n j : \u2115\nij : i < j + 1\njn : j + 1 < n\n\u22a2 xn a1 j < xn a1 (j + 1)\n[PROOFSTEP]\nexact strictMono_x _ (Nat.lt_succ_self _)\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : 2 * xn a1 n = xn a1 (n + 1)\n\u22a2 a = 2 \u2227 n = 0\n[PROOFSTEP]\nrw [xn_succ, mul_comm] at h \n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\n\u22a2 a = 2 \u2227 n = 0\n[PROOFSTEP]\nhave : n = 0 :=\n  n.eq_zero_or_pos.resolve_right fun np =>\n    _root_.ne_of_lt\n      (lt_of_le_of_lt (Nat.mul_le_mul_left _ a1) (Nat.lt_add_of_pos_right <| mul_pos (d_pos a1) (strictMono_y a1 np))) h\n[GOAL]\na : \u2115\na1 : 1 < a\nn : \u2115\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\nthis : n = 0\n\u22a2 a = 2 \u2227 n = 0\n[PROOFSTEP]\ncases this\n[GOAL]\ncase refl\na : \u2115\na1 : 1 < a\nh : xn a1 0 * 2 = xn a1 0 * a + Pell.d a1 * yn a1 0\n\u22a2 a = 2 \u2227 0 = 0\n[PROOFSTEP]\nsimp at h \n[GOAL]\ncase refl\na : \u2115\na1 : 1 < a\nh : 2 = a\n\u22a2 a = 2 \u2227 0 = 0\n[PROOFSTEP]\nexact \u27e8h.symm, rfl\u27e9\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\n[PROOFSTEP]\nlet k2nl :=\n  lt_of_add_lt_add_right <|\n    show 2 * n - k + k < n + k by\n      rw [tsub_add_cancel_of_le]\n      rw [two_mul]; exact add_lt_add_left kn n\n      exact k2n\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 2 * n - k + k < n + k\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 2 * n < n + k\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 k \u2264 2 * n\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 n + n < n + k\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 k \u2264 2 * n\n[PROOFSTEP]\nexact add_lt_add_left kn n\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\n\u22a2 k \u2264 2 * n\n[PROOFSTEP]\nexact k2n\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\n\u22a2 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\n[PROOFSTEP]\nhave xle : xn a1 (2 * n - k) \u2264 xn a1 n := le_of_lt <| strictMono_x a1 k2nl\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\n\u22a2 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\n[PROOFSTEP]\nsuffices xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k) by rw [this, Int.ofNat_sub xle]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\nthis : xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)\n\u22a2 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\n[PROOFSTEP]\nrw [this, Int.ofNat_sub xle]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\n\u22a2 xn a1 k % xn a1 n = xn a1 n - xn a1 (2 * n - k)\n[PROOFSTEP]\nrw [\u2190 Nat.mod_eq_of_lt (Nat.sub_lt (x_pos a1 n) (x_pos a1 (2 * n - k)))]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\n\u22a2 xn a1 k % xn a1 n = (xn a1 n - xn a1 (2 * n - k)) % xn a1 n\n[PROOFSTEP]\napply ModEq.add_right_cancel' (xn a1 (2 * n - k))\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\n\u22a2 xn a1 k + xn a1 (2 * n - k) \u2261 xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le xle]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\n\u22a2 xn a1 k + xn a1 (2 * n - k) \u2261 xn a1 n [MOD xn a1 n]\n[PROOFSTEP]\nhave t := xn_modEq_x2n_sub_lem a1 k2nl.le\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\nt : xn a1 (2 * n - (2 * n - k)) + xn a1 (2 * n - k) \u2261 0 [MOD xn a1 n]\n\u22a2 xn a1 k + xn a1 (2 * n - k) \u2261 xn a1 n [MOD xn a1 n]\n[PROOFSTEP]\nrw [tsub_tsub_cancel_of_le k2n] at t \n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nk : \u2115\nkn : k > n\nk2n : k \u2264 2 * n\nk2nl : 2 * n - k < n :=\n  lt_of_add_lt_add_right\n    (let_fun this :=\n      Eq.mpr (id (tsub_add_cancel_of_le k2n \u25b8 Eq.refl (2 * n - k + k < n + k)))\n        (Eq.mpr (id (two_mul n \u25b8 Eq.refl (2 * n < n + k))) (add_lt_add_left kn n));\n    this)\nxle : xn a1 (2 * n - k) \u2264 xn a1 n\nt : xn a1 k + xn a1 (2 * n - k) \u2261 0 [MOD xn a1 n]\n\u22a2 xn a1 k + xn a1 (2 * n - k) \u2261 xn a1 n [MOD xn a1 n]\n[PROOFSTEP]\nexact t.trans dvd_rfl.zero_modEq_nat\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn : j = n\n\u22a2 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\ncases jn\n[GOAL]\ncase refl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\n\u22a2 xn a1 i % xn a1 n < xn a1 (n + 1) % xn a1 n\n[PROOFSTEP]\napply Int.lt_of_ofNat_lt_ofNat\n[GOAL]\ncase refl.a\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\n\u22a2 \u2191(xn a1 i % xn a1 n) < \u2191(xn a1 (n + 1) % xn a1 n)\n[PROOFSTEP]\nrw [lem2 (n + 1) (Nat.lt_succ_self _) j2n,\n  show 2 * n - (n + 1) = n - 1 by rw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\n\u22a2 2 * n - (n + 1) = n - 1\n[PROOFSTEP]\nrw [two_mul, tsub_add_eq_tsub_tsub, add_tsub_cancel_right]\n[GOAL]\ncase refl.a\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\n\u22a2 \u2191(xn a1 i % xn a1 n) < \u2191(xn a1 n) - \u2191(xn a1 (n - 1))\n[PROOFSTEP]\nrefine' lt_sub_left_of_add_lt (Int.ofNat_lt_ofNat_of_lt _)\n[GOAL]\ncase refl.a\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\n\u22a2 xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n\n[PROOFSTEP]\ncases' lt_or_eq_of_le <| Nat.le_of_succ_le_succ ij with lin ein\n[GOAL]\ncase refl.a.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\n\u22a2 xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (strictMono_x _ lin)]\n[GOAL]\ncase refl.a.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\nhave ll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n :=\n  by\n  rw [\u2190 two_mul, mul_comm, show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)], xn_succ]\n  exact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\n\u22a2 xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\n[PROOFSTEP]\nrw [\u2190 two_mul, mul_comm, show xn a1 n = xn a1 (n - 1 + 1) by rw [tsub_add_cancel_of_le (succ_le_of_lt npos)], xn_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\n\u22a2 xn a1 n = xn a1 (n - 1 + 1)\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le (succ_le_of_lt npos)]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\n\u22a2 xn a1 (n - 1) * 2 \u2264 xn a1 (n - 1) * a + Pell.d a1 * yn a1 (n - 1)\n[PROOFSTEP]\nexact le_trans (Nat.mul_le_mul_left _ a1) (Nat.le_add_right _ _)\n[GOAL]\ncase refl.a.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\nhave npm : (n - 1).succ = n := Nat.succ_pred_eq_of_pos npos\n[GOAL]\ncase refl.a.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\nhave il : i \u2264 n - 1 := by\n  apply Nat.le_of_succ_le_succ\n  rw [npm]\n  exact lin\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\n\u22a2 i \u2264 n - 1\n[PROOFSTEP]\napply Nat.le_of_succ_le_succ\n[GOAL]\ncase a\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\n\u22a2 succ i \u2264 succ (n - 1)\n[PROOFSTEP]\nrw [npm]\n[GOAL]\ncase a\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\n\u22a2 succ i \u2264 n\n[PROOFSTEP]\nexact lin\n[GOAL]\ncase refl.a.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\ncases' lt_or_eq_of_le il with ill ile\n[GOAL]\ncase refl.a.inl.inl\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nill : i < n - 1\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\nexact lt_of_lt_of_le (Nat.add_lt_add_left (strictMono_x a1 ill) _) ll\n[GOAL]\ncase refl.a.inl.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\n\u22a2 xn a1 (n - 1) + xn a1 i < xn a1 n\n[PROOFSTEP]\nrw [ile]\n[GOAL]\ncase refl.a.inl.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\n\u22a2 xn a1 (n - 1) + xn a1 (n - 1) < xn a1 n\n[PROOFSTEP]\napply lt_of_le_of_ne ll\n[GOAL]\ncase refl.a.inl.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\n\u22a2 xn a1 (n - 1) + xn a1 (n - 1) \u2260 xn a1 n\n[PROOFSTEP]\nrw [\u2190 two_mul]\n[GOAL]\ncase refl.a.inl.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\n\u22a2 2 * xn a1 (n - 1) \u2260 xn a1 n\n[PROOFSTEP]\nexact fun e =>\n  ntriv <|\n    by\n    let \u27e8a2, s1\u27e9 := @eq_of_xn_modEq_lem2 _ a1 (n - 1) (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])\n    have n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos\n    rw [ile, a2, n1]; exact \u27e8rfl, rfl, rfl, rfl\u27e9\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\n\u22a2 a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2\n[PROOFSTEP]\nlet \u27e8a2, s1\u27e9 := @eq_of_xn_modEq_lem2 _ a1 (n - 1) (by rwa [tsub_add_cancel_of_le (succ_le_of_lt npos)])\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\n\u22a2 2 * xn a1 (n - 1) = xn a1 (n - 1 + 1)\n[PROOFSTEP]\nrwa [tsub_add_cancel_of_le (succ_le_of_lt npos)]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\n\u22a2 a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2\n[PROOFSTEP]\nhave n1 : n = 1 := le_antisymm (tsub_eq_zero_iff_le.mp s1) npos\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n\u22a2 a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2\n[PROOFSTEP]\nrw [ile, a2, n1]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nlin : i < n\nll : xn a1 (n - 1) + xn a1 (n - 1) \u2264 xn a1 n\nnpm : succ (n - 1) = n\nil : i \u2264 n - 1\nile : i = n - 1\ne : 2 * xn a1 (n - 1) = xn a1 n\na2 : a = 2\ns1 : n - 1 = 0\nn1 : n = 1\n\u22a2 2 = 2 \u2227 1 = 1 \u2227 1 - 1 = 0 \u2227 1 + 1 = 2\n[PROOFSTEP]\nexact \u27e8rfl, rfl, rfl, rfl\u27e9\n[GOAL]\ncase refl.a.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nein : i = n\n\u22a2 xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n\n[PROOFSTEP]\nrw [ein, Nat.mod_self, add_zero]\n[GOAL]\ncase refl.a.inr\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\nij : i < n + 1\nj2n : n + 1 \u2264 2 * n\njnn : n + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 n + 1 = 2)\no : n = n \u2228 n < n\nein : i = n\n\u22a2 xn a1 (n - 1) < xn a1 n\n[PROOFSTEP]\nexact strictMono_x _ (Nat.pred_lt npos.ne')\n[GOAL]\na : \u2115\na1\u271d : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1\u271d k % xn a1\u271d n) = \u2191(xn a1\u271d n) - \u2191(xn a1\u271d (2 * n - k))\no : j = n \u2228 n < j\njn\u271d : j > n\njn : j \u2260 n\ns : xn a1\u271d j % xn a1\u271d n < xn a1\u271d (j + 1) % xn a1\u271d n\nh : i < j\nx\u271d : a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n\u22a2 False\n[PROOFSTEP]\nrw [n1, j2] at j2n \n[GOAL]\na : \u2115\na1\u271d : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : 2 + 1 \u2264 2 * 1\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1\u271d k % xn a1\u271d n) = \u2191(xn a1\u271d n) - \u2191(xn a1\u271d (2 * n - k))\no : j = n \u2228 n < j\njn\u271d : j > n\njn : j \u2260 n\ns : xn a1\u271d j % xn a1\u271d n < xn a1\u271d (j + 1) % xn a1\u271d n\nh : i < j\nx\u271d : a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n\u22a2 False\n[PROOFSTEP]\nexact absurd j2n (by decide)\n[GOAL]\na : \u2115\na1\u271d : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : 2 + 1 \u2264 2 * 1\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1\u271d k % xn a1\u271d n) = \u2191(xn a1\u271d n) - \u2191(xn a1\u271d (2 * n - k))\no : j = n \u2228 n < j\njn\u271d : j > n\njn : j \u2260 n\ns : xn a1\u271d j % xn a1\u271d n < xn a1\u271d (j + 1) % xn a1\u271d n\nh : i < j\nx\u271d : a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2\na1 : a = 2\nn1 : n = 1\ni0 : i = 0\nj2 : j = 2\n\u22a2 \u00ac2 + 1 \u2264 2 * 1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn\u271d : j > n\njn : j \u2260 n\ns : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n\u22a2 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nrw [h]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn\u271d : j > n\njn : j \u2260 n\ns : xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\nh : i = j\n\u22a2 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n\n[PROOFSTEP]\nexact s\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn : j > n\nlem1 : j \u2260 n \u2192 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n \u2192 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n\u22a2 \u2191(xn a1 j % xn a1 n) < \u2191(xn a1 (j + 1) % xn a1 n)\n[PROOFSTEP]\nrw [lem2 j jn (le_of_lt j2n), lem2 (j + 1) (Nat.le_succ_of_le jn) j2n]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn : j > n\nlem1 : j \u2260 n \u2192 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n \u2192 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n\u22a2 \u2191(xn a1 n) - \u2191(xn a1 (2 * n - j)) < \u2191(xn a1 n) - \u2191(xn a1 (2 * n - (j + 1)))\n[PROOFSTEP]\nrefine' sub_lt_sub_left (Int.ofNat_lt_ofNat_of_lt <| strictMono_x _ _) _\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn : j > n\nlem1 : j \u2260 n \u2192 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n \u2192 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n\u22a2 2 * n - (j + 1) < 2 * n - j\n[PROOFSTEP]\nrw [Nat.sub_succ]\n[GOAL]\na : \u2115\na1 : 1 < a\ni n : \u2115\nnpos : 0 < n\nj : \u2115\nij : i < j + 1\nj2n : j + 1 \u2264 2 * n\njnn : j + 1 \u2260 n\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j + 1 = 2)\nlem2 : \u2200 (k : \u2115), k > n \u2192 k \u2264 2 * n \u2192 \u2191(xn a1 k % xn a1 n) = \u2191(xn a1 n) - \u2191(xn a1 (2 * n - k))\no : j = n \u2228 n < j\njn : j > n\nlem1 : j \u2260 n \u2192 xn a1 j % xn a1 n < xn a1 (j + 1) % xn a1 n \u2192 xn a1 i % xn a1 n < xn a1 (j + 1) % xn a1 n\n\u22a2 pred (2 * n - j) < 2 * n - j\n[PROOFSTEP]\nexact Nat.pred_lt (_root_.ne_of_gt <| tsub_pos_of_lt j2n)\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : n = 0\n\u22a2 i = j\n[PROOFSTEP]\nsimp_all\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\n\u22a2 False\n[PROOFSTEP]\nrefine' _root_.ne_of_gt _ h\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\n\u22a2 xn a1 j % xn a1 n < xn a1 i % xn a1 n\n[PROOFSTEP]\nrw [jn, Nat.mod_self]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\n\u22a2 0 < xn a1 i % xn a1 n\n[PROOFSTEP]\nhave x0 : 0 < xn a1 0 % xn a1 n :=\n  by\n  rw [Nat.mod_eq_of_lt (strictMono_x a1 (Nat.pos_of_ne_zero npos))]\n  exact Nat.succ_pos _\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\n\u22a2 0 < xn a1 0 % xn a1 n\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt (strictMono_x a1 (Nat.pos_of_ne_zero npos))]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\n\u22a2 0 < xn a1 0\n[PROOFSTEP]\nexact Nat.succ_pos _\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nij : i \u2264 j\nj2n : j \u2264 2 * n\nh : xn a1 i \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 i = 0 \u2227 j = 2)\nnpos : \u00acn = 0\nij' : i < j\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\n\u22a2 0 < xn a1 i % xn a1 n\n[PROOFSTEP]\ncases' i with i\n[GOAL]\ncase zero\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\nij : zero \u2264 j\nh : xn a1 zero \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 zero = 0 \u2227 j = 2)\nij' : zero < j\n\u22a2 0 < xn a1 zero % xn a1 n\ncase succ\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : succ i < j\n\u22a2 0 < xn a1 (succ i) % xn a1 n\n[PROOFSTEP]\nexact x0\n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : succ i < j\n\u22a2 0 < xn a1 (succ i) % xn a1 n\n[PROOFSTEP]\nrw [jn] at ij' \n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : succ i < n\n\u22a2 0 < xn a1 (succ i) % xn a1 n\n[PROOFSTEP]\nexact\n  x0.trans\n    (eq_of_xn_modEq_lem3 _ (Nat.pos_of_ne_zero npos) (Nat.succ_pos _) (le_trans ij j2n) (_root_.ne_of_lt ij')\n      fun \u27e8_, n1, _, i2\u27e9 => by rw [n1, i2] at ij' ; exact absurd ij' (by decide))\n[GOAL]\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : succ i < n\nx\u271d : a = 2 \u2227 n = 1 \u2227 0 = 0 \u2227 succ i = 2\nleft\u271d\u00b9 : a = 2\nn1 : n = 1\nleft\u271d : 0 = 0\ni2 : succ i = 2\n\u22a2 False\n[PROOFSTEP]\nrw [n1, i2] at ij' \n[GOAL]\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : 2 < 1\nx\u271d : a = 2 \u2227 n = 1 \u2227 0 = 0 \u2227 succ i = 2\nleft\u271d\u00b9 : a = 2\nn1 : n = 1\nleft\u271d : 0 = 0\ni2 : succ i = 2\n\u22a2 False\n[PROOFSTEP]\nexact absurd ij' (by decide)\n[GOAL]\na : \u2115\na1 : 1 < a\nj n : \u2115\nj2n : j \u2264 2 * n\nnpos : \u00acn = 0\njn : j = n\nx0 : 0 < xn a1 0 % xn a1 n\ni : \u2115\nij : succ i \u2264 j\nh : xn a1 (succ i) \u2261 xn a1 j [MOD xn a1 n]\nntriv : \u00ac(a = 2 \u2227 n = 1 \u2227 succ i = 0 \u2227 j = 2)\nij' : 2 < 1\nx\u271d : a = 2 \u2227 n = 1 \u2227 0 = 0 \u2227 succ i = 2\nleft\u271d\u00b9 : a = 2\nn1 : n = 1\nleft\u271d : 0 = 0\ni2 : succ i = 2\n\u22a2 \u00ac2 < 1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\n\u22a2 i \u2264 2 * n\n[PROOFSTEP]\napply le_trans hin\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\n\u22a2 n \u2264 2 * n\n[PROOFSTEP]\nrw [two_mul]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\n\u22a2 n \u2264 n + n\n[PROOFSTEP]\napply Nat.le_add_left\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\nj0 : j = 0\ni2 : i = 2\n\u22a2 False\n[PROOFSTEP]\nrw [n1, i2] at hin \n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : 2 \u2264 1\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\nj0 : j = 0\ni2 : i = 2\n\u22a2 False\n[PROOFSTEP]\nexact absurd hin (by decide)\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : 2 \u2264 1\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\nj0 : j = 0\ni2 : i = 2\n\u22a2 \u00ac2 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\n\u22a2 4 * n - j + ?m.157805 ipos hin j4n h i2n j2n \u2264 2 * n + ?m.157805 ipos hin j4n h i2n j2n\n[PROOFSTEP]\nrw [tsub_add_cancel_of_le j4n, show 4 * n = 2 * n + 2 * n from right_distrib 2 2 n]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\n\u22a2 2 * n + 2 * n \u2264 2 * n + j\n[PROOFSTEP]\nexact Nat.add_le_add_left (le_of_lt j2n) _\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nj42n : 4 * n - j \u2264 2 * n\n\u22a2 xn a1 j \u2261 xn a1 (4 * n - j) [MOD xn a1 n]\n[PROOFSTEP]\nlet t := xn_modEq_x4n_sub a1 j42n\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nj42n : 4 * n - j \u2264 2 * n\nt : xn a1 (4 * n - (4 * n - j)) \u2261 xn a1 (4 * n - j) [MOD xn a1 n] := xn_modEq_x4n_sub a1 j42n\n\u22a2 xn a1 j \u2261 xn a1 (4 * n - j) [MOD xn a1 n]\n[PROOFSTEP]\nrwa [tsub_tsub_cancel_of_le j4n] at t \n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nj42n : 4 * n - j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\ni2 : i = 2\n\u22a2 4 * n - j \u2260 0\n[PROOFSTEP]\nrw [n1, i2] at hin \n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : 2 \u2264 1\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nj42n : 4 * n - j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\ni2 : i = 2\n\u22a2 4 * n - j \u2260 0\n[PROOFSTEP]\nexact absurd hin (by decide)\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : 2 \u2264 1\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nj42n : 4 * n - j \u2264 2 * n\na2 : a = 2\nn1 : n = 1\ni2 : i = 2\n\u22a2 \u00ac2 \u2264 1\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nj4n : j \u2264 4 * n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\ni2n : i \u2264 2 * n\nj2n : 2 * n < j\nthis : i = 4 * n - j\n\u22a2 j + i = 4 * n\n[PROOFSTEP]\nrw [this, add_tsub_cancel_of_le j4n]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\n\u22a2 0 < 4\n[PROOFSTEP]\ndecide\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\n\u22a2 j \u2261 j' [MOD 4 * n]\n[PROOFSTEP]\ndelta ModEq\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\n\u22a2 j % (4 * n) = j' % (4 * n)\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt jl]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\n\u22a2 \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\n[PROOFSTEP]\nintro j q\n[GOAL]\na : \u2115\na1 : 1 < a\ni j\u271d n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j\u271d \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j\u271d % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j\u271d \u2261 j' [MOD 4 * n]\nj q : \u2115\n\u22a2 xn a1 (j + 4 * n * q) \u2261 xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\ninduction' q with q IH\n[GOAL]\ncase zero\na : \u2115\na1 : 1 < a\ni j\u271d n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j\u271d \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j\u271d % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j\u271d \u2261 j' [MOD 4 * n]\nj : \u2115\n\u22a2 xn a1 (j + 4 * n * zero) \u2261 xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\nsimp [ModEq.refl]\n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\ni j\u271d n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j\u271d \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j\u271d % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j\u271d \u2261 j' [MOD 4 * n]\nj q : \u2115\nIH : xn a1 (j + 4 * n * q) \u2261 xn a1 j [MOD xn a1 n]\n\u22a2 xn a1 (j + 4 * n * succ q) \u2261 xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\nrw [Nat.mul_succ, \u2190 add_assoc, add_comm]\n[GOAL]\ncase succ\na : \u2115\na1 : 1 < a\ni j\u271d n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j\u271d \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j\u271d % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j\u271d \u2261 j' [MOD 4 * n]\nj q : \u2115\nIH : xn a1 (j + 4 * n * q) \u2261 xn a1 j [MOD xn a1 n]\n\u22a2 xn a1 (4 * n + (j + 4 * n * q)) \u2261 xn a1 j [MOD xn a1 n]\n[PROOFSTEP]\nexact (xn_modEq_x4n_add _ _ _).trans IH\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\nthis : \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\nji : j' = i\n\u22a2 j \u2261 i [MOD 4 * n]\n[PROOFSTEP]\nrwa [\u2190 ji]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\nthis : \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\nji : j' + i = 4 * n\n\u22a2 j' + i \u2261 0 [MOD 4 * n]\n[PROOFSTEP]\nrw [ji]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\nthis : \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\nji : j' + i = 4 * n\n\u22a2 4 * n \u2261 0 [MOD 4 * n]\n[PROOFSTEP]\nexact dvd_rfl.modEq_zero_nat\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\nthis : \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\n\u22a2 xn a1 j \u2261 xn a1 j' [MOD xn a1 n]\n[PROOFSTEP]\nrw [\u2190 Nat.mod_add_div j (4 * n)]\n[GOAL]\na : \u2115\na1 : 1 < a\ni j n : \u2115\nipos : 0 < i\nhin : i \u2264 n\nh : xn a1 j \u2261 xn a1 i [MOD xn a1 n]\nj' : \u2115 := j % (4 * n)\nn4 : 0 < 4 * n\njl : j' < 4 * n\njj : j \u2261 j' [MOD 4 * n]\nthis : \u2200 (j_1 q : \u2115), xn a1 (j_1 + 4 * n * q) \u2261 xn a1 j_1 [MOD xn a1 n]\n\u22a2 xn a1 (j % (4 * n) + 4 * n * (j / (4 * n))) \u2261 xn a1 j' [MOD xn a1 n]\n[PROOFSTEP]\nexact this j' _\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\n\u22a2 xn a1 0 \u2261 xn b1 0 [MOD c] \u2227 yn a1 0 \u2261 yn b1 0 [MOD c]\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase left\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\n\u22a2 xn a1 0 \u2261 xn b1 0 [MOD c]\n[PROOFSTEP]\nrfl\n[GOAL]\ncase right\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\n\u22a2 yn a1 0 \u2261 yn b1 0 [MOD c]\n[PROOFSTEP]\nrfl\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\n\u22a2 xn a1 1 \u2261 xn b1 1 [MOD c] \u2227 yn a1 1 \u2261 yn b1 1 [MOD c]\n[PROOFSTEP]\nsimp\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\n\u22a2 a \u2261 b [MOD c] \u2227 1 \u2261 1 [MOD c]\n[PROOFSTEP]\nexact \u27e8h, ModEq.refl 1\u27e9\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\nn : \u2115\n\u22a2 xn a1 (n + 2) + xn a1 n \u2261 xn b1 (n + 2) + xn b1 n [MOD c]\n[PROOFSTEP]\nrw [xn_succ_succ a1, xn_succ_succ b1]\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\nn : \u2115\n\u22a2 2 * a * xn a1 (n + 1) \u2261 2 * b * xn b1 (n + 1) [MOD c]\n[PROOFSTEP]\nexact (h.mul_left _).mul (xy_modEq_of_modEq _ _ h (n + 1)).left\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\nn : \u2115\n\u22a2 yn a1 (n + 2) + yn (_ : 1 < a) n \u2261 yn b1 (n + 2) + yn (_ : 1 < b) n [MOD c]\n[PROOFSTEP]\nrw [yn_succ_succ a1, yn_succ_succ b1]\n[GOAL]\na b c : \u2115\na1 : 1 < a\nb1 : 1 < b\nh : a \u2261 b [MOD c]\nn : \u2115\n\u22a2 2 * a * yn a1 (n + 1) \u2261 2 * b * yn b1 (n + 1) [MOD c]\n[PROOFSTEP]\nexact (h.mul_left _).mul (xy_modEq_of_modEq _ _ h (n + 1)).right\n[GOAL]\na k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n\u22a2 1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\n[PROOFSTEP]\nrw [\u2190 hx, \u2190 hy]\n[GOAL]\na k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n\u22a2 1 < a \u2227\n    k \u2264 yn a1 k \u2227\n      (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])\n[PROOFSTEP]\nrefine' \u27e8a1, (Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact \u27e8le_rfl, Or.inl \u27e8rfl, rfl\u27e9\u27e9) fun kpos => _\u27e9\n[GOAL]\na k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nk0 : k = 0\n\u22a2 k \u2264 yn a1 k \u2227\n    (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])\n[PROOFSTEP]\nrw [k0]\n[GOAL]\na k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nk0 : k = 0\n\u22a2 0 \u2264 yn a1 0 \u2227\n    (xn a1 0 = 1 \u2227 yn a1 0 = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 0 * xn a1 0 - (a * a - 1) * yn a1 0 * yn a1 0 = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 0] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 0 * yn a1 0 \u2223 v \u2227 s \u2261 xn a1 0 [MOD u] \u2227 t \u2261 0 [MOD 4 * yn a1 0])\n[PROOFSTEP]\nexact \u27e8le_rfl, Or.inl \u27e8rfl, rfl\u27e9\u27e9\n[GOAL]\na k x y : \u2115\nx\u271d : \u2203 a1, xn a1 k = x \u2227 yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n\u22a2 k \u2264 yn a1 k \u2227\n    (xn a1 k = 1 \u2227 yn a1 k = 0 \u2228\n      \u2203 u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 \u2227\n          u * u - (a * a - 1) * v * v = 1 \u2227\n            s * s - (b * b - 1) * t * t = 1 \u2227\n              1 < b \u2227\n                b \u2261 1 [MOD 4 * yn a1 k] \u2227\n                  b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 k * yn a1 k \u2223 v \u2227 s \u2261 xn a1 k [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 k])\n[PROOFSTEP]\nexact\n  let x := xn a1 k\n  let y := yn a1 k\n  let m := 2 * (k * y)\n  let u := xn a1 m\n  let v := yn a1 m\n  have ky : k \u2264 y := yn_ge_n a1 k\n  have yv : y * y \u2223 v := (ysq_dvd_yy a1 k).trans <| (y_dvd_iff _ _ _).2 <| dvd_mul_left _ _\n  have uco : Nat.coprime u (4 * y) :=\n    have : 2 \u2223 v := modEq_zero_iff_dvd.1 <| (yn_modEq_two _ _).trans (dvd_mul_right _ _).modEq_zero_nat\n    have : Nat.coprime u 2 := (xy_coprime a1 m).coprime_dvd_right this\n    (this.mul_right this).mul_right <| (xy_coprime _ _).coprime_dvd_right (dvd_of_mul_left_dvd yv)\n  let \u27e8b, ba, bm1\u27e9 := chineseRemainder uco a 1\n  have m1 : 1 < m :=\n    have : 0 < k * y := mul_pos kpos (strictMono_y a1 kpos)\n    Nat.mul_le_mul_left 2 this\n  have vp : 0 < v := strictMono_y a1 (lt_trans zero_lt_one m1)\n  have b1 : 1 < b :=\n    have : xn a1 1 < u := strictMono_x a1 m1\n    have : a < u := by simp at this ; exact this\n    lt_of_lt_of_le a1 <| by\n      delta ModEq at ba ; rw [Nat.mod_eq_of_lt this] at ba ; rw [\u2190 ba]\n      apply Nat.mod_le\n  let s := xn b1 k\n  let t := yn b1 k\n  have sx : s \u2261 x [MOD u] := (xy_modEq_of_modEq b1 a1 ba k).left\n  have tk : t \u2261 k [MOD 4 * y] :=\n    have : 4 * y \u2223 b - 1 := Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n    (yn_modEq_a_sub_one _ _).of_dvd this\n  \u27e8ky, Or.inr \u27e8u, v, s, t, b, pell_eq _ _, pell_eq _ _, pell_eq _ _, b1, bm1, ba, vp, yv, sx, tk\u27e9\u27e9\n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis : xn a1 1 < u\n\u22a2 a < u\n[PROOFSTEP]\nsimp at this \n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis : a < xn a1 (2 * (k * yn a1 k))\n\u22a2 a < u\n[PROOFSTEP]\nexact this\n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b\n[PROOFSTEP]\ndelta ModEq at ba \n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a % u\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrw [Nat.mod_eq_of_lt this] at ba \n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 a \u2264 b\n[PROOFSTEP]\nrw [\u2190 ba]\n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b % u = a\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nthis\u271d : xn a1 1 < u\nthis : a < u\n\u22a2 b % u \u2264 b\n[PROOFSTEP]\napply Nat.mod_le\n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nb1 : 1 < b\ns : \u2115 := xn b1 k\nt : \u2115 := yn b1 k\nsx : s \u2261 x [MOD u]\n\u22a2 \u2191(4 * y) \u2223 \u2191(b - 1)\n[PROOFSTEP]\nrw [Int.ofNat_sub (le_of_lt b1)]\n[GOAL]\na k x\u271d\u00b9 y\u271d : \u2115\nx\u271d : \u2203 a1, xn a1 k = x\u271d\u00b9 \u2227 yn a1 k = y\u271d\na1 : 1 < a\nhx : xn a1 k = x\u271d\u00b9\nhy : yn a1 k = y\u271d\nkpos : k > 0\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nm : \u2115 := 2 * (k * y)\nu : \u2115 := xn a1 m\nv : \u2115 := yn a1 m\nky : k \u2264 y\nyv : y * y \u2223 v\nuco : coprime u (4 * y)\nb : \u2115\nba : b \u2261 a [MOD u]\nbm1 : b \u2261 1 [MOD 4 * y]\nm1 : 1 < m\nvp : 0 < v\nb1 : 1 < b\ns : \u2115 := xn b1 k\nt : \u2115 := yn b1 k\nsx : s \u2261 x [MOD u]\n\u22a2 \u2191(4 * y) \u2223 \u2191b - \u21911\n[PROOFSTEP]\nexact bm1.symm.dvd\n[GOAL]\na k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 y\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 k = x \u2227 yn a1 k = y\n[PROOFSTEP]\nrw [y0] at ky \n[GOAL]\na k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 k = x \u2227 yn a1 k = y\n[PROOFSTEP]\nrw [Nat.eq_zero_of_le_zero ky, x1, y0]\n[GOAL]\na k x y : \u2115\nx\u271d :\n  1 < a \u2227\n    k \u2264 y \u2227\n      (x = 1 \u2227 y = 0 \u2228\n        \u2203 u v s t b,\n          x * x - (a * a - 1) * y * y = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y])\na1 : 1 < a\nky : k \u2264 0\no :\n  x = 1 \u2227 y = 0 \u2228\n    \u2203 u v s t b,\n      x * x - (a * a - 1) * y * y = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227 b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n\u22a2 xn a1 0 = 1 \u2227 yn a1 0 = 0\n[PROOFSTEP]\nexact \u27e8rfl, rfl\u27e9\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i\n[PROOFSTEP]\nsimp [i0] at ky \n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i\n[PROOFSTEP]\nrw [i0, ky]\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\ni0 : i = 0\nky : k = 0\n\u22a2 xn a1 0 = xn a1 0 \u2227 yn a1 0 = yn a1 0\n[PROOFSTEP]\nexact \u27e8rfl, rfl\u27e9\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i\n[PROOFSTEP]\nsuffices i = k by rw [this]; exact \u27e8rfl, rfl\u27e9\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\nthis : i = k\n\u22a2 xn a1 k = xn a1 i \u2227 yn a1 k = yn a1 i\n[PROOFSTEP]\nrw [this]\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\nthis : i = k\n\u22a2 xn a1 k = xn a1 k \u2227 yn a1 k = yn a1 k\n[PROOFSTEP]\nexact \u27e8rfl, rfl\u27e9\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\nrem : b \u2261 1 [MOD 4 * y] \u2227 b \u2261 a [MOD u] \u2227 0 < v \u2227 y * y \u2223 v \u2227 s \u2261 x [MOD u] \u2227 t \u2261 k [MOD 4 * y]\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\no :\n  xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n    \u2203 u v s t b,\n      xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n        u * u - (a * a - 1) * v * v = 1 \u2227\n          s * s - (b * b - 1) * t * t = 1 \u2227\n            1 < b \u2227\n              b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i]\nxy : xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1\nuv : xn a1 n * xn a1 n - (a * a - 1) * yn a1 n * yn a1 n = 1\nst : xn b1 j * xn b1 j - (b * b - 1) * yn b1 j * yn b1 j = 1\nipos : i > 0\n\u22a2 i = k\n[PROOFSTEP]\nclear o rem xy uv st\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\n\u22a2 i = k\n[PROOFSTEP]\nhave iln : i \u2264 n :=\n  le_of_not_gt fun hin => not_lt_of_ge (Nat.le_of_dvd vp (dvd_of_mul_left_dvd yv)) (strictMono_y a1 hin)\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\n\u22a2 i = k\n[PROOFSTEP]\nhave yd : 4 * yn a1 i \u2223 4 * n := mul_dvd_mul_left _ <| dvd_of_ysq_dvd a1 yv\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 i = k\n[PROOFSTEP]\nhave jk : j \u2261 k [MOD 4 * yn a1 i] :=\n  have : 4 * yn a1 i \u2223 b - 1 := Int.coe_nat_dvd.1 <| by rw [Int.ofNat_sub (le_of_lt b1)]; exact bm1.symm.dvd\n  ((yn_modEq_a_sub_one b1 _).of_dvd this).symm.trans tk\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 \u2191(4 * yn a1 i) \u2223 \u2191(b - 1)\n[PROOFSTEP]\nrw [Int.ofNat_sub (le_of_lt b1)]\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\n\u22a2 \u2191(4 * yn a1 i) \u2223 \u2191b - \u21911\n[PROOFSTEP]\nexact bm1.symm.dvd\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 i = k\n[PROOFSTEP]\nhave ki : k + i < 4 * yn a1 i :=\n  lt_of_le_of_lt (_root_.add_le_add ky (yn_ge_n a1 i)) <|\n    by\n    rw [\u2190 two_mul]\n    exact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 yn a1 i + yn a1 i < 4 * yn a1 i\n[PROOFSTEP]\nrw [\u2190 two_mul]\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 2 * yn a1 i < 4 * yn a1 i\n[PROOFSTEP]\nexact Nat.mul_lt_mul_of_pos_right (by decide) (strictMono_y a1 ipos)\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\n\u22a2 2 < 4\n[PROOFSTEP]\ndecide\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n\u22a2 i = k\n[PROOFSTEP]\nhave ji : j \u2261 i [MOD 4 * n] :=\n  have : xn a1 j \u2261 xn a1 i [MOD xn a1 n] := (xy_modEq_of_modEq b1 a1 ba j).left.symm.trans sx\n  (modEq_of_xn_modEq a1 ipos iln this).resolve_right fun ji : j + i \u2261 0 [MOD 4 * n] =>\n    not_le_of_gt ki <|\n      Nat.le_of_dvd (lt_of_lt_of_le ipos <| Nat.le_add_left _ _) <|\n        modEq_zero_iff_dvd.1 <| (jk.symm.add_right i).trans <| ji.of_dvd yd\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\n\u22a2 i = k\n[PROOFSTEP]\nhave : i % (4 * yn a1 i) = k % (4 * yn a1 i) := (ji.of_dvd yd).symm.trans jk\n[GOAL]\na k x y : \u2115\na1 : 1 < a\nky\u271d : k \u2264 y\nu v s t b : \u2115\nb1 : 1 < b\ni n j : \u2115\nbm1 : b \u2261 1 [MOD 4 * yn a1 i]\nba : b \u2261 a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i \u2223 yn a1 n\nsx : xn b1 j \u2261 xn a1 i [MOD xn a1 n]\ntk : yn b1 j \u2261 k [MOD 4 * yn a1 i]\nky : k \u2264 yn a1 i\nx\u271d :\n  1 < a \u2227\n    k \u2264 yn a1 i \u2227\n      (xn a1 i = 1 \u2227 yn a1 i = 0 \u2228\n        \u2203 u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 \u2227\n            u * u - (a * a - 1) * v * v = 1 \u2227\n              s * s - (b * b - 1) * t * t = 1 \u2227\n                1 < b \u2227\n                  b \u2261 1 [MOD 4 * yn a1 i] \u2227\n                    b \u2261 a [MOD u] \u2227 0 < v \u2227 yn a1 i * yn a1 i \u2223 v \u2227 s \u2261 xn a1 i [MOD u] \u2227 t \u2261 k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i \u2264 n\nyd : 4 * yn a1 i \u2223 4 * n\njk : j \u2261 k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j \u2261 i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n\u22a2 i = k\n[PROOFSTEP]\nrwa [Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_left _ _) ki),\n  Nat.mod_eq_of_lt (lt_of_le_of_lt (Nat.le_add_right _ _) ki)] at this \n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\n\u22a2 \u2191a \u2264 \u2191a ^ 2 - (\u2191a - 1) ^ 2 - 1\n[PROOFSTEP]\nrw [sub_sq, mul_one, one_pow, sub_add, sub_sub_cancel, two_mul, sub_sub, \u2190 add_sub, le_add_iff_nonneg_right, sub_nonneg,\n  Int.add_one_le_iff]\n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\n\u22a2 1 < \u2191a\n[PROOFSTEP]\nnorm_cast\n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\n\u22a2 1 < a\n[PROOFSTEP]\nexact lt_of_le_of_lt (Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)) hya\n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\n\u22a2 \u2191a ^ 2 - (\u2191a - 1) ^ 2 - 1 \u2264 \u2191a ^ 2 - (\u2191a - \u2191y) ^ 2 - 1\n[PROOFSTEP]\nhave := hya.le\n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\n\u22a2 \u2191a ^ 2 - (\u2191a - 1) ^ 2 - 1 \u2264 \u2191a ^ 2 - (\u2191a - \u2191y) ^ 2 - 1\n[PROOFSTEP]\nmono*\n[GOAL]\ncase hab.hab.ha\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 0 \u2264 \u2191a\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hab.hcd.ha\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 0 \u2264 \u2191a - \u2191y\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hab.hcd.hab.hcd\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 1 \u2264 \u2191y\n[PROOFSTEP]\nnorm_cast\n[GOAL]\ncase hab.hab.ha\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 0 \u2264 a\n[PROOFSTEP]\nsimp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]\n[GOAL]\ncase hab.hcd.ha\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 0 \u2264 a - y\n[PROOFSTEP]\nsimp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]\n[GOAL]\ncase hab.hcd.hab.hcd\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\nthis : y \u2264 a\nhy0_symm : 0 \u2260 y\nhk0_symm : 0 \u2260 k\n\u22a2 1 \u2264 y\n[PROOFSTEP]\nsimp [Nat.zero_le, Nat.succ_le_of_lt (Nat.pos_of_ne_zero hy0)]\n[GOAL]\na y k : \u2115\nhy0 : y \u2260 0\nhk0 : k \u2260 0\nhyk : y ^ k < a\nhya : y < a\n\u22a2 \u2191a ^ 2 - (\u2191a - \u2191y) ^ 2 - 1 = 2 * \u2191a * \u2191y - \u2191y * \u2191y - 1\n[PROOFSTEP]\nring\n[GOAL]\nm n k : \u2115\n\u22a2 n ^ k = m \u2194\n    k = 0 \u2227 m = 1 \u2228\n      0 < k \u2227\n        (n = 0 \u2227 m = 0 \u2228\n          0 < n \u2227\n            \u2203 w a t z a1,\n              xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t] \u2227\n                2 * a * n = t + (n * n + 1) \u2227\n                  m < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nm n k : \u2115\n\u22a2 n ^ k = m \u2192\n    k = 0 \u2227 m = 1 \u2228\n      0 < k \u2227\n        (n = 0 \u2227 m = 0 \u2228\n          0 < n \u2227\n            \u2203 w a t z a1,\n              xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t] \u2227\n                2 * a * n = t + (n * n + 1) \u2227\n                  m < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase mp\nn k : \u2115\n\u22a2 k = 0 \u2227 n ^ k = 1 \u2228\n    0 < k \u2227\n      (n = 0 \u2227 n ^ k = 0 \u2228\n        0 < n \u2227\n          \u2203 w a t z a1,\n            xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n              2 * a * n = t + (n * n + 1) \u2227\n                n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)\n[PROOFSTEP]\nrefine' k.eq_zero_or_pos.imp (fun k0 : k = 0 => k0.symm \u25b8 \u27e8rfl, rfl\u27e9) fun hk => \u27e8hk, _\u27e9\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\n\u22a2 n = 0 \u2227 n ^ k = 0 \u2228\n    0 < n \u2227\n      \u2203 w a t z a1,\n        xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n          2 * a * n = t + (n * n + 1) \u2227\n            n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nrefine' n.eq_zero_or_pos.imp (fun n0 : n = 0 => n0.symm \u25b8 \u27e8rfl, zero_pow hk\u27e9) fun hn => \u27e8hn, _\u27e9\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nset w := max n k\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave nw : n \u2264 w := le_max_left _ _\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave kw : k \u2264 w := le_max_right _ _\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave wpos : 0 < w := hn.trans_le nw\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave w1 : 1 < w + 1 := Nat.succ_lt_succ wpos\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nset a := xn w1 w\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave a1 : 1 < a := strictMono_x w1 wpos\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave na : n \u2264 a := nw.trans (n_lt_xn w1 w).le\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nset x := xn a1 k\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nset y := yn a1 k\n[GOAL]\ncase mp\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nobtain \u27e8z, ze\u27e9 : w \u2223 yn w1 w\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\n\u22a2 w \u2223 yn w1 w\ncase mp.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 w = w * z\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nexact modEq_zero_iff_dvd.1 ((yn_modEq_a_sub_one w1 w).trans dvd_rfl.modEq_zero_nat)\n[GOAL]\ncase mp.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 w = w * z\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave nt : (\u2191(n ^ k) : \u2124) < 2 * a * n - n * n - 1 :=\n  by\n  refine' eq_pow_of_pell_lem hn.ne' hk.ne' _\n  calc\n    n ^ k \u2264 n ^ w := Nat.pow_le_pow_of_le_right hn kw\n    _ < (w + 1) ^ w := (Nat.pow_lt_pow_of_lt_left (Nat.lt_succ_of_le nw) wpos)\n    _ \u2264 a := xn_ge_a_pow w1 w\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 w = w * z\n\u22a2 \u2191(n ^ k) < 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\n[PROOFSTEP]\nrefine' eq_pow_of_pell_lem hn.ne' hk.ne' _\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 w = w * z\n\u22a2 n ^ k < a\n[PROOFSTEP]\ncalc\n  n ^ k \u2264 n ^ w := Nat.pow_le_pow_of_le_right hn kw\n  _ < (w + 1) ^ w := (Nat.pow_lt_pow_of_lt_left (Nat.lt_succ_of_le nw) wpos)\n  _ \u2264 a := xn_ge_a_pow w1 w\n[GOAL]\ncase mp.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 w\nkw : k \u2264 w\nwpos : 0 < w\nw1 : 1 < w + 1\na : \u2115 := xn w1 w\na1 : 1 < a\nna : n \u2264 a\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 w = w * z\nnt : \u2191(n ^ k) < 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nlift (2 * a * n - n * n - 1 : \u2124) to \u2115 using (Nat.cast_nonneg _).trans nt.le with t te\n[GOAL]\ncase mp.intro.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave tm : x \u2261 y * (a - n) + n ^ k [MOD t] := by\n  apply modEq_of_dvd\n  rw [Int.ofNat_add, Int.ofNat_mul, Int.ofNat_sub na, te]\n  exact x_sub_y_dvd_pow a1 n k\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\n\u22a2 x \u2261 y * (a - n) + n ^ k [MOD t]\n[PROOFSTEP]\napply modEq_of_dvd\n[GOAL]\ncase a\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\n\u22a2 \u2191t \u2223 \u2191(y * (a - n) + n ^ k) - \u2191x\n[PROOFSTEP]\nrw [Int.ofNat_add, Int.ofNat_mul, Int.ofNat_sub na, te]\n[GOAL]\ncase a\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\n\u22a2 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1 \u2223 \u2191y * (\u2191(xn w1 (max n k)) - \u2191n) + \u2191(n ^ k) - \u2191x\n[PROOFSTEP]\nexact x_sub_y_dvd_pow a1 n k\n[GOAL]\ncase mp.intro.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave ta : 2 * a * n = t + (n * n + 1) := by\n  zify\n  rw [te]\n  ring_nf\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\n\u22a2 2 * a * n = t + (n * n + 1)\n[PROOFSTEP]\nzify\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\n\u22a2 2 * \u2191(xn w1 (max n k)) * \u2191n = \u2191t + (\u2191n * \u2191n + 1)\n[PROOFSTEP]\nrw [te]\n[GOAL]\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\n\u22a2 2 * \u2191(xn w1 (max n k)) * \u2191n = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1 + (\u2191n * \u2191n + 1)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase mp.intro.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\nta : 2 * a * n = t + (n * n + 1)\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nhave zp : a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1 := ze \u25b8 pell_eq w1 w\n[GOAL]\ncase mp.intro.intro\nn k : \u2115\nhk : 0 < k\nhn : 0 < n\nw : \u2115 := max n k\nnw : n \u2264 max n k\nkw : k \u2264 max n k\nwpos : 0 < max n k\nw1 : 1 < max n k + 1\na : \u2115 := xn w1 w\na1 : 1 < xn w1 (max n k)\nna : n \u2264 xn w1 (max n k)\nx : \u2115 := xn a1 k\ny : \u2115 := yn a1 k\nz : \u2115\nze : yn w1 (max n k) = max n k * z\nt : \u2115\nte : \u2191t = 2 * \u2191a * \u2191n - \u2191n * \u2191n - 1\nnt\u271d nt : \u2191(n ^ k) < \u2191t\ntm : x \u2261 y * (a - n) + n ^ k [MOD t]\nta : 2 * a * n = t + (n * n + 1)\nzp : a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 \u2203 w a t z a1,\n    xn a1 k \u2261 yn a1 k * (a - n) + n ^ k [MOD t] \u2227\n      2 * a * n = t + (n * n + 1) \u2227 n ^ k < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n[PROOFSTEP]\nexact \u27e8w, a, t, z, a1, tm, ta, Nat.cast_lt.1 nt, nw, kw, zp\u27e9\n[GOAL]\ncase mpr\nm n k : \u2115\n\u22a2 k = 0 \u2227 m = 1 \u2228\n      0 < k \u2227\n        (n = 0 \u2227 m = 0 \u2228\n          0 < n \u2227\n            \u2203 w a t z a1,\n              xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t] \u2227\n                2 * a * n = t + (n * n + 1) \u2227\n                  m < t \u2227 n \u2264 w \u2227 k \u2264 w \u2227 a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1) \u2192\n    n ^ k = m\n[PROOFSTEP]\nrintro (\u27e8rfl, rfl\u27e9 | \u27e8hk0, \u27e8rfl, rfl\u27e9 | \u27e8hn0, w, a, t, z, a1, tm, ta, mt, nw, kw, zp\u27e9\u27e9)\n[GOAL]\ncase mpr.inl.intro\nn : \u2115\n\u22a2 n ^ 0 = 1\n[PROOFSTEP]\nexact _root_.pow_zero n\n[GOAL]\ncase mpr.inr.intro.inl.intro\nk : \u2115\nhk0 : 0 < k\n\u22a2 0 ^ k = 0\n[PROOFSTEP]\nexact zero_pow hk0\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw a t z : \u2115\na1 : 1 < a\ntm : xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t]\nta : 2 * a * n = t + (n * n + 1)\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nzp : a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave hw0 : 0 < w := hn0.trans_le nw\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw a t z : \u2115\na1 : 1 < a\ntm : xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t]\nta : 2 * a * n = t + (n * n + 1)\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nzp : a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhw0 : 0 < w\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave hw1 : 1 < w + 1 := Nat.succ_lt_succ hw0\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw a t z : \u2115\na1 : 1 < a\ntm : xn a1 k \u2261 yn a1 k * (a - n) + m [MOD t]\nta : 2 * a * n = t + (n * n + 1)\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nzp : a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhw0 : 0 < w\nhw1 : 1 < w + 1\n\u22a2 n ^ k = m\n[PROOFSTEP]\nrcases eq_pell hw1 zp with \u27e8j, rfl, yj\u27e9\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave hj0 : 0 < j := by\n  apply Nat.pos_of_ne_zero\n  rintro rfl\n  exact lt_irrefl 1 a1\n[GOAL]\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 0 < j\n[PROOFSTEP]\napply Nat.pos_of_ne_zero\n[GOAL]\ncase a\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 j \u2260 0\n[PROOFSTEP]\nrintro rfl\n[GOAL]\ncase a\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nyj : w * z = yn hw1 0\na1 : 1 < xn hw1 0\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 0 - n) + m [MOD t]\nta : 2 * xn hw1 0 * n = t + (n * n + 1)\nzp : xn hw1 0 * xn hw1 0 - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\n\u22a2 False\n[PROOFSTEP]\nexact lt_irrefl 1 a1\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave wj : w \u2264 j :=\n  Nat.le_of_dvd hj0 (modEq_zero_iff_dvd.1 <| (yn_modEq_a_sub_one hw1 j).symm.trans <| modEq_zero_iff_dvd.2 \u27e8z, yj.symm\u27e9)\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave hnka : n ^ k < xn hw1 j\n[GOAL]\ncase hnka\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\n\u22a2 n ^ k < xn hw1 j\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\n\u22a2 n ^ k = m\n[PROOFSTEP]\ncalc\n  n ^ k \u2264 n ^ j := Nat.pow_le_pow_of_le_right hn0 (le_trans kw wj)\n  _ < (w + 1) ^ j := (Nat.pow_lt_pow_of_lt_left (Nat.lt_succ_of_le nw) hj0)\n  _ \u2264 xn hw1 j := xn_ge_a_pow hw1 j\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave nt : (\u2191(n ^ k) : \u2124) < 2 * xn hw1 j * n - n * n - 1 := eq_pow_of_pell_lem hn0.ne' hk0.ne' hnka\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave na : n \u2264 xn hw1 j := (Nat.le_self_pow hk0.ne' _).trans hnka.le\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave te : (t : \u2124) = 2 * xn hw1 j * n - n * n - 1 :=\n  by\n  rw [sub_sub, eq_sub_iff_add_eq]\n  exact_mod_cast ta.symm\n[GOAL]\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\n\u22a2 \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n[PROOFSTEP]\nrw [sub_sub, eq_sub_iff_add_eq]\n[GOAL]\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\n\u22a2 \u2191t + (\u2191n * \u2191n + 1) = 2 * \u2191(xn hw1 j) * \u2191n\n[PROOFSTEP]\nexact_mod_cast ta.symm\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + n ^ k [MOD t] :=\n  by\n  apply modEq_of_dvd\n  rw [te, Nat.cast_add, Nat.cast_mul, Int.ofNat_sub na]\n  exact x_sub_y_dvd_pow a1 n k\n[GOAL]\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n\u22a2 xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + n ^ k [MOD t]\n[PROOFSTEP]\napply modEq_of_dvd\n[GOAL]\ncase a\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n\u22a2 \u2191t \u2223 \u2191(yn a1 k * (xn hw1 j - n) + n ^ k) - \u2191(xn a1 k)\n[PROOFSTEP]\nrw [te, Nat.cast_add, Nat.cast_mul, Int.ofNat_sub na]\n[GOAL]\ncase a\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\n\u22a2 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1 \u2223 \u2191(yn a1 k) * (\u2191(xn hw1 j) - \u2191n) + \u2191(n ^ k) - \u2191(xn a1 k)\n[PROOFSTEP]\nexact x_sub_y_dvd_pow a1 n k\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nthis : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + n ^ k [MOD t]\n\u22a2 n ^ k = m\n[PROOFSTEP]\nhave : n ^ k % t = m % t := (this.symm.trans tm).add_left_cancel' _\n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nthis\u271d : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + n ^ k [MOD t]\nthis : n ^ k % t = m % t\n\u22a2 n ^ k = m\n[PROOFSTEP]\nrw [\u2190 te] at nt \n[GOAL]\ncase mpr.inr.intro.inr.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nm n k : \u2115\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : \u2115\nmt : m < t\nnw : n \u2264 w\nkw : k \u2264 w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : \u2115\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1\nhj0 : 0 < j\nwj : w \u2264 j\nhnka : n ^ k < xn hw1 j\nnt : \u2191(n ^ k) < \u2191t\nna : n \u2264 xn hw1 j\nte : \u2191t = 2 * \u2191(xn hw1 j) * \u2191n - \u2191n * \u2191n - 1\nthis\u271d : xn a1 k \u2261 yn a1 k * (xn hw1 j - n) + n ^ k [MOD t]\nthis : n ^ k % t = m % t\n\u22a2 n ^ k = m\n[PROOFSTEP]\nrwa [Nat.mod_eq_of_lt (Nat.cast_lt.1 nt), Nat.mod_eq_of_lt mt] at this \n", "meta": {"mathlib_filename": "Mathlib.NumberTheory.PellMatiyasevic", "llama_tokens": 97530, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624890918021, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.501401652750449}}
{"text": "[GOAL]\nx y : \u211d\n\u22a2 MonotoneOn (fun x => log x * x) {x | 1 \u2264 x}\n[PROOFSTEP]\nsimp only [MonotoneOn, mem_setOf_eq]\n[GOAL]\nx y : \u211d\n\u22a2 \u2200 \u2983a : \u211d\u2984, 1 \u2264 a \u2192 \u2200 \u2983b : \u211d\u2984, 1 \u2264 b \u2192 a \u2264 b \u2192 log a * a \u2264 log b * b\n[PROOFSTEP]\nintro x hex y hey hxy\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : 1 \u2264 x\ny : \u211d\nhey : 1 \u2264 y\nhxy : x \u2264 y\n\u22a2 log x * x \u2264 log y * y\n[PROOFSTEP]\nhave x_pos : 0 < x := lt_of_lt_of_le zero_lt_one hex\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : 1 \u2264 x\ny : \u211d\nhey : 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\n\u22a2 log x * x \u2264 log y * y\n[PROOFSTEP]\nhave y_pos : 0 < y := lt_of_lt_of_le zero_lt_one hey\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : 1 \u2264 x\ny : \u211d\nhey : 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\n\u22a2 log x * x \u2264 log y * y\n[PROOFSTEP]\nrefine' mul_le_mul ((log_le_log x_pos y_pos).mpr hxy) hxy (le_of_lt x_pos) _\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : 1 \u2264 x\ny : \u211d\nhey : 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\n\u22a2 0 \u2264 log y\n[PROOFSTEP]\nrwa [le_log_iff_exp_le y_pos, Real.exp_zero]\n[GOAL]\nx y : \u211d\n\u22a2 AntitoneOn (fun x => log x / x) {x | exp 1 \u2264 x}\n[PROOFSTEP]\nsimp only [AntitoneOn, mem_setOf_eq]\n[GOAL]\nx y : \u211d\n\u22a2 \u2200 \u2983a : \u211d\u2984, exp 1 \u2264 a \u2192 \u2200 \u2983b : \u211d\u2984, exp 1 \u2264 b \u2192 a \u2264 b \u2192 log b / b \u2264 log a / a\n[PROOFSTEP]\nintro x hex y hey hxy\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\n\u22a2 log y / y \u2264 log x / x\n[PROOFSTEP]\nhave x_pos : 0 < x := (exp_pos 1).trans_le hex\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\n\u22a2 log y / y \u2264 log x / x\n[PROOFSTEP]\nhave y_pos : 0 < y := (exp_pos 1).trans_le hey\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\n\u22a2 log y / y \u2264 log x / x\n[PROOFSTEP]\nhave hlogx : 1 \u2264 log x := by rwa [le_log_iff_exp_le x_pos]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\n\u22a2 1 \u2264 log x\n[PROOFSTEP]\nrwa [le_log_iff_exp_le x_pos]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\n\u22a2 log y / y \u2264 log x / x\n[PROOFSTEP]\nhave hyx : 0 \u2264 y / x - 1 := by rwa [le_sub_iff_add_le, le_div_iff x_pos, zero_add, one_mul]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\n\u22a2 0 \u2264 y / x - 1\n[PROOFSTEP]\nrwa [le_sub_iff_add_le, le_div_iff x_pos, zero_add, one_mul]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\nhyx : 0 \u2264 y / x - 1\n\u22a2 log y / y \u2264 log x / x\n[PROOFSTEP]\nrw [div_le_iff y_pos, \u2190 sub_le_sub_iff_right (log x)]\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\nhyx : 0 \u2264 y / x - 1\n\u22a2 log y - log x \u2264 log x / x * y - log x\n[PROOFSTEP]\ncalc\n  log y - log x = log (y / x) := by rw [log_div y_pos.ne' x_pos.ne']\n  _ \u2264 y / x - 1 := (log_le_sub_one_of_pos (div_pos y_pos x_pos))\n  _ \u2264 log x * (y / x - 1) := (le_mul_of_one_le_left hyx hlogx)\n  _ = log x / x * y - log x := by ring\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\nhyx : 0 \u2264 y / x - 1\n\u22a2 log y - log x = log (y / x)\n[PROOFSTEP]\nrw [log_div y_pos.ne' x_pos.ne']\n[GOAL]\nx\u271d y\u271d x : \u211d\nhex : exp 1 \u2264 x\ny : \u211d\nhey : exp 1 \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nhlogx : 1 \u2264 log x\nhyx : 0 \u2264 y / x - 1\n\u22a2 log x * (y / x - 1) = log x / x * y - log x\n[PROOFSTEP]\nring\n[GOAL]\nx y a : \u211d\nha : 0 < a\n\u22a2 AntitoneOn (fun x => log x / x ^ a) {x | exp (1 / a) \u2264 x}\n[PROOFSTEP]\nsimp only [AntitoneOn, mem_setOf_eq]\n[GOAL]\nx y a : \u211d\nha : 0 < a\n\u22a2 \u2200 \u2983a_1 : \u211d\u2984, exp (1 / a) \u2264 a_1 \u2192 \u2200 \u2983b : \u211d\u2984, exp (1 / a) \u2264 b \u2192 a_1 \u2264 b \u2192 log b / b ^ a \u2264 log a_1 / a_1 ^ a\n[PROOFSTEP]\nintro x hex y _ hxy\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\n\u22a2 log y / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nhave x_pos : 0 < x := lt_of_lt_of_le (exp_pos (1 / a)) hex\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\n\u22a2 log y / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nhave y_pos : 0 < y := by linarith\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\n\u22a2 0 < y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\n\u22a2 log y / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nhave x_nonneg : 0 \u2264 x := le_trans (le_of_lt (exp_pos (1 / a))) hex\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\n\u22a2 log y / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nhave y_nonneg : 0 \u2264 y := by linarith\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\n\u22a2 0 \u2264 y\n[PROOFSTEP]\nlinarith\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 log y / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nnth_rw 1 [\u2190 rpow_one y]\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 log (y ^ 1) / y ^ a \u2264 log x / x ^ a\n[PROOFSTEP]\nnth_rw 1 [\u2190 rpow_one x]\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 log (y ^ 1) / y ^ a \u2264 log (x ^ 1) / x ^ a\n[PROOFSTEP]\nrw [\u2190 div_self (ne_of_lt ha).symm, div_eq_mul_one_div a a, rpow_mul y_nonneg, rpow_mul x_nonneg,\n  log_rpow (rpow_pos_of_pos y_pos a), log_rpow (rpow_pos_of_pos x_pos a), mul_div_assoc, mul_div_assoc,\n  mul_le_mul_left (one_div_pos.mpr ha)]\n[GOAL]\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 log (y ^ a) / y ^ a \u2264 log (x ^ a) / x ^ a\n[PROOFSTEP]\nrefine' log_div_self_antitoneOn _ _ _\n[GOAL]\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 x ^ a \u2208 {x | exp 1 \u2264 x}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 \u2264 x ^ a\n[PROOFSTEP]\nconvert rpow_le_rpow _ hex (le_of_lt ha) using 1\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 = exp (1 / a) ^ a\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nrw [\u2190 exp_mul]\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 = exp (1 / a * a)\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nsimp only [Real.exp_eq_exp]\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 1 = 1 / a * a\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nfield_simp [(ne_of_lt ha).symm]\n[GOAL]\ncase refine'_1\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nexact le_of_lt (exp_pos (1 / a))\n[GOAL]\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 y ^ a \u2208 {x | exp 1 \u2264 x}\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq]\n[GOAL]\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 \u2264 y ^ a\n[PROOFSTEP]\nconvert rpow_le_rpow _ (_root_.trans hex hxy) (le_of_lt ha) using 1\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 = exp (1 / a) ^ a\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nrw [\u2190 exp_mul]\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 exp 1 = exp (1 / a * a)\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nsimp only [Real.exp_eq_exp]\n[GOAL]\ncase h.e'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 1 = 1 / a * a\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nfield_simp [(ne_of_lt ha).symm]\n[GOAL]\ncase refine'_2\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 0 \u2264 exp (1 / a)\n[PROOFSTEP]\nexact le_of_lt (exp_pos (1 / a))\n[GOAL]\ncase refine'_3\nx\u271d\u00b9 y\u271d a : \u211d\nha : 0 < a\nx : \u211d\nhex : exp (1 / a) \u2264 x\ny : \u211d\nx\u271d : exp (1 / a) \u2264 y\nhxy : x \u2264 y\nx_pos : 0 < x\ny_pos : 0 < y\nx_nonneg : 0 \u2264 x\ny_nonneg : 0 \u2264 y\n\u22a2 x ^ a \u2264 y ^ a\n[PROOFSTEP]\nexact rpow_le_rpow x_nonneg hxy (le_of_lt ha)\n[GOAL]\nx y : \u211d\n\u22a2 AntitoneOn (fun x => log x / sqrt x) {x | exp 2 \u2264 x}\n[PROOFSTEP]\nsimp_rw [sqrt_eq_rpow]\n[GOAL]\nx y : \u211d\n\u22a2 AntitoneOn (fun x => log x / x ^ (1 / 2)) {x | exp 2 \u2264 x}\n[PROOFSTEP]\nconvert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)\n[GOAL]\nx y : \u211d\n\u22a2 0 < 1 / 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase h.e'_6.h.e'_2.h.h.e'_3.h.e'_1\nx y x\u271d : \u211d\n\u22a2 2 = 1 / (1 / 2)\n[PROOFSTEP]\nnorm_num\n", "meta": {"mathlib_filename": "Mathlib.Analysis.SpecialFunctions.Log.Monotone", "llama_tokens": 6697, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7826624688140728, "lm_q2_score": 0.6406358685621719, "lm_q1q2_score": 0.5014016504997173}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 T R (n + 2) = 2 * X * T R (n + 1) - T R n\n[PROOFSTEP]\nrw [T]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 T R 2 = 2 * X ^ 2 - 1\n[PROOFSTEP]\nsimp only [T, sub_left_inj, sq, mul_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 n\n\u22a2 T R n = 2 * X * T R (n - 1) - T R (n - 2)\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := Nat.exists_eq_add_of_le h\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 T R (2 + n) = 2 * X * T R (2 + n - 1) - T R (2 + n - 2)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 T R (n + 2) = 2 * X * T R (n + 2 - 1) - T R (n + 2 - 2)\n[PROOFSTEP]\nexact T_add_two R n\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 U R (n + 2) = 2 * X * U R (n + 1) - U R n\n[PROOFSTEP]\nrw [U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 U R 2 = 4 * X ^ 2 - 1\n[PROOFSTEP]\nsimp only [U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 2 * X * (2 * X) - 1 = 4 * X ^ 2 - 1\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 n\n\u22a2 U R n = 2 * X * U R (n - 1) - U R (n - 2)\n[PROOFSTEP]\nobtain \u27e8n, rfl\u27e9 := Nat.exists_eq_add_of_le h\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 U R (2 + n) = 2 * X * U R (2 + n - 1) - U R (2 + n - 2)\n[PROOFSTEP]\nrw [add_comm]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh : 2 \u2264 2 + n\n\u22a2 U R (n + 2) = 2 * X * U R (n + 2 - 1) - U R (n + 2 - 2)\n[PROOFSTEP]\nexact U_add_two R n\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 U R (0 + 1) = X * U R 0 + T R (0 + 1)\n[PROOFSTEP]\nsimp only [T, U, two_mul, mul_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 U R (1 + 1) = X * U R 1 + T R (1 + 1)\n[PROOFSTEP]\nsimp only [T, U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 2 * X * (2 * X) - 1 = X * (2 * X) + (2 * X * X - 1)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1))\n[PROOFSTEP]\nrw [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1), U_eq_X_mul_U_add_T n]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) =\n    X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) = X * U R (n + 2) + T R (n + 2 + 1)\n[PROOFSTEP]\nsimp only [U_add_two, T_add_two]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 T R (n + 1) = U R (n + 1) - X * U R n\n[PROOFSTEP]\nrw [U_eq_X_mul_U_add_T, add_comm (X * U R n), add_sub_cancel]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 T R (0 + 2) = X * T R (0 + 1) - (1 - X ^ 2) * U R 0\n[PROOFSTEP]\nsimp only [T, U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 2 * X * X - 1 = X * X - (1 - X ^ 2) * 1\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 T R (1 + 2) = X * T R (1 + 1) - (1 - X ^ 2) * U R 1\n[PROOFSTEP]\nsimp only [T, U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 2 * X * (2 * X * X - 1) - X = X * (2 * X * X - 1) - (1 - X ^ 2) * (2 * X)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 2 * X * T R (n + 2 + 1) - T R (n + 2) =\n    2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n)\n[PROOFSTEP]\nsimp only [T_eq_X_mul_T_sub_pol_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 2 * X * (X * T R (n + 2) - (1 - X ^ 2) * U R (n + 1)) - (X * T R (n + 1) - (1 - X ^ 2) * U R n) =\n    X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 X * (2 * X * T R (n + 2) - T R (n + 1)) - (1 - X ^ 2) * (2 * X * U R (n + 1) - U R n) =\n    X * T R (n + 2 + 1) - (1 - X ^ 2) * U R (n + 2)\n[PROOFSTEP]\nrw [T_add_two _ (n + 1), U_add_two]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (1 - X ^ 2) * U R n = X * T R (n + 1) - T R (n + 2)\n[PROOFSTEP]\nrw [T_eq_X_mul_T_sub_pol_U, \u2190 sub_add, sub_self, zero_add]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (T R 0) = T S 0\n[PROOFSTEP]\nsimp only [T_zero, Polynomial.map_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (T R 1) = T S 1\n[PROOFSTEP]\nsimp only [T_one, map_X]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f (T R (n + 2)) = T S (n + 2)\n[PROOFSTEP]\nsimp only [T_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add, Polynomial.map_one,\n  Polynomial.map_ofNat, map_T f (n + 1), map_T f n]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (U R 0) = U S 0\n[PROOFSTEP]\nsimp only [U_zero, Polynomial.map_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\n\u22a2 map f (U R 1) = U S 1\n[PROOFSTEP]\nsimp [U_one, map_X, Polynomial.map_mul, Polynomial.map_add, Polynomial.map_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f (U R (n + 2)) = U S (n + 2)\n[PROOFSTEP]\nsimp only [U_add_two, Polynomial.map_mul, Polynomial.map_sub, map_X, Polynomial.map_add, Polynomial.map_one,\n  map_U f (n + 1), map_U f n]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nf : R \u2192+* S\nn : \u2115\n\u22a2 map f 2 * X * U S (n + 1) - U S n = 2 * X * U S (n + 1) - U S n\n[PROOFSTEP]\nnorm_num\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2191derivative (T R (0 + 1)) = (\u21910 + 1) * U R 0\n[PROOFSTEP]\nsimp only [T_one, U_zero, derivative_X, Nat.cast_zero, zero_add, mul_one]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2191derivative (T R (1 + 1)) = (\u21911 + 1) * U R 1\n[PROOFSTEP]\nsimp [T_two, U_one, derivative_sub, derivative_one, derivative_mul, derivative_X_pow, add_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191derivative (T R (n + 2 + 1)) = 2 * T R (n + 2) + 2 * X * \u2191derivative (T R (n + 1 + 1)) - \u2191derivative (T R (n + 1))\n[PROOFSTEP]\nrw [T_add_two _ (n + 1), derivative_sub, derivative_mul, derivative_mul, derivative_X, derivative_ofNat]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (0 * X + 2 * 1) * T R (n + 1 + 1) + 2 * X * \u2191derivative (T R (n + 1 + 1)) - \u2191derivative (T R (n + 1)) =\n    2 * T R (n + 2) + 2 * X * \u2191derivative (T R (n + 1 + 1)) - \u2191derivative (T R (n + 1))\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 2 * T R (n + 2) + 2 * X * \u2191derivative (T R (n + 1 + 1)) - \u2191derivative (T R (n + 1)) =\n    2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * ((\u2191n + 1 + 1) * U R (n + 1)) - (\u2191n + 1) * U R n\n[PROOFSTEP]\nrw_mod_cast [T_derivative_eq_U (n + 1), T_derivative_eq_U n, T_eq_U_sub_X_mul_U _ (n + 1)]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 2 * (U R (n + 1 + 1) - X * U R (n + 1)) + 2 * X * ((\u2191n + 1 + 1) * U R (n + 1)) - (\u2191n + 1) * U R n =\n    (\u2191n + 1) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 1) * (2 * X * U R (n + 1) - U R n) + 2 * U R (n + 2) = (\u2191n + 1) * U R (n + 2) + 2 * U R (n + 2)\n[PROOFSTEP]\nrw [U_add_two]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 1) * U R (n + 2) + 2 * U R (n + 2) = (\u2191n + 2 + 1) * U R (n + 2)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 2 + 1) * U R (n + 2) = (\u2191(n + 2) + 1) * U R (n + 2)\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (1 - X ^ 2) * \u2191derivative (T R (n + 1)) = (1 - X ^ 2) * ((\u2191n + 1) * U R n)\n[PROOFSTEP]\nrw [T_derivative_eq_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (1 - X ^ 2) * ((\u2191n + 1) * U R n) = (\u2191n + 1) * ((1 - X ^ 2) * U R n)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 1) * ((1 - X ^ 2) * U R n) = (\u2191n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n))\n[PROOFSTEP]\nrw [one_sub_X_sq_mul_U_eq_pol_in_T, T_add_two]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 1) * (X * T R (n + 1) - (2 * X * T R (n + 1) - T R n)) = (\u2191n + 1) * (T R n - X * T R (n + 1))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 (\u2191n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nhave h :\n  derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) :=\n  by\n  conv_lhs => rw [T_eq_X_mul_T_sub_pol_U]\n  simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow, one_mul, T_derivative_eq_U]\n  rw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]\n  ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nconv_lhs => rw [T_eq_X_mul_T_sub_pol_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n| \u2191derivative (T R (n + 2))\n[PROOFSTEP]\nrw [T_eq_X_mul_T_sub_pol_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n| \u2191derivative (T R (n + 2))\n[PROOFSTEP]\nrw [T_eq_X_mul_T_sub_pol_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n| \u2191derivative (T R (n + 2))\n[PROOFSTEP]\nrw [T_eq_X_mul_T_sub_pol_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 \u2191derivative (X * T R (n + 1) - (1 - X ^ 2) * U R n) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nsimp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow, one_mul, T_derivative_eq_U]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 T R (n + 1) + X * ((\u2191n + 1) * U R n) - ((0 - \u2191C \u21912 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * \u2191derivative (U R n)) =\n    U R (n + 1) - X * U R n + X * ((\u2191n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nrw [T_eq_U_sub_X_mul_U, C_eq_nat_cast]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\n\u22a2 U R (n + 1) - X * U R n + X * ((\u2191n + 1) * U R n) -\n      ((0 - \u21912 * X ^ (2 - 1)) * U R n + (1 - X ^ 2) * \u2191derivative (U R n)) =\n    U R (n + 1) - X * U R n + X * ((\u2191n + 1) * U R n) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh :\n  \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n\u22a2 (\u2191n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\ncalc\n  ((n : R[X]) + 1) * T R (n + 1) =\n      ((n : R[X]) + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((n + 1 : R[X]) * U R n) - (X * U R n + T R (n + 1)) :=\n    by ring\n  _ = derivative (T R (n + 2)) - X * derivative (T R (n + 1)) - U R (n + 1) := by\n    rw [\u2190 U_eq_X_mul_U_add_T, \u2190 T_derivative_eq_U, \u2190 Nat.cast_one, \u2190 Nat.cast_add, Nat.cast_one, \u2190\n      T_derivative_eq_U (n + 1)]\n  _ =\n      U R (n + 1) - X * U R n + X * derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * derivative (U R n) -\n          X * derivative (T R (n + 1)) -\n        U R (n + 1) :=\n    by rw [h]\n  _ = X * U R n - (1 - X ^ 2) * derivative (U R n) := by ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh :\n  \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n\u22a2 (\u2191n + 1) * T R (n + 1) = (\u2191n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((\u2191n + 1) * U R n) - (X * U R n + T R (n + 1))\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh :\n  \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n\u22a2 (\u2191n + 1 + 1) * (X * U R n + T R (n + 1)) - X * ((\u2191n + 1) * U R n) - (X * U R n + T R (n + 1)) =\n    \u2191derivative (T R (n + 2)) - X * \u2191derivative (T R (n + 1)) - U R (n + 1)\n[PROOFSTEP]\nrw [\u2190 U_eq_X_mul_U_add_T, \u2190 T_derivative_eq_U, \u2190 Nat.cast_one, \u2190 Nat.cast_add, Nat.cast_one, \u2190\n  T_derivative_eq_U (n + 1)]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh :\n  \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n\u22a2 \u2191derivative (T R (n + 2)) - X * \u2191derivative (T R (n + 1)) - U R (n + 1) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n) -\n        X * \u2191derivative (T R (n + 1)) -\n      U R (n + 1)\n[PROOFSTEP]\nrw [h]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2115\nh :\n  \u2191derivative (T R (n + 2)) =\n    U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n\u22a2 U R (n + 1) - X * U R n + X * \u2191derivative (T R (n + 1)) + 2 * X * U R n - (1 - X ^ 2) * \u2191derivative (U R n) -\n        X * \u2191derivative (T R (n + 1)) -\n      U R (n + 1) =\n    X * U R n - (1 - X ^ 2) * \u2191derivative (U R n)\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (k : \u2115), 2 * T R 0 * T R (0 + k) = T R (2 * 0 + k) + T R k\n[PROOFSTEP]\nsimp [two_mul, add_mul]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (k : \u2115), 2 * T R 1 * T R (1 + k) = T R (2 * 1 + k) + T R k\n[PROOFSTEP]\nsimp [add_comm]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm : \u2115\n\u22a2 \u2200 (k : \u2115), 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k\n[PROOFSTEP]\nintro k\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\n\u22a2 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k\n[PROOFSTEP]\nsuffices 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n  by\n  have h_nat\u2081 : 2 * (m + 2) + k = 2 * m + k + 4 := by ring\n  have h_nat\u2082 : m + 2 + k = m + k + 2 := by ring\n  simpa [h_nat\u2081, h_nat\u2082] using this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n\u22a2 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k\n[PROOFSTEP]\nhave h_nat\u2081 : 2 * (m + 2) + k = 2 * m + k + 4 := by ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n\u22a2 2 * (m + 2) + k = 2 * m + k + 4\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat\u2081 : 2 * (m + 2) + k = 2 * m + k + 4\n\u22a2 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k\n[PROOFSTEP]\nhave h_nat\u2082 : m + 2 + k = m + k + 2 := by ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat\u2081 : 2 * (m + 2) + k = 2 * m + k + 4\n\u22a2 m + 2 + k = m + k + 2\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nthis : 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\nh_nat\u2081 : 2 * (m + 2) + k = 2 * m + k + 4\nh_nat\u2082 : m + 2 + k = m + k + 2\n\u22a2 2 * T R (m + 2) * T R (m + 2 + k) = T R (2 * (m + 2) + k) + T R k\n[PROOFSTEP]\nsimpa [h_nat\u2081, h_nat\u2082] using this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nhave H\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1) :=\n  by\n  have h_nat\u2081 : m + 1 + (k + 1) = m + k + 2 := by ring\n  have h_nat\u2082 : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring\n  simpa [h_nat\u2081, h_nat\u2082] using\n    mul_T (m + 1)\n      (k + 1)\n        -- clean up the `T` nat indices in the inductive hypothesis applied to `m` and `k + 2`\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\n\u22a2 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n[PROOFSTEP]\nhave h_nat\u2081 : m + 1 + (k + 1) = m + k + 2 := by ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\n\u22a2 m + 1 + (k + 1) = m + k + 2\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nh_nat\u2081 : m + 1 + (k + 1) = m + k + 2\n\u22a2 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n[PROOFSTEP]\nhave h_nat\u2082 : 2 * (m + 1) + (k + 1) = 2 * m + k + 3 := by ring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nh_nat\u2081 : m + 1 + (k + 1) = m + k + 2\n\u22a2 2 * (m + 1) + (k + 1) = 2 * m + k + 3\n[PROOFSTEP]\nring\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nh_nat\u2081 : m + 1 + (k + 1) = m + k + 2\nh_nat\u2082 : 2 * (m + 1) + (k + 1) = 2 * m + k + 3\n\u22a2 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n[PROOFSTEP]\nsimpa [h_nat\u2081, h_nat\u2082] using\n  mul_T (m + 1)\n    (k + 1)\n      -- clean up the `T` nat indices in the inductive hypothesis applied to `m` and `k + 2`\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nhave H\u2082 : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2) :=\n  by\n  have h_nat\u2081 : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]\n  have h_nat\u2082 : m + (k + 2) = m + k + 2 := by simp [add_assoc]\n  simpa [h_nat\u2081, h_nat\u2082] using\n    mul_T m\n      (k + 2)\n        -- state the `T` recurrence relation for a few useful indices\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n\u22a2 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n[PROOFSTEP]\nhave h_nat\u2081 : 2 * m + (k + 2) = 2 * m + k + 2 := by simp [add_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\n\u22a2 2 * m + (k + 2) = 2 * m + k + 2\n[PROOFSTEP]\nsimp [add_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat\u2081 : 2 * m + (k + 2) = 2 * m + k + 2\n\u22a2 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n[PROOFSTEP]\nhave h_nat\u2082 : m + (k + 2) = m + k + 2 := by simp [add_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat\u2081 : 2 * m + (k + 2) = 2 * m + k + 2\n\u22a2 m + (k + 2) = m + k + 2\n[PROOFSTEP]\nsimp [add_assoc]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nh_nat\u2081 : 2 * m + (k + 2) = 2 * m + k + 2\nh_nat\u2082 : m + (k + 2) = m + k + 2\n\u22a2 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n[PROOFSTEP]\nsimpa [h_nat\u2081, h_nat\u2082] using\n  mul_T m\n    (k + 2)\n      -- state the `T` recurrence relation for a few useful indices\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH\u2082 : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nhave h\u2081 := T_add_two R m\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH\u2082 : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh\u2081 : T R (m + 2) = 2 * X * T R (m + 1) - T R m\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nhave h\u2082 : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2) := T_add_two R (2 * m + k + 2)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH\u2082 : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh\u2081 : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh\u2082 : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nhave h\u2083 := T_add_two R k\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm k : \u2115\nH\u2081 : 2 * T R (m + 1) * T R (m + k + 2) = T R (2 * m + k + 3) + T R (k + 1)\nH\u2082 : 2 * T R m * T R (m + k + 2) = T R (2 * m + k + 2) + T R (k + 2)\nh\u2081 : T R (m + 2) = 2 * X * T R (m + 1) - T R m\nh\u2082 : T R (2 * m + k + 4) = 2 * X * T R (2 * m + k + 3) - T R (2 * m + k + 2)\nh\u2083 : T R (k + 2) = 2 * X * T R (k + 1) - T R k\n\u22a2 2 * T R (m + 2) * T R (m + k + 2) = T R (2 * m + k + 4) + T R k\n[PROOFSTEP]\nlinear_combination 2 * T R (m + k + 2) * h\u2081 + 2 * (X : R[X]) * H\u2081 - H\u2082 - h\u2082 - h\u2083\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (n : \u2115), T R (0 * n) = comp (T R 0) (T R n)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\n\u22a2 \u2200 (n : \u2115), T R (1 * n) = comp (T R 1) (T R n)\n[PROOFSTEP]\nsimp\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm : \u2115\n\u22a2 \u2200 (n : \u2115), T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)\n[PROOFSTEP]\nintro n\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)\n[PROOFSTEP]\nhave : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n) := by\n  convert mul_T R n (m * n) using 1 <;> ring_nf\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)\n[PROOFSTEP]\nconvert mul_T R n (m * n) using 1\n[GOAL]\ncase h.e'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 2 * T R n * T R ((m + 1) * n) = 2 * T R n * T R (n + m * n)\n[PROOFSTEP]\nring_nf\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\n\u22a2 T R ((m + 2) * n) + T R (m * n) = T R (2 * n + m * n) + T R (m * n)\n[PROOFSTEP]\nring_nf\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nm n : \u2115\nthis : 2 * T R n * T R ((m + 1) * n) = T R ((m + 2) * n) + T R (m * n)\n\u22a2 T R ((m + 2) * n) = comp (T R (m + 2)) (T R n)\n[PROOFSTEP]\nsimp [this, T_mul m, \u2190 T_mul (m + 1)]\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Polynomial.Chebyshev", "llama_tokens": 13573, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.5010852675954145}}
{"text": "[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 ((evaluation C D).obj X).map (kernel.\u03b9 \u03b1) \u226b (Iso.refl (((evaluation C D).obj X).obj F)).hom =\n    (PreservesKernel.iso ((evaluation C D).obj X) \u03b1).hom \u226b kernel.\u03b9 (NatTrans.app \u03b1 X)\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app (kernel.\u03b9 \u03b1) X \u226b \ud835\udfd9 (F.obj X) =\n    (PreservesKernel.iso ((evaluation C D).obj X) \u03b1).hom \u226b kernel.\u03b9 (NatTrans.app \u03b1 X)\n[PROOFSTEP]\nsimp only [Category.comp_id, PreservesKernel.iso_hom]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app (kernel.\u03b9 \u03b1) X = kernelComparison \u03b1 ((evaluation C D).obj X) \u226b kernel.\u03b9 (NatTrans.app \u03b1 X)\n[PROOFSTEP]\nexact (kernelComparison_comp_\u03b9 _ ((evaluation C D).obj X)).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 ((evaluation C D).obj X).map (cokernel.\u03c0 \u03b1) \u226b (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).hom =\n    (Iso.refl (((evaluation C D).obj X).obj G)).hom \u226b cokernel.\u03c0 (NatTrans.app \u03b1 X)\n[PROOFSTEP]\napply (cancel_mono (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).inv).1\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 (((evaluation C D).obj X).map (cokernel.\u03c0 \u03b1) \u226b (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).hom) \u226b\n      (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).inv =\n    ((Iso.refl (((evaluation C D).obj X).obj G)).hom \u226b cokernel.\u03c0 (NatTrans.app \u03b1 X)) \u226b\n      (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).inv\n[PROOFSTEP]\nsimp only [Category.assoc, Iso.hom_inv_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 ((evaluation C D).obj X).map (cokernel.\u03c0 \u03b1) \u226b \ud835\udfd9 (((evaluation C D).obj X).obj (cokernel \u03b1)) =\n    (Iso.refl (((evaluation C D).obj X).obj G)).hom \u226b\n      cokernel.\u03c0 (NatTrans.app \u03b1 X) \u226b (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).inv\n[PROOFSTEP]\ndsimp\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app (cokernel.\u03c0 \u03b1) X \u226b \ud835\udfd9 ((cokernel \u03b1).obj X) =\n    \ud835\udfd9 (G.obj X) \u226b cokernel.\u03c0 (NatTrans.app \u03b1 X) \u226b (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).inv\n[PROOFSTEP]\nsimp only [PreservesCokernel.iso_inv, Category.id_comp, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app (cokernel.\u03c0 \u03b1) X = cokernel.\u03c0 (NatTrans.app \u03b1 X) \u226b cokernelComparison \u03b1 ((evaluation C D).obj X)\n[PROOFSTEP]\nexact (\u03c0_comp_cokernelComparison _ ((evaluation C D).obj X)).symm\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 coimageImageComparison (NatTrans.app \u03b1 X) =\n    (coimageObjIso \u03b1 X).inv \u226b NatTrans.app (coimageImageComparison \u03b1) X \u226b (imageObjIso \u03b1 X).hom\n[PROOFSTEP]\next\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 (coequalizer.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) 0 \u226b coimageImageComparison (NatTrans.app \u03b1 X)) \u226b\n      equalizer.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X)) 0 =\n    (coequalizer.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) 0 \u226b\n        (coimageObjIso \u03b1 X).inv \u226b NatTrans.app (coimageImageComparison \u03b1) X \u226b (imageObjIso \u03b1 X).hom) \u226b\n      equalizer.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X)) 0\n[PROOFSTEP]\ndsimp\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 (cokernel.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) \u226b coimageImageComparison (NatTrans.app \u03b1 X)) \u226b\n      kernel.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X)) =\n    (cokernel.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) \u226b\n        (coimageObjIso \u03b1 X).inv \u226b NatTrans.app (coimageImageComparison \u03b1) X \u226b (imageObjIso \u03b1 X).hom) \u226b\n      kernel.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X))\n[PROOFSTEP]\ndsimp [imageObjIso, coimageObjIso, cokernel.map]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 (cokernel.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) \u226b coimageImageComparison (NatTrans.app \u03b1 X)) \u226b\n      kernel.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X)) =\n    (cokernel.\u03c0 (kernel.\u03b9 (NatTrans.app \u03b1 X)) \u226b\n        (cokernel.desc (kernel.\u03b9 (NatTrans.app \u03b1 X)) (\ud835\udfd9 (F.obj X) \u226b cokernel.\u03c0 (NatTrans.app (kernel.\u03b9 \u03b1) X))\n              (_ : kernel.\u03b9 (NatTrans.app \u03b1 X) \u226b \ud835\udfd9 (F.obj X) \u226b cokernel.\u03c0 (NatTrans.app (kernel.\u03b9 \u03b1) X) = 0) \u226b\n            (PreservesCokernel.iso ((evaluation C D).obj X) (kernel.\u03b9 \u03b1)).inv) \u226b\n          NatTrans.app (coimageImageComparison \u03b1) X \u226b\n            (PreservesKernel.iso ((evaluation C D).obj X) (cokernel.\u03c0 \u03b1)).hom \u226b\n              kernel.map (NatTrans.app (cokernel.\u03c0 \u03b1) X) (cokernel.\u03c0 (NatTrans.app \u03b1 X)) (\ud835\udfd9 (G.obj X))\n                (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).hom\n                (_ :\n                  ((evaluation C D).obj X).map (cokernel.\u03c0 \u03b1) \u226b (PreservesCokernel.iso ((evaluation C D).obj X) \u03b1).hom =\n                    (Iso.refl (((evaluation C D).obj X).obj G)).hom \u226b cokernel.\u03c0 (NatTrans.app \u03b1 X))) \u226b\n      kernel.\u03b9 (cokernel.\u03c0 (NatTrans.app \u03b1 X))\n[PROOFSTEP]\nsimp only [coimage_image_factorisation, PreservesKernel.iso_hom, Category.assoc, kernel.lift_\u03b9, Category.comp_id,\n  PreservesCokernel.iso_inv, cokernel.\u03c0_desc_assoc, Category.id_comp]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app \u03b1 X =\n    cokernel.\u03c0 (NatTrans.app (kernel.\u03b9 \u03b1) X) \u226b\n      cokernelComparison (kernel.\u03b9 \u03b1) ((evaluation C D).obj X) \u226b\n        NatTrans.app (coimageImageComparison \u03b1) X \u226b\n          kernelComparison (cokernel.\u03c0 \u03b1) ((evaluation C D).obj X) \u226b kernel.\u03b9 (NatTrans.app (cokernel.\u03c0 \u03b1) X)\n[PROOFSTEP]\nerw [kernelComparison_comp_\u03b9 _ ((evaluation C D).obj X), \u03c0_comp_cokernelComparison_assoc _ ((evaluation C D).obj X)]\n[GOAL]\ncase h.h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app \u03b1 X =\n    ((evaluation C D).obj X).map (cokernel.\u03c0 (kernel.\u03b9 \u03b1)) \u226b\n      NatTrans.app (coimageImageComparison \u03b1) X \u226b ((evaluation C D).obj X).map (kernel.\u03b9 (cokernel.\u03c0 \u03b1))\n[PROOFSTEP]\nconv_lhs => rw [\u2190 coimage_image_factorisation \u03b1]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n| NatTrans.app \u03b1 X\n[PROOFSTEP]\nrw [\u2190 coimage_image_factorisation \u03b1]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n| NatTrans.app \u03b1 X\n[PROOFSTEP]\nrw [\u2190 coimage_image_factorisation \u03b1]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n| NatTrans.app \u03b1 X\n[PROOFSTEP]\nrw [\u2190 coimage_image_factorisation \u03b1]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 NatTrans.app (coimageImageComparison \u03b1) X =\n    (coimageObjIso \u03b1 X).hom \u226b coimageImageComparison (NatTrans.app \u03b1 X) \u226b (imageObjIso \u03b1 X).inv\n[PROOFSTEP]\nsimp only [coimageImageComparison_app, Iso.hom_inv_id_assoc, Iso.hom_inv_id, Category.assoc, Category.comp_id]\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 IsIso (coimageImageComparison \u03b1)\n[PROOFSTEP]\nhave : \u2200 X : C, IsIso ((Abelian.coimageImageComparison \u03b1).app X) :=\n  by\n  intros\n  rw [coimageImageComparison_app']\n  infer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\n\u22a2 \u2200 (X : C), IsIso (NatTrans.app (coimageImageComparison \u03b1) X)\n[PROOFSTEP]\nintros\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX X\u271d : C\n\u22a2 IsIso (NatTrans.app (coimageImageComparison \u03b1) X\u271d)\n[PROOFSTEP]\nrw [coimageImageComparison_app']\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX X\u271d : C\n\u22a2 IsIso ((coimageObjIso \u03b1 X\u271d).hom \u226b coimageImageComparison (NatTrans.app \u03b1 X\u271d) \u226b (imageObjIso \u03b1 X\u271d).inv)\n[PROOFSTEP]\ninfer_instance\n[GOAL]\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nD : Type w\ninst\u271d\u00b9 : Category.{z, w} D\ninst\u271d : Abelian D\nF G : C \u2964 D\n\u03b1 : F \u27f6 G\nX : C\nthis : \u2200 (X : C), IsIso (NatTrans.app (coimageImageComparison \u03b1) X)\n\u22a2 IsIso (coimageImageComparison \u03b1)\n[PROOFSTEP]\napply NatIso.isIso_of_isIso_app\n", "meta": {"mathlib_filename": "Mathlib.CategoryTheory.Abelian.FunctorCategory", "llama_tokens": 4193, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506635289835, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.5008257137797205}}
{"text": "[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 IsPeriodicPt ((fun x x_1 => x * x_1) x) n 1 \u2194 x ^ n = 1\n[PROOFSTEP]\nrw [IsPeriodicPt, IsFixedPt, mul_left_iterate]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 (fun x_1 => x ^ n * x_1) 1 = 1 \u2194 x ^ n = 1\n[PROOFSTEP]\ndsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 x ^ n * 1 = 1 \u2194 x ^ n = 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 IsOfFinOrder x \u2194 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nsimp [IsOfFinOrder, mem_periodicPts, isPeriodicPt_mul_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\nh : Injective fun n => x ^ n\n\u22a2 \u00acIsOfFinOrder x\n[PROOFSTEP]\nsimp_rw [isOfFinOrder_iff_pow_eq_one, not_exists, not_and]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\nh : Injective fun n => x ^ n\n\u22a2 \u2200 (x_1 : \u2115), 0 < x_1 \u2192 \u00acx ^ x_1 = 1\n[PROOFSTEP]\nintro n hn_pos hnx\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\nx : G\nh : Injective fun n => x ^ n\nn : \u2115\nhn_pos : 0 < n\nhnx : x ^ n = 1\n\u22a2 False\n[PROOFSTEP]\nrw [\u2190 pow_zero x] at hnx \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\nx : G\nh : Injective fun n => x ^ n\nn : \u2115\nhn_pos : 0 < n\nhnx : x ^ n = x ^ 0\n\u22a2 False\n[PROOFSTEP]\nrw [h hnx] at hn_pos \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\nx : G\nh : Injective fun n => x ^ n\nn : \u2115\nhn_pos : 0 < 0\nhnx : x ^ n = x ^ 0\n\u22a2 False\n[PROOFSTEP]\nexact irrefl 0 hn_pos\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Submonoid G\nx : { x // x \u2208 H }\n\u22a2 IsOfFinOrder x \u2194 IsOfFinOrder \u2191x\n[PROOFSTEP]\nrw [isOfFinOrder_iff_pow_eq_one, isOfFinOrder_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Submonoid G\nx : { x // x \u2208 H }\n\u22a2 (\u2203 n, 0 < n \u2227 x ^ n = 1) \u2194 \u2203 n, 0 < n \u2227 \u2191x ^ n = 1\n[PROOFSTEP]\nnorm_cast\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\ninst\u271d : Monoid H\nf : G \u2192* H\nx : G\nh : IsOfFinOrder x\n\u22a2 \u2203 n, 0 < n \u2227 \u2191f x ^ n = 1\n[PROOFSTEP]\nrcases(isOfFinOrder_iff_pow_eq_one _).mp h with \u27e8n, npos, hn\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\ninst\u271d : Monoid H\nf : G \u2192* H\nx : G\nh : IsOfFinOrder x\nn : \u2115\nnpos : 0 < n\nhn : x ^ n = 1\n\u22a2 \u2203 n, 0 < n \u2227 \u2191f x ^ n = 1\n[PROOFSTEP]\nexact \u27e8n, npos, by rw [\u2190 f.map_pow, hn, f.map_one]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\ninst\u271d : Monoid H\nf : G \u2192* H\nx : G\nh : IsOfFinOrder x\nn : \u2115\nnpos : 0 < n\nhn : x ^ n = 1\n\u22a2 \u2191f x ^ n = 1\n[PROOFSTEP]\nrw [\u2190 f.map_pow, hn, f.map_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\n\u03b7 : Type u_6\nGs : \u03b7 \u2192 Type u_7\ninst\u271d : (i : \u03b7) \u2192 Monoid (Gs i)\nx : (i : \u03b7) \u2192 Gs i\nh : IsOfFinOrder x\n\u22a2 \u2200 (i : \u03b7), IsOfFinOrder (x i)\n[PROOFSTEP]\nrcases(isOfFinOrder_iff_pow_eq_one _).mp h with \u27e8n, npos, hn\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn\u271d m : \u2115\n\u03b7 : Type u_6\nGs : \u03b7 \u2192 Type u_7\ninst\u271d : (i : \u03b7) \u2192 Monoid (Gs i)\nx : (i : \u03b7) \u2192 Gs i\nh : IsOfFinOrder x\nn : \u2115\nnpos : 0 < n\nhn : x ^ n = 1\n\u22a2 \u2200 (i : \u03b7), IsOfFinOrder (x i)\n[PROOFSTEP]\nexact fun _ => (isOfFinOrder_iff_pow_eq_one _).mpr \u27e8n, npos, (congr_fun hn.symm _).symm\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 x ^ orderOf x = 1\n[PROOFSTEP]\nrefine\n  Eq.trans ?_\n    (isPeriodicPt_minimalPeriod (x * \u00b7) 1)\n      -- porting note: we need a `dsimp` in the middle of the rewrite to do beta reduction\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 x ^ orderOf x = (fun x_1 => x * x_1)^[minimalPeriod (fun x_1 => x * x_1) 1] 1\n[PROOFSTEP]\nrw [orderOf, mul_left_iterate]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 x ^ minimalPeriod (fun x_1 => x * x_1) 1 = (fun x_1 => x ^ minimalPeriod (fun x_2 => x * x_2) 1 * x_1) 1\n[PROOFSTEP]\ndsimp\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\n\u22a2 x ^ minimalPeriod (fun x_1 => x * x_1) 1 = x ^ minimalPeriod (fun x_1 => x * x_1) 1 * 1\n[PROOFSTEP]\nrw [mul_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : \u00acIsOfFinOrder x\n\u22a2 orderOf x = 0\n[PROOFSTEP]\nrwa [orderOf, minimalPeriod, dif_neg]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\n\u22a2 orderOf x = 0 \u2194 \u2200 (n : \u2115), 0 < n \u2192 x ^ n \u2260 1\n[PROOFSTEP]\nsimp_rw [orderOf_eq_zero_iff, isOfFinOrder_iff_pow_eq_one, not_exists, not_and]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\n\u22a2 orderOf x = n \u2194 x ^ n = 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 x ^ m \u2260 1\n[PROOFSTEP]\nsimp_rw [Ne, \u2190 isPeriodicPt_mul_iff_pow_eq_one, orderOf, minimalPeriod]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\n\u22a2 (if h : 1 \u2208 periodicPts fun x_1 => x * x_1 then Nat.find h else 0) = n \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nsplit_ifs with h1\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : 1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 Nat.find h1 = n \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nclassical\nrw [find_eq_iff]\nsimp only [h, true_and]\npush_neg\nrfl\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : 1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 Nat.find h1 = n \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nrw [find_eq_iff]\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : 1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 ((n > 0 \u2227 IsPeriodicPt (fun x_1 => x * x_1) n 1) \u2227\n      \u2200 (n_1 : \u2115), n_1 < n \u2192 \u00ac(n_1 > 0 \u2227 IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nsimp only [h, true_and]\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : 1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 (IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227\n      \u2200 (n_1 : \u2115), n_1 < n \u2192 \u00ac(n_1 > 0 \u2227 IsPeriodicPt (fun x_1 => x * x_1) n_1 1)) \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\npush_neg\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : 1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 (IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (n_1 : \u2115), n_1 < n \u2192 n_1 > 0 \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) n_1 1) \u2194\n    IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nrfl\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : \u00ac1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 0 = n \u2194 IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1\n[PROOFSTEP]\nrw [iff_false_left h.ne]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : \u00ac1 \u2208 periodicPts fun x_1 => x * x_1\n\u22a2 \u00ac(IsPeriodicPt (fun x_1 => x * x_1) n 1 \u2227 \u2200 (m : \u2115), m < n \u2192 0 < m \u2192 \u00acIsPeriodicPt (fun x_1 => x * x_1) m 1)\n[PROOFSTEP]\nrintro \u27e8h', -\u27e9\n[GOAL]\ncase neg.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : 0 < n\nh1 : \u00ac1 \u2208 periodicPts fun x_1 => x * x_1\nh' : IsPeriodicPt (fun x_1 => x * x_1) n 1\n\u22a2 False\n[PROOFSTEP]\nexact h1 \u27e8n, h, h'\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\n\u22a2 0 < orderOf x \u2194 IsOfFinOrder x\n[PROOFSTEP]\nrw [iff_not_comm.mp orderOf_eq_zero_iff, pos_iff_ne_zero]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx y\u271d : G\na b : A\nn m : \u2115\ninst\u271d : Monoid \u03b2\ny : \u03b2\nhx : IsOfFinOrder x\nh : orderOf y \u2223 orderOf x\n\u22a2 IsOfFinOrder y\n[PROOFSTEP]\nrw [\u2190 orderOf_pos_iff] at hx \u22a2\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx y\u271d : G\na b : A\nn m : \u2115\ninst\u271d : Monoid \u03b2\ny : \u03b2\nhx : 0 < orderOf x\nh : orderOf y \u2223 orderOf x\n\u22a2 0 < orderOf y\n[PROOFSTEP]\nexact Nat.pos_of_dvd_of_pos h hx\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nhn : 0 < n\nh : x ^ n = 1\n\u22a2 IsPeriodicPt (fun x_1 => x * x_1) n 1\n[PROOFSTEP]\nrwa [isPeriodicPt_mul_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\n\u22a2 orderOf 1 = 1\n[PROOFSTEP]\nrw [orderOf, \u2190 minimalPeriod_id (x := (1 : G)), \u2190 one_mul_eq_id]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\n\u22a2 orderOf x = 1 \u2194 x = 1\n[PROOFSTEP]\nrw [orderOf, minimalPeriod_eq_one_iff_isFixedPt, IsFixedPt, mul_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\n\u22a2 x ^ n = x ^ (n % orderOf x + orderOf x * (n / orderOf x))\n[PROOFSTEP]\nrw [Nat.mod_add_div]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\n\u22a2 x ^ (n % orderOf x + orderOf x * (n / orderOf x)) = x ^ (n % orderOf x)\n[PROOFSTEP]\nsimp [pow_add, pow_mul, pow_orderOf_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\nh : orderOf x \u2223 n\n\u22a2 x ^ n = 1\n[PROOFSTEP]\nrw [pow_eq_mod_orderOf, Nat.mod_eq_zero_of_dvd h, _root_.pow_zero]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn\u271d m n : \u2115\n\u22a2 orderOf (x ^ n) \u2223 orderOf x\n[PROOFSTEP]\nrw [orderOf_dvd_iff_pow_eq_one, pow_right_comm, pow_orderOf_eq_one, one_pow]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nx : G\nhn : n < orderOf x\nhm : m < orderOf x\neq : x ^ n = x ^ m\n\u22a2 (fun x_1 => x * x_1)^[n] 1 = (fun x_1 => x * x_1)^[m] 1\n[PROOFSTEP]\nsimpa only [mul_left_iterate, mul_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\n\u22a2 x ^ n = 1 \u2194 n \u2261 0 [MOD orderOf x]\n[PROOFSTEP]\nrw [modEq_zero_iff_dvd, orderOf_dvd_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Type u_6\ninst\u271d : Monoid H\n\u03c8 : G \u2192* H\nx : G\n\u22a2 orderOf (\u2191\u03c8 x) \u2223 orderOf x\n[PROOFSTEP]\napply orderOf_dvd_of_pow_eq_one\n[GOAL]\ncase h\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Type u_6\ninst\u271d : Monoid H\n\u03c8 : G \u2192* H\nx : G\n\u22a2 \u2191\u03c8 x ^ orderOf x = 1\n[PROOFSTEP]\nrw [\u2190 map_pow, pow_orderOf_eq_one]\n[GOAL]\ncase h\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Type u_6\ninst\u271d : Monoid H\n\u03c8 : G \u2192* H\nx : G\n\u22a2 \u2191\u03c8 1 = 1\n[PROOFSTEP]\napply map_one\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nby_cases h0 : orderOf x = 0\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\nh0 : orderOf x = 0\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nrw [h0, coprime_zero_right] at h \n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : n = 1\nh0 : orderOf x = 0\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nexact \u27e81, by rw [h, pow_one, pow_one]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : n = 1\nh0 : orderOf x = 0\n\u22a2 (x ^ n) ^ 1 = x\n[PROOFSTEP]\nrw [h, pow_one, pow_one]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nby_cases h1 : orderOf x = 1\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\nh1 : orderOf x = 1\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nexact \u27e80, by rw [orderOf_eq_one_iff.mp h1, one_pow, one_pow]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\nh1 : orderOf x = 1\n\u22a2 (x ^ n) ^ 0 = x\n[PROOFSTEP]\nrw [orderOf_eq_one_iff.mp h1, one_pow, one_pow]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\nh1 : \u00acorderOf x = 1\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nobtain \u27e8m, h\u27e9 := exists_mul_emod_eq_one_of_coprime h (one_lt_iff_ne_zero_and_ne_one.mpr \u27e8h0, h1\u27e9)\n[GOAL]\ncase neg.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m\u271d : \u2115\nh\u271d : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\nh1 : \u00acorderOf x = 1\nm : \u2115\nh : n * m % orderOf x = 1\n\u22a2 \u2203 m, (x ^ n) ^ m = x\n[PROOFSTEP]\nexact \u27e8m, by rw [\u2190 pow_mul, pow_eq_mod_orderOf, h, pow_one]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m\u271d : \u2115\nh\u271d : coprime n (orderOf x)\nh0 : \u00acorderOf x = 0\nh1 : \u00acorderOf x = 1\nm : \u2115\nh : n * m % orderOf x = 1\n\u22a2 (x ^ n) ^ m = x\n[PROOFSTEP]\nrw [\u2190 pow_mul, pow_eq_mod_orderOf, h, pow_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\n\u22a2 orderOf x = n\n[PROOFSTEP]\ncases' exists_eq_mul_right_of_dvd (orderOf_dvd_of_pow_eq_one hx) with a ha\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\n\u22a2 orderOf x = n\n[PROOFSTEP]\nsuffices a = 1 by\n  simp [this, ha]\n    -- Assume `a` is not one...\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nthis : a = 1\n\u22a2 orderOf x = n\n[PROOFSTEP]\nsimp [this, ha]\n  -- Assume `a` is not one...\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\n\u22a2 a = 1\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\n\u22a2 False\n[PROOFSTEP]\nhave a_min_fac_dvd_p_sub_one : a.minFac \u2223 n :=\n  by\n  obtain \u27e8b, hb\u27e9 : \u2203 b : \u2115, a = b * a.minFac := exists_eq_mul_left_of_dvd a.minFac_dvd\n  rw [hb, \u2190 mul_assoc] at ha \n  exact Dvd.intro_left (orderOf x * b) ha.symm\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\n\u22a2 minFac a \u2223 n\n[PROOFSTEP]\nobtain \u27e8b, hb\u27e9 : \u2203 b : \u2115, a = b * a.minFac := exists_eq_mul_left_of_dvd a.minFac_dvd\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b\u271d : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\nb : \u2115\nhb : a = b * minFac a\n\u22a2 minFac a \u2223 n\n[PROOFSTEP]\nrw [hb, \u2190 mul_assoc] at ha \n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b\u271d : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nh : \u00aca = 1\nb : \u2115\nha : n = orderOf x * b * minFac a\nhb : a = b * minFac a\n\u22a2 minFac a \u2223 n\n[PROOFSTEP]\nexact Dvd.intro_left (orderOf x * b) ha.symm\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\na_min_fac_dvd_p_sub_one : minFac a \u2223 n\n\u22a2 False\n[PROOFSTEP]\nrefine' hd a.minFac (Nat.minFac_prime h) a_min_fac_dvd_p_sub_one _\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\na_min_fac_dvd_p_sub_one : minFac a \u2223 n\n\u22a2 x ^ (n / minFac a) = 1\n[PROOFSTEP]\nrw [\u2190 orderOf_dvd_iff_pow_eq_one, Nat.dvd_div_iff a_min_fac_dvd_p_sub_one, ha, mul_comm,\n  Nat.mul_dvd_mul_iff_left (orderOf_pos' _)]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\na_min_fac_dvd_p_sub_one : minFac a \u2223 n\n\u22a2 minFac a \u2223 a\n[PROOFSTEP]\nexact Nat.minFac_dvd a\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\na_min_fac_dvd_p_sub_one : minFac a \u2223 n\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\nrw [isOfFinOrder_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na\u271d b : A\nn m : \u2115\nhn : 0 < n\nhx : x ^ n = 1\nhd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 n \u2192 x ^ (n / p) \u2260 1\na : \u2115\nha : n = orderOf x * a\nh : \u00aca = 1\na_min_fac_dvd_p_sub_one : minFac a \u2223 n\n\u22a2 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nexact Exists.intro n (id \u27e8hn, hx\u27e9)\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx y\u271d : G\na b : A\nn m : \u2115\nH : Type u_6\ninst\u271d : Monoid H\ny : H\n\u22a2 orderOf x = orderOf y \u2194 \u2200 (n : \u2115), x ^ n = 1 \u2194 y ^ n = 1\n[PROOFSTEP]\nsimp_rw [\u2190 isPeriodicPt_mul_iff_pow_eq_one, \u2190 minimalPeriod_eq_minimalPeriod_iff, orderOf]\n[GOAL]\nG : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\ninst\u271d\u00b9 : AddMonoid A\nx\u271d y : G\na b : A\nn m : \u2115\nH : Type u_6\ninst\u271d : Monoid H\nf : G \u2192* H\nhf : Injective \u2191f\nx : G\n\u22a2 orderOf (\u2191f x) = orderOf x\n[PROOFSTEP]\nsimp_rw [orderOf_eq_orderOf_iff, \u2190 f.map_pow, \u2190 f.map_one, hf.eq_iff, forall_const]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : n \u2260 0\n\u22a2 orderOf (x ^ n) = orderOf x / gcd (orderOf x) n\n[PROOFSTEP]\nunfold orderOf\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : n \u2260 0\n\u22a2 minimalPeriod (fun x_1 => x ^ n * x_1) 1 =\n    minimalPeriod (fun x_1 => x * x_1) 1 / gcd (minimalPeriod (fun x_1 => x * x_1) 1) n\n[PROOFSTEP]\nrw [\u2190 minimalPeriod_iterate_eq_div_gcd h, mul_left_iterate]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : IsOfFinOrder x\n\u22a2 orderOf (x ^ n) = orderOf x / gcd (orderOf x) n\n[PROOFSTEP]\nunfold orderOf\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : IsOfFinOrder x\n\u22a2 minimalPeriod (fun x_1 => x ^ n * x_1) 1 =\n    minimalPeriod (fun x_1 => x * x_1) 1 / gcd (minimalPeriod (fun x_1 => x * x_1) 1) n\n[PROOFSTEP]\nrw [\u2190 minimalPeriod_iterate_eq_div_gcd' h, mul_left_iterate]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime (orderOf y) m\n\u22a2 orderOf (y ^ m) = orderOf y\n[PROOFSTEP]\nby_cases hg : orderOf y = 0\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime (orderOf y) m\nhg : orderOf y = 0\n\u22a2 orderOf (y ^ m) = orderOf y\n[PROOFSTEP]\nrw [m.coprime_zero_left.mp (hg \u25b8 h), pow_one]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : coprime (orderOf y) m\nhg : \u00acorderOf y = 0\n\u22a2 orderOf (y ^ m) = orderOf y\n[PROOFSTEP]\nrw [orderOf_pow'' y m (hg.imp_symm orderOf_eq_zero), h.gcd_eq_one, Nat.div_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 orderOf (x * y) \u2223 lcm (orderOf x) (orderOf y)\n[PROOFSTEP]\nrw [orderOf, \u2190 comp_mul_left]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 minimalPeriod ((fun x_1 => x * x_1) \u2218 fun x => y * x) 1 \u2223 lcm (orderOf x) (orderOf y)\n[PROOFSTEP]\nexact Function.Commute.minimalPeriod_of_comp_dvd_lcm h.function_commute_mul_left\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\n\u22a2 orderOf y \u2223 lcm (orderOf x) (orderOf (x * y))\n[PROOFSTEP]\nby_cases h0 : orderOf x = 0\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : orderOf x = 0\n\u22a2 orderOf y \u2223 lcm (orderOf x) (orderOf (x * y))\n[PROOFSTEP]\nrw [h0, lcm_zero_left]\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : orderOf x = 0\n\u22a2 orderOf y \u2223 0\n[PROOFSTEP]\napply dvd_zero\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : \u00acorderOf x = 0\n\u22a2 orderOf y \u2223 lcm (orderOf x) (orderOf (x * y))\n[PROOFSTEP]\nconv_lhs =>\n  rw [\u2190 one_mul y, \u2190 pow_orderOf_eq_one x, \u2190 succ_pred_eq_of_pos (Nat.pos_of_ne_zero h0), _root_.pow_succ', mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : \u00acorderOf x = 0\n| orderOf y\n[PROOFSTEP]\nrw [\u2190 one_mul y, \u2190 pow_orderOf_eq_one x, \u2190 succ_pred_eq_of_pos (Nat.pos_of_ne_zero h0), _root_.pow_succ', mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : \u00acorderOf x = 0\n| orderOf y\n[PROOFSTEP]\nrw [\u2190 one_mul y, \u2190 pow_orderOf_eq_one x, \u2190 succ_pred_eq_of_pos (Nat.pos_of_ne_zero h0), _root_.pow_succ', mul_assoc]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : \u00acorderOf x = 0\n| orderOf y\n[PROOFSTEP]\nrw [\u2190 one_mul y, \u2190 pow_orderOf_eq_one x, \u2190 succ_pred_eq_of_pos (Nat.pos_of_ne_zero h0), _root_.pow_succ', mul_assoc]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nh0 : \u00acorderOf x = 0\n\u22a2 orderOf (x ^ pred (orderOf x) * (x * y)) \u2223 lcm (orderOf x) (orderOf (x * y))\n[PROOFSTEP]\nexact\n  (((Commute.refl x).mul_right h).pow_left _).orderOf_mul_dvd_lcm.trans\n    (lcm_dvd_iff.2 \u27e8(orderOf_pow_dvd _).trans (dvd_lcm_left _ _), dvd_lcm_right _ _\u27e9)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhco : coprime (orderOf x) (orderOf y)\n\u22a2 orderOf (x * y) = orderOf x * orderOf y\n[PROOFSTEP]\nrw [orderOf, \u2190 comp_mul_left]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhco : coprime (orderOf x) (orderOf y)\n\u22a2 minimalPeriod ((fun x_1 => x * x_1) \u2218 fun x => y * x) 1 = orderOf x * orderOf y\n[PROOFSTEP]\nexact h.function_commute_mul_left.minimalPeriod_of_comp_eq_mul_of_coprime hco\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) = orderOf y\n[PROOFSTEP]\nhave hoy := orderOf_pos' hy\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\n\u22a2 orderOf (x * y) = orderOf y\n[PROOFSTEP]\nhave hxy := dvd_of_forall_prime_mul_dvd hdvd\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) = orderOf y\n[PROOFSTEP]\napply orderOf_eq_of_pow_and_pow_div_prime hoy\n[GOAL]\ncase hx\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 (x * y) ^ orderOf y = 1\n[PROOFSTEP]\nsimp only [Ne, \u2190 orderOf_dvd_iff_pow_eq_one]\n[GOAL]\ncase hd\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf y \u2192 (x * y) ^ (orderOf y / p) \u2260 1\n[PROOFSTEP]\nsimp only [Ne, \u2190 orderOf_dvd_iff_pow_eq_one]\n[GOAL]\ncase hx\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 orderOf (x * y) \u2223 orderOf y\n[PROOFSTEP]\nexact h.orderOf_mul_dvd_lcm.trans (lcm_dvd hxy dvd_rfl)\n[GOAL]\ncase hd\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\n\u22a2 \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf y \u2192 \u00acorderOf (x * y) \u2223 orderOf y / p\n[PROOFSTEP]\nrefine' fun p hp hpy hd => hp.ne_one _\n[GOAL]\ncase hd\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p = 1\n[PROOFSTEP]\nrw [\u2190 Nat.dvd_one, \u2190 mul_dvd_mul_iff_right hoy.ne', one_mul, \u2190 dvd_div_iff hpy]\n[GOAL]\ncase hd\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 orderOf y \u2223 orderOf y / p\n[PROOFSTEP]\nrefine' (orderOf_dvd_lcm_mul h).trans (lcm_dvd ((dvd_div_iff hpy).2 _) hd)\n[GOAL]\ncase hd\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\n\u22a2 p * orderOf x \u2223 orderOf y\n[PROOFSTEP]\nby_cases h : p \u2223 orderOf x\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh\u271d : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : p \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m : \u2115\nh\u271d : Commute x y\nhy : IsOfFinOrder y\nhdvd : \u2200 (p : \u2115), Nat.Prime p \u2192 p \u2223 orderOf x \u2192 p * orderOf x \u2223 orderOf y\nhoy : 0 < orderOf y\nhxy : orderOf x \u2223 orderOf y\np : \u2115\nhp : Nat.Prime p\nhpy : p \u2223 orderOf y\nhd : orderOf (x * y) \u2223 orderOf y / p\nh : \u00acp \u2223 orderOf x\n\u22a2 p * orderOf x \u2223 orderOf y\n[PROOFSTEP]\nexacts [hdvd p hp h, (hp.coprime_iff_not_dvd.2 h).mul_dvd_of_dvd_of_dvd hpy hxy]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nhg : x ^ p = 1\nhg1 : x \u2260 1\n\u22a2 \u00acIsFixedPt (fun x_1 => x * x_1) 1\n[PROOFSTEP]\nrwa [IsFixedPt, mul_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nhnot : \u00acx ^ p ^ n = 1\nhfin : x ^ p ^ (n + 1) = 1\n\u22a2 orderOf x = p ^ (n + 1)\n[PROOFSTEP]\napply minimalPeriod_eq_prime_pow\n[GOAL]\ncase hk\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nhnot : \u00acx ^ p ^ n = 1\nhfin : x ^ p ^ (n + 1) = 1\n\u22a2 \u00acIsPeriodicPt (fun x_1 => x * x_1) (p ^ n) 1\n[PROOFSTEP]\nrwa [isPeriodicPt_mul_iff_pow_eq_one]\n[GOAL]\ncase hk1\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nhnot : \u00acx ^ p ^ n = 1\nhfin : x ^ p ^ (n + 1) = 1\n\u22a2 IsPeriodicPt (fun x_1 => x * x_1) (p ^ (n + 1)) 1\n[PROOFSTEP]\nrwa [isPeriodicPt_mul_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d : \u2203 k, orderOf x = p ^ k\nk : \u2115\nhk : orderOf x = p ^ k\n\u22a2 x ^ p ^ k = 1\n[PROOFSTEP]\nrw [\u2190 hk, pow_orderOf_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d : \u2203 m, x ^ p ^ m = 1\nw\u271d : \u2115\nhm : x ^ p ^ w\u271d = 1\n\u22a2 \u2203 k, orderOf x = p ^ k\n[PROOFSTEP]\nobtain \u27e8k, _, hk\u27e9 := (Nat.dvd_prime_pow hp.elim).mp (orderOf_dvd_of_pow_eq_one hm)\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid G\ninst\u271d : AddMonoid A\nx y : G\na b : A\nn m p : \u2115\nhp : Fact (Nat.Prime p)\nx\u271d : \u2203 m, x ^ p ^ m = 1\nw\u271d : \u2115\nhm : x ^ p ^ w\u271d = 1\nk : \u2115\nleft\u271d : k \u2264 w\u271d\nhk : orderOf x = p ^ k\n\u22a2 \u2203 k, orderOf x = p ^ k\n[PROOFSTEP]\nexact \u27e8k, hk\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm n : \u2115\n\u22a2 x ^ n = x ^ m \u2194 n \u2261 m [MOD orderOf x]\n[PROOFSTEP]\nwlog hmn : m \u2264 n generalizing m n\n[GOAL]\ncase inr\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm n : \u2115\nthis : \u2200 {m n : \u2115}, m \u2264 n \u2192 (x ^ n = x ^ m \u2194 n \u2261 m [MOD orderOf x])\nhmn : \u00acm \u2264 n\n\u22a2 x ^ n = x ^ m \u2194 n \u2261 m [MOD orderOf x]\n[PROOFSTEP]\nrw [eq_comm, ModEq.comm, this (le_of_not_le hmn)]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm\u271d n\u271d m n : \u2115\nhmn : m \u2264 n\n\u22a2 x ^ n = x ^ m \u2194 n \u2261 m [MOD orderOf x]\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Nat.exists_eq_add_of_le hmn\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm\u271d n m k : \u2115\nhmn : m \u2264 m + k\n\u22a2 x ^ (m + k) = x ^ m \u2194 m + k \u2261 m [MOD orderOf x]\n[PROOFSTEP]\nrw [\u2190 mul_one (x ^ m), pow_add, mul_left_cancel_iff, pow_eq_one_iff_modEq]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm\u271d n m k : \u2115\nhmn : m \u2264 m + k\n\u22a2 k \u2261 0 [MOD orderOf x] \u2194 m + k \u2261 m [MOD orderOf x]\n[PROOFSTEP]\nexact \u27e8fun h => Nat.ModEq.add_left _ h, fun h => Nat.ModEq.add_left_cancel' _ h\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\n\u22a2 (Injective fun n => x ^ n) \u2194 \u00acIsOfFinOrder x\n[PROOFSTEP]\nrefine' \u27e8fun h => not_isOfFinOrder_of_injective_pow h, fun h n m hnm => _\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm\u271d n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\nn m : \u2115\nhnm : (fun n => x ^ n) n = (fun n => x ^ n) m\n\u22a2 n = m\n[PROOFSTEP]\nrwa [pow_eq_pow_iff_modEq, orderOf_eq_zero_iff.mpr h, modEq_zero_iff] at hnm \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx y : G\nm\u271d n\u271d : \u2115\nh : orderOf x = 0\nn m : \u2115\n\u22a2 x ^ n = x ^ m \u2194 n = m\n[PROOFSTEP]\nrw [pow_eq_pow_iff_modEq, h, modEq_zero_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\n\u22a2 Set.Infinite {y | \u00acIsOfFinOrder y}\n[PROOFSTEP]\nlet s := {n | 0 < n}.image fun n : \u2115 => x ^ n\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n\u22a2 Set.Infinite {y | \u00acIsOfFinOrder y}\n[PROOFSTEP]\nhave hs : s \u2286 {y : G | \u00acIsOfFinOrder y} :=\n  by\n  rintro - \u27e8n, hn : 0 < n, rfl\u27e9 (contra : IsOfFinOrder (x ^ n))\n  apply h\n  rw [isOfFinOrder_iff_pow_eq_one] at contra \u22a2\n  obtain \u27e8m, hm, hm'\u27e9 := contra\n  exact \u27e8n * m, mul_pos hn hm, by rwa [pow_mul]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\n\u22a2 s \u2286 {y | \u00acIsOfFinOrder y}\n[PROOFSTEP]\nrintro - \u27e8n, hn : 0 < n, rfl\u27e9 (contra : IsOfFinOrder (x ^ n))\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : \u2115\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n\u22a2 False\n[PROOFSTEP]\napply h\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : \u2115\nhn : 0 < n\ncontra : IsOfFinOrder (x ^ n)\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\nrw [isOfFinOrder_iff_pow_eq_one] at contra \u22a2\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : \u2115\nhn : 0 < n\ncontra : \u2203 n_1, 0 < n_1 \u2227 (x ^ n) ^ n_1 = 1\n\u22a2 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nobtain \u27e8m, hm, hm'\u27e9 := contra\n[GOAL]\ncase intro.intro.intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm\u271d n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : \u2115\nhn : 0 < n\nm : \u2115\nhm : 0 < m\nhm' : (x ^ n) ^ m = 1\n\u22a2 \u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nexact \u27e8n * m, mul_pos hn hm, by rwa [pow_mul]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm\u271d n\u271d : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nn : \u2115\nhn : 0 < n\nm : \u2115\nhm : 0 < m\nhm' : (x ^ n) ^ m = 1\n\u22a2 x ^ (n * m) = 1\n[PROOFSTEP]\nrwa [pow_mul]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\n\u22a2 Set.Infinite {y | \u00acIsOfFinOrder y}\n[PROOFSTEP]\nsuffices s.Infinite by exact this.mono hs\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nthis : Set.Infinite s\n\u22a2 Set.Infinite {y | \u00acIsOfFinOrder y}\n[PROOFSTEP]\nexact this.mono hs\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\nh : \u00acIsOfFinOrder x\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\n\u22a2 Set.Infinite s\n[PROOFSTEP]\ncontrapose! h\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nh : \u00acSet.Infinite ((fun n => x ^ n) '' {n | 0 < n})\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\nhave : \u00acInjective fun n : \u2115 => x ^ n :=\n  by\n  have := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)\n  contrapose! this\n  exact Set.injOn_of_injective this _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nh : \u00acSet.Infinite ((fun n => x ^ n) '' {n | 0 < n})\n\u22a2 \u00acInjective fun n => x ^ n\n[PROOFSTEP]\nhave := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nh : \u00acSet.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nthis : \u00acSet.InjOn (fun n => x ^ n) (Set.Ioi 0)\n\u22a2 \u00acInjective fun n => x ^ n\n[PROOFSTEP]\ncontrapose! this\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nh : \u00acSet.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nthis : Injective fun n => x ^ n\n\u22a2 Set.InjOn (fun n => x ^ n) (Set.Ioi 0)\n[PROOFSTEP]\nexact Set.injOn_of_injective this _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LeftCancelMonoid G\nx\u271d y : G\nm n : \u2115\nx : G\ns : Set G := (fun n => x ^ n) '' {n | 0 < n}\nhs : s \u2286 {y | \u00acIsOfFinOrder y}\nh : \u00acSet.Infinite ((fun n => x ^ n) '' {n | 0 < n})\nthis : \u00acInjective fun n => x ^ n\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\nrwa [injective_pow_iff_not_isOfFinOrder, Classical.not_not] at this \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx\u271d y : G\ni : \u2124\nx : G\nhx : IsOfFinOrder x\n\u22a2 \u2203 n, 0 < n \u2227 x\u207b\u00b9 ^ n = 1\n[PROOFSTEP]\nrcases(isOfFinOrder_iff_pow_eq_one x).mp hx with \u27e8n, npos, hn\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx\u271d y : G\ni : \u2124\nx : G\nhx : IsOfFinOrder x\nn : \u2115\nnpos : 0 < n\nhn : x ^ n = 1\n\u22a2 \u2203 n, 0 < n \u2227 x\u207b\u00b9 ^ n = 1\n[PROOFSTEP]\nrefine' \u27e8n, npos, by simp_rw [inv_pow, hn, inv_one]\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx\u271d y : G\ni : \u2124\nx : G\nhx : IsOfFinOrder x\nn : \u2115\nnpos : 0 < n\nhn : x ^ n = 1\n\u22a2 x\u207b\u00b9 ^ n = 1\n[PROOFSTEP]\nsimp_rw [inv_pow, hn, inv_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\n\u22a2 \u2191(orderOf x) \u2223 i \u2194 x ^ i = 1\n[PROOFSTEP]\nrcases Int.eq_nat_or_neg i with \u27e8i, rfl | rfl\u27e9\n[GOAL]\ncase intro.inl\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2115\n\u22a2 \u2191(orderOf x) \u2223 \u2191i \u2194 x ^ \u2191i = 1\n[PROOFSTEP]\nrw [Int.coe_nat_dvd, orderOf_dvd_iff_pow_eq_one, zpow_ofNat]\n[GOAL]\ncase intro.inr\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2115\n\u22a2 \u2191(orderOf x) \u2223 -\u2191i \u2194 x ^ (-\u2191i) = 1\n[PROOFSTEP]\nrw [dvd_neg, Int.coe_nat_dvd, zpow_neg, inv_eq_one, zpow_ofNat, orderOf_dvd_iff_pow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx\u271d y : G\ni : \u2124\nx : G\n\u22a2 orderOf x\u207b\u00b9 = orderOf x\n[PROOFSTEP]\nsimp [orderOf_eq_orderOf_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\n\u22a2 x ^ i = ?m.965169\n[PROOFSTEP]\nrw [\u2190 Int.emod_add_ediv i (orderOf x)]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\n\u22a2 x ^ (i % \u2191(orderOf x) + \u2191(orderOf x) * (i / \u2191(orderOf x))) = x ^ (i % \u2191(orderOf x))\n[PROOFSTEP]\nsimp [zpow_add, zpow_mul, pow_orderOf_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\n\u22a2 (x ^ i) ^ orderOf x = 1\n[PROOFSTEP]\nby_cases h : IsOfFinOrder x\n[GOAL]\ncase pos\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\nh : IsOfFinOrder x\n\u22a2 (x ^ i) ^ orderOf x = 1\n[PROOFSTEP]\nrw [\u2190 zpow_ofNat, \u2190 zpow_mul, mul_comm, zpow_mul, zpow_ofNat, pow_orderOf_eq_one, one_zpow]\n[GOAL]\ncase neg\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\nh : \u00acIsOfFinOrder x\n\u22a2 (x ^ i) ^ orderOf x = 1\n[PROOFSTEP]\nrw [orderOf_eq_zero h, _root_.pow_zero]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\nh : IsOfFinOrder x\nh' : y \u2208 Subgroup.zpowers x\n\u22a2 IsOfFinOrder y\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Subgroup.mem_zpowers_iff.mp h'\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx : G\ni : \u2124\nh : IsOfFinOrder x\nk : \u2124\nh' : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 IsOfFinOrder (x ^ k)\n[PROOFSTEP]\nexact h.zpow\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\ni : \u2124\nh : y \u2208 Subgroup.zpowers x\n\u22a2 orderOf y \u2223 orderOf x\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Subgroup.mem_zpowers_iff.mp h\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx : G\ni k : \u2124\nh : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 orderOf (x ^ k) \u2223 orderOf x\n[PROOFSTEP]\nrw [orderOf_dvd_iff_pow_eq_one]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx : G\ni k : \u2124\nh : x ^ k \u2208 Subgroup.zpowers x\n\u22a2 (x ^ k) ^ orderOf x = 1\n[PROOFSTEP]\nexact zpow_pow_orderOf\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1\u271d : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\ni : \u2124\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\nhx : x \u2208 Subgroup.zpowers y\na : \u03b1\nhs : y \u2022 a = a\n\u22a2 x \u2022 a = a\n[PROOFSTEP]\nobtain \u27e8k, rfl\u27e9 := Subgroup.mem_zpowers_iff.mp hx\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1\u271d : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\ny : G\ni : \u2124\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\na : \u03b1\nhs : y \u2022 a = a\nk : \u2124\nhx : y ^ k \u2208 Subgroup.zpowers y\n\u22a2 y ^ k \u2022 a = a\n[PROOFSTEP]\nrw [\u2190 MulAction.toPerm_apply, \u2190 MulAction.toPermHom_apply, MonoidHom.map_zpow _ y k, MulAction.toPermHom_apply]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1\u271d : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\ny : G\ni : \u2124\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\na : \u03b1\nhs : y \u2022 a = a\nk : \u2124\nhx : y ^ k \u2208 Subgroup.zpowers y\n\u22a2 \u2191(MulAction.toPerm y ^ k) a = a\n[PROOFSTEP]\nexact Function.IsFixedPt.perm_zpow (by exact hs) k\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1\u271d : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\ny : G\ni : \u2124\n\u03b1 : Type u_6\ninst\u271d : MulAction G \u03b1\na : \u03b1\nhs : y \u2022 a = a\nk : \u2124\nhx : y ^ k \u2208 Subgroup.zpowers y\n\u22a2 IsFixedPt (\u2191(MulAction.toPerm y)) a\n[PROOFSTEP]\nexact hs\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\n\u22a2 \u2200 (x : \u2115),\n    x \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n)) \u2192\n      \u2200 (y : \u2115),\n        y \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n)) \u2192\n          x \u2260 y \u2192\n            Disjoint (Finset.filter (fun x_1 => orderOf x_1 = x) Finset.univ)\n              (Finset.filter (fun x => orderOf x = y) Finset.univ)\n[PROOFSTEP]\nintros\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx\u271d : \u2115\na\u271d\u00b2 : x\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\ny\u271d : \u2115\na\u271d\u00b9 : y\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\na\u271d : x\u271d \u2260 y\u271d\n\u22a2 Disjoint (Finset.filter (fun x => orderOf x = x\u271d) Finset.univ) (Finset.filter (fun x => orderOf x = y\u271d) Finset.univ)\n[PROOFSTEP]\napply Finset.disjoint_filter.2\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx\u271d : \u2115\na\u271d\u00b2 : x\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\ny\u271d : \u2115\na\u271d\u00b9 : y\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\na\u271d : x\u271d \u2260 y\u271d\n\u22a2 \u2200 (x : G), x \u2208 Finset.univ \u2192 orderOf x = x\u271d \u2192 \u00acorderOf x = y\u271d\n[PROOFSTEP]\nrintro _ _ rfl\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\ny\u271d : \u2115\na\u271d\u00b3 : y\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\nx\u271d : G\na\u271d\u00b2 : x\u271d \u2208 Finset.univ\na\u271d\u00b9 : orderOf x\u271d \u2208 Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))\na\u271d : orderOf x\u271d \u2260 y\u271d\n\u22a2 \u00acorderOf x\u271d = y\u271d\n[PROOFSTEP]\nassumption\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\n\u22a2 \u2200 (a : G),\n    (a \u2208\n        Finset.biUnion (Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))) fun m =>\n          Finset.filter (fun x => orderOf x = m) Finset.univ) \u2194\n      a \u2208 Finset.filter (fun x => x ^ n = 1) Finset.univ\n[PROOFSTEP]\nintro x\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx\u271d : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx : G\n\u22a2 (x \u2208\n      Finset.biUnion (Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))) fun m =>\n        Finset.filter (fun x => orderOf x = m) Finset.univ) \u2194\n    x \u2208 Finset.filter (fun x => x ^ n = 1) Finset.univ\n[PROOFSTEP]\nsuffices orderOf x \u2264 n \u2227 orderOf x \u2223 n \u2194 x ^ n = 1 by simpa [Nat.lt_succ_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx\u271d : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx : G\nthis : orderOf x \u2264 n \u2227 orderOf x \u2223 n \u2194 x ^ n = 1\n\u22a2 (x \u2208\n      Finset.biUnion (Finset.filter (fun x => x \u2223 n) (Finset.range (succ n))) fun m =>\n        Finset.filter (fun x => orderOf x = m) Finset.univ) \u2194\n    x \u2208 Finset.filter (fun x => x ^ n = 1) Finset.univ\n[PROOFSTEP]\nsimpa [Nat.lt_succ_iff]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx\u271d : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx : G\n\u22a2 orderOf x \u2264 n \u2227 orderOf x \u2223 n \u2194 x ^ n = 1\n[PROOFSTEP]\nexact\n  \u27e8fun h => by\n    let \u27e8m, hm\u27e9 := h.2\n    rw [hm, pow_mul, pow_orderOf_eq_one, one_pow], fun h =>\n    \u27e8orderOf_le_of_pow_eq_one hn.bot_lt h, orderOf_dvd_of_pow_eq_one h\u27e9\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx\u271d : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx : G\nh : orderOf x \u2264 n \u2227 orderOf x \u2223 n\n\u22a2 x ^ n = 1\n[PROOFSTEP]\nlet \u27e8m, hm\u27e9 := h.2\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Monoid G\nx\u271d : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nhn : n \u2260 0\nx : G\nh : orderOf x \u2264 n \u2227 orderOf x \u2223 n\nm : \u2115\nhm : n = orderOf x * m\n\u22a2 x ^ n = 1\n[PROOFSTEP]\nrw [hm, pow_mul, pow_orderOf_eq_one, one_pow]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\nhave : (Set.univ : Set G).Finite := Set.univ.toFinite\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\nthis : Set.Finite Set.univ\n\u22a2 IsOfFinOrder x\n[PROOFSTEP]\ncontrapose! this\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\nthis : \u00acIsOfFinOrder x\n\u22a2 \u00acSet.Finite Set.univ\n[PROOFSTEP]\nexact Set.Infinite.mono (Set.subset_univ _) (infinite_not_isOfFinOrder this)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx\u271d y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nx : G\nn : \u2115\nhn : \u2203 m, x ^ m = x ^ n\n\u22a2 \u2191(finEquivPowers x).symm { val := x ^ n, property := hn } =\n    { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) }\n[PROOFSTEP]\nrw [Equiv.symm_apply_eq, finEquivPowers_apply, Subtype.mk_eq_mk, pow_eq_mod_orderOf, Fin.val_mk]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nh : orderOf x = orderOf y\nn : \u2115\n\u22a2 \u2191(powersEquivPowers h) { val := x ^ n, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) x y = x ^ n) } =\n    { val := y ^ n, property := (_ : \u2203 y_1, (fun x x_1 => x ^ x_1) y y_1 = y ^ n) }\n[PROOFSTEP]\nrw [powersEquivPowers, Equiv.trans_apply, Equiv.trans_apply, finEquivPowers_symm_apply, \u2190 Equiv.eq_symm_apply,\n  finEquivPowers_symm_apply]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : LeftCancelMonoid G\nx y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nh : orderOf x = orderOf y\nn : \u2115\n\u22a2 \u2191(Fin.castIso h).toEquiv { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } =\n    { val := n % orderOf y, isLt := (_ : n % orderOf y < orderOf y) }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\n\u22a2 \u2203 i x_1, x ^ i = 1\n[PROOFSTEP]\nrcases exists_pow_eq_one x with \u27e8w, hw1, hw2\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\nw : \u2115\nhw1 : w > 0\nhw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1\n\u22a2 \u2203 i x_1, x ^ i = 1\n[PROOFSTEP]\nrefine' \u27e8w, Int.coe_nat_ne_zero.mpr (_root_.ne_of_gt hw1), _\u27e9\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\nw : \u2115\nhw1 : w > 0\nhw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1\n\u22a2 x ^ \u2191w = 1\n[PROOFSTEP]\nrw [zpow_ofNat]\n[GOAL]\ncase intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn : \u2115\ninst\u271d : Finite G\nx : G\nw : \u2115\nhw1 : w > 0\nhw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1\n\u22a2 x ^ w = 1\n[PROOFSTEP]\nexact (isPeriodicPt_mul_iff_pow_eq_one _).mp hw2\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nx\u271d : y \u2208 Submonoid.powers x\nn : \u2115\nhn : (fun x x_1 => x ^ x_1) x n = y\n\u22a2 (fun x x_1 => x ^ x_1) x \u2191n = y\n[PROOFSTEP]\nsimp_all\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Finite G\nx\u271d : y \u2208 zpowers x\ni : \u2124\nhi : (fun x x_1 => x ^ x_1) x i = y\n\u22a2 (fun x x_1 => x ^ x_1) x (Int.natAbs (i % \u2191(orderOf x))) = y\n[PROOFSTEP]\ndsimp only\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Finite G\nx\u271d : y \u2208 zpowers x\ni : \u2124\nhi : (fun x x_1 => x ^ x_1) x i = y\n\u22a2 x ^ Int.natAbs (i % \u2191(orderOf x)) = y\n[PROOFSTEP]\nrwa [\u2190 zpow_ofNat, Int.natAbs_of_nonneg (Int.emod_nonneg _ (Int.coe_nat_ne_zero_iff_pos.2 (orderOf_pos x))), \u2190\n  zpow_eq_mod_orderOf]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\nx y : G\nn : \u2115\ninst\u271d\u00b9 : Finite G\ninst\u271d : DecidableEq G\n\u22a2 y \u2208 zpowers x \u2194 y \u2208 Finset.image ((fun x x_1 => x ^ x_1) x) (Finset.range (orderOf x))\n[PROOFSTEP]\nrw [\u2190 mem_powers_iff_mem_zpowers, mem_powers_iff_mem_range_orderOf]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn\u271d : \u2115\nn : \u2124\n\u22a2 x ^ n = 1 \u2194 n \u2261 0 [ZMOD \u2191(orderOf x)]\n[PROOFSTEP]\nrw [Int.modEq_zero_iff_dvd, orderOf_dvd_iff_zpow_eq_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn\u271d : \u2115\nm n : \u2124\n\u22a2 x ^ m = x ^ n \u2194 m \u2261 n [ZMOD \u2191(orderOf x)]\n[PROOFSTEP]\nrw [\u2190 mul_inv_eq_one, \u2190 zpow_sub, zpow_eq_one_iff_modEq, Int.modEq_iff_dvd, Int.modEq_iff_dvd, zero_sub, neg_sub]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn : \u2115\n\u22a2 (Injective fun n => x ^ n) \u2194 \u00acIsOfFinOrder x\n[PROOFSTEP]\nrefine' \u27e8_, fun h n m hnm => _\u27e9\n[GOAL]\ncase refine'_1\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn : \u2115\n\u22a2 (Injective fun n => x ^ n) \u2192 \u00acIsOfFinOrder x\n[PROOFSTEP]\nsimp_rw [isOfFinOrder_iff_pow_eq_one]\n[GOAL]\ncase refine'_1\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn : \u2115\n\u22a2 (Injective fun n => x ^ n) \u2192 \u00ac\u2203 n, 0 < n \u2227 x ^ n = 1\n[PROOFSTEP]\nrintro h \u27e8n, hn, hx\u27e9\n[GOAL]\ncase refine'_1.intro.intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn\u271d : \u2115\nh : Injective fun n => x ^ n\nn : \u2115\nhn : 0 < n\nhx : x ^ n = 1\n\u22a2 False\n[PROOFSTEP]\nexact Nat.cast_ne_zero.2 hn.ne' (h <| by simpa using hx)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn\u271d : \u2115\nh : Injective fun n => x ^ n\nn : \u2115\nhn : 0 < n\nhx : x ^ n = 1\n\u22a2 (fun n => x ^ n) \u2191n = (fun n => x ^ n) 0\n[PROOFSTEP]\nsimpa using hx\n[GOAL]\ncase refine'_2\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : Group G\nx y : G\nn\u271d : \u2115\nh : \u00acIsOfFinOrder x\nn m : \u2124\nhnm : (fun n => x ^ n) n = (fun n => x ^ n) m\n\u22a2 n = m\n[PROOFSTEP]\nrwa [zpow_eq_zpow_iff_modEq, orderOf_eq_zero_iff.2 h, Nat.cast_zero, Int.modEq_zero_iff] at hnm \n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nx : G\nn : \u2115\nhn : \u2203 m, x ^ m = x ^ n\n\u22a2 \u2191(finEquivZpowers x).symm { val := x ^ n, property := hn } =\n    { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) }\n[PROOFSTEP]\nrw [finEquivZpowers, Equiv.symm_trans_apply]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx\u271d y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nx : G\nn : \u2115\nhn : \u2203 m, x ^ m = x ^ n\n\u22a2 \u2191(finEquivPowers x).symm\n      (\u2191(Equiv.Set.ofEq (_ : \u2191(Submonoid.powers x) = \u2191(zpowers x))).symm { val := x ^ n, property := hn }) =\n    { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) }\n[PROOFSTEP]\nexact finEquivPowers_symm_apply x n\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nh : orderOf x = orderOf y\nn : \u2115\n\u22a2 \u2191(zpowersEquivZpowers h) { val := x ^ n, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) x y = x ^ n) } =\n    { val := y ^ n, property := (_ : \u2203 y_1, (fun x x_1 => x ^ x_1) y y_1 = y ^ n) }\n[PROOFSTEP]\nrw [zpowersEquivZpowers, Equiv.trans_apply, Equiv.trans_apply, finEquivZpowers_symm_apply, \u2190 Equiv.eq_symm_apply,\n  finEquivZpowers_symm_apply]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn\u271d : \u2115\ninst\u271d : Finite G\nh : orderOf x = orderOf y\nn : \u2115\n\u22a2 \u2191(Fin.castIso h).toEquiv { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } =\n    { val := n % orderOf y, isLt := (_ : n % orderOf y < orderOf y) }\n[PROOFSTEP]\nsimp [h]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nclassical\nhave ft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 zpowers x) := Fintype.ofEquiv G groupEquivQuotientProdSubgroup\nhave ft_s : Fintype (zpowers x) := @Fintype.prodRight _ _ _ ft_prod _\nhave ft_cosets : Fintype (G \u29f8 zpowers x) := @Fintype.prodLeft _ _ _ ft_prod \u27e8\u27e81, (zpowers x).one_mem\u27e9\u27e9\nhave eq\u2081 : Fintype.card G = @Fintype.card _ ft_cosets * @Fintype.card _ ft_s :=\n  calc\n    Fintype.card G = @Fintype.card _ ft_prod := @Fintype.card_congr _ _ _ ft_prod groupEquivQuotientProdSubgroup\n    _ = @Fintype.card _ (@instFintypeProd _ _ ft_cosets ft_s) := (congr_arg (@Fintype.card _) <| Subsingleton.elim _ _)\n    _ = @Fintype.card _ ft_cosets * @Fintype.card _ ft_s := @Fintype.card_prod _ _ ft_cosets ft_s\nhave eq\u2082 : orderOf x = @Fintype.card _ ft_s :=\n  calc\n    orderOf x = _ := orderOf_eq_card_zpowers\n    _ = _ := congr_arg (@Fintype.card _) <| Subsingleton.elim _ _\nexact Dvd.intro (@Fintype.card (G \u29f8 Subgroup.zpowers x) ft_cosets) (by rw [eq\u2081, eq\u2082, mul_comm])\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nhave ft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 zpowers x) := Fintype.ofEquiv G groupEquivQuotientProdSubgroup\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nhave ft_s : Fintype (zpowers x) := @Fintype.prodRight _ _ _ ft_prod _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\nft_s : Fintype { x_1 // x_1 \u2208 zpowers x }\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nhave ft_cosets : Fintype (G \u29f8 zpowers x) := @Fintype.prodLeft _ _ _ ft_prod \u27e8\u27e81, (zpowers x).one_mem\u27e9\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\nft_s : Fintype { x_1 // x_1 \u2208 zpowers x }\nft_cosets : Fintype (G \u29f8 zpowers x)\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nhave eq\u2081 : Fintype.card G = @Fintype.card _ ft_cosets * @Fintype.card _ ft_s :=\n  calc\n    Fintype.card G = @Fintype.card _ ft_prod := @Fintype.card_congr _ _ _ ft_prod groupEquivQuotientProdSubgroup\n    _ = @Fintype.card _ (@instFintypeProd _ _ ft_cosets ft_s) := (congr_arg (@Fintype.card _) <| Subsingleton.elim _ _)\n    _ = @Fintype.card _ ft_cosets * @Fintype.card _ ft_s := @Fintype.card_prod _ _ ft_cosets ft_s\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\nft_s : Fintype { x_1 // x_1 \u2208 zpowers x }\nft_cosets : Fintype (G \u29f8 zpowers x)\neq\u2081 : Fintype.card G = Fintype.card (G \u29f8 zpowers x) * Fintype.card { x_1 // x_1 \u2208 zpowers x }\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nhave eq\u2082 : orderOf x = @Fintype.card _ ft_s :=\n  calc\n    orderOf x = _ := orderOf_eq_card_zpowers\n    _ = _ := congr_arg (@Fintype.card _) <| Subsingleton.elim _ _\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\nft_s : Fintype { x_1 // x_1 \u2208 zpowers x }\nft_cosets : Fintype (G \u29f8 zpowers x)\neq\u2081 : Fintype.card G = Fintype.card (G \u29f8 zpowers x) * Fintype.card { x_1 // x_1 \u2208 zpowers x }\neq\u2082 : orderOf x = Fintype.card { x_1 // x_1 \u2208 zpowers x }\n\u22a2 orderOf x \u2223 Fintype.card G\n[PROOFSTEP]\nexact Dvd.intro (@Fintype.card (G \u29f8 Subgroup.zpowers x) ft_cosets) (by rw [eq\u2081, eq\u2082, mul_comm])\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nft_prod : Fintype ((G \u29f8 zpowers x) \u00d7 { x_1 // x_1 \u2208 zpowers x })\nft_s : Fintype { x_1 // x_1 \u2208 zpowers x }\nft_cosets : Fintype (G \u29f8 zpowers x)\neq\u2081 : Fintype.card G = Fintype.card (G \u29f8 zpowers x) * Fintype.card { x_1 // x_1 \u2208 zpowers x }\neq\u2082 : orderOf x = Fintype.card { x_1 // x_1 \u2208 zpowers x }\n\u22a2 orderOf x * Fintype.card (G \u29f8 zpowers x) = Fintype.card G\n[PROOFSTEP]\nrw [eq\u2081, eq\u2082, mul_comm]\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx\u271d y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nx : G\n\u22a2 orderOf x \u2223 Nat.card G\n[PROOFSTEP]\ncases' fintypeOrInfinite G with h h\n[GOAL]\ncase inl\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx\u271d y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nx : G\nh : Fintype G\n\u22a2 orderOf x \u2223 Nat.card G\n[PROOFSTEP]\nsimp only [Nat.card_eq_fintype_card, orderOf_dvd_card_univ]\n[GOAL]\ncase inr\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx\u271d y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nx : G\nh : Infinite G\n\u22a2 orderOf x \u2223 Nat.card G\n[PROOFSTEP]\nsimp only [card_eq_zero_of_infinite, dvd_zero]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\n\u22a2 x ^ Fintype.card G = 1\n[PROOFSTEP]\nrw [\u2190 Nat.card_eq_fintype_card, pow_card_eq_one']\n[GOAL]\nG\u271d : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b3 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b2 : Fintype G\u271d\nG : Type u_6\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\ng : G\n\u22a2 g ^ index H \u2208 H\n[PROOFSTEP]\nrw [\u2190 eq_one_iff, QuotientGroup.mk_pow H, index, pow_card_eq_one']\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn\u271d : \u2115\ninst\u271d : Fintype G\nn : \u2115\n\u22a2 x ^ n = x ^ (n % Fintype.card G)\n[PROOFSTEP]\nrw [pow_eq_mod_orderOf, \u2190 Nat.mod_mod_of_dvd n orderOf_dvd_card_univ, \u2190 pow_eq_mod_orderOf]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn\u271d : \u2115\ninst\u271d : Fintype G\nn : \u2124\n\u22a2 x ^ n = x ^ (n % \u2191(Fintype.card G))\n[PROOFSTEP]\nrw [zpow_eq_mod_orderOf, \u2190 Int.emod_emod_of_dvd n (Int.coe_nat_dvd.2 orderOf_dvd_card_univ), \u2190 zpow_eq_mod_orderOf]\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\n\u22a2 (fun g => g ^ gcdB (Nat.card G) n) ((fun g => g ^ n) g) = g\n[PROOFSTEP]\nhave key := congr_arg ((\u00b7 ^ \u00b7) g) ((Nat.card G).gcd_eq_gcd_ab n)\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\nkey :\n  (fun x x_1 => x ^ x_1) g \u2191(gcd (Nat.card G) n) =\n    (fun x x_1 => x ^ x_1) g (\u2191(Nat.card G) * gcdA (Nat.card G) n + \u2191n * gcdB (Nat.card G) n)\n\u22a2 (fun g => g ^ gcdB (Nat.card G) n) ((fun g => g ^ n) g) = g\n[PROOFSTEP]\ndsimp only at key \n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\nkey : g ^ \u2191(gcd (Nat.card G) n) = g ^ (\u2191(Nat.card G) * gcdA (Nat.card G) n + \u2191n * gcdB (Nat.card G) n)\n\u22a2 (fun g => g ^ gcdB (Nat.card G) n) ((fun g => g ^ n) g) = g\n[PROOFSTEP]\nrwa [zpow_add, zpow_mul, zpow_mul, zpow_ofNat, zpow_ofNat, zpow_ofNat, h.gcd_eq_one, pow_one, pow_card_eq_one',\n  one_zpow, one_mul, eq_comm] at key \n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\n\u22a2 (fun g => g ^ n) ((fun g => g ^ gcdB (Nat.card G) n) g) = g\n[PROOFSTEP]\nhave key := congr_arg ((\u00b7 ^ \u00b7) g) ((Nat.card G).gcd_eq_gcd_ab n)\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\nkey :\n  (fun x x_1 => x ^ x_1) g \u2191(gcd (Nat.card G) n) =\n    (fun x x_1 => x ^ x_1) g (\u2191(Nat.card G) * gcdA (Nat.card G) n + \u2191n * gcdB (Nat.card G) n)\n\u22a2 (fun g => g ^ n) ((fun g => g ^ gcdB (Nat.card G) n) g) = g\n[PROOFSTEP]\ndsimp only at key \n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b9 : Fintype G\u271d\nG : Type u_6\ninst\u271d : Group G\nh : coprime (Nat.card G) n\ng : G\nkey : g ^ \u2191(gcd (Nat.card G) n) = g ^ (\u2191(Nat.card G) * gcdA (Nat.card G) n + \u2191n * gcdB (Nat.card G) n)\n\u22a2 (fun g => g ^ n) ((fun g => g ^ gcdB (Nat.card G) n) g) = g\n[PROOFSTEP]\nrwa [zpow_add, zpow_mul, zpow_mul', zpow_ofNat, zpow_ofNat, zpow_ofNat, h.gcd_eq_one, pow_one, pow_card_eq_one',\n  one_zpow, one_mul, eq_comm] at key \n[GOAL]\nG\u271d : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2074 : Group G\u271d\nx y : G\u271d\nn : \u2115\ninst\u271d\u00b3 : Fintype G\u271d\nG : Type u_6\ninst\u271d\u00b2 : Group G\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype { x // x \u2208 K }\nh : coprime (Fintype.card { x // x \u2208 H }) (Fintype.card { x // x \u2208 K })\n\u22a2 H \u2293 K = \u22a5\n[PROOFSTEP]\nrefine' (H \u2293 K).eq_bot_iff_forall.mpr fun x hx => _\n[GOAL]\nG\u271d : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2074 : Group G\u271d\nx\u271d y : G\u271d\nn : \u2115\ninst\u271d\u00b3 : Fintype G\u271d\nG : Type u_6\ninst\u271d\u00b2 : Group G\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype { x // x \u2208 K }\nh : coprime (Fintype.card { x // x \u2208 H }) (Fintype.card { x // x \u2208 K })\nx : G\nhx : x \u2208 H \u2293 K\n\u22a2 x = 1\n[PROOFSTEP]\nrw [\u2190 orderOf_eq_one_iff, \u2190 Nat.dvd_one, \u2190 h.gcd_eq_one, Nat.dvd_gcd_iff]\n[GOAL]\nG\u271d : Type u_1\nH\u271d : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2074 : Group G\u271d\nx\u271d y : G\u271d\nn : \u2115\ninst\u271d\u00b3 : Fintype G\u271d\nG : Type u_6\ninst\u271d\u00b2 : Group G\nH K : Subgroup G\ninst\u271d\u00b9 : Fintype { x // x \u2208 H }\ninst\u271d : Fintype { x // x \u2208 K }\nh : coprime (Fintype.card { x // x \u2208 H }) (Fintype.card { x // x \u2208 K })\nx : G\nhx : x \u2208 H \u2293 K\n\u22a2 orderOf x \u2223 Fintype.card { x // x \u2208 H } \u2227 orderOf x \u2223 Fintype.card { x // x \u2208 K }\n[PROOFSTEP]\nexact\n  \u27e8(congr_arg (\u00b7 \u2223 Fintype.card H) (orderOf_subgroup \u27e8x, hx.1\u27e9)).mpr orderOf_dvd_card_univ,\n    (congr_arg (\u00b7 \u2223 Fintype.card K) (orderOf_subgroup \u27e8x, hx.2\u27e9)).mpr orderOf_dvd_card_univ\u27e9\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\nx y : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\n\u22a2 Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = Set.toFinset \u2191(zpowers x)\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b2 : Group G\nx\u271d y : G\nn : \u2115\ninst\u271d\u00b9 : Fintype G\ninst\u271d : DecidableEq G\nx : G\n\u22a2 x \u2208 Finset.image (fun i => x\u271d ^ i) (Finset.range (orderOf x\u271d)) \u2194 x \u2208 Set.toFinset \u2191(zpowers x\u271d)\n[PROOFSTEP]\nrw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nh : x ^ gcd n (Fintype.card G) = 1\n\u22a2 x ^ n = 1\n[PROOFSTEP]\nlet \u27e8m, hm\u27e9 := gcd_dvd_left n (Fintype.card G)\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Group G\nx y : G\nn : \u2115\ninst\u271d : Fintype G\nh : x ^ gcd n (Fintype.card G) = 1\nm : \u2115\nhm : n = gcd n (Fintype.card G) * m\n\u22a2 x ^ n = 1\n[PROOFSTEP]\nrw [hm, pow_mul, h, one_pow]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nM : Type u_6\ninst\u271d\u00b9 : LeftCancelMonoid M\ninst\u271d : Fintype M\nS : Set M\nhS1 : Set.Nonempty S\nhS2 : S * S = S\na : M\nha : a \u2208 S\n\u22a2 a ^ (zero + 1) \u2208 S\n[PROOFSTEP]\nrwa [Nat.zero_eq, zero_add, pow_one]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nM : Type u_6\ninst\u271d\u00b9 : LeftCancelMonoid M\ninst\u271d : Fintype M\nS : Set M\nhS1 : Set.Nonempty S\nhS2 : S * S = S\npow_mem : \u2200 (a : M), a \u2208 S \u2192 \u2200 (n : \u2115), a ^ (n + 1) \u2208 S\n\u22a2 1 \u2208 { carrier := S, mul_mem' := (_ : \u2200 {a b : M}, a \u2208 S \u2192 b \u2208 S \u2192 a * b \u2208 S) }.carrier\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := hS1\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nM : Type u_6\ninst\u271d\u00b9 : LeftCancelMonoid M\ninst\u271d : Fintype M\nS : Set M\nhS2 : S * S = S\npow_mem : \u2200 (a : M), a \u2208 S \u2192 \u2200 (n : \u2115), a ^ (n + 1) \u2208 S\na : M\nha : a \u2208 S\n\u22a2 1 \u2208 { carrier := S, mul_mem' := (_ : \u2200 {a b : M}, a \u2208 S \u2192 b \u2208 S \u2192 a * b \u2208 S) }.carrier\n[PROOFSTEP]\nrw [\u2190 pow_orderOf_eq_one a, \u2190 tsub_add_cancel_of_le (succ_le_of_lt (orderOf_pos a))]\n[GOAL]\ncase intro\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nM : Type u_6\ninst\u271d\u00b9 : LeftCancelMonoid M\ninst\u271d : Fintype M\nS : Set M\nhS2 : S * S = S\npow_mem : \u2200 (a : M), a \u2208 S \u2192 \u2200 (n : \u2115), a ^ (n + 1) \u2208 S\na : M\nha : a \u2208 S\n\u22a2 a ^ (orderOf a - succ 0 + succ 0) \u2208 { carrier := S, mul_mem' := (_ : \u2200 {a b : M}, a \u2208 S \u2192 b \u2208 S \u2192 a * b \u2208 S) }.carrier\n[PROOFSTEP]\nexact pow_mem a ha (orderOf a - 1)\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS1 : Set.Nonempty S\nhS2 : S * S = S\nsrc\u271d : Submonoid G := submonoidOfIdempotent S hS1 hS2\na : G\nha :\n  a \u2208\n    {\n          toSubsemigroup :=\n            { carrier := S, mul_mem' := (_ : \u2200 {a b : G}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a * b \u2208 src\u271d.carrier) },\n          one_mem' := (_ : 1 \u2208 src\u271d.carrier) }.toSubsemigroup.carrier\n\u22a2 a\u207b\u00b9 \u2208 submonoidOfIdempotent S hS1 hS2\n[PROOFSTEP]\nrw [\u2190 one_mul a\u207b\u00b9, \u2190 pow_one a, \u2190 pow_orderOf_eq_one a, \u2190 pow_sub a (orderOf_pos a)]\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS1 : Set.Nonempty S\nhS2 : S * S = S\nsrc\u271d : Submonoid G := submonoidOfIdempotent S hS1 hS2\na : G\nha :\n  a \u2208\n    {\n          toSubsemigroup :=\n            { carrier := S, mul_mem' := (_ : \u2200 {a b : G}, a \u2208 src\u271d.carrier \u2192 b \u2208 src\u271d.carrier \u2192 a * b \u2208 src\u271d.carrier) },\n          one_mem' := (_ : 1 \u2208 src\u271d.carrier) }.toSubsemigroup.carrier\n\u22a2 a ^ (orderOf a - succ 0) \u2208 submonoidOfIdempotent S hS1 hS2\n[PROOFSTEP]\nexact pow_mem ha (orderOf a - 1)\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS : Set.Nonempty S\n\u22a2 1 \u2208 S ^ Fintype.card G\n[PROOFSTEP]\nobtain \u27e8a, ha\u27e9 := hS\n[GOAL]\ncase intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\na : G\nha : a \u2208 S\n\u22a2 1 \u2208 S ^ Fintype.card G\n[PROOFSTEP]\nrw [\u2190 pow_card_eq_one]\n[GOAL]\ncase intro\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\na : G\nha : a \u2208 S\n\u22a2 ?m.1815695 ^ Fintype.card G \u2208 S ^ Fintype.card G\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\na : G\nha : a \u2208 S\n\u22a2 G\n[PROOFSTEP]\nexact Set.pow_mem_pow ha (Fintype.card G)\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS : Set.Nonempty S\none_mem : 1 \u2208 S ^ Fintype.card G\n\u22a2 S ^ Fintype.card G * S ^ Fintype.card G = S ^ Fintype.card G\n[PROOFSTEP]\nclassical!\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS : Set.Nonempty S\none_mem : 1 \u2208 S ^ Fintype.card G\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 S ^ Fintype.card G * S ^ Fintype.card G = S ^ Fintype.card G\n[PROOFSTEP]\napply (Set.eq_of_subset_of_card_le (Set.subset_mul_left _ one_mem) (ge_of_eq _)).symm\n[GOAL]\nG\u271d : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nG : Type u_6\ninst\u271d\u00b9 : Group G\ninst\u271d : Fintype G\nS : Set G\nhS : Set.Nonempty S\none_mem : 1 \u2208 S ^ Fintype.card G\nem\u271d : (a : Prop) \u2192 Decidable a\n\u22a2 Fintype.card \u2191(S ^ Fintype.card G) = Fintype.card \u2191(S ^ Fintype.card G * S ^ Fintype.card G)\n[PROOFSTEP]\nsimp_rw [\u2190 pow_add, Group.card_pow_eq_card_pow_card_univ S (Fintype.card G + Fintype.card G) le_add_self]\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\n\u22a2 orderOf x = 0\n[PROOFSTEP]\nrw [orderOf_eq_zero_iff']\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\n\u22a2 \u2200 (n : \u2115), 0 < n \u2192 x ^ n \u2260 1\n[PROOFSTEP]\nintro n hn hx\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\nn : \u2115\nhn : 0 < n\nhx : x ^ n = 1\n\u22a2 False\n[PROOFSTEP]\nreplace hx : |x| ^ n = 1 := by simpa only [abs_one, abs_pow] using congr_arg abs hx\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\nn : \u2115\nhn : 0 < n\nhx : x ^ n = 1\n\u22a2 |x| ^ n = 1\n[PROOFSTEP]\nsimpa only [abs_one, abs_pow] using congr_arg abs hx\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\nn : \u2115\nhn : 0 < n\nhx : |x| ^ n = 1\n\u22a2 False\n[PROOFSTEP]\ncases' h.lt_or_lt with h h\n[GOAL]\ncase inl\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh\u271d : |x| \u2260 1\nn : \u2115\nhn : 0 < n\nhx : |x| ^ n = 1\nh : |x| < 1\n\u22a2 False\n[PROOFSTEP]\nexact ((pow_lt_one (abs_nonneg x) h hn.ne').ne hx).elim\n[GOAL]\ncase inr\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh\u271d : |x| \u2260 1\nn : \u2115\nhn : 0 < n\nhx : |x| ^ n = 1\nh : 1 < |x|\n\u22a2 False\n[PROOFSTEP]\nexact ((one_lt_pow h hn.ne').ne' hx).elim\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\n\u22a2 orderOf x \u2264 2\n[PROOFSTEP]\ncases' ne_or_eq |x| 1 with h h\n[GOAL]\ncase inl\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| \u2260 1\n\u22a2 orderOf x \u2264 2\n[PROOFSTEP]\nsimp [orderOf_abs_ne_one h]\n[GOAL]\ncase inr\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nx : G\nh : |x| = 1\n\u22a2 orderOf x \u2264 2\n[PROOFSTEP]\nrcases eq_or_eq_neg_of_abs_eq h with (rfl | rfl)\n[GOAL]\ncase inr.inl\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nh : |1| = 1\n\u22a2 orderOf 1 \u2264 2\n[PROOFSTEP]\nsimp\n[GOAL]\ncase inr.inr\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nh : |(-1)| = 1\n\u22a2 orderOf (-1) \u2264 2\n[PROOFSTEP]\napply orderOf_le_of_pow_eq_one\n[GOAL]\ncase inr.inr.hn\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nh : |(-1)| = 1\n\u22a2 0 < 2\n[PROOFSTEP]\nnorm_num\n[GOAL]\ncase inr.inr.h\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d : LinearOrderedRing G\nh : |(-1)| = 1\n\u22a2 (-1) ^ 2 = 1\n[PROOFSTEP]\nnorm_num\n[GOAL]\nG : Type u_1\nH : Type u_2\nA : Type u_3\n\u03b1 : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u00b9 : Monoid \u03b1\ninst\u271d : Monoid \u03b2\nx : \u03b1 \u00d7 \u03b2\na : \u03b1\nb : \u03b2\n\u22a2 IsOfFinOrder a \u2192 IsOfFinOrder b \u2192 IsOfFinOrder (a, b)\n[PROOFSTEP]\nsimpa only [\u2190 orderOf_pos_iff, Prod.orderOf] using Nat.lcm_pos\n", "meta": {"mathlib_filename": "Mathlib.GroupTheory.OrderOfElement", "llama_tokens": 42156, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.5008257066347745}}
{"text": "[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Nontrivial R\ni j : \u2115\n\u22a2 \u2191(monomial i) 1 = \u2191(monomial j) 1 \u2194 i = j\n[PROOFSTEP]\nsimp_rw [\u2190 ofFinsupp_single, ofFinsupp.injEq]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Nontrivial R\ni j : \u2115\n\u22a2 Finsupp.single i 1 = Finsupp.single j 1 \u2194 i = j\n[PROOFSTEP]\nexact AddMonoidAlgebra.of_injective.eq_iff\n[GOAL]\nR : Type u\na b : R\nm\u271d n\u271d : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\ninst\u271d : Nontrivial R\nm n : \u2115\nh : (fun i => \u2191(monomial i) 1) m = (fun i => \u2191(monomial i) 1) n\n\u22a2 m = n\n[PROOFSTEP]\nsimpa [monomial_one_eq_iff] using h\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\n\u22a2 Finset.card (support f) \u2264 1 \u2194 \u2203 n a, f = \u2191(monomial n) a\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\n\u22a2 Finset.card (support f) \u2264 1 \u2192 \u2203 n a, f = \u2191(monomial n) a\n[PROOFSTEP]\nintro H\n[GOAL]\ncase mp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nH : Finset.card (support f) \u2264 1\n\u22a2 \u2203 n a, f = \u2191(monomial n) a\n[PROOFSTEP]\nrw [Finset.card_le_one_iff_subset_singleton] at H \n[GOAL]\ncase mp\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nH : \u2203 x, support f \u2286 {x}\n\u22a2 \u2203 n a, f = \u2191(monomial n) a\n[PROOFSTEP]\nrcases H with \u27e8n, hn\u27e9\n[GOAL]\ncase mp.intro\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\n\u22a2 \u2203 n a, f = \u2191(monomial n) a\n[PROOFSTEP]\nrefine' \u27e8n, f.coeff n, _\u27e9\n[GOAL]\ncase mp.intro\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\n\u22a2 f = \u2191(monomial n) (coeff f n)\n[PROOFSTEP]\next i\n[GOAL]\ncase mp.intro.a\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\n\u22a2 coeff f i = coeff (\u2191(monomial n) (coeff f n)) i\n[PROOFSTEP]\nby_cases hi : i = n\n[GOAL]\ncase pos\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\nhi : i = n\n\u22a2 coeff f i = coeff (\u2191(monomial n) (coeff f n)) i\n[PROOFSTEP]\nsimp [hi, coeff_monomial]\n[GOAL]\ncase neg\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\nhi : \u00aci = n\n\u22a2 coeff f i = coeff (\u2191(monomial n) (coeff f n)) i\n[PROOFSTEP]\nhave : f.coeff i = 0 := by\n  rw [\u2190 not_mem_support_iff]\n  exact fun hi' => hi (Finset.mem_singleton.1 (hn hi'))\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\nhi : \u00aci = n\n\u22a2 coeff f i = 0\n[PROOFSTEP]\nrw [\u2190 not_mem_support_iff]\n[GOAL]\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\nhi : \u00aci = n\n\u22a2 \u00aci \u2208 support f\n[PROOFSTEP]\nexact fun hi' => hi (Finset.mem_singleton.1 (hn hi'))\n[GOAL]\ncase neg\nR : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\nn : \u2115\nhn : support f \u2286 {n}\ni : \u2115\nhi : \u00aci = n\nthis : coeff f i = 0\n\u22a2 coeff f i = coeff (\u2191(monomial n) (coeff f n)) i\n[PROOFSTEP]\nsimp [this, Ne.symm hi, coeff_monomial]\n[GOAL]\ncase mpr\nR : Type u\na b : R\nm n : \u2115\ninst\u271d : Semiring R\np q r f : R[X]\n\u22a2 (\u2203 n a, f = \u2191(monomial n) a) \u2192 Finset.card (support f) \u2264 1\n[PROOFSTEP]\nrintro \u27e8n, a, rfl\u27e9\n[GOAL]\ncase mpr.intro.intro\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 Finset.card (support (\u2191(monomial n) a)) \u2264 1\n[PROOFSTEP]\nrw [\u2190 Finset.card_singleton n]\n[GOAL]\ncase mpr.intro.intro\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 Finset.card (support (\u2191(monomial n) a)) \u2264 Finset.card {n}\n[PROOFSTEP]\napply Finset.card_le_of_subset\n[GOAL]\ncase mpr.intro.intro.a\nR : Type u\na\u271d b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nn : \u2115\na : R\n\u22a2 support (\u2191(monomial n) a) \u2286 {n}\n[PROOFSTEP]\nexact support_monomial' _ _\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\n\u22a2 f = g\n[PROOFSTEP]\nset f' := f.comp (toFinsuppIso R).symm.toRingHom with hf'\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 f = g\n[PROOFSTEP]\nset g' := g.comp (toFinsuppIso R).symm.toRingHom with hg'\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 f = g\n[PROOFSTEP]\nhave A : f' = g' := by\n  -- Porting note: Was `ext; simp [..]; simpa [..] using h\u2082`.\n  ext : 1\n  \u00b7 ext\n    simp [h\u2081, RingEquiv.toRingHom_eq_coe]\n  \u00b7 refine MonoidHom.ext_mnat ?_\n    simpa [RingEquiv.toRingHom_eq_coe] using h\u2082\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 f' = g'\n[PROOFSTEP]\next : 1\n[GOAL]\ncase h\u2081\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 RingHom.comp f' AddMonoidAlgebra.singleZeroRingHom = RingHom.comp g' AddMonoidAlgebra.singleZeroRingHom\n[PROOFSTEP]\next\n[GOAL]\ncase h\u2081.a\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nx\u271d : R\n\u22a2 \u2191(RingHom.comp f' AddMonoidAlgebra.singleZeroRingHom) x\u271d = \u2191(RingHom.comp g' AddMonoidAlgebra.singleZeroRingHom) x\u271d\n[PROOFSTEP]\nsimp [h\u2081, RingEquiv.toRingHom_eq_coe]\n[GOAL]\ncase h_of\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 MonoidHom.comp (\u2191f') (AddMonoidAlgebra.of R \u2115) = MonoidHom.comp (\u2191g') (AddMonoidAlgebra.of R \u2115)\n[PROOFSTEP]\nrefine MonoidHom.ext_mnat ?_\n[GOAL]\ncase h_of\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\n\u22a2 \u2191(MonoidHom.comp (\u2191f') (AddMonoidAlgebra.of R \u2115)) (\u2191Multiplicative.ofAdd 1) =\n    \u2191(MonoidHom.comp (\u2191g') (AddMonoidAlgebra.of R \u2115)) (\u2191Multiplicative.ofAdd 1)\n[PROOFSTEP]\nsimpa [RingEquiv.toRingHom_eq_coe] using h\u2082\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\n\u22a2 f = g\n[PROOFSTEP]\nhave B : f = f'.comp (toFinsuppIso R) := by\n  rw [hf', RingHom.comp_assoc]\n  ext x\n  simp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply, RingHom.coe_comp,\n    RingEquiv.coe_toRingHom]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\n\u22a2 f = RingHom.comp f' \u2191(toFinsuppIso R)\n[PROOFSTEP]\nrw [hf', RingHom.comp_assoc]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\n\u22a2 f = RingHom.comp f (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R))) \u2191(toFinsuppIso R))\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nx : R[X]\n\u22a2 \u2191f x = \u2191(RingHom.comp f (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R))) \u2191(toFinsuppIso R))) x\n[PROOFSTEP]\nsimp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply, RingHom.coe_comp,\n  RingEquiv.coe_toRingHom]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nB : f = RingHom.comp f' \u2191(toFinsuppIso R)\n\u22a2 f = g\n[PROOFSTEP]\nhave C' : g = g'.comp (toFinsuppIso R) := by\n  rw [hg', RingHom.comp_assoc]\n  ext x\n  simp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply, RingHom.coe_comp,\n    RingEquiv.coe_toRingHom]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nB : f = RingHom.comp f' \u2191(toFinsuppIso R)\n\u22a2 g = RingHom.comp g' \u2191(toFinsuppIso R)\n[PROOFSTEP]\nrw [hg', RingHom.comp_assoc]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nB : f = RingHom.comp f' \u2191(toFinsuppIso R)\n\u22a2 g = RingHom.comp g (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R))) \u2191(toFinsuppIso R))\n[PROOFSTEP]\next x\n[GOAL]\ncase a\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nB : f = RingHom.comp f' \u2191(toFinsuppIso R)\nx : R[X]\n\u22a2 \u2191g x = \u2191(RingHom.comp g (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R))) \u2191(toFinsuppIso R))) x\n[PROOFSTEP]\nsimp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply, RingHom.coe_comp,\n  RingEquiv.coe_toRingHom]\n[GOAL]\nR : Type u\na b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\np q r : R[X]\nS : Type u_1\ninst\u271d : Semiring S\nf g : R[X] \u2192+* S\nh\u2081 : \u2200 (a : R), \u2191f (\u2191C a) = \u2191g (\u2191C a)\nh\u2082 : \u2191f X = \u2191g X\nf' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhf' : f' = RingHom.comp f (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\ng' : AddMonoidAlgebra R \u2115 \u2192+* S := RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nhg' : g' = RingHom.comp g (RingEquiv.toRingHom (RingEquiv.symm (toFinsuppIso R)))\nA : f' = g'\nB : f = RingHom.comp f' \u2191(toFinsuppIso R)\nC' : g = RingHom.comp g' \u2191(toFinsuppIso R)\n\u22a2 f = g\n[PROOFSTEP]\nrw [B, C', A]\n", "meta": {"mathlib_filename": "Mathlib.Data.Polynomial.Monomial", "llama_tokens": 8128, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506418255927, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.5008257045882094}}
{"text": "[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\nha : a \u2260 \u22a5\nhab : Disjoint a b\nh : a = b\n\u22a2 Disjoint a a\n[PROOFSTEP]\nrwa [\u2190 h] at hab \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\n\u22a2 Disjoint (a \u2293 b) c \u2194 Disjoint a (b \u2293 c)\n[PROOFSTEP]\nrw [disjoint_iff_inf_le, disjoint_iff_inf_le, inf_assoc]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\n\u22a2 Disjoint a (b \u2293 c) \u2194 Disjoint b (a \u2293 c)\n[PROOFSTEP]\nsimp_rw [disjoint_iff_inf_le, inf_left_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SemilatticeInf \u03b1\ninst\u271d : OrderBot \u03b1\na b c d : \u03b1\n\u22a2 Disjoint (a \u2293 b) c \u2194 Disjoint (a \u2293 c) b\n[PROOFSTEP]\nsimp_rw [disjoint_iff_inf_le, inf_right_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na b c : \u03b1\n\u22a2 Disjoint (a \u2294 b) c \u2194 Disjoint a c \u2227 Disjoint b c\n[PROOFSTEP]\nsimp only [disjoint_iff, inf_sup_right, sup_eq_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na b c : \u03b1\n\u22a2 Disjoint a (b \u2294 c) \u2194 Disjoint a b \u2227 Disjoint a c\n[PROOFSTEP]\nsimp only [disjoint_iff, inf_sup_left, sup_eq_bot_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na b c : \u03b1\nh : a \u2264 c \u2294 b\nhd : Disjoint a c\n\u22a2 a \u2264 b \u2294 c\n[PROOFSTEP]\nrwa [sup_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : PartialOrder \u03b1\ninst\u271d : OrderTop \u03b1\na b c d : \u03b1\nha : a \u2260 \u22a4\nhab : Codisjoint a b\nh : a = b\n\u22a2 Codisjoint a a\n[PROOFSTEP]\nrwa [\u2190 h] at hab \n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderTop \u03b1\na b c : \u03b1\n\u22a2 Codisjoint (a \u2293 b) c \u2194 Codisjoint a c \u2227 Codisjoint b c\n[PROOFSTEP]\nsimp only [codisjoint_iff, sup_inf_right, inf_eq_top_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderTop \u03b1\na b c : \u03b1\n\u22a2 Codisjoint a (b \u2293 c) \u2194 Codisjoint a b \u2227 Codisjoint a c\n[PROOFSTEP]\nsimp only [codisjoint_iff, sup_inf_left, inf_eq_top_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderTop \u03b1\na b c : \u03b1\nh : b \u2293 a \u2264 c\nhd : Codisjoint b c\n\u22a2 a \u2293 b \u2264 c\n[PROOFSTEP]\nrwa [inf_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b c : \u03b1\nhab : Disjoint a b\nhbc : Codisjoint b c\n\u22a2 a \u2264 c\n[PROOFSTEP]\nrw [\u2190 @inf_top_eq _ _ _ a, \u2190 @bot_sup_eq _ _ _ c, \u2190 hab.eq_bot, \u2190 hbc.eq_top, sup_inf_right]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b c : \u03b1\nhab : Disjoint a b\nhbc : Codisjoint b c\n\u22a2 a \u2293 (b \u2294 c) \u2264 (a \u2294 c) \u2293 (b \u2294 c)\n[PROOFSTEP]\nexact inf_le_inf_right _ le_sup_left\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z : \u03b1\nh : IsCompl x y\nhle : a \u2264 b \u2294 y\n\u22a2 b \u2293 x \u2294 y \u2293 x = b \u2293 x\n[PROOFSTEP]\nrw [h.symm.inf_eq_bot, sup_bot_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z : \u03b1\nh : IsCompl y z\n\u22a2 x \u2293 y = \u22a5 \u2194 x \u2264 z\n[PROOFSTEP]\nrw [\u2190 le_bot_iff, \u2190 h.le_sup_right_iff_inf_left_le, bot_sup_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z : \u03b1\nh : IsCompl y z\n\u22a2 Disjoint x y \u2194 x \u2264 z\n[PROOFSTEP]\nrw [disjoint_iff]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z : \u03b1\nh : IsCompl y z\n\u22a2 x \u2293 y = \u22a5 \u2194 x \u2264 z\n[PROOFSTEP]\nexact h.inf_left_eq_bot_iff\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z x' y' : \u03b1\nh : IsCompl x y\nh' : IsCompl x' y'\n\u22a2 (x \u2294 x') \u2293 (y \u2293 y') = \u22a5\n[PROOFSTEP]\nrw [inf_sup_right, \u2190 inf_assoc, h.inf_eq_bot, bot_inf_eq, bot_sup_eq, inf_left_comm, h'.inf_eq_bot, inf_bot_eq]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b x y z x' y' : \u03b1\nh : IsCompl x y\nh' : IsCompl x' y'\n\u22a2 x \u2294 x' \u2294 y \u2293 y' = \u22a4\n[PROOFSTEP]\nrw [sup_inf_left, @sup_comm _ _ x, sup_assoc, h.sup_eq_top, sup_top_eq, top_inf_eq, sup_assoc, sup_left_comm,\n  h'.sup_eq_top, sup_top_eq]\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Disjoint x y \u2194 Disjoint x.fst y.fst \u2227 Disjoint x.snd y.snd\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Disjoint x y \u2192 Disjoint x.fst y.fst \u2227 Disjoint x.snd y.snd\n[PROOFSTEP]\nintro h\n[GOAL]\ncase mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\n\u22a2 Disjoint x.fst y.fst \u2227 Disjoint x.snd y.snd\n[PROOFSTEP]\nrefine' \u27e8fun a hx hy \u21a6 (@h (a, \u22a5) \u27e8hx, _\u27e9 \u27e8hy, _\u27e9).1, fun b hx hy \u21a6 (@h (\u22a5, b) \u27e8_, hx\u27e9 \u27e8_, hy\u27e9).2\u27e9\n[GOAL]\ncase mp.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\na : \u03b1\nhx : a \u2264 x.fst\nhy : a \u2264 y.fst\n\u22a2 (a, \u22a5).snd \u2264 x.snd\ncase mp.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\na : \u03b1\nhx : a \u2264 x.fst\nhy : a \u2264 y.fst\n\u22a2 (a, \u22a5).snd \u2264 y.snd\ncase mp.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\nb : \u03b2\nhx : b \u2264 x.snd\nhy : b \u2264 y.snd\n\u22a2 (\u22a5, b).fst \u2264 x.fst\ncase mp.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\nb : \u03b2\nhx : b \u2264 x.snd\nhy : b \u2264 y.snd\n\u22a2 (\u22a5, b).fst \u2264 y.fst\n[PROOFSTEP]\nall_goals exact bot_le\n[GOAL]\ncase mp.refine'_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\na : \u03b1\nhx : a \u2264 x.fst\nhy : a \u2264 y.fst\n\u22a2 (a, \u22a5).snd \u2264 x.snd\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase mp.refine'_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\na : \u03b1\nhx : a \u2264 x.fst\nhy : a \u2264 y.fst\n\u22a2 (a, \u22a5).snd \u2264 y.snd\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase mp.refine'_3\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\nb : \u03b2\nhx : b \u2264 x.snd\nhy : b \u2264 y.snd\n\u22a2 (\u22a5, b).fst \u2264 x.fst\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase mp.refine'_4\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nh : Disjoint x y\nb : \u03b2\nhx : b \u2264 x.snd\nhy : b \u2264 y.snd\n\u22a2 (\u22a5, b).fst \u2264 y.fst\n[PROOFSTEP]\nexact bot_le\n[GOAL]\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 Disjoint x.fst y.fst \u2227 Disjoint x.snd y.snd \u2192 Disjoint x y\n[PROOFSTEP]\nrintro \u27e8ha, hb\u27e9 z hza hzb\n[GOAL]\ncase mpr.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : OrderBot \u03b1\ninst\u271d : OrderBot \u03b2\nx y : \u03b1 \u00d7 \u03b2\nha : Disjoint x.fst y.fst\nhb : Disjoint x.snd y.snd\nz : \u03b1 \u00d7 \u03b2\nhza : z \u2264 x\nhzb : z \u2264 y\n\u22a2 z \u2264 \u22a5\n[PROOFSTEP]\nrefine' \u27e8ha hza.1 hzb.1, hb hza.2 hzb.2\u27e9\n[GOAL]\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : PartialOrder \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b2\ninst\u271d\u00b9 : BoundedOrder \u03b1\ninst\u271d : BoundedOrder \u03b2\nx y : \u03b1 \u00d7 \u03b2\n\u22a2 IsCompl x y \u2194 IsCompl x.fst y.fst \u2227 IsCompl x.snd y.snd\n[PROOFSTEP]\nsimp_rw [isCompl_iff, Prod.disjoint_iff, Prod.codisjoint_iff, and_and_and_comm]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b : Complementeds \u03b1\n\u22a2 \u2191a \u2264 \u2191b \u2194 a \u2264 b\n[PROOFSTEP]\nsimp\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b : Complementeds \u03b1\n\u22a2 Disjoint \u2191a \u2191b \u2194 Disjoint a b\n[PROOFSTEP]\nrw [disjoint_iff, disjoint_iff, \u2190 coe_inf, \u2190 coe_bot, coe_inj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b : Complementeds \u03b1\n\u22a2 Codisjoint \u2191a \u2191b \u2194 Codisjoint a b\n[PROOFSTEP]\nrw [codisjoint_iff, codisjoint_iff, \u2190 coe_sup, \u2190 coe_top, coe_inj]\n[GOAL]\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : BoundedOrder \u03b1\na b : Complementeds \u03b1\n\u22a2 IsCompl \u2191a \u2191b \u2194 IsCompl a b\n[PROOFSTEP]\nsimp_rw [isCompl_iff, disjoint_coe, codisjoint_coe]\n", "meta": {"mathlib_filename": "Mathlib.Order.Disjoint", "llama_tokens": 4251, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.7025300636233416, "lm_q1q2_score": 0.500364521707979}}
{"text": "[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2203 p, p \u2208 Ideal.minimalPrimes I \u2227 p \u2264 J\n[PROOFSTEP]\nsuffices\n  \u2203 m \u2208 {p : (Ideal R)\u1d52\u1d48 | Ideal.IsPrime p \u2227 I \u2264 OrderDual.ofDual p},\n    OrderDual.toDual J \u2264 m \u2227 \u2200 z \u2208 {p : (Ideal R)\u1d52\u1d48 | Ideal.IsPrime p \u2227 I \u2264 p}, m \u2264 z \u2192 z = m\n  by\n  obtain \u27e8p, h\u2081, h\u2082, h\u2083\u27e9 := this\n  simp_rw [\u2190 @eq_comm _ p] at h\u2083 \n  exact \u27e8p, \u27e8h\u2081, fun a b c => le_of_eq (h\u2083 a b c)\u27e9, h\u2082\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nthis :\n  \u2203 m,\n    m \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2227\n      \u2191OrderDual.toDual J \u2264 m \u2227 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 m \u2264 z \u2192 z = m\n\u22a2 \u2203 p, p \u2208 Ideal.minimalPrimes I \u2227 p \u2264 J\n[PROOFSTEP]\nobtain \u27e8p, h\u2081, h\u2082, h\u2083\u27e9 := this\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\np : (Ideal R)\u1d52\u1d48\nh\u2081 : p \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nh\u2082 : \u2191OrderDual.toDual J \u2264 p\nh\u2083 : \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 p \u2264 z \u2192 z = p\n\u22a2 \u2203 p, p \u2208 Ideal.minimalPrimes I \u2227 p \u2264 J\n[PROOFSTEP]\nsimp_rw [\u2190 @eq_comm _ p] at h\u2083 \n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\np : (Ideal R)\u1d52\u1d48\nh\u2081 : p \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nh\u2082 : \u2191OrderDual.toDual J \u2264 p\nh\u2083 : \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 p \u2264 z \u2192 p = z\n\u22a2 \u2203 p, p \u2208 Ideal.minimalPrimes I \u2227 p \u2264 J\n[PROOFSTEP]\nexact \u27e8p, \u27e8h\u2081, fun a b c => le_of_eq (h\u2083 a b c)\u27e9, h\u2082\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2203 m,\n    m \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2227\n      \u2191OrderDual.toDual J \u2264 m \u2227 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 m \u2264 z \u2192 z = m\n[PROOFSTEP]\napply zorn_nonempty_partialOrder\u2080\n[GOAL]\ncase ih\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2200 (c : Set (Ideal R)\u1d52\u1d48),\n    c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2192\n      IsChain (fun x x_1 => x \u2264 x_1) c \u2192\n        \u2200 (y : (Ideal R)\u1d52\u1d48),\n          y \u2208 c \u2192 \u2203 ub, ub \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2227 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 c \u2192 z \u2264 ub\ncase hxs\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2191OrderDual.toDual J \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\n[PROOFSTEP]\nswap\n[GOAL]\ncase hxs\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2191OrderDual.toDual J \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\n[PROOFSTEP]\nrefine' \u27e8show J.IsPrime by infer_instance, e\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 IsPrime J\n[PROOFSTEP]\ninfer_instance\n[GOAL]\ncase ih\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\n\u22a2 \u2200 (c : Set (Ideal R)\u1d52\u1d48),\n    c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2192\n      IsChain (fun x x_1 => x \u2264 x_1) c \u2192\n        \u2200 (y : (Ideal R)\u1d52\u1d48),\n          y \u2208 c \u2192 \u2203 ub, ub \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2227 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nrintro (c : Set (Ideal R)) hc hc' J' hJ'\n[GOAL]\ncase ih\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\n\u22a2 \u2203 ub, ub \u2208 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p} \u2227 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 c \u2192 z \u2264 ub\n[PROOFSTEP]\nrefine' \u27e8OrderDual.toDual (sInf c), \u27e8Ideal.sInf_isPrime_of_isChain \u27e8J', hJ'\u27e9 hc'.symm fun x hx => (hc hx).1, _\u27e9, _\u27e9\n[GOAL]\ncase ih.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\n\u22a2 I \u2264 \u2191OrderDual.ofDual (\u2191OrderDual.toDual (sInf c))\n[PROOFSTEP]\nrw [OrderDual.ofDual_toDual, le_sInf_iff]\n[GOAL]\ncase ih.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\n\u22a2 \u2200 (b : Ideal R), b \u2208 c \u2192 I \u2264 b\n[PROOFSTEP]\nexact fun _ hx => (hc hx).2\n[GOAL]\ncase ih.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\n\u22a2 \u2200 (z : (Ideal R)\u1d52\u1d48), z \u2208 c \u2192 z \u2264 \u2191OrderDual.toDual (sInf c)\n[PROOFSTEP]\nrintro z hz\n[GOAL]\ncase ih.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\nz : (Ideal R)\u1d52\u1d48\nhz : z \u2208 c\n\u22a2 z \u2264 \u2191OrderDual.toDual (sInf c)\n[PROOFSTEP]\nrw [OrderDual.le_toDual]\n[GOAL]\ncase ih.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime J\ne : I \u2264 J\nc : Set (Ideal R)\nhc : c \u2286 {p | IsPrime p \u2227 I \u2264 \u2191OrderDual.ofDual p}\nhc' : IsChain (fun x x_1 => x \u2264 x_1) c\nJ' : (Ideal R)\u1d52\u1d48\nhJ' : J' \u2208 c\nz : (Ideal R)\u1d52\u1d48\nhz : z \u2208 c\n\u22a2 sInf c \u2264 \u2191OrderDual.ofDual z\n[PROOFSTEP]\nexact sInf_le hz\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 Ideal.minimalPrimes (radical I) = Ideal.minimalPrimes I\n[PROOFSTEP]\nrw [Ideal.minimalPrimes, Ideal.minimalPrimes]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p} =\n    minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\ncongr\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p} =\n    minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\next p\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\n\u22a2 p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p} \u2194\n    p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\nrefine' \u27e8_, _\u27e9\n[GOAL]\ncase h.refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\n\u22a2 p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p} \u2192\n    p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\nrintro \u27e8\u27e8a, ha\u27e9, b\u27e9\n[GOAL]\ncase h.refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\n\u22a2 p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p} \u2192\n    p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p}\n[PROOFSTEP]\nrintro \u27e8\u27e8a, ha\u27e9, b\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\nb : \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 radical I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\na : IsPrime p\nha : radical I \u2264 p\n\u22a2 p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\nrefine' \u27e8\u27e8a, a.radical_le_iff.1 ha\u27e9, _\u27e9\n[GOAL]\ncase h.refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\nb : \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 radical I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\na : IsPrime p\nha : radical I \u2264 p\n\u22a2 \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, and_imp] at *\n[GOAL]\ncase h.refine'_1.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\na : IsPrime p\nha : radical I \u2264 p\nb : \u2200 \u2983b : Ideal R\u2984, IsPrime b \u2192 radical I \u2264 b \u2192 b \u2264 p \u2192 p \u2264 b\n\u22a2 \u2200 \u2983b : Ideal R\u2984, IsPrime b \u2192 I \u2264 b \u2192 b \u2264 p \u2192 p \u2264 b\n[PROOFSTEP]\nexact fun _ h2 h3 h4 => b h2 (h2.radical_le_iff.2 h3) h4\n[GOAL]\ncase h.refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\nb : \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\na : IsPrime p\nha : I \u2264 p\n\u22a2 p \u2208 minimals (fun x x_1 => x \u2264 x_1) {p | IsPrime p \u2227 radical I \u2264 p}\n[PROOFSTEP]\nrefine' \u27e8\u27e8a, a.radical_le_iff.2 ha\u27e9, _\u27e9\n[GOAL]\ncase h.refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\nb : \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\na : IsPrime p\nha : I \u2264 p\n\u22a2 \u2200 \u2983b : Ideal R\u2984, b \u2208 {p | IsPrime p \u2227 radical I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b p \u2192 (fun x x_1 => x \u2264 x_1) p b\n[PROOFSTEP]\nsimp only [Set.mem_setOf_eq, and_imp] at *\n[GOAL]\ncase h.refine'_2.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J p : Ideal R\na : IsPrime p\nha : I \u2264 p\nb : \u2200 \u2983b : Ideal R\u2984, IsPrime b \u2192 I \u2264 b \u2192 b \u2264 p \u2192 p \u2264 b\n\u22a2 \u2200 \u2983b : Ideal R\u2984, IsPrime b \u2192 radical I \u2264 b \u2192 b \u2264 p \u2192 p \u2264 b\n[PROOFSTEP]\nexact fun _ h2 h3 h4 => b h2 (h2.radical_le_iff.1 h3) h4\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 sInf (Ideal.minimalPrimes I) = radical I\n[PROOFSTEP]\nrw [I.radical_eq_sInf]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 sInf (Ideal.minimalPrimes I) = sInf {J | I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\napply le_antisymm\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 sInf (Ideal.minimalPrimes I) \u2264 sInf {J | I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nx : R\nhx : x \u2208 sInf (Ideal.minimalPrimes I)\n\u22a2 x \u2208 sInf {J | I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nrw [Ideal.mem_sInf] at hx \u22a2\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 Ideal.minimalPrimes I \u2192 x \u2208 I_1\n\u22a2 \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 IsPrime J} \u2192 x \u2208 I_1\n[PROOFSTEP]\nrintro J \u27e8e, hJ\u27e9\n[GOAL]\ncase a.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J\u271d : Ideal R\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 Ideal.minimalPrimes I \u2192 x \u2208 I_1\nJ : Ideal R\ne : I \u2264 J\nhJ : IsPrime J\n\u22a2 x \u2208 J\n[PROOFSTEP]\nobtain \u27e8p, hp, hp'\u27e9 := Ideal.exists_minimalPrimes_le e\n[GOAL]\ncase a.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J\u271d : Ideal R\nx : R\nhx : \u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 Ideal.minimalPrimes I \u2192 x \u2208 I_1\nJ : Ideal R\ne : I \u2264 J\nhJ : IsPrime J\np : Ideal R\nhp : p \u2208 Ideal.minimalPrimes I\nhp' : p \u2264 J\n\u22a2 x \u2208 J\n[PROOFSTEP]\nexact hp' (hx hp)\n[GOAL]\ncase a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 sInf {J | I \u2264 J \u2227 IsPrime J} \u2264 sInf (Ideal.minimalPrimes I)\n[PROOFSTEP]\napply sInf_le_sInf _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 Ideal.minimalPrimes I \u2286 {J | I \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nintro I hI\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J I : Ideal R\nhI : I \u2208 Ideal.minimalPrimes I\u271d\n\u22a2 I \u2208 {J | I\u271d \u2264 J \u2227 IsPrime J}\n[PROOFSTEP]\nexact hI.1.symm\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\n\u22a2 \u2203 p', IsPrime p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave := H.1.1\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis : IsPrime p\n\u22a2 \u2203 p', IsPrime p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave : Nontrivial (Localization (Submonoid.map f p.primeCompl)) :=\n  by\n  refine' \u27e8\u27e81, 0, _\u27e9\u27e9\n  convert\n    (IsLocalization.map_injective_of_injective p.primeCompl (Localization.AtPrime p)\n          (Localization <| p.primeCompl.map f) hf).ne\n      one_ne_zero\n  \u00b7 rw [map_one]\n  \u00b7 rw [map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis : IsPrime p\n\u22a2 Nontrivial (Localization (Submonoid.map f (primeCompl p)))\n[PROOFSTEP]\nrefine' \u27e8\u27e81, 0, _\u27e9\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis : IsPrime p\n\u22a2 1 \u2260 0\n[PROOFSTEP]\nconvert\n  (IsLocalization.map_injective_of_injective p.primeCompl (Localization.AtPrime p) (Localization <| p.primeCompl.map f)\n        hf).ne\n    one_ne_zero\n[GOAL]\ncase h.e'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis : IsPrime p\n\u22a2 1 =\n    \u2191(IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n      1\n[PROOFSTEP]\nrw [map_one]\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis : IsPrime p\n\u22a2 0 =\n    \u2191(IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n      0\n[PROOFSTEP]\nrw [map_zero]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\n\u22a2 \u2203 p', IsPrime p' \u2227 comap f p' = p\n[PROOFSTEP]\nobtain \u27e8M, hM\u27e9 := Ideal.exists_maximal (Localization (Submonoid.map f p.primeCompl))\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\n\u22a2 \u2203 p', IsPrime p' \u2227 comap f p' = p\n[PROOFSTEP]\nrefine' \u27e8M.comap (algebraMap S <| Localization (Submonoid.map f p.primeCompl)), inferInstance, _\u27e9\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\n\u22a2 comap f (comap (algebraMap S (Localization (Submonoid.map f (primeCompl p)))) M) = p\n[PROOFSTEP]\nrw [Ideal.comap_comap, \u2190\n  @IsLocalization.map_comp _ _ _ _ _ _ _ _ Localization.isLocalization _ _ _ _ p.primeCompl.le_comap_map _\n    Localization.isLocalization,\n  \u2190 Ideal.comap_comap]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\n\u22a2 comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M) =\n    p\n[PROOFSTEP]\nsuffices _ \u2264 p by exact this.antisymm (H.2 \u27e8inferInstance, bot_le\u27e9 this)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d\u00b9 : IsPrime p\nthis\u271d : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nthis : ?m.48761 \u2264 p\n\u22a2 comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M) =\n    p\n[PROOFSTEP]\nexact this.antisymm (H.2 \u27e8inferInstance, bot_le\u27e9 this)\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\n\u22a2 comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M) \u2264\n    p\n[PROOFSTEP]\nintro x hx\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nx : R\nhx :\n  x \u2208\n    comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M)\n\u22a2 x \u2208 p\n[PROOFSTEP]\nby_contra h\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nx : R\nhx :\n  x \u2208\n    comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M)\nh : \u00acx \u2208 p\n\u22a2 False\n[PROOFSTEP]\napply hM.ne_top\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nx : R\nhx :\n  x \u2208\n    comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M)\nh : \u00acx \u2208 p\n\u22a2 M = \u22a4\n[PROOFSTEP]\napply M.eq_top_of_isUnit_mem hx\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nx : R\nhx :\n  x \u2208\n    comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M)\nh : \u00acx \u2208 p\n\u22a2 IsUnit\n    (\u2191(IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n      (\u2191(algebraMap R (Localization (primeCompl p))) x))\n[PROOFSTEP]\napply IsUnit.map\n[GOAL]\ncase intro.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nf : R \u2192+* S\nhf : Function.Injective \u2191f\np : Ideal R\nH : p \u2208 minimalPrimes R\nthis\u271d : IsPrime p\nthis : Nontrivial (Localization (Submonoid.map f (primeCompl p)))\nM : Ideal (Localization (Submonoid.map f (primeCompl p)))\nhM : IsMaximal M\nx : R\nhx :\n  x \u2208\n    comap (algebraMap R (Localization (primeCompl p)))\n      (comap\n        (IsLocalization.map (Localization (Submonoid.map f (primeCompl p))) f\n          (_ : primeCompl p \u2264 Submonoid.comap f (Submonoid.map f (primeCompl p))))\n        M)\nh : \u00acx \u2208 p\n\u22a2 IsUnit (\u2191(algebraMap R (Localization (primeCompl p))) x)\n[PROOFSTEP]\napply IsLocalization.map_units _ (show p.primeCompl from \u27e8x, h\u27e9)\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave := H.1.1\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nlet f' := (Ideal.Quotient.mk I).comp f\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave e : RingHom.ker f' = I.comap f := by\n  ext1\n  exact Submodule.Quotient.mk_eq_zero _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\n\u22a2 RingHom.ker f' = comap f I\n[PROOFSTEP]\next1\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\nx\u271d : R\n\u22a2 x\u271d \u2208 RingHom.ker f' \u2194 x\u271d \u2208 comap f I\n[PROOFSTEP]\nexact Submodule.Quotient.mk_eq_zero _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave : RingHom.ker (Ideal.Quotient.mk <| RingHom.ker f') \u2264 p :=\n  by\n  rw [Ideal.mk_ker, e]\n  exact H.1.2\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\n\u22a2 RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n[PROOFSTEP]\nrw [Ideal.mk_ker, e]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\n\u22a2 comap f I \u2264 p\n[PROOFSTEP]\nexact H.1.2\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nsuffices _\n  by\n  have \u27e8p', hp\u2081, hp\u2082\u27e9 :=\n    Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective (RingHom.kerLift_injective f')\n      (p.map <| Ideal.Quotient.mk <| RingHom.ker f') this\n  refine' \u27e8p'.comap <| Ideal.Quotient.mk I, Ideal.IsPrime.comap _, _, _\u27e9\n  \u00b7 exact Ideal.mk_ker.symm.trans_le (Ideal.comap_mono bot_le)\n  \u00b7 convert congr_arg (Ideal.comap <| Ideal.Quotient.mk <| RingHom.ker f') hp\u2082\n    rwa [Ideal.comap_map_of_surjective (Ideal.Quotient.mk <| RingHom.ker f') Ideal.Quotient.mk_surjective, eq_comm,\n      sup_eq_left]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d\u00b9 : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : ?m.53321\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nhave \u27e8p', hp\u2081, hp\u2082\u27e9 :=\n  Ideal.exists_comap_eq_of_mem_minimalPrimes_of_injective (RingHom.kerLift_injective f')\n    (p.map <| Ideal.Quotient.mk <| RingHom.ker f') this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d\u00b9 : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : IsPrime p'\nhp\u2082 : comap (RingHom.kerLift f') p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 \u2203 p', IsPrime p' \u2227 I \u2264 p' \u2227 comap f p' = p\n[PROOFSTEP]\nrefine' \u27e8p'.comap <| Ideal.Quotient.mk I, Ideal.IsPrime.comap _, _, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d\u00b9 : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : IsPrime p'\nhp\u2082 : comap (RingHom.kerLift f') p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 I \u2264 comap (Quotient.mk I) p'\n[PROOFSTEP]\nexact Ideal.mk_ker.symm.trans_le (Ideal.comap_mono bot_le)\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d\u00b9 : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : IsPrime p'\nhp\u2082 : comap (RingHom.kerLift f') p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap f (comap (Quotient.mk I) p') = p\n[PROOFSTEP]\nconvert congr_arg (Ideal.comap <| Ideal.Quotient.mk <| RingHom.ker f') hp\u2082\n[GOAL]\ncase h.e'_3\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d\u00b9 : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis\u271d : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nthis : map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\np' : Ideal (S \u29f8 I)\nhp\u2081 : IsPrime p'\nhp\u2082 : comap (RingHom.kerLift f') p' = map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p = comap (Quotient.mk (RingHom.ker f')) (map (Quotient.mk (RingHom.ker f')) p)\n[PROOFSTEP]\nrwa [Ideal.comap_map_of_surjective (Ideal.Quotient.mk <| RingHom.ker f') Ideal.Quotient.mk_surjective, eq_comm,\n  sup_eq_left]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2208 minimalPrimes (R \u29f8 RingHom.ker f')\n[PROOFSTEP]\nrefine' \u27e8\u27e8_, bot_le\u27e9, _\u27e9\n[GOAL]\ncase refine'_1\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 IsPrime (map (Quotient.mk (RingHom.ker f')) p)\n[PROOFSTEP]\napply Ideal.map_isPrime_of_surjective _ this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 Function.Surjective \u2191(Quotient.mk (RingHom.ker f'))\n[PROOFSTEP]\nexact Ideal.Quotient.mk_surjective\n[GOAL]\ncase refine'_2\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\n\u22a2 \u2200 \u2983b : Ideal (R \u29f8 RingHom.ker f')\u2984,\n    b \u2208 {p | IsPrime p \u2227 \u22a5 \u2264 p} \u2192\n      (fun x x_1 => x \u2264 x_1) b (map (Quotient.mk (RingHom.ker f')) p) \u2192\n        (fun x x_1 => x \u2264 x_1) (map (Quotient.mk (RingHom.ker f')) p) b\n[PROOFSTEP]\nrintro q \u27e8hq, -\u27e9 hq'\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264 q\n[PROOFSTEP]\nrw [\u2190\n  Ideal.map_comap_of_surjective (Ideal.Quotient.mk (RingHom.ker ((Ideal.Quotient.mk I).comp f)))\n    Ideal.Quotient.mk_surjective q]\n[GOAL]\ncase refine'_2.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 map (Quotient.mk (RingHom.ker f')) p \u2264\n    map (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f)))\n      (comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) q)\n[PROOFSTEP]\napply Ideal.map_mono\n[GOAL]\ncase refine'_2.intro.h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2264 comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) q\n[PROOFSTEP]\napply H.2\n[GOAL]\ncase refine'_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) q \u2208 {p | IsPrime p \u2227 comap f I \u2264 p}\n[PROOFSTEP]\nrefine' \u27e8inferInstance, (Ideal.mk_ker.trans e).symm.trans_le (Ideal.comap_mono bot_le)\u27e9\n[GOAL]\ncase refine'_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) q \u2264 p\n[PROOFSTEP]\nrefine' (Ideal.comap_mono hq').trans _\n[GOAL]\ncase refine'_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) (map (Quotient.mk (RingHom.ker f')) p) \u2264 p\n[PROOFSTEP]\nrw [Ideal.comap_map_of_surjective]\n[GOAL]\ncase refine'_2.intro.h.a\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 p \u2294 comap (Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f))) \u22a5 \u2264 p\ncase refine'_2.intro.h.a.hf\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\nthis\u271d : IsPrime p\nf' : R \u2192+* S \u29f8 I := RingHom.comp (Quotient.mk I) f\ne : RingHom.ker f' = comap f I\nthis : RingHom.ker (Quotient.mk (RingHom.ker f')) \u2264 p\nq : Ideal (R \u29f8 RingHom.ker f')\nhq : IsPrime q\nhq' : q \u2264 map (Quotient.mk (RingHom.ker f')) p\n\u22a2 Function.Surjective \u2191(Quotient.mk (RingHom.ker (RingHom.comp (Quotient.mk I) f)))\n[PROOFSTEP]\nexacts [sup_le rfl.le this, Ideal.Quotient.mk_surjective]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\n\u22a2 \u2203 p', p' \u2208 Ideal.minimalPrimes I \u2227 comap f p' = p\n[PROOFSTEP]\nobtain \u27e8p', h\u2081, h\u2082, h\u2083\u27e9 := Ideal.exists_comap_eq_of_mem_minimalPrimes f p H\n[GOAL]\ncase intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh\u2081 : IsPrime p'\nh\u2082 : I \u2264 p'\nh\u2083 : comap f p' = p\n\u22a2 \u2203 p', p' \u2208 Ideal.minimalPrimes I \u2227 comap f p' = p\n[PROOFSTEP]\nobtain \u27e8q, hq, hq'\u27e9 := Ideal.exists_minimalPrimes_le h\u2082\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh\u2081 : IsPrime p'\nh\u2082 : I \u2264 p'\nh\u2083 : comap f p' = p\nq : Ideal S\nhq : q \u2208 Ideal.minimalPrimes I\nhq' : q \u2264 p'\n\u22a2 \u2203 p', p' \u2208 Ideal.minimalPrimes I \u2227 comap f p' = p\n[PROOFSTEP]\nrefine' \u27e8q, hq, Eq.symm _\u27e9\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh\u2081 : IsPrime p'\nh\u2082 : I \u2264 p'\nh\u2083 : comap f p' = p\nq : Ideal S\nhq : q \u2208 Ideal.minimalPrimes I\nhq' : q \u2264 p'\n\u22a2 p = comap f q\n[PROOFSTEP]\nhave := hq.1.1\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh\u2081 : IsPrime p'\nh\u2082 : I \u2264 p'\nh\u2083 : comap f p' = p\nq : Ideal S\nhq : q \u2208 Ideal.minimalPrimes I\nhq' : q \u2264 p'\nthis : IsPrime q\n\u22a2 p = comap f q\n[PROOFSTEP]\nhave := (Ideal.comap_mono hq').trans_eq h\u2083\n[GOAL]\ncase intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nI : Ideal S\nf : R \u2192+* S\np : Ideal R\nH : p \u2208 Ideal.minimalPrimes (comap f I)\np' : Ideal S\nh\u2081 : IsPrime p'\nh\u2082 : I \u2264 p'\nh\u2083 : comap f p' = p\nq : Ideal S\nhq : q \u2208 Ideal.minimalPrimes I\nhq' : q \u2264 p'\nthis\u271d : IsPrime q\nthis : comap f q \u2264 p\n\u22a2 p = comap f q\n[PROOFSTEP]\nexact (H.2 \u27e8inferInstance, Ideal.comap_mono hq.1.2\u27e9 this).antisymm this\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\n\u22a2 comap f J \u2208 Ideal.minimalPrimes (comap f I)\n[PROOFSTEP]\nhave := h.1.1\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis : IsPrime J\n\u22a2 comap f J \u2208 Ideal.minimalPrimes (comap f I)\n[PROOFSTEP]\nrefine' \u27e8\u27e8inferInstance, Ideal.comap_mono h.1.2\u27e9, _\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis : IsPrime J\n\u22a2 \u2200 \u2983b : Ideal R\u2984,\n    b \u2208 {p | IsPrime p \u2227 comap f I \u2264 p} \u2192 (fun x x_1 => x \u2264 x_1) b (comap f J) \u2192 (fun x x_1 => x \u2264 x_1) (comap f J) b\n[PROOFSTEP]\nrintro K \u27e8hK, e\u2081\u27e9 e\u2082\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\n\u22a2 comap f J \u2264 K\n[PROOFSTEP]\nhave : RingHom.ker f \u2264 K := (Ideal.comap_mono bot_le).trans e\u2081\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis\u271d : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\nthis : RingHom.ker f \u2264 K\n\u22a2 comap f J \u2264 K\n[PROOFSTEP]\nrw [\u2190 sup_eq_left.mpr this, RingHom.ker_eq_comap_bot, \u2190 Ideal.comap_map_of_surjective f hf]\n[GOAL]\ncase intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis\u271d : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\nthis : RingHom.ker f \u2264 K\n\u22a2 comap f J \u2264 comap f (map f K)\n[PROOFSTEP]\napply Ideal.comap_mono _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis\u271d : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\nthis : RingHom.ker f \u2264 K\n\u22a2 J \u2264 map f K\n[PROOFSTEP]\napply h.2 _ _\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis\u271d : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\nthis : RingHom.ker f \u2264 K\n\u22a2 map f K \u2208 {p | IsPrime p \u2227 I \u2264 p}\n[PROOFSTEP]\nexact \u27e8Ideal.map_isPrime_of_surjective hf this, Ideal.le_map_of_comap_le_of_surjective f hf e\u2081\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nh : J \u2208 Ideal.minimalPrimes I\nthis\u271d : IsPrime J\nK : Ideal R\nhK : IsPrime K\ne\u2081 : comap f I \u2264 K\ne\u2082 : K \u2264 comap f J\nthis : RingHom.ker f \u2264 K\n\u22a2 map f K \u2264 J\n[PROOFSTEP]\nexact Ideal.map_le_of_le_comap e\u2082\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal S\n\u22a2 Ideal.minimalPrimes (comap f I) = comap f '' Ideal.minimalPrimes I\n[PROOFSTEP]\next J\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal S\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes (comap f I) \u2194 J \u2208 comap f '' Ideal.minimalPrimes I\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal S\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes (comap f I) \u2192 J \u2208 comap f '' Ideal.minimalPrimes I\n[PROOFSTEP]\nintro H\n[GOAL]\ncase h.mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal S\nJ : Ideal R\nH : J \u2208 Ideal.minimalPrimes (comap f I)\n\u22a2 J \u2208 comap f '' Ideal.minimalPrimes I\n[PROOFSTEP]\nobtain \u27e8p, h, rfl\u27e9 := Ideal.exists_minimalPrimes_comap_eq f J H\n[GOAL]\ncase h.mp.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI p : Ideal S\nh : p \u2208 Ideal.minimalPrimes I\nH : comap f p \u2208 Ideal.minimalPrimes (comap f I)\n\u22a2 comap f p \u2208 comap f '' Ideal.minimalPrimes I\n[PROOFSTEP]\nexact \u27e8p, h, rfl\u27e9\n[GOAL]\ncase h.mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI : Ideal S\nJ : Ideal R\n\u22a2 J \u2208 comap f '' Ideal.minimalPrimes I \u2192 J \u2208 Ideal.minimalPrimes (comap f I)\n[PROOFSTEP]\nrintro \u27e8J, hJ, rfl\u27e9\n[GOAL]\ncase h.mpr.intro.intro\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI\u271d J\u271d : Ideal R\nf : R \u2192+* S\nhf : Function.Surjective \u2191f\nI J : Ideal S\nhJ : J \u2208 Ideal.minimalPrimes I\n\u22a2 comap f J \u2208 Ideal.minimalPrimes (comap f I)\n[PROOFSTEP]\nexact Ideal.mimimal_primes_comap_of_surjective hf hJ\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\n\u22a2 Ideal.minimalPrimes I = comap (Quotient.mk I) '' minimalPrimes (R \u29f8 I)\n[PROOFSTEP]\nrw [minimalPrimes, \u2190 Ideal.comap_minimalPrimes_eq_of_surjective Ideal.Quotient.mk_surjective, \u2190\n  RingHom.ker_eq_comap_bot, Ideal.mk_ker]\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nhI : IsPrimary I\n\u22a2 Ideal.minimalPrimes I = {radical I}\n[PROOFSTEP]\next J\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J\u271d : Ideal R\nhI : IsPrimary I\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes I \u2194 J \u2208 {radical I}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J\u271d : Ideal R\nhI : IsPrimary I\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes I \u2192 J \u2208 {radical I}\n[PROOFSTEP]\nexact fun H =>\n  let e := H.1.1.radical_le_iff.mpr H.1.2\n  (H.2 \u27e8Ideal.isPrime_radical hI, Ideal.le_radical\u27e9 e).antisymm e\n[GOAL]\ncase h.mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J\u271d : Ideal R\nhI : IsPrimary I\nJ : Ideal R\n\u22a2 J \u2208 {radical I} \u2192 J \u2208 Ideal.minimalPrimes I\n[PROOFSTEP]\nrintro (rfl : J = I.radical)\n[GOAL]\ncase h.mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nI J : Ideal R\nhI : IsPrimary I\n\u22a2 radical I \u2208 Ideal.minimalPrimes I\n[PROOFSTEP]\nexact \u27e8\u27e8Ideal.isPrime_radical hI, Ideal.le_radical\u27e9, fun _ H _ => H.1.radical_le_iff.mpr H.2\u27e9\n[GOAL]\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J : Ideal R\ninst\u271d : IsPrime I\n\u22a2 Ideal.minimalPrimes I = {I}\n[PROOFSTEP]\next J\n[GOAL]\ncase h\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J\u271d : Ideal R\ninst\u271d : IsPrime I\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes I \u2194 J \u2208 {I}\n[PROOFSTEP]\nconstructor\n[GOAL]\ncase h.mp\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J\u271d : Ideal R\ninst\u271d : IsPrime I\nJ : Ideal R\n\u22a2 J \u2208 Ideal.minimalPrimes I \u2192 J \u2208 {I}\n[PROOFSTEP]\nexact fun H => (H.2 \u27e8inferInstance, rfl.le\u27e9 H.1.2).antisymm H.1.2\n[GOAL]\ncase h.mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nI J\u271d : Ideal R\ninst\u271d : IsPrime I\nJ : Ideal R\n\u22a2 J \u2208 {I} \u2192 J \u2208 Ideal.minimalPrimes I\n[PROOFSTEP]\nrintro (rfl : J = I)\n[GOAL]\ncase h.mpr\nR : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nJ\u271d J : Ideal R\ninst\u271d : IsPrime J\n\u22a2 J \u2208 Ideal.minimalPrimes J\n[PROOFSTEP]\nrefine' \u27e8\u27e8inferInstance, rfl.le\u27e9, fun _ h _ => h.2\u27e9\n", "meta": {"mathlib_filename": "Mathlib.RingTheory.Ideal.MinimalPrime", "llama_tokens": 22450, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.5002657491951908}}
{"text": "[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2074 : PseudoEMetricSpace X\ninst\u271d\u00b3 : SMul M X\ninst\u271d\u00b2 : SMul M\u1d50\u1d52\u1d56 X\ninst\u271d\u00b9 : IsCentralScalar M X\ninst\u271d : IsometricSMul M X\nc : M\u1d50\u1d52\u1d56\nx y : X\n\u22a2 edist ((fun x => c \u2022 x) x) ((fun x => c \u2022 x) y) = edist x y\n[PROOFSTEP]\nsimpa only [\u2190 op_smul_eq_smul] using isometry_smul X c.unop x y\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2076 : PseudoEMetricSpace X\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G X\ninst\u271d\u00b3 : IsometricSMul G X\ninst\u271d\u00b2 : DivInvMonoid M\ninst\u271d\u00b9 : PseudoEMetricSpace M\ninst\u271d : IsometricSMul M\u1d50\u1d52\u1d56 M\na b c : M\n\u22a2 edist (a / c) (b / c) = edist a b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, edist_mul_right]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2076 : PseudoEMetricSpace X\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G X\ninst\u271d\u00b3 : IsometricSMul G X\ninst\u271d\u00b2 : PseudoEMetricSpace G\ninst\u271d\u00b9 : IsometricSMul G G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\n\u22a2 edist a\u207b\u00b9 b\u207b\u00b9 = edist a b\n[PROOFSTEP]\nrw [\u2190 edist_mul_left a, \u2190 edist_mul_right _ _ b, mul_right_inv, one_mul, inv_mul_cancel_right, edist_comm]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2076 : PseudoEMetricSpace X\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G X\ninst\u271d\u00b3 : IsometricSMul G X\ninst\u271d\u00b2 : PseudoEMetricSpace G\ninst\u271d\u00b9 : IsometricSMul G G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\nx y : G\n\u22a2 edist x\u207b\u00b9 y = edist x y\u207b\u00b9\n[PROOFSTEP]\nrw [\u2190 edist_inv_inv, inv_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2076 : PseudoEMetricSpace X\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : MulAction G X\ninst\u271d\u00b3 : IsometricSMul G X\ninst\u271d\u00b2 : PseudoEMetricSpace G\ninst\u271d\u00b9 : IsometricSMul G G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b c : G\n\u22a2 edist (a / b) (a / c) = edist b c\n[PROOFSTEP]\nrw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : PseudoEMetricSpace X\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\ninst\u271d : IsometricSMul G X\nc : G\nx : X\nr : \u211d\u22650\u221e\n\u22a2 (fun x x_1 => x \u2022 x_1) c \u207b\u00b9' ball x r = ball (c\u207b\u00b9 \u2022 x) r\n[PROOFSTEP]\nrw [preimage_smul, smul_ball]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : PseudoEMetricSpace X\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\ninst\u271d : IsometricSMul G X\nc : G\nx : X\nr : \u211d\u22650\u221e\n\u22a2 (fun x x_1 => x \u2022 x_1) c \u207b\u00b9' closedBall x r = closedBall (c\u207b\u00b9 \u2022 x) r\n[PROOFSTEP]\nrw [preimage_smul, smul_closedBall]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoEMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoEMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\u22650\u221e\n\u22a2 (fun x => x * a) \u207b\u00b9' ball b r = ball (b / a) r\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoEMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoEMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\u22650\u221e\n\u22a2 (fun x => x * a) \u207b\u00b9' ball b r = ball (b * a\u207b\u00b9) r\n[PROOFSTEP]\nexact preimage_smul_ball (MulOpposite.op a) b r\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoEMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoEMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\u22650\u221e\n\u22a2 (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b / a) r\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoEMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoEMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\u22650\u221e\n\u22a2 (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b * a\u207b\u00b9) r\n[PROOFSTEP]\nexact preimage_smul_closedBall (MulOpposite.op a) b r\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b2 : DivInvMonoid M\ninst\u271d\u00b9 : PseudoMetricSpace M\ninst\u271d : IsometricSMul M\u1d50\u1d52\u1d56 M\na b c : M\n\u22a2 dist (a / c) (b / c) = dist a b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, dist_mul_right]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b2 : DivInvMonoid M\ninst\u271d\u00b9 : PseudoMetricSpace M\ninst\u271d : IsometricSMul M\u1d50\u1d52\u1d56 M\na b c : M\n\u22a2 nndist (a / c) (b / c) = nndist a b\n[PROOFSTEP]\nsimp only [div_eq_mul_inv, nndist_mul_right]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : PseudoMetricSpace G\ninst\u271d\u00b9 : IsometricSMul G G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b c : G\n\u22a2 dist (a / b) (a / c) = dist b c\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : PseudoMetricSpace G\ninst\u271d\u00b9 : IsometricSMul G G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b c : G\n\u22a2 nndist (a / b) (a / c) = nndist b c\n[PROOFSTEP]\nsimp [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : PseudoMetricSpace X\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\ninst\u271d : IsometricSMul G X\nc : G\nx : X\nr : \u211d\n\u22a2 (fun x x_1 => x \u2022 x_1) c \u207b\u00b9' ball x r = ball (c\u207b\u00b9 \u2022 x) r\n[PROOFSTEP]\nrw [preimage_smul, smul_ball]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : PseudoMetricSpace X\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\ninst\u271d : IsometricSMul G X\nc : G\nx : X\nr : \u211d\n\u22a2 (fun x x_1 => x \u2022 x_1) c \u207b\u00b9' closedBall x r = closedBall (c\u207b\u00b9 \u2022 x) r\n[PROOFSTEP]\nrw [preimage_smul, smul_closedBall]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u00b3 : PseudoMetricSpace X\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : MulAction G X\ninst\u271d : IsometricSMul G X\nc : G\nx : X\nr : \u211d\n\u22a2 (fun x x_1 => x \u2022 x_1) c \u207b\u00b9' sphere x r = sphere (c\u207b\u00b9 \u2022 x) r\n[PROOFSTEP]\nrw [preimage_smul, smul_sphere]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\n\u22a2 (fun x => x * a) \u207b\u00b9' ball b r = ball (b / a) r\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\n\u22a2 (fun x => x * a) \u207b\u00b9' ball b r = ball (b * a\u207b\u00b9) r\n[PROOFSTEP]\nexact preimage_smul_ball (MulOpposite.op a) b r\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\n\u22a2 (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b / a) r\n[PROOFSTEP]\nrw [div_eq_mul_inv]\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\ninst\u271d\u2075 : PseudoMetricSpace X\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G X\ninst\u271d\u00b2 : IsometricSMul G X\ninst\u271d\u00b9 : PseudoMetricSpace G\ninst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G\na b : G\nr : \u211d\n\u22a2 (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b * a\u207b\u00b9) r\n[PROOFSTEP]\nexact preimage_smul_closedBall (MulOpposite.op a) b r\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace X\ninst\u271d\u00b2 : PseudoEMetricSpace Y\ninst\u271d\u00b9 : SMul M X\ninst\u271d : IsometricSMul M X\nc : M\nx y : X\u1d50\u1d52\u1d56\n\u22a2 edist ((fun x => c \u2022 x) x) ((fun x => c \u2022 x) y) = edist x y\n[PROOFSTEP]\nsimpa only using edist_smul_left c x.unop y.unop\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace X\ninst\u271d\u00b2 : PseudoEMetricSpace Y\ninst\u271d\u00b9 : SMul M X\ninst\u271d : IsometricSMul M X\nc : ULift M\n\u22a2 Isometry fun x => c \u2022 x\n[PROOFSTEP]\nsimpa only using isometry_smul X c.down\n[GOAL]\nM : Type u\nG : Type v\nX : Type w\nY : Type u_1\ninst\u271d\u00b3 : PseudoEMetricSpace X\ninst\u271d\u00b2 : PseudoEMetricSpace Y\ninst\u271d\u00b9 : SMul M X\ninst\u271d : IsometricSMul M X\nc : M\nx y : ULift X\n\u22a2 edist ((fun x => c \u2022 x) x) ((fun x => c \u2022 x) y) = edist x y\n[PROOFSTEP]\nsimpa only using edist_smul_left c x.1 y.1\n", "meta": {"mathlib_filename": "Mathlib.Topology.MetricSpace.IsometricSMul", "llama_tokens": 3755, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185318, "lm_q2_score": 0.6992544335934766, "lm_q1q2_score": 0.5002634094362479}}